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\begin{document}
\title[Quasiconformal Mappings and Neumann Eigenvalues]{Quasiconformal Mappings and Neumann Eigenvalues of Divergent Elliptic Operators}
\author{V.~Gol'dshtein, V.~Pchelintsev, A.~Ukhlov}
\begin{abstract}
We study spectral properties of divergence form elliptic operators $-\textrm{div} [A(z) \nabla f(z)]$ with the Neumann boundary condition in planar domains (including some fractal type domains), that satisfy to the quasihyperbolic boundary conditions. Our method is based on an interplay between quasiconformal mappings, elliptic operators and composition operators on Sobolev spaces.
\end{abstract}
\maketitle
\footnotetext{\textbf{Key words and phrases:} Elliptic equations, Sobolev spaces, quasiconformal mappings.}
\footnotetext{\textbf{2010
Mathematics Subject Classification:} 35P15, 46E35, 30C60.}
\section{Introduction}
In this paper we apply methods of the (quasi)conformal geometry to spectral problems for $A$-divergent form elliptic operators
with the Neumann boundary condition
\begin{equation}\label{EllDivOper}
L_{A}=-\textrm{div} [A(z) \nabla f(z)], \quad z=(x,y)\in \Omega, \quad
\left\langle A(z) \nabla f, n \right\rangle\big|_{\partial \Omega}=0,
\end{equation}
in the large class of (non)convex domains $\Omega \subset \mathbb C$ that satisfy the quasihyperbolic boundary condition \cite{KOT01,KOT02}. Here matrix functions $A(z)=\left\{a_{kl}(z)\right\}$ with measurable entries $a_{kl}(z)$ belongs to a class $M^{2 \times 2}(\Omega)$ of all $2 \times 2$ symmetric matrix functions that satisfy to an additional condition $\textrm{det} A=1$ a.e. and to the uniform ellipticity condition:
\begin{equation}\label{UEC}
\frac{1}{K}|\xi|^2 \leq \left\langle A(z) \xi, \xi \right\rangle \leq K |\xi|^2 \,\,\, \text{a.e. in}\,\,\, \Omega,
\end{equation}
for every $\xi \in \mathbb C$ and for some $1\leq K< \infty$.
Such type of elliptic operators arise in various problems of mathematical physics (see, for example, \cite{AIM}).
The suggested method is based on a (quasi)conformal representation of a non smooth Riemannian metric in the domain $\Omega$:
\[
ds^2=a_{11}(x,y)dx^2+2a_{12}(x,y)dxdy+a_{22}(x,y)dy^2
\]
induced by the matrix $A$. The complex dilatation $\mu$ of the corresponding quasiconformal mapping $\varphi$ from $\Omega$ to the unit disc $\mathbb D\subset\mathbb C$ can be calculated using the matrix $A$ (see, for example, \cite[p. 412]{AIM}). Inverse, if the complex dilatation $\mu$ is given then the matrix $A$ can be reproduced. It means that there is one to one correspondence between the matrices and the complex dilatations. By the construction the quasiconformal mapping $\varphi:\Omega\to\mathbb D$ is an isometry of the domain $\Omega$ with this new Riemannian metric $ds$ (induced by the matrix $A$) and the unit disc $\mathbb{D}$ with the hyperbolic metric. It is reasonable to call such metric $ds$ as an $A$-quasiconformal metric and the quasiconformal mapping $\varphi$ as an $A$-quasiconformal mapping (i.e. quasiconformal mapping agreed with the matrix $A$).
Hence, we can conclude that an $A$-quasiconformal mapping is conformal (i.e. preserve angles) in this $A$-quasiconformal Riemannian metric.
Let us remind that conformal homeomorphisms induce isometries of uniform Sobolev spaces $L^{1,2}(\Omega)$ and $L^{1,2}(\mathbb D)$. In the present article we prove that $A$-quasi\-con\-for\-mal mappings induce isometries of an uniform Sobolev space $L_A^{1,2}(\Omega)$ (Sobolev space agreed with the matrix $A$) and the uniform Sobolev space $L^{1,2}(\mathbb D)$.
\vskip 0.2cm
{\bf Conjecture.} {\it Spectral properties of the $A$-divergent form elliptic operators with the Neumann boundary condition depends on
$A$-quasiconformal geometry of domains only.}
\vskip 0.2cm
The suggested method is based on connections between composition operators on Sobolev spaces, elliptic operators and quasiconformal mappings. We prove that $\varphi:\Omega\to\Omega'$ is a quasiconformal mapping agreed with the matrix $A$ (i.e. induced by $A$ via the Beltrami equation) if and only if
\[
\iint\limits_\Omega \left\langle A(z)\nabla f(\varphi(z)),\nabla f(\varphi(z))\right\rangle\,dxdy=\iint\limits_{\Omega'} \left\langle \nabla f(w),\nabla f(w)\right\rangle\,dudv,
\]
for all $f\in L^{1,2}(\Omega')$.
This result permits us to introduce an $A$-norm in the corresponding uniform Sobolev space $L^{1,2}_A$. For this norm the $A$-quasiconformal mappings play a role similar to conformal mappings for the Laplace operator and uniform Sobolev spaces $L^{1,2}$. In particular, we prove that the $A$-quasiconformal mappings generalize the well known property of conformal mappings to generate isometries of uniform Sobolev spaces $L^{1,2}(\Omega)$ and $L^{1,2}(\Omega')$ (see, for example, \cite{C50}). In terms of the $A$-quasiconformal mappings we also refine the functional characterization of quasiconformal mappings, obtained in the article \cite{VG75} in the terms of isomorphisms of uniform Sobolev spaces $L^{1,2}$.
Short historical remarks. Spectral estimates of elliptic operators eigenvalues represent an important part of the modern spectral theory (see, for example,
\cite{A98,AB06,BLL,BCT15,EP15,ENT,FNT,LM98}).
The classical upper estimate for the first non-trivial Neumann eigenvalue of the Laplace operator (the $A$-divergent form elliptic operator with the matrix $A=I$)
\begin{equation*}
\mu_1(I,\Omega):=\mu_1(\Omega)\leq \mu_1(\Omega^{\ast})=\frac{{j_{1,1}'^2}}{R^2_{\ast}}
\end{equation*}
was proved by Szeg\"o \cite{S54} for simply connected planar domains via a conformal mappings technique ("the method of conformal normalization"). In this inequality $j_{1,1}'$ denotes the first positive zero of the derivative of the Bessel function $J_1$ and $\Omega^{\ast}$ is a disc of the same area as $\Omega$ with $R_{\ast}$ as its radius.
In convex domains $\Omega\subset\mathbb R^n$, $n\geq 2$, the classical lower estimates of the Neumann eigenvalues of the Laplace operator \cite{PW} state that
\begin{equation}
\label{PW}
\mu_1(\Omega)\geq \frac{\pi^2}{d(\Omega)^2},
\end{equation}
where $d(\Omega)$ is a diameter of a convex domain $\Omega$. Similar estimates for the non-linear $p$-Laplace operator, $p\ne 2$, were obtained much later in \cite{ENT}.
Unfortunately, for non-convex domains $\mu_1(\Omega)$ can not be characterized in the terms of its Euclidean diameters. It can be seen by considering a domain consisting of two identical squares connected by a thin corridor \cite{BCDL16}.
Let us return to our studies. In the previous works \cite{GPU17_2,GPU19,GU16} we returned to applications of a (quasi)conformal mappings techniques to such estimates in rough (non-convex) domains. Let us remind that some applications of a conformal mappings to this problem can be found in \cite{S54}). We used (quasi)conformal mappings in a framework of composition operators on Sobolev spaces \cite{U93,VG75,VU02}. This method permitted us to obtain lower estimates of the first non-trivial Neumann-Laplace eigenvalue $\mu_1(\Omega)$ in the terms of the hyperbolic (conformal) radius of $\Omega$ for a large class of domains that includes some fractal domains.
In this paper we use the $A$-quasiconformal mappings via the composition operator theory. The corresponding composition operators (isometries for the norm induced by matrices $A$) allows us reduce the spectral problem for the divergence form elliptic operator \eqref{EllDivOper} defined in a simply connected domain $\Omega\subset\mathbb C$ to a weighted spectral problem for the Laplace operator in the unit disc $\mathbb D\subset\mathbb C$.
Roughly speaking, by the chain rule applied to a function $f(z)=g \circ \varphi(z)$ \cite{GNR18}, we have
\begin{equation}\label{QCE}
-\textrm{div} [A(z) \nabla f(z)] = -\textrm{div} [A(z) \nabla g(\varphi(z))]= -\left|J(w,\varphi^{-1})\right|^{-1} \Delta g(w),
\end{equation}
where the weight $J(w,\varphi^{-1})$ is the Jacobian of the inverse mapping $\varphi^{-1}:\mathbb D\to\Omega$.
As an example we consider the divergent form operator $-\textrm{div} [A(z) \nabla f(z)]$ with the matrix
$$
A(z)=\begin{pmatrix} \frac{a+b}{a-b} & 0 \\ 0 & \frac{a-b}{a+b} \end{pmatrix},\,\,a>b\geq 0,
$$
defined in the interior of ellipse $\Omega_e$ with semi-axes $a+b$ and $a-b$. By Theorem~\ref{T4.7} we have
$$
\mu_1(A,\Omega_e) \geq \frac{(j'_{1,1})^2}{a^2-b^2},
$$
what is better (Example~\ref{example1}) than the lower estimate obtained by using the classical estimate (\ref{PW}) and the uniform ellipticity condition:
$$
\mu_1(A,\Omega_e) \geq \frac{\pi^2}{4(a+b)^2} \frac{a-b}{a+b}.
$$
For thin ellipses, i.e $a+b$ fixed and $(a-b)$ tends to zero an asymptotic of our estimate is $\infty$ when the classical asymptotic is $0$.
The application of the composition operators theory to spectral problems of the $A$-divergent form elliptic operators is based on reducing of a positive quadratic form
\[
ds^2=a_{11}(x,y)dx^2+2a_{12}(x,y)dxdy+a_{22}(x,y)dy^2
\]
defined in a planar domain $\Omega$, by means of a quasiconformal change of variables, to the canonical form
\[
ds^2=\Lambda(du^2+dv^2),\,\, \Lambda\neq 0,\,\, \text{a.e. in}\,\, \Omega',
\]
given that $a_{11}a_{22}-a^2_{12} \geq \kappa_0>0$, $a_{11}>0$, almost everywhere in $\Omega$ \cite{Ahl66, AIM, BGMR}. Note that this fact can be extended to linear operators of the form $\textrm{div} [A(z) \nabla f(z)]$, $z=x+iy$, for matrix function $A \in M^{2 \times 2}(\Omega)$.
Let $\xi(z)=\Re(\varphi(z))$ be a real part of a quasiconformal mapping $\varphi(z)=\xi(z)+i \eta(z)$, which satisfies to the Beltrami equation:
\begin{equation}\label{BelEq}
\varphi_{\overline{z}}(z)=\mu(z) \varphi_{z}(z),\,\,\, \text{a.e. in}\,\,\, \Omega,
\end{equation}
where
$$
\varphi_{z}=\frac{1}{2}\left(\frac{\partial \varphi}{\partial x}-i\frac{\partial \varphi}{\partial y}\right) \quad \text{and} \quad
\varphi_{\overline{z}}=\frac{1}{2}\left(\frac{\partial \varphi}{\partial x}+i\frac{\partial \varphi}{\partial y}\right),
$$
with the complex dilatation $\mu(z)$ is given by
\begin{equation}\label{ComDil}
\mu(z)=\frac{a_{22}(z)-a_{11}(z)-2ia_{12}(z)}{\det(I+A(z))},\quad I= \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}.
\end{equation}
We call this quasiconformal mapping (with the complex dilatation $\mu$ defined by (\ref{ComDil})) as an $A$-quasiconformal mapping.
Note that the uniform ellipticity condition \eqref{UEC} can be reformulated as
\begin{equation}\label{OVCE}
|\mu(z)|\leq \frac{K-1}{K+1},\,\,\, \text{a.e. in}\,\,\, \Omega.
\end{equation}
Conversely, using the complex dilatation $\mu$ we can obtain from \eqref{ComDil} (see, for example, \cite[p.412]{AIM}) the following representation of the matrix $A$ :
\begin{equation}\label{Matrix-F}
A(z)= \begin{pmatrix} \frac{|1-\mu|^2}{1-|\mu|^2} & \frac{-2 \Imag \mu}{1-|\mu|^2} \\ \frac{-2 \Imag \mu}{1-|\mu|^2} & \frac{|1+\mu|^2}{1-|\mu|^2} \end{pmatrix},\,\,\, \text{a.e. in}\,\,\, \Omega.
\end{equation}
So, given any $A \in M^{2 \times 2}(\Omega)$, one produced, by \eqref{OVCE}, the complex dilatation $\mu(z)$, for which, in turn, the Beltrami equation \eqref{BelEq} induces a quasiconformal homeomorphism $\varphi:\Omega \to \varphi(\Omega)$ as its solution, by the Riemann measurable mapping theorem (see, for example, \cite{Ahl66}). We will say that the matrix function $A$ induces the corresponding $A$-quasiconformal homeomorphism $\varphi$ or that $A$ and $\varphi$ are agreed. The $A$-quasiconformal mapping $\psi: \Omega \to \mathbb D$ of simply connected domain $\Omega \subset \mathbb C$
onto the unit disc $\mathbb D \subset \mathbb C$ can be obtained as a composition of $A$-quasiconformal homeomorphism $\varphi:\Omega \to \varphi(\Omega)$ and a conformal mapping $\omega : \varphi(\Omega) \to \mathbb D$.
So, by the given an $A$-divergent form elliptic operator defined in a domain $\Omega\subset\mathbb C$ we construct an $A$-quasiconformal mapping $\psi: \Omega \to \mathbb D$ with a metric quasiconformality coefficient
$$
K=\frac{1+\|\mu\mid L^{\infty}(\Omega)\|}{1-\|\mu\mid L^{\infty}(\Omega)\|},
$$
where $\mu$ defined by (\ref{ComDil}).
We prove that any $A$-quasiconformal mapping $\varphi: \Omega \to \Omega'$ induces an isometry of the spaces $L^{1,2}_A(\Omega)$ and $L^{1,2}(\Omega')$. This is the main technical result of this paper about Sobolev spaces. Using applications of quasiconformal mappings to the Sobolev type embedding theorems \cite{GG94,GU09}, we prove discreteness of the spectrum of the divergence form elliptic operators $-\textrm{div} [A(z) \nabla f(z)]$ with the Neumann boundary condition. Well-known estimates of constants in the Sobolev-Poincar\'e inequality for the unit disc and the previous isometry result in the framework of the composition operator theory permit us to obtain lower estimates of Neumann eigenvalues in the terms of integrals of derivatives of $A$-quasiconformal mappings for a large class of rough domains that includes a subclass of domains with fractal boundaries (quasidiscs).
From geometrical point of view it means that we study a class domains $\Omega\subset\mathbb C$ equipped with the corresponding quasiconformal geometry. Any such domain can be considered as a Riemannian manifold and we suppose that our estimates of Neumann eigenvalues are closely connected to spectral estimates of the Beltrami-Laplace operator.
\section{Sobolev spaces and $A$-quasiconformal mappings}
Let $E \subset \mathbb C$ be a measurable set on the complex plane and $h:E \to \mathbb R$ be a positive a.e. locally integrable function i.e. a weight. The weighted Lebesgue space $L^p(E,h)$, $1\leq p<\infty$,
is the space of all locally integrable functions endowed with the following norm:
$$
\|f\,|\,L^{p}(E,h)\|= \left(\iint\limits_E|f(z)|^ph(z)\,dxdy \right)^{\frac{1}{p}}< \infty.
$$
The two-weighted Sobolev space $W^{1,p}(\Omega,h,1)$, $1\leq p< \infty$, is defined
as the normed space of all locally integrable weakly differentiable functions
$f:\Omega\to\mathbb{R}$ endowed with the following norm:
\[
\|f\mid W^{1,p}(\Omega,h,1)\|=\|f\,|\,L^{p}(\Omega,h)\|+\|\nabla f\mid L^{p}(\Omega)\|.
\]
In the case $h=1$ this weighted Sobolev space coincides with the classical Sobolev space $W^{1,p}(\Omega)$.
The seminormed Sobolev space $L^{1,p}(\Omega)$, $1\leq p< \infty$,
is the space of all locally integrable weakly differentiable functions $f:\Omega\to\mathbb{R}$ endowed
with the following seminorm:
\[
\|f\mid L^{1,p}(\Omega)\|=\|\nabla f\mid L^p(\Omega)\|, \,\, 1\leq p<\infty.
\]
We also need a weighted seminormed Sobolev space $L_{A}^{1,2}(\Omega)$ (associated with the matrix $A$), defined
as the space of all locally integrable weakly differentiable functions $f:\Omega\to\mathbb{R}$ endowed with the following norm:
\[
\|f\mid L_{A}^{1,2}(\Omega)\|=\left(\iint\limits_\Omega \left\langle A(z)\nabla f(z),\nabla f(z)\right\rangle\,dxdy \right)^{\frac{1}{2}}.
\]
The corresponding Sobolev space $W^{1,2}_{A}(\Omega)$ is defined
as the normed space of all locally integrable weakly differentiable functions
$f:\Omega\to\mathbb{R}$ endowed with the following norm:
\[
\|f\mid W^{1,2}_{A}(\Omega)\|=\|f\,|\,L^{2}(\Omega)\|+\|f\mid L^{1,2}_{A}(\Omega)\|.
\]
These Sobolev spaces are closely connected with quasiconformal mappings.
Recall that a homeomorphism $\varphi: \Omega\to \Omega'$, where $\Omega,\, \Omega'\subset\mathbb C$, is called a $K$-quasiconformal mapping if $\varphi\in W^{1,2}_{\loc}(\Omega)$ and there exists a constant $1\leq K<\infty$ such that
$$
|D\varphi(z)|^2\leq K |J(z,\varphi)|\,\,\text{for almost all}\,\,z\in\Omega.
$$
An important subclass of quasiconformal mappings represent the class of bi-Lipschitz mappings. Note that a homeomorphism $\varphi: \Omega\to \Omega'$ is said to be an $L$-bi-Lipschitz if it satisfies the double inequality
\begin{equation}\label{Bi-Lip}
\frac{1}{L}|z-z'| \leq |\varphi(z)- \varphi(z')| \leq L|z-z'|,
\end{equation}
whenever $z, z' \in \Omega$. The smallest $L \geq 1$ for which \eqref{Bi-Lip} holds is called the isometric distortion of $\varphi$. It is known (see, for example, \cite{VGR}) that each $L$-bi-Lipschitz mapping $\varphi$ is $L^2$-quasiconformal.
Conversely we have the following connection between quasiconformal and bi-Lipschitz mappings:
\begin{lemma}
Let $\varphi: \Omega\to \Omega'$ be a $K$-quasiconformal mapping such that $|J(z, \varphi)|=1$ for almost all $z \in \Omega$. Then $\varphi$ is locally $\sqrt{K}$-bi-Lipschitz a.e. in $\Omega$.
\end{lemma}
\begin{proof}
Since $\varphi: \Omega\to \Omega'$ is a $K$-quasiconformal mapping then $\varphi$ is differentiable almost everywhere in $\Omega$ and we have
$$
|D\varphi(z)|^2\leq K |J(z,\varphi)|\,\,\,\text{for almost all}\,\,\,z\in\Omega.
$$
Because $|J(z, \varphi)|=1$ a.e. in $\Omega$ we obtain
$$
\lim\limits_{z'\to z}\frac{|\varphi(z)- \varphi(z')|}{|z-z'|}=|D\varphi(z)|\leq \sqrt{K}\,\,\, \text{for almost all}\,\,\, z\in\Omega.
$$
Hence, $\varphi$ is locally $L$-Lipschitz a.e. in $\Omega$ with $L \leq \sqrt{K}$.
On the other hand, it is known that the inverse mapping to $\varphi$ is again $K$-quasiconformal. So, $\varphi^{-1}$ is also locally $L$-Lipschitz a.e. in $\Omega$ with $L \leq \sqrt{K}$. Hence, $\varphi$ is locally $\sqrt{K}$-bi-Lipschitz a.e. in $\Omega$.
\end{proof}
Now we study a connection between composition operators on Sobolev spaces and the $A$-quasiconformal mappings that refine (in the case $n=2$) the corresponding assertion for quasiconformal mappings (\cite{VG75}).
\begin{theorem}\label{IsomSS}
Let $\Omega,\Omega'$ be domains in $\mathbb C$. Then a homeomorphism $\varphi :\Omega \to \Omega'$ is an $A$-quasiconformal mapping
if and only if $\varphi$ induces, by the composition rule $\varphi^{*}(f)=f \circ \varphi$,
an isometry of Sobolev spaces $L^{1,2}_A(\Omega)$ and $L^{1,2}(\Omega')$:
\[
\|\varphi^{*}(f)\,|\,L^{1,2}_A(\Omega)\|=\|f\,|\,L^{1,2}(\Omega')\|
\]
for any $f \in L^{1,2}(\Omega')$.
\end{theorem}
\begin{proof}
Sufficiency. We prove that if $\varphi :\Omega \to \Omega'$ is an $A$-quasiconformal mapping
then the composition operator
\[
\varphi^{*}:L^{1,2}(\Omega') \to L^{1,2}_A(\Omega),\,\,\, \varphi^{*}(f)=f \circ \varphi
\]
is an isometry.
Let $f \in L^{1,2}(\Omega')$ be a smooth function. Then the composition $g(z)=f \circ \varphi(z)$
is defined on $\Omega$ and is weakly differentiable almost everywhere in $\Omega$ \cite{VG75}. Let us check that $g(z)=f \circ \varphi(z)$ belongs to the Sobolev space $L^{1,2}_A(\Omega)$. By the chain rule \cite{VGR} we have
\begin{multline*}
\|g\,|\,L^{1,2}_A(\Omega)\| = \left(\iint\limits_{\Omega} \left\langle A(z) \nabla (f \circ \varphi(z)), \nabla (f \circ \varphi(z)) \right\rangle dxdy\right)^{\frac{1}{2}} \\
= \left(\iint\limits_{\Omega} |\nabla f|^{2}(\varphi(z)) |J(z,\varphi)| dxdy\right)^{\frac{1}{2}}
\\
=\left(\iint\limits_{\Omega'} |\nabla f|^{2}(w) dudv\right)^{\frac{1}{2}} =\|f\,|\,L^{1,2}(\Omega')\|.
\end{multline*}
Let $f \in L^{1,2}(\Omega')$ be an arbitrary function. Then there exists a sequence $\{f_k\}$, $k=1,2,...$ of smooth functions such that $f_k \in L^{1,2}(\Omega')$,
$$
\lim\limits_{k\to\infty}\|f-f_k\mid L^{1,2}(\Omega')\|=0
$$
and $\{f_k\}$ converges to $f$ a.e. in $\Omega'$.
Denote by $g_k=f_k\circ\varphi$, $k=1,2,...\,$.
Then
$$
\|g_k-g_l\,|\,L^{1,2}_A(\Omega)\|=\|f_k-f_l\,|\,L^{1,2}(\Omega')\|,\,\,k,l\in\mathbb N,
$$
and because the sequence $\{f_k\}$ converges in $L^{1,2}(\Omega')$ then the sequence $\{g_k\}$ converges in $L^{1,2}_A(\Omega)$.
Note that quasiconformal mappings possess the $N^{-1}$-Luzin property. It means that the preimage of a set of measure zero has measure zero.
So, the sequence $g_k=f_k\circ\varphi$ converges to $g=f\circ\varphi$ a.e. in $\Omega$ and hence in $L^{1,2}_A(\Omega)$.
Therefore
\[
\|\varphi^{*}(f)\,|\,L^{1,2}_A(\Omega)\|=\|f\,|\,L^{1,2}(\Omega')\|
\]
for any $f \in L^{1,2}(\Omega')$.
Necessity. Suppose that the composition operator
\[
\varphi^{*}:L^{1,2}(\Omega') \to L^{1,2}_A(\Omega)
\]
is an isometry, i.e.
\begin{equation}
\label{eqA}
\iint\limits_{\Omega} \left\langle A(z) \nabla (f \circ \varphi(z)), \nabla (f \circ \varphi(z)) \right\rangle dxdy
= \iint\limits_{\Omega'} |\nabla f|^{2}(w) dudv.
\end{equation}
Because the matrix $A$ satisfies the uniform ellipticity condition (\ref{UEC}) then by \eqref{eqA} we have
\begin{multline*}
\frac{1}{K}\iint\limits_{\Omega} |\nabla (f \circ \varphi(z))|^2~dxdy\leq
\iint\limits_{\Omega} \left\langle A(z) \nabla (f \circ \varphi(z)), \nabla (f \circ \varphi(z)) \right\rangle dxdy\\
= \iint\limits_{\Omega'} |\nabla f|^{2}(w) dudv.
\end{multline*}
Hence the following inequality
$$
\left(\iint\limits_{\Omega} |\nabla (f \circ \varphi(z))|^2~dxdy\right)^{\frac{1}{2}}
\leq K^{\frac{1}{2}} \left(\iint\limits_{\Omega'} |\nabla f|^{2}(w)~dudv\right)^{\frac{1}{2}}
$$
holds for any $f\in L^{1,2}(\Omega')$.
So, by \cite{VG75} we can conclude that the mapping $\varphi :\Omega \to \Omega'$ will be a $K$-quasiconformal mapping. Hence, by \cite{Ahl66} $\varphi$ will be a solution of the Beltrami equation
\begin{equation}\label{Beltr}
\varphi_{\overline{z}}(z)=\nu(z)\varphi_{z}(z),\,\,\, \text{a.e. in}\,\,\, \Omega
\end{equation}
with some complex dilatation $\nu(z)$, $|\nu(z)|<1$ a.e. in $\Omega$.
Now we consider the matrix $B$ generated by the complex dilatation $\nu(z)$:
\[
B(z)=\begin{pmatrix} \frac{|1-\nu|^2}{1-|\nu|^2} & \frac{-2 \Imag \nu}{1-|\nu|^2} \\ \frac{-2 \Imag \nu}{1-|\nu|^2} & \frac{|1+ \nu|^2}{1-|\nu|^2} \end{pmatrix},\,\,\, \text{a.e. in}\,\,\, \Omega.
\]
Then $\varphi$ is a $B$-quasiconformal mapping. Because $\varphi$ defined by (\ref{Beltr}) we have finally
\begin{equation}
\label{eqB}
\iint\limits_{\Omega} \left\langle B(z) \nabla (f \circ \varphi(z)), \nabla (f \circ \varphi(z)) \right\rangle dxdy
= \iint\limits_{\Omega'} |\nabla f|^{2}(w) dudv
\end{equation}
for any $f \in L^{1,2}(\Omega')$.
Now using the equalities (\ref{eqA}) and (\ref{eqB}) we obtain
\[
\iint\limits_{\Omega} \left\langle A(z) \nabla g(z), \nabla g(z) \right\rangle dxdy =
\iint\limits_{\Omega} \left\langle B(z) \nabla g(z), \nabla g(z) \right\rangle dxdy
\]
for any $g \in L^{1,2}_{A}(\Omega)$.
It means that Hilbert spaces $W^{1,2}_A(\Omega)$ and $W^{1,2}_B(\Omega)$ coincide. Therefore $A=B$ and $\mu=\nu$ a.e. in $\Omega$.
\end{proof}
Next, we set the following property for $A$-quasiconformal mappings.
\begin{lemma}
Let $\varphi : \Omega \to \mathbb D$ be an $A$-quasiconformal mapping. Then the inverse mapping $\psi=\varphi^{-1} : \mathbb D \to \Omega$ is $A^{-1}$-quasiconformal.
\end{lemma}
\begin{proof}
Let $\varphi : \Omega \to \mathbb D$ be an $A$-quasiconformal mapping with the matrix $A$ defined by the formula \eqref{Matrix-F}, i.e.
$$
A(z)= \begin{pmatrix} \frac{|1-\mu(z)|^2}{1-|\mu(z)|^2} & \frac{-2 \Imag \mu(z)}{1-|\mu(z)|^2} \\ \frac{-2 \Imag \mu(z)}{1-|\mu(z)|^2} & \frac{|1+\mu(z)|^2}{1-|\mu(z)|^2} \end{pmatrix},\,\,\, \text{a.e. in}\,\,\, \Omega.
$$
By \cite{Ahl66} it is known that the complex dilatation for the inverse mapping $\varphi^{-1} : \mathbb D \to \Omega$ satisfies
\[
\mu_{\varphi^{-1}}(w)=-\nu_{\varphi} \circ \varphi^{-1}(w)\,\,\,\text{for almost all}\,\,\,w\in \mathbb D,
\]
where
\[
\nu_{\varphi}=\frac{\varphi_{\overline{z}}}{\overline{\varphi_z}}=\left(\frac{\varphi_z}{|\varphi_z|}\right)^2\mu_{\varphi},\,\,\, \text{a.e. in}\,\,\, \Omega
\]
is called the second complex dilatation of $\varphi$.
Hence, the matrix $B$ induces by the complex dilatation $\mu_{\varphi^{-1}}$ of the inverse mapping $\varphi^{-1}$ has the form
$$
B(w)= \begin{pmatrix} \frac{|1+\nu_{\varphi} \circ \varphi^{-1}|^2}{1-|\nu_{\varphi} \circ \varphi^{-1}|^2} & \frac{2 \Imag (\nu_{\varphi} \circ \varphi^{-1})}{1-|\nu_{\varphi} \circ \varphi^{-1}|^2} \\ \frac{2 \Imag (\nu_{\varphi} \circ \varphi^{-1})}{1-|\nu_{\varphi} \circ \varphi^{-1}|^2} & \frac{|1-\nu_{\varphi} \circ \varphi^{-1}|^2}{1-|\nu_{\varphi} \circ \varphi^{-1}|^2} \end{pmatrix},
\,\,\,\text{a.e. in}\,\,\,\mathbb D.
$$
Because $\det B=1$, $|\mu_{\varphi}(z)|=|\mu_{\varphi^{-1}}(w)|$, $\Imag \mu_{\varphi} = -\Imag (\nu_{\varphi} \circ \varphi^{-1})$ for almost all $z\in \Omega$ and almost all $w=\varphi(z)\in \mathbb D$ we have
$$
A(z)B(\varphi(z))=I\,\,\,\text{for almost all}\,\,\, z\in \Omega.
$$
Therefore we conclude that $B(w)=A^{-1}(\varphi^{-1}(w))$ for almost all $w\in \mathbb D$ and $A^{-1}(z)=B(\varphi(z))$ for almost all $z\in \Omega$.
\end{proof}
\section{ Weighted Sobolev-Poincar\'e inequalities}
Denote by $B_{r,2}(\mathbb D)$, $1<r<\infty$, the best constant in the (non-weighted) Sobolev-Poincar\'e inequality in the unit disc $\mathbb D$. Exact calculations of $B_{r,2}(\mathbb D)$, $r\ne 2$, is an open problem and we use the upper estimate (see, for example, \cite{GT77,GU16}):
$$
B_{r,2}(\mathbb D) \leq \left(2^{-1} \pi\right)^{\frac{2-r}{2r}}\left(r+2\right)^{\frac{r+2}{2r}}.
$$
On the basis of Theorem~\ref{IsomSS} we prove an universal weighted Sobolev-Poincar\'e inequality which holds in any simply connected planar domain with non-empty boundary. Denote by $h(z) =|J(z,\varphi)|$ the quasihyperbolic weight defined by an $A$-quasiconformal mapping $\varphi : \Omega \to \mathbb D$.
\begin{theorem}\label{Th4.1}
Let $A$ belongs to a class $M^{2 \times 2}(\Omega)$ and $\Omega$ be a simply connected planar domain.
Then for any function $f \in W^{1,2}_{A}(\Omega)$ the following weighted Sobolev-Poincar\'e inequality
\[
\inf\limits_{c \in \mathbb R}\left(\iint\limits_\Omega |f(z)-c|^rh(z)dxdy\right)^{\frac{1}{r}} \leq B_{r,2}(h,A,\Omega)
\left(\iint\limits_\Omega \left\langle A(z) \nabla f(z), \nabla f(z) \right\rangle dxdy\right)^{\frac{1}{2}}
\]
holds for any $r \geq 1$ with the constant $B_{r,2}(h,A,\Omega) = B_{r,2}(\mathbb D)$.
\end{theorem}
\begin{proof}
Because $\Omega$ is a simply connected planar domain, then there exists \cite{Ahl66} an $\mu$-quasiconformal homeomorphism $\varphi : \Omega \to \mathbb D$ with
\begin{equation*}
\mu(z)=\frac{a_{22}(z)-a_{11}(z)-2ia_{12}(z)}{\det(I+A(z))},
\end{equation*}
which is an $A$-quasiconformal mapping.
Hence by Theorem~\ref{IsomSS} the equality
\begin{equation}\label{IN2.1}
||f \circ \varphi^{-1} \,|\, L^{1,2}(\mathbb D)|| = ||f \,|\, L^{1,2}_{A}(\Omega)||
\end{equation}
holds for any function $f \in L^{1,2}_{A}(\Omega)$.
Denote by $h(z):=|J(z,\varphi)|$ the quasihyperbolic weight in $\Omega$.
Now using the change of variable formula for the quasiconformal mappings \cite{VGR}, the equality \eqref{IN2.1} and the classical Sobolev-Poincar\'e inequality for the unit disc $\mathbb D$ \cite{M}
\begin{equation*}\label{IN2.3}
\inf\limits_{c \in \mathbb R}\left(\iint\limits_{\mathbb D} |f \circ \varphi^{-1}(w)-c|^rdudv\right)^{\frac{1}{r}} \\
\leq B_{r,2}(\mathbb D)
\left(\iint\limits_{\mathbb D} \nabla (f \circ \varphi^{-1}(w))dudv\right)^{\frac{1}{2}}
\end{equation*}
that holds for any $r \geq 1$, we obtain that for any smooth function $f\in L^{1,2}_{A}(\Omega)$
\begin{multline*}
\inf\limits_{c \in \mathbb R}\left(\iint\limits_\Omega |f(z)-c|^rh(z)dxdy\right)^{\frac{1}{r}}
= \inf\limits_{c \in \mathbb R}\left(\iint\limits_\Omega |f(z)-c|^r |J(z,\varphi)| dxdy\right)^{\frac{1}{r}} \\
= \inf\limits_{c \in \mathbb R}\left(\iint\limits_{\mathbb D} |f \circ \varphi^{-1}(w)-c|^rdudv\right)^{\frac{1}{r}}
\leq B_{r,2}(\mathbb D)
\left(\iint\limits_{\mathbb D} \nabla (f \circ \varphi^{-1}(w))dudv\right)^{\frac{1}{2}} \\
= B_{r,2}(\mathbb D)
\left(\iint\limits_{\Omega} \left\langle A(z) \nabla g(z), \nabla f(z) \right\rangle dxdy\right)^{\frac{1}{2}}.
\end{multline*}
Approximating an arbitrary function $f \in W^{1,2}_{A}(\Omega)$ by smooth functions we obtain finally
$$
\inf\limits_{c \in \mathbb R}\left(\iint\limits_\Omega |f(z)-c|^rh(z)dxdy\right)^{\frac{1}{r}} \leq
B_{r,2}(h,A,\Omega) \left(\iint\limits_{\Omega} \left\langle A(z) \nabla f(z), \nabla f(z) \right\rangle dxdy\right)^{\frac{1}{2}},
$$
with the constant
$$
B_{r,2}(h,A,\Omega)=B_{r,2}(\mathbb D) \leq \left(2^{-1} \pi\right)^{\frac{2-r}{2r}}\left(r+2\right)^{\frac{r+2}{2r}}.
$$
\end{proof}
\section{ Estimates of Sobolev-Poincar\'e constants}
In this section we consider (sharp) upper estimates of Sobolev-Poincar\'e constants in domains that satisfy the quasihyperbolic boundary condition.
Recall that a domain $\Omega$ satisfy the $\gamma$-quasihyperbolic boundary condition \cite{KOT01,KOT02} with some $\gamma>0$, if the growth condition on the quasihyperbolic metric
$$
k_{\Omega}(x_0,x)\leq \frac{1}{\gamma}\log\frac{\dist(x_0,\partial\Omega)}{\dist(x,\partial\Omega)}+C_0
$$
is satisfied for all $x\in\Omega$, where $x_0\in\Omega$ is a fixed base point and $C_0=C_0(x_0)<\infty$.
This quasihyperbolic boundary condition is equivalent to integrability of Jacobians of corresponding quasiconformal mappings with some exponent $\beta>1$. Let us reformulate a theorem about integrability of the Jacobians from \cite{AK} in the convenient for our study form. Firstly, recall that for quasiconformal mappings $\psi:\mathbb D\to\Omega$
the volume derivative
$$
J_{\psi}(w):=\lim\limits_{r\to 0}\frac{|\psi(B(w,r))|}{|B(w,r)|}=|J(w, \psi)|
$$
is defined for almost all $w\in\mathbb D$.
\begin{theorem} \cite{AK} Let $\psi: \mathbb{D} \to \Omega$ be a quasiconformal mapping. Then $J_{\psi} \in L^{\beta}(\mathbb{D})$ for some $\beta>1$ if and only if $\Omega$ satisfy to a $\gamma$-quasihyperbolic boundary conditions for some $\gamma$.
\end{theorem}
Let us remark that the degree of integrability $\beta$ depends only on
$\Omega$ and the quasiconformility coefficient $K(\psi)$.
Our main goal is a reduction of weighted Sobolev-Poincar\'e inequalities to non-weighted embedding theorems. This goal requires the exact value of integrability exponent of Jacobians. It leads us to a following new definition.
Namely, we say that a simply connected domain $\Omega \subset \mathbb C$ is called an $A$-quasiconformal $\beta$-regular domain, $\beta >1$, if
$$
\iint\limits_\mathbb D |J(w, \varphi^{-1})|^{\beta}~dudv < \infty,
$$
where $\varphi: \Omega\to\mathbb D$ is a corresponding $A$-quasiconformal mapping.
Since (see, for example, \cite{Ahl66}) $A$-quasiconformal mappings $\varphi: \Omega\to\mathbb D$ are defined up to conformal automorphisms of $\mathbb D$, a property of quasiconformal $\beta$-regularity doesn't depend on a choice of $\varphi$ and depends on the "quasihyperbolic geometry" of $\Omega$ only.
Of course, any quasiconformal $\beta$-regular domain satisfies to some $\gamma$-quasihyperbolic boundary conditions and the class of all quasiconformal regular domains coincide with the class of all domains satisfying to the quasihyperbolic boundary conditions.
In \cite{GU14} it was proved that if $\Omega \subset \mathbb C$ is an $A$-quasiconformal $\beta$-regular domain, $\beta >1$, then $\Omega$ has a finite geodesic diameter. Hence, "maze-like" domains \cite{GPU18,KOT02} are not $A$-quasiconformal $\beta$-regular domains.
Ahlfors domains \cite{Ahl66} (quasidiscs \cite{VGR}) represent an important subclass of $A$-quasi\-con\-for\-mal $\beta$-regular domains. Moreover, in these domains spectral estimates can be specified in terms of the "quasiconformal geometry" of domains (Section~6).
The following theorem represents a Sobolev type embedding theorem with estimates of the norm of the embedding operator in quasiconformal regular domains.
\begin{theorem}\label{Th4.3}
Let $A$ belongs to a class $M^{2 \times 2}(\Omega)$ and $\Omega$ be an $A$-quasiconformal $\beta$-regular domain. Then:
\begin{enumerate}
\item the embedding operator
\[
i_{\Omega}:W^{1,2}_{A}(\Omega) \hookrightarrow L^s(\Omega)
\]
is compact for any $s \geq 1$;
\item for any function $f \in W^{1,2}_{A}(\Omega)$ and for any $s \geq 1$, the Sobolev-Poincar\'e inequality
\[
\inf\limits_{c \in \mathbb R}\|f-c\mid L^s(\Omega)\| \leq B_{s,2}(A,\Omega)
\|f\mid L^{1,2}_{A}(\Omega)\|
\]
holds with the constant
$$
B_{s,2}(A,\Omega) \leq B_{\frac{\beta s}{\beta-1},2}(\mathbb D) \|J_{\varphi^{-1}}\mid L^{\beta}(\mathbb D)\|^{\frac{1}{s}},
$$
where $J_{\varphi^{-1}}$ is a Jacobian of the quasiconformal mapping $\varphi^{-1}:\mathbb D\to\Omega$.
\end{enumerate}
\end{theorem}
\begin{proof}
Let $s \geq 1$. Since $\Omega$ is an $A$-quasiconformal $\beta$-regular domain,
then there exists an $A$-quasiconformal mapping $\varphi: \Omega \to \mathbb D$ satisfies the condition of $\beta$-regularity:
\[
\iint\limits_\mathbb D \big|J(w,\varphi^{-1})\big|^{\beta}~dudv < \infty.
\]
Hence \cite{VU04} the composition operator for Lebesgue spaces
\[
\varphi^{*}:L^r(\mathbb D) \to L^s(\Omega)
\]
is bounded for $r/(r-s)=\beta$ i.e. for $r=\beta s/(\beta -1)$.
By Theorem~\ref{IsomSS} $A$-quasiconformal mappings $\varphi: \Omega \to \mathbb D$ generate a bounded composition operator on seminormed Sobolev spaces
\[
(\varphi^{-1})^{*}:L^{1,2}_{A}(\Omega) \to L^{1,2}(\mathbb D).
\]
Because the matrix $A$ satisfies to the uniform ellipticity condition~\eqref{UEC} then the norm of Sobolev space $W^{1,2}_{A}(\Omega)$ is equivalent to the norm of Sobolev space $W^{1,2}(\Omega)$ and by \cite{GPU19} we obtain that the composition operator on normed Sobolev spaces
\[
(\varphi^{-1})^{*}:W^{1,2}_{A}(\Omega) \to W^{1,2}(\mathbb D),\,\,\, (\varphi^{-1})^{*}(f)=f \circ \varphi^{-1} ,
\]
is bounded.
Therefore according to the "transfer" diagram \cite{GG94} we obtain that the embedding operator
\[
i_{\Omega}:W^{1,2}_{A}(\Omega) \hookrightarrow L^s(\Omega)
\]
is compact as a composition of three operators: the bounded composition operator on Sobolev spaces
$(\varphi^{-1})^{*}:W^{1,2}_{A}(\Omega) \to W^{1,2}(\mathbb D)$, the compact embedding operator
\[
i_{\mathbb D}:W^{1,2}(\mathbb D) \hookrightarrow L^r(\mathbb D)
\]
and the bounded composition operator on Lebesgue spaces $\varphi^{*}:L^r(\mathbb D) \to L^s(\Omega)$.
Let $s=\frac{\beta -1}{\beta}r$ then by \cite{GPU19} the inequality
\begin{equation}
\label{Weight}
||f\,|\,L^s(\Omega)|| \leq \left(\iint\limits_\mathbb D \big|J(w,\varphi^{-1})\big|^{\beta}~dudv \right)^{{\frac{1}{\beta}} \cdot \frac{1}{s}} ||f\,|\,L^r(\Omega,h)||
\end{equation}
holds for any function $f\in L^{r}(\Omega,h)$.
Using Theorem \ref{Th4.1} and inequality~\eqref{Weight} we have
\begin{multline*}
\inf_{c \in \mathbb R} \left(\iint\limits_{\Omega} |f(z)-c|^s dxdy\right)^{\frac{1}{s}} \\
{} \leq \left(\iint\limits_\mathbb D \big|J(w,\varphi^{-1})\big|^{\beta}dudv \right)^{{\frac{1}{\beta}} \cdot \frac{1}{s}}
\inf_{c \in \mathbb R} \left(\iint\limits_{\Omega} |f(z)-c|^r h(z)dxdy\right)^{\frac{1}{r}} \\
{} \leq B_{r,2}(\mathbb D)
\left(\iint\limits_\mathbb D \big|J(w,\varphi^{-1})\big|^{\beta}dudv \right)^{{\frac{1}{\beta}} \cdot \frac{1}{s}}
\left(\iint\limits_\Omega \left\langle A(z) \nabla f(z), \nabla f(z) \right\rangle dxdy\right)^{\frac{1}{2}}
\end{multline*}
for $s\geq 1$.
\end{proof}
The following theorem gives compactness of the embedding operator in the limit case $\beta = \infty$:
\begin{theorem}\label{T4.5}
Let $A$ belongs to a class $M^{2 \times 2}(\Omega)$ and $\Omega$ be an $A$-quasiconformal $\infty$-regular domain. Then:
\begin{enumerate}
\item The embedding operator
\[
i_{\Omega}:W^{1,2}_{A}(\Omega) \hookrightarrow L^2(\Omega),
\]
is compact.
\item For any function $f \in W^{1,2}_{A}(\Omega)$, the Poincar\'e--Sobolev inequality
\[
\inf\limits_{c \in \mathbb R}\|f-c\mid L^2(\Omega)\| \leq B_{2,2}(A,\Omega)
\|f\mid L^{1,2}_{A}(\Omega)\|
\]
holds with the constant $B_{2,2}(A,\Omega) \leq B_{2,2}(\mathbb D) \big\|J_{\varphi^{-1}}\mid L^{\infty}(\mathbb D)\big\|^{\frac{1}{2}}$,
where $J_{\varphi^{-1}}$ is a Jacobian of the quasiconformal mapping $\varphi^{-1}:\mathbb D\to\Omega$.
\end{enumerate}
\end{theorem}
\begin{remark}
The constant $B_{2,2}^2(\mathbb D)=1/\mu_1(\mathbb D)$, where $\mu_1(\mathbb D)=j'^2_{1,1}$ is the first non-trivial Neumann eigenvalue of Laplacian in the unit disc $\mathbb D\subset\mathbb C$.
\end{remark}
\begin{proof}
Since $\Omega$ is an $A$-quasiconformal $\infty$-regular domain,
then there exists an $A$-quasiconformal mapping $\varphi: \Omega \to \mathbb D$ that
generates a bounded composition operator
\[
(\varphi^{-1})^{*}:L^{1,2}_{A}(\Omega) \to L^{1,2}(\mathbb D).
\]
Using the embedding $L^{1,2}(\mathbb D)\subset L^2(\mathbb D)$ (see, for example, \cite{M}) we obtain that the composition operator on normed Sobolev spaces
\[
(\varphi^{-1})^{*}:W^{1,2}_{A}(\Omega) \to W^{1,2}(\mathbb D)
\]
is bounded also.
Because $\Omega$ is an $A$-quasiconformal $\infty$-regular domain, then the $A$-quasiconformal mapping $\varphi: \Omega \to \mathbb D$
satisfies the following condition:
\[
\big\|J_{\varphi^{-1}}\mid L^{\infty}(\mathbb D)\big\|=\esssup\limits_{|w|<1}|J(w,\varphi^{-1})|<\infty,
\]
and we have that the composition operator
$$
\varphi^{*}:L^2(\mathbb D) \to L^2(\Omega)
$$
is bounded \cite{VU04}.
Finally, note that in the unit disc $\mathbb D$ the embedding operator
$$
i_{\mathbb D}:W^{1,2}(\mathbb D) \hookrightarrow L^2(\mathbb D)
$$
is compact (see, for example, \cite{M}).
Therefore the embedding operator
$$
i_{\Omega}:W^{1,2}_A(\Omega)\to L_2(\Omega)
$$
is compact as a composition of bounded composition operators $\varphi^{*}$, $(\varphi^{-1})^{*}$ and the compact embedding operator $i_{\mathbb D}$.
Let a function $f \in L^{2}(\Omega)$. Because quasiconformal mappings possess the Luzin $N$-property, then
$|J(z,\varphi)|^{-1}=|J(w,\varphi^{-1})|$ for almost all $z\in\Omega$ and for almost all $w=\varphi(z)\in \mathbb D$. Hence
the following inequality is correct:
\begin{multline*}
\inf\limits_{c \in \mathbb R}\left(\iint\limits_\Omega |f(z)-c|^2dxdy\right)^{\frac{1}{2}} =
\inf\limits_{c \in \mathbb R}\left(\iint\limits_\Omega |f(z)-c|^2 |J(z,\varphi)|^{-1} |J(z,\varphi)|~dxdy\right)^{\frac{1}{2}} \\
{} \leq \big\|J_{\varphi}\mid L^{\infty}(\Omega)\big\|^{-\frac{1}{2}}
\inf\limits_{c \in \mathbb R}\left(\iint\limits_\Omega |f(z)-c|^2 |J(z,\varphi)|~dxdy\right)^{\frac{1}{2}}.
\end{multline*}
By Theorem~\ref{Th4.1} we obtain
\begin{multline*}
\inf\limits_{c \in \mathbb R}\left(\iint\limits_\Omega |f(z)-c|^2dxdy\right)^{\frac{1}{2}}
\leq \big\|J_{\varphi^{-1}}\mid L^{\infty}(\mathbb D)\big\|^{\frac{1}{2}}
\inf\limits_{c \in \mathbb R}\left(\iint\limits_\mathbb D |g(w)-c|^2~dudv\right)^{\frac{1}{2}} \\
{} \leq B_{2,2}(\mathbb D) \big\|J_{\varphi^{-1}}\mid L^{\infty}(\mathbb D)\big\|^{\frac{1}{2}}
\left(\iint\limits_\Omega \left\langle A(z) \nabla f(z), \nabla f(z) \right\rangle dxdy\right)^{\frac{1}{2}},
\end{multline*}
for any $f\in L_A^{1,2}(\Omega)$.
\end{proof}
\section{Eigenvalue Problem for Neumann Divergence Form Elliptic Operator}
We consider the weak formulation of the Neumann eigenvalue problem~(\ref{EllDivOper}):
\begin{equation}\label{WFWEP}
\iint\limits_\Omega \left\langle A(z )\nabla f(z), \nabla \overline{g(z)} \right\rangle dxdy
= \mu \iint\limits_\Omega f(z)\overline{g(z)}~dxdy, \,\,\, \forall g\in W_{A}^{1,2}(\Omega).
\end{equation}
By the Min--Max Principle (see, for example, \cite{Henr}) the first non-trivial Neumann eigenvalue $\mu_1(\Omega)$ of the divergence form elliptic operator $L_{A}=-\textrm{div} [A(z) \nabla f(z)]$ can be characterized as
$$
\mu_1(A,\Omega)=\min\left\{\frac{\|f \mid L^{1,2}_{A}(\Omega)\|^2}{\|f \mid L^{2}(\Omega)\|^2}:
f \in W^{1,2}_{A}(\Omega) \setminus \{0\},\,\, \iint\limits _{\Omega}f\, dxdy=0 \right\}.
$$
Hence $\mu_1(A,\Omega)^{-\frac{1}{2}}$ is the best constant $B_{2,2}(A,\Omega)$ in the following Poincar\'e inequality
$$
\inf\limits _{c \in \mathbb R} \|f-c \mid L^2(\Omega)\| \leq B_{2,2}(A,\Omega) \|f \mid L^{1,2}_{A}(\Omega)\|, \quad f \in W^{1,2}_{A}(\Omega).
$$
\begin{theorem}\label{Th5.1}
Let $A$ belongs to a class $M^{2 \times 2}(\Omega)$ and $\Omega$ be an $A$-quasiconformal $\beta$-regular domain. Then the spectrum of the Neumann divergence form elliptic operator $L_{A}$ in $\Omega$ is discrete,
and can be written in the form of a non-decreasing sequence:
\[
0=\mu_0(A,\Omega)<\mu_1(A,\Omega)\leq \mu_2(A,\Omega)\leq \ldots \leq \mu_n(A,\Omega)\leq \ldots ,
\]
and
\[
\frac{1}{\mu_1(A,\Omega)} \leq B_{\frac{2\beta}{\beta -1},2}(\mathbb D)
\|J_{\varphi^{-1}}\mid L^{\beta}(\mathbb D)\|
{} \leq
\frac{4}{\sqrt[\beta]{\pi}} \left(\frac{2\beta -1}{\beta -1}\right)^{\frac{2 \beta-1}{\beta}} \big\|J_{\varphi^{-1}}\mid L^{\beta}(\mathbb D)\big\|,
\]
where $J_{\varphi^{-1}}$ is a Jacobian of the quasiconformal mapping $\varphi^{-1}:\mathbb D\to\Omega$.
\end{theorem}
\begin{proof}
By Theorem~\ref{Th4.3} in the case $s=2$, the embedding operator
$$
i_{\Omega}:W^{1,2}_{A}(\Omega)\to L_2(\Omega)
$$
is compact. Hence the spectrum of the Neumann divergence form elliptic operator $L_{A}$ is discrete and can be written in the form of a non-decreasing sequence
\[
0=\mu_0(A,\Omega)<\mu_1(A,\Omega)\leq \mu_2(A,\Omega)\leq \ldots \leq \mu_n(A,\Omega)\leq \ldots .
\]
By the Min-Max principle and Theorem~\ref{Th4.3} we have
\[
\inf_{c \in \mathbb R} \left(\iint\limits_{\Omega} |f(z)-c|^2 dxdy\right) \leq B^2_{2,2}(A,\Omega)
\iint\limits_\Omega \left\langle A(z) \nabla f(z), \nabla f(z) \right\rangle dxdy,
\]
where
\[
B_{2,2}(A,\Omega) \leq B_{r,2}(\mathbb D)
\left(\iint\limits_\mathbb D \big|J(w,\varphi^{-1})\big|^{\beta}~dudv \right)^{{\frac{1}{2\beta}}}.
\]
Hence
\[
\frac{1}{\mu_1(A,\Omega)} \leq B^2_{r,2}(\mathbb D)
\left(\iint\limits_\mathbb D \big|J(w,\varphi^{-1})\big|^{\beta}~dudv \right)^{{\frac{1}{\beta}}}.
\]
Using the upper estimate of the $(r,2)$-Poincar\'e constant in the unit disc (see, for example, \cite{GT77,GU16})
\[
B_{r,2}(\mathbb D) \leq \left(2^{-1} \pi\right)^{\frac{2-r}{2r}}\left(r+2\right)^{\frac{r+2}{2r}},
\]
where by Theorem~\ref{Th4.3}, $r=2\beta /(\beta -1)$,
we obtain
\[
\frac{1}{\mu_1(A,\Omega)} \leq
\frac{4}{\sqrt[\beta]{\pi}} \left(\frac{2\beta -1}{\beta -1}\right)^{\frac{2 \beta-1}{\beta}} \big\|J_{\varphi^{-1}}\mid L^{\beta}(\mathbb D)\big\|.
\]
\end{proof}
In the case of $A$-quasiconformal $\infty$-regular domains we have:
\begin{theorem}\label{T4.7}
Let $A$ belongs to a class $M^{2 \times 2}(\Omega)$ and $\Omega$ be an $A$-quasiconformal $\infty$-regular domain. Then the spectrum of the Neumann divergence form elliptic operator $L_{A}$ in $\Omega$ is discrete,
and can be written in the form of a non-decreasing sequence:
\[
0=\mu_0(A,\Omega)<\mu_1(A,\Omega)\leq \mu_2(A,\Omega)\leq \ldots \leq \mu_n(A,\Omega)\leq \ldots ,
\]
and
\begin{equation}
\frac{1}{\mu_1(A,\Omega)} \leq B^2_{2,2}(\mathbb D) \big\|J_{\varphi^{-1}}\mid L^{\infty}(\mathbb D)\big\|
= \frac{\big\|J_{\varphi^{-1}}\mid L^{\infty}(\mathbb D)\big\|}{(j'_{1,1})^2},
\end{equation}
where $j'_{1,1}\approx 1.84118$ denotes the first positive zero of the derivative of the Bessel function $J_1$, and
$J_{\varphi^{-1}}$ is a Jacobian of the quasiconformal mapping $\varphi^{-1}:\mathbb D\to\Omega$.
\end{theorem}
As an applications of Theorem~\ref{T4.7} we consider some examples.
\begin{example}
\label{example1}
The homeomorphism
\[
\varphi(z)= \frac{a}{a^2-b^2}z- \frac{b}{a^2-b^2} \overline{z}, \quad z=x+iy, \quad a>b\geq 0,
\]
is an $A$-quasiconformal and maps the interior of ellipse
$$
\Omega_e= \left\{(x,y) \in \mathbb R^2: \frac{x^2}{(a+b)^2}+\frac{y^2}{(a-b)^2}=1\right\}
$$
onto the unit disc $\mathbb D.$ The mapping $\varphi$ satisfies the Beltrami equation with
\[
\mu(z)=\frac{\varphi_{\overline{z}}}{\varphi_{z}}=-\frac{b}{a}
\]
and the Jacobian $J(z,\varphi)=|\varphi_{z}|^2-|\varphi_{\overline{z}}|^2=1/(a^2-b^2)$.
It is easy to verify that $\mu$ induces, by formula \eqref{Matrix-F}, the matrix function $A(z)$ form
$$
A(z)=\begin{pmatrix} \frac{a+b}{a-b} & 0 \\ 0 & \frac{a-b}{a+b} \end{pmatrix}.
$$
Given that $|J(w,\varphi^{-1})|=|J(z,\varphi)|^{-1}=a^2-b^2$. Then by Theorem~\ref{T4.7} we have
$$
\frac{1}{\mu_1(A,\Omega_e)} \leq
\frac{1}{(j'_{1,1})^2} \esssup\limits_{|w|<1}|J(w,\varphi^{-1})| = \frac{a^2-b^2}{(j'_{1,1})^2}.
$$
The classical estimate~\ref{PW} with the uniform ellipticity condition states that
$$
\mu_1(A,\Omega_e) \geq \frac{\pi^2}{4(a+b)^2} \frac{a-b}{a+b}
$$
and we have that
$$
\frac{\pi^2}{4(a+b)^2} \frac{a-b}{a+b}< \frac{(j'_{1,1})^2}{a^2-b^2}.
$$
\end{example}
\begin{example}
The homeomorphism
\[
\varphi(z)= \frac{z^{\frac{3}{2}}}{\sqrt{2} \cdot \overline{z}^{\frac{1}{2}}}-1,\,\, \varphi(0)=-1, \quad z=x+iy,
\]
is an $A$-quasiconformal and maps the interior of the ``rose petal"
$$
\Omega_p:=\left\{(\rho, \theta) \in \mathbb R^2:\rho=2\sqrt{2}\cos(2 \theta), \quad -\frac{\pi}{4} \leq \theta \leq \frac{\pi}{4}\right\}
$$
onto the unit disc $\mathbb D$.
The mapping $\varphi$ satisfies the Beltrami equation with
\[
\mu(z)=\frac{\varphi_{\overline{z}}}{\varphi_{z}}=-\frac{1}{3}\frac{z}{\overline{z}}
\]
and the Jacobian $J(z,\varphi)=|\varphi_{z}|^2-|\varphi_{\overline{z}}|^2=1$.
We see that $\mu$ induces, by formula \eqref{Matrix-F}, the matrix function $A(z)$ form
$$
A(z)=\begin{pmatrix} \frac{|3\overline{z}+z|^2}{8|\overline{z}|^2} & \frac{3}{4}\Imag \frac{z}{\overline{z}} \\ \frac{3}{4}\Imag \frac{z}{\overline{z}} & \frac{|3\overline{z}-z|^2}{8|\overline{z}|^2} \end{pmatrix}.
$$
Given that $|J(w,\varphi^{-1})|=|J(z,\varphi)|^{-1}=1$. Then by Theorem~\ref{T4.7} we have
$$
\frac{1}{\mu_1(A,\Omega_p)} \leq
\frac{1}{(j'_{1,1})^2} \esssup\limits_{|w|<1}|J(w,\varphi^{-1})| = \frac{1}{(j'_{1,1})^2}.
$$
The classical estimate~\ref{PW} with the uniform ellipticity condition states that
$$
\mu_1(A,\Omega_p) \geq \left(\frac{\pi}{4}\right)^2
$$
and we have that
$$
\left(\frac{\pi}{4}\right)^2<(j'_{1,1})^2 \quad \text{or} \quad \frac{\pi}{4}<j'_{1,1}.
$$
\end{example}
\begin{example}
The homeomorphism
\[
\varphi(z)= \frac{2 \cdot z^{\frac{3}{8}}}{\overline{z}^{\frac{1}{8}}}-1,\,\, \varphi(0)=-1, \quad z=x+iy,
\]
is an $A$-quasiconformal and maps the interior of the non-convex domain
$$
\Omega_c:=\left\{(\rho, \theta) \in \mathbb R^2:\rho=\cos^{4}\left(\frac{\theta}{2}\right), \quad - \pi \leq \theta \leq \pi\right\}
$$
onto the unit disc $\mathbb D$.
The mapping $\varphi$ satisfies the Beltrami equation with
\[
\mu(z)=\frac{\varphi_{\overline{z}}}{\varphi_{z}}=-\frac{1}{3}\frac{z}{\overline{z}}
\]
and the Jacobian
$$J(z,\varphi)=|\varphi_{z}|^2-|\varphi_{\overline{z}}|^2=\frac{1}{2\cdot |z|^{\frac{3}{2}}}.
$$
We see that $\mu$ induces, by formula \eqref{Matrix-F}, the matrix function $A(z)$ form
$$
A(z)=\begin{pmatrix} \frac{|3\overline{z}+z|^2}{8|\overline{z}|^2} & \frac{3}{4}\Imag \frac{z}{\overline{z}} \\ \frac{3}{4}\Imag \frac{z}{\overline{z}} & \frac{|3\overline{z}-z|^2}{8|\overline{z}|^2} \end{pmatrix}.
$$
Given that $|J(w,\varphi^{-1})|=|J(z,\varphi)|^{-1}=2\cdot |z|^{\frac{3}{2}}$. Then by Theorem~\ref{T4.7} we have
$$
\frac{1}{\mu_1(A,\Omega_c)} \leq
\frac{1}{(j'_{1,1})^2} \esssup\limits_{|w|<1}|J(w,\varphi^{-1})| \leq \frac{2}{(j'_{1,1})^2}.
$$
\end{example}
\section{Spectral estimates in quasidiscs}
In this section we precise Theorem~\ref{Th5.1} for Ahlfors-type domains (i.e. quasidiscs) using the weak inverse H\"older inequality and the sharp estimates of the constants in doubling conditions for measures generated by Jacobians of quasiconformal mappings \cite{GPU17_2}.
Recall that a domain $\Omega$ is called a $K$-quasidisc if it is the image of the unit disc $\mathbb D$ under a $K$-quasicon\-for\-mal homeomorphism of the plane onto itself. A domain $\Omega$ is a quasidisc if it is a $K$-quasidisc for some $K \geq 1$.
According to \cite{GH01}, the boundary of any $K$-quasidisc $\Omega$
admits a $K^{2}$-quasi\-con\-for\-mal reflection and thus, for example,
any quasiconformal homeomorphism $\psi:\mathbb{D}\to\Omega$ can be
extended to a $K^{2}$-quasiconformal homeomorphism of the whole plane
to itself.
Recall that for any planar $K$-quasiconformal homeomorphism $\psi:\Omega\rightarrow \Omega'$
the following sharp result is known: $J(w,\psi)\in L^p_{\loc}(\Omega)$
for any $1 \leq p<\frac{K}{K-1}$ (\cite{Ast,G81}).
In \cite{GPU17_2} was proved but not formulated the result concerning an estimate of the constant in the inverse H\"older inequality for Jacobians of quasiconformal mappings.
\vskip 0.2cm
\begin{theorem}
\label{thm:IHIN}
Let $\psi:\mathbb R^2 \to \mathbb R^2$ be a $K$-quasiconformal mapping. Then for every disc $\mathbb D \subset \mathbb R^2$ and
for any $1<\kappa<\frac{K}{K-1}$ the inverse H\"older inequality
\begin{equation*}\label{RHJ}
\left(\iint\limits_{\mathbb D} |J(w,\psi)|^{\kappa}~dudv \right)^{\frac{1}{\kappa}}
\leq \frac{C_\kappa^2 K \pi^{\frac{1}{\kappa}-1}}{4}
\exp\left\{{\frac{K \pi^2(2+ \pi^2)^2}{2\log3}}\right\}\iint\limits_{\mathbb D} |J(w,\psi)|~dudv
\end{equation*}
holds. Here
$$
C_\kappa=\frac{10^{6}}{[(2\kappa -1)(1- \nu)]^{1/2\kappa}}, \quad \nu = 10^{8 \kappa}\frac{2\kappa -2}{2\kappa -1}(24\pi^2K)^{2\kappa}<1.
$$
\end{theorem}
If $\Omega$ is a $K$-quasidisc, then given the previous theorem and that a quasiconformal mapping $\psi:\mathbb{D}\to\Omega$ allows $K^2$-quasiconformal reflection \cite{Ahl66, GH01}, we obtain the following assertion.
\begin{corollary}\label{Est_Der}
Let $\Omega\subset\mathbb R^2$ be a $K$-quasidisc and $\varphi:\Omega \to \mathbb D$ be an $A$-quasiconformal mapping. Assume that $1<\kappa<\frac{K}{K-1}$.
Then
\begin{equation*}\label{Ineq_2}
\left(\iint\limits_{\mathbb D} |J(w,\varphi^{-1})|^{\kappa}~dudv \right)^{\frac{1}{\kappa}}
\leq \frac{C_\kappa^2 K^2 \pi^{\frac{1}{\kappa}-1}}{4}
\exp\left\{{\frac{K^2 \pi^2(2+ \pi^2)^2}{2\log3}}\right\}\cdot |\Omega|.
\end{equation*}
where
$$
C_\kappa=\frac{10^{6}}{[(2\kappa -1)(1- \nu)]^{1/2\kappa}}, \quad \nu = 10^{8 \kappa}\frac{2\kappa -2}{2\kappa -1}(24\pi^2K^2)^{2\kappa}<1.
$$
\end{corollary}
Combining Theorem~\ref{Th5.1} and Corollary~\ref{Est_Der} we obtain
spectral estimates of linear elliptic operators in divergence form
with Neumann boundary conditions in Ahlfors-type domains.
\begin{theorem}\label{Quasidisk}
Let $\Omega$ be a $K$-quasidisc. Then
\begin{equation*}
\mu_1(A,\Omega) \geq \frac{M(K)}{|\Omega|}=\frac{M^{*}(K)}{R^2_{*}},
\end{equation*}
where $R_{*}$ is a radius of a disc $\Omega^{*}$ of the same area as $\Omega$ and
$M^{*}(K)=M(K)\pi^{-1}$.
\end{theorem}
The quantity $M(K)$ depends only on a quasiconformality
coefficient K of $\Omega$:
\[
M(K):= \frac{\pi}{K^2}
\exp\left\{{-\frac{K^2 \pi^2(2+ \pi^2)^2}{2\log3}}\right\}
\inf\limits_{1< \beta <\beta^{*}}
\Biggl\{
\left(\frac{2\beta -1}{\beta -1}\right)^{-\frac{2 \beta-1}{\beta}} C^{-2}_{\beta}
\Biggr\},
\]
\[
C_\beta=\frac{10^{6}}{[(2\beta -1)(1- \nu(\beta))]^{1/2\beta}},
\]
where $\beta^{*}=\min{\left(\frac{K}{K-1}, \widetilde{\beta}\right)}$, and $\widetilde{\beta}$ is the unique solution of the equation
$$\nu(\beta):=10^{8 \beta}\frac{2\beta -2}{2\beta -1}(24\pi^2K^2)^{2\beta}=1.
$$
The function $\nu(\beta)$ is a monotone increasing function. Hence for
any $\beta < \beta^{*}$ the number $(1- \nu(\beta))>0$ and $C_\beta > 0$.
\begin{proof}
Given that, for $K\geq 1$, $K$-quasidiscs are $A$-quasiconformal $\beta$-regular domains if $1<\beta<\frac{K}{K-1}$. Therefore, by Theorem~\ref{Th5.1} for $1<\beta<\frac{K}{K-1}$ we have
\begin{equation}\label{Inequal_1}
\frac{1}{\mu_1(A,\Omega)} \leq
\frac{4}{\sqrt[\beta]{\pi}} \left(\frac{2\beta -1}{\beta -1}\right)^{\frac{2 \beta-1}{\beta}} \big\|J_{\varphi^{-1}}\mid L^{\beta}(\mathbb D)\big\|.
\end{equation}
Now, using Corollary~\ref{Est_Der} we estimate the quantity $\|J_{\varphi^{-1}}\,|\,L^{\beta}(\mathbb D)\|$.
Direct calculations yield
\begin{multline}\label{Inequal_2}
\|J_{\varphi^{-1}}\,|\,L^{\beta}(\mathbb D)\| =
\left(\iint\limits_{\mathbb D} |J(w,\varphi^{-1})|^{\beta}~dudv \right)^{\frac{1}{\beta}} \\
\leq \frac{C^2_{\beta} K^2 \pi^{\frac{1-\beta}{\beta}}}{4} \exp\left\{{\frac{K^2 \pi^2(2+ \pi^2)^2}{2\log3}}\right\} \cdot |\Omega|.
\end{multline}
Finally, combining inequality \eqref{Inequal_1} with inequality \eqref{Inequal_2} after some computations, we obtain
\[
\frac{1}{\mu_1(A,\Omega)} \leq
\frac{C^2_{\beta} K^2}{\pi} \left(\frac{2\beta -1}{\beta -1}\right)^{\frac{2 \beta-1}{\beta}} \exp\left\{{\frac{K^2 \pi^2(2+ \pi^2)^2}{2\log3}}\right\} \cdot |\Omega|.
\]
\end{proof}
Let $\varphi:\Omega \to \Omega'$ be quasiconformal mappings. We note that there exist so-called volume-preserving maps, i.e. $|J(z,\varphi)|=1$, $z \in \Omega$. Examples of such maps were considered in the previous section.
Now we construct another examples of such maps.
Let $f\in L^{\infty}(\mathbb R)$. Then $\varphi(x,y)=(x+f(y),\,y)$ is a quasiconformal mapping with a quasiconformality coefficient $K=\lambda/J_{\varphi}(x,y)$.
Here $\lambda$ is the largest eigenvalue of the matrix $Q=DD^T$, where
$D=D\varphi(x,y)$ is Jacobi matrix of mapping $\varphi=\varphi(x,y)$ and $J_{\varphi}(x,y)=\det D\varphi(x,y)$ is its Jacobian.
It is easy to see that the Jacobi matrix corresponding to the mapping $\varphi=\varphi(x,y)$ has the form
\[
D=\left(\begin{array}{cc}
1 & f'(y)\\
0 & 1
\end{array}\right).
\]
A basic calculation implies $J_{\varphi}(x,y)=1$ and
\[
\lambda=\left(1+\frac{\left(f'(y)\right)^{2}}{2}\right)\left(1+\sqrt{1-\frac{4}{\left(2+\left(f'(y)\right)^{2}\right)^{2}}}\right)\,.
\]
Therefore any mapping $\varphi=\varphi(x,y)$ is a quasiconformal mapping
from $\mathbb R^{2}\to \mathbb R^{2}$ with $J_{\varphi}(x,y)=1$ and arbitrary large quasiconformality coefficient.
We can use their restrictions $\varphi|_{\mathbb D}$ to the unit disc $\mathbb D$. Images
can be very exotic quasidiscs.
If $a>0$ then mappings $\varphi(x,y)=(ax+f(y),\,\frac{1}{a}y)$ have
similar properties.
In this case we obtain lower estimates of the first non-trivial Neumann eigenvalues of the divergent form elliptic operator $L_A$ in $A$-quasiconformal $\beta$-regular domains via the Sobolev-Poincar\'e constant for the unit disc $\mathbb D$.
\vskip 0.2cm
\textbf{Acknowledgements.} The first author was supported by the United States-Israel Binational Science Foundation (BSF Grant No. 2014055).
\vskip 0.3cm
\vskip 0.3cm
Department of Mathematics, Ben-Gurion University of the Negev, P.O.Box 653, Beer Sheva, 8410501, Israel
\emph{E-mail address:} \email{[email protected]} \\
Division for Mathematics and Computer Sciences, Tomsk Polytechnic University,
634050 Tomsk, Lenin Ave. 30, Russia; Regional Scientific and Educational Mathematical
Center, Tomsk State University, 634050 Tomsk, Lenin Ave. 36, Russia
\emph{E-mail address:} \email{[email protected]} \\
Department of Mathematics, Ben-Gurion University of the Negev, P.O.Box 653, Beer Sheva, 8410501, Israel
\emph{E-mail address:} \email{[email protected]}
\end{document} |
\begin{document}
\title{Reversible perturbations of conservative H\'enon-like maps}
\begin{abstract}
For area-preserving H\'enon-like maps and their compositions, we consider smooth perturbations
that keep the reversibility of the initial maps but destroy their conservativity. For constructing such perturbations, we use two methods, the original method based on reversible properties of maps written in the so-called cross-form, and the classical Quispel-Roberts method based on a variation of involutions of the initial map. We study symmetry breaking bifurcations of symmetric periodic points in reversible families containing quadratic conservative orientable and nonorientable H\'enon maps as well as the product of two asymmetric H\'enon maps (with the Jacobians $b$ and $b^{-1}$).
\end{abstract}
\section{Introduction}
Among dynamical systems of various classes, the so-called {\em reversible systems}, which are characterized by invariance with respect to time reversal, are of special interest that can be explained by two main circumstances: first, such systems often appear in applications \cite{LR98}, and second, they form a class of systems with special symmetries, and therefore require the development of very specific mathematical methods for their study \cite{D76,RQ92}. To date, much is already known about the dynamics of reversible systems and the list of related papers is very vast, the reader can find it, for example, in the review paper \cite{LR98}. Especially a lot of fundamental results were obtained in the case of two-dimensional reversible maps, which are the main object of our article, see e.g. \cite{RQ92}--\cite{LT12}.
Recall that a $C^r$-map (diffeomorphism) $f$ is said to be \emph{reversible} if it is conjugate to its inverse $f^{-1}$ by an involution ${\cal G}$, i.e. the following condition holds
$f = {\cal G} \circ f^{-1} \circ {\cal G}$, where ${\cal G}^2=Id$ and the diffeomorphism ${\cal G}$ is also at least $C^r$-smooth.
Recently, the study of dynamics of reversible systems has got a new motivation
due to the discovery of the new, third, form of dynamical chaos,
the so-called {\em mixed dynamics} \cite{Gon16,GT17}, which
is characterized by the
the principal inseparability of dissipative elements of dynamics (attractors and repellers)
from conservative ones.
The above property makes the mixed dynamics fundamentally different from the two other classical forms of dynamical chaos, the conservative and dissipative chaos.
The most known type of conservative dynamics is demonstrated by Hamiltonian systems
or, more generally, systems preserving the phase volume. From the point of view of topological
dynamics, the conservative dynamics is characterized by the fact that the entire phase space of the
corresponding system is chain transitive, i.e., any two points can be connected by $\varepsilon$-orbits
for any $\varepsilon >0 $. Recall that a sequence of points $x_1,...,x_n$ is called an $\varepsilon$-orbit (of length $n$) for a map $f: y_{i+1} = f(y_i)$
if $\mbox{{\rm dist}}\left(f(x_j), x_{j+1}\right) < \varepsilon$ for all $j=1,...,n-1$. We will say that an $\varepsilon$-orbit $x_1,...,x_n$ connects the
points $x_1$ and $x_n$.
The dissipative dynamics has a completely different nature: it is associated with the existence of
``holes'' (absorbing and repelling domains) in the phase space $M$. Recall that an open domain $D$
is said to be absorbing (repelling) if its image under the action of a map $T$ (a map $T^{-1}$)
lies strictly inside it.
By definition, a dissipative attractor, closed stable invariant set, resides in some absorbing domain $D_a$,
analogously, a (dissipative) repeller resides in some repelling domain $D_r$.
Accordingly, we have $D_a\cap D_r = \emptyset$ here.
As for the mixed dynamics, unlike the conservative case, the phase space is not chain transitive, since infinitely many dissipative attractors and repellers exist here, which intersect along closed invariant sets, the so-called reversible cores, having neutral (conservative-like) type of stability.
The latter means that
the reversible core itself attracts nothing and repels nothing -- for any nearby point, its forward orbit tends to the nearest attractor and backward orbit tends to the nearest repeller.\footnote{Note that the reversible core can be very vast and occupy very big part of the phase space that certain examples show, see e.g. \cite{GGK13,GGKT17}.}
In other words, here, unlike the dissipative case, it is
impossible to construct a set of disjoint absorbing and repelling domains. The explanation of this phenomenon
from the topological point of view
was given in \cite{GT17} based on the concept of attractor going back to D. Ruelle \cite{R81}.
Note that one of the main fundamental properties of systems with mixed dynamics, which
can also be considered as a criterion and even as its definition (from the mathematical point of view), is the existence of so-called {\em absolute Newhouse regions} \cite{GST97,T11,T15}.
Recall that Newhouse regions are open regions in
the space of dynamical systems (or in the parameter space) in which systems with homoclinic
tangencies are dense (or values of parameters corresponding to systems with homoclinic tangencies
are dense) \cite{N79,GST93b,PV94,R95}.
It was shown by Newhouse himself \cite{N74} that, in the dissipative case,
there may exist Newhouse regions in which systems with infinitely many stable and saddle periodic
orbits are dense and, moreover, generic, i.e., they form subsets of the second Baire category (see also \cite{GST08}).
The absolute Newhouse regions are characterized
by the following property: systems with infinitely many periodic orbits of all possible types (sinks,
sources, and saddles)
are generic in such regions, and
these orbits are inseparable from each other, i.e., the closures of the sets of orbits of different types have nonempty intersections.
The absolute Newhouse regions exist for two-dimensional reversible maps as well \cite{LS04,DGGLS13,GGS20}.
However, the dynamics of systems from reversible absolute Newhouse regions is much richer than for those in general case. In particular, as shown in \cite{LS04}, diffeomorphisms with
infinitely many coexisting periodic sinks, sources, and symmetric elliptic periodic orbits are generic in reversible absolute
Newhouse regions.
A periodic orbit of a reversible map $f$ is called {\em symmetric} if it is invariant
with respect to the involution ${\mathcal G}$, i.e., if its points are posed ${\mathcal G}$-symmetrically around the set $\mbox{{\rm Fix}}(h) = \{x: {\mathcal G}(x) = x\}$ of fixed points of the
involution, thus, if $Q$ is such an orbit, then $Q = {\mathcal G}(Q)$. In the two-dimensional case, a symmetric periodic orbit of an orientable reversible map $f$
has multipliers $\lambda$ and $\lambda^{-1}$. In general case $\lambda\neq \lambda^{-1}$, symmetric periodic points can be divided into two
types: saddle points, if $\lambda\neq \pm 1$ is real, and elliptic points, if $\lambda_{1,2} = e^{\pm i\varphi}$ and $0<\varphi<\pi$. The saddle
points are rough (structurally stable). As for elliptic points, although they are very similar to
conservative elliptic points \cite{Sevruyk}, they can differ greatly from the latter, as shown by the following results \cite{GT17,GLRT14}:
\begin{itemize}
\item
{\em all symmetric elliptic periodic orbits of a $C^r$-generic $(r = 1,...,\infty )$
two-dimensional reversible map are limits of periodic sinks and sources.}\footnote{The genericity is understood here in the sense that reversible maps with the indicated properties form a subset of the second Baire category in the space of $C^r$-smooth reversible maps having symmetric elliptic periodic orbits.}
\item
{\em all symmetric elliptic periodic orbits of a $C^r$-generic ($r = 1,...,\infty$) two-dimensional reversible
map are reversible cores;}
\item
{\em the generic elliptic periodic point of a reversible map is totally stable, i.e., stable under permanently acting perturbations (Lyapunov stable for $\varepsilon$-orbits), while in the case of area-preserving maps any such orbit is unstable (although it is Lyapunov orbitally stable).}
\end{itemize}
Thus, the reversible mixed dynamics manifests itself locally, but wherever symmetric elliptic points exist.
This important circumstance, of course, testifies to the fact that mixed dynamics should be viewed
as one of the fundamental properties of reversible systems. Moreover, the above result shows that symmetric (elliptic and other) orbits form a skeleton of global mixed dynamics, as they compose naturally that invariant set, reversible core, which simultaneously separates and connects the attractor and repeller.
The other important circumstance testifying to the universality of mixed dynamics in reversible
systems is what can be called the conjecture on Reversible Mixed Dynamics:
\begin{itemize}
\item
{\em Near any reversible map with a symmetric homoclinic tangency or a symmetric
nontransversal heteroclinic cycle, there are absolute reversible Newhouse regions.}
\end{itemize}
This RMD-conjecture was formulated in \cite{DGGLS13} and was almost immediately proved in \cite{GLRT14} for Newhouse
regions from the space of reversible systems in the $C^r$-topology with $2\leq r \leq\infty$. In the analytical case, as well as in the case of parameter
families, the RMD-conjecture was proved only for the so-called a priori non-conservative reversible diffeomorphisms \cite{LS04,GGS20,DGGL18},
when the (heteroclinic) cycle contains non-conservative elements (for example, saddles with the
Jacobians greater and less than one, or pairs of nonsymmetric homoclinic tangencies of a symmetric
saddle point, as in \cite{DGGL18}), as well as for reversible maps with symmetric heteroclinic cycles of conservative
type \cite{DGGLS13}.
Essentially, it remains to consider only two most interesting cases:
{\em reversible maps with symmetric quadratic and cubic homoclinic tangencies}.
However, these two cases are also the most difficult, since, in the principle plan, the main problem of this topic is connected
with the study of symmetry breaking bifurcations in first return maps constructing near orbits of symmetric homoclinic tangencies.
In the main order, these maps coincide with the conservative H\'enon-like maps: the standard H\'enon map $\bar x = y,\; \bar y = M - x - y^2$ (in the case of symmetric quadratic homoclinic tangency) and the reversible cubic H\'enon maps $\bar x = y,\; \bar y = -x + My \pm y^3$ (appearing near symmetric cubic homoclinic tangencies of two different types).
Both these maps are only certain truncated normal forms for the complete first return maps, and they demonstrate exclusively conservative dynamics.
What can be said about the dynamics of these maps under perturbations that keep the reversibility, and how can dissipative dynamics elements appear here, such as periodic sinks, sources, or saddles with a Jacobian other than 1?
This is still open problem which requires solving the following issues.
\begin{itemize}
\item
How to construct perturbations of area-preserving H\'enon-like maps
which maintain their reversibility, but destroy the conservativity?
\item
What is the structure of symmetry breaking bifurcations under such perturbations?
\end{itemize}
We consider these questions to be relevant and interesting
not only because their solution will give an addition to the theory of H\'enon-like maps \cite{Henon76,S79,Bir87,DM00,GGO17}, but also it will make a certain contribution to the theory of mixed dynamics of reversible systems.
In the current paper we deal with these questions. Accordingly, the paper is divided into two parts. In the first one, Sections~\ref{sec:revpert} and~\ref{sec:QR}, we consider two types of methods for the construction of reversible perturbations for conservative H\'enon-like maps.
The first method looks to be new: we call it ``cross-form perturbations'', see Section~\ref{sec:crossform}. We apply this method for the conservative H\'enon-like maps
(\ref{eq:HenonMap}), see Section~\ref{sec:crossformH1}, and for compositions of two H\'enon-like maps, see Sections~\ref{sec:crossformH-2} and~\ref{sec:crossform2H}.
The second method is the classical method proposed in the paper \cite{RQ92} by Quispel and Roberts. We apply this Quispel-Roberts method for map (\ref{eq:HenonMap}) in Section~\ref{sec:QR} and for the nonorientable conservative H\'enon-like map of the form $\bar x = - y, \bar y = -x + F(y)$ in Section~\ref{sec:asymH-}.\footnote{Note that,
formally, the cross-form perturbations method and Quispel-Roberts method give different results, at first sight. Of course, the Quispel-Roberts method is more general, since it can be apply to any reversible maps, however, it is not very clear how certain perturbations can be obtained by means of it, in particular, those that the cross-form method gives.}
In the second part of the paper, Section~\ref{sec:sbrbif}, we study symmetry breaking bifurcations in one-parameter families of reversible non-conservative H\'enon-like maps, using those perturbations that were constructed in the first part of the paper. We show that the simplest bifurcations of this type are reversible pitchfork bifurcations of periodic orbits. We consider such families in the cases of the product of two (quadratic) H\'enon maps (Section~\ref{sec:sbrbif_dggls13}), the nonorientable conservative H\'enon map (Section~\ref{sec:bifnor}) and the orientable conservative H\'enon map (Sections~\ref{sec:sborH} and~\ref{sec:FO6}). In the first two cases we show that even symmetric fixed points can undergo pitchfork bifurcations and recover their structure. It is interesting that, in the case of orientable conservative H\'enon map, this bifurcation occurs starting only with an orbit of period 6 (no such bifurcation takes place for orbits of less period), that is very surprising.
\section{On construction of reversible perturbations for H\'enon-like maps and their compositions} \label{sec:revpert}
The conservative H\'enon-like maps are the two-dimensional planar diffeomorphisms that can be represented in the form
\begin{equation}
H: \;
\bar x = y,\;\;\;
\bar y = - x + F(y),
\label{eq:HenonMap}
\end{equation}
where $F(y)$ is some nonlinear function (e.g. a polynomial).
Map (\ref{eq:HenonMap}) is area-preserving, with the Jacobian equal to 1, and reversible with respect to the linear involution $h: (x, y) \to (y, x)$.
Indeed, $H^{-1}$ takes the form $x=\bar y, y = - \bar x + F(\bar y)$; the relation $h \circ H^{-1} \circ h$ means, due to the simplicity of $h$, that we need to make interchanges
$x \leftrightarrow y, \bar x \leftrightarrow \bar y$ in the formula for $H^{-1}$, after which we get (\ref{eq:HenonMap}).
In this section we consider two methods for the construction of such sufficiently smooth (analytic) perturbations of H\'enon-like maps (\ref{eq:HenonMap}) and their compositions that destroy the conservativity of these maps but keep their reversibility with respect to the involution $h$.
\subsection{Cross-form perturbations} \label{sec:crossform}
The first method to obtain reversible perturbations is based on the following cross-form map
\begin{equation}
g: (x,y)\to (\bar x, \bar y)\;: \;\;\; \bar x = G(x, \bar y),\;\;\;
y = G(\bar y, x).
\label{eq:RevMap}
\end{equation}
Note that the map~(\ref{eq:RevMap}) is reversible with respect to the involution $h: (x,y)\to (y,x)$.
The proof is immediate: the map $g^{-1}$ has the form $x = G(\bar x, y),\;\;\; \bar y = G(y, \bar x)$, and the composition $h\circ g^{-1}\circ h$ means that we need to make interchanges
$x \leftrightarrow y$ and $\bar x \leftrightarrow \bar y$ in $g^{-1}$, which leads to (\ref{eq:RevMap}).
We introduce certain notations for the derivatives of functions:
\begin{itemize}
\item
$F^\prime(\rho)$ denotes the first derivative of the function $F(y)$ at the point $y=\rho$;
\item
for a smooth function $s(x,y)$, we denote
$$
u(x,y) = \frac{\partial s(x,y)}{\partial x}, \;\; v(x,y) = \frac{\partial s(x,y)}{\partial y}.
$$
and
$$
s_x(\xi,\eta) = u(\xi,\eta),\; s_y(\xi,\eta) = v(\xi,\eta).
$$
Thus, the subscripts $x$ and $y$ means the differentiation with respect to the first and second variables, respectively.
\end{itemize}
\begin{lemma} \label{lm:crosjac}
The Jacobian of map (\ref{eq:RevMap}) takes the form
\begin{equation}
J \; = \; \frac{G_x(x, \bar y)}{G_x(\bar y, x)}.
\label{eq:JRevMap}
\end{equation}
\end{lemma}
\begin{proof}
It follows from (\ref{eq:RevMap}) that
$$
\frac{\partial\bar x}{\partial x} = G_x(x,\bar y) + G_y(x,\bar y)\frac{\partial\bar y}{\partial x}, \;\;\;
\frac{\partial\bar x}{\partial y} = G_y(x,\bar y) \frac{\partial\bar y}{\partial y},
$$
$$
0 = G_x(\bar y,x) \frac{\partial\bar y}{\partial x} + G_y(\bar y,x), \;\;\; 1 = G_x(\bar y,x) \frac{\partial\bar y}{\partial y}
$$
Then we get
$$
\frac{\partial\bar y}{\partial y} = \frac{1}{G_x(\bar y,x)}, \; \frac{\partial\bar y}{\partial x} = - \frac{G_y(\bar y,x)}{G_x(\bar y,x)},\;
\frac{\partial\bar x}{\partial x} = G_x(x,\bar y) - \frac{G_y(x,\bar y)G_y(\bar y,x)}{G_x(\bar y,x)}, \; \frac{\partial\bar x}{\partial y} = \frac{G_y(x,\bar y)}{G_x(\bar y,x)},
$$
and, as a result, we deduce formula (\ref{eq:JRevMap}) for the Jacobian
$
J = \partial\bar x /\partial x \cdot\partial\bar y/\partial y \; - \;\partial\bar x/\partial y \cdot \partial\bar y/\partial x .
$
\end{proof}
Therefore, once having a conservative map written in the implicit form~(\ref{eq:RevMap}), we can simply add a perturbation in such a way that the cross-form is preserved, and the perturbed system will be reversible.
\subsection{Cross-form perturbation of~(\ref{eq:HenonMap})} \label{sec:crossformH1}
The idea to write a perturbation for the map~$H$, given in~~(\ref{eq:HenonMap}), comes from the formal solution of the second equation of~(\ref{eq:HenonMap}) for~$y$:
$
y=F^{-1}(x+\bar y)
$. Then the map~$H$ is rewritten in the cross-form
$$
H: \;
\bar x = F^{-1}(x+\bar y),\;\;\;
y = F^{-1}(\bar y+x).
$$
Thus, the perturbation of the form
$$
\tilde H: \;
\bar x = F^{-1}(x+\bar y) + \varepsilon(x,\bar y),\;\;\;
y = F^{-1}(\bar y+x) + \varepsilon(\bar y, x)
$$
is formally reversible. For this map we obtain from the second equation that $F^{-1}(\bar y+x) = y - \varepsilon(\bar y, x)$ and $\bar y + x = F(y- \varepsilon(\bar y, x))$. Then map~$\tilde H$ takes the following form
\begin{equation}
\tilde H: \;
\bar x = y + \varepsilon(x, \bar y) - \varepsilon(\bar y, x),\;\;\;
\bar y = - x + F(y-\varepsilon(\bar y, x)).
\label{eq:HenonMapPert}
\end{equation}
By construction, map (\ref{eq:HenonMapPert}) should be reversible, however, the operator $F^{-1}$ is only formal, therefore the reversibility of $\tilde H$ must be proved directly. This is done in the following lemma.
\begin{lemma}
The map $\tilde H$, defined in~(\ref{eq:HenonMapPert}), is reversible with respect to the involution $h: (x,y)\to (y,x)$.
\end{lemma}
\begin{proof} To prove the reversibility of $\tilde{H}$, we have to show that $\tilde{H}=h\circ \tilde{H}^{-1}\circ h$.
The inverse map $\tilde{H}^{-1}$ is obtained after swapping the bar and no-bar variables $\bar x \leftrightarrow x, \bar y \leftrightarrow y$, i.e.,
\begin{equation}\label{eq:tldHm1}
\tilde{H}^{-1}\;: \;\;\; \bar x = - y + F(\bar y-\varepsilon(y, \bar x)), \;\;\; \bar y = x - \varepsilon(\bar x, y) + \varepsilon(y, \bar x).
\end{equation}
After exchanging $x \leftrightarrow y$ and $\bar x \leftrightarrow \bar y$ in (\ref{eq:tldHm1}), due to the involution $h$, we get the expression for $h\circ \tilde{H}^{-1}\circ h$ which coincides with~(\ref{eq:HenonMapPert}).
\end{proof}
\begin{lemma}\label{lm:J_Htld}
The Jacobian of the map~(\ref{eq:HenonMapPert}) takes the following formula
\begin{equation}\label{eq:J_Htld}
J=\frac{1+ F'(y-\varepsilon(\bar y,x)) \varepsilon_x(x,\bar y)}{1+F'(y-\varepsilon(\bar y, x)) \varepsilon_x(\bar y, x)}.
\end{equation}
\end{lemma}
\begin{proof}
Differentiating the first equation of~(\ref{eq:HenonMapPert}) with respect to $x$ and $y$ we get
$$
\frac{\partial \bar x}{\partial x} = \varepsilon_x(x,\bar y) +\varepsilon_y(x, \bar y) \frac{\partial \bar y}{\partial x} - \varepsilon_x (\bar y, x) \frac{\partial \bar y}{\partial x} - \varepsilon_y (\bar y, x), \;\;\; \frac{\partial \bar x}{\partial y} = 1 + \varepsilon_y(x,\bar y) \frac{\partial \bar y}{\partial y} - \varepsilon_x(\bar y, x) \frac{\partial \bar y}{\partial y}.
$$
Therefore, we have
\begin{equation}\label{eq:Jl3}
J= \frac{\partial \bar x}{\partial x} \frac{\partial \bar y}{\partial y} -\frac{\partial \bar x}{\partial y} \frac{\partial \bar y}{\partial x} = \left( \varepsilon_x(x,\bar y) - \varepsilon_y(\bar y, x)\right) \frac{\partial \bar y}{\partial y} - \frac{\partial \bar y}{\partial x}.
\end{equation}
We find the derivatives ${\partial \bar y}/{\partial x}$ and ${\partial \bar y}/{\partial y}$ from the second equation of~(\ref{eq:HenonMapPert}) by its implicit differentiation
$$
\frac{\partial \bar y}{\partial x} = \frac{-1-F'(y-\varepsilon(\bar y, x)) \varepsilon_y(\bar y, x)}{1+F'(y-\varepsilon(\bar y, x)) \varepsilon_x(\bar y, x)}, \;\;\;
\frac{\partial \bar y}{\partial y} = \frac{F'(y-\varepsilon(\bar y,x))}{1+F'(y-\varepsilon(\bar y, x)) \varepsilon_x(\bar y, x)}.
$$
After substituting these into~(\ref{eq:Jl3}), we obtain~(\ref{eq:J_Htld}).
\end{proof}
It is worth mentioning that if the perturbation $\varepsilon(x,y)$ in~(\ref{eq:HenonMapPert}) is a symmetric function, i.e. $\varepsilon(x,y)=\varepsilon(y,x)$, the perturbed map~(\ref{eq:HenonMapPert}) takes the simpler form
\begin{equation}\label{eq:HMPertSymm}
\tilde H: \;
\bar x = y,\;\;\;
\bar y = - x + F(y-\varepsilon(x, \bar y)),
\end{equation}
and the same formula~(\ref{eq:J_Htld}) holds for the Jacobian.
Note that the perturbed systems~(\ref{eq:HenonMapPert}) and (\ref{eq:HMPertSymm}) contain perturbing terms inside a nonlinear function $F$, and, hence, it is hard to iterate the maps --
one needs to solve the second equations for $\bar y$ and calculate $\bar y = f(x,y)$. In the following subsections we show that using cross-form~(\ref{eq:RevMap}) it is possible to construct reversibility preserving perturbations of another kind which allows to iterate the maps directly. We also show that such perturbations can be constructed by the Quispel-Roberts method, see Section~\ref{sec:QR}.
\subsection{Perturbations of $H^{-2}$} \label{sec:crossformH-2}
The cross-form reversible perturbations can be easily constructed for the map $H^{-2}$ that is the square of $H^{-1}$, i.e., the inverse map to the conservative H\'enon-like map $H$.
We obtain from (\ref{eq:HenonMap}) that map $H^{-1}$ takes the form
$$
H^{-1}: \;\;\; \bar x = -y + F(x), \;\;\; \bar y =x.
$$
Then the map $H^{-2}$ is written as
\begin{equation}\label{eq:Hm-2}
H^{-2}=H^{-1}\circ H^{-1}: \;\;\; \bar x = -x + F(-y + F(x)), \;\;\; \bar y = - y + F(x),
\end{equation}
\begin{lemma}
The map of the form
\begin{equation}\label{eq:tldHm2}
\tilde{H}^{-2}: \;\;\; \bar x = -x + F(\bar y) + \varepsilon(x,\bar y), \;\;\; \bar y= -y + F(x) + \varepsilon(\bar y, x)
\end{equation}
is reversible with respect to the involution $h:(x,y)\to(y,x)$. The Jacobian of $\tilde{H}^{-2}$ is
$$
J=\frac{1-\varepsilon_x(x,\bar y)}{1-\varepsilon_x(\bar y, x)}.
$$
\end{lemma}
\begin{proof} Map (\ref{eq:Hm-2}) can be presented in the cross-form as follows
$$
H^{-2}: \;\;\; \bar x = -x + F(\bar y), \;\;\; y = - \bar y + F(x),
$$
that have form (\ref{eq:RevMap}) with $G(x,y) = - x + F(y)$. Thus, the perturbation
$\bar x = -x + F(\bar y) + \varepsilon(x,\bar y), \;\; y = - \bar y + F(x) + \varepsilon(\bar x,y)$ is what we need, and it takes the form~(\ref{eq:tldHm2}). The desired formula for the Jacobian $J(\tilde H^{-2})$ is obtained from (\ref{eq:JRevMap}) for $G(x,y) = - x + F(y) + \varepsilon(x,y)$.
\end{proof}
The form of the map~(\ref{eq:tldHm2}) allows to write the map explicitly for some perturbations. For example, if $\varepsilon(x,y)$ is linear in $x$, i.e. $\varepsilon(x,y)=\alpha(y)+x \beta(y)$, then the map yields
$$
\tilde{H}^{-2}: \;\;\; \bar x = -x + F(\bar y) + \alpha(\bar y) + x \beta(\bar y), \;\;\; \bar y= \frac{-y + F(x) + \alpha(x) }{1- \beta(x)}
$$
and its Jacobian is
$$
J= \frac{1- \beta(\bar y)}{ 1- \beta(x)}.
$$
Hence, the new map is a diffeomorphism in some ball $\{x\in \mathbb{R} : \|(x,y)\|\leq R_\beta\}$, where $R_\beta\to\infty$ as $|\beta|\to 0$.
Besides, for some special functions $\beta(x)$ (for instance, $\beta(x)=\mu \arctan(x)$ with sufficiently small $\mu$), the map is an analytical diffeomorphism in the whole plane $\mathbb{R}^2$.
\subsection{Perturbations of conservative compositions of two non-conservative H\'enon-ilke maps} \label{sec:crossform2H}
The next approach is connected with perturbations of the product of two asymmetric non-conservative H\'enon-like maps $H_1$ and $H_2$ of the form
\begin{equation}
H_1: \;\;\; \bar x = y, \;\;\; \bar y = b x + F(y)\;\; \mbox{{\rm and}}\;\; H_2: \;\;\; \bar x = y, \;\;\; \bar y = \frac{1}{b} x - \frac{1}{b} F(y).
\label{eq:Hlms}
\end{equation}
These maps have the Jacobians $b$ and $1/b$, respectively, and their nonlinearities are asymmetric.
Their inverse maps are
$$
H_1^{-1}: \;\;\; \bar x = \frac{1}{b} y - \frac{1}{b} F(x), \;\;\; \bar y = x \;\; \mbox{{\rm and}}\;\; H_2^{-1}: \;\;\; \bar x = b y + F(x), \;\;\; \bar y = x,
$$
respectively. The composition $ H_{12}^{-1} = H_1^{-1}\circ H_2^{-1}$ of the last two maps can be written as
$$
H_{12}^{-1}: \;\;\; \bar x = \frac{1}{b} x - \frac{1}{b} F\left( b y + F(x)\right), \;\; \bar y = b y + F(x),
$$
or, in the cross-form, as
\begin{equation}
H_{12}^{-1}: \;\;\; \bar x = \frac{1}{b} x - \frac{1}{b} F\left(\bar y \right), \;\; y = \frac{1}{b} \bar y - \frac{1}{b} F(x).
\label{eq:H12-1}
\end{equation}
Thus, the composition $H_1^{-1}\circ H_2^{-1}$ has the cross-form ~(\ref{eq:RevMap}) with $G(x,y)= b^{-1}(x - F(y))$. This implies the following result
\begin{lemma}\label{lm:H12tld}
The map
\begin{equation}
\tilde{H}_{12}^{-1}: \;\;\; \bar x = \frac{1}{b} x - \frac{1}{b} F(\bar y) + \varepsilon(x,\bar y), \;\;\; y = \frac{1}{b} \bar y - \frac{1}{b} F(x) + \varepsilon(\bar y, x)
\label{eq:H12-1e}
\end{equation}
is a reversible perturbation of $H_1^{-1}\circ H_2^{-1}$ that keeps the involution $h:(x,y)\to (y,x)$, and
\begin{equation} \displaystyle
J\left(\tilde{H}_{12}^{-1}\right)=\frac{1+b\varepsilon_x(x,\bar y)}{1+b\varepsilon_x(\bar y, x)}.
\label{eq:J(H12)}
\end{equation}
\end{lemma}
\section{Quispel-Roberts method for construction of reversible perturbations.} \label{sec:QR}
The basic elements of the theory of reversible systems were developed in the famous paper \cite{RQ92} by Quispel and Roberts. In particular, in this paper general methods for the construction of reversible perturbations of reversible maps were proposed. One of such methods is based on the following two facts:
\begin{itemize}
\item[1)] Any reversible map can be represented as a composition of its two involutions.
\item[2)] If $g$ is an involution, then $\tilde g = T^{-1}\circ g \circ T$ is also involution, if the map $T$ is a diffeomorphism.
\end{itemize}
Indeed, for item 1), if $g$ is an involution of a map $f$,
we have $f = g\circ f^{-1}\circ g = g\circ \left(f^{-1}\circ g\right)$ and the map $f^{-1}\circ g$ is also involution, since
$$
\left(f^{-1}\circ g\right)^2 = f^{-1}\circ g \circ f^{-1}\circ g = f^{-1}\circ \left( g \circ f^{-1}\circ g \right) = f^{-1}\circ f = \;\mbox{{\rm id}}.
$$
For item 2), we obtain
$\tilde g^2 = T^{-1}\circ g \circ \left( T \circ T^{-1} \right) \circ g \circ T = T^{-1}\circ \left( g\circ g \right) \circ T = T^{-1}\circ T = \;\mbox{{\rm id}}. $
The conservative H\'enon-like map $H$, given by (\ref{eq:HenonMap}), can be also presented as the product $H = h_1\circ h_2$ of two involutions:
\begin{equation}
h_1 = h = \left\{\begin{array}{l} \bar x = y, \\ \bar y = x \end{array}\right. \;\;\mbox{{\rm and}}\;\;
h_2 = \left\{\begin{array}{l} \bar x = - x + F(y), \\ \bar y = y \end{array}\right.
\label{eq:2invH}
\end{equation}
Thus, we can construct reversible perturbations of $H$ by means of changing their involutions. For our goals,
we keep the involution $h_1 = h$ and take new involution $\tilde h_2$ as the perturbation $\tilde h_2 = T^{-1}\circ h_2 \circ T$ of the involution $h_2$ by means of a map
$T$ that is close to the identity map $\bar x = x, \bar y = y$. The following lemma summarizes results of the corresponding calculations.
\begin{lemma}\label{lm:til_H}
The map
\begin{equation}
\hat H : \left\{
\begin{array}{l}
\bar x = y + \varepsilon_2(x,y) - \varepsilon_2(\bar y,\bar x), \\
\bar y = -x + F \left(y + \varepsilon_2(x,y)\right) - \varepsilon_1(x,y) - \varepsilon_1(\bar y,\bar x)
\end{array}
\right.
\label{eq:tildeH}
\end{equation}
is a reversible perturbation of the conservative H\'enon-like map $H$, given in (\ref{eq:HenonMap}), that is constructed in the form $\hat H = h_1\circ \tilde h_2$, where $\tilde h_2 = T^{-1}\circ h_2 \circ T$ and the map
$T: \; \bar x = x + \varepsilon_1 (x,y), \bar y = y + \varepsilon_2 (x,y)$ is assumed to a near identity map.
\end{lemma}
\begin{proof}
We will find first the new involution $\tilde h_2 = T^{-1}\circ h_2 \circ T$.
By~(\ref{eq:2invH}), the composition $h_2 \circ T : (x,y)\to (x^\prime,y^\prime)$ can be written as
$$
h_2 \circ T : \left\{\begin{array}{l} x^\prime = -x - \varepsilon_1(x,y) + F(y + \varepsilon_2(x,y)), \\ y^\prime = y + \varepsilon_2(x,y). \end{array}\right.
$$
We can write the map $T^{-1}: (x^\prime,y^\prime)\to (\bar x,\bar y)$ as follows
$\bar x + \varepsilon_1 (\bar x,\bar y) = x^\prime, \; \bar y + \varepsilon_2 (\bar x,\bar y)=y^\prime$.
Then for the new involution $\tilde h_2$, we get the following expression
$$
\tilde h_2 = T^{-1}\circ h_2 \circ T : \left\{\begin{array}{l} \bar x + \varepsilon_1(\bar x, \bar y) = -x - \varepsilon_1(x,y) + F(y + \varepsilon_2(x,y)), \\
\bar y + \varepsilon_2(\bar x, \bar y) = y + \varepsilon_2(x,y). \end{array}\right.
$$
After this, formula (\ref{eq:tildeH}) for the map $\hat H = h_1\circ \tilde h_2$ is easy obtained: we only need to replace $\bar x \leftrightarrow \bar y$ in this expression for $\tilde h_2$ ($x$ and $y$ are not changed).
\end{proof}
\begin{lemma}\label{lm:Jtil_H}
The Jacobian of the perturbed map $\hat H$ is
\begin{equation} \label{eq:QRJac}
J(\hat {H})=\frac { \left( 1 +\varepsilon_{2y}(x,y)\right) \left( 1 +\varepsilon_{1x}(x,y)\right) - \varepsilon_{2x}(x,y)\varepsilon_{1y}(x,y)}
{\left(1 +\varepsilon_{2y}(\bar y,\bar x)\right) \left( 1 +\varepsilon_{1x}(\bar y,\bar x)\right) - \varepsilon_{2x}(\bar y,\bar x)\varepsilon_{1y}(\bar y,\bar x)}.
\end{equation}
\end{lemma}
\begin{proof}
We calculate the derivatives ${\partial \bar x}/{\partial x}$, ${\partial \bar y}/{\partial x}$, ${\partial \bar x}/{\partial y}$ and ${\partial \bar y}/{\partial y}$ from (\ref{eq:tildeH}):
$$
\begin{array}{l}
\displaystyle \left(1 + \varepsilon_{2y} (\bar y, \bar x) \right) \frac{\partial \bar x}{\partial x} = \varepsilon_{2x} (x,y) - \varepsilon_{2x}(\bar y,\bar x) \frac{\partial \bar y}{\partial x}, \;\;\;
\left(1 + \varepsilon_{2y} (\bar y, \bar x) \right) \frac{\partial \bar x}{\partial y} = 1 + \varepsilon_{2y} (x,y) -
\varepsilon_{2x}(\bar y,\bar x) \frac{\partial \bar y}{\partial y}, \\ \\
\displaystyle \left(1 + \varepsilon_{1x} (\bar y, \bar x) \right) \frac{\partial \bar y}{\partial x} = - 1 + F^\prime \left(y + \varepsilon_2(x,y)\right)\cdot \varepsilon_{2x} (x,y)
- \varepsilon_{1x} (x,y) - \varepsilon_{1y}(\bar y,\bar x) \frac{\partial \bar x}{\partial x}, \\ \\
\displaystyle \left(1 + \varepsilon_{1x} (\bar y, \bar x) \right) \frac{\partial \bar y}{\partial y} = F^\prime \left(y + \varepsilon_2(x,y)\right)\cdot (1+\varepsilon_{2y} (x,y)) -
\varepsilon_{1y} (x,y) - \varepsilon_{1y}(\bar y,\bar x) \frac{\partial \bar x}{\partial y}.
\end{array}
$$
Solving this system for the partial derivatives, we get the formula (\ref{eq:QRJac}) for the Jacobian.
\end{proof}
Notice that among the perturbations in the form~(\ref{eq:tildeH}) we can select more simple ones which preserve reversibility and destroy conservativity. Let us consider two examples.
\textbf{Example 1.} We consider the case with $\varepsilon_2\equiv 0$. Then the map~(\ref{eq:tildeH}) takes the form
\begin{equation}
\hat H :
\begin{array}{l}
\bar x = y, \;\; \bar y = -x + F(y) - \varepsilon_1(x,y) - \varepsilon_1(\bar y,\bar x)
\end{array}
\label{eq:tildeHe1}
\end{equation}
and the Jacobian of this map
\begin{equation}\label{eq:J_eps3y}
J = \frac{1 + \varepsilon_{1x}(x,y)}{1 + \varepsilon_{1x}(\bar y,\bar x)}
\end{equation}
is not 1 generally. Moreover, if, for example, $\varepsilon_{1}(x,y) = a_{20} x^2 + a_{11} xy + a_{02}y^2$, then (since $\bar x = y$)
$$
J = \frac{1 + a_{11} y + 2 a_{20} x}{1 + a_{11} \bar x + 2 a_{20} \bar y} = \frac{1 + a_{11} y + 2 a_{20} x}{1 + a_{11} y + 2 a_{20} \bar y},
$$
i.e. including quadratic terms $x y$ and $x^2$ into the perturbation $\varepsilon_1$ makes the Jacobian non-constant.
Other particular case of the function $\varepsilon_1(x,y)$ includes, for example, $\varepsilon_1(x,y) = x f_1(y) + f_2(y)$, where $f_1(0)=0, f_2(0)=f^\prime_2(0)=0$, i.e. $\varepsilon_1(x,y)$ being linear in $x$. Then $J = (1 + f_1(y))(1 + f_1(\bar x))^{-1} \equiv 1$.
Let us consider a perturbation with $\varepsilon_1(x,y) = p(x) + q(y)$ and $p'(x) = v(x)$. Then
$\varepsilon_{1x}(x,y) = v(x)$, $\varepsilon_{1x}(\bar y,\bar x) = v(\bar y)$ and, by (\ref{eq:J_eps3y}),
$$
J = \frac{1+v(x)}{1+v(\bar y)}
$$
Formally, it means that $J\not\equiv 1$. However, for any periodic orbit, the Jacobian $J_n$ of its first return map will be equal to 1.
Indeed, let $M_i(x_i,y_i)$, $i=1,...,n$, be the points of an $n$-periodic orbit $P$. Then, since $x_i = y_{i-1}$, we obtain that
\begin{equation} \label{eq:ruleJ}
J_n = \left. J(\hat{H}^n) \right|_{M_1} = \prod\limits_{i=1}^n \frac{1+v(y_{i-1})}{1+v(y_{i+1})} \equiv 1
\end{equation}
since the nominator and denominator of this product contain the same factors. This means that any periodic orbit is conservative, any invariant sets with dense subsets of periodic orbits (for instance, horseshoes) are also conservative etc.
Moreover, we can claim that the dynamics of map $\hat H$ in the form (\ref{eq:tildeHe1}) with $\varepsilon_1(x,y) = p(x)+q(y)$ is totally conservative,
since this map possesses a smooth invariant measure.
Indeed, as known \cite{R78}, a measure $d\mu=\rho(x,y)dxdy$ is invariant if and only if the density $\rho(x,y)$ is a fixed point of the Ruelle-Perron-Frobenius operator, i.e.
$$
\rho(x,y)=\frac{\rho \circ \hat{H}^{-1}(x,y)}{|J|}=\frac{1+v(\bar{y})}{1+v(x)} \cdot \rho \circ \hat{H}^{-1}(x,y).
$$
Let us check that the function $\rho(x,y)=(1+v(y))\cdot (1+v(\bar{y}))$ satisfies this relation. For simplicity, take a point $(x_0,y_0)$ and denote its image by $(x_1,y_1)=\tilde{H}(x_0,y_0)$ and the preimage by $(x_{-1},y_{-1})=\tilde{H}^{-1}(x_0,y_0)$. Then we have
$$\rho(x_0,y_0)=(1+v(y_0))(1+v(y_1)), \;\;\; \rho(x_{-1},y_{-1})=(1+v(y_{-1}))(1+v(y_0)).$$
Since $x_0=y_{-1}$ we obtain
$$
\frac{1+v(y_1)}{1+v(x_0)} \cdot \rho(x_{-1},y_{-1})=\frac{1+v(y_1)}{1+v(x_0)} \cdot (1+v(x_0))(1+v(y_0))=\rho(x_0,y_0).
$$
Therefore, the measure
$$\mu(A)=\int_A (1+v(y))\cdot (1+v(\bar{y}))dxdy$$
is invariant for the map $\tilde{H}$.
\textbf{Example 2.} Consider the case with $\varepsilon_1(x,y) \equiv 0$.
Then map~(\ref{eq:tildeH}) takes the form
\begin{equation}\label{eq:PertMarina1}
\hat{H}^{(2)}: \;\;\; \bar x = y + \varepsilon_2(x,y) - \varepsilon_2(\bar y, \bar x), \;\;\; \bar y = -x + F(y+\varepsilon_2(x,y)),
\end{equation}
and
$$
J(\hat{H}^{(2)}) = \frac {1 + \varepsilon_{2y}(x,y)}{1 + \varepsilon_{2y}(\bar y, \bar x)},
$$
i.e., $J$ is not 1 generally. However, not any perturbation $\varepsilon_2$ is suitable. For example, let the function $\varepsilon_{2y}(x,y)= v(x,y)$ be symmetric, i.e.
$v(x,y) = v(y,x)$ (as for $\varepsilon_2 = xy^2$). In this case, $J(\hat{H}^{(2)}|_{(x_i,y_i)}) = (1+v(x_i,y_i))(1+v(x_{i+1},y_{i+1}))^{-1}$, and when calculating the Jacobian of a periodic orbit $\{(x_i,y_i): i=1,\ldots,n, \hat{H}^{(2)}(x_i,y_i)= (x_{i+1}, y_{i+1}), x_{n}=x_1, y_{n}=y_1\}$ as in~(\ref{eq:ruleJ}), we get $J_n = 1$. At the same time, the perturbation $\varepsilon_2 = \alpha xy$ is quite suitable. Indeed, the Jacobian
$J = (1+\alpha x)(1 + \alpha \bar y)^{-1}$ is not constant for $\alpha\neq 0$ and, moreover, since the function $\varepsilon_2(x,y)$ is linear in $y$, the map (\ref{eq:PertMarina1}) can be represented in the explicit form. Note that such perturbations were considered in \cite{GKSS19} while studying effects of reversible perturbations on 1:3 resonance in the conservative cubic H\'enon maps.
\subsection{Perturbations of nonorientable conservative H\'enon-like maps.} \label{sec:asymH-}
In this section we show that nonorientable conservative H\'enon-like maps also admit reversible perturbations of the same types that have been considered in the previous sections for orientable maps.
We consider the following nonorientable conservative H\'enon-like map of the form
\begin{equation}
H_{-1}: \;\;\; \bar x = -y, \;\; \bar y = -x + F(y).
\label{eq:TH-0}
\end{equation}
It is easy to show that this map is reversible with respect to the involution $h: x\to y, y\to x$, if function $F(y)$ is even, i.e. $F(-y) = F(y)$ (in particular, it follows from Lemma~\ref{lm:Hninveps} below). By analogy with Lemma~\ref{lm:til_H}, we consider the following perturbation
\begin{equation}
\hat{H}_{-1}: \;\;\; \bar x = -y, \;\; \bar y = -x + F(y) - \varepsilon(x,y) - \varepsilon(\bar y,\bar x),
\label{eq:TH-eps}
\end{equation}
where $\varepsilon(x,y)$ is some smooth function.
\begin{lemma}
If $F(y)$ is an even function, $F(y) =F(-y)$,
then the map $\hat{H}_{-1}$, given in~(\ref{eq:TH-eps}), is reversible with respect to the involution $h: x\to y, y\to x$, and
\begin{equation}
J(\hat H_{-1}) = - \frac{1+\varepsilon_{x}(x,y)}{1+\varepsilon_{x}(\bar y,\bar x)}.
\label{eq:J-eps0}
\end{equation}
\label{lm:Hninveps}
\end{lemma}
\begin{proof} The inverse map $(\hat{H}_{-1})^{-1}$ takes the form
\begin{equation}\label{eq:TH_inv}
(\hat{H}_{-1})^{-1}: \;\;\; \bar x = - y + F(-x) - \varepsilon(\bar x,\bar y) - \varepsilon(y,x), \;\;\; \bar y = - x.
\end{equation}
After the interchange $x\leftrightarrow y, \bar x \leftrightarrow \bar y$ in (\ref{eq:TH_inv}) we obtain the map $h \circ \hat{H}_{-1} \circ h$ that coincides with map (\ref{eq:TH-eps}), if $F(-y) = F(y)$, i.e., $F(y)$ is an even function.
For map $\hat H_{-1}$ in the form (\ref{eq:TH-eps}), we have that $J(\hat H_{-1}) = - \partial \bar x /\partial y \cdot \partial \bar y /\partial x$, since
$\partial \bar x /\partial x \equiv 0$. Then we have
$$
\frac{\partial \bar x}{\partial y} = -1, \;\; \frac{\partial \bar y}{\partial x} = -1 - \varepsilon_x(x,y) - \varepsilon_x(\bar y,\bar x) \frac{\partial \bar y}{\partial x}.
$$
This gives us the desired formula (\ref{eq:J-eps0}).
\end{proof}
\section{Symmetry breaking bifurcations in reversible perturbations of
H\'enon-like maps} \label{sec:sbrbif}
In this section we consider several examples of two-dimensional reversible maps that are perturbations of H\'enon-like maps and demonstrate reversible symmetry breaking bifurcations \cite{LT12} of fixed points or periodic orbits. Even with arbitrarily small perturbations,
such bifurcations lead to the appearance of dissipative elements of dynamics, although these bifurcations closely follow the corresponding bifurcations in the unperturbed area-preserving maps.
For example, a symmetric couple of elliptic or saddle orbits for the area-preserving map is transformed into a symmetric couple containing sink and source or saddles with the Jacobians greater and less than 1 in the perturbed map, etc.
The knowledge of these bifurcations and conditions of their realization is very relevant to understand such phenomenon as the appearance of mixed dynamics at (reversible) perturbations of conservative systems \cite{Gon16,GT17,Tur15}.
\subsection{Symmetry breaking bifurcations in the product of two quadratic H\'enon maps}\label{sec:sbrbif_dggls13}
Note that the product of two non-conservative asymmetric H\'enon maps $H_1$ and $H_2$ of the form~(\ref{eq:Hlms}) with $F(y) = M - y^2$ appears naturally as the normal forms of first return maps near symmetric quadratic homoclinic or heteroclinic tangencies to symmetric periodic orbits of reversible diffeomorphisms~\cite{DGGLS13,DGGL18}. Accordingly, their local bifurcations under reversible perturbations can play a role of global symmetry breaking bifurcations leading to the onset of reversible mixed dynamics.
In this section we consider bifurcations of this type. They are bifurcations of fixed points in some one-parameter family of reversible maps that unfolds the product of two non-conservative H\'enon maps
$$ H_1: \bar x =y, \; \bar y = M - y^2 \text{ and } H_2: \bar x = y, \; b \bar y = M - y^2$$
with the Jacobians equal to $b$ and $b^{-1}$, respectively. Their compositions $H_2\circ H_1$ and $H_1^{-1}\circ H_2^{-1}$ are both area-preserving maps, and, moreover, the latter map $T_2 = H_1^{-1}\circ H_2^{-1}$ can be written in the following cross-form (see formula (\ref{eq:H12-1e}))
$$
T_2 : \;\;\; \bar x = \frac{1}{b} x - \frac{M}{b} + \frac{1}{b}\bar y^2 , \;\; y = \frac{1}{b} \bar y - \frac{M}{b} + \frac{1}{b} x^2.
$$
To study symmetry breaking bifurcations appearing at reversible perturbations of this map we embed it in the following one-parameter family
\begin{equation}
{T}_{2\mu}: \;\;\;\;
\displaystyle \bar x = - \frac{M}{b} + \frac{1}{b} x + \frac{1}{b} \bar y^2 +\mu x \bar y, \;\;\;
\displaystyle y = - \frac{M}{b} + \frac{1}{b} \bar y + \frac{1}{b} x^2 + \mu \bar x y,
\label{eq:Teps}
\end{equation}
where $\mu$ is a small parameter. This family is a representative of the class (\ref{eq:H12-1e}) of reversible perturbations given by Lemma~\ref{lm:H12tld}, and, thus, it preserves the reversibility with respect to the involution $h: x\to y, y\to x$.
In Figure~\ref{fig-bdiag8} the main elements of bifurcation diagrams for fixed points of the maps $T_2$ and $T_{2\mu}$ are represented in the $(b,M)$-parameter plane for $\mu=0$ in Figure~\ref{fig-bdiag8}(a) and for a sufficiently small fixed $\mu$ in Figure~\ref{fig-bdiag8}(b). We exclude a small strip containing the axis $b=0$ from the consideration since the maps $T_2$ and $T_{2\mu}$ are not defined for $b=0$. The main bifurcation curves are the following: the fold bifurcation curves $F_1$ and $F_2$, the reversible pitchfork bifurcation curves $PF_1$ and $PF_2$ as well as several period-doubling curves $PD$ that are shown as gray dashed lines.
The equations of the curves are as follows:
$$
\begin{array}{rl}
F_1: & 4 (1+b\mu) M= - (b-1)^2, \text{ where } b<0, \\
F_2: & 4 (1+b\mu) M= - (b-1)^2, \text{ where } b>0, \\
PF_1: & 4 M = ({3}+b\mu) (b-1)^2, \text{ where } b<0 \\
PF_2: & 4 M = ({3}+b\mu) (b-1)^2, \text{ where } b>0.
\end{array}
$$
In the conservative case $\mu=0$, the curves $F_1$ and $F_2$ correspond to the creation of fixed points of $T_2$. There appears a symmetric parabolic fixed point which is nondegenerate for all parameter values in $F_1$ and $F_2$, except for the point $Q^*(b=1,M=0)\in F_2$. The parabolic point bifurcates into 2 symmetric elliptic and saddle fixed points. The transition through the point $Q^*(b=1,M=0)$ corresponds to a codimension 2 bifurcation which consists in the emergence of 4 fixed points: 2 symmetric elliptic and 2 asymmetric saddle fixed points which compose a symmetric couple of points.
\begin{figure}
\caption{Elements of the bifurcation diagram in the $(b,M)$-parameter plane for the maps (a) $T_2$ and (b) $T_{2\mu}
\label{fig-bdiag8}
\end{figure}
In the perturbed case $\mu \neq 0$, the character of the fold and period-doubling bifurcations is not changed qualitatively if $\mu$ is sufficiently small.
However, the pitchfork bifurcations can give rise to non-conservative fixed points.
It follows from Lemma~\ref{lm:H12tld} that the Jacobian of~(\ref{eq:Teps}) is
\begin{equation}
J= \frac{1+ b\mu\bar y}{1+ b\mu x}
\label{eq:JTepsfp}
\end{equation}
In order to find the fixed points of~(\ref{eq:Teps}), we equate $\bar x=x, \bar y =y$ and obtain the following system
\begin{equation}
\begin{array}{l}
x(b-1) = y^2 - M + b\mu xy, \;\;\; y(b-1) = x^2 - M + b\mu xy.
\end{array}
\label{eq:Tepsfp}
\end{equation}
Subtracting and adding up the equations and taking into account that $x\neq y$ (for asymmetric fixed points) gives us
$$
x+y = 1-b, \;\;\;
xy = \frac {(1-b)^2 -M}{1-b\mu}.
$$
Thus, if
$$
D = \frac{ 4 M - (1-b)^2(3+b\mu)}{4(1-b\mu)}>0,
$$
then two asymmetric fixed points $M_1$ and $M_2$ appear for $T_{2\mu}$:
$$
M_1=\left(\frac{1-b}{2} + \sqrt{D}, \frac{1-b}{2} - \sqrt{D}\right)\;\;\mbox{and}\;\; M_2=\left(\frac{1-b}{2} - \sqrt{D}, \frac{1-b}{2} + \sqrt{D}\right).
$$
These two points are symmetric with respect to the line $y=x$ and they merge with the corresponding symmetric fixed point under a reversible pitchfork bifurcation (which occurs when $D=0$).
It follows from~(\ref{eq:JTepsfp}) that the Jacobian at the fixed points $M_1$ and $M_2$ is
$$
J_{1} = 1 - \frac{4b\mu\sqrt{D}}{2 + b\mu(1-b + 2\sqrt{D})}, \;\;\;
J_{2} = 1 + \frac{4b\mu\sqrt{D}}{2 + b\mu(1-b - 2\sqrt{D})},
$$
respectively. Thus, if $b\mu>0$, then $J_1<1$ and $J_2 = J_1^{-1}> 1$.
The topological type of these points (for small $\mu$) is easily determined from the conservative approximation $\mu=0$, see Figure~\ref{fig-bdiag8}a. In the case $b<0$, the points $M_1$ and $M_2$ compose a symmetric couple of elliptic fixed points for $\mu=0$. For $\mu\neq 0$, they are transformed into a symmetric couple of ``sink-source'' fixed points: the points $M_1$ and $M_2$ become stable and unstable foci, respectively, if $\mu>0$.
In the case $b>0$, the points $M_1$ and $M_2$ become non-conservative saddles for $\mu\neq 0$: with the Jacobians $J_1<1$ and $J_2>1$, respectively, if $\mu>0$.
We also note that in the case $|b|=1$, the map $T_{2\mu}$ of the form (\ref{eq:Teps}) gives an example of a reversible perturbation for the second iteration of the conservative H\'enon map, the orientable one at $b=-1$ and nonorientable at $b=+1$. However, these perturbations are not suitable for the H\'enon maps themselves. Thus, at $b=-1$, the curve $PF_1$ is, in fact, the period-doubling curve for a symmetric fixed point which means that proper reversible perturbations can not lead to symmetry breaking. In the next sections, we consider the questions on correct reversible perturbations for the H\'enon maps, nonorientable and orientable, and on the structure of the accompanying symmetry breaking bifurcations.
\subsection{Symmetry breaking bifurcations in the nonorientable reversible H\'enon maps. } \label{sec:bifnor}
As an example we consider now the nonorientable H\'enon map $H_{-1}$ of the form
\begin{equation}
\bar x = -y, \bar y = -M -x + y^2 ,
\label{eq:HM-}
\end{equation}
that is a particular case of the map~(\ref{eq:TH-0}).
\begin{figure}
\caption{{\footnotesize Main bifurcations in maps (\ref{eq:HM-}
\label{fig-bdiagH2}
\end{figure}
For $M<0$, the map~(\ref{eq:HM-}) has no fixed points or periodic orbits. However, they appear immediately for $M>0$ under the so-called fold-flip bifurcation occurring at $M=0$ when the map has a fixed point $P(0,0)$ with eigenvalues $+1$ and $-1$. For $M>0$, this point splits into 4 points, see Figure~\ref{fig-bdiagH2}, two of them are the fixed points $S_1$ and $S_2$, and the other two points form a 2-periodic orbit $(Q_1,Q_2)$, i.e. $H_{-1}(Q_1)=Q_2, \; H_{-1}(Q_2)=Q_1$. Note that points $Q_1$ and $Q_2$ are elliptic 2-periodic orbits, and they are also symmetric since they both belong to the symmetry line $x=y$. In contrast, the points $S_1$ and $S_2$ are saddles and they compose a symmetric couple of points, i.e. $h(S_1)=S_2$ and $h(S_2)=S_1$. The coordinates of these points are
$$
Q_1 = (-\sqrt{M},-\sqrt{M}),\; Q_2 = (\sqrt{M},\sqrt{M}),\; S_1 = (-\sqrt{M},\sqrt{M}),\; S_2 = (\sqrt{M},-\sqrt{M}).
$$
All these points are conservative points of the map~(\ref{eq:HM-}).
However, due to Lemma~\ref{lm:Hninveps}, adding reversible perturbations we can destroy the conservativity of the fixed points $S_1$ and $S_2$.
For example, let us consider the following perturbed map
\begin{equation}
\tilde{H}_{-1\mu}: \;\;\; \bar x = -y, \;\;\; \bar y + \mu \bar x \bar y = -M -x + y^2 - \mu xy,
\label{eq:Hen-mu}
\end{equation}
where we have chosen the perturbation $\varepsilon(x,y)=\mu x y$, being $\mu$ a small parameter.
By Lemma~\ref{lm:Hninveps}, this map is reversible with respect to the involution $h$, however, it is no longer conservative for $\mu \neq 0$. Indeed, formula~(\ref{eq:J-eps0}) reads as
$$
J = - \frac{1+\mu y}{1+\mu \bar x} = - \frac{1+\mu y}{1-\mu y}.
$$
The fixed points of map (\ref{eq:Hen-mu}) are easily found:
$S_1 = (-a(\mu), a(\mu)),\; S_2 = (a(\mu), -a(\mu))$, where $
a(\mu) = \sqrt{{M}/{(1+2\mu)}}.
$
Then
we have that the Jacobian at the points $S_1$ and $S_2$ are
$$
J_1= - 1 - \frac{2\mu \sqrt{M}}{\sqrt{1+2\mu} - \mu \sqrt{M}},\;\;\; J_2= - 1 + \frac{2\mu \sqrt{M}}{\sqrt{1+2\mu} + \mu \sqrt{M}},
$$
respectively.
Thus, if $M>0$ and $\mu >0$ is not very large, the points $S_1$ and $S_2$ compose a symmetric couple of (nonorientable) saddles with the Jacobians $J_1<-1$ and $-1<J_2=J_1^{-1}<0$.
\subsection{Symmetry breaking bifurcations in the orientable reversible H\'enon maps.} \label{sec:sborH}
As an example we consider the standard area-preserving and orientable H\'enon map $H_{+1}$ of the form
\begin{equation}
\bar x = y, \;\; \bar y = M - x - y^2, \;\; .
\label{eq:HM+ }
\end{equation}
Bifurcations of its fixed points are well-known and include a parabolic bifurcation at $M=-1$, giving rise to symmetric elliptic and saddle fixed points, and a conservative period-doubling bifurcation of the fixed elliptic point at $M=3$, after which the elliptic fixed point becomes saddle and an elliptic 2-periodic orbits are born. Besides, when $M$ changes from $M=-1$ to $M=3$, the symmetric elliptic fixed point undergoes infinitely many bifurcations related to the appearance of resonant periodic orbits of period $q$ in its neighbourhood -- whenever the eigenvalues $e^{\pm i\varphi}$ pass through the values $\varphi = 2\pi \frac{p}{q}$, where $p$ and $q$ are mutually prime natural numbers and $p<q$.
However, as it is well-known, the fixed points and 2-periodic orbits in the H\'enon map are symmetric. The resonant periodic points are also symmetric if the resonances are nondegenerate. Thus, 3-periodic and 5-periodic resonant orbits are symmetric.
Although, the 1:4 resonance (related to eigenvalues $e^{\pm i\pi/2}=\pm i$) is degenerate in the H\'enon map \cite{Bir87, SimoVieiro09} (the so-called Arnold degeneracy \cite{Arn} takes place here), this bifurcation is not of symmetry breaking type \cite{LT12}.
A simple calculation of the number of points in periodic orbits of periods 1, 2, 3 and 4 (this number cannot be greater than $2^n$, by Bezout's theorem, even if we
include all points of periodic orbits of periods divisors of $n$)
\begin{itemize}
\item
period 1 (fixed points) -- two points;
\item
period 2 -- two points (we exclude 2 fixed points) that compose one 2-periodic orbit appearing after the period-doubling bifurcation of the elliptic fixed point;
\item
period 3 -- 6 points (we exclude 2 fixed points and the remaining 6 points compose two (elliptic and saddle) 3-periodic orbits accompanying the 1:3 resonance);
\item
period 4 -- 12 points (we exclude 2 fixed points and 2 points of the 2-periodic orbits and, thus, the remaining 12 points form two 4-periodic orbits born from the 1:4 resonance and one 4-periodic orbit appearing after a period-doubling bifurcation of the elliptic 2-periodic orbit);
\end{itemize}
shows that there are no asymmetric periodic orbits of these periods.
The case of period 5 points is more delicate. If two fixed points are excluded, then 30 more points remain. 20 such points compose four 5-periodic orbits born from the 1:5 and 2:5 resonances. Concerning remaining 10 points, they appear at a symmetric parabolic bifurcation of 5-periodic orbit. The last bifurcation we have found numerically at $M=5.5517$. The $y$-coordinates of the corresponding symmetric parabolic point is $y_1=y_2=-2.243751084, y_3=y_5=2.761032157, y_4=0.172152512$.
The calculation of the number of points of 6-periodic orbits shows the following. There are 64 such points in total. They include 10 points of smaller periods: 2 fixed points, 2 points of the 2-periodic orbit and 6 points of the 3-periodic orbits. The remaining 54 points form 9 orbits of period 6. Among them, 5 orbits are symmetric -- two 6-periodic orbits are born from the 1:6 resonance, one periodic orbit appears via the period-doubling bifurcation of the elliptic 3-periodic orbit, and two orbits arise due to the 1:3 resonance of a 2-periodic elliptic orbit.
The remaining 4 orbits of period 6 may be asymmetric.
For example, some of these orbits can appear as a result of a symmetry breaking bifurcation when a symmetric couple of two parabolic 6-periodic orbits appears and then splits into
two symmetric couples of elliptic and saddle 6-periodic orbits. Other possible cases can be related to two bifurcations of symmetric 6-periodic orbits and at least one of these bifurcations is a pitchfork bifurcation. We show below, see Section~\ref{sec:FO6}, that the second possibility is indeed realized in the H\'enon map $H_{+1}$.
We found numerically one couple of such orbits $O_6^1$ and $O_6^2$. In particular, for $M=4$, the orbit $O_6^1 =\{(x_i,y_i)\}$, $i=1,...,6$, where
$x_{i+1} = y_i$, has the following $y$-coordinates
$$
\begin{array}{l}
y_1=2.114907541,\;\;\; y_2=-1.935432332, \;\;\; y_3=-1.860805853, \\
y_4=2.472833909, \;\;\; y_5=-0.254101688, \; \;\; y_6=1.462598423.
\end{array}
$$
The orbit $O_6^2$ is symmetric to $O_6^1$ and, thus, has coordinates $\tilde x_i = y_i$ and $\tilde y_i = \tilde x_{i+1}$.
Now we consider the reversible perturbation of the H\'enon map $H_{+1}$ as follows
\begin{equation}
\tilde{H}_{+1\mu}: \;\;\; \bar x = y, \bar y + \mu (\bar x\bar y + \bar y^2)= M -x - y^2 - \mu (xy + x^2),
\label{eq:revHen+}
\end{equation}
that preserve reversibility of the H\'enon map due to Lemma~\ref{lm:til_H}, see also Example~1 for $\varepsilon_1(x,y) = \mu (xy + y^2)$.
We note that in the perturbed map (\ref{eq:revHen+}) the orbit $O_6^1$ has at $\mu = 0.01$ the following $y_i$-coordinates (here again $x_{i+1}=y_i$)
$$
\begin{array}{l}
y_1=2.107429699, \;\;\; y_2=-1.911473368, \;\;\; y_3=-1.833980679, \\
y_4=2.460965013, \;\;\; y_5=-0.2062196180, \;\;\; y_6=1.423687035.
\end{array}
$$
We calculate the Jacobian of map~(\ref{eq:revHen+}) at $O_6^1$
$$
J= \prod_{i=1}^{6} \frac{1+\mu y_i + 2\mu y_{i-1}}{1 + \mu y_i + 2\mu y_{i+1}}
$$
and obtain that $J= 0.9999999555$.
\subsubsection{Search of the asymmetric 6-periodic orbit} \label{sec:FO6}
It is very surprising that the main bifurcations related to the appearance of 6-periodic orbits $O_6^1$ and $O_6^2$ can be studied analytically due to the fact that these orbits are born as a result of a symmetry breaking bifurcation of a symmetric 6-periodic orbit.
The corresponding bifurcation scenario starts at the value $M=M_1=\frac{5}{4}$ when an elliptic 3-periodic orbit $O_3$ undergoes a supercritical period-doubling bifurcation after which the orbit $O_3$ becomes a symmetric saddle 3-periodic orbit and a symmetric elliptic 6-periodic orbit $\tilde O_6$ is born in its neighbourhood. Then increasing $M$, two successive period-doubling bifurcations of the orbit $\tilde O_6$, supercritical (at $M=M_2\approx 1.2813$) and subcritical (at $M=M_3\approx 2.98038$), take place. For $M_2<M<M_3$ the orbit $\tilde O_6$ is saddle, and for $M>M_2$ it becomes elliptic again. An important bifurcation occurs at $M=M_4=3$ when the orbit $\tilde O_6$ undergoes a pitchfork bifurcation after which the orbit $\tilde O_6$ becomes a symmetric saddle 6-periodic orbit and a symmetric couple of elliptic 6-periodic orbits $O_6^1$ and $O_6^2$ is born, see Figure~\ref{Henon2D_PF}.
\begin{figure}
\caption{{\footnotesize A schematic tree for the bifurcation scenario of the appearance of a symmetric couple of 6-periodic orbits in the H\'enon map $H_{+1\mu}
\label{Henon2D_PF}
\end{figure}
Let us explain this scenario in more detail. The 3-periodic orbits appear in the H\'enon map $H_{+1}$ at $M=1$ under a symmetric parabolic bifurcation leading to the birth of elliptic and saddle orbits $O_3$ and $S_3$. At $M=M_1=\frac{5}{4}$ the saddle orbit $S_3$ merges with the fixed point $O_{1:3}(\frac{1}{2},\frac{1}{2})$ with eigenvalues $e^{\pm 2\pi/3}$ (the 1:3 resonance) and simultaneously (exactly at this moment $M=M_1=\frac{5}{4}$) the elliptic orbit $O_3$ undergoes a supercritical period-doubling bifurcation giving rise to elliptic and saddle 6-periodic orbits, see Figure~\ref{Henon2D_PF_portr} (a)-(d). The elliptic orbit is the orbit $\tilde O_6$. As this orbit is symmetric, it has two intersection points with the line $x=y$. Let $\tilde O_6 = \{P_i(x_i,y_i)\},\; i=1,2,...,6$. Then the coordinates $(x_i,y_i)$ satisfy the following equations
$$
x_{i+1} = y_i,\; y_{i+1} = M - x_i - y_i^2, \;\;
$$
where $i = 1,..., 6$.
Since $x_{i+1} = y_i$, we can reduce this system to the following system of 6 quadratic equations
\begin{equation}
y_{2} = M - y_{6} - y_1^2, \; y_{3} = M - y_{1} - y_2^2, \; y_{4} = M - y_{2} - y_3^2, \\
y_{5} = M - y_{3} - y_4^2, \; y_{6} = M - y_{4} - y_5^2, \; y_{1} = M - y_{5} - y_6^2 .
\label{eq:6eqy}
\end{equation}
\begin{figure}
\caption{{\footnotesize Phase portraits of 3- and 6-periodic orbitsfor the H\'enon map $H_{+1}
\label{Henon2D_PF_portr}
\end{figure}
Assume that the point $P_1(x_1,y_1)$ of $\tilde O_6$ belongs to the symmetry line $x=y$, i.e. $x_1=y_1$. Then the point $P_4=H_{+1}^3(P_1)$ is also symmetric, i.e. $x_4=y_4$. Since $x_1=y_6$ and $x_4 =y_3$ and, hence,
$y_6=y_1$ and $y_3=y_4$. Then we get from the first and the last equations of (\ref{eq:6eqy}) that $y_2 = y_5$ and, thus, the system (\ref{eq:6eqy}) is reduced to the following system of three equations
\begin{equation}
y_{1} + y_2 = M - y_1^2, \; y_{1} + y_3 = M - y_2^2, \; y_{2} + y_3 = M - y_3^2, \\
\label{eq:3eqy}
\end{equation}
From the first and last equations of (\ref{eq:3eqy}) we obtain the relation
$y_1-y_3 = (y_3-y_1)(y_3 + y_1)$. If $y_1=y_3$, then the corresponding orbit has period 3. It follows that $y_1+y_3 = -1$. The second equation of (\ref{eq:3eqy}) gives us that $y_2^2 = M+1$. Then $y_1$ and $y_3$ satisfy the relations
$$
y_1^2 +y_{1} - M \pm \sqrt{M+1} =0, \; y_3^2 +y_{3} - M \pm \sqrt{M+1} =0.
$$
These two equations have the same solution, but $y_1$ and $y_3$ should take different values. Then, assuming for more definiteness that $y_3>y_1$, we find the following $y_i$-coordinates for the two symmetric 6-periodic orbits $\tilde O_6^1$ and $\tilde O_6^2$: for $\tilde O_6^1$
$$
y_1=y_6 = \frac{1}{2}\left(-1-\sqrt{1-4\sqrt{M+1}+4M} \right), y_2=y_5 =\sqrt{M+1}, y_3=y_4 = \frac{1}{2}\left(-1+\sqrt{1-4\sqrt{M+1}+4M} \right);
$$
for $\tilde O_6^2$
$$
y_1=y_6 = \frac{1}{2}\left(-1-\sqrt{1+4\sqrt{M+1}+4M} \right), y_2=y_5 =-\sqrt{M+1}, y_3=y_4 = \frac{1}{2}\left(-1+\sqrt{1+4\sqrt{M+1}+4M} \right).
$$
We stress that the orbit $\tilde O_6^1$ is born at $M = \frac{5}{4}$ (when $1-4\sqrt{M+1}+4M = 0$) under a supercritical period-doubling bifurcation of the elliptic 3-periodic orbit, while the orbit $\tilde O_6^2$ appears at $M = \frac{-3}{4}$ (when $1 + 4\sqrt{M+1}+4M = 0$) via a bifurcation of the 1:6 resonance fixed point.
Further we consider only the orbit $\tilde O_6^1$. In the analysis of its bifurcations we find the trace $Tr$ of the characteristic matrix for the map $H_{+1}^6$ at some point of $\tilde O_6^1$. As a result, we obtain that
$$
Tr = 86+24\sqrt{M+1}+116M-128\sqrt{M+1}M+96M^2-128\sqrt{M+1}M^2+64M^3.
$$
If we denote $\sqrt{M+1}=x$ ($x>0$), we obtain the polynomial
$$
Tr(x) = 2 + 24x + 116 x^2 + 128 x^3 -96x^4 -128x^5 + 64 x^6 = 2+ 4 x (2 x + 1)^3 (2 x - 3) (x - 2)
$$
Thus, the equation $Tr(x)=2$ has the solutions $x=0$, $x=-\frac{1}{2}$ (the triple root) and two positive solutions $x =\frac{3}{2}$ and $x=2$. The root $x=\frac{3}{2}$ corresponds to the value $M = \frac{5}{4}$ when the orbit $\tilde O_6^1$ is born.
The root $x=2$ corresponds to the value $M = 3$ when the symmetric elliptic orbit $\tilde O_6^1$ undergoes a pitchfork bifurcation -- the elliptic orbit becomes symmetric saddle and a symmetric couple of elliptic 6-periodic orbits $O_6^1$ and $O_6^2$ emerges, see Figure~\ref{Henon2D_PF_portr}(e)-(h). Namely, the orbits $O_6^1$ and $O_6^2$ are considered in Section~\ref{sec:sborH}. \\~\\
\textbf{Acknowledgements.} The authors thank D.V. Turaev and A.O. Kazakov for very useful remarks. This work is supported by the Russian Science Foundation under grants 19-11-00280 (Sections 1, 2 and 3) and 19-71-10048 (subsection 4.3). The authors thank also the Russian Foundation for Basic Research, project nos. 19-01-00607 (subsection 3.3) and 18-29-10081 (subsection 2.5) for support of scientific researches.
Numerical experiments in Section 4 were supported by the Laboratory of Dynamical Systems and Applications NRU HSE, of the Russian Ministry of Science and Higher Education (Grant No. 075-15-2019-1931). S.G. and K.S. acknowledge the financial support of the Ministry of Science and Higher Education of Russian Federation (Project \#0729-2020-0036).
M.G. was partially supported by the Spanish grants Juan de la Cierva-Incorporaci\'on IJCI-2016-29071, PGC2018-098676-B-I00 (AEI/FEDER/UE) and the Catalan grant 2017SGR1374.
\end{document} |
\begin{document}
\title{Arnold diffusion for a complete family of perturbations with two independent
harmonics\footnote{This work has been partially supported by
the Spanish MINECO-FEDER grant MTM2015-65715 and
the Catalan grant 2014SGR504. AD has been also partially supported by
the Russian Scientific Foundation grant 14-41-00044
at the Lobachevsky University of Nizhny Novgorod. RS has been also partially supported by CNPq, Conselho Nacional de Desenvolvimento Cient\'{i}fico e Tecnol\'{o}gico - Brasil.}}
\author{Amadeu Delshams\thanks{[email protected]}}
\author{Rodrigo G. Schaefer\thanks{[email protected]}}
\affil{Departament de Matem\`atiques and Lab of Geometry and Dynamical Systems\\
Universitat Polit\`ecnica de Catalunya, Barcelona}
\text{m}aketitle
\centerline{\emph{To Rafael de la Llave on the occasion of his 60th birthday}}
\begin{abstract}
We prove that for any non-trivial perturbation depending on any two independent harmonics of a pendulum and a rotor there is global instability.
The proof is based on the geometrical method and relies on
the concrete computation of several scattering maps. A complete description of the
different kinds of scattering maps taking place as well as the existence of piecewise smooth global scattering maps
is also provided.
\par
\noindent MSC2010 numbers: 37J40
\noindent\emph{Keywords}:
Arnold diffusion,
Normally hyperbolic invariant manifolds,
Scattering maps
\end{abstract}
\text{s}ection{Introduction}
\text{s}ubsection{Main result}
We consider an \emph{a priori unstable} Hamiltonian with $2+1/2$ degrees of freedom
\begin{equation}
H_{\varepsilon}(p , q , I , \varphi , s) = \pm\left( \frac{p^2}{2} + \cos q -1 \right) + \frac{I^2}{2} + \varepsilon h(q,\varphi,s)
\label{eq:hamil_system}
\end{equation}
consisting of a pendulum and a rotor plus a time periodic perturbation depending on two harmonics in the variables $(\varphi,s)$:
\begin{equation}
\begin{gathered}
h(q,\varphi,s)=f(q)g(\varphi, s),\\
f(q) = \cos q, \qquad g(\varphi , s) = a_1 \cos(k_1\varphi + l_1 s) + a_2\cos(k_2\varphi + l_2 s),\label{eq:g_general_case}
\end{gathered}
\end{equation}
with $k_1,\,k_2,\, l_1,\,l_2\in\text{m}athbb{Z}$.
The goal of this paper is to prove that for \emph{any} non-trivial perturbation $a_1a_2 \neq 0$ depending on \emph{any} two \emph{independent} harmonics
$\arraycolsep=0.8pt\def0.8}\left|\begin{array}{ll}k_1 & k_2 \\ l_1 & l_2\end{array{0.8}\left|\begin{array}{ll}k_1 & k_2 \\ l_1 & l_2\end{array}\right|\neq 0$,
there is global instability of the action $I$ for any $\varepsilon>0$ small enough.
\begin{theorem}\label{theo:main_theo}
Assume that $a_1a_2\neq 0$ and $k_1 l_2-k_2 l_1\neq 0$ in Hamiltonian \text{\eqref{eq:hamil_system}-\eqref{eq:g_general_case}}.
Then, for any $I^* > 0$, there exists $\varepsilon^* = \varepsilon^*(I^*, a_1, a_2)>0$ such that for any $\varepsilon$, $0<\varepsilon<\varepsilon^*$,
there exists a trajectory $\left(p(t),q(t),I(t), \varphi(t)\right)$ such that for some $T>0$
\begin{equation*}
I(0)\leq - I^*<I^*\leq I(T).
\end{equation*}
\end{theorem}
\begin{remark}
For a rough estimate of $\varepsilon^* \text{s}im \exp( - \pi I^* /2 ) $ at least for $\left|a_1/a_2\right| < 0.625$, $k_1 = l_2 = 1$ and $l_1 = k_2 = 0$, and of the diffussion time $T = T(\varepsilon^* , I^* , a_1 , a_2) \text{s}im (\text{T}_s(I^* , a_1 , a_2)/\varepsilon)\log( C(I^* , a_1 , a_2) /\varepsilon)$ the reader is referred to \cite{Delshams2017}.
Analogous estimates could be obtained for all the other values of the parameters.
\end{remark}
The proof is based on the geometrical method introduced in~\cite{Seara2006} and relies on
the concrete computation of several \emph{scattering maps}. A scattering map is a map
of transverse homoclinic orbits to a \emph{normally hyperbolic invariant manifold} (NHIM).
For Hamiltonian~\eqref{eq:hamil_system}, the NHIM turns out to be simply
\begin{equation}
\tilde{\Lambda}_{\varepsilon}=\tilde{\Lambda} = \left\{( 0 , 0 , I , \varphi , s ) : ( I , \varphi , s ) \in \text{m}athbb{R}\times \text{m}athbb{T}^2\right\}.\label{eq:NHIM_manifold}
\end{equation}
In the unperturbed case, i.e., $\varepsilon = 0$, for any $I^* >0$ the NHIM $\tilde{\Lambda}$ possesses a 4D \emph{separatrix},
that is to say, coincident stable and unstable invariant manifolds
\begin{equation*}
W^0\tilde{\Lambda} = \left\{ ( p_0( \tau ) , q_0( \tau ) , I , \varphi , s ) : \tau\in\text{m}athbb{R} , I\in\left[-I^* , I^* \right] , ( \varphi , s ) \in \text{m}athbb{T}^2 \right\},
\end{equation*}
where $( p_0, q_0)$ are the separatrices to the saddle equilibrium point of the pendulum
\begin{equation*}
\left(p_0(t) , q_0(t)\right) = \left(\frac{\pm 2}{\cosh t} , 4 \arctan\textrm{e}^{ \pm t}\right).
\end{equation*}
In the perturbed case, i.e., for small $\varepsilon > 0 $, $W^u(\tilde{\Lambda}_{\varepsilon})$
and $W^s(\tilde{\Lambda}_{\varepsilon})$ do not coincide (this is the so-called
splitting of separatrices), and every local transversal intersection between them gives
rise to a (local) scattering map which is simply the correspondence between a
past asymptotic motion in the NHIM to the corresponding future asymptotic motion following a homoclinic orbit.
Since the NHIM has also an inner dynamics, an adequate combination of these two dynamics on the NHIM, the inner one and the outer one provided by the scattering map, generates the global instability (also called in short \emph{Arnold diffusion}) as long as the outer dynamics does not preserve the invariant objects of the inner dynamics.
The inner motion is described in Section~\ref{sec:inner}, the scattering maps in Section~\ref{sec:scattering} and the absence of
invariant sets in both dynamics is checked in Section~\ref{sec:diffusion},
which also includes the proof of Theorem~\ref{theo:main_theo}.
Section~\ref{sec:piecewise} deals with the construction of a piecewise smooth global scattering map
which is introduced as a possible new tool to design fast and simple paths of global instability.
We finish this Introduction with some remarks
about the necessity of the assumptions, as well as other features of the scattering map and
a discussion of the model chosen and related work.
\text{s}ubsection{Necessity of the assumptions}
If the determinant $\Delta:=k_1 l_2-k_2 l_1$ or some coefficient $a_1$, $a_2$ vanishes, for instance, if there is only one harmonic in $g$, there is no
global instability for the action $I$. Indeed, looking at the equations associated to Hamiltonian~\eqref{eq:hamil_system}
\begin{align}
\label{eq:hamil_equations}
\dot{q} &= \pm p&\dot{p} &= \left[ \pm 1 + \varepsilon\left(a_1\cos(k_1\varphi + l_1 s) + a_2\cos(k_2\varphi + l_2 s)\right)\right]\text{s}in q\nonumber \\
\dot{\varphi} &= I&\dot{I} &= \varepsilon\cos q \left(k_1 a_1\text{s}in(k_1\varphi + l_1 s) + k_2 a_2\text{s}in(k_2\varphi + l_2 s)\right)\\
\dot{s} &= 1\nonumber&&
\end{align}
this is clear for $k_1=k_2=0$, since in this case $I$ is a constant of motion.
If $k_1$ or $k_2\neq 0$, say $k_1\neq 0$,
the change of variables
\begin{equation*}
\bar{\varphi} = k_1\varphi + l_1 s, \quad\quad r\bar{\varphi} - \bar{s} = k_2\varphi + l_2s, \quad\quad \bar{I}= k_1 I + l_1,
\end{equation*}
where $r = k_2/k_1$ can be assumed to satisfy $0\leq r \leq 1$ without loss of generality, casts system~\eqref{eq:hamil_equations} into
\begin{align*}
\dot{q} &=\pm p& \dot{p} &= \left[\pm 1 + \varepsilon \left( a_1 \cos\bar{\varphi} + a_2\cos(r\bar{\varphi} - \bar{s})\right)\right] \text{s}in q\\
\dot{\bar{\varphi}} &= \bar{I}&\dot{\bar{I}} &= \varepsilon k_1^2\cos q\left(a_1\text{s}in \bar{\varphi} + r a_2 \text{s}in(r\bar{\varphi} - \bar{s})\right)\\
\dot{\bar{s}} &= \Delta/k_1&&
\end{align*}
which is a Hamiltonian system with the Hamiltonian given by
\begin{equation}
\label{eq:bar_hamil_system}
\bar{H}_{\varepsilon}(p,q,\bar{I},\bar{\varphi},\bar{s}) = \pm \left(\frac{p^2}{2} + \cos q - 1 \right) + \frac{\bar{I}^2}{2} + \varepsilon k_1^2\cos q\left(a_1\cos\bar{\varphi} + a_2\cos(r\bar{\varphi} - \bar{s})\right).
\end{equation}
If $\Delta=0$ Hamiltonian~\eqref{eq:bar_hamil_system} is autonomous with 2 degrees of freedom,
and therefore a global drift for the action $I$ is not possible. Only drifts of size $\text{s}qrt{\varepsilon}$ are possible due to KAM theorem.
Analogously one easily checks that for $a_1a_2=0$ Hamiltonian~\eqref{eq:hamil_system} is integrable or autonomous.
\text{s}ubsection{Reduction of the harmonic types}
Under the hypothesis $\left(k_1 l_2-k_2 l_1\right) a_1 a_2\neq 0$ of Theorem~\ref{theo:main_theo}, we first
notice that the case $k_2 = 0 $ of Theorem~\ref{theo:main_theo} is already proven in~\cite{Delshams2017}.
Indeed, $k_2 = 0 $ implies $r:=k_2/k_1 = 0$ and it turns out from~\eqref{eq:bar_hamil_system} that Hamiltonian~\eqref{eq:hamil_system}
is equivalent to the one with $k_1=1, k_2 = 0,l_1=0,l_2= 1$:
\begin{equation}
H_{\varepsilon}(p , q , I , \varphi , t) = \pm\left( \frac{p^2}{2} + \cos q -1 \right) + \frac{I^2}{2}
+ \varepsilon \cos q \left(a_1 \cos \varphi + a_2\cos s\right),
\label{eq:old_hamil_system}
\end{equation}
which is just the Hamiltonian studied in \cite{Delshams2017}.
Therefore, we only need to prove
Theorem~\ref{theo:main_theo} for $k_1 k_2 \neq 0$ or equivalently for $r\in (0,1]$.
For the sake of clarity we will explain in full detail and prove Theorem~\ref{theo:main_theo}
along Sections~\ref{sec:inner}-\ref{sec:diffusion} just for $r=1$, which by~\eqref{eq:bar_hamil_system}
is equivalent to the case $k_1=1, k_2 = 1,l_1=0,l_2 = -1$:
\begin{equation}
H_{\varepsilon}(p , q , I , \varphi , t) = \pm\left( \frac{p^2}{2} + \cos q -1 \right) + \frac{I^2}{2}
+ \varepsilon \cos q \left(a_1 \cos \varphi + a_2\cos (\varphi - s)\right).
\label{eq:new_hamil_system}
\end{equation}
To finish the proof of Theorem~\ref{theo:main_theo}, in Section~\ref{sec:diffusion} we will sketch the
modifications needed for the case $r\in (0,1)$.
\text{s}ubsection{Scattering map types}
By the definition given at the beginning of Section~\ref{sec:scattering}, a scattering map is
in principle only \emph{locally} defined, that is, for a small ball of values of the variables
$(I,\varphi,s)$ or $(I,\theta=\varphi-Is)$, since it depends on a non-degenerate critical point
$\tau^*=\tau^*(I,\varphi,s)$ of a real function~\eqref{eq: SM_critical_point}, depending smoothly
on the variables $(I,\varphi,s)$, already introduced in~\cite{Seara2006}.
In the study carried out in Section~\ref{sec:scattering}, it
will be described whether, in terms of the parameter $\text{m}u:=a_1/a_2$ and the variable $I$, a local
scattering map can or cannot be smoothly defined for all the values of the angles $(\varphi,s)$ or
$\theta=\varphi-Is$, becoming thus a \emph{global} or \emph{extended} scattering map. This description will depend essentially
on a geometrical characterization of the function~$\tau^*(I,\varphi,s)$ in terms of the
intersection of \emph{crests} and \emph{NHIM lines}, following~\cite{Delshams2011}.
Any degeneration of the critical point $\tau^*=\tau^*(I,\varphi,s)$ may give rise to more
non-degenerate critical points and a bifurcation to \emph{multiple} local scattering maps
or to a non global scattering map. Different critical points $\tau^*=\tau^*(I,\varphi,s)$
give rise to different local scattering maps, and putting together different local scattering maps,
one can sometimes obtain \emph{piecewise smoth} global scattering maps, which are very useful
to design paths of instability for the action $I$, and are simply called diffusion paths.
For instance, in the paper~\cite{Delshams2017} devoted to the Hamiltonian~\eqref{eq:old_hamil_system},
it was proven that for $0<\text{m}u=a_1/a_2<0.625$, there exist two different global scattering maps.
Among the different kinds of associated orbits of these scattering maps, there appeared
two of them called \emph{highways}, where the drift of the action $I$ was very fast and simple.
As will be described in Section~\ref{sec:scattering}, such highways do not appear for
Hamiltonian~\eqref{eq:new_hamil_system}. Nevertheless, as will be proven in Section~\ref{sec:piecewise},
there exist piecewise smooth global scattering maps, and the possible diffusion along
the discontinuity sets opens the possibility of applying the theory of
piecewise smooth dynamical systems~\cite{Filippov88}.
\text{s}ubsection{About the model chosen and related work}
Hamiltonian~\eqref{eq:hamil_system} is a standard example of an \emph{a priori unstable} Hamiltonian
system~\cite{ChierchiaGallavotti} formed by a pendulum, a rotor and a perturbation. It is usual
in the literature to choose a perturbation depending periodically only on the
positions---which turn out to be angles in our case---and time. Our perturbation $h(q,\varphi,s)$ \eqref{eq:g_general_case}
is a little bit special since it is a product of a function $f(q)$ times a function $g(\varphi,s)$. This
choice makes easier the computations of the Poincar\'e-Melnikov potential~\eqref{eq:our_meln_potential},
which is based on the Cauchy's residue theorem.
Theorem~\ref{theo:main_theo} can be easily generalized to any trigonometric polynomial or meromorphic function $f(q)$,
although the computations of poles of high order become more complicated. In the same way, it could also be
generalized to more general perturbations $h(q,\varphi,s)$, as long that $h$ is a trigonometric polynomial or meromorphic in $q$.
The dependence on more than two harmonics gives rise to the appearance of more resonances in the inner dynamics,
which requires more control of their sizes, see for instance~\cite{DelshamsS97,Delshams2009}.
Apart from more difficulty in the computations of the Poincar\'e-Melnikov
potential and the inner Hamiltonian, we do not foresee substantial changes, so we believe that
Hamiltonian~\eqref{eq:hamil_system} could be considered as a paradigmatic example
of an \emph{a priori unstable} Hamiltonian system.
This paper is a natural culmination of~\cite{Delshams2017}, which dealt
with the simpler Hamiltonian~\eqref{eq:old_hamil_system}, and where a detailed description of NHIM lines
and crests was carried out. An ``optimal'' estimate of the diffusion time close to some special orbits of the scattering map,
called \emph{highways}, was also given there. The study in this paper of Hamiltonian~\eqref{eq:new_hamil_system} is more
complicated, due to a greater complexity of the evolution of the NHIM lines and crests with respect to the action $I$
and the parameters of the system. In particular, the absence of highways prevents us of showing an estimate
of diffusion time close to them.
The paper~\cite{Delshams2017} also contains a fairly extensive list of references about global instability. Let us
simply mention some new references that are not there, like~\cite{Davletshin2016} which contains a similar approach
to the function~$\tau^*$ of~\cite{Seara2006} and the crests of~\cite{Delshams2011}, and the recent preprints
\cite{GelfreichT14,LazzariniMS15,Marco16,GideaM17,Cheng17}
involving the geometrical method or variational methods.
We finish this introduction by noticing that in this paper we stress the interaction between {\text{NHIM}} lines and crests,
since this allows us to describe the diverse scattering maps, as well as their domains, that appear in our problem.
In more complicated models of Celestial Mechanics the Melnikov potential is not available.
In these cases the computations of scattering maps rely on the numerical computation of invariant manifolds of a NHIM
or some of its selected invariant objects, and the search of diffusion orbits is performed in a more crafted way (see \cite{Canalias2006,delshams2008b,delshams2013,capinski2017,delshams201629}).
\text{s}ection{Inner dynamics}
\label{sec:inner}
The inner dynamics is derived from the restriction of $H_{\varepsilon}$ in \eqref{eq:new_hamil_system} and its equations to $\tilde{\Lambda}$, that is,
\begin{equation}
K(I,\varphi,s) = \frac{I^{2}}{2} + \varepsilon\left(a_{1}\cos\varphi + a_{2}\cos(\varphi -s)\right), \label{eq:hamiltonian_inner}
\end{equation}
and differential equations
\begin{equation}
\dot{\varphi} = I \quad\quad \dot{s} = 1\quad\quad\dot{I} =\varepsilon\left(a_{1}\text{s}in\varphi + a_{2}\text{s}in(\varphi-s)\right).\label{eq:equations_inner}
\end{equation}
Note that in this case the inner dynamics is slightly more complicated to describe than in \cite{Delshams2017} where there was just one resonance, namely, in $I=0$.
In the current case we have two resonant regions of size $\text{m}athcal{O}(\text{s}qrt{\varepsilon})$ where secondary KAM tori appear.
To describe these regions, we use normal forms as in \cite{Seara2006}.
Consider the autonomous extended Hamiltonian
\begin{equation}
\overline{K}(I,A,\varphi,s) = \frac{I^2}{2} + A + \varepsilon\left(a_{1}\cos\varphi + a_{2}\cos(\varphi -s)\right),\label{eq:bar_K}
\end{equation}
with the differential equations
\begin{align*}
\dot{\varphi} =& I &\dot{I} =&\varepsilon\left(a_{1}\text{s}in\varphi + a_{2}\text{s}in(\varphi-s)\right)\\
\dot{s} =& 1 &\dot{A} =& -\varepsilon a_{2}\text{s}in(\varphi-s).
\end{align*}
This system is equivalent to the system represented by \eqref{eq:hamiltonian_inner}+\eqref{eq:equations_inner}.
We wish to eliminate the dependence on the angle variables.
Consider a change of variables $\varepsilon$-close to the identity $$(\varphi,s , I , A ) = g(\phi , \text{s}igma , J , B) = (\phi , \text{s}igma , J , B) + \text{m}athcal{O}(\varepsilon)$$
such that it is the one-time flow for a Hamiltonian $\varepsilon G$, i.e., $g = g_{t=1}$, where $g_t$ is solution of
\begin{equation*}
\frac{dg_t}{dt} = J_0\nabla\varepsilon G\circ g_t, \text{ where } J_0 \text{ is the symplectic matrix }\begin{pmatrix}
0&1 \\
-1&0
\end{pmatrix}.
\end{equation*}
Composing $\overline{K}$ with $g$ and expanding in a Taylor series around $t=0$, one obtains
\begin{equation*}
\overline{K}\circ g = \overline{K} + \left\{\overline{K},\varepsilon G\right\} + \frac{1}{2}\left\{\left\{\overline{K},\varepsilon G\right\} , \varepsilon G\right\} + \dots,
\end{equation*}
where $\left\{\cdot\right\}$ is the Poisson bracket.
Using the expansion \eqref{eq:bar_K} of $\overline{K}$, the equation above can be written as
\begin{equation}
\begin{split}
\overline{K}\circ g = \frac{J^2}{2} + B + \varepsilon\left( a_{1}\cos\phi + a_{2}\cos(\phi-\text{s}igma) + \left\{\frac{J^2}{2} + B ,G\right\} \right) \\+ \frac{\varepsilon^2}{2}\left\{\left\{\frac{J^2}{2} + B,G\right\},G\right\} + \text{m}athcal{O}(\varepsilon^3).\label{eq:exp_K}
\end{split}
\end{equation}
We want to find $G$ such that $a_{1}\cos\phi + a_{2}\cos(\phi-\text{s}igma) + \left\{\frac{J^2}{2} + B ,G\right\}=0$, or equivalently,
\begin{equation*}
J\frac{\partial G}{\partial\phi} + \frac{\partial G}{\partial \text{s}igma } = a_{1}\cos\phi + a_{2}\cos(\phi-\text{s}igma).
\end{equation*}
Given $a <b<1$, consider any function $\Psi \in C^{\infty}(\text{m}athbb{R})$ satisfying $\Psi(x) = 1$ for $x\in \left[-a,a\right]$ and $\Psi(x) = 0$ for $\left|x\right| > b$ and introduce
\begin{equation*}
G(J,B,\phi,\text{s}igma) := \frac{a_{1}}{J}\left(1-\Psi(J)\right)\text{s}in\phi + \frac{a_{2}}{J-1}\left(1-\Psi(J-1)\right)\text{s}in(\phi - \text{s}igma),
\end{equation*}
Substituting the above function $G(J,B,\phi,\text{s}igma)$ in \eqref{eq:exp_K} we have
\begin{eqnarray}
\overline{K}\circ g =\frac{J^2}{2} + B + \text{m}athcal{O}(\varepsilon^2) , \label{eq:invariant_tori_nr}
\end{eqnarray}
for $J,J-1 \notin [-b,b]$.
For $J\in [-a,a]$,
\begin{equation}
\overline{K}\circ g = \frac{J^2}{2} + B + \varepsilon a_{1}\cos\phi + \text{m}athcal{O}(\varepsilon^2). \label{eq:ressonance_1}
\end{equation}
Finally, for $J-1 \in [-a,a]$,
\begin{equation}
\overline{K}\circ g = \frac{J^2}{2} + B + \varepsilon a_{2}\cos(\phi-\text{s}igma) + \text{m}athcal{O}(\varepsilon^2). \label{eq:ressonance_2}
\end{equation}
From \eqref{eq:ressonance_1} and \eqref{eq:ressonance_2}, one sees that on $J = 0$ and $J = 1$ there are resonances of first order in $\varepsilon$ with a pendulum-like behavior.
Coming back to the original variables, three kinds of invariant tori are obtained.
For the first order resonance $ I = 0 $, there is a positive $\overline{a}$ such that the invariant tori are given by $F^{0}(I,\varphi,s)=~\text{constant}$ with
\begin{equation}
F^0(I,\varphi,s) = \frac{I^2}{2} + \varepsilon a_{1}\cos\varphi + \text{m}athcal{O}(\varepsilon^2).\label{eq:invariant_tori_i=0}
\end{equation}
for $I \in \left[-\overline{a},\overline{a}\right]$
Analogously, for the first order resonance $ I = 1 $, with
\begin{equation*}
F^1(I,\varphi,s) = \frac{(I-1)^2}{2} + \varepsilon a_{2} \cos(\varphi-s) + \text{m}athcal{O}(\varepsilon^2),\label{eq:invariant_tori_i=1}
\end{equation*}
for $I-1 \in \left[-\overline{a},\overline{a}\right]$.
\begin{remark}
As commented in \cite{Seara2006}, there exists a \emph{secondary} resonance in $I = 1/2$, but the size of the gap in its resonant region is much smaller than the size of gaps in resonant regions associated to $I = 0$ and $I= 1$.
\end{remark}
\begin{remark}In a more general case with $r\neq 1$, the resonances take place in $I = 0$ and $I = 1/r$.\label{rem:r_1}
\end{remark}
From \eqref{eq:invariant_tori_nr}, on the non-resonant region the invariant tori has equations $F^{\text{nr}}(I) =~\text{constant}$ with
\begin{equation*}
F^{\text{nr}}(I) = \frac{I^2}{2} + \text{m}athcal{O}(\varepsilon^2).\label{eq:invariant_tori_nr_b}
\end{equation*}
An illustration of the inner dynamics is displayed in Figure \ref{fig:inner_dynamics}.
\begin{figure}
\caption{Plane $\varphi \times I$ of inner dynamics for $\text{m}
\label{fig:inner_dynamics}
\end{figure}
\text{s}ection{Scattering map}
\label{sec:scattering}
\text{s}ubsection{Definition of scattering map }
We are going to explore the properties of the scattering maps of Hamiltonian \eqref{eq:new_hamil_system}.
The notion of scattering map on a {\text{NHIM}} was introduced in \cite{Delshams2000}.
Let $W$ be an open set of $\left[-I^* , I^*\right] \times \text{m}athbb{T}^2$ such that the invariant manifolds of the {\text{NHIM}} $\tilde{\Lambda}$ introduced in~\eqref{eq:NHIM_manifold} intersect transversally along a homoclinic manifold $\Gamma = \left\{\tilde{z}(I,\varphi,s;\varepsilon) , (I,\varphi,s)\in W\right\}$ so that for any $\tilde{z}\in\Gamma$ there exist unique $\tilde{x}_{+,-} = \tilde{x}_{+,-}(I,\varphi,s;\varepsilon)\in\tilde{\Lambda}$ such that $\tilde{z}\in W_{\varepsilon}^{s}(x_-)\cap W_{\varepsilon}^{u}(\tilde{x}_+)$.
Let
$$H_{+,-} = \bigcup\left\{\tilde{x}_{+,-}(I,\varphi , s ; \varepsilon) : (I,\varphi,s)\in W\right\}.$$
The scattering map associated to $\Gamma$ is the map
\begin{eqnarray*}
S: H_- & \longrightarrow & H_+\\
\tilde{x}_- &\longmapsto & S(\tilde{x}_-) = \tilde{x}_+.
\end{eqnarray*}
For the characterization of the scattering maps, it is required to select the homoclinic manifold $\Gamma$ and this is done using the Poincar\'{e}-Melnikov theory. From \cite{Seara2006,Delshams2011}, we have the following proposition
\begin{proposition}\label{prop:melnpot}
Given $(I,\varphi,s)\,\in\,\left[-I^{*},I^{*}\right]\,\times\,\text{m}athbb{T}^{2}$, assume that the real function
\begin{equation}\label{eq: SM_critical_point}
\tau\,\in\,\text{m}athbb{R}\,\longmapsto\,\text{m}athcal{L}(I,\varphi-I\,\tau,s-\tau)\,\in\,\text{m}athbb{R}
\end{equation}
has a non degenerate critical point $\tau^{*}\, =\, \tau^*(I,\varphi,s)$, where
\begin{equation*}
\text{m}athcal{L}(I,\varphi,s):=\int_{-\infty}^{+\infty}\left(f(q_{0}(\text{s}igma)) - f(0)\right)g(\varphi+I\text{s}igma,s+\text{s}igma;0)d\text{s}igma.
\end{equation*}
Then, for $0\,<\,\varepsilon$ small enough, there exists a unique transversal homoclinic point $\tilde{z}$ to $\tilde{\Lambda}_{\varepsilon}$ of Hamiltonian~\eqref{eq:hamil_system}, which is $\varepsilon$-close to the point
$\tilde{z}^{*}(I,\varphi,s)\,=\,(p_{0}(\tau^{*}),q_{0}(\tau^{*}),I,\varphi,s)\,\in\,W^{0}(\tilde{\Lambda})$:
\begin{equation*}
\tilde{z}=\tilde{z}(I,\varphi,s)=(p_{0}(\tau^{*})+O(\varepsilon), q_{0}(\tau^{*})+O(\varepsilon),I,\varphi,s)\,\in\,W^{u}(\tilde{\Lambda_ {\varepsilon}})\,\pitchfork\,W^{s}(\tilde{\Lambda_{\varepsilon}}).
\end{equation*}
\end{proposition}
The function $\text{m}athcal{L}$ is called the \emph{Melnikov potential} of Hamiltonian \eqref{eq:hamil_system}.
For the concrete Hamiltonian~\eqref{eq:new_hamil_system} it takes the form
\begin{equation}
\text{m}athcal{L}(I,\varphi,s) = A_{1}(I)\cos\varphi + A_{2}(I)\cos(\varphi - s),\label{eq:our_meln_potential}
\end{equation}
where
\begin{equation*}
A_{1}(I) = \frac{2\pi I a_{1}}{\text{s}inh(\pi I/2)} \quad \text{ and }\quad A_{2}(I) = \frac{2\pi(I-1)a_{2}}{\text{s}inh(\pi(I-1)/2)}.
\end{equation*}
The homoclinic manifold $\Gamma$ is characterized by the function $\tau^*(I,\varphi,s)$.
Once a $\tau^*(I,\varphi,s)$ is chosen, which under the conditions of Proposition~\ref{prop:melnpot}, is locally smoothly well defined, by the geometric properties of the scattering map, see \cite{Delshams2008,Delshams2009,Delshams2011}, the scattering map has the explicit local form
\begin{equation*}
S(I,\varphi,s) = \left(I + \varepsilon\frac{\partial L^*}{\partial \varphi}(I,\varphi,s) + \text{m}athcal{O}(\varepsilon^2) , \varphi - \varepsilon\frac{\partial L^*}{\partial I}(I,\varphi,s) +\text{m}athcal{O}(\varepsilon^2) , s \right),
\end{equation*}
where
\begin{equation}
L^*(I,\varphi,s) = \text{m}athcal{L}(I,\varphi - I\tau^*(I,\varphi,s) , s-\tau^*(I,\varphi,s)).\label{eq:L^*-def}
\end{equation}
Notice that the variable $s$ is fixed under the scattering map.
As a consequence \cite{Delshams2011,Delshams2017}, introducing the variable
\begin{equation*}
\theta = \varphi - I s
\end{equation*}
and defining the \emph{reduced Poincar\'{e} function}
\begin{equation}\label{eq:reduced_poincare_function}
\text{m}athcal{L}^{*}(I,\theta) := L^*(I,\varphi - Is , 0) = L^*(I,\varphi,s),
\end{equation}
in the variables $(I,\theta)$, the scattering map has the simple form
\begin{equation*}
\text{m}athcal{S}(I,\theta) = \left( I + \varepsilon\frac{\partial \text{m}athcal{L}^*}{\partial\theta}(I,\theta) + \text{m}athcal{O}(\varepsilon^2) , \theta - \varepsilon\frac{\partial \text{m}athcal{L}^*}{\partial I}(I,\theta) + \text{m}athcal{O}(\varepsilon^2) \right),
\end{equation*}
so up to $\text{m}athcal{O}(\varepsilon^2)$ terms, $\text{m}athcal{S}(I,\theta)$ is the $\varepsilon$ times flow of the \emph{autonomous} Hamiltonian $-\text{m}athcal{L}^*(I,\theta)$.
In particular, the iterates under the scattering map follow the level curves of $\text{m}athcal{L}^*$ up to $\text{m}athcal{O}(\varepsilon^2)$.
\text{s}ubsection{Crests and {\text{NHIM}} lines}
We have seen that the function $\tau^*$ plays a central role in our study.
Therefore, we are interested in finding the critical points $\tau^* = \tau^*(I,\varphi,s) $ of function \eqref{eq: SM_critical_point}.
For our concrete case \eqref{eq:our_meln_potential}, $\tau^*$ is a solution of
\begin{equation}
I A_{1}(I)\text{s}in(\varphi - I\tau^*) + (I-1)A_{2}(I)\text{s}in(\varphi - s - (I-1)\tau^*)=0.\label{eq:tau^*}
\end{equation}
This equation can be viewed from two equivalently geometrical viewpoints.
The first one is that to find $\tau^* = \tau^*(I,\varphi,s)$ satisfying \eqref{eq:tau^*} for any $(I,\varphi,s)\in \left[-I^* , I^* \right]\times\text{m}athbb{T}^2$ is the same as to look for the extrema of $\text{m}athcal{L}$ on the \emph{{\text{NHIM}} line}
\begin{equation}
R(I,\varphi,s) = \left\{(I,\varphi - I\tau , s-\tau) : \tau \in \text{m}athbb{R}\right\}.\label{eq:nhim_line}
\end{equation}
\begin{remark}
Since $(\varphi,s)\in\text{m}athbb{T}^2$, $R(I,\varphi,s)$ is a closed line if $I \in \text{m}athbb{Q}$ and it is a dense line on
$\{I\}\times\text{m}athbb{T}^2$ if $I \notin \text{m}athbb{Q}$.
\end{remark}
The other viewpoint is that, fixing $(I,\varphi,s)$, a solution $\tau^*$ of \eqref{eq:tau^*} is equivalent to finding intersections between a {\text{NHIM}} line \eqref{eq:nhim_line} and a curve defined by
\begin{equation}
IA_1( I ) \text{s}in\varphi + (I - 1)A_2(I)\text{s}in(\varphi - s) = 0.\label{eq:def_crests_2}
\end{equation}
These curves are called \emph{crests}, and in a general way can be defined as follows.
\begin{definition}\label{def:crest}
\cite{Delshams2011} We define by \emph{Crests} $\text{m}athcal{C}(I)$ the curves on $(I,\varphi,s)$, $(\varphi,s)\in \text{m}athbb{T}^2$, such that
\begin{equation}
\frac{\partial \text{m}athcal{L}}{\partial \tau}(I , \varphi - I \tau ,s - \tau)|_{\tau = 0} = 0,\label{eq:def_crests}
\end{equation}
or equivalently,
$$I\frac{\partial \text{m}athcal{L}}{\partial\varphi}(I,\varphi,s) + \frac{\partial\text{m}athcal{L}}{\partial s}(I,\varphi,s)= 0.$$
\end{definition}
As in our case $\text{m}athcal{L}(I , \varphi - I\tau , s - \tau ) = A_1(I)\cos(\varphi - I\tau) + A_2( I ) \cos( \varphi - s - (I - 1) \tau )$, equation \eqref{eq:def_crests} takes the form
\eqref{eq:def_crests_2}.
Introducing
\begin{equation}
\text{s}igma = \varphi - s, \label{eq:def_sigma}
\end{equation}
equation \eqref{eq:def_crests_2} can be rewritten as
\begin{equation}
\text{m}u \alpha( I ) \text{s}in\varphi + \text{s}in\text{s}igma =0,\label{eq:crests_equation}
\end{equation}
for $I \neq 1$, where
\begin{equation}
\text{m}u = \frac{a_{1}}{a_{2}}\quad\text{and}\quad \alpha(I) =\frac{I^2\text{s}inh(\frac{\pi}{2}(I-1))}{(I-1)^2\text{s}inh(\frac{\pi I}{2})}.\label{eq:mu_alpha}
\end{equation}
From now on, when we refer to crests $\text{m}athcal{C}(I)$ we mean the set of points $(I,\varphi , \text{s}igma)$ satisfying equation~\eqref{eq:crests_equation}.
See an illustration in Fig.~\ref{fig:qual_plot}.
\begin{remark}
In \cite{Delshams2017} the crests were described on the plane $(\varphi , s )$, whereas now such curves lie on the plane $(\varphi , \text{s}igma)$.
Besides, differently from the cases studied in \cite{Delshams2011,Delshams2017}, the function $\alpha(I)$ is not defined for all $I$.
More precisely, it is not defined for $I = 1$. For this value of $I$, equation \eqref{eq:crests_equation} is not adequate,
and one has to use \eqref{eq:def_crests_2} to check that for $I = 1$ the crests are just two vertical straight lines on the plane $(\varphi,\text{s}igma)$ given by $\varphi = 0$ and $\varphi = \pi$.
\end{remark}
\begin{remark}
For Hamiltonian \eqref{eq:bar_hamil_system} and $r\in (0,1)$, $\alpha_{r}(I)$ is not defined for $I = 1/r$ and is given by
\begin{equation*}
\alpha_r(I) = \frac{I^2\text{s}inh\left(\frac{\pi}{2}(rI - 1)\right)}{(rI-1)^2\text{s}inh\left(\frac{\pi I}{2}\right)}.
\end{equation*}
\label{rem:r_2}
\end{remark}
We are interested in understanding the behavior of these crests because, as we have seen in previous works \cite{Delshams2011,Delshams2017}, their intersection with the {\text{NHIM}} lines determine the existence and behavior of scattering maps.
From \eqref{eq:crests_equation}, when $\left|\alpha(I)\right| <1/\left|\text{m}u\right|$, $\text{s}igma$ can be written as a function of $\varphi$ for all $\varphi\in\text{m}athbb{T}$ on the crest $\text{m}athcal{C}(I)$.
On the other hand, if $\left|\alpha(I)\right| > 1/\left|\text{m}u\right|$, $\varphi$ can be written as a function of $\text{s}igma$ for all $\text{s}igma\in\text{m}athbb{T}.$
These two conditions give us two kinds of crests: \emph{horizontal} for $\left|\alpha(I)\right| <1/\left|\text{m}u\right|$ and \emph{vertical} for $\left|\alpha(I)\right| > 1/\left|\text{m}u\right|$.
These names are due to their forms on the plane $(\varphi , \text{s}igma)$.
We consider the same characterization used in \cite{Delshams2017}:
\begin{itemize}
\item For $\left|\alpha(I)\right|\, <\, 1/\left|\text{m}u\right|$, there are two horizontal crests $\text{s}igma = \xi_{\text{M},\text{m}}(I,\varphi)$
$$\text{m}athcal{C}_{\text{M},\text{m}}(I)=\{(I,\varphi,\xi_{\text{M},\text{m}}(I,\varphi)):\varphi\in\text{m}athbb{T}\},$$
\begin{eqnarray}\label{eq:cristas_06}
\xi_{\text{M}}(I,\varphi)&=&-\arcsin(\text{m}u\alpha(I)\text{s}in \varphi ) \quad\quad\text{m}od{2\pi}\\
\xi_{\text{m}}(I,\varphi)&=&\arcsin(\text{m}u\alpha(I)\text{s}in \varphi )+\pi \quad\quad\text{m}od{2\pi}.\nonumber
\end{eqnarray}
\item For $\left|\alpha(I)\right|\, >\, 1/\left|\text{m}u\right|$, there are two vertical crests $\varphi = \eta_{\text{M},\text{m}}(I,\text{s}igma)$
$$\text{m}athcal{C}_{M,m}(I)=\{(I,\eta_{M,m}(I,\text{s}igma),\text{s}igma):\text{s}igma\in\text{m}athbb{T}\},$$
\begin{eqnarray*}
\eta_{M}(I,\text{s}igma)&=&-\arcsin(\text{s}in \text{s}igma/\left(\text{m}u \alpha(I)\right))\quad\quad\text{m}od{2\pi} \label{eq:eta_definition}\\
\eta_{m}(I,\text{s}igma)&=&\arcsin(\text{s}in \text{s}igma/\left(\text{m}u \alpha(I)\right))+\pi\quad\quad\text{m}od{2\pi}.\nonumber
\end{eqnarray*}
\end{itemize}
\begin{remark}
\label{rmk:singular}
$\left|\alpha(I)\right| = 1/\left|\text{m}u\right|$ is a singular or bifurcation case.
In this case, the crests are straight lines and are not differentiable in $\varphi = \pi/2$ and $\varphi = 3\pi/2$.
See Fig.~6 of~\cite{Delshams2017}.
\end{remark}
\begin{remark}
The crest containing the point $(\varphi , \text{s}igma) = (0,0)$ will be denoted by $\text{m}athcal{C}_{\text{M}}(I)$ and the crest containing the point $(\varphi , \text{s}igma) = (\pi , \pi)$ by $\text{m}athcal{C}_{\text{m}}(I)$.
\end{remark}
Note that the function $\left|\alpha(I)\right|$ is not bounded, indeed
\begin{equation*}
\lim_{I\rightarrow 1}\left|\alpha(I)\right| = +\infty.
\end{equation*}
This implies that for any $\text{m}u$ there exists a neighborhood $U$ of $I = 1$ such that for all $I\in U$ the crests are vertical.
On the other hand, since $\alpha( 0 ) = 0 $ there exists a neighborhood $V$ of $I = 0$ such that for all $I\in V$ the crests are horizontal.
We notice here a remarkable difference with the Hamiltonians studied in \cite{Delshams2011,Delshams2017}, where, for $\left|\text{m}u\right| \leq 0.97$, all the crests are horizontal for all $I$.
Now take a look at the properties of the function $\alpha(I)$ introduced in \eqref{eq:mu_alpha} to describe under which conditions in $\text{m}u$ the crests are horizontal or vertical.
First of all, observe that for $I\neq 1$, $\alpha(I)$ is smooth and $\alpha'(I)\neq 0, $ and for $I = 1$ $\alpha(I)$ is not bounded, indeed it has a vertical asymptote
\begin{equation*}
\lim_{I\rightarrow 1^-} \alpha(I) = -\infty \quad\quad \text{ and }\quad\quad\lim_{I\rightarrow 1^+}\alpha(I) = +\infty.
\end{equation*}
Given a $\text{m}u\neq0$, since $\alpha(0) = 0$, there exists a unique $I_c \in (0,1)$ such that $\left|\alpha(I)\right| = 1/\left|\text{m}u\right|$.
So, the crests are horizontal for $I\in\left[ 0 , I_c\right)$ and vertical for $I\in(I_c , 1)$.
Others important limits are
\begin{equation*}
\lim_{I\rightarrow -\infty}\alpha(I) = \exp(\pi/2) \quad\quad \text{ and }\quad\quad \lim_{I\rightarrow + \infty } \alpha(I) = \exp(-\pi/2).
\end{equation*}
The first limit implies that $\left|\alpha(I)\right| < \exp( \pi/2)$ for $I\in\left(-\infty , 0\right)$.
Thus, if $\exp(\pi/2)\leq 1/\left|\text{m}u\right|$ the crests are horizontal for $I\in(-\infty , 0)$.
Otherwise, if $1/\left|\text{m}u\right| < \exp(\pi/2 )$, there exists a unique $I_{\text{l}}\in(-\infty , 0)$ such that $\left|\alpha(I)\right| = 1/\left|\text{m}u\right|$ and the crests are vertical for $I\in(-\infty , I_{\text{l}})$ and horizontal for $I\in\left(I_{\text{l}} ,0\right)$.
The second limit implies that $\left|\alpha(I)\right| > \exp(-\pi/2)$ for $I\in(1 , +\infty)$.
Then, if $\exp( -\pi/2 )\geq 1/\left|\text{m}u\right|$, the crests are vertical for $I\in\left[1 , +\infty\right)$.
if $\exp( -\pi/2 )< 1/\left|\text{m}u\right|$, there exists a unique $I_{\text{r}}\in(1 , + \infty)$, such that the crests are vertical for any $I$ in $\left[1 , I_{\text{r}}\right)$ and horizontal for $I\in(I_\text{r} , +\infty)$.
Summarizing, for $1/\left|\text{m}u\right| \geq \exp(\pi/2)$, crests are horizontal for $I\in\left(-\infty , I_{\text{c}}\right)\cup(I_{\text{r}} , +\infty)$ and vertical for $I\in\left(I_{\text{c}},I_{\text{r}}\right)$.
For $\exp(-\pi/2) < 1/\left|\text{m}u\right| < \exp( \pi/2)$, crests are horizontal for $I\in(I_{\text{l}}, I_{\text{c}})\cup(I_{\text{r}} , +\infty)$ and vertical for $I\in(-\infty , I_{\text{l}})\cup(I_{\text{c}} , I_{\text{r}})$.
Finally, if $ 1/\left|\text{m}u\right| < \exp( -\pi/2)$, crests are horizontal for $I\in(I_{\text{l}} , I_{\text{c}})$ and vertical for $I\in(-\infty , I_{\text{l}})\cup(I_{\text{c}, + \infty})$.
\begin{remark}
For $r\in (0,1)$, $\alpha_r(I)$ is not bounded on a neighbourhood of the resonance $I = 1/r$, i.e., $\lim_{I \rightarrow 1/r^{-}} \alpha_r(I) = -\infty $ and $\lim_{I\rightarrow 1/r^{+}}\alpha_r(I) = +\infty$.
The same behavior takes place for $r = 1$ and close to $I = 1$.
On the other hand, for $I\rightarrow \pm \infty$, $\alpha_r(I)$ has the same behavior as in the case for $r = 0$, $\lim_{I \rightarrow \pm\infty}\alpha_r(I) = 0$.
This implies that for any value of $\text{m}u$, for $I$ close enough to $I = 1/r$ the crests are vertical, and for $\left|I\right| $ large enough the crests are horizontal. \label{rem:r_3}
\end{remark}
\paragraph{Example} To illustrate this discussion, we present a concrete example.
Taking $\text{m}u = 0.5$, we have $\exp(-\pi/2)< 1/\text{m}u = 2<\exp(\pi/2)$.
In this case we have $I_{\text{l}}\approx -1.807$, $I_{\text{c}} \approx 0.701$ and $I_{\text{r}} \approx 1.367$.
The crests are horizontal in $(-1.807 , 0.701)\cup(1,367 , + \infty)$ and vertical in $(-\infty , -1.807)\cup( 0.701 , 1.367 )$.
We emphasize that this scenario is very different from the case in \cite{Delshams2017}.
There, for $\text{m}u = 0.5$ the crests are horizontal for all $I$.
\\
Now, we are going to focus on the transversality of the intersection between {\text{NHIM}} lines $R(I,\varphi , s)$ and crests $\text{m}athcal{C}(I)$.
On the plane $(\varphi , \text{s}igma)$ the {\text{NHIM}} lines can be written as
\begin{equation}
R_{I}(\varphi, \text{s}igma) = \{( \varphi - I\tau , \text{s}igma -(I-1)\tau) , \tau\in\text{m}athbb{R}\},\label{eq:nhim_line_sigma}
\end{equation}
so that its slope is $(I-1)/I$ in such plane.
Therefore, there exists an intersection between {\text{NHIM}} lines and crests that is not transversal if, and only if, there exists a tangent vector of $\text{m}athcal{C}(I)$ at a point that is parallel to $(I,I-1)$, or, using the parameterizations,
\begin{equation*}
\frac{\partial\xi}{\partial \varphi}(I,\varphi) = \frac{I-1}{I} \quad\quad\text{ or } \quad\quad \frac{\partial \eta}{\partial\text{s}igma}(I,\text{s}igma) = \frac{I}{I-1}.
\end{equation*}
Considering a \emph{horizontal} parameterization of $\text{m}athcal{C}(I)$, the tangency condition is equivalent to
\begin{equation*}
\frac{\pm\alpha(I)\text{m}u\cos\varphi}{\text{s}qrt{1-\text{m}u^2\alpha^2(I)\text{s}in^2\varphi}} = \frac{I-1}{I}.
\end{equation*}
Therefore, there exists a $\varphi$ satisfying the above condition if, and only if,
\begin{equation*}
\left|\beta(I)\right| \geq \frac{1}{\left|\text{m}u\right|},\quad\text{ where }\quad\beta(I) = \frac{I\alpha(I)}{I-1}\label{eq:condition_tan_hor}
\end{equation*}
and $\varphi$ takes the form
\begin{equation*}
\varphi =\pm \arctan\left(\text{s}qrt{\frac{\beta(I)^2-(1/\text{m}u)^2}{(1/\text{m}u)^2-\alpha(I)^2}}\right).
\end{equation*}
In an analogous way, for a \emph{vertical} parameterization $\eta(I,\text{s}igma)$, there are tangencies if, and only if,
\begin{equation*}
\left|\beta(I)\right|\leq \frac{1}{\left|\text{m}u\right|} \quad\quad \text{ with }\quad\quad \text{s}igma = \pm \arctan\left(\left|\frac{I-1}{I}\right|\text{s}qrt{\frac{(1/\text{m}u)^2 - \beta(I)^2}{\alpha(I)^2-(1/\text{m}u)^2}}\right). \label{eq:condition_tan_ver}
\end{equation*}
\begin{remark}
Observe that in both cases, horizontal and vertical crests, there are tangencies if, and only if,
\begin{equation*}
\left(\left|\alpha(I)\right| - \frac{1}{\left|\text{m}u\right|}\right)\left( \left|\beta(I)\right| - \frac{1}{\left|\text{m}u\right|} \right) < 0.
\end{equation*}
\end{remark}
The function $\left|\beta(I)\right|$ is smooth in $\text{m}athbb{R}\text{s}etminus\left\{1\right\}$ and $d\left|\beta(I)\right|/dI = 0$ only for $I=0$.
Besides, we have
\begin{equation*}
\lim_{I\rightarrow 1}\left|\beta(I)\right| = +\infty, \quad\quad \lim_{I\rightarrow - \infty}\left|\beta(I)\right| = \exp(\pi/2)\quad\text{ and }\quad \lim_{I\rightarrow +\infty}\left|\beta(I)\right| = \exp( -\pi/2).
\end{equation*}
Therefore, there are three possibilities:
\begin{itemize}
\item for $1/\left|\text{m}u\right| \geq \exp(\pi/2)$, there exist $I_0\in(1/2 , 1)$ and $I_+\in(1,+\infty)$ such that $I_0$ and $I_+$ are solutions of $\left|\beta(I)\right| - 1/\left|\text{m}u\right|=0$.
Besides, $\left|\beta(I)\right|<1/\left|\text{m}u\right|$ for $I\in(-\infty , I_0)\cup(I_+,+\infty)$ and $\left|\beta(I)\right|> 1/\left|\text{m}u\right|$ for $I\in(I_0 , 1)\cup(1 , I_+)$.
\item for $\exp(-\pi/2 )< 1/\left|\text{m}u\right| < \exp(\pi/2)$, there exist $I_-\in(-\infty , 0)$, $I_0\in(0 , 1)$ and $I_+\in(1,+\infty)$ such that $I_-$, $I_0$ and $I_+$ are solutions of $\left|\beta(I)\right| - 1/\left|\text{m}u\right|=0$.
Besides, $\left|\beta(I)\right|<1/\left|\text{m}u\right|$ for $I\in(I_- , I_0)\cup(I_+,+\infty)$ and $\left|\beta(I)\right|> 1/\left|\text{m}u\right|$ for $I\in(-\infty,I_-)\cup(I_0 , 1)\cup(1 , I_+)$.
\item For $ 1/\left|\text{m}u\right| \leq \exp(-\pi/2)$, there exist $I_-\in(-\infty , 0)$ and $I_0\in(0 , 1/2)$ such that $I_-$ and $I_0$ are solutions of $\left|\beta(I)\right| - 1/\left|\text{m}u\right|=0$.
Besides, $\left|\beta(I)\right|<1/\left|\text{m}u\right|$ for $I\in(I_- , I_0)$ and $\left|\beta(I)\right|> 1/\left|\text{m}u\right|$ for $I\in(-\infty,I_-)\cup(I_0 , 1)\cup(1 , \infty)$.
\end{itemize}
Putting together this description of $\left|\beta(I)\right|$ with the study about vertical and horizontal crests and adding that
\begin{eqnarray*}
\left|\beta(I)\right| < \left|\alpha(I)\right| & \forall I\in(-\infty , 0)\cup(0,1/2);\\
\left|\beta(I)\right| > \left|\alpha(I)\right| & \forall I\in(1/2 , 1)\cup(1,+\infty);\\
\left|\beta(0)\right| = \left|\alpha(0)\right| = 0& \left|\beta(1/2)\right| = \left|\alpha(1/2)\right| = 1
\end{eqnarray*}
we can state the proposition below.
\begin{proposition}\label{prop:cristas_behave}
Consider the two crests $\text{m}athcal{C}(I)$ defined by \eqref{eq:crests_equation} and the {\text{NHIM}} line $R_{I}(\varphi,\text{s}igma)$ defined in \eqref{eq:nhim_line} for Hamiltonian~\eqref{eq:new_hamil_system}.
\begin{itemize}
\item For $\left|\text{m}u\right| \leq \exp(-\pi/2)$, there exist $I_{\text{b}} < I_{\text{a}}< I_{\text{A}} <I_{\text{B}}$ such that
\begin{itemize}
\item for $I<I_{\text{b}}$ or $I_{\text{B}}<I$, $\text{m}athcal{C}(I)$ are horizontal and intersect transversally any $R_{I}(\varphi , \text{s}igma)$;
\item for $I_{\text{b}} \leq I < I_{\text{a}}$ or $I_{\text{A}} < I\leq I_{\text{B}}$, the crests $\text{m}athcal{C}(I)$ are horizontal, but now, there exist tangencies between $\text{m}athcal{C}(I)$ and two {\text{NHIM}} lines $R_{I}(\varphi , \text{s}igma)$;
\item for $I_{\text{a}} < I < I_{\text{A}}$, the crests $\text{m}athcal{C}(I)$ are vertical and intersect transversally any $R_{I}(\varphi , \text{s}igma)$.
\end{itemize}
\item For $\exp(-\pi/2) < \left| \text{m}u \right| < \exp( \pi/2)$ there exist $I_{\text{b}} < I_{\text{a}} < I _{\text{c}} \leq I_{\text{C}}< I_{\text{A}} <I_{\text{B}}$ such that
\begin{itemize}
\item for $I < I_{\text{b}}$ or $I_{\text{C}} < I < I_{\text{A}}$, $\text{m}athcal{C}(I)$ are vertical and intersect transversally any $R_{I}(\varphi , \text{s}igma)$;
\item for $I_{\text{b}}\leq I < I_{\text{a}}$, the crests $\text{m}athcal{C}(I)$ are vertical and there exist tangencies between $\text{m}athcal{C}(I)$ and two {\text{NHIM}} lines $R_{I}(\varphi , \text{s}igma)$;
\item for $I_{\text{a}} < I < I_{\text{c}}$ or $I_{\text{B}} < I$, $\text{m}athcal{C}(I)$ are horizontal and intersect transversally any $R_{I}(\varphi , \text{s}igma)$;
\item for $I_{\text{A}} \leq I \leq I_{\text{B}}$, the crests $\text{m}athcal{C}(I)$ are horizontal and there exist tangencies between $\text{m}athcal{C}(I)$ and two {\text{NHIM}} lines $R_{I}(\varphi , \text{s}igma)$;
\item for $I_{\text{c}} \leq I \leq I_{\text{C}}$, if $I_{\text{c}} <1/2$, the crests $\text{m}athcal{C}(I)$ are vertical and there
exist tangencies between $\text{m}athcal{C}(I)$ and $R_{I}(\varphi,\text{s}igma)$.
If $I_{\text{c}} = 1/2$, from the properties of $\alpha(I)$ and $\beta(I)$ this interval is just one point.
If $I_{\text{c}}> 1/2$, the crests $\text{m}athcal{C}(I)$ are horizontal and there exist tangencies.
\end{itemize}
\item For $\left|\text{m}u\right| \geq \exp( \pi/2 )$ there exist $I_{\text{b}} < I_{\text{a}} < I_{\text{A}} < I_{\text{B}}$ such that
\begin{itemize}
\item for $I < I_{\text{b}}$ or $I_{\text{B}} < I$, $\text{m}athcal{C}(I)$ are vertical and intersect transversally any $R_{I}(\varphi , \text{s}igma)$;
\item for $ I_{\text{b}} \leq I < I_{\text{a}}$ or $ I_{\text{A}} < I \leq I_{\text{B}}$, the crests $\text{m}athcal{C}(I)$ are vertical and there exist tangencies between $\text{m}athcal{C}(I)$ and two {\text{NHIM}} lines $R_{I}(\varphi , \text{s}igma)$;
\item for $I_{\text{a}} < I < I_{\text{A}}$, the crests $\text{m}athcal{C}(I)$ are horizontal and intersect transversally any $R_{I}(\varphi , \text{s}igma)$.
\end{itemize}
\end{itemize}
\end{proposition}
\begin{remark}
Note that we are not considering the singular case $\left|\alpha(I)\right| = 1/\left|\text{m}u\right|$ described in Remark~\ref{rmk:singular}.
\end{remark}
\paragraph{Example} Again, to illustrate this proposition, we take the case with $\text{m}u = 0.5$, see Fig.~\ref{fig:alpha_and_beta}.
In this case, we have $\left|\beta(I)\right| = 1/\text{m}u$ for $I \approx -2.942,\, 0.595,\, 1.85$
and
\begin{itemize}
\item for $I \in (-\infty , -2.942)\cup \left(0.701 , 1\right)\cup(1 , 1.367)\Rightarrow \left\{\begin{matrix}
\left|\alpha(I)\right| > 1/\left|\text{m}u\right|& \Rightarrow & \text{vertical crests} \\
\left|\beta(I)\right| > 1/\left|\text{m}u\right| & \Rightarrow & \text{no tangencies}
\end{matrix}\right. $
\item for $I \in \left[ -2.942 , -1.807 \right) \Rightarrow \left\{\begin{matrix}
\left|\alpha(I)\right| > 1/\left|\text{m}u\right|& \Rightarrow & \text{vertical crests} \\
\left|\beta(I)\right| \leq 1/\left|\text{m}u\right| & \Rightarrow & \text{tangencies}
\end{matrix}\right.$
\item for $I \in (-1.807 , 0.595 )\cup( 1.85 ,+\infty)\Rightarrow \left\{\begin{matrix}
\left|\alpha(I)\right| < 1/\left|\text{m}u\right|& \Rightarrow & \text{horizontal crests} \\
\left|\beta(I)\right| < 1/\left|\text{m}u\right| & \Rightarrow & \text{no tangencies}
\end{matrix}\right.$
\item for $I \in \left[0.595 , 0.701 \right)\cup \left( 1.367 , 1.85\right]\Rightarrow \left\{\begin{matrix}
\left|\alpha(I)\right| < 1/\left|\text{m}u\right|& \Rightarrow & \text{horizontal crests} \\
\left|\beta(I)\right| \geq 1/\left|\text{m}u\right| & \Rightarrow & \text{tangencies}
\end{matrix}\right.$
\end{itemize}
Once more, we compare with the Hamiltonian~\eqref{eq:old_hamil_system} studied in \cite{Delshams2017}.
For Hamiltonian~\eqref{eq:old_hamil_system} and $\text{m}u = 0.5$ there is no tangency, but for Hamiltonian~\eqref{eq:new_hamil_system}
we can find tangencies for horizontal and vertical crests.
Indeed, for Hamiltonian~\eqref{eq:old_hamil_system} and any $0<\left|\text{m}u\right| < 0.625$ there is no tangency,
whereas for any $\text{m}u\neq 0$ there are tangencies for Hamiltonian~\eqref{eq:new_hamil_system}.
\begin{figure}
\caption{$\left|\alpha(I)\right|$ and $\left|\beta(I)\right|$ : Behavior of the crests and tangencies.}
\label{fig:alpha_and_beta}
\label{fig:alpha_and_beta_r_0.5}
\end{figure}
\begin{remark}
For $r\in (0 ,1 ),$ $\beta_r(I)$ is defined by $\beta_r(I) = I\alpha_r(I)/(rI - 1)$.
In this case, $\lim_{I\rightarrow 1/r}\left|\beta_r(I)\right| = +\infty$ and $\lim_{I \rightarrow \pm \infty}\left|\beta_r(I)\right| = 0$.
In Fig.~\ref{fig:alpha_and_beta_r_0.5}, a comparison between the functions $\alpha_r(I)$, $\beta_r(I)$ and the straight line $1/\left|\text{m}u\right|$ for $r=1/2$ is displayed.\label{rem:r_4}
\end{remark}
For each crest, where it is well defined, there exists, at least, a value $\tau^*$ such that
\begin{equation*}
(\varphi - I\tau^* , \text{s}igma - (I - 1)\tau^*) = (\varphi - I\tau^* , \xi(I , \varphi - I\tau^*))\text{ or } (\eta(I ,\text{s}igma - (I-1)\tau^* ) , \text{s}igma - (I-1)\tau^*),
\end{equation*}
which means that $R_{I}(\varphi , \text{s}igma)\cap \text{m}athcal{C}(I)\neq \emptyset$.
This intersection is intrinsically associated to a homoclinic orbit to the NHIM.
To make a choice about how to take such $\tau^*$ is to choose in which homoclinic manifold $\Gamma$
the homoclinic points $\tilde{z}^*$ lie.
Even more, it is to choose what scattering map we are going to use.
\text{s}ubsection{Construction of scattering maps}
We have now several goals. First, to explain, given $(I,\theta)$, how to find the intersection between one of
the {\text{NHIM}} lines and one of the two crests, and consequently, to define the function $\tau^*$.
Second, to show how each crest can give rise to many scattering maps.
And third, to explain the different scattering maps or combinations of them that can be defined.
Let us first study the intersection between {\text{NHIM}} lines and crests.
From the definition of the function $\tau^* = \tau^*(I,\varphi,s)$ in equation \eqref{eq:tau^*} and the definition of a {\text{NHIM}} line $R(I,\varphi,s)$ in \eqref{eq:nhim_line} and a crest $\text{m}athcal{C}(I)$ in Definition~\ref{def:crest}, it turns out that
\begin{equation*}
R(I,\varphi,s)\cap\text{m}athcal{C}(I) = \left\{\left(I,\varphi- I\tau^*(I,\varphi,s),s - \tau^*(I,\varphi,s) \right)\right\}.
\end{equation*}
Moreover, from the equation satisfied by the function $\tau^*$, one can get (see Eq.~(3.12) in \cite{Delshams2017}) that for any $\gamma$
\begin{equation*}
\tau^*(I,\varphi - I\gamma, s -\gamma) = \tau^*(I,\varphi,s) - \gamma.
\end{equation*}
In particular, for the change \eqref{eq:def_sigma} $s = \varphi - \text{s}igma$ and $\gamma = \varphi - \text{s}igma$ one gets
\begin{equation}\label{eq:tau_sigma}
\tau^*(I,\varphi,\varphi-\text{s}igma) = \tau^*(I,\theta) + \varphi - \text{s}igma,
\end{equation}
where $\theta = \varphi - Is = (1-I)\varphi + I\text{s}igma$.
In the variables $(I,\varphi,\text{s}igma)$, taking into account the expression \eqref{eq:nhim_line_sigma} for the {\text{NHIM}} lines $R(I,\varphi,\varphi - \text{s}igma)$ and again equation \eqref{eq:tau^*} satisfied by the $\tau^*(I,\varphi,s)$, we have that
\begin{align*}
R(I,\varphi, \varphi-\text{s}igma)\cap\text{m}athcal{C}(I) &= \left\{\left(I,\varphi - I\tau^*(I,\varphi,\varphi-\text{s}igma) , \text{s}igma - (I-1)\tau^*(I,\varphi,\varphi -\text{s}igma )\right)\right\}\\
&=\left\{\left(I,\theta - I\tau^*(I,\theta),\theta - (I-1)\tau^*(I,\theta)\right)\right\},
\end{align*}
where \eqref{eq:tau_sigma} has been used, and $\theta = (1-I)\varphi + I\text{s}igma$.
From a geometrical point of view, to find an intersection between a {\text{NHIM}} line and a crest, one throws from a point $(\theta , \theta)$ on the plane $(\varphi,\text{s}igma)$ a straight line with slope $(I-1)/I$, until it touches the crest $\text{m}athcal{C}(I)$. The function $\tau^*(I,\theta)$ is the time spent to go from a point $(\theta,\theta)$ in the diagonal $\text{s}igma = \varphi$ up to $\text{m}athcal{C}(I)$ with a velocity vector $\text{m}athbf{v} = -(I , I-1)$, see Fig.~\ref{fig:qual_plot}.
\begin{figure}
\caption{Finding $\tau^*(I,\theta)$ using the straight line $\text{s}
\label{fig:qual_plot}
\end{figure}
One has to decide the direction for $\tau^*$ using the idea explained above.
For example, if we are on a point on the straight line $\text{s}igma = \varphi$, we have to decide if we go up or go down along the {\text{NHIM}} line, i.e., to look for a negative or a positive $\tau^*(I,\theta)$ (to look at the past or the future).
In both cases we are going to detect an intersection with the desired crest, but, in general, different choices give rise to different scattering maps, because we are looking for different homoclinic invariant manifolds $\Gamma$.
To show another difference between scattering maps from the choice of $\tau^*$ we begin by introducing each kind of scattering map.
The first one is inspired in \cite{Delshams2011} and \cite{Delshams2017} for $\left|\text{m}u\right| < 0.97$.
In these cited cases all scattering maps studied were associated to one of the horizontal crests like in \eqref{eq:cristas_06}.
In the same way, we can separate completely the scattering maps associated to the horizontal crests and the scattering maps associated to the vertical crests.
Notice that the scattering maps associated to horizontal crests are defined only for values of $I$ satisfying $\left|\alpha(I)\right|< 1/\left|\text{m}u\right|$ whereas the scattering maps associated to the vertical crests are defined only for values of $I$ satisfying $\left|\alpha(I)\right| > 1/\left|\text{m}u\right|$.
As noted previously, crests are vertical in a neighborhood of $I = 1$ for any value of $\text{m}u$.
Therefore, there is no scattering map associated to a horizontal crest close to $I=1$.
Analogously, since $\left|\alpha(0)\right| = 0$, crests are horizontal in a neighborhood of $I = 0$ for any value of $\text{m}u$ and, therefore, there is no scattering map associated to a vertical crest close to $I = 0$.
This implies that these ``horizontal'' or ``vertical'' scattering maps are just locally defined, in other words, they are not defined on the whole plane $(\theta , I)$.
This motivates to define \emph{global scattering maps}.
Global scattering maps are important because they describe the outer dynamics for large intervals of $I$ and are defined as follows
\begin{definition}
A scattering map $\text{m}athcal{S}(I,\theta)$ is called a \emph{global scattering map} if it is defined on all $\theta\in\text{m}athbb{T}$ for any fixed $I$.
\end{definition}
Note that $\text{m}athcal{S}(I,\theta)$ is a global scattering map as long as $\tau^*(I , \theta)$ is a global function, i.e., defined on all $\theta\in\text{m}athbb{T}$ for any fixed $I$.
If $\tau^*(I,\theta)$ is smoothly defined, the same will happen to $\text{m}athcal{S}(I,\theta)$.
Tangencies between \text{NHIM}\, lines and crests, as well as discontinuities in their intersections give rise to non-smooth scattering maps.
\begin{remark}
For instance, in the paper~\cite{Delshams2017} devoted to the Hamiltonian~\eqref{eq:old_hamil_system},
it was proven that for $0<\text{m}u=a_1/a_2<0.625$, there exist two different global scattering maps.
Let us add that for $0.625\leq \text{m}u < 0.97$, due to the existence of tangencies between the {\text{NHIM}} lines and the crests, there appear two or six scattering maps.
Such \emph{multiple} scattering maps are indeed piecewise smooth global scattering maps, see Figs.~9--11 of~\cite{Delshams2017}.
Their discontinuities lie along the \emph{tangency locus} and were avoided there to construct diffusion paths,
just for the sake of simplicity.
\end{remark}
For Hamiltonian \eqref{eq:new_hamil_system}, to extend scattering maps which are in principle only locally defined we have now two options: to combine a scattering map associated to a horizontal crest with a scattering map associated to a vertical crest or to extend the previously called ``horizontal'' or ``vertical'' scattering maps.
Although the first option may provide a global scattering map, they may appear complex discontinuity sets which give rise to a complicated phase space.
The second option is to apply the same idea used in \cite{Delshams2017} when we defined the scattering map ``with holes''.
When $\left|\alpha(I)\right| >1/\left|\text{m}u\right|$, the horizontal crests are no longer defined for all $\varphi\in\text{m}athbb{T}$, indeed, they become vertical crests defined for all $\text{s}igma\in\text{m}athbb{T}$. Nevertheless, the vertical crests are formed by pieces of horizontal crests.
This implies that even for these values of $I$ we can use $\xi$ given in \eqref{eq:cristas_06} to parameterize some intersections between $R(I,\varphi ,\text{s}igma)$ and $\text{m}athcal{C}(I)$.
As we can see in Fig.~\ref{fig:hor_vert_prox}, the vertical and horizontal crest $\text{m}athcal{C}_{\text{M}}$ are very close in a neighbourhood of $\varphi = 0$.
When we have a bifurcation from horizontal to vertical crests (or vice versa), it is natural just to change the parameterization from $\xi_{\text{M}}$ to $\eta_{\text{M}}$ for these values of $\varphi$.
With this choice the orbits of the scattering map are continuous for $\theta$ close to $0$ or $2\pi$.
The same happens with $\xi_{\text{m}}$ and $\eta_{\text{m}}$ for values of $\varphi$ close to $\pi$.
Scattering maps associated to horizontal crests for values of $I$ satisfying $\left|\text{m}u\alpha(I)\right| < 1$ are defined for all $\varphi\in\text{m}athbb{T}$.
The extension of them to values of $I$ for $\varphi\in\text{m}athbb{T}$ such that $\left|\text{m}u\alpha(I)\text{s}in\varphi\right| < 1$ are called \emph{extended scattering maps}.
\begin{definition}
A scattering map $\text{m}athcal{S}(I,\theta)$ is called an \emph{extended scattering map} if it is associated to horizontal crests for which $\left|\text{m}u\alpha(I)\right|<1$, and is continuously extended to the pieces of the vertical crests where they behave as horizontal crests, that is, for the values $\varphi$ such that $\left|\text{m}u\alpha(I)\text{s}in\varphi\right| < 1$.
\end{definition}
Since we have already seen in Proposition~\ref{prop:cristas_behave} that there exist tangencies between {\text{NHIM}} lines and crests for any value of $\text{m}u$, there are no global scattering maps for Hamiltonian \eqref{eq:new_hamil_system}. However, there exist
extended scattering maps with a domain large enough to provide diffusion paths.
\begin{figure}
\caption{Comparison between $\xi_{\text{M}
\label{fig:hor_crest_part}
\label{fig:vert_crest_part}
\label{fig:hor_vert_prox}
\end{figure}
To illustrate the current scenario we will display the level curves of the reduced Poincar\'{e} function $\text{m}athcal{L}^*(I,\theta)$ defined in \eqref{eq:reduced_poincare_function}, which up to $\text{m}athcal{O}(\varepsilon^2)$ contain orbits of the scattering map $\text{m}athcal{S}(I,\theta)$.
We begin by considering $\text{m}u = 0.6$ and the horizontal crest $\text{m}athcal{C}_{\text{M}}(I)$. In Fig.~\ref{fig:go_down} we display the scattering map built using $\tau^*$ defined by the first intersection between $R_I(\varphi,\text{s}igma)$ and $\text{m}athcal{C}_{\text{M}}(I)$ from $\text{s}igma =\varphi$ going down along $R_I(\varphi,\text{s}igma)$.
In Fig.\ref{fig:go_up}, we use a similar idea, but now, form $\text{s}igma =\varphi$ going up along $R_I(\varphi,\text{s}igma)$.
Alternatively, if we choose $\tau^*$ with minimal absolute value, independently of going up or down, we obtain the scattering map plotted on Fig.~\ref{fig:min_tau}.
In this last case, there are orbits of the scattering maps that are not smooth in $\theta = \pi$.
This happens because we change the homoclinic manifold $\Gamma$, so we are using, indeed, two different scattering maps.
In \cite{Delshams2017} we chose scattering maps associated to a function $\tau^*$ with the minimal absolute value,
which were called \emph{primary} scattering maps.
This example show us that is not enough to say what crest is associated to a scattering map, but it is also necessary to make
explicit the criterion used for $\tau^*$ (going up or down along the {\text{NHIM}} lines, or choosing a minimal $\left|\tau^*\right|)$.
\begin{figure}
\caption{Different phase space of scattering maps $\text{m}
\label{fig:go_down}
\label{fig:go_up}
\label{fig:min_tau}
\end{figure}
The next lemma is a good example about the criteria for $\tau^*(I,\theta)$ and its consequences, and is used to prove Proposition \ref{prop:orb_cres}.
Before, a new notation is introduced.
An \emph{even} subindex $k$ is assigned to the branches $\text{m}athcal{C}_k(I)$ of $\text{m}athcal{C}_{\text{M}}(I)$ when considering $\text{s}igma,\,\varphi \in\text{m}athbb{R}$
\begin{equation*}
\xi_{k}(I,\varphi) = -\arcsin\left(\alpha(I)\text{m}u\text{s}in\varphi\right) + k\pi\quad\text{ and }\quad\eta_{k} = -\arcsin\left(\frac{\text{s}in\text{s}igma}{\alpha(I)\text{m}u}\right) + k\pi
\end{equation*}
and an \emph{odd} subindex $k$ to the branches $\text{m}athcal{C}_k(I)$ of $\text{m}athcal{C}_{\text{m}}(I)$ when considering $\text{s}igma,\,\varphi \in\text{m}athbb{R}$
\begin{equation*}
\xi_{k}(I,\varphi) = \arcsin\left(\alpha(I)\text{m}u\text{s}in\varphi\right) + k\pi\quad\text{ and }\quad\eta_{k} = \arcsin\left(\frac{\text{s}in\text{s}igma}{\alpha(I)\text{m}u}\right) + k\pi.
\end{equation*}
We notice that the crests $\text{m}athcal{C}(I)$ are naturally defined for $(\varphi,\text{s}igma)\in\text{m}athbb{T}^2$ and give rise to two different crests $\text{m}athcal{C}_{\text{M}}(I)$, $\text{m}athcal{C}_{\text{m}}(I)$ (except for the singular case $\left|\text{m}u\alpha(I)\right| = 1$).
When we run now over real values of $\varphi,\,\text{s}igma$, we may have an \emph{infinite} number of crests $\text{m}athcal{C}_k(I)$, where even (odd) values of $k$ are assigned to the branches of $\text{m}athcal{C}_{\text{M}}(I)$ ($\text{m}athcal{C}_{\text{m}}(I)$). Among them,
we are going to use $\text{m}athcal{C}_{0}(I)$, $\text{m}athcal{C}_{1}(I)$ and $\text{m}athcal{C}_{2}(I)$.
\begin{lemma}Let $\text{m}athcal{L}_{\text{0}}^*$ and $\text{m}athcal{L}_{\text{2}}^*$ be reduced Poincar\'{e} functions associated to the same crest $\text{m}athcal{C}(I)$, where for $\text{m}athcal{L}_{\text{0}}^*$ we look at the first intersection points ``under'' $\text{s}igma = \varphi$, that is, with $\text{m}athcal{C}_0(I)$, and for $\text{m}athcal{L}_{\text{2}}^*$ we look at the first intersection points ``over'' $\text{s}igma = \varphi$, that is, with $\text{m}athcal{C}_2(I)$.
Then we have
\begin{equation}
\frac{\partial\text{m}athcal{L}_{\text{0}}^*}{\partial\theta}\left(I,\theta\right) = -\frac{\partial\text{m}athcal{L}_{\text{2}}^*}{\partial\theta}\left(I,2\pi - \theta\right).\label{eq:lemma}
\end{equation}
\end{lemma}
\begin{remark}
We say ``under" $\text{s}igma = \varphi$ and ``over" $\text{s}igma = \varphi$ for intersection points going down or up along $R_I(\varphi,\text{s}igma)$, respectively on $(\varphi , \xi_{0}(I,\varphi))$ and $(\varphi,\xi_{2}(I,\varphi))$, because when the horizontal crest $\text{m}athcal{C}_{\text{M}}(I)$ is defined for all $\varphi\in\text{m}athbb{T}$ the graphs $(\varphi , \xi_{0}(I,\varphi))$ of $\text{m}athcal{C}_0(I)$ and $(\varphi,\xi_{2}(I,\varphi))$ of $\text{m}athcal{C}_2(I)$ are under and over the straight line $\text{s}igma = \varphi$.
\end{remark}
\begin{proof}
Let $\text{m}athcal{L}^*$ be a reduced Poincar\'{e} function~\eqref{eq:reduced_poincare_function}-\eqref{eq:our_meln_potential}, then
\begin{equation*}
\frac{\partial\text{m}athcal{L}^*}{\partial\theta}\left(I,\theta\right) = \frac{A_1(I)\text{s}in(\theta - I\tau^*(I,\theta))}{I-1}.\label{eq:diff_L__d_theta}
\end{equation*}
So, equation \eqref{eq:lemma} is satisfied if, and only if
\begin{equation}
\text{s}in(\theta - I\tau_{\text{0}}^*(I,\theta)) = \text{s}in(\theta -I(-\tau_{\text{2}}^*(I,2\pi-\theta))).\label{eq:equi_lemma}
\end{equation}
We assume that the crest is horizontal and given by the graph of $\xi_{\text{M}}$, the other cases are analogous.
Indeed, we are going to use
\begin{equation}
\xi_{\text{0}}(I,\varphi) = -\arcsin(\text{m}u\alpha(I)\text{s}in\varphi)\quad \text{ and }\quad \xi_{\text{2}}(I,\varphi) = \xi_{\text{0}}(I,\varphi) + 2\pi.\label{eq:new_notation}
\end{equation}
This implies that the intersection point ``under'' $\text{s}igma = \varphi$ is a point on the curve parameterized by $\xi_{\text{0}}(I,\varphi)$.
Otherwise, the intersection ``over'' $\text{s}igma = \varphi $ is a point on the curve parameterized by $\xi_{\text{2}}(I,\varphi)$.
As the slope of the {\text{NHIM}} lines is $(I-1)/I$, given a point $(\theta,\theta)$, we obtain
\begin{equation*}
\frac{\xi_{\text{2}}(I,\theta-I\tau_{\text{2}}^*(I,\theta)) - \theta}{\theta - I\tau_{\text{2}}^*(I,\theta) - \theta} = \frac{I-1}{I},
\end{equation*}
which can be rewritten as
\begin{equation*}
\frac{2\pi + \xi_{\text{0}}(I,\theta - I\tau_{\text{2}}^*(I,\theta))- \theta}{- I\tau_{\text{2}}^*(I,\theta)} = \frac{I-1}{I}.
\end{equation*}
From this equation, we obtain an expression for $\tau_{\text{2}}^*(I,\theta$)
\begin{equation*}
\tau_{\text{2}}^*(I,\theta) = \frac{-\left(2\pi + \xi_{\text{0}}(I,\theta - I\tau_{\text{2}}^*(I,\theta)) - \theta\right)}{I-1}.
\end{equation*}
From the expressions of $\tau_{\text{2}}^*(I,\theta)$ above and \eqref{eq:new_notation},
\begin{equation*}
\tau_{\text{2}}^*(I,2\pi-\theta)= \frac{\left(\xi_{\text{0}}(I, \theta -I\left(- \tau_{\text{2}}^*(I,2\pi -\theta)\right))+ \theta\right)}{I-1},
\end{equation*}
and therefore
\begin{equation*}
\theta - (I-1)(-\tau_{\text{2}}^*(I,2\pi - \theta)) = \xi_{\text{0}}(I,\theta - (I-1)(-\tau_{\text{2}}^*(I,2\pi-\theta))),
\end{equation*}
which implies that $-\tau_{\text{2}}^*(I,2\pi- \theta)$ is a time of intersection between the {\text{NHIM}} line and the curve parameterized by $\xi_{\text{0}}$.
In the case that there exists only one intersection point, this implies
\begin{equation*}
\tau_{\text{0}}^*(I,\theta) = \tau_{\text{2}}^*(I,2\pi- \theta).
\end{equation*}
So, condition \eqref{eq:equi_lemma} is satisfied.
\end{proof}
\begin{proposition}
Let $\text{m}athcal{S}_{\text{1}}(I,\theta)$ be the scattering map associated to the graphs of $\xi_{\text{1}}$ and $\eta_{\text{1}}$ of $\text{m}athcal{C}_1(I)$.
Assuming $a_1,a_2 >0$, for any $I$ there exists a $\theta_{+}$ such that $\dot{I}>0$ for $\theta \in (\pi, \theta_{+})$.
Moreover, $\theta_{+}\geq 3\pi/2$ for $I\notin(-1/2 , 1/2)$. \label{prop:orb_cres}
\end{proposition}
\begin{proof}
A proof is given in Appendix~\ref{app:A}.
\end{proof}
\begin{remark}
If $a_1 < 0$, we have that there exists a $\theta_-$ such that $\dot{I}> 0$ for any $\theta \in (\theta_- , \pi)$.
\end{remark}
\begin{remark}
An analogous proposition holds for $\text{m}athcal{S}_{\text{2}}(I,\theta)$, the scattering map associated to the graphs of $\xi_{\text{2}}$ and $\eta_{\text{2}}$ of $\text{m}athcal{C}_2(I)$.
In such case, there is a $\theta_+$ such that $\dot{I}\geq 0$ for any $\theta\in(\theta_+ , 2\pi)$ where $\theta \geq 3\pi/2$ for $I\in(1/2 , 3/2)$.\end{remark}
Note that this proposition leads us to ensure the diffusion in an analogous way to the one used to prove Theorem 4 in \cite{Delshams2017}.
Next, the diffusion mechanism is stated and the Arnold diffusion is proven.
\text{s}ection{ Arnold Diffusion }
\label{sec:diffusion}
In this section we are going to complete our goal proving the existence of global instability or Arnold diffusion, that is, Theorem~\ref{theo:main_theo}.
We begin by presenting some general geometrical properties of the scattering maps that we have to take into account to prove the theorem of diffusion.
The first one reduces the study of scattering maps to positive values of $\text{m}u$.
More precisely, we have the lemma below
\begin{lemma}\label{lem:geometrical_lemmas}
The scattering map for a value of $\text{m}u$ and $s = \pi$, associated to the intersection between $R(I,\varphi,s)$ and $C_{\text{m}}(I)$ ($C_{\text{M}}(I)$) has the same geometrical properties as the scattering map for $-\text{m}u$ and $s = 0$, associated to the intersection between $R_{\theta}(I)$ and $C_{\text{M}}(I)$ ($C_{\text{m}}(I)$), i.e.,
\begin{equation*}\label{eq:equivalence_sign_mu}
S^{\text{m}u}_{\text{m}(\text{M})}(I,\varphi,\pi) = S^{-\text{m}u}_{\text{M}(\text{m})}(I,\varphi,0) = \text{m}athcal{S}^{-\text{m}u}_{\text{M}(\text{m})}(I,\theta)
\end{equation*}
\end{lemma}
\begin{proof}
First, we look for $\tau^*_{\text{m}}$ such that the {\text{NHIM}} segment $R(I,\varphi,s)$ intersects the crest $C_{\text{m}}(I)$.
If we fix $s=\pi$, we have from \eqref{eq:our_meln_potential} and \eqref{eq:L^*-def}:
\begin{equation}
L_{\text{m}u,\text{m}}^*(I,\varphi,\pi) = A_{1}(I)\cos(\varphi-I\tau_{\text{m}}^*(I,\varphi,\pi))+A_{2}(I)\cos(\varphi-\pi-(I-1)\tau_{\text{m}}^*(I,\varphi,\pi)).\label{eq:mel_pot_phi_mu_neg}
\end{equation}
Besides, $\tau^*$ satisfies
\begin{equation*}\label{eq:def_tau_mu_neg}
\text{m}u\alpha(I)\text{s}in(\varphi - I\tau_{\text{m}}^*) + \text{s}in(\varphi -\pi-(I-1)\tau_{\text{m}}^*) =0,
\end{equation*}
or
\begin{equation*}
-\text{m}u\alpha(I)\text{s}in(\varphi - I\tau_{\text{m}}^*) +\text{s}in(\varphi-(I-1)\tau_{\text{m}}^*) =0.\label{eq:equ_def_tau_mu_neg_and_pos}
\end{equation*}
We have that $\varphi - \pi -(I-1) \tau_{\text{m}}^* \pmod{2\pi} = \xi_{\text{m}}(I,\varphi - I\tau_{\text{m}}^*)$ with $\pi/2 \leq\xi_{\text{m}}\leq 3\pi/2$.
Then, for each $\tau^*_{\text{m}}$ there exists a $K\in\text{m}athbb{Z}$ such that
$$\frac{\pi}{2} < \varphi-\pi - (I-1)\tau^*_{\text{m}} + 2\pi K <\frac{3\pi}{2}.$$
This implies
$$\frac{3\pi}{2} < \varphi - (I-1)\tau^*_{\text{m}} + 2\pi K \quad\text{ and }\quad \varphi - (I-1)\tau^*_{\text{m}} + 2\pi (K-1) < \frac{\pi}{2}.$$
Therefore,
$$\varphi - (I - 1)\tau^*_{m} \pmod{2\pi} < \frac{\pi}{2}\quad\text{ or }\quad \varphi - (I - 1)\tau^*_{m} \pmod{2\pi} > \frac{3\pi}{2}.$$
We can conclude that $\varphi - (I - 1)\tau^*_{m} \pmod{2\pi} = \xi_{\text{M}}(I,\varphi - I\tau^*_{\text{m}})$.
Therefore $\tau_{\text{m}}^*(I,\varphi,\pi)$ for $\text{m}u$ is equal to $\tau_{\text{M}}^*(I,\varphi,0)$ for $-\text{m}u$.
From (\ref{eq:mel_pot_phi_mu_neg}), $L^*_{\text{m}u,\text{m}}(I,\varphi,\pi)$ satisfies
\begin{eqnarray*}
L_{\text{m}u,\text{m}}^*(I,\varphi,\pi) &=& A_{1}(I)\cos(\varphi - \tau_{\text{M}}^*(I,\varphi,0)) +(-A_{2}(I))\cos(\varphi -(I-1)\tau_{\text{M}}^*(I,\varphi,0))\\
&=& L_{-\text{m}u,\text{M}}^*(I,\varphi,0).
\end{eqnarray*}
Since $L^*_{\text{m}u,\text{m}}(\cdot,\cdot,\pi)$ and $L^*_{-\text{m}u,\text{M}}(\cdot,\cdot,0)$ coincide, their derivatives too and this implies that $$S^{\text{m}u}_{\text{m}}(I,\varphi,\pi)=S^{-\text{m}u}_{\text{M}}(I,\varphi,0) = \text{m}athcal{S}^{-\text{m}u}_{\text{M}}(I,\theta).$$
\end{proof}
From now on, just to simplify the exposition, $a_1$ and $a_2$ are considered positive.
The same strategy used in \cite{Delshams2017} is applied to prove the existence the diffusion:
we combine the scattering map in an interval of $\theta$ where $\dot{I}>0$ and the inner map to build a diffusion pseudo-orbit.
Then we apply shadowing results to get the existence of a diffusion orbit.
Since $I = 0$ and $I= 1$ are resonance values, the application of the inner map must be more careful, because in these resonance regions, for some orbits, the value of $I$ decreases in order $\text{m}athcal{O}(\text{s}qrt{\varepsilon})$, i. e., the tori cannot be considered flat.
We study the transversality between the foliations of invariant sets of the inner and the scattering map in resonant
and non-resonant regions and its image under the scattering map $\text{m}athcal{S}$.
For more details and a more general case, the reader is referred to \cite{Delshams2009}.
Consider the resonant region associated to $I = 0$.
In such region, the tori can be approximated by $F^{0}(I,\varphi)$ given in \eqref{eq:invariant_tori_i=0}.
The tranversality between invariant sets of the inner and the scattering map holds if the gradient vectors of the level curves of $F^0$ and $\text{m}athcal{L}^*$ are not parallel vectors, or equivalently,
$$\left\{F^0(I,\theta) , \text{m}athcal{L}^*(I,\theta)\right\} \neq 0,$$
where $\left\{ , \right\}$ is the Poisson bracket,
$$\left\{F^0 , \text{m}athcal{L^*}\right\} = \frac{\partial F^0}{\partial \theta}\frac{\partial \text{m}athcal{L}}{\partial I}-\frac{\partial F^0}{\partial I}\frac{\partial \text{m}athcal{L}}{\partial \theta}.$$
From \eqref{eq:invariant_tori_i=0}, the partial derivatives of $F^0$ are
\begin{eqnarray*}
\frac{\partial F^0}{\partial I} = I &\text{ and }&\frac{\partial F^0}{\partial \theta} = -\varepsilon a_1\text{s}in\theta,
\end{eqnarray*}
and since $\text{m}athcal{L}^*(I,\theta) = A_1(I)\cos(\theta - I\tau^*(I,\theta)) + A_2(I)\cos(\theta-(I-1)\tau^*(I,\theta))$, we have the partial derivatives given by
$$\frac{\partial \text{m}athcal{L}^*}{\partial \theta} = \frac{A_1(I)\text{s}in(\theta - I\tau^*)}{I-1},$$
$$\frac{\partial \text{m}athcal{L}^*}{\partial I} = A_1'(I)\cos(\theta-I\tau^*) + A_2'(I)\cos(\theta- (I-1)\tau^*) + A_1(I)\tau^* \text{s}in(\theta - I\tau^*) + A_2(I)\tau^*\text{s}in(\theta - (I-1)\tau^*).$$
Note that if $\left|I\right|> \text{m}athcal{O}(\varepsilon)$, $\partial F^0/\partial I$ dominates $\partial F^0/\partial \theta$, so the Poisson bracket above can be reduced to
$$\left\{F^0 , \text{m}athcal{L^*}\right\} \text{s}imeq -\frac{\partial F^0}{\partial I}\frac{\partial \text{m}athcal{L}}{\partial \theta} = \frac{-IA_1(I)\text{s}in(\theta - I\tau^*)}{I-1}$$
Expanding $\text{s}in(\theta - I\tau^*)$ in Taylor's series around $I = 0$, we have
$$\text{s}in(\theta - I\tau^*) = \text{s}in\theta + \text{m}athcal{O}(I),$$
which implies $\left\{F^0,\text{m}athcal{L}^*\right\}=0$ if, and only if, $\theta \approx 0,\pi$, assuming that $\text{m}athcal{O}(I)$ is small enough.
Now, we consider $I = \text{m}athcal{O}(\varepsilon)$ and look at the intersections between the {\text{NHIM}} lines and the graph of $\xi_1$.
Note that as the value of $I$ is close to $0$ we can assume that the crests are horizontal.
Using Taylor's series we can write
\begin{eqnarray*}
\text{s}in(\theta - I\tau^*) = \text{s}in\theta + \text{m}athcal{O}(I) && \cos(\theta - I\tau^*) = \cos\theta + \text{m}athcal{O}(I)\\
\text{s}in(\theta - (I-1)\tau^*) = \text{m}athcal{O}(I) &&\cos(\theta - (I-1)\tau^*) = -1 +\text{m}athcal{O}(I).
\end{eqnarray*}
This implies
\begin{equation}
\left\{F^0 , \text{m}athcal{L}^*\right\}= -\frac{IA_1(I)\text{s}in\theta}{I-1} -\varepsilon a_1\text{s}in\theta\left(A_1'(I)\cos\theta - A_2'(I) + A_1(I)\tau^*\text{s}in\theta \right) + \text{m}athcal{O}(I^2 ,\varepsilon I).\label{eq:transv_pois_b}
\end{equation}
Taylor expanding the functions $A_1(I)$, $A_1'(I)$ and $A_2'(I)$ around $I = 0$, we obtain
\begin{equation*}
A_1(I) = 4a_1 + \text{m}athcal{O}(I²), \quad A_1'(I) = \text{m}athcal{O}(I) \text{ and } A_2'(I) = a_2\pi(\pi\coth\pi/2 - 2)\text{csch}(\pi/2) + \text{m}athcal{O}(I)
\end{equation*}
Plugging these expressions in \eqref{eq:transv_pois_b}, we set
$$\left\{F^0 , \text{m}athcal{L^*}\right\} = -\frac{4a_1I\text{s}in\theta}{I-1} -\varepsilon a_1\text{s}in\theta\left[a_2\pi(\pi\coth\pi/2 - 2)\text{csch}(\pi/2) + 4a_1(\pi - \theta)\text{s}in\theta\right] + \text{m}athcal{O}(I^2 , I\varepsilon).$$
So, $\left\{F^0 , \text{m}athcal{L^*}\right\} = 0 \Leftrightarrow a_1\text{s}in\theta\left[\frac{-4I}{I-1}-\varepsilon a_2\pi (\pi\coth\pi/2 - 2)\text{csch}(\pi/2) + \varepsilon4(\pi - \theta)\text{s}in\theta\right]= 0$.
In other words, we do not have transversality if, and only if, $\theta = 0,\pi$ or satisfies
$$(\pi - \theta)\text{s}in\theta = \frac{I}{\varepsilon a_1} + \frac{\pi(\coth\pi/2 - 2)\text{csch}\pi/2)}{4},$$
which is not an horizontal curve in the plane $(\theta , I)$ and is transversal to an invariant torus of the inner dynamics.
For the other resonant region $I = 1$, $F^1$ is very similar.
Assuming $I-1 = \text{m}athcal{O}(\varepsilon)$, we have
\begin{equation*}
\left\{F^1 , \text{m}athcal{L}^*\right\} = a_2\text{s}in\theta \left\{4\left(\frac{I-1}{I}\right) - \varepsilon\left[\pi a_1(2-\pi\coth(\pi/2))\text{csch}(\pi/2) + 4 a_2 \text{s}in\theta \right]\right\}.
\end{equation*}
Applying the same methodology, we obtain an analogous result for the other resonant region $F^1$.
In short, we conclude that the image $\text{m}athcal{S}(\text{m}athcal{T}_i)$ of an invariant torus $\text{m}athcal{T}_i$ of
the inner map under the scattering map intersects tranversally another invariant torus $\text{m}athcal{T}_{i+1}$
of the inner map.
Finally, in the non-resonant region, we notice that
$$\left\{F^{\text{nr}} , \text{m}athcal{L}^*\right\} = -\frac{\partial F^{\text{nr}}}{\partial I}\frac{\partial \text{m}athcal{L}^*}{\partial \theta} = -\frac{IA_{1}(I)\text{s}in(\theta - I\tau^*)}{I-1},$$
just the same expression as the one for the resonance $I = 0$, so the transversality between invariant sets of the inner and the scattering map follows.
Now, a constructive proof of Theorem~\ref{theo:main_theo} is presented.
This proof is similar to the proof presented in \cite{Delshams2017}, but now, there is no any piece of ``highway'' or fast vertical lines where $\left|I\right|$ is large.
So, the inner map is applied more times.
\text{s}ubsection{Proof of Theorem~\ref{theo:main_theo}}
\begin{proof}
First of all we have to choose what scattering map we use.
This choice depends on the sign of $\text{m}u$ as explained in Lemma~\ref{lem:geometrical_lemmas}.
Assuming $\text{m}u >0$, we take $\text{m}athcal{S}_{\text{1}}(I,\theta)$, the global scattering map associated to the graphs of $\xi_{\text{1}}$ and $\eta_{\text{1}}$.
If $a_1 >0$, by Proposition~\ref{prop:orb_cres} for any $I$ there exists an interval $\theta \in (\pi,\theta_+)$ where $\dot{I}>0$.
Define $H_{\text{r}}$ the set $\left(\rho,\theta_+\right)\times\left[-I^*,I^*\right]$, where $\rho = \pi + \delta$ is such that $\pi <\rho<\theta_+$ and the transversality between {\text{NHIM}} lines and $\text{m}athcal{L}_1^*$ holds.
We first construct a pseudo-orbit $\{(I_i,\theta_i): i = 0,\dots,N_1\}\text{s}ubset H_{\text{r}}$ with $I_0 = -I^*$ and $\theta_{N_1}$ as close as possible to $\rho$.
Note that all these points lie in the same level curve of $\text{m}athcal{L}_{\text{1}}^*$, that is, $\text{m}athcal{L}_{\text{1}}^*(I_0,\theta_0) = \text{m}athcal{L}_{\text{1}}^*(I_i,\theta_i)$, $i = 1,\dots,N_1$.
Applying the inner dynamics, we get $(I_{\text{N}_1 + 1},\theta_{\text{N}_1 + 1}) = \phi_{t_{N_1}}(I_{\text{N}_1},\theta_{\text{N}_1} )$ with $\theta_{\text{N}_1 + 1}\in (\rho , \theta_+)$ and then we construct a pseudo-orbit $\{(I_i,\theta_i): i = N_1 +1,\dots,N_1+M_1\}\text{s}ubset \text{m}athcal{L}_{\text{1}}^*(I_{\text{N}_1+1},\theta_{\text{N}_1+1}) = l_{\text{N}_1 +1}$ with $\theta_{i}\in(\rho,\theta_{\text{N}_1 +1})$, $\theta_+ -\theta_{\text{N}_1 + \text{M}_{1}} = \text{m}athcal{O}(\varepsilon^2)$.
Applying the inner dynamics, we get $(I_{\text{N}_1 + \text{M}_1+1}, \theta_{\text{N}_1 + \text{M}_1+1})=\phi_{t_{\text{N}_1 + \text{M}_1}}(I_{\text{N}_1 + \text{M}_1}, \theta_{\text{N}_1 + \text{M}_1})$ with $\theta_{\text{N}_1 + \text{M}_1 +1}\in (\rho,\theta_+)$.
Recursively, we construct a pseudo-orbit $\{(I_i,\theta_i): i = \text{N}_1 +1,\dots,\text{N}_2\}$ such that $I_{\text{N}_2}\geq I^*$.
In the same ways as in \cite{Delshams2017} (Theorem 4), we can apply shadowing techniques of \cite{fontich2000,fontich2003,Gidea2014}, due to the fact that the inner dynamics is simple enough to satisfy the required hypothesis of these references, to prove the existence of a diffusion trajectory.
If $a_{10}<0$, changing $H_{\text{r}}$ to $H_{\text{l}}=\left(\theta_+,\pi\right)$ all the previous reasoning applies.
\end{proof}
\begin{remark}Considering Remark~\ref{rem:r_1}, Remark~\ref{rem:r_2}, Remark~\ref{rem:r_3} and Remark~\ref{rem:r_4}, for any $r\in(0,1)$, an equivalent diffusion result is readily obtained.
\end{remark}
\text{s}ection{Piecewise smooth global scattering maps}
\label{sec:piecewise}
In this section, the geometric freedom of the choice of $\tau^*$ is explored.
Until now, only two different scattering maps have been used to build a global one, and this was enough to ensure diffusion.
But, with this approach, finding a diffusion pseudo-orbit is not always easy enough and this pseudo-orbit can be also complicated.
This depends simply on the ``aspect" of the scattering map obtained.
We now suggest a new criterion to choose $\tau^*$: to take the minimal value for $\left|\tau^*\right|$ for any $(\theta , I)$.
This provides us with a piecewise smooth global scattering map with a good property: the phase space of this scattering map which is $\text{m}athcal{O}(\varepsilon^2)$-close to the level sets of the reduced Poincar\'{e} function $\text{m}athcal{L}^*(I,\theta)$ associated to the chosen $\tau^*$ is simpler and ``cleaner'' than the phase spaces of other scattering maps displayed up to now.
By a cleaner scattering map, we mean that we can easily identify and understand the orbits of the scattering maps, except for a small region which contains the tangency locus.
\begin{figure}
\caption{Examples of piecewise smooth global scattering maps. The orbits of scattering maps are represented by the blue lines.
In the red zones the values of $I$ on such orbits decrease, in the green one the values of $I$ increase. \label{fig:global_sm_piece}
\label{fig:pw_03}
\label{fig:pw_05}
\label{fig:pw_09}
\label{fig:pw_15}
\label{fig:global_sm_piece}
\end{figure}
Besides, the zones where the value of $I$ is increased or decreased under the scattering map is well behaved.
$I$ decreases for $\theta \in (0,\pi)$ (the red region on all pictures in Fig.~\ref{fig:global_sm_piece}) and $I$ increases for $\theta \in (\pi , 2\pi)$ (the green region on all pictures in Fig.~\ref{fig:global_sm_piece}).
So it is easy to infer that for finding a diffusion pseudo-orbit it is enough to build a combination between the inner map and this scattering map restricted to $(\pi , 2\pi)$, for example if an increased value of $I$ is wished.
The same idea used in the proof of Theorem~\ref{theo:main_theo}.
Observe that the scattering maps we are now considering are a mix of the scattering maps studied previously.
As an example, we illustrate the scattering map obtained for $\text{m}u = 0.9$.
Such scattering map can be divided into three regions and in each region, the scattering map coincides with a scattering map studied before.
\begin{figure}
\caption{A piecewise smooth global scattering map divided into 3 regions. The vertical black lines are the boundaries of
the domains of smooth scattering maps.\label{fig:3_regions}
\label{fig:3_regions}
\end{figure}
In Fig.~\ref{fig:3_regions}, for regions I ($0<\theta<\pi/2$), II ($\pi/2<\theta<3\pi/2$) and III ($3\pi/2<\theta<2\pi$) the scattering map has the following correspondence:
\begin{itemize}
\item[I] Extended scattering map $\text{m}athcal{S}_0(I,\theta)$ associated to the horizontal $\text{m}athcal{C}_{\text{M}}(I)$ ``under" $\text{s}igma = \varphi$.
\item[II] Extended scattering map $\text{m}athcal{S}_1(I,\theta)$ associated to the horizontal $\text{m}athcal{C}_{\text{m}}(I)$.
\item[III] Extended scattering map $\text{m}athcal{S}_{2}(I,\theta)$ associated to the horizontal $\text{m}athcal{C}_{\text{M}}(I)$ ``over" $\text{s}igma = \varphi$.
\end{itemize}
If extended scattering maps are not considered and we just use scattering maps associated to horizontal and vertical crests, one can see that these scattering maps can be divided into 6 regions, i.e., they can be viewed as a combination of up to 6 scattering maps.
Another property of these scattering maps is the loss of differentiability on the straight lines $\theta = \pi/2$ and $\theta = 3\pi/2$.
The vector field associated to the Hamiltonian $-\text{m}athcal{L}_i^*$ defined around these discontinuity lines behaves as the vector fields studied in non-smooth dynamics theory.
More precisely, we can find regions with slide and unstable slide behavior \cite{Filippov88}.
In a future work, we envisage to design special pseudo-orbits along these discontinuity lines using such theory.
Note that these pseudo-orbits would be very similar to the ``highways" defined in \cite{Delshams2017}, so in principle, one can expect fast and simple diffusion along these discontinuity lines.
\text{s}ubsection*{Acknowledgments}
The authors would like to express their gratitude to the
anonymous referees for their comments and suggestions which
have contributed to improved the final form of this paper.
We also thank C. Sim\'o for several discussions and comments.
\appendix
\text{s}ection{Proof of Proposition~\ref{prop:orb_cres} \label{app:A}}
\begin{repproposition}{prop:orb_cres}
Let $\text{m}athcal{S}_{ \text{1}}(I,\theta)$ be the scattering map associated to the graphs $\xi_{\text{1}}$ and $\eta_{\text{1}}$.
Assuming $a_1,a_2 >0$, then for any $I$, there exists a $\theta_{+}$ such that $\dot{I}>0$ for $\theta \in (\pi, \theta_{+})$.
Moreover, $\theta_{+}\geq 3\pi/2$ for $I\notin(-1/2 , 1/2)$. \end{repproposition}
\begin{proof}
We have
\begin{equation}
\dot{I} =\frac{\partial \text{m}athcal{L}^*}{\partial\theta}(I,\theta) = \frac{A_1(I)\text{s}in(\theta - I\tau^*(I,\theta))}{I-1} = -\frac{A_2(I)\text{s}in(\theta - (I-1)\tau^*(I,\theta)}{I}.\label{eq:dot_I}
\end{equation}
where $A_{1}(I)$ and $A_{2}(I)$ are positive, because $a_1,\,a_{2}>0$.
Notice that $\text{m}u = a_1/a_2 >0$.
Note that as $(I , \varphi = \pi,\theta = \pi)$ is always on the crest $\text{m}athcal{C}_{\text{m}}(I)$, $\tau^*(I,\pi)= 0 $ for all $I$.
Consider first the case of horizontal crests ($\left|\alpha(I)\text{m}u\right|< 1$).
\begin{itemize}
\item[a)] For $I< 0$, the function $\alpha(I)$ introduced in \eqref{eq:mu_alpha} satisfies $\alpha(I)> 0 $, and from \eqref{eq:cristas_06}, $\text{s}in(\xi_{\text{1}}(I,\varphi))\text{s}in\varphi = -\text{m}u \alpha(I)\text{s}in\varphi\leq 0$.
Take $\theta = \frac{3\pi}{2}$; since $I<0$, the slope $m = (I-1)/I$ of the {\text{NHIM}} lines is greater than 1.
Therefore, $3\pi/2 - I\tau_{\text{1}}^*(I,3\pi/2)\in(\pi,3\pi/2)$.
This implies that for any $\theta \in (\pi, 3\pi/2)$, $\theta - I\tau_{\text{1}}^*(I,\theta) \in (\pi,3\pi/2)$, so $\text{s}in(\theta - I\tau_{\text{1}}^*)<0$.
From \eqref{eq:dot_I}, $\dot{I}>0$.
\item[b)] For $0<I<1$, $\alpha(I)< 0$, so $\text{s}in\xi_{\text{1}}(I,\varphi)\text{s}in\varphi\geq 0$.
Besides, $m<0$, so if we look for $\theta_*$ satisfying
\begin{eqnarray}
\theta - I\tau = 2\pi\label{eq:lem_sys_1}\\
\theta - (I-1)\tau = \pi,\nonumber
\end{eqnarray}
we have that for any $\theta\in(\pi,\theta_*)$, $\theta - I\tau^*_{\text{1}}\in(\pi,2\pi)$.
By solving \eqref{eq:lem_sys_1} and defining $\theta_+ := \theta_*$, we obtain $\theta_+ = (2-I)\pi$.
Then, $\text{s}in(\theta - I\tau^*_{\text{1}}(I,\theta))<0$ and therefore $\dot{I}>0$ for any $\theta \in (\pi,\theta_+ = (2-I)\pi)$.
In particular, $\theta_+ <3\pi/2$ if, and only if, $I\in(1/2,1)$.
\item[c)] For $I >1$, one more time $\alpha(I) > 0$ and $\text{s}in\xi_{\text{1}}(I,\varphi)\text{s}in(\varphi)< 0$, but now $ 0 < m = 1 - 1/I < 1 $.
We first fix $ \theta = 3 \pi/ 2 $ and search for $I$ such that
\begin{eqnarray*}
\frac{3\pi}{2} -I\tau^*(I,3\pi/2) = 0\\
\frac{3\pi}{2} - (I - 1)\tau^*(I,3\pi/2) = \pi.
\end{eqnarray*}
We obtain $I = 3/2$, so $\theta - I\tau^*_{\text{1}}(I,\theta) \in (0,\pi)$ for any $I \geq 3/2$ and $\theta \in (\pi , \theta_+ = 3\pi/2)$.
Consequently, $\text{s}in(\theta - I\tau^*_{\text{1}}(I,\theta))>0$ and $\dot{I}>0$.
For the values of $I\in(1,3/2)$ we change the strategy.
We look for $\theta_*$ such that
\begin{eqnarray*}
\theta - I\tau^*(I,\theta) = 0\\
\theta - (I-1)\tau^*(I,\theta) = \pi.
\end{eqnarray*}
We have $\theta_* = \pi I$ and $\theta - I\tau^*_{\text{1}}(I,\theta_*)\in(0,\pi)$ for any $I\in(1,3/2)$ and $\theta\in(\pi, \theta_*)$, so $\dot{I}>0$.
Note that $\theta_* < 3\pi/2$ and we can define $\theta_+ := \theta_*$.
\end{itemize}
Observe that for $I = 1 $ the crests are vertical, and for $I=0$, $\theta = \theta - I\tau^*_{\text{1}}(I,\theta)$, and $\dot{I}>0$ for $\theta \in (\pi,3\pi/2)$.
Consider now the case of vertical crests ($\left|\alpha(I)\text{m}u\right|>1$).
\begin{itemize}
\item[a)]For $I< 0$, $\text{s}in\eta_{\text{1}}(I,\text{s}igma)\text{s}in\text{s}igma = -\text{m}u\alpha(I)\text{s}in^2\text{s}igma \leq 0 $ and $m>1$.
We fix $\theta = 3\pi/2$ and look for $I$ such that
\begin{align*}
3\pi/2\pi - I\tau^* &= \pi\\
3\pi/2 - (I-1)\tau^*(I,3\pi/2) &= 0.
\end{align*}
We obtain $I = -1/2$ and therefore, $\text{s}in(\theta - (I-1)\tau^*_{\text{1}}(I,\theta))> 0$ for $I\in(-\infty,-1/2)$ and $\theta \in(\pi,3\pi/2)$.
Consequently, $\dot{I} > 0$ from \eqref{eq:dot_I}.
For $I\in(-1/2 , 0)$, we have that $\theta_+ = (1-I)\pi$ satisfies
\begin{align*}
\theta - I\tau^*(I,\theta_+) & = \pi\\
\theta_+ -(I-1)\tau^*(I,\theta_+) & = 0.
\end{align*}
Therefore, $\text{s}in(\theta - (I-1)\tau^*_{\text{1}})(I,\theta) >0$ and $\dot{I}>0$ for any $\theta \in (\pi,\theta_+)$.
\item[b)] For $ 0 < I < 1$ $\text{s}in\eta_{\text{1}}(I,\text{s}igma)\text{s}in\text{s}igma \geq 0 $ and $m<0$.
$\theta_+ = (I + 1)\pi$ satisfies
\begin{align*}
\theta - I\tau^*(I,\theta_+) &= \pi\\
\theta_+ -(I-1)\tau^*(I,\theta_+)& = 2\pi.
\end{align*}
So, $\text{s}in(\theta - (I-1)\tau^*_{\text{1}}(I,\theta))>0$ and $\dot{I}>0$ for any $\theta \in(\pi,\theta_+)$.
Note that $\theta_{+} <3\pi/2$ for $I \in (0,1/2)$.
\item[c)] Finally, for $I > 1$, $\text{s}in\eta_{\text{1}}(I,\text{s}igma) \text{s}in\text{s}igma \leq 0$.
We have that $\theta - (I-1)\tau^*_{\text{1}}(I,\theta)\in(\pi,2\pi$), so $\text{s}in(\theta - (I-1)\tau^*_{\text{1}}(I,\theta))<0$ and $\dot{I}> 0$ for any $\theta\in(\pi,3\pi/2)$.
\end{itemize}
For $I = 0$ the crests are horizontal. For $I = 1$, $\theta = \theta - (I-1)\tau^*_{\text{1}}(I,\theta)$, so $\dot{I}>0$ for $\theta \in(\pi,2\pi)$.
\end{proof}
\end{document} |
\begin{document}
\title{Tight Bounds for Sketching the Operator Norm, Schatten Norms, and Subspace Embeddings ootnote{This paper appeared in the Proceedings of RANDOM/APPROX 2016, LIPIcs Vol.\ 60, 39:1--39:11. The current version corrects the proof of Corollary 7.
Yi Li was supported by ONR grant N00014-14-1-0632 when he was at Harvard University, where the major part of this work was done. David P.\ Woodruff was at IBM Research Almaden when this work was done.}
\begin{abstract}
We consider the following oblivious sketching problem: given $\varepsilonilon \in (0,1/3)$
and $n \geq d/\varepsilonilon^2$,
design a distribution $\mathcal{D}$ over $\mathbb{R}^{k \times nd}$ and a function
$f: \mathbb{R}^k \times \mathbb{R}^{nd} \rightarrow \mathbb{R}$, so that for any $n \times d$ matrix $A$,
$$\Pr_{S \sim \mathcal{D}} [(1-\varepsilonilon) \|A\|_{op} \leq f(S(A),S) \leq (1+\varepsilonilon)\|A\|_{op}] \geq 2/3,$$
where $\|A\|_{op} = \sup_{x:\|x\|_2 = 1} \|Ax\|_2$ is the operator norm of $A$ and $S(A)$ denotes
$S \cdot A$, interpreting $A$ as a vector in $\mathbb{R}^{nd}$. We show a tight lower
bound of $k = \Omega(d^2/\varepsilonilon^2)$ for this problem. Previously, Nelson and Nguyen (ICALP, 2014)
considered the problem of finding a distribution $\mathcal{D}$ over $\mathbb{R}^{k \times n}$ such
that for any $n \times d$ matrix $A$,
$$\Pr_{S \sim \mathcal{D}}[\forall x, \ (1-\varepsilonilon)\|Ax\|_2 \leq \|SAx\|_2 \leq (1+\varepsilonilon)\|Ax\|_2]
\geq 2/3,$$ which is called an oblivious subspace embedding (OSE). Our result considerably strengthens
theirs, as it (1) applies only to estimating the operator norm, which can be estimated given
any OSE, and (2) applies to distributions
over general linear operators $S$ which treat $A$ as a vector and compute $S(A)$,
rather than the restricted class of linear operators corresponding to
matrix multiplication.
Our technique also implies the first tight bounds for approximating the Schatten $p$-norm
for even integers $p$ via general linear sketches, improving the previous lower bound
from $k = \Omega(n^{2-6/p})$ [Regev, 2014]
to $k = \Omega(n^{2-4/p})$.
Importantly, for sketching the operator norm up to a factor of $\alpha$, where $\alpha - 1 = \Omega(1)$,
we obtain a tight $k = \Omega(n^2/\alpha^4)$ bound, matching the upper bound of
Andoni and Nguyen (SODA, 2013), and improving the previous $k = \Omega(n^2/\alpha^6)$ lower bound.
Finally, we also obtain the first lower bounds for approximating Ky Fan norms.
\end{abstract}
\title{Tight Bounds for Sketching the Operator Norm, Schatten Norms, and Subspace Embeddings ootnote{This paper appeared in the Proceedings of RANDOM/APPROX 2016, LIPIcs Vol.\ 60, 39:1--39:11. The current version corrects the proof of Corollary 7.
Yi Li was supported by ONR grant N00014-14-1-0632 when he was at Harvard University, where the major part of this work was done. David P.\ Woodruff was at IBM Research Almaden when this work was done.}
\section{Introduction}\label{sec:intro}
Understanding the sketching complexity
of estimating matrix norms \cite{BS15,LNW14,Regev14,w14} has been a goal of recent work, generalizing
a line of work on estimating frequency moments in the sketching
model~\cite{anpw13,lw13,pw12}, and in the somewhat related streaming model of computation~\cite{ams99}.
In the sketching model, one fixes a distribution
$\mathcal{D}$ over $k\times (nd)$ matrices $S$, and is then given an $n \times d$ matrix $A$ which, without loss of generality,
satisfies $n \geq d$.
One then samples $S$ from $\mathcal{D}$, and computes $S(A)$, which denotes the operation of treating $A$ as a
column vector in $\mathbb{R}^{nd}$ and left-multiplying that vector by the matrix $S$.
Any linear transformation
applied to $A$ can be expressed in this form, and therefore we sometimes refer to such a distribution $\mathcal{D}$
as a {\it general linear sketch}. There is also the related notion of a {\it bilinear sketch}, in which one fixes a
distribution $\mathcal{D}$ over $k \times n$ matrices $S$, and is then given an $n \times d$ matrix $A$. One
samples $S$ from $\mathcal{D}$ and computes $S \cdot A$. Bilinear sketches are special cases of general linear sketches since they form a subclass of all possible linear transformations of $A$,
and general linear sketches can be much more powerful than bilinear sketches.
For example, to compute the trace exactly of an $n \times n$
matrix $A$, setting $k = 1$ suffices for a general linear sketch, while we do not know how to compute the trace
with a small value $k$ for bilinear sketches, and several lower bounds on $k$ are known even to approximate the trace
\cite{wwz14}.
The goal
in the sketching model
is to minimize the {\it sketching dimension} $k$ so that $S(A)$ can be used to approximate a property of $A$ with constant probability. Associated with distribution $\mathcal{D}$ is an estimation procedure, which we model as a function $f$,
for which $f(S(A), S)$ outputs a correct answer to the problem at hand with constant probability. For numerical properties, such as estimating a norm of $A$, this probability can be amplified to $1-\delta$, by creating a distribution $\mathcal{D}'$ corresponding to taking $O(\log (1/\delta))$ independent copies $S^1, \ldots, S^{\log(1/\delta)}$ from $\mathcal{D}$, and outputting the median of
$$f(S^1(A), S^1), f(S^2(A), S^2), \ldots, f(S^{\log(1/\delta)}(A), S^{\log(1/\delta)}).$$
Notice that the mapping $S$ is linear and
oblivious, both of which are important for a number of applications such as merging sketches in distributed
computation, or for approximately recovering a signal in compressed sensing.
Minimizing $k$ is crucial for these applications, as it corresponds to the communication
or number of observations of the underlying algorithm.
A quantity of interest is the operator norm. Given a matrix $A$, the operator norm $\|A\|_{op}$
is defined to be $\|A\|_{op} =\sup_{x:\|x\|_2=1} \|Ax\|_2$.
The operator norm arises in several applications; for example
one sometimes approximates a matrix $A$ by another matrix $\hat{A}$ for which
$\|A -\hat{A}\|_2$ is small. Often $\hat{A}$ has low rank, in which case this is the low rank approximation
problem with spectral error, see, e.g., recent work on this \cite{mm15}.
If one had an estimator for the operator norm of
$A - \hat{A}$, one could
use it to verify if $\hat{A}$ is a good approximation to $A$. Given the linearity in the sketching model,
if $S$ is sampled from a distribution $D$, one can compute $S(A) - S(\hat{A}) = S(A - \hat{A})$, from
which one then has an estimation procedure to estimate
$\|A-\hat{A}\|_2$ as $f(S(A-\hat{A}), S)$.
In the sketching model, it was first
shown that approximating the operator norm up to a constant factor requires $k = \Omega(d^{3/2})$ \cite{LNW14},
which was later improved by Regev to the tight $k = \Omega(d^2)$ \cite[Section 6.2]{w14}. Note that these
lower bounds rule out {\it any} possible function $f$ as the estimation procedure.
It is also implicit in \cite[Section 6.2]{w14} that approximating the operator norm up to a factor
$\alpha$, where $\alpha - 1 = \Omega(1)$, requires $k = \Omega(d^2/\alpha^6)$.
Andoni and Nguyen showed an upper bound of $k = O(d^2/\alpha^4)$ \cite{AN13}, that is, they constructed a distribution $\mathcal{D}$ and corresponding estimation procedure $f$ for which it suffices to set $k = O(d^2/\alpha^4)$. This follows by Theorem 1.2 of \cite{AN13}.
A wide class of matrix norms is the Schatten $p$-norms, which are the analogues of
$\ell_p$-norms of vectors
and contain the operator norm as a special case. The Schatten $p$-norm of matrix $A$ is denoted
by $\|A\|_p$ and defined to be
$\|A\|_p = (\sum_{i=1}^n (\sigma_i(A))^p)^{1/p}$, where $\sigma_1,\dots,\sigma_n$
are the singular values of $A$. When $p < 1$, $\|A\|_p$ is not a norm but still a well-defined quantity. For $p=0$,
viewing $\|A\|_0$ as the limit $\lim_{p\to 0^+} \|A\|_p$ recovers exactly the {\it rank} of $A$, which has
been studied in the data stream~\cite{BS15,cw09} and property testing models~\cite{ks03,lww14}.
When $p = 1$, it is the nuclear or trace norm\footnote{The trace norm is not to be confused with
the trace. These two quantities only coincide if $A$ is positive semidefinite.}, which has applications
in differential privacy~\cite{hlm10,lm12}
and non-convex optimization~\cite{cr12,dtv11}. When $p = 2$, it is the Frobenius norm, and when $p\to\infty$,
it holds that $\|A\|_p$ tends to $\|A\|_{op}$.
Such norms are useful in geometry and linear algebra, see, e.g.,~\cite{w14}.
A $k = \Omega(\sqrt{d})$ lower bound for every $p \geq 0$ was shown in~\cite{lnw14b}. For
$p > 2$ a lower bound on the sketching dimension of $k = \Omega(d^{2/3-3/p})$, and an upper bound of
$k = O(d^{2-4/p})$ were shown in~\cite{lnw14b}.
The upper bound is only known to hold when $p$ is an even integer. The lower bound was
improved by Regev to
$k = \Omega(d^{2-6/p})$ for $p > 6$ \cite[Section 6.2]{w14}\footnote{The section discusses only the case of $p=\infty$, i.e., the operator norm, but the same method can be used for general $p$ and gives the bound claimed here.}.
Other related work includes that on {\it oblivious subspace embeddings} (OSEs), which fall into the category
of bilinear sketches.
Here one seeks a distribution $\mathcal{D}$ over $\mathbb{R}^{k \times n}$ such
that for any $n \times d$ matrix $A$,
$$\Pr_{S \sim \mathcal{D}}[\forall x, \ (1-\varepsilonilon)\|Ax\|_2 \leq \|SAx\|_2 \leq (1+\varepsilonilon)\|Ax\|_2]
\geq 2/3.$$
This notion has proved important in numerical linear algebra, and has led to the fastest
known algorithms for low rank approximation and regression \cite{cw13,mm13,nn13}. Since an OSE
has the property that $\|SAx\|_2 = (1 \pm \varepsilonilon) \|Ax\|_2$ for all $x$, it holds in particular
that $\|SA\|_{op} = (1 \pm \varepsilonilon) \|A\|_{op}$, where the notation $a = (1 \pm \varepsilonilon) b$ means
$(1-\varepsilonilon) b \leq a \leq (1+\varepsilonilon)b$. When $n \geq d/\varepsilonilon^2$, Nelson and Nguyen show the
tight bound that any OSE requires $k = \Omega(d/\varepsilonilon^2)$ \cite{NN14}.
Finally, we mention recent related work in the data stream model on approximation of matrix norms
\cite{BS15,lw16}.
Here one sees
elements of $A$ one at a time and the goal is to output an approximation to $\|A\|_p$.
It is important to note that the data stream model and sketching models are incomparable. The main
reason for this is that unlike in the data stream model,
the bit complexity is not accounted for in the sketching model,
and both $S$ and $A$ are assumed to have entries which are real numbers. The latter is the common
model adopted in compressed sensing. In the data stream model, if one wants to output a vector
$v \in \{0, 1, \ldots, M-1, M\}^n$, one needs $n \log M$ bits of space. On the other hand, if
$u$ is the vector $(1, (M+1), (M+1)^2, (M+1)^3, \ldots, (M+1)^n)$, then from $\langle u, v \rangle$,
one can output $v$, so the sketching dimension $k$ is only equal to $1$. The sketching complexity
thus gives a meaningful measure of complexity in the real RAM model. Conversely, lower bounds in
the sketching model do not translate into lower bounds in the data stream model.
This statement holds even given the
work of \cite{LNW14} which characterizes turnstile streaming algorithms as linear sketches. The
problem is that lower bounds in the sketching model
involve continuous distributions and after discretizing the distributions it is no longer clear if the
lower bounds hold.
\subsection{Our Contributions}
In this paper we strengthen known sketching lower bounds for the operator norm,
Schatten $p$-norms, and subspace embeddings. Our lower bounds are optimal for any approximation to
the operator norm, for subspace embeddings, and for Schatten $p$-norms for even integers $p$.
We first describe our results for the operator norm, as the results for Schatten $p$-norms and
subspace embeddings follow from them.
We consider the following problem: given $\varepsilonilon \in (0,1/3)$
and $n \geq d/\varepsilonilon^2$,
design a distribution $\mathcal{D}$ over $\mathbb{R}^{k \times nd}$ and a function $f: \mathbb{R}^k \times \mathbb{R}^{k \times nd} \rightarrow \mathbb{R}$, so that for any $n \times d$ matrix $A$,
$$\Pr_{S \sim \mathcal{D}} [(1-\varepsilonilon) \|A\|_{op} \leq f(S(A), S) \leq (1+\varepsilonilon)\|A\|_{op}] \geq 2/3,$$
For this problem, we show a tight $k = \Omega(d^2/\varepsilonilon^2)$ lower bound.
Our result considerably strengthens
the result of Nelson and Nguyen \cite{NN14} as it
(1) applies only to estimating the operator norm, which can be estimated given
any OSE, and (2) applies to general linear sketches rather than only to bilinear sketches.
Regarding (1), this
shows that designing a general linear sketch for approximating the operator norm of a matrix is
{\it as hard as designing an oblivious subspace embedding}.
Regarding (2), we lower bound a much larger class of data structures than OSEs that
one could
use to approximate $\|Ax\|_2$ for all vectors $x$.
We then generalize the argument above to handle approximation factors $\alpha$, with $\alpha - 1 = \Omega(1)$,
for approximating the operator norm. In this case we consider $n = d$, which is without loss of generality
since by first applying an OSE $S$ to $A$ with $k = O(d)$, replacing $A$ with $S \cdot A$, all singular
values of $A$ are preserved up to a constant factor (we can also pad $SA$ with zero columns to make
$SA$ be a square matrix) - see Appendix C of \cite{lnw14b}. We can then apply our general linear
sketch to $SA$ (the composition of linear sketches is a general linear sketch). We show a lower
bound of $k = \Omega(n^2/\alpha^4)$, improving the previous $k = \Omega(n^2/\alpha^6)$ bound,
and maching the $k = O(n^2/\alpha^4)$ upper bound. This answers Open Question 2 in \cite{lnw14b}.
The proof shows the problem is already hard
to distinguish between the two cases: (1) $A$ has one singular value of value $\Theta(\alpha)$
and remaining singular values of
value $\Theta(1)$, versus (2) all singular values of $A$ are of value $\Theta(1)$.
By setting $\alpha = n^{1/p}$, we are able to obtain a constant factor gap
in the Schatten-$p$ norm in the two cases, and therefore additionally obtain an $\Omega(n^{2-4/p})$
lower bound for Schatten $p$-norms for constant factor approximation. This improves the previous
$\Omega(n^{2-6/p})$ lower bound, and matches the known upper bound for even integers $p$. Our proof
also establishes a lower bound of $k = \Omega(n^2/s^2)$
for estimating the Ky-Fan $s$-norm of an $n \times n$
matrix $A$ up to a constant factor, whenever $s \leq .0789\sqrt{n}$.
Our main technical novelty is avoiding a deep
theorem of Lata\l{}a~\cite{latala} concerning tail bounds for Gaussian chaoses
used in the prior lower bounds for sketching the operator norm and Schatten $p$-norms.
Instead we prove a simple lemma (Lemma~\ref{lem:Eexp(x^TAy)}) allowing us
to bound $\mathbb{E}_{x,y}[e^{x^T Ay}]$ for Gaussian vectors $x$ and $y$ and a
matrix $A$,
in terms of the Frobenius norm
of $A$. Surprisingly, this lemma
suffices for directly upper-bounding the $\chi^2$-distance between the distributions
considered in previous works, and without losing any additional factors.
Our technical arguments
are thus arguably more elementary and simpler than those given in previous work.
\section{Preliminaries}\label{sec:prelim}
\subsection*{Notation.} Let $\ensuremath\mathbb{R}^{n\times d}$ be the set of $n\times d$ real matrices and $N(\mu,\Sigma)$ denote the (multi-variate) normal distribution of mean $\mu$ and covariance matrix $\Sigma$.
We write $X\sim \mathcal{D}$ for a random variable $X$ subject to a probability distribution $\mathcal{D}$.
Denote by $\ensuremath\mathcal{G}(n,n)$ the ensemble of random matrices with entries i.i.d.\ $N(0,1)$.
\subsection*{Singular values and matrix norms.} Consider a matrix $A\in \ensuremath\mathbb{R}^{n \times n}$. Then $A^TA$ is a positive semi-definite matrix. The eigenvalues of $\sqrt{A^TA}$ are called the singular values of $A$, denoted by $\sigma_1(A)\geq \sigma_2(A)\geq \cdots \geq\sigma_n(A)$ in decreasing order. Let $r=\rk(A)$. It is clear that $\sigma_{r+1}(A) = \cdots = \sigma_n(A) = 0$.
Define $
\|A\|_p = (\sum_{i=1}^r (\sigma_i(A))^p)^{1/p}
$ ($p>0$).
For $p\geq 1$, it is a norm over $\ensuremath\mathbb{R}^{n\times d}$, called the $p$-th \textit{Schatten norm}, over $\ensuremath\mathbb{R}^{n\times n}$ for $p\geq 1$. When $p=1$, it is also called the trace norm or nuclear norm. When $p=2$, it is exactly the Frobenius norm $\|A\|_F$.
Let $\|A\|_{op}$ denote the operator norm of $A$ when treating $A$ as a linear operator from $\ell_2^n$ to $\ell_2^n$. It holds that $\lim_{p\to\infty} \|A\|_p = \sigma_1(A) = \|A\|_{op}$.
The Ky-Fan $s$-norm of $A$, denoted by $\|A\|_{F_s}$, is defined as the sum of the largest $s$ singular values: $\|A\|_{F_s} = \sum_{i=1}^s \sigma_i(A)$. Note that $\|A\|_{F_1} = \|A\|_{op}$ and $\|A\|_{F_s} = \|A\|_1$ for $s\geq r$.
\subsection*{Distance between probability measures.} Suppose $\mu$ and $\nu$ are two probability measures over some Borel algebra $\mathcal{B}$ on $\ensuremath\mathbb{R}^n$ such that $\mu$ is absolutely continuous with respect to $\nu$. For a convex function $\phi:\ensuremath\mathbb{R}\to \ensuremath\mathbb{R}$ such that $\phi(1)=0$, we define the $\phi$-divergence
\[
D_\phi(\mu || \nu) = \int \phi\left(\frac{d\mu}{d\nu}\right)d\nu.
\]
In general $D_\phi(\mu||\nu)$ is not a distance because it is not symmetric.
The \textit{total variation distance} between $\mu$ and $\nu$, denoted by $d_{TV}(\mu,\nu)$, is defined as $D_\phi(\mu||\nu)$ for $\phi(x) = |x-1|$. It can be verified that this is indeed a distance.
The \textit{$\chi^2$-divergence} between $\mu$ and $\nu$, denoted by $\chi^2(\mu||\nu)$, is defined as $D_\phi(\mu||\nu)$ for $\phi(x) = (x-1)^2$ or $\phi(x) = x^2-1$. It can be verified that these two choices of $\phi$ give exactly the same value of $D_\phi(\mu||\nu)$.
\begin{proposition}[{\cite[p90]{Tsybakov}}] \label{prop:TV_chi^2}
$d_{TV}(\mu,\nu) \leq \sqrt{\chi^2(\mu||\nu)}$.
\end{proposition}
\begin{proposition}[{\cite[p97]{IS}}] \label{prop:chi^2}
$\chi^2(N(0,I_n)\ast \mu||N(0,I_n)) \leq \E e^{\langle x,x'\rangle}-1$, where $x,x'\sim \mu$ are independent.
\end{proposition}
\section{Sketching Lower Bound for $p > 2$}\label{sec:sketching_p>2}
We follow the notations in \cite{lnw14b} throughout this section, though
the presentation here is self-contained. To start, we present the following lemma.
\begin{lemma}[\footnote{A similar result holds for subgaussian vectors $x$ and $y$ with the right-hand side replaced with $\exp(c\|A\|_F^2)$ for some absolute constant $c>0$, whose proof requires heavier machinery. We only need the elementary variant here by our choice of hard instance.}]\label{lem:Eexp(x^TAy)}
Suppose that $x\sim N(0,I_m)$ and $y\sim N(0,I_n)$ are independent and $A\in \ensuremath\mathbb{R}^{m\times n}$ satisfies $\|A\|_F < 1$. It holds that
\[
\E_{x,y} e^{x^T Ay} \leq \frac{1}{\sqrt{1-\|A\|_F^2}}.
\]
\end{lemma}
\begin{proof}
First, it is easy to verify that
\begin{align*}
\E_{x,y\sim N(0,1)} e^{axy} &= \frac{1}{2\pi}\iint_{\ensuremath\mathbb{R}\times \ensuremath\mathbb{R}} e^{axy-\frac{x^2+y^2}{2}} dxdy\\
&= \frac{1}{2\pi}\int_{\ensuremath\mathbb{R}}\int_{\ensuremath\mathbb{R}} e^{-\frac{1}{2} (x-ay)^2}e^{-\frac{1}{2}(1-a^2)y^2} dx dy\\
&= \frac{1}{\sqrt{2\pi}}\int_{\ensuremath\mathbb{R}} e^{-\frac{1}{2}(1-a^2)y^2} dy\\
& = \frac{1}{\sqrt{1-a^2}},\quad a\in [0,1).
\end{align*}
Without loss of generality, assume that $m\geq n$. Consider the singular value decomposition $A = U\Sigma V^T$ where $U$ and $V$ are orthogonal matrices of dimension $m$ and $n$ respectively and $\Sigma = \diag\{\sigma_1,\dots,\sigma_n\}$ with $\sigma_1,\dots,\sigma_n$ being the non-zero singular values of $A$. We know that $\sigma_i \in [0,1)$ for all $i$ by the assumption that $\|A\|_F < 1$. By rotational invariance of the Gaussian distribution, we may assume that $m = n$ and thus
\begin{align*}
\E_{x,y\sim N(0,I_n)} e^{x^TAy} &= \E_{x,y\sim N(0,I_n)} e^{x^T\Sigma y}\\
&= \frac{1}{(2\pi)^{n}}\iint_{\ensuremath\mathbb{R}^n\times \ensuremath\mathbb{R}^n} \exp\left\{\sum_{i=1}^n \left(\sigma_i x_i y_i - \frac{x_i^2+y_i^2}{2}\right)\right\} dxdy\\
&= \prod_{i=1}^n \frac{1}{\sqrt{1-\sigma_i^2}}\\
&\leq \frac{1}{\sqrt{1-\sum_{i=1}^n \sigma_i^2}}\\
&= \frac{1}{\sqrt{1- \|A\|_F^2}}.\qedhere
\end{align*}
\end{proof}
Next we consider the problem of distinguishing two distributions $\ensuremath\mathcal{D}_1 = \ensuremath\mathcal{G}(m,n)$ and $\ensuremath\mathcal{D}_2$ as defined below. Let $u_1,\dots,u_r$ be i.i.d.~$N(0,I_m)$ vectors and $v_1,\dots,v_r$ i.i.d.~$N(0,I_n)$ vectors and further suppose that $\{u_i\}$ and $\{v_i\}$ are independent. Let $s\in \ensuremath\mathbb{R}^r$ and define the distribution $\ensuremath\mathcal{D}_2$ as $\ensuremath\mathcal{G}(m,n) + \sum_{i=1}^r s_i u^i (v^i)^T$. We take $k$ linear measurements and denote the corresponding rows (measurements) of the sketching matrix by $L^1,\dots,L^k$. Without loss of generality we may assume that $\tr((L^i)^T L^i))=1$ and $\tr((L^i)^T L^j))=0$ for $i\neq j$, since this corresponds to the rows of the sketching matrix being orthonormal, which we can assume since we can always change the basis of the row space of the sketching matrix in a post-processing step. Let $\ensuremath\mathcal{L}_1$ and $\ensuremath\mathcal{L}_2$ be the corresponding distribution of the linear sketch of dimension $k$ on $\ensuremath\mathcal{D}_1$ and $\ensuremath\mathcal{D}_2$, respectively. The main result is the following theorem.
\begin{theorem}\label{thm:sketch_lower_bound} There exists an absolute constant $c>0$ such that $d_{TV}(\ensuremath\mathcal{L}_1,\ensuremath\mathcal{L}_2)\leq 1/10$ whenever $k\leq c/\|s\|_2^4$.
\end{theorem}
\begin{proof}
It is not difficult to verify that $\ensuremath\mathcal{L}_1 = N(0,I_k)$ and $\ensuremath\mathcal{L}_2 = N(0,I_k) + \mu$, where $\mu$ is the distribution of
\[
\begin{pmatrix}
\sum_{i=1}^r s_i (u^i)^T L^1 v^i\\
\sum_{i=1}^r s_i (u^i)^T L^2 v^i\\
\vdots\\
\sum_{i=1}^r s_i (u^i)^T L^k v^i\\
\end{pmatrix}.
\]
Consider a random variable (we shall see in a moment where it comes from)
\[
\xi = \sum_{i=1}^k \sum_{j, l=1}^r \sum_{a,c=1}^m \sum_{b,d=1}^n s_j s_l (L^i)_{ab}(L^i)_{cd} (u^j)_a (v^j)_b (u^l)_c (v^l)_d.
\]
Take expectation on both sides and notice that the non-vanishing terms on the right-hand side must have $j=l$, $a=c$ and $b=d$,
\[
\E\xi = \sum_{i=1}^k \sum_{j=1}^r \sum_{a=1}^m \sum_{b=1}^n s_j^2 (L^i)_{ab}^2 \E(u^j)_a^2 \E(v^j)_a^2 = k\|s\|_2^2.
\]
Define an event $\mathcal{E} = \{ \|s\|^2\xi < 1/2\}$ and it follows from our assumption and Markov's inequality that $\Pr(\mathcal{E}) \geq 1-2c$. Restrict $\mu$ to this event and denote the induced distribution by $\tilde\mu$. Let $\tilde \ensuremath\mathcal{L}_2 = N(0,I_n) + \tilde\mu$.
Then the total variation distance between $\ensuremath\mathcal{L}_1$ and $\ensuremath\mathcal{L}_2$ can be upper bounded as
\begin{align*}
d_{TV}(\ensuremath\mathcal{L}_1,\ensuremath\mathcal{L}_2) &\leq d_{TV}(\ensuremath\mathcal{L}_1,\tilde\ensuremath\mathcal{L}_2) + d_{TV}(\ensuremath\mathcal{L}_2,\tilde\ensuremath\mathcal{L}_2)\\
&\leq \sqrt{\E_{z_1,z_2\sim \tilde\mu}e^{\langle z_1,z_2\rangle} - 1} + d_{TV}(\mu,\tilde\mu)\\
&\leq \sqrt{\frac{1}{\Pr(\mathcal{E})}(\E_{z_1\sim \tilde\mu, z_2\sim\mu}e^{\langle z_1,z_2\rangle} - 1)} + \frac{1}{\Pr(\mathcal{E})}-1
\end{align*}
and we shall bound $\E e^{\langle z_1,z_2\rangle}$ in the rest of the proof.
\begin{align*}
\E_{z_1\sim\tilde\mu,z_2\sim\mu} e^{\langle z_1,z_2\rangle} &= \E \exp\left\{ \sum_{i=1}^k \sum_{j,a,b} \sum_{j',a',b'} s_j (L^i)_{ab} (u^j)_a (v^j)_b \cdot s_{j'} (L^i)_{a'b'} (x^{j'})_{a'} (y^{j'})_{b'} \right\}\\
&= \E_{u^1,\dots,u^r,v^1\dots,v^r|\tilde\mu} \prod_{j'=1}^r \E_{\substack{x_{j'}\sim N(0,I_m)\\y_{j'}\sim N(0,I_n)}} \exp\left\{ \sum_{a',b'} Q^{j'}_{a',b'} (x^{j'})_{a'} (y^{j'})_{b'} \right\},
\end{align*}
where
\[
Q^{j'}_{a',b'} = s_{j'}\sum_{i=1}^k \sum_{j,a,b} (L^i)_{ab}(L^i)_{a'b'}\cdot s_j (u^j)_a (v^j)_b.
\]
In order to apply the preceding lemma, we need to verify that $\|Q^{j'}\|_F^2 < 1$. Indeed,
\begin{align*}
\|Q^{j'}\|_F^2 &= \sum_{a',b'} (Q^{j'})_{a',b'}^2\\
&= s_{j'}^2 \sum_{a',b'} \sum_{i,i'} \sum_{j,a,b}\sum_{\ell,c,d} s_j(L^i)_{ab}(L^i)_{a'b'}(u^j)_a (v^j)_b \cdot s_\ell(L^{i'})_{cd}(L^{i'})_{a'b'}(u^\ell)_c (v^\ell)_d\\
&= s_{j'}^2 \sum_{a',b'} \sum_{i} (L^i)^2_{a'b'} \sum_{j,a,b}\sum_{\ell,c,d} s_j(L^i)_{ab}(u^j)_a (v^j)_b \cdot s_\ell(L^{i})_{cd}(u^\ell)_c (v^\ell)_d \quad (i\text{ must equal to }i')\\
&= s_{j'}^2 \sum_{i} \sum_{j,a,b}\sum_{\ell,c,d} s_j(L^i)_{ab}(u^j)_a (v^j)_b \cdot s_\ell(L^{i})_{cd}(u^\ell)_c (v^\ell)_d\\
&= s_{j'}^2 \xi < 1
\end{align*}
since we have conditioned on $\mathcal{E}$. Now it follows from the preceding lemma that
\begin{align*}
\E_{u^1,\dots,u^r,v^1\dots,v^r} \prod_{i=1}^r \E_{x_{j'},y_{j'}} \exp\left\{ \sum_{a',b'} Q^{j'}_{a',b'} (x^{j'})_{a'} (y^{j'})_{b'} \right\}
&\leq \E_{u^1,\dots,u^r,v^1\dots,v^r} \prod_{j'=1}^r \frac{1}{\sqrt{1-s_{j'}^2\xi}}\\
&\leq \E_{u^1,\dots,u^r,v^1\dots,v^r} \frac{1}{\sqrt{1-\|s\|^2\xi}} \\
&\leq 1+\|s\|^2\E\xi \\
&\leq 1 + k\|s\|^4,
\end{align*}
where, in the third inequality, we used the fact that $1/\sqrt{1-x}\leq 1+x$ for $x\in [0,1/2]$.
Therefore,
\[
d_{TV}(\ensuremath\mathcal{L}_1,\ensuremath\mathcal{L}_2) \leq \sqrt{\frac{k\|s\|^4}{1-2c}} + \frac{2c}{1-2c}
\leq \sqrt{\frac{c}{1-2c}} + \frac{2c}{1-2c}
\leq \frac{1}{10}
\]
when $c > 0$ is small enough.
\end{proof}
We will apply the preceding theorem to obtain our lower bounds for the
applications. To do so, notice that by Yao's minimax principle, we can
fix the rows of our sketching matrix, and show that the resulting distributions
$\mathcal{L}_1$ and $\mathcal{L}_2$ above have small total variation distance.
By standard properties of the variation distance,
this implies that no estimation procedure $f$ can be used to distinguish
the two distributions with sufficiently large probability, thereby establishing
our lower bound.
\begin{corollary}[$\alpha$-approximation to operator norm] Let $c>0$ be an arbitrarily small constant. For $\alpha \geq 1+c$, any sketching algorithm that estimates $\|X\|_{op}$ for $X\in \ensuremath\mathbb{R}^{n\times n}$ within a factor of $\alpha$ with error probability $\leq 1/6$ requires sketching dimension $\Omega(n^2/\alpha^4)$.
\end{corollary}
\begin{proof}
Let $m=n$ and take $r=1$ and $s_1 = C\alpha/\sqrt{n}$ for some constant $C$ large enough in $\ensuremath\mathcal{D}_2$ and apply the preceding theorem.
\end{proof}
\begin{corollary}[Schatten norms] There exists an absolute constant $c>0$ such that any sketching algorithm that estimates $\|X\|_p^p$ ($p>2$) for $X\in \ensuremath\mathbb{R}^{n\times n}$ within a factor of $1+c$ with error probability $\leq 1/6$ requires sketching dimension $\Omega(n^{2(1-2/p)})$.
\end{corollary}
\begin{proof}
Let $m=n$ and take $r=1$ and $s_1 = 5/n^{1/2-1/p}$ in $\ensuremath\mathcal{D}_2$. Note that $\|X\|_p^p$ differs by a constant factor with high probability when $X\sim \ensuremath\mathcal{D}_1$ and $X\sim \ensuremath\mathcal{D}_2$ (the same hard distribution as in \cite{lnw14b}), apply the preceding theorem.
\end{proof}
\begin{corollary} Let $\varepsilonilon\in (0,1/3)$. For any matrix $X\in \ensuremath\mathbb{R}^{(d/\varepsilonilon^2)\times d}$, any sketching algorithm that estimates $\|X\|_{op}$ within a factor of $1+\varepsilonilon$ with error probability $\leq 1/6$ requires sketching dimension $\Omega(d^2/\varepsilonilon^2)$.
\end{corollary}
\begin{proof}
Let $m = d/\varepsilonilon^2$ and $n = d$. Take $r=1$ and $s_1 = \alpha\sqrt{\varepsilonilon/d}$ for some constant $\alpha > 0$ large enough and apply Theorem~\ref{thm:sketch_lower_bound}. Next we shall justify this choice of parameters, that is,
\[
G \qquad\text{ and }\qquad G + \alpha\sqrt{\frac{\varepsilonilon}{d}} uv^T
\]
differ in operator norm by a factor of $1+\varepsilonilon$ for some constant $\alpha$. This is de facto proved in the proof of Theorem 7.3 of~\cite{LWW21}; nevertheless, we include a full proof below for completeness. First, it follows from the standard result~\cite{V12} that
\[
\|G\|_{op} \leq \frac{\sqrt{d}}{\varepsilonilon} + 1.1\sqrt{d} = (1+1.1\varepsilonilon)\frac{\sqrt{d}}{\varepsilonilon}
\]
with probability at least $1-e^{-\Omega(d)}$. Next we shall show that
\[
\left\|G + \alpha\sqrt{\frac{\varepsilonilon}{d}} uv^T\right\|_{op}\geq (1+2\varepsilonilon)\frac{\sqrt{d}}{\varepsilonilon}
\]
with high probability. Observe that (denoting the unit sphere in $\ensuremath\mathbb{R}^d$ by $\ensuremath\mathbb{S}^{d-1}$)
\begin{align*}
\left\|G + \alpha\sqrt{\frac{\varepsilonilon}{d}} uv^T\right\|_{op} = \sup_{x\in \ensuremath\mathbb{S}^{d-1}} \left\|\left(G+ \alpha\sqrt{\frac{\varepsilonilon}{d}} uv^T\right)x\right\|_2
&\geq \left\|\left(G + \alpha\sqrt{\frac{\varepsilonilon}{d}} uv^T \right)\frac{v}{\|v\|_2}\right\|_2 \\
&= \left\|G\frac{v}{\|v\|_2} + \alpha\sqrt{\frac{\varepsilonilon}{d}} u \|v\|_2 \right\|_2.
\end{align*}
Since $v\sim N(0,I_d)$, the direction $v/\|v\|_2 \sim \Unif(\ensuremath\mathbb{S}^{d-1})$ and the magnitude $\|v\|_2$ are independent, and by rotational invariance of the Gaussian distribution, $Gx\sim N(0,I_d)$ for any $x\in \ensuremath\mathbb{S}^{d-1}$. Hence
\[
\left\|G\frac{v}{\|v\|_2} + \alpha\sqrt{\frac{\varepsilonilon}{d}} u \|v\|_2 \right\|_2 \eqdist \left\|u_1 + \alpha\sqrt{\frac{\varepsilonilon}{d}} t u_2\right\|_2 \eqdist \sqrt{1+\frac{\alpha^2 \varepsilon^2 t^2}{d}}\|u\|_2,
\]
where $t$ follows the distribution of $\|v\|_2$, $u_1,u_2,u\sim N(0,I_m)$, $t, u_1, u_2$ are independent, $t$ and $u$ are independent. By the standard results of standard gaussian vectors, with probability at least $1-e^{-\Omega(d)}$, it holds that $t\geq \sqrt{d/2}$ and $\|u\|_2\geq (1-\varepsilon)\sqrt{d}/\varepsilon$. Therefore, with probability at least $1-e^{-\Omega(d)}$, we have
\[
\left\|G + \alpha\sqrt{\frac{\varepsilonilon}{d}} uv^T\right\|_{op}\geq \sqrt{1 + \frac{\alpha^2 \varepsilon}{2}}(1-\varepsilon)\frac{\sqrt{d}}{\varepsilon}\geq (1+2\varepsilon)\frac{\sqrt d}{\varepsilon}
\]
for all $\varepsilon\in(0,1/3)$, provided that $\alpha\geq 3\sqrt{7/2}$.
\end{proof}
\begin{corollary}[Ky-fan norm]\label{cor:ky-fan}
There exists an absolute constant $c>0$ such that any sketching algorithm that estimates $\|X\|_{F_s}$ for $X\in \ensuremath\mathbb{R}^{n\times n}$ and $s\leq 0.0789\sqrt{n}$ within a factor of $1+c$ with error probability $\leq 1/6$ requires sketching dimension $\Omega(n^2/s^2)$.
\end{corollary}
\begin{proof}
Take $r=s$ and $s_1 = s_2 = \cdots = s_r = 5/\sqrt{n}$ in $\ensuremath\mathcal{D}_2$ and apply Theorem~\ref{thm:sketch_lower_bound}, for which we shall show the KyFan $s$-norms are different with high probability in the two cases.
When $X\sim \mathcal{D}_1$, we know that $\sigma_1(X)\leq 2.1\sqrt{n}$ with high probability and thus $\|X\|_{F_s} \leq 2.1s\sqrt{n}$ with high probability.
When $X\sim \mathcal{D}_2$, we can write $X = G + \frac{5}{\sqrt n}P$, where $P = u_1v_1^T + \cdots + u_s v_s^T$. We claim that with high probability $\|P\|_1 \geq 0.9sn$ and thus $\|X\|_{F_s}\geq \frac{5}{\sqrt n}\|P\|_{F_s} - \|G\|_{F_s} \geq 4.5s\sqrt{n} - 2.1s\sqrt{n}\geq 2.4s\sqrt{n}$, evincing a multiplicative gap of $\|X\|_{F_s}$ between the two cases.
Now we prove the claim. With high probability, it holds that $0.99\sqrt{n} \leq \|u_i\|\leq 1.01\sqrt{n}$ for all $i$ and $|\sum_{i\neq j}\langle u_i,u_j\rangle|\leq 1.01s\sqrt{n}$.
We shall condition on these events below.
By the min-max theorem for singular values,
\[
\sigma_\ell^2(P) = \max_{H:\dim H = \ell} \min_{\substack{x\in H\\ \|x\|_2=1}} x^TP^TPx,
\]
where
\begin{align*}
x^TP^TPx &= \sum_{i,j} x^T v_i u_i^T u_j v_j^T x\\
&= \sum_i x^T v_i u_i^T u_i v_i^T x + \sum_{i\neq j} x^T v_i (u_i^T u_j) v_j^T x\\
&\geq 0.99^2 n\sum_i x^T v_i^T v_i x - 1.01\sqrt{n} \cdot 1.01 k\sqrt{n}\cdot 1.01\sqrt{n}\\
&= 0.99^2 n\sum_i x^T v_i^T v_i x - 1.01^3 k n^{\frac{3}{2}}
\end{align*}
and thus,
\begin{align*}
\sigma_\ell^2(P) &\geq 0.99^2 n \max_{H:\dim H = \ell} \min_{\substack{x\in H\\ \|x\|_2=1}} \sum_i x^T v_i^T v_i x - 1.01^3 k n^{\frac{3}{2}}\\
&= 0.99^2 n\sigma_\ell^2(V) - 1.01^3 k n^{\frac{3}{2}},
\end{align*}
where $V$ is a $k\times n$ matrix with rows $v_1^T,\dots, v_k^T$. Therefore
\[
\|P\|_1 \geq 0.99\sqrt{n}\|V\|_1 - 1.01^{\frac32} s^{\frac32} n^{\frac34}.
\]
Since $V$ is a Gaussian random matrix, the classical results imply that $\|V\|_1\geq 0.99s\sqrt{n}$ with high probability \cite{tao}. The claim follows from our assumption on $s$.
\end{proof}
\section{Conclusion}
We have presented a simple, surprisingly
powerful new analysis which gives optimal bounds
on the sketching dimension for a number of previously studied sketching problems,
including approximating
the operator norm, Schatten norms, and subspace embeddings. We have also presented
the first lower bounds for estimating Ky Fan norms.
It would be interesting to see if there are other applications of this method to
the theory of linear sketches.
\end{document} |
\begin{document}
\title{Corrado Segre and nodal cubic threefolds}
\author{Igor Dolgachev}
\begin{abstract} We discuss the work of Corrado Segre on nodal cubic hypersurfaces with emphasis on the cases of 6-nodal and 10-nodal cubics. In particular we discuss the Fano surface of lines and conic bundle structures on such threefolds. We review some of the modern research in algebraic geometry related to Segre's work.
\end{abstract}
\address{Department of Mathematics, University of Michigan, 525 E. University Av., Ann Arbor, Mi, 49109, USA}
\email{[email protected]}
\maketitle
\text{ss}ction{Introduction}
The following is a detailed exposition of my talk devoted to two memoirs of Corrado Segre on irreducible cubic hypersurfaces in $\mathbb{P}^4$ with $d$ ordinary double points (nodes) \cite{Segre1}, \cite{Segre2}. We will show in this article how Segre's work is related to the current research in algebraic geometry.
Let $X$ be an irreducible cubic hypersurface in $\mathbb{P}^4$ with a node $q$. Choose a hyperplane $H$ that intersects $X$ transversally along a nonsingular cubic surface $F$ and consider the projective coordinates in $\mathbb{P}^4$ such that $H = V(t_0)$ and $q$ is equal to the point $[1,0,0,0,0]$. Then the equation of $X$ can be written in the form
\begin{equation}\langlebel{cubic1}
X:t_0a_2(t_1,t_2,t_3,t_4)+a_3(t_1,t_2,t_3,t_4) = 0,
\end{equation}
where $a_2$ and $a_3$ are homogeneous forms of degrees 2 and 3, respectively, such that $Q = V(a_2)$ is a nonsingular quadric surface in $H$ and $F = V(a_3)$. The equations
$a_2 = a_3= 0$ define a curve of degree 6 in the
hyperplane $H$. Following \cite{Fink1} we call it the \emph{associated curve} of $X$ with respect to $q$, it will be denoted by $C(X,q)$. The curve $C(X,q)$ is
a curve of bidegree $(3,3)$ lying on $Q$. We will assume that it is reduced. Let $X'$ be the proper transform of $X$ under the blow-up of $\mathbb{P}^4$ at the point $q$. Then the projection map $\textrm{pr}_q:X'\to \mathbb{P}^3$ defines an isomorphism between $X'$ and the blow-up of $\mathbb{P}^3$ with center at $C(X,q)$.
The inverse rational map
$$\alpha:\mathbb{P}^3\dasharrowsharrow \mathbb{P}^4$$
is given by the linear system of cubic containing $C(X,q)$. The
latter is spanned by the cubic $F = V(a_3)$ and any cubic of the form $V(a_2l)$, where $l$ is a linear form in $t_1,\ldots,t_4$. It follows that the rational map $\alpha$ is given by the formula
$(t_1,\ldots,t_4) \mapsto (a_3,t_1a_2,t_2a_2,t_3a_2,t_4a_2)$, and hence $Q$
is contracted to the point $q = [1,0,0,0,0]$. Also, it is clear that any singular point $q'\ne q$ of $X$ is projected to a singular point of $C(X,q)$. Since we assume that all singular points are ordinary double points, their images are ordinary double points of $C(X,q)$.
The arithmetical genus of a curve of bidegree $(3,3)$ on $Q$ is equal to 4. If the curve has more than four double points, it must be reducible. A simple analysis
shows that the largest possible number $k$ of double points of $C(X,q)$ is equal to $9$. Moreover, $k = 9$ happens if and only if $C(X,q)$ is the union of 6 lines on $Q$. This gives the following.
\begin{proposition} The number $d = k+1$ of ordinary double points of an irreducible cubic hypersurface in $\mathbb{P}^4$ is less than or equal to 10.
\end{proposition}
It follows from the proof of the previous proposition that the curve $C(X,q)$ is reducible if $d > 5$. The number of its irreducible components is equal to the fourth Betti number $b_4(X)$.\footnote{This fact was essentially known to G. Fano, see \cite{Fano1}.}
The number $b_4(X)-1$ is called the \emph{defect} $\textrm{def}(X)$ of $X$. The maximal number of
linearly independent homology classes of exceptional curves in any small resolution of $X$ is equal to $d-\textrm{def}(X)$ (see \cite{FW}). Note that a small resolution of $X$ may not be a projective variety. The number of projective small resolutions of a nodal cubic threefold $X$ can be found in \cite{FW}.
In this article we will restrict ourselves with two most interesting, in my view, cases when $d = 10$ or $d = 6$. We will start with the case $d = 10$,
I am grateful to the referee and Alessandro Verra for many useful remarks that allowed me to improve the exposition of this paper.
\text{ss}ction{Segre 10-nodal cubic primer}\langlebel{S:2}
By a projective transformation, we can fix the equation $a_2 = 0$ of the quadric $Q$ given in \eqref{cubic1}. The curve $C(X,q)$ has 9 singular points, this forces it to be the union of 9 lines on $Q$ as in the following picture:
\xy (-60,10)*{};(-30,-30)*{};
(0,0)*{};(30,0)*{}**\dir{-};
(0,-10)*{};(30,-10)*{}**\dir{-};
(0,-20)*{};(30,-20)*{}**\dir{-};
(5,5)*{};(5,-25)*{}**\dir{-};
(15,5)*{};(15,-25)*{}**\dir{-};
(25,5)*{};(25,-25)*{}**\dir{-};
(-10,-10)*{C(X,q):};
\endxy
Also, by an automorphism of the quadric, we can fix the curve $C(X,q)$. Two cubic forms defining $F= 0$ and $F' = 0$ that cut out $C(X,q)$ in $Q$, differ by $a_2l$, where $l$ is a linear form in $t_1,\ldots,t_4$. Applying the transformation
$t_0\mapsto t_0+l$, we fix the equation $a_3 = 0$ of the cubic. This shows that
\begin{proposition}
Two 10-nodal cubic hypersurfaces in $\mathbb{P}^4$ are projectively isomorphic.
\end{proposition}
We choose one representative of the isomorphism class and denote it by $\mathsf{S}_3$. Any cubic threefold isomorphic to $\mathsf{S}_3$ is called a \emph{Segre cubic primal}. We refer to \cite{CAG}, 9.4.4 for many beautiful classical facts about such threefolds. Here we add some more.
The polar quadric $V(\frac{\partial}{\partial t_0})$ of $X$ at a node $q$ cuts out in $X$ a surface of degree 6 with a point of multiplicity 4 at $q$ which is projected isomorphically outside $q$ onto the quadric $V(a_2)$ and passes through the nodes. This shows that the pre-image of each line component of $C(X,q)$ is a plane in $\mathsf{S}_3$ containing 4 nodes. Also it shows that no three nodes are collinear. Note that a cubic threefold containing a plane can be written in appropriate projective coordinates by equation $xQ_1+yQ_2 = 0$, where $Q_1$ and $Q_2$ are quadratic forms and the plane is given by $x = y = 0$. It follows immediately that the points given by $x= y=Q_1 =Q_2 = 0$ are singular points of the threefold. Their number is less than or equal to 4. Thus we obtain that each node is contained in 6 planes and each plane contains 4 nodes. This gives the following.
\begin{proposition} The Segre cubic primal $\mathsf{S}_3$ contains 10 nodes and 15 planes that form an abstract configuration $(15_4,10_6).$
\end{proposition}
This synthetic argument belongs to Segre. Guido Castelnuovo \cite{Cast} and later, but independently, H. W. Richmond \cite{Richmond}, were able to find the following
$\mathfrak{S}_6$-symmetric equations of the Segre cubic (see \cite{CAG}, 9.4.4).
\begin{equation}\langlebel{segre2}
\sum_{i=0}^5t_i^3 = \sum_{i=0}^5t_i = 0.
\end{equation}
It is checked that the singular points form the $\mathfrak{S}_6$-orbit of the point
$[1,1,1,-1,-1,-1]$ and the planes form the $\mathfrak{S}_6$-orbit of the plane
$t_0+t_1 = 0, t_2+t_3= 0, t_4+t_5 = 0.$
In his paper Castelnuovo studies linear systems of complexes of lines in $\mathbb{P}^4$, i.e. linear systems of hyperpane sections of the Grassmannian variety $G_1(\mathbb{P}^4)$ in its
Pl\"ucker embedding in $\mathbb{P}^9$ (see \cite{CAG}, 10.2). A 3-dimensional linear system of such complexes defines a rational map
$$f:\mathbb{P}^3\dasharrowsharrow \mathbb{P}^4$$
which is given by pfaffians of principal $4\times 4$-matrices of a skew-symmetric matrix of size $5\times 5$ whose entries are linear forms in 4 variables.
There are five points of indeterminacy $q_1,\ldots,q_5$ of this map corresponding to skew-symmetric matrices of rank 2. The linear system defining the map $
f$ is equal to the linear system $|2h-q_1-\cdots-q_5|$ of quadrics passing through the points $q_1,\ldots,q_5$, where $h$ is the class of a plane in $\mathbb{P}^3$. If one chooses projective coordinates such that
the points $q_i$ become $[1,0,0,0],\ldots,[0,0,0,1], [1,1,1,1]$, then the quadrics from the linear system acquire equations of the following type
$$\sum_{0\le i<j\le 3}a_{ij}u_iu_j = 0,$$
where $\sum a_{ij} = 0.$
The 5-dimensional vector space of such quadrics form the fundamental irreducible representation of the group $\mathfrak{S}_6$. This shows that the image of the
map $f$ admits a $\mathfrak{S}_6$-symmetry. Castelnuovo and Richmond find a special basis in the linear system of quadrics to show that the image of the map $f$ can be given by the equations \eqref{segre2}. The images of the 10 lines
$\overline{q_i,q_j}$ are the ten nodes of the cubic. The images $\Pi_{i,j,k}$ of the ten planes $\overline{q_i,q_j,q_k}$
and the images $\Pi_i$ of the five exceptional divisors $E_i$ blown-up from the points $q_i$ are the 15 planes of the cubic.
In an earlier work of P. Joubert \cite{Joubert}, the equations \eqref{segre2} appear as the relations between
certain six polynomials in roots of a general equation of degree 6. It shows that $\mathsf{S}_3$ should be considered as
the set of ordered sets of 6 points in $\mathbb{P}^1$ modulo projective equivalence. Later A. Coble \cite{Coble1} made it more
precise by proving that the geometric invariant quotient of $(\mathbb{P}^1)^6$ by the group $\textrm{SL}(2)$ is isomorphic
to the Segre cubic
$$\mathsf{S}_3\cong \mathsf{P}_1^6:= (\mathbb{P}^1)^6/\!/\textrm{SL}_2.$$
The rational map $f:\mathbb{P}^3\dasharrow \mathsf{S}_3$ can be extended to a regular map $\tilde{f}:X\to \mathsf{S}_3$, where $X\to \mathbb{P}^3$ is the
composition of the blow-up of the five points $q_1,\ldots,q_5$ and the blow-up of the proper transforms of the lines
$\overline{q_i,q_j}$.
\begin{equation}\langlebel{kap1}
\xymatrix{\bar{M}_{0,6}\ar[dr]\ar[rr]&&\mathbb{P}^3\ar[dl]^f\\
&\mathsf{S}_3&}.
\end{equation}
In modern times, the variety $X$ appears as the special case of M. Kapranov's realization of the Knudsen-Mumford moduli space $\overline{\mathcal{M}}_{0,6}$ of stable rational curves with $6$ marked ordered points \cite{Kapranov}. For any $n\ge 4$, one chooses $n-1$ points in general linear position in $\mathbb{P}^{n-3}$, then start blowing up the points, then the proper transforms of lines joining two points, then proper transforms of planes joining 3 points, and so on. The result is isomorphic to the Knudsen-Mumford moduli space $\overline{\mathcal{M}}_{0,n}$ of stable rational curves with $n$ marked ordered points. If one chooses a general point $q$ in $\mathbb{P}^n$ and passes through it and the points $q_1,\ldots,q_{n+2}$ the unique normal rational
curve $C(q)$ of degree $n$, then one finds $n$ points on $C(q)\cong \mathbb{P}^1$, namely, the points $q_1,\ldots,q_n$ and the point $q$.
It follows from Kapranov's realization of the variety $\overline{M}_{0,n}$ as a Chow quotient that we have the following commutative triangle of regular maps
$$
\xymatrix{\bar{M}_{0,n}\ar[rr]^{\pi}\ar[dr]&&\mathbb{P}^{n-3}\ar[dl]_{f_n}\\
&\mathsf{P}_1^{n}:= (\mathbb{P}^1)^n/\!/\textrm{SL}(2)&},
$$
If $n = 6$, the image of $f_n$ is the Segre cubic $\mathsf{S}_3$.
In the case when $n = 2g+2$ is even, the geometric invariant theory quotient
$\mathsf{P}_1^{2g+2}:= (\mathbb{P}^1)^{2g+2}/\!/\textrm{SL}(2)$ is isomorphic to a compactification of the moduli space of hyperelliptic curves of genus $g$ together with a full 2-level structure, i.e. a choice of a standard symplectic basis in the group of 2-torsion divisor classes. In \cite{Coble2} A. Coble shows that the map $f_{2g+2}:\mathbb{P}^{2g-1}\dasharrow \mathsf{P}_1^{2g+2}$ is given by the linear system
\begin{equation}\langlebel{coble3}
|gh-(g-1)(q_1+\cdots+q_{2g+1})|.
\end{equation}
It maps $\mathbb{P}^{2g-1}$ to the projective space $\mathbb{P}^{N-1}$, where $N = \binom{2g}{g}-\binom{2g}{2g-2}$.
The image of this
map is isomorphic to $\mathsf{P}_1^{2g+2}$.
Also, Coble shows that the projection of $\mathsf{P}_1^{2g+2}$ from a general point $p$ defines a degree 2 map onto the Kummer variety
$\textrm{Kum}(\textrm{Jac}(C_p))$ associated with the Jacobian variety of the hyperelliptic curve $C_p$ corresponding to the point $p$.
It lies in $\mathbb{P}^{2^g-1}$ and embeds there by the map $\textrm{Jac}(C_p)\to \textrm{Kum}(\textrm{Jac}(C_p)) \to \mathbb{P}^{2^g-1}$ given by the linear system $|2\Theta|$, where $\Theta$ is the theta divisor of the Jacobian. The locus of singular points of the hypersurfaces from the linear system $|gh-(g-1)(q_1+\cdots+q_{2g+1})-q|$, where $f_{2g+2}(q) = p$, is the \emph{Weddle variety} $W_g$. It maps birationally onto the Kummer variety of $C_p$. We refer for a modern exposition of Coble's results to \cite{Dolg} and C. Kumar's paper \cite{Kumar}. Kumar calls the variety
$\mathsf{P}_1^{2g+2}$ the \emph{generalized Segre variety}. One finds in Kumar's paper a nice relationship between the generalized Segre variety and the theory of vector bundles on hyperelliptic curves.
Note that Segre himself was aware of the relationship between the cubic $\mathsf{S}_3$ and the Kummer quartic surface associated to curves of genus 2. In fact, he shows that the projection of $\mathsf{S}_3$ to $\mathbb{P}^3$ from its nonsingular point is the quartic Kummer surface $\textrm{Kum}(\textrm{Jac}(C_p)))$. He also shows that the set of nodes of quadrics in $\mathbb{P}^3$ passing through the points $q_1,\ldots,q_5$ and the additional point $q$ is the Weddle quartic surface.
A nice relationship between the generalized Segre variety $\mathsf{P}_1^{2g+2}$ and the theory of stable rank 2 vector bundles on not-necessary hyperelliptic curves was recently studied by A. Alzati and M. Bolognesi \cite{Alzati}.
Let
$C_g$ be a non-hyperelliptic smooth projective curve of genus $g\ge 2$ and let $\textrm{SU}_{C_g}(2)$ be the moduli space of semi-stable rank 2 bundles on $C_g$ with trivial determinant. Alzati and Bolognesi prove that there exists a rational map
$$\textrm{SU}_{C_g}(2)\dasharrowsharrow \mathbb{P}^g$$
whose fibers are birationally isomorphic to $\mathsf{P}_1^{2g+2}$. If $g = 3$, they are isomorphic to $\mathsf{S}_3$.
Finally note that, according to \cite{Fink1}, the Segre cubic primal has 1024 small resolutions with 13 isomorphism classes, among them are 332 projective varieties which are divided into 6 isomorphism classes. Since $C(\mathsf{S}_3,q)$ has 6 irreducible components, the rank of the Picard group of a projective small resolution is equal to 5.
\text{ss}ction{Cubic threefolds with 6 nodes}
Let $X$ be a cubic hypersurface in $\mathbb{P}^4$ with six nodes. Choosing one of the nodes of $X$, we can get the equation of $X$ as in \eqref{cubic1}. Let $Q = V(a_2)$ and $F = V(a_3)$ be the quadric and a cubic surface in the hyperplane $t_0 = 0$ defined by the coefficients $a_2$ and $a_3$ of the equation.
\begin{proposition}\langlebel{nondeg} Assume that any five of the nodes of $X$ span $\mathbb{P}^4$. Then $C(X,q) = Q\cap F$ is the union of two rational curves of degree 3 intersecting transversally at 5 points.
\end{proposition}
\begin{proof} We know that remaining 5 nodes of $X$ are projected to the double points $p_1,\ldots,p_5$ of the curve
$C(X,q)$. By assumption, any four of them span $\mathbb{P}^3$. It implies that no four of the points $p_i$ lie on a conic or on a line contained in $Q$, and hence the curve $C(X,q)$ has no irreducible components of degree $\le 2$. Then an easy computation with the formula for the arithmetic genus of $C$ gives that $C$ consists of two components $\gamma$ and $\gamma'$ of degree 3. Since the quadric $Q$ is nonsingular (otherwise $q$ is not an ordinary double point), the curves are curves of types $(2,1)$ and $(1,2)$ intersecting at the points $p_1,\ldots,p_5$. These points are the projections of the five nodes of $X$ and if two of them coincide, together with $q$, they would lie on a line. This contradicts our assumption.
\end{proof}
The following example shows that the condition on the six points is necessary for $C(X,q)$ to be the union of two curves of degree 3.
\begin{example} Let $Q = V(a_2)$ be a nonsingular quadric and $C$ be a curve on $Q$ equal to the union of a nonsingular conic $C_1$ and an irreducible curve $C_2$ of type $(2,2)$ with a node. Let $H= V(l)$ be a plane intersecting $Q$ along $C_1$ and let $Q' = V(q)$ be a quadric intersecting $Q$ along $C_2$. Pick up a general plane $H' = V(l')$ such that $F = V(a_2l'+ql)$ is a nonsingular cubic. Then the cubic threefold given by equation \eqref{cubic1} with $a_2,a_3$ as above, has 6 nodes with 5 of them spanning $\mathbb{P}^3$ (four of them are projected to the intersection points $C_1\cap C_2$).
Let $X$ be any 6-nodal cubic such that five of its nodes $q_1,\ldots,q_5$ span a hyperplane $H$. The intersection $H\cap X$ is a cubic surface with 5 nodes, hence it must be reducible. It is easy to see that this implies that four of the nodes are coplanar. In particular, any five nodes of $X$ span a hyperplane.
\end{example}
We say that a 6-nodal cubic threefold is \emph{nondegenerate} if any subset of five nodes span $\mathbb{P}^4$. Thus Proposition \ref{nondeg} applies, and we obtain that the projection from any node gives an equation \eqref{cubic1}, where $V(a_2)\cap V(a_3)$ is the union of two rational cubics intersecting at 5 points.
Note that any 5-nodal complete intersection of a quadric and a cubic surfaces in $\mathbb{P}^3$ defines a 6-nodal cubic threefold, and the case from Proposition \ref{nondeg} is the only case which gives a nondegenerate 6-nodal cubic threefold.
We shall use the following well-known fact about cubic surfaces several times, so better let us record it.
\begin{lemma}\langlebel{fact} Let $F$ be a smooth cubic surface and $\gamma_1,\gamma_2$ be two rational smooth curves of degree 3 on $F$ such that $\gamma_1+\gamma_2\in |-2K_F|$. Then the set of 27 lines on $F$ is divided into three disjoint sets:
\begin{itemize}
\item 6 skew lines that do not intersect $\gamma_1$ but intersect $\gamma_2$ with multiplicity 2,
\item 6 skew lines that do not intersect $\gamma_2$ but intersect $\gamma_1$ with multiplicity 2,
\item 15 lines that intersect both $\gamma_1$ and $\gamma_2$ at one point.
\end{itemize}
\end{lemma}
\begin{proof} We have $\gamma_i\cdot K_F = -3$, hence $\gamma_i^2 = 1$ and the linear system $|\gamma_i|$ defines a birational morphism $\pi_i:F\to \mathbb{P}^2$. Each of these morphisms blows down six lines that together form a double-six. In the standard (geometric) basis
$(e_0,e_1,\ldots,e_6)$ in $\text{Pic}(F)$ defined by $\pi_1$, we have $\gamma_1\sim e_0$ and $\gamma_2\in |5e_0-2(e_1+\cdots+e_6)|$. The double-six is represented by the classes $e_i$ and
$2e_0-(e_1+\cdots+e_6)+e_i, i = 1,\ldots,6$. Each line in the class $e_i$ is blown down by $\pi_1$ and intersects $\gamma_2$ with multiplicity 2. The remaining lines are represented by the classes $e_0-e_i-e_j$ which intersect $\gamma_1$ and $\gamma_2$ with multiplicity 1.
\end{proof}
Let $\textrm{Bis}(C_i)$ be the surface of bisecant lines of $C_i$. It is naturally isomorphic to the symmetric product of $C_i$, and hence to $\mathbb{P}^2$. In the Pl\"ucker embedding of the Grassmann variety $G_1(\mathbb{P}^3) \subset \mathbb{P}^5$, it is isomorphic to a Veronese surface. Let $\ell_1,\ldots,\ell_6$ (resp. $\ell_1',\ldots,\ell_6$) be the set of lines in $\textrm{Bis}(C_1)$ (resp. $\textrm{Bis}(C_2)$) corresponding to a double-six of lines on $F$ via the previous lemma. For any $x\in F$,
there exists a unique bisecant line of $C_1$ (resp. $C_2$) passing through $x$. This defines two maps $\pi_1:F\to \textrm{Bis}(C_1), \pi_2:F\to \textrm{Bis}(C_2)$ that blows down the lines
$\ell_1,\ldots,\ell_6$ (resp. $\ell_1',\ldots,\ell_6'$).
\begin{proposition}\langlebel{nonsing} Let $X$ be a nondegenerate 6-nodal cubic hypersurface and let $C(X,q) = Q\cap F$ be the fundamental curve associated to $q$. Let $S$ be the blow-up of $Q$ at the five nodes $q_1,\ldots,q_5$ of $C(X,q)$. Then $S$ is isomorphic to a nonsingular cubic surface.
\end{proposition}
\begin{proof} The linear system of curves $|D| = |\mathcal{O}_Q(2)-p_1-\cdots-p_5|$ of bidegree $(2,2)$ containing the five nodes $p_1,\ldots,p_5$ of $C(X,q)$ is 3-dimensional and defines a regular map $f:S\to S'$ onto a cubic surface $S' \subset |D|^* \cong \mathbb{P}^3$. Since $X$ is nondegenerate, the images of the 5 exceptional curves $E(q_i)$, the 10 lines each passing through one of the points $p_i$ (obviously, no two of them lie on one line), 10 conics passing through three of the points $q_i$, and the curves $C_1,C_2$ are the 27 lines on $S'$. This implies that $S'\cong S$ is a nonsingular cubic surface.
\end{proof}
Let is add some remarks to the previous construction. The lines $\ell_1 = f(C_1)$ and $\ell_2 = f(C_2)$ are skew lines on $S'$, and the images of the exceptional curves $E(q_i)$ is the set of 5 skew lines that intersect
both $\ell_1$ and $\ell_2$. The composition
$f' = \sigma\circ f^{-1}:S'\to Q \cong \mathbb{P}^1\times \mathbb{P}^1$ is given by the two pencils of conics cut out by the pencils of planes $\mathcal{P}_1,\mathcal{P}_2$ in $\mathbb{P}^3$ containing $\ell_1,\ell_2$, respectively. This map factors through an isomorphism $f^{-1}:S'\to S$ which is inverse of the morphism $f:S\to S'$.
For any $x\in S'$ there exists a unique line in $\mathbb{P}^3$ intersecting $\ell_1$ and $\ell_2$. It is obvious, if $x\not\in \ell_1\cup \ell_2$. If $x\in \ell_1$, we choose the line contained in the plane spanned by $\ell_2$ and $x$ and in the plane containing $\ell_1$ and tangent to $S'$ at $x$. Assigning to $x$ the intersection points of this line with $\ell_1$ and $\ell_2$, we obtain that the surface $S'$, and hence $S$, is isomorphic to the irreducible surface $S''$ in $G_1(\mathbb{P}^3)$ of lines in $\mathbb{P}^3$ that intersects both components of $C(X,q)$ The lines passing through the singular points of $C(X,q)$ must be tangent to the quadric at these points. Such lines are 5 lines on $S''$ that correspond to the skew lines intersecting $\ell_1$ and $\ell_2$.
Segre proves that any nondegenerate 6-nodal cubic hypersurface can be projectively generated. This means that one can find three projectively equivalent nets of hyperplanes.
$$H(\langlembda)_j:= \sum_{i=0}^4a_i^{(j)}(\langlembda)t_i = 0, \quad j = 0,1,2,$$
such that
$$X = \{x\in \mathbb{P}^4:x\in H_1(\langlembda)\cap H_2(\langlembda)\cap H_3(\langlembda)\ \textrm{for some $\langlembda$}\}.$$
Let us rewrite these equations in the following form
$$\langlembda_0l_{0j}(t)+\langlembda_1l_{1j}(t)+\langlembda_2l_{2j}(x) = 0, \ j = 0,1,2,$$
where $l_{ij}(t)$ are linear forms in variables $t_0,t_1,t_2,t_3,t_4$. Then
\begin{equation}\langlebel{det1}
X = \{x\in \mathbb{P}^4:\det (l_{ij}(x)) = 0\}.
\end{equation}
The six nodes of $X$ are the points $x$ such that $\ranglenk (l_{ij}(x)) = 1.$
We see from formula \eqref{det1} that a projective generation gives a determinantal representation of $X$. Conversely, the determinantal representation defines a projective generation.
Segre's proof is rather cumbersome and I had a difficulty to follow it. A modern proof was given by B. Hassett and Yu. Tschinkel \cite{Hassett}. They deduce a determinantal representation of $X$ from a determinantal representation of a certain cubic surface associated to $X$. Let us reproduce a modified version of their proof that, in my opinion, is more straightforward and constructive.
\begin{theorem} Let $X$ be a nondegenerate 6-nodal cubic hypersurface in $\mathbb{P}^4$. Then $X$ is isomorphic to the hypersurface $V(\det(A))$, where $A$ is a $3\times 3$-matrix with linear form in coordinates on $\mathbb{P}^4$.
\end{theorem}
\begin{proof}
A normal cubic surface has only double rational points as its singularities. Each cubic surface without singular point of type $E_6$ is determinantal. This means that there is an embedding of $\mathbb{P}^3$ in the projective space $\mathbb{P}^8$ of $3\times 3$ matrices such that the pre-image of the determinantal cubic hypersurface $\mathsf{D}_3$ in $\mathbb{P}^8$ is equal to $X$.
This was proved first by L. Cremona in 1868 (with a gap related to the
assumption on the singularities). C. Segre had filled the gap in 1906. We refer to \cite{CAG}, 9.3 for the details.
So, our cubic surface $F = V(a_3)$, being nonsingular by our choice of projective coordinates, admits a determinantal representation. Let us recall its construction. We assume that $F$ is a smooth cubic surface in the projective space $|W|$ of lines in a
linear vector space $W$ of dimension 4. A determinantal equation of $F$ is defined by a choice of a linear system $|\gamma_1|$ of curves of degree 3 with $\gamma_1^2 = 1$. Let $|\gamma_2| = |-2K_F-\gamma_1|$ be represented by a smooth rational curve $\gamma_2$ with $\gamma_2^2 = 1$. The pair of smooth curves $(\gamma_1,\gamma_2)$ is a pair from Lemma \ref{fact}. Let $\pi_i:F\to
|\gamma_i|^*$ be the corresponding birational morphisms. The birational map
$\pi_2\circ \pi_1^{-1}:|\gamma_1|^*\dasharrowsharrow |\gamma_2|^*$ is defined by the linear system of
curves of degree 5 with double points at the points $p_i = \pi_1(\ell_i)$, where $(\ell_1,\ldots,\ell_6)$ is the sixer of lines blown down by $\pi_1$. Consider the natural map defined by adding the divisors
\begin{equation}\langlebel{schur}
|\gamma_1|\times |\gamma_2| \to |6e_0-2e_1-\cdots-2e_6| = |-2K_F| \cong |\mathcal{O}_{|W|}(2)|.
\end{equation}
The image of this map is a hyperplane in $|\mathcal{O}_{|W|}(2)|$ orthogonal to a quadric $\mathcal{Q}^*$ in the dual space $|W^\vee|$. It is the dual quadric of the Schur quadric $\mathcal{Q}$ associated to the double-six of lines blown-down by $\pi_1$ and $\pi_2$ (see \cite{CAG}, 9.1.3). Composing \eqref{schur} with a linear function defined by $\mathcal{Q}^*$, we can identify the plane $|\gamma_2|$ with the plane $|\gamma_1|^*$, the dual plane of $|\gamma_1|$. Let $|\gamma_1| = |U|$ for some 3-dimensional linear space $U$. Then the map
$\pi_1\times \pi_2:F\to |\gamma_1|^*\times |\gamma_2|^*$ can be identified with the linear map
$$j:F\to |U^*|\times |U| \hookrightarrow |U^*\otimes U| = |\text{End}(U)| \cong \mathbb{P}^8.$$
Let $\mathsf{D}_3$ be the determinantal hypersurface in $|\text{End}(U)|$. It is a cubic hypersurface with singular locus equal to $|U^*|\times |U|$ (it parameterizes endomorphisms of rank 1). The determinantal representation of $F$ is defined by the embedding $|W|\hookrightarrow |\text{End}(U^*)| \cong |\text{End}(U)|^*$ such that the pre-image of the determinantal hypersurface is equal to $S$ and the map $\pi_1$ (resp. $\pi_2$) is defined by taking the kernel of the corresponding endomorphism (resp. its transpose) of $U^*$.
All of this is well-known and can be found in \cite{CAG}.
Now let $X$ be a nondegenerate $6$-nodal cubic hypersurface given by equation \eqref{cubic1} and $C_1+C_2 = C(X,q)$ be the associated curve with respect to a node $q$. It follows from the above discussion that the linear systems $|C_1|$ and $|C_2|$ define two determinantal representation of $F$, each is obtained from another by taking the transpose of the matrix.
Choose a basis in the linear space $U = H^0(F,\mathcal{O}_F(\gamma_1))$ and the dual basis in $U^*$, then we can identify $\text{End}(U)$ with the space of $3\times 3$-matrices and the determinantal representation of $F$ gives a matrix $B = (b_{ij})$ whose entries are linear forms in $t_1,\ldots,t_4$ such that
$$F = V(\det B).$$
We are looking for a matrix $\tilde{B} = (\tilde{b}_{ij})$ whose entries are linear forms in $t_0,\ldots,t_4$ such that
$$X = V(\det \tilde{B}).$$
If we plug in $t_0 = 0$ in the entries of $\tilde{B}$, we should obtain a matrix equal to $B$, up to a scalar multiple. This shows that
$\tilde{B} = t_0A+B$, where $A$ is a constant matrix. Write $A = [A_1 A_2 A_3]$ and $B = [B_1 B_2 B_3]$ as the collection of its columns. The usual formula for the determinant of the sum of the matrices shows that
\begin{equation}\langlebel{matrix}
\det \tilde{B} = t_0^3\det A+t_0^2 (\det [B_1 A_2 A_3]+\ldots)+t_0(\det [B_1 B_2 A_3]+\ldots)+\det B.
\end{equation}
To make this expression equal to $t_0a_2+a_3$ from \eqref{cubic1}, we have to take $A$ with rank equal to 1. So we may assume that the columns of $A$ are equal to some nonzero vector $\mathbf{v} = (\alpha_0,\alpha_1,\alpha_2)$.
Let $\pi_1:F\to \mathbb{P}^2 = |U|$ and $\pi_2:F\to |U^*|$ be the two maps defined by the right and the left kernels of the matrix $B$. Let $\ell = \pi(\gamma)$ and $\ell' = \pi'(\gamma')$. The curve $\ell$ is a line in $\mathbb{P}^2$, the curve $\ell'$ is a line in the dual plane. Observe that, for any $y = [t_1,\ldots,t_4]\in F$, the adjugate matrix $\text{adj} B$ of $B$ is of rank 1. Thus, for any $x\in F$, the equations $\det [B_1(x) B_2(x) \mathbf{v}] = 0,
\det [B_1(x) \mathbf{v} B_3(x)] = 0, \det [\mathbf{v} B_2(x) B_3(x)] = 0$ with unknown vector $\mathbf{v}$, are the equations of the same line $\ell(x)$ in $\mathbb{P}^2$ which we consider as a point in the dual plane. When $x$ runs $C_1$, the image $\pi_1(C_1)$ is a line in the plane, hence the set of lines $\ell(x), x\in C_1,$ is a line in the dual plane. If we take it to be equal to the line equal to $\pi_2(C_2)$, we obtain that the coefficient at $t_0$ in \eqref{matrix} is equal to zero for any $x\in C_1\cup C_2$. Thus the quadric $Q'$ defined by this coefficient coincides with the quadric $Q$, and we are done. \end{proof}
\begin{remark}\langlebel{R3.6} Suppose equation \eqref{cubic1} of a nodal cubic threefold can be brought to the form $\det A(t) = 0$. Then, plugging in $t_0 = 0$, we obtain a determinantal representation of the cubic surface $F = V(a_3)$. This shows that $F$ has at most rational double points of type different from $E_6$. Also, since the discriminant variety $\mathsf{D}_3$ has the double locus of degree 6, we obtain that the singular locus of $X$ is either of dimension $\ge 1$, or consists of isolated singular points whose Milnor numbers add up to 6.
\end{remark}
Let $X$ be a nondegenerate 6-nodal cubic threefold with nodes $q_1,\ldots,q_6$. The linear system of cubics $|\mathcal{O}_{\mathbb{P}^4}(3)-2q_1-\cdots-2q_6|$ with double points at $q_1,\ldots,q_6$ defines a rational map $f:\mathbb{P}^4\dasharrowsharrow \mathsf{S}_3$ to the Segre cubic primal $\mathsf{S}_3$ in $\mathbb{P}^4$. Its fibers are quartic rational normal curves passing through the nodes. In Kapranov's realization of $\mathcal{M}_{0,7}$ this corresponds to the composition of the
projection $\mathcal{M}_{0,7}\to \mathcal{M}_{0,6}$ and the map $\mathcal{M}_{0,6}\to \mathsf{S}_3$ from \eqref{kap1}. This shows that $X$ is birationally isomorphic to the pre-image of a hyperplane section of $\mathsf{S}_3$. Let $X'$ be the blow-up of $X$ at the nodes, followed by the blow-up of the proper transforms of lines joining two nodes. Then $f$ extends to a regular map $X'\to S$, where $S$ is a hyperplane section of $\mathsf{S}_3$. If we use equation \eqref{segre2} of $\mathsf{S}_3$, then the additional equation
$\sum_{i=0}a_it_i = 0$ defines a cubic surface $S$ given by Cremona's hexahedral equations (see \cite{CAG}, 9.4.3). By Theorem 9.4.8 from loc.cit., if $S$ is nonsingular, these equations determine uniquely an ordered double-six of lines on $S$. Conversely, a choice of an ordered double-six of lines defines Cremona's hexahedral equations.
It is an obvious guess that the cubic surface $S$ is isomorphic to the cubic surface from Proposition \ref{nondeg}. To see this, we consider the blow-up $X'$ of $X$ at any of its singular point $q$. The exceptional divisor is identified with the quadric $Q_i$ containing $C(X,q_i)$. The pre-image of the linear system $|\mathcal{O}_{\mathbb{P}^4}(3)-2(q_1+\cdots+q_6)|$ to $X'$ restricted to the exceptional divisor $E(q)$ consists of quadrics through the 5 points on $Q_i$ corresponding to the lines joining $q_i$ with other nodes of $X$. It maps $E(q_i)$ to the hyperplane section of $\mathsf{S}_3$ corresponding to $X$.
We can easily see the double-six of lines on $S$ identified with the blow-up of $Q_i$ at the singular points of $C(X,q_i)$. It consists of the images of the ten lines on $Q$ passing through the singular points and the curves $C_1$ and $C_2$. The linear systems $|\gamma_1$ and $|\gamma_2|$ defining a determinantal representation of $S$ are the images of the curves of bi-degree $(3,1)$ and $(1,3)$ passing through the singular points of $C(X,q)$. The order on the 6 nodes of $X$ defines an order on the set of singular points of $C(X,q)$, hence an order of the lines in a sixer, hence a basis of the Picard group of the cubic surface $S$.
Finally, let us look at the moduli space of nondegenerate 6-nodal cubic hypersurfaces. By a projective transformation we can fix the nodes, to assume that their coordinates form the reference points in $\mathbb{P}^4$. Then the space of cubics with singular points at the reference points consists of cubics with equations
$$\sum_{0\le i < j<k \le 4}a_{ijk}t_it_jt_k = 0,$$
where the coefficients $a_{ijk}$ satisfy
$$\sum a_{0jk} = 0,\ \sum a_{i1k} = 0,\ldots, \sum a_{ij4} = 0.$$
The dimension of the projective space $|V|$ of such cubics is equal to 4. The permutation group $\mathfrak{S}_6$ acts linearly on $V$ via its natural $5$-dimensional irreducible permutation. By above, the linear system of such cubics map $\mathbb{P}^4$ onto the Segre cubic primal $\mathsf{S}_3$. Thus its dual space is identified with the space of hyperplane sections of $\mathsf{S}_3$. The action of $\mathfrak{S}_6$ agrees with the natural action of $\mathfrak{S}_6$ on $\mathsf{S}_3$. We know that the orbit of a hyperplane section of $\mathsf{S}_3$ corresponds to the moduli space of cubic surfaces together with an unordered double-six. It is the cover of degree 6 of the moduli space of cubic surface. It is known that such variety is rational (see \cite{Bauer}).
This gives us the following result.
\begin{theorem} The moduli space of nondegenerate 6-nodal cubic threefolds is naturally birationally isomorphic to the moduli space of cubic surfaces together with an unordered double-six of lines. It is a rational variety of dimension 4.
\end{theorem}
\begin{remark} Let $X \subset |W| \cong \mathbb{P}^4$ and $W\to \text{End}(U)$ be a linear map defined by a determinantal representation of $X$. Consider the nondegenerate bilinear form on $\text{End}(U)$ defined by the trace. It allows one to identify $\text{End}(U)$ with its dual space $\text{End}(U)^\vee$. The orthogonal space $W^\perp$ is of dimension 4, and the linear embedding $W^\perp \hookrightarrow \text{End}(U)$ defines a determinantal representation of the cubic surface $S' = |W^\perp|\cap \mathsf{D}_3$. It is proven in \cite{Hassett} that this cubic surface is isomorphic to the cubic surface $S$.
Conversely, starting with a determinantal representation of a nonsingular cubic $S \subset |V|$ defined by
a linear map $V\to \text{End}(U)$, we obtain a determinantal 6-nodal cubic $X$ by taking the intersection of $|V^\perp|$ with the determinantal hypersurface $\mathsf{D}_3$. This is the approach to determinantal representations of 6-nodal cubic threefolds taken in \cite{Hassett}. It is analogous to the earlier construction of A. Beauville and R. Donagi \cite{Beauville} that uses pfaffian hypersurface to pair K3 surfaces of genus 8 and 4-dimensional pfaffian cubic hypersurfaces.
\end{remark}
The number of small resolutions of a nondegenerate 6-nodal cubic threefold is equal to 64. Among them there are two projective resolutions \cite{FW}. Note that a degenerate irreducible 6-nodal cubic does not admit a small projective resolution.
\text{ss}ction{The Fano surface of lines}\langlebel{S:1}
Corrado Segre also studied the surface $F(\mathsf{S}_3)$ of lines in $\mathsf{S}_3$. Later on, Gino Fano wrote two papers which establish the basic facts about the surface of lines in any nodal cubic threefold \cite{Fano1}, \cite{Fano2}. For a modern exposition of some of Fano's result see \cite{Altman}). Here we remind some known facts about the Fano surface $F(X)$ of lines in a nodal cubic threefold, nowadays called the \emph{Fano surface} of $X$. In the Pl\"ucker embedding, the surface $F(X) \subset G_1(\mathbb{P}^4)$ is a locally complete intersection projectively normal surface of degree 45 canonically embedded in the Pl\"ucker space $\mathbb{P}^9$. Its class in $H^4(G_1(\mathbb{P}^4),\mathbb{Z})\cong \mathbb{Z}^2$ with respect to the basis given by the Schubert cycles $\sigma_1$ of lines intersecting a general line and $\sigma_2$ of lines in a general hyperplane is equal to $(18,27)$.
The Fano surface $F(X)$ is smooth at any point representing a line that does not pass through a singular point of $X$. The union of lines passing through a node $q\in X$ is projected from $q$ to the curve $C(X,q)$. In particular, $F(X)$ is a non-normal surface. If the number $d$ of nodes is less than or equal to 5, it is an irreducible surface birationally isomorphic to the surface of bisecant lines of the associated curve $C(X,q)$. Starting from $d = 6$, the curve $C$ become reducible, and $F(X)$ becomes reducible too.
Let us start with the Segre cubic primal $\mathsf{S}_3$.
Under the map $f:\mathbb{P}^3\dasharrowsharrow \mathsf{S}_3$ given by the linear system of quadrics through 5 points $p_1,\ldots,p_5$, the image of a line $\ell$ containing the point $p_i$ is a line in $\mathsf{S}_3$. The set $D_i$ of such lines is isomorphic to a del Pezzo surface of degree 5, the blow-up of 4 points in the plane (the points are of course the lines joining $p_i$ with $p_j, j\ne i$). This gives us five isomorphic irreducible components of $F(\mathsf{S}_3)$. The image of a twisted cubic passing through the points $p_1,\ldots,p_5$ is also a line in $\mathsf{S}_3$. The set of such lines is also isomorphic to a del Pezzo surface of degree 5. To see this, we use the Kapranov's realization of the projection map
$\overline{\mathcal{M}}_{0,6}\to \overline{\mathcal{M}}_{0,5}$. So we have 15 components isomorphic to $\mathbb{P}^2$, they are lines containing in a plane in $\mathsf{S}_3$. Each of these planes are the images of a conic in the plane $ \langle p_i,p_j,p_k\rangle$ through the points $p_i,p_j,p_k$ or the lines in one of the exceptional divisor $E_i$ over the point $q_i$.
Each line intersects five planes. For example, a line from $D_1$ intersects the plane corresponding to the exceptional divisor $E_i$ and four planes
$\langle p_a,p_b,p_c\rangle, 1\not\in \{a,b,c\}.$ So, we see that $F(\mathsf{S}_3)$ is highly reducible, it consists of 6 components isomorphic to a del Pezzo surface of degree 5, and 15 components isomorphic to the projective planes. In the Pl\"ucker embedding of $G_1(\mathbb{P}^4)$ they are anti-canonically embedded surfaces of degree 5, and the planes.
The group $\mathfrak{S}_6$ of automorphisms of $\mathsf{S}_3$ acts transitively on the set of 6 components $D_i$ of the Fano surface. The stabilizer subgroup is isomorphic to the group of automorphisms of $D_i$. It also acts transitively on the 15 plane components of the Fano surface. The stabilizer group is isomorphic to $2^3\rtimes \mathfrak{S}_3$ of order 48.
\begin{remark} The description of conic bundle structures on the Segre cubic primal can be also found in \cite{Gwena}.
\end{remark}
Next we consider the case when $X$ is a nondegenerate 6-nodal cubic threefold.
Since $X$ has 6 nodes, the Fano surface is singular along the curve parameterizing lines passing through a node. Fix a node $q$ of $X$ and let $C(X,q) = C_1+C_2 = Q\cap F$ be the associated curve of degree 6. Each line passing through $q$ is projected to a point on $C(X,q)$. Since $C(X,q)$ consists of two curves of degree 3, we see that the lines through $q$ are parameterized by $C(X,q)$ and their union consists of two cubic cones intersecting along 5 common lines.
Let $\textrm{Bis}(C)$ be the set of lines in $\mathbb{P}^3$ that intersect $C(X,q)$ at $\ge 2$ points counting with multiplicity. For any two points $x,y\in C(X,q)$, let $\ell = \overline{x,y}$ be the line spanned by $x,y$ or tangent to $Q = V(a_2)$ if $x =y$. Suppose $\ell$ intersects $C$ at two points $x,y$. The linear system of cubics through $C(X,q)$ maps $\mathbb{P}^3$ to $X$ and the image of $\ell$ is a line $\ell_{x,y}$ on $X$. If $\ell_x,\ell_y$ are the lines through $q$ which are projected to $x,y$, then $\ell_{x,y}$ is the residual line of the intersection of the plane spanned by $\ell_x,\ell_y$ with $X$ (if $x = y$ we take the tangent plane of $X$ along the line $\ell_x$). If $\ell$ intersects $C(X,q)$ at three point, then it must belong to one of the two rulings of $Q$. Let $C_i$ be the component of $C(X,q)$ such that this ruling intersects $C_i$ at one point $z$. Then we assign to it the line $\ell_z$ passing through $q$. For any line $\ell$ in $X$, let $\Pi(\ell)$ be the unique plane containing $q$ and cutting $X$ in 3 lines. If $q\not\in \ell$, then $\Pi(\ell)$ is spanned by $\ell$ and $q$. Otherwise, $\Pi(\ell)$ is the unique plane cutting $X$ along three lines passing through $q$. The projection of $\Pi(\ell)$ to $\mathbb{P}^3$ from the point $q$ is a line from $\textrm{Bis}(C(X,q))$. It is contained in $Q$ if $\Pi(\ell)\cap X$ consists of three lines passing through $q$. This defines an isomorphism
$$ \textrm{Bis}(C(X,q))\cong F(X).$$
Obviously, $\textrm{Bis}(C(X,q))$ consists of three irreducible components, two of which are isomorphic to $P_i = \textrm{Bis}(C_i)$. The third component $P_3$ consists of lines intersecting both $C_1$ and $C_2$.
For any general point in $\mathbb{P}^3$ there exists a unique bisecant line of a normal rational cubic. Since a general line intersects $X$ at three points, we see that the Schubert cycle $\sigma_1$ intersects the components $P_1$ and $P_2$ of $F(X)$ at three points. A general hyperplane $H$ in $\mathbb{P}^4$ intersects $X$ along a nonsingular cubic surface. It also intersects each of the cubic cones of lines passing through $q$ along a cubic rational curve $\gamma_i$. The curve $\gamma_i$ can be identified (under the projection map) with a component $C_i$ of $C(X,q)$. Using Lemma \ref{fact}, we see that this defines a set of 6 skew lines, each intersecting one of the curves $\gamma_i$ with multiplicity 2. These lines correspond to 6 bisecants of $C_i$. This shows that the bidegree of the surface $\textrm{Bis}(C_i)$ in $G_1(\mathbb{P}^4)$ is equal to $(3,6)$, it is isomorphic to $\mathbb{P}^2$ embedded by the third Veronese map.
The third component $P_3$ of $F(X)$ must be of bidegree $(12,15)$. As above, 15 corresponds to the 15 lines in a general hyperplane that intersect $\gamma_1$ and $\gamma_2$, and 12 corresponds to the fact that through each point $x$ on the intersection of a general line with $X$ passes 4 lines whose projection to $\mathbb{P}^3$ intersects $C_1$ and $C_2$ at one point.
It follows from Proposition \ref{nonsing} that $P_3$ is isomorphic to the cubic surface $S$ isomorphic to the blow-up of $Q$ at the singular points of $C(X,q)$. Thus we have a map
$$\phi:P_1\sqcup P_2\sqcup P_3 \cong \mathbb{P}^2\sqcup \mathbb{P}^2\sqcup S \to F(X)$$
that coincides with the normalization map.
Note that $P_1\cap P_2$ consists of 10 points corresponding to lines $\overline{q_i,q_j}$. The intersection $P_i\cap P_3$ consists of 5 lines corresponding to the lines through $q_i$ intersecting $C_i$ at some other point.
\begin{remark} It follows from the above that the isomorphism class of the cubic surface isomorphic to the blow-up of $Q$ at singular points of $C(X,q)$ is independent of a choice of a node.
\end{remark}
\begin{remark} The following nice observation is due to A. Verra. Take any line $\ell$ on $X$ and consider its image $f(\ell)$ under the map $f:\mathbb{P}^4\to \mathbb{P}^4$ given by the linear system of cubics through the nodes of $X$. As we saw in above, the image of $X$ is isomorphic to the cubic surface $S$. The three components of $F(X)$ are distinguished by the following different kinds of the curves $f(\ell)$. The image of a line from the components $P_1$ or $P_2$ is a twisted cubics $\gamma_i$ such that $|\gamma_i|$ gives one of the two blowing down structures of $S$ defined by a double-six of lines. The image of a line from the component $P_3$ is cut out by a tangent plane section of the cubic surface.
\end{remark}
One can easily describe the Fano surface $F(X)$ of lines on $X$ using a determinant representation $X = V(\det A)$ of $X$. Let $q_1,\ldots,q_6$ be the nodes of $X$ and $X^{\textrm{ns}} = X\text{ss}tminus \{q_1,\ldots,q_6\}$. The nodes correspond to the points at which the matrix $X$ is of rank 1. The right and the left kernel of $A$ define two regular map $\mathfrak{r}:X^{\textrm{ns}}\to \mathbb{P}^2$ and $\mathfrak{l}:X^{\textrm{ns}}\to \mathbb{P}^2$. The fiber $\mathfrak{r}^{-1}(p)$ (resp. $\mathfrak{l}^{-1}(p)$) consists of the points $x\in X'$ such that the right kernel (resp. the left kernel) of the matrix $A(x)$ is equal to $p$. The restriction of these maps to the hyperplane $t_0 = 0$ in equation \eqref{cubic1} define the two blowing-down structures on the cubic surface $F = V(a_3)$ defined by the linear systems $|C_1|, |C_2|$. These are two irreducible components of $F(X)$ isomorphic to $\mathbb{P}^2$. At each node $X$ contains two scrolls $\mathcal{S}_1, \mathcal{S}_2$ of lines passing through $q_i$. We may assume that $\mathfrak{r}(\mathcal{S}_1)$ and $\mathfrak{l}(\mathcal{S}_2)$ are lines. For any pair of lines
$\ell_i \subset\mathcal{S}_i$, the plane spanned by $\ell_1$ and $\ell_2$ intersects $X$ at the union of $\ell_1,\ell_2$ and a third line $\ell$. It belongs to the third irreducible component of $F(X)$. The pair of points
$(y_1,y_2) = (\ell\cap\ell_1,\ell\cap\ell)$ corresponds to a point on the cubic surface
$S$ associated to $X$ via Proposition \ref{nondeg}.
Since the components $P_1,P_2$ of $F(X)$ do not depend on the determinant representation
of $X$, we obtain the following.
\begin{corollary} A nondegenerate 6-nodal cubic threefold admits two non-equivalent determinant representations, one is obtained from another by taking the transpose of the matrix.
\end{corollary}
\text{ss}ction{Conic bundles}
Let $X$ be a $d$-nodal cubic hypersurface in $\mathbb{P}^4$ and $\ell$ be a line on $X$. We assume that $\ell$ is not contained in any plane on $X$. The projection from $\ell$ defines a conic bundle structure on the blow-up $X'$ of $X$ along $\ell$. The discriminant curve $\Delta_\ell$ of this bundle is of degree 5 and has $\ge d$ ordinary nodes or cusps. To see this, one takes a general hyperplane section $F$ of $X$ that contains $\ell$. It is know that a line in a nonsingular cubic surface intersects ten more lines forming 5 pairs of lines that are coplanar with $\ell$. The restriction of the projection to $F$ defines a conic bundle on $F$ with 5 singular fibers. The projection from $\ell$ defines a conic bundle on $F$ equal to the pre-image of the conic bundle on $X'$ over a line. This shows that the degree of the discriminant curve is equal to 5.
Let $\Gamma_\ell$ be the curve of lines in $X$ intersecting $\ell$. For any line $\ell'\in \Gamma$, we have the line $\ell''$ such that $\ell,\ell',\ell''$ are coplanar. This defines an involution
$\iota:\Gamma_\ell\to \Gamma_\ell$, and the quotient by this involution is isomorphic to $\Delta_\ell$. If $X$ is a nonsingular, G. Fano showed in \cite{Fano2} that $\Gamma$ is a nonsingular curve of genus 11 and degree 15 in $G_1(\mathbb{P}^4)$ (see a modern proof in \cite{CG}).
We start with the case $d = 10$. A line on $X$ which is not contained in a plane in $X$ belongs to one of the six del Pezzo components $D_i$. We assume also that $\ell$ is a general line from $D_i$. Since $\mathfrak{S}_6$ permutes these components, we may assume that $\ell\in D_1$. Consider the projection map $\textrm{Bl}_{\mathsf{S}_3}(\ell)\to \mathbb{P}^2$,
where $\textrm{Bl}_{\mathsf{S}_3}(\ell)$ is the blow-up of the line $\ell$. We consider $\mathsf{S}_3$ as the image of the rational map $\mathbb{P}^3\dasharrowsharrow \mathsf{S}_3$ given by quadrics through 5 points $p_1,\ldots,p_5$. The curve $\Gamma_\ell$ consists of 10 components. Four of the components are conics $K_i$ of lines in $D_i, i\ne 1,6,$ represented by lines in $\mathbb{P}^3$ intersecting $\ell$. Another four components are lines $L_{ijk}$ represented by conics in the planes $\Pi_{ijk}, 1\not\in \{i,j,k\}$. Also we have the conic $K_5$ of lines in $D_6$ represented by rational normal cubics through $p_1,\ldots,p_5$ intersecting $\ell$ and the pencil $L_1$ of lines in the exceptional divisor $E(p_1)$ passing through the point corresponding to the line $\ell$. The involution $\iota:\Gamma_\ell\to \Gamma_\ell$ pairs
the conic $K_i$ with the line $L_{abc}, i\not \in \{a,b,c\}$ and the conic $K_5$ with the line $L_1$. The following picture shows a component of $\Delta_\ell$ represented by the pair
$K_2$ and $L_{345}$.
\xy (-50,-15)*{};(-50,25)*{};
(0,0)*{};(37,17.5)*{}**\dir{-};
(50,0)*{};(15,17.5)*{}**\dir{-};
(2,-3)*{p_1};(48,-3)*{p_2};(13,-5)*{p_3};(38,-5)*{p_4};(25,18)*{p_5};
(39,18)*{\ell};(13,18)*{\ell'};(25,-6)*{C};
(2,1)*{\bullet};(48,1)*{\bullet};
(16,-5)*{\bullet};(34.5,-5)*{\bullet};(25,15)*{\bullet};
(25,3)*\cir<35pt>{};
\endxy
Next we consider the case of a nondegenerate $6$-nodal cubic threefold.
Let $F$ be a nonsingular cubic surface obtained by blowing up $6$ points $x_1,\ldots,x_6$ in the plane. Let $(e_0,e_1,\ldots,e_6)$ be the corresponding geometric basis of $\text{Pic}(F)$. Let $|\gamma_1| = |e_0|$ and $|\gamma_2| = |5e_0-2(e_1+\ldots+e_6)$ as in Lemma \ref{fact}. We divided all 27 lines on $F$ in three disjoint subsets. The first set of 6 lines intersecting $\gamma_1$ will be called lines of type $(0,2)$, they are represented by the exceptional curves $E(x_i)$. The second set of 6 lines will be called lines of type $(2,0)$, they are represented by the conics through 5 points. The remaining 15 lines will be called lines of type $(1,1)$, they are represented by lines joining a pair of the six points.
Let $H$ be a hyperplane in $\mathbb{P}^4$ that cuts out $X$ along a nonsingular surface $F$. We fix a node $q$ to identify the linear system of hyperplanes with the linear system of cubics in $H$ containing the fundamental curve $C(X,q) = C_1+C_2$. Thus, we can divide all lines in $F$ in three sets as above.
Take a general line $\ell$ in the component $P_1$ of $F(X)$. Let $\Gamma_\ell$ be the curve of lines in $X$ intersecting $\ell$. Since no curve from $P_1$ different from $\ell$ intersects $\ell$, we see that $\Gamma_1$ consists of two components $\Gamma_\ell(2)$ and $\Gamma_\ell(3)$ of curves from $P_2$ or $P_3$ intersecting $\ell$. Take another general curve $\gamma'$ from $P_1$ and let $F$ be a cubic surface in $X$ that contains $\ell$ and $\ell'$. We see that $\ell,\ell'$ are lines on $F$ of types $(2,0)$. We may represent then by two exceptional curves $E(q_i),E(q_j)$ of the blow-up of 6 points. The number of lines on $F$ of type $(0,2)$ intersecting $\ell$ and $\ell'$ is equal to 4, they are represented by conics passing through 5 points including $p_i$ and $p_j$. This implies that $\Gamma_\ell(2)$ has self-intersection in $P_1\cong \mathbb{P}^2$ equal to 4. Similarly, we compute the self-intersection of $\Gamma_\ell(3)$. It is equal to 1. We also obtain that $\Gamma_\ell(2)\cdot \Gamma_\ell(3) = 2$. Thus $\Gamma_\ell(2)$ is a conic on $\mathbb{P}^2$ and $\Gamma_\ell(3)$ is a line. In the Pl\"ucker embedding of $G_1(\mathbb{P}^4)$, they are curves of degrees 6 and 3.\footnote{To get expected degree 15 one has to add six lines here that come from a choice of one ruling in each exceptional divisor of the resolution $X'\to X$.} The involution $\iota_\Gamma$ switches the two components. The discriminant curve $\Delta_\ell$ is an irreducible rational curve of degree 5 with 6 nodes.
Next, we take $\ell$ to be a general line in the component $P_3$. The curve $\Gamma_\ell$ consists of three components $\Gamma_\ell(1),\Gamma_\ell(2), \Gamma_\ell(3)$ of lines from $P_1,P_2,P_3$, respectively, intersecting $\ell$. By similar argument as above, we find that two disjoint lines of type $(1,1)$ on $F$ are intersected by one line of type $(0,2)$, one line of type $(2,0)$ and three lines of type $(1,1)$. This intersection matrix of these curves is equal to
$$\begin{pmatrix}1&5&3\\
5&1&3\\
3&3&3\end{pmatrix}.$$
The curves $\Gamma_\ell(1)$ and $\Gamma_\ell(2)$ are rational curves of degree 3 on the cubic surface $P_3$, and the curve $\Gamma_\ell(3)$ is an elliptic curve of degree 3 on $P_3$. In the Pl\"ucker embedding of $G_1(\mathbb{P}^4)$ the curves are of degrees $3,3$ and $6$, respectively, and their sum is a hyperplane section equal to $-3K_{P_3}$. The involution $\iota$ switches the first two components and defines a fixed-point free involution on the third component. The discriminant curve $\Delta_\ell$ consists of the union of a conic and a cubic intersecting at 6 points.
\begin{remark} It follows that, for any general line from the component $P_3$, there exists a quadric singular along $\ell$ that contains the six nodes of $X$. Note that, counting parameters, we see that it is not true if we replace $\ell$ with a general line in $\mathbb{P}^4$.
Also, $X$ contains an elliptic curve parameterizing the singular points of the conics corresponding to the degree 3 component of the discriminant curve $\Delta_\ell$, where $\ell\in P_3$. This elliptic curve contains the six nodes of $X$. When $\ell$ varies in $P_3$, we get a family of elliptic curves (or their degenerations) parameterized by the cubic surface $P_3$. Is it an elliptic fibration on the blow-up of $X$ at 6 nodes?
\end{remark}
\begin{remark} It is known that the intermediate Jacobian of a smooth cubic threefold is a principally polarized abelian variety of dimension 5.
It is also known that it is isomorphic to the Prym variety associated to the discriminant curve of degree 5 of the conic bundle on $X$ defined by the projection from a line on $X$ (see, for example, \cite{Beauville1}). In the case when $X$ is a nondegenerate 6-nodal cubic threefold, the intermediate Jacobian $J(X)$ degenerates to a 5-dimensional algebraic torus which is isomorphic to the generalized Prym variety $\textrm{Prym}(\tilde{\Delta}/\Delta)$, where $\Delta$ is the discriminant curve of the conic bundle on $X$ defined by the projection from a line on $X$ (as was recently shown in a joint work of S. Casalaina-Martin, S. Grushevsky, K. Hulek and R. Laza, this isomorphism does not depend on a choice of a line).
The projection from a line $\ell$ on $X$ corresponding to the bisecant of $C(X,q)$ joining two points $x,y$ on different component of $C(X,q)$ can be viewed as the conic bundle structure of the blow-up $\tilde{\mathbb{P}}^3$ of $\mathbb{P}^3$ at the reducible curve $C(X,q)\cup \ell$ of degree 7 and arithmetic genus 5 which is given by the linear system of cubics through the curve. In the case of a connected smooth curve of degree 7 of genus 5 in $\mathbb{P}^3$ this conic bundle was considered by V. Iskovskikh (it is a Fano variety with the Picard number equal to 2, No 9 in the list of such varieties that can be found in \cite{Iskovskikh}).
\end{remark}
\text{ss}ction{6-nodal cubic threefolds and nonsingular cubic fourfolds}
Let $Y$ be a nonsingular cubic hypersurface in $\mathbb{P}^5$. Suppose $Y$ contains a normal cubic scroll $T$ spanning a hyperplane $H$ in $\mathbb{P}^5$. It is isomorphic to the blow-up of $\mathbb{P}^2$ at one point $p$ and it is embedded in $\mathbb{P}^4$ by the linear system of
conics through the point $p$ (see \cite{CAG}, 8,1,1). Choosing the point $p$ to be $[1,0,0]$ and the basis of the linear system in the form
$(x_2^2,x_2x_3,x_1x_2,x_1x_3,x_1^2)$, we can
express the equations of the scroll as
$$\ranglenk\begin{pmatrix}t_0&t_1&t_2\\
t_2&t_3&t_4\end{pmatrix} \le 1.$$
A cubic hypersurface in $\mathbb{P}^4$ containing $T$ can be given by the equation
$$l_1(t_1 t_4-t_2 t_3)+l_2(t_2^2-t_0 t_4)+l_3(t_0 t_3-t_1 t_2) = 0,$$
For some linear forms $l_i$ in variables $t_0,\ldots,t_4$. For appropriate $l_1,l_2,l_3$, we find the intersection $X = H\cap X$. Clearly the previous equation gives a determinantal representation of $X$. As was explained in Remark \ref{R3.6}, we expect that $X$ is a nondegenerate 6-nodal cubic threefold.
Assume now that $Y$ is a nonsingular cubic 4-fold that contains a cubic scroll $T$ whose linear span intersects $Y$ along a nondegenerate 6-nodal cubic 3-fold $X$. We know that the degree of the curve in $G_1(\mathbb{P}^4)$ parameterizing the ruling of $T$ is equal to 3
(\cite{CAG}, 10,4,1.) This implies that $T$ is the pre-image of a line under under the two rulings of $X$ parameterized by the components $P_1,P_2$ of $F(X)$. In the determinant representation of $X$, $T$ is the pre-image of a line under the left or the right kernel maps.
Assume that $T$ is the pre-image of a line in the component $P_1$. We know that
$\dim H^4(X,\mathbb{Q}) = $2. Let $\mathcal{Q}$ be a quadric in $\mathbb{P}^4$ that contains $T$ (it could be taken as one of the minors defining $T$), it intersects $X$ along the union of $T$ and another cubic scroll $T'$ corresponding to a line in the component $P_2$.
Let
$$\xymatrix{&Z\ar[dl]^{\sigma}\ar[dr]^{\tau}&\\
Y&&F(Y)}
$$
Be the incidence correspondence of lines and points in $Y$, where $F(Y)$ is the Fano fourfold of lines in $Y$. It is known to be an irreducible holomorphic symplectic 4-fold. The Abel-Jacobi map of integral Hodge structures
$$\Phi:\tau_*\sigma^*:H^4(Y,\mathbb{Z})[2] \to H^2(F(Y))$$
defines an isomorphism of free abelian groups of rank 21:
$$H^{2,2}(Y)\cap H^4(Y,\mathbb{Z}) \to H^{11}(F(Y),\mathbb{Z})\cap H^2(F(Y),\mathbb{Z})).$$
The group $H^2(F(Y),\mathbb{Z})$ is equipped with the Beauville-Bogomolov quadratic form $q_{BB}$ of signature $(1,20)$. The isomorphism $\Phi$ is a compatible (with the change of the sign) with the cup-product on primitive cohomology of $H^4(Y,\mathbb{Z})$ and the Beauville-Bogomolov quadratic form restricted to the primitive cohomology of $H^2(F(Y),\mathbb{Z})$. Let
$\sigma$ be the class of a hyperplane section of $F(Y)$ in its Pl\"ucker embedding in $G_1[\mathbb{P}^5)$. It is known that the degree of $F(Y)$ is equal to 36, so that
$\sigma^4 = $36. However, $q_{BB}(h) = $6.
For a nonsingular cubic 4-fold $Y$, we have
$$\text{Pic}(F(Y)) = H^2(F(Y),\mathbb{Z})\cap H^{11}(F(Y),\mathbb{Z}) \cong \mathbb{Z}^r,$$
If $Y$ is general in the sense of moduli (the number of them is equal to 20), then $r = 1$.
Let $h$ be the class of a hyperplane section of $Y$. Then
$[X] = h$ and $2h^2 = [T]+[T']$. This shows that $\Phi([T]),\Phi[T']$ belong to $\text{Pic}(F(Y))$ and give $r \ge $2. We assume that $r = $2 and the classes
$\tau = \Phi([T])$ and $\sigma$ freely generate $\text{Pic}(F(Y))$. The quadratic lattice
$(H^4(Y,\mathbb{Z}),\cup)$ has a basis $(h^2,[T])$ and is defined in this basis by the matrix
$\left(\begin{smallmatrix}3&6\\
3&7\end{smallmatrix}\right).$
The quadratic lattice $(\text{Pic}(F(Y)),q_{BB})$ has a basis $(\sigma,\tau)$ and is defined in this basis by the matrix
$\left(\begin{smallmatrix}6&6\\
6&2\end{smallmatrix}\right).$
In their paper \cite{Hassett}, B. Hassett and Yu. Tschinkel compute the nef cone of $F(Y)$ in $\text{Pic}(F(Y))_\mathbb{R}$ and find that it equals the dual cone of the cone
spanned by the classes $\alpha_1 = 7\sigma-3\tau$ and $\sigma+3\tau$.
Recall that $F(X)$ contains two planes $P_1,P_2$. Since, obviously, $F(X)\subset F(Y)$, we see that $F(Y)$ contains two planes (embedded in the Pl\"ucker space of $G_1(\mathbb{P}^5)$ as
Veronese surfaces spanning $\mathbb{P}^5$).
We may perform Mukai flop $P_1$ (resp. $P_2$) to obtain new holomorphic symplectic manifolds $V_1,V_2$ (\cite{Mukai}). The birational maps between $F(Y)$ and $V_1$ (resp. $V_2$) identify their Picard groups but induce an orthogonal transformation $R_1$ (resp $R_2$) of their quadratic forms.
The following beautiful result is proven in \cite{Hassett}.
\begin{theorem} There is an infinite sequence of flops
$$\cdots F_{-2}\dasharrowsharrow F_{-1}^\vee \dasharrowsharrow F(Y)\dasharrowsharrow F_1\dasharrowsharrow F_2\cdots$$
with isomorphisms $F_i\cong F_{i+2}$. The cone of effective divisors on $F(Y)$ can be expressed as the non-overlapping union of nef cones of the $F_i$'s. The birational pseudo-automorphism of $F(Y)$ defined by $F(Y)\dasharrowsharrow F_1\dasharrowsharrow F_2\cong F(Y)$ acts on $F(Y)$ by the matrix $\left(\begin{smallmatrix}-1&-2\\
6&11\end{smallmatrix}\right)$ in the basis $\sigma,\tau$. The following picture copied from \cite{Hassett}:
\end{theorem}
\end{document} |
\begin{document}
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\title{ Shortcuts to adiabaticity for open quantum systems and a mixed-state inverse engineering scheme}
\author{S. L. Wu }
\email{[email protected]}
\affiliation{School of Physics and Materials Engineering,
Dalian Nationalities University, Dalian 116600 China}
\author{W. Ma }
\affiliation{School of Physics and Materials Engineering,
Dalian Nationalities University, Dalian 116600 China}
\author{X. L. Huang}
\affiliation{School of Physics and Electronic Technology,
Liaoning Normal University, Dalian 116029, China}
\author{Xuexi Yi }
\email{[email protected]}
\affiliation{Center for Quantum Sciences and School of Physics,
Northeast Normal University, Changchun 130024, China}
\date{\today}
\begin{abstract}
We propose a fast mixed-state control scheme to transfer the quantum state along designable trajectories in Hilbert space, which
is robust to multiple decoherence noises. Starting with the dynamical invariants of open quantum systems, we present the
shortcuts to adiabaticity (STAs) of open quantum systems at first, then apply the STAs to speed up the adiabatic steady process. Our
scheme drives open systems from a initial steady state to a target steady state by a controlled Liouvillian that possesses the same form
as the reference (original) one which is accessible in present-day experiments. The experimental observation with current available
parameters for the nitrogen-vacancy (NV) center in diamond is suggested and discussed.
\end{abstract}
\pacs{03.67.-a, 03.65.Yz, 05.70.Ln, 05.40.Ca}
\maketitle
{\mathcal S}ection{Introduction}
Controlling quantum systems to accomplish special task is at the heart of emerging quantum technologies\cite{Koch2019,
Osnaghi2001,Hacohen2018}. The ideal control scheme needs to satisfy three important issues: (i) High-speed: The quantum
state transfers to the target state within desired control time length; (ii) High-fidelity: The control process should be with
an admissible error; (iii) High-controllability: The trajectory from an initial state to a target state must be completely controllable.
The most selected schemes in experiment are based on the unitary evolution of closed systems\cite{Thanopulos2007,Berry2009,
Chen2010,Chen2016}. Yet unwanted couplings to the environment reduce the fidelity severely\cite{Huneke2013, Zhou2017},
and the final state can not be steadied on the target state after the control is done. This shackles the quantum sciences and technologies
to realize efficient and scalable devices beyond the current circumstances of proof-of-principle demonstrations. To overcome such
shackles, a straightforward thinking is to formulate a scheme based on the theory of open quantum systems. Due to its steadiness, the
steady state becomes an important candidate to achieve the quantum control task\cite{Sarandy2005}. The adiabatic steady state
engineering scheme transfers the quantum state into the target state along instantaneous steady state\cite{Lorenzo2016,Wu2017}.
But the control process needs to be very slow so that the adiabatic condition can be satisfied. Also, the fast steady state engineering
schemes are proposed to accelerate the adiabatic evolution of the equilibrium state of quantum thermodynamic system, which can only
be applied to the Gibbs states or the Gaussian states \cite{Dupays2020,Dann2019}. On the other hand, the transitionless quantum driving
method of open quantum systems is also proposed, but a feasible control protocol is hard to be presented\cite{Vacanti2014,Wu2019}.
Therefore, up to present, none of schemes satisfies the requirements including high-accuracy, high-controllability, and high-speed at the
same time.
In this paper, we propose a fast control scheme of open quantum systems, named the mixed-state inverse engineering scheme (MIE), which
allows a robust and precise transfer to a given target state with a designable mixed state trajectory. Firstly, by analysing the spectral features
of dynamical invariant superoperators of open quantum systems\cite{Sarandy2007}, we present general solutions of quantum states governed
by the master equation
\begin{eqnarray}
\partial_t |\rho(t)\rangle\rangle=\hat{\mathcal L}_c (t)|\rho(t)\rangle\rangle.\label{deq1}
\end{eqnarray}
with the Liouvillian superoperator $\hat{\mathcal L}_c (t)$. Based on this general solution, the STAs of open quantum systems are established.
Then, we apply the STAs of open quantum systems on the steady state engineering (i.e., the MIE scheme), and show that the target state can be
reached with extremely high fidelity and within desired control time length. The central advantage of our scheme is that the control tasks are, in
general, achieved with fruitful feastible control protocols in experiment by selecting different quantum state trajectories. More importantly, the
selectable trajectories can bring the ideal final fidelity even multiple noise sources are involved. In fact, the pure-state inverse engineering scheme
of closed quantum systems is a particular case of the STAs scheme of open quantum systems \cite{Chen2011}, when the trajectories of quantum
states are the pure-state trajectories. And the MIE scheme can overcome the difficulties in the earlier STAs methods for closed system due to
the designable mixed-state trajectory..
The rest of this paper is organized as follows. In Sec.\ref{method}, we present the STAs scheme of open quantum systems, and propose the
MIE scheme to accelerate the adiabatic steady state process. In Sec. \ref{stirap}, we apply the MIE scheme to NV center system, which provides
simple, practical, robust control protocols to transfer the population from one ground state to the other\cite{Chen2012,Baksic2016}. It is shown
that the MIE scheme is far better at transfer efficiency than any STAs scheme of closed systems\cite{Zhou2017}, especially in the case that
the three level system suffers from dissipation and dephasing at the same time. In Sec.\ref{comparison}, we discuss the relationship between the
mixed-state inverse engineering scheme and the pure-state inverse engineering scheme of closed systems\cite{Chen2011}. Conclusions
are presented in Sec. \ref{conclusion}.
{\mathcal S}ection{ Methods} \label{method}
{\mathcal S}ubsection{STAs of Open Quantum Systems}
For an open quantum system governed by Eq.(\ref{deq1}), a dynamical invariant of the open quantum system is defined as a superoperator
$\hat{\mathcal I} (t) $ which satisfies\cite{Sarandy2007}
\begin{eqnarray}
\partial_t \hat{\mathcal I} (t)- [\hat{\mathcal L}_c (t),\hat{\mathcal I} (t)] =0.\label{di}
\end{eqnarray}
In general, invariants are non-Hermitian. Thus $\hat{\mathcal I} (t) $ needs to be expressed as the Jordan canonical form. Consider that there are $m$ Jordan
blocks, and the $\alpha$-th Jordan block is $n_\alpha$-dimensional. According to the Jordan decomposition of $\hat{\mathcal I} (t)$, we introduce
right vectors $\{|D_\alpha^{(i)}\rangle\rangle\}$ and left vectors $\{\langle\langle E_\alpha^{(i)}|\}$ in the Hilbert-Schmidt space. The left and right
vectors always satisfy
\begin{eqnarray*}
\hat {\mathcal I}\,|D_\alpha^{(i)}\rangle\rangle=\lambda_\alpha|D_\alpha^{(i)}\rangle\rangle+|D_\alpha^{(i-1)}\rangle\rangle,\\
\langle\langle E_\alpha^{(i)}| \hat {\mathcal I}=\lambda_\alpha\langle\langle E_\alpha^{(i)}|+\langle\langle E_\alpha^{(i+1)}|,
\end{eqnarray*}
with $|D_\alpha^{(-1)}\rangle\rangle\equiv0,\,\langle\langle E_\alpha^{(n_\alpha)}|\equiv0$ for $i=0,1,...,n_\alpha-1$. Thus, the right and left vectors
$|D_\alpha^{(0)}\rangle\rangle$ and $\langle\langle E_\alpha^{(n_\alpha-1)}|$ are the right and left eigenstates of $\hat {\mathcal I}(t)$ with the eigenvalue
$\lambda_\alpha$. Here we assume that all of eigenvalues are nondegenerate, i.e., $\lambda_\alpha\neq\lambda_\beta$ for $\forall\, \alpha\neq\beta$.
And the left and right vectors satisfy the orthonormality condition
\begin{eqnarray*}
\langle\langle E_\alpha^{(i)}|D_\beta^{(j)}\rangle\rangle=\delta_{\alpha\beta}\delta_{ij}
\end{eqnarray*}
It can be verified that the eigenvalues of the dynamical invariants are time-independent, and
\begin{eqnarray*}
\langle\langle E_\beta^{(j)}|\hat O|D_\alpha^{(i)}\rangle\rangle=0,\,\,\forall\,i,\,j,\label{a12}
\end{eqnarray*}
with $ \hat O=\hat{\mathcal L}-\partial_t$ for $\alpha\neq\beta$. Therefore, it can be verified that the general solution of Eq.(\ref{deq1}) reads
\begin{eqnarray}
|\rho(t)\rangle\rangle={\mathcal S}um_{\alpha=0}^{m-1} c_\alpha\,\exp(\eta_\alpha(t))\,|\Phi_\alpha(t)\rangle\rangle,\label{fs1m}
\end{eqnarray}
in which $\eta_\alpha(t)$ is a complex phase, $c_\alpha$ is a time-independent expansion efficient. $|\Phi_\alpha(t)\rangle\rangle$ is a right vector
in the $\alpha$-th Jordan block, which can be written as
\begin{eqnarray}
|\Phi_\alpha(t)\rangle\rangle={\mathcal S}um_{i=0}^{n_\alpha-1} b^\alpha_i(t)|D_\alpha^{(i)}(t)\rangle\rangle,
\end{eqnarray}
with coefficients $b^\alpha_i(t)$. The details of the derivation of the general solution Eq.(\ref{fs1m}) can be found in Appendix. \ref{AA}.
Since the adiabaticity of open quantum systems requires only forbidding the transition between different Jordan blocks\cite{Sarandy2005}, the general solution
Eq.(\ref{fs1m}) is enough to establish STAs of open quantum systems. Suppose that our aim is to drive the quantum system from an initial Liouvillian $\hat
{\mathcal L}_c(0)$ to a final one $\hat {\mathcal L}_c(t_f)$, such that the ``population" in the initial and final instantaneous Jordan blocks are same but admitting
transitions at the intermediate times. Based on the general solution Eq.(\ref{fs1m}), the complex phases $\eta_\alpha(t)$ are chosen as arbitrary functions to write
down the time-evolution superoperator $\hat {\mathcal E }(t)$ as
\begin{eqnarray*}
\hat {\mathcal E}(t)={\mathcal S}um_{\alpha=0}^{m-1} \exp\left(\eta_\alpha(t)\right)|\Phi_\alpha(t)\rangle\rangle \langle\langle\Psi_\alpha(0)|,
\end{eqnarray*}
where $\langle\langle\Psi_\alpha(t)|$ is a left vector of the $\alpha$-th Jordan block, which satisfies
$\langle\langle\Psi_\beta(t)|\Phi_\alpha(t)\rangle\rangle=\delta_{\alpha\beta}$. The evolution superoperator obeys
\begin{eqnarray*}
\partial_t \hat{ \mathcal E}(t)=\hat{\mathcal L}_c (t)\hat{\mathcal E}(t),
\end{eqnarray*}
which we formally solve for the control Liouvillian
\begin{eqnarray*}
\hat{\mathcal L}_c (t)=\partial_t \hat{ \mathcal E}(t)\hat{\mathcal E}^{-1}(t),
\end{eqnarray*}
with $$\hat{\mathcal E}^{-1}(t)={\mathcal S}um_{\alpha=0}^{m-1} \exp\left(-\eta_\alpha(t)\right)|\Phi_\alpha(0)\rangle\rangle \langle\langle\Psi_\alpha(t)|.$$
Thus, we can express the control Liouvillian superoperator as
\begin{eqnarray}
&&\hat{\mathcal L}_c (t)=\nonumber\\
&&{\mathcal S}um_{\alpha=0}^{m-1} \left(|\partial_t \Phi_\alpha(t)\rangle\rangle \langle\langle\Psi_\alpha(t)|+\partial_t\eta_\alpha(t)
|\Phi_\alpha(t)\rangle\rangle \langle\langle\Psi_\alpha(t)|\right).\label{contliou}
\end{eqnarray}
Note that for a given dynamical invariant, there are many possible Liouvillians corresponding to different choices of complex phases $\eta_\alpha(t)$.
In general, $\hat{\mathcal I}(0)$ does not commute with $\hat{\mathcal L}_c (0)$, which implies that the Jordan blocks of
$\hat{\mathcal I}(0)$ do not coincide with the Jordan blocks of $\hat{\mathcal L}_c (0)$. $\hat{\mathcal L}_c (t_f)$ does not necessarily commute with
$\hat{\mathcal I}(t_f)$ either. We impose $[\hat{\mathcal I}(0),\hat{\mathcal L}_c (0)]=[\hat{\mathcal I}(t_f),\hat{\mathcal L}_c (t_f)]=0$, such that
the Jordan blocks coincide and then the quantum state transfer from the initial block to the final one is guaranteed. Here, we must emphasize
that $|\Phi_\alpha(t)\rangle\rangle$ can be arbitrary superposition of the right basis vectors $\{|D_\alpha^{(i)}(t)\rangle\rangle\}_{i=0}^{n_\alpha-1}$
for the $\alpha$-th Jordan block. In other words, if the quantum state is prepared in a given Jordan block of $ \hat{\mathcal L}_c (0)$ at the beginning
and the final state is still in the same block of $\hat{\mathcal L}_c (t_f)$, the shortcuts to adiabaticity of open quantum systems is established, which is
the control Liouvillian given by Eq.(\ref{contliou}) designed to. This is the first result of this paper.
In Appendix. \ref{AB}, we present a detailed comparison between the STAs scheme and the transitionless quantum driving scheme of open quantum
systems\cite{Vacanti2014}. It is shown that, if the trajectory of the STAs scheme of open quantum systems is chosen
as the adiabatic trajectory, the STAs scheme is coincident with the transitionless quantum driving method proposed in Ref.\cite{Vacanti2014}. However, the
adiabatic trajectory is not the only choice of the trajectories in our scheme. There are many trajectories can be used to inversely engineer the open quantum
system. Proper trajectories always provide reasonable and applicable control protocols, which helps us to overcome the difficulties met in the
control of microscopic or/and mesoscopic systems.
{\mathcal S}ubsection{Mixed-state inverse engineering}
In a practical application, the general control Liouvillian presented in Eq.(\ref{contliou}) will meets difficulties in giving practical and affirmatory
control protocols. In addition, most of eigenvectors of the control Liouvllian $ \hat{\mathcal L}_c(t)$ are unphysical quantum states, except the eigenvectors
with zero eigenvalues, which corresponds to steady states of open quantum systems. Therefore, for practical applications, we focus our attention on the steady
states engineering of open quantum systems. Even if we restricts our discussion on speeding up the adiabatic steady state process, it is
still difficult to obtain feasible control protocols \cite{Wu2019}, since the control Liouvillian Eq.(\ref{contliou}) is in form of the superoperator. In the following, we
propose an effective and practical method to obtain feasible control protocols, which is easy to be used in experiment.
We consider a quantum system with $N$-dimensional Hilbert space govern by a linear, time-local
master equation
\begin{eqnarray*}
\partial _t \rho(t)&=&\hat{ \mathcal L_0} (t)\mathcal [ \rho(t)]\nonumber\\
&=& -\frac{i}{\hbar}[{H}_0(t),{\rho}]+{\mathcal S}um_\alpha\hat {\mathcal D}[L_\alpha](\rho),
\end{eqnarray*}
where $\hat{\mathcal L_0}(t)$ is the reference Liouvillian in the Lindblad form, $H_0(t)$ is the
Hamiltonian, and
\begin{eqnarray}
\hat{\mathcal D}[L_\alpha](\rho)=L_{\alpha}(N_\alpha){\rho}L_{\alpha}^{\dagger}(N_\alpha)-\frac{1}{2}\{L_{\alpha}^{\dagger}(N_\alpha)
L_{\alpha}(N_\alpha),{\rho}\}\nonumber\\\label{lindblad}
\end{eqnarray}
is the Lindbladian. The Lindblad operators $L_\alpha(N_\alpha)$ are related to some parameters $\{N_\alpha\}$, such as
the decoherence rates and the temperatures of the environments. Here, we do not limit the master equation to be
Markovian, but the corresponding evolution must be a completely positive trace-preserving map. Further, we assume that
$\hat{\mathcal L_0}(t)$ admits an unique (instantaneous) steady state $\rho_0(t)$, which satisfies $$\hat{\mathcal L_0}(t)[\rho_0(t)]=0.$$
For practical applications, we focus our attention on speeding up the adiabatic steady state process\cite{Lorenzo2016}, and seek the control
Liouvillian from Eq.(\ref{di}) directly. Concretely, the control task is to drive the open quantum system from the steady state of an initial Liouvillian
$\hat{\mathcal L_0} (0)$ to the target one $\hat{\mathcal L_0} (t_f)$\cite{Lorenzo2016,Sarandy2005}. In order to achieve such purpose, we consider
that the invariant has only one 1-dimensional Jordan block with non-zero eigenvalue $\Omega_\text{I}$. If we set $| \Phi_0 (0)\rangle\rangle=
|\rho_0(0)\rangle\rangle$ and $| \Phi_0 (t_f)\rangle\rangle=|\rho_0(t_f)\rangle\rangle$, the eigenvector $|\Phi_0 (t)\rangle\rangle$ corresponds
to the trajectory connecting the initial steady state and the target steady state. In general, we need to parameterize $\hat{\mathcal I} (t) $ in
$N^2$-dimensional Hilbert-Schmidt space with $N^4$ independent coefficients \cite{Wu2019}. For the MIE scheme, since $| \Phi_0(t)\rangle\rangle$
should be a quantum state of the open quantum system, we can expand $| \Phi_0(t)\rangle\rangle$ by right vectors $\{|T_\mu\rangle\rangle\}
_{\mu=1}^{N^2-1}$ which correspond to the SU($N$) Hermitian generators $\{T_\mu\}_{\mu=1}^{N^2-1}$ , i.e.,
\begin{eqnarray}
| \Phi_0 (t)\rangle\rangle=\frac{1}{N}\left ( | \text I\rangle\rangle+{\mathcal S}qrt{\frac{N (N-1)}{2}}{\mathcal S}um_{\mu=1}^{N^2-1} r_\mu |T_\mu\rangle\rangle\right ),
\end{eqnarray}
where $\vec r= (r_1, r_2, ..., r_{N^2-1}) $ is the generalized Bloch vector with ${\mathcal S}um_\mu |r_\mu|^2<1$, and $|\text{I}\rangle\rangle$ is the right vector
corresponding to a $N\times N$ identical operator. Thus, the dynamical invariants used in the MIE scheme can be defined as
\begin{eqnarray}
\hat{\mathcal{I}}(t)=\Omega_\text{I}|\Phi_0(t)\rangle\rangle\langle\langle \text{I}|,\label{di11}
\end{eqnarray}
where $\Omega_\text{I}$ is an arbitrary nonzero constant and $\langle\langle \text{I}|$ is the left vector corresponding to a $N\times N$ identity matrix.
With this notation, we can parameterize the dynamical invariant $\hat{\mathcal I}(t)$ with only $N^2-1$
parameters, which greatly simplifies the procedure in formulating control protocols.
To formulate feasible control protocols, we impose the control Liouvillians take the form as
\begin{eqnarray}
\hat{ \mathcal L}_c[\bullet]=-\frac{i}{\hbar}[{H}(t),{\bullet}]+{\mathcal S}um_\alpha\hat {\mathcal D}[L_\alpha](\bullet),
\end{eqnarray}
in which the Lindbladians and the Hamiltonian are chosen according to the following principles: (i) The Lindbladians in the control Liouvillian
$\hat{\mathcal L}_c (t) $ have the same form as Eq.(\ref{lindblad}), which controls the system through control parameters $\{N_\alpha\}$.
(ii) The control can also exert on the system via the Hamiltonian in $\hat{\mathcal L}_c(t).$ As it can be written
in terms of SU (N) Hermitian generators $\{T_k\}$, i.e., $$H (t) ={\mathcal S}um_{k=1}^{N^2-1}c_k(t)T_k,$$ we might choose $\{c_k (t)\}$ as the
control parameters to manipulate the system. In this way, the control Liouvillians can always present feastible control protocols.
Substituting $\hat{\mathcal L}_c(t)$ and $\hat{\mathcal I} (t) $ into Eq.(\ref{di}), we can express the control parameters $\{c_k, N_\alpha\}$
as a function of the generalized Bloch vector and its derivative $\{r_\mu, \partial_t r_\mu\}$. On the other hand, the control Liouvillian is
always the same as the reference Liouvillian at the initial and final moment, which leads to boundary conditions for $\{r_\mu,\partial_t r_\mu\}$.
By utilizing the control parameters $\{c_k, N_\alpha\}$ and setting proper boundary conditions for $\{r_\mu, \partial_t r_\mu\}$, the
open quantum system can be transferred from the initial steady state into the target steady state along an exact trajectory given by $|\Phi_0 (t)
\rangle\rangle$. This is the second result of this paper.
If some control parameters, say $\{\tilde c_k, \tilde N_\alpha\}$, are difficult to implement in a real setting, we can single out the equations for
those parameters, and force them to be some values which are available in the experimental setting. Notice that the equations for $\{\tilde c_k,
\tilde N_\alpha\}$ are about the components of the trajectory $\{r_\mu, \partial_t r_\mu\}$. Thus picking up proper $\{\tilde r_\mu\}$ to be
free components in trajectory, we obtain a set of
differential equations about $\{\tilde r_\mu\}$. Solving those differential equations is equivalent to choose a trajectory with particular components
$\{\tilde r_\mu\}$. Therefore the MIE scheme can avoid those difficulties encountered in the control process, such as the negative decoherence
rate\cite{Alipour2020} and the impractical energy-levels couplings\cite{Chen2010}. As a result, the MIE offers a practical
method to engineer an open quantum system with an exact trajectory which can be realized in laboratory with current technology.
{\mathcal S}ection{Example: The stimulated Raman adiabatic passage (STIRAP)}\label{stirap}
Consider a single nitrogen-vacancy (NV) center in diamond, which hosts a solid-state $\Lambda$ system.
The NV center has a spin-triplet, orbital-singlet ground state ($^3 A_2$) that is coupled optically to a spin-triplet,
orbital-doublet excited state ($^3 E$), as shown in Fig. \ref{il3} (a). The experiments have had identified the three singlet
states ($\ket{^1E}$, $\ket{^1A_1}$) \cite{Rogers2008}, where $\ket{^1E}$ is double degenerate.
\begin{figure}
\caption{The level structure of the NV center. (a) A schematic illustration of the level structure of the NV centers.
The optical zero photon line (ZPL) at 637 nm is related to the transition from $^3E$ to $^3 A_2$ and
the ZPL at 3371 nm corresponds to the transition from $^1E$ to $^3 A_2$.
(b) State transfer in a NV centre $\Lambda$ system by the protocol with initial-to-final state coupling. (c)
State transfer in a NV centre $\Lambda$ system by the protocol without the initial-to-final state coupling.}
\label{il3}
\end{figure}
For the negative charged NV center with electron spin $S=1$, the ground state is a spin-triplet state with a zero-field
splitting $D_0=2.87$ GHz between spin sublevels $\ket{m_s=0}$ and $\ket{m_s=\pm 1}$ due to electronic spin-spin interaction.
Applying a static magnetic field $B_{NV}$ along the NV axis splits the $\ket{m_s=-1}$ and $\ket{m_s=+1}$
ground states by $2\gamma_{NV}B_{NV}$ with $\gamma_{NV}=2.8\,\text{MHz G}^{-1}$.
Passing single tunable laser (637.2 nm) through a phase electro-optic modulator
produce frequency harmonics to resonantly excite both $\ket{m_s=-1}$ and $\ket{m_s=+1}$ to the spin-orbit excited
state $\ket{A_2}$, which is used as the intermediate state for STIRAP. After the modulation of an amplitude electro-optic
modulator with a 10 GHz arbitrary wave form generator produces the control fields $\Omega_s(t)$
and $\Omega_p(t)$ used in STIRAP. The Hamiltonian within the rotating wave approximation can be expressed in
the basis $\{\ket{m_s=-1},\ket{A_2},\ket{m_s=+1}\}$ by a matrix\cite{Carroll1988},
\begin{equation}
H_0(t)=\frac{\hbar}{2}
\left(
\begin{array}{ccc}
0& \Omega_p(t)& 0\\
\Omega_p(t)& 0& \Omega_s(t)\\
0& \Omega_s(t)& 0\\
\end{array}
\right).\label{rh3}
\end{equation}
For simplify our discussion, the ``one-photon resonance'' case is considered. The shortcuts of the open STIRAP for a general case can
be obtained with the same procedure. Here we assume that the adiabatic pulses satisfy
\begin{equation}
\Omega_s(t)=\Omega(t)\cos\theta(t), \,\Omega_p(t)=\Omega(t){\mathcal S}in\theta(t), \label{ap3}
\end{equation}
with $\tan\theta(t)=\Omega_p(t)/\Omega_s(t)$ and $\Omega(t)={\mathcal S}qrt{\Omega_s^2(t)+\Omega_p^2(t)}$.
Consider that the $\Lambda$ system couples to a bosonic heat reservoir at finite temperature $T$.
The effect of the heat reservoir is to induce decay from $\ket{A_2}$ to $\ket{m_s=\pm1}$. The decay
rate are $\Gamma_{-1}=2\pi\times4.3$ MHz (from $\ket{A_2}$ to $\ket{m_s=-1}$) and $\Gamma_{+1}=2\pi\times8.5$ MHz (from $\ket{A_2}$
to $\ket{m_s=+1}$), as shown in FIG. \ref{il3} (b). Moreover, the orbital dephasing of the level $\ket{A_2}$ can not
be ignored \cite{Issoufa2014}, where the dephasing rate is $\Gamma_d=2\pi\times8.8$ MHz.
All of decoherence rates mentioned here are selected from the measurement in recent experiment\cite{Zhou2017}. The dynamics of the $\Lambda$
system is governed by
\begin{equation}
\partial _t \rho(t)=\hat{ \mathcal L_0}\rho(t)+\Gamma_d\hat{ \mathcal D}[L_d]\rho(t),\label{me3}
\end{equation}
where
\begin{eqnarray*}
\hat{ \mathcal L_0}\rho(t)&=&-\frac{i}{\hbar}[{H}_0(t),{\rho}]\\&+&{\mathcal S}um_{\alpha=0,+1}\Gamma_\alpha((N_\alpha +1)\hat {\mathcal D}[L_\alpha](\rho)
+N_\alpha\hat {\mathcal D}[L_\alpha^\dagger](\rho)). \nonumber
\end{eqnarray*}
and $\hat{\mathcal D}[L_\alpha](\rho)=
L_{\alpha}(t){\rho}L_{\alpha}^{\dagger}(t)-\frac{1}{2}\{L_{\alpha}^{\dagger}(t)L_{\alpha}(t),{\rho}\}$. For the decay form $\ket{A_2}$,
the lindblad operators can be expressed as $L_{\pm1}=\ket{m_s=\pm1}\bra{A_2}$; and $L_d=\ket{A_2}\bra{A_2}$ for the orbital dephasing.
$N_\alpha=[\exp(\hbar \omega_{2\rightarrow\alpha}/kT)-1]^{-1}$ denote the mean excitation numbers. In the following discussion, we choose
$\hat{ \mathcal L_0}$ as the reference Liouvillian, and the dephasing is the key obstacle for the performance of the protocol.
{\mathcal S}ubsection{The Adiabatic Trajectory}
In this subsection, we present the control protocol where the quantum state transfers along the adiabatic trajectory given by the instantaneous
steady state of $\hat{ \mathcal L}_0(t)$. We parameterize the instantaneous steady state of $\hat{ \mathcal L}_0(t)$ via the generalized Bloch
vector $\{r_k\}_{k=1}^8$. The density matrix of the three-level system can be written as,
\begin{equation}
\rho(t)=\frac{1}{3}\left(\text{I}+{\mathcal S}qrt{3}{\mathcal S}um_{k=1}^8 r_k(t) T_k\right),\label{is3}
\end{equation}
where $\text{I}$ is a $3\times 3$ identity matrix, and $T_k$ denotes the regular Gellmann matrix. These $\{T_k\}$ span all traceless Hermitian
matrices of the Lie algebra su(3). If $\Gamma_{+1}=\Gamma_{-1}\equiv\Gamma$ and $N_{+1}=N_{-1}\equiv N$, the components of the Bloch
vector corresponding to the instantaneous steady state of $\hat{ \mathcal L}_0(t)$ are
\begin{eqnarray}
&&r_2={\mathcal S}qrt{3}\,\mathrm{N}\, \mathrm{\Gamma}\, \mathrm{\Omega_p}/z,\nonumber\\
&&r_3={\mathcal S}qrt{3}\,\left(\left(3\, \mathrm{N}^2 + 2\mathrm{N}\right)\, {\mathrm{\Gamma}}^2 + {\mathrm{\Omega_s}}^2 \right)/(2z),\nonumber\\
&&r_4=-{\mathcal S}qrt{3}\,\mathrm{\Omega_p}\, \mathrm{\Omega_s}/z,\nonumber\\
&&r_7=-{\mathcal S}qrt{3}\,\mathrm{N}\, \mathrm{\Gamma}\, \mathrm{\Omega_s}/z,\nonumber\\
&&r_8=-\left(\left(3\, \mathrm{N}^2 + 2\mathrm{N}\right)\, {\mathrm{\Gamma}}^2+2\,{\mathrm{\Omega_p}}^2 -{\mathrm{\Omega_s}}^2 \right)
/(2z),\label{r38m}
\end{eqnarray}
with $z={\left(3\, \mathrm{N} + 1\right)\, {\mathrm{\Omega}}^2 + \mathrm{N}\, {\mathrm{\Gamma}}^2\, {\left(3\, \mathrm{N} + 2\right)}^2}$,
and the other components are zeros. The details for obtaining the instantaneous steady state can be found in Appendix. \ref{AC}. Correspondingly,
the dynamical invariants can be expressed by the Bloch vector according to the MIE scheme (see Eq.(\ref{di11})),
\begin{eqnarray}
\hat{\mathcal{I}}(t)=\Omega_{I}|\rho_0\rangle\rangle\langle\langle \text I|,\label{di1}
\end{eqnarray}
where $\Omega_I$ is an arbitrary nonzero constant and $\langle\langle \text I|$ is the left vector corresponding to a $3\times 3$ identity matrix.
Assume that the total Liouvillian reads
\begin{eqnarray}
\hat{ \mathcal L}=\hat{ \mathcal L_c}+\Gamma_d\hat{ \mathcal D}[L_d].\label{cliou3}
\end{eqnarray}
The control Liouvillian has the same form as Eq.(\ref{me3}),
\begin{eqnarray}
\hat{ \mathcal L_c}\rho(t)&=&-i[{H}_c(t),{\rho}]\label{lc}\\
&&+{\mathcal S}um_{\alpha=\pm1}\Gamma_\alpha((N_\alpha^i+1)\hat {\mathcal D}[L_\alpha](\rho)
+N_\alpha^i\hat {\mathcal D}[L_\alpha^\dagger](\rho)),\nonumber
\end{eqnarray}
where the Hamiltonian is
\begin{eqnarray}
H_c(t)=\frac{\hbar}{2}
\left(
\begin{array}{ccc}
0& \Omega_p^i(t)&i \,\Omega_c^i(t)\\
\Omega_p^i(t)& 0& \Omega_s^i(t)\\
- i\, \Omega_c^i(t)& \Omega_s^i(t)& 0\\
\end{array}
\right).
\end{eqnarray}
Here we also assume that the mean excitation numbers $N_{\pm 1}^i$ are tunable independently, which can be achieved by properly engineering the temperature
of the environment \cite{Shabani2016} or shifting the energy difference between $\ket{A_2}$ and $\ket{m_s=\pm 1}$. Substituting Eqs. (\ref{di1}) and (\ref{lc}) into
Eq.(\ref{di}), we can determine all the control parameters in the control Liouvillian $\hat{ \mathcal L_c}$. The analytical expression of these control parameters are
presented in Appendix. \ref{AD}. Alternatively, these control parameters can be obtained numerically, especially for more complex situation such as $\Gamma_{-1}
\neq\Gamma_{+1}$. Taking Eq.(\ref{ap3}) and Eqs.(\ref{r38m}) into the analytical expressions, we obtain the control parameters for the adiabatic trajectory with a
time-independent $\Omega$,
\begin{eqnarray*}
&&\Omega_{s}^i=\Omega\,{\mathcal S}in(\theta(t)),\,\Omega_{p}^i=\Omega\,\cos(\theta(t)),\,\Omega_{c}^i=\partial_t \theta(t),\nonumber\\
&&N_{-1}^i=N,\,N_{+1}^i=N.\label{cpa3}
\end{eqnarray*}
These control parameters are exactly the parameters obtained in the transitionless quantum driving scheme of closed systems for the STIRAP\cite{Chen2010}.
Despite the MIE scheme and the transitionless driving scheme of closed systems provide similar control protocols for an adiabatic trajectory, their essences are
quite different. The two schemes give similar control protocols only if $\Omega={\mathcal S}qrt{\Omega_s^2(t)+\Omega_p^2(t)}$ is time-independent. This can be
illustrated by the spectrum decomposition of the instantaneous steady state of $\hat{ \mathcal L_0}$. The eigenvalues of $\rho_0$ are
\begin{eqnarray}
p_1&=&\frac{1}{z}\left(\left(2\mathrm{N}+1\right)\left(3\mathrm{N}+2\right)\mathrm{N}\mathrm{\Gamma}^{2}+2\mathrm{N}\mathrm{\Omega}^{2}\right.\nonumber\\
&&\left.-\mathrm{N}\mathrm{\Gamma}{\mathcal S}qrt{\left(3\mathrm{N}+2\right)^{2}\mathrm{\Gamma}^{2}+4\mathrm{\Omega}^{2}}\right),\nonumber\\
p_2&=&\frac{1}{z}\left(\left(2\mathrm{N}+1\right)\left(3\mathrm{N}+2\right)\mathrm{N}\mathrm{\Gamma}^{2}+2\mathrm{N}\mathrm{\Omega}^{2}\right.\nonumber\\
&&\left.+\mathrm{N}\mathrm{\Gamma}{\mathcal S}qrt{\left(3\mathrm{N}+2\right)^{2}\mathrm{\Gamma}^{2}+4\mathrm{\Omega}^{2}}\right), \nonumber\\
p_3&=&\frac{\left(\mathrm{N}+1\right)}{z}\left(\left(3\mathrm{N}+2\right)\mathrm{N}\mathrm{\Gamma}^{2}+\mathrm{\Omega}^{2}\right).\nonumber
\end{eqnarray}
with $z={\left(3\, \mathrm{N} + 1\right)\, {\mathrm{\Omega}}^2 + \mathrm{N}\, {\mathrm{\Gamma}}^2\, {\left(3\, \mathrm{N} + 2\right)}^2}$,
which denote the populations on corresponding eigenstates. If $\Omega$ is constant, the population on every eigenstate is invariant, so that a
unitary evolution is enough to transfer the quantum state into the target steady state. But if $\Omega$ is time-dependent, the incoherent controls
are required for the STIRAP process along the adiabatic trajectory because of time-varying purity of the quantum state.
\begin{figure}
\caption{(a) The control field and (b) the population on $\ket{m_s=-1}
\label{pp3}
\end{figure}
Figures \ref{pp3} (a) and (b) show the control fields and the corresponding population for the open $\Lambda$ system at room temperature (300K). The corresponding
mean excitation number is $N=1.9\times10^{-33}$. The boundary conditions of $\theta(t)$ are set to be $\theta(0)=0$ and $\theta(\tau)=\pi/2$. We choose the simplest
STIRAP pulses, i.e., $\theta(t)=\pi t/(2 \tau),$ where $\tau$ is the pulse length. The numerical results of the control fields show that, for nonadiabatic state transfer along the
adiabatic trajectory, the control protocol is completely the same as the protocol given by the transitionless quantum driving method \cite{Chen2010}. This illustrates that the
transitionless driving method \cite{Vepsalainen2019,Zhang2013} will present a better transfer efficiency than any other scheme else\cite{Zhou2017,Du2016}, if $\Omega$
is set to be constant. As shown in FIG.\ref{pp3} (b), the quantum state is transferred into the target state with a perfect final population. In FIG. \ref{pp3} (c), we plot the final
population not on $\ket{m_s=+1}$ (denoted by $\log_{10}(1-P_3)$ with $P_3=\bra{m_s=+1}\rho(\tau)\ket{m_s=+1}$) as a function of the pulse length $\tau$ for both the
adiabatic scheme (the red dots and blue dot-dash lines) and the MIE scheme (green dash and black solid lines). Comparing to the adiabatic scheme, our scheme can transfer
the population into $\ket{m_s=+1}$ with transfer efficiency very close to 1(at least $10^{-6}$, see the figure). Since $\ket{A_2}$ is unoccupied, our protocol is immune to the
orbital dephasing encountered in the earlier proposals.
In practical applications, the initial-to-final state coupling $\Omega_c^i$ can be implemented in some but not in all systems, e.g., because of selection rules due to
symmetry of the states or the necessary phase of the term. In nitrogen-vacancy electronic spins, this additional coupling was implemented mechanically via a strain
field in the recent experiment \cite{Barfuss2015,Kolbl2019}. The strain field to drive NV spins is based on the sensitive response of the NV spin states to strain in the
diamond host lattice. For uniaxial strain applied transverse to the NV axis, the transverse strain field couples the electronic spin states $\ket{m_s=-1}$ and
$\ket{m_s=+1}$. To realize such strain field for efficient coherent driving, a mechanical resonator is required in the form of a singly clamped, single-crystalline diamond
cantilever, in which the NV center is directly embedded \cite{Barfuss2015}. The cantilever is actuated at its mechanical resonance frequency $\omega_m=2\pi\times6.84$
MHz. Thus, for resonance driving, the external magnetic field satisfies $B_{NV}=2.42$ G. And the corresponding Rabi frequency is charactered by $\Omega_c=\gamma_
T x_{\text{c}}/x_{\text{zpf}}$, where $\gamma_T$ is the transverse single-phonon strain-coupling strength, $ x_{\text{c}}$ and $x_{\text{zpf}}$ are the cantilever
zero-point fluctuation and peak amplitude, respectively (with $\gamma_T{\mathcal S}im2\pi\times0.08$ MHz and $x_{\text{zpf}}{\mathcal S}im7.7\times10^{-15}$ m).
Therefore, we can tune the cantilever’s amplitude $ x_{\text{c}}$ to adjust the Rabi frequency of the strain field.
{\mathcal S}ubsection{The Trajectory without Initial-to-Final State Couplings} \label{secIIIB}
In experiment, a initial-to-final state coupling induced by the strain field $\Omega_c^i(t)$ can be realized in artificial structure but not in real atoms via dipole-dipole coupling
due to the selection rule. The pure-state inverse engineering scheme solves this by providing alternative shortcuts that do not couple directly levels $\ket{m_s=-1}$ and
$\ket{m_s=+1}$ \cite{Chen2012}. In the following, we show that the MIE scheme has the same quality via selecting a proper trajectory of the quantum state. And this particular
trajectory can be obtained only by solving differential equations about the components of the Bloch vector, but not to design a complex transformation of the entire trajectory as
done in the STAs schemes of closed quantum systems. More importantly, our protocol is robust to the decay and dephasing noise at the same time.
We consider the case where the Stocks pulse and the pumping pulse resonantly excite both $\ket{m_s=0}$ and $\ket{m_s=+1}$ to the excited-state spin singlet
$\ket{^1 E}$, as shown in FIG. \ref{il3} (c). The singlet-ground ($\ket{m_s=0}\rightarrow\ket{^1E}$) splitting is about 89 THz. The decay from the singlet state to
the ground states is spin-nonpreservingnonradiative decay. The corresponding decay rates depend on the static magnetic field $\mathbf{B}_{NV}$ with an angle
$\eta$ with respect to the NV defect axis. It has been shown that the decay rates are approximatively equal for $\eta=\pi/10$, and have been measured in the lab with
the results $\Gamma_0\approx\Gamma_{+1}=2\pi\times1.5$ MHz \cite{Tetienne2012}. Moreover, due to spin-nonpreserving decay, the effective rate of the population
excitation from $\ket{m_s=0}$ to $\ket{^1 E}$ is much lower than the decay rate from $\ket{^1 E}$ to $\ket{m_s=0}$, while the effective rate from the $\ket{m_s=+1}$ to
$\ket{^1E}$ is almost equal to its corresponding decay rate. This results that the $\ket{m_s=+1}-\ket{^1E}$ and $\ket{m_s=0}-\ket{^1E}$ subsystems can be seen as
coupling to two different thermal reservoirs \cite{Klatzow2019}. It has also been shown in Ref. \cite{Klatzow2019} that tuning the strength of $\mathbf{B}_{NV}$ can
effectively engineer temperatures of the thermal reservoirs.
In order to cancel the initial-to-final state coupling, the control field $\Omega_c^i(t)$ must be zero at any point in time. A direct manner is to modify the trajectory of
the quantum state. Here, we consider the case where the decay rates are equal. As illustrated in the analytical expressions of the control parameters (see Appendix. \ref{AD}),
$\Omega_c^i(t)$ can be written as a function of the Bloch vector and its time derivative $\{r_i,\,\partial_t r_i\}_{i=1}^8$. What we may do is to force $r_4(t)$ to vary in
a proper way such that $\Omega_c^i(t)=0$. To be specific, we consider the requirement $\Omega_c^i(t)=0$ as a restriction to solve the differential equation of $r_4(t)$,
keeping other components of the Bloch vector unchanged. This would finally leads to a proper trajectory that cancels the initial-to-final state coupling.
By choosing a trajectory according to the instantaneous steady state of $\hat{ \mathcal L_0}$ (see Appendix. \ref{AE}), we plot the control parameters and the populations as
a function of time in FIG. \ref{Lam3}. We notice that the additional coupling $\Omega_c^i(t)$ is eliminated in the engineering process (the green solid line in FIG. \ref{Lam3} (a)),
and the population is transferred from $\ket{m_s=0}$ into the target state $\ket{m_s=+1}$ with high transfer efficiency as shown in FIG. \ref{Lam3} (c). In addition,
the excitation numbers of the reservoirs need to be engineered accordingly (see FIG. \ref{Lam3} (b)).
\begin{figure}
\caption{(a) The control fields, (b) the mean excitation numbers, and (c) the population on states $\ket{m_s=0}
\label{Lam3}
\end{figure}
Although FIG. \ref{Lam3} presents positive main excitation numbers, these main excitation
numbers may still negative at some points of the time. In experiment, we may restrict the main excitation numbers to be within the regime $N_{0}^i,\,
N_{+1}^i\geq6.55\times10^{-7}$ which corresponds to $T_{\text{cutoff}}\approx 300$ K. In FIG. \ref{PO3}(a), we plot the final
population which is not on $\ket{m_s=+1}$ as a function of the pulse length $\tau$ for the control field $\Omega=154$ KHz. Both dynamical processes
with (the red dash line) and without (the blue solid line) the dephasing noise are considered in FIG.\ref{PO3}.
As we see, the MIE scheme fails to transfer the population into $\ket{m_s=+1}$ for short pulse length
due to the cut-off on the main excited number $N_{+1}^i$. But it performs good
by prolonging pulse length, and the final population on $\ket{m_s=+1}$ approach asymptotically to a predicted value give by
the instantaneous steady state of $\hat{ \mathcal L_0}(\tau)$. On the other hand, since the coherence between $\ket{^1E}$ and
$\ket{m_s=0}$ ($\ket{m_s=+1}$) in our designed trajectory is negligible, the orbital dephasing noise of $\ket{^1 E}$ will not
affect the state transfer process evidently. The numerical result confirms our analysis as illustrated by the blue solid line and the red dash line in
FIG. \ref{PO3} (a). We also present the results given by the superadiabatic scheme\cite{Barfuss2015} with (the green dots line) and without (the black dash-dot line)
the dephasing noise. As a pure-state STAs scheme, the superadiabatic scheme needs to keep states as pure states. When $\ket{^1 E}$ is populated,
the strong coherence between the excited and ground states are required, so that pure-state STAs schemes are sensitive to the dephasing noise.
Therefore, the MIE scheme is more robust to the orbital dephasing noise than the STAs schemes of closed quantum systems.
We also plot the population on all states except $\ket{m_s=+1}$(blue solid line) as a function of the control field $\Omega$ with a pulse length
$\tau=5\, \mu\text{s}$ in FIG. \ref{PO3} (c). As expected, the final population on $\ket{m_s=+1}$ increases with the control field. The population
not on $\ket{m_s=+1}$ given by the steady state $\rho_0(\tau)$ is also plotted, see the green dash line. We find that the final population for the inverse engineering
can not reach the predicted value due to the main excitation number cutoff, see FIG. \ref{PO3} (d).
As illustrated by the red dot line in FIG. \ref{PO3} (c), the worst deviation of the final population is no more than $10^{-4}$. Therefore, although the
final population deviates from the predicted value, we can still obtain a satisfactory transfer efficiency in the regime of strong control field.
\begin{figure}
\caption{ The final population not on $\ket{m_s=+1}
\label{PO3}
\end{figure}
{\mathcal S}ection{Comparison with the Pure-state Inverse Engineering Scheme} \label{comparison}
{\mathcal S}ubsection{Pure-state Engineering}
In this section, we compare the MIE scheme to the pure-state inverse engineering (PIE) scheme. Firstly, we show that the STAs scheme of open quantum systems
includes the PIE scheme. To inversely engineering a pure-state, the dynamical invariants $I(t)$ of closed systems are needed\cite{Chen2011}, which satisfies
\begin{eqnarray*}
i\,\partial_t I(t)=[H(t),I(t)],
\end{eqnarray*}
with the Hamiltonian $H(t)$. The general solution of closed systems can be written as
\begin{eqnarray}
\ket{\Psi(t)}={\mathcal S}um_n c_n \exp(i\alpha_n(t))\ket{\psi_n(t)}, \label{gs}
\end{eqnarray}
where $c_n$ are time-independent amplitudes, $\ket{\psi_n(t)}$ are orthonormal eigenvectors of the invariant $I(t)$, and $\alpha_n(t)$ are the Lewis-Riesenfeld phases.
According to the general solution Eq.(\ref{gs}), we can write down the time-dependent unitary evolution operator $U$ as
\begin{eqnarray*}
U(t)={\mathcal S}um_n \exp(i\alpha_n(t))\ket{\psi_n(t)}\bra{\psi_n(0)},
\end{eqnarray*}
which obeys
\begin{eqnarray*}
i\partial_t U(t)=H(t)U(t).
\end{eqnarray*}
Thus, we obtain the formal expression of the Hamiltonian as
\begin{eqnarray}
H(t)={\mathcal S}um_n\left(i\,\ket{\partial_t\psi_n(t)}\bra{\psi_n(t)}-\partial_t\alpha_n(t)\ket{\psi_n(t)}\bra{\psi_n(t)}\right).\nonumber\\\label{ht}
\end{eqnarray}
If we impose $[H(0),I(0)]=[H(\tau),I(\tau)]=0$, the eigenstates coincide and then a state transfer from a initial eigenstate
of the Hamiltonian to the final one is guaranteed.
For applying the STAs scheme of open quantum systems to a pure-state engineering task, we can formulate the dynamical invariant
superoperator $\hat{\mathcal I}$ by the eigenstates of closed systems' dynamical invariant. The time-dependent eigenstates of
$\hat{\mathcal I}$ are chosen as
\begin{eqnarray}
|\Psi_{mn}\rangle\rangle=\ket{\psi_m}\otimes\ket{\psi_n^*},\label{pmn}
\end{eqnarray}
which corresponds to the time-independent eigenvalues $\lambda_{mn}$. Substituting $\{|\Psi_{mn}\rangle\rangle\}$ into Eq.(\ref{contliou}),
the control Liouvillian superoperator can be obtained with arbitrary phases $\eta_{mn}(t)$. Thus, $\hat{\mathcal L}_c(t)$ transfers the
quantum state from $\ket{\psi_n(0)}$ to $\ket{\psi_n(\tau)}$ along with $\ket{\Psi_{nn}(t)}\rangle$. According to Eq.(\ref{contliou}), the control
Liouvillian can be written as
\begin{eqnarray*}
\hat{\mathcal L}_c(t)={\mathcal S}um_{mn}\left(|\partial_t\Psi_{mn}\rangle\rangle\langle\langle\Psi_{mn}|+\partial_t\eta_{mn}|\Psi_{mn}\rangle
\rangle\langle\langle\Psi_{mn}|\right).
\end{eqnarray*}
By considering Eq.(\ref{pmn}), it yields
\begin{eqnarray*}
\hat{\mathcal L}_c(t)&=&{\mathcal S}um_{mn}\left(|\partial_t\psi_{m}\rangle\langle\psi_{m}|\otimes(|\psi_{n}\rangle\langle\psi_{n}|)^*\right.\nonumber\\
&&+|\psi_{m}\rangle\langle\psi_{m}|\otimes(\partial_t|\psi_{n}\rangle\langle\psi_{n}|)^*\nonumber\\
&&\left.+\partial_t\eta_{mn}|\Psi_{mn}\rangle\rangle\langle\langle|\Psi_{mn}|\right).
\end{eqnarray*}
Since $\{\ket{\psi_m}\}$ is a complete set of the Hilbert space, we have
\begin{eqnarray*}
&\hat{\mathcal L}_c(t)&=-i\left(\right.i\,{\mathcal S}um_{mn}\partial_t\eta_{mn}|\Psi_{mn}\rangle\rangle\langle\langle|\Psi_{mn}|\\
&&\left.+({\mathcal S}um_{m}i\,|\partial_t\psi_{m}\rangle\langle\psi_{m}|)\otimes\text I
-\text I\otimes({\mathcal S}um_{n}i\,\partial_t|\psi_{n}\rangle\langle\psi_{n}|)^*\right).\nonumber
\end{eqnarray*}
where $\text I$ is an identity matrix. If we choose the phase satisfies $\eta_{mn}=i\,(\alpha_m-\alpha_n)$,
it can be obtained that
\begin{eqnarray*}
\hat{\mathcal L}_c(t)&&=-i\left((i\,{\mathcal S}um_{m}|\partial_t\psi_{m}\rangle\langle\psi_{m}|-\partial_t \alpha_m|\partial_t\psi_{m}
\rangle\langle\psi_{m}|)\otimes \text I\right.\nonumber\\
&&\left.-\text I\otimes(i\,{\mathcal S}um_{n}\partial_t|\psi_{n}\rangle\langle\psi_{n}|-\partial_t \alpha_n|\partial_t\psi_{n}\rangle\langle\psi_{n}|)^*\right).
\end{eqnarray*}
Considering the map from the superoperator to operators: $A\otimes B^*|\rho\rangle\rangle\mapsto A\rho B^\dagger$ for arbitrary operators
$A$ and $B$, we immediately have
\begin{eqnarray}
\hat{\mathcal L}_c(t)|\rho(t)\rangle\rangle=-i[H(t),\rho(t)],
\end{eqnarray}
where $H(t)$ is just the Hamiltonian involved in the PIE scheme as shown in Eq.(\ref{ht}). Therefore, the STAs scheme of open quantum systems includes
the PIE scheme.
In the following, we take the STIRAP as an example to shown that the STAs scheme gives same
control protocol as the PIE scheme. Following the step of Ref.\cite{Chen2012}, the eigenstates
of dynamical invariant are given by
\begin{eqnarray*}
&&\ket{\psi_0}=\left(
\begin{array}{c}
\cos\gamma\,\cos\beta \\
-i{\mathcal S}in\gamma \\
-\cos\gamma\,{\mathcal S}in\beta \\
\end{array}
\right),\,\nonumber\\
&&\ket{\psi_\pm}=\frac{{\mathcal S}qrt{2}}{2}\left(
\begin{array}{c}
{\mathcal S}in\gamma\,\cos\beta\pm i {\mathcal S}in\beta \\
i\cos\gamma \\
-{\mathcal S}in\gamma\,{\mathcal S}in\beta \pm i\cos\beta\\
\end{array}
\right).
\end{eqnarray*}
Thus, we can formulate the eigenstate of the dynamical invariant superoperator according to Eq.(\ref{pmn}). Substituting Eq.(\ref{pmn}) into
Eq.(\ref{contliou}), we obtain the control Liouviilian $\hat{\mathcal L}_c$ as a function of
$\beta$, $\gamma$, and phases $\eta_{mn}$. Comparing $\eta_{mn}$ with the Lewis-Riesenfeld phases $\alpha_n$, we immediately find
that $\eta_{mn}=i(\alpha_m-\alpha_n)$. Thus, we have $\eta_{nn}=0$, $\eta_{+-}=-\eta_{-+}\equiv\eta_1$, $\eta_{+0}=-\eta_{0+}\equiv\eta_2$,
and $\eta_{0-}=-\eta_{-0}\equiv\eta_3$. The control Liouvillian can be expanded by SU(3) generators ($T_i$ - the regular Gellmann matrixes Eq.(\ref{Tx})), i.e.,
\begin{eqnarray*}
\hat{\mathcal L}_c(t)={\mathcal S}um_{i,j=1}^9 c_{ij} T_i\otimes T_j^*.
\end{eqnarray*}
where $T_9$ is a $3\times 3$ identity matrix. $c_{ij}$ is time-dependent expanding coefficients, which can be determined by
\begin{eqnarray*}
c_{ij}=\text{Tr}(\hat{\mathcal L}(t) T_i\otimes T_j^*).
\end{eqnarray*}
Moreover, if the phase $\eta_k\,(k=1,2,3)$ have following relation: $\eta_2=\eta_1/2$ and $\eta_3=-\eta_1/2$, it yields the coefficient matrix
\begin{widetext}
\begin{eqnarray}
&&c=\frac{i}{2}\times\nonumber\\&&\left(\begin{array}{ccccccccc} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 2\partial_t \gamma\cos\beta - \eta_1 \cos\gamma{\mathcal S}in\beta\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\partial_t \beta - \eta_1{\mathcal S}in\gamma\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 \partial_t \gamma{\mathcal S}in\beta - \eta_1\cos\beta \cos\gamma\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 2\partial_t \gamma\cos\beta + \eta_1\cos\gamma{\mathcal S}in\beta & 0 & 0 & 0 & - 2\partial_t \beta + \eta_1{\mathcal S}in\gamma & -2\partial_t \gamma {\mathcal S}in\beta+ \eta_1 \cos\beta \cos\gamma & 0 & 0 & 0 \end{array}\right).\nonumber
\end{eqnarray}
\end{widetext}
In other words, the control Liouvillian reads
\begin{eqnarray*}
\hat{\mathcal L}_c(t)=&&-i\left((2\partial_t \gamma\cos\beta+\eta_1 \cos\gamma{\mathcal S}in\beta)(T_1\otimes T_9^*-T_9\otimes T_1^*)\right.\nonumber\\
&&+(2\partial_t \beta - \eta_1{\mathcal S}in\gamma)(T_5\otimes T_9^*-T_9\otimes T_5^*)\nonumber\\
&&\left.+(\eta_1\cos\beta \cos\gamma-2 \partial_t \gamma{\mathcal S}in\beta)(T_6\otimes T_9^*-T_9\otimes T_6^*)\right).
\end{eqnarray*}
Thus, the control Liouvillian can be transformed as
\begin{eqnarray}
\hat{\mathcal L}_c(t)|\rho(t)\rangle\rangle\mapsto -i[H(t),\rho(t)]
\end{eqnarray}
with $H(t)=\Omega_p/2\, T_1+\Omega_c/2\, T_5+\Omega_s/2\, T_6$, in which the control fields are
\begin{eqnarray}
\Omega_p&=&\eta_1 \cos\gamma{\mathcal S}in\beta+2\partial_t \gamma\cos\beta,\nonumber\\
\Omega_c&=&2\partial_t \beta - \eta_1{\mathcal S}in\gamma,\nonumber\\
\Omega_s&=&\eta_1\cos\beta \cos\gamma-2 \partial_t \gamma{\mathcal S}in\beta.
\end{eqnarray}
In order to cannel initial-to-final state couplings, the phase $\eta_1$ has to selected as $\eta_1=2\partial_t \beta/{\mathcal S}in\gamma$. Taking $\eta_1$ into
above equations, we immediately obtain the same control protocol given in Ref.\cite{Chen2012}, i.e.,
\begin{eqnarray}
\Omega_p&=&2\partial_t \beta \cot\gamma{\mathcal S}in\beta+2\partial_t \gamma\cos\beta,\nonumber\\
\Omega_s&=&2\partial_t \beta\cot \gamma\cos\beta-2 \partial_t \gamma{\mathcal S}in\beta,\nonumber\\
\Omega_c&=&0.\label{pcp}
\end{eqnarray}
Therefore, the STAs scheme of open quantum systems is equivalent to the PIE scheme if the control task is to transfer pure states with a pure-state trajectory.
{\mathcal S}ubsection{Pure-state Engineering with Mixed-state Trajectories}
The MIE scheme provides more feasible control protocols than the PIE scheme, because the trajectory does not has to be a pure-state
trajectory. As illustrated by the STRIAP of open quantum systems, the robustness to the dephasing noise attributes to the mixed-state
trajectory with weak coherence between the energy levels. In this subsection, we show that the MIE scheme overcomes the difficulties
meeting in the PIE scheme. As shown in Eq.(\ref{pcp}), $\ket{^1E}$ needs to be populated for avoiding infinitely large $\Omega_{p,s}$.
The strengths of control fields satisfy $\Omega_{p,s}\propto 1/{\mathcal S}qrt{P_2}$ where $P_2={\mathcal S}in^2\gamma$ is the population on $\ket{^1E}$.
For the control protocol given by the MIE scheme, reasonable and feasible $\Omega_{p,s}$ are needed instead of infinitely large control fields,
which is illustrated in FIG. \ref{mpc} (a).
\begin{figure}
\caption{ (a) The control field $\Omega_p$, (b) the population on $\ket{^1E}
\label{mpc}
\end{figure}
For the control protocol given by the PIE scheme, the parameters in Eq.(\ref{pcp}) is chosen as: $\gamma=(\gamma_m-\gamma_0)
{\mathcal S}in^3(\pi t/\tau)+\gamma_0$ with constants $\gamma_m$ and $\gamma_0$; $\beta=\pi/2{\mathcal S}in(\pi t/(2\tau))$. For the mixed-state protocol,
we set a analogous trajectory with $r_3=-{\mathcal S}qrt{3}({\mathcal S}in^2\gamma- \cos^2\beta\cos^2\gamma)/2$, $r_8=(3{\mathcal S}in^2\gamma + 3\cos^2\beta\cos^2
\gamma - 2)/2$. Since the mixed-state protocol does not need the population on $\ket{^1 E}$, we set $\gamma_0=0$ in the mixed-state trajectory.
The other components of the bloch vector are $r_{2}=-{\mathcal S}in^2\xi\cos\beta$ and $r_7={\mathcal S}in^2\xi{\mathcal S}in\beta$ with $\xi=\xi_0{\mathcal S}in^2(2\pi t/\tau)$;
$r_4$ is used to cannel the initial-final state coupling, which is determined by the equation $\Omega^i_c(r_4)=0$.
In FIGs. \ref{mpc} (a) and (b), we plot the control field $\Omega_p$ and the population on $\ket{^1 E}$ as a function of the dimensionless time
$t/\tau$. The red dash lines are the results given by the mixed-state protocol, and the green dot-dash lines and the blue solid lines are the numerical
results given by the pure-state protocol with $\gamma_0=0.1$ and $\gamma_0=0.01$ respectively. As shown in FIG.\ref{mpc} (a), the mixed-state
protocol only requires finite strength of the control field $\Omega_p^i$, even if the population on $\ket{^1 E}$ is zero. But, for the PIE scheme, the strength of
$\Omega_p$ tends to infinity when the population on $\ket{^1 E}$ goes to zero at $t/\tau=1$. The finite control field strength is the contribution from
designable $r_2$ and $r_7$. For $t/\tau=1$, it can be obtained that the final state satisfies $r_3=0$, $r_8=-1$ and $r_4=0$, and the time
derivative of these components of the general Bloch vector are zeros. Then, we can obtain the control field $\Omega_p^i$ as a function of
$r_2$ and $r_7$ from Eq.(\ref{cp3}), which reads
\begin{eqnarray}
\Omega_p^i=-\frac{2{\mathcal S}qrt{3}\,\partial_tr_7\,r_2 -{\mathcal S}qrt{3}\,\partial_tr_2\,r_7 + 2\,{\mathcal S}qrt{3}\,\Gamma\,r_2\,r_7}{2\,r_2\,(- r_2^2 + r_7^2 + 3)}.\label{limit1}
\end{eqnarray}
This equation illustrates that, even if $|^1E\rangle$ is unpopulated, we can still obtain a finite control field by selecting proper relation
between $r_2$ and $r_7$. Substituting the concrete parameterized $r_2$ and $r_7$ into Eq.(\ref{limit1}), we finally obtain $\Omega_p^i(\tau)=0$.
To illustrate this clearly, we further plot the ratio between the control fields for the mixed-state protocol $\Omega_p^i(t)$ and the pure-state protocol
$\Omega_p(t)$ with $\gamma_0=0.01$ in FIG. \ref{mpc} (c). When $t/\tau\rightarrow0$ and $t/\tau\rightarrow1$ (the
minimal population on $\ket{^1 E}$ for the pure-state protocol), the ratio goes to infinity, which verifies that the mixed-state protocol does not
require a extremely strong control field in the control process. As shown in FIG.\ref{mpc} (d), the MIE scheme also presents a better transfer efficiency
than the pure-state inverse engineering scheme.
{\mathcal S}ection{Conclusion}\label{conclusion}
To achieve feasible control of open quantum systems with high-accuracy, high-controllability, and high-speed, we propose a fast and robust
control scheme. After presenting the STAs based on dynamical invariants of open quantum system, we apply STAs
of open quantum systems to accelerate the adiabatic steady process. As a result, with the same form as the reference Liouvillian,
the control Liouvillian can drive the open quantum system from an initial steady state into a target steady state along a designed trajectory with
desired fidelity and pulse length. We highlight the high-controllability of the trajectory, which leads to robustness of control protocols to some
particular noises and eliminates untunable control manners.
Our scheme opens several promising avenues for further developments. Theoretically, it would be interesting to explore possible speed-limits
and trade-off relations for the open quantum systems \cite{Funo2019,Pietzonka2018}. Experimentally, due to the feasible of the MIE protocol,
the present protocol can be realized in various systems, such as cavity quantum electrodynamical systems\cite{Bason2012}, superconducting circuits
\cite{Vepsalainen2019}, nitrogen-vacancy centers \cite{Zhang2013} and spin-chains\cite{Zhou2020}.
This work is supported by National Natural Science Foundation of China (NSFC) under Grants No.
12075050 and 11775048.
\appendix
{\mathcal S}ection{The General Solution of Eq.(\ref{deq1})} \label{AA}
{The dynamical invariants $\hat{\mathcal I}(t)$ are defined as superoperators which satisfy the dynamical equation
\begin{eqnarray}
\partial_t \hat{\mathcal I}(t)-[\hat{\mathcal L}_c(t),\hat{\mathcal I}(t)]=0,\label{a10}
\end{eqnarray}
where $\hat{\mathcal L}(t)$ is the control Liouvillian superoperator.
Generally speaking, the superoperator $\hat {\mathcal I}(t)$ is non-Hermitian. We can introduce
a right basis $\{|D_\alpha^{(i)}\rangle\rangle\}$ and left basis $\{\langle\langle E_\alpha^{(i)}|\}$
in Hilbert-Schmidt space based on the Jordan canonical form. The left and right
basis always satisfy
\begin{eqnarray}
\hat {\mathcal I}\,|D_\alpha^{(i)}\rangle\rangle=\lambda_\alpha|D_\alpha^{(i)}\rangle\rangle+|D_\alpha^{(i-1)}\rangle\rangle,\label{a11}\\
\langle\langle E_\alpha^{(i)}| \hat {\mathcal I}=\lambda_\alpha\langle\langle E_\alpha^{(i)}|+\langle\langle E_\alpha^{(i+1)}|,\nonumber
\end{eqnarray}
where
\begin{eqnarray}
|D_\alpha^{(-1)}\rangle\rangle\equiv0,\label{rmax}\\
\langle\langle E_\alpha^{(n_\alpha)}|\equiv0\label{lmax},
\end{eqnarray}
with $i=0,1,...,n_\alpha-1$ ($n_\alpha$ is the dimension of the block $\alpha$). Here we assume that all of eigenvalues are nondegenerate, i.e.,
$\lambda_\alpha\neq\lambda_\beta$ for $\forall\, \alpha\neq\beta$. Moreover, the left and right bases satisfy the orthonormality condition
\begin{eqnarray*}
\langle\langle E_\alpha^{(i)}|D_\beta^{(j)}\rangle\rangle=\delta_{\alpha\beta}\delta_{ij}
\end{eqnarray*}
The right and left basis $|D_\alpha^{(0)}\rangle\rangle$ and $\langle\langle E_\alpha^{(n_\alpha-1)}|$ are the right and left eigenvector of $\hat {\mathcal I}(t)$
with the eigenvalue $\lambda_\alpha$. Taking the first derivative of Eq.(\ref{a11}) with respect to time, we have
\begin{eqnarray}
\partial_t\,\hat {\mathcal I}\,|D_\alpha^{(i)}\rangle\rangle+&\hat {\mathcal I}&\,|\partial_t\,D_\alpha^{(i)}\rangle\rangle
=\partial_t\lambda_\alpha|D_\alpha^{(i)}\rangle\rangle\nonumber\\
&+&\lambda_\alpha|\partial_tD_\alpha^{(i)}\rangle\rangle+|\partial_t
D_\alpha^{(i-1)}\rangle\rangle.
\end{eqnarray}
Multiplying above equation and $\langle\langle E_\beta^{(j)}|$ yields
\begin{eqnarray}
\langle\langle &E_\beta^{(j)}&|\partial_t\,\hat {\mathcal I}\,|D_\alpha^{(i)}\rangle\rangle=\partial_t\lambda_\alpha \delta_{\alpha\beta}
\delta_{ij}\nonumber\\
&&+(\lambda_\alpha-\lambda_\beta)\langle\langle E_\beta^{(j)}| \partial_tD_\alpha^{(i)}\rangle\rangle+\langle\langle E_\beta^{(j)}
|\partial_tD_\alpha^{(i-1)}\rangle\rangle\nonumber\\
&&-\langle\langle E_\beta^{(j+1)}|\partial_tD_\alpha^{(i)}\rangle\rangle.
\end{eqnarray}
Substituting Eq.(\ref{a10}) into the last equation, we obtain
\begin{eqnarray}
\partial_t\lambda_\alpha &\delta_{\alpha\beta}&\delta_{ij}=(\lambda_\alpha-\lambda_\beta)\langle\langle E_\beta^{(j)}| \hat O|D_\alpha^{(i)}\rangle\rangle\nonumber\\
&&+\langle\langle E_\beta^{(j)}|\hat O|D_\alpha^{(i-1)}\rangle\rangle-\langle\langle E_\beta^{(j+1)}|\hat O|D_\alpha^{(i)}\rangle\rangle,\label{lambdat}
\end{eqnarray}
where $\hat O\equiv\hat{\mathcal L}_c-\partial_t$.}
For $\alpha\neq\beta$, the above equation yields
\begin{eqnarray}
(\lambda_\alpha-&\lambda_\beta&)\langle\langle E_\beta^{(j)}| \hat O|D_\alpha^{(i)}\rangle\rangle
+\langle\langle E_\beta^{(j)}|\hat O|D_\alpha^{(i-1)}\rangle\rangle\nonumber\\
&&-\langle\langle E_\beta^{(j+1)}|\hat O|D_\alpha^{(i)}\rangle\rangle=0,\label{recurrence}
\end{eqnarray}
In the following, we illustrate that $\langle\langle E_\beta^{(j)}| \hat O|D_\alpha^{(i)}\rangle\rangle=0$ for $\forall\,i,\,j$. Firstly, we notice that
above equation is a recurrence equation about the indexes $i$ and $j$. Setting $i=0$ and $j=n_\beta-1$, we obtain
\begin{eqnarray}
\langle\langle E_\beta^{(n_\beta-1)}| \hat O|D_\alpha^{(0)}\rangle\rangle=0,\label{n0x}
\end{eqnarray}
where Eqs. (\ref{rmax}) and (\ref{lmax}) are used. Then, using the recurrence equation Eq.(\ref{recurrence}) again and considering $i=1$ and $j=n_\beta-1$, it results in
\begin{eqnarray*}
(\lambda_\alpha-\lambda_\beta)\langle\langle E_\beta^{(n_\beta-1)}| \hat O|D_\alpha^{(1)}\rangle\rangle
+\langle\langle E_\beta^{(n_\beta-1)}|\hat O|D_\alpha^{(0)}\rangle\rangle=0,
\end{eqnarray*}
Placing Eq.(\ref{n0x}) into above equation, we have
\begin{eqnarray}
\langle\langle E_\beta^{(n_\beta-1)}| \hat O|D_\alpha^{(1)}\rangle\rangle=0,\label{n1}
\end{eqnarray}
Repeating this procedure and checking every index $i=1,...,n_\alpha-1$, we conclude that the follow relations are insured,
\begin{eqnarray}
\langle\langle E_\beta^{(n_\beta-1)}| \hat O|D_\alpha^{(i)}\rangle\rangle=0,\,\forall i=0,...,n_\alpha-1.\label{nxx}
\end{eqnarray}
Secondly, we check $\langle\langle E_\beta^{(j)}| \hat O|D_\alpha^{(0)}\rangle\rangle$ by the same procedure as before. Setting $j=n_\beta-2$, Eq.(\ref{recurrence}) is turning into
\begin{eqnarray*}
(\lambda_\alpha-\lambda_\beta)\langle\langle E_\beta^{(n_\beta-2)}| &\hat O&|D_\alpha^{(0)}\rangle\rangle
+\langle\langle E_\beta^{(n_\beta-2)}|\hat O|D_\alpha^{(-1)}\rangle\rangle\nonumber\\
&&-\langle\langle E_\beta^{(n_\beta-1)}|\hat O|D_\alpha^{(0)}\rangle\rangle=0.
\end{eqnarray*}
Due to the relation Eqs. (\ref{rmax}) and (\ref{n0x}), we immediately have
\begin{eqnarray*}
\langle\langle E_\beta^{(n_\beta-2)}| \hat O|D_\alpha^{(0)}\rangle\rangle=0.
\end{eqnarray*}
Thus, reducing the index $j$ step by step and check all terms by Eq.(\ref{recurrence}), it can be concluded that
\begin{eqnarray}
\langle\langle E_\beta^{(j)}| \hat O|D_\alpha^{(0)}\rangle\rangle=0.\label{nyy}
\end{eqnarray}
Equipping with Eqs. (\ref{nxx}) and (\ref{nyy}), we check the other indexes $i$ and $j$ by the recurrence equation. For instance,
setting $i=1$ and $j=n_\beta-2$, we have
\begin{eqnarray*}
(\lambda_\alpha-\lambda_\beta)\langle\langle E_\beta^{(n_\beta-2)}|& \hat O&|D_\alpha^{(1)}\rangle\rangle
+\langle\langle E_\beta^{(n_\beta-2)}|\hat O|D_\alpha^{(0)}\rangle\rangle\nonumber\\
&&-\langle\langle E_\beta^{(n_\beta-1)} |\hat O|D_\alpha^{(1)}\rangle\rangle=0,
\end{eqnarray*}
It yields
\begin{eqnarray}
\langle\langle E_\beta^{(n_\beta-2)}| \hat O|D_\alpha^{(1)}\rangle\rangle=0,
\end{eqnarray}
by using Eqs. (\ref{nxx}) and (\ref{nyy}). As a result, when all of the indexes $i$ and $j$ are iterated, we conclude that
\begin{eqnarray}
\langle\langle E_\beta^{(j)}| \hat O|D_\alpha^{(i)}\rangle\rangle=0,\,\,\forall\,i,\,j,\label{a12}
\end{eqnarray}
for $\lambda_\alpha\neq\lambda_\beta$ for $\forall\, \alpha\neq\beta$.
In case of $\alpha=\beta$, Eq.(\ref{lambdat}) can be written as
\begin{eqnarray*}
\partial_t\lambda_\alpha\delta_{ij}=\langle\langle E_\alpha^{(j)}|\hat O|D_\alpha^{(i-1)}\rangle\rangle-
\langle\langle E_\alpha^{(j+1)}|\hat O|D_\alpha^{(i)}\rangle\rangle,
\end{eqnarray*}
Therefore, we can obtain the dynamical equation of the eigenvalues by taking sum over all indexes $i$ and $j$ of the Jordan block $\alpha$,
\begin{eqnarray*}
\partial_t\lambda_\alpha=\frac{1}{n_\alpha}{\mathcal S}um_{i,j=0}^{n_\alpha-1}\left(\langle\langle E_\alpha^{(j)}
|\hat O|D_\alpha^{(i-1)}\rangle\rangle-\langle\langle E_\alpha^{(j+1)}|\hat O|D_\alpha^{(i)}\rangle\rangle\right),
\end{eqnarray*}
If $i\neq j$, we find that
\begin{eqnarray*}
\langle\langle E_\alpha^{(j)}|\hat O|D_\alpha^{(i-1)}\rangle\rangle=\langle\langle E_\alpha^{(j+1)}
|\hat O|D_\alpha^{(i)}\rangle\rangle,
\end{eqnarray*}
Thus, we can write the dynamical equation of eigenvalues as
\begin{eqnarray*}
&\partial_t&\lambda_\alpha=\nonumber\\
&&\frac{1}{n_\alpha}\left({\mathcal S}um_{i=0}^{n_\alpha-1}\langle\langle E_\alpha^{(i)}
|\hat O|D_\alpha^{(i-1)}\rangle\rangle-{\mathcal S}um_{j=0}^{n_\alpha-1}\langle\langle E_\alpha^{(j+1)}|\hat O
|D_\alpha^{(j)}\rangle\rangle\right),
\end{eqnarray*}
Replacing the index $j$ by $k=j+1$, it yields
\begin{eqnarray*}
&\partial_t&\lambda_\alpha\nonumber\\
&=&\frac{1}{n_\alpha}\left({\mathcal S}um_{i=0}^{n_\alpha-1}\langle\langle E_\alpha^{(i)}
|\hat O|D_\alpha^{(i-1)}\rangle\rangle-{\mathcal S}um_{k=1}^{n_\alpha}\langle\langle E_\alpha^{(k)}|\hat O|D_\alpha^{(k-1)}
\rangle\rangle\right)\nonumber\\
&=&\frac{1}{n_\alpha}\left(\langle\langle E_\alpha^{(0)}|\hat O|D_\alpha^{(-1)}\rangle\rangle-\langle\langle
E_\alpha^{(n_\alpha)}|\hat O|D_\alpha^{(n_\alpha-1)}\rangle\rangle\right),
\end{eqnarray*}
Then, by considering Eqs. (\ref{rmax}) and (\ref{lmax}) , we finally obtain
\begin{eqnarray}
\partial_t\lambda_\alpha=0,
\end{eqnarray}
which implies that the dynamical invariants have indeed time-independent eigenvalues.
Let us consider now the solution of the master equation with the Liouvillian $\hat{ \mathcal L}_c(t)$, i.e.,
\begin{eqnarray}
\partial_t|\rho(t)\rangle\rangle=\hat{ \mathcal L}_c(t)|\rho(t)\rangle\rangle.\label{mel}
\end{eqnarray}
We expand the density matrix vector by the left basis vectors of the dynamical invariant $\hat{ \mathcal I }(t)$,
\begin{eqnarray}
|\rho(t)\rangle\rangle={\mathcal S}um_{\alpha=0}^{m-1} c_\alpha(t)|\Phi_\alpha(t)\rangle\rangle,\label{dmv}
\end{eqnarray}
with
\begin{eqnarray}
|\Phi_\alpha(t)\rangle\rangle={\mathcal S}um_{i=0}^{n_\alpha-1} b^\alpha_i(t)|D_\alpha^{(i)}(t)\rangle\rangle,\label{jex1}
\end{eqnarray}
where $m$ is the number of Jordan blocks. Inserting Eq.(\ref{dmv}) into Eq.(\ref{mel}), it yields
\begin{eqnarray}
{\mathcal S}um_{\alpha=0}^{m-1} \partial_tc_\alpha(t)|\Phi_\alpha(t)\rangle\rangle&+&{\mathcal S}um_{\alpha=0}^{m-1}
c_\alpha(t)\partial_t|\Phi_\alpha(t)\rangle\rangle\nonumber\\
&=&{\mathcal S}um_{\alpha=0}^{m-1} c_\alpha(t)\hat{ \mathcal L}_c(t)|\Phi_\alpha(t)\rangle\rangle.\label{effeq1}
\end{eqnarray}
Here we define a left vector $\langle\langle\Psi_\beta(t)|$ of the Jordan block $\beta$, which satisfy
$\langle\langle\Psi_\beta(t)|\Phi_\alpha(t)\rangle\rangle=\delta_{\alpha\beta}$. This left vector can be
expanded by the left basis vectors of the Jordan block $\beta$, i.e., $\langle\langle\Psi_\beta(t)|=
{\mathcal S}um_{j=0}^{n_\beta-1}a_j^{\beta}(t)\langle\langle E_\beta^{(j)}(t)|$. Projecting Eq.(\ref{effeq1}) in
$\langle\langle\Psi_\beta(t)|$, we obtain
\begin{eqnarray}
\partial_t c_\beta(t)={\mathcal S}um_{\alpha=0}^{m-1} c_\alpha(t)\langle\langle\Psi_\beta(t)|\hat{ O}(t)
|\Phi_\alpha(t)\rangle\rangle,\label{effeq2}
\end{eqnarray}
with $$\langle\langle\Psi_\beta(t)|\hat{ O}(t)|\Phi_\alpha(t)\rangle\rangle={\mathcal S}um_{i,j}a_j^{\beta}(t)b^\alpha_i(t)
\langle\langle E_\beta^{(j)}| \hat O|D_\alpha^{(i)}\rangle\rangle.$$ By making use of Eq.(\ref{a12}), we have
\begin{eqnarray}
\partial_t c_\beta(t)= c_\beta(t)\langle\langle\Psi_\beta(t)|\hat{ O}(t)
|\Phi_\beta(t)\rangle\rangle.\label{effeq3}
\end{eqnarray}
This results in a formal solution of the density matrix vector
\begin{eqnarray}
|\rho(t)\rangle\rangle={\mathcal S}um_{\alpha=0}^{m-1} c_\alpha(0)|\tilde\Phi_\alpha(t)\rangle\rangle,\label{fs1}
\end{eqnarray}
with ``dynamical modes'' $|\tilde\Phi_\alpha(t)\rangle\rangle=\exp(\eta_\alpha(t))|\Phi_\alpha(t)\rangle\rangle$,
where the phases are defined as
\begin{eqnarray}
\eta_\alpha(t)=\int_0^t d\tau \langle\langle\Psi_\alpha(\tau)|\hat{ O}(\tau)|\Phi_\alpha(\tau)\rangle\rangle.
\end{eqnarray}
In fact, we do not obtain a complete solution of the master equation Eq.(\ref{mel}), since the coefficients in
$|\phi_\alpha(t)\rangle\rangle$ have not been determined. This will be an interesting and open question for further\
investigation. But the formal solution in Eq.(\ref{fs1}) is enough to establish the shortcuts to adiabaticity of
open quantum systems, since the adiabatic theorem of open quantum systems just requires that the transition
between different Jordan blocks are forbidden \cite{Sarandy2005}. Putting undetermined coefficients in $|\phi_\alpha(t)\rangle\rangle$
aside, we only need to ensure the quantum state in the same Jordan block of $\hat{ \mathcal L}_c(t)$ at the beginning and end of
control process.
{\mathcal S}ection{Comparison with the Transitionless Driving Scheme of Open Quantum Systems}\label{AB}
In the following, we present a proof that our scheme is as general as, in some cases it is beyond, the G. Vacanti's method \cite{Vacanti2014}.
For the reference Liouvillian superoperator $\hat{ \mathcal L}_0(t)$, it is always possible to find a similarity transformation $C(t)$ such that
$\hat {\mathcal L}_0(t)$ is written in the canonical Jordan form
\begin{eqnarray}
\hat{\mathcal L}_J(t)=C^{-1}(t)\hat {\mathcal L}_0(t)C(t)=\text{diag}[J_1(t),...,J_N(t)],\label{lj}
\end{eqnarray}
where $J_\alpha(t )$ represents the Jordan block (of dimension $n_\alpha$) corresponding to the eigenvalue $\zeta_\alpha(t)$
of $\hat {\mathcal L}_0(t)$. The number $N$ of Jordan blocks is equal to the number of linear independent eigenvectors of $\hat {\mathcal L}_0(t)$ and the
similarity transformation is given by
\begin{eqnarray*}
C(t)={\mathcal S}um_{\alpha=1}^{N}{\mathcal S}um_{i=1}^{n_\alpha} |D_\alpha^{(i)}(t)\rangle\rangle\langle\langle {\mathcal S}igma_\alpha^{(i)}|,
\end{eqnarray*}
where $\{|D_\alpha^{(i)}(t)\rangle\rangle\}$ is a right instantaneous quasi-eigenbases of $\hat {\mathcal L}_0(t)$ associated
with the eigenvalues $\{\zeta_\alpha(t)\}$ which satisfies
\begin{eqnarray*}
\hat {\mathcal L}_0\,|D_\alpha^{(i)}\rangle\rangle=\zeta_\alpha|D_\alpha^{(i)}\rangle\rangle+|D_\alpha^{(i-1)}\rangle\rangle.
\end{eqnarray*}
And $\{ |{\mathcal S}igma_\alpha^{(i)}\rangle\rangle\}$ is a set of time-independent bases which is used to calculate the matrix form of $\hat {\mathcal L}_0(t)$.
Here we construct a dynamical invariant superoperator $\hat {\mathcal I}(t)$ which have same Jordan blocks structure. In other words,
$\hat {\mathcal I}(t)$ can be diagonal by the same similarity transformation $C(t)$, i.e.,
\begin{eqnarray}
\hat{\mathcal I}_J=C^{-1}(t)\hat{ \mathcal I}(t)C(t)=\text{diag}[I_1,...,I_N],\label{Ij}
\end{eqnarray}
where $I_\alpha$ is the Jordan block (of dimension $n_\alpha$) corresponding to the eigenvalue $\lambda_\alpha$ of $\hat {\mathcal I}(t)$.
The eigenvalues $\{\lambda_\alpha\}$ are time-independent. In fact,
we choose a dynamical invariant with the same Jordan blocks structure is equivalent to choose the trajectory of the mixed-state inverse
engineering as the adiabatic trajectory. Therefore, $\hat {\mathcal L}_0(t)$ and ${\hat{ \mathcal I}}(t)$ share common quasi-eigenvectors
$|D_{\alpha}^{(i)}(t)\rangle\rangle$, i.e.,
\begin{eqnarray*}
\hat {\mathcal I}\,|D_\alpha^{(i)}\rangle\rangle=\lambda_\alpha|D_\alpha^{(i)}\rangle\rangle+|D_\alpha^{(i-1)}\rangle\rangle.
\end{eqnarray*}
The dynamical invariants $\hat{\mathcal I}(t)$ satisfy the dynamical equation Eq.(\ref{a10})
Following the G. Vacanti method, we set that the control Liouvillian superoperator
can be written as $\hat{\mathcal L}(t)=\hat{\mathcal L}_0(t)+\hat{\mathcal L}_c(t)$, where $\hat{\mathcal L}_c(t)$ is the counterdiabatic
superoperator. Substituting Eq.(\ref{Ij}) into Eq.(\ref{a10}), it yields
\begin{eqnarray}
\partial_{t}\left(C\hat{\mathcal{I}}_{J}C^{-1}\right)&=&[C\hat{\mathcal{L}}_{J}C^{-1}+\hat{\mathcal{L}}_{c},C\hat{\mathcal{I}}_{J}C^{-1}]
\end{eqnarray}
where Eq.(\ref{lj}) has been used. Considering that $\hat{\mathcal{I}}_{J}$ is time-independent and taking the same similarity transformation
$C(t)$ on above equation, we obtain
\begin{eqnarray}
\left[C^{-1}\partial_{t}C,\hat{\mathcal{I}}_{J}\right]=\left[\hat{\mathcal{L}}_{J},\hat{\mathcal{I}}_{J}\right]
+\left[C^{-1}\,\hat{\mathcal{L}}_{c}C,\hat{\mathcal{I}}_{J}\right].\label{a11}
\end{eqnarray}
in which the fact $C^{-1}\partial_{t}C=-\partial_{t}C^{-1}\,C$ are considered. Since $\hat{\mathcal I}(t)$ and $\hat{\mathcal L}_0(t)$ have
same Jordan blocks structure, we have $ [\hat{\mathcal{L}}_{J},\hat{\mathcal{I}}_{J}]=0$. It is not difficult to see that,
if $ \hat{\mathcal{L'}}_{c}\equiv C^{-1}\,\hat{\mathcal{L}}_{c}C=C^{-1}\partial_{t}C$, Eq.(\ref{a11}) holds. We expand $ \hat{\mathcal{L'}}_{c}$
by the bases $\{ |{\mathcal S}igma_\alpha^{(i)}\rangle\rangle\}$, and separate it into two parts, $ \hat{\mathcal{L'}}_{c}= \hat{\mathcal{L'}}_{J}+
\hat{\mathcal{L'}}_{nd}$, where
\begin{eqnarray}
&&\hat{\mathcal{L'}}_{J}={\mathcal S}um_{\alpha,i,j}C_\alpha^{i,j} |{\mathcal S}igma_\alpha^{(i)}\rangle\rangle\langle\langle{\mathcal S}igma_\alpha^{(j)}|,\nonumber\\
&&\hat{\mathcal{L'}}_{nd}={\mathcal S}um_{\alpha\neq\beta,i,j}C_{\alpha,\beta}^{i,j} |{\mathcal S}igma_\alpha^{(i)}\rangle\rangle\langle\langle{\mathcal S}igma_\beta^{(j)}|,\nonumber
\end{eqnarray}
with $C_{\alpha,\beta}^{i,j}=\langle\langle{\mathcal S}igma_\alpha^{(j)}|C^{-1}\partial_{t}C|{\mathcal S}igma_\beta^{(i)}\rangle\rangle$. The superoperator
$\hat{\mathcal{L'}}_{nd}$ is used to forbid the transitions from $J_\alpha$ to $J_\beta$. Therefore, the counterdiabatic superoperator which
really required for STAs is
\begin{eqnarray}
\hat{\mathcal{L}}_{\text{tqd}}(t)=C(t)\hat{\mathcal{L'}}_{nd}(t)C^{-1}(t).\label{tqd}
\end{eqnarray}
Comparing Eq.(\ref{tqd}) with Eq. (15) in Ref.\cite{Vacanti2014}, we immediately find that $\hat{\mathcal{L}}_{\text{tqd}}(t)$
is the very counterdiabatic superoperator given by G. Vacanti and his co-authors \cite{Vacanti2014}.
Above simple proof illustrates that, if the trajectory of the general STAs based on the invariant theory of open quantum systems is chosen as the adiabatic trajectory,
our method is coincident with the transitionless quantum driving method proposed in Ref.\cite{Vacanti2014}.
However, the adiabatic trajectory is not the only choice of the trajectories in our scheme. There are many trajectories can be used
to inversely engineer the open quantum system. As shown by the example in Sec. \ref{secIIIB}, proper
trajectories can always produce reasonable and applicable control protocols, which helps us to overcome the difficulties met in the
control of microscopic or/and mesoscopic systems. Hence, the mixed-state inverse engineering is more general than the
transitionless quantum driving method of open quantum systems proposed by G. Vacanti \cite{Vacanti2014}.
The main difficulty in the G. Vacanti's method is how to realize the counterdiabatic
superoperator into a practical control. The MIE scheme solves those problems by selecting proper trajectories. According to the
symmetry of the open quantum systems, we can choose flexibly the form of the control Liouvillian, this make the
proposal widely applicable in the control of open quantum systems.
{\mathcal S}ection{The Instantaneous Steady State of $\hat{\mathcal L}_0(t)$} \label{AC}
We use the ``bra-ket'' notation for the superoperator to rewrite the master equation Eq.(\ref{me3}),
and reshape the density matrix into a $1\times 9$ complex vector. The density matrix vector can be written as $$\ket{\rho}\rangle=
(\rho_{-1\,-1},\rho_{-1\,2},\rho_{-1\,1},\rho_{2\,-1},\rho_{2\,2},\rho_{2\,1},\rho_{1\,-1},\rho_{1\,2},\rho_{1\,1})^{\text{T}}$$ with $\rho_{i\, j}=\bra{i}\hat \rho\ket{j}$.
In order to present an analytic result, we assume that $\Gamma_{-1}=\Gamma_{+1}\equiv\Gamma$ and $\omega_{2\rightarrow-1}=\omega_{2\rightarrow+1}
\equiv\omega_0$. At room temperature (T=300 K), the mean excitation number is $N=1.9\times10^{-33}$. For practice application, we choose the instantaneous steady state of the reference Liouvillian $\hat{ \mathcal L_0}$ as the trajectory
of inverse engineering, in this case the dephasing is the key obstacle for the performance of the protocol. The reference
Liouvillian superoperator $\hat{ \mathcal L_0}$ can be expressed as a $9\times 9$ matrix,
\begin{widetext}
\begin{equation}
\hat{ \mathcal L_0}=\hbar\Gamma\left(\begin{array}{ccccccccc} - 2\, \mathrm{N}\, & \mathrm{i}\mathrm{\Omega_p/\Gamma} & 0 & -\mathrm{i} \mathrm{\Omega_p/\Gamma} &
2\, \, \left(\mathrm{N} + 1\right) & 0 & 0 & 0 & 0\\ \mathrm{i} \mathrm{\Omega_p/\Gamma}& - \, \left(3\, \mathrm{N} + 2\right) &
\mathrm{i}\mathrm{\Omega_s/\Gamma} & 0 & - \mathrm{i} \mathrm{\Omega_p/\Gamma}& 0 & 0 & 0 & 0\\ 0 & \mathrm{i} \mathrm{\Omega_s/\Gamma}&
- 2\, \mathrm{N}\, & 0 & 0 & - \mathrm{i}\mathrm{\Omega_p/\Gamma} & 0 & 0 & 0\\ - \mathrm{i} \mathrm{\Omega_p/\Gamma} & 0
& 0 & - \, \left(3\, \mathrm{N} + 2\right) &\mathrm{i} \mathrm{\Omega_p/\Gamma} & 0 & - \mathrm{i}\mathrm{\Omega_s/\Gamma} &
0 & 0\\ 2\, \mathrm{N}\, & - \mathrm{i} \mathrm{\Omega_p/\Gamma}& 0 &\mathrm{i} \mathrm{\Omega_p/\Gamma} &
- 4\, \, \left(\mathrm{N} + 1\right) & \mathrm{i}\mathrm{\Omega_s/\Gamma} & 0 & - \mathrm{i}\mathrm{\Omega_s/\Gamma} &
2\, \mathrm{N}\, \\ 0 & 0 & - \mathrm{i} \mathrm{\Omega_p/\Gamma} & 0 &\mathrm{i} \mathrm{\Omega_s/\Gamma} &
- \, \left(3\, \mathrm{N} + 2\right) & 0 & 0 & - \mathrm{i}\mathrm{\Omega_s/\Gamma}\\ 0 & 0 & 0 &
-\mathrm{i} \mathrm{\Omega_s/\Gamma}& 0 & 0 & - 2\, \mathrm{N}\, &\mathrm{i} \mathrm{\Omega_p/\Gamma}
& 0\\ 0 & 0 & 0 & 0 & -\mathrm{i} \,\mathrm{\Omega_s/\Gamma} & 0 & \mathrm{i}\mathrm{\Omega_p/\Gamma} &
- \, \left(3\, \mathrm{N} + 2\right) &\mathrm{i} \mathrm{\Omega_s/\Gamma}\\ 0 & 0 & 0 & 0
& 2\, \, \left(\mathrm{N} + 1\right) & -\mathrm{i} \mathrm{\Omega_s/\Gamma} & 0 &
\mathrm{i} \mathrm{\Omega_s/\Gamma}& - 2\, \mathrm{N}\, \end{array}\right).\nonumber
\end{equation}
The steady state is obtained immediately by considering $\hat{ \mathcal L_0}\ket{\rho_0}\rangle=0$,
\begin{equation}
\ket{\rho_0}\rangle=\frac{1}{z}\left(\begin{array}{ccccccccc}
{\left(3\, \mathrm{N}^2 + 2\mathrm{N}\right)\, \left(\mathrm{N} + 1\right)\, {\mathrm{\Gamma}}^2 + \mathrm{N}\, {\mathrm{\Omega}}^2 + {\mathrm{\Omega_p}}^2}\\
{-{i}\, \mathrm{N}\, \mathrm{\Gamma}\, \mathrm{\Omega_p}}\\
{-\mathrm{\Omega_p}\, \mathrm{\Omega_s}}\\
{{i}\, \mathrm{N}\, \mathrm{\Gamma}\, \mathrm{\Omega_p}}\\
{\mathrm{N}^2\, \left(3\, \mathrm{N} + 2\right)\, {\mathrm{\Gamma}}^2 + \mathrm{N}\, {\mathrm{\Omega}}^2 }\\
{{i}\, \mathrm{N}\, \mathrm{\Gamma}\, \mathrm{\Omega_s}}\\
{-\mathrm{\Omega_p}\, \mathrm{\Omega_s}}\\
{-{i}\, \mathrm{N}\, \mathrm{\Gamma}\, \mathrm{\Omega_s}}\\
{\left(3\, \mathrm{N}^2 + 2\mathrm{N}\right)\, \left(\mathrm{N} + 1\right)\, {\mathrm{\Gamma}}^2 + \mathrm{N}\, {\mathrm{\Omega}}^2 + {\mathrm{\Omega_s}}^2}\\
\end{array}\right).\label{is3}
\end{equation}
with the normalized factor $$z={\left(3\, \mathrm{N} + 1\right)\, {\mathrm{\Omega}}^2 + \mathrm{N}\, {\mathrm{\Gamma}}^2\, {\left(3\, \mathrm{N} + 2\right)}^2}.$$
For $N=0$, the instantaneous steady state is the dark state of the Hamiltonian Eq.(\ref{rh3}), i.e. $\ket{\rho_0}\rangle=(\cos^2\theta,0,{\mathcal S}in\theta\cos\theta,0,0,0,
{\mathcal S}in\theta\cos\theta,0,{\mathcal S}in^2\theta)^{\text{T}}$.
We parameterize the adiabatic trajectory given by the instantaneous steady state of $\hat{ \mathcal L_0}$ via the generalized Bloch vector $\{r_k\}_{k=1}^8$, which expands the density matrix of the
three-level system as follows,
\begin{equation}
\rho(t)=\frac{1}{3}\left(\text{I}+{\mathcal S}qrt{3}{\mathcal S}um_{k=1}^8 r_k(t) T_k\right),
\end{equation}
where $\text{I}$ is a $3\times 3$ identity matrix, and $T_k$ denotes the regular Gellmann matrix
\begin{eqnarray}
&T_1=\left(\begin{array}{ccc}
0 & 1 & 0\\
1 & 0 & 0\\
0 & 0 & 0
\end{array}\right),\,
T_2=\left(\begin{array}{ccc}
0 & -i & 0\\
i & 0 & 0\\
0 & 0 & 0
\end{array}\right),\,
T_3=\left(\begin{array}{ccc}
1 & 0 & 0\\
0 & -1 & 0\\
0 & 0 & 0
\end{array}\right),\,
T_4=\left(\begin{array}{ccc}
0 & 0 & 1\\
0 & 0 & 0\\
1 & 0 & 0
\end{array}\right),\,\nonumber\\
&T_5=\left(\begin{array}{ccc}
0 & 0 & -i\\
0 & 0 & 0\\
i & 0 & 0
\end{array}\right),\,
T_6=\left(\begin{array}{ccc}
0 & 0 & 0\\
0 & 0 & 1\\
0 & 1 & 0
\end{array}\right),\,
T_7=\left(\begin{array}{ccc}
0 & 0 & 0\\
0 & 0 & -i\\
0 & i & 0
\end{array}\right),\,
T_8=\frac{1}{{\mathcal S}qrt{3}}\left(\begin{array}{ccc}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & -2
\end{array}\right).\label{Tx}
\end{eqnarray}
These $\{T_\mu\}$ span all traceless Hermitian matrices of the Lie algebra su(3). Thus, the Bloch vectors corresponding to
the instantaneous steady state
Eq.(\ref{is3}) are
\begin{eqnarray}
&&r_2={\mathcal S}qrt{3}\,\mathrm{N}\, \mathrm{\Gamma}\, \mathrm{\Omega_p}/z,\nonumber\\
&&r_3={\mathcal S}qrt{3}\,\left(\left(3\, \mathrm{N}^2 + 2\mathrm{N}\right)\, {\mathrm{\Gamma}}^2 + {\mathrm{\Omega_s}}^2 \right)/(2z),\nonumber\\
&&r_4=-{\mathcal S}qrt{3}\,\mathrm{\Omega_p}\, \mathrm{\Omega_s}/z,\nonumber\\
&&r_7=-{\mathcal S}qrt{3}\,\mathrm{N}\, \mathrm{\Gamma}\, \mathrm{\Omega_s}/z,\nonumber\\
&&r_8=-\left(\left(3\, \mathrm{N}^2 + 2\mathrm{N}\right)\, {\mathrm{\Gamma}}^2+2\,{\mathrm{\Omega_p}}^2 -{\mathrm{\Omega_s}}^2 \right)/(2z),\label{r38}
\end{eqnarray}
and the other components are zeros. Correspondingly, the dynamical invariants can be parameterized by the Bloch vector according to our proposal (see Eq.(\ref{di11})),
\begin{eqnarray}
\hat{\mathcal{I}}(t)=\frac{{\mathcal S}qrt{3}}{3}\,\Omega_{I}\left(\begin{array}{ccccccccc}
\left(\mathrm{r_{3}}+\frac{{\mathcal S}qrt{3}\,\mathrm{r_{8}}}{3}\right)+\frac{{\mathcal S}qrt{3}}{3} & 0 & 0 & 0 & \left(\mathrm{r_{3}}+\frac{{\mathcal S}qrt{3}\,\mathrm{r_{8}}}{3}\right)+\frac{{\mathcal S}qrt{3}}{3} & 0 & 0 & 0 & \left(\mathrm{r_{3}}+\frac{{\mathcal S}qrt{3}\,\mathrm{r_{8}}}{3}\right)+\frac{{\mathcal S}qrt{3}}{3}\\
\mathrm{i}\,\mathrm{r_{2}} & 0 & 0 & 0 & \mathrm{i}\,\mathrm{r_{2}} & 0 & 0 & 0 & \mathrm{i}\,\mathrm{r_{2}}\\
\mathrm{r_{4}} & 0 & 0 & 0 & \mathrm{r_{4}} & 0 & 0 & 0 & \mathrm{r_{4}}\\
-\mathrm{i}\,\mathrm{r_{2}} & 0 & 0 & 0 & -\mathrm{i}\,\mathrm{r_{2}} & 0 & 0 & 0 & -\mathrm{i}\,\mathrm{r_{2}}\\
\frac{{\mathcal S}qrt{3}}{3}\,-\left(\mathrm{r_{3}}-\frac{{\mathcal S}qrt{3}\,\mathrm{r_{8}}}{3}\right) & 0 & 0 & 0 & \frac{{\mathcal S}qrt{3}}{3}\,-\left(\mathrm{r_{3}}-\frac{{\mathcal S}qrt{3}\,\mathrm{r_{8}}}{3}\right) & 0 & 0 & 0 & \frac{{\mathcal S}qrt{3}}{3}\,-\left(\mathrm{r_{3}}-\frac{{\mathcal S}qrt{3}\,\mathrm{r_{8}}}{3}\right)\\
\mathrm{i}\,\mathrm{r_{7}} & 0 & 0 & 0 & \mathrm{i}\,\mathrm{r_{7}} & 0 & 0 & 0 & \mathrm{i}\,\mathrm{r_{7}}\\
\mathrm{r_{4}} & 0 & 0 & 0 & \mathrm{r_{4}} & 0 & 0 & 0 & \mathrm{r_{4}}\\
-\mathrm{i}\,\mathrm{r_{7}} & 0 & 0 & 0 & -\mathrm{i}\,\mathrm{r_{7}} & 0 & 0 & 0 & -\mathrm{i}\,\mathrm{r_{7}}\\
\frac{{\mathcal S}qrt{3}}{3}-\frac{2\,{\mathcal S}qrt{3}\,\mathrm{r_{8}}}{3} & 0 & 0 & 0 & \frac{{\mathcal S}qrt{3}}{3}-\frac{2\,{\mathcal S}qrt{3}\,\mathrm{r_{8}}}{3} & 0 & 0 & 0 & \frac{{\mathcal S}qrt{3}}{3}-\frac{2\,{\mathcal S}qrt{3}\,\mathrm{r_{8}}}{3}
\end{array}\right)
\end{eqnarray}
where $\Omega_I$ is an arbitrary nonzero constant.\
{\mathcal S}ection{The Control Parameters in the Control Liouvillian} \label{AD}
Considering the dynamical equation of the dynamical invariants Eq.(\ref{a10}), we can determine all of control parameters in the control Liouvillian $\hat{ \mathcal L_c}$,
\begin{eqnarray}
\Omega_{s}^i=n_{s}/d,\,\Omega_{p}^i=n_{p}/d,\,\Omega_{c}^i=n_{c}/d,\,N_{-1}^i=n_{-1}/d,\,N_{+1}^i=n_{+1}/d, \label{cp3}
\end{eqnarray}
in which
\begin{eqnarray*}
d&=&2\,{\mathcal S}qrt{3}\,\mathrm{r_4}^{4}\,C_{2}+4\,{\mathcal S}qrt{3}\,\mathrm{r_2}^{4}\,C_{1}+\mathrm{r_2}^{2}\,\left(12\,\mathrm{r_8}\,\left(\mathrm{r_4}^{2}+\mathrm{r_7}^{2}\right)+12\,{\mathcal S}qrt{3}\,\left(\mathrm{r_3^{2}}+\mathrm{r_8^{2}}\right)\,C_{1}\right)\\
&&+\mathrm{r_4}^{2}\,\left(6\,\mathrm{r_7}^{2}\,C_{2}-8\,{\mathcal S}qrt{3}\,\mathrm{r_3}\,C_{1}C_{2}\right)+8\,\mathrm{r_3}^{2}\,C_{2}\,C_{1}^{2}+4\,{\mathcal S}qrt{3}\,\mathrm{r_3}\,\mathrm{r_7}^{2}\,\left(\mathrm{r_2}^{2}-2\,\mathrm{r_7}^{2}\right)\\
&&-8\,\mathrm{r_2}\,\mathrm{r_4}\,\mathrm{r_7}\,\left({\mathcal S}qrt{3}\,\mathrm{r_7}^{2}+3\,C_{2}\,C_{4}\right)+8\,{\mathcal S}qrt{3}\,\mathrm{r_2}^{3}\,\mathrm{r_4}\,\mathrm{r_7}+12\,\mathrm{r_3}\,\mathrm{r_7}^{2}\,\left(\mathrm{r_3}-\left({\mathcal S}qrt{3}-2\right)\,\mathrm{r_8}\right)\,\left(\left(2\,{\mathcal S}qrt{3}+3\right)\,\mathrm{r_8}+{\mathcal S}qrt{3}\,\mathrm{r_3}\right),
\end{eqnarray*}
\begin{eqnarray*}
n_{s}&=&2\,{\mathcal S}qrt{3}\,\partial_{t}r_{2}\left(\mathrm{r_{4}}^{2}\,\left(\mathrm{r_{7}}^{2}-C_{1}C_{2}\right)+2\,\mathrm{r_{2}}^{2}\,\mathrm{r_{3}}\,C_{1}+\mathrm{r_{2}}\,\mathrm{r_{4}}\,\mathrm{r_{7}}\,\left(C_{2}+4\,\mathrm{r_{3}}\right)+2\,\mathrm{r_{3}}\,C_{2}\,\left(2\,\mathrm{r_{7}}^{2}+C_{1}^{2}\right)\right)\nonumber\\
&&+{\mathcal S}qrt{3}\,\partial_{t}r_{3}\left(-2\,\mathrm{r_{2}}^{3}\,C_{1}+2\,\mathrm{r_{2}}\,\left(-C_{1}^{2}C_{2}-2\,\mathrm{r_{7}}^{2}\,\left(C_{2}+\mathrm{r_{3}}\right)\right)-\mathrm{r_{4}}\,\mathrm{r_{7}}\,\left(4\,\mathrm{r_{2}}^{2}+\mathrm{r_{4}}^{2}-\mathrm{r_{7}}^{2}+2\,{\mathcal S}qrt{3}\,\mathrm{r_{8}}\,C_{1}\right)\right)\\
&&+\partial_{t}r_{4}\left(-\mathrm{r_{7}}\,\left(4\,\mathrm{r_{3}}\,C_{1}^{2}-6\,\mathrm{r_{4}}^{2}\,C_{3}\right)-4\,{\mathcal S}qrt{3}\,\mathrm{r_{3}}\,\mathrm{r_{7}}\,\left(\mathrm{r_{2}}^{2}+2\,\mathrm{r_{7}}^{2}\right)-2\,{\mathcal S}qrt{3}\,\mathrm{r_{2}}\,\mathrm{r_{4}}\,\left(2\,C_{1}C_{2}+3\,\mathrm{r_{7}}^{2}\right)\right)\\&&
+2\,{\mathcal S}qrt{3}\,\partial_{t}r_{7}\left(2\,\mathrm{r_{3}}\,\mathrm{r_{4}}\,C_{1}C_{2}-\mathrm{r_{4}}^{3}\,C_{2}-\left(2\,{\mathcal S}qrt{3}\,\mathrm{r_{2}}\,\mathrm{r_{8}}-\mathrm{r_{4}}\,\mathrm{r_{7}}\right)\,\left(\mathrm{r_{2}}\,\mathrm{r_{4}}+2\,\mathrm{r_{3}}\,\mathrm{r_{7}}\right)\right)\\
&&+\partial_{t}r_{8}\,\left(\mathrm{r_{2}}\,\left(2\,C_{1}C_{2}^{2}+2\,\mathrm{r_{4}}^{2}\,C_{2}+2\,\mathrm{r_{7}}^{2}\,\left(\mathrm{r_{3}}+C_{1}\right)\right)+2\,\mathrm{r_{2}}^{3}\,C_{2}+\mathrm{r_{4}}\,\mathrm{r_{7}}\,\left(4\,C_{2}^{2}+2\,\mathrm{r_{2}}^{2}+\mathrm{r_{4}}^{2}+\mathrm{r_{7}}^{2}+2\,{\mathcal S}qrt{3}\,\mathrm{r_{8}}\,C_{1}\right)\right)\\
&&+\Gamma\,\mathrm{r_{2}}^{2}\,\left(14\,\mathrm{r_{4}}\,\mathrm{r_{7}}+2\,\mathrm{r_{4}}\,\mathrm{r_{7}}\,\left(C_{3}+3\,{\mathcal S}qrt{3}\,\mathrm{r_{3}}\right)\right)-\mathrm{r_{7}}^{3}\,\left(2\,\mathrm{r_{4}}+2\,\mathrm{r_{4}}\,\left(C_{3}-6\,{\mathcal S}qrt{3}\,\mathrm{r_{3}}\right)\right)\\
&&-\Gamma\,\mathrm{r_{7}}\,\left(\mathrm{r_{4}}^{3}\,\left(8\,C_{3}-4\right)-\mathrm{r_{4}}\,\left(4\,{\mathcal S}qrt{3}\,\mathrm{r_{3}}^{3}-44\,\mathrm{r_{3}}^{2}\,\mathrm{r_{8}}+4\,\mathrm{r_{3}}^{2}+4\,{\mathcal S}qrt{3}\,\mathrm{r_{3}}\,\mathrm{r_{8}}^{2}+16\,{\mathcal S}qrt{3}\,\mathrm{r_{3}}\,\mathrm{r_{8}}-12\,\mathrm{r_{8}}^{3}-12\,\mathrm{r_{8}}^{2}\right)\right)\\
&&-\Gamma\,\mathrm{r_{2}}^{3}\,\left(4\,{\mathcal S}qrt{3}\,\mathrm{r_{8}}-8\,\mathrm{r_{3}}+4\,\mathrm{r_{8}}\,C_{2}\right)+\Gamma\,\mathrm{r_{2}}\,\mathrm{r_{7}}^{2}\,\left(16\,\mathrm{r_{3}}+4\,{\mathcal S}qrt{3}\,\mathrm{r_{8}}+4\,\mathrm{r_{8}}\,C_{2}\right)\\
&&+\Gamma\,\mathrm{r_{2}}\,\mathrm{r_{4}}^{2}\,\left(8\,{\mathcal S}qrt{3}\,\mathrm{r_{7}}^{2}+\left(2-6\,{\mathcal S}qrt{3}\mathrm{r_{3}}+14\,\mathrm{r_{8}}\right)\,C_{2}\right)-\Gamma\,\mathrm{r_{2}}\,C_{1}C_{2}\,\left(4\,{\mathcal S}qrt{3}\,\mathrm{r_{8}}-8\,\mathrm{r_{3}}+4\,\mathrm{r_{8}}\,C_{2}\right),
\end{eqnarray*}
\begin{eqnarray*}
n_{p}&=&\partial_{t}r_{2}\,\left(6\,\mathrm{r_4}^{3}\,C_{2}-\mathrm{r_4}\,\left(12\,\mathrm{r_3}\,C_{1}C_{2}-6\,{\mathcal S}qrt{3}\,\mathrm{r_7}^{2}\,C_{3}\right)+6\,\mathrm{r_2}\,\mathrm{r_7}\,\left(\mathrm{r_4}^{2}+{\mathcal S}qrt{3}\,C_{1}C_{3}\right)+6\,\mathrm{r_2}^{2}\,\mathrm{r_4}\,C_{1}\right)\\
&&+\partial_{t}r_{3}\,\left(3\,\mathrm{r_4}^{2}\,\mathrm{r_7}\,C_{2}-6\,\mathrm{r_2}^{2}\,\mathrm{r_7}\,C_{1}-3\,\mathrm{r_2}\,\mathrm{r_4}\,\left(\mathrm{r_4}^{2}+3\,\mathrm{r_7}^{2}-2\,C_{1}\,\left(\mathrm{r_3}+C_{2}\right)\right)+6\,\mathrm{r_7}\,-C_{1}\,\left(\mathrm{r_3}\,C_{2}-\frac{\mathrm{r_7}^{2}}{2}\right)\right)\\
&&+6\,\partial_{t}r_{4}\,\left(-2\,C_{1}\,\mathrm{r_2}^{3}-3\,\mathrm{r_4}\,\mathrm{r_7}\,\mathrm{r_2}^{2}+\left(-2\,{\mathcal S}qrt{3}\,\mathrm{r_8}\,\mathrm{r_4}^{2}-C_{1}\,\left(4\,\mathrm{r_3}^{2}+\mathrm{r_7}^{2}\right)\right)\,\mathrm{r_2}-4\,\mathrm{r_3}\,\mathrm{r_4}\,\mathrm{r_7}\,C_{2}\right)\\
&&+\partial_{t}r_{7}\,\left(\mathrm{r_2}^{2}\,\left(-12\,C_{1}C_{2}+6\,\mathrm{r_4}^{2}\right)-24\,\mathrm{r_3}^{2}\,C_{1}C_{2}+12\,\mathrm{r_3}\,\mathrm{r_4}^{2}\,C_{2}+12\,\mathrm{r_3}\,\mathrm{r_7}^{2}\,C_{1}+6\,\mathrm{r_2}\,\mathrm{r_4}\,\mathrm{r_7}\,\left(3\,C_{1}-2\,\mathrm{r_3}\right)\right)\\
&&+\partial_{t}r_{8}\,\left({\mathcal S}qrt{3}\,\mathrm{r_4}^{2}\,\mathrm{r_7}\,C_{2}-4\,{\mathcal S}qrt{3}\,\mathrm{r_2}^{2}\,\mathrm{r_7}\,C_{2}-{\mathcal S}qrt{3}\,\mathrm{r_2}\,\mathrm{r_4}\,\left(2\,\left(\mathrm{r_3}+C_{2}\right)^{2}-5\,\mathrm{r_7}^{2}\right)-{\mathcal S}qrt{3}\,\mathrm{r_2}\,\mathrm{r_4}\,\left(2\,\mathrm{r_2}^{2}-\mathrm{r_4}^{2}\right)\right)\\
&&+\partial_{t}r_{8}\,\left(-7\,{\mathcal S}qrt{3}\,\mathrm{r_3}\,\mathrm{r_7}\,\left(2\,\mathrm{r_3}^{2}-\mathrm{r_7}^{2}\right)+2\,{\mathcal S}qrt{3}\,\mathrm{r_3}\,\mathrm{r_8}\,\left(\mathrm{r_2}\,\mathrm{r_4}-3\,\mathrm{r_7}\,\mathrm{r_8}\right)-3\,\mathrm{r_7}\,\mathrm{r_8}\,\left(4\,\mathrm{r_3}-\mathrm{r_7}\right)\,\left(4\,\mathrm{r_3}+\mathrm{r_7}\right)\right)\\
&&+\Gamma\,\mathrm{r_2}\,\left(2\,\mathrm{r_7}^{2}\,\left(-2\,\mathrm{r_3}+5\,C_{1}+7\,{\mathcal S}qrt{3}\right)-4\,{\mathcal S}qrt{3}\,\left(\mathrm{r_8}+1\right)\,\left(4\,\mathrm{r_3}^{2}+C_{1}C_{2}\right)+36\,\mathrm{r_3}\,\mathrm{r_4}\,\left(\mathrm{r_3}^{2}+\mathrm{r_8}^{2}\right)\right)\\
&&+\Gamma\,\left(-\mathrm{r_7}^{3}\,\left(-4\,\mathrm{r_8}-10\,C_{4}+2\,{\mathcal S}qrt{3}\,C_{2}C_{3}\right)+\mathrm{r_2}^{2}\,\left(2\,\mathrm{r_7}\,C_{2}^{2}-12\,\mathrm{r_7}\,\left(3\,\mathrm{r_8}-2\,\mathrm{r_4}^{2}\right)+2\,\mathrm{r_7}\,C_{2}\,\left(2\,\mathrm{r_3}+{\mathcal S}qrt{3}\right)\right)\right)\\
&&+\Gamma\,\left(\mathrm{r_2}^{3}\,\left(18\,\mathrm{r_4}\,C_{1}-2\,{\mathcal S}qrt{3}\,\mathrm{r_4}\,\left(2\,\mathrm{r_8}+1\right)\right)+4\,{\mathcal S}qrt{3}\,\left(4\,\mathrm{r_8}+1\right)\,\mathrm{r_2}\,\mathrm{r_4}^{3}\right)\\
&&-\Gamma\,\mathrm{r_7}\,\left(\mathrm{r_4}^{2}\,C_{2}\,\left(2\,{\mathcal S}qrt{3}-30\,\mathrm{r_3}+2{\mathcal S}qrt{3}\,\mathrm{r_8}\right)-4\,\mathrm{r_3}\,\left(C_{2}\,\left(2\,\mathrm{r_8}-5\,C_{4}\right)+\mathrm{r_3}\,C_{2}\,\left(5\,C_{3}-2\,\mathrm{r_3}\right)-{\mathcal S}qrt{3}\,\mathrm{r_8}\,C_{1}C_{2}\right)\right),
\end{eqnarray*}
\begin{eqnarray*}
n_{c}&=&\partial_{t}r_{2}\,\left(6\,\mathrm{r_2}\,\mathrm{r_4}\,\left(\mathrm{r_7}^{2}+{\mathcal S}qrt{3}\,C_{1}C_{4}\right)+12\,\mathrm{r_3}\,\mathrm{r_7}\,\left(\mathrm{r_4}^{2}+2\,\mathrm{r_7}^{2}\right)-6\,\mathrm{r_2}^{2}\,\mathrm{r_7}\,C_{1}+12\,\mathrm{r_3}\,\mathrm{r_7}\,C_{1}^{2}\right)\\
&&+\partial_{t}r_{3}\,\left(\mathrm{r_4}\,\left(6\,\mathrm{r_3}\,C_{1}^{2}+3\,\mathrm{r_7}^{2}\,C_{2}\right)-3\,\mathrm{r_4}^{3}\,C_{1}-6\,\mathrm{r_2}^{2}\,\mathrm{r_4}\,C_{1}-\mathrm{r_2}\,\mathrm{r_7}\,\left(3\,\mathrm{r_4}^{2}+9\,\mathrm{r_7}^{2}+6\,C-1\,\left(\mathrm{r_3}-C_{1}\right)\right)\right)\\
&&+\partial_{t}r_{4}\,\left(12\,\mathrm{r_3}\,\mathrm{r_4}^{2}\,C_{1}-\mathrm{r_2}^{2}\,\left(12\,C_{1}^{2}+18\,\mathrm{r_7}^{2}\right)-24\,\mathrm{r_3}^{2}\,\left(C_{1}^{2}+2\,\mathrm{r_7}^{2}\right)-6\,{\mathcal S}qrt{3}\,\mathrm{r_2}\,\mathrm{r_4}\,\mathrm{r_7}\,C_{4}\right)\\
&&+\partial_{t}r_{7}\,\left(6\,\mathrm{r_7}\,\left(\mathrm{r_2}\,\left(\mathrm{r_2}\,\mathrm{r_4}-2\,\mathrm{r_3}\,\mathrm{r_7}\right)-2\,{\mathcal S}qrt{3}\,\mathrm{r_3}\,\mathrm{r_4}\,C_{4}\right)+12\,\mathrm{r_2}^{3}\,C_{1}+6\,\mathrm{r_2}\,C_{1}\,\left(4\,\mathrm{r_3}^{2}+\mathrm{r_4}^{2}\right)\right)\\
&&+\partial_{t}r_{8}\,\left(\left(3\,\mathrm{r_8}+3\,{\mathcal S}qrt{3}\,\mathrm{r_3}\right)\,\mathrm{r_4}^{3}+\left(3\,\mathrm{r_7}^{2}\,\left(\mathrm{r_8}+3\,{\mathcal S}qrt{3}\,\mathrm{r_3}\right)+12\,\mathrm{r_2}^{2}\,\mathrm{r_8}-6\,\mathrm{r_3}\,\left(\mathrm{r_8}+{\mathcal S}qrt{3}\,\mathrm{r_3}\right)\,\left(\mathrm{r_3}-{\mathcal S}qrt{3}\,\mathrm{r_8}\right)\right)\,\mathrm{r_4}\right)\\
&&+\partial_{t}r_{8}\,\left(\mathrm{r_2}\,\mathrm{r_7}\,\left(6\,{\mathcal S}qrt{3}\,\mathrm{r_2}^{2}+4\,{\mathcal S}qrt{3}\,\mathrm{r_3}^{2}+{\mathcal S}qrt{3}\,\mathrm{r_4}^{2}-3\,{\mathcal S}qrt{3}\,\mathrm{r_7}^{2}\right)+6\,\frac{{\mathcal S}qrt{3}}{3}\,\mathrm{r_2}\,\mathrm{r_7}\,C_{4}\,\left(\mathrm{r_8}+2\,C_{3}\right)\right)\\
&&+\Gamma\,\left(\mathrm{r_2}^{2}\,\left(24\,\mathrm{r_4}\,\mathrm{r_7}^{2}-6\,\mathrm{r_4}\,\left(\mathrm{r_8}-{\mathcal S}qrt{3}\,\mathrm{r_3}\right)\,\left({\mathcal S}qrt{3}\,\mathrm{r_3}-5\,\mathrm{r_8}+1\right)\right)-6\,\mathrm{r_2}^{3}\,\mathrm{r_7}\,\left(\mathrm{r_3}+{\mathcal S}qrt{3}\,\mathrm{r_8}-{\mathcal S}qrt{3}\right)\right)\\
&&+\Gamma\,\left(\mathrm{r_2}\,\left(\left(6\,C_{2}+6\,{\mathcal S}qrt{3}\right)\,\mathrm{r_7}^{3}+\left(\mathrm{r_4}^{2}\,\left(8\,{\mathcal S}qrt{3}\,C_{4}+4\,{\mathcal S}qrt{3}\right)+4\,{\mathcal S}qrt{3}\,\left({\mathcal S}qrt{3}\,C_{1}C_{4}+4\,\mathrm{r_3}^{2}\right)+4\,{\mathcal S}qrt{3}\,C_{2}^{2}\,C_{1}\right)\,\mathrm{r_7}\right)\right)\\
&&+\Gamma\,\left(\mathrm{r_4}\,\mathrm{r_7}^{2}\,\left(6\,C_{4}+{\mathcal S}qrt{3}\,C_{4}\,\left(18\,\mathrm{r_3}+2\,{\mathcal S}qrt{3}\,\mathrm{r_8}\right)\right)-6\,\mathrm{r_4}^{3}\,C_{4}\left(C_{4}+2\right)+12\,\mathrm{r_3}\,C_{1}\,C_{4}\,\mathrm{r_4}\,\left(C_{4}-1\right)\right),
\end{eqnarray*}
\begin{eqnarray*}
n_{-1}&=&\partial_{t}r_{2}\,\left(6\,{\mathcal S}qrt{3}\,\mathrm{r_4}\,\mathrm{r_7}\,\left(C_{1}^{2}-3\,\mathrm{r_2}^{2}+\mathrm{r_4}^{2}+2\,\mathrm{r_7}^{2}\right)-6\,{\mathcal S}qrt{3}\,\mathrm{r_2}^{3}\,C_{1}-{\mathcal S}qrt{3}\,\mathrm{r_2}\,\left(6\,C_{1}^{2}C_{2}-12\,\mathrm{r_4}^{2}\,C_{1}+12\,\mathrm{r_7}^{2}\,C_{2}\right)\right)\\
&&+\partial_{t}r_{3}\,\left(-3\,{\mathcal S}qrt{3}\,\mathrm{r_2}^{2}\,\left(\left(\mathrm{r_4}^{2}-\mathrm{r_7}^{2}\right)+6\,\mathrm{r_3}\,C_{1}\right)-{\mathcal S}qrt{3}\,\mathrm{r_7}^{2}\,\left(9\,\mathrm{r_3}+\left({\mathcal S}qrt{3}+2\right)\,\mathrm{r_8}\right)\,\left(\mathrm{r_3}-\left({\mathcal S}qrt{3}-2\right)\,\mathrm{r_8}\right)\right)\\
&&+\partial_{t}r_{3}\,\left(3\,{\mathcal S}qrt{3}\,\left(3\,\mathrm{r_7}^{4}-\mathrm{r_4}^{4}\right)-6\,{\mathcal S}qrt{3}\,\mathrm{r_3}\,C_{2}\,C_{1}^{2}+9\,\mathrm{r_4}^{2}\,C_{1}C_{3}-24\,{\mathcal S}qrt{3}\,\mathrm{r_2}\,\mathrm{r_3}\,\mathrm{r_4}\,\mathrm{r_7}\right)\\
&&+\partial_{t}r_{4}\,\left(6\,{\mathcal S}qrt{3}\,\mathrm{r_7}\,\mathrm{r_2}\,\left(C_{1}^{2}-3\,\mathrm{r_4}^{2}+2\,\mathrm{r_7}^{2}+\mathrm{r_2}^{2}\right)+{\mathcal S}qrt{3}\,\mathrm{r_4}\,\left(6\,C_{1}\left(\mathrm{r_4}^{2}-2\,\mathrm{r_2}^{2}\right)-12\,\mathrm{r_3}\,\left(C_{1}^{2}+2\,\mathrm{r_7}^{2}\right)\right)\right)\\
&&+\partial_{t}r_{7}\,\left({\mathcal S}qrt{3}\,\left(6\,\mathrm{r_7}\,\left(2\,{\mathcal S}qrt{3}\,\mathrm{r_8}\,\mathrm{r_2}^{2}-{\mathcal S}qrt{3}\,\mathrm{r_4}^{2}C_{4}\right)+6\,\mathrm{r_2}\,\mathrm{r_4}\,\left(C_{1}^{2}+\mathrm{r_2}^{2}+\mathrm{r_4}^{2}-2\,\mathrm{r_7}^{2}\right)\right)\right)\\
&&+\partial_{t}r_{8}\,\left(-\mathrm{r_2}^{2}\,\left(-\mathrm{r_2}^{2}+3\,\mathrm{r_4}^{2}+9\,\mathrm{r_7}^{2}+6\,C_{1}\,\left(\mathrm{r_3}+C_{2}\right)\right)-6\,\mathrm{r_3}\,C_{2}\,C_{1}^{2}\right)\\
&&+\partial_{t}r_{8}\,\left(3\,\mathrm{r_4}^{2}\,\left(\mathrm{r_4}^{2}+3\,\mathrm{r_7}^{2}-C_{1}^{2}\right)+6\,\mathrm{r_7}^{4}-\mathrm{r_7}^{2}\,\left(9\,\mathrm{r_3}+\left({\mathcal S}qrt{3}+2\right)\,\mathrm{r_8}\right)\,\left(\mathrm{r_3}-\left({\mathcal S}qrt{3}-2\right)\,\mathrm{r_8}\right)-24\,{\mathcal S}qrt{3}\,\mathrm{r_2}\,\mathrm{r_4}\,\mathrm{r_7}\,\mathrm{r_8}\right)\\
&&+\Gamma\,\left({\mathcal S}qrt{3}\,\mathrm{r_2}^{4}\,\left(2\,{\mathcal S}qrt{3}\,\left(2\,\mathrm{r_8}-1\right)-6\,C_{1}\right)-24\,\mathrm{r_2}\,\mathrm{r_4}\,\mathrm{r_7}\,\left({\mathcal S}qrt{3}\,\left(\mathrm{r_2}^{2}-\mathrm{r_4}^{2}\right)-{\mathcal S}qrt{3}\,C_{4}+2\,{\mathcal S}qrt{3}\,\left(\mathrm{r_3}^{2}-\mathrm{r_8}^{2}\right)\right)\right)\\
&&+\Gamma\,\left(-\mathrm{r_2}^{2}\,\left(6\,C_{1}^{2}\,\left(3\,C_{3}+2\,\mathrm{r_8}-1\right)+{\mathcal S}qrt{3}\,\mathrm{r_4}^{2}\,\left(2\,{\mathcal S}qrt{3}\,\left(2\,\mathrm{r_8}-1\right)-18\,C_{1}\right)+6\,{\mathcal S}qrt{3}\,\mathrm{r_7}^{2}\,\left(C_{2}+{\mathcal S}qrt{3}\right)\right)\right)\\
&&+\Gamma\,\left(12\,\left(1-C_{1}\right)\,\mathrm{r_4}^{4}+\left(18\,\mathrm{r_7}^{2}\,\left(C_{4}+1\right)-2\,{\mathcal S}qrt{3}\,C_{1}\,\left(4\,\mathrm{r_3}+C_{2}\right)\,\left(-2\,C_{1}-{\mathcal S}qrt{3}\,\left(\mathrm{r_8}-1\right)\right)\right)\,\mathrm{r_4}^{2}\right)\\
&&+\Gamma\,\left(12\,\mathrm{r_3}\,C_{2}\,C_{1}^{2}-12\,\mathrm{r_7}^{4}+18\,\mathrm{r_7}^{2}\,\left(\mathrm{r_3}+\left({\mathcal S}qrt{3}+2\right)\,\mathrm{r_8}\right)\,\left(\mathrm{r_3}-\left({\mathcal S}qrt{3}-2\right)\,\mathrm{r_8}\right)\right)\,\left(1-C_{4}\right),
\end{eqnarray*}
\begin{eqnarray*}
n_{+1}&=&\partial_{t}r_{2}\,\left(6\,\mathrm{r_2}\,\left(\mathrm{r_7}^{2}\,C_{3}-\mathrm{r_4}^{2}\,C_{4}\right)-2\,{\mathcal S}qrt{3}\,\mathrm{r_4}\,\mathrm{r_7}\,\left(-2\,\mathrm{r_2}^{2}+4\,\mathrm{r_3}^{2}+\mathrm{r_4}^{2}+\mathrm{r_7}^{2}\right)\right)\\
&&+\partial_{t}r_{3}\,\left(\left(\mathrm{r_4}^{2}+\mathrm{r_7}^{2}\right)\,\left(6\,\mathrm{r_3}\,\mathrm{r_8}+{\mathcal S}qrt{3}\,\left(\mathrm{r_4}^{2}-\mathrm{r_7}^{2}\right)\right)+2\,{\mathcal S}qrt{3}\,\left(\mathrm{r_4}^{2}\,\left(\mathrm{r_2}+\mathrm{r_3}\right)^{2}-\mathrm{r_7}^{2}\,\left(\mathrm{r_2}-\mathrm{r_3}\right)^{2}\right)\right)\\
&&+\partial_{t}r_{4}\,\left(4\,{\mathcal S}qrt{3}\,\left(\mathrm{r_2}^{2}+2\,\mathrm{r_3}^{2}\right)\,\mathrm{r_4}\,C_{1}-4\,{\mathcal S}qrt{3}\,\mathrm{r_3}\,\mathrm{r_4}\,\left(\mathrm{r_4}^{2}-2\,\mathrm{r_7}^{2}\right)-2\,{\mathcal S}qrt{3}\,\mathrm{r_2}\,\mathrm{r_7}\,\left(2\,\mathrm{r_2}^{2}+4\,\mathrm{r_3}^{2}-3\,\mathrm{r_4}^{2}+\mathrm{r_7}^{2}\right)\right)\\
&&+\partial_{t}r_{7}\,\left(-4\,{\mathcal S}qrt{3}\,\mathrm{r_2}^{2}\,\mathrm{r_7}\,C_{2}-4\,{\mathcal S}qrt{3}\,\mathrm{r_3}\,\mathrm{r_7}\,\left(2\,\mathrm{r_4}^{2}-\mathrm{r_7}^{2}+2\,\mathrm{r_3}\,C_{2}\right)-2\,{\mathcal S}qrt{3}\,\mathrm{r_2}\,\mathrm{r_4}\,\left(2\,\mathrm{r_2}^{2}+4\,\mathrm{r_3}^{2}+\mathrm{r_4}^{2}-3\,\mathrm{r_7}^{2}\right)\right)\\
&&+\partial_{t}r_{8}\,\left(4\,\left(\mathrm{r_2}^{2}+C_{1}C_{2}\right)\,\left(\mathrm{r_2}^{2}+2\,\mathrm{r_3}^{2}\right)-\left(2\,\mathrm{r_3}^{2}+\mathrm{r_4}^{2}+\mathrm{r_7}^{2}\right)\,\left(\mathrm{r_4}^{2}+\mathrm{r_7}^{2}\right)+8\,{\mathcal S}qrt{3}\,\mathrm{r_2}\,\mathrm{r_4}\,\mathrm{r_7}\,\mathrm{r_8}-6\,{\mathcal S}qrt{3}\,\mathrm{r_3}\,\mathrm{r_8}\,\left(\mathrm{r_4}^{2}-\mathrm{r_7}^{2}\right)\right)\\
&&+\Gamma\,\left(\left(2\,\mathrm{r_8}+6\,{\mathcal S}qrt{3}\,\mathrm{r_3}+2\right)\,\mathrm{r_7}^{4}+4\,\left(\mathrm{r_2}^{4}-\mathrm{r_4}^{4}\right)\,\left(1-C_{4}\right)\right)\\
&&+\Gamma\,\left(-\mathrm{r_2}^{2}\,\left(6\,\mathrm{r_4}^{2}\,\left(C_{4}+1\right)+2\,\mathrm{r_7}^{2}\,\left({\mathcal S}qrt{3}\,C_{2}-3\right)-12\,\left(\mathrm{r_3}^{2}-\mathrm{r_8}^{2}\right)\,(1-C_{4})\right)\right)\\
&&+\Gamma\,\left(4\,\mathrm{r_3}\,\mathrm{r_4}^{2}\,\left(\mathrm{r_3}-3\,{\mathcal S}qrt{3}\,\mathrm{r_8}\right)\,\left(1-C_{4}\right)-8\,{\mathcal S}qrt{3}\,\mathrm{r_2}\,\mathrm{r_4}\,\mathrm{r_7}\,\left(\mathrm{r_3}^{2}-\mathrm{r_8}^{2}+\mathrm{r_4}^{2}-\mathrm{r_7}^{2}+2\,{\mathcal S}qrt{3}\,\mathrm{r_3}\,\mathrm{r_8}+C_{4}\right)\right)\\
&&+\Gamma\,\left(8\,\mathrm{r_3}^{2}\,C_{1}C_{2}\,\left(1-C_{4}\right)-\mathrm{r_7}^{2}\,\left(2\,\mathrm{r_4}^{2}\,\left(\mathrm{r_8}+7\,{\mathcal S}qrt{3}\,\mathrm{r_3}+1\right)+8\,\mathrm{r_3}^{2}\,\left(7\,\mathrm{r_8}+{\mathcal S}qrt{3}\,\mathrm{r_3}+1\right)\right)\right),
\end{eqnarray*}
with $C_{1}=\mathrm{r_{3}}-{\mathcal S}qrt{3}\,\mathrm{r_{8}},\,C_{2}=\mathrm{r_{3}}+{\mathcal S}qrt{3}\,\mathrm{r_{8}},\,C_{3}={\mathcal S}qrt{3}\,
\mathrm{r_{3}}+\mathrm{r_8},\,C_{4}={\mathcal S}qrt{3}\,\mathrm{r_3}-\mathrm{r_8}.$
\end{widetext}
{\mathcal S}ection{The Bloch vectors of the Trajectory without Initial-to-Final State Couplings}\label{AE}
To obtain a reasonable control parameters, the intermediate state $\ket{^1E}$ is allowed to be occupied. Without loss of generality, the trajectory charactered by the Bloch vector is designed as follows:
\begin{eqnarray*}
r_3&=&{\mathcal S}qrt{3}\,\left(\left(2\,\mathrm{N}\,\left(3\,\mathrm{N}+2\right)\,\mathrm{\Gamma}^{2}+\mathrm{\Omega_{p}}^{2}-6\,\mathrm{\Omega_{p}}\,\mathrm{\Omega_{s}}+\mathrm{\Omega_{s}}^{2}\right)\right.\nonumber\\
&&+\cos\!\left(2\phi(t)\right)\,\left(3\,\left(\mathrm{\Omega_{p}}+\mathrm{\Omega_{s}}\right)^{2}+6\,\mathrm{N}\,\mathrm{\Gamma}^{2}\,\left(3\,\mathrm{N}+2\right)\right).\nonumber\\
&&-4\,\cos\!\left(\phi(t)\right)\,\left(\mathrm{\Omega_{p}}^{2}-\mathrm{\Omega_{s}}^{2}\right)\nonumber\\
&&+4\,{\mathcal S}qrt{2}\,\mathrm{N}\,\mathrm{\Gamma}\,{\mathcal S}in\!\left(\phi(t)\right)\,\left(\mathrm{\Omega_{p}}+\mathrm{\Omega_{s}}\right)\nonumber\\
&&\left.+6\,{\mathcal S}qrt{2}\,\mathrm{N}\,\mathrm{\Gamma}\,{\mathcal S}in\!\left(2\,\phi(t)\right)\,\left(\mathrm{\Omega_{p}}-\mathrm{\Omega_{s}}\right)\right)/(16z).\\
\end{eqnarray*}
\begin{eqnarray*}
r_8&=&\left(-\left(2\,\mathrm{N}\,\left(3\,\mathrm{N}+2\right)\,\mathrm{\Gamma}^{2}+\mathrm{\Omega_{p}}^{2}-6\,\mathrm{\Omega_{p}}\,\mathrm{\Omega_{s}}+\mathrm{\Omega_{s}}^{2}\right)\right.\nonumber\\
&&-12\,\cos\!\left(\phi(t)\right)\,\left(\mathrm{\Omega_{p}}^{2}-\mathrm{\Omega_{s}}^{2}\right)\nonumber\\
&&-\cos\!\left(2\,\phi(t)\right)\,\left(3\,\left(\mathrm{\Omega_{p}}+\mathrm{\Omega_{s}}\right)^{2}+6\,\mathrm{N}\,\mathrm{\Gamma}^{2}\,\left(3\,\mathrm{N}+2\right)\right)\nonumber\\
&&+12\,{\mathcal S}qrt{2}\,\mathrm{N}\,\mathrm{\Gamma}\,{\mathcal S}in\!\left(\phi(t)\right)\,\left(\mathrm{\Omega_{p}}+\mathrm{\Omega_{s}}\right)\nonumber\\
&&\left.-6\,{\mathcal S}qrt{2}\,\mathrm{N}\,\mathrm{\Gamma}\,{\mathcal S}in\!\left(2\,\phi(t)\right)\,\left(\mathrm{\Omega_{p}}-\mathrm{\Omega_{s}}\right)\right)/(16z),
\end{eqnarray*}
in which $\phi(t)$ determines the population on $\ket{^1E}$, i.e.,
\begin{eqnarray}
P_2&=&\left(2\, \mathrm{N}\, {\mathrm{\Omega}}^2 + 2\, {\mathrm{N}}^2\, {\mathrm{\Gamma}}^2\, \left(3\, \mathrm{N} + 2\right)\right.\nonumber\\
&& + {{\mathcal S}in\!\left(\phi(t)\right)}^2\, \left({\left(\mathrm{\Omega_p} + \mathrm{\Omega_s}\right)}^2 + 2\, \mathrm{N}\, {\mathrm{\Gamma}}^2\, \left(3\, \mathrm{N} + 2\right)\right) \nonumber\\
&&\left. -{\mathcal S}qrt{2}\, \mathrm{N}\, \mathrm{\Gamma}\, {\mathcal S}in\!\left(2\,\phi(t)\right)\, \left( \mathrm{\Omega_p} - \mathrm{\Omega_s}\right)\right)/(2z).
\end{eqnarray}
$r_4$ is determined by the implicit differential equation $\Omega_c^i(t)=0$; $r_2$ and $r_7$ are the same as in the adiabatic trajectory ( $r_2$ and $r_7$ in Eq.(\ref{r38})).
We consider the adiabatic pulses $\Omega_{p,s}$ like Eq.(\ref{ap3}) with $$\theta(t)=\frac{\pi}{2}{\mathcal S}in\!\left(\frac{\pi t}{2\tau}\right)^2,$$ and the trajectory with $$\phi(t)=\frac{\pi}{9}{\mathcal S}in\!\left(\frac{\pi t}{\tau}\right)^2 .$$
Thus, we can determine all of the control parameters numerically.
\end{document} |
{\bf e}gin{document}
\title{A simple finite element method for the Stokes equations}
{\bf e}gin{abstract}
The goal of this paper is to introduce a simple finite element method
to solve the Stokes and the Navier-Stokes equations. This method is in primal velocity-pressure formulation and is so simple such that both velocity and pressure are approximated by piecewise constant functions. Implementation issues as well as error analysis are investigated. A basis for a divergence free subspace of the velocity field is constructed so that the original saddle point problem can be reduced to a symmetric and positive definite system with much fewer unknowns. The numerical experiments indicate that the method is accurate and robust.
\end{abstract}
{\bf e}gin{keywords}
finite element methods, Stokes problem, weak Galerkin methods
\end{keywords}
{\bf e}gin{AMS}
Primary, 65N15, 65N30, 76D07; Secondary, 35B45, 35J50
\end{AMS}
\pagestyle{myheadings}
\section{Introduction}
The Stokes problem is to
seek a pair of unknown functions $({\bf u}; p)$ satisfying
{\bf e}gin{eqnarray}
-\nu\Delta{\bf u}+\nabla p &=&{\bf f}\quad \mbox{in}\;\Omega,{\langle}abel{moment}\\
\nabla\cdot{\bf u}&=&0\quad\mbox{in}\;\Omega,{\langle}abel{cont}\\
{\bf u} &=& {\bf 0}\quad\mbox{on}\;\partial\Omega,{\langle}abel{bc}
\end{eqnarray}
where $\nu$ denotes the fluid viscosity; $\Delta$, $\nabla$, and
$\nabla\cdot$ denote the Laplacian, gradient, and divergence
operators, respectively; $\Omega \subset \mathbb{R}^d$ is the
region occupied by the fluid; ${\bf f}={\bf f}({\bf x})\in ([L^2(\Omega))]^d$
is the unit external volumetric force acting on the fluid at
${\bf x}\in\Omega$. For simplicity, we let $\nu=1$.
The weak formulation of the Stokes equations seeks ${\bf u}\in [H_0^1(\Omega)]^d$ and $p\in L_0^2(\Omega)$ satisfying
{\bf e}gin{eqnarray}
(\nabla{\bf u},\;\nabla{\bf v})-(\nabla\cdot{\bf v},\;p)&=&({\bf f},\;{\bf v}),\quad {\bf v}\in [H_0^1(\Omega)]^d{\langle}abel{wf-m} \\
(\nabla\cdot{\bf u},\;q)&=&0,\quad\quad\quad q\in L_0^2(\Omega).{\langle}abel{wf-c}
\end{eqnarray}
The linear Stokes equations are the limiting case of zero Reynolds number for the Navier-Stokes equations. The Stokes equations have attracted a substantial attention from researchers because of its close relation with the Navier-Stokes equations. Numerical solutions of the Stokes equations have been investigated intensively and many different numerical schemes have been developed such as conforming/noconforming finite element methods, MAC method and finite volume methods. It is impossible to cite all the references. Therefore we just cite some classic ones \cite{cr,gr,gun,ht}.
In this paper, we present a finite element scheme for the Stokes equations and its equivalent divergence free formulation. In this method, velocity is approximated by weak Galerkin element of degree $k=0$ and pressure is approximated by piecewise polynomials of degree $k=0$. Weak Galerkin refers to a general finite element technique for
partial differential equations in which differential operators are approximated by weak forms as distributions for generalized functions. The weak Galerkin finite element method first introduced in \cite{wy, wy-mixed} is a natural extension of the standard Galerkin finite element method for functions with discontinuity.
One of the main difficulties in solving the the Stokes and the Navier-Stokes equations
is that the velocity and the pressure variables are coupled in a saddle point system. Many methods are developed to overcome
this difficulty. Divergence free finite element methods are such methods by approximating velocity from weakly or exactly divergence free subspaces. As a consequence, pressure is eliminated from a saddle point system, along with the incompressibility
constraint resulting in a symmetric and positive definite system with a significantly
smaller number of unknowns. For this simple finite element formulation, a divergence free basis is constructed explicitly.
The rest of the paper is organized as follows.
The finite element formulation of this weak Galerkin method is introduced in Section {\rangle}ef{section-wg}. Implementation issues of the method are discussed in Section {\rangle}ef{section-implementation}. In Section {\rangle}ef{section-error}, we prove optimal order convergence rate of the method. Divergence free basis functions are constructed in Section {\rangle}ef{section-divfree}. Using these basis functions, we can derive a divergence free weak Galerkin finite element formulation that will reduce a saddle point problem to a symmetric and positive definite system. Numerical examples are presented in Section {\rangle}ef{section-ne} to demonstrate the robustness and accuracy of the method.
\section{Finite Element Scheme}{\langle}abel{section-wg}
Let ${\cal T}_h$ be a shape-regular triangulation of the domain
$\Omega$ with mesh size $h$. Denote by ${\cal E}_h$
the set of all edges or faces in ${\cal T}_h$, and let ${\cal
E}_h^0={\cal E}_h\backslash\partial\Omega$ be the set of all
interior edges or faces. let ${\cal V}_h$ be the set of all interior vertices in ${\mathcal T}_h$. Define $N_E=card ({\mathcal E}_h^0)$, $N_V=card ({\cal V}_h)$ and $N_T=card ({\mathcal T}_h)$. For every element $T\in {\mathcal T}_h$, we
denote by $h_T$ its diameter and mesh size $h=\max_{T\in{\mathcal T}_h} h_T$
for ${\cal T}_h$.
The weak Galerkin methods create a new way to define a function $v$ that allows $v$ taking different forms in the interior and on the boundary of the element:
$$
v=
{\langle}eft\{
{\bf e}gin{array}{l}
\displaystyle
v_0,\quad {{\rangle}m in}\; T^0
\\ [0.08in]
\displaystyle
v_b,\quad {{\rangle}m on}\;\partial T
\end{array}
{\rangle}ight.
$$
where $T_0$ denotes the interior of $T$.
Since weak function $v$ is formed by two parts $v_0$ and $v_b$, we write $v$ as $v=\{v_0,v_b\}$ in short without confusion. Let $P_k(T)$ denote the set consisting all the polynomials of degree less or equal to $k$.
Associated with ${\mathcal T}_h$, we define finite element spaces $V_h$ for velocity
{\bf e}gin{equation}{\langle}abel{vhspace}
V_h=\{{\bf v}=\{{\bf v}_0,{\bf v}_b\}:\; {\bf v}_0|_T\in [P_0(T)]^d,\ {\bf v}_b|_e\in [P_0(e)]^d,\ e\in{\partial T}, T\in {\mathcal T}_h\}
\end{equation}
and $W_h$ for pressure
{\bf e}gin{equation}{\langle}abel{phspace}
W_h=\{q\in L^2_0(\Omega):\; q|_T\in P_0(T),\; T\in{\mathcal T}_h\},
\end{equation}
where $L^2_0(\Omega)$ is the subspace of $L_2(\Omega)$ consisting of functions with mean value zero.
We define $V_h^0$ a subspace of $V_h$ as
{\bf e}gin{equation}{\langle}abel{vh0space}
V^0_h=\{{\bf v}=\{{\bf v}_0,{\bf v}_b\}\in V_h:\ {\bf v}_b=0 \mbox{ on } \partial\Omega\}.
\end{equation}
We would like to emphasize that any function $v\in V_h$ has a single
value $v_b$ on each edge $e\in{\mathcal E}_h$.
Since the functions in $V_h$ are discontinuous polynomials, gradient operator $\nabla$ and divergence operator $\nabla\cdot$ in ({\rangle}ef{wf-m})-({\rangle}ef{wf-c}) cannot be applied to them. Therefore we defined weak gradient and weak divergence for the functions in $V_h$. Let $RT_0(T)=[P_0(T)]^d+{\bf x}P_0(T)$ introduced in \cite{rt}. Let ${\bf n}$ denote the unit outward normal.
For ${\bf v}\in V_h$ and $T\in{\mathcal T}_h$, we define weak gradient
$\nabla_{w}{\bf v} \in [RT_0(T)]^d$ as the unique polynomial
satisfying the following
equation
{\bf e}gin{equation}{\langle}abel{dwg}
(\nabla_{w}{\bf v}, {\tilde a}u)_T = -({\bf v}_0,\nabla\cdot {\tilde a}u)_T+ {\langle}angle {\bf v}_b,
{\tilde a}u\cdot{\bf n}{\rangle}angle_{\partial T},\qquad \forall {\tilde a}u\in [RT_0(T)]^d,
\end{equation}
and define weak divergence $\nabla_{w}\cdot{\bf v} \in P_0(T)$ as the unique polynomial
satisfying
{\bf e}gin{equation}{\langle}abel{dwd}
(\nabla_{w}\cdot{\bf v}, q)_T = -({\bf v}_0,\nabla q)_T+ {\langle}angle {\bf v}_b,
q{\bf n}{\rangle}angle_{\partial T},\qquad \forall q\in P_0(T).
\end{equation}
Define two bilinear forms as
{\bf e}gin{eqnarray*}
a({\bf v},{\bf w})=\sum_{T\in{\mathcal T}_h}(\nabla_w{\bf v},\nabla_w{\bf w})_T, \quad b({\bf v},q)=\sum_{T\in{\mathcal T}_h}(\nabla_w\cdot{\bf v},q)_T.
\end{eqnarray*}
For each element $T\in {\mathcal T}_h$, denote by $Q_0$ and ${\bf Q}_0$ the $L^2$ projections from $L^2(T)$ to $P_0(T)$ and
from $[L^2(T)]^d$ to $[P_0(T)]^d$ respectively. Denote by ${\bf Q}_b$ the $L^2$ projection from
$[L^2(e)]^d$ to $[P_{0}(e)]^d$.
{\bf e}gin{algorithm}
A weak Galerkin method for the Stokes equations seeks
${\bf u}_h=({\bf u}_0,{\bf u}_b)\in V_h^0$ and $p_h\in W_h$ satisfying the following equation:
{\bf e}gin{eqnarray}
a({\bf u}_h,{\bf v})-b({\bf v},p_h)&=&({\bf f},\;{\bf v}_0), \quad\forall {\bf v}=\{{\bf v}_0, {\bf v}_b\}\in V_h^0{\langle}abel{wg-m}\\
b({\bf u}_h,q)&=&0,\quad\quad\quad\forall q\in W_h.{\langle}abel{wg-c}
\end{eqnarray}
\end{algorithm}
\section{Implementation of the method}{\langle}abel{section-implementation}
The linear system associated with the algorithm ({\rangle}ef{wg-m})-({\rangle}ef{wg-c}) is a saddle point problem with the form,
{\bf e}gin{equation}{\langle}abel{matrix}
{\langle}eft(
{\bf e}gin{array}{cc}
A & -B \\
B^T&0
\end{array}
{\rangle}ight)
{\langle}eft({\bf e}gin{array}{c}U \\P \end{array}{\rangle}ight)={\langle}eft({\bf e}gin{array}{c}
F_1 \\ F_2
\end{array}
{\rangle}ight).
\end{equation}
The methodology of implementing this weak Galerkin method is the same as that for continuous
Galerkin finite element method except that computing standard gradient $\nabla$ and divergence $\nabla\cdot$ are replaced by computing weak gradient $\nabla_w$ and weak divergence $\nabla_w\cdot$. For basis function ${\mathcal T}heta_l$, we will show that $\nabla_w\cdot{\mathcal T}heta_l$ and $\nabla_w{\mathcal T}heta_l$ can be calculated explicitly.
The procedures of implementing the method ({\rangle}ef{wg-m})-({\rangle}ef{wg-c}) can be described as following steps. Here we let $d=2$ for simplicity.
1. Find basis functions for $V_h$ and $W_h$. First we define two types of scalar piecewise constant basis functions $\phi_i$ associated with the interior of the triangle $T_i\in{\mathcal T}_h$ and $\psi_j$ associated with an edge $e_j\in {\mathcal E}_h$ respectively,
{\bf e}gin{equation}{\langle}abel{phi}
\phi_i={\langle}eft\{
{\bf e}gin{array}{l}
1\quad\mbox{on} \;\; T_i^0,\\ [0.08in]0\quad\mbox{therwise},
\end{array}
{\rangle}ight.
\psi_j={\langle}eft\{
{\bf e}gin{array}{l}
1\quad\mbox{on} \;\; e_j.\\ [0.08in]0\quad\mbox{therwise},
\end{array}
{\rangle}ight.
\end{equation}
Please note that $\phi_i$ and $\psi_j$ are functions defined over the whole domain $\Omega$. Then we can define the vector basis functions for velocity as
{\bf e}gin{equation}{\langle}abel{b1}
\Phi_{i,1}={\langle}eft({\bf e}gin{array}{c}\phi_i \\0 \end{array}{\rangle}ight),\;\;\Phi_{i,2}={\langle}eft({\bf e}gin{array}{c}0 \\\phi_i \end{array}{\rangle}ight),\;\; i=1,\cdots,N_T,
\end{equation}
and
{\bf e}gin{equation}{\langle}abel{b2}
\Psi_{i,1}={\langle}eft({\bf e}gin{array}{c}\psi_i \\0 \end{array}{\rangle}ight),\;\;\Psi_{i,2}={\langle}eft({\bf e}gin{array}{c}0 \\\psi_i \end{array}{\rangle}ight), \;\; i=1,\cdots,N_E.
\end{equation}
Let $n=N_T+N_E$. These $2n$ vector functions will form a basis for $V_h$,
{\bf e}gin{eqnarray}
V_h&=&{{\rangle}m span} \{\Phi_{i,j}, i=1,\cdots, N_T,\;\Psi_{k,j}, k=1,\cdots,N_E, \;j=1,2\}\nonumber\\
&=&{{\rangle}m span} \{{\mathcal T}heta_1,\cdots,{\mathcal T}heta_{2n}\}.{\langle}abel{vh}
\end{eqnarray}
Define $\overline{W}_h$ as
\[
\overline{W}_h={{\rangle}m span} \{\phi_1,\cdots,\phi_{N_T}\}.
\]
The pressure space is a subspace of $\overline{W}_h$,
\[
W_h=\{q\in \overline{W}_h, \int_\Omega qdx=0\}.
\]
2. Compute weak gradient $\nabla_w$ and weak divergence $\nabla_w\cdot$ for the basis function ${\mathcal T}heta_l$ defined in ({\rangle}ef{vh}). By the definition of ${\mathcal T}heta_l$, to compute $\nabla_w\cdot{\mathcal T}heta_l$, we compute $\nabla_w\cdot\Phi_{i,j}$ and $\nabla_w\cdot\Psi_{i,j}$ instead. To find $\nabla_w{\mathcal T}heta_l$, we just need to compute $\nabla_w\phi_i$ and $\nabla_w\psi_i$.
{\bf e}gin{itemize}
\item Computing $\nabla_w\cdot\Phi_{i,j}$.\\
Using ({\rangle}ef{dwd}), we have $\nabla_w\cdot\Phi_{i,j}|_T=0$ for all $T\in{\mathcal T}_h$.\\
\item Computing $\nabla_w\cdot\Psi_{i,j}$.\\
Assume that $i^{th}$ edge $e_i$ is on ${\partial T}$ and $\Psi_{i,j}$ is defined in ({\rangle}ef{b2}). Then
{\bf e}gin{equation}{\langle}abel{cwd}
\nabla_w\cdot\Psi_{i,j}|_T=\frac1{|T|}\int_{e_i}\Psi_{i,j,b}\cdot{\bf n} ds,
\end{equation}
where $|T|$ is the area of $T$ and $\Psi_{i,j}=\{\Psi_{i,j,0},\Psi_{i,j,b}\}$.
Note that $\nabla_w\cdot\Psi_{i,j}$ is only nonzero on two triangles that share $e_i$.\\
\item Computing $\nabla_w\phi_i$.\\
Let $T$ be the $i^{th}$ triangle in ${\mathcal T}_h$ and $\phi_i$ be defined as in ({\rangle}ef{phi}). Then $\nabla_w\phi_i$ is only nonzero on $T$ and can be calculated by
{\bf e}gin{eqnarray*}
\nabla_w\phi_i|_T&=&-C_T({\bf x}-{\bf x}_T),
\end{eqnarray*}
where ${\bf x}_T$ is the centroid of $T$ and $C_T= \frac{2|T|}{\|{\bf x}-{\bf x}_T\|_T^2}$.\\
\item Computing $\nabla_w\psi_i$.\\
Assume that $i^{th}$ edge $e_i$ is on ${\partial T}$ and $\psi_{i}$ is defined in ({\rangle}ef{phi}). Then
{\bf e}gin{eqnarray*}
\nabla_w\psi_i|_T&=&\frac{C_T}{3}({\bf x}-{\bf x}_T)+\frac{|e_i|}{|T|}{\bf n},
\end{eqnarray*}
Note that $\nabla_w\psi_i$ is only nonzero on two triangles that share $e_i$.
\end{itemize}
3. Form the stiffness matrix ({\rangle}ef{matrix}) with
\[
A=(a_{ij})=(a({\mathcal T}heta_i,{\mathcal T}heta_j)),\quad B=(b_{ij})=(b({\mathcal T}heta_i,\phi_j)).
\]
Note that
$$
a({\mathcal T}heta_i,{\mathcal T}heta_j)=\sum_{T\in{\mathcal T}_h}(\nabla_w{\mathcal T}heta_i,\nabla_w{\mathcal T}heta_j)_T,\;\; b({\mathcal T}heta_i,\phi_j)=\sum_{T\in{\mathcal T}_h}(\nabla_w\cdot{\mathcal T}heta_i,\phi_j)_T.
$$
\section{Error estimate}{\langle}abel{section-error}
Denote by $\pi_h$ a $L^2$ projection from $[L^2(T)]^{d\times d} $ to $[RT_0(T)]^d$.
We also define a projection $\Pi_h$ such that
$\Pi_h{\bf q}\in [H({{\rangle}m div},\Omega)]^d$, and on each $T\in {\cal T}_h$,
one has $\Pi_h{\bf q} \in [RT_0(T)]^d$ and the following equation satisfied:
$$
(\nabla\cdot{\bf q},\;{\bf v}_0)_T=(\nabla\cdot\Pi_h{\bf q},\;{\bf v}_0)_T, \qquad
\forall {\bf v}_0\in [P_0(T)]^d.
$$
For any ${\tilde a}u\in [H({{\rangle}m div},\Omega)]^d$, we have (see \cite{bf})
{\bf e}gin{equation}{\langle}abel{4.200}
\sum_{T\in {\cal T}_h}(-\nabla\cdot{\tilde a}u, \;{\bf v}_0)_T=\sum_{T\in {\cal T}_h}(\Pi_h{\tilde a}u, \;\nabla_w{\bf v})_T,\quad\forall {\bf v}=\{{\bf v}_0,{\bf v}_b\}\in V_h.
\end{equation}
The following two identities can be verified easily and also can be found in \cite{wy, wy-mixed}.
{\bf e}gin{eqnarray}
\nabla_w {\bf Q}_h {\bf u} &=& \pi_h (\nabla {\bf u}),{\langle}abel{4.88}\\
\nabla_w\cdot {\bf Q}_h {\bf u} &=& Q_0 (\nabla \cdot{\bf u}).{\langle}abel{4.99}
\end{eqnarray}
We introduce two semi-norms ${|\hspace{-.02in}|\hspace{-.02in}|} {\bf v}{|\hspace{-.02in}|\hspace{-.02in}|}$ and $\|\cdot\|_{1,h}$ as follows:
{\bf e}gin{eqnarray}
{|\hspace{-.02in}|\hspace{-.02in}|} {\bf v}{|\hspace{-.02in}|\hspace{-.02in}|}^2 &=& a({\bf v},{\bf v}), {\langle}abel{norm2}\\
\|{\bf v}\|_{1,h}^2&=&\sum_{T\in {\mathcal T}_h}{\langle}eft(\|\nabla{\bf v}_0\|_T^2+h_T^{-1}\|{\bf v}_0-{\bf v}_b\|_{{\partial T}}^2{\rangle}ight).{\langle}abel{norm3}
\end{eqnarray}
The following norm equivalences is proved in \cite{mwwy} that there exist two constants $C_1$ and $C_2$ independent of $h$ satisfying
{\bf e}gin{equation}{\langle}abel{norm-e}
C_1\|{\bf v}\|_{1,h}{\langle}e {|\hspace{-.02in}|\hspace{-.02in}|}{\bf v}{|\hspace{-.02in}|\hspace{-.02in}|}{\langle}e C_2 \|{\bf v}\|_{1,h}.
\end{equation}
{\bf e}gin{lemma}
The semi-nome ${|\hspace{-.02in}|\hspace{-.02in}|}\cdot{|\hspace{-.02in}|\hspace{-.02in}|}$ defined in ({\rangle}ef{norm2}) is a norm in $V_h^0$.
\end{lemma}
{\bf e}gin{proof}
We only need to prove $v=0$ if ${|\hspace{-.02in}|\hspace{-.02in}|} v{|\hspace{-.02in}|\hspace{-.02in}|}=0$ for all $v\in V_h^0$. Let $v\in V_h^0$ and $\|v\|_{1,h}=0$. Then we have $\nabla v_0=0$ on each $T\in{\mathcal T}_h$ , $v_0=v_b$ on $e\in{\mathcal E}_h^0$ and $v_b=0$ for $e$ on $\partial\Omega$. $\nabla v_0=0$ on $T$ implies that $v_0$ is a constant on each $T$. $v_0=v_b$ on $e$ means that $v_0$ is continuous. With $v_0=v_b=0$ on $\partial\Omega$, we conclude $v=0$ and prove that $\|\cdot\|_{1,h}$ is a norm in $V_h$. Combining it with ({\rangle}ef{norm-e}), we have proved that ${|\hspace{-.02in}|\hspace{-.02in}|}\cdot{|\hspace{-.02in}|\hspace{-.02in}|}$ is a norm in $V_h^0$.
\end{proof}
Define two linear functionals on $V_h^0$ by
{\bf e}gin{eqnarray*}
\ell_{{\bf u}}({\bf v})=\sum_{T\in{\mathcal T}_h}(\Pi_h\nabla {\bf u}-\pi_h\nabla{\bf u}, \nabla_w{\bf v})_T,\;\;\;
\ell_p({\bf v})=\sum_{T\in{\mathcal T}_h}{\langle} {\bf v}_0-{\bf v}_b,\;(p-Q_0p){\bf n}{\rangle}_{\partial T}.
\end{eqnarray*}
{\bf e}gin{lemma}{\langle}abel{e-e}
Let ${\bf e}_h={\bf Q}_h{\bf u}-{\bf u}_h=\{{\bf e}_0,\;{\bf e}_b\}=\{{\bf Q}_0{\bf u}-{\bf u}_0,\;{\bf Q}_b{\bf u}-{\bf u}_b\}$ and $\varepsilon_h=Q_0p-p_h$.
Then, the following equations hold
true
{\bf e}gin{eqnarray}
a({\bf e}_h,{\bf v})-b({\bf v},\varepsilon_h)&=&-\ell_{{\bf u}}({\bf v})-\ell_p({\bf v}),\;\forall{\bf v}\in V_h^0,{\langle}abel{ee-m}\\
b({\bf e}_h,q)&=&0,\quad\qquad\qquad\forall q\in W_h.{\langle}abel{ee-c}
\end{eqnarray}
\end{lemma}
{\bf e}gin{proof}
Testing ({\rangle}ef{moment}) by ${\bf v}=\{{\bf v}_0,{\bf v}_b\}\in V_h^0$ gives
{\bf e}gin{equation}{\langle}abel{mmm1}
(-\nabla\cdot(\nabla{\bf u}), \;{\bf v}_0)+(\nabla p,\;{\bf v}_0)=({\bf f}, {\bf v}_0).
\end{equation}
Equations ({\rangle}ef{4.200}) and ({\rangle}ef{4.88}) imply
{\bf e}gin{eqnarray*}
\sum_{T\in{\mathcal T}_h}(-\nabla\cdot(\nabla{\bf u}), \;{\bf v}_0)_T&=&\sum_{T\in{\mathcal T}_h}((\Pi_h\nabla{\bf u}, \nabla_w{\bf v})_T=\sum_{T\in{\mathcal T}_h}((\Pi_h\nabla {\bf u}-\pi_h\nabla{\bf u}, \nabla_w{\bf v})_T+(\nabla_w{\bf Q}_h{\bf u}, \nabla_w{\bf v})_T).
\end{eqnarray*}
It follows from ({\rangle}ef{dwd}), the integration by parts and the fact $\sum_{K\in{\mathcal T}_h}{\langle}angle{\bf v}_b,\; p\;{\bf n}{\rangle}angle_{\partial K}=0$ that for $v\in V_h^0$
{\bf e}gin{eqnarray*}
(\nabla p,\;{\bf v}_0)&&=\sum_{T\in{\mathcal T}_h}(-(p,\; \nabla\cdot{\bf v}_0)_T +{\langle} p{\bf n},\; {\bf v}_0{\rangle}_T)\\
&&=\sum_{T\in{\mathcal T}_h}(-(Q_0p,\; \nabla\cdot{\bf v}_0)_T +{\langle} p{\bf n},\; {\bf v}_0-{\bf v}_b{\rangle}_T)\\
&&=\sum_{T\in{\mathcal T}_h}(({\bf v}_0,\; \nabla Q_0p)_T -{\langle} Q_0p{\bf n},\; {\bf v}_0{\rangle}_T+{\langle} p{\bf n},\; {\bf v}_0-{\bf v}_b{\rangle}_T)\\
&&=-b({\bf v}, Q_0p)+\sum_{T\in{\mathcal T}_h}{\langle} {\bf v}_0-{\bf v}_b,\;(p-Q_0p){\bf n}{\rangle}_{\partial T}.
\end{eqnarray*}
Combining two equations above with ({\rangle}ef{mmm1}), we have
{\bf e}gin{eqnarray}
a({\bf Q}_h{\bf u},{\bf v})-b({\bf v},Q_0p)=({\bf f}, {\bf v}_0)-\ell_{{\bf u}}({\bf v})-\ell_p({\bf v}).{\langle}abel{t-m}
\end{eqnarray}
Using ({\rangle}ef{4.99}) and ({\rangle}ef{cont}), we have that for any $q\in W_h$
{\bf e}gin{eqnarray}
b({\bf Q}_h{\bf u},q)&=&\sum_{T\in{\mathcal T}_h}(\nabla_w\cdot{\bf Q}_h{\bf u}, q)_T=\sum_{T\in{\mathcal T}_h}(Q_0(\nabla\cdot{\bf u}), q)_T\nonumber\\
&=&\sum_{T\in{\mathcal T}_h}(\nabla\cdot{\bf u}, q)_T=b({\bf u},q)=0,{\langle}abel{t-c}
\end{eqnarray}
The differences of ({\rangle}ef{wg-m})-({\rangle}ef{wg-c}) and ({\rangle}ef{t-m})-({\rangle}ef{t-c}) yield ({\rangle}ef{ee-m})-({\rangle}ef{ee-c}). The proof is completed.
\end{proof}
{\bf e}gin{lemma}
For any ${\rangle}ho\in W_h$, then there exists a constant $C$ independent of $h$ such that
{\bf e}gin{equation}{\langle}abel{inf-sup}
\sup_{{\bf v}\in V_h^0}\frac{(\nabla_w\cdot{\bf v},\;{\rangle}ho)}{{|\hspace{-.02in}|\hspace{-.02in}|}{\bf v}{|\hspace{-.02in}|\hspace{-.02in}|}}\ge C\|{\rangle}ho\|.
\end{equation}
\end{lemma}
{\bf e}gin{proof}
For a given ${\rangle}ho\in W_h\subset L_0^2(\Omega)$, it is well known that there exists $\tilde{\bf v}\in [H_0^1(\Omega)]^d$ such that
{\bf e}gin{equation}{\langle}abel{c-inf-sup}
\frac{(\nabla\cdot\tilde{\bf v},{\rangle}ho)}{\|\nabla\tilde{\bf v}\|}\ge C\|{\rangle}ho\|.
\end{equation}
Let ${\bf v}={\bf Q}_h\tilde{{\bf v}}$.
({\rangle}ef{4.88}) implies
{\bf e}gin{equation}{\langle}abel{m2}
{|\hspace{-.02in}|\hspace{-.02in}|}{\bf v}{|\hspace{-.02in}|\hspace{-.02in}|}=\|\nabla_w{\bf v}\|=\|\nabla_w({\bf Q}_h\tilde{{\bf v}})|=\|\pi_h\nabla \tilde{{\bf v}}\|{\langle}e \|\nabla\tilde{\bf v}\|.
\end{equation}
It follows from ({\rangle}ef{4.99}) and the definition of $\pi_h$
{\bf e}gin{eqnarray*}
(\nabla_w\cdot{\bf v},\;{\rangle}ho)&=&(\nabla_w\cdot {\bf Q}_h\tilde{\bf v},\;{\rangle}ho)=(Q_0(\nabla\cdot\tilde{\bf v}),\;{\rangle}ho)=(\nabla\cdot\tilde{\bf v},\;{\rangle}ho).
\end{eqnarray*}
Using the equation above, ({\rangle}ef{c-inf-sup}) and ({\rangle}ef{m2}), we have
{\bf e}gin{eqnarray*}
\frac{(\nabla_w{\bf v},{\rangle}ho)} {{|\hspace{-.02in}|\hspace{-.02in}|}{\bf v}{|\hspace{-.02in}|\hspace{-.02in}|}} &\ge & \frac{(\nabla\cdot\tilde{\bf v},{\rangle}ho)}{\|\nabla\tilde{\bf v}\|}\ge C\|{\rangle}ho\|.
\end{eqnarray*}
We proved the lemma.
\end{proof}
For any function $\varphi\in H^1(T)$, the following trace
inequality holds true
{\bf e}gin{equation}{\langle}abel{trace}
\|\varphi\|_{e}^2 {\langle}eq C {\langle}eft( h_T^{-1} \|\varphi\|_T^2 + h_T
\|\nabla \varphi\|_{T}^2{\rangle}ight).
\end{equation}
{\bf e}gin{theorem}{\langle}abel{h1-bd}
Let $({\bf u}, p)\in [H_0^1(\Omega)\cap H^{2}(\Omega)]^d\times L_0^2(\Omega)\cap H^1(\Omega)$ and $({\bf u}_h,p_h)\in V_h\times W_h$ be the solutions of ({\rangle}ef{moment})-({\rangle}ef{bc}) and ({\rangle}ef{wg-m})-({\rangle}ef{wg-c}) respectively. Then
{\bf e}gin{eqnarray}
\|\nabla_w ({\bf Q}_h{\bf u}- {\bf u}_h)\|+\|Q_0p-p_h\|&{\langle}e& Ch(\|{\bf u}\|_{2}+\|p\|_1),{\langle}abel{error1}\\
\|{\bf Q}_0{\bf u}-{\bf u}_0\|&{\langle}e& Ch^2(\|{\bf u}\|_{2}+\|p\|_1).{\langle}abel{error2}
\end{eqnarray}
\end{theorem}
{\bf e}gin{proof}
Letting ${\bf v}={\bf e}_h$ and $q=\varepsilon_h$ in ({\rangle}ef{ee-m})-({\rangle}ef{ee-c}) and adding the two equations yield
{\bf e}gin{equation}{\langle}abel{eee1}
{|\hspace{-.02in}|\hspace{-.02in}|}{\bf e}_h{|\hspace{-.02in}|\hspace{-.02in}|}^2=a({\bf e}_h,{\bf e}_h)=|\ell_{{\bf u}}({\bf e}_h)+\ell_{p}({\bf e}_h)|.
\end{equation}
It follows from the definitions of $\Pi_h$ and $\pi_h$ that
{\bf e}gin{eqnarray}
|\ell_{{\bf u}}({\bf e}_h)|&=&|\sum_{T\in{\mathcal T}_h}(\Pi_h\nabla {\bf u}-\pi_h\nabla{\bf u}, \nabla_w{\bf e}_h)_T|\nonumber\\
&{\langle}e& \sum_{T\in{\mathcal T}_h}\|\Pi_h\nabla {\bf u}-\pi_h\nabla{\bf u}\|_T\|\nabla_w{\bf e}_h\|_T\|{\langle}e Ch\|{\bf u}\|_2{|\hspace{-.02in}|\hspace{-.02in}|}{\bf e}_h{|\hspace{-.02in}|\hspace{-.02in}|}.{\langle}abel{eee2}
\end{eqnarray}
Using the trace inequality ({\rangle}ef{trace}), the definition of $Q_0$ and norm equivalence ({\rangle}ef{norm-e}), we have
{\bf e}gin{eqnarray}
|\ell_{p}({\bf e}_h)|&=&|\sum_{T\in{\mathcal T}_h}{\langle} {\bf e}_0-{\bf e}_b,\;(p-Q_0p){\bf n}{\rangle}_{\partial T}|\nonumber\\
&{\langle}e&\sum_{T\in{\mathcal T}_h}\|p-Q_0p\|_{{\partial T}}\|{\bf e}_0-{\bf e}_b\|\nonumber\\
&{\langle}e&(\sum_{T\in{\mathcal T}_h}h\|p-Q_0p\|_{{\partial T}}^2)^{1/2}(\sum_{T\in{\mathcal T}_h}h^{-1}\|{\bf e}_0-{\bf e}_b\|_{\partial T}^2)^{1/2}\nonumber\\
&{\langle}e& Ch\|p\|_1{|\hspace{-.02in}|\hspace{-.02in}|}{\bf e}_h{|\hspace{-.02in}|\hspace{-.02in}|}.{\langle}abel{eee3}
\end{eqnarray}
Combining the estimates ({\rangle}ef{eee2})-({\rangle}ef{eee3}) and ({\rangle}ef{eee1}) gives
{\bf e}gin{equation}{\langle}abel{eee4}
{|\hspace{-.02in}|\hspace{-.02in}|}{\bf e}_h{|\hspace{-.02in}|\hspace{-.02in}|}{\langle}e Ch(\|{\bf u}\|_2+\|p\|_1).
\end{equation}
Using ({\rangle}ef{ee-m}) and ({\rangle}ef{eee2})-({\rangle}ef{eee4}), we have
{\bf e}gin{equation}{\langle}abel{eee5}
|b({\bf v},\varepsilon_h)|=|a({\bf e}_h,{\bf v})+\ell_{{\bf u}}({\bf v})+\ell_p({\bf v})|{\langle}e Ch(\|{\bf u}\|_2+\|p\|_1){|\hspace{-.02in}|\hspace{-.02in}|}{\bf v}{|\hspace{-.02in}|\hspace{-.02in}|}.
\end{equation}
It follows from ({\rangle}ef{inf-sup}) and ({\rangle}ef{eee5})
\[
\|Q_0p-p_h\|{\langle}e Ch(\|{\bf u}\|_{2}+\|p\|_1).
\]
The estimate ({\rangle}ef{error2}) can be derived by using the standard duality argument. The main goal of this paper is about introducing a new method and how to implement it. We skip the proof of ({\rangle}ef{error2}).
\end{proof}
\section{Discrete Divergence Free Basis}{\langle}abel{section-divfree}
The finite element formulations ({\rangle}ef{wg-m})-({\rangle}ef{wg-c}) lead to a large saddle point system ({\rangle}ef{matrix})
for which most existing numerical solvers are less effective and robust than for definite
systems. Such a saddle-point system can be reduced to a definite problem
for the velocity unknown defined in a divergence-free subspace $D_h$ of $V_h^0$,
{\bf e}gin{equation}{\langle}abel{D_h}
D_h=\{{\bf v}\in V_h;\;b({\bf v},q) =0,\quad\forall q\in W_h\}.
\end{equation}
By taking the test function from $D_h$,
the discrete formulation ({\rangle}ef{wg-m})-({\rangle}ef{wg-c}) is equivalent to the following divergence-free finite element scheme:
{\bf e}gin{algorithm}{\langle}abel{algorithm2}
A discrete divergence free approximation for ({\rangle}ef{moment})-({\rangle}ef{bc}) with homogeneous boundary condition is to find
${\bf u}_h=\{{\bf u}_0,{\bf u}_b\}\in D_h$
{\bf e}gin{eqnarray}
a({\bf u}_h,\ {\bf v})&=&(f,\;{\bf v}_0),\quad \forall {\bf v}=\{{\bf v}_0,{\bf v}_b\}\in D_h.{\langle}abel{df-wg}
\end{eqnarray}
\end{algorithm}
The system ({\rangle}ef{df-wg}) is symmetric and positive definite with much fewer unknowns. To implement Algorithm {\rangle}ef{algorithm2}, we need to construct basis functions for the divergence free subspace $D_h$. There are three types of functions in $V_h$ that are divergence free.
{\bf Type 1}. All $\Phi_{i,j}$ defined in ({\rangle}ef{b1}) are divergence free. This can be verified easily. Since all the functions $\Phi_{i,j}$ defined in ({\rangle}ef{b1}) take zero value on ${\partial T}$ for all $T\in{\mathcal T}_h$, it follows from ({\rangle}ef{cwd}) that $\nabla_w\cdot\Phi_{i,j}|_T=\frac1{|T|}\int_{\partial T}\Phi_{i,j,b}\cdot{\bf n} ds=0$.
{\bf Type 2}. For any $e_i\in{\mathcal E}_h^0$, let ${\bf n}_{e_i}$ and ${\bf t}_{e_i}$ be a normal vector and a tangential vector to $e_i$ respectively. Define $\Upsilon_i=C_1\Psi_{i,1}+C_2\Psi_{i,2}$ such that $\Upsilon_i|_{e_i}={\bf t}_{e_i}$. Thus $\Upsilon_i$ is only nonzero on $e_i$. It is easy to see that $\nabla_w\cdot\Upsilon_i|_T=\frac1{|T|}\int_{\partial T}\Upsilon_{i,b}\cdot{\bf n} ds=0$.
{\bf Type 3}. For a given interior vertex $P_i\in {\cal V}_h$, there are $r$ elements having $P_i$ as a vertex which form a hull ${\cal H}_{P_i}$ as shown in Figure {\rangle}ef{fig1}. Then there are $r$ interior edges $e_j$ ($j=1,\cdots, r$) associated with ${\cal H}_{P_i}$. Let ${\bf n}_{{e_j}}$ be a normal vector on $e_j$ such that normal vectors ${\bf n}_{e_j}$ $j=1,\cdots,r$ are counterclockwise around vertex $P_i$ as shown in Figure {\rangle}ef{fig1}. For each $e_j$, let $\Psi_{j,1}$ and $\Psi_{j,2}$ be the two basis functions of $V_h$ which is only nonzero on $e_j$. Define ${\mathcal T}heta_j=C_1\Psi_{j,1}+C_2\Psi_{j,2}\in V_h$ such that ${\mathcal T}heta_j|_{e_j}={\bf n}_{e_j}$. Define $\Lambda_i=\sum_{j=1}^r \frac{1}{|e_j|}{\mathcal T}heta_j$. It can be shown that $\nabla_w\cdot\Lambda_i=0$ (see \cite{mwy-divfree} for the details)
{\bf e}gin{figure}
{\bf e}gin{center}
\includegraphics[width=4cm]{div-free_stokes}
\caption{Hull ${\cal H}_{P_i}$ .} {\langle}abel{fig1}
\end{center}
\end{figure}
{\bf e}gin{lemma}
These three types of divergence free functions form a basis for $D_h$, i.e.
{\bf e}gin{equation}{\langle}abel{b5}
D_h={{\rangle}m span}\{\Phi_{i,j},i=1,\cdots,N_T,j=1,2;\Upsilon_i, i=1,\cdots, N_E; \Lambda_i, i=1,\cdots, N_V\}.
\end{equation}
\end{lemma}
{\bf e}gin{proof}
It is easy to check that all the weakly divergence free functions in ({\rangle}ef{b5}) are linearly independent. It is left to check that the number of the basis functions in ({\rangle}ef{b5}) is equal to the dimension of $D_h$. Obviously, the number of the functions in ({\rangle}ef{b5}) is $2N_T+N_E+N_V$. On the other hand we have
$${\rangle}m{dim} (D_h)={\rangle}m{dim} (V_h)-{\rangle}m{dim}(W_h)=2N_T+2N_E-N_T+1.$$
For ${\mathcal T}_h$, it is well known as Euler formula that
{\bf e}gin{equation}{\langle}abel{key}
N_E+1=N_V+N_K.
\end{equation}
Using ({\rangle}ef{key}), we have
$${\rangle}m{dim} (D_h)={\rangle}m{dim} (V_h)-{\rangle}m{dim}(W_h)=2N_T+2N_E-N_T+1=2N_T+N_E+N_V.$$
We have proved the lemma.
\end{proof}
\section{Numerical Examples}{\langle}abel{section-ne}
In this section, six numerical examples are tested for
the two dimensional Stokes equations ({\rangle}ef{moment})-({\rangle}ef{bc}). The numerical experiments indicate that the weak Galerkin methods are robust, accurate and easy to implement.
\subsection{Example 1}{\langle}abel{Num_ex1}
We first consider the Stokes equations with homogeneous boundary condition defined on a square $(0, 1)\times (0, 1)$.
The exact solutions are given by
$$
{\bf u} = {\bf e}gin{pmatrix}2\pi\sin(\pi x)\sin(\pi x)\cos(\pi y)\sin(\pi y) \\
-2\pi\sin(\pi x)\sin(\pi y)\cos(\pi x)\sin(\pi y)
\end{pmatrix},
$$
and
\[
p=\cos(\pi x)\cos(\pi y).
\]
The uniform triangular mesh is used for testing. Denote mesh size by $h.$ The numerical results of Algorithm 1 are presented in Table {\rangle}ef{ex1_1}. These results show the $O(h)$ error of the velocity in the $H^1-$norm and pressure in the $L^2-$norm as predicted by Theorem {\rangle}ef{h1-bd}. Convergence rate of $O(h^2)$ for velocity in the $L^2-$norm is observed.
Furthermore, the divergence free weak Galerkin Algorithm 2 is tested for this example. The weakly divergence-free subspace $D_h$ is constructed as described in previous section. By using the basis functions in $D_h$, the saddle-point system ({\rangle}ef{wg-m})-({\rangle}ef{wg-c}) is reduced to a definite system ({\rangle}ef{df-wg}) only depending on velocity unknowns. The numerical performance of velocity measured in $H^1-$norm and $L^2-$norm is shown in Table {\rangle}ef{ex1_2}.
{\bf e}gin{table}
\caption{Example {\rangle}ef{Num_ex1}. Numerical results of Algorithm 1.}
{\langle}abel{ex1_1}
\center
{\bf e}gin{tabular}{||c||ccc||}
\hline\hline
$h$ & ${|\hspace{-.02in}|\hspace{-.02in}|} {\bf u}_h-{\bf Q}_{h}{\bf u}{|\hspace{-.02in}|\hspace{-.02in}|}$ & $\|{\bf u}_0-{\bf Q}_{0}{\bf u}\|$ & $||Q_0p-p_h||$ \\
\hline\hline
1/4 &4.0478 &3.7181e-1 &1.7906 \\ \hline
1/8 &1.8723 &9.8624e-2 &8.7513e-1 \\ \hline
1/16 &9.1907e-1 &2.5276e-2 &4.1211e-1 \\ \hline
1/32 &4.5785e-1 &6.3793e-3 &2.0019e-1 \\ \hline
1/64 &2.2874e-1 &1.5992e-3 &9.9207e-2 \\ \hline
1/128 &1.1435e-1 &4.0009e-4 &4.9486e-2 \\ \hline\hline
$O(h^r),r=$ & 1.0238 & 1.9750 & 1.0386 \\ \hline\hline
\end{tabular}
\end{table}
{\bf e}gin{table}
\caption{Example {\rangle}ef{Num_ex1}. Numerical results of Algorithm 2.}
{\langle}abel{ex1_2}
\center
{\bf e}gin{tabular}{||c||cc||}
\hline\hline
$h$ &${|\hspace{-.02in}|\hspace{-.02in}|}{\bf u}_h-{\bf Q}_{h}{\bf u}{|\hspace{-.02in}|\hspace{-.02in}|}$ & $\|{\bf u}_0-{\bf Q}_{0}{\bf u}\|$ \\
\hline\hline
1/4 &6.3120 &2.6300e-1 \\ \hline
1/8 &3.3499 &6.9789e-2 \\ \hline
1/16 &1.7174 &1.7890e-2 \\ \hline
1/32 &8.6696e-1 &4.5154e-3 \\ \hline
1/64 &4.3468e-1 &1.1320e-3 \\ \hline
1/128 &2.1750e-1 &2.8320e-4 \\ \hline\hline
$O(h^r),r=$ &9.7484e-01 &1.9748 \\ \hline\hline
\end{tabular}
\end{table}
\subsection{Example 2}{\langle}abel{Num_ex2}
The purpose of this example is to test the robustness and accuracy of this WG method for handling non-homogeneous boundary condition and complicated geometry.
Consider the Stokes equations with non-homogeneous boundary condition that have the exact solutions
$$
{\bf u} = {\bf e}gin{pmatrix}x+x^2-2xy+x^3-3xy^2+x^2y \\
-y-2xy+y^2-3x^2y+y^3-xy^2
\end{pmatrix},
$$
and
\[
p=xy+x+y+x^3y^2-4/3.
\]
In this example, domain $\Omega$ is derived from a square $(0,1)\times(0,1)$ by taking out three circles centered at $(0.5,0.5),\ (0.2,0.8),\ (0.8,0.8) $ with radius $0.1$.
We start the weak Galerkin simulation on the initial mesh as shown in Figure {\rangle}ef{IniMesh_ex2}. Then each refinement is obtained by dividing each triangle into four congruent triangles.
Table {\rangle}ef{tab:inhomogeneous0} displays the errors and convergence rate of the numerical solution of Algorithm 1. Optimal order convergence rates for velocity and pressure are observed in corresponding norms.
{\bf e}gin{table}
\caption{Example {\rangle}ef{Num_ex2}. Numerical results of Algorithm 1.}
{\langle}abel{tab:inhomogeneous0}
\center
{\bf e}gin{tabular}{||c||ccc||}
\hline\hline
$h$ & ${|\hspace{-.02in}|\hspace{-.02in}|} {\bf u}_h-{\bf Q}_{h}{\bf u}{|\hspace{-.02in}|\hspace{-.02in}|}$ & $\|{\bf u}_0-{\bf Q}_{0}{\bf u}\|$ & $||Q_0p-p_h||$ \\
\hline\hline
Level 1 &1.5123e-1 &6.5055e-3 &2.2512e-1\\ \hline
Level 2 &7.6271e-2 &1.7736e-3 &9.6785e-2\\ \hline
Level 3 &3.8276e-2 &4.6167e-4 &4.0673e-2\\ \hline
Level 4 &1.9168e-2 &1.1707e-4 &2.3700e-2\\ \hline
Level 5 &9.5900e-3 &2.9394e-5 &1.9385e-2\\ \hline\hline
$O(h^r),r=$ & 9.9506e-1 & 1.9501 & 9.1053e-1\\ \hline\hline
\end{tabular}
\end{table}
{\bf e}gin{figure}[!tb]
\centering
{\bf e}gin{tabular}{c}
{\rangle}esizebox{2.45in}{2.1in}{\includegraphics{holesquare_1_mesh.pdf}}
\end{tabular}
\caption{Example {\rangle}ef{Num_ex2}: Initial mesh.}
{\langle}abel{IniMesh_ex2}
\end{figure}
\subsection{Example 3}{\langle}abel{Num_ex3}
Let $({\bf u},p)$ as follows
{\bf e}gin{eqnarray*}
{\bf u} = {\bf e}gin{pmatrix}1-e^{{\langle}ambda x}\cos(2\pi y) \\
\frac{{\langle}ambda}{2\pi}e^{{\langle}ambda x}\sin(2\pi y)
\end{pmatrix},
p=\frac{1}{2}e^{2{\langle}ambda x}+C,
\end{eqnarray*}
be the exact solution of the Stokes equations,
{\bf e}gin{eqnarray}
-\frac{1}{\mathrm {Re}}\Delta{\bf u}+\nabla p={\bf f},\quad \nabla\cdot{\bf u}=0\mbox{ in }\Omega,
\end{eqnarray}
where ${\langle}ambda=\mathrm {Re}/2-\sqrt{\mathrm {Re}^2/4+4\pi^2}$ and $\mathrm {Re}$ is the Reynolds number.
Let $\Omega=(-1/2,3/2)\times(0,2).$ The Dirichlet boundary condition for velocity is considered in this example.
In Table {\rangle}ef{ex3}, we demonstrate the error profiles and convergence rates of the numerical solution of Algorithm 1 with $\mathrm {Re}=1,\ 10,\ 100,\ 1000.$ The streamlines of velocity and color contour of pressure for $\mathrm {Re}=1,\ 10,\ 100,\ 1000$ are plotted in Figure {\rangle}ef{fig:ex3}.
{\bf e}gin{table}
\caption{Example {\rangle}ef{Num_ex3}: Numerical results of Algorithm 1.}{\langle}abel{ex3}
\center
{\bf e}gin{tabular}{|c|cc|cc|cc|}
\hline\hline
$h$ & ${|\hspace{-.02in}|\hspace{-.02in}|}{\bf u}_h-{\bf Q}_{h}{\bf u}{|\hspace{-.02in}|\hspace{-.02in}|}$ & order& $\|{\bf u}_0-{\bf Q}_{0}{\bf u}\|$ & order &$||Q_0p-p_h||$ & order\\
\hline
\multicolumn{7}{|c|}{$\mathrm {Re}=1$}\\ \hline
1/8 &4.2375e+1 & &4.2372 & &2.9223e+1& \\
1/16 &2.4722e+1 &7.7742e-1 &1.3963 &1.6015 &1.2713e+1&1.2008\\
1/32 &1.3100e+1 &9.1623e-1 &3.9686e-1 &1.8149 &5.2018 &1.2892\\
1/64 &6.6667e &9.7452e-1 &1.0374e-1 &1.9357 &2.3142 &1.1685\\
1/128 &3.3504e &9.9264e-1 &2.6294e-2 &1.9802 &1.1550 &1.0026\\
\hline
\multicolumn{7}{|c|}{$\mathrm {Re}=10$}\\ \hline
1/8 &6.0606 & &2.0457 & &7.6173e-1 & \\
1/16 &3.2851 &8.8352e-1 &5.9724e-1 &1.7762 &2.9472e-1 &1.3699\\
1/32 &1.6896 &9.5926e-1 &1.5926e-1 &1.9069 &1.1379e-1 &1.3730\\
1/64 &8.5296e-1 &9.8613e-1 &4.0832e-2 &1.9636 &4.5851e-2 &1.3113\\
1/128 &4.2787e-1 &9.9531e-1 &1.0306e-2 &1.9862 &1.9770e-2 &1.2136\\
\hline
\multicolumn{7}{|c|}{$\mathrm {Re}=100$}\\ \hline
1/8 &5.5209e-1 & &6.7127e-1 & &1.5818e-2 & \\
1/16 &2.7981e-1 &9.8046e-1 &1.7946e-1 &1.9032 &6.7914e-3 &1.2198\\
1/32 &1.4063e-1 &9.9254e-1 &4.5955e-2 &1.9654 &2.9102e-3 &1.2226\\
1/64 &7.0434e-2 &9.9756e-1 &1.1575e-2 &1.9892 &1.3386e-3 &1.1204\\
1/128 &3.5235e-2 &9.9926e-1 &2.9001e-3 &1.9968 &6.4795e-4 &1.0468\\
\hline
\multicolumn{7}{|c|}{$\mathrm {Re}=1000$}\\ \hline
1/8 &2.0636e-1 & &8.1461e-1 & &1.8694e-3 & \\
1/16 &1.0436e-1 &9.8359e-1 &2.1625e-1 &1.9134 &7.6097e-4 &1.2967\\
1/32 &5.2395e-2 &9.9407e-1 &5.5149e-2 &1.9713 &3.2176e-4 &1.2419\\
1/64 &2.6230e-2 &9.9821e-1 &1.3868e-2 &1.9916 &1.4850e-4 &1.1155\\
1/128 &1.3120e-2 &9.9945e-1 &3.4726e-3 &1.9977 &7.2192e-5 &1.0406\\ \hline\hline
\end{tabular}
\end{table}
{\bf e}gin{figure}[!tb]
\centering
{\bf e}gin{tabular}{cc}
{\rangle}esizebox{2.45in}{2.1in}{\includegraphics{Kovasznay_80-1.pdf}} \quad
{\rangle}esizebox{2.45in}{2.1in}{\includegraphics{Kovasznay_80-10.pdf}} \\
{\rangle}esizebox{2.45in}{2.1in}{\includegraphics{Kovasznay_80-100.pdf}}\quad
{\rangle}esizebox{2.45in}{2.1in}{\includegraphics{Kovasznay_80-1000.pdf}}
\end{tabular}
\caption{Example {\rangle}ef{Num_ex3} Streamlines of velocity and color contour of pressure for $\mathrm {Re}=1,\ 10,\ 100,\ 1000$ (top left, top right, bottom left, bottom right.)}{\langle}abel{fig:ex3}
\end{figure}
\subsection{Example 4}{\langle}abel{Num_ex4}
Two dimensional channel flow around a circular obstacle is simulated in this problem.
We consider the Stokes equations with non-homogeneous boundary condition:
{\bf e}gin{eqnarray*}
{\bf u}|_{\partial\Omega}={\bf g}={\bf e}gin{cases}
(1,0)^t,&\mbox{ if } x=0;\\
(1,0)^t,&\mbox{ if } x=1;\\
(0,0)^t,&\mbox{ else. }
\end{cases}
\end{eqnarray*}
Let $\Omega=(0,1)\times(0,1)$ with one circular hole centered at $(0.5,0.5),$ with radius 0.1.
We start with initial mesh and then refined the mesh uniformly. Level 1 mesh and level 2 mesh are shown in Figure {\rangle}ef{IniMesh_Ex4}. The velocity fields of Algorithm 1 on level 1 and level 2 meshes are shown in Figures {\rangle}ef{Num_Ex4_1}. The streamlines of velocity and color contour of pressure are plotted in Figure {\rangle}ef{Num_Ex4_2}.
{\bf e}gin{figure}[!tb]
\centering
{\bf e}gin{tabular}{cc}
{\rangle}esizebox{2.45in}{2.1in}{\includegraphics{holes1quare_1_mesh.pdf}} \quad
{\rangle}esizebox{2.45in}{2.1in}{\includegraphics{holes1quare_2_mesh.pdf}}
\end{tabular}
\caption{Example {\rangle}ef{Num_ex4} Meshes of Level 1 (Left) and Level 2 (Right)}
{\langle}abel{IniMesh_Ex4}
\end{figure}
{\bf e}gin{figure}[!tb]
\centering
{\bf e}gin{tabular}{cc}
{\rangle}esizebox{2.45in}{2.1in}{\includegraphics{holes1quare_1_quiver.pdf}} \quad
{\rangle}esizebox{2.45in}{2.1in}{\includegraphics{holes1quare_2_quiver.pdf}}
\end{tabular}
\caption{Example {\rangle}ef{Num_ex4}: Vector fields of Velocity on Mesh Level 1 (Left) and Mesh Level 2 (Right)}
{\langle}abel{Num_Ex4_1}
\end{figure}
{\bf e}gin{figure}[!tb]
\centering
{\bf e}gin{tabular}{c}
{\rangle}esizebox{2.7in}{2.45in}{\includegraphics{squarehole.pdf}}
\end{tabular}
\caption{Example {\rangle}ef{Num_ex4}: Streamlines of velocity and color contour of pressure.}
{\langle}abel{Num_Ex4_2}
\end{figure}
\subsection{Example 5}{\langle}abel{Num_ex5}
This example is used to test the backward facing step problem. This example is a benchmark problem. Let $\Omega=(-2,8)\times(-1,1)\backslash [-2,0]\times[-1,0]$, consider the Stokes problem with ${\bf f}=0,$ and Dirichlet boundary condition as:
{\bf e}gin{eqnarray*}
{\bf u}|_{\partial\Omega}={\bf g}={\bf e}gin{cases}
(-y(y-1)/10,0)^t,&\mbox{ if } x=-2;\\
(-(y+1)(y-1)/80,0)^t,&\mbox{ if } x=8;\\
(0,0)^t, &\mbox{else}.
\end{cases}
\end{eqnarray*}
{\bf e}gin{figure}[!tb]
\centering
{\bf e}gin{tabular}{c}
{\rangle}esizebox{4.45in}{1.1in}{\includegraphics{backstepe.pdf}} \\
{\rangle}esizebox{4.45in}{1.1in}{\includegraphics{backstepe_zoom2.pdf}}
\end{tabular}
\caption{Example {\rangle}ef{Num_ex5}: Streamlines of velocity and color contour of pressure (top); Zoom in plot.}
{\langle}abel{Num_Ex5_1}
\end{figure}
Figure {\rangle}ef{Num_Ex5_1} plots the streamlines of velocity and color contour of pressure. The plot shows that the pressure is high on the left and low on the right. A zoom figure of lower left corner $[0,0.5]\times[-1,-0.65]$ is plotted in Figure {\rangle}ef{Num_Ex5_1}, which shows one eddy.
\subsection{Example 6}{\langle}abel{Num_ex6}
The lid-driven cavity flow is considered in this example with $\Omega=(0,1)\times(0,1)$, ${\bf f}=0$, and the Dirichlet boundary condition as:
{\bf e}gin{eqnarray*}
{\bf u}|_{\partial\Omega}={\bf g}={\bf e}gin{cases}
(1,0)^t,&\mbox{ if } y=1;\\
(0,0)^t, &\mbox{else}.
\end{cases}
\end{eqnarray*}
{\bf e}gin{figure}[!tb]
\centering
{\bf e}gin{tabular}{ccc}
{\rangle}esizebox{2.45in}{2.1in}{\includegraphics{lidcavity_80_p.pdf}} \quad
{\rangle}esizebox{2.45in}{2.1in}{\includegraphics{lidcavity_80.pdf}} \\
{\rangle}esizebox{2.45in}{2.1in}{\includegraphics{lidcavity_80_left.pdf}}\quad
{\rangle}esizebox{2.45in}{2.1in}{\includegraphics{lidcavity_80_right.pdf}}
\end{tabular}
\caption{Example {\rangle}ef{Num_ex6}: Color contour of pressure (top left); Streamlines of velocity (top right); Zoom in plot of two bottom corners.}{\langle}abel{Num_ex6_1}
\end{figure}
Figure {\rangle}ef{Num_ex6_1} displays the color contour of pressure $p_h$ and streamlines of velocity ${\bf u}_h$ on a uniform mesh. Two Moffat eddies at the bottom corners are detected in Figure {\rangle}ef{Num_ex6_1}.
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\end{document} |
\begin{document}
\begin{center}
{\large\bf STURM--LIOUVILLE-TYPE OPERATORS WITH FROZEN ARGUMENT AND CHEBYSHEV POLYNOMIALS}
\\[0.5cm]
{\large\bf Tzong-Mo Tsai}
\\[0.1cm]
\footnotesize{General Education Center, Ming Chi University of Technology, New Taipei City, 24301, Taiwan
\\[0.1cm]
\it [email protected]}
\\[0.2cm]
{\large\bf Hsiao-Fan Liu}
\\[0.1cm]
\footnotesize{Department of Mathematics, Tamkang University, New Taipei City, 25137, Taiwan
\\[0.1cm]
\it [email protected]}
\\[0.2cm]
{\large\bf Sergey
Buterin}
\\[0.1cm]
\footnotesize{Department of Mathematics, Saratov State University, Astrakhanskaya 83, Saratov 410012, Russia
\\[0.1cm]
\it [email protected]}
\\[0.2cm]
{\large\bf Lung-Hui Chen}
\\[0.1cm]
\footnotesize{General Education Center, Ming Chi University of Technology, New Taipei City, 24301, Taiwan
\\[0.1cm]
\it [email protected]}
\\[0.2cm]
{\large\bf Chung-Tsun Shieh}
\footnotesize{Department of Mathematics, Tamkang University, New Taipei City, 25137, Taiwan
\\[0.1cm]
\it [email protected]}\\[0.2cm]
\end{center}
{\bf Abstract.} The paper deals with Sturm--Liouville-type operators with frozen argument of the form $\ell
y:=-y''(x)+q(x)y(a),$ $y^{(\alpha)}(0)=y^{(\beta)}(1)=0,$ where $\alpha,\beta\in\{0,1\}$ and $a\in[0,1]$ is
an arbitrary fixed rational number. Such nonlocal operators belong to the so-called loaded differential
operators, which often appear in mathematical physics. We focus on the inverse problem of recovering the
potential $q(x)$ from the spectrum of the operator $\ell.$ Our goal is two-fold. Firstly, we establish a deep
connection between the so-called main equation of this inverse problem and Chebyshev polynomials of the first
and the second kinds. This connection gives a new perspective method for solving the inverse problem. In
particular, it allows one to completely describe all non-degenerate and degenerate cases, i.e. when the
solution of the inverse problem is unique or not, respectively. Secondly, we give a complete and convenient
description of iso-spectral potentials in the space of complex-valued integrable functions.
{\it Key words}: Sturm--Liouville operator, functional-differential operator, frozen argument, Chebyshev
polynomials, Jacobi matrices, inverse spectral problem, iso-spectral potentials.
{\it 2010 Mathematics Subject Classification}: 34A55 34K29 47B36
\\
{\large\bf 1. Introduction}
\\
Consider the boundary value problem ${\cal L}:={\cal L}(q(x),a,\alpha,\beta)$ of the form
\begin{equation}\label{1.1}
\ell y:=-y''(x)+q(x)y(a)=\lambda y(x), \quad 0<x<1,
\end{equation}
\begin{equation}\label{1.2}
y^{(\alpha)}(0)=y^{(\beta)}(1)=0,
\end{equation}
where $\lambda$ is the spectral parameter, $q(x)$ is a complex-valued function in $L(0,1),$ to which we refer
as {\it potential}, and $\alpha,\beta\in\{0,1\},$ while $a\in[0,1].$ The operator $\ell$ is called the
Sturm--Liouville-type operator with {\it frozen argument}.
Denote by $\{\lambda_n\}_{n\ge1}$ the spectrum of ${\cal L}$ and consider the following inverse problem.
{\bf Inverse Problem 1.} Given $\{\lambda_n\}_{n\ge1},$ $a,$ $\alpha$ and $\beta;$ find $q(x).$
Nonlocal operators of the form (\ref{1.1}), (\ref{1.2}) belong to the so-called loaded differential operators
(see, e.g., \cite{Isk, Kral, Nakh76, NakhBor77, Nakh82, Nakh12, Lom14, Lom15, BH21}), which often appear in
mathematical physics.
For example, some models of physical systems with feedback leading to nonlocal differential operators with
frozen argument were described in \cite{BH21}. The presence of a feedback means that the external affect on
the system depends on its current state. If this state is taken into account only at some fixed physical
point of the system, then mathematically this corresponds to an operator with frozen argument.
Among purely mathematical applications, we illustrate here the so-called {\it method of reduction to loaded
equations} (see, \cite{Nakh82, Nakh12}). For this purpose, let us aim to study the boundary value problem for
the integro-differential equation
\begin{equation}\label{1.2-1}
-y''(x)+\int\limits_0^1H(x,t)y(t)\,\mathrm{d}t=\lambda y(x), \quad 0<x<1,
\end{equation}
subject to boundary conditions (\ref{1.2}) for some $\alpha$ and $\beta.$ The method consists in replacing
equation~(\ref{1.2-1}) with the loaded one
\begin{equation}\label{1.2-2}
-y''(x)+\sum_{\nu=1}^N q_\nu(x)y(a_\nu)=\lambda y(x), \quad 0<x<1,
\end{equation}
possessing frozen arguments $a_1,\ldots,a_N,$ where the sum is an appropriate quadrature formula for
approximating the integral in (\ref{1.2-1}). For example, Simpson's rule (see, e.g., \cite{Atk89})
$$
\int\limits_{x_1}^{x_2} f(x)\,\mathrm{d}x\approx\frac{x_2-x_1}6\Big(f(x_1)+4f\Big(\frac{x_1+x_2}2\Big)
+f(x_2)\Big)
$$
in the case $\alpha=\beta=0$ leads to equation (\ref{1.1}) with $q(x)=2H(x,a)/3$ and $a=1/2.$
Various aspects of Inverse Problem~1 in the case $q(x)\in L_2(0,1)$ were studied in \cite{BBV, BV, BK,
Wang20}. In \cite{BBV, BV, BK}, diverse cases of the triple $(a,\alpha,\beta)$ with rational $a$'s were
considered. In particular, it was established that the inverse problem may be uniquely solvable or not
depending on the parameters $\alpha,\,\beta$ and also on the parity of the integers $k,$ $j$ or $j+k$ taken
from the representation $a=j/k$ under the assumption that $j$ and $k$ are {\it mutually prime}. According to
this, there were highlighted two cases: {\it non-degenerate} and {\it degenerate}~ones, respectively.
Moreover, a complete characterization of the spectrum $\{\lambda_n\}_{n\ge1}$ was given, which includes the
asymptotics for large modulus eigenvalues
along with a special additional condition in the degenerate case.
Specifically, it was established that, in the degenerate case, asymptotically $k$-th part of the spectrum
degenerates, i.e. each $k$-th eigenvalue carries no information on the potential.
For example, in the case when $\alpha=\beta=0$ and $q(x)\in L_2(0,1),$ the spectrum is completely
characterized by the relations
$$
\lambda_n=(\pi n)^2+\varkappa_n, \quad \{\varkappa_n\}\in l_2, \quad \lambda_{kn}=(\pi kn)^2, \quad
n\in{\mathbb N},
$$
i.e. each $k$-th eigenvalue $\lambda_{kn}$ degenerates. In this case, for the unique solvability of Inverse
Problem~1, one should specify the potential on one of the subintervals $((\nu-1)/k,\nu/k),$
$\nu=\overline{1,k}.$ Thus, the smaller part of the spectrum degenerates, the less additional information on
the potential is required, while in the non-degenerate case no extra information is required at all (see
(\ref{deg}) and (\ref{non-deg}) below for a complete description of degenerate and non-degenerate subcases).
This causes instable informativity of the spectrum with respect to~$a,$ which was first revealed in \cite{BV}
(see also \cite{BK}). For example, while a half of the spectrum degenerates as soon as $a=1/2,$ for
$a=a_k:=(k-1)/(2k)$ with even $k$ so does only its $2k$-th part, but $a_k\to1/2$ as $k\to\infty.$
Thus, returning to the method of reduction to loaded equations mentioned above, one can note that this method
is sensitive to choosing the nodes $a_1,\ldots,a_N$ in (\ref{1.2-2}) at least
when approximating the spectrum of the initial integro-differential operator (\ref{1.2-1}) subject to
(\ref{1.2}).
Concerning irrational values of $a,$ it was established in \cite{Wang20} that all they correspond to the
non-degenerate case for all pairs $(\alpha,\beta),$ i.e. the solution of Inverse Problem~1 is always unique
as soon as $a\notin{\mathbb Q}.$ However, the question of the spectrum characterization
still remains open.
In \cite{Nizh-09, AlbHryNizh
} and other works, in connection with the theory of diffusion processes, the case $a=1$ was investigated but
with the special nonlocal boundary conditions
\begin{equation}\label{1.3}
y(0)-\alpha y(1)=y'(1)-\alpha y'(0)+\int\limits_0^1y(t)\overline{q(t)}\,\mathrm{d}t=0, \quad
\alpha\in\{0,1\},
\end{equation}
guarantying the self-adjointness of the corresponding operator generated by (\ref{1.1}) and (\ref{1.3}).
However, such settings never entail the uniqueness of recovering the function $q(x)$ from the spectrum.
In \cite{BH21}, the case of the quasi-periodic boundary conditions of the form
$$
y(0)-\gamma y(1)=y'(0)-\gamma y'(1)=0
$$
for any possible $\gamma\in{\mathbb C}\setminus\{0\}$ was studied, and a complete solution was obtained for
the inverse problem of recovering the potential $q(x)$ from the corresponding spectrum (spectra). Further
aspects of recovering the operator $\ell$ as well as its spectral properties were studied in \cite{XY19-1,
XY19-2, HBY, Hu20}.
In the present paper, we return to Inverse Problem~1. In \cite{BBV, BV, BK}, this problem was reduced to some
linear functional equation with respect to the potential $q(x)$ (see equation~(\ref{dd:1}) in the next
section), which was referred to as {\it main equation} of the inverse problem. For rational values of~$a,$
the main equation was reduced to linear system (\ref{2.7}) with a special $k\times
k$-matrix~$A_{j,k}^{(\alpha,\beta)},$ whose rank appeared to be ranging between $k-1$ and $k.$ These two
possibilities, in turn, correspond to the degenerate and non-degenerate cases, respectively. In the works
\cite{BBV, BV, BK}, various approaches to studying this matrix and calculating its determinant were
developed.
Here, we establish a deep connection between the matrix $A_{j,k}^{(\alpha,\beta)}$ and Chebyshev polynomials
of the first and the second kinds, which gives another approach for studying Inverse~Problem~1. Using this
new approach, we obtain a complete and convenient description of all iso-spectral complex-valued potentials
in $L(0,1)$ for the degenerate case.
The paper is organized as follows. In the next section, we provide some necessary information on the boundary
value problem ${\cal L}.$ In Section~3, we represent characteristic determinants of the matrices
$A_{1,k}^{(\alpha,\beta)}$ via Chebyshev polynomials. In Section~4, we establish that the matrices
$A_{j,k}^{(\alpha,\beta)}$ can be obtained after substituting $A_{1,k}^{(\alpha,\beta)}$ into appropriate
Chebyshev polynomials, which allows one to study their spectra for $j>0.$ In Section~5, we construct
iso-spectral potentials in the degenerate case. In Section~6, we provide some illustrative examples.
\\
{\large\bf 2. Preliminary information}
\\
Let $C(x,\lambda)$ and $S(x,\lambda)$ be solutions of equation (\ref{1.1}) under the initial conditions
\begin{equation}\label{2.2}
C(a,\lambda)=S'(a,\lambda)=1, \quad S(a,\lambda)=C'(a,\lambda)=0.
\end{equation}
By substitution, it can be easily checked that
\begin{equation}\label{2.1}
C(x,\lambda)=\cos\rho(x-a)+\int\limits_a^x\frac{\sin\rho(x-t)}{\rho}q(t)\,\mathrm{d}t, \quad
S(x,\lambda)=\frac{\sin\rho(x-a)}{\rho}, \quad \rho^2:=\lambda.
\end{equation}
Since these solutions are uniquely determined by conditions~(\ref{2.2}), eigenvalues of the problem~${\cal
L}$ coincide with zeros of the entire function
\begin{equation}\label{2.3}
\Delta_{\alpha,\beta}(\lambda)=
\begin{vmatrix}
C^{(\alpha)}(0,\lambda) & S^{(\alpha)}(0,\lambda) \\
C^{(\beta)}(1,\lambda) & S^{(\beta)}(1,\lambda)
\end{vmatrix},
\end{equation}
which is called {\it characteristics function} of ${\cal L}.$
Without loss of generality, we always assume that $0\le a\le1/2$ since the spectrum of the problem ${\cal
L}(q(x),a,\alpha,\beta),$ obviously, coincides with the one of ${\cal L}(q(1-x),1-a,\beta,\alpha).$
Substituting (\ref{2.1}) into (\ref{2.3}) one can obtain the following representations (see, e.g.,
\cite{BK}):
\begin{equation} \label{2.5}
\Delta_{\alpha,\alpha}(\lambda)=\rho^{2\alpha}\Big(\frac{\sin\rho}{\rho}+
\int\limits_0^1W_{\alpha,\alpha}(x)\frac{\cos\rho x}{\rho^2}\,\mathrm{d}x\Big),\;\;
\Delta_{\alpha,\beta}(\lambda)=(-1)^\alpha\cos\rho+\int\limits_0^1W_{\alpha,\beta}(x)\frac{\sin\rho
x}{\rho}\,\mathrm{d}x
\end{equation}
for $\alpha\ne\beta,$ where the functions $W_{\alpha,\beta}(x)$ have the form
\begin{equation}\label{dd:1}
\displaystyle W_{\alpha,\beta}(x)=\frac{(-1)^{\alpha\beta}}{2} \left\{
\begin{array}{cl}
\displaystyle q(1-a+x)+dq(1-a-x), & x\in(0,a),\\[3mm]
\displaystyle cq(1+a-x)+dq(1-a-x), & x\in(a,1-a),\\[3mm]
\displaystyle c\Big(q(1+a-x)+q(x-1+a)\Big), & x\in(1-a,1),
\end{array}\right.
\end{equation}
while the numbers $c$ and $d$ are determined by the formulae
\begin{equation}\label{2.8.1}
c = (-1)^{\beta + 1}, \quad d=(-1)^{\alpha+\beta}.
\end{equation}
Assuming the function $W_{\alpha,\beta}(x)$ to be known, one can consider (\ref{dd:1}) as a linear functional
equation with respect to $q(x).$ Since each function $\Delta_{\alpha,\beta}(\lambda)$ is uniquely determined
by its zeros:
\begin{equation}\label{EQ}
\Delta_{\alpha,\beta}(\lambda) =(-1)^\alpha(\lambda_1-\lambda)^{\alpha\beta}
\prod\limits_{n=1+\alpha\beta}^\infty\frac{\lambda_n-\lambda}{\Big(n-\frac{\alpha+\beta}2\Big)^2\pi^2},
\end{equation}
Inverse Problem~1 is equivalent to this functional equation (\ref{dd:1}), which is called {\it main equation}
of the inverse problem.
If $a$ is rational, i.e. there exist {\it mutually prime} integers $j$ and $k$ such that $a=j/k,$ then the
main equation can be represented in the following way:
\begin{equation}\label{2.7}
W_{\alpha,\beta}(x)=\frac{(-1)^{\alpha\beta}}{2} Q^{-1}A_{j,k}^{(\alpha,\beta)} Rq(x), \quad 0<x<1,
\end{equation}
where $A_{j,k}^{(\alpha,\beta)}$ is a square matrix of order $k,$ while $Q$ and $R$ are bijective operators
mapping $L(0,1)$ onto $(L(0,b))^k,$ $b:=1/k,$ and acting by the formulae
\begin{equation} \label{2.4.1}
Qf:=(f,Q_2f,\ldots,Q_kf)^T, \quad Rf:=(R_1f,R_2f,\ldots,R_kf)^T.
\end{equation}
Here, $T$ is the transposition sign, while $Q_\nu$ and $R_\nu$ are shift and involution operators mapping
$L(0,1)$ onto $L(0,b),$ which are determined by the formulae
\begin{equation}\label{2.3.1}
Q_\nu f(x)=\left\{
\begin{array}{ll}
\displaystyle\!\! f((\nu-1)b+x) \;\; \text{for odd} \; \nu,\\[3mm]
\displaystyle\!\! f(\nu b-x) \;\; \text{for even} \; \nu,
\end{array}\right.
R_\nu f(x)=\left\{
\begin{array}{l}
\displaystyle\!\! f((k-\nu)b+x) \;\; \text{for even} \; j+\nu,\\[3mm]
\displaystyle\!\! f((k-\nu+1)b-x) \;\; \text{for odd} \; j+\nu,
\end{array}\right.
\!\!
\end{equation}
where $x\in(0,b)$ and $\nu=\overline{1,k}.$
The matrix $A_{j,k}^{(\alpha,\beta)}=(a_{m,n})_{m,n=\overline{1,k}},$ in turn, is constructed in the
following way. For $k=1,$ it consists of a single element $a_{1,1}=2(-1)^{\beta+1}\alpha$ as soon as $j=0,$
while, for $k\ge2j\ge2,$ its elements are determined by the formulae
\begin{equation}\label{2.8}
\left.\begin{array}{rll}
(i) & a_{m,j-m+1}= 1, & m=\overline{1,j},\\[3mm]
(ii) & a_{m,m+j}= d, & m = \overline{1,k-j}, \\[3mm]
(iii) & a_{m,m-j}= c, & m=\overline{j +1,k},\\[3mm]
(iv) & a_{m,2k-m-j+1}= c, & m=\overline{k-j + 1,k},\\[3mm]
(v) & a_{m,n}=0 & \text{for the remaining pairs}\;\; (m,n).
\end{array}\right\}
\end{equation}
The items $(i)$--$(iv)$ in \eqref{2.8} correspond to subdiagonals, consisting of equal elements: $1,$ $c$
and~$d.$ For example, the the matrix $A_{3,7}^{(\alpha,\beta)}$ has the form
\begin{equation*}
\begin{pmatrix}
\cdot & \cdot & 1 & d & \cdot & \cdot & \cdot \\
\cdot & 1 & \cdot & \cdot & d & \cdot & \cdot \\
1 & \cdot & \cdot & \cdot & \cdot & d & \cdot \\
c & \cdot & \cdot & \cdot & \cdot & \cdot & d \\
\cdot & c & \cdot & \cdot & \cdot & \cdot & c \\
\cdot & \cdot &c & \cdot & \cdot & c & \cdot \\
\cdot & \cdot & \cdot & c & c & \cdot & \cdot
\end{pmatrix},
\end{equation*}
where dots indicate zero elements.
In papers \cite{BBV, BV, BK}, various approaches were used for calculating the determinant and the rank of
$A_{j,k}^{(\alpha,\beta)},$ depending on generality of the situation. In particular, in \cite{BV}, a
reduction-type algorithm was suggested for the case $\alpha=\beta=0.$ This algorithm appeared to be
equivalent to the Euclidean algorithm for finding the greatest common devisor of the numbers $j$ and $k.$
Since $j$ and $k$ are mutually prime, the algorithm gave consecutive relations leading to the result:
$$
\rank A_{j,k}^{(0,0)}=\rank A_{j,k-j}^{(0,0)}+j= \rank A_{k-2j,k-j}^{(0,0)}+j=\ldots=\rank
A_{0,1}^{(0,0)}+k-1 =k-1.
$$
Later, in \cite{BK}, it was established that the rank of $A_{j,k}^{(\alpha,\beta)}$ cannot be less than
$k-1.$ For this purpose, a combinatorial approach for calculating the determinant of
$A_{j,k}^{(\alpha,\beta)}$ was suggested, which was based on studying properties of an undirected graph $G$
corresponding to a special traversal of nonzero elements of $A_{j,k}^{(\alpha,\beta)}.$ It was established
that $G$ was a bipartite Eulerian cycle, which has led to representing $\det A_{j,k}^{(\alpha,\beta)}$ as a
sum of precisely two products consisting of nonzero elements of $A_{j,k}^{(\alpha,\beta)}.$ This gave a
complete classification of degenerate and non-degenerate cases corresponding to non-unique and unique
solvability of Inverse Problem~1, respectively. Specifically, the degenerate case occurs when one of the
following groups of conditions is fulfilled:
\begin{equation}\label{deg}
\left.\begin{array}{rl}
\text{(I)}& \alpha=\beta=0;\\[3mm]
\text{(II)}& \alpha=0, \ \beta=1 \text{ and } j \text{ is even;}\\[3mm]
\text{(III)}& \alpha=1,\ \beta=0 \text{ and } k+j \text{ is even;}\\[3mm]
\text{(IV)}& \alpha=\beta=1 \text{ and } k \text{ is even};
\end{array}\right\}
\end{equation}
while the non-degenerate case includes the remaining groups of conditions:
\begin{equation}\label{non-deg}
\left.\begin{array}{rl}
\text{(V)}& \alpha=0, \;\beta=1 \text{ and } j \text{ is odd;}\\[3mm]
\text{(VI)}& \alpha=1,\ \beta=0 \text{ and } k+j \text{ is odd;}\\[3mm]
\text{(VII)}& \alpha=\beta=1 \text{ and } k \text{ is odd.}
\end{array}\right\}
\end{equation}
This classification remains valid also for $a>1/2,$ i.e. it holds for all relevant $j\in\{0,\ldots,k\}.$
In the subsequent sections, we establish a deep connection between the matrix $A_{j,k}^{(\alpha,\beta)}$ and
Chebyshev polynomials of the first and the second kinds. This connection gives, in particular, another
approach for studying the main equation~(\ref{2.7}). Using this approach one can easily give a complete
description of iso-spectral potentials in the degenerate case (see Section~5).
\\
{\large\bf 3. Chebyshev polynomials and the case $j=1$}
\\
First, we give some necessary information about Chebyshev polynomials $T_{n}(z)$ and $U_{n}(z)$ of the first
and the second kinds, respectively, which can be defined by the formulae
\begin{equation}\label{3.10,1}
T_{n}(\cos\theta)=\cos n\theta, \quad U_{n}(\cos\theta)=\frac{\sin(n+1)\theta}{\sin\theta}, \quad
n+1\in{\mathbb N}.
\end{equation}
Alternatively, one can use the following recurrent relation
\begin{equation}\label{3.8,1}
Y_{n+1}(z)=2z Y_{n}(z)- Y_{n-1}(z).
\end{equation}
Then the polynomials of the first kind $T_n(z)=Y_n(z)$ are determined by the initial conditions
\begin{equation}\label{3.8,2}
T_{0}(z)=1, \quad T_{1}(z)=z,
\end{equation}
while the initial conditions
\begin{equation}\label{3.9,1}
U_{0}(z)=1, \quad U_{1}(z)=2z
\end{equation}
determine the polynomials of the second kind $Y_n(z)=U_n(z).$ For more details, see, e.g., \cite{Riv90}.
It is well known and also can be easily seen that Chebyshev polynomials may possess only simple zeros, and
they are always odd or even functions in accordance with the parity of $n.$ In particular, we have
$T_{n}(0)=U_{n}(0)=0$ as soon as $n$ is odd, and $T_{n}(0)U_{n}(0)\ne0$ for even $n.$
Let us proceed with studying the matrix $A_{j,k}^{(\alpha,\beta)}$ for $k\ge2.$ In this section, we focus on
the case~$j=1.$ Consider the characteristic polynomial
\begin{equation}\label{2.4}
p_k(z):=\det(zI-A_{1,k}^{(\alpha,\beta)}),
\end{equation}
where $I$ is the unit matrix. The following lemma holds.
{\bf Lemma 1. }{\it The characteristic polynomial of the matrix $A_{1,k}^{(\alpha,\beta)}$ has the form
\begin{equation}\label{2.3.2}
p_k(z)=(z-c)q_{k-1}(z)-cdq_{k-2}(z),
\end{equation}
where $c$ and $d$ are determined by (\ref{2.8.1}), while the polynomials $q_n(z)$ can be found from the
recurrent relations
\begin{equation}\label{2.3.3}
q_0(z)=1, \quad q_1(z)=z-1, \quad q_{n+1}(z)=z q_n(z)- cdq_{n-1}(z), \quad n=\overline{1,k-2}.
\end{equation}
}
{\it Proof.} First, we note that $q_\nu(z)$ is the characteristic polynomial of the three-diagonal matrix
$B_\nu$ that is obtained from $A_{1,\nu+1}^{(\alpha,\beta)}$ by removing the last column along with the last
row, i.e.
\begin{equation}\label{2.3.4}
q_\nu(z)=\det(zI-B_\nu)= \left|\begin{array}{ccccc}
z-1& -d & & & \\[1mm]
-c & z & -d & & \\[1mm]
& -c & \ddots & \ddots & \\[1mm]
& & \ddots & z & -d \\[1mm]
& & & -c & z \\[1mm]
\end{array}\right|,
\end{equation}
where each of both subdiagonals consists of equal elements, while all elements of the main diagonal starting
from the second position are equal too. Indeed, for $\nu=1$ formula (\ref{2.3.4}) is obvious. Further, let it
hold for any $\nu\le n.$ Then expanding the determinant in (\ref{2.3.4}) for $\nu=n+1$ with respect to the
elements of the last row we obtain the last equality in (\ref{2.3.3}).
Finally, expanding the determinant
$$
\det(zI-A_{1,k}^{(\alpha,\beta)})= \left|\begin{array}{ccccc}
z-1& -d & & & \\[1mm]
-c & z & -d & & \\[1mm]
& -c & \ddots & \ddots & \\[1mm]
& & \ddots & z & -d \\[1mm]
& & & -c & z-c \\[1mm]
\end{array}\right|
$$
with respect to the last row, we obtain representation (\ref{2.3.2}). $
\Box$
The following corollary gives the classifications (\ref{deg}) and (\ref{non-deg}) for $j=1.$
{\bf Corollary 1. }{\it The determinant of the matrix $A_{1,k}^{(\alpha,\beta)}$ can be calculated by the
formula}
\begin{equation}\label{3.5}
\det A_{1,k}^{(\alpha,\beta)} = \left\{
\begin{array}{l}
\displaystyle (-cd)^{(k-1)/2}(1+c) \quad \text{if}\;\;k\;\;\text{is odd},\\[1mm]
\displaystyle c(-cd)^{k/2-1}(1-d) \quad \text{if}\;\;k\;\;\text{is even}.
\end{array}\right.
\end{equation}
{\it Proof.} According to (\ref{2.4}) and (\ref{2.3.2}), we have $\det A_{1,k}^{(\alpha,\beta)}=(-1)^{k+1}c
(q_{k-1}(0)+dq_{k-2}(0)).$ The first two formulae in (\ref{2.3.3}) give $q_0(0)=1$ and $q_1(0)=-1.$ Assume
that
\begin{equation}\label{3.6}
q_{2\nu}(0)=(-cd)^\nu, \quad q_{2\nu+1}(0)=-(-cd)^\nu, \quad 0\le\nu\le l,
\end{equation}
for some $l\in{\mathbb N}.$ Then the last relation in (\ref{2.3.3}) implies
$$
q_{2(l+1)}(0)=-cdq_{2l}(0)=(-cd)^{l+1}, \quad q_{2(l+1)+1}(0)=-cdq_{2l+1}(0)=-(-cd)^{l+1}.
$$
Hence, (\ref{3.6}) holds for all $\nu\ge0.$ Substituting (\ref{3.6}) into the first formula of this proof, we
arrive~at
$$
\det A_{1,k}^{(\alpha,\beta)} = \left\{
\begin{array}{l}
\displaystyle c(q_{2\nu}(0)+dq_{2\nu-1}(0))=(-cd)^\nu(1+c) \quad \text{for}\;\;k=2\nu+1,\\[2mm]
\displaystyle -c(q_{2\nu+1}(0)+dq_{2\nu}(0))=c(-cd)^\nu(1-d) \quad \text{for}\;\;k=2\nu+2,
\end{array}\right.
$$
which finalizes the proof. $
\Box$
The main result of this section is contained in the following theorem.
{\bf Theorem 1. }{\it The following representations hold:}
\begin{equation}\label{3.6.1}
\det(zI-A_{1,k}^{(0,0)})=i^{k-1}zU_{k-1}\Big(\frac{z}{2i}\Big),
\end{equation}
\begin{equation}\label{3.7}
\det(zI-A_{1,k}^{(0,1)})=2i^kT_k\Big(\frac{z}{2i}\Big)-2i^{k-1}U_{k-1}\Big(\frac{z}{2i}\Big),
\end{equation}
\begin{equation}\label{3.8}
\det(zI-A_{1,k}^{(1,0)})=2T_k\Big(\frac{z}2\Big),
\end{equation}
\begin{equation}\label{3.9}
\det(zI-A_{1,k}^{(1,1)})=(z-2)U_{k-1}\Big(\frac{z}2\Big).
\end{equation}
{\it Proof.} First, we note that, for $Y_n(z)=T_n(z)$ and $Y_n(z)=U_n(z),$ relations
(\ref{3.8,1})--(\ref{3.9,1}) give
\begin{equation}\label{3.14}
2zU_n(z)=\left\{\begin{array}{r} 2z, \quad n=0,\\[2mm]
4z^2,\quad n=1,
\end{array}\right. \quad
2T_{n+1}(z)+2iU_n(z)=\left\{\begin{array}{r} 2z+2i, \quad n=0,\\[2mm]
4z^2+4iz-2,\quad n=1,
\end{array}\right.
\end{equation}
\begin{equation}\label{3.15}
2T_{n+1}(z)=\left\{\begin{array}{r} 2z, \quad n=0,\\[2mm]
4z^2-2,\quad n=1,
\end{array}\right. \quad
(2z-2)U_n(z)=\left\{\begin{array}{r} 2z-2, \quad n=0,\\[2mm]
4z^2-4z,\quad n=1.
\end{array}\right.
\end{equation}
Let $\alpha=0.$ Then formulae (\ref{2.8.1}) give $cd=-1.$ Then (\ref{2.4}) and (\ref{2.3.2}) imply
\begin{equation}\label{3.10}
\det(zI-A_{1,k}^{(0,\beta)})=(z-c)q_{k-1}(z)+q_{k-2}(z).
\end{equation}
Put $Y_n(z):=i^{-n}q_n(2iz),$ $n=\overline{0,k-1}.$ Using (\ref{2.3.3}), one can easily check that the
polynomials $Y_n(z)$ satisfy the recurrent relations~(\ref{3.8,1}). Substituting $q_n(z)=i^nY_n(z/(2i))$ into
(\ref{3.10}), we get
\begin{equation}\label{3.12}
\det(zI-A_{1,k}^{(0,\beta)})=i^k\Big(\Big(2\frac{z}{2i}+ ic\Big)Y_{k-1}\Big(\frac{z}{2i}\Big)
-Y_{k-2}\Big(\frac{z}{2i}\Big)\Big).
\end{equation}
Using (\ref{2.3.3}), we calculate: $Y_{-1}(z)=-i,$ $Y_0(z)=1$ and $Y_1(z)=2z+i.$ Hence, we obtain
$$
(2z+ic)Y_n(z)-Y_{n-1}(z)=\left\{\begin{array}{r} \left.\begin{array}{r} 2z, \quad n=0,\\[2mm]
4z^2,\quad n=1,
\end{array}\right\} \quad \beta=0,\\[5mm]
\left.\begin{array}{r} 2z+2i, \quad n=0,\\[2mm]
4z^2+4iz-2,\quad n=1,
\end{array}\right\} \quad \beta=1.
\end{array}\right.
$$
Comparing this with (\ref{3.14}), we get
$$
(2z+ic)Y_n(z)-Y_{n-1}(z)=\left\{\begin{array}{r} 2zU_n(z), \quad \beta=0,\\[5mm]
2T_{n+1}(z)+2iU_n(z), \quad \beta=1,
\end{array}\right. \quad n=\overline{0,k-1},
$$
which along with (\ref{3.12}) gives (\ref{3.6.1}) and (\ref{3.7}).
Further, let $\alpha=1.$ Then $cd=1,$ and formulae (\ref{2.4}) and (\ref{2.3.2}) imply
\begin{equation}\label{2.10.1}
\det(zI-A_{1,k}^{(1,\beta)})=(z-c)q_{k-1}(z)-q_{k-2}(z).
\end{equation}
Put $Y_n(z):=q_n(2z),$ $n=\overline{0,k-1}.$ By virtue of (\ref{2.3.3}), these polynomials $Y_n(z)$ satisfy
the recurrent relations~(\ref{3.8,1}). Substituting $q_n(z)=Y_n(z/2)$ into (\ref{2.10.1}), we get
\begin{equation}\label{2.12.1}
\det(zI-A_{1,k}^{(1,\beta)})=\Big(2\frac{z}2 -c\Big)Y_{k-1}\Big(\frac{z}2\Big) -Y_{k-2}\Big(\frac{z}2\Big).
\end{equation}
By virtue of (\ref{2.3.3}), we have
$$
(2z-c)Y_n(z)-Y_{n-1}(z)=\left\{\begin{array}{r} \left.\begin{array}{r} 2z, \quad n=0,\\[2mm]
4z^2-2,\quad n=1,
\end{array}\right\} \quad \beta=0,\\[5mm]
\left.\begin{array}{r} 2z-2, \quad n=0,\\[2mm]
4z^2-4z,\quad n=1,
\end{array}\right\} \quad \beta=1.
\end{array}\right.
$$
Comparing this with (\ref{3.15}) and using (\ref{2.12.1}), we arrive at (\ref{3.8}) and (\ref{3.9}).
$
\Box$
{\bf Corollary 2. }{\it Denote by $\sigma(A)$ the spectrum of the matrix $A.$ Then
\begin{equation}\label{3.16}
\sigma(A_{1,k}^{(0,0)})=\{0\}\cup\Big\{2i\cos\frac{\nu\pi}{k}\Big\}_{\nu=\overline{1,k-1}},
\end{equation}
\begin{equation}\label{3.17}
0\notin \sigma(A_{1,k}^{(0,1)}),
\end{equation}
\begin{equation}\label{3.18}
\sigma(A_{1,k}^{(1,0)})=\Big\{2\cos\frac{(2\nu+1)\pi}{2k}\Big\}_{\nu=\overline{0,k-1}},
\end{equation}
\begin{equation}\label{3.19}
\sigma(A_{1,k}^{(1,1)})=\Big\{2\cos\frac{\nu\pi}{k}\Big\}_{\nu=\overline{0,k-1}}.
\end{equation}
}
{\it Proof.} It is well known and also can be obtained as a simple corollary from (\ref{3.10,1}) that the
sets of zeros of the polynomials $T_n(z)$ and $U_n(z)$ have the forms
\begin{equation}\label{3.19-1}
{\cal T}_n:=
\Big\{\cos\frac{(2\nu+1)\pi}{2n} \Big\}_{\nu=\overline{0,n-1}}, \quad {\cal
U}_n:=
\Big\{\cos\frac{\nu\pi}{n+1} \Big\}_{\nu=\overline{1,n}},
\end{equation}
respectively. Thus, (\ref{3.16}), (\ref{3.18}) and (\ref{3.19}) follow directly from (\ref{3.6.1}),
(\ref{3.8}) and (\ref{3.9}). Concerning~(\ref{3.17}), it is sufficient to recall that, in (\ref{3.7}),
$T_k(0)U_{k-1}(0)=0,$ while $T_k(0)\ne U_{k-1}(0).$ $
\Box$
\\
{\large\bf 4. The case $j>1$}
\\
In this section, we establish connections between the matrices $A_{j,k}^{(\alpha,\beta)}$ and
$A_{1,k}^{(\alpha,\beta)},$ which allow one to reduce studying the case $j>1$ to the case $j=1.$ Namely, the
following theorem holds.
{\bf Theorem 2. }{\it For $\beta=0,1$ and $j=\overline{1,n_k},$ where $n_k=[k/2],$ the following relations
hold:
\begin{equation}\label{4.1}
A_{j,k}^{(0,\beta)}=U_{j-1}\Big(-\frac{c}2A_{1,k}^{(1,1-\beta)}\Big)A_{1,k}^{(0,\beta)},
\end{equation}
\begin{equation}\label{4.2}
A_{j,k}^{(1,\beta)}=2c T_j\Big(\frac{c}2A_{1,k}^{(1,\beta)}\Big).
\end{equation}
Here, $[\,x\,]$ denotes the integer part of $x$ and, as before, $c=(-1)^{1+\beta}.$ }
{\it Proof.} For $j=1,$ the assertion is obvious. According to the formulae $U_1(z)=2z$ and $T_2(z)=2z^2-1,$
each of relations (\ref{4.1}) and (\ref{4.2}) for $j=2$ is equivalent to the common relation
\begin{equation}\label{4.3}
A_{2,k}^{(\alpha,\beta)}=dA_{1,k}^{(1,\gamma)}A_{1,k}^{(\alpha,\beta)} -2\alpha cI, \quad
\gamma:=\left\{\begin{array}{r}1-\beta, \;\alpha=0,\\[1mm]
\beta,\;\alpha=1,
\end{array}\right. \quad
\alpha, \beta=0,1.
\end{equation}
Consider a column vector $X=(x_1,\ldots,x_k)^T$ and denote $[X]_n:=x_n$ for $n=\overline{1,k}.$ Then, by
virtue of (\ref{2.8}), we have
\begin{equation}\label{4.6}
[A_{j,k}^{(\alpha,\beta)}X]_m=\left\{\begin{array}{cl}
x_{j-m+1} +dx_{j+m}, & m=\overline{1,j},\\[2mm]
cx_{m-j} +dx_{j+m}, & m=\overline{j+1,k-j},\\[2mm]
c(x_{m-j} +x_{2k-m-j+1}), & m=\overline{k-j+1,k}.
\end{array}\right.
\end{equation}
In particular, this gives the formulae
\begin{equation}\label{4.6.1}
[A_{1,k}^{(\alpha,\beta)}X]_m=\left\{\begin{array}{cl}
x_1 +dx_2, & m=1,\\[2mm]
cx_{m-1} +dx_{m+1}, & m=\overline{2,k-1},\\[2mm]
c(x_{k-1} +x_k), & m=k,
\end{array}\right.
\end{equation}
\begin{equation}\label{4.9}
[A_{1,k}^{(1,\gamma)}X]_m=\left\{\begin{array}{cl}
x_1 +dx_2, & m=1,\\[2mm]
d(x_{m-1} +x_{m+1}), & m=\overline{2,k-1},\\[2mm]
d(x_{k-1} +x_k), & m=k,
\end{array}\right.
\end{equation}
since $\gamma=\alpha\beta+(1-\alpha)(1-\beta)=2\alpha\beta-\alpha-\beta+1$ and, hence,
$(-1)^{1+\gamma}=(-1)^{\alpha+\beta}=d.$ Substituting $A_{1,k}^{(\alpha,\beta)}X$ given by (\ref{4.6.1}) into
(\ref{4.9}) instead of $X,$ we get the relation
$$
[A_{1,k}^{(1,\gamma)}A_{1,k}^{(\alpha,\beta)}X]_m=\left\{\begin{array}{ll}
(1+cd)x_m +dx_{3-m}+x_{2+m}, & m=1,2,\\[2mm]
(1+cd)x_m +cdx_{m-2} +x_{m+2}, & m=\overline{3,k-2},\\[2mm]
(1+cd)x_m +cd(x_{m-2} +x_{2k-m-1}), & m=k-1,k.
\end{array}\right.
$$
Comparing this with (\ref{4.6}) for $j=2$ and taking into account that $c+d=2\alpha c,$ we arrive at
(\ref{4.3}).
Assume now that (\ref{4.1}) and (\ref{4.2}) are valid when $j=\overline{1,\nu}$ for some $\nu\in\{2,\ldots,
n_k-1\}.$ Then, according to (\ref{3.8,1}), relation (\ref{4.1}) for $j=\nu+1$ is equivalent to
$$
A_{\nu+1,k}^{(0,\beta)}=-cA_{1,k}^{(1,1-\beta)}U_{\nu-1}\Big(-\frac{c}2A_{1,k}^{(1,1-\beta)}\Big)
A_{1,k}^{(0,\beta)} -U_{\nu-2}\Big(-\frac{c}2A_{1,k}^{(1,1-\beta)}\Big) A_{1,k}^{(0,\beta)}
\qquad\qquad\qquad
$$
\begin{equation}\label{4.10}
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\;\,
=-cA_{1,k}^{(1,1-\beta)}A_{\nu,k}^{(0,\beta)} -A_{\nu-1,k}^{(0,\beta)},
\end{equation}
while (\ref{4.2}) for $j=\nu+1$ takes the form
\begin{equation}\label{4.11}
A_{\nu+1,k}^{(1,\beta)}=2A_{1,k}^{(1,\beta)}T_\nu\Big(\frac{c}2A_{1,k}^{(1,\beta)}\Big) -2c
T_{\nu-1}\Big(\frac{c}2A_{1,k}^{(1,\beta)}\Big) =cA_{1,k}^{(1,\beta)}A_{\nu,k}^{(1,\beta)}
-A_{\nu-1,k}^{(1,\beta)}.
\end{equation}
Using the relation $(-1)^{\alpha+1} c=d$ along with the definition of $\gamma$ in~(\ref{4.3}), one can
rewrite~(\ref{4.10}) and~(\ref{4.11}) in the following common form:
\begin{equation}\label{4.12}
A_{\nu+1,k}^{(\alpha,\beta)}=dA_{1,k}^{(1,\gamma)}A_{\nu,k}^{(\alpha,\beta)} -A_{\nu-1,k}^{(\alpha,\beta)},
\quad \alpha,\beta=0,1.
\end{equation}
Thus, it remains to prove relation (\ref{4.12}). Using (\ref{4.6}), we calculate
$$
[(A_{\nu+1,k}^{(\alpha,\beta)}+A_{\nu-1,k}^{(\alpha,\beta)})X]_m
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
$$
\begin{equation}\label{4.13}
\qquad\qquad\quad\;\,=\left\{\begin{array}{cl}
x_{\nu-m+2} +dx_{\nu+1+m}+x_{\nu-m} +dx_{\nu-1+m}, & m=\overline{1,\nu-1},\\[2mm]
x_{\nu-m+2} +dx_{\nu+1+m}+cx_{m-\nu+1} +dx_{\nu-1+m}, & m=\nu,\nu+1,\\[2mm]
cx_{m-\nu-1} +dx_{\nu+1+m}+cx_{m-\nu+1} +dx_{\nu-1+m}, & m=\overline{\nu+2,k-\nu-1},\\[2mm]
c(x_{m-\nu-1} +x_{2k-m-\nu})+cx_{m-\nu+1} +dx_{\nu-1+m}, & m=k-\nu,k-\nu+1,\\[2mm]
c(x_{m-\nu-1} +x_{2k-m-\nu}+x_{m-\nu+1} +x_{2k-m-\nu+2}), & m=\overline{k-\nu+2,k}.
\end{array}\right.
\end{equation}
Further, substituting $A_{\nu,k}^{(\alpha,\beta)}X$ given by (\ref{4.6}) into (\ref{4.9}) instead of $X,$ we
get the relation
$$
[A_{1,k}^{(1,\gamma)}A_{\nu,k}^{(\alpha,\beta)}X]_m=\left\{\begin{array}{cl}
d(x_{\nu-m+2} +dx_{\nu-1+m}+x_{\nu-m} +dx_{\nu+1+m}), & m=\overline{1,\nu-1},\\[2mm]
d(x_{\nu-m+2} +dx_{\nu-1+m}+cx_{m-\nu+1} +dx_{\nu+1+m}), & m=\nu,\nu+1,\\[2mm]
d(cx_{m-\nu-1} +dx_{\nu-1+m}+cx_{m-\nu+1} +dx_{\nu+1+m}), & m=\overline{\nu+2,k-\nu-1},\\[2mm]
d(cx_{m-\nu-1} +dx_{\nu-1+m}+c(x_{m-\nu+1} +x_{2k-m-\nu})), & m=k-\nu,k-\nu+1,\\[2mm]
dc(x_{m-\nu-1} +x_{2k-m-\nu+2}+x_{m-\nu+1} +x_{2k-m-\nu}), & m=\overline{k-\nu+2,k}.
\end{array}\right.
$$
Comparing this with (\ref{4.13}), we arrive at (\ref{4.12}). $
\Box$
{\bf Corollary 3. }{\it For $1\le j\le[k/2],$ the matrix $A_{j,k}^{(\alpha,\beta)}$ is degenerate, i.e. $\det
A_{j,k}^{(\alpha,\beta)}=0,$ if and only if one of the four conditions in (\ref{deg}) is fulfilled.
Equivalently, $\det A_{j,k}^{(\alpha,\beta)}\ne0$ if and only if one of the three conditions in
(\ref{non-deg}) is fulfilled.}
{\it Proof.} Consider $\alpha=\beta=0$ first. Then (\ref{3.16}) and (\ref{4.1})
imply $\det A_{j,k}^{(0,0)}=0$ for any possible~$j$ and~$k.$ Thus, the assertion of the corollary is proven
for condition (I) in (\ref{deg}).
The rest part of the proof is based on the following well-known fact, which is valid both for Hermitian and
non-Hermitian square matrices $A,$ being a particular case of the corresponding abstract assertion (see,
e.g., Theorem 3.3 on p.~16 in \cite{GGK90} or Theorem 10.28 on p.~263 in \cite{Rud91}).
{\bf Proposition 1. }{\it Let $P(z)$ be an algebraic polynomial and $A$ be a square matrix. Then
$$
\sigma(P(A))=\{P(z)\}_{z\in\sigma(A)}.
$$
Moreover, if $X$ is an eigenvector corresponding to an eigenvalue $z_0$ of the matrix $A,$ then~$X$ is an
eigenvector related to the eigenvalue $P(z_0)$ of $P(A).$
}
Let us return to the proof of Corollary~3. For $\alpha=0$ and $\beta=1,$ according to (\ref{2.8.1}),
(\ref{3.17}) and (\ref{4.1}),
we have $\det A_{j,k}^{(0,1)}=0$ if and only if $0\in\sigma(U_{j-1}((-1/2)A_{1,k}^{(1,0)})).$ By virtue of
Proposition~1, this inclusion is equivalent to the relation ${\cal
U}_{j-1}\cap\sigma((-1/2)A_{1,k}^{(1,0)})\ne\emptyset,$ where, as in the proof of Corollary~2, we use the
designation ${\cal U}_n=\{z:U_n(z)=0\}$ similarly to ${\cal T}_n=\{z:T_n(z)=0\}.$ Thus, according to
(\ref{3.18}) and (\ref{3.19-1}), the latter intersection is not empty if and only if
$$
\cos\frac{\nu\pi}j+\cos\frac{(2l+1)\pi}{2k}=0
$$
for a certain choice of $\nu\in\{1,\ldots,j-1\}$ and $l\in\{0,\ldots,k-1\}.$ The latter, in turn, is
equivalent to the relation
\begin{equation}\label{4.14}
\frac\nu{j}+(-1)^s\frac{2l+1}{2k}=1+2m
\end{equation}
for some integers $s$ and $m.$ Obviously, (\ref{4.14}) implies the evenness of $j.$ Conversely, let $j$ be
even. Then $k$ is odd, and (\ref{4.14}) holds for $s=m=0$ as soon as $\nu=j/2$ and $l=(k-1)/2.$
For $\alpha=1$ and $\beta=0,$ relations (\ref{2.8.1}) and (\ref{4.2}) imply that $\det A_{j,k}^{(1,0)}=0$ is
equivalent to ${\cal T}_j\cap\sigma((-1/2)A_{1,k}^{(1,0)})\ne\emptyset.$ By virtue of (\ref{3.18}) and
(\ref{3.19-1}), this intersection is not empty if and only~if
$$
\cos\frac{(2\nu+1)\pi}{2j}+\cos\frac{(2l+1)\pi}{2k}=0
$$
for some $\nu\in\{0,\ldots,j-1\}$ and $l\in\{0,\ldots,k-1\},$ which is equivalent to the relation
\begin{equation}\label{4.15}
\frac{2\nu+1}{2j}+(-1)^s\frac{2l+1}{2k}=1+2m
\end{equation}
with $s,\,m\in{\mathbb Z}.$ In its turn, (\ref{4.15}) implies the evenness of $j+k.$ Conversely, let $j+k$ be
even. Then $j$ and $k$ are odd, and (\ref{4.15}) holds for $s=m=0$ with $\nu=(j-1)/2$ and $l=(k-1)/2.$
Finally, let $\alpha=\beta=1.$ Then (\ref{2.8.1}) and (\ref{4.2}) imply that $\det A_{j,k}^{(1,1)}=0$ is
equivalent to ${\cal T}_j\cap\sigma((1/2)A_{1,k}^{(1,1)})\ne\emptyset.$ By virtue of (\ref{3.19}) and
(\ref{3.19-1}), the latter holds if and only if
$$
\cos\frac{(2\nu+1)\pi}{2j}=\cos\frac{l\pi}k
$$
for some $\nu\in\{0,\ldots,j-1\}$ and $l\in\{0,\ldots,k-1\},$ which, in turn, is equivalent to the relation
\begin{equation}\label{4.16}
\frac{2\nu+1}{2j}+(-1)^s\frac{l}k=2m
\end{equation}
for some $s,\,m\in{\mathbb Z}.$ Obviously, (\ref{4.16}) implies the evenness of $k.$ Conversely, let $k$ be
even. Then~$j$ is odd, and (\ref{4.16}) holds for $s=1$ and $m=0$ with $\nu=(j-1)/2$ and $l=k/2.$
$
\Box$
\\
{\large\bf 5. Iso-spectral potentials}
\\
In this section, we return to Inverse Problem~1. The above results give an easy and convenient way for
constructing iso-spectral potentials in the degenerate case.
Let the parameters $\alpha,$ $\beta$ and $a=j/k$ with mutually prime $j$ and $k$ satisfy one of conditions
(I)--(IV) in (\ref{deg}) and, for definiteness, also let $a\in(0,1/2].$ Fix a model complex-valued potential
$q_0(x)\in L(0,1)$ and consider the corresponding eigenvalue problem ${\cal L}(q_0(x),\alpha,\beta,a)$ with
the spectrum $\Lambda:=\{\lambda_n\}_{n\ge1}.$ Consider the set ${\cal M}_\Lambda$ of all corresponding
iso-spectral potentials~$q(x),$ i.e. of such ones for which the spectrum of the problem ${\cal
L}(q(x),\alpha,\beta,a)$ coincides with $\Lambda.$
By virtue of (\ref{2.5}), (\ref{EQ}) and (\ref{2.7}), we have the representation
\begin{equation}\label{5.0}
{\cal M}_\Lambda=\Big\{q_0(x)+g(x): g(x)\in{\cal R}_{j,k}^{(\alpha,\beta)}\Big\},
\end{equation}
where
$$
{\cal R}_{j,k}^{(\alpha,\beta)}=\Big\{R^{-1}F(x): F(x)\in(L(0,b))^k\;\; {\rm and}\;\;
A_{j,k}^{(\alpha,\beta)}F(x)=0 \;\; {\rm a.e.\;\, on}\;\, (0,b)\Big\}, \quad b=\frac1k,
$$
i.e. the supplement $g(x)$ in (\ref{5.0})
is independent of $q_0(x).$
Thus, the question of describing all iso-spectral potentials is reduced to studying the kernel of the matrix
$A_{j,k}^{(\alpha,\beta)}.$ The following lemma answers this question for $j=1.$
{\bf Lemma 2. }{\it Each eigenvalue $z_0$ of the matrix $A_{1,k}^{(\alpha,\beta)}$ has the geometric
multiplicity one, while the corresponding eigenvector has the form
\begin{equation}\label{5.1}
X_0=\Big(1,dq_1(z_0),d^2q_2(z_0),\ldots,d^{k-1}q_{k-1}(z_0)\Big)^T.
\end{equation} }
{\it Proof.} According to (\ref{4.6.1}), relation $A_{1,k}^{(\alpha,\beta)}X_0=z_0X_0$ is equivalent to the
system
\begin{equation}\label{5.2}
\left.\begin{array}{l}
[X_0]_1 +d[X_0]_2=z_0[X_0]_1,\\[2mm]
c[X_0]_{m-1} +d[X_0]_{m+1}=z_0[X_0]_m, \quad m=\overline{2,k-1},\\[2mm]
c([X_0]_{k-1} +[X_0]_k)=z_0[X_0]_k.
\end{array}\right\}
\end{equation}
Thus, we have $[X_0]_1\ne0$ as soon as $X_0$ is an eigenvector, otherwise (\ref{5.2}) would imply $X_0=0.$
Without loss of generality, we put $[X_0]_1=1.$ Then the first two lines in (\ref{5.2}) give the relations
\begin{equation}\label{5.3}
[X_0]_2=d(z_0-1), \quad [X_0]_{m+1}=dz_0[X_0]_m-cd[X_0]_{m-1}, \quad m=\overline{2,k-1}.
\end{equation}
Substituting $[X_0]_m=:d^{m-1}Y_m,$ $m=\overline{1,k},$ into (\ref{5.3}), we arrive at the relations
$$
Y_1=1, \quad Y_2=z_0-1, \quad Y_{m+1}=z_0Y_m-cdY_{m-1}, \quad m=\overline{2,k-1}.
$$
Comparing this with (\ref{2.3.3}), we get $Y_m=q_{m-1}(z_0)$ and, hence, $[X_0]_m=d^{m-1}q_{m-1}(z_0)$ for
$m=\overline{1,k},$ which finalizes the proof. $
\Box$
It can be easily seen that the last relation in (\ref{5.2}) is fulfilled automatically as soon as $z_0$ is an
eigenvalue of the matrix $A_{1,k}^{(\alpha,\beta)}.$ Indeed, by virtue of (\ref{5.1}), this relation is
equivalent to the relation $cdq_{k-2}(z_0)=(z_0-c)q_{k-1}(z_0),$ which, according to (\ref{2.3.2}), is
equivalent to $p_k(z_0)=0.$
We also note that, according to (\ref{3.6.1}), the algebraic multiplicity of the zero eigenvalue of the
matrix $A_{1,k}^{(0,0)}$ may be equal to $2,$ while, by virtue of Lemma~2, the geometric one cannot.
{\bf Lemma 3. }{\it Let the values $\alpha,$ $\beta,$ $j$ and $k$ obey one of conditions~(I)--(IV)
in~(\ref{deg}). Then the kernel of the matrix $A_{j,k}^{(\alpha,\beta)}$ coincides with a linear hull of the
vector $X=(x_1,\ldots, x_k)^T$ determined in the following way:
\begin{equation}\label{5.4}
\;\;\alpha= \beta=0: \quad x_\nu=(-1)^{\nu-1}, \quad \nu=\overline{1,k};
\end{equation}
\begin{equation}\label{5.5}
\alpha=0,\;\; \beta=1: \quad x_\nu=(-1)^{[\frac\nu2]}, \quad \nu=\overline{1,k}; \quad\;
\end{equation}
\begin{equation}\label{5.6}
\alpha=1,\;\; \beta=0: \quad x_\nu=(-1)^{[\frac{\nu-1}2]}, \quad \nu=\overline{1,k}; \;\;\,
\end{equation}
\begin{equation}\label{5.7}
\alpha=
\beta=1: \quad x_\nu=(-1)^{[\frac\nu2]}, \quad \nu=\overline{1,k}.
\end{equation}
}
{\it Proof.} According to Remark~2 in \cite{BK}, in the degenerate case, we have $\rank
A_{j,k}^{(\alpha,\beta)}=k-1,$ i.e.
$\ker A_{j,k}^{(\alpha,\beta)}$ is always one-dimensional. By virtue of (\ref{2.8.1}) and~(\ref{3.5}), the
matrix $A_{1,k}^{(0,\beta)}$ is degenerate if and only if $\beta=0.$ Hence, relation (\ref{4.1}) along with
Lemma~2 implies that
$\ker A_{j,k}^{(0,0)}$ is a linear hull of the vector $X=X_0$ determined by~(\ref{5.1}) for
$\alpha=\beta=z_0=0.$ Moreover, by virtue of (\ref{4.2}) along with Proposition~1 and Lemma~2, the kernel of
$A_{j,k}^{(1,\beta)}$ for $\beta\in\{0,1\}$ is a linear hull of the vector $X=X_0$ determined by~(\ref{5.1})
for $\alpha=1$ and the corresponding $\beta$ as well as $z_0=0$ since~$j$ is odd in both subcases (III), (IV)
and, hence, $T_j(0)=0.$ Thus, formulae (\ref{5.4}), (\ref{5.6}) and (\ref{5.7}) for components of $X$ can be
easily obtained using (\ref{2.8.1}), (\ref{3.6}) and~(\ref{5.1}).
Now let $(\alpha,\beta)=(0,1).$ Then representation (\ref{4.1}) takes the form
$$
A_{j,k}^{(0,1)}=U_{j-1}\Big(-\frac12A_{1,k}^{(1,0)}\Big)A_{1,k}^{(0,1)}.
$$
According to (II) in (\ref{deg}) as well as (V) in (\ref{non-deg}), we have $\det A_{j,k}^{(0,1)}=0$ if and
only if~$j$ is even. In the degenerate case, since $\det A_{1,k}^{(0,1)}\ne0,$ we have $\det
U_{j-1}((-1/2)A_{1,k}^{(1,0)})=0.$ Moreover, since $j$ and $k$ are mutually prime, the value $k$ is odd.
Thus, by virtue of Proposition~1 along with the relation $U_{j-1}(0)=0,$ a unique up to a multiplicative
constant eigenvector of the matrix~$A_{j,k}^{(0,1)}$ corresponding to the zero eigenvalue satisfies the
linear equation
\begin{equation}\label{5.9}
A_{1,k}^{(0,1)}X=X_0,
\end{equation}
where $X_0$ is an eigenvector of the matrix $A_{1,k}^{(1,0)}$ related to the zero eigenvalue. By virtue of
(\ref{5.6}), we have $[X_0]_\nu=(-1)^{[\frac{\nu-1}2]},$ $\nu=\overline{1,k}.$ Thus, according to
(\ref{4.6.1}), equation (\ref{5.9}) is equivalent to the system of scalar equations
\begin{equation}\label{5.10}
x_1-x_2=1, \quad x_{\nu-1}-x_{\nu+1}=(-1)^{[\frac{\nu-1}2]}, \;\;\nu=\overline{2,k-1}, \quad
x_{k-1}+x_k=(-1)^{[\frac{k-1}2]}.
\end{equation}
Summing up all equations in (\ref{5.10}), we get $2x_1=s_k,$ where
$$
s_n=\sum_{\nu=1}^n(-1)^{[\frac{\nu-1}2]}.
$$
Obviously, $s_{4l+1}=s_{4l+3}=1,$ $s_{4l+2}=2$ and $s_{4l+4}=0$ for all $l\ge0.$ Thus, since $k$ is odd, we
have $s_k=1$ and, hence, $x_1=1/2.$ Rewrite the first $k-1$ equations in (\ref{5.10}) in the following way:
$$
x_1-x_2=x_1-x_3=1, \quad
\left.\begin{array}{r} x_{2\nu}-x_{2\nu+2}=(-1)^\nu,\\[2mm] x_{2\nu+1}-x_{2\nu+3}=(-1)^\nu,
\end{array}\right\}
\quad\nu=\overline{1,\frac{k-3}2},
$$
whence relations (\ref{5.5}) can be easily established by induction and multiplication with $2.$ $
\Box$
{\bf Corollary 5. }{\it Let the values $\alpha,$ $\beta,$ $j$ and $k$ obey one of
conditions~(I)--(IV) in~(\ref{deg}). Then $F(x)=Xf(x)$ is a general solution of the functional equation
$A_{j,k}^{(\alpha,\beta)}F(x)=0$ in $(L(0,b))^k,$ where components of the vector $X=(x_1,\ldots, x_k)^T$ are
determined by the corresponding formula in (\ref{5.4})--(\ref{5.7}), while the function $f(x)$ ranges over
$L(0,b).$}
Thus, we arrive at the following procedure for constructing an iso-spectral potential $q(x)$ that is
different from~$q_0(x).$
{\bf Algorithm 1. }{\it Let $q_0(x)\in L(0,1)$ as well as appropriate $\alpha,$ $\beta$ and $j,$ $k$ be
given. Then
(i) Choose a nonzero function $f(x)\in L(0,b);$
(ii) Construct the vector $X=(x_1,\ldots, x_k)^T$ by the corresponding formula in (\ref{5.4})--(\ref{5.7});
(ii) Calculate $q(x)$ by the formulae
\begin{equation}\label{5.11}
q(x)=q_0(x)+R^{-1}F(x), \quad F(x)=Xf(x).
\end{equation}
}
Obviously, the obtained function $q(x)$ ranges over ${\cal M}_\Lambda$ as soon as so does $f(x)$ over
$L(0,b),$ where~$\Lambda$ is the spectrum of the problem ${\cal L}(q_0(x),\alpha,\beta,j/k).$
According to (\ref{2.4.1}) and (\ref{2.3.1}), we have the following formulae for $R^{-1}F(x),$ $x\in(0,1),$
with $F(t)=(f_1(t),\ldots, f_k(t))^T,$ $t\in(0,b):$
$$
R^{-1}F(x)=\left\{\begin{array}{cl} f_\nu(x-(k-\nu)b) \;\; \text{for even} \; j+\nu,\\[3mm]
f_\nu((k-\nu+1)b-x) \;\; \text{for odd} \; j+\nu,
\end{array}\right. \; x\in((k-\nu)b,(k-\nu+1)b), \;\; \nu=\overline{1,k}.
$$
Thus, we arrive at the following representations depending on the parities of~$j$ and~$k:$
\begin{equation}\label{5.12}
R^{-1}F(x)=\left\{\begin{array}{cl} f_k(x), & x\in(0,b),\\
f_{k-1}(2b-x), & x\in(b,2b),\\
f_{k-2}(x-2b), & x\in(2b,3b),\\
f_{k-3}(4b-x), & x\in(3b,4b),\\
f_{k-4}(x-4b), & x\in(4b,5b),\\
\ldots & \\
f_2(1-b-x), & x\in(1-2b,1-b),\\
f_1(x-1+b), & x\in(1-b,1),
\end{array}\right.
\end{equation}
for odd $j$ and odd $k;$
\begin{equation}\label{5.13}
R^{-1}F(x)=\left\{\begin{array}{cl} f_k(b-x), & x\in(0,b),\\
f_{k-1}(x-b), & x\in(b,2b),\\
f_{k-2}(3b-x), & x\in(2b,3b),\\
f_{k-3}(x-3b), & x\in(3b,4b),\\
f_{k-4}(5b-x), & x\in(4b,5b),\\
\ldots & \\
f_2(x-1+2b), & x\in(1-2b,1-b),\\
f_1(1-x), & x\in(1-b,1),
\end{array}\right.
\end{equation}
for even $j$ and odd $k;$
\begin{equation}\label{5.14}
R^{-1}F(x)=\left\{\begin{array}{cl} f_k(b-x), & x\in(0,b),\\
f_{k-1}(x-b), & x\in(b,2b),\\
f_{k-2}(3b-x), & x\in(2b,3b),\\
f_{k-3}(x-3b), & x\in(3b,4b),\\
f_{k-4}(5b-x), & x\in(4b,5b),\\
\ldots & \\
f_3(x-1+3b), & x\in(1-3b,1-2b),\\
f_2(1-b-x), & x\in(1-2b,1-b),\\
f_1(x-1+b), & x\in(1-b,1),
\end{array}\right.
\end{equation}
for odd $j$ and even $k.$
\\
{\large\bf 6. Illustrative examples}
\\
Finally, we give some examples illustrating the term $R^{-1}F(x)$ in (\ref{5.11}) for all degenerate subcases
(I)--(IV) in (\ref{deg}). We also provide the corresponding graphs of $R^{-1}F(x)$ taking the model function
$f(x)$ in (\ref{5.11}) of the following form:
$$
f(x)=\frac{10x}{3b}-\frac{25x^2}{9b^2}, \quad b=\frac1k.
$$
{\bf Example I.} Let $\alpha=\beta=0.$ Then, for $(j,k)=(3,7),$ formulae (\ref{5.4}) and (\ref{5.12}) give
$$
F(x)=\left[\begin{array}{r}1\\-1\\1\\-1\\1\\-1\\1\end{array}\right]f(x),\quad
R^{-1}F(x)=\left\{\begin{array}{rl}f(x), & x\in(0,1/7),\\[2mm]
-f(2/7-x), & x\in(1/7,2/7),\\[2mm]
f(x-2/7), & x\in(2/7,3/7),\\[2mm]
-f(4/7-x), & x\in(3/7,4/7),\\[2mm]
f(x-4/7), & x\in(4/7,5/7),\\[2mm]
-f(6/7-x), & x\in(5/7,6/7),\\[2mm]
f(x-6/7), & x\in(6/7,1),
\end{array}\right.
$$
while, for $(j,k)=(3,8),$ formulae (\ref{5.4}) and (\ref{5.14}) give the representations
$$
F(x)=\left[\begin{array}{r}1\\-1\\1\\-1\\1\\-1\\1\\-1\end{array}\right]f(x),\quad
R^{-1}F(x)=\left\{\begin{array}{rl} -f(1/8-x), & x\in(0,1/8),\\[2mm]
f(x-1/8), & x\in(1/8,1/4),\\[2mm]
-f(3/8-x), & x\in(1/4,3/8),\\[2mm]
f(x-3/8), & x\in(3/8,1/2),\\[2mm]
-f(5/8-x), & x\in(1/2,5/8),\\[2mm]
f(x-5/8), & x\in(5/8,3/4),\\[2mm]
-f(7/8-x), & x\in(3/4,7/8),\\[2mm]
f(x-7/8), & x\in(7/8,1).
\end{array}\right.
$$
\begin{center}
\begin{figure}\end{figure}
\end{center}
{\bf Example II.} Let $\alpha=0$ and $\beta=1.$ Then, for $(j,k)=(2,7),$ formulae (\ref{5.5}) and
(\ref{5.13}) give
$$
F(x)=\left[\begin{array}{r}1\\-1\\-1\\1\\1\\-1\\-1\end{array}\right]f(x),\quad
R^{-1}F(x)=\left\{\begin{array}{rl}-f(1/7-x), & x\in(0,1/7),\\[2mm]
-f(x-1/7), & x\in(1/7,2/7),\\[2mm]
f(3/7-x), & x\in(2/7,3/7),\\[2mm]
f(x-3/7), & x\in(3/7,4/7),\\[2mm]
-f(5/7-x), & x\in(4/7,5/7),\\[2mm]
-f(x-5/7), & x\in(5/7,6/7),\\[2mm]
f(1-x), & x\in(6/7,1).
\end{array}\right.
$$
{\bf Example III.} Let $\alpha=1$ and $\beta=0.$ Then, for $(j,k)=(3,7),$ formulae (\ref{5.6}) and
(\ref{5.12}) give
$$
F(x)=\left[\begin{array}{r}1\\1\\-1\\-1\\1\\1\\-1\end{array}\right]f(x),\quad
R^{-1}F(x)=\left\{\begin{array}{rl}-f(x), & x\in(0,1/7),\\[2mm]
f(2/7-x), & x\in(1/7,2/7),\\[2mm]
f(x-2/7), & x\in(2/7,3/7),\\[2mm]
-f(4/7-x), & x\in(3/7,4/7),\\[2mm]
-f(x-4/7), & x\in(4/7,5/7),\\[2mm]
f(6/7-x), & x\in(5/7,6/7),\\[2mm]
f(x-6/7), & x\in(6/7,1).
\end{array}\right.
$$
\begin{center}
\begin{figure}\end{figure}
\end{center}
{\bf Example IV.} Let $\alpha=1$ and $\beta=1.$ Then, for $(j,k)=(3,8),$ formulae (\ref{5.7}) and
(\ref{5.14}) give
$$
F(x)=\left[\begin{array}{r}1\\-1\\-1\\1\\1\\-1\\-1\\1\end{array}\right]f(x),\quad
R^{-1}F(x)=\left\{\begin{array}{rl} f(1/8-x), & x\in(0,1/8),\\[2mm]
-f(x-1/8), & x\in(1/8,1/4),\\[2mm]
-f(3/8-x), & x\in(1/4,3/8),\\[2mm]
f(x-3/8), & x\in(3/8,1/2),\\[2mm]
f(5/8-x), & x\in(1/2,5/8),\\[2mm]
-f(x-5/8), & x\in(5/8,3/4),\\[2mm]
-f(7/8-x), & x\in(3/4,7/8),\\[2mm]
f(x-7/8), & x\in(7/8,1).
\end{array}\right.
$$
\begin{center}
\begin{figure}\end{figure}
\end{center}
{\bf Acknowledgements.} Sergey Buterin is supported by Grant 20-31-70005 of the Russian Foundation for Basic
Research. Chung-Tsun Shieh is partially supported by the Ministry of Science and Technology, Taiwan under
Grant no. 109-2115-M-032-004-.
\end{document} |
\begin{document}
\title{Genealogy of the extremal process of the~branching~random~walk}
\begin{abstract}
The extremal process of a branching random walk is the point measure recording the position of particles alive at time $n$, shifted around the expected position of the minimal position. Madaule~\cite{Mad15} proved that this point measure converges, as $n \to \infty$, toward a randomly shifted, decorated Poisson point process. In this article, we study the joint convergence of the extremal process together with its genealogical informations. This result is then used to describe the law of the decoration in the limiting process, as well as to study the supercritical Gibbs measures of the branching random walk.
\mathbf{e}nd{abstract}
\section{Introduction}
A branching random walk on $\mathbb{R}$ is a discrete time particle system on the real line, which can be defined as follows. It starts with a unique particle positioned at 0 at time 0. At each new time $n \in \mathbb{N}$, each particle alive at time $(n-1)$ dies, giving birth to children that are positioned according to i.i.d. versions of a random point measure, shifted by the position of their parent. We denote by $\mathbf{T}$ the genealogical tree of the branching random walk. For $u \in \mathbf{T}$, we write $V(u)$ for the position of the particle $u$ and $|u|$ for the time at which $u$ is alive. The branching random walk is the random marked tree $(\mathbf{T},V)$.
We assume that the process is supercritical:
\begin{equation}
\label{eqn:supercritical}
\E\left( \#\{u \in \mathbf{T} : |u|=1\} \right) > 1.
\mathbf{e}nd{equation}
It is a well-known result from Galton-Watson processes theory that this assumption is equivalent to the fact that the surviving event $S = \{ \#\mathbf{T} = \infty\}$, in which the process never dies out, occurs with positive probability. Moreover, we assume the branching random walk to be in the boundary case:
\begin{equation}
\label{eqn:boundary}
\E\left( \sum_{|u|=1} e^{-V(u)} \right) = 1 \quad \text{and} \quad \E\left( \sum_{|u|=1} V(u)e^{-V(u)} \right) = 0,
\mathbf{e}nd{equation}
and that the reproduction law is non-lattice. This assumption guarantees that the minimal position $M_n = \min_{|u|=n} V(u)$ satisfies $\lim_{n \to \infty} \frac{M_n}{n} = 0$ a.s. (c.f. Biggins \cite{Big76}). Any branching random walk satisfying mild assumptions can be reduced to this case by an affine transformation (see e.g. the discussion in~\cite{BeG11}). We set
\[
\forall n \geq 0, \quad W_n = \sum_{|u|=n} e^{-V(u)} \quad \text{and} \quad Z_n = \sum_{|u|=n} V(u)e^{-V(u)}.
\]
By \mathbf{e}qref{eqn:boundary} and the branching property of the branching random walk, the processes $(W_n)$ and $(Z_n)$ are martingales, which are called the critical martingale and derivative martingale of the branching random walk respectively.
We introduce the following additional integrability conditions:
\begin{align}
&\sigma^2 := \E\left( \sum_{|u|=1} V(u)^2 e^{-V(u)} \right) \in (0,\infty)\label{eqn:variance}\\
\text{and} \quad &\E\left(\sum_{|u|=1} e^{-V(u)} \log_+\left(\sum_{|u|=1} (1 + V(u)_+)e^{-V(u)}\right)^2 \right) < \infty\label{eqn:integrability},
\mathbf{e}nd{align}
where $x_+ = \max(x,0)$ and $\log_+(x) = \max(\log x,0)$. Under these assumptions, is is well-known (see \cite{Aid13,BiK04}) there exists a random variable $Z_\infty$, which is a.s. positive on the survival event $S$, such that
\begin{equation}
\lim_{n \to \infty} Z_n = Z_\infty \quad \text{and} \quad \lim_{n \to \infty} W_n = 0 \quad \text{a.s.}
\label{eqn:cvMartingales}
\mathbf{e}nd{equation}
Assumption \mathbf{e}qref{eqn:integrability} is a rephrasing of \cite[Equation (1.4)]{Aid13} (see Lemma~\ref{lem:equivalent} for the proof of the equivalence of these two integrability conditions). However, this version appears directly in our computations, cf Lemma~\ref{lem:entangled}.
Recall that $M_n = \min_{|u|=n} V(u)$ is the minimal position at time $n$ occupied by a particle. We set $m_n = \frac{3}{2} \log n$. Under the above integrability assumptions, Addario-Berry and Reed \cite{ABR09} observed that $(M_n - m_n)$ is tight, and Hu and Shi \cite{HuS09} proved this sequence has almost sure logarithmic size fluctuations. Finally, Aïdékon \cite{Aid13} obtained the convergence in law of $M_n - m_n$, and Chen \cite{Che15} proved the above integrability assumptions to be optimal for this convergence in law. We take interest in all particles that are at time $n$ in a $O(1)$ neighborhood of the minimal displacement $M_n$.
We introduce some notation on point measures, the Radon measures on $\mathbb{R}$ that takes values in $\mathbb{Z}_+ \cup \{ \infty\}$. Given a point measure $\varrho$, we denote by $\mathcal{P}(\varrho)$ the multiset of the atoms of the point measure $\varrho$, that satisfy
\[
\varrho = \sum_{r \in \mathcal{P}(\varrho)} \delta_r.
\]
For any $x \in \mathbb{R}$, we write $\theta_x \varrho = \sum_{r \in \mathcal{P}(\varrho)} \delta_{r + x}$ the shift of the measure $\varrho$ by $x$. The space of point measures is endowed with the topology of the vague convergence, meaning that we write $\lim_{n \to \infty} \varrho_n = \varrho_\infty$ if $\lim_n \varrho_n(f) = \varrho_\infty(f)$ for any continuous function $f$ on $\mathbb{R}$ with compact support. As observed in \cite[Theorem A2.3]{Kal02}, the set of random point measures endowed with the topology of the vague convergence is a Polish space.
We use the extremal process of the branching random walk to record the positions of particles close to the maximal displacement at time $n$, defined by
\begin{equation}
\label{eqn:defineExtremalProcess}
\gamma_n = \sum_{|u|=n} \delta_{V(u) - m_n}.
\mathbf{e}nd{equation}
Madaule \cite{Mad15} proved the convergence in law of the extremal process toward a decorated Poisson point process with exponential intensity, or more precisely the following result.
\begin{fact}[Theorem 1.1 in \cite{Mad15}]
\label{fct:Madaule}
We assume \mathbf{e}qref{eqn:supercritical}, \mathbf{e}qref{eqn:boundary}, \mathbf{e}qref{eqn:variance} and \mathbf{e}qref{eqn:integrability}. There exist $c_*>0$ and a point measure $D$ satisfying $\min D = 0$ a.s. such that
\[
\lim_{n \to \infty} \left(\gamma_n,Z_n\right) = \left( \gamma_\infty, Z_\infty \right) \quad \text{in law on $S$,}
\]
for the topology of the vague convergence, where $(\xi_n)$ are the atoms of a Poisson point process with intensity $c_* e^x dx$, $(D_n, n \geq 1)$ are i.i.d. copies of $D$ and
\begin{equation}
\label{eqn:rhoInfinity}
\gamma_\infty = \sum_{n=1}^{\infty} \theta_{\xi_n- \log Z_\infty} D_n.
\mathbf{e}nd{equation}
We denote by $\mathcal{D}$ the law of the point measure $D$.
\mathbf{e}nd{fact}
The point measure $\gamma_\infty$ is called a shifted decorated Poisson point process with shift $-\log Z_\infty$ and decoration law $\mathcal{D}$ (or SDPPP($c_*e^{x}dx$,$-\log Z_\infty$, $\mathcal{D}$) for short). These point measures have been studied in particular by Subag and Zeitouni \cite{SuZ15}. The proof of Fact~\ref{fct:Madaule} gives little information on the law $\mathcal{D}$ of the point measure $D$ used for the decoration of $\gamma_\infty$. Indeed, the convergence of $(\gamma_n)$ is obtained through the study of its Laplace transform, and the law of the limiting point measure $\gamma_\infty$ is identified using its superposability property\footnote{Cf. Maillard \cite{Mai13} for the characterization of point measures occurring as the limits of the extremal processes of branching random walks.}.
A result similar to Fact~\ref{fct:Madaule} was previously obtained for the branching Brownian motion independently by Arguin, Bovier and Kistler \cite{ABK13}, and Aïdékon, Berestycki, Brunet and Shi \cite{ABBS13}. In this model as well, the extremal process converges toward a decorated Poisson point process. However, the decoration law is explicitly described in both these articles. In \cite{ABBS13}, the point measure $D$ corresponds to positions of the close relatives of the particle realizing the minimal displacement. In \cite{ABK13}, it is described as the extremal process of the branching random walk conditioned on having an unusually small minimal displacement.
In this article, we observe that using the branching property as well as an enriched version of the extremal process, Fact~\ref{fct:Madaule} immediately implies a stronger version of itself. More precisely, thanks to a careful encoding of the genealogy of the branching random walk, which is presented in Section~\ref{sec:notation}, we can prove the joint convergence in law of the extremal process with some genealogical informations in Section~\ref{sec:cv}. This convergence yields the observation that in the point process $\gamma_\infty$, the Poisson point process correspond to leaders realizing independently a small displacement, while each decoration comes from the family of the close relatives to a leader. This is reminiscent of the result obtained in \cite{ABK12} in the context of branching Brownian motion.
Similar results of convergence of enriched extremal processes have been recently obtained by Biskup and Louidor \cite{BiL} for the 2 dimensional Gaussian free field, and by Bovier and Hartung \cite{BoH} for the branching Brownian motion. Cortines, Hartung and Louidor \cite{CHL17} obtained refined results on the law of the decoration of the branching Brownian motion using among other things enrichment of the extremal process techniques. This method thus seems promising. For example, it might be used to proved simultaneous convergence in law of the rescaled trajectories of extremal particles toward Brownian excursions, as conjectured in \cite{CMM}, i.e. that for all $\beta > 1$
\begin{equation}
\label{eqn:conjCMM}
\lim_{n \to \infty} \frac{1}{\sum_{|u|=n} e^{-\beta V(u)}} \sum_{|u|=n} e^{-\beta V(u)}\delta_{H_n(u)} = \sum_{k \in \mathbb{N}} p_k \delta_{\mathbf{e}_k} \quad \text{ in law},
\mathbf{e}nd{equation}
where $H_n(u) : t \in [0,1] \mapsto n^{-1/2}V(u_{\floor{nt}})$, $(p_k, k \geq 1)$ is a Poisson-Dirichlet distribution with parameters ($\beta^{-1},0$), and $(\mathbf{e}_k, k \geq 1)$ are i.i.d. standard Brownian excursions.
We use here the convergence of the extremal process with genealogical informations to obtain simple proofs for a few additional results. We study the weak convergence of the so-called supercritical Gibbs measure of the branching random walk, as obtained in \cite{BRV12}. We also prove a conjecture of Derrida and Spohn on the asymptotic behavior of the so-called overlap of the branching random walk. More precisely, conditionally on the branching random walk $(\mathbf{T},V)$ we select two particles $u^{(n)}$, $v^{(n)}$ at the $n$th generation with probability proportional to $e^{-\beta (V(u) + V(v))}$, and denote by $\omega_{n,\beta}$ the law of the age of their most recent common ancestors, rescaled by a factor $n$. We prove that $(\omega_{n,\beta})$ converges, as $n \to \infty$ toward the probability measure $(1-\pi_\beta)\delta_0 + \pi_\beta \delta_1$, where $\pi_\beta$ is a random variable whose law depend on $\beta$.
As an other application of the convergence of the enriched extremal process, we finally obtain a description of the law $\mathcal{D}$ of the decoration of this process as the limit of the position of close relatives of the minimal displacement at time $n$. This result mimics the one proved in \cite{ABBS13} for the decoration of the branching Brownian motion. We expect a result similar to \cite{ABK13} would also holds, i.e. that the law $\mathcal{D}$ could be obtained as the limit in distribution of the extremal process conditioned on having a very small minimum.
\paragraph*{Outline.}
In the next section, we precise the encoding of the branching random walk, and use it to define the so-called critical measure: a measure on the boundary of the tree $\mathbf{T}$ of the branching random walk, whose distribution is related to the derivative martingale. We prove in Section~\ref{sec:cv} the convergence of the enriched extremal process. In Section~\ref{sec:applications}, we use this enriched convergence to prove the weak convergence of the supercritical Gibbs measure and the Derrida--Spohn conjecture. The expression of the law of the decoration $\mathcal{D}$ as the position of close relatives of the minimal displacement at time $n$ is obtained in Section~\ref{sec:decoration}.
\section{The critical measure of the branching random walk}
\label{sec:notation}
In this section, we first introduce the so-called Ulam-Harris notation for trees, that is used for a precise definition of the the branching random walk. In a second time we define the so-called critical measure of the branching random walk and study some of its properties. This measure is defined on the boundary of the tree of the branching random walk, and its distribution is related to the derivative martingale.
\subsection{Construction of the branching random walk}
\label{subsec:ulam}
We introduce the sets
\[
\mathcal{U} = \bigcup_{n \geq 0} \mathbb{N}^n ,\quad \partial \mathcal{U} = \mathbb{N}^\mathbb{N} \quad \text{and} \quad \bar{\mathcal{U}} = \mathcal{U} \cup \partial \mathcal{U},
\]
with the convention $\mathbb{N}^0 = \{ \mathbf{e}mptyset \}$. In the Ulam-Harris notation, a (plane, rooted) tree is constructed as a subset of $\mathcal{U}$, each element $u \in \mathcal{U}$ representing a potential individual.
Let $u \in \bar{\mathcal{U}}$, which is a finite or infinite sequence of integers. We denote by $|u|$ the length of the sequence $u$ and, for $k \leq |u|$ by $u_k$ the sequence consisting of the $k$ first values of $u$. If $u \in \mathcal{U} \backslash \{\mathbf{e}mptyset\}$, we write $\pi u = u_{|u|-1}$ the sequence obtained by erasing the last element. For $u \in \mathcal{U}$ and $v \in \bar{\mathcal{U}}$, we denote by $u.v$ the concatenation of the sequences. For $u,v \in \bar{\mathcal{U}}$, we write $u \leq v$ if $v_{|u|}=u$, which define a partial order on $\bar{\mathcal{U}}$. We then define $|u \wedge v| = \max\{ k \in \mathbb{N} : u_k = v_k\}$ and $u\wedge v = u_{|u \wedge v|} = v_{|u \wedge v|}$.
The genealogical tree $\mathbf{T}$ of the branching random walk is encoded as a subset of $\mathcal{U}$ in the following way. The root is encoded by the empty sequence $\mathbf{e}mptyset$, while $u = (u(1), \ldots u(n)) \in \mathcal{U}$ represents the $u(n)$th child of the $u(n-1)$th child of the ... of the $u(1)$th child of the root. With this encoding, $\pi u$ is the parent of $u$, $|u|$ the generation to which $u$ belongs, $u_k$ the ancestor of $u$ at generation $k$. We write $u<v$ if $u$ is an ancestor of $v$, and $u \wedge v$ is the most recent common ancestor of $u$ and $v$.
The family of positions $(V(u), u \in \mathbf{T})$ is then a random map from $\mathbf{T}$ to $\mathbb{R}$, which can be extended as a random map $\mathcal{U} \to \mathbb{R} \cup\{-\infty\}$, by setting $V(u) = -\infty$ for $u \in \mathcal{U} \backslash \mathbf{T}$. We then call $V : \mathcal{U} \to \mathbb{R} \cup\{-\infty\}$ the branching random walk, which can be constructed as follows. Let $\{(\mathbf{e}ll^u_j, j \in \mathbb{N}), u \in \mathcal{U}\}$ be a family of i.i.d. random variables in $(\mathbb{R} \cup\{-\infty\})^\mathbb{N}$, we set
\[
V(u) = \sum_{j=1}^{|u|} \mathbf{e}ll^{u_{j-1}}_{u(j)},
\]
with the convention $-\infty + x = x - \infty = -\infty$ for all $x \in \mathbb{R} \cup \{-\infty\}$. The law of $(\mathbf{e}ll^\mathbf{e}mptyset_j, j \in \mathbb{N})$ is called the reproduction law of the branching random walk $V$. Note that one can recover $\mathbf{T}$ from $V$ as $\{ u \in \mathcal{U} : V(u) > -\infty\}$.
\subsection{A topology on the set of leaves of infinite trees}
With the above notation, the set $\partial \mathcal{U}$ represents the set of possible leaves in the tree $\mathbf{T}$, infinite non-backtracking paths starting from the root. The critical measure of the branching random walk that we now describe is constructed as a Radon measure on the set of leaves. In this section, we introduce a topology on $\bar{\mathcal{U}}$ that makes it a compact space, and observe that finite measures on that space are identified with flows on the tree $\mathcal{U}$.
We embed $\bar{\mathcal{U}}$ in $[0,1]$, observing that the application
\[\mathbb{P}si : u \in \bar{\mathcal{U}} \longmapsto 2\sum_{j=1}^{|u|} 3^{-\sum_{i=1}^j u(i)}\]
is a bijection between $\bar{\mathcal{U}}$ and the Cantor ternary set $K$, depicted in Figure~\ref{fig:Cantor}.
\begin{figure}
\label{fig:Cantor}
\centering
\begin{tikzpicture}
\draw [line width=0.3cm] (0.0,0) -- (0.099,0);
\draw [line width=0.3cm] (0.198,0) -- (0.296,0);
\draw [line width=0.3cm] (0.593,0) -- (0.691,0);
\draw [line width=0.3cm] (0.79,0) -- (0.889,0);
\draw [line width=0.3cm] (1.778,0) -- (1.877,0);
\draw [line width=0.3cm] (1.975,0) -- (2.074,0);
\draw [line width=0.3cm] (2.37,0) -- (2.469,0);
\draw [line width=0.3cm] (2.568,0) -- (2.667,0);
\draw [line width=0.3cm] (5.333,0) -- (5.432,0);
\draw [line width=0.3cm] (5.531,0) -- (5.63,0);
\draw [line width=0.3cm] (5.926,0) -- (6.025,0);
\draw [line width=0.3cm] (6.123,0) -- (6.222,0);
\draw [color=red, line width=0.34cm] (7.085,0) -- (7.23,0);
\draw [color=red, line width=0.34cm] (7.279,0) -- (7.427,0);
\draw [color=red, line width=0.34cm] (7.674,0) -- (7.822,0);
\draw [color=red, line width=0.34cm] (7.871,0) -- (8.02,0);
\draw [line width=0.3cm] (7.111,0) -- (7.21,0);
\draw [line width=0.3cm] (7.309,0) -- (7.407,0);
\draw [line width=0.3cm] (7.704,0) -- (7.802,0);
\draw [line width=0.3cm] (7.901,0) -- (8.0,0);
\draw [color = red] (8.7,0) node {$B(1,1)$};
\draw [->,thick,color=blue] (0,1.8) -- (0,0.25);
\draw [color=blue] (-0.0,1.8) node[above] {$\mathbb{P}si(\mathbf{e}mptyset)$};
\draw [->,thick,color=blue] (5.333,1.2) node[above] {$\mathbb{P}si(1)$} -- (5.333,0.25);
\draw [->,thick,color=blue] (1.778,1.2) node[above] {$\mathbb{P}si(2)$} -- (1.778,0.25);
\draw [->,thick,color=blue] (0.593,1.2) -- (0.593,0.25);
\draw [color=blue] (0.593,1.2) node[above] {$\mathbb{P}si(3)$};
\draw [->,thick,color=blue] (7.111,0.7) node[above] {$\mathbb{P}si(1,1)$} -- (7.111,0.25);
\draw [->,thick,color=blue] (5.926,0.7) node[above] {$\mathbb{P}si(1,2)$} -- (5.926,0.25);
\draw [line width=0.3cm] (0.0,-0.5) -- (0.296,-0.5);
\draw [line width=0.3cm] (0.593,-0.5) -- (0.889,-0.5);
\draw [line width=0.3cm] (1.778,-0.5) -- (2.074,-0.5);
\draw [line width=0.3cm] (2.37,-0.5) -- (2.667,-0.5);
\draw [line width=0.3cm] (5.333,-0.5) --(5.63,-0.5);
\draw [line width=0.3cm] (5.926,-0.5) -- (6.222,-0.5);
\draw [line width=0.3cm] (7.111,-0.5) -- (7.407,-0.5);
\draw [line width=0.3cm] (7.704,-0.5)-- (8.0,-0.5);
\draw [line width=0.3cm] (0.0,-1) -- (0.889,-1);
\draw [line width=0.3cm] (1.778,-1) -- (2.667,-1);
\draw [line width=0.3cm] (5.333,-1)-- (6.222,-1);
\draw [line width=0.3cm] (7.111,-1) -- (8.0,-1);
\draw [line width=0.3cm] (0.0,-1.5) -- (2.667,-1.5);
\draw [line width=0.3cm] (5.333,-1.5)-- (8.0,-1.5);
\draw [line width=0.3cm] (0.0,-2) -- (8.0,-2);
\mathbf{e}nd{tikzpicture}
\caption{Mapping between $\bar{\mathcal{U}}$ and the Cantor ternary set}
\mathbf{e}nd{figure}
Using this bijection, we define a distance on $\bar{\mathcal{U}}$ by
\[
\forall u, v \in \bar{\mathcal{U}}, \quad d(u,v) = 2^{1-\min\{ n \in \mathbb{N} : 3^n |\mathbb{P}si(u)-\mathbb{P}si(v)| \geq 1\}},
\]
with the convention that $\min \mathbf{e}mptyset = \infty$ and $2^{-\infty} = 0$. Note this distance can be rewritten as
\[
d(u,v) = 2^{1- \min(u(|u \wedge v|+1), v(u \wedge v|+1))- \sum_{j=1}^{|u \wedge v|} u(j)},
\]
with the convention that if $|u|=n$, then $u(n+1) = 0$. This distance measures to which depth of construction of the Cantor set one should go before the images of $u$ and $v$ by $\mathbb{P}si$ are in distinct blocs. It is thus straightforward that $\bar{\mathcal{U}}$ is a compact ultrametric space when endowed by this distance. Informally, the topology of $(\bar{\mathcal{U}},d)$ can be described as the topology of pointwise convergence for infinite sequences of integers, with the addition that $\lim_{n \to \infty} u.n = u$, or in other words, identification between the sequence $(u(1),\ldots u(n), \infty, v(1),\ldots)$ with $u$, where $u \in \mathbb{N}^n$ and $v \in (\mathbb{N} \cup \infty)^\mathbb{N}$.
Note that $\mathcal{U}$ is a dense countable subset of $\bar{\mathcal{U}}$ for this topology. For any $u \in \mathcal{U}$, we denote by
\[
B(u) = \left\{ v \in \bar{\mathcal{U}} : u \wedge v =u \right\} = \left\{ u.w, w \in \bar{\mathcal{U}} \right\}/
\]
Observe that $\{B(u), u \in \mathcal{U}\}$ is a family of open and close balls of $\bar{\mathcal{U}}$ for the distance $d$. We also set, for $u \in \mathcal{U}$ and $n \in \mathbb{N}$
\begin{equation}
\label{def:countableBase}
C(u,n) = B(u) \backslash \left( \cup_{j=1}^{n-1} B(u.j)\right) = \{u\} \cup \bigcup_{j = n}^{\infty} B(u.j).
\mathbf{e}nd{equation}
\begin{lemma}
\label{lem:countableBase}
The family $\{C(u,j), u \in \mathcal{U}, j \in \mathbb{N}\}$ forms a countable base of open sets for $(\bar{\mathcal{U}},d)$.
\mathbf{e}nd{lemma}
\begin{proof}
Let $\mathcal{O}$ be an open subset of $\bar{\mathcal{U}}$. We define
\begin{equation}
\Gamma(\mathcal{O}) = \left\{ u \in \mathcal{U} : C(\pi u, u(|u|)) \not \subset \mathcal{O}, \mathbf{e}xists j \in \mathbb{N} : C(u,j) \subset \mathcal{O}\right\}
\mathbf{e}nd{equation}
as well as $j_u = \inf\{ j \in \mathbb{N} : C(u,j) \subset \mathcal{O} \}$ for $u \in \Gamma(\mathcal{O})$. We observe that
\begin{equation}
\label{eqn:countableBase}
\mathcal{O} = \bigcup_{u \in \Gamma(\mathcal{O})} C(u,j_u),
\mathbf{e}nd{equation}
and that the union is of pairwise disjoint elements.
Indeed, for any $v \in \mathcal{O}$, there exists $n \in \mathbb{N}$ such that $B(v_n) \subset \mathcal{O}$. We denote by $n_0 = \inf\{ n \in \mathbb{N} : B(v_n) \subset \mathcal{O}\}$. Then either $v_{n_0} \in \Gamma(\mathcal{O})$ and $j_{v_0}=1$, or $v_{n_0-1} \in \Gamma(\mathcal{O})$ and $j_{v_{n_0-1}} \leq v(n_0)$. Reciprocally, it follows from definition that $C(u,j_u) \subset \mathcal{O}$ for all $u \in \Gamma(\mathcal{O})$.
We now observe that $u \in C(u,n)$ for all $n \in \mathbb{N}$. Hence, as $C(u,n) \subset B(u)$, if $C(u,n) \cap C(v,m) \neq \mathbf{e}mptyset$, then either $u$ is an ancestor of $v$, or $v$ is an ancestor of $u$, or $u=v$. Moreover, if $u$ is an ancestor of $v$, then $v \in C(u,n)$.
We assume there exists $u \neq v \in \Gamma(\mathcal{O})$ such that $C(u,j_u) \cap C(v,j_v) \neq \mathbf{e}mptyset$. We can assume without loss of generality that $u$ is an ancestor of $v$, hence $v = u. l . w$, with $l \in \mathbb{N}$ and $w \in \mathcal{U}$. As $v \in C(u,j_u) $, we have $l \geq j_u$. Hence, by definition, we have $B(u.l) \subset \mathcal{O}$, which is in contradiction with the fact that $v \in \Gamma(\mathcal{O})$.
\mathbf{e}nd{proof}
Note that \mathbf{e}qref{eqn:countableBase} can be rewritten as the disjoint union of elements belonging to the families $\{B(u), u \in \mathcal{U}\}$ and $\{\{u\}, u \in \mathcal{U}\}$.
In the rest of the section, we study finite measures on the space $(\bar{\mathcal{U}},d)$. We first identify them with pseudo-flows on $\mathcal{U}$, which we now define.
\begin{definition}
A function $f : \mathcal{U} \to \mathbb{R}_+$ is called a pseudo-flow on $\mathcal{U}$ if it satisfies
\begin{equation}
\label{eqn:pseudoflow}
\forall u \in \mathcal{U}, \quad f(u) \geq \sum_{j \in \mathbb{N}} f(u.j).
\mathbf{e}nd{equation}
A function $f : \mathcal{U} \to \mathbb{R}_+$ is called a flow on $\mathcal{U}$ if it satisfies
\begin{equation}
\forall u \in \mathcal{U}, \quad f(u) = \sum_{j \in \mathbb{N}} f(u.j).
\mathbf{e}nd{equation}
\mathbf{e}nd{definition}
To each Radon measure $\mu$ on $(\bar{\mathcal{U}},d)$ we can associate a pseudo-flow on $\mathcal{U}$ defined as
\[
\forall u \in \mathcal{U}, \quad f_\mu(u) = \mu(B(u)).
\]
Note that the function $f_\mu$ is a flow if and only if $\mu(\mathcal{U}) = 0$. We now observe that $\mu \mapsto f_\mu$ realizes a bijection between the Radon measures and the pseudo-flows.
\begin{proposition}
\label{prop:flows}
For each pseudo-flow $f$ on $\mathcal{U}$, there exists a unique finite measure $\mu$ such that $f=f_\mu$.
A measure $\mu$ is atomless if and only if $f_\mu$ is a flow and
\[
\lim_{n \to \infty} \max_{|u|=n} f_\mu(u) = 0.
\]
\mathbf{e}nd{proposition}
\begin{proof}
Let $\mu$ and $\nu$ be two finite measures such that $f_\mu = f_\nu$. By definition, this indicates that
\[
\forall u \in \mathcal{U}, \quad \mu(B(u)) = \nu(B(u)).
\]
Therefore, by sigma-additivity, one successively deduces that
\[
\forall u \in \mathcal{U}, n \in \mathbb{N}, \quad \mu(C(u,n)) = \sum_{j=n}^\infty \mu(B(u.j)) = \sum_{j=n}^\infty \nu(B(u.j)) = \nu(C(u,n)),
\]
and, thanks to \mathbf{e}qref{eqn:countableBase}, that $\mu(\mathcal{O}) = \nu(\mathcal{O})$ for all open subset of $\bar{\mathcal{U}}$. By monotone classes theorem, we deduce that $\mu = \nu$.
Let $f$ be a pseudo-flow on $\mathcal{U}$, we now construct a measure on $\bar{\mathcal{U}}$ associated to that pseudo-flow. We first observe that if $f(\mathbf{e}mptyset) = 0$, then $f(u) = 0$ for all $u \in \mathcal{U}$, hence the null measure is associated to that flow.
We now assume that $f(\mathbf{e}mptyset) \neq 0$. Up to replacing $f$ by $f/f(\mathbf{e}mptyset)$ we can assume without loss of generality that $f(\mathbf{e}mptyset) = 1$. We observe that for all $n \in \mathbb{N}$, we can define the law $\mu_n$ on $\{ u \in \mathcal{U} : |u| \leq n\}$ by
\begin{align*}
\forall u \in \mathcal{U} : |u| < n, \quad \mu_n(u) = f(u) - \sum_{j =1}^\infty f(u.j)\\
\forall u \in \mathcal{U} : |u|=n, \quad \mu_n(u) = f(u).
\mathbf{e}nd{align*}
Observe that thanks to the pseudo-flow property, the family of probability distributions $(\mu_n, n \geq 1)$ is consistent under the family of projections $\pi_n : u \mapsto u_n$. Hence, thanks to Kolmogorov extension theorem, there exists a probability measure $\mu$ on $\bar{\mathcal{U}}$ such that its image measure by $\pi_n$ is $\mu_n$ for all $n \in \mathbb{N}$. Hence, one has straightforwardly $f_\mu = f$.
The second point is a straightforward consequence as if $\mu$ has an atom of mass $x$ at point $u \in \bar{\mathcal{U}}$, then
\[
f_\mu(u) = x + \sum_{j=1}^\infty f_\mu(u.j),
\]
if $u \in \mathcal{U}$, by sigma-additivity, and
\[
\lim_{n \to \infty} f_\mu(u_n) = x
\]
if $u \in \partial \mathcal{U}$, by dominated convergence.
\mathbf{e}nd{proof}
\subsection{The critical measure of the branching random walk}
\label{subsec:critical}
Let $V$ be a branching random walk satisfying \mathbf{e}qref{eqn:supercritical}, \mathbf{e}qref{eqn:boundary}, \mathbf{e}qref{eqn:variance} and \mathbf{e}qref{eqn:integrability}. We recall that we set $\mathbf{T} = \{ u \in \mathcal{U} : V(u) \neq -\infty\}$. For any $n \in \mathbb{N}$, we denote by $\mathbf{T}(n) = \{ u \in \mathbf{T} : |u|=n\}$ the set of individuals alive at generation $n$. We introduce the filtration $(\mathcal{F}_n)$, defined by
\[
\mathcal{F}_n = \sigma\left( V(u), u \in \mathcal{U} : |u| \leq n \right).
\]
Note that by definition of the branching random walk, if we set, for all $u \in \mathbf{T}$
\begin{equation}
\label{eqn:subBrw}
V^u : v \in \mathcal{U} \mapsto V(u.v) - V(u)
\mathbf{e}nd{equation}
the branching random walk issued from particle $u$, then by definition, for all $n \in \mathbb{N}$, $\{V^u, u \in \mathbf{T}(n)\}$ is a family of i.i.d. branching random walks, with same law as $V$, that are independent from $\mathcal{F}_n$. This fact is often called the branching property of the branching random walk.
We denote the boundary of the branching random walk by
\[
\partial \mathbf{T} = \left\{ u \in \partial \mathcal{U} : \forall n \in \mathbb{N}, u_n \in \mathbf{T}\right\}.
\]
An element of $\partial \mathbf{T}$ represent a spine of the tree: a semi-infinite path starting at and going away from the root in the tree $\mathbf{T}$.
The critical measure of the branching random walk has been introduced by the physicists Derrida and Spohn in \cite{DeS88}. Its existence is a consequence the precise study of the derivative martingale in \cite{AiS14}. This measure has been the subject of multiple studies \cite{BKNSW,Bur09,BDK16+}.
To define the critical measure, for any $u \in \mathbf{T}$, we set
\[
Z^u_n = \sum_{|v|=n, v > u} (V(v)-V(u))e^{V(u)-V(v)} \quad \text{and} \quad Z^u_\infty = \liminf_{n \to \infty} Z^u_n.
\]
Thanks to the branching property, we observe $\left(Z^u_\infty, u \in \mathbf{T}(k)\right)$ are i.i.d. copies of $Z_\infty$, which are independent of $\mathcal{F}_k$. Moreover, for any $k \leq n$, we have
\[
Z_n = \sum_{|u|=k} e^{-V(u)} Z^u_{n} + \sum_{|u|=k} V(u)e^{-V(u)} \sum_{|v|=n,v>u} e^{V(u)-V(v)}.
\]
Letting $n \to \infty$ and using \mathbf{e}qref{eqn:cvMartingales}, we deduce $Z_\infty = \sum_{|u|=k} e^{-V(u)} Z^u_\infty$ a.s. More generally, almost surely, for all $u \in \mathbf{T}$ we have
\begin{equation}
\label{eqn:flowDerivative}
e^{-V(u)} Z_\infty^u = \sum_{j \in \mathbb{N}} e^{-V(u.j)}Z_\infty^{u.j}.
\mathbf{e}nd{equation}
In other words, the function
\[
f^* : u \in \mathcal{U} \mapsto \begin{cases} e^{-V(u)} Z_\infty^u & \quad \text{ if } u \in \mathbf{T}\\ 0 & \quad \text{otherwise,}\mathbf{e}nd{cases}
\]
is a.s. a flow on $\mathcal{U}$. The critical measure of the branching random walk is the unique measure $\nu$ on $\bar{\mathcal{U}}$ associated to the flow $f^*$, i.e.
\begin{equation}
\label{eqn:defineCriticalMeasure}
\forall u \in \mathcal{U}, \quad \nu\left( B(u) \right) = \ind{u \in \mathbf{T}} e^{-V(u)} Z_\infty^u \quad \text{a.s.}
\mathbf{e}nd{equation}
Existence and uniqueness of $\nu$ are proved in Proposition~\ref{prop:flows}. Moreover, note that as $Z_\infty >0$ a.s. on the survival event $S$ of the branching random walk, the support of $\nu$ is a.s. the adherence of the boundary of the tree $\partial \mathbf{T}$, for the distance $d$.
\begin{remark}
\label{rem:cvCritical}
Note that the following convergence holds
\[
\nu = \lim_{n \to \infty} \sum_{|u|=n} V(u) e^{-V(u)} \delta_{u} \quad \text{a.s.}
\]
for the topology of weak convergence of measures on $\bar{\mathcal{U}}$, as for all $v \in \mathcal{U}$, we have
$ \displaystyle \lim_{n \to \infty} \sum_{|u|=n} \ind{u \in B(v)} V(u) e^{-V(u)} = \nu(B(v))$ a.s. We conclude using the Portmanteau theorem.
\mathbf{e}nd{remark}
We end this section with a short proof that $\nu$ is non-atomic. We first note that that as $f^*$ is a (proper) flow, we have immediately $\nu(\{u\}) = 0$ a.s for all $u \in \mathcal{U}$, therefore $\nu(\mathcal{U})=0$ a.s. As a result, the fact that $\nu$ is non-atomic is a consequence of the following lemma.
\begin{lemma}
\label{lem:nonAtomic}
Under assumptions \mathbf{e}qref{eqn:supercritical}, \mathbf{e}qref{eqn:boundary}, \mathbf{e}qref{eqn:variance} and \mathbf{e}qref{eqn:integrability}, we have
\[
\lim_{n \to \infty} \max_{|u|=n} \nu(B(u)) = 0 \quad \text{a.s.}
\]
\mathbf{e}nd{lemma}
\begin{proof}
We first recall the precise estimate on the tail of $Z_\infty$ obtained by Madaule \cite{Mad16}: there exists $c_1>0$ such that for any $x \geq 0$, $\mathbb{P}(Z_\infty \geq x) \leq \frac{c_1}{x}$.
We now observe that for any $\mathbf{e}psilon>0$ and $n\in \mathbb{N}$, we have
\begin{align*}
\mathbb{P}\left( \max_{|u|=n} \nu(B(u)) \geq \mathbf{e}psilon \middle| \mathcal{F}_n \right) &\leq \sum_{|u|=n} \mathbb{P}\left( e^{-V(u)} Z^u_\infty \geq \mathbf{e}psilon \middle| \mathcal{F}_n \right)\\
&\leq \frac{c_1}{\mathbf{e}psilon} \sum_{|u|=n} e^{-V(u)} \text{ a.s,}
\mathbf{e}nd{align*}
thus $\lim_{n \to \infty} \mathbb{P}\left( \max_{|u|=n} \nu(B(u)) \geq \mathbf{e}psilon \middle| \mathcal{F}_n \right) = 0$ a.s. by \mathbf{e}qref{eqn:cvMartingales}. As the sequence $(\max_{|u|=n} \nu(B(u)), n \geq 0)$ is non-increasing in $n$, we conclude that
\[\lim_{n \to \infty} \max_{|u|=n} B(u) = 0 \quad \text{ a.s.} \tag*{\qedhere}\]
\mathbf{e}nd{proof}
\section{Convergence in law of the extremal process with genealogical informations}
\label{sec:cv}
Using the notation of the previous section, we can now state the main result of this paper, namely the convergence of the point measure
\begin{equation}
\label{eqn:genalogicalExtremalProcess}
\mu_n = \sum_{|u|=n} \delta_{u,V(u)-m_n},
\mathbf{e}nd{equation}
on $\bar{\mathcal{U}} \times \mathbb{R}$. The sketch of proof is the following: we first define a candidate for the limiting measure, then observe that Fact~\ref{fct:Madaule} implies that $\mu_n$ converges toward this well-chosen limiting measure.
Independently from the branching random walk $V$, let $(\xi_n, n \geq 1)$ be the atoms of a Poisson point process with intensity $c_* e^{x}dx$, $(u^{(n)}, n \geq 1)$ be i.i.d. random variables with law $\bar{\nu}$ and $(D_n, n \geq 1)$ be i.i.d. point measures with law~$\mathcal{D}$, with $c_*$ and $\mathcal{D}$ defined in Fact~\ref{fct:Madaule}. For any $n \in \mathbb{N}$, we set
\begin{equation}
\label{eqn:limitlExtremalProcess}
\mu_\infty = \sum_{n=1}^{\infty} \sum_{d \in \mathcal{P}(D_n)} \delta_{u^{(n)},\xi_n + d-\log Z_\infty}.
\mathbf{e}nd{equation}
By classical properties of Poisson point processes, $( u^{(n)},\xi_n-\log Z_\infty)$ are the atoms of a Poisson point process with intensity $c_*\nu\otimes e^x dx$ on $\bar{\mathcal{U}}\times\mathbb{R}$. Hence $\mu_\infty$ can alternatively be described as a Poisson point process with intensity $c_*\nu \otimes e^x dx$, with an i.i.d. decoration on the second coordinate. The main result of the article is the following convergence.
\begin{theorem}
\label{thm:main}
Assuming \mathbf{e}qref{eqn:supercritical}, \mathbf{e}qref{eqn:boundary}, \mathbf{e}qref{eqn:variance} and \mathbf{e}qref{eqn:integrability}, we have
\[\lim_{n \to \infty} (\mu_n,Z_n) = (\mu_\infty,Z_\infty) \quad \text{ in law on $S$},\]
for the topology of the vague convergence.
\mathbf{e}nd{theorem}
\begin{remark}
The genealogical informations encoded in $\mu_n$ only concern the local behavior in a neighborhood of the root of the process. Informally, we say that two individuals do not belong to the same family if the age of their most recent common ancestor if $O(1)$. However, we know that with high probability, for individuals close to the minimal displacement at time $n$, the age of their most recent common ancestor is either $O(1)$ or $n-O(1)$ with high probability (see e.g. \mathbf{e}qref{eqn:genealogy}). But to obtain informations on the genealogy within the group of the followers, different quantities should be considered, such as the branching random walk seen from the local leader, for the topology of local convergence.
\mathbf{e}nd{remark}
The convergence in Theorem~\ref{thm:main} can be interpreted as follows. We can decompose the extremal process at time $n$, near position $m_n$ into families of individuals whose common ancestor was alive at generation $n-O(1)$. In each of these families, there is a leader, a particle whose position is the smallest within the family. The point process of the leaders converge toward a Poisson point process with exponential intensity, and the relative positions of their relatives converge toward i.i.d. copies of a point process of law $\mathcal{D}$. The fact that $\nu$ has no atom proves that with high probability, the most recent common ancestor between two individuals of two distinct families was alive at time $O(1)$.
This convergences gives some informations on the genealogical relationships for particles close to the smallest position at time $n$. For example, in the non-lattice case, if two particles $u$ and $v$ are at position $M_n$ at time $n$, they are close relatives with high probability. Note that this result would not hold in the lattice case, as observed by Pain \cite[Footnote 3]{Pai17+}.
\begin{proof}[Proof of Theorem~\ref{thm:main}]
This is a direct consequence of Madaule's convergence in law for the extremal process of the branching random walk with its genealogy. For any $v \in \mathbf{T}$, we denote by
\[\mu^v_\infty(.) = \int_{B(v)} \mu_\infty(du,.) = \sum_{k=1}^{\infty} \ind{u^{(k)} > v} \theta_{\xi_k-\log Z_\infty} D_k,\]
For every $k \in \mathbb{N}$, conditionally on $\mathcal{F}_k$ and $(Z^v_\infty, |v|=k)$, we observe that $(\theta_{-V(v)} \mu^v_\infty, v \in \mathbf{T}(k))$ are independent SDPPP($c_*e^{x}dx,-\log Z^v_\infty,\mathcal{D}$).
In other words, $\mu^v_\infty$ has the same law as the limit of the extremal process of the branching random walk $V^v$ issued from particle $v$, defined in \mathbf{e}qref{eqn:subBrw}. Thus, by Fact~\ref{fct:Madaule}, conditionally on $\mathcal{F}_k$, for any $v \in \mathbf{T}(k)$, we have
\[
\lim_{n \to \infty} \left(\sum_{|u|=n,u>v} \delta_{V(u)-V(v)-m_n}, Z^v_n \right) = \left(\theta_{-V(v)}\mu^v_\infty, Z^v_\infty\right) \text{ in law on $S$.}
\]
We denote by $f$ a continuous non-negative function on $\mathbb{R}$ with compact support and $k \in \mathbb{N}$. By the branching property, conditionally on $\mathcal{F}_k$, the subtrees of the branching random walk rooted at points $v \in \mathbf{T}(k)$ behave as independent branching random walk. Therefore
\begin{align*}
&\lim_{n \to \infty} \left(\left(\mu_n(\mathbf{1}_{B(v)}f), v \in \mathbf{T}(k)\right),Z_n \right)\\
= &\lim_{n \to \infty} \left(\left(\sum_{|u|=n, u > v} f(V(u)-m_n),v \in \mathbf{T}(k)\right), \sum_{|v|=k} e^{-V(v)} Z_n^v \right)\\
= &\left(\left(\mu^v_\infty(f), v \in \mathbf{T}(k)\right), Z_\infty\right) \quad \text{ in law on $S$}.
\mathbf{e}nd{align*}
By \cite[Theorem 14.16]{Kal02}, we conclude that $\lim_{n \to \infty} (\mu_n,Z_n) = (\mu_\infty,Z_\infty)$ in law on the survival event $S$.
\mathbf{e}nd{proof}
Using \cite[Theorem 10]{SuZ15}, we observe that writing $\hat{\mu}^x$ for a point measure with distribution $\theta_{-\min \mu_\infty} \mu_\infty$ conditionally on $\{\min \mu_\infty < -x\}$, we have
\[
\lim_{x \to \infty} \hat{\mu}^x = D_1 \quad \text{ in law}.
\]
This result can be seen as a (weaker form of the) characterization of \cite{ABK13} of the law $\mathcal{D}$. We provide an alternative characterization of this law in Section~\ref{sec:decoration}.
A straightforward consequence of Theorem~\ref{thm:main} is the convergence for the extremal process seen from the smallest position.
\begin{corollary}
\label{cor:main}
Under the same assumptions as Theorem~\ref{thm:main}, we set $\mathbf{e}$ a standard exponential random variable, $\zeta_1=0$ and $(\zeta_n, n \geq 2)$ a Poisson point process with intensity $\mathbf{e}e^{x}\ind{x>0}dx$. We have
\[
\lim_{n \to \infty} \left(M_n - m_n, \sum_{|u|=n} \delta_{u,V(u)-M_n}\right) = \left(\log(\mathbf{e}/Z_\infty), \sum_{d \in \mathcal{P}(D_n)} \delta_{u^{(n)},\zeta_n + d}\right),
\]
in law, on the survival event $S$.
\mathbf{e}nd{corollary}
\begin{remark}
If the law of the decoration $\mathcal{D}$ is explicit, it becomes possible to compute the asymptotic probability for two particles within $O(1)$ distance from the minimal displacement $M_n$ to belong to distinct families. For example, setting $u^{1,n}$ and $u^{2,n}$ the labels of the smallest two individuals at generation $n$, we have
\begin{align*}
\lim_{n \to \infty} \mathbb{P}(|u^{1,n} \wedge u^{2,n}| \geq n/2 ) = \mathbb{P}( d_2 \leq \zeta_2) = \E(e^{-d_2}),
\mathbf{e}nd{align*}
where $d_2$ the second smallest point of $D_1$, as $\zeta_2$ is distributed as an exponential random variable with parameter $1$.
\mathbf{e}nd{remark}
In the next sections, we derive some additional informations of the genealogy of particles close to the minimal displacement at time $n$, that can be extracted from the convergence in Theorem~\ref{thm:main}.
\section{The supercritical Gibbs measure}
\label{sec:applications}
In this section, we use Theorem~\ref{thm:main} to give a simple construction of the so-called supercritical Gibbs measures on $\bar{\mathcal{U}}$, as obtained in \cite{BRV12}. More precisely, the aim is to mimic the construction of the critical measure describe in Section~\ref{subsec:critical}, but instead of using the derivative martingale $(Z_n)$ one use the supercritical additive martingale with parameter $\beta >1$.
For any $\beta \geq 0$, we denote by
\[
\kappa(\beta) = \log \E\left( \sum_{|u|=1} e^{-\beta V(u)} \right) \in (-\infty,\infty].
\]
For all $n \in \mathbb{N}$ and $\beta \geq 0$ such that $\kappa(\beta) < \infty$, we denote by
\[
W_n^{(\beta)} = \sum_{|u|=n} e^{-\beta V(u) - n \kappa(\beta)}.
\]
By the branching property, $(W_n^{(\beta)}, n \geq 0)$ is a non-negative martingale.
If we assume that $W_\infty^{(\beta)}>0$ a.s. on the survival event of the branching random walk, then one can use the same techniques as in Section~\ref{subsec:critical} to define a finite measure on $\bar{\mathcal{U}}$ such that $\nu_\beta(B(u)) = e^{-\beta V(u)} W_\infty^{(\beta),u}$, which we call the Gibbs measure of the branching random walk. To justify this name, observe that
\begin{equation}
\label{eqn:gibbsMeasure}
\lim_{n \to \infty} \frac{\sum_{|u|=n} e^{-\beta V(u)}\delta_u}{\sum_{|u|=n} e^{-\beta V(u)}} = \frac{\nu_\beta }{W_\infty(\beta)} \text{ a.s.}
\mathbf{e}nd{equation}
for the topology of weak convergence.
However, under assumption \mathbf{e}qref{eqn:boundary}, it is well-known that $\lim_{n \to \infty} W^{(\beta)}_n = 0$ a.s. for all $\beta > 1$ (see e.g. \cite{Lyo97}). Nevertheless, the aim of this section is to obtain a convergence similar to \mathbf{e}qref{eqn:gibbsMeasure} for $\beta > 1$, thus defining the supercritical Gibbs measure on $\bar{\mathcal{U}}$.
Let $\beta > 1$, as $\lim_{n \to \infty} W_n^{(\beta)}=0$, one has to choose a different renormalization in order to obtain a non-degenerate limit. We set
\[
W_{n,\beta} = n^{3\beta/2} e^{n \kappa(\beta)} W_n^{(\beta)} = \sum_{|v|=n} e^{\beta (m_n - V(v))}.
\]
Madaule \cite[Theorem 2.3]{Mad15} proved there exists a random variable $W_{\beta,\infty}$ defined on the same probability space as $Z_\infty$ such that
\[\lim_{n \to \infty} (W_{n,\beta},Z_n) = (W_{\infty, \beta},Z_\infty) \text{ in law,}\]
with $W_{\infty,\beta}$ and $Z_\infty$ being a.s. either both positive or both null. For all $u \in \mathbf{T}$, we set
\[
W_{n,\beta}^u = \sum_{|v|=n, v > u} e^{\beta (m_n + V(u) - V(v))},
\]
We construct a measure which gives mass $W_{\infty,\beta}^u$ to the ball $B(u)$ for all $u \in \mathbf{T}$. This measure is then used to study the so-called overlap of the branching random walk.
We recall that $(u^{(n)})$ are i.i.d. random elements of $\bar{\mathcal{U}}$ sampled with law~$\bar{\nu}$. We denote by $(\xi^\beta_n, n \geq 1)$ the atoms of a Poisson point process with intensity $c_\beta e^x dx$, where we write~$c_\beta = c_* \E\left( \sum_{d \in D} e^{-\beta d} \right)$. We introduce the random measures on $\bar{\mathcal{U}}$ defined by
\begin{equation}
\label{eqn:supercriticalMeasure}
\nu_{\beta,n} = \sum_{|u|=n} e^{\beta (m_n - V(u))} \delta_u \quad \text{and} \quad \nu_{\beta,\infty} = \sum_{n \in \mathbb{N}} Z_\infty^\beta e^{-\beta \xi^\beta_n} \delta_{u^{(n)}}.
\mathbf{e}nd{equation}
\begin{theorem}
\label{thm:cvSupercritical}
Assuming \mathbf{e}qref{eqn:supercritical}, \mathbf{e}qref{eqn:boundary}, \mathbf{e}qref{eqn:variance} and \mathbf{e}qref{eqn:integrability}, for any $\beta > 1$, we have
\[
\lim_{n \to \infty} \nu_{\beta,n} = \nu_{\beta,\infty}\quad \text{ in law,}
\]
for the topology of the weak convergence of measures.
\mathbf{e}nd{theorem}
Note that this convergence is similar to the one observed for the critical measure in Remark~\ref{rem:cvCritical}. However, the convergence holds in distribution, and not almost surely. We also have $\nu_{\beta,\infty}(B(u)) = W_{\infty, \beta}^u$ in law, for all $u \in \mathbf{T}$.
\begin{proof}
By \cite[Theorem 2.3]{Mad15}, $\nu_{\beta,n}(B(u))$ converges in law for any $u \in \mathbf{T}$ as $n \to\infty$. Consequently, using Theorem~\ref{thm:main}, for any $u \in \mathbf{T}$, we can identify the law of the limit as
\begin{align*}
\lim_{n \to \infty} \nu_{\beta,n}(B(u)) &= \sum_{k=1}^{\infty} \ind{u^{(k)}>u} \sum_{d \in \mathcal{P}(D_k)} Z_\infty^\beta e^{-\beta (\xi_k + d)}\quad \text{ in law,}\\
&= Z_\infty^\beta \sum_{k=1}^{\infty} \ind{u^{(k)}>u} e^{-\beta \xi_k}\sum_{d \in \mathcal{P}(D_k)} e^{-\beta d}.
\mathbf{e}nd{align*}
Setting $X^\beta_k = -\frac{1}{\beta} \log \sum_{d \in \mathcal{P}(D_k)} e^{-\beta d}$, we have
\[
\lim_{n \to \infty} \nu_{\beta,n}(B(u)) = \sum_{k=1}^{\infty} \ind{u^{(k)}>u} Z_\infty^\beta e^{-\beta (\xi_k + X^\beta_k)} \quad \text{ in law}.
\]
Moreover, as $(\xi_k+X^\beta_k, k \in \mathbb{N})$ are the atoms of a Poisson point process with intensity $c_\beta e^{x} dx$ independent of $(u^{(k)})$, we conclude that for any $j \in \mathbb{N}$,
\[
\lim_{n \to \infty} \left(\nu_{\beta,n}(B(u)), u \in \mathbf{T}(j) \right) = \left( \nu_{\beta,\infty}(B(u)), u \in \mathbf{T}(j)\right) \quad \text{ in law},
\]
which concludes the proof.
\mathbf{e}nd{proof}
\begin{remark}
A straightforward consequence of this result is that under assumptions \mathbf{e}qref{eqn:supercritical}, \mathbf{e}qref{eqn:boundary}, \mathbf{e}qref{eqn:variance} and \mathbf{e}qref{eqn:integrability}, the solution $Y_\beta$ of the supercritical smoothing transform (see e.g. \cite{ABM})
\[
Y_\beta \mathbf{e}galdistr \sum_{|u|=1} e^{-\beta V(u)} Y^{(u)}_\beta
\]
can be written $Y_\beta = Z_\infty^\beta\sum_{k=1}^{\infty} e^{-\beta \xi^\beta_k}$.
\mathbf{e}nd{remark}
Theorem~\ref{thm:cvSupercritical} indirectly implies a proof of the conjecture of Derrida and Spohn \cite{DeS88}: the rescaled distribution of the genealogy of the most recent common ancestor of two particles chosen independently at random according to the measure $\nu_{n,\beta}/\nu_{n,\beta}(\bar{\mathcal{U}})$ converges in law toward a random measure on $[0,1]$ with no mass on $(0,1)$.
\begin{theorem}
\label{thm:DeS88}
For any $n \in \mathbb{N}$ and $\beta > 1$, we set
\begin{equation}
\omega_{n,\beta} = W_{n,\beta}^{-2}\sum_{|u|=|v|=n} e^{\beta (2 m_n - V(u)-V(v))} \delta_{|u\wedge v|/n}.
\mathbf{e}nd{equation}
Assuming \mathbf{e}qref{eqn:supercritical}, \mathbf{e}qref{eqn:boundary}, \mathbf{e}qref{eqn:variance} and \mathbf{e}qref{eqn:integrability}, conditionally on $S$, we have
\[
\lim_{n \to \infty} \omega_{n,\beta} = (1-\pi_\beta) \delta_0 + \pi_\beta \delta_1 \quad \text{in law,}
\]
where $\pi_\beta = \sum_{k = 1}^{\infty} p^2_k$ and $(p_k, k \geq 1)$ is a Poisson-Dirichlet mass partition with parameters $(\beta^{-1},0)$.
\mathbf{e}nd{theorem}
A similar result was already known for multiple types of Gaussian processes with a logarithmic correlation structure, such as the Generalized Random Energy Model \cite{BoK}, log-correlated Gaussian fields such as the Gaussian Free Field \cite{ArZ}, or the binary branching random walk with Gaussian increments \cite{Jag}. More precisely, it is proved that a measure similar to $\nu_{n,\beta}/\nu_{n,\beta}(\bar{\mathcal{U}})$ converges in law toward a Ruelle probability cascade. Ouimet \cite{Oui} recently extended this family of results to Gaussian fields with scale-dependent variance. Theorems~\ref{thm:cvSupercritical} and~\ref{thm:DeS88} represent extensions of these results to branching random walks with non-Gaussian increments. Contrarily to what was done in this past literature, the proof relies on the study of the extremal point process instead of proving Ghirlanda-Guerra type identities. Thus Poisson-Dirichlet distributions appear as simple functional of a Poisson point process instead of the application of Talagrand's identity (see \cite[Remark 3.8]{Jag}).
\begin{proof}
We first observe that it is enough to prove that conditionally on $S$,
\begin{equation}
\label{eqn:target} \forall t \in (0,1) \quad \lim_{n \to \infty} \omega_{n,\beta}((t,1]) = \pi_\beta\quad \text{in law}.
\mathbf{e}nd{equation}
Indeed, the function $t \mapsto \omega_{n,\beta}((t,1])$ is decreasing on $[0,1]$, therefore \mathbf{e}qref{eqn:target} and Slutsky's lemma imply the convergence of the finite-dimensional distributions of the tail of $\omega_{n,\beta}$, which concludes the proof.
For $k \leq n $ and $t \in [0,1]$, we set
\begin{align*}
\Lambda_n^k &= W_{n,\beta}^{-2}\sum_{|u|=|v|=n} e^{2\beta m_n-\beta V(u)-\beta V(v)} \ind{|u\wedge v| \geq k}\\
\text{and}\quad \Delta_n^{k,t} &= W_{n,\beta}^{-2}\sum_{|u|=|v|=n} e^{2\beta m_n-\beta V(u)-\beta V(v)} \ind{|u\wedge v| \in [k,tn]}.
\mathbf{e}nd{align*}
We observe that for every $k \in [1,tn) \cap \mathbb{N}$, we have
\begin{equation}
\label{eqn:gendarme}
\Lambda_n^k -\Delta_n^{k,t} \leq \omega_{n,\beta}((t,1]) \leq \Lambda_n^k \quad \text{a.s. on S.}
\mathbf{e}nd{equation}
By Theorem~\ref{thm:main}, as $S = \{ Z_\infty > 0\}$ a.s. we have
\[
\lim_{n \to \infty} \Lambda_n^k = \Lambda_\infty^k := \frac{ \sum_{|u|=k} \left(\sum_{j=1}^{\infty} \ind{u^{(j)}>u} e^{-\beta \xi^\beta_j} \right)^2 }{\left( \sum_{j=1}^{\infty} e^{-\beta \xi^\beta_j} \right)^2} \quad \text{in law on $S$.}
\]
Moreover, as $\nu$ is non-atomic by Lemma~\ref{lem:nonAtomic}, letting $k \to \infty$ we have
\[ \lim_{k \to \infty} \Lambda_\infty^k = \frac{\sum_{j=1}^{\infty} e^{-2\beta \xi^\beta_j}}{\left( \sum_{j=1}^{\infty} e^{-\beta \xi^\beta_j} \right)^2} \quad \text{ a.s.}\]
Using \cite[Proposition 10]{PiY97}, we have $\lim_{k \to \infty} \Lambda_\infty^k = \pi_\beta$ a.s.
We now study the asymptotic behaviour of $\Delta_n^{k,t}$, more precisely we prove that for any $\delta > 0$,
\begin{equation}
\label{eqn:deltato0}
\lim_{k \to \infty} \limsup_{n \to \infty} \mathbb{P}\left( \Delta_n^{k,t} > \delta, S \right) = 0.
\mathbf{e}nd{equation}
By \cite[Theorem 2.3]{Mad15}, $\displaystyle \lim_{\mathbf{e}psilon \to 0} \lim_{n \to \infty} \mathbb{P}\left( n^{3\beta/2} W_{n,\beta} \leq \mathbf{e}psilon, S \right) = 0$, therefore it is enough to prove that for any $\mathbf{e}psilon > 0$,
\begin{equation}
\label{eqn:epsilonto0}
\lim_{k \to \infty} \limsup_{n \to \infty} \mathbb{P}\left( \sum_{|u|=|v|=n} \ind{|u \wedge v| \in [k,tn]} e^{\beta(m_n-V(u)) + \beta(m_n-V(v))} > \delta \mathbf{e}psilon^2 \right)= 0.
\mathbf{e}nd{equation}
The proof of this result, rather technical, is postponed to Lemma~\ref{lem:entangled}.
Let $x \in [0,1]$ and $\delta > 0$, using \mathbf{e}qref{eqn:gendarme}, we have
\[
\mathbb{P}(\Lambda_n^k \leq x + \delta, S) + \mathbb{P}(\Delta_n^{k,t} \geq \delta,S) \geq \mathbb{P}\left( \omega_{n,\beta}((t,1]) \leq x,S \right) \geq \mathbb{P}\left( \Lambda_n^k \leq x,S \right).
\]
Thus, letting $n$ then $k$ grows to $\infty$ and using \mathbf{e}qref{eqn:deltato0}, for any $t \in (0,1)$, $\omega_{n,\beta}((t,1])$ converges in law toward $\pi_\beta$ on $S$, proving \mathbf{e}qref{eqn:target}.
\mathbf{e}nd{proof}
Note that this proof can easily be adapted to the convergence of the overlap measure of more than two particles.
\begin{remark}
With similar computations, we can obtain a ``local limit'' convergence for the genealogy of two particles sampled according to the Gibbs measure. In effect, if we consider the non-rescaled measure
\[
\lambda_{n,\beta} = W_{n,\beta}^{-2} \sum_{|u|=|v|=n} e^{\beta (2 m_n - V(u)-V(v))} \delta_{|u \wedge v|},
\]
we obtain $\lim_{n \to \infty} \lambda_{n,\beta} = \lambda_{\infty,\beta}$ in law on $S$, where $(p_k)$ is a Poisson-Dirichlet distribution with parameters $(\beta^{-1},0)$ and $\lambda_{\infty,\beta} = \sum_{k,k'=1}^{\infty} p_k p_{k'} \delta_{|u^{(k)}\wedge u^{(k')}|}$. Note that $\lambda_{\infty,\beta}(\{\infty\}) = \pi_\beta$.
\mathbf{e}nd{remark}
\begin{remark}
\label{rem:gibbssub(cri)tical}
Chauvin and Rouault \cite{CR} studied similarly the overlap of subcritical measures, such that $\beta < 1$. They proved that in this case, the measure $\omega_{n,\beta}$ converges toward $\delta_0$, and the measure $\lambda_{n,\beta}$ converges toward a proper probability measure on $\mathbb{N}$. For the critical case, Pain \cite{Pai17+} proves that if $(\beta_n)$ is a sequence converging to $1$, then $\lim_{n \to \infty} \omega_{n,\beta_n} = \delta_0$ in probability.
\mathbf{e}nd{remark}
\section{The decoration as the close relatives of minimal displacement}
\label{sec:decoration}
In this section, we prove that the law $\mathcal{D}$ is the limiting distribution of the relative positions of the family of the particle that realizes the minimal displacement at time $n$. This result is similar to the one obtained in \cite{ABBS13} for branching Brownian motion. For any $n \in \mathbb{N}$, we denote by $\hat{u}_n$ a particle alive at time $n$ such that $V(u) = M_n$, for example the one which is the smallest for the lexicographical order on $\mathcal{U}$.
\begin{theorem}
\label{thm:abbs}
For any $n \in \mathbb{N}$ and $k < n$, we set
\begin{equation}
\varrho_{n,k} = \sum_{|u|=n} \ind{|u \wedge \hat{u}_n| \geq k} \delta_{V(u)-M_n}.
\mathbf{e}nd{equation}
Under the assumptions of Theorem~\ref{thm:main}, we have
\[\lim_{k \to \infty} \lim_{n \to \infty} \varrho_{n,k} = \lim_{k \to \infty} \tilde{\lim_{n \to \infty}} \varrho_{n,n-k} = D_1 \quad \text{in law,}\]
where $\displaystyle \tilde{\lim_{n \to \infty}} \varrho_{n,n-k}$ represents any accumulation point for the sequence $(\varrho_{n,n-k})$ as $n \to \infty$.
\mathbf{e}nd{theorem}
Observe that by Corollary~\ref{cor:main}, the triangular array $(\varrho_{n,k}, n \geq 1, k \leq n)$ is tight. Indeed, for any continuous positive function $f$, we have
\[
\varrho_{n,k}(f) \leq \varrho_{n,0}(f) = \sum_{|u|=n} f(V(u)-M_n).
\]
A straightforward consequence of Theorem~\ref{thm:abbs} is the following, more intuitive convergence.
\begin{corollary}
\label{cor:abbs}
Let $(k_n)$ be such that $\lim_{n \to \infty} k_n = \lim_{n \to \infty} n - k_n = \infty$. Under the assumptions of Theorem~\ref{thm:main},
\begin{equation}
\lim_{n \to \infty} \varrho_{n,k_n} = D_1 \quad \text{ in law}.
\mathbf{e}nd{equation}
\mathbf{e}nd{corollary}
\begin{proof}
We observe that for any $i \leq j \leq k$, and any continuous positive function $f$, we have $\varrho_{n,i}(f) \geq \varrho_{n,j}(f) \geq \varrho_{n,k}(f)$. Consequently, for any $k \in \mathbb{N}$, for all $n \geq 1$ large enough, we have $\varrho_{n,k}(f) \geq \varrho_{n,k_n}(f) \geq \varrho_{n,n-k}(f)$. Applying Theorem~\ref{thm:abbs}, we have
\[
\lim_{k \to \infty} \limsup_{n \to \infty} \mathbb{P}(\varrho_{n,k_n}(f)-\varrho_{n,k}(f) > \mathbf{e}psilon) = 0 \quad \text{ for any } \mathbf{e}psilon > 0,
\]
which concludes the proof.
\mathbf{e}nd{proof}
The first limit in distribution for Theorem~\ref{thm:abbs} is a straightforward consequence of Fact~\ref{fct:Madaule}.
\begin{lemma}
\label{lem:oneSideabbs}
We have
\begin{equation}
\displaystyle \lim_{k \to \infty} \lim_{n \to \infty} \varrho_{n,k} = D \quad \text{ in law.}
\mathbf{e}nd{equation}
\mathbf{e}nd{lemma}
\begin{proof}
Using Fact~\ref{fct:Madaule}, we observe that for any $k \in \mathbb{N}$, conditionally on $\mathcal{F}_k$,
\[
\lim_{n \to \infty} \left( \sum_{|v|=n,v > u} \delta_{V(v)-m_n}, Z_n^u, u \in \mathbf{T}(k)\right) = \left( \mu^u_\infty, Z^u_\infty, u \in \mathbf{T}(k) \right) \text{ in law on $S$.}
\]
Therefore, setting $M_n^u = \min_{|v|=n, v > u} V(v)$, we have in particular
\[
\lim_{n \to \infty} \sum_{|u|=k} \ind{M_n^u = M_n} \sum_{|v|=n,v>u} \delta_{V(v)-m_n} = \sum_{|u|=k} \ind{u^{(1)}>u} \mu^u_\infty \quad \text{ in law on $S$.}
\]
Observe that $\sum_{|u|=k} \ind{u^{(1)}>u} \mu^u_\infty = \sum_{n=1}^{\infty}\ind{u^{(n)}_k = u^{(1)}_k} \theta_{\xi_n -\log Z_\infty} D_n$.
Let $f$ be a continuous positive function with compact support, we prove that
\begin{equation}
\label{eqn:convergenceSupplementaire}
\lim_{k \to \infty} \sum_{n=2}^{\infty} \ind{u^{(n)}_k = u^{(1)}_k} \sum_{d \in \mathcal{P}(D_n)} f(\xi_n + d-\log Z_\infty) = 0 \quad \text{ in probability}.
\mathbf{e}nd{equation}
In effect, for any $k \in \mathbb{N}$, we have
\begin{multline*}
\E\left( \sum_{n=2}^{\infty} \sum_{d \in \mathcal{P}(D_n)} f(\xi_n+d-\log Z_\infty) \ind{u^{(n)}_k = u^{(1)}_k} \middle| \mathcal{F}_k \right)\\
= \sum_{|u|=k} \bar{\nu}(B(u)) \sum_{n=2}^{\infty} \E\left(\ind{u^{(n)}_k = u} g(\xi_n - \log Z_\infty)\middle|\mathcal{F}_k\right),
\mathbf{e}nd{multline*}
where $g : x \mapsto \E\left( \sum_{d \in \mathcal{P}(D)} f(x + d) \right)$. Therefore,
\begin{multline*}
\E\left( \sum_{n=2}^{\infty} \sum_{d \in \mathcal{P}(D_n)} f(\xi_n+d-\log Z_\infty) \ind{u^{(n)}_k = u^{(1)}_k} \middle| \mathcal{F}_k \right)\\ = \left( \sum_{|u|=k} \bar{\nu}(B(u))^2 \right) \sum_{n=2}^{\infty} \E(g(\xi_n-\log Z_\infty)|\mathcal{F}_k).
\mathbf{e}nd{multline*}
As $\lim_{k \to \infty} \max_{|u|=k} \bar{\nu}(B(u)) = 0$ a.s. (see Lemma~\ref{lem:nonAtomic}), we conclude that \mathbf{e}qref{eqn:convergenceSupplementaire} holds. This result yields that $\lim_{k \to \infty} \sum_{|u|=k} \ind{u^{(1)}>u} \mu^u_\infty(f) = \theta_{\xi_1-\log Z_\infty} D(f)$ in law. We conclude the proof observing that we chose the law of the decoration such that $\min D = 0$ a.s.
\mathbf{e}nd{proof}
To complete the proof of Theorem~\ref{thm:abbs}, we first observe that the genealogy of particles close to the minimal displacement at time $n$ in the branching random walk are either close relatives, or their most recent common ancestor is a close relative to the root. This well-known estimate can be found for example in \cite[Theorem 4.5]{Mal16+}. For any $z \geq 1$, we have
\begin{equation}
\label{eqn:genealogy}
\lim_{k \to \infty} \limsup_{n \to \infty} \mathbb{P}\left( \mathbf{e}xists u,v \in \mathbf{T} : \begin{array}{l}|u|=|v|=n, V(u),V(v) \leq m_n + z,\\ |u\wedge v| \in [k,n-k]\mathbf{e}nd{array} \right) = 0.
\mathbf{e}nd{equation}
\begin{lemma}
\label{lem:otherSideabbs}
For any $k \in \mathbb{N}$, we set $(n^k_p, p \geq 1)$ an increasing sequence such that $(\varrho_{n^k_p,n^k_p-k})$ converges. We have $\lim_{k \to \infty} \lim_{p \to \infty} \varrho_{n^k_p,n^k_p-k} = D$ in law.
\mathbf{e}nd{lemma}
\begin{proof}
For any positive continuous function $f$ with compact support and $k \in \mathbb{N}$, we have
\[
\varrho_{n,k}(f) - \varrho_{n,n-k}(f) = \sum_{|u|=n} \ind{|u\wedge \hat{u}_n| \in [k,n-k]} f(V(u)-M_n).
\]
We write $z = \sup\{x \geq 0 : f(x) > 0 \}$, for any $y \geq 0$, we have
\begin{multline*}
\mathbb{P}\left( \varrho_{n,k}(f) - \varrho_{n,n-k}(f)>0 \right)
\leq \mathbb{P}\left( \mathbf{e}xists u \in \mathbf{T}(n) : \begin{array}{l}|u\wedge \hat{u}_n| \in [k,n-k],\\ V(u)-M_n \leq z\mathbf{e}nd{array} \right)\\
\leq \mathbb{P}(M_n - m_n \geq y) + \mathbb{P}\left( \mathbf{e}xists u,v \in \mathbf{T}(n) : \begin{array}{l}|u\wedge v| \in [k,n-k],\\ V(u),V(v) \leq m_n + y + z \mathbf{e}nd{array} \right).
\mathbf{e}nd{multline*}
Letting $n$ then $k \to \infty$, we have by \mathbf{e}qref{eqn:genealogy},
\begin{equation*}
\limsup_{k \to \infty} \limsup_{n \to \infty} \mathbb{P}\left( \varrho_{n,k}(f) - \varrho_{n,n-k}(f)>0 \right)\leq \sup_{n \in \mathbb{N}} \mathbb{P}(M_n \geq m_n + y).
\mathbf{e}nd{equation*}
Moreover, $(M_n-m_n)$ is tight, by \cite{Aid13}, thus letting $y \to \infty$, we conclude that
\[
\lim_{k \to \infty} \limsup_{n \to \infty} \mathbb{P}\left( \varrho_{n,k}(f) - \varrho_{n,n-k}(f)>0 \right) = 0.
\]
Using Lemma~\ref{lem:oneSideabbs}, we conclude the proof.
\mathbf{e}nd{proof}
We were not able to study the limiting distribution of $\varrho_{n,n-k}$, but this law can probably be constructed, similarly to the point process $\lim_{t \to \infty} \mathscr{Q}(t,\zeta)$ defined in \cite{ABBS13} for the branching Brownian motion.
\begin{conjecture}
For any $k \in \mathbb{N}$, there exists a point measure $\varrho_k$ such that $\lim_{n \to \infty} \varrho_{n,n-k} = \varrho_k$.
\mathbf{e}nd{conjecture}
\appendix
\section{Some technical results}
In this section, we provide some technical estimates on the branching random walks. We first prove that \mathbf{e}qref{eqn:integrability} is equivalent to the usual integrability conditions for the branching random walk.
\begin{lemma}
\label{lem:equivalent}
Under assumptions \mathbf{e}qref{eqn:supercritical}, \mathbf{e}qref{eqn:boundary} and \mathbf{e}qref{eqn:variance}, the condition \mathbf{e}qref{eqn:integrability} is equivalent to
\begin{align*}
&\E\left( \sum_{|u|=n} e^{-V(u)} \log_+ \left(\sum_{|u|=n} e^{-V(u)} \right)^2 \right) < \infty\\
&\E\left( \sum_{|u|=n} V(u)_+ e^{-V(u)} \log_+ \left(\sum_{|u|=n} V(u)_+ e^{-V(u)} \right) \right) < \infty
\mathbf{e}nd{align*}
\mathbf{e}nd{lemma}
\begin{proof}
The reciprocal part is a direct consequence of \cite[Lemma B.1]{Aid13}. To prove the direct part, we first observe that by \mathbf{e}qref{eqn:integrability},
\begin{multline*}
\qquad \qquad \E\left( \sum_{|u|=n} e^{-V(u)} \log_+ \left(\sum_{|u|=n} e^{-V(u)} \right)^2 \right)\\
\leq \E\left( \sum_{|u|=n} e^{-V(u)} \log_+ \left(\sum_{|u|=n} (1 + V(u)_+)e^{-V(u)} \right)^2 \right)< \infty.\qquad \qquad
\mathbf{e}nd{multline*}
We now use the celebrated spinal decomposition of the branching random walk, introduced by Lyons \cite{Lyo97}. Loosely speaking, it is an alternative description of the law of the branching random walk biased by the martingale $(W_n)$, as the law of a branching random walk $(\mathbf{T},V)$ with a distinguished spine $w \in \partial \mathbf{T}$ that makes more children than usual. For any $u \in \mathbf{T}$, we write $\xi(u) = \log_+ \left(\sum_{v \in \Omega(\pi u)} \sum_{|u|=1} V(u)_+ e^{-V(u)}\right)$. We denote by $\hat{\mathbb{P}}=W_n.\mathbb{P}$ the size-biased distribution, and refer to \cite{Lyo97} for more details on the spinal decomposition. We have
\begin{align*}
&\E\left( \sum_{|u|=n} V(u)_+ e^{-V(u)} \log_+ \left(\sum_{|u|=n} V(u)_+ e^{-V(u)} \right) \right)\\
= &\hat{\E}\left( \xi(w_1) V(w_1)_+ \right)\\
\leq &\hat{\E}\left( V(w_1)^2 \right)^{1/2} \hat{\E}\left( \xi(w_1)^2 \right)^{1/2} < \infty,
\mathbf{e}nd{align*}
by Cauchy-Schwarz inequality, using \mathbf{e}qref{eqn:variance} and \mathbf{e}qref{eqn:integrability} to conclude.
\mathbf{e}nd{proof}
We now prove that \mathbf{e}qref{eqn:epsilonto0} holds.
\begin{lemma}
\label{lem:entangled}
For any $\beta > 1$ and $k \leq n$, we set
\[
R_{n,k}^\beta = \sum_{|u|=|v|=n} \ind{|u \wedge v| \in [k,n-k]} e^{\beta(m_n-V(u)) + \beta(m_n-V(v))}.
\]
For any $\mathbf{e}psilon>0$, we have $\displaystyle \lim_{k \to \infty} \limsup_{n \to \infty} \mathbb{P}\left( R_{n,k}^\beta \geq \mathbf{e}psilon \right) = 0$.
\mathbf{e}nd{lemma}
\begin{proof}
To prove this result, we first introduce some notation. For any $u \in \mathbf{T}$, we set
\[
\xi(u) = \log \sum_{|v|=|u|+1,v>u} (1 + (V(v)-V(u))_+ ) e^{V(u)-V(v)}.
\]
For any $n \in \mathbb{N}$ and $k \leq n$, we write $f_n(k) = \frac{3}{2} \log \frac{n+1}{n-k+1}$ and, for $y,z,h \geq 0$,
\begin{align*}
\mathcal{A}_n(y)& = \left\{ |u| \leq n : V(u_j) \geq f_n(j) - y, j \leq |u| \right\},\\
\bar{\mathcal{A}}_n(y,h) &= \left\{ |u| = n : u \in \mathcal{A}_n(y), V(u) - f_n(n) + y \in [h-1,h] \right\}\\
\mathcal{B}_n(y,z)& = \left\{ |u| \leq n : \xi(u_j) \leq z + (V(u_j) - f_n(j) + y)/2 , j \leq |u| \right\}.
\mathbf{e}nd{align*}
We introduce branching random walk estimates obtained in \cite{Mal16+}. There exist $C>0$ and a function $\chi$ such that $\lim_{z \to \infty} \chi(z)=0$ such that for any $k \leq n$ and $y,z,h \geq 1$ we have
\begin{align}
&\mathbb{P}\left( \mathbf{e}xists u, v \in \bar{\mathcal{A}}_n(y,h) \cap \mathcal{B}_n(y,z) : |u \wedge v| \in [k,n-k] \right) \leq C \frac{zyh^2e^{2h-y}}{k^{1/2}},\nonumber\\
&\mathbb{P}\left( \bar{\mathcal{A}}_n(y,h) \cap \mathcal{B}^c_n(y,z) \neq \mathbf{e}mptyset \right) \leq \chi(z)yhe^{h-y},\nonumber \\
&\mathbb{P}\left( \mathcal{A}_n(y) \neq \mathbf{e}mptyset \right) \leq C y e^{-y} \quad \text{ and } \quad \E\left( \# \bar{\mathcal{A}}_n(y,h) \right) \leq C yhe^{h-y}. \label{eqn:listestimates}
\mathbf{e}nd{align}
In the rest of this proof, $C$ is a large positive constant, that depends only on the law of the branching random walk, and may change from line to line.
We decompose $R_{n,k}^\beta$ into three parts, that we bound separately. For any $h \geq 0$, we have
\[
R_{n,k}^\beta \leq \tilde{R}^\beta_{n,k}(h) + 2 W_{n,\beta}(h) W_{n,\beta},
\]
where we write
\[\tilde{R}^\beta_{n,k}(y,h) = \sum_{|u|=|v|=n} \ind{|u \wedge v| \in [k,n-k]} \ind{V(u)-m_n \leq h} e^{\beta(m_n-V(u)) + \beta(m_n-V(v))},\]
and $W_{n,\beta}(h) = \sum_{|u|=n} \ind{V(u)-m_n \geq h} e^{\beta(m_n-V(u))}$.
By \mathbf{e}qref{eqn:listestimates}, for any $y,h \geq 0$, we have
\begin{align*}
&\E\left( \sum_{|u|=n} \ind{u \in \mathcal{A}_n(y),V(u) \geq m_n + h} e^{\beta (m_n - V(u))} \right) \\
= &\sum_{j=h+1}^{\infty} e^{-\beta(j-1)}\E\left( \# \bar{\mathcal{A}}_n(y,j+y) \right)\\
\leq &C y e^{-y} \sum_{j=h+1}^{\infty} (j+y)e^{(1-\beta)j} \leq C y (h+y) e^{(1-\beta)h}.
\mathbf{e}nd{align*}
We have $E\left( \sum_{|u|=n} \ind{u \in \mathcal{A}_n(y)} e^{\beta (m_n - V(u))} \right) \leq C y e^{(\beta-1)y}$ by similar computations. Using the Markov inequality, there exists $C>0$ such that for any $\mathbf{e}psilon \geq 0$ and $y,h \geq 0$, we have
\begin{align*}
\mathbb{P}\left( W_{n,\beta}(h) \geq \mathbf{e}psilon \right) &\leq \mathbb{P}\left( \mathcal{A}_n(y) \neq \mathbf{e}mptyset \right) + \frac{C}{\mathbf{e}psilon} y (h+y) e^{(1-\beta)h}\\
&\leq C y e^{-y}+ \frac{C}{\mathbf{e}psilon} y(h+y)e^{(1-\beta)h},
\mathbf{e}nd{align*}
and similarly for any $A>0$, $\mathbb{P}(W_{n,\beta} \geq A) \leq C ye^{-y} + C y^2 e^{(\beta-1)y}/A$. Thus, for any $\delta > 0$, we have
\begin{align*}
\mathbb{P}\left( W_{n,\beta}(h) W_{n,\beta} \geq \delta \right)
&\leq \mathbb{P}(W_{n,\beta}(h) \geq \delta\mathbf{e}psilon) + \mathbb{P}(W_{n,\beta} \geq 1/\mathbf{e}psilon)\\
&\leq C y e^{-y} + C y (h+y) e^{(1-\beta)h}/(\delta \mathbf{e}psilon) + C \mathbf{e}psilon y e^{(\beta-1)y}.
\mathbf{e}nd{align*}
Choosing $y \geq 1$ large enough, then $\mathbf{e}psilon>0$ small enough and $h$ large enough, we obtain
\[
\sup_{n \in \mathbb{N}} \mathbb{P}\left( W_{n,\beta}(h) W_{n,\beta} \geq 2\delta \right) \leq \delta.
\]
In a second time,we bound $\tilde{R}^\beta_{n,k}$, by observing that for any $y,z \geq 0$,
\begin{align*}
\mathbb{P}(\tilde{R}^\beta_{n,k}(h) \neq 0) &\leq \mathbb{P}(\mathcal{A}_n(y) \neq \mathbf{e}mptyset) + \sum_{j=0}^{h+y}\mathbb{P}\left( \bar{\mathcal{A}}_n(y,j) \cap \mathcal{B}^c_n(y,z) \neq \mathbf{e}mptyset \right)\\
&\qquad + \sum_{j=0}^{h+y}\mathbb{P}\left( \mathbf{e}xists u, v \in \bar{\mathcal{A}}_n(y,j) \cap \mathcal{B}_n(y,z) : |u \wedge v| \in [k,n-k] \right)\\
\leq & C y e^{-y} + \chi(z) y (h+y)e^{h} + C \frac{zy(y+h)^2e^{2h+y}}{k^{1/2}},
\mathbf{e}nd{align*}
using again \mathbf{e}qref{eqn:listestimates}.
As a consequence, for any $\delta > 0$, we can choose $y \geq 1$, $\mathbf{e}psilon>0$, and $h \geq 0$ large enough such that for any $k,z \geq 0$ and $n \geq k$, we have
\[
\mathbb{P}\left( R_{n,k}^\beta \geq \delta \right) \leq \delta + \chi(z)y (h+y)e^{h} + C \frac{zy(y+h)^2e^{2h+y}}{k^{1/2}}.
\]
Setting $z = k^{1/4}$, we obtain
$
\limsup_{k \to \infty} \limsup_{n \to \infty} \mathbb{P}\left( R_{n,k}^\beta \geq \delta \right) \leq \delta,
$
which concludes the proof.
\mathbf{e}nd{proof}
\paragraph*{Acknowledgements.} I wish to thank Thomas Madaule and Julien Barral for many useful discussions, as well as pointing me references \cite{SuZ15} and \cite{BKNSW} respectively.
\newcommand{\mathbf{e}talchar}[1]{$^{#1}$}
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\mathbf{e}nd{document} |
\begin{document}
\baselineskip=17pt
\title{Counting points of fixed degree and bounded height}
\author{Martin Widmer}
\address{Department of Mathematics\\
The University of Texas at Austin\\
1 University Station C1200\\
Austin, Texas 78712\\
U.S.A.}
\email{[email protected]}
\date{May 15, 2009}
\subjclass[2000]{Primary 11R04; Secondary 11G50, 11G35}
\keywords{Height, Northcott's Theorem, counting}
\begin{abstract}
We consider the set of points in projective $n$-space that generate an extension of degree $e$
over given number field $k$, and deduce an asymptotic formula for the number of such points of absolute
height at most $X$, as $X$ tends to infinity. We deduce a similar such formula with instead of the
absolute height, a so-called adelic-Lipschitz height.
\end{abstract}
\maketitle
\section{Introduction}
Let $k$ be a number field of degree $\m=[k:\IQ]$ in a fixed algebraic closure $\kbar$ of $k$
and let $n$ be a positive integer. Write $\IP^n(\kbar)$ for the projective space of dimension $n$ over the field $\kbar$ and
denote by $H(\cdot)$ the non-logarithmic absolute Weil height on $\IP^n(\kbar)$ as defined in \cite{BG} p.16.
A fundamental property of the height, usually associated with the name of Northcott due to his result Theorem 1 in \cite{26},
states that subsets of $\IP^n(\kbar)$ of bounded degree and bounded height are finite.
This raises the question of estimating the cardinality of such a set as the height bound gets large.
Schanuel proved in \cite{25} that for the counting function
\begin{alignat*}1
Z_H(\IP^n(k),X)=|\{P\in \IP^n(k);H(P)\leq X\}|
\end{alignat*}
one has an asymptotic formula
\begin{alignat}1
\label{Scha1}
Z_H(\IP^n(k),X)=S_k(n)X^{\m(n+1)}+O(X^{\m(n+1)-1}\log X)
\end{alignat}
as $X$ tends to infinity
where $S_k(n)$ is a positive constant involving all classical field
invariants (see (\ref{Schanuelconst}) for its definition) and the constant implied
by the Landau $O$-symbol depends on $k$ and $n$.
The logarithm can be omitted in all cases except for
$n=\m=1$.\\
A projective point $P=(\alpha_0:...:\alpha_n)$ in $\IP^n(\kbar)$ has a natural degree
defined as
\begin{alignat*}1
[k(P):k]
\end{alignat*}
where $k(P)$ denotes the
extension we get by adjoining all ratios $\alpha_i/\alpha_j$
$(0\leq i,j\leq n, \alpha_j\neq 0)$ to $k$.
In 1993 Schmidt drew attention to the more general set
\begin{alignat*}1
\IP^n(k;e)=\{P\in \IP^n(\kbar);[k(P):k]=e\}
\end{alignat*}
of points with relative degree $e$.
Clearly $\IP^n(k;1)=\IP^n(k)$ and so Schanuel deals with the case $e=1$.
For the counting function
\begin{alignat*}1
Z_H(\IP^n(k;e),X)=|\{P\in \IP^n(k;e);H(P)\leq X\}|
\end{alignat*}
Schmidt \cite{22} proved the following general estimates
\begin{alignat}1
\label{ThSchm1'}
\cS X^{\m e(\max\{e,n\}+1)}\leq Z_H(\IP^n(k;e),X)&\leq \CS X^{\m e(e+n)}
\end{alignat}
where $c=c(k,e,n)$ and $C=C(k,e,n)$ are positive constants depending solely
on $k,e$ and $n$.
The upper bound holds for $X\geq 0$ and the lower
bound holds for $X\geq X_0(k,e,n)$ depending also on $k,e$ and $n$.
Moreover one can choose $C=2^{\m e(e+n+3)+e^2+n^2+10e+10n}$.\\
For $k=\IQ$ more is known.
Schmidt \cite{14} investigated the quadratic case.
Here he provided not only the correct order of magnitude but he found also the
precise asymptotics and this in all dimensions $n$.
As $X$ tends to infinity one has
\begin{alignat}3\label{ThSchm2}
Z_H(\IP^n(\IQ;2),X)=
\left\{ \begin{array}{llc}
\Ce_1X^{6}+O(X^4\log X)& \mbox{if $n=1$} \\
\Ce_2X^{6}\log X+O(X^6\sqrt{\log X})&
\mbox{if $n=2$}\\
\Ce_nX^{2(n+1)}+O(X^{2n+1})& \mbox{if $n>2$}
\end{array}\right..
\end{alignat}
The constant implied by the $O$-symbol depends only on $n$.
In fact Schmidt's result was more precise since it gave the
asymptotics for real and imaginary quadratic points separately.
Here $\Ce_1=\frac{8}{\zeta(3)}$,
$\Ce_2=\frac{8(12+\pi^2)}{\zeta(3)^2}$ and $\Ce_n=\Ce(\IQ,2,n)$ is given by
\begin{alignat*}1
\Ce(\IQ,2,n)=\sum_K S_K(n)
\end{alignat*}
where the sum runs over
all quadratic fields $K$.
Schmidt proved also an analogue to the above result for a more general kind
of height and showed that this leads to asymptotic formulae for
the number of decomposable quadratic forms (i.e. product of two linear forms)
$f(x_0,...,x_n)=\sum_{0\leq i\leq j\leq n}a_{ij}x_ix_j$
with coefficients $a_{ij}$ in $\IZ$ having $|a_{ij}|\leq X$ and moreover for the number of symmetric $(n+1)\times(n+1)$
matrices with rank $\leq 2$ such that $b_{ii}\in \IZ$, $|b_{ii}|\leq X$
and $2b_{ij}\in \IZ$, $2|b_{ij}|\leq X$ for $i\neq j$.
Already way back in 1967 Schmidt \cite{86} introduced more general classes
of heights where the maximum norm in (\ref{height}) at the infinite places is replaced by an arbitrary but fixed distance function.
More recently Thunder \cite{ThMiHl} and Roy-Thunder \cite{79} introduced ``twisted heights''
which allow also modifications at the finite places.\\
One year after Schmidt's article on quadratic points Gao \cite{7} made further progress.
He proved that if $n>e>2$ one has
\begin{alignat}1
\label{ThGao}
Z_H(\IP^n(\IQ;e),X)=\Ce(\IQ,e,n)X^{e(n+1)}+O(X^{e(n+1)-1})
\end{alignat}
as $X$ tends to infinity. The constant implied by the $O$-symbol depends on
$e$ and $n$ and the constant $\Ce(\IQ,e,n)$ is given by $\sum_K S_K(n)$ where the sum runs over
all extensions $K$ of $\IQ$ of degree $e$.
For $1\leq n\leq e$ Gao showed that the correct order
of magnitude of $Z_H(\IP^n(\IQ;e),X)$ is $X^{e(e+1)}$.
Here the asymptotics are still unknown, even in the case
$e=3$ and $n=2$ of cubic points in two dimensions.\\
Schmidt's and Gao's results are restricted to the ground field $k=\IQ$.
A more recent result of Masser and Vaaler \cite{1}
gives the asymptotics for the number of points of fixed degree over an arbitrary
fixed number field $k$,
but only in dimension $n=1$. Masser and Vaaler established the asymptotic formula
\begin{alignat}1\label{ThMV1}
Z_H(\IP^1(k;e),X)=
eV_{\IR}(e)^{r_k}V_{\IC}(e)^{s_k}S_k(e)X^{\m e(e+1)}+O(X^{\m e(e+1)-e}\log X)
\end{alignat}
as $X$ tends to infinity.
The constants $V_{\IR}(e), V_{\IC}(e)$ have their origins
in \cite{43}. Moreover the logarithm
can be omitted in all cases except $(\m,e)=(1,1)$ and
$(\m,e)=(1,2)$ and the constant implied by the $O$-symbol depends on
$k$ and $e$. Unfortunately the proof of Masser and Vaaler's
theorem shed no light on the case $n>1$.
Very roughly speaking Masser and Vaaler's idea was to interpret the height of the root of an irreducible polynomial in $k[x]$ of fixed degree
$e$ as a suitable height of the coefficient vector
of this polynomial and to proceed by counting minimal
polynomials with respect to this modified height.
To carry out this plan they had to generalize the
class of heights introduced by Schmidt \cite{86}, allowing now
different distance functions at the infinite places
instead of only one for all infinite places as Schmidt did.
On the other hand Masser and Vaaler
had to impose a technical condition, associated
with the name of Lipschitz, on the
boundaries of the unit balls given by the respective
distance function. They therefore introduced so-called
Lipschitz systems, giving what one could call
Lipschitz heights.\\
In the present article we establish the
asymptotics for $\IP^n(k;e)$ if $n$ is slightly larger than $5e/2$.
Let us write
\begin{alignat*}1
\Ce=\Ce(k,e,n)=\sum_K S_K(n)
\end{alignat*}
for the formal sum taken over all extensions of $K$ of $k$ in $\kbar$ of degree $e$.
We have the following result.
\begin{theorem}\label{mainthintro}
Let $e,n$ be positive integers and $k$ a number field
of degree $\m$ and suppose that
$n>5e/2+4+2/(\m e)$.
Then the sum defining $\Ce$ converges and
as $X$ tends to infinity we have
\begin{alignat*}1
Z_{H}(\IP^n(k;e),X)=\Ce X^{\m e(n+1)}
+O(X^{\m e(n+1)-1}\log X).
\end{alignat*}
The logarithm can be omitted unless $(\m e,n)=(1,1)$
and the constant implied by the $O$-symbol depends on
$k,e$ and $n$.
\end{theorem}
If $e$ and $n$ are both larger than one there is a considerable gap between the exponents of the lower and
the upper bound in (\ref{ThSchm1'}). Schmidt mentioned that the lower bound is likely to be nearer the truth
than the upper bound. Our Theorem \ref{mainthintro} confirms Schmidt's conjecture at least if $n$ is large enough.
We will prove a more general result involving adelic-Lipschitz heights.\\
Let us give a single new example of our theorem.
We take $n=11$, $k=\IQ(i)$, $e=2$, so that we
are counting points in eleven dimensions quadratic
over $\IQ(i)$. For the number
$Z=Z_H(\IP^{11}(\IQ(i);2),X)$ of points of height at most $X$, the Schmidt bounds are
$X^{48}\ll Z\ll X^{52}$
for $X\geq X_0$,
with absolute implied constants. Our result implies
that
\begin{alignat*}1
Z=DX^{48}+O(X^{47})
\end{alignat*}
with
\begin{alignat*}1
D=12\cdot(2\pi)^{24}\sum_{K\atop [K:\IQ(i)]=2}\frac{h_KR_K}{\wK \zeta_K(12)|\Delta_K|^6}.
\end{alignat*}
\indent Our proof follows the general strategy of Schmidt and Gao.
Their audacious idea was
to prove a result similar to (\ref{Scha1})
but with $\IP^n(K)$ replaced by $\IP^n(K/\IQ)$ the subset
of primitive points in
$\IP^n(K)$; by definition these satisfy $K=\IQ(P)$. Now $\IP^n(\IQ;e)$ is a disjoint union
of the sets $\IP^n(K/\IQ)$ where $K$ runs over all
number fields of degree $e$.
For each $\IP^n(K/\IQ)$ the
main term is the same as that in (\ref{Scha1}) with $K$ instead of $k$, but for $e=2$
Schmidt obtained a more precise error term
\begin{alignat}1\label{Schmerror1}
O\left(\frac{\sqrt{h_KR_K\log(3+h_KR_K)}}{|\Delta_K|^{n/2}}X^{2n+1}\right)
\end{alignat}
where the constant in $O$ depends only on $n$ but is independent
of the field $K$. This is the major step of the proof
and involves many new ideas.
Now one can sum over all quadratic number fields and the
Theorem of Siegel-Brauer ensures that the
sum over the main terms $S_K(n)$ as well as over
the error terms converges provided $n>2$. For similar reasons the restriction $n>e$ in Gao's result appears.\\
We close the introduction with some remarks about the structure of the paper.
First we take up the definition
of an adelic-Lipschitz system from \cite{art1} on a number field
and we define a uniform adelic-Lipschitz system
on the collection of all extensions of $k$ of degree $e$.
This then gives
rise to a class of heights $\hen$ defined
on $\IP^n(k;e)$.
The main result asymptotically
estimates the counting function of $\IP^n(k;e)$
with respect to the height $\hen$.
In Theorem \ref{mainthintro} we used only the simplest formulation by
choosing a special uniform adelic-Lipschitz system
with maximum norms at all places so that the corresponding adelic-Lipschitz height $\hen$ becomes just the Weil height $H$.
In Section \ref{introchap4} we state our main theorem for general adelic-Lipschitz heights. It is in \cite{art3} and \cite{art4}
where we see the advantage of working in such generality.
In \cite{art3} we are concerned with counting points of fixed degree on linear subvarieties
of projective space. In \cite{art4} we prove the following result:
let $m,n$ be positive integers with $n>\max\{6m+2+2/m,m^2+m\}$. Then
as $X$ tends to infinity the number
of algebraic numbers $\alpha$ of degree $mn$ such that $\IQ(\alpha)$
contains a subfield of degree $m$ and $H(1,\alpha)\leq X$ is
asymptotically equal to
\begin{alignat*}1
\Ce'(m,n) X^{mn(n+1)}
\end{alignat*}
where $\Ce'(m,n)=\sum_K nV_{\IR}(n)^{r_k}V_{\IC}(n)^{s_k}S_k(n)$
and the sum runs over all number fields of degree $m$.
Note that the subfield condition reduces the order of magnitude
from $X^{mn(mn+1)}$ to $X^{mn(n+1)}$.\\
In Section \ref{2subsec5} we prove the main result Theorem \ref{main theorem} which is a
generalization of Theorem \ref{mainthintro} to adelic-Lipschitz heights.
Section \ref{countingfields} is devoted to some simple lower and upper bounds for the number of extensions
$K/k$ of fixed degree with $\delta(K/k)\leq T$, where $\delta(K/k)=\underset{\alpha}\inf\{H(1,\alpha);K=k(\alpha)\}$. The invariant $\delta(K/k)$
was already introduced in \cite{art1}. Our bounds are essentially by-products of the
proof of Theorem \ref{main theorem}.
\section{Adelic-Lipschitz heights revisited}\label{2subsec4}
The Subsections \ref{prelim1}, \ref{subsecdefALS} and \ref{2subsec2} of this section are entirely contained in \cite{art1}.
But adelic-Lipschitz heights are crucial for the entire paper and thus, for convenience of the reader,
we introduce this notion here once again.
Before we can define adelic-Lipschitz heights we have to fix some basic notation.
For a detailed account on heights we refer to \cite{BG} and \cite{3}.\\
\subsection{Preliminaries}\label{prelim1}
Let $K$ be a finite extension
of $\IQ$ of degree $d=[K:\IQ]$. By a place $v$ of $K$ we mean an equivalence class of non-trivial absolute values on $K$.
The set of all places of $K$ will be denoted by $M_K$.
For each $v$ in $M_K$ we write $K_v$ for the completion
of $K$ at the place $v$ and $d_v$
for the local degree defined by
$d_v=[K_v:\IQ_v]$
where $\IQ_v$ is a completion with respect to the place which extends to $v$. A place $v$ in $M_K$ corresponds either to a non-zero prime ideal $\pw_v$
in the ring of integers $\Oseen_K$ or to an embedding
$\sigma$ of $K$ into $\IC$.
If $v$ comes from a prime ideal we call $v$
a finite or non-archimedean place and denote this by $v\nmid \infty$ and if $v$ corresponds to
an embedding we say $v$ is an infinite or archimedean
place and denote this by $v\mid \infty$.
For each place in $M_K$ we choose a representative
$|\cdot|_v$,
normalized in the following way:
if $v$ is finite and $\alpha\neq 0$ we set by convention
\begin{alignat*}3
|\alpha|_{v}=N\pw_v^{-\frac{\ord_{\pw_v}(\alpha\Oseen_K)}{d_v}}
\end{alignat*}
where $N\pw_v$ denotes the norm of $\pw_v$
from $K$ to $\IQ$ and $\ord_{\pw_v}(\alpha\Oseen_K)$
is the power of $\pw_v$ in the prime ideal decomposition
of the fractional ideal $\alpha\Oseen_K$.
Moreover we set
\begin{alignat*}3
|0|_{v}=0.
\end{alignat*}
And if $v$ is infinite and corresponds to an embedding $\sigma:K \hookrightarrow \IC$ we define
\begin{alignat*}3
|\alpha|_{v}=|\sigma(\alpha)|.
\end{alignat*}
If $\alpha$ is in $K^*=K\backslash\{0\}$ then
$|\alpha|_v\neq 1$
holds for only a finite number of places $v$.\\
Throughout this article $n$ will denote a positive rational integer. The height
on $K^{n+1}$ is defined by
\begin{alignat}3
\label{height}
H(\alpha_0,...,\alpha_n)=\prod_{M_K}\max\{|\alpha_0|_v,...,|\alpha_n|_v\}^{\frac{d_v}{d}}.
\end{alignat}
Due to the remark above this is in fact a finite product.
Furthermore this definition is independent of the field $K$ containing
the coordinates (see \cite{BG} Lemma 1.5.2 or \cite{3} pp.51-52) and therefore
defines a height on $\Qbar^{n+1}$ for an algebraic closure
$\Qbar$ of $\IQ$.
The well-known \it product formula \em (see \cite{BG} Proposition 1.4.4) asserts that
\begin{alignat*}3
\prod_{M_K}|\alpha|_v^{d_v}=1 \text{ for each $\alpha$ in $K^*$}.
\end{alignat*}
This implies in particular that the value of the height in (\ref{height}) does not change if we multiply each coordinate with a fixed element of $K^*$.
Therefore one can define a height on points
$P=(\alpha_0:...:\alpha_n)$ in $\IP^n(\Qbar)$ by
\begin{alignat}3
\label{heightproj}
H(P)=H(\alpha_0,...,\alpha_n).
\end{alignat}
Moreover, to evaluate the height, we can assume that one of the coordinates is $1$ which shows that $H(\balf)\geq 1$ for $\balf\in \Qbar^{n+1}\backslash\{\vnull\}$.
The equations (\ref{height}) and (\ref{heightproj}) define the absolute non-logarithmic
projective Weil height or just Weil height.\\
\subsection{Adelic-Lipschitz systems on a number field}\label{subsecdefALS}
Let $r$ be the number of
real embeddings and $s$ the number of pairs of complex conjugate embeddings
of $K$ so that $d=r+2s$.
Recall that $M_K$ denotes the set of places of $K$.
For every place $v$ we fix a
completion $K_v$ of $K$ at $v$. The value set of $v$, $\Gamma_v:=\{|\alpha|_v;\alpha \in K_v\}$
is equal to $[0,\infty)$ if $v$ is archimedean,
and to
\begin{alignat*}3
\{0,(N\pw_v)^{0},(N\pw_v)^{\pm 1/d_v},(N\pw_v)^{\pm 2/d_v},...\}
\end{alignat*}
if $v$ is non-archimedean.
For $v \mid \infty$ we identify $K_v$ with $\IR$ or
$\IC$ respectively and we identify $\IC$ with
$\IR^2$ via $\xi\longrightarrow (\Re(\xi),\Im(\xi))$
where we used $\Re$ for the real and $\Im$ for the
imaginary part of a complex number.\\
Before we can introduce adelic-Lipschitz systems we have to
give a technical definition.
For a vector $\vx$ in $\IR^n$ we write $|\vx|$ for the euclidean length of $\vx$.
\begin{definition}
Let $\M$ and $\Da>1$ be positive integers and let $L$ be a non-negative real.
We say that a set $S$ is in Lip$(\Da,\M,L)$ if
$S$ is a subset of $\IR^\Da$, and
if there are $\M$ maps
$\phi_1,...,\phi_M:[0,1]^{\Da-1}\longrightarrow \IR^\Da$
satisfying a Lipschitz condition
\begin{alignat}3
\label{lipcond1}
|\phi_i(\vx)-\phi_i(\vy)|\leq L|\vx-\vy| \text{ for } \vx,\vy \in [0,1]^{\Da-1}, i=1,...,M
\end{alignat}
such that $S$ is covered by the images
of the maps $\phi_i$.
\end{definition}
We call $L$ a Lipschitz constant for the maps $\phi_i$. By definition the empty set
lies in Lip$(\Da,\M,L)$ for any positive integers $\M$ and $\Da>1$ and any
non-negative $L$.
\begin{definition}[Adelic-Lipschitz system]\label{defALS}
An adelic-Lipschitz system ($\ALS$)
$\en_K$ or simply $\en$ on $K$ (of dimension $n$) is
a set of continuous maps
\begin{alignat}3
\label{Abb1}
N_v: K_v^{n+1}\rightarrow \Gamma_v \quad v \in M_K
\end{alignat}
such that for $v \in M_K$ we have
\begin{alignat*}3
(i)&\text{ } N_v({\vz })=0 \text{ if and only if } {\vz} ={\vnull},\\
(ii)&\text{ } N_v(\omega {\vz})=|\omega|_v N_v({\vz}) \text{ for all
$\omega$ in $K_v$ and all ${\vz}$ in $K_v^{n+1}$},\\
(iii)&\text{ if $v \mid \infty$: }\{{\vz};N_v({\vz})=1\} \text{
is in $Lip(d_v(n+1),\M_v,L_v)$ for some $\M_v, L_v$},\\
(iv)&\text{ if $v \nmid \infty$: }N_v({\vz_1}+{\vz_2})
\leq \max\{N_v({\vz}_1),N_v({\vz}_2)\} \text{ for all
${\vz}_1,{\vz}_2$
in $K_v^{n+1}$}.
\end{alignat*}
Moreover we assume that
\begin{alignat}3
\label{Nvmaxnorm}
N_v(\vz)=\max\{|z_0|_v,...,|z_n|_v\}
\end{alignat}
\end{definition}
for all but a finite number
of $v \in M_K$.
If we consider only the functions $N_v$ for $v\mid\infty$
then we get an $(r,s)$-Lipschitz system (of dimension $n$)
in the sense of Masser and Vaaler \cite{1}.
With $\M_v$ and $L_v$ from $(iii)$ we define
\begin{alignat*}1
\M_{\en}&=\max_{v\mid \infty}\M_v,\\
L_{\en}&=\max_{v\mid \infty}L_v.
\end{alignat*}
We say that $\en$
is an $\ALS$ with associated constants $\M_{\en}, L_{\en}$.
The set defined in $(iii)$ is the boundary
of the set ${\bf B}_v=\{{\vz};N_v({\vz})<1\}$
and therefore ${\bf B}_v$ is a bounded symmetric open star-body
in $\IR^{n+1}$ or $\IC^{n+1}$ (see also \cite{1} p.431). In particular ${\bf B}_v$ has a finite volume $V_v$.\\
Let us consider the system
where $N_v$ is as in (\ref{Nvmaxnorm}) for all places $v$.
If $v$ is an infinite place then
${\bf B}_v$ is a
cube for $d_v=1$ and the complex analogue
if $d_v=2$. Their boundaries are clearly
in Lip$(d_v(n+1),M_v,L_v)$ most naturally
with $M_v=2n+2$ maps and $L_v=2$
if $d_v=1$ and
with $M_v=n+1$ maps and for example $L_v=2\pi\sqrt{2n+1}$
if $d_v=2$.
This system is called the standard
adelic-Lipschitz system.\\
We return to arbitrary adelic-Lipschitz systems.
We claim that for any $v\in M_K$ there is a $c_v$ in the value group
$\Gamma_v^*=\Gamma_v\backslash\{0\}$ with
\begin{alignat}3
\label{Nineq1}
N_v({\vz})\geq c_v\max\{|z_0|_v,...,|z_n|_v\}
\end{alignat}
for all $\vz=(z_0,...,z_n)$ in $K_v^{n+1}$.
For if $v$ is archimedean then ${\bf B}_v$ is
bounded open and it contains the origin.
Since $\Gamma_v^*$ contains arbitrary small
positive numbers the
claim follows by $(ii)$.
Now for $v$ non-archimedean $N_v$ and $\max\{|z_0|_v,...,|z_n|_v\}$ define norms on
the vector space $K_v^{n+1}$ over the complete field $K_v$.
But on a finite dimensional vector space over a complete
field all norms are equivalent (\cite{2} Corollary 5. p.93)
hence (\ref{Nineq1}) remains true for a suitable choice
of $c_v$.\\
So let $\en$ be an $\ALS$ on $K$ of dimension $n$. For
every $v$ in $M_K$ let
$c_v$ be an element of $\Gamma_v^*$,
such that $c_v\leq 1$ and (\ref{Nineq1}) holds.
Due to (\ref{Nvmaxnorm}) we can assume that $c_v\neq 1$ only for a finite number of places $v$.
We define
\begin{alignat}3
\label{defcfin}
C^{fin}_{\en}&=\prod_{v\nmid \infty}c_v^{-\frac{d_v}{d}}\geq 1
\end{alignat}
and
\begin{alignat}3
\label{defcinf}
C^{inf}_{\en}&=\max_{v\mid \infty}\{c_v^{-1}\}\geq 1.
\end{alignat}
Multiplying the finite and the infinite part
gives rise to another constant
\begin{alignat}3
\label{defc}
C_{\en}&=C^{fin}_{\en}C^{inf}_{\en}.
\end{alignat}
It will turn out that besides $\M_{\en}$ and $L_{\en}$ this is another important quantity for an $\ALS$. So we say that \it $\en$
is an $\ALS$ with associated constants $C_{\en},\M_{\en},L_{\en}$.\rm
\begin{remark}\label{normconvex}
Let $v$ be an infinite place.
Suppose $N_v:K_v^{n+1}\longrightarrow [0,\infty)$
defines a norm, so that
$N_v({\vz_1}+{\vz_2})
\leq N_v({\vz}_1)+N_v({\vz}_2)$. Then
${\bf B}_v$ is convex and (\ref{Nineq1}) combined with
(\ref{defcfin}), (\ref{defcinf}) and (\ref{defc}) shows that ${\bf B}_v$ lies in
$B_0(C_{\en}\sqrt{n+1})$.
This implies (see Theorem A.1 in \cite{WiThesis}) that $\partial{\bf B}_v$ lies in Lip$(d_v(n+1),1,8{d_v}^2(n+1)^{5/2}C_{\en})$.
\end{remark}
We denote by $\sigma_1,...,\sigma_d$
the embeddings from $K$ to $\IR$ or
$\IC$ respectively, ordered such that
$\sigma_{r+s+i}=\overline{\sigma}_{r+i}$ for
$1\leq i \leq s$.
We define
\begin{alignat}3
\label{sigd}
&\sigma:K\longrightarrow \IR^r\times\IC^s\\
\nonumber&\sigma(\alpha)=(\sigma_1(\alpha),...,\sigma_{r+s}(\alpha)).
\end{alignat}
Sometimes it will be more readable to omit the brackets
and simply to write $\sigma\alpha$.
We identify $\IC$ in the usual way with $\IR^2$
and extend $\sigma$ componentwise to get a map
\begin{alignat}3
\label{sigD}
\sigma:K^{n+1}\longrightarrow \IR^{D}
\end{alignat}
where $D=d(n+1)$.
On $\IR^{D}$ we use $|\cdot|$ for the
usual euclidean norm. For $v\in M_K$
let $\sigma_v$ be the canonical embedding of $K$ in $K_v$,
again extended componentwise on $K^{n+1}$.
\begin{definition}
Let $\D\neq 0$ be a fractional ideal in $K$ and $\en$
an $\ALS$ of dimension $n$. We define
\begin{alignat}3
\label{defLamen}
{\mathfrak{L}}amen(\D)=\{\sigma(\balf); \balf \in K^{n+1},
N_v(\sigma_v\balf)\leq |\D|_v \text{ for all finite }v \}
\end{alignat}
where $|\D|_v=N\pw_v^{-\frac{\ord_{\pw_v}\D}{d_v}}$.
\end{definition}
It is easy to see that
${\mathfrak{L}}amen(\D)$ is an additive subgroup of $\IR^D$.
Now assume $B\geq 1$ and
$|\sigma(\balf)|\leq B$; then (\ref{Nineq1}) implies
$H(\balf)^d\leq (BC_{\en}^{fin})^d N\D^{-1}$ and by Northcott's Theorem
we deduce that ${\mathfrak{L}}amen(\D)$ is discrete.
The same argument as for (\ref{Nineq1}) yields positive real numbers $C_v$, one for each non-archimedean place $v\in M_K$,
with $N_v({\vz})\leq C_v\max\{|z_0|_v,...,|z_n|_v\}$ for all $\vz=(z_0,...,z_n)$ in $K_v^{n+1}$
and $C_v=1$ for all but finitely many non-archimedean $v\in M_K$.
Thus there exists an ideal $\C_1\neq 0$ in $\Oseen_K$ with
$|\C_1|_v\leq 1/C_v$ for all non-archimedean places $v\in M_K$.
This means that $\sigma(\C_1\D)^{n+1}\subseteq {\mathfrak{L}}amen(\D)$.
It is well-known that the additive group $\sigma(\C_1\D)^{n+1}$ has maximal rank in $\IR^D$.
Therefore ${\mathfrak{L}}amen(\D)$ is a discrete additive subgroup of $\IR^D$
of maximal rank. Hence ${\mathfrak{L}}amen(\D)$
is a lattice.
Notice that for $\varepsilon$ in $K^*$ one has
\begin{alignat}3
\label{deltawelldef1}
\det{\mathfrak{L}}amen((\varepsilon)\D)=
|N_{K/\IQ}(\varepsilon)|^{n+1}\det{\mathfrak{L}}amen(\D).
\end{alignat}
Therefore
\begin{alignat}3
\label{idkl}
\Delta_{\en}(\mathcal{D})=\frac{\det{\mathfrak{L}}ambda_{\en}(\D)}
{N\D^{n+1}}
\end{alignat}
is independent of the choice of the representative $\D$
but depends only on the ideal class $\mathcal{D}$ of $\D$.
Let $\Cl_K$ denote the ideal class group of $K$.
We define
\begin{alignat}3
\label{defVfin1}
V_{\en}^{fin}=2^{-s(n+1)}|\Delta_K|^{\frac{n+1}{2}}h_K^{-1}
\sum_{\mathcal{D} \in \Cl_K}\Delta_{\en}(\mathcal{D})^{-1}
\end{alignat}
for the finite part, where as usual,
$s$ denotes the number of pairs of complex conjugate
embeddings of $K$, $h_K$ the class number
of $K$ and $\Delta_K$ the discriminant of $K$.
The infinite part is defined by
\begin{alignat*}3
V_{\en}^{inf}=\prod_{v \mid \infty}V_{v}.
\end{alignat*}
By virtue of (\ref{Nineq1}) we observe that
\begin{alignat}3
\label{Vinfbou1}
V_{\en}^{inf}=\prod_{v|\infty} V_v\leq
\prod_{v|\infty}(2 C^{inf}_{\en})^{d_v(n+1)}=
(2 C^{inf}_{\en})^{d(n+1)}.
\end{alignat}
We multiply the finite and the infinite part
to get a global volume
\begin{alignat}3
\label{defVen}
V_{\en}=V_{\en}^{inf}V_{\en}^{fin}.
\end{alignat}
\subsection{Adelic-Lipschitz heights on $\mathbb{P}^n(K)$}\label{2subsec2}
Let $\en$ be an $\ALS$ on $K$
of dimension $n$. Then the height $\hen$
on $K^{n+1}$ is defined by
\begin{alignat*}3
\hen(\balf)=\prod_{v \in M_K} N_v(\sigma_v(\balf))^{\frac{d_v}{d}}.
\end{alignat*}
Thanks to the product formula
and $(ii)$ from Subsection \ref{subsecdefALS} $\hen(\balf)$ does not change
if we multiply each coordinate of $\balf$ with a fixed element of $K^*$.
Therefore $\hen$ is well-defined on $\IP^n(K)$ by setting
\begin{alignat*}3
\hen(P)=\hen(\balf)
\end{alignat*}
where $P=(\alpha_0:...:\alpha_n) \in \IP^n(K)$ and $\balf=(\alpha_0,...,\alpha_n) \in K^{n+1}$.
\begin{remark}
Multiplying (\ref{Nineq1}) over all places with
suitable multiplicities yields
\begin{alignat}3
\label{HAquiv}
\hen(P)\geq C_{\en}^{-1} H(P)
\end{alignat}
for $P\in \IP^n(K)$.
Thanks to Northcott's Theorem it follows that
$\{P\in \IP^n(K); \hen(P)\leq X\}$ is a finite set for
each $X$ in $[0,\infty)$.
\end{remark}
\subsection{Adelic-Lipschitz systems on a collection of number fields}\label{subsec1.1}
Recall that $k$ is a number field of degree $\m$ and $\kbar$ is an algebraic closure of $k$.
We fix $k$ and $\kbar$ throughout and assume finite extensions of $k$ to lie in $\kbar$.
Let $\coll$ be a collection of finite extensions of $k$. We
are especially interested in the set of all extensions
of fixed relative degree. We denote it by
\begin{alignat*}1
\coll_e=\coll_e(k)=\{K\subseteq \kbar;[K:k]=e\}.
\end{alignat*}
Let
$\en$ be a collection of adelic-Lipschitz systems $\en_K$ of dimension $n$ - one for each $K$ of $\coll$. Then we call $\en$ an \em adelic-Lipschitz system $(\ALS)$ on
$\coll$ of dimension $n$. \rm
We say $\en$ is a \em uniform \rm $\ALS$ on $\coll$
of dimension $n$
with associated constants $C_{\en},\M_{\en},L_{\en}$ in $\IR$ if
the following holds:
for each $\ALS$ $\en_K$ of the collection $\en$
we can choose associated constants
$C_{\en_K},\M_{\en_K},L_{\en_K}$ satisfying
\begin{alignat*}1
C_{\en_K}\leq C_{\en},\quad
\M_{\en_K}\leq \M_{\en},\quad
L_{\en_K}\leq L_{\en}.
\end{alignat*}
Notice that a uniform $\ALS$ $\en$ (of dimension $n$) on the collection consisting only of a single field $K$ with associated constants $C_{\en},\M_{\en},L_{\en}$ is simply an $\ALS$
$\en$ (of dimension $n$) on $K$ with associated constants
$C_{\en},\M_{\en},L_{\en}$ in the sense of Subsection \ref{subsecdefALS}.\\
A standard example for a uniform $\ALS$ on $\coll_e$ (of dimension $n$) is given as
follows: for each $K$ in $\coll_e$ choose the standard $\ALS$ on $K$ (of dimension $n$)
so that $N_v$ is as in (\ref{Nvmaxnorm}) for each
$v$ in $M_K$. For this system we may choose
$C_{\en}=1$,
$\M_{\en}=2n+2$ and $L_{\en}=2\pi\sqrt{2n+1}$.
Choosing $l^2$-norms at all infinite places
and $N_v$ as in (\ref{Nvmaxnorm}) for all finite places yields another important uniform $\ALS$.
\subsection{Adelic-Lipschitz heights on $\mathbb{P}^n(k;e)$}\label{ALHoncoll}
Let $\coll$ be a collection of finite extensions of $k$ and let $\en$ be an $\ALS$ of dimension $n$ on $\coll$.
Now we can define heights on $\IP^n(K/k)$ (the set of points $P\in\IP^n(K)$ with $k(P)=K$)
for $K$ in $\coll$.
Let $P\in \IP^n(K/k)$.
According to Subsection \ref{2subsec2} we know that $H_{\en_{K}}(\cdot)$
defines a projective height on $\IP^n(K)$.
Now we define
\begin{alignat}1
\label{defALH2}
\hen(P)=H_{\en_{K}}(P).
\end{alignat}
From Subsection \ref{2subsec2} we know
\begin{alignat}1
\label{defALH3}
H_{\en_{K}}(P)=\prod_{v\in M_{K}} N_v(\sigma_v(\balf))^{\frac{d_v}{d}}
\end{alignat}
for the functions $N_v$ of $\en_{K}$ and
$[K:\IQ]=d$, $[K_v:\IQ_v]=d_v$. Starting with $\coll=\coll_e$
we get a height defined on $\IP^n(k;e)$.
\section{The main result}\label{introchap4}
Let $\en$ be an $\ALS$ on $\coll_e$ of dimension $n$.
Then $\hen(\cdot)$ defines a height on $\IP^n(k;e)$, the set of points
$P=(\alpha_0:...:\alpha_n)$ in $\IP^n(\kbar)$ with $[k(P):k]=e$
where $k(P)=k(...,\alpha_i/\alpha_j,...)$
for $0\leq i,j\leq n$, $\alpha_j\neq 0$.
The associated counting function $Z_{\en}(\IP^n(k;e),X)$ denotes the number of points
$P$ in $\IP^n(k;e)$ with $\hen(P)\leq X$. Assume $\en$ is a
uniform $\ALS$ on $\coll_e$
(of dimension $n$). Then due to Northcott and
(\ref{HAquiv}) $Z_{\en}(\IP^n(k;e),X)$
is finite for all $X$ in $[0,\infty)$.
The Schanuel constant $S_K(n)$ is defined as follows
\begin{alignat}1\label{Schanuelconst}
S_K(n)=\frac{h_KR_K}{w_K\zeta_K(n+1)}
(\frac{2^{r_K}(2\pi)^{s_K}}{\sqrt{|\Delta_K|}})^{n+1}
(n+1)^{r_K+s_K-1}.
\end{alignat}
Here $h_K$ is the class number, $R_K$ the regulator,
$w_K$ the number of roots of unity in $K$, $\zeta_K$
the Dedekind zeta-function of $K$, $\Delta_K$ the discriminant,
$r_K$ is the number of real embeddings of $K$ and $s_K$ is
the number of pairs of distinct complex conjugate embeddings of $K$.
Recall also the definition of $V_{\en_K}$ (see (\ref{defVen})).
Now we define the sum
\begin{alignat}1
\label{kkonst}
\Ce_{\en}=\Ce_{\en}(k,e,n)=\sum_{K}2^{-r_K(n+1)}\pi^{-s_K(n+1)}V_{\en_K}S_K(n)
\end{alignat}
where the sum runs over all extensions of $k$
with relative degree $e$.
We will prove that the sum in (\ref{kkonst})
converges if $n$ is large enough compared to $e$.
It will be convenient to use Landau's $O$-notation.
For non-negative real functions $f(X), g(X), h(X)$ we say that
$f(X)=g(X)+O(h(X))$ as $X>X_0$ tends to infinity if there is a constant $C_0$ such that
$|f(X)-g(X)|\leq C_0h(X)$ for each $X>X_0$.\\
After all this we are ready to state the main result.
\begin{theorem}\label{main theorem}
Let $e,n$ be positive integers and $k$ a number field
of degree $\m$. Suppose $\en$ is a uniform adelic-Lipschitz system of dimension $n$
on $\coll_e$, the collection of all finite extensions of $k$ of relative degree $e$, with associated constants $C_{\en},\M_{\en}$ and $L_{\en}$. Write
\begin{alignat}1\label{defAN}
A_{\en}&=\M_{\en}^{\m e}(C_{\en}(L_{\en}+1))^{\m e(n+1)-1}.
\end{alignat}
Suppose that either $e=1$ or
\begin{alignat}1\label{necond}
n>{5e}/{2}+4+2/(\m e).
\end{alignat}
Then the sum in (\ref{kkonst}) converges and
as $X>0$ tends to infinity we have
\begin{alignat*}1
Z_{\en}(\IP^n(k;e),X)=\Ce_{\en}X^{\m e(n+1)}
+O(A_{\en}X^{\m e(n+1)-1}{\mathfrak{L}}),
\end{alignat*}
where ${\mathfrak{L}}=\log\max\{2,2C_{\en}X\}$ if
$(\m e,n)=(1,1)$ and ${\mathfrak{L}}=1$ otherwise.
The constant in $O$ depends only on
$k,e$ and $n$.
\end{theorem}
In subsequent papers \cite{art3} and \cite{art4} we will explore some
applications of Theorem \ref{main theorem}. Here we are content
with some immediate consequences.
For $e=1$ we recover a version of the
Proposition in \cite{1}, which allows more general norms at the finite places (this generalization will
be essential to deduce the results of \cite{art3}).
Now choose the standard uniform $\ALS$ as described at the end of Subsection \ref{subsec1.1} so that $\hen$ becomes the Weil height. Schanuel's Theorem
implies $S_K(n)=\Ce_{\en}(K,1,n)=2^{-r_K(n+1)}\pi^{-s_K(n+1)}V_{\en_K}S_K(n)$. We can verify
\begin{alignat}1
\label{VenWeilheight}
V_{\en_K}=2^{r_K(n+1)}\pi^{s_K(n+1)}
\end{alignat}
directly by noting
that ${\mathfrak{L}}amen(\D)=(\sigma\D)^{n+1}$ in (\ref{defLamen}),
so that $\det {\mathfrak{L}}amen(\D)=(2^{-s_K}N\D\sqrt{|\Delta_K|})^{n+1}$
(see \cite{1} Lemma 5).
Inserting the latter in definition (\ref{defVfin1}) we get $V_{\en_K}^{fin}=1$ and it is clear that
$V_{\en_K}^{inf}=2^{r_K(n+1)}\pi^{s_K(n+1)}$.
Then (\ref{VenWeilheight}) follows from $V_{\en_K}=V_{\en_K}^{inf}V_{\en_K}^{fin}$
and so we find Theorem \ref{mainthintro} from the introduction.
For $k=\IQ$ and $e=2$ we recover essentially Schmidt's theorem (\ref{ThSchm2}) but only for $n>10$ while Schmidt does it for all
$n\geq 3$ and even (in a modified form) for $n=1,2$. For $k=\IQ$ and $e>2$ we find Gao's result (\ref{ThGao})
but again with the stronger restriction $n>5e/2+4+2/(\m e)$
instead of Gao's $n>e$.\\
It is likely that Theorem \ref{main theorem} is valid for $n>e$ instead of (\ref{necond}).
Gao showed, at least for his definition of height (see also
\cite{WiThesis} Appendix B),
that for $k=\IQ$
the bound $n>e$ suffices. On the other hand Schmidt's
lower bound in (\ref{ThSchm1'}) implies that
Theorem \ref{main theorem} cannot hold for $e>1$ and $n<e$.
However, there is a good possibility of obtaining
the asymptotics for $e>1$ and $n=1$ using a kind
of generalized Mahler measure.
\section{Proof of the main result}\label{2subsec5}
The major part of the work was already done in \cite{art1} where
we proved estimates for $Z_{\en_K}(\IP^n(K/k),X)$. These estimates will be essential to deduce
Theorem \ref{main theorem}.
\subsection{Preliminaries}\label{prelim}
Let $K$ be in $\coll_e$. Then by definition $\hen(P)=\henK(P)$
for all $P$ in $\IP^n(K/k)$.
Since
\begin{alignat}1
\label{Gl1.5.1}
\IP^n(k;e)=\bigcup_{K\in \coll_e}\IP^n(K/k)
\end{alignat}
where the right hand side is a disjoint union,
we get
\begin{alignat}1
\label{Gl1.5.2}
Z_{\en}(\IP^n(k;e),X)=
\sum_{K \in \coll_e}Z_{\en_K}(\IP^n(K/k),X).
\end{alignat}
To state the estimates for $Z_{\en_K}(\IP^n(K/k),X)$ from \cite{art1} we are forced to introduce some more notation.
For fields $k,K$ with $k\subseteq K$ and $[K:k]=e$ we define
\begin{alignat*}3
G(K/k)=
\{[K_0:k]; \text{$K_0$ is a field with $k\subseteq K_0\subsetneq K$}\}
\end{alignat*}
if $k\neq K$, and we define
\begin{alignat*}3
G(K/k)=\{1\}
\end{alignat*}
if $k=K$. Clearly $|G(K/k)|\leq e$.
Then for an integer $g\in G(K/k)$ we define
\begin{alignat}3
\label{delta2}
\delta_g(K/k)=
\underset{\alpha,\beta}\inf\{H(1,\alpha,\beta); k(\alpha,\beta)=K,
[k(\alpha):k]=g\}
\end{alignat}
(which is $\geq 1$) and
\begin{alignat}3
\label{mu2}
\mu_g=m(e-g)(n+1)-1.
\end{alignat}
In \cite{art1} the author proved the following result.
\begin{theorem}\label{prop3}
Let $k,K$ be number fields with $k\subseteq K$ and $[K:k]=e$, $[k:\IQ]=m$, $[K:\IQ]=d$. Let $\en_K$ be an adelic-Lipschitz system of dimension $n$
on $K$ with associated constants $C_{\en_K},L_{\en_K},\M_{\en_K}$.
Write
\begin{alignat}3
\label{defANK}
A_{\en_K}&=\M_{\en_K}^{d}(C_{\en_K}(L_{\en_K}+1))^{d(n+1)-1},\\
\label{mainterm1}
\Mainterm&=2^{-r_K(n+1)}\pi^{-s_K(n+1)}V_{\en_K}S_K(n),\\
\label{errorterm1}
\Errorterm&=A_{\en_K}R_Kh_K\sum_{g\in G(K/k)}\delta_{g}(K/k)^{-\mu_g}.
\end{alignat}
Then as $X>0$ tends to infinity we have
\begin{alignat*}3
Z_{\en_K}(\IP^n(K/k),X)=
\Mainterm X^{d(n+1)}
+O(\Errorterm X^{d(n+1)-1}{\mathfrak{L}}),
\end{alignat*}
where
\begin{alignat*}3
{\mathfrak{L}}&=\log\max\{2,2C_{\en_K}X\} \text{ if }(n,d)=(1,1)\text{ and }{\mathfrak{L}}=1 \text{ otherwise}
\end{alignat*}
and the implied constant in $O$ depends only on $n$ and $d$.
\end{theorem}
Thanks to (\ref{Gl1.5.2}) and Theorem \ref{prop3} it
suffices to show that $\sum \Mainterm$ and $\sum \Errorterm$
are convergent
(here the sum runs over the same fields as in (\ref{kkonst}) and
(\ref{Gl1.5.2})).\\
We will also deal with $\delta(\cdot)$, a simplified version of $\delta_g(\cdot)$
\begin{alignat*}3
\delta(K/k)=\underset{\alpha}\inf\{H(1,\alpha);K=k(\alpha)\}.
\end{alignat*}
The quantity $\delta(K/\IQ)$ was already introduced by Roy and Thunder \cite{8}.\\
It will be extremely convenient to use Vinogradov's $\ll,\gg$-notation.
The constants involved in $\ll$ and $\gg$ will depend only on
$k,n,e$ unless we indicate the dependence on additional parameters by an index.\\
The case $e=1$ of Theorem \ref{main theorem} is already covered by Theorem \ref{prop3} by choosing $K=k$.
For the rest of this article we assume
\begin{alignat*}1
e>1.
\end{alignat*}
For a non-zero ideal $\A$ in $K$ let $D_{K/k}(\A)$
be the discriminant-ideal of $\A$ relative to $k$ (for definitions see \cite{23} p.212 or \cite{13})
and write $D_{K/k}$ for $D_{K/k}(\Oseen_K)$ where $\Oseen_K$ denotes the ring of integers in $K$.
By assumption we have $\IQ\subseteq k \subseteq K$ and hence by \cite{23} (2.10) Korollar p.213
\begin{alignat}1
\label{dis}
|\Delta_{K/\IQ}|=|\Delta_{k/\IQ}|^{[K:k]}N_{k/\IQ}(D_{K/k})
\end{alignat}
where $N_{k/\IQ}(a)$ denotes the absolute norm of an ideal $a\neq 0$ of the ring of integers
$\Oseen_k$, i.e. $N_{k/\IQ}(a)=|\Oseen_k/a|$.
Let $P$ be in $\IP^n(K/k)$, so
$K=k(P)$. We use a theorem of Silverman (\cite{9} Theorem 2)
with Silverman's $S_F$ (for $F=k$) as the set of archimedean absolute values.
Then Silverman's $L_F(\cdot)$ is simply the usual norm
$N_{k/\IQ}(\cdot)$. Hence we deduce
\begin{alignat}1
\label{sil}
H(P)^{\m }\geq \exp \left(-\frac{\delta_k\log e}{2(e-1)}\right)
N_{k/\IQ}(D_{K/k})^{\frac{1}{2e(e-1)}}
\end{alignat}
where $\delta_k$ is the number of archimedean places
in $M_k$. Since Silverman uses not an absolute height
but rather an ``absolute height relative to $k$'', we had
to take the $\m $-th power on the left hand side of
(\ref{sil}).
Combining (\ref{dis}) and (\ref{sil}) yields
\begin{alignat}1
\label{sil2}
H(P)&\geq \exp \left(-\frac{\delta_k\log e}{2(e-1)\m}\right)
|\Delta_k|^{-\frac{1}{2(e-1)\m }}
|\Delta_K|^{\frac{1}{2e(e-1)\m }}\\
\nonumber&\gg |\Delta_K|^{\frac{1}{2e(e-1)\m }}.
\end{alignat}
Recalling the definitions of $\delta$, $\delta_g$ and
$G(K/k)$ and taking $P=(1:\alpha_1:\alpha_2)$ in $\IP^2(K/k)$ we get
\begin{alignat}1
\label{de1}
\delta_{g}(K/k)
\gg |\Delta_K|^{\frac{1}{2e(e-1)\m }}
\end{alignat}
for any $g\in G(K/k)$; and similarly
\begin{alignat}1
\label{de11}
\delta(K/k)
\gg |\Delta_K|^{\frac{1}{2e(e-1)\m }}.
\end{alignat}
Here it might be worthwile to point out that
(\ref{sil2}) can be used to prove
a version of Theorem \ref{prop3} where
$\Errorterm$ is redefined in terms of the discriminants; namely
\begin{alignat}1
\label{discerrorterm}
\Errorterm=A_{\en_K}R_Kh_K\sum_{g\in G(K/k)}(|\Delta_k|^{-e}|\Delta_K|)^{-\frac{\mu_g}{2e(e-1)\m}}.
\end{alignat}
At a first glance this error term looks more appropriate
due to the unavoidable appearance of $\Delta_K$ in the main term.
But as it turns out, the summation over $\Delta_K$ instead of over $\delta_g(K/k)$ leads to a result
weaker than Theorem \ref{main theorem}, in which we have to assume that $n$ exceeds some quadratic function of $e$ instead
of (\ref{necond}). The reason for this is, that we have estimates for the number of number fields $K$ with $\delta_g(K/k)\leq T$
which are more accurate than the best available estimates for the number of number fields with $|\Delta_K|\leq T$, see Section \ref{countingfields}
for a discussion on this.
Thanks to the well-known Theorem of Siegel-Brauer
(\cite{13} p.328 Corollary or \cite{24} p.67 Satz 1 for
a more precise version) we
can use the inequalities (\ref{de1}) and (\ref{de11}) to bound the product of regulator and class number.
More precisely we have
\begin{alignat}1
\label{de2}
R_Kh_K \ll_{\beta} \delta_{g}(K/k)^{\beta}
\end{alignat}
and
\begin{alignat}1
\label{de22}
R_Kh_K \ll_{\beta} \delta(K/k)^{\beta}.
\end{alignat}
for any $\beta>e(e-1)\m $ and any $g\in G(K/k)$.
\subsection{Three preparatory lemmas}
We start with a very simple argument, known as
dyadic summation. Since it will be frequently used we state it as a lemma.
\begin{lemma}[Dyadic summation]\label{dyadicsum}
Let $\coll$ be a non-empty subset of $\coll_e$ and let $\f$ be a map $\f:\coll \longrightarrow [1,\infty)$.
Write $N_{\f}(T)=|\{K\in \coll;\f(K)\leq T\}|$ and
suppose there
are nonnegative real numbers $b,c$ (independent of $T$)
with
\begin{alignat*}1
N_{\f}(T)\leq cT^b
\end{alignat*}
for every $T>0$.
Let $\coll'$ be a non-empty subset of $\coll$.
Set $\mathfrak{M}=[\log_2\max_{\coll'}\f(K)]+1$ if
$\coll'$ is finite and $\mathfrak{M}=\infty$
otherwise.
Moreover
suppose $\alpha$ is a real number such that
$\sum_{i=1}^{\mathfrak{M}}2^{i(\alpha+b)}$ converges.
Then we have
\begin{alignat*}1
\sum_{K\in \coll'}\f(K)^{\alpha}\leq c 2^{|\alpha|}\sum_{i=1}^{\mathfrak{M}}2^{i(\alpha+b)}.
\end{alignat*}
\end{lemma}
\begin{rproof}
From the definition of $\mathfrak{M}$ and since $\coll'\subseteq \coll$ we have
\begin{alignat*}1
\sum_{K\in \coll'}\f(K)^{\alpha}=
\sum_{i=1}^{\mathfrak{M}}\sum_{K\in \coll' \atop 2^{i-1}\leq \f(K)<2^i}
\f(K)^{\alpha}\leq \sum_{i=1}^{\mathfrak{M}}\sum_{K\in \coll \atop 2^{i-1}\leq \f(K)<2^i}
\f(K)^{\alpha}.
\end{alignat*}
First suppose $\alpha<0$. Then the latter is
\begin{alignat*}1
\leq \sum_{i=1}^{\mathfrak{M}}2^{(i-1)\alpha}N_{\f}(2^i)
\leq
c 2^{-\alpha}\sum_{i=1}^{\mathfrak{M}}2^{i(\alpha+b)}.
\end{alignat*}
If $\alpha\geq 0$ then we even get
\begin{alignat*}1
\sum_{K\in \coll'}\f(K)^{\alpha}\leq
c\sum_{i=1}^{\mathfrak{M}}2^{i(\alpha+b)}.
\end{alignat*}
This proves the lemma.
\end{rproof}
Recall the definition of $G(K/k)$ from Subsection \ref{prelim}.
In our applications $\f$ will be $\delta_g$ and
$\coll$ will be
\begin{alignat*}1
\collg=\{K\in \coll_e; g\in G(K/k)\}
\end{alignat*}
the set of extensions
$K$ of $k$ of relative degree $e$
containing an intermediate field $K_0\subsetneq K$ with $[K_0:k]=g$.
Let $G_u$ be the union of all sets $G(K/k)$
\begin{alignat*}1
G_u=\bigcup_{K\in \coll_e}G(K/k),
\end{alignat*}
so that $\collg$ is non-empty if and only if $g \in G_u$.
In fact $G_u$ is simply the set of positive, proper
divisors of $e$ but we need only
\begin{alignat*}1
\{1\}\subseteq G_u \subseteq \{1,...,[e/2]\}.
\end{alignat*}
To apply the dyadic summation lemma we need information
about the growth rate of $N_{\delta_g}(T)$.
In accordance with the notation in Lemma \ref{dyadicsum} we define for an integer $g\in G_u$
and real positive $T$
\begin{alignat*}1
N_{\delta_g}(T)=|\{K\in \collg;\delta_g(K/k)\leq T\}|.
\end{alignat*}
The set on the right-hand side is finite. More precisely
we have the following lemma.
\begin{lemma}\label{lemmaNdeltas}
Set $\gamma_g=\m (g^2+g+e^2/g+e)$. Then for real positive
$T$ and $g$ in $G_u$ we have
\begin{alignat*}1
N_{\delta_g}(T)\ll T^{\gamma_g}.
\end{alignat*}
\end{lemma}
\begin{rproof}
Since $H(1,\alpha_1,\alpha_2)\geq \max\{H(1,\alpha_1),H(1,\alpha_2)\}$ it suffices to show that the number of tuples $(\alpha_1,\alpha_2)\in \kbar^2$ with
\begin{alignat}1
\label{conddeg1}
&[k(\alpha_1):k]=g\\
\label{conddeg2}
&[k(\alpha_1,\alpha_2):k(\alpha_1)]=e/g\\
\label{condheight}
&H(1,\alpha_1), H(1,\alpha_2)\leq T
\end{alignat}
is $\ll T^{\gamma_g}$.
The number of projective points
in $\IP(k;g)$ with height not exceeding $T$ is an upper
bound for the number of $\alpha_1$ in $\kbar$ of relative degree
$g$ with $H(1,\alpha_1)\leq T$.
Thus by (\ref{ThSchm1'}) we get the upper bound
\begin{alignat}1
\label{ubalpha1}
\ll T^{\m g(g+1)}
\end{alignat}
for the number of $\alpha_1$.
Next for each $\alpha_1$ we count the number of $\alpha_2$.
By (\ref{conddeg2}) we have
$[k(\alpha_1,\alpha_2):k(\alpha_1)]=e/g$
and moreover $H(1,\alpha_2)\leq T$. Applying (\ref{ThSchm1'})
(note that the constant
$\CS(k,e,n)$ in (\ref{ThSchm1'}) depends only on $[k:\IQ],e,n$)
once more yields the upper bound
\begin{alignat}1
\label{ubalpha2}
\ll T^{[k(\alpha_1):\IQ](e/g)(e/g+1)}=T^{\m e(e/g+1)}
\end{alignat}
for the number of $\alpha_2$ provided $\alpha_1$ is fixed.
Multiplying the bound (\ref{ubalpha1}) for the number
of $\alpha_1$ and (\ref{ubalpha2}) gives the upper bound
\begin{alignat*}1
\ll T^{\m(g^2+g+e^2/g+e)}
\end{alignat*}
for the number of tuples $(\alpha_1,\alpha_2)$ and thereby
proves the lemma.
\end{rproof}
Recall that $\delta_1=\delta$
and that $N_{\delta}(T)$ denotes the number of number fields $K$
in $\kbar$ of relative degree $e$ with $\delta(K/k)\leq T$.
So Lemma \ref{lemmaNdeltas} with $g=1$ yields an upper bound for the growth rate of
$N_{\delta}(T)$ but applying (\ref{ThSchm1'}) directly gives a slightly better result.
\begin{lemma}\label{lemmagamma}
Set $\gamma=\m e(e+1)$ and let $C_{\delta}=\CS(k,e,1)$ be as in
(\ref{ThSchm1'}). Then for $T>0$ we have
\begin{alignat}1
\label{Ndelta}
N_{\delta}(T)\leq C_{\delta} T^{\gamma}.
\end{alignat}
\end{lemma}
\begin{rproof}
The number of points in $\IP(k;e)$ with height not
larger than $T$ is clearly an upper bound for
$N_{\delta}(T)$. Thus the lemma follows from
the upper bound in (\ref{ThSchm1'}).
\end{rproof}
In fact Lemma \ref{lemmaNdeltas}
would suffice to prove the full Theorem \ref{main theorem}, so
one could omit Lemma \ref{lemmagamma}. We did not
because $\gamma$ looks nicer than $\gamma_1$
and the proof above is essentially simply a reference.
\subsection{Proof of Theorem \ref{main theorem}}
Recall the definition of $\Errorterm$ and $\Mainterm$ from (\ref{errorterm1}) and (\ref{mainterm1}).
We have seen that it
suffices to show that $\sum \Errorterm$ and $\sum \Mainterm$
are convergent where the sums run over all fields in $\coll_e$.\\
Since $\en$ is a uniform $\ALS$ on $\coll_e$ with associated constants
$C_{\en},\M_{\en}$ and $L_{\en}$ we can assume that
\begin{alignat}1
\label{CenCenk}
C_{\en_K}&\leq C_{\en},\\
\label{MenMenk}
\M_{\en_K}&\leq \M_{\en},\\
\label{LenLenk}
L_{\en_K}&\leq L_{\en}.
\end{alignat}
Hence by definition (\ref{defAN}) and (\ref{defANK})
\begin{alignat}1
\label{Aen}
A_{\en_K}\leq A_{\en}.
\end{alignat}
Let us now prove that $\sum_K \Errorterm$ converges.
We set $\beta=e(e-1)\m +1/8$. Using (\ref{de2})
and (\ref{Aen}) we get
\begin{alignat*}1
\sum_{K\in \coll_e} \Errorterm \ll A_{\en}\sum_{K\in \coll_e} \sum_{g\in G(K/k)}\delta_{g}(K/k)^{\beta-\mu_g}.
\end{alignat*}
Recall that $G_u=\bigcup_{\coll_e} G(K/k)$.
So the term on the right-hand side above is
\begin{alignat}1
\nonumber
&A_{\en}\sum_{g\in G_u}\sum_{K \in \coll_e \atop g\in G(K/k)}\delta_{g}(K/k)^{\beta-\mu_g}\\
\label{EKsum1}=&A_{\en}\sum_{g\in G_u}\sum_{K\in \collg}
\delta_{g}(K/k)^{\beta-\mu_g}
\end{alignat}
provided the sum converges. This will be verified in a
moment (see (\ref{confirmconv})).
Applying the dyadic summation lemma with
$\f=\delta_g$ and $b=\gamma_g$ from
Lemma \ref{lemmaNdeltas} we see that the latter is
\begin{alignat*}1
\ll A_{\en}\sum_{g\in G_u}\sum_{i=1}^{\infty}2^{i(\gamma_g+\beta-\mu_g)}.
\end{alignat*}
The next lemma will tell us that
$\gamma_{g}+\beta-\mu_g\leq -1/8$. Assuming this for
a moment we see that
the inner sum above
is $\ll 1$. Thus we derive the upper bound
\begin{alignat}1
\label{confirmconv}
\ll A_{\en}\sum_{g\in G_u}1
\ll A_{\en},
\end{alignat}
confirming that the whole sum in (\ref{EKsum1}) converges. This verifies the convergence
of $\sum_K \Errorterm$ under the hypothesis $\gamma_{g}+\beta-\mu_g\leq -1/8$ for all $g\in G_u$.
The following lemma shows that this hypothesis holds true.
Recall that we assume $e>1$ and therefore by our assumption
in Theorem \ref{main theorem}
$n>5e/2+4+2/(\m e)$.
\begin{lemma}\label{gabemuisneg}
Let $g$ be in $G_u$. Then
\begin{alignat}1
\label{ineqgabemu}
\gamma_{g}+\beta-\mu_g\leq -\frac{1}{8}.
\end{alignat}
\end{lemma}
\begin{rproof}
Recall that $G_u\subseteq \{1,...,[e/2]\}$ and $\mu_g=\m (e-g)(n+1)-1$.
Write
\begin{alignat*}1
\IG(g)=\frac{1}{\m(e-g)}(\gamma_{g}+\beta+1).
\end{alignat*}
So (\ref{ineqgabemu}) claims that $m(e-g)(\IG(g)-(n+1))\leq -1/8$
for all $g\in G_u$.
Hence it suffices to show that
\begin{alignat*}1
\IG(g)-(n+1)\leq -\frac{1}{4\m e}
\end{alignat*}
for $1\leq g\leq e/2$. By definition
\begin{alignat*}1
\IG(g)=\frac{g^2+g+e^2/g+e}{e-g}+
\frac{e(e-1)}{e-g}+\frac{1+1/8}{\m (e-g)}.
\end{alignat*}
We claim that the second derivative $\IG''(g)$ is positive
for $1\leq g\leq e/2$. One finds
\begin{alignat*}1
\IG''(g)=&\frac{2(e^2/g^3+1)(e-g)+2(2g+1-e^2/g^2)}{(e-g)^2}+
\frac{2e(e-1)}{(e-g)^3}\\
\nonumber+&\frac{2(g^2+g+e^2/g+e)}{(e-g)^3}+
\frac{2(1+1/8)}{\m (e-g)^3}.
\end{alignat*}
For $1\leq g\leq e/2$ the last three fractions are certainly
positive and so we may focus on the numerator of the
first fraction. Now if $2g+1-e^2/g^2\geq 0$ the claim
follows at once. If $2g+1-e^2/g^2<0$ it suffices to show
that
\begin{alignat*}1
(e^2/g^3+1)(e-g)\geq e^2/g^2-2g-1.
\end{alignat*}
With $u=e/g$ the latter is equivalent to
$u^3-u^2+e-g\geq u^2-2g-1$ and this is equivalent
to $u^2(u-2)+e+g+1\geq 0$, which is certainly true
since $1\leq g\leq e/2$ and therefore $2\leq u \leq e$.\\
Thus we have shown that $\IG''(g)>0$ for $1\leq g\leq e/2$
so that $\IG$ is here concave. It suffices to prove $\IG(g)-(n+1)\leq -1/(4\m e)$
for $g=1$, $g=e/2$.
First we use a simple arithmetic argument. Since $n$ is an integer and $n>E=5e/2+4+2/(\m e)$ with denominator dividing
$2\m e$ we see that
\begin{alignat}1
\label{nest}
n+1\geq E+1+1/(2\m e).
\end{alignat}
Now $\IG(e/2)=5e/2+5+2/(\m e)+1/(4\m e)=E+1+1/(4\m e)$ and
thus
\begin{alignat*}1
\IG(e/2)-(n+1)\leq 1/(4\m e)-1/(2\m e)=-1/(4\m e).
\end{alignat*}
Finally
\begin{alignat*}1
\IG(1)=2e+2+4/(e-1)+9/(8\m (e-1)).
\end{alignat*}
Using (\ref{nest}) again yields
\begin{alignat}1
\label{IG1est}
\IG(1)-(n+1)\leq \frac{4}{e-1}+\frac{9}{8\m(e-1)}-\frac{e}{2}-3-
\frac{2}{\m e}-\frac{1}{2\m e}.
\end{alignat}
First suppose $e=2$. Then (\ref{IG1est}) says
$\IG(1)-(n+1)\leq-1/(8\m)=-1/(4e\m)$. Next suppose $e>2$. Then
the right-hand side in (\ref{IG1est}) is $\leq 4/2+9/(16\m)-e/2-3-5/(2\m e)<
-5/(2\m e)<-1/(4e\m)$.
This completes the proof of the lemma.
\end{rproof}
To show convergence for $\sum_K \Mainterm$ we may use
similar arguments but here we use only $\delta=\delta_1$ instead of $\delta_g$. Let $d=\m e$ so that $[K:\IQ]=d$.
To estimate $V_{\en_K}$ in (\ref{mainterm1}) recall that $V_{\en_K}=V_{\en_K}^{inf}V_{\en_K}^{fin}$.
By (\ref{Vinfbou1}) we have
\begin{alignat*}1
V_{\en_K}^{inf}\ll (C_{\en_K}^{inf})^{d(n+1)}.
\end{alignat*}
To estimate $V_{\en_K}^{fin}$ we define the non-zero ideal $\C_0$ by
\begin{alignat}3
\label{C0}
\C_0=\prod_{v\nmid \infty}\pw_v^{-\frac{d_v\log c_v}{\log N\pw_v}}
\end{alignat}
with $c_v$ as in (\ref{defcfin}).
Thus $|\C_0|_v=c_v$ and
\begin{alignat}3
\label{NC0}
N\C_0=(C_{\en_K}^{fin})^d.
\end{alignat}
Let $\D\neq 0$ be a fractional ideal.
Clearly $|\alpha|_v\leq |\C_0^{-1}\D|_v$ for all
non-archimedean $v$ is equivalent to $\alpha \in \C_0^{-1}\D$.
By (\ref{Nineq1}) we conclude
\begin{alignat}3
\label{OG}
{\mathfrak{L}}amenK(\D) \subseteq \sigma(\C_0^{-1}\D)^{n+1}
\end{alignat}
(where $\sigma$ is given by (\ref{sigD})) and therefore
\begin{alignat*}1
\det {{\mathfrak{L}}amenK}(\D)\geq \det \sigma(\C_0^{-1}\D)^{n+1}.
\end{alignat*}
It is well-known (see \cite{23} p.33 (5.2) Satz) that
\begin{alignat*}1
\det \sigma(\C_0^{-1}\D)=
2^{-s_K}\sqrt{|\Delta_K|}N(\D)N(\C_0)^{-1},
\end{alignat*}
where $s_K$ is the number of pairs of complex conjugate embeddings of $K$.
Combining the latter with (\ref{NC0}) we see that
\begin{alignat*}1
\det \sigma(\C_0^{-1}\D)^{n+1}=2^{-s_K(n+1)}|\Delta_K|^{(n+1)/2}N\D^{n+1}(C_{\en_K}^{fin})^{-d(n+1)}.
\end{alignat*}
Inserting
the latter in definition (\ref{defVfin1}) yields
\begin{alignat*}1
V_{\en_K}^{fin}\ll (C_{\en_K}^{fin})^{d(n+1)}.
\end{alignat*}
Now on recalling that $C_{\en_K}=C_{\en_K}^{inf}C_{\en_K}^{fin}$
and using (\ref{CenCenk}) we conclude
\begin{alignat*}1
V_{\en_K}\ll C_{\en_K}^{d(n+1)}\leq C_{\en}^{d(n+1)}.
\end{alignat*}
The number of roots of unity $\wK$ in (\ref{Schanuelconst}) is at least $2$.
Furthermore $\zeta_K(n+1)>1$. Hence
$S_K(n)\ll R_Kh_K|\Delta_K|^{-\frac{n+1}{2}}$.
This together with the above estimate for $V_{\en_K}$
implies $\Mainterm\ll C_{\en}^{d(n+1)}R_Kh_K|\Delta_K|^{-\frac{n+1}{2}}$
and since by Siegel-Brauer $R_Kh_K\ll_{\epsilon} |\Delta_K|^{\frac{1}{2}+\epsilon}$ for any positive $\epsilon$
we get
\begin{alignat}1
\label{Maintermdelta}
\Mainterm \ll_{\epsilon} C_{\en}^{d(n+1)}|\Delta_K|^{-\frac{n}{2}+\epsilon}.
\end{alignat}
\begin{remark}
Let $N_{\Delta}(T)=|\{K\in \coll_e;|\Delta_K|\leq T\}|$.
Schmidt \cite{55} showed
\begin{alignat}1
\label{Schmidtdisc}
N_{\Delta}(T)\ll T^{\frac{e+2}{4}}.
\end{alignat}
Thus we could apply the dyadic summation lemma
with $\f=|\Delta_K|$ and $b=(e+2)/4$ to see that $\sum_K \Mainterm$
converges for $n>e/2+1$.
\end{remark}
Instead of using Schmidt's bound (\ref{Schmidtdisc}) we will prove a lower bound for $|\Delta_K|$ in terms of $\delta(K/k)$
which might be of interest for its own sake. Then we can apply Lemma \ref{lemmagamma} instead of (\ref{Schmidtdisc}).
\begin{lemma}\label{lemmade4}
We have
\begin{alignat}1
\label{de4}
\delta(K/k)\leq \delta(K/\IQ)\ll |\Delta_K|^{\frac{1}{d}}.
\end{alignat}
\end{lemma}
\begin{rproof}
The lemma is trivially true for $K=k=\IQ$.
However we have by assumption $e\geq 2$
and so $[K:\IQ]=e\m\geq 2$.
The first inequality follows immediately from the definition.
Let $\sigma$ be as in (\ref{sigd}) and suppose
$\alpha$ is a non-zero integer of $K$.
One gets
\begin{alignat}3\label{heightboundalgint}
\nonumber H(1,\alpha)&=\prod_{i=1}^{d}\max\{1,|\sigma_i(\alpha)|\}^{1/d}\\
\nonumber &\leq \max\{1,\max_{1\leq i\leq d}\{|\sigma_i(\alpha)|\}\}\\
&\leq|\sigma(\alpha)|
\end{alignat}
because $\prod_{1\leq i \leq d}|\sigma_i(\alpha)|\geq 1$.
Let $v_1=\sigma(\alpha_1),...,v_d=\sigma(\alpha_d)$ be linearly independent vectors of the lattice $\sigma \Oseen_K$ with
$|v_i|=\lambda_i$ for the successive minima
$\lambda_i$ ($i=1,...,[K:\IQ]=d$).
Let us temporarily denote by $b$ the maximum of the degrees of the proper
subfields of $K$. Therefore
$K=\IQ(\alpha_1,...,\alpha_{b+1})$. Next we need to construct
a primitive element in $\Oseen_K$ with small height.
A standard argument (see \cite{AlgLa} p.244 or Lemma 3.3 in \cite{art1}) yields a primitive
$\alpha=\sum_{j=1}^{b+1}m_j\alpha_j$ with rational integers $0\leq m_j<e$.
Hence by (\ref{heightboundalgint}) we get
\begin{alignat*}1
H(1,\alpha)\leq |\sigma(\sum_{j=1}^{b+1}m_j\alpha_j)|\leq
\sum_{j=1}^{b+1}m_j|\sigma(\alpha_j)|\ll \lambda_{b+1}.
\end{alignat*}
We shall estimate $\lambda_{b+1}$:
\begin{alignat*}2
\lambda_{b+1} &=
\left(\frac{\lambda_1...\lambda_{b}\lambda_{b+1}^{d-b}}
{\lambda_1...\lambda_{b}}\right)^{\frac{1}{d-b}}\\
&\leq
\left(\frac{\lambda_1...\lambda_d}
{\lambda_1...\lambda_{b}}\right)^{\frac{1}{d-b}}\\
&\ll
\left(\frac{\det(\sigma\Oseen_K)}
{\lambda_1...\lambda_{b}}\right)^{\frac{1}{d-b}}
\\
&=
\left(\frac{ |\Delta_K|^{\frac{1}{2}}}
{2^{s_K}\lambda_1...\lambda_{b}}\right)^{\frac{1}{d-b}}
\\
&\ll |\Delta_K|^{\frac{1}{2(d-b)}}
\end{alignat*}
where we used that $\lambda_1=|\sigma(\alpha_1)|\geq H(1,\alpha_1)\geq 1$.
So all this together implies
\begin{alignat}1
\label{de3}
\delta(K/\IQ)\ll |\Delta_K|^{\frac{1}{2(d-b)}}.
\end{alignat}
Now $b$ is the degree of a proper subfield. Thus
$b\leq d/2$ and we get (\ref{de4}).
\end{rproof}
Using Lemma \ref{lemmade4} and (\ref{Maintermdelta}) with
$\epsilon$ replaced by $\epsilon/d$ we deduce
\begin{alignat}1
\label{deltaDelta1}
\Mainterm &\ll_{\epsilon} C_{\en}^{d(n+1)}\delta(K/k)^{-\frac{dn}{2}+\epsilon}
\end{alignat}
for any positive $\epsilon$.
Choosing $\epsilon=1/2$ we get
\begin{alignat}1
\nonumber \Mainterm &\ll C_{\en}^{d(n+1)}\delta(K/k)^{-\frac{dn}{2}+\frac{1}{2}}.
\end{alignat}
Applying the dyadic summation lemma with
$\f=\delta$ and $b=\gamma$ from Lemma \ref{lemmagamma} we
conclude
\begin{alignat*}2
\sum_{K\in \coll_e} \Mainterm &\ll
C_{\en}^{d(n+1)} \sum_{K\in \coll_e}\delta(K/k)^{-\frac{dn}{2}+\frac{1}{2}}\\
&\ll C_{\en}^{d(n+1)}\sum_{i=1}^{\infty}2^{(-\frac{dn}{2}+\frac{1}{2}+\gamma)i}\\
&\ll C_{\en}^{d(n+1)}
\end{alignat*}
provided $-\frac{dn}{2}+\frac{1}{2}+\gamma<0$, which is
equivalent to
$n>2e+2+1/d$. But the latter holds since $n>5e/2+4+2/(\m e)$.
This completes the proof of Theorem \ref{main theorem}.
\section{Counting number fields}\label{countingfields}
Using results of the previous section we give simple lower bounds for the growth rate of $N_{\delta}(T)$ and $N_{\Delta}(T)$, the number of field extensions $K/k$ of degree $e$ with $\delta(K/k)\leq T$ or $|\Delta_K|\leq T$. The following corollary shows that the estimates for $N_{\delta}(T)$ are more precise than those available for $N_{\Delta}(T)$. Recall that $e>1$.
\begin{korollar}\label{corlowerdeltabound}
With $\cS=\cS(k,e,1), \CS=\CS(k,e,1)$ and $X_0(k,e,1)$
from (\ref{ThSchm1'}) set
\begin{alignat*}1
c_{\delta}=2^{-5e\m-22}\cS,\text{ } C_{\delta}=\CS \text{ and } T_0=X_0(k,e,1).
\end{alignat*}
Then we have
\begin{alignat*}1
c_{\delta}T^{\m e(e-1)}\leq N_{\delta}(T)\leq C_{\delta}T^{\m e(e+1)}
\end{alignat*}
where the upper bounds holds for $T>0$ and the lower
bound holds for $T\geq T_0$.
\end{korollar}
\begin{rproof}
From the definition it is clear that $Z_H(\IP(K/k),T)>0$ if and only if
$\delta(K/k)\leq T$. Therefore we have
\begin{alignat}1
\label{Ndeltaexpr}
N_{\delta}(T)=\sum_{K \in \coll_e \atop \delta(K/k)\leq T}1=\sum_{K \in \coll_e \atop \delta(K/k)\leq T}\frac{Z_{H}(\IP(K/k),T)}{Z_{H}(\IP(K/k),T)}.
\end{alignat}
Using the equivalence above once again, we see that
the term on the far right-hand side of (\ref{Ndeltaexpr}) is
\begin{alignat*}1
&\geq (\sup_{K \in \coll_e}\{Z_{H}(\IP(K/k),T)\})^{-1}\sum_{K \in \coll_e \atop \delta(K/k)\leq T}Z_{H}(\IP(K/k),T)\\
&= (\sup_{K \in \coll_e}\{Z_{H}(\IP(K/k),T)\})^{-1}\sum_{K \in \coll_e}Z_{H}(\IP(K/k),T)\\
&=(\sup_{K \in \coll_e}\{Z_{H}(\IP(K/k),T)\})^{-1}
Z_{H}(\IP(k;e),T).
\end{alignat*}
Now $Z_{H}(\IP(K/k),T)\leq Z_{H}(\IP(K;1),T)$
and by the upper bound in (\ref{ThSchm1'})
and recalling that $[K:\IQ]=e\m$ we get
\begin{alignat*}1
Z_{H}(\IP(K;1),T)\leq \CS(K,1,1) T^{2\m e}=2^{5e\m+22}T^{2\m e}.
\end{alignat*}
The lower bound in (\ref{ThSchm1'})
with $\cS=\cS(k,e,1)$
yields
\begin{alignat*}1
Z_{H}(\IP(k;e),T)\geq \cS T^{\m e(e+1)}
\end{alignat*}
for $T\geq X_0(k,e,1)=T_0$.
Hence
\begin{alignat*}1
N_{\delta}(T)\geq (2^{5e\m+22}T^{2\m e})^{-1}\cS T^{\m e(e+1)}=c_{\delta}T^{\m e(e-1)}
\end{alignat*}
for $T\geq T_0$. On the other hand Lemma \ref{lemmagamma} tells us that
\begin{alignat*}1
N_{\delta}(T)\leq C_{\delta} T^{\m e(e+1)}
\end{alignat*}
for $T>0$.
\end{rproof}
Corollary \ref{corlowerdeltabound} combined with the lower bound
(\ref{de11}) for $\delta$ in terms of $|\Delta_K|$ yields
\begin{korollar}\label{corlowerDeltabound}
There are positive constants $\cDe=\cDe(k,e)$ and $T_1=T_1(k,e)$
depending only on $k,e$ such that
\begin{alignat*}1
N_{\Delta}(T)\geq \cDe T^{1/2}
\end{alignat*}
for $T\geq T_1$.
\end{korollar}
\begin{rproof}
From (\ref{de11}) we know that there is a positive constant
$\ceins=\ceins(k,e)$ depending only on $k,e>1$ such that
$\delta(K/k)\geq \ceins|\Delta_K|^{1/(2e(e-1)\m)}$.
Using Corollary \ref{corlowerdeltabound} and setting
$\cDe=c_{\delta}\ceins^{\m e(e-1)}$, $T_1=(T_0/\ceins)^{2e(e-1)\m}$
we conclude
\begin{alignat*}1
N_{\Delta}(T)\geq N_{\delta}(\ceins T^{1/(2e(e-1)\m)})
\geq \cDe T^{1/2}
\end{alignat*}
provided $T\geq T_1$.
\end{rproof}
Ellenberg and Venkatesh's Theorem 1.1 in \cite{56}
shows that the exponent $1/2$ in Corollary \ref{corlowerDeltabound}
can be replaced by $1/2+1/e^2$ and according to Linnik's Conjecture (see \cite{56} p.723) the correct exponent is $1$. Although the general Linnik Conjecture is known
to be true only for $e\leq 3$
the exponent $1/2$ can always be increased to $1$ if $e$ is even
or a multiple of $3$ (see \cite{56} pp. 723,724).\\
What about upper bounds for $N_{\Delta}(T)$?
From (\ref{de4}) we know that there is a positive
constant $\czwei=\czwei(d)$ depending only on $d=e\m$
such that
\begin{alignat*}1
\delta(K/k)\leq
\czwei|\Delta_K|^{\frac{1}{e\m}}.
\end{alignat*}
Thus we get
\begin{alignat*}1
N_{\Delta}(T)\leq N_{\delta}(\czwei T^{\frac{1}{e\m}})
\leq C_{\delta}\czwei^{\m e(e+1)}T^{e+1}
\end{alignat*}
for $T>0$.
But Schmidt's bound (\ref{Schmidtdisc}) has the much better exponent $(e+2)/4$ on $T$.
\end{document} |
\begin{document}
\title{Rigidity and Non-recurrence along Sequences}
\author{V. Bergelson, A. del Junco, M. Lema\'nczyk, J. Rosenblatt}
\date{February, 2011}
\begin{abstract}
Two properties of a dynamical system, rigidity and non-recurrence,
are examined in detail. The ultimate aim is to characterize the sequences
along which these properties do or
do not occur for different classes of transformations. The
main focus in this article is to characterize explicitly the
structural properties of sequences which can be rigidity sequences
or non-recurrent sequences for some weakly mixing dynamical system.
For ergodic transformations generally and for weakly mixing transformations
in particular there are both parallels and distinctions between the class of
rigid sequences and the class of non-recurrent sequences.
A variety of classes of sequences with various properties are considered
showing the complicated and rich structure of rigid and
non-recurrent sequences.
\end{abstract}
\maketitle
\section{\bf Introduction}
\lambdabel{intro}
Let $(X,\mathcal B,p,T)$ be a dynamical system: that is, we have
a non-atomic probability space $(X,\mathcal B,p)$ and an invertible measure-preserving transformation
$T$ of $(X,\mathcal B,p)$. We consider here two properties of the dynamical system
$(X,\mathcal B,p,T)$, rigidity and non-recurrence. Ultimately we would
like to characterize the sequences along which these properties do, or
do not occur, for different classes of transformations. The
main focus here is to characterize which subsequences $(n_m)$ in
$\mathbb Z^+$ can be a sequence for rigidity, and which
can be a sequence for non-recurrence, for some
weakly mixing dynamical system. In the process of doing this, we will
see that there are parallels and distinctions between the class of
rigid sequences and the class of non-recurrent sequences, both for
ergodic transformations in general and for weakly mixing transformations
in particular.
The properties of rigidity and non-recurrence along a given sequence $(n_m)$
are opposites of one another. By {\em rigidity along the
sequence} $(n_m)$ we mean that the powers $(T^{n_m})$ are converging in
the strong operator topology to the identity; that is, $\|f{\mathcal I}rc
T^{n_m} - f\|_2 \to 0$ as $m \to \infty$, for all $f \in L_2(X,p)$.
So rigidity along $(n_m)$ means that $p(T^{n_m}A{\mathcal A}p A) \to p(A)$
as $m\to \infty$ for all $A \in \mathcal B$. On the other hand,
{\em non-recurrence along the sequence} $(n_m)$ means that for some
$A \in \mathcal B$ with $p(A) > 0$, we have $p(T^{n_m}A {\mathcal A}p A) = 0$
for all $m \ge 1$. Nonetheless, there are structural parallels between these
two properties of a sequence $(n_m)$. For example, neither property can occur
for an ergodic transformation unless
the sequence is sparse. Also, these two properties
cannot occur without the sequence $(n_m)$ having (or avoiding)
various combinatorial or algebraic structures.
These properties can occur simultaneously for a given transformation
if the sequences are disjoint. For example, we are
able to use Baire category results to show that the generic
transformation $T$ is weak mixing and rigid along some sequence $(n_m)$
such that it is also non-recurrent along $(n_m-1)$. In proving this,
one sees a connection between rigidity and non-recurrence. The non-recurrence
along $(n_m-1)$ is created by first using rigidity to take a rigid sequence
$(n_m)$ for $T$ and a set $A, p(A) > 0$,
such that $\sum\limits_{m=1}^\infty p(T^{n_m}A\Delta A) \le \frac 1{100}p(A)$.
This allows one to prove that $T$ is non-recurrent along $(n_m-1)$ for some
subset $C$ of $TA$. One can extend this argument somewhat and show that
for every whole number $K$, there is a weakly mixing transformation $T$ that is rigid
along a sequence $(n_m)$, such that also for some set $C$, $p(C) > 0$,
$T$ is non-recurrent for $C$ along $(n_m+k)$ for all $k \not= 0, |k| \le K$.
First, in Section~\ref{rigidity}, we discuss some generalities about
rigidity and weak mixing. We also consider the more restrictive property
of IP-rigidity. We will see that both rigidity and IP-rigidity can be viewed as a
spectral property and therefore characterized in terms of the behavior of
the Fourier transforms $\widehat \nu$ of the positive
Borel measures $\nu$ on $\mathbb T$ that are the spectral measures
of the dynamical system. We will see that rigidity
sequences must be sparse, but later in Section~\ref{ratesofgrowth},
it is made clear that they are not necessarily very sparse.
In addition, we show that rigidity sequences cannot
have certain types of
algebraic structure for rigidity to occur even for an ergodic
transformations, let alone a weakly mixing one.
After this in Section~\ref{methods}, we prove a variety of results
about rigidity that serve to demonstrate how rich and complex is
the structure of rigid sequences. Here is a sample of what we prove:
\noindent {\bf a)} In Proposition~\ref{ratiogrows} we show that
if $\lim\limits_{m \to \infty} \frac {n_{m+1}}{n_m} = \infty$, then
$(n_m)$ is a rigidity sequence for some weakly mixing transformation $T$.
This result uses the Gaussian measure space construction.
Also, by a cutting and stacking construction, we construct an infinite measure preserving
rank one transformation $S$
for which $(n_m)$ is a rigidity sequence.
Under some additional assumptions on $(n_m)$, we can use the cutting and
stacking construction to produce a
weakly mixing rank one transformation $T$ on a probability space for which $(n_m)$
is a rigidity sequence. See specifically
Proposition~\ref{specialinfrankone} and generally Section~\ref{rankone}.
\noindent {\bf b)} In contrast, we show that sequences like $(a^m: m \ge 1)$, and $a
\in \mathbb N, a\not= 1$,
are also rigidity sequences for weakly mixing transformations. However,
perturbations of them, like
$(a^m+p(m): m \ge 1)$ with $p \in \mathbb Z[x], p \not=0$, are never rigidity
sequences for ergodic transformations,
let alone weakly mixing transformations. See Proposition~\ref{integerratios}
and Remark~\ref{linformeg} c).
\noindent {\bf c)} We prove a number of results in
Section~\ref{ratesofgrowth} that show that rigidity
sequences do not necessarily have to grow quickly, but rather can have
their density decreasing to zero
infinitely often slower than any given rate. One consequence may
illustrate what this tells us: we
show that there are rigidity sequences for weakly mixing transformations
that are not Sidon sets.
See Proposition~\ref{ergodicratewm} and Corollary~\ref{notSidon}.
\noindent {\bf d)} In Section~\ref{secdisjoint}, we show that there is
no universal rigid sequence. That is, we show that given a weakly
mixing transformation $T$ that is rigid along some sequence, there is
another weakly mixing transformation $S$ which is rigid along some other
sequence such that $T\times S$ is not rigid along any sequence.
\noindent {\bf e)} In Section~\ref{seccocycle}, we show how cocycle
construction can be used to construct rigidity sequences
for weakly mixing transformations. One particular result is
Corollary~\ref{denominators}: if $(\frac {p_n}{q_n}:n\ge 1)$
are the convergents associated with the continued fraction expansion of
an irrational number, then
$(q_n)$ is a rigidity sequence for a weakly mixing transformation.
We then consider non-recurrence in Section~\ref{nonrecurrence}.
We show that the sequences exhibiting non-recurrence must be
sparse and cannot have certain types of algebraic structure for there to be
non-recurrence even for ergodic transformation, let alone a
weakly mixing one.
We conjecture that any lacunary sequence is a sequence of non-recurrence
for some
weakly mixing transformation, but we have not been able to prove this
result at this time.
Here are some specific results on non-recurrence that we prove:
\noindent {\bf a)}
It is well-known that the generic transformation $T$ is weakly mixing and
rigid. We show that in addition, there is a
rigidity sequence $(n_m)$ for such a generic $T$, so that for any whole
number $K$, each
$(n_m+k), 0 < |k|\le K$, is a
non-recurrent sequence for $T$. See Proposition~\ref{revise} and Remark~\ref{notrecshift}.
\noindent {\bf b)} We observe in Proposition~\ref{Chacon} that some
weakly mixing transformations,
like Chacon's transformation, are non-recurrent along a lacunary sequence with
bounded ratios.
\noindent {\bf c)} We also show that for any increasing sequence $(n_m)$
with $\sum\limits_{m=1}^\infty \frac {n_m}{n_{m+1}} < \infty$,
and a whole number $K$, there is
a weakly mixing
transformation $T$ and
a set $C, p(C) > 0$, such that $(n_m)$ is a rigidity sequence for $T$ and $T$
non-recurrent for $C$ along $(n_m +k)$ for all $k, 0 < |k| \le K$.
See Proposition~\ref{fastworks} and Remark~\ref{notrecshiftagain}.
When considering both rigidity and non-recurrence of measure-preserving transformations,
there are often unitary versions of the results that are either almost identical in
statement and proof, or worth more consideration. When possible, we will take note of
this. See Krengel~{\mathcal I}te{Krengel} for a general reference on this and other aspects of
ergodic theory used in this article.
There is also a larger issue of considering both rigidity and non-recurrence for
general groups of invertible measure-preserving transformations. This
will require a careful look at general spectral issues, including the irreducible
representations of the groups. We plan to pursue this in a later
paper.
\section{\bf Generalities on Rigidity and Weak Mixing}
\lambdabel{rigidity}
Suppose we consider a dynamical system $(X,\mathcal B,p,T)$. Unless it
is noted otherwise, we will be assuming that $(X,\mathcal B,p)$ is a
standard Lebesgue probability space i.e. it is measure theoretically
isomorphic to $[0,1]$ in Lebesgue measure. In particular it is non-atomic
and $L_2(X,p)$ has a countable dense subset in the norm topology. We say that the
dynamical system is separable in this case. We also assume that
$T$ is an invertible measure-preserving transformation
$(X,\mathcal B,p)$.
This section provides the background information needed in this article.
First, in Section~\ref{rigidwmboth} we deal with the basic properties of rigidity and
weak mixing in order to give a general version of the well-known fact that the generic transformation
is both weakly mixing and rigid along some sequence. Second, in Section~\ref{weakmixsec} we
look at weak mixing and aspects of it that are important to this article. Third, in Section~\ref{rigidonly} we
consider rigidity itself in somewhat more detail. See Furstenberg and Weiss~{\mathcal I}te{FurstWeiss}
and Queffelec~{\mathcal I}te{Queff}, especially Section 3.2.2,
for background information about rigidity as we consider it, and other types of rigidity
that have been considered by other authors.
\subsection{\bf Rigidity and Weak Mixing in General}\lambdabel{rigidwmboth}
Given an increasing sequence $(n_m)$ of integers we consider the
family $$
{\mathcal A}(n_m)=\{A\in\mathcal B:\:p\left(T^{n_m}A\triangle
A\right)\to 0\}.$$ We now recall some basic and well-known facts
about ${\mathcal A}(n_m)$. See Walters~{\mathcal I}te{Walters2} for the following
result.
\begin{prop}
${\mathcal A}(n_m)\subset\mathcal B$ is a sub-$\sigmagma$-algebra which is also
$T$-invariant. ${\mathcal A}(n_m)$ is the maximal $\sigmagma$-algebra
${\mathcal A}\subset\mathcal B$ such that
$$
T^{n_m}|_{{\mathcal A}}\to Id|_{{\mathcal A}}\;\;\mbox{as}\;\;m\to\infty.$$
Moreover \begin{equation}\lambdabel{ww1} L_2(X,{\mathcal A}(n_m),p)=\{f\in L_2(X,\mathcal
B,p):\:f{\mathcal I}rc
T^{n_m}\to f\;\;\mbox{in}\;\;L_2(X,\mathcal B,p)\}.\end{equation}
\end{prop}
\begin{rem} An approach to the above result different than in ~{\mathcal I}te{Walters2}
begins by observing that $\{f\in L_\infty(X,p):
\|f{\mathcal I}rc T^{n_m} - f\|_2 \to 0\}$ is an algebra. So there is a corresponding
factor map of $(X,\mathcal B,p,T)$ for which there is an associated
$T$ \-invariant sub-$\sigmagma$-algebra, namely
${\mathcal A}(n_m)\subset\mathcal B$.
\end{rem}
If ${\mathcal A}(n_m)=\mathcal B$ then one says that $(n_m)$ is a {\em
rigidity sequence} for $(X,\mathcal B,p,T)$. Systems possessing rigidity
sequences are called {\em rigid}. The
fact that $(n_m)$ is a rigidity sequence for $T$ is a {\em spectral property}; that is, it is a
unitary invariant of the associated Koopman operator $U_T$ on
$L_2(X,p)$ given by the formula $U_T(f)=f{\mathcal I}rc T$. The following
discussion should make this clear.
First, recall some basic notions of spectral theory (see e.g.
{\mathcal I}te{CFS}, {\mathcal I}te{Ka-Th}, {\mathcal I}te{Pa}). For each $f\in L_2(X,p)$, the
function $\rho(n) = \lambdangle f{\mathcal I}rc T^n,f\rangle$ is a positive-definite
function and hence, by the Herglotz Theorem, is the Fourier
transform of a positive Borel measure on the circle $\mathbb T$. So
for each $f \in L_2(X,p)$, there is a unique positive Borel
measure $\nu_f^T$ on $\mathbb T$, called the
{\em spectral measure} for $T$ corresponding to $f$
which is determined by $\widehat {\nu_f^T}(n) = \lambdangle f{\mathcal I}rc
T^{n},f\rangle$ for all $n \in \mathbb Z$. Spectral measures are
non-negative and have $\nu_f^T(\mathbb T) =\|f\|_2^2$.
We will also need to use the adjoint $\nu^*$ given by $\nu^*(E) = \overlineerline
{\nu(E^{-1})}$ for all Borel sets $E\subset \mathbb T$. The
adjoint has $\widehat {\nu^*}(n) =\overlineerline {\widehat {\nu}(-n)}$
for all $n \in \mathbb Z$.
Absolute continuity of measures is important
here: given two positive Borel measures $\nu_1$ and $\nu_2$ on $\mathbb T$,
we say $\nu_1$ is absolutely continuous with respect to $\nu_2$,
denoted by $\nu_1 \ll \nu_2$, if $\nu_1(E) = 0$ for all Borel sets
$E$ such that $\nu_2(E) = 0$. Now, among all spectral
measures there exist measures $\nu_F^T$ such that all other
spectral measures are absolutely continuous with respect to
$\nu_F^T$. Any one of these is called a {\em maximal spectral measure}
of $T$. These measures are all mutually
absolutely continuous with respect to one another.
The equivalence class of the maximal spectral measures is denoted by $\nu^T$.
By abuse of notation, we refer to $\nu^T$ as a measure too.
Recall that the type of a finite positive measure (e.g. whether the
measure is singular, absolutely continuous with respect to Lebesgue measure, etc.)
is a property of the equivalence
class of all finite positive measures $\omega$ such that $\omega \ll \nu$
and $\nu \ll \omega$. The type of $\nu^T$ (that is, of a maximal spectral measure $\nu^T_F$)
has a special role in the structure of the transformation. For this
reason the type of $\nu^T$ is called the {\em
maximal spectral type} of $T$. For example, rigid transformations must
have singular maximal spectral type; see Remark~\ref{Rokhlinsingular}. Also, Bernoulli transformations
must have Lebesgue type i.e. their maximal spectral measures are
equivalent to
Lebesgue measure. In general, a strongly mixing transformation does not need to be
of Lebesgue type. It could be of singular type (this occurs when every
maximal spectral measure is singular but yet has the Fourier transform
tending to zero at infinity).
For a given sequence $(n_m)$, a transformation $T$
and a function $f \in L_2(X,p)$, we say {\em $(n_m)$ is
a rigidity sequence of $T$ for $f$} if
$f{\mathcal I}rc T^{n_m}\to f$ in $L_2$-norm. Recall that
$\widehat{\nu^T_f}(n_m)=\lambdangle f{\mathcal I}rc
T^{n_m},f\rangle$.
\begin{prop} \lambdabel{rigidfacts} Fix the transformation $T$. The
following are equivalent for $f \in L_2(X,p)$:
\begin{enumerate}
\item The sequence $(n_m)$ is
a rigidity sequence for the function $f$.
\item $\lambdangle f{\mathcal I}rc T^{n_m},f\rangle=\int_X f{\mathcal I}rc T^{n_m}{\mathcal D}ot
\overlineerline{f}\,dp\to \|f\|_2^2$.
\item $\widehat{\nu^T_f}(n_m)\to\|f\|_2^2$.
\item $z^{n_m} \to 1$ in
$L_2(\mathbb T,\nu_f^T)$.
\item $z^{n_m} \to 1$ in measure with respect to
$\nu_f^T$.
\end{enumerate}
\end{prop}
\begin{proof} We have $\|f {\mathcal I}rc T^{n_m} - f\|_2^2 =2\|f\|_2^2 -
2\text{Re}\lambdangle f{\mathcal I}rc T^{n_m},f\rangle$.
Since $|\lambdangle f{\mathcal I}rc T^{n_m},f\rangle| \le \|f\|_2^2$, we see that (1)
is equivalent to (2). Now
(2) is equivalent to (3) by the definition of the spectral measure
$\nu_f^T$. We also have
$\int |z^{n_m}-1|^2\, d\nu_f^T(z) = 2\|f\|_2^2 - 2\text{Re}(\widehat
{\nu_f^T}(n_m))$. Since
$|\widehat {\nu_f^T}(n_m)|\le \|f\|_2^2$, we see that (3) is equivalent
to (4). It is clear
that (4) is equivalent to (5) because $|1-z^{n_m}| \le 2$ and
$\nu_f^T$ is a finite, positive measure.
\end{proof}
\begin{rem} This result is really a fact about a unitary operator
$U$ on a Hilbert space $H$. That is, a sequence $(n_m)$ and
vector $v \in H$ satisfy
$\lim\limits_{m \to \infty} ||U^{n_m} v - v \|_H = 0$ if and only
if the spectral measure $\nu_v^U$ determined by $\widehat {\nu_v^U}(k)
= \lambdangle U^kv,v\rangle $ for all $k \in \mathbb Z$ has the property that
$\lim\limits_{m \to \infty} \widehat {\nu_v^U}(n_m) = \|v\|_H^2$.
\end{rem}
Proposition ~\ref{rigidfacts} shows that
if $(n_m)$ is a rigidity sequence for $T$ for
a given function $F$, then for any spectral measure
$\nu_f^T \ll \nu_F^T$, we would also have $z^{n_m} \to 1$
in measure with respect to $\nu_f^T$. Hence, $(n_m)$ would
be a rigidity sequence for $T$ for the function $f$ too. It
follows then easily that $T$ is rigid and has a rigidity
sequence $(n_m)$ if and only if $(n_m)$ is a rigidity sequence for
$F$ where $\nu_F^T$ is a maximal spectral measure for
$T$.
\begin{cor}\lambdabel{thouvenot}
$T$ is rigid if and only if for each
function $f\in L_2(X,p)$ there exists $(n_m)=(n_m(f))$ such that
$f{\mathcal I}rc T^{n_m}\to f$ in $L_2(X,p)$.
\end{cor}
\begin{rem} It is clear that an argument like this works equally
well for a unitary transformation $U$ of a separable Hilbert space
$H$. That is, there is one sequence $(n_m)$ such that for all $v \in H$,
$\|U^{n_m}v - v\|_H \to 0$ as $m \to \infty$ if and only if
for every vector $v \in H$, there exists a
sequence $(n_m)$ such that
$\|U^{n_m}v - v\|_H \to 0$ as $m \to \infty$
\end{rem}
\begin{rem} J.-P.
Thouvenot was the first to observe that $T$ is rigid if and only if
for each $f \in L_2(X,p)$ (or just for each characteristic
function $f = 1_A, A \in \mathcal B$), there exists $(n_m)$ depending on
$f$ such
that $\|f{\mathcal I}rc T^{n_m} - f\|_2 \to 0$ as $m \to \infty$.
There are a number of different ways to prove this. We have
given one such argument above. Another argument
would use the characterization up to isomorphism of
unitary operators as multiplication operators. Here is
an interesting approach via Krieger's Generator Theorem;
see Krieger~{\mathcal I}te{Krieger}.
It is sufficient to prove rigidity holds assuming that one has the weaker
condition of there being rigidity sequences for each characteristic
function. Suppose that an automorphism $T$ has the property
that for each set
$A\in\mathcal B$ there exists $(n_m)=(n_m(A))$ such that
$p\left(T^{-n_m}A\triangle A\right)\to 0$.
Then all spectral measures of functions of the
form $1_A$, $A\in\mathcal B$ are singular, and since the family of
such functions is linearly dense, the maximal spectral type of $T$
is singular. It follows that $T$ has zero entropy;
see Remark~\ref{Rokhlinsingular}
for an explanation of this point. Hence, by Krieger's
Generator Theorem, there exists a two element partition
$P=\{A,A^c\}$ which generates $\mathcal B$. Now, let
$(n_m)=(n_m(A))$ and notice that for each $k\geq1$ and for each
$B\in\bigvee_{i=0}^{k-1}T^iP$ we have $p(T^{n_m}B\triangle
B)\to0$. Hence by approximating the $L_2(X,p)$ functions by
simple functions, $(n_m)$ is a rigidity sequence for $T$.
\end{rem}
\begin{rem}\lambdabel{Rokhlinsingular}
From Proposition~\ref{rigidfacts}, we see that a maximal spectral
measure $\nu^T$ of a rigid transformation is
a {\em Dirichlet measure}. This means that for some increasing sequence
$(n_m)$, we have $\gamma^{n_m} \to 1$ in measure with respect to $\nu^T$
as $m \to \infty$. Hence, as in Proposition~\ref{rigidfacts},
we have $\widehat{\nu^T}(n_m)\to \nu^T(\mathbb T)$ as $m \to \infty$. A measure
with this property is also sometimes called a {\em rigid measure}. Note that a
measure absolutely continuous with respect to a Dirichlet measure
is a Dirichlet measure. So by the Riemann-Lebesgue Lemma,
there is no
non-zero positive measure $\nu$ which is absolutely continuous with
respect to Lebesgue measure
such that $\nu \ll\nu_f^T$ for a non-zero spectral measure $\nu_f^T$ of a
rigid transformation.
Therefore, for a rigid transformation, all spectral measures, and
$\nu^T$ itself, are Dirichlet measures and hence
singular measures. So $T$ has singular maximal spectral type.
Rokhlin shows in his
classical paper ~{\mathcal I}te{Rokhlin} that if $T$ has positive
entropy, then for every maximal spectral measure $\nu_F^T$, there is a
non-zero spectral
measure $\nu_f^T \ll \nu_F^T$ that is equivalent
to (mutually absolutely continuous with respect to) Lebesgue measure.
Therefore, all rigid transformations have zero entropy.
\end{rem}
One can often use Baire category arguments to distinguish the
behavior of transformations. For this we use the Polish group
$Aut(X,\mathcal B,p)$ of invertible measure-preserving transformations
on $(X,\mathcal B,p)$, with the topology of strong operator convergence. That
is, a sequence $(S_n)$ in $Aut(X,\mathcal B,p)$ converges to $S \in
Aut(X,\mathcal B,p)$ if and only if $\|f{\mathcal I}rc S_n- f{\mathcal I}rc S\|_2 \to 0$
as $n\to \infty$ for all $f \in L_2(x,p)$. By a {\em generic property}, we mean that the
property holds on at least a dense $G_\delta$ subset of $Aut(X,\mathcal
B,p)$, and any set containing a dense $G_\delta$
set is called a {\em generic set}. So a generic property is one that holds on a set whose
complement is first category.
For example, it is well-known that the generic dynamical system is
weakly mixing. See Halmos~{\mathcal I}te{Halmos} where this was used to give
a Baire category argument for the existence of weakly mixing transformations
that are not strongly mixing. Also, the generic transformation has
a rigidity sequence. See Katok and Stepin~{\mathcal I}te{Ka-St} and
Walters~{\mathcal I}te{Walters2}.
Hence, the generic transformation is
weakly mixing, rigid, and has zero entropy (see Remark~\ref{Rokhlinsingular}).
We will show this in a slightly more
general setting.
First, in order to see that a generic transformation is rigid we will
prove the following stronger result. This result may be well-known,
but we provide a proof because we could not find
a good reference for it. In this proof, and then later in
Section~\ref{rankone},
{\em rank one transformations} arise. These are transformations obtained
by cutting and stacking where at each inductive stage only one Rokhlin
tower is used. See Nadkarni~{\mathcal I}te{Nadkarni} and
Ferenczi~{\mathcal I}te{Ferenczi} for background information about rank one transformations.
\begin{prop}\lambdabel{folklore1} Given an increasing sequence $(n_m)$ of
natural
numbers, let ${\mathcal G}_{(n_m)}$ be the set consisting of all
$S\in Aut(X,\mathcal B,p)$ such that
$S^{n_{m_k}}\to Id$ in the strong operator topology
for some subsequence $(n_{m_k})$ of
$(n_m)$. Then ${\mathcal G}_{(n_m)}$ is a generic subset of
$Aut(X,\mathcal B,p)$.
\end{prop}
\begin{proof}
We can obtain a metric $d$ for $Aut(X,\mathcal B,p)$ that is
compatible with the strong topology as follows. Take
$\{A_i:\:i\geq1\}$ which is a dense subset
in $(\mathcal B,p)$. Let $d$ be given by
\begin{equation}\lambdabel{metric}d(R,S)=\sum_{i=1}^\infty\frac1{2^i}
(p\left(RA_i\triangle SA_i\right)+p\left(R^{-1}A_i\triangle
S^{-1}A_i\right))\end{equation}
It follows that given $n\in{\mathbb{Z}}$ and $\varphirepsilon>0$ the set
$$
\{S\in Aut(X,\mathcal B,p):\:d\left(S^n,Id\right)<\varphirepsilon\}$$ is open, and
therefore the set
$$
A_{k,\varphirepsilon}:=\{S\in Aut(X,\mathcal B,p):\: d\left(S^{n_q},Id\right)<\varphirepsilon \
\text{for some}\ q \ge k\}$$ is open as well.
Also, $A_{k,\varphirepsilon}$ is dense.
Indeed, given $A_1,\ldots,A_m$ and $R\in Aut(X,\mathcal B,p)$, we can construct $S\in
A_{k,\varphirepsilon}$ so that \begin{equation}\lambdabel{p1} p(SA_i\triangle RA_i)\;\mbox{is
as close to zero as we like,}\end{equation} and also for some $q\geq k$
\begin{equation}\lambdabel{p2}p(S^{n_q}A_i\triangle A_i)\;\mbox{is as close to
zero as we like}\end{equation} for $i=1,\ldots,m$. Actually, as needed,
the argument below will show the
same facts hold if we replace $R$ and $S$ by their inverses. With no loss of
generality, we can assume that $R$ is of rank one as this family
is dense in $Aut(X,\mathcal B,p)$; see ~{\mathcal I}te{Nadkarni} and ~{\mathcal I}te{Ferenczi}.
This allows us to approximate
the sets $A_1,\ldots,A_m$ by unions of levels of large Rokhlin towers for $R$. Now fix
$n_q$. We will see that the only condition on $n_q$ will be that
$n_q\to\infty$. Let $h_s$ be the height of a Rokhlin tower for
$R$ so that the levels of the tower can be used to
approximate the sets $A_1,\ldots,A_m$. We also assume
without loss of generality that $h_s$ is a (large) multiple of
$n_q$. We now divide
this tower into consecutive subtowers (without changing the
levels) of height $n_q$. This is done by taking the first $n_q$ levels, then
the next $n_q$ levels, etc. Then define $S$ in the following way:
inside each subtower of height $n_q$, the automorphism $S$ acts as
$R$ except on the top level of the subtower on which we require that $S$ sends
this level into the bottom level of that subtower. For example,
for the first
subtower, the first level is sent into the second, the second to the third,
and so on, but the $n_q$-th level is mapped to the first. This same
pattern is used on the rest of the subtowers. Now, if $h_s/n_q$ is sufficiently
large, then we can see that we can well approximate each $A_i$ by a union of levels
of some of the $h_s/n_q$ Rokhlin subtowers (of height
$n_q$). Taking this approach, the errors in
(\ref{p1}),~(\ref{p2}) come only from the fact that these subtowers are
cyclically permutated by $S$. The total error is hence of order
$$
\frac{h_s}{n_q}{\mathcal D}ot\frac1{h_s}=\frac1{n_q}.$$
Hence, to get the approximations
we need, we only need to
know that $n_q\to \infty$.
Now take $0<\varphirepsilon_l\to0$ and notice that the set
$$
\bigcap_{l=1}^\infty\bigcap_{k=1}^\infty A_{k,\varphirepsilon_l}
$$
is included in ${\mathcal G}_{(n_m)}$.
\end{proof}
\begin{rem} This argument should be compared with the beginning of
the proof of Proposition~\ref{revise}. Also, see the end of
Section~\ref{contfrac} for another
approach using continued fractions that gives generic results.
\end{rem}
\begin{rem}\lambdabel{metric1} We also notice that the metric $d$
defined in~(\ref{metric}) has the following properties:
$d(R,S)=d(R^{-1},S^{-1})$ and $d(TR,TS)=d(R,S)$ once $T$ commutes
with $R$ and $S$. Denote $\|T\|=d(T,Id)$. Then
$$
\|T^{n+m}\|=d(T^{n},T^{-m}\|\leq
d(T^n,Id)+d(Id,T^{-m})=\|T^n\|+\|T^m\|.$$
\end{rem}
\subsection{Weak Mixing Specifically} \lambdabel{weakmixsec}
Now we consider weakly mixing transformations. Recall
that $T$ is {\em weakly mixing} if and only if for all $A,B \in \mathcal B$,
we have
$$\lim\limits_{N\to\infty}
\frac 1N\sum\limits_{n=1}^N |p(T^nA{\mathcal A}p B) - p(A)p(B)| = 0.$$
So $T$ is weakly mixing if and only if
for all mean-zero $f \in L_2(X,p)$, we have
$$\lim\limits_{N\to \infty} \frac 1N
\sum\limits_{n=1}^N |\lambdangle f{\mathcal I}rc T^{n},f\rangle| = 0.$$
Now recall Wiener's Lemma: given a positive Borel measure
$\nu$ on ${\mathbb{T}}$ we have
$$\lim\limits_{N \to \infty} \frac 1{2N+1}
\sum\limits_{n=-N}^N |\widehat{\nu}(n)|^2 =
\sum_{\gamma \in{\mathbb{T}}}\nu^2(\{\gamma\}).$$
It follows that $\nu$ is continuous (i.e. has no point masses)
if and only if $\lim\limits_{N \to \infty} \frac 1{2N+1}
\sum\limits_{n=-N}^N |\widehat{\nu}(n)|^2 =0$. The latter
condition is well-known to be equivalent to the fact that
$\widehat{\nu}(n)$ tends to zero along a subsequence of
density $1$. Here we say that a set $S \subset \mathbb N$
of density one if
\[\lim\limits_{N\to\infty} \frac 1N\#(S{\mathcal A}p
\{1,2,\ldots,N\}) = 1.\]
So a transformation $T$ is weakly mixing if and only if $\nu_f^T$ is
continuous for each $f \in L_2(X,p)$ which is mean-zero, that is
(by Wiener's Lemma) we have $\lambdangle f{\mathcal I}rc T^{n},f\rangle$ tends
to zero along a sequence of density one. Denote a given such
density one sequence by $\mathcal N^T_f$. As $f$ changes, this
sequence generally might need to change. However, it is a well-known
fact that because $(X,\mathcal B,p)$ is separable,
we can choose a subsequence of density~$1$ which works for
all $L_2$-functions. We give a proof of this fact for the reader's
convenience. This proof is different than the one in
Petersen~{\mathcal I}te{Petersen}. See also Jones~{\mathcal I}te{LeeJones}.
\begin{prop} \lambdabel{wmone} Assume that $T$
is weakly mixing. Then
there is a sequence $(n_m)$ in $\mathbb Z^+$ of density one
such that for all mean-zero $f \in L_2(X,p)$, one has
$\lim\limits_{m \to \infty} \lambdangle f{\mathcal I}rc T^{n_m},f\rangle = 0$.
\end{prop}
\begin{proof} Let $(f_s)$ be a sequence of non-zero mean-zero functions
which is dense in the subspace of $L_2(X,p)$
consisting of the mean-zero functions. Consider the measure
$\omega = \sum\limits_{s=1}^\infty \frac 1{2^s\|f_s\|_2^4}\,
\nu_{f_s}^T\ast(\nu_{f_s}^T)^*$. This is a continuous measure with a
positive Fourier transform.
Hence, there is a sequence $(n_m)$ in $\mathbb Z^+$
of density one such that $\widehat {\omega}(n_m) \to 0$ as $m \to
\infty$. For every $s\geq1$, we have $2^s\|f_s\|_2^4 \,\widehat
{\omega}(n)\ge |\widehat {\nu_{f_s}^T}(n)|^2$ for all
$n \in \mathbb Z$. So it follows that for every $s\geq1$, we also
have $\widehat {\nu_{f_s}^T}(n_m) \to 0$ as $m \to \infty$.
Then, by a standard approximation argument, for any mean-zero function
$f \in L_2(X,p)$, we have $\widehat
{\nu_f^T}(n_m) \to 0$ as $m \to \infty$.
\end{proof}
\begin{rem} This result also holds for a unitary operator $U$ on
a separable Hilbert space $H$. That is, if all the spectral measures
$\nu_v^U$ for $v \in H$ are continuous, then there exists a sequence
$(n_m)$ of density one such that $\widehat {\nu_v^U}(n_m) \to 0$ as
$m \to \infty$.
\end{rem}
Let $L_{2,0}(X,p)$ denote the mean-zero functions in
$L_2(X,p)$. We can rewrite the assertion of Proposition~\ref{wmone} as
\begin{equation}\lambdabel{ww3} \mbox{$U_T^{n_m}\to 0$ weakly in the space
$L_{2,0}(X,p)$}.\end{equation} Each sequence $(n_m)$ (not necessarily of
density~$1$) of integers for which~(\ref{ww3}) holds is called a
{\em mixing sequence} for $T$. Any
transformation possessing a mixing sequence is weakly mixing.
The following result about mixing subsequences is also folklore.
\begin{prop}\lambdabel{mixresidual}Given an increasing sequence $(n_m)$ of
natural
numbers, consider the set $\mathcal M_{(n_m)}$ that consists of all
$S\in Aut(X,\mathcal B,p)$
such that $S^{n_{m_k}}\to 0$ weakly in $L_{2,0}(X,p)$, for some
subsequence $(n_{m_k})$
of $(n_m)$. Then $\mathcal M_{(n_m)}$ is a generic subset of
$Aut(X,\mathcal B,p)$.\end{prop}
\begin{proof} Let $\{A_i:\:i\geq1\}$ be a dense family in
$(\mathcal B,d)$. Take $\varphirepsilon>0$ and set
$$
{\mathcal M}(k,\varphirepsilon)=\{S\in Aut(X,\mathcal B,p):\:\sum_{i,j=1}^\infty
\frac1{2^{i+j}}\left|p(S^{-n_k}A_i{\mathcal A}p
A_j)-p(A_i)p(A_j)\right|<\varphirepsilon\}.
$$
Notice that ${\mathcal M}(k,\varphirepsilon)$ is open and, for
$0<\varphirepsilon_i\to0$, consider the set
$$
{\mathcal M}= \bigcap_{i=1}^\infty\bigcup_{k=i}^\infty {\mathcal
M}(k,\varphirepsilon_i).$$ It is not hard to check that $\bigcup_{k=i}^\infty
{\mathcal M}(k,\varphirepsilon_i)$ is dense (each mixing
transformation belongs to it), so $\mathcal M$ is a $G_\delta$ and
dense. If $T\in{\mathcal M}$ then for each $i\geq1$ there exists
$k_i\geq i$ such that
$$
\sum\limits_{r,s=1}^\infty\frac1{2^{r+s}}\left| p(T^{-n_{k_i}}A_r{\mathcal A}p
A_s)-p(A_r)p(A_s)\right|<\varphirepsilon_i.$$ Hence for each $r,s\geq1$
$$
\left| p(T^{-n_{k_i}}A_r{\mathcal A}p
A_s)-p(A_r)p(A_s)\right|\to0\;\mbox{when}\;i\to\infty$$ and
therefore $(n_{k_i})$ is a mixing sequence for $T$.
\end{proof}
\begin{rem}\lambdabel{wmisgeneric} It is easy to see that
Proposition~\ref{mixresidual} shows that weakly mixing transformations $\mathcal W$
are generic set because
they can be characterized as having only the trivial eigenvalue $1$ with
the eigenvectors being the constant functions. Now taking $n_m = m$ for
all $m$, we have any transformation with a non-trivial eigenvalue
must be in $\mathcal M(n_m)^c$. So $\mathcal W^c \subset \mathcal M(n_m)^c$,
and $\mathcal M(n_m) \subset \mathcal W$. Actually, it is also well-known
that $\mathcal W$ itself is a $G_\delta$ set.
\end{rem}
Combining our two basic category results, Proposition~\ref{folklore1} and
Proposition~\ref{mixresidual}, gives the following.
\begin{prop} \lambdabel{together} Given an increasing sequence $(n_m)$ of
natural
numbers, consider the set $\mathcal B_{(n_m)}$ that consists of all
$S\in Aut(X,\mathcal B,p)$
such that $S^{n_{m_k(1)}}\to Id$ weakly in $L_{2,0}(X,p)$, for some
subsequence $(n_{m_k(1)})$
of $(n_m)$ and
such that $S^{n_{m_k(2)}}\to 0$ weakly in $L_{2,0}(X,p)$, for some
subsequence $(n_{m_k(2)})$
of $(n_m)$. Then $\mathcal B_{(n_m)}$ is a generic subset of
$Aut(X,\mathcal B,p)$.
\end{prop}
\begin{rem}\lambdabel{wmandrigid} The category result in Proposition~\ref{together}
also holds if we ask
for the stronger property that $S^{\sigmagma}\to Id$ weakly in $L_{2,0}(X,p)$,
as $\sigmagma \to \infty(IP)$ for the IP set generated by some
subsequence $(n_{m_k})$ of $(n_m)$. See Proposition~\ref{spectralIP} and the
discussion before it for the definition and basic characterization of IP rigidity.
\end{rem}
\begin{rem} We can also formulate unitary versions of Proposition~\ref{folklore1},
Proposition~\ref{mixresidual}, and Proposition~\ref{together}.
\end{rem}
\subsection{Rigidity Specifically} \lambdabel{rigidonly}
In addition to the examples given inherently by Proposition~\ref{together}, each
ergodic transformation with discrete spectrum is rigid. One
can see this in several ways. One way is to note that $T$ is rigid
for each eigenfunction $f$ since if $\gamma \in \mathbb T$ there is
a sequence $(n_m)$ such that $\gamma^{n_m} \to 1$ in $\mathbb T$. Then
use the principle of Corollary~\ref{thouvenot}. Alternatively, in
order to see this via the Halmos-von Neumann Theorem, consider an ergodic
rotation $Tx=x+x_0$ where $X$ is a compact metric monothetic
group, $x_0$ is its topological cyclic generator, and $p$ stands
for Haar measure of $X$. Take any increasing sequence $(n_t)$ of
integers, and consider $(n_t{\mathcal D}ot x_0)$. By passing to a
subsequence if necessary, we can assume that $n_tx_0\to y\in X$.
This is equivalent to saying that $T^{n_t}\to S$, where $Sx=x+y$.
Because the convergence is taking part in the strong operator
topology, it is not hard to see that we will obtain
$$
T^{n_{t_{k+1}}-n_{t_k}}\to S{\mathcal I}rc S^{-1}=Id,
$$
and therefore $T$ is rigid (indeed, $n_{t_{k+1}}-n_{t_k}\to\infty$ by
Proposition~\ref{sparse} below). These arguments show that
each purely atomic measure is a Dirichlet measure.
Moreover, we have also shown that in the discrete spectrum case
the closure of $\{T^n: n \in \mathbb Z\}$ in the strong
operator topology is compact. The converse is also true.
See for example Bergelson and Rosenblatt~{\mathcal I}te{bergros} and
Ku\v{s}hnirenko~{\mathcal I}te{Ku}. It also is not difficult to see that
the centralizer of $T$ in $Aut(X,\mathcal B,p)$, denoted by $C(T)$, can be identified with
this closure and so is compact in the strong operator topology.
The converse of this is also true (see again, e.g.\ {\mathcal I}te{Ku}).
Moreover, $T$ is isomorphic to the translation by $T$ on $C(T)$ considered
with Haar measure.
We will see
later that ergodic transformations with discrete spectrum are completely determined by their
rigidity sequences (see Corollary~\ref{AA2} below). There we will be
using the information summarized here.
A positive finite Borel measure $\nu$ on ${\mathbb{T}}$ is called a {\em
Rajchman measure} if its Fourier transform vanishes at infinity,
that is
\begin{equation}\lambdabel{ww7} \widehat{\nu}(n)\to 0\;\;\mbox{when}\;
|n|\to \infty.\end{equation}
So the spectral measures of a strongly mixing transformation
are Rajchman measures, and $T$ is strongly mixing if and only if the
maximal spectral type $\nu^T$ is a
Rajchman measure. Moreover, by the {\em Gaussian measure space construction}
(GMC) discussed in Remark~\ref{GMC}, any
Rajchman measure is one of the spectral measures for some strongly mixing
transformation. It is not hard to
see that a measure absolutely continuous with respect to a
Rajchman measure is Rajchman. Also, Rajchman measures and
Dirichlet measures are mutually singular.
\begin{rem} It would be interesting to characterize
the sets $\mathcal N = \{n_m\}$ of density one that occur in
Proposition~\ref{wmone}. This means
characterizing sets $\mathcal L$ of density zero that are the
complements of such sets. Characterizing rigidity
sequences for weakly mixing transformations means
characterizing certain types of sets $\mathcal L$.
However, this may not capture all sets in $\mathcal N$.
For example, it may be possible for a set $\mathcal N$
to fail to have a rigidity sequence in its complement,
but contain a set of the form $\{n\ge 1: |\widehat {\mu}(n)| \ge \delta \}$
for some $\delta > 0$, e.g. with $\mu$ that is a spectral measure
for a mildly mixing, not strongly mixing, transformation.
\end{rem}
Proposition~\ref{wmone} certainly shows that rigidity
sequences for weakly mixing transformations are density zero.
Proposition~\ref{sparse} below shows also that more than this is
true without the assumption that $T$ is weakly mixing.
There is a general principle in play here, but the
argument has to be different when there are eigenfunctions. If the
system is not ergodic, or if some power $T^n$ is not ergodic, then
there can exist a non-zero, mean-zero function $f \in L_2(X,p)$ and
a periodic sequence $(n_m)$ such that $f{\mathcal I}rc T^{n_m} = f$ for all
$m \ge 1$. Otherwise, the only way a sequence can exhibit
rigidity for a function, or for the whole dynamical system, is
when the sequence has gaps tending to $\infty$, and hence is
certainly of density zero. We recall that our probability spaces
are standard Lebesgue spaces and so have no atoms. This is important
in the next result where the Rokhlin Lemma is used.
\begin{prop}
\lambdabel{sparse} Let $(n_m)$ be an increasing sequence of integers.
\noindent a) Let $T$ be totally ergodic. If $\|f_0{\mathcal I}rc T^{n_m} -f_0
\|_2 \to 0$ as $m \to \infty$ for some non-zero, mean-zero $f_0 \in
L_2(X,p)$, then the sequence $(n_m)$ has gaps
tending to $\infty$ and hence has zero density.
\noindent b) Suppose $T$ is ergodic. If $\|f{\mathcal I}rc T^{n_m} -f \|_2
\to 0$ as $m \to \infty$ for all $f \in L_2(X,p)$, then $(n_m)$ has
gaps tending to $\infty$ and hence has zero density.
\end{prop}
\begin{proof} In a) we claim that $n_{m+1} - n_m
\to \infty$ as $m \to \infty$. Otherwise, there would be a value
$d \ge 1$ such that $d = n_{m+1} - n_m$ infinitely often. It
follows that $f_0{\mathcal I}rc T^d = f_0$. This is not possible since
$f_0$ is non-zero and mean-zero, and $T$ is totally ergodic. To
prove b), one again argues that $n_{m+1} - n_m \to \infty$ as $m
\to \infty$ since otherwise there exists $d \ge 1$ such that $d =
n_{m+1} - n_m$ for infinitely many $m$, and hence $f {\mathcal I}rc T^d =
f$ for all $f \in L_2(X,p)$. But this is impossible since our
system is ergodic. Indeed, for any $d_0$, using the Rokhlin Lemma,
there is a set $B$ of positive measure such that $T^jB$ are
pairwise disjoint for all $j, 1 \le j \le d_0$. Take $f_0$
supported on $B$ that is non-zero and mean-zero. Then $f_0{\mathcal I}rc
T^j \not= f_0$ for all $j, 1 \le j \le d_0$. Hence, once $d_0 > d$,
we cannot have $f_0{\mathcal I}rc T^d = f_0$.
\end{proof}
\begin{rem} Consider part b) above in the case of suitable unitary
operators. Since we used the Rokhlin Lemma, we would need a different
proof to show that a rigidity sequence for a unitary operator
has gaps tending to infinity. This can be seen by the above if
the operator has an infinite discrete spectrum. An
additional argument is needed in case all the non-trivial spectral
measures are continuous. Then using the GMC (see Remark~\ref{GMC})
and the result in part a) gives
the result in this case too.
\end{rem}
We will be constructing various examples of rigidity sequences
in Section~\ref{methods}. To have some contrast with these
constructions, it is worthwhile to make some remarks now about
sequences that cannot be rigidity sequences. We have seen from the
above, that rigidity sequences must have gaps growing to infinity.
But much more structural information is needed to guarantee that the
sequence can be a rigidity sequence.
For example, we have the following basic result.
\begin{prop} \lambdabel{unifdist}
Suppose $(n_m)$ is an increasing sequence such that
$(n_mx\mod 1)$
is uniformly distributed for all but a countable set of values $x \in \mathbb R`$. Then
$(n_m)$ cannot be a rigidity sequence for a weakly mixing transformation.
\end{prop}
\begin{proof} For any continuous measure $\nu$ on $\mathbb T$, we would
have $\int \frac 1M\sum\limits_{m=1}^M \gamma^{n_m} \,d\nu(\gamma) \to 0$ as
$M \to \infty$ because $\frac 1M\sum\limits_{m=1}^M \gamma^{n_m} \to 0$
as $M\to \infty$ for all but countable many $\gamma$. Hence, we cannot
have $\frac 1M\sum\limits_{m=1}^M \gamma^{n_m} \to 1$ in measure with
respect
to $\nu$ as $M \to \infty$.
\end{proof}
\begin{rem} \lambdabel{UDeg} See Kuipers and
Niederreiter~{\mathcal I}te{KN} for information about uniform distribution of sequences.
For example, Vinogradov proved that the prime numbers $(p_m)$ in
increasing order satisfy the hypothesis in Proposition~\ref{unifdist}.
So the prime numbers cannot
be a rigidity sequence for a weakly mixing dynamical system. Of
course, the property in Proposition~\ref{unifdist} also shows that
they cannot be a rigidity sequence for an ergodic rotation
of $\mathbb T$. Actually, they cannot be
a rigidity sequence for any ergodic transformation
with discrete spectrum on a Lebesgue space by Proposition~\ref{linform}. However,
the property in Proposition~\ref{unifdist} cannot be used to argue this
because there are ergodic transformations with discrete spectrum
whose spectral measures are supported on the roots of unity. Other examples
using Proposition~\ref{unifdist} include polynomial sequences $(p(m): m \ge 1)$, with $p$
a non-zero polynomial with integer coefficients.
However, we can reach the same conclusion for
polynomial sequences $(p(m): m \ge 1)$ by a simpler argument using successive
differences. See Remark~\ref{linformeg} b) below.
\end{rem}
The proof that certain sequences, like the prime numbers, satisfy the
hypothesis in Proposition~\ref{unifdist} is linked to another property
that prohibits rigidity. First, we consider what happens when a linear
form on the sequence has bounded values. Recall that our underlying
probability space is a standard Lebesgue space.
\begin{prop}\lambdabel{linform} Suppose $F(x_1,\ldots,x_K) = \sum\limits_{k=1}^K c_kx_k$
where $c_1,\ldots,c_K \in \mathbb Z\backslash \{0\}$. Suppose $(n_m)$ is
a sequence of whole numbers. Assume that for some non-zero $d\in \mathbb Z$,
we know that for any $M \ge 1$, there are
$m_k \ge M$ for all $k=1,\ldots,K$, such that
$F(n_{m_1},\ldots,n_{m_K}) = d$. Then $(n_m)$ is not a rigidity sequence
for an ergodic transformation.
\end{prop}
\begin{proof} Suppose $\nu$ is a Borel probability measure on $\mathbb T$
such that $\widehat {\nu}(n_m) \to 1$ as $m \to \infty$. Hence, $\gamma^{n_m} \to 1$
in measure with respect to $\nu$ as $m\to \infty$. Then also
$\gamma^d = \gamma^{F(n_{m_1},\ldots,n_{m_K})} \to 1$ in measure with respect
to $\nu$ as $m_k \to \infty$, for all $k=1,\ldots,K$. That is, $\nu$ is
supported on the $d$-th roots of unity. However, if $T$ is ergodic, we cannot have all
the spectral measures supported in the $d$-th roots of unity.
\end{proof}
\begin{rem}\lambdabel{linformeg} a) The proof is showing that given the
hypothesis of Proposition~\ref{linform}, $(n_m)$ cannot
be a rigidity sequence for a transformation $T$ and a specific function $f\in L_2(X,p)$
unless $f {\mathcal I}rc T^d = f$. So if $T$ is totally ergodic, $(n_m)$ cannot be
a rigidity sequence for any non-trivial function, let alone a rigidity sequence
for all functions.
\noindent b) It is easy to see that polynomial sequences $(p(n): n \ge 1)$ satisfy
the hypothesis of Proposition~\ref{linform} and therefore cannot be rigidity
sequences for weakly mixing transformations. The easiest way to see this is to
note that the difference $q(n) = p(n+1)-p(n)$ is a polynomial of less degree than $p$.
So successive differences will eventually lead to a constant. For example,
if $p(n) = n^2$, then let $q(n) = p(n+1) - p(n)$. Then $q(n+1)- q(n) = 2$.
So $p(n+2) - 2p(n+1) + p(n) = p(n+2)-p(n+1) - (p(n+1) - p(n)) = q(n+1) - q(n) = 2$. So
let $F(x,y,z) = x - 2y +z$. Then for all (large) $n$, $F(p(n+2),p(n+1),p(n)) = 2$.
Hence, $(n^2)$ cannot be a rigidity sequence for a weakly mixing transformation.
\noindent c) Here are other simple examples of the above. Suppose $n_m= 2^m+1$.
Then $2n_m - n_{m+1} = 1$. So if $F(x,y) = 2x-y$, then
$F(n_m,n_{m+1}) =1$ for all $m$. Also, suppose $n_m = 2^m+m$.
Then $2n_m - n_{m+1} = m -1$. Hence, $2n_{m+1} - n_{m+2} - (2n_m - n_{m+1}) = 1$
for all $m$. So with $F(x,y,z) = 2y-z-(2x - y) = 3y - z -2x$, we have
$F(n_m,n_{m+1},n_{m+2}) = 1$ for all $m$.
Therefore, both $(2^n +1)$ and $(2^n+n)$ are not
rigidity sequences for a weakly mixing transformation. It is not hard to see that
a calculation of this sort can also be carried out for any sequence $(2^n+p(n))$ where
$p$ is a non-zero polynomial with integer coefficients. Hence, such sequences
are generally not rigidity sequences for weakly mixing transformations.
See Proposition~\ref{integerratios} which shows that $(2^m)$ itself
is a rigidity sequence for a weakly mixing transformation.
\noindent d) Another example will give some idea of other issues that can arise
in using Proposition~\ref{linform}. Take the sequence $(2^n+p_n)$ where $(p_n)$
is the prime numbers in increasing order. If this is a rigidity sequence, then
so is $(a_n) = (p_{n+1} - 2p_n)$ since $a_n = 2^{n+1} + p_{n+1} - 2(2^n +p_n)$.
It is not clear what holds for this resulting
sequence. For example, does this $(a_n)$ satisfy the hypothesis in Proposition~\ref{unifdist}?
\noindent e) Proposition~\ref{linform} can also be used in a positive way.
For example, start with the fact that $(2^n)$ is a rigidity sequence for a weakly
mixing transformation proved in Proposition~\ref{integerratios}. It follows
that for any $m_1,\ldots,m_K$ and $N_1,\ldots,N_K$, the sequence $(n_m) =
(\sum\limits_{k=1}^K m_k2^{N_k+m})$ is also a rigidity sequence.
\end{rem}
Proposition~\ref{linform} in turn can be used to prove the following result.
This result was suggested by the fact that certain sequences, like the prime
numbers, were originally seen to satisfy the hypothesis of Proposition~\ref{unifdist}
by using analytic number theory arguments which also gave the hypothesis
of Proposition~\ref{sumset}. We will use in the next proposition the notion
of upper density: given a set $A = \{a_n:n \ge 1\}$ of integers, the
upper density of $A$ is
\[\limsup\limits_{N\to \infty} \frac {\#(\{a_n: n \ge 1\}{\mathcal A}p \{-N,\ldots,N\})}{2N+1}.\]
\begin{prop}\lambdabel{sumset} Suppose that $\mathbf a$ is a sequence of whole numbers. Assume that
the set $A = \{a_n:n\ge 1\}$ has the property
that for some integers $c_1,\ldots,c_L$ the set of sums $c_1A+\ldots+c_LA =
\{\sum\limits_{l=1}^L c_la_{n_l}: n_1,\ldots,n_L \in A\}$ has
positive upper density. Then $\mathbf a$ cannot be a rigidity sequence for an
ergodic transformation.
\end{prop}
\begin{proof}
Suppose that the condition on the set of sums holds and $L$ is the smallest possible value
for which this holds for some $c_1,\ldots,c_L$. Let $B = c_1A+\ldots+c_LA$ and
assume it has upper density $D > 0$.
Consider the set of sums $B_N =\{\sum\limits_{l=1}^L c_l a_{n(l)}:
n(1),\ldots,n(L) \ge N\}$. $B_N$ can be obtained from $B$ by deleting a finite set and
a finite number of translates
of sets of sums of the form $c_{i_1}A+\ldots+c_{i_{L^\prime}}A$ with $L^\prime < L$. Since these sets of sums
are assumed to all be of upper density zero, $B_N$ also has upper density $D$.
Then there are infinitely many pairs $\sigmagma_1 < \sigmagma_2$ with $\sigmagma_1,\sigmagma_2 \in B_N$
and $\sigmagma_2 = d + \sigmagma_1$ for some non-zero $d \le 2\frac 1D$. So let $K = 2L$ and $F(x_1,\ldots,x_K) =
\sum\limits_{k=1}^Lc_kx_k - \sum\limits_{k=L+1}^Kc_{k-L}x_k$. With this linear form $F$, we have shown
that $A$ satisfies the hypothesis of Proposition~\ref{linform}.
\end{proof}
\begin{rem} \lambdabel{sumseteg}
Proposition~\ref{sumset}
clearly applies to squares $S = \{n^2: n \ge 0\}$. Indeed, $(n+1)^2 - n^2 = 2n+1$,
so the odd numbers are a subset of $S - S$. It also applies to the prime numbers $P$,
because of the well-known fact that for some whole number $K$, the sum of $K$
copies of $P$ contains all of the whole numbers $n \ge 3$.
\end{rem}
By using spectral measures and the {\it Gaussian measure space
construction} (denoted here by GMC), we can see that our basic
desire to characterize rigidity sequences
is equivalent to a fact about Fourier transforms of
measures.
\begin{rem}\lambdabel{GMC} The GMC is a standard method of creating a weakly mixing transformation $G_{\nu}$
such that one of its spectral measures is a given continuous measure $\nu$.
The transformation $G_{\nu}$ itself is a coordinate shift on an infinite product space,
but the probability measure on the product space that it leaves invariant must be constructed
specifically with $\nu$ in mind. See Cornfeld, Fomin, and Sinai~{\mathcal I}te{CFS} for details
about the GMC. We will use the notation $G_{\nu}$ for the transformation
obtained by applying GMC to the positive Borel measure $\nu$. One
can actually see that this method applies to all measures by using
complex scalars, but it is traditional to apply it to symmetric
measures whose Fourier transform is real-valued so that GMC gives
a real centered stationary Gaussian process. In all of our
applications, we can symmetrize the measures $\nu$ that we construct
by replacing them with $\nu_s = \nu\star\nu^*$, or with $\nu_s = \nu +
\nu^*$. This
will allow us to use the GMC in its traditional form, while still
preserving the properties that we need the GMC to give us.
\end{rem}
\begin{rem}\lambdabel{Poisson} We will also have occasion to use another
general method, that of Poisson suspensions. See
Cornfeld, Fomin, and Sinai~{\mathcal I}te{CFS},
Kingman~{\mathcal I}te{Poisson}, and
Neretin~{\mathcal I}te{Poisson1}
for information about Poisson suspensions.
Here is briefly the idea.
Let $T$ be a transformation of a standard Lebesgue space $(X,{\mathcal
B},\mu)$, where $\mu$ is a $\sigmagma$-finite, infinite positive measure. We define a
probability space $(\widetilde{X},\widetilde{\mathcal B},
\widetilde{\mu})$. The points of the configuration space
$\widetilde{X}$ are infinite countable subsets
$\widetilde{x}=\{x_n:\:n\geq1\}$ of $X$. Given a set $A\in{\mathcal
B}$ of finite measure we define $N_A:\widetilde{X}\to{\mathbb{N}}{\mathcal U}p \{\infty\}$ by
setting
$$
N_A(\widetilde{x})=\#\{n\in {\mathbb{N}}:\; x_n\in A\}.
$$
Then $\widetilde{{\mathcal B}}$ is defined as the smallest $\sigmagma$-algebra
of subsets of $\widetilde{X}$ making all variables $N_A$,
$\mu(A)<+\infty$, measurable. The measure $\widetilde{\mu}$ is the
only probability measure (see {\mathcal I}te{Poisson} for details) such
that \begin{itemize}\item the variables $N_A$ satisfy the Poisson
law with parameter $\mu(A)$;
\item
for each family $A_1,\ldots,A_k$ of pairwise disjoint subsets of
$X$ of finite measure the corresponding variables
$N_{A_1},\ldots,N_{A_k}$ are independent.
\end{itemize}
The space $(\widetilde{X},\widetilde{\mathcal B}, \widetilde{\mu})$ is
a standard Lebesgue probability space. Then, we define
$\widetilde{T}$ on $\widetilde{X}$ by setting
$$\widetilde{T}(\{x_n\})= (\{Tx_n\})$$ and obtain a transformation
of $(\widetilde{X},\widetilde{\mathcal B}, \widetilde{\mu})$ which is
called the Poisson suspension of $T$. Then $\widetilde{T}$ is
ergodic if and only if $T$ has no non-trivial invariant sets of
finite measure. In this case, $\widetilde{T}$ turns out to be
weakly mixing and moreover $\widetilde{T}$ is spectrally
isomorphic to the GMC transformation $G$ given by the unitary
operator $U_T$ acting on $L_2(X,{\mathcal B},\mu)$, i.e.\ the unitary
operators $U_{\widetilde{T}}$ and $U_G$ are equivalent.
\end{rem}
\begin{prop} \lambdabel{spectral}
The sequence $(n_m)$ is a rigidity sequence for some
weakly mixing dynamical system if and only if there is a
continuous Borel probability measure $\nu$ on $\mathbb T$ such
that $\lim\limits_{m \to \infty} \widehat {\nu}(n_m)= 1$.
\end{prop}
\begin{proof}
First, if $f \in L_2(X,p)$ is norm one, then rigidity along the
sequence $(n_m)$ for $f$ means that we will have $\lim\limits_{m \to
\infty} \widehat {\nu_f^T}(n_m) = 1$. So when $T$ is weakly
mixing and $f$ is mean-zero, then $\nu_f^T$ is continuous.
Conversely, if there is a continuous Borel probability measure
$\nu$ on $\mathbb T$ such that $\widehat {\nu}(n_m) \to 1$ as $m
\to \infty$, then the GMC gives us a weakly mixing
dynamical system $(X,\mathcal B,p, T)$, with $T = G_{\nu}$, and a mean-zero function $f \in
L_2(X,p)$ with $\|f\|_2^2 = 1$ and $\widehat {\nu}(n) = \lambdangle
f{\mathcal I}rc T^{n},f\rangle$ for all $n$. Indeed, it is not hard to see
that this construction gives us a weakly mixing dynamical system
such that $T$ is rigid along the sequences $(n_m)$.
\end{proof}
\begin{rem} The second part of this proof is saying that
the GMC preserves rigidity. In particular, if $\nu=\nu^S$
for some $S\in Aut(Y,\mathcal B_Y,p_Y)$ then $S$ and $T:=G_\nu$ have
the same rigidity sequences.
\end{rem}
\begin{rem} The unitary version of this result is that there is a
continuous Borel probability measure $\nu$ on $\mathbb T$ such that
$\lim\limits_{m \to \infty} \widehat {\nu}(n_m) = 1$ if and only
if there is a unitary operator $U$ on a Hilbert space $H$ such
that 1) for no non-zero $v \in H$ is the orbit $\{U^kv: k \mathbb Z\}$
precompact in the strong operator topology, and
2) for all $v \in H$, $\lim\limits_{m\to \infty} \|U^{n_m} v -v \|_H = 0$.
The first property here 1) is saying that $U$ is called a {\em weakly mixing unitary
operator}. See Bergelson and Rosenblatt~{\mathcal I}te{bergros}.
\end{rem}
The result in Proposition~\ref{spectral} holds
with appropriate changes if we ask for the
stronger property that we have rigidity along the IP
set generated by a sequence. Some notation is useful for
understanding this. The {\em IP set generated by a sequence} consists
of all finite sums of elements with distinct indices in the sequence. So the
notation $\Sigma = FS(n_m)$ for this IP set makes sense. Here
given
$m_1 < \ldots < m_k$, let $\sigmagma
= \sigmagma(n_{m_1},\ldots,n_{m_k}) = n_{m_1}+\ldots+n_{m_k}$. We say
$\sigmagma \to \infty (IP)$ if $\sigmagma = m_1$ tends to $\infty$.
Also, given the dynamical system, we say that $T$ is {\em
IP-rigid} along $\Sigma$ if $T^{\sigmagma} \to Id$ in the strong
operator topology as $\sigmagma \to \infty (IP)$. With this
understood, it is easy to see the following.
\begin{prop}\lambdabel{spectralIP} There is a weakly mixing
dynamical system that is IP-rigid along $\Sigma$ if and only if there
is a continuous Borel probability measure $\nu$ on $\mathbb T$
such that $\widehat {\nu}(\sigmagma) \to 1$ as $\sigmagma \in FS(n_m: m
\ge 1)$ tends to $\infty$ (IP).
\end{prop}
There is another way of phrasing the Fourier transform condition
above for rigidity, and IP-rigidity, that is very useful. We have
already pointed this out in terms of spectral measures in Proposition
~\ref{rigidfacts}. We leave
the routine proof to the reader.
\begin{prop} \lambdabel{inmeasure} Given a sequence $(n_m)$ and a
positive Borel measure on $\mathbb T$, we have $\widehat
{\nu}(n_m)$ tends to $\nu(\mathbb T)$ if and only if $\gamma^{n_m} \to
1$ in measure with respect to $\nu$ as $m \to \infty$. Also,
we have $\widehat {\nu}(\sigmagma)$ tends to $1$ as $\sigmagma \to \infty$ (IP)
if and only if $\gamma^{\sigmagma}\to 1$ in measure
with respect to $\nu$ as $\sigmagma \to \infty$ (IP).
\end{prop}
\begin{rem}\lambdabel{moreIP} a) The Fourier transform characterizations of rigidity sequences
above suggests that we might be able to take this further by
finding the correct growth/sparsity condition on a strictly
increasing sequence $(n_m)$ to be a rigidity sequence or
IP-rigidity sequence for a weakly mixing dynamical system. First,
consider the property of IP-rigidity. It is easy to see, from the
spectral measure characterization of IP-rigidity above, that it is
sufficient to have a criterion that guarantees there is an
uncountable Borel set of points $K \subset \mathbb T$ such that
for all $\gamma \in K$, we have $\gamma^{\sigmagma}\to 1$ as $\sigmagma
\to \infty (IP)$. See the beginning of the proof of Proposition~\ref{ABCD}
where the same point is made. Let us consider this in the parametrization of
$\mathbb T$ where $\gamma = \exp(2\pi i x)$ for $x \in [0,1)$. We
also denote by $K$ the set of $x$ corresponding to $\exp(2\pi ix)
\in K$. Our pointwise criterion then means that for all $x \in K$,
we have both ${\mathcal O}s(2\pi \sigmagma x) \to 1$ and $\sigman(2 \pi \sigmagma x)
\to 0$ as $\sigmagma \to \infty\,(IP)$. It is enough to just have
$\sigman(2 \pi \sigmagma x) \to 0$ as $\sigmagma \to \infty\,(IP)$. Indeed,
with the usual notation that
$\{z\} = z - \Lambda_\varphirphiloor z \rfloor$ is the fractional
part of a real number $z$, the $\{y = \{2x\}: x \in K\}$ will give
an uncountable set of values $y$ such that both ${\mathcal O}s(2\pi \sigmagma
y) \to 1$ and $\sigman(2 \pi \sigmagma y) \to 0$ as $\sigmagma \to
\infty\,(IP)$
\noindent b) The pointwise spectral property above is equivalent to
having rigidity along $\Sigma$ for ALL functions whose spectral
measure in a given dynamical system is supported in $K$. This is
stronger than what is needed for rigidity along $\Sigma$ for some
weakly mixing dynamical system. This weaker notion is equivalent
to having a continuous positive measure $\nu$ supported on $K$
such that $\exp(2\pi i\sigmagma x) \to 1$ in measure with respect to
$\nu$ as $\sigmagma \to \infty\, (IP)$.
\end{rem}
\section{\bf Constructions of Rigid Sequences for Weakly Mixing
Transformations}
\lambdabel{methods}
We give a number of different approaches here for constructing weakly
mixing transformations that have a specific type of sequence as a rigidity
sequence. These methods are sometimes overlapping, but the different
approaches give us insights into the issues nonetheless. Also, there
are a variety of number theoretic and harmonic analysis connections
with some of these methods; these are also explored in this section.
When this paper was in the final draft, we learned of the work of
Eisner and Grivaux~{\mathcal I}te{EG}. Our papers are largely complementary,
although we do cover some of the same basic issues.
We will cite their work more in place later in this section.
Besides the question of the structure of rigidity sequences in general, we would
like to be able to answer the following questions:
\noindent{\bf Questions}: Which rigidity sequences of an ergodic
transformation with discrete spectrum can be rigidity sequences
for weakly mixing transformation? Which rigidity sequences for
weakly mixing transformations can be rigidity sequences for an
ergodic transformation with discrete spectrum?
\begin{rem}\lambdabel{whyquestion} a) At this time, we do not know
if there is a counterexample to either of the questions above.
\noindent b) As discussed later in this section,
one viewpoint to answering these questions is to consider, for
fixed $(n_m)$, the group $\mathcal R(n_m) =
\{\gamma \in \mathbb T: \lim\limits_{m\to \infty} \gamma^{n_m} =1\}$.
We will see that if $\mathcal R(n_m)$ is uncountable, then $(n_m)$ is a rigidity
sequence for an ergodic rotation of $\mathbb T$ and for some weakly
mixing transformation. If $\mathcal R(n_m)$ is countably infinite,
then there is an ergodic transformation $T$ with discrete spectrum such that $(n_m)$
is a rigidity sequence for $T$, but we do not know in general if there
is a weakly mixing transformation with $(n_m)$ as a rigidity sequence.
For example, sequences
like $(2^n)$ cannot be a rigidity sequence for an ergodic
rotation of the circle, but can be a rigidity sequence for a weakly
mixing transformation. This does not make this sequence a
counterexample to the second question above because this sequence is a rigidity sequence for
the ergodic generator of another compact abelian group $G$.
Just take $G$ to be the inverse limit of the finite groups
$\mathcal R_n = \{\gamma \in \mathbb T: \gamma^{2^n} = 1\}$.
\noindent c) Our results, in particular Proposition~\ref{integerratios} or Proposition~\ref{intlac},
and the structure of subgroups of $\mathbb T$ in the discrete topology,
show that all rigidity sequences for transformations
with discrete spectrum are rigidity sequences for some weakly mixing transformation
if the following is true: given an element $\gamma \in \mathbb T$
of infinite order and a sequence $(n_m)$ such that $\gamma^{n_m} \to 1$
as $m \to \infty$, the sequence $(n_m)$ is a rigidity sequence for some
weakly mixing transformations.
\noindent d) If $\mathcal R(n_m)$ is finite, then no ergodic
transformation with discrete spectrum has this sequence as a rigidity
sequence, but it might be possible to construct weakly mixing transformations
with $(n_m)$ as a rigidity sequence. This is not at all clear yet. A good
example of a candidate sequence for this case is $n_m = 2^m+3^m$ for
which $\mathcal R(n_m) = \{1\}$. To see this, note that
$n_{m+1} - 2n_m = 3^m$ and $3n_m-n_{m+1} = 2^m$. So any $\gamma \in \mathcal R(n_m)$
must have both $\gamma^{3^m}$ and $\gamma^{2^m}$ tending to $1$
as $m \to \infty$. Hence, $\gamma$ is simultaneously a root of unity
for a power of $2$ and a power of $3$, and hence $\gamma = 1$.
\end{rem}
A sequence being just density zero is clearly not
enough for rigidity for an ergodic transformation
because a sequence can be density zero and
have infinitely many pairs of terms $(n_m,n_{m+1})$ with, say, $n_{m+1}
- n_m \le 10$. More sparsity is needed than just density zero. In this
direction, lacunary sequences might seem to be good candidates to be
rigidity sequences for some weakly mixing transformations because
they certainly have the necessary sparseness. We take lacunarity
here to be as usual: $(n_m)$ is lacunary if and only if there exists
$\rho >1$ such that $n_{m+1}/n_m \ge \rho$ for all $m \ge 1$.
However, even in this class of sequences the situation is not
clear as there are lacunary sequences which cannot even be rigidity
sequences for any ergodic transformations. For example, $(2^n+1)$
is lacunary but cannot be a rigidity sequence
even for an ergodic transformation with discrete spectrum.
See Remark~\ref{linformeg} c). This answers the question in
Eisner and Grivaux~{\mathcal I}te{EG}, p. 5.
\subsection{\bf Diophantine approach}
\lambdabel{diophantine}
\subsubsection{\bf Existence of Supports}\lambdabel{existenceofsupports}
Denote by $[x]$ the nearest integer to $x \in \mathbb R$, choosing
$\Lambda_\varphirphiloor x \rfloor$ if $\{x\} = 1/2$. So $|x - [x]|$ would be the
distance of $x$ to $\mathbb Z$. We will denote this by $\|x\|$.
The distance $\|x\| = \{x\}$ if $\{x\} \le \frac 12$ and $\|x\| =
1 - \{x\}$ if $\frac 12 \le \{x\}$. We have the following result.
\begin{prop} \lambdabel{ABCD} The following are equivalent:
\noindent a) there exists some
infinite perfect compact set $K \subset [0,1)$, such that
$\sigman(2\pi\sigmagma x) \to 0$ as $\sigmagma \to \infty
(IP)$ for all $x \in K$,
\noindent b) for some uncountable set of $x$ values, we have
\begin{equation}\lambdabel{ABS}
\sum\limits_{m=1}^\infty \|n_mx\| < \infty.
\end{equation}
\end{prop}
\begin{proof}
The condition $\sum\limits_{m=1}^\infty \|n_mx\| < \infty$ describes
a Borel set of values $x$. Hence, the set theoretic aspects of this proposition work
because any uncountable Borel set in $[0,1)$ contains an
infinite perfect compact subset. See Sierpi\'nski~{\mathcal I}te{Sierpinski},
p. 228.
Assume that
$\sum\limits_{m=1}^\infty \|n_mx\| < \infty$
for an uncountable set of $x$ values.
Consider separately the
values of $\|n_mx\|$ where it is $\{n_mx\}$ or it is $1 - \{n_mx\}$.
In the first case, as $m \to \infty$, $0\le \sigman(2\pi n_mx) \sigmam
2\pi \{n_mx\} = 2\pi\|n_mx\|$.
In the second case, as $m \to \infty$, $0\ge \sigman(2\pi n_mx) \sigmam
-2\pi (1 -\{n_mx\}) = -2\pi\|n_mx\|$.
So Equation\,(\ref{ABS}) implies that
$\sum\limits_{m=1}^\infty \sigman(2\pi n_mx)$ converges absolutely
for an uncountable set. Using the formula
\[\sigman(\alphapha+\beta) = \sigman(\alphapha){\mathcal O}s(\beta)+{\mathcal O}s(\alphapha)\sigman(\beta)\]
repeatedly, we see that $\sigman(2\pi x\sigmagma) = \sum\limits_{j=1}^k
f_j \sigman(2\pi x n_{m_j})$ with coefficients $f_j$ that
suitable products of cosines. Here $|f_j| \le 1$. So the
convergence in Equation\,(\ref{ABS}) tells us that for an
uncountable set we have $\sigman(2 \pi \sigmagma x) \to 0$ as $\sigmagma
\to \infty\ (IP)$.
Conversely, suppose we have for an uncountable set of $x$ such that
$\sigman(2 \pi
\sigmagma x) \to 0$ as $\sigmagma \to \infty\,(IP)$. By doubling the $x$
values, we see that this means that for an uncountable set $K$, if
$x \in K$, and $\epsilon > 0$, we can choose $M_{\epsilon} \ge 1$
such that for all finite sets $F$ of whole numbers, all no smaller
than $M_{\epsilon}$, we have $ \|\sum\limits_{m \in F} n_mx \|
\le \epsilon$. In particular, for all $x\in K$, we have $\|n_mx\|
\to 0$ as $m \to \infty$. It follows that there is an uncountable
set $K_0 \subset K$ and some $M_0$ such that for all $x \in K_0$
and all finite sets $F$ of whole numbers all no smaller than
$M_0$, we have $ \| \sum\limits_{m\in F} n_mx \| \le \frac
1{100}$. In particular, if $m \ge M_0$, $\| n_mx_0\| \le \frac
1{100}$. Now suppose $\sum\limits_{m=1}^\infty \| n_mx_0\| =
\infty$ for some $x_0 \in K_0$. Then also
$\sum\limits_{m=M_0}^\infty \| n_mx_0\| = \infty$. Each
$\|n_mx_0\|$ is either $\{n_mx_0\}$ or it is $1 - \{n_mx_0\}$.
Say $I_1$ is the set of $m \ge M_0$ where the first formula holds,
and $I_2$ is the set of $m \ge M_0$ where the second formula
holds. Then either $\sum\limits_{m\in I_1} \|n_mx_0\| = \infty$
or $\sum\limits_{m\in I_2} \|n_mx_0\| = \infty$. Assume it is the
first case. Then by an upcrossing argument, we can choose a
finite set $I \subset I_1$ such that $\sum\limits_{m\in I}\|
n_mx_0 \|\in (\frac 18,\frac 38)$. But then $\sigman(2\pi
\sum\limits_{m \in I} n_mx_0) = \sigman(2\pi \sum\limits_{m \in I} \|
n_mx_0 \|) \ge \frac 1{\sqrt 2}$. This is not possible for $x\in
K_0$ because we know that $ \|\sum\limits_{m\in I} n_m x_0 \|$ is
small and so $|\sigman(2\pi \sum\limits_{m\in I} n_mx_0)| =
|\sigman(2\pi \| \sum\limits_{m\in I}n_mx_0 \|)| \le 2\pi
\|\sum\limits_{m\in I} n_mx_0 \| \le 2\pi \frac 1{100}$. In the
second case, again by an upcrossing argument, we can choose a
finite set $I \subset I_2$ such that $\sum\limits_{m\in I}\|
n_mx_0 \|\in (\frac 18,\frac 38)$. So then $\sigman(2\pi
\sum\limits_{m \in I} n_mx_0) = -\sigman(2\pi \sum\limits_{m \in I}
\| n_mx_0 \|) \le -\frac 1{\sqrt 2}$. Again this is not possible
for $x\in K_0$ because we know that $ \|\sum\limits_{m\in I} n_m
x_0 \|$ is small and so $|\sigman(2\pi \sum\limits_{m\in I} n_mx_0)|
= |\sigman(2\pi \| \sum\limits_{m\in I}n_mx_0 \|)| \le 2\pi
\|\sum\limits_{m\in I} n_mx_0 \| \le 2\pi \frac 1{100}$.
\end{proof}
This result, and Remark~\ref{moreIP} a) at the end of Section~\ref{rigidity}, give
this basic result.
\begin{prop}\lambdabel{OKforIP} If there exists an uncountable set of
$x$ values such that $\sum\limits_{m=1}^\infty \|n_mx\| < \infty$,
then there is a weakly mixing transformation that is rigid along the
$IP$ set generated by $(n_m)$.
\end{prop}
\begin{rem}\lambdabel{IPinfo}
It is not clear what growth property for $(n_m)$ corresponds to
Equation\,(\ref{ABS}) holding for an uncountable set of points. At
least this analysis shows that we can use results from
Erd\H{o}s and Taylor~{\mathcal I}te{ET}. In particular, they show that a
sufficient condition for the absolute convergence on an
uncountable set that we need is that $\sum\limits_{m=1}^\infty
n_m/n_{m+1}$ converges. See the beginning of the proof of
Proposition~\ref{fastworks}. Sometimes much less of a growth
condition is needed. For example, if one knows $n_{m+1}/n_m$
tends to infinity and $n_{m+1}/n_m$ is eventually a whole number,
then we have $\sum\limits_{m=1}^\infty \|n_mx\|$
converging for an uncountable set of $x$ values.
However, Erd\H{o}s and Taylor~{\mathcal I}te{ET} also give an example where $\lim\limits_{m\to \infty}
n_{m+1}/n_m = \infty$, but the ratio is infinitely often not a whole number, and yet
$\sum\limits_{m=1}^\infty \|n_mx\|$ converges only
for a countable set of values.
\end{rem}
Erd\H{o}s and Taylor~{\mathcal I}te{ET} observe that an earlier result
of Eggleston~{\mathcal I}te{Egg} is relevant here. Eggleston~{\mathcal I}te{Egg}
showed that it is necessary to have some hypothesis on $(n_m)$
that prevents $n_{m+1}/n_m$ from being bounded because
if these ratios are bounded then one
has $\exp(2\pi i n_mx) \to 1$ as $m \to \infty$ for at most a
countable set of values $x$. This fact is related to the weaker
version of the result above that is worth observing here, the
pointwise criterion needed for rigidity along the sequence itself.
The question is: what growth condition on a strictly increasing
sequence $(n_m)$ is needed for the sequence to admit a weakly
mixing dynamical system for which there is a non-trivial rigid
function along $(n_m)$? The same method as above, using the GMC,
shows that it is sufficient to know when there is an uncountable
set of points $K \subset [0,1]$ such that for all $x \in K$, we
have $\exp(2\pi in_mx) \to 1$ as $m \to \infty$. Hence, this
property can be characterized by having
\begin{equation}\lambdabel{termwise}
\lim\limits_{m \to \infty} \|n_mx\| = 0
\end{equation}
for an uncountable set of $x$ values. As with convergence along
an IP set, this property is stronger than what is needed to
produce just one weakly mixing dynamical system with rigidity
along $(n_m)$. This property guarantees that any dynamical
system, weakly mixing or not, whose non-trivial spectral measures
are supported on a subset of $K$, would have $(n_m)$ as a rigidity sequence.
These characterizations,
Equation\,(\ref{termwise}) and Equation\,(\ref{ABS}), show
the difference between having rigidity along a sequence
versus having rigidity along the IP set that the sequence
generates for all spectral measures supported in the set.
Eggleston shows the following in ~{\mathcal I}te{Egg}.
\begin{prop}\lambdabel{ratiogrows}
If $\lim\limits_{m\to \infty} n_{m+1}/n_m = \infty$, then
Equation\,(\ref{termwise}) holds for an uncountable set. Hence,
if $\lim\limits_{m\to \infty} n_{m+1}/n_m = \infty$, then $(n_m)$
is a rigidity sequence for some weakly mixing transformation.
\end{prop}
\begin{rem}\lambdabel{boundedornot}
It is not necessary to have $\lim\limits_{m\to \infty} n_{m+1}/n_m = \infty$
for Equation\,(\ref{termwise}) to hold on an uncountable set.
Indeed, it is not hard to construct examples of
strictly increasing sequences $(n_m)$ with Equation\,(\ref{termwise})
holding on an uncountable set, and yet there are
arbitrarily long pairwise disjoint blocks $B_k \subset \mathbb N$
such that $n_{m+1}/n_m = 2$
for all $m \in B_k$ and all $k$. However, as commented above,
Eggleston also shows in ~{\mathcal I}te{Egg}
that for Equation\,(\ref{termwise}) to hold on an uncountable set,
it is certainly necessary to know that the ratios $n_{m+1}/n_m$ are
not uniformly bounded. Also, Eisner and Grivaux~{\mathcal I}te{EG} show in Proposition 3.8
that one can weaken the hypothesis of Proposition~\ref{ratiogrows}
to just $\limsup\limits_{m\to\infty} \frac {n_{m+1}}{n_m}= \infty$
if $\frac {n_{m+1}}{n_m}$ is always a whole number. This also follows
from using Proposition~\ref{integerratios}.
\end{rem}
\begin{rem}\lambdabel{egdisjoint} a) Proposition~\ref{ratiogrows} allows us
to give interesting examples of disjoint weakly mixing dynamical systems with common
rigidity sequences. Here disjointness is the standard disjointness from Furstenberg~{\mathcal I}te{Furstdisjoint},
their product is their only non-trivial joining. This property means they also do not have
any common factors, and so of course are not isomorphic. We construct weakly mixing $T$ and $S$ as follows.
Let $(a_m) = (2^{m^2})$, and let $(b_m)$ be the sequence which is $2^{m^2}$
for even $m$ and $2^{m^2}+1$ for odd $m$. Since $a_{m+1}/a_m \to \infty$ and
$b_{m+1}/b_m \to \infty$ as $m \to \infty$, by Proposition~\ref{ratiogrows}
there exists $T$ which is weakly mixing and rigid along $(a_m)$ and $S$ which is weakly
mixing and rigid along $(b_m)$. We see that $T$ and $S$ are rigid along the sequence
$(2^{(2m)^2})$. Now take $\nu^T$ and $\nu^S$ to be the maximal spectral
types of $T$ and $S$ on $L_{2,0}(X,p)$. If we show that $\nu^T$ and $\nu^S$ are
mutually singular, then by Hahn and Parry~{\mathcal I}te{HP}, $T$ and
$S$ are disjoint. But if $\omega \ll \nu^T$ and $\omega \ll \nu^S$,
we have $\omega$ rigid along both $(2^{(2m+1)^2})$ and $(2^{(2m+1)^2}+1)$. It follows
that $\omega$ would have to be concentrated at $\{1\}$, which means $\omega = 0$
because $\nu^T(\{1\}) = \nu^S(\{1\}) = 0$.
\noindent b) Given ergodic transformations $T$ and $S$, with $T$ rigid along $(a_m)$ and
$S$ rigid along $(b_m)$, such that $b_m = a_m + p(m)$ for a non-zero polynomial, one can
argue in the style above, by taking successive differences, that the only spectral overlap of $T$ and $S$ can be with
eigenvalues that are $d$-th roots of unity where $d = p(0)$. So if either $T$ or $S$
is totally ergodic (or even say weakly mixing), then $T$ and $S$ are disjoint.
\end{rem}
We want to make some general observations about the values of $x$ such
that $\|n_mx\| \to 0$ as $m\to \infty$. Alternatively, consider this set
in its
representation in $\mathbb T$; we are then looking for all $\gamma \in
\mathbb T$
such that $\gamma^{n_m} \to 1$ as $m \to \infty$. The first important point
is that this is a subgroup of $\mathbb T$, which we have denoted by
$\mathcal R(n_m)$. It
is easy to see that it is a Borel set, indeed it is clearly an
$\mathcal F_{\delta\sigmagma}$ because
\[\mathcal R(n_m) = \bigcap\limits_{k=1}^\infty \bigcup\limits_{M=1}^\infty
\bigcap\limits_{m=M}^\infty \{\gamma: |\gamma^{n_m} -1| \le 1/k\}.\]
There is quite a bit of literature
about such subgroups, and there is some interesting descriptive set
theory involved
in the study of the structure of this set too. First, consider this set
in the situation
that the $n_m = q_m$ are the denominators $q_m$ of the convergents
$\frac {p_m}{q_m}$
of the continued fraction
expansion of some fixed $\alphapha \in [0,1]$, $\alphapha$ irrational. Sometimes the ratios $\frac
{q_{m+1}}{q_m}$ are bounded.
In this case Larcher~{\mathcal I}te{Larcher} showed that $\mathcal R(n_m)$ is
just $\mathbb Z\alphapha +
\mathbb Z$ i.e. in $\mathbb T$, we have $\mathcal R(n_m)$ just the
circle group generated
by $\exp(2\pi i\alphapha)$. See also Kraaikamp and Liardet~{\mathcal I}te{Kr-Li} who
discuss issues of
speed of approach of $\|q_m\alphapha\|$ to $0$. See also Host, M\'ela, and
Parreau~{\mathcal I}te{Ho-Me-Pa}
where this subgroup is considered extensively in the context of spectral
analysis of
dynamical systems.
Also, one can reverse the question of the structure of subgroups
$\mathcal R(n_m)$, by asking
which subgroups of $\mathbb T$ can be realized as such subgroups. It is
generally known that
there are subgroups of the circle that are not even Lebesgue measurable,
let alone Borel measurable.
In communication with S. Solecki, we learned of references that are very
thorough in evaluating
the descriptive set theoretic structure of subgroups of the circle. For
example, see the
articles by Klee~{\mathcal I}te{Klee}, Mauldin~{\mathcal I}te{Mauldin},
Solecki~{\mathcal I}te{Solecki1}, and Farah and Solecki~{\mathcal I}te{Farah-Solecki}.
In addition, Solecki~{\mathcal I}te{Solecki2}
has pointed out that there are even subgroups that are $\mathcal
F_{\delta\sigmagma}$ sets
which are not of the form $\mathcal R(n_m)$ for some $(n_m)$. Moreover,
subgroups like
$\mathcal R(n_m)$ are {\em Polishable} (see Farah and
Solecki~{\mathcal I}te{Farah-Solecki}), and not all
subgroups that are $\mathcal F_{\delta\sigmagma}$ sets are Polishable. In
addition, he points out
that there are Polishable $\mathcal F_{\delta\sigmagma}$ subgroups which
are not of the
form $\mathcal R(n_m)$ for some $(n_m)$. These results suggest that
there is unlikely to
be a descriptive set theoretic characterization of the class of
subgroups of the form
$\mathcal R(n_m)$.
\subsubsection{\bf Rank One Constructions} \lambdabel{rankone}
In this section, given an increasing sequence $(n_m)$
such that either $\frac {n_{m+1}} {n_m} \to \infty$ or
$\frac {n_{m+1}} {n_m}$ is a whole number for each $m$,
we will explicitly construct an infinite
measure-preserving rank one map $T$ such that $T^{n_m} \to Id$
in the strong operator topology. As observed
in Remark~\ref{Poisson}, the Poisson suspension gives an example
of a weakly mixing finite measure-preserving transformation $S$ such
that $S^{n_m} \to Id$ in the strong operator topology.
The Poisson suspension is an appealingly natural
construction in that
no spectral measure intervenes. However, it is also
worth noting that by the following lemma
$U_{T}$ automatically has continuous spectrum
so that one may apply the GMC to its maximal spectral type to
obtain the desired $S$. We note that any rank one $T$ is
necessarily ergodic.
\begin{lem}\lambdabel{wmnoeigen}
If $T$ is an ergodic measure-preserving automorphism of an
infinite measure space $(X,\mathcal B,\mu)$ then $U_{T}$ has
continuous spectrum.
\end{lem}
\begin{proof}
Suppose $f \in L_{2}(X,\mu)$ and $f {\mathcal I}rc T = \lambdambda f$. Then $|\lambdambda|
= 1$ so $|f|$ is $T$-invariant and it follows that $|f|$ is constant. Since $\mu$ is
infinite it follows that $f=0$.
\end{proof}
We use below the notation $a:=b$ or $b=:a$ to mean that $a$ is defined to be $b$.
We assume that the reader has some familiarity with
rank one constructions but
the following is a quick refresher. For more details see ~{\mathcal I}te{Nadkarni}
or ~{\mathcal I}te{Ferenczi}.
Suppose $T$ is a rank one map preserving a finite or infinite
measure $\mu$ and $\{\tau_N\}$ is a refining sequence of rank one
towers for $T$. This means that $\tau_{N+1}$ may be viewed as
constructed from $\tau_N$ by
cutting $\tau_N$ into columns of equal width and
stacking them above each other, with the possible addition of
spacer levels between the columns. We will refer to these
columns of $\tau_N$ as copies of $\tau_N$. The crucial condition
that makes $T$ rank one is that the towers
$\{\tau_N\}$ are required to
converge to the full sigma-algebra of the space in
the sense that for any measurable set $E$ of finite measure
and $\epsilon > 0$ there is
an $E'$ which is a union of levels of some $\tau_N$ (and hence of
all $\tau_N$ for $N$ sufficiently large) such that $\mu(E \triangle
E') < \epsilon$. We let $X_N$ denote the union of the levels of $\tau_N$.
Any such $T$ may be realized concretely as a map of
an interval $I \subset {\mathbb{R}}$ as follows. We take
$X_{0}$ to be a finite sub-interval of ${\mathbb{R}}$ and let $\tau_{0}$
be the tower of height $1$ consisting of the single level $X_{0}$.
Now suppose that $\tau_{1}, \ldots , \tau_N$ have been constructed,
each $\tau_{i}$ a tower whose levels are intervals and the union
of the levels of each $\tau_{i}$ is an interval $X_{i}$. At this point $T$ is
partially defined on $X_N$ by mapping each level of $\tau_N$ to
the level directly above it by the appropriate translation, except
for the top level, where $T$ remains undefined as yet.
Divide the base of $\tau_N$ into $q$ subintervals of equal width $w$
and denote the columns of $\tau_N$ over these by $C_{1}, \ldots,
C_{q}$.
Let $r \geq 0$, take $S$ to be an
interval of width $rw$ adjacent to the interval $X_N$,
divide $S$ into $r$ spacer
intervals of width $w$,
stack $C_{1}, \ldots, C_{q}$ in order above each other and
interleave the $r$ spacer intervals in any way between, below and
above the columns $C_{1}, \ldots, C_{q}$.
We then define $T$ partially on $X_{N+1}$ using $\tau_{N+1}$ in the
same way it was defined on $X_N$ and this is evidently consistent
with the definition of $T$ on $X_N$. Thus, in the limit $T$ is
almost everywhere defined on $I = \bigcup\limits_{N=1}^\infty X_N$ and is evidently
rank one. Note that in the concrete model the convergence of
$\tau_N$ to the Borel $\sigmagma$-algebra of $I$ is automatic.
We let $S_N = X_{N+1} \backslash X_N$ and
$\epsilon_N = \frac {\mu(S_N)} {\mu(X_{N+1})}$, the fraction of the
levels of $\tau_{N+1}$ which are not contained in a level of
$\tau_N$; that is, they are spacers added at stage $n$ of the
construction. We observe that $\mu$ is infinite precisely when
$\sum\limits_{N=1}^\infty \epsilon_N = \infty$.
For a fixed $\tau_N$ we will say a time $N > 0$ is
$\epsilon$-rigid for $\tau_N$ if for each level $E$ of $\tau_N$ we have
$\mu(T^{N}E \triangle E) < \epsilon \mu(E)$. Note that one then has the
same inequality for any $E$ which is a union of levels of $\tau_N$.
Consequently if $N$ is $\epsilon$-rigid for $\tau_N$
then it is also $\epsilon$-rigid for any
$\tau_{M}, \ M \leq N$.
We will say that the sequence $(n_m)$ is {\em rigid for a set $E$
of finite measure} if
$\mu(T^{n_{m}}E \triangle E) \to 0$; and that {\em $(n_{m})$ is rigid
for $\tau_N$, $N$ fixed} if $(n_{m})$ is rigid for each level
of $\tau_N$ (equivalently, for the base of $\tau_N$).
Finally note that if $(n_{m})$ is rigid for every
$\tau_N$ then $T$ is rigid along $(n_m)$.
\begin{prop} \lambdabel{infrankone}
Suppose $\frac {n_{m+1}} {n_m} \to \infty $ as $m \to \infty$ or $\frac {n_{m+1}}
{n_m} $ is a whole number, $\frac {n_{m+1}}
{n_m} \ge 2$ for all $m$. Then there exists an
infinite measure-preserving, weakly mixing, rank one transformation $T$ such that $T$ is rigid along $(n_m)$.
\end{prop}
\begin{proof}
Suppose first that $\frac {n_{m+1}} {n_m} \to \infty $ as $m \to \infty$.
We introduce the notation $h_{m}:=n_m$ as, in this case, these will be the
heights of the rank one towers we construct.
Write $h_{m+1} = q_m h_m + r_m$, $0 \leq r_m < h_m$.
Define $p_m < q_m$ to be the least integer $l \geq 0$
such that, $ \frac { r_m + lh_m} {h_{m+1}} > \frac 1 m$ ($p_m$ may be
zero) and let $\epsilon_{m} = \frac { r_m + p_mh_m} {h_{m+1}}$.
Thus we have $\sum\limits_{m=1}^\infty \epsilon_m = \infty$. Moreover $\epsilon_m -
\frac 1m < \frac {h_m} {h_{m+1 }} \to 0$ so $\epsilon_m \to 0$.
Construct $T$ as follows.
Start with a tower $\tau_{1}$ of height
$h_{1}$ and suppose the towers $\tau_{1}, \ldots, \tau_m$
have been constructed. Form $\tau_{m+1}$ by slicing $\tau_m$ into
$s_m: = q_m - p_m$ columns, stacking these directly above
each other and then following them
by ${ h_{m+1}- s_mh_m} = r_m + p_m h_m$ spacers
to create a tower $\tau_{m+1}$ of height
$h_{m+1}$. Thus $\frac {\mu(S_m)} {\mu(X_{m+1})} = \epsilon_m$,
and since $\sum\limits_{m=1}^\infty \epsilon_m = \infty$ we see that the measure
of the space we have constructed is infinite. By Lemma~\ref{wmnoeigen}, $T$
is weakly mixing.
We now check that $T$ is rigid along $(h_m)$. If $E$ is a
level of $\tau_m$ and $E_{s_1}, \ldots, E_{s_m}$ are its pieces
in $\tau_{m+1}$ then $T^{h_m}E_{i} = E_{i+1}$, except for $i =
s_m$.
It follows that $\mu(T^{h_m}E \backslash E) < \frac 1 {s_m} \mu(E) =:
\delta_m\mu(E)/2$ so
$h_m$ is $\delta_m$-rigid for $\tau_m$. Since this holds for
every $m$ it follows that, for any fixed $n$, $h_m$ is $\delta_m$
rigid for $\tau_{n}$, for each $m > n$. Since $\delta_m \to 0$, it
follows that for each fixed $m$,
$(h_m)$ is rigid for $\tau_{m}$. So $T$ is rigid along $(h_m)$.
This concludes the argument in case $\frac {n_{m+1}} {n_m}
\to \infty$.
Now suppose that $\frac {n_{m+1}} {n_m} $ is a whole number as large as $2$ for all
$m$. For simplicity we will consider only the case $n_m = 2^{m}$.
The general case is no more difficult. Let $h_m = 2^{m^{2}}$
so $q_m:= h_{m+1} / h_m= 2^{2m+1} \to \infty$. Let
$p_{m} \geq 0$ be the least
integer $r$ so that
$\frac {r h_m} {h_{m+1}} \geq \frac 1 m$
and let $\epsilon_m = \frac {p_m h_m} {h_{m+1}}$.
As before we have $\sum\limits_{m=1}^\infty \epsilon_m = \infty$ and $\epsilon_m \to 0$.
We construct the towers $\tau_m$ for $T$ as before,
by concatenating $s_m: = q_m - p_m$ copies
of $\tau_m$ and adding $p_mh_m$ spacers to get the tower
$\tau_{m+1}$ of height $h_{m+1}$. As before the space on which $T$
acts has infinite measure and we need only check the rigidity of
the sequence
$(n_{m}) = (2^{m})$.
Now suppose $E$ is a level of
$\tau_m$ and $E_{1}, \ldots, E_{l}$, $l=s_ms_{m+1}$, are its pieces in
$\tau_{m+2}$. These occur with period $h_m$ in
$\tau_{m+2}$, except for gaps corresponding to the spacers in $S_m$
and $S_{m+1}$. More precisely,
let us divide $\tau_{m+2}$ into $q_mq_{m+1}$
blocks of length $h_m$ and also into $q_{m+1}$
blocks of length $h_{m+1}$ and refer to these as
$m$-blocks and $(m+1)$-blocks respectively. Each $m$-block is
contained in either $X_m$, $S_m$ or $ S_{m+1}$.
Call these three types $X_m$-blocks, $S_m$-blocks and
$S_{m+1}$-blocks
and let
the numbers of the three types be
$a= s_ms_{m+1}, b = p_{m}s_{m+1}$ and $c=q_mp_{m+1}$.
Now suppose that $M >0$ and let $m = m_M > 0$ be the integer such
that $m^{2} \leq M < (m+1)^{2} $. Since there is at
least one $(m+1)$-block at the top of
$\tau_{m+2}$ which is contained in $S_{m+1}$ we see that for each
$i$,
$1 \leq i \leq l$,
$T^{2^{M}} E_{i}$ is
still a level of $\tau_{m+2}$. Thus, if it is not contained in $E$ it
must lie in an $S_m$-block or an $S_{m+1}$-block. It follows that
\begin{align*}
\frac {\mu(T^{2^{M}} E \backslash E)} {\mu(E)} \leq \frac {b+c} {a}
= & \frac {p_ms_{m+1} + q_mp_{m+1}} {s_ms_{m+1}} \\
= & \frac{p_m} {s_m} + (\frac {q_m} {s_m})
(\frac {p_{m+1}} {s_{m+1}})
= \epsilon_m + \frac 1 {1-\epsilon_m} \epsilon_{m+1} =: \delta_m/2.
\end{align*}
This shows $2^{M}$ is $\delta_{m_{M}}$-rigid for
$\tau_{m_{M}}$. Fixing any $k \leq n_{M}$, it follows that
$2^{M}$ is $\delta_{m_{M}}$-rigid for $\tau_{k}$. Letting $M \to
\infty$ we have $m_{M} \to \infty$ and $\delta_{m_{M}} \to 0$.
So we see
that the sequence $\{2^{M}\}$ is rigid for $\tau_{k}$, and since $k$
is arbitrary it follows that $T$ is rigid along $(2^{M})$, as desired.
\end{proof}
We also note that there is a special case of Proposition~\ref{infrankone}
where we can
provide a direct construction of a finite measure-preserving rank one
$T$ such that $T^{n_m} \to Id$.
\begin{prop}\lambdabel{specialinfrankone}
Suppose that $ {n_{m+1}} = q_{m}{n_m} + r_{m}$, $0 \leq
r_{m} < n_m$, $q_{m} \to \infty$, $\sum\limits_{n}^{\infty} \frac {r_{m}}
{n_{m+1}} < \infty$ and $r_{m} \neq 0$ infinitely often.
Then there is a finite measure-preserving, weakly mixing, rank one transformation
$T$ that is rigid along $(n_m)$.
\end{prop}
\begin{proof}
We construct the rank
one towers $\tau_{m}$ of height $h_{m}: = n_m$ for $T$ as follows.
Start with a tower of height $h_{1}$. When $r_{m} = 0$,
we construct $\tau_{m+1}$ from $\tau_{m}$ by simply
concatenating $q_{m}$ copies of $\tau_{m}$ to obtain the tower
$\tau_{m+1}$ of height
$h_{m+1}$. When $r_{m} \neq 0$, we
place $a_{m}:= [q_m/3]$ consecutive copies of $\tau_{m}$ followed
by one spacer, followed by $q_m - a_{m}$ consecutive copies
of $\tau_{m}$, followed by $r_{m} - 1$ spacers, again giving
$\tau_{m+1}$ of height $h_{m+1}$.
The resulting $T$ is finite measure-preserving because we have
assumed $\sum\limits_{m=1}^{\infty} \frac {r_{m}}
{h _{m+1}} < \infty$ and it is very easy see that
$T$ is rigid along $(h_m)$.
We now check that $T$ is weakly mixing. Suppose that $f \in
L_{2}(X,\mu)$ and $f {\mathcal I}rc T = \lambdambda f$. Without loss of generality
$|f| = 1$. Given $\epsilon > 0$, find $n = n_{m}$ such that
$r_m \neq 0$, and $f'$ which is a
linear combination of the characteristic functions of the
levels of $\tau_{m}$ such that $\|f -f'\|_{2} < \epsilon$.
In addition we may assume that $|f(x)| = 1$ for all $x
\in X_{m}$.
\def\close#1{\ {\mathop{\sigmam}\limits^{#1}}\ }
We agree to write $g \close \delta h$ whenever $g,h \in L_{2}(X,\mu)$ and
$\| g- h\|_{2} < \delta $. Note that for any $k$ we have
\begin{equation*}
f' {\mathcal I}rc T^{k} \close \epsilon f {\mathcal I}rc T^{k} = \lambdambda^{k} f \close
\epsilon \lambdambda^{k} f'.
\end{equation*}
Let $E_{1}$ denote the union of the first $s_{m}:= a_{m}h_{m}$ levels of
$\tau_{m+1}$ and $E_{2}$ the union of the $s_{m}$ levels
after the first spacer in $\tau_{m+1}$. We observe that
$$
f'|_{E_{1}} = (f' {\mathcal I}rc T^{s_{m}+1})|_{E_{1}} \close {2\epsilon}
\lambdambda^{s_{m}+1} f'|_{E_{1}}.
$$
It follows that $\lambdambda^{s_{m} + 1}
\close {2\epsilon/ \|f'|_{E_{1}}\|_{2}} 1$. By taking $m$
sufficiently large we may assume that $\mu(E_{1}) \geq \frac 1
4$ and so
$$\|f'|_{E_{1}}\|_{2} = \sqrt {\mu(E_{1})} > \frac 1 2.$$
Thus, $\lambdambda^{s_{m} + 1} \close {4\epsilon} 1$. A similar argument with
$E_{2}$ replacing $E_{1}$ shows that $\lambdambda^{s_{m}} \close {4\epsilon}
1$ so we get $\lambdambda^{s_{m} + 1} \close {8\epsilon} \lambdambda^{s_{m}}$.
Since $|\lambdambda| = 1$ it follows that $\lambdambda \close {8\epsilon} 1$ and
since $\epsilon > 0$ is arbitrary we conclude that $\lambdambda = 1$.
\end{proof}
\subsubsection{\bf Rates of Growth}\lambdabel{ratesofgrowth}
Here is some information on the question of rates of growth
of the gaps in a rigidity sequence. These results show in various ways that
although rigidity sequences $(n_m)$ have the gaps $n_{m+1} - n_m$
tending to infinity, they do not need to have these gaps growing quickly.
Indeed, there is no rate, no matter how slow, that these gaps must
grow for either ergodic rotations of the circle or weakly mixing
transformations.
Suppose we have an increasing sequence $\mathbf{n} = (n_m)$. We let
\[D(N,\mathbf{n}) =\frac {\#(\{n_m:m \ge 1\}{\mathcal A}p \{1,\ldots,N\})}N.\]
We say that $\mathbf{n}$ has {\em density zero} if
$D(N,\mathbf{n}) \to 0$ as $N \to \infty$. The following result
can be improved, see Corollary~\ref{bestdecay} below. We prove
this here because it gives insight into the ideas in
Proposition~\ref{ergodicratewm}.
\begin{prop} \lambdabel{ergodicrate}
Given any sequence $(d_N:N\ge 1)$ such that $d_N \to 0$
as $N \to \infty$, and any ergodic rotation $T$ of $\mathbb T$, there
exists a rigidity sequence $\mathbf{n} = (n_m)$ for $T$ such that
$D(N,\mathbf{n}) > d_N$ for infinitely many $N \ge 1$.
\end{prop}
\begin{proof}
We have some
$\gamma \in \mathbb T$ of infinite order such that $T(\alphapha) =
\gamma \alphapha$ for all $\alphapha \in \mathbb T$.
Because $d_N\to 0$
as $N\to \infty$, we can choose an increasing sequence $(N_k)$ such
that for all $N \ge N_k$, we have
$d_N \le \frac 1{4^k}$. Now we construct a
suitable $(n_m)$ that is rigid for $T$. First, we can inductively
choose an increasing sequence $(M_k)$ so that we have $\#\{n \in [1,M_k]:
|\gamma^n-1| \le \frac 1{2^k}\} \ge \frac {M_k}{2^k}$. This is possible
because
the Lebesgue measure of the arc $\{\alphapha: |\alphapha - 1|\le \frac 1{2^k}\}$
is $\frac 2{2^k}$ and $(\gamma^n:n\ge 1)$ is uniformly distributed in
$\mathbb T$. In the process
of choosing $(M_k)$, there is no obstruction to taking each $M_k \ge N_k$.
Now let $(n_m)$ be the increasing sequence whose terms are
$\bigcup\limits_{k=1}^\infty \{n \in [1,M_k]: |\gamma^n-1| \le \frac
1{2^k}\}$.
By the construction, $(n_m)$ is rigid for $T$. Now
we claim that $D(N,\mathbf{n}) \ge d_N$ for infinitely many $N$.
Indeed, $D(M_k,\mathbf{n}) \ge \frac 1{2^k}$ by the choice
of $M_k$ and the definition of $\mathbf {n}$.
However, since $M_k \ge N_k$, we have $d_{M_k} \le \frac 1{4^k}$.
\end{proof}
\begin{cor} \lambdabel{rotslow} Given any sequence $G(m)$ tending to infinity
and any ergodic rotation $T$ of $\mathbb T$,
there exists a rigidity sequence $(n_m)$ for $T$ such that
$\limsup\limits_{m\to \infty} \frac {G(m)}{n_{m+1} - n_m} = \infty$.
\end{cor}
\begin{proof}
We have some
$\gamma \in \mathbb T$ of infinite order such that $T(\alphapha) =
\gamma \alphapha$ for all $\alphapha \in \mathbb T$.
Take $g(m)$ tending to infinity.
\noindent {\bf Claim}: There is a sequence $(d_N)$ tending to zero,
determined
by $g$ alone, such that for any
sequence $\mathbf {n} =(n_m)$ that has $n_{m+1} - n_m \ge g(m)$ for all
$m \ge 1$,
we would have $D(N,\mathbf{n}) \le d_N$ for all $N \ge 1$.
\noindent{\bf Proof of Claim}: Observe that among sequences with
$n_{m+1} - n_m \ge g(m)$ for all $m \ge 1$,
$\#\{n_m \le N\}$ is largest in the case that we take the explicit sequence
$n_1 =1$ and $n_{m+1} = n_m + \lceil g(m)\rceil$ for all $m\ge 1$. So,
take this
as our sequence. Let $g(0) = 0$. Then
\[\#\{n_m \le N\} \le \sup\{m \ge 1: 1+\sum\limits_{k=0}^{m-1} \lceil
g(k)\rceil \le N\}.\]
Thus, let $d_N = \frac {\sup\{m: 1+\sum\limits_{k=0}^{m-1} \lceil
g(k)\rceil \le N\}}N$,
which tends to zero as $N \to \infty$ because $g(m) \to \infty$ as $m
\to \infty$.
We have $D(N,\mathbf n) \le d_N$ for all $N \ge 1$.
\noindent Continuing now with our proof, for any sequence $\mathbf {n}
=(n_m)$ that has $n_{m+1} - n_m \ge g(m)$ for all $m \ge M$,
then there exists $N_M$ such that $D(N,\mathbf{n}) \le 2d_N$ for all $N
\ge N_M$.
Using $(2d_N)$ in place of $(d_N)$, Proposition~\ref{ergodicrate} and
Claim~\ref{rotslow} above
how that we can construct a rigidity
sequence $(n_m)$ for $T$ such that $n_{m+1} - n_m < g(m)$ for infinitely
many $m$.
Now, for any $G(m)$ increasing to $\infty$, we
can construct $g(m)$ tending to $\infty$ so that
$\lim\limits_{m\to\infty} \frac {G(m)}{g(m)} =\infty$.
Using this $g$ above, we have shown that
there is a rigid sequence $(n_m)$ for $T$ such that
$\limsup\limits_{m\to\infty}
\frac {G(m)}{n_{m+1} - n_m} = \infty$.
\end{proof}
\begin{cor} \lambdabel{notexp} For any ergodic rotation $T$ of $\mathbb T$,
there is a rigidity sequence
$(n_m)$ such that $\liminf\limits_{m\to \infty} \frac {n_{m+1}}{n_m} = 1$.
\end{cor}
\begin{proof} Let $G(m) = \sqrt m$. Then use Corollary~\ref{rotslow} to
construct
a rigidity sequence for $T$ such that $n_{m+1} - n_m \le \sqrt m$
infinitely often. Since
$n_m \ge m$, we have $\liminf\limits_{m\to \infty} \frac {n_{m+1}}{n_m}
= 1$.
\end{proof}
Our next result, and some that follow, show that although rigidity
sequences are sparse sets,
they are not always {\em thin sets}
in certain senses that are commonly used in harmonic analysis. See Lopez
and Ross~{\mathcal I}te{LR}
for background information on thin sets in harmonic analysis. In
particular, we will see that
rigidity sequences are not
always {\em Sidon sets}. By a Sidon set here we mean a subset $\mathcal
S$ of the integers
such that given any bounded complex-valued function $\psi$ on $\mathcal
S$, there exists a complex-valued Borel
measure $\nu$ on $\mathbb T$ such that $\widehat {\nu} = \psi$ on
$\mathcal S$. Originally, this
property was observed for lacunary sets, and finite unions of lacunary
sets, but the general notion of Sidon sets gives a larger class of sets
to work with that in a general sense will have similar harmonic analysis properties.
\begin{cor} \lambdabel{notlac}
Given any ergodic rotation $T$ of $\mathbb T$, there exists a rigidity
sequence for $T$ which is not a Sidon set, and so is not the
union of a finite number of lacunary sequences.
\end{cor}
\begin{proof}
It is a standard fact that finite unions of lacunary sequences are Sidon
sets. See
for example ~{\mathcal I}te{LR}. Also in ~{\mathcal I}te{LR}, Corollary 6.11, is the proof
that if $\mathbf n = (n_m)$ is a Sidon set then there is a constant $C$
such that
$D(N,\mathbf n) \le \frac {C\log N}N$. By the argument above, there exists
some $d_n \to 0$ as $n \to \infty$ such that for all $C$, eventually
$d_N \ge C \frac {\log N}N$. Using this $(d_N)$, construct a rigidity
sequence $\mathbf {n}$
for $T$ as in Proposition~\ref{ergodicrate}. This choice of $(d_N)$
shows that $\mathbf {n}$ is not a Sidon set.
\end{proof}
\begin{rem} We did not need it here, but sometimes when dealing with
classes of
sequences, it is good to have a result as follows. Suppose we have a
sequence of sequences $(d_N(s):N\ge 1)$ where
$d_N(s) \to 0$ as $N \to \infty$ for every $s$. Then there exists a sequence
$(d_N)$ which also has $d_N \to 0$ as $N \to \infty$, but also for all
$s$, $d_N \ge d_N(s)$
for large enough $N$. This is a standard result. First, let $d_N^*(k) =
\max(d_N(1),\ldots,d_N(k))$. Then for all $k$, again $d_N^* \to 0$ as
$N\to \infty$.
Choose an increasing sequence $(N_k)$ such that $d_N^*(k) \le \frac 1{2^k}$
all $N \ge N_k$. Let $d_N = 1$ for all $1 \le N < N_1$, and for $k \ge
1$, let $d_N =
d_N^*(k)$ for $N_k \le N < N_{k+1}$. Then $d_N \le \frac 1{2^k}$ for $N
\ge N_k$. Also,
for all $j$, $d_N \ge d_N^*(k) \ge d_N(j)$ for any $N \ge N_k$ with $k
\ge j$.
\end{rem}
Now we extend the construction above to give weakly mixing
transformations with
rigidity sequences that satisfy similar slow decay properties.
\begin{prop}\lambdabel{ergodicratewm}
Given any sequence $(d_N:N\ge 1)$ such that $d_N \to 0$
as $N \to \infty$, there exists a weakly mixing transformation
and a rigidity sequence $\mathbf {n} =(n_m)$ for $T$ such that
$D(N,\mathbf{n}) > d_N$ for infinitely many $N$.
\end{prop}
\begin{proof}
To carry out this construction, we
start with a closed perfect set $\mathcal K$ in $\mathbb T$ such that
every finite set $F\subset \mathcal K$
generates a free abelian group of order $\#F$. Except that this is in
the circle, it is the
same as constructing a closed perfect set $\mathcal K$ in $[0,1]$ of
rationally independent
real numbers modulo $1$ (i.e. $\mathcal K{\mathcal U}p \{1\}$ is rationally
independent in the real numbers).
See Rudin~{\mathcal I}te{rudin}.
The rational independence tells us that for any $\gamma_1,\ldots,\gamma_L \in \mathcal K$, the
sequence of powers $(\gamma_1^n,\ldots,\gamma_L^n), n \ge 1$
is uniformly distributed in $\mathbb T^L$.
Because $d_N\to 0$
as $N\to \infty$, we can choose an increasing sequence $(N_k)$ such
that for all $N \ge N_k$, we have
$d_N \le (\frac 1{4^k})^{4^k}$.
We now inductively construct $K$, a closed subset of $\mathcal K$, and
$(M_k)$ with certain properties. This
will be a Cantor set type of construction. First,
choose distinct $\gamma(i_1), i_1 = 1,2$, in $\mathcal K$. Then choose
$M_1 \ge N_1$
so that $\#\{n \in [1,M_1]: \text {for}\, i_1=1,2,\, |\gamma(i_1)^n - 1| \le
\frac 13\} \ge (\frac 23)^2 \frac {M_1}2$. This is possible because the
set in $\mathbb T^2$
consisting of $(\alphapha_1(1),\alphapha_1(2))$ with $|\alphapha_1(i) - 1| \le
\frac 13$ for both $i=1,2$ has
Lebesgue measure $(\frac 23)^2$, and $((\gamma(1),\gamma(2))^n:n\ge 1)$
is uniformly distributed
in $\mathbb T^2$. We then choose two disjoint closed arcs $B(i_1),
i_1=1,2$ with $\gamma(i_1) \in \text{int}(B(i_1))$
for $i_1=1,2$ and such that for any $\omega(i_1) \in B(i_1)$, and
$n \in [1,M_1]$ such that $|\gamma(i_1)^n - 1| \le \frac 13$ for
$i_1=1,2$, we have the
somewhat weaker inequality $|\omega(i_1)^n - 1| \le \frac 23$. Let
$A(i_1) = B(i_1){\mathcal A}p \mathcal K$ for $i_1=1,2$.
Let $A(i_0) = K_0 = \mathcal K$.
We have to continue this inductively. Suppose for fixed $k \ge 1$,
we have constructed $K_l$ and
$M_l \ge N_l$ for all $l=1,\ldots,k$ with the following properties. Each
$K_l \subset K_{l-1}$
and each $K_l$ is a union of a finite number
of pairwise disjoint, non-empty, closed perfect sets $A(i_1,\ldots,i_l)=
B(i_1,\ldots,i_l){\mathcal A}p \mathcal K$,
given by $i_j=1,2$ for all $j=1,\ldots,l$. Here the sets
$B(i_1,\ldots,i_l)$ are closed arcs , with $\text
{int}(B(i_1,\ldots,i_l)){\mathcal A}p\mathcal K$
not empty, for all $i_j=1,2, j=1,\ldots,l$.
For each $(i_1,\ldots,i_l)$, we have $A(i_1,\ldots,i_l) \subset
A(i_1,\ldots,i_{l-1})$.
In addition, consider the set $E_l$ of $n \in [1,M_l]$ such that for all
$\omega(i_1,\ldots,i_l)
\in B(i_1,\ldots,i_l)$, we have $|\omega(i_1,\ldots,i_l)^n - 1| \le \frac
2{3^l}$. We assume
inductively that $\#E_l \ge (\frac 2{3^l})^{2^l} \frac {M_l}2$.
Now, for each $(i_1,\ldots,i_k)$, choose two distinct
$\gamma(i_1,\ldots,i_k,i_{k+1}) \in A(i_1,\ldots,i_k)$
where $i_{k+1} =1,2$. We can choose $\gamma(i_1,\ldots,i_k,i_{k+1})
\in \text {int}(B(i_1,\ldots,i_k))$. Then the
point $p$ in $\mathbb T^{2^{k+1}}$ with coordinates
$\gamma(i_1,\ldots,i_k,i_{k+1})$, listed in any order,
has $(p^n: n\ge 1)$ uniformly distributed in $\mathbb T^{2^{k+1}}$. So,
there exists $M_{k+1} \ge
N_{k+1}$ such that $\#E_{k+1} \ge (\frac 2{3^{k+1}})^{2^{k+1}} \frac
{M_{k+1}}2$ where $E_{k+1}$ is the set of
$n \in [1,M_{k+1}]$ such that for all $(i_1,\ldots,i_{k+1})$ we have
$|\gamma(i_1,\ldots,i_{k+1})^n-1|
\le \frac 1{3^{k+1}}$. Choose pairwise disjoint closed arcs
$B(i_1,\ldots,i_{k+1})$
with $\gamma(i_1,\ldots,i_{k+1})\in \text{int}(B(i_1,\ldots,i_{k+1}))$
such that we have
the following. Consider any $\omega(i_1,\ldots,i_{k+1})\in
B(i_1,\ldots,i_{k+1})$,
and any $n \in [1,M_{k+1}]$ such that $|\gamma(i_1,\ldots,i_{k+1})^n-1|
\le \frac 1{3^{k+1}}$. Then we have the somewhat weaker inequality
$|\omega(i_1,\ldots,i_{k +1})^n-1|
\le \frac 2{3^{k+1}}$. There is no difficulty in also having
$B(i_1,\ldots,i_{k+1}) \subset B(i_1,\ldots,i_k)$ because we chose
$\gamma(i_1,\ldots,i_k,i_{k+1})
\in \text {int}(B(i_1,\ldots,i_k))$.
We now let
\[A(i_1,\ldots,i_{k+1}) = B(i_1,\ldots,i_{k+1}){\mathcal A}p \mathcal K.\]
Since $B(i_1,\ldots,i_{k+1}) \subset B(i_1,\ldots,i_k)$, we have
$A(i_1,\ldots,i_{k+1}) \subset A(i_1,\ldots,i_k)$.
Let $K_{k+1} = \bigcup\limits_{(i_1,\ldots,i_{k+1})}A(i_1,\ldots,i_{k+1})$.
This completes the inductive step.
To finish this construction, let $K = \bigcap\limits_{k=1}^\infty K_k$.
Then $K$
is a closed perfect subset of $\mathcal K$. Let $\mathbf {n} = (n_m)$ be the
increasing sequence whose terms are all values of $n$
such that $n \in [1,M_k]$ for some $k$, and for all $\omega(i_1,\ldots,i_k)
\in A(i_1,\ldots,i_k)$, we have $|\omega(i_1,\ldots,i_k)^n - 1| \le \frac
2{3^k}$. By the construction,
for all $\omega \in K$, we have $\omega^{n_m} \to 1$ as $m \to \infty$.
Indeed, given $m\ge 1$ choose $k_m$ to be the largest $k$ so that $n_m >
M_k$.
Notice that $k_m\to\infty$ when $m\to \infty$, and the term $n_m$ was
chosen from $[1,M_{s_m}]$ with $s_m > k_m$ and satisfying
$|\omega(i_1,\ldots,i_{s_m})^{n_m}-1|< \frac 2{3^{s_m}}$
for all $\omega(i_1,\ldots,i_{s_m}) \in B(i_1,\ldots,i_{s_m})$. In
particular, if $\omega \in \mathcal K$,
then $|\omega^{n_m}-1| \le \frac 2{3^{s_m}} \le \frac 2{3^{k_m}}$
because $\omega \in B(i_1,\ldots,i_{s_m})$ for some
choice of $(i_1,\ldots,i_{s_m})$.
Hence, for any Borel probability measure $\nu$ supported in $K$, we have
$\widehat {\nu}(n_m) \to 1$
as $m \to \infty$. Since $K$ is a closed perfect set, there are continuous
Borel probability measures $\nu$ supported in $K$. Take the
symmetrization of any such measure and use
the GMC to construct the corresponding
weakly mixing transformation $T$. Then $\mathbf {n}$ is a rigidity
sequence for $T$.
But also we claim that $D(N,\mathbf{n}) > d_N$ for infinitely many $N$.
Indeed, our construction guarantees that
$D(M_k,\mathbf{n}) = \#\{m: n_m \in [1,M_k]\} \ge \frac 12(\frac
2{3^k})^{2^k}$ for
all $k$. But also, because we have chosen $M_k \ge N_k$ for all $k$,
we have $d_{M_k} \le (\frac 1{4^k})^{4^k}$. Therefore,
$d_{M_k}<D(M_k,{\bf n})$.
\end{proof}
\begin{cor} \lambdabel{wmslow} Given any sequence $G(m)$ tending to
infinity, there exists a weakly mixing transformation
and a rigidity sequence $(n_m)$ for $T$ such that
$\limsup\limits_{m\to \infty} \frac {G(m)}{n_{m+1} - n_m} = \infty$.
\end{cor}
\begin{proof}
The proof proceeds in the same manner as Corollary~\ref{rotslow}.
\end{proof}
Using the same argument as given in Corollary~\ref{notexp}, one can see from
Corollary~\ref{wmslow} that there is a weakly mixing transformation $T$
and a rigidity sequence
$(n_m)$ for $T$ such that $\liminf\limits_{m\to \infty} \frac
{n_{m+1}}{n_m} = 1$.
See also Remark~\ref{wazna} for another construction of this type.
However, this is actually
a pervasive principle.
\begin{prop} \lambdabel{rationear1} Given any rigid weakly mixing transformation
$T$, there is a rigidity sequence $(n_m)$ for $T$ such that
$\liminf\limits_{m\to \infty} \frac {n_{m+1}}{n_m} = 1$.
\end{prop}
\begin{proof} Take a rigidity sequence $(N_m)$ for $T$. We can replace
this by a subsequence so that the
IP set it generates when written in increasing order is also a rigidity
sequence $(n_m)$ for $T$. We can also
arrange that this IP set is sufficiently rarified (by excluding more
terms from $(N_m)$ if necessary)
so that $(n_m)$ has $\liminf\limits_{m\to \infty} \frac {n_{m+1}}{n_m} = 1$.
\end{proof}
\begin{rem} The syndetic nature of recurrence noted at the beginning of
the proof of Proposition~\ref{bestgrowth} below
shows that the above can be modified to give a rigidity sequence $(n_m)$
for $T$, either in the
ergodic rotation case, or the weakly mixing case, for which the ratios
$\frac {n_{m+1}}{n_m}$
are near one for arbitrarily long blocks of values $m$, infinitely often.
\end{rem}
\begin{rem}\lambdabel{EisnerGrivaux} In Example 3.18, Eisner and Grivaux~{\mathcal I}te{EG}
construct a weakly mixing transformation $T$ and a rigidity sequence $(n_m)$ for $T$
such that $\lim\limits_{m\to \infty} \frac {n_{m+1}}{n_m} = 1$. However,
their example does not necessarily give the type of upper density rate
results of Proposition~\ref{ergodicratewm} and Proposition~\ref{wmslow}.
\end{rem}
The techniques used in the above constructions give the following
important consequence in Corollary~\ref{notSidon}. If one looks at
all of the other constructions of rigidity sequences given in this
article, one might expect
the opposite of what Corollary~\ref{notSidon} gives us. Also, given this type of example of a
rigidity sequence
for a weakly mixing transformation, and the others in this
article, it seems that it may be very difficult to characterize these
sequences
in any simple structural fashion. See also Remark~\ref{linformeg}, c) for a
different viewpoint
on the issue of characterizing rigidity sequences.
\begin{cor} \lambdabel{notSidon} There is a weakly mixing transformation $T$
and a rigidity sequence
$(n_m)$ for $T$ such that $(n_m)$ is not a Sidon set, and so is not
the union of a finite number of lacunary sequences.
\end{cor}
\begin{proof} The proof is just like the proof of
Corollary~\ref{notlac}, only here we use Proposition~\ref{ergodicratewm} instead of
Proposition~\ref{ergodicrate}.
\end{proof}
\begin{rem} Again, Example 3.18 in Eisner and Grivaux~{\mathcal I}te{EG} gives a rigidity sequence
for a weakly mixing transformation such that $\lim\limits_{m\to \infty} \frac {n_{m+1}}{n_m} = 1$. Such a sequence cannot
be a Sidon set because its density $D(N,\mathbf {n})$ is not bounded by $C\log N/N$
for any constant $C$.
\noindent There are other types of sequences that our technique here could apply
to, and show that they
cannot characterize rigidity sequences. For example, consider the
sequences studied by Erd\H{o}s
and Tur\'an~{\mathcal I}te{ETuran}, which they call Sidon sets but are now given a
different name (they are
called $\mathcal B_2$ sequences) because of
the current use of the term Sidon sets mentioned above. They show
there sets have density at most $C\frac {N^{1/4}}N$.
So again, we can construct rigidity sequences that cannot be a finite
union of such $\mathcal B_2$
sequences.
\end{rem}
Another direction we can seek for constructing special rigidity sequences
$(n_m)$ is to try and construct them with $n_m \le \Psi(m)$ where
$\Psi(m)$ is growing slowly, or to prove instead that this would force
$\Psi(m)$ to grow quickly. First, we have this result.
\begin{prop} \lambdabel{bestgrowth}
Suppose $\Psi(m)\ge m$ for all $m \ge 1$,
and $\lim\limits_{m\to \infty} \frac {\Psi(m)}m = \infty$. Then for any
ergodic rotation $T$ of $\mathbb T$, there
is a rigidity sequence $(n_m)$ and a constant $C$ such that $n_m \le
C\Psi(m)$
for all $m \ge 1$.
\end{prop}
\begin{proof} First choose $\gamma \in \mathbb T$ of infinity order so that
$T(\alphapha) = \gamma \alphapha$ for all $\alphapha \in \mathbb T$.
Fix open arcs $A_s$ centered on $1$ with
$\lambdambda_{\mathbb T}(A_s) = \epsilon_s$, where $(\epsilon_s)$
is a sequence decreasing to $0$ as $s \to \infty$. For each
$s$, the sequence $(n\ge 1: \gamma^n \in A_s)$ is syndetic. That
is, there is some $N_s \ge 1$ such that for all
$M$, there exists $n\in [M+1,\ldots,M+N_s]$ such that $\gamma^n \in A_s$.
The syndetic property here is automatic from minimality. But it is easy to see
this explicitly in this case.
Indeed, fix an open arc $B_s$ centered at $1$ with $\lambdambda_{\mathbb
T}(B_s) = \epsilon_s/4$.
Then $\{\gamma^nB_s: n\ge 1\}$ covers $\mathbb T$ and so by compactness
there exists
$(\gamma^{n_1}B_s,\ldots,\gamma^{n_K}B_s\}$ which also covers $\mathbb
T$. But
then, for all $M$, there exists some $n_k$ such that $\gamma^{n_k}B_s
{\mathcal A}p \gamma^{-M}B_s
\not= \emptyset$. Hence, $\gamma^{n_k+M} \in B_sB_s^{-1} \subset A_s$. So
if we let $N_s = \max\{n_1,\ldots,n_K\}$, then there exists
$n\in [M+1,\ldots,M+N_s]$ such that $\gamma^n \in A_s$, i.e. $|\gamma^n
-1|< \epsilon_s/2$.
By replacing $\Psi$ by a more slowly growing function, we can arrange
without loss of generality for the additional property that $\Psi(m)/m$
is increasing.
We write $\Psi(m) = m\theta(m)$. Let $C \ge N_1$ and take an
increasing sequence of whole numbers $(K_L: L \ge 1)$ such that
$\theta(K_L) \ge
N_{L+1}$ for all $L \ge 1$. Let $K_0 = N_0 = 0$.
Consider the blocks in the integers, for $L \ge 1$,
of the form
\[B(L,j) =
[K_0N_0+\ldots+K_{L-1}N_{L-1}+jN_L+1,K_0N_0+\ldots+K_{L-1}N_{L-1}+(j+1)N_L]\]
where $j=0,\ldots,K_L-1$. Then we have a sequence of $K_L$ blocks of length
$N_L$. So we can choose $n(L,j) \in B(L,j)$ such that
$|\gamma^{n(L,j)} -1| < \epsilon_L/2$. Let $(n_m)$ be the increasing
sequence
consisting of all such choices $n(L,j)$ where $L\ge 1$ and
$j=0,\ldots,K_L-1$.
By construction, $(n_m)$ is a rigidity sequence for $T$.
We want to show that $n_m \le Cm\theta(m)$ for all $m$. But the values of
$m$ here are of the form $K_0+\ldots+K_{L-1}+j+1$.
For each such $m$, the corresponding $n_m$ is being chosen in
the interval $B(L,j)$. So, it is sufficient for us to prove that
\[K_0N_0+\ldots+K_{L-1}N_{L-1}+(j+1)N_L \le C\Psi(K_0+\ldots+K_{L-1}+j+1).\]
For $L=1$, we need to have $(j+1)N_1 \le C\Psi(j+1)$.
But this follows since $\Psi(m) \ge m$ for all $m\ge 1$, and $C \ge N_1$.
For $L \ge 2$, we see that the inequality is
\[K_1N_1+\ldots+K_{L-1}N_{L-1}+(j+1)N_L \le
C(K_1+\ldots+K_{L-1}+j+1)\theta(K_1+\ldots+K_{L-1}+j+1).\]
But we have
\[C(K_1+\ldots+K_{L-1}+j+1)\theta(K_1+\ldots+K_{L-1}+j+1)\ge\]
\[(K_1+\ldots+K_{L-1}+j+1)\theta(K_{L-1}) \ge (K_1+\ldots+K_{L-1}+j+1)N_L\]
\[\ge K_1N_1+\ldots+K_LN_{L-1}+(j+1)N_L.\]
\end{proof}
\begin{cor}\lambdabel{bestdecay}
Given any sequence $(d_N:N\ge 1)$ such that $d_N \to 0$
as $N \to \infty$, and any ergodic rotation $T$ of $\mathbb T$, there
exists a rigidity sequence $\mathbf{n} = (n_m)$ for $T$ and
a constant $c > 0$ such that
$D(N,\mathbf{n}) > cd_N$ all $N \ge 1$.
\end{cor}
\begin{proof} We assume without loss of generality that $(d_N)$
is decreasing. Take the sequence $(n_m)$ constructed in
Proposition~\ref{bestgrowth}
with $\Psi(m) = \frac m{d_{m-1}}$ for $m \ge 2$.
Fix $N$ with $n_m \le N < n_{m+1}$. Then
\begin{eqnarray*} \frac {\#\{n_k:k\ge 1\}{\mathcal A}p\{1,\ldots,N\}}N &=& \frac mN
\ge \frac m{n_{m+1}} \\
&\ge& \frac m{C\Psi(m+1)}
\ge \frac {d_m}{2C}\\
&\ge& \frac {d_{n_m}}{2C}
\ge \frac {d_N}{2C}.
\end{eqnarray*}
\end{proof}
\begin{rem} The arguments above for an ergodic rotation of the circle
can easily be
generalized to an ergodic rotation of any compact metric abelian group
$G$. The details of this
are a straightforward generalization of the arguments given here.
Indeed, an ergodic rotation $T$
of $G$ is given by $T(g)=g_0g$ for some $g_0 \in G$ that generates a
dense subgroup of $G$. But then also for
any $U$, an open neighborhood of the identity in $G$, we have $\{n \ge
1: g_0^n \in U\}$ is a
syndetic sequence. Again, the syndetic property here
is automatic from minimality. But it is easy to see
this explicitly in this general case by a proof similar to the one given at the
beginning of
Proposition~\ref{bestgrowth}. Another approach to
Proposition~\ref{bestgrowth} and Corollary~\ref{bestdecay},
and their generalizations to compact metric abelian groups, would be to
use discrepancy estimates and
the fact the $\{g_0^n: n \ge 1\}$ is uniformly distributed in $G$.
See Kuipers and Niederreiter~{\mathcal I}te{KN} for background information about
uniform distribution in
compact abelian groups.
\end{rem}
\begin{rem} It has not yet been possible (and may not be) to carry out a construction
as in Proposition~\ref{bestgrowth} or Corollary~\ref{bestdecay} for some
weakly mixing transformation.
\end{rem}
\subsection{\bf Symbolic approach}
\lambdabel{symbolic}
In this section, we use shifts on products of finite spaces,
a standard model that appears in symbolic dynamics. We look at a construction
of rigid sequences for weakly mixing transformations that is given by taking the usual
coordinate shift on the product of certain finite sets, and giving a careful construction of a
measure on this product space. This will allow us to show that
certain types of sequences $(n_m)$ can be rigidity sequences for weakly mixing
transformations, even though the sequences do not have the pointwise
behavior of the sequences in Section~\ref{diophantine} i.e. the set $\mathcal R(n_m)$
contains no elements at all in $\mathbb T$ of infinite order, let alone an infinite
perfect set of points.
After this, in Section~\ref{products}, we take an approach to
the basic construction in this section, but we use Riesz products to get the results.
We consider these Riesz product constructions here partly because the issues that one needs to handle there anticipate the
results in Section~\ref{secdisjoint} where we show there are no universal rigid sequences.
We have seen that if $n_{m+1}/n_m$ is bounded then a pointwise approach
to the rigidity question will not work. So when this happens, the next
result
is giving us weakly mixing transformations that are rigid along $(n_m)$ even
though $\mathcal R(n_m)=\{\gamma \in \mathbb T: \lim\limits_{m\to \infty}
\gamma^{n_m} =1\}$ is countable.
\begin{prop} \lambdabel{integerratios} Given an increasing sequence
$(n_m)$ such that $n_{m+1}/n_m$ is always a whole number, there is a weakly
mixing dynamical system for which there is rigidity along $(n_m)$.
\end{prop}
\begin{proof}
let $a_1 = n_1$ and $a_{m+1} = n_{m+1}/n_m$ for all $m \ge 1$.
So $n_m = \prod\limits_{k=1}^m a_k$ and
$\frac {n_m}{n_M} = \prod\limits_{k=M+1}^m a_k$ for all $m \ge M+1$.
Consider the series representations for $x \in [0,1)$ of the form
$x =\sum\limits_{m=1}^\infty \frac {b_m}{n_m}$ where $b_m$ is
a whole number such that $0 \le b_m < a_m$. Except for a set
of Lebesgue measure zero, such series are uniquely determined
by $x$, and vice versa. We will write $b_m(x)$ to indicate
the dependence of $(b_m)$ on $x$. Let $\Pi =
\prod\limits_{m=1}^\infty \{0,\ldots,a_m-1\}$ and let
$\pi = \prod\limits_{m=1}^\infty \pi_m$ where $\pi_m$
is the uniform counting
measure on $\{0,\ldots,a_m-1\}$. It is easy to see that
there is a one-to-one, onto Borel mapping $\Phi$ of
$[0,1)$ to $\Pi$ such that $\pi{\mathcal I}rc \Phi^{-1}$ is Lebesgue
measure.
Now we use a block construction to build a positive
continuous Borel measure $\nu$ such that $\|n_mx\| \to 0$ in
measure with respect to $\nu$ as $m \to \infty$.
Consider disjoint intervals $I_k = [N_k+1,\ldots,N_{k+1}]$, with
$0 = N_0 < N_1 < N_2 < \ldots$, and $N_{k+1} - N_k = |I_k|$, the length
of $I_k$, increasing to $\infty$. Fix $(\epsilon_k)$ with
$0 < \epsilon_k \le \frac 12$ such that $\sum\limits_{k=1}^\infty \epsilon_k
= \infty$ and $\lim\limits_{k \to \infty} \epsilon_k = 0$.
Define $\nu_k$ on $\Pi$ as follows. Let $\overlineerline {0_k}$ be
the element in $\prod_{I_k} \{0,\ldots,a_m-1\}$ whose entries are
all $0$. Let $\nu_k(\overlineerline {0_k})
= 1 - \epsilon_k$ and $\nu_k$ uniformly distributed over the points
in $\prod_{I_k} \{0,\ldots,a_m-1\}\backslash \{\overlineerline {0_k}\}$ with
the total mass $\nu_k(\prod_{I_k} \{0,\ldots,a_m-1\}
\backslash \{\overlineerline {0_k}\}) =
\epsilon_k$. Then let
$\nu = \prod\limits_{k=1}^\infty \nu_k$ and consider
the measure $\nu{\mathcal I}rc \Phi$ on $[0,1)$.
We know that $\nu$ and $\nu{\mathcal I}rc \Phi$ are
regular Borel probability measures because all finite Borel measures
are regular in this situation. Moreover, $\nu$
and $\nu{\mathcal I}rc \Phi$ are
continuous, i.e. they have no point masses. Indeed, $\nu(\{\overlineerline 0\})
= \prod\limits_{k=1}^\infty (1 - \epsilon_k) = 0$ because
$\sum\limits_{k=1}^\infty \epsilon_k = \infty$, and by the definition
of $\nu$ for every other point $\overlineerline x \in \Pi$, we have
$\nu(\{\overlineerline x\}) \le \nu(\{\overlineerline 0\})$.
We claim that $\|n_Mx\| \to 0$ in measure
with respect to $\nu{\mathcal I}rc \Phi$ as $M \to \infty$.
We see that $\|n_Mx\| = \sum\limits_{m=M+1}^\infty \frac
{b_m}{n_m/n_M} = \sum\limits_{m=M+1}^\infty
\frac {b_m}{a_{M+1}\ldots a_m}$.
For any $M \ge 1$, we have
$M \in I_k$ for a unique $k = k(M)$.
Clearly, as $M \to \infty$, we have $k(M) \to \infty$.
Consider the set $D_k$ of vectors
$\overlineerline b = (b_m) \in \Pi$ such that $b_m = 0$ for
$m \in I_k {\mathcal U}p I_{k+1}$. We have $\nu(D_k) =
(1 - \epsilon_k)(1 - \epsilon_{k+1}) \to 1$ as $k \to \infty$.
But also for $\overlineerline b \in D_{k(M)}$, we have
$b_m = 0$ for all
$m, N_{k(M)}\le m \le N_{k(M)+2}$. Hence, if $\Phi(x) \in D_{k(M)}$,
then
$\|n_Mx\| = \sum\limits_{m=N_{k(M)+2}+1}^\infty
\frac {b_m}{a_{M+1}\ldots a_m} \le \sum\limits_{m=N_{k(M)+2}+1}^\infty
\frac 1{a_{M+1}\ldots a_{m-1}} \le \sum\limits_{m=N_{k(M) +2}+1}\frac
1{2^{m-M-1}} = 2^{M+1}\frac 1{2^{N_{k(M)+2}}}$.
But because we chose
$|I_k|$ increasing to $\infty$, we have
$2^{M+1}\frac 1{2^{N_{k(M)+2}}} \to 0$ as $M \to \infty$.
This means that, as $M \to \infty$,
we have $\nu{\mathcal I}rc \Phi(\Phi^{-1}D_{k(M)}) \to 1$
and for $x \in \Phi^{-1} D_{k(M)}$, we have $\|n_Mx\| \to 0$.
\end{proof}
This result leads to the following important special case.
\begin{cor} \lambdabel{powers} Given any whole number $a \ge 2$, there is a
weakly
mixing dynamical system for which there is rigidity along
$(a^m: m \ge 1)$. However, there is never rigidity along $\sigmagma
\in FS(a^m: m \ge 1)$ as $\sigmagma \to \infty (IP)$.
\end{cor}
\begin{proof} The first statement follows
from Proposition~\ref{integerratios}.
For the second part, see Proposition~\ref{ipodometer}.
But here is at least the idea. First
consider the sequence $(2^j: j \ge 1)$. When we take all finite
sums $2^{j_1}+\ldots+2^{j_k}$ with $m \le j_1 < \ldots < j_k$, then
we get all whole numbers $2^ms, s \ge 1$. Hence, to have rigidity
for $f {\mathcal I}rc T^{\sigmagma}$ as $\sigmagma \to \infty (IP)$, with $\sigmagma
\in \Sigma = FS(2^j:j \ge 1)$ would imply that $\|f {\mathcal I}rc T^{2^ms}
- f\|_2 \to 0$ as $m \to \infty$, independently of $s \ge 1$.
Assume that $f$ is mean-zero and not zero. Then we fix $m$ such
that $\|f{\mathcal I}rc T^{2^ms} - f\|_2 \le \frac 12$ for all $s\ge 1$.
But if $T$ is weakly mixing, so is $T^{2^m}$ and hence
\[\lim\limits_{s \to \infty} \lambdangle f{\mathcal I}rc T^{2^ms},f\rangle = 0.\]
This is not possible. The same argument works with any sequence
$(a^j: j \ge 1)$ in place of $(2^j)$ with $a \in \mathbb Z^+, a
\ge 2$. The only difference is that we would need to use some
fixed number of finite sums of elements in $\Sigma$ when $a \ge
3$.
\end{proof}
\begin{rem} \lambdabel{notIP} a) See Proposition~\ref{ipodometer} for
a different approach to a version of the above result.
\noindent b) The contrast of this result with the examples
in Remark~\ref{linformeg} c) is clear. While $(2^n)$ is a
rigidity sequence for a weakly mixing transformation, certain
simple perturbations of it, that are still lacunary sequences,
like $(2^n+1)$ are not rigidity sequences for weakly mixing
transformations. This leads to the obvious question:
if we assume that $(n_m)$ is lacunary, but not
necessarily as above a power of a fixed whole number $a \ge 2$,
when is there still a continuous Borel probability measure on
$[0,1)$ with $\widehat {\nu}(n_m) \to 1$ as $m \to \infty$? This
seems to be a difficult problem because it is not just about the
growth rate inherent in lacunarity, but also about the algebraic
nature of the sequence $(n_m)$.
\noindent c) On the other hand, it is not clear what happens for
the lacunary sequence $(2^m+3^m)$. That is, can
there be a continuous Borel probability measure
$\nu$ on $\mathbb T$ such that $\widehat {\nu}(2^m+3^m) \to 1$
as $m \to \infty$? By Proposition~\ref{rigidfacts},
if this happens, then $z^{2^m+3^m} \to 1$
in measure with respect to $\nu$. So $z^{2(2^m+3^m)}\to 1$ and
$z^{2^{m+1}+3^{m+1}} \to 1$ in measure with respect to $\nu$.
Taking the ratio gives $z^{3^m}\to 1$ in measure with respect
to $\nu$, and so again taking the appropriate ratio both $z^{2^m} \to 1$
and $z^{3^m} \to 1$ in measure with respect to $\nu$. But the
converse is also true by taking the product of these. So Proposition
~\ref{rigidfacts} shows that $\widehat {\nu}(2^m+3^m) \to 1$
as $m \to \infty$ if and only if both $\widehat {\nu}(2^m) \to 1$
and $\widehat {\nu}(3^m) \to 1$ as $m \to \infty$. Therefore, using
the GMC, what we are seeking is a weakly mixing
transformation which has both $(2^m)$ and $(3^m)$ as rigidity sequences.
\noindent d) The example of Eisner and Grivaux~{\mathcal I}te{EG} of an increasing sequence $(n_m)$ such
that $\lim\limits_{m \to \infty} \frac {n_{m+1}}{n_m} = 1$, for
which there is rigidity along $(n_m)$ for some weakly mixing
dynamical system, is one that is constructed inductively.
Concretely, is there rigidity along $(n_m)$ for
some weakly mixing dynamical system if $(n_m)$ is the sequence
obtained by writing $\{2^k3^l: k,l \ge 1\}$ in increasing order?
Note: results of Ajtai, Havas, and
Koml\'os~{\mathcal I}te{AHK} show that for any sequence $(\epsilon_m) $
decreasing to $0$, no matter how slowly, there exists an
increasing sequence $(n_m)$ such that $\frac {n_{m+1}}{n_m} \ge 1
+ \epsilon_m$ but there is not rigidity along $(n_m)$. Indeed,
they give examples where $\lim\limits_{M \to \infty} \frac 1M
\sum\limits_{m=1}^M \exp(2\pi i
n_m x) = 0$ for all $x \in (0,1)$, and hence for any positive
Borel measure $\nu \not= \delta_0$, $\lim\limits_{M \to \infty} \frac 1M
\sum\limits_{m=1}^M \widehat
{\nu}(n_m) = 0$.
\end{rem}
\subsection{\bf Riesz Products}
\lambdabel{products}
In this section, we will use products of discrete measures
to give an alternative approach to the results in Section~\ref{symbolic}.
There are several benefits to looking at this approach. One
is that it may with more work end up being more flexible than
the method in Proposition~\ref{integerratios}. Also, it gives
us interesting examples that relate to the question of disjointness: the construction of
two weakly mixing dynamical systems with rigid sequences for
which the product action has no rigid sequences. Disjointness
turns out to be a very difficult fact to prove in explicit cases, requiring analytical
methods that are not yet fully developed. However, in Section~\ref{secdisjoint},
we will see how to use random methods to create this disjointness in a
very general way.
The approach we are taking here to defining continuous Borel probability measures $\nu$
with $\widehat {\nu}(n_m) \to 1$ as $m \to \infty$ currently
works just in cases like $(n_m) = (a^m)$ for any integer $a \ge 2$. We have
some indication that this method can be made more flexible. This
method is like the classical method of Riesz products. There
are many references for Riesz products, but one that is closely
related to the considerations in this article is Host, M\'ela,
and Parreau~{\mathcal I}te{Ho-Me-Pa}. But unlike the classical approach,
where one is taking Riesz products
in the frequency variable, we are taking Riesz products
in the space variable, using finitely supported
discrete measures. Of course, this too has been used
by others to give constructions of measures.
See Brown and Moran~{\mathcal I}te{BM1, BM2} and
Graham and McGehee~{\mathcal I}te{GMcG}. This method may possibly give a way
to handle cases that we have not been able to handle before, but
that is not clear. However, some other harmonic analysis issues
come into play here that lead to interesting conclusions in
ergodic theory.
The basic approach is as follows. Choose $(a_k)$ and $(b_k)$ positive,
with $a_k + b_k =1$. We assume that $\lim\limits_{k \to \infty}
b_k = 0$ and $\prod\limits_{k=1}^{\infty} a_k =
\prod\limits_{k=1}^\infty (1 - b_k) = 0$. So we assuming
$\sum\limits_{k=1}^\infty b_k = \infty$.
Take a sequence of points $(x_k)$ from $[0,1)$ and let $\omega_k =
a_k\delta_1 + b_k\delta_{\exp(2\pi ix_k)}$. Let $\nu_k = \omega_k
\ast \omega_k^* = (a_k^2 + b_k^2)\delta_1 +
a_kb_k\delta_{\exp(2\pi ix_k)} + a_kb_k\delta_{\exp(-2\pi ix_k)}$.
So we have $\widehat {\nu_k}(j) = |\widehat {\omega_k}(j)|^2 = 1 -
2a_kb_k(1 - {\mathcal O}s(2\pi jx_k))$. In particular, $0 \le \widehat
{\nu_k}(j) \le 1$ for all $k$ and $j$. Consider the infinite
product $\nu = \prod\limits_{k=1}^\infty \nu_k$. We take this as
defined in the weak$^*$ sense; that is, we know that we have the
Fourier transforms of the partial products $\Pi_K =
\prod\limits_{k=1}^K \nu_k$ converging and hence they give the
Fourier transform of some Borel probability $\nu$ in the limit.
Then we evaluate $\widehat {\nu}(n_m)$ by looking at what the
infinite product does.
Here is an example of this approach. Take $n_m = 2^m$ for all $m
\ge 1$ and let $x_k = \frac 1{2^k}$ for all $k \ge 1$. We
specifically choose $b_k = \frac 1{k+1}$, so $a_k = 1 -\frac
1{k+1}$. We defined $\nu$ by the spectral condition
\[\widehat {\nu}(j) = \prod\limits_{k=1}^\infty
\left (1 - 2\frac {(1 - \frac 1{k+1})}{k+1}(1 - {\mathcal O}s (2\pi \frac
j{2^k}))\right ).\]
Now consider the values $\widehat {\nu}(2^m)$. We always have
\[\widehat {\nu}(2^m) = \prod\limits_{k=m+1}^\infty
\left (1 - 2\frac {(1 - \frac 1{k+1})}{k+1}(1 - {\mathcal O}s (2\pi \frac
{2^m}{2^k}))\right ).\]
So we need to know what this does as $m \to \infty$.
But we claim that $\widehat {\nu}(2^m) \to 1$ as $m \to \infty$.
That is, by taking the natural logarithm of the expression, we
need to show that
\[\lim\limits_{m \to \infty} \sum\limits_{k=m+1}^\infty 2\frac {(1 -
\frac 1{k+1})}{k+1}(1 -{\mathcal O}s(2\pi \frac 1{2^{k-m}})) = 0.\]
However, this is the same as showing
\[\lim\limits_{m \to \infty} \sum\limits_{k=m+1}^\infty 2\frac {(1 -
\frac 1{k+1})}{k+1} \, \frac 1{2^{2(k-m)}} = 0.\]
But $\sum\limits_{k=m+1}^\infty 2\frac {(1 -
\frac 1{k+1})}{k+1} \, \frac 1{2^{2(k-m)}} \le \frac {2^{2m+1}}m
\sum\limits_{k=m+1}^\infty \frac 1{2^{2k}} \le \frac {2^{2m}}m
\frac 1{2^{2m}} = \frac 1m$. So we have the estimate that we
needed.
We also need to make an argument that $\nu$ has no atoms. First,
each of the partial products $\Pi_K = \prod\limits_{k=1}^K \nu_k$
is a purely atomic measure. The choice of $a_k$ and $b_k$
guarantees that for $K \ge 1$, these partial products have a
maximal mass at $1$ with the value $\prod\limits_{k=1}^K
(a_k^2+b_k^2) = \prod\limits_{k=1}^K (1 - 2a_kb_k)$. Suppose that
$\nu$ is suspected of having a non-trivial point mass at some
point $\gamma_0$. For a fixed value of $K$, choose an open dyadic
arc $A = \{\exp(2\pi i x): \frac {j-1}{2^K} < x < \frac
{j+1}{2^K}\}$ which contains $\gamma_0$. Then there is a
continuous function $h_K, 0 \le h_K \le 1$, with $h_K(\gamma_0)
=1$ and the support of $h_K$, that is the closure of $\{h_K >
0\}$, contained in $A$. We have for any $M$, $\int h_K \, d\Pi_M
\le \Pi_M(A)$. We can easily see that for $M \ge K$, $\Pi_M(A) \le
\epsilon_K$ with $\lim\limits_{K \to \infty} \epsilon_K = 0$. But
then $\int h_K\, d\Pi_M \le \epsilon_K$ too, so letting $M$ tend
to $\infty$ shows that $\int h_K \, d\nu \le \epsilon_K$. Then
letting $K \to \infty$, this proves that $\nu(\{\gamma_0\}) = 0$
because for all $K$, $\nu(\{\gamma_0\}) \le \int h_K \, d\nu$.
But the estimate we need is easy because, for
$M \ge K$, we can write the partial product $\Pi_M$ as a sum
$\sum\limits_{t=1}^{2^M} c_t(M) \delta_{\exp(2\pi i \frac
{t-1}{2^M})}$ with positive coefficients $c_t(M)$ such that
$\sum\limits_{t=1}^{2^M} c_t(M) = 1$. The value of $\Pi_M(A)$ can
be seen to be no larger than the value that the measure $\mu$
given by
\[ \left (c_{j-1}(K)\delta_{\exp(2\pi i\frac {j-1}{2^K})}+
c_j(K)\delta_{\exp(2\pi i \frac j{2^K})} +
c_{j+1}(K)\delta_{\exp(2\pi i \frac {j+1}{2^K})}\right )
\prod\limits_{k={K+1}}^M
\nu_k\]
gives to $A$ because for the other point masses $\delta_{\exp(2\pi
i \frac {t-1}{2^K})}$, the measure $\delta_{\exp(2\pi i \frac
{t-1}{2^K})}\prod\limits_{k=K+1}^M \nu_k$ has support disjoint
from $A$. Since $\prod\limits_{k={K+1}}^M \nu_k$ itself is a
probability measure, this shows that $\Pi_M(A) \le c_{j-1}(K) +
c_j(K) + c_{j+1}(K) \le 3\prod\limits_{k=1}^K (a_k^2+b_k^2)$. We
take $\epsilon_K= 3\prod\limits_{k=1}^K (a_k^2+b_k^2) =
3\prod\limits_{k=1}^K(1 - 2a_kb_k)$. The choice of $(a_k)$ shows
that $\epsilon_K \to 0$ as $K \to \infty$. Indeed,
$\sum\limits_{k=1}^\infty b_k = \infty$ and $\lim\limits_{k \to
\infty} a_k = 1$, so $\prod\limits_{k=1}^\infty (1 - 2a_kb_k) =
0$.
\begin{rem} a) The method above suggests that we might be able
to choose $(x_k)$ for more general sequences, including perhaps
ones that are not lacunary. But the method implicitly needs
inductively chosen good choices of $x$ where $\exp(2\pi i n_mx)
\to 1$ as $m \to \infty$. It is not clear for a fixed $(n_m)$ how
to characterize when the set of $x$ such that $\lim\limits_{m \to
\infty} \exp(2\pi i n_mx) = 1$ is empty, when it is finite, when
it is countably infinite, or when it is uncountable!
\noindent b) We believe that what is needed generally for the
construction as above of a continuous Borel probability measure
$\nu$ with $\lim\limits_{m \to \infty}
\widehat {\nu}(n_m) = 1$ as $m \to \infty$ is
\begin{itemize}
\item[(a)] $\lim\limits_{k \to \infty} b_k = 0$,
\item [(b)] $\sum\limits_{k=1}^\infty b_k = \infty$, and
\item[(c)] there is a sequence $(x_k)$ such that
$\lim\limits_{m \to \infty} \sum\limits_{k=1}^\infty b_k
\|n_mx_k\|^2=0$
\end{itemize}
For example, this method would work not only for the sequences
above, but also for sequences as in Proposition~\ref{integerratios}.
\noindent c) We have tried to prove directly that the examples
above have a Fourier transform converging to $0$ at $\infty$ along
a sequence of positive density. For example, one would want to
show that for some sequence of values $j$ with positive density,
we have
$$\lim\limits_{j \to \infty} \sum\limits_{k=1}^\infty b_k(1
-{\mathcal O}s(2\pi \frac j{2^k})) = \infty.$$
But currently we do not see how to prove this.
\end{rem}
\subsubsection{\bf Disjoint Rigidity}
\lambdabel{secdisjoint}
The method above can perhaps give us some good examples to look at
for which we have two systems that are rigid but not
simultaneously, indeed when the
convolution of their maximal spectral types is strongly mixing.
For example, take the measures $\nu(2)$ and $\nu(3)$ where
\[\widehat {\nu(2)}(j) = \prod\limits_{k=1}^\infty
\left (1 - 2a_kb_k(1 - {\mathcal O}s (2\pi \frac
j{2^k}))\right )\]
and
\[\widehat {\nu(3)}(j)= \prod\limits_{k=1}^\infty
\left (1 - 2a_kb_k(1 - {\mathcal O}s (2\pi \frac
j{3^k}))\right ).\]
We have $\widehat {\nu(2)}(2^m) \to 1$ and $\widehat {\nu(3)}(3^m)
\to 1$ as $m \to \infty$. But does
\begin{equation}\lambdabel{disjoint}
\widehat {\nu(2)}(j)\widehat {\nu(3)}(j) \to 0 \ \ \text {as} \ \ j
\to \infty?\end{equation}
This is the same as asking if the series
\[\sum\limits_{k=1}^\infty b_k\left (
1 - {\mathcal O}s (2\pi \frac j{2^k}) + 1 - {\mathcal O}s (2\pi \frac j{3^k})\right
) \] tends to $\infty$ as $j$ tends to $\infty$. This will not
work with $b_k = \frac 1k$; one can see this by taking $j = 6^k$
for large values of $k$. But it may work with $b_k = \frac
1{\sqrt k}$.
While we cannot answer the question in Equation~\ref{disjoint} at this time, we can
modify a construction of LaFontaine~{\mathcal I}te{lafontaine} to
obtain Proposition~\ref{AA0} below. We will need two procedural lemmas.
First, given a finite Borel measure $\omega$ on $\mathbb R$, we
let $FT_B(\omega)(n) = \int\limits_0^B \exp(-2\pi i n\frac xB)\ d\omega(x)$.
We use the notation $\widehat {\omega}$ for the
Fourier transform on $\mathbb R$ given by $\widehat {\omega}(t) =
\int \exp(-2\pi itx)\ d\omega(x)$.
Let $m=m_{\mathbb R}$ denote the Lebesgue measure on $\mathbb R$, and let
$L_2(m)$ denote the Lebesgue space $L_2(\mathbb R,\mathcal B_m,m)$.
\begin{lem}\lambdabel{transforms} Suppose we have a positive
Borel measure $\omega$ on $\mathbb R$ that has compact
support. Then $\frac {d\omega}{dm} \in L_2(m)$ if and only
if $\sum\limits_{n=-\infty}^\infty |\widehat {\omega}(n)|^2 < \infty$.
\end{lem}
\begin{proof}
By translating $\omega$, we may assume without loss of generality
that the support of $\omega$ is a subset $[0,B]$ for some whole number
$B$. Then
\begin{eqnarray*}
\widehat {\omega}(n) &=& \int\ \exp(-2\pi i n x)\ d\omega(x)
= \sum\limits_{k=0}^{B-1} \int\limits_{k}^{k+1} \exp(-2\pi i n x) \
d\omega(x)\\
&=& \sum\limits_{k=0}^{B-1} \int\limits_0^1\exp(-2\pi i n (x+k)) \
d\omega(x+k)\\
&=& \sum\limits_{k=0}^{B-1} \int\limits_0^1\exp(-2\pi i n x) \
d\omega(x+k)
=\int\limits_0^1 \exp(-2\pi i n x)\ d\Omega(x).
\end{eqnarray*}
where $d\Omega(x) = 1_{[0,1)}(x)\sum\limits_{k=0}^{B-1} \omega(x+k)$
is a positive Borel measure supported on $[0,1]$. That is,
$\widehat {\omega}(n) = FT_1(\Omega)(n)$ for all $n$. The
usual classical argument shows that $\frac {d\Omega}{dm} \in L_2(m)$
if and only if $\sum\limits_{n=-\infty}^\infty |FT_1(\Omega)(n)|^2 < \infty$
because $\Omega$ is supported on $[0,1)$.
Therefore, $\sum\limits_{n= -\infty}^\infty |\widehat {\omega}(n)|^2 <
\infty$
is equivalent to knowing that $\frac {d\Omega}{dm} \in L_2([0,1),m)$.
So we conclude that if $\frac {d\Omega}{dm} \in L_2([0,1),m)$, then
for each $k=0,\ldots,B-1$, the positive measure
$1_{[0,1]} d\omega(x+k)$ satisfies
$0 \le 1_{[0,1]} d\omega(x+k) \le d\Omega(x)$, and so it also has a
density in $L_2([0,1),m)$.
Hence, by translating the terms back again and adding them together, we
also know that
$\frac {d\omega}{dm}$ is in $L_2(\mathbb R,m)$. Of course also
conversely, if
$\frac {d\omega}{dm}$ is in $L_2(\mathbb R,m)$, then $\frac
{d\Omega}{dm} \in L_2([0,1),m)$.
This proves that $\frac {d\omega}{dm} \in L_2(m)$ if and only if
$\sum\limits_{n= -\infty}^\infty |\widehat {\omega}(n)|^2 < \infty$.
\end{proof}
\begin{rem}\lambdabel{othertransforms} This remark and
Lemma~\ref{transforms} are related to the ideas behind
Shannon sampling and the Nyquist frequency for band-limited signals.
In Lemma~\ref{transforms}, if we replace $\omega$ by a dilation of it,
then one can see that more generally for a positive Borel measure
$\omega$ on $\mathbb R$ with compact support, we have $\frac {d\omega}{dm}
\in L_2(m)$ if and only if for some $b > 0$ (or for all $b > 0$),
we have $\sum\limits_{n=-\infty}^\infty |\widehat {\omega}(bn)|^2 < \infty$.
The assumption that $\omega$ is positive is necessary here.
Indeed, suppose we have a compactly
supported complex-valued Borel measure $\omega$ on $\mathbb R$. Suppose
that the support of $\omega$ is a subset of $[0,B]$.
Then by the usual classical argument, we know that
$\frac {d\omega}{dm} \in L_2(m)$ if and only if
$\sum\limits_{n= -\infty}^\infty |FT_B(\omega)(n)|^2 < \infty$.
But $\widehat {\omega}(\frac nB) = FT_B(\omega)(n)$
because $\omega$ is supported on $[0,B]$.
So we have in this case, $\frac {d\omega}{dm} \in L_2(m)$ if and only if
$\sum\limits_{n= -\infty}^\infty |\widehat{\omega}(\frac nB)|^2 < \infty$.
Here, $B$ can be replaced by any larger value, but not necessarily
by a smaller value. For example, if we take $\omega_0$ supported
on $[0,1)$ and define $d\omega(x) = d\omega_0(x) - d\omega_0(x-1)$,
then our value of $B = 2$, but
$\widehat {\omega}(n)=0$ for all $n$. However, the measure $\omega_0$
could be singular to $m$ and hence $\omega$ might not
have an $L_2(\mathbb R,m)$-density.
\end{rem}
We also want to make a few observations about the differences between
convolving measures on $\mathbb T$ and convolving their associated
measures on $\mathbb R$. Given a Borel measure $\omega$
on $\mathbb T$, let $\omega_{\mathbb R}$ denote the Borel measure on
$\mathbb R$ given by $\omega_{\mathbb R}=\omega{\mathcal I}rc E$
where $E:[0,1)\to\mathbb T$ by $E(x) = \exp(2\pi i x)$. Let
$m_{\mathbb T}$ denote the usual Lebesgue measure on $\mathbb T$
i.e. $m_{\mathbb T}{\mathcal I}rc E = 1_{[0,1)}m_{\mathbb R}$.
\begin{lem}\lambdabel{linetocircle} If $\mu$ and $\nu$ are Borel
measures on $\mathbb T$, then $\mu\ast\nu$
is absolutely continuous with respect to $m_{\mathbb T}$ if
$\mu_{\mathbb R}\ast\nu_{\mathbb R}$ is absolutely continuous with
respect to $m_{\mathbb R}$.
\end{lem}
\begin{proof} For $f \in C(\mathbb T)$, we have
\begin{eqnarray*}
\int f(\gamma) d(\mu\ast\nu)(\gamma) &=& \int\int
f(\gamma_1\gamma_2)d\mu(\gamma_1)d\nu(\gamma_2)\\
&=& \int\int f(E(x)E(y)) d\mu_{\mathbb R}(x)d\nu_{\mathbb R}(y)\\
&=& \int f{\mathcal I}rc E(x+y) d\mu_{\mathbb R}(x)d\nu_{\mathbb R}(y)\\
&=& \int f{\mathcal I}rc E(z) d(\mu_{\mathbb R}\ast\nu_{\mathbb R})(z).
\end{eqnarray*}
Hence, it is clear that if $\mu_{\mathbb R}\ast \nu_{\mathbb R}$
is absolutely continuous with respect to $m_{\mathbb R}$ with
density $F$, then $\mu\ast\nu$ is absolutely continuous with respect
to $m_{\mathbb T}$ with density $F{\mathcal I}rc E^{-1}$.
\end{proof}
\begin{rem} The converse statement to Lemma~\ref{linetocircle} is
not true without additional assumption, for example the assumption
that both measures are positive. For example, take a non-zero measure
$\omega_0$ on $\mathbb T$ supported in the arc $E([0,1/2))$. Let
$\omega = \omega_0 - \omega_0\ast \delta_{-1}$. Then let $\nu = \delta_1+
\delta_{-1}$, a positive discrete measure on $\mathbb T$. We have
\begin{eqnarray*}
\omega\ast\nu &=& \omega_0 -
\omega_0\ast\delta_{-1}+\omega_0\ast\delta_{-1} -
\omega_0\ast\delta_1\\
&=&\omega_0-\omega\ast\delta_1\\
&=&0.
\end{eqnarray*}
However, $\omega_{\mathbb R} = (\omega_0)_{\mathbb R}
-(\omega_0)_{\mathbb R}\ast\delta_{1/2}$ and $\nu_{\mathbb R} =
\delta_0+\delta_{1/2}$. So
\begin{eqnarray*}
\omega_{\mathbb R} \ast \nu_{\mathbb R}
&=&(\omega_0)_{\mathbb R} -(\omega_0)_{\mathbb R}\ast\delta_{1/2}+
(\omega_0)_{\mathbb R}\ast\delta_{1/2}-(\omega_0)_{\mathbb R}\ast\delta_1\\
&=& (\omega_0)_{\mathbb R} -(\omega_0)_{\mathbb R}\ast\delta_1.
\end{eqnarray*}
This measure is not zero as a measure on $\mathbb R$.
If also $\omega_0$ is singular to $m_{\mathbb T}$, then we
have $\omega\ast\nu$ absolutely continuous with respect to
$m_{\mathbb T}$, since it is $0$, while
$\omega_{\mathbb R} \ast \nu_{\mathbb R}$ is not absolutely continuous
with respect to $m_{\mathbb R}$.
\end{rem}
These two lemmas will help in proving the following.
\begin{prop}\lambdabel{AA0} Assume that
$(X,\mathcal B_X,p_X,T)$ is a rigid, weakly mixing dynamical system.
Then there is a weakly mixing, rigid dynamical system
$(Y,\mathcal B_Y,p_Y,S)$ such that the maximal spectral type of
$U_{T\times S}$ in the orthocomplement of
$F:=L_2(X,p_X)\otimesimes1_Y\oplus1_X\otimesimes L_2(Y,p_Y)$ is Rajchman.
In other words, for all $f \in L_2(X\times Y,p_X \otimesimes p_Y)$
that are orthogonal to both the $X$-measurable functions and the
$Y$-measurable functions, we have $\lambdangle f {\mathcal I}rc (T \times
S)^n,f\rangle \to 0$ as $n \to \infty$. In fact, $U_{T\times S}$
has absolutely continuous spectrum on $F^\perp$.
\end{prop}
\begin{proof} From the spectral point of view we want
to show that given a continuous Dirichlet measure $\mu$,
there is a continuous Dirichlet probability measure $\nu$ on
$\mathbb T$ such that $\mu\ast \nu$ is an absolutely continuous
(hence a Rajchman measure). Indeed, we then can take $\mu=\nu^T$ and we
let $(Y,\mathcal B_Y,p_Y,S)$ be given by $S=G_\nu$. As $\mu\ast\nu^{\ast
k}=(\mu\ast\nu)\ast\nu^{\ast(k-1)}$ we easily check that
$\mu\ast\sum\limits_{k=1}^\infty\frac1{2^k}\nu^{\ast k}$ is still
absolutely continuous.
Given the references we are using,
it is better to carry out our construction in $\mathbb R$.
So consider first the measure $\mu{\mathcal I}rc E$ on $\mathbb R$.
Lemma~\ref{linetocircle} shows that, to get our result, it will
be enough to construct a suitable positive Borel measure $\nu{\mathcal I}rc E$
on $\mathbb R$, with support in $[0,1)$,
such that $(\mu{\mathcal I}rc E)\ast(\nu{\mathcal I}rc E)$ is absolutely continuous.
For notational convenience we will denote $\mu{\mathcal I}rc E$
and $\nu{\mathcal I}rc E$ by $\mu$ and $\nu$ in the rest of this proof.
Hence, suppose we have a continuous positive Borel measure $\mu$ on
${\mathbb{R}}$, which is
compactly supported. Consider the dilation $\mu_{\lambdambda}$,
$\lambdambda > 0$, given by $\mu_{\lambdambda}(E) = \mu(\frac 1{\lambdambda}
E)$ for all Borel sets $E\subset{\mathbb{R}}$. Then $\widehat
{\mu_{\lambdambda}}(t) = \int \exp(-2\pi itx) \, d\mu_{\lambdambda}(x) = \int
\exp(-2\pi it\lambdambda x) \, d\mu(x) = \widehat {\mu}(\lambdambda t)$.
We would like to construct a suitable continuous positive Borel
measure $\nu$ supported in $[0,1]$ such that $J(\lambdambda) =
\sum\limits_{n=-\infty}^\infty |\widehat {\mu_{\lambdambda}}(n)|^2
|\widehat {\nu}(n)|^2$ is finite. But
\[\int\limits_0^1 J(\lambdambda) \, dm(\lambdambda)
= \sum\limits_{n=-\infty}^\infty |\widehat {\nu}(n)|^2 \left
(\int\limits_0^1 |\widehat {\mu}(\lambdambda n)|^2 \,
dm(\lambdambda)\right ).\] Using Wiener's lemma for measures on ${\mathbb{R}}$
(e.g.\ {\mathcal I}te{Ka}, Chapter VI.2) and the fact that $\mu$ is
continuous
$$a_n := \int\limits_0^1 |\widehat {\mu}(\lambdambda n)|^2
\, dm(\lambdambda)=\frac1n\int_0^n|\widehat{\mu}(t)|^2\,dt \to 0$$ as
$|n| \to \infty$. So, we are seeking a suitable $\nu$ such that
\begin{equation}\lambdabel{ww10}\sum\limits_{n = -\infty}^\infty |\widehat
{\nu}(n)|^2a_n<+\infty.\end{equation}
Clearly, if $\nu$ were actually
absolutely continuous with respect to $m$ with a square integrable
density, then we would have this condition. But $\nu$ could not
be rigid in this situation. However, as LaFontaine points out,
the two articles of Salem~{\mathcal I}te{salem1,salem2} give a construction
of a Borel probability measure $\nu$ with support in $[0,1)$ with
this property, and which is also continuous and rigid. See
LaFontaine~{\mathcal I}te{lafontaine} and Salem~{\mathcal I}te{salem1,salem2} for
the details. It follows that there exists a continuous probability
measure $\nu$ supported on $[0,1)$ such that for some increasing sequence
$(n_m)$ of integers \begin{equation}\lambdabel{ww11} \widehat{\nu}(n_m)\to 1\end{equation}
and (\ref{ww10}) holds.
It follows from (\ref{ww10}) that for $m$-a.e.\ $\lambdambda\in[0,1]$
\begin{equation}\lambdabel{ww12} \sum\limits_{n=-\infty}^\infty
|\widehat{\mu_\lambdambda\ast\nu}(n)|^2=
\sum\limits_{n=-\infty}^\infty |\widehat
{\mu_\lambdambda}(n)|^2|\widehat {\nu}(n)|^2 =J(\lambdambda) < \infty.\end{equation}
The measure $\mu_\lambdambda\ast\nu$
is supported in $[0,1+\frac1\lambdambda]$ and in view
of Equation~(\ref{ww12}) and Lemma~\ref{transforms}, it must be
absolutely continuous with respect to $m$, with
$\frac {d(\mu_\lambdambda\ast\nu)}{dm} \in L_2(m)$.
Thus, if we choose any value of $\lambdambda$
satisfying~(\ref{ww12}), we have
\[\int\limits_{-\infty}^\infty
|\widehat {\mu}(\lambdambda t)|^2 |\widehat {\nu}(t)|^2 \,dm(t) =
\frac 1{\lambdambda}\int\limits_{-\infty}^\infty |\widehat
{\mu}(t)|^2|\widehat {\nu}(\frac t{\lambdambda})|^2\, dm(t)\]
is finite. It follows that the measure $\mu\ast \nu_{1/\lambdambda}$ is
compactly supported and absolutely continuous. In view of~(\ref{ww11}),
$\widehat
{\nu_{1/\lambdambda}}(\lambdambda n_m)$ converges to $1$ as $m\to \infty$.
Moreover, with respect to $m$, for almost every
$\lambdambda$, we would know that $(n_k\lambdambda: k \ge 1)$ is uniformly
distributed modulo $1$. Hence, for some subsequence $(n_{k_j})$
and some sequence of integers $(m_j)$, we have $\lim\limits_{j
\to\infty}\left( n_{k_j}\lambdambda - m_j\right) = 0$. Then by the
uniform continuity of $\widehat {\nu_{1/\lambdambda}}$, we would also
have $\widehat {\nu_{1/\lambdambda}}(m_j) \to 1$ as $j \to \infty$.
Hence, with respect to $m$, almost every choice of $0<\lambdambda < 1$
gives $\nu_{1/\lambdambda}$ supported on $[0,\lambdambda]\subset [0,1]$, which is a
continuous rigid Borel probability measure, such that $\mu\ast
\nu_{1/\lambdambda}$ is absolutely continuous.
\end{proof}
\begin{rem}\lambdabel{notjtrigid} The transformations $T$ and $S$ in Proposition~\ref{AA0}
must be disjoint in the sense that their only joining is their product.
See Example~\ref{egdisjoint} a). As a first step, assume that $T^{n_m}\to Id$
in the strong operator topology. By passing to a subsequence if
necessary we can assume that $S^{n_m}\to \Phi$ in the
weak operator topology, where $\Phi:L_2(Y,p_Y)\to L_2(Y,p_Y)$
is a Markov operator. We claim that
$\Phi(g)\neq g$ for each non-zero $g\in L_{2,0}(Y,p_Y)$.
Indeed suppose for some non-zero $g$, $\Phi(g)=g$. Take any non-zero $f\in L_{2,0}(X,p_X)$. Then
$\lambdangle (T\times S)^{n_m}(f\otimesimes g),f\otimesimes g\rangle \to \lambdangle f\otimesimes g,f\otimesimes g\rangle$.
So the spectral
measure of $f\otimesimes g$ is a Dirichlet measure, contrary to construction
in Proposition~\ref{AA0}. Now as a second step,
take a joining $J$ of $T$ and $S$. On the operator level, this means that
we have a Markov operator $W=W_J$ corresponding to $J$ such that $W: L_2(X,p_X)\to L_2(Y,p_Y)$ and $WT=SW$.
Then $WT^{n_m}=S^{n_m}W$,
and by passing to limits we obtain $W=\Phi W$. So by our first step,
we have $W(L_{2,0}(Y,p_Y))= \{0\}$. But this means
that $W$ is the Markov operator for the product joining, and so $J$
is the product measure $p_T\otimesimes p_S$.
Thus, $T$ and $S$ are disjoint. For background,
see Glasner~{\mathcal I}te{Glasner}, Chapter 6, Section 2.
\end{rem}
\begin{rem} The abstract argument used in Proposition~\ref{AA0}
gives some motivation for finding concrete examples of
continuous rigid measures whose convolution has a
Fourier transform vanishing at infinity. For example, we were considering
this in Equation~\ref{disjoint} with the Riesz product constructions for powers of $2$ and
powers of $3$.
\end{rem}
\begin{rem} We can argue differently that if a continuous
probability measure $\rho$ supported on $[0,1]\subset{\mathbb{R}}$ satisfies
$\widehat{\rho}(r_k)\to1$ for some sequence of reals
$r_k\to\infty$, then, as a circle measure, $\rho$ is Dirichlet.
Indeed, consider the flow $V_t(f)(x)=e^{itx}f(x)$ on
$L_2({\mathbb{R}},\rho)$. Our assumption says that $V_{r_k}\to Id$. Consider
then $V_{\{r_k\}}$, $k\geq1$, which replaces $r_k$ by its
fractional part $\{r_k\}$. By passing to a subsequence if
necessary and using the continuity of the unitary representation
${\mathbb{R}}\ni t\mapsto V_t$, we have $V_{\{r_k\}}\to V_s$ for some
$s\in[0,1]$. It follows that $V_{[r_k]}\to Id{\mathcal I}rc
(V_s)^{-1}=V_{-s}$. Then, $([r_{k+1}]-[r_k])$ is a rigidity
sequence for $\rho$.\end{rem}
\begin{rem} Clearly, there is an IP version of Proposition~\ref{AA0}
that follows by passing to a subsequence of the rigidity sequence
for $S$.
\end{rem}
This result shows that whenever we have a weakly mixing rigid
transformation $T$, then there is a weakly mixing rigid transformation
$S$ such that $T\times S$ is not rigid for any sequence. More
generally, we can prove the following result. We again use
the closure of $\{T^n: n \in \mathbb Z\}$ in the strong operator
topology, which can be identified with the centralizer of $T$ in
case $T$ has discrete spectrum.
\begin{cor}\lambdabel{AA1} Assume that $(X,\mathcal B,p,T)$
is an ergodic dynamical system. Then $T$ has discrete spectrum if
and only if for each weakly mixing rigid system
$(Y,\mathcal B_Y,p_Y,S)$ the Cartesian product system $T\times S$
remains rigid.
\end{cor}
\begin{proof} Assume that $T$ is an ergodic rotation and let $S$
be weakly mixing, $S^{n_m}\to Id$. By passing to a subsequence if
necessary, $T^{n_m}\to R\in C(T)$, and so $T^{n_{m+1}-n_m}\to Id$
and still $S^{n_{m+1}-n_m}\to Id$; thus $T\times S$ is rigid (see
also Proposition~\ref{sparse}).
To prove the converse, suppose that $T$ does not have discrete spectrum,
but $T\times S$ is rigid for each $S$ which is rigid and weakly
mixing. Then there is some continuous $\nu$ with $\nu\ll\nu^T$. It
follows that $\nu$ is a Dirichlet measure, and if
$\widehat{\nu^T}(n_m)\to\nu^T(\mathbb T)$ then
$\widehat{\nu}(n_m)\to \nu(\mathbb T)$. Consider the Gaussian system
$G_\nu$ given by $\nu$. Then $G_\nu$ is weakly mixing and each
sequence which is a rigid sequence for $T$ is also a rigid
sequence for $G_\nu$. It follows that $G_\nu\times S$ is rigid for
each weakly mixing rigid $S$ which is in conflict with from
Proposition~\ref{AA0}.
\end{proof}
\begin{rem} Using the viewpoint in Bergelson and Rosenblatt~{\mathcal I}te{bergros},
it is clear that Corollary~\ref{AA1} has a unitary version. That is,
a unitary operator $U$ on a Hilbert space $H_U$ has discrete spectrum if and only if for every
weakly mixing rigid unitary operator $V$ on a Hilbert space $H_V$, the
product $U\times V$ on $H_U\times H_V$ is a rigid unitary operator.
\end{rem}
The following is a folklore result.
\begin{prop}\lambdabel{A2} Assume that $T$ and $S$ are ergodic
transformations with discrete spectrums. Then they are isomorphic if and only if they have the same
rigidity sequences.
\end{prop}
\begin{proof} Consider $C(T)$ and $C(S)$ respectively. Both are
given as weak closure of powers. Take the map:
$$
F:C(T)\to C(S), \;\;F(T^n)=S^n,\;n\in{\mathbb{Z}}.$$
We easily show that
this extends to a homeomorphism equivariant with rotation by $T$
and $S$ respectively.
Now refer back to the comments about centralizers at the beginning
of Section~\ref{rigidonly}. We know that $T$ is isomorphic to the translation by $T$
on $C(T)$ considered with Haar measure and
$S$ is isomorphic to the translation by $S$
on $C(S)$ considered with Haar measure. Hence, $T$ and $S$ are
isomorphic.
\end{proof}
\begin{cor}\lambdabel{AA2} Assume that $(X,\mathcal B_X,p_X,T)$
is an ergodic transformation with a discrete spectrum, and $(Y,\mathcal B_Y,p_Y,S)$ is ergodic and has
the same rigidity sequences as $T$. Then $S$ is isomorphic to $T$.
\end{cor}
\begin{proof} It follows directly from
Corollary~\ref{AA1} that $S$ has discrete spectrum. The result
follows from Proposition~\ref{A2}.
\end{proof}
\subsection{\bf Cocycle Methods}
\lambdabel{seccocycle}
\subsubsection{\bf Tools} \lambdabel{tools}
We will now describe tools using cocycles over rotations to
produce weakly mixing transformations with a prescribed sequence
as rigidity sequences. We will start with a transformation $T$ having
discrete spectrum and its sequence of rigidity. (In fact, for
applications, we will consider one dimensional
rotations by irrational $\alphapha$ and the sequence given the denominators of
$\alphapha$.) Then we will consider cocycles over $T$ with values in
locally compact Abelian groups. We will then pass to the
associated unitary operators (weighted operators) and we will try
to ``lift'' some rigidity sequences for the rotation to the
weighted operator. Once such an operator has continuous spectrum
we apply the GMC which preserves rigidity. Another option to
obtain ``good'' weakly mixing transformations will be to pass to
Poisson suspensions (in case we extend by a locally compact and
not compact group) - which in a sense will be even easier as
ergodicity of Poisson suspension is closely related to the
fact that the cocycles are not coboundaries. See Remark~\ref{Poisson},
and also {\mathcal I}te{CFS}, ~{\mathcal I}te{De-Fr-Le-Pa}
and ~{\mathcal I}te{Ro}, for details concerning
ergodic properties of Poisson suspensions. Note: at times
in this section $\mathbb T$ denotes the unit circle in $\mathbb C$
with multiplicative notation, and at times it will mean $[0,1)$
with addition modulo one. The reader will be able to distinguish
which model for the circle is being used by the context of the
discussion.
\subsubsection{\bf Compact Group Extensions and Weighted Operators}
Assume that $T$ is an ergodic transformation acting on a standard
Lebesgue space $(X,\mathcal B,p)$. Let $G$ be a compact metric
Abelian group with Haar measure $\lambda_G$. We take the $\lambda_G$
measurable sets, denoted by $\mathcal G$ as our measurable sets
for $(G,\mathcal B_G,\lambda_G)$. A measurable map
$\varphi:X\to G$ generates a
{\em cocycle} $\varphi^{({\mathcal D}ot)}({\mathcal D}ot)$ which is given by $\varphi^{(n)}({\mathcal D}ot):X\to G$,
$n \in \mathbb Z$, by the formula, for $x \in X$,
\begin{equation}\lambdabel{cocycle}
\varphi^{(n)}(x)=\left\{\begin{array}{ccc}\varphi(x)+\varphi(Tx)+\ldots+\varphi(T^{n-1}x)&\mbox{if}&
n>0,\\
0&\mbox{if}& n=0,\\
-(\varphi(T^{-n}x)+\ldots+\varphi(T^{-1}x))&\mbox{if}&n<0.\end{array}\right.\end{equation}
Using $T$ and $\varphi$ we define a {\em compact group extension}
$T_\varphirphi$ of $T$ which acts on the space $(X\times
G,{\mathcal B}\otimes{\mathcal B}_G,p\otimes\lambda_G)$ by the formula \begin{equation}\lambdabel{gext}
T_\varphirphi(x,g)=(Tx,\varphi(x)+g)\;\;\mbox{for}\;\;(x,g)\in X\times G.
\end{equation} Notice that for each $n\in {\mathbb{Z}}$ and $(x,g)\in X\times G$
\begin{equation}\lambdabel{gext0} (T_\varphirphi)^n(x,g)=(T^nx,\varphi^{(n)}(x)+g).\end{equation}
The natural decomposition of $L_2(G,\lambda_G)$ using the character
group $\widehat{G}$ yields the decomposition
\begin{equation}\lambdabel{gext1}
L_2(X\times G,p\otimes\lambda_G)=\bigoplus_{{\mathcal H}i\in\widehat{G}}L_2(X,p)\otimes
({\mathbb{C}}{\mathcal H}i).\end{equation}
Here ${\mathbb{C}}{\mathcal H}i$ is the one-dimensional subspace spanned by the
character ${\mathcal H}i$.
To understand ergodic and other mixing properties of
$T_\varphirphi$ we need to study the associated Koopman operator
$U_{T_\varphirphi}$,
$$U_{T_\varphirphi}F=F{\mathcal I}rc T_\varphirphi,\;\mbox{for}\;\;F\in L_2(X\times
G,p\otimes\lambda_G).$$ As the (closed) subspaces $L_2(X,p)\otimes({\mathbb{C}}{\mathcal H}i)$
in~(\ref{gext1}) are $U_{T_\varphirphi}$-invariant, we can examine
those mixing properties separately on all such subspaces (notice
that for ${\mathcal H}i=1$ we consider the original Koopman operator
$U_T$). It is well-known and not hard to see that the map
$f\otimes{\mathcal H}i\mapsto f$ provides a spectral equivalence of
$U_{T_\varphirphi}|_{L_2(X,p)\otimes({\mathbb{C}}{\mathcal H}i)}$ and the operator
$V_{{\mathcal H}i{\mathcal I}rc\varphi}^T$ acting on $L_2(X,\mathcal B,p)$ by the formula
\begin{equation}\lambdabel{gext2} V^T_{{\mathcal H}i{\mathcal I}rc\varphi}f={\mathcal H}i{\mathcal I}rc\varphi{\mathcal D}ot f{\mathcal I}rc
T\;\;\mbox{for}\;\;f\in L_2(X,p).\end{equation} Each such operator is an
example of a {\em weighted operator} $V^T_\xi$ over $T$, where
$\xi:X\to \mathbb T$ is a measurable function with values in
the (multiplicative) circle $\mathbb T$ and
$V_\xi^Tf=\xi{\mathcal D}ot f{\mathcal I}rc T$.
Assume now that $T$ is an ergodic transformation with discrete
spectrum, i.e.\ without loss of generality, we can assume that $X$
is a compact monothetic metric group with $p=\lambda_X$ Haar measure
on $X$, and $Tx=x+x_0$ where $x_0$ and $\{nx_0:\:n\in{\mathbb{Z}}\}$ is
dense in $X$. Assume that $\xi:X\to \mathbb T$ is measurable.
By Helson's analysis {\mathcal I}te{He} (see also e.g.\ {\mathcal I}te{Iw-Le-Ru}):
\begin{thm}[{\mathcal I}te{He}]\lambdabel{helson} For $T$ as above, the maximal
spectral type of $V^T_\xi$ is either discrete or continuous. If it
is continuous then either it is singular or it is
Lebesgue.\end{thm}
We will use only the first part of this theorem. An important
practical point that comes from this theorem is that once we find
a function $f\in L_2(X,\mathcal B,p)$ such that the spectral measure $\sigmagma_f =
\sigmagma_f^{V_\xi^T}$ is continuous, then
$V^T_\xi$ has purely continuous spectrum. Consider $f=1$.
Using~(\ref{gext0}), for each $n\in{\mathbb{Z}}$, we obtain
\begin{equation}\lambdabel{gext3} \lambdangle
\left(V^T_\xi\right)^n1,1\rangle=\int_X\xi^{(n)}(x)\,dp(x).\end{equation} It
follows that if there is a subsequence $(n_m)_{m\geq1}$ such that
\begin{equation}\lambdabel{gext4} \int_X\xi^{(n_m)}(x)\,dp(x)\to 0\;{\mathbb{R}}ightarrow\;
V^T_\xi\;\mbox{has continuous spectrum}.\end{equation} It is also nice to
note in passing that we have a bit stronger
result:\begin{equation}\lambdabel{gext5}
\int_X\xi^{(n_m)}(x)\,dp(x)\to 0\;{\mathbb{R}}ightarrow\;
\left(V^T_\xi\right)^{n_m}\to 0\;\mbox{weakly},\end{equation} which follows
directly from~(\ref{gext4}); indeed,
$$\int_X\xi^{(n_m)}(x)f(T^{n_m}x)\overline{f(x)}\,dp(x)\to 0$$
whenever $f$ is a character of $X$.
It is well-known and easy to check that if $(n_m)$ is a rigidity
sequence for $T$
\begin{equation}\lambdabel{gext6}
\xi^{(n_m)}\to1\;\mbox{in measure}{\mathbb{R}}ightarrow\;
\left(V^T_{\xi}\right)^{n_m}\to Id\;\mbox{strongly}.\end{equation}
\subsubsection{\bf ${\mathbb{R}}$-extensions, Weighted Operators and Poisson
Suspensions} We will now consider the case $G={\mathbb{R}}$. We assume now
that $f:X\to{\mathbb{R}}$ is a cocycle for $T$ acting ergodically on a
standard probability Borel space $(X,\mathcal B,p)$. We consider $T_f$
$$
T_f(x,r)=(Tx,f(x)+r)$$ acting on $(X\times{\mathbb{R}},p\otimes\lambda_{{\mathbb{R}}})$. Note
that we are now on a standard Lebesgue space with a $\sigmagma$-finite
(and not finite) measure. In particular, constants are not
integrable.
To study spectrally $T_f$ we will write it slightly differently,
namely
$$
T_f=T_{f,\tau}$$ where $\tau$ is the natural action of ${\mathbb{R}}$ on
itself by translations: $\tau_t(r)=r+t$ and
$$
T_{f,\tau}(x,r)=(Tx,\tau_{f(x)}(r)).$$ This transformation is a
special case of so called {\em Rokhlin extension} (of $T$), see for
example {\mathcal I}te{Le-Le}, and the spectral analysis below is similar to the
one in {\mathcal I}te{Le-Le}. So let us just imagine a slightly more
general situation
$$T_{f,{\mathcal S}}(x,y)=(Tx,S_{f(x)}(y))$$ where ${\mathcal S}=(S_t)$ is a flow
acting on $(Y,{\mathcal C},\nu)$ ($\nu$ can be finite or infinite). We will denote
the spectral measure of $a\in L_2(Y,{\mathcal C},\nu)$ (for the Koopman
representation $t\mapsto U_{S_t}$ on $L_2(Y,{\mathcal C},\nu)$) by
$\sigmagma_{a,{\mathcal S}}$.
The space $L_2(X\times Y,p\otimes\nu)$ is nothing but a tensor product
of two Hilbert subspaces, so to understand spectral measures we
only need to study spectral measures for tensors $a\otimes b$ and we
have
$$
\int_{X\times Y}(a\otimes b){\mathcal I}rc (T_{f,{\mathcal S}})^n{\mathcal D}ot\overline{a\otimes
b}\,dp\,d\nu$$
$$=\int_X\int_Y a(T^nx)\overline{a(x)}b(S_{f^{(n)}(x)}y)
\overline{b(y)}\,dp(x)\,d\nu(y)$$
$$=\int_Xa(T^nx)\overline{a(x)}\left(\int_Y e^{2\pi
itf^{(n)}(x)}\,d\sigmagma_{b,{\mathcal S}}(t)\right)\,dp(x)$$
$$= \int_Y\left(\int_X e^{2\pi itf^{(n)}(x)}
a(T^nx)\overline{a(x)}\,dp(x)\right)\,d\sigmagma_{b,{\mathcal S}}(t).$$
\begin{prop}\lambdabel{ll1}
If $T^{n_k}\to Id$ and $f^{(n_k)}\to0$ in measure then $(n_k)$ is
a rigidity sequence for $T_{\varphirphi}s$.\end{prop}
\begin{proof}Take $a\in L^\infty(X,p)$ and notice that
by assumption for each $t\in{\mathbb{R}}$
$$
\int_X e^{2\pi
itf^{(n_k)}(x)}a(T^{n_k}x)\overline{a(x)}\,dp(x)\to\int_X|a|^2\,dp.$$ By
the Lebesgue Dominated theorem
$$\int_Y\left(\int_X e^{2\pi itf^{(n_k)}(x)}
a(T^{n_k}x)\overline{a(x)}\,dp(x)\right)d\sigmagma_{b,{\mathcal S}}(t)\to
\int_Y\left(\int_X|a|^2\,dp\right)d\sigmagma_{b,{\mathcal S}}=\|a\otimes
b\|_{L_2(p\otimes\lambda_{{\mathbb{R}}})}^2.$$
\end{proof}
We need more information about sequences of the form $$\int_X
e^{2\pi itf^{(n)}(x)} a(T^nx)\overline{a(x)}\,dp(x),\,\,n\in{\mathbb{Z}}.$$ In
fact, they turn out to be again Fourier coefficients of some
spectral measures. Indeed, consider $V_t$ acting on $L_2(X,\mathcal B,p)$ by
the formula
$$
(V_ta)(x)=e^{2\pi itf(x)}a(Tx).$$ This is nothing but a weighted
unitary operator and
$$
\lambdangle V_t^n a,a\rangle=\int_X e^{2\pi itf^{(n)}(x)}
a(T^nx)\overline{a(x)}\,dp(x).$$ (Notice that we came back to the finite
measure-preserving case.)
Clearly, for $b\in L_2({\mathbb{R}})$ with compact support and $\mathcal S = \tau$
$$
\widehat{\sigmagma}_{b,{\mathcal S}}(t)=\int_{{\mathbb{R}}}b{\mathcal I}rc S_t{\mathcal D}ot\overline{b}\,dr
$$
$$=\int_{{\mathbb{R}}}b(r+t)\overline{b(r)}\,dr=(b\ast\overline{b})(-t).$$
Hence, the Fourier transform of $\sigmagma_{b,{\mathcal S}}$ is square
summable, and therefore this spectral measure is absolutely
continuous. In fact, the maximal spectral type of ${\mathcal S}$ is
Lebesgue, and we can see the maximal spectral type of $U_{T_f}$ as
an integral (against ``Lebesgue'' measure) of the maximal spectral
types of the family indexed by $t\in{\mathbb{R}}$ of weighted operators.
Suppose now that $U_{T_f}$ has an eigenvalue $c$, $|c|=1$. Then we
cannot have that all spectral measures $\sigmagma_{a\otimes b,T_f}$ are
continuous. In fact, we must have that $c$ appears as an
eigenvalue for ``many'' $V_t$ (on a set of $t\in{\mathbb{R}}$ of positive
Lebesgue measure), and the following result is well-known (it is
an exercise).
\begin{lem}
The scalar $c$ is an eigenvalue of $V_t$ if and only if we can solve the
following functional equation:
$$
e^{2\pi itf}=\frac {c{\mathcal D}ot \xi{\mathcal I}rc T}{\xi}$$ in measurable functions
$\xi:X\to\mathbb T$.
\end{lem}
It follows that having an eigenvalue for $U_{T_f}$ means that we
can solve the above multiplicative equations on a set of positive
Lebesgue measure of $t\in{\mathbb{R}}$. We are now in the framework of the
classical Helson's problem (e.g.\ {\mathcal I}te{Mo-Sch}) of passing from
multiplicative coboundaries to additive coboundaries. Using known
results in this area ({\mathcal I}te{Mo-Sch}, and see also the appendix in
{\mathcal I}te{Le-Pa}) we obtain the following (remember that constant
functions are not elements of $L_2(X\times{\mathbb{R}},p\otimesimes\lambda_{{\mathbb{R}}})$).
\begin{prop}\lambdabel{He}
If $U_{T_f}$ has an eigenvalue then $f$ is an additive
quasi-coboundary, that is there exist a measurable $g:X\to{\mathbb{R}}$ and
$r\in{\mathbb{R}}$ such that $f(x)=r+g(x)-g(Tx)$ for $p$-a.e.\ $x\in X$.
\end{prop}
Therefore, if $f$ is a non-trivial cocycle then automatically
$U_{T_f}$ has continuous spectrum and classically the Poisson
suspension over $T_f$ is ergodic, hence weakly mixing (see Remark~\ref{Poisson}). Recall
that from spectral point of view Poisson suspension over
$(X\times {\mathbb{R}},p\otimes\lambda_{{\mathbb{R}}},T_f)$ will be the same as Gaussian
functor over $(L_2(X\times{\mathbb{R}},p\otimes\lambda_{{\mathbb{R}}}),U_{T_f})$. In
particular, if $(T_f)^{n_t}\to Id$ on
$L_2(X\times{\mathbb{R}},p\otimes\lambda_{{\mathbb{R}}})$ then $(n_t)$ will be a rigidity
sequence for the suspension (in view of Proposition~\ref{ll1}).
From the above discussion, it follows that to have a weakly mixing
transformation $\widetilde {T }_f$ with a rigidity sequence $(N_t)$, we need:
(i) $f$ is not an additive coboundary,\, (ii) $T^{N_t}\to Id$,\, (iii) $f^{(N_t)}\to 0$ in measure.
\subsubsection{\bf Denominators of $\alphapha$ and Rigidity} \lambdabel{contfrac}
We have already seen that the sequence $(2^n)$ is a rigidity
sequence for a weakly mixing transformation. We can construct some
other explicit examples of rigidity sequences by using known
results from the theory of ``smooth'' cocycles over
one-dimensional rotations. This will allow us to show that if
$\alphapha$ is irrational, and $(q_n)$ stands for its sequence of
denominators then $(q_n)$ is also a rigidity sequence for a weakly
mixing transformation. The most interesting case is of course the
bounded partial quotient case (for example for the Golden Mean).
So assume that $\varphi:{\mathbb{T}}\to{\mathbb{R}}$ is a smooth mean-zero cocycle. We use
the term ``smooth'' here in a not very precise way; it may refer
to a good speed of decaying of the Fourier transform of $\varphi$.
We recall first that one of consequences of the Denjoy-Koksma
Inequality for $AC_0$ (absolutely continuous mean-zero) cocycles
is that \begin{equation}\lambdabel{DK} \varphi^{(q_n)}\to 0\;\;\mbox{uniformly}\end{equation}
for every irrational rotation by $\alphapha$, see {\mathcal I}te{Herman}.
Another type of Denjoy-Koksma inequality has been proved in
{\mathcal I}te{Aa-Le-Ma-Na} for functions $\varphi$ whose Fourier transform is
of order $\mbox{O}(1/|n|)$ -- as its consequence we have the
following: \begin{equation}\lambdabel{DK1} \mbox{If
$\widehat{\varphi}(n)=\mbox{o}(\frac1{|n|})$, $\widehat{\varphi}(0)=0$
then $\varphi^{(q_n)}\to0$ in measure}\end{equation} for every
rotation by an irrational $\alphapha$.
We would like also to recall another (unpublished) result by M.
Herman {\mathcal I}te{Herman1}. While this may not be available, one
can see also Krzy\.zewski~{\mathcal I}te{Kr} for generalizations of Herman's result.
\begin{thm}\lambdabel{MH} Assume that a mean-zero $\varphi:{\mathbb{T}}\to{\mathbb{R}}$ is in
$L_2({\mathbb{T}},\lambda_{{\mathbb{T}}})$ and its Fourier transform is concentrated on a
lacunary subset of ${\mathbb{Z}}$. Suppose that
$$
\varphi(x)=g(x)-g(x+\alphapha), \;\;\lambda_{{\mathbb{T}}}-\mbox{a.e.}$$ for some
irrational $\alphapha\in[0,1)$. Then $g\in L_2({\mathbb{T}},\lambda_{{\mathbb{T}}})$.
\end{thm}
Fix $\alphapha\in[0,1)$ irrational, and let
$\alphapha=[0:a_1,a_2,\ldots]$ stand for the continued fraction
expansion of $\alphapha$. Denote by $(q_n)$ the sequence of
denominators of $\alphapha$: $q_0=1$, $q_1=a_1$ and
$q_{n+1}=a_{n+1}q_n+q_{n-1}$ for $n\geq2$. Then
$$
\frac{q_{n+2}}{q_{n}}=\frac{a_{n+2}q_{n+1}+q_n}{q_{n}}\geq
a_{n+2}+1\geq2.$$ It follows that \begin{equation}\lambdabel{lac1}
(q_{2n})\;\mbox{is lacunary}.\end{equation} Moreover, \begin{equation}\lambdabel{lac2}
q_n\|q_n\alphapha\|\leq 1\;\mbox{for each}\;n\geq1.\end{equation} We define
$\varphi:{\mathbb{T}}\to{\mathbb{R}}$ by \begin{equation}\lambdabel{lac3} \varphi(x)=\sum\limits_{n=0}^\infty
a_{q_{2n}}{\mathcal O}s2\pi iq_{2n}x\end{equation} where for $n\geq 1$
\begin{equation}\lambdabel{lac3a} a_{q_{2n}}=\frac1{\sqrt n}\|q_{2n}\alphapha\|.\end{equation}
We then have $\varphi:{\mathbb{T}}\to{\mathbb{R}}$, $\widehat{\varphi}(n)=\mbox{o}(1/|n|)$ and
$(a_{q_{2n}})\in l_2$ in view of~(\ref{lac2}). Now, suppose that
\begin{equation}\lambdabel{lac4}\varphi(x)=g(x)-g(x+\alphapha)\end{equation} for a measurable
$g:{\mathbb{T}}\to{\mathbb{R}}$. In view of Theorem~\ref{MH} and~(\ref{lac1}), $g\in
L_2({\mathbb{T}},\lambda_{{\mathbb{T}}})$. Hence,
$$
g(x)=\sum\limits_{k=-\infty}^\infty b_k e^{2\pi ikx}.$$ Furthermore, by
comparing Fourier coefficients on both sides in~(\ref{lac4}),
$$
b_k=0\;\;\mbox{if}\;k\neq
q_{2n}\;\mbox{and}\;\;b_{q_{2n}}=a_{q_{2n}}/(1-e^{2\pi
iq_{2n}\alphapha})$$ with $(b_{q_{2n}})\in l_2$. However
$$
|b_{q_{2n}}|=a_{q_{2n}}/\|q_{2n}\alphapha\|=1/\sqrt n$$ which is a
contradiction. We hence proved the following.
\begin{prop}\lambdabel{lac6} For each irrational $\alphapha\in[0,1)$
there is a mean-zero $\varphi:{\mathbb{T}}\to{\mathbb{R}}$ such that
$\widehat{\varphi}(n)=\mbox{o}(1/|n|)$ and $\varphi$ is not an additive
coboundary.\end{prop}
Using~(\ref{DK1}) and Proposition~\ref{ll1} we hence obtain the
following.
\begin{prop}\lambdabel{lac7} For $\varphi$ satisfying the assertion of
Proposition~\ref{lac6} the sequence $(q_n)$ of denominators of
$\alphapha$ is a rigidity sequence of $T_\varphi$ on
$L_2({\mathbb{T}}\times{\mathbb{R}},\lambda_{{\mathbb{T}}}\otimes\lambda_{{\mathbb{R}}})$ and $U_{T_{\varphi}}$ has
continuous spectrum.\end{prop}
By using GMC method or by passing to the relevant Poisson
suspension we obtain:
\begin{cor}\lambdabel{denominators} For each sequence $(q_n)$ of
denominators there exists a
weakly mixing transformation $R$ such that $R^{q_n}\to
Id$.\end{cor}
\begin{rem} \lambdabel{wazna}(i) We would like to emphasize that in general the
sequence $(q_n)$ of denominators of $\alphapha$ is not lacunary.
Indeed, assume that $\alphapha=[0:a_1,a_2,\ldots]$ stands for the
continued fraction expansion of $\alphapha$. Suppose that for a
subsequence $(n_k)$ we have $a_{n_k+1}=1$, $a_{n_k}\to\infty$.
Then by the recurrence formula $q_{m+1}=a_{m+1}q_m+q_{m-1}$ we
obtain that
$$
\liminf_{n\to\infty}\frac{q_{n+1}}{q_n}=1.$$ So the
sequences of denominators are another type of non-lacunary
sequences which can be realized as rigidity sequences for weakly
mixing transformations, besides the ones in Section~\ref{ratesofgrowth}.
\noindent (ii) As the above shows $\{q_n:\:n\geq1\}$ is always a Sidon set
(see~{\mathcal I}te{Ka},{\mathcal I}te{Ru}); indeed,
$$
\{q_n:\:n\geq1\}=\{q_{2n}:n\geq1\}{\mathcal U}p\{q_{2n+1}:\:n\geq 0\}.$$ It
follows that the set of denominators is the union of two lacunary
sets and hence is a Sidon set ({\mathcal I}te{Ka},{\mathcal I}te{Ru}).
\noindent (iii) The assertion of Theorem~\ref{MH} is true for functions
whose Fourier transform is concentrated on a Sidon set and when
$T$ is an arbitrary ergodic rotation on a compact metric Abelian
group (by the proof of the main result in {\mathcal I}te{He} or by
{\mathcal I}te{Kr}).
\noindent (iv) It follows that to prove Proposition~\ref{lac6} we could have
used all denominators, with (for example) $a_{q_n}=\frac1{\sqrt
n}\|q_n\alphapha\|$.
\noindent (v) Eisner and Grivaux~{\mathcal I}te{EG} obtain some results in the
direction of Corollary~\ref{denominators}, but their examples
are restricted to badly approximated irrational numbers.
\end{rem}
There are also more complicated constructions showing that for
each irrational $\alphapha$ there is an absolutely continuous
mean-zero $\varphi:{\mathbb{T}}\to{\mathbb{R}}$ which is not a coboundary --
see~{\mathcal I}te{Li-Vo}. We can then use such cocycles and~(\ref{DK1})
for another proof of the above corollary.
Here is a concrete example of Corollary~\ref{denominators}
using the continued fraction expansion of the Golden Mean.
\begin{cor} The Fibonacci
sequence is a rigidity sequence for some weakly mixing transformation .
\end{cor}
\begin{rem} Suppose we take a
increasing sequence like the Fibonacci sequence, which is obtained by recursion.
That is, we have
$z = F(x_1,\ldots,x_K) = \sum\limits_{k=1}^K c_kx_k$ where $c_k$
are whole numbers, and we have
$n_{m+1} = F(n_m,\ldots,n_{m-K+1})$ for all $m$.
Is this always a rigidity sequence for a weakly mixing transformation?
\end{rem}
\begin{rem} It is not clear how to characterize rigidity sequences
that cannot be IP rigidity sequences. See
Proposition~\ref{powers}, and its generalization
Proposition~\ref{ipodometer} for examples of this phenomenon. In
reference to the above, it would be interesting to show that
$FS((q_n))$ is not a rigidity net.
\end{rem}
\begin{rem} The above results can be used to answer positively the
following question: {\em Given an increasing sequence $(n_m)$ of
integers is there a weakly mixing transformation $R$ such that
$R^{n_{m_k}}\to Id$ for some subsequence $(n_{m_k})$ of $(n_m)$?}
In fact, we have already answered this question (see
Proposition~\ref{folklore1}), but we will now take a very different
approach. We start with the following well-known lemma; see
for example ~{\mathcal I}te{Kw-Le-Ru}.
\begin{lem} Given an increasing sequence $(n_m)$ of natural
numbers, consider the set of $\alphapha \in [0,1)$ such that a
subsequence of $(n_m)$ is a subsequence of denominators of
$\alphapha$. This is a generic subset of $[0,1)$.
\end{lem}
\noindent Now, given $(n_m)$ choose any irrational $\alphapha$ so that for some
subsequence $(n_{m_k})$ we have all numbers $n_{m_k}$ being
denominators of $\alphapha$. Then use previous arguments to construct a weakly
mixing ``realization'' of the whole sequence of denominators of
$\alphapha$.
\end{rem}
\subsubsection{\bf Integer Lacunarity Case}
We will now give an alternative proof of Proposition~\ref{integerratios}
using the cocycle methods that have been developed here.
Assume that $(n_m)_{m\geq0}$ is an increasing sequence of positive
integers such that $n_0=1$, $n_{m+1}/n_m\in{\mathbb{Z}}$ with
\begin{equation}\lambdabel{odo1} \rho_m:=n_{m+1}/n_m\geq2\;\;\mbox{for}\;m\geq0.
\end{equation} Notice that in view of~(\ref{odo1}) there exists a constant
$C>0$ such that \begin{equation}\lambdabel{odo2}
\frac1{n_{m+1}^2}+\frac1{n_{m+2}^2}+\ldots\leq \frac
C{n_{m}^2}\;\;\mbox{for each $m\geq0$}.\end{equation}
Let $X=\Pi_{m=1}^\infty\{0,1,\ldots,\rho_m-1\}$ which is a
metrizable compact group when we consider the product topology and
the addition is meant coordinatewise with carrying the remainder
to the right. On $X$ we consider Haar measure $p_X$ which is the
usual product measure of uniform measures. Define $Tx=x+\widehat{A}t 1$,
where
$$
\widehat{A}t 1=(1,0,0,\ldots).$$ The resulting dynamical system is called
the $(n_m)$-{\em odometer}.
For each $t\geq 0$ set
$$
D_0^{n_m}=\{x\in X:\:x_0=x_1=\ldots=x_{m-1}=0\}.$$ Note that
$\{D^{n_m}_0,TD^{n_m}_0,\ldots,T^{n_m-1}D^{n_m}_0\}$ is a Rokhlin
tower fulfilling the whole space $X$ and \begin{equation}\lambdabel{odo3}\widehat{A}t 1\in
TD^{n_m}_0 \;\;\mbox{for each $m\geq0$}.\end{equation}
The character group $\widehat X$ of $X$ is discrete and is
isomorphic to the (discrete) group of roots of unity of
degree~$n_m$, $m\geq0$. More precisely, for $m\geq0$ set
$$
1_{n_m}(x)=\varphirepsilon^j_{n_m}:=e^{2\pi ij/n_m}\;\;\mbox{for $x\in
T^jD^{n_m}_0$,}\;j=0,1,\ldots, n_m-1.$$ Then $\widehat
X=\{1_{n_m}^j:\:j=0,1,\ldots,n_m-1,m\geq0\}$.
From now on we will consider $f\in L_{2,0}(X,p_X)$ whose Fourier
transform is ``concentrated'' on $\{1_{n_m}:\:m\geq0\}$. We
have \begin{equation}\lambdabel{odo4} f(x)=\sum\limits_{m=1}^\infty
a_{n_m}1_{n_m}(x),\;\sum\limits_{m=1}^\infty|a_{n_m}|^2<+\infty.\end{equation}
{\bf A) Small divisors.} Assume that $f$ satisfies~(\ref{odo4})
and suppose that \begin{equation}\lambdabel{odo5}
f(x)=g(x)-g(x+\widehat{A}t 1)\;\;\mbox{for $p_X$-a.e.}\;x\in X.\end{equation} Suppose
moreover that $g\in L_2(X,p_X)$. Hence
$g(x)=\sum\limits_{{\mathcal H}i\in\widehat{X}}b_{{\mathcal H}i}{\mathcal H}i(x)$ and by comparison
of Fourier coefficients on both sides in~(\ref{odo4}) we obtain
$$
b_{\mathcal H}i=0\;\mbox{whenever ${\mathcal H}i\neq 1_{n_m}$ and
$a_{n_m}=b_{n_m}(1-1_{n_m}(\widehat{A}t 1))$ for $m\geq1$}.$$
Using~(\ref{odo3}) we obtain that
$b_{n_m}=\frac{a_{n_m}}{1-\varphirepsilon_{n_m}}$, so
$$
|b_{n_m}|^2=\frac{|a_{n_m}|^2}{|1-\varphirepsilon_{n_m}|^2}=n_m^2|
a_{n_m}|^2\;\;\mbox{for $m\geq1$}.$$ We have proved the following
\begin{equation}\lambdabel{odo6} \mbox{(\ref{odo5}) has an $L_2$-solution if and
only if $(n_ma_{n_m})_m\in l_2$.}\end{equation}
{\bf B) Estimate of $L_2$-norms for the cocycle.} Assume that
$n\in{\mathbb{N}}$ then
$$
f^{(n)}(x)=\sum\limits_{m=1}^\infty a_{n_m}(1+1_{n_m}(\widehat{A}t 1)+\ldots+
1_{n_m}((n-1)\widehat{A}t 1))1_{n_m}(x)
$$$$=\sum\limits_{m=1}^\infty a_{n_m}\left(\sum_{j=0}^{n-1}
\varphirepsilon_{n_m}^j\right)1_{n_m}(x).$$ Fix $n=n_{m_0}$. We then have
$$f^{(n_{m_0})}(x)=\sum_{m=1}^{m_0} a_{n_m}\left(
\sum\limits_{j=0}^{n_{m_0}-1} \varphirepsilon_{n_m}^j\right)1_{n_m}(x)+
\sum\limits_{m=m_0+1}^\infty a_{n_m}\left(\sum_{j=0}^{n_{m_0}-1}
\varphirepsilon_{n_m}^j\right)1_{n_m}(x)$$$$ =\sum\limits_{m=m_0+1}^\infty
a_{n_m}\frac{1-\varphirepsilon_{n_m}^{n_{m_0}}}{1-\varphirepsilon_{n_m}}1_{n_m}(x).$$
Since $|1-\varphirepsilon_{n_m}|=1/n_m$ and $|1-\varphirepsilon^{n_{m_0}}_{n_m}|\leq
n_{m_0}/n_t$, \begin{equation}\lambdabel{odo7}
\|f^{(n_{m_0})}\|^2_{L_2(X,p_X)}\leq
n_{m_0}^2\sum\limits_{m=m_0+1}^\infty |a_{n_m}|^2.\end{equation}
{\bf C) Sidon sets and a ``good'' function.} According to
{\mathcal I}te{Ru} (see Example 5.7.6 therein) every infinite subset of a
discrete group contains an infinite Sidon set. Hence we can choose
a subsequence $(n_{m_k})$ of $(n_m)$ so that \begin{equation}\lambdabel{odo8}
\mbox{$\{1_{n_{m_k}}:\:k\geq1\}$ is a Sidon subset of $\widehat
X$}.\end{equation} We set \begin{equation}\lambdabel{odo9} f(x)=\sum\limits_{k=1}^\infty
\frac1{\sqrt kn_{m_k}}1_{n_{m_k}}(x).\end{equation} Suppose now
that~(\ref{odo5}) has a measurable solution $g$ (we should
consider $f$ real valued, so in fact we should consider
$f+\overlineerline{f}$ below). In view of~(\ref{odo8}) and~{\mathcal I}te{Kr},
$g\in L_2(X,p_X)$. But
$$\sum\limits_{m=1}^\infty|n_m a_{n_m}|^2=\sum\limits_{k\geq1}\frac1k,$$
so by~(\ref{odo6}), we cannot obtain an $L_2$-solution. This
means that $f$ is not a measurable coboundary. According
to~(\ref{odo7}), the definition of $f$ and~(\ref{odo1}) for each
$s\geq1$
$$
\|f^{(n_{s})}\|^2_{L_2(X,p_X)}\leq n_s^2\sum\limits_{m=s+1}^\infty
|a_{n_m}|^2= n_s^2\sum\limits_{k\geq1:\:n_{m_k}\geq n_{s+1}}^\infty
|a_{n_{m_k}}|^2=n_s^2\sum\limits_{k=k_s}^\infty |a_{n_{m_k}}|^2
$$$$
\leq \frac{n_s^2}{k_s}\sum\limits_{j=s+1}^\infty
\frac1{n_{j}^2}\leq\frac C{k_s}\to0$$ as clearly $k_s\to\infty$
when $s\to\infty$.
Using our general method we hence proved the following.
\begin{prop}\lambdabel{intlac} Assume that $(n_m)$ is an increasing
sequence of integers with $n_{m+1}/n_m$ being an integer at
least~2. Then there exists a weakly mixing transformation $R$ such
that $R^{n_m}\to Id$.\end{prop}
We will now discuss the problem of $IP$-rigidity along $(n_m)$.
First of all notice that $(n_m)$ is a sequence of $IP$ rigidity
for the $(n_m)$-odometer (indeed, $\sum\limits_{m=1}^\infty
|1_{n_m}(\widehat{A}t 1)-1|<+\infty$).
Let us also notice that if $R$ is weakly mixing and $R^{n_m}\to
Id$ then by passing to a subsequence, we will get
$IP-R^{n_{m_k}}\to Id$. But if we then set $m_k=n_{m_k}$ then
$m_{k}$ divides $m_{k+1}$ and $(m_k)$ is a sequence of $IP$-rigidity
for $R$. It means that if the sequence $(n_{m+1}/n_m)$ is
unbounded then, at least in some cases, it is a sequence of
$IP$-rigidity for a weakly mixing transformation. On the other
hand we have already seen (Corollary~\ref{powers}) that when
$\rho_t=a$, $t\geq1$ then $IP$-rigidity does not take place. The
proposition below generalizes that result and shows that in the
bounded case an $IP$-rigidity is excluded.
\begin{prop}\lambdabel{ipodometer} Assume that
$n_{t+1}/n_t\in{\mathbb{N}}\setminus\{0,1\}$, $t\geq0$, and
$\sup_{t\geq0}n_{t+1}/n_t=:C<+\infty$. Then for any weakly mixing
transformation $R$ for which $R^{n_t}\to Id$, the sequence $(n_t)$
is not a sequence of $IP$-rigidity.
\end{prop}
\begin{proof} Recall that $Tx=x+\widehat{A}t 1$ where $X$ stands for the
$(n_t)$-odometer. Each natural number $r\geq1$ can be expressed in
a unique manner as
$$
r=\sum\limits_{t=0}^Na_tn_t,\; 0\leq a_t<\rho_t=n_{t+1}/n_t.$$ Assume
that $T^{r_m}\to Id_X$. Write
$$
r_m=\sum\limits_{t=0}^{N_m}a^{(m)}_tn_t,\; 0\leq a^{(m)}_t<\rho_t$$ and
set $k_m=\max\{t\geq0:\:a^{(m)}_0=\ldots=a^{(m)}_t=0\}$ (so
$r_m=\sum\limits_{t=k_m}^{N_m}a^{(m)}_tn_t$). We claim that
\begin{equation}\lambdabel{cla1} k_m\to\infty\;\;\mbox{whenever}\;m\to\infty.\end{equation}
Indeed, suppose that the claim does not hold. Then without loss of
generality we can assume that there exists $t_0\geq0$ such that
$k_m=t_0$ for all $m\geq1$, that is
$$
r_m=a^{(m)}_{t_0}n_{t_0}+\sum\limits_{t=t_0+1}^{N_m}a^{(m)}_tn_t
\;\;\mbox{with}\;1\leq a_{t_0}^{(m)}<\rho_{t_0}. $$ Consider the
tower $\{D^{n_{t_0+1}}_0,\ldots,D^{n_{t_0+1}}_{n_{t_0+1}-1}\}$ and let
$A=D^{n_{t_0+1}}_0$. Notice that for each $i\geq0$ and $j\geq1$ we
have $T^{in_{t_0+j}}A=A$. It follows that
$$
T^{r_m}A=T^{a^{(m)}_{t_0}n_{t_0}}\left(T^{
\sum\limits_{t=t_0+1}^{N_m}a^{(m)}_tn_t}(A)\right)$$$$=
T^{a^{(m)}_{t_0}n_{t_0}}(A)\in
\{D^{n_{t_0+1}}_1,\ldots,D^{n_{t_0+1}}_{n_{t_0+1}-1}\},$$ where the latter
follows from the fact that $1\leq a_{t_0}^{(m)}<\rho_{t_0}$. So
$p_X(T^{r_m}(A)\triangle A)=2/n_{t_0+1}$ and hence
$(r_m)$ is not a rigidity sequence for $T$, a contradiction.
Thus~(\ref{cla1}) has been shown.
Assume now that $R$ is a weakly mixing transformation for which
$(n_t)$ is its $IP$-rigidity sequence. It follows that we have a
convergence along the net \begin{equation}\lambdabel{cla2}
R^{\sum\limits_{t=k}^N\eta_tn_t}\to
Id\;\;\mbox{whenever}\;\;k\to\infty\;\mbox{and}\;\eta_k=1,
0\leq\eta_t\leq1,t\geq k+1.\end{equation} We claim that also
\begin{equation}\lambdabel{cla3} R^{\sum_{t=k}^Na_tn_t}\to
Id\;\;\mbox{whenever}\;\;k\to\infty\;\;\mbox{and}\;1\leq
a_k\leq\rho_k-1, 0\leq a_t\leq\rho_t-1, t\geq k+1.\end{equation} Indeed, we
write $R^{\sum\limits_{t=k}^Na_tn_t}$ as the composition of at most
$S_1{\mathcal I}rc\ldots {\mathcal I}rc S_D$ with $D\leq C=\max_t\rho_t$
automorphisms of the form $R^{\sum\limits_{t=k}^N\delta_tn_t}$ with
$\delta_t\in\{0,1\}$ (to define the first automorphism $S_1$ we
put $\delta_t=1$ as soon as $a_t\geq1$ and $\delta_t=0$ elsewhere,
for the second automorphism $S_2$ we put $\delta_t=1$ as soon as
$a_t\geq 2$ and $\delta_t=0$ elsewhere, etc.). Notice that for
each $i=1,\ldots,D$
$$
S_i=R^{\sum_{t=k_i}^Nn_t}\;\;\mbox{with}\;\;k_i\geq k.$$
Therefore, if we assume that in~(\ref{cla2}),
$\|R^{\sum_{t=k}^Nn_t}\|<\varphirepsilon$ for $k\geq K$ then
$\|S_1{\mathcal I}rc\ldots{\mathcal I}rc S_D\|<D\varphirepsilon$ (see Remark~\ref{metric1}) and
thus~(\ref{cla3}) follows.
Combining (\ref{cla3}) and (\ref{cla1}) we see that each rigidity
sequence for $T$ is also a rigidity sequence for $R$. This
however contradicts Corollary~\ref{AA2} (or rather to its
proof).
\end{proof}
{\bf Question.} If $(\rho_t)$ is bounded, but not always a whole
number, can it still $(n_t)$ be a sequence of $IP$-rigidity for some weakly
mixing transformation? How fast does $(\rho_t)$ have to grow for
$(n_t)$ to be a sequence of $IP$-rigidity for some weakly
mixing transformation?
\section{\bf Non-Recurrence} \lambdabel{nonrecurrence}
In the previous sections, we have seen that the characterization
of which sequences $(n_m)$ exhibit rigidity for some ergodic, or
more specifically weakly mixing dynamical system will, most
certainly be difficult. The only aspect that is totally clear at this time is that
these sequences must have density zero because their gaps tend to
infinity. Lacunary sequences are always candidates for
consideration in such a situation. However, we have seen in Remark~\ref{linformeg} d)
that there are lacunary sequences which cannot be rigidity sequences for
even ergodic transformations, let alone weakly mixing ones.
In a similar vein, we would like to characterize which increasing
sequences $(n_m)$ in $\mathbb Z^+$ are not recurrent for some
ergodic dynamical system i.e. $(n_m)$ has the property that for some
ergodic system $(X,\mathcal B,p,T)$ and some set $A$ of positive
measure, the sets $T^{n_m}A$ are disjoint from $A$ for all $m \ge
1$. So we are taking {\em recurrence along $(n_m)$} here to mean that $p(T^{n_m}A {\mathcal A}p A) > 0$
for some $m$. A central unanswered question is the following
\noindent {\bf Question:} Is it the case that any lacunary sequence is
a sequence of non-recurrence for a weakly mixing system?
\begin{rem} \lambdabel{unions} It is not hard to see that any lacunary sequence
fails to be a recurrent sequence for
some ergodic dynamical system. Indeed, this happens even with
ergodic rotations of $\mathbb T$. See Pollington~{\mathcal I}te{Poll},
de Mathan~{\mathcal I}te{deM}, and Furstenberg~{\mathcal I}te{Furst}, p. 220. They show that for any lacunary
sequence $(n_m)$ there is some $\gamma\in \mathbb T$ of infinite
order, and some $\delta > 0$, such $|\gamma^{n_m} -1| \ge \delta$
for all $m \ge 1$. The arguments there also give information
about the size of the set of rotations that work for a given
lacunary sequence. The constructions in these articles are made
more difficult, as with a number of other results about lacunary
sequences, by not knowing the degree or nature of lacunarity. But
if one just wants some ergodic dynamical system to exhibit
non-recurrence, then the construction is easier. This was observed
by Furstenberg~{\mathcal I}te{Furst}. In short, his argument goes like
this. Suppose
$(n_k)$ is lacunary, say $\frac {n_{k+1}}{n_k} \ge \lambdambda > 1$
for all $k \ge 1$. Depending only on $\lambdambda$, we can choose $K$
so that the subsequences $(p_{m,j}:m\ge 1)$ given by $p_{m,j} = n_{j+Km},
j = 0,\ldots,K-1$ each have lacunary constant $\inf\limits_{m \ge
1} \frac {p_{m+1,j}}{p_{m,j}} \ge 5$. Then for each $j$, a
standard argument shows that there is a closed perfect set $C_j$
such that for all $\gamma \in C_j$, we have $|\gamma^{p_{m,j}}- 1|
\ge \frac 1{100}$ for all $m\ge 1$. We can choose
$(\gamma_0,\ldots,\gamma_{K-1})$ with each $\gamma_j \in C_j$ and
such that $\gamma_0,\ldots,\gamma_{K-1}$ are independent. Then we
would know that the transformation $T$ of the $K$-torus $\mathbb
T^K$ given by $T(\alphapha_0,\ldots,\alphapha_{K-1}) =
(\gamma_0\alphapha_0,\ldots,\gamma_{K-1}\alphapha_{K-1})$ is ergodic
Also, for a sufficiently small $\epsilon$, the arc $I$ of radius
$\epsilon$ around $1$ in $\mathbb T$ will give a set $C =
I\times\ldots\times I \subset T^K$ such that for all $n_k$, we have
$T^{n_k}C$ and $C$ disjoint. Indeed, each $n_k$ is some $p_{m,j}$
and so $T^{n_k}C$ and $C$ are disjoint because in the $j$-th
coordinate $T^{n_k}$ corresponds to the rotation
$\gamma_j^{p_{m,j}} I$ which is disjoint from $I$.
\end{rem}
The main idea in Remark~\ref{unions} that appears in Furstenberg
~{\mathcal I}te{Furst} gives us this basic principle.
\begin{prop} \lambdabel{wmunions} If a sequence $\mathbf n$ is a finite
union of sequences $\mathbf n_i,i=1,\ldots, I$, each of which is a
sequence of
non-recurrence for some weakly mixing transformation $T_i$, then
$\mathbf n$ is also a sequence of non-recurrence for a weakly mixing
transformation.
\end{prop}
\begin{proof} Write $\mathbf n_i = (\mathbf n_i(j):j\ge 1)$.
We take $T$ = $T_1\times\ldots\times T_I$. This is a weakly
mixing transformation since each $T_i$ is weakly mixing. There is
a set $C_i$ such that $T_i^{\mathbf n_i(j)}C_i$ is disjoint from $C_i$
for all $j$. So $C = C_1\times\ldots\times C_i$ has the property
that $T^{\mathbf n_i(j)}C$ is disjoint from $C$ for all $i$ and all
$j$. That is, the sequence $T_1\times\ldots\times T_I$ is not recurrent
along $\mathbf n$ for the set $C$.
\end{proof}
\begin{rem} \lambdabel{different} a) This property of non-recurrent sequences
does not hold for rigidity sequences. For example, consider $\mathbf n_1 = (2^{m^2})$ and
$\mathbf n_2 = (2^{m^2}+1)$. By Proposition~\ref{fastworks} below, these
are both sequences of non-recurrence for a weakly mixing transformation
and hence by the above their union is too. These two sequences are
also rigidity sequences for ergodic rotations and weakly
mixing transformations
by Proposition~\ref{ratiogrows}.
But the union of these two sequences is not a rigidity
sequence for an ergodic transformation
because a rigidity sequence for an ergodic transformation cannot have infinitely many terms differing
by $1$.
\noindent b) Here is a related example that shows how rigidity sequences
and non-recurrent sequences behave differently. The sequence $A = (p: p\,\,\text {prime})$
is not recurrent but the sequence $A = (p - 1: p\,\,\text {prime})$ is recurrent. See
S\'ark\"ozy~{\mathcal I}te{sark} and apply the Furstenberg Correspondence Principle. But we see from either
Proposition~\ref{unifdist} or Proposition~\ref{sumset}
that neither sequence is a rigidity sequence for an ergodic transformation.
\end{rem}
We see that the non-recurrence phenomenon is both pervasive
and not, depending on how one chooses the quantifiers. As usual,
the measure-preserving transformations of a non-atomic separable
probability space $(X,\mathcal B, p)$ can be given the weak topology,
and become a complete pseudo-metric group $\mathcal G$ in this
topology. By a generic transformation, we mean an element in
a set that contains some dense $G_\delta$ set in $\mathcal G$.
\begin{prop} \lambdabel{revise} The generic transformation is both weakly
mixing and rigid,
and moreover is not recurrent along some increasing sequence
in $\mathbb Z^+$.
\end{prop}
\begin{proof} Remark~\ref{wmandrigid} pointed out that
the generic transformation is
weakly mixing and rigid. Fix such a transformation $T$ and
some $(n_m)$ such that $\|f{\mathcal I}rc
T^{n_m} - f\|_2 \to 0$ for all $f \in L_2(X,p)$. Then it
follows that for any set $A$, $\lim\limits_{m \to \infty}
p(T^{n_m}A \Delta A) = 0$. Since $T$ is ergodic, we can choose a
set $A$ with $p(A) > 0$ and $T A$ and $A$ disjoint. Let $B = T
A$. We have $B$ and $A$ disjoint, and $\lim\limits_{m\to \infty}
p(T^{n_m-1}B \Delta A) = 0$. Hence, we can pass to a subsequence
$(m_s)$ so that $\sum\limits_{s=1}^\infty p(T^{n_{m_s}-1}B \Delta
A) \le \frac 1{100} p(A)= \frac 1{100}p(B)$. So
$$C = B\backslash
\bigcup\limits_{s=1}^\infty T^{-(n_{m_s}-1)}(T^{n_{m_s}-1}B\Delta
A)$$ will have $p(C) > 0$. Also, $C \subset B=TA$ is disjoint
from $A$. But at the same time
\[T^{n_{m_s}-1}C\subset T^{n_{m_s}-1}B\setminus\left(T^{n_{m_s}-1}B\triangle
A\right)=T^{n_{m_s}-1}B{\mathcal A}p A\subset A\]
for all $s\geq 1$.
Thus, the generic transformation is weakly mixing and rigid, and
additionally for some
sequence of powers $T^{n_{m_s}}$ and some set $C$ of positive measure,
we have non-recurrence because $C$ is disjoint from all
$T^{n_{m_s}}C$.
\end{proof}
\begin{rem}\lambdabel{notrecshift} a) This argument can easily be
used to show that if $T$ is rigid along $(n_m)$, then for any
$K$, by passing to a subsequence $(n_{m_s})$, we can have for
each $k \not= 0, |k| \le K$, the transformation $T$ is non-recurrent
along $(n_{m_s} + k)$ for some set $C_k, p(C_k) > 0$. By taking $S$ to be
a product of $T$ with itself $2K$ times, we can arrange that the
weakly mixing transformation $S$ is rigid along $(n_m)$ and
there is one set $C, p(C) > 0$, such that $S$ is non-recurrent for $C$
along each of the sequences $(n_{m_s} + k)$ with $0 < |k| \le K$.
\noindent b) One cannot restrict the sequence along which the non-recurrence
is to occur. It is not hard to see that the class of transformations that is
non-recurrent for some set along a fixed sequence is a meager set of transformations.
\end{rem}
We do have some specific, interesting examples of the failure of
recurrence for a weakly mixing dynamical system. See Chacon~{\mathcal I}te{Chacon}
for the construction of the rank one Chacon transformation. The important point
here is that the non-recurrence occurs along a lacunary sequence $(n_m)$ with
ratios $n_{m+1}/n_m$ bounded. See Remark~\ref{moreChacon} a) for more information
about this example.
\begin{prop}\lambdabel{Chacon}
The Chacon transformation is not recurrent for sequence $(n_m) = (\frac
{3^{m+1} -1}2 -1)$.
\end{prop}
\begin{proof} Let $n_m = \frac
{3^{m+1} -1}2 -1$. The transformation $T$ that we are using here is
constructed
inductively as follows. Take the current stack (single tower)
$T_m$ of interval and cut it in thirds $T_{m,1}$, $T_{m,2}$, and
$T_{m,3}$. Add a spacer $s$ of the size of levels above the
middle third $T_{m,2}$, and let the new stack $T_{m+1}$ consists of
$T_{m,1}$, $T_{m,2}$, $s$, and $T_{m,3}$ in that order from bottom
to top. We take $T_0$ to be $[0,1)$ to start this construction.
So it is easy to see that the height $h_m$ of our $m$-th tower
is $\frac {3^{m+1} -1}2$. To see the failure of recurrence, use the standard
symbolic dynamics for $T$ i.e. assign the symbol $1$ to all the
spacer levels and $0$ to the rest of the levels. Then let $B_m$
be the name of length $h_m$ of a point in the base of the $m$-th
tower $T_m$. Then $B_0 = 0$, $B_1 = 0010$, and $B_{m+1} = B_m\
B_m\ 1\ B_m$ in general. It is a routine check that when one
shifts $B_k$ by $n_m = h_m-1$ for $k$ larger than $m$, and compares this
with $B_k$, then they have no common occurrences of $1$. So if $A$
is the first added spacer level, then we have $T^{n_m}A$ and $A$
disjoint for all $m$.
\end{proof}
\begin{rem} \lambdabel{moreChacon} a) The Chacon transformation is mildly mixing so it
cannot have rigidity sequences at all. But there is partial
rigidity in that $T^{h_m} \to \frac 12(Id +T^{-1})$
in the strong operator topology. So $(h_m)$ and $(h_m +1)$ are
recurrent sequences for $T$ in a strong sense, while the above is
showing that $(h_m -1)$ is not recurrent for $T$.
\noindent b)The obvious question here is what other sequences, besides ones
like the one above for the Chacon transformation, can be show
to be sequences of non-recurrence for weakly mixing transformations
via classical cutting and stacking constructions?
\noindent b) Friedman and King~{\mathcal I}te{FK} consider a class of weakly
mixing, but not strongly mixing, transformations constructed by
Chacon; they prove that these, unlike the Chacon transformation
above, are lightly mixing (see {\mathcal I}te{FK} for the definition)
and so there is always recurrence for
these transformations along any increasing sequence. Hence, these
transformations form a meager set by the category result in Proposition~\ref{revise}.
\end{rem}
If we have a sufficient growth rate assumed for $(n_m)$, we can
give a construction of a weakly mixing transformation which
exhibits non-recurrence along the sequence. The argument here
starts like the constructions in Section~\ref{diophantine}
\begin{prop}
\lambdabel{fastworks}
Suppose we have a sequence $(n_m)$ such that
$\sum\limits_{k=1}^\infty n_m/n_{m+1} < \infty$. Then there is a
weakly mixing transformation $T$ for which $(n_m)$ is an IP
rigidity sequence and such that $T$ is not recurrent for
$(n_m-1)$.
\end{prop}
\begin{proof}
Choose a non-decreasing sequence of whole numbers $(h_m)$ with
$1/h_m \ge 10(n_m/n_{m+1})$ and $\sum\limits_{m=1}^\infty
1/h_m < \infty$. We construct a Cantor set $\mathcal C$ with constituent intervals
at each level that are arcs of size $1/(n_mh_m)$ around some of
the $n_m$-th roots of unity (determined as part of the induction).
Our conditions allow us to find in each such constituent interval
many points $j/n_{m+1}$ because $1/n_{m+1}$ is sufficiently
smaller than $1/n_mh_m$, and then select in these constituent
intervals new ones of length $1/(n_{m+1}h_{m+1})$ around some of
the $n_{m+1}$-th roots of unity for $m \ge M$. The resulting
Cantor set $\mathcal C$ has the property that for all points $x$
in the set $n_mx$ is within
$1/h_m$ of an integer
for $m \ge M$. Also, it follows that if we take a continuous, positive
measure $\nu_0$ on $\mathcal C$, with $\nu_0([0,1)) = 1$, then
$|1- \widehat {\nu_0}(n_m)| \le 1/h_m$ for $m \ge M$.
Now we take the GMC construction corresponding to the
symmetrization $\omega$ of $\nu_0$. We can use Proposition~\ref{OKforIP} and the
result from Erd\H{o}s and Taylor~{\mathcal I}te{ET} cited in Remark~\ref{IPinfo}
to conclude that $(n_m)$ is an IP rigidity sequence for the weakly
mixing transformation $T = G_{\omega}$. This
gives us a weakly mixing dynamical system $(X,\mathcal B,p,T)$ and a
function $f$ of norm one in $L_2(X,p)$ such that $\|f{\mathcal I}rc T^{n_m} -
f\|_2^2 \le C/h_m$ for $m \ge M$. The function $f$ here is the
one in the GMC such that $\nu_f^T = \omega$.
It follows immediately, by the fact that $\omega$ is symmetric and
by the symmetric Fock space construction in the GMC,
that $f$ is a Gaussian variable from the first chaos. So $f$ is real-valued,
and being a Gaussian random variable it takes both positive and negative values.
So we have a non-constant, real-valued function of norm
one such that $\|f{\mathcal I}rc T^{n_m} - f\|_2^2 \le 4/h_m$ for $m \ge
M$.
Now we claim that both the positive part $f^+$ and the negative
part $f^-$ of $f$ satisfies the same inequality. To see this, write
\[\int |f{\mathcal I}rc T^{n_m} - f|^2\,dp =
\int |f^+{\mathcal I}rc T^{n_m} -f^-{\mathcal I}rc T^{n_m} - f^+ + f^-|^2\, dp.\]
Expand this into the sixteen terms involved. Use that fact that
the terms $f^+f^-$ and $(f^+{\mathcal I}rc T^{n_m})(f^-{\mathcal I}rc T^{n_m})$ are
zero, and regroup terms to see that
\begin{eqnarray*} \int |f{\mathcal I}rc T^{n_m} - f|^2\,dp &=&
\int |f^+{\mathcal I}rc T^{n_m} - f^+|^2\,dp\\
&+&\int |f^-{\mathcal I}rc T^{n_m} - f^-|^2\,dp\\
&+& \int 2(f^+{\mathcal I}rc T^{n_m})f^- + 2(f^-{\mathcal I}rc T^{n_m})f^+ \,dp.
\end{eqnarray*}
Because $2(f^+{\mathcal I}rc T^{n_m})f^- + 2(f^-{\mathcal I}rc T^{n_m})f^+ $ is
positive, we have
\[\int |f{\mathcal I}rc T^{n_m} - f|^2\,dp
\ge \int |f^+{\mathcal I}rc T^{n_m} - f^+|^2\,dp\] and
\[\int |f{\mathcal I}rc T^{n_m} - f|^2\,dp
\ge \int |f^-{\mathcal I}rc T^{n_m} - f^-|^2\,dp.\]
In addition, the same argument above shows that for every constant
$L$, the function $(f-L)^+$ also would satisfy this last estimate
too. Hence, taking $L = 1/2$, we would have $F= f1_{\{f \ge L\}}$
satisfies this inequality too and not being the zero function. But
let $A = \{f \ge L\}$. On $T^{n_k}A \backslash A$, we would have
$|F{\mathcal I}rc T^{n_m} - F|^2\ge 1/4$. So $p(T^{n_m}A\backslash A) \le
64/h_m$. But similarly, on $A\backslash T^{n_m}A$, we would have
$|F{\mathcal I}rc T^{n_m} - F|^2\ge 1/4$. So $p(A\backslash T^{n_m}A) \le
64/h_m$. The result is that from our original GMC construction, we
can infer the existence of a proper set $A$ of positive measure,
which depends on the original function $f$ and not on $m$, such
that $p(T^{n_m}A\Delta A) \le 64/h_m$ for all $m \ge M$. Hence,
$\sum\limits_{m=1}^\infty p(T^{n_m}A\Delta A) < \infty$.
We do have to also make certain that the set $A$ here is a proper
set i.e. $p(A) < 1$. But $f$ is a Gaussian random variable and
so both
$f^+$ and $f^-$ are non-trivial, and so it is easy to choose a value of $L$
so that the above construction gives us a proper set $A$ of positive measure.
Now we can use the convergence of $\sum\limits_{m=1}^\infty
p(T^{n_m}A\Delta A)$ to construct a set of positive measure $B$
for which $T^{n_m-1}B$ and $B$ are disjoint for all $m$. The
argument is a variation on the one given in
Proposition~\ref{revise}. There is the issue that $TA$ and $A$ are
not necessarily disjoint. But there is some subset $A_0$ of $A$
of positive measure such that $T A_0$ and $A$ are disjoint. So if
we take $B = T A_0\backslash \bigcup\limits_{m=M}^\infty T^{-(n_m
-1)}(T^{n_m}A\Delta A)$, for suitably large $M$, then we would
have $p(B) > 0$, $B \subset T A_0\backslash A$, and $T^{n_m-1}B
\subset A$ for all $m \ge M$. Hence $T^{n_m-1}B$ and $B$ disjoint
for all $m \ge M$.
Now we need to revise the result above so that we get the same
disjointness for all $m$. But here we know that $T$ is weakly
mixing, and consequently all of its powers are ergodic. So one
can inductively revise $B$ as follows. One takes a subset $B_1$
of $B$ such that $T^{n_1-1}B_1$ and $B_1$ are disjoint, then one
takes a subset $B_2$ of $B_1$ such that $T^{n_2 -1}B_2$ and $B_2$
are disjoint, and so on. After a finite number of steps one ends
up with a subset $B_{M-1}$ of $B$ such that $T^{n_m -1}B_{M-1}$
and $B_{M-1}$ are disjoint for all $m \ge 1$. Now, with $B_{M-1}$
replacing $B$, we have $T^{n_m-1}B$ and $B$ disjoint for all $m
\ge 1$. So this construction gives a weakly mixing transformation
that is not only IP rigid along $(n_m)$, but such that along
$(n_m-1)$ it is not recurrent.
\end{proof}
\begin{rem}\lambdabel{notrecshiftagain} With the hypothesis of
Proposition~\ref{fastworks}, by taking $S$ to be
a product of $T$ with itself $2K$ times, we can arrange that the
weakly mixing transformation $S$ is rigid along $(n_m)$ and
there is one set $C, p(C) > 0$, such that $S$ is non-recurrent for $C$
along each of the sequences $(n_m + k)$ with $0 < |k| \le K$.
\end{rem}
\begin{rem} Proposition~\ref{wmunions} allows us to use Proposition~\ref{fastworks}
to give other examples of non-recurrent sequences. Again, as in Remark~\ref{different} a),
both $(2^{n^2})$
and $(2^{n^2}+1)$ satisfy the hypothesis of Proposition~\ref{fastworks},
so there is a weakly mixing transformation for which $\mathbf n =
(\ldots,2^{n^2}, 2^{n^2}+1,\ldots)$ is a sequence of non-recurrence,
even though $\mathbf n$ does not satisfy the hypothesis of
Proposition~\ref{fastworks}.
\end{rem}
\begin{rem} Using Proposition~\ref{specialinfrankone}, we can construct
examples of non-recurrence along $(n_m - 1)$ for $T$ which is weakly mixing
and rank one. For
example, take $(n_m)$ such that $n_{m+1}/n_m \ge 2$ is a whole
number for all $m$ and such that $\sum\limits_{m=1}^\infty \frac {n_m}{n_{m+1}}
< \infty$.
\end{rem}
\begin{rem} a) In Proposition~\ref{fastworks},
replacing our original sequence by $(n_m +1)$, we
conclude this fact: whenever $(n_m)$ is increasing and
$\sum\limits_{m=1}^\infty \frac {n_m}{n_{m+1}} < \infty$, there exists a
weakly mixing transformation $T$ and a set of positive measure $B$
such that $T^{n_m}B$ and $B$ are disjoint for all $m$. Of course
now the transformation is IP rigid along $(n_m+1)$.
\noindent b) The condition we are using of course will not hold
for lacunary sequences like $n_m = 2^m$. But it is the case that
if $(k_m)$ is lacunary, and $\delta > 1$, then the subsequence
$(n_m) = (k_{\Lambda_\varphirphiloor
m^\delta\rfloor})$ will have our series property. So speeding
up the exponent for a lacunary sequence slightly will give us the
type of non-recurrence that we want.
\end{rem}
\begin{rem} Consider the series
$\sum\limits_{m=1}^\infty p(T^{n_m}A\Delta A)$. Can this be convergent for all
$A$? This is not obviously impossible, although it seems to us unlikely.
But if one replaces this by the corresponding functional version, then
it cannot be convergent for all functions. First,
one can see that $\sum\limits_{m=1}^\infty p(T^{n_m}A\Delta A)=
\sum\limits_{m=1}^\infty\|f{\mathcal I}rc T^{n_m} - f\|_2^2$ if we take $f
= 1_A$. So the question is, can we have a dynamical system in
which the square function $Sf = \left
(\sum\limits_{m=1}^\infty|f{\mathcal I}rc T^{n_m} - f|^2\right )^{1/2}$ is
always an $L_2$-function? But it is clear that this is not
possible. If it were, then one can show there is a homogeneous
inequality $\|Sf\|_2 \le K\|f\|_2$ for some constant $K$. Then
one takes again $f=1_A$, and sees that $\sum\limits_{m=1}^\infty
p(T^{n_m}A\Delta A)\le K^2 p(A)$. But this cannot be. Indeed,
just take a very long Rokhlin tower with $A$ as the base and the
left-hand side could exceed the right-hand side.
\end{rem}
\noindent {\bf Acknowledgements}: \,We would like to think S. Eigen, R. Kaufman,
J. King, V. Ryzhikov, S. Solecki, Y. Son, and B. Weiss for their input.
\scriptsize
{\small
\parbox[t]{3.5in}
{V. Bergelson\\
Department of Mathematics\\
Ohio State University\\
Columbus, OH 43210, USA\\
E-mail: [email protected]}
{\small
\parbox[t]{3.5in}
{A. del Junco\\
Department of Mathematics\\
University of Toronto\\
Toronto, M5S 3G3, Canada\\
E-mail: [email protected]}
{\small
\parbox[t]{5in}
{M. Lema\'nczyk\\
Faculty of Mathematics and Computer Science\\
Nicolaus Copernicus University,
Toru\'n, Poland, and\\
Institute of Mathematics\\
Polish Academy of Sciences, Warsaw, Poland\\
E-mail:
[email protected]}
{\small
\parbox[t]{5in}
{J. Rosenblatt\\
Department of Mathematics\\
University of Illinois at Urbana-Champaign\\
Urbana, IL 61801, USA\\
E-mail: [email protected]}
\end{document} |
\begin{document}
\title{A Short Proof that the List Packing Number of any Graph is Well Defined}
\author{Jeffrey A. Mudrock$^1$}
\footnotetext[1]{Department of Mathematics, College of Lake County, Grayslake, IL 60030. E-mail: {\tt {[email protected]}}}
\maketitle
\begin{abstract}
List packing is a notion that was introduced in 2021 (by Cambie et al.). The list packing number of a graph $G$, denoted $\chi_{\ell}^*(G)$, is the least $k$ such that for any list assignment $L$ that assigns $k$ colors to each vertex of $G$, there is a set of $k$ proper $L$-colorings of $G$, $\{f_1, \ldots, f_k \}$, with the property $f_i(v) \neq f_j(v)$ whenever $1 \leq i < j \leq k$ and $v \in V(G)$. We present a short proof that for any graph $G$, $\chi_{\ell}^*(G) \leq |V(G)|$. Interestingly, our proof makes use of Galvin's celebrated result that the list chromatic number of the line graph of any bipartite multigraph equals its chromatic number.
\noindent {\bf Keywords.} list coloring, list packing, list coloring conjecture.
\noindent \textbf{Mathematics Subject Classification.} 05C15
\end{abstract}
\section{Introduction}\label{intro}
In this note all graphs are nonempty, finite, simple graphs unless otherwise noted. Generally speaking we follow West~\cite{W01} for terminology and notation. The set of natural numbers is $\mathbb{N} = \{1,2,3, \ldots \}$. For $m \in \mathbb{N}$, we write $[m]$ for the set $\{1, \ldots, m \}$. The chromatic number of a graph $G$ is denoted $\chi(G)$. We write $K_{n,m}$ for complete bipartite graphs with partite sets of size $n$ and $m$. The Cartesian product of graphs $G$ and $H$, denoted $G \square H$, is the graph with vertex set $V(G) \times V(H)$ and edges created so that $(u,v)$ is adjacent to $(u',v')$ if and only if either $u=u'$ and $vv' \in E(H)$ or $v=v'$ and $uu' \in E(G)$. We will use the well-known fact that $\chi(G \square H) = \max \{\chi(G), \chi(H) \}$. The line graph of $G$, denoted $L(G)$, is the graph with vertex set $E(G)$ such that distinct edges $e_1, e_2 \in E(G)$ are adjacent in $L(G)$ when $e_1$ and $e_2$ are incident in $G$. Note that $L(K_{n,m}) = K_n \square K_m$.
\subsection{List Packing and the Main Theorem} \label{basic}
In classical vertex coloring we wish to color the vertices of a graph $G$ with up to $m$ colors from $[m]$ so that adjacent vertices receive different colors, a so-called \emph{proper $m$-coloring}. List coloring is a well-known variation on classical vertex coloring that was introduced independently by Vizing~\cite{V76} and Erd\H{o}s, Rubin, and Taylor~\cite{ET79} in the 1970s. For list coloring, we associate a \emph{list assignment} $L$ with a graph $G$ such that each vertex $v \in V(G)$ is assigned a list of colors $L(v)$ (we say $L$ is a list assignment for $G$). Then, $G$ is \emph{$L$-colorable} if there exists a proper coloring $f$ of $G$ such that $f(v) \in L(v)$ for each $v \in V(G)$ (we refer to $f$ as a \emph{proper $L$-coloring} of $G$)~\footnote{A coloring $f$ of $G$ satisfying $f(v) \in L(v)$ for each $v \in V(G)$ that is not necessarily proper is called an \emph{$L$-coloring} of $G$}. A list assignment $L$ for $G$ is said to be a \emph{$k$-assignment} if $|L(v)|=k$ for each $v \in V(G)$. The \emph{list chromatic number} of $G$, denoted $\chi_{\ell}(G)$, is the smallest $k$ such that $G$ is $L$-colorable whenever $L$ is a $k$-assignment for $G$. It is obvious that for any graph $G$, $\chi(G) \leq \chi_{\ell}(G)$.
One of the most famous open questions about list coloring is the List Coloring Conjecture which was formulated by many different researchers and has received considerable attention in the literature (see~\cite{HC92}).
\begin{conj}[{\bf List Coloring Conjecture}] \label{conj: Edge List Coloring Conjecture} If $G$ is a loopless multigraph, then $\chi(L(G)) = \chi_{\ell}(L(G))$.
\end{conj}
In 1995, Galvin famously proved the following.
\begin{thm}[\cite{G95}] \label{thm: Galvin} If $G$ is a bipartite multigraph, then $\chi(L(G)) = \chi_{\ell}(L(G))$.
\end{thm}
Galvin's proof of Theorem~\ref{thm: Galvin} is regarded by many as one of the most beautiful proofs on the topic of list coloring (see~\cite{AZ18}).
List packing is a relatively new notion that was first suggested by Alon, Fellows, and Hare in 1996~\cite{AF96}. This suggestion was not formally embraced until a recent paper of Cambie et al.~\cite{CC21}. We now mention some important definitions. Suppose $L$ is a list assignment for a graph $G$. An \emph{$L$-packing of $G$ of size $k$} is a set of $k$ $L$-colorings of $G$, $\{f_1, \ldots, f_k \}$, such that $f_i(v) \neq f_j(v)$ whenever $i, j \in [k]$, $i \neq j$, and $v \in V(G)$. Moreover, we say that $\{f_1, \ldots, f_k \}$ is \emph{proper} if $f_i$ is a proper $L$-coloring of $G$ for each $i \in [k]$. The \emph{list packing number} of $G$, denoted $\chi_{\ell}^*(G)$, is the least $k$ such that $G$ has a proper $L$-packing of size $k$ whenever $L$ is a $k$-assignment for $G$. Clearly, for any graph $G$, $\chi(G) \leq \chi_{\ell}(G) \leq \chi_{\ell}^*(G)$.
In~\cite{CC21}, while defining $\chi_{\ell}^*(G)$, the authors remark that ``The reader might already find it interesting that such a minimal $k$ is well defined.". The authors go on to show that for any graph $G$, $\chi_{\ell}^*(G) \leq |V(G)|$ and equality holds if and only if $G$ is complete, and they conjecture that there is a $C > 0$ such that $\chi_{\ell}^*(G) \leq C \chi_{\ell}(G)$. In this note we present a short proof that for any graph $G$, $\chi_{\ell}^*(G) \leq |V(G)|$ which makes use of Theorem~\ref{thm: Galvin}. In particular, we prove the following.
\begin{thm} \label{thm: simpleproof}
$\chi_{\ell}^*(K_n) = n$.
\end{thm}
Note that Theorem~\ref{thm: simpleproof} implies that for any graph $G$, $\chi_{\ell}^*(G) \leq |V(G)|$ since $\chi_{\ell}^*(H) \leq \chi_{\ell}^*(G)$ whenever $H$ is a subgraph of $G$.
\section{Proof of Theorem~\ref{thm: simpleproof}} \label{main}
\begin{proof}
Since $\chi(K_n) = n$, $\chi_{\ell}^*(K_n) \geq n$. Suppose $G=K_n$ and $V(G) = \{v_1, \ldots, v_n\}$. Let $L$ be an arbitrary $m$-assignment for $G$ with $m \geq n$. To prove the desired result, we will show that there is a proper $L$-packing of $G$ of size $m$.
Suppose $H = G \square K_m$ and the vertices of the copy of $K_m$ used to form $H$ are $\{u_1, \ldots, u_m \}$. Let $L_H$ be the $m$-assignment for $H$ given by $L_H(v_i,u_j) = L(v_i)$ for each~$(i,j) \in [n] \times [m]$. By Theorem~\ref{thm: Galvin} and properties of the Cartesian product of graphs, $m = \chi(K_n \square K_m) = \chi(L(K_{n,m})) = \chi_{\ell}(L(K_{n,m})) = \chi_{\ell}(H)$. Consequently, there is a proper $L_H$-coloring of $H$ which we will name $f$. Now, for each $j \in [m]$ let $f_j$ be the proper $L$-coloring of $G$ given by $f_j(v_i) = f(v_i,u_j)$ for each $i \in [n]$. Finally, notice that $\{f_1, \dots, f_m \}$ is a proper $L$-packing of $G$ of size $m$.
\end{proof}
{\bf Acknowledgment.} The author would like to thank Hemanshu Kaul for helpful conversations regarding the contents of this note. The author would also like to thank the anonymous referees for their helpful comments on this note.
\end{document} |
\begin{document}
\date{}
\begin{abstract}
The adjoint method, recently introduced by Evans,
is used to study obstacle problems, weakly coupled systems,
cell problems for weakly coupled systems of Hamilton--Jacobi equations,
and weakly coupled systems of obstacle type.
In particular, new results about the speed of convergence of some approximation procedures are derived.
\end{abstract}
\maketitle
{\small
\keywords{\noindent {\bf Keywords:}
adjoint methods, cell problems,
Hamilton--Jacobi equations, obstacle problems,
weakly coupled systems, weak KAM theory.}
\subjclass{\noindent {\bf 2010 Mathematics Subject Classification:}
35F20, 35F30, 37J50, 49L25.}}
\begin{section}{Introduction}
{We study} the speed of convergence of certain approximations
for obstacle problems and weakly coupled systems of Hamilton--Jacobi equations,
using the Adjoint Method.
This technique, recently introduced by Evans
(see \cite{E2}, and also \cite{T1} and \cite{CGT1}),
is a very successful tool to understand several types of degenerate PDEs.
It can be applied, for instance, to Hamilton--Jacobi equations with non convex Hamiltonians,
e.g. time dependent (see \cite{E2}) and time independent (see \cite{T1}),
to weak KAM theory (see \cite{CGT1}), and to the infinity Laplacian equation (see \cite{E3}).
We address here several new applications, and propose some new open questions.
Further results, which will not be discussed here, can be found in \cite{E2} and \cite{CGT1}.
\begin{subsection}{Outline of the paper}
{The} paper contains four further sections concerning
obstacle problems, weakly coupled systems, effective Hamiltonian
for weakly coupled systems of Hamilton--Jacobi equations,
and weakly coupled systems of obstacle type, respectively.
We use a common strategy to study all these problems.
Note, however, that each of them presents different challenges, which
are described in the corresponding sections.
Also, we believe that the applications we present here illustrate
how to face the difficulties that can be encountered in the study
of other systems of PDEs and related models.
In particular, we show how to control singular terms arising from the switching
to an obstacle (Lemma \ref{obs_lem2}), random switching (Lemma \ref{kiz}),
or optimal switching (Lemma \ref{gamprimeb}).
In order to clarify our approach, let us give the details
of its application to the obstacle problem (see Section \ref{sectobs}):
\begin{equation} \label{qw}
\left\{ \begin{aligned}
\max \{ u- \psi, u+H(x,Du)\}&=0 \quad \mbox{in}~ U,
\\
u &= 0 \quad \mbox{on}~ \partial U,\\
\end{aligned} \right.
\end{equation}
where $\psi: \overline{U} \to \mathbb{R}$ and $H: \mathbb R^n \times \overline{U} \to \mathbb R$ are smooth,
with $\psi \geq 0$ on $\partial U$.
Here and in all the paper, $U$ is an open bounded domain in $\mathbb R^n$ with smooth boundary,
and $n \geq 2$.
Moreover, we will denote with $\nu$ the \textit{outer} unit normal to $\partial U$.
This equation arises naturally in optimal control theory, in the study of optimal stopping (see \cite{L}).
See also \cite{BP,IY}.
Classically, in order to study \eqref{qw} one first modifies the equation,
by adding a perturbation term that penalizes the region where $u > \psi$.
Then, a solution is obtained as a limit of the solutions of the penalized problems.
More precisely, let $\gamma: \mathbb R \to \mathbb [0, +\infty)$ be smooth
such that
\begin{equation}
\left\{ \begin{aligned}
\gamma(s)=0~\mbox{for}~s\le 0, \quad \gamma(s)>0~\mbox{for}~s>0,
\\
0<\gamma'(s) \le 1~\mbox{for}~s>0,~\mbox{and}~ \lim_{s \to + \infty} \gamma(s)= + \infty,\\
\end{aligned} \right.
\notag
\end{equation}
and define $\gamma^\varepsilon: \mathbb R \to \mathbb [0, +\infty)$ as
\begin{equation} \label{gammaep}
\gamma^\varepsilon(s) := \gamma \left( \dfrac{s}{\varepsilon} \right),~\mbox{for all}~s \in \mathbb R,
\quad \mbox{for all}~\varepsilon > 0.
\end{equation}
In some of the problems we discuss we also require $\gamma$ to
be convex in order to obtain improved results, but that will be pointed out where necessary.
For every $\varepsilon >0$, one can introduce the penalized PDE
\begin{equation}
\left\{ \begin{aligned}
u^\varepsilon+H(x,Du^\varepsilon)+\gamma^\varepsilon(u^\varepsilon-\psi)
&= \varepsilon \Delta u^\varepsilon \quad \mbox{in}~ U,
\\
u^\varepsilon &= 0 \qquad \mbox{on}~ \partial U.\\
\end{aligned} \right.
\label{obs_regintr}
\end{equation}
To avoid confusion, we stress the fact that here $\gamma^\varepsilon(u^\varepsilon-\psi)$
stands for the composition of the function $\gamma^{\varepsilon}$ with $u^\varepsilon-\psi$.
Unless otherwise stated, we will often simply write
$\gamma^{\varepsilon}$ and $\left( \gamma^{\varepsilon} \right)'$
to denote $\gamma^\varepsilon(u^\varepsilon-\psi)$
and $\left( \gamma^\varepsilon \right)' (u^\varepsilon-\psi)$, respectively.
Thanks to \cite{L}, for every $\varepsilon > 0$
there exists a smooth solution $u^\varepsilon$ to \eqref{obs_regintr}.
It is also well known that, up to subsequences, $u^{\varepsilon}$
converges uniformly to a viscosity solution $u$ of \eqref{qw}
(see also Section \ref{sectobs} for further details).
We face here the problem requiring a coercivity assumption
on $H$ and a compatibility condition for equation \eqref{qw} (see hypotheses (H\ref{sectobs}.1) and (H\ref{sectobs}.2), respectively),
and show that the speed of convergence in the general case is $O(\varepsilon^{1/2})$.
Notice that we do not require the Hamiltonian $H$ to be convex in $p$.
\begin{Theorem}
\label{obs_speed}
Suppose conditions (H\ref{sectobs}.1) and (H\ref{sectobs}.2) in Section~\ref{sectobs} hold.
Then, there exists a positive constant $C$, independent of
$\varepsilon$, such that
\begin{equation}
\|u^\varepsilon - u\|_{L^\infty} \le C \varepsilon^{1/2}.
\notag
\end{equation}
\end{Theorem}
{The proof of Theorem \ref{obs_speed} consists of three steps.}
\textbf{Step I: Preliminary Estimates.} We first show that
\begin{equation} \label{hfj}
\max_{x \in \overline{U}} \frac{u^{\varepsilon} (x) - \psi (x)}{\varepsilon} \leq C,
\end{equation}
for some constant $C > 0$ independent of $\varepsilon$ (see Lemma \ref{obs_lem4}).
This allows us to prove that
\begin{equation*}
\| u^{\varepsilon} \|_{L^{\infty}}, \| D u^{\varepsilon} \|_{L^{\infty}} \leq C,
\end{equation*}
see Proposition \ref{obs1}.
\textbf{Step II: Adjoint Method.}
We consider the formal linearization of \eqref{obs_regintr},
and then introduce the correspondent adjoint equation (see equation~\eqref{obs_adj}).
The study of this last equation for different values of the right-hand side
allows us to obtain several useful estimates (see Lemma~\ref{obs_lem2} and Lemma~\ref{obs_lem3}).
\textbf{Step III: Conclusion.}
We conclude the proof of Theorem \ref{obs_speed} by showing that
\begin{equation} \label{hf}
\max_{x \in \overline{U}} \left| u^\varepsilon_{\varepsilon} (x) \right| \le
\dfrac{C}{\varepsilon^{1/2}}, \qquad \qquad u^\varepsilon_{\varepsilon} (x) := \frac{\partial u^\varepsilon}{\partial \varepsilon} (x),
\end{equation}
for some constant $C > 0$ independent of $\varepsilon$ (see Lemma \ref{obs_thm1}).
The most delicate part of the proof of \eqref{hf} consists in controlling the term
(see relation \eqref{obsk})
\begin{equation} \label{dvar}
\gamma^{\varepsilon}_{\varepsilon} (s)
:= \dfrac{\partial \gamma^\varepsilon }{\partial \varepsilon} (s)
=-\dfrac{s}{\varepsilon^2}
\gamma ' \left( \dfrac{s}{\varepsilon} \right),
\qquad \text{for } s \in \mathbb{R} .
\end{equation}
We underline that getting a bound for \eqref{dvar}
can be extremely hard in general.
In this context, this is achieved by differentiating equation \eqref{obs_regintr} w.r.t. $\varepsilon$
(see equation \eqref{obs5}), and then by using {inequality} \eqref{hfj},
Lemma \ref{obs_lem2} and Lemma \ref{obs_lem3}.
This means that we overcome the problem by essentially using
the Maximum Principle and the monotonicity of $\gamma^\varepsilon$
(see estimates \eqref{obs6} and \eqref{uj}).
We were not able to obtain such a bound
when dealing with homogenization or singular perturbation, where also similar terms appear.
We believe it would be very interesting to find the correct way
to apply the Adjoint Method in these situations.
In Section \ref{weakcou} we study
monotone weakly coupled systems of Hamilton--Jacobi equations
\begin{equation}
\left\{ \begin{aligned}
c_{11} u_1 +c_{12} u_2 + H_1(x,Du_1) &=0,
\\
c_{21} u_1 +c_{22} u_2 + H_2(x,D u_2) &=0, \\
\end{aligned} \right.
\quad \mbox{in}~ U,
\label{syst}
\end{equation}
with boundary conditions $u_1 = u_2=0$ on $\partial U$,
by considering the following approximation:
\begin{equation*}
\left\{ \begin{aligned}
c_{11} u^\varepsilon_1 +c_{12} u^\varepsilon_2 + H_1(x,D u^\varepsilon_1) &=\varepsilon \Delta u^\varepsilon_1
\\
c_{21}u^\varepsilon_1+c_{22}u^\varepsilon_2 + H_2(x,Du^\varepsilon_2) &=\varepsilon \Delta u^\varepsilon_2 \\
\end{aligned} \right.
\quad \mbox{in}~ U,
\end{equation*}
with $u^\varepsilon_1 = u^\varepsilon_2=0$ on $\partial U$.
\noindent
Under some coupling assumptions on the coefficients
(see conditions (H\ref{weakcou}.2) and (H\ref{weakcou}.3)),
Engler and Lenhart \cite{EL}, Ishii and Koike \cite{IK1}
prove existence, uniqueness and stability
for the viscosity solutions $(u_1,u_2)$ of \eqref{syst}, but they do not consider any approximation of the system.
We observe that these coupling assumptions
are similar to monotone conditions of single equations,
and play a crucial role in the establishment of
the comparison principle and uniqueness result, and thus cannot be removed.
We show that,
under the same assumptions of \cite{EL}, the speed of convergence
of $(u^{\varepsilon}_1,u^{\varepsilon}_2)$
to $(u_1,u_2)$ is $O(\varepsilon^{1/2})$ (see Theorem \ref{wc_thm1}).
For the sake of simplicity, we just focus on a system of two equations,
but the general case can be treated in a similar way.
Section \ref{secweak} is devoted to an analog of the {\it cell problem}
introduced by Lions, Papanicolaou and Varadhan \cite{LPV}.
More precisely, we consider the following quasi-monotone
weakly coupled system of Hamilton--Jacobi equations:
\begin{equation}
\left\{ \begin{aligned}
c_1 u_1- c_1 u_2 + H_1(x,Du_1) &=\overline{H}_1
\\
-c_2 u_1 + c_2 u_2 + H_2(x,Du_1) &=\overline{H}_2 \\
\end{aligned} \right.
\quad \mbox{in}~ {\mathbb T^n},
\label{tripz}
\end{equation}
also called the \textit{cell problem}.
Here $c_{1}$ and $c_{2}$ are positive constants
and $H_1, H_2 : \mathbb T^n \times \mathbb{R}^n \to \mathbb{R}$ are smooth,
while $u_1, u_2 : \mathbb T^n \to \mathbb R$ and $\overline{H}_1,\overline{H}_2 \in \mathbb{R}$ are unknowns.
Systems of this type have been studied by Camilli, Loreti and Yamada in \cite{CL} and \cite{CLY},
for uniformly convex Hamiltonians in a bounded domain.
They arise naturally in optimal control and in large deviation theory for random
evolution processes.
Under a coercivity-like assumption on $H_1, H_2$ (see condition (H\ref{secweak}.1)),
we obtain the following new result.
\begin{Theorem} \label{cell_const}
Assume that (H\ref{secweak}.1) holds.
Then, there exists a pair of constants $(\overline{H}_1, \overline{H}_2)$
such that \eqref{tripz} admits a viscosity solution $(u_1,u_2) \in C(\mathbb T^n)^2$.
\end{Theorem}
One can easily see that the pair $(\overline{H}_1, \overline{H}_2)$ is not unique (see Remark \ref{nouniq}).
Nevertheless, we have the following.
\begin{Theorem} \label{cell_const2}
There exists a unique $\mu \in \mathbb R$ such that
\begin{equation*}
c_2 \overline{H}_1 + c_1\overline{H}_2 = \mu,
\end{equation*}
for every pair $(\overline{H}_1, \overline{H}_2) \in \mathbb{R}^2$ such that
\eqref{tripz} admits a viscosity solution $(u_1,u_2) \in C(\mathbb T^n)^2$.
\end{Theorem}
Theorem \ref{cell_const2} can be rephrased
by saying that there exists a unique $\overline{H} \in \mathbb{R}$ such that the system
\begin{equation} \label{falsj}
\left\{ \begin{aligned}
c_1 u_1- c_1 u_2 + H_1(x,Du_1) &=\overline{H}
\\
-c_2 u_1 + c_2 u_2 + H_2(x,Du_1) &=\overline{H} \\
\end{aligned} \right.
\quad \mbox{in}~ {\mathbb T^n},
\end{equation}
admits viscosity solutions $u_1,u_2 \in C(\mathbb T^n)$, with (see Remark \ref{Hbar})
$$
\overline{H}= \frac{\mu}{c_1+c_2} .
$$
Thus, this is the analogous to the uniqueness result of the effective Hamiltonian for the single equation case in \cite{LPV}.
Notice that this {\it cell problem} is the important basis for the study
of homogenization and
large time behavior of weakly coupled systems
of Hamilton--Jacobi equations.
Besides, we also consider the regularized system
\begin{equation}
\left\{ \begin{aligned}
(c_1+\varepsilon)u^\varepsilon_1 - c_1 u^\varepsilon_2 + H_1(x,Du^\varepsilon_1)
& = \varepsilon^2 \Delta u^\varepsilon_1
\\
(c_2+\varepsilon) u^\varepsilon_2 - c_2 u^\varepsilon_1 + H_2(x,D u^\varepsilon_2 )
&=\varepsilon^2 \Delta u^\varepsilon_2 \\
\end{aligned} \right.
\quad \mbox{in}~ \mathbb T^n,
\label{trip}
\end{equation}
and prove that both $\varepsilon u_1^{\varepsilon}$ and $\varepsilon u_2^{\varepsilon}$
converge uniformly to $-\overline{H}$ with speed of convergence $O (\varepsilon)$ (see Theorem~\ref{cell_thm2}).
We call $\overline{H}$ the effective Hamiltonian of the {\it cell problem}
for {the} weakly coupled system of Hamilton--Jacobi equations.
In Section \ref{kilo}, we conclude the paper with the study of weakly coupled systems of obstacle type, namely
\begin{equation} \label{jk}
\left\{ \begin{aligned}
\max\{ u_1 - u_2 -\psi_1 , u_1 +H_1(x,D u_1) \}&=0 \quad\mbox{in}~ U,
\\
\max\{ u_2-u_1-\psi_2 , u_2+H_2(x,Du_2) \}&=0 \quad\mbox{in}~ U, \\
\end{aligned} \right.
\end{equation}
{with boundary conditions $u_1 = u_2=0$ on $\partial U$.}
Problems of this type appeared in \cite{DE} and \cite{CLY}.
Here $H_1, H_2: \overline{U} \times \mathbb{R}^n \to \mathbb{R}$
and $\psi_1, \psi_2 : \overline{U} \to \mathbb{R}$ are smooth, with $\psi_1, \psi_2 \geq \alpha > 0$.
In this case, although the two equations in \eqref{jk} are coupled just through the difference $u_1~-~u_2$
(\textit{weakly} coupled system), the problem turns out to be considerably more difficult
than the corresponding scalar equation \eqref{qw}.
Indeed, we cannot show now the analogous of estimate \eqref{hfj}
as in Section~\ref{sectobs}.
For this reason, the hypotheses we require are stronger than in the scalar case.
Together with the usual hypotheses of coercivity and compatibility
(see conditions (H\ref{kilo}.2) and (H\ref{kilo}.4)),
we have to assume that $H_1 (x,\cdot)$ and $H_2 (x,\cdot)$ are convex (see (H\ref{kilo}.1)),
and we also ask that $D_x H_1$ and $D_x H_2$ are bounded (see (H\ref{kilo}.3)).
Under these hypotheses, that are natural in optimal switching problems,
we are able to establish several delicate estimates
by crucially employing the adjoint method (Lemmas from \ref{gamprimeb} to \ref{dggt}),
which then yield a rate of convergence (Theorem \ref{obs_speedfin}).
\end{subsection}
\begin{subsection}{Acknowledgments}
The authors are grateful to Craig Evans and
Fraydoun Rezakhanlou for very useful discussions on the subject of the paper.
We thank the anonymous referee for many valuable comments and suggestions.
Diogo Gomes was partially supported by CAMGSD/IST through FCT Program POCTI - FEDER
and by grants PTDC/EEA-ACR/67020/2006, PTDC/MAT/69635/2006, and UTAustin/MAT/0057/2008.
Filippo Cagnetti was partially supported by FCT through the CMU$|$Portugal program.
Hung Tran was partially supported by VEF fellowship.
\end{subsection}
\end{section}
\begin{section}{Obstacle problem} \label{sectobs}
In this section, we study the following obstacle problem
\begin{equation}
\left\{ \begin{aligned}
\max \{ u- \psi, u+H(x,Du)\}&=0 \quad \mbox{in}~ U
\\
u &= 0 \quad \mbox{on}~ \partial U,\\
\end{aligned} \right.
\label{obs_eqn}
\end{equation}
where $\psi: \overline{U} \to \mathbb{R}$ and $H: {\overline{U} \times \mathbb R^n} \to \mathbb R$
are smooth, with $\psi \ge 0$ on $\partial U$.
We also assume that
\begin{itemize}
\item[(H\ref{sectobs}.1)] {there exists $\beta >0$ such that}
$$
\lim_{|p| \to + \infty}
\left( {\beta}|H(x,p)|^2 + D_x H(x,p) \cdot p \right)=
\lim_{|p| \to + \infty} \frac{H(x,p)}{|p|}=+\infty
\text{ uniformly in } x \in \overline U;
$$
\item[(H\ref{sectobs}.2)] there exists a function $\Phi \in C^2 (U) \cap C^1 (\overline U)$
such that $\Phi \leq \psi$ on $\overline U$, $\Phi=0$ on $\partial U$ and
$$
\Phi+H(x,D\Phi) <0\quad \mbox{in}~ \overline{U}.
$$
\end{itemize}
We observe that in the classical case $H(x,p) = \mathcal{H} (p)+V(x)$ with
$$
\lim_{|p| \to + \infty} \frac{\mathcal{H} (p)}{|p|} = + \infty,
$$
or when $H$ is superlinear in $p$ and $|D_xH(x,p)| \le C(1+|p|)$, then we immediately have (H\ref{sectobs}.1).
Assumption (H\ref{sectobs}.2) (stating, in particular, that $\Phi$ is a sub-solution of \eqref{obs_eqn}),
will be used to derive the existence of solutions of \eqref{obs_eqn},
and to give a uniform bound for the gradient of solutions of the penalized equation below.
\begin{subsection}{The classical approach}
For every $\varepsilon >0$, the \textit{penalized} equation
of \eqref{obs_eqn} is given by
\begin{equation}
\left\{ \begin{aligned}
u^\varepsilon+H(x,Du^\varepsilon)+\gamma^\varepsilon(u^\varepsilon-\psi)
&= \varepsilon \Delta u^\varepsilon \quad \mbox{in}~ U,
\\
u^\varepsilon &= 0 \qquad \mbox{on}~ \partial U,\\
\end{aligned} \right.
\label{obs_reg}
\end{equation}
where $\gamma^{\varepsilon}$ is defined by \eqref{gammaep}.
From \cite{L} it follows that under conditions (H\ref{sectobs}.1) and (H\ref{sectobs}.2),
for every $\varepsilon > 0$ there exists
a smooth solution $u^\varepsilon$ to \eqref{obs_reg}.
The first result we establish is a uniform bound
for the $C^1$-norm of the sequence $\{u^{\varepsilon}\}$.
\begin{Proposition} \label{obs1}
There exists a positive constant $C$, independent of $\varepsilon$, such that
\begin{equation*}
\|u^\varepsilon\|_{L^\infty}, \|Du^\varepsilon\|_{L^\infty} \le C.
\end{equation*}
\end{Proposition}
In order to prove Proposition \ref{obs1}, we need the following fundamental lemma:
\begin{Lemma} \label{obs_lem4}
There exists a constant $C>0$, independent of $\varepsilon$, such that
\begin{equation}
\max_{x \in \overline{U}} \gamma^\varepsilon (u^\varepsilon -\psi) \le C, \quad \quad \quad \max_{x \in \overline{U}}
\dfrac{u^\varepsilon -\psi}{\varepsilon} \le C.
\notag
\end{equation}
\end{Lemma}
\begin{proof}
We only need to show that $\max_{x \in \overline{U}} \gamma^\varepsilon (u^\varepsilon -\psi) \le C$,
since then the second estimate follows directly by the definition of $\gamma^\varepsilon$.
Since $u^\varepsilon - \psi \le 0$ on $\partial U$, we have $\max_{x \in \partial U} \gamma^\varepsilon
(u^\varepsilon -\psi) = 0$.
Now, if $\max_{x \in \overline{U}} \gamma^\varepsilon (u^\varepsilon -\psi)=0$, then we are done.
Thus, let us assume that there exists $x_1 \in U$ such that
$\max_{x \in \overline{U}} \gamma^\varepsilon (u^\varepsilon -\psi)
=\gamma^\varepsilon (u^\varepsilon -\psi)(x_1) >0$.
Since $\gamma^\varepsilon$ is increasing, we also have
$\max_{x \in U} (u^\varepsilon -\psi) = u^\varepsilon (x_1)- \psi (x_1)$.
Thus, using \eqref{obs_reg}, by the Maximum principle
\begin{align*}
&(u^\varepsilon(x_1) - \psi(x_1)) + \gamma^\varepsilon (u^\varepsilon(x_1) - \psi(x_1))
= \varepsilon \Delta u^{\varepsilon} (x_1) - H(x_1, D u^{\varepsilon} (x_1)) - \psi(x_1) \\
& \le \varepsilon \Delta \psi(x_1) - H(x_1, D \psi(x_1)) - \psi(x_1).
\end{align*}
Since $u^\varepsilon(x_1) - \psi(x_1)>0$,
\begin{equation}
\gamma^\varepsilon (u^\varepsilon(x_1) - \psi(x_1)) \le \max_{x \in \overline{U}} (|\Delta \psi| + |H(x,D \psi)|+
|\psi(x)|) \le C,
\notag
\end{equation}
for any $\varepsilon<1$, and this concludes the proof.
\end{proof}
\begin{proof}[Proof of Proposition \ref{obs1}]
Suppose there exists $x_0 \in U$ such that $u^{\varepsilon} (x_0) = \max_{x \in \overline{U}} u^{\varepsilon} (x)$.
Then, since $\Delta u^\varepsilon (x_0)\leq~0$ and using the fact that $\gamma^{\varepsilon} \geq 0$
\begin{equation*} \begin{split}
u^\varepsilon (x_0) &= \varepsilon \Delta u^\varepsilon (x_0)
- H(x_0, 0 ) - \gamma^\varepsilon(u^\varepsilon (x_0) -\psi (x_0)) \\
&\leq - H(x_0, 0 ) \leq \max_{x \in \overline{U}} \left( - H(x, 0 ) \right) \leq C.
\end{split} \end{equation*}
Let now $x_1 \in U$ be such that $u^{\varepsilon} (x_1) = \min_{x \in \overline{U}} u^{\varepsilon} (x_1)$.
Then, using Lemma~\ref{obs_lem4},
\begin{equation*} \begin{split}
u^\varepsilon (x_1) &= \varepsilon \Delta u^\varepsilon (x_1)
- H(x_1, 0 ) - \gamma^\varepsilon(u^\varepsilon (x_1) -\psi (x_1)) \\
&\geq - H(x_1, 0 ) - \gamma^\varepsilon(u^\varepsilon (x_1) -\psi (x_1)) \\
&\geq \min_{x \in \overline{U}}
\left( - H(x, 0 ) - \gamma^\varepsilon(u^\varepsilon (x) -\psi (x)) \right) \geq -C.
\end{split} \end{equation*}
This shows that $\| u^{\varepsilon} \|_{L^{\infty}}$ is bounded.
To prove that $\| Du^{\varepsilon} \|_{L^{\infty}}$ is bounded independently of $\varepsilon$,
we first need to prove that $\| Du^{\varepsilon} \|_{L^{\infty}(\partial U)}$ is bounded
by constructing appropriate barriers.
Let $\Phi$ be as in (H2.2). For $\varepsilon$ small enough, we have that
$$
\Phi+H(x,D\Phi)+\gamma^\varepsilon(\Phi-\psi) <\varepsilon \Delta \Phi,
$$
and $\Phi=0$ on $\partial U$.
Therefore, $\Phi$ is a sub-solution of \eqref{obs_reg}. By the comparison principle, $u^\varepsilon \ge \Phi$ in~$U$.
Let now $d(x)=\text{dist}(x,\partial U)$.
It is well-known that for some $\delta>0$ $d \in C^2(U_\delta)$ and $|Dd|=1$ in $U_\delta$,
where $U_\delta :=\{x\in U:~d(x) <\delta\}$.
For $\mu>0$ large enough, the uniform bound on $\|u^\varepsilon\|_{L^\infty}$ yields
$v:=\mu d \ge u^\varepsilon$ on $\partial U_\delta $. Assumption (H\ref{sectobs}.1) then implies
$$
v+H(x,Dv)+\gamma^\varepsilon(v-\psi) - \varepsilon \Delta v \ge H(x,\mu Dd) -C \mu \ge 0,
$$
for $\mu$ is sufficiently large.
So the comparison principle gives us that $\Phi \le u^\varepsilon \le v$ in $ U_\delta$.
Thus, since $\nu$ is the \textit{outer} unit normal to $\partial U$, and $\Phi= u^\varepsilon= v = 0$ on $\partial U$, we have
$$
\dfrac{\partial v}{\partial \nu}(x) \le \dfrac{\partial u^\varepsilon}{\partial \nu}(x) \le \dfrac{\partial \Phi}{\partial \nu}(x),\quad \text{for } x\in \partial U.
$$
Hence, we obtain $\| Du^{\varepsilon} \|_{L^{\infty}(\partial U)} \leq C$.
Next, let us set $w^\varepsilon = \dfrac{|Du^{\varepsilon}|^2}{2}$.
By a direct computation one can see that
\begin{equation} \label{obs2}
2(1+(\gamma^\varepsilon) ' )w^\varepsilon + D_p H \cdot D w^\varepsilon
+ D_xH\cdot Du^\varepsilon-(\gamma^\varepsilon)' Du^\varepsilon\cdot D\psi = \varepsilon \Delta w^\varepsilon -
\varepsilon |D^2 u^\varepsilon|^2.
\end{equation}
If $\| Du^{\varepsilon} \|_{L^{\infty}} \leq \max (\| D \psi \|_{L^{\infty}},\| Du^{\varepsilon} \|_{L^{\infty}(\partial U)})$ then we are done.\\
Otherwise, $ \max (\| D \psi \|_{L^{\infty}},\| Du^{\varepsilon} \|_{L^{\infty}(\partial U)}) < \| D u^{\varepsilon} \|_{L^{\infty}}$.
We can choose $x_2 \in U$ such that $w^{\varepsilon} (x_2) = \max_{x \in \overline{U}} w^{\varepsilon} (x)$.
Then, using \eqref{obs2}
\begin{equation} \label{1k} \begin{split}
\varepsilon |D^2 u^\varepsilon|^2 (x_2)
&= \varepsilon \Delta w^\varepsilon (x_2)
- 2 w^\varepsilon (x_2) - D_xH (x_2 , Du^\varepsilon (x_2)) \cdot Du^\varepsilon (x_2) \\
&+ (\gamma^\varepsilon)'
\left( Du^\varepsilon (x_2) \cdot D\psi (x_2) - |D u^{\varepsilon}|^2 (x_2) \right) \\
&\leq - D_xH (x_2 , Du^\varepsilon (x_2)) \cdot Du^\varepsilon (x_2).
\end{split} \end{equation}
Moreover, for $\varepsilon$ sufficiently small we have
$$
\varepsilon |D^2 u^\varepsilon|^2 (x_2)
\geq 2 \beta \varepsilon^2 |\Delta u^{\varepsilon} (x_2)|^2
= 2 \beta \left[ u^\varepsilon (x_2) + H(x_2 ,Du^\varepsilon (x_2))
+\gamma^\varepsilon(u^\varepsilon (x_2) - \psi (x_2)) \right]^2.
$$
{By Young's inequality,
\begin{equation} \label{2k} \begin{split}
\varepsilon |D^2 u^\varepsilon|^2 (x_2)
&\geq \beta | H(x_2 ,Du^\varepsilon (x_2)) |^2
- 2 \beta \left[ u^\varepsilon (x_2) + \gamma^\varepsilon(u^\varepsilon (x_2) - \psi (x_2)) \right]^2 \\
&\geq \beta | H(x_2 ,Du^\varepsilon (x_2)) |^2 - C_{\beta},
\end{split} \end{equation}
for some positive constant $C_{\beta}$,
where we used Lemma \ref{obs_lem4} for the last inequality.
Collecting \eqref{1k} and \eqref{2k}
\begin{equation*}
\beta | H(x_2 ,Du^\varepsilon (x_2)) |^2
+ D_xH (x_2 , Du^\varepsilon (x_2)) \cdot Du^\varepsilon (x_2) \leq C_{\beta}.
\end{equation*}
Recalling hypothesis (H\ref{sectobs}.1),} we must have
\begin{equation*}
\| D u^{\varepsilon} \|_{L^{\infty}} = | Du^\varepsilon (x_2) | \leq C.
\end{equation*}
\end{proof}
Thanks to Proposition \ref{obs1} one can show that, up to subsequences,
$u^{\varepsilon}$ converges uniformly
to a viscosity solution $u$ of the obstacle problem (\ref{obs_eqn}).
\end{subsection}
\begin{subsection}{Proof of Theorem \ref{obs_speed}}
We now study the speed of convergence.
\noindent
To prove our theorem we need several steps.
{\bf Adjoint method:}
The formal linearized operator $L^\varepsilon: C^2 (U) \to C (U)$
corresponding to \eqref{obs_reg} is given by
$$
L^\varepsilon z :=(1+(\gamma^\varepsilon)') z + D_pH\cdot Dz -\varepsilon \Delta z.
$$
We will now introduce the adjoint PDE corresponding to $L^\varepsilon$.
Let $x_0 \in U$ be fixed.
We denote by $\sigma^\varepsilon$ the solution of:
\begin{equation}
\left\{ \begin{aligned}
(1+(\gamma^\varepsilon)')\sigma^\varepsilon-\mbox{div}(D_pH\sigma^\varepsilon)
&=\varepsilon \Delta \sigma^\varepsilon+ \delta_{x_0},\ \qquad \qquad&\mbox{in}~ U ,
\\
\sigma^\varepsilon &= 0, \qquad \qquad\qquad &\mbox{on}~ \partial U, \\
\end{aligned} \right.
\label{obs_adj}
\end{equation}
where $\delta_{x_0}$ stands for the Dirac measure concentrated in $x_0$.
In order to show existence and uniqueness of $\sigma^\varepsilon$,
we have to pass to a further adjoint equation.
Let $f \in C (U)$ be fixed.
Then, we denote by $v$ the solution to
\begin{equation}
\left\{ \begin{aligned}
(1+(\gamma^\varepsilon)')v+D_pH\cdot Dv&=\varepsilon \Delta v+f, \qquad \qquad&\mbox{in}~ U ,
\\
v &= 0, \qquad \qquad\qquad &\mbox{on}~ \partial U. \\
\end{aligned} \right.
\label{obs3}
\end{equation}
When $f \equiv 0$, by using the Maximum Principle one can show that $v \equiv 0$
is the unique solution to \eqref{obs3}.
Thus, by the Fredholm Alternative we infer that \eqref{obs_adj}
admits a unique solution $\sigma^\varepsilon$.
Moreover, one can also prove that $\sigma^\varepsilon \in C^\infty(U\setminus \{x_0\})$.
Some additional properties of $\sigma^\varepsilon$ are given by the following lemma.
\begin{Lemma}[Properties of $\sigma^\varepsilon$] \label{obs_lem2}
Let $\nu$ denote the outer unit normal to $\partial U$. Then,
\begin{itemize}
\item[(i)] $\sigma^\varepsilon \ge 0$ on $\overline{U}$.
In particular, $\dfrac{\partial \sigma^\varepsilon}{\partial \nu} \le 0$ on $\partial U$.
\item[(ii)] The following equality holds:
\begin{equation}
\int_U (1+(\gamma^\varepsilon)') \, \sigma^\varepsilon \,dx =1+\varepsilon \int_{\partial U} \dfrac{\partial
\sigma^\varepsilon}{\partial \nu}\, dS.
\notag
\end{equation}
In particular,
$$
\varepsilon \int_{\partial U} \left| \dfrac{\partial \sigma^\varepsilon}{\partial \nu} \right| \,dS \le 1.
$$
\end{itemize}
\end{Lemma}
\begin{proof}
First of all, consider equation \eqref{obs3} and observe that
\begin{equation} \label{vpos}
f \geq 0 \Longrightarrow v \geq 0.
\end{equation}
Indeed, assume $f \geq 0$ and let $\overline{x} \in \overline{U}$ be such that
\begin{equation*}
v (\overline{x}) = \min_{x \in \overline{U}} v (x).
\end{equation*}
We can assume that $\overline{x} \in U$, since otherwise clearly $v \geq 0$.
Then, for every $x \in U$
\begin{equation*}
((1+(\gamma^\varepsilon)') v (\overline{x}) = \varepsilon \Delta v (\overline{x}) + f (\overline{x}) \geq 0,
\end{equation*}
and \eqref{vpos} follows, since $1+(\gamma^\varepsilon)' > 0$ .
Now, multiply equation \eqref{obs_adj} by $v$ and integrate by parts, obtaining
$$
\int_U f \sigma^\varepsilon\, dx = v(x_0).
$$
Taking into account \eqref{vpos}, from last relation we infer that
\begin{equation*}
\int_U f \sigma^\varepsilon\, dx \geq 0 \qquad \text{ for every }f \geq 0,
\end{equation*}
and this implies $\sigma^\varepsilon \ge 0$.\\
To prove (ii), we integrate \eqref{obs_adj} over $U$, to get
\begin{align}
\int_U (1+(\gamma^\varepsilon)')\sigma^\varepsilon\, dx
= & \int_U \mbox{div}(D_pH \sigma^\varepsilon) \, dx +\varepsilon \int_U \Delta \sigma^\varepsilon \,dx+1\notag
\\
=& \int_{\partial U} (D_p H \cdot \nu) \sigma^\varepsilon \,dS
+ \varepsilon \int_{\partial U} \dfrac{\partial \sigma^\varepsilon}{\partial \nu} \,dS+1
=\varepsilon \int_{\partial U} \dfrac{\partial \sigma^\varepsilon}{\partial \nu} \,dS + 1 ,\notag
\end{align}
where we used the fact that $\sigma^\varepsilon=0$ on $\partial U$.
\end{proof}
Using the adjoint equation, we have the following new estimate.
\begin{Lemma}
\label{obs_lem3}
There exists $C>0$, independent of $\varepsilon > 0$, such that
\begin{equation}
\frac{1}{2} \int_U (1 + ( \gamma^{\varepsilon} )')|D u^\varepsilon|^2 \sigma^\varepsilon \,dx
+ \varepsilon \int_U |D^2 u^\varepsilon|^2 \sigma^\varepsilon \,dx\le C.
\label{obs4}
\end{equation}
\end{Lemma}
\begin{proof}
Multiplying \eqref{obs2} by $\sigma^\varepsilon$ and integrating by parts, using equation \eqref{obs_adj} we
get
\begin{align*}
&\frac{1}{2} \int_U (1 + ( \gamma^{\varepsilon} )')|D u^\varepsilon|^2 \sigma^\varepsilon\, dx
+ \varepsilon \int_U |D^2 u^\varepsilon|^2 \sigma^\varepsilon \,dx \\
&\hspace{.5cm}= - w^\varepsilon (x_0)
- \int_{U} \left[ D_x H \cdot D u^{\varepsilon} - ( \gamma^{\varepsilon} )' D \psi \cdot D u^{\varepsilon}
\right] \sigma^{\varepsilon} \, dx
- \varepsilon \int_{\partial U} w^{\varepsilon} \frac{\partial \sigma^{\varepsilon}}{\partial \nu} \, d S.
\end{align*}
Thanks to Lemma \ref{obs_lem2} and Proposition \ref{obs1}
(which, in particular, implies
$\| D_x H (\cdot , D u^{\varepsilon} (\cdot)) \|_{L^{\infty} (U)} \leq C$)
the conclusion follows.
\end{proof}
\noindent
Relation (\ref{obs4}) shows that we have a good control of the Hessian $D^2 u^\varepsilon$
in the support of $\sigma^\varepsilon$.
We finally have the following result, which immediately implies Theorem \ref{obs_speed}.
\begin{Lemma}
\label{obs_thm1}
There exists $C>0$, independent of $\varepsilon$, such that
\begin{equation}
\max_{x \in \overline{U}} \left| u^\varepsilon_{\varepsilon} (x) \right| \le
\dfrac{C}{\varepsilon^{1/2}}.
\notag
\end{equation}
\end{Lemma}
\begin{proof}
By standard elliptic estimates, the solution $u^\varepsilon$ is smooth in the parameter $\varepsilon$ for $\varepsilon>0$
(see \cite{E2, T1} for similar arguments).
Differentiating (\ref{obs_reg}) w.r.t. $\varepsilon$ we get
\begin{equation}
(1+(\gamma^\varepsilon)')u_\varepsilon^\varepsilon +D_pH\cdot Du_\varepsilon^\varepsilon
+ \gamma_\varepsilon^\varepsilon = \varepsilon \Delta u_\varepsilon^\varepsilon + \Delta u^\varepsilon,
\quad \quad \text{ in }U.
\label{obs5}
\end{equation}
In addition, we have $u_\varepsilon^\varepsilon(x)=0$ for all $x\in \partial U$,
since $u^\varepsilon(x)=0$ on $\partial U$ for every $\varepsilon$.
So, we may assume that there exists $x_2 \in U$ such that
$|u_\varepsilon^\varepsilon(x_2)|=\max_{x \in \overline{U}} |u_\varepsilon^\varepsilon (x)|$.
Consider the adjoint equation \eqref{obs_adj}, and choose $x_0=x_2$.
Multiplying by $\sigma^\varepsilon$ both sides of (\ref{obs5}) and integrating by parts,
$$
u^\varepsilon_\varepsilon(x_2) = - \int_U \gamma_\varepsilon^\varepsilon \sigma^\varepsilon \,dx+ \int_U \Delta u^\varepsilon \sigma^\varepsilon \,dx.
$$
Hence,
\begin{equation}
|u_\varepsilon^\varepsilon(x_2)| \le \int_U |\gamma_\varepsilon^\varepsilon |\sigma^\varepsilon \,dx+ \int_U |\Delta u^\varepsilon| \sigma^\varepsilon \,dx.
\label{obsk}
\end{equation}
By Lemma \ref{obs_lem4},
\begin{equation}
|\gamma_\varepsilon^\varepsilon|
= \left| -\dfrac{u^\varepsilon - \psi}{\varepsilon^2} \gamma' \left( \dfrac{u^\varepsilon-\psi}{\varepsilon}
\right) \right|
= \left| \dfrac{u^\varepsilon - \psi}{\varepsilon} (\gamma^\varepsilon)'(u^\varepsilon-\psi) \right| \le C
(\gamma^\varepsilon)'.
\label{obs6}
\end{equation}
Hence, thanks to Lemma \ref{obs_lem2}
\begin{equation}
\int_U |\gamma_\varepsilon^\varepsilon| \sigma^\varepsilon \,dx \le C \int_U (\gamma^\varepsilon)'
\sigma^\varepsilon \,dx\le C,
\label{uj}
\end{equation}
while using \eqref{obs4}
\begin{equation} \label{obs7} \begin{split}
\int_U |\Delta u^\varepsilon| \sigma^\varepsilon \,dx
&\le \left( \int_U | \Delta u^\varepsilon|^2 \sigma^\varepsilon \,dx \right)^{1/2}
\left( \int_U\sigma^\varepsilon \,dx \right)^{1/2} \\
&\le C\left( \int_U |D^2 u^\varepsilon|^2 \sigma^\varepsilon \,dx \right)^{1/2}
\left( \int_U\sigma^\varepsilon \,dx \right)^{1/2} \le \dfrac{C}{\varepsilon^{1/2}}.
\end{split} \end{equation}
Thus, by (\ref{obsk}), \eqref{uj} and (\ref{obs7})
\begin{equation}
|u_\varepsilon^\varepsilon(x_2)| \le \dfrac{C}{ \varepsilon^{1/2}}, ~\mbox{for}~ \varepsilon<1.
\label{obs8}
\end{equation}
\end{proof}
\end{subsection}
\end{section}
\begin{section}{Weakly coupled systems of Hamilton--Jacobi equations} \label{weakcou}
We study now the model of monotone weakly coupled systems of Hamilton-Jacobi equations
considered by Engler and Lenhart \cite{EL}, and by Ishii and Koike \cite{IK1}.
For the sake of simplicity, we will just focus on the following system of two equations:
\begin{equation}
\left\{ \begin{aligned}
c_{11} u_1 +c_{12} u_2 + H_1(x,Du_1) &=0
\\
c_{21} u_1 +c_{22} u_2 + H_2(x,D u_2) &=0 \\
\end{aligned} \right.
\quad \mbox{in}~ U,
\label{w_coupled}
\end{equation}
with boundary conditions $u_1 = u_2 =0$ on $\partial U$.
The general case of more equations can be treated in a similar way.
We assume that the Hamiltonians
$H_1, H_2: \overline{U} \times \mathbb R^n \rightarrow \mathbb R$ are smooth satisfying
{the following hypotheses.}
\begin{itemize}
\item[(H\ref{weakcou}.1)] {There exists $\beta_1,\beta_2 >0$ such that for every $j=1,2$
$$
\lim_{|p| \to +\infty} \left( \beta_j |H_j(x,p)|^2+D_xH_j(x,p)\cdot p \right)
=\lim_{|p| \to+\infty} \frac{H_j(x,p)}{|p|} =+\infty \text{ uniformly in } x\in \overline{U}.
$$}
\end{itemize}
Following \cite{EL} and \cite{IK1}, we suppose further that
\begin{itemize}
\item[(H\ref{weakcou}.2)] $\displaystyle c_{12},\, c_{21} {\leq} 0$;
\item[(H\ref{weakcou}.3)] there exists $\alpha>0$ such that
$\displaystyle c_{11}+c_{12}, ~c_{21}+c_{22} \geq \alpha>0$.
\end{itemize}
We observe that, as a consequence, we also have $c_{11}, c_{22}>0$.
Finally, we require that
\begin{itemize}
\item[(H\ref{weakcou}.4)] There exist $\Phi_1, \Phi_2 \in C^2 (U) \cap C^1 (\overline U)$
with $\Phi_j =0$ on $\partial U$ ($j=1,2$), and such that
\begin{equation*}
\left\{ \begin{aligned}
c_{11} \Phi_1 + c_{12} \Phi_2 +H_1 (x,D\Phi_1 )
< 0 \quad\mbox{in}~U,
\\
c_{22} \Phi_2 + c_{21} \Phi_1 +H_2 (x,D\Phi_2 )
< 0 \quad\mbox{in}~U.
\end{aligned} \right.
\end{equation*}
\end{itemize}
Thanks to these conditions, the Maximum Principle can be applied and existence,
comparison and uniqueness results hold true, as stated in \cite{EL}.
We consider now the following regularized system (here $\varepsilon >0$):
\begin{equation}
\left\{ \begin{aligned}
c_{11} u^\varepsilon_1 +c_{12} u^\varepsilon_2 + H_1(x,D u^\varepsilon_1) &=\varepsilon \Delta u^\varepsilon_1
\\
c_{21}u^\varepsilon_1+c_{22}u^\varepsilon_2 + H_2(x,Du^\varepsilon_2) &=\varepsilon \Delta u^\varepsilon_2 \\
\end{aligned} \right.
\quad \mbox{in}~ U,
\label{wc_reg}
\end{equation}
with boundary conditions $u^\varepsilon_1 = u^\varepsilon_2=0$ on $\partial U$.\\
Conditions (H\ref{weakcou}.1), (H\ref{weakcou}.2), and (H\ref{weakcou}.3) yield {existence} and uniqueness
of the pair of solutions $(u^\varepsilon_1, u^\varepsilon_2)$ in \eqref{wc_reg}.
Next lemma gives a uniform bound for the $C^1$ norm
of the sequences $\{ u^{\varepsilon}_i \}$, $i=1,2$.
Its proof, which is very similar to that one of Proposition \ref{obs1},
is still presented for readers' convenience.
\begin{Lemma} \label{lemu}
There exists a positive constant $C$, independent of $\varepsilon$, such that
\begin{equation}
\|u^\varepsilon_i \|_{L^\infty}, \| Du^\varepsilon_i \|_{L^\infty} \le C,
\quad \text{for } i=1,2.
\notag
\end{equation}
\end{Lemma}
\begin{proof}
\textbf{Step I: Bound on $\|u^\varepsilon_j \|_{L^\infty}, j=1,2$.}
{First of all observe that $u^{\varepsilon}_1 = u^{\varepsilon}_2 = 0$ on $\partial U$ for every $\varepsilon$.
Thus, it will be sufficient to show that $u^{\varepsilon}_1$ and $u^{\varepsilon}_2$
are bounded in the interior of $U$.
Without loss of generality, we can assume that there exists $\overline{x} \in U$ such that
\begin{equation*}
\max_{\substack{ j=1,2 \\ x \in \overline{U} }} u^{\varepsilon}_j (x) = u^{\varepsilon}_1 (\overline{x}).
\end{equation*}
We have
\begin{equation*}
\alpha u^{\varepsilon}_1 (\overline{x}) \leq c_{11} u^{\varepsilon}_1 (\overline{x}) + c_{12} u^{\varepsilon}_2 (\overline{x})
\leq - H_1 (\overline{x}, 0) \leq \max_{x \in \overline{U}} \left( - H_1 (x,0) \right),
\end{equation*}
where we used (H\ref{weakcou}.3) and equation \eqref{wc_reg}.
Analogously, if $\widehat{x} \in U$ is such that
\begin{equation*}
\min_{\substack{ j=1,2 \\ x \in \overline{U} }} u^{\varepsilon}_j (x) = u^{\varepsilon}_1 (\widehat{x}),
\end{equation*}
then
\begin{equation*}
u^{\varepsilon}_1 (\widehat{x}) \geq \frac{c_{11}}{c_{11} + c_{12} } u^{\varepsilon}_1 (\widehat{x})
+ \frac{c_{12}}{c_{11} + c_{12} } u^{\varepsilon}_2 (\widehat{x})
\geq - \frac{H_1 (\widehat{x}, 0)}{c_{11} + c_{12} } \geq \frac{1}{c_{11} + c_{12} } \min_{x \in \overline{U}} \left( - H_1 (x,0) \right).
\end{equation*}}
Concerning the bounds on the gradients, we will argue as in the proof of Proposition \ref{obs1}.
\textbf{Step II: Bound on $\| D u^\varepsilon_j \|_{L^\infty (\partial U)}, j=1,2$.}
{We now show that
\begin{equation*}
\max_{\substack{ j=1,2 \\ x \in \partial U }} | D u^{\varepsilon}_j (x) | \leq C,
\end{equation*}
for some constant $C$ independent of $\varepsilon$.
As it was done in Section \ref{sectobs}, we are going to construct appropriate barriers.
For $\varepsilon$ small enough, assumption (H\ref{weakcou}.4) implies that
\begin{equation*}
\left\{ \begin{aligned}
c_{11} \Phi_1 + c_{12} \Phi_2 +H_1 (x,D\Phi_1 )
< \varepsilon \Delta \Phi_1 \quad\mbox{in}~U,
\\
c_{22} \Phi_2 + c_{21} \Phi_1 +H_2 (x,D\Phi_2 )
< \varepsilon \Delta \Phi_2 \quad\mbox{in}~U,
\end{aligned} \right.
\end{equation*}
and $\Phi_1 = \Phi_2 =0$ on $\partial U$.
Therefore, $(\Phi_1,\Phi_2)$ is a sub-solution of \eqref{wc_reg}.
By the comparison principle, $u^\varepsilon_j \ge \Phi_j$ in~$U$, $j=1,2$.
Let $d(x)$, $\delta$, and $U_{\delta}$ be as in the proof of Proposition \ref{obs1}.
For $ \mu > 0 $ large enough, the uniform bounds on $\|u^\varepsilon_1\|_{L^\infty}$ and $\|u^\varepsilon_2\|_{L^\infty}$ yield
$v := \mu d \ge u^\varepsilon_j$ on $\partial U_\delta $, $j=1,2$, so that
\begin{equation*}
\left\{ \begin{aligned}
(c_{11} + c_{12} ) v +H_1(x,Dv) - \varepsilon \Delta v
\ge H_1(x,\mu Dd) - \mu C \quad\mbox{in}~U,
\\
(c_{21} + c_{22} ) v +H_2(x,Dv) - \varepsilon \Delta v
\ge H_2(x,\mu Dd) - \mu C \quad\mbox{in}~U.
\\
\end{aligned} \right.
\end{equation*}
Now, we have $\Phi_j= u^\varepsilon_j= v = 0$ on $\partial U$.
Also, thanks to assumption (H\ref{weakcou}.1), for $\mu >0$ large enough
\begin{equation*}
\left\{ \begin{aligned}
(c_{11} + c_{12} ) v +H_1(x,Dv) - \varepsilon \Delta v
\ge 0 \quad\mbox{in}~U,
\\
(c_{21} + c_{22} ) v +H_2(x,Dv) - \varepsilon \Delta v
\ge 0 \quad\mbox{in}~U,
\\
\end{aligned} \right.
\end{equation*}
that is, the pair $(v,v)$ is a super-solution for the system \eqref{wc_reg}.
Thus, the comparison principle gives us that $\Phi_j \le u^\varepsilon_j \le v$ in $ U_\delta$, $j=1,2$.
Then, from the fact that $\Phi_j= u^\varepsilon_j= v = 0$ on $\partial U$ we get
$$
\dfrac{\partial v}{\partial \nu}(x) \le \dfrac{\partial u^\varepsilon_j}{\partial \nu}(x) \le \dfrac{\partial \Phi_j}{\partial \nu}(x),
\quad \text{for } x\in \partial U, \quad j=1,2.
$$
Hence, we obtain $\| Du^{\varepsilon}_j \|_{L^{\infty}(\partial U)} \leq C$, $j=1,2$.}
\textbf{Step III: Conclusion.}
{Setting $w^\varepsilon_j=\dfrac{|Du^\varepsilon_j|^2}{2}$, $j=1,2$, by a direct computation we have that
\begin{equation} \label{weqn}
\left\{ \begin{aligned}
2 c_{11} w^\varepsilon_1 + D_p H_1\cdot Dw^\varepsilon_1 + c_{12} Du^\varepsilon_1 \cdot Du^\varepsilon_2
+D_x H_1\cdot Du^\varepsilon_1 = \varepsilon \Delta w^\varepsilon_1 - \varepsilon |D^2 u^\varepsilon_1|^2,
\\
2 c_{22} w^\varepsilon_2 + D_p H_2\cdot D w^\varepsilon_2 + c_{21} Du^\varepsilon_1 \cdot D u^\varepsilon_2
+D_x H_2\cdot Du^\varepsilon_2 = \varepsilon \Delta w^\varepsilon_2 - \varepsilon |D^2 u^\varepsilon_2 |^2.
\end{aligned} \right.
\end{equation}
Assume now that there exists $\widehat{x} \in U$ such that
\begin{equation*}
\max_{\substack{ j=1,2 \\ x \in \overline{U} }} w^{\varepsilon}_j (x) = w^{\varepsilon}_1 (\widehat{x}).
\end{equation*}
Then, we have
\begin{equation*} \begin{split}
\varepsilon |D^2 u^\varepsilon_1|^2 (\widehat{x})
&= \varepsilon \Delta w^\varepsilon_1 (\widehat{x}) - 2 c_{11} w^\varepsilon_1 (\widehat{x})
- c_{12} Du^\varepsilon_1 (\widehat{x}) \cdot Du^\varepsilon_2 (\widehat{x})
- D_x H_1\cdot Du^\varepsilon_1 (\widehat{x}) \\
&\leq - 2 ( c_{11} + c_{12} ) w^{\varepsilon}_1 (\widehat{x}) - D_x H_1\cdot Du^\varepsilon_1 (\widehat{x})
\leq - D_x H_1\cdot Du^\varepsilon_1 (\widehat{x}).
\end{split} \end{equation*}
Now, arguing as in the proof of Proposition \ref{obs1}, for $\varepsilon$ sufficiently small
\begin{equation*} \begin{split}
\varepsilon |D^2 u^\varepsilon_1 (\widehat{x}) |^2
&\geq 2 \beta_1 \varepsilon^2 | \Delta u^\varepsilon_1 (\widehat{x})|^2
=2 \beta_1 \left[ c_{11} u^\varepsilon_1 (\widehat{x}) +c_{12} u^\varepsilon_2 (\widehat{x}) + H_1( \widehat{x} ,D u^\varepsilon_1(\widehat{x})) \right]^2 \\
&\geq \beta_1 | H_1( \widehat{x} ,D u^\varepsilon_1(\widehat{x})) |^2 - C.
\end{split} \end{equation*}
Collecting the last two relations we have
\begin{equation*}
\beta_1 | H_1( \widehat{x} ,D u^\varepsilon_1(\widehat{x})) |^2
+ D_x H_1( \widehat{x} ,D u^\varepsilon_1(\widehat{x})) \cdot Du^\varepsilon_1 (\widehat{x}) \leq C.
\end{equation*}
Recalling condition (H\ref{weakcou}.1) the conclusion follows.}
\end{proof}
\noindent
{\bf Adjoint method.} At this point, we introduce the
adjoint of the linearization of system \eqref{wc_reg}.
Let us emphasize that the adjoint equations
we introduce
form a system of weakly coupled type, which is very natural
in this setting, and create a systematic way to
the study of weakly coupled systems.
The linearized operator corresponding to \eqref{wc_reg} is
\begin{equation*}
L^{\varepsilon} (z_1,z_2) : =
\left\{ \begin{aligned}
D_p H_1 (x,Du_1^\varepsilon) \cdot D z_1
+ c_{11} z_1 + c_{12} z_2- \varepsilon \Delta z_1 ,
\\
D_p H_2 (x,Du_2^\varepsilon) \cdot D z_2
+ c_{22} z_2 + c_{21} z_1- \varepsilon \Delta z_2.
\end{aligned} \right.
\end{equation*}
Let us now identify the adjoint operator $(L^{\varepsilon})^*$.
For every $\nu^1,\nu^2 \in C^{\infty}_c (U)$ we have
\begin{equation*} \begin{split}
&\langle (L^{\varepsilon})^* (\nu^1, \nu^2), (z_1, z_2) \rangle
:= \langle (\nu^1, \nu^2), L^{\varepsilon} (z_1, z_2) \rangle \\
&= \langle \, \nu^1, \left[ L^{\varepsilon} (z_1, z_2) \right]_1 \, \rangle
+ \langle \, \nu^2, \left[ L^{\varepsilon} (z_1, z_2) \right]_2 \, \rangle \\
&= \int_U \left[ D_p H_1 (x,Du_1^\varepsilon) \cdot D z_1
+ c_{11} z_1 + c_{12} z_2- \varepsilon \Delta z_1 \right] \, \nu^1 \, dx \\
&+ \int_U \left[ D_p H_2 (x,Du_2^\varepsilon) \cdot D z_2
+ c_{22} z_2 + c_{21} z_1- \varepsilon \Delta z_2 \right] \, \nu^2 \, dx \\
&= \int_U \left[ - \mbox{div}(D_pH_1 \nu^1 ) + c_{11} \nu^1 + c_{21} \nu^2
- \varepsilon \Delta \nu^1 \right] \, z_1 \, dx \\
&+ \int_U \left[ - \mbox{div}(D_pH_2 \nu^2 ) + c_{22} \nu^2 + c_{12} \nu^1
- \varepsilon \Delta \nu^2 \right] \, z_2 \, dx.
\end{split} \end{equation*}
Then, the adjoint equations are:
\begin{equation} \label{rfd0}
\left\{ \begin{aligned}
- \mbox{div}(D_pH_1 \sigma^{1,\varepsilon}) + c_{11} \sigma^{1,\varepsilon} + c_{21} \sigma^{2,\varepsilon}
&= \varepsilon \Delta
\sigma^{1,\varepsilon}+ (2 - i) \delta_{x_0} \quad &\mbox{in}~ U,
\\
- \mbox{div}(D_pH_2 \sigma^{2,\varepsilon}) + c_{22} \sigma^{2,\varepsilon} + c_{12} \sigma^{1,\varepsilon}
&= \varepsilon \Delta
\sigma^{2,\varepsilon}+ (i - 1) \delta_{x_0} \quad &\mbox{in}~ U,
\end{aligned} \right.
\end{equation}
with boundary conditions
\begin{equation*}
\left\{ \begin{aligned}
\sigma^{1,\varepsilon}&=0 \qquad \quad&\mbox{on}~\partial U,
\\
\sigma^{2,\varepsilon}&=0 \qquad \quad&\mbox{on}~\partial U ,\\
\end{aligned} \right.
\end{equation*}
where $i \in \{ 1, 2 \} $ and $x_0 \in U$ will be chosen later.
{Observe that, once $x_0$ is given, the choice $i=1$ ($i=2$) corresponds to an adjoint system
of two equations
where a Dirac delta measure concentrated at $x_0$ appears only on the right-hand side
of the first (second) equation.}
Existence and uniqueness of $\sigma^{1,\varepsilon}$ and $\sigma^{2,\varepsilon}$
follow by Fredholm alternative, by arguing as in Section \ref{sectobs},
and we have $\sigma^{1,\varepsilon}, \sigma^{2,\varepsilon} \in C^{\infty} (U \setminus \{ x_0 \})$.
We study now further properties of $\sigma^{1,\varepsilon}$ and $\sigma^{2,\varepsilon}$.
\begin{Lemma}[Properties of $\sigma^{1,\varepsilon}, \sigma^{2,\varepsilon}$] \label{kiz}
Let $\nu$ be the outer unit normal to $\partial U$. Then
\begin{itemize}
\item[(i)] $\sigma^{j,\varepsilon} \ge 0$ on $\overline{U}$.
In particular, $\dfrac{\partial \sigma^{j,\varepsilon}}{\partial \nu} \le 0$ on $\partial U \,\, (j=1,2)$.
\item[(ii)] The following equality holds:
\begin{equation}
\sum_{j=1}^2 \left( \int_U (c_{j1} + c_{j2}) \sigma^{j,\varepsilon} \,dx
- \varepsilon \int_{\partial U} \dfrac{\partial \sigma^{j,\varepsilon}}{\partial \nu} \,dS \right) = 1 .
\notag
\end{equation}
In particular,
$$
\sum_{j=1}^2 \int_U (c_{j1} + c_{j2}) \sigma^{j,\varepsilon} \,dx \le 1.
$$
\end{itemize}
\end{Lemma}
\begin{proof}
First of all, we consider the adjoint of equation \eqref{rfd0}:
\begin{equation} \label{adjadj50}
\left\{ \begin{aligned}
D_p H_1 (x,Du_1^\varepsilon) \cdot D z_1
+ c_{11} z_1 + c_{12} z_2- \varepsilon \Delta z_1 = f_1,
\\
D_p H_2 (x,Du_2^\varepsilon) \cdot D z_2
+ c_{22} z_2 + c_{21} z_1- \varepsilon \Delta z_2 = f_2,
\end{aligned} \right.
\end{equation}
where $f_1,f_2 \in C (U)$, with boundary conditions $z_1=z_2 =0$ on $\partial U$.
Note that
\begin{equation} \label{posit0}
f_1,f_2 \geq 0 \Longrightarrow
\min_{\substack{ j=1,2 \\ x \in \overline{U} }} z_j (x) \geq 0.
\end{equation}
Indeed, if the minimum is achieved for some $\overline{x} \in \partial U$, then clearly $z_1, z_2 \geq 0$.
Otherwise, assume
\begin{equation*}
\min_{\substack{ j=1,2 \\ x \in \overline{U} }} z_j (x) = z_1 (\overline{x}),
\end{equation*}
for some $\overline{x} \in U$.
Using condition (H\ref{weakcou}.2)
\begin{equation*}
(c_{11} + c_{12} ) z_1 (\overline{x}) \geq c_{11} z_1 (\overline{x}) + c_{12} z_2 (\overline{x})
= \varepsilon \Delta z_1 (\overline{x}) + f_1 (\overline{x}) \geq 0.
\end{equation*}
Thanks to (H\ref{weakcou}.3), \eqref{posit0} follows.
Let us now multiply \eqref{rfd0}$_1$ and \eqref{rfd0}$_2$ by the solutions $z_1$ and $z_2$ of \eqref{adjadj50}.
Adding up the relations obtained we have
\begin{equation*}
\int_U f_1 \sigma^{1,\varepsilon} \, dx + \int_U f_2 \sigma^{2,\varepsilon} \, dx
= (2 - i) z_1 (x_0) + (1 - i) z_2 (x_0).
\end{equation*}
Thanks to \eqref{posit0}, from last relation we conclude that\begin{equation*}
\int_U f_1 \sigma^{1,\varepsilon} \, dx + \int_U f_2 \sigma^{2,\varepsilon} \, dx \geq 0,
\quad \text{ for every }f_1,f_2 \geq 0,
\end{equation*}
and this implies that $\sigma^{1,\varepsilon}, \sigma^{2,\varepsilon} \geq 0$.
To prove (ii), it is sufficient to integrate equations \eqref{rfd0}$_1$ and \eqref{rfd0}$_2$ over $U$,
and to add up the two relations obtained.
\end{proof}
{The proof of the next lemma can be obtained by arguing as in the proof of Lemma \ref{obs_lem3}.}
\begin{Lemma}
\label{wc_lem1}
There exists a constant $C>0$, independent of $\varepsilon$, such that
\begin{equation*}
\varepsilon \int_U |D^2 u^\varepsilon_1 |^2 \sigma^{1,\varepsilon}\, dx
+ \varepsilon \int_U |D^2 u^\varepsilon_2 |^2 \sigma^{2,\varepsilon} \,dx \le C.
\end{equation*}
\end{Lemma}
We now give the last lemma needed to estimate the speed of convergence.
Here we use the notation $u^{\varepsilon}_{j, \varepsilon} (x) := \partial u^{\varepsilon}_j (x) / \partial \varepsilon$, $j=1,2$.
\begin{Lemma}
\label{wc_lem2}
There exists a constant $C>0$, independent of $\varepsilon$, such that
\begin{equation}
\max_{\substack{ j=1,2 \\ x \in \overline{U} }} | u^{\varepsilon}_{j, \varepsilon} (x) | \leq \dfrac{C}{\varepsilon^{1/2}}.
\notag
\end{equation}
\end{Lemma}
\begin{proof}
Differentiating (\ref{wc_reg}) w.r.t $\varepsilon$ we obtain the system
\begin{equation} \label{juh}
\left\{ \begin{aligned}
c_{11}u^\varepsilon_{1,\varepsilon} +c_{12}u^\varepsilon_{2,\varepsilon} + D_p H_1\cdot Du^\varepsilon_{1,\varepsilon}
&=\varepsilon \Delta u^\varepsilon_{1,\varepsilon} + \Delta u^\varepsilon_1,
\\
c_{21}u^\varepsilon_{1,\varepsilon} + c_{22} u^\varepsilon_{2,\varepsilon} + D_p H_2\cdot Du^\varepsilon_{2,\varepsilon}
&=\varepsilon \Delta u^\varepsilon_{2,\varepsilon} +\Delta u^\varepsilon_2. \\
\end{aligned} \right.
\end{equation}
Since $u^{\varepsilon}_{1,\varepsilon} =u^{\varepsilon}_{2,\varepsilon} = 0$ on $\partial U$,
we have
\begin{equation*}
\max_{x \in \partial U} u^{\varepsilon}_{1,\varepsilon} (x)
= \max_{x \in \partial U} u^{\varepsilon}_{2,\varepsilon} (x) = 0.
\end{equation*}
Assume now that there exists $\widehat{x} \in U$ such that
\begin{equation*}
\max_{\substack{ j=1,2 \\ x \in \overline{U} }} | u^{\varepsilon}_{j, \varepsilon} (x) | = | u^{\varepsilon}_{1, \varepsilon} (\widehat{x}) |,
\end{equation*}
and let $\sigma^{1,\varepsilon}, \sigma^{2,\varepsilon}$ be the solutions of system \eqref{rfd0}
with $i=1$ and $x_0 = \widehat{x}$.
Multiplying equations \eqref{juh}$_1$ and \eqref{juh}$_2$ by
$\sigma^{1,\varepsilon}$ and $\sigma^{2,\varepsilon}$ respectively and adding up,
thanks to \eqref{rfd0} we obtain
\begin{equation*}
u^{\varepsilon}_{1, \varepsilon} (\widehat{x})
= \int_U \Delta u^\varepsilon_1 \, \sigma^{1,\varepsilon} \, dx
+ \int_U \Delta u^\varepsilon_2 \, \sigma^{2,\varepsilon} \, dx.
\end{equation*}
Thanks to Lemma \ref{wc_lem1}, and repeating the chain of inequalities in \eqref{obs7} one can show that
\begin{equation*}
\left| \int_U \Delta u^\varepsilon_j \, \sigma^{j,\varepsilon} \, dx \right|
\leq \frac{C}{\varepsilon^{1/2}}, \qquad j=1,2,
\end{equation*}
and from this the conclusion follows.
\end{proof}
We can now prove the following result on the speed of convergence.
\begin{Theorem}
\label{wc_thm1}
There exists $C>0$, independent of $\varepsilon$, such that
$$
\|u^\varepsilon_1 - u_1 \|_{L^\infty}, \|u^\varepsilon_2 - u_2 \|_{L^\infty} \le C \varepsilon^{1/2}.
$$
\end{Theorem}
\begin{proof}
The theorem is a direct consequence of Lemma \ref{wc_lem2}.
\end{proof}
\end{section}
\begin{section}{Cell problem for Weakly coupled {systems} of Hamilton--Jacobi equations} \label{secweak}
In this {section we study the following class of weakly coupled systems} of Hamilton--Jacobi equations:
\begin{equation}
\left\{ \begin{aligned}
c_1u_1 - c_1 u_2 + H_1(x,Du_1) &=\overline{H}_1
\\
- c_2 u_1 +c_2 u_2 + H_2(x,Du_2) &=\overline{H}_2 \\
\end{aligned} \right.
\quad \mbox{in}~ \mathbb T^n,
\qquad \qquad \overline{H}_1, \overline{H}_2 \in \mathbb{R},
\label{cell}
\end{equation}
which is the analog of the cell problem for single equation introduced by Lions, Papanicolaou, and Varadhan \cite{LPV}.
We will assume that $H_1, H_2\in C^\infty(\mathbb T^n \times \mathbb R^n )$, and
\begin{itemize}
\item[(H\ref{secweak}.1)] {there exist $\omega_1, \omega_2>0$ such that for every $j=1,2$
$$
\lim_{|p| \to +\infty} \left[ \omega_j |H_j(x,p)|^2+D_x H_j (x,p)\cdot p - 16 n \omega_j c_j^2 |p|^2 \right] =+\infty \
\text{ uniformly in } x\in \mathbb T^n;
$$}
\item[(H\ref{secweak}.2)] $c_1, c_2>0$.
\end{itemize}
It is easy to see that the coefficients of $u_1 , u_2$ in this system do not satisfy
the coupling assumptions of the previous section.
Indeed, as it happens for the cell problem in the context of weak KAM theory,
there is no hope of a uniqueness result for (\ref{cell}).
{
\begin{Remark}
The presence of the term $16 n \omega_j c_j^2$ in condition (H\ref{secweak}.1)
will be justified by later computations.
Nevertheless, we observe that (H\ref{secweak}.1) is weaker than (H\ref{weakcou}.1).
Indeed, if (H\ref{weakcou}.1) holds, then for every $\omega_j >\beta_j$
\begin{align*}
&\lim_{|p| \to +\infty} \left[ \omega_j |H_j(x,p)|^2+D_x H_j (x,p)\cdot p - 16 n \omega_j c_j^2 |p|^2 \right] \\
&= \lim_{|p| \to +\infty} \left[ \beta_j |H_j(x,p)|^2+D_x H_j (x,p) \cdot p + (\omega_j - \beta_j) |H_j(x,p)|^2 -16 n \omega_j c_j^2 |p|^2 \right] = +\infty,
\end{align*}
uniformly in $x$, and hence (H\ref{secweak}.1) is satisfied.
\end{Remark}}
To find the constants
$\overline{H}_1, \overline{H}_2$
we use the same arguments as in \cite{T1}. See also \cite{GomesSto, CGT1}.
First, for every $\varepsilon > 0$, let us consider the following regularized system:
\begin{equation}
\left\{ \begin{aligned}
(c_1+\varepsilon)u^\varepsilon_1 - c_1 u^\varepsilon_2 + H_1(x,Du^\varepsilon_1)
& = \varepsilon^2 \Delta u^\varepsilon_1
\\
(c_2+\varepsilon) u^\varepsilon_2 - c_2 u^\varepsilon_1 + H_2(x,D u^\varepsilon_2 )
&=\varepsilon^2 \Delta u^\varepsilon_2 \\
\end{aligned} \right.
\quad \mbox{in}~ \mathbb T^n.
\label{cell_reg}
\end{equation}
For every $\varepsilon>0$ fixed,
the coefficients of this new system satisfy the coupling assumptions
(H\ref{weakcou}.2) and (H\ref{weakcou}.3) of the previous section.
Thus, (\ref{cell_reg}) admits a unique pair of smooth solutions $(u^\varepsilon_1 , u^\varepsilon_2)$.
In particular, this implies that $u^\varepsilon_1$ and $u^\varepsilon_2$ are $\mathbb T^n$-periodic.
The following result gives some a priori estimates.
\begin{Theorem}
\label{cell_thm1}
There exists $C>0$, independent of $\varepsilon$, such that
$$
\| \varepsilon u^\varepsilon_1 \|_{L^\infty}, \| \varepsilon u^\varepsilon_2 \|_{L^\infty},
\|D u^\varepsilon_1 \|_{L^\infty}, \| D u^\varepsilon_2 \|_{L^\infty} \le C.
$$
\end{Theorem}
\begin{proof}
Our proof is based on the Maximum Principle.
Without loss of generality, we may assume that
\begin{equation*}
\max_{\substack{ j=1,2 \\ x \in \mathbb T^n }} \left\{ \varepsilon u^{\varepsilon}_j (x) \right\} = \varepsilon u^{\varepsilon}_1 (x^\varepsilon_0){,}
\end{equation*}
for some $x^\varepsilon_0 \in \mathbb T^n$.
Applying the Maximum Principle to the first equation of (\ref{cell_reg}),
\begin{equation} \label{ol}
\varepsilon u^\varepsilon_1 (x^\varepsilon_0) \le (c_1+\varepsilon) u^\varepsilon_1 (x^\varepsilon_0)
-c_1u^\varepsilon_2 (x^\varepsilon_0) \le -H^1(x^\varepsilon_0,0) \le C,
\end{equation}
and this shows the existence of a bound from above for $\varepsilon u^{\varepsilon}_1$ and $\varepsilon u^{\varepsilon}_2$.
Using a similar argument one can show that there is also a bound from below, so that
\begin{equation} \label{po}
\|\varepsilon u^\varepsilon_1 \|_{L^\infty}, \|\varepsilon u^\varepsilon_2 \|_{L^\infty} \le C.
\end{equation}
The previous inequality allows us to prove a bound for the difference
$u^\varepsilon_1 (x^\varepsilon_0) - u^\varepsilon_2 (x^\varepsilon_0)$.
Indeed, thanks to {\eqref{ol} and} \eqref{po} we have
\begin{equation} \label{cell_inequ}
| u^\varepsilon_1 (x^\varepsilon_0) - u^\varepsilon_2 (x^\varepsilon_0) |
= u^\varepsilon_1 (x^\varepsilon_0) - u^\varepsilon_2 (x^\varepsilon_0)
\le - \frac{1}{c_1} H_1(x^\varepsilon_0,0) - \frac{\varepsilon}{c_1} u^\varepsilon_1 (x^\varepsilon_0) \le C.
\end{equation}
In order to find a bound for the gradients,
let us set $w^\varepsilon_j = \dfrac{|Du^\varepsilon_j|^2}{2}$, $j=1,2$.
Then, by a direct computation one can see that
\begin{equation*}
\left\{ \begin{aligned}
2(c_1+\varepsilon)w^\varepsilon_1 +D_p H_1 \cdot Dw^\varepsilon_1 - c_1 Du^\varepsilon_1 \cdot Du^\varepsilon_2 + D_x H_1 \cdot Du^\varepsilon_1
&= \varepsilon^2 \Delta w^\varepsilon_1 -\varepsilon^2 |D^2 u^\varepsilon_1|^2
\\
2(c_2+\varepsilon)w^\varepsilon_2 +D_p H_2\cdot Dw^\varepsilon_2 - c_2 Du^\varepsilon_1 \cdot Du^\varepsilon_2 + D_xH_2\cdot Du^\varepsilon_2
&= \varepsilon^2 \Delta w^\varepsilon_2 -\varepsilon^2 |D^2 u^\varepsilon_2|^2
\end{aligned} \right.
\quad \mbox{in}~ \mathbb T^n.
\end{equation*}
Without loss of generality, we may assume that there exists $x^\varepsilon_1 \in \mathbb T^n$ such that
\begin{equation*}
\max_{\substack{ j=1,2 \\ x \in \mathbb T^n }} \left\{ w^{\varepsilon}_j (x) \right\} = w^{\varepsilon}_1 (x^\varepsilon_1).
\end{equation*}
Then, by the Maximum Principle
\begin{equation} \label{kjh} \begin{split}
\varepsilon^2 |D^2u^\varepsilon_1 (x^\varepsilon_1)|^2
& \le -2(c_1+\varepsilon) w^\varepsilon_1 (x^\varepsilon_1)
+ c_1Du^\varepsilon_1 (x^\varepsilon_1)\cdot Du^\varepsilon_2 (x^\varepsilon_1) - D_xH_1\cdot Du^\varepsilon_1 (x^\varepsilon_1) \\
& \le - D_xH_1\cdot Du^\varepsilon_1 (x^\varepsilon_1) .
\end{split} \end{equation}
Moreover, for $\varepsilon$ sufficiently small
\begin{equation} \label{gft}
\varepsilon^2 |D^2u^\varepsilon_1 (x^\varepsilon_1)|^2 \ge2 \omega_1 \varepsilon^4 (\Delta u^\varepsilon_1 (x^\varepsilon_1))^2 =
2 \omega_1 \left[ H_1(x^\varepsilon_1,Du^\varepsilon_1 (x^\varepsilon_1)) + (c_1+\varepsilon)u^\varepsilon_1 (x^\varepsilon_1)-c_1u^\varepsilon_2 (x^\varepsilon_1) \right]^2.
\end{equation}
Also, thanks to \eqref{po} and \eqref{cell_inequ}
\begin{align}
&|(c_1+\varepsilon)u^\varepsilon_1 (x^\varepsilon_1)- c_1 u^\varepsilon_2 (x^\varepsilon_1)| \notag\\
&\hspace{.2cm} \leq \varepsilon |u^\varepsilon_1 (x^\varepsilon_1)|+c_1 |u^\varepsilon_1 (x^\varepsilon_1)- u^\varepsilon_1 (x^\varepsilon_0)|
+c_1 |u^\varepsilon_2 (x^\varepsilon_1) - u^\varepsilon_2 (x^\varepsilon_0)|
+c_1 |u^\varepsilon_1 (x^\varepsilon_0)- u^\varepsilon_2(x^\varepsilon_0)| \notag\\
&\hspace{.2cm}\leq C+c_1|u^\varepsilon_1 (x^\varepsilon_1)- u^\varepsilon_1 (x^\varepsilon_0)|
+c_1 | u^\varepsilon_2 (x^\varepsilon_1) - u^\varepsilon_2 (x^\varepsilon_0)|\notag\\
&\hspace{.2cm}\leq C+ 2c_1 |Du^\varepsilon_1 (x^\varepsilon_1)| |x^\varepsilon_1-x^\varepsilon_0|
\leq C+2c_1\sqrt{n} |Du^\varepsilon_1 (x^\varepsilon_1)|, \notag
\end{align}
where we used the fact that the diameter of $\mathbb T^n$ is $\sqrt{n}$.
{Last relation, thanks to \eqref{gft} and Young's inequality, gives that
\begin{align}
\varepsilon^2 |D^2u^\varepsilon_1 (x^\varepsilon_1)|^2
&\ge 2 \omega_1 \left[ H_1(x^\varepsilon_1,Du^\varepsilon_1 (x^\varepsilon_1)) + (c_1+\varepsilon)u^\varepsilon_1 (x^\varepsilon_1)-c_1u^\varepsilon_2 (x^\varepsilon_1) \right]^2 \notag\\
&\geq \omega_1 |H_1(x^\varepsilon_1,Du^\varepsilon_1 (x^\varepsilon_1)) |^2
- 2 \omega_1 \left[ (c_1+\varepsilon)u^\varepsilon_1 (x^\varepsilon_1)-c_1u^\varepsilon_2 (x^\varepsilon_1) \right]^2 \notag\\
&\geq \omega_1 |H_1(x^\varepsilon_1,Du^\varepsilon_1 (x^\varepsilon_1)) |^2
- 2 \omega_1 \left[ C+2c_1\sqrt{n} |Du^\varepsilon_1 (x^\varepsilon_1)| \right]^2 \notag\\
&\geq \omega_1 |H_1(x^\varepsilon_1,Du^\varepsilon_1 (x^\varepsilon_1)) |^2
-C-16 n \omega_1 c_1^2 |Du^\varepsilon_1 (x^\varepsilon_1)|^2, \notag
\end{align}
Using last inequality and \eqref{kjh} we have
$$
\omega_1 |H_1(x_1,Du^\varepsilon_1 (x^\varepsilon_1)) |^2 +D_xH_1\cdot Du^\varepsilon_1 (x^\varepsilon_1)
- 16 n \omega_1 c_1^2 |Du^\varepsilon_1 (x^\varepsilon_1)|^2 \le C.
$$
Thanks to condition (H\ref{secweak}.1), we obtain the conclusion.}
\end{proof}
In the sequel, all the functions will be regarded as functions defined in the whole $\mathbb{R}^n$
and $\mathbb{Z}^n$-periodic.
Next lemma provides some a priori bounds on $u^\varepsilon_1$ and $u^\varepsilon_2$.
\begin{Lemma}
\label{cell_lem3}
There exists a constant $C>0$, independent of $\varepsilon$, such that
$$
|u^\varepsilon_1 (x) - u^\varepsilon_1 (y)| , | u^\varepsilon_2 (x) - u^\varepsilon_2 (y) |
, |u^\varepsilon_1 (x) - u^\varepsilon_2 (y)| \le C,\quad x,y \in \mathbb R^n.
$$
\end{Lemma}
\begin{proof}
The first two inequalities follow from the periodicity of
$u^\varepsilon_1$ and $u^\varepsilon_2$, and from the fact that
$Du^\varepsilon_1$ and $Du^\varepsilon_2$ are bounded.
Let us now show the last inequality.
{As in the previous proof, without loss of generality} we may assume that
there exists ${x_0^{\varepsilon}} \in \mathbb T^n$ such that
\begin{equation*}
\max_{\substack{ j=1,2 \\ x \in \mathbb T^n }} \left\{ u^{\varepsilon}_j (x) \right\}
= u^{\varepsilon}_1 (x^\varepsilon_0).
\end{equation*}
Combining the second inequality of the lemma with \eqref{ol},
$$
u^\varepsilon_1 (x) - u^\varepsilon_2 (y)
\leq u^\varepsilon_1 (x^\varepsilon_0) - u^\varepsilon_2 (x^\varepsilon_0)
+ u^\varepsilon_2 (x^\varepsilon_0) - u^\varepsilon_2 (y) \le C, \qquad x,y \in \mathbb R^n.
$$
The proof can be concluded by repeating the same argument for
$\min_{\substack{ j=1,2 \\ x \in \mathbb T^n }} \left\{ u^{\varepsilon}_j (x) \right\}$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{cell_const}]
{Thanks to Theorem \ref{cell_thm1} and Lemma \ref{cell_lem3},
\begin{equation} \label{Cbar}
\varepsilon u^\varepsilon_i \rightarrow \overline{C}
\quad \text{ uniformly in }\mathbb{T}^n,
\ \text{for } i=1,2,
\end{equation}
for some constant $\overline{C} \in \mathbb{R}$.
Furthermore, still up to subsequences,
\begin{equation} \label{convergences}
\left\{ \begin{aligned}
u^\varepsilon_1 - \min_{\mathbb T^n} u^\varepsilon_1 &\rightarrow u_1,
\\
u^\varepsilon_2 - \min_{\mathbb T^n} u^\varepsilon_2 &\rightarrow u_2, \\
\end{aligned} \right.
\quad \mbox{ and } \quad
\quad
\left\{ \begin{aligned}
-\varepsilon u^\varepsilon_1+
c_1( \min_{\mathbb T^n} u^\varepsilon_1- \min_{\mathbb T^n} u^\varepsilon_2) &\to \overline{H}_1,
\\
-\varepsilon u^\varepsilon_2+
c_2( \min_{\mathbb T^n} u^\varepsilon_2- \min_{\mathbb T^n} u^\varepsilon_1) &\to \overline{H}_2, \\
\end{aligned} \right.
\end{equation}
uniformly in $\mathbb T^n$, for some functions $ {u_1, u_2} \in C(\mathbb{T}^n)$
and some constants $\overline{H}_1, \overline{H}_2 \in \mathbb{R}$.
From \eqref{convergences} it follows that the functions $(u_1,u_2)$ and the constants $(\overline{H} _1, \overline{H} _2)$
are such that (\ref{cell}) holds, in the viscosity sense.}
\end{proof}
{\begin{Remark} \label{nouniq}
In general, $\overline{H}_1$ and $\overline{H}_2$ are not unique.
Indeed, let $(u_1,u_2)$ be a viscosity solution of (\ref{cell}).
Then, for every pair of constants $(C_1, C_2)$, the pair of functions
$(\widetilde{u}_1,\widetilde{u}_2)$ where
$\widetilde{u}_1 := u_1 +C_1$ and $\widetilde{u}_2 := u_2+C_2$ is still a viscosity solution
of (\ref{cell}), {with}
$$
\widetilde{H}_1 := \overline{H}_1 +c_1 (C_1-C_2), \quad \quad \quad
~\widetilde{H}_2 := \overline{H}_2 +c_2(C_2-C_1),
$$
in place of $\overline{H}_1$ and $\overline{H}_2$, respectively.
Anyway, we have $c_2 \overline{H}_1 + c_1 \overline{H}_2
=c_2 \widetilde{H}_1 + c_1 \widetilde{H}_2$.
This suggests that, although $\overline{H}_1$ and $\overline{H}_2$ may vary,
the expression $c_2 \overline{H}_1 + c_1 \overline{H}_2$ is unique.
Theorem \ref{cell_const2} shows that this is the case.
\end{Remark}}
\begin{proof}[Proof of Theorem \ref{cell_const2}]
Without loss of generality, we may assume $c_1=c_2=1$.
Suppose, by contradiction, that there exist two pairs $(\lambda_1, \lambda_2) \in \mathbb R^2$
and $(\mu_1, \mu_2) \in \mathbb R^2$,
and four functions $u_1, u_2, \widetilde{u}_1, \widetilde{u}_2\in C(\mathbb T^n)$
such that $\lambda_1+\lambda_2<\mu_1+\mu_2$ and
\begin{equation}
\left\{ \begin{aligned}
u_1-u_2 + H_1(x,Du_1) &=\lambda_1
\\
-u_1+u_2 + H_2(x,Du_2) &=\lambda_2 \\
\end{aligned} \right.
\quad \mbox{in}~ \mathbb T^n,
\notag
\end{equation}
and
\begin{equation}
\left\{ \begin{aligned}
\widetilde{u}_1 - \widetilde{u}_2 + H_1(x,D \widetilde{u}_1) &=\mu_1
\\
- \widetilde{u}_1 + \widetilde{u}_2 + H_2(x,D \widetilde{u}_2) &=\mu_2 \\
\end{aligned} \right.
\quad \mbox{in}~ \mathbb T^n.
\notag
\end{equation}
By possibly substituting $u_1$ and $u_2$
with functions $\widehat{u}_1 := u_1 + C_1$ and $\widehat{u}_2 := u_2 + C_2$,
for suitable constants $C_1$ and $C_2$, we may always assume that
$\lambda_1<\mu_1, \lambda_2<\mu_2$.
In the same way, by a further substitution
$\overline{u}_1 := u_1 + C_3$, $\overline{u}_2 := u_2 + C_3$, with $C_3 > 0$ large enough,
we may assume that $u_1 > \widetilde{u}_1, u_2 > \widetilde{u}_2$.
Then, there exists $\varepsilon>0$ small enough such that
\begin{equation}
\left\{ \begin{aligned}
(\varepsilon+1) u_1-u_2 + H_1(x,Du_1) & < (\varepsilon+1) \widetilde{u}_1 - \widetilde{u}_2 + H_1(x,D\widetilde{u}_1)
\\
(\varepsilon+1) u_2 - u_1 + H_2 (x,Du_2) & < (\varepsilon+1)\widetilde{u}_2-\widetilde{u}_1+ H_2(x,D\widetilde{u}_2) \\
\end{aligned} \right.
\quad \mbox{in}~ \mathbb T^n.
\notag
\end{equation}
Observe that the coefficients of the last system satisfy the coupling assumptions
(H\ref{weakcou}.2) and (H\ref{weakcou}.3).
Hence, applying the Comparison Principle in \cite{EL} and \cite{IK1},
we conclude that $u_1 < \widetilde{u}_1$ and $u_2 < \widetilde{u}_2$, which gives a contradiction.
\end{proof}
{\begin{Remark} \label{Hbar}
Multiplying the two convergences in the right in \eqref{convergences} by $c_2$ and $c_1$,
respectively, one can see that
$$
- \overline{C} = \overline{H}= \dfrac{\mu}{c_1+c_2}.
$$
Here, $\overline{C}$ is defined in \eqref{Cbar}, $\mu$ given by Theorem \ref{cell_const2},
and $\overline{H}$ is the unique constant such that \eqref{falsj} has viscosity solutions.
We call $\overline{H}$ the \textit{effective Hamiltonian} of
the cell problem for the weakly coupled system of Hamilton--Jacobi equations.
\end{Remark}}
The following is the main theorem of the section.
See also \cite{T1} for similar results.
\begin{Theorem}
\label{cell_thm2}
There exists a constant $C>0$, independent of $\varepsilon$, such that
$$
\|\varepsilon u^\varepsilon_i + \overline{H} \|_{L^\infty} \le C \varepsilon,
\quad \text{for } i=1,2.
$$
\end{Theorem}
{\bf Adjoint method:} Also in this case, we introduce the adjoint equations
associated to the linearization of the original problem.
We look for $\sigma^{1,\varepsilon}, \sigma^{2,\varepsilon}$
which are $\mathbb{T}^n$-periodic and such that
\begin{equation}\label{cell_adj1}
\left\{ \begin{aligned}
- \mbox{div}(D_pH_1 \sigma^{1,\varepsilon}) + ( c_{1} + \varepsilon ) \sigma^{1,\varepsilon} - c_{2} \sigma^{2,\varepsilon}
&= \varepsilon^2 \Delta \sigma^{1,\varepsilon}+ \varepsilon (2 - i) \delta_{x_0} \quad &\mbox{in}~ \mathbb T^n,
\\
- \mbox{div}(D_pH_2 \sigma^{2,\varepsilon}) + ( c_{2} + \varepsilon ) \sigma^{2,\varepsilon} - c_{1} \sigma^{1,\varepsilon}
&= \varepsilon^2 \Delta \sigma^{2,\varepsilon}+ \varepsilon (i - 1) \delta_{x_0} \quad &\mbox{in}~ \mathbb T^n,
\end{aligned} \right.
\end{equation}
where $i \in \{ 1, 2 \} $ and $x_0 \in \mathbb T^n$ will be chosen later.
The argument used in Section \ref{sectobs} gives also in this case
existence and uniqueness for $\sigma^{1,\varepsilon}$ and $\sigma^{2,\varepsilon}$.
As before, we also have $\sigma^{1,\varepsilon}, \sigma^{2,\varepsilon} \in C^{\infty} (\mathbb T^n \setminus \{ x_0 \})$.
{The next two lemmas can be proven
as Lemma \ref{kiz} and Lemma \ref{wc_lem1}, respectively.}
\begin{Lemma}[Properties of $\sigma^{1,\varepsilon}, \sigma^{2,\varepsilon}$] \label{kir}
The functions $\sigma^{1,\varepsilon}, \sigma^{2,\varepsilon}$ satisfy the following:
\begin{itemize}
\item[(i)] $\sigma^{j,\varepsilon} \ge 0$ on $\mathbb{T}^n$ \,\, $(j=1,2)$;
\item[(ii)] Moreover, the following equality holds:
$$
\sum_{j=1}^2 \int_{\mathbb{T}^n} \sigma^{j,\varepsilon} \,dx = 1.
$$
\end{itemize}
\end{Lemma}
\begin{Lemma}
\label{cell_lem1}
There exists a constant $C>0$, independent of $\varepsilon$, such that
$$
\varepsilon^2\int_{\mathbb R^n} |D^2 u^\varepsilon_1|^2 \sigma^{1,\varepsilon} \,dx \le C,
$$
$$
\varepsilon^2 \int_{\mathbb R^n} |D^2 u^\varepsilon_2|^2 \sigma^{2,\varepsilon} \,dx \le C.
$$
\end{Lemma}
Finally, next lemma allows us to prove Theorem \ref{cell_thm2}.
\begin{Lemma}
\label{cell_lem2}
There exists a constant $C>0$, independent of $\varepsilon$, such that
$$
\max_{\mathbb T^n} |(\varepsilon u^\varepsilon_1)_\varepsilon|,
\quad \quad \max_{\mathbb T^n} |(\varepsilon u^\varepsilon_2)_\varepsilon| \le C.
$$
\end{Lemma}
\begin{proof}
Differentiating (\ref{cell_reg}) w.r.t. $\varepsilon$,
\begin{equation}
\left\{ \begin{aligned}
D_pH_1\cdot Du^\varepsilon_{1,\varepsilon} + (c_1+\varepsilon) u^\varepsilon_{1,\varepsilon}
+ u^\varepsilon_1 - c_1 u^\varepsilon_{2,\varepsilon}
&=\varepsilon^2 \Delta u^\varepsilon_{1,\varepsilon}
+ 2 \varepsilon \Delta u^\varepsilon_1,
\\
D_pH_2\cdot Du^\varepsilon_{2,\varepsilon} + (c_2+\varepsilon) u^\varepsilon_{2,\varepsilon}
+ u^\varepsilon_2 - c_2 u^\varepsilon_{1,\varepsilon}
&= \varepsilon^2 \Delta u^\varepsilon_{2,\varepsilon} + 2\varepsilon \Delta u^\varepsilon_2, \\
\end{aligned} \right.
\notag
\end{equation}
where we set $u^\varepsilon_{j,\varepsilon} := \partial u^\varepsilon_{j} / \partial \varepsilon$, $j=1,2$.
Without loss of generality, we may assume that there exists $x_2\in {\mathbb T^n}$ such that
\begin{equation*}
\max_{\substack{ j=1,2 \\ x \in \mathbb T^n }} ( \varepsilon u^{\varepsilon}_j (x))_{\varepsilon}
= \max_{\substack{ j=1,2 \\ x \in \mathbb T^n }}
\left\{ \varepsilon u^{\varepsilon}_{j, \varepsilon} (x) + u^{\varepsilon}_{j} (x) \right\}
= \varepsilon u^\varepsilon_{1,\varepsilon} (x_2)
+ u^{\varepsilon}_{1} (x_2).
\end{equation*}
Choosing $x_0=x_2$ in the adjoint equation (\ref{cell_adj1}),
and repeating the {steps in the proof of} Theorem~\ref{obs_thm1}, we get
\begin{equation}
\label{cell2} \begin{split}
&\varepsilon u^\varepsilon_{1,\varepsilon} (x_2)
+\int_{\mathbb T^n} u^\varepsilon_1 \sigma^{1,\varepsilon}\,dx
+ \int_{\mathbb T^n} u^\varepsilon_2 \sigma^{2,\varepsilon}\,dx \\
&\hspace{.2cm}\le 2\varepsilon \int_{\mathbb T^n} |\Delta
u^\varepsilon_1| \sigma^{1,\varepsilon}\,dx
+ 2\varepsilon \int_{\mathbb T^n} |\Delta
u^\varepsilon_2| \sigma^{2,\varepsilon}\,dx\le C,
\end{split} \end{equation}
where the latter inequality follows by repeating the chain of inequalities in \eqref{obs7}
and thanks to Lemma \ref{cell_lem1}.
Using Lemma \ref{cell_lem3} and property (ii) of Lemma \ref{kir} we have
\begin{equation*} \begin{split}
&\left| \int_{\mathbb T^n} u^\varepsilon_1 (x) \sigma^{1,\varepsilon}\,dx
+ \int_{\mathbb T^n} u^\varepsilon_2 (x) \sigma^{2,\varepsilon}\,dx
- u^{\varepsilon}_{1} (x_2) \right| \\
&\hspace{.5cm}= \left| \int_{\mathbb T^n} ( u^\varepsilon_1 (x) - u^{\varepsilon}_{1} (x_2) ) \sigma^{1,\varepsilon}\,dx
+ \int_{\mathbb T^n} ( u^\varepsilon_2 (x) - u^{\varepsilon}_{1} (x_2) ) \sigma^{2,\varepsilon}\,dx \right| \leq C.
\end{split} \end{equation*}
In view of the previous inequality, \eqref{cell2} becomes
\begin{equation*}
\varepsilon u^\varepsilon_{1,\varepsilon} (x_2)
+ u^{\varepsilon}_{1} (x_2) \leq C,
\end{equation*}
thus giving the bound from above.
The same argument, applied to $\min_{\substack{ j=1,2 \\ x \in \mathbb T^n }} ( \varepsilon u^{\varepsilon}_j (x))_{\varepsilon}$,
allows to prove the bound from below.
\end{proof}
\begin{proof}[Proof of Theorem \ref{cell_thm2}]
The theorem immediately follows by using Lemma \ref{cell_lem2}.
\end{proof}
\begin{Remark}
(i). In order to achieve existence and uniqueness of the effective Hamiltonian $\overline{H}$
one can require either (H4.1) or the usual coercive assumption (i.e.
$H_1, H_2$ are uniformly coercive in $p$).
Indeed one can consider the regularized system
\begin{equation}
\left\{ \begin{aligned}
(c_1+\varepsilon)u^\varepsilon_1 - c_1 u^\varepsilon_2 + H_1(x,Du^\varepsilon_1)
& = 0
\\
(c_2+\varepsilon) u^\varepsilon_2 - c_2 u^\varepsilon_1 + H_2(x,D u^\varepsilon_2 )
&=0 \\
\end{aligned} \right.
\quad \mbox{in}~ \mathbb T^n,
\label{cell_rem}
\end{equation}
and derive the results similarly to what we did above by using the coercivity of $H_1, H_2$.
We require {\rm (H4.1)} in order to get
the speed of convergence as in Theorem \ref{cell_thm2}. \\
(ii). By using the same arguments, we can show that for any $P \in \mathbb R^n$,
there exist a pair of constants $(\overline{H}_1(P), \overline{H}_2(P))$ such that the system
\begin{equation*}
\left\{ \begin{aligned}
c_1u_1 - c_1 u_2 + H_1(x,P+Du_1)
& = \overline{H}_1(P)
\\
c_2 u_2 - c_2 u_1 + H_2(x,P+D u_2 )
&=\overline{H}_2(P) \\
\end{aligned} \right.
\quad \mbox{in}~ \mathbb T^n,
\end{equation*}
admits a solution $(u_1(\cdot,P),u_2(\cdot,P)) \in C(\mathbb T^n)^2$.
Moreover $\overline{H}(P)$, the effective Hamiltonian, is unique and
$$
\overline{H}(P)=\dfrac{c_2\overline{H}_1(P)+c_1\overline{H}_2(P)}{c_1+c_2}.
$$
\end{Remark}
\end{section}
\begin{section}{weakly coupled systems of obstacle type} \label{kilo}
In this last section we apply the Adjoint Method
to weakly coupled systems of obstacle type.
Let $H_1, H_2: \overline{U} \times \mathbb{R}^n \to \mathbb{R}$ be smooth Hamiltonians,
and let $\psi_1, \psi_2 : \overline{U} \to \mathbb{R}$ be smooth functions describing the obstacles.
We assume that there exists $\alpha > 0$ such that
\begin{equation} \label{strpos}
\psi_1, \, \, \psi_2 \geq \alpha \quad \text{ in }\overline{U},
\end{equation}
and consider the system
\begin{equation} \label{ki}
\left\{ \begin{aligned}
\max\{ u_1 - u_2 - \psi_1 , u_1+H_1(x,Du_1) \}&=0 \quad\mbox{in}~ U,
\\
\max\{ u_2 - u_1 - \psi_2 , u_2 +H_2(x, D u_2 ) \}&=0 \quad\mbox{in}~ U, \\
\end{aligned} \right.
\end{equation}
with boundary conditions $u_1 \mid_{\partial U}= u_2 \mid_{\partial U}= 0$.
We observe that \eqref{strpos} guarantees the compatibility of the boundary conditions,
since $ \psi_1, \psi_2 > 0$ on $\partial U$.
Although the two equations in \eqref{ki} are coupled just through the difference $u_1 - u_2$,
this problem turns out to be more difficult that the correspondent scalar equation \eqref{obs_eqn} studied in Section \ref{sectobs}.
For this reason, the hypotheses we require now are stronger.
We assume that
\begin{itemize}
\item[(H\ref{kilo}.1)] $H_j (x, \cdot)$ is convex for every $x \in \overline{U}$, $j=1,2$.
\item[(H\ref{kilo}.2)] Superlinearity in $p$:
\begin{equation*}
\lim_{|p| \to \infty} \frac{H_j (x,p)}{|p|} = + \infty \quad \quad \text{ uniformly in }x, \quad j=1,2.
\end{equation*}
\item[(H\ref{kilo}.3)] $|D_x H_j (x,p)| \leq C$ for each $(x,p) \in \overline{U} \times \mathbb{R}^n$, $j=1,2$.
\item[(H\ref{kilo}.4)] There exist $\Phi_1, \Phi_2 \in C^2 (U) \cap C^1 (\overline U)$
with $\Phi_j =0$ on $\partial U$ ($j=1,2$), $- \psi_2 \leq \Phi_1 - \Phi_2 \leq \psi_1$, and such that
$$
\Phi_j +H_j (x,D\Phi_j) < 0\quad \mbox{in}~ \overline{U} \quad (j=1,2).
$$
\end{itemize}
Let $\varepsilon > 0$ and let $\gamma^{\varepsilon}: \mathbb{R} \to [0, + \infty)$
be the function defined by \eqref{gammaep}.
We make in this section the additional assumption that $\gamma$ is convex.
We approximate \eqref{ki} by the following system
\begin{equation} \label{tg}
\left\{ \begin{aligned}
u^\varepsilon_1 +H_1 (x,Du^\varepsilon_1 )
+\gamma^\varepsilon(u^\varepsilon_1 - u^{\varepsilon}_2 - \psi_1 )
= \varepsilon \Delta u^\varepsilon_1 \quad\mbox{in}~U,
\\
u_2^\varepsilon+H_2 (x,D u_2^\varepsilon)
+\gamma^\varepsilon(u_2^\varepsilon- u_1^{\varepsilon} - \psi_2 )
= \varepsilon \Delta u_2^\varepsilon \quad\mbox{in}~U.
\end{aligned} \right.
\end{equation}
We are now ready to state the main result of the section.
\begin{Theorem}
\label{obs_speedfin}
There exists a positive constant $C$, independent of
$\varepsilon$, such that
\begin{equation}
\|u^\varepsilon_i - u_i\|_{L^\infty} \le C \varepsilon^{1/2},
\quad \text{for } i=1,2.
\notag
\end{equation}
\end{Theorem}
In order to prove the theorem we need several lemmas.
In the sequel, we shall use the notation
\begin{equation*}
\theta^\varepsilon_1:= u^\varepsilon_1- u_2^{\varepsilon} - \psi_1 ,
\quad \quad
\theta^\varepsilon_2:= u_2^\varepsilon- u_1^{\varepsilon} - \psi_2 .
\end{equation*}
The linearized operator corresponding to \eqref{tg} is
\begin{equation*}
L^{\varepsilon} (z_1,z_2) : =
\left\{ \begin{aligned}
z_1 + D_p H_1 (x,Du_1^\varepsilon) \cdot D z_1
+ (\gamma^\varepsilon)' \mid_{\theta^\varepsilon_1} (z_1 - z_2)
- \varepsilon \Delta z_1 ,
\\
z_2 + D_p H_2 (x,Du_2^\varepsilon) \cdot D z_2
+ (\gamma^\varepsilon)' \mid_{\theta^\varepsilon_2} (z_2 - z_1)
- \varepsilon \Delta z_2.
\end{aligned} \right.
\end{equation*}
Then, the adjoint equations are:
\begin{equation} \label{rfd}
\left\{ \begin{aligned}
(1 + (\gamma^\varepsilon)' \mid_{\theta^\varepsilon_1})
\sigma^{1,\varepsilon}- \mbox{div}(D_pH_1 \sigma^{1,\varepsilon})
- (\gamma^\varepsilon)' \mid_{\theta^\varepsilon_2} \sigma^{2,\varepsilon}
&= \varepsilon \Delta
\sigma^{1,\varepsilon}+ (2 - i) \delta_{x_0} \quad &\mbox{in}~ U,
\\
(1 + (\gamma^\varepsilon)' \mid_{\theta^\varepsilon_2})
\sigma^{2,\varepsilon}- \mbox{div}(D_pH_2 \sigma^{2,\varepsilon})
- (\gamma^\varepsilon)' \mid_{\theta^\varepsilon_1} \sigma^{1,\varepsilon}
&= \varepsilon \Delta
\sigma^{2,\varepsilon}+ (i - 1) \delta_{x_0} \quad &\mbox{in}~ U,
\end{aligned} \right.
\end{equation}
with boundary conditions
\begin{equation*}
\left\{ \begin{aligned}
\sigma^{1,\varepsilon}&=0 \qquad \quad&\mbox{on}~\partial U,
\\
\sigma^{2,\varepsilon}&=0 \qquad \quad&\mbox{on}~\partial U ,\\
\end{aligned} \right.
\end{equation*}
where $i \in \{ 1, 2 \} $ and $x_0 \in U$ will be chosen later.
By repeating what was done in Section \ref{sectobs},
we get the existence and uniqueness of $\sigma^{1,\varepsilon}$ and $\sigma^{2,\varepsilon}$ by Fredholm alternative.
Furthermore, $\sigma^{1,\varepsilon}$ and $\sigma^{2,\varepsilon}$ are well defined
and $\sigma^{1,\varepsilon}, \sigma^{2,\varepsilon} \in C^{\infty} (U \setminus \{ x_0 \})$.
In order to derive further properties of $\sigma^{1,\varepsilon}$ and $\sigma^{2,\varepsilon}$,
we need the following useful formulas.
\begin{Lemma}
For every $\varphi_1, \varphi_2 \in C^2 (\overline{U})$ we have
\begin{equation} \label{gbh} \begin{split}
(2 - i) \varphi_1 (x_0)
&= - \varepsilon \int_{\partial U} \frac{\partial \sigma^{1,\varepsilon} }{\partial \nu} \varphi_1 \, dS
- \int_U (\gamma^\varepsilon)' \mid_{\theta^\varepsilon_2} \varphi_1 \, \sigma^{2,\varepsilon} \, dx \\
& + \int_U \left[ (1 + (\gamma^\varepsilon)' \mid_{\theta^\varepsilon_1}) \, \varphi_1
+ D_pH_1 \cdot D \varphi_1 - \varepsilon \Delta \varphi_1 \right] \, \sigma^{1,\varepsilon} \, dx,
\end{split} \end{equation}
and
\begin{equation} \label{gbh2} \begin{split}
( i - 1 ) \varphi_2 (x_0)
&= - \varepsilon \int_{\partial U} \frac{\partial \sigma^{2,\varepsilon} }{\partial \nu} \varphi_2 \, dS
- \int_U (\gamma^\varepsilon)' \mid_{\theta^\varepsilon_1} \varphi_2 \, \sigma^{1,\varepsilon} \, dx \\
& + \int_U \left[ (1 + (\gamma^\varepsilon)' \mid_{\theta^\varepsilon_2}) \, \varphi_2
+ D_pH_2 \cdot D \varphi_2 - \varepsilon \Delta \varphi_2 \right] \, \sigma^{2,\varepsilon} \, dx,
\end{split} \end{equation}
where $\nu$ is the outer unit normal to $\partial U$.
\end{Lemma}
\begin{proof}
The conclusion follows by simply multiplying by $\varphi_j$ $(j=1,2)$
the two equations in \eqref{rfd} and integrating by parts.
\end{proof}
{From the previous lemma, the analogous of Lemma \ref{obs_lem2} follows}.
\begin{Lemma}[Properties of $\sigma^{1,\varepsilon}, \sigma^{2,\varepsilon}$] \label{kiUJ}
Let $\nu$ be the outer unit normal to $\partial U$. Then
\begin{itemize}
\item[(i)] $\sigma^{j,\varepsilon} \ge 0$ on $\overline{U}$.
In particular, $\dfrac{\partial \sigma^{j,\varepsilon}}{\partial \nu} \le 0$ on $\partial U \,\, (j=1,2)$.
\item[(ii)] The following equality holds:
\begin{equation}
\sum_{j=1}^2 \left( \int_U \sigma^{j,\varepsilon} \,dx
- \varepsilon \int_{\partial U} \dfrac{\partial \sigma^{j,\varepsilon}}{\partial \nu} \,dS \right) = 1 .
\notag
\end{equation}
In particular,
$$
\sum_{j=1}^2 \int_U \sigma^{j,\varepsilon} \,dx \le 1.
$$
\end{itemize}
\end{Lemma}
We are now able to prove a uniform bound on $u^{\varepsilon}_1$ and $u^{\varepsilon}_2$.
{The proof is skipped, since it is analogous to those of the previous sections.}
\begin{Lemma} \label{bd}
There exists a positive constant $C$, independent of $\varepsilon$, such that
\begin{equation*}
\| u^{\varepsilon}_1 \|_{L^{\infty}}, \| u_2^{\varepsilon} \|_{L^{\infty}} \leq C.
\end{equation*}
\end{Lemma}
Next lemma will be used to give a uniform bound for $D u^{\varepsilon}_1$ and $D u^{\varepsilon}_2$.
\begin{Lemma} \label{gamprimeb}
We have
\begin{equation*}
\int_U ( \gamma^\varepsilon )' \mid_{\theta^1_{\varepsilon}} \, \sigma^{1,\varepsilon} \, dx
+ \int_U ( \gamma^\varepsilon )' \mid_{\theta^2_{\varepsilon}} \, \sigma^{2,\varepsilon} \, dx
\leq C,
\end{equation*}
where $C$ is a positive constant independent of $\varepsilon$.
\end{Lemma}
\begin{proof}
First of all, observe that condition (H\ref{kilo}.1) implies that
\begin{equation} \label{conv12}
H_j (x,p) - D_p H_j (x,p) \cdot p \leq H_j (x,0), \qquad \text{ for every }(x,p) \in \overline{U} \times \mathbb{R}^n, \qquad j=1,2.
\end{equation}
In the same way, the convexity of $\gamma$ implies
\begin{equation} \label{convgam}
\gamma^{\varepsilon} (s) - \left[ (\gamma^{\varepsilon})' (s) \right] s
= \gamma \left( \frac{s}{\varepsilon} \right) -
\left[ (\gamma') \left( \frac{s}{\varepsilon} \right) \right] \frac{s}{\varepsilon} \leq \gamma (0) = 0.
\end{equation}
Equation \eqref{tg}$_1$ gives
\begin{equation*} \begin{split}
0&=u^\varepsilon_1 + H_1 (x,Du_1^\varepsilon)
+\gamma^\varepsilon \mid_{\theta^1_{\varepsilon}} - \varepsilon \Delta u_1^\varepsilon \\
&= u^\varepsilon_1 + D_p H_1 (x,Du^\varepsilon_1) \cdot D u^{\varepsilon}_1
- \varepsilon \Delta u^\varepsilon_1 +H_1 (x,Du^\varepsilon_1) - D_p H_1 (x,Du^\varepsilon_1) \cdot D u^{\varepsilon}_1 \\
& \hspace{.3cm} +\gamma^\varepsilon \mid_{\theta^1_{\varepsilon}}
- ( \gamma^\varepsilon )' \mid_{\theta^1_{\varepsilon}} \theta^1_{\varepsilon}
+ ( \gamma^\varepsilon )' \mid_{\theta^1_{\varepsilon}} (u^{\varepsilon}_1 - u_2^{\varepsilon}) - ( \gamma^\varepsilon )' \mid_{\theta^1_{\varepsilon}} \psi_1.
\end{split} \end{equation*}
Multiplying last relation by $\sigma^{1,\varepsilon}$, integrating and using \eqref{conv12} and \eqref{convgam}
\begin{equation*} \begin{split}
&\int_U ( \gamma^\varepsilon )' \mid_{\theta^1_{\varepsilon}} \psi_1 \, \sigma^{1,\varepsilon} \, dx
= \int_U \left[ H_1 (x,Du^\varepsilon_1) - D_p H_1 (x,Du^\varepsilon_1) \cdot D u^{\varepsilon}_1 \right] \, \sigma^{1,\varepsilon} \, dx \\
&+ \int_U \left[ \gamma^\varepsilon \mid_{\theta^1_{\varepsilon}}
- ( \gamma^\varepsilon )' \mid_{\theta^1_{\varepsilon}} \theta^1_{\varepsilon}
\right] \, \sigma^{1,\varepsilon} \, dx \\
&+ \int_U \left[ \left( 1 + ( \gamma^\varepsilon )' \mid_{\theta^1_{\varepsilon}} \right) u^\varepsilon_1
+ D_p H_1 (x,Du^\varepsilon_1) \cdot D u^{\varepsilon}_1
- \varepsilon \Delta u^\varepsilon_1
- ( \gamma^\varepsilon )' \mid_{\theta^1_{\varepsilon}} u_2^{\varepsilon}
\right] \, \sigma^{1,\varepsilon} \, dx \\
&\leq \int_U H_1 (x,0) \, \sigma^{1,\varepsilon} \, dx \\
&+\int_U \left[ \left( 1 + ( \gamma^\varepsilon )' \mid_{\theta^1_{\varepsilon}} \right) u^\varepsilon_1
+ D_p H_1 (x,Du^\varepsilon_1) \cdot D u^{\varepsilon}_1
- \varepsilon \Delta u^\varepsilon_1 - ( \gamma^\varepsilon )' \mid_{\theta^1_{\varepsilon}} u_2^{\varepsilon}
\right] \, \sigma^{1,\varepsilon} \, dx.
\end{split} \end{equation*}
Analogously,
\begin{equation*} \begin{split}
&\int_U ( \gamma^\varepsilon )' \mid_{\theta^2_{\varepsilon}} \psi_2 \, \sigma^{2,\varepsilon} \, dx
\leq \int_U H_2 (x,0) \,
\sigma^{2,\varepsilon} \, dx \\
&\hspace{1cm}+ \int_U \left[ \left( 1 + ( \gamma^\varepsilon )' \mid_{\theta^2_{\varepsilon}} \right)
u_2^\varepsilon
+ D_p H_2 (x,Du_2^\varepsilon) \cdot D u_2^{\varepsilon}
- \varepsilon \Delta u_2^\varepsilon - ( \gamma^\varepsilon )' \mid_{\theta^2_{\varepsilon}} u_2^{\varepsilon}
\right] \, \sigma^{2,\varepsilon} \, dx.
\end{split} \end{equation*}
Summing up the last two relations and using \eqref{gbh} and \eqref{gbh2}
\begin{equation*} \begin{split}
& \int_U ( \gamma^\varepsilon )' \mid_{\theta^1_{\varepsilon}} \psi_1 \, \sigma^{1,\varepsilon} \, dx
+ \int_U ( \gamma^\varepsilon )' \mid_{\theta^2_{\varepsilon}} \psi_2 \, \sigma^{2,\varepsilon} \, dx
\leq (2 - i) u^{\varepsilon}_1 (x_0) + (i-1) u^{\varepsilon}_2 (x_0) \\
&\hspace{.3cm}+ \| H_1 (\cdot ,0)\|_{L^{\infty}} \int_U \, \sigma^{1,\varepsilon} \, dx
+ \| H_2 (\cdot ,0)\|_{L^{\infty}} \int_U \, \sigma^{2,\varepsilon} \, dx.
\end{split} \end{equation*}
Thus,
\begin{equation*} \begin{split}
& \int_U ( \gamma^\varepsilon )' \mid_{\theta^1_{\varepsilon}} \, \sigma^{1,\varepsilon} \, dx
+ \int_U ( \gamma^\varepsilon )' \mid_{\theta^2_{\varepsilon}} \, \sigma^{2,\varepsilon} \, dx
\leq \frac{2-i}{\alpha} u_1^{\varepsilon} (x_0) + \frac{i-1}{\alpha} u_2^{\varepsilon} (x_0) \\
&\hspace{.3cm}+ \frac{ \| H_1 (\cdot ,0)\|_{L^{\infty}} }{\alpha} \int_U \, \sigma^{1,\varepsilon} \, dx
+ \frac{ \| H_2 (\cdot ,0)\|_{L^{\infty}} }{\alpha} \int_U \, \sigma^{2,\varepsilon} \, dx \leq C ,
\end{split} \end{equation*}
where we used \eqref{strpos}, Lemma \ref{kiUJ} and Lemma \ref{bd}.
\end{proof}
We can finally show the existence of a uniform bound for the gradients
of $u^{\varepsilon}_1$ and $u^{\varepsilon}_2$.
\begin{Lemma} \label{boundgrad}
There exists a positive constant $C$, independent of $\varepsilon$, such that
\begin{equation*}
\| D u^{\varepsilon}_1 \|_{L^{\infty}}, \, \, \| D u^{\varepsilon}_2 \|_{L^{\infty}} \leq C.
\end{equation*}
\end{Lemma}
\begin{proof}
\textbf{Step I: Bound on $\partial U$.}
As it was done in Section \ref{sectobs}, we are going to construct appropriate barriers.
For $\varepsilon$ small enough, assumption (H\ref{kilo}.4) implies that
\begin{equation*}
\left\{ \begin{aligned}
\Phi_1 +H_1 (x,D\Phi_1 )
+\gamma^\varepsilon(\Phi_1 - \Phi_2 - \psi_1 )
< \varepsilon \Delta \Phi_1 \quad\mbox{in}~U,
\\
\Phi_2 +H_2 (x,D \Phi_2 )
+\gamma^\varepsilon(\Phi_2 - \Phi_1 - \psi_2 )
< \varepsilon \Delta \Phi_2 \quad\mbox{in}~U,
\end{aligned} \right.
\end{equation*}
and $\Phi_1 = \Phi_2 =0$ on $\partial U$.
Therefore, $(\Phi_1,\Phi_2)$ is a sub-solution of \eqref{tg}.
By the comparison principle, $u^\varepsilon_j \ge \Phi_j$ in~$U$, $j=1,2$.
Let $d(x)$, $\delta$, and $U_{\delta}$ be as in the proof of Proposition \ref{obs1}.
For $ \mu > 0 $ large enough, the uniform bounds of $\|u^\varepsilon_1\|_{L^\infty}$ and $\|u^\varepsilon_2\|_{L^\infty}$ yield
$v := \mu d \ge u^\varepsilon_j$ on $\partial U_\delta $, $j=1,2$, so that
\begin{equation*}
\left\{ \begin{aligned}
v +H_1(x,Dv)+\gamma^\varepsilon( v - v - \psi_1 ) - \varepsilon \Delta v
= v+H_1(x,Dv) - \varepsilon \Delta v
\ge H_1(x,\mu Dd) - \mu C \quad\mbox{in}~U,
\\
v +H_2(x,Dv)+\gamma^\varepsilon( v - v - \psi_2 ) - \varepsilon \Delta v
= v +H_2(x,Dv) - \varepsilon \Delta v
\ge H_2(x,\mu Dd) - \mu C \quad\mbox{in}~U.
\\
\end{aligned} \right.
\end{equation*}
Now, we have $\Phi_j= u^\varepsilon_j= v = 0$ on $\partial U$.
Also, thanks to assumption (H\ref{kilo}.2), for $\mu >0$ large enough
\begin{equation*}
\left\{ \begin{aligned}
v +H_1(x,Dv)+\gamma^\varepsilon(v- v - \psi_1 ) - \varepsilon \Delta v \ge 0 \quad\mbox{in}~U,
\\
v +H_2(x,Dv)+\gamma^\varepsilon(v - v - \psi_2 ) - \varepsilon \Delta v \ge 0 \quad\mbox{in}~U,
\\
\end{aligned} \right.
\end{equation*}
that is, the pair $(v,v)$ is a super-solution for the system \eqref{tg}.
Thus, the comparison principle gives us that $\Phi_j \le u^\varepsilon_j \le v_j$ in $ U_\delta$.
Then, from the fact that $\Phi_j= u^\varepsilon_j= v = 0$ on $\partial U$ we get
$$
\dfrac{\partial v}{\partial \nu}(x) \le \dfrac{\partial u^\varepsilon_j}{\partial \nu}(x) \le \dfrac{\partial \Phi_j}{\partial \nu}(x),\quad \text{for } x\in \partial U.
$$
Hence, we obtain $\| Du^{\varepsilon}_j \|_{L^{\infty}(\partial U)} \leq C$, $j=1,2$.
\textbf{Step II: Bound on $U$.}
Assume now that there exists $\widehat{x} \in U$ such that
\begin{equation*}
\max_{\substack{ j=1,2 \\ x \in \overline{U} }} w^{\varepsilon}_j (x) = w^{\varepsilon}_1 ( \widehat{x} ),
\hspace{1.5cm} \text{ where }
w^{\varepsilon}_j (x):= \frac{1}{2} | D u^{\varepsilon}_j |^2, \quad j=1,2.
\end{equation*}
By a direct computation one can see that
\begin{equation*}
2(1+(\gamma^\varepsilon) ' \mid_{\theta^{\varepsilon}_1})w^\varepsilon_1 + D_p H_1 \cdot D w^\varepsilon_1
+ D_xH_1 \cdot Du^\varepsilon_1 -(\gamma^\varepsilon)' \mid_{\theta^{\varepsilon}_1} Du^\varepsilon_1 \cdot ( D\psi_1 + D u^{\varepsilon}_2 )
= \varepsilon \Delta w^\varepsilon_1 - \varepsilon |D^2 u^\varepsilon_1|^2.
\end{equation*}
Multiplying last relation by $\sigma^{1,\varepsilon}$ and integrating over $U$
\begin{equation} \label{obs25}
\begin{split}
&2 \int_U w^\varepsilon_1 \sigma^{1,\varepsilon} \, dx
+ \int_U D_p H_1 \cdot D w^\varepsilon_1 \sigma^{1,\varepsilon} \, dx
- \varepsilon \int_U \Delta w^\varepsilon_1\sigma^{1,\varepsilon} \, dx
+ \int_U \varepsilon |D^2 u^\varepsilon_1|^2 \sigma^{1,\varepsilon} \, dx \\
& + \int_U D_x H_1 \cdot Du^\varepsilon_1 \sigma^{1,\varepsilon} \, dx
+ \frac{1}{2} \int_U (\gamma^\varepsilon)' \mid_{\theta^{\varepsilon}_1}
\left[ | Du^\varepsilon_1 |^2 + | Du^\varepsilon_1 - Du^\varepsilon_2 |^2
- | Du^\varepsilon_2 |^2 \right] \, \sigma^{1,\varepsilon} \, dx \\
&- \int_U (\gamma^\varepsilon)' \mid_{\theta^{\varepsilon}_1} Du^\varepsilon_1 \cdot D\psi_1 \sigma^{1,\varepsilon} \, dx
= 0.
\end{split} \end{equation}
Then, using equation \eqref{gbh}$_1$ with $i=1$ and $x_0 = \widehat{x}$
\begin{equation} \label{obs26}
\begin{split}
&\int_U w^\varepsilon_1 \sigma^{1,\varepsilon} \, dx
+ \int_U \varepsilon |D^2 u^\varepsilon_1|^2 \sigma^{1,\varepsilon} \, dx
- \int_U (\gamma^\varepsilon)' \mid_{\theta^{\varepsilon}_1} Du^\varepsilon_1 \cdot D\psi_1 \sigma^{1,\varepsilon} \, dx \\
& + \int_U D_x H_1 \cdot Du^\varepsilon_1 \sigma^{1,\varepsilon} \, dx
+ \frac{1}{2} \int_U (\gamma^\varepsilon)' \mid_{\theta^{\varepsilon}_1}
\left[ | Du^\varepsilon_1 - Du^\varepsilon_2 |^2
- | Du^\varepsilon_2 |^2 \right] \, \sigma^{1,\varepsilon} \, dx \\
& w^\varepsilon_1 ( \widehat{x})
+ \varepsilon \int_{\partial U} \frac{\partial \sigma^{1,\varepsilon} }{\partial \nu} w^\varepsilon_1 \, dS
+ \int_U (\gamma^\varepsilon)' \mid_{\theta^\varepsilon_2} w^\varepsilon_1 \, \sigma^{2,\varepsilon} \, dx = 0,
\end{split} \end{equation}
which implies
\begin{equation*}
\begin{split}
& w^\varepsilon_1 ( \widehat{x}) - \int_U (\gamma^\varepsilon)' \mid_{\theta^{\varepsilon}_1} w^\varepsilon_2 \, \sigma^{1,\varepsilon} \, dx
+ \int_U (\gamma^\varepsilon)' \mid_{\theta^\varepsilon_2} w^\varepsilon_1 \, \sigma^{2,\varepsilon} \, dx\\
& \leq \int_U (\gamma^\varepsilon)' \mid_{\theta^{\varepsilon}_1} Du^\varepsilon_1 \cdot D\psi_1 \sigma^{1,\varepsilon} \, dx
- \int_U D_x H_1 \cdot Du^\varepsilon_1 \sigma^{1,\varepsilon} \, dx
- \varepsilon \int_{\partial U} \frac{\partial \sigma^{1,\varepsilon} }{\partial \nu} w^\varepsilon_1 \, dS.
\end{split} \end{equation*}
Let now $\eta > 0$ be a constant to be chosen later.
Using Step I and Lemmas \ref{kiUJ} and \ref{gamprimeb}, thanks to Young's inequality
\begin{equation} \label{fc1}
\begin{split}
& w^\varepsilon_1 ( \widehat{x} ) - \int_U (\gamma^\varepsilon)' \mid_{\theta^{\varepsilon}_1} w^\varepsilon_2 \, \sigma^{1,\varepsilon} \, dx
+ \int_U (\gamma^\varepsilon)' \mid_{\theta^\varepsilon_2} w^\varepsilon_1 \, \sigma^{2,\varepsilon} \, dx\\
& \leq \int_U (\gamma^\varepsilon)' \mid_{\theta^{\varepsilon}_1}
\left[ \eta^2 w^\varepsilon_1 (\widehat{x}) + \frac{\|D\psi_1\|^2_{L^{\infty}}}{2 \, \eta^2}
\right] \sigma^{1,\varepsilon} \, dx \\
&\hspace{.4cm} + \int_U \left[ \eta^2 w^\varepsilon_1 (\widehat{x}) + \frac{\|D_x H_1\|^2_{L^{\infty}}}{2 \, \eta^2}
\right] \sigma^{1,\varepsilon} \, dx +C \\
&\leq \eta^2 (C + 1) w^\varepsilon_1 (\widehat{x}) + C \left( 1 + \frac{1}{\eta^2} \right).
\end{split} \end{equation}
In the same way, considering the analogous of equation \eqref{obs25} for the function $w^{\varepsilon}_2$
(recalling that in \eqref{gbh} we chose $i=1$) we can obtain the following inequality :
\begin{equation} \label{fc2}
- \int_U (\gamma^\varepsilon)' \mid_{\theta^{\varepsilon}_2} w^\varepsilon_1 \, \sigma^{2,\varepsilon} \, dx
+ \int_U (\gamma^\varepsilon)' \mid_{\theta^\varepsilon_1} w^\varepsilon_2 \, \sigma^{1,\varepsilon} \, dx
\leq \eta^2 (C + 1) w^\varepsilon_1 (\widehat{x}) + C \left( 1 + \frac{1}{\eta^2} \right),
\end{equation}
where we also used the fact that $\| w^{\varepsilon}_2 \|_{L^{\infty}} \leq w^\varepsilon_1 (\widehat{x})$.
Summing inequalities \eqref{fc1} and \eqref{fc2} and choosing $\eta >0$ small enough the conclusion follows.
\end{proof}
Next lemma gives a control of the Hessians $D^2 u^{\varepsilon}_1$
and $D^2 u^{\varepsilon}_2$ in the support of $\sigma^{1,\varepsilon}$ and $\sigma^{2,\varepsilon}$
respectively.
\begin{Lemma} \label{dggt}
There exists a positive constant $C$, independent of $\varepsilon$, such that
\begin{equation*}
\sup_{j=1,2} \int_U \varepsilon |D^2 u^\varepsilon_j|^2 \sigma^{j,\varepsilon} \, dx \leq C.
\end{equation*}
\end{Lemma}
\begin{proof}
The bound of the Hessian of $D^2 u^{\varepsilon}_1$ comes from identity \eqref{obs26}, together with Lemma \ref{boundgrad}.
The other bound can be obtained in an similar way.
\end{proof}
We can finally prove the analogous of Lemma \ref{obs_lem4}.
\begin{Lemma}
There exists a positive constant, independent of $\varepsilon$, such that
\begin{equation*}
\max_{\substack{ j=1,2 \\ x \in \overline{U} }} \dfrac{\theta^{\varepsilon}_j (x)}{\varepsilon} \leq C,
\quad \quad \max_{\substack{ j=1,2 \\ x \in \overline{U} }} \gamma^{\varepsilon} ( \theta^{\varepsilon}_j (x) ) \leq C.
\end{equation*}
\end{Lemma}
\begin{proof}
It will be enough to prove the second inequality, since the first one will follow by the definition of $\gamma^{\varepsilon}$.
If the maximum is attained at the boundary, then
\begin{equation*}
\max_{\substack{ j=1,2 \\ x \in \overline{U} }} \gamma^{\varepsilon} ( \theta^{\varepsilon}_j (x) )
= \max_{\substack{ j=1,2 \\ x \in \partial U }} \gamma^{\varepsilon} ( - \psi_j (x) ) =0.
\end{equation*}
Otherwise, let us assume that there exists $x_1 \in U$ such that
\begin{equation*}
\max_{j=1,2} \max_{x \in \overline{U}} \gamma^\varepsilon (\theta^\varepsilon_j)=
\gamma^\varepsilon (\theta^\varepsilon_1) (x_1) > 0, \quad \quad \gamma^{\varepsilon} (\theta^{\varepsilon}_2 (x_1)) = 0.
\end{equation*}
Since $\gamma^\varepsilon$ is increasing and $\gamma^{\varepsilon} (z) > 0$ if and only if $z > 0$, we also have
$\max_{x \in \overline{U}} (\theta^\varepsilon_1(x)) = \theta^\varepsilon_1 (x_1) > 0$.
Evaluating the two equations in \eqref{tg} at $x_1$ and subtracting the second one from the first one
\begin{align*}
&\theta^\varepsilon_1 (x_1) + \gamma^\varepsilon (\theta^\varepsilon_1 (x_1) )
= \varepsilon \Delta u^{\varepsilon}_1 (x_1) - \varepsilon \Delta u^{\varepsilon}_2 (x_1)
- H_1 (x_1, D u_1^{\varepsilon} (x_1)) + H_2 (x_1, D u_2^{\varepsilon} (x_1)) - \psi_1 (x_1) \\
& \le \varepsilon \Delta \psi_1 (x_1) - H_1 (x_1, D u_1^{\varepsilon} (x_1)) + H_2 (x_1, D u_2^{\varepsilon} (x_1)) - \psi_1 (x_1) \\
&\leq \| \Delta \psi_1 (\cdot) \|_{L^{\infty}} + \| H_1 (\cdot, D u_1^{\varepsilon} (\cdot)) \|_{L^{\infty}}
+ \| H_2 (\cdot, D u_2^{\varepsilon} (\cdot) ) \|_{L^{\infty}} + \| \psi_1 (\cdot ) \|_{L^{\infty}} \leq C,
\end{align*}
where we the last inequality follows from Lemma \ref{boundgrad}.
\end{proof}
We now set for every $\varepsilon \in (0,1)$
\begin{equation*}
u_{j,\varepsilon}^\varepsilon (x) := \frac{\partial u^{\varepsilon}_j}{\partial \varepsilon} (x),
\quad x \in \overline{U}, j=1,2.
\end{equation*}
The next lemma gives a uniform bound for $u_{1,\varepsilon}^\varepsilon$
and $u_{2,\varepsilon}^\varepsilon$, thus concluding the proof of Theorem~\ref{obs_speedfin}.
\begin{Lemma}
There exists a positive constant $C > 0$ such that
\begin{equation*}
\max_{\substack{ j=1,2 \\ x \in \overline{U} }} | u_{j,\varepsilon}^\varepsilon (x) | \leq \frac{C}{\varepsilon^{1/2}}.
\end{equation*}
\end{Lemma}
\begin{proof}
If the above maximum is attained at the boundary, then
\begin{equation*}
\max_{\substack{ j=1,2 \\ x \in \overline{U} }} | u_{j,\varepsilon}^\varepsilon (x) |=
\max_{\substack{ j=1,2 \\ x \in \partial U }} | u_{j,\varepsilon}^\varepsilon (x) | = 0,
\end{equation*}
since $u_{1,\varepsilon}^\varepsilon = u_{2,\varepsilon}^\varepsilon = 0$ on $\partial U$.
Otherwise, assume that there exists $\overline{x} \in U$ such that
\begin{equation*}
\max_{\substack{ j=1,2 \\ x \in \overline{U} }} | u_{j,\varepsilon}^\varepsilon (x) |
= | u_{1,\varepsilon}^\varepsilon (\overline{x}) |.
\end{equation*}
Differentiating \eqref{tg} w.r.t. $\varepsilon$ we have
\begin{equation} \label{tg4}
\left\{ \begin{aligned}
(1+(\gamma^\varepsilon)' \mid_{\theta^{\varepsilon}_1} )u_{1,\varepsilon}^\varepsilon
+D_p H_1\cdot Du_{1,\varepsilon}^\varepsilon
- (\gamma^\varepsilon)' \mid_{\theta^{\varepsilon}_1} u_{2,\varepsilon}^\varepsilon
+ \gamma_\varepsilon^\varepsilon \mid_{\theta^{\varepsilon}_1}
= \varepsilon \Delta u_{1,\varepsilon}^\varepsilon
+ \Delta u^\varepsilon_1 \quad\mbox{in}~U,
\\
(1+(\gamma^\varepsilon)' \mid_{\theta^{\varepsilon}_2} )u_{2,\varepsilon}^\varepsilon
+D_p H_2\cdot Du_{2,\varepsilon}^\varepsilon
- (\gamma^\varepsilon)' \mid_{\theta^{\varepsilon}_2} u_{1,\varepsilon}^\varepsilon
+ \gamma_\varepsilon^\varepsilon \mid_{\theta^{\varepsilon}_2}
= \varepsilon \Delta u_{2,\varepsilon}^\varepsilon + \Delta u^\varepsilon_2 \quad\mbox{in}~U.
\end{aligned} \right.
\end{equation}
Let $\sigma^{1,\varepsilon}$ and $\sigma^{2,\varepsilon}$ be the solutions
to system \eqref{rfd} with $i=1$ and $x_0 = \overline{x}$.
Multiplying \eqref{tg4}$_1$ and \eqref{tg4}$_2$ by $\sigma^{1,\varepsilon}$
and $\sigma^{2,\varepsilon}$ respectively, integrating by parts
and adding up the two relations obtained we have
\begin{equation*}
u_{1,\varepsilon}^\varepsilon (\overline{x})
= \sum_{j=1}^2
\left( \int_U \Delta u^{\varepsilon}_j \, \sigma^{j,\varepsilon} \, dx
- \int_U \gamma^{\varepsilon}_{\varepsilon} \mid_{\theta^{\varepsilon}_j} \sigma^{j,\varepsilon} \, dx \right).
\end{equation*}
Thus,
\begin{equation*}
|u_{1,\varepsilon}^\varepsilon (\overline{x})|
\leq \sum_{j=1}^2
\left( \int_U |\Delta u^{\varepsilon}_j | \, \sigma^{j,\varepsilon} \, dx
+ \int_U | \gamma^{\varepsilon}_{\varepsilon} \mid_{\theta^{\varepsilon}_j} | \, \sigma^{j,\varepsilon} \, dx \right).
\end{equation*}
At this point, the proof can be easily concluded by repeating what was done in Section \ref{sectobs}
showing relations \eqref{obs6}--\eqref{obs8}.
\end{proof}
\end{section}
\end{document} |
\begin{document}
\title{Improving the estimation of the odds ratio \\using auxiliary information}
\author{{\sc Camelia Goga }$^1$ \quad and \quad {\sc Anne Ruiz-Gazen}$^2$ \\
$^1$Institut de Math\'ematiques de Bourgogne, Universit\'e de Bourgogne,
Dijon, France \\
$^2$ Toulouse School of Economics, Universit\'e Toulouse 1 Capitole,
Toulouse, France\\
email : [email protected], [email protected]}
\maketitle
\baselineskip=24pt
\centerline{Abstract}
The odds ratio measure is used in health and social surveys where the odds
of a certain event is to be compared between two populations. It is defined using logistic regression, and requires that data from surveys are accompanied by their weights.
A nonparametric estimation method that incorporates survey weights and auxiliary information may improve the precision of the odds ratio estimator.
It consists in $B$-spline calibration which can handle the nonlinear structure of the parameter. The variance is estimated through linearization.
Implementation is possible through standard survey softwares. The gain in precision depends on the data as shown on two examples.\\
\noindent \textbf{Key Words}: $B$-spline functions, calibration, estimating equation,
influence function, linearization, logistic regression.\\
\noindent {\bf Running title:} Odds ratio estimation in surveys
\section{Introduction }
We study the use of nonparametric weights for estimating the odds ratio when the risk variable, which is the explanatory variable in the logistic regression, is either a continuous or a binary variable.
The odds ratio is used to describe the strength of association
or non-independence between two binary variables defining two groups experiencing a particular event. One binary variable defines a group at risk and a group not at risk; the second binary variable defines the presence or absence of an event related to health. The odds ratio is the ratio of the odds of the event occurring in one group to
the odds of the same event occurring in the other group. An odds ratio equal to 1 means that the
event has the same odds in both groups; an odds ratio greater than 1 means that the event
has a larger odds in the first group; an odds ratio under 1 means that the event
has a smaller odds in the first group.
When both variables are categorical, the odds ratio estimator is obtained from a contingency table, as the ratio of the estimated row ratios, then, as a function of four numbers. As suggested by a reviewer, this definition leads to an estimator which takes survey weights into account and yields confidence intervals after linearization. However, this simple definition is not adapted to a continuous risk variable. In this case, the odds ratio measures the change in the odds for an increase of one unit in the risk variable. It is defined through the logistic regression.
For a binary risk variable, the odds ratio is the exponential of the difference of two logits, the logit function being the link function in the logistic regression.
So the logistic regression coefficient for a binary risk variable
corresponds to the logarithm of the odds ratio associated with this risk variable, net the effect of the other variables.
When the risk variable is continuous, the regression coefficient represents the logarithm of the odds ratio associated with a change
in the risk variable of one unit, net the effect of the other variables. The regression coefficient is a solution of a population estimating equation using the theory developed in Binder (1983) for making inference.
The sampling design must not be neglected especially for cluster sampling (Lohr, 2010).
Korn and Graubard (1999) and Heeringa et al. (2010) give details and examples of estimating an odds ratio but ignore auxiliary information. Korn and Graubard (1999: 169-170) advocate the use of weighted odds ratios contrary to Eideh and Nathan (2006). Rao et al. (2002) suggest using post-stratification
information to estimate parameters of interest obtained as solution of an estimating equation. The vector of parameters in the logistic regression is an example. Deville (1999) suggested ``allocating a weight $w_k$ to any point in the sample and zero to any other point, regardless of the origin of the weights (Horvitz-Thompson or calibration).'' Goga and Ruiz-Gazen (2014) use auxiliary information to estimate nonlinear parameters through nonparametric weights. The solutions of estimating equations are nonlinear but Goga and Ruiz-Gazen (2014) give no detail. Our project is the estimation of the odds ratio with auxiliary information.
In Section 2, we recall the definition
of the odds ratio and express the $B$-spline calibration estimator. In Section 3, we use linearization to derive the asymptotic variance of the estimator under broad assumptions. We infer a variance estimator together with asymptotic normal confidence intervals. In Section 4, we draw guidelines for practical implementation and show the properties of our estimator on two case studies.
\section{Estimation of the odds ratio with survey data}
\subsection{Definition of the parameter}
The odds ratio, denoted by OR, is used to quantify the association between the levels of a response variable
$Y$ and a risk variable $X.$ The value taken by $Y$ is $y_i$ and the value taken by $X$ is $x_i$ for the $i$-th individual in a population $U=\{1, \ldots, N\}.$
The logistic regression
\begin{equation}
\mbox{logit}(p_i)=\log \frac{p_i}{1-p_i}={\beta}_0+{\beta}_1x_i,
\end{equation}
where $p_i=P(Y=1|X=x_i)$ implies that
\begin{equation}
p_i=\exp({\beta}_0+{\beta}_1x_i)(1+\exp({\beta}_0+{\beta}_1x_i))^{-1}.
\end{equation}
The odds ratio is (Agresti, 2002):
\begin{equation}
\mbox{OR} = \frac{\mbox{odds}(Y=1|X=x_i+1)}{\mbox{odds}(Y=1|X=x_i)} = \exp{\beta_1}. \label{expbeta}
\end{equation}
With a binary variable $X$, the OR has a simpler form and can be derived from a contingency table. The OR is equal to
\begin{eqnarray}
\mbox{OR}=\frac{N_{00}N_{11}}{N_{01}N_{10}},\label{or_quali}
\end{eqnarray}
where $N_{00},$ $N_{01},$ $N_{10}$, and $N_{11}$ are the population counts associated with the contingency table. In order to estimate the OR of Eq.~(\ref{expbeta}), we estimate first the regression coefficient ${\beta}'=(\beta_0,\beta_1)$ by $\hat{{\beta}}'=(\hat{\beta}_0,\hat{\beta}_1)$, where $\mbox{x}'$ denotes the transpose of $\mbox{x}. $ Eq.~(\ref{expbeta}) yields the estimator of OR:
\begin{equation}
\widehat{\mbox{OR}}=\exp{\hat{\beta}_1}.
\end{equation}
The regression parameters $\beta_0$ and $\beta_1$ are
obtained by maximization of the population likelihood:
\begin{equation}
L(y_1, \ldots, y_N;{\beta})=\prod_{i \in U}\, p_i^{y_i}\, (1-p_i)^{1-y_i}.
\end{equation}
The maximum likelihood estimator of ${\beta}$ satisfies:
\begin{eqnarray}
\sum_{i\in U}(y_i-p_i) & = & 0\, , \label{eq:ml1}\\
\sum_{i \in U}(y_i-p_i)x_i & = & \, 0. \label{eq:ml2}
\end{eqnarray}
Let $\mbox{x}_i=(1 \quad x_i)'$ and $\mu(\mbox{x}'_i{\beta})=\exp(\mbox{x}'_i{\beta})
(1+\exp(\mbox{x}'_i{\beta}))^{-1}. $ We write Eq.~(\ref{eq:ml1}) and (\ref{eq:ml2}) in the equivalent form
\begin{eqnarray}
\sum_{i \in U}\mbox{x}_i(y_i-\mu(\mbox{x}'_i{\beta}))=0\label{maxvrais}
\end{eqnarray}
or, with $\mbox{t}_i({\beta})=\mbox{x}_i(y_i-\mu(\mbox{x}'_i{\beta})),$
\begin{eqnarray}
\sum_{i \in U}\mbox{t}_i({\beta})=0\label{def_t}.
\end{eqnarray}
The regression estimator of ${\beta}$ is defined as an implicit solution of the estimating Eq.~(\ref{maxvrais}). We use iterative methods to compute it.
\subsection{The $ B$-spline nonparametric calibration}\label{b_spline_calibration}
\noindent For $s$ a sample selected from the population $U$ according to a sample design $p(\cdot)$ , we denote by
$\pi_i >0$ the probability of unit $i$ to be selected in the sample and $\pi_{ij}>0$ the joint probability of units $i$ and $j$ to be selected in the sample with $\pi_{ii}=\pi_i$. We look for an estimator of ${\beta}$ and of OR taking the auxiliary variable $Z$, with values $z_1, \ldots, z_N$, into account.
Deville and S\"arndal (1992) suggest deriving the calibration weights $w_{ks}$ as close as possible to the Horvitz-Thompson sampling weights $d_i=1/\pi_i$ while satisfying the calibration
constraints on known totals $Z$:
\begin{equation}
\sum_{i \in s} w_{is}z_i = \sum_{i \in U}z_i.
\end{equation}
This method works well for a linear
relationship between the main and the auxiliary variables. When this relationship is no longer linear,
the calibration constraints must be changed while keeping the property that
the obtained weights do not depend on the main variable.
Basis functions that are more general than the ones defined by constants and $z_i$, include $B$-splines, which are simple to use (Goga and Ruiz-Gazen, 2013), truncated polynomial basis functions, and wavelets.
\subsubsection{$B$-spline functions}\label{sec:bsp}
Spline functions are used to model nonlinear trends.
A spline function of degree $m$ with $K$ interior knots is a piecewise polynomial
of degree $m-1$ on the intervals between knots, smoothly connected at knots.
The $B$-spline functions $B_1, \ldots, B_q$ of degree $m$ with $K$ interior knots, $q=m+K$ are among the possible basis functions (Dierckx, 1993).
Other basis functions
exist such as the truncated power basis (Ruppert et al., 2003).
For $m=1,$ the $B$-spline basis functions are step functions with jumps at the knots; for $m=2, $
they are piecewise linear polynomials, and so on. Figure \ref{base_Bspline} shows the six $B$-spline basis functions obtained for $m=3$
and $K=3$. Figure \ref{approx_sinus} gives the approximation of the curve $f(x)=x+\sin(4\pi x)$ taking the noisy data points into account and using the $B$-spline basis.
Even if the function $f$ is nonlinear, the $B$-spline approximation almost coincides with it.
The user chooses
the spline degree $m$ and the total number $K$ of knots. There is no general rule giving the total number of knots but Ruppert al. (2003) recommend $m=3$ or $m=4$ and no more than
30 to 40 knots. They also give a simple rule for choosing $K$ (Ruppert et al., 2003: 126).
Usually, the knots are located at the quantiles of the explanatory variable
(Goga and Ruiz-Gazen, 2013).
\begin{figure}
\caption{$B$-spline basis functions with $K=3$ interior knots and $m=3$.}
\label{base_Bspline}
\end{figure}
\begin{figure}
\caption{$B$-spline approximation of $f(x)=x+\sin(4\pi x)$ with $K=3$ interior knots and $m=3$. The crosses correspond to the noisy data.
The solid line is the true function $f$; the dashed line is the $B$-spline approximation. }
\label{approx_sinus}
\end{figure}
\subsubsection{Nonparametric calibration with $B$-spline functions}
The $B$-splines calibration weights $w_{is}^b$ are solution of the optimization problem:
\begin{equation}
(w_{is}^b)_{i\in s} = \mbox{argmin}_{w}\,\sum_{i \in s}\frac{(w_i-d_i)^2}{q_id_i}
\end{equation}
subject to
\begin{equation}
\sum_{i \in s} w_{is}^b\mbox{b}(z_i) = \sum_{i \in U}\mbox{b}(z_i),
\end{equation}
where $\mbox{b}(z_i)=(B_1(z_i), \ldots, B_q(z_i))'$ and $q_i$ is a positive constant. They are given by
\begin{eqnarray}
w_{is}^b=d_i\left(1-q_i\mbox{b}'(z_i)(\sum_{i\in s}d_iq_i\mbox{b}(z_i)\mbox{b}'(z_i))^{-1}(\hat t_{b,d}-t_{b})\right)\label{weightcalageB}
\end{eqnarray}
with $\hat t_{b,d}=\sum_{i\in s}d_i\mbox{b}(z_i), $
$t_{b}=\sum_{i \in U}\mbox{b}(z_i). $ The weights $w_{is}^b$ depend only on the auxiliary variable and are similar to
Deville and S\"arndal's weights. The calibration equation implies $\sum_{i \in s}w_{is}^b=N$ and
$\sum_{i \in s}w_{is}^bz_i=\sum_{i \in U}z_i.$
If $q_i=1$ for all $i\in U,$ we obtain (Goga, 2005):
\begin{eqnarray}
w_{is}^b=d_i t_{b}'\left(\sum_{k\in s}d_k\mbox{b}(z_k)\mbox{b}'(z_k)\right)^{-1}\mbox{b}(z_i).\label{poidsNP}
\end{eqnarray}
Goga and Ruiz-Gazen (2014) use these weights to estimate totals for variables, which are related
nonlinearly to the auxiliary information and to estimate nonlinear parameters such as a Gini index.
We use $w_{is}^b$ to estimate the logistic regression coefficient and the odds ratio efficiently.
\subsection{Estimation of OR using $B$-spline nonparametric calibration}\label{section_np}
The regression coefficient ${\beta}$ is a nonlinear finite population function defined by the implicit Eq.~(\ref{maxvrais}).
The functional method by Deville (1999), specified for the nonparametric case by Goga and Ruiz-Gazen (2014), is used
to build a nonparametric estimator of ${\beta}$ defined through the weights of Eq.~(\ref{poidsNP}).
$M$ is the finite measure assigning the unit mass to each $y_i$, $i\in U$, and zero elsewhere:
\begin{equation}
M=\sum_{i \in U}\delta_{y_i}
\end{equation}
where $\delta_{y_i}$ is the Dirac function at $y_i,$ $\delta_{y_i}(y)=1$ for $y=y_i$ and zero elsewhere.
The functional $T$ defined with respect to the measure $M$ and depending on the parameter ${\beta}$ defined by
\begin{equation}
T(M; {\beta})=\sum_{i \in U}\mbox{x}_i(y_i-\mu(\mbox{x}'_i{\beta})).
\end{equation}
The regression coefficient ${\beta}$ is the solution of the implicit equation
\begin{eqnarray}
T(M; {\beta})=0\label{betafunct}. \label{eq:sco}
\end{eqnarray}
Eq.~(\ref{eq:sco}) is called the score equation.
\noindent The measure $M$ may be estimated using the Horvitz-Thompson weights $d_k=1/\pi_k$ or the linear calibration weights (Deville, 1999). We suggest
using the nonparametric weights derived in Eq.~(\ref{poidsNP}):
\begin{eqnarray}
w_{is}^b=d_i\left(\sum_{k\in U}\mbox{b}(z_k)\right)'\left(\sum_{k\in s}d_k\mbox{b}(z_k)\mbox{b}'(z_k)\right)^{-1}\mbox{b}(z_i)
\end{eqnarray}
and estimate $M$ by
\begin{eqnarray}
\widehat M=\sum_{i\in s}w^b_{is}\delta_{y_i}.
\end{eqnarray}
\noindent Plugging $\widehat M$ into the functional expression of $ {\beta}$ given by Eq.~(\ref{betafunct}) yields the $B$-spline
calibrated estimator $\widehat{{\beta}}$ of ${\beta}$:
\begin{eqnarray}
T(\widehat M;\widehat{{\beta}})=0, \label{estimbetafunct}
\end{eqnarray}
which means that $\widehat{{\beta}}$ is the solution of the implicit equation:
\begin{eqnarray}
\sum_{i \in s}w_{is}^b\mbox{x}_i(y_i-\mu(\mbox{x}'_i\widehat{{\beta}}))=0.\label{maxvraisech}
\end{eqnarray}
The functional method allows us to incorporate auxiliary information for estimating the logistic regression coefficient and any parameter ${\beta}$ defined
as a solution of estimating equations.
\noindent The functional $T$ is differentiable with respect to ${\beta}$ and
\begin{equation}
\frac{\partial T}{\partial{{\beta}}}=-\sum_{i\in U}\nu(\mbox{x}'_i{\beta})\mbox{x}_i\mbox{x}'_i=\mbox{X}'
{\Lambda({\beta})}\mbox{X}:=\mbox{J}({\beta}), \label{jacobian_pop}
\end{equation}
with $\mbox{X}=(\mbox{x}'_i)_{i\in U}$ and ${\Lambda({\beta})}=-\mbox{diag}(\nu(\mbox{x}'_i{\beta}))$
with $\nu(\mbox{x}'_i{\beta})=\mu(\mbox{x}'_i{\beta})(1-\mu(\mbox{x}'_i{\beta}))$ the derivative of $\mu$. The $2\times 2$ matrix $\mbox{X}'{\Lambda({\beta})}\mbox{X}$ is invertible and $\mbox{J}({\beta})$
is definite negative. From Eq.~(\ref{jacobian_pop}), the matrix $\mbox{J}({\beta})$ is a total estimated using the nonparametric weights $w_{is}^b$ by:
\begin{eqnarray}
\widehat{\mbox{J}}_w({\beta})=-\sum_{i \in s}w_{is}^b\nu(\mbox{x}'_i{\beta})\mbox{x}_i\mbox{x}'_i=
\mbox{X}_s'\widehat{{\Lambda}}({\beta})\mbox{X}_s, \label{estimJ}
\end{eqnarray}
where $\widehat{{\Lambda}}({\beta})=-\mbox{diag}(w_{is}^b\nu(\mbox{x}'_i{\beta}))_{i\in s}$ and
$\mbox{X}_s=(\mbox{x}'_i)_{i\in s}$.
An iterative Newton-Raphson method is used to compute $\widehat{{\beta}}.$ The $r$-th step of the Newton-Raphson
algorithm is:
\begin{equation}
\widehat{{\beta}}_r=\widehat{{\beta}}_{r-1}-\widehat{\mbox{J}}_w(\widehat{{\beta}}_{r-1})T(\widehat M;
\widehat{{\beta}}_{r-1}),
\end{equation}
where $\widehat{{\beta}}_{r-1}$ is the value of $\widehat{{\beta}}$ obtained at the $(r-1)$-th step.
$\widehat{\mbox{J}}_w(\widehat{{\beta}}_{r-1})$ is the value of
$\widehat{\mbox{J}}_w({\beta})$ and $T(\widehat M; \widehat{{\beta}}_{r-1})$ the value of $T(\widehat M;{\beta})$ evaluated at
${\beta}=\widehat{{\beta}}_{r-1}. $ Iterating to convergence produces the nonparametric estimator
$\widehat{{\beta}}$ and the estimated Jacobian matrix $\widehat{\mbox{J}}_w(\widehat{{\beta}}).$
The odds ratio is estimated by $\widehat{\mbox{OR}}=\exp(\hat{\beta_1})$ and $\widehat{\mbox{J}}_w(\widehat{{\beta}})$
is used in section \ref{sec:var} to estimate the variance of $\hat{{\beta}}.$
\section{Variance estimation and confidence intervals}\label{sec:var}
\subsection{Variance estimation}
The coefficient ${\beta}$ of the logistic regression is nonlinear and nonparametric weights $w^b_{is}$ to estimate ${\beta}$ add more nonlinearity. We approximate $\widehat{{\beta}}$ in Eq.~(\ref{estimbetafunct}) by a linear estimator in two steps: we first treat the nonlinearity due to ${\beta}$, and second the nonlinearity due to the nonparametric estimation.
This procedure is different from Deville (1999).
From the implicit function theorem, there exists a unique functional $\widetilde T$ such that
\begin{equation}
\widetilde T(M)={\beta}\;\;\;\mbox{ and } \;\;\;
\widetilde T(\widehat M)=\widehat{{\beta}}.
\end{equation}
Moreover, the functional $\widetilde T$ is also Fr\'echet differentiable with respect to $M$. The derivative of $\widetilde T$
with respect to $M$, called the influence function, is defined by
\begin{eqnarray} \label{if}
I\widetilde T(M,\xi)=\lim_{\lambda\rightarrow 0}\,\frac{\widetilde T(M+\lambda \delta_{\xi})-\widetilde T(M)}{\lambda},
\end{eqnarray}
\noindent where $\delta_{\xi}$ is the Dirac function at $\xi . $ We give a first-order expansion of $\tilde T$ in $\widehat M/N$ around $M/N,$
\begin{eqnarray}
\widetilde T\left(\frac{\widehat M}{N}\right)=\widetilde T\left(\frac{M}{N}\right)+\int_{-\infty}^{+\infty} I\widetilde T\left(\frac{M}{N},\xi\right)d
\left(\frac{\widehat M}{N}-\frac{M}{N}\right)(\xi)+o_p(n^{-1/2}),\label{vonmises1}
\end{eqnarray}
which is also:
\begin{eqnarray}
\widetilde T(\widehat M)=\widetilde T(M)+\int_{-\infty}^{+\infty} I\widetilde T\left(M,\xi\right)d(\widehat M-M)(\xi)+o_p(n^{-1/2}), \label{vonmises2}
\end{eqnarray}
because $\widetilde T$ is a functional of degree zero, namely $\widetilde T(M/N)=\widetilde T(M)$ and $I\widetilde T
\left(M/N,\xi\right)=N I\widetilde T\left(M,\xi\right)$ (Deville, 1999).
For all $i\in U$, the linearized variable $\mbox{u}_i$ of $\widetilde T(M)={\beta}$ is defined as the value
of the influence function $I\widetilde T$ at $\xi=y_i$:
\begin{eqnarray}
\mbox{u}_i & = & I\widetilde T(M,y_i)=-\left(\frac{\partial T}{\partial{{\beta}}}\right)^{-1} IT(M,y_i;
{\beta})\nonumber\\
& = & -\left(\mbox{X}'{\Lambda({\beta})}\mbox{X}\right)^{-1}\mbox{x}_i(y_i-\mu(\mbox{x}'_i{\beta}))
=-\mbox{J}^{-1}({\beta})\cdot\mbox{x}_i(y_i-\mu(\mbox{x}'_i{\beta})).\label{eq_lin}
\end{eqnarray}
The linearized variable $\mbox{u}_i=(u_{i,0},u_{i,1})'$ is a two-dimensional vector depending on the unknown parameter ${\beta}$ and on totals contained in the matrix $\mbox{J}({\beta})$.
\noindent Eq.~(\ref{vonmises2}) becomes:
\begin{eqnarray}
\hat {{\beta}}-{\beta}\simeq \sum_{i\in s}w_{is}^b\mbox{u}_i-\sum_{i\in U}\mbox{u}_i.\label{linear1}
\end{eqnarray}
The second component $u_{i,1}$ of $\mbox{u}_i,$ is the linearized variable of $\beta_1$. With binary data, the odds ratio is given by Eq.~(\ref{or_quali}), which implies that
\begin{equation}
\ln(\mbox{OR})=\ln(N_{00})+\ln(N_{11})-\ln(N_{01})-\ln(N_{10}).
\end{equation}
In this case, the linearized variable of $\beta_1$ has the expression:
\begin{eqnarray}
u_{i,1}=\frac{\mbox{1}_{\{x_i=0,y_i=0\}}}{N_{00}}+\frac{\mbox{1}_{\{x_i=1,y_i=1\}}}{N_{11}}-\frac{\mbox{1}_{\{x_i=1,y_i=0\}}}{N_{10}}-\frac{\mbox{1}_{\{x_i=0,y_i=1\}}}{N_{01}}
\label{lin1}
\end{eqnarray}
and the same expression is obtained from Eq.~(\ref{eq_lin}) after some algebra.
When the weights $w_{is}^b$ are equal to the sampling weights, namely $w_{is}^b=1/\pi_i$, Eq.~(\ref{linear1}) implies that the asymptotic variance
of $\hat {{\beta}}$ is:
\begin{eqnarray}
\mbox{AV}(\hat {{\beta}})=\mbox{Var}\left(\sum_{i\in s}d_i\mbox{u}_i\right)=\mbox{J}^{-1}({\beta})\,
\mbox{\mbox{V}}_{\mbox{\sc ht}}(\hat{\mbox{t}}_d({\beta}))\, \mbox{J}^{-1}
({\beta}), \label{var_sans_inf}
\end{eqnarray}
where $\displaystyle \mbox{\mbox{V}}_{\mbox{\sc ht}}(\hat{\mbox{t}}_d({\beta}))$ is the Horvitz-Thompson variance of
$\hat{\mbox{t}}_d({\beta})=\sum_{i\in s} \mbox{t}_i({\beta}) / \pi_i$ with $\mbox{t}_i({\beta})=
\mbox{x}_i(y_i-\mu(\mbox{x}'_i{\beta}))$:
\begin{equation}
\mbox{\mbox{V}}_{\mbox{\sc ht}}(\hat{\mbox{t}}_d({\beta}))=
\mbox{Var}\left(\sum_{i\in s}\frac{\mbox{t}_i({\beta})}{\pi_i}\right)=
\sum_{i\in U}\sum_{i\in U}(\pi_{ij}-\pi_i\pi_j)\frac{\mbox{t}_i({\beta})}{\pi_i}\frac{\mbox{t}_j({\beta})}{\pi_j}.
\end{equation}
Binder (1983) gives the same asymptotic expression for the variance.
For $B$-spline basis functions formed by step functions on intervals between knots ($m=1$), the weights $w_{is}^b$ yield the post-stratified estimator of ${\beta}$
(Rao et al., 2002). Linear calibration weights lead to the case treated by Deville (1999).
For the general case of nonparametric calibration weights $w_{is}^b$, a supplementary linearization step is necessary.
The right hand side of Eq.~(\ref{linear1}) is a nonparametric calibration estimator for the total of the linearized variable $\mbox{u}_i$. It can be written
as a generalized regression estimator (GREG):
\begin{eqnarray}
\sum_{i\in s}w_{is}^b\mbox{u}_i-\sum_{i\in U}\mbox{u}_i= \sum_{i\in s}\frac{\mbox{u}_i-\widehat{{\theta}}_{u}'\mbox{b}(z_i)}{\pi_i}+\sum_{i\in U}\widehat{{\theta}}_{u}'\mbox{b}(z_i)-
\sum_{i\in U}\mbox{u}_i, \label{greg_var_lin}
\end{eqnarray}
where $\widehat{{\theta}}_{u}=(\sum_{i\in s}d_i\mbox{b}(z_i)\mbox{b}'(z_i))^{-1}(\sum_{i\in s}d_i\mbox{b}(z_i)\mbox{u}'_i). $
We explain the linearized variable by means of a piecewise polynomial function. This fitting allows more flexibility and implies that the residuals $\mbox{u}_i-\widehat{{\theta}}_{u}'\mbox{b}(z_i)$
have a smaller dispersion than with a
linear fitting regression.
In order to derive the asymptotic variance of the nonparametric calibrated estimator, we assume
that $||\mbox{x}_i||< C$ for all $i\in U$ with $C$ a positive constant independent of $i$ and $N$.
The Euclidian norm is denoted $||\cdot||$. The matrix norm $||\cdot||_2$ is defined by $||\mbox{A}||_2^2=\mbox{tr}(\mbox{A}'\mbox{A}). $ The linearized variable verifies $N||\mbox{u}_i||=O(1)$ uniformly in $i,$ because
\begin{equation}
N||\mbox{u}_i||\leq ||N\mbox{J}^{-1}({\beta})||_2\, ||\mbox{x}_i||\, |y_i-\mu(\mbox{x}_i'{\beta}))|=O(1),
\end{equation}
where the Jacobian matrix $\mbox{J}({\beta}) $ contains totals
\begin{equation}
\mbox{J}({\beta})=-\left(\begin{array}{cc}
\sum_{i\in U}\nu(\mbox{x}_i'{\beta}) & \sum_{i\in U}x_i\nu(\mbox{x}_i'{\beta})\\
\sum_{i\in U}x_i\nu(\mbox{x}_i'{\beta}) & \sum_{i\in U}x^2_i\nu(\mbox{x}_i'{\beta})\end{array}\right)
\end{equation}
and
\begin{equation}
\left(\frac{1}{N}\sum_{i\in U}\nu(\mbox{x}_i'{\beta})\right)^2\leq \frac{1}{N}\sum_{i\in U}(\nu(\mbox{x}_i'{\beta}))^2=O(1)
\end{equation} because $\nu(\mbox{x}_i'{\beta})<1.$
Under the assumptions of theorem 7 in Goga and Ruiz-Gazen (2014), the
nonparametric calibrated estimator $\sum_{i\in s}w_{is}^b\mbox{u}_i$ is asymptotically equivalent to
\begin{eqnarray}
\sum_{i\in s}w_{is}^b\mbox{u}_i-\sum_{i\in U}\mbox{u}_i\simeq \sum_{i\in s}\frac{\mbox{u}_i-\widetilde{{\theta}}_{u}'\mbox{b}(z_i)}{\pi_i}+\sum_{i\in U}\widetilde{{\theta}}_{u}'\mbox{b}(z_i)-
\sum_{i\in U}\mbox{u}_i, \label{linear2}
\end{eqnarray}
where $\widetilde{{\theta}}_{u}=(\sum_{i\in U}\mbox{b}(z_i)\mbox{b}'(z_i))^{-1}\sum_{i\in U}\mbox{b}(z_i)\mbox{u}'_i.$
The variance of $\hat {{\beta}}$ is approximated by the Horvitz-Thompson variance of the residuals
$\mbox{u}_i-\widetilde{{\theta}}_{u}'\mbox{b}(z_i),$
\begin{eqnarray}
\mbox{AV}(\hat {{\beta}})=\mbox{Var}\left(\sum_{i\in s}\frac{\mbox{u}_i-\widetilde{{\theta}}_{u}'\mbox{b}(z_i)}{\pi_i}\right)=\sum_{i\in U}\sum_{i\in U}(\pi_{ij}-\pi_i\pi_j)\frac{\mbox{u}_i-\widetilde{{\theta}}_{u}'\mbox{b}(z_i)}{\pi_i}\frac{\mbox{u}_j-\widetilde{{\theta}}_{u}'\mbox{b}(z_j)}{\pi_j}.\label{var_asymt}
\end{eqnarray}
Eq.~(\ref{linear2}) states that the $B$-spline nonparametric calibration estimator of $\sum_{i\in U}\mbox{u}_i$ is asymptotically equivalent
to the generalized difference estimator. We interpret this result as fitting a nonparametric model on the linearized variable $\mbox{u}_i$
taking into account the auxiliary information $z_i$. Nonparametric models are a good choice when the linearized variable obtained from the first linearization step
does not depend linearly on $z_i$, as it is the case in the logistic regression, which implies a second linearization step.
We write the asymptotic variance in Eq.~(\ref{var_asymt}) in a matrix form similar to Eq.~(\ref{var_sans_inf}). Consider again
$\mbox{t}_i({\beta})=\mbox{x}_i(y_i-\mu(\mbox{x}'_i{\beta}))$ and let $\widetilde{{\theta}}_{\mathbf t}=(\sum_{i\in s}\mbox{b}(z_i)\mbox{b}'(z_i))^{-1}\sum_{i\in s}\mbox{b}(z_i)\mbox{t}'_i({\beta}).$ We have
\begin{equation}
\mbox{u}_i-\widetilde{{\theta}}_{u}'\mbox{b}(z_i)=-\mbox{J}^{-1}({\beta})\left(\mbox{t}_i({\beta})-
\widetilde{{\theta}}'_{\mathbf t}\mbox{b}(z_i)\right),
\end{equation}
and the asymptotic variance of $\hat {{\beta}}$ is:
\begin{equation}
\mbox{AV}(\hat {{\beta}}) = \mbox{J}^{-1}({\beta})\, \mbox{V}_{\mbox{\sc ht}}(\hat{{e}}_d({\beta}))\,
\mbox{J}^{-1}({\beta})\label{varasymbeta}
\end{equation}
where $\hat{{e}}_d({\beta})=\displaystyle\sum_{i\in s}\frac{\mbox{e}_i({\beta})}{\pi_i}$ is the
Horvitz-Thompson estimator of the residual ${e}_i({\beta})=\mbox{t}_i({\beta})-
\widetilde{{\theta}}'_{\mathbf t}\mbox{b}(z_i)$ of $\mbox{t}_i({\beta})$ using $B$-spline calibration.
Eq.~(\ref{varasymbeta}) shows that improving the estimation of ${\beta}$ is equivalent to improving the estimation
of the score equation $\mbox{t}_i=\mbox{x}_i(y_i-\mu(\mbox{x}'_i{\beta})). $\\
The quantity of interest is the asymptotic variance of $\hat{\beta_1}$. It is the $(2,2)$ element of the matrix $\mbox{AV}(\hat{{\beta}})$ given by
Eq.~(\ref{var_asymt}). We have $\mbox{u}_i=(u_{i,0},u_{i,1})'$ and
\begin{eqnarray}
\mbox{u}_i-\widetilde{{\theta}}'_{u}\mbox{b}(z_i)=\left(\begin{array}{c}
u_{i,0}-\widetilde{{\theta}}'_{u_0}\mbox{b}(z_i)\\
u_{i,1}-\widetilde{{\theta}}'_{u_1}\mbox{b}(z_i)\end{array}
\right)
\end{eqnarray}
where $\widetilde{{\theta}}_{u_0}=(\sum_{i\in U}\mbox{b}(z_i)\mbox{b}'(z_i))^{-1}\sum_{i\in U}\mbox{b}(z_i)u_{i,0}$ and $\widetilde{{\theta}}_{u_1}=(\sum_{i\in U}\mbox{b}(z_i)\mbox{b}'(z_i))^{-1}\sum_{i\in U}\mbox{b}(z_i)u_{i,1}.$
We obtain
\begin{equation}
\mbox{AV}(\hat{\beta}_1)=\mbox{Var}\left(\sum_{i\in s}\frac{u_{i,1}-\widetilde{{\theta}}_{u_1}'\mbox{b}(z_i)}{\pi_i}\right).
\end{equation}
\noindent The linearized variable $\mbox{u}_i$ is unknown and is estimated by:
\begin{eqnarray}
\hat{\mbox{u}}_i & = &-\widehat{\mbox{J}}_w^{-1}(\widehat{{\beta}})\, \mbox{x}_i(y_i-\mu(\mbox{x}'_i
\widehat{{\beta}})) \label{varlin_estimator}\\
& = & -\widehat{\mbox{J}}_w^{-1}(\widehat{{\beta}})\, \hat{\mbox{t}}_i
\end{eqnarray}
where the matrix $\widehat{\mbox{J}}_w$ is computed according to Eq.~(\ref{estimJ}) and $\hat{\mbox{t}}_i$ is the estimation of $\mbox{t}_i({\beta})$ for ${\beta}=\widehat{{\beta}}$.
The asymptotic variance $\mbox{AV}(\widehat {{\beta}})$ given in Eq.~(\ref{var_asymt}) is estimated by the Horvitz-Thompson variance estimator
with $\mbox{u}_i$ replaced by $\hat{\mbox{u}}_i$ given in Eq.~(\ref{varlin_estimator}):
\begin{eqnarray}
\widehat{V}(\widehat {{\beta}}) & = & \widehat{V}_{\mbox{\sc ht}}\left(\sum_{i\in s}\frac{\hat{\mbox{u}}_i-
\widehat{{\theta}}'_{\widehat{\mbox{u}}}\mbox{b}(z_i)}{\pi_i}\right)\label{estimator_variance}
\end{eqnarray}
where $\widehat{{\theta}}_{\widehat{\mbox{u}}}=(\sum_{i\in s}d_i\mbox{b}(z_i)\mbox{b}'(z_i))^{-1}\sum_{i\in s}d_i\mbox{b}(z_i)\hat{\mbox{u}}'_i. $ The variance estimator of $\hat{\beta}_1$ is given by
\begin{eqnarray}
\hat{V}(\hat{\beta}_1)=\mbox{Var}\left(\sum_{i\in s}\frac{\hat{u}_{i,1}-\widehat{{\theta}}_{\hat{u}_1}'\mbox{b}(z_i)}{\pi_i}\right).
\end{eqnarray}
The variance estimator given in Eq.~(\ref{estimator_variance}) can be written in a matrix form. Let $\widehat{{\theta}}_{\widehat{\mathbf t}}=(\sum_{i\in s}d_i\mbox{b}(z_i)\mbox{b}'(z_i))^{-1}\sum_{i\in s}d_i\mbox{b}(z_i)\hat{\mbox{t}}'_i$ and $ \widehat{V}(\widehat {{\beta}})$ is written as:
\begin{eqnarray}
\widehat{V}(\widehat {{\beta}}) &= & \widehat{\mbox{J}}_w^{-1}(\widehat{{\beta}})\, \widehat{\mbox{V}}_{\mbox{\sc ht}}
(\hat{{e}}_d(\widehat{{\beta}}))\, \widehat{\mbox{J}}_w^{-1}(\widehat{{\beta}})\label{variance_estim}
\end{eqnarray}
where $\widehat{\mbox{V}}_{\mbox{\sc ht}}(\hat{{e}}_d)$
is the Horvitz-Thompson variance estimator of $\hat{{e}}_d(\hat{{\beta}})$ obtained by replacing
${e}_i({\beta})$ with $\hat{{e}}_i(\hat{{\beta}})=\hat{\mbox{t}}_i-
\widehat{{\theta}}'_{\hat{\mathbf t}}\mbox{b}(z_i), $
\begin{eqnarray}
\widehat{\mbox{V}}_{\mbox{\sc ht}}(\hat{{e}}(\hat{{\beta}}))=\sum_{i\in s}\sum_{i\in s}\frac{\pi_{ij}-\pi_i\pi_j}{\pi_{ij}}
\frac{\hat{{e}}_i(\hat{{\beta}})}{\pi_i}\frac{\hat{{e}}_j(\hat{{\beta}})}{\pi_j}.
\end{eqnarray}
\subsection{Confidence interval for the odds ratio}
The variance estimator of $\hat{\beta}_1$ is obtained from Eq.~(\ref{variance_estim}) as:
\begin{equation}
\hat{V}(\hat{\beta}_1)=\widehat{\mbox{J}}_w^{-1}(\widehat{{\beta}})\, \mbox{V}_{\mbox{\sc ht}}(\hat{e}_2(\widehat{{\beta}}))
\, \widehat{\mbox{J}}_w^{-1}(\widehat{{\beta}}),
\end{equation}
where $\hat{e}_2(\widehat{{\beta}})$ is the second component of $\hat{{e}}(\widehat{{\beta}})$ so that,
under regularity conditions, the $(1-\alpha)\%$ normal interval for $\beta_1$ is:
\begin{equation}
\mbox{CI}_{1-\alpha}({\beta_1})=\left[\hat{\beta}_1-z_{\alpha/2}\left(\hat{V}(\hat{\beta}_1)\right)^{1/2},
\hat{\beta}_1+z_{\alpha/2}\left(\hat{V}(\hat{\beta}_1)\right)^{1/2}\right],
\end{equation}
where $z_{\alpha/2}$ is the upper $\alpha/2$-quantile of a $\mathcal{N}(0,1)$ variable. Then the confidence interval for OR is:
\begin{equation}
\mbox{CI}_{1-\alpha}(\mbox{OR})=\left[\exp{\left(\hat{\beta}_1-z_{\alpha/2}\left({\hat{V}(\hat{\beta}_1)}\right)^{1/2}\right)}, \exp{\left(\hat{\beta}_1+z_{\alpha/2}
\left({\hat{V}(\hat{\beta}_1)}\right)^{1/2}\right)}\right],
\end{equation}
which is not symmetric around the estimated odds ratio but provides more accurate coverage rates of the true population value for a specified
$\alpha$ (Heeringa et al., 2010).
\section{Implementation and case studies}
\subsection{Implementation}
\begin{enumerate}
\item Compute the $B$-spline basis functions $B_j,$ for $j=1, \ldots, q. $ The $B$-spline basis functions are obtained using \textrm{SAS} or \textrm{R}. The user has only to specify the degree $m$ and the total number of knots.
\item Use the sampling weights $d_i=1/\pi_i$ and the $B$-spline functions to derive the nonparametric weights $w_{is}^b$ and the estimated
parameter ${\beta}$.
\item Compute the linearized variable $\mbox{u}_i$ estimated by $\hat{\mbox{u}}_i.$
\item Compute the estimated predictions $\widehat{{\theta}}_{\widehat{\mbox{u}}}'\mbox{b}(z_i)$ with
\begin{equation}
\widehat{{\theta}}_{\widehat{\mbox{u}}}=\left(\sum_{i\in s}d_i\mbox{b}(z_i)\mbox{b}'(z_i)\right)^{-1}\left(\sum_{i\in s}d_i\mbox{b}(z_i)
\widehat{\mbox{u}}'_i\right)
\end{equation}and the associated residuals $\hat{\mbox{u}}_i-
\widehat{{\theta}}'_{\widehat{\mbox{u}}}\mbox{b}(z_i)$.
\item Use a standard computer software able to compute variance estimators and apply it to the previously computed residuals.
\end{enumerate}
\subsection{Case studies}
We compare the asymptotic variance of different estimators of the odds ratio in the simple case
of one binary risk variable for two data sets.
In this case, the odds ratio is a simple function of four counts given by Eq.~(\ref{or_quali}).
We focus on the simple random sampling without replacement and compare three estimators.
The first one is the Horvitz-Thompson estimator which
does not use the auxiliary variable and whose asymptotic variance is given by Eq.~(\ref{var_sans_inf}). The second estimator is the generalized regression estimator
which takes the auxiliary variable into account through a linear model fitting the linearized variable against the auxiliary variable.
The third estimator is the $B$-spline calibration estimator with an asymptotic variance given by Eq.~(\ref{varasymbeta}).
In order to gain efficiency, the auxiliary variable is related to the linearized variable.
In the context of one binary factor, the linearized variable is given by Eq.~(\ref{lin1}) and takes four different values, which depend on the values of the variables $ X$ and $ Y$.
In order to be related to the linearized variable, the auxiliary variable is related to the product of the two variables
$X$ and $Y$, which is a strong property.
Moreover, because $u_{i,1}$, $X$, and $ Y$ are discrete, using auxiliary information
does not necessarily lead to an important gain in efficiency as the first health survey example will show.
The gain in efficiency however is significant in some cases.
In the second example using labor survey data, the gain in using the $B$-splines calibration estimator
compared to the Horvitz-Thompson estimator is significant because the auxiliary variable is related to
the variable $ Y$ but also to the factor $ X$; $X$ and $Y$ being related to one another, too.
\noindent \textit{Example from the California Health Interview Survey}
The data set comes from the Center for Health Policy Research at the University of California. It was extracted
from the adult survey data file of the California Health Interview Survey in 2009 and consists of 11074 adults.
The response dummy variable equals one if the person is currently insured; the binary factor equals one
if the person is currently a smoker. The auxiliary variable is age and we consider people who are less than 60 years old.
The data are presented in detail in Lumley (2010).
We compare the Horvitz-Thompson, the generalized regression, and the $B$-splines calibration estimators in terms of
asympotic variance. In order to calculate the $B$-splines functions, we use the SAS procedure {\em transreg} and take $K=15$ knots and $B$-splines of degree $m=3$.
The gain in using the generalized regression estimator compared to the Horvitz-Thompson estimator is only 0.01\%. It is 1.5\% when using
$B$-splines instead of the generalized regression.
When changing the number of knots and the degree of the $B$-spline functions, the results remain similar and the gain remains under 2\%.
In this example, there is no gain in using auxiliary information even with flexible $B$-splines, because the auxiliary variable
is not related enough to the linearized variable. The linearized variable
takes negative values for smokers without insurance and non smokers with insurance, positive values for smokers with insurance and non smokers without insurance. Age is not a good predictor for this variable,
because we expect to find sufficient people of any age in each of the four categories (smokers/non smokers $\times$ insurance/no insurance). Incorporating this auxiliary information brings no gain.
\noindent \textit{Example from the French Labor Survey}
We consider 14621
wage-earners under 50 years of age, from the French labour force survey. The initial data set consists of monthly wages in 2000 and 1999.
A dummy variable W00 equals one if the monthly wage in 2000
exceeds 1500 euros and zero otherwise. The same for W99 in 1999.
The population is divided in lower and upper education groups.
The value of the categorical factor DIP equals one for people with a university degree and zero otherwise.
W00 corresponds to the binary response variable $ Y$ while
the diploma variable DIP corresponds to the risk variable $ X$. The variable W99 is the auxiliary variable $Z$.
To compare the Horvitz-Thompson estimator with the generalized regression estimator and the $B$-splines calibration estimator, we calculate the gain in terms of
asympotic variance. We consider $K=15$ knots and the degree $m=3$.
The gain in using the generalized estimator compared to the Horvitz-Thompson estimator is now 20\%. It is 33\% when using $B$-splines.
The result is independent of the number of knots and, of the degree of $B$-spline functions. When the total number of knots varies
from 5 to 50 and the degree varies from 1 to 5, the gain is between 32\% and 34\%.
The nonlinear link between the linearized variable of a complex parameter with the
auxiliary variable explains the gain in using a nonparametric estimator compared to an estimator based on a linear model
(Goga and Ruiz-Gazen, 2013).
For the odds ratio with one binary factor, the linearized variable is discrete and the linear model does
not fit the data.
\section{Conclusion}
Estimating the variance of parameter estimators in a logistic regression is not straightforward especially if auxiliary
information is available. We applied the method of Goga and Ruiz-Gazen (2014) to the case of parameters defined through estimating equations.
The method relies on a linearization principle. The asymptotic variance of the estimator incorporates residuals of the model that we assume between
the linearized variable and the auxiliary variable. The gain in using auxiliary information is thus based on the fitting
quality of the model for the linearized variable. Because of the complexity of linearized variables, linear models
that incorporate auxiliary information seldom fit linearized variables and we use nonparametric $B$-spline estimators. A particular case is post-stratification. Using the influence function defined by Eq.~(\ref{if}), we derive the asymptotic variance of the estimators together with confidence intervals for the odds ratio.\\
\noindent {\bf \large Acknowledgement:} we thank Beno\^it Riandey for drawing our attention to the odds ratio and one rewiever for his/her constructive comments.
\section*{Bibliography}
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\newblock{\em Curves and Surfaces Fitting with Splines}.
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\newblock{\em Journal of the Royal Statistical Society series B}, \textbf{76}, 113-140.
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\end{description}
\end{document} |
\begin{document}
\title[Converging expansions]{\bf Converging expansions for Lipschitz\\ self-similar perforations of a plane sector}
\author{Martin Costabel}
\address{IRMAR UMR 6625 du CNRS, Universit\'{e} de Rennes 1, Campus de Beaulieu,
35042 Rennes Cedex, France}
\email{[email protected]}
\author{Matteo Dalla Riva}
\address{Department of Mathematics, The University of Tulsa, 800 South Tucker Drive, Tulsa, Oklahoma 74104, USA}
\email{[email protected]}
\author{Monique Dauge}
\address{IRMAR UMR 6625 du CNRS, Universit\'{e} de Rennes 1, Campus de Beaulieu,
35042 Rennes Cedex, France}
\email{[email protected]}
\author{Paolo Musolino}
\address{Department of Mathematics, Aberystwyth University, Ceredigion SY23 3BZ, Wales, UK}
\email{[email protected]}
\keywords{Dirichlet problem, corner singularities, perforated domain, double layer potential, Diophantine approximation}
\subjclass{35J05, 45A05, 31A10, 35B25, 35C20, 11J99}
\begin{abstract}
In contrast with the well-known methods of matching asymptotics and multiscale (or compound) asymptotics, the ``functional analytic approach'' of Lanza de Cristoforis (Analysis 28, 2008) allows to prove convergence of expansions around interior small holes of size $\varepsilon$ for solutions of elliptic boundary value problems. Using the method of layer potentials, the asymptotic behavior of the solution as $\varepsilon$ tends to zero is described not only by asymptotic series in powers of $\varepsilon$, but by convergent power series.
Here we use this method to investigate the Dirichlet problem for the Laplace operator where holes are collapsing at a polygonal corner of opening $\omega$. Then in addition to the scale $\varepsilon$ there appears the scale $\eta=\varepsilon^{\pi/\omega}$. We prove that when $\pi/\omega$ is irrational, the solution of the Dirichlet problem is given by convergent series in powers of these two small parameters.
Due to interference of the two scales, this convergence is obtained, in full generality, by grouping together integer powers of the two scales that are very close to each other. Nevertheless, there exists a dense subset of openings $\omega$ (characterized by Diophantine approximation properties), for which real analyticity in the two variables $\varepsilon$ and $\eta$ holds and the power series converge unconditionally. When $\pi/\omega$ is rational, the series are unconditionally convergent, but contain terms in $\log\varepsilon$.
\end{abstract}
\maketitle
{
\parskip 1pt
\tableofcontents
}
\section*{Introduction}
\label{s:1}
Domains with small holes are fundamental examples of singularly perturbed domains. The analysis of the asymptotic behavior of elliptic boundary value problems in such perforated domains as the size of the holes tends to zero lays the basis for numerous applications in more involved situations that can be found in the classical monographs \cite{Il92}, \cite{MaNaPl00}, \cite{KoMaMo99} and the more recent \cite{AmKa07}. The two methods that are most widely spread are the {\em matching of asymptotic expansions} as exposed by Il'in \cite{Il92}, and the method of {\em multiscale} (or {\em compound}) expansions as in Maz'ya, Nazarov, and Plamenevskij \cite{MaNaPl00} or Kozlov, Maz'ya, and Movchan \cite{KoMaMo99}. Ammari and Kang \cite{AmKa07} use the method of layer potentials to construct asymptotic expansions.
When the holes are shrinking to the corner of a polygonal domain, one encounters the class of self-similar singular perturbations, a case that has been treated by Maz'ya, Nazarov, and Plamenevskij \cite[Ch.2]{MaNaPl00} with the method of compound expansions, and by Dauge, Tordeux, and Vial \cite{DaToVi10} with both methods of matched and compound expansions.
The common feature of these methods is their algorithmic and constructive nature: The terms of the asymptotic expansions are constructed according to a sequential order. At each step of the construction, remainder estimates are proved, but there is no uniform control of the remainders,
and this does therefore not lead to a convergence proof.
Another method appeared recently, based on the ``functional analytic approach'' introduced by Lanza de Cristoforis \cite{La02}. This method has so far mainly been applied to the Laplace equation on domains with holes collapsing to interior points. The core feature is a description of the solution as a real analytic function of one or several variables depending on the small parameter $\varepsilon$ that characterizes the size of the holes. This is proved by a reduction to the boundary via integral equations, and after a careful analysis of the boundary integral operators the analytic implicit function theorem can be invoked, providing the expansion of the solutions into convergent series. Analytic functions of several variables appear for example in \cite{La08,DaMu15}, where the two-dimensional Dirichlet problem leads to the introduction of the scale $1/\log\varepsilon$ besides $\varepsilon$, or in \cite{DaMu16}, where a boundary value problem in a domain with moderately close holes is studied
and the size of the holes and their distance are defined by small parameters that may be of different size.
Our aim in this paper is to understand how this method would apply to the Dirichlet problem in a polygonal domain when holes are shrinking to the corner in a self-similar manner. In the limit $\varepsilon\to0$, the singular behavior of solutions at corners without holes will combine with the singular perturbation of the geometry.
In contrast to what happens in the case of holes collapsing at interior points or at smooth boundary points \cite{BoDaDaMu16}, we find that the series expansions in powers of $\varepsilon$ that correspond to the asymptotic expansions of \cite{DaToVi10} are only ``stepwise convergent''. For corner opening angles $\omega$ that are rational multiples of $\pi$, the series will be unconditionally convergent, but in general for irrational multiples of $\pi$, certain pairs of terms in the series may have to be grouped together in order to achieve convergence. This is a peculiar feature similar to, and in the end caused by, the stepwise convergence of the asymptotic expansion of the solution of boundary value problems near corners when the data are analytic \cite{BraDau82,Dauge84}.
\subsection{Geometric setting}\label{ss:geomset}
We consider perforated domains where the holes are shrinking towards a point of the boundary that is the vertex of a plane sector. For the sake of simplicity, we try to concentrate on the essential features and avoid unnecessary generality. Therefore we consider only one corner, but we admit several holes.
We denote by $t=(t_1,t_2)$ the Cartesian coordinates in the plane $\R^2$, and by $({\mathsf{g}}o=|t|,\vartheta=\arg(t))$ the polar coordinates. The open ball with center $0$ and radius ${\mathsf{g}}o_0$ is denoted by ${\mathscr B}(0,{\mathsf{g}}o_0)$. Let the opening angle $\omega$ be chosen in $(0,2\pi)$ and denote by ${\mathsf S}_\omega$ the infinite sector
\begin{equation}
\label{eq:1E1}
{\mathsf S}_\omega = \{t\in\R^2,\quad \vartheta\in(0,\omega)\}.
\end{equation}
The case $\omega=\pi$ is degenerate and corresponds to a half-plane.
The perforated domains ${\mathsf{A}}_{\varepsilon}$ are determined by an unperforated domain ${\mathsf{A}}$, a hole pattern ${\mathsf{P}}$ and scale factors $\varepsilon$, about which we make some hypotheses.
The \emph{unperforated domain} ${\mathsf{A}}$ satisfies the following assumptions, see Fig.{\mathrm e}f{fig:1} left,
\begin{enumerate}
\item ${\mathsf{A}}$ is a subset of the sector ${\mathsf S}_\omega$ and coincides with it near its vertex:
\begin{equation}
\label{eq:1E2}
\exists{\mathsf{g}}o_0>0 \quad\mbox{such that}\quad {\mathscr B}(0,{\mathsf{g}}o_0)\cap {\mathsf{A}} = {\mathscr B}(0,{\mathsf{g}}o_0)\cap {\mathsf S}_\omega,
\end{equation}
\item ${\mathsf{A}}$ is bounded, simply connected, and has a Lipschitz boundary,
\item ${\mathsf S}_\omega\setminus{\mathsf{A}}$ has a Lipschitz boundary,
\item $\partial{\mathsf{A}}\cap\partial{\mathsf S}_\omega$ is connected.
\end{enumerate}
The \emph{hole pattern} ${\mathsf{P}}$ satisfies, see Fig.{\mathrm e}f{fig:1} right,
\begin{enumerate}
\item ${\mathsf{P}}$ is a subset of the sector ${\mathsf S}_\omega$ and its complement ${\mathsf S}_\omega\setminus{\mathsf{P}}$ coincides with ${\mathsf S}_\omega$ at infinity:
\begin{equation}
\label{eq:1E3}
\exists{\mathsf{g}}o'_0>0 \quad\mbox{such that}\quad
{\mathsf{P}}\subset{\mathsf S}_\omega\cap{\mathscr B}(0,{\mathsf{g}}o'_0).
\end{equation}
\item ${\mathsf{P}}$ is a finite union of bounded simply connected Lipschitz domains ${\mathsf{P}}_j$, $j=1,\ldots,J$,
\item ${\mathsf S}_\omega\setminus{\mathsf{P}}$ has a Lipschitz boundary,
\item For any $j\in\{1,\ldots,J\}$, $\partial{\mathsf{P}}_j\cap\partial{\mathsf S}_\omega$ is connected.
\end{enumerate}
\begin{figure}
\caption{Limit domain ${\mathsf{A}
\label{fig:1}
\end{figure}
Let $\varepsilon_0={\mathsf{g}}o_0/{\mathsf{g}}o'_0$.
The family of perforated domains $\big({\mathsf{A}}_\varepsilon\big)_{0<\varepsilon<\varepsilon_0}$ is defined by, see Fig.{\mathrm e}f{fig:2},
\begin{equation}
\label{eq:Aeps}
{\mathsf{A}}_\varepsilon = {\mathsf{A}} \setminus \varepsilon\overline{\mathsf{P}},\quad\mbox{for}\quad
0<\varepsilon<\varepsilon_0.
\end{equation}
The family $\varepsilon{\mathsf{P}}$ can be seen as a self-similar collection of holes concentrating at the vertex of the sector.
Here, in contrast with \cite{DaMu15} we do not assume that $0$ belongs to ${\mathsf{P}}$. We do not even assume that $0$ does not belong to $\partial{\mathsf{P}}$.
Our assumptions \eqref{eq:1E2}, \eqref{eq:1E3} exclude some classes of self-similar perturbations of corner domains that are also interesting to study and have been analyzed using different methods, see \cite{DaToVi10}. For example, condition (2) in \eqref{eq:1E3} excludes the case of the approximation of a sharp corner by rounded corners constructed with circles of radius $\varepsilon$.
Condition (3) in \eqref{eq:1E3} excludes holes touching the boundary in a point.
The Lipschitz regularity conditions (2) and (3) in \eqref{eq:1E2}, \eqref{eq:1E3} are essential for our boundary integral equation approach.
On the other hand, the condition that ${\mathsf{A}}$ is simply connected and the related connectivity conditions (4) in \eqref{eq:1E2}, \eqref{eq:1E3} are not essential, they are merely made for simplicity of notation.
\begin{figure}
\caption{Perforated domain ${\mathsf{A}
\label{fig:2}
\end{figure}
\subsection{Dirichlet problems and mutiscale expansions}
We are interested in the collective behavior of solutions of the family of Poisson problems
\begin{equation}
\label{eq:poisson}
\begin{cases}
\begin{array}{rcll}
\Delta u_\varepsilon &=& f \quad& \mbox{in}\quad {\mathsf{A}}_\varepsilon,\\
u_\varepsilon &=& 0 \quad& \mbox{on}\quad \partial{\mathsf{A}}_\varepsilon.
\end{array}
\end{cases}
\end{equation}
We assume that the common right hand side $f$ is an element of $L^2({\mathsf{A}})$, which, by restriction to ${\mathsf{A}}_\varepsilon$, defines an element of $L^2({\mathsf{A}}_\varepsilon)$ and provides a unique solution $u_\varepsilon\in H^1_0({\mathsf{A}}_\varepsilon)$ to problem \eqref{eq:poisson}.
If moreover $f$ is infinitely smooth on $\overline{\mathsf{A}}$ {in a neighborhood of the origin}, then a description of the $\varepsilon$-behavior of $u_\varepsilon$ can be performed in terms of {\em multiscale asymptotic expansions}. We refer to \cite{MaNaPl00,DaToVi10} which apply to the present situation. As a result of this approach, cf \cite[Th.\,4.1 \& Sect.\,7.1]{DaToVi10}, $u_\varepsilon$ can be described by an asymptotic expansion containing two sorts of terms:
\begin{itemize}
\item Slow terms $u^\beta(t)$, defined in the standard variables $t$
\item Rapid terms, or {\em profiles}, $U^\beta(\frac{t}{\varepsilon})$, defined in the rapid variable $\frac{t}{\varepsilon}$.
\end{itemize}
Here the exponent $\beta$ runs in the set
$\,\N+\frac{\pi}{\omega}\N=\{\ell+k\frac\pi\omega,\: k,\ell\in\N\}$.
If $\frac{\pi}{\omega}$ is not a rational number, $u_\varepsilon$ can be expanded in powers of $\varepsilon$
\begin{equation}
\label{eq:multisca}
u_\varepsilon(t) \simeq \sum_{\beta\in\N +\frac{\pi}{\omega}\N} \varepsilon^\beta\,u^\beta(t)
+ \sum_{\beta\in\N +\frac{\pi}{\omega}\N} \varepsilon^\beta\,U^\beta(\tfrac{t}{\varepsilon})\,.
\end{equation}
The sums are asymptotic series, which means the following here:
Let $(\beta_n)_{n\in\N}$ be the strictly increasing enumeration of $\N+\frac{\pi}{\omega}\N$ and define the $N$th partial sum by
\begin{equation}
\label{eq:multiscN}
u_\varepsilon^{[N]}(t) = \sum_{n=0}^N \varepsilon^{\beta_n}\,u^{\beta_n}(t)
+ \sum_{n=0}^N \varepsilon^{\beta_n}\,U^{\beta_n}(\tfrac{t}{\varepsilon})
\,.
\end{equation}
Then for all $ N\in\N$ there exists $C_{N}$ such that for all $\varepsilon\in(0,\varepsilon_1]$
\begin{equation}
\label{eq:multiscb}
\big\Vert u_\varepsilon - u_\varepsilon^{[N]} \big\Vert_{H^1({\mathsf{A}}_\varepsilon)} \le C_N\,\varepsilon^{\beta_{N+1}}
\end{equation}
where we have chosen $\varepsilon_1<\varepsilon_0$.
If $\frac{\pi}{\omega}$ is a rational number, the terms corresponding to $\beta$ in the intersection $\beta\in\N\cap\frac{\pi}{\omega}\N_*$ contain a $\log\varepsilon$ and the estimate \eqref{eq:multiscb} has to be modified accordingly.
\subsection{Convergence analysis}
{If we want to have convergence of the series \eqref{eq:multisca}, it is not enough that the right hand side $f$ belongs to $L^2({\mathsf{A}})$ and not even that it is infinitely smooth near the origin, but in addition its asymptotic expansion (Taylor series) at the origin needs to be a convergent series converging to $f$. Thus we have to assume, and we will do this from now on,
that $f$
}
has an extension as a real analytic function in a neighborhood of the origin.
More specifically, we assume that there exist two positive constants $M_f$ and $C$ so that {$f\in L^2({\mathsf{A}})$ and}
\begin{equation}
\label{eq:f}
f(t) = \sum_{\alpha\in\N^2} f_\alpha\, t_1^{\alpha_1}t_2^{\alpha_2} ,
\quad \forall t\in {\mathscr B}(0,M_f^{-1})\cap{\mathsf{A}},\quad\mbox{with}\quad
|f_\alpha| \le C M_f^{|\alpha|}.
\end{equation}
{A simple special case would be a right hand side $f\in L^2({\mathsf{A}})$ that vanishes in a neighborhood of the origin. Likewise, one could consider, as in \cite{La08,DaMu15,BoDaDaMu16}, a variant of the boundary value problem \eqref{eq:poisson} that is driven not by a domain force $f$, but by a given trace on the boundary.}
In the present work we address the question of the convergence of the series \eqref{eq:multisca}
{under the assumption \eqref{eq:f}. In the above references \cite{MaNaPl00,DaToVi10} the recursive construction of the terms $u^\beta$ and $U^\beta$ of \eqref{eq:multisca} is performed without control of the constants $C_N$ in function of $N$, thus without providing any information on the convergence of the asymptotic series.
We will exploit the ``functional analytic approach'' to obtain this convergence.
}
It follows from general properties of power series that convergence of \eqref{eq:multisca} in the sense that
\[
\lim_{N\to\infty} \big\Vert u_\varepsilon - u_\varepsilon^{[N]} \big\Vert =0
\]
in some norm and for some $\varepsilon=\varepsilon_1>0$ implies that the series converges absolutely and unconditionally for any $\varepsilon\in(-\varepsilon_1,\varepsilon_1)$.
It will follow from our analysis that there exists a set $\Lambda_{\mathsf{s}}$ of real irrational numbers (super-exponential Liouville numbers, see Definition~{\mathrm e}f{def:Liouville}) with the property that whenever the opening angle $\omega$ does not belong to $\pi\Lambda_{\mathsf{s}}$, then such an $\varepsilon_1>0$ does indeed exist. For $\omega\in \pi\Lambda_{\mathsf{s}}$ on the other hand, in general the series \eqref{eq:multisca} does not converge for any $\varepsilon\ne0$. It is known from classical number theory that both $\Lambda_{\mathsf{s}}$ and its complement are uncountable and dense in $\R$ and $\Lambda_{\mathsf{s}}$ is of Lebesgue measure zero and even of Hausdorff dimension zero.
The series can be made convergent, however, for any $\omega\in(0,2\pi)$ by grouping together certain pairs of terms in the sums for which $\beta_{n+1}-\beta_n$ is small. This situation can also be expressed by the fact that there exists a subsequence $(N_k)_{k\in\N}$ of $\N$ such that for any $\varepsilon\in(-\varepsilon_1,\varepsilon_1)$
\[
\lim_{k\to\infty} \big\Vert u_\varepsilon - u_\varepsilon^{[N_k]} \big\Vert =0 \,.
\]
We call this kind of convergence ``stepwise convergence'', and the main result of this paper is the construction of a convergent series in this sense.
Our analysis relies on four main steps, developed in the four sections of this paper.
{\sc Step 1.} We set ${\tilde{u}}_\varepsilon = u_\varepsilon-u_0\on{{\mathsf{A}}_\varepsilon}$, where
$u_0\in H^1_0({\mathsf{A}})$ is the solution of the limit problem
\begin{equation}
\label{eq:u0}
\begin{cases}
\begin{array}{rcll}
\Delta u_0 &=& f \quad& \mbox{in}\quad {\mathsf{A}},\\
u_0 &=& 0 \quad& \mbox{on}\quad \partial{\mathsf{A}}\,.
\end{array}
\end{cases}
\end{equation}
Doing this, we reduce our investigation to the harmonic function ${\tilde{u}}_\varepsilon$, solution of the problem
\begin{equation}
\label{eq:tue}
\begin{cases}
\begin{array}{rcll}
\Delta {\tilde{u}}_\varepsilon &=& 0 \quad& \mbox{in}\quad {\mathsf{A}}_\varepsilon,\\
{\tilde{u}}_\varepsilon &=& -u_0 \quad& \mbox{on}\quad \partial{\mathsf{A}}_\varepsilon,
\end{array}
\end{cases}
\end{equation}
Since $u_0$ is zero on $\partial{\mathsf{A}}$, the trace of $u_0$ on $\partial{\mathsf{A}}_\varepsilon$ can be nonzero only on the boundary of the holes $\varepsilon\partial{\mathsf{P}}$. In order to analyze this trace, we expand $u_0$ near the origin in quasi-homogeneous terms with respect to the distance ${\mathsf{g}}o$ to the vertex according to the classical Kondrat'ev theory \cite{Kondratev67}. The investigation of the possible convergence of this series is far less classical, see \cite{BraDau82}, and it may involve stepwise convergent series.
This issue is also related to the stability of the terms in the expansion with respect to the opening, cf \cite{CostabelDauge93c,CostabelDauge94}. We provide rather explicit formulas for such expansions in complex variable form.
{\sc Step 2.} We transform problem \eqref{eq:tue} into a similar problem on a perforated domain for which the holes shrink to an \emph{interior} point of the limit domain, a situation studied in \cite{La08,DaMu15,DaMu16}. To get there, we compose two transformations that are compatible with the Dirichlet Laplacian,
\begin{itemize}
\item A conformal map of power type,
\item An odd reflection.
\end{itemize}
In this way the unperturbed sector domain ${\mathsf{A}}$ is transformed into a bounded simply connected Lipschitz domain ${\mathsf{B}}$ that contains the origin, and the hole pattern ${\mathsf{P}}$ is transformed into another hole pattern ${\mathsf{Q}}$ that is a finite union of simply connected bounded Lipschitz domains ${\mathsf{Q}}_j$. The small parameter is transformed into another small parameter $\eta$ by the power law
\[
\eta = \varepsilon^{\pi/\omega},
\]
and the new perforated domains ${\mathsf{B}}_\eta$ have the form
\[
{\mathsf{B}}_\eta = {\mathsf{B}}\setminus\eta\overline{\mathsf{Q}},\quad \eta\in(0,\eta_0).
\]
The boundary of ${\mathsf{B}}_\eta$ is the disjoint union of the external part $\partial{\mathsf{B}}$ and the boundary $\eta \hskip0.15ex\partial{\mathsf{Q}}$ of the holes $\eta{\mathsf{Q}}$. The holes shrink to the origin $0$, which now lies in the interior of the unperforated domain ${\mathsf{B}}$.
In this way problems \eqref{eq:tue} are transformed into Dirichlet problems on ${\mathsf{B}}_\eta$
\begin{equation}
\label{eq:pB}
\begin{cases}
\begin{array}{rcll}
\Delta v_\eta &=& 0 \quad& \mbox{in}\quad {\mathsf{B}}_\eta,\\
v_\eta &=& \mu_\eta \quad& \mbox{on}\quad \partial{\mathsf{B}}_\eta\,.
\end{array}
\end{cases}
\end{equation}
The family of Dirichlet traces $\mu_\eta$ are determined by the trace of $u_0$ on the family of boundaries $\varepsilon\partial{\mathsf{P}}$ of the holes. They have a special structure due to the mirror symmetry.
{\sc Step 3.} We study analytic families of model problems of this type where $\mu_\eta$ depends on $\eta$ as follows
\begin{equation}
\label{eq:psi}
\begin{cases}
\mu_\eta(x) = \psi(x) & \mbox{if $x\in\partial{\mathsf{B}}$,} \\
\mu_\eta(x) = \Psi(\frac{x}{\eta}) & \mbox{if $x\in\eta \hskip0.15ex\partial{\mathsf{Q}}$.}\\
\end{cases}
\end{equation}
Here we have a clear separation between the external boundary $\partial{\mathsf{B}}$ where $\mu_\eta$ does not depend on $\eta$, and the internal boundary $\eta \hskip0.15ex\partial{\mathsf{Q}}$ of $\partial{\mathsf{B}}_\eta$ that is the boundary of the scaled holes $\eta{\mathsf{Q}}$. Via representation formulas involving the double layer potential,
we transform the problem \eqref{eq:pB} with right hand side \eqref{eq:psi} into an equivalent system of boundary integral equations \eqref{eq:Meta} with a matrix of boundary integral operators $\cM(\eta)$ depending analytically on $\eta$ in a neighborhood of zero and such that $\cM(0)$ is invertible, see Theorem~{\mathrm e}f{thm:Meta=0}.
The crucial property making this possible is the homogeneity of the double layer kernel, which allows to write $\cM(\eta)$ in such a form that its diagonal terms are independent of $\eta$ and the off-diagonal terms vanish at $\eta=0$. The problem corresponding to the boundary integral operator $\cM(0)$ can be interpreted as a decoupled system of Dirichlet problems, one (in slow variables $x$) on the unperturbed domain ${\mathsf{B}}$ and a second one (in rapid variables $X=\frac{x}\eta$) on the complement $\R^2\setminus\overline{\mathsf{Q}}$ of the holes at $\eta=1$.
{\sc Step 4.}
From the formulation via an analytic family of boundary integral equations in Step~ 3 follows that there exists $\eta_1>0$ such that the solutions $v_\eta$ of problems \eqref{eq:pB}-\eqref{eq:psi} depend on the data $\psi\in H^{1/2}(\partial{\mathsf{B}})$ and $\Psi\in H^{1/2}(\partial{\mathsf{Q}})$ via a solution operator $\cL(\eta)$ that is analytic in $\eta$ for $\eta\in(-\eta_1,\eta_1)$ and, therefore, is given by a convergent series around $0$
\begin{equation}
\label{eq:gL}
\cL(\eta) = \sum_{n=0}^\infty \eta^n\cL_n,\quad
|\eta|\le\eta_1\,.
\end{equation}
Combining this with the results of Steps 1 and 2, we obtain expansions of the solutions $u_\varepsilon$ of problem \eqref{eq:poisson} in slow and rapid variables similar to \eqref{eq:multisca} that are not only asymptotic series as $\varepsilon\to0$ like in \eqref{eq:multiscb}, but convergent for $\varepsilon$ in a neighborhood of zero. The convergence is shown in weighted Sobolev norms and it is, in general, ``stepwise'' in the same sense as had been known for the convergence of the expansion in corner singular functions of the solution $u_0$ of the unperturbed problem \eqref{eq:u0}.
The main results on this kind of convergent expansions are given in Theorems~{\mathrm e}f{thm:ueps}, {\mathrm e}f{thm:outer} and {\mathrm e}f{thm:inner}.
While the series in powers of $\varepsilon$ are, under our general conditions, not unconditionally convergent due to the interaction of integer powers coming from the Taylor expansion of the right hand side $f$ in our problem \eqref{eq:poisson} and the powers of the form $k\pi/\omega$, $k\in\N$, coming from the corner singularities, there are two situations where the convergence is, in fact, unconditional. \\[0.5ex]
The first such situation is met when the opening angle $\omega$ is such that $\pi/\omega$ is either a rational number or, conversely, is not approximated too fast by rational numbers, namely not a super-exponential Liouville number as defined in Definition~{\mathrm e}f{def:Liouville}. In this case, the right hand side $f$ can be arbitrary, as long as it is analytic in a neighborhood of the corner,
see Corollaries {\mathrm e}f{cor:omegaQ} and {\mathrm e}f{cor:irrat}.
\\[0.5ex]
The second situation where we find unconditional convergence is met for arbitrary opening angles $\omega$ when the right hand side $f$ in \eqref{eq:poisson} vanishes in a neighborhood of the corner: Then we have the {\em converging expansion} in $L^\infty(\Omega)$
\begin{equation}
\label{eq:conv}
u_\varepsilon(t) = u_0(t)+\sum_{\beta\in \frac{\pi}{\omega}\N_*} \varepsilon^\beta\,u^\beta(t)
+ \sum_{\beta\in\frac{\pi}{\omega}\N_*} \varepsilon^\beta\,U^\beta(\tfrac{t}{\varepsilon}).
\end{equation}
Thus both parts of this two-scale decomposition of $u_\varepsilon$ are given by functions that are real analytic near zero in the variable $\eta=\varepsilon^{\pi/\omega}$, see Corollary~{\mathrm e}f{cor:f=0}.
\section{Unperturbed problem on a plane sector}
\label{s:2}
We are going to analyze the solution $u_0$ of problem \eqref{eq:u0} when the right hand side satisfies the assumption \eqref{eq:f}. We represent $u_0$ as the sum of three series converging in a neighborhood of the vertex:
\[
u_0 = u_{f} + u_{\partial} + u_{\sf rm}
\]
where
\begin{enumerate}
\item $u_{f}$ is a particular solution of $\Delta u = f$,
\item $u_{\partial}$ is a particular solution of $\Delta u = 0$, with $u_{\partial}+u_{f}=0$ on the sides $\vartheta = 0$ or $\omega$,
\item $u_{\sf rm}$ is the remaining part of $u_0$.
\end{enumerate}
We use the complex variable form of Cartesian coordinates
\begin{equation}
\label{eq:zeta}
\zeta = t_1 + it_2,\ \ \bar \zeta = t_1-it_2\quad\mbox{i.e.}\quad
\zeta = {\mathsf{g}}o e^{i\vartheta}.
\end{equation}
In particular, instead of \eqref{eq:f}, we write the Taylor expansion at origin of $f$ in the form
\begin{equation}
\label{eq:fcomp}
f(t) = \sum_{\alpha\in\N^2} \tilde f_\alpha\, \zeta^{\alpha_1}\bar\zeta{}^{\alpha_2}
\ \mbox{ in } \ {\mathscr B}(0,M_f^{-1}),\quad\mbox{with}\quad
|\tilde f_\alpha| \le C_M M^{|\alpha|},\ (M>M_f).
\end{equation}
\subsection{Interior particular solution}
The existence of a real analytic particular solution to the equation $\Delta u = f$ is a consequence of classical regularity results (cf.~Morrey and Nirenberg \cite{MoNi57}). Nevertheless, we can also provide an easy direct proof by an explicit formula using the complex variable representation \eqref{eq:fcomp}: It suffices to set
\begin{equation}
\label{eq:uf}
u_{f}(t) = \sum_{\alpha\in\N^2} \,
\frac{\tilde f_\alpha}{4(\alpha_1+1)(\alpha_2+1)}\,\zeta^{\alpha_1+1}\bar\zeta{}^{\alpha_2+1}
\ \mbox{ in } \ {\mathscr B}(0,M_f^{-1}).
\end{equation}
to obtain a particular real analytic solution to the equation $\Delta u = f$ in ${\mathscr B}(0,M_f^{-1})$.
\subsection{Lateral particular solution}\label{ss:1.2}
Set ${\mathsf{g}}o_1=\min\{{\mathsf{g}}o_0,M^{-1}\}$ for a chosen $M>M_f$. In the finite sector ${\mathsf{A}}\cap{\mathscr B}(0,{\mathsf{g}}o_1)$, the difference $\tilde u\equiv u_0-u_f$ is a harmonic function and its traces on the sides $\vartheta=0$ and $\vartheta=\omega$ coincide with $-u_f$.
Denote by $g^0$ and $g^\omega$ the restriction of $-u_f$ on the rays $\vartheta=0$ and $\vartheta=\omega$. These two functions are analytic in the variable ${\mathsf{g}}o$:
\begin{equation}
\label{eq:estG}
g^0 = \sum_{\ell\in\N_*} g^0_\ell {\mathsf{g}}o^\ell,\quad
g^\omega = \sum_{\ell\in\N_*} g^\omega_\ell {\mathsf{g}}o^\ell,\quad
|g^0_\ell| + |g^\omega_\ell| \le C {\mathsf{g}}o_1^{-\ell}.
\end{equation}
The constant ${\mathsf{g}}o_1>0$, which is a lower bound for the convergence radius of the power series \eqref{eq:estG}, is by construction less than $M_f^{-1}$, where $M_f$ is the constant in the assumption \eqref{eq:f} on the analyticity of the right hand side $f$.
Note that, by construction, $g^0$ and $g^\omega$ vanish at the origin.
As a next step in the analysis of $u_0$, we now construct a particular solution $u_{\partial}$ of the problem satisfied by $\tilde u$
\begin{equation}
\label{eq:Dirlat}
\left\{
\begin{array}{rcll}
\Delta u_{\partial}(t)&=&0 \quad& \forall t \in {\mathsf{A}}\cap {\mathscr B}(0,{\mathsf{g}}o_1)\,,\\
u_{\partial}(t)&=&-u_{f}(t) \quad& \forall t \in ({\mathsf{T}}_0\cup {\mathsf{T}}_\omega) \cap {\mathscr B}(0,{\mathsf{g}}o_1)\,. \\
\end{array}
\right.
\end{equation}
This solution uses the convergent series expansion \eqref{eq:estG} and will be given as a convergent series, too.
Following \cite{BraDau82}, for any positive integer $\ell\in\N_*$ we can write explicit particular solutions $w_\ell$ to the Dirichlet problem in the infinite sector ${\mathsf S}_\omega$
\begin{equation}
\label{eq:bvpellsect}
\left\{
\begin{array}{rcll}
\Delta w_\ell(t)&=&0 \quad& \forall t \in {\mathsf S}_\omega\,,\\
w_\ell(t)&=&g^0_\ell {\mathsf{g}}o^\ell \quad& \forall t \in {\mathsf{T}}_0 \,, \\
w_\ell(t)&=&g^\omega_\ell {\mathsf{g}}o^\ell \quad& \forall t \in {\mathsf{T}}_\omega \,,
\end{array}
\right.
\end{equation}
where ${\mathsf{T}}_0$ and ${\mathsf{T}}_\omega$ are the two sides of the sector ${\mathsf S}_\omega$.
The idea is then to give estimates of the $w_\ell$ that show convergence of the series
$ u_\partial=\sum_{\ell\in\N_*}w_\ell\,.
$
There exists always a (quasi-)homogeneous solution of degree $\ell$. The harmonic functions that are homogeneous of degree $\ell$ are
\[
\operatorname{Im}\zeta^\ell\quad\mbox{and}\quad\operatorname{Re}\zeta^\ell
\]
They are given in polar coordinates by ${\mathsf{g}}o^\ell\sin\ell\vartheta$ and ${\mathsf{g}}o^\ell\cos\ell\vartheta$. The determinant of their boundary values is $\sin \ell\omega$. If this is zero, we cannot solve \eqref{eq:bvpellsect} in homogeneous functions (except in the smooth case, i.e. when $\omega=\pi$, where we find that $w_\ell = b_\ell \, \operatorname{Re}\zeta^\ell$ with $b_\ell = g^0_\ell$ is a solution), but we need the quasihomogeneous function
\[
\operatorname{Im}(\zeta^\ell\log\zeta)\,.
\]
We find the solution
{\em(i)} If $\sin \ell\omega\neq0$, i.e. if $\ell\omega\not\in \pi \N$
\begin{equation}
\label{eq:solel}
\left\{\
\begin{aligned}
& w_\ell(t) = a_\ell \,\operatorname{Im}\zeta^\ell + b_\ell \, \operatorname{Re}\zeta^\ell
\quad \\
& \mbox{with}\quad
a_\ell = \frac{g^\omega_\ell - g^0_\ell\,\cos\ell\omega}{\sin \ell\omega}
\quad\mbox{and}\quad
b_\ell = g^0_\ell\,.
\end{aligned}
\right.
\end{equation}
In this case the solution to \eqref{eq:bvpellsect} is unique in the space of homogeneous functions of degree $\ell$.
{\em(ii)} If $\sin \ell\omega=0$, i.e. if $\ell\omega = k\pi$ with $k\in \N$, so $\cos \ell\omega=(-1)^k$,
\begin{equation}
\label{eq:solellog}
\left\{\
\begin{aligned}
& w_\ell(t) = a_\ell \,\operatorname{Im}(\zeta^\ell\log\zeta)
+ b_\ell \, \operatorname{Re}\zeta^\ell \\
& \mbox{with}\quad
a_\ell = \frac{g^\omega_\ell - g^0_\ell\,\cos\ell\omega}{\omega\cos \ell\omega}
\quad\mbox{and}\quad
b_\ell = g^0_\ell\,.
\end{aligned}
\right.
\end{equation}
We draw the following consequences according to whether $\frac\pi\omega$ is rational or not:
{\em(a)} If $\frac\pi\omega\in\Q$, then the coefficients $a_\ell$ and $b_\ell$ in \eqref{eq:solel}-\eqref{eq:solellog} are controlled since $\sin\ell\omega$ spans a finite set of values: There exists $C'$ such that
\begin{equation}
\label{eq:estaell}
|a_\ell| + |b_\ell| \le C' {\mathsf{g}}o_1^{-\ell},\quad\ell\in\N_*\,.
\end{equation}
{\em(b)} If $\frac\pi\omega\not\in\Q$,
estimating $a_\ell$ is hindered by the possible appearance of small denominators $\sin\ell\omega$. In Appendix~{\mathrm e}f{app:Liouville} we show that there exists a dense set of angles $\omega$ such that $\sin\ell\omega$ takes such small values that the series with $w_\ell$ defined by \eqref{eq:solel} will not converge, in general. The criterion is that $\pi/\omega$ belongs to the set $\Lambda_{\mathsf{s}}$ of super-exponential Liouville numbers, defined in Definition~{\mathrm e}f{def:Liouville} by their very fast approximability by rational numbers. We can restore the control of $w_\ell$ by modifying it as proposed in \cite{BraDau82,Dauge84}. For this, we ``borrow'' a term from the expansion \eqref{eq:urm} of $u_{\sf rm}$ in Section~{\mathrm e}f{ss:urm} below, namely a solution of the problem with zero lateral boundary conditions, a Laplace-Dirichlet singularity
\[
\operatorname{Im}\zeta^{k\pi/\omega} \quad\mbox{(harmonic in ${\mathsf S}_\omega$, zero on ${\mathsf{T}}_0$ and ${\mathsf{T}}_\omega$)}.
\]
Using this with $k
=\round{\ell\omega/\pi}
\in\N_*$ such that $|\ell\omega-k\pi|$ is minimal, we introduce a variant of $w_\ell$ from \eqref{eq:solel} by defining
\[
\widetilde w_\ell(t) =
a_\ell \,\big(\operatorname{Im}\zeta^\ell - \operatorname{Im}\zeta^{k\pi/\omega} \big)
+ b_\ell \, \operatorname{Re}\zeta^\ell
\]
with $a_\ell$ and $b_\ell$ as in \eqref{eq:solel}.
We note that
\[
a_\ell \,\big(\operatorname{Im}\zeta^\ell - \operatorname{Im}\zeta^{k\pi/\omega} \big) =
(g^\omega_\ell - g^0_\ell\,\cos\ell\omega) \operatorname{Im}\,
\frac{\zeta^\ell- \zeta^{k\pi/\omega}}{\sin\ell\omega}
\;.
\]
The quotient on the right is \emph{stable} because it can be expressed by {\em divided differences}:
\[
\frac{\zeta^\ell - \zeta^{k\pi/\omega}}{\sin\ell\omega} = \
\frac{\zeta^\ell - \zeta^{k\pi/\omega}}{\ell - k\pi/\omega}
\ \frac{\ell - k\pi/\omega}{\sin\ell\omega - \sin k\pi}\;.
\]
For fixed $\ell$, this is continuous in $\omega$, even if $\ell\omega\to k\pi$, and we recover the logarithmic term from \eqref{eq:solellog}:
\[
\lim_{\ell\omega\to k\pi} \operatorname{Im} \frac{\zeta^\ell - \zeta^{k\pi/\omega}}{\sin\ell\omega} =
\operatorname{Im}(\zeta^\ell\log\zeta) \ \frac{1}{\omega\cos\ell\omega}\;.
\]
For fixed $\omega$, we find a bound for the coefficient uniformly in $\ell$ if $|\ell\omega-k\pi|\le\pi/2$:
\[
\left| \frac{\ell - k\pi/\omega}{\sin\ell\omega} \right|
= \frac1\omega
\, \left| \frac{\ell\omega - k\pi}{\sin(\ell\omega- k\pi)} \right|
\le \frac\pi{2\omega}\,.
\]
The stable variant of \eqref{eq:solel}, which contains the logarithmic expressions \eqref{eq:solellog}, is therefore
\begin{equation}
\label{eq:solelk}
\left\{\
\begin{aligned}
& \widetilde w_\ell(t) = \tilde a_\ell \,\operatorname{Im} \frac{\zeta^\ell - \zeta^{k\pi/\omega}}{\ell - k\pi/\omega}
+ b_\ell \, \operatorname{Re}\zeta^\ell
\quad \\
& \mbox{with}\quad
\tilde a_\ell = (g^\omega_\ell - g^0_\ell\,\cos\ell\omega)
\ \frac{\ell - k\pi/\omega}{\sin\ell\omega}
\quad\mbox{and}\quad
b_\ell = g^0_\ell\,.
\end{aligned}
\right.
\end{equation}
We need this variant only when $|\ell\omega-k\pi|$ is small. We fix a threshold
\begin{equation}
\label{eq:thresh}
0<\delta_\omega<\tfrac12\,\min\{\omega,\pi\}
\end{equation}
and replace $w_\ell$ by $\widetilde w_\ell$ if there exists $k\in\N_*$ such that $|\ell\omega-k\pi|\le \delta_\omega$.
The bounds on $\delta_\omega$ imply on one hand that in this definition $k$ is defined uniquely by $\ell$, but $\ell$ is also uniquely determined by $k$. On the other hand, we can check that the coefficients $a_\ell$ and $\tilde a_\ell$ are uniformly controlled: There exists $C'$ independent of $\ell$ such that
\begin{equation}
\label{eq:esttaell}
|a^\flat_\ell| \le C' {\mathsf{g}}o_1^{-\ell}\,,\quad
\mbox{ where }
a^\flat_\ell=\Big\{\begin{array}{ll}a_\ell\\
\widetilde a_\ell\end{array}\;
\mbox{ if }\; \displaystylest(\ell\omega,\pi\N)\,
\Big\{\begin{array}{ll}>\delta_\omega\,,\\
\le \delta_\omega\,.\end{array}
\end{equation}
Thus, choosing for each value of $\ell$ solutions $w_\ell$ or $\widetilde w_\ell$, cf \eqref{eq:solel}-\eqref{eq:esttaell}, we obtain a convergent series expansion
for a particular solution $u_{\partial}$ of the (partial) Dirichlet problem
\eqref{eq:Dirlat}.
\subsection{Remaining boundary condition and convergence}\label{ss:urm}
Let us write in ${\mathsf{A}}\cap {\mathscr B}(0,{\mathsf{g}}o_1)$:
\begin{equation}
\label{eq:tu3}
u_0 = u_{f} + u_{\partial} + u_{\sf rm}\,.
\end{equation}
Now the function $u_{\sf rm}$ resolves the remaining boundary condition (here we choose ${\mathsf{g}}o'_1\in(0,{\mathsf{g}}o_1)$)
\begin{equation}
\label{bvpremain}
\left\{
\begin{array}{rcll}
\Delta u_{\sf rm}(t)&=&0 & \forall t \in {\mathsf{A}}\cap {\mathscr B}(0,{\mathsf{g}}o'_1)\,,\\
u_{\sf rm}(t)&=&0 & \forall t \in ({\mathsf{T}}_0\cup {\mathsf{T}}_\omega) \cap {\mathscr B}(0,{\mathsf{g}}o'_1)\,, \\
u_{\sf rm}(t)&=&g(t) & \forall t \in {\mathsf S}_\omega, \ \ |t|={\mathsf{g}}o'_1\,,
\end{array}
\right.
\end{equation}
where
\[
g(t) \equiv u_0(t)-u_{f}(t)-u_{\partial}(t) \quad\mbox{for}\quad |t|={\mathsf{g}}o'_1 \,.
\]
Denoting by $\Pi$ the arc $\vartheta\in(0,\omega)$, ${\mathsf{g}}o=1$, we can see that the trace $g$ belongs to $H^{1/2}_{00}({\mathsf{g}}o'_1\Pi)$. By partial Fourier expansion with respect to the eigenfunction basis $\big(\sin\frac{k\pi}{\omega}\vartheta\big)_{k\in\N_*}$ we find
\[
g(t) = \sum_{k\ge1} g_k \sin\frac{k\pi}{\omega}\vartheta,\quad t\in {\mathsf{g}}o'_1\Pi,
\]
with a bounded\footnote{In fact the sequence $\big(\sqrt{k}\,g_k\big)_{k\in\N_*}$ belongs to $\ell^2(\N_*)$.} sequence $\big(g_k\big)_{k\in\N_*}$, and we deduce the representation
\[
u_{\sf rm}(t) = \sum_{k\ge1} g_k \Big(\frac{{\mathsf{g}}o}{{\mathsf{g}}o'_1}\Big)^{k\pi/\omega}
\sin\frac{k\pi}{\omega}\vartheta,
\quad t\in\overline{\mathsf{A}}\cap {\mathscr B}(0,{\mathsf{g}}o'_1).
\]
Setting $c_{k\pi/\omega}=g_k({\mathsf{g}}o'_1)^{-k\pi/\omega}$, we find that the expansion for the remaining term can be written as the converging series
\begin{equation}
\label{eq:urm}
u_{\sf rm}(t) = \sum_{\gamma\in\frac\pi\omega\N_*}
c_\gamma \operatorname{Im}\zeta^{\gamma},
\quad t\in\overline{\mathsf{A}}\cap {\mathscr B}(0,{\mathsf{g}}o'_1),
\end{equation}
with the estimates
\begin{equation}
\label{eq:estcg}
|c_\gamma| \le C ({\mathsf{g}}o'_1)^{-\gamma}.
\end{equation}
The collection of formulas and estimates \eqref{eq:uf}, \eqref{eq:solel}-\eqref{eq:esttaell}, and \eqref{eq:urm}-\eqref{eq:estcg} motivates the following unified notation.
\begin{notation}
\label{not:1}
Let $\gA$ be the set of indices (here $\N^2_*$ denotes $\N^2\setminus\{(0,0)\}$)
\[
\gA = \N^2_* \cup \tfrac\pi\omega\N_*,
\]
and let $\gA_0$ be the subset of $\gA$ of elements of the form $(\ell,0)$ with $\ell\in\N_*$ such that there exists
\begin{equation}
\label{eq:ellk}
k\in\N_*\quad\mbox{with}\quad
|\ell\omega-k\pi| \le \delta_\omega
\quad\mbox{ (see \eqref{eq:thresh})}.
\end{equation}
For any $\gamma\in\gA$, we define the function $t\mapsto{\mathscr Z}_\gamma(t)$ as follows
\begin{enumerate}
\item If $\gamma\in\frac\pi\omega\N_*$, set ${\mathscr Z}_\gamma(t) = \zeta^\gamma$,
\item If $\gamma=(\alpha_1,\alpha_2)\in\N^2_*$ and $\gamma\not\in\gA_0$, set ${\mathscr Z}_\gamma(t) = \zeta^{\alpha_1}\bar\zeta{}^{\alpha_2}$,
\item If $\gamma=(\ell,0)\in\gA_0$, let $k$ be the unique integer such that \eqref{eq:ellk} holds. Set
\begin{equation}
\label{eq:Zgam}
\begin{cases}
{\mathscr Z}_\gamma(t) = \zeta^{\ell}\log \zeta & \mbox{if } \ell=\frac{k\pi}{\omega}, \\
\displaystylesplaystyle {\mathscr Z}_\gamma(t) = \frac{\zeta^\ell - \zeta^{k\pi/\omega}}{\ell - k\pi/\omega}
& \mbox{if } \ell\neq\frac{k\pi}{\omega}. \\
\end{cases}
\end{equation}
\end{enumerate}
\end{notation}
We are ready to prove the main result of this section:
\begin{theorem}
\label{th:tu}
Let $u_0$ be the solution of the unperturbed problem \eqref{eq:u0} with right hand side $f\in L^2({\mathsf{A}})$ satisfying \eqref{eq:f}.
We can represent $u_0$ as the sum of a convergent series in a neighborhood of the vertex $0$
\begin{equation}
\label{eq:tu}
u_0(t) = \operatorname{Im} \sum_{\gamma\in\gA} a_\gamma {\mathscr Z}_\gamma(t), \quad t\in{\mathsf{A}}\cap{\mathscr B}(0, {\mathsf{g}}o_1)
\end{equation}
where the set $\gA$ and the special functions ${\mathscr Z}_\gamma$ are introduced in Notation {\mathrm e}f{not:1}, and the coefficients $a_\gamma$ satisfy the analytic type estimates: for all $M>{\mathsf{g}}o_1^{-1}$ there exists $C$ such that
\begin{equation}
\label{eq:tuest}
|a_\gamma| \le C M^{|\gamma|},\quad \gamma\in\gA,
\end{equation}
where $|\gamma|=\alpha_1+\alpha_2$ if $\gamma=(\alpha_1,\alpha_2)\in(\N_*)^2$, and $|\gamma|=\gamma$ if $\gamma\in\frac\pi\omega\N_*$. The coefficients $a_\gamma$ are real if $\gamma\in\frac{\pi}{\omega}\N_*$ or if $\gamma=(\ell,0)\in\N^2_*$.
\end{theorem}
\begin{proof}
We start from the representation \eqref{eq:tu3} of $u_0$ in the three parts $u_{f}$, $u_{\partial}$ and $u_{\sf rm}$.
1) $u_{f}$ has the explicit expression \eqref{eq:uf} that can be written as\\
$\sum_{\alpha_1\in\N_*}\sum_{\alpha_2\in\N_*} b_\alpha \zeta^{\alpha_1}\bar\zeta{}^{\alpha_2}$ with suitable estimates for the coefficients $b_\alpha$:
\[
|b_\alpha| \le C M^{|\alpha|}.
\]
We notice that the set of indices $(\N_*)^2$ has an empty intersection with $\gA_0$. So
\[
u_{f}(t) = \sum_{\gamma\in(\N_*)^2} b_\gamma {\mathscr Z}_\gamma(t).
\]
Since $u_{f}$ is real, we can
set $a_\gamma=ib_\gamma$ and get
\[
u_{f}(t) = \operatorname{Im} \sum_{\gamma\in(\N_*)^2} a_\gamma {\mathscr Z}_\gamma(t).
\]
2) $u_{\partial}$ is equal to $\sum_{\ell\in\N_*}w^\flat_\ell$ with
\begin{itemize}
\item[{\em i)}] $w^\flat_\ell=w_\ell$ with $w_\ell$ given by \eqref{eq:solel} if $(\ell,0)\not\in\gA_0$,
\item[{\em ii)}] $w^\flat_\ell=w_\ell$ with $w_\ell$ given by \eqref{eq:solellog} if $\ell=\frac{k\pi}{\omega}$ for some $k\in\N_*$,
\item[{\em iii)}] $w^\flat_\ell=\widetilde w_\ell$ with $\widetilde w_\ell$ given by \eqref{eq:solelk} if $(\ell,0)\in\gA_0$ and $\ell\neq\frac{k\pi}{\omega}$, where $k$ is the integer such that \eqref{eq:ellk} holds.
\end{itemize}
We parse each of these three cases
{\em i)} $(\ell,0)\not\in\gA_0$: Then ${\mathscr Z}_{(\ell,0)}=\zeta^\ell$ and ${\mathscr Z}_{(0,\ell)}=\bar\zeta{}^\ell$. We use formula \eqref{eq:solel} to obtain
\begin{equation}
\label{eq:well}
w_\ell = \operatorname{Im} ( a_\ell {\mathscr Z}_{(\ell,0)} + ib_\ell {\mathscr Z}_{(0,\ell)})
\end{equation}
The coefficients $a_{(\ell,0)}=a_\ell$ and $a_{(0,\ell)}=ib_\ell$ satisfy the desired estimates, because $(\ell,0)\not\in\gA_0$ implies $|\sin\ell\omega|\ge\sin\frac{\omega}{2}$ .
{\em ii)} There exists $k\in\N_*$ such that $\ell=\frac{k\pi}{\omega}$: Then ${\mathscr Z}_{(\ell,0)}=\zeta^\ell\log\zeta$ and ${\mathscr Z}_{(0,\ell)}=\bar\zeta{}^\ell$. We use formula \eqref{eq:solellog} to obtain the representation \eqref{eq:well} again.
{\em iii)} $(\ell,0)\in\gA_0$ and $\ell\neq\frac{k\pi}{\omega}$, with the integer $k$ for which \eqref{eq:ellk} holds. Now we start from formula \eqref{eq:solelk} and find once more the representation \eqref{eq:well} with $a_\ell$ replaced by $\tilde a_\ell$.
3) Finally $u_{\sf rm}$ given by \eqref{eq:urm} is already written in the desired form.
\end{proof}
\begin{remark}
\label{rem:imre}
1) Examining the structure of the terms in \eqref{eq:tu} we can see that a real valued basis for the expansion of $u_0$ is the union of
\begin{itemize}
\item $\operatorname{Im}{\mathscr Z}_\gamma$ if $\gamma\in\frac\pi\omega\N_*$ or if $\gamma=(\alpha_1,\alpha_2)\in\N^2_*$ with $\alpha_1>\alpha_2$,
\item $\operatorname{Re}{\mathscr Z}_\gamma$ if $\gamma=(\alpha_1,\alpha_2)\in\N^2_*$ with $\alpha_1\le\alpha_2$.
\end{itemize}
2) The traces of the function $\operatorname{Im}{\mathscr Z}_{k\pi/\omega}$ are zero on $\partial{\mathsf S}_\omega$ for all $k\in\N_*$. If we write the expansion \eqref{eq:tu} in the form
\begin{equation}
\label{eq:tup}
u_0 = \sum_{k\in\N_*} a_{k\pi/\omega} \operatorname{Im}{\mathscr Z}_{k\pi/\omega}
+ \sum_{\ell\in\N_*} \Big(\operatorname{Im} \sum_{\substack {\gamma\in\N^2\\ |\gamma|=\ell}} a_\gamma {\mathscr Z}_\gamma\Big)
\end{equation}
we obtain terms $\operatorname{Im}{\mathscr Z}_{k\pi/\omega}$, or packets of terms $\operatorname{Im}\sum_{|\gamma|=\ell} a_\gamma {\mathscr Z}_\gamma$ that have zero traces on $\partial{\mathsf S}_\omega$.
\end{remark}
\begin{remark}
\label{rem:pi}
If $\omega=\pi$, then $u_0$ has a converging Taylor expansion at the origin.
\end{remark}
\begin{remark}
\label{rem:f=0}
If $f=0$ in a neigborhood of the origin, then in the above construction we find
that $u_{f}$ and $u_{\partial}$ vanish identically, hence $u_0=u_{\sf rm}$. For the latter we have the convergent expansion \eqref{eq:urm}, and therefore $u_0$ has an expansion in terms of $\operatorname{Im}\zeta^{k\frac\pi\omega}$, $k\in\N_*\,$, that is convergent in a neighborhood of the origin.
\end{remark}
\begin{remark}
\label{rem:other}
The definition of $\gA_0$ depends on the choice of the threshold $\delta_\omega$, see \eqref{eq:thresh}. This influences which pairs of terms $\zeta^\ell$ and $\zeta^{k\pi/\omega}$ are grouped together into ${\mathscr Z}_\gamma$ in the sum \eqref{eq:tu}, but changing $\gA_0$ does not change the sum. One can also omit a finite number of indices from $\gA_0$ without changing the sum. From Appendix~{\mathrm e}f{app:Liouville} follows that we can even set $\delta_\omega=0$ and therefore reduce $\gA_0$ to the empty set if $\pi/\omega$ is irrational, but not a super-exponential Liouville number. The resulting series in which no pairs of terms are regrouped will then converge, with a possibly smaller convergence radius than ${\mathsf{g}}o_1$ if $\pi/\omega$ is an exponential, but not super-exponential Liouville number. The full convergence radius ${\mathsf{g}}o_1$ is retained if $\pi/\omega$ is not an exponential Liouville number, in particular if it is not a Liouville number. If $\pi/\omega$ is a super-exponential Liouville number, then there exist right hand sides $f$ such that the unmodified series does not converge for $t\ne0$.
\end{remark}
\subsection{Residual problem on the perforated domain}
Setting $\tilde u_\varepsilon=u_\varepsilon-u_0$ with $u_\varepsilon$ and $u_0$ the solutions of problems \eqref{eq:poisson} and \eqref{eq:u0}, respectively, we obtain that $\tilde u_\varepsilon$ solves the residual problem
\begin{equation}
\label{eq:pepsgen}
\begin{cases}
\begin{array}{rcll}
\Delta \tilde u_\varepsilon &=& 0 \quad& \mbox{in}\quad {\mathsf{A}}_\varepsilon,\\
\tilde u_\varepsilon &=& -u_0 \quad& \mbox{on}\quad \partial{\mathsf{A}}_\varepsilon,
\end{array}
\end{cases}
\end{equation}
By construction, $u_0$ is zero on $\partial{\mathsf{A}}$, therefore on $\partial{\mathsf{A}}_\varepsilon\cap\partial{\mathsf{A}}$. Thus the trace of $u_0$ on $\partial{\mathsf{A}}_\varepsilon$ can be nonzero only on the part $\varepsilon\partial{\mathsf{P}}\cap{\mathsf S}_\omega$ of the boundary of the perforations, compare Fig.{\mathrm e}f{fig:2}. The converging expansion \eqref{eq:tu} allows us to interpret traces of $u_0$ on $\varepsilon\partial{\mathsf{P}}\cap{\mathsf S}_\omega$ as a series of traces on $\partial{\mathsf{P}}\cap{\mathsf S}_\omega$ with coefficients depending on $\varepsilon$.
To describe this dependence, we recall Notation {\mathrm e}f{not:1} and introduce corresponding combinations of powers of $\varepsilon$.
\begin{notation}
\label{not:2}
Let $\gA$ and $\gA_0$ be the sets of indices introduced in Notation {\mathrm e}f{not:1}. For any $\gamma\in\gA$ we define the function $\varepsilon\mapsto{\mathscr E}_\gamma(\varepsilon)$ as follows
\begin{enumerate}
\item If $\gamma\in\frac\pi\omega\N_*$, set ${\mathscr E}_\gamma(\varepsilon) = \varepsilon^\gamma$,
\item If $\gamma=(\alpha_1,\alpha_2)\in\N^2_*$ and $\gamma\not\in\gA_0$, set ${\mathscr E}_\gamma(\varepsilon) = \varepsilon^{|\gamma|}$,
\item If $\gamma=(\ell,0)\in\gA_0$, let $k$ be the unique integer such that \eqref{eq:ellk} holds. Set
\begin{equation}
\label{eq:Egam}
\begin{cases}
{\mathscr E}_\gamma(\varepsilon) = \varepsilon^{\ell}\log \varepsilon & \mbox{if } \ell=\frac{k\pi}{\omega}, \\
\displaystylesplaystyle {\mathscr E}_\gamma(\varepsilon) =
\frac{\varepsilon^\ell - \varepsilon^{k\pi/\omega}}{\ell - k\pi/\omega}
& \mbox{if } \ell\neq\frac{k\pi}{\omega}. \\
\end{cases}
\end{equation}
\end{enumerate}
\end{notation}
The functions ${\mathscr Z}_\gamma$ \eqref{eq:Zgam} are pseudo-homogeneous in the following sense.
\begin{lemma}
\label{lem:homo}
Let $\gamma\in\gA$ and $T\in{\mathsf S}_\omega$.
\begin{itemize}
\item If $\gamma\not\in\gA_0$, then
\[
{\mathscr Z}_\gamma(\varepsilon T) = \varepsilon^{|\gamma|} {\mathscr Z}_\gamma(T) = {\mathscr E}_\gamma(\varepsilon){\mathscr Z}_\gamma(T)\,.
\]
\item If $\gamma=(\ell,0)\in\gA_0$, let $k$ be the unique integer such that \eqref{eq:ellk} holds. We set $\gamma'=\frac{k\pi}{\omega}$ and we have
\[
{\mathscr Z}_\gamma(\varepsilon T) =
\varepsilon^{|\gamma|} {\mathscr Z}_{\gamma}(T) + {\mathscr E}_\gamma(\varepsilon) {\mathscr Z}_{\gamma'}(T)\,.
\]
\end{itemize}
\end{lemma}
\begin{corollary}
\label{cor:tu}
Under the conditions of Theorem {\mathrm e}f{th:tu}, using the packet expansion \eqref{eq:tup}, we find
\begin{multline}
\label{eq:tupe}
u_0(\varepsilon T) = \!\!
\sum_{\gamma\in\frac{\pi}{\omega}\N_*} \!\! a_{\gamma} \varepsilon^{\gamma} \operatorname{Im}{\mathscr Z}_{\gamma}(T)\\
+ \sum_{\ell\in\N_*} \varepsilon^\ell
\Big(\operatorname{Im} \sum_{\substack {\gamma\in\N^2\\ |\gamma|=\ell}} a_\gamma {\mathscr Z}_\gamma(T)\Big)
+ \sum_{\gamma\in\gA_0} a_\gamma {\mathscr E}_\gamma(\varepsilon)\operatorname{Im}{\mathscr Z}_{\gamma'}(T).
\end{multline}
Each of the terms or packets has zero trace on $\partial{\mathsf S}_\omega$.
\end{corollary}
\section{From a perforated sector to a domain with interior holes}
\label{s:3}
In this section, we transform the residual Dirichlet problem \eqref{eq:pepsgen} into a problem on a perforated domain with holes shrinking towards an interior point, so that to be able to use integral representations for its solution.
A suitable transformation is obtained as the composition of two operations, see Fig.{\mathrm e}f{fig:3}:
\begin{itemize}
\item A conformal map $\cG_\kappa$: $\zeta\mapsto z = \zeta^{\kappa}$ with $\kappa=\frac\pi\omega$ that transforms the sector ${\mathsf S}_\omega$ into the upper half-plane ${\mathsf S}_\pi=\R\times\R_+$,
\item The odd reflection operator $\cE$ that extends domains and functions from ${\mathsf S}_\pi$ to $\R^2$.
\end{itemize}
\begin{figure}
\caption{Conformal map and symmetry acting on hole pattern ${\mathsf{P}
\label{fig:3}
\end{figure}
We introduce these two operations and list some of their properties before composing them in view of the transformation of problem \eqref{eq:pepsgen}.
\subsection{Conformal mapping of power type}\label{ss:conf}
Let $\omega\in(0,2\pi)$ and $\kappa>0$ be chosen so that $\kappa\omega<2\pi$. The conformal map $\cG_\kappa$: $\zeta\mapsto z = \zeta^{\kappa}$ transforms Cartesian coordinates $t$ into Cartesian coordinates $x$ with
\[
\zeta = t_1+it_2\quad\mbox{and}\quad z = x_1+ix_2
\]
and polar coordinates $({\mathsf{g}}o,\vartheta)$ into $(r,\theta)$ with
\[
r = {\mathsf{g}}o^\kappa\quad\mbox{and}\quad \theta = \kappa\vartheta.
\]
\begin{lemma}
\label{lem:LipGk}
Assume that $\Omega\subset{\mathsf S}_\omega$ and that $\Omega$ and ${\mathsf S}_\omega\setminus\overline\Omega$ have a Lipschitz boundary. Then $\cG_\kappa\Omega\subset{\mathsf S}_{\kappa\omega}$ and, moreover, $\cG_\kappa\Omega$ and ${\mathsf S}_{\kappa\omega}\setminus\overline{\cG_\kappa\Omega}$ have a Lipschitz boundary.
\end{lemma}
A function $u$ defined on such a domain $\Omega\subset{\mathsf S}_\omega$ is transformed into a function $\cG_\kappa^*u$ defined on $\cG_\kappa\Omega$ through the composition
\[
\cG_\kappa^*u = u\circ\cG^{-1}_\kappa = u\circ\cG_{1/\kappa}\,.
\]
If $\partial\Omega$ is disjoint from the origin, then for any real $s$, the transformation $\cG_\kappa^*$ defines an isomorphism from the standard Sobolev space $H^s(\Omega)$ onto $H^s(\cG_\kappa\Omega)$.
If, on the contrary, $0\in\partial\Omega$, there is no such simple transformation law for standard Sobolev spaces. Nevertheless, weighted Sobolev spaces of Kondrat'ev type can be equivalently expressed using polar coordinates and support such transformation: For real $\beta$ and natural integer $m$, the space $K^m_\beta(\Omega)$ is defined as
\begin{equation}
\label{eq:Kmb}
K^m_\beta(\Omega) = \{u\in L^2_{\rm loc}(\Omega),\quad
{\mathsf{g}}o^{\beta+|\alpha|}\partial^\alpha_t u\in L^2(\Omega),\ \ \forall\alpha\in\N^2,\ |\alpha|\le m\}.
\end{equation}
We have the equivalent definition in polar coordinates
\[
K^m_\beta(\Omega) = \{u\in L^2_{\rm loc}(\Omega),\quad
{\mathsf{g}}o^{\beta}({\mathsf{g}}o\partial_{\mathsf{g}}o)^{\alpha_1} \partial^{\alpha_2}_\vartheta u\in L^2(\Omega),\ \
\forall\alpha\in\N^2,\ |\alpha|\le m\}.
\]
\begin{lemma}
\label{lem:Km}
The conformal map $\cG_\kappa$ defines an isomorphism
\[
\cG^*_\kappa : K^m_\beta(\Omega) \quad\mbox{onto}\quad K^m_{\frac{1+\beta}{\kappa}-1}(\cG_\kappa\Omega).
\]
\end{lemma}
The proof is based on the formulas
\[
{\mathsf{g}}o\partial_{\mathsf{g}}o = \kappa\, r\partial_r \quad\mbox{and}\quad
{\mathsf{g}}o {\mathrm d}{\mathsf{g}}o{\mathrm d}\vartheta = r^{\frac2\kappa-2} r{\mathrm d} r{\mathrm d}\theta.
\]
Details are left to the reader.
A relation between standard Sobolev spaces $H^s$ for real positive $s$ and the weighted scale $K^m_\beta$ is the following \cite[Appendix A]{Dauge88}
\begin{equation}
\label{eq:KmHs}
K^m_\beta(\Omega) \subset H^s(\Omega)\quad\mbox{if}\quad m\ge s\quad\mbox{and}\quad \beta<-s.
\end{equation}
Coming back to the solution $u_0$ of problem \eqref{eq:u0}, we check that as a consequence of \eqref{eq:tu} and of Lemma {\mathrm e}f{lem:Km}, there holds:
\begin{lemma}
\label{lem:u0Km}
Let $m\ge1$ be an integer. Then the solution $u_0$ of problem \eqref{eq:u0} satisfies
\begin{equation}
\label{eq:u0Km}
u_0\in K^m_\beta({\mathsf{A}})\quad\forall\beta\quad\mbox{such that}\quad
1+\beta>-\min\{\tfrac{\pi}{\omega},2\}.
\end{equation}
Let $\kappa>0$. Then
\begin{equation}
\label{eq:u0KmGk}
\cG_\kappa u_0\in K^m_{\beta'}(\cG_\kappa{\mathsf{A}})\quad\forall\beta'\quad\mbox{such that}\quad
1+\beta'>-\tfrac{1}{\kappa}\,\min\{\tfrac{\pi}{\omega},2\}.
\end{equation}
\end{lemma}
\subsection{Reflection and odd extension}\label{Ss:Reflection}
We first denote by $\cR$ the mapping from $\mathbb{R}^2$ to itself defined as the reflection across the $x_1$ axis
\[
\cR(x_1,x_2)\equiv (x_1,-x_2) \qquad \forall x=(x_1,x_2) \in \mathbb{R}^2\, .
\]
Then if $\Omega$ is a subset of $\mathbb{R}^2$ and $g$ a function defined on $\Omega$, we denote by $\cR^\ast[g]$ the function on $\cR(\Omega)$ defined by
\[
\cR^\ast[g](x)=g(\cR(x)) \qquad \forall x \in \cR(\Omega)\, .
\]
Let $\Omega$ be a subdomain of the half-plane ${\mathsf S}_\pi$. Let us set
\[
\Gamma = {\rm interior}\,(\partial\Omega\cap\partial{\mathsf S}_\pi).
\]
We denote by $\cE(\Omega)$ the symmetric extension of $\Omega$ across the $x_1$ axis
\begin{equation}
\label{eq:EOm}
\cE(\Omega) = \Omega \cup \cR(\Omega) \cup \Gamma.
\end{equation}
Since we want to stay within the category of Lipschitz domains, we need here the assumption that both $\Omega$ and its complement in ${\mathsf S}_\pi$ are Lipschitz.
Note that whereas for a Lipschitz domain its complement in $\R^2$ is automatically Lipschitz, too, this is not the case, in general, for the complement in ${\mathsf S}_\pi$.
This is the reason why we had to make corresponding assumptions in Subsection~{\mathrm e}f{ss:geomset}, see assumptions (2) and (3) on the domain ${\mathsf{A}}$ and the perforations ${\mathsf{P}}$. Under these assumptions $\cE(\Omega)$ is a Lipschitz domain. Since the proof of this fact is rather technical, we present it in Appendix~{\mathrm e}f{app:symextlip}, Lemma~{\mathrm e}f{lem:LipE}.
If $g$ is a function defined on $\Omega$, the \emph{odd extension} of $g$ to $\cE(\Omega)$ is defined as
\[
\cE^*[g](x)\equiv
\left\{
\begin{array}{ll}
g(x)\qquad & \forall x\in\Omega \\
-g(\cR(x))\qquad & \forall x\in\cR(\Omega)\\
0 & \forall x\in\Gamma\,.
\end{array}
\right.
\]
Let us denote by $H^1_{0,\Gamma}(\Omega)$ the following subspace of $H^1(\Omega)$
\[
H^1_{0,\Gamma}(\Omega) = \{u\in H^1(\Omega),\quad u\on{\Gamma}=0\}.
\]
\begin{lemma}
\label{lem:KmE}
Assume that $\Omega\subset{\mathsf S}_\pi$ and that $\Omega$ and ${\mathsf S}_\pi\setminus\overline\Omega$ have a Lipschitz boundary. Then the odd extension $\cE^*$ defines a bounded embedding
\[
\cE^*: K^2_\beta(\Omega)\cap H^1_{0,\Gamma}(\Omega) \longrightarrow K^2_\beta(\cE(\Omega))\quad
\forall \beta\in\R.
\]
\end{lemma}
\begin{proof}
If $u$ belongs to $K^2_\beta(\Omega)\cap H^1_{0,\Gamma}(\Omega)$, the jumps of $\cE^*[u]$ and of $\partial_2\cE^*[u]$ across $\Gamma$ are zero. Hence for all multiindices $\alpha$, $|\alpha|\le2$, the partial derivative $\partial^\alpha \cE^*[u]$ has no density across $\Gamma$ and
\[
\DNormc{r^{|\alpha|+\beta}\partial^\alpha \cE^*[u]}{L^2({\cE(\Omega)})} =
2\DNormc{r^{|\alpha|+\beta}\partial^\alpha u}{L^2(\Omega)}.
\]
\end{proof}
\subsection{Transformation of the residual problem}\label{ss:residual}
We come back to our main setting, with unperforated domain ${\mathsf{A}}$, hole pattern ${\mathsf{P}}$, and family of perforated domains ${\mathsf{A}}_\varepsilon$. We denote by $\cT$ and $\cT^*$ the composition of the conformal map $\cG_{\pi/\omega}$ and the odd extension acting on domains and functions respectively
\begin{equation}
\label{eq:TT}
\cT = \cE\circ\cG_{\pi/\omega}\quad\mbox{and}\quad \cT^* = \cE^*\circ\cG^*_{\pi/\omega}\,.
\end{equation}
Then we denote
\[
{\mathsf{B}} = \cT({\mathsf{A}}),\quad
{\mathsf{Q}} = \cT({\mathsf{P}}).
\]
As a consequence of assumptions on ${\mathsf{A}}$ and ${\mathsf{P}}$, and of Lemmas {\mathrm e}f{lem:LipGk} and {\mathrm e}f{lem:LipE}, ${\mathsf{B}}$ is a bounded simply connected Lipschitz domain containing the origin, and ${\mathsf{Q}}$ is a finite union of bounded simply connected Lipschitz domains.
The perforated sector ${\mathsf{A}}_\varepsilon$ is transformed by $\cT$ into the perforated domain ${\mathsf{B}}_\eta$ with
\begin{equation}
\label{eq:eta}
\eta = \varepsilon^{\pi/\omega}\quad\mbox{and}\quad {\mathsf{B}}_\eta = {\mathsf{B}}\setminus\eta\overline{\mathsf{Q}}.
\end{equation}
We note that the boundary of ${\mathsf{B}}_\eta$ is the disjoint union of $\partial{\mathsf{B}}$ and $\eta \hskip0.15ex\partial{\mathsf{Q}}$, see Fig.{\mathrm e}f{fig:4}:
\begin{equation}
\label{eq:dBe}
\partial{\mathsf{B}}_\eta = \partial{\mathsf{B}}\cup\eta \hskip0.15ex\partial{\mathsf{Q}}.
\end{equation}
\begin{figure}
\caption{Transformed perforated domain ${\mathsf{B}
\label{fig:4}
\end{figure}
The residual problem \eqref{eq:pepsgen} on ${\mathsf{A}}_\varepsilon$ is transformed into the Dirichlet problem on ${\mathsf{B}}_\eta$
\begin{equation*}
\begin{cases}
\begin{array}{rcll}
\Delta v_\eta &=& 0 \quad& \mbox{in}\quad {\mathsf{B}}_\eta,\\
v_\eta &=& -\cT^*[u_0] \quad& \mbox{on}\quad \partial{\mathsf{B}}_\eta,
\end{array}
\end{cases}
\end{equation*}
where we have set $v_\eta=\cT^*[\widetilde u_\varepsilon]$. We note that $v_\eta$ belongs to $H^1({\mathsf{B}}_\eta)$ and that its trace is zero on $\partial{\mathsf{B}}$. We analyze now the structure of the trace of $\cT^*[u_0]$ on $\eta \hskip0.15ex\partial{\mathsf{Q}}$. We take advantage of the converging expansion \eqref{eq:tu} and of the pseudo-homogeneity of its terms.
We recall from Notation {\mathrm e}f{not:1} that the set of indices $\gA$ is the union of $\frac{\pi}{\omega}\N_*$ and $\N^2_*$, and from Notation {\mathrm e}f{not:2} that the pseudo-homogeneous functions ${\mathscr E}_\gamma(\varepsilon)$ are defined as $\varepsilon^{|\gamma|}$ if $\gamma$ does not belong to the set of exceptional indices $\gA_0$, and by a divided difference or a logarithmic term in the opposite case.
\begin{theorem}
\label{th:Tu0}
Let $u_0$ be the solution of the unperturbed problem \eqref{eq:u0} with right hand side $f\in L^2({\mathsf{A}})$ satisfying \eqref{eq:f}. The residual problem \eqref{eq:pepsgen} on ${\mathsf{A}}_\varepsilon$ is transformed by the transformation $\cT$ \eqref{eq:TT} into the Dirichlet problem on ${\mathsf{B}}_\eta$, with $\eta=\varepsilon^{\pi/\omega}$:
\begin{equation}
\label{eq:petagen}
\begin{cases}
\begin{array}{rcll}
\Delta v_\eta &=& 0 \quad& \mbox{in}\quad {\mathsf{B}}_\eta,\\
v_\eta &=& 0 \quad& \mbox{on}\quad \partial{\mathsf{B}},\\
v_\eta &=& -\cT^*[u_0] \quad& \mbox{on}\quad \eta\partial{\mathsf{Q}}\,.
\end{array}
\end{cases}
\end{equation}
The trace of $\cT^*[u_0]$ can be written as a convergent sum for $\varepsilon\in(0,\varepsilon_1]$ for some positive $\varepsilon_1$
\begin{equation}
\label{eq:Tu0}
\cT^*[u_0](\eta X) = \sum_{\gamma\in\gA} {\mathscr E}_\gamma(\varepsilon)\Psi_\gamma(X), \quad X\in\partial{\mathsf{Q}}\,,
\end{equation}
where the set $\gA$ and the functions ${\mathscr E}_\gamma$ are introduced in Notations {\mathrm e}f{not:1} and {\mathrm e}f{not:2}.
There exists a positive number $\tau\in(0,1/2)$ such that the convergence takes place in the trace Sobolev space $H^{\tau+1/2}(\partial{\mathsf{Q}})$: There exist positive constants $C$ and $M$ such that
\begin{equation}
\label{eq:Tu0est}
\DNorm{\Psi_\gamma}{H^{\tau+1/2}(\partial{\mathsf{Q}})} \le C M^{|\gamma|},\quad \gamma\in\gA\,.
\end{equation}
\end{theorem}
\begin{proof}
We use the expansion of $u_0(\varepsilon T)$ as written by packets in \eqref{eq:tupe}. Applying the transformation $\cT^*$ we find
\begin{equation}
\label{eq:tupeT}
\begin{aligned}
\cT^*[u_0](\eta X) =&
\sum_{\gamma\in\frac{\pi}{\omega}\N_*} a_{\gamma} \varepsilon^{\gamma} \cT^*[\operatorname{Im}{\mathscr Z}_{\gamma}](X) \\
+& \sum_{\ell\in\N_*} \varepsilon^\ell \cT^*
\Big[\operatorname{Im} \sum_{\substack {\gamma\in\N^2\\ |\gamma|=\ell}} a_\gamma {\mathscr Z}_\gamma\Big](X)
+ \sum_{\gamma\in\gA_0} a_\gamma {\mathscr E}_\gamma(\varepsilon) \cT^*[\operatorname{Im}{\mathscr Z}_{\gamma'}](X).
\end{aligned}
\end{equation}
We define for $\gamma\in\gA$
\begin{equation}
\label{eq:Phi}
\Phi_\gamma =
\begin{cases}
\ a_{\gamma} \operatorname{Im}{\mathscr Z}_{\gamma}
\quad & \mbox{if}\quad \gamma\in\frac{\pi}{\omega}\N_*,\\
\ \operatorname{Im} \sum_{\tilde\gamma\in\N^2,\ |\tilde\gamma|=\ell} a_{\tilde\gamma} {\mathscr Z}_{\tilde\gamma}
\quad & \mbox{if}\quad \gamma =(0,\ell),\ \ell\in\N_*,\\
\ a_\gamma \operatorname{Im}{\mathscr Z}_{\gamma'}
\quad & \mbox{if}\quad \gamma =(\ell,0)\in\gA_0,\\
\ 0 \quad & \mbox{for remaining $\gamma$'s , }
\end{cases}
\end{equation}
and set
\begin{equation}
\label{eq:Psi}
\Psi_\gamma = \cT^*[\Phi_\gamma],\quad\forall\gamma\in\gA.
\end{equation}
Thus \eqref{eq:tupeT}-\eqref{eq:Psi} imply \eqref{eq:Tu0}.
Let us prove estimates \eqref{eq:Tu0est}. Let us choose $\beta<-1$ such that $1+\beta>-\min\{\tfrac{\pi}{\omega},2\}$, cf \eqref{eq:u0Km}. Relying on the explicit form of the functions $\Phi_\gamma$ and on the boundedness of the domain ${\mathsf{P}}$, we find that there exist constants $C$ and $M$ such that
\begin{equation}
\label{eq:PhiNorm}
\DNorm{\Phi_\gamma}{K^2_\beta({\mathsf{P}})} \le C M^{|\gamma|},\quad \gamma\in\gA\,.
\end{equation}
Let $\beta'=\frac{\omega}{\pi}(1+\beta)-1$. Then $\beta'<-1$ and by Lemma {\mathrm e}f{lem:Km} the conformal map $\cG^*_{\pi/\omega}$ is bounded from $K^2_{\beta}({\mathsf{P}})$ to $K^2_{\beta'}(\cG_{\pi/\omega}{\mathsf{P}})$. Then by Lemma {\mathrm e}f{lem:KmE}, the odd extension $\cE^*$ is bounded from $K^2_{\beta'}(\cG_{\pi/\omega}{\mathsf{P}})$ to $K^2_{\beta'}({\mathsf{Q}})$. If we choose $\tau$ such that
\[
\tau \le -(1+\beta') \quad\mbox{and}\quad \tau\in(0,\tfrac{1}{2}),
\]
we find that by \eqref{eq:KmHs}, the space $K^2_{\beta'}(\cG_{\pi/\omega}{\mathsf{P}})$ is continuously embedded in $H^{\tau+1}({\mathsf{Q}})$. The trace theorem for Lipschitz domains then yields the continuity of the trace from $H^{\tau+1}({\mathsf{Q}})$ to $H^{\tau+1/2}(\partial{\mathsf{Q}})$. Hence there exists $C'$ such that
\[
\DNorm{\Psi_\gamma}{H^{\tau+1/2}(\partial{\mathsf{Q}})} \le C'\DNorm{\Phi_\gamma}{K^2_\beta({\mathsf{P}})},
\quad\forall\gamma\in\gA \,.
\]
Combining this with the previous estimate of $\DNorm{\Phi_\gamma}{K^2_\beta({\mathsf{P}})}$ gives estimates \eqref{eq:Tu0est}.
We finally notice that estimates \eqref{eq:Tu0est} imply the convergence of the series \eqref{eq:Tu0} for $\varepsilon\in[0,\varepsilon_1]$ if $\varepsilon_1$ is chosen such that $\varepsilon_1M<1$.
\end{proof}
\begin{remark}
\label{rem:tau}
From the proof one finds a bound for the regularity index $\tau$
\begin{equation}
\label{eq:boundontau}
\tau<\min\{\tfrac12,\tfrac{2\omega}\pi\}\,.
\end{equation}
This inequality, which is restrictive for small angles $\omega< \tfrac\pi4$, is mainly due to the use of the embedding \eqref{eq:KmHs} of the weighted Sobolev spaces into unweighted Sobolev spaces. It is needed only in the special situation where the boundary of the holes touches the origin. If, conversely, $0\not\in\partial{\mathsf{P}}$, then we can use the trace theorem directly without passing by the embedding \eqref{eq:KmHs}, and the statement of the theorem is true for all $\tau\in(0,1/2)$.
\end{remark}
\section{Symmetric perforated Lipschitz domains}
\label{s:symperf}
In this section we investigate the asymptotic behavior of the solution of a Dirichlet problem in a symmetric Lipschitz domain with small holes. The analysis here performed will allow to study the behavior of the solution of problem \eqref{eq:petagen}.
More precisely, we will consider the case where
the domain and its holes are symmetric with respect to the horizontal axis and the boundary data are antisymmetric.
We use the technique with which the behavior of harmonic functions in perforated planar domains was studied in Lanza de Cristoforis \cite{La08} and in Dalla Riva and Musolino \cite{DaMu15}. As in \cite{DaMu15}, we employ boundary integral equations, but there are some differences in the assumptions:
In \cite{DaMu15}, perforations were of class $C^{1,\alpha}$ and connected, whereas we consider here perforations with Lipschitz boundaries and a finite number of connected components. This generalization is naturally implied by the construction of the perforation ${\mathsf{Q}}$ from ${\mathsf{P}}$ by the conformal transformations and reflections described in the previous sections, because even for a smooth and connected hole ${\mathsf{P}}$ in the sector ${\mathsf S}_\omega$, the resulting perforation ${\mathsf{Q}}$ in $\R^2$ may have corners or several connected components.
On the other hand, our symmetry assumptions will allow to simplify notably the treatment of the problem. In particular, we find that we do not have to deal with the logarithmic behavior which arises in the general setting for two-dimensional perforated domains.
\subsection{Some notions of potential theory on Lipschitz domains}\label{ss:prelLip}
We collect here some known results about harmonic double layer potentials on Lipschitz domains in the plane. Main references for these facts are the paper by Costabel \cite{Co88} and the books by Folland \cite{Fo95} and McLean \cite{Mc00}.
We assume that $\Omega\subset\R^2$ is a bounded Lipschitz domain (a role that will mainly be played by the perforated domain ${\mathsf{B}}\setminus\eta{\mathsf{Q}}$). Furthermore, $\Omega$ will be connected, but its complement
$\Omega^\mathrm{comp}lement=\mathbb{R}^2 \setminus \overline{\Omega}$
may be not connected.
Let $\Omega^\mathrm{comp}lement_{(1)}, \dots, \Omega^\mathrm{comp}lement_{(m)}$ be the bounded connected components of $\Omega^\mathrm{comp}lement$ and $\Omega^\mathrm{comp}lement_{(0)}$ the unbounded connected component of $\Omega^\mathrm{comp}lement$. Thus the boundary $\partial\Omega$ has the $m+1$ connected components
$\partial\Omega^\mathrm{comp}lement_{(0)}, \dots, \partial\Omega^\mathrm{comp}lement_{(m)}$.
Let $E $ be the function from $\mathbb{R}^2\setminus\{0\}$ to $\mathbb{R}$ defined by
\[
E (x) \equiv
-\frac{1}{2\pi}\log |x| \qquad \forall x\in
{\mathbb{R}}^{2}\setminus\{0\}.
\]
As is well known, $E $ is a fundamental solution of $-\Delta$ on $\mathbb{R}^2$.
If $\phi$ is an integrable function on $\partial \Omega$, we define the double layer potential $\mathcal{D}_{\partial\Omega}[\phi]$ by setting
\[
\mathcal{D}_{\partial\Omega}[\phi] (x)\equiv
-\int_{\partial\Omega}\phi(y)\, n(y)\cdot \nabla E (x-y)\,\mathrm{d}s_y
\qquad\forall x\in\mathbb{R}^2 \setminus \partial \Omega\, ,
\]
where $\mathrm{d}s$ denotes the length element on $\partial\Omega$ and $n$ denotes the outward unit normal to $\Omega$, which exists almost everywhere on $\partial\Omega$.
In $\R^2\setminus\partial\Omega$, the double layer potential $\mathcal{D}_{\partial\Omega}[\phi]$ is a harmonic function, vanishing at infinity.
By Costabel \cite[Thm.~1]{Co88}, if $\tau \in [-1/2,1/2]$ and $\phi \in H^{1/2+\tau}(\partial \Omega)$ then
\begin{equation}
\label{eq:dblreptau}
\mathcal{D}_{\partial\Omega}[\phi]\on{\Omega} \in H^{1+\tau}(\Omega)\, , \qquad \mathcal{D}_{\partial\Omega}[\phi]\on{\Omega^\mathrm{comp}lement} \in H^{1+\tau}_{\mathrm{loc}}(\Omega^\mathrm{comp}lement)\, .
\end{equation}
We denote by $\gamma_{0}$ and $\gamma_{0}^\mathrm{comp}lement$ the interior and exterior traces on $\partial \Omega$, respectively, and by $\gamma_{1}$ and $\gamma_{1}^\mathrm{comp}lement$ the interior and exterior normal derivatives on $\partial \Omega$, respectively, (both taken with respect to the exterior normal $n$). Then we have the jump relations \cite[Lem.~4.1]{Co88}
\[
\gamma_{0}^\mathrm{comp}lement\mathcal{D}_{\partial\Omega}[\phi]-\gamma_{0}\mathcal{D}_{\partial\Omega}[\phi]=\phi \ , \qquad \gamma_{1} \mathcal{D}_{\partial\Omega}[\phi]=\gamma_{1}^\mathrm{comp}lement\mathcal{D}_{\partial\Omega}[\phi]\, ,
\]
for all $\phi \in H^{1/2+\tau}(\partial \Omega)$. We introduce the boundary operators $\mathcal{K}_{\partial\Omega}$ and $\mathcal{W}_{\partial \Omega}$ by setting
\[
\mathcal{K}_{\partial \Omega}[\phi]\equiv \frac{1}{2}\bigg(\gamma_{0}\mathcal{D}_{\partial\Omega}[\phi]+\gamma_{0}^\mathrm{comp}lement\mathcal{D}_{\partial\Omega}[\phi]\bigg)\, , \quad
\mathcal{W}_{\partial \Omega}[\phi]\equiv-\gamma_{1}\mathcal{D}_{\partial\Omega}[\phi] =-\gamma_{1}^\mathrm{comp}lement\mathcal{D}_{\partial\Omega}[\phi]
\]
for all $\phi \in H^{1/2+\tau}(\partial \Omega)$. As a consequence,
\begin{equation}\label{eq:jump}
\gamma_{0}\mathcal{D}_{\partial \Omega}[\phi]=- \tfrac{1}{2}\phi + \mathcal{K}_{\partial \Omega}[\phi]\, , \quad \gamma_{0}^\mathrm{comp}lement\mathcal{D}_{\partial\Omega} [\phi]= \tfrac{1}{2}\phi + \mathcal{K}_{\partial \Omega}[\phi]\, , \quad \forall \phi \in H^{1/2+\tau}(\partial \Omega)\, .
\end{equation}
Thus the boundary integral operator associated with the Dirichlet problem in $\Omega$ is $-\frac{1}{2}I + \mathcal{K}_{\partial \Omega}$, whose mapping properties we therefore want to summarize in the sequel.
From Costabel and Wendland \cite[Remark 3.15]{CoWe86} (see also Steinbach and Wendland \cite{StWe01} and Mayboroda and Mitrea \cite{MaMi06}), we deduce the validity of the following.
\begin{lemma}\label{thm:Fred}
For any $\tau \in [-1/2,1/2]$,
the operators $\pm\frac{1}{2}I + \mathcal{K}_{\partial \Omega} \colon H^{1/2+\tau}(\partial \Omega)\to H^{1/2+\tau}(\partial \Omega)$ are Fredholm operators of index 0.
\end{lemma}
The kernels and cokernels of $\pm\frac{1}{2}I + \mathcal{K}_{\partial \Omega}$ are also known; they are independent of $\tau$. They are described in terms of the characteristic functions of the connected components of $\partial \Omega$. Here, if $\mathcal{O}$ is a subset of $\partial \Omega$ we denote by $\chi_{\mathcal{O}}$ the function from $\partial \Omega$ to $\mathbb{R}$ defined by
\[
\chi_{\mathcal{O}}(x)\equiv \left\{
\begin{array}{ll}
1 &\text{if $x \in \mathcal{O}$}\, ,\\
0 &\text{if $x \in \partial \Omega \setminus \mathcal{O}$}\, .
\end{array}
\right.
\]
The value of the double layer potential of a constant density is well known.
\[
\mathcal{D}_{\partial\Omega}[\chi_{\partial\Omega}](x) = \left\{
\begin{array}{ll}
0 &\text{if $x \in \Omega^\mathrm{comp}lement$}\\
-1 &\text{if $x \in \Omega $}
\end{array}
\right.\quad
\text{ hence } \quad
\mathcal{K}_{\partial \Omega}[\chi_{\partial\Omega}]= -\tfrac12 \chi_{\partial\Omega}\,.
\]
Applying this to the components $\Omega^\mathrm{comp}lement_{(j)}$, $j=1,\dots,m$, we see that the characteristic functions $\chi_{\partial\Omega^\mathrm{comp}lement_{(j)}}$ generate double layer potentials that vanish in $\Omega$ and that they are therefore in the kernel of the operator $-\frac{1}{2}I + \mathcal{K}_{\partial \Omega}$. In fact, by arguing as in Folland \cite[Ch.~3]{Fo95} one can prove the following.
\begin{lemma}\label{lem:V-}
Let
\[
\mathfrak{V}_{\pm}\equiv \bigg\{\phi \in H^{1/2}(\partial \Omega) \colon \pm \tfrac{1}{2}\phi + \mathcal{K}_{\partial \Omega}[\phi] =0 \bigg\}\, .
\]
Then $\mathfrak{V}_+$ has dimension $1$ and consists of constant functions on $\partial\Omega$.
The space $\mathfrak{V}_-$ has dimension $m$ and is generated by $\{\chi_{\partial \Omega^\mathrm{comp}lement_{(j)}}\}_{j=1}^{m}$.
\end{lemma}
In order to characterize the range of the double layer potential operator, we use the description of the mapping properties of the operator of the normal derivative of the double layer potential
$\mathcal{W}_{\partial \Omega}=-\gamma_1\mathcal{D}_{\partial \Omega}$
as given in McLean \cite[Thm.~8.20]{Mc00}.
\begin{lemma}\label{lem:kerW}
The operator $\mathcal{W}_{\partial \Omega}$ is a bounded selfadjoint operator from $H^{1/2}(\partial \Omega)$ to its dual space $H^{-1/2}(\partial \Omega)$. The kernel of $\mathcal{W}_{\partial \Omega}$ consists of locally constant functions in $H^{1/2}(\partial \Omega)$. Its dimension is $m+1$, and it is generated by $\{\chi_{\partial \Omega^\mathrm{comp}lement_{(j)}}\}_{j=0}^{m}$.
\end{lemma}
For a bounded selfadjoint operator, the kernel determines the range, the latter being the orthogonal complement of the former. We thus obtain the following description of the range of the double layer potential operator.
\begin{corollary}\label{cor:ranD}
Let $\tau\in[-1/2,1/2]$. Let $u \in H^{1+\tau}(\Omega)$ be such that $\Delta u=0$ in $ \Omega$. Then there exists $\mu \in H^{1/2+\tau}(\partial\Omega)$ such that $u=\mathcal{D}_{\partial \Omega}[\mu]\on{\Omega}$ if and only if
\begin{equation}\label{eq:ranD}
\big\langle \gamma_{1}u \,,\, \chi_{\partial \Omega^\mathrm{comp}lement_{(j)}} \big\rangle =0
\qquad \forall j\in \{1,\dots,m\}\, .
\end{equation}
\end{corollary}
Here we use the brackets $\big\langle\cdot,\cdot\big\rangle$ to denote the natural duality between a Sobolev space $H^{s}(\partial \Omega)$ and $H^{-s}(\partial \Omega)$. Thus the condition in \eqref{eq:ranD} means that the integral of the normal derivative
\[
\partial_n u = n\cdot\nabla u
\]
of $u$ over each component of $\partial\Omega$ vanishes. The condition for $j=0$ is implied by the others, because by Green's formula any harmonic function $u$ in $\Omega$ satisfies
\[
\sum_{j=0}^m \big\langle \gamma_{1}u \,,\,
\chi_{\partial \Omega^\mathrm{comp}lement_{(j)}} \big\rangle =
\int_{\partial\Omega} \partial_n u\, \mathrm{d}s = 0\,.
\]
\begin{remark}\label{rem:ranK}
The function $\mu$ represents $u$ as a double layer potential if and only if $\mu$ is a solution of the boundary integral equation
\begin{equation}\label{eq:BIE}
-\tfrac12\mu + \mathcal{K}_{\partial \Omega}[\mu] = \gamma_{0} u \qquad\text{ on } \partial\Omega\,.
\end{equation}
This follows from the jump relation \eqref{eq:jump} and from the uniqueness of the solution of the Dirichlet problem. Thus Corollary~{\mathrm e}f{cor:ranD} can be seen as a statement on the solvability of the boundary integral equation~\eqref{eq:BIE}, and conditions~\eqref{eq:ranD} characterize the cokernel of the operator
$-\frac12I + \mathcal{K}_{\partial \Omega}$.
\end{remark}
\subsection{The double layer potential for symmetric planar domains}\label{ss:prelsym}
We now come back to the geometric situation found at the end of Section~{\mathrm e}f{s:3}. This means that in this subsection $\Omega={\mathsf{B}}_\eta={\mathsf{B}}\setminus\eta\overline{\mathsf{Q}}$, where ${\mathsf{B}}$ is a simply connected bounded Lipschitz domain in $\R^2$ containing the origin, $\eta\in(0,\eta_0)$ is a small positive real number, and ${\mathsf{Q}}$ is a finite union of bounded simply connected Lipschitz domains such that $\eta_0{\mathsf{Q}}$ is contained in ${\mathsf{B}}$. In addition, ${\mathsf{B}}$ and ${\mathsf{Q}}$ are symmetric with respect to reflection at the horizontal axis. From Subsection~{\mathrm e}f{Ss:Reflection} we recall the notation for the reflection and the corresponding pullback
\[
\cR(x_1,x_2)\equiv (x_1,-x_2) \qquad \forall x=(x_1,x_2) \in \mathbb{R}^2\,
\qquad\text{and}\quad
\cR^\ast[g]=g\circ\cR\,.
\]
Thus we assume
\[
{\mathsf{B}} = \cR({\mathsf{B}}) \qquad\text{and}\quad {\mathsf{Q}} = \cR({\mathsf{Q}})\,.
\]
The symmetry of ${\mathsf{Q}}$ implies that there exist two natural numbers $m^{\times}$ and $m^{\#}$, such that ${\mathsf{Q}}$ has $m=m^{\times}+2m^{\#}>0$ connected components
\[
{\mathsf{Q}}^{\times}_{1}, \dots, {\mathsf{Q}}^{\times}_{m^{\times}}, {\mathsf{Q}}^{+}_{1}, \dots, {\mathsf{Q}}^{+}_{m^{\#}}, {\mathsf{Q}}^{-}_{1}, \dots, {\mathsf{Q}}^{-}_{m^{\#}}\,,
\]
satisfying
\begin{equation}\label{eq:symass}
\begin{aligned}
& {\mathsf{Q}}^{\times}_i = \cR( {\mathsf{Q}}^{\times}_i)
\qquad &\forall i \in \{1,\dots,m^{\times}\}\,,\\
& {\mathsf{Q}}^{+}_j = \cR({\mathsf{Q}}^{-}_j)\,\quad\text{and}\quad
\overline{{\mathsf{Q}}^{+}_{j}}\subset {\mathsf S}_\pi \equiv \R\times\R_+
\qquad &\forall j \in \{1,\dots,m^{\#}\}\,.
\end{aligned}
\end{equation}
See Figure~{\mathrm e}f{fig:3} for an example with $m^{\times}=2$ and $m^{\#}=1$.
We introduce the following definition.
\begin{definition}\label{def:odd}
Let $\tau\in[-1/2,1/2]$. If $\partial \Omega = \cR(\partial \Omega)$, then $ H^{1/2+\tau}_{\mathrm{odd}}(\partial\Omega)$ denotes the closed subspace of $H^{1/2+\tau}(\partial \Omega)$ defined by
\[
H^{1/2+\tau}_{\mathrm{odd}}(\partial\Omega)\equiv\left\{g \in H^{1/2+\tau}(\partial \Omega)\,:\; g=-\cR^\ast[g] \quad \text{on $\partial \Omega$} \right\}\,.
\]
\end{definition}
The mapping properties of the boundary integral operator $-\frac12I+\mathcal{K}_{\partial{\mathsf{B}}_\eta}$ on odd functions can be summarized as follows.
\begin{lemma}\label{lem:Kodd}
Let $\tau\in[-1/2,1/2]$ and $\eta\in(0,\eta_0)$.
\begin{itemize}
\item[(i)]
The operator $-\frac12I+\mathcal{K}_{\partial{\mathsf{B}}_\eta}$ defines a Fredholm operator of index zero from $H^{1/2+\tau}_{\mathrm{odd}}(\partial{\mathsf{B}}_\eta)$ to itself.
\item[(ii)]
If $m^{\#}=0$, this operator is an isomorphism. More generally,
its kernel has dimension $m^{\#}$ and is generated by the functions
$\{\chi_{\eta\partial{\mathsf{Q}}^+_j}-\chi_{\eta\partial{\mathsf{Q}}^-_j}\}_{j=1}^{m^{\#}}$, where
\[
(\chi_{\eta\partial{\mathsf{Q}}^+_j}-\chi_{\eta\partial{\mathsf{Q}}^-_j})(x) =
\left\{\begin{array}{cl}
+1, &\qquad x\in\eta\hskip0.15ex\partial{\mathsf{Q}}^+_j\,,\\
-1, &\qquad x\in\eta \hskip0.15ex\partial{\mathsf{Q}}^-_j\,,\\
0, &\qquad x\in \partial{\mathsf{B}}_\eta\setminus(\eta \hskip0.15ex\partial{\mathsf{Q}}^+_j\cup\eta \hskip0.15ex\partial{\mathsf{Q}}^-_j) \,.
\end{array}\right.
\]
\item[(iii)]
If $u\in H^{1+\tau}({\mathsf{B}}_\eta)$ is such that $\Delta u=0$ in ${\mathsf{B}}_\eta$ and
$\cR^\ast[u]=-u$ , then there exists $\mu \in H^{1/2+\tau}_{\mathrm{odd}}(\partial{\mathsf{B}}_\eta)$ such that $u=\mathcal{D}_{\partial {\mathsf{B}}_\eta}[\mu]\on{{\mathsf{B}}_\eta}$ if and only if
\begin{equation}\label{eq:ranDodd}
\big\langle \gamma_{1}u \,,\, \chi_{\eta\partial{\mathsf{Q}}^+_j} \big\rangle =0
\qquad \forall j\in \{1,\dots,m^{\#}\}\, .
\end{equation}
\item[(iv)]
If $u \in H^{1+\tau}_{\mathrm{loc}}({\mathsf{Q}}^\mathrm{comp}lement)$ is such that $\Delta u=0$ in ${\mathsf{Q}}^\mathrm{comp}lement$, $u$ is harmonic at infinity, and $\cR^\ast[u]=-u$ ,
then there exists $\mu \in H^{1/2+\tau}_{\mathrm{odd}}(\partial{\mathsf{Q}})$ such that $u=\mathcal{D}_{\partial{\mathsf{Q}}}[\mu]\on{{\mathsf{Q}}^\mathrm{comp}lement}$ if and only if
\[
\big\langle \gamma_{1}u \,,\, \chi_{\partial{\mathsf{Q}}^+_j} \big\rangle =0
\qquad \forall j\in \{1,\dots,m^{\#}\}\, .
\]
\end{itemize}
\end{lemma}
\begin{proof}
By the rule of change of variables in integrals, we have
$\mathcal{D}_{\partial{\mathsf{B}}_\eta}\big[\cR^\ast[\psi]\big]=\cR^\ast\big[\mathcal{D}_{\partial{\mathsf{B}}_\eta}[\psi]\big]$, hence
\begin{equation}\label{eq:reflK}
\quad
\mathcal{K}_{\partial{\mathsf{B}}_\eta}\big[\cR^\ast[\psi]\big]=\cR^\ast\big[\mathcal{K}_{\partial{\mathsf{B}}_\eta}[\psi]\big]\quad \forall \psi \in H^{1/2+\tau}(\partial{\mathsf{B}}_\eta) \, .
\end{equation}
Thus $\mathcal{K}_{\partial{\mathsf{B}}_\eta}$ maps odd functions to odd functions.
By Lemma~{\mathrm e}f{lem:V-} the kernel consists of odd functions in $\mathfrak{V}_-$, that is, odd linear combinations of the characteristic functions $\chi_{\eta\partial{\mathsf{Q}}^\times_i}$ and $\chi_{\eta\partial{\mathsf{Q}}^\pm_j}$. This space is generated by
$\chi_{\eta\partial{\mathsf{Q}}^+_j}-\chi_{\eta\partial{\mathsf{Q}}^-_j}$, $j=1,\dots,m^{\#}$. If $m^{\#}=0$, the operator is therefore injective.\\
Let us now show (iii). For an odd function $u$, the $m^{\#}$ conditions \eqref{eq:ranDodd} are equivalent to the whole set of $m^\times+2m^{\#}$ conditions \eqref{eq:ranD}, because the integrals over $\eta \hskip0.15ex\partial{\mathsf{Q}}^\times_i$ vanish by symmetry, and the integrals over $\eta \hskip0.15ex\partial{\mathsf{Q}}^-_j$ equal the negatives of the integrals over $\eta \hskip0.15ex\partial{\mathsf{Q}}^+_j$. Thus \eqref{eq:ranDodd} is equivalent to the existence of $\mu\in H^{1/2+\tau}(\partial{\mathsf{B}}_\eta)$ such that $u=\mathcal{D}_{\partial {\mathsf{B}}_\eta}[\mu]\on{{\mathsf{B}}_\eta}$. If $\mu$ is not yet odd, we can replace it by its odd part
\[
\tilde\mu \equiv \tfrac12\big(\mu-\mathcal{R}^\ast[\mu]\big)\,,
\]
which will also represent $u$, that is $u=\mathcal{D}_{\partial {\mathsf{B}}_\eta}[\tilde\mu]\on{{\mathsf{B}}_\eta}$. Statement (iii) is proved.\\
To prove that $-\tfrac12I+\mathcal{K}_{\partial{\mathsf{B}}_\eta}$ is a Fredholm operator of index zero, it remains to show that its range has codimension $m^{\#}$ (we recall that $\mathcal{K}_{\partial{\mathsf{B}}_\eta}$ is not compact, in general, when $\partial{\mathsf{B}}_\eta$ is a only required to be Lipschitz). We use the observation noted in Remark~{\mathrm e}f{rem:ranK}.
For $g\in H^{1/2+\tau}_{\mathrm{odd}}(\partial{\mathsf{B}}_\eta)$, let $u\in H^{1+\tau}({\mathsf{B}}_\eta)$ be its harmonic extension, that is, the unique harmonic function in ${\mathsf{B}}_\eta$ satisfying $\gamma_{0}u=g$. It is clear that $u$ is an odd function. The $m^{\#}$ conditions \eqref{eq:ranDodd} that guarantee the representability of $u$ as a double layer potential with a density $\mu \in H^{1/2+\tau}_{\mathrm{odd}}(\partial{\mathsf{B}}_\eta)$ can be considered as $m^{\#}$ continuous linear functionals acting on $g$ and defining solvability conditions for the boundary integral equation
\[
(-\tfrac12I+\mathcal{K}_{\partial{\mathsf{B}}_\eta})[\mu]=g\,.
\]
We have shown that the cokernel of $-\frac12I+\mathcal{K}_{\partial{\mathsf{B}}_\eta}$ in $H^{1/2+\tau}_{\mathrm{odd}}(\partial{\mathsf{B}}_\eta)$ has dimension $m^{\#}$, and thus the remaining statements of (i) and (ii) follow.\\
Statement (iv) is proved in the same way as (iii) by relying on the characterization of the cokernel of the operator $\mathcal{W}_{\partial{\mathsf{Q}}}$ as given by McLean \cite[Thm.~8.20]{Mc00}.
\end{proof}
In addition to the boundary integral operators, we will also need mapping properties of the double layer potential restricted to some subsets of the domain.
The first result is global and concerns the entire interior of $\partial{\mathsf{B}}$ or exterior of $\partial{\mathsf{Q}}$. For this we use the weighted Sobolev spaces of Kondrat'ev type $K^s_\beta$ introduced in Section~{\mathrm e}f{ss:conf} and defined for integer regularity exponent $s=m\in\N$ in \eqref{eq:Kmb}. For non-integer $s$ there exist several equivalent ways to define this space (see \cite{Dauge88}), the shortest (and for us, easiest to use) is by Hilbert space interpolation:
If $s=m+\tau$, $m\in\N$ and $\tau\in[0,1]$, then
\[
K^s_\beta(\Omega) = \big[K^m_\beta(\Omega),K^{m+1}_\beta(\Omega)\big]_\tau\,.
\]
It is clear that if $\Omega$ is a bounded domain with a positive distance to the origin, then the norm in $K^s_\beta(\Omega)$ is equivalent to the norm in the standard Sobolev space $H^s(\Omega)$, and the weight exponent $\beta$ influences only the behavior near the origin and at infinity. Thus let $\chi\in C^\infty_0(\R^2)$ be a cutoff function that is equal to $1$ on the ball ${\mathscr B}(0,1/2)$ and $0$ outside of ${\mathscr B}(0,1)$ and for $R>0$ define
\[
\chi_R(x)=\chi(\tfrac xR)\,;
\]
let further $R_0,R_1$ be such that $0<2R_0\le R_1$ and let $\beta_0,\beta_1\in\R$ be two weight indices. Then if we define
\begin{equation}
\label{eq:Ksb0b1}
\DNorm{u}{K^s_{\beta_0\beta_1}(\Omega)} \equiv
\DNorm{\chi_{R_0}u}{K^s_{\beta_0}(\Omega)} +
\DNorm{(1-\chi_{R_0})\chi_{R_1}u}{H^s(\Omega)} +
\DNorm{(1-\chi_{R_1})u}{K^s_{\beta_1}(\Omega)} \,,
\end{equation}
we see that for any Lipschitz domain $\Omega$
the norms $\DNorm{u}{K^s_\beta(\Omega)}$ and $\DNorm{u}{K^s_{\beta\beta}(\Omega)}$ are equivalent;
for any bounded $\Omega$
the norm $\DNorm{u}{K^s_{\beta_0\beta_1}(\Omega)}$ is equivalent to
$\DNorm{u}{K^s_{\beta_0}(\Omega)}$, and
for any $\Omega$ that has a positive distance to the origin
it is equivalent to
$\DNorm{u}{K^s_{\beta_1}(\Omega)}$.
With this preparation, we can now prove the boundedness of the double layer representation.
\begin{lemma}
\label{lem:Drep}
For any $\tau\in[-1/2,1/2]$, $\beta_0>-2$ and $\beta_1<0$, the following operators are bounded:
\begin{align}
\label{eq:DdB}
\cD_{\partial{\mathsf{B}}} &: H^{1/2+\tau}_{\mathrm{odd}}(\partial{\mathsf{B}})
\to K^{1+\tau}_{\beta_0}({\mathsf{B}}) \,, \\
\label{eq:DdQ}
\cD_{\partial{\mathsf{Q}}} &: H^{1/2+\tau}_{\mathrm{odd}}(\partial{\mathsf{Q}})
\to K^{1+\tau}_{\beta_0\beta_1}({\mathsf{Q}}^\mathrm{comp}lement)
\quad\text{ if }\quad 0\not\in\partial{\mathsf{Q}} \,,\\
\label{eq:DdQ0}
\cD_{\partial{\mathsf{Q}}} &: H^{1/2+\tau}_{\mathrm{odd}}(\partial{\mathsf{Q}})
\to K^{1+\tau}_{\beta_0\beta_1}({\mathsf{Q}}^\mathrm{comp}lement)
\quad\text{ if }\quad 0\in\partial{\mathsf{Q}} \;\,\text{ and }\;
\beta_0\ge-1-\tau\,.
\end{align}
\end{lemma}
\begin{proof}
In \eqref{eq:dblreptau} we already quoted from \cite[Thm.~1]{Co88} that for a bounded domain $\Omega$, $\cD_{\partial\Omega}$ maps $H^{1/2+\tau}(\partial\Omega)$ boundedly into $H^{1+\tau}(\Omega)$. This implies that
\[
\cD_{\partial{\mathsf{B}}} : H^{1/2+\tau}_{\mathrm{odd}}(\partial{\mathsf{B}})
\to H^{1+\tau}({\mathsf{B}})
\]
is bounded. Let $R_0$ be such that ${\mathscr B}(0,R_0)\subset{\mathsf{B}}$ and let
$g$ be the trace of $u\equiv\cD_{\partial{\mathsf{B}}}[\psi]$ on $\partial{\mathscr B}(0,R_0)$.
Since $u$ is harmonic in ${\mathsf{B}}$ and odd, it can be expanded in a Fourier series of the form
\[
u(x)= \sum_{k=1}^\infty g_k \Big(\frac r{R_0}\Big)^k \sin k\theta \,.
\]
Here $g_k$ are the Fourier coefficients of $g$, and the Sobolev norms of $g$ can equivalently be expressed by weighted norms of the sequence $g_k$.
\[
\DNormc{g}{H^s(\partial{\mathscr B}(0,R_0))} = \sum_{k=1}^\infty k^{2s} |g_k|^2\,.
\]
It can be verified by explicit computation that for $\beta>-2$ and any $m\in\N$
\[
\DNormc{u}{K^m_{\beta}({\mathscr B}(0,R_0))} =
\sum_{k\ge1}c_{k,m}|g_k|^2
\qquad\text{ with }\quad
c\,k^{2m-1} \le c_{k,m} \le C\,k^{2m-1}\,.
\]
The constants here depend on $R_0$ and $\beta$, but not on $u$.
Thus $\DNorm{u}{K^m_{\beta}({\mathscr B}(0,R_0))}$ is equivalent to
$\DNorm{g}{H^{m-1/2}(\partial{\mathscr B}(0,R_0))}$. By interpolation it follows that this is also true for $m$ replaced by $1+\tau$. Thus
\[
\DNorm{u}{K^{1+\tau}_{\beta_0}({\mathscr B}(0,R_0))} \le
C\, \DNorm{g}{H^{1/2+\tau}(\partial{\mathscr B}(0,R_0))} \le
C\, \DNorm{u}{H^{1+\tau}({\mathsf{B}})}\,.
\]
Adding $\DNorm{u}{H^{1+\tau}({\mathsf{B}})}$, we find
\[
\DNorm{u}{K^{1+\tau}_{\beta_0}({\mathsf{B}})} \le
C\, \DNorm{g}{H^{1/2+\tau}(\partial{\mathsf{B}})}\,,
\]
hence \eqref{eq:DdB}.
For the proof of \eqref{eq:DdQ}, we use a similar argument:
Let $U=\cD_{\partial{\mathsf{Q}}}[\Psi]$ in ${\mathsf{Q}}^\mathrm{comp}lement$ and let $G$ be the trace of $U$ on some $\partial{\mathscr B}(0,R_1)$ with $R_1$ chosen such that ${\mathsf{Q}}\subset{\mathscr B}(0,R_1)$. Then
\[
\DNorm{G}{H^{1/2+\tau}(\partial{\mathscr B}(0,R_1))} \le
\DNorm{U}{H^{1+\tau}({\mathsf{Q}}^\mathrm{comp}lement\cap{\mathscr B}(0,R_1))}\le
C\, \DNorm{\Psi}{H^{1/2+\tau}(\partial{\mathsf{Q}})}\,.
\]
Now we write $U$ in ${\mathscr B}(0,R_1)^\mathrm{comp}lement$ as a Fourier series, using that it is harmonic in ${\mathsf{Q}}^\mathrm{comp}lement$ and vanishes at infinity (for this we do not even need that $U$ is odd), and prove by explicit calculation of weighted Sobolev norms and interpolation that for any $\beta<0$ there is an estimate
\[
\DNorm{U}{K^{1+\tau}_{\beta}({\mathscr B}(0,R_1)^\mathrm{comp}lement)}
\le C\, \DNorm{G}{H^{1/2+\tau}(\partial{\mathscr B}(0,R_1))} \,.
\]
If $0\in{\mathsf{Q}}$, we do not need to estimate $U$ in a neighborhood of $0$. If $0\in{\mathsf{Q}}^\mathrm{comp}lement$, we can get an estimate of $U$ in a neighborhood of $0$ as above for $u$.
Together, this implies \eqref{eq:DdQ}.
For \eqref{eq:DdQ0}, we use the previous estimate outside of a neighborhood of the origin, but now we additionally have to estimate
$\DNorm{U}{K^{1+\tau}_{\beta_0}({\mathsf{Q}}^\mathrm{comp}lement\cap{\mathscr B}(0,R_0))}$
for some $R_0>0$. We cannot apply the same argument as for $u$ above, because $0$ is on the boundary of ${\mathsf{Q}}$, and $U$ is not harmonic in a whole neighborhood of $0$. Instead we will use the fact that if $\beta_0\ge-1-\tau$ then there is a continuous inclusion
\[
H^{1+\tau}_{\mathrm{odd}}({\mathsf{Q}}^\mathrm{comp}lement\cap{\mathscr B}(0,R_0))
\subset
K^{1+\tau}_{\beta_0}({\mathsf{Q}}^\mathrm{comp}lement\cap{\mathscr B}(0,R_0))\,.
\]
For $\tau\ne0$ this follows from Dauge \cite[Theorem (AA.7)]{Dauge88}.
It is also true for $\tau=0$ as follows easily from the well known Hardy inequality
\[
\DNorm{\frac{U(\cdot)}{x_2}}{L^2({\mathsf{Q}}^\mathrm{comp}lement)}
\le 2\, \DNorm{\partial_{x_2}U}{L^2({\mathsf{Q}}^\mathrm{comp}lement)}
\]
for all $U\in H^1({\mathsf{Q}}^\mathrm{comp}lement)$ satisfying $U=0$ for $x_2=0$.
This inclusion together with the previous estimates that led to \eqref{eq:DdQ} proves \eqref{eq:DdQ0} and ends the proof of the lemma.
\end{proof}
\begin{remark}
\label{rem:DLinf}
If $\tau\in(0,1/2]$, then one also has bounded mappings
\begin{equation}
\label{eq:Dlinf}
\cD_{\partial{\mathsf{B}}} : H^{1/2+\tau}(\partial{\mathsf{B}})
\to L^\infty({\mathsf{B}})
\;\text{ and }\;
\cD_{\partial{\mathsf{Q}}} : H^{1/2+\tau}(\partial{\mathsf{Q}})
\to L^\infty({\mathsf{Q}}^\mathrm{comp}lement) \,.
\end{equation}
This follows for $\cD_{\partial{\mathsf{B}}}$ from the Sobolev inclusion
$
H^{1+\tau}({\mathsf{B}})\subset L^\infty({\mathsf{B}})
$
and for $\cD_{\partial{\mathsf{Q}}}$ from the Sobolev inclusion on
${\mathsf{Q}}^\mathrm{comp}lement\cap{\mathscr B}(0,R_1)$ combined with Fourier series (or simply the maximum principle) on ${\mathscr B}(0,R_1)^\mathrm{comp}lement$.
\end{remark}
The second class of results is local in nature and describes the analyticity of the double layer potential near the origin and near infinity in a form that is suitable for our situation of a symmetric domain with small perforations. The result can be considered simply to be a consequence of the analyticity of the fundamental solution $E$ on $\R^2\setminus\{0\}$, and it is similar to the subject studied in Lanza de Cristoforis and Musolino \cite{LaMu13}, but there are some particularities related to the symmetry and weak smoothness of the domains studied here.
\begin{lemma}
\label{lem:Danal}
Let $\tau\in[-1/2,1/2]$.
\begin{itemize}
\item[(i)]
Let $\Omega\subset\R^2$ be a bounded Lipschitz domain.
For positive $\eta$ sufficiently small so that $\eta\Omega\subset{\mathsf{B}}$, we define the restriction $\cD_{\partial{\mathsf{B}},\Omega}(\eta)$ of the double layer potential $\cD_{\partial{\mathsf{B}}}$ to $\eta\Omega$ written in ``fast'' variables
\[
\cD_{\partial{\mathsf{B}},\Omega}(\eta)[\psi](X)
\equiv \cD_{\partial{\mathsf{B}}}[\psi]\on{\eta\Omega}(\eta X)
\quad (X\in\Omega) \qquad \forall \psi \in H^{1/2+\tau}_{\mathrm{odd}}(\partial{\mathsf{B}})\,.
\]
Then there exists $\eta_1>0$ such that the function
$\eta\mapsto\cD_{\partial{\mathsf{B}},\Omega}(\eta)$ has for any $s\in\R$ a continuation to $\eta\in(-\eta_1,\eta_1)$ as an analytic function with values in
${\mathscr L}\big(H^{1/2+\tau}_{\mathrm{odd}}(\partial{\mathsf{B}}),\,H^s(\Omega)\big)$.
\item[(ii)]
Let $\Omega\subset\R^2$ be a bounded Lipschitz domain such that $0\not\in\overline\Omega$.
For positive $\eta$ sufficiently small so that $(1/\eta)\Omega\subset{\mathsf{Q}}^\mathrm{comp}lement$, we define the restriction $\cD^\mathrm{comp}lement_{\partial{\mathsf{Q}},\Omega}(\eta)$ of the double layer potential $\cD_{\partial{\mathsf{Q}}}$ to $(1/\eta)\Omega$ written in ``slow'' variables
\[
\cD^\mathrm{comp}lement_{\partial{\mathsf{Q}},\Omega}(\eta)[\Psi](x)
\equiv \cD_{\partial{\mathsf{Q}}}[\Psi]\on{(1/\eta)\Omega}(\tfrac{x}{\eta})
\quad (x\in\Omega) \qquad \forall \Psi \in H^{1/2+\tau}_{\mathrm{odd}}(\partial{\mathsf{Q}}) \,.
\]
Then there exists $\eta_1>0$ such that the function
$\eta\mapsto\cD^\mathrm{comp}lement_{\partial{\mathsf{Q}},\Omega}(\eta)$ has for any $s\in\R$ a continuation to $\eta\in(-\eta_1,\eta_1)$ as an analytic function with values in
${\mathscr L}\big(H^{1/2+\tau}_{\mathrm{odd}}(\partial{\mathsf{Q}}),\,H^s(\Omega)\big)$.
\item[(iii)]
In addition, $\cD_{\partial{\mathsf{B}},\Omega}(0)=0$ and $\cD^\mathrm{comp}lement_{\partial{\mathsf{Q}},\Omega}(0)=0$.
\end{itemize}
\end{lemma}
\begin{proof}
The proofs for (i) and (ii) are similar. Both use the fact that the double layer potential is analytic outside of the boundary and vanishes at infinity, and for an odd density it vanishes at the origin. We give the proof of (ii) and leave the proof of (i) and (iii) to the reader. \\
Let $\Psi\in H^{1/2+\tau}_{\mathrm{odd}}(\partial{\mathsf{Q}})$ and define
$W=\cD_{\partial{\mathsf{Q}}}[\Psi]$ in ${\mathsf{Q}}^\mathrm{comp}lement$. We can choose $R_0$ such that
$\overline{\mathsf{Q}}\subset{\mathscr B}(0,R_0)$ and $R_1,R_2$ such that
$\Omega\subset{\mathscr B}(0,R_2)\cap{\mathscr B}(0,R_1)^\mathrm{comp}lement$.
For $|X|\ge R_0$, we can expand the harmonic and odd function $W$ in a Fourier series
\begin{equation}
\label{eq:Wexp}
W(X)= \sum_{k=1}^\infty w_k \Big(\frac R{R_0}\Big)^{-k} \sin k\theta \,.
\end{equation}
Here $(R,\theta)$ denote polar coordinates for $X$, and from the fact that $\cD_{\partial{\mathsf{Q}}}$ maps $H^{1/2+\tau}_{\mathrm{odd}}(\partial{\mathsf{Q}})$ to $H^{1+\tau}_{\mathrm{odd}}({\mathsf{Q}}^\mathrm{comp}lement\cap{\mathscr B}(0,R_0))$ we deduce the (crude) estimate that the $w_k$ are bounded and satisfy an estimate
\[
\sup_k |w_k| \le C\, \DNorm{\Psi}{H^{1/2+\tau}(\partial{\mathsf{Q}})}\,.
\]
If $\eta\in(0,R_1/R_0)$, then
$X\in(1/\eta)\Omega \subset{\mathscr B}(0,R_2/\eta)\cap{\mathscr B}(0,R_1/\eta)^\mathrm{comp}lement$ implies
$|X|>R_0$, so that we can use the expansion
\eqref{eq:Wexp} for the restriction of $W$ to $(1/\eta)\Omega$. Writing this in slow variables $x=\eta X$, or in polar coordinates with $|x|=r=\eta R$, we get
\[
\cD^\mathrm{comp}lement_{\partial{\mathsf{Q}},\Omega}(\eta)[\Psi](x) = W(\frac x\eta)
= \sum_{k=1}^\infty \eta^{k}\, w_k\, p_k(x)
\quad\text{ with }\;
p_k(x)=\Big(\frac r{R_0}\Big)^{-k} \sin k\theta
\]
By explicit computation for any chosen $m\in\N$, we can estimate the $H^m$ norm of $p_k$
\[
\DNorm{p_k}{H^m(\Omega)}\le \DNorm{p_k}{H^m({\mathscr B}(0,R_2)\cap{\mathscr B}(0,R_1)^\mathrm{comp}lement)}
\le C\, \big(\frac{R_0}{R_1}\big)^{k}k^{2m-1}\,,
\]
with $C$ independent of $k$.
We conclude that $\cD^\mathrm{comp}lement_{\partial{\mathsf{Q}},\Omega}(\eta)$ has a convergent expansion
\begin{equation}
\label{eq:Dexp}
\cD^\mathrm{comp}lement_{\partial{\mathsf{Q}},\Omega}(\eta) =
\sum_{k=1}^\infty \eta^{k} \, D_k \,,
\end{equation}
where the $D_k$ are bounded linear operators from
$H^{1/2+\tau}_{\mathrm{odd}}(\partial{\mathsf{Q}})$ to $H^m(\Omega)$ satisfying
\[
\DNorm{D_k}{{\mathscr L}\big(H^{1/2+\tau}_{\mathrm{odd}}(\partial{\mathsf{Q}}),H^m(\Omega)\big)}
\le C\, \big(\frac{R_0}{R_1}\big)^{k} k^{2m-1}\,.
\]
It follows that the expansion \eqref{eq:Dexp} converges for $|\eta|<R_1/R_0$ and
this proves the analyticity as claimed in (ii).
\end{proof}
\subsection{The Dirichlet problem in a symmetric perforated domain}\label{ss:Dir}
In this subsection we apply the double layer representation to the solution of the Dirichlet problem in our perforated symmetric domain ${\mathsf{B}}_\eta$.
Thus we assume that we are given odd functions
$\psi\in H^{1/2+\tau}_{\mathrm{odd}}(\partial{\mathsf{B}})$ and
$\Psi\in H^{1/2+\tau}_{\mathrm{odd}}(\partial{\mathsf{Q}})$, and
we denote by $u[\eta,\psi,\Psi]$ the unique solution in $H^{1+\tau}({\mathsf{B}}_\eta)$ of the boundary value problem
\begin{equation}\label{eq:symdir}
\left\{
\begin{array}{ll}
\Delta u=0&\text{ in }\;{\mathsf{B}}_\eta\,,\\
\gamma_{0}u=\psi &\text{ on }\;\partial{\mathsf{B}}\,,\\
\gamma_{0}u=\Psi(\cdot/\eta) &\text{ on } \;\eta \hskip0.15ex\partial{\mathsf{Q}}\,.\\
\end{array}
\right.
\end{equation}
We would like to represent $u[\eta,\psi,\Psi]$ as a double layer potential.
It is clear that $u$ is an odd function. The conditions \eqref{eq:ranDodd} will, however, not be satisfied, in general, if the number $m^{\#}$ of ``paired holes'' is non-zero. As a remedy for this problem, we introduce harmonic functions $\Xi_1,\dots,\Xi_{m^{\#}}$ that span a complement of the range of the double layer potential operator. We define $\Xi_j$ as the
unique function in $H^{1+\tau}_{\mathrm{loc}}\big(\mathbb{R}^2 \setminus( \overline{{\mathsf{Q}}^+_j}\cup\overline{{\mathsf{Q}}^-_j})\big)$ such that
\begin{equation}\label{eq:Xi}
\left\{
\begin{array}{ll}
\Delta \Xi_j=0&\text{ in }\mathbb{R}^2 \setminus\big( \overline{{\mathsf{Q}}^+_j}\cup\overline{{\mathsf{Q}}^-_j}\big)\,,\\
\gamma_{0}\Xi_j=\pm 1&\text{ on }\partial{\mathsf{Q}}_j^\pm\,,\\
\|\Xi_j\|_\infty<+\infty\, .&
\end{array}
\right.
\end{equation}
A simple argument for the existence of such functions $\Xi_j$ is to use the Kelvin transformation with origin in ${\mathsf{Q}}^+_j$ that reduces the exterior Dirichlet problem problem~\eqref{eq:Xi} to a Dirichlet problem on a bounded domain (see Folland \cite[Ch. 2.I]{Fo95}), and then invoke the existence and uniqueness of solution of the Dirichlet problem on a bounded domain.
The uniqueness of $\Xi_j$ implies in particular that $\Xi_j$ is odd,
\begin{equation}\label{eq:symphi}
\Xi_j(X)=-\cR^\ast[\Xi_j](X) \qquad \text{for $X \in\mathbb{R}^2 \setminus\big( \overline{{\mathsf{Q}}^+_j}\cup\overline{{\mathsf{Q}}^-_j}\big)$.}
\end{equation}
Then by \eqref{eq:symphi} and by the harmonicity in $\mathbb{R}^2 \setminus\big( \overline{{\mathsf{Q}}^+_j}\cup\overline{{\mathsf{Q}}^-_j}\big)$ and at infinity of $\Xi_j$ it follows that
\begin{equation}\label{eq:limphi}
\lim_{X\to\infty}\Xi_j(X)=0\,.
\end{equation}
Concerning the integrals of the normal derivative of $\Xi_j$ over the boundaries of the connected components of ${\mathsf{Q}}$, it follows from the harmonicity that they vanish except for the components ${\mathsf{Q}}_j^\pm$, in particular
\begin{equation}\label{eq:phiQpm0}
\big\langle \gamma_{1}\Xi_j\,,\, \chi_{\partial{\mathsf{Q}}^\pm_k} \big\rangle =0
\quad \forall k\in \{1,\dots,m^{\#}\}\setminus\{j\}
\, .
\end{equation}
From the harmonicity at infinity and \eqref{eq:limphi} follows that
$\nabla \Xi_j \in L^2\Big(
\mathbb{R}^2 \setminus\big( \overline{{\mathsf{Q}}^+_j}\cup\overline{{\mathsf{Q}}^-_j}\big)\Big)$
and that we can use the Divergence Theorem, which gives
\begin{multline}\label{eq:phiQpm}
0<\int_{\mathbb{R}^2 \setminus\big( \overline{{\mathsf{Q}}^+_j}\cup\overline{{\mathsf{Q}}^-_j}\big)}
|\nabla\Xi_j(X)|^2\, \mathrm{d}X\\
=-\int_{\partial {\mathsf{Q}}^{+}_j} \partial_n \Xi_j\, \mathrm{d}s
+\int_{\partial {\mathsf{Q}}^{-}_j} \partial_n \Xi_j\, \mathrm{d}s
= -2\big\langle \gamma_{1}\Xi_j\,,\, \chi_{\partial{\mathsf{Q}}^+_j} \big\rangle\,.
\end{multline}
We can now show the following augmented double layer representation for the solution $u[\eta,\psi,\Psi]$ of problem \eqref{eq:symdir}.
\begin{lemma}\label{lem:equiv}
Let $\tau\in[-1/2,1/2]$. Let $\eta \in (0,\eta_0)$. Then the following statements hold.
\begin{itemize}
\item[(i)] If $m^{\#}=0$, then there exists a unique function $\mu\in H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{B}}_\eta)$ such that
\begin{equation}\label{eq:equiv=}
u[\eta,\psi,\Psi]=\mathcal{D}_{\partial {\mathsf{B}}_\eta}[\mu] \qquad \text{in ${\mathsf{B}}_\eta$.}
\end{equation}
\item[(ii)] If $m^{\#}>0$, then there exists a unique
$m^{\#}$-tuple $\boldsymbol{c}=(c_1,\dots,c_{m^{\#}})\in\mathbb{R}^{m^{\#}}$ and
a unique function $\mu\in H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{B}}_\eta)$
satisfying
\begin{equation}\label{eq:equiv>}
\left\{
\begin{array}{ll}
u[\eta,\psi,\Psi]=\mathcal{D}_{\partial {\mathsf{B}}_\eta}[\mu]+\sum_{j=1}^{m^{\#}}c_j\Xi_j(\cdot/\eta) & \text{in ${\mathsf{B}}_\eta$}\\
\int_{\eta \partial {\mathsf{Q}}^{+}_j}\mu\, \mathrm{d}s=0 & \forall j \in \{1,\dots,m^{\#}\}\, .
\end{array}
\right.
\end{equation}
\end{itemize}
\end{lemma}
\begin{proof}
Statement (i) follows from Lemma {\mathrm e}f{lem:Kodd} (ii). We now consider statement (ii). We first note that by \eqref{eq:phiQpm} for each $j \in \{1,\dots,m^{\#}\}$ there exists a unique $c_j \in \mathbb{R}$ such that
\[
\big\langle \gamma_{1} u[\eta,\psi,\Psi]\,,\, \chi_{\eta\partial{\mathsf{Q}}^+_j} \big\rangle
-c_j\, \big\langle \gamma_{1} \Xi_j(\cdot/\eta) \,,\, \chi_{\eta\partial{\mathsf{Q}}^+_j} \big\rangle =0\,.
\]
Using \eqref{eq:phiQpm0}, it follows that the function
$u[\eta,\psi,\Psi]-\sum_{j=1}^{m^{\#}}c_j\Xi_j(\cdot/\eta)$ satisfies the conditions of Lemma~{\mathrm e}f{lem:Kodd} for the existence of a representation as a double layer potential.
As a consequence, there exists
$\tilde{\mu} \in H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{B}}_\eta)$ such that
\[
\mathcal{D}_{\partial \Omega_\eta}[\tilde{\mu}]=u[\eta,\psi,\Psi]-\sum_{j=1}^{m^{\#}}c_j\Xi_j(\cdot/\eta) \qquad \text{in ${\mathsf{B}}_\eta$}\, .
\]
Recalling from Lemma~{\mathrm e}f{lem:Kodd}(ii) that the kernel
$\mathfrak{V}_{-,\mathrm{odd}} \equiv \mathfrak{V}_{-}\cap H^{1/2+\tau}_\mathrm{odd}(\partial{\mathsf{B}}_\eta)$ of the operator
$-\frac{1}{2}I+\mathcal{K}_{\partial {\mathsf{B}}_\eta}$ acting on odd functions is spanned by
the functions
$\{\chi_{\eta\partial{\mathsf{Q}}^+_j}-\chi_{\eta\partial{\mathsf{Q}}^-_j}\}_{j=1}^{m^{\#}}$, we find that among the functions
$\mu \in \tilde\mu + \mathfrak{V}_{-,\mathrm{odd}}$ that satisfy the first line of \eqref{eq:equiv>} there is exactly one satisfying the side conditions of the second line of \eqref{eq:equiv>}.
\end{proof}
With the help of the augmented double layer potential representation \eqref{eq:equiv>} we can now rewrite our Dirichlet problem \eqref{eq:symdir} as an equivalent boundary integral equation on $\partial{\mathsf{B}}_\eta$. This is still a problem on an $\eta$-dependent domain, but it is possible to interpret it as a system of boundary integral equations in the function space $H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{B}})\times H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{Q}})$ defined on the fixed domain $\partial{\mathsf{B}}\times \partial{\mathsf{Q}}$. Owing to the special form of the double layer kernel, this system has a simple form that makes it natural to study the dependence on $\eta$ in the limit $\eta\to0$ and even to extend it in an analytic way to a neighborhood of $\eta=0$.
The formulation \eqref{eq:symdir} of our Dirichlet problem already makes use of the identification of a function defined on $\partial{\mathsf{B}}_\eta$ with a pair $\big(\psi,\Psi(\cdot/\eta)\big)$ of functions, the first one defined on $\partial{\mathsf{B}}$ and depending on standard or ``slow'' variables $x$, the second one defined on $\partial{\mathsf{Q}}$ and depending on ``fast'' variables $X=x/\eta$. Let $\cJ_\eta$ denote this mapping from $H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{B}})\times H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{Q}})$ to $H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{B}}_\eta)$, which obviously is an isomorphism.
\begin{equation}\label{eq:cJ}
\cJ_\eta[\phi,\Phi](x) \equiv \left\{
\begin{array}{ll}
\phi(x) & \text{on $\partial {\mathsf{B}}$}\, ,\\
\Phi(x/\eta) & \text{on $\eta \partial {\mathsf{Q}}$}\, .
\end{array}
\right.
\end{equation}
The boundary integral equation for \eqref{eq:symdir} is obtained from the representation formula \eqref{eq:equiv>} by taking traces on $\partial{\mathsf{B}}_\eta$.
In order to treat simultaneously the case $m^{\#}=0$ and the case $m^{\#}>0$, from now on we will assume that the symbols $c_1,\dots,c_{m^{\#}}$ and $\sum_{j=1}^{m^{\#}}c_j\phi_j$ are omitted if $m^{\#}=0$.
In addition we find it convenient to set
\[
H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{Q}})_{\#}\equiv \Big\{\mu \in H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{Q}}) \colon \int_{\partial {\mathsf{Q}}^{+}_j}\mu\, \mathrm{d}s=0\;\ \forall j \in \{1,\dots,m^{\#}\}\Big\}\, .
\]
Clearly, if $m^{\#}=0$ then $H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{Q}})_{\#}= H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{Q}})$.
We can then write the trace of \eqref{eq:equiv>} as the problem of finding
$\mu\in\cJ_\eta\left[H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{B}})\times H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{Q}})_{\#}\right]$ and
$\boldsymbol{c}\in\mathbb{R}^{m^{\#}}$
such that
\begin{equation}\label{eq:BIEdBeta}
(-\tfrac12I + \mathcal{K}_{\partial{\mathsf{B}}_\eta})[\mu]
+\sum_{j=1}^{m^{\#}}c_j\,\gamma_{0}\,\Xi_j(\cdot/\eta) = g
\quad\text{ on } \partial{\mathsf{B}}_\eta
\end{equation}
where $g=\cJ_\eta[\psi,\Psi]$. With $\mu=\cJ_\eta[\phi,\Phi]$ we find a first form of the equivalent system of boundary integral equations on $\partial{\mathsf{B}}\times \partial{\mathsf{Q}}$.
\begin{equation}\label{eq:BIEdBdQ}
\cJ_\eta^{-1}\circ\biggl[(-\tfrac12I + \mathcal{K}_{\partial{\mathsf{B}}_\eta})
\circ\cJ_\eta[\phi,\Phi] +\sum_{j=1}^{m^{\#}}c_j\,\gamma_{0}\,\Xi_j(\cdot/\eta)\biggr] = \big(\psi,\Psi\big).
\end{equation}
We will now describe this system in more detail.
Changing variables $y\mapsto\eta Y$ in the double layer integral and using the fact that
\[
\nabla E(x) = -\frac{x}{2\pi|x|^2}
\]
is a function homogeneous of degree $-1$, we can write
\[
\begin{split}
\mathcal{D}_{\partial {\mathsf{B}}_\eta}\Big[\cJ_\eta[\phi,\Phi]\Big]&=
\mathcal{D}_{\partial{\mathsf{B}}}[\phi]
-\int_{\eta\partial{\mathsf{Q}}} \Phi(y/\eta)
\partial_{n(y)} E (\cdot-y)\, \mathrm{d}s_y\\
&=\mathcal{D}_{\partial{\mathsf{B}}}[\phi]
+\eta\int_{\partial{\mathsf{Q}}}\Phi(Y)\, n(Y)\cdot \nabla E(\cdot-\eta Y)\, \mathrm{d}s_Y\quad \text{in ${\mathsf{B}}_\eta$}\, ,
\end{split}
\]
and then express the representation formula \eqref{eq:equiv>} both in ``slow'' variables:
\begin{equation}\label{eq:rep}
u[\eta,\psi,\Psi]=\mathcal{D}_{\partial {\mathsf{B}}}[\phi]
+\eta\int_{\partial {\mathsf{Q}}}\Phi(Y)\, n(Y)\cdot \nabla E (\cdot-\eta Y)\, \mathrm{d}s_Y
+\sum_{j=1}^{m^{\#}}c_j\Xi_j(\cdot/\eta) \quad \text{in ${\mathsf{B}}_\eta$}\,
\end{equation}
and in ``fast'' variables:
\begin{equation}\label{eq:rep-bis}
\begin{split}
u[\eta,\psi,\Psi](\eta X)&=\mathcal{D}_{\partial{\mathsf{B}}}[\phi](\eta X)\\ &\quad
+\eta\int_{\partial{\mathsf{Q}}}\Phi(Y)\, n(Y)\cdot \nabla E(\eta(X-Y))\, \mathrm{d}s_Y
+\sum_{j=1}^{m^{\#}}c_j\Xi_j(X) \\
&=\mathcal{D}_{\partial{\mathsf{B}}}[\phi](\eta X)\\ &\quad
+\int_{\partial{\mathsf{Q}}}\Phi(Y)\, n(Y)\cdot \nabla E(X-Y)\, \mathrm{d}s_Y
+\sum_{j=1}^{m^{\#}}c_j\Xi_j(X)\\
&=-\mathcal{D}_{\partial{\mathsf{Q}}}[\Phi](X)+\mathcal{D}_{\partial{\mathsf{B}}}[\phi](\eta X)
+\sum_{j=1}^{m^{\#}}c_j\Xi_j(X)\, .
\end{split}
\end{equation}
We obtain the concrete form of the system \eqref{eq:BIEdBdQ} by taking traces of the
equalities \eqref{eq:rep} and \eqref{eq:rep-bis} on $\partial {\mathsf{B}}$ and on $\partial {\mathsf{Q}}$, respectively. We deduce with \eqref{eq:symdir} that the unique element
$(\phi,\Phi,\boldsymbol{c})$ of $H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{B}})\times H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{Q}})_{\#}\times \mathbb{R}^{m^{\#}}$ such that \eqref{eq:rep} holds is the (unique) solution in
$ H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{B}})\times H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{Q}})_{\#}\times \mathbb{R}^{m^{\#}}$ of
\begin{equation}\label{eq:sys}
\begin{split}
(-\tfrac{1}{2}I+\mathcal{K}_{\partial {\mathsf{B}}})[\phi](x)
+\eta\int_{\partial {\mathsf{Q}}}\!\!\Phi(Y)\, n(Y)\cdot \nabla E(x-\eta Y)\, \mathrm{d}s_Y
&\\
+\sum_{j=1}^{m^{\#}}c_j\Xi_j(x/\eta)&=\psi(x)
\, \text{ , $\,x \in \partial {\mathsf{B}}$} ,\\
-(\tfrac{1}{2}I+\mathcal{K}_{\partial {\mathsf{Q}}})[\Phi](X)
+\mathcal{D}_{\partial {\mathsf{B}}}[\phi](\eta X)
+\sum_{j=1}^{m^{\#}}c_j\Xi_j(X)&=\Psi(X)
\, \text{, $X \in \partial {\mathsf{Q}}$}\, .\\
\end{split}
\end{equation}
Problem \eqref{eq:symdir} is now converted into the equivalent system of boundary integral equations \eqref{eq:sys}, which we can write as an $\eta$-dependent family of problems
\begin{equation}\label{eq:Meta}
\cM(\eta)\begin{pmatrix}\phi\\\Phi\\\boldsymbol{c}\end{pmatrix}
= \begin{pmatrix}\psi\\\Psi\end{pmatrix}
\end{equation}
with a block $(2\times3)$ operator
\[
\cM(\eta)\equiv\
\begin{pmatrix}
\cM_{11}(\eta)&\cM_{12}(\eta)&\cM_{13}(\eta)\\
\cM_{21}(\eta)&\cM_{22}(\eta)&\cM_{23}(\eta)
\end{pmatrix}
\]
acting from
$
H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{B}})\times H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{Q}})_{\#}\times \mathbb{R}^{m^{\#}}
$ to
$ H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{B}})\times H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{Q}})$.
In this form \eqref{eq:sys}, it is now possible to extend the problem to $\eta=0$ and to analyze the analyticity of its dependence on $\eta$. The main result in this section is the following.
\begin{theorem}\label{thm:Meta=0}
Let $\tau\in[-1/2,1/2]$ and let $\cM(\eta)$ be defined by \eqref{eq:sys}, \eqref{eq:Meta}.
\begin{itemize}
\item[(i)]
There exists $\eta_1\in(0,\eta_0)$ such that the operator valued function
$\cM$ mapping $\eta$ to
\[
\cM(\eta)\in {\mathscr L}\Big(
H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{B}})\times H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{Q}})_{\#}\times \mathbb{R}^{m^{\#}}
,\;
H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{B}})\times H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{Q}})
\Big)
\]
admits a real analytic continuation to $(-\eta_1,\eta_1)$.
\item[(ii)]
For each $\eta\in(-\eta_1,\eta_1)$, the operator $\cM(\eta)$ is an isomorphism.
\end{itemize}
\end{theorem}
\begin{proof}
We will show first that the matrix elements $\cM_{mn}(\eta)$ are operator functions of $\eta$ that can be extended as real analytic functions in a neighborhood of $\eta=0$, and then that for $\eta=0$ the operator $\cM(0)$ is an isomorphism. From this it will follow that $\cM(\eta)$ is an isomorphism for $\eta$ in a neighborhood of $0$. Note that our previous arguments leading to \eqref{eq:sys} already showed that $\cM(\eta)$ is an isomorphism for $\eta\in(0,\eta_0)$.
We will begin by analyzing the dependence of the matrix elements $\cM_{mn}(\eta)$ on $\eta$, in particular at $\eta=0$. Of these, $\cM_{11}$, $\cM_{22}$ and $\cM_{23}$ are independent of $\eta$.
The matrix elements
\[
\cM_{12}(\eta): \left\{
\begin{array}{l}
H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{Q}})_{\#} \to
H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{B}})\\
\Phi\mapsto
\eta\int_{\partial {\mathsf{Q}}}\Phi(Y)\, n(Y)\cdot \nabla E(\cdot-\eta Y)\, \mathrm{d}s_Y
\end{array} \right.
\]
and
\[
\cM_{21}(\eta): \left\{
\begin{array}{l}
H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{B}})\to
H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{Q}})
\\
\phi\mapsto
\mathcal{D}_{\partial {\mathsf{B}}}[\phi](\eta\,\cdot\,)
\end{array} \right.
\]
depend analytically on $\eta$ as long as $\partial{\mathsf{B}}$ and $\eta \hskip0.15ex\partial{\mathsf{Q}}$ do not intersect, and both vanish for $\eta=0$. This follows from Lemma~{\mathrm e}f{lem:Danal} by taking traces.
For the remaining matrix element
\[
\cM_{13}(\eta): \left\{
\begin{array}{l}
\mathbb{R}^{m^{\#}} \to
H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{B}})\\
\boldsymbol{c}\mapsto
\sum_{j=1}^{m^{\#}}c_j\Xi_j(\cdot/\eta)
\end{array} \right.
\]
the analyticity and the vanishing at $\eta=0$ follow from the analyticity of $\Xi_j$ at infinity. In fact, we could simply make the same argument as for $\cM_{12}$, based on Lemma~{\mathrm e}f{lem:Danal}(ii), if we could represent $\Xi_j$ as a double layer potential
$\mathcal{D}_{\partial{\mathsf{Q}}}[\mu]$
with odd density $\mu$. But the $\Xi_j$ were precisely constructed to be in the complement of the range of $\mathcal{D}_{\partial{\mathsf{Q}}}$, so that this is impossible. One can, however, choose a connected smooth domain $\Omega_\#$ that satisfies ${\mathsf{Q}}\subset\Omega_\#$ and $\eta_0\Omega_\#\subset{\mathsf{B}}$ and is symmetric with respect to the reflection $\cR$. Then $\Xi_j$ will be representable as a double layer potential
$\mathcal{D}_{\partial \Omega_\#}[\mu_j]$ on $\Omega_\#^\mathrm{comp}lement$, because
$\int_{\partial\Omega_\#}\partial_n \Xi_j\,\mathrm{d}s=0$ by symmetry and we can apply Lemma~{\mathrm e}f{lem:Kodd}(iv).
We now turn to the proof that $\cM(0)$ is an isomorphism. From the description of the solvability of the exterior Dirichlet problem with data on $\partial{\mathsf{Q}}$ implied by Lemma~{\mathrm e}f{lem:Kodd}(iv) one can deduce, by taking traces on $\partial{\mathsf{Q}}$, the unique solvability of the corresponding augmented boundary integral equation. This follows from the same arguments that led to the unique solvability of the augmented boundary integral equation \eqref{eq:BIEdBeta} on $\partial{\mathsf{B}}_\eta$ associated with the interior Dirichlet problem in ${\mathsf{B}}_\eta$. Taking into account that the normal vector on $\partial{\mathsf{Q}}$ is by our convention exterior to ${\mathsf{Q}}$ and therefore interior to ${\mathsf{Q}}^\mathrm{comp}lement$, we find a change in the sign of the operator $\cK_{\partial{\mathsf{Q}}}$ and can state the following lemma that provides the remaining argument for the completion of the proof of Theorem~{\mathrm e}f{thm:Meta=0}.
\begin{lemma}\label{lem:BIEdQ}
Let $\tau\in[-1/2,1/2]$. For any $\Psi\in H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{Q}}) $ there exist unique $\Phi\in H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{Q}})_{\#}$, $\boldsymbol{c}\in \mathbb{R}^{m^{\#}}$ such that
\begin{equation}\label{eq:BIEdQ}
(-\tfrac12I - \mathcal{K}_{\partial{\mathsf{Q}}})[\Phi]
+\sum_{j=1}^{m^{\#}}c_j\,\gamma_{0}\,\Xi_j = \Psi
\quad\text{ on } \partial{\mathsf{Q}} \, .
\end{equation}
\end{lemma}
Now if we use the values at $\eta=0$ of the matrix elements of $\cM(\eta)$, we find
\[
\cM(0)=
\begin{pmatrix}
\cM_{11}&0&0\\
0&\cM_{22}&\cM_{23}
\end{pmatrix} \,.
\]
Here $\cM_{11}=-\tfrac12I + \mathcal{K}_{\partial{\mathsf{B}}}$ is an isomorphism from
$H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{B}})$ to itself according to Lemma~{\mathrm e}f{lem:Kodd}(ii). And the fact that the operator
$(\cM_{22},\,\cM_{23})$ is an isomorphism from
$H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{Q}})_{\#}\times \mathbb{R}^{m^{\#}}$ to
$H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{Q}})$ is precisely the statement of Lemma~{\mathrm e}f{lem:BIEdQ}. Together this shows that $\cM(0)$ is an isomorphism as claimed, and the proof of the theorem is complete.
\end{proof}
\section{The convergent expansion}
\label{s:convexp}
As a consequence of Theorem~{\mathrm e}f{thm:Meta=0}, we obtain that the unique solution of the boundary integral system \eqref{eq:Meta} depends analytically on $\eta\in(-\eta_1,\eta_1)$. Namely,
\begin{equation}
\label{eq:solEqInt}
\begin{pmatrix}\phi\\\Phi\\\boldsymbol{c}\end{pmatrix} =
\cM(\eta)^{-1}\, \begin{pmatrix}\psi\\\Psi\end{pmatrix} \, ,
\end{equation}
and the operator function $\eta\mapsto\cM(\eta)^{-1}$ is real analytic for $\eta\in(-\eta_1,\eta_1)$.
Inserting this form of the solution of \eqref{eq:Meta} into the representation formula \eqref{eq:equiv>}, we find that the solution $u$ of the Dirichlet problem \eqref{eq:symdir} depends analytically on $\eta\in(-\eta_1,\eta_1)$, too, and therefore has a convergent expansion in powers of $\eta$ in a neighborhood of $\eta=0$.
From this, by comparing \eqref{eq:symdir} with the form \eqref{eq:petagen} of the Dirichlet data in the residual problem found in Section~{\mathrm e}f{ss:residual}, we will obtain a convergent double series for the solution $v_\eta$ of the residual problem. In this way we will then be able to complete the construction of a convergent expansion of the solution of the original problem \eqref{eq:poisson}.
\subsection{Analytic parameter dependence for the auxiliary Dirichlet problem \protect\eqref{eq:symdir}}
\label{ss:4.1}
In this subsection, we consider the Dirichlet problem \eqref{eq:symdir} on the perforated domain ${\mathsf{B}}_\eta$, where the Dirichlet data are given by $(\psi,\Psi)$ independent of $\eta$.
Let us first note that the auxiliary functions $\Xi_j$ introduced in \eqref{eq:Xi} can be considered in the same weighted Sobolev spaces that appear in Lemma~{\mathrm e}f{lem:Drep}.
\begin{equation}
\label{eq:XiK}
\Xi_j \in K^{1+\tau}_{\beta_0\beta_1}({\mathsf{Q}}^\mathrm{comp}lement)
\quad \text{ for all } \tau\in[-1/2,1/2]\,,\;
\beta_0>-2, \beta_1<0\,.
\end{equation}
We can now prove the first result about the solution of the boundary value problem \eqref{eq:symdir}. It is a global decomposition of the solution into two terms that are analytic with respect to $\eta$ if the first one is written in slow coordinates and the second one in fast variables.
\begin{theorem}
\label{thm:u=w+W}
Let $\tau\in[-1/2,1/2]$,
$\beta_0>-2$ with $\beta_0\ge-1-\tau$ if $0\in\partial{\mathsf{Q}}$,
$\beta_1<0$. Then there exists a sequence of bounded linear operators
\[
\cL_n \,:\;
H^{1/2+\tau}_{\mathrm{odd}}(\partial{\mathsf{B}}) \times
H^{1/2+\tau}_{\mathrm{odd}}(\partial{\mathsf{Q}})
\,\to\;
K^{1+\tau}_{\beta_0}({\mathsf{B}}) \times K^{1+\tau}_{\beta_0\beta_1}({\mathsf{Q}}^\mathrm{comp}lement)
\,,\quad j\in\N
\]
such that the solution $u[\eta,\psi,\Psi]$ of the Dirichlet problem \eqref{eq:symdir} in ${\mathsf{B}}_\eta$ has the following form
\[
u[\eta,\psi,\Psi](x) = w(x) + W(x/\eta)
\]
with $w\in K^{1+\tau}_{\beta_0}({\mathsf{B}})$ and $W\in K^{1+\tau}_{\beta_0\beta_1}({\mathsf{Q}}^\mathrm{comp}lement)$ given by the convergent series
\begin{equation}
\label{eq:wWsum}
\binom wW = \sum_{n=0}^\infty \eta^n\, \cL_n \binom\psi\Psi \,.
\end{equation}
There exists $\eta_1>0$ such that for any $\eta\in(-\eta_1,\eta_1)$ the series
\[
\cL(\eta) \equiv \sum_{n=0}^\infty \eta^n\, \cL_n
\]
converges in the operator norm of
${\mathscr L}\big(
H^{1/2+\tau}_{\mathrm{odd}}(\partial{\mathsf{B}}) \times
H^{1/2+\tau}_{\mathrm{odd}}(\partial{\mathsf{Q}})
\,,\;
K^{1+\tau}_{\beta_0}({\mathsf{B}}) \times K^{1+\tau}_{\beta_0\beta_1}({\mathsf{Q}}^\mathrm{comp}lement)
\big)\,.
$
\end{theorem}
\begin{proof}
According to the representation formula \eqref{eq:equiv>} in the form \eqref{eq:rep},
we have
\[
u[\eta,\psi,\Psi](x) = w(x) + W(x/\eta) \text{ with }
w=\mathcal{D}_{\partial {\mathsf{B}}}[\phi] \text{ , }
W=-\mathcal{D}_{\partial{\mathsf{Q}}}[\Phi]+\sum_{j=1}^{m^{\#}}c_j\Xi_j\,,
\]
where $(\phi,\Phi,\boldsymbol{c})$ is the solution of our system of boundary integral equations \eqref{eq:Meta}.
Now we use the solution formula \eqref{eq:solEqInt} of this system, which involves the analytic resolvent
\[
\cM(\eta)^{-1} = \sum_{n=0}^\infty \eta^n \cM_n\,,
\]
where $\cM_n$ is a sequence of operators such that, after possibly shrinking $\eta_1>0$, this series converges for
$\eta\in(-\eta_1,\eta_1)$ in the operator norm of
${\mathscr L}\big(
H^{1/2+\tau}_{\mathrm{odd}}(\partial{\mathsf{B}}) \times
H^{1/2+\tau}_{\mathrm{odd}}(\partial{\mathsf{Q}})
\,,\;
H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{B}})\times H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{Q}})_{\#}\times \mathbb{R}^{m^{\#}}
\big)\,.
$
We combine this with the mapping
\[
\cD:(\phi,\Phi,\boldsymbol{c})\mapsto(w,W)
=\big(\mathcal{D}_{\partial {\mathsf{B}}}[\phi],-\mathcal{D}_{\partial{\mathsf{Q}}}[\Phi]+\sum c_j\Xi_j\big),
\]
which thanks to Lemma~{\mathrm e}f{lem:Drep} and \eqref{eq:XiK} is known to be continuous from
$H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{B}})\times H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{Q}})\times \mathbb{R}^{m^{\#}}$ to
$K^{1+\tau}_{\beta_0}({\mathsf{B}}) \times K^{1+\tau}_{\beta_0\beta_1}({\mathsf{Q}}^\mathrm{comp}lement)$.
This gives the desired representation \eqref{eq:wWsum} as a convergent series where we set
$\cL_n = \cD\,\cM_n$.
\end{proof}
\begin{remark}
\label{rem:wWLinf}
Replacing Lemma {\mathrm e}f{lem:Drep} by Remark {\mathrm e}f{rem:DLinf} in this proof, we conclude that the series expansions \eqref{eq:wWsum} for the functions $w$ and $W$ converge also uniformly, the series for $w$ in $L^\infty({\mathsf{B}})$ and the series for $W$ in $L^\infty({\mathsf{Q}}^\mathrm{comp}lement)$.
\end{remark}
The second result gives a convergent series expansion for the whole solution $u[\eta,\psi,\Psi]$ when written in slow variables and thus shows that it is an analytic function of $\eta$ in a neighborhood of $0$. It is valid outside of a neighborhood of the origin (``outer expansion'').
\begin{theorem}
\label{thm:uexpslow}
Let $\tau\in[-1/2,1/2]$ and $s\in\R$. Let $\Omega$ be a Lipschitz subdomain of ${\mathsf{B}}$ such that $0\not\in\overline\Omega$. If $\overline\Omega\subset{\mathsf{B}}$, then $s$ can be any real number. If, however, $\partial\Omega\cap\partial{\mathsf{B}}\ne\emptyset$, then we assume $s\le1+\tau$.
Let $\eta_\Omega>0$ be such that $\overline{\Omega}\cap \eta\overline{{\mathsf{Q}}}=\emptyset$ for all $\eta\in(0,\eta_\Omega)$.
Then there exist $\eta_1 \in (0,\eta_\Omega)$ and a real analytic map ${\cU}_{\mathsf S}$ from $(-\eta_1,\eta_1)$ to
${\mathscr L}\big(H^{1/2+\tau}_{\mathrm{odd}}(\partial{\mathsf{B}})\times H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{Q}}),\, H^s(\Omega)\big)$
such that
\begin{equation}
\label{eq:rep:1}
u[\eta,\psi,\Psi]\on{\Omega}=\cU_{\mathsf S}(\eta)\binom\psi\Psi
\quad\forall (\eta,\psi,\Psi) \in (0,\eta_1)\times H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{B}})\times H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{Q}})\,.
\end{equation}
Moreover,
\begin{equation}\label{eq:rep:2}
\cU_{\mathsf S}(0)\binom\psi\Psi=w_{0,\psi}\on{|\Omega}\qquad \forall(\psi,\Psi)\in H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{B}})\times H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{Q}})\, ,
\end{equation}
where $w_{0,\psi}$ is the unique solution in $H^{1+\tau}({\mathsf{B}})$ of the Dirichlet problem
\begin{equation}\label{eq:rep:3}
\left\{
\begin{array}{ll}
\Delta w_{0,\psi}=0&\text{ in }{\mathsf{B}}\,,\\
\gamma_{0}w_{0,\psi}=\psi&\text{ on }\partial{\mathsf{B}}\,.
\end{array}
\right.
\end{equation}
\end{theorem}
\begin{proof}
We write $u[\eta,\psi,\Psi](x) = w(x) + W(x/\eta)$ as in Theorem~{\mathrm e}f{thm:u=w+W} and use the analytic dependency on $\eta$ of
$\cM(\eta)^{-1}: (\psi,\Psi)\mapsto (\phi,\Phi,\boldsymbol{c})$ as in the proof of that theorem. The map
$(\phi,\Phi,\boldsymbol{c})\mapsto w\on{\Omega}$ being independent of $\eta$, only its range is of interest. This is contained in $H^{1+\tau}(\Omega)$ for any $\Omega\subset{\mathsf{B}}$, and since $w=\cD_{\partial{\mathsf{B}}}[\phi]$ is harmonic in ${\mathsf{B}}$, it is contained in $C^\infty(\overline\Omega)\subset H^s(\Omega)$ for any $s$ if $\overline\Omega\subset{\mathsf{B}}$.\\
For the map $(\Phi,\boldsymbol{c})\mapsto
W(\cdot/\eta)\on{\Omega} = -\cD^\mathrm{comp}lement_{\partial{\mathsf{Q}},\Omega}(\eta)[\Phi]
+ \sum_{j=1}^{m^{\#}}c_j\Xi_j (\cdot/\eta)$
we invoke Lemma~{\mathrm e}f{lem:Danal} (ii) to get the desired analyticity (see also the argument for the analyticity of $\cM_{13}$ in the proof of Theorem {\mathrm e}f{thm:Meta=0} ).
\end{proof}
The third result shows the analytic dependence on $\eta$ for the solution $u[\eta,\psi,\Psi]$ when written in fast variables. It concerns the solution on a subdomain of size $\eta$ (``inner expansion''). The proof is similar to the proof of Theorem~{\mathrm e}f{thm:uexpslow}, but simpler, because it is based on the formula
\[
u[\eta,\psi,\Psi](\eta X)=w(\eta X)+W(X)
\]
and it therefore simply invokes the
harmonicity, hence analyticity of $w=\cD_{\partial{\mathsf{B}}}[\phi]$ near the origin, see Lemma~{\mathrm e}f{lem:Danal}(i).
\begin{theorem}
\label{thm:uexpfast}
Let $\tau\in[-1/2,1/2]$ and $s\in\R$. Let $\Omega\subset{\mathsf{Q}}^{\mathrm{comp}lement}=\R^2\setminus\overline{\mathsf{Q}}$ be a bounded Lipschitz domain. If $\overline\Omega\subset{\mathsf{Q}}^{\mathrm{comp}lement}$, then $s$ can be any real number. If $\partial\Omega\cap\partial{\mathsf{Q}}\ne\emptyset$, then we assume $s\le1+\tau$.
Let $\tilde\eta_\Omega>0$ be such that $\eta\Omega\subset{\mathsf{B}}$ for all $\eta\in(0,\tilde\eta_\Omega)$.
Then there exist $\eta_1 \in (0,\tilde\eta_\Omega)$ and a real analytic map ${\cU}_{\mathsf{F}}$ from $(-\eta_1,\eta_1)$ to
${\mathscr L}\big(H^{1/2+\tau}_{\mathrm{odd}}(\partial{\mathsf{B}})\times H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{Q}}),\, H^s(\Omega)\big)$ such that
\begin{multline}
\label{eq:repF:1}
u[\eta,\psi,\Psi](\eta\,\cdot\,)\on{\Omega}=\cU_{\mathsf{F}}(\eta)\binom\psi\Psi
\\
\quad\forall (\eta,\psi,\Psi) \in (0,\eta_1)\times H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{B}})\times H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{Q}})\,.
\end{multline}
Moreover,
\begin{equation}\label{eq:repF:2}
\cU_{\mathsf{F}}(0)\binom\psi\Psi=W_{0,\Psi}\on{\Omega}\qquad \forall(\psi,\Psi)\in H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{B}})\times H^{1/2+\tau}_{\mathrm{odd}}(\partial {\mathsf{Q}})\, ,
\end{equation}
where $W_{0,\Psi}$ is the unique solution in $K^{1+\tau}_{\beta_0\beta_1}({\mathsf{Q}}^\mathrm{comp}lement)$ for $\beta_0\in(-2,0)$, $\beta_1\in(-1,0)$ of the exterior Dirichlet problem
\begin{equation}\label{eq:repF:3}
\left\{
\begin{array}{ll}
\Delta W_{0,\Psi}=0&\text{ in }{\mathsf{Q}}^\mathrm{comp}lement\,,\\
\gamma_{0}W_{0,\Psi}=\Psi&\text{ on }\partial{\mathsf{Q}}\,.
\end{array}
\right.
\end{equation}
\end{theorem}
\subsection{Convergent expansion of the solution of the original problem}
\label{ss:4.2}
In this subsection, we first insert into the expansion of the solution $u[\eta,\psi,\Psi]$ of the Dirichlet problem on the perforated domain ${\mathsf{B}}_\eta$ obtained in the preceding subsection the knowledge about the Dirichlet data from Theorem~{\mathrm e}f{th:Tu0}, namely $\psi=0$ and
$\Psi=-\cT^*[u_0]$. The latter is given by a convergent series in \eqref{eq:Tu0}.
We then have to write the resulting double series as a series in $\varepsilon$ by using $\eta=\varepsilon^{\pi/\omega}$ and we have to interpret the result as a series that converges in function spaces defined on the original domain ${\mathsf{A}}_\varepsilon$, by
undoing the conformal map $\cG^*_{\pi/\omega}$. This will give a convergent expansion for the solution $\tilde u_\varepsilon$ of the residual problem \eqref{eq:pepsgen}.
The final step is to add the function $u_0$ as described in Theorem~{\mathrm e}f{th:tu}, in order to find a convergent expansion for the solution $u_\varepsilon$ of the original problem \eqref{eq:poisson}.
Corresponding to the three results about the Dirichlet problem in the perforated domain ${\mathsf{B}}_\eta$, Theorems~{\mathrm e}f{thm:u=w+W}, {\mathrm e}f{thm:uexpslow} and {\mathrm e}f{thm:uexpfast}, we prove three different results about the solution of the original problem \eqref{eq:poisson}.
For the notation describing the convergent series in powers of $\varepsilon$, we refer to Sections~{\mathrm e}f{s:2} and {\mathrm e}f{s:3}, in particular to Notation~{\mathrm e}f{not:1} for the definition of the index set $\gA$ and to Notation~{\mathrm e}f{not:2} for the powers and divided differences of powers of $\varepsilon$ abbreviated by the symbol ${\mathscr E}_\gamma(\varepsilon)$.
In view of Theorem {\mathrm e}f{th:Tu0} and Remark {\mathrm e}f{rem:tau}, we introduce a maximal regularity index
\begin{equation}
\label{eq:tau0}
\tau_{0} = \tfrac12 \;\mbox{ if } 0\not\in\partial{\mathsf{P}}\,,\quad
\tau_{0} = \min\{\tfrac12,\tfrac{2\omega}\pi\} \;\mbox{ if } 0\in\partial{\mathsf{P}}\,.
\end{equation}
The first result is a globally valid two-scale splitting of the solution $u_{\varepsilon}$, where the slow-variable part and the fast-variable part have separate convergent expansions, when written in their respective variables.
\begin{theorem}
\label{thm:ueps}
There exist $\varepsilon_1>0$ such that the solution $u_\varepsilon$ of Problem \eqref{eq:poisson} has the following structure.
\begin{equation}
\label{eq:u=u+U}
u_\varepsilon(t) = u_0(t) + u(\varepsilon)(t) + U(\varepsilon)(\tfrac t\varepsilon)
\qquad \forall\, t\in {\mathsf{A}}_\varepsilon, \; \varepsilon\in(0,\varepsilon_1)\,.
\end{equation}
Here $u_0$ is the solution of the limit problem \eqref{eq:u0} on the unperforated corner domain ${\mathsf{A}}$. Its singular behavior near the corner is described by the convergent series \eqref{eq:tu} in Theorem~{\mathrm e}f{th:tu}.\\
The functions $u(\varepsilon)(t)$ and $U(\varepsilon)(T)$ are defined for $t\in{\mathsf{A}}$ and $T\in{\mathsf{P}}^\mathrm{comp}lement$, respectively, and have a convergent series expansion of the following form.
\begin{equation}
\label{eq:U=sumV}
\binom uU
= \sum_{(n,\gamma)\in\N\times\gA} \varepsilon^{n\pi/\omega}{\mathscr E}_\gamma(\varepsilon)
\binom{v_{n\gamma}}{V_{n\gamma}}\,.
\end{equation}
Let $\tau\in(0,\tau_{0})$ and $\beta_0>-1-\pi/\omega$ with $\beta_0>-1-\tau\pi/\omega$ if\/ $0\in\partial{\mathsf{P}}$, and let $\beta_1<-1+\pi/\omega$. The series converges in the weighted Sobolev spaces
$
K^{1+\tau}_{\beta_0}({\mathsf{A}})\times K^{1+\tau}_{\beta_0\beta_1}({\mathsf{P}}^\mathrm{comp}lement)\,,
$
and there exist constants $C$ and $M$ such that
\[
\DNorm{v_{n\gamma}}{K^{1+\tau}_{\beta_0}({\mathsf{A}})}+
\DNorm{V_{n\gamma}}{K^{1+\tau}_{\beta_0\beta_1}({\mathsf{P}}^\mathrm{comp}lement)}
\le C M^{n+|\gamma|},\quad (n,\gamma)\in\N\times\gA\,.
\]
The series converge also uniformly, in
$L^\infty({\mathsf{A}})\times L^\infty({\mathsf{P}}^\mathrm{comp}lement)$.
\end{theorem}
\begin{proof}
We have $u_\varepsilon=u_0+\tilde u_\varepsilon$, where $\tilde u_\varepsilon$ solves the residual problem \eqref{eq:pepsgen}. After applying the conformal mapping $\cG_{\pi/\omega}$ and the odd reflection, this was rewritten in Theorem~{\mathrm e}f{th:Tu0} as the problem \eqref{eq:petagen}, a special case of the boundary value problem \eqref{eq:symdir}. Thus we have the identification
$\tilde u_\varepsilon = u[\eta,\psi,\Psi]\circ \cG_{\pi/\omega}$, where $\psi=0$ and $\Psi=-\cT^*[u_0](\eta \cdot)$.
Combining the convergent expansion \eqref{eq:Tu0} for $\cT^*[u_0]$ with the power series \eqref{eq:wWsum} for the solution operator of problem \eqref{eq:symdir}, we thus find a convergent expansion that has the form \eqref{eq:U=sumV}
\[
\binom uU
= \sum_{n=0}^\infty \sum_{\gamma\in\gA} \varepsilon^{n\pi/\omega}{\mathscr E}_\gamma(\varepsilon)
\cL_n \binom{0}{\Psi_\gamma} \circ \cG_{\pi/\omega}\,.
\]
The right choice of weighted Sobolev spaces for the convergence follows from Theorem~{\mathrm e}f{thm:u=w+W} with the transformation rule of Lemma~{\mathrm e}f{lem:Km}. Note that this transformation rule motivates the use of weighted Sobolev spaces instead of non-weighted spaces. For the uniform convergence finally, we notice that $L^\infty$ remains invariant under the conformal mappings $\cG^*_\kappa$.
\end{proof}
The second result is a convergent expansion of the whole solution $u_{\varepsilon}$ written in slow (macroscopic) variables. It is valid in any fixed subdomain of ${\mathsf{A}}_{\varepsilon}$ that has a positive distance to the corner and thus is free of holes for sufficiently small $\varepsilon$. This corresponds to the outer expansion in the method of matched asymptotic expansions, compare \cite[Section 5]{DaToVi10}.
\begin{theorem}
\label{thm:outer}
Let $\Omega$ be a Lipschitz subdomain of ${\mathsf{A}}$ such that $0\not\in\overline\Omega$.
Let $\varepsilon_\Omega>0$ be such that $\overline{\Omega}\cap \varepsilon\overline{{\mathsf{P}}}=\emptyset$ for all $\varepsilon\in(0,\varepsilon_\Omega)$.
Then there exists $\varepsilon_1 \in (0,\varepsilon_\Omega)$ such that for $\varepsilon\in(0,\varepsilon_{1})$ the solution $u_{\varepsilon}$ of Problem \eqref{eq:poisson} has the following expansion in $\Omega$:
\begin{equation}
\label{eq:outer}
u_{\varepsilon}(t) = u_{0}(t) +
\sum_{(n,\gamma)\in\N_{*}\times\gA} \varepsilon^{n\pi/\omega}{\mathscr E}_\gamma(\varepsilon) u^{{\mathsf S}}_{n\gamma}(t)\,,
\qquad t\in\Omega\,.
\end{equation}
Let $\tau<\tau_0$. Then the series converges for $|\varepsilon|<\varepsilon_{1}$ in $H^{1+\tau}(\Omega)$, and there exist constants $C$ and $M$ such that
\[
\DNorm{u^{{\mathsf S}}_{n\gamma}}{H^{1+\tau}(\Omega)}
\le C M^{n+|\gamma|},\quad (n,\gamma)\in\N_{*}\times\gA\,.
\]
The series converges also uniformly in $\Omega$.
\end{theorem}
\begin{proof}
In the multiscale decomposition \eqref{eq:u=u+U}
$u_\varepsilon=u_0+u(\varepsilon)+U(\varepsilon)\big(\tfrac\cdot\varepsilon\big)$, the term $u(\varepsilon)$ has the required expansion according to Theorem~{\mathrm e}f{thm:ueps}. For $U(\varepsilon)$ we write it as
\[
U(\varepsilon)=W\circ \cG_{\pi/\omega}\,,
\]
where $W$ is the function defined in Theorem~{\mathrm e}f{thm:u=w+W} in the special case where $\psi=0$ and $\Psi=-\cT^*[u_0](\eta \cdot)$. The analyticity of $W(\cdot/\eta)$ with respect to $\eta$ at $\eta=0$ in the case of $\eta$-independent $\Psi$ has been deduced in the proof of Theorem~{\mathrm e}f{thm:uexpslow} from Lemma~{\mathrm e}f{lem:Danal}. We have to combine this, as in the proof of Theorem~{\mathrm e}f{thm:ueps}, with the expansion
\eqref{eq:Tu0} for $\cT^*[u_0]$ and set $\eta=\varepsilon^{\pi/\omega}$, ending up with the expansion required for \eqref{eq:outer}. The coefficient functions $u^{{\mathsf S}}_{n\gamma}$ are the sum of the corresponding terms of the expansion of $u(\varepsilon)$ and of $U(\varepsilon)(\cdot/\varepsilon)$. For $n=0$ both of these terms vanish, because they correspond to $u[\eta,\psi,\Psi]$ in \eqref{eq:rep:1} at $\eta=0$ and $\psi=0$, and according to \eqref{eq:rep:2}--\eqref{eq:rep:3}, this is zero. Therefore the sum over $n$ in \eqref{eq:outer} starts with $n\ge1$.
\end{proof}
The third result is a convergent expansion of the whole solution $u_{\varepsilon}$ written in fast (microscopic) variables. It is valid outside of the holes in a scaled family $\varepsilon\Omega$ of subdomains of ${\mathsf{A}}_{\varepsilon}$. This corresponds to the inner expansion in the method of matched asymptotic expansions, compare \cite[Section 5]{DaToVi10}.
\begin{theorem}
\label{thm:inner}
Let $\Omega\subset{\mathsf S}_{\omega}\setminus\overline{\mathsf{P}}$ be a bounded Lipschitz domain.
Let $\tilde\varepsilon_\Omega>0$ be such that $\varepsilon\Omega\subset{\mathsf{A}}$ for all $\varepsilon\in(0,\tilde\varepsilon_\Omega)$.
Then there exists $\varepsilon_1 \in (0,\tilde\varepsilon_\Omega)$ such that for $\varepsilon\in(0,\tilde\varepsilon_{1})$ the solution $u_{\varepsilon}$ of Problem \eqref{eq:poisson} has the following expansion in $\varepsilon\Omega$:
\begin{equation}
\label{eq:inner}
u_{\varepsilon}(\varepsilon T) =
\sum_{(n,\gamma)\in\N\times\gA} \varepsilon^{n\pi/\omega}{\mathscr E}_\gamma(\varepsilon) U^{{\mathsf{F}}}_{n\gamma}(T)\,,
\qquad T\in\Omega\,.
\end{equation}
Let $\tau<\tau_0$.
Then the series converges for $|\varepsilon|<\tilde\varepsilon_{1}$ in $H^{1+\tau}(\Omega)$, and there exist constants $C$ and $M$ such that
\[
\DNorm{U^{{\mathsf{F}}}_{n\gamma}}{H^{1+\tau}(\Omega)}
\le C M^{n+|\gamma|},\quad (n,\gamma)\in\N_{*}\times\gA\,.
\]
The series converges also uniformly in $\Omega$.
\end{theorem}
\begin{proof}
As in the proof of Theorem~{\mathrm e}f{thm:ueps} we use the identity
$u_\varepsilon=u_0+\tilde u_\varepsilon\equiv u_0+u[\eta,\psi,\Psi]\circ \cG_{\pi/\omega}$, where $\psi=0$ and $\Psi=-\cT^*[u_0](\eta \cdot)$. Together with Theorem~{\mathrm e}f{thm:uexpfast} for $u[\eta,\psi,\Psi](\eta\,\cdot\,)$, this gives the desired form \eqref{eq:inner} of the expansion for the second term $\tilde u_\varepsilon(\varepsilon\cdot)$. Here, as in the outer expansion \eqref{eq:outer}, the sum over $n$ lacks the term $n=0$.
It remains to analyze the first term $u_0(\varepsilon\cdot)$.
Here we need the asymptotic behavior (expansion into corner singular functions) of $u_0$ that was described in \eqref{eq:tup} and used for expanding $u_0(\varepsilon T)$ into a convergent series in \eqref{eq:tupe}. With the notation introduced in \eqref{eq:Phi} in the proof of Theorem~{\mathrm e}f{th:Tu0}, this series can be written as
\[
u_0(\varepsilon T) = \sum_{\gamma\in\gA} {\mathscr E}_\gamma(\varepsilon)\Phi_\gamma(T)\,.
\]
This is a series of the form \eqref{eq:inner} with $n=0$. Explicitly estimating norms of the functions $\Phi_\gamma$ or relying on the estimate \eqref{eq:PhiNorm}, we see that the series converges in $H^{1+\tau}(\Omega)$.
\end{proof}
The fact that the series expansions in the last three theorems
are only stepwise convergent, that is convergent when pairs of powers of $\varepsilon$ are grouped together into the terms ${\mathscr E}_\gamma(\varepsilon)$ from Notation~{\mathrm e}f{not:2}(3), is caused entirely by the corresponding fact for the expansion of $u_0$ studied in Section~{\mathrm e}f{s:2}, see in particular Remarks~{\mathrm e}f{rem:pi}--{\mathrm e}f{rem:other}. Thus if we assume that one of the conditions mentioned in these Remarks is satisfied, we find convergent power series, and it is then possible to reformulate the statements of Theorems~{\mathrm e}f{thm:ueps}--{\mathrm e}f{thm:inner} in terms of analytic functions of $\varepsilon$ and $\varepsilon^{\pi/\omega}$.
\begin{corollary}
\label{cor:f=0}
Suppose that the right hand side $f$ vanishes in a neighborhood of the corner $0$. Denote by $u_\varepsilon$ the solution of Problem \eqref{eq:poisson}.\\
\emph{(i)}
Let the parameters $\tau$, $\beta_0$ and $\beta_1$ be chosen as in Theorem~{\mathrm e}f{thm:ueps}. Then there exists $\eta_1>0$ and a real analytic function
\[
(-\eta_1,\eta_1) \ni \eta\mapsto
\mathcal{V}[\eta]=\binom{v[\eta]}{V[\eta]} \in
K^{1+\tau}_{\beta_0}({\mathsf{A}})\times K^{1+\tau}_{\beta_0\beta_1}({\mathsf{P}}^\mathrm{comp}lement)
\]
such that in the two-scale decomposition \eqref{eq:u=u+U}
$u_\varepsilon=u_0+u(\varepsilon)+U(\varepsilon)(\tfrac\cdot\varepsilon)$ we have
\[
u(\varepsilon) = v[\varepsilon^{\pi/\omega}]\,,\qquad
U(\varepsilon) = V[\varepsilon^{\pi/\omega}]
\qquad \forall\, \varepsilon\in(0,\eta_1^{\omega/\pi})\,.
\]
\emph{(ii)}
Let $\Omega$ be a Lipschitz subdomain of ${\mathsf{A}}$ such that $0\not\in\overline\Omega$ and let $\tau$ be chosen as in Theorem~{\mathrm e}f{thm:outer}. Then there exists $\eta_1>0$ and a real analytic function
\[
(-\eta_1,\eta_1) \ni \eta\mapsto
u_{\mathsf S}[\eta] \in H^{1+\tau}(\Omega)
\]
such that we have
$u_{{\mathsf S}}[0] = u_0$ and
\[
u_\varepsilon = u_{\mathsf S}[\varepsilon^{\pi/\omega}]
\qquad \mbox{ in }\Omega\,,\quad\forall\, \varepsilon\in(0,\eta_1^{\omega/\pi})\,.
\]
\emph{(iii)}
Let $\Omega\subset{\mathsf S}_{\omega}\setminus\overline{\mathsf{P}}$ be a bounded Lipschitz domain and let $\tau$ be chosen as in Theorem~{\mathrm e}f{thm:inner}. Then there exists $\eta_1>0$ and a real analytic function
\[
(-\eta_1,\eta_1) \ni \eta\mapsto
U_{\mathsf{F}}[\eta] \in H^{1+\tau}(\Omega)
\]
such that we have
\[
u_\varepsilon(\varepsilon T) = U_{\mathsf{F}}[\varepsilon^{\pi/\omega}](T)
\qquad \forall \,T\in\Omega\,,\quad \varepsilon\in(0,\eta_1^{\omega/\pi})\,.
\]
\end{corollary}
\begin{proof}
As we have seen in Remark~{\mathrm e}f{rem:f=0}, if $f$ vanishes in a neighborhood of the corner, then in the series expansion of $u_0$ there appear only exponents that are of the form $k\pi/\omega$ with integer $k$, and the series is unconditionally convergent. In the resolution of the residual problem in Section~{\mathrm e}f{ss:Dir}, integer powers of $\eta=\varepsilon^{\pi/\omega}$ were incorporated, so that the final convergent series expansions \eqref{eq:U=sumV}, \eqref{eq:outer} and \eqref{eq:inner} also contain only exponents that are integer multiples of $\pi/\omega$. It follows that these series are convergent power series, hence analytic functions, in the variable $\eta=\varepsilon^{\pi/\omega}$.
\end{proof}
Let now $\omega$ be a rational multiple of $\pi$, i.e. $\pi/\omega=p/q$, where $p$ and $q$ are relatively prime positive integers. In this case, all the exponents of $\varepsilon$ appearing in the
convergent series expansions \eqref{eq:U=sumV}, \eqref{eq:outer} and \eqref{eq:inner} can be seen to be integer multiples of $1/q$. The expressions ${\mathscr E}_\gamma(\varepsilon)$ as defined in Notation~{\mathrm e}f{not:2} are now either integer powers of $\delta=\varepsilon^{1/q}$ or of the form $\varepsilon^{\ell}\log \varepsilon$ with integer $\ell$. They can therefore be expressed via two real analytic functions of one variable. We formulate this observation for the two-scale decomposition \eqref{eq:u=u+U} of Theorem~{\mathrm e}f{thm:ueps} and its convergent series expansion \eqref{eq:U=sumV} and leave the corresponding reformulations of Theorems~{\mathrm e}f{thm:outer} and {\mathrm e}f{thm:inner} to the reader.
\begin{corollary}
\label{cor:omegaQ}
Let $\pi/\omega=p/q$. With the notations of Theorem~{\mathrm e}f{thm:ueps}, there exist $\delta_1>0$ and two real analytic functions (we set $\varepsilon_1\equiv(\delta_1)^q$)
\[
\begin{aligned}
(-\delta_1,\delta_1) \ni \delta &\mapsto
\mathcal{V}_0[\delta] \in
K^{1+\tau}_{\beta_0}({\mathsf{A}})\times K^{1+\tau}_{\beta_0\beta_1}({\mathsf{P}}^\mathrm{comp}lement)\\
(-\varepsilon_1,\varepsilon_1) \ni \varepsilon &\mapsto
\mathcal{V}_1[\varepsilon] \in
K^{1+\tau}_{\beta_0}({\mathsf{A}})\times K^{1+\tau}_{\beta_0\beta_1}({\mathsf{P}}^\mathrm{comp}lement)\\
\end{aligned}
\]
such that
\[
\binom{u(\varepsilon)}{U(\varepsilon)} =
\mathcal{V}_0[\varepsilon^{1/q}] + \mathcal{V}_1[\varepsilon^{p}]\log\varepsilon
\qquad \forall\, \varepsilon\in(0,\varepsilon_1)\,.
\]
\end{corollary}
The third case where we find absolutely convergent expansions in powers of $\varepsilon$ is when $\omega$ is not a rational multiple of $\pi$ but is such that we can choose $\gA_0=\emptyset$. According to the discussion in Section~{\mathrm e}f{ss:1.2} and in Appenix~{\mathrm e}f{app:Liouville}, this is the case if and only if
$\frac\pi\omega$ is not a super-exponential Liouville number. In this case we do not need the divided differences of Notation~{\mathrm e}f{not:2}(3), and the terms in the convergent expansions \eqref{eq:U=sumV}, \eqref{eq:outer} and \eqref{eq:inner} are simply monomials in the two variables $\varepsilon$ and $\varepsilon^{\pi/\omega}$, and the series therefore define real analytic functions of two variables. We formulate again the corresponding result for the two-scale expansion of Theorem~{\mathrm e}f{thm:ueps} and leave the reformulations of Theorems~{\mathrm e}f{thm:outer} and {\mathrm e}f{thm:inner} to the reader.
\begin{corollary}
\label{cor:irrat}
Suppose that $\pi/\omega$ is irrational and not a super-exponential Liouville number in the sense of Definition~{\mathrm e}f{def:Liouville}. Then there exist $\varepsilon_1>0$ and a real analytic function of two variables (we set $\eta_1=\varepsilon_1^{\pi/\omega}$)
\[
(-\varepsilon_1,\varepsilon_1)\times(-\eta_1,\eta_1)
\ni (\varepsilon,\eta) \mapsto
\mathcal{V}[\varepsilon,\eta] \in
K^{1+\tau}_{\beta_0}({\mathsf{A}})\times K^{1+\tau}_{\beta_0\beta_1}({\mathsf{P}}^\mathrm{comp}lement)
\]
such that
\[
\binom{u(\varepsilon)}{U(\varepsilon)} =
\mathcal{V}[\varepsilon, \varepsilon^{\pi/\omega}]
\qquad \forall\, \varepsilon\in(0,\varepsilon_1)\,.
\]
\end{corollary}
\appendix
\section{Symmetric extension of Lipschitz domains}\label{app:symextlip}
In this section we use the objects defined in Section~{\mathrm e}f{Ss:Reflection}, in particular the upper half-plane ${\mathsf S}_\pi$ and the operation $\cE$ of symmetric extension of a subset of ${\mathsf S}_\pi$ by reflection at the horizontal axis.
In general, the symmetric extension of a Lipschitz domain is not Lipschitz, and therefore the following result is not entirely obvious and merits a complete proof.
\begin{lemma}
\label{lem:LipE}
Assume that $\Omega$ is a bounded subdomain of\/ ${\mathsf S}_\pi$ and that $\Omega$ and ${\mathsf S}_\pi\setminus\overline\Omega$ have Lipschitz boundaries. Then
$\cE(\Omega)$ has a Lipschitz boundary.
\end{lemma}
\begin{proof}
As a characterization of a Lipschitz boundary we use the property that it is locally congruent to the graph of a Lipschitz continuous function.
A simple consequence of this property is that in 2 dimensions, each point of the boundary has a 2-dimensional neighborhood in which the boundary is a simple curve, in particular it is homeomorphic to an interval.
Let us now first show that $\partial\Omega\cap\partial{\mathsf S}_\pi$ has no isolated points.
Suppose there were such a point $\boldsymbol{x}_0=(x_0,0)$. We show that then ${\mathsf S}_\pi\setminus\overline\Omega$ cannot be a Lipschitz domain, contrary to the hypothesis. Since $\Omega$ is Lipschitz, there is a neighborhood $\cU$ of $\boldsymbol{x}_0$ in which $\partial\Omega$ coincides with a simple curve $\Gamma_0$ and such that $\cU\cap\partial\Omega\cap\partial{\mathsf S}_\pi=\{\boldsymbol{x}_0\}$. This neighborhood can be chosen such that $\cU\cap \partial{\mathsf S}_\pi$ is an interval $\Gamma_1$. Since
$(\partial\Omega\cup \partial{\mathsf S}_\pi)\setminus(\partial\Omega\cap \partial{\mathsf S}_\pi)$
is contained in the boundary of ${\mathsf S}_\pi\setminus\overline\Omega$, the latter coincides in $\cU$ with the union of the two curves $\Gamma_0$ and
$\Gamma_1$ that intersect in the interior point $\boldsymbol{x}_0$. Such a union is clearly not homeomorphic to an interval.
We will now use the following equivalent reformulation of the above definition of a Lipschitz boundary $\partial\Omega$ in two dimensions: To each of its points there is a neighborhood $\cU$ and a convex cone $\cC_{\alpha\beta}$ with the following property:
If the curve $\Gamma_0=\partial\Omega\cap\cU$ is parametrized by an interval,
\[
\gamma: (t_0,t_1)\to \Gamma_0\subset \cU \,,
\]
then for $\boldsymbol{x}=\gamma(s)$, $\boldsymbol{y}=\gamma(t)$ with
$s<t$ (we say ``$\boldsymbol{x}$ precedes $\boldsymbol{y}$'' or $\boldsymbol{x}\prec\boldsymbol{y}$) we have
$\boldsymbol{y}\in \boldsymbol{x} + \cC_{\alpha\beta}$.
Here the cone $\cC_{\alpha\beta}$ is defined by two angles $\alpha$, $\beta$ with $\alpha<\beta<\alpha+\pi$,
\[
\cC_{\alpha\beta} =
\{({\mathsf{g}}o\cos\theta, {\mathsf{g}}o\sin\theta) \colon 0<{\mathsf{g}}o<\infty, \alpha<\theta<\beta \}\,.
\]
One can observe that the rotation angles $\omega$ (modulo $2\pi$) of coordinate axes that allow the representation of $\Gamma_0$ as a graph are given by the complement of $\cC_{\alpha\beta}$, the condition being
\[
\omega-\tfrac\pi2\in(\beta-\pi,\alpha)\cup(\beta,\alpha+\pi)\,.
\]
Let now $\cU$ be such a neighborhood of a point of $\partial\Omega$. If $\partial\Omega\cap\cU$ is entirely contained either in the upper half-plane ${\mathsf S}_\pi$ or in the axis of symmetry $\partial{\mathsf S}_\pi$, then there is nothing to prove, because in this case (after possibly choosing a smaller neighborhood), the set
$\cU\cup\cR(\cU)$ will be a suitable neighborhood for the boundary of $\cE(\Omega)$.
The nontrivial case is when $\cU$ is a neighborhood of a point
$\boldsymbol{x}_0\in\partial\Omega\cap\partial{\mathsf S}_\pi$ and both
$\cU\cap\partial\Omega\cap\partial{\mathsf S}_\pi$ and
$\cU\cap\partial\Omega\cap{\mathsf S}_\pi$ are non-empty.
Since, as we have seen, $\boldsymbol{x}_0$ is not an isolated point of $\partial\Omega\cap\partial{\mathsf S}_\pi$, the structure of $\partial\Omega\cap\cU$ is (after possibly choosing a smaller neighborhood) the following:
\[
\partial\Omega\cap\cU = \Gamma_1\cup\Gamma_0\,,
\]
where $\Gamma_1$ is an interval $I_1\times\{0\}\subset\partial{\mathsf S}_\pi$, and $\Gamma_0$ is a Lipschitz curve contained in ${\mathsf S}_\pi$. Locally, the boundary of the complement has the form
\[
\partial({\mathsf S}_\pi\setminus\overline\Omega)\cap\cU = \Gamma_1'\cup\Gamma_0\,,
\]
where $\Gamma_1'$ is another interval $I_1'\times\{0\}\subset\partial{\mathsf S}_\pi$.
The intervals have one point in common, which we can assume to be $\boldsymbol{x}_0$
\[
\Gamma_1\cap\Gamma_1' = \{\boldsymbol{x}_0\} = \overline\Gamma_0\cap\partial{\mathsf S}_\pi\,.
\]
Since now $\Omega$ and ${\mathsf S}_\pi\setminus\Omega$ play symmetric roles, it is no restriction to assume that $I_1=[x_0-\delta,x_0]$ and $I_1'=[x_0,x_0+\delta]$ with some $\delta>0$. We can also assume that the two parametrizations of $\partial\Omega\cap\cU$ and of $\partial({\mathsf S}_\pi\setminus\overline\Omega)\cap\cU$ are oriented such that in both cases the segment $\Gamma_1$ or $\Gamma_1'$, respectively, precedes the curve $\Gamma_0$.
Now from our definition of a Lipschitz boundary, we get a cone $\cC_{\alpha\beta}$ that satisfies
\[
\boldsymbol{x},\boldsymbol{y}\in\partial\Omega\cap\cU
\:\text{ and }\: \boldsymbol{x}\prec\boldsymbol{y}
\quad\Longrightarrow\quad
\boldsymbol{y}-\boldsymbol{x}\in \cC_{\alpha\beta} \,.
\]
In particular, this holds for $\boldsymbol{x},\boldsymbol{y}\in \Gamma_1$, and this implies that we have $-\pi<\alpha<0$ and $0<\beta<\alpha+\pi$.
Likewise, there is a cone $\cC_{\alpha'\beta'}$ that satisfies
\[
\boldsymbol{x},\boldsymbol{y}\in\partial({\mathsf S}_\pi\setminus\overline\Omega)\cap\cU
\:\text{ and }\: \boldsymbol{x}\prec\boldsymbol{y}
\quad\Longrightarrow\quad
\boldsymbol{y}-\boldsymbol{x}\in \cC_{\alpha'\beta'} \,.
\]
Since this holds for $\boldsymbol{x},\boldsymbol{y}\in \Gamma_1'$, we must have $0<\alpha'<\pi$ and $\pi<\beta'<\alpha'+\pi$.
For the curve $\Gamma_0$ we have both conditions,
\[
\boldsymbol{x},\boldsymbol{y}\in\Gamma_0
\:\text{ and }\: \boldsymbol{x}\prec\boldsymbol{y}
\quad\Longrightarrow\quad
\boldsymbol{y}-\boldsymbol{x}\in
\cC_{\alpha\beta}\cap \cC_{\alpha'\beta'} = \cC_{\alpha'\beta}\,.
\]
The latter cone $\cC_{\alpha'\beta}$ is contained in the upper half-plane ${\mathsf S}_\pi$, and this implies that the curve $\Gamma_0$ can be represented as a graph in a coordinate system rotated by a right angle $\omega=\pi/2$. This means that there is a Lipschitz continuous function $\phi:(0,y_0)\to \R$ such that
\[
\Gamma_0=\partial\Omega\cap\cU\cap{\mathsf S}_\pi=
\{(x,y)\in\R^2 \colon x=\phi(y), 0<y<y_0 \} \,.
\]
Now we can execute our symmetric extension and find that the point
$\boldsymbol{x}_0\in\partial(\cE(\Omega))$ has $\cE(\cU)$ as a neighborhood in which the boundary
\[
\partial(\cE(\Omega))\cap\cE(\cU) =
\Gamma_0\cup\{\boldsymbol{x}_0\}\cup\cR(\Gamma_0)
\]
is represented as the graph $\{x=\tilde\phi(y)\}$ of a Lipschitz continuous function $\tilde\phi$, namely the even extension of $\phi$, $\tilde\phi(y)=\phi(|y|)\}$, $-y_0 < y < y_0$, completed by the choice $\phi(0)=x_0$.
\end{proof}
\section{Convergence of the corner expansion for the Dirichlet problem and Diophantine approximation}\label{app:Liouville}
In this section, we use the notation of Section~{\mathrm e}f{ss:1.2}. We find conditions on the opening angle $\omega$ for the convergence of the series of particular solutions constructed according to \eqref{eq:solel}
\begin{equation}
\label{eq:sumupartial}
u_\partial(t) = \sum_{\ell\in\N_*} w_\ell(t)
= \sum_{\ell\in\N_*}
\big(\frac{g^\omega_\ell - g^0_\ell\,\cos\ell\omega}{\sin \ell\omega}
\,\operatorname{Im}\zeta^\ell
+ g^0_\ell \, \operatorname{Re}\zeta^\ell
\big)\,,
\end{equation}
provided the two power series with coefficients $g^0_\ell$ and $g^\omega_\ell$ have a nonzero convergence radius as in \eqref{eq:estG}.
We will assume here that the number
$\kappa=\tfrac\pi\omega$ is irrational, so that the coefficients in the sum \eqref{eq:sumupartial} are well defined.
As was observed already in \cite{BraDau82,Dauge84}, for certain angles $\omega$ for which $\kappa$ is irrational the small denominators $\sin\ell\omega$ pose a problem for the convergence of the series \eqref{eq:sumupartial}, and a procedure for reestablishing the convergence was found. The convergence of the sum depends on the rate of approximability of $\kappa$ by rational numbers, a question that has been a classical subject of number theory for a long time, see for example \cite[Chapter XI]{HardyWright2008}. A classical theorem by Liouville states that irrationals that can be fast approximated by rationals in a certain way are transcendental, and it was shown by Greenfield and Wallach in 1972 \cite{GreenfieldWallach72} that these Liouville numbers play a role in the study of global hypoellipticity of differential operators on manifolds.
More recently, Himonas \cite{Himonas2001} and Bergamasco \cite{Bergamasco1999} introduced a subset of Liouville numbers, the exponential Liouville numbers, in the context of questions of global analytic hypoellipticity.
For the situation in our present paper, it turns out that we need to consider an even smaller subset of irrationals that have a fast approximation by rationals. We call them super-exponential Liouville numbers.
\begin{definition}
\label{def:Liouville} Let $a\in\R\setminus\Q$. Then $a$ is said to be\\
(i) a \emph{Liouville number} if for every $n\in\N_*$, there exist $p\in\Z$ and $q\in\N_*$ such that
\[
0<\left|a- \tfrac {p}{q} \right| < \tfrac {1}{q^{n}} \,,
\]
(ii) an \emph{exponential Liouville number} if there exists $c\in\R$, $c>0$, and infinitely many $p\in\Z$ and $q\in\N_*$ such that
\[
0<\left|a- \tfrac {p}{q} \right| < e^{-cq} \,,
\]
(iii) a \emph{super-exponential Liouville number} if for any $c\in\R$, $c>0$, there exist $p\in\Z$ and $q\in\N_*$ such that
\[
0<\left|a- \tfrac {p}{q} \right| < e^{-cq} \,.
\]
We denote the sets of Liouville, exponential Liouville and super-exponential
Liouville numbers by $\Lambda$, $\Lambda_{\mathrm e}$ and $\Lambda_{\mathsf{s}}$, respectively.
\end{definition}
It is clear that $\Lambda_{\mathsf{s}}\subset\Lambda_{\mathrm e}\subset\Lambda$. It is known that $\Lambda$ is dense in $\R$, uncountable and of measure zero \cite[Theorem 198]{HardyWright2008}. Using the same arguments, one can see that these properties are valid for $\Lambda_{\mathrm e}$ and $\Lambda_{\mathsf{s}}$, too. Finally, it is worth noting that each of these sets is invariant with respect to taking inverses, addition of rational numbers and multiplication by nonzero rational numbers.
\begin{proposition}
\label{pro:Liouville}
Let $\kappa=\pi/\omega$ be irrational.
Let the lateral boundary data $g^0$ and $g^\omega$ be given by series
\[
g^0({\mathsf{g}}o) = \sum_{\ell\in\N_*} g^0_\ell {\mathsf{g}}o^\ell,\quad
g^\omega({\mathsf{g}}o) = \sum_{\ell\in\N_*} g^\omega_\ell {\mathsf{g}}o^\ell
\]
that converge for $|{\mathsf{g}}o|<{\mathsf{g}}o_0$.
Then the following two statements are equivalent:\\
\emph{(i)} For any such $g^0$ and $g^\omega$, the series \eqref{eq:sumupartial} for the particular solution $u_\partial$ of the Dirichlet problem in the sector converges for $|\zeta|<{\mathsf{g}}o_0$.\\
\emph{(ii)} $\kappa$ is not an exponential Liouville number.\\
Likewise, the following two statements are equivalent:\\
\emph{(iii)} There exists ${\mathsf{g}}o_1>0$ such that for any such $g^0$ and $g^\omega$, the series \eqref{eq:sumupartial} for the particular solution $u_\partial$ of the Dirichlet problem in the sector converges for $|\zeta|<{\mathsf{g}}o_1$.\\
\emph{(iv)} $\kappa$ is not a super-exponential Liouville number.
\end{proposition}
For the proof, we use the following elementary observation about power series: Let the series
$\sum_{\ell\ge1} a_\ell\,x^\ell$ and $\sum_{\ell\ge1} b_\ell\,x^\ell$ have convergence radii ${\mathsf{g}}o_a$ and ${\mathsf{g}}o_b$, respectively. Then the series
$\sum_{\ell\ge1} a_\ell\,b_\ell\,x^\ell$ has convergence radius ${\mathsf{g}}o_a{\mathsf{g}}o_b$ or greater, with equality if, for example, $b_\ell={\mathsf{g}}o_b^{-\ell}$ for all $\ell$. Applying this with $a_\ell=1/\sin\ell\omega$, we see that the proof of the proposition is achieved if we prove the following lemma.
\begin{lemma}
\label{lem:Liouville}
Let $\pi/\omega$ be irrational and let ${\mathsf{g}}o_s$ be the convergence radius of the power series
$$
\sum_{\ell\in\N_*} \frac{x^\ell}{\sin\ell\omega}\,.
$$
Then ${\mathsf{g}}o_s=1$ if and only if $\pi/\omega$ is not an exponential Liouville number, and ${\mathsf{g}}o_s>0$ if and only if $\pi/\omega$ is not a super-exponential Liouville number.
\end{lemma}
\begin{proof}
We use Hadamard's characterization
\[
{\mathsf{g}}o_s^{-1} = \limsup_{\ell\to\infty}|\sin\ell\omega|^{-1/\ell},
\]
and we freely use that
\[
\limsup_{\ell\to\infty}(c\,\ell^d)^{1/\ell}=1 \;\mbox{ for any $c>0$, $d\in\R$. }
\]
Rational approximations of $\kappa=\pi/\omega$ appear because for all $k\in\N$: $|\sin\ell\omega|=|\sin(\ell\omega-k\pi)|$, and we can choose $k$ such that the difference is minimal:
$$
k=k(\ell)\equiv\round{\tfrac{\ell\omega}\pi}\in (\tfrac{\ell\omega}\pi-\tfrac12,\tfrac{\ell\omega}\pi+\tfrac12]
\quad\Longrightarrow\quad
\ell\omega- k\pi \in [-\tfrac\pi2,\tfrac\pi2)\,.
$$
Then, using $\tfrac2\pi\le\tfrac{\sin x}x\le1$ for $|x|\le\tfrac\pi2$, we get with the $k$ chosen as above,
$$
\tfrac2\pi|\ell\omega- k\pi| \le |\sin\ell\omega| \le |\ell\omega- k\pi|\,.
$$
Thus $|\sin\ell\omega|\simeq |\ell\omega- k\pi|
= k\omega |\tfrac\ell k - \tfrac\pi\omega|
\simeq k\, |\tfrac\ell k - \tfrac\pi\omega|$, implying
\[
\limsup_{\ell\to\infty}|\sin\ell\omega|^{-1/\ell}
= \limsup_{\ell\to\infty}|\tfrac\ell{k(\ell)} - \kappa|^{-1/\ell}\,.
\]
Therefore the condition ${\mathsf{g}}o_s=1$ is equivalent to (note that ${\mathsf{g}}o_s\le1$ in any case)
$$
\begin{aligned}
&\qquad \quad \forall M>1\; \exists \ell_M \;:\;
\ell\ge\ell_M\Rightarrow |\tfrac\ell{k(\ell)} - \kappa|^{-1/\ell}\le M\\
&\Longleftrightarrow\quad
\forall M>1\; \exists \ell_M \;:\;
\ell\ge\ell_M\Rightarrow |\tfrac\ell{k(\ell)} - \kappa|\ge M^{-\ell}\\
\quad&\Longleftrightarrow\quad
\forall c>0\; \mbox{ the inequality } |\tfrac\ell{k(\ell)} - \kappa|< e^{-c\ell}\\
&\qquad\qquad\qquad
\mbox{ has only finitely many solutions }\ell\in\N_*\\
\quad&\Longleftrightarrow\quad
\forall c>0\; \mbox{ the inequality } |\tfrac\ell k - \tfrac\pi\omega|< e^{-ck}\\
&\qquad\qquad\qquad
\mbox{ has only finitely many solutions }k,\ell\in\N_* \\
\end{aligned}
$$
The last condition means, according to Definition~{\mathrm e}f{def:Liouville}, that $\kappa$ is not an exponential Liouville number.
Likewise, ${\mathsf{g}}o_s>0$ is equivalent to
$$
\begin{aligned}
\limsup_{\ell\to\infty}|\tfrac\ell{k(\ell)} - \kappa|^{-1/\ell}<\infty
\quad&\Longleftrightarrow\quad
\sup_{\ell}|\tfrac\ell{k(\ell)} - \kappa|^{-1/\ell}<\infty\\
\quad&\Longleftrightarrow\quad
\exists c>0\;:\;
\forall \ell \;:\; |\tfrac\ell{k(\ell)} - \kappa|\ge e^{-c\ell}\\
\quad&\Longleftrightarrow\quad
\exists c>0\;:\; \forall k,\ell\in\N_* \;:\; |\tfrac\ell k - \kappa|\ge e^{-ck}\,.
\end{aligned}
$$
Again comparing the negation of the last condition with Definition~{\mathrm e}f{def:Liouville}, we see that this is equivalent to the fact that that $\kappa$ is not a super-exponential Liouville number.
\end{proof}
Let us finally note that if $\kappa$ is a super-exponential Liouville number, one can give explicit examples for the right hand side $f$ such that the series \eqref{eq:sumupartial} for $u_\partial$ {diverges for almost all} $t\ne0$. One such example is
$f(t)=1/({\mathsf{g}}o_0-t_1)$.
\end{document} |
\begin{document}
\title{Push-sum on random graphs}
\author{Pouya Rezaienia, Bahman Gharesifard,\\ Tam{\'a}s Linder, and Behrouz Touri\thanks{The first three authors are with the Department of Mathematics and Statistics at Queen's University, Kingston, ON, Canada. The last author is with the Department of Electrical and Computer Engineering at the University of California, San Diego.}}
\maketitle
\begin{abstract}
In this paper, we study the problem of achieving average consensus over a random time-varying sequence of directed graphs by extending the class of so-called push-sum algorithms to such random scenarios. Provided that an ergodicity notion, which we term the directed infinite flow property, holds and the auxiliary states of agents are uniformly bounded away from zero infinitely often, we prove the almost sure convergence of the evolutions of this class of algorithms to the average of initial states. Moreover, for a random sequence of graphs generated using a time-varying $B$-irreducible probability matrix, we establish convergence rates for the proposed push-sum algorithm.
\end{abstract}
\section{Introduction}\label{sec:intro}
Many distributed algorithms, executed with limited information over a network of agents, rely on estimating the average value of the initial state of the individual agents. These include the distributed optimization protocols~\cite{JNT-DPB-MA:86, MR-RN:04,LX-SB:06, AN-AO:09,PW-MDL:09,AN-AO-PAP:10,BJ-MR-MJ:09,JW-NE:10,JW-NE:11, BG-JC:14-tac,AN-AO:15-tac}, distributed regret minimization algorithms in machine learning~\cite{MA-BG-TL:15-tcsn}, and dynamics for fusion of information in sensor networks~\cite{KIT-SL-MGR:12}. There is a large body of work devoted to the average consensus problem, starting with the pioneering work~\cite{DK-AD-JG:03}, where the so-called \emph{push-sum algorithm} is first introduced. The key differentiating factor of the push-sum algorithm from consensus dynamics is that it takes advantage of a paralleled scalar-valued agreement dynamics, initiated uniformly across the agents, that tracks the imbalances of the network and adjusts for them when estimating the consensus value.
In addition to the earlier work~\cite{DK-AD-JG:03}, several recent papers have studied the problem of average consensus, see for example~\cite{ADDG-CNH:13}, where other classes of algorithms based on weight adaptation are considered, ensuring convergence to the average on fixed directed graphs. The study of convergence properties of push-sum algorithms on time-varying deterministic sequences of directed graphs, to best of our knowledge, was initiated in~\cite{FB-VB-PT-JT-MV:10} and extended in~\cite{AN-AO:15-tac}, where push-sum protocols are intricately utilized to prove the convergence of a class of distributed optimization protocols on a sequence of time-varying directed graphs. The key assumption in~\cite{AN-AO:15-tac} is the $ B $-connectedness of the sequence, which means that in any window of size $ B $ the union of the underlying directed graphs over time is strongly connected. As we demonstrate, a by product of our work in deterministic settings is the generalization of the sequences on which the convergence of the push-sum algorithms is valid to the ones which satisfy the infinite flow property; in this sense, this extension mimics the properties required for the convergence of consensus dynamics, along the lines of~\cite{BT:12-book}.
This paper is concerned with the problem of average consensus for scenarios where communication between nodes is time-varying and possibly random. The convergence properties of consensus dynamics on random sequences of directed graphs are by this time well-established, see for example~\cite{BT:12-book,touri2014endogenous,touri2014product}. Average consensus on random graphs has also been studied in~\cite{FB-VB-PT-JT-MV:10}, under the assumption that the corresponding random sequence of stochastic matrices is \emph{stationary} and ergodic with positive diagonals and irreducible expectation. One of our main objectives in this work is to extend these result to more general sequences of random stochastic matrices, in particular, beyond stationary. More importantly, to best of our knowledge, we establish for the first time convergence rates for the push-sum algorithms on random sequences of directed graphs.
The remainder of this paper is organized as follows. Section~\ref{sec:prelim} contains mathematical preliminaries. In Section~\ref{sec:statement}, we give a formal description of our consensus problem. In Section~\ref{sec:randompushsum}, we describe the push-sum algorithm. Section~\ref{sec:ergodicity} studies the ergodicity of row-stochastic matrices, and Section~\ref{sec:convergenceofpushsum} contains our main convergence results. In Section~\ref{sec:convergencerate}, we derive convergence rates for the push-sum algorithm for a class of random column-stochastic matrices. Finally, we gather our conclusions and ideas for future directions in Section~\ref{sec:conclusion}.
\section{Mathematical Preliminaries}\label{sec:prelim}
We start with introducing some notational conventions. Let $ {\mathbb{R}} $ and $ \mathbb{Z} $ denote the set of real and integer numbers, respectively, and let $ {\mathbb{R}}_{\mathbf{g}eq 0} $ and $ \mathbb{Z}_{\mathbf{g}eq 0} $ denote the set of non-negative real numbers and integers, respectively. For a set $\mathbb{A}$, we write $S\subset \mathbb{A}$ if $S$ is a proper subset of $\mathbb{A}$, and we call the empty set and $\mathbb{A}$ trivial subsets of $\mathbb{A}$. The complement of $S$ is denoted by $\bar{S}$. Let $ |S| $ denote the cardinality of a finite set $ S $. We view all vectors in ${\mathbb{R}}^n$ as column vectors, where $n\in \mathbb{Z}_{\mathbf{g}eq 0}$. We denote by $ \|\cdot\| $, $ \|\cdot\|_1 $ and $ \|\cdot\|_{\infty} $, the standard Euclidean norm, the $1$-norm, and the infinity norm on $ {\mathbb{R}}^n $, respectively. The $i$th unit vector in $ {\mathbb{R}}^n $, whose $ i $th component is $ 1 $ and all other components are $0$, is denoted by $ e_i $. We will also use the short-hand notation $ \mathbf{1}_n=(1,\ldots,1)^T $ and $ \mathbf{0}_n=(0,\ldots,0)^T \in {\mathbb{R}}^n $. A vector $v$ is stochastic if its elements are nonnegative real numbers that sum to $1$. We use $ {\mathbb{R}}_{\mathbf{g}eq 0}^{n\times n}$ to denote the set of $n\times n$ non-negative real-valued matrices. A matrix $A\in {\mathbb{R}}_{\mathbf{g}eq 0}^{n\times n}$ is row-stochastic (column-stochastic) if each of its rows (columns) sums to 1. For a given $A\in {\mathbb{R}}_{\mathbf{g}eq 0}^{n\times n}$ and any nontrivial $S\subset [n]$, we let ${A_{S\bar{S}} = \sum_{i\in S, j\in \bar{S}}A_{ij}}$. The notation $A'$ and $v'$ will refer to the transpose of the matrix $A$ and the vector $v$, respectively. A positive matrix is a real matrix all of whose elements are positive. Finally, $A_i$ denotes the $i$th row of matrix $A$ and $A^j$ denotes the $j$th column of $A$.
\subsection{Graph theory}
A (weighted) \emph{directed graph} $\mathcal{G}=(\mathcal{V},\mathcal{E},A) $ consists of a node set \sloppy ${\mathcal{V} = \{v_1,v_2,\ldots,v_n\}}$, an edge set $ \mathcal{E} \subseteq \mathcal{V}\times \mathcal{V}$, and a weighted \emph{adjacency matrix} $ {A \in {\mathbb{R}}^{n\times n}_{\mathbf{g}eq0}} $, with $ a_{ji}>0 $ if and only if $ (v_i,v_j)\in \mathcal{E} $, in which case we say that $v_i$ is connected to $v_j$. Similarly, given a matrix $ A \in {\mathbb{R}}^{n\times n}_{\mathbf{g}eq0} $, one can associate to $ A $ a directed graph $ \mathcal{G}=(\mathcal{V},\mathcal{E}) $, where $ (v_i,v_j)\in \mathcal{E} $ if and only if $ a_{ji}>0 $, and hence $ A $ is the corresponding adjacency matrix for $ \mathcal{G} $. The in-neighbors and the out-neighbors of $v_i$ are the set of nodes ${\Nin_i= \{j\in [n]: a_{ij}>0\}}$ and ${\Nout_i= \{j\in [n]: a_{ji}>0\}}$, respectively. The out-degree of $v_i$ is ${\dout_i=|\Nout_i|}$. A path is a sequence of nodes connected by edges. A directed graph is \emph{strongly connected} if there is a path between any pair of nodes. A directed graph is \emph{complete} if every pair of distinct vertices is connected by an edge. If the directed graph $\mathcal{G}=(\mathcal{V},\mathcal{E},A) $ is strongly connected, we say that $A$ is irreducible.
\subsection{Sequences of random stochastic matrices}
Let $\mathcal{S}^{+}_{n}$ be the set of $n\times n$ column-stochastic matrices that have positive diagonal entries, and let $\mathcal{F}_{\mathcal{S}^{+}_{n}}$ denote the Borel $\sigma$-algebra on $\mathcal{S}^{+}_{n}$. Given a probability space $(\Omega, \mathcal{B}, \mu)$, a measurable function \sloppy $ {\map{W}{(\Omega, \mathcal{B}, \mu)}{(\mathcal{S}^{+}_n, \mathcal{F}_{\mathcal{S}^{+}_{n}})}} $ is called a random column-stochastic matrix, and a sequence $ \{W(t)\} $ of such measurable functions on $(\Omega, \mathcal{B}, \mu)$ is called a random column-stochastic matrix sequence; throughout, we assume that $t\in \mathbb{Z}_{\mathbf{g}eq 0}$. Note that for any $ \omega \in \Omega $, one can associate a sequence of directed graphs $\{\mathcal{G}(t)(\omega)\}$ to $\{W(t)(\omega)\} $, where $(v_i,v_j)\in \mathcal{E}(t)(\omega)$ if and only if $W_{ji}(t)(\omega)>0$. This in turn defines a sequence of random directed graphs on $ \mathcal{V}=\{v_1,\ldots, v_n\} $, which we denote by $\{\mathcal{G}(t)\}$.
\section{Problem Statement}\label{sec:statement}
Consider a network of nodes $ \mathcal{V}=\{v_1,v_2,\ldots, v_n\} $, where node $ v_i \in \mathcal{V}$ has an initial state (or opinion) $x_i(0)\in {\mathbb{R}}$; the assumption that this initial state is a scalar is without loss of generality, and our treatment can easily be extended to the vector case. The objective of each node is to achieve \emph{average consensus}; that is to compute the average $ {\bar{x} =\frac{1}{n}\sum_{i=1}^{n}x_i(0)} $ with the constraint that only limited exchange of information between nodes is permitted. The communication layer between nodes at each time $t\mathbf{g}eq 0$ is specified by a sequence of random directed graphs $ \{\mathcal{G}(t)\} $, where $ \mathcal{G}(t) = (\mathcal{V}, \mathcal{E}(t), W(t)) $. Specifically, at each time $ t $, node $ v_i $ updates its value based on the values of its in-neighbors $ v_j\in \Nin_i (t)$, where ${\Nin_i(t)=\{v_j\in \mathcal{V}:W_{ij}(t)>0\}}$.
One standing assumption throughout this paper is that each node knows its out-degree at every time $ t $; this assumption is indeed necessary, as shown in~\cite{JH-JT:15}.
Our main objective is to show that the class of so-called push-sum algorithms can be used to achieve average consensus at every node, under the assumption that the communication network is random. This key point distinguishes our work from the existing results in the literature~\cite{DK-AD-JG:03},~\cite{AN-AO:15-tac},~\cite{ADDG-CNH:13}. Another key objective that we pursue in this paper is to obtain rates of convergence for such algorithms. We start our treatment with reviewing the push-sum algorithm.
\section{Random Push-Sum}\label{sec:randompushsum}
Consider a network of nodes $ \mathcal{V}=\{v_1,v_2,\ldots, v_n\} $, where node $ v_i \in \mathcal{V}$ has an initial state (or opinion) $x_i(0)\in {\mathbb{R}}$. The push-sum algorithm, proposed originally in~\cite{DK-AD-JG:03}, is defined as follows. Each node $ v_i $ maintains and updates, at each time ${t\mathbf{g}eq 0}$, two state variables $ x_i(t) $ and $ y_i(t) $. The first state variable is initialized to $ x_i(0) $ and the second one is initialized to $ y_i(0)=1 $, for all $i\in [n]$. At time $t\mathbf{g}eq0$, node $v_i$ sends $\frac{x_i(t)}{\dout_i(t)}$ and $\frac{y_i(t)}{\dout_i(t)}$ to its out-neighbors in the random directed graph $\mathcal{G}(t) = (\mathcal{V}, \mathcal{E}(t), W(t))$, which we assume to contain self-loops at each node for all $t\mathbf{g}eq 0$. At time $ (t+1) $, node $ v_i $ updates its state variables according to
\begin{align}\label{eqn:main-algo}
x_i(t+1) &=\sum_{j\in N_i^{in}(t)}\frac{x_j(t)}{\dout_j(t)},\cr
y_i(t+1) &= \sum_{j\in N_i^{in}(t)}\frac{y_j(t)}{\dout_j(t)}.
\end{align}
It is useful to define another auxiliary variable ${z_i(t+1) = \frac{x_i(t+1)}{y_i(t+1)}}$; as we will show later, $z_i(t+1) $ is the estimate by node $ v_i $ of the average $ \bar{x} $. One can rewrite this algorithm in a vector form; let the column-stochastic matrix $W(t)$ to be a function of $\mathcal{E}(t)$ with entries
\begin{equation}\label{eqn:W}
W_{ij}(t) =
\begin{cases} \frac{1}{\dout_j(t)} &\text{if } j\in \Nin_i(t),\\
0&\text{otherwise.}
\end{cases}
\end{equation}
Using these weighted adjacency matrices, for every $t\mathbf{g}eq 0$, we can rewrite the dynamics~\eqref{eqn:main-algo} as
\begin{align}\label{eqn:mtxalgorithm}
x(t+1) &= W(t)x(t),\cr
y(t+1) &= W(t)y(t),
\end{align}
where
\begin{align*}
x(t)&= (x_1(t), \ldots, x_n(t))', \cr
y(t)&= (y_1(t), \ldots, y_n(t))'.
\end{align*}
\section{Ergodicity}\label{sec:ergodicity}
In this section, we establish some important auxiliary results regarding the convergence of products of matrices which satisfy the so-called directed infinite flow property (c.f. Definition~\ref{def:inf-flow}). We study the products of a class of matrices in a deterministic setting, which we then use to study the push-sum algorithm in the next section. We start by some definitions.
\begin{definition}[Ergodicity~\cite{SC-ES:77},~\cite{BT:12-book}]
Let $ \{A(t)\} $ be a sequence of row-stochastic matrices, and for $ t \mathbf{g}eq s \mathbf{g}eq 0$, let $ A(t:s)$ denote the product
\begin{align}\label{eqn:matrixprod}
A(t:s) = A(t)A(t-1)\cdots A(s),
\end{align}
where $ A(s:s) =A(s) $. The sequence $ \{A(t)\} $ is said to be \textit{weakly ergodic}, if for all $i,j,l \in [n]$ and any $s \mathbf{g}eq 0$, $\lim_{t\rightarrow \infty} \left(A_{il}(t:s) - A_{jl}(t:s)\right) =0$. The sequence is said to be \textit{strongly ergodic} if $\lim_{t\rightarrow \infty} A(t:s) = \mathbf{1}_n v'(s)$ for any $ s\mathbf{g}eq 0 $, where $ v(s) \in \real^n $ is a stochastic vector.
\end{definition}
It can be shown that weak ergodicity and strong ergodicity are equivalent~\cite[Theorem~1]{SC-ES:77}. We will simply call such a sequence of row-stochastic matrices ergodic.
We first establish a sufficient condition for ergodicity of a sequence of row-stochastic matrices, Proposition~\ref{prop:shrink}, which we subsequently use in our convergence result for the push-sum algorithm. For this reason, we consider the following dynamical system:
\begin{align}\label{eqn:dynamicalsystem}
x(t+1) = A(t)x(t), \qquad \text{for all } t\mathbf{g}eq 0.
\end{align}
Let us start by two key definitions.
\begin{definition}[Strong Aperiodicity~\cite{BT:12-book}]\label{def:fdbk}
We say that a sequence of matrices $\{A(t)\}$ is \textit{strongly aperiodic} if there exists $\mathbf{g}amma>0$ such that ${A_{ii}(t)\mathbf{g}eq \mathbf{g}amma}$, for all $t\mathbf{g}eq0$ and $i\in [n]$.
\end{definition}
Motivated by the \textit{infinite flow property}~\cite[Definition 3.2.]{BT:12-book}, we provide the following definition.
\begin{definition}[Directed Infinite Flow Property]\label{def:inf-flow}
We say that a sequence of matrices $ \{A(t)\} $ has the \textit{directed infinite flow property} if for any non-trivial $ S\subset[n] $, $ {\sum_{t=0}^{\infty}A_{S\bar{S}}(t) = \infty} $.
\end{definition}
Consider now a sequence of matrices $\{A(t)\}$ that is strongly aperiodic and has the directed infinite flow property. Let ${k_0=0}$, and for any $q\mathbf{g}eq1$, define
\begin{align}\label{def:kq}
k_q = \argmin_{t'> k_{q-1}} \left(\min_{S\subset [n]}\sum_{t=k_{q-1}}^{t'-1}A_{S\bar{S}}(t)> 0\right).
\end{align}
Note that $k_q$ is the minimal time instance after $k_{q-1}$, such that there is nonzero information flow between any non-trivial subset of $\mathcal{V}$ and its complement; consequently, the directed graph associated with the product ${A(k_{q}-1)A(k_{q}-2)\cdots A(k_{q-1})}$ is strongly connected.
\begin{proposition}
If a sequence of matrices $\{A(t)\}$ has the directed infinite flow property, $k_q$ is finite for all $ q \mathbf{g}eq 0 $.
\end{proposition}
\begin{proof}
Suppose that $k_q$ is not finite for some ${q\mathbf{g}eq 0}$. Then, using~\eqref{def:kq}, there exists a non-trivial subset $S\subset [n]$ such that ${\sum_{t=k_{q-1}}^{\infty}A_{S\bar{S}}(t)= 0}$. This implies that $\sum_{t=0}^{\infty}A_{S\bar{S}}(t)< \infty$, which contradicts the assumption that $\{A(t)\}$ has the directed infinite flow property.
\end{proof}
To establish convergence results for the products of row-stochastic matrices satisfying Proposition~\ref{def:inf-flow}, we argue that in each time window where the underlying directed graph becomes strongly connected for $n$ times, i.e., after $k_{qn} - k_{(q-1)n}$ time steps for some $q$, \emph{significant mixing} will occur. To formalize this statement, let $\ell_0 =0$ and
\begin{align}\label{def:lq}
\ell_q = k_{qn}-k_{(q-1)n},
\end{align}
for $q\mathbf{g}eq 1$. For $t>s\mathbf{g}eq 0$, we also define
\[\mathbb{Q}_{t,s}= \{q: s\leq k_{(q-1)n},k_{qn}\leq t\}.\]
We are now ready to state our first result.
\begin{proposition}\label{prop:shrink}
Consider the dynamics~\eqref{eqn:dynamicalsystem}, where the sequence of row-stochastic matrices $\{A(t)\}$ is such that $A'(t)$ satisfies~\eqref{eqn:W}. Suppose, additionally, that $\{A(t)\}$
is strongly aperiodic and has the directed infinite flow property.
Then,
\begin{enumerate}[(i)]
\item
there is a vector $\phi(s) \in {\mathbb{R}}^{n}$ such that, for all $i,j \in [n]$ and $t\mathbf{g}eq s$,
\begin{align*}
\biggl|[A(t:s)]_{ij} - \phi_j(s)\biggl| \leq \Lambda_{t,s},
\end{align*}
where $\Lambda_{t,s}=\prod_{q\in \mathbb{Q}_{t,s}} \lambda_q$ and $\lambda_q = \left(1-\frac{1}{n^{\ell_q}}\right) \in (0,1)$;
\item
if, for the sequence $\{\ell_q\}$ associated with $\{A(t)\}$, we have
\begin{align}\label{eqn:suminfty}
\sum_{q=1}^{\infty}\frac{1}{n^{\ell_q}}= \infty,
\end{align}
then the sequence $\{A(t)\}$ is ergodic.
\end{enumerate}
\end{proposition}
\begin{proof}
We start by proving the first statement.
By definition of $k_q$, we know that for all $q\mathbf{g}eq 0$, ${A(k_{q+1}-1:k_q)}$ is irreducible. Since each $A(t)$ is strongly aperiodic, by Lemma~\ref{lemma:compgraph}, the matrix
\begin{multline*}
A(k_{n(q+1)} -1:k_{nq})\\
= A(k_{n(q+1)}-1:k_{n(q+1)-1})\times\cdots\times A(k_{nq+2}-1:k_{nq+1})\\
\times A(k_{nq+1}-1:k_{nq}),
\end{multline*}
which is the product of $n$ irreducible matrices, is positive for all $q\mathbf{g}eq 0$.
Hence, by Lemma~\ref{lemma:Asu}~(ii), for all $i,j\in [n]$, we have
\begin{align*}
[A(k_{n(q+1)} -1:k_{nq})]_{ij}\mathbf{g}eq \frac{1}{n^{k_{n(q+1)} -k_{nq}} }= \frac{1}{n^{l_{q+1}}}.
\end{align*}
Now, since $A(t:s) = A(t:s) I_n$ and for all $j\in [n]$, \sloppy ${\max_{i\in[n]}[I_n]_{ij}-\min_{i\in[n]}[I_n]_{ij}= 1 }$, using~\cite[Lemma~3]{JH-MSB:58}, we obtain
\begin{align}\label{eqn:m-m}
\max_{i\in [n]} [A(t:s)]_{ij} - \min_{i\in [n]} [A(t:s)]_{ij} \leq \Lambda_{t,s}.
\end{align}
Note that if we let $ \phi_j(s) = \min_{i\in [n]}A_{ij}(t:s) $ for all $ j \in [n] $, we have
\begin{align}\label{eqn:A-phi-ineq}
\biggl| [A(t:s)]_{ij} - \phi_j(s) \biggl| \leq \max_{i\in [n]} [A(t:s)]_{ij} - \min_{i\in [n]} [A(t:s)]_{ij}.
\end{align}
Using~\eqref{eqn:m-m} and \eqref{eqn:A-phi-ineq}, we conclude that
\begin{align*}
\biggl|[A(t:s)]_{ij} - \phi_j(s)\biggl| \leq \Lambda_{t,s},
\end{align*}
for all $ i,j\in [n] $.
We next prove part (ii); since $\lambda_q \in \left(0,1\right)$ for all $q\mathbf{g}eq 1$, we have that $\ln\left(\lambda_q\right) \leq\frac{-1}{n^{\ell_q}}$, where we have used the fact that $\ln(\mathbf{z}eta)\leq\mathbf{z}eta-1$ for all $\mathbf{z}eta>0$. This implies
\begin{align}
\sum_{q=1}^{\infty}\ln\left(\lambda_q\right)&\leq-\sum_{q=1}^{\infty}\frac{1}{n^{\ell_q}}.\label{eqn:aux-a}
\end{align}
On the other hand, we have
\begin{align*}
\lim_{t\rightarrow\infty}\Lambda_{t,0}=\lim_{t\rightarrow\infty}\prod_{q\in \mathbb{Q}_{t,0}}\lambda_q= \lim_{t\rightarrow\infty}\exp\left(\sum_{q\in \mathbb{Q}_{t,0}}\ln\left(\lambda_q\right) \right).
\end{align*}
The definition of the sets $\mathbb{Q}_{t,s}$ implies that we can write the right hand side as $\exp\left({\sum_{q=1}^{\infty}\ln\left(\lambda_q\right)}\right)$, which gives
\begin{align*}
\lim_{t\rightarrow\infty}\Lambda_{t,0} = \exp\left(\sum_{q=1}^{\infty}\ln\left(\lambda_q\right)\right) = 0,
\end{align*}
where the last equality follows from~\eqref{eqn:aux-a} and the assumption $\sum_{q=0}^{\infty}\frac{1}{n^{\ell_q}} = \infty$. Using the fact that $ \lim_{t\rightarrow\infty}\Lambda_{t,0} = 0$, we have that $ {\lim_{t\rightarrow\infty}\Lambda_{t,s} = 0}$, for any $s>0$. Hence, by Proposition~\ref{prop:shrink}, part (i), we conclude that $\{A(t)\}$ is weakly (and thus strongly) ergodic.
\end{proof}
Following similar steps as in Proposition~\ref{prop:shrink} we obtain the following result for sequences of column-stochastic matrices of the form~\eqref{eqn:W}.
\begin{proposition}\label{prop:sh2}
Consider the dynamics~\eqref{eqn:dynamicalsystem} and assume that sequence of matrices $\{A(t)\}$ is strongly aperiodic and has the directed infinite flow property, where the $A(t)$ are weighted adjacency matrices in the form of~\eqref{eqn:W}.
Then,
\begin{enumerate}[(i)]
\item
there is a vector $\phi(t) \in {\mathbb{R}}^{n}$ such that, for all $i,j \in [n]$ and $t\mathbf{g}eq s$,
\begin{align*}
\biggl|[A(t:s)]_{ij} - \phi_i(t)\biggl| \leq \Lambda_{t,s},
\end{align*}
where $\Lambda_{t,s} = \prod_{q\in \mathbb{Q}_{t,s}}\lambda_q$ and $\lambda_q = \left(1-\frac{1}{n^{\ell_q}}\right)$;
\item
for the sequence $\{\ell_q\}$ associated with $\{A(t)\}$, if
\begin{align*}
\sum_{q=1}^{\infty}\frac{1}{n^{\ell_q}}= \infty,
\end{align*}
then for all $j\in [n]$, $\lim_{t\rightarrow \infty}\biggl|[A(t:s)]_{ij} - \phi_i(t)\biggl| =0$.
\end{enumerate}
\end{proposition}
It is worth pointing out that in Proposition~\ref{prop:shrink}, since the $A(t)$ are row-stochastic, $x(t)$ approaches a vector with identical entries. However, in Proposition~\ref{prop:sh2} the $x(t)$ does not necessarily approach a fixed vector.
\section{Convergence of Push-Sum}\label{sec:convergenceofpushsum}
With all the pieces in place, we are now ready to study the behavior of the push-sum algorithm in a random setting.
\begin{theorem}\label{push}
Consider the push-sum algorithm~\eqref{eqn:mtxalgorithm} and suppose that the sequence of random column-stochastic matrices $\{W(t)\}$ has the directed infinite flow property, almost surely. Then, we have
\begin{align*}
\left| z_i(t+1)- \bar{x}\right|\leq \frac{2\|x(0)\|_1}{{y_{i}(t+1)}}\Lambda_{t,0},
\end{align*}
where $\Lambda_{t,0} = \prod_{q\in \mathbb{Q}_{t,0}}\lambda_q$ and $\lambda_q= \left(1-\frac{1}{n^{\ell_q}}\right)\in (0,1)$.
\end{theorem}
\begin{proof}
Define
\begin{align*}
D(t:s) \triangleq W(t:s) - \phi(t)\mathbf{1}_n',
\end{align*}
where $\phi(t)$ is a (random) vector from part (i) of Proposition~\ref{prop:sh2}. In addition, under the push-sum algorithm we have that
\begin{align*}
x(t+1)=W(t:0)x(0),\cr
y(t+1)=W(t:0)y(0),
\end{align*}
for all $t\mathbf{g}eq0$. Hence, for every $t\mathbf{g}eq 0$ and all $i\in [n]$, we have
\begin{align*}
z_i(t+1) - \bar{x}&= \frac{x_i(t+1)}{y_i(t+1)} - \frac{\mathbf{1}_n' x(0)}{n}\cr
&=\frac{[W(t:0)x(0)]_i}{[W(t:0)y(0)]_i} - \frac{\mathbf{1}_n' x(0)}{n}\cr
&=\frac{[D(t:0)x(0)]_i + \phi_i(t)\mathbf{1}_n' x(0)}{[D(t:0)y(0)]_i + \phi_i(t)\mathbf{1}_n' y(0)} - \frac{\mathbf{1}_n' x(0)}{n}.
\end{align*}
Using the fact that $y(0) = \mathbf{1}_n$ and by bringing the fractions to a common denominator, we have
\begin{align*}
z_i(t+1) - \bar{x}=&\frac{[D(t:0)x(0)]_i + \phi_i(t)\mathbf{1}_n' x(0)}{[D(t:0)\mathbf{1}_n]_i + n\phi_i(t)} - \frac{\mathbf{1}_n' x(0)}{n}\cr
=&\frac{n[D(t:0)x(0)]_i + n\phi_i(t)\mathbf{1}_n' x(0)}{n([D(t:0)\mathbf{1}_n]_i + n\phi_i(t))}\cr
&- \frac{[D(t:0)\mathbf{1}_n]_i\mathbf{1}_n' x(0)+n\phi_i(t)\mathbf{1}_n' x(0)}{n([D(t:0)\mathbf{1}_n]_i + n\phi_i(t))}\cr
=&\frac{n[D(t:0)x(0)]_i + [D(t:0)\mathbf{1}_n]_i\mathbf{1}_n' x(0)}{n([D(t:0)\mathbf{1}_n]_i + n\phi_i(t))}.
\end{align*}
Note that the denominator in the last equation is equal to $ny_i(t+1)$. Hence, for all $i\in[n]$ and $t\mathbf{g}eq 1$ we have
\begin{align*}
\left|z_i(t+1) - \bar{x}\right|\leq&\frac{\|x(0)\|_1}{{y_{i}(t+1)}}\left(\max_{j}|[D(t:0)]_{ij}|\right)\cr
&+\frac{|\mathbf{1}_n' x(0)|}{n{y_{i}(t+1)}}\left(\max_{j}|[D(t:0)]_{ij}|\right)n\cr
=&\frac{|\mathbf{1}_n' x(0)|+\|x(0)\|_1}{{y_{i}(t+1)}} \left(\max_{j}\left|[D(t:0)]_{ij}\right| \right),
\end{align*}
where the inequality follows from the triangle inequality. Since $ |\mathbf{1}_n'x(0)|\leq \|x(0)\|_1$, we have that
\begin{align*}
\left|z_i(t+1) - \bar{x}\right|\leq \frac{2\|x(0)\|_1}{{y_{i}(t+1)}}\left(\max_{j}\left|[D(t:0)]_{ij}\right|\right).
\end{align*}
Using the upper bound in part (i) of Proposition~\ref{prop:sh2}, we obtain
\begin{align}\label{eqn:T1}
\left|z_i(t+1) - \bar{x}\right|\leq \frac{2\|x(0)\|_1}{{y_{i}(t+1)}} \Lambda_{t,0}.
\end{align}
\end{proof}
\begin{proposition}\label{prop:pushconv}
Consider the push-sum algorithm~\eqref{eqn:mtxalgorithm} and suppose that the sequence of random column-stochastic matrices $\{W(t)\}$ has the directed infinite flow property, almost surely. Moreover, suppose that the sequence $\{\ell_q\}$ associated with $\{W(t)\}$ satisfies~\eqref{eqn:suminfty}, almost surely. If there exists $\delta>0$, such that for any $t\mathbf{g}eq0 $, there is $t'\mathbf{g}eq t$ such that $y_i(t')\mathbf{g}eq \delta$ for all $i\in [n]$, then
\begin{align*}
\lim_{t\rightarrow\infty}&\left|z_i(t+1) - \bar{x}\right| = 0,\quad\text{almost surely}.
\end{align*}
\end{proposition}
\begin{remark}
In the next section we exhibit a class of random matrix sequences $\{W(t)\}$ that satisfy the conditions of Proposition~\ref{prop:pushconv} and thus admit average consensus almost surely.
\end{remark}
\begin{proof}
Proof of this proposition is similar to the proof of Theorem 4.1 in~\cite{FB-VB-PT-JT-MV:10}, where the sequence $\{W(t)\}$ is assumed to be stationary; however, since we do not assume stationarity, we provide a proof. By Proposition~\ref{prop:sh2} part (ii), for any $\varepsilon>0 $ there is a time $t_{\varepsilon}$ such that for all $t\mathbf{g}eq t_{\varepsilon}$ and $i\in [n]$,
\begin{align*}
\sum_{j=1}^{n}|[W(t:0)]_{ij}-\frac{1}{n}\sum_{k=1}^{n}[W(t:0)]_{ik}|<{\delta\varepsilon}.
\end{align*}
By assumption, there exists $t_{\varepsilon}'\mathbf{g}eq t_{\varepsilon}$ such that $y(t_{\varepsilon}')\mathbf{g}eq \delta$, which implies that $ f(t_{\varepsilon}')<{\varepsilon}$, where $f(t)$ is defined as in Lemma~\ref{lemma:WG}. Since by Lemma~\ref{lemma:WG}, $f(t)$ is non-increasing, $f(t)<{\varepsilon}$ for all $t\mathbf{g}eq t_{\varepsilon}'$, meaning that $f(t)$ converges to zero as $t\rightarrow\infty$ and hence,
$ \lim_{t\rightarrow\infty}\left|z_i(t+1) - \bar{x}\right| = 0 $, almost surely.
\end{proof}
\section{B-Irreducible Sequences}\label{sec:convergencerate}
In this section we characterize a class of random column-stochastic matrices that admits average consensus and we provide a rate of convergence of the push-sum algorithm for this class. To achieve this, we restrict the class of random matrices that we consider; as we will point out later, this restricted class still includes many interesting sequences of random matrices.
In the following discussion, we assume that the push-sum dynamics is generated by a column-stochastic matrix sequence $\{W(t)\}$ where
\begin{align}\label{eqn:wt}
W_{ij}(t) = \frac{R_{ij}(t)}{\sum_{i=1}^{n}R_{ij}(t)},
\end{align}
for all $ i,j \in [n] $, where $ R_{ij}(t)$ is $ 1 $ with probability $ P_{ij}(t) $, and is $ 0 $ with probability $ 1-P_{ij}(t) $ such that ${\{R_{ij}(t):i,j\in [n],t\mathbf{g}eq0\}}$ are independent random variables. In other words, there is a random communication link between node $v_j$ and $v_i$ at time $t$ with probability $P_{ij}(t)$. Note that $\{W(t)\}$ is a sequence of independent random column-stochastic matrices.
Furthermore, for the probability matrix sequence $\{P(t)\}_{t\mathbf{g}eq 0}$, we assume that the following holds.
\begin{assumption}\label{assum:Pt}
$ \{P (t)\}_{t\mathbf{g}eq 0}$ is a sequence of $n\times n$ matrices with $ {P_{ij}(t) \in [0,1] }$. Additionally, we assume that $P_{ii}(t)=1$, for all $ v_i \in \mathcal{V}$. Also, for some constant $\epsilon>0$, we assume that $P_{ij}(t)\mathbf{g}eq \epsilon$ for all $i,j\in[n]$ and all $t\mathbf{g}eq 0$ such that $P_{ij}(t)\not=0$. Finally, we assume that the sequence $\{P(t)\}_{t\mathbf{g}eq 0}$ is $B$-irreducible, i.e.\ for some integer $B>0$, \[\sum_{t'=tB}^{(t+1)B-1}P(t)\]
is irreducible for all $t\mathbf{g}eq 0$.
\end{assumption}
We next state the main result of this section.
\begin{theorem}\label{theorem:main-rate}
Consider the push-sum algorithm~\eqref{eqn:mtxalgorithm} and let $\{W(t)\}$ be a sequence of random column-stochastic matrices defined by \eqref{eqn:wt}, where $\{P(t)\}$ satisfies Assumption~\ref{assum:Pt}. Let $p = \epsilon^{2(n-1)}$. Then, for any \sloppy ${t\mathbf{g}eq B+\frac{2nB}{p} }$, where $n\mathbf{g}eq 2$
\begin{align*}
\mathbb{E}\left[ \ln\left(\left|z_i(t+1) - \bar{x}\right|\right)\right]\leq c_0 -c_1t
\end{align*}
where
\begin{align*}
c_0 =& \ln\left( 2\|x(0)\|_1 \right) + \ln (n)\left(\frac{nB}{p}+B\right) +\ln(15),\cr
c_1 = &- \frac{p}{2nB}\ln \left(1-\frac{1}{n^{\frac{4nB}{p}}}\right).
\end{align*}
\end{theorem}
The proof relies on the following results.
\begin{lemma}\label{lm:PtDIFP}
Let $\{W(t)\}$ be a sequence of random column-stochastic matrices defined by \eqref{eqn:wt}, where $\{P(t)\}$ satisfies Assumption~\ref{assum:Pt}. Let $ \{k_q\} $ and $\{\ell_q\} $ be the sequences defined, respectively, in~\eqref{def:kq} and~\eqref{def:lq} along each sample path. Then
\begin{enumerate}[(i)]
\item the sequence $\{W(t)\}$ has the directed infinite flow property almost surely, and
\item for the sequence $\{\ell_q\}$, we have
\begin{align*}
\sum_{q=0}^{\infty}\frac{1}{n^{\ell_q}} = \infty,\quad\text{almost surely.}
\end{align*}
\end{enumerate}
\end{lemma}
\begin{proof}
We start by proving~(i). For any $ t\mathbf{g}eq0$, let us define the sequence of events
\begin{align}\label{eqn:At}
\mathcal{A}_t=\mathcal{B}igl\{\sum_{t'=tB}^{(t+1)B-1}W(t') \text{ is irreducible}\mathcal{B}igr\}.
\end{align}
Note that for all $t\mathbf{g}eq 0$, the events $\{\mathcal{A}_t\}_{t\mathbf{g}eq 0}$ are independent and that $\mathcal{A}_t$ implies $\sum_{t'=tB}^{(t+1)B-1} W_{S\bar{S}}(t')>0$, for any non-trivial $S\subset [n]$. Since $\min_{i,j\in [n]:P_{ij}(t)>0}P_{ij}(t) >\epsilon>0$, for all $t\mathbf{g}eq 0$, we have
\begin{align*}
\mathbf{P}r(\mathcal{A}_t)\mathbf{g}eq\epsilon^{2(n-1)}.
\end{align*}
This follows from~\cite[Corollary 5.3.6]{digraph} and the fact that $\{P(t)\}$ is \sloppy \mbox{$B$-irreducible} and hence, there is at least a subset of size $2(n-1)$ of the edges $(v_j,v_i)$ that form a strongly connected graph and $P_{ij}(t')\mathbf{g}eq \epsilon$ for some $t'\in [tB,(t+1)B-1]$.
Since the events $\mathcal{A}_t$ are independent, hence, by the second Borel-Contelli lemma~\cite[Theorem 2.3.6]{RD:10}, ${\sum_{t'=tB}^{(t+1)B-1}W_{S\bar{S}}(t') >0}$ infinitely often, almost surely. Moreover, since every positive entry of $W(t)$ is bounded below by~$\frac{1}{n}$, for any non-trivial ${S\subset [n]}$, $\sum_{t=0}^{\infty}W_{S\bar{S}}(t) = \infty$, almost surely, implying that $\{W(t)\}$ has the directed infinite flow property, almost surely. This also implies that $k_q$ and $\ell_q$ are finite for all $q$, almost surely. This completes the proof of~(i).
To prove~(ii), let us define, for all $t\mathbf{g}eq 0$,
\begin{align}\label{eqn:Ct}
\mathcal{C}_t=\bigcap_{t'=tn}^{(t+1)n-1}\mathcal{A}_{t'},
\end{align}
where $\mathcal{A}_t$ is defined in \eqref{eqn:At}. Since the $\mathcal{A}_t$ are independent, $\mathbf{P}r(\mathcal{C}_t) = \prod_{t'=tn}^{(t+1)n-1} \mathbf{P}r(\mathcal{A}_{t'})\mathbf{g}eq \epsilon^{2n(n-1)}$ for all ${t\mathbf{g}eq 0}$. This implies that ${\sum_{t=0}^{\infty}\mathbf{P}r(\mathcal{C}_t) =\infty}$. Again, since the $\mathcal{C}_t$ are independent, by the Borel-Contelli lemma, $\mathcal{C}_t$ occurs infinitely often, almost surely. This implies that ${\ell_q\leq nB}$ infinitely often, almost surely. Hence, $\sum_{q=1}^{\infty}\frac{1}{n^{\ell_q}} = \infty$, almost surely.
\end{proof}
\begin{lemma}
In the push-sum algorithm~\eqref{eqn:mtxalgorithm} let $\{W(t)\}$ be a sequence of random column-stochastic matrices corresponding to the sequence $\{P(t)\}$ satisfying Assumption~\ref{assum:Pt}. Then for all $t\mathbf{g}eq 0$ there exists $t'\mathbf{g}eq t$ such that for all $i\in [n]$, $y_i(t')\mathbf{g}eq \frac{1}{n^{nB}}$.
\end{lemma}
\begin{proof}
Consider the event $\mathcal{C}_t$ defined in \eqref{eqn:Ct}. At any time $\mathcal{C}_t$ occurs, by Lemma~\ref{lemma:compgraph}, the product $W(tnB+nB-1:tnB)$ is positive; moreover, by Lemma~\ref{lemma:Asu}, $W_{ij}(tnB+nB-1:tnB)\mathbf{g}eq \frac{1}{n^{nB}}$ for all $i,j\in [n]$. Since $W(t)$ is column-stochastic, we have ${W_{ij}(tnB+nB-1:0)\mathbf{g}eq \frac{1}{n^{nB}}}$. By Lemma~\ref{lm:PtDIFP}, $\mathcal{C}_t$ occurs infinitely often, almost surely; therefore, for all $t\mathbf{g}eq 0$ there exists $t'\mathbf{g}eq t$ such that for all $i\in [n]$, $y_i(t')\mathbf{g}eq \frac{1}{n^{nB}}$.
\end{proof}
The preceding two lemmas and Proposition~\ref{prop:pushconv} imply the following.
\begin{corollary}
Let $\{W(t)\}$ be a sequence of random column-stochastic matrices corresponding to the sequence $\{P(t)\}$ satisfying Assumption~\ref{assum:Pt}. Then $ \{W(t)\} $ admits average consensus, almost surely.
\end{corollary}
\begin{lemma}\label{Lemma:product-rate}
Let $\{W(t)\}$ be a sequence of random column-stochastic matrices corresponding to the sequence $\{P(t)\}$ satisfying Assumption~\ref{assum:Pt}. Let $ \{\ell_q\} $ be the sequence defined in~\eqref{def:lq} along each sample path. For all ${t\mathbf{g}eq B + \frac{2nB}{p}}$, we have
\begin{align*}
\mathbb{E}\left[\Lambda_{t,0}\right] \leq \exp\left(-\beta_t^2\left(\frac{t}{B}-2\right) \right) + 2\left(1-\frac{1}{n^{\frac{4nB}{p}}}\right)^{ \frac{pt}{2nB}},
\end{align*}
where $\Lambda_{t,0} = \prod_{q\in \mathbb{Q}_{t,0}}(1-\frac{1}{n^{l_q}})$, $\beta_t = \frac{p}{2}-\frac{2pB}{t}$, and ${p=\epsilon^{2(n-1)}}$.
\end{lemma}
\begin{proof}
Let $X_B(t)$ be the indicator of the event $\mathcal{A}_t$, i.e.,
\begin{align*}
X_B(t) =
\begin{cases}
1 &\text{if }\sum_{t'=tB}^{(t+1)B-1} W(t') \text{ is irreducible},\cr
0 &\text{otherwise.}
\end{cases}
\end{align*}
By the preceding argument, we have $\mathbf{P}r(X_B(t)=1)\mathbf{g}eq p= \epsilon^{2(n-1)}>0$. Note that the $X_B(t)$ are independent. We let $H_B(T) = \sum_{t=0}^{T}X_B(t)$ for all $T\mathbf{g}eq 0$, and define
\begin{align*}
q_t \triangleq \max\{q:k_q\leq t\}.
\end{align*}
By definition of $H_B(\cdot)$ and $q_t$, we have that
\begin{align}\label{eqn:qgeqH}
q_t\mathbf{g}eq H_B\left(\left\lfloor\frac{t}{B}\right\rfloor-1\right).
\end{align}
Now, we have that
\begin{align*}
\mathbb{E} \left[\Lambda_{t,0}\right]
=& \mathbb{E}\left[ \Lambda_{t,0}\ \biggl| \ q_t \leq \frac{pt}{2B} \right] \mathbf{P}r\left(q_t \leq \frac{pt}{2B} \right) \\
&+ \mathbb{E}\left[\Lambda_{t,0}\ \biggl|\ q_t > \frac{pt}{2B} \right] \mathbf{P}r\left(q_t > \frac{pt}{2B}\right).
\end{align*}
Since all terms on the right-hand side are less than or equal to $1$, we have
\begin{align*}
\mathbb{E}\left[\Lambda_{t,0}\right] \leq \mathbf{P}r\left(q_t \leq \frac{pt}{2B} \right) + \mathbb{E}\left[\Lambda_{t,0}\ \biggl|\ q_t > \frac{pt}{2B} \right].
\end{align*}
Using~\eqref{eqn:qgeqH}, we have
\begin{align*}
\mathbb{E}\left[\Lambda_{t,0}\right] \leq& \mathbf{P}r\left(H_B\left(\left\lfloor\frac{t}{B}\right\rfloor-1\right) \leq \frac{pt}{2B} \right) \\
&+ \mathbb{E}\left[\Lambda_{t,0}\ \biggl| \ q_t > \frac{pt}{2B} \right].
\end{align*}
Let us consider the second term on the right-hand side. When $ q_t > \frac{pt}{2B} $, we have $\left| \mathbb{Q}_{t,0}\right|\mathbf{g}eq \left\lfloor\frac{pt}{2nB}\right\rfloor$. Using Lemma~\ref{lemma:opt1} to maximize the second term on the right-hand side over the choices of $\ell_q$, we obtain
\begin{align}\label{eqn:partB}
\mathbb{E}\left[\Lambda_{t,0}\ \biggl|\ q_t > \frac{pt}{2B} \right] \leq& \left(1-\frac{1}{n^{\frac{t}{\left\lfloor \frac{pt}{2nB}\right\rfloor}}}\right)^{\left\lfloor \frac{pt}{2nB}\right\rfloor}\cr \leq& 2\left(1-\frac{1}{n^{\frac{t}{\left\lfloor \frac{pt}{2nB}\right\rfloor}}}\right)^{ \frac{pt}{2nB}}.
\end{align}
To further simplify the above inequality, we show that $\frac{t}{\left\lfloor \frac{pt}{2nB}\right\rfloor}\leq \frac{4nB}{p}$. To show this, we note that for all $t\mathbf{g}eq\frac{2nB}{p}+B$, we have $\frac{pt}{2nB}>1$ and hence, ${\left\lfloor \frac{pt}{2nB}\right\rfloor}\mathbf{g}eq 1$. Now, assume that $\mathbf{x}i={\left\lfloor \frac{pt}{2nB}\right\rfloor}\mathbf{g}eq 1$. We have $2nB\mathbf{x}i\leq pt\leq 2nB(\mathbf{x}i+1)$. Therefore,
\begin{align*}
\frac{t}{\left\lfloor \frac{pt}{2nB}\right\rfloor}\leq \frac{2nB}{p}\left(\frac{\mathbf{x}i+1}{\mathbf{x}i}\right)\leq \frac{4nB}{p},
\end{align*}
where the last inequality follows from the fact that $\mathbf{x}i\mathbf{g}eq 1$.
Using this inequality in \eqref{eqn:partB}, we get
\begin{align}\label{eqn:rhsfirst}
\mathbb{E}\left[\Lambda_{t,0}\ \biggl|\ q_t > \frac{pt}{2B} \right] \leq&2 \left(1-\frac{1}{n^{\frac{t}{\left\lfloor \frac{pt}{2nB}\right\rfloor}}}\right)^{ \frac{pt}{2nB}}\cr
\leq& 2\left(1-\frac{1}{n^{\frac{4nB}{p}}}\right)^{ \frac{pt}{2nB}}.
\end{align}
On the other hand, since $\mathbb{E}[X_B(t)]\mathbf{g}eq p$ for all $t\mathbf{g}eq B$, we have
\begin{align*}
&\mathbf{P}r\left(H\left(\left\lfloor\frac{t}{B}\right\rfloor-1\right) \leq \frac{pt}{2B} \right) \cr
&= \mathbf{P}r\left(\sum_{t'=0}^{\lfloor{t/B}\rfloor-1}X_B(t') - p \left(\left\lfloor\frac{t}{B}\right\rfloor-1\right) \leq - \alpha_t \left(\left\lfloor\frac{t}{B}\right\rfloor-1\right)\right)\cr
&\leq \mathbf{P}r\left(\sum_{t'=0}^{\lfloor{t/B}\rfloor-1}\left(X_B(t') - \mathbb{E}[X_B(t')] \right)\leq -\alpha_t \left(\left\lfloor\frac{t}{B}\right\rfloor-1\right)\right),
\end{align*}
where
\begin{align}\label{eqn:alpha}
\alpha_t = \frac{p\left(\left\lfloor\frac{t}{B}\right\rfloor-1\right)-\frac{pt}{2B}}{\left\lfloor\frac{t}{B}\right\rfloor-1}.
\end{align}
When $t\mathbf{g}eq B+\frac{2nB}{p}$, $\alpha_t>0$ and hence, by Lemma~\ref{lemma:Hoeffding}, we obtain
\begin{align}\label{eqn:rhssecond}
\mathbf{P}r\left(H\left(\left\lfloor\frac{t}{B}\right\rfloor-1\right) \leq \frac{pt}{2B} \right) \leq& \exp\left(-\alpha_t^2\left(\left\lfloor\frac{t}{B}\right\rfloor-1\right)\right)\cr
\leq& \exp\left(-\alpha_t^2\left(\frac{t}{B}-2\right)\right).
\end{align}
From~\eqref{eqn:alpha}, we have
\begin{align*}
\alpha_t > \frac{p\left(\frac{t}{B}-2\right)-\frac{pt}{2B}}{\frac{t}{B}}=\frac{p}{2}-\frac{2pB}{t}.
\end{align*}
If we let $\beta_t =\frac{p}{2}-\frac{2pB}{t}$, using~\eqref{eqn:rhsfirst} and \eqref{eqn:rhssecond}, we conclude that
\begin{align*}
\mathbb{E}\left[\Lambda_{t,0}\right] \leq \exp\left(-\beta_t^2\left(\frac{t}{B}-2\right) \right) + 2\left(1-\frac{1}{n^{\frac{4nB}{p}}}\right)^{ \frac{pt}{2nB}},
\end{align*}
finishing the proof.
\end{proof}
\begin{lemma}\label{Lemma:El1}
In the push-sum algorithm~\eqref{eqn:mtxalgorithm} let $\{W(t)\}$ be a sequence of random column-stochastic matrices corresponding to the sequence $\{P(t)\}$ satisfying Assumption~\ref{assum:Pt}. We have, for all $i\in [n]$ and $t\mathbf{g}eq 0$,
\begin{align*}
\mathbb{E}\left[\ln\left(\frac{1}{y_{i}(t)}\right)\right]\leq \ln(n)\left(B\frac{n}{p}+B\right).
\end{align*}
\end{lemma}
\begin{proof}
By Lemma~\ref{lemma:Asu}, for all $t< \frac{Bn}{p}+B$ and $i\in [n]$,
\begin{align*}
[W(t:0)]_{ii}\mathbf{g}eq \frac{1}{n^{B\frac{n}{p}+B}} ,
\end{align*}
almost surely. This implies that
\begin{align*}
\mathbb{E}\left[\ln\left(\frac{1}{y_i(t)}\right)\right]\leq \ln(n)\left(B\frac{n}{p}+B\right),
\end{align*}
for all $t< \frac{Bn}{p}+B$ and $i\in [n]$.
If $t\mathbf{g}eq \frac{Bn}{p}+B$, let $t = aB+b$, where $a, b\in \mathbb{Z}_{\mathbf{g}eq 0}$ and $b<B$. Define
\begin{equation*}
\tau_t = \begin{cases}
\min\{T:\sum_{t=a-T}^{a-1} X_B(t)=n\}, & \text{if }\sum_{t=0}^{a-1}X_B(t)\mathbf{g}eq n\\
a& \text{otherwise}.
\end{cases}
\end{equation*}
When $\tau_t=a$, ${W_{ij}(t:0)\mathbf{g}eq \frac{1}{n^{\tau_t B+B}}}$, for all $i,j\in [n]$. When $\tau_t\neq a$, by Lemma~\ref{lemma:compgraph}, ${W(aB-1:(a-\tau_t) B)}$ is a positive matrix and consequently by Lemma~\ref{lemma:Asu}, ${W_{ij}(t:(a-\tau_t) B)\mathbf{g}eq \frac{1}{n^{\tau_t B+B}}}$ for all ${i,j\in [n]}$; in addition, since the $W(t)$ are column-stochastic, we have ${W_{ij}(t:0)\mathbf{g}eq \frac{1}{n^{\tau_t B+B}}}$. Therefore, for all $t\mathbf{g}eq 0$ we have
\begin{align*}
\ln\left(\frac{1}{W_{ij}(t:0)}\right)\leq \ln(n)(\tau_t B+B)\quad\text{for all } i,j\in[n].
\end{align*}
Consider a sequence of independent Bernoulli trials $Y_t$, where in each trial the probability of success is $p$. The number of trials until $n$ successes occur is a negative binomial random variable $Z$ having parameters $n$ and $p$. Since ${\mathbf{P}r(\tau_t \leq i)\mathbf{g}eq\mathbf{P}r(Z\leq i)}$ for all $i\mathbf{g}eq n $, we have $\mathbb{E}[\tau_t]\leq \mathbb{E}[Z]$. Since ${\mathbb{E}[Z] = \frac{n}{p}}$, we obtain
$ \mathbb{E}[\tau_t]\leq\frac{n}{p} $,
and hence the result follows.
\end{proof}
We are now in a position to prove Theorem~\ref{theorem:main-rate}.
\begin{proof}[Proof of Theorem~\ref{theorem:main-rate}]
In \eqref{eqn:T1}, since both sides are positive, we have
\begin{align*}
\ln\left(\left|z_i(t+1) - \bar{x}\right|\right)\leq& \ln\left(\frac{2\|x(0)\|_1}{{y_{i}(t+1)}} \Lambda_{t,0}\right)\cr
=& \ln\left( 2\|x(0)\|_1 \right) + \ln\left(\frac{1}{{y_{i}(t+1)}} \right) \cr
&+ \ln\left(\Lambda_{t,0}\right).
\end{align*}
By taking expectations and using Lemma~\ref{Lemma:El1}, we obtain
\begin{align}\label{eqn:Eineq}
\mathbb{E}\left[ \ln\left(\left|z_i(t+1) - \bar{x}\right|\right)\right]\leq& \ln\left( 2\|x(0)\|_1 \right) + \ln (n)\left(\frac{nB}{p}+B\right)\cr
&+ \mathbb{E} \left[\ln\left(\Lambda_{t,0}\right)\right]\cr
\leq& \ln\left( 2\|x(0)\|_1 \right) + \ln (n)\left(\frac{nB}{p}+B\right)\cr
&+ \ln\left(\mathbb{E} \left[\Lambda_{t,0}\right]\right),
\end{align}
where the last inequality follows from Jensen's inequality. Now by Lemma~\ref{Lemma:product-rate}, we have
\begin{align*}
\mathbb{E} \left[\Lambda_{t,0}\right]\leq& \exp\left(-\beta_t^2\left(\frac{t}{B}-2\right)\right) +2\left(1-\frac{1}{n^{\frac{4nB}{p}}}\right)^{\frac{pt}{2nB}},
\end{align*}
where $\beta_t = \frac{p}{2}-\frac{2pB}{t}$. Let us consider the first term on the right hand side; since $\beta_t\leq \frac{1}{2}$ we have
\begin{align*}
\exp\left(-\beta_t^2\left(\frac{t}{B}-2\right)\right)\leq& \exp\left(-\beta_t^2\frac{t}{B}+\frac{1}{2}\right) \cr
= & \exp\left(-\frac{p^2t}{4B}+2p^2+\frac{1}{2}-\frac{4p^2B}{t}\right) \cr
\leq & \exp\left(-\frac{p^2t}{4B}+\frac{5}{2}\right) \cr
\leq & 13\exp\left(-\frac{p^2t}{4B}\right) \cr
=& 13\left(\exp\left(-\frac{pn}{2}\right)\right)^{\frac{pt}{2nB}}.
\end{align*}
Since $n\mathbf{g}eq 2$, $\exp\left(-\frac{pn}{2}\right) \leq \exp\left(-p\right)$. On the other hand, ${\left(1-\frac{1}{n^{\frac{4nB}{p}}}\right) \mathbf{g}eq \left(1-\frac{1}{2^{\frac{8}{p}}}\right) }$ for all $n\mathbf{g}eq 2 $ and $B\mathbf{g}eq 1$. It can be seen the for $p\in [0,1]$, $\exp\left(-p\right) \leq \left(1-\frac{1}{2^{\frac{8}{p}}}\right) $, and consequently $ {\exp\left(-\frac{pn}{2}\right) \leq \left(1-\frac{1}{n^{\frac{4nB}{p}}}\right)}$. Hence
\begin{align}\label{eqn:Elambda}
\mathbb{E} \left[\Lambda_{t,0}\right]\leq 15 \left(1-\frac{1}{n^{\frac{4nB}{p}}}\right)^{\frac{pt}{2nB}}.
\end{align}
The result now follows using \eqref{eqn:Eineq} and \eqref{eqn:Elambda}.
\end{proof}
\section{Conclusion}\label{sec:conclusion}
We have studied the convergence properties of the push-sum algorithm for average consensus on sequences of random directed graphs. We have proved that this dynamics is convergent almost surely when some mild connectivity assumptions are met and the auxiliary states of agents are uniformly bounded away from zero infinitely often. We have shown that the latter assumption holds for sequences of random matrices constructed using a sequence of time-varying $ B $-irreducible probability matrices. We have also obtained convergence rates for the proposed push-sum algorithm. Future work include studying the implications in scenarios with link-failure and in distributed optimization on random time-varying graphs.
\section{Appendix}
\newcounter{mycounter}
\renewcommand{A.\arabic{mycounter}}{A.\arabic{mycounter}}
\newtheorem{propositionappendix}[mycounter]{Proposition}
\newtheorem{lemmaappendix}[mycounter]{Lemma}
\newtheorem{remarkappendix}[mycounter]{Remark}
\begin{lemmaappendix}\label{lemma:compgraph}
For $n\mathbf{g}eq 2$, let $\{A(i)\}_{i=1}^{n-1}$ be a sequence of weighted adjacency matrices associated with the strongly connected directed graphs $\{\mathcal{G}(i)\}_{i=1}^{n-1}$ on the node set ${\mathcal{V} = \{v_1,v_2,\ldots,v_n\}}$, where ${\mathcal{G}(i) = (\mathcal{V},\mathcal{E}(i),A(i))}$ and $A(i)\in S_n^{+}$ for all $i\in [n-1]$. Then the matrix product $A(n-1:1)$ is positive.
\end{lemmaappendix}
\begin{proof}
Let $\mathcal{G}(k:1)= (\mathcal{V},\mathcal{E}(k:1))$ indicate the directed graph associated with the product $A(k:1)$, where $k\in [n-1]$. Let $\Nout_i(k:1)$ and $\dout_i(k:1)$ indicate the set of out-neighbors and out-degree of node $i\in [n]$ in directed graph $\mathcal{G}(k:1)$, respectively. Consider an arbitrary but fixed node $i\in [n]$. Since $A(1)\in \mathcal{S}^{+}_n$ and $\mathcal{G}(1)$ is strongly connected, we have
\begin{align}\label{eqn:d1}
\dout_i(1)\mathbf{g}eq 2.
\end{align}
Now consider the directed graph $\mathcal{G}(k:1)$ and assume that ${\dout_i(k:1)\leq n-1}$ for some $k\in[n-1]$; we show that ${\dout_i(k+1:1)>\dout_i(k:1)}$. By Lemma~\ref{lemma:Asu}(ii), we have ${\Nout_i(k:1)\subseteq \Nout_i(k+1:1)}$. Moreover, since $\mathcal{G}(k+1)$ is strongly connected and $\dout_i(k:1)\leq n-1$, there is ${l\notin \Nout_i(k:1)}$ such that $l\in \Nout_j(k+1)$ for some $j\in \Nout_i(k:1)$; otherwise, there is no path between $i$ and $l$ in $\mathcal{G}(k+1)$, contradicting the strong connectivity of $\mathcal{G}(k+1)$. Hence, by Lemma~\ref{lemma:Asu} (iii) $l\in \Nout_i(k+1:1)$, implying that \begin{align*}\label{eqn:dk+1}
\dout_i(k+1:1)>\dout_i(k:1).
\end{align*}
This along with \eqref{eqn:d1} imply that
\begin{align*}
\dout_i(k:1)\mathbf{g}eq k+1,,
\end{align*}
for all $k\in [n-1]$, which implies that $\dout_i(n-1:1) = n$. Since this statement holds for any $i\in [n]$, the matrix product $A(n-1:1)$ is positive.
\end{proof}
\begin{lemmaappendix}[Lemma 1 \cite{AN-AO:09}]\label{lemma:Asu}
Consider a sequence of directed graphs $\{\mathcal{G}(t)\}$, which we assume to contain all the self-loops, with a corresponding sequence of weighted adjacency matrices $\{A(t)\}$. In addition, assume that $A_{ij}(t)\mathbf{g}eq \mathbf{g}amma$ whenever $A_{ij}(t)>0$, for some $\mathbf{g}amma>0$.
Then the following statements hold:
\begin{enumerate}[(i)]
\item $[A(t:s)]_{ii}\mathbf{g}eq \mathbf{g}amma^{t-s+1}$, for all $i\in[n]$ and $t\mathbf{g}eq s\mathbf{g}eq 0$;
\item if $[A(r)]_{ij}>0$ for some $t\mathbf{g}eq r\mathbf{g}eq s\mathbf{g}eq 0$ and $i,j\in[n]$, then ${[A(t:s)]_{ij}\mathbf{g}eq \mathbf{g}amma^{t-s+1}}$;
\item if $[A(s)]_{ik}>0$ and $[A(r)]_{kj}>0$ for some ${t\mathbf{g}eq r>s\mathbf{g}eq 0}$, then $[A(t:s)]_{ij}\mathbf{g}eq \mathbf{g}amma^{t-s+1}$.
\end{enumerate}
\end{lemmaappendix}
\begin{lemmaappendix}[Lemma 4.3 \cite{FB-VB-PT-JT-MV:10}]\label{lemma:WG}
Consider the push-sum algorithm \eqref{eqn:mtxalgorithm}. Define
\begin{align*}
f(t) = \max_{i\in [n]}\frac{\sum_{j=1}^{n}|[W(t:0)]_{ij}-\frac{1}{n}\sum_{k=1}^{n}[W(t:0)]_{ik}|}{y_i(t)}.
\end{align*}
Then, $f(t)$ is non-increasing and
\begin{align*}
\|z(t)-\bar{x}\mathbf{1}_n\|_{\infty}\leq \|x(0)\|_{\infty}f(t).
\end{align*}
\end{lemmaappendix}
\begin{lemmaappendix}[Hoeffding's inequality~\cite{WH:63}]\label{lemma:Hoeffding}
If $X_1,X_2,\ldots,X_n $ are independent random variables and $0\leq X_i\leq 1$, for all $i\in [n]$, then for any $\alpha>0$, we have
\begin{align*}
\mathbf{P}r\left(\sum_{i=1}^{n}\left(X_i - \mathbb{E}[X_i]\right)\leq -\alpha n\right)\leq \exp\left(-2\alpha^2n\right).
\end{align*}
\end{lemmaappendix}
\begin{lemmaappendix}\label{lemma:opt1}
For $n>1$ and for all $l_1, l_2, \ldots, l_q\in \mathbb{Z}_{\mathbf{g}eq 0}$, ${q>0}$, we have
\begin{align*}
\prod_{i=1}^{q}\left(1-\frac{1}{n^{l_i}}\right)\leq\left(1-\frac{1}{n^{\frac{t}{q}}}\right)^{q},
\end{align*}
where $ t= l_1+l_2+\cdots +l_q$.
\end{lemmaappendix}
\begin{proof}
It suffices to show that
\begin{align*}
\frac{1}{q}\sum_{i=1}^{q}\ln\left(1-\frac{1}{n^{l_q}}\right)\leq\ln\left(1-\frac{1}{n^{\frac{t}{q}}}\right),
\end{align*}
which simply follows from Jensen's inequality, since the function ${g(\mathbf{z}eta) = \ln\left(1-\frac{1}{n^{\mathbf{z}eta}}\right)}$ is concave.
\end{proof}
\end{document} |
\begin{document}
\title{Levels of distribution for sieve problems in prehomogeneous vector spaces}
\begin{abstract}
In our companion paper \cite{TT_orbital}, we developed an efficient algebraic method for computing the Fourier transforms
of certain functions defined on prehomogeneous vector spaces over finite fields, and we carried out these computations in a variety of cases.
Here we develop a method, based on Fourier analysis and algebraic geometry, which exploits these Fourier transform formulas
to yield {\itshape level of distribution} results, in the sense of analytic number theory. Such results are of the shape typically
required for a variety of sieve methods. As an example of such an
application we prove that there are $\gg \frac{X}{\log X}$ quartic fields whose discriminant is squarefree, bounded above by $X$,
and has at most eight prime factors.
\end{abstract}
In this paper we will develop a general technique sufficient to prove the following:
\begin{theorem}\langlebel{thm:ap3}
There is an absolute constant $C_3 > 0$ such that for each $X > 0$,
there exist $\geq (C_3 + o_X(1)) \frac{X}{\log X}$ cubic fields $K$
whose discriminant is squarefree, bounded above by $X$, and has at most
$3$ prime factors.
\end{theorem}
\begin{theorem}\langlebel{thm:ap4}
There is an absolute constant $C_4 > 0$ such that for each $X > 0$,
there exist $\geq (C_4 + o_X(1)) \frac{X}{\log X}$ quartic fields $K$
whose discriminant is squarefree, bounded above by $X$, and has at most
$8$ prime factors.
\end{theorem}
Theorem \ref{thm:ap3} improves on a result of Belabas and Fouvry \cite{BF} (which in turn improved upon Belabas \cite{bel_sieve}),
where they obtained the same result with $7$ in place of our $3$, and
our methods are in large part a further development
of their ideas.
In \cite{BF} they remarked that introducing a {\itshape weighted sieve} would lower this $7$ to $4$; the further improvement to $3$ comes from an
improvement in the corresponding {\itshape level of distribution}.
The application to quartic fields is, to our knowledge, new.
Besides the weighted sieve, the main ingredients of our method are unusually strong estimates
for the relevant
exponential sums (which we obtained in \cite{TT_orbital}), and a suitably adapted version of the recently established Ekedahl-Bhargava geometric sieve \cite{B_geosieve}.
The method is designed to yield strong {\itshape level of distribution estimates} as inputs to {\itshape sieve methods for prehomogeneous vector spaces}. We consider the following setup:
\begin{enumerate}
\item
A sieve begins with a set of objects $\mathcal{A}$ to be sieved. Here this is defined by a representation $V$
of an algebraic group $G$ which we assume to be {\itshape prehomogeneous}:
the action of $G(\mathbb{C})$ on $V(\mathbb{C})$ has a Zariski-open orbit, defined by the nonvanishing
of a polynomial which we call the {\itshape discriminant}.
We assume here that $\Disc(gx) = \Disc(x)$ identically
for all $g \in G(\mathbb{Z})$, and take as our set $\mathcal{A} = \mathcal{A}(X)$ the set of $G(\mathbb{Z})$-orbits $x \in V(\mathbb{Z})$ with
$0 < |\Disc(x)| < X$.
Prehomogeneous vector spaces are the subject of a wealth of parametrization theorems (see, e.g., \cite{WY, HCL1, HCL2, HCL3, HCL4})
and corresponding theoerems concerning the arithmetic objects being parametrized. We refer also to \cite{BH} for a large
number of interesting {\itshape coregular} (not prehomogneous) examples, for which the methods of this paper (and perhaps also
\cite{TT_orbital}) are likely to apply.
\item\langlebel{foo2}
For each prime $p$ we define a notion of an object $x \in \mathcal{A}$ being `bad at $p$'; a typical application of sieve methods is to
estimate or bound the number of $x \in \mathcal{A}$ which are not bad at any prime $p < Y$,
for some parameter $Y$. For Theorems \ref{thm:ap3} and \ref{thm:ap4} we will take
`bad at $p$' to mean `has discriminant divisible by $p$'. Other defintions of `bad' are also important,
for example the Davenport-Heilbronn nonmaximality condition of \cite{DH, BBP, BST, TT_rc}.
We may work with any definition of `bad' meeting the following technical condition: there is an integer $a \geq 1$ and
a $G(\mathbb{Z}/p^a\mathbb{Z})$-invariant
subset $S_p \subseteq V(\mathbb{Z}/p^a \mathbb{Z})$, such that $x \in V(\mathbb{Z})$ is bad if and only if its reduction $\pmod {p^a}$ is in $S_p$.
\item\langlebel{foo}
For each squarefree integer $q$, we require estimates for the number of $x \in \mathcal{A}(X)$ bad at each prime dividing $q$.
These may be obtained via the geometry of numbers (see, among other references, \cite{DH, bel_sieve, BF, B_quintic, BST, EPW}) or using Shintani zeta functions \cite{TT_rc},
and we develop a third (simpler) method here.
Here these estimates will be of the form $C \omega(q) X^{r/d} + E(X, q)$ for a fixed constant $C$ and multiplicative function $\omega$,
and an error term $E(X, q)$ which we want to bound.
\item
In cases where $\mathcal{A}$ depends on a parameter $X$,
a {\itshape level of distribution} is any $\alpha > 0$ for which the sum $\sum_{q \leq X^{\alpha}} |E(X, q)|$ is adequately small.
Typically, and here, a cumulative error $\ll_A X (\log X)^{-A}$ (for each $A > 0$) is more than sufficient -- but see \cite{BST, TT_rc, ST5}
for examples where a power savings is relevant.
\end{enumerate}
We refer to books such as \cite{CM} and \cite{ODC} for examples of sieve methods and applications; many of the methods can be used essentially
as black boxes, for which the level of distribution is the most important input. {\itshape It is the goal of this paper to develop a method
for proving levels of distribution for prehomogeneous vector spaces which are quantitatively as strong as possible.}
Typically, an important ingredient is finite Fourier analysis. Let $\mathbb{P}si_p : V(\mathbb{Z}/p^a \mathbb{Z}) \rightarrow \mathbb{C}$
be the characteristic function of the subset $S_p$ described above, and let $\widehat{\mathbb{P}si_p}$ be its Fourier transform
(defined by \eqref{eqn:ft_intro} below). We expect upper bounds on $E(X, p)$ to follow from statements to the effect that
that the function $\mathbb{P}si_p$ is equidistributed, and upper bounds on the $L_1$-norm of $\widehat{\mathbb{P}si_p}$ constitute a strong quantitatve
statement of this equidistribution.
The basic heuristic of this paper is that {\itshape $L_1$ norm bounds for Fourier transforms over finite fields should lead to level of distribution statements for arithmetic
objects}. For examples carried out via the geometry of numbers we refer to \cite{BF}, \cite[(80)-(83)]{BST}, and \cite[Proposition 9.2]{FK};
for examples using the Shintani zeta function method we refer to \cite{TT_rc} (espectially Section 3) or the forthcoming work \cite{PT}.
In the present paper
we develop a simpler version of this idea, not requiring any knowledge of the geometry of the `cusps' or of the analytic
behavior of the zeta function, in the context of a {\itshape lower bound sieve}. For any Schwartz function $\phi$, the Poisson summation formula
takes the form
\begin{align}\langlebel{eq:poisson_error_0}
\sum_{x \in V(\mathbb{Z})} \mathbb{P}si_q(x) \phi(xX^{-1/d}) & =
X^{\dim(V)/d} \sum_{w \in V^*(\mathbb{Z})} \widehat{\mathbb{P}si_q}(w) \widehat{\phi}\left(\frac{w X^{1/d}}{q} \right),
\end{align}
and for suitable $\phi$ the left side will be a smoothed undercount of the number of
$G(\mathbb{Z})$-orbits $x \in V(\mathbb{Z})$ with $0 < |\Disc(x)| < X$, satisfying the `local conditions' described by $\mathbb{P}si_q$.
The left side of \eqref{eq:poisson_error_0} takes the role of (the counting function of) $\mathcal{A}(X)$, and the function
$\mathbb{P}si_q$ may be used to detect those $x$ which are `bad at $q$'.
On the right side of \eqref{eq:poisson_error}, the $w = 0$ term the expected main term, and
the rapid decay of $\widehat{\phi}$ will imply that the error term is effectively bounded by a sum of $|\widehat{\mathbb{P}si_q}(w)|$
over a box of side length $\ll q X^{-1/d}$.
Therefore we are reduced to bounding sums of
$|\widehat{\mathbb{P}si_q}(w)|$ over boxes and over ranges of squarefree $q$. Obviously we require bounds on the individual
$|\widehat{\mathbb{P}si_q}(w)|$ to proceed. At this point we could incorporate the general bounds of
Fouvry and Katz \cite{FK}; indeed, our method would quite directly exploit the geometric structure of their results.
However, in the cases of interest much stronger bounds hold for the $|\widehat{\mathbb{P}si_q}(w)|$ than are proved in \cite{FK}.
We proved this in \cite{TT_orbital} as a special case of exact formulas for $\widehat{\mathbb{P}si_q}(w)$, for any
of five prehomogeneous vector spaces $V$, any squarefree integer $q$, and any $G(\mathbb{Z}/q\mathbb{Z})$-invariant function
$\mathbb{P}si_q$.
Our $L_1$-norm heuristic arises by replacing each $|\widehat{\mathbb{P}si_q}(w)|$ by its average over $V^*(\mathbb{Z}/q\mathbb{Z})$.
In practice there are limits to the equidistribution of $|\widehat{\mathbb{P}si_q}(w)|$, and
algebraic geometry now takes center stage. For $q = p \neq 2$ prime, the largest values of
$|\widehat{\mathbb{P}si_p}(w)|$ will be confined to $w \in \mathfrak{X}(\mathbb{F}_p)$ for a {\itshape scheme} $\mathfrak{X}$, defined over $\mathbb{Z}$, and of high
codimension. (The same is also true of the Fouvry-Katz bounds.)
We now apply the {\itshape Ekedahl-Bhargava} geometric sieve \cite{B_geosieve}, which essentially bounds the number of
pairs $(w, p)$ with $w$ in our box and $p$ prime. As we must work with general squarefree integers $q$, and a filtration of schemes
$\mathfrak{X}_i$ rather than just one fixed $\mathfrak{X}$, we develop a variation of the geometric
sieve adapted to this counting problem.
{\itshape Organization of this paper.}
For simplicity we structure our paper around the proof of Theorem \ref{thm:ap4}, even though our more important aim is to present a method
which works in much greater generality. We begin in Section \ref{sec:properties} by introducing the prehomogeneous vector
space $(G, V) = (\GL_2 \times \GL_3, 2\co_{T2}mes \Sym_2(3))$ and describing
its relevant properties. We take some care to delineate which of its
properties are relevant to the proof, so that the reader can see what is required to adapt our method to other prehomogeneous
vector spaces and to other sieve problems.
In Section \ref{sec:explain_ld} we introduce the {\itshape weighted sieve} of Richert \cite{richert} and Greaves \cite{greaves}, used to conclude Theorems \ref{thm:ap3} and \ref{thm:ap4},
and we also precisely formulate the level of distribution statement which it will require.
In Section \ref{sec:smoothing} we introduce our smoothing method. Our main result is Proposition \ref{prop:gen_ld_simp}, which states that a level
of distribution follows from an essentially combinatorial estimate. The proof is a fairly typical application of Poisson summation, and follows along
lines that should be familiar to experts. The proof is carried out in a general setting, and we state all of our assumptions at the beginning of the section.
Section \ref{sec:ap3} proves Theorem \ref{thm:ap3}, and may be skipped without loss of continuity. Here the geometry is simple enough that
we may conclude without introducing any algebro-geometric machinery, and the argument may be read as a prototype for the generalities which follow.
In Section \ref{subsec:2sym23-subschemes} we introduce some algebraic geometry to describe the $G(\mathbb{F}_p)$-orbits on $V(\mathbb{F}_p)$
in terms of schemes defined over $\mathbb{Z}$.
This sets up an application of the Ekedahl-Bhargava geometric sieve \cite{B_geosieve}, of which we develop
a variation in Section \ref{sec:geom}.
Section \ref{sec:ld} is the heart of the proof, further developing the geometric sieve to prove
our level of distribution statement. The proof is specalized
to the particular $(G, V)$ and $\mathbb{P}hi_q$ being studied, but the generalization to other cases should be immediate.
Finally, in Section \ref{sec:ld_implies} we prove Theorem \ref{thm:ap4}. Essentially the
proof is a formal consequence of our level of distribution, although there are a few technicalities pertaining to this particular
$(G, V)$ and its arithmetic interpretation.
{\itshape Remark.} The quartic fields produced by Theorem \ref{thm:ap4} will all have Galois group $S_4$, and the theorem still holds
if we specify that $\Disc(K)$ should be positive or negative, or indeed that $K$ has a fixed number of real embeddings.
There are other methods for producing almost-prime quartic field discriminants,
One, suggested to us by Bhargava, is to specialize $11$ of the $12$ variables in $V$ to particular values and then
apply results on polynomials in one variable; a second is to apply results \cite{CT4} on quartic fields with fixed cubic resolvent.
In either case we obtain quartic field discriminants with fewer than $8$ prime factors. but it is unclear how to get, say,
$\gg X^{9/10}$ of them, let alone $\gg X (\log X)^{-1}$. Moreover, proving that one obtains {\itshape fundamental} discriminants seems to be
nontrivial with the first method, and the second method intrinsically produces non-fundamental discriminants.
{\itshape Notation.} We observe the following conventions in this paper. $x$ will denote a general element of $V(\mathbb{Z})$ (sometimes only up
to $G(\mathbb{Z})$-equivalence). $r$ will denote the dimension of $V$ and $d$ will denote the degree of its (homogeneous) `discriminant' polynomial.
$X$ will indicate a discriminant bound, and in Section \ref{sec:explain_ld} we write $X$ and $Y$ in place of the $x$ and $X$ of \cite{ODC}.
$p$ will denote a prime and $q$ a squarefree integer, in contrast to \cite{TT_orbital} where $q$ was used for the cardinality of a finite field.
\section{The `quartic' representation and its essential properties}\langlebel{sec:properties}
For each ring $R$ (commutative, with unit), let $V(R) = R^2\co_{T2}mes\Sym_2(R^3)$ be
the set of pairs of ternary quadratic forms with coefficients in $R$; when $2$ is not a zero divisor in $R$,
we also regard $V(R)$ as the set of pairs of $3 \times 3$ symmetric matrices. Let
$G(R) := \GL_2(R) \times \GL_3(R)$;
there is an action of $G$ on $V$, defined by
\[
(g_2, g_3) \cdot (A, B) = (r \cdot g_3 A g_3^T + s \cdot g_3 B g_3^T, t \cdot g_3 A g_3^T +
u \cdot g_3 B g_3^T),
\]
where $g_2 := \begin{pmatrix} r & s \\ t & u \end{pmatrix}$.
The {\itshape discriminant} is defined (see \cite[p. 1340]{HCL3}) by the equation
\begin{equation}\langlebel{def:disc}
\Disc((A, B)) := \Disc(4 \det(Ax + By)),
\end{equation}
where $4 \det(Ax + By)$ is a binary cubic form in the variables $x$ and $y$, and the second `$\Disc$' above is its
discriminant.
It was proved by
Bhargava \cite{HCL3} that the $G(\mathbb{Z})$-orbits on $V(\mathbb{Z})$ parametrize quartic rings, in a sense
that we recall precisely in Section \ref{sec:ld_implies}.
(This parametrization, together with a geometry of numbers argument, allowed Bhargava to prove
\cite{B_quartic} an asymptotic formula for the number of quartic fields
of bounded discriminant.)
We note the following additional facts about this representation. Although we will not attempt to axiomatize our method here,
these are the inputs required to establish a lower bound sieve for $G(\mathbb{Z})$-orbits on $V(\mathbb{Z})$. (Arithmetic applications, such as passing from quartic rings to
quartic {\itshape fields} as we do
in Section \ref{sec:ld_implies}, may in some cases require extra steps which will not generalize as readily.)
\begin{enumerate}
\item ({\itshape Homogeneity of the discriminant; definitions of $r$ and $d$.})
By \eqref{def:disc}, $\Disc$ is a homogeneous polynomial of degree $12$ on $V$.
We write $d = 12$ for the degree of this polynomial, and $r := \dim(V) = 12$; these quantities coincide in this and other interesting cases, but not always.
\item ({\itshape Approximation of the fundamental domain.})
Let $\mathcal{F}$ be a fundamental domain for the action of $G(\mathbb{Z})$ on $V(\mathbb{R})$, and let
$\mathcal{F}^1$ be the subset of $x \in \mathcal{F}$ with $0 < |\Disc(x)| < 1$.
We approximate $\mathcal{F}^1$
by choosing a smooth (Schwartz class) function $\phi: V(\mathbb{R}) \rightarrow [0, 1]$ compactly supported within $\mathcal{F}^1$.
Since the discriminant is homogeneous of degree $d$,
the weighting function $\phi(x X^{-1/d})$ is a smoothed undercount of those $x \in G(\mathbb{Z}) \backslash V(\mathbb{Z})$
with $0 < |\Disc(x)| < X$. That is,
the role of the (counting function of the) set $\mathcal{A}(X)$ described in the introduction is taken
by the expression
\begin{equation}\langlebel{eqn:smoothed_AX}
\sum_{\substack{x \in V(\mathbb{Z})}} \phi(x X^{-1/d}).
\end{equation}
Although it won't be necessary here, it is possible to approximate $G(\mathbb{Z}) \backslash V(\mathbb{R})$ as closely as we wish in the sense that,
for any $\beta < 1$, we may additionally require the locus $\mathcal{F}_\beta$ of $x$ with $\phi(x) = 1$ to satisfy
\begin{equation}
\frac{ \Vol(\mathcal{F}_\beta) }{ \Vol(\mathcal{F}^1)} > \beta.
\end{equation}
A number of variations are possible; for example we could restrict $\mathcal{F}^1$ to those
$x \in \mathcal{F}$ with a particular sign, or choose $\phi$
to be supported away from any algebraic subset of $V(\mathbb{R})$ defined by the vanishing of one or more homogeneous
equations.
\item ({\itshape Fourier transform formulas.}) For a squarefree integer $q$ we let $\mathbb{P}si_q$ be the characteristic function of those
$x \in V(\mathbb{Z})$ with $q \mid \Disc(x)$; this function factors through the reduction map
$V(\mathbb{Z}) \rightarrow V(\mathbb{Z}/q\mathbb{Z})$. Its Fourier transform
$\widehat{\mathbb{P}si_q}\mathcal{O}lon V^\ast(\mathbb{Z}/q\mathbb{Z}) \rightarrow \mathbb{C}$
is defined by the usual formula
\begin{equation}\langlebel{eqn:ft_intro}
\widehat{\mathbb{P}si_q}(x) = q^{-r} \sum_{x' \in V(\mathbb{Z}/q \mathbb{Z})} \mathbb{P}si_q(x') \exp\left(\frac{2 \pi i [x', x]}{q}\right).
\end{equation}
The Fourier transform $\widehat{\mathbb{P}si_q}$ is easily seen
to be multiplicative in $q$, and for $q = p \neq 2$ we
proved the following explicit formula
in \cite{TT_orbital}:
\begin{equation}\langlebel{eq:Psi_fourier_ternary}
\widehat{\mathbb{P}si_p}(x)=
\begin{cases}
p^{-1} + 2p^{-2} - p^{-3} - 2 p^{-4} - p^{-5} + 2p^{-6} + p^{-7} - p^{-8}
& x \in \mathcal{O}_0,\\
p^{-3} - p^{-4} - 2p^{-5} + 2p^{-6} + p^{-7} - p^{-8}
& x \in \co_{D1^2},\\
2p^{-4} - 5p^{-5} + 3p^{-6} + p^{-7} - p^{-8}
& x \in \co_{D11},\\
p^{-4} - 3p^{-5} + 2p^{-6} + p^{-7} - p^{-8}
& x \in \co_{C\rm{s}},\\
- p^{-5} + p^{-6} + p^{-7} - p^{-8}
& x \in \co_{D2}, \co_{D\rm{ns}}, \co_{C\rm{ns}}, \co_{T11}, \co_{T2},\\
-p^{-6} + 2p^{-7} - p^{-8}
& x \in \mathcal{O}_{1^2 1^2}, \\
p^{-6} - p^{-8}
& x \in \mathcal{O}_{2^2}, \\
p^{-7} - p^{-8}
& x \in \mathcal{O}_{1^4}, \mathcal{O}_{1^3 1}, \mathcal{O}_{1^2 11}, \mathcal{O}_{1^2 2}, \\
-p^{-8}
& x \in \mathcal{O}_{1111}, \mathcal{O}_{112}, \mathcal{O}_{22}, \mathcal{O}_{13}, \mathcal{O}_{4}.
\end{cases}
\end{equation}
When $p \neq 2$ there are $20$ orbits for the action of
$G(\mathbb{F}_p)$ on $V^\ast(\mathbb{F}_p)$, and each has a description that is essentially uniform in $p$; these are denoted by
the $\mathcal{O}$ above, or by $\mathcal{O}(p)$ when we indicate the prime $p$ explicitly.
We refer to \cite{TT_orbital} for descriptions of each of the $\mathcal{O}$,
together with computations of their cardinalities.
The $L_1$ norm of $\widehat{\mathbb{P}si_p}(x)$ is $O(p^4)$ -- better than square root cancellation! In particular the larger contributions come from
the more singular orbits, and {\itshape our methods are designed to exploit this structure.}
What is required in general is that $\mathbb{P}si_p$ be any bounded function,
which factors through the reduction map $V(\mathbb{Z}) \rightarrow V(\mathbb{Z}/p^a\mathbb{Z})$
for some $a \geq 1$, for which we can compute or bound the Fourier transform.
(Incorporating the trivial bound $|\widehat{\mathbb{P}si_p}(x)| \ll 1$ yields results which are in some sense nontrivial, but our interest is in doing better.)
In \cite{TT_orbital}, explicit formulas like \eqref{eq:Psi_fourier_ternary} are computed for any function $\mathbb{P}si_p$ for which $\mathbb{P}si_p(gx) = \mathbb{P}si_p(x)$ for all
$g \in G(\mathbb{F}_p)$ when $a=1$.
\item ({\itshape Orbits in geometric terms.})
For each orbit description $\mathcal{O}$,
there exists an integer $i = i(\mathcal{O}) \in [0, d]$ such that $\# \mathcal{O}(p) \asymp_{\mathcal{O}} p^i$ as $p$ ranges; we call this integer the
dimension of $\mathcal{O}$.
We will show in Section \ref{subsec:2sym23-subschemes}
that there also exists a closed {\itshape subscheme}
$\mathfrak{X} \subseteq V$ defined over $\mathbb{Z}$
of the same dimension $i(\mathcal{O})$,
such that $\mathcal{O}(p) \subseteq \mathfrak{X}(\mathbb{F}_p)$ for all but (possibly) finitely many primes
$p$. As we will see, this will allow lattice point counting methods which use algebraic geometry.
\begin{remark} The `schemes' in question are simply
the vanishing loci of systems of polynomials defined over $\mathbb{Z}$, and the algebraic geometry to be invoked will be fairly elementary.
However, one can study related problems using very sophisticated algebro-geometric tools; see for example \cite{DG, FK}.
\end{remark}
\end{enumerate}
\section{Levels of distribution and the weighted sieve}\langlebel{sec:explain_ld}
We begin by discussing this sieve machinery we will apply.
In some (but not complete) generality, a {\itshape level of distribution} describes the following. Suppose that $a(n) : \mathbb{Z}^+ \rightarrow [0, \infty)$
is a function for which we can prove, for each squarefree integer $q$ (including $q = 1$), an estimate of the shape
\begin{equation}\langlebel{eq:sieve2}
\sum_{\substack{n < X \\ q \mid n}} a(n) = \omega(q) C Y + E(X, q)
\end{equation}
for some constant $C$, multiplicative function $\omega(q)$ satisfying $0 \leq \omega(q) < 1$ for all $q$, function $Y$ of $X$,
and
error term $E(X, q)$.
With the setup described in Section \ref{sec:properties} we will have $Y = X^{r/d}$ with
\begin{equation}\langlebel{eq:construct_weights}
a(n) = \sum_{\substack{x \in V(\mathbb{Z}) \\ |\Disc(x)| = n}} \phi(x X^{-1/d}).
\end{equation}
We say that the function $a(n)$ has {\itshape level of distribution}
$\alpha$ if for any $\epsilon > 0$ we have
\begin{equation}\langlebel{eq:sieve3}
\sum_{q < X^{\alpha}} |E(X, q)| \ll_{\epsilon} Y^{1 - \epsilon},
\end{equation}
where the sum is over squarefree integers $q$ only.
Bounds of the shape \eqref{eq:sieve3} are required for essentially all sieve methods, and also in many other
analytic number theory techniques.
For our formulation of the {\itshape weighted sieve} we will
also demand a {\itshape one-sided linear sieve inequality}
\begin{equation}\langlebel{eq:ls}
\prod_{w \leq p < z} (1 - \omega(p))^{-1} \leq
K \left( \frac{ \log z}{ \log w} \right)
\end{equation}
for all $2 \leq w < z$ and some fixed constant $K \geq 1$; the product is over primes. A familiar computation (see, for example,
\cite[(5.34)-(5.37)]{ODC}) shows that \eqref{eq:ls} holds if we assume for all prime $p$ that $w(p) < 1$ and that
\begin{equation}\langlebel{eq:ls2}
\bigg|w(p) - \frac{1}{p}\bigg| < \frac{C}{p^2}
\end{equation}
for a fixed constant $C$, on which the constant
$K$ of \eqref{eq:ls} depends. (Conditions such as \eqref{eq:ls} and \eqref{eq:ls2} are often required in sieve methods,
and
may appear in a variety of guises.)
The {\itshape weighted sieve}, developed principally by Richert \cite{richert} and Greaves \cite{greaves}, and described here in the
formulation of Friedlander and Iwaniec \cite[Theorem 25.1]{ODC}, detects almost prime values of $n$ in the sequence $a(n)$.
\begin{theorem}[The weighted sieve \cite{richert, greaves}]\langlebel{thm:ls}
Assume \eqref{eq:sieve2}, \eqref{eq:sieve3}, and \eqref{eq:ls}, and let $t \geq \frac{1}{\alpha} + \frac{\log 4}{\log 3} - 1$ be a positive integer.
Then we have
\begin{equation}\langlebel{eq:ls_result}
\sum_{\substack{n \leq X \\ p \mid n \mathbb{R}ightarrow p > X^{\alpha/4} \\ \nu(n) \leq t}} a(n) \gg \frac{Y}{\log X},
\end{equation}
where $\nu(n)$ denotes the number of prime divisors of $n$.
\end{theorem}
This is one of many sieve methods which establish various consequences from
hypotheses of the form \eqref{eq:sieve2}-\eqref{eq:ls}. We refer to
\cite{ODC} for a nice overview of many different sieve methods and their applications, and
to \cite{BBP, BF, B_quartic, B_quintic, B_geosieve,
BCT, BST, FK, ST5, TT_rc} for applications concerning
prehomogeneous vector spaces. The papers \cite{BBP, B_quartic, B_quintic, BST, TT_rc} sieve rings for maximality, where the
analogue of $\omega(q)$ is roughly $1/q^2$; conversely, \cite{BCT, ST5} illustrate sieves where
$\omega(q)$ is not a decreasing function of $q$.
{\itshape Technical notes.}
Theorem \ref{thm:ls} may be deduced from the precise statement of Theorem 25.1 of \cite{ODC} as follows.
We take $N = 1$ in (25.7), corresponding to \eqref{eq:sieve2}. We choose $u = 1$
so that $\delta(u) = \frac{\log 4}{\log 3}$ (this is the limit as $u \rightarrow 1$ of the expression in (25.17)), and we
have $V(X) \gg_{\alpha} \log X$ by \eqref{eq:ls}.
As mentioned in \cite{ODC}, Greaves proved \cite[Chapter 5]{greaves} a related result with $\frac{\log 4}{\log 3} = 1.261\dots$ replaced with
$1.124\dots$ Since we will eventually obtain $\alpha = \frac{7}{48}$ for the representation $\mathbb{Z}^2\co_{T2}mes \Sym_2(\mathbb{Z}^3)$, this would yield
quartic field discriminants with only $7$ prime factors in Theorem \ref{thm:ap4}. But since our main goal is to showcase our sieve method,
we have chosen to apply a form of the weighted sieve
that is easier to extract from the literature. Note that the lower bound $p > X^{\alpha / 4}$ will be important in Section \ref{sec:ld_implies}.
\section{Smoothing and the Poisson summation formula}\langlebel{sec:smoothing}
{\itshape Assumptions.} Until Section \ref{sec:reform}, the analysis in this section is quite general (and very standard). $V(\mathbb{Z})$ will
denote a complete lattice in a vector space $V(\mathbb{R})$ of dimension $r$. ($V$ itself will denote an $r$-dimensional affine space over $\mathbb{Z}$.)
In what follows $d$ will be the (homogeneous) degree of the discriminant polynomial, but in this section (where such a polynomial need not be defined)
$d$ may be any positive real constant.
$X$ will be
a positive real number; and for each squarefree
integer $q$, $\mathbb{P}si_q : V(\mathbb{Z}) \rightarrow \mathbb{C}$ is any function which factors through the reduction map $V(\mathbb{Z}) \rightarrow V(\mathbb{Z}/q\mathbb{Z})$.
All sums over $q$ will implicitly be over {\itshape squarefree} positive integers $q$ only.
We assume for simplicity that $|\mathbb{P}si_q(x)| \leq 1$ for all $q$ and $x$. Finally, $\phi$ will denote any fixed smooth, Schwartz class function,
on which all implied constants below are allowed to depend.
The aim of this section is to estimate the values of the sum
\begin{equation}\langlebel{eq:intro_sum}
\sum_{x \in V(\mathbb{Z})} \mathbb{P}si_\ldmod(x) \phi(xX^{-1/d}),
\end{equation}
and in particular to prove upper bounds for the error terms, summed over $q$.
In the general setting of Section \ref{sec:properties} this is a smoothed undercount of
those $x \in G(\mathbb{Z}) \backslash V(\mathbb{Z})$ with $0 < \pm \Disc(x) < X$ satisfying the
congruence conditions implied by the function $\mathbb{P}si_q$.
In the more specific setting
of the proof of Theorem \ref{thm:ap4}, $\mathbb{P}si_q$ is the characteristic function of those
$x$ with $q \mid \Disc(x)$, so that \eqref{eq:intro_sum} counts discriminants divisible by $q$.
By Poisson summation and a standard unfolding argument, we may check that
\begin{align}\langlebel{eq:poisson_error}
\sum_{x \in V(\mathbb{Z})} \mathbb{P}si_\ldmod(x) \phi(xX^{-1/d}) & = X^{r/d} \sum_{x \in V^*(\mathbb{Z})} \widehat{\mathbb{P}si_\ldmod}(x) \widehat{\phi}\left(\frac{x X^{1/d}}{\ldmod} \right)\\
& = \widehat{\mathbb{P}si_q}(0) \widehat{\phi}(0) X^{r/d} + E(X, \mathbb{P}si_\ldmod, \phi),
\end{align}
where the error term $E(X, \mathbb{P}si_\ldmod, \phi)$ is defined by
\begin{equation}\langlebel{eq:def_error}
E(X, \mathbb{P}si_\ldmod, \phi) := X^{r/d} \sum_{0 \neq x \in V^\ast(\mathbb{Z})} \widehat{\mathbb{P}si_\ldmod}(x) \widehat{\phi}\left(\frac{x X^{1/d}}{\ldmod} \right),
\end{equation}
and $\widehat{\phi}$ satisfies the rapid decay property
\begin{equation}\langlebel{eq:rapid_decay}
|\widehat{\phi}(y)| \ll_{A} (1 + |y|)^{-A}
\end{equation}
for every $A > 0$ and every
$y \in V^\ast(\mathbb{R})$.
(Here $|y|^2 := y_1^2 + \cdots + y_{d}^2$.)
With an eye to \eqref{eq:sieve3}, we desire the following conclusion:
\begin{conclusion}[Level of distribution $\alpha$]\langlebel{prop:gen_ld}
We have, for a parameters $\alpha > 0$ to be determined,
that the following inequality holds for
some $c < r/d$:
\begin{equation}\langlebel{eq:gen_ld}
\sum_{\ldmod < X^{\alpha}} |E(X, \mathbb{P}si_\ldmod, \phi)| \ll X^c.
\end{equation}
\end{conclusion}
We will now prove that this conclusion is implied by the
more combinatorial statements of \eqref{eq:ld_simp} in
Proposition \ref{prop:gen_ld_simp} or \eqref{eq:ld_simp2} in Proposition \ref{prop:gen_ld_simp2}.
For a parameter $Z > 0$, denote by $E_{\leq Z}(X, \mathbb{P}si_\ldmod, \phi)$ the contribution to $E(X, \mathbb{P}si_\ldmod, \phi)$ from those $x$ whose coordinates
are all bounded by $Z$, and write $E_{> Z}(X, \mathbb{P}si_\ldmod, \phi)$ for the remaining contribution.
\begin{lemma}
For any $Z > 0$ and $A > d$ we have
\begin{equation}\langlebel{eq:error_bound1}
E_{> Z}(X, \mathbb{P}si_\ldmod, \phi)
\ll_{A}
X^{r/d} \left( \frac{\ldmod}{X^{1/d}} \right)^A Z^{-A + d},
\end{equation}
and if $Z := \ldmod X^{-1/d + \eta}$ for a fixed constant $\eta > 0$ and $\ldmod < X$ we have, for any $B > 0$,
\begin{equation}\langlebel{eq:error_bound2}
E_{> Z}(X, \mathbb{P}si_\ldmod, \phi)
\ll_{B, \eta} X^{-B}.
\end{equation}
\end{lemma}
\begin{proof}
By \eqref{eq:rapid_decay}, we have
\begin{equation}
E_{> Z}(X, \mathbb{P}si_\ldmod, \phi)
\ll_{A} X^{r/d} \sum_{\substack{x \\ \exists i \ |x_i| > Z}}
\left( 1 + \frac{|x| X^{1/d}}{\ldmod} \right)^{-A}
\leq
X^{r/d} \left( \frac{\ldmod}{X^{1/d}} \right)^A \sum_{\substack{x \\ |x| > Z}}
|x|^{-A}.
\end{equation}
There are $\ll R^{d}$ elements $x$ with $|x| \in [R, 2R]$ for any $R > 0$.
Therefore, assuming
that $A > d$ this sum
is
\begin{equation}
\ll
X^{r/d} \left( \frac{\ldmod}{X^{1/d}} \right)^A \sum_{j = 0}^{\infty} (2^j Z)^{-A + d}
\ll
X^{r/d} \left( \frac{\ldmod}{X^{1/d}} \right)^A Z^{-A + d},
\end{equation}
proving \eqref{eq:error_bound1}. With $Z := \ldmod X^{-1/d + \eta}$
this simplifies to
$X^{\frac{r}{d} - 1 + (d - A) \eta} \ldmod^{d} \leq X^{\frac{r}{d} - 1 + d + (d - A) \eta}$, and the result follows by choosing
$A = \frac{B + d + \frac{r}{d} - 1}{\eta} + d.$
\end{proof}
In what follows we will choose $Z = Z(\ldmod) := \ldmod X^{-1/d + \eta}$ for a fixed small $\eta > 0$ so as to guarantee
\eqref{eq:error_bound2}, so that we have $E(X, \mathbb{P}si_\ldmod, \phi) = O_{B, \eta, \phi}(X^{-B}) + E_{\leq Z}(X, \mathbb{P}si_\ldmod, \phi)$,
with
\begin{equation}\langlebel{eq:poisson_simp}
|E_{\leq Z}(X, \mathbb{P}si_\ldmod, \phi)| \leq
X^{r/d} \widehat{\phi}(0) \sum_{\substack{0 \neq x \in V^\ast(\mathbb{Z}) \\ |x_i| \leq Z \ \forall i }}
|\widehat{\mathbb{P}si_\ldmod}(x)|,
\end{equation}
with $\widehat{\phi}(0)$ being a convenient upper bound for $|\widehat{\phi}(t)|$.
We remark that if $\ldmod < X^{1/d - \eta}$ then the sum in \eqref{eq:poisson_simp} is
empty and $E(X, \mathbb{P}si_\ldmod, \phi) \ll_B X^{-B}$; i.e., the error is essentially zero. In general, we conclude the following:
\begin{proposition}[Level of distribution $\alpha$, simplified version]\langlebel{prop:gen_ld_simp}
Conclusion \ref{prop:gen_ld} follows if we have, for the same
$\alpha > 0$, some $c < {r/d}$ and $\eta > 0$, every $N < X^{\alpha}$,
and with $Z := 2 N X^{\eta - 1/d}$, that
\begin{equation}\langlebel{eq:ld_simp}
X^{r/d}
\sum_{\ldmod \in [N, 2N]}
\sum_{\substack{0 \neq x \in V^\ast(\mathbb{Z}) \\ |x_i| \leq Z \ \forall i}}
|\widehat{\mathbb{P}si_\ldmod}(x)|
\ll X^c.
\end{equation}
\end{proposition}
\begin{proof}
We divide the sum in \eqref{eq:gen_ld}
into $\ll \log X$ dyadic intervals $[N, 2N]$ and apply \eqref{eq:poisson_simp} to each $E(X, \mathbb{P}si_\ldmod, \phi)$,
for each $q$ expanding the condition $|x_i| \leq Z(q)$ to $|x_i| \leq Z(N) = 2 N X^{-1/d + \eta}$. The term
$\widehat{\phi}(0)$ may be subsumed (for fixed $\phi$) into constants implied by the notation $\ll$ and $O(-)$,
and Conclusion \ref{prop:gen_ld} follows (with any $c$ strictly larger than that in \eqref{eq:ld_simp}, so as to incorporate
a contributions of $O(\log X)$ from the number of intervals).
\end{proof}
This statement may initially look more complicated than Conclusion \ref{prop:gen_ld}, but it is simpler in that it
lends
itself naturally to geometric proofs. Moreover the sums over $\ldmod$ and $x$ are independent
and can be interchanged.
{\itshape The $L_1$ norm heuristic.} In the introduction, we said that
`$L_1$ norm bounds for Fourier transforms over finite fields should lead to level of distribution statements for arithmetic objects.'
Such a heuristic arises from \eqref{eq:ld_simp} by assuming that $|\widehat{\mathbb{P}si_q}(x)|$ has the same average value
in the box defined by $|x_i| \leq Z$ as it does in all of $V^*(\mathbb{Z}/q\mathbb{Z})$. Such a statement cannot be proved in general,
and indeed it is not always true: for example, in $\mathbb{Z}^2 \co_{T2}mes \Sym_2 \mathbb{Z}^3$ there are disproportionately
many
doubled forms $x = (x_1, x_1) \in V^*(\mathbb{Z}) \cap [-Z, Z]^{12}$ near the origin. That said, this heuristic is the motivation behind
our geometric sieve method, and it also provides a target for what we may hope to prove.
\subsection{Reformulation in terms of $V(\mathbb{Z})$}\langlebel{sec:reform}
In practice it will be convenient to describe the Fourier transforms $\widehat{\mathbb{P}si_q}(x)$ in terms
of $V(\mathbb{Z})$ instead of $V^*(\mathbb{Z})$. To do this, assume we have a linear map $\rho : V^* \rightarrow V$,
defined by equations over $\mathbb{Z}$, satisfying the following properties\footnote{Formally we may define $\rho$ as a
morphism of schemes over $\mathbb{Z}$ (which is an isomorphism over $\mathbb{Z}[1/m]$); what we need is that
$\rho$ defines maps $V^*(\mathbb{Z}) \rightarrow V(\mathbb{Z})$, $V^*(R) \rightarrow V(R)$ for each ring $R$ containing $\mathbb{Z}$,
and $V^*(\mathbb{Z}/q\mathbb{Z}) \rightarrow V(\mathbb{Z}/q\mathbb{Z})$ for each quotient $\mathbb{Z}/q\mathbb{Z}$ of $\mathbb{Z}$,
all defined
by the same equations and hence compatible with the appropriate ring homomorphisms.} for some integer $m$:
(1) We have
$m V(\mathbb{Z}) \subseteq \rho(V^*(\mathbb{Z})) \subseteq V(\mathbb{Z})$; (2) $\rho$ defines an isomorphism
$V^*(\mathbb{Z}/q\mathbb{Z}) \rightarrow V(\mathbb{Z}/q\mathbb{Z})$ for all integers $q$ coprime to $m$; (3) for each $x \in V^*(\mathbb{Z})$, the coefficients
of $\rho(x) \in V(\mathbb{Z})$ are bounded above by $m$ times those of $x$. Note that (2) is implied by (1), since
$m V(\mathbb{Z}/q\mathbb{Z}) \subseteq \rho(V^*(\mathbb{Z}/q\mathbb{Z})) \subseteq V(\mathbb{Z}/q\mathbb{Z})$ for each $q$.
We then define $\widehat{\mathbb{P}si_q}$ on $V(\mathbb{Z}/q\mathbb{Z})$ by writing
$\widehat{\mathbb{P}si_q}(x) = \widehat{\mathbb{P}si_q}(\rho^{-1}(x))$, and we
lift this definition of $\widehat{\mathbb{P}si_q}$ to all of $V(\mathbb{Z})$.
Finally, by abuse of notation we write
$\widehat{\mathbb{P}si_q}(x) = \widehat{\mathbb{P}si_{\frac{q}{(q, m)}}}(x)$
for an arbitrary squarefree $q$,
so that we have defined $\widehat{\mathbb{P}si_q}(x)$
for all squarefree $q$ and all $x \in V(\mathbb{Z})$.
The following is then immediate:
\begin{proposition}[Level of distribution $\alpha$, simplified version in terms of $V(\mathbb{Z})$]\langlebel{prop:gen_ld_simp2}
Given the constructions above, Conclusion \ref{prop:gen_ld}
follows if we have, for the same
$\alpha > 0$, some $c < {r/d}$ and $\eta > 0$, every $N < X^{\alpha}$,
and with $Z := N X^{\eta - 1/d}$, that
\begin{equation}\langlebel{eq:ld_simp2}
X^{r/d}
\sum_{\ldmod \in [N, 2N]}
\sum_{\substack{0 \neq x \in V(\mathbb{Z}) \\ |x_i| \leq Z \ \forall i}}
|\widehat{\mathbb{P}si_\ldmod}(x)|
\ll X^c.
\end{equation}
\end{proposition}
In fact this conclusion is immediate only with $Z = 2m N X^{\eta - 1/d}$, but we observe that we may divide all
our previous choices of $Z$ by $2m$, with identical results holding at each step; alternatively we may take $\eta$ larger than
that of Proposition \ref{prop:gen_ld_simp}. The implied constant in \eqref{eq:ld_simp2} is independent of $N$ and $X$
but may depend on the other variables.
For each of the two specific representations
$(G,V)$ we treat in this paper,
as well as many other cases of interest,
such a $\rho$ is naturally induced by a non-degenerate symmetric bilinear form
$[-,-]$ on $V$, defined over $\mathbb{Z}[1/m]$, for which
$[gx, g^{\iota} y] = [x, y]$ identically for an involution $\iota$ of $G$. Whenever $(q, m) = 1$, this implies that
$\rho : V^*(\mathbb{Z}/q\mathbb{Z}) \rightarrow V(\mathbb{Z}/q\mathbb{Z})$ defines an isomorphism of $G(\mathbb{Z}/q\mathbb{Z})$-modules.
These facts are important to our evaluation of the Fourier transforms $\widehat{\mathbb{P}si_q}$
in \cite{TT_orbital}, which we describe as functions on $V(\mathbb{F}_q)$ rather than on $V^*(\mathbb{F}_q)$.
We refer to \cite{TT_orbital}, especially Sections 2 and A, for further details and explicit constructions.
For example, let $V$ be the space of binary cubic forms
with $G=\gl_2$.
Then $V(\mathbb{Z})$ is the lattice of all integral binary cubic forms and
$m=3$.
The bilinear form on $V$ is defined by
\[
[x,x']=aa'+\frac13bb'+\frac13cc'+dd'
\]
for
$x=au^3+bu^2v+cuv^2+dv^3$
and
$x'=a'u^3+b'u^2v+c'uv^2+d'v^3$; the involution $\iota$ is defined by $g \mapsto g^{-T}$;
and
$\rho$ is the inverse of the map $V\ni x\mapsto [\cdot,x]\in V^\ast$, which is an isomorphism
over $\mathbb{Z}[1/3]$. Since $V^\ast(\mathbb{Z})=\{\phi\in V^\ast(\mathbb{Q})\mid \phi(V(\mathbb{Z}))\subset\mathbb{Z}\}$,
under the identification $V(\mathbb{Q})=V^\ast(\mathbb{Q})$,
$V^\ast(\mathbb{Z})$ is the lattice of integral binary cubic forms
whose two middle coefficients are multiples of $3$,
and thus $V^\ast(\mathbb{Z}) \subset V(\mathbb{Z})$.
For the space $V$ of pairs of ternary quadratic forms,
$V(\mathbb{Z})$ is the lattice of all pairs of integral quadratic forms,
$V^\ast(\mathbb{Z})\subset V(\mathbb{Z})$
is the lattice of pairs of integral quadratic forms
whose off-diagonal coefficients are multiples of $2$, and $m = 2$.
\section{Proof of Theorem \ref{thm:ap3}}\langlebel{sec:ap3}
We now prove Theorem \ref{thm:ap3} by obtaining a level
of distribution of $\frac{1}{2} - \epsilon$ for a smoothed subset of integral orbits of binary cubic forms (where the level of distribution is again taken
with respect to the
property of the discriminant being divisible by $q$). Although we could appeal to the geometric sieve method of Section \ref{sec:ld}, we instead
give an easier proof whose idea is roughly equivalent to a special case of this method.
In this section $V(\mathbb{Z}) := \Sym_3 \mathbb{Z}^2$ is the lattice of integral binary cubic forms, $r = d = 4$, $Z := NX^{-1/4 + \eta}$, $\mathbb{P}si_q$ is the characteristic function of $x \in V(\mathbb{Z})$ with
$q \mid \Disc(x)$, and we argue that we can prove the bound \eqref{eq:ld_simp2} for any $\alpha < \frac{1}{2}$, i.e. that
\begin{equation}\langlebel{ld:cubic}
\sum_{\ldmod \in [N, 2N]}
\sum_{\substack{0 \neq x \in V(\mathbb{Z}) \\ |x_i| \leq Z \ \forall i}}
|\widehat{\mathbb{P}si_\ldmod}(x)|
\ll X^{-1 + c}
\end{equation}
for each $N < X^\alpha$ for some $c = c(\alpha) < 1$.
The Fourier transform $\widehat{\mathbb{P}si_q}$ is multiplicative in $q$, and satisfies
\begin{equation}\langlebel{eq:Psi_fourier_cubic}
\widehat{\mathbb{P}si_p}(x)=
\begin{cases}
p^{-1} + p^{-2} - p^{-3}
& \text{if } x \in pV(\mathbb{Z}),\\
p^{-2} - p^{-3}
& \text{if } x \not \in pV(\mathbb{Z}) \text{ but } p \mid \Disc(x),\\
- p^{-3} & \text{if } p \nmid \Disc(x).
\end{cases}
\end{equation}
For each positive divisor $q_0$ of $q$ and $x \in q_0 V(\mathbb{Z})$, we have
\[
\widehat{\mathbb{P}si_q}(x) = \widehat{\mathbb{P}si_{q_0}}(x) \cdot \widehat{\mathbb{P}si_{q / q_0}}(x)
= \widehat{\mathbb{P}si_{q_0}}(x) \cdot \widehat{\mathbb{P}si_{q / q_0}}(x / q_0).
\]
Therefore
the summation of \eqref{ld:cubic} is equal to
\begin{equation}\langlebel{ld:cubic2}
\sum_{q_0 \leq Z} \Big( \prod_{p \mid q_0} (p^{-1} + p^{-2} - p^{-3}) \Big)
\sum_{\substack{\frac{N}{q_0} \leq q_1 \leq \frac{2N}{q_0} \\ (q_0, q_1) = 1}}
\sump_{\substack{x \\ |x_i| \leq \frac{Z}{q_0} \ \forall i}}
|\widehat{\mathbb{P}si_{q_1}}(x)|,
\end{equation}
where the inner sum is over those $x \in V(\mathbb{Z})$ which are not in
$p V(\mathbb{Z})$ for any prime divisor $p$ of $q_0 q_1$.
We split the sum of \eqref{ld:cubic2} into two pieces: a sum over those $x$ for
which $\Disc(x) = 0$, and a sum over those $x$ for which $\Disc(x) \neq 0$.
{\itshape Those $x$ with $\Disc(x) = 0$.} The number of such $x$ with all coordinates bounded by $Y$,
is $O(Y^2)$ for any $Y$. To see this, note that any such $x$ can be written
as $(ax + by)^2(cx + dy)$ for some $a, b, c, d \in \mathbb{Z}$. The number of possibilities with
$a = 0$ is $O(Y^2)$, as this forces the $x^3$ and $x^2 y$ coefficients to both be zero.
Similarly there are $O(Y^2)$ possibilities with $b = 0$. We are therefore left with the number
of integer quadruples $(a, b, c, d)$ with $a b \neq 0$, $|a^2 c| \leq Y$, and $|b^2 d| \leq Y$,
which is the square of the number of integer pairs $(a, c)$ with $a \neq 0$, $|a^2 c| \leq Y$.
This latter quantity is easily seen to be $O(Y)$, as needed.
The inner sum is therefore over $O(Z/q_0)^2$ elements, and
for each $x$ we have $|\widehat{\mathbb{P}si_{q_1}}(x)| \leq {q_1}^{-2}$. Therefore,
this portion of the sum in \eqref{ld:cubic2} is
\begin{equation}\langlebel{ld:cubic2a}
\ll_{\epsilon}
\sum_{q_0 \leq Z} q_0^{-1 + \epsilon} \cdot
\frac{N}{q_0} \cdot \left( \frac{Z}{q_0} \right)^2 \cdot \left( \frac{N}{q_0} \right)^{-2}
\ll_{\epsilon} N^{-1} Z^2 \sum_{q_0 \leq Z} q_0^{-2 + \epsilon}
\ll X^{2\eta} N X^{-1/2},
\end{equation}
which satisfies the bound \eqref{ld:cubic}.
{\itshape Those $x$ with $\Disc(x) \neq 0$.}
The contribution of these is bounded above by
\begin{align*}\langlebel{ld:cubic2b}
& \sum_{q_0 \leq Z} \Big( \prod_{p \mid q_0} (p^{-1} + p^{-2} - p^{-3}) \Big)
\sum_{\substack{\frac{N}{q_0} \leq q_1 \leq \frac{2N}{q_0} \\ (q_0, q_1) = 1}} q_1^{-3}
\sump_{\substack{x \\ |x_i| \leq \frac{Z}{q_0} \ \forall i}}
\gcd(\Disc(x), q_1)
\\
\leq
& \sum_{q_0 \leq Z} \Big( \prod_{p \mid q_0} (p^{-1} + p^{-2} - p^{-3}) \Big)
\left( \frac{N}{q_0} \right)^{-3}
\sum_{\substack{x \\ |x_i| \leq Z/q_0 \ \forall i \\ \Disc(x) \neq 0}}
\sum_{\substack{\frac{N}{q_0} \leq q_1 \leq \frac{2N}{q_0}}}
\gcd(\Disc(x), q_1).
\end{align*}
Now, in general, whenever $m \neq 0$ we have
\[
\sum_{n \in [N, 2N]} \gcd(m, n)
\leq
\sum_{\substack{ f \mid m \\ f \leq 2N}} f
\sum_{\substack{ n \in [N, 2N] \\ f \mid n}} 1
\leq
\sum_{\substack{ f \mid m \\ f \leq 2N}} f
\left( \frac{N}{f} + 1 \right)
\ll N m^{\epsilon}.
\]
Therefore, using that the discriminant of any $x$ in the sum is $\ll N^4$, we see that the previous quantity
is
\begin{align*}\langlebel{ld:cubic2c}
\ll_{\epsilon}
& \sum_{q_0 \leq Z} \Big( \prod_{p \mid q_0} (p^{-1} + p^{-2} - p^{-3}) \Big)
\left( \frac{N}{q_0} \right)^{-3}
\cdot
\left( \frac{Z}{q_0} \right)^4
\frac{N}{q_0} \cdot N^{\epsilon}
\\
\ll_{\epsilon} &
N^{\epsilon} Z^4 N^{-2}
\sum_{q_0 \leq Z} q_0^{-3 + \epsilon}
\\
\ll_{\epsilon} &
N^{\epsilon} N^2 X^{-1 + 4 \eta},
\end{align*}
again satisfying \eqref{ld:cubic}.
Applying the weighted sieve of Theorem \ref{thm:ls}, and following the beginning of the proof of Proposition \ref{prop:ld_gives_main},
we obtain $\gg \frac{X}{\log X}$ elements $x \in V(\mathbb{Z})$ whose discriminants
have
at most three prime factors. Of these, at most $O_\epsilon(X^{3/4 + \epsilon})$ can be reducible. (For a simple proof see \cite[Lemma 21]{BST}; only
the second paragraph of the proof there is relevant, as we are counting points in a box of side length $\ll X^{1/4}$.) As the weighted sieve produces $x$ with
each prime factor
$> X^{\alpha/4}$, the number of $x \in V(\mathbb{Z})$ with any repeated prime factor is
$\ll \sum_{p > X^{\alpha/4}} \frac{X}{p^2} \ll X^{1 - \alpha/4}$ by \cite[Lemma 3.4]{BBP}. Accordingly we produce $\gg \frac{X}{\log X}$ irreducible elements $x \in V(\mathbb{Z})$ with
squarefree (and hence fundamental) discriminants, which must therefore correspond to (distinct) maximal cubic orders and hence to cubic fields.
\begin{remark}
In place of our estimate of $O(Y^2)$ for reducible elements $x$ in boxes of side length $O(Y)$,
the method of Section \ref{sec:ld} would implicitly incorporate
a bound of $O(Y^3)$, as $3$ is the dimension of the variety $\Disc(x) = 0$. This proof illustrates that
counting elements more directly may yield improvements in the end results.
\end{remark}
\section{Closed subschemes containing singular orbits}
\langlebel{subsec:2sym23-subschemes}
Let $V$ be the space of pairs of ternary quadratic forms, together with its
action of $G = \GL_2 \times \GL_3$. Recall from \cite[Proposition 21]{TT_orbital}
that, for each $p \neq 2$ there are $20$ orbits for the action of $G(\mathbb{F}_p)$ on $V(\mathbb{F}_p)$.
We gave twenty `orbit descriptions' $\mathcal{O}$ which were essentially uniform in $p$, and for each $p$
we write $\mathcal{O}(p)$ for the associated orbit over $\mathbb{F}_p$.
\begin{proposition}\langlebel{prop:same_dim}
For each of the orbit descriptions $\mathcal{O}$ described above there exists a closed subscheme
$\mathfrak{X} \subset V$, defined over $\mathbb{Z}$, such that $\mathcal{O}(p) \subseteq \mathfrak{X}(\mathbb{F}_p)$ for each prime $p \neq 2$, and of the same
`dimension' as
$\mathcal{O}$ in the sense that $\# \mathcal{O}(p) \asymp \# \mathfrak{X}(\mathbb{F}_p)$.
\end{proposition}
We will prove this statement, with `$p \neq 2$' replaced with `$p \not \in S$ for some finite set $S$',
for any finite dimensional $(G, V)$ satisfying the following two properties:
\begin{itemize}
\item
There exist finitely many elements $x_\sigma\in V(\mathbb{Z})$
such that
for any algebraically closed field $K$ with ${\rm char}(K)\not \in S$,
the images of the $x_\sigma$ in $V(K)$ via the
canonical map $V(\mathbb{Z})\rightarrow V(K)$
form a set of complete representatives
for $G(K)\backslash V(K)$.
\item
There exists a constant $c > 0$ such that for each $p \not \in S$ and $G(\mathbb{F}_p)$-orbit $\mathcal{O} \subseteq V(\mathbb{F}_p)$
we have $\# \mathcal{O} > c \# \widetilde{\mathcal{O}}$, with $\widetilde{\mathcal{O}} := G(\overline{\mathbb{F}_p}) \mathcal{O} \cap V(\mathbb{F}_p)$.
\end{itemize}
These properties hold for all of the $(G, V)$ studied in \cite{TT_orbital}, as we explain now in the case of
pairs of ternary quadratic forms.
We group the $20$ orbit descriptions of \cite[Proposition 21]{TT_orbital} as follows:
\begin{multline*}
\{ \mathcal{O}_0 \},
\{ \mathcal{O}_{D1^2} \},
\{ \mathcal{O}_{D11}, \mathcal{O}_{D2} \},
\{ \mathcal{O}_{D\rm ns} \},
\{ \mathcal{O}_{C\rm s} \},
\{ \mathcal{O}_{C\rm ns} \},
\{ \mathcal{O}_{B11}, \mathcal{O}_{B2} \}, \\
\{ \mathcal{O}_{1^4} \},
\{ \mathcal{O}_{1^3 1} \},
\{ \mathcal{O}_{1^2 1^2}, \mathcal{O}_{2^2} \},
\{ \mathcal{O}_{1^2 11}, \mathcal{O}_{1^2 2} \},
\{ \mathcal{O}_{1111}, \mathcal{O}_{112,} \mathcal{O}_{22},
\mathcal{O}_{13}, \mathcal{O}_4 \}.
\end{multline*}
Within each of these twelve sets, the orbital representative of the {\itshape first-listed} $\mathcal{O}$ is described
in \cite[Proposition 21]{TT_orbital} as the reduction $\pmod p$ of a fixed element $x_\sigma \in V(\mathbb{Z})$, and
when $K$ is algebraically closed with ${\rm{char}}(K) \neq 2$, the proof
in \cite[Section 7.1]{TT_orbital} establishes that the images of these $x_\sigma$ in $V(K)$ are a set of representatives for
$G(K) \backslash V(K)$. Moreover, for each $p \not \in S$ and $G(\mathbb{F}_p)$-orbit $\mathcal{O} \subseteq V(\mathbb{F}_p)$, the associated
$\widetilde{\mathcal{O}}$ is the union of the $\mathcal{O}$ in the grouping described above, and the second property above may
be deduced from the point counts in \cite[Proposition 21]{TT_orbital}.
To conclude Proposition \ref{prop:same_dim} from these two properties, write
$Y_\sigma := G(\overline{\mathbb{Q}}) x_\sigma \subseteq V(\overline{\mathbb{Q}})$ for each $x_\sigma$. By
\cite[Propositions I.1.8 and II.6.7]{borel} we may write each $Y_\sigma$ in the form
$Y_\sigma = \mathfrak{X}_\sigma - \cup_j W_{\sigma, j}$ where the $\mathfrak{X}_\sigma$ and $W_{\sigma, j}$ are (finitely many)
closed varieties, defined over $\mathbb{Q}$, and with $\dim(W_{\sigma, j}) < \dim(\mathfrak{X}_\sigma)$ for all $\sigma$ and $j$.
(Each $\mathfrak{X}_\sigma$ is the closure of $Y_\sigma$, and the $W_{\sigma, j}$ are defined by the closures of other $G(\overline{\mathbb{Q}})$-orbits, of which there are finitely many,
and since each of the $x_\sigma$ is defined over $\mathbb{Q}$ their orbits are as well.)
We choose (arbitrary) integral models for the
$\mathfrak{X}_\sigma$ so as to regard them as closed subschemes of $V$.
For all but finitely many $p$,
these equations reduce $\pmod p$ and define varieties of the same dimension over $\mathbb{F}_p$, and we conclude
by Lang-Weil \cite{LW} that $\#\mathfrak{X}_\sigma(\mathbb{F}_p) \asymp \# Y_\sigma(\mathbb{F}_p) \asymp p^{\dim(\mathfrak{X}_\sigma)}.$ The second bullet point above then gives
the desired conclusion.
\section{A version of the geometric sieve}\langlebel{sec:geom}
The {\itshape Ekedahl-Bhargava geometric sieve}, in the form of \cite[Theorem 3.3]{B_geosieve}, asserts the following.
Let $B$ be a compact region in $\mathbb{R}^r$, let $\mathfrak{X}$ be a closed subscheme of $\mathbb{A}_\mathbb{Z}^r$ of codimension
$a \geq 1$, and let $\langlembda$ and $P$ be positive real numbers. Then, we have
\begin{equation}\langlebel{eq:b_geo}
\# \{ x \in \langlembda B \cap \mathbb{Z}^r \ | \ x \pmod p \in \mathfrak{X}(\mathbb{F}_p)
\textnormal{ for some prime } p > P \}
= O_{B, \mathfrak{X}} \Bigg( \frac{\langlembda^r}{P^{a - 1} \log P} + \langlembda^{r - a + 1} \Bigg).
\end{equation}
We introduce a variation with two modifications. Firstly, we count each $x$ with multiplicity, given by
the number of pairs $(x, p)$ for which $x \pmod p \in \mathfrak{X}(\mathbb{F}_p)$ and $p \in [P, 2P]$.
Secondly, we introduce an `arithmetic progression' condition $x - x_0 \in m V(\mathbb{Z})$, allowing for additional flexibility
in applications (as we will see in Section \ref{sec:ld}.)
We refer also to \cite{BCT} where the same generalization is presented concurrently; the application there
replaces {\itshape primes} in \eqref{eq:b_geo} with squarefree integers, which amounts roughly to a simpler version of the argument in Section \ref{sec:ld} here.
\begin{theorem}\langlebel{lem:geosieve}
Let $B$, $\mathfrak{X}$, $a$, and $\langlembda$ be as in the statement of \eqref{eq:b_geo}, let $m$ be a positive integer, let $x_0 \in V(\mathbb{Z})$,
and let $P > \langlembda/m > 1$ be any real number.
Then, we have
\begin{equation}\langlebel{eq:beo}
\# \{ (x, p) \ | \ x \in \langlembda B \cap (x_0 + m\mathbb{Z}^r), \ \ \textnormal{$p$ is a prime in $[P, 2P]$}, \ p \nmid m, \
x \pmod p \in \mathfrak{X}(\mathbb{F}_p) \}
\ll_{B, \mathfrak{X}, \epsilon} \Big( \frac{\langlembda}{m} \Big)^{r - a} P \langlembda^{\epsilon}.
\end{equation}
\end{theorem}
\begin{proof} This closely follows the proof of Theorems 3.1 and 3.3 of Bhargava \cite{B_geosieve}. In \cite{B_geosieve} (with $m = 1$),
the quantity $\langlembda$ appears only as an upper bound for the number of lattice points in $rB$ lying on a line defined by fixing all
but one of the coordinates. Therefore, with $m > 1$ we can replace $\langlembda$ with $\frac{\langlembda}{m}$ at each occurrence.
The analogue of \cite[Lemma 3.1]{B_geosieve}, proved identically, thus reads that
\begin{equation}\langlebel{eq:vz}
\# \{ x \in \langlembda B \cap (x_0 + m\mathbb{Z}^r) \cap \mathfrak{X}(\mathbb{Z}) \}
\ll_{B, \mathfrak{X}} \Big( \frac{\langlembda}{m} \Big)^{r - a},
\end{equation}
and we obtain the bound of \eqref{eq:beo} for those $x \in \mathfrak{X}(\mathbb{Z})$.
For those $x \not \in \mathfrak{X}(\mathbb{Z})$, it suffices to prove that
\begin{equation}\langlebel{eq:beo2}
\# \left\{ (x, p) \ \Bigg|
\begin{array} {l}
x \in \langlembda B \cap (x_0 + m\mathbb{Z}^r), \ x \not \in \mathfrak{X}(\mathbb{Z}), \\
p > \frac{\langlembda}{m}, \ p \nmid m, \
x \pmod p \in \mathfrak{X}(\mathbb{F}_p)
\end{array}
\right\}
\ll_{B, \mathfrak{X}, \epsilon} \Big( \frac{\langlembda}{m} \Big)^{r - a + 1} \cdot \langlembda^{\epsilon},
\end{equation}
the exact analogue of \cite[(17)]{B_geosieve}.
This follows \cite{B_geosieve} exactly. The condition $p \nmid m$ is
needed in the last paragraph of \cite[Theorem 3.3]{B_geosieve}, to conclude that if $f_k(x)$ is a polynomial in one variable
with $f_k \not \equiv 0 \pmod p$, then it has $O_{\deg(f_k)}(1)$ roots $x$ in an interval of length $O(\langlembda)$, and with
$x \equiv x_0 \pmod m$ for any fixed $x_0$. The factor of $\langlembda^{\epsilon}$ arises in adapting the argument immediately after \cite[(17)]{B_geosieve}:
any nonzero $f_i(x)$ can have only $O_{f_i}(1)$ prime factors $p > \langlembda$, but it may have $O_{f_i, \epsilon}(\langlembda^{\epsilon})$
prime factors $p > \langlembda/m$.
\end{proof}
\section{Application of the geometric sieve: Proof of Proposition \ref{prop:gen_ld_simp2}}\langlebel{sec:ld}
In this section we prove the bound \eqref{eq:ld_simp2} for each $\alpha < \frac{7}{48}$ for the `quartic' $(G, V)$ of Section \ref{sec:properties}.
For each $p \neq 2$ and
$i \in \{0, 4, 7, 8, 10, 11, 12 \}$ we define sets
$U_i(p)$,
each of which is a union of $G(\mathbb{F}_p)$-orbits on $V(\mathbb{F}_p)$, as follows.
\[
\begin{array}{c | c | c c}
\textnormal{Label} & \textnormal{Consists of}
& \text{Dimension} \ i & \text{Fourier contribution} \ \text{fc}(i)
\\ \hline
U_0 &
\mathcal{O}_0 & 0 & -1 \\
U_4&
\mathcal{O}_{D1^2}
& 4 & -3 \\
U_7
&
\mathcal{O}_{D11}, \mathcal{O}_{C\rm s}
& 7 & -4 \\
U_8
&\mathcal{O}_{T11}, \mathcal{O}_{T2}, \mathcal{O}_{D2}, \mathcal{O}_{D\rm ns}, \mathcal{O}_{C\rm ns} &
8 & -5 \\
U_{10}&
\mathcal{O}_{1^2 1^2}, \mathcal{O}_{2^2}, \mathcal{O}_{1^3 1}, \mathcal{O}_{1^4}
& 10 & -6 \\
U_{11}&
\mathcal{O}_{1^2 11}, \mathcal{O}_{1^2 2}
& 11 & -7 \\
U_{12}&
\text{nonsingular orbits}
& 12 & -8 \\
\end{array}
\]
In Section \ref{subsec:2sym23-subschemes} we proved that for each $i \in \{0, 4, 7, 8, 10, 11, 12 \}$
there are subschemes $\mathfrak{X}_i$ of $\mathbb{A}^r_{\mathbb{Z}} = V$ of dimension $i$, defined over $\mathbb{Z}$,
for which $U_i(p) \subseteq \mathfrak{X}_i(\mathbb{F}_p)$ and $\# \mathfrak{X}_i(\mathbb{F}_p) \ll p^i$ for all $p \not \in S$.
The function $\fc(i)$ is chosen such that
$|\widehat{\mathbb{P}si_p}(x)| \leq 2 p^{\fc(i)}$ for each $x \in U_i(p)$.
For every squarefree $n \in [N, 2N]$ (with $N < X^{\alpha}$ for $\alpha$ to be determined)
we consider the contribution to \eqref{eq:ld_simp2} from
every factorization
\begin{equation}\langlebel{eq:factor_n}
n = n_0 n_4 n_7 n_8 n_{10} n_{11} n_{12}
\end{equation}
and those
$x$ with $x \in U_i(p)$ for each $p \mid n_i$.
When $n$ is even we will assume as a bookkeeping device that
$n_{0}$ is as well, but we will never
demand any geometric condition on $x$ modulo $2$.
The contribution of each such $x$ is bounded above by
$2^{\omega(n) + 1} \prod_i n_i^{\fc(i)}$, where the $2^1$ factor reflects the trivial
bound $|\widehat{\mathbb{P}si_2}(x)| \leq 1$,
and we write
$2^{\omega(n)} = O_{\epsilon}(X^{\epsilon})$, uniformly in $n$.
We consider the following choices of parameters:
\begin{itemize}
\item Squarefree and pairwise coprime integers $n_i$ for $i \in \{ 0, 4, 7, 8, 10, 11, 12 \}$, with $\prod_i n_i \in [N, 2N]$.
\item A parameter $j \in \{ 4, 7, 8, 10, 11, 12 \}$, and a factorization $n_j = n_j' p n_j''$, where $p$ is a prime.
\item Writing $m := n_j' \prod_{i < j} n_i$, these choices are subject to the condition that
$m \leq Z < mp$.
\end{itemize}
We claim that every factorization \eqref{eq:factor_n} corresponds to at least one choice of the above data,
with $n_j = n_j' p n_j''$. First of all, note that $n_0 \leq Z$ for each
nonzero $x \in [-Z, Z]^{r}$.
Thus, given any factorization \eqref{eq:factor_n}, we let $j \geq 4$ be the minimal index
with $\prod_{i \leq j} n_i > Z$, choose $n_j'$ to be the largest divisor of $n_j$ less than or equal to
$Z \prod_{i < j} n_i^{-1}$, and choose $p$ to be any prime divisor of $n_j / n_j'$.
The conditions modulo $m$, namely that $x \in U_i(q)$ for each odd prime $q \mid m$ with $i = i(q)$ determined by the factorization
above, are equivalent to demanding that $x$ lie in one of $O_{\epsilon}(X^{\epsilon} n_j'^{j} \prod_{i < j} n_i^i)$
residue classes $\pmod{m V(\mathbb{Z})}$.
(Here
$X^{\epsilon}$ is a simple upper
bound for $C^{\omega(n)}$, the product of the implied
constants occurring in the point counts for the $U_i(p)$.)
We must have $x \in \mathfrak{X}_{j}(\mathbb{F}_p)$, and for each of the residue classes
$\pmod{m V(\mathbb{Z})}$ determined above we
use Bhargava's geometric sieve
(Theorem \ref{lem:geosieve}) to bound the number
of pairs $(x, p)$ where $x \in V(\mathbb{Z})$ lies in this residue class,
has all coefficients bounded by $Z$, and lies in $\mathfrak{X}_j(\mathbb{F}_p)$, and where
$p \not \in S$ lies in a dyadic interval $[P, 2P]$. By the theorem, the number of such pairs is
$\ll Z^\epsilon (Z/m)^{j} P$.
(Any contribution of $(x, p)$ with $p$ in the exceptional set $S$ of Proposition \ref{prop:same_dim} trivially satisfies the same bound, as in this case
$Z/m \ll_S 1$.)
The Fourier contribution of each $x$ being counted is
$\ll X^{\epsilon} \prod_i n_i^{\fc(i)}$, and for each choice of
$j$, $n_i$ ($i < j$), and $n_j'$, and for each fixed dyadic interval
$[P, 2P]$, we multiply: the number of residue classes modulo $mV(\mathbb{Z})$;
the number of pairs $(x, p)$ in each; the Fourier contribution of each $x$ being counted;
and the $N^{1 + \epsilon}/mP$ choices of $n_j''$ and $n_i$ ($i > j$).
Recalling that $Z = NX^{-1/d + \eta}$, we conclude that the contribution to
\eqref{eq:ld_simp2} from the choices previously determined is
\[
\ll_{\epsilon} X^{\epsilon} \cdot X \cdot
n_j'^{j} \prod_{i < j} n_i^i
\cdot
\left( \frac{NX^{-1/d + \eta}}{m} \right)^j P \cdot
\prod_i n_i^{\fc(i)} \cdot \frac{N^{1 + \epsilon}}{mP}.
\]
Using the fact that $\fc(i)$ is a decreasing function of $i$, and summing over the $\ll X^{\epsilon}$ choices of dyadic interval
$[P, 2P]$, we see that this is
\[
\ll_{\epsilon} X^{\epsilon + r \eta} \cdot X^{1 - j/d} \cdot
\left(
\prod_{i < j} n_i^{i + \fc(i)}
\right)
\cdot
(n_{j}')^{j + \fc(j)}
\cdot
\left( \frac{N}{m} \right)^{j + \fc(j) + 1}
\]
Now, since $i + \fc(i)$ is an increasing function of $i$ this is bounded above by
\begin{align*}
\ll_{\epsilon} & X^{\epsilon + r \eta} \cdot X^{1 - j/d} \cdot
m^{j + \fc(j)}
\cdot
\left( \frac{N}{m} \right)^{j + \fc(j) + 1}
\\
\ll_{\epsilon} & X^{\epsilon + r \eta} \cdot X^{1 - j/d}
\cdot m^{-1} N^{j + \fc(j) + 1},
\end{align*}
and, now fixing only the parameter $j$,
we sum over all $m \leq Z$ and (for each $m$)
the $\ll N^{\epsilon}$ choices of factorizations of $m$ to
obtain a total contribution
\begin{equation}\langlebel{eq:final_cont}
\ll_{\epsilon} X^{\epsilon + r \eta} \cdot X^{1 - j/d}
\cdot N^{j + \fc(j) + 1}
\end{equation}
from all choices of \eqref{eq:factor_n} associated to this factor $j$. Up to an implied constant,
the total error is bounded above by the maximum of \eqref{eq:final_cont} over the six admissible values of $j$.
The quantity in \eqref{eq:final_cont} is:
\[
\begin{array}{c | c}
\textnormal{$j$} & \textnormal{Bound} (\times X^{\epsilon + r \eta})
\\ \hline
j = 4 & X^{2/3} N^2\\
j = 7 & X^{5/12} N^4\\
j = 8 & X^{1/3} N^4\\
j = 10 & X^{1/6} N^5\\
j = 11 & X^{1/12} N^5\\
\ j = 12 & N^5\\
\end{array}
\]
The case $j = 7$ turns out to be
the bottleneck, and choosing $N = X^{\alpha}$ with any $\alpha < \frac{7}{48}$ we may choose $\eta$ and $\epsilon$ with $c := \frac{5}{12} + 4 \alpha + \epsilon + 12 \eta < 1$,
so that \eqref{eq:ld_simp2} holds with this value of $c$.
\section{Conclusion of the proof of Theorem \ref{thm:ap4}}\langlebel{sec:ld_implies}
We give a slightly more general statement, which illustrates how improvements to Theorem \ref{thm:ap4} would automatically
follow from improvements in the level of distribution.
\begin{proposition}\langlebel{prop:ld_gives_main}
Assume, for some integer $t \geq 1$, that Proposition \ref{prop:gen_ld_simp2} (and hence
Conclusion \ref{prop:gen_ld}) holds for some $c < 1$ and $\alpha > \Big(t + 1 - \frac{\log 4}{\log 3}\Big)^{-1}$.
Then there are $\gg_{t, \alpha, c} \frac{X}{\log X}$ $S_4$-quartic field discriminants $K$
with $|\Disc(K)| < X$, such that $\Disc(K)$ has at most $t$ prime factors.
\end{proposition}
Since we proved Proposition \ref{prop:gen_ld_simp2} with any $\alpha < \frac{7}{48}$, we thus obtain
Theorem \ref{thm:ap4} with any $t > \frac{48}{7} - 1 + \frac{\log 4}{\log 3} = 7.119\dots$, and in particular with
$t = 8$.
\begin{proof}
We apply the weighted sieve of Theorem \ref{thm:ls}, with $Y = X$ and
\begin{equation}\langlebel{eqn:ws_sum}
a(n) := \sum_{\substack{x \in V(\mathbb{Z}) \\ |\Disc(x)| = n}} \phi(x X^{-1/12}).
\end{equation}
Each sum is finite because $\phi$ is compactly supported. We then have
\[
\sum_{\substack{n < X \\ \ldmod \mid n}} a(n) = \sum_{\substack{x \in V(\mathbb{Z})}} \mathbb{P}si_\ldmod(x) \phi(x X^{-1/12}),
\]
where $\mathbb{P}si_\ldmod$ is the characteristic function of $x \in V(\mathbb{Z})$ with $\ldmod \mid \Disc(x)$.
By \eqref{eq:poisson_error}-\eqref{eq:def_error} the sequence satisfies the sieve axiom
\eqref{eq:sieve2}, and by assumption Proposition \ref{prop:gen_ld_simp2} and therefore Conclusion \ref{prop:gen_ld} and
\eqref{eq:sieve3} hold. The linearity conditions \eqref{eq:ls} and \eqref{eq:ls2} follow from the first line of \eqref{eq:Psi_fourier_ternary}.
Theorem \ref{thm:ls} therefore implies that the sum of $\phi(x X^{-1/12})$, over all $x$
whose discriminants have at most $t$ prime factors, is $\gg \frac{X}{\log X}$. By construction
the count of such $x$ satisfies the same lower bound, and
these discriminants are all in $(-X, 0) \cup (0, X)$ and are $G(\mathbb{Z})$-inequivalent in $V(\mathbb{Z})$.
By Bhargava \cite[Theorem 1]{HCL3}, these are in bijection with pairs $(Q, R)$, where $Q$ is a quartic ring
and $R$ is a cubic resolvent ring of $R$, and in case $Q$ is maximal then it has
exactly one cubic resolvent \cite[Corollary 5]{HCL3}.
Moreover, if $x \in V(\mathbb{Z})$ corresponds to $(Q, R)$, then
$\Disc(x) = \Disc(Q)$.
As described on \cite[p. 1037]{HCL3},
$Q$ is an order in an $S_4$- or $A_4$-field
if and only if the corresponding $x \in V(\mathbb{Z})$ is absolutely irreducible.
The number of $x$ which are not absolutely irreducible is $\ll X^{11/12 + \epsilon}$ and hence negligible; this is proved in
\cite[Lemmas 12 and 13]{HCL3}. In our case these proofs simplify
because we may ignore the cusp: the compact support of $\phi$ ensures that we are only counting points in a box of side length $O(X^{1/12})$, and that the number of points
with $a_{11} = 0$ is $O(X^{11/12})$.
We must then bound the number of pairs $(Q, R)$ where $Q$ is a nonmaximal $S_4$- or $A_4$-quartic order.
By Theorem \ref{thm:ls} the discriminants of
$x \in V(\mathbb{Z})$ being counted have all of their prime factors $> X^{\alpha / 4}$,
and in particular any nonmaximal $Q$ which survives the sieve must be nonmaximal at
some prime $p > X^{\alpha/4}$. By \cite[Proposition 23]{B_quartic}, the
number of such $x$ is
\[
\ll \sum_{p > X^{\alpha/4}} X/p^2 \ll X^{1 - \alpha/4},
\]
negligible for any $\alpha > 0$.
This leaves the maximal $Q$ whose discriminants are divisible by $p^2$ for some $p$ in the same range. These can be handled
by the geometric sieve, precisely as Bhargava did in \cite{B_geosieve}. We apply the geometric sieve in its original formulation
directly to \eqref{eqn:ws_sum}, in contrast
to Section \ref{sec:ld} where we applied our variation after an application of Poisson summation.
Any maximal $Q$ whose discriminant is divisible by $p^2$ must be, in the language of Bhargava \cite{B_geosieve}, a {\itshape strong multiple}
of $p$; and hence (as in Section \ref{subsec:2sym23-subschemes})
the corresponding $x$ must be in $\mathfrak{X}(\mathbb{F}_p)$ for a suitably defined subscheme $\mathfrak{X} \subseteq V$ of codimension $2$.
By \cite[Theorem 3.3]{B_geosieve},
the number of such $x$ which satisfy this criterion for any
$p > X^{\alpha/4}$ is again $\ll X^{1 - \alpha/4}$.
In conclusion, the contributions of everything other than maximal orders in $S_4$-quartic fields to our sieve result is negligible,
and hence we obtain $\gg \frac{X}{\log X}$ $S_4$-quartic fields with the stated properties.
\end{proof}
\section*{Acknowledgments}
We would like to thank Theresa Anderson, Manjul Bhargava, Alex Duncan, \'Etienne Fouvry, Yasuhiro Ishitsuka, Kentaro Mitsui, Arul Shankar, Ari Shnidman, Jack Thorne, Jerry (Xiaoheng) Wang,
Melanie Matchett Wood and Kota Yoshioka for helpful discussions, comments and feedback.
Duncan, in particular, explained to us the application of the lemma from \cite{borel} to
the proof of Proposition \ref{prop:same_dim}.
Taniguchi's work was partially supported by the JSPS, KAKENHI Grant Numbers JP24654005, JP25707002, and JP17H02835.
Thorne's work was partially supported by the National Science Foundation under Grant No. DMS-1201330 and by the National Security Agency under a Young Investigator Grant.
\end{document}
\end{document} |
\betagin{document}
\title{Bases of random unconditional convergence in Banach spaces}
\author[J. Lopez-Abad]{J. Lopez-Abad}
\address{J. Lopez-Abad \\ Instituto de Ciencias Matem\'aticas (ICMAT). CSIC-UAM-UC3M-UCM. C/ Nicol\'{a}s Cabrera 13-15, Campus Cantoblanco, UAM
28049 Madrid, Spain.}
\address{Instituto de Matem\'atica e Estat\'{\i}stica - IME/USP
Rua do Mat\~{a}o, 1010 - Cidade Universit\'aria, S\~{a}o Paulo - SP, 05508-090, Brasil.}
\email{[email protected]}
\author[P. Tradacete]{P. Tradacete}
\address{P. Tradacete\{\mathbb M}athematics Department\\ Universidad Carlos III de Madrid\\ 28911, Legan\'es, Madrid, Spain.}
\email{[email protected]}
\thanks{Both authors have been partially supported by the Spanish Government Grant MTM2012-31286 and Grupo UCM 910346. The first author acknowledges the support of Fapesp, Grant 2013/24827-1. The second author has also been partially supported by the Spanish Government Grant MTM2010-14946.}
\subjclass[2010]{46B09, 46B15}
\keywords{Unconditional basis, Random unconditional convergence.}
\betagin{abstract}
We study random unconditional convergence for a basis in a Banach space. The connections between this notion and classical unconditionality are explored. In particular, we analyze duality relations, reflexivity, uniqueness of these bases and existence of unconditional subsequences.
\itemd{abstract}
\maketitle
\succeqction{Introduction}
A series $\sum_n x_n$ in a Banach space is \emph{randomly unconditionally convergent} when $\sum_n \varepsilon_n x_n$ converges almost surely on signs $(\varepsilon_n)_n$ (with respect to the Haar probability measure on $\{-1,1\}^{\mathbb N}$). P. Billard, S. Kwapie\'n, A. Pelczy\'nski and Ch. Samuel introduced in \cite{BKPS} the notion of random unconditionally convergent (RUC) coordinate systems $(e_i)_i$ in a Banach space, which have the property that the expansion of every element is randomly unconditionally convergent. Equivalently, a RUC system $(e_i,e_i^*)_i$ in a Banach space satisfies that for a certain constant $K$ and every $x$ in the span of $(e_i)_i$
$$
\sup_n\int_0^1\mathbb{B}ig\|\sum_{i=1}^n r_i(t)e_i^*(x)e_i\mathbb{B}ig\|dt\leq K\|x\|
$$
where $(r_i)$ is the sequence of Rademacher function on $[0,1]$. For a RUC Schauder basis $(e_n)$, this is equivalent to
$$
\int_0^1\mathbb{B}ig\|\sum_{i=1}^m r_i(t)a_je_i\mathbb{B}ig\|dt\leq K\mathbb{B}ig\|\sum_{i=1}^ma_ie_i\mathbb{B}ig\|
$$
for some constant $K$ independent of the scalars $(a_i)_{i=1}^m$.
It is therefore natural to consider also bases (or more generally, systems) satisfying a converse inequality, i.e.
$$
\|x\|\leq K\int_0^1\mathbb{B}ig\|\sum_{i=1}^m r_i(t)e_i^*(x)e_i\mathbb{B}ig\|dt.
$$
These will be called random unconditionally divergent (RUD) and satisfy a natural duality relation with RUC systems. These two notions, weaker than that of unconditional basis, are the central objects for our research in this paper.
The search for bases or more general coordinate systems in Banach spaces is a major theme both within the theory and its applications to other areas (signal processing, harmonic analysis...) A basis allows us to represent a space as a space of sequences of scalars via the coordinate expansion of each element. Several interesting properties for bases have been investigated as they provide better, or more efficient ways, to approximate an element in a Banach space. Recall that a a sequence $(x_n)$ of vectors in a Banach space $X$ is called a basis (or Schauder basis) if every $x\in X$ can be written in a unique way as $x=\sum_{n=1}^\infty a_n x_n$, where $(a_n)$ are scalars. It is well-known that this is equivalent to the fact that the projections $P_n(x)=\sum_{i=1}^n a_i x_i$ are uniformly bounded. Among bases, the unconditional ones play a relevant role, as they provide certain extra structure to the space. A basis $(x_n)$ is called unconditional when the corresponding expansions $\sum_{n=1}^\infty a_n x_n$ converge unconditionally. This is equivalent to the fact that for every choice of signs $\epsilonsilon=(\epsilonsilon_n)$ we have a bounded linear operator $M_{\epsilonsilon}(\sum_{n=1}^\infty a_n x_n)=\sum_{n=1}^\infty \epsilonsilon_na_n x_n$.
There has been considerable interest in finding unconditional basic sequences in Banach spaces. Since the celebrated paper of W. T. Gowers and B. Maurey \cite{Gowers-Maurey}, we know that not every Banach space contains an unconditional basic sequence. In order to remedy this, weaker versions of unconditionality, such as Elton-unconditionality or Odell-unconditionality, have been considered in the literature \cite{Elton, Odell}. RUC and RUD bases also provide a weakening of unconditionality so several questions arise in a natural way. We will study the relation of these two notions with reflexivity in the spirit of the classical James' theorem \cite{James}, we will investigate the uniqueness of RUC (respectively, RUD) bases in a Banach space, as in \cite{LP}, and several other questions related to unconditionality of subsequences and blocks of a given sequence.
Our approach will begin with some probabilistic observations to illustrate the definition of RUC and RUD bases. We will see how the two notions are related by duality and that they also complement each other, in the sense that a basis which is both RUC and RUD must be unconditional. After this grounding discussion, a list of examples of classical bases which are RUC and/or RUD will be given.
Let us point out a major difference with unconditionality: every block-subsequence of an unconditional basis is also unconditional, whereas this stability may fail for RUC and RUD bases. Actually, every separable Banach space can be linearly embedded in a space with an RUC basis (namely, $C[0,1]$). This follows from \cite{Wojtaszczyk} where it is shown that if a space with a basis contains $c_0$, then it has a RUC basis.
This fact also provides a justification for the hypothesis in our version of James reflexivity theorem in this context (Theorem \ref{James_thm}): Suppose that every block-subsequence of a basis $(x_n)$ is RUD, then $(x_n)$ is shrinking if and only if $X$ does not contain a subspace isomorphic to $\ell_1$. Similarly, if every block subsequence of $(x_n)$ is RUC, then $(x_n)$ is boundedly complete if and only if $X$ does not contain a subspace isomorphic to $c_0$.
Another point worth dwelling on is motivated by the classical theorem of J. Lindenstrauss and A. Pe\l czynski: the only Banach spaces with a unique, up to equivalence, unconditional basis are $c_0$, $\ell_1$ and $\ell_2$ \cite[2.b]{LT1}. In this respect, it was shown in \cite{BKPS} that all RUC bases of $\ell_1$ are equivalent, and they must be then unconditional; since it is known that conditional RUC bases of $c_0$ and $\ell_2$ exist, $\ell_1$ stands as the only space with this property. However, the situation for the uniqueness of RUD bases is more involved. Of course, the standard argument leaves $c_0$ as the only possible candidate, nevertheless, using a well-known construction of $\mathcal{L}_\infty$ spaces by J. Bourgain and F. Delbaen \cite{BD}, we will provide a RUD basis of $c_0$ which is not equivalent to the unit vector basis. As a consequence, every Banach space with a RUD basis has another non-equivalent RUD basis.
Let us also recall the first example of a weakly null sequence with no unconditional subsequences due to B. Maurey and H. P. Rosenthal \cite{MR}. It can be seen that this construction also produces an example of a weakly null sequence with no RUD subsequence (Theorem \ref{Maurey-Rosenthal}). Based on this we can provide a weakly null RUC basis without unconditional subsequences (Theorem \ref{MR-RUC}). Using equi-distributed sequences of signs, a modification of Maurey-Rosenthal construction can be given to build a RUD basis without unconditional subsequences (Theorem \ref{MR-RUD}). Moreover, this example also shows that normalized blocks of a RUD basis need not be RUD. Incidentally, the construction of a weakly null sequence in the space $L_1$ without unconditional subsequences given in \cite{JMS} by W. Johnson, B. Maurey and G. Schechtman, can also be taken to be RUD. In fact, it will be shown that on r.i. spaces which are separated from $L_\infty$ (in the sense that the upper Boyd index is finite) every weakly null sequence has an RUD subsequence (Theorem \ref{ri RUD}).
The research on RUC and RUD bases gives rise to a number of natural questions concerning unconditionality in Banach spaces. Among them, the fundamental question of whether every Banach space contains an RUD or an RUC basic sequence remains open.
Throughout the paper we follow standard terminology concerning Banach spaces as in the monographs \cite{LT1, LT2}, and for questions related to probability the reader is referred to \cite{Ledoux-Talagrand} and \cite{Loeve}.
\succeqction{RUC and RUD bases}
\betagin{definition}
A series $\sum_n x_n$ in a Banach space is \emph{randomly unconditionally convergent} when $\sum_n \varepsilon_n x_n$ converges
almost surely on signs $(\varepsilon_n)_n$ with respect to the Haar probability measure on $\{-1,1\}^{\mathbb N}$, or,
equivalently, when the series $\sum_n r_n(t)x_n$ converges almost surely with respect to the Lebesgue measure
on $[0,1]$, where $(r_n(t))_n$ is the \emph{Rademacher} sequence in $[0,1]$.
\itemd{definition}
Since the convergence does not
depend on finitely many changes, it follows from the corresponding 0-1 law that either $\sum_n \varepsilon_n x_n$ converges a.s. or $\sum_n \varepsilon_n x_n$ diverges a.s. (see \cite[pp 7]{Ka} for more details).
Recall the following fact, know as the {\it contraction principle}.
\betagin{propo}
Suppose that $\sum_n r_n(t)x_n$ converges a.s. Then for every sequence $(a_n)_n$, $\sup_n |a_n|\le 1$, one
has that $\sum_n a_n r_n(t)x_n$ also converges a.s.
\qed
\itemd{propo}
Consequently, the sequence $(r_n x_n)_n$ in the Bochner space $L_1([0,1],X)$ is a 1-unconditional basic sequence.
We recall the corresponding expected value
\betagin{equation}\lambdabel{oiioe4joigfofg}
\mathbb{E} \mathbb{B}ig(\mathbb{B}ig\|\sum_{n=1}^m \epsilonsilon_n x_n \mathbb{B}ig\| \mathbb{B}ig)=\frac{1}{2^m}\sum_{(\epsilonsilon_n)\in\{-1,+1\}^m}\mathbb{B}ig\|\sum_{n=1}^m \epsilonsilon_n x_n \mathbb{B}ig\|=\int_0^1\mathbb{B}ig\|\sum_{n=1}^m r_n(t)x_n\mathbb{B}ig\|_Xdt.
\itemd{equation}
It is shown by J. P. Kahane \cite[Theorem 4]{Ka} that if $\sum_n r_n(t) x_n$ converges a.s., then $\mathbb{E} \nrm{\sum_{n}r_n(t) x_n}<\infty$, i.e.
the $X$-vector valued function $\sum_n r_n(t) x_n$ belongs to the \emph{Bochner} space $L_1([0,1],X)$. The converse is also true when $(x_n)_n$ is basic.
\betagin{propo}
Suppose that $(x_n)_n$ is basic and $(a_n)_n$ is a sequence such that the series $\sum_n a_n r_n(t) x_n $
Bochner-converges. Then $\sum_n r_n(t) a_n x_n$ converges almost surely.
\itemd{propo}
\betagin{proof}
Suppose that $(s_n(t))_{n}$ converges to an $X$-valued Bochner measurable function $f$, where $s_n(t):=\sum_{i=1}^n a_i r_i(t) x_i$ for every $n$. This means that $\int_0^1 \nrm{s_n(t)-f(t)}_X\to_n 0$.
Hence, $\nrm{s_n(t)-f(t)}_X\to_n 0$ in probability. It follows that there is a subsequence $\nrm{s_{n_k}(t)-f(t)}_X\to_k 0$ almost surely.
In particular, $(s_{n_k}(t))_k$ is a Cauchy sequence almost surely. We prove that $(s_{n}(t))_n$ is in fact a Cauchy
sequence almost surely: Let
$$A:=\ensuremath{c_{00}(\omega_1)}nj{t\in [0,1]}{(s_{n_k}(t))_k\text{ is a Cauchy sequence}}.$$
By hypothesis, $\lambda(A)=1$. Then $(s_n(t))_n$ is Cauchy for every $t\in A$: Let $C$ be the basic constant
of $(x_n)_n$, and given $\varepsilon>0$, let $k_\varepsilon$ be such that $\nrm{s_{n_k}(t)-s_{n_l}}\le \varepsilon/(2C)$ for every
$k,l \ge k_\varepsilon$. Then, using that $(x_n)_n$ is $C$-basic, if $n_{k_\varepsilon}\le m\le n$, it follows that
\betagin{align*}
\nrm{s_m(t)-s_n(t)}\le 2C\nrm{s_{n_{k_\varepsilon}}(t)- s_{n_l}(t)}\le \varepsilon,
\itemd{align*}
where $l$ is such that $n_l\ge n$. We have just proved that $\sum_n r_n(t) a_n x_n$ converges almost surely to $f(t)$.
\itemd{proof}
A complete account on series of the form $\sum_n \epsilonsilon_n x_n$, also referred to as Rademacher averages, can be found in \cite[Chapter 4]{Ledoux-Talagrand}.
\subsection{Definition and basic properties}
\betagin{definition}
A basic sequence $(x_n)_n$ in a Banach space $X$ is of \emph{Random Unconditional Convergence} (a RUC basis in short)
when every convergent series $\sum_n a_n x_n$ is randomly unconditionally convergent.
A basic sequence $(x_n)_n$ of $X$ is called of \emph{Random Unconditional Divergence} (RUD basis in short) when
whenever a series $\sum_n a_n x_n$ is randomly unconditional, the series $\sum_n a_n x_n$ is convergent, or equivalently,
the only randomly unconditional series $\sum_n a_n x_n$ are the unconditional ones.
\itemd{definition}
It is clear that the definition extends to biorthogonal systems in a natural way. The terminology is justified by the 0-1 law implying that $(x_n)_n$ is RUD if and only if for every
divergent series $\sum_n a_n x_n$ the signed series $\sum_n \varepsilon_n a_n x_n$ diverges almost surely. RUC
bases are those with the maximal number of random unconditionally convergent series, while RUD bases are those with the minimal number of them, only the unconditional ones.
\betagin{propo}
A basic sequence is unconditional if and only if it is RUC and RUD.
\itemd{propo}
\begin{proof}
Suppose that $(x_n)_n$ is a RUC and RUD basic sequence, suppose that $\sum_n a_n x_n$ converges and let $(\rightarrowgma_n)_n$ be a sequence of signs. We have to prove that $\sum_n \rightarrowgma_n a_n x_n$ also converges. Suppose otherwise that $\sum_n \rightarrowgma_n a_n x_n$ diverges. Since $(x_n)_n$ is RUC, it follows that $\sum_n \varepsilon_n\rightarrowgma_n a_n x_n$ diverges a.s. in $(\varepsilon_n)_n$, or equivalently, $\sum_n \varepsilon_n a_n x_n$ diverges a.s. Since $(x_n)_n$ is RUC, it follows that $\sum_n a_n x_n$ diverges, a contradiction.
\end{proof}
RUC sequences
were introduced by P. Billard, S. Kwapie\'n, A. Pelczy\'nski and Ch. Samuel in \cite{BKPS}, where they prove
the following quantitative characterization for RUC biorthogonal systems.
\betagin{propo}\lambdabel{ljejrejriedfd}
For a basic sequence $(x_n)_n$ in $X$ the following are equivalent.
\betagin{enumerate}
\item[(a)] $(x_n)_n$ is RUC.
\item[(b)] There is a constant $C$ such that for every $n\in {\mathbb N}$ and every sequence of scalars $(a_i)_{i=1}^n$ one has that
\betagin{equation}
\lambdabel{ljjgijfgf} \mathbb E \nrm{\sum_{i=1}^n \varepsilon_i a_i x_i}\le C\nrm{\sum_{i=1}^n a_i x_i}.
\itemd{equation}
\itemd{enumerate}
\itemd{propo}
In a similar way, we have the following.
\betagin{propo}
Let $(x_n)_n$ be a basic sequence of $X$. The following are equivalent.
\betagin{enumerate}
\item[(a)] $(x_n)_n$ is RUD.
\item[(b)] There is a constant $C$ such that for every $n\in {\mathbb N}$ and every sequence of scalars $(a_i)_{i=1}^n$ one has that
\betagin{equation}\lambdabel{knlknlknjr5565f}
\nrm{\sum_{i=1}^n a_i x_i}\le C\mathbb E \nrm{\sum_{i=1}^n \varepsilon_i a_i x_i}.
\itemd{equation}
\itemd{enumerate}
\itemd{propo}
\betagin{proof}
Suppose that $(x_n)_n$ is RUD. This implies that $\sum_n a_n x_n$ converges whenever $\sum_n \varepsilon_n a_n x_n$
a.s. converges. Let $Y$ be the closed subspace of the Bochner space $L_1([0,1],X)$ spanned by
$(r_n(t)x_n)_{n\in {\mathbb N}}$. Since $(r_n(t)x_n)_n$ is a 1-unconditional basis of $Y$, for each $n\in {\mathbb N}$, the
linear operator $S_n:Y\to X$ defined by $S_n(\sum_i a_i r_i(t)x_i)=\sum_{i=1}^n a_ix_i$ is well defined and bounded. Now, for
a fixed $y=\sum_i a_i r_i(t)x_i\in Y$ we know by hypothesis that $\sum_i a_i x_i$ converges; since $(x_i)_i$ is a basic sequence, with basic constant $K$, it follows that
\betagin{equation}
\nrm{S_n(y)}=\nrm{\sum_{i=1}^n a_i x_i}\le K
\nrm{\sum_{i=1}^\infty a_i x_i}
\itemd{equation}
for every $n$. Hence, by the Banach-Steinhaus
principle, it follows that that $C:=\sup_n \nrm{S_n}<\infty$, that is,
\betagin{equation}
\nrm{\sum_{i=1}^n a_i x_i}\le C\nrm{\sum_{i=1}^\infty r_i(t) a_i x_i}_{L_1([0,1],X)}.
\itemd{equation}
For a fixed $n$, if we replace $(a_i)_i$ by $(b_i)_i$ where $b_i=a_i$ for $i\le n$ and $b_i=0$ otherwise, we
obtain the inequality in \eqref{knlknlknjr5565f}.
Suppose now that \eqref{knlknlknjr5565f} holds for every $n$ and every $(a_i)_{i=1}^n$. Suppose that
$\sum_{n}a_n x_n$ diverges. If $\sum_{n}\varepsilon_n a_n x_n$ does not diverge a.s., by the 0-1 Law, it converges
a.s. Hence, by Kahane's result, it follows that $\mathbb E_{(\varepsilon_i)}\nrm{\sum_i \varepsilon_i a_i x_i}<\infty$, or equivalently,
$\sum_i a_i r_i(t)x_i $ converges in $L_1([0,1],X)$. It follows that $(\sum_{i=1}^n a_i r_i(t)x_i)_{n}$ is a
Cauchy sequence. Now, the inequality in \eqref{knlknlknjr5565f} implies that $(\sum_{i=1}^na_i x_i)_{n}$ is also
Cauchy, a contradiction.
\itemd{proof}
\betagin{remark}{\rm
\betagin{enumerate}
\item[(a)] Sequences that satisfy the inequality in \eqref{ljjgijfgf} are obviously biorthogonal, and in fact the characterization in Proposition \ref{ljejrejriedfd} is still valid for biorthogonal sequences.
\item[(b)] On the other hand, an arbitrary semi normalized sequence satisfying the inequality in \eqref{knlknlknjr5565f} must have basic subsequences: By applying Rosenthal's $\ell_1$ Theorem to $(r_n(t) x_n)_n$, there are two cases to consider: suppose first that there is a subsequence $(r_n(t) x_n)_{n\in M}$ equivalent to the unit basis of $\ell_1$. It follows then that there is a subsequence $(x_n)_{n\in N}$ equivalent to the unit basis of $\ell_1$ (see Proposition \ref{iuuiuiere}), hence basic. Otherwise, there is a weakly-Cauchy subsequence $(r_n(t) x_n)_{n\in M}$. Since this sequence is 1-unconditional, it must be weakly-null: otherwise, $(r_n(t) x_n)_{n\in M}$ is not weakly-convergent, hence it has a basic subsequence $(r_n(t) x_n)_{n\in N}$ which dominates the summing basis of $c_0$; since $(r_n(t)x_n)_{n\in N}$ is unconditional and bounded, it will be equivalent to the unit basis of $\ell_1$, so it cannot be weakly-Cauchy. Now from the fact that $(r_n(t)x_n)_{n\in M}$ is weakly-null and the inequality in \eqref{knlknlknjr5565f} it follows that $(x_n)_{n\in M}$ is also weakly-null, and consequently it has a further basic subsequence.
\item[(c)]There is a significant difference if almost everywhere convergence of the series $\sum_{i=1}^n \epsilonsilon_i a_i x_i$ is replaced by quasi-everywhere convergence, that is when the set of signs for which the series converges contains a dense $G_\deltalta$. This last condition is equivalent to the unconditionality of the basic sequence $(x_i)_i$, as it has been proved by P. Lefevre in \cite{Lefevre}.
\itemd{enumerate}}
\itemd{remark}
\betagin{defin}\rm
A RUC (RUD) basic sequence $(x_n)_n$ is $C$-RUC ($C$-RUD) when the inequality in \eqref{ljjgijfgf} (resp. \eqref{knlknlknjr5565f}) holds. The corresponding RUC and RUD constants are defined naturally as
\betagin{align*}
\mathrm{RUC}((x_n)_n):= &\inf\{C>0: \|\sum_{i=1}^n a_i e_i\|\ge \frac 1 C \mathbb{E} \mathbb{B}ig(\mathbb{B}ig\|\sum_{i=1}^n \epsilonsilon_i a_i x_i \mathbb{B}ig\| \mathbb{B}ig)\},\\
\mathrm{RUD}((x_n)_n)):= &\inf\{C>0: \|\sum_{i=1}^n a_i x_i\|\leq C\mathbb{E} \mathbb{B}ig(\mathbb{B}ig\|\sum_{i=1}^n \epsilonsilon_i a_i x_i \mathbb{B}ig\| \mathbb{B}ig)\},
\itemd{align*}
where the infimums are taken over all finite sequences $(a_i)_{i=1}^n$ of scalars.
\itemd{defin}
It is also clear from the definition is that if $(e_n)$ is RUC (RUD), then for any choice of scalars
$\lambdambda_n$, the sequence $(\lambdambda_n e_n)$ is also RUC (resp. RUD) (with the same constant).
Since we always have the inequalities
\betagin{equation}
\min_{\tau_n=\pm1}\mathbb{B}ig\|\sum_{n=1}^m \tau_n a_n x_n \mathbb{B}ig\| \leq\mathbb{E} \mathbb{B}ig(\mathbb{B}ig\|\sum_{n=1}^m \epsilonsilon_n a_n x_n \mathbb{B}ig\| \mathbb{B}ig)\leq \max_{\tau_n=\pm1}\mathbb{B}ig\|\sum_{n=1}^m \tau_n a_n x_n \mathbb{B}ig\|
\itemd{equation}
it follows that the RUC and RUD constants, if they exist, are at least 1. In fact, we have the following simple characterizations.
\betagin{propo}
\lambdabel{characterization_ruc}
Let $(x_n)_n$ be a basic sequence. The following are equivalent:
\betagin{enumerate}
\item $(x_n)_n$ is $C$-RUC.
\item For any sequence of scalars $(a_n)_{n=1}^m$ we have
$$
\min_{\tau_n=\pm1}\mathbb{B}ig\|\sum_{n=1}^m \tau_n a_n x_n \mathbb{B}ig\| \leq \mathbb{E} \mathbb{B}ig(\mathbb{B}ig\|\sum_{n=1}^m \epsilonsilon_n a_n x_n \mathbb{B}ig\| \mathbb{B}ig)\leq C
\min_{\tau_n=\pm1}\mathbb{B}ig\|\sum_{n=1}^m \tau_n a_n x_n \mathbb{B}ig\|.
$$
\itemd{enumerate}
Consequently, $(x_n)_n$ is 1-RUC if and only if $(x_n)_n$ is 1-unconditional.
\itemd{propo}
\betagin{propo}
\lambdabel{characterization_sub}
Let $(x_n)_n$ be a basic sequence. The following are equivalent:
\betagin{enumerate}
\item $(x_n)_n$ is $C$-RUD.
\item For any sequence of scalars $(a_n)_{n=1}^m$ we have
$$
\mathbb{E} \mathbb{B}ig(\mathbb{B}ig\|\sum_{n=1}^m \epsilonsilon_n a_n x_n \mathbb{B}ig\| \mathbb{B}ig)\leq \max_{\tau_n=\pm1}\mathbb{B}ig\|\sum_{n=1}^m \tau_n a_n x_n \mathbb{B}ig\|\leq C \mathbb{E} \mathbb{B}ig(\mathbb{B}ig\|\sum_{n=1}^m \epsilonsilon_n a_n x_n \mathbb{B}ig\| \mathbb{B}ig).
$$
\itemd{enumerate}
Consequently, $(x_n)_n$ is 1-RUD if and only if $(x_n)_n$ is 1-unconditional.
\itemd{propo}
In the case of RUC basic sequences, we can always
renorm the space to get RUC-constant as close to one as desired. We do not know if the same is true for RUD basic sequences.
\betagin{propo}
Let $(x_n)$ be a RUC basic sequence in $X$. For every $\deltalta>0$ there is an equivalent norm in $X$ such
that $(x_n)$ is $(1+\deltalta)$-RUC, although there are examples for every $\delta>0$ of $(1+\delta)$-RUD sequences without unconditional subsequences (see Theorem \ref{MR-RUD}).
\itemd{propo}
\betagin{proof}
Without loss of generality, we may assume that $(x_n)_n$ is a basis of $X$. Let $\|\cdot\|$ denote the norm in $X$ such that for some $C>1$
$$
\mathbb{E}\mathbb{B}ig\|\sum_{n}a_n\varepsilon_n x_n\mathbb{B}ig\|\leq C \mathbb{B}ig\|\sum_{n}a_n x_n\mathbb{B}ig\|.
$$
Given $\deltalta>0$, let us define a new norm
$$
\mathbb{B}ig\|\sum_{n}a_n x_n\mathbb{B}ig\|_{\deltalta}=\mathbb{E}\mathbb{B}ig\|\sum_{n}a_n\varepsilon_n x_n\mathbb{B}ig\|+\deltalta\mathbb{B}ig\|\sum_{n}a_n x_n\mathbb{B}ig\|.
$$
It is clear that
$$
\deltalta\|\cdot\|\leq\|\cdot\|_\deltalta\leq(C+\deltalta)\|\cdot\|,
$$
while we have
\betagin{align*}
\mathbb{E}\mathbb{B}ig\|\sum_{n}a_n\varepsilon_n x_n\mathbb{B}ig\|_\deltalta= &\mathbb{E}\mathbb{B}ig(\mathbb{E}\mathbb{B}ig\|\sum_{n}a_n\varepsilon_n x_n\mathbb{B}ig\|+\deltalta\mathbb{B}ig\|\sum_{n}a_n\varepsilon_n x_n\mathbb{B}ig\|\mathbb{B}ig)=(1+\deltalta)\mathbb{E}\mathbb{B}ig\|\sum_{n}a_n\varepsilon_n x_n\mathbb{B}ig\|\leq \\
\leq & (1+\deltalta)\mathbb{B}ig\|\sum_{n}a_n x_n\mathbb{B}ig\|_\deltalta.
\itemd{align*}
\itemd{proof}
The signs-average given above is equivalent (i.e. up to a universal constant) to the following
subsets-average.
$$
\mathbb{E}_0 \mathbb{B}ig(\mathbb{B}ig\|\sum_{n=1}^m \theta_n x_n \mathbb{B}ig\|\mathbb{B}ig)=\frac{1}{2^m}\sum_{(\theta_n)\in\{0,1\}^m}\mathbb{B}ig\|\sum_{n=1}^m \theta_n x_n \mathbb{B}ig\|=\frac{1}{2^m}\sum_{A\subset\{1,\ldots,m\}}\mathbb{B}ig\|\sum_{n\in A}x_n \mathbb{B}ig\|.
$$
More precisely,
$$
\mathbb{E}_0 \mathbb{B}ig(\mathbb{B}ig\|\sum_{n=1}^m \theta_n x_n \mathbb{B}ig\|\mathbb{B}ig) \leq \mathbb{E} \mathbb{B}ig(\mathbb{B}ig\|\sum_{n=1}^m \epsilonsilon_n x_n \mathbb{B}ig\| \mathbb{B}ig) \leq 2 \mathbb{E}_0 \mathbb{B}ig(\mathbb{B}ig\|\sum_{n=1}^m \theta_n x_n \mathbb{B}ig\|\mathbb{B}ig).
$$
It is also natural to consider random versions of symmetric bases. For instance, if $Pi_n$ denotes the group of permutations of $\{1,\dots,n\}$, and we consider a finite basis $(x_i)_{i=1}^n$ and scalars $(a_i)_{i=1}^n$, we can define
$$
\mathbb E_\pi \nrm{\sum_{i=1}^n a_{\pi(i)} x_{i}}:=\frac{1}{n!}\sum_{\pi\in Pi_n}\nrm{\sum_{i=1}^n a_{\pi(i)} x_{i}}.
$$
Hence, we say that a basis $(x_i)$ is of Random Symmetric Convergence (RSC in short) with constant $C$ when for every $n\in\mathbb N$ and scalars $(a_i)_{i=1}^n$
\betagin{equation}
\mathbb E_\pi \nrm{\sum_{i=1}^n a_{\pi(i)} x_{i}}\le C \nrm{\sum_{i=1}^n a_i x_{i}}.
\itemd{equation}
Similarly, $(x_i)$ is of Random Symmetric Divergence
(RSD in short) with constant $C$ when
\betagin{equation}
\nrm{\sum_{i=1}^n a_i x_{i}} \le C \mathbb E_\pi \nrm{\sum_{i=1}^n a_{\pi(i)} x_{i}}
\itemd{equation}
for every choice of $n$ and scalars $(a_i)_{i=1}^n$. The research of these notions will be carried out elsewhere.
Recall that given an integer $k$ and a property $\mc P$ of sequences in a given space $X$ we say that a sequence $(x_n)_n$ has the $k-$skipping property $\mc P$ when every subsequence $(x_{n_i})_{i}$ of $(x_n)_n$ has the property $\mc P$ provided that $n_{i+1}-n_{i}\ge k$.
\betagin{propo}\lambdabel{hohriogihohgfhg}
Let $(x_n)_{n\in I}$ be a basic sequence in $X$, $I$ finite or infinite.
\betagin{enumerate}
\item[(a)] If $(x_n)_n$ is $k$-skipping RUD for some $k\in {\mathbb N}$, then it is RUD.
In fact, suppose that $I=P_1\cup \cdots \cup P_k$ is a partition of $I$ such that each subsequence $(x_n)_{n\in P_i}$ is RUD with constant $C_i$,
$i=1,\dots,k$, then $(x_n)_{n\in I}$ is RUD with constant $\le \sum_{i=1}^kC_i$.
\item[(b)] Suppose that $(x_n)_n$ is a RUC basis of $X$. Then every unconditional subsequence of it generates a complemented
subspace of $X$.
\itemd{enumerate}
\itemd{propo}
\betagin{proof}
(a): Suppose that $\sum_n r_n(t) a_n x_n$ converges a.s. It follows from the contraction principle that each $\sum_{n\in P_i}r_n(t) a_n x_n$, $i=1,\dots,n$, is also convergent a.s. Hence each series $\sum_{n\in P_i} a_n x_n$ converges, $i=1,\dots,n$, and consequently also $\sum_n a_n x_n$ converges. As for the constants: Fix $n$ and scalars $(a_i)_{i=1}^n$. Then
\betagin{align*}
\nrm{\sum_{i=1}^n a_i x_i}\le & \sum_{j=1}^k\nrm{\sum_{i\in P_j\cap \{1,\dots, n\} } a_i x_i } \le \sum_{j=1}^k C_j\mathbb E_{\varepsilon}
\nrm{\sum_{i\in P_j\cap \{1,\dots, n\} } \varepsilon_i a_i x_i }\le \\
\le &\sum_{j=1}^k C_j \mathbb E_{\varepsilon}
\nrm{\sum_{i=1 }^n \varepsilon_i a_i x_i}.
\itemd{align*}
(b): Suppose that $(x_n)_{n\in M}$ is unconditional. We claim that the boolean projection $\sum_{n}a_n x_n
\mapsto \sum_{n\in M }a_n x_n$ is bounded:
\betagin{align*}
\nrm{\sum_{n\in M}a_n x_n}\approx \mathbb E_\varepsilon \nrm{\sum_{n\in M}\varepsilon_n a_n x_n} \le \mathbb E_\varepsilon \nrm{\sum_{n}\varepsilon_n a_n x_n} \lesssim \nrm{\sum_{n} a_n x_n} .
\itemd{align*}
\itemd{proof}
\betagin{corollary}\lambdabel{ufdd}
Suppose $X$ is a Banach space with an unconditional f.d.d. $(F_n)_n$ such that
$$
\sup_n \overline{d}m F_n<\infty.
$$
Then $X$ has a RUD basis.
\itemd{corollary}
\betagin{proof}
Choose for each $n$ a basis $(x_i^{(n)})_{i<k_n}$, $k_n:=\overline{d}m F_n$ with basic constant $\le C$, independent of $n$. Then $(x_i^{(n)})_{i<k_n,n\in {\mathbb N}}$ ordered naturally $(x_j)_j$ is a Schauder basis of $X$, and it is $k$-skipping RUD.
\itemd{proof}
Let us establish now some duality relation between RUC and RUD bases. Recall that a functional $x^*\in X^*$ and a function $f\in L_2(0,1)$ always define an element in
$L_2((0,1),X)^*$ as follows: for any $g\in L_2((0,1),X)$
$$
f\leftarrowimes x^* (g):=\int_0^1 \lambdangle x^*,g(t)\text{Rang }le f(t) dt.
$$
\betagin{propo}\lambdabel{duality} Let $(x_n)_n$ be a basis of $X$.
\betagin{enumerate}
\item If $(x_n)$ is $C$-RUC then every biorthogonal sequence $(x_n^*)$ is $2C$-RUD.
\item If $(x_n^*)_n$ is $C$-RUC, then $(x_n)_n$ is $C\cdot D$-RUD, where $D$ is the basic constant of $(x_n)_n$.
\itemd{enumerate}
\itemd{propo}
\betagin{proof}
Suppose that $(x_n)$ is a RUC basis of the space $X$ with RUC constant $C$, and let $(x_n^*)\subset X^*$ be its sequence of biorthogonal functionals.
Now, fix $\sum_{i=1}^n b_i x_i^*\in X^*$, and let $x=\sum_{i=1}^n a_i x_i$ be such that $\|x\|=1$ and
$$
\sum_{i=1}^n a_i b_i=\lambdangle x, \sum_{i=1}^n b_i x_i^*\text{Rang }le=\|\sum_{i=1}^n b_ix_i^*\|.
$$
Since $(x_n)_n$ is RUC with RUC constant $C$, it follows from Khintchine-Kahane that
$$ \nrm{\sum_{ i=1}^n a_i r_i(t) x_i^*}_{L_2([0,1],X)} \le \sqrt{2}\nrm{\sum_{ i=1}^n a_i r_i(t) x_i^*}_{L_1([0,1],X)}\le \sqrt{2}C\nrm{x}=\sqrt{2}C.$$
Hence,
\betagin{align*}
\nrm{\sum_{i=1}^n b_i r_i(t) x_i^*}_{L_1([0,1],X^*)} \ge & \frac1{\sqrt{2}}
\nrm{\sum_{i=1}^n b_i r_i(t) x_i^*}_{L_2([0,1],X^*)} \ge \\
\ge & \frac1{2C}\lambdangle \sum_{i=1}^n a_i r_i(t) x_i, \sum_{i=1}^n b_i r_i(t) x_i^* \text{Rang }le= \frac1{2C} \sum_{i=1}^n a_i b_i = \\
= & \frac1{2C} \|\sum_{i=1}^n b_ix_i^*\|
\itemd{align*}
Hence, $(x_n^*)$ is RUD with basic constant $\le 2C$.
The proof of (2) is done similarly now observing that the unit sphere of $\lambdangle x_n^*\text{Rang }le_n$ is $1/D$-norming, where $D$ is the basic constant of $(x_n)_n$.
\itemd{proof}
The corresponding duality result for RUD bases is not true in general (see Example \ref{summing}).
We will give now a version of James theorem characterizing shrinking and boundedly complete unconditional
basis in terms of subspaces isomorphic to $\ell_1$ and $c_0$.
\betagin{teore}\lambdabel{James_thm} Let $(x_n)_n$ be a basis of a Banach space $X$.
\betagin{enumerate}
\item Suppose that every block subsequence of $(x_n)$ is RUD. Then $(x_n)$ is shrinking if and only if $X$ does not contain a subspace isomorphic to $\ell_1$.
\item Suppose that every block subsequence of $(x_n)$ is RUC. Then $(x_n)$ is boundedly complete if and only if $X$ does not contain a subspace isomorphic to $c_0$.
\itemd{enumerate}
\itemd{teore}
\betagin{proof}
$(1)$ Clearly, if $X$ contains a subspace isomorphic to $\ell_1$, then $\ell_\infty$ is a quotient of $X^*$.
Thus, $X^*$ is non-separable, and $(x_n)$ cannot be shrinking. Conversely, suppose that $(x_n)$ fails to be
shrinking. This means that for some $\varepsilon>0$ and $x^*\in X^*$ with $\|x^*\|=1$ we can find blocks
$(u_j)$ of the basis $(x_n)$ such that $x^*(u_j)\geq\varepsilon$ for every $j\in\mathbb{N}$. Since $(u_j)$ is
RUD, given scalars $(a_j)_{j=1}^m$ we have
\betagin{align*}
\mathbb{E} \mathbb{B}ig(\mathbb{B}ig\|\sum_{j=1}^m \epsilonsilon_j a_j u_j \mathbb{B}ig\| \mathbb{B}ig)= & \mathbb{E} \mathbb{B}ig(\mathbb{B}ig\|\sum_{j=1}^m \epsilonsilon_j |a_j| u_j \mathbb{B}ig\| \mathbb{B}ig)\geq C \mathbb{B}ig\|\sum_{j=1}^m |a_j| u_j\mathbb{B}ig\|\geq Cx^*\big(\sum_{j=1}^m |a_j| u_j\big)\geq \\
\geq & C\varepsilon\sum_{j=1}^m |a_j|.
\itemd{align*}
Therefore, we have the equivalence
$$
\mathbb{E} \mathbb{B}ig(\mathbb{B}ig\|\sum_{j=1}^m \epsilonsilon_j a_j u_j \mathbb{B}ig\| \mathbb{B}ig)\approx\sum_{j=1}^m |a_j|
$$
which, by a Result of Bourgain in \cite{Bo2} (see also Proposition \ref{iuuiuiere} below), implies that there
is a further subsequence $(u_{j_k})$ equivalent to the unit basis of $\ell_1$.
$(2)$: If $X$ has a subspace isomorphic to $c_0$, then it is easy to see that the basis $(x_n)$ cannot be
boundedly complete. Conversely, let us assume that $(x_n)$ is not boundedly complete. Thus, there exist
scalars $(\lambdambda_n)$ such that
$$
\sup_m\mathbb{B}ig\|\sum_{n=1}^m \lambdambda_nx_n\mathbb{B}ig\|\leq1,
$$
but the series
$$
\sum_{n=1}^\infty \lambdambda_nx_n
$$
does not converge. This means that for some increasing sequence of natural numbers $(p_k)_{k\in\mathbb{N}}$
and some $\varepsilon>0$ we have
$$
u_k=\sum_{j=p_{2k}+1}^{p_{2k+1}}\lambdambda_j x_j,
$$
with $\|u_k\|\geq\varepsilon$, for $k\in\mathbb{N}$. Hence, since $(u_k)$ is a block sequence, then it is
RUC, and we have
$$
\sup_m\int_0^1\mathbb{B}ig\|\sum_{i=1}^m r_i(t)u_i\mathbb{B}ig\|dt=\sup_m \mathbb{E} \mathbb{B}ig(\mathbb{B}ig\|\sum_{i=1}^m \epsilonsilon_i u_i \mathbb{B}ig\| \mathbb{B}ig)\leq C\sup_m \mathbb{B}ig\|\sum_{i=1}^m u_i\mathbb{B}ig\|\leq C<\infty.
$$
By a result of Kwapien in \cite{Kw} (see Theorem \ref{oiio3rere} for more details), $(u_k)$ has a
subsequence equivalent to the unit basis of $c_0$.
\itemd{proof}
\betagin{problem}
Suppose that $(x_n)_n$ is a basis of $X$ such that every block-subsequence of $(x_n)_n$ is RUC (equiv. RUD).
Is $(x_n)_n$ unconditional? More generally, does there exist an unconditional block-subsequence of $(x_n)_n$?
\itemd{problem}
We will see in Section \ref{Sec_RUD_Banach} that there exist conditional basis (namely, the Haar basis in $L_1$) such that every block subsequence is RUD.
\subsection{Examples}
We will present next a list of examples of classical bases in Banach spaces, illustrating the notions of RUC and RUD bases. Let us begin with an example of a basis without RUC nor RUD subsequences.
\betagin{ejemplo}\lambdabel{summing}
The summing basis $(s_n)$ in $c_0$ does not have RUD or RUC subsequences, but its biorthogonal sequence in $\ell_1$ is RUD.
\itemd{ejemplo}
\betagin{proof}
Recall that the $n^\mathrm{th}$ term $s_n$ of the summing basis is the sequence
$$ s_n:=\sum_{i=1}^n u_i=(\overset{(n)}{\overbrace{1,\dots,1}},0,0,\dots),$$
where $(u_n)_n$ is the unit basis of $c_0$. It follows that for any finite subset $s$ of $\mathbb N$ and any
sequence of scalars $(a_i)_{i\in s}$ it holds that
$$
\mathbb{B}ig\|\sum_{i\in s} a_i s_i\mathbb{B}ig\|=\max_{m\in s}\mathbb{B}ig|\sum_{i \in s,\, i\ge m} a_i\mathbb{B}ig|.
$$
We claim that
$$
\mathbb{E}_\varepsilon \mathbb{B}ig(\mathbb{B}ig\|\sum_{i\in s} \epsilonsilon_i a_i s_i \mathbb{B}ig\| \mathbb{B}ig)\approx\mathbb{B}ig(\sum_{i\in s} a_i^2\mathbb{B}ig)^{\frac12}.
$$
Indeed, we have that
$$
\mathbb{E}_\varepsilon \mathbb{B}ig(\mathbb{B}ig\|\sum_{i\in s} \epsilonsilon_i a_i s_i \mathbb{B}ig\| \mathbb{B}ig)=
\int_0^1\max_{m\in s}\mathbb{B}ig|\sum_{i\in s,\, i\ge m} a_ir_i(t)\mathbb{B}ig|dt.
$$
Now, Levy's inequality (cf. \cite[2.3]{Ledoux-Talagrand}, \cite[p. 247]{Loeve}) yields
$$
\mu\{t\in[0,1]:\max_{m\in s}\mathbb{B}ig|\sum_{i\in s,\, i\ge m} a_ir_i(t)\mathbb{B}ig|\geq s\} \leq 2\, \mu\{t\in[0,1]:\mathbb{B}ig|\sum_{i\in s} a_ir_i(t)\mathbb{B}ig|\geq s\}.
$$
Hence, this fact together with Khintchine's inequality give that
\betagin{align*}
\frac1{\sqrt{2}}\mathbb{B}ig(\sum_{i\in s} a_i^2\mathbb{B}ig)^{\frac12} \le & \int_0^1 \mathbb{B}ig|\sum_{i\in s} a_ir_i(t)\mathbb{B}ig|dt \le
\int_0^1\max_{ m\in s}\mathbb{B}ig|\sum_{i\in s,\, i\ge m} a_ir_i(t)\mathbb{B}ig|dt\leq \\
\leq &
2\int_0^1\mathbb{B}ig|\sum_{i\in s} a_ir_i(t)\mathbb{B}ig|dt\leq 2\mathbb{B}ig(\sum_{i\in s}a_i^2\mathbb{B}ig)^{\frac12}.
\itemd{align*}
In particular, there is no constant $K\ge 1$ such that for every finite subset $s$ of a given infinite
$N\subseteq \mathbb N$ we could have
$$
\sharp s=\mathbb{B}ig\|\sum_{i\in s} s_i\mathbb{B}ig\|\leq K \mathbb{E}_\varepsilon \mathbb{B}ig(\mathbb{B}ig\|\sum_{i\in s }^m \epsilonsilon_i s_i \mathbb{B}ig\| \mathbb{B}ig)\leq 2K\sqrt{\sharp s},
$$
and there is no constant $K\ge 1$ such that for every $n_1<\dots <n_k$ in $ N$,
$$
1=\mathbb{B}ig\|\sum_{i=1}^k (-1)^i s_{n_i}\mathbb{B}ig\|\ge \frac1K \mathbb{E}_\varepsilon \mathbb{B}ig(\mathbb{B}ig\|\sum_{i=1 }^k \epsilonsilon_i (-1)^i s_i \mathbb{B}ig\| \mathbb{B}ig)\ge
\frac{1}{K \sqrt{2}}
\sqrt{k}.
$$
The biorthogonal sequence $(s_n^*)_n$ in $\ell_1$ to $(s_n)_n$ is RUD: To see this, notice that
$s_n^*=u_n-u_{n+1}$ for every $n$, where $(u_n)_n$ is the unit basis of $\ell_1$. Hence, for every sequence
of scalars $(a_i)_{i=1}^n$ one has that $\nrm{\sum_{i=1}^n a_i
s_i^*}_1=|a_1|+\sum_{i=1}^{n-1}|a_i-a_{i+1}|+|a_n|$. Consequently,
\betagin{align*}
\mathbb E_\varepsilon \nrm{\sum_{i=1}^n a_i \varepsilon_i s_i^*}_1= |a_1|+|a_n|+\sum_{i=1}^{n-1}\frac{1}{2}( | a_i + a_{i+1}|+|a_i-a_{i+1}|)\ge \sum_{i=1}^n |a_i|.
\itemd{align*}
Since $\nrm{s_n^*}=2$ for every $n$, it follows that
\betagin{equation}
\nrm{\sum_{i=1}^n a_i s_i^*}\le 2\mathbb E_\varepsilon \nrm{\sum_{i=1}^n a_i \varepsilon_i s_i^* }.
\itemd{equation}
\itemd{proof}
Note that proving the conditionality of $(s_n)$ is considerably simpler than showing that it is not RUC nor RUD, for which some probability technology is employed. In this case, Levy's inequality makes the trick, but for slightly more general situations other estimates like H\`ajek-R\'enyi inequality can be helpful \cite{HR}: If $X_1,\ldots, X_n$ are independent centered random variables, $S_k=\sum_{i=1}^k X_i$, and $c_1\geq c_2\geq\ldots\geq c_n\geq0$, then we have
$$
\mu\{\max_{1\leq k\leq n}c_k|S_k|\geq\varepsilon\}\leq\varepsilon^{-2}\int c_n^2S_n^2+\sum_{i=1}^{n-1}(c_i^2-c_{i+1}^2)S_i^2d\mu.
$$
Let us provide now an example of a RUD basis which is not unconditional. Recall first that James space $J$
\cite{James} is the completion of the space of eventually null sequences $c_{00}$ under the norm
$$
\|(a_n)_n\|_J=\sup\{\big(\sum_{k=1}^m(a_{p_k}-a_{p_{k+1}})^2\big)^{\frac12}:p_1<p_2<\cdots<p_{m+1}\}.
$$
\betagin{ejemplo}\lambdabel{james}
The unit vector basis $(u_n)$ of James space $J$ is RUD. In fact, it is a conditional RUD basis whose expected value is the unit basis of $\ell_2$.
\itemd{ejemplo}
\betagin{proof}
Let us consider an arbitrary sequence of scalars $(a_i)_{i=1}^m$ and let $p_1<p_2<\cdots<p_n$ be such that
$$
\|\sum_{i=1}^ma_iu_i\|_J=\big(\sum_{j=1}^n(a_{p_j}-a_{p_{j+1}})^2\big)^{\frac12}.
$$
It follows that
\betagin{equation}
\nrm{\sum_i a_i u_i}_J^2= \sum_{j=1}^n(a_{p_j}-a_{p_{j+1}})^2 \le \sum_{j=1}^n(a_{p_j}^2+a_{p_{j+1}}^2)\le \sum_{i} a_i^2
\itemd{equation}
Hence,
\betagin{equation}\lambdabel{dfsfdsdfddd333ds}
\nrm{\sum_i a_i u_i}_J \le \nrm{\sum_i a_i u_i}_{\ell_2}
\itemd{equation}
On the other hand, if $(u_{n_i})_i$ is such that $n_{i+1}-n_{i}>1$, then it follows that
\betagin{equation}
\nrm{\sum_i a_i u_{n_i}}_J\ge (\sum_i a_i^2)^{1/2}.
\itemd{equation}
Since the unit basis of $\ell_2$ is spreading it follows that every such subsequence is 1-equivalent to the unit basis of $\ell_2$. Hence,
\betagin{equation}
\mathbb E \nrm{\sum_i \varepsilon_i a_i u_{2i}}_J=\mathbb E \nrm{\sum_i \varepsilon_i a_i u_{2i+1}}_J=(\sum_{i} a_i^2)^{1/2}.
\itemd{equation}
Consequently,
\betagin{equation} \lambdabel{dfsssfdsdfddd333ds}
\mathbb E \nrm{\sum_i a_i u_i}_J=
\mathbb E \nrm{\sum_i \varepsilon_i a_{2i} u_{2i}}_J+ \mathbb E \nrm{\sum_i \varepsilon_i a_{2i+1} u_{2i+1}}_J \approx (\sum_i a_i^2)^{1/2}.
\itemd{equation}
Now it follows from \eqref{dfsfdsdfddd333ds} and \eqref{dfsssfdsdfddd333ds} that the unit basis of $J$ is RUD with constant $\sqrt{2}$.
\itemd{proof}
This fact also shows that spaces with RUD bases need not be embeddable into a space with unconditional basis. Note that there is an analogous situation if we replace the role of the $\ell_2$ in the construction of James
space by an unconditional basis.
\betagin{ejemplo}
Let $(x_n)$ be an unconditional basis for the space $X$, and let $J_X$ be the generalized James space, which
is the completion of $c_{00}$ under the norm
$$
\|(a_n)\|_{J_X}=\sup\mathbb{B}ig\{\mathbb{B}ig\|\sum_{k=1}^{m-1}(a_{p_k}-a_{p_{k+1}})x_k\mathbb{B}ig\|_X:p_1<\cdots<p_m\mathbb{B}ig\}.
$$
The unit vector basis $(u_n)$ of $J_X$ is RUD. If $(x_n)$ is not equivalent to the $c_0$-basis, then $(u_n)$
is not unconditional. If in addition the basis $(x_n)_n$ is spreading, then
\betagin{equation}
\mathbb E\nrm{\sum_n a_n u_n}_{J_X}\approx \nrm{\sum_n a_n x_n}_X.
\itemd{equation}
\itemd{ejemplo}
\betagin{proof}
Fix a sequence of scalars $(a_i)_{i=1}^m$ and let $p_1<p_2<\cdots<p_n$ be such that
$$
\|\sum_{i=1}^ma_iu_i\|_{J_X}=\nrm{\sum_{j=1}^n(a_{p_j}-a_{p_{j+1}})x_j}_{X}.
$$
Now let $\tau:\{-1,+1\}^m\rightarrow \{-1,+1\}^m$ be defined in the following way: For
$\Theta=(\theta_i)_{i=1}^m$, let $\tau(\Theta)=(\theta'_i)_{i=1}^m$ be given by
$$
\theta'_i=
\left\{
\betagin{array}{cl}
\theta_i &\textrm{ if }i\notin\{p_1,\ldots,p_{n+1}\} \\
(-1)^j\theta_{p_j} & \textrm{ if } i=p_j.
\itemd{array}
\right.
$$
Now using that
$$
|a_{p_j}-a_{p_{j+1}}|\leq \max\{|\theta_{p_j}a_{p_j}-\theta_{p_{j+1}}a_{p_{j+1}}|, |\theta'_{p_j}a_{p_j}-\theta'_{p_{j+1}}a_{p_{j+1}}|\}
$$
and the fact that $(x_n)_n$ is $C$-unconditional, we have
\betagin{align*}
\|\sum_{i=1}^m a_iu_i\|_{J_X} = & \nrm{\sum_{j=1}^n(a_{p_j}-a_{p_{j+1}})x_j}_X \leq C\nrm{\sum_{j=1}^n(\theta_{p_j}a_{p_j}-\theta_{p_{j+1}}a_{p_{j+1}})x_j}_X+ \\
+& C \nrm{\sum_{j=1}^n(\theta'_{p_j}a_{p_j}-\theta'_{p_{j+1}}a_{p_{j+1}})x_j}_X \le\\
\le & C( \nrm{\sum_{i=1}^ma_i\theta_iu_i }_{J_X}+\nrm{\sum_{i=1}^ma_i\theta'_iu_i}_{J_X}).
\itemd{align*}
Since this holds for every choice of $(\theta_i)_{i=1}^m$ and $\tau$ is an involution
($\tau(\tau(\Theta))=\Theta$), taking averages at both sides gives us
$$
\|\sum_{i=1}^m a_iu_i\|_J \leq 2C\, \mathbb{E} \mathbb{B}ig(\mathbb{B}ig\|\sum_{i=1}^m \theta_i a_i u_i \mathbb{B}ig\|_J \mathbb{B}ig).
$$
Now, to
check that $(u_n)$ is not unconditional, note that for every $k\in\mathbb N$, we have $\|\sum_{n=1}^k
u_n\|_{J_X}=1$, while
$$
\mathbb{B}ig\|\sum_{n=1}^{2k} (-1)^n u_n\mathbb{B}ig\|_{J_X}\geq \mathbb{B}ig\|\sum_{n=1}^k x_n\mathbb{B}ig\|_X.
$$
Hence, if $(u_n)$ were unconditional, then there would be a constant $C>0$ such that $\|\sum_{n=1}^k
x_n\|_X\leq C$. This is imposible because $(x_n)$ is not equivalent to the unit basis of $c_0$.
\itemd{proof}
\betagin{ejemplo}
The well-known twisted sum of $\ell_2$ with $\ell_2$ by N. Kalton and N. Peck \cite{KP} has a natural 2-dimensional unconditional f.d.d. but it does not have an unconditional basis. Hence, by Corollary \ref{ufdd} it has a conditional RUD basis. In general, the non-trivial twisted sum of two spaces with unconditional bases gives also examples of conditional RUD bases.
\itemd{ejemplo}
We will see later (Theorem \ref{ri RUD}) that every block sequence of the Haar system on a rearrangement
invariant space with finite upper Boyd index is RUD. In particular, the Haar basis in $L_1(0,1)$ is another
example of a RUD basis which is not unconditional. We also have the following:
\betagin{ejemplo}
The Walsh basis in $L_1[0,1]$ is RUD.
\itemd{ejemplo}
\betagin{proof}
Recall that the Walsh basis is the canonical extension of the sequence of Rademacher functions $(r_n)$ to an
orthonormal basis of $L_2[0,1]$. Namely, for every finite set $s\subset\mathbb{N}$ we denote
$$
w_s=Pi_{j\in s}r_j.
$$
Since, $(w_s)_s$ are orthonormal in $L_2[0,1]$, it follows that
$$
\mathbb{B}ig\|\sum_s a_sw_s\mathbb{B}ig\|_1\leq\mathbb{B}ig\|\sum_s a_sw_s\mathbb{B}ig\|_2=\mathbb{B}ig(\sum_s a_s^2\mathbb{B}ig)^\frac12.
$$
Now, since $(w_s)_s$ are also normalized in $L_1[0,1]$, and this space has cotype 2, it follows that
$$
\mathbb{E}\mathbb{B}ig\|\sum_s a_s\varepsilon_s w_s\mathbb{B}ig\|_1\gtrsim \mathbb{B}ig(\sum_s a_s^2\mathbb{B}ig)^\frac12\geq \mathbb{B}ig\|\sum_s a_sw_s\mathbb{B}ig\|_1.
$$
Hence, $(w_s)_s$ is RUD.
\itemd{proof}
\betagin{ejemplo}
The Rademacher functions in $BMO[0,1]$ are a RUC basic sequence.
\itemd{ejemplo}
\betagin{proof}
Recall the norm of the space $BMO[0,1]$ is given by
$$
\|f\|_{BMO[0,1]}=\sup_{I\subset[0,1]}\frac{1}{\lambdambda(I)}\int_I \mathbb{B}ig|f-\frac{1}{\lambdambda(I)}\int_I fd\lambdambda\mathbb{B}ig|d\lambdambda,
$$
where $\lambdambda$ denotes Lebesgue's measure on $[0,1]$. It is easy to check that for the Rademacher functions
$(r_n)$ we have
$$
\mathbb{B}ig\|\sum_n a_nr_n\mathbb{B}ig\|_{BMO[0,1]}=\mathbb{B}ig(\sum_n a_n^2\mathbb{B}ig)^\frac12+\sup_n\mathbb{B}ig|\sum_{k=1}^n a_k\mathbb{B}ig|.
$$
Hence, using the computations given in the proof of Example \ref{summing}, we have
$$
\mathbb{E}\mathbb{B}ig\|\sum_n a_n\varepsilon_n r_n\mathbb{B}ig\|_{BMO[0,1]}\leq 3\mathbb{B}ig(\sum_n a_n^2\mathbb{B}ig)^\frac12\leq 3\mathbb{B}ig\|\sum_n a_n r_n\mathbb{B}ig\|_{BMO[0,1]}.
$$
\itemd{proof}
\betagin{ejemplo}
A conditional RUC basis of $\ell_p$ and a conditional RUD basis of $\ell_p$ for $1<p<\infty$.
\itemd{ejemplo}
\betagin{proof}
Let $(x_n)_n$ and $(y_n)_n$ be a Besselian non-Hilbertian, and Hilbertian non-Besselian bases of $\ell_2$,
respectively. Find a sequence of successive intervals $(I_k)_k$ such that $\bigcup_k I_k=\mathbb N$ and that
$(x_i)_{i\in I_k}$ and $(y_i)_{i\in I_k}$ are not $k$-Hilbertian and not $k$-Besselian, respectively. Since
$\lambdangle x_i \text{Rang }le_{i\in I_k}$ and $\lambdangle y_i \text{Rang }le_{i\in I_k}$ are (isometrically) finite dimensional
Hilbert spaces, of dimensions $d_k$ and $l_k$ respectively, and since $(\bigoplus_k\ell_2^{d_k})_{\ell_p},
(\bigoplus_k\ell_2^{l_k})_{\ell_p}$ are isomorphic to $\ell_p$, for $1<p<\infty$, the sequences $(x_i)_{i }$
and $(y_i)_{i}$ are, in the natural ordering, bases of $\ell_p$. On the other hand, given scalars $(a_i)_i$,
one has that
\betagin{align*}
\mathbb E_\varepsilon \nrm{\sum_{k}\sum_{j\in I_k}\varepsilon_j a_j x_j}\approx &(\mathbb E_\varepsilon \nrm{\sum_{k}\sum_{j\in I_k}\varepsilon_j a_j x_j}^p)^{\frac1p}=
(\mathbb E_\varepsilon \sum_{k}(\nrm{\sum_{j\in I_k}\varepsilon_j a_j x_j}_{2})^p)^{\frac1p}=\\
=&( \sum_{k}\mathbb E_\varepsilon (\nrm{\sum_{j\in I_k}\varepsilon_j a_j x_j}_{2})^p)^{\frac1p}\approx
(\sum_{k} (\mathbb E_\varepsilon \nrm{\sum_{j\in I_k}\varepsilon_j a_j x_j}_{2})^p)^{\frac1p}=\\
=&(\sum_k (\sum_{j\in I_k}a_j^2)^{\frac p2})^{\frac1p}
\itemd{align*}
and similarly
\betagin{align*}
\mathbb E_\varepsilon \nrm{\sum_{k}\sum_{j\in I_k}\varepsilon_j a_j y_j}\approx &(\sum_k (\sum_{j\in I_k}a_j^2)^{\frac p2})^{\frac1p}
\itemd{align*}
Hence, since $(x_n)_n$ is a Besselian basis of $\ell_2$ , it follows that
\betagin{align*}
\mathbb E_\varepsilon \nrm{\sum_{k}\sum_{j\in I_k}\varepsilon_j a_j x_j}\approx &(\sum_k (\sum_{j\in I_k}a_j^2)^{\frac p2})^{\frac1p} \lesssim \nrm{\sum_{k}\sum_{j\in I_k} a_j x_j}
\itemd{align*}
So, $(x_i)_i$ is a conditional RUC basis of $\ell_p$. And since $(y_n)_n$ is a Hilbertian basis of $\ell_2$,
it follows that
\betagin{align*}
\mathbb E_\varepsilon \nrm{\sum_{k}\sum_{j\in I_k}\varepsilon_j a_j x_j}\approx &(\sum_k (\sum_{j\in I_k}a_j^2)^{\frac p2})^{\frac1p} \gtrsim \nrm{\sum_{k}\sum_{j\in I_k} a_j x_j}
\itemd{align*}
So, $(y_i)_i$ is a conditional RUD basis of $\ell_p$.
\itemd{proof}
There are further examples that have been considered in the literature. For instance, in \cite{KS} it is shown that the Olevskii system, an orthonormal system which is simultaneously a basis in $L_1[0,1]$ and a basic sequence in $L_\infty[0,1]$, forms an RUC basis in $L_p[0,1]$ if and only if $2\leq p<\infty$. In fact, this is an RUC basis of every rearrangement invariant (r.i.) space $X$ with finite cotype and upper Boyd index $\betata_X<1/2$ \cite[Theorem 1]{KS}. These results are extended in \cite{DSS} where the authors study conditions for an r.i. space to have a complete orthonormal uniformly bounded RUC system.
In the non-commutative setting there are also interesting examples of RUC bases. For instance, in the space $C^p$ (compact operators $a:\ell_2\rightarrow\ell_2$ such that $\rightarrowgmama_p(a)=(tr(aa^*)^{p/2})^{1/p}<\infty$) it is well-known that the canonical basis $(e_n\leftarrowimes e_m)_{n,m=1}^\infty$ is not unconditional for $p\neq2$. However, for $2\leq p<\infty$, $(e_n\leftarrowimes e_m)_{n,m=1}^\infty$ is a RUC basis \cite[Theorem 3.1]{BKPS}. Hence, by Proposition \ref{duality} and the duality between $C^p$ and $C^{p/p-1}$, it follows that for $1<p\leq2$, $(e_n\leftarrowimes e_m)_{n,m=1}^\infty$ is a RUD basis (which of course cannot be RUC). Surprisingly enough, in \cite{Garling-Tomczak} it was shown that the space $C^p$ also has a RUC basis for $1\leq p\leq2$.
More examples in the non-commutative context can be found in \cite{DS}. Also, in \cite{Witvliet}, the connection between R-boundedness, UMD spaces and RUC Schauder decompositions is explored.
\succeqction{Uniqueness of bases}
Another point worth dwelling on is the uniqueness of RUD or RUC basis on some Banach spaces. Concerning
unconditionallity, it is well known that the only Banach spaces with a unique unconditional basis (up to
equivalence) are $\ell_1$, $\ell_2$ and $c_0$ (cf. \cite{LT1}). Using \cite[Prop. 2.1]{BKPS}, one can see
that every RUC basis in $\ell_1$ must be equivalent to the unit vector basis (See Theorem \ref{iueiuhuitrtr}
below).
Note also that there are RUC basis of $c_0$ which are not RUD (see \cite[Prop. 2.2]{BKPS}, or use the
construction of \cite{Wojtaszczyk} starting with the summing basis of $c_0$).
In $\ell_2$ we can find bases which are RUD but not RUC, or viceversa. Indeed, for every basis $(e_n)$ in
$\ell_2$, using the parallelogram law we know that
$$
\mathbb{E} \mathbb{B}ig\|\sum_{i=1}^m \epsilonsilon_i a_i e_i \mathbb{B}ig\|^2 =\sum_{i=1}^m a_i^2.
$$
\betagin{definition}
A basis $(x_n)_n$ is called Besselian if there is a constant $K>0$ such that
\betagin{equation}
(\sum_n a_n^2)^{\frac12} \le K\nrm{\sum_n a_n x_n} \text{ for every sequence of scalars $(a_n)_n$.}
\itemd{equation}
A basis $(x_n)_n$ is called Hilbertian if there is a constant $K>0$ such that
\betagin{equation}
\nrm{\sum_n a_n x_n}\le K (\sum_n a_n^2)^{\frac12} \text{ for every sequence of scalars $(a_n)_n$.}
\itemd{equation}
\itemd{definition} Thus, every non-Besselian (respectively non-Hilbertian) basis of $\ell_2$ is not RUC (resp.
RUD). A combination of a non RUD basis with a non RUC one yields a basis of $\ell_2$ which fails both
properties.
\betagin{teore}[P. Billard, S. Kwapie\'n, A. Pelczy\'nski and Ch. Samuel \cite{BKPS}] \lambdabel{iueiuhuitrtr}
Every RUC basis of $\ell_1$ is equivalent to the unit basis of $\ell_1$.
\itemd{teore}
\betagin{proof}
Fix a RUC basis $(x_n)_n$ of $\ell_1$ with constant $C$. Let $(x_n^*)_n$ be the biorthogonal sequence to
$(x_n)_n$. Let $K$ be the cotype constant of $\ell_1$. Define the operator $T:L_1([0,1],\ell_1)\to \ell_2$
defined by
$$ T(f):=\sum_{n=1}^\infty (\int_0^1 x_n^*(f(t))r_n(t) dt)u_n $$
for every $f\in L_1([0,1],\ell_1)$. It is well-defined and bounded:
\betagin{align*}
\nrm{T(f)}_2 = &\left( \sum_n \left(\int_0^1 x_n^*(f(t)) r_n(t)dt \right)^2 \right)^\frac12\le \int_0^1 \left( \sum_n \left( x_n^*(f(t)) r_n(t)\right)^2dt \right)^\frac12 = \\
=& \int_0^1 \left( \sum_n \left( x_n^*(f(t)) \right)^2dt \right)^\frac12 \le K \int_0^1 \mathbb E_\varepsilon \nrm{\sum_n \varepsilon_n x_n^*(f(t)) x_n}dt \le \\
\le & C\cdot K \int_0^1 \nrm{\sum_n x_n^*(f(t)) x_n}dt = C\cdot K \int_0^1 \nrm{f(t)}dt =C\cdot K \nrm{f}
\itemd{align*}
Since $L_1([0,1],\ell_1)$ is a $\mathcal L_1$-space, it follows that the operator $T$ is absolutely summing,
with absolutely summing constant $K_G\nrm{T}$. It follows that for every sequence of scalars $(a_i)_{i=1}^n$
one has that
\betagin{align*}
\sum_{i=1}^n |a_i|=&\sum_{i=1}^n \nrm{T(a_i r_i(\cdot) x_i)}\le K_G \nrm{T} \max_{\varepsilon} \nrm{\sum_{i=1}^n \varepsilon_i a_i r_i(\cdot) x_i}\le\\
\le & K_G \cdot C \cdot K \nrm{\sum_{i=1}^n a_i r_i(\cdot) x_i}
\le K_G \cdot C^2 \cdot K \nrm{\sum_{i=1}^n a_i x_i}.
\itemd{align*}
\itemd{proof}
\betagin{corollary}
A Banach space has a unique (up to equivalence) RUC basis if an only if it is isomorphic to $\ell_1$.
\itemd{corollary}
\betagin{proof}
The previous Theorem \ref{iueiuhuitrtr} proves that $\ell_1$ has a unique RUC basis. Suppose now that $X$ is
a space with the same property. Fix a RUC basis $(x_n)_n$ of $X$. It follows that $(\varepsilon_n x_n)_n$ is a RUC
sequence of $X$ for every sequence $(\varepsilon_n)_n$ of signs. Hence, by hypothesis, it is equivalent to $(x_n)_n$
a simple uniform boundedness principle shows that there is a constant $K$ such that
$$\nrm{\sum_n a_n x_n}\le K \nrm{\sum_n \varepsilon_n a_n x_n}$$
for every sequence of scalars. Hence, $(x_n)_n$ is the unique unconditional basis of $X$. It follows then
that $X$ is isomorphic to either $c_0$, $\ell_1$ or $\ell_2$. We have already said that $c_0$ and $\ell_2$
have conditional RUC bases.
\itemd{proof}
Theorem \ref{iueiuhuitrtr} also motivates the following question: Is every basis of $\ell_1$ a RUD basis? It is not hard to check that every triangular basis of $\ell_1$ is RUD (in particular, every Bourgain-Delbaen basis of $\ell_1$ is RUD). The same question for $L_1$ is also open.
\subsection{Uniqueness of RUD bases}
\betagin{teore}
Every Banach space with an RUD basis has two non-equivalent RUD bases.
\itemd{teore}
The proof has two parts.
\betagin{lem}
Suppose that $X$ is a space with an RUD basis and not isomorphic to $c_0$. Then $X$ has two non-equivalent
RUD bases.
\itemd{lem}
\betagin{proof}As in the proof of the previous corollary such
space has a unique unconditional basis; hence it must be isomorphic to $c_0$, $\ell_1$ or $\ell_2$. It
cannot be $c_0$ by hypothesis, or $\ell_2$ as this space has a Hilbertian conditional basis; in $\ell_1$ the
sequence $(x_n)_n$ defined by $x_0=u_0$, $x_{n+1}=u_{n+1}-u_n$ is a conditional basis of $\ell_1$ such that
$\mathbb E_\varepsilon\nrm{\sum_n a_n x_n}\approx \sum_n |a_n|$, hence RUD.
\itemd{proof}
The next is the key result
\betagin{lem}\lambdabel{iuuirtgbjbjkgf322}
$c_0$ has two non-equivalent RUD bases.
\itemd{lem}
The proof of this Lemma is based on the Bourgain-Delbaen construction of $\mathcal L_\infty$ spaces with the Schur property in \cite{Bourgain-Lp, BD}, and we follow the exposition and notation of \cite{Haydon}. In fact, the authors construct for arbitrarily large $n$ a basis $(d_i)_{i=1}^n:=(d_i^{(n)})_{i=1}^n$ of $\ell_\infty^n$ a partition $A\cup B\cup C=\{1,\dots,n\}$ and a constant $K$ independent on $n$ such that
\betagin{enumerate}
\item[(i)] $(d_i)_{i\in A}$, $(d_i)_{i\in B}$ and $(d_i)_{i\in C}$ are $K$-equivalent to the unit basis of $(\oplus_{j=1}^k \ell_\infty^{n_j})_{\ell_1}$, $(\oplus_{j=1}^l \ell_\infty^{m_j})_{\ell_1}$ and of $\ell_\infty(r)$ respectively.
\item[(ii)] $k$ and $l$ grow to infinity as $n$ grows to infinite.
\itemd{enumerate}
It follows then from (i) that the basis $(d_i)_{i=1}^n$ is at most $K$-equivalent to the unit basis of $\ell_\infty^n$. Now the canonical basis $(d_i)_i$ of $c_0=(\oplus_n \ell_\infty^n)_{c_0}$ extending each $(d_i^{(n)})_{i=1}^n$ cannot be, by the condition (ii), equivalent to the unit basis of $c_0$. On the other hand, it will follow from Proposition \ref{hohriogihohgfhg} that $(e_i)_i$ is RUD.
We will begin by recalling the badly unconditional RUD-bases of $\ell_\infty^n$. Fix $\lambda>1$, and $b<1/2$ such that
$$ 1+2b\lambda\le \lambda.$$
Let ${\mathcal D}elta_0:=\{0\}$; Suppose defined ${\mathcal D}elta_{n}$, and set $\Gamma_n:=\bigcup_{k\le n}{\mathcal D}elta_n$. Let ${\mathcal D}elta_{n+1}$ be
the collection of all quintuples $(m,\varepsilon_0,\varepsilon_1,\rightarrowgma_0,\rightarrowgma_1)$ such that $\varepsilon_0,\varepsilon_1\in \{-1,1\}$,
$\rightarrowgma_0\in \Gamma_m$ and $\rightarrowgma_1\notin \Gamma_m$. Let $\Gamma_{n+1}:=\Gamma_n\cup {\mathcal D}elta_{n+1}$. For every $n$, fix a total
ordering $\prec_n$ of the finite set ${\mathcal D}elta_{n}$, and let $\prec^n$ be the total ordering on $\Gamma_n$ extending
the fix orderings $\prec_n$ on each ${\mathcal D}elta_m$, and such that each element of ${\mathcal D}elta_{m}$ is strictly smaller than
each element of ${\mathcal D}elta_{m+1}$.
We define vectors $(d_{\rightarrowgma}^*)_{\rightarrowgma\in \Gamma_{n}}\subset \ell_1(\Gamma_{n+1})$ and $(d_{\rightarrowgma}^{(n)})_{\rightarrowgma\in
\Gamma_{n}}\subset \ell_\infty(\Gamma_n)$ with the following properties:
\betagin{enumerate}
\item[(a)] $(d_{\rightarrowgma}^*,d_{\rightarrowgma}^{(n)})_{\rightarrowgma\in \Gamma_n}$ is a biorthogonal
sequence with $1\le \nrm{d_{\rightarrowgma}^*}_{\ell_1}, \nrm{d_{\rightarrowgma}^{(n)}}_\infty\le \lambda$.
\item[(b)] $(d_\rightarrowgma^*)_{\rightarrowgma\in \Gamma_n}$, ordered by $\prec_n$, is a Schauder basis of $\ell_1(\Gamma_n)$ with basis constant $\le \lambda$.
\itemd{enumerate}
The construction of $(d_\rightarrowgma^*,d_{\rightarrowgma}^{(n)})$, $\rightarrowgma\in \Gamma_n$, is done inductively on $n$: For $n=0$, let
$d_{0}^*=d_{0}^{(0)}:=u_0$. Suppose all done for $n$, and for each $m\le n$, let $P_m^*: \ell_1(\Gamma_n)\to
\ell_1(\Gamma_m)$ be the canonical projection
$$P_m^*:=\sum_{\tau \in \Gamma_m} d_\tau^{(n)}\leftarrowimes d_\tau^*$$
of norm $\le\lambda$ associated to the basis $(d_\tau^*)_{\tau\in \Gamma_n}$. For $\rightarrowgma\in {\mathcal D}elta_{n+1}$,
$\rightarrowgma=(m,\varepsilon_0,\varepsilon_1,\rightarrowgma_0,\rightarrowgma_1)$, let
\betagin{align*}
d_\rightarrowgma^*:= & u_\rightarrowgma^*-c_{\rightarrowgma}^* \\
c_\rightarrowgma^*:= & \varepsilon_0 u_{\rightarrowgma_0}^*+ \varepsilon_1 b(u_{\rightarrowgma_1}^*-P_{m}^* u_{\rightarrowgma_1}^*)\\
d_\rightarrowgma^{(n+1)}:= & u_{\rightarrowgma}.
\itemd{align*}
Let $\tau\in \Gamma_{n}$. Then let
\betagin{align*}
d_\tau^{(n+1)}:=& d_\tau^{(n)}+\sum_{\rightarrowgma\in {\mathcal D}elta_{n+1}}\lambdangle c_{\rightarrowgma}^*, d_\tau^{(n)} \text{Rang }le u_\rightarrowgma^*.
\itemd{align*}
Observe that for $\rightarrowgma\in {\mathcal D}elta_m$,
\betagin{equation}\lambdabel{nmfngjfngd}
d_\rightarrowgma^{(n)}\upharpoonright {\mathcal D}elta_m=u_\rightarrowgma.
\itemd{equation}
For each $m\le n$, let $D_m^{(n)}:=\lambdangle d_{\rightarrowgma}^{(n}\text{Rang }le_{\rightarrowgma\in {\mathcal D}elta_m}$.
\betagin{propo}\lambdabel{ioiuohoi4h543fdbf} For every $m\le n$ and every sequence of scalars $(a_\rightarrowgma)_{\rightarrowgma\in {\mathcal D}elta_m}$ one has that
\betagin{equation}\lambdabel{jnjkruiuiee}
\max_{\rightarrowgma\in {\mathcal D}elta_m} |a_\rightarrowgma|\le\nrm{\sum_{\rightarrowgma \in {\mathcal D}elta_m}a_\rightarrowgma d_\rightarrowgma^{(n)}\upharpoonright {\mathcal D}elta_m }_\infty \le \nrm{\sum_{\rightarrowgma \in {\mathcal D}elta_m}a_\rightarrowgma d_\rightarrowgma^{(n)} }_\infty\le \lambda\max_{\rightarrowgma\in {\mathcal D}elta_m} |a_\rightarrowgma|.
\itemd{equation}
\itemd{propo}
\betagin{proof}
Set $x:=\sum_{\rightarrowgma \in {\mathcal D}elta_m}a_\rightarrowgma d_\rightarrowgma^{(n)}$. It follows from \eqref{nmfngjfngd} that $(x)_\rightarrowgma= a_\rightarrowgma$
for every $\rightarrowgma\in {\mathcal D}elta_m$; hence, $\nrm{x\upharpoonright {\mathcal D}elta_m}_\infty\ge \max_{\rightarrowgma\in {\mathcal D}elta_m}|a_\rightarrowgma|$.
Let $\tau\in {\mathcal D}elta_k$ with $k\le n$. If $k<m$, then $ (x)_\tau= 0$ because $(d_\rightarrowgma^{(n)})_\tau=0$ because
$(d_\rightarrowgma^{(n)})_\tau=0$ for every $\rightarrowgma\in {\mathcal D}elta_m$. If $k=m$, then $(x)_\tau=a_\tau$ because of
\eqref{nmfngjfngd}. Suppose that $m<k\le n$. We prove by induction on $k$ that
\betagin{equation}
|(x)|_\tau\le \lambda\max_{\rightarrowgma\in {\mathcal D}elta_m}|a_\rightarrowgma|.
\itemd{equation}
$\tau=(l,\varepsilon_0,\varepsilon_1,\tau_0,\tau_1)$ with $l<k$. Then
$$(x)_\tau= \lambdangle d_\tau^*+c_\tau^*,x\text{Rang }le=\lambdangle c_\tau^*, x\text{Rang }le.$$
Suppose first that $l< m$. Then
\betagin{align*}
|\lambdangle c_\tau^*, x\text{Rang }le|\le & | (x)_{\tau_0}|+ b |((I_n-P_{l,n})x)_{\tau_1}|= b |(x)_{\tau_1}|\le \lambda b \max_{\rightarrowgma\in {\mathcal D}elta_m}|a_\rightarrowgma| \le \lambda \max_{\rightarrowgma \in {\mathcal D}elta_m} |a_\rightarrowgma|.
\itemd{align*}
If $l\ge m$, then
\betagin{align*}
|\lambdangle c_\tau^*, x\text{Rang }le|= & |(x)_{\tau_0}|+ 0 \le \lambda\max_{\rightarrowgma\in {\mathcal D}elta_m}|a_\rightarrowgma|.
\itemd{align*}
\itemd{proof}
\betagin{propo} \lambdabel{ioiodsdggfss}
Let $m_0<m_1<\cdots<m_{l}<n$. Then
$$\frac1\lambda\nrm{\sum_{i=0}^l\sum_{\rightarrowgma\in {\mathcal D}elta_{m_i}}a_\rightarrowgma d_{\rightarrowgma}^{(n)} }_\infty \le
\sum_{i=0}^l\max_{\rightarrowgma\in {\mathcal D}elta_m}|a_\rightarrowgma| \le \frac{1}{b} \nrm{\sum_{i=0}^l\sum_{\rightarrowgma\in {\mathcal D}elta_{m_i}}a_\rightarrowgma d_{\rightarrowgma}^{(n)} }_\infty, $$
for every sequence of scalars $(a_\rightarrowgma)_{\rightarrowgma\in \bigcup_{i\le l}{\mathcal D}elta_{m_i}}$.
\itemd{propo}
\betagin{proof}
The first inequality: Using \eqref{jnjkruiuiee} in Proposition \ref{ioiuohoi4h543fdbf},
\betagin{align*}
\nrm{\sum_{i=0}^l\sum_{\rightarrowgma\in {\mathcal D}elta_{m_i}}a_\rightarrowgma d_{\rightarrowgma}^{(n)} }_\infty \le \sum_{i=0}^l\nrm{\sum_{\rightarrowgma\in {\mathcal D}elta_{m_i}}a_\rightarrowgma d_{\rightarrowgma}^{(n)} }_\infty \le \lambda \sum_{i=0}^l\max_{\rightarrowgma\in {\mathcal D}elta_{m_i}}|a_\rightarrowgma|.
\itemd{align*}
For the second inequality: For each $i\le l$, let $\rightarrowgma_i\in {\mathcal D}elta_{m_i}$ and $\varepsilon_i\in \{-1,1\}$ be such
that $ \varepsilon_i a_{\rightarrowgma_i}=\max_{\rightarrowgma\in {\mathcal D}elta_{m_i}}|a_\rightarrowgma|$. We also suppose that $l\geq 1$, since otherwise
there is nothing to prove. For each $0<i\le l$ We define recursively $\tau_i\in {\mathcal D}elta_{m_i+1}$ as follows.
Let $\tau_1:= (m_0,\varepsilon_0,\varepsilon_1,\rightarrowgma_0,\rightarrowgma_1)\in {\mathcal D}elta_{m_1+1} $. Let
$\tau_2:=(m_1,1,\varepsilon_2,\tau_1,\rightarrowgma_2){\mathcal D}elta_{m_2+1}$; in general, let $\tau_{i}:=(m_{i-1}
,1,\varepsilon_{m_i},\tau_{i-1},\rightarrowgma_{i})$. Set $x_{i}:=\sum_{\rightarrowgma\in {\mathcal D}elta_{m_i}}a_\rightarrowgma d_{\rightarrowgma}^{(n)}$ for each
$i\le l$, and $x:=\sum_{i=0}^lx_i$. Let us prove inductively that for every $0<i\le l$ one has that
$$ (x)_{\tau_{i}}= |a_{\rightarrowgma_0}|+b\sum_{j=1}^{i}|a_{\rightarrowgma_{j}}|= \max_{\rightarrowgma\in {\mathcal D}elta_{m_0}}|a_{\rightarrowgma}|+b \sum_{j=1}^{i}\max_{\rightarrowgma\in {\mathcal D}elta_{m_j}}|a_{\rightarrowgma}|: $$
Suppose that $i=1$. Then, using that $\tau_{1}\in {\mathcal D}elta_{m_1+1}$ implies that $\lambdangle
d_{\tau_1}^*,x\text{Rang }le=0$, it follows that
\betagin{align*}
(x)_{\tau_{1}}=& \lambdangle u_{\tau_1}^*,x\text{Rang }le=\lambdangle d_{\tau_1}^*+c_{\tau_1}^*,x\text{Rang }le =\lambdangle c_{\tau_1}^*,x\text{Rang }le= \varepsilon_0 (x)_{\rightarrowgma_0}+ b \varepsilon_1 (x-P_{m_0}x)_{\rightarrowgma_1}= \\
= & |a_{\rightarrowgma_0}|+ b\varepsilon_1( \sum_{j=0}^lx_j)_{\rightarrowgma_1}= |a_{\rightarrowgma_0}|+ b\varepsilon_1( x_1)_{\rightarrowgma_1}= |a_{\rightarrowgma_0}|+ b|a_{\rightarrowgma_1}|.
\itemd{align*}
Suppose that $\tau_{i}\in {\mathcal D}elta_{m_i+1}$ is such that
$(x)_{\tau_{i}}=|a_{\rightarrowgma_0}|+b\sum_{j=1}^{i}|a_{\rightarrowgma_{j}}|$. Then,
\betagin{align*}
(x)_{\tau_{i+1}}=& \lambdangle u_{\tau_{i+1}}^*,x\text{Rang }le=\lambdangle d_{\tau_{i+1}}^*+c_{\tau_{i+1}}^*,x\text{Rang }le =\lambdangle c_{\tau_{i+1}}^*,x\text{Rang }le= (x)_{\tau_{i}}+ b \varepsilon_{i+1} (x-P_{m_{i+1}}x)_{\rightarrowgma_{i+1}}= \\
= & |a_{\rightarrowgma_0}|+ b\sum_{j=1}^{i}|a_{\rightarrowgma_{j}}|+\varepsilon_{i+1}b( x_{i+1})_{\rightarrowgma_{i+1}}= |a_{\rightarrowgma_0}|+ b\sum_{j=1}^{i+1}|a_{\rightarrowgma_{j}}|.
\itemd{align*}
\itemd{proof}
\betagin{propo}
The basis $(d_\rightarrowgma^{(n)})_{\rightarrowgma\in \Gamma_n}$ of $\ell_\infty(\Gamma_n)$ is RUD with constant $\le \lambda(2/b+1)$.
\itemd{propo}
\betagin{proof}
Let $A:=\bigcup_{m<n,m\text{ even}}{\mathcal D}elta_m$, $B:=\bigcup_{m<n, m\text{ odd}}{\mathcal D}elta_m$ and $C:={\mathcal D}elta_n$. Then, by
Proposition \ref{ioiodsdggfss}, $(d_\rightarrowgma^{(n)})_{\rightarrowgma \in A}$ and $(d_\rightarrowgma^{(n)})_{\rightarrowgma \in B}$ are $\lambda/b$
equivalent to the unit vector basis of $(\sum_{m\in A}\ell_\infty({\mathcal D}elta_m))_{\ell_1}$ and of $(\sum_{m\in
B}\ell_\infty({\mathcal D}elta_m))_{\ell_1}$ respectively. Since these two unit vector bases are 1-unconditional, the
subsequences $(d_\rightarrowgma^{(n)})_{\rightarrowgma \in A}$ and $(d_\rightarrowgma^{(n)})_{\rightarrowgma \in B}$ are unconditional with
constant $\le \lambda/b$. Also, it follows from Proposition \ref{ioiuohoi4h543fdbf} that $(d_\rightarrowgma^{(n)})_{\rightarrowgma
\in C}$ is $\lambda$-equivalent to the unit vector basis of $\ell_\infty(C)$, hence unconditional with constant
$\le \lambda$. The desired result follows from Proposition \ref{hohriogihohgfhg} (1).
\itemd{proof}
We are ready to prove Lemma \ref{iuuirtgbjbjkgf322}.
\betagin{proof}
For each $n$ let $\Gamma_n$ be the finite sets defined above, and let $\Gamma:=\bigcup_n \Gamma_n$, disjoint union.
Then $(\sum_{n\in {\mathbb N}} \ell_\infty(\Gamma_n))_\infty$ is isometric to $c_0(\Gamma)$, which in turn is isometric to
$c_0$. We order $\Gamma$ canonically by first consider the total ordering $\prec_n$ as above and then declaring
that each element of $\Gamma_m$ strictly smaller than each element of $\Gamma_n$ for $m<n$. Then
$(d_\rightarrowgma^{(n)})_{n\in {\mathbb N},\rightarrowgma\in \Gamma_n}$ is a basis of $(\sum_{n\in {\mathbb N}} \ell_\infty(\Gamma_n))_\infty$ which is
RUD with constant $\le\lambda(2/b+1)$. On the other hand, this basis has arbitrary long subsequences
$\lambda/b$-equivalent to the unit vector basis of $\ell_1$, hence it cannot be equivalent to the unit vector
basis of $c_0$.
\itemd{proof}
Note this construction also provides an example of a basis $(x_n)$ such that both $(x_n)$ and its biortogonal functionsl $(x_n^*)$ are RUD, but $(x_n)$ is not unconditional.
\succeqction{RUC, RUD and unconditional bases}\lambdabel{nounc}
It is not true that every basic sequence has a RUC or a RUD subsequence as the summing basis of $c_0$ shows. However, it is well-known that weakly-null sequences have always subsequences with some sort of partial unconditionality such as Elton's or Odell's unconditionality (see \cite{Elton}, \cite{Odell}). It is natural then to ask if weakly-null sequences have subsequences with partial random unconditionality RUC or RUD. We are going to prove that the Maurey-Rosenthal example of a weakly-null basis without unconditional subsequences has the stronger property of not having RUD subsequences.
Secondly, we will see that RUC or RUD basic sequences do not necessarily have unconditional subsequences. Interestingly, the Johnson, Maurey, and Schechtman example of a weakly-null sequence in $L_1[0,1]$ without unconditional subsequences have a RUD subsequence as this is the case not only for $L_1[0,1]$ but also for many rearrangement invariant spaces on $[0,1]$ (see Theorem
\ref{ri RUD}). Observe that this subsequence gives an example of a weakly-null sequence without RUC subsequences. And a simple modification of the Maurey-Rosenthal example gives a RUC sequence without unconditional subsequences.
Finally, we will give an example of a RUD sequence that has a non-RUD block-subsequence; the analogue for RUC sequences can be found by taking a RUC basis of $C[0,1]$, that always exist by a result of Wojtaszczyk in \cite{Wojtaszczyk}.
Let us first introduce some useful notation, which we will use to introduce not only the Maurey-Rosenthal example but also the ulterior examples. Given any finite set $s\subset\mathbb{N}$ of even cardinality,
let
$$
\mathcal{E}(s)=\{(\varepsilon_i)_{i\in s}\in\{-1,1\}^s:\sharp\{i\in s:\varepsilon_i=1\}=\sharp\{i\in s:\varepsilon_i=-1\}\}.
$$
This set consists of all equi-distributed signs indexed on a given set $s$. Let
$k_m=\sharp\mathcal{E}(\{1,\ldots,m\})$. Notice that the cardinality of a set $\mathcal{E}(s)$ only depends
on the cardinality of $s$, so $\sharp\mathcal{E}(s)=k_m$ for any set $s$ with $\sharp s=m$. From the central
limit theorem it follows that
$$
\lim_{m\rightarrow\infty}\frac{k_m}{2^m}=1.
$$
Maurey-Rosenthal's space $Z_{MR}$ can be described as follows: Given $\deltalta\in(0,1)$, take an increasing
sequence $M=\{m_n\}$ so that
\betagin{equation} \lambdabel{oiojihsdfoijsdswebl}
\sum_j\sum_{k\neq j}\sqrt{\min\{\frac{m_j}{m_k},\frac{m_k}{m_j}\}}\leq\deltalta,
\itemd{equation}
and fix a one-to-one function
$$
\rightarrowgmama:\mathbb{N}^{<\infty}\rightarrow \{m_k\}_{k\ge 2}
$$
such that $\rightarrowgmama(s)>\sharp s$. Let
$$
\mathcal{B}_0=\{(s_1,\ldots,s_n):s_1\in\mathcal{S},\,s_1<\ldots<s_n,\,\sharp s_j\in M,\, \sharp s_{i+1}=\rightarrowgmama(s_1\cup\ldots\cup s_i)>\sharp s_i\}.
$$
Let $u_n$ denote the n-th unit vector in $c_{00}$ and $u_n^*$ its bi-orthogonal functional. Let us consider
the set
$$
\mathcal{N}_0=\{\sum_{i=1}^n\frac{1}{\sqrt{\sharp s_i}}\sum_{j\in s_i}u_j^*: (s_1,\ldots, s_n)\in\mathcal{B}_0\}
$$
and define $Z_{MR}$ as the space of scalar sequences $(a_n)_{n=0}^\infty$ such that
$$
\|(a_n)\|_{Z_{MR}}=\sup\{|\lambdangle \phi,\sum_n a_n u_n\text{Rang }le|:\phi\in\mathcal{N}_0\}\vee\sup_{n\in\mathbb{N}}|a_n|<\infty.
$$
\betagin{lem}
Let $(s_1,\dots,s_n)\in \mathcal B_0$. Then
\betagin{equation} \lambdabel{jkmncsaeerer}
\mathbb E_\varepsilon \nrm{\sum_{i=1}^n\frac1{(\sharp s_i)^\frac12}\sum_{k\in s_i}\varepsilon_k u_k}\le 3.
\itemd{equation}
\itemd{lem}
\betagin{proof}
Fix $(s_1,\dots,s_n)\in \mathcal B_0$, fix $(t_1,\dots,t_m)\in \mathcal B_0$, and set
$\varphi:=\sum_{j=1}^n (\sharp t_j)^{-1/2}\sum_{k\in t_j}u_k$, $x_\varepsilon:=\sum_{i=1}^n (\sharp
s_i)^{-1/2}\sum_{k\in s_i}\varepsilon_ku_k$, where $\varepsilon=(\varepsilon_k)_{k\in \bigcup_{i}s_i}$ is a
sequence of signs. Let
$$i_0=\min\{i\in\{1,\ldots,n\}:s_i\neq t_i\}. $$
Then
\betagin{align*}
|\lambdangle \varphi,x_\varepsilon \text{Rang }le|=& \sum_{j=1}^m\sum_{i=1}^n\frac1{(\sharp s_i \sharp t_j)^\frac12}\sum_{k\in s_i\cap t_j} \varepsilon_k=
\sum_{j=1}^m\left(\sum_{\sharp s_i =\sharp t_j}\frac1{\sharp s_i}\sum_{k\in s_i\cap t_j} \varepsilon_k+ \sum_{\sharp s_i \neq\sharp t_j}
\frac1{\sharp s_i}\sum_{k\in s_i\cap t_j} \varepsilon_k \right)=\\
=&\sum_{j<i_0} \frac1{\sharp s_j}\sum_{k\in s_j}\varepsilon_k +\frac{1}{\sharp s_{i_0}}\sum_{k\in s_{i_0}\cap t_{i_0}}\varepsilon_k +\sum_{j\ge i_0}\sum_{i>i_0}
\frac1{(\sharp s_i \sharp t_j)^{\frac12}}\sum_{k\in s_i\cap t_j}\varepsilon_k.
\itemd{align*}
It follows from \eqref{oiojihsdfoijsdswebl} that
\betagin{equation}
|\sum_{j\ge i_0}\sum_{i>i_0}
\frac1{(\sharp s_i \sharp t_j)^{\frac12}}\sum_{k\in s_i\cap t_j}\varepsilon_k|\le \sum_{j\ge i_0}\sum_{i>i_0}
\frac{\sharp(s_i\cap t_j)}{(\sharp s_i \sharp t_j)^{\frac12}}\le \delta.
\itemd{equation}
Hence,
\betagin{align*}
\nrm{x_\varepsilon} \le \max_{m=1}^n |\sum_{i=1}^m \frac{1}{\sharp s_i}\sum_{k\in s_i}\varepsilon_k |+1+\delta.
\itemd{align*}
Using this inequality and Levy's inequality, we obtain that
\betagin{align*}
\mathbb E_\varepsilon \nrm{x_\varepsilon} \le & \mathbb E_\varepsilon(\max_{m=1}^n
|\sum_{i=1}^n \frac{1}{\sharp s_i}\sum_{k\in s_i}\varepsilon_k |)+1+\delta \le 2\mathbb E_\varepsilon(
|\sum_{i=1}^m \frac{1}{\sharp s_i}\sum_{k\in s_i}\varepsilon_k |)+1+\delta = \\
= & 2 \int_0^1 |\sum_{i=1}^n \frac{1}{\sharp s_i}\sum_{k\in s_i}r_k(t) |dt +1+\delta \le
2 \nrm{\sum_{i=1}^n\frac{1}{\sharp s_i}\sum_{k\in s_i}\varepsilon_k u_k }_2 +1+\delta = \\
= & (\sum_{i=1}^n
\frac1{\sharp s_i})^\frac12\le 1+2\delta \le 3.
\itemd{align*}
\itemd{proof}
\betagin{teore}\lambdabel{Maurey-Rosenthal}
The unit vector basis $(u_n)$ in the space $Z_{MR}$ is a weakly null sequence with no RUD subsequences.
\itemd{teore}
\betagin{proof}
Let $N\subset\mathbb{N}$ be any infinite set. Given any $K>0$ we will see that $(u_n)_{n\in N}$ is not
$K$-RUD. Let $n>3K$ and $s_1,\ldots,s_n\subset N$ such that $(s_1,\ldots,s_n)\in\mathcal{B}_0$. Then
\betagin{equation}
\nrm{\sum_{i=1}^n \frac1{(\sharp s_i)^\frac12}\sum_{k\in s_i}u_k}\ge \lambdangle \sum_{i=1}^n \frac1{(\sharp s_i)^\frac12}\sum_{k\in s_i}u_k,\sum_{i=1}^n \frac1{(\sharp s_i)^\frac12}\sum_{k\in s_i}u_k
\text{Rang }le=n,
\itemd{equation}
while from \eqref{jkmncsaeerer} it we have that
\betagin{equation}
\mathbb E_{\varepsilon}\nrm{\sum_{i=1}^n\frac1{(\sharp s_i)^\frac12}\sum_{k\in s_i}\varepsilon_k u_k} \le 3.
\itemd{equation}
Hence $(u_n)_{n\in N}$ is not $K$-RUD.
\itemd{proof}
We present now a RUC sequence without unconditional subsequences. Given $(a_i)_{i=1}^n\in c_{00}$, let
$$\nrm{(a_i)_i}_{\mathrm{RUC}}:= \nrm{(a_i)_i}_{\mathrm{MR}}+ \mathbb E_{\varepsilon}\nrm{(\varepsilon_i a_i)_i}_{\mathrm{MR}}.$$
Let $Z_{\mathrm{RUC}}$ be the completion of $c_{00}$ under this norm.
\betagin{teore}\lambdabel{MR-RUC}
The unit basis $(u_n)_n$ of $Z_{\mathrm{RUC}}$ is a weakly-null RUC basis without unconditional subsequences.
\itemd{teore}
\betagin{proof}
It is RUC: We have that
$$\nrm{(a_i)_i}_{\mathrm{RUC}}\le \nrm{(a_i)_i}_{\mathrm{MR}}+\mathbb{E}_\varepsilon \nrm{(\varepsilon_i a_i)_i}_{\mathrm{MR}}\le 2 \nrm{(a_i)_i}_{\mathrm{RUC}}.$$
Hence,
$$\mathbb E_\varepsilon\nrm{(\varepsilon_i a_i)_i}_{\mathrm{RUC}}\le
2\mathbb E_\varepsilon\nrm{(\varepsilon_i a_i)_i}_{\mathrm{MR}} \le 2 \mathbb
E_\varepsilon\nrm{(\varepsilon_i a_i)_i}_{\mathrm{RUC}}.$$ It follows that
$$\mathbb E_\varepsilon\nrm{(\varepsilon_i a_i)_i}_{\mathrm{RUC}}\le
2\mathbb E_\varepsilon\nrm{(\varepsilon_i a_i)_i}_{\mathrm{MR}} \le 2\nrm{(a_i)_i}_{\mathrm{RUC}} $$
Hence $(u_i)_i$ is 2-RUC. On the other hand, given $(s_i)_{i=1}^n\mathcal B_0$, we have from
\eqref{jkmncsaeerer} that
\betagin{align*}
\nrm{\sum_{i=1}^n \frac{1}{(\sharp s_i)^\frac12}(-1)^i \sum_{k\in s_i}u_k}_{\mathrm{RUC}}=&
\nrm{\sum_{i=1}^n \frac{1}{(\sharp s_i)^\frac12}(-1)^i \sum_{k\in s_i}u_k}_{\mathrm{MR}} + \\
+ &
\mathbb E_{\varepsilon}\nrm{\sum_{i=1}^n \frac{1}{(\sharp s_i)^\frac12}(-1)^i \sum_{k\in s_i}\varepsilon_ku_k}_{\mathrm{MR}} \le
6.
\itemd{align*}
On the other hand $\nrm{\sum_{i=1}^n (\sharp s_i)^{-1/2}\sum_{k\in s_i}u_k}_{\mathrm{RUC}}\ge n$. Thus, it
has no unconditional subsequence.
\itemd{proof}
\subsection{RUD basis without unconditional subsequences}
We present now a weakly-null RUD basis without unconditional subsequences. Given a finite set $s$, let
$\mathcal{E}(s)$ be the collection of equi-distributed signs in $s$.
Let us fix $\deltalta\in (0,1)$. We will take an increasing sequence of even numbers $M=\{m_n\}$ so that
\betagin{enumerate}
\item[(i)] $\sum_j\sum_{k\neq j}\sqrt{\min\{\frac{m_j}{m_k},\frac{m_k}{m_j}\}}\leq\deltalta$, and
\item[(ii)] $\prod_{n=1}^\infty\frac{k_{m_n}}{2^{m_n}}\geq 1-\deltalta$.
\itemd{enumerate}
Fix a one-to-one function
$$
\rightarrowgmama:\mathbb{N}^{<\infty}\rightarrow M
$$
such that $\rightarrowgmama(s)>\sharp s$. Let
$$
\mathcal{B}=\{(s_1,\ldots,s_n):s_1\in\mathcal{S},\,s_1<\ldots<s_n,\,\sharp s_j\in M,\, \sharp s_{i+1}=\rightarrowgmama(s_1\cup\ldots\cup s_i)>\sharp s_i\}.
$$
Let $u_n$ denote the $n^{\mathrm{th}}$ unit vector in $c_{00}$ and $u_n^*$ its bi-orthogonal functional. Let
us consider the set
$$
\betagin{array}{ll}
\mathcal{N}=&\{\sum_{i=1}^n\frac{1}{\sqrt{\sharp s_i}}\sum_{j\in s_i}\varepsilon_i(j)u_j^*: (s_1,\ldots, s_n)\in\mathcal{B}, \textrm{ with }\varepsilon_i\in\mathcal{E}(s_i), \,\forall i=1,\ldots,n\}\\
&\cup\{\pm\sum_{i=1}^n\frac{1}{\sqrt{\sharp s_i}}\sum_{j\in s_i}u_j^*: (s_1,\ldots, s_n)\in\mathcal{B}\}\cup\{\pm u_k^*:k\in \mathbb{N}\}.
\itemd{array}
$$
Now, we define $Z_{\mathrm{RUD}}$ as the space of scalar sequences $(a_n)_{n=0}^\infty$ such that
$$
\|(a_n)\|_{Z_{\mathrm{RUD}}}=\sup\{\lambdangle \phi,\sum_n a_n u_n\text{Rang }le:\phi\in\mathcal{N}\}<\infty.
$$
\betagin{teore}\lambdabel{MR-RUD}
The unit basis $(u_n)$ is a RUD basis of the space $Z_{RUD}$ endowed with the norm $\|\cdot\|_{Z_{RUD}}$
without unconditional subsequences. In addition, given any infinite set $N\subset\mathbb{N}$, if for every
$n\in \mathbb{N}$ we take $s_n\subset N$ such that $(s_1,\ldots, s_n)\in\mathcal{B}$, and let
$$
x_n=\frac1{\sqrt{\sharp s_n}}\sum_{j\in s_n}u_j,
$$
then $(x_n)$ is a normalized block sequence of $(u_n)_{n\in N}$ which is not RUD.
\itemd{teore}
\betagin{proof}
Let us see first that $(u_n)_{n\in\mathbb{N}}$ is RUD. To this end, we take arbitrary scalars $(a_k)_{k=1}^l$
and let us prove that
$$
\mathbb{B}ig\|\sum_{k=1}^l a_ku_k\mathbb{B}ig\|=\sup_{\phi\in\mathcal{N}}\lambdangle\phi,\sum_{k=1}^l a_ku_k\text{Rang }le\leq C \mathbb{E} \mathbb{B}ig(\mathbb{B}ig\|\sum_{k=1}^l \epsilonsilon_k |a_k| e_k \mathbb{B}ig\| \mathbb{B}ig),
$$
for some constant $C$ independent on $(a_k)_{k=1}^l$. First, for $\phi=\pm u_k$ we clearly have
$$
\lambdangle \phi,\sum_{k=1}^l a_ku_k\text{Rang }le\leq\max_k |a_k|\leq \mathbb{E} \mathbb{B}ig(\mathbb{B}ig\|\sum_{k=1}^l \epsilonsilon_k |a_k| e_k \mathbb{B}ig\| \mathbb{B}ig).
$$
Now, suppose $\phi$ has the form
$$
\phi=\pm\sum_{i=1}^n\frac{1}{\sqrt{\sharp s_i}}\sum_{j\in s_i}u_j^*, \hspace{1cm}\textrm{ or }\hspace{1cm} \phi=\sum_{i=1}^n\frac{1}{\sqrt{\sharp s_i}}\sum_{j\in s_i}\varepsilon_i(j)u_j^*,
$$
for some fixed $(s_1,\ldots,s_n)\in\mathcal{B}$ and $\varepsilon_i\in\mathcal{E}(s_i)$. Let us consider the
set
$$
A=\{\varepsilon\in\{-1,1\}^l:\varepsilon|_{s_i}\in\mathcal{E}(s_i) \,\forall i=1,\ldots,n\}.
$$
Hence, for $(\theta_k)_{k=1}^l\in A$, we have that $\sum_{i=1}^n\frac{1}{\sqrt{\sharp s_i}}\sum_{j\in
s_i}\theta_j u_j^*\in\mathcal{N}$ so we get that, for both cases of $\phi$,
\betagin{align*}
\mathbb{B}ig\|\sum_{k=1}^l\theta_k|a_k|u_k\mathbb{B}ig\|\geq &\lambdangle\sum_{i=1}^n\frac{1}{\sqrt{\sharp s_i}}\sum_{j\in s_i}\theta_j u_j^*,
\sum_{k=1}^l\theta_k|a_k|u_k\text{Rang }le= \sum_{i=1}^n\frac{1}{\sqrt{\sharp s_i}}\sum_{j\in s_i\cap\{1,\ldots,l\}}
|a_j|\geq \\
\geq & \lambdangle\phi,\sum_{k=1}^l a_ku_k\text{Rang }le.
\itemd{align*}
In particular, we have
$$
\mathbb{E} \mathbb{B}ig(\mathbb{B}ig\|\sum_{k=1}^l \epsilonsilon_k |a_k| e_k \mathbb{B}ig\| \mathbb{B}ig)\geq \mathbb{E} \mathbb{B}ig(\mathbb{B}ig\|\sum_{k=1}^l \epsilonsilon_k |a_k| e_k \mathbb{B}ig\|\chi_A((\epsilonsilon_k)_{k=1}^l) \mathbb{B}ig)\geq \lambdangle\phi,\sum_{k=1}^l a_ku_k\text{Rang }le\frac{\sharp A}{2^l}.
$$
Notice that if we denote $m_{j_i}=\sharp s_i\in M$, then the cardinality of $A$ is given by
$$
\sharp A=\prod_{i=1}^n \sharp \mathcal{E}(s_i) \times 2^{l-\sharp(s_1\cup\ldots\cup s_n)}=2^l\prod_{i=1}^n\frac{k_{m_{j_i}}}{2^{m_{j_i}}}\geq2^l\prod_{i=1}^\infty\frac{k_{m_i}}{2^{m_i}}\geq 2^l (1-\deltalta),
$$
because of condition $(ii)$ in the definition of the sequence $M$. Thus, we have that
$$
\mathbb{E} \mathbb{B}ig(\mathbb{B}ig\|\sum_{k=1}^l \epsilonsilon_k |a_k| e_k \mathbb{B}ig\| \mathbb{B}ig)\geq (1-\deltalta)\lambdangle\phi,\sum_{k=1}^l a_ku_k\text{Rang }le.
$$
Therefore, we finally get that for any scalars $(a_k)_{k=1}^l$
$$
\mathbb{E} \mathbb{B}ig(\mathbb{B}ig\|\sum_{k=1}^l \epsilonsilon_k a_k e_k \mathbb{B}ig\| \mathbb{B}ig)=\mathbb{E} \mathbb{B}ig(\mathbb{B}ig\|\sum_{k=1}^l \epsilonsilon_k |a_k| e_k \mathbb{B}ig\| \mathbb{B}ig)\geq (1-\deltalta)\mathbb{B}ig\|\sum_{k=1}^l a_ku_k\mathbb{B}ig\|,
$$
so $(u_n)_{n\in\mathbb{N}}$ is RUD.
For the second part, given an infinite set $N\subset\mathbb{N}$, let
$$
x_n=\frac1{\sqrt{\sharp s_n}}\sum_{j\in s_n}u_j,
$$
where for every $n\in \mathbb{N}$, $s_n\subset N$ is such that $(s_1,\ldots, s_n)\in\mathcal{B}$. We claim
that for any scalars $(a_j)_{j=1}^n$ we have that
$$
\mathbb{B}ig\|\sum_{j=1}^n a_j x_j\mathbb{B}ig\|\approx\sup_{1\leq l\leq n}\mathbb{B}ig|\sum_{k=1}^l a_k\mathbb{B}ig|,
$$
independently of the scalars and $n\in\mathbb{N}$. In particular, by Theorem \ref{summing}, $(x_n)$ cannot be
RUD. Besides, since this holds for any $N\subset\mathbb{N}$, no subsequence of $(u_n)_{n\in\mathbb{N}}$ can
be unconditional.
First, for $l=1,\ldots,n$ let
$$
\phi_l=\sum_{i=1}^l\frac1{\sqrt{\sharp s_i}}\sum_{k\in s_i}u_k^*\in\mathcal{N}.
$$
Hence, we have that
$$
\mathbb{B}ig\|\sum_{j=1}^n a_j x_j\mathbb{B}ig\|\geq\sup_{1\leq l\leq n}\lambdangle\pm\phi_l,\sum_{j=1}^n a_j x_j\text{Rang }le=\sup_{1\leq l\leq n}\pm\sum_{i=1}^l \frac{a_i}{\sharp s_i}\sharp s_i=\sup_{1\leq l\leq n}\mathbb{B}ig|\sum_{i=1}^l a_i\mathbb{B}ig|.
$$
For the converse inequality, first, if $\phi$ has the form $\phi=\pm u_k^*$ for $k\in s_i$, then we have that
$$
\lambdangle \phi, \sum_{j=1}^n a_j x_j\text{Rang }le=\frac{a_i}{\sqrt{\sharp s_i}}\leq\sup_{1\leq i\leq n}|a_i|\leq2\sup_{1\leq l\leq n}\mathbb{B}ig|\sum_{k=1}^l a_k\mathbb{B}ig|.
$$
Now, suppose $\phi$ has the form
$$
\phi= \sum_{i=1}^m\frac{1}{\sqrt{\sharp t_i}}\sum_{l\in t_i}\varepsilon_i(l)u_l^*,
$$
for some $(t_1,\ldots,t_m)\in\mathcal{B}$ and $\varepsilon_i\in\mathcal{E}(t_i)$ for every $i=1,\ldots,m$, or
$\varepsilon_i=\varepsilon 1_{t_i}$ for every $i=1,\ldots,m$, with $\varepsilon\in\{-1,1\}$. Let
$$
j_0=\min \{j\leq n:s_j\neq t_j\}.
$$
Hence, we can write
$$
\lambdangle \phi,\sum_{j=1}^n a_j x_j\text{Rang }le=\overbrace{\lambdangle \phi,\sum_{j=1}^{j_0-1} a_j x_j\text{Rang }le}^{(A)}+\overbrace{\overset{\,}{\overset{\,}{\overset{\,}{\lambdangle \phi, a_{j_0} x_{j_0}\text{Rang }le}}}}^{(B)}+\overbrace{\lambdangle \phi,\sum_{j=j_0+1}^n a_j x_j\text{Rang }le}^{(C)}.
$$
Since for any $i\geq j_0>j$, we have $t_i\cap s_j=\emptyset$, and $s_k=t_k$ for $k<j_0$, we get that
$$
(A)=\lambdangle\sum_{i=1}^{j_0-1}\frac{1}{\sqrt{\sharp t_i}}\sum_{l\in t_i}\varepsilon_i(l)u_l^*,\sum_{j=1}^{j_0-1} a_j x_j\text{Rang }le=\sum_{j=1}^{j_0-1} \frac{a_j}{\sharp s_j}\sum_{k\in s_j}\varepsilon_j(k).
$$
Thus, depending on the form of $\phi$ we either have $\sum_{k\in s_j}\varepsilon_j(k)=0$ for every
$j=1,\ldots, j_0-1$, or $\sum_{k\in s_j}\varepsilon_j(k)=\varepsilon\sharp s_j$ for every $j=1,\ldots,
j_0-1$, and some $\varepsilon\in\{-1,1\}$. In any case we get
$$
(A)\leq\mathbb{B}ig|\sum_{j=1}^{j_0-1}a_j\mathbb{B}ig|\leq\sup_{1\leq l\leq n}\mathbb{B}ig|\sum_{i=1}^l a_i\mathbb{B}ig|.
$$
Now, since for $i<j_0$, we have $t_i\cap s_{j_0}=\emptyset$, we get that
$$\betagin{array}{lll}
(B)&=&\lambdangle\sum_{i=j_0}^m\frac{1}{\sqrt{\sharp t_i}}\sum_{l\in t_i}\varepsilon_i(l)u_l^*,a_{j_0} \frac{1}{\sqrt{\sharp s_{j_0}}}\sum_{k\in s_{j_0}}u_k\text{Rang }le \\
&\leq &|a_{j_0}|\mathbb{B}ig(\frac{1}{\sqrt{\sharp t_{j_0}\sharp s_{j_0}}}\sum_{l\in t_{j_0}\cap s_{j_0}}\varepsilon_{j_0}(l)+\sum_{i>j_0}\frac{1}{\sqrt{\sharp t_i\sharp s_{j_0}}}\sum_{l\in t_i\cap s_{j_0}}\varepsilon_{j_0}(l)\mathbb{B}ig)\\
&\leq&|a_{j_0}|(1+\sum_j\sum_{k\neq j}\sqrt{\min\{\frac{m_j}{m_k},\frac{m_k}{m_j}\}})\leq(1+\deltalta)|a_{j_0}|.
\itemd{array}
$$
So we also get that
$$
(B)\leq 2(1+\deltalta)\sup_{1\leq l\leq n}\mathbb{B}ig|\sum_{i=1}^l a_i\mathbb{B}ig|.
$$
And, finally we have
$$
\betagin{array}{lll}
(C)&=&\lambdangle\sum_{i=j_0}^m\frac{1}{\sqrt{\sharp t_i}}\sum_{l\in t_i}\varepsilon_i(l)u_l^*,\sum_{j=j_0+1}^n a_j \frac1{\sqrt{\sharp s_j}}\sum_{k\in s_j}u_k\text{Rang }le\\
&=&\sum_{i=j_0}^m\sum_{j>j_0}\frac{a_j}{\sqrt{\sharp t_i \sharp s_j}}\sum_{l\in t_i\cap s_j}\varepsilon_i(l)\\
&\leq&\sup_{j_0<j\leq n}|a_j|\sum_{i=j_0}^m\sum_{j>j_0}\frac{\min\{\sharp t_i, \sharp s_j\}}{\sqrt{\sharp t_i \sharp s_j}}\\
&\leq&\sup_{j_0<j\leq n}|a_j|\sum_i\sum_{k\neq i}\sqrt{\min\{\frac{m_i}{m_k},\frac{m_k}{m_i}\}}\leq2\deltalta\sup_{1\leq l\leq n}\mathbb{B}ig|\sum_{i=1}^l a_i\mathbb{B}ig|.
\itemd{array}
$$
Thus, we have seen that for every $\phi\in\mathcal{N}$
$$
\lambdangle \phi,\sum_{j=1}^n a_j x_j\text{Rang }le\leq(3+4\deltalta)\sup_{1\leq l\leq n}\mathbb{B}ig|\sum_{i=1}^l a_i\mathbb{B}ig|.
$$
Therefore, we get that
$$
\sup_{1\leq l\leq n}\mathbb{B}ig|\sum_{i=1}^l a_i\mathbb{B}ig|\leq\mathbb{B}ig\|\sum_{j=1}^n a_j x_j\mathbb{B}ig\|\leq(3+4\deltalta)\sup_{1\leq l\leq n}\mathbb{B}ig|\sum_{i=1}^l a_i\mathbb{B}ig|.
$$
\itemd{proof}
\betagin{problem}
Does every Banach space have a RUC or RUD basic sequence?
\itemd{problem}
\succeqction{RUD sequences in rearrangement invariant spaces}\lambdabel{Sec_RUD_Banach}
In the framework of Banach lattices, Krivine's functional calculus (cf. \cite[Section 1.d.]{LT2}) allows us to
give a meaning to expressions like $\big(\sum_{i=1}^n |x_i|^p\big)^{\frac1p}$, which coincides with the
corresponding pointwise operation when we deal with a Banach lattice of functions. Using Khintchine's
inequality we get a constant $C>0$ such that for any $(x_i)_{i=1}^n$ in an arbitrary Banach lattice $X$ we
have
$$
\int_0^1\mathbb{B}ig\|\sum_{i=1}^n r_i(t)x_i\mathbb{B}ig\|dt\geq\frac1C\mathbb{B}ig\|\mathbb{B}ig(\sum_{i=1}^n|x_i|^2\mathbb{B}ig)^{\frac12}\mathbb{B}ig\|.
$$
Moreover, if $X$ is $q$-concave for some $q<\infty$ (equivalently, if $X$ has finite cotype) then there is a
constant $C(q)>0$ such that a converse estimate holds:
$$
\int_0^1\mathbb{B}ig\|\sum_{i=1}^n r_i(t)x_i\mathbb{B}ig\|dt\leq C(q)\mathbb{B}ig\|\mathbb{B}ig(\sum_{i=1}^n|x_i|^2\mathbb{B}ig)^{\frac12}\mathbb{B}ig\|.
$$
In particular, a sequence $(x_n)_{n=1}^\infty$ in a Banach lattice $X$ with finite cotype is RUD if and only
if there is $K>0$ such that for any scalars $(a_k)_{k=1}^n$
$$
\mathbb{B}ig\|\sum_{k=1}^n a_k x_k\mathbb{B}ig\|\leq K \mathbb{B}ig\|\mathbb{B}ig(\sum_{k=1}^n |a_k x_k|^2\mathbb{B}ig)^{\frac12}\mathbb{B}ig\|.
$$
It is reasonable to expect that if the lattice structure has a lot of symmetry, then it is easier to find RUD
sequences. This is precisely stated in the next result for rearrangement invariant spaces which makes use of
the estimates for martingale difference sequences given in \cite{JS}.
\betagin{teore}\lambdabel{ri RUD}
Let $X$ be a separable rearrangement invariant space on $[0,1]$ with non-trivial upper Boyd index. Every block
sequence of the Haar basis in $X$ is RUD. In particular, every weakly null sequence $(x_n)$ in $X$ has a
subsequence which is basic RUD.
\itemd{teore}
\betagin{proof}
Let $(h_j)$ denote the Haar system on $[0,1]$. That is, for $j=2^k+l$, with $k\in\mathbb{N}$ and $1\leq l\leq
2^k$, we have
$$
h_j=\chi_{[\frac{2l-2}{2^{k+1}},\frac{2l-1}{2^{k+1}})}-\chi_{[\frac{2l-1}{2^{k+1}},\frac{2l}{2^{k+1}})}.
$$
By \cite[Proposition 2.c.1]{LT2}, $(h_j)$ is a monotone basis of $X$. Let us take a block sequence
$$
y_k=\sum_{j=p_k}^{q_k}b_j h_j
$$
(with $p_k\leq q_k<p_{k+1}$). Given scalars $(a_k)_{k=1}^m$ we can consider the sequence
$$
f_n=
\left\{
\betagin{array}{cc}
\sum_{k=1}^n a_ky_k & n<m \\
&\\
\sum_{k=1}^m a_ky_k & n\geq m.
\itemd{array}
\right.
$$ It holds that $(f_n)$ is a martingale with respect to the filtration $(\mathcal{D}_{q_n})$, where $\mathcal{D}_{q_n}$ is the smallest $\rightarrowgmama$-algebra $\mathcal{A}$ for which the functions $\{h_1,\ldots,h_{q_n}\}$ are $\mathcal{A}$-measurable.
By \cite[Theorem 3]{JS} there is $C>0$, which is independent of the scalars $(a_k)_{k=1}^m$, such that
$$
\mathbb{B}ig\|\sum_{k=1}^m a_ky_k\mathbb{B}ig\| \leq \|\sup_n |f_n|\| \leq C \mathbb{B}ig\|\mathbb{B}ig(\sum_{k=1}^\infty |f_k-f_{k-1}|^2\mathbb{B}ig)^{\frac12}\mathbb{B}ig\|= C \mathbb{B}ig\|\mathbb{B}ig(\sum_{k=1}^m |a_k y_k|^2\mathbb{B}ig)^{\frac12}\mathbb{B}ig\|.
$$
Now, by \cite[Theorem 1.d.6]{LT2} there is a universal constant $A>0$ such that
$$
\mathbb{B}ig\|\mathbb{B}ig(\sum_{k=1}^m |a_k y_k|^2\mathbb{B}ig)^{\frac12}\mathbb{B}ig\|\leq A\int_0^1\mathbb{B}ig\|\sum_{k=1}^m r_k(s)a_ky_k\mathbb{B}ig\|ds.
$$
Hence, since $(x_{n_k})$ is equivalent to $y_k$ we have that
$$
\mathbb{B}ig\|\sum_{k=1}^m a_kx_{n_k}\mathbb{B}ig\| \leq K \mathbb{B}ig\|\sum_{k=1}^m a_ky_k\mathbb{B}ig\| \leq CAK\int_0^1\mathbb{B}ig\|\sum_{k=1}^m r_k(s)a_ky_k\mathbb{B}ig\|ds \leq CAK^2\mathbb{E}\mathbb{B}ig(\mathbb{B}ig\|\sum_{k=1}^m \varepsilon_n a_k x_{n_k} \mathbb{B}ig\| \mathbb{B}ig).
$$
\itemd{proof}
A similar idea has been used in \cite{AKS} to show that if a separable r.i. space $X$ on $[0,1]$ is p-convex
for some $p>1$ and has strictly positive lower Boyd index, then $X$ has the Banach-Saks property.
\betagin{corollary}\lambdabel{jms}
There is in $L_1$ a RUD basic sequence without unconditional subsequences.
\itemd{corollary}
\betagin{proof}
Let $(f_n)_n$ be the weakly-null basic sequence in $L_1$ without unconditional subsequences given in
\cite{JMS}. Then any RUD subsequence of $(f_n)_n$ (existing by Theorem \ref{ri RUD}) fulfills the desired
requirements.
\itemd{proof}
Note that the Haar basis in $L_1[0,1]$ is a conditional basis such that every block is RUD (compare with
Theorem \ref{James_thm}). We do not know if a basis with the property that every block subsequence is RUD has some unconditional block subsequence. The sequence given in Corollary \ref{jms} satisfies that every block subsequence is RUD, and fails to have an unconditional subsequence (although it has unconditional blocks).
\succeqction{average norms}
The motivating question here is the following: Given an unconditional basic sequence $(x_n)_n$, find an RUC or RUD
basis $(y_n)_n$ such that $(r_n \leftarrowimes y_n)_n$ is equivalent to $(x_n)_n$ but
\betagin{enumerate}
\item $(y_n)_n$ is not equivalent to $(x_n)_n$, or
\item $(y_n)_n$ does not contain subsequences equivalent to subsequences of $(x_n)$, or
\item $(y_n)_n$ does not contain unconditional subsequences.
\itemd{enumerate}
\betagin{problem}
Characterize unconditional sequences $(x_n)_n$ under one of the previous criteria.
\itemd{problem} In the case for the unit vector basis of $c_0$ or $\ell_1$ it is not possible to find such a basis as the
following well-known theorems show. By the sake of completeness, we will reproduce the original proofs.
\betagin{teore}[S. Kwapien \cite{Kw}]\lambdabel{oiio3rere}
Suppose that $(x_n)_n$ is a seminormalized basic sequence in a Banach space such that $\sup_n \mathbb
E_{\varepsilon}\nrm{\sum_{i=1}^n \varepsilon_i x_i}<\infty$. Then $(x_n)_n$ has a subsequence equivalent to the unit basis
of $c_0$.
\itemd{teore}
\betagin{proof}
For every measurable set $B\subseteq [0,1]$ one has that $\lim_{n\to \infty}\lambda (\{t\in B\, : \,
r_n(t)=1\})=\lim_{n\to \infty}\lambda (\{t\in B\, : \, r_n(t)=-1\})=(1/2)\lambda(B)$. Let $M>0$ be such that
$A=\ensuremath{c_{00}(\omega_1)}nj{t\in [0,1]}{\sup_n \nrm{\sum_{i=1}^n r_i(t)x_i}\le M}$ has Lebesgue measure $\lambda(A)>1/2$. Now let
$n_1\in {\mathbb N}$ be such that
\betagin{equation}
\lambda(\ensuremath{c_{00}(\omega_1)}nj{t\in A}{r_{n_1}(t)=1})= \lambda(\ensuremath{c_{00}(\omega_1)}nj{t\in A}{r_{n_1}(t)=-1})>\frac{1}{2^2}.
\itemd{equation}
In general, let $(n_k)_k$ be a strictly increasing sequence of integers such that for every $k$ and every
sequence of signs $(\varepsilon_i)_{i=1}^k$ one has that
\betagin{equation} \lambdabel{lkjdflksldkfsjdfe}
\lambda(A({(\varepsilon_i)_{i=1}^k}))> \frac1{2^{k+1}}.
\itemd{equation}
where $A({(\varepsilon_i)_{i=1}^k})= \ensuremath{c_{00}(\omega_1)}nj{t\in A}{r_{n_i}(t)=\varepsilon_i \text{ for every $1\le i\le k$}} $. Let now
$s_i=r_i$ if $i\in \{n_j\}_j$ and $s_i=-r_i$ if $i\notin \{n_j\}_j$. Let
$$B=\ensuremath{c_{00}(\omega_1)}nj{t\in [0,1]}{\sup_n\nrm{\sum_{i=1}^n s_i(t)x_i}\le M}.$$ Since $(r_i)_i$ and $(s_i)_i$ are
equidistributed, it follows that
\betagin{equation}
\lambda(B((\varepsilon_i)_{i=1}^k))=\lambda(A((\varepsilon_i)_{i=1}^k))>\frac1{2^{k+1}},
\itemd{equation}
where $B((\varepsilon_i)_{i=1}^k)=\ensuremath{c_{00}(\omega_1)}nj{t\in B}{s_{n_i}=\varepsilon_i \text{ for every $1\le i \le k$}}$. Since the set
$$
\bigcap_{i=1}^k \{r_{n_i}=\varepsilon_i\}=\ensuremath{c_{00}(\omega_1)}nj{t\in [0,1]}{r_{n_i}(t)=\varepsilon_i\text{ for every $1\le i\le k$}}
$$
has measure $2^{-k}$, and since
$$
A((\varepsilon_i)_{i=1}^k),B((\varepsilon_i)_{i=1}^k)\subseteq \bigcap_{i=1}^k \{r_{n_i}=\varepsilon_i\}
$$
it follows that
$$
A((\varepsilon_i)_{i=1}^k)\cap B((\varepsilon_i)_{i=1}^k) \neq \emptyset.
$$
Let $t_0\in A((\varepsilon_i)_{i=1}^k)\cap B((\varepsilon_i)_{i=1}^k) $. Hence,
\betagin{equation}
\nrm{\sum_{i=1}^k\varepsilon_i x_{n_i}}=\frac{1}{2}\nrm{\sum_{j=1}^{n_k} r_{j}(t_0) x_{j} +\sum_{j=1}^{n_k}s_j(t_0)x_j}\le M.
\itemd{equation}
Now it is easy to deduce from here that $\nrm{\sum_{i=1}^ka_i x_{n_i}}\le M \max_{i=1}^k |a_i|$.
\itemd{proof}
\betagin{propo}[J. Bourgain \cite{Bo2}]\lambdabel{iuuiuiere}
Suppose that $(x_n)_n$ is a bounded sequence in a Banach space $X$ such that for some constant $\delta>0$ one
has that
\betagin{equation}\lambdabel{nnnkfkjgdf}
\text{$\mathbb E_\varepsilon \nrm{\sum_{i=1}^n \varepsilon_i a_i x_i}\ge \delta\sum_{i=1}^n|a_i|$ for
every sequence of scalars $(a_i)_{i=1}^n$.}
\itemd{equation}Then $(x_n)_n$ has a subsequence equivalent to the unit basis of
$\ell_1$.
\itemd{propo}
\betagin{proof}
By Rosenthal's $\ell_1$ theorem, we may assume otherwise that $(x_n)_n$ has a subsequence which is
weakly-Cauchy. Since our hypothesis \eqref{nnnkfkjgdf} passes to subsequences, we may assume without loss of
generality that $(x_n)_n$ is weakly-convergent to $x^{**}\in X^{**}$. It is well known that for every
$\gammamma>0$ there is a convex combination $(a_i)_{i=1}^n$ such that
\betagin{enumerate}
\item[(a)]
$\nrm{\sum_{i=1}^n a_i\varepsilon_i x_i-(\sum_{i=1}^n \varepsilon_i a_i)x^{**}}\le \gammamma$ for every sequence of
signs $(\varepsilon_i)_{i=1}^n$.
\item[(b)] $\nrm{(a_i)_{i=1}^n}_2\le \gammamma$.
\itemd{enumerate}
Indeed, for the first part, think of each $x_n-x^{**}$ as a function in $C[0,1]$, and use Mazur's result for the
weakly-null sequence $(|x_n-x^{**}|)_n$; once (a) is established for each $\gammamma$, let $n$ be such that
$\gammamma\sqrt{n}\ge 1$, and find $s_1<\dots <s_n$ and convex combinations $\sum_{j\in s_i}a_ju_j$ fulfilling
(a) for $\gammamma/n$; then the convex combination $(1/n)\sum_{i=1}^n \sum_{j\in s_i}a_j u_j$ satisfies that
$$
\nrm{(1/n)\sum_{i=1}^n\sum_{j\in s_i}\varepsilon_j a_j (x_j-x^{**})}\le \gammamma
$$
for every choice of signs $(\varepsilon_i)_i$, and
$$
\nrm{(1/n)\sum_{i=1}^n \sum_{j\in s_i}a_j u_j}_2\le 1/\sqrt{n}\le\gammamma.
$$
Now let $(a_i)_{i=1}^n$ be the corresponding combination for $\gammamma$ such that $\gammamma(1+\nrm{x^{**}})<\delta$.
Then
\betagin{equation}
\delta \le \mathbb E_\varepsilon \nrm{\sum_{i=1}^n \varepsilon_i a_i x_i}\le \gammamma + \nrm{x^{**}}\mathbb E_\varepsilon |\sum_{i=1}^n a_i\varepsilon_i| \le \gammamma +
\nrm{x^{**}}(\sum_{i=1}^n a_i^2)^\frac12<\delta,
\itemd{equation}
a contradiction.
\itemd{proof}
\betagin{ejemplo}
For each $1<p\le 2$, on $c_{00}$ define the norm
$$\nrm{(a_i)_{i=1}^n}_{s,p}:=\max\{ \nrm{\sum_{i=1}^n a_i
s_i}_\infty,\nrm{(a_i)_{i=1}^n}_p\},$$
where $(s_i)_i$ is the summing basis of $c_{0}$; let $X$
be the completion of $c_{00}$ under this norm. Then the unit Hamel basis $(u_n)_n$ is RUC and satisfies that
$(r_n \leftarrowimes u_n)_n$ is equivalent to the unit basis of $\ell_p$. It is easy to see that $(u_i)_{i}$ in $X$
does not have unconditional subsequences.
\itemd{ejemplo}
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\itemd{document} |
\begin{document}
\title{Genuine multipartite system-environment correlations in decoherent
dynamics}
\author{Jonas Maziero}
\email{[email protected]}
\selectlanguage{english}
\author{F\'{a}bio M. Zimmer}
\email{[email protected]}
\selectlanguage{english}
\address{Departamento de F\'{i}sica, Universidade Federal de Santa Maria, 97105-900,
Santa Maria, RS, Brazil}
\begin{abstract}
We propose relative entropy-based quantifiers for genuine multipartite total, quantum, and classical correlations. These correlation measures are applied to investigate the generation of genuine multiparticle correlations in decoherent dynamics induced by the interaction of two qubits with local-independent environments. We consider amplitude- and phase-damping channels and compare their capabilities to spread information through the creation of many-body correlations. We identify changes in behavior for the genuine $4$- and $3$-partite total correlations
and show that, contrary to amplitude environments, phase-noise channels
transform the bipartite correlation initially shared between the qubits
into genuine multiparticle system-environment correlations.
\end{abstract}
\maketitle
\section{Introduction}
The nonlocality \cite{Aspect-Review,Mermin-RMP}, nonseparability
\cite{Horodecki-RMP,Guhne-PR}, and quantumness \cite{Celeri-Review,Modi-Review}
of correlations in composite systems are currently recognized as important
ingredients for the efficiency of protocols in quantum information
science. Substantial progress has been achieved latterly regarding
the characterization, identification, and quantification of total,
classical, and quantum correlations \cite{Groisman-PRA,Henderson-JPA,Ollivier-QD,Horodecki-Defict,Piani-NLB,Cavalcanti-SMerg,Madhok-SMerg,Ferraro,Zurek-MD,Adesso-CV,Giorda-CV,Cornelio-EIrrev,Streltsov-DEM,Maziero-Wit,Aguilar-WPRL}.
The nonclassicality originated from local indistinguishability of
quantum states is the more recent paradigm for analyzing correlations.
This kind of correlation was studied mostly in the case of bipartite
systems, with novel and interesting results already obtained. By its
turn, the investigation of (genuine) multipartite quantum and classical
correlations has been receiving its deserved attention only very recently
\cite{kaszlikowski,walczak,grudka-arxiv,Saitoh-IJQI,Rulli-Sarandy-GQD,Chakrabarty-MQD,Okrasa-EPL,Parashar-Rana,Campbell-IJQI,Giorgi-GMTC-XY,Byczuk-PRL,Ranzan-Dynamics-GQD,Comment-Paper-Giorgi,Square-D-&-Monogamy,Average-GMQD,Mahdian-MQD-Dec,Xu-GQD,GMQD-Detection,Saguia-Wit,Modi-reldis,Giorgi-genuine,Bennett-genuine}.
In this article we are interested in this last scenario. We shall
define measures for genuine $n$-partite correlations (Section \ref{sec:quantifiers})
and study how these correlations develop during decoherent system-environment
dynamics (Section \ref{sec:dynamics}).
\subsection{The decoherence process}
The thorough investigation of the decoherence process \cite{Zurek-RMP,Breuer-Petruccione,Schlosshauer}
is an important issue to be addressed towards large scale implementations
of, for instance, quantum computers, quantum simulators, and communication
protocols. The decoherence phenomenon is a result of the inevitable
interaction between a quantum system and its surroundings. Let us
consider a system $s$ and its environment $E$ in an initial product
state $\rho_{sE}=\rho_{s}\otimes|0_{E}\rangle\langle0_{E}|$, where $|0_{E}\rangle$ is the vacuum state of the environment. The dynamics of the whole closed system is unitary, i.e., $\rho_{sE}(t)=U_{sE}(t)\rho_{sE}U_{sE}^{\dagger}(t)$
with $U_{sE}(t)U_{sE}^{\dagger}(t)=\mathbb{I}$, where $\mathbb{I}$
is the identity operator. Then, by tracing out the environmental variables
we write down the system's evolved state in the Kraus or operator-sum
representation: $\mathcal{E}(\rho_{s})=\sum_{i}K_{i}\rho_{s}K_{i}^{\dagger}$,
where $K_{i}:=\langle i_{E}|U_{sE}(t)|0_{E}\rangle$ are linear operators
on the state space of the system, $\mathcal{H}_{s}$, such that $\sum_{i}K_{i}^{\dagger}K_{i}\le\mathbb{I}$ and $\{|i_{E}\rangle\}$ is a basis for the environment with $i$ specifying number of excitations that is distributed in all its modes.
We observe that the quantum operation $\mathcal{E}$ can be used to describe general
transformations between quantum states \cite{Nielsen-Chuang,Preskill}.
One can verify that the dynamical map for the evolution of the system-environment
state:
\begin{equation}
U_{sE}|\psi_{s}\rangle\otimes|0_{E}\rangle=\sum_{i}K_{i}|\psi_{s}\rangle\otimes|i_{E}\rangle,
\end{equation}
leads to the same motion equation for the system state as shown above.
Thus, we can obtain the Kraus' operators describing the noise channel
acting on the system using a phenomenological approach \cite{Almeida-POA,Diogo-Dec-Control}
or via quantum process tomography \cite{Chuang.Nielsen-PT,Vandersypen.Chuang-RMP}
and use this information to investigate, for example, the system-environment
correlations, without worry about the usually complicated structure
of the environment.
\subsection{A partial classification of quantum states}
\label{state-classification}
A possible, partial, classification of multipartite quantum states
concerning its correlations, or with regard to the operations needed
to generate such correlations, can be introduced as follows. For an
$n$-partite system prepared in a product state vector $|\psi_{\mathrm{init}}\rangle=|\psi_{01}\cdots\psi_{0n}\rangle,$
any uncorrelated $n$-partite product state of this system,
\begin{equation}
\rho_{1\cdots n}^{p}=\rho_{1}\otimes\cdots\otimes\rho_{n},
\end{equation}
can be created by means of local quantum operations (LQO). The sub-index
$is$ in $|\psi_{\mathrm{init}}\rangle$ specifies the state and subsystem,
respectively, and we use throughout this article the notation $|\psi_{01}\cdots\psi_{0n}\rangle:=|\psi_{01}\rangle\otimes\cdots\otimes|\psi_{0n}\rangle$.
Starting with the system in state $|\psi_{\mathrm{init}}\rangle$,
any $n$-partite classically-correlated state can be prepared via
local classical operations (LCO) coordinated by the exchange of classical
communication (CC) and has the form:
\begin{equation}
\rho_{1\cdots n}^{c}=\sum_{i1,\cdots,in}p_{i1,\cdots,in}|\psi_{i1}\cdots\psi_{in}\rangle\langle\psi_{i1}\cdots\psi_{in}|,
\end{equation}
where the states $\{|\psi_{is}\rangle\}\in\mathcal{H}_{s}$ form a
complete ($\sum_{is}|\psi_{is}\rangle\langle\psi_{is}|=\mathbb{I}_{s}$)
orthonormal ($\langle\psi_{is}|\psi_{js}\rangle=\delta_{ij}$), and
therefore distinguishable, basis for the subsystem $s$. Above, by
LCO we mean (complete) transformations between the pointer basis states
\cite{Zurek-RMP}. It can be seen that if the probability distribution
in the state $\rho_{1\cdots n}^{c}$ factorizes, i.e., $p_{i1,\cdots,in}=p_{i1}\cdots p_{in}$,
there is no correlation at all in the system, that is to say, it is an $n$-partite
product state.
An $n$-partite separable but quantumly correlated state
needs LQO and CC to be generated from $|\psi_{\mathrm{init}}\rangle$.
This kind of state has the general form
\begin{equation}
\rho_{1\cdots n}^{s}={\textstyle \sum_{i}}p_{i}\rho_{i1}\otimes\cdots\otimes\rho_{in},
\end{equation}
with $\{p_{i}\}$ being a joint probability distribution and $\{\rho_{is}\}$
being noncommuting density operators. We note that if the density
operators $\rho_{is}$ commute for different $i$, then the state
$\rho_{1\cdots n}^{s}$ is an $n$-partite classically-correlated state.
If in addition the probability distribution $\{p_{i}\}$ factorizes,
then $\rho_{1\cdots n}^{s}$ is an $n$-partite product state.
The $n$-partite entangled, or non-separable, states $\rho_{1\cdots n}^{e}$
cannot be prepared locally, requiring direct or mediated interaction
for its generation. One famous example of entangled state is the GHZ
state \cite{GHZ,GHZ-E}:
\begin{equation}
2^{-1/2}(|0_{1}\cdots0_{n}\rangle+|1_{1}\cdots1_{n}\rangle),
\end{equation}
where $\{|0_{s}\rangle,|1_{s}\rangle\}$ is the
one-qubit computational basis.
\subsection{Relative entropy-based measures of correlation}
\label{subsec-rebq}
Considering the operational state classification of the last subsection and with the aim of quantifying the different kinds of correlation in an unified manner, Modi \emph{et al.} \cite{Modi-reldis} introduced measures of correlation using the relative entropy \cite{Vedral-RE},
\begin{equation}
S(\rho||\sigma)=\mathrm{tr}(\rho(\log_{2}\rho-\log_{2}\sigma)),
\end{equation}
to estimate the ``distance'' between two states $\rho$ and $\sigma$.
The total correlation in an $n$-partite state $\rho_{1\cdots n}$
is quantified by how distinguishable or how distant it is from an
uncorrelated $n$-partite product state \cite{Modi-reldis}, i.e.,
\begin{eqnarray}
I(\rho_{1\cdots n}) & = & \min_{\rho_{1\cdots n}^{p}}S(\rho_{1\cdots n}||\rho_{1\cdots n}^{p})\nonumber \\
& = & S(\rho_{1\cdots n}||\rho_{1}\otimes\cdots\otimes\rho_{n}).
\end{eqnarray}
The last equality was established in Ref. \cite{Modi-reldis} and
shows that the closest $n$-partite product state of any state $\rho_{1\cdots n}$
is obtained from its marginal states in the product form. Recalling
the state classification presented in the previous paragraph, the
quantum part of the correlation in $\rho_{1\cdots n}$ can be defined
as its minimal distance from $n$-partite classically-correlated states \cite{Modi-reldis}:
\begin{eqnarray}
Q(\rho_{1\cdots n}) & = & \min_{\rho_{1\cdots n}^{c}}S(\rho_{1\cdots n}||\rho_{1\cdots n}^{c})\nonumber \\
& = & S(\rho_{1\cdots n}||\chi_{1\cdots n}^{\rho}),
\end{eqnarray}
with $\chi_{1\cdots n}^{\rho}=\sum_{i1,\cdots,in}p_{i1,\cdots,in}|\psi_{i1}\cdots\psi_{in}\rangle\langle\psi_{i1}\cdots\psi_{in}|$
and $p_{i1,\cdots,in}=\langle\psi_{i1}\cdots\psi_{in}|\rho_{1\cdots n}|\psi_{i1}\cdots\psi_{in}\rangle$
\cite{Modi-reldis}. In the second equality for $Q$, we leave implicit
the minimization over local basis needed to find the closest classically-correlated
state $\chi_{1\cdots n}^{\rho}$. Finally, the classical part of the
correlations in $\rho_{1\cdots n}$ is given by the total correlation
of $\chi_{1\cdots n}^{\rho}$ \cite{Modi-reldis}:
\begin{eqnarray}
C(\rho_{1\cdots n}) & = & \min_{\rho_{1\cdots n}^{p}}S(\chi_{1\cdots n}^{\rho}||\rho_{1\cdots n}^{p}),
\end{eqnarray}
and quantifies how distant $\chi_{1\cdots n}^{\rho}$ is from being an uncorrelated state.
\subsection{Genuine multipartite correlations}
\label{subsec-gmc}
Though the state classification and the correlation quantifiers presented
above are very useful in several contexts, they are restricted in
a certain sense because they classify the system state without considering
the possibility of a more complex distribution for the correlations.
For instance, in studying multipartite spin systems, which may have
a far more intricate density operator with groups or cluster of spins
in different families of states, it would be desirable to generalize
the results presented in the last two paragraphs in order to include
and quantify such complexity. As we will discuss in the sequence,
the concept of genuine multipartite correlation fits well for the
study of these more general scenarios and can be utilized as the starting
point to define generalizations of the correlation quantifiers
discussed above.
Addressing this subject, Bennett \emph{et al.} (see
Ref. \cite{Bennett-genuine})\emph{ }postulated that if an $n$-partite
state has no genuine $n$-partite correlation, then we cannot create
genuine $n$-partite correlation by adding a subsystem in a product
state or by performing trace non-increasing local quantum operations.
Also, we cannot create genuine ($n+1$)-partite correlation by splitting
a subsystem in two. They proved that the following definitions satisfy
these requirements. An $n$-partite state has genuine $n$-partite
correlation only if it is non-product under any bipartite cut of the
system. For $k\le n$, an $n$-partite state has genuine $k$-partite
correlation only if there exists a subset of $k$ subsystems presenting
genuine $k$-partite correlation. Bennett and collaborators also defined
the degree of correlation of an $n$-partite state as the maximum number
$k$ of subsystems possessing genuine $k$-partite correlation.
\section{Quantifiers for Genuine Multipartite Correlations}
\label{sec:quantifiers}
The investigation of the different types of correlation presented in quantum states is one of the main problems in quantum information science. Recently the identification and quantification of genuine multipartite correlations (which are those correlations we cannot account looking only to a part of the system) has been receiving its first studies. In this section we shall introduce definitions and quantifiers for genuine multipartite correlations by using the concepts reviewed in Section \ref{subsec-gmc} to generalize the measures discussed in Section \ref{subsec-rebq}.
From the definition given in Section \ref{subsec-gmc}, it follows that an $n$-partite state
has genuine $n$-partite correlation only if it is non-product under
any bipartite cut of the system, i.e., if $\rho_{1\cdots n}\ne\rho_{c_{1}}\otimes\rho_{c_{2}},$
where $c_{1}\mbox{ }(c_{2})$ indicates a group with a number $n_{1}\mbox{ }(n_{2})$
of subsystems, and $n_{1}+n_{2}=n$. In other words, if $\rho_{1\cdots n}$
possesses genuine $n$-partite correlation then there does not exist
two completely uncorrelated clusters of particles in the system. Thus,
a measure for \textit{genuine $n$-partite total correlation} of an $n$-partite
state can be logically defined as
\begin{equation}
I_{n}(\rho_{1\cdots n}):=\min_{(c_{1},c_{2})}S(\rho_{1\cdots n}||\rho_{c_{1}}\otimes\rho_{c_{2}}),
\end{equation}
where the minimum is taken over all possible bi-partitions $(c_{1},c_{2})$
of the system. Therefore the quantity $I_{n}$ measures the minimum
distance between $\rho_{1\cdots n}$ and states that do not possess
genuine $n$-partite correlation.
Starting from a possible generalization of Shannon's mutual information for tripartite systems, the quantifier for genuine $n$-partite total correlation $I_{n}(\rho_{1\cdots n})$ was also noticed by Giorgi
and colleagues in Ref. \cite{Giorgi-genuine}. They also defined measures for genuine classical and quantum correlations using a quantum discord-like approach, i.e., assuming additivity of mutual information in classical
and quantum correlations. As shown in Ref. \cite{Comment-Paper-Giorgi}, the complicated alternative definitions presented in Ref. \cite{Giorgi-genuine} for genuine $n$-partite quantum and classical correlation of $n$-partite states do not coincide in general.
In sequence of this article, we shall start from a direct extension of the operational classification of states discussed in Section \ref{state-classification} and use relative entropy-based distinguishability measures to define simple and general correlation quantifiers for genuine multipartite correlations that are valid for any number of subsystems and for any dimension of its Hilbert spaces. It is also worthwhile to mention that, although we choose the relative entropy as a measure of distance, our quantifiers for correlation can be defined in a similar manner using other distance measures.
Regarding total correlations, we can also look to the structure of the system's correlation by defining a measure for \textit{genuine $k$-partite total correlation} of an $n$-partite state:
\begin{equation}
I_{k}(\rho_{1\cdots n}):=\max_{\rho_{k}}I_{k}(\rho_{k}).
\end{equation}
In the last equation, the maximum is taken over all states $\rho_{k}$ comprising $k$ subsystems and aims to identify the most correlated $k$-partite group of particles in the system. For example, if $k=n-1$, $I_{n-1}(\rho_{1\cdots n})$ is the maximal genuine $(n-1)$-partite total correlation of the states obtained by tracing out one subsystem of $\rho_{1\cdots n}$.
This last quantifier, $I_{k}(\rho_{1\cdots n})$, can be applied to study the \textit{degree of correlation} (as defined in Section \ref{subsec-gmc}) of a system and can find application, for instance, in the study of quantum phase transitions in critical systems
\cite{Sachdev}.
Based on the discussion about classically-correlated states and of the quantum correlation present in non-classical states, addressed in the introductory section, we propose the following definition.
\textit{Definition 1.} An $n$-partite state
$\rho_{1\cdots n}$ has genuine $n$-partite quantum correlation only
if it is not a classical state under any bipartite cut of the system,
viz., $\rho_{1\cdots n}\ne\chi_{c_{1}c_{2}}^{\rho},$ with $\chi_{c_{1}c_{2}}^{\rho}=\sum_{ic_{1},ic_{2}}p_{ic_{1},ic_{2}}|\psi_{ic_{1}}\psi_{ic_{2}}\rangle\langle\psi_{ic_{1}}\psi_{ic_{2}}|,$
where $\{p_{ic_{1},ic_{2}}=\langle\psi_{ic_{1}}\psi_{ic_{2}}|\rho_{1\cdots n}|\psi_{ic_{1}}\psi_{ic_{2}}\rangle\}$
is a probability distribution and $\{|\psi_{ic_{s}}\rangle\}$ is
an orthonormal basis for $\mathcal{H}_{c_{s}}$.
From this definition, a quantifier of \textit{genuine $n$-partite quantum correlation} of an $n$-partite state follows as
\begin{equation}
Q_{n}(\rho_{1\cdots n}):=\min_{(c_{1},c_{2})}S(\rho_{1\cdots n}||\chi_{c_{1}c_{2}}^{\rho}).
\end{equation}
$Q_{n}(\rho_{1\cdots n})$ measures the minimum distance between $\rho_{1\cdots n}$ and the bipartite classical states $\chi_{c_{1}c_{2}}^{\rho}$. By definition, $Q_{n}(\rho_{1\cdots n})\ge0$ with equality only in cases where $\rho_{1\cdots n}=\chi_{c_{1}c_{2}}^{\rho}$.
Now a measure for \textit{genuine $k$-partite quantum correlation} of an $n$-partite system
is defined in an analogous manner to the genuine $k$-partite total correlation introduced above, namely:
\begin{equation}
Q_{k}(\rho_{1\cdots n}):=\max_{\rho_{k}}Q_{k}(\rho_{k}).
\end{equation}
Using this quantifier, we define the \textit{degree of quantumness}, or degree of quantum correlation, of an $n$-partite system as the maximum number $k$ of its subsystems that presents genuine $k$-partite quantum correlation.
One possible definition for genuine $n$-partite classical correlation
of an $n$-partite state is given as follows.
\textit{Definition 2.} An $n$-partite state
$\rho_{1\cdots n}$ possesses genuine $n$-partite classical correlation
only if its closest $n$-partite classical state $\chi_{1\cdots n}^{\rho}$
has genuine $n$-partite total correlation, namely, if the classical
probability distribution $p_{i1,\cdots,in}$ does not factorizes under
any bipartite cut of $\chi_{1\cdots n}^{\rho}$, i.e., $p_{i1,\cdots,in}\ne p_{c_{1}}p_{c_{2}}.$
Based on this definition, we propose the following quantifiers for
\textit{genuine $n$-partite classical correlation}:
\begin{equation}
C_{n}(\rho_{1\cdots n}):=I_{n}(\chi_{1\cdots n}^{\rho}),
\end{equation}
and for \textit{genuine $k$-partite classical correlation}:
\begin{equation}
C_{k}(\rho_{1\cdots n})=\max_{\chi_{k}^{\rho}}I_{k}(\chi_{k}^{\rho}),
\end{equation}
of an $n$-partite state. The states $\chi_{k}^{\rho}$ in the last
equation are obtained by tracing out all but the chosen $k$ subsystems
of $\chi_{1\cdots n}^{\rho}$ and the maximization is made over all
possibilities for $\chi_{k}^{\rho}$.
Now, using $C_{k}$, we can define the \textit{degree of classical correlation} of an $n$-partite state as the maximum number $k$ of subsystems possessing genuine $k$-partite classical correlation.
In the next section we will
apply some of the quantifiers of genuine $n$-partite correlations
introduced here to study the generation of genuine multipartite system-environment correlations
in some decoherent dynamics.
\section{System-Environment Correlations in Decoherent Dynamics}
\label{sec:dynamics}
Let us consider a two-qubit system initially prepared in a Werner's state
\begin{equation}
\rho_{ab}^{w}=(1-c)\mathbb{I}_{ab}/4+c|\psi_{ab}^{-}\rangle\langle\psi_{ab}^{-}|,
\end{equation}
where $|\psi_{ab}^{-}\rangle=(|0_{a}1_{b}\rangle-|1_{a}0_{b}\rangle)/\sqrt{2}$ and $0\le c\le1$. This state has a rich structure with respect to correlations. It violates the CHSH inequality \cite{CHSH} for $c\ge 1/2$, it violates the Peres' criterion for separability \cite{PeresC} when $c> 1/3$, and it has nonzero quantum discord \cite{Ollivier-QD} for all $c\ne 0$. So, any possible qualitative difference in the multipartite system-environment correlations due to system's initial correlations (i.e., if the system state is nonlocal, nonseparable, discordant or classical) would be observed using the Werner's state as the system's initial state.
Now these two sub-systems are let to interact locally with two independent
environments in the vacuum state $|0_{E_{s}}\rangle$, where $s=a,b$.
So, the initial state of the whole system is
\begin{equation}
\rho_{abE_{a}E_{b}}=\rho_{ab}^{w}\otimes|0_{E_{a}}\rangle\langle0_{E_{a}}|\otimes|0_{E_{b}}\rangle\langle0_{E_{b}}|.
\end{equation}
For this system, it was shown in Ref. \cite{Maziero-PRA} that, in
contrast to dissipative interactions, the decoherent dynamics of the
two qubits under phase-damping or Pauli channels does not generate
entanglement between the systems and its respective environments or
between the two environments. The bipartite quantum discord created
in such a dynamics was then indicated as the mechanism for the leakage
of quantum information out of the systems. Moreover, the initial quantum
correlation between the two qubits was shown to be not transfered
to the environments, seeming to evaporate.
Here we extend these results
by studying the generation of genuine multipartite system-environment
correlations. We show that the quantum correlations initially shared
between the qubits do not disappear, but are transformed into genuine
multipartite correlations between systems and environments. Furthermore,
these correlations present an interesting dynamics with sudden changes
in behavior. For more results related to the sudden-change phenomenon
see Refs. \cite{Maziero-SC,Auccaise-SC,Xu-SC-Nature,Mazzola-SC,Cornelio-SC}.
Further works considering the dynamics of system-environment correlations
can be found in Refs. \cite{Lopez-ESB,Zhang-SE,Pernice-SE,Luo-Dec-Cap,Man-SE,Fanchini-SE}.
\subsection{Amplitude-damping channels}
We begin by studying the situation in which the two qubits, $a$ and
$b$, evolve under the influence of local-independent amplitude-damping
channels. The Kraus' operators for a dissipative reservoir at zero
temperature are $K_{0}=|0_{s}\rangle\langle0_{s}|+\sqrt{1-p_{s}}|1_{s}\rangle\langle1_{s}|$
and $K_{1}=\sqrt{p_{s}}|0_{s}\rangle\langle1_{s}|$, where $p_{s}$
is a parametrization of time for the subsystem $s$, with $p=0$ corresponding
to $t=0$ and $p=1$ being equivalent to $t\rightarrow\infty$ \cite{Salles-PRA}.
Throughout this article we consider identical environments and consequently
$p_{a}=p_{b}:=p$. Thus, by using the Kraus operators shown above,
we obtain the dynamical map for the system-environment evolution:
\begin{eqnarray}
U_{sE_{s}}|0_{s}0_{E_{s}}\rangle & = & |0_{s}0_{E_{s}}\rangle,\\
U_{sE_{s}}|1_{s}0_{E_{s}}\rangle & = & \sqrt{1-p_{s}}|1_{s}0_{E_{s}}\rangle+\sqrt{p_{s}}|0_{s}1_{E_{s}}\rangle.\nonumber
\end{eqnarray}
Utilizing this map, we find the evolved state for the whole system:
\begin{equation}
\rho_{abE_{a}E_{b}}^{ad}(p)=(1-c)\iota_{ad}(p)+c|\Upsilon_{ad}(p)\rangle\langle\Upsilon_{ad}(p)|,\label{eq:Global-Evol-state}
\end{equation}
with $\iota_{ad}(p)$ and $|\Upsilon_{ad}(p)\rangle$ given in Appendix
\ref{appendix-A}.
From $\rho_{abE_{a}E_{b}}^{ad}(p)$ we can access
the state of any partition of the system and calculate numerically
its entropy and correlations.
\begin{figure}
\caption{Genuine $4$- and $3$-partite total correlation of the whole system
$aE_{a}
\label{gtc-ad}
\end{figure}
\begin{figure}
\caption{Multipartite system-environment quantum correlations for local amplitude-damping
channels. For $p=1$ the systems state is $|0_{s}
\label{qc-ad}
\end{figure}
\begin{figure}
\caption{Genuine multipartite system-environment classical correlation for
local amplitude-damping channels. The generic behavior of these correlations
is similar to that of genuine multipartite total correlation.}
\label{gcc-ad}
\end{figure}
Here we will concentrate on multipartite correlations. A detailed
analysis of the bipartite correlations can be found in Ref. \cite{Maziero-PRA}.
We observe that, due to the assumed symmetries of the systems and
environments, the following partitions are equivalent in what refer
to correlations: $aE_{a}\equiv bE_{b}$, $aE_{b}\equiv bE_{a}$, $abE_{a}\equiv abE_{b}$,
and $aE_{a}E_{b}\equiv bE_{a}E_{b}$. Considering these symmetries we have,
from the definition of genuine $n$-partite total correlation introduced
in Section II, that
\begin{equation}
I_{4}(\rho_{aE_{a}bE_{b}})=\min_{(c_{1},c_{2})}(S(\rho_{c_{1}})+S(\rho_{c_{2}}))-S(\rho_{aE_{a}bE_{b}}^{ad}),
\end{equation}
with the following possible bi-partitions of the system: $(c_{1},c_{2})=(a,E_{a}bE_{b})$,$(E_{a},abE_{b})$,$(ab,E_{a}E_{b})$,$(aE_{a},bE_{b})$,
$(aE_{b},bE_{a})$. In the last equation $S(\rho)=-\mathrm{tr}(\rho\log_{2}\rho)$ is the von Neumann entropy.
The $3$-partite genuine total correlation of
the whole system is given by
\begin{equation}
I_{3}(\rho_{aE_{a}bE_{b}}^{ad})=\max(I_{3}(\rho_{aE_{a}b}^{ad}),I_{3}(\rho_{aE_{a}E_{b}}^{ad}))
\end{equation}
with analogous bi-partitions used to compute the genuine $3$-partite
correlation of the $3$-partite states. All genuine total correlations
discussed above are shown in Fig. \ref{gtc-ad}. We see that, in general,
multipartite correlations are generated during the dissipative interaction
between the qubits and its respective reservoirs up to a certain instant
of time from which such correlations \textit{suddenly} begin to decrease going
to zero in the asymptotic time $p=1$.
We also calculated the $3$-
and $4$-partite quantum correlations and genuine classical correlations
of the system, which are presented in Figs. \ref{qc-ad} and \ref{gcc-ad},
respectively. As expected, the asymptotic behavior of the genuine
3- and 4-partite classical correlations is similar to that of genuine
multipartite total correlation. We observe that the quantum correlation
remaining at $p=1$ is due solely to the environments, once the systems
state in this limit is $|0_{s}\rangle$.
It is worthwhile mentioning that,
although explicit parametrizations for states and unitary operators
of systems with dimension greater than two are possible in principle
\cite{Bruning-Par}, all the important aspects we want to emphasize
here can be addressed without computing the genuine multipartite quantum
correlations. We leave related issues for future investigations.
\subsection{Phase-damping channels}
\begin{figure}
\caption{Genuine $4$- and $3$-partite total correlation of the whole system
$aE_{a}
\label{gtc-pd}
\end{figure}
Let us consider the dynamics of two qubits under local phase-damping
channels. This kind of noise environment causes loss of phase relations
in the system without exchange of energy. The Kraus' operators for
these channels are given by: $K_{0}=|0_{s}\rangle\langle0_{s}|+\sqrt{1-p}|1_{s}\rangle\langle1_{s}|$
and $K_{1}=\sqrt{p}|1_{s}\rangle\langle1_{s}|$. Thus, the following
map for the system-environment evolution is obtained
\begin{eqnarray}
U_{sE}|0_{s}0_{E_{s}}\rangle & = & |0_{s}0_{E_{s}}\rangle,\\
U_{sE}|1_{s}0_{E_{s}}\rangle & = & |1_{s}\rangle\otimes\left(\sqrt{1-p}|0_{E_{s}}\rangle+\sqrt{p}|1_{E_{s}}\rangle\right).\nonumber
\end{eqnarray}
In a correspondent manner as we did for amplitude-damping channels,
we use the map shown in the last equation to compute the global evolved
state and then calculate its correlations. In this case, $\rho_{aE_{a}bE_{b}}^{pd}(p)$
is given as in Eq. (\ref{eq:Global-Evol-state}) but with $\iota_{pd}(p)$
and $|\Upsilon_{pd}(p)\rangle$ presented in Appendix \ref{Appendix-B}.
The genuine total correlations generated in the evolution under local
phase environments are shown in Fig. \ref{gtc-pd}. These correlations also exhibit the sudden-change phenomenon.
\begin{figure}
\caption{Dynamics of $3$- and $4$-partite system-environment quantum correlations
for local phase-damping channels. Tripartite quantum correlations
are generated during the system's evolution but decay to zero at $p=1$
while there still exists multipartite quantum correlation involving
the whole system in this asymptotic limit. }
\label{qc-pd}
\end{figure}
\begin{figure}
\caption{Genuine $3$- and $4$-partite system-environment classical correlation
for local phase-damping channels. By comparing these results with
the genuine total correlations in Fig. \ref{gtc-pd}
\label{gcc-pd}
\end{figure}
In sharp contrast to the case of amplitude-damping channels, we see
that the dynamics induced by phase environments does generate genuine
multipartite correlations in the asymptotic limit $p=1$. In this
limit, the whole system will be correlated in a degree that is proportional
to the purity of the two-qubit initial state.
We also calculated the
multipartite quantum correlations and the genuine $3$- and $4$-partite
classical correlations, which are presented in Figs. \ref{qc-pd}
and \ref{gcc-pd}, respectively. It is interesting that the genuine
$3$-partite classical and total correlations range from zero to one
for this kind of environment. This fact indicates that the system presents
no or small genuine $3$-partite quantum correlation. As can be seen
in Figs. \ref{gtc-pd}-\ref{gcc-pd}, this situation changes in the
case of $4$-partite correlations. In fact, we observe in Fig. \ref{qc-pd}
that tripartite quantum correlations are created during the system's
evolution but disappear in the asymptotic limit while there are $4$-partite
quantum correlation remnant at $p=1$.
\section{Concluding Remarks}
Summing up, we introduced operational definitions and quantifiers for genuine $n$- and $k$-partite total, quantum, and classical correlations of an $n$-partite state. We also used our correlation quantifiers to define the degree of correlation, the degree of quantumness, and the degree of classical correlation of a physical system.
Using these correlation measures, we showed that, in contrast to amplitude-damping
channels, for which the initial correlations between the qubits are
simply transfered to the environments, phase noise channels turn such
bipartite correlations into genuine multipartite system-environment
total, quantum, and classical correlations.
Now, in order to get a better grasp of these results, let us look
at the kind of state generated during the evolution of the system
under different kinds of noise environment. It is straightforward
to verify that if the system initial state is $|\psi_{ab}^{-}\rangle$
(i.e., if $c=1$), the whole system state for amplitude-damping channels
and $p=1/2$ is
\begin{eqnarray}
|\Upsilon_{ad}(1/2)\rangle & = & (|0001\rangle+|0010\rangle-|0100\rangle-|1000\rangle)/2\nonumber \\
& := & |\psi_{W}\rangle,
\end{eqnarray}
which is equivalent, modulo local rotations, to the well known four-qubit
$W$ state \cite{W-state}. For phase noise environments and $p=1$,
it follows that
\begin{eqnarray}
|\Upsilon_{pd}(1)\rangle & = & (|0011\rangle-|1100\rangle)/\sqrt{2}\nonumber \\
& := & |\psi_{GHZ}\rangle,
\end{eqnarray}
which is, also modulo local unitaries, equivalent to a $GHZ$ state
(see Section I).
\begin{figure}
\caption{Dependence of the fidelities (a) $F(|\psi_{W}
\label{fidelities}
\end{figure}
For any $c$ and $p$, we obtain the following expressions for the fidelity,
\begin{equation}
F(|\psi\rangle,\rho):=\sqrt{\langle\psi|\rho|\psi\rangle},
\end{equation}
between these states and its corresponding evolved density operators:
\begin{equation}
F(|\psi_{W}\rangle,\rho_{aE_{a}bE_{b}}^{ad}(p))=\sqrt{\frac{(1+3c)(1+2\sqrt{p(1-p)})}{8}}
\end{equation}
and
\begin{equation}
F(|\psi_{GHZ}\rangle,\rho_{aE_{a}bE_{b}}^{pd}(p))=\sqrt{(1+3c)p}/2.
\end{equation}
These fidelities are shown in Fig. \ref{fidelities}.
We also observe that the mutual information is invariant under local unitary
transformations and, as one can easily verify, the genuine total correlations
for the $W$ and $GHZ$ states are given by: $I_{4}(|W\rangle)=I_{4}(|GHZ\rangle)=2$,
$I_{3}(|W\rangle)=0.81$, and $I_{3}(|GHZ\rangle)=1$. Moreover, if
we trace out a subsystem of a $GHZ$ state, the obtained tripartite
density operator has zero quantum correlations. The same action in
a $W$ state produces a $3$-partite system possessing quantumness
in its correlations. Thus, the values of the fidelities (shown in
Fig. \ref{fidelities}) and the structure of $W$ and $GHZ$ states
with respect to its correlations help us to partially explain the
general behavior of the correlations that we presented and discussed
in the last section. In reality, both kinds of decoherent dynamics generated
genuine multipartite correlations. The main difference is that for
the amplitude-damping channel these correlations are null for $p=1$
while for phase environments they generally reach its maximum value
in this limit.
Thorough investigations about the decoherence process are essential for us to obtain a better understanding of the phenomenon per se and also for the development of methods to circumvent it in the path for large scale implementations of protocols in quantum information science. From the fundamental point of view, before 2010 one believed that the flow of coherent information from the system to the environment was caused by the creation of entanglement between the two. Nevertheless, considering bipartite correlations, one of us and colleagues showed, in Ref. [57], that this is not the case in general. For some composite systems interacting with local-independent phase-damping channels (or Pauli channels), it was shown that the systems lose its initial coherent phase relations but only bipartite non-classical correlation of separable states are generated during such decoherent dynamics. Another interesting finding reported in this article was the fact that the initial bipartite quantum correlation between the
systems (non-local, non-separable, and discordant) evaporated, i.e., only bipartite classical correlations was present at the asymptotic time of evolution. In the present manuscript, besides proposing definitions and introducing quantifiers for genuine multipartite correlations, we studied the many-body correlations generated for some important decoherent processes. Our investigation extended the previous ones in several directions, helping us to understand the global structure of the states generated during these evolutions and also explaining why the bipartite quantum correlations disappeared in the asymptotic time, by showing that they are transformed into genuine multipartite correlations. Moreover, we showed that the genuine multipartite total correlation may also exhibits a sudden change in its evolution rate.
As continuation of the present work, besides the actual calculation of the genuine multipartite quantum correlations, other interesting topic for future research is considering the dynamics of such correlations for the composition of both phase and (generalized) amplitude channels, which is a common situation in nature \cite{Auccaise-SC,Diogo-Dec-Control}.
In this case, the global state generated during the evolution under local environments will be a mixture involving W and GHZ components. The extension of these results for global and correlated environments \cite{Diogo-quad} and to continuous variables systems \cite{Eisert-Plenio} is also worth pursuing.
It is also important to mention that the first experiment with complete
tomography of the environment's state has been successfully performed
recently using an optical system \cite{Tom-Env}. Therefore, all the
theoretical results presented here can be verified experimentally
with current technology.
\begin{acknowledgments}
The authors acknowledge financial support from the Brazilian funding agencies Coordena\c{c}\~ao de Aperfei\c{c}oamento de Pessoal de N\'ivel Superior, Conselho Nacional de Desenvolvimento Cient\'ifico e Tecnol\'ogico, and Funda\c{c}\~ao de Amparo \`a Pesquisa do Estado do Rio Grande do Sul. J.M. thanks Marcelo S. Sarandy and Gabriel H. Aguilar for helpful discussions. We acknowledge a referee for his (her) constructive comments.
\end{acknowledgments}
\appendix
\section{Evolved State for Local Amplitude-Damping Channels}
\label{appendix-A}
For this channels, the global evolved state is given by Eq. (\ref{eq:Global-Evol-state})
with
\begin{eqnarray}
4\iota_{ad}(p) & = & |0\rangle\langle0|+(1-p)(|2\rangle\langle2|+|8\rangle\langle8|)\nonumber \\
& & +p^{2}|5\rangle\langle5|+(1-p)^{2}|10\rangle\langle10|\nonumber \\
& & +p(1-p)(|6\rangle\langle6|+|9\rangle\langle9|)+p(|4\rangle\langle4|+|1\rangle\langle1|)\nonumber \\
& & +\sqrt{p(1-p)}(|4\rangle\langle8|+|1\rangle\langle2|+\mathrm{h.c.})\nonumber \\
& & +\sqrt{p(1-p)^{3}}((|6\rangle+|9\rangle)\langle10|+\mathrm{h.c.})\nonumber \\
& & +p(1-p)(|5\rangle\langle10|++|6\rangle\langle9|+\mathrm{h.c.})\nonumber \\
& & +\sqrt{p^{3}(1-p)}(|5\rangle(\langle6|+\langle9|)+\mathrm{h.c.})
\end{eqnarray}
and
\begin{equation}
|\Upsilon_{ad}(p)\rangle=\sqrt{\frac{1-p}{2}}(|2\rangle-|8\rangle)+\sqrt{\frac{p}{2}}(|1\rangle-|4\rangle).
\end{equation}
Above, and in the next appendix, $\mathrm{h.c.}$ refers to the Hermitian
conjugate and we use the decimal representation for the indexes of
the computational bases states.
\section{Evolved State for Local Phase-Damping Channels}
\label{Appendix-B}
For phase environments, the global evolved state is given as in Eq.
(\ref{eq:Global-Evol-state}) but with
\begin{eqnarray}
4\iota_{pd}(p) & = & |0\rangle\langle0|+(1-p)(|2\rangle\langle2|+|8\rangle\langle8|)\nonumber \\
& & +p(|3\rangle\langle3|+|12\rangle\langle12|)+(1-p)^{2}|10\rangle\langle10|\nonumber \\
& & +p(1-p)(|11\rangle\langle11|+|14\rangle\langle14|)+p^{2}|15\rangle\langle15|\nonumber \\
& & +\sqrt{p(1-p)}(|2\rangle\langle3|+|8\rangle\langle12|+\mathrm{h.c.})\nonumber \\
& & +\sqrt{p(1-p)^{3}}((|14\rangle+|11\rangle)\langle10|+\mathrm{h.c.})\nonumber \\
& & +p(1-p)(|11\rangle\langle14|+|10\rangle\langle15|+\mathrm{h.c.})\nonumber \\
& & +\sqrt{p^{3}(1-p)}(|15\rangle(\langle11|+\langle14|)+\mathrm{h.c.})
\end{eqnarray}
and
\begin{equation}
|\Upsilon_{pd}(p)\rangle=\sqrt{\frac{1-p}{2}}(|2\rangle-|8\rangle)+\sqrt{\frac{p}{2}}(|3\rangle-|12\rangle).
\end{equation}
\end{document} |
{\mathfrak m}athfrak{b}egin{document}
\subjclass[2010]{13A18 (12J10, 12J20, 14E15)}
{\mathfrak m}athfrak{b}egin{abstract}
For an arbitrary valued field $(K,v)$ and a given extension $v(K^*){\mathfrak m}athfrak{h}k \Lambda$ of ordered groups, we analyze the structure of the tree formed by all $\Lambda$-valued extensions of $v$ to the polynomial ring $\op{Ker}x$. As an application, we find a model for the tree of all equivalence classes of valuations on $\op{Ker}x$ (without fixing their value group), whose restriction to $K$ is equivalent to $v$.
{\mathfrak m}edskipnd{abstract}
{\mathfrak m}aketitle
\section*{Introduction}
A valuation on a commutative ring $A$ is a mapping ${\mathfrak m}u{\mathfrak m}athfrak{c}olon A\thetao \Lambda\infty$,
where $\Lambda$ is an ordered abelian group, satisfying the following conditions:
(0) \ ${\mathfrak m}u(1)=0$, \ ${\mathfrak m}u(0)=\infty$,
(1) \ ${\mathfrak m}u(ab)={\mathfrak m}u(a)+{\mathfrak m}u(b),\mathfrak{q}uad{\mathfrak m}athfrak{f}orall\,a,b\in A$,
(2) \ ${\mathfrak m}u(a+b)\Gammae{\mathfrak m}in\{{\mathfrak m}u(a),{\mathfrak m}u(b)\},\mathfrak{q}uad{\mathfrak m}athfrak{f}orall\,a,b\in A$.{\mathfrak m}edskip
The {\mathfrak m}edskipmph{support} of ${\mathfrak m}u$ is the prime ideal $\mathfrak{p}=\op{supp}({\mathfrak m}u)={\mathfrak m}u^{-1}(\infty)\in\operatorname{Spec}(A)$.
The {\mathfrak m}edskipmph{value group} of ${\mathfrak m}u$ is the subgroup $\Gammam\subset \Lambda$ generated by ${\mathfrak m}u{\mathfrak m}athfrak{l}eft(A\setminus\mathfrak{p}\rho_ight)$.
Two valuations ${\mathfrak m}u$, $\mathbf{n}u$ on $A$ are {\mathfrak m}edskipmph{equivalent} if there is an isomorphism of ordered groups $\iota{\mathfrak m}athfrak{c}olon \Gammam {\mathfrak m}athfrak{l}ower.3ex\hbox{\ars{.08}$\begin{array}{c}\,\thetao\\{\mathfrak m}box{\thetainy $\sim\,$}\end{array}$}\Gamman$ such that $\mathbf{n}u=\iota{\mathfrak m}athfrak{c}irc {\mathfrak m}u$.
In this case, we write ${\mathfrak m}u\sim\mathbf{n}u$.
The {\mathfrak m}edskipmph{valuative spectrum} of $A$ is the set $\op{Spec}v(A)$ of equivalence classes of valuations on $A$. We denote by $[{\mathfrak m}u]\in\op{Spec}v(A)$ the equivalence class of ${\mathfrak m}u$.
Any ring homomorphism $A\thetao B$ induces a restriction of valuations which behaves well on equivalence classes and determines a mapping $\op{Spec}v(B)\thetao\op{Spec}v(A)$.
For any field $K$ we may consider the relative affine line $\op{Spec}v(\op{Ker}x)\thetao\op{Spec}v(K)$.
Given any valuation $v$ on $K$, the fiber $\thetatt_v$ of the equivalence class $[v]\in\op{Spec}v(K)$ is called the {\mathfrak m}edskipmph{valuative tree} over the valued field $(K,v)$.
$$
{\mathfrak m}athfrak{a}rs{1.2}
{\mathfrak m}athfrak{b}egin{array}{ccc}
\thetatt_v&{\mathfrak m}athfrak{h}ra &\op{Spec}v(\op{Ker}x)\\
\Deltaownarrow&&\Deltaownarrow\\
{\mathfrak m}box{$[v]$}&{\mathfrak m}athfrak{h}ra &\op{Spec}v(K)
{\mathfrak m}edskipnd{array}
$$
This terminology is borrowed from Favre and Jonsson's book {\mathfrak m}athfrak{c}ite{FJ}, where valuations of certain 2-dimensional local rings are studied.
In the case $\op{rk}(\Gamma)=1$ and $K$ algebraically closed, the valuative tree admits a structure of a Berkovich space and has relevant analytical properties {\mathfrak m}athfrak{c}ite{Bch}.
The main aim of this paper is to obtain a thorough description of the tree $\thetatt_v$ for an arbitrary valued field $(K,v)$.
In the first part of the paper, composed of sections 1--5, we fix an extension $\Gamma{\mathfrak m}athfrak{h}k\Lambda$ of ordered abelian groups, and we describe the tree $\thetatt=\thetatt(\Lambda)$ formed by all $\Lambda$-valued extensions of $v$ to $\op{Ker}x$.
Section 1 includes some background on key polynomials of valued fields. For any valuation ${\mathfrak m}u$ on $\op{Ker}x$, let $\op{Ker}pm$ be the set of Maclane-Vaqui\'e key polynomials for ${\mathfrak m}u$. If $\op{Ker}pm\mathbf{n}e{\mathfrak m}edskipmptyset$, then ${\mathfrak m}u$ has a {\mathfrak m}edskipmph{degree} and a {\mathfrak m}edskipmph{singular value}, defined as
$\Deltaeg({\mathfrak m}u)=\Deltaeg(\mathfrak{p}hi)$, $\op{SV}al({\mathfrak m}u)={\mathfrak m}u(\mathfrak{p}hi)$,
for any key polynomial $\mathfrak{p}hi\in\op{Ker}pm$ of minimal degree.
Section 2 discusses tangent directions and the tangent space of $\thetatt$. The leaves of $\thetatt$ (maximal nodes) are characterized by the property $\op{Ker}pm={\mathfrak m}edskipmptyset$. The set of tangent directions of an inner node ${\mathfrak m}u\in\thetatt$ is parametrized by the set $\op{Ker}pm/\!\sim_\mu$ of ${\mathfrak m}u$-equivalence classes of key polynomials.
Section 3 describes the set ${\mathfrak m}athfrak{l}fin(\thetatt)$ of {\mathfrak m}edskipmph{finite leaves} of the tree, determined by all valuations with non-trivial support. There is a bijection between ${\mathfrak m}athfrak{l}fin(\thetatt)$ and the set of monic irreducible polynomials in $\op{Ker}h[x]$, where $\op{Ker}h$ is a henselization of $K$. This result is just a reformulation of classical valuation-theoretic results.
Section 4 describes the set of {\mathfrak m}edskipmph{infinite leaves} of $\thetatt$, which are a kind of limit of certain totally ordered families of inner nodes of $\thetatt$. This section contains a detailed analysis of {\mathfrak m}edskipmph{limit augmentations} of valuations too. This concept was introduced by Vaqui\'e in his fundamental papers {\mathfrak m}athfrak{c}ite{Vaq0,Vaq} extending to arbitrary valued fields the pioneering work of Maclane for discrete rank-one valued fields {\mathfrak m}athfrak{c}ite{mcla}.
Limit augmentations are based on {\mathfrak m}edskipmph{continuous families} of iterated augmentations. In the literature, we find different conditions imposed on these families, serving different purposes. We define a continuous family as a totally ordered family of valuations in $\thetatt$, containing no maximal element, and having a stable degree. There is a natural equivalence relation between these families and we show, in Lemma \ref{specialCont}, that every equivalence class of continuous families contains a family satisfying all relevant conditions that are attributed to these families in the literature.
Section 5 gives a detailed description of $\thetatt$. Section 5.1 reviews the fundamental result of Maclane-Vaqui\'e describing how to reach all nodes of $\thetatt$ by a combination of ordinary augmentations, limit augmentations and stable limits. Every node ${\mathfrak m}u\in\thetatt$ may be linked to some degree-one node in $\thetatt$ by an essentially unique {\mathfrak m}edskipmph{Maclane-Vaqui\'e (MLV) chain}, supporting data intrinsically associated to ${\mathfrak m}u$ {\mathfrak m}athfrak{c}ite{MLV}. For instance, each node ${\mathfrak m}u\in\thetatt$ has a {\mathfrak m}edskipmph{depth}, defined as the length of its MLV chain, which is either a natural number or infinity. In Section 5.2, we show that these intrinsic data encode arithmetic or geometric invariants of ${\mathfrak m}u$, depending on the context in which the base valued field $(K,v)$ is considered. Sections 5.3--5.5, describe the different kinds of paths we may find in $\thetatt$. In Section 5.6, we prove that every two nodes of $\thetatt$ have a greatest common lower node in $\thetatt$ and relate our description of $\thetatt$ to the notion of $\Lambdambda$-tree.
In the second part of the paper, composed of sections 6--7, we find a concrete model for the valuative tree $\thetatt_v$.
For any valuation ${\mathfrak m}u$ on $\op{Ker}x$ extending $v$, the embedding $\Gamma{\mathfrak m}athfrak{h}k\Gammam$ is a {\mathfrak m}edskipmph{small extension} of ordered groups; that is, if $\Gamma'\subset\Gammam$ is the relative divisible closure of $\Gamma$ in $\Gammam$, then the quotient $\Gammam/\Gamma'$ is a cyclic group {\mathfrak m}athfrak{c}ite[Thm. 1.5]{Kuhl}.
In {\mathfrak m}athfrak{c}ite{csme}, a universal extension $\Gamma{\mathfrak m}athfrak{h}k\rho_ii$ of ordered groups is constructed, which contains all small extensions of $\Gamma$ up to $\Gamma$-isomorphism as ordered groups. On a certain subset $\rho_\elll\subset\rho_ii$,
an equivalence relation $\sim_{\mbox{\thetainy $\op{sme}$}}$ is defined such that the quotient set $\rho_\elll/\!\sim_{\mbox{\thetainy $\op{sme}$}}$ parametrizes the quasi-cuts of the divisible hull of $\Gamma$. Also, there is a canonical subset $\Gammasme\subset\rho_\elll$ which faithfully represents all $\sim_{\mbox{\thetainy $\op{sme}$}}$ classes.
In Section 6, we consider the subtree $\thetaz\subset \thetatt(\rho_ii)$ formed by all nodes ${\mathfrak m}u$ such that $\Gammam\subset \rho_\elll$. Then, we characterize equivalence of valuations in $\thetaz$ as follows. {\mathfrak m}edskip
\mathbf{n}oindent{{\mathfrak m}athfrak{b}f Proposition 6.3. }{\it Let ${\mathfrak m}u,\mathbf{n}u\in\thetaz$ be two inner nodes. Then, ${\mathfrak m}u\sim\mathbf{n}u$ if and only if the following three conditions hold:
(a) \ The valuations ${\mathfrak m}u$, $\mathbf{n}u$ admit a common key polynomial of minimal degree.
(b) \ For all \,$a\in\op{Ker}x\,$ such that \,$\Deltaeg(a)<\Deltaeg({\mathfrak m}u)$, we have $\,{\mathfrak m}u(a)=\mathbf{n}u(a)$.
(c) \ $\op{SV}al({\mathfrak m}u)\sim_{\mbox{\thetainy $\op{sme}$}} \op{SV}al(\mathbf{n}u)$.
In this case, we have $\op{Ker}pm=\op{Ker}pn$.}{\mathfrak m}edskip
In Section 7, we consider the subtree $\thetas\subset\thetaz$ formed by all leaves of $\thetaz$, and all inner nodes ${\mathfrak m}u$ such that $\op{SV}al({\mathfrak m}u)$ belongs to $\Gammasme$. Then, we use Proposition 6.3 to obtain our main theorem.{\mathfrak m}edskip
\mathbf{n}oindent{{\mathfrak m}athfrak{b}f Theorem 7.1 }{\it The mapping ${\mathfrak m}u{\mathfrak m}apsto[{\mathfrak m}u]$ induces a bijection between $\thetas$ and $\thetatt_v$.}{\mathfrak m}edskip
In the rest of the section, we discuss special features of the paths in $\thetas$ and we show the existence of {\mathfrak m}edskipmph{primitive nodes}, leading to a certain stratification of the tree by {\mathfrak m}edskipmph{limit-depth}, which is the number of limit augmentations in the MLV chains.{\mathfrak m}edskip
The techniques of this paper have been applied in two different contexts {\mathfrak m}athfrak{c}ite{AGNR,Rig}. Let $(\op{Ker}h,v^h)$ be a henselization of $(K,v)$. In {\mathfrak m}athfrak{c}ite{AGNR}, we use the primitive nodes of the valuative tree to establish a complete parallelism between the arithmetic properties of irreducible polynomials $F\in \op{Ker}h[x]$, encoded by their Okutsu frames, and the valuation-theoretic properties of their induced valuations $v_F$ on $\op{Ker}h[x]$, encoded by their MLV chains.
In {\mathfrak m}athfrak{c}ite{Rig}, it is shown that the natural restriction mapping $\thetatt_{v^h}\thetao\thetatt_v$ is an isomorphism of posets.
\section{Key polynomials over valued fields}{\mathfrak m}athfrak{l}abel{secKP}
In this section we introduce notation and well-known facts on key polynomials. Proofs and a more detailed exposition can be found in the survey {\mathfrak m}athfrak{c}ite{KP}.
For any field $L$, let $\op{Irr}(L)$ be the set of monic irreducible polynomials in $L[x]$.
Let $(K,v)$ be a valued field. Let $k$ be the residue class field, $\Gamma=v(K^*)$ the value group and $\Gammaq=\Gamma\otimes{\mathfrak m}athbb Q$ the divisible hull of $\Gamma$.
Let $\Gamma{\mathfrak m}athfrak{h}k\Lambdambda$ be an extension of ordered abelian groups. We write simply $\Lambdambda\infty$ instead of $\Lambdambda{\mathfrak m}athfrak{c}up\{\infty\}$.
Consider a valuation on the polynomial ring $\op{Ker}x$
$$
{\mathfrak m}u{\mathfrak m}athfrak{c}olon \op{Ker}x{\mathfrak m}athfrak{l}ra \Lambdambda\infty
$$
whose restriction to $K$ is $v$.
Let $\mathfrak{p}={\mathfrak m}u^{-1}(\infty)$ be the support of ${\mathfrak m}u$.
The valuation ${\mathfrak m}u$ induces in a natural way a valuation ${\mathfrak m}athfrak{b}ar{{\mathfrak m}u}$ on the field of fractions of $\op{Ker}x/\mathfrak{p}$; that is, $K(x)$ if $\mathfrak{p}=0$, or $\op{Ker}x/(f)$ if $\mathfrak{p}=f\op{Ker}x$ for some $f\in\op{Irr}(K)$.
The residue field $\op{Ker}m$ of ${\mathfrak m}u$ is, by definition, the residue field of ${\mathfrak m}athfrak{b}ar{{\mathfrak m}u}$.
We say that ${\mathfrak m}u$ is {\mathfrak m}edskipmph{commensurable} (over $v$) if $\Gamma_{\mathfrak m}u/\Gamma$ is a torsion group. In this case, there is a canonical embedding $\Gamma_{\mathfrak m}u{\mathfrak m}athfrak{h}ookrightarrow \Gammaq$.
All valuations with nontrivial support are commensurable.
For any ${\mathfrak m}athfrak{a}lpha\in\Gamma_{\mathfrak m}u$, consider the abelian groups:
$$
\mathfrak{p}pa=\{g\in \op{Ker}x{\mathfrak m}id {\mathfrak m}u(g)\Gammae {\mathfrak m}athfrak{a}lpha\}\supset
\mathfrak{p}pa^+=\{g\in \op{Ker}x{\mathfrak m}id {\mathfrak m}u(g)> {\mathfrak m}athfrak{a}lpha\}.
$$
The {\mathfrak m}edskipmph{graded algebra of ${\mathfrak m}u$} is the integral domain:
$$
\Gammagm:=\Gammar_{{\mathfrak m}u}(\op{Ker}x)={\mathfrak m}athfrak{b}igoplus\mathbf{n}olimits_{{\mathfrak m}athfrak{a}lpha\in\Gamma_{\mathfrak m}u}\mathfrak{p}pa/\mathfrak{p}pa^+.
$$
There is a natural {\mathfrak m}edskipmph{initial term} mapping $\op{in}_{\mathfrak m}u{\mathfrak m}athfrak{c}olon \op{Ker}x\thetao \Gammagm$, given by $\op{in}_{\mathfrak m}u\mathfrak{p}=0$ and
$$
\op{in}_{\mathfrak m}u g= g+\mathfrak{p}set_{{\mathfrak m}u(g)}^+\in\mathfrak{p}set_{{\mathfrak m}u(g)}/\mathfrak{p}set_{{\mathfrak m}u(g)}^+, \mathfrak{q}quad{\mathfrak m}box{if }\ g\in \op{Ker}x\setminus\mathfrak{p}.
$$
There is a natural embedding of graded algebras \ $\Gammag_v:=\operatorname{gr}_v(K){\mathfrak m}athfrak{h}ookrightarrow \Gammagm$.
The following definitions translate properties of the action of ${\mathfrak m}u$ on $\op{Ker}x$ into algebraic relationships in the graded algebra $\Gammagm$.{\mathfrak m}edskip
\Deltaefn Let $g,\,h\in \op{Ker}x$.
We say that $g,h$ are {\mathfrak m}edskipmph{${\mathfrak m}u$-equivalent}, and we write $g\sim_\mu h$, if $\op{in}_{\mathfrak m}u g=\op{in}_{\mathfrak m}u h$.
We say that $g$ is {\mathfrak m}edskipmph{${\mathfrak m}u$-divisible} by $h$, and we write $h{\mathfrak m}mu g$, if $\op{in}_{\mathfrak m}u h{\mathfrak m}id \op{in}_{\mathfrak m}u g$ in $\Gammagm$.
We say that $g$ is ${\mathfrak m}u$-irreducible if $(\op{in}_{\mathfrak m}u g)\Gammagm$ is a nonzero prime ideal.
We say that $g$ is ${\mathfrak m}u$-minimal if $g\mathbf{n}mid_{\mathfrak m}u f$ for all nonzero $f\in \op{Ker}x$ with $\Deltaeg(f)<\Deltaeg(g)$.{\mathfrak m}edskip
The ${\mathfrak m}u$-minimality condition admits a relevant characterization {\mathfrak m}athfrak{c}ite[Prop. 2.3]{KP}.
{\mathfrak m}athfrak{b}egin{lemma}{\mathfrak m}athfrak{l}abel{minimal0}
A polynomial $g\in \op{Ker}x\setminus K$ is ${\mathfrak m}u$-minimal if and only if ${\mathfrak m}u$ acts as follow on $g$-expansions:
$$f=\sum\mathbf{n}olimits_{0{\mathfrak m}athfrak{l}e s}a_s g^s,\mathfrak{q}uad \Deltaeg(a_s)<\Deltaeg(g)\ \ \Longrightarrow\ \ {\mathfrak m}u(f)={\mathfrak m}in{\mathfrak m}athfrak{l}eft\{{\mathfrak m}u{\mathfrak m}athfrak{l}eft(a_sg^s\rho_ight){\mathfrak m}id 0{\mathfrak m}athfrak{l}e s\rho_ight\}.$$
{\mathfrak m}edskipnd{lemma}
\Deltaefn A {\mathfrak m}edskipmph{(Maclane-Vaqui\'e) key polynomial} for ${\mathfrak m}u$ is a monic polynomial in $\op{Ker}x$ which is simultaneously ${\mathfrak m}u$-minimal and ${\mathfrak m}u$-irreducible.
The set of key polynomials for ${\mathfrak m}u$ is denoted $\op{Ker}pm$. {\mathfrak m}edskip
All $\mathfrak{p}hi\in\op{Ker}pm$ are irreducible in $\op{Ker}x$. Let $[\mathfrak{p}hi]_{\mathfrak m}u\subset\op{Ker}pm$ be the subset of all key polynomials ${\mathfrak m}u$-equivalent to $\mathfrak{p}hi$. Two ${\mathfrak m}u$-equivalent key polynomials have the same degree {\mathfrak m}athfrak{c}ite[Prop. 6.6]{KP}; hence, it makes sense to consider the degree $\Deltaeg\, [\mathfrak{p}hi]_{\mathfrak m}u$ of a class.
The existence of key polynomials can be characterized as follows {\mathfrak m}athfrak{c}ite[Thm. 4.4]{KP}.
{\mathfrak m}athfrak{b}egin{theorem}{\mathfrak m}athfrak{l}abel{KPempty}
The following conditions are equivalent.
{\mathfrak m}athfrak{b}egin{enumerate}
\item $\op{Ker}pm={\mathfrak m}edskipmptyset$.
\item ${\mathfrak m}u$ is commensurable and $\op{Ker}m/k$ is an algebraic extension of fields.
\item $\Gammagm$ is a simple algebra (all nonzero homogeneous elements are units).
{\mathfrak m}edskipnd{enumerate}
{\mathfrak m}edskipnd{theorem}
\Deltaefn
Suppose that $\op{Ker}pm\mathbf{n}e{\mathfrak m}edskipmptyset$ and take $\mathfrak{p}hi\in\op{Ker}pm$ of minimal degree. The {\mathfrak m}edskipmph{degree} and {\mathfrak m}edskipmph{singular value} of ${\mathfrak m}u$ are defined as
$$
\Deltaeg({\mathfrak m}u)=\Deltaeg(\mathfrak{p}hi),\mathfrak{q}quad \op{SV}al({\mathfrak m}u)={\mathfrak m}u(\mathfrak{p}hi).
$$
The singular value is well defined by {\mathfrak m}athfrak{c}ite[Thm. 3.9]{KP}. {\mathfrak m}edskip
Another relevant invariant of a valuation ${\mathfrak m}u$ on $\op{Ker}x$ is its field $\op{Ker}a=\op{Ker}am$ of {\mathfrak m}edskipmph{algebraic residues}, defined as the relative algebraic closure of $k$ in the residue field $\op{Ker}m$ of ${\mathfrak m}u$.
Let $\Delta=\Deltam\subset\Gammagm$ be the subring of homogeneous elements of degree zero in the graded algebra. There are natural embeddings
$$k{\mathfrak m}athfrak{h}k\op{Ker}a{\mathfrak m}athfrak{h}k\Delta{\mathfrak m}athfrak{h}k\op{Ker}m.$$
The structure of $\Delta$ as a $\op{Ker}a$-algebra plays an essential role in the description of the branches of a node in the valuative tree.
{\mathfrak m}athfrak{b}egin{theorem}{\mathfrak m}athfrak{l}abel{Delta}
Let ${\mathfrak m}u$ be a valuation on $\op{Ker}x$, whose restriction to $K$ is $v$.
{\mathfrak m}athfrak{b}egin{enumerate}
\item If $\op{Ker}pm={\mathfrak m}edskipmptyset$, then $\op{Ker}a=\Delta=\op{Ker}m$ is a countably generated extension of $k$.
\item If ${\mathfrak m}u$ is incommensurable, then $\op{Ker}a=\Delta=\op{Ker}m$ is a finite extension of $k$.
\item If ${\mathfrak m}u$ is commensurable and $\op{Ker}pm\mathbf{n}e{\mathfrak m}edskipmptyset$, then $\Delta=\op{Ker}a[\xi]$ and $\op{Ker}m=\op{Ker}a(\xi)$, for some $\xi\in\Delta$ which is transcendental over $\op{Ker}a$.
{\mathfrak m}edskipnd{enumerate}
{\mathfrak m}edskipnd{theorem}
The valuations ${\mathfrak m}u$ falling in case (3) of Theorem \ref{Delta} are said to be {\mathfrak m}edskipmph{residually transcendental}.
There is a tight link between $\Delta$ and the set $\op{Ker}pm$ {\mathfrak m}athfrak{c}ite[Thm. 6.7]{KP}
{\mathfrak m}athfrak{b}egin{theorem}{\mathfrak m}athfrak{l}abel{DeltaKP}
If $\op{Ker}pm\mathbf{n}e{\mathfrak m}edskipmptyset$, the residual ideal mapping $$\op{Ker}pm{\mathfrak m}athfrak{l}ra\operatorname{Max}(\Delta),\mathfrak{q}quad \mathfrak{p}hi{\mathfrak m}athfrak{l}ongmapsto {\mathfrak m}athfrak{l}eft(\op{in}_{\mathfrak m}u(\mathfrak{p}hi)\Gammagm\rho_ight){\mathfrak m}athfrak{c}ap \Delta$$
induces a bijection between $\op{Ker}pm/\!\sim_\mu$ and the maximal spectrum of $\Delta$.
{\mathfrak m}edskipnd{theorem}
If ${\mathfrak m}u$ is incommensurable, then $\Delta$ is a field and $\operatorname{Max}(\Delta)$ is a one-element set. In this case, $\op{Ker}pm=[\mathfrak{p}hi]_{\mathfrak m}u$, for any monic polynomial $\mathfrak{p}hi\in\op{Ker}x$ of minimal degree such that ${\mathfrak m}u(\mathfrak{p}hi)$ is torsion-free over $\Gamma$.
If ${\mathfrak m}u$ is residually transcendental, Theorems \ref{Delta} and \ref{DeltaKP} yield a bijection
$$\op{Ker}pm/\!\sim_\mu\mathfrak{q}uad {\mathfrak m}athfrak{l}ongleftrightarrow\mathfrak{q}uad\operatorname{Max}(\Delta)\mathfrak{q}uad {\mathfrak m}athfrak{l}ongleftrightarrow\mathfrak{q}uad \op{Irr}(\op{Ker}a),$$ which depends on the choice of a generator $\xi$ for $\Delta$, as shown in Theorem \ref{Delta}(3).
\section{Tree of valuations with values in a fixed group}{\mathfrak m}athfrak{l}abel{secTreeLa}
Let $\thetatt=\thetala$ be the set of all valuations ${\mathfrak m}u{\mathfrak m}athfrak{c}olon \op{Ker}x\thetao\Lambda\infty$, whose restriction to $K$ is $v$.
This set admits a partial ordering.
For ${\mathfrak m}u,\mathbf{n}u\in \thetatt$ we say that ${\mathfrak m}u{\mathfrak m}athfrak{l}e\mathbf{n}u$ if
$$
{\mathfrak m}u(f){\mathfrak m}athfrak{l}e \mathbf{n}u(f),\mathfrak{q}quad {\mathfrak m}athfrak{f}orall\,f\in\op{Ker}x.
$$
As usual, we write ${\mathfrak m}u<\mathbf{n}u$ to indicate that ${\mathfrak m}u{\mathfrak m}athfrak{l}e\mathbf{n}u$ and ${\mathfrak m}u\mathbf{n}e\mathbf{n}u$.
If ${\mathfrak m}u{\mathfrak m}athfrak{l}e\mathbf{n}u$, there is a canonical homomorphism of graded $\Gammag_v$-algebras:
$$\Gammagm{\mathfrak m}athfrak{l}ra\Gammagn,\mathfrak{q}quad \op{in}_{\mathfrak m}u f{\mathfrak m}athfrak{l}ongmapsto
{\mathfrak m}athfrak{b}egin{cases}\op{in}_\mathbf{n}u f,& {\mathfrak m}box{ if }{\mathfrak m}u(f)=\mathbf{n}u(f),\\ 0,& {\mathfrak m}box{ if }{\mathfrak m}u(f)<\mathbf{n}u(f).{\mathfrak m}edskipnd{cases}
$$
This poset $\thetatt$ has the structure of a {\mathfrak m}edskipmph{tree}. By this, we simply mean that all intervals
$$
(-\infty,{\mathfrak m}u\,]:={\mathfrak m}athfrak{l}eft\{\rho\in\thetatt{\mathfrak m}id \rho{\mathfrak m}athfrak{l}e{\mathfrak m}u\rho_ight\}
$$
are totally ordered {\mathfrak m}athfrak{c}ite[Thm. 2.4]{MLV}. {\mathfrak m}edskip
\Deltaefn A node ${\mathfrak m}u\in\thetatt$ is a {\mathfrak m}edskipmph{leaf} if it is a maximal element with respect to the ordering ${\mathfrak m}athfrak{l}e$. Otherwise, we say that ${\mathfrak m}u$ is an {\mathfrak m}edskipmph{inner node}.
{\mathfrak m}athfrak{b}egin{theorem}{\mathfrak m}athfrak{c}ite[Thm. 2.3]{MLV}{\mathfrak m}athfrak{l}abel{maximal}
A node ${\mathfrak m}u\in\thetatt$ is a leaf if and only it $\op{Ker}pm={\mathfrak m}edskipmptyset$.
{\mathfrak m}edskipnd{theorem}
All valuations with nontrivial support are leaves of $\thetatt$, because they satisfy condition (2) of Theorem \ref{KPempty}. We call them {\mathfrak m}edskipmph{finite leaves}.
The leaves of $\thetatt$ having trivial support are {\mathfrak m}edskipmph{valuation-algebraic} in Kuhlmann's terminology {\mathfrak m}athfrak{c}ite{Kuhl}. We call them {\mathfrak m}edskipmph{infinite leaves}.
We denote the set of leaves and subsets of finite and infinite leaves as follows:
$$
{\mathfrak m}athfrak{l}l(\thetatt)={\mathfrak m}athfrak{l}fin(\thetatt)\sqcup{\mathfrak m}athfrak{l}i(\thetatt),
$$
\Deltaefn For a leaf ${\mathfrak m}u\in{\mathfrak m}athfrak{l}l(\thetatt)$ we define its {\mathfrak m}edskipmph{degree} as:
$$
\Deltaeg({\mathfrak m}u)=\sup{\mathfrak m}athfrak{l}eft\{\Deltaeg(\rho){\mathfrak m}id \rho\in\thetatt,\ \rho<{\mathfrak m}u\rho_ight\}\in{\mathfrak m}athbb N\infty.
$$\vskip0.2cm
A finite leaf ${\mathfrak m}u\in{\mathfrak m}athfrak{l}fin(\thetatt)$ has $\op{supp}({\mathfrak m}u)=f\op{Ker}x$ for some monic irreducible $f\in\op{Ker}x$ and $\Deltaeg({\mathfrak m}u)=\Deltaeg(f)$. The infinite leaves may have finite or infinite degree.
\subsection{Tangent directions and augmentations}{\mathfrak m}athfrak{l}abel{subsecTanDir}
Let ${\mathfrak m}u,\,\mathbf{n}u$ be two nodes in $\thetatt$ such that ${\mathfrak m}u<\mathbf{n}u$.
Let $\thetamn$ be the (nonempty) set of monic polynomials $\mathfrak{p}hi\in\op{Ker}x$ of minimal degree satisfying ${\mathfrak m}u(\mathfrak{p}hi)<\mathbf{n}u(\mathfrak{p}hi)$.
We say that $\thetamn$ is the {\mathfrak m}edskipmph{tangent direction} of ${\mathfrak m}u$, determined by $\mathbf{n}u$. This terminology will be justified in section \ref{subsecTanSpace}, when we study the tangent space of $\thetatt$.
The following properties of $\thetamn$ were proven by Maclane for discrete rank-one valued fields, and generalized by Vaqui\'e to arbitrary valued fields {\mathfrak m}athfrak{c}ite[Thm. 1.15]{Vaq}, {\mathfrak m}athfrak{c}ite[Prop. 2.2, Cor. 2.5]{MLV}.
{\mathfrak m}athfrak{b}egin{lemma}{\mathfrak m}athfrak{l}abel{propertiesTMN}
Let ${\mathfrak m}u<\mathbf{n}u$ be two nodes in $\thetatt$ and let $\mathfrak{p}hi\in\thetamn$.
{\mathfrak m}athfrak{b}egin{enumerate}
\item The polynomial $\mathfrak{p}hi$ belongs to $\op{Ker}pm$ and $\thetamn=[\mathfrak{p}hi]_{\mathfrak m}u$. Also, $\Deltaeg({\mathfrak m}u){\mathfrak m}athfrak{l}e \Deltaeg(\mathbf{n}u)$.
\item For all nonzero $f\in\op{Ker}x$ the equality ${\mathfrak m}u(f)=\mathbf{n}u(f)$ holds if and only if $\mathfrak{p}hi\mathbf{n}mid_{\mathfrak m}u f$.
\item If ${\mathfrak m}u<\mathbf{n}u<\rho$ in $\thetatt$, then $\thetamr=\thetamn$. In particular,
$$
{\mathfrak m}u(f)=\rho(f)\ \Longleftrightarrow\ {\mathfrak m}u(f)=\mathbf{n}u(f),\mathfrak{q}quad {\mathfrak m}athfrak{f}orall\,f\in\op{Ker}x.
$$
{\mathfrak m}edskipnd{enumerate}
{\mathfrak m}edskipnd{lemma}
On the other hand, for any inner node ${\mathfrak m}u\in\thetatt$, all ${\mathfrak m}u$-equivalence classes in $\op{Ker}pm$ are the tangent direction of ${\mathfrak m}u$ with respect to some $\mathbf{n}u\in\thetatt$ such that ${\mathfrak m}u<\mathbf{n}u$.
Indeed, for any $\mathfrak{p}hi\in\op{Ker}pm$ and any $\Gammaa\in\Lambdambda\infty$ such that ${\mathfrak m}u(\mathfrak{p}hi)<\Gammaa$, we may construct the {\mathfrak m}edskipmph{augmented valuation} $\mathbf{n}u=[{\mathfrak m}u;\mathfrak{p}hi,\Gammaa]$,
defined in terms of $\mathfrak{p}hi$-expansions as
$$
f=\sum\mathbf{n}olimits_{0{\mathfrak m}athfrak{l}e s}a_s\mathfrak{p}hi^s,\mathfrak{q}uad \Deltaeg(a_s)<\Deltaeg(\mathfrak{p}hi)\ \Longrightarrow\ \mathbf{n}u(f)={\mathfrak m}in\{{\mathfrak m}u(a_s)+s\Gammaa{\mathfrak m}id 0{\mathfrak m}athfrak{l}e s\}.
$$
Note that $\mathbf{n}u(\mathfrak{p}hi)=\Gammaa$.
The following properties of this augmented valuation are also due to Maclane and Vaqui\'e {\mathfrak m}athfrak{c}ite[Prop. 2.1]{MLV}.
{\mathfrak m}athfrak{b}egin{lemma}{\mathfrak m}athfrak{l}abel{propertiesAug}
Let $\mathbf{n}u=[{\mathfrak m}u;\mathfrak{p}hi,\Gammaa]$ be an augmented valuation of ${\mathfrak m}u$.
{\mathfrak m}athfrak{b}egin{enumerate}
\item We have ${\mathfrak m}u<\mathbf{n}u$ and $\,\thetamn=[\mathfrak{p}hi]_{\mathfrak m}u$.
\item The value group of $\mathbf{n}u$ is $\Gamman=\Gammaen{\Gamma_{{\mathfrak m}u,\mathfrak{p}hi},\Gammaa}$, where $\Gamma_{{\mathfrak m}u,\mathfrak{p}hi}$ is the subgroup $$\Gamma_{{\mathfrak m}u,\mathfrak{p}hi}={\mathfrak m}athfrak{l}eft\{{\mathfrak m}u(a){\mathfrak m}id a\in\op{Ker}x,\ 0{\mathfrak m}athfrak{l}e \Deltaeg(a)<\Deltaeg(\mathfrak{p}hi)\rho_ight\}.$$
\item If $\Gammaa=\infty$, then $\op{supp}(\mathbf{n}u)=\mathfrak{p}hi\op{Ker}x$. If $\Gammaa<\infty$, then $\mathfrak{p}hi$ is a key polynomial for $\mathbf{n}u$ of minimal degree. In both cases, $\Deltaeg(\mathbf{n}u)=\Deltaeg(\mathfrak{p}hi)$.
{\mathfrak m}edskipnd{enumerate}
{\mathfrak m}edskipnd{lemma}
For all $\mathfrak{p}hi_*\in\op{Ker}pm$, $\Gammaa_*\in\Lambda\infty$ such that ${\mathfrak m}u(\mathfrak{p}hi_*)<\Gammaa_*$, {\mathfrak m}athfrak{c}ite[Lem. 2.8]{MLV} shows that
{\mathfrak m}athfrak{b}egin{equation}{\mathfrak m}athfrak{l}abel{eqAug}
[{\mathfrak m}u;\,\mathfrak{p}hi,\Gammaa]=[{\mathfrak m}u;\,\mathfrak{p}hi_*,\Gammaa_*] \ \Longleftrightarrow\ {\mathfrak m}u(\mathfrak{p}hi_*-\mathfrak{p}hi)\Gammae \Gammaa=\Gammaa_*\ \Longrightarrow\ \mathfrak{p}hi\sim_\mu\mathfrak{p}hi_*.
{\mathfrak m}edskipnd{equation}
\subsection{The tangent space of $\thetatt$}{\mathfrak m}athfrak{l}abel{subsecTanSpace}
For any inner node ${\mathfrak m}u\in\thetatt$, consider the quotient set
$$
\thetadm={\mathfrak m}athfrak{l}eft\{\mathbf{n}u\in\thetatt{\mathfrak m}id {\mathfrak m}u<\mathbf{n}u\rho_ight\}/\!\thetaan,
$$
with respect to the equivalence relation $\thetaan$ which considers $\mathbf{n}u\thetaan\mathbf{n}u'$ if and only if $ ({\mathfrak m}u,\mathbf{n}u]{\mathfrak m}athfrak{c}ap({\mathfrak m}u,\mathbf{n}u']\mathbf{n}e{\mathfrak m}edskipmptyset$.
The transitivity of $\thetaan$ follows easily from the fact that $\thetatt$ is a tree. We denote by $[\mathbf{n}u]_{\op{tan}}$ the class of $\mathbf{n}u$.
The elements of $\thetadm$ can be identified with the tangent directions of ${\mathfrak m}u$ defined in the last section.
{\mathfrak m}athfrak{b}egin{proposition}{\mathfrak m}athfrak{l}abel{td=td}
For all inner nodes ${\mathfrak m}u\in\thetatt$, the association
$$
\mathfrak{p}hi{\mathfrak m}athfrak{l}ongmapsto t_{\mathfrak m}u(\mathfrak{p}hi):={\mathfrak m}athfrak{l}eft[\,[{\mathfrak m}u;\,\mathfrak{p}hi,\Gammaa]\,\rho_ight]_{\op{tan}},\mathfrak{q}uad \Gammaa\in\Lambda\infty, \ \Gammaa>{\mathfrak m}u(\mathfrak{p}hi),
$$
is independent of the choice of $\Gammaa$ and so it defines a mapping $t_{\mathfrak m}u{\mathfrak m}athfrak{c}olon\op{Ker}pm\thetao\thetadm$, which induces a bijection between $\op{Ker}pm/\!\sim_\mu$ and $\thetadm$.
{\mathfrak m}edskipnd{proposition}
{\mathfrak m}athfrak{b}egin{proof}
If ${\mathfrak m}u(\mathfrak{p}hi)<\Gammaa<\Gammaa_*$, then $\mathbf{n}u<[{\mathfrak m}u;\,\mathfrak{p}hi,\Gammaa]<[{\mathfrak m}u;\,\mathfrak{p}hi,\Gammaa_*]$. Thus, $[{\mathfrak m}u;\,\mathfrak{p}hi,\Gammaa]\thetaan[{\mathfrak m}u;\,\mathfrak{p}hi,\Gammaa_*]$, so that the mapping $t_{\mathfrak m}u$ is well defined.
Take $\mathbf{n}u \in \thetatt$ such that ${\mathfrak m}u<\mathbf{n}u$. For all $\mathfrak{p}hi\in\thetamn$ we have
$$
{\mathfrak m}u<[{\mathfrak m}u;\,\mathfrak{p}hi,\mathbf{n}u(\mathfrak{p}hi)]{\mathfrak m}athfrak{l}e\mathbf{n}u,
$$
by comparing their actions on $\mathfrak{p}hi$-expansions. Thus, $t_{\mathfrak m}u(\mathfrak{p}hi)=[\mathbf{n}u]_{\op{tan}}$. This proves that $t_{\mathfrak m}u$ is onto.
Finally, let us show that, for all $\mathfrak{p}hi,\mathfrak{p}hi_*\in\op{Ker}pm$, the equality $t_{\mathfrak m}u(\mathfrak{p}hi)=t_{\mathfrak m}u(\mathfrak{p}hi_*)$ holds if and only if $\mathfrak{p}hi\sim_\mu\mathfrak{p}hi_*$.
If $\mathfrak{p}hi\sim_\mu\mathfrak{p}hi_*$ and $\Gammaa={\mathfrak m}u(\mathfrak{p}hi-\mathfrak{p}hi_*)>{\mathfrak m}u(\mathfrak{p}hi)$, then $[{\mathfrak m}u;\,\mathfrak{p}hi,\Gammaa]=[{\mathfrak m}u;\,\mathfrak{p}hi_*,\Gammaa]$, by (\ref{eqAug}); thus $t_{\mathfrak m}u(\mathfrak{p}hi)=t_{\mathfrak m}u(\mathfrak{p}hi_*)$.
Conversely, if $t_{\mathfrak m}u(\mathfrak{p}hi)=t_{\mathfrak m}u(\mathfrak{p}hi_*)$, there exists $\rho\in\thetatt$ such that
$$
{\mathfrak m}u<\rho{\mathfrak m}athfrak{l}e [{\mathfrak m}u;\,\mathfrak{p}hi,\Gammaa]\mathfrak{q}quad{\mathfrak m}box{and}\mathfrak{q}quad {\mathfrak m}u<\rho{\mathfrak m}athfrak{l}e [{\mathfrak m}u;\,\mathfrak{p}hi_*,\Gammaa_*],
$$
for some $\Gammaa>{\mathfrak m}u(\mathfrak{p}hi)$, $\Gammaa_*>{\mathfrak m}u(\mathfrak{p}hi_*)$. By {\mathfrak m}athfrak{c}ite[Lem. 2.7]{MLV}, there exist $\Deltata,\Deltata_*\in\Lambda$ such that
$[{\mathfrak m}u;\,\mathfrak{p}hi,\Deltata]=\rho=[{\mathfrak m}u;\,\mathfrak{p}hi_*,\Deltata_*]$. By (\ref{eqAug}), we have $\mathfrak{p}hi\sim_\mu\mathfrak{p}hi_*$.
{\mathfrak m}edskipnd{proof}{\mathfrak m}edskip
By the remarks following Theorem \ref{DeltaKP}, $\thetadm$ is a one-element set if ${\mathfrak m}u$ is incommensurable, while there is a (non-canonical) bijection between $\thetadm$ and $\op{Irr}(\op{Ker}am)$, if ${\mathfrak m}u$ is commensurable.{\mathfrak m}edskip
\Deltaefn The {\mathfrak m}edskipmph{tangent space} of $\thetatt$ is the set ${\mathfrak m}athbb T(\thetatt)$ containing all pairs $({\mathfrak m}u,t)$, where ${\mathfrak m}u$ is an inner node in $\thetatt$ and
$t\in\thetadm$ is a tangent direction of ${\mathfrak m}u$.
\section{Finite leaves}{\mathfrak m}athfrak{l}abel{secFinLeaves}
For any field $L$ and a monic irreducible polynomial $F\in\op{Irr}(L)$, we denote by $L_F$ the simple field extension $L[x]/(F)$.
In this section, we assume that $\Lambda$ contains the divisible closure of $\Gamma$; that is, $\Gammaq\subset\Lambda$.
Under this assumption, the set ${\mathfrak m}athfrak{l}fin(\thetatt)$ of finite leaves of $\thetatt$ may be parametrized as
$$
{\mathfrak m}athfrak{l}fin(\thetatt)={\mathfrak m}athfrak{l}eft\{(\mathfrak{p}hi,\mathbf{n}b){\mathfrak m}id \mathfrak{p}hi\in\op{Irr}(K),\ \mathbf{n}b\ {\mathfrak m}box{ valuation on $K_\mathfrak{p}hi$ extending }v \rho_ight\},
$$
where we identify each pair $(\mathfrak{p}hi,\mathbf{n}b)$ with the following valuation with support $\mathfrak{p}hi\op{Ker}x$:
$$
\mathbf{n}u{\mathfrak m}athfrak{c}olon \op{Ker}x{\mathfrak m}athfrak{l}ongtwoheadrightarrow K_\mathfrak{p}hi\stackrel{\mathbf{n}b}{\mathfrak m}athfrak{l}ra\Gammaq\infty
$$
Every simple field extension $L/K$ admits a finite number of extensions of $v$ to $L$.
Any such extension determines an infinite number (if $K$ is infinite) of finite leaves of $\thetatt$, one for each $\mathfrak{p}hi\in\op{Irr}(K)$ such that $K_\mathfrak{p}hi$ is $K$-isomorphic to $L$.
For instance, the valuation $v$ on $K$ determines the finite leaves $(x-a,v)$, for $a$ running in $K$.
Let us recall the description of all extensions of $v$ to simple finite extensions of $K$, which can be found (for instance) in {\mathfrak m}athfrak{c}ite[Sec. 17]{endler}.
Let us first describe all extensions of $v$ to an arbitrary algebraic extension $L$ of $K$. These extensions are commensurable over $v$; thus, we aim to describe the set:
$$
{\mathfrak m}edskipe(L)={\mathfrak m}athfrak{l}eft\{w{\mathfrak m}athfrak{c}olon L{\mathfrak m}athfrak{l}ra \Gammaq\infty\,{\mathfrak m}id\, w\, {\mathfrak m}box{ valuation extending }v\rho_ight\}.
$$
Consider $K\subset \op{Ker}s\subset\op{Ker}b$, the separable closure of $K$ in a fixed algebraic closure $\op{Ker}b$.
Let $\bar{v}$ be a fixed extension of $v$ to $\op{Ker}b$.
Let $K\subset \op{Ker}h\subset \op{Ker}s$ be the henselization of $K$ determined by the choice of $\bar{v}$. Thus, $\op{Ker}h$ is the fixed field of the decomposition group of the restriction of $\bar{v}$ to $\op{Ker}s$.
On the set ${\mathfrak m}on$ of all $K$-morphisms from $L$ to $\op{Ker}b$, we define the following equivalence relation
$$
{\mathfrak m}athfrak{l}a\sim_{\op{Ker}h}{\mathfrak m}athfrak{l}a' \ \ \Longleftrightarrow\ \ {\mathfrak m}athfrak{l}a'=\sigma{\mathfrak m}athfrak{c}irc{\mathfrak m}athfrak{l}a\mathfrak{q}uad{\mathfrak m}box{for some}\mathfrak{q}uad \sigma\in{\mathfrak m}athfrak{a}ut(\op{Ker}b/\op{Ker}h).
$$
{\mathfrak m}athfrak{b}egin{theorem}{\mathfrak m}athfrak{l}abel{endler}
The mapping ${\mathfrak m}on\thetao{\mathfrak m}edskipe(L)$, defined by ${\mathfrak m}athfrak{l}a{\mathfrak m}apsto \bar{v}{\mathfrak m}athfrak{c}irc{\mathfrak m}athfrak{l}a$,
induces a bijection between the quotient set ${\mathfrak m}on{}/\!\sim_{\op{Ker}h}$ and ${\mathfrak m}edskipe(L)$.
{\mathfrak m}edskipnd{theorem}
{\mathfrak m}athfrak{b}egin{center}
\setlength{\unitlength}{4mm}
{\mathfrak m}athfrak{b}egin{picture}(8,9)
\mathfrak{p}ut(4,0){$K$}\mathfrak{p}ut(3.6,4){${\mathfrak m}athfrak{l}a(L)$}\mathfrak{p}ut(4,8){$\op{Ker}b$}
\mathfrak{p}ut(0,2){$L$}\mathfrak{p}ut(8,4.3){$\op{Ker}h$}\mathfrak{p}ut(8,6.8){$\op{Ker}s$}
\mathfrak{p}ut(4.4,1.1){{\mathfrak m}athfrak{l}ine(0,1){2.3}}\mathfrak{p}ut(4.4,5.1){{\mathfrak m}athfrak{l}ine(0,1){2.4}}\mathfrak{p}ut(8.4,5.2){{\mathfrak m}athfrak{l}ine(0,1){1.3}}
\mathfrak{p}ut(3.8,0.4){\vector(-2,1){2.9}}\mathfrak{p}ut(1,2.5){\vector(2,1){2.5}}
\mathfrak{p}ut(5,.3){{\mathfrak m}athfrak{l}ine(4,5){2.9}}\mathfrak{p}ut(5,8.2){{\mathfrak m}athfrak{l}ine(3,-1){2.7}}
\mathfrak{p}ut(1.6,3.2){{\mathfrak m}athfrak{f}ootnotesize{${\mathfrak m}athfrak{l}a$}}
{\mathfrak m}edskipnd{picture}
{\mathfrak m}edskipnd{center}{\mathfrak m}edskip
For instance, ${\mathfrak m}edskipe(\op{Ker}b)$ is in bijection with ${\mathfrak m}athfrak{a}ut(\op{Ker}b/\op{Ker}h){\mathfrak m}athfrak{b}ackslash{\mathfrak m}athfrak{a}ut(\op{Ker}b/K)$. Every right coset ${\mathfrak m}athfrak{a}ut(\op{Ker}b/\op{Ker}h)\,\sigma$ determines the valuation $\bar{v}{\mathfrak m}athfrak{c}irc\sigma$.
Suppose now that $L/K$ is a simple finite extension; that is, $L=K_\mathfrak{p}hi$ for some $\mathfrak{p}hi\in\op{Irr}(K)$. Since $\op{Ker}h/K$ is a separable extension, the factorization of $\mathfrak{p}hi$ into a product of monic irreducible polynomials in $\op{Ker}h[x]$ takes the form
$$
\mathfrak{p}hi=F_1{\mathfrak m}athfrak{c}dots F_r,
$$
with pairwise different $F_1,\Deltaots,F_r\in\op{Irr}(\op{Ker}h)$.
Let $Z(\mathfrak{p}hi)\subset \op{Ker}b$ be the set of zeros of $\mathfrak{p}hi$, avoiding multiplicities. We have a natural bijection
$$
Z(\mathfrak{p}hi){\mathfrak m}athfrak{l}ra{\mathfrak m}on,\mathfrak{q}quad \theta{\mathfrak m}athfrak{l}ongmapsto {\mathfrak m}athfrak{l}a_\theta,
$$
where ${\mathfrak m}athfrak{l}a_\theta$ is determined by ${\mathfrak m}athfrak{l}a_\theta{\mathfrak m}athfrak{l}eft(x+\mathfrak{p}hi\op{Ker}x\rho_ight)=\theta$.
Clearly,
$$
{\mathfrak m}athfrak{b}egin{array}{rccl}
{\mathfrak m}athfrak{l}a_\theta \sim_{\op{Ker}h} {\mathfrak m}athfrak{l}a_{\theta'}&\ \Longleftrightarrow\ &{\mathfrak m}athfrak{l}a_{\theta'}=\sigma{\mathfrak m}athfrak{c}irc {\mathfrak m}athfrak{l}a_\theta&\mathfrak{q}uad{\mathfrak m}box{for some}\mathfrak{q}uad\sigma\in{\mathfrak m}athfrak{a}ut(\op{Ker}b/\op{Ker}h)\\
&\ \Longleftrightarrow\ &\theta'=\sigma(\theta)&\mathfrak{q}uad{\mathfrak m}box{for some}\mathfrak{q}uad\sigma\in{\mathfrak m}athfrak{a}ut(\op{Ker}b/\op{Ker}h)
{\mathfrak m}edskipnd{array}
$$
Therefore, ${\mathfrak m}athfrak{l}a_\theta \sim_{\op{Ker}h} {\mathfrak m}athfrak{l}a_{\theta'}$ if and only if $\theta$ and $\theta'$ are roots of the same irreducible factor of $\mathfrak{p}hi$ over $\op{Ker}h[x]$.
Let us choose an arbitrary root $\theta_i\in Z(F_i)$ for each irreducible factor of $\mathfrak{p}hi$. By Theorem \ref{endler}, the set of valuations $\overline{w}_{F_i}=\bar{v}{\mathfrak m}athfrak{c}irc{\mathfrak m}athfrak{l}a_{\theta_i}$ does not depend on the chosen roots and contains all extensions of $v$ to $L$.
{\mathfrak m}athfrak{b}egin{theorem}{\mathfrak m}athfrak{l}abel{CorEndler}
There are $r$ extensions of $v$ to $L=K_\mathfrak{p}hi$, given by $\overline{w}_{F_1},\Deltaots,\overline{w}_{F_r}$.
{\mathfrak m}edskipnd{theorem}
This description of the extensions of $v$ to simple finite extensions of $K$ yields a parametrization of the finite leaves by the set $\op{Irr}(\op{Ker}h)$.
For all $F\in\op{Irr}(\op{Ker}h)$ consider the finite leaf $w_F\in{\mathfrak m}athfrak{l}fin$ given by
$$
w_F(g)=\bar{v}(g(\theta))\mathfrak{q}uad {\mathfrak m}box{for all }g\in\op{Ker}x,
$$
where $\theta\in\op{Ker}b$ is any root of $F$ in $\op{Ker}b$. By the henselian property, this valuation $w_F$ is independent on the choice of $\theta$.
Clearly, $\op{supp}(w_F)=N(F)\op{Ker}x$, where the ``norm" polynomial $N(F)\in\op{Irr}(K)$ is uniquely determined by the equality ${\mathfrak m}athfrak{l}eft(F \op{Ker}hx\rho_ight){\mathfrak m}athfrak{c}ap \op{Ker}x=N(F)\op{Ker}x$.
Since the valuation induced by $w_F$ on $K_{N(F)}$ is precisely $\overline{w}_F=\bar{v}{\mathfrak m}athfrak{c}irc{\mathfrak m}athfrak{l}a_\theta$, Theorem \ref{CorEndler} implies the following result.
{\mathfrak m}athfrak{b}egin{theorem}{\mathfrak m}athfrak{l}abel{Lfin=Hensel}
If $\,\Gammaq\subset \Lambda$, we have a bijection
$$
\op{Irr}(\op{Ker}h){\mathfrak m}athfrak{l}ra {\mathfrak m}athfrak{l}fin(\thetatt),\mathfrak{q}quad F{\mathfrak m}athfrak{l}ongmapsto w_F=(N(F),\overline{w}_F).
$$
{\mathfrak m}edskipnd{theorem}
The inverse mapping associates to each pair $(\mathfrak{p}hi,\mathbf{n}b)\in{\mathfrak m}athfrak{l}fin(\thetatt)$, the irreducible factor of $\mathfrak{p}hi$ over $\op{Ker}h[x]$ canonically associated to $\mathbf{n}b$ by Theorem \ref{CorEndler}.
\section{Infinite leaves and limit nodes}
In this section, we study the nodes of $\thetatt$ which cannot be obtained by a finite chain of ordinary augmentations starting with a degree-one valuation.
These nodes will be a kind of limit of certain totally ordered families of valuations in $\thetatt$.
\subsection{Totally ordered families of valuations}{\mathfrak m}athfrak{l}abel{subsecTOF}
Consider a totally ordered family of inner nodes of $\thetatt$, not containing a maximal element:
$$
{\mathfrak m}athfrak{a}a={\mathfrak m}athfrak{c}fa,\mathfrak{q}uad \rho_i\in\thetatt.
$$
We assume that ${\mathfrak m}athfrak{a}a$ is parameterized by a totally ordered set set $A$ of indices such that the mapping $A\thetao{\mathfrak m}athfrak{a}a$ determined by $i{\mathfrak m}apsto \rho_i$ is an isomorphism of totally ordered sets.
By Lemma \ref{propertiesTMN}, the degree function $\Deltaeg{\mathfrak m}athfrak{c}olon {\mathfrak m}athfrak{a}a\thetao{\mathfrak m}athbb N$
is order-preserving. Hence, these families fall into two radically different cases:{\mathfrak m}edskip
(a) \ The set $\Deltaeg({\mathfrak m}athfrak{a}a)$ is unbounded in ${\mathfrak m}athbb N$. We say that ${\mathfrak m}athfrak{a}a$ has {\mathfrak m}edskipmph{unbounded degree}.{\mathfrak m}edskip
(b) \ There exists $i_0\in A$ such that $\Deltaeg(\rho_i)=\Deltaeg(\rho_{i_0})$ for all $i\Gammae i_0$.
We say that ${\mathfrak m}athfrak{a}a$ is a {\mathfrak m}edskipmph{continuous family} of stable degree $m({\mathfrak m}athfrak{a}a)=\Deltaeg(\rho_{i_0})$.{\mathfrak m}edskip
In any case, ${\mathfrak m}athfrak{a}a$ determines a unique tangent direction of every valuation $\rho_i\in{\mathfrak m}athfrak{a}a$. Indeed, Lemma \ref{propertiesTMN} shows that $\thetay(\rho_i,\rho_j)=\thetay(\rho_i,\rho_k)$ for all $i<j<k$ in $A$.
We denote by $\thetay(\rho_i,{\mathfrak m}athfrak{a}a)$ this common tangent direction. By Lemma \ref{propertiesTMN}, there exists a key polynomial $\varphi_i\in\op{Ker}p(\rho_i)$ such that $\thetay(\rho_i,{\mathfrak m}athfrak{a}a)=[\varphi_i]_{\rho_i}$,
and for any nonzero polynomial $f\in\op{Ker}x$ we have
$$
\varphi_i{\mathfrak m}id_{\rho_i}f \ \ \Longleftrightarrow\ \ \rho_i(f)<\rho_j(f)\ {\mathfrak m}box{ for all }\ j>i \ {\mathfrak m}box{ in }A.
$$
$$
\varphi_i\mathbf{n}mid_{\rho_i}f \ \ \Longleftrightarrow\ \ \rho_i(f)=\rho_j(f)\ {\mathfrak m}box{ for all }\ j>i \ {\mathfrak m}box{ in }A.
$$\vskip0.2cm
\Deltaefn
We say that a nonzero $f\in\op{Ker}x$ is {\mathfrak m}edskipmph{${\mathfrak m}athfrak{a}a$-stable} if it satisfies $\varphi_i\mathbf{n}mid_{\rho_i}f$, for some index $i\in A$.
In this case, we denote its stable value by $$\rho_\aa(f)={\mathfrak m}ax\{\rho_i(f){\mathfrak m}id i\in A\}.$$
We obtain in this way a {\mathfrak m}edskipmph{stability function}
$\rho_\aa{\mathfrak m}athfrak{c}olon \op{Ker}x_{\mathfrak m}athfrak{a}a\thetao \Lambda\infty$, defined only on the subset $\op{Ker}x_{\mathfrak m}athfrak{a}a\subset \op{Ker}x$ formed by the ${\mathfrak m}athfrak{a}a$-stable polynomials.{\mathfrak m}edskip
The following basic properties of the function $\rho_\aa$ are obvious:
{\mathfrak m}athfrak{b}egin{itemize}
\item $(\rho_\aa)_{{\mathfrak m}id K}=v$,
\item $f,g\in \op{Ker}x_{\mathfrak m}athfrak{a}a\ \ \Longrightarrow\ \ fg\in\op{Ker}x_{\mathfrak m}athfrak{a}a\ {\mathfrak m}box{ and } \ \rho_\aa(fg)=\rho_\aa(f)+\rho_\aa(g)$,
\item $f,g,f+g\in \op{Ker}x_{\mathfrak m}athfrak{a}a\ \ \Longrightarrow\ \ \rho_\aa(f+g)\Gammae{\mathfrak m}in\{\rho_\aa(f),\rho_\aa(g)\}$.
{\mathfrak m}edskipnd{itemize}
In particular, if $\op{Ker}x_{\mathfrak m}athfrak{a}a=\op{Ker}x$, then the function $\rho_\aa$ is a valuation in $\thetatt$.{\mathfrak m}edskip
\Deltaefn
If all the polynomials in $\op{Ker}x$ are ${\mathfrak m}athfrak{a}a$-stable, we say that the valuation $\rho_\aa$ is the {\mathfrak m}edskipmph{stable limit} of ${\mathfrak m}athfrak{a}a$.
In this case, we write $\rho_\aa={\mathfrak m}athfrak{l}im({\mathfrak m}athfrak{a}a)={\mathfrak m}athfrak{l}im_{i\in A}\rho_i$.{\mathfrak m}edskip
{\mathfrak m}athfrak{b}egin{proposition}{\mathfrak m}athfrak{c}ite[Prop. 3.1]{MLV}{\mathfrak m}athfrak{l}abel{leaves}
If ${\mathfrak m}athfrak{a}a={\mathfrak m}athfrak{c}fa$ has a stable limit, then $\rho_\aa$ has trivial support and $\op{Ker}p(\rho_\aa)={\mathfrak m}edskipmptyset$. In particular, $\rho_\aa$ is an infinite leaf of the tree $\thetatt$.
{\mathfrak m}edskipnd{proposition}
Let us see a necessary condition for a polynomial to be ${\mathfrak m}athfrak{a}a$-unstable.
{\mathfrak m}athfrak{b}egin{lemma}{\mathfrak m}athfrak{l}abel{aunstab}
All ${\mathfrak m}athfrak{a}a$-unstable polynomials $f$ satisfy $\Deltaeg(f)\Gammae \Deltaeg(\rho_i)$ for all $i\in A$.
{\mathfrak m}edskipnd{lemma}
{\mathfrak m}athfrak{b}egin{proof}
Let $\thetay(\rho_i,{\mathfrak m}athfrak{a}a)=[\varphi_i]_{\rho_i}$ for some $i\in A$. If $f\in\op{Ker}x$ has $\Deltaeg(f)<\Deltaeg(\rho_i)$, then $\Deltaeg(f)<\Deltaeg(\varphi_i)$. Hence, $\varphi_i\mathbf{n}mid_{\rho_i}f$, contradicting the unstability of $f$.
{\mathfrak m}edskipnd{proof}
{\mathfrak m}athfrak{b}egin{corollary}{\mathfrak m}athfrak{l}abel{unbIsStab}
Every totally ordered family of unbounded degree has a stable limit.
{\mathfrak m}edskipnd{corollary}
\subsection{Continuous families and limit augmentations}{\mathfrak m}athfrak{l}abel{subsecCont}
Let ${\mathfrak m}athfrak{c}c={\mathfrak m}athfrak{c}fa$ be a {\mathfrak m}edskipmph{continuous family} of valuations in $\thetatt$ of stable degree $m=m({\mathfrak m}athfrak{c}c)$.
The following result shows that the valuations of maximal degree $m$ in ${\mathfrak m}athfrak{c}c$ are ``close" one to another in a certain sense.
{\mathfrak m}athfrak{b}egin{lemma}{\mathfrak m}athfrak{l}abel{samedegree}
Let ${\mathfrak m}u<\mathbf{n}u$ be two inner nodes of $\thetatt$ of degree $m$. Take any $\mathfrak{p}hi\in\op{Ker}pn$ of degree $m$. Then, $\mathfrak{p}hi\in\op{Ker}pm$ and $\mathbf{n}u= [{\mathfrak m}u;\,\mathfrak{p}hi, \mathbf{n}u(\mathfrak{p}hi)]$.
{\mathfrak m}edskipnd{lemma}
{\mathfrak m}athfrak{b}egin{proof}
For all $a \in\op{Ker}x$ with $\Deltaeg(a)<m$, we have ${\mathfrak m}u(a)=\mathbf{n}u(a)$, because any $\varphi\in \thetay({\mathfrak m}u,\mathbf{n}u)$ satisfies $\varphi\mathbf{n}mid_{\mathfrak m}u a$.
Necessarily ${\mathfrak m}u(\mathfrak{p}hi)<\mathbf{n}u(\mathfrak{p}hi)$, because the equality ${\mathfrak m}u(\mathfrak{p}hi)=\mathbf{n}u(\mathfrak{p}hi)$ leads to a contradiction. Indeed, for all $f\in\op{Ker}x$ with $\mathfrak{p}hi$-expansion $f=\sum_{0{\mathfrak m}athfrak{l}e s}a_s\mathfrak{p}hi^s$, we would have
$$
{\mathfrak m}u(f)\Gammae{\mathfrak m}in_{0{\mathfrak m}athfrak{l}e s}\{{\mathfrak m}u{\mathfrak m}athfrak{l}eft(a_s\mathfrak{p}hi^s\rho_ight)\}={\mathfrak m}in_{0{\mathfrak m}athfrak{l}e s}\{\mathbf{n}u{\mathfrak m}athfrak{l}eft(a_s\mathfrak{p}hi^s\rho_ight)\}=\mathbf{n}u(f),
$$
showing that ${\mathfrak m}u\Gammae\mathbf{n}u$, against our assumption.
By Lemma \ref{propertiesTMN}, $\mathfrak{p}hi$ is a key polynomial for ${\mathfrak m}u$ such that $\thetay({\mathfrak m}u,\mathbf{n}u)=[\mathfrak{p}hi]_{\mathfrak m}u$. The equality $\mathbf{n}u= [{\mathfrak m}u;\,\mathfrak{p}hi, \mathbf{n}u(\mathfrak{p}hi)]$ follows then from the action of both valuations on $\mathfrak{p}hi$-expansions.
{\mathfrak m}edskipnd{proof}
\subsubsection{Stable group of a continuous family}{\mathfrak m}athfrak{l}abel{subsubsecG0} {\mathfrak m}box{\mathbf{n}ull}
Let $\rho\in\thetatt$ be an inner node. By Lemma \ref{minimal0}, applied to any key polynomial for $\rho$ of minimal degree, we have $\Gammarh=\Gammaen{\Gammarh^0,\op{SV}al(\rho)}$, where $\Gammarh^0$ is the subgroup:
$$
\Gammarh^0={\mathfrak m}athfrak{l}eft\{\rho(a){\mathfrak m}id a\in\op{Ker}x,\ 0{\mathfrak m}athfrak{l}e\Deltaeg(a)<\Deltaeg(\rho)\rho_ight\}.
$$
Moreover, $\Gammarh^0$ is commensurable over $\Gamma$. Indeed, for all $a\in\op{Ker}x$ of degree less than $\Deltaeg(\rho)$, the initial term $\op{in}_\rho a\in\Gammag_\rho$ is algebraic over the graded algebra $\Gammag_v$ {\mathfrak m}athfrak{c}ite[Prop. 3.5]{KP}; thus, $n\rho(a)$ belongs to $\Gamma$ for some $n\in{\mathfrak m}athbb N$.
The index ${\mathfrak m}edskiprel(\rho)={\mathfrak m}athfrak{l}eft(\Gammarh{\mathfrak m}athfrak{c}olon \Gammarh^0\rho_ight)$ is called the {\mathfrak m}edskipmph{relative ramification index} of $\rho$.
If $\rho$ is incommensurable over $v$, then ${\mathfrak m}edskiprel(\rho)=\infty$.
Let us denote the set of stable values of nonzero ${\mathfrak m}athfrak{c}c$-stable polynomials by
$$\Gammac=\rho_\cc{\mathfrak m}athfrak{l}eft(\op{Ker}x_{\mathfrak m}athfrak{c}c\setminus\{0\}\rho_ight)\subset\Lambda.$$
The following lemma shows that $\Gammac$ is a subgroup of $\Lambda$. It is called
the {\mathfrak m}edskipmph{stable group} of the continuous family ${\mathfrak m}athfrak{c}c$.
{\mathfrak m}athfrak{b}egin{lemma}{\mathfrak m}athfrak{l}abel{GAcomm}
Let ${\mathfrak m}athfrak{c}c={\mathfrak m}athfrak{c}fa$ be a {\mathfrak m}edskipmph{continuous family} in $\thetatt$ of stable degree $m$. Then, $\Gamma_{\rho_i}^0=\Gammac$ \,for all $\rho_i\in {\mathfrak m}athfrak{c}c$ of degree $m$.
{\mathfrak m}edskipnd{lemma}
{\mathfrak m}athfrak{b}egin{proof}
Suppose that $\Deltaeg(\rho_i)=m$ for some $i\in A$.
Since ${\mathfrak m}athfrak{c}c$ contains no maximal element, there exists $j\in A$ such that $i<j$. Since the degree function preserves the ordering, we have $\Deltaeg(\rho_i)=\Deltaeg(\rho_j)=m$.
By Lemma \ref{samedegree}, $\thetay(\rho_i,\rho_j)=[\varphi]_{\rho_i}$, for any $\varphi\in\op{Ker}p(\rho_j)$ of minimal degree $m$.
Now, for all $a\in\op{Ker}x$ with $0{\mathfrak m}athfrak{l}e\Deltaeg(a)<m$, we have $\rho_i(a)=\rho_j(a)=\rho_\cc(a)$ because $\varphi\mathbf{n}mid_{\rho_i}a$. This proves $\Gamma_{\rho_i}^0=\Gamma_{\rho_j}^0\subset \Gammac$ already.
Let us show the inclusion $\Gammac\subset \Gamma_{\rho_i}^0$. Let $\Gammaa=\rho_\cc(f)\in\Gammac$ be the stable value of a nonzero ${\mathfrak m}athfrak{c}c$-stable $f\in\op{Ker}x$. Let $i{\mathfrak m}athfrak{l}e k<{\mathfrak m}edskipll$ in $A$ such that $\Deltaeg(\rho_k)=\Deltaeg(\rho_\ell)=m$ and $\rho_k(f)=\rho_\ell(f)$. By {\mathfrak m}athfrak{c}ite[Cor. 2.6]{MLV}, the element $\op{in}_{\rho_\ell} f$ is a unit. By {\mathfrak m}athfrak{c}ite[Prop. 3.5]{KP}, there exists $a\in\op{Ker}x$ of degree less than $m$ such that $\op{in}_{\rho_\ell} f=\op{in}_{\rho_\ell} a$. In particular, $\rho_\ell(f)=\rho_\ell(a)$ belongs to $\Gamma_{\rho_\ell}^0=\Gamma_{\rho_i}^0$.
{\mathfrak m}edskipnd{proof}
\subsubsection{Limit key polynomials and limit augmentations}{\mathfrak m}athfrak{l}abel{subsubsecLimit}{\mathfrak m}box{\mathbf{n}ull}
The set $\op{Ker}pi({\mathfrak m}athfrak{c}c)$ of {\mathfrak m}edskipmph{limit key polynomials} for ${\mathfrak m}athfrak{c}c$ is the set of all monic ${\mathfrak m}athfrak{c}c$-unstable polynomials of minimal degree. Since the product of stable polynomials is stable, all limit key polynomials are irreducible in $ \op{Ker}x$.
The {\mathfrak m}edskipmph{unstable degree} ${\mathfrak m}i={\mathfrak m}i({\mathfrak m}athfrak{c}c)$ is the common degree of all limit key polynomials for ${\mathfrak m}athfrak{c}c$.
If all polynomials in $\op{Ker}x$ are ${\mathfrak m}athfrak{c}c$-stable, then $\op{Ker}pi({\mathfrak m}athfrak{c}c)={\mathfrak m}edskipmptyset$ and we agree that ${\mathfrak m}i=\infty$.
By Lemma \ref{aunstab}, ${\mathfrak m}i\Gammae m$.
Take any limit key polynomial $\mathfrak{p}hi\in\op{Ker}pi{\mathfrak m}athfrak{l}eft({\mathfrak m}athfrak{c}c\rho_ight)$, and choose $\Gammaa\in\Lambda\infty$ such that $$\rho_i(\mathfrak{p}hi)<\Gammaa\mathfrak{q}uad {\mathfrak m}box{for all } \ i\in A.$$
The {\mathfrak m}edskipmph{limit augmentation} $\mathbf{n}u=[{\mathfrak m}athfrak{c}c;\mathfrak{p}hi,\Gammaa]$ acts as follows on $\mathfrak{p}hi$-expansions:
$$f=\sum\mathbf{n}olimits_{0{\mathfrak m}athfrak{l}e s}a_s\mathfrak{p}hi^s,\mathfrak{q}uad\Deltaeg(a_s)<{\mathfrak m}i\ \ \Longrightarrow\ \ \mathbf{n}u(f)={\mathfrak m}in\{\rho_\cc(a_s)+s\Gammaa{\mathfrak m}id 0{\mathfrak m}athfrak{l}e s\}.
$$
The following properties of $\mathbf{n}u$ can be found in
{\mathfrak m}athfrak{c}ite[Sec. 1.4]{Vaq} and {\mathfrak m}athfrak{c}ite[Sec. 7]{KP}.
{\mathfrak m}athfrak{b}egin{proposition}{\mathfrak m}athfrak{l}abel{extensionlim}
The augmentation $\mathbf{n}u=[{\mathfrak m}athfrak{c}c;\mathfrak{p}hi,\Gammaa]$ is a valuation in $\thetatt$ such that $\rho_i<\mathbf{n}u$ for all $i\in A$.
If $\Gammaa<\infty$, then $\mathbf{n}u$ is an inner node, $\Gamman=\Gammaen{\Gammac,\Gammaa}$ and $\mathfrak{p}hi$ is a key polynomial for $\mathbf{n}u$, of minimal degree; thus, $\Deltaeg(\mathbf{n}u)={\mathfrak m}i$ and $\,\op{SV}al(\mathbf{n}u)=\Gammaa$.
If $\Gammaa=\infty$, then $\Gamman=\Gammac$ and the support of $\mathbf{n}u$ is $\mathfrak{p}hi\op{Ker}x$; thus, $\mathbf{n}u$ is a finite leaf of $\thetatt$.
{\mathfrak m}edskipnd{proposition}
If ${\mathfrak m}i=m$, then for all $i\in A$ with maximal degree $\Deltaeg(\rho_i)=m$, Lemma \ref{samedegree} shows that
$\mathbf{n}u=[\rho_i;\,\mathfrak{p}hi,\mathbf{n}u(\mathfrak{p}hi)]$.
Thus, any limit augmentation of ${\mathfrak m}athfrak{c}c$ is equal to an ordinary augmentation of some $\rho_i$.
Therefore, we may discard the families ${\mathfrak m}athfrak{c}c$ of stable degree $m={\mathfrak m}i$ because they do not contribute to produce new nodes in $\thetatt$ by limit augmentations.
Summing up, a continuous family ${\mathfrak m}athfrak{c}c$ of valuations has three possibilities:
(a) \ If $m<{\mathfrak m}i=\infty$, then ${\mathfrak m}athfrak{c}c$ has a stable limit.
(b) \ If $m={\mathfrak m}i<\infty$, we say that ${\mathfrak m}athfrak{c}c$ is {\mathfrak m}edskipmph{inessential}.
(c) \ If $m<{\mathfrak m}i<\infty$, we say that ${\mathfrak m}athfrak{c}c$ is {\mathfrak m}edskipmph{essential}.{\mathfrak m}edskip
The continuous families having a stable limit determine infinite leaves of the tree $\thetatt$, by Proposition \ref{leaves}.
The essential continuous families determine inner {\mathfrak m}edskipmph{limit nodes} of $\thetatt$ as limit augmentations of the family.
By the uniqueness of {\mathfrak m}lv chains {\mathfrak m}athfrak{c}ite[Thm. 4.7]{MLV}, these limit nodes cannot be obtained by chains of ordinary augmentations starting with valuations that are smaller than some $\rho_i\in{\mathfrak m}athfrak{c}c$.
The limit key polynomials are an essential ingredient in the construction of these limit nodes. Let us describe them in more detail.
{\mathfrak m}athfrak{b}egin{lemma}{\mathfrak m}athfrak{l}abel{allLKP}
Let ${\mathfrak m}athfrak{c}c$ be an essential continuous family and let $\mathfrak{p}hi\in\op{Ker}pi({\mathfrak m}athfrak{c}c)$. Then,
$$
\op{Ker}pi({\mathfrak m}athfrak{c}c)={\mathfrak m}athfrak{l}eft\{\mathfrak{p}hi+a{\mathfrak m}id a\in\op{Ker}x,\ \Deltaeg(a)<{\mathfrak m}i, \ \rho_\cc(a)>\rho_i(\mathfrak{p}hi) \ {\mathfrak m}box{ for all }i\in A\rho_ight\}.
$$
{\mathfrak m}edskipnd{lemma}
{\mathfrak m}athfrak{b}egin{proof}
Let $\varphi\in\op{Ker}x$ be a monic polynomial of degree ${\mathfrak m}i$. Then,
$\varphi=\mathfrak{p}hi+a$ for some $a\in\op{Ker}x$ with $\Deltaeg(a)<{\mathfrak m}i$.
Since $a$ is ${\mathfrak m}athfrak{c}c$-stable, there exists $i_0\in A$ such that
$$
\rho_\cc(a)=\rho_j(a)\mathfrak{q}uad{\mathfrak m}box{ for all }\ i_0{\mathfrak m}athfrak{l}e j.
$$
For all indices $i_0{\mathfrak m}athfrak{l}e j<k$ we have $\rho_j(\mathfrak{p}hi)<\rho_k(\mathfrak{p}hi)$ and $\rho_j(a)=\rho_k(a)=\rho_\cc(a)$.
Suppose that $\rho_\cc(a)>\rho_i(\mathfrak{p}hi)$ for all $i\in A$. From $\rho_\cc(a)>\rho_k(\mathfrak{p}hi)$ we deduce that $\rho_j(\varphi)=\rho_j(\mathfrak{p}hi)<\rho_k(\mathfrak{p}hi)=\rho_k(\varphi)$. Thus, $\varphi$ belongs to $\op{Ker}pi({\mathfrak m}athfrak{c}c)$.
Suppose that $\rho_\cc(a){\mathfrak m}athfrak{l}e \rho_i(\mathfrak{p}hi)$ for some $i\in A$. Then, for all ${\mathfrak m}ax\{i_0,i\}<j<k$ we have $\rho_j(\varphi)=\rho_\cc(a)=\rho_k(\varphi)$. Thus, $\varphi$ is ${\mathfrak m}athfrak{c}c$-stable.
{\mathfrak m}edskipnd{proof}
\subsection{Equivalence of totally ordered families of valuations}{\mathfrak m}athfrak{l}abel{subsecEquivCont}
Let ${\mathfrak m}athfrak{a}a={\mathfrak m}athfrak{c}fa$, ${\mathfrak m}athfrak{b}b={\mathfrak m}athfrak{c}fb$ be totally ordered families in $\thetatt$, not containing a maximal element.
We say that ${\mathfrak m}athfrak{a}a$ and ${\mathfrak m}athfrak{b}b$ are {\mathfrak m}edskipmph{equivalent}, and we write ${\mathfrak m}athfrak{a}a\sim{\mathfrak m}athfrak{b}b$, if they are cofinal in each other.
Obviously, two equivalent families either both have unbounded degree, or both have stable degree. Also, in the latter case they have the same stable and unstable degrees, and the same stable group.
Two totally ordered families in the same class have the same limit behavior.
{\mathfrak m}athfrak{b}egin{proposition}{\mathfrak m}athfrak{l}abel{equiv=lim}
Let ${\mathfrak m}athfrak{a}a$, ${\mathfrak m}athfrak{b}b$ be totally ordered families in $\thetatt$, not containing a maximal element. Suppose that none of them is a continuous inessential family. Then,
$$
{\mathfrak m}athfrak{a}a\sim{\mathfrak m}athfrak{b}b \ \ \Longleftrightarrow\ \ \op{Ker}x_{\mathfrak m}athfrak{a}a=\op{Ker}x_{\mathfrak m}athfrak{b}b \ {\mathfrak m}box{ and }\ \rho_\aa=\rho_\bb.
$$
{\mathfrak m}edskipnd{proposition}
{\mathfrak m}athfrak{b}egin{proof}
Suppose ${\mathfrak m}athfrak{a}a\sim{\mathfrak m}athfrak{b}b$. Take any $f\in\op{Ker}x_{\mathfrak m}athfrak{a}a$, so that there exists $i_0\in A$ such that
$$
\rho_\aa(f)=\rho_i(f)\mathfrak{q}uad {\mathfrak m}box{for all }\ i\Gammae i_0.
$$
Since ${\mathfrak m}athfrak{b}b$ is cofinal in ${\mathfrak m}athfrak{a}a$, there exists $j_0\in B$ such that $\rho_{i_0}<\zeta_{j_0}$. Take any $j\in B$, $j>j_0$; since ${\mathfrak m}athfrak{a}a$ is cofinal in ${\mathfrak m}athfrak{b}b$, there exists $i\in A$ such that
$$
\rho_{i_0}<\zeta_{j_0}<\zeta_j<\rho_i.
$$
Necessarily, $\rho_{i_0}(f)=\zeta_{j_0}(f)=\zeta_j(f)=\rho_\bb(f)$. Hence, $f$ is ${\mathfrak m}athfrak{b}b$-stable and $\rho_\aa(f)=\rho_\bb(f)$.
The symmetry of the argument shows that $\op{Ker}x_{\mathfrak m}athfrak{a}a=\op{Ker}x_{\mathfrak m}athfrak{b}b$ and $\rho_\aa=\rho_\bb$.
Conversely, suppose that $\op{Ker}x_{\mathfrak m}athfrak{a}a=\op{Ker}x_{\mathfrak m}athfrak{b}b$ and $\rho_\aa=\rho_\bb$.
Let us first assume that both families have a stable limit; that is, $\op{Ker}x_{\mathfrak m}athfrak{a}a=\op{Ker}x_{\mathfrak m}athfrak{b}b=\op{Ker}x$.
Since all valuations $\rho_i, \zeta_j$ are bounded above by the valuation $\rho_\aa=\rho_\bb$, the set ${\mathfrak m}athfrak{a}a{\mathfrak m}athfrak{c}up{\mathfrak m}athfrak{b}b$ is totally ordered. If ${\mathfrak m}athfrak{a}a$ and ${\mathfrak m}athfrak{b}b$ were not cofinal in each other, there would exist (for instance) some $j\in B$ such that $\zeta_j>\rho_i$ for all $i\in A$. But this implies $\zeta_j\Gammae\rho_\aa=\rho_\bb$, leading to $\zeta_k>\rho_\bb$ for all $k>j$ in $B$. This is a contradiction.
Suppose now $\op{Ker}x_{\mathfrak m}athfrak{a}a=\op{Ker}x_{\mathfrak m}athfrak{b}b\subsetneq\op{Ker}x$. Take any $\mathfrak{p}hi\in\op{Ker}pi({\mathfrak m}athfrak{a}a)=\op{Ker}pi({\mathfrak m}athfrak{b}b)$. Since $\rho_\aa=\rho_\bb$, the following limit augmentations coincide
$$
\mathbf{n}u:=[{\mathfrak m}athfrak{a}a;\,\mathfrak{p}hi, \infty]=[{\mathfrak m}athfrak{b}b;\,\mathfrak{p}hi, \infty],
$$
because they have the same action on $\mathfrak{p}hi$-expansions. As above, the set ${\mathfrak m}athfrak{a}a{\mathfrak m}athfrak{c}up{\mathfrak m}athfrak{b}b$ is totally ordered and ${\mathfrak m}athfrak{a}a$ and ${\mathfrak m}athfrak{b}b$ are
cofinal in each other, unless there exists (for instance) some $j\in B$ such that $\zeta_j>\rho_i$ for all $i\in A$. This leads to a contradiction too.
Indeed, let $\thetay(\zeta_j,\zeta_k)=[\varphi]_{\zeta_j}$ for some $k\in B$ such that $k>j$. Since $\zeta_j(\varphi)<\zeta_k(\varphi)$, the augmentation ${\mathfrak m}u=[\zeta_j;\,\varphi,\zeta_k(\varphi)]$ satisfies $\zeta_j<{\mathfrak m}u{\mathfrak m}athfrak{l}e\zeta_k$.
Since
$$
\Deltaeg(\varphi)=\Deltaeg({\mathfrak m}u){\mathfrak m}athfrak{l}e\Deltaeg(\zeta_k){\mathfrak m}athfrak{l}e m({\mathfrak m}athfrak{b}b)<{\mathfrak m}i({\mathfrak m}athfrak{b}b),
$$
we deduce that $\varphi$ is ${\mathfrak m}athfrak{b}b$-stable. By our hypothesis, $\varphi$ is ${\mathfrak m}athfrak{a}a$-stable too, and this implies $\zeta_j(\varphi)=\rho_\aa(\varphi)=\zeta_k(\varphi)$, which is a contradiction.
{\mathfrak m}edskipnd{proof}{\mathfrak m}edskip
{\mathfrak m}athfrak{b}egin{corollary}{\mathfrak m}athfrak{l}abel{sameLKP}
Let ${\mathfrak m}athfrak{c}c$, ${\mathfrak m}athfrak{c}c'$ be two equivalent essential continuous families in $\thetatt$. Then, $\op{Ker}pi({\mathfrak m}athfrak{c}c)=\op{Ker}pi({\mathfrak m}athfrak{c}c')$.
{\mathfrak m}edskipnd{corollary}
Since all totally ordered sets admit cofinal well-ordered subsets, in every class of totally ordered families of valuations in $\thetatt$ there are well-ordered families.
In this vein, for any given class, we want to study the existence of ``nice" families in the class, having special properties.
{\mathfrak m}athfrak{b}egin{lemma}{\mathfrak m}athfrak{l}abel{numerable}
Let ${\mathfrak m}athfrak{a}a={\mathfrak m}athfrak{c}fa$ be a totally ordered family of unbounded degree. Then, there is a countable equivalent family ${\mathfrak m}athfrak{b}b={\mathfrak m}athfrak{l}eft(\zeta_n\rho_ight)_{n\in {\mathfrak m}athbb N}$ such that all $\zeta_n$ are commensurable over $v$ and $\Deltaeg(\zeta_{m})<\Deltaeg(\zeta_{n})$ for all $m<n$.
{\mathfrak m}edskipnd{lemma}
{\mathfrak m}athfrak{b}egin{proof}
By Corollary \ref{unbIsStab}, ${\mathfrak m}athfrak{a}a$ has a stable limit ${\mathfrak m}u={\mathfrak m}athfrak{l}im_{i\in A}\rho_i$. The Maclane--Vaqui\'e theorem (Theorem \ref{main}) shows that ${\mathfrak m}u$ is the stable limit of a countable infinite Maclane--Vaqui\'e chain ${\mathfrak m}athfrak{b}b={\mathfrak m}athfrak{l}eft(\zeta_n\rho_ight)_{n\in {\mathfrak m}athbb N}$ such that all $\zeta_n$ are commensurable over $v$ and $\Deltaeg(\zeta_{m})<\Deltaeg(\zeta_{n})$ for all $m<n$.
By Proposition \ref{equiv=lim}, ${\mathfrak m}athfrak{a}a$ and ${\mathfrak m}athfrak{b}b$ are equivalent.
{\mathfrak m}edskipnd{proof}
{\mathfrak m}athfrak{b}egin{lemma}{\mathfrak m}athfrak{l}abel{specialCont}
Let ${\mathfrak m}athfrak{c}c={\mathfrak m}athfrak{c}fa$ be a continuous family such that $m({\mathfrak m}athfrak{c}c)<{\mathfrak m}i({\mathfrak m}athfrak{c}c)$. Then, there is an equivalent family ${\mathfrak m}athfrak{b}b={\mathfrak m}athfrak{l}eft(\zeta_j\rho_ight)_{j\in B}$ satisfying the following properties:
{\mathfrak m}athfrak{b}egin{enumerate}
\item ${\mathfrak m}athfrak{b}b$ is well-ordered and all valuations $\zeta_i\in{\mathfrak m}athfrak{b}b$ have degree $m({\mathfrak m}athfrak{c}c)$.
\item All valuations $\zeta_j\in{\mathfrak m}athfrak{b}b$ have relative ramification index equal to one.
\item All valuations $\zeta_j\in{\mathfrak m}athfrak{b}b$ have value group $\Gamma_{\zeta_j}=\Gammab$.
\item For all $j\in B$, there exist ${\mathfrak m}athfrak{c}hi_j\in\op{Ker}p(\zeta_j)$ of minimal degree, such that
$$
j<k\ {\mathfrak m}box{in }B \ \ \Longrightarrow\ \ {\mathfrak m}athfrak{c}hi_j\mathbf{n}ot\sim_{\zeta_j}{\mathfrak m}athfrak{c}hi_k\ {\mathfrak m}box{ and }\ \zeta_k=[\zeta_j;\,{\mathfrak m}athfrak{c}hi_k,\op{SV}al(\zeta_k)].
$$
{\mathfrak m}edskipnd{enumerate}
{\mathfrak m}edskipnd{lemma}
{\mathfrak m}athfrak{b}egin{proof}
By replacing ${\mathfrak m}athfrak{c}c$ with a cofinal subfamily, we may assume that ${\mathfrak m}athfrak{c}c$ is well-ordered and $\Deltaeg(\rho_i)=m({\mathfrak m}athfrak{c}c)$ for all $i\in A$. By Lemma \ref{GAcomm}, $\Gamma_{\rho_i}^0=\Gammac$ for all $i\in A$.
Let us construct an equivalent family ${\mathfrak m}athfrak{b}b={\mathfrak m}athfrak{l}eft(\zeta_j\rho_ight)_{j\in B}$ all whose valuations have degree $m({\mathfrak m}athfrak{c}c)$, are commensurable and have relative ramification index equal to one; that is, $\op{SV}al(\zeta_j)\in \Gammac=\Gammab$ for all $j\in B$.
Let ${\mathfrak m}athfrak{c}cb$ be the subset of ${\mathfrak m}athfrak{c}c$ formed by all $\rho_i$ such that $\op{SV}al(\rho_i)\mathbf{n}ot\in \Gammac$.
Let us first construct, for each $\rho_i\in{\mathfrak m}athfrak{c}cb$, a valuation $\rho'_i$ such that $\op{SV}al(\rho'_i)\in\Gammac$ and $\rho_i<\rho'_i{\mathfrak m}athfrak{l}e \rho_k$ for some $k$ in $A$.
For any given $\rho_i\in{\mathfrak m}athfrak{c}cb$, let $\varphi\in\op{Ker}p(\rho_i)$ be a key polynomial of minimal degree. Then, $\op{SV}al(\rho_i)=\rho_i(\varphi)\mathbf{n}ot\in\Gammac$.
Since $\Deltaeg(\varphi)=\Deltaeg(\rho_i)<{\mathfrak m}i({\mathfrak m}athfrak{c}c)$, the polynomial $\varphi$ is ${\mathfrak m}athfrak{c}c$-stable. Take $i<j<k$ in $A$, such that $\rho_j(\varphi)=\rho_k(\varphi)=\rho_\cc(\varphi)$. Since $\rho_\cc(\varphi)\in \Gammac$, the inequality $\rho_i(\varphi)< \rho_k(\varphi)$ must be strict.
Now, take ${\mathfrak m}athfrak{c}hi\in\op{Ker}p(\rho_k)$ a key polynomial of minimal degree. By {\mathfrak m}athfrak{c}ite[Thm. 3.9]{KP}, we have $\rho_k(\varphi){\mathfrak m}athfrak{l}e\rho_k({\mathfrak m}athfrak{c}hi)=\op{SV}al(\rho_k)$. By Lemma \ref{samedegree},
$\rho_k=[\rho_i;\, {\mathfrak m}athfrak{c}hi,\op{SV}al(\rho_k)]$. Since $\rho_i(\varphi)< \rho_k(\varphi){\mathfrak m}athfrak{l}e \rho_k({\mathfrak m}athfrak{c}hi)$, the augmented valuation
$$
\rho'_i:=[\rho_i;\,{\mathfrak m}athfrak{c}hi,\rho_k(\varphi)]
$$
satisfies $\op{SV}al(\rho'_i)=\rho'_i({\mathfrak m}athfrak{c}hi)=\rho_k(\varphi)\in\Gammac$ and $\rho_i<\rho'_i{\mathfrak m}athfrak{l}e \rho_k$.
Consider the totally ordered family ${\mathfrak m}athfrak{c}c'={\mathfrak m}athfrak{c}c{\mathfrak m}athfrak{c}up \{\rho'_i{\mathfrak m}id \rho_i\in{\mathfrak m}athfrak{c}cb\}$. Obviously, ${\mathfrak m}athfrak{c}c$ and ${\mathfrak m}athfrak{c}c'$ are equivalent. Also, by construction, the subfamily ${\mathfrak m}athfrak{b}b$ of ${\mathfrak m}athfrak{c}c'$ formed by all valuations $\rho$ such that $\op{SV}al(\rho)\in\Gammac$ is cofinal in ${\mathfrak m}athfrak{c}c'$. Thus, the family ${\mathfrak m}athfrak{b}b$ is equivalent to ${\mathfrak m}athfrak{c}c$ and satisfies conditions (1) and (2). Since $\Gamma_{\zeta_j}=\Gammac=\Gamma_{\zeta_j}^0$ for all $\zeta_j\in{\mathfrak m}athfrak{b}b$, condition (3) follows from Lemma \ref{GAcomm}.
Let us prove that condition (4) holds too, for this family ${\mathfrak m}athfrak{b}b$.
Let us choose arbitrary key polynomials ${\mathfrak m}athfrak{c}hi_j\in\op{Ker}p(\zeta_j)$ of minimal degree, for all valuations $\zeta_j\in{\mathfrak m}athfrak{b}b$. By Lemma \ref{samedegree}, $\zeta_k=[\zeta_j;\,{\mathfrak m}athfrak{c}hi_k,\op{SV}al(\zeta_k)]$ for all $j<k$ in $B$.
The class $\thetay(\zeta_j,{\mathfrak m}athfrak{b}b)=\thetay(\zeta_j,\zeta_k)=[{\mathfrak m}athfrak{c}hi_k]_{\zeta_j}$ depends only on $\zeta_j$ and ${\mathfrak m}athfrak{b}b$.
Thus, the condition ${\mathfrak m}athfrak{c}hi_j\mathbf{n}ot\sim_{\zeta_j}{\mathfrak m}athfrak{c}hi_k$, which is equivalent to ${\mathfrak m}athfrak{c}hi_j\mathbf{n}ot\in \thetay(\zeta_j,{\mathfrak m}athfrak{b}b)$, depends only on ${\mathfrak m}athfrak{c}hi_j$.
Let us show that there exists a key polynomial ${\mathfrak m}athfrak{c}hi'_j\in\op{Ker}p(\zeta_j)$ of minimal degree, satisfying ${\mathfrak m}athfrak{c}hi'_j\mathbf{n}ot\in \thetay(\zeta_j,{\mathfrak m}athfrak{b}b)$. Since $\zeta_j$ has relative ramification index equal to one, we have $\op{SV}al(\zeta_j)\in\Gamma_{\zeta_j}^0$ and there exists $a\in\op{Ker}x$ of degree less than $m$ such that $\zeta_j(a)=\op{SV}al(\zeta_j)=\zeta_j({\mathfrak m}athfrak{c}hi_j)$. By {\mathfrak m}athfrak{c}ite[Prop. 6.3]{KP}, ${\mathfrak m}athfrak{c}hi'_j={\mathfrak m}athfrak{c}hi_j+a$ is a key polynomial for $\zeta_j$ of minimal degree $m$, such that ${\mathfrak m}athfrak{c}hi'_j\mathbf{n}ot\sim_{\zeta_j}{\mathfrak m}athfrak{c}hi_j$.
Therefore, at least one of the two key polynomials ${\mathfrak m}athfrak{c}hi_j$, ${\mathfrak m}athfrak{c}hi'_j$ does not fall in the class $\thetay(\zeta_j,{\mathfrak m}athfrak{b}b)$.
{\mathfrak m}edskipnd{proof}
\section{Paths in the tree $\thetatt$}{\mathfrak m}athfrak{l}abel{secCtDepth}
\subsection{{\mathfrak m}lv chains}{\mathfrak m}athfrak{l}abel{subsecMLV}
In this section, we review the fundamental theorem of {\mathfrak m}lv describing how to reach all nodes in $\thetatt$ by a combination of ordinary augmentations, limit augmentations and stable limits {\mathfrak m}athfrak{c}ite{mcla,Vaq}. All results are extracted from the survey {\mathfrak m}athfrak{c}ite{MLV}.
Consider a finite, or countably infinite, chain of nodes in $\thetatt$
{\mathfrak m}athfrak{b}egin{equation}{\mathfrak m}athfrak{l}abel{depthMLV}
{\mathfrak m}u_0\ \stackrel{\mathfrak{p}hi_1,\Gammaa_1}{\mathfrak m}athfrak{l}ra\ {\mathfrak m}u_1\ \stackrel{\mathfrak{p}hi_2,\Gammaa_2}{\mathfrak m}athfrak{l}ra\ {\mathfrak m}athfrak{c}dots
\ {\mathfrak m}athfrak{l}ra\ {\mathfrak m}u_{n-1}
\ \stackrel{\mathfrak{p}hi_{n},\Gammaa_{n}}{\mathfrak m}athfrak{l}ra\ {\mathfrak m}u_{n} \ {\mathfrak m}athfrak{l}ra\ {\mathfrak m}athfrak{c}dots
{\mathfrak m}edskipnd{equation}
in which every node is an augmentation of the previous node, of one of the following two types:{\mathfrak m}edskip
{\mathfrak m}edskipmph{Ordinary augmentation}: \ ${\mathfrak m}u_{n+1}=[{\mathfrak m}u_n;\, \mathfrak{p}hi_{n+1},\Gammaa_{n+1}]$, for some $\mathfrak{p}hi_{n+1}\in\op{Ker}p({\mathfrak m}u_n)$.{\mathfrak m}edskip
{\mathfrak m}edskipmph{Limit augmentation}: \ ${\mathfrak m}u_{n+1}=[{\mathfrak m}athfrak{a}a_n;\, \mathfrak{p}hi_{n+1},\Gammaa_{n+1}]$, for some $\mathfrak{p}hi_{n+1}\in\op{Ker}pi({\mathfrak m}athfrak{a}a_n)$, where ${\mathfrak m}athfrak{a}a_n$ is an essential continuous family whose first valuation is ${\mathfrak m}u_n$.{\mathfrak m}edskip
We consider an implicit choice of a key polynomial $\mathfrak{p}hi_0\in\op{Ker}p({\mathfrak m}u_0)$ of minimal degree, and we denote $\Gammaa_0={\mathfrak m}u_0(\mathfrak{p}hi_0)$.
Therefore, for all $n$ such that ${\mathfrak m}u_n$ is an inner node of $\thetatt$, the polynomial $\mathfrak{p}hi_n$ is a key polynomial for ${\mathfrak m}u_n$ of minimal degree, and we have
$$
m_n:=\Deltaeg({\mathfrak m}u_n)=\Deltaeg(\mathfrak{p}hi_n),\mathfrak{q}quad \op{SV}al({\mathfrak m}u_n)={\mathfrak m}u_n(\mathfrak{p}hi_n)=\Gammaa_n.
$$
\Deltaefn
A chain of mixed augmentations as in (\ref{depthMLV}) is said to be a {\mathfrak m}edskipmph{{\mathfrak m}lv (abbreviated MLV) chain} if $\Deltaeg({\mathfrak m}u_0)=1$ and every augmentation step satisfies:
{\mathfrak m}athfrak{b}egin{itemize}
\item If $\,{\mathfrak m}u_n\thetao{\mathfrak m}u_{n+1}\,$ is ordinary, then $\ \Deltaeg({\mathfrak m}u_n)<\Deltaeg\,\thetay({\mathfrak m}u_n,{\mathfrak m}u_{n+1})$.
\item If $\,{\mathfrak m}u_n\thetao{\mathfrak m}u_{n+1}\,$ is limit, then $\ \Deltaeg({\mathfrak m}u_n)=m({\mathfrak m}athfrak{a}a_n)$ and $\ \mathfrak{p}hi_n\mathbf{n}ot \in\thetay({\mathfrak m}u_n,{\mathfrak m}u_{n+1})$.
{\mathfrak m}edskipnd{itemize}{\mathfrak m}edskip
Let $0{\mathfrak m}athfrak{l}e r{\mathfrak m}athfrak{l}e \infty$ be the length of a MLV chain. For $n<r$, all nodes ${\mathfrak m}u_n$ are residually transcendental valuations. Indeed, in all augmentations of the chain, either ordinary or limit, we have
$$
\mathfrak{p}hi_n\in\op{Ker}p({\mathfrak m}u_n),\mathfrak{q}uad \mathfrak{p}hi_n\mathbf{n}ot\in\thetay({\mathfrak m}u_n,{\mathfrak m}u_{n+1}).
$$
Hence, $\op{Ker}p({\mathfrak m}u_n)$ contains at least two different ${\mathfrak m}u_n$-equivalence classes. By the remarks at the end of Section \ref{secKP}, this implies that ${\mathfrak m}u_n$ is commensurable.
{\mathfrak m}athfrak{b}egin{theorem}{\mathfrak m}athfrak{l}abel{main}
Every node $\mathbf{n}u\in\thetatt$ falls in one, and only one, of the following cases. {\mathfrak m}edskip
(a) \ It is the last valuation of a finite MLV chain.
$$ {\mathfrak m}u_0\ \stackrel{\mathfrak{p}hi_1,\Gammaa_1}{\mathfrak m}athfrak{l}ra\ {\mathfrak m}u_1\ \stackrel{\mathfrak{p}hi_2,\Gammaa_2}{\mathfrak m}athfrak{l}ra\ {\mathfrak m}athfrak{c}dots\ {\mathfrak m}athfrak{l}ra\ {\mathfrak m}u_{r-1}\ \stackrel{\mathfrak{p}hi_{r},\Gammaa_{r}}{\mathfrak m}athfrak{l}ra\ {\mathfrak m}u_{r}=\mathbf{n}u.$$
(b) \ It is the stable limit of a continuous family ${\mathfrak m}athfrak{a}a={\mathfrak m}athfrak{c}fa$ of augmentations whose first valuation ${\mathfrak m}u_r$ falls in case (a):
$$ {\mathfrak m}u_0\ \stackrel{\mathfrak{p}hi_1,\Gammaa_1}{\mathfrak m}athfrak{l}ra\ {\mathfrak m}u_1\ \stackrel{\mathfrak{p}hi_2,\Gammaa_2}{\mathfrak m}athfrak{l}ra\ {\mathfrak m}athfrak{c}dots\ {\mathfrak m}athfrak{l}ra\ {\mathfrak m}u_{r-1}\ \stackrel{\mathfrak{p}hi_{r},\Gammaa_{r}}{\mathfrak m}athfrak{l}ra\ {\mathfrak m}u_{r}\ \stackrel{{\mathfrak m}athfrak{c}fa}{\mathfrak m}athfrak{l}ra\ \rho_\aa=\mathbf{n}u.
$$
Moreover, we may assume that $\Deltaeg({\mathfrak m}u_r)=m({\mathfrak m}athfrak{a}a)$ and $\mathfrak{p}hi_r\mathbf{n}ot\in\thetay({\mathfrak m}u_r,\mathbf{n}u)$.{\mathfrak m}edskip
(c) \ It is the stable limit, $\mathbf{n}u={\mathfrak m}athfrak{l}im_{n\in{\mathfrak m}athbb N}\,{\mathfrak m}u_n$, of an infinite MLV chain.
{\mathfrak m}edskipnd{theorem}
The inner nodes and the finite leaves of $\thetatt$ fall in case (a). These are the ``bien-specifi\'ees" valuations in Vaqui\'e's terminology.
We denote by ${\mathfrak m}athfrak{l}ci(\thetatt),\,{\mathfrak m}athfrak{l}ui(\thetatt)\subset{\mathfrak m}athfrak{l}i(\thetatt)$ the subsets of infinite leaves falling in cases (b), (c), respectively. The infinite leaves in ${\mathfrak m}athfrak{l}ci(\thetatt)$ have finite degree and those in ${\mathfrak m}athfrak{l}ui(\thetatt)$ have infinite degree.
Also, Lemma \ref{propertiesTMN} shows that in all cases displayed in Theorem \ref{main}, we have
$$
\mathfrak{p}hi_n\mathbf{n}ot\in\thetay({\mathfrak m}u_n,\mathbf{n}u) \mathfrak{q}uad {\mathfrak m}box{ and }\mathfrak{q}uad \mathbf{n}u(\mathfrak{p}hi_n)={\mathfrak m}u_n(\mathfrak{p}hi_n)=\Gammaa_n=\op{SV}al({\mathfrak m}u_n)\mathfrak{q}uad {\mathfrak m}box{for all }\ n.
$$
The main advantage of MLV chains is that their nodes are essentially unique, so that we may read in them several data intrinsically associated to the valuation ${\mathfrak m}u$.
For instance the sequence $(m_n)_{n\Gammae0}$ and the character ``ordinary" or ``limit" of each augmentation step ${\mathfrak m}u_n\thetao{\mathfrak m}u_{n+1}$ are intrisic features of $\mathbf{n}u$ {\mathfrak m}athfrak{c}ite[Thm. 4.7]{MLV}.
Thus, we may define order preserving functions
$$
\Deltaep,{\mathfrak m}athfrak{l}dp{\mathfrak m}athfrak{c}olon\thetatt{\mathfrak m}athfrak{l}ra{\mathfrak m}athbb N\infty,
$$
where $\Deltaep(\mathbf{n}u)$ is the length of the MLV chain underlying $\mathbf{n}u$, and
${\mathfrak m}athfrak{l}dp(\mathbf{n}u)$ counts the number of limit augmentations in this MLV chain
{\mathfrak m}athfrak{f}ootnote{Thus, all valuations of both types (a) and (b) have a finite depth. At this point, we are not following the convention of {\mathfrak m}athfrak{c}ite{MLV}, where the valuations of type (a) were said to have {\mathfrak m}edskipmph{finite depth} while those of type (b) were said to have {\mathfrak m}edskipmph{quasi-finite depth}}.
The arguments in the proof of {\mathfrak m}athfrak{c}ite[Lem. 4.2]{MLV} show that these functions preserve the ordering.
\subsection{Decoding MLV chains for arithmetic and geometric applications}
{\mathfrak m}athfrak{l}abel{subsecAMdata}{\mathfrak m}box{\mathbf{n}ull}
Besides their intrinsic theoretical interest, MLV chains encode a large amount of information which can
be useful in several contexts. In this section, we describe a concrete MLV chain in full
generality, and then we interpret it from both the number theoretic and the geometric
perspective. In the former case, we will see how to describe the decomposition of primes in number
fields, while in the geometric context we will provide the desingularization of a curve.
Let $(\mathcal{O},v)$ be a valuation ring with fraction field $K$ and value group $\Gamma_v={\mathfrak m}athbb Z$. Let $p\in \mathcal{O}$ be a uniformizing element.
Consider the following polynomials in $\mathcal{O}[x]$:
{\mathfrak m}athfrak{b}egin{equation}{\mathfrak m}athfrak{l}abel{poli}
{\mathfrak m}athfrak{a}rs{1.2}
{\mathfrak m}athfrak{b}egin{array}{rl}
\mathfrak{p}hi_0=&x,\\
\mathfrak{p}hi_1=&x^5+p^3,\\
\mathfrak{p}hi_2=&\mathfrak{p}hi_1^3+p^{10}=x^{15}+3p^3x^{10}+3p^6x^5+p^9+p^{10}, \\
\mathfrak{p}hi_3=&\mathfrak{p}hi_2^2+p^{11}\mathfrak{p}hi_0^4\mathfrak{p}hi_1^2, \\
= & x^{30}+ 6p^3x^{25}+ 15p^6x^{20} + 2p^{9}(p+10)x^{15}+ p^{11}x^{14} +\\
&(6p+15)p^{12}x^{10}+2p^{14}x^9 +6(p+1)p^{15}x^5+p^{17}x^4+p^{18}(p+1)^2.
{\mathfrak m}edskipnd{array}
{\mathfrak m}edskipnd{equation}
Let us build a MLV chain of valuations on $K[x]$. We start with the valuation:
$$
\Deltaisplaystyle{\mathfrak m}u_0{\mathfrak m}athfrak{l}eft(\sum\mathbf{n}olimits_i a_i x^i\rho_ight):={\mathfrak m}in_{i} \{ v(a_i)+(3/5)i\},
$$
and consider the augmentations:
$$
\Deltaisplaystyle{\mathfrak m}u_1=[{\mathfrak m}u_0; \mathfrak{p}hi_1,10/3], \mathfrak{q}quad
\Deltaisplaystyle{\mathfrak m}u_2=[{\mathfrak m}u_1; \mathfrak{p}hi_2, 301/30], \mathfrak{q}quad
\Deltaisplaystyle{\mathfrak m}u_3=[{\mathfrak m}u_2; \mathfrak{p}hi_3, \infty].
$$
Note that these valuations are distributed along a path of the valuative tree $\thetatt({\mathfrak m}athbb Q)$, reaching the finite leave ${\mathfrak m}u_3$. We get the following MLV for ${\mathfrak m}u_3$:
{\mathfrak m}athfrak{b}egin{equation}{\mathfrak m}athfrak{l}abel{exampleMLVchain}
{\mathfrak m}u_0 \stackrel{\mathfrak{p}hi_1,10/3}{\mathfrak m}athfrak{l}ra {\mathfrak m}u_1
\stackrel{\mathfrak{p}hi_2,301/30}{\mathfrak m}athfrak{l}ra {\mathfrak m}u_2 \stackrel{\mathfrak{p}hi_3,\infty}{\mathfrak m}athfrak{l}ra\ {\mathfrak m}u_3.
{\mathfrak m}edskipnd{equation}
Let us look at this MLV chain in an arithmetic context, by taking $K={\mathfrak m}athbb Q$ as our base field, fixing a rational prime $p\in{\mathfrak m}athbb Z$ and considering the $p$-adic valuation as our valuation $v$. In this setting, the MLV chain (\ref{exampleMLVchain}) encodes an important amount of information on the ring of integers ${\mathfrak m}athbb Z_L$ of the number field $L={\mathfrak m}athbb Q[x]/(\mathfrak{p}hi_3)$. For instance, the prime ideal decomposition of $p$ in ${\mathfrak m}athbb Z_L$ is completely described by (\ref{exampleMLVchain}). This can be checked by applying the OM-algorithm {\mathfrak m}athfrak{c}ite{gen} to the pair $K,p$. The algorithm yields an OM-representation of $\mathfrak{p}hi_3(x)$ consisting of the unique order 3 type:
$$
\thetay=(y;(x,3/5,y+1);(\mathfrak{p}hi_2,5/3,y+1); (\mathfrak{p}hi_3,1/2,y+1)),
$$
which can be seen as a computational representation of the MLV chain. It exhibits key polynomials, slopes of Newton polygons and {\mathfrak m}edskipmph{residual polynomials}.
Since the OM-algorithm returns a unique type, we know that
$p{\mathfrak m}athbb Z_L$ is divided by a unique prime ideal ${\mathfrak m}athfrak{P}$ whose ramification index is the product of the denominators of the slopes in $\thetay$: $e({\mathfrak m}athfrak{P}/p)=5{\mathfrak m}athfrak{c}dot3{\mathfrak m}athfrak{c}dot2=30$. The residual degree $f({\mathfrak m}athfrak{P}/p)=1$ is the product of the degrees of the residual polynomials in $\thetay$.
For any root $\thetaheta\in\overline{{\mathfrak m}athbb Q}$ of $\mathfrak{p}hi_3$, we can derive from this data the following values:
{\mathfrak m}athfrak{b}egin{equation}{\mathfrak m}athfrak{l}abel{val-Okutsu}
{\mathfrak m}athfrak{b}ar{v}(\thetaheta)=3/5,\mathfrak{q}quad {\mathfrak m}athfrak{b}ar{v}(\mathfrak{p}hi_1(\thetaheta))=10/3, \mathfrak{q}quad {\mathfrak m}athfrak{b}ar{v}(\mathfrak{p}hi_2(\thetaheta))=301/30,
{\mathfrak m}edskipnd{equation}
where ${\mathfrak m}athfrak{b}ar{v}$ is the unique extension of $v$ to $L$.
Now, suppose that $p$ is an indeterminate, so that (\ref{poli}) can be thought as the equations of the germs at the origin $(0,0)$ of plane curves $f(p,x)=\mathfrak{p}hi_3(x)=0$, $f_i(p,x)=\mathfrak{p}hi_i(x)=0$ with $i=1,2$, $f_0(p,x)=p=0$, $f_{-1}(p,x)=\mathfrak{p}hi_0(x)=0$, over ${\mathfrak m}athbb C$. Take $K={\mathfrak m}athbb C(p)$ as our base field, equipped with the $p$-adic valuation.
The very same OM-algorithm shows that $f$ is irreducible in ${\mathfrak m}athbb C((p))[x]$. There is a unique finite leaf ${\mathfrak m}u_3\in\thetatt({\mathfrak m}athbb Q)$ with support $f{\mathfrak m}athbb C(p)[x]$, and a MLV chain of ${\mathfrak m}u_3$ is given in (\ref{exampleMLVchain}). Let $\mathbf{n}u = 30 {\mathfrak m}u _3$. Then, for any $\mathfrak{p}hi \in {\mathfrak m}athbb C((p))[x]$, $\mathbf{n}u (\mathfrak{p}hi )$ is the intersection multiplicity between the germs of curve $f=0$ and $\mathfrak{p}hi = 0$.
The data supported by this chain contains completely analogous arithmetic information about $f$, but these data have an added geometric perspective.
Indeed, the equation (\ref{poli}) has the property that the line $p=0$ cuts the curve $f=0$ only in its
singular point $(0,0)$, being tangent at it. In this case, the OM algorithm parallels the Newton-Puiseux algorithm for desingularization {\mathfrak m}athfrak{c}ite[Sec. 1.2]{casas}, and the slopes and key polynomials can be reinterpreted in terms of the sequence of blow-ups involved in the desingularization process. The number of finite leaves detected by the algorithm (one in our case) is the number of points in the normalized curve lying above $(0,0)$.
Any point $P$ blown-up in the desingularization process of $f=0$ gives rise to an exceptional divisor $E_P$. For the sake of simplicity, we will use the same notation $E_P$ for any strict transform of this divisor. Any such $P$ lies either on the intersection of two exceptional divisors and $P$ is called {\mathfrak m}edskipmph{satellite}, or it lies just on only one exceptional divisor and $P$ is called {\mathfrak m}edskipmph{free}.
We say that a satellite point is {\mathfrak m}edskipmph{satellite of} the last free point preceding it.
Among all the points on $f=0$, there is a first satellite point, satellite of a free point $P_1$, which is followed by a sequence of satellite points, being $Q_1$ the last of them. Now, let $P_2$ ($P_3$) be the second (third) free point followed by some satellite point, and let $Q_2$ ($Q_3$) be last point in the sequence of satellite points following $P_2$ ($P_3$).
The points $Q_i$ are special points in the desingularization of the curve, since they are {\mathfrak m}edskipmph{rupture} points, that is, the exceptional divisor $E_{Q_i}$ intersects three or more other components in the pull-back of the curve.
Consider the divisorial valuation $\mathbf{n}u_i $ whose last centre is $Q_i$ for any $3 \Gammaeq i \Gammaeq 1$.
It turns out that $\mathbf{n}u _1= 5 {\mathfrak m}u_0$, $\mathbf{n}u_2 = 15 {\mathfrak m}u _1$, $\mathbf{n}u_3= 30 {\mathfrak m}u_2$ and moreover
$$
\mathbf{n}u (\mathfrak{p}hi_0)=18,\mathfrak{q}quad \mathbf{n}u (\mathfrak{p}hi_1)=100, \mathfrak{q}quad \mathbf{n}u (\mathfrak{p}hi_2)=301,
$$
which are the values appearing at (\ref{val-Okutsu}) multiplied by $\mathbf{n}u (f_0) = \mathbf{n}u (p) = 30$.
Furthermore, the germ of curve $f_{i-1}(p,x)=0$ shares with $f=0$ all its singular points and some more free simple points until $P_i$, for each $3 \Gammaeq i \Gammaeq 1$.
\subsection{Nodes of depth zero}{\mathfrak m}athfrak{l}abel{subsecDepth0}
For given $a\in K$ and $\Deltata\in \Lambda\infty$, the depth-zero node $\mathbf{n}u=\omega_{a,\Deltata}$ is defined as
$$
\mathbf{n}u{\mathfrak m}athfrak{l}eft(\sum\mathbf{n}olimits_{0{\mathfrak m}athfrak{l}e s}a_s(x-a)^s\rho_ight) = {\mathfrak m}in{\mathfrak m}athfrak{l}eft\{v(a_s)+s\Deltata{\mathfrak m}id0{\mathfrak m}athfrak{l}e s\rho_ight\}.
$$
Clearly, $\omega_{a,\infty}$ is a finite leaf of $\thetatt$ with support $(x-a)\op{Ker}x$, while for $\Deltata<\infty$ the valuation $\omega_{a,\Deltata}$ is an inner node admitting $x-a$ as a key polynomial.
Besides these (well specified) inner nodes and finite leaves, $\thetatt$ may have depth-zero infinite leaves which are the stable limit of a continuous family of augmentations of stable degree one:
$$
{\mathfrak m}u_0\stackrel{(\rho_i)_{i\in A}}{\mathfrak m}athfrak{l}ra \mathbf{n}u,
$$
where ${\mathfrak m}u_0$ is an inner node of depth zero. By Theorem \ref{main}, all depth-zero nodes in $\thetatt$ arise from either of these two ways.
For any fixed $a \in K$, the set $\Lambda\infty$ parametrizes a certain path in $\thetatt$, containing all depth-zero nodes $\omega_{a,\Deltata}$:
{\mathfrak m}athfrak{b}egin{center}
\setlength{\unitlength}{4mm}
{\mathfrak m}athfrak{b}egin{picture}(20,3)
\mathfrak{p}ut(0.25,1.2){{\mathfrak m}athfrak{l}ine(1,0){16}}
\mathfrak{p}ut(6,0.9){${\mathfrak m}athfrak{b}ullet$}\mathfrak{p}ut(20,0.9){${\mathfrak m}athfrak{b}ullet$}
\mathfrak{p}ut(-2.5,1){{\mathfrak m}athfrak{b}egin{footnotesize}${\mathfrak m}athfrak{c}dots{\mathfrak m}athfrak{c}dots${\mathfrak m}edskipnd{footnotesize}}
\mathfrak{p}ut(17,1){{\mathfrak m}athfrak{b}egin{footnotesize}${\mathfrak m}athfrak{c}dots{\mathfrak m}athfrak{c}dots${\mathfrak m}edskipnd{footnotesize}}
\mathfrak{p}ut(21,1.1){{\mathfrak m}athfrak{b}egin{footnotesize}$\omega_{a,\infty}${\mathfrak m}edskipnd{footnotesize}}
\mathfrak{p}ut(5.8,2){{\mathfrak m}athfrak{b}egin{footnotesize}$\omega_{a,\Deltata}${\mathfrak m}edskipnd{footnotesize}}
{\mathfrak m}edskipnd{picture}
{\mathfrak m}edskipnd{center}
The node $\omega_{a,\Deltata}$ is commensurable if and only if $\Deltata\in\Gammaq\infty$.
The relative position of the paths corresponding to two different elements $a,b\in K$ is completely determined by the following easy observation:
{\mathfrak m}athfrak{b}egin{equation}{\mathfrak m}athfrak{l}abel{balls}
\omega_{a,\Deltata}{\mathfrak m}athfrak{l}e\omega_{b,{\mathfrak m}edskipp}\ \ \Longleftrightarrow\ \ {\mathfrak m}in\{v(b-a),{\mathfrak m}edskipp\}\Gammae\Deltata.
{\mathfrak m}edskipnd{equation}
Thus, the depth-zero paths in $\thetatt$ determined by any two $a,b\in K$ coincide for all parameters $\Deltata\in\Lambda$ such that
$\Deltata{\mathfrak m}athfrak{l}e v(b-a)$.
{\mathfrak m}athfrak{b}egin{center}
\setlength{\unitlength}{4mm}
{\mathfrak m}athfrak{b}egin{picture}(22,7.5)
\mathfrak{p}ut(4.75,1){${\mathfrak m}athfrak{b}ullet$}\mathfrak{p}ut(11,1){${\mathfrak m}athfrak{b}ullet$}\mathfrak{p}ut(11,3.16){${\mathfrak m}athfrak{b}ullet$}
\mathfrak{p}ut(18,1){${\mathfrak m}athfrak{b}ullet$}\mathfrak{p}ut(18,5.4){${\mathfrak m}athfrak{b}ullet$}
\mathfrak{p}ut(0.25,1.3){{\mathfrak m}athfrak{l}ine(1,0){16}}\mathfrak{p}ut(5,1.3){{\mathfrak m}athfrak{l}ine(3,1){11.3}}
\mathfrak{p}ut(16.66,4.48){$\Deltaot{}$}\mathfrak{p}ut(16.99,4.59){$\Deltaot{}$}\mathfrak{p}ut(17.32,4.7){$\Deltaot{}$}
\mathfrak{p}ut(-2.4,1.05){{\mathfrak m}athfrak{b}egin{footnotesize}${\mathfrak m}athfrak{c}dots{\mathfrak m}athfrak{c}dots${\mathfrak m}edskipnd{footnotesize}}
\mathfrak{p}ut(16.5,1.05){{\mathfrak m}athfrak{b}egin{footnotesize}${\mathfrak m}athfrak{c}dots${\mathfrak m}edskipnd{footnotesize}}
\mathfrak{p}ut(17.6,0){{\mathfrak m}athfrak{b}egin{footnotesize}$\omega_{a,\infty}${\mathfrak m}edskipnd{footnotesize}}
\mathfrak{p}ut(17.6,6.4){{\mathfrak m}athfrak{b}egin{footnotesize}$\omega_{b,\infty}${\mathfrak m}edskipnd{footnotesize}}
\mathfrak{p}ut(2.6,2){{\mathfrak m}athfrak{b}egin{footnotesize}$\omega_{b,v(a-b)}${\mathfrak m}edskipnd{footnotesize}}
\mathfrak{p}ut(2.6,0){{\mathfrak m}athfrak{b}egin{footnotesize}$\omega_{a,v(a-b)}${\mathfrak m}edskipnd{footnotesize}}
\mathfrak{p}ut(10.6,0){{\mathfrak m}athfrak{b}egin{footnotesize}$\omega_{a,\Deltata}${\mathfrak m}edskipnd{footnotesize}}
\mathfrak{p}ut(10.6,4.2){{\mathfrak m}athfrak{b}egin{footnotesize}$\omega_{b,\Deltata}${\mathfrak m}edskipnd{footnotesize}}
{\mathfrak m}edskipnd{picture}
{\mathfrak m}edskipnd{center}{\mathfrak m}edskip
In particular, for all depth-zero nodes ${\mathfrak m}u_0,\mathbf{n}u_0\in \thetatt$, there is a depth-zero node $\rho_0$ such that $\rho_0<{\mathfrak m}u_0$ and $\rho_0<\mathbf{n}u_0$. By Theorem \ref{main}, for all nodes ${\mathfrak m}u,\mathbf{n}u\in\thetatt$ we have
{\mathfrak m}athfrak{b}egin{equation}{\mathfrak m}athfrak{l}abel{intersect}
(-\infty,{\mathfrak m}u\,]{\mathfrak m}athfrak{c}ap (-\infty,\mathbf{n}u\,]\mathbf{n}e{\mathfrak m}edskipmptyset.
{\mathfrak m}edskipnd{equation}
\subsection{Paths of constant depth obtained by ordinary augmentations}{\mathfrak m}athfrak{l}abel{subsecConstDepthOrd}
Let us fix an inner node ${\mathfrak m}u\in\thetatt$.
For all $\mathfrak{p}hi\in\op{Ker}pm$, we define the {\mathfrak m}edskipmph{constant-depth path} beyond ${\mathfrak m}u$ as the set:
$$
\mathfrak{p}mph={\mathfrak m}athfrak{l}eft\{{\mathfrak m}u(\mathfrak{p}hi,\Gammaa){\mathfrak m}id {\mathfrak m}u(\mathfrak{p}hi)<\Gammaa{\mathfrak m}athfrak{l}e\infty\rho_ight\},\mathfrak{q}quad {\mathfrak m}u(\mathfrak{p}hi,\Gammaa):=[{\mathfrak m}u;\,\mathfrak{p}hi,\Gammaa],
$$
containing all ordinary augmentations of ${\mathfrak m}u$ with respect to $\mathfrak{p}hi$.
This path joins ${\mathfrak m}u$ with the finite leaf ${\mathfrak m}u(\mathfrak{p}hi,\infty)$. Actually, by {\mathfrak m}athfrak{c}ite[Lem. 2.7]{MLV}, $\mathfrak{p}mph$ coincides with the semiopen interval $(\,{\mathfrak m}u,{\mathfrak m}u(\mathfrak{p}hi,\infty)\,]$ in $\thetatt$.
{\mathfrak m}athfrak{b}egin{center}
\setlength{\unitlength}{4mm}
{\mathfrak m}athfrak{b}egin{picture}(18,4)
\mathfrak{p}ut(-2,1){${\mathfrak m}athfrak{b}ullet$}\mathfrak{p}ut(-1.6,1.3){{\mathfrak m}athfrak{l}ine(1,0){16}}\mathfrak{p}ut(18,1){${\mathfrak m}athfrak{b}ullet$}
\mathfrak{p}ut(6,1){${\mathfrak m}athfrak{b}ullet$}
\mathfrak{p}ut(-2,2){{\mathfrak m}athfrak{b}egin{footnotesize}${\mathfrak m}u${\mathfrak m}edskipnd{footnotesize}}
\mathfrak{p}ut(16.5,1){{\mathfrak m}athfrak{b}egin{footnotesize}${\mathfrak m}athfrak{c}dots${\mathfrak m}edskipnd{footnotesize}}
\mathfrak{p}ut(15.2,1){{\mathfrak m}athfrak{b}egin{footnotesize}${\mathfrak m}athfrak{c}dots${\mathfrak m}edskipnd{footnotesize}}
\mathfrak{p}ut(16.8,2){{\mathfrak m}athfrak{b}egin{footnotesize}${\mathfrak m}u(\mathfrak{p}hi,\infty)${\mathfrak m}edskipnd{footnotesize}}
\mathfrak{p}ut(4.8,2){{\mathfrak m}athfrak{b}egin{footnotesize}${\mathfrak m}u(\mathfrak{p}hi,\Gammaa)${\mathfrak m}edskipnd{footnotesize}}
{\mathfrak m}edskipnd{picture}
{\mathfrak m}edskipnd{center}{\mathfrak m}edskip
Regardless of the commensurability or incommensurability of ${\mathfrak m}u$, Lemma \ref{propertiesAug},(2) shows that
${\mathfrak m}u(\mathfrak{p}hi,\Gammaa)$ is commensurable if and only if $\Gammaa\in\Gammaq\infty$.{\mathfrak m}edskip
\Deltaefn A key polynomial $\mathfrak{p}hi\in\op{Ker}pm$ is said to be {\mathfrak m}edskipmph{strong} if $\Deltaeg(\mathfrak{p}hi)>\Deltaeg({\mathfrak m}u)$.
We say that $\mathfrak{p}mph$ is {\mathfrak m}edskipmph{strong} if $\mathfrak{p}hi$ is strong, and $\mathfrak{p}mph$ is {\mathfrak m}edskipmph{weak} otherwise. {\mathfrak m}edskip
All the nodes in this path have the same degree: $\Deltaeg{\mathfrak m}athfrak{l}eft({\mathfrak m}u(\mathfrak{p}hi,\Gammaa)\rho_ight)=\Deltaeg(\mathfrak{p}hi)$, and the same depth too:
$$
\Deltaep{\mathfrak m}athfrak{l}eft({\mathfrak m}u(\mathfrak{p}hi,\Gammaa)\rho_ight)=
{\mathfrak m}athfrak{b}egin{cases}
\Deltaep({\mathfrak m}u),&{\mathfrak m}box{ if the path is weak},\\
\Deltaep({\mathfrak m}u)+1,&{\mathfrak m}box{ if the path is strong}.
{\mathfrak m}edskipnd{cases}
$$
Actually, for any given MLV chain of ${\mathfrak m}u$:
$$ {\mathfrak m}u_0\ \stackrel{\mathfrak{p}hi_1,\Gammaa_1}{\mathfrak m}athfrak{l}ra\ {\mathfrak m}u_1\ \stackrel{\mathfrak{p}hi_2,\Gammaa_2}{\mathfrak m}athfrak{l}ra\ {\mathfrak m}athfrak{c}dots\ {\mathfrak m}athfrak{l}ra\ {\mathfrak m}u_{r-1}\ \stackrel{\mathfrak{p}hi_{r},\Gammaa_{r}}{\mathfrak m}athfrak{l}ra\ {\mathfrak m}u_{r}={\mathfrak m}u,$$
we may obtain a MLV chain of ${\mathfrak m}u(\mathfrak{p}hi,\Gammaa)$ as follows.
If the path is weak and ${\mathfrak m}u$ has depth zero, then all ${\mathfrak m}u(\mathfrak{p}hi,\Gammaa)$ have depth zero too.
If the path is weak and ${\mathfrak m}u$ has a positive depth, we may consider
$$ {\mathfrak m}u_0\ \stackrel{\mathfrak{p}hi_1,\Gammaa_1}{\mathfrak m}athfrak{l}ra\ {\mathfrak m}u_1\ \stackrel{\mathfrak{p}hi_2,\Gammaa_2}{\mathfrak m}athfrak{l}ra\ {\mathfrak m}athfrak{c}dots\ {\mathfrak m}athfrak{l}ra\ {\mathfrak m}u_{r-1}\ \stackrel{\mathfrak{p}hi,\Gammaa}{\mathfrak m}athfrak{l}ra\ {\mathfrak m}u(\mathfrak{p}hi,\Gammaa),$$
regardless of the fact that ${\mathfrak m}u_{r-1}\thetao{\mathfrak m}u_r$ is an ordinary or a limit augmentation.
If the path is strong, we may just add one more (ordinary) augmentation:
$$ {\mathfrak m}u_0\ \stackrel{\mathfrak{p}hi_1,\Gammaa_1}{\mathfrak m}athfrak{l}ra\ {\mathfrak m}u_1\ \stackrel{\mathfrak{p}hi_2,\Gammaa_2}{\mathfrak m}athfrak{l}ra\ {\mathfrak m}athfrak{c}dots\ {\mathfrak m}athfrak{l}ra\ {\mathfrak m}u_{r-1}\ \stackrel{\mathfrak{p}hi_{r},\Gammaa_{r}}{\mathfrak m}athfrak{l}ra\ {\mathfrak m}u\ \stackrel{\mathfrak{p}hi,\Gammaa}{\mathfrak m}athfrak{l}ra\ {\mathfrak m}u(\mathfrak{p}hi,\Gammaa).$$
Therefore, the nodes in a strong path are ``properly" derived from ${\mathfrak m}u$, while the nodes in weak paths are derived from lower nodes.
Let us analyze the intersection of two paths of constant depth beyond the same node ${\mathfrak m}u$, determined by different key polynomials $\mathfrak{p}hi,\mathfrak{p}hi_*\in\op{Ker}pm$.
Obviously,
$$\thetay({\mathfrak m}u,\rho)=[\mathfrak{p}hi]_{\mathfrak m}u\mathfrak{q}uad {\mathfrak m}box{for all}\mathfrak{q}uad \rho\in \mathfrak{p}mph.
$$
Therefore, if $\mathfrak{p}hi\mathbf{n}ot\sim_{\mathfrak m}u\mathfrak{p}hi_*$, Proposition \ref{td=td} shows that $\mathfrak{p}mph{\mathfrak m}athfrak{c}ap\mathfrak{p}set_{\mathfrak m}u(\mathfrak{p}hi_*)={\mathfrak m}edskipmptyset$.
If $\mathfrak{p}hi\sim_\mu\mathfrak{p}hi_*$, then ${\mathfrak m}u(\mathfrak{p}hi-\mathfrak{p}hi_*)>{\mathfrak m}u(\mathfrak{p}hi)={\mathfrak m}u(\mathfrak{p}hi_*)$. Thus, (\ref{eqAug}) shows that
$$\mathfrak{p}mph{\mathfrak m}athfrak{c}ap\mathfrak{p}set_{\mathfrak m}u(\mathfrak{p}hi_*)={\mathfrak m}athfrak{l}eft({\mathfrak m}u,{\mathfrak m}u(\mathfrak{p}hi,\Gammaa_0)\rho_ight],\mathfrak{q}quad \Gammaa_0={\mathfrak m}u(\mathfrak{p}hi-\mathfrak{p}hi_*).
$$
{\mathfrak m}athfrak{b}egin{center}
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\mathfrak{p}ut(-2,1){${\mathfrak m}athfrak{b}ullet$}\mathfrak{p}ut(4.75,1){${\mathfrak m}athfrak{b}ullet$}
\mathfrak{p}ut(11,1){${\mathfrak m}athfrak{b}ullet$}\mathfrak{p}ut(11,3.16){${\mathfrak m}athfrak{b}ullet$}
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\mathfrak{p}ut(-1.6,1.3){{\mathfrak m}athfrak{l}ine(1,0){19}}\mathfrak{p}ut(5,1.3){{\mathfrak m}athfrak{l}ine(3,1){12.3}}
\mathfrak{p}ut(17.76,4.88){$\Deltaot{}$}\mathfrak{p}ut(18.09,4.99){$\Deltaot{}$}\mathfrak{p}ut(18.42,5.11){$\Deltaot{}$}
\mathfrak{p}ut(18.9,5.26){$\Deltaot{}$}\mathfrak{p}ut(19.23,5.37){$\Deltaot{}$}\mathfrak{p}ut(19.56,5.5){$\Deltaot{}$}
\mathfrak{p}ut(-3,1.1){{\mathfrak m}athfrak{b}egin{footnotesize}${\mathfrak m}u${\mathfrak m}edskipnd{footnotesize}}
\mathfrak{p}ut(17.7,1){{\mathfrak m}athfrak{b}egin{footnotesize}${\mathfrak m}athfrak{c}dots${\mathfrak m}edskipnd{footnotesize}}
\mathfrak{p}ut(18.9,1){{\mathfrak m}athfrak{b}egin{footnotesize}${\mathfrak m}athfrak{c}dots${\mathfrak m}edskipnd{footnotesize}}
\mathfrak{p}ut(21,1.2){{\mathfrak m}athfrak{b}egin{footnotesize}${\mathfrak m}u(\mathfrak{p}hi,\infty)${\mathfrak m}edskipnd{footnotesize}}
\mathfrak{p}ut(21,6.2){{\mathfrak m}athfrak{b}egin{footnotesize}${\mathfrak m}u(\mathfrak{p}hi_*,\infty)${\mathfrak m}edskipnd{footnotesize}}
\mathfrak{p}ut(2.5,2){{\mathfrak m}athfrak{b}egin{footnotesize}${\mathfrak m}u(\mathfrak{p}hi_*,\Gammaa_0)${\mathfrak m}edskipnd{footnotesize}}
\mathfrak{p}ut(2.6,0){{\mathfrak m}athfrak{b}egin{footnotesize}${\mathfrak m}u(\mathfrak{p}hi,\Gammaa_0)${\mathfrak m}edskipnd{footnotesize}}
\mathfrak{p}ut(9.6,0){{\mathfrak m}athfrak{b}egin{footnotesize}${\mathfrak m}u(\mathfrak{p}hi,\Gammaa)${\mathfrak m}edskipnd{footnotesize}}
\mathfrak{p}ut(9.6,4.2){{\mathfrak m}athfrak{b}egin{footnotesize}${\mathfrak m}u(\mathfrak{p}hi_*,\Gammaa)${\mathfrak m}edskipnd{footnotesize}}
{\mathfrak m}edskipnd{picture}
{\mathfrak m}edskipnd{center}{\mathfrak m}athfrak{b}s
\subsection{Paths of constant depth obtained by limit augmentations}{\mathfrak m}athfrak{l}abel{subsecConstDepthLim}
Let us fix an essential continuous family ${\mathfrak m}athfrak{a}a={\mathfrak m}athfrak{l}eft(\rho_i\rho_ight)_{i\in A}$. By taking a cofinal family, if necessary, we may assume that ${\mathfrak m}athfrak{a}a$ contains a minimal valuation ${\mathfrak m}u$.
For all limit key polynomials $\mathfrak{p}hi\in\op{Ker}pi({\mathfrak m}athfrak{a}a)$, we may consider the {\mathfrak m}edskipmph{constant depth path} beyond ${\mathfrak m}athfrak{a}a$:
$$
\mathfrak{p}aph={\mathfrak m}athfrak{l}eft\{{\mathfrak m}athfrak{a}a(\mathfrak{p}hi,\Gammaa){\mathfrak m}id \rho_i(\mathfrak{p}hi)<\Gammaa{\mathfrak m}athfrak{l}e\infty \ {\mathfrak m}box{ for all }i\in A\rho_ight\},\mathfrak{q}quad {\mathfrak m}athfrak{a}a(\mathfrak{p}hi,\Gammaa):=[{\mathfrak m}athfrak{a}a;\,\mathfrak{p}hi,\Gammaa],
$$
containing all possible limit augmentations determined by $\mathfrak{p}hi$.
{\mathfrak m}athfrak{b}egin{center}
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\mathfrak{p}ut(-2,1){${\mathfrak m}athfrak{b}ullet$}\mathfrak{p}ut(26,1){${\mathfrak m}athfrak{b}ullet$}\mathfrak{p}ut(12,1){${\mathfrak m}athfrak{b}ullet$}
\mathfrak{p}ut(-1.6,1.25){{\mathfrak m}athfrak{l}ine(1,0){3.4}}\mathfrak{p}ut(2.8,1){${\mathfrak m}athfrak{c}dots$}\mathfrak{p}ut(5,1){${\mathfrak m}athfrak{c}dots$}\mathfrak{p}ut(6.5,1.25){{\mathfrak m}athfrak{l}ine(1,0){16}}
{\mathfrak m}ultiput(4.5,0.1)(0,.25){10}{\vrule height1pt}
\mathfrak{p}ut(-1,2){$(\rho_i)_{i\in A}$}
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\mathfrak{p}ut(24,1){{\mathfrak m}athfrak{b}egin{footnotesize}${\mathfrak m}athfrak{c}dots${\mathfrak m}edskipnd{footnotesize}}
\mathfrak{p}ut(24.8,2){{\mathfrak m}athfrak{b}egin{footnotesize}${\mathfrak m}athfrak{a}a(\mathfrak{p}hi,\infty)${\mathfrak m}edskipnd{footnotesize}}
\mathfrak{p}ut(10.8,2){{\mathfrak m}athfrak{b}egin{footnotesize}${\mathfrak m}athfrak{a}a(\mathfrak{p}hi,\Gammaa)${\mathfrak m}edskipnd{footnotesize}}
{\mathfrak m}edskipnd{picture}
{\mathfrak m}edskipnd{center}{\mathfrak m}edskip
As in the previous cases, the last node of the path, ${\mathfrak m}athfrak{a}a(\mathfrak{p}hi,\infty)$, is a finite leaf.
By Proposition \ref{extensionlim}, ${\mathfrak m}athfrak{a}a(\mathfrak{p}hi,\Gammaa)$ is commensurable if and only if $\Gammaa\in\Gammaq\infty$.
By {\mathfrak m}athfrak{c}ite[Lem. 3.8]{MLV}, we have $\mathfrak{p}aph={\mathfrak m}athfrak{b}igcap\mathbf{n}olimits_{i\in A}(\,\rho_i,{\mathfrak m}athfrak{a}a(\mathfrak{p}hi,\infty)\,]$.
All the nodes in this path have the same degree and the same depth:
$$\Deltaeg{\mathfrak m}athfrak{l}eft({\mathfrak m}athfrak{a}a(\mathfrak{p}hi,\Gammaa)\rho_ight)=\Deltaeg(\mathfrak{p}hi),\mathfrak{q}quad\Deltaep{\mathfrak m}athfrak{l}eft({\mathfrak m}athfrak{a}a(\mathfrak{p}hi,\Gammaa)\rho_ight)=\Deltaep({\mathfrak m}u)+1.
$$
For any $\mathfrak{p}hi,\mathfrak{p}hi_*\in\op{Ker}pi({\mathfrak m}athfrak{a}a)$, let $\Deltata=\rho_{\mathfrak m}athfrak{a}a(\mathfrak{p}hi-\mathfrak{p}hi_*)$. By {\mathfrak m}athfrak{c}ite[Lem. 3.7]{MLV}, we have
$$
{\mathfrak m}athfrak{a}a(\mathfrak{p}hi,\Gammaa)={\mathfrak m}athfrak{a}a(\mathfrak{p}hi_*,\Gammaa_*)\ \Longleftrightarrow\ \Gammaa=\Gammaa_*\Gammae\Deltata.
$$
By Lemma \ref{allLKP}, ${\mathfrak m}athfrak{a}a(\mathfrak{p}hi,\Deltata)={\mathfrak m}athfrak{a}a(\mathfrak{p}hi_*,\Deltata)$ belongs to $\mathfrak{p}aph{\mathfrak m}athfrak{c}ap\mathfrak{p}set_{\mathfrak m}athfrak{a}a(\mathfrak{p}hi_*)$.
Therefore, the intersection of the paths determined by $\mathfrak{p}hi$ and $\mathfrak{p}hi_*$ is completely analogous to the case of depth-zero valuations.
\subsection{Greatest common lower node}{\mathfrak m}athfrak{l}abel{subsecGCN}
Given ${\mathfrak m}u,\mathbf{n}u\in\thetatt$, their {\mathfrak m}edskipmph{greatest common lower node} is defined as
$$
{\mathfrak m}u\wedge\mathbf{n}u={\mathfrak m}ax{\mathfrak m}athfrak{l}eft( (-\infty,{\mathfrak m}u\,]{\mathfrak m}athfrak{c}ap (-\infty,\mathbf{n}u\,]\rho_ight)\in \thetatt,
$$
provided that this maximal element exists.
{\mathfrak m}athfrak{b}egin{proposition}{\mathfrak m}athfrak{l}abel{GCN}
For all ${\mathfrak m}u,\mathbf{n}u\in\thetatt$, their greatest common lower node ${\mathfrak m}u\wedge\mathbf{n}u$ exists
{\mathfrak m}edskipnd{proposition}
{\mathfrak m}athfrak{b}egin{proof}
If ${\mathfrak m}u{\mathfrak m}athfrak{l}e\mathbf{n}u$, then obviously ${\mathfrak m}u\wedge\mathbf{n}u={\mathfrak m}u$. Suppose that neither ${\mathfrak m}u{\mathfrak m}athfrak{l}e\mathbf{n}u$ nor ${\mathfrak m}u\Gammae\mathbf{n}u$.
As we saw in (\ref{intersect}),
$
(-\infty,{\mathfrak m}u\,]{\mathfrak m}athfrak{c}ap (-\infty,\mathbf{n}u\,]\mathbf{n}e{\mathfrak m}edskipmptyset$.
Let us prove that this totally ordered set always contains a maximal element.
Consider a MLV chain of ${\mathfrak m}u$
$$
{\mathfrak m}u_0\ \stackrel{\mathfrak{p}hi_{1},\Gammaa_{1}}{\mathfrak m}athfrak{l}ra\ {\mathfrak m}u_{1}\ {\mathfrak m}athfrak{l}ra\ {\mathfrak m}athfrak{c}dots
\ {\mathfrak m}athfrak{l}ra\ {\mathfrak m}u_{r-1}
\ \stackrel{\mathfrak{p}hi_{r},\Gammaa_{r}}{\mathfrak m}athfrak{l}ra\ {\mathfrak m}u_{r}={\mathfrak m}u,
$$
Since ${\mathfrak m}u={\mathfrak m}u_r\mathbf{n}ot\in (-\infty,\mathbf{n}u\,]$, there exists a minimal index $i$ such that ${\mathfrak m}u_i\mathbf{n}ot\in (-\infty,\mathbf{n}u\,]$.
We need only to show that ${\mathfrak m}u_i\wedge\mathbf{n}u$ exists, because this clearly implies ${\mathfrak m}u\wedge\mathbf{n}u={\mathfrak m}u_i\wedge\mathbf{n}u$.
Suppose that $i=0$. If ${\mathfrak m}u_0=\omega_{a,\Gammaa}$, we have {\mathfrak m}athfrak{c}ite[Sec. 2.2]{MLV}
$$
{\mathfrak m}athfrak{l}eft(-\infty,{\mathfrak m}u_0\,\rho_ight]={\mathfrak m}athfrak{l}eft\{\omega_{a,\Deltata}{\mathfrak m}id\Deltata\in\Lambda,\ \Deltata{\mathfrak m}athfrak{l}e\Gammaa\rho_ight\}.
$$
On the other hand, by comparing their action of $(x-a)$-expansions, we see that $\omega_{a,\Deltata}{\mathfrak m}athfrak{l}e\mathbf{n}u$ if and only if $\Deltata{\mathfrak m}athfrak{l}e\mathbf{n}u(x-a)$. Since ${\mathfrak m}u_0\mathbf{n}ot{\mathfrak m}athfrak{l}e\mathbf{n}u$, necessarily $\mathbf{n}u(x-a)<\Gammaa$, Thus, there is a maximal element in ${\mathfrak m}athfrak{l}eft(-\infty,{\mathfrak m}u_0\,\rho_ight]{\mathfrak m}athfrak{c}ap{\mathfrak m}athfrak{l}eft(-\infty,\mathbf{n}u\,\rho_ight]$, namely
$${\mathfrak m}u_0\wedge\mathbf{n}u=\omega_{a,\mathbf{n}u(x-a)}.$$
Suppose that $i>0$, so that ${\mathfrak m}u_{i-1}<\mathbf{n}u$, ${\mathfrak m}u_{i}\mathbf{n}ot{\mathfrak m}athfrak{l}e\mathbf{n}u$.
If $\thetay({\mathfrak m}u_{i-1},{\mathfrak m}u_i)\mathbf{n}e\thetay({\mathfrak m}u_{i-1},\mathbf{n}u)$, then Proposition \ref{td=td} shows that $({\mathfrak m}u_{i-1},{\mathfrak m}u_i\,]{\mathfrak m}athfrak{c}ap ({\mathfrak m}u_{i-1},\mathbf{n}u]={\mathfrak m}edskipmptyset$. Hence, ${\mathfrak m}u_i\wedge\mathbf{n}u={\mathfrak m}u_{i-1}$. Suppose that $\thetay({\mathfrak m}u_{i-1},{\mathfrak m}u_i)=\thetay({\mathfrak m}u_{i-1},\mathbf{n}u)$.
If ${\mathfrak m}u_{i-1}\thetao{\mathfrak m}u_i$ is an ordinary augmentation, then {\mathfrak m}athfrak{c}ite[Lem. 2.7]{MLV} shows that
$$
{\mathfrak m}athfrak{l}eft({\mathfrak m}u_{i-1},{\mathfrak m}u_i\,\rho_ight]={\mathfrak m}athfrak{l}eft\{[{\mathfrak m}u_{i-1};\,\mathfrak{p}hi_i,\Deltata]\,{\mathfrak m}id\,{\mathfrak m}u_{i-1}(\mathfrak{p}hi_i)<\Deltata{\mathfrak m}athfrak{l}e\Gammaa_i\rho_ight\}.
$$
Since $\thetay({\mathfrak m}u_{i-1},\mathbf{n}u)=\thetay({\mathfrak m}u_{i-1},{\mathfrak m}u_i)=[\mathfrak{p}hi_i]_{{\mathfrak m}u_{i-1}}$, we have ${\mathfrak m}u_{i-1}(\mathfrak{p}hi_i)<\mathbf{n}u(\mathfrak{p}hi_i)$ and
$$
{\mathfrak m}u_{i-1}(f)=\mathbf{n}u(f) \ \Longleftrightarrow\ \mathfrak{p}hi_i\mathbf{n}mid_{{\mathfrak m}u_{i-1}}f.
$$
In particular, ${\mathfrak m}u_{i-1}(a)=\mathbf{n}u(a)$ for all $a\in\op{Ker}x$ with $\Deltaeg(a)<\Deltaeg(\mathfrak{p}hi_i)$. Hence, by comparing their action on $\mathfrak{p}hi_i$-expansions, we have
$$
[{\mathfrak m}u_{i-1};\,\mathfrak{p}hi_i,\Deltata]{\mathfrak m}athfrak{l}e\mathbf{n}u\ \Longleftrightarrow\ \Deltata{\mathfrak m}athfrak{l}e \mathbf{n}u(\mathfrak{p}hi_i).
$$
Since ${\mathfrak m}u_i\mathbf{n}ot{\mathfrak m}athfrak{l}e\mathbf{n}u$, we have $\mathbf{n}u(\mathfrak{p}hi_i)<\Gammaa_i$. Thus, there is a maximal element in ${\mathfrak m}athfrak{l}eft({\mathfrak m}u_{i-1},{\mathfrak m}u_i\,\rho_ight]{\mathfrak m}athfrak{c}ap{\mathfrak m}athfrak{l}eft({\mathfrak m}u_{i-1},\mathbf{n}u\,\rho_ight]$, namely
$${\mathfrak m}u_i\wedge\mathbf{n}u=[{\mathfrak m}u_{i-1};\,\mathfrak{p}hi_i,\mathbf{n}u(\mathfrak{p}hi_i)].$$
Suppose that ${\mathfrak m}u_{i-1}\thetao{\mathfrak m}u_i$ is a limit augmentation with respect to an essential continuous family ${\mathfrak m}athfrak{a}a=(\rho_j)_{j\in A}$ admitting ${\mathfrak m}u_{i-1}$ as its first element. Then, $\mathfrak{p}hi_i\in\op{Ker}pi({\mathfrak m}athfrak{a}a)$. By Lemma \ref{specialCont} we may assume that, for all $j\in A$,
$$
\rho_j=[{\mathfrak m}u_{i-1};\,{\mathfrak m}athfrak{c}hi_j,{\mathfrak m}athfrak{b}e_j],\mathfrak{q}quad {\mathfrak m}athfrak{c}hi_j\in\op{Ker}p({\mathfrak m}u_{i-1}),\ {\mathfrak m}athfrak{b}e_j=\op{SV}al(\rho_j).
$$
If $\rho_j\mathbf{n}ot{\mathfrak m}athfrak{l}e\mathbf{n}u$ for some $j\in A$, then we can mimic the arguments of the ordinary-augmentation case to conclude that
$${\mathfrak m}u\wedge\mathbf{n}u=\rho_j\wedge\mathbf{n}u=[{\mathfrak m}u_{i-1};\,{\mathfrak m}athfrak{c}hi_j,\mathbf{n}u({\mathfrak m}athfrak{c}hi_j)].$$
Suppose that $\rho_j<\mathbf{n}u$ for all $j\in A$. By Lemma \ref{propertiesTMN}(3), we see that
$\mathbf{n}u$ coincides with $\rho_\aa$ on all ${\mathfrak m}athfrak{a}a$-stable polynomials.
Let $V={\mathfrak m}athfrak{l}eft\{\rho_j(\mathfrak{p}hi_i){\mathfrak m}id j \in A\rho_ight\}$. By {\mathfrak m}athfrak{c}ite[Lem. 3.8]{MLV},
$$
{\mathfrak m}athfrak{l}eft\{\rho\in({\mathfrak m}u_{i-1},{\mathfrak m}u_i\,]\,{\mathfrak m}id\,\rho_j<\rho{\mathfrak m}box{ for all }j\in A\rho_ight\}=
{\mathfrak m}athfrak{l}eft\{[{\mathfrak m}athfrak{a}a;\,\mathfrak{p}hi_i,\Deltata]{\mathfrak m}id V<\Deltata{\mathfrak m}athfrak{l}e\Gammaa_i\rho_ight\}.
$$
By comparing their action on $\mathfrak{p}hi_i$-expansions, we have
$$
[{\mathfrak m}athfrak{a}a;\,\mathfrak{p}hi_i,\Deltata]{\mathfrak m}athfrak{l}e\mathbf{n}u\ \Longleftrightarrow\ \Deltata{\mathfrak m}athfrak{l}e\mathbf{n}u(\mathfrak{p}hi_i).
$$
On the other hand, for all ${\mathfrak m}athfrak{a}a$-unstable polynomials, we have $\rho_j(f)<\rho_{\mathfrak m}edskipll(f)$ for all $j<{\mathfrak m}edskipll$ in $A$. Thus,
$$\rho_j(f)<\rho_{\mathfrak m}edskipll(f){\mathfrak m}athfrak{l}e\mathbf{n}u(f),\mathfrak{q}uad{\mathfrak m}box{for all }j\in A. $$
In particular, $\mathbf{n}u(\mathfrak{p}hi_i)>V$.
As a consequence, there is a maximal valuation in $(-\infty,{\mathfrak m}u_i\,]$ which is less than $\mathbf{n}u$, namely
$$
{\mathfrak m}u_i\wedge\mathbf{n}u=[{\mathfrak m}athfrak{a}a;\,\mathfrak{p}hi_i,\mathbf{n}u(\mathfrak{p}hi_i)].
$$
This ends the proof of the proposition.
{\mathfrak m}edskipnd{proof}{\mathfrak m}edskip
Suppose that ${\mathfrak m}u,\mathbf{n}u\in\thetatt$ are incomparable; that is, ${\mathfrak m}u\mathbf{n}ot{\mathfrak m}athfrak{l}e\mathbf{n}u$ and $\mathbf{n}u\mathbf{n}ot{\mathfrak m}athfrak{l}e{\mathfrak m}u$. Then,
their greatest common lower node, $\rho={\mathfrak m}u\wedge\mathbf{n}u$, has at least two different tangent directions: $\thetay(\rho,{\mathfrak m}u)\mathbf{n}e \thetay(\rho,\mathbf{n}u)$. By Theorems \ref{Delta} and \ref{DeltaKP}, $\rho$ is an inner commensurable node; in other words, $\rho$ is a residually transcendental valuation.
\subsubsection{$\thetatt$ as a $\Lambdambda$-tree}
Given an ordered abelian group $\Lambdambda$, a $\Lambdambda$-tree is defined {\mathfrak m}athfrak{c}ite{lambdatrees} as a geodesic $\Lambdambda$-metric space $T$ such that
{\mathfrak m}athfrak{b}egin{enumerate}
\item If two geodesics of $T$ intersect in a single point, which is an endpoint of both, then their union is a geodesic;
\item The intersection of two geodesics with a commond endpoint is also a geodesic.
{\mathfrak m}edskipnd{enumerate}
The existence of a greatest common lower node can be used to define a $\Lambdambda$-metric on the subtree ${\mathfrak m}athcal{H}$ of $\thetatt$ consisting of all inner nodes.
Namely, we set
\[
d({\mathfrak m}u,\mathbf{n}u)=
\op{SV}al({\mathfrak m}u)+\op{SV}al(\mathbf{n}u)-2\,\op{SV}al({\mathfrak m}u\wedge\mathbf{n}u).
\]
Note that $d({\mathfrak m}u,\mathbf{n}u)={\mathfrak m}athfrak{l}eft|\op{SV}al({\mathfrak m}u) - \op{SV}al(\mathbf{n}u)\rho_ight|$ if ${\mathfrak m}u$ and $\mathbf{n}u$ are comparable.
It is easy to see that with this definition, ${\mathfrak m}athcal{H}$ is a geodesic $\Lambdambda$-metric space, and the unique geodesic with endpoints ${\mathfrak m}u$, $\mathbf{n}u$ is the union of the segments $[{\mathfrak m}u\wedge\mathbf{n}u,{\mathfrak m}u]$ and $[{\mathfrak m}u\wedge\mathbf{n}u,\mathbf{n}u]$; the two properties above follow.
We are not going to use any metric properties of the tree ${\mathfrak m}athcal{H}$, noting only that this is a hyperbolic space. This fact, along with a plethora of additional information, can be found in the monograph {\mathfrak m}athfrak{c}ite{lambdatrees}.
\section{Equivalence classes of valuations and small extensions of groups}{\mathfrak m}athfrak{l}abel{secSME}
For our given valued field $(K,v)$, consider a valuation ${\mathfrak m}u{\mathfrak m}athfrak{c}olon \op{Ker}x\thetao\Lambda\infty$,
whose restriction to $K$ is equivalent to $v$. That is, there exists an order-preserving embedding $j{\mathfrak m}athfrak{c}olon \Gamma{\mathfrak m}athfrak{h}k\Lambda$, fitting into a commutative diagram
$$
{\mathfrak m}athfrak{a}rs{1.3}
{\mathfrak m}athfrak{b}egin{array}{ccc}
\op{Ker}x&\stackrel{{\mathfrak m}u}{\mathfrak m}athfrak{l}ra&\Lambda\infty\\
\uparrow&&\ \uparrow{\mathfrak m}box{\thetainy$j$}\\
K&\stackrel{v}{\mathfrak m}athfrak{l}ra&\Gamma\infty
{\mathfrak m}edskipnd{array}
$$
The induced embedding $j{\mathfrak m}athfrak{c}olon \Gamma{\mathfrak m}athfrak{h}k\Gammam$ is necessarily a {\mathfrak m}edskipmph{small extension} of ordered abelian groups. That is, if $\Gamma'\subset\Gammam$ is the relative divisible closure of $\Gamma$ in $\Gammam$, then $\Gammam/\Gamma'$ is a cyclic group {\mathfrak m}athfrak{c}ite[Thm. 1.5]{Kuhl}.
Not all small extensions of $\Gamma$ arise from valuations on a polynomial ring. In {\mathfrak m}athfrak{c}ite{Kuhl} it is shown that the divisible closure of $\Gamma$ in $\Gammam$ must be countably generated over $\Gamma$, and it must be finitely generated over $\Gamma$, if $\Gammam/\Gamma$ is not a torsion group.
Our aim is to describe the tree $\thetatt_v$ whose nodes are all equivalence classes of va\-lua\-tions on $\op{Ker}x$ whose restriction to $K$ is equivalent to $v$. The first natural step is to build up some universal ordered group $\Lambda$ containing all small extensions of $\Gamma$ up to order-preserving $\Gamma$-isomorphism.
\subsection{Maximal rank-preserving extension of $\Gamma$}{\mathfrak m}athfrak{l}abel{subsecRlex}
From now on, an {\mathfrak m}edskipmph{embedding} of totally ordered sets is a mapping which strictly preserves the ordering. Also, an {\mathfrak m}edskipmph{embedding} $\Lambda{\mathfrak m}athfrak{h}k \Lambda'$ of totally ordered abelian groups is a group homomorphism which is an embedding as totally ordered sets.
A subgroup $H\subset \Gamma$ is {\mathfrak m}edskipmph{convex} if for all positive $h\in H$, it holds $[-h,h]\subset H$. For all $a\in \Gamma$, the intersection of all convex subgroups of $\Gamma$ containing $a$, is a {\mathfrak m}edskipmph{principal} convex subgroup of $\Gamma$.
Let $\mathfrak{p}cv(\Gamma)$ be the totally ordered set of {{\mathfrak m}athfrak{b}f nonzero} principal convex subgroups of $\Gamma$, ordered by {{\mathfrak m}athfrak{b}f decreasing} inclusion.
Any embedding $j{\mathfrak m}athfrak{c}olon \Gamma{\mathfrak m}athfrak{h}k\Lambda$ induces an embedding of ordered sets$$\mathfrak{p}cv(\Gamma){\mathfrak m}athfrak{h}k\mathfrak{p}cv(\Lambda),$$
which maps the principal convex subgroup generated by $a\in\Gamma$ to the principal convex subgroup generated by $j(a)$ in $\Lambda$.{\mathfrak m}edskip
\Deltaefn
We say that $j$ {\mathfrak m}edskipmph{preserves the rank} if this mapping is bijective. {\mathfrak m}edskip
For instance, the canonical embeddding $\Gamma{\mathfrak m}athfrak{h}k\Gammaq$ preserves the rank. From now on, we shall consider the bijection between $\mathfrak{p}cv(\Gamma)$ and $\mathfrak{p}cv(\Gammaq)$ as an identity:
$$
I:=\mathfrak{p}cv(\Gamma)=\mathfrak{p}cv(\Gammaq).
$$
We may identify $I\infty$ with a set of indices parameterizing all principal convex subgroups of $\Gammaq$. For all $i\in I$ we denote by $H_i$ the corresponding principal convex subgroup. We agree that $H_\infty=\{0\}$.
Then, according to our convention, for any pair of indices $i,j\in I\infty$, we have
$$
i<j \ \Longleftrightarrow\ H_i\supsetneq H_j.
$$
Let ${\mathfrak m}athfrak{l}eft\{I,(Q_i)_{i\in I}\rho_ight\}$ be the skeleton of $\Gammaq$. That is, $Q_i=H_i/H_i^*$ for all $i\in I$, where $H_i^*\subset H_i$ is the maximal proper convex subgroup of $H_i$. That is, if $a\in H_i$ generates $H_i$ as a convex subgroup, then $H_i^*$ is the union of all convex subgroups of $\Gammaq$ not containing $a$.
The convex subgroup $H_i^*$ is not necessarily principal.
Consider the respective Hahn's products:
$$
{\mathfrak m}athfrak{h}q\subset \mathfrak{p}rod\mathbf{n}olimits_{i \in I}Q_i,\mathfrak{q}quad \rho_\ellx\subset {\mathfrak m}athbb R^I,
$$
equipped with the lexicographical order. That is, ${\mathfrak m}athfrak{h}q,\,\rho_\ellx$ are the subsets of $\mathfrak{p}rod_{i \in I}Q_i,\,{\mathfrak m}athbb R^I$, respectively, formed by all elements $x=(x_i)_{i\in I}$ whose support
$$
\op{supp}(x)=\{i\in I{\mathfrak m}id x_i\mathbf{n}e0\}
$$is a well-ordered subset of $I$, with respect to the ordering induced by that of $I$.
By Hahn's embedding theorem {\mathfrak m}athfrak{c}ite[Sec. A]{Rib}, there exists a (non-canonical) ${\mathfrak m}athbb Q$-linear embedding $$\Gammaq{\mathfrak m}athfrak{h}ra{\mathfrak m}athfrak{h}q$$ which induces an isomorphism between the respective skeletons.
On the other hand, the ordered groups $Q_i$ have rank one; that is, they have only two convex subgroups: $\{0\}$ and $Q_i$.
Hence, the choice of positive elements $1_i\in Q_i$ determines ${\mathfrak m}athbb Q$-linear embeddings for all $i\in I$:
$$
Q_i{\mathfrak m}athfrak{h}ra {\mathfrak m}athbb R,\mathfrak{q}quad 1_i{\mathfrak m}athfrak{l}ongmapsto 1,
$$
Therefore, we have a natural embedding ${\mathfrak m}athfrak{h}q{\mathfrak m}athfrak{h}k \rho_\ellx$.
All in all, we obtain a rank-preserving extension
$$
\thetaau{\mathfrak m}athfrak{c}olon \Gamma{\mathfrak m}athfrak{h}ra\Gammaq{\mathfrak m}athfrak{h}ra{\mathfrak m}athfrak{h}q{\mathfrak m}athfrak{h}ra\rho_\ellx,
$$
which is maximal among all rank-preserving extensions of $\Gamma$ {\mathfrak m}athfrak{c}ite[Sec. A]{Rib}.
{\mathfrak m}athfrak{b}egin{theorem}{\mathfrak m}athfrak{l}abel{MaxEqRk}
For any rank-preserving extension $\Gamma{\mathfrak m}athfrak{h}k \Lambda$, there exists an embedding $\Lambda{\mathfrak m}athfrak{h}k\rho_\ellx$ fitting into a commutative diagram
$$
{\mathfrak m}athfrak{a}rs{1.2}
{\mathfrak m}athfrak{b}egin{array}{ccc}
\Lambda&&\\
\uparrow&\raise.5ex{\mathfrak m}athfrak{h}box{$\searrow$}&\\
\Gamma&\stackrel{\thetaau}{\mathfrak m}athfrak{l}ra&\rho_\ellx
{\mathfrak m}edskipnd{array}
$$
{\mathfrak m}edskipnd{theorem}
The embedding $\Lambda{\mathfrak m}athfrak{h}k\rho_\ellx$ is not unique. Thus, every rank-preserving extension of $\Gamma$ is $\Gamma$-equivalent to some subgroup of $\rho_\ellx$, but not to a unique one.
The nonzero principal convex subgroups of $\rho_\ellx$ are parametrized by $I$ via:
$$
H_i=\{(x_j)_{j\in I}{\mathfrak m}id x_j=0{\mathfrak m}box{ for all }j<i\}\mathfrak{q}uad{\mathfrak m}box{for all }i\in I.
$$
The convex subgroups of $\rho_\ellx$ are parametrized by the set $\op{Init}i$ of initial segments of $I$, as follows:
{\mathfrak m}athfrak{b}egin{equation}{\mathfrak m}athfrak{l}abel{HSHahn}
S\in \op{Init}i\mathfrak{q}uad \rho_ightsquigarrow\mathfrak{q}uad H_S=\{(x_j)_{j\in I}{\mathfrak m}id x_j=0{\mathfrak m}box{ for all }j\in S\}.
{\mathfrak m}edskipnd{equation}
\subsection{A universal group for small extensions of $\Gamma$}{\mathfrak m}athfrak{l}abel{subsecRii}
For all $S\in\op{Init}i$, let $i_S$ be a formal symbol and consider the ordered set
$$
I_S=S+\{i_S\}+(I\setminus S),
$$
where $+$ is the usual addition of totally ordered sets.
We define the {\mathfrak m}edskipmph{one-added-element hull} of $I$ as the set
$$
{\mathfrak m}athbb I:=I{\mathfrak m}athfrak{c}up{\mathfrak m}athfrak{l}eft\{i_S{\mathfrak m}id S\in\op{Init}i\rho_ight\},
$$
equipped with the total ordering determined by
{\mathfrak m}athfrak{b}egin{enumerate}
\item[(i)] For all $S\in\op{Init}i$, the inclusion $I_S{\mathfrak m}athfrak{h}k {\mathfrak m}athbb I$ preserves the order.
\item[(ii)] For all $S,T\in\op{Init}i$, we have $i_S<i_T$ if and only if $S\subsetneq T$.
{\mathfrak m}edskipnd{enumerate}
Consider the Hahn product $\rho_ii\subset {\mathfrak m}athbb R^{{\mathfrak m}athbb I}$, defined as above, just by replacing $I$ with ${\mathfrak m}athbb I$.
The inclusions $I\subset I_S\subset {\mathfrak m}athbb I$ determine canonical embeddings
$$
\rho_\ellx{\mathfrak m}athfrak{h}ra \rho_\ellxs{\mathfrak m}athfrak{h}ra \rho_ii,\mathfrak{q}uad{\mathfrak m}box{for all }S\in\op{Init}i.
$$
Altogether, we obtain an embedding
$$
{\mathfrak m}edskipll{\mathfrak m}athfrak{c}olon \Gamma{\mathfrak m}athfrak{h}ra\Gammaq{\mathfrak m}athfrak{h}ra{\mathfrak m}athfrak{h}q{\mathfrak m}athfrak{h}ra\rho_\ellx{\mathfrak m}athfrak{h}ra\rho_ii.
$$
As shown in {\mathfrak m}athfrak{c}ite[Prop. 5.1]{csme}, $\rho_ii$ is the universal ordered abelian group we are looking for.
{\mathfrak m}athfrak{b}egin{proposition}{\mathfrak m}athfrak{l}abel{riiUniverse}
Let ${\mathfrak m}u$ be a valuation on $\op{Ker}x$ whose restriction to $K$ is equivalent to $v$. Then, there exists an embedding $j{\mathfrak m}athfrak{c}olon\Gammam{\mathfrak m}athfrak{h}k \rho_ii$ satisfying the following properties:
{\mathfrak m}athfrak{b}egin{enumerate}
\item[(i)] The following diagram commutes:
$$
{\mathfrak m}athfrak{a}rs{1.3}
{\mathfrak m}athfrak{b}egin{array}{ccc}
\op{Ker}x&\stackrel{j{\mathfrak m}athfrak{c}irc{\mathfrak m}u}{\mathfrak m}athfrak{l}ra&\rho_ii\infty\\
\uparrow&&\ \uparrow{\mathfrak m}box{\thetainy${\mathfrak m}edskipll$}\\
K&\stackrel{v}{\mathfrak m}athfrak{l}ra&\Gamma\infty
{\mathfrak m}edskipnd{array}
$$
\item[(ii)] There exists $S\in\op{Init}i$ such that $j(\Gammam)\subset \rho_\ellxs$.
{\mathfrak m}edskipnd{enumerate}
Moreover, $\Gammam/\Gamma$ is commensurable if and only if $j(\Gammam)\subset{\mathfrak m}edskipll(\Gamma)_{\mathfrak m}athbb Q$. Also, $\Gammam/\Gamma$ preserves the rank if and only if $j(\Gammam)\subset\rho_\ellx$.
{\mathfrak m}edskipnd{proposition}
Clearly, ${\mathfrak m}u$ is equivalent to the valuation $j{\mathfrak m}athfrak{c}irc {\mathfrak m}u$ on $\op{Ker}x$, and $v$ is equivalent to the valuation ${\mathfrak m}edskipll{\mathfrak m}athfrak{c}irc v$ on $K$. Also, the valuation $j{\mathfrak m}athfrak{c}irc {\mathfrak m}u$ restricted to $K$ is equal to ${\mathfrak m}edskipll{\mathfrak m}athfrak{c}irc v$.
As a consequence, in order to describe all equivalence classes of valuations ${\mathfrak m}u$ on $\op{Ker}x$ whose restriction to $K$ is equivalent to $v$, we may assume that $v$ and ${\mathfrak m}u$ satisfy the following conditions:{\mathfrak m}edskip
(V1) \ The valuation $v$ takes values in $\rho_\ellx$. That is, $\Gamma=v(K^*)\subset \rho_\ellx$.{\mathfrak m}edskip
(V2) \ The valuation ${\mathfrak m}u$ satisfies ${\mathfrak m}u_{{\mathfrak m}id K}=v$ and takes values in $\rho_\ellxs$ for some $S\in\op{Init}i$.
\subsection{Small-extensions equivalence on a subset of the universal value group}{\mathfrak m}athfrak{l}abel{subsecSME}
From now on, we assume that our valuation $v$ on $K$ satisfies (V1), so that the embedding ${\mathfrak m}edskipll$ of the last section is the canonical inclusion.
Consider the subset
$$\rho_\elll=\rho_\elll(I):={\mathfrak m}athfrak{b}igcup_{S\in\op{Init}i}\rho_\ellxs\subset \rho_ii.$$
For all ${\mathfrak m}athfrak{b}e\in\rho_\elll$, we denote by $\Gammagb\subset \rho_ii$ the subgroup generated by $\Gamma$ and ${\mathfrak m}athfrak{b}e$.
Let $\Lambda=\rho_ii$ and $\thetatt=\thetala$. Consider the subtree
$$\thetaz={\mathfrak m}athfrak{l}eft\{\rho\in\thetatt{\mathfrak m}id \Gammarh\subset\rho_\ellxs\ {\mathfrak m}box{ for some }\ S\in\op{Init}i \rho_ight\}\subset \thetatt.$$
Note that all valuations in $\thetaz$ satisfy the condition (V2).
On the set $\rho_\elll$ we define the following equivalence relation.{\mathfrak m}edskip
\mathbf{n}oindent{{\mathfrak m}athfrak{b}f Definition.} We say that ${\mathfrak m}athfrak{b}e,\Gammaa\in \rho_\elll$ are $\,\operatorname{sme}$-equivalent if there exists an isomorphism of ordered groups
$$
\Gammagb\ {\mathfrak m}athfrak{l}ower.3ex\hbox{\ars{.08}$\begin{array}{c}{\mathfrak m}athfrak{l}ra\\{\mathfrak m}box{\thetainy $\sim\,$}\end{array}$}\ \Gammaga,
$$
which acts as the identity on $\Gamma$ and sends ${\mathfrak m}athfrak{b}e$ to $\Gammaa$.
In this case, we write ${\mathfrak m}athfrak{b}e\sim_{\mbox{\thetainy $\op{sme}$}}\Gammaa$.
We denote by $[{\mathfrak m}athfrak{b}e]_{\mbox{\thetainy $\op{sme}$}}\subset \rho_\elll$ the class of ${\mathfrak m}athfrak{b}e$.{\mathfrak m}edskip
The motivation for this definition lies in the following result.
{\mathfrak m}athfrak{b}egin{proposition}{\mathfrak m}athfrak{l}abel{motivation}
Let ${\mathfrak m}u,\mathbf{n}u\in\thetaz$ be two inner nodes. Then, ${\mathfrak m}u\sim\mathbf{n}u$ if and only if the following three conditions hold:
(a) \ The valuations ${\mathfrak m}u$, $\mathbf{n}u$ admit a common key polynomial of minimal degree.
(b) \ For all \,$a\in\op{Ker}x\,$ such that \,$\Deltaeg(a)<\Deltaeg({\mathfrak m}u)$, we have $\,{\mathfrak m}u(a)=\mathbf{n}u(a)$.
(c) \ $\op{SV}al({\mathfrak m}u)\sim_{\mbox{\thetainy $\op{sme}$}} \op{SV}al(\mathbf{n}u)$.
In this case, we have $\op{Ker}pm=\op{Ker}pn$.
{\mathfrak m}edskipnd{proposition}
{\mathfrak m}athfrak{b}egin{proof}
Suppose that ${\mathfrak m}u\sim\mathbf{n}u$. Then, there exists an isomorphism of ordered groups $\iota{\mathfrak m}athfrak{c}olon \Gammam{\mathfrak m}athfrak{l}ower.3ex\hbox{\ars{.08}$\begin{array}{c}\,\thetao\\{\mathfrak m}box{\thetainy $\sim\,$}\end{array}$}\Gamman$ such that $\mathbf{n}u=\iota{\mathfrak m}athfrak{c}irc{\mathfrak m}u$.
The isomorphism $\iota$ induces an isomorphim between the graded algebras:
$$
\Gammagm\ {\mathfrak m}athfrak{l}ower.3ex\hbox{\ars{.08}$\begin{array}{c}{\mathfrak m}athfrak{l}ra\\{\mathfrak m}box{\thetainy $\sim\,$}\end{array}$}\ \Gammagn,\mathfrak{q}quad f+\mathfrak{p}set^+_{\mathfrak m}athfrak{a}l({\mathfrak m}u){\mathfrak m}athfrak{l}ongmapsto f+\mathfrak{p}set^+_{\iota({\mathfrak m}athfrak{a}l)}(\mathbf{n}u),
$$
for all $f\in\mathfrak{p}set_{\mathfrak m}athfrak{a}l({\mathfrak m}u)$, ${\mathfrak m}athfrak{a}l\in\Gammam$. Since key polynomials are characterized by algebraic properties of their initial terms in the graded algebra, this implies that both valuations have the same key polynomials: \,$\op{Ker}pm=\op{Ker}pn$.
Let $\mathfrak{p}hi$ be a common key polynomial of minimal degree.
Since ${\mathfrak m}u_{{\mathfrak m}id K}=\mathbf{n}u_{{\mathfrak m}id K}$, the isomorphism $\iota$ restricted to $\Gamma$ is the identity and $\iota({\mathfrak m}u(\mathfrak{p}hi))=\mathbf{n}u(\mathfrak{p}hi)$. Hence,
$$\op{SV}al({\mathfrak m}u)={\mathfrak m}u(\mathfrak{p}hi)\sim_{\mbox{\thetainy $\op{sme}$}}\mathbf{n}u(\mathfrak{p}hi)=\op{SV}al(\mathbf{n}u).
$$
Finally, since $\iota$ restricted to $\Gamma$ is the identity, then $\iota$ acts as the identity on any torsion element in $\Gammam/\Gamma$. Now, for all $a\in \op{Ker}x$ of degree less than $\Deltaeg(\mathfrak{p}hi)$, the values ${\mathfrak m}u(a)$, $\mathbf{n}u(a)$ belong to $\Gammaq$ {\mathfrak m}athfrak{c}ite[Lem. 1.3]{MLV}. Thus,
$$
\mathbf{n}u(a)=\iota({\mathfrak m}u(a))={\mathfrak m}u(a).
$$
Conversely, suppose that ${\mathfrak m}u$ and $\mathbf{n}u$ satisfy conditions (a), (b) and (c). Take $\mathfrak{p}hi\in\op{Ker}pm{\mathfrak m}athfrak{c}ap\op{Ker}pn$ of minimal degree in both sets. Let us denote $${\mathfrak m}athfrak{b}e=\op{SV}al({\mathfrak m}u)={\mathfrak m}u(\mathfrak{p}hi), \mathfrak{q}quad \Gammaa=\op{SV}al(\mathbf{n}u)=\mathbf{n}u(\mathfrak{p}hi).$$ By condition (c), there is an order-preserving $\Gamma$-isomor\-phism $\iota{\mathfrak m}athfrak{c}olon\Gammaen{\Gamma,{\mathfrak m}athfrak{b}e}{\mathfrak m}athfrak{l}ower.3ex\hbox{\ars{.08}$\begin{array}{c}\,\thetao\\{\mathfrak m}box{\thetainy $\sim\,$}\end{array}$}\Gammaen{\Gamma,\Gammaa}$, mapping ${\mathfrak m}athfrak{b}e$ to $\Gammaa$.
As we saw in Section \ref{subsubsecG0}, the subgroup
$$
H:=\Gammam^0={\mathfrak m}athfrak{l}eft\{{\mathfrak m}u(a){\mathfrak m}id 0{\mathfrak m}athfrak{l}e \Deltaeg(a)<\Deltaeg({\mathfrak m}u)\rho_ight\}\subset\Gammam
$$
is commensurable over $\Gamma$ and satisfies $\Gammam=\Gammaen{H,{\mathfrak m}u(\mathfrak{p}hi)}=\Gammaen{H,{\mathfrak m}athfrak{b}e}$. By condition (b), $H=\Gamman^0$ and $\Gamman=\Gammaen{H,\mathbf{n}u(\mathfrak{p}hi)}=\Gammaen{H,\Gammaa}$ too. Since $H/\Gamma$ is a torsion abelian group, the $\Gamma$-isomorphism $\iota$ induces an order-preserving isomorphism
$$\iota{\mathfrak m}athfrak{c}olon\Gammam=\Gammaen{H,{\mathfrak m}athfrak{b}e}\ {\mathfrak m}athfrak{l}ower.3ex\hbox{\ars{.08}$\begin{array}{c}{\mathfrak m}athfrak{l}ra\\{\mathfrak m}box{\thetainy $\sim\,$}\end{array}$}\ \Gamman=\Gammaen{H,\Gammaa},$$ which acts as the identity on $H$ and maps ${\mathfrak m}athfrak{b}e$ to $\Gammaa$. Let us check that $\mathbf{n}u=\iota{\mathfrak m}athfrak{c}irc{\mathfrak m}u$.
For $f\in\op{Ker}x$, consider its $\mathfrak{p}hi$-expansion $f=\sum_{0{\mathfrak m}athfrak{l}e s}a_s\mathfrak{p}hi^s$, where $\Deltaeg(a_s)<\Deltaeg(\mathfrak{p}hi)=\Deltaeg({\mathfrak m}u)$. Since $\mathfrak{p}hi$ is ${\mathfrak m}u$-minimal and $\mathbf{n}u$-minimal, Lemma \ref{minimal0} shows that
$$
{\mathfrak m}u(f)={\mathfrak m}in{\mathfrak m}athfrak{l}eft\{{\mathfrak m}u(a_s\mathfrak{p}hi^s){\mathfrak m}id 0{\mathfrak m}athfrak{l}e s\rho_ight\},\mathfrak{q}quad \mathbf{n}u(f)={\mathfrak m}in{\mathfrak m}athfrak{l}eft\{\mathbf{n}u(a_s\mathfrak{p}hi^s){\mathfrak m}id 0{\mathfrak m}athfrak{l}e s\rho_ight\}.
$$
Let us denote $\Deltata_s={\mathfrak m}u(a_s)=\mathbf{n}u(a_s)\in H$, for all $s\Gammae0$. Clearly,
$$
\iota{\mathfrak m}athfrak{l}eft({\mathfrak m}u(a_s\mathfrak{p}hi^s)\rho_ight)=\iota(\Deltata_s+s{\mathfrak m}athfrak{b}e)=\Deltata_s+s\Gammaa=\mathbf{n}u(a_s\mathfrak{p}hi^s).
$$
Since $\iota$ preserves the ordering, for arbitrary indices $s,t$ we have
$$
{\mathfrak m}u(a_s\mathfrak{p}hi^s){\mathfrak m}athfrak{l}e{\mathfrak m}u(a_t\mathfrak{p}hi^t)\ \Longrightarrow\ \mathbf{n}u(a_s\mathfrak{p}hi^s){\mathfrak m}athfrak{l}e\mathbf{n}u(a_t\mathfrak{p}hi^t).
$$
Thus, there is a common index $s$ for which ${\mathfrak m}u(f)={\mathfrak m}u(a_{s}\mathfrak{p}hi^{s})$ and $\mathbf{n}u(f)=\mathbf{n}u(a_{s}\mathfrak{p}hi^{s})$, simultaneously. Therefore,
$
\iota{\mathfrak m}athfrak{l}eft({\mathfrak m}u(f)\rho_ight)=\iota{\mathfrak m}athfrak{l}eft({\mathfrak m}u(a_s\mathfrak{p}hi^s)\rho_ight)=\mathbf{n}u(a_s\mathfrak{p}hi^s)=\mathbf{n}u(f)
$.
{\mathfrak m}edskipnd{proof}
{\mathfrak m}athfrak{b}egin{corollary}{\mathfrak m}athfrak{l}abel{motivation2}
Take ${\mathfrak m}athfrak{b}e,\,\Gammaa\in \rho_\elll$.
{\mathfrak m}athfrak{b}egin{enumerate}
\item[(i)] For all $a\in K$ we have $ \omega_{a,{\mathfrak m}athfrak{b}e}\sim\omega_{a,\Gammaa}$ if and only if ${\mathfrak m}athfrak{b}e\sim_{\mbox{\thetainy $\op{sme}$}}\Gammaa$.
\item[(ii)] Let ${\mathfrak m}u\in\thetaz$ be an inner node and let $\mathfrak{p}hi\in\op{Ker}pm$. If ${\mathfrak m}athfrak{b}e,\Gammaa>{\mathfrak m}u(\mathfrak{p}hi)$, then
$$
[{\mathfrak m}u;\,\mathfrak{p}hi,{\mathfrak m}athfrak{b}e]\sim [{\mathfrak m}u;\,\mathfrak{p}hi,\Gammaa]\ \Longleftrightarrow\ {\mathfrak m}athfrak{b}e\sim_{\mbox{\thetainy $\op{sme}$}}\Gammaa.
$$
\item[(iii)] Let ${\mathfrak m}athfrak{a}a={\mathfrak m}athfrak{l}eft(\rho_i\rho_ight)_{i\in A}$ be an essential continuous family in $\thetaz$ and let $\mathfrak{p}hi\in\op{Ker}pi({\mathfrak m}athfrak{a}a)$. If ${\mathfrak m}athfrak{b}e,\Gammaa>\rho_i(\mathfrak{p}hi)$ for all $i\in A$, then
$$
[{\mathfrak m}athfrak{a}a;\,\mathfrak{p}hi,{\mathfrak m}athfrak{b}e]\sim [{\mathfrak m}athfrak{a}a;\,\mathfrak{p}hi,\Gammaa]\ \Longleftrightarrow\ {\mathfrak m}athfrak{b}e\sim_{\mbox{\thetainy $\op{sme}$}}\Gammaa.
$$
{\mathfrak m}edskipnd{enumerate}
{\mathfrak m}edskipnd{corollary}
{\mathfrak m}athfrak{b}egin{proof}
All items follow immediately from Proposition \ref{motivation}, once we see that for the two involved valuations, conditions (a) and (b) hold in each case.
In case (i), the common key polynomial of minimal degree is $\mathfrak{p}hi=x-a$.
In cases (ii) and (iii), $\mathfrak{p}hi$ is a common key polynomial of minimal degree for both augmentations by Lemma \ref{propertiesAug} and Proposition \ref{extensionlim}, respectively.
{\mathfrak m}edskipnd{proof}
\subsection{Quasi-cuts in $\Gammaq$ and small-extensions closure of $\Gamma$}{\mathfrak m}athfrak{l}abel{subsecQcuts}
Consider any subset $\Gammasme\subset \rho_\elll$ which is a set of representatives of the quotient set $\rho_\elll/\!\sim{\mbox{\thetainy $\op{sme}$}}$.
The only $\Gamma$-automorphism of $\Gammaq$ as an ordered group is the identity. Thus, for all ${\mathfrak m}athfrak{b}e\in\Gammaq\subset\rho_\elll$, we have ${\mathfrak m}athfrak{l}eft[{\mathfrak m}athfrak{b}e\rho_ight]_{\mbox{\thetainy $\op{sme}$}}=\{{\mathfrak m}athfrak{b}e\}$. Therefore, we have necessarily
$$\Gammaq\subset\Gammasme\subset \rho_\elll.$$
Any such ``small-extensions closure" $\Gammasme$ contains generators of all small extensions of $\Gamma$, up to the relative divisible closure of $\Gamma$. Let us give a precise explanation of this statement, which follows easily from the definition of $\sim_{\mbox{\thetainy $\op{sme}$}}$.
{\mathfrak m}athfrak{b}egin{proposition}{\mathfrak m}athfrak{l}abel{smallSme}Let $\Gamma{\mathfrak m}athfrak{h}k\Omega$ be a small extension and let $\Gammaa\in\Omega$ such that $\Omega=\Gammaen{\Delta,\Gammaa}$, where $\Delta$ is the relative divisible closure of $\Gamma$ in $\Omega$. Let $\Delta\!{\mathfrak m}athfrak{l}ower.3ex\hbox{\ars{.08}$\begin{array}{c}\,\thetao\\{\mathfrak m}box{\thetainy $\sim\,$}\end{array}$}\Delta_0\subset\Gammaq$ be the canonical embedding of $\Delta$ into $\Gammaq$. Then, for a unique ${\mathfrak m}athfrak{b}e\in\Gammasme$ there exists an isomorphism of ordered groups $$\Omega\ {\mathfrak m}athfrak{l}ower.3ex\hbox{\ars{.08}$\begin{array}{c}{\mathfrak m}athfrak{l}ra\\{\mathfrak m}box{\thetainy $\sim\,$}\end{array}$}\ \Gammaen{\Delta_0,{\mathfrak m}athfrak{b}e},\mathfrak{q}quad \Gammaa{\mathfrak m}athfrak{l}ongmapsto {\mathfrak m}athfrak{b}e,$$ which maps $\Gammaa$ to ${\mathfrak m}athfrak{b}e$, and whose restriction to $\Delta$ is the canonical isomorphism $\Delta\!{\mathfrak m}athfrak{l}ower.3ex\hbox{\ars{.08}$\begin{array}{c}\,\thetao\\{\mathfrak m}box{\thetainy $\sim\,$}\end{array}$} \Delta_0$.
{\mathfrak m}edskipnd{proposition}
In {\mathfrak m}athfrak{c}ite{csme}, a real model for the set of quasi-cuts of $\Gammaq$ is constructed, which serves as a canonical choice for $\Gammasme$. Let us recall this construction.
A {\mathfrak m}edskipmph{quasi-cut} in $\Gammaq$ is a pair $D={\mathfrak m}athfrak{l}eft(D^L,D^R\rho_ight)$ of subsets such that $D^L{\mathfrak m}athfrak{l}e D^R$ and $D^L{\mathfrak m}athfrak{c}up D^R=\Gammaq$.
Then, $D^L$ is an initial segment of $\Gammaq$, $D^R$ is a final segment of $\Gammaq$ and $D^L{\mathfrak m}athfrak{c}ap D^R$ consists of at most one element.
If $D^L{\mathfrak m}athfrak{c}ap D^R=\{a\}$, we say that $D$ is the {\mathfrak m}edskipmph{principal quasi-cut} determined by $a\in \Gammaq$. If $D^L{\mathfrak m}athfrak{c}ap D^R={\mathfrak m}edskipmptyset$, we say that $D$ is a {\mathfrak m}edskipmph{cut} in $\Gammaq$.
The set $\operatorname{Qcuts}(\Gammaq)$ of all quasi-cuts in $\Gammaq$ admits a total ordering:
$$
D={\mathfrak m}athfrak{l}eft(D^L,D^R\rho_ight){\mathfrak m}athfrak{l}e E={\mathfrak m}athfrak{l}eft(E^L,E^R\rho_ight) \ \ \Longleftrightarrow\ \ D^L\subset E^L\mathfrak{q}uad{\mathfrak m}box{and}\mathfrak{q}uad D^R\supset E^R.
$$
For all $x\in\rho_\elll$ we consider the folllowing quasi-cut $D_x$ in $\Gammaq$:
$$
D_x^L=\{a\in\Gammaq{\mathfrak m}id a{\mathfrak m}athfrak{l}e x\},\mathfrak{q}quad D_x^R=\{a\in\Gammaq{\mathfrak m}id a\Gammae x\}.
$$
We say that $x$ {\mathfrak m}edskipmph{realizes} the cut $D_x$. The set $\rho_\elll$ contains realizations of all quasi-cuts in $\Gammaq$ {\mathfrak m}athfrak{c}ite[Sec. 4]{csme}.
Moreover, these quasi-cuts provide the following reinterpretation of the equivalence relation $\sim_{\mbox{\thetainy $\op{sme}$}}$ {\mathfrak m}athfrak{c}ite[Lem. 5.4]{csme}.
{\mathfrak m}athfrak{b}egin{lemma}
For all $x,y\in\rho_\elll$, we have $x\sim_{\mbox{\thetainy $\op{sme}$}} y$ if and only if $D_x=D_y$.
{\mathfrak m}edskipnd{lemma}
As a consequence, if we consider on $\Gammasme$ the total ordering induced by that of $\rho_\elll$,
we derive a natural isomorphism of ordered sets:
$$\Gammasme{\mathfrak m}athfrak{l}ra \operatorname{Qcuts}(\Gammaq),\mathfrak{q}quad x{\mathfrak m}athfrak{l}ongmapsto D_x.$$
{\mathfrak m}athfrak{b}egin{corollary}{\mathfrak m}athfrak{l}abel{complete}
Equipped with the order topology, $\Gammasme$ is complete and contains $\Gammaq$ as a dense subset.
{\mathfrak m}edskipnd{corollary}
Indeed, it is well known that the ordered set $\operatorname{Qcuts}(\Gammaq)$ has these properties. We recall that being complete with respect to the order topology means that every non-empty subset of $\Gammasme$ has a supremum and an infimum.
In {\mathfrak m}athfrak{c}ite[Sec. 4]{csme}, a canonical choice for $\Gammasme$ is described as
$$
\Gammasme=\Gammaq\sqcup \Gammanbc\sqcup \Gammabc,
$$
for certain subsets $\Gammanbc\subset \rho_\ellx\setminus\Gammaq$ and $\Gammabc\subset \rho_\elll\setminus\rho_\ellx$.
The elements $x\in \Gammaq$ parametrize the principal quasi-cuts. The elements $x\in\Gammanbc$, $x\in\Gammabc$ correspond to $D_x$ being a {\mathfrak m}edskipmph{non-ball cut}, or a {\mathfrak m}edskipmph{ball cut}. Equivalently, the small extension $\Gamma{\mathfrak m}athfrak{h}k\Gammaen{\Gammaq,x}$ preserves, or increases the rank, respectively.
Let us briefly describe $\Gammanbc$.
For all $S\in\op{Init}i$, consider the truncation by $S$:
$$
\mathfrak{p}i_S{\mathfrak m}athfrak{c}olon \rho_\ellx{\mathfrak m}athfrak{l}ra\rho_\ellx,\mathfrak{q}quad x=(x_i)_{i\in I}{\mathfrak m}athfrak{l}ongmapsto x_S=(y_i)_{i\in I},
$$
where $y_i=x_i$ for all $i\in S$ and $y_i=0$ otherwise.
Note that $\mathfrak{p}i_S^{-1}({\mathfrak m}athfrak{b}e_S)={\mathfrak m}athfrak{b}e+H_S$, where $H_S$ is the convex subgroup defined in (\ref{HSHahn}).
The set $\Gammanbc$ is stratified as:
$$
\Gammanbc={\mathfrak m}athfrak{b}igsqcup\mathbf{n}olimits_{S\in\op{Init}i}\operatorname{nbc}(S),
$$
where \ $\operatorname{nbc}(S)={\mathfrak m}athfrak{l}eft\{x\in\mathfrak{p}i_S{\mathfrak m}athfrak{l}eft(\rho_\ellx\rho_ight)\setminus \Gammaq{\mathfrak m}id x_T\in\Gammaq {\mathfrak m}box{ for all } T\in\op{Init}i,\ T\subsetneq S \rho_ight\}$.
Now, let us describe $\Gammabc$.
For all $b=(b_i)_{i\in I}\in\Gammaq$, $S\in\op{Init}i$, denote
$$
b_S^-=((b_j)_{j\in S}{\mathfrak m}id-1{\mathfrak m}id0{\mathfrak m}athfrak{c}dots0)\in{\mathfrak m}athbb R^{I_S}_{{\mathfrak m}athfrak{l}x},\mathfrak{q}quad b_S^+=((b_j)_{j\in S}{\mathfrak m}id1{\mathfrak m}id0{\mathfrak m}athfrak{c}dots0)\in{\mathfrak m}athbb R^{I_S}_{{\mathfrak m}athfrak{l}x},
$$
where $\mathfrak{p}m1$ is placed at the $i_S$-th coordinate.
Then, $\Gammabc$ is constructed as:
$$
\Gammabc={\mathfrak m}athfrak{b}igsqcup\mathbf{n}olimits_{S\in\op{Init}i}{\mathfrak m}athfrak{l}eft\{b_S^-,\,b_S^+{\mathfrak m}id b\in\Gammaq\rho_ight\}.
$$
The elements determined by the initial segment $S={\mathfrak m}edskipmptyset$ deserve a special notation:
$$
-\infty=(-1{\mathfrak m}id0{\mathfrak m}athfrak{c}dots 0)={\mathfrak m}in(\Gammasme),\mathfrak{q}quad \infty^-=(1{\mathfrak m}id0{\mathfrak m}athfrak{c}dots 0)={\mathfrak m}ax(\Gammasme),
$$
where $\mathfrak{p}m1$ is placed at the $i_{\mathfrak m}edskipmptyset$-th coordinate; that is, the first coordinate of $I_{\mathfrak m}edskipmptyset=\{i_{\mathfrak m}edskipmptyset\}+I$.
The notation for $\infty^-$ is motivated by the fact that this element is the immediate predecessor of $\infty$ in the set $\Gammasme\infty$.
\section{Construction of the valuative tree}{\mathfrak m}athfrak{l}abel{secConstruct}
We keep with the notation of the previous section
$$\rho_\elll\subset\Lambda:=\rho_ii, \mathfrak{q}quad \thetaz\subset\thetatt:=\thetala,$$
and we assume that the valuation $v$ takes values in a subgroup of $\Lambda$.
Since $\Gammasme$ is complete, we may extend the singular value function $\op{SV}al$ to the leaves of $\thetaz$. For a finite leaf $\mathbf{n}u\in{\mathfrak m}athfrak{l}fin(\thetaz)$, we agree that $\op{SV}al(\mathbf{n}u)=\infty$, while for an infinite leaf $\mathbf{n}u\in{\mathfrak m}athfrak{l}i(\thetaz)$ we define
$$
\op{SV}al(\mathbf{n}u)=\sup{\mathfrak m}athfrak{l}eft\{\op{SV}al(\rho){\mathfrak m}id \rho\in(-\infty,\mathbf{n}u)\rho_ight\}\in\Gammasme.
$$
\subsection{Equivalence classes of commensurable extensions}
Let $\thetatt_v$ be the set of equivalence classes of valuations on $\op{Ker}x$ whose restriction to $K$ is equivalent to $v$.
It is well-known how to describe the subset $\thetatt_v^{\mathfrak m}athfrak{c}om \subset\thetatt_v$ of the equivalence classes which are commensurable over $[v]$.
By Proposition \ref{riiUniverse}, any such valuation is equivalent to some commensurable node ${\mathfrak m}u\in\thetatt$; that is,
a node belonging to the subtree:
$\thetaq:=\thetagq\subset\thetatt$.
Finally, it is easy to classify the nodes of $\thetaq$ up to equivalence.
Two commensurable valuations ${\mathfrak m}u,{\mathfrak m}u'\in\thetaq$ are equivalent if and only if ${\mathfrak m}u={\mathfrak m}u'$. Indeed,
if two subgroups
$$\Gamma\subset\Delta\subset\Gammaq,\mathfrak{q}quad \Gamma\subset\Delta'\subset\Gammaq,$$
admit an order-preserving isomorphism $\iota{\mathfrak m}athfrak{c}olon \Delta {\mathfrak m}athfrak{l}ower.3ex\hbox{\ars{.08}$\begin{array}{c}\,\thetao\\{\mathfrak m}box{\thetainy $\sim\,$}\end{array}$}\Delta'$ which is the identity on $\Gamma$, then necessarily $\Delta=\Delta'$ and $\iota$ is the identity mapping.
Therefore, we have a natural bijective mapping
$$
\thetaq{\mathfrak m}athfrak{l}ra\thetatt_v^{\mathfrak m}athfrak{c}om,\mathfrak{q}quad {\mathfrak m}u{\mathfrak m}athfrak{l}ongmapsto[{\mathfrak m}u].
$$
Since all leaves of $\thetatt$ are commensurable, they are leaves of $\thetaq$ too.
Therefore, both trees have the same leaves. More precisely, with the notation of Section \ref{subsecMLV}, we have:
{\mathfrak m}athfrak{b}egin{equation}{\mathfrak m}athfrak{l}abel{EqualLeaves}
{\mathfrak m}athfrak{l}fin(\thetaq)={\mathfrak m}athfrak{l}fin(\thetatt),\mathfrak{q}uad
{\mathfrak m}athfrak{l}ui(\thetaq)={\mathfrak m}athfrak{l}ui(\thetatt),\mathfrak{q}uad
{\mathfrak m}athfrak{l}ci(\thetaq)={\mathfrak m}athfrak{l}ci(\thetatt).
{\mathfrak m}edskipnd{equation}
By Lemmas \ref{equiv=lim}, \ref{numerable} and \ref{specialCont}, every leaf in ${\mathfrak m}athfrak{l}ui(\thetaq)$ is the stable limit of a countable family of nodes in $\thetaq$ with unbounded degree, and every leaf in ${\mathfrak m}athfrak{l}ci(\thetaq)$ is the stable limit of an essential continuous family of nodes in $\thetaq$.
\subsection{Description of the valuative tree}
Consider the subtree
$$\thetas={\mathfrak m}athfrak{l}eft\{\rho\in\thetaz{\mathfrak m}id \op{SV}al(\rho)\in\Gammasme\rho_ight\}.$$
Since $\Gammaq\subset\Gammasme$, we have $\thetaq\subset\thetas\subset\thetaz\subset\thetatt$. In particular, from (\ref{EqualLeaves}) we deduce
$$
{\mathfrak m}athfrak{l}fin(\thetaq)={\mathfrak m}athfrak{l}fin(\thetas),\mathfrak{q}uad
{\mathfrak m}athfrak{l}ui(\thetaq)={\mathfrak m}athfrak{l}ui(\thetas),\mathfrak{q}uad
{\mathfrak m}athfrak{l}ci(\thetaq)={\mathfrak m}athfrak{l}ci(\thetas).
$$
{\mathfrak m}athfrak{b}egin{theorem}{\mathfrak m}athfrak{l}abel{main2}
The mapping ${\mathfrak m}u{\mathfrak m}apsto[{\mathfrak m}u]$ induces a bijection between $\thetas$ and $\thetatt_v$.
{\mathfrak m}edskipnd{theorem}
{\mathfrak m}athfrak{b}egin{proof}
Let ${\mathfrak m}u$ be a valuation on $\op{Ker}x$ whose restriction to $K$ is equivalent to $v$. By Proposition \ref{riiUniverse}, ${\mathfrak m}u$ is equivalent to some valuation in $\thetaz$. Thus, we may suppose ${\mathfrak m}u\in\thetaz$. If ${\mathfrak m}u$ is commensurable, then ${\mathfrak m}u\in\thetaq\subset\thetas$, so that $[{\mathfrak m}u]$ is the image of some node of $\thetas$.
Suppose that ${\mathfrak m}u$ is incommensurable. Then, it is the last node of a finite MLV chain
$$ {\mathfrak m}u_0\ \stackrel{\mathfrak{p}hi_1,\Gammaa_1}{\mathfrak m}athfrak{l}ra\ {\mathfrak m}u_1\ {\mathfrak m}athfrak{l}ra\ {\mathfrak m}athfrak{c}dots\ {\mathfrak m}athfrak{l}ra\ {\mathfrak m}u_{r-1}\ \stackrel{\mathfrak{p}hi_{r},\Gammaa_{r}}{\mathfrak m}athfrak{l}ra\ {\mathfrak m}u_{r}=\mathbf{n}u.$$
If ${\mathfrak m}u={\mathfrak m}u_0=\omega_{a,\Gammaa}$ has depth zero, then Corollary \ref{motivation2} shows that ${\mathfrak m}u$ is equivalent to $\omega_{a,{\mathfrak m}athfrak{b}e}\in\thetas$, where ${\mathfrak m}athfrak{b}e\in\Gammasme$ is the representative of the class $[\Gammaa]_{\mbox{\thetainy $\op{sme}$}}$.
If ${\mathfrak m}u$ has a positive depth and ${\mathfrak m}u_{r-1}\thetao{\mathfrak m}u$ is an ordinary augmentation, then Corollary \ref{motivation2} shows that ${\mathfrak m}u$ is equivalent to $[{\mathfrak m}u_{r-1};\,\mathfrak{p}hi_r,{\mathfrak m}athfrak{b}e]\in\thetas$, where ${\mathfrak m}athfrak{b}e\in\Gammasme$ is the representative of the class $[\Gammaa_r]_{\mbox{\thetainy $\op{sme}$}}$.
If ${\mathfrak m}u$ has a positive depth and ${\mathfrak m}u_{r-1}\thetao{\mathfrak m}u$ is a limit augmentation, then ${\mathfrak m}u=[{\mathfrak m}athfrak{a}a;\,\mathfrak{p}hi_r,\Gammaa_r]$ for some essential continuous family in $\thetaq$ and Corollary \ref{motivation2} shows that ${\mathfrak m}u$ is equivalent to $[{\mathfrak m}athfrak{a}a;\,\mathfrak{p}hi_r,{\mathfrak m}athfrak{b}e]\in\thetas$, where ${\mathfrak m}athfrak{b}e\in\Gammasme$ is the representative of the class $[\Gammaa_r]_{\mbox{\thetainy $\op{sme}$}}$.
This proves that the mapping ${\mathfrak m}u{\mathfrak m}apsto[{\mathfrak m}u]$ is onto.
Finally, let us show that the mapping ${\mathfrak m}u{\mathfrak m}apsto [{\mathfrak m}u]$ is injective.
Suppose that ${\mathfrak m}u,\mathbf{n}u\in\thetas$ are equivalent. Then $\op{SV}al({\mathfrak m}u)\sim_{\mbox{\thetainy $\op{sme}$}}\op{SV}al(\mathbf{n}u)$ by Proposition \ref{motivation}.
Since ${\mathfrak m}u$ and $\mathbf{n}u$ belong to $\thetas$, we have $\op{SV}al({\mathfrak m}u),\,\op{SV}al(\mathbf{n}u)\in\Gammasme$, so that necessarily $\op{SV}al({\mathfrak m}u)=\op{SV}al(\mathbf{n}u)$.
Also, Proposition \ref{motivation} shows that $\op{Ker}pm=\op{Ker}pn$ and
$\mathbf{n}u(a)={\mathfrak m}u(a)$ for all $a\in\op{Ker}x$ of degree less than $\Deltaeg({\mathfrak m}u)$.
This implies ${\mathfrak m}u=\mathbf{n}u$ by comparison of their action on $\mathfrak{p}hi$-expansions, for any common key polynomial $\mathfrak{p}hi$ of minimal degree, having in mind that ${\mathfrak m}u(\mathfrak{p}hi)=\op{SV}al({\mathfrak m}u)=\op{SV}al(\mathbf{n}u)=\mathbf{n}u(\mathfrak{p}hi)$.
{\mathfrak m}edskipnd{proof}{\mathfrak m}edskip
This subtree $\thetas\subset\thetatt$ shares many of the properties of $\thetatt$ discussed in Sections \ref{secTreeLa} and \ref{secCtDepth}. Let us explicitely quote some of them.{\mathfrak m}edskip
${\mathfrak m}athfrak{b}ullet$\mathfrak{q}uad For all ${\mathfrak m}u\in \thetas$, the nodes of a MLV chain of ${\mathfrak m}u$, except for (eventually) ${\mathfrak m}u$ itself, are commensurable.
Thus, these nodes belong to $\thetas$ and the depth of ${\mathfrak m}u$ can be described solely in terms of $\thetas$. {\mathfrak m}edskip
${\mathfrak m}athfrak{b}ullet$\mathfrak{q}uad If ${\mathfrak m}u\in \thetas$ is an inner node and $\mathfrak{p}hi\in\op{Ker}pm$, then we may build up ordinary augmentations in $\thetas$:
$$
\mathbf{n}u=[{\mathfrak m}u;\,\mathfrak{p}hi,\Gammaa]\in \thetas,\mathfrak{q}uad \Gammaa\in\Gammasme,\ \Gammaa>{\mathfrak m}u(\mathfrak{p}hi).
$$
For any such augmentation, the interval $({\mathfrak m}u,\mathbf{n}u]\subset \thetas$ may be described as
$$
({\mathfrak m}u,\mathbf{n}u]={\mathfrak m}athfrak{l}eft\{[{\mathfrak m}u;\,\mathfrak{p}hi,\Deltata]{\mathfrak m}id \Deltata\in\Gammasme,\ {\mathfrak m}u(\mathfrak{p}hi)<\Deltata{\mathfrak m}athfrak{l}e\Gammaa\rho_ight\}.
$$
${\mathfrak m}athfrak{b}ullet$\mathfrak{q}uad In particular, Proposition \ref{td=td} holds in $\thetas$ too. There is a canonical bijection between $\op{Ker}pm/\!\sim_\mu$ and the set of tangent directions of ${\mathfrak m}u$ in the tree $\thetas$.{\mathfrak m}edskip
${\mathfrak m}athfrak{b}ullet$\mathfrak{q}uad Let ${\mathfrak m}athfrak{a}a=(\rho_i)_{i\in A}$ be an essential continuous family in $\thetatt$, and $\mathfrak{p}hi\in\op{Ker}pi({\mathfrak m}athfrak{a}a)$ a limit key polynomial. Then, we may build up limit augmentations in $\thetas$:
$$
\mathbf{n}u=[{\mathfrak m}athfrak{a}a;\,\mathfrak{p}hi,\Gammaa]\in \thetas,\mathfrak{q}uad \Gammaa\in\Gammasme,\ \Gammaa>\rho_i(\mathfrak{p}hi)\ {\mathfrak m}box{ for all }i\in A.
$$
By Lemma \ref{specialCont}, we may assume that all $\rho_i$ are commensurable. Thus, we may think that these limit augmentations are constructed solely from objects in the tree $\thetas$.
For any such augmentation, we may describe the following interval in $\thetas$:
$$
{\mathfrak m}athfrak{b}igcap\mathbf{n}olimits_{i\in A}(\rho_i,\mathbf{n}u]={\mathfrak m}athfrak{l}eft\{[{\mathfrak m}athfrak{a}a;\,\mathfrak{p}hi,\Deltata]{\mathfrak m}id \Deltata\in\Gammasme,\ \rho_i(\mathfrak{p}hi)<\Deltata{\mathfrak m}athfrak{l}e\Gammaa\ {\mathfrak m}box{ for all }i\in A\rho_ight\}.
$$
${\mathfrak m}athfrak{b}ullet$\mathfrak{q}uad Every two nodes ${\mathfrak m}u,\,\mathbf{n}u\in\thetas$ have a greatest common lower node ${\mathfrak m}u\wedge\mathbf{n}u\in\thetas$.
Indeed, as remarked after Proposition \ref{GCN}, if neither ${\mathfrak m}u{\mathfrak m}athfrak{l}e\mathbf{n}u$ nor ${\mathfrak m}u\Gammae\mathbf{n}u$, the greatest common lower node ${\mathfrak m}u\wedge\mathbf{n}u\in\thetatt$ is commensurable; thus, it belongs to $\thetas$.
\subsection{Paths of constant depth in $\thetas$}{\mathfrak m}athfrak{l}abel{subsecConstTs}
The main difference between $\thetas$ and $\thetatt$ lies in the fact that the paths of constant depth in $\thetas$ are ``compact", thanks to the completeness of $\Gammasme$.
\subsubsection{Inner depth-zero nodes}
With the notation in Section \ref{subsecDepth0}, the inner depth-zero nodes of $\thetas$ are of the form $\omega_{a,\Gammaa}$ for $a\in K$ and $\Gammaa\in\Gammasme$. By (\ref{balls}), we have
$$
\omega_{a,-\infty}=\omega_{b,-\infty}{\mathfrak m}athfrak{l}e\omega_{c,\Gammaa} \mathfrak{q}uad{\mathfrak m}box{ for all}\mathfrak{q}uad a,b,c\in K,\ \Gammaa\in \Gammasme.
$$
Let us denote by ${\mathfrak m}inf:=\omega_{a,-\infty}$ this minimal depth-zero valuation, which is independent of $a$.
By Theorem \ref{main}, this node ${\mathfrak m}inf$ is an absolute minimal node of $\thetas$. We say that ${\mathfrak m}inf$ is the {\mathfrak m}edskipmph{root node} of $\thetas$. As a valuation, it works as follows:
$$
{\mathfrak m}inf{\mathfrak m}athfrak{c}olon\op{Ker}x{\mathfrak m}athfrak{l}ra {\mathfrak m}athfrak{l}eft({\mathfrak m}athbb Z\thetaimes\Gamma\rho_ight)\infty,\mathfrak{q}quad f{\mathfrak m}athfrak{l}ongmapsto {\mathfrak m}athfrak{l}eft(-\Deltaeg(f),v({\mathfrak m}athfrak{l}c(f))\rho_ight),
$$
where ${\mathfrak m}athfrak{l}c(f)$ is the leading coefficient of a nonzero polynomial $f$.
All valuations ${\mathfrak m}u$ on $\op{Ker}x$ satisfying ${\mathfrak m}u(x)<\Gammaq$ are equivalent to ${\mathfrak m}inf$ {\mathfrak m}athfrak{c}ite[Thm. 2.4]{RPO}.
Since ${\mathfrak m}inf$ is incommensurable, it has a unique tangent direction. Actually, $$\op{Ker}p({\mathfrak m}inf)=\{x-a{\mathfrak m}id a\in K\}=[x]_{{\mathfrak m}inf}.$$
All inner depth-zero nodes in $\thetas$ are obtained as a single ordinary augmentation of the root node ${\mathfrak m}inf$:
$$
\omega_{a,\Gammaa}=[{\mathfrak m}inf;\,x-a,\Gammaa]\mathfrak{q}uad{\mathfrak m}box{ for all }a\in K,\ \Gammaa\in\Gammasme,\ \Gammaa>-\infty.
$$
In particular, the set of all inner depth-zero nodes is:
$$
{\mathfrak m}athfrak{l}eft\{{\mathfrak m}inf\rho_ight\}{\mathfrak m}athfrak{c}up{\mathfrak m}athfrak{b}igcup\mathbf{n}olimits_{a\in K}\mathfrak{p}set_{{\mathfrak m}inf}(x-a).
$$
For any key polynomial $x-a\in \op{Ker}p({\mathfrak m}inf)$, the constant-depth path $\mathfrak{p}set_{{\mathfrak m}inf}(x-a)$ is parametrized by the interval $(-\infty,\infty]\subset\Gammasme\infty$:
{\mathfrak m}athfrak{b}egin{center}
\setlength{\unitlength}{4mm}
{\mathfrak m}athfrak{b}egin{picture}(20,3.5)
\mathfrak{p}ut(-1,1){${\mathfrak m}athfrak{b}ullet$}\mathfrak{p}ut(-0.8,1.3){{\mathfrak m}athfrak{l}ine(1,0){15.6}}\mathfrak{p}ut(18,1){${\mathfrak m}athfrak{b}ullet$}
\mathfrak{p}ut(6,1){${\mathfrak m}athfrak{b}ullet$}\mathfrak{p}ut(20,1){${\mathfrak m}athfrak{b}ullet$}
\mathfrak{p}ut(-3.2,1){{\mathfrak m}athfrak{b}egin{footnotesize}${\mathfrak m}inf${\mathfrak m}edskipnd{footnotesize}}
\mathfrak{p}ut(15.4,1){{\mathfrak m}athfrak{b}egin{footnotesize}${\mathfrak m}athfrak{c}dots{\mathfrak m}athfrak{c}dots${\mathfrak m}edskipnd{footnotesize}}
\mathfrak{p}ut(17.4,2){{\mathfrak m}athfrak{b}egin{footnotesize}$\omega_{a,\infty^-}${\mathfrak m}edskipnd{footnotesize}}
\mathfrak{p}ut(21,1){{\mathfrak m}athfrak{b}egin{footnotesize}$\omega_{a,\infty}${\mathfrak m}edskipnd{footnotesize}}
\mathfrak{p}ut(5.6,2){{\mathfrak m}athfrak{b}egin{footnotesize}$\omega_{a,\Gammaa}${\mathfrak m}edskipnd{footnotesize}}
{\mathfrak m}edskipnd{picture}
{\mathfrak m}edskipnd{center}
Moreover, $\omega_{a,\Gammaa}$ is commensurable if and only if $\Gammaa\in\Gammaq\infty$, and it preserves the rank if and only if $\Gammaa\in\Gammanbc\infty$.
The finite leaf $\omega_{a,\infty}$ has an immediate predecessor node $\omega_{a,\infty^-}$, represented by the valuation
$$
\omega_{a,\infty^-}{\mathfrak m}athfrak{c}olon\op{Ker}x{\mathfrak m}athfrak{l}ra {\mathfrak m}athfrak{l}eft({\mathfrak m}athbb Z\thetaimes\Gamma\rho_ight)\infty,\mathfrak{q}quad f{\mathfrak m}athfrak{l}ongmapsto {\mathfrak m}athfrak{l}eft(\op{ord}_{x-a}(f),v(\op{Init}t(f))\rho_ight),
$$
where $\op{Init}t(f)$ is the first nonzero coefficient of the $(x-a)$-expansion of $f\in\op{Ker}x$.
The intersection of the depth-zero paths in $\thetas$ determined by any two $a,b\in K$ may be computed as in Section \ref{subsecDepth0}:
$$
\mathfrak{p}set_{{\mathfrak m}inf}(x-a){\mathfrak m}athfrak{c}ap\mathfrak{p}set_{{\mathfrak m}inf}(x-b)=[{\mathfrak m}inf,\omega_{a,v(b-a)}].
$$
\subsubsection{Ordinary augmentations}
Let ${\mathfrak m}u\in\thetas$ be an inner node and let $\mathfrak{p}hi\in\op{Ker}pm$ be a key polynomial. The constant-depth path $\mathfrak{p}mph\subset\thetas$, of all nodes in $\thetas$ determined by an ordinary augmentation of ${\mathfrak m}u$ with respect to $\mathfrak{p}hi$, is parametrized by all $\Gammaa\in\Gammasme\infty$ such that $\Gammaa>{\mathfrak m}u(\mathfrak{p}hi)$:
{\mathfrak m}athfrak{b}egin{center}
\setlength{\unitlength}{4mm}
{\mathfrak m}athfrak{b}egin{picture}(22,3.5)
\mathfrak{p}ut(-2,1){${\mathfrak m}athfrak{b}ullet$}\mathfrak{p}ut(-1.6,1.3){{\mathfrak m}athfrak{l}ine(1,0){16}}\mathfrak{p}ut(18,1){${\mathfrak m}athfrak{b}ullet$}\mathfrak{p}ut(20,1){${\mathfrak m}athfrak{b}ullet$}
\mathfrak{p}ut(6,1){${\mathfrak m}athfrak{b}ullet$}
\mathfrak{p}ut(-3,1){{\mathfrak m}athfrak{b}egin{footnotesize}${\mathfrak m}u${\mathfrak m}edskipnd{footnotesize}}
\mathfrak{p}ut(16.5,1){{\mathfrak m}athfrak{b}egin{footnotesize}${\mathfrak m}athfrak{c}dots${\mathfrak m}edskipnd{footnotesize}}
\mathfrak{p}ut(15.2,1){{\mathfrak m}athfrak{b}egin{footnotesize}${\mathfrak m}athfrak{c}dots${\mathfrak m}edskipnd{footnotesize}}
\mathfrak{p}ut(16.5,2){{\mathfrak m}athfrak{b}egin{footnotesize}${\mathfrak m}u(\mathfrak{p}hi,\infty^-)${\mathfrak m}edskipnd{footnotesize}}
\mathfrak{p}ut(21,1){{\mathfrak m}athfrak{b}egin{footnotesize}${\mathfrak m}u(\mathfrak{p}hi,\infty)${\mathfrak m}edskipnd{footnotesize}}
\mathfrak{p}ut(4.8,2){{\mathfrak m}athfrak{b}egin{footnotesize}${\mathfrak m}u(\mathfrak{p}hi,\Gammaa)${\mathfrak m}edskipnd{footnotesize}}
{\mathfrak m}edskipnd{picture}
{\mathfrak m}edskipnd{center}{\mathfrak m}edskip
Moreover, ${\mathfrak m}u(\mathfrak{p}hi,\Gammaa)$ is commensurable if and only if $\Gammaa\in\Gammaq\infty$, and it preserves the rank if and only if $\Gammaa\in\Gammanbc\infty$.
The finite leaf ${\mathfrak m}u(\mathfrak{p}hi,\infty)$ has an immediate predecessor node ${\mathfrak m}u(\mathfrak{p}hi,\infty^-)$, represented by the valuation
$$
{\mathfrak m}u(\mathfrak{p}hi,\infty^-){\mathfrak m}athfrak{c}olon\op{Ker}x{\mathfrak m}athfrak{l}ra {\mathfrak m}athfrak{l}eft({\mathfrak m}athbb Z\thetaimes\Gamma\rho_ight)\infty,\mathfrak{q}quad f{\mathfrak m}athfrak{l}ongmapsto {\mathfrak m}athfrak{l}eft(\op{ord}_\mathfrak{p}hi(f),{\mathfrak m}u(\op{Init}t(f))\rho_ight),
$$
where $\op{Init}t(f)$ is the first nonzero coefficient of the $\mathfrak{p}hi$-expansion of $f\in\op{Ker}x$.
The intersection of the constant-depth paths in $\thetas$ determined by any two $\mathfrak{p}hi,\mathfrak{p}hi_*\in \op{Ker}pm$ may be computed as in Section \ref{subsecConstDepthOrd}:
$$
\mathfrak{p}mph{\mathfrak m}athfrak{c}ap\mathfrak{p}set_{\mathfrak m}u(\mathfrak{p}hi_*)={\mathfrak m}athfrak{b}egin{cases}
{\mathfrak m}edskipmptyset,&\mathfrak{q}uad{\mathfrak m}box{ if }\mathfrak{q}uad\mathfrak{p}hi\mathbf{n}ot\sim_{\mathfrak m}u\mathfrak{p}hi_*,\\
{\mathfrak m}athfrak{l}eft({\mathfrak m}u,{\mathfrak m}u(\mathfrak{p}hi,\Gammaa_0)\rho_ight],&\mathfrak{q}uad{\mathfrak m}box{ if }\mathfrak{q}uad\Gammaa_0={\mathfrak m}u(\mathfrak{p}hi-\mathfrak{p}hi_*)>{\mathfrak m}u(\mathfrak{p}hi).
{\mathfrak m}edskipnd{cases}
$$
\subsubsection{Limit augmentations}
Finally, let ${\mathfrak m}athfrak{a}a=(\rho_i)_{i\in A}$ be an essential continuous family in $\thetas$, and let $\mathfrak{p}hi\in\op{Ker}pi({\mathfrak m}athfrak{a}a)$ be a limit key polynomial. Let ${\mathfrak m}u\in{\mathfrak m}athfrak{a}a$ be the first valuation in the family. The completeness of $\Gammasme$ implies the existence of a minimal limit augmentation of ${\mathfrak m}athfrak{a}a$ in $\thetas$ with respect to $\mathfrak{p}hi$; namely
$$
{\mathfrak m}u_{\mathfrak m}athfrak{a}a:=[{\mathfrak m}athfrak{a}a;\,\mathfrak{p}hi,\Gammaa_{\mathfrak m}athfrak{a}a],\mathfrak{q}quad \Gammaa_{\mathfrak m}athfrak{a}a:=\sup{\mathfrak m}athfrak{l}eft\{\rho_i(\mathfrak{p}hi){\mathfrak m}id i\in A\rho_ight\}\in\Gammasme.
$$
Note that $\Gammaa_{\mathfrak m}athfrak{a}a>\rho_i(\mathfrak{p}hi)$ for all $i$, because ${\mathfrak m}athfrak{a}a$ has no maximal element. Also, $\Gammaa_{\mathfrak m}athfrak{a}a<\infty$.
The following result follows immediately from Lemma \ref{allLKP}.
{\mathfrak m}athfrak{b}egin{lemma}{\mathfrak m}athfrak{l}abel{muaa}
The value $\Gammaa_{\mathfrak m}athfrak{a}a\in\Gammasme$ and the valuation ${\mathfrak m}u_{\mathfrak m}athfrak{a}a\in\thetas$ are independent of the choice of the limit key polynomial $\mathfrak{p}hi$ in $\op{Ker}pi({\mathfrak m}athfrak{a}a)$.
{\mathfrak m}edskipnd{lemma}
The constant-depth path $\mathfrak{p}aph\subset\thetas$, of all nodes determined by a limit augmentation of ${\mathfrak m}athfrak{a}a$ with respect to $\mathfrak{p}hi$, is parametrized by all $\Gammaa\in\Gammasme\infty$ such that $\Gammaa\Gammae\Gammaa_{\mathfrak m}athfrak{a}a$:
{\mathfrak m}athfrak{b}egin{center}
\setlength{\unitlength}{4mm}
{\mathfrak m}athfrak{b}egin{picture}(28,4)
\mathfrak{p}ut(-2,0.9){${\mathfrak m}athfrak{b}ullet$}\mathfrak{p}ut(4.25,1){${\mathfrak m}athfrak{b}ullet$}\mathfrak{p}ut(25,1){${\mathfrak m}athfrak{b}ullet$}\mathfrak{p}ut(27,1){${\mathfrak m}athfrak{b}ullet$}\mathfrak{p}ut(12,1){${\mathfrak m}athfrak{b}ullet$}
\mathfrak{p}ut(-1.6,1.25){{\mathfrak m}athfrak{l}ine(1,0){4}}\mathfrak{p}ut(2.8,1){${\mathfrak m}athfrak{c}dots$}\mathfrak{p}ut(4.5,1.3){{\mathfrak m}athfrak{l}ine(1,0){17}}
{\mathfrak m}ultiput(4.5,0.1)(0,.25){10}{\vrule height1pt}
\mathfrak{p}ut(-1,2){$(\rho_i)_{i\in A}$}
\mathfrak{p}ut(-3,1.1){{\mathfrak m}athfrak{b}egin{footnotesize}${\mathfrak m}u${\mathfrak m}edskipnd{footnotesize}}
\mathfrak{p}ut(5,.4){{\mathfrak m}athfrak{b}egin{footnotesize}${\mathfrak m}u_{\mathfrak m}athfrak{a}a${\mathfrak m}edskipnd{footnotesize}}
\mathfrak{p}ut(21.8,1){{\mathfrak m}athfrak{b}egin{footnotesize}${\mathfrak m}athfrak{c}dots${\mathfrak m}edskipnd{footnotesize}}
\mathfrak{p}ut(23,1){{\mathfrak m}athfrak{b}egin{footnotesize}${\mathfrak m}athfrak{c}dots${\mathfrak m}edskipnd{footnotesize}}
\mathfrak{p}ut(23.2,2){{\mathfrak m}athfrak{b}egin{footnotesize}${\mathfrak m}athfrak{a}a(\mathfrak{p}hi,\infty^-)${\mathfrak m}edskipnd{footnotesize}}
\mathfrak{p}ut(28,1){{\mathfrak m}athfrak{b}egin{footnotesize}${\mathfrak m}athfrak{a}a(\mathfrak{p}hi,\infty)${\mathfrak m}edskipnd{footnotesize}}
\mathfrak{p}ut(10.8,2){{\mathfrak m}athfrak{b}egin{footnotesize}${\mathfrak m}athfrak{a}a(\mathfrak{p}hi,\Gammaa)${\mathfrak m}edskipnd{footnotesize}}
{\mathfrak m}edskipnd{picture}
{\mathfrak m}edskipnd{center}{\mathfrak m}edskip
Note that ${\mathfrak m}u_{\mathfrak m}athfrak{a}a\in\mathfrak{p}aph$. Again, ${\mathfrak m}athfrak{a}a(\mathfrak{p}hi,\Gammaa)$ is commensurable if and only if $\Gammaa\in\Gammaq\infty$, and it preserves the rank if and only if $\Gammaa\in\Gammanbc\infty$.
The finite leaf ${\mathfrak m}athfrak{a}a(\mathfrak{p}hi,\infty)$ has an immediate predecessor node ${\mathfrak m}athfrak{a}a(\mathfrak{p}hi,\infty^-)$, represented by the valuation
$$
{\mathfrak m}athfrak{a}a(\mathfrak{p}hi,\infty^-){\mathfrak m}athfrak{c}olon\op{Ker}x{\mathfrak m}athfrak{l}ra {\mathfrak m}athfrak{l}eft({\mathfrak m}athbb Z\thetaimes\Gamma\rho_ight)\infty,\mathfrak{q}quad f{\mathfrak m}athfrak{l}ongmapsto {\mathfrak m}athfrak{l}eft(\op{ord}_\mathfrak{p}hi(f),\rho_\aa(\op{Init}t(f))\rho_ight),
$$
where $\op{Init}t(f)$ is the first nonzero coefficient of the $\mathfrak{p}hi$-expansion of $f\in\op{Ker}x$.
The intersection of the constant-depth paths in $\thetas$ determined by any two $\mathfrak{p}hi,\mathfrak{p}hi_*\in \op{Ker}pi({\mathfrak m}athfrak{a}a)$ is an interval in $\thetas$ which may be computed as in Section \ref{subsecConstDepthLim}:
$$
\mathfrak{p}aph{\mathfrak m}athfrak{c}ap\mathfrak{p}set_{\mathfrak m}athfrak{a}a(\mathfrak{p}hi_*)=[{\mathfrak m}u_{\mathfrak m}athfrak{a}a,{\mathfrak m}athfrak{a}a(\mathfrak{p}hi,\rho_\aa(\mathfrak{p}hi-\mathfrak{p}hi_*)]\subset\thetas.
$$
Since the set ${\mathfrak m}athfrak{l}eft\{\rho_i(\mathfrak{p}hi){\mathfrak m}id i\in A\rho_ight\}\subset\Gammaq$ contains no maximal element, its supremum $\Gammaa_{\mathfrak m}athfrak{a}a$ in $\Gammasme$ is incommensurable. Indeed, if $\Gammaa_{\mathfrak m}athfrak{a}a\in\Gammaq$, then it would admit an immediate predecessor $\Gammaa_{\mathfrak m}athfrak{a}a^-<\Gammaa_{\mathfrak m}athfrak{a}a$, defined as $\Gammaa_{\mathfrak m}athfrak{a}a^-=b_S^-$, for $b=\Gammaa_{\mathfrak m}athfrak{a}a$ and $S=I$ (cf. Section \ref{subsecQcuts}). Since $\Gammaa_{\mathfrak m}athfrak{a}a^-\in\Gammanbc$ is incommensurable, it is still an upper bound of the set ${\mathfrak m}athfrak{l}eft\{\rho_i(\mathfrak{p}hi){\mathfrak m}id i\in A\rho_ight\}$. This contradicts the minimality of $\Gammaa_{\mathfrak m}athfrak{a}a$ as an upper bound of this set.
Thus, ${\mathfrak m}u_{\mathfrak m}athfrak{a}a$ is incommensurable. In particular, it has a unique tangent direction.
Since $\Gammaa_{\mathfrak m}athfrak{a}a<\infty$, Proposition \ref{extensionlim} shows that
$\mathfrak{p}hi$ is a key polyomial for ${\mathfrak m}u_{\mathfrak m}athfrak{a}a$ of minimal degree. Actually, by {\mathfrak m}athfrak{c}ite[Thm. 4.2]{KP} and Lemma \ref{allLKP}, we have
$$
\op{Ker}p({\mathfrak m}u_{\mathfrak m}athfrak{a}a)=[\mathfrak{p}hi]_{{\mathfrak m}u_{\mathfrak m}athfrak{a}a}= {\mathfrak m}athfrak{l}eft\{\mathfrak{p}hi+a{\mathfrak m}id a\in\op{Ker}x,\ \Deltaeg(a)<{\mathfrak m}i, \ \rho_{\mathfrak m}athfrak{a}a(a)>\Gammaa_{\mathfrak m}athfrak{a}a \rho_ight\}=\op{Ker}pi({\mathfrak m}athfrak{a}a).
$$
Also, all limit augmentations ${\mathfrak m}athfrak{a}a(\mathfrak{p}hi,\Gammaa)$ are ordinary augmentations of ${\mathfrak m}u_{\mathfrak m}athfrak{a}a$:
$$
{\mathfrak m}athfrak{a}a(\mathfrak{p}hi,\Gammaa)=[{\mathfrak m}athfrak{a}a;\,\mathfrak{p}hi,\Gammaa]=[{\mathfrak m}u_{\mathfrak m}athfrak{a}a;\,\mathfrak{p}hi,\Gammaa]\mathfrak{q}uad {\mathfrak m}box{ for all }\Gammaa\in\Gammasme,\ \Gammaa>\Gammaa_{\mathfrak m}athfrak{a}a,
$$
by comparing the action of both valuations on $\mathfrak{p}hi$-expansions. Indeed, for all polynomials $a\in\op{Ker}x$ of degree less than ${\mathfrak m}i=\Deltaeg(\mathfrak{p}hi)$, we have ${\mathfrak m}u_{\mathfrak m}athfrak{a}a(a)=\rho_\aa(a)$, by the definition of a limit augmentation.
The above picture might suggest that the interval $({\mathfrak m}u,{\mathfrak m}u_A)$ is contained in a single constant-depth path beyond ${\mathfrak m}u$. This is not the case.
By Lemma \ref{specialCont}, we may suppose that ${\mathfrak m}athfrak{a}a={\mathfrak m}athfrak{l}eft(\rho_i\rho_ight)_{i\in A}$, with $\rho_i=[{\mathfrak m}u;\,{\mathfrak m}athfrak{c}hi_i,{\mathfrak m}athfrak{b}e_i]$. Then, for each $i\in A$, the interval $({\mathfrak m}u,\rho_i]$ is contained in $\mathfrak{p}set_{\mathfrak m}u({\mathfrak m}athfrak{c}hi_i)$; however, for $j>i$, the valuation $\rho_j$ belongs to $\mathfrak{p}set_{\mathfrak m}u({\mathfrak m}athfrak{c}hi_j)$, but it does not belong to $\mathfrak{p}set_{\mathfrak m}u({\mathfrak m}athfrak{c}hi_i)$. Therefore, a more appropriate picture of this interval would be the following one:
{\mathfrak m}athfrak{b}egin{center}
\setlength{\unitlength}{4mm}
{\mathfrak m}athfrak{b}egin{picture}(20,9.5)
\mathfrak{p}ut(-2,0.9){${\mathfrak m}athfrak{b}ullet$}\mathfrak{p}ut(-1.6,1.25){{\mathfrak m}athfrak{l}ine(1,0){8.5}}\mathfrak{p}ut(7.3,1){${\mathfrak m}athfrak{c}dots$}
\mathfrak{p}ut(8.5,8){${\mathfrak m}athfrak{b}ullet$}\mathfrak{p}ut(8.7,8.3){{\mathfrak m}athfrak{l}ine(1,0){3}}\mathfrak{p}ut(12,8){${\mathfrak m}athfrak{c}dots$}
{\mathfrak m}ultiput(8.8,1.2)(0,.21){33}{\vrule height1pt}
\mathfrak{p}ut(1,1.2){{\mathfrak m}athfrak{l}ine(3,1){6}}
\mathfrak{p}ut(7.4,2.7){$\Deltaot{}$}\mathfrak{p}ut(7.82,2.85){$\Deltaot{}$}\mathfrak{p}ut(8.24,3){$\Deltaot{}$}
\mathfrak{p}ut(3,1.85){{\mathfrak m}athfrak{l}ine(2,1){4}}
\mathfrak{p}ut(7.4,3.45){$\Deltaot{}$}\mathfrak{p}ut(7.82,3.66){$\Deltaot{}$}\mathfrak{p}ut(8.24,3.87){$\Deltaot{}$}
\mathfrak{p}ut(5,2.85){{\mathfrak m}athfrak{l}ine(1,1){2}}
\mathfrak{p}ut(7.4,4.64){$\Deltaot{}$}\mathfrak{p}ut(7.82,5.06){$\Deltaot{}$}\mathfrak{p}ut(8.24,5.49){$\Deltaot{}$}
\mathfrak{p}ut(-3,1.1){{\mathfrak m}athfrak{b}egin{footnotesize}${\mathfrak m}u${\mathfrak m}edskipnd{footnotesize}}
\mathfrak{p}ut(7,8.2){{\mathfrak m}athfrak{b}egin{footnotesize}${\mathfrak m}u_{\mathfrak m}athfrak{a}a${\mathfrak m}edskipnd{footnotesize}}
{\mathfrak m}edskipnd{picture}
{\mathfrak m}edskipnd{center}
\subsection{Primitive nodes}
The constant-depth paths beyond a limit augmentation have completely analogous properties as the depth-zero paths. For the ease of the reader we include the depth-zero paths as a special case of the limit augmentation paths.{\mathfrak m}edskip
\mathbf{n}n{{\mathfrak m}athfrak{b}f Convention. }We admit the empty set ${\mathfrak m}athfrak{a}a={\mathfrak m}edskipmptyset$ as an essential continuous family in $\thetas$. We agree that this family has $\Gammaa_{\mathfrak m}athfrak{a}a=-\infty$, ${\mathfrak m}u_{\mathfrak m}athfrak{a}a={\mathfrak m}inf$, and
$$
\op{Ker}pi({\mathfrak m}athfrak{a}a)=\op{Ker}p({\mathfrak m}u_{\mathfrak m}athfrak{a}a)={\mathfrak m}athfrak{l}eft\{x-a{\mathfrak m}id a\in K\rho_ight\},\mathfrak{q}uad
\mathfrak{p}set_{{\mathfrak m}athfrak{a}a}(x-a)=\{{\mathfrak m}inf\}\,{\mathfrak m}athfrak{c}up\,\mathfrak{p}set_{{\mathfrak m}inf}(x-a).
$$
\Deltaefn A {\mathfrak m}edskipmph{primitive-limit} node in $\thetas$ is the inner limit node ${\mathfrak m}u_{\mathfrak m}athfrak{a}a$ associated to an essential continuous family ${\mathfrak m}athfrak{a}a$ in $\thetas$. The set of primitive-limit nodes is in bijection with the set of equivalence classes of essential continuous families.
A {\mathfrak m}edskipmph{primitive-ordinary\,} node in $\thetas$ is an inner node ${\mathfrak m}u\in\thetas$ admitting strong constant-depth paths (cf. Section \ref{subsecConstDepthOrd}). That is, $\op{Ker}pmz\mathbf{n}e{\mathfrak m}edskipmptyset$, where
$$
\op{Ker}pmz:=\{\mathfrak{p}hi\in\op{Ker}pm{\mathfrak m}id\Deltaeg(\mathfrak{p}hi)>\Deltaeg({\mathfrak m}u)\}.
$$
Since ${\mathfrak m}u$ has key polynomials of different degrees, it is necessarily commensurable.
A {\mathfrak m}edskipmph{primitive} node in $\thetas$ is a node which is either primitive-limit or primitive-ordinary. Let us denote by $\mathfrak{p}rim(\thetas)$ the set of all primitive nodes. {\mathfrak m}edskip
By our convention, the root node ${\mathfrak m}inf$ is a primitive-limit node.
By Theorems \ref{main} and {\mathfrak m}athfrak{c}ite[Thm. 4.7]{MLV}, the primitive-limit nodes cannot be obtained as an ordinary augmentation of a lower node.{\mathfrak m}edskip
\Deltaefn Let $\rho\in \thetas$ be a primitive node. Then, we define
$$
\mathfrak{p}rh={\mathfrak m}athfrak{b}egin{cases}
{\mathfrak m}athfrak{b}igcup_{\mathfrak{p}hi\in\op{Ker}p_{\operatorname{str}}(\rho)}\mathfrak{p}set_\rho(\mathfrak{p}hi),&{\mathfrak m}box{ if $\rho$ is primitive-ordinary},\\
{\mathfrak m}athfrak{b}igcup_{\mathfrak{p}hi\in\op{Ker}pi({\mathfrak m}athfrak{a}a)}\mathfrak{p}set_{\mathfrak m}athfrak{a}a(\mathfrak{p}hi),&{\mathfrak m}box{ if $\rho={\mathfrak m}u_{\mathfrak m}athfrak{a}a$ is primitive-limit}.
{\mathfrak m}edskipnd{cases}
$$
We emphasize that $\rho\in\mathfrak{p}rh$ if $\rho$ is a primitive-limit node, but $\rho\mathbf{n}ot\in\mathfrak{p}rh$ if $\rho$ is primitive-ordinary. However, in both cases, the arguments in Section \ref{subsecConstTs} show that
{\mathfrak m}athfrak{b}egin{equation}{\mathfrak m}athfrak{l}abel{remind}
{\mathfrak m}u\in\mathfrak{p}rh,\ \rho<{\mathfrak m}u\ \ \Longrightarrow\ \ {\mathfrak m}u=[\rho; \mathfrak{p}hi,\op{SV}al({\mathfrak m}u)],
{\mathfrak m}edskipnd{equation}
for some $\mathfrak{p}hi\in\op{Ker}pm$. If $\rho$ is primitive-ordinary, then necessarily $\mathfrak{p}hi\in\op{Ker}pmz$.
{\mathfrak m}athfrak{b}egin{theorem}{\mathfrak m}athfrak{l}abel{previous}
Let $\mathbf{n}u\in\thetas$ be either an inner node, or a finite leaf. There exists a unique primitive node $\rho\in\mathfrak{p}rim(\thetas)$ such that
$\mathbf{n}u\in\mathfrak{p}rh$. In other words,
$$
\thetas\setminus{\mathfrak m}athfrak{l}i(\thetas)={\mathfrak m}athfrak{b}igsqcup\mathbf{n}olimits_{\rho\in\mathfrak{p}rim(\thetas)}\mathfrak{p}rh.
$$
{\mathfrak m}edskipnd{theorem}
{\mathfrak m}athfrak{b}egin{proof}
If $\mathbf{n}u$ has depth zero, then $\mathbf{n}u$ belongs to $\mathfrak{p}mi$, as we saw in Section \ref{subsecConstTs}.
If $\mathbf{n}u$ has a positive depth, then it is the last node of a finite MLV chain
$$ {\mathfrak m}u_0\ \stackrel{\mathfrak{p}hi_1,\Gammaa_1}{\mathfrak m}athfrak{l}ra\ {\mathfrak m}u_1\ {\mathfrak m}athfrak{l}ra\ {\mathfrak m}athfrak{c}dots\ {\mathfrak m}athfrak{l}ra\ {\mathfrak m}u_{r-1}\ \stackrel{\mathfrak{p}hi_{r},\Gammaa_{r}}{\mathfrak m}athfrak{l}ra\ {\mathfrak m}u_{r}=\mathbf{n}u.$$
If the last augmentation step ${\mathfrak m}u_{r-1}\thetao\mathbf{n}u$ is ordinary, then ${\mathfrak m}u_{r-1}$ is a primitive-ordinary node and $\mathbf{n}u\in\mathfrak{p}set({\mathfrak m}u_{r-1})$. Indeed,
$\mathbf{n}u=[{\mathfrak m}u_{r-1};\,\mathfrak{p}hi_r,\Gammaa_r]\in\mathfrak{p}set_{{\mathfrak m}u_{r-1}}(\mathfrak{p}hi)$ and $\Deltaeg(\mathfrak{p}hi_r)>\Deltaeg({\mathfrak m}u_{r-1})$ by the definition of a MLV chain.
If ${\mathfrak m}u_{r-1}\thetao\mathbf{n}u$ is a limit augmentation, then $\mathbf{n}u=[{\mathfrak m}athfrak{a}a;\,\mathfrak{p}hi_r,\Gammaa_r]\in\mathfrak{p}ma$.
Therefore, the union of all sets $\mathfrak{p}rh$, for $\rho$ running on all the primitive nodes in $\thetas$, covers $\thetas\setminus{\mathfrak m}athfrak{l}i(\thetas)$. It remains only to show that
$$
\rho,{\mathfrak m}edskipta\in\mathfrak{p}rim(\thetas), \ \ \rho\mathbf{n}e{\mathfrak m}edskipta\ \ \Longrightarrow\ \ \mathfrak{p}rh{\mathfrak m}athfrak{c}ap \mathfrak{p}set({\mathfrak m}edskipta)={\mathfrak m}edskipmptyset.
$$
Since $\thetas$ is a tree, this is obvious if $\rho\mathbf{n}ot{\mathfrak m}athfrak{l}e{\mathfrak m}edskipta$ and ${\mathfrak m}edskipta\mathbf{n}ot{\mathfrak m}athfrak{l}e\rho$.
Suppose that $\rho<{\mathfrak m}edskipta$ and there exists ${\mathfrak m}u\in\mathfrak{p}rh{\mathfrak m}athfrak{c}ap\mathfrak{p}set({\mathfrak m}edskipta)$. By (\ref{remind}), the valuation ${\mathfrak m}u\in\mathfrak{p}rh$ may be obtained after a single ordinary augmentation step: ${\mathfrak m}u=[\rho;\,\mathfrak{p}hi,\op{SV}al({\mathfrak m}u)]$, for a certain $\mathfrak{p}hi\in\op{Ker}pr$. Since ${\mathfrak m}edskipta$ belongs to the interval $(\rho,{\mathfrak m}u)$, {\mathfrak m}athfrak{c}ite[Lem. 2.7]{MLV} shows that ${\mathfrak m}edskipta=[\rho;\,\mathfrak{p}hi,\op{SV}al({\mathfrak m}edskipta)]$ too. By Lemma \ref{propertiesAug}, $\Deltaeg({\mathfrak m}edskipta)=\Deltaeg(\mathfrak{p}hi)=\Deltaeg({\mathfrak m}u)$.
This leads to a contradiction. Indeed, ${\mathfrak m}edskipta$ cannot be a primitive-limit node
because it is an ordinary augmentation of a lower node. Hence, $\mathfrak{p}set({\mathfrak m}edskipta)$ is the union of strong constant-depth paths and this implies $\Deltaeg({\mathfrak m}edskipta)<\Deltaeg({\mathfrak m}u)$.
{\mathfrak m}edskipnd{proof}
\subsection{Stratification of $\thetas$ by limit-depth}{\mathfrak m}athfrak{l}abel{subsecStrat}
Let $\rho\in\thetas$ be a primitive-limit node. The {\mathfrak m}edskipmph{inductive tree} with root $\rho$ is the subset
$\thetaind(\rho)\subset\thetas$
formed by all inner nodes, or finite leaves in $\thetas$, which may be obtained by a finite chain of {{\mathfrak m}athfrak{b}f ordinary} augmentations starting from $\rho$:
$$
\rho\ \stackrel{\mathfrak{p}hi_1,\Gammaa_1}{\mathfrak m}athfrak{l}ra\ {\mathfrak m}u_1\ {\mathfrak m}athfrak{l}ra\ {\mathfrak m}athfrak{c}dots
\ {\mathfrak m}athfrak{l}ra\ {\mathfrak m}u_{n-1}
\ \stackrel{\mathfrak{p}hi_{n},\Gammaa_{n}}{\mathfrak m}athfrak{l}ra\ {\mathfrak m}u_{n}={\mathfrak m}u.
$$
We may consider the stratification by limit-depth
$$
\thetas\setminus{\mathfrak m}athfrak{l}i(\thetas)={\mathfrak m}athfrak{b}igsqcup\mathbf{n}olimits_{n\in{\mathfrak m}athbb N_0}\thetatt_n,
$$
where $\thetatt_n$ is the subtree of all nodes in $\thetas\setminus{\mathfrak m}athfrak{l}i(\thetas)$ whose limit-depth is equal to $n$.
These subtrees may be recursively constructed as:
$$
\thetatt_0=\thetaind({\mathfrak m}inf),\mathfrak{q}quad \thetatt_{n+1}={\mathfrak m}athfrak{b}igsqcup\mathbf{n}olimits_{[{\mathfrak m}athfrak{a}a]\in\mathbf{n}ni(\thetatt_n)}\thetaind({\mathfrak m}u_{\mathfrak m}athfrak{a}a),
$$
where $\mathbf{n}ni(\thetatt_n)$ is the set of equivalence classes of essential continuous families in $\thetatt_n$.
We could stratify ${\mathfrak m}athfrak{l}i(\thetas)$ in a similar way, but we must add a stratum corresponding to the infinite leaves with an infinite limit-depth. In {\mathfrak m}athfrak{c}ite{ILD} we showed that such infinite leaves do exist.
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{\mathfrak m}edskipnd{document} |
\begin{document}
\begin{frontmatter}
\title{Error estimates for splitting methods based on AMF-Runge-Kutta formulas for the time
integration of advection diffusion reaction PDEs.}
\author{
S. Gonzalez-Pinto, D. Hernandez-Abreu and S. Perez-Rodriguez}
\thanks[thanks1]{This work has been supported by projects
MTM2010-21630-C02-02 and MTM2013-47318-C2-2}
\corauth[cor1]{Corresponding author: S. Gonzalez-Pinto
([email protected])}
\baselineskip=0.9\mbox{$|||$}lbaselineskip
\maketitle
\address{
{ \footnotesize Departamento de An\'{a}lisis Matem\'{a}tico. Universidad de
La Laguna.\\
38071. La Laguna, Spain. \\
email: spinto\symbol{'100}ull.es, dhabreu\symbol{'100}ull.es}}
\begin{abstract}
The convergence of a family of AMF-Runge-Kutta methods (in short
AMF-RK) for the time integration of evolutionary Partial
Differential Equations (PDEs) of Advection Diffusion Reaction type
semi-discretized in space is considered. The methods are based on
very few inexact Newton Iterations of Aproximate Matrix
Factorization splitting-type (AMF) applied to the Implicit
Runge-Kutta formulas, which allows very cheap and inexact
implementations of the underlying Runge-Kutta formula. Particular
AMF-RK methods based on Radau IIA formulas are considered. These
methods have given very competitive results when compared with
important formulas in the literature for multidimensional systems of
non-linear parabolic PDE problems. Uniform bounds for the global
time-space errors on semi-linear PDEs when simultaneously the time
step-size and the spatial grid resolution tend to zero are derived.
Numerical illustrations supporting the theory are presented.
\end{abstract}
\begin{keyword}
Evolutionary Advection-Diffusion-Reaction Partial Differential
equations, Approximate Matrix Factorization,
Runge-Kutta Radau IIA methods, Finite Differences, Stability and Convergence.\\
{\sl AMS subject classifications: 65M12, 65M15, 65M20.}
\end{keyword}
\end{frontmatter}
\section{Introduction}
We consider numerical methods for the time integration of a family
of Initial Value Problems in ODEs
\begin{equation}\langlebel{ode}
y_h'(t) = f_h(t, y_h(t)),\;\;\;y_h(0) = {u}^*_{0,h}, \;\;\; 0 \le t
\le t^*, \quad y_h, f_h \in \mathbb{R}^{m(h)}, \quad h\rightarrow
0^+,
\end{equation}
coming from the spatial semi-discretization of an $l-$dimensional
Advection Diffusion Reaction problem in time
dependent Partial Differential Equations (PDEs), with prescribed
Boundary Conditions and an Initial Condition. Here $h$ denotes a
small positive parameter associated with the spatial resolution and usually $l=2,3,\ldots$ .
The typical PDE problem with Dirichlet
boundary conditions is given by ($\Omega$ is
a bounded open connected region in $\mathbb{R}^l$, $\partial \Omega$ its boundary and $\nabla$ is the
gradient operator)
\begin{equation}\langlebel{pde} \begin{array}{c}
u_t(x,t) = - \nabla \cdot (a(x,t) u(x,t)) + \nabla \cdot
(\bar{d}(x,t)\cdot \nabla u(x,t)) + r(u,x,t),\\[0.5pc] x \in \Omega , \; t\in [0,t^*]; \; a(x,t)=(a_j(x,t))_{j=1}^l \in \mathbb{R}^l, \;\bar{d}(x,t)=(\bar{d}_j(x,t))_{j=1}^l \in \mathbb{R}^l,\\[0.3pc]
u(x,t) =g_1(x,t), \:(x,t)\in \partial \Omega\times[0,t^*];
\qquad u(x,0)=g_2(x), \; x \in \Omega,\end{array}
\end{equation}
which is assumed to have some diffusion ($\bar{d}_j(x,t)\ge
d_0>0,\;j=1,\ldots,l$), namely that it is not of pure hyperbolic
type, and it is also assumed that some adequate spatial
discretization based on Finite Differences or Finite Volume is
applied to obtain the system (\ref{ode}). Some stiffnes in the
reaction part $r(u,x,t)$ is also allowed. The treatment of Systems
of PDEs do not involve more difficulty for our analysis but for
simplicity of presentation we prefer to confine ourselves to the
case of one PDE.
We denote by $u_h(t)$ the solution of the PDE problem confined to
the spatial grid (or well to the $h$-space related). It will be
tacitly assumed that the PDE problem admits a smooth solution
$u(x,t)$ in the sense that continuous partial derivatives in all variables up to some order $p$
exist and are continuous and uniform bounded on $\Omega\times[0,t^*]$ and that $u(x,t)$ is continuous
on $\bar{\Omega}\times[0,t^*]$ ($\bar{\Omega}=\Omega \bigcup \partial \Omega$).
It is also assumed that the spatial discretization errors \begin{equation}\langlebel{spatialerrors}
\sigma_h(t):= u_h'(t)-f_h(t,u_h(t)),
\end{equation}
satisfy in the norm considered,
\begin{equation}\langlebel{norm}
\Vert \sigma_h(t) \Vert \le C \:h^r , \quad (C\ge 0,\;r>0), \quad 0
\le t \le t^*, \quad h\rightarrow 0.
\end{equation}
In general $C$, $C'$ or $C^*$ will refer to some constants that
maybe different at each occurrence but that all of them are
independent of $h\rightarrow 0$ and from the time-stepsize
$\tau\rightarrow 0$. The vector norm used is arbitrary as long as it
is defined for vectors of any dimension. For square matrices the
norm used is the induced operator norm, $\Vert A\Vert = \sup_{v\ne
0} \Vert A v \Vert / \Vert v \Vert. $
In spite of most of our results apply in general, we will provide
specific results for weighted Euclidean norms of type
$$\Vert (v_j)_{j=1}^N \Vert=N^{-1/2} \Vert (v_j)_{j=1}^N \Vert_2. $$
It should be noted that in this case we have for any square matrix
$A$ that,
$$ \Vert A \Vert= \Vert A \Vert_2, \quad \forall \:A \in \mathbb{R}^{N,N},\;N=1,2,3,\ldots .$$
We assume some natural
splitting for $f_h$ (directional or other),
\begin{equation}\langlebel{splitting}
f_h(t,y)=\sum_{j=1}^d f_{j,h}(t,y),
\end{equation}
which provides some natural splitting for the Jacobian matrix at the current point $(t_n,y_n)$,
\begin{equation}\langlebel{split}
J_h=\sum_{j=1}^d J_{j,h}, \quad J_h :=
\displaystyle{\frac{\partial f_h(t_n,y_n)}{\partial
y}}, \quad J_{j,h} :=\displaystyle{\frac{\partial f_{j,h}(t_n,y_n)}{\partial y}}.
\end{equation}
This goal of the paper is to analyze the convergence order of
the Method of Lines (MoL) approach for time-dependent PDEs of Advection Reaction Diffusion
PDEs, with the main focuss on the time integration of the large ODE systems
resulting of the spatial PDE-semidiscretization, where some
stiffness is assumed (parabolic dominant problems with stiff
reaction terms) and the time integrators are based on very few
iterations of splitting type (Approximate Matrix Factorization and
Newton-type schemes) applied to highly stable Implicit Runge-Kutta
methods. It should be remarked that the underlying Implicit
Runge-Kutta method is never solved up to convergence, hence the
convergence study does not follows from the results collected in
classical references about finite difference methods such as
\cite{Rich-Morton67,Bur-Hun-Ver86,Marchuk90,Thomee90,Trefethen92,HV}.
The kind of approach to be considered here has interest since it is
easily applicable to general systems of PDEs as we will see later on
and it is
reasonably cheap for non-linear problems in general
(although we give convergence results for semilinear problems only) when some splitting of the function $f_h$ and
its Jacobian is available and the split terms can be handled efficiently. In particular a method based on
three AMF-iterations of the two-stage Radau IIA method
\cite{Axelsson69} has shown to be competitive \cite{apnum-sevsole10} when compared with some
standard PDE-solvers such as VODPK \cite{Brown-Byrne-Hindmarsh-SISC89,Brown}
in some interesting non-linear diffusion reaction problems
widely considered in the literature. We also present two new
methods based on the 2-stage Radau IIA, by performing just one or
two iterations of splitting type, respectively. The method based on
two iterations is one of the very few one-step methods of splitting
type we have seen in the literature that has order three in
PDE-sense for the time integration.
The rest of the paper is organized as follows. In section 2 we
introduce the {\sf AMF$_q$-RK} methods, and special attention is
paid to some methods based on Radau IIA formulas. In section 3, the
convergence for semilinear PDEs is studied in detail. The local and
global errors are studied for the {\sf AMF$_q$-RK} splitting
methods based on some general Runge-Kutta methods. Section 4 is
devoted to some applications of the convergence results to 2D and
3D-parabolic PDEs.
{\rm Henceforth, for simplicity in the
notations, we omit in many cases the $h$-dependence
of some vectors such as $f_h,\;f_{j,h}$ and of some matrices such as $J_h$
and $J_{j,h}$ ($j=1,\ldots,d$). It should be clear from the context which ones are $h$-dependent.
Besides, we will refer to the identity matrix as $I$ when its dimension is clear from the context. }
\section{AMF-IRK methods}
For the integration of the ODEs (\ref{ode}), we consider as a first
step an implicit s-stage Runge-Kutta method with a nonsingular
coefficient matrix $A=(a_{ij})_{i,j=1}^s$ and a weight vector
$b=(b_j)_{j=1}^s$. The method is given by the compact formulation
(below $\otimes$ denotes the Kronecker product of matrices $A\otimes
B=(a_{ij}B), \;A=(a_{ij}), \;B=(b_{ij})$)
\begin{equation}\langlebel{IRK} \begin{array}{c}
Y_n=e\otimes y_n +\tau (A\otimes I_m)F(Y_n), \\
y_{n+1}= \varpi y_n + (\ss^T\otimes I_m)Y_n, \\ c\end{equation}uiv (c_j)_{j=1}^s:=A e, \quad e=(1,\ldots,1)^T \in \mathbb{R}^{s}, \quad \ss^T:=b^TA^{-1},\quad \varpi=1-\ss^Te, \\
Y_n=(Y_{n,j})_{j=1}^s \in \mathbb{R}^{ms}, \qquad F(Y_n)=(f(t_n+\tau c_j,Y_{n,j}))_{j=1}^s \in \mathbb{R}^{ms}.
\end{array}
\end{equation}
It should be noted that we have replaced the usual formulation at the stepping point $y_{n+1}= y_n + \tau (b^T\otimes I_m) F(Y_n)$ by the equivalent in (\ref{IRK}), which has some computational advantages for stiff problems when the algebraic system for the stages is not exactly solved.
A typical Quasi-Newton iteration to solve the stage equations above
is given by (below, $J=\partial{f}/\partial y\:(t_n,y_n)$ is the
exact Jacobian at the step-point $(t_n,y_n)$),
\begin{equation}\langlebel{newt}
[I_{ms}- A\otimes \tau J]\Delta^\nu=D_n^{\nu-1},\quad
Y_n^\nu=Y_n^{\nu-1}+\Delta^\nu, \quad \nu=1,2,\dots,
\end{equation}
where
\begin{equation}\langlebel{residual}
D_n^{\nu-1}\end{equation}uiv D(t_n,\tau,y_n,Y_n^{\nu-1}):=
e\otimes y_n - Y_n^{\nu-1} + \tau ( A\otimes I_m)
F(Y_n^{\nu-1}).
\end{equation}
A cheaper iteration of Newton-type when the matrix $A$ has a
multipoint spectrum has been considered in
\cite{APNUM95-seve,PGS09} (denoted as Single-Newton iteration)
\begin{equation}\langlebel{s-newt}
[I_{ms}- T_\nu\otimes \tau J]\Delta^\nu=D_n^{\nu-1},\quad
Y_n^\nu=Y_n^{\nu-1}+\Delta^\nu, \quad \nu=1,2,\dots,q
\end{equation}
where
\begin{equation}\langlebel{T} \begin{array}{c}
T_\nu=\gamma S_\nu (I-L_\nu)^{-1} S_\nu^{-1}, \quad \gamma>0,\\
\quad S_\nu \in \mathbb{R}^{s,s} \;\mbox{\rm are regular matrices and }
\\ L_\nu \in \mathbb{R}^{s,s} \; \mbox{\rm are strictly lower triangular matrices.}
\end{array}
\end{equation}
After some simple manipulations, by using standard properties of the Kronecker product,
this iteration can be rewritten in the equivalent form,
\begin{equation}\langlebel{s-newt1}
\begin{array}{rl}
[I_s\otimes(I_{m}-\gamma \tau J)] E^\nu&=((I_s-L_\nu)S_\nu^{-1} \otimes I_m)D_n^{\nu-1}+ (L_\nu\otimes I_m)E^\nu, \\
Y_n^\nu &=Y_n^{\nu-1}+(S_\nu\otimes I_m)E^\nu, \qquad \nu=1,2,\dots,q.
\end{array}
\end{equation}
To reduce the algebra cost, we use the Approximate Matrix
Factorization \cite{How-Som-JCAM2001} in short AMF, with $J\end{equation}uiv
J_h$ and $J_j\end{equation}uiv J_{j,h}$ given in (\ref{split}),
\begin{equation}\langlebel{product}
\Pi_d := \prod_{j=1}^d (I_m-\gamma \tau J_j)= (I_{m}-\gamma \tau J) +
\mathcal{O}(\tau^2),
\end{equation}
and replace in (\ref{s-newt1}) $(I_{m}-\gamma \tau J)$ by $\Pi_d$,
which yields the {\sf AMF$_q$-RK} method based on the underlying
Runge-Kutta method
\begin{equation}\langlebel{AMF-RK}
\begin{array}{rl}
[I_s\otimes\Pi_d] E^\nu&=((I_s-L_\nu)S_\nu^{-1} \otimes I_m)D_n^{\nu-1}+ (L_\nu\otimes I_m)E^\nu, \\
Y_n^\nu &=Y_n^{\nu-1}+(S_\nu\otimes I_m)E^\nu, \qquad \nu=1,2,\dots,q \\
Y_n^0&=e\otimes y_n \qquad \mbox{\rm (Predictor)} \\ y_{n+1}&= \varpi y_n + (\ss^T\otimes I_m)Y^q_n
\qquad \mbox{\rm (Corrector).}
\end{array}
\end{equation}
Our starting point for the convergence analysis in the next section
takes into account that the {\sf AMF$_q$-RK} method can be rewritten
in the equivalent form \cite{sevedom-AMFestab}
\begin{equation}\langlebel{AMF-RK1} \begin{array}{c}
[I\otimes I-
T_\nu\otimes \tau P](Y_n^\nu-Y_n^{\nu-1})=
D(t_n,\tau,y_n,Y_n^{\nu-1}), \;\; 1\leq \nu\leq q \\
Y_n^0=e\otimes y_n, \qquad y_{n+1}= \varpi y_n + (\ss^T\otimes I_m)Y^q_n,
\end{array}
\end{equation}
where the matrix $P$ plays a primary role
\begin{equation}\langlebel{matrixP}
\begin{array}{lll} P&:=&(\gamma
\tau)^{-1}(I-\Pi_d)\\&=& J+(-\gamma
\tau)\displaystyle{\sum_{j<k}} J_jJ_k +(-\gamma
\tau)^2 \displaystyle{\sum_{j<k<l}} J_jJ_kJ_l
+\ldots+(-\gamma \tau)^{d-1} J_1J_2\cdots
J_d.\end{array}
\end{equation}
\subsection{AMF$_q$-RK methods based on the 2 stage Radau IIA
formula}\langlebel{sec-2.1}
We are going to deserve special attention to AMF$_q$-RK methods
based on the 2 stage Radau IIA formula \cite{Axelsson69}. This formula has
coefficient Butcher tableau given by
$$\begin{array}{c|c} c&A\\ \hline \\[-2pc] & b^T\end{array} \quad \end{equation}uiv
\quad \begin{array}{c|cc} 1/3& 5/12 & \;-1/12\\ 1 & 3/4&\; 1/4
\\[0.3pc] \hline \\[-2pc]
& 3/4&\; 1/4\end{array}$$
This is a collocation method (stage order is two) possessing good stability properties, such as
$L$-stability (i.e. $A$-stability plus $R(\infty)=0$, with $R(z)$
being the linear stability function of the method), and has order of
convergence three (in ODE sense), not only on non-stiff problems but
also in many kinds of stiff problems \cite{Bur-Hun-Ver86}. These
properties for the underlying Runge-Kutta method are convenient,
since the family of ODEs (\ref{ode}) involves stiffness in most of
cases, due to the diffusion terms and possibly to the reaction part,
and it is expected that the methods to be built on inherit part of
the good properties of the original Runge-Kutta method.
The next three {\sf AMF$_q$-Rad} methods have coefficient matrices
($L_\nu$, $S_\nu$ and $T_\nu$) and eigenvalue $\gamma$ of the form
\begin{equation}\langlebel{T}
T_\nu=\gamma S_\nu (I_2-L_\nu)^{-1} S_\nu^{-1}, \;
S_\nu=\left(\begin{array}{cc} 1 & s_\nu
\\ 0&1\end{array}\right),\; L_\nu=\left(\begin{array}{cc} 0 & 0 \\
l_\nu &0\end{array}\right),\; \gamma=\sqrt{\det(A)}=1/\sqrt{6}.
\end{equation}
{\sf AMF$_1$-Rad} was derived in \cite{sevedom-AMFestab} by looking
for good stability properties and order two (ODE sense). In
particular the method is A($\pi/2$)-stable for a $2$-splitting (see
in Definition \ref{def-estability} below, the concept of stability
for a $d$-splitting), A($0$)-stable for any $d$-splitting and has
stability wedges close to $\theta_d=\pi/(2(d-1))$ for $d=3,4$. The
method is based on one iteration ($q=1$) and was required to fulfil
$(A-T_1)c=0$ and it has coefficients given by
\begin{equation}\langlebel{met1}
s_1= -\frac{3+2 \sqrt{6}}{9}, \quad l_1=\frac{3}{4}(-12+5\sqrt{6}).
\end{equation}
{\sf AMF$_2$-Rad} was derived in \cite{sevedom-AMFestab} by looking
for good stability properties and order three (ODE sense). The
method is A($\pi/2$)-stable for a $2$-splitting, A($0$)-stable for
any $d$-splitting and A($\pi/6$)-stable for $d=3,4$. The method is
based on two iterations ($q=2$) and their matrices $T_1$ and $T_2$
were required to satisfy $(A-T_1)c=0$ and $e_2^T
T_2^{-1}(A-T_2)=0^T, \;e_2^T=(0,1)$, respectively. Its coefficients
are uniquely given by
\begin{equation}\langlebel{met2} \begin{array}{c}
\displaystyle{s_1= -\frac{3+2 \sqrt{6}}{9}, \quad
l_1=\frac{3}{4}(-12+5\sqrt{6})}\\[0.3pc]
\displaystyle{s_2= \frac{5-2\sqrt{6}}{9}, \quad
l_2=\frac{3\sqrt{6}}{4}}.\end{array}
\end{equation}
{\sf AMF$_3$-Rad} was derived in \cite{PGS09,apnum-sevsole10} by
looking for good stability properties and order three (ODE sense).
The method is A($\pi/2$)-stable for a $2$-splitting, A($0$)-stable
for any $d$-splitting and close to A($\theta_d$)-stable for
$d=3,4$ with $\theta_d=\pi/(2(d-1))$. The method is based on three
iterations ($q=3$) and their matrices $T=T_1=T_2=T_3$ were required
to satisfy $e_2^T T^{-1}(A-T)=0^T$. Its coefficients are uniquely
given by
\begin{equation}\langlebel{met3} \begin{array}{c}
\displaystyle{s_1=s_2=s_3= \frac{5-2\sqrt{6}}{9}, \quad
l_1=l_2=l_3=\frac{3\sqrt{6}}{4}}.\end{array}
\end{equation}
In \cite{apnum-sevsole10}, a variable-stepsize integrator based
on the {\sf AMF$_3$-Rad} method was successfully tested on several
interesting $2D$ and $3D$ advection diffusion reaction PDEs by
exhibiting good performances in comparison with state-of-the-art
codes like {\sf VODPK} \cite{Brown-Byrne-Hindmarsh-SISC89,Brown} and
{\sf RKC} \cite{RKC,VSH-JcompPhys2004} and its implicit-explicit
counterpart, {\sf IRKC} \cite{IRKC,IMEXRKC}. The other two methods,
{\sf AMF$_q$-Rad} ($q=1,2$), were introduced later
\cite{sevedom-AMFestab} after carefully analyzing the PDE errors on
semilinear problems and with the purpose of reducing the number of
iterations w.r.t. {\sf AMF$_3$-Rad}.
\section{Convergence for semilinear problems}
For our convergence analysis we consider {\sf AMF$_q$-RK} methods
applied to the ODE problems coming from the spatial
discretizations of semilinear PDE problems of type (\ref{pde}) where
the advection and diffusion vectors $a(x,t)$ and $\bar{d}(x,t)$ are
both constant and the reaction part has the form
\begin{equation}\langlebel{reaction} r(u,x,t)=\kappa\: u + g(x,t), \quad \kappa \;
\mbox{\rm being a constant},\quad x\in\Omega \subseteq \mathbb{R}^l.
\end{equation} In this way, the ODE systems have the form
\begin{equation}\langlebel{lin-system}
\begin{array}{c}
y_h'(t)=f_h(t,y_h):=J_h y_h(t)+g_h(t),\quad
y_h(0)=u^*_{0,h},\quad h\rightarrow 0^+,\\
J_h=\sum_{j=1}^d
J_{j,h}, \qquad t\in [0,t^*].
\end{array}
\end{equation}
Here, the exact solution of the PDE confined to the spatial grid
$u_h(t)=u(x,t)$, is assumed to satisfy (\ref{spatialerrors}) and
(\ref{norm}). Thus, we focus on the global errors of the MoL
approach, where the spatial discretization is carried out first by
using finite differences (or finite volumes) and then the time
discretization is performed by using {\sf AMF$_q$-RK} methods. It
is important to remark that we will not pursue the details of the
spatial semidiscretizations but rather it is assumed that the
spatial semidiscretizations are stable and provides spatial
discretization errors satisfying (\ref{norm}). We shall provide
uniform bounds for the global errors of the MoL approach ($y_h(t)$
henceforth denotes the numerical solution of the MoL approach) in
the sense
\begin{equation}\langlebel{global-errors}
\varepsilonilon_{n,h}:=u_h(t_n)-{y}_h(t_n)=
\mathcal{O}(\tau)^{p_1}
+\mathcal{O}(h^{\alpha}\tau^{p_2}) ,\quad h\rightarrow 0^+,\tau
\rightarrow 0^+,
\end{equation}
which is meant that there exist constants $C_1,\:C_2,\:p_1,\:p_2,\:\alpha$
(all of them independent on $h$ and $\tau$) so that in the norm considered,
$$\Vert \varepsilonilon_{n,h}\Vert \le C_1 \tau^{p_1} + C_2 h^{\alpha} \tau^{p_2},\quad h \rightarrow 0^+,\tau
\rightarrow 0^+ \quad \mbox{\rm holds. }$$ In our convergence
analysis we need that all the matrices $J_{j,h}$ pairwise conmute
and that they can be brought to the following decomposition (it has
some resemblance with the Jordan's decomposition, but it is a little
more general)
\begin{equation}\langlebel{jordan}
\begin{array}{c}
J_{j,h}= \Theta_{h} \Lambda_{j,h}
\Theta_{h}^{-1}, \quad \mbox{\rm Cond}
(\Theta_{h}):=\Vert \Theta_{h}\Vert\cdot \Vert
\Theta_{h}^{-1}\Vert \le C,
\;h\rightarrow 0^+, \;1\leq j\leq d,\\
\Lambda_{j,h}=\mbox{\rm
BlockDiag}(\Lambda^{(1)}_{j,h},\Lambda^{(2)}_{j,h},\ldots,\Lambda^{(\vartheta_h)}_{j,h}),
\quad \Lambda^{(l)}_{j,h}=\langlembda^{(l)}_{j,h} I + E^{(l)}_{h} , \quad \mbox{\rm Re } \langlembda^{(l)}_{j,h} \le 0,\\
\mbox{\rm dim}(E^{(l)}_{h})\le N, \quad \Vert E^{(l)}_{h} \Vert_\infty \le C',
\; l=1,2,\ldots,\vartheta_h \quad (h\rightarrow 0^+).\\
E^{(l)}_{h} \quad \mbox{\rm are all of them strictly lower triangular matrices. }
\end{array}
\end{equation}
Another important approach for the convergence analysis of the MoL
method (mainly concerned with the time integration) is based on the
pseudo-spectra analysis of the matrix $J_{h}$ \cite{Trefethen92} and
the related matrices $J_{j,h}$. That analysis is of more general
scope but it is much more difficult to make and as we will see
below, our analysis is enough for some interesting kind of
semilinear problems and it is expected that the results extend to
most of the non-linear problems of parabolic dominant type.
Next, we consider a standard 3D semilinear-PDEs problem where the
assumptions in (\ref{jordan}) are fulfilled.
\subsection{An example}
Consider the semilinear PDE-problem (\ref{pde}) with $x\in
\Omega=(0,1)^3$, with constant vectors, $a(x,t)=(a_j)_{j=1}^3, \;
\bar{d}(x,t)=(\bar{d}_j)_{j=1}^3, \;\bar{d}_j>0 \;(j=1,2,3)$ and
$r(x,u,t)$ as in (\ref{reaction}). Consider the spatial
semidiscretization by using second order central differences and
spatial resolution $h=1/(N+1)$. This yields a semilinear ODE systems
of dimension $m=N^3$ of the form (\ref{lin-system}) for $d=3$. The
matrices $J_{j,h}$ are given by
\begin{equation}\langlebel{J-example} \begin{array}{c}
J_{1,h}=I_N\otimes I_N\otimes \mathcal{T}_1,\quad J_{2,h}=I_N\otimes
\mathcal{T}_2\otimes I_N,\quad J_{3,h}= \mathcal{T}_3\otimes I_N
\otimes I_N\\ \mathcal{T}_l =\mbox{\rm
Tridiag}(\alpha_l,\delta_l,\beta_l)\in \mathbb{R}^{N,N},\quad l=1,2,3,\\
\alpha_l=h^{-2}( \bar{d}_l- 2^{-1}h \: a_l ), \quad \beta_l=h^{-2}(
\bar{d}_l+ 2^{-1}h \:a_l ), \quad \delta_l=h^{-2}( -2\bar{d}_l+h^2
\kappa),
\end{array}
\end{equation}
and the vector $g_h(t)$ includes the reaction part $g(x,t)$ plus the
boundary conditions. It is straightforward to see that the $J_{l,h}$
pairwise commute. Moreover, by assuming Cell-P\'{e}clet numbers \cite[p.
67, formula (3.42) ]{HV}
$$ h|a_l|/\bar{d}_l< 2 ,\quad l=1,2,3,$$ from \cite[section
2]{Pasquini13} it follows that their spectral decomposition has the
form
\begin{equation}\langlebel{Tj} \begin{array}{c}
\mathcal{T}_l =\mbox{\rm
Tridiag}(\alpha_l,\delta_l,\beta_l) =V_l \Lambda_lV_l^{-1},\quad V_l= D_l U, \quad l=1,2,3,\\[0.2pc]
\Lambda_l=\mbox{\rm Diag}\displaystyle{\left(
\langlembda_{l,k}\right)_{k=1}^N,\quad \langlembda_{l,k}= \delta_l +
2\sqrt{\alpha_l\beta_l} \cos{\frac{k\pi}{N+1}},}
\\[0.2pc] U=(\frac{2}{N+1})^{1/2}\displaystyle{\left(\sin{\frac{kj\pi}{N+1}}\right)_{k=1,N
\atop j=1,N}}
\; \mbox{\rm is an orthogonal matrix and } \\[0.2pc]
D_l= (\frac{N+1}{2})^{1/2}\mbox{\rm
Diag}\displaystyle{\left((\alpha_l/\beta_l)^{k/2}\right)_{k=1}^N.}
\end{array}
\end{equation}
From here we conclude that all the matrices can be brought to the
spectral decomposition in (\ref{jordan}) having negative
eigenvalues and with matrix $\Theta_{h}=V_3\otimes V_2\otimes V_1. $
Observe that
$$\begin{array}{rcl} \Vert \Theta_{h}\Vert_2\Vert
\Theta_{h}^{-1}\Vert_2&=& \prod_{l=1}^3 \Vert V_l \Vert_2 \Vert
V_l^{-1} \Vert_2= \prod_{l=1}^3 \Vert D_l \Vert_2 \Vert D_l^{-1}
\Vert_2\\ &=& \displaystyle{ \prod_{l=1}^3 \left(\frac{2\bar{d}_l
+h|a_l|}{2\bar{d}_l -h|a_l|}\right)^{N/2}\le \prod_{l=1}^3
\left(\frac{2\bar{d}_l +h|a_l|}{2\bar{d}_l -h|a_l|}\right)^{1/(2h)}}
\\ & \simeq & \displaystyle{\exp\left(\sum_{l=1}^3 \frac{|a_l|}{2\bar{d}_l}\right)}
\quad \mbox{\rm as } h\rightarrow 0.
\end{array}
$$
\subsection{Analysis of the Truncation Errors}
The {\sf AMF$_q$-RK} method applied on problem (\ref{ode}) can be
expressed in the simple one-step format
$y_{n+1}=\phi_f(t_n,y_n,\tau),\;n\geq 0$. Thus, the time-space
global errors $\varepsilonilon_{n}=u_h(t_n)-y_n$ satisfy
$$
\begin{array}{lll}\varepsilonilon_{n+1}&:=&
u_h(t_{n+1})-\phi_f(t_n,y_n,\tau)\\
&=& (u_h(t_{n+1})-\phi_f(t_n,u_h(t_n),\tau)) +
(\phi_f(t_n,u_h(t_n),\tau)-\phi_f(t_n,y_n,\tau))\\
&=&l(t_n,\tau,h)+[\partial \phi_f/\partial y]_n (u_h(t_n)-y_n),
\end{array}
$$ where
$$
[\partial \phi_f/\partial y]_n=\int_0^1 \frac{\partial
\phi_f}{\partial
y}(t_n,u_h(t_n)+(\theta-1)\varepsilonilon_{n},\tau)d\theta,
$$
and the {\it time-space local errors} are defined by
\begin{equation}\langlebel{local-errors}
l_n\end{equation}uiv l(t_n,\tau,h) :=u_h(t_{n+1})-\phi_f(t_n,u_h(t_n),\tau).
\end{equation}
Then, we have for the {\it time-space global errors} $\varepsilonilon_{n}$
the recurrence
\begin{equation}\langlebel{global-error}
\varepsilonilon_{n+1}= [\partial \phi_f/\partial y]_n \cdot \varepsilonilon_{n}+
l_n,\quad n=0,1,2,\ldots,t^*/\tau-1.
\end{equation}
In order to get a better understanding of the latter recurrence, we
next introduce the following matrix operators ($P$ is defined in
(\ref{matrixP}))
\begin{equation}\langlebel{Mnu-Qnu}
Q_\nu=(I\otimes I- T_\nu\otimes \tau P)^{-1}, \quad M_\nu=Q_\nu
(A\otimes \tau J - T_\nu \otimes \tau P),\quad \nu\geq 1, \;Q_0=I.
\end{equation}
\begin{lemma}\langlebel{GE-recursion}
The time-space global errors provided by the {\sf AMF$_q$-RK} method
when applied to the problem (\ref{lin-system})
satisfy the recurrence
\begin{equation}\langlebel{global-errors1}
\varepsilonilon_{n+1}= R_q(\tau J,\tau P)\cdot \varepsilonilon_{n}+
l_n,\quad n=0,1,2,\ldots,t^*/\tau-1,
\end{equation}
where $l_n$ stands for the time-space local error defined in
(\ref{local-errors}) and
\begin{equation}\langlebel{estab}\begin{array}{l}
R_q(\tau J,\tau P)= \varpi I + \displaystyle{(\ss^T\otimes
I)\left( Q_q + \sum_{j=q}^1 (\prod_{i=q}^{j}
M_i)Q_{j-1}\right) (e\otimes I)},
\end{array} \end{equation}
with $Q_\nu, M_\nu$ ($\nu\geq 1$) given by
(\ref{Mnu-Qnu}). Moreover, the function $R_q(\tau J,\tau P)$ fulfils
\begin{equation}\langlebel{estab-1} R_q(\tau J,\tau P)-I= (\ss^T\otimes I)\left(Q_q + \sum_{j=q}^1
(\prod_{i=q}^{j} M_i)Q_{j-1}-\prod_{i=q}^{1} M_i\right) (c\otimes \tau J).
\end{equation}
\end{lemma}
\begin{remark} {\rm It must be observed that commutativity does not hold in
general, thus $\prod_{j=q}^1 M_j\end{equation}uiv M_q M_{q-1}\cdots M_1.$
On the other hand, $R_q(\cdot)$ can be
seen as the linear stability function of the method. The identity
(\ref{estab-1}) for the function $R_q(\cdot)-I$ will play a major
role in a favourable propagation of the local errors in a similar
way as indicated in Lemma 2.3 in \cite[p.162]{HV}.}
\end{remark}
\noindent {\bf Proof of Lemma \ref{GE-recursion}.} Our first step is
to analyze the operator $[\partial \phi_f/\partial y]_n$ for the
semilinear problem (\ref{lin-system}). Taking into account that the
method is defined by (\ref{AMF-RK1}), then we are led to compute
$\displaystyle{\frac{\partial y_{n+1}}{\partial y_n}}$ with $
y_{n+1}= \varpi y_n + (\ss^T\otimes) Y_n^q$. At this end, by taking
derivatives with regard to $y_n$ in the iteration (\ref{AMF-RK1}),
it holds that
$$
\begin{array}{lll}
(I\otimes I- T_\nu\otimes \tau P)\left(\dfrac{\partial
Y_n^\nu}{\partial y_n}-\dfrac{\partial Y_n^{\nu-1}}{\partial
y_n}\right)&=& \dfrac{\partial
D(t_n,\tau,y_n,Y_n^{\nu-1})}{\partial y_n}\\
&=& e\otimes I +(-I\otimes I + A\otimes \tau J) \dfrac{\partial
Y_n^{\nu-1}}{\partial y_n}.
\end{array}
$$
From here, after some simple manipulations it follows that,
\begin{equation}\langlebel{deriv}\begin{array}{l}
\displaystyle{\frac{\partial Y_n^\nu}{\partial y_n}}=
\displaystyle{Q_\nu (e\otimes I) + M_\nu \frac{\partial
Y_n^{\nu-1}}{\partial y_n}}, \quad (\nu=1,2,\ldots,q), \quad
\displaystyle{\frac{\partial Y_n^0}{\partial y_n}=e\otimes I.}
\end{array} \end{equation}
From an inductive argument, it is not difficult to see that
\begin{equation}\langlebel{sol1}\begin{array}{l}
\displaystyle{\frac{\partial Y_n^q}{\partial y_n}}=
\displaystyle{\left( Q_q + \sum_{j=q}^1
(\prod_{i=q}^{j} M_i)Q_{j-1}\right) (e\otimes I).}
\end{array} \end{equation}
Then, by denoting $R_q(\tau J,\tau P):=\displaystyle{\frac{\partial
y_{n+1}}{\partial y_n}}$ it follows $$R_q(\tau J,\tau P)= \varpi I +
(\ss^T\otimes)\displaystyle{\frac{\partial Y_n^q}{\partial y_n}},$$
and we deduce both (\ref{estab}) and (\ref{global-errors1}) from
(\ref{global-error}) and (\ref{sol1}).
In order to prove (\ref{estab-1}), we first take into account that
$R_q(\cdot)-I= (\ss^T\otimes I)Z_n^q$, where
$Z_n^\nu= \partial Y_n^\nu/\partial y_n-e\otimes I$. Then, from the
recurrence (\ref{deriv}), it follows after some simple calculations
that $Z_n^\nu = M_\nu Z_n^{\nu-1} + Q_\nu (c\otimes \tau J)$,
$(\nu=1,2,\ldots,q)$, with $Z_n^0=0$. From here, we deduce
$Z_n^q=\displaystyle{\left( Q_q + \sum_{j=q}^1 (\prod_{i=q}^{j}
M_i)Q_{j-1}-\prod_{i=q}^{1}
M_i\right) (c\otimes \tau J)},$ and this directly gives
(\ref{estab-1}).
$\Box$
\begin{remark}\langlebel{sev-remark-0} {\rm For a given rational function of two complex variables
\begin{equation}\langlebel{sev-res0} \displaystyle{\zeta(z,w)=\frac{\sum_{i,j=0}^{m_1} \alpha_{ij}
z^iw^j}{\sum_{i,j=0}^{m_2} \beta_{ij}z^iw^j}\end{equation}uiv \left(
{\sum_{i,j=0}^{m_1} \alpha_{ij} z^iw^j}\right)
\left({\sum_{i,j=0}^{m_2}
\beta_{ij}z^iw^j}\right)^{-1}},\end{equation} we define the
associated mapping $\zeta(Z,W)$ for two arbitrary commuting matrices
$Z$ and $W$ just by replacing $z$ by $Z$ and $w$ by $W$ whenever the
denominator yields a regular matrix. Sometimes we are given the
rational mapping $\zeta(Z,W)$ first and then we define the rational
complex function just by replacing the matrices $Z$ and $W$ by the
complex variables $z$ and $w$, respectively. The above definitions
are straightforward extended to functions and mappings of more than
two complex variables.
We will be mainly concerned with the case in which $z=\tau J$ and $w= \tau P$, where $J$ and $P$
are defined in (\ref{lin-system}) and (\ref{matrixP}), respectively. It should be noticed that for instance
the $(i,j)$-element of the matrix $M_\nu$, see (\ref{Mnu-Qnu}), would be given by (observe that it is a matrix itself)
$$ M_{ij}(\tau J, \tau P)= (e_i^T\otimes I) (I_s\otimes I_m- T_\nu\otimes \tau P)^{-1}
(A\otimes \tau J - T_\nu \otimes \tau P)(e_j\otimes I),$$
where $e_j$ denotes the $j$-vector of the canonical basis in $\mathbb{R}^s$ and the corresponding
complex function is $$M_{ij}(z,w)=e_i^T(I_s-wT_\nu)^{-1}(zA-wT_\nu)e_j.$$
Another important point is that despite of we are considering cases
with a $d$-splitting for $J$ as indicated in (\ref{lin-system}), the
replacement of every $\tau J_j$ by the complex variable $z_j$ and
the definition of
\begin{equation}\langlebel{z-w}
z:=\sum_{k=1}^d z_k, \qquad w:=
\gamma^{-1}\left(1-\prod_{k=1}^d (1-\gamma
z_k)\right),
\end{equation}
simplifies the study to the case of two complex variables $z$ and
$w$ or well to the case of mappings acting on the two matrices
$\tau J$ and $\tau P$.
It is worth to mention that our rational mappings and related
complex functions are all well defined whenever Re $z_k \le 0$ for
$k=1,2,\ldots,d$ and $d$ arbitrary, because the existence of the
matrix inverse $(I-wT_\nu)^{-1}$ is guaranteed if and only if
$\displaystyle{(1-\gamma w)^{-1}=\prod_{k=1}^d (1-\gamma z_k)^{-1}}$
exists. It is easily seen the existence of the late expression by
virtue of $\gamma>0$ and that all the eigenvalues of the matrices
$J_j,\;(j=1,\ldots,d)$ have a non-positive real part. Moreover, for
any $\nu=1,\ldots,q$ and any $d\ge 1$, we next prove that
\begin{equation}\langlebel{sev-res0a}\begin{array}{c}
\displaystyle{\sup_{\mbox{\scriptsize Re}\:z_k\:\le 0,
\;k=1,\ldots,d} |Q_\nu(z,w)|< + \infty, \quad
\sup_{\mbox{\scriptsize Re}\:z_k\:\le 0, \;k=1,\ldots,d}
|M_\nu(z,w)|< + \infty,}\\ z \;\mbox{\rm and } w\; \mbox{\rm defined
in (\ref{z-w}).} \end{array}
\end{equation} This see this, observe that $(T_\nu-\gamma I)$ is a
nilpotent matrix fulfilling $(T_\nu-\gamma I)^s=0$ and that
$$ \begin{array}{rcl} Q_\nu(z,w)&=& (I-w T_\nu)^{-1}=\left((1-w\gamma) I-w (T_\nu-\gamma I)\right)^{-1}\\
&=& (1-w\gamma)^{-1}\left( I-\frac{w}{1-w \gamma} (T_\nu-\gamma
I)\right)^{-1}=(1-w\gamma)^{-1}\sum_{j=0}^{s-1}\left(\frac{w}{1-w\gamma}\right)^j
(T_\nu-\gamma I)^{j}
\end{array}
$$
and
$$ \begin{array}{rcl} M_\nu(z,w)&=& Q_\nu(z,w)(z A-w T_\nu)= \frac{z}{1-w\gamma}\left(\sum_{j=0}^{s-1}
(\frac{w}{1-w\gamma})^j (T_\nu-\gamma I)^{j}\right)A \\[0.2pc] &-&
\frac{w}{1-w\gamma}\left(\sum_{j=0}^{s-1}(\frac{w}{1-w\gamma})^j
(T_\nu-\gamma I)^{j}\right)T_\nu.
\end{array}
$$
Hence the boundedness of $Q_\nu(z,w)$ and $M_\nu(z,w)$ follows from
the boundedness of
$$ \begin{array}{rcl} |\frac{1}{1-w\gamma}|&=&|\prod_{k=1}^d (1-\gamma
z_k)^{-1}| \le 1, \\
|\frac{w}{1-w\gamma}|&=&\gamma^{-1}|1-\frac{1}{1-w\gamma}| \le
\gamma^{-1}(1+1)=2\gamma^{-1},
\end{array}
$$
and from the next lemma.
$\Box$ }
\end{remark}
\begin{lemma}\langlebel{sev-lema-0} For any $d=2,3,\ldots$, and
$z$ and $w$ defined in (\ref{z-w}), we have that
$$\begin{array}{c}
\displaystyle{\sup_{\mbox{\scriptsize Re}\:z_k\:\le 0 \atop
k=1,\ldots,d} \left|\frac{z}{1-\gamma w}\right|=\gamma^{-1}\left(
\frac{(d-1)^{d-1}}{d^{d-2}}\right)^{1/2}.}
\end{array}
$$
\end{lemma}
\noindent {\bf Proof.} The third equality below follows from the
Maximum Modulus principle, which says that the Maximum Modulus is
reached at the boundary of the open region for complex analytical
functions,
$$\begin{array}{rcl}
\displaystyle{\sup_{\mbox{\scriptsize Re}\:z_k\:\le 0 \atop
k=1,\ldots,d} \left|\frac{z}{1-\gamma w}\right|}&=&
\gamma^{-1}\displaystyle{\sup_{\mbox{\scriptsize Re}\:z_k\:\le 0
\atop k=1,\ldots,d} \left|\frac{\gamma z}{1-\gamma w}\right|}=
\gamma^{-1}\displaystyle{\sup_{\mbox{\scriptsize Re}\:u_k\:\le 0
\atop k=1,\ldots,d}
\left|\frac{u_1+u_2+\ldots+u_d}{\prod_{k=1}^d(1-u_k)}\right|}\\[0.5pc]
&=&
\gamma^{-1}\displaystyle{\left|\frac{(y_1+y_2+\ldots+y_d)i}{\prod_{k=1}^d\sqrt{1+(y_k)^2}}\right|}=
\gamma^{-1}\displaystyle{\max_{x_k\:\ge 0 \atop
k=1,\ldots,d}\left(\frac{(x_1+x_2+\ldots+x_d)^2}{\prod_{k=1}^d(1+(x_k)^2)}\right)^{1/2}.}
\end{array}
$$
The computation of the extrema by making zero the gradient of the
real function of several variables ($x_1,\ldots,x_d$) gives the
maximum for $x_1=x_2=\ldots=x_d=(d-1)^{-1/2}.$ The proof follows
after substituting above this value.
$\Box$
\begin{definition}\langlebel{def-estability} {\rm A method of the form (\ref{AMF-RK1}) is said to be
$A(\theta)$-stable for a $d$-splitting, if and only if
$$
|R_q(z,w)| \le 1,\quad \forall z,w \; \mbox{\rm given by (\ref{z-w}) whenever }
z_k \in \mathcal{W}(\theta),\;k=1,2,\ldots,d,
$$
where (we consider that the argument of a no-null complex number
ranges in $[-\pi,\pi)$)
\begin{equation}\langlebel{ss1-1}
\mathcal{W}(\theta):= \{ u \in \mathbb{C}: u=0 \; \mbox{or}\;
|\mbox{arg}(-u)| \le \theta \}.\end{equation}}
\end{definition}
\subsection{Analysis of the Local Errors}
Next, we study the {\it time-space local errors} $l_n$ given by
(\ref{local-errors}). We will see that the time-space local error
$l_n$ is composed of two terms, $l_n^{[2]}$ related to the predictor
used in the {\sf AMF$_q$-RK} method and $l_n^{[1]}$ related to the
quadrature associated with the underlying Runge-Kutta method.
\begin{lemma}\langlebel{lem-loc-err}
If the linear system has continuous derivatives $u_h^{(k)}(t)$ up to
order $p+1$ in $[0,t^*]$ and the underlying RK method has stage
order $\ell\ge 1$ ($\ell \le p$), i.e.
$$Ac^{j-1}=j^{-1}c^j,\qquad b^Tc^{j-1}=j^{-1},\quad j=1,2,\ldots,\ell.$$ Then, the local
error $l_n$ in (\ref{local-errors}) of the {\sf AMF$_q$-RK} method
is given by
\begin{equation}\langlebel{local-errors1}
\begin{array}{lll} l_n&=&l_n^{[1]}+
l_n^{[2]},\\ l_n^{[1]}&:=&(\ss^T\otimes I)\left( Q_q + \sum_{j=q}^1
(\prod_{i=q}^{j} M_i)Q_{j-1}-\prod_{i=q}^{1} M_i \right) \hat{D}_n + \delta_n,\\
l_n^{[2]}&:=&(\ss^T\otimes I) (\prod_{i=q}^{1} M_i)\: \hbox{tr}iangle
u_h(t_n),
\end{array}
\end{equation}
with
\begin{equation}\langlebel{triangle} \begin{array}{rcl}\hbox{tr}iangle
u_h(t_n)&:=&(u_h(t_n+c_i\tau)-u_h(t_n))_{i=1}^s=\sum_{j=1}^p\frac{\tau^j}{j!}(c^j\otimes
I) u^{(j)}_h(t_n)\\[0.3pc] & + & \displaystyle{\frac{\tau^{p+1}}{p!}\left(\int_0^1 (c_i-\theta)_+^p
u^{(p+1)}_h(t_n+\theta \tau) d\theta\right)_{i=1}^s.}
\end{array}
\end{equation}
\end{lemma}
and (we use, $(x)_+:=x$ if $x\ge 0$ and $(x)_+:=0$ otherwise)
\begin{equation}\langlebel{resid-2} \begin{array}{rcl}
\hat{D}_n &=& \displaystyle{\sum_{j=\ell+1}^p \frac{\tau^j}{j!}
\left((c^j-jAc^{j-1})\otimes u^{(j)}_h(t_n) \right)+ \tau^{p+1}
\int_0^1 \left(\varphi(\theta)
\otimes u^{(p+1)}_h(t_n+ \theta \tau)\right) d\theta + }\\[0.5pc] & & \tau
(A\otimes I) \left(\sigma_h(t_n+c_i\tau)\right)_{i=1}^s;\quad
\varphi(\theta)=\displaystyle{\frac{1}{p!}\left((c_i-\theta)_+^p-p\sum_{j=1}^s
a_{ij}(c_j-\theta)_+^{p-1}\right)_{i=1}^s} \\[0.8pc]
\delta_n &=& \displaystyle{\sum_{j=\ell+1}^p \frac{\tau^j}{j!}
(1-\ss^T c^j) u^{(j)}_h(t_n)+ \tau^{p+1} \int_0^1\phi(\theta)\:
u^{(p+1)}_h(t_n+ \theta
\tau) d\theta},\\[0.3pc]
\phi(\theta)&=&\displaystyle{\frac{1}{p!}\left((1-\theta)^p-\sum_{j=1}^s
\ss_{j}(c_j-\theta)_+^{p}\right)}.
\end{array}
\end{equation}
\noindent {\bf Proof.} Let us define
\begin{equation}\langlebel{sev-res1}
\hat{D}_n:= (u_h(t_n+c_i\tau))_{i=1}^s-e\otimes u_h(t_n) - \tau
(A\otimes I) (f_h(t_n+c_i
\tau,u_h(t_n+c_i\tau))_{i=1}^s.\end{equation} From
(\ref{spatialerrors}), it follows that
\begin{equation}\langlebel{sev-res1a}\hat{D}_n=
(u_h(t_n+c_i\tau))_{i=1}^s-(u_h(t_n))_{i=1}^s - \tau (A\otimes I)
(u'_h(t_n+c_i\tau)-\sigma_h(t_n+c_i\tau))_{i=1}^s.\end{equation}
Now, by using the Taylor expansion with integral remainder (below
$\zeta(x)$ denotes a generic function having $r+1$-continuous
derivatives in an adequate interval)
\begin{equation}\langlebel{sev-res2}
\displaystyle{\zeta(t_n+x)=\sum_{l=0}^r \frac{x^l}{l!}
\zeta^{(l)}(t_n) + \frac{x^{r+1}}{r!}\int_0^1(1-\theta)^r
\zeta^{(r+1)}(t_n+\theta x) d\theta},
\end{equation}
and applying it conveniently to $u_h(t_n+c_i\tau)$ and
$u'_h(t_n+c_i\tau)$ in (\ref{sev-res1a}) with $r=p$ and $r=p-1$
respectively, we deduce after some computations, the expression for
$\hat{D}_n$ in (\ref{resid-2}). Observe that order stage $\ell$ for
the Runge-Kutta method implies that $c^j-jAc^{j-1}=0,
\;\ss^Tc^j-1=0,\;j=1,\ldots,\ell$. The expression for $\delta_n$ is
obtained in a similar way, but taking into account that this time we
define,
\begin{equation}\langlebel{sev-res3} \delta_n:= u_h(t_n+\tau)- \varpi u_h(t_n)
- \sum_{j=1}^s \ss_j u_h(t_n+c_j\tau).\end{equation} Let us now take
$\hat{U}_n:=(u_h(t_n+c_i\tau))_{i=1}^s$ and
$\Delta_n^\nu:=\hat{U}_n- U_n^\nu,$ where $ U_n^\nu$ are the
iterates obtained by the scheme (\ref{AMF-RK1}) when the predictor
$U_n^0=e\otimes u_h(t_n)$ is taken on the exact solution of the PDE
at $t_n$, i.e. $y_n=u_h(t_n)$. This gives as solution, see
(\ref{AMF-RK1})
\begin{equation}\langlebel{sev-res4}
y_{n+1}= \varpi u_h(t_n) + (\ss^T\otimes I)U_n^q.
\end{equation}
From (\ref{sev-res3}) and (\ref{sev-res4}) it follows
\begin{equation}\langlebel{sev-res5}
l_n=u_h(t_{n+1})-y_{n+1}= (\ss^T\otimes I)\Delta_n^q + \delta_n.
\end{equation}
In order to compute $\Delta_n^q$ we insert the expression for
$U_n^\nu$ in (\ref{AMF-RK1}). It follows for the semi-linear
problem (\ref{lin-system}) that
$$
\begin{array}{lll}
(I\otimes I- T_\nu\otimes \tau P)(\Delta_n^\nu-\Delta_n^{\nu-1})&=&
-D(t_n,\tau,u_h(t_n),U_n^{\nu-1})\\&=&-(I\otimes I- A\otimes \tau J)
\Delta_n^{\nu-1}+ \hat{D}_n,\end{array} \quad (\nu=1,2,\dots,q).
$$
This implies that $ \Delta_n^\nu = M_\nu \Delta_n^{\nu-1}+ Q_\nu
\hat{D}_n$, $1\leq \nu\leq q$, and from this recurrence
$$
\Delta_n^q=\displaystyle{\left( Q_q + \sum_{j=q}^1 (\prod_{i=q}^{j}
M_i)Q_{j-1} - \prod_{i=q}^{1} M_i\right) \hat{D}_n+ (\prod_{i=q}^{1}
M_i)\: \Delta_n^0},$$ with $\Delta_n^0=\hbox{tr}iangle u_h(t_n)$ in
(\ref{triangle}). Now, from this expression and from
(\ref{sev-res5}) the formula (\ref{local-errors1}) follows.
$\Box$
\begin{theorem}\langlebel{th3-0}
Consider a family of matrices $\{J_{k,h}\}_{k=1}^d$ and $P_h$,
$h\rightarrow 0^+$, as given in (\ref{lin-system}) and
(\ref{matrixP}), respectively. Assume that (\ref{jordan}) holds and
that
\begin{equation}\langlebel{spect-1} \bigcup_{k=1}^d \mbox{\rm
Spect}(J_{k,h}) \subseteq \mathcal{W}(\theta),\qquad (h\rightarrow
0^+)\end{equation} is fulfilled for some $\theta\in[0,\pi/2]$. Let
$L(z,w)$ be a complex rational function satisfying
$$\sup_{z_k \in \mathcal{W}(\theta), \;k=1,2,\ldots,d} |L(z,w)| \le 1, \quad
\mbox{\rm $z$ and $w$ given by (\ref{z-w}).} $$
Then, we have that
$$\begin{array}{c}
\Vert L(\tau J, \tau P)^n \Vert \le C^*, \quad 0\leq
n\tau\leq t^*, \qquad (\tau,h\rightarrow 0^+).\end{array}$$
\end{theorem}
{\bf Proof.} For simplicity of notations, we omit the sub-index $h$
in the matrices. By virtue of (\ref{lin-system}), (\ref{jordan}) and
(\ref{sev-res0}) it follows that
$$
\Vert \left(L(\tau J, \tau P)\right)^n \Vert=
\Vert \Theta \cdot \left(L(\tau \Lambda, \tau
\Upsilon)\right)^n \cdot \Theta^{-1} \Vert \le C
\Vert \left(L (\tau \Lambda, \tau \Upsilon
)\right)^n\Vert, \quad n\geq 1,
$$
where
$$
\begin{array}{l}
\tau \Lambda:=\sum_{k=1}^d \tau \Lambda_k, \quad
\Lambda_k=\mbox{\rm
Block-Diag}(\Lambda^{(1)}_k,\Lambda^{(2)}_k,\ldots,
\Lambda^{(\vartheta)}_k),
\\ \tau \Upsilon := \gamma^{-1}\left(I-\prod_{k=1}^d (I-\gamma \tau
\Lambda_k)\right).
\end{array}
$$
By defining $\tau \Lambda^{(l)}:= \sum_{k=1}^d \tau \Lambda^{(l)}_k$
and $\tau \Upsilon^{(l)}:= \gamma^{-1}\left(I-\prod_{k=1}^d
(I-\gamma \tau \Lambda^{(l)}_k)\right)$, for the norm considered it
follows that
$$
\Vert \left(L(\tau \Lambda, \tau
\Upsilon)\right)^n \Vert = \max_{
l=1,\ldots,\vartheta} \Vert \left(L(\tau
\Lambda^{(l)} , \tau \Upsilon^{(l)})\right)^n
\Vert, \quad n\geq 1.
$$
Consider any diagonal block $\Lambda^{(l)}_k=\langlembda_k^{(l)}I + E$
($E\end{equation}uiv E^{(l)}$ for simplicity of notation. Observe that all the
matrices $E$ are strictly lower triangular and they have uniform
bounded entries and uniform bounded dimensions, hence all of them
are nilpotent with nilpotency index $\le N$) and define
$$z_k= \tau \langlembda^{(l)}_k,\;1\leq k\leq d,\; z=\sum_{k=1}^d z_k,
\; w= \gamma^{-1}\left(1-\prod_{k=1}^d (1-\gamma z_k)\right),$$ it
follows that,
$$ L(\tau \Lambda^{(l)} , \tau \Upsilon^{(l)}) = L\left(\sum_{k=1}^d
(z_k I+\tau E),\gamma^{-1}(I-\prod_{k=1}^d (I-\gamma (z_k I+\tau
E))\right).
$$
By defining the function of $d$ complex
variables,
$$
\psi(w_1,\ldots,w_d):= L\left(\sum_{k=1}^d w_k
,\gamma^{-1}(1-\prod_{k=1}^d (1-\gamma
w_k))\right),
$$
we get that $ L(\tau \Lambda^{(l)} , \tau \Upsilon^{(l)})=\psi(z_1
I+\tau E,\ldots,z_d I+\tau E).$ Then, by using the Taylor expansion
for $\psi$ around $\tau=0$ and taking into the nilpotency of the
matrix $E$,
we
deduce that,
$$ \begin{array}{l}
\psi(z_1 I+\tau E,\ldots,z_d I+\tau E)= \psi(z_1,\ldots,z_d) I +
\\ \qquad \qquad \displaystyle{ \sum_{l=1}^{N-1}
\frac{\tau^l}{l!} E^l \sum_{i_1+i_2+\ldots+i_d=l}
\frac{\partial^{l}\psi }{\partial^{i_1} z_1\ldots\partial^{i_d} z_d}
(z_1,z_2,\ldots,z_d)}.\end{array}
$$
Now, since $L(z,w)\end{equation}uiv L(z_1,\ldots,z_d)$ and all its partial
derivatives up to order $N$ are uniformly bounded on the wedge
$\mathcal{W}(\theta)$, we can write that
$$ \psi(z_1 I+\tau E,\ldots,z_d I+\tau E) = \psi(z_1,\ldots,z_d ) I + \tau L^*_{\tau,h},\quad
\Vert L^*_{\tau,h}\Vert \le C^*, \;(\tau,\:h\rightarrow 0^+).$$
From here we get for $0\le
\tau n \le t^*$ that
$$ \Vert \left(\psi(z_1 I+\tau E,\ldots,z_d I+\tau
E)\right)^n \Vert = \Vert \left (L(z,w) I + \tau
L^*_{\tau,h}\right)^n\Vert \le (1+\tau C^*)^n \le \exp(t^*C^*).$$
$\Box$
\subsection{Some mappings and definitions}
For a given mapping $\zeta(X,Y) \in \mathbb{C}^{m,m}$ where $X$ and
$Y$ are two arbitrary square complex matrices of order $m$ we
define some associated mappings in the following way,
\begin{equation}\langlebel{sev-res7}\begin{array}{c}
\zeta^{[1]}(X,Y):=\left(\zeta(X,Y)-\zeta(X,X)\right)(Y-X)^{-1},
\quad \mbox{\rm
whenever } \det(Y-X)\ne 0,\\
\zeta^{[1]}(X,X):=\lim_{\varepsilonilon\rightarrow 0}
\zeta^{[1]}(X,X+\varepsilonilon I), \quad \mbox{\rm whenever the limit
exists. }
\end{array}
\end{equation}
In a recursive form, when $\det(Y-X)\ne 0$ and $\zeta^{[l]}(X,X)$
exists, we continue by defining
\begin{equation}\langlebel{sev-res8}\begin{array}{c}
\zeta^{[l+1]}(X,Y):=\left(\zeta^{[l]}(X,Y)-\zeta^{[l]}(X,X)\right)(Y-X)^{-1},
\\
\zeta^{[l+1]}(X,X):=\lim_{\varepsilonilon\rightarrow 0}
\zeta^{[l+1]}(X,X+\varepsilonilon I), \quad l=1,2,\ldots,l^*.
\end{array}
\end{equation}
By assuming $\det(Y-X)\ne 0$ and the existence of
$\zeta^{[l]}(X,X),\;l=1,2,\ldots,l^*$, it is straightforward to show
by induction that
\begin{equation}\langlebel{sev-res9}
\zeta(X,Y)=\sum_{l=0}^{l^*}\zeta^{[l]}(X,X)(Y-X)^{l} +
\zeta^{[l^*+1]}(X,Y)(Y-X)^{l^*+1}.
\end{equation}
We have considered for convenience that
$\zeta^{[0]}(X,Y):=\zeta(X,Y).$ It should be noted that the
commutativity of the matrices $X$ and $Y$ is neither necessary in
the definitions above nor in the formula (\ref{sev-res9}).
To have a practical meaning of the mapping $\zeta^{[l]}(X,Y)$ we
show next that assuming $\zeta(x,y)$ has $l^*$ continuous partial
derivatives regarding the second variable, then it holds that
\begin{equation}\langlebel{sev-res10}
\zeta^{[l]}(X,X)=\frac{1}{l!} \frac{\partial^l \zeta(x,y)}{\partial
y^l} (X,X),\quad l=1,2,\ldots,l^*.
\end{equation}
To see (\ref{sev-res10}), we use the induction. For $l=0$ it is
true for convenience. For $l=1$ it is true since
$$ \zeta^{[l]}(X,X)=\lim_{\varepsilonilon \rightarrow 0} \zeta^{[l]}(X,X+\varepsilonilon
I)=\lim_{\varepsilonilon \rightarrow 0} \varepsilonilon^{-1}
\left(\zeta(X,X+\varepsilonilon I)-\zeta(X,X)\right)= \frac{\partial
\zeta}{\partial y} (X,X).$$ Assume it is true up to $l$, we show it
for $l+1$ by using (\ref{sev-res9}) in the second equality and the
induction in the third equality below. The L'Hospital formula for
limits (for the indetermination $0/0$) is used $l+1$ times in the
fourth equality,
$$\begin{array}{rcl}
\zeta^{[l+1]}(X,X)&=&\displaystyle{\lim_{\varepsilonilon\rightarrow 0}
\zeta^{[l+1]}(X,X+\varepsilonilon I)=\lim_{\varepsilonilon\rightarrow
0}\frac{\zeta(X,X+\varepsilonilon
I)-\sum_{j=0}^{l}\zeta^{[j]}(X,X)(\varepsilonilon I)^{j}}{(\varepsilonilon
I)^{l+1}}}\\[0.5pc] &= & \displaystyle{\lim_{\varepsilonilon\rightarrow 0}
\frac{\zeta(X,X+\varepsilonilon I)-\sum_{j=0}^{l}\frac{\varepsilonilon^j}{j!}
\frac{\partial^j \zeta(x,y)}{\partial y^l}(X,X)}{\varepsilonilon^{l+1}}}\\[0.5pc]
& = &\displaystyle{\lim_{\varepsilonilon\rightarrow 0} \frac{1}{(l+1)!}
\frac{\partial^{l+1} \zeta(x,y)}{\partial y^{l+1}}(X,X+\varepsilonilon I) =
\frac{1}{(l+1)!} \frac{\partial^{l+1} \zeta}{\partial y^{l+1}}(X,X)}
\end{array}$$
These results can be trivially extended to vectors (and matrices),
namely $(\zeta_{ij}(X,Y))\in \mathbb{C}^{q_1m,q_2m}$, by applying
them to each component $\zeta_{ij}(X,Y)\in \mathbb{C}^{m,m}$.
Sometimes we will make use of this kind of vectors as we will see in
the next section.
\subsection{Bounds for the local errors}
The forthcoming convergence results for {\sf AMF$_q$-RK} methods
are based in the Lemma II.2.3 \cite[p. 162]{HV},
which can be stated as follows
\begin{lemma}\langlebel{sev-lema-glob-err}
Assume that the global errors $\varepsilonilon_n\end{equation}uiv\varepsilonilon_n(\tau;h)$,
of a one-step method satisfy the recursion (\ref{global-errors1}),
where the local errors $l_n$ can be split (uniformly on $h$ and
$\tau$) as
\begin{equation}\langlebel{ln-domi}
l_n=\left(R_q(\tau J,\tau P)-I\right) \phi(t_n)\tau^\mu h^\alpha +
\tau \mathcal{O}(\tau^\nu h^\beta),\quad n=0,1,\ldots, t^*/\tau-1,
\end{equation}
where the function $\phi(t)$ and its first derivative regarding $t$
are uniformly bounded, then the stability condition
\begin{equation}\langlebel{sev-estab}
\sup_{1\le n\le t^*/\tau \atop \tau\rightarrow 0^+, \;h\rightarrow
0^+} \Vert R_q(\tau J,\tau P)^n \Vert\le C,
\end{equation}
implies that the global errors uniformly fulfil
\begin{equation}\langlebel{sev-global-err} \varepsilonilon_{n} = \mathcal{O}(\tau^\mu h^\alpha) +
\mathcal{O}(\tau^\nu h^\beta),\quad n=1,\ldots, t^*/\tau, \quad
\tau\rightarrow 0^+, h\rightarrow 0^+.\end{equation}
\end{lemma}
{\sf General Assumptions on the semilinear problem.}
{\it To bound the local errors and consequently the global errors
we henceforth assume that the exact PDE solution $u_h(t)$ confined
to the spatial grid and the semilinear problem (\ref{lin-system})
fulfil (\ref{spatialerrors})-(\ref{norm}),
(\ref{jordan}) and (\ref{spect-1}) for some $\theta\in [0,\pi/2]$,
and that the following hypotheses (related the matrices $J$ and $P$)
hold for some constants (not necessarily positive) $\alpha_l,\;
\beta_l $ and $\eta$ and some nonnegative integer $l^*$, whenever
$h\rightarrow 0^+$ and $\tau\rightarrow 0^+$,
\begin{equation}\langlebel{P-hypo}
\begin{array}{rcl} {\bf (P1)} & & \left\{ \begin{array}{l}
(P-J)^lu_h^{(k)}(t)=\tau^l h^{\alpha_l} \:
\mathcal{O}(1), \\
(P-J)^{l+1} u_h^{(k)}(t)=\tau^{l+1} h^{\beta_{l+1}} J \:
\mathcal{O}(1) \end{array} \right\} \quad {l=0,1,\ldots,l^* \atop
k=1,2,\ldots,p+1.} \\[0.5pc]
{\bf (P2)} & & J^\eta u_h^{(k)}(t)=\mathcal{O}(1), \quad
k=1,2,\ldots,p+1, \;\mbox{\rm for some } \eta.
\end{array}
\end{equation}
It should be noticed that always $\alpha_0=0$, because the
derivatives (up to some order) of the exact solution are uniformly
bounded, i.e. $u_h^{(k)}(t)=\mathcal{O}(1), \;t\in
[0,t^*],\;k=0,1,\ldots,p+1$.}
\begin{theorem}\langlebel{sev-th-1} Assume that the
Runge-Kutta method has stage order $\ell$ and that
\begin{equation}\langlebel{sev-equ-1}\sup_{z_k\in \mathcal{W}(\theta),\atop k=1,2,\ldots,d} |z/(R_q(z,w)-1)|\le C, \quad z\;\mbox{\rm and } w \; \mbox{\rm
given by } (\ref{z-w}). \end{equation} Then for the {\sf AMF$_q$-RK}
method we have that,
$$l_n^{[1]}= \mathcal{O}(\tau h^r) + \mathcal{O}(\tau^{\ell+1}),
\qquad (\tau\rightarrow 0, \; h\rightarrow 0),$$ and
$$l_n^{[1]}= \tau h^r\mathcal{O}(1) + \tau^{\ell+1}(R_q(\tau J, \tau P)-I)\left(\mathcal{O}(1) +
\tau h^{\beta_1} \mathcal{O}(1)\right), \; \tau\rightarrow 0, \;
h\rightarrow 0.
$$
\end{theorem}
{\bf Proof.} According to Lemma \ref{lem-loc-err} the term
$l_n^{[1]}$ of the local error is given by,
\begin{equation}\langlebel{sev-res11}
l_n^{[1]}= \xi(\tau J, \tau P) \hat{D}_n + \delta_n,
\end{equation}
where
\begin{equation}\langlebel{sev-res12}
\xi(\tau J, \tau P):=(\ss^T\otimes I)\left( Q_q(\tau J, \tau P) +
\sum_{j=q}^1 (\prod_{i=q}^{j} M_i(\tau J, \tau P))Q_{j-1}(\tau J,
\tau P)-\prod_{i=q}^{1} M_i(\tau J, \tau P) \right)
\end{equation}
From Remark \ref{sev-remark-0} we have that ($e_j$ denotes the
$j$-vector of the canonical basis)
$$ \sup_{\mbox {\tiny Re } z_k \le 0\atop k=1,\ldots,d}|\xi(z,w)e_j|\le C, \quad (j=1,\ldots,s),\quad
z,\; w \;\mbox{\rm given by (\ref{z-w})}. $$ From Theorem
\ref{th3-0} this implies that
$$ \max_{j=1,\ldots,s} \Vert \xi(\tau J, \tau P)(e_j\otimes I) \Vert \le C', \quad
\tau \rightarrow 0^+, \quad h \rightarrow 0^+.$$ Then, from
(\ref{resid-2}) in Lemma \ref{lem-loc-err} the first bound for
$l_n^{[1]}$ follows.
For the second bound, we separate in (\ref{sev-res11}) the
$\tau^{\ell+1}$-term from the others, take into account
(\ref{sev-res12}) and Lemma \ref{lem-loc-err}, we get
\begin{equation}\langlebel{sev-res13}\begin{array}{rcl}
l_n^{[1]}&=& \frac{\tau^{\ell+1}}{(\ell+1)!} \left(\xi(\tau J, \tau
P)\left((c^{\ell+1}-(\ell+1)Ac^\ell)\otimes I \right) + (1-\ss^T
c^{\ell+1}) I
\right) u_h^{(\ell+1)}(t_n) + \circledR,\\[0.3pc]
\mbox{\rm where }& &\circledR= \mathcal{O}(\tau^{\ell+2}) +
\mathcal{O}(\tau h^r).
\end{array}
\end{equation}
Next, we define the mapping (assume that $J$ is regular only to
simplify the proof)
\begin{equation}\langlebel{sev-res14}
\upsilon(\tau J, \tau P):=(R_q(\tau J, \tau P) - I)^{-1}\left(
\xi(\tau J, \tau P)\left((c^{\ell+1}-(\ell+1)Ac^\ell)\otimes
I\right) + (1-\ss^T c^{\ell+1}) I\right).
\end{equation}
By using the assumption (\ref{sev-equ-1}), the bounds in Remark
\ref{sev-remark-0} and Lemma \ref{sev-lema-0}, it is not very
difficult to see that
\begin{equation}\langlebel{sev-res14a} \begin{array}{c}
\displaystyle{\sup_ {z \in \mathcal{W}(\theta)}|\upsilon(z, z)|<
+\infty. \quad \sup_{z_k \in \mathcal{W}(\theta) \atop
k=1,2,\ldots,d }|z\upsilon^{[1]}(z, w)|< +\infty, \; \mbox{\rm $z$
and $w$ given by (\ref{z-w}).}}\end{array}
\end{equation} Then, from (\ref{sev-res13}) it follows that,
\begin{equation}\langlebel{sev-res15}\begin{array}{rcl}
l_n^{[1]}&=& \frac{\tau^{\ell+1}}{(\ell+1)!} \left(R(\tau J, \tau
P)-I\right)\upsilon(\tau J, \tau P) u_h^{(\ell+1)}(t_n) +
\circledR \\[0.3pc] &=& \frac{\tau^{\ell+1}}{(\ell+1)!} \left(R(\tau J, \tau
P)-I\right)\left(\upsilon(\tau J, \tau J)+ \upsilon^{[1]}(\tau J,
\tau P)(\tau P-\tau J)\right)u_h^{(\ell+1)}(t_n) +
\circledR \\[0.3pc] &=& \circledR + \frac{\tau^{\ell+1}}{(\ell+1)!} \left(R(\tau J, \tau
P)-I\right)\upsilon(\tau J, \tau J)u_h^{(\ell+1)}(t_n) \\[0.3pc] &+& \frac{\tau^{\ell+1}}
{(\ell+1)!} \left(R(\tau J, \tau
P)-I\right)\upsilon^{[1]}(\tau J, \tau P)(\tau J)(J^{-1}(P-J))
u_h^{(\ell+1)}(t_n)\\[0.3pc] &=& \circledR + \frac{\tau^{\ell+1}}{(\ell+1)!}
\left(R(\tau J, \tau P)-I\right) \mathcal{O}(1) +
\frac{\tau^{\ell+1}}{(\ell+1)!}\left(R(\tau J, \tau P)-I\right)
\mathcal{O}(\tau h^{\beta_1})\quad \mbox{\rm
$\Box$}
\end{array}
\end{equation}
For the analysis of the local error term $l_n^{[2]}$ in
(\ref{local-errors1}), we define the mappings
\begin{equation} \begin{array}{lll}\langlebel{H-eq}
\psi_{q}(\tau J,\tau X)&:=& (\ss^T\otimes I)\prod_{j=q}^1 M_j(\tau
J,\tau X) \in \mathbb{C}^{m,sm}, \\[0.3pc] \zeta_{q}(\tau J,\tau X)&:=& \left(R_q(\tau J,\tau
X)-I\right)^{-1} \psi_{q}(\tau J,\tau X) \in \mathbb{C}^{m,sm},
\end{array}
\end{equation}
and their associated vector complex functions
\begin{equation}
\begin{array}{lll}\langlebel{H-eq1} \psi_{q}(z,w)&:=&
\ss^T\prod_{j=q}^1 (I-wT_j)^{-1}(zA-wT_j) \in \mathbb{C}^{1,s}, \\[0.3pc]
\zeta_{q}(z,w)&:=& \left(R_q(z,w)-1\right)^{-1} \psi_{q}(z,w) \in
\mathbb{C}^{1,s}.
\end{array}
\end{equation}
These mappings will play a mayor role in the proof of the
convergence results. It must be remarked that whereas $\Vert
\psi_{q}(z,w)\Vert_2$ is uniformly bounded when $z$ and $w$ are
given by (\ref{z-w}), the vector
$\zeta_{q}(z,w)=\mathcal{O}(z^{-1})$ as $z\rightarrow 0$ due to the
fact that (see (\ref{estab-1}))
\begin{equation}\langlebel{sev-eqq2}
R_q(z,w)-1= \ss^T\left(Q_q(z,w) + \sum_{j=q}^1 (\prod_{i=q}^{j}
M_i(z,w))Q_{j-1}(z,w)-\prod_{i=q}^{1} M_i(z,w)\right) c z.
\end{equation}
Hence $\zeta_{q}(z,w)$ is not bounded in general for $z$ and $w$
given by (\ref{z-w}). However, $\zeta_{q}(z,z)$ is uniformly
bounded as long as $R_q(z,z)-1\ne 0$ for $z\in \mathcal{W}(\theta)
\backslash \{0\}$.
From (\ref{local-errors1}), by using (\ref{sev-res9}), we
deduce that,
\begin{equation}\langlebel{sev-res16} \begin{array}{rcl}l_n^{[2]}&=& (R_q(\tau J, \tau P)-I)
\sum_{j=0}^{l^*} \zeta_q^{[j]}(\tau J,
\tau J) (I_s\otimes (\tau(P-J))^j)\Delta_h(t_n)\\[0.3pc] & + & (R_q(\tau J, \tau
P)-I) \zeta_q^{[l^*+1]}(\tau J, \tau P)(I_s\otimes
(\tau(P-J))^{l^*+1})\Delta_h(t_n).\end{array}
\end{equation}
Next, we provide some convergence results for different
kind of {\sf AMF$q$-RK} methods, which depends on the Runge-Kutta
method on which the {\sf AMF$_q$-RK} is based on. We start with
Theorem \ref{sev-th-2} that meets applications for DIRK methods
(Diagonally Implicit Runge-Kutta) and SIRK methods (Single Implicit
Runge-Kutta) and then with Theorems \ref{sev-th-3}, \ref{sev-th-4}
and \ref{sev-th-5} which meet applications in the {\sf AMF$q$-Rad}
methods presented in section two. Of course, the assumptions {\bf
(P1)-(P2)} will be always assumed for some integers $l^*\ge 0$,
$\ell\ge 1, \:p\ge 1$.
\begin{theorem}\langlebel{sev-th-2} If $T_\nu=A, \;\nu=1,\ldots,q$, with the Runge-Kutta coefficient matrix $A$
having unique eigenvalue $\gamma>0$ (with multiplicity $s$), then
the local errors ($l_n=l_n^{[1]}+ l_n^{[2]}$) fulfil
$$\left. \begin{array}{rcl}
l_n^{[1]}&=& \mathcal{O}(\tau h^r) + \tau^{\ell+1}(R(\tau J,\tau
P)-I) (\mathcal{O}(1)+ \mathcal{O}(\tau h^{\beta_1})),
\\
l_n^{[2]}&=& \tau^{2l+2}h^{\beta_{l+1}}(R(\tau J,\tau P)-I)
\mathcal{O}(1),\; l=0,1,\ldots,\tilde{l}, \\
\tilde{l}&=&\max\{0,\min\{q-2,l^*\}\}.
\end{array} \right\} \; (\tau\rightarrow 0^+, \; h\rightarrow 0^+). $$
If the method is A$(\theta)$-stable for a $d$-splitting and
(\ref{spect-1}) holds, then for any $l=0,1,\ldots,\tilde{l}$,
the global errors fulfil (whenever $\tau\rightarrow 0^+$ and $h\rightarrow 0^+$) that,
$$\varepsilonilon_{n,h} = \mathcal{O}( h^r)+
\tau^{\ell}\min\{1,\max\{\tau,\tau^2 h^{\beta_1}\}\} \mathcal{O}(1)
+\mathcal{O}(\tau^{2l+2} h^{\beta_{l+1}}); \;
n=1,2,\ldots,t^*/\tau.$$
\end{theorem}
{\bf Proof.} The expression of $l_n^{[1]}$ was seen in Theorem
\ref{sev-th-1}. In order to show the expression for $l_n^{[2]}$, we
start by deducing from (\ref{H-eq}) and (\ref{estab-1}) that
\begin{equation}\langlebel{sev-eqq-1} \begin{array}{rcl}
\zeta_q(z,w)&=&(R_q(z,w)-1)^{-1} \ss^T\left((I-wA)^{-1}A\right)^q
(z-w)^q,\\[0.3pc]
R_q(z,w)-1&=&\ss^T\left(\left((I-wA)^{-1}A\right)^q
(z-w)^q-I\right)(zA-I)c z.
\end{array}
\end{equation}
From (\ref{sev-res10}) we have that
$\displaystyle{\zeta_q^{[l]}(z,z)=\frac{1}{l!}\frac{\partial^l
\zeta_q}{\partial w^l}}(z,z)$. From here and from (\ref{sev-eqq-1})
it follows that
$$\zeta_q^{[l]}(z,z)=0,\quad l=0,1,\ldots,\tilde{l}.$$
From (\ref{sev-res16}) by taking $\tilde{l}$ as upper index, for any
$l=0,1,\ldots,\tilde{l}$, we have that
$$ \begin{array}{rcl}
\begin{array}{rcl}l_n^{[2]}&=& (R_q(\tau J, \tau
P)-I) \zeta_q^{[l+1]}(\tau J, \tau P)(I_s\otimes
(\tau(P-J))^{l+1}\Delta_h(t_n) \\[0.3pc] &=& \tau^{l}(R_q(\tau J, \tau
P)-I) \left( \zeta_q^{[l+1]}(\tau J, \tau P)(I_s\otimes \tau
J)\right)(I_s\otimes J^{-1}(P-J)^{l+1})
\Delta_h(t_n) \\[0.3pc] &=& \tau^{l}(R(\tau J,\tau P)-I)
\mathcal{O}(1) (I_s\otimes J^{-1}(P-J)^{l+1})(\tau c\otimes
u_h'(t_n) + \tau^2\mathcal{O}(1))\\[0.3pc]
&=&\tau^{2l+2}h^{\beta_{l+1}}(R(\tau J,\tau P)-I) \mathcal{O}(1).
\end{array}
\end{array} $$
To see the bound for the global errors we apply Lemma
\ref{sev-lema-glob-err}. The bounds for the local errors $l_n$ have
been obtained above (see also Theorem \ref{sev-th-1} for
$l_n^{[1]}$). The boundedness of the powers of $R_q(\tau J, \tau P)$
as indicated in (\ref{sev-estab}) follows from Theorem \ref{th3-0}
by taking into account the A$(\theta)$-stability of the method for
the $d$-splitting and that (\ref{spect-1}) holds. Now from Lemma
\ref{sev-lema-glob-err} the proof is accomplished.
$\Box$
\begin{theorem}\langlebel{sev-th-3} For AMF$_q$-RK methods with $\gamma>0$ and
satisfying $(A-T_1)c=0$, we have that
$$
l_n^{[2]}= \tau^{2}(R(\tau J,\tau P)-I)\left( \mathcal{O}(1) +
h^{\beta_1}\mathcal{O}(1)\right), \; (\tau\rightarrow 0^+, \;
h\rightarrow 0^+). $$ Additionally if the method is
A$(\theta)$-stable for a $d$-splitting and (\ref{spect-1}) holds,
then for $\tau\rightarrow 0^+$ and $h\rightarrow 0^+$, the global
errors fulfil
$$\varepsilonilon_{n,h} = \mathcal{O}(
h^r)+ \tau^{\ell}\min\{1,\max\{\tau,\tau^2 h^{\beta_1}\}\}
\mathcal{O}(1) +\tau^{2}\left(\mathcal{O}(1)+
h^{\beta_1}\mathcal{O}(1) \right); \; n=1,2,\ldots,t^*/\tau.$$
\end{theorem}
{\bf Proof.} The expression of $l_n^{[1]}$ was seen in Theorem
\ref{sev-th-1}. In order to show the expression for $l_n^{[2]}$,
from (\ref{sev-res16}) by setting $l^*=0$ we get that (observe that
$\zeta_q(z,z)c=0$ because $(A-T_1)c=0$. This expression is used in
the third equality below)
$$ \begin{array}{rcl}
\begin{array}{rcl}l_n^{[2]}&=& (R_q(\tau J, \tau
P)-I) \zeta_q(\tau J, \tau J)\Delta_h(t_n) \\[0.3pc] &+&(R_q(\tau J, \tau
P)-I) \zeta_q^{[1]}(\tau J, \tau P)(I_s\otimes
(\tau(P-J)))\Delta_h(t_n) \\[0.3pc] &=&
(R_q(\tau J, \tau
P)-I) \zeta_q(\tau J, \tau J)\left(\tau c \otimes I + \tau^2 \mathcal{O}(1)\right) \\[0.3pc] &+&
(R_q(\tau J, \tau P)-I) \left(\zeta_q^{[1]}(\tau J, \tau P)(I\otimes
\tau J)\right)(I_s\otimes
(J^{-1}(P-J)))\left(\tau \mathcal{O}(1)\right) \\[0.3pc] &=&
(R(\tau J,\tau P)-I) \left(\tau^{2} \mathcal{O}(1)\right) + (R(\tau
J,\tau P)-I)
\mathcal{O}(1)\:\left(\tau^{2}h^{\beta_1}\mathcal{O}(1)\right).
\end{array}
\end{array} $$
This provides the bound for the local errors $l_n^{[2]}$. The
boundedness of the powers of $R_q(\tau J, \tau P)$ as indicated in
(\ref{sev-estab}) follows from Theorem \ref{th3-0} by taking into
account the A$(\theta)$-stability of the method for the
$d$-splitting and that (\ref{spect-1}) holds. Now, from the bounds
for the local error and from Lemma \ref{sev-lema-glob-err} the
proof follows.
$\Box$
\begin{theorem}\langlebel{sev-th-4} For AMF$_q$-RK methods with $\gamma>0$ and
satisfying $$\sup_{\mbox{\tiny Re$\:z$} \:\le\: 0, \:z\ne 0}\Vert
z^{-\eta} \zeta_q(z,z)\Vert_2 < +\infty,$$ with $\eta$ given in
{\bf (P2)} we have that
$$
l_n^{[2]}= (R(\tau J,\tau P)-I)\left( \mathcal{O}( \tau^{1+\eta}) +
\mathcal{O}( \tau^2 h^{\beta_1})\right), \; (\tau\rightarrow 0^+, \;
h\rightarrow 0^+). $$ Additionally if the method is
A$(\theta)$-stable for a $d$-splitting and (\ref{spect-1}) holds,
then for $\tau\rightarrow 0^+$ and $h\rightarrow 0^+$, the global
errors fulfil
$$\varepsilonilon_{n,h} = \mathcal{O}(
h^r)+ \min\{1,\max\{\tau,\tau^2 h^{\beta_1}\}\} \mathcal{O}(
\tau^{\ell}) + \mathcal{O}(\tau^{1+\eta})+ \mathcal{O}(\tau^2
h^{\beta_1}); \; n=1,2,\ldots,t^*/\tau.$$
\end{theorem}
{\bf Proof.} The expression of $l_n^{[1]}$ was seen in Theorem
\ref{sev-th-1}. In order to show the expression for $l_n^{[2]}$,
from (\ref{sev-res16}) by setting $l^*=0$ we get that
$$ \begin{array}{rcl}
\begin{array}{rcl}l_n^{[2]}&=& (R_q(\tau J, \tau
P)-I) \zeta_q(\tau J, \tau J)\Delta_h(t_n) \\[0.3pc] &+&(R_q(\tau J, \tau
P)-I) \zeta_q^{[1]}(\tau J, \tau P)(I_s\otimes
(\tau(P-J)))\Delta_h(t_n) \\[0.3pc] &=&
(R_q(\tau J, \tau
P)-I) \zeta_q(\tau J, \tau J)\left(\tau \mathcal{O}(1)\right) \\[0.3pc] &+&
(R_q(\tau J, \tau P)-I) \left(\zeta_q^{[1]}(\tau J, \tau P)(I\otimes
\tau J)\right)(I_s\otimes
(J^{-1}(P-J)))\left(\tau \mathcal{O}(1)\right) \\[0.3pc] &=&
(R_q(\tau J, \tau P)-I) \left(\zeta_q(\tau J, \tau J)(I\otimes (\tau
J)^{-\eta})\right)
\left(I\otimes (\tau J)^{\eta}\right)\left(\tau \mathcal{O}(1)\right) \\[0.3pc] &+&
(R(\tau J,\tau P)-I)
\mathcal{O}(1)\:\left(\tau^{2}h^{\beta_1}\mathcal{O}(1)\right)\\[0.3pc] &=&
R_q(\tau J, \tau P)-I) \left(\mathcal{O}(1) \right)
\left(\tau^{\eta+1} I\otimes J^{\eta} \mathcal{O}(1)\right) \\[0.3pc] &+&
(R(\tau J,\tau P)-I)
\left(\tau^{2}h^{\beta_1}\mathcal{O}(1)\right)\\[0.3pc] &=&
(R(\tau J,\tau P)-I) \left( \mathcal{O}(\tau^{1+\eta}) +
\mathcal{O}(\tau^{2}h^{\beta_1})\right).
\end{array}
\end{array} $$
This provides the bound for the local errors $l_n^{[2]}$. The rest
of the proof follows as in the previous theorems.
$\Box$
\begin{theorem}\langlebel{sev-th-5} For AMF$_q$-RK methods with $\gamma>0$ and
$$\begin{array}{c} (A-T_1)c=0,\quad \sup_{\mbox{\tiny Re$\:z$} \:\le\: 0, \:z\ne 0}\Vert z^{-\eta}
\zeta_q(z,z)\Vert_2 < +\infty,\end{array}$$ with $\eta$ given in
{\bf (P2)} and assuming {\bf (P1)} for $l^*=1$, we have that
$$
l_n^{[2]}= (R(\tau J,\tau P)-I)\left( \mathcal{O}( \tau^{2+\eta}) +
\mathcal{O}( \tau^3 h^{\alpha_1}) + \mathcal{O}( \tau^4
h^{\beta_2})\right), \; (\tau\rightarrow 0^+, \; h\rightarrow 0^+).
$$ Additionally if the method is A$(\theta)$-stable for a
$d$-splitting and (\ref{spect-1}) holds, then the global errors
fulfil
$$\begin{array}{c} \varepsilonilon_{n,h} = \mathcal{O}(
h^r)+ \min\{1,\max\{\tau,\tau^2 h^{\beta_1}\}\} \mathcal{O}(
\tau^{\ell}) + \mathcal{O}(\tau^{2+\eta})+ \mathcal{O}(\tau^3
h^{\alpha_1})+ \mathcal{O}(\tau^4 h^{\beta_2}),\\
n=1,2,\ldots,t^*/\tau,\qquad (\tau\rightarrow 0^+,\; h\rightarrow
0^+).\end{array}$$
\end{theorem}
{\bf Proof.} In order to show the expression for $l_n^{[2]}$, from
(\ref{sev-res16}) by setting $l^*=1$ we get that
$$ \begin{array}{rcl}
\begin{array}{rcl}l_n^{[2]}&=& (R_q(\tau J, \tau
P)-I)\left( \zeta_q(\tau J, \tau J)+ \zeta^{[1]}_q(\tau J, \tau J)(I\otimes \tau(P-J))\right)
\Delta_h(t_n) \\[0.3pc] &+&(R_q(\tau J, \tau
P)-I) \zeta_q^{[2]}(\tau J, \tau P)(I_s\otimes
\tau^2(P-J)^2)\Delta_h(t_n) \\[0.3pc] &=&
(R_q(\tau J, \tau P)-I)\left( \zeta_q(\tau J, \tau J)+
\zeta^{[1]}_q(\tau J, \tau J)(I\otimes \tau(P-J))\right)
\left((\tau c\otimes I)u_h'(t_n)+ \tau^2\mathcal{O}(1)\right)\\[0.3pc] &+&(R_q(\tau J, \tau
P)-I) \zeta_q^{[2]}(\tau J, \tau P)(I_s\otimes
\tau^2(P-J)^2)(\tau \mathcal{O}(1)) \\[0.3pc] &=&
(R_q(\tau J, \tau P)-I)\left( \tau^2 \zeta_q(\tau J, \tau
J)\mathcal{O}(1)+ \tau \zeta^{[1]}_q(\tau J, \tau J)(I\otimes
\tau(P-J)\mathcal{O}(1))\right)
\\[0.3pc] &+&(R_q(\tau J, \tau
P)-I) \left(\zeta_q^{[2]}(\tau J, \tau P)(I_s\otimes \tau
J)\right)(I_s\otimes \tau J^{-1}(P-J)^2)(\tau \mathcal{O}(1))
\\[0.3pc] &=&
(R_q(\tau J, \tau P)-I)\left(\tau^2\left(\zeta_q(\tau J, \tau
J)(\tau J)^{-\eta}\right)(\tau^\eta J^\eta \mathcal{O}(1))+
\mathcal{O}(\tau^3 h^{\alpha_1})\right)
\\[0.3pc] &+&
(R_q(\tau J, \tau P)-I) \left(\mathcal{O}(1)\: \tau^2 J^{-1}(P-J)^2
\mathcal{O}(1)\right)
\\[0.3pc] &=&
(R(\tau J,\tau P)-I)\left( \mathcal{O}( \tau^{2+\eta}) +
\mathcal{O}( \tau^3 h^{\alpha_1}) + \mathcal{O}( \tau^4
h^{\beta_2})\right).
\end{array}
\end{array} $$
This provides the bound for the local errors $l_n^{[2]}$. The rest
of the proof follows as in the previous theorems.
$\Box$
\section{Application of the convergence results for Dirichlet Boundary Conditions in parabolic problems}
Let us next consider the $2D$ semi-linear diffusion-reaction model
($\varepsilon$ is a positive constant)
\begin{equation}\langlebel{2D-dif-reac}
u_t=\varepsilon(u_{xx}+u_{yy}) + g(x,y,t),\; (x,y)\in
(0,1)^2,\,t\in[0,1],\;\varepsilon>0,
\end{equation}
with prescribed Dirichlet boundary conditions and an initial
condition. The PDE is discretized on uniform spatial meshes
$(x_i,y_j)=(ih,jh)$, $h= N^{-1}$, $1\le i,j\le N-1$, where $N-1$ is
the number of interior grid-points for each spatial variable. We
shall assume that the exact solution of the PDE (\ref{2D-dif-reac})
is regular enough when $(x,y,t)\in [0,1]^2\times [0,t^*]$.
Let us denote
$u_h(t):=(u_{i,j}(t))_{i,j=1}^{N-1}$ with a row-wise ordering, where
$u_{i,j}(t):=u(x_i,y_j,t)$ for $0\leq i,j\leq N$. Then, by using
second-order central differences, we obtain for the exact solution
of (\ref{2D-dif-reac}) on the grid a semi-discrete system
(\ref{pde}) with dimension $m=(N-1)^2$
\begin{equation}\langlebel{semilin-exact-2D}
u_h'(t)=\varepsilon J u_h(t)+g_h(t)+\sigma_h(t)+\varepsilon
h^{-2}u_{\Gamma_h}(t),
\end{equation}
where
\begin{equation}\langlebel{J-2D}\begin{array}{c}
J:=J_1+J_2, \;\;J_1=I_{N-1}\otimes B_{N-1},\;\; J_2=B_{N-1}\otimes
I_{N-1},\\ B_{N-1}=h^{-2}TriDiag(1,-2,1)\in
\mathbb{R}^{(N-1)\times(N-1)},\quad h=1/N. \end{array}
\end{equation}
Moreover, $g_h(t)=(g(x_i,y_j,t))_{i,j=1}^{N-1}$, $\Vert \sigma_h(t)
\Vert_{2,h} =\mathcal{O}(h^2)$ ($0 \le t \le t^*$), whereas
$u_{\Gamma_h}(t)$ contains the values of the exact solution on the
boundary, i.e.,
\begin{equation}\langlebel{uh-boundary}
u_{\Gamma_h}(t)=u_h^{(0,y)}(t)\otimes e_1+u_h^{(1,y)}(t)\otimes
e_{N-1}+e_1\otimes u_h^{(x,0)}(t)+e_{N-1}\otimes u_h^{(x,1)}(t),
\end{equation}
with $u_h^{(0,y)}(t)=(u_{0,j}(t))_{j=1}^{N-1}$,
$u_h^{(1,y)}(t)=(u_{N,j}(t))_{j=1}^{N-1}$,
$u_h^{(x,0)}(t)=(u_{i,0}(t))_{i=1}^{N-1}$ and
$u_h^{(x,1)}(t)=(u_{i,N}(t))_{i=1}^{N-1}$. Above,
$\{e_1,\ldots,e_{N-1}\}$ denotes the canonical
basis in $\mathbb{R}^{N-1}$.
For the proof of the convergence results we need the lemma
\ref{lema-valP2} and the lemma \ref{lema-Pgeneral} given below.
These lemmas can be derived from the material in \cite[pp.
96-300]{HV} (see from Lemma 6.1 to Lemma 6.5). Lemmas
\ref{lema-valP2} and \ref{lema-Pgeneral} supply sharp values for the
constants $\alpha_l,\;\beta_l$ and $\eta$ appearing in the {\bf
P}-assumptions of section 3. These constants together with the
convergence theorems provide specific orders of convergence of the
MoL approach for several AMF$_q$-RK methods, in particular for the
{\sf AMF$_q$-Rad} methods presented in section 2.
The norm considered here for vectors, is the weighed Euclidean norm
$$ \displaystyle{\Vert (v_{ij})_{i,j=1}^{N-1}\Vert_{2,h}
:=\sqrt{\frac{1}{N^2}\sum_{i,j=1}^{N-1}|v_{ij}|^2}=h\Vert
(v_{ij})_{ij=1}^{N-1}\Vert_2, }$$ and for matrices the corresponding
operator norm.
\begin{lemma}\langlebel{lema-valP2}
Assume that exact solution $u(x,y,t)$ of the 2D-PDE problem
(\ref{2D-dif-reac}) has as many continuous partial derivatives as
needed in the analysis in $(x,y,t)\in [0,1]^2\times [0,t^*]$. Then
for $ k=1,2\ldots$ and $\omega<\frac{1}{4}$ we have that,
$$\begin{array}{rcl}\norm{J^{\omega}u_h^{(k)}(t)}_{2,h}&=&\mathcal{O}(1),
\quad \mbox{\rm and moreover } \\
\norm{J^{1+\omega}u_h^{(k)}(t)}_{2,h}&=&\mathcal{O}(1), \; \mbox{\rm
whenever } u^{(1)}_{\Gamma_h}(t)\end{equation}uiv 0.
\end{array}$$
\end{lemma}
\begin{lemma}\langlebel{lema-Pgeneral}
Assume that exact solution $u(x,y,t)$ of the 2D-PDE problem
(\ref{2D-dif-reac}) has as many continuous partial derivatives as
needed in the analysis in $(x,y,t)\in [0,1]^2\times [0,t^*]$.
Then, for $l=0,1,...$ we have that,
\begin{equation}\langlebel{P-hypo-gen}
\norm{(P-J)^l u_h^{(k)}(t)}_{2,h}=\mathcal{O}(\tau^l
h^{\alpha_l}),\qquad \norm{J^{-1}(P-J)^l
u_h^{(k)}(t)}_{2,h}=\mathcal{O}(\tau^l h^{\beta_l}),
\end{equation}
where
\begin{equation}\langlebel{P-hypo-alpha}
\alpha_l= \left\{\begin{array}{ll}-\max\{0,3+4(l-2)\}, &\quad {\rm
if} \; u^{(1)}_{\Gamma_h}(t)\end{equation}uiv 0,\\-\max\{0,3+4(l-1)\}, & \quad
{\rm otherwise},\end{array}\right.
\end{equation}
and
\begin{equation}\langlebel{P-hypo-beta}
\beta_l= \left\{\begin{array}{ll}-\max\{0,1+4(l-2)\}, &\quad {\rm
if} \; u^{(1)}_{\Gamma_h}(t)\end{equation}uiv 0,\\-\max\{0,1+4(l-1)\}, & \quad
{\rm otherwise}.\end{array}\right.
\end{equation}
\end{lemma}
$\Box$
We next give a convergence theorem for 2D-parabolic PDEs when the
MoL approach with {\sf AMF$_q$-Rad} methods in section 2 are
applied to the time discretization. The results still hold for
3D-parabolic problems (even for $d$D-parabolic problems and $d\ge
3$) and Time-Independent Dirichlet boundary conditions, but the
proof requires some extra length to be included here.
\begin{theorem}\langlebel{sev-th-7} The global errors (GE) in the weighted Euclidean norm of the MoL
approach for the 2D-PDE (\ref{2D-dif-reac}) when the spatial
semi-discretization is carried out with second order central
differences and the time integration is performed with AMF$_q$-RK
methods, are given in Table \ref{estimates1D}. There,
$\varrho=\min\{1,\tau^2h^{-1}\}$ and $\mathcal{O}(\tau^{2.25^*})$ is
meant for $\mathcal{O}(\tau^{\mu})$ where $\mu<2.25$ is any
constant.
\begin{table}[h!]
\centering
\begin{tabular}{|c|c|c|} \hline
$(\tau\rightarrow 0^+,\:h\rightarrow 0^+)$ & GE (Time-Indep.) & GE (Time-Dep.) \\[0.2pc]
\hline
{\sf AMF$_1$-Rad} & $\mathcal{O}(h^2)+\mathcal{O}(\tau^2)$ &
$\mathcal{O}(h^2) + \mathcal{O}(\varrho)$ \\[0.2pc]\hline {\sf AMF$_2$-Rad} &
$\mathcal{O}(h^2)+\mathcal{O}(\tau^3)+\tau^2\mathcal{O}(\varrho)$ &
$\mathcal{O}(h^2) + \mathcal{O}(\varrho)$ \\[0.2pc]\hline {\sf AMF$_3$-Rad} &
$\mathcal{O}(h^2)+\mathcal{O}(\tau^{2.25^*})$ &
$\mathcal{O}(h^2)+\mathcal{O}(\varrho)$ \\[0.2pc]
\hline
\end{tabular}\caption{\scriptsize Global error estimates in the weighted Euclidean norm
for Time-Dependent Dirichlet boundary conditions (in short
Time-Dep.) and Time-Independent Dirichlet boundary conditions (in
short Time-Indep.).} \langlebel{estimates1D}
\end{table}
\end{theorem}
{\bf Proof.} In all cases we have that the stage order of the
underlying Runge-Kutta Radau IIA method is $\ell=2$ and the order
of the spatial semi-discretization is $r=2$. Moreover, all the
three methods {\sf AMF$_q$-Rad} ($q=1,\:2,\:3$) are
A($\pi/2$)-stable for a 2-splitting as it is shown in
\cite{sevedom-AMFestab} for the cases $q=1$ and $q=2$ and in
\cite{apnum-sevsole10} for the case $q=3$. Also, it should be
noticed that (\ref{spect-1}) holds.
We start with the {\sf AMF$_1$-Rad} method. We have for the case of
Time-Independent Dirichlet Boundary conditions that the derivative
regarding $t$ vanishes on boundary points $(x,y)\in \Gamma_h$, i.e.
$ u^{(1)}_{\Gamma_h}(t)\end{equation}uiv 0$. From Lemma \ref{lema-Pgeneral} we
get that $\alpha_1=0$ and $\beta_1=0$. Then the bound for the global
errors follows from Theorem \ref{sev-th-3}. For the case of
Time-Dependent Dirichlet Boundary conditions, from Lemma
\ref{lema-Pgeneral}, we have that $\alpha_1=-3$ and $\beta_1=-1$.
Then, the bound for the global errors follows from Theorem
\ref{sev-th-3}. The bound also applies to the {\sf AMF$_2$-Rad}
method for Time-Dependent Dirichlet BCs, because this method fulfils
the assumptions in Theorem \ref{sev-th-3}.
For the case of the {\sf AMF$_2$-Rad} method and Time-Independent
Dirichlet BCs we apply Theorem \ref{sev-th-3} for the case
$\varrho=1$ and Theorem \ref{sev-th-5} with $l^*=1$ for the case
$\varrho=\tau^2h^{-1}$. Observe that from Lemma \ref{lema-Pgeneral}
we have that $\alpha_1=0$ and $\beta_1=0$ and $\beta_2=-1$. Moreover
the {\sf AMF$_2$-Rad} method fulfils all the assumptions in Theorem
\ref{sev-th-5} by taking $\eta=1$, see also Lemma \ref{lema-valP2}.
For the case of the {\sf AMF$_3$-Rad} method and Time-Independent
Dirichlet BCs we apply Theorem \ref{sev-th-4} with any $\eta<1.25$,
see Lemma \ref{lema-valP2}. Observe that in this case
$\alpha_1=0,\;\beta_1=0$. Then from Theorem \ref{sev-th-4} the
global errors are of size $\mathcal{O}(h^2)+\mathcal{O}(\tau^{2})$.
The proof that the order can be increased up to
$\mathcal{O}(h^2)+\mathcal{O}(\tau^{2.25^*})$ requires some extra
technical details that we have omitted for simplicity. The case of
Time-Dependent Dirichlet BCs follows from Theorem \ref{sev-th-4}
too, but in this case $\beta_1=-1$.
$\Box$
\subsection{Numerical Experiments}
We have performed some numerical experiments on two 2D-PDE and
3D-PDE problems of parabolic type in order to illustrate the
convergence results presented in former sections for the {\sf
AMF$_q$-Rad} methods.
\begin{enumerate}
\item {\sf Problem 1} is the 2D-PDE problem (\ref{2D-dif-reac}) with
diffusion parameter $\varepsilon=0.1$ and Dirichlet Boundary
Conditions and an Initial Condition so that
\begin{equation}\langlebel{2D-solution} u(x,y,t)= 10x(1-x)y(1-y)e^t + \beta
e^{2x-y-t}, \end{equation} is the exact solution. The case $\beta=0$
provides Time-Independent Boundary conditions and no spatial error
($\sigma_h(t)\end{equation}uiv 0$, due to the polynomial nature of the exact
solution). The case $\beta=1$ provides Time-Dependent boundary
conditions and spatial discretizations errors of order two.
\item {\sf Problem 2} is the 3D-PDE problem (\ref{3D-dif-reac}) with
diffusion parameter $\varepsilon=0.1$
\begin{equation}\langlebel{3D-dif-reac}\begin{array}{c}
u_t(\overrightarrow{x},t)= \varepsilon\:
\Delta u(\overrightarrow{x},t)+g(\overrightarrow{x},t),\\
t\in[0,1],\;\;\overrightarrow{x}=(x,y,z)\in (0,1)^3 \in
\mathbb{R}^3,
\end{array}
\end{equation}
and Dirichlet Boundary Conditions and an Initial Condition so that
\begin{equation}\langlebel{2D-solution} u(x,y,t)= 64x(1-x)y(1-y)z(1-z)e^t + \beta
e^{2x-y-z-t}, \end{equation} is the exact solution. Again, the case
$\beta=0$ provides Time-Independent Boundary conditions and no
spatial error and the case $\beta\ne 0$ provides Time-Dependent
boundary conditions and spatial discretizations errors of order two.
\end{enumerate}
On the end-point of the time interval $t^*=1$, in the weighted
Euclidean norm we have computed as specified in (\ref{sev-equa-1}),
the global errors $\varepsilonilon_2(h,\tau)$ ($y_{\rm met}(t^*)$ denotes
the numerical solution at $t^*$ by the method considered), the
number of significant figures of the global errors
$\delta_2(h,\tau)$ and the estimated order of the global errors
$p(h,\tau)$ as powers of $h$ when $r=\tau/h$ is kept constant and
both $\tau$ and $h$ tend to zero.
\begin{equation}\langlebel{sev-equa-1}\begin{array}{c}
\varepsilonilon_2(h,\tau):=\norm{u_h(t^*)-y_{\rm met}(t^*)}_{2,h}, \quad
\delta_2(h,\tau)=-\log_{10} \varepsilonilon_2(h,\tau)\\
p(h,\tau)= (\delta_2(h/2,\tau/2)- \delta_2(h,\tau))/\log_{10}
2.
\end{array}
\end{equation}
In the Tables \ref{table-linear2D-1}, \ref{table-linear2D-2} and
\ref{table-linear3D-1} we have considered for each $h$ the
time-stepsize $\tau=q h$ for the corresponding {\sf AMF$_q$-Rad}
method ($q=1,2,3$), so that all the methods make use of the same
number of $f$-evaluations and similar CPU times in the computations.
In those tables we have displayed the number of significant figures
in the global errors $\delta_2(h,\tau)$ and in brackets the
estimated orders $p(h,\tau)$ of each method.
From Theorem \ref{sev-th-7}, the global errors are expected to be
of size $h^\mu$ (observe that $\tau/h$ is kept constant) where:
\begin{enumerate}
\item for the {\sf AMF$_1$-Rad} method, $\mu=2$ if Time-Independent BCs are considered and $\mu=1$ if Time-Dependent
BCs are imposed. This nicely fits with the results displayed in
Table \ref{table-linear2D-1} (Time-Independent BCs) and in Table
\ref{table-linear2D-2} (Time-Dependent BCs) for the 2D-PDE problem.
Moreover, the convergence order is still $\mu=2$ in the 3D-PDE
problem for Time-Independent BCs as it can be seen in Table
\ref{table-linear3D-1}.
\item For the {\sf AMF$_2$-Rad} method, $\mu=3$ if Time-Independent BCs are considered and $\mu=1$ if Time-Dependent
BCs are imposed. This fits well with the results displayed in Table
\ref{table-linear2D-1} (Time-Independent BCs) and in Table
\ref{table-linear2D-2} (Time-Dependent BCs) for the 2D-PDE problem.
Moreover, the convergence order is also $\mu=3$ in the 3D-PDE
problem for Time-Independent BCs as it can be observed in Table
\ref{table-linear3D-1}.
\item For the {\sf AMF$_3$-Rad} method, $\mu=2.25^*$ if Time-Independent BCs are considered
and $\mu=1$ if Time-Dependent BCs are imposed. This can be
observed in Table \ref{table-linear2D-1} (Time-Independent BCs) and
in Table \ref{table-linear2D-2} (Time-Dependent BCs) for the 2D-PDE
problem. Moreover, the convergence order also approaches to
$\mu=2.3$ in the 3D-PDE problem for Time-Independent BCs as shown in
Table \ref{table-linear3D-1}.
\end{enumerate}
\begin{table}[h!]
\centering
\begin{tabular}{|l|l|l|l|} \hline
$h$ & $\begin{array}{c}\mbox{\sf AMF$_1$-Rad} \;({p})\\ \tau/h= 1\end{array}$ &
$\begin{array}{c}\mbox{\sf AMF$_2$-Rad} \;({p})\\ \tau/h= 2\end{array}$ &
$\begin{array}{c}\mbox{\sf AMF$_3$-Rad} \;({p})\\ \tau/h= 3\end{array}$ \\ \hline
$\frac{1}{24}$ & $ \delta_2=3.74 \; (2.03)$ & $ \delta_2=4.94 \;
(2.82)$ & $ \delta_2=4.90 \; (3.56)$
\\\hline
$\frac{1}{48}$ & $ \delta_2=4.35 \; (2.03)$ & $ \delta_2=5.79 \;
(2.89)$ & $ \delta_2=5.67 \; (2.42)$
\\\hline$\frac{1}{96}$ & $ \delta_2=4.96
\; (1.99)$ & $ \delta_2=6.66 \; (2.92)$ & $ \delta_2=6.40 \; (2.36)$
\\\hline$\frac{1}{192}$ & $ \delta_2=5.56
\; (1.99)$ & $ \delta_2=7.54 \; (2.93)$ & $ \delta_2=7.11 \; (2.29)$
\\\hline$\frac{1}{384}$ & $ \delta_2=6.16
\; (2.03)$ & $ \delta_2=8.42 \; (2.96)$ & $ \delta_2=7.80 \; (2.29)$
\\\hline$\frac{1}{768}$ & $ \delta_2=6.77
\; (--)$ & $ \delta_2=9.31 \; (--)$ & $ \delta_2=8.49 \; (--)$
\\\hline
\end{tabular}
\caption{\scriptsize Significant correct digits ($l_{2,h}$-norm)
for the 2D-PDE problem with Time-Independent Dirichlet BCs
($\beta=0$). In brackets the estimated orders of convergence (by
halving both the spatial resolution $h$ and the time-stepizes $\tau$
and taking ratio $r=\tau/h$).}\langlebel{table-linear2D-1}
\end{table}
\begin{table}[h!]
\centering
\begin{tabular}{|l|l|l|l|} \hline
$h$ & $\begin{array}{c}\mbox{\sf AMF$_1$-Rad} \;({p})\\ \tau/h= 1\end{array}$ &
$\begin{array}{c}\mbox{\sf AMF$_2$-Rad} \;({p})\\ \tau/h= 2\end{array}$ &
$\begin{array}{c}\mbox{\sf AMF$_3$-Rad} \;({p})\\ \tau/h= 3\end{array}$ \\ \hline
$\frac{1}{24}$ & $ \delta_2=3.02 \; (1.00)$ & $ \delta_2=2.79 \;
(0.76)$ & $ \delta_2=2.52 \; (0.66)$
\\\hline
$\frac{1}{48}$ & $ \delta_2=3.32 \; (0.97)$ & $ \delta_2=3.02 \;
(0.83)$ & $ \delta_2=2.72 \; (0.76)$
\\\hline$\frac{1}{96}$ & $ \delta_2=3.61
\; (1.00)$ & $ \delta_2=3.27 \; (0.90)$ & $ \delta_2=2.95 \; (0.86)$
\\\hline$\frac{1}{192}$ & $ \delta_2=3.91
\; (1.00)$ & $ \delta_2=3.54 \; (0.93)$ & $ \delta_2=3.21 \; (0.91)$
\\\hline$\frac{1}{384}$ & $ \delta_2=4.21
\; (1.03)$ & $ \delta_2=3.82 \; (0.97)$ & $ \delta_2=3.48 \; (0.97)$
\\\hline$\frac{1}{768}$ & $ \delta_2=4.52
\; (--)$ & $ \delta_2=4.11 \; (--)$ & $ \delta_2=3.77 \; (--)$
\\\hline
\end{tabular}
\caption{\scriptsize Significant correct digits ($l_{2,h}$-norm)
for the 2D-PDE problem with Time-Dependent Dirichlet BCs
($\beta=1$). In brackets the estimated orders of convergence (by
halving both the spatial resolution $h$ and the time-stepizes $\tau$
and taking ratio $r=\tau/h$).}\langlebel{table-linear2D-2}
\end{table}
\begin{table}[h!]
\centering
\begin{tabular}{|l|l|l|l|} \hline
$h$ & $\begin{array}{c}\mbox{\sf AMF$_1$-Rad} \;({p})\\ \tau/h= 1\end{array}$ &
$\begin{array}{c}\mbox{\sf AMF$_2$-Rad} \;({p})\\ \tau/h= 2\end{array}$ &
$\begin{array}{c}\mbox{\sf AMF$_3$-Rad} \;({p})\\ \tau/h= 3\end{array}$ \\ \hline
$\frac{1}{24}$ & $ \delta_2=3.40 \; (2.03)$ & $ \delta_2=4.31 \;
(2.96)$ & $ \delta_2=4.53 \; (2.69)$
\\\hline
$\frac{1}{48}$ & $ \delta_2=4.01 \; (2.03)$ & $ \delta_2=5.20 \;
(2.96)$ & $ \delta_2=5.34 \; (2.59)$
\\\hline$\frac{1}{96}$ & $ \delta_2=4.62
\; (--)$ & $ \delta_2=6.09 \; (--)$ & $ \delta_2=6.12 \; (--)$
\\\hline
\end{tabular}
\caption{\scriptsize Significant correct digits ($l_{2,h}$-norm)
for the 3D-PDE problem with Time-Independent Dirichlet BCs
($\beta=0$). In brackets the estimated orders of convergence (by
halving both the spatial resolution $h$ and the time-stepizes $\tau$
and taking ratio $r=\tau/h$).}\langlebel{table-linear3D-1}
\end{table}
As a conclusion we can say that the convergence results presented in
Theorem \ref{sev-th-7} seem to be sharp for 2D-parabolic problems
and that they still hold for $d$D-parabolic problems ($d > 2$) when
Time-Independent boundary conditions are considered. The proof of
this fact requires some additional work and is not presented here.
On the other hand, the convergence results are very poor when
Time-Dependent Boundary conditions are considered. However, in such
a situation we have developed a very simple technique (Boundary
Correction Technique) to recover the convergence order as if
Time-Independent Boundary conditions were considered. The
explanation of the Boundary Correction Technique and the proof of
the convergence orders requires some extra length and will be the
objective of another paper.
It is also important to remark that although we have considered in
Theorem \ref{sev-th-7}, second-order central differences for the
spatial discretization, the convergence results also hold for most
of the usual spatial discretizations as long as they are stable and
consistent with order $r\ge 1$. Numerical experiments carried by the authors
seem to indicate that the convergence results also hold for many classes of non-linear problems.
\end{document} |
\begin{document}
\preprint{PRE/003}
\title{Entanglement properties of optical coherent states under amplitude damping}
\author{Ricardo Wickert}
\email{[email protected]}
\author{Nadja Kolb Bernardes}
\author{Peter van Loock}
\affiliation{Optical Quantum Information Theory Group, Max Planck Institute for the Science of Light, G\"unther-Scharowsky-Str. 1/Bau 24, 91058 Erlangen, Germany}
\affiliation{Institute of Theoretical Physics I, Universit\"at Erlangen-N\"urnberg, Staudttr. 7/B2, 91058 Erlangen, Germany}
\date{\today}
\pacs{42.50.Dv, 03.67.Mn, 03.67.Pp}
\begin{abstract}
Through concurrence, we characterize the entanglement properties of optical coherent-state qubits subject to an amplitude damping channel. We investigate the distillation capabilities of known error correcting codes and obtain upper bounds on the entanglement depending on the non-orthogonality of the coherent states and the channel damping parameter. This work provides a first, full quantitative analysis of these photon-loss codes which are naturally reminiscent of the standard qubit codes against Pauli errors.
\end{abstract}
\maketitle
\section{Introduction}
Entanglement has been established as a necessary resource for the implementation of many useful quantum primitives - teleportation of unknown states \cite{BennettTeleport}, key distribution \cite{BB84} and computational speed-up in classically exponential-time calculations \cite{Deutsch, Shor}, to cite but a few examples. This valued resource must, however, be protected against undesired interactions with the environment which lead, ultimately, to the decoherence of the quantum system.
There are two proven ways of safeguarding the fragile quantum states from decohering: Quantum Error Correction (QEC) codes protect the information through encoding in a larger Hilbert space; Entanglement Purification (EP) protocols aim to distill entanglement from a number of identically prepared copies. It is known that QEC codes can be recast as EP schemes and vice-versa; the connection for discrete-variable (DV) systems was established in the seminal paper by Bennett \textit{et al} \cite{Bennett}, whereas the corresponding bridge for continuous-variables (CV) has been demonstrated in Refs. \cite{choi,Giedke,CerfNoGo}.
In the optical context, amplitude damping - the absorption of transmitted photons - is a dominant source of decoherence. It can be appropriately modeled by having the signal interact with a vacuum mode in a beam splitter with the appropriately chosen transmissivity parameter. Amplitude damping appears thus as a \textit{Gaussian} error, and as such it is known that Gaussian resources - the set of operations which can easily be implemented through Gaussian ancilla, beam-splitters, phase-shifters and homodyne measurements - are of no use in protecting the signal state \cite{CerfNoGo,Giedke,Eisert,Fiurasek}, and more elaborate (and possibly non-deterministic) non-Gaussian operations must be accounted for.
In the present paper we review such a QEC scheme as proposed by Glancy \textit{et al.}\cite{GVR}, which protects arbitrary coherent-state superpositions (CSS, also known as ``cat states" \cite{RalphLund}) through the use of non-Gaussian encoding operations. We provide a hitherto inexistent quantitative analysis of this code's performance, examining its entanglement distillation capabilities through Wootters' concurrence \cite{Wooters}. This study is carried both directly and through the use of entanglement evolution equations \cite{konrad}, thus demonstrating their prowess over non-Gaussian CV carriers - even though the logical information may be codified in discrete qubits.
Entangled coherent states subjected to an amplitude damping channel were also investigated by P. Munhoz \textit{et al.} with similar methods \cite{Munhoz}, however, employing a different class of states and aiming towards distinct applications from those presented in this contribution \footnote{In fact, the cluster-type entangled states of \cite{Munhoz} employ a bit-flip redundancy, whereas the amplitude damping channel manifests itself in the coherent-state basis as a phase-flip. The code employed here is designed with this particular error model in mind.}.
A coherent-state $ |\alpha \rangle$, defined as an eigenvalue of the annihilation operator ($\hat{a} |\alpha\rangle = \alpha |\alpha\rangle$), can be expressed in the Fock (number) basis as
\begin{equation}
|\alpha\rangle = e^{-|\alpha|^2} \sum_{n=0}^{\infty} \frac{\alpha^n}{\sqrt{n!}}|n\rangle \quad .
\end{equation}
A detailed introduction to the properties of coherent states can be found for instance in \cite{Mandel,Scully}.
Following \cite{GVR}, we identify the logical qubits as $|0\rangle_L = |-\alpha\rangle$ and $|1\rangle_L = |\alpha\rangle$, in the so-called $(-,+)$ encoding. Qubits can equally be defined in the $(0,\alpha)$ encoding as $|0\rangle_L = |0\rangle$ and $|1\rangle_L = |2\alpha\rangle$, but it can be shown that both are equivalent in their decoherence properties and can easily be translated via displacement operations, so the first convention will be adopted here. An arbitrary qubit superposition is therefore represented as
\begin{equation}
|Q_\alpha\rangle = \frac{1}{\sqrt{N(\alpha)}} ( a |-\alpha\rangle + b |\alpha\rangle ) \quad
\end{equation}
where $|a|^2 + |b|^2 = 1$ and $N(\alpha)$ is a normalization constant, $N(\alpha) = 1 + e^{-2|\alpha|^2}(ab^*+a^*b)$. It is argued that, for sufficiently large values of $\alpha$, $|-\alpha\rangle$ and $|\alpha\rangle$ are approximately orthogonal, and $N(\alpha) \approx 1$; however, present-day technologies only achieve limited $\alpha$ sizes (``Schr\"odinger Kittens" \cite{GrangierScience}). Therefore, a significant amount of non-orthogonality must be considered (fig. \ref{fig:overlap}).
\begin{figure}
\caption{\label{fig:overlap}
\label{fig:overlap}
\end{figure}
\section{Amplitude Damping}
Photon loss is considered to be the predominant source of errors to affect qubits in the optical context \cite{Yamamoto}. We model such loss by interacting the signal with a vacuum mode $|0\rangle_l$ in a beam splitter of transmissivity $\eta$, resulting in
\begin{eqnarray}
\label{bssup}
|Q\rangle_T &=& \frac{1}{\sqrt{N(\alpha)}} ( a |-\alpha\sqrt{\eta}\rangle|-\alpha\sqrt{1-\eta}\rangle_l + \nonumber \\
&& b |\alpha\sqrt{\eta}\rangle|\alpha\sqrt{1-\eta}\rangle_l )
\end{eqnarray}
The final state after transmission is obtained by integrating over the loss mode (denoted here by $|\beta\rangle_l$):
\begin{equation}
\label{trace}
\rho = \frac{1}{\pi}\int {} d^2\beta \,\: _l\langle\beta|Q\rangle_T {} _T\langle Q| \beta \rangle_l
\end{equation}
For a single coherent state, this integration is trivial and amounts to an amplitude contraction, remaining in a pure state. For a superposition, though, the resulting state after tracing out the loss mode is now mixed. One obtains
\begin{eqnarray}
\label{tracemixed}
\rho &=& (1-p_e)\frac{1}{\sqrt{N(\alpha)}}(a |-\alpha \sqrt{\eta}\rangle + b |\alpha \sqrt{\eta}\rangle ) \times H.c. \nonumber \\
&& + p_e \frac{1}{\sqrt{N^\prime(\alpha)}}(a |-\alpha \sqrt{\eta}\rangle - b |\alpha \sqrt{\eta}\rangle ) \times H.c. ,
\end{eqnarray}
where $H.c.$ is the Hermitian conjugate of the previous term and $N^\prime(\alpha) = 1 - e^{-2|\alpha|^2}(ab^*+a^*b)$. In order to simplify the analysis of the decohered state, Eq. (\ref{tracemixed}) can be cast into a more convenient form \cite{GVR}, namely,
\begin{equation}
\label{tracemixed2}
\rho = (1-p_e)|Q_{\alpha\sqrt{\eta}}\rangle \langle Q_{\alpha\sqrt{\eta}}| + p_e Z|Q_{\alpha\sqrt{\eta}}\rangle \langle Q_{\alpha\sqrt{\eta}}|Z
\end{equation}
where $p_e = \frac{1}{2}(1-e^{-2(1-\eta)|\alpha|^2})$ is the probability that the Pauli Z operator ($Z(a|0\rangle_L + b|1\rangle_L) = a|0\rangle_L - b|1\rangle_L$) was applied (fig. \ref{fig:flipprob}). With this expression, photon loss can be seen as having a two-fold effect: first, the amplitude of the states is unconditionally reduced from $\alpha$ to $\alpha \sqrt{\eta}$; second, with probability $p_e$, the qubit suffers a phase flip.
\begin{figure}
\caption{\label{fig:flipprob}
\label{fig:flipprob}
\end{figure}
\section{Error Correction}
Having identified the effect of amplitude damping as a phase flip, a traditional 3-mode error-correcting code \cite{NielsenChuang} can be used to protect the qubit. Such a code can be implemented \cite{GVR} in the optical setting by sending the input state through a sequence of three beam-splitters followed by Hadamard gates - a highly non-Gaussian operation which implements, up to a normalization constant, $|0\rangle_L \rightarrow |0\rangle_L + |1\rangle_L$, $|1\rangle_L \rightarrow |0\rangle_L - |1\rangle_L$. The (unnormalized) encoded state which results is
\begin{equation}
\label{encoded}
a( |-\alpha\rangle + |\alpha\rangle )^{\otimes 3} + b( |-\alpha\rangle - |\alpha\rangle )^{\otimes 3}
\end{equation}
After transmission through the loss channels, another Hadamard gate is applied to each of the modes, which are then recombined through an inverted sequence of beam-splitters. The two ancilla modes are measured to provide syndrome information, from which the appropriate correcting operation can be applied to return the signal to its ``unflipped" state. Finally, teleporting the state into an appropriately prepared Bell state $|-\alpha\sqrt{\eta}\rangle|-\alpha\rangle + |\alpha\sqrt{\eta}\rangle|\alpha\rangle$, the amplitude can be restored to its original value.
The three-way redundant encoding achieved by the procedure outlined above can correct up to one error; therefore, the probability of achieving an error-free transmission is given by
\begin{equation}
\label{Psuccess}
p_{success,3} = 1 - 3 p_e^2 + 2 p_e^3 \quad .
\end{equation}
This can be increased by encoding the input state with a higher number of repetitions. We will thus also analyze codes with 5, 11 and 51 repetitions. 5-way redundancy increases the success probability to $1-10 p_e^3 + 15 p_e^4 - 6 p_e^5$; a n-repetition achieves
\begin{equation}
\label{pgeneral}
p_{success,n} = \sum_{k=0}^{\frac{n-1}{2}} \binom{n}{n-k} (1-p_e)^{n-k} p_e^k \quad .
\end{equation}
\section{Entanglement}
The main focus on the QEC literature cited above lies on the achievement of the encodings - for instance, the implementation of the Hadamard gates or the teleportation strategy. The scheme's overall performance, though, is not quantified except for certain success probabilities or the fidelities of the involved operations. However, it has been noted that ``fidelity is insufficient to quantify quantum processes and protocols" \cite{bjork}. As such, we will explore the known fact that QEC codes can be recast as EP protocols \cite{Bennett,Aschauer} and, employing the entanglement as a figure of merit, provide quantitative benchmarks for this codification.
\subsection{Direct calculation}
We will consider initial cat states of the form
\begin{eqnarray}
&& |\chi_{\alpha_1,\alpha_2}\rangle = \\
&& \frac{1}{\sqrt{\tilde{N}(\alpha_1,\alpha_2)}} \left( \sqrt{w}|\alpha_1,\alpha_2\rangle + e^{i\theta}\sqrt{1-w}|-\alpha_1,-\alpha_2\rangle \right) \nonumber
\end{eqnarray}
with $\tilde{N}(\alpha_1,\alpha_2) = 1 + 2 \cos{\theta} \sqrt{w(1-w)} e^{-2|\alpha_1|^2-2|\alpha_2|^2}$ and $|\alpha_1,\alpha_2\rangle$ a shorthand notation for $|\alpha_1\rangle|\alpha_2\rangle$.
In the case of direct transmission, the first mode is kept, while the second is sent through the photon-loss channel, resulting in
\begin{eqnarray}
\label{rhodirect}
\rho_{direct} &=& (1-P_e) |\chi_{\alpha,\alpha\sqrt{\eta}} \rangle \langle \chi_{\alpha,\alpha\sqrt{\eta}}| \nonumber \\
&& + P_e Z |\chi_{\alpha,\alpha\sqrt{\eta}} \rangle \langle \chi_{\alpha,\alpha\sqrt{\eta}}| Z \quad ,
\end{eqnarray}
with $P_e$ being the phase flip probability for a two-mode state, adjusted to preserve the normalization of each component, defined as
\begin{equation} P_e=\frac{1-e^{4\left|\alpha\right|^2}-e^{2\left|\alpha\right|^2(1-\eta)}+e^{2\left|\alpha\right|^2(1+\eta)}}{2(1-e^{4\left|\alpha\right|^2})}.
\end{equation}
If the sender and receiver make use of the 3-mode repetition, the entangled pair they will share after encoding, transmission and decoding is given by
\begin{eqnarray}
\label{rhoenc3}
\rho_{final,3} && = (1 - 3 P_e^2 + 2 P_e^3) |\chi_{\alpha,\alpha\sqrt{\eta}} \rangle \langle \chi_{\alpha,\alpha\sqrt{\eta}}| \nonumber \\
&& + (3 P_e^2 - 2 P_e^3) Z |\chi_{\alpha,\alpha\sqrt{\eta}} \rangle \langle\chi_{\alpha,\alpha\sqrt{\eta}}| Z ,
\end{eqnarray}
with the $Z$ operator taken to act, in both (\ref{rhodirect}) and (\ref{rhoenc3}), on the transmitted mode. The use of higher order n-repetitions results in
\begin{eqnarray}
\label{rhoencn}
\rho_{final,n} && = P_{success,n} |\chi_{\alpha,\alpha\sqrt{\eta}} \rangle \langle \chi_{\alpha,\alpha\sqrt{\eta}}| \\
&& + (1 - P_{success,n}) Z |\chi_{\alpha,\alpha\sqrt{\eta}}\rangle \langle\chi_{\alpha,\alpha\sqrt{\eta}}| Z .\nonumber
\end{eqnarray}
with $P_{success,n}$ defined as in (\ref{pgeneral}), but depending on $P_e$ instead of $p_e$.
We adopt Wootters' concurrence to quantify the entanglement; in the case of a bipartite qubit system, the concurrence is given by \cite{Wooters}
\begin{equation}
\label{conceq}
C = max\{0,\sqrt{\lambda_1}-\sqrt{\lambda_2}-\sqrt{\lambda_3}-\sqrt{\lambda_4} \}
\end{equation}
where $\lambda_i$ are the eigenvalues, listed in decreasing order, of $\rho \tilde{\rho}$. $\tilde{\rho}$ is the time-reversed density operator,
\begin{equation}
\label{magictr}
\tilde{\rho} = ( \sigma_{y,1} \otimes \sigma_{y,2} ) \rho^{*} ( \sigma_{y,1} \otimes \sigma_{y,2} ) \quad ,
\end{equation}
where $\sigma_{y,i}$ is the Pauli $Y$ operator in the $i$-th mode. In order to cast the density matrix in the appropriate form, we make use of an orthogonal basis $\{ |u_{\alpha}\rangle,|v_{\alpha}\rangle \}$ (see for instance \cite{wang}) such that
\begin{align}
|\alpha\rangle = \mu_\alpha |u_{\alpha}\rangle& + \nu_\alpha |v_{\alpha}\rangle \\
|-\alpha\rangle = \mu_\alpha |u_{\alpha}\rangle& - \nu_\alpha |v_{\alpha}\rangle \nonumber \\
\mbox{with } \mu_\alpha = \left(\frac{1+e^{-2|\alpha|^2}}{2}\right)^{\frac{1}{2}}& \quad \textrm{and} \quad \nu_\alpha = \left(\frac{1-e^{-2|\alpha|^2}}{2}\right)^{\frac{1}{2}} \nonumber
\end{align}
This way, the entangled pair can be written, up to a normalization factor of $1/\sqrt{\tilde{N}(\alpha,\alpha)}$ (omitted below for clarity), as
\begin{align}
|\chi_{\alpha,\alpha}\rangle &= \sqrt{w}|\alpha\rangle|\alpha\rangle + e^{i\theta}\sqrt{1-w}|-\alpha\rangle|-\alpha\rangle \\
&= (\sqrt{w} + e^{i\theta}\sqrt{1-w}) \left( \mu_\alpha^2 |u_{\alpha}\rangle|u_{\alpha}\rangle + \nu_\alpha^2 |v_{\alpha}\rangle|v_{\alpha}\rangle \right) \nonumber \\
&\ +(\sqrt{w} - e^{i\theta}\sqrt{1-w}) \mu_\alpha \nu_\alpha \left( |u_{\alpha}\rangle|v_{\alpha}\rangle + |v_{\alpha}\rangle|u_{\alpha}\rangle \right) \nonumber
\end{align}
and thus
\begin{equation}
\label{matrixunflip}
|\chi_{\alpha,\alpha}\rangle \langle \chi_{\alpha,\alpha}| = \left(
\begin{array}{cccc}
|a|^2 & ab^* & ac^* & ad^* \\
ba^* & |b|^2 & bc^* & bd^* \\
ca^* & cb^* & |c|^2 & cd^* \\
da^* & db^* & dc^* & |d|^2
\end{array}
\right)
\end{equation}
where
\begin{eqnarray}
a &=& (\sqrt{w}+e^{i\theta}\sqrt{1-w})\mu_\alpha^2 \nonumber \\
b = c &=& (\sqrt{w}-e^{i\theta}\sqrt{1-w})\mu_\alpha \nu_\alpha \nonumber \\
d &=& (\sqrt{w}+e^{i\theta}\sqrt{1-w})\nu_\alpha^2 \quad .\nonumber
\end{eqnarray}
After transmission, a similar matrix is also obtained for the unflipped state ($|\chi_{\alpha,\alpha\sqrt{\eta}}\rangle$); the phase-flipped states after transmission are described by
\begin{equation}
\label{matrixflip}
Z |\chi_{\alpha,\alpha\sqrt{\eta}}\rangle \langle \chi_{\alpha,\alpha\sqrt{\eta}} | Z = \left(
\begin{array}{cccc}
|\tilde{a}|^2 & \tilde{a}\tilde{b}^* & \tilde{a}\tilde{c}^* & \tilde{a}\tilde{d}^* \\
\tilde{b}\tilde{a}^* & |\tilde{b}|^2 & \tilde{b}\tilde{c}^* & \tilde{b}\tilde{d}^* \\
\tilde{c}\tilde{a}^* & \tilde{c}\tilde{b}^* & |\tilde{c}|^2 & \tilde{c}\tilde{d}^* \\
\tilde{d}\tilde{a}^* & \tilde{d}\tilde{b}^* & \tilde{d}\tilde{c}^* & |\tilde{d}|^2
\end{array}
\right)
\end{equation}
with $Z$ taken to act on mode 2, and
\begin{eqnarray}
\tilde{a} &=& (\sqrt{w}-e^{i\theta}\sqrt{1-w})\mu_\alpha\mu_{\alpha\sqrt{\eta}}\nonumber \\
\tilde{b} &=& (\sqrt{w}+e^{i\theta}\sqrt{1-w})\mu_\alpha \nu_{\alpha\sqrt{\eta}}\nonumber \\
\tilde{c} &=& (\sqrt{w}+e^{i\theta}\sqrt{1-w})\mu_{\alpha\sqrt{\eta}}\nu_\alpha\nonumber \\
\tilde{d} &=& (\sqrt{w}-e^{i\theta}\sqrt{1-w})\nu_\alpha\nu_{\alpha\sqrt{\eta}}\quad .\nonumber
\end{eqnarray}
With (\ref{matrixunflip}) and (\ref{matrixflip}), one can construct the matrices (\ref{rhodirect}), (\ref{rhoenc3}) or (\ref{rhoencn}) and, through the use of (\ref{magictr}), calculate the concurrence as defined in (\ref{conceq}). Fig. \ref{fig:blabla3d} plots the entanglement of the initial state with $w=1/2$, corresponding to ``genuine" Bell states. It is seen that $\theta=0$ and $\theta=\pi$ (respectively ``even" and ``odd" due to the parity of the number states in the corresponding superposition) result in maximum entanglement, the former asymptotically for large superpositions, the latter independently of the superposition size.
\begin{figure}
\caption{\label{fig:blabla3d}
\label{fig:blabla3d}
\end{figure}
The above method, however, is not convenient for calculating the entanglement of different levels of encoding, as it requires the eigenvalues to be computed for each variation, which can be a time-consuming task if done analytically. Thus we present, in the following section, an alternative way to obtain the state's concurrence.
\subsection{Entanglement Evolution}
The effect of an arbitrary quantum channel $\$$ to a state can also be described in the dual picture \cite{choi}, interchanging the roles of the channel and the initial state. An evolution equation is obtained \cite{konrad}, which equates the entanglement (and in particular the concurrence) of the final state to the product of the concurrence of a maximally entangled state $\ket{\phi^{+}}$ subjected to the same channel times the concurrence of the initial state $\ket{\chi}$. The problem is thus reduced to the calculation of two, possibly simpler, concurrences.
\begin{equation}
C\left[\left(1\otimes\$\right)\ket{\chi}\bra{\chi}\right] = C\left[\left(1\otimes\$\right)\ket{\phi^{+}}\bra{\phi^{+}}\right]C\left[\ket{\chi}\right].
\label{ent.evo}
\end{equation}
As in the previous section, the states will be written in the orthogonal basis $\{\ket{u_{\alpha}},\ket{v_{\alpha}}\}$. One possible maximally entangled Bell state in this basis is given as $\ket{\phi^{+}_{\alpha,\alpha}}=\left(\ket{u_\alpha}\ket{u_\alpha}+\ket{v_\alpha}\ket{v_\alpha}\right)/\sqrt{2}$, but any state carrying exactly one ebit will have its entanglement affected in the same way. The resulting state after transmission through the amplitude damping channel can be described by an X matrix
\begin{equation}
\left(1\otimes\$\right)\ket{\phi^{+}_{\alpha,\alpha}}\bra{\phi^{+}_{\alpha,\alpha}}=
\left(
\begin{array}{cccc}
a & 0 & 0 & f \\
0 & b & z & 0 \\
0 & z* & c & 0 \\
f* & 0 & 0 & d \\
\end{array} \right),
\label{matx}
\end{equation}
where
\begin{eqnarray}
a &=& \frac{(1+e^{-2\left|\alpha\right|^2(1-\eta)})\mu^2_{\tiny{\alpha\sqrt{\eta}}}}{4\mu_{\alpha}^2} \nonumber \\
b &=& -\frac{(-1+e^{-2\left|\alpha\right|^2(1-\eta)})\nu^2_{\tiny{\alpha\sqrt{\eta}}}}{4\mu_{\alpha}^2} \nonumber \\
c &=& \frac{-(-1+e^{-2\left|\alpha\right|^2(1-\eta)})\mu^2_{\tiny{\alpha\sqrt{\eta}}}}{4\mu_{\alpha}^2} \nonumber \\
d &=& \frac{(-1+e^{-2\left|\alpha\right|^2(1-\eta)})\mu_{\tiny{\alpha\sqrt{\eta}}}\nu_{\tiny{\alpha\sqrt{\eta}}}}{4\mu_{\alpha}\nu_{\alpha}} \nonumber \\
f &=& -\frac{(1+e^{-2\left|\alpha\right|^2(1-\eta)})\mu_{\tiny{\alpha\sqrt{\eta}}}\nu_{\tiny{\alpha\sqrt{\eta}}}}{4\mu_{\alpha}\nu_{\alpha}} \nonumber \\
z &=& \frac{(-1+e^{-2\left|\alpha\right|^2(1-\eta)})\mu_{\tiny{\alpha\sqrt{\eta}}}\nu_{\tiny{\alpha\sqrt{\eta}}}} {4\mu_{\alpha}\nu_{\alpha}} \nonumber .
\end{eqnarray}
The outer and inner elements of the (\ref{matx}) represent, respectively, ``unflipped" and ``flipped" Bell states of reduced, $\alpha\sqrt{\eta}$, amplitude. Just as in (\ref{tracemixed}), this allows one to rewrite the resulting state as
\begin{eqnarray}
\left(1\otimes\$\right)\ket{\phi^{+}_{\alpha,\alpha}}\bra{\phi^{+}_{\alpha,\alpha}}&=&
(1-P_e) \ket{\phi^{+}_{\alpha,\alpha\sqrt{\eta}}}\bra{\phi^{+}_{\alpha,\alpha\sqrt{\eta}}} \nonumber \\
&& + P_e Z\ket{\phi^{+}_{\alpha,\alpha\sqrt{\eta}}}\bra{\phi^{+}_{\alpha,\alpha\sqrt{\eta}}}Z
\label{bellflipanaoflipa} \quad ,
\end{eqnarray}
Equation (\ref{ent.evo}) can also be extended to encoded states, such as those obtained by the QEC repetition codes described in Sec. III. In this case, the channel $\$$ will include not only the lossy transmission channels itself, but also the encoding, syndrome measurement, error correction and decoding operations. Nevertheless, the resulting density matrix for the Bell state is still an X matrix,
\begin{eqnarray}
\left(1\otimes\$_{enc}\right)\ket{\phi^{+}_{\alpha,\alpha}}\bra{\phi^{+}_{\alpha,\alpha}} = P_{success,n}\ket{\phi^{+}_{\alpha,\alpha\sqrt{\eta}}}\bra{\phi^{+}_{\alpha,\alpha\sqrt{\eta}}} \nonumber \\
+(1-P_{success,n})Z\ket{\phi^{+}_{\alpha,\alpha\sqrt{\eta}}}\bra{\phi^{+}_{\alpha,\alpha\sqrt{\eta}}}Z \quad,
\label{mat.bell.final}
\end{eqnarray}
where $P_{success,n}$, as in the Sec. III, is the probability of achieving error-free transmission.
The concurrence of a state described by an X matrix can easily be found by \cite{eberly}
\begin{equation}
C\left(\rho\right) = 2 \mbox{ max} \left[0,|z|-\sqrt{ad},|f|-\sqrt{bc}\right].
\label{conX}
\end{equation}
The only remaining step is the calculation of the second term of the RHS of (\ref{ent.evo}). Through the methods outlined in the previous subsection, one finds:
\begin{equation}
C\left[\ket{\chi}\right]=
\frac{2 \left( 1 - e^{-4 \left| \alpha \right|^2} \right) \sqrt{w(w-1)} }{1 + 2 \sqrt{w(w-1)} e^{-4 \left| \alpha \right|^2} \cos \theta} \; ,
\label{con.state}
\end{equation}
This method matches the results found in the previous subsection, but allows for a computationally significant speed-up in calculation times. The concurrences for different encodings are compared below:
\begin{figure}
\caption{\label{fig:condut1}
\label{fig:condut1}
\end{figure}
Two distinct observations can be made from fig. \ref{fig:condut1}. First, for the even cat states (a) and (b), the non-orthogonality of the basis states prevents the pair from achieving high entanglement for low values of $\alpha$; higher encodings are of little advantage in this regime. The choice of the phase $\theta$ is of paramount importance when one compares to the odd states (c) and (d), where even for very small $\alpha$ there is still a significant amount of shared entanglement, and encoding in more qubits still improves on this amount. Second, for sufficiently large sizes of $\alpha$, the flip probability approaches $0.5$ and thus it dephases the qubit entirely, independently of the phase or encoding. However, for a certain range, the different codifications are seen to help achieve and sustain higher entanglement between the shared pairs.
Finally, in fig. \ref{fig:concfix15} one can observe that, all the way to zero transmissivity, the pair has some residual entanglement. Higher redundancy improves this figure; we also note that, in the context of \cite{bjork}, the encoding does not induce entanglement sudden death (ESD) \cite{eberly}.
\begin{figure}
\caption{\label{fig:concfix15}
\label{fig:concfix15}
\end{figure}
\section{Discussion and Conclusions}
In this work, we have studied the decoherence process of entangled coherent-state superpositions in the amplitude damping channel, exploring QEC codes originated from the discrete-variable regime in an optical, continuous-variable setting. The quantitative analysis was reminiscent of EP protocols, however, this translation must be carried out carefully: a true distillation would, in addition to the non-Gaussian Hadamard gates necessary for the encoding, require teleportations to be performed via non-Gaussian two-mode Bell measurements \cite{Bennett,vanenk}. In this picture, multiple copies of the entangled resource $\ket{\phi^{+}}$ are shared across the amplitude damping channel. The use of the encoding and decoding circuits of the original error-correcting code then results in one pair with higher entanglement (see fig. \ref{fig:pdfeppecc}), which effectively characterizes this scheme as a one-way, deterministic entanglement distillation for noisy, non-Gaussian CV states.
Alternatively, in the context of \cite{Aschauer}, the translation to an entanglement purification protocol could be performed through an elaborate non-Gaussian multi-mode projective measurement in the encoding basis. The non-Gaussian elements are, in any case, a required condition, enabling the circumvention of known No-Go theorems \cite{CerfNoGo,Giedke,Eisert,Fiurasek}.
\begin{figure}
\caption{\label{fig:pdfeppecc}
\label{fig:pdfeppecc}
\end{figure}
The protection granted against phase-flip (or $Z$) errors by different repetition codes was investigated and quantified through concurrence; however, possible physical limitations in the implementation of the gates necessary for these encodings have not been been taken into account, therefore establishing the above results as an upper bound. In particular, we note that both Hadamard and teleportation operations in \cite{GVR} only approach unit fidelity and/or high success probabilities for large superposition sizes. However, as we showed here, in this regime these codes would cease to work. We conclude that, in order to incorporate the actual encoding operations into the above protocol, gates that succeed with high fidelity on not too-large $\alpha$ cat states will be needed. One possibility to achieve this could be the scheme of \cite{RalphLund}, where $\alpha \approx 1.2$ is enough for performing fault-tolerant quantum computing.
We have observed a trade-off between the orthogonality of the coherent-state basis and the probability of a phase-flip error. The former imposes a necessity for a maximum size of coherent-state superposition, while the latter prevents the use of arbitrarily large superpositions, thus hinting at an optimal regime depending on channel parameters.
The generation of optical cat states remains an experimental challenge, though much progress has been recently achieved. Arbitrarily large squeezed cat states can be obtained through photon number states and homodyne detection \cite{Ourjoumtsev}, however such technique is highly probabilistic and presents only moderate output fidelities. A different approach involves tapping squeezed vacuum in a beam splitter (BS), with one output mode directed to a number-resolving photon counter. Conditional on the number of photons detected, the other mode is projected into an odd or even cat state \cite{GrangierScience, Takahashi}; an n-photon subtraction is described as $\hat{a}^n\hat{S}(\epsilon_0)|0\rangle$, where $\hat{a}$ is the annihilation operator, $\hat{S}$ is the squeezing operator and $\epsilon_0$ amounts to the degree of squeezing applied. Superpositions with $\alpha \sim 1.2-1.3$ can be created through this technique; it is also worth noting that the more ``valuable", 1-ebit odd CSS resource is more readily obtained than its less entangled even state counterpart.
The choice of ``odd" cat states (with $\theta=\pi$) could be seen as a viable alternative to employ small-amplitude superpositions to distribute entanglement while minimizing the amplitude damping effects. Still, $\alpha$ may not be chosen too small so that the resulting qubits become impractical for information transfer or computational purposes. Nonetheless, in the context of optical, realistic quantum communication, a combination of ``odd" CSS resources, linear optics and nonlinear measurements for encoding may lead to efficient error correction and entanglement distillation strategies.
$\quad$
\subsection*{Acknowledgments}
Support from the Emmy Noether Program of the Deutsche Forschungsgemeinschaft is gratefully acknowledged.
\end{document} |
\begin{document}
\title{Priorities Without Priorities: Representing Preemption in Psi-Calculi}
\begin{abstract}
Psi-calculi is a parametric framework for extensions of the pi-calculus with data terms and arbitrary logics. In this framework there is no direct way to represent action priorities, where an action can execute only if all other enabled actions have lower priority.
We here demonstrate that the
psi-calculi parameters can be chosen such that the effect of action priorities can be encoded.
To accomplish this we define an extension of psi-calculi with action priorities, and show that for every calculus in the extended framework there is a corresponding ordinary psi-calculus, without priorities, and a translation between them that satisfies strong operational correspondence. This is a significantly stronger result than for most encodings between process calculi in the literature.
We also formally prove in Nominal Isabelle that the standard congruence and structural laws about strong bisimulation hold in psi-calculi extended with priorities.
\end{abstract}
\section{Introduction}
Priorities in process calculi allow certain actions to take precedence over others. This is useful when modelling systems because it admits more fine-grained control over the model's behaviour. Phenomena that exhibit prioritised behaviour include eg.~interrupts in operating systems, and exception handling in programming languages. In this paper we demonstrate how priorities can be represented in the psi-calculi framework, by encoding them into the logical theory that determines how actions are generated by process syntax.
Psi-calculi~\cite{bengtson.johansson.ea:psi-calculi-long} is a family of applied process calculi that generalises the {\pic} in three ways. First, the subjects (designating the communication channels) and objects (designating the communicated data) of input and output actions may be \emph{terms} taken from an arbitrary set, and not just single names. Second, equality tests on names are replaced by tests of predicates called \emph{conditions}, taken from an arbitrary logic. Finally, the process syntax is extended with \emph{assertions}, which can be seen as introducing new facts about the environment in which a process executes. The unguarded assertions of a process influence the evaluation of conditions and the connectivity between channel terms, and can change as the process executes.
In this paper, we show that the psi-framework is sufficiently expressive to represent action priorities derived from a priority order on the communication channels. We are interested in priorities for two reasons. First, previous work on priorities indicate that they are highly expressive: Jeffrey defines a process calculus with time and priority where timed processes can be encoded in the untimed fragment of the calculus~\cite{DBLP:conf/ftrtft/Jeffrey92}; Jensen shows that CCS augmented with priority choice can encode broadcast communication~\cite{Torp94interpretingbroadcast}; and Versari et~al.~shows that CCS with priority and only the prefix and parallel operators can solve both leader election (unlike the {\pic}), and the ``last man standing''-problem (unlike the broadcast {\pic})~\cite{DBLP:conf/concur/VersariBG07}. Second, we are not aware of another process calculus (without priorities) where adding priorities has been shown to yield no increased expressiveness. The prevailing methods to introduce priorities in process algebras are through semantic rules with negative premises or new auxiliary relations to express the absence of higher-priority actions; we shall need none of those.
We accomplish our result as follows. First we define an extension of the psi-calculi framework with explicit channel priorities, where the priority level of a channel can change dynamically during process execution, as defined by an auxiliary relation representing absence of actions. We formally prove, using the interactive theorem prover Isabelle~\cite{nipkow:isabelle}, that in this setting strong bisimilarity satisfies the usual algebraic laws and congruence properties familiar from the {\pic}. We proceed to show that for every psi-calculus with priorities, separate choice and prefix-guarded replication, it can be encoded in a standard psi-calculus without priorities. This encoding satisfies particularly strong quality criteria, namely strong operational correspondence, meaning that the translation does not introduce any protocol in the target language. The main idea is that we use a non-monotonic logic for the assertions, where the appearance of enabled high-priority channels can temporarily prevent lower priority channels from resulting in actions.
The rest of the paper is structured as follows. In Section~2 we briefly recapitulate the essentials of psi-calculi, and in Section~3 we define the extension with explicit channel priorities. Section~4 contains an encoding into standard psi-calculi. In Section~5 we establish strong operational correspondence and briefly discuss other criteria for encodings, among them full abstraction, and Section~6 contains conclusions with future work.
Full proofs of all theorems presented in this paper are available online at \url{http://www.it.uu.se/research/group/mobility/prio-proofs.pdf}.
\section{Psi-calculi}\label{sec:psi}
The following is a quick recapitulation of the psi-calculi framework. For an in-depth introduction with motivations and examples we refer the reader to \cite{bengtson.johansson.ea:psi-calculi-long}.
We assume a countably infinite set of atomic \emph{names} $\,\nameset$ ranged over by $a,b,\ldots,z$. Intuitively, names are the symbols that can be scoped and be subject to substitution.
A \emph{nominal set}~\cite{PittsAM:nomlfo-jv,Gabbay01anew} is a set equipped with a formal notion of what it means to swap names in an element; this leads to a notion of when a name $a$ occurs in an element $X$, written $a \in \names{X}$ (pronounced ``$a$ is in the support of $X$'').
We write $a \freshin X$, pronounced ``$a$ is fresh for $X$'', for $a \not\in \names{X}$, and if $A$ is a set of names we write $A\freshin X$ to mean $\forall a \in A \;.\; a \freshin X$.
In the following $\tilde{a}$ is a finite sequence of names. The empty sequence is written $\epsilon$ and the concatenation of $\tilde{a}$ and $\tilde{b}$ is written $\tilde{a} \tilde{b}$.
We say that a function symbol is \emph{equivariant} if all name swappings distribute over it.
A \emph{nominal datatype} is a nominal set together with a set of
functions on it. In particular we shall consider substitution functions that
substitute elements for names. If $X$ is an element of a datatype,
$\tilde{a}$ is a sequence of names without duplicates and $\tilde{Y}$ is an
equally long sequence of elements of possibly another datatype,
the \emph{substitution}
$X\subst{\tilde{a}}{\tilde{Y}}$ is an element of the same datatype as
$X$.
The substitution function can be chosen freely, but must satisfy certain natural laws regarding the treatment of names; it must be equivariant, the names $\vec{a}$ in $X[\vec{a}:=\vec{T}]$ must be alpha-convertible as if they were binding in $X$. See~\cite{bengtson.johansson.ea:psi-calculi-long} for details.
A psi-calculus is defined by instantiating three nominal data types and four equivariant operators; formally it is a tuple $(\terms,\assertions,\conditions,\vdash,\otimes,\sch,\unit)$ as follows.
\begin{definition}[Psi-calculus parameters]
\label{def:parameters}
A psi-calculus requires the three (not necessarily disjoint) nominal data types:
\iffalse
\[\begin{array}{ll}
\terms & \mbox{the (data) terms, ranged over by $M,N$} \\
\conditions & \mbox{the conditions, ranged over by $\varphi$}\\
\assertions & \mbox{the assertions, ranged over by $\Psi$}
\end{array}\]
\else
the (data) terms $\terms$, ranged over by $M,N$,
the conditions $\conditions$, ranged over by $\varphi$,
the assertions $\assertions$, ranged over by $\Psi$,
\fi
and the four operators:
\[\begin{array}{llcll}
{\sch}\in \terms \times \terms \to \conditions & \mbox{Channel Equivalence} & \quad &
{\ftimes}\in \assertions \times \assertions \to \assertions& \mbox{Composition} \\
\one: \assertions& \mbox{Unit} & \quad &
{\vdash}\subseteq \assertions \times \conditions & \mbox{Entailment} \\
\end{array}
\]
\end{definition}
\begin{comment}and substitution functions $\subst{\ve{a}}{\ve{M}}$, substituting
terms for names, on each of~\textbf{T},~\textbf{C} and~\textbf{A}, where the
substitution function on \terms{}, in addition to the alpha-conversion-like law above, satisfies the following name
preservation laws:
if $\; \vec{a}\subseteq\supp M$ and $b \in \supp{\vec{N}}$ then $b \in \supp{M[\vec{a}:=\vec{N}]}$; and
if $\; b\in\supp M$ and $b \freshin \vec{a},\vec{N}$ then $b \in \supp{M[\vec{a}:=\vec{N}]}$.\end{comment}
The binary functions above will be written in infix.
Thus, $M \sch N$ is a condition, pronounced ``$M$ and $N$ are channel equivalent''.
We write $\Psi \vdash \varphi$, pronounced ``$\Psi$ entails $\varphi$'', for $(\Psi, \varphi)\in\; \vdash$,
and if $\Psi$ and $\Psi'$ are assertions then so is $\Psi \ftimes \Psi'$, which intuitively represents the conjunction of the information in $\Psi$ and $\Psi'$.
We say that two assertions are \emph{statically equivalent}, written $\Psi \sequivalent \Psi'$ if they entail the same conditions, i.e. for all $\varphi$ we have that $\Psi \vdash \varphi$ iff $\Psi' \vdash \varphi$.
We impose certain requisites on the sets and operators: channel equivalence must be symmetric and transitive, $\ftimes$ must be
compositional with regard to $\sequivalent$, and the assertions with
$(\ftimes,\one)$ form an abelian monoid modulo $\sequivalent$. Finally, substitution $M\subst{\ve{a}}{\ve{T}}$ on terms must be such that if the names $\ve{a}$ are in the support of $M$, the support of $\ve{T}$ must be in the support of $M\subst{\ve{a}}{\ve{T}}$.
A \emph{frame} is an assertion together with a sequence of names that bind into it: it is of the form $\framepair{\frnames{}}{\Psi}$ where $\frnames{}$ binds into the assertion~$\Psi$. We use $F,G$ to range over frames.
We overload $\Psi$ to also mean~$\framepair{\epsilon}{\Psi}$
and $\ftimes$ to composition on frames defined by $\framepair{\frnames{1}}{\Psi_1} \ftimes \framepair{\frnames{2}}{\Psi_2} =
\framepair{\frnames{1} \frnames{2}}{(\Psi_1 \ftimes \Psi_2)}$ where
$\frnames{1}\freshin\frnames{2},\Psi_2$
and vice versa.
We write
$\Psi \ftimes F$ to mean $\framepair{\epsilon}{\Psi} \ftimes F$, and
$(\nu
c)(\framepair{\frnames{}}{\Psi})$ for~$\framepair{c\frnames{}}{\Psi}$.
We define $F \vdash \varphi$ to mean that there exists an alpha variant $\framepair{\frnames{}}{\Psi}$ of $F$ such that $\frnames{} \freshin \varphi$ and $\Psi \vdash \varphi$. We also define~\mbox{$F\sequivalent G$} to mean that for all $\varphi$ it holds that $ F \vdash \varphi$ iff $ G \vdash \varphi$.
\begin{definition}[Psi-calculus agents]\label{def:agents}
Given a psi-calculus $\mathcal{P}$ with parameters as in Definition~\ref{def:parameters},
the \emph{agents} $\processes(\mathcal{P})$, ranged over by $P,Q,\ldots$, are of the following forms.
\begin{comment}
\[\begin{array}{lclclclcl}
\nil
& \qquad &
\outlabel{M}{N} \sdot P
& \qquad &
\inprefix{M}{\ve{x}}{N} \sdot P
& \qquad &
P \pll Q
& \qquad &
\pass{\Psi}
\\
(\nu a)P
& \qquad &
!P
& \qquad &
\multicolumn{5}{l}{\case{\ci{\varphi_1}{P_1}\casesep\cdots\casesep\ci{\varphi_n}{P_n}}}
\end{array}\]
\end{comment}
\begin{comment}{\rm
\[
\begin{array}{llcllcll}
\nil & \quad\mbox{Nil} & \qquad &
\outlabel{M}{N} \sdot P & \quad\mbox{Output} & \qquad &
\inprefix{M}{\ve{x}}{N} \sdot P & \quad\mbox{Input} \\
(\nu a)P & \quad\mbox{Restriction} & &
P \pll Q & \quad\mbox{Parallel} & &
! P & \quad\mbox{Replication} \\
\pass{\Psi} & \quad\mbox{Assertion} & &
\multicolumn{4}{l}{\case{\ci{\varphi_1}{P_1}\casesep\cdots\casesep\ci{\varphi_n}{P_n}}} & \quad\mbox{Case}
\end{array}\]
}\end{comment}
{\rm
\[
\begin{array}{ll}
\nil & \mbox{Nil} \\
\outlabel{M}{N} \sdot P & \mbox{Output}\\
\inprefix{M}{\ve{x}}{N} \sdot P & \mbox{Input}\\
\case{\ci{\varphi_1}{P_1}\casesep\cdots\casesep\ci{\varphi_n}{P_n}}\quad
&\mbox{Case} \\
(\nu a)P & \mbox{Restriction}\\
P \pll Q & \mbox{Parallel}\\
! P & \mbox{Replication} \\
\pass{\Psi} & \mbox{Assertion}
\end{array}\]
}
\noindent Restriction $(\nu a)P$ binds $a$ in $P$ and input $\inprefix{M}{\ve{x}}{N} \sdot P$ binds $\vec{x}$ in both $N$ and $P$.
An occurrence of a subterm in an agent is \emph{guarded} if it is a proper subterm of an input or output term.
An agent is \emph{assertion guarded} if it contains no unguarded assertions.
An agent is \emph{well-formed} if in $\inprefix{M}{\ve{x}}N.P$ it holds that $\ve{x} \subseteq \names{N}$ is a sequence without duplicates,
that in a replication $!P$ the agent $P$ is assertion guarded,
and that in $\case{\ci{\varphi_1}{P_1}\casesep\cdots\casesep\ci{\varphi_n}{P_n}}$ the agents $P_i$ are assertion guarded.
\end{definition}
The agent
$\case{\ci{\varphi_1}{P_1}\casesep\cdots\casesep\ci{\varphi_n}{P_n}}$
is sometimes abbreviated as
\mbox{\rm $\case{\ci{\ve{\varphi}}{\ve{P}}}$}.
We sometimes write $\inprefixempty{M}(x).P$ for $\inprefix{M}{x}{x}.P$.
From this point on, we only consider well-formed agents.
The \emph{frame $\frameof{P}$ of an agent} P is defined inductively as follows:
\begin{mathpar}
\frameof{\inprefix{M}{\ve{x}}{N} \sdot P} =
\frameof{\outprefix{M}{N} \sdot P} =
\frameof{\nil} =
\frameof{\case{\ci{\ve{\varphi}}{\ve{P}}}} =
\frameof{!P} = \one \and
\frameof{\pass{\Psi}} = \framepair{\epsilon}{\Psi} \and
\frameof{P \pll Q} = \frameof{P} \ftimes \frameof{Q} \and
\frameof{\res{b}P} = (\nu b)\frameof{P}
\end{mathpar}
The \emph{actions} ranged over by $\alpha, \beta$ are of the following three kinds:\\
\emph{Output} $\outlabel{M}{(\nu \tilde{a})N}$,
\emph{input} $\inlabel{M}{N}$, and \emph{silent} $\tau$. Here we refer to $M$ as the \emph{subject} and $N$ as the \emph{object}. We define
$\bn{\outlabel{M}{(\nu \tilde{a})N}} = \tilde{a}$, and $\bn{\alpha}=\emptyset$
if $\alpha$ is an input or $\tau$.
As in the pi-calculus, the output $\outlabel{M}{(\nu \tilde{a})N}$ represents
an action sending $N$ along $M$ and opening the scopes of the names
$\tilde{a}$. \begin{comment}Note in particular that the support of this action includes
$\tilde{a}$. Thus $\outlabel{M}{(\nu a)a}$ and $\outlabel{M}{(\nu b)b}$ are
different actions.\end{comment}
\begin{table*}[tb]
\begin{mathpar}
\inferrule*[Left=\textsc{In}]
{\Psi \vdash K \sch M}
{\framedtrans{\Psi}{\inprefix{M}{\ve{y}}{N} \sdot
P}{\inlabel{K}{N}\subst{\ve{y}}{\ve{L}}}{P\subst{\ve{y}}{\ve{L}}}}
\inferrule*[left=\textsc{Out}]
{\Psi \vdash M \sch K}
{\framedtrans{\Psi}{\outprefix{M}{N} \sdot P}{\outlabel{K}{N}}{P}}
\inferrule*[left={\textsc{Case}}]
{\framedtrans{\Psi}{P_i}{\alpha}{P'} \\ \Psi \vdash \varphi_i}
{\framedtrans{\Psi}{\case{\ci{\ve{\varphi}}{\ve{P}}}}{\alpha}{P'}}
\inferrule*[left=\textsc{Par}, right={$
\bn{\alpha} \freshin Q
$}]
{\framedtrans{\frass{Q} \ftimes \Psi}{P} {\alpha}{P'}}
{\framedtrans{\Psi}{P \pll Q}{\alpha}{P' \pll Q}}
\inferrule*[left=\textsc{Com}, right={$
\ve{a} \freshin Q
$}]
{\Psi \ftimes \frass{P} \ftimes \frass{Q} \vdash M \sch K \\
\framedtrans{\frass{Q} \ftimes \Psi}{P}{\outlabel{M}{(\nu \ve{a})N}}{P'} \\
\framedtrans{\frass{P} \ftimes \Psi}{Q}{\inlabel{K}{N}}{Q'}
}
{\framedtrans{\Psi}{P \pll Q}{\tau}{(\nu \ve{a})(P' \pll Q')}}
\inferrule*[left=\textsc{Rep}]
{\framedtrans{\Psi}{P \pll !P}{\alpha}{P'}}
{\framedtrans{\Psi}{!P}{\alpha}{P'}}
\inferrule*[left=\textsc{Scope}, right={$b \freshin \alpha,\Psi$}]
{\framedtrans{\Psi}{P}{\alpha}{P'}}
{\framedtrans{\Psi}{(\nu b)P}{\alpha}{(\nu b)P'}}
\inferrule*[left=\textsc{Open}, right={$\inferrule{}{b \freshin \ve{a},\Psi,M\\\\b \in \names{N}}$}]
{\framedtrans{\Psi}{P}{\outlabel{M}{(\nu \ve{a})N}}{P'}}
{\framedtrans{\Psi}{(\nu b)P}{\outlabel{M}{(\nu \ve{a} \cup \{b\})N}}{P'}}
\end{mathpar}
\caption{\rm Structured operational semantics. Symmetric versions of \textsc{Com} and \textsc{Par} are elided. In the rule $\textsc{Com}$ we assume that
$\frameof{P} =
\framepair{\frnames{P}}{\Psi_P}$ and $\frameof{Q} =
\framepair{\frnames{Q}}{\Psi_Q}$ where $\frnames{P}$ is fresh for all of
$\Psi, \frnames{Q}, Q, M$ and $P$, and that $\frnames{Q}$ is similarly fresh. In the rule
\textsc{Par} we assume that $\frameof{Q} = \framepair{\frnames{Q}}{\Psi_Q}$
where $\frnames{Q}$ is fresh for
$\Psi, P$ and $\alpha$.
In $\textsc{Open}$ the expression $\tilde{a} \cup \{b\}$ means the sequence
$\tilde{a}$ with $b$ inserted anywhere.
}
\label{table:full-struct-free-labeled-operational-semantics}
\end{table*}
\FloatBarrier
\begin{definition}[Transitions]
\label{transitions}
A \emph{transition} is written \mbox{$\framedtrans{\Psi}{P}{\alpha}{P'}$},
meaning that in the environment $\Psi$, $P$ can do $\alpha$ to become $P'$. The transitions are defined inductively in
Table~\ref{table:full-struct-free-labeled-operational-semantics}.
We abbreviate $\unit \frames \trans{P}{\alpha}{P'}$ as $\trans{P}{\alpha}{P'}$.
\end{definition}
We identify alpha-equivalent agents, frames and transitions. In a transition the names in $\bn{\alpha}$ bind into both the action object and the derivative, therefore $\bn{\alpha}$ is in the support of $\alpha$ but not in the support of the transition.
\begin{comment}This means that the bound names can be chosen fresh, substituting each
occurrence in both the object and the derivative.\end{comment}
\FloatBarrier
\begin{definition}[Strong bisimulation]\label{defn:strongbisim}
A \emph{strong bisimulation}
$\cal R$ is a ternary relation on assertions and pairs of agents such that
${\cal R}(\Psi,P,Q)$ implies
\begin{enumerate}\addtolength{\itemsep}{0.3\baselineskip}
\item Static equivalence:
$\Psi \ftimes \frameof{P} \sequivalent \Psi \ftimes \frameof{Q}$; and
\item Symmetry: ${\cal R}(\Psi,Q,P)$; and
\item Extension of arbitrary assertion:
$\forall \Psi'.\ {\cal R}(\Psi \ftimes \Psi',P,Q)$; and
\item Simulation:
for all $\alpha, P'$ such that $\framedtrans{\Psi}{P}{\alpha}{P'}$ and
$\bn{\alpha}\freshin \Psi,Q$,\\there exists $Q'$ such
that $\framedtrans{\Psi}{Q}{\alpha}{Q'}$ and ${\mathcal R}(\Psi , P', Q')$.
\end{enumerate}
\label{def:bisim}
We define $\Psi \frames P \bisim Q$ to mean that there exists a bisimulation ${\cal R}$ such that
${\cal R}(\Psi,P,Q)$, and write $P \bisim Q$, pronounced $P$ and $Q$ are {\em (strongly) bisimilar}, for $\unit \rhd P \bisim Q$.
\end{definition}
\begin{definition}[Strong congruence]\label{def:strongcong}
We define $P \sim_\Psi Q$ to mean that for all substitution sequences $\sigma$, $\Psi \frames P\sigma \bisim Q\sigma$ holds. We write $P \sim Q$, pronounced $P$ is {\em (strongly) congruent} to $Q$, to mean $P \sim_\unit Q$.
\end{definition}
We have shown~\cite{bengtson.johansson.ea:psi-calculi-long} that strong bisimilarity preserves all operators except input, and that strong congruence is a congruence and satisfies the expected algebraic laws for structural congruence.
\section{Extension: Psi-calculi with priorities}
The most common approaches to implementing priorities in process calculi are (1) to add a priority operator $\Theta$ such that $\Theta(P)$ may only take the highest-priority actions of $P$ as defined by some ordering on actions~\cite{BBK86}, and (2) to alway enforce priorities, rather than only at special operators~\cite{DBLP:conf/lics/CleavelandH88,HPA2001}. In order to avoid introducing a new operator, we follow the second approach.
We associate a priority level to actions that may depend on the assertion environment, and hence change dynamically as a process evolves.
The intuition is that we write $\Psi \vdash \Prio{M}{p}$ to mean that the priority level of communication on the channel $M$ in the environment $\Psi$ is $p$, where lower values of $p$ indicate \emph{higher} priority.
Priorities are subject to some natural constraints: they must be equivariant, and in a given assertion, channel equivalent terms must have the same unique priority level.
\begin{definition}[Psi-calculi with priorities]\label{defn:psiwithprio} A \emph{psi-calculus with priorities}, ranged over by $\mathcal{P},\mathcal{Q}$, is a tuple $(\terms,\assertions,\conditions,\vdash,\otimes,\sch,\unit,\PRIO)$ such that
\begin{enumerate}
\item $(\terms,\assertions,\conditions,\vdash,\otimes,\sch,\unit)$ is a psi-calculus, and
\item${\PRIO}\;$ of type $\terms\times\numbers \Rightarrow \conditions$ is an equivariant operator written in infix, i.e., we write $\Prio M p$ for $\PRIO(M,p)$,
such that for all $\Psi,M,N$, if $\Psi \vdash M \sch N$ then there is a unique $p \in \numbers$ such that $\Psi \vdash \Prio M p$ and $\Psi \vdash \Prio N p$.
\end{enumerate}
\end{definition}
The semantics of psi-calculi with priorities is as the semantics of psi-calculi, but with two changes. The first is that $\tau$ actions are replaced with $\tau:p$ actions, where $p$ is the priority level of the transition. The second is that the rules are augmented with side conditions that prevent a process from taking low priority actions. This has a natural formulation in terms of negative premises~\cite{DBLP:journals/jacm/BolG96}, but in order to make implementation in Isabelle easier we instead define the semantics in two layers, following~\cite{DBLP:conf/lics/CleavelandH88,HPA2001,DBLP:conf/esop/Versari07}.
The bottom layer is denoted with the transition arrow $\prionegtransarrow{}$ and is used to determine which transitions would be available, disregarding priorities. The semantics of $\prionegtransarrow{}$ is exactly as in Table~\ref{table:full-struct-free-labeled-operational-semantics} with the sole extension that the \textsc{Com} rule generates an action of kind $\tau:p$, where $p$ is derived from the priority of the channel.
We then define a predicate $\priook(\alpha,\Psi,P)$, which intuitively means that no $\tau$ transition whose priority is higher than that of $\alpha$ can be derived from $P$ in $\Psi$. Finally we define $\priotransarrow{}$ to represent
transitions respecting priorities, where the \textsc{Case}, \textsc{Par}, and \textsc{Com} rules get side conditions using $\priook$.
\begin{definition}
\[\priook(\alpha,\Psi,P) \defn
\neg \exists n\;P'. (\Psi\frames {P\prionegtransarrow{\tau:n}}\; P'\land n <\Prioof{(\Psi\otimes\frameof{P},\alpha)})
\]
where $\Prioof{(F,\alpha)}$ is defined to be $p$ if either $\alpha = \tau:p$ or $F \vdash \Prio {\subj{\alpha}} p$, and $\prionegtransarrow{}$ is defined in
Definition~\ref{priotransitions} below.
\end{definition}
\begin{definition}[Transitions with priorities]\label{priotransitions} The transitions of psi-calculi with priorities are defined inductively by the same rules as in Table~\ref{table:full-struct-free-labeled-operational-semantics}, but with all occurrences of $\apitransarrow{}$ replaced with $\priotransarrow{}$, and the $\textsc{Case}$, $\textsc{Com}$ and $\textsc{Par}$ rules replaced by the following:
\begin{mathpar}
\inferrule*[left={\textsc{Case}},right={$\priook (\alpha,\Psi,\case{\ci{\ve{\varphi}}{\ve{P}}})$}]
{\prioframedtrans{\Psi}{P_i}{\alpha}{P'} \\ \Psi \vdash \varphi_i}
{\prioframedtrans{\Psi}{\case{\ci{\ve{\varphi}}{\ve{P}}}}{\alpha}{P'}}
\inferrule*[left=\textsc{Par}, right={$
{\begin{array}{l}
\priook (\alpha,\Psi,P\mid Q)\\
\bn{\alpha} \freshin Q
\end{array}
}$}]
{\prioframedtrans{\frass{Q} \ftimes \Psi}{P} {\alpha}{P'}}
{\prioframedtrans{\Psi}{P \pll Q}{\alpha}{P' \pll Q}}
\inferrule*[left=\textsc{Com}, right={$
{\begin{array}{l}
\priook (\tau:p,\Psi,P\mid Q)\\
\ve{a} \freshin Q
\end{array}
}$}]
{\Psi \ftimes \frass{P} \ftimes \frass{Q} \vdash M \sch K\qquad\Psi \otimes \frass{P} \otimes \frass{Q}\vdash\Prio M p \\
\prioframedtrans{\frass{Q} \ftimes \Psi}{P}{\outlabel{M}{(\nu \ve{a})N}}{P'} \\
\prioframedtrans{\frass{P} \ftimes \Psi}{Q}{\inlabel{K}{N}}{Q'}
}
{\prioframedtrans{\Psi}{P \pll Q}{\tau:p}{(\nu \ve{a})(P' \pll Q')}}
\end{mathpar}
The transition relation $\prionegtransarrow{}$ is defined by the same rules as $\priotransarrow{}$, but with all side conditions involving $\priook$ omitted.
\end{definition}
Strong bisimulation and strong congruence on psi-calculi with priorities can be obtained from Definitions~\ref{defn:strongbisim}-\ref{def:strongcong} by replacing all occurrences of $\apitransarrow{}$ with $\priotransarrow{}$. The meta-theory pertaining to strong bisimulation from the original psi-calculi carries over to psi-calculi with priorities, and formal proofs in Isabelle have been carried out:
\begin{theorem}\label{thm:priobisim} Strong congruence $\sim$ on psi-calculi with priorities is a congruence, and satisfies
\[\begin{array}{rcll}
{P} &\sim& {P \parop \nil} & \\
{P\parop ( Q \parop R)}&\sim&{(P \parop Q) \parop R} & \\
{P \parop Q}&\sim&{Q \parop P} & \\
{(\nu a)\nil}&\sim&{\nil} & \\
{(\nu a)(\nu b)P}&\sim&{(\nu b)(\nu a)P} & \\
{!P}&\sim&{P \parop !P} & \\
{P \parop (\nu a) Q} &\sim& {(\nu a)(P \parop Q)} & \mbox{if $a \freshin P$}\\
{\outprefix{M}{N}.(\nu a)P} &\sim& {(\nu a)\outprefix{M}{N}.P} & \mbox{if $a \freshin M,N$}\\
{\inprefix{M}{\vec{x}}{N}.(\nu a)P} &\sim& {(\nu a)\inprefix{M}{\vec{x}}{N}.P} & \mbox{if $a \freshin M,\ve{x},N$} \\
{\caseonly\;{\ci{\vec{\varphi}}{\vec{(\nu a)P}}}} &\sim& {(\nu a)\caseonly\;{\ci{\vec{\varphi}}{\vec{P}}}} & \mbox{if $a \freshin \ve{\varphi}$}
\end{array}\]
\begin{comment}
\begin{mathpar}
\bicong{P \parop (\nu a) Q}{(\nu a)(P \parop Q)}
\and \bicong{\outprefix{M}{N}.(\nu a)P}{(\nu a)\outprefix{M}{N}.P}
\and \bicong{\inprefix{M}{\vec{x}}{N}.(\nu a)P}{(\nu a)\inprefix{M}{\vec{x}}{N}.P}
\and \bicong{\caseonly\;{\ci{\vec{\varphi}}{\vec{(\nu a)P}}}}{(\nu a)\caseonly\;{\ci{\vec{\varphi}}{\vec{P}}}}
\end{mathpar}
\end{comment}
\end{theorem}
As an example, Versari's $\pi @$~\cite{DBLP:conf/esop/Versari07} is an extension of the {\pic} with priorities. Input and output prefixes in $\pi @$ are of form $\mu : k(y)$ and $\overline{\mu} : k\langle z\rangle$, where $\mu$ is the subject, $k$ is the priority level and $y$ and $z$ are the objects. The semantics is the standard reduction semantics of the {\pic}, augmented with side conditions stating that no higher-priority reduction is possible, similar to our use of the $\priook$ predicate.
$\pi @$ can be recovered in our framework as follows. For simplicity we consider only monadic synchronisation. Let the terms be the union of $\nameset$ (corresponding to objects in $\pi @$) and $\{a:n | a \in \nameset, n \in \numbers\}$ (corresponding to subjects annotated with their priority level), let the conditions be the booleans and the assertions be $\{\unit\}$. Define channel equivalence and $\PRIO$ so that $a:n$ is equivalent to itself and has priority $n$.
As an immediate consequence, we equip $\pi @$ with a labelled semantics and a theory of strong bisimulation; no labelled semantics or bisimulation theory has been previously developed for $\pi @$.
Note that in our representation of $\pi @$, it is possible to write agents where the term $a:n$ occurs in object position. We can rule out such ill-formed agents by using the sort system for psi-calculi described in~\cite{Borgstr_m_2014}, the details of which are beyond the scope of the present paper.
For a slightly more involved example, we consider dynamic priorities. We define a psi-calculus with priorities based on the \pic, with the addition that channels may have one of two priority levels: $0$ (high) and $1$ (low). Rather than annotating prefixes with a priority level, we let channels have high priority by default, and let our assertions be the set of channels whose priority have been flipped to low priority. If a channel is asserted to be flipped twice, the assertions cancel each other and the channel is flipped back to high priority. Thus we may flip the priority of a channel $a$ dynamically by asserting $\Set{a}$. Similarly, asserting $\Set{a,b}$ flips the priorities of both $a$ and $b$. Composition of assertions is exclusive or, e.g.~$\Set{a}\otimes\Set{a,b} = \Set{b}$. To illustrate how this calculus can be used, suppose we want to enforce a fairness scheme such that synchronisations on two channels $x$ and $y$ are guaranteed to interleave. This can be achieved by swapping the priorities of $x$ and $y$ after every such synchronisation, as in the following derivation sequence, where for all $z\in\Set{x,y}$ we let $P_z = \pass{\Set z} \pll !\overline{x}.\pass{\Set{x,y}} \pll !\overline{y}.\pass{\Set{x,y}}$.
\[
\begin{array}{rclcl}
\one & \frames &
P_y \pll x\sdot x\sdot x \pll y
& \priotransarrow{\tau:0} & P_x \pll x\sdot x \pll y \\
& & & \priotransarrow{\tau:0} & P_y \pll x\sdot x \\
& & & \priotransarrow{\tau:0} & P_x \pll x \\
& & & \priotransarrow{\tau:1} & P_y
\end{array}
\]
Note that the above $\tau$ sequence is the only possible $\tau$ sequence --- as long as both $x$ and $y$ are available they are guaranteed to be consumed alternatingly.
Formally, we define this psi-calculus by letting
$\terms = \nameset$, $\conditions = \Set{x = y \;|\; x, y \in \terms} \cup \Set{\Prio M n \;|\; M \in \terms \wedge n \in \numbers}$ and by letting $\assertions$ be the finite sets of names. Moreover, let $\one$ be the empty set and $A \ftimes B = (A \cup B) - (A \cap B)$. Entailment is defined so that $\Psi \vdash x=y$ iff $x=y$, $\Psi \vdash \Prio x 1$ iff $x \in \Psi$, and $\Psi \vdash \Prio x 0$ iff $x \not\in \Psi$. Finally, we let channel equivalence be syntactic equality on names.
The definition of composition as the pairwise exclusive or on the elements of its arguments achieves the priority flip in a manner that is associative, commutative and compositional. This is a useful general technique for constructing psi-calculi where facts can be retracted.
\section{Encoding priorities}
In this section we present a translation from psi-calculi with priorities to the original psi-calculi. The main idea is that we augment the assertions with information about prefixes, and ensure that the frame of a process records precisely its enabled prefixes. The $\priook$ predicate is thus obtained from the entailment relation.
The main technical complication with this idea is that when $P$ takes a transition to $P'$, some of the top-level prefixes of $P$ may be absent in $P'$. The frame of $P'$ will always be the frame of $P$ composed with assertions that are guarded in $P$ and unguarded in $P'$; in other words $\frameof{P'} \simeq (\nu \frnames{P'})(\frass{P} \otimes \Psi)$. It follows that composing with this $\Psi$ must in effect retract the prefixes no longer available in $P'$ from $\frass{P}$. For this purpose we use a non-monotonic logic, where assertions contain
multisets with negative occurrence~\cite{blizard1990}.
\subsection{Preliminaries: integer-indexed multisets}
Intuitively, an integer-indexed multiset is like a regular multiset, except that the number of occurrences of an element may be negative. We use \emph{finite integer-indexed multisets with a maximum element} (henceforth abbreviated \emph{FIMM}), ranged over by $E$. Let $\znumbers^\infty$ denote $\znumbers \cup \Set{\infty}$. Formally, the FIMMs over a set $S$ is the set of functions $E : \msetof{S}$ such that for all but finitely many elements $s \in S$, $E(s) = 0$. We define some of the usual operations on sets as follows:
\begin{mathpar}
x \in E \;\defn\; E(x) > 0 \and
\emptyset \;\defn\; \lambda x. 0 \and
E \cup E' \;\defn\; \lambda x. (E(x) + E'(x))
\end{mathpar}
The maximal element $\infty$ will be used to represent prefixes under a replication operator (these are permanently enabled and cannot ever be retracted). We will write $\{(z_0)x_0,\,\dots,\,(z_n)x_n\}$ for the multiset $E$ such that $E(x_i) = z_i$ if $0 \leq i \leq n$, and $E(x_i) = 0$ otherwise. We will sometimes write $x_i$ to mean $(1)x_i$ and $-x_i$ to mean $(-1)x_i$.
\subsection{Preliminaries: Requisites and guarding elements}\label{sec:guardelm}
From this point in the paper, we restrict attention to psi-calculi with separate choice and prefix-guarded replication. In other words, case statements have the form $\caseonly\;{\ci{\ve{\varphi}}{\ve{\alpha. P}}}$, where either every $\alpha_i$ is an input, or every $\alpha_i$ is an output. Moreover, replications are of the form $\bang \alpha.P$. These restrictions significantly simplify our definitions and proofs. In the conclusion we briefly discuss what would be involved to lift them.
We also require that substitution has no effect on terms where the names being substituted do not occur, i.e. that if $\ve{x} \freshin M$ then $M\subst{\ve x}{\ve T} = M$. This natural requirement on substitution is found in the original publication on psi-calculi~\cite{LICS2009:Psi-calculi}, but is often omitted since it is not needed for the standard structural and congruence properties of bisimulation.
Further, for convenience we will assume that the psi-calculus under consideration has a condition $\top$ that is always true in every context, i.e. it is such that $\forall \Psi. \Psi \vdash \top$, $\forall \sigma. \top\sigma = \top$ and $\supp{\top} = \emptyset$. If such a condition is absent, it can simply be added.
A \emph{guarding element} is simply a prefix guarded by a condition. Enriching the assertions with FIMMs of guarding elements will provide all the information necessary to encode $\priook$ in the entailment relation.
\begin{definition}[guarding elements]
The set of \emph{guarding elements} of a psi-calculus $\mathcal{P} = (\terms,\assertions,\conditions,\vdash,\otimes,\sch,\unit)$ is denoted $\prefixesof{\mathcal{P}}$ and defined as
\[
\prefixesof{\mathcal P} = \conditions \times (\{\outprefix{M}{N}: M,N \in \terms\} \cup \{\inprefix{M}{\vec{x}}{X}: M,N \in \terms\})
\]
We consider guarding elements as implicitly quotiented by alpha-equivalence, where the names $\vec{x}$ in the input prefix $\inprefix{M}{\vec{x}}{X}$ bind into $N$. We will sometimes write $\alpha$ to mean $(\top,\alpha)$.
\end{definition}
\subsection{The encoding}\label{sec:cuteencoding}
Assume a psi-calculus with priorities $\mathcal{P} = (\terms,\assertions,\conditions,\vdash,\otimes,\sch,\unit,\PRIO)$. We shall encode it in the psi-calculus $\mathcal{Q} = (\terms,\assertions',\conditions',\vdash',\otimes',\sch',(\unit,\emptyset))$, whose parameters are defined as follows:
\[
\begin{array}{rcl}
\assertions' & = & \assertions \times (\msetof{\prefixesof{\mathcal{P}}}) \\
\conditions' & = & \conditions \uplus (\znumbers^\infty \times \prefixesof{\mathcal{P}}) \uplus \{M \sch' N: M,N \in \terms\} \\
(\Psi,E) \otimes' (\Psi',E') & = & (\Psi \otimes \Psi', E \cup E') \\
\end{array}
\]
\[
\begin{array}{rcl}
(\Psi,E) \vdash' \varphi & = & \Psi \vdash \varphi \qquad \mbox{if $\varphi \in \conditions$}\\
(\Psi,E) \vdash' (z)(\varphi, \alpha) & = & E(\varphi, \alpha) = z \\
(\Psi,E) \vdash' M \sch' N & = & \Psi \vdash M \sch N \wedge \neg\exists M'\;N'\;n\;m\;X\;K\;\ve{x}\;\ve{L}\;\varphi\;\varphi'. \Psi \vdash M' \sch N' \\
& & \wedge \Psi \vdash \Prio{M}{m} \wedge \Psi \vdash \Prio{M'}{n} \wedge n < m \wedge (\varphi,\inprefix{M'}{\ve x}X) \in E \\
& & \wedge (\varphi',\outprefix{N'}{K}) \in E \wedge K = X\subst{\ve x}{\ve L} \wedge \Psi \vdash \varphi \wedge \Psi \vdash \varphi'\\
\end{array}
\]
Assertions in $\assertions'$ augment the original assertions with FIMMs of guarding elements, representing the top-level prefixes of a process. The conditions are augmented with multiplicity tests on elements of the FIMMs (only needed for technical reasons concerning the compositionality of $\otimes'$), as well as channel equivalence statements. Composition and entailment of multiplicity tests and conditions in $\conditions$ are straightforward. The definition of entailment of channel equivalence statements intuitively means that two channels $M,N$ are equivalent in $(\Psi,E)$ if (1) they are equivalent in $\Psi$, and (2) $E$ does not contain prefixes that can communicate with each other with a priority higher than that of $M,N$. This is the mechanism by which we prevent lower-priority actions in the translations: those actions that would be ruled out by $\priook$ in $\mathcal{P}$ are ruled out in $\mathcal{Q}$ by not being channel equivalent to anything.
In order to avoid bogging down the notation with brackets, we introduce some syntactic sugar for assertions in $\assertions'$. We will sometimes write $\Psi$ for $(\Psi,\emptyset)$ and $E$ for $(\unit,E)$. Further, we will sometimes write single-element multisets without the curly brackets, ie. $(z)x$ for $\{(z)x\}$. For an example, combined with the previously introduced syntactic sugar for multisets and guarding elements, we may write $(\unit,\{(1)(\top,\alpha)\})$ as simply $\alpha$, and $(\unit,\{(-1)(\top,\alpha)\})$ as $-\alpha$.
\begin{lemma} $\mathcal{Q}$ is a psi-calculus, meaning that it satisfies the requisites outlined in Section~\ref{sec:psi}.
\end{lemma}
The translation of agents from $\mathcal{P}$ to $\mathcal{Q}$ is defined by the function $\semb{\_} : \processesof{\mathcal P} \Rightarrow \processesof{\mathcal Q}$. The main idea is that in parallel to every prefix, we add the prefix as an assertion (recall that $\pass{\Psi}$ denotes the assertion $\Psi$ occurring as a process), so that it can be used when deciding channel equivalences. The continuation after the prefix contains the same prefix negatively, and since $\{\alpha\} \cup \{-\alpha\}=\emptyset$ the effect is to retract the prefix from the frame once it has been used,
and thus ensures that the frame of an agent $\semb{P}$ contains an up-to-date copy of the top-level prefixes of $P$. Since replicated prefixes are permanently enabled, a replicated prefix is asserted with infinite multiplicity to ensure that it is never retracted. For $\caseonly$ statements, we make sure to retract the guarding elements associated with the other branches after a particular branch has been chosen.
\[
\begin{array}{rcl}
\semb{\nil} & = & \nil \\
\semb{\pass{\Psi}} & = & \pass{(\Psi,\emptyset)}\\
\semb{P \parop Q} & = & \semb{P} \parop \semb{Q} \\
\semb{(\nu x)P} & = & (\nu x)\semb{P} \\
\semb{\alpha. P} & = & \pass{\alpha} \parop \alpha. (\semb{P} \parop \pass{-\alpha}) \\
\semb{\bang \alpha. P} & = & \pass{(\infty)\alpha} \parop \bang \alpha. (\semb{P} \parop \pass{-\alpha}) \\
\semb{\caseonly\;{\ci{\ve{\varphi}}{\ve{\alpha. P}}}} & = & \pass{(\ve{\varphi},\ve{\alpha})} \parop \caseonly\;{\ci{\ve{\varphi}}{\ve{\alpha}. (\ve{\semb{P}} \parop \pass{(-1)(\ve{\varphi},\ve{\alpha})})}} \\
\end{array}
\]
Recall that we require that substitution has no effect on terms where the names being substituted do not occur. To see why, consider the encoding of the input prefix $\alpha = \inprefix{M}{\ve{x}}{N}$, where $\ve{x}$ is chosen to be fresh in $M$. If the encoding takes a transition $\trans{\semb{\alpha}}{\inlabel{M}{N\subst{\ve{x}}{\ve{L}}}}{\pass{\alpha} \parop \pass{-\alpha\subst{\ve{x}}{\ve{L}}}}$, we need that $\alpha\subst{\ve{x}}{\ve{L}} = \alpha$ to achieve a retraction of $\alpha$. This follows from our requirement since $\ve{x}$ does not occur freely in $\alpha$.
\section{Quality of the encoding}\label{sec:quality}
In this section, we show that the encoding presented in Section~\ref{sec:cuteencoding} satisfies strong operational correspondence, and briefly discuss two other quality criteria: Gorla's framework~\cite{DBLP:conf/concur/Gorla08} and full abstraction.
Let $\equiv$, pronounced \emph{structural congruence}, be the smallest congruence on processes that satisfies the commutative monoid laws with respect to $(\parop,\nil)$ and the rules $!P \equiv P \parop !P$ and $\nil \equiv \pass{\one}$ and $\pass{\Psi} \parop \pass{\Psi'} \equiv \pass{\Psi \otimes \Psi'}$.
The main result of this paper is a one-to-one transition correspondence between agents in $\mathcal{P}$ and their encodings in $\mathcal{Q}$:
\begin{theorem}[Strong operational correspondence]\label{thm:stopcon}$\,$
\begin{enumerate}
\item If $\prioframedtrans{\Psi}{P}{\alpha}{P'}$ and $\bn\alpha \freshin P$ and $\alpha \neq \tau:p$,
then there exists $P''$ such that $\framedtrans{(\Psi,\emptyset)}{\semb{P}}{\alpha}{P''}$ and $\semb{P'} \equiv P''$.
\item If $\prioframedtrans{\Psi}{P}{\tau:p}{P'}$,
then there exists $P''$ such that $\framedtrans{(\Psi,\emptyset)}{\semb{P}}{\tau}{P''}$ and $\semb{P'} \equiv P''$.
\item If $\framedtrans{(\Psi,\emptyset)}{\semb{P}}{\alpha}{P'}$ and $\bn\alpha \freshin P$ and $\alpha \neq \tau$, then there exists $P''$ such that $\semb{P''} \equiv P'$ and $\prioframedtrans{\Psi}{P}{\alpha}{P''}$.
\item If $\framedtrans{(\Psi,\emptyset)}{\semb{P}}{\tau}{P'}$, then there exists $p$ and $P''$ such that $\semb{P''} \equiv P'$ and $\prioframedtrans{\Psi}{P}{\tau:p}{P''}$.
\end{enumerate}
\end{theorem}
Note that a simplification of the encoding with $\semb{\bang \alpha. P} = \pass{(\infty)\alpha} \parop \bang \alpha. \semb{P} $ would render the above theorem false, since we would then lose the property that $\semb{\bang \alpha. P} \equiv \semb{\alpha. P \parop \bang\alpha. P}$, and transitions may unfold replications.
Gorla~\cite{DBLP:conf/concur/Gorla08} proposes a unified approach to encodability results, wherein a translation function is considered an encoding if it satisfies the five properties \emph{compositionality},
\emph{name invariance},
\emph{operational correspondence},
\emph{divergence reflection},
and
\emph{success sensitiveness}.
Because our encoding satisfies strong operational correspondence, the three last criteria follow immediately. Name invariance is immediate since our encoding is equivariant, and compositionality holds with the caveat that we must consider replicated prefixes $!\alpha.P$ as an operator in itself, rather than considering the replication and the prefix as separate operators, and likewise for $\caseonly$-guarded prefixes.
Full abstraction means that two agents are equivalent iff their translations are equivalent.
The encoding presented in Section~\ref{sec:cuteencoding} is not fully abstract with respect to strong bisimilarity. This is because we require bisimilar agents to be statically equivalent, but the translation function introduces assertions such that the translation of bisimilar agents may not be statically equivalent. For a simple example, consider the agents $P = \alpha. \nil$ and $Q = \alpha. P$, where $\alpha$ is an output prefix. Clearly $P \parop P \priobisim Q$ holds, but for $\semb{P} = \pass{\alpha} \parop \alpha. (\nil \parop \pass{-\alpha})$ and $\semb{Q} = \pass{\alpha} \parop {\alpha. (\semb{P} \parop \pass{-\alpha})}$, we have $\frameof{\semb{P \parop P}} \vdash' (2)\alpha$ but $\frameof{\semb{Q}} \not\vdash' (2)\alpha$ and hence $\semb{P \parop P} \not\bisim \semb{Q}$.
At first glance, this difference between $\semb{P \parop P}$ and $\semb{Q}$ seems to be an unimportant technicality: the conditions $(2)\alpha$ and $(1)\alpha$ are not intended to be used as guards in $\caseonly$-statements. Their only use is in the evaluation of channel equivalences, but $\frameof{\semb{P \parop P}}$ and $\frameof{\semb{Q}}$ entail the same channel equivalences since the set of prefixes available coincides.
To motivate that they must be considered different, consider the distinguishing context $R = \pass{-\alpha} \parop \pass{\beta} \parop \gamma.0$, where $\beta$ is an input that can synchronise with $\alpha$, and $\gamma$ has lower priority than $\alpha$; we have that $R \parop \semb{Q}$ can take an action on $\gamma$, but $R \parop \semb{P \parop P}$ cannot. This highlights an interesting difference between $\mathcal{P}$ and $\mathcal{Q}$: in $\mathcal{P}$, a prefix describes both an interaction possibility and a constraint on other (lower-priority) interactions; in $\mathcal{Q}$, the interaction possibility and interaction constraint are two separate syntactical elements. This means that in $\mathcal{Q}$ we may write $\pass{\alpha}$, which is a process with no transitions that blocks lower-priority transitions as though it had an $\alpha$-transition; conversely $\alpha. P$ has a non-blocking $\alpha$-transition that may be blocked by higher-priority transitions.
Note that in the counterexample to full abstraction presented above, the context $R$ is not in the range of $\semb{\cdot}$. Thus our encoding may well satisfy weak full abstraction~\cite{Parrow_2008}, meaning that full abstraction holds if we restrict attention to contexts in the range of $\semb{\cdot}$. An investigation of this is deferred to future work.
A related question is whether a fully abstract encoding of $\mathcal{P}$ into some psi-calculus exists. The following theorem, inspired by recent work by Gorla and Nestmann~\cite{GorlaNestmann:Abstraction} and Parrow~\cite{Parrow:Abstraction}
, shows that because of the generality of the psi-calculi framework a trivial fully abstract ``encoding'' with strong bisimilarity as the target equivalence always exists, regardless of the source language and source equivalence under consideration.
Let $\mathbf{S}$ be a set ranged over by $s$, and $\sim$ be an equivalence on $\mathbf{S}$. Then
there is a psi-calculus $\mathcal S$ with no terms, with elements of $\mathbf{S}$ as assertions and conditions, where entailment is $\sim$. Define the encoding $\semb{\_}_\mathbf{S} : \mathbf{S} \Rightarrow \processesof{\mathcal S}$ by $\semb{s}_\mathbf{S} \defn \pass{s}$.
\begin{theorem}
$s \sim s'$ iff $\semb{s}_\mathbf{S} \bisim \semb{s'}_\mathbf{S}$
\end{theorem}
This ``encoding'' simply embeds both the source language and source equivalence into a target language with no transition behaviour at all.
We conclude that
a meaningful approach to full abstraction would have to impose additional criteria.
For an example, if we consider Gorla's criteria presented earlier, this ``encoding'' satisfies name invariance and divergence reflection, but fails to satisfy compositionality, operational correspondence and success sensitiveness.
\section{Conclusion}
In this paper, we have defined an extension of the psi-calculi framework with dynamic action priorities, and translated it to the original framework.
This illustrates the high expressiveness of the assertion mechanism in psi-calculi: usually, it is necessary to introduce negative premises or define a multi-layered transition system in order to obtain action priorities in a given calculus; for psi-calculi, what is already there suffices.
The extension with explicit priorities is interesting in its own right despite the encoding. Expressiveness is not usefulness. Modelling a system with priorities in terms of the translation would be more cumbersome than representing priorities directly.
Also, strong bisimulation in the extension is useful for proving equivalences that fail to hold in the encoding.
The most closely related development to psi-calculi with priorities is the \emph{attributed {\pic} with priorities}, written $\pi(\mathcal{L})$~\cite{articlereference201003038754456007}. It is designed as a generalisation of $\pi @$~\cite{DBLP:conf/esop/Versari07} and the stochastic {\pic}. Input and output prefixes take the form $e_1[e'_1]?\ve{x}$ and $e_2[e'_2]!\ve{y}$, where $e_1$ and $e_2$ are subjects, $\ve{x}$ and $\ve{y}$ are objects and $e'_1$ and $e'_2$ are interaction constraints, which may be instantiated to priorities or stochastic rates. $e$ ranges over expressions in an \emph{attribute language}, which is a kind of call-by-value $\lambda$-calculus equipped with a big-step reduction relation. The idea in the case of priorities is that if the expressions $e_1$ and $e_2$ reduce to the same channel name, and $\ve{e}$ reduces to some values $\ve{v}$, and the application $e'_1 e'_2$ reduces to the priority level $r$, then $e_1[e'_1]?\ve{x}.P \parop e_2[e'_2]!\ve{e}.Q$ reduces to $P\subst{\ve{x}}{\ve{v}} \parop Q$, unless another pair of prefixes can similarly communicate on a higher priority level. The focus is on developing type systems to prevent mismatches, on showing how the calculus can be applied to model phenomena in systems biology, and on the development and implementation of a stochastic simulation algorithm.
While $\pi(\mathcal{L})$ and our approach both generalise $\pi @$, the way the priorities are set up have several interesting differences that suggest incomparable expressive power in general. Priority levels in $\pi(\mathcal{L})$ are taken from an arbitrary partial order, whereas our priorities are natural numbers. Thus in $\pi(\mathcal{L})$ we may have systems where actions have mutually incomparable priority levels, unlike psi-calculi with priorities. The reason we use natural numbers is that the proof of Theorem~\ref{thm:priobisim} uses induction and successor arithmetic on the priority level; for future work we would like to investigate alternative proof strategies that would permit a generalised notion of priorities. In psi-calculi, priority levels are associated to communication channels, whereas in $\pi(\mathcal{L})$ they are associated with a particular pair of prefixes. The priority level of a particular pair in $\pi(\mathcal{L})$ is however static and cannot be influenced by the environment in any way, whereas in our approach priorities are dynamic and may change arbitrarily as the assertion environment evolves. While psi-calculi has no explicit notion of computation on data such as that given by the attribute language, the substitution function can be chosen so that it performs explicit computation on data, or implicit computation can be performed during the evaluation of entailments. For a detailed discussion of how to express computation on data in psi-calculi we refer to~\cite{Borgstr_m_2014}.
The translation assumes separate choice and prefix-guarded replication. An interesting question is if these assumptions can be relaxed. Allowing mixed choice is possible, but a different definition of guarding elements must be made, that records which prefixes occur in different branches of the same $\caseonly$-statement. With the current definition, $\semb{\caseonly\;{\ci{\top}{\overline{M}}\casesep \ci{\top}{\underline{M}}}}$ has the same guarding elements as $\semb{\overline{M} \parop \underline{M}}$, meaning that the former erroneously blocks other transitions as if a communication on $M$ could be derived. Allowing unguarded choice and replication would be more difficult, but we conjecture that it is possible at the expense of compositionality. The solution would involve extending the guarding elements to contain whole syntax trees, including binders. We then lose compositionality since if e.g. $\semb{\caseonly\;{\ci{\top}{P}\casesep\ci{\top}{Q}}}$ takes a transition from $Q$, the derivative must contain an assertion that retracts all interaction possibilities offered by $P$. Hence the translation of $Q$ depends on $P$, violating compositionality.
Another way to introduce priorities in process calculi is with a \emph{priority choice} operator $P +\rangle Q$, as is done for CCS in~\cite{Camilleri1991}. It is like the standard choice operator, with two exceptions. First, $P$ and $Q$ may for technical reasons not contain unguarded output prefixes. Second, transitions from $P$ take precedence over $Q$. More precisely, its semantics is defined so that it may always act as $P$, but may act as $Q$ only if no synchronisation on the prefixes of $P$ is possible in the current environment. This operator could be encoded in psi-calculi using techniques similar to those presented in this paper. The main idea is to augment the assertions with information about output prefixes as in Section~\ref{sec:cuteencoding}, and to translate priority choice as $\semb{P +\rangle Q} = \caseonly\;{\ci{\top}{\semb{P}} \casesep \ci{\varphi_P}{\semb{Q}}}$, where $\varphi_P$ is a condition that holds if no output prefixes matching the inputs of $P$ are enabled in the current environment. A more detailed investigation of this idea is deferred to future work.
We would also like to investigate if a result by Jensen~\cite{Torp94interpretingbroadcast}, that broadcast communication can be encoded in CCS with priority choice up-to weak bisimulation, can be adapted to our setting. If broadcast psi-calculi~\cite{borgstroem.huang.ea:broadcast-psi-sefm} can be encoded in psi-calculi with priorities, then by transitivity so can the original psi-calculi. This would contrast with the situation in the {\pic}, where broadcast communication cannot be encoded~\cite{ene.muntean:expressiveness-point}.
Since both the original psi-calculi and their extension with priorities have been formalised in Nominal Isabelle, we aim to formalise the correspondence results in this paper, in order to be more certain of their correctness. As a first step, it would be necessary to develop a formalisation of FIMMs in Isabelle, and integrate it with the nominal logic package.
\end{document} |
\begin{document}
\def\colon\thinspace{\colon\thinspacelon\thinspace}
\def\colon\thinspaceeq{\colon\thinspaceloneqq\thinspace}
\title{Examples of contact mapping classes \ of infinite order in all dimensions}
\begin{abstract}
We give examples of tight high dimensional contact manifolds admitting a contactomorphism whose powers are all smoothly isotopic but not contact-isotopic to the identity.
This is a generalization of an observation in dimension $3$ by Gompf, also reused by Ding and Geiges.
\end{abstract}
\section{Introduction}
\label{SecIntro}
In this paper we study the topology of the space of contactomorphisms $\diff{V,\xi}$ of a given contact manifold $(V,\xi)$, comparing it with the one of the space of diffeomorphisms $\diff{V}$ of the underlying smooth manifold $V$.
\\
It is known that the space $\ContStr{V}$ of contact structures on $V$ plays an important role in the study of the relations between $\diff{V,\xi}$ and $\diff{V}$.
Indeed, if $V$ is a closed manifold, then the map $\diff{V}\rightarrow\ContStr{V}$, defined by $\phi\mapsto \phi_*\xi$, is a locally-trivial fibration with fiber $\diff{V,\xi}$; this essentially follows from (the proof of) Gray's theorem, as explained for instance in \cite{MasGir15,MasFibrNotes}. (See also \cite{GeiGon04}, in which it is proved that the map is a Serre fibration, which is enough for this discussion.)
This fibration induces a long exact sequence of homotopy groups
\begin{equation}
\label{eq:long_exact_seq}
\ldots \rightarrow \pi_{k+1}\left(\ContStr{V}\right)\rightarrow \pi_k\left(\diff{V,\xi}\right) \xrightarrow{j_*} \pi_k\left(\diff{V}\right)\rightarrow \pi_k\left(\ContStr{V}\right)\rightarrow\ldots
\end{equation}
where $j_*:\pi_k\left(\diff{V,\xi}\right)\rightarrow\pi_k\left(\diff{V}\right)$ is the map induced on the homotopy groups by the natural inclusion $j:\diff{V,\xi}\rightarrow\diff{V}$.
(Here, the homotopy groups of $\diff{V}$ and $\diff{V,\xi}$ are considered with base point $\Id$, whereas those of $\ContStr{V}$ are considered based at $\xi$; for notational ease, this natural choice of base points is suppressed from the notation.)
The higher (i.e.\ $k\geq2$) homotopy groups $\pi_k(\ContStr{V},\xi)$ have not been studied much in the literature.
To our knowledge, the only works dealing with these higher homotopy groups are \cite{Bou06,CasSpa16,FerGir19},
which use, respectively, contact homology, some algebraic topology and the $h$-principle from \cite{BorEliMur15} in order to study the groups $\pi_k\left(\ContStr{V},\xi\right)$ for some particular contact manifolds $(V,\xi)$ and some $k>1$.
In the rest of the literature, the focus is typically on the study of the map $j_*:\pi_0\left(\diff{V,\xi}\right)\rightarrow\pi_0\left(\diff{V}\right)$ induced by $j$ on the space of connected components, which is of course deeply related, via the exact sequence above, to the study of $\pi_1\left(\ContStr{V},\xi\right)$.
The results available so far in this direction also consist in concrete examples of contact manifolds $(V,\xi)$ where, thanks to the specific geometry of the underlying manifold $V$, one can use techniques from both contact geometry, such as convex surface theory in the tight and overtwisted $3$-dimensional case and holomorphic curves in the tight high-dimensional case, and from algebraic topology, in the overtwisted high dimensional case, to obtain results on the map $j_*\vert_{\pi_{0}}$ and on the fundamental group of $\ContStr{V}$.
\\
For instance, one can find in the literature several examples of contact manifolds $(V,\xi)$ for which $\ker\left(j_*\vert_{\pi_{0}}\right)$ is non-trivial;
the interested reader can consult \cite{Gom98,Gir01,GeiGon04,Bou06,DinGei10,GeiKlu14,MasGir15} and \cite{Dym01,Vog16} for, respectively the tight and overtwisted $3$-dimensional cases, and \cite{Bou06,MasNie16,LanZap15} and \cite{My17a} for, respectively, the tight and overtwisted higher-dimensional cases.
Notice that the examples in \cite{Bou06,MasNie16,LanZap15} are tight according to the definition of overtwistedness in higher dimensions in \cite{BorEliMur15}, which generalizes the $3$-dimensional one given in \cite{Eli89}.
This paper focuses more precisely on the problem of the existence of infinite cyclic subgroups in $\ker(j_*\vert_{\pi_{0}})$.
To our knowledge, the only known example of such phenomenon is given in \cite{Gom98,DinGei10}.
More precisely, in \cite[Remark at page 642]{Gom98}, using rotation numbers, Gompf argues that $\sphere{2}\times\sphere{1}$, equipped with its unique (up to isotopy) tight contact structure $\xi_{std}$, has a contact mapping class of infinite order.
Then, starting from Gompf's remark, Ding and Geiges prove in \cite{DinGei10} that $\ker(j_*\vert_{\pi_{0}})$ and $\pi_1(\ContStr{\sphere{2}\times\sphere{1}},\xi_{std})$ are actually both isomorphic to $\mathbb{Z}$.
Our aim is to give explicit examples of high-dimensional tight manifolds that admit an element of infinite order in $\ker(j_*\vert_{\pi_{0}})$.
This is achieved by first exhibiting elements of infinite order in $\pi_0(\diff{V,\xi})$ for $V$ given by the product of the double $\mathcal{D}W$ of a stabilized Weinstein domain $W$ and the circle $\sphere{1}$, equipped with a natural fillable contact structure $\xi$ on it.
For some particular choices of $W$, these infinite-order elements of $\pi_0(\diff{V,\xi})$ can moreover be shown to be in $\ker(j_*\vert_{\pi_{0}})$.
\\
More precisely, we start by analyzing the following general situation.
Consider a Weinstein manifold $(F^{2n-2},\omega_F,Z_F,\psi_F)$; namely, $F$ has only positive boundary component $\partial F = \partial_+ F$, $\omega_F$ is a symplectic form on $F$, $\psi_F\colon\thinspacelon F \rightarrow \mathbb{R}$ is an exhausting (i.e.\ proper and bounded from below) Morse function, such that $\psi_F$ is constant on $\partial_+ F$ and $Z_F$ is a complete Liouville vector field for $\omega_F$, which is also gradient-like for $\psi_F$.
Consider then the stabilization $(F\times\mathbb{C},\omega_F\oplus\omega_0,Z_F+Z_0,\psi_F+\abs{.}^2_\mathbb{C})$, where $\omega_0=rdr\wedge d\varphi$ and $Z_0 = \frac{1}{2}r\partial_r$, using coordinates $z=re^{i\varphi}\in\mathbb{C}$.
Suppose that $c> \min \psi_F$ is a regular value of $\psi\colon\thinspaceeq\psi_F+\abs{.}^2_\mathbb{C}$ and let $W$ be the compact domain $\psi^{-1}((-\infty,c])$.
We also assume that there is an almost complex structure $J_F$ on $F$, which is tamed by $\omega_F$ and such that $(TF,J_F)$ is trivial as complex bundle over $F$.
\\
Consider now the Weinstein manifold $(F\times\mathbb{C}\times\mathbb{R}\times\sphere{1},\omega',Z',\psi')$, where, using coordinates $(s,\theta)\in\mathbb{R}\times\sphere{1}$, $\omega' = \omega_F + \omega_0 + ds\wedge d\theta$, $Z' = Z_F + Z_0 + s\partial_s$ and $\psi'(p,z,s,\theta) = \psi(p,z) + s^2$.
The preimage $(\psi')^{-1}(c)$, which is diffeomorphic to the product of the double $\mathcal{D}W\colon\thinspaceeq W\cup_{\partial W} \overline{W}$ of $W$ and $\sphere{1}$, is naturally equipped with the contact structure $\xi=\ker\alpha$, where $\alpha = (\iota_{Z'}\omega')\vert_{\mathcal{D}W\times\sphere{1}}$.
Moreover, the diffeomorphism of $F\times\mathbb{C}\times\mathbb{R}\times\sphere{1}$ given by $(q,z,s,\theta)\mapsto (q,e^{i\theta}z,s,\theta)$ restricts to a well defined diffeomorphism $\Psi$ of $\mathcal{D}W\times\sphere{1}$.
\\
We will then prove the following result:
\begin{thm}
\label{ThmMain}
In the setting described above (recall, in particular, that $(TF,J_F)$ is assumed to be trivial), the diffeomorphism $\Psi$ of $\mathcal{D}W\times\sphere{1}$ is smoothly isotopic to a contactomorphism $\Psi_c$ of $(\mathcal{D}W\times\sphere{1},\xi)$ such that, for each integer $k\neq0$, its $k$-th iterate is not contact-isotopic to the identity.
\end{thm}
A direct application of \Cref{ThmMain} with $F=\mathbb{C}^{n-1}$, $\omega_F= \sum_{i=1}^{n-1}r_i dr_i\wedge d\varphi_i$, $Z_F =\frac{1}{2} \sum_{i=1}^{n-1} r_i\partial_{r_{i}}$, $\psi_F(z_1,\ldots,z_{n-1})=r_1^2+\ldots + r_{n-1}^2$ and $c=1$, where we use polar coordinates $(z_1=r_1e^{i\varphi_{1}},\ldots,z_{n-1}=r_{n-1}e^{i\varphi_{n-1}})$ on $F=\mathbb{C}^{n-1}$, gives the following generalization of the observation in \cite{Gom98} to higher dimensions:
\begin{cor}
\label{CorMain}
Let $(z_1,\ldots,z_n,s,\theta)$ be coordinates on $\mathbb{C}^n\times\mathbb{R}\times\sphere{1}=\mathbb{R}^{2n+1}\times\sphere{1}$ and $\xi$ be the tight contact structure on $V\colon\thinspaceeq\mathbb{S}^{2n}\times\sphere{1}$ defined by the restriction of $\lambda = sd\theta + \frac{1}{2}\sum_{i=1}^{n} r_i^2 d\varphi_i $ on $\mathbb{R}^{2n+1}\times\sphere{1}$ to $\mathbb{S}^{2n}\times\sphere{1}=\{s^2 + \sum_{i=1}^{n}\vert z_i \vert^2=1\}$.
Consider now the diffeomorphism $\Psi$ of $\mathbb{S}^{2n}\times\sphere{1}$ given by the restriction of
\begin{align*}
\mathbb{R}^{2n+1}\times\sphere{1} & \longrightarrow\mathbb{R}^{2n+1}\times\sphere{1} \\
(z_1,\ldots,z_n,s,\theta)& \mapsto (z_1,\ldots,z_{n-1},e^{i\theta}z_n,s,\theta)
\end{align*}
\\
Then, $\Psi$ is smoothly isotopic to a contactomorphism $\Psi_c$ of $(V,\xi)$ such that $[\Psi_c^2]$ generates an infinite cyclic subgroup of $\ker\left(\pi_0\diff{V,\xi}\rightarrow\pi_0\diff{V}\right)$.
\end{cor}
Notice that each even power of $\Psi_c$ in \Cref{CorMain} is indeed smoothly isotopic to the identity:
because the fundamental group of $SO(m)$ is isomorphic to $\mathbb{Z}_2$ for all $m\geq3$,
there is, for all $k\in\mathbb{N}$, a smooth isotopy of $\mathbb{S}^{2n}\times\sphere{1}$, (globally) preserving each submanifold $\mathbb{S}^{2n}\times\{pt\}$, from $\Psi^{2k}$ to the identity; in particular, $\Psi_c^{2k}$ is also smoothly isotopic to the identity.
Analogously to \Cref{CorMain}, \Cref{ThmMain} can be applied to the case of $F=T^*\mathbb{T}^n$, $\omega_F=\sum_{i=1}^n dp_i\wedge dq_i$,
$Z_F=\sum_{i=1}^n p_i\partial_{p_{i}}$, $c=1$ and $\psi_F(q_i,p_i)=\sum_{i=1}^n p_i^2$ (perturbed to a Morse function with a perturbation supported on a neighborhood of $\psi_F^{-1}(0)$).
This gives, for each $n\geq1$, another explicit example of tight $(V^{2n+1},\xi)$ such that $\ker\left(\pi_0\diff{V,\xi}\rightarrow\pi_0\diff{V}\right)$ has an infinite cyclic subgroup.
\\
Indeed, each even power of the diffeomorphism $\Psi$ is smoothly isotopic to the identity. This follows from the facts that $T^*\mathbb{T}^n \simeq \mathbb{T}^n\times\mathbb{R}^n$, that $\mathcal{D}W\times\sphere{1}\simeq\mathbb{T}^n\times \sphere{n+2}\times\sphere{1}$ and that, for each $\theta\in\sphere{1}$, $\Psi\colon\thinspace\mathbb{T}^n\times \sphere{n+2}\times\sphere{1}\rightarrow\mathbb{T}^n\times \sphere{n+2}\times\sphere{1}$ acts trivially on the first and thirds factors and as a rotation of angle $\theta$ around a given axis on each $\{pt\}\times\sphere{n+2}\times\{\theta\}$; because $\pi_1(SO(m))\simeq\mathbb{Z}_2$ for each $m\geq3$, we can then conclude, as done in the case of \Cref{CorMain}, that $\Psi_c^{2k}$ is smoothly isotopic to the identity, for each $k\neq1$.
\begin{remark}
\label{rmk:formal}
To be precise, the actual content of \Cref{ThmMain}, and hence of \Cref{CorMain}, is that $\Psi_c$ is not homotopic to $\Id$ among \emph{formal contactomorphisms} (cf.\ \Cref{rmk:formal_bis}).
In analogy with the case of formal isocontact embeddings in \cite[Section 12.3]{EliMisBook}, by \emph{formal contactomorphism} we mean here a pair $(f,F_t)$ where $f\in\diff{V}$ and $(F_t)_{t\in[0,1]}$ is a path of fiber-wise injective morphisms $TV\to TV$, each lifting $f$, which starts at $F_0=d f$ and ends at a certain $F_1$ preserving both $\xi$ and the conformal symplectic structure $\CS_\xi$, i.e.\ $F_1(\xi)=\xi$ and $(F_1)_*\CS_{\xi} = \CS_{\xi}$.
We denote the space of such pairs by $\diff{V,\xi}f$.
(Notice that there is an alternative, weaker but equally natural, notion of formal contactomorphism in the literature, which gives a long exact sequence analogous to the one in \Cref{eq:long_exact_seq} but with the space of almost contact structures; see \cite[Appendix A]{CKS18}.)
In particular, we point out that the question of finding (explicit) examples of contactomorphisms which are of infinite order in $\pi_0(\diff{V,\xi})$ but are formally trivial, i.e.\ trivial in $\pi_0(\diff{V,\xi}f)$, is still open.
\end{remark}
\section{Preliminaries}
In \Cref{SubSecProdDoubleLiouvDomWithCircle} we describe how, given a Liouville domain $W$, one can naturally construct an explicit Liouville manifold having $\mathcal{D}W\times\sphere{1}$ as convex boundary, as well as contactomorphisms of the latter; this will then be used in the case of Weinstein domains in the proof of \Cref{ThmMain}.
\\
\Cref{SubSecFamLegBases} introduces a simple invariant of homotopical nature for contact isotopy (actually, formal contact isotopy) classes of contactomorphisms, namely \emph{families of Lagrangian frames}.
\subsection{Product of doubled Liouville domains and $\sphere{1}$}
\label{SubSecProdDoubleLiouvDomWithCircle}
Let $\widehat{W}$ be a smooth manifold of dimension $m$ and $f\colon\thinspace \widehat{W}\rightarrow \mathbb{R}$ be a proper and bounded from below function which is also a \emph{regular equation} of a (cooriented) hypersurface $M\subset \widehat{W}$, i.e. $f$ is transverse to $0$ and satisfies $M=f^{-1}(0)$ (with coorientation).
We denote by $W$ the compact submanifold $f^{-1}((-\infty,0])$ of $\widehat{W}$.
\begin{definition}
\label{DefDoubleFun}
We denote by $\mathcal{D}funW{f}$ the smooth manifold given by $\{(p,s) \in \widehat{W}\times\mathbb{R} \,\vert\, s^2+f(p)=0\}$, and we call it \emph{$f$-double} of $W$.
\end{definition}
Notice that $\mathcal{D}funW{f}$ is just obtained by gluing two copies of $W$, embedded as graphs of $W$ inside $W\times\mathbb{R}$, one in the region $\{s\geq0\}$ and one in the region $\{s\leq0\}$, along their common boundary, which lies in $\{s=0\}=W\times\{0\}$.
This justifies the nomenclature ``$f$-double''.
We also point out that $\mathcal{D}funW{f}$ is indeed a smooth submanifold of $\widehat{W}\times\mathbb{R}$, because the function $\widehat{W}\times\mathbb{R}\rightarrow\mathbb{R}$ given by $(p,s)\mapsto s^2 + f(p)$ is transverse to $0$.
\\
Indeed, one can always find a vector field $Z$ on $\widehat{W}$ which is \emph{boundary-gradient-like} for $f$, i.e which satisfies $df(Z)\geq 0$ everywhere on $\widehat{W}$ and $df(Z)>0$ along $M=f^{-1}(0)$.
More precisely, there is a vector field $Z'$ on a neighborhood $U$ of the (cooriented) hypersurface $M$ such that $df(Z')>0$ on $U$, and we can choose $Z$ to be $Z'$ multiplied by a non-negative cutoff function $\chi$ supported in $U$.
Then, $d(s^2+f)(s\partial_s + Z) = 2s^2 + df(Z)>0$ along $\mathcal{D}funW{f}\subset \widehat{W}\times \mathbb{R}$, which shows that $\mathcal{D}funW{f}$ is a regular hypersurface.
\begin{notation}
\label{NotDouble}
If $f\colon\thinspace\widehat{W}\rightarrow\mathbb{R}$, we denote by $f^\mathcal{D}\colon\thinspace \widehat{W}\times\mathbb{R}\rightarrow\mathbb{R}$ the function $f^\mathcal{D}(p,s)=s^2+f(p)$.
In particular, if $f$ is an equation of the hypersurface $M\subset \widehat{W}$, then $f^\mathcal{D}$ defines the hypersurface $\mathcal{D}W\subset\widehat{W}\times\mathbb{R}$, as shown above.
\\
In a similar way, if $Z$ is a vector field on $\widehat{W}$, we denote by $Z^\mathcal{D}$ the vector field $Z+s\partial_s$ on $\widehat{W}\times\mathbb{R}_s$; if $Z$ is boundary-gradient-like for $f$, then so is $Z^\mathcal{D}$ for $f^\mathcal{D}$.
\end{notation}
We now describe how $\mathcal{D}funW{f}$ actually depends on the choice of $f$.
For this, we first state a general uniqueness lemma.
\begin{lemma}
\label{LemmaGenUniqSmoothDouble}
Let $Y$ be a smooth manifold and $(g_t)_{t\in[0,1]}$ be a smooth family of functions $g_t\colon\thinspace Y\to\mathbb{R}$ such that $0$ is a regular value for each $g_t$.
Additionally, assume that each $N_t\colon\thinspaceeq g_t^{-1}(0)$ is compact and that there is a vector field $V$ on $Y$ which is boundary-gradient-like for every $g_t$ with respect to $N_t$ (i.e.\ $dg_t(Z)>0$ along $N_t$).
Then the flow $\psi_{X_t}^t$ of
\begin{equation*}
X_t\colon\thinspaceeq\frac{\dot{g_t}}{d g_t(V)}\, V
\end{equation*}
satisfies $\psi_{X_{t}}^t(N_t)=N_0$.
In particular, all the $N_t$'s are diffeomorphic.
\end{lemma}
\begin{proof}
The smooth function $G\colon\thinspace Y\times [0,1]\rightarrow \mathbb{R}$, given by $G(y,t)= g_t(y)$, is transverse to $0$: indeed, $dG(V)>0$ along $G^{-1}(0)=\bigcup_t N_t\times\{t\}$.
Then, $G^{-1}(0)$ is a smooth submanifold of $Y\times [0,1]$, which is moreover contained in $Y\times[0,1]\setminus\{(y,t)\vert dg_t(V)=0\}$.
Moreover, the (well defined on $\image G$) vector field $-\partial_t + X_t$
is tangent to $G^{-1}(0)$ and its flow at time $1$ restricts to a diffeomorphism from $G^{-1}(0)\cap \left(Y\times\{1\}\right)=N_1$ to $G^{-1}(0)\cap \left(Y\times\{0\}\right)=N_0$, as desired.
\end{proof}
\noindent
In our setting of the $f$-double, we get the following:
\begin{cor}
\label{LemmaUniqSmoothDouble}
Let $f_0,f_1\colon\thinspace \widehat{W}\rightarrow\mathbb{R}$ be proper, bounded from below and both regular equations for $M$, and $Z$ be a vector field on $\widehat{W}$ which is boundary-gradient-like for both $f_0$ and $f_1$.
Then, $\mathcal{D}funW{f_0}$ and $\mathcal{D}funW{f_1}$ are diffeomorphic.
\end{cor}
\begin{proof}
Apply \Cref{LemmaGenUniqSmoothDouble} with $Y=\widehat{W}\times\mathbb{R}_s$, $g_t=tf_1^\mathcal{D} +(1-t)f_0^\mathcal{D}$ and $V=Z$.
\end{proof}
\noindent
Notice that if $f_0,f_1\colon\thinspace \widehat{W}\rightarrow\mathbb{R}$ are two regular equations for $M$, then there always is a vector field $Z$ on $\widehat{W}$ which is boundary-gradient-like for both $f_0$ and $f_1$; this can be proven as done above in the case of a single regular equation.
\Cref{LemmaUniqSmoothDouble} then tells that $\mathcal{D}funW{f}$ does not depend on $f$, up to diffeomorphism.
By a slight abuse of notation, we may hence write $\mathcal{D}W$ and simply talk about the \emph{double} of $W$.
\\
Let now $(\widehat{W}^{2n},\lambda)$ be a Liouville manifold and denote by $Z$ its Liouville vector field. Consider also a smooth proper function $f\colon\thinspace \widehat{W} \rightarrow \mathbb{R}$, bounded from below and such that $Z$ is boundary-gradient-like for $f$.
Denote by $W$ the (compact) submanifold $f^{-1}((-\infty,0])$ of $\widehat{W}$.
Notice that $(M,\eta=\ker(\lambda\vert_M))$ is a contact manifold and that $(W,\lambda)$ is a Liouville filling of it.
\\
Consider now the Liouville manifold $(\widehat{W}\times\mathbb{R}_s\times\sphere{1}_\theta,\lambda+sd\theta)$, where $\mathbb{R}_s$ and $\sphere{1}_\theta$ denote the manifolds $\mathbb{R}$ and $\sphere{1}$ with coordinates $s$ and $\theta$ respectively.
Notice that the vector field $Z^\mathcal{D}=Z+s\partial_s$ and the function $f^\mathcal{D}$ can naturally be seen on $\widehat{W}\times\mathbb{R}\times\sphere{1}$.
Moreover, $Z^\mathcal{D}$ is Liouville for $\lambda + sd\theta$ and transverse to $\mathcal{D}funW{f}\times\sphere{1}=\{f^\mathcal{D}=0\}\subset \widehat{W}\times\mathbb{R}\times\sphere{1}$; in particular, $\alpha_f\colon\thinspaceeq (\lambda + sd\theta)\vert_{\mathcal{D}funW{f}\times\sphere{1}}$ is a contact form on $\mathcal{D}funW{f}\times\sphere{1}$.
In analogy with \Cref{NotDouble}, we will also denote the Liouville form $\lambda+sd\theta$ on $\widehat{W}\times\mathbb{R}\times\sphere{1}$ by $\lambda^\mathcal{D}$ in the following.
As in the case of the smooth double, we now describe how $(\mathcal{D}funW{f}\times\sphere{1},\ker\alpha_f)$ depends on the specific choice of $f$.
For this, we first state a more general uniqueness property:
\begin{lemma}
\label{LemmaGenUniqContStrDouble}
Let $(Y,\lambda_Y)$ be a Liouville manifold and denote by $V$ the Liouville vector field associated to $\lambda_Y$.
Consider also a smooth family $(g_t)_{t\in[0,1]}$ of functions $g_t\colon\thinspace Y\to\mathbb{R}$ such that $0$ is a regular value for each $g_t$, $N_t\colon\thinspaceeq g_t^{-1}(0)$ is compact, and $V$ is boundary-gradient-like for every $g_t$ with respect to $N_t$;
in particular, $\xi_t\colon\thinspaceeq\ker(\lambda_Y\vert_{N_t})$ is a contact form on $N_t$.
Then the flow $\psi_{X_t}^t$ of
\begin{equation*}
X_t\colon\thinspaceeq\frac{\dot{g_t}}{d g_t(V)}\, V
\end{equation*}
satisfies $\psi_{X_{t}}^t(N_t)=N_0$ and $(\psi_{X_{t}}^t\vert_{N_t})_*\xi_t=\xi_0$.
In particular, all the $(N_t,\xi_t)$'s are contactomorphic.
\end{lemma}
\begin{proof}
According to \Cref{LemmaGenUniqSmoothDouble}, it's enough to prove that the flow $\psi_{X_t}^t$ of $X_t$ preserves $\ker(\lambda)$ (on its domain of definition).
An explicit computation shows that $\mathcal{L}_{X_{t}}(\lambda) = \frac{\dot{g_t}}{d g_t(V)} \lambda$.
In particular, one can check that $(\psi_{X_t}^t)^{*}\lambda = h_t \lambda$ with
\begin{equation*}
h_t= \exp\left(\int_{0}^{t}\frac{\dot{g_s}}{d g_s(V)}\circ \psi_{X_{s}}^s ds\right) \text{ ,}
\end{equation*}
well defined on a neighborhood of $N_0$ and with values in $\mathbb{R}_{>0}$, as desired.
\end{proof}
\noindent
Going back to our particular setting, we get:
\begin{cor}
\label{LemmaUniqContStrDouble}
Let $f_0,f_1\colon\thinspace \widehat{W}\rightarrow\mathbb{R}$ be proper, bounded from below and both regular equations for $M$, and $Z$ be a Liouville vector field on $\widehat{W}\times\mathbb{R}\times\sphere{1}$ which is boundary-gradient-like for both $f_0$ and $f_1$.
Then, $(\mathcal{D}funW{f_0}\times\sphere{1},\ker(\alpha_{f_{0}}))$ and $(\mathcal{D}funW{f_1}\times\sphere{1},\ker(\alpha_{f_{1}}))$ are contactomorphic.
\end{cor}
\begin{proof}
Simply apply \Cref{LemmaGenUniqContStrDouble} with $(Y,\lambda_Y)=(\widehat{W}\times\mathbb{R}_s\times\sphere{1}_\theta, \lambda + sd\theta)$, $g_t=tf_1^\mathcal{D} +(1-t)f_0^\mathcal{D}$ and $V=Z$.
\end{proof}
\noindent
We will hence drop the $f$ from the notation and just denote $(\mathcal{D}W\times\sphere{1},\ker\alpha)$.
\begin{remark*}
In \cite{GeiSti10}, Geiges and Stipsicz construct, more generally, contact forms on $(W_1\cup_M W_2)\times \sphere{1}$, where $(W_1,\lambda_1)$ and $(W_2,\lambda_2)$ are Liouville domains with the same (strict) contact boundary $(M,\alpha)$.
The contact structure they obtain in the particular case where $W_1=W_2$ and $\lambda_1=\lambda_2$ (and $\partial W_1$ identified with $\partial W_2$ via the identity) is the same, up to isotopy, as the contact structure on $\mathcal{D}W\times\sphere{1}$ that we described above.
\\
Even though the construction described in this paper is less general (as it only covers the case $W_1=W_2$), it has the advantage of involving a natural Liouville filling of the strict contact manifold $(\mathcal{D}W\times\sphere{1},\alpha)$, which will be useful in \Cref{SecProofs}.
We also point out that one cannot always expect a presentation involving a symplectic filling for the construction in \cite{GeiSti10}.
For instance, in the case $W_1=D^2$ and $W_2=\Sigma_g\setminus D^2$, where $\Sigma_g$ is a closed surface with genus $g\neq0$, the theory of convex surfaces by Giroux tells that the contact structure on $(W_1\cup_{\sphere{1}}W_2)\times\sphere{1}$ obtained as in \cite{GeiSti10} is overtwisted: indeed, it is the unique $\sphere{1}$-invariant contact structure on $\Sigma_g\times\sphere{1}$ such that each $\Sigma_g\times\{pt\}$ is a convex surface with dividing set consisting of a homotopically trivial circle.
\end{remark*}
We now describe an explicit natural way to construct (strict) contactomorphisms of $(\mathcal{D}W\times\sphere{1},\xi\colon\thinspaceeq\ker\alpha)$.
\\
Consider a smooth action $\rho\colon\thinspace \sphere{1}_\theta\times\widehat{W}\rightarrow\widehat{W}$ by diffeomorphisms $\varphi_\theta\colon\thinspaceeq \rho(\theta,.)$ of $\widehat{W}$, each of which preserves both $\lambda$ and $f\colon\thinspace\widehat{W}\rightarrow\mathbb{R}$.
(Notice that we do not assume that $\rho$ restricts to the trivial action by the identity on $M=\partial W$.)
Let also $X$ be the infinitesimal generator of $\rho$, i.e.\ the vector field on $\widehat{W}$ given by
\begin{equation*}
X(p)=\left.\frac{d}{d\theta}\right\vert_{\theta=0}\varphi_{\theta}(p) \text{ ,}
\end{equation*}
where we interpret $\sphere{1}=\mathbb{R}/\mathbb{Z}$ and $\varphi_.(p)\colon\thinspace(-\epsilon,\epsilon)\to\widehat{W}$ as a curve on $\widehat{W}$ passing through $p$ at time $\theta=0$.
Lastly, consider the diffeomorphism $\Psi\colon\thinspace \mathcal{D}W \times\sphere{1} \rightarrow\mathcal{D}W\times\sphere{1}$ given by the restriction of
\begin{equation*}
\widehat{\Psi}\colon\thinspace\widehat{W}\times\mathbb{R}\times\sphere{1}\rightarrow\widehat{W}\times\mathbb{R}\times\sphere{1} \text{ ,} \quad
(p,s,\theta)\mapsto(\varphi_\theta(p),s,\theta)
\end{equation*}
to $\mathcal{D}W \times\sphere{1}$; notice that this restriction is well defined because $\varphi_\theta$ preserves $f$.
\begin{lemma}
\label{LemmaContactomorphismProdWithCircle}
The flow $\psi_Y^t$ of the vector field
\begin{equation*}
Y = \frac{\lambda(X)}{ d f^\mathcal{D}(Z^\mathcal{D})}\, \left( 2s \, Z - df(Z)\, \partial_s\right)
\end{equation*}
gives a smooth isotopy $\Psi\circ\psi_Y^t$ from $\Psi = \Psi\circ\psi_Y^0$ to a contactomorphism $\Psi_c\colon\thinspaceeq \Psi\circ\psi_Y^1$ of $(\mathcal{D}W\times\sphere{1},\xi=\ker\alpha)$.
\end{lemma}
Notice that $Y$, well defined as a vector field on $\widehat{W}\times\mathbb{R}\times\sphere{1}\setminus \{s=0,df(Z)=0\}$, is indeed tangent to (the $\mathcal{D}W$ factor of) $\mathcal{D}W\times\sphere{1}$.
\begin{proof}[Proof (\Cref{LemmaContactomorphismProdWithCircle})]
For notational ease, let $h\colon\thinspaceeq\lambda(X)$, defined on $\widehat{W}$.
Notice that $\mathcal{L}_{X}\lambda = 0$, as $\varphi_\theta^*\lambda=\lambda$ for each $\theta\in\sphere{1}$.
In particular, $dh = - \iota_{X}d\lambda$ and,
evaluating on the Liouville vector field $Z$, one gets $dh(Z)=h$.
We now want to compute the pullback $\Psi^*\left[\lambda^\mathcal{D}\vert_{\mathcal{D}W\times\sphere{1}}\right]$.
\\
First, notice that $(\widehat{\Psi}^*\lambda^\mathcal{D})(\partial_s)=\lambda^\mathcal{D}_{\widehat{\Psi}(.)}(d\widehat{\Psi}(\partial_s))=\lambda^\mathcal{D}_{\widehat{\Psi}(.)}(\partial_s)=0$.
Moreover, for any vector field $V$ on $\widehat{W}\times\mathbb{R}_s\times\sphere{1}_\theta$ tangent to the $\widehat{W}$ factor, one can compute
\begin{align*}
(\widehat{\Psi}^*\lambda^\mathcal{D})_{(p,s,\theta)}(V(p,s,\theta)) & = \lambda^\mathcal{D}_{\widehat{\Psi}(p,s,\theta)}(d_{(p,s,\theta)}\widehat{\Psi}(V(p,s,\theta))) \\
& = \lambda_{\varphi_\theta(p)}(d_{p}\varphi_\theta(V(p,s,\theta))) \\
& = (\varphi_\theta^*\lambda)_p(V(p,s,\theta)) \\
& = \lambda_p(V(p,s,\theta))\text{ .}
\end{align*}
Lastly, one can similarly compute:
\begin{align*}
(\widehat{\Psi}^*\lambda^\mathcal{D})_{(p,s,\theta)}(\partial_\theta) & = \lambda^\mathcal{D}_{\widehat{\Psi}(p,s,\theta)}(d_{(p,s,\theta)}\widehat{\Psi}(\partial_\theta)) \\
& = (\lambda + s d\theta)_{(\varphi_\theta(p),s,\theta)}(X(\varphi_\theta(p)) + \partial_\theta) \\
& = \lambda_{\varphi_\theta(p)}(X(\varphi_\theta)) + s \\
& \overset{(*)}{=} \lambda_p(X(p)) + s\text{ ,}
\end{align*}
where $(*)$ used the fact that $\varphi_\theta^*(\iota_X\lambda)= \iota_{(\varphi_\theta^*X)}(\varphi_\theta^*\lambda) = \iota_X \lambda$, as $\varphi_\theta$ preserves both $X$ and $\lambda$ for each $\theta\in\sphere{1}$.
In conclusion, $\widehat{\Psi}^*\lambda^\mathcal{D}=\lambda +\left(s+h\right)d\theta$, hence
\begin{equation*}
\Psi^*\left[\lambda^\mathcal{D}\vert_{T(\mathcal{D}W\times\sphere{1})}\right]=\left[\lambda +\left(s+h\right)d\theta\right]\vert_{T(\mathcal{D}W\times\sphere{1})} \text{ .}
\end{equation*}
We now describe a simple homotopy of contact forms between $\alpha$ and $\Psi^*\alpha$.
\\
For all $t\in[0,1]$, denote the $1$-form $\lambda + (s+th)d\theta$ on $\widehat{W}\times\mathbb{R}\times\sphere{1}$ by $\lambda^\mathcal{D}_t$, and its restriction to $\mathcal{D}W\times\sphere{1}$ by $\alpha_t$.
An explicit computation shows that, for each $t\in[0,1]$, $\lambda^\mathcal{D}_t$ is a Liouville form with corresponding Liouville vector field $Z^\mathcal{D}$ (independent of $t$), which is hence transverse to $\mathcal{D}W\times\sphere{1}$.
In particular, for each $t\in[0,1]$, $\alpha_t$ is a contact form on $\mathcal{D}W\times\sphere{1}$.
Lastly, we need to prove that this homotopy of contact forms $\alpha_t$ is realized by the vector field $Y$ as in the statement.
According to (the proof of) Gray's theorem, the flow of the (a priori time-dependent) vector field $X_t$ such that $\alpha_t(X_t)=0$ and $\iota_{X_{t}}d\alpha_t\vert_{\ker\alpha_t}=-\dot{\alpha}_t\vert_{\ker\alpha_t}$ gives an isotopy that pulls back $\ker\alpha_t$ to $\ker\alpha_0$ (see for instance the discussion after \cite[Theorem 2.2.2]{Gei08}).
It's hence enough to show that the vector field $Y$ in the statement verifies these two conditions.
\\
An explicit computation gives that $d f^\mathcal{D} (Y) = 0$ and $\lambda^\mathcal{D}_t(Y)=0$, i.e. that $Y\in \ker\alpha_t=\ker\lambda^\mathcal{D}_t\cap T(\mathcal{D}W\times\sphere{1})$.
Moreover, we can compute
\begin{align*}
\iota_Y d\lambda^\mathcal{D}_t & = \iota_Y(d\lambda + ds\wedge d\theta + tdh \wedge d\theta) \\
& = \frac{h}{d f^\mathcal{D}(Z^\mathcal{D})}\left[2s\lambda + 2ts \,d h(Z)\, d\theta - df(Z)d\theta\right] \\
& \overset{(i)}{=} \frac{h}{d f^\mathcal{D}(Z^\mathcal{D})}\left[2s \lambda^\mathcal{D}_t - 2s^2 d\theta - df(Z)d\theta \right]\\
& \overset{(ii)}{=} \frac{2s h}{d f^\mathcal{D}(Z^\mathcal{D})} \lambda^\mathcal{D}_t -\frac{d}{dt}\lambda^\mathcal{D}_t \text{ ,}
\end{align*}
where for $(i)$ we used that $dh(Z)=h$ and for $(ii)$ we used that $df^\mathcal{D} (Z^\mathcal{D})= 2s^2 + df(Z)$ and $\frac{d}{dt}\lambda^\mathcal{D}_t=h d\theta$.
In particular $\iota_{Y}d\alpha_t\vert_{\ker\alpha_t}=-\dot{\alpha}_t\vert_{\ker\alpha_t}$, as desired.
\end{proof}
\subsection{Families of Lagrangian frames}
\label{SubSecFamLegBases}
Let $V$ be a smooth $(2n+1)$-manifold and $\xi$ a contact structure on $V$.
Given a compact manifold $Y^m$, we call \emph{family of Lagrangian frames} of $\xi$ indexed by $Y$ the data of a smooth map $\gamma\colon\thinspace Y\rightarrow V$ and, for $j=1,\ldots,n$, of smooth maps $X_j\colon\thinspace Y \rightarrow \xi$ such that the following diagram commutes
\begin{equation*}
\begin{tikzcd}
& \xi \ar[d] \\
Y \ar["\gamma", r] \ar["X_j",ru] & V
\end{tikzcd}
\end{equation*}
and such that, for each $q\in Y$, the $X_1(q),\ldots, X_n(q) $ are $\mathbb{R}$-linearly independent and generate a Lagrangian subspace of $(\xi_p, (\CS_\xi)_p)$.
Here, $\CS_\xi$ is the natural conformal symplectic structure on $\xi$;
in particular, $(\CS_\xi)_p$ is a conformal class of symplectic alternating forms on $\xi_p$ and, hence, has a well defined class of (isotropic and) Lagrangian subspaces.
In the following, we denote families of Lagrangian frames $(\gamma,X_1,\ldots,X_n)$ simply by $\mathfrak{L}$.
We point out that if $f\colon\thinspace (V,\xi) \rightarrow (V,\xi)$ is a contactomorphism, then $f_*\mathfrak{L}\colon\thinspaceeq (f\circ\gamma, df (X_1),\ldots, df (X_n))$ is also a $Y$-family of Lagrangian frames of $\xi$: indeed, $f$ preserves the conformal symplectic structure $\CS_\xi$ on $\xi$.
\\
Moreover, if $f_t\colon\thinspace (V,\xi) \rightarrow (V,\xi)$ is a contact-isotopy from $f_0=\Id$ to $f_1=f$, then $(f_t)_*\mathfrak{L}$ is a path of $Y$-families of Lagrangian frames of $\xi$ from $\mathfrak{L}$ to $f_*\mathfrak{L}$.
In other words, we have the following obstruction to contact-isotopies:
\begin{lemma}
\label{LemmaObstr}
Let $f\colon\thinspace (V,\xi) \rightarrow (V,\xi)$ be a contactomorphism, and $\mathfrak{L}$ a $Y$-family of Lagrangian frames for $\xi$.
If $f_*\mathfrak{L}$ is not homotopic (among families of Lagrangian frames) to $\mathfrak{L}$, then $f$ is not contact-isotopic to the identity.
\end{lemma}
What we will actually use in the proof of \Cref{ThmMain} is a stabilized version of \Cref{LemmaObstr}; here are the details.
Let $\varepsilon_\mathbb{R}$ be the trivial real line bundle $V\times \mathbb{R} \rightarrow V$, and suppose that there exists a conformal symplectic structure $\CS_{\xi_{stab}}$ on the real vector bundle $\xi_{stab}\colon\thinspaceeq\xi\oplus\varepsilon_\mathbb{R}^{2k}$ extending $\CS_{\xi}$ on $\xi$ and such that $\varepsilon_\mathbb{R}^{2k}$ coincides with the $\CS_{\xi_{stab}}$-orthogonal complement of $\xi$ in $\xi_{stab}$;
here, $\varepsilon_\mathbb{R}^{2k}$ denotes the direct sum of $\varepsilon_\mathbb{R}$ with itself $2k$ times.
\\
Assume also that there is a complex structure $J$ on $\xi_{stab}$, which is tamed by $\CS_{\xi_{stab}}$ and such that there is an isomorphism of complex vector bundles $\Phi\colon\thinspace (\xi_{stab},J) \xrightarrow{\sim} \varepsilon_\mathbb{C}^{n+k}$,
where $\varepsilon_\mathbb{C}$ is the trivial complex line bundle $V\times \mathbb{C} \rightarrow V$.
\\
Notice that the property that $(\xi_{stab},J)$ is trivial is independent of the specific choice of $J$ tamed by $\CS_{\xi_{stab}}$.
Indeed, the space of complex structures on $\xi_{stab}$ which are tamed by $\CS_{\xi_{stab}}$ is contractible, hence $(\xi,J)$ and $(\xi,J')$ are isomorphic as complex vector bundles if $J,J'$ are both tamed by it.
Let now $\mathfrak{L} = (\gamma,X_1,\ldots,X_n)$ be a $Y$-family of Lagrangian frames for $\xi$.
Fix also a \emph{global} Lagrangian frame $(e_1,\ldots,e_k)$ for $(\varepsilon_\mathbb{R}^{2k},\CS_{\xi_{stab}}\vert_{\varepsilon_\mathbb{R}^{2k}})$ (i.e. a $V$-family of Lagrangian frames lifting the identity map $V\to V$).
Then, the stabilization
\begin{equation*}
\mathfrak{L}_{stab}\colon\thinspaceeq(\gamma,X_1,\ldots,X_n,e_1\circ\gamma,\ldots,e_k\circ\gamma)
\end{equation*}
is a $Y$-family of Lagrangian frames for $(\xi_{stab},\CS_{\xi_{stab}})$.
In the following, we say that the family $\mathfrak{L}_{stab}$ is the \emph{stabilization} of $\mathfrak{L}$ \emph{via $(e_1,\ldots,e_k)$} (sometimes omitting the sections $(e_1,\ldots,e_k)$ of $\varepsilon_\mathbb{R}^{2k}$ if there is no ambiguity), and denote it more concisely by $\mathfrak{L}\oplus (e_1,\ldots,e_k)$.
As $J$ is tamed by $\CS_{\xi_{stab}}$, for each $y\in Y$ the real subspace spanned by
\[\left(X_1(y),\ldots, X_n(y), e_1\circ\gamma(y),\ldots,e_k\circ\gamma(y)\right)\]
is totally real in $((\xi_{stab})_{\gamma(y)},J_{\gamma(y)})$.
Hence, the family of Lagrangian frames $\mathfrak{L}_{stab}$ gives, via $\Phi$, a \emph{$Y$-family of complex frames} $\Phi_*\mathfrak{L}_{stab}$ for $\varepsilon_\mathbb{C}^{n+k}$, i.e.\ for all $y\in Y$ one has
\begin{equation*}
\left<\,\Phi(X_1(y)),\ldots, \Phi (X_n(y)), \Phi(e_1\circ\gamma(y)),\ldots,\Phi(e_k\circ\gamma(y))\,\right>_{\mathbb{C}}=(\varepsilon_\mathbb{C}^{n+k})_{\gamma(y)} = \mathbb{C}^{n+k} \text{ .}
\end{equation*}
\noindent
In particular, considering, for each $y\in Y$, the linear endomorphism of $(\varepsilon_\mathbb{C}^{n+k})_{\gamma(y)}=\mathbb{C}^{n+k}$ obtained by sending the canonical complex framing of $\mathbb{C}^{n+k}$ over $\gamma(y)$ to the one given by $\Phi_*\mathfrak{L}_{stab}$, we obtain a smooth map $M\colon\thinspace Y \rightarrowGL_{n+k}(\mathbb{C})$.
In the following, we will say that the map $M$ is the \emph{$Y$-family of (invertible) matrices} associated (via $\Phi$) to $\mathfrak{L}_{stab}$.
We also point out that, given a contactomorphism $f$ of $(V,\xi)$, the family
\begin{equation*}
(f_*\mathfrak{L})_{stab}=\left(\,f\circ\gamma,\, df \left(X_1\right),\ldots,\,df \left(X_n\right),\,e_1\circ f \circ \gamma,\,\ldots,\,e_k\circ f \,\circ \gamma\right)
\end{equation*}
gives another $Y$-family of invertible matrices, denoted $f_* M\colon\thinspace Y \rightarrowGL_{n+k}(\mathbb{C})$.
As this can be done parametrically, we get this stabilized version of \Cref{LemmaObstr}:
\begin{lemma}
\label{LemmaObstrStabTriv}
Let $(V^{2n+1},\xi)$ be a contact manifold and $\CS_{\xi_{stab}}$ a conformal symplectic structure on $\xi_{stab}=\xi\oplus\varepsilon_\mathbb{R}^{2k}$ that extends $\CS_{\xi}$ on $\xi$ and such that $\varepsilon_\mathbb{R}^{2k}$ is the $\CS_{\xi_{stab}}$-orthogonal complement of $\xi$ in $\xi_{stab}$.
Consider also $J$ an almost complex structure on $\xi_{stab}$ which is tamed by $\CS_{\xi_{stab}}$ and such that $(\xi_{stab},J)$ is trivial, via an isomorphism $\Phi\colon\thinspace (\xi_{stab},J) \rightarrow \varepsilon_\mathbb{C}^{n+k}$ of complex vector bundles over $V$.
Let finally $f\colon\thinspace (V,\xi) \rightarrow (V,\xi)$ be a contactomorphism, $\mathfrak{L}=(\gamma,X_1,\ldots,X_n)$ a $Y$-family of Lagrangian frames for $\xi$ and $(e_1,\cdots,e_k)$ a global Lagrangian frame $\varepsilon_\mathbb{R}^{2k}$.
\\
If the $Y$-family of matrices associated via $\Phi$ to the stabilization $(f_*\mathfrak{L})_{stab}$ is not homotopic, as map $Y\rightarrow GL_{n+k}(\mathbb{C})$, to the $Y$-family of matrices associated via $\Phi$ to the stabilization $\mathfrak{L}_{stab}$, then $f$ is not contact-isotopic to the identity.
\end{lemma}
\begin{remark}
\label{rmk:formal_bis}
Notice that the invariant just described is of a completely formal nature.
More precisely, given a reference $Y$-family of Lagrangian frames $\mathfrak{L}=(\gamma,X_1\ldots,X_n)$, the space $\diff{V,\xi}f$ of formal contactomorphisms $(f,F_t)$ acts on the space of families of Lagrangian frames by pushforward (at time $1$) as follows:
\begin{equation*}
(f,F_t)_*\mathfrak{L} = (f\circ\gamma, F_1(X_1),\ldots,F_1(X_n)) \text{ .}
\end{equation*}
Moreover, under the same hypothesis as in \Cref{LemmaObstrStabTriv} but with $(f,F_t)\in\diff{V,\xi}f$ instead of $f\in\diff{V,\xi}$, one can prove that if the $Y$-family of matrices associated to $((f,F_t)_*\mathfrak{L})_{stab}$ is not homotopic to the one associated to $\mathfrak{L}_{stab}$, then $[(f,F_t)]$ is not trivial in $\pi_0(\diff{V,\xi}f)$.
\noindent
As a consequence, such invariant detects non--triviality of contactomorphisms directly in $\pi_0(\diff{V,\xi}f)$ (cf.\ \Cref{rmk:formal}), hence is not able to detect \emph{rigid} contactomorphisms, i.e.\ those which are non--trivial in $\pi_0(\diff{V,\xi})$ but trivial in $\pi_0(\diff{V,\xi}f)$.
\end{remark}
\section{Contact mapping classes of infinite order}
\label{SecProofs}
The aim of this section is to prove \Cref{ThmMain}, stated in the introduction.
We briefly recall the setting, reinterpreting it in terms of the construction of the double of a Liouville manifold from \Cref{SubSecProdDoubleLiouvDomWithCircle} and specifying explicitly who is the candidate contactomorphism $\Psi_c$.
The starting data is that of a Weinstein manifold $(F^{2n-2},\omega_F,Z_F,\psi_F)$ such that there exists a $J_F$ tamed by $\omega_F$ for which $(TF,J_F)$ is trivial as complex vector bundle over $F$.
We then consider the stabilization $(F\times\mathbb{C},\omega=\omega_F\oplus\omega_0,Z=Z_F+Z_0,\psi=\psi_F+\abs{.}^2_\mathbb{C})$, with $\omega_0=rdr\wedge d\varphi$ and $Z_0 = \frac{1}{2}r\partial_r$, where $z=re^{i\varphi}\in\mathbb{C}$.
Then, for a regular value $c> \min \psi_F$ of $\psi\colon\thinspaceeq\psi_F+\abs{.}^2_\mathbb{C}$, we denote $W=\psi^{-1}((-\infty,c])$.
\\
One then considers the Weinstein manifold $(F\times\mathbb{C}\times\mathbb{R}_s\times\sphere{1}_\theta,\omega^\mathcal{D},Z^\mathcal{D},\psi^\mathcal{D})$, where $\omega^\mathcal{D} = \omega_F + \omega_0 + ds\wedge d\theta$, $Z^\mathcal{D} = Z_F + Z_0 + s\partial_s$ and $\psi^\mathcal{D}(q,z,s,\theta) = \psi(q,z) + s^2$.
Let also $\lambda=\iota_Z\omega$ on $\widehat{W}=F\times\mathbb{C}$ and $\lambda^\mathcal{D} = \lambda + s d\theta$ on $\widehat{W}\times\mathbb{R}\times\sphere{1}$.
The manifold of interest is then $(\psi^\mathcal{D})^{-1}(c)=\mathcal{D}funW{\psi-c}\times\sphere{1}$, equipped with the contact structure $\xi=\ker\alpha_{\psi-c}$, where $\alpha_{\psi-c} = \lambda^\mathcal{D}\vert_{(\psi^\mathcal{D})^{-1}(c)}$.
We also have a simple diffeomorphism $\Psi$ of $\mathcal{D}W\times\sphere{1}$, given by the restriction of the diffeomorphism $\widehat{\Psi}$ of $F\times\mathbb{C}\times\mathbb{R}\times\sphere{1}$ defined as $\widehat{\Psi}(q,z,s,\theta)= (q,e^{i\theta}z,s,\theta)$.
\\
What we are going prove is that $\Psi$, is isotopic, via the flow $\psi_{Y}^t$ of the vector field $Y$ described in \Cref{LemmaContactomorphismProdWithCircle}, to a contactomorphism $\Psi_c=\Psi \circ \psi_Y^1$ which is of infinite order in $\pi_0\left(\diff{\mathcal{D}funW{\psi-c}\times\sphere{1},\alpha_{\psi-c}}\right)$.
\paragraph*{A complex trivialization.}
We start by extending the trivialization of $(TF,J_F)$ to a natural complex trivialization of the tangent bundle of $F\times\mathbb{C}\times\mathbb{R}\times\sphere{1}$.
More precisely, the almost complex structure $J\colon\thinspaceeq J_F\oplus i$ on $F\times\mathbb{C}$ can be further extended to $J^\mathcal{D}$ on $F\times\mathbb{C}\times\mathbb{R}_s\times\sphere{1}_\theta$ by defining $J^\mathcal{D}(\partial_s)\colon\thinspaceeq \partial_\theta$ on $T(\mathbb{R}_s\times\sphere{1}_\theta)$.
Notice in particular that $J^\mathcal{D}$ is tamed by $\omega^\mathcal{D}$.
Let now $\varepsilon_F^{n-1}$ be the trivial complex vector bundle $(F\times\mathbb{C}^{n-1},J_{std})$ over $F$ and denote $\nu\colon\thinspace(TF,J_F)\xrightarrow{\sim} \varepsilon_F^{n-1}$ the complex trivialization from \Cref{ThmMain}.
Then, $\nu$ extends to a trivialization
\begin{equation}
\label{EqTriv}
\mu\colon\thinspace\left(T\left(F\times\mathbb{C}\times\mathbb{R}\times\sphere{1}\right),J^\mathcal{D}\right)\xrightarrow{\sim}\varepsilon_{F\times\mathbb{C}\times\mathbb{R}\times\sphere{1}}^{n+1}
\end{equation}
such that, for each $(q,z,s,\theta)\in F\times\mathbb{C}\times\mathbb{R}\times\sphere{1}$, one has:
\begin{itemize}
\item the following diagram commutes
\begin{equation*}
\begin{tikzcd}
(T_qF,J_F) \ar["\nu_q",d] \ar["i",r]& \left(T_{(q,z,s,\theta)}\left(F\times\mathbb{C}\times\mathbb{R}\times\sphere{1}\right),J^\mathcal{D}\right) \ar["\mu_{(q,z,s,\theta)}", dd] \\
(\varepsilon_F^{n-1})_q=\mathbb{C}^{n-1} \ar["j", d] &\\
(\varepsilon_F^{n+1})_{q}=\mathbb{C}^{n+1} \ar["\Id", r] &
(\varepsilon_{F\times\mathbb{C}\times\mathbb{R}\times\sphere{1}}^{n+1})_{(q,z,s,\theta)}=\mathbb{C}^{n+1}
\end{tikzcd}
\end{equation*}
where $i$ and $j$ are the natural inclusions given by $T_q F = T_qF \oplus\{(0,0,0)\}\subset T_{(q,z,s,\theta)}\left(F\times\mathbb{C}\times\mathbb{R}\times\sphere{1}\right)$ and $\mathbb{C}^{n-1}=\mathbb{C}^{n-1}\times\{(0,0)\}\subset \mathbb{C}^{n+1}$;
\item $\mu_{(q,z,s,\theta)}(\partial_x)=(0,\ldots,0,1,0)\in \mathbb{C}^{n+1}$,
where we use here coordinates $(x,y)$ on the factor $\mathbb{C}$ of $F\times\mathbb{C}\times\mathbb{R}\times\sphere{1}$,
\item $\mu_{(q,z,s,\theta)}(\partial_s)=(0,\ldots,0,1)\in \mathbb{C}^{n+1}$, where $s$ is the coordinate on the factor $\mathbb{R}$ of $F\times\mathbb{C}\times\mathbb{R}\times\sphere{1}$.
\end{itemize}
\paragraph*{Flattening the hypersurface.}
We now need to modify the equation $\psi^\mathcal{D} - c = 0$ defining $\mathcal{D}W\times\sphere{1}$, in order to ``flatten'' a portion of this hypersurface.
The reason for such modification is that we want to understand the flow of the vector field $Y$ from \Cref{LemmaContactomorphismProdWithCircle} in order to use the invariant defined in \Cref{SubSecFamLegBases} (cf. \Cref{rmk:flattening}): having partially flattened the hypersurface will allow us to give an explicit and easy expression for such flow, at least in the flattened region.
Of course, we also need to prove that one can indeed do such modification without altering the conclusions of \Cref{ThmMain}; this is done in \Cref{RmkChoiceRegEq} below.
Let then $a >0 $ be very small; in particular, it can be chosen to be smaller than $\frac{c- \min(\psi_F)}{2}>0$ (this parameter will intervene later in the proof) and such that $\psi$ has no critical value between $c-2a$ and $c$.
Consider also a non-decreasing smooth cut-off function $\chi\colon\thinspace \mathbb{R}\rightarrow[-1,1]$, equal to $1$ exactly on $(2a,+\infty)$, equal to $-1$ exactly on $(-\infty,-2a)$, and such that $\chi(x)=x$ for $x\in(-a,a)$.
Then, the function $f\colon\thinspace F\times\mathbb{C}\rightarrow \mathbb{R}$ defined by $f\colon\thinspaceeq \chi (\psi-c)$ is a regular equation of $M=f^{-1}(0)=\psi^{-1}(c)$; in particular, $Z^\mathcal{D}=Z+s\partial_s$ on $\widehat{W}\times\mathbb{R}_s$ is transverse to $\mathcal{D}funW{f}$ too.
See \Cref{FigTwoDoubles}.
\begin{figure}
\caption{$\mathcal{D}
\label{FigTwoDoubles}
\end{figure}
As announced above, one can indeed prove \Cref{ThmMain} using the equation $f$ instead of $\psi-c$ without loss of generality:
\begin{lemma}
\label{RmkChoiceRegEq}
If the conclusion of \Cref{ThmMain} holds with the special choice of equation $f$ for $\mathcal{D}funW{f}\subset F\times\mathbb{C}\times\mathbb{R}$, then it holds also for $\mathcal{D}funW{\psi-c}$ defined by $\psi-c$ (i.e. as in the statement of \Cref{ThmMain}).
\end{lemma}
\begin{proof}[Proof (\Cref{RmkChoiceRegEq})]
Let $f_1\colon\thinspaceeq\psi-c$, $f_0\colon\thinspaceeq f=\chi(\psi-c)$ and $f_t=tf_1+(1-t)f_0$.
According to \Cref{LemmaUniqContStrDouble} (and the expression for the vector field from \Cref{LemmaGenUniqContStrDouble}), the flow $\psi_{X_{t}}^1$ of the vector field $X_t=\frac{f_1-f_0}{df_t^\mathcal{D}(Z^\mathcal{D})} Z^\mathcal{D}$
gives a contactomorphism from $(\mathcal{D}funW{f_1}\times\sphere{1},\ker(\alpha_{f_1}))$ to $(\mathcal{D}funW{f_0}\times\sphere{1},\ker(\alpha_{f_0}))$.
In order to prove \Cref{RmkChoiceRegEq}, it's then enough to show that the diffeomorphism $\psi_{X_{t}}^1\circ\Psi\circ (\psi_{X_{t}}^1)^{-1}$ of $\mathcal{D}funW{f_0}$ is still induced by the diffeomorphism of $F\times\mathbb{C}\times\mathbb{R}\times\sphere{1}$ given by $(q,z,s,\theta)\mapsto (q,e^{i\theta}z,s,\theta)$.
But this is indeed the case, because the flow $\psi_{X_{t}}^1$ preserves the angular component of the $\mathbb{C}$-factor as well as the $\sphere{1}$-factor of the product $F\times\mathbb{C}\times\mathbb{R}\times\sphere{1}$, and hence commutes with $(q,z,s,\theta)\mapsto (q,e^{i\theta}z,s,\theta)$.
\end{proof}
\paragraph*{Proving non-triviality of $\Psi_c^k$.}
We know from \Cref{SubSecProdDoubleLiouvDomWithCircle} that, inside the Liouville manifold $(F\times\mathbb{C}\times \mathbb{R}_s\times\sphere{1}_\theta, \lambda^\mathcal{D}=\lambda+sd\theta)$, the preimage of $(-\infty,0]$ via $F\times\mathbb{C}\times \mathbb{R}_s\times\sphere{1}_\theta\rightarrow \mathbb{R}$, $(q,z,s,\theta)\mapsto s^2+f(q,z)$, gives a Liouville filling of $(V=\mathcal{D}funW{f}\times\sphere{1}_\theta,\alpha_f)$.
In particular, we have a natural isomorphism of real vector bundles $\xi_{stab}=\xi\oplus\varepsilon_V^2 \simeq T(F\times\mathbb{C}\times\mathbb{R}\times\sphere{1})\vert_V$ over $V=\mathcal{D}funW{f}\times\sphere{1}$, where $\varepsilon_V$ is the trivial real line bundle $V\times\mathbb{R}\to V$, given by the natural inclusion on $\xi$ and by sending the two canonical sections $e_1,e_2$ of $\varepsilon_V^2$ to the Liouville vector field and Reeb vector field respectively.
Moreover, under this isomorphism, $\omega^\mathcal{D}$ and $J^\mathcal{D}$ on $F\times\mathbb{C}\times\mathbb{R}\times\sphere{1}$ give, respectively, a conformal symplectic structure $\CS_{\xi_{stab}}$ on $\xi_{stab}$, which restricts to $\CS_{\xi}$ on $\xi$ and such that $\varepsilon_V^{2}$ is the $\CS_{\xi_{stab}}$-orthogonal complement of $\xi$, and a complex structure $J$ on $\xi_{stab}$ tamed by $\CS_{\xi_{stab}}$.
Lastly, the trivialization $\mu$ described in \Cref{EqTriv} gives, via the isomorphism just described, a trivialization of $\xi_{stab}$, which we still denote by $\mu$.
We are then in the setting of \Cref{SubSecFamLegBases} and can use of the invariant described there in order to prove the non-triviality of the contactomorphism $\Psi_c^k$.
More precisely, as already announced at the beginning of the section, the candidate contactomorphism is $\Psi_c = \Psi \circ \psi_Y^1$, with $\psi_Y^t$ the flow at time $t$ of $Y$ defined in \Cref{LemmaContactomorphismProdWithCircle}.
In order to show that, for each $k\neq 0$, $\Psi_c^k$ is not contact isotopic to the identity, we are going to proceed by steps as follows:
\begin{enumerate}[align=left, leftmargin=*, label=Step \arabic*.]
\item \label{Step1} Let $W_-\colon\thinspaceeq \mathcal{D}funW{f}\cap\{s=-1\}\subset F\times\mathbb{C}\times\mathbb{R}$ (see \Cref{FigTwoDoubles}).
Notice that, by definition of $a$ and construction of $f$, $W_-$ is just a (slightly shrinked) copy of $W$ inside $F\times\mathbb{C}\times\mathbb{R}$, namely $\{\psi\leq c-a, s=-1\}\simeq W=\{\psi\leq c\}$ (recall $\psi$ has no critical values between $c-a$ and $c$).
We then describe an explicit $\sphere{1}$-family of Lagrangian frames $\mathfrak{L}$ for $\ker(\alpha_f)$ on $W_-\times\sphere{1}$.
\item \label{Step2} We remark that, for all $t\geq0$, $\psi_{Y}^{t}(W_-\times\sphere{1})\subset W_-\times\sphere{1}$, and we describe the behavior of the restriction of $\Psi_c$, and its iterates, to $W_-\times\sphere{1}$.
This allows us to describe, for all $k\geq 1$, the pushforward $(\Psi_c^k)_*\mathfrak{L}$ of $\mathfrak{L}$ via the $k$-th iterate of $\Psi_c$.
\item \label{Step3} We describe, for each $k\geq 0$, the family of matrices $B_k\colon\thinspace \sphere{1}\rightarrowGL_{n+1}(\mathbb{C})$ associated, via the trivialization $\mu$, to the stabilization $(\Psi_c^k)_*\mathfrak{L}\oplus Z^\mathcal{D}$.
We then show that, if $k\geq1$, $B_k$ is not homotopically trivial as map $\sphere{1}\rightarrowGL_{n+1}(\mathbb{C})$.
\end{enumerate}
According to \Cref{LemmaObstrStabTriv}, this proves that, for all $k\geq 1$, the $k$-th iterate of the contactomorphism $\Psi_c$ is not contact isotopic to the identity. The space of contactomorphisms being a group, this implies the same conclusion for all $k<0$.
\paragraph{\ref{Step1}}
Let $q_0\in F$ be the global minimum of $\psi_F$, and $(v_1,\ldots,v_{n-1})$ be a Lagrangian frame of $(T_{q_0}F,\omega_F)$.
Let also $z_0=0\in\mathbb{C}$ and $s_0=-1$.
Notice in particular that, by the choice of $0<a<\frac{c-\min\psi_F}{2}$ and $\chi$ in the definition of $f$, the point $(q_0,z_0,s_0,\theta)\in F\times\mathbb{C}\times\mathbb{R}\times\sphere{1}$ actually lies in $W_-\times\sphere{1}\subset\mathcal{D}funW{f}\times\sphere{1}$.
Moreover, $\lambda^\mathcal{D}= - d\theta$ at the point $\gamma(\theta)$ (recall $Z_F$ is gradient-like for $d\psi_F$), so that
$\xi_f=\ker(\alpha_f)$ coincides, over the point $(q_0,z_0=0,s_0=-1,\theta)$, with the subspace $T_{q_0}F\oplus T_0\mathbb{C}$ of $T_{(q_0,z_0,s_0,\theta)}\left(F\times\mathbb{C}\times\mathbb{R}\times\sphere{1}\right)$.
In particular, if one considers the loop
\begin{align*}
\gamma\colon\thinspace\sphere{1} &\rightarrow W_-\times\sphere{1}\subset\mathcal{D}funW{f}\times\sphere{1} \\
\theta&\mapsto(q_0,z_0=0,s_0=-1,\theta)
\end{align*}
then $\mathfrak{L}\colon\thinspaceeq\left(\gamma, v_1,\ldots,v_{n-1},\partial_x\right)$ is a $\sphere{1}$-family of Lagrangian frames for $\ker(\alpha_f)$ over $\gamma$, where we denoted $z=(x,y)\in\mathbb{C}$.
\paragraph{\ref{Step2}}
Here, we give explicit expressions for $\Psi_c^k$ and $(\Psi_c^k)_*\mathfrak{L}$.
\begin{lemma}
\label{Lemma1Step2}
The time$-1$ flow $\psi_Y^1$ of $Y$ satisfies $\psi_Y^1(W_-\times\sphere{1})\subset W_-\times\sphere{1}$ and, for each $k\geq 0$, $(\psi_Y^1)^k\vert_{W_-\times\sphere{1}}$ has the following form:
\begin{align*}
(\psi_Y^1)^k \colon\thinspace & W_-\times\sphere{1} \rightarrow W_-\times\sphere{1}\\
&(q,re^{i\varphi},-1,\theta) \mapsto (Q_k(q,r),R_k(r) e^{i\varphi},-1,\theta)
\end{align*}
for some functions $Q\colon\thinspace F\times\mathbb{C} \rightarrow F$ and $R\colon\thinspace \mathbb{C} \rightarrow\mathbb{R}$ depending, respectively, only on $(q,r)$ and $r$, where we use polar coordinates $z=re^{i\varphi}\in\mathbb{C}$.
Moreover, $R_k(0)=0$ and $Q_k(q_0,r)=q_0$ for every $r\geq0$
\\
In particular, $\Psi_c^k\circ\gamma=\gamma$ and, moreover, $\psi_Y^1$ commutes with $\Psi$ on the set $W_-\times\sphere{1}$ (which is obviously preserved by $\Psi$ too), hence $\Psi_c^k = \Psi^k\circ(\psi_Y^1)^k$ on $W_-\times\sphere{1}$.
\end{lemma}
\begin{lemma}
\label{Lemma2Step2}
Let $\gamma, (v_1,\cdots,v_{n-1})$ and $\mathfrak{L}$ be as in \ref{Step1}.
Then, for each $k\geq 0$, there are a Lagrangian frame $(v_1^{k},\ldots,v_{n-1}^k)$ of $T_{q_0}F$ and a real number $s_k\neq 0$ such that $(\Psi_c^k)_*\mathfrak{L}$
is given by
\begin{equation*}
\left(\,\gamma,\, v^k_1, \,\ldots,\,v_{n-1}^k,\, s_k \left[\colon\thinspaces(k\theta) \partial_x+\sin\left(k\theta\right)\partial_y\right]\,\right) \text{ .}
\end{equation*}
\end{lemma}
\begin{proof}[Proof (\Cref{Lemma1Step2})]
We give a proof by induction on $k$.
The case $k=0$ is trivial.
Moreover, it is not hard to check that if the statement of \Cref{Lemma1Step2} holds for both $k=N$ and $k=1$, then it also holds for $k=N+1$.
In other words, it's actually enough to show that the lemma holds for $k=1$.
We then show that $\psi_Y^1$ can be written in the desired form, with $Q_1,R_1$ satisfying the desired properties.
For this, we use the formula for $Y$ given in \Cref{LemmaContactomorphismProdWithCircle}.
Notice that, on $W_-\times\sphere{1}$, the function $f$ is constant and the coordinate $s$ is constant at $-1$.
Moreover, on $\widehat{W} = F\times\mathbb{C}$, the Liouville vector field $Z$ for $\lambda=\lambda_F + \frac{1}{2}r d\varphi$ is $Z_F+\frac{1}{2}r\partial_r$ and the vector field $X$ generating the circle action on the $\mathbb{C}$-factor is just $\partial_\varphi$, using polar coordinates $z=re^{i\varphi}\in\mathbb{C}$; in particular $\lambda(X)=\frac{1}{2}r^2$.
Hence, for all $(q,z,-1,\theta)\in W_-\times\sphere{1} $, we have
\begin{equation}
\label{eq:vect_field_Y}
Y (q,z,-1,\theta)
\; = \;
-\frac{r^2}{2} Z_F(q)-\frac{r^3}{4}\, \partial_r (re^{i\varphi})
\; = \;
-\frac{r^2}{2} \,Z (q,z) \text{ .}
\end{equation}
\\
In particular, the flow $\psi_Y^t\colon\thinspace \mathcal{D}W\times\sphere{1}\rightarrow\mathcal{D}W\times\sphere{1}$ of $Y$ at time $t\geq 0$ then satisfies $\psi_Y^t(W_-\times\sphere{1})\subset W_-\times\sphere{1}$.
Indeed, by the choice of the parameter $a$ in the definition of $f$, the restriction of $Y$ to $\partial (W_-\times\sphere{1})=\{\psi=c-a,s=-1\}$ is either $0$ (where $r=0$) or transverse to $\partial (W_-\times\sphere{1})$ (where $r\neq0$) and pointing in the direction of $W_-\times\sphere{1}$.
In other words, the orbit via the flow of $Y$ of each point of $W_-\times\sphere{1}$ stays inside $W_-\times\sphere{1}$ at all positive times.
\\
It also follows from \Cref{eq:vect_field_Y} that, at time $t=1$, the embedding $\psi_Y^1\colon\thinspace W_-\times\sphere{1}\rightarrow W_-\times\sphere{1}$ can be written as $\psi_Y^1(q,re^{i\varphi},-1,\theta)=(Q_1(q,r),R_1(r)e^{i\varphi},-1,\theta)$, for some functions $Q_1\colon\thinspace F\times\mathbb{C} \rightarrow F$ and $R_1\colon\thinspace \mathbb{C} \rightarrow\mathbb{R}$, with $Q_1$ and $R_1$ both independent of the angular component $\varphi$ on $\mathbb{C}$.
Moreover, $\psi_Y^1$ clearly commutes with $\Psi$ on $W_-\times\sphere{1}$ and, as $Y(q_0,z_0=0,-1,\theta)=0$ and $Z_F(q_0)=0$ (recall $Z_F$ is gradient--like for $\psi_F$), we also have $R_1(0)=0$ and $Q_1(q_0,r)=q_0$ for each $r\geq0$, as desired.
This concludes the proof of the base case $k=1$ of \Cref{Lemma1Step2}.
\end{proof}
\begin{proof}[Proof (\Cref{Lemma2Step2})]
According to \Cref{Lemma1Step2}, $\Psi_c^k=\Psi^k\circ(\psi_Y^1)^k$ and $\Psi_c\circ\gamma=\gamma$.
Notice that $d_{\gamma(\theta)}\Psi$ acts on $T_{\gamma(\theta)}(F\times\mathbb{C}\times\mathbb{R}\times\sphere{1})$ as $\Id$ on the subspace $T_{q_0}F\oplus T_{s_0}\mathbb{R}\oplus T_{\theta}\sphere{1}$ and as the rotation of angle $\theta$ on the subspace $T_{z_0=0}\mathbb{C}$.
It's then enough to prove that there are $v_1^k,\ldots,v_{n-1}^k \in T_{q_0}F$ and $s_k>0$ such that
\begin{equation*}
(\psi_Y^1)^k_{\;*}\mathfrak{L} = \left(\,\gamma,\, v^k_1, \,\ldots,\,v_{n-1}^k,\, s_k \partial_x\,\right) \text{ .}
\end{equation*}
\noindent
Using the expression for $\psi_Y^1$ from \Cref{Lemma1Step2} (recall in particular that $Q_k(q_0,r)=q_0$ for $r\geq0$), an explicit computation shows that $d_{\gamma(\theta)}\psi_{Y}^1 (\partial_x) = R_k'(0) \partial_x$; notice that $R_k'(0)$ is necessarily non-zero as $\psi_Y^1$ is a diffeomorphism.
In particular, choosing $s_k\colon\thinspaceeq R_k'(0)$ and $v_j^k\colon\thinspaceeq d_{(q_0,0)}Q_k (v_j)$ for each $j=1,\ldots,{n-1}$, we get the desired expression for $(\psi_Y^1)^k_{\;*}\mathfrak{L}$.
\end{proof}
\begin{remark}
\label{rmk:flattening}
The flattening procedure from $\mathcal{D}funW{\psi-c}\times\sphere{1}$ to $\mathcal{D}funW{f}\times\sphere{1}$ done at the beginning of the section finds its motivation in \Cref{Lemma1Step2,Lemma2Step2}.
Indeed, they both rely on the very explicit formula for $Y$ in the flattened picture, namely \Cref{eq:vect_field_Y}.
\end{remark}
\paragraph{\ref{Step3}}
We now consider the stabilization $(\Psi_c^k)_*\mathfrak{L} \oplus Z^\mathcal{D}$ given, at each point $\gamma(\theta)$, by the following Lagrangian frame of $T_{\gamma(\theta)}(F\times\mathbb{C}\times\mathbb{R}\times\sphere{1})$:
\begin{equation*}
\left(v^k_1,\ldots,v_{n-1}^k, s_k \left[\colon\thinspaces(k\theta) \partial_x+\sin\left(k\theta\right)\partial_y\right], Z^\mathcal{D} (\gamma(\theta))\right) \text{ .}
\end{equation*}
Notice that $\colon\thinspaces(k\theta) \,\partial_x+\sin(k\theta)\,\partial_y = \left[\colon\thinspaces(k\theta)+\sin(k\theta)\,J^\mathcal{D}\right]\partial_x$ and $Z(\gamma(\theta))=-\partial_s$.
In particular, the family of matrices $B_k\colon\thinspace \sphere{1} \rightarrow GL_{n+1}(\mathbb{C})$ associated via the trivialization $\mu$ (defined in \Cref{EqTriv}) has the form
\begin{equation*}
B_k(\theta) = \left(
\begin{array}{r@{}c|c@{}l}
& B_{0,k} & \mbox{0} \\\hline
& \mbox{0} &
\begin{matrix}\rule{0pt}{2ex}
s_k e^{ik\theta} & 0 \\
0 & -1
\end{matrix}
\end{array}
\right)
\end{equation*}
where $B_{0,k}\inGL_{n-1}(\mathbb{C})$ does not depend on $\theta$.
Thus, $B_k$ is homotopically trivial as map $ \sphere{1} \rightarrow GL_{n+1}(\mathbb{C})$ if and only if $k = 0$.
Indeed, $B_0$ is a constant map and $\det(B_k(\theta))=b_k e^{ik\theta}$, for a certain $b_k\in\mathbb{C}\setminus\{0\}$ (notice that $b_k\neq 0$ necessarily because $B_k(\theta)\in GL_{n+1}(\mathbb{C})$), hence the map $\sphere{1}\to\mathbb{C}\setminus\{0\}$ defined as $\theta\mapsto \det(B_k(\theta))$ is homotopically non-trivial if $k\geq 1$.
This concludes \ref{Step3}, hence the proof of \Cref{ThmMain}.
\end{document} |
\begin{document}
\title{ ``Identifiability and Observability in Epidemiological Models -- a survey --''\
a book proposal for the BCAM SpringerBriefs Series}
\thispagestyle{empty}
{\bf Abstract.}
In this monograph, the mathematical concepts of Identifiability and Observability of dynamical systems are first recalled, and then analyzed in the framework of Mathematical Epidemiology. It is shown that, even for simple and well known models of the literature, these properties are not always fulfilled. The concept of Observers, which allow the online reconstructuction of state variable of the model that are not observed, is also recalled and it is shown how it can used with epidemiological models.
Classical and less classical results are presented, illustrated on several examples, some of which are original. The presentation mixes in an original way identifiability and observability.
The questions of practical identifiability and observability, which are connected to sensitivity and numerical condition numbers, are addressed in a second part, illustrated on concrete examples with data.
The book is intended for all those who are interested in mathematical modeling for epidemiology.
{\bf Key-words.}
Dynamical systems, Identifiability, Observability, Mathematical Epidemiology, Sensitivity Analysis.
\end{document} |
\begin{document}
\title {Ordinary and symbolic powers are Golod}
\author {J\"urgen Herzog and Craig Huneke}
\address{J\"urgen Herzog, Fachbereich Mathematik, Universit\"at Duisburg-Essen, Campus Essen, 45117
Essen, Germany}
\email{[email protected]}
\address{Craig Huneke, Department of Mathematics, University of Virginia,
1404 University Ave,
Charlottesville, VA 22903-2600,
United States}
\email{[email protected]}
\subjclass[2000]{13A02, 13D40}
\keywords{Powers of ideals, Golod rings, Koszul cycles}
\begin{abstract}
Let $S$ be a positively graded polynomial ring over a field of characteristic $0$, and $I\subset S$ a proper graded ideal. In this note it is shown that $S/I$ is Golod if $\partial(I)^2\subset I$. Here $\partial(I)$ denotes the ideal generated by all the partial derivatives of elements of $I$. We apply this result to find large classes of Golod ideals, including powers, symbolic powers, and saturations of ideals.
\end{abstract}
\thanks{Part of the paper was written while the authors were visiting MSRI at Berkeley. They wish to acknowledge the support, the hospitality and the inspiring atmosphere of this institution. The second author was partially suppported by NSF grant 1259142.}
\maketitle
\section*{Introduction}
Let $(R,{\frk m})$ be a Noetherian local ring with residue class field $K$, or a standard graded $K$-algebra with graded maximal ideal ${\frk m}$. The formal power series $P_R(t)= \sum_{i \geq 0} \dim_K \Tor_i^{R} (R/{\frk m},R/{\frk m}) t^i$ is called the {\em Poincar\'{e} series} of $R$. Though the ring is Noetherian, in general the Poincar\'{e} series of $R$ is not a rational function. The first example that showed that $P_R(t)$ is not
necessarily rational was given by Anik \cite{An}. In the meantime more such examples are known, see \cite{Ro} and its references. On the other hand, Serre showed that $P_R(t)$ is coefficientwise bounded above by the rational series
\[
\frac{(1+t)^n}{1-t\sum_{i\geq 1}\dim_K H_i({\bold x};R)t^i},
\]
where ${\bold x}=x_1,\ldots,x_n$ is a minimal system of generators of ${\frk m}$ and where $H_i({\bold x};R)$ denotes the $i$th Koszul homology of the sequence ${\bold x}$.
The ring $R$ is called {\em Golod}, if $P_R(t)$ coincides with this upper bound given by Serre. There is also a relative version of Golodness which is defined for local homomorphisms as an obvious extension of the above concept of Golod rings. We refer the reader for details regarding Golod rings and Golod homomorphism to the survey article \cite{Av} by Avramov. Here we just want to quote the following result of Levin \cite{Le} which says that for any Noetherian local ring $(R,{\frk m})$, the canonical map $R\to R/{\frk m}^k$ is a Golod homomorphism for all $k\gg 0$. It is natural to ask whether in this statement ${\frk m}$ could be replaced by any other proper ideal of $R$. Some very recent results indicate that this question may have a positive answer. In fact, in \cite{HWY} it is shown that if $R$ is regular, then for any proper ideal $I\subset R$ the residue class ring $R/I^k$ is Golod for $k\gg 0$, which, since $R$ is regular, is equivalent to saying that the residue class map $R\to R/I^k$ is a Golod homomorphism for $k\gg 0$. But how big $k$ has to be chosen to make sure that $R/I^k$ is Golod? In the case that $R$ is the polynomial ring and $I$ is a proper monomial ideal, the surprising answer is that $R/I^k$ is Golod for all $k\geq 2$, as has been shown by Fakhari and Welker in \cite{FW}. The authors show even more: if $I$ and $J$ a proper monomial ideals, then $R/IJ$ is Golod. Computational evidence suggests that $R/IJ$ is Golod for any two proper ideals $I,J$ in a local ring (or graded ideals in a graded ring). This is supported by a result of Huneke \cite{Hu} which says that for an unramified regular local ring $R$, the residue class ring $R/IJ$ is never Gorenstein, unless $I$ and $J$ are principal ideals: Indeed, being Golod implies in particular that the Koszul homology $H({\bold x};R)$ admits trivial multiplication, while for a Gorenstein ring, by a result of Avramov and Golod \cite{LAv}, the multiplication map induces for all $i$ a non-degenerate pairing $H_i({\bold x};R)\times H_{p-i}({\bold x};R) \to H_p({\bold x}:R)$ where $p$ is place of the top non-vanishing homology of the Koszul homology. In the case that $I$ and $J$ are not necessarily monomial ideals, it is only known that $R/IJ$ is Golod if $IJ=I\sect J$, see \cite{HSt}.
In the present note we consider graded ideals in the graded polynomial ring $S=K[x_1,\ldots,x_n]$ over a field $K$ of characteristic $0$ with $\deg x_i=a_i>0$ for $i=1,\ldots,n$. The main result of Section~\ref{diff} is given in Theorem~\ref{muchbetter} which says that $S/I$ is Golod if $\partial(I)^2\subset I$. Here $\partial(I)$ denotes the ideal which is generated by all partial derivatives of the generators of $I$. This is easily seen to be independent of the generators of $I$ chosen. We call an ideal {\em strongly Golod} if $\partial(I)^2\subset I$. In Section~\ref{classes} it is shown that the class of strongly Golod ideals is closed under several important ideal operations like products, intersections and certain colon ideals. In particular it is shown that for any $k\geq 2$, the $k$th power of a graded ideal, as well as its $k$th symbolic power and its $k$th saturated power, is strongly Golod. We also prove the surprising fact that all the primary components of a graded ideal $I$ which belong to the minimal prime ideals of $I$ are strongly Golod, if $I$ is so. Even more, we prove that every strongly Golod ideal has a primary decomposition in which each
primary component is strongly Golod. We are also able to prove that the integral closure of a strongly Golod monomial ideal
is again strongly Golod; in particular, the integral closure of $I^k$ is always strongly Golod if $I$ is monomial and $k\geq 2$. We
do not know if this last assertion is true for general ideals.
It should be noted that our results, though quite general, do not imply the result of Fakhari and Welker concerning products of monomial ideals. One can easily find products of monomial ideals which are not strongly Golod. Moreover we have to require that the base field is of characteristic $0$. Actually as the proof will show, it is enough to require in Theorem~\ref{muchbetter} that the characteristic of $K$ is big enough compared with the shifts in the graded free resolution of the ideal.
A preliminary version of this paper by the first author was posted on ArXiv, proving that powers are Golod. After some discussions with the second author, this final
version emerged.
\section{A differential condition for Golodness}
\label{diff}
Let $K$ be a field of characteristic $0$, and $S=K[x_1,\ldots,x_n]$ the graded polynomial ring over $K$ with $\deg x_i=a_i>0$ for $i=1,\ldots,n$, and let $I\subset S$ be a graded ideal different from $S$. We denote by $\partial(I)$ the ideal which is generated by the partial derivatives $\partial f/\partial x_i$ with $f\in I$ and $i=1,\ldots,n$.
\begin{Theorem}
\label{muchbetter}
Suppose that $\partial(I)^2\subset I$. Then $S/I$ is Golod.
\end{Theorem}
\begin{proof}
We set $R=S/I$, and denote by $K(R)$ the Koszul complex of $R$ with respect to the sequence ${\bold x}=x_1,\ldots,x_n$. Furthermore we denote by $Z(R)$, $B(R)$ and $H(R)$ the module of cycles, boundaries and the homology of $K(R)$.
Golod \cite{Go} showed that Serre's upper bound for the Poincar\'{e} series
is reached if and only if all Massey operations of $R$ vanish. By definition, this is the case (see \cite[Def. 5.5 and 5.6]{AKM}), if for each subset $\mathcal{S}$ of
homogeneous elements of ${\mathcal D}irsum_{i=1}^nH_i(R)$ there exists a function $\gamma$, which is defined on the set of finite
sequences of elements from ${\mathcal S}$ with values in ${\frk m}\dirsum{\mathcal D}irsum_{i=1}^nK_i(R)$, subject to the following conditions:
\begin{enumerate}
\item[(G1)] if $h\in {\mathcal S}$, then $\gamma(h)\in Z(R)$ and $h=[\gamma(h)]$;
\item[(G2)] if $h_1,\ldots,h_m$ is a sequence in $\mathcal{S}$ with $m>1$, then
\[
\partial\gamma(h_1,\ldots,h_m)=\sum_{\ell=1}^{m-1}\overline{\gamma(h_1,\ldots,h_\ell)}\gamma(h_{\ell+1},\ldots,h_m),
\]
where $\bar{a} = (-1)^{i+1}a$ for $a\in K_i(R)$.
\end{enumerate}
Note that (G2) implies, that $\gamma(h_1)\gamma(h_2)$ is a boundary for all $h_1,h_2\in {\mathcal S}$ (which in particular implies that the Koszul homology of a Golod ring has trivial multiplication). Suppose now that for each ${\mathcal S}$ we can choose a functions $\gamma$ such that $\gamma(h_1)\gamma(h_2)$ is not only a boundary but that $\gamma(h_1)\gamma(h_2)=0$ for all $h_1,h_2\in {\mathcal S}$. Then obviously we may set $\gamma(h_1,\ldots h_r)=0$ for all $r\geq 2$, so that in this case (G2) is satisfied and $R$ is Golod.
The proof of Theorem~\ref{muchbetter} follows, once we have shown that $\gamma$ can be chosen that $\gamma(h_1)\gamma(h_2)=0$ for all $h_1,h_2\in {\mathcal S}$. For the proof of this fact we use the following result from \cite{H}: Let
\[
0\to F_p\to F_{n-1}\to \cdots \to F_1\to F_0\to S/J\to 0
\]
be the graded minimal free $S$-resolution of $S/J$, and for each $i$ let $f_{11},\ldots, f_{ib_i}$ a homogeneous basis of $F_i$. Let $\varphi_i\: F_i\to F_{i-1}$ denote the chain maps in the resolution, and let
\[
\varphi_i(f_{ij})=\sum_{k=1}^{b_{i-1}}\alpha_{jk}^{(i)}f_{i-1,k},
\]
where the $\alpha_{jk}^{(i)}$ are homogeneous polynomials.
In \cite[Corollary 2]{H} it is shown that for all $l=1,\ldots,p$ the elements
\[
\sum_{1\leq i_1<i_2<\cdots <i_l\leq n}a_{i_1}a_{i_2}\cdots a_{i_l}\sum_{j_2=1}^{b_{l-1}}\cdots \sum_{j_l=1}^{b_1}c_{j_1,\ldots,j_l}\frac{\partial(\alpha_{j_1,j_2}^{(l)},\alpha_{j_2,j_3}^{(l-1)},\ldots, \alpha_{j_l,1}^{(1)})}{\partial(x_{i_1},\ldots,x_{i_l})}e_{i_1}\wedge \cdots \wedge e_{i_l},
\]
$j_1=1,\ldots,b_l$ are cycles of $K(R)$ whose homology classes form a $K$-basis of $H_l(R)$.
Thus we see that a $K$-basis of $H_l(R)$ is given by cycles which are linear combinations of Jacobians determined by the entries $\alpha_{jk}^{(i)}$ of the matrices describing the resolution of $S/J$. The coefficients $c_{j_1,\ldots,j_l}$ which appear in these formulas are rational numbers determined by the degrees of the $\alpha_{jk}^{(i)}$, and the elements $e_{i_1}\wedge \cdots \wedge e_{i_l}$ form the natural $R$-basis of the free module $K_l(R)=\bigwedge^l({\mathcal D}irsum_{i=1}^nRe_i)$.
From this result it follows that any homology class of $H_l(R)$ can be represented by a cycle which is a linear combination of Jacobians of the form
\begin{eqnarray}
\label{jacobian}
\frac{\partial(\alpha_{j_1,j_2}^{(l)}\alpha_{j_2,j_3}^{(l-1)},\ldots, \alpha_{j_l,1}^{(1)})}{\partial(x_{i_1},\ldots,x_{i_l})}.
\end{eqnarray}
We choose such representatives for the elements of the set ${\mathcal S}$. Thus we may choose the map $\gamma$ in such a way that it assigns to each element of ${\mathcal S}$ a cycle which is a linear combination of Jacobians as in (\ref{jacobian}).
The elements $\alpha_{j_l,1}^{(1)}$, generate $I$. Thus it follows that $\gamma(h)\in Z(R)\sect (\partial I) K(R)$ for all $h\in {\mathcal S}$, and hence our assumption, $\partial(I)^2\subset I$, implies that $\gamma(h_1)\gamma(h_2)=0$ for any two elements $h_1,h_2\in {\mathcal S}$.
\end{proof}
\section{Classes of Golod ideals}
\label{classes}
We keep the notation and the assumptions of Section~\ref{diff} and apply Theorem~\ref{muchbetter} to exhibit new classes of Golod rings.
It is customary to call a graded ideal $I\subset S$ a {\em Golod ideal}, if $S/I$ is Golod. For convenience, we call a graded ideal $I\subset S$ {\em strongly Golod}, if $\partial(I)^2\subset I$. As we have shown in Theorem~\ref{muchbetter}, any strongly Golod ideal is Golod.
As usual we denote by $I^{(k)}$ the symbolic powers and by $\widetilde{I^k}$ the saturated powers of $I$. Recall that $I^{(k)}=\Union_{t\geq 1} I^k: L^t$, where $L$ is the intersection of all associated, non-minimal prime ideals of $I^k$, while $\widetilde{I^k}=\Union_{t\geq 1} I^k: {\frk m}^t$ where ${\frk m}$ is the graded maximal ideal of $S$.
Since $S$ is Noetherian, there exists an integer $t_0$ such that $I^{(k)}= I^k: L^t$ for all $t\geq t_0$. In particular, if we let $J=L^t$ for some $t\geq t_0$, then $I^{(k)}=I:J=I:J^2$.
The next result shows that strongly Golod ideals behave well with respect to several important ideal operations; in particular combining the various parts of
the theorem yields a quite large class of Golod ideals.
\begin{Theorem}
\label{twodaysbeforemy71thbirthday}
Let $I,J\subset S$ be graded ideals. Then the following hold:
\begin{enumerate}
\item[(a)] if $I$ and $J$ are strongly Golod, then $I\sect J$ and $IJ$ are strongly Golod;
\item[(b)] if $I$ and $J$ are strongly Golod and $\partial(I)\partial(J)\subset I+J$, then $I+J$ is strongly Golod;
\item[(c)] if $I$ is strongly Golod, $J$ is arbitrary, and $I: J=I:J^2$, then $I:J$ is strongly Golod;
\item[(d)] $I^k$, $I^{(k)}$ and $\widetilde{I^k}$ are strongly Golod for all $k\geq 2$.
\end{enumerate}
\end{Theorem}
\begin{proof}
To simplify notation we write $\partial f$ to mean anyone of the partials $\partial f/\partial x_i$.
(a) Let $f,g\in I\sect J$. Since $I$ and $J$ are strongly Golod, it follows that $(\partial f)(\partial g)\in I$ and $(\partial f)(\partial g)\in J$, and hence $(\partial f)(\partial g)\in I\sect J$. This shows that $I\sect J$ is strongly Golod.
Due to the product rule for partial derivatives it follows that $\partial(IJ)\subset \partial(I)J+I\partial(J)$. This implies that
\[
\partial(IJ)^2\subset \partial(I)^2J^2 + \partial(I)\partial(J)IJ+\partial(J)^2I^2.
\]
Obviously, the middle term is contained in $IJ$, while $\partial(I)^2J^2\subset IJ^2\subset IJ$, since $I$ is strongly Golod. Similarly, $\partial(J)^2I^2\subset IJ$. This shows that $\partial(IJ)^2\subset IJ$, and proves that $IJ$ is strongly Golod.
(b) is proved in the same manner as (a).
(c) Let $f,g\in I:J$. Then for all $h_1,h_2\in J$ one has $fh_1\in I$ and $gh_2\in I$. This implies that $(\partial f)h_1+f\partial h_1\in \partial(I)$ and
$(\partial g)h_2+g\partial h_2\in \partial(I)$. Thus
\begin{eqnarray*}
&&((\partial f)h_1+f\partial h_1)((\partial g)h_2+g\partial h_2)\\
&=&(\partial f)(\partial g)h_1h_2+(\partial f)(\partial h_2)gh_1+(\partial g)(\partial h_1)fh_2 +fg (\partial h_1)(\partial h_2)\in (\partial I)^2\subset I.
\end{eqnarray*}
Since $gh_1,fh_2\in I$, it follows that $(\partial f)(\partial h_2)gh_1+(\partial g)(\partial h_1)fh_2\in I$. Moreover, $Jfg (\partial h_1)(\partial h_2)\subset I$. Hence $J(\partial f)(\partial g)h_1h_2\subset I$. Since $h_1,h_2$ were
arbitrary in $J$, it then follows that $(\partial(I:J))^2\subset I:J^3=I:J$, as desired.
(d) Let $k\geq 2$. Then $\partial(I^k)\subset I^{k-1}\partial(I)$. It follows that $\partial(I^k)^2\subset I^{2k-2}\partial(I)^2\subset I^k$. Thus $I^k$ is strongly Golod.
As explained above, for appropriately chosen $J$, $I^{(k)}=I^k:J=^kI:J^2$. Now it follows from (c) that $I^{(k)}$ is strongly Golod.
The same arguments show that $\widetilde{I^k}$ is strongly Golod.
\end{proof}
\begin{Corollary}
\label{mm}
Let $P$ be a homogeneous prime ideal of $S$ containing $I$, a strongly Golod ideal. Then $I+P^k$ is strongly Golod for all $k\geq 2$.
\end{Corollary}
\begin{proof}
First note that $\partial(I)\subset P$. If not, then $\partial(I)^2$ will also not be in $P$, and therefore not in $I$, a contradiction. We also
observe that $\partial(P^k)\subset P^{k-1}$. It follows that $(\partial I)(\partial P^k)\subset P^k\subset I+P^k$. Thus the assertion follows from Theorem~\ref{twodaysbeforemy71thbirthday}(b) and (d) (we apply (d) to ensure that $P^k$ is strongly Golod). \end{proof}
Let $R=S/I$ and ${\frk n}$ the graded maximal ideal of $R$, and suppose that $I$ is strongly Golod. Then Corollary~\ref{mm} implies that $R/{\frk n}^k$ is Golod for all $k\geq 2$. Also note that by the theorem of Zariski-Nagata (see, e.g., \cite[p.143]{N}), the fact that $\partial(I)$ is contained in every homogeneous prime containing
$I$ implies that $I$ is in the second symbolic power of every such prime. In particular, self-radical ideals are never strongly Golod, though they may be Golod.
\begin{Corollary}
\label{components}
The (uniquely determined) primary components belonging to the minimal prime ideals of a strongly Golod ideal are strongly Golod.
\end{Corollary}
\begin{proof}
Let $I$ be strongly Golod and $P_1,\ldots,P_s$ its minimal prime ideals. Let $Q_i$ be the primary component of $I$ with $\Ass(S/Q_i)=\{P_i\}$, and set $L_i=\Sect_{j\neq i}P_j$. Then there exists an integer $r>1$ such that $Q_i=I:L_i^r=I:L_i^{2r}$. It follows from Theorem~\ref{twodaysbeforemy71thbirthday}(c) that $Q_i$ is strongly Golod.
\end{proof}
\begin{Corollary}
\label{components1}
Every strongly Golod ideal has a primary decomposition with strongly Golod primary ideals.
\end{Corollary}
\begin{proof} Let $P$ be an associated prime of $I$. By Corollary \ref{mm}, $I+P^k$ is strongly Golod for all $k\geq 2$, and then by Corollary \ref{components}
the unique $P$-primary minimal component of $I+P^k$ is also strongly Golod. We denote this component by $P_k$. The Corollary now follows from the
general fact that $I = \cap P_k$ for large $k$, where the intersection is taken over all associated primes of $I$. To prove this, fix a primary decomposition
of $I$, say $I = \cap Q_P$, where $Q_P$ is $P$-primary, and the intersection runs over all associated primes of $I$. It suffices to prove that
$P_k\subset Q_P$ for large $k$, since then $I\subset \cap P_k\subset \cap Q_P = I$. To check that $P_k\subset Q_P$, we may localize at $P$, since
both of these ideals are $P$-primary. Then the claim is clear.
\end{proof}
Statement (d) of Theorem~\ref{twodaysbeforemy71thbirthday} can be substantially generalized follows.
\begin{Theorem}
\label{inbetween}
Let $I\subset S$ be a homogeneous ideal and suppose suppose that $(I^{(k-1)})^2\subset I^k$ for some $k\geq 2$. Then all homogeneous ideals $J$ with $I^k\subset J\subset I^{(k)}$ are strongly Golod. In particular, any homogeneous ideal $J$ with $I^2\subset J\subset I^{(2)}$ is strongly Golod.
\end{Theorem}
\begin{proof}
We use a theorem of Zariski-Nagata \cite[p.143]{N} according to which $f\in S$ belongs to $I^{(k)}$ if and only if all partials of $f$ of order $<k$ belong to $I$. It follows from this characterization of the $k$th symbolic power of $I$ that $\partial(I^{(k)})\subset I^{(k-1)}$.
Now assume that $I^k\subset J\subset I^{(k)}$. Then
\[
\partial(J)^2\subset \partial(I^{(k)})^2\subset (I^{(k-1)})^2\subset I^k\subset J.
\]
This proves that $J$ is strongly Golod.
\end{proof}
\begin{Example} {\rm Let $X$ be a generic set of points in $\mathbb P^2$, and let $I$ be the ideal of polynomials vanishing at $X$. Bocci and Harbourne \cite{BH} proved that $I^{(3)}\subset I^2$. In this case, $(I^{(3)})^2\subset I^4$, so Theorem \ref{inbetween} applies to conclude that every homogeneous ideal
$J$ between $I^3$ and $I^{(3)}$ is strongly Golod.} \end{Example}
\begin{Example} {\rm The hypothesis of Theorem \ref{inbetween} is certainly not always satisfied. For example, let $X$ be a generic $4$ by $4$ matrix,
and let $I$ be the ideal generated by the $3$ by $3$ minors of $X$. It is well-known that $I$ is a height 4 prime ideal. Moreover, if ${\mathcal D}elta$ is the
determinant of $X$, then ${\mathcal D}elta\in I^{(2)}$. However, ${\mathcal D}elta^2$ cannot be in $I^3$ since this element has degree $8$, and the generators of
$I^3$ have degree $9$. Thus $(I^{(2)})^2$ is not contained in $I^3$.} \end{Example}
Another case to consider when the hypotheses of Theorem~\ref{inbetween} are satisfied is the following: Let $G$ e a finite simple graph on the vertex set $[n]$. A vertex cover of $G$ is a subset $C\subset [n]$ such that $C\sect \{i,j\}\neq \emptyset$ for all edges $\{i,j\}$ of $G$. The vertex cover ideal $I$ of $G$ is the ideal generated by all monomials $\prod_{i\in C}x_i\subset S=K[x_1,\ldots,x_n]$ where $C$ is a vertex cover of $G$. It is obvious that
\[
I= \Sect_{\{i,j\}\in E(G)}(x_i,x_j),
\]
where $E(G)$ denotes the set of edges of $G$.
It follows that $I^{(k)}=\Sect_{\{i,j\}\in E(G)}(x_i,x_j)^k$ for all $k$.
\begin{Proposition}
\label{onecase}
Let $I$ be the vertex cover ideal a finite simple graph $G$, and suppose that $(I^{(2)})^2\subset I^3$. Then $(I^{(k-1)})^2\subset I^k$ for all $k$. In particular,
every homogeneous ideal $J$ between $I^k$ and $I^{(k)}$ is strongly Golod.
\end{Proposition}
In \cite[Theorem~5.1]{HHT} it is shown that the graded $S$-algebra ${\mathcal D}irsum_{k\geq 0}I^{(k)}t^k\subset S[t]$ is generated in degree $1$ and $2$. Thus
the proposition follows from the more general proposition:
\begin{Proposition}
\label{onecase}
Let $I$ be an unmixed ideal of height two, and suppose that ${\mathcal D}irsum_{k\geq 0}I^{(k)}t^k$ is generated in degrees $1$ and $2$, and $(I^{(2)})^2\subset I^3$. Then $(I^{(k-1)})^2\subset I^k$ for all $k$.
\end{Proposition}
\begin{proof} The assumption that ${\mathcal D}irsum_{k\geq 0}I^{(k)}t^k\subset S[t]$ is generated in degree $1$ and $2$ is equivalent to
the statement that for all $p\geq 0$ one has
\[
I^{(2p+1)}=I(I^{(2)})^{p}\quad \text{and}\quad I^{(2p)} = (I^{(2)})^{p}.
\]
Since $I$ is an ideal of height $2$ it follows by a theorem of Ein, Lazarsfeld, and Smith \cite{ELS} that $I^{(2p)}\subset I^p$ for all $p\geq 0$.
This implies that $(I^{(2)})^p\subset I^p$ for all $p$, since $(I^{(2)})^p\subset I^{(2p)}$.
We want to prove that $(I^{(k-1)})^2\subset I^k$. We consider the two cases where $k$ is even or odd.
We have that
\[
(I^{(2p+1)})^2=I^{2}((I^{(2)})^p)^2\subset I^{2}(I^{p})^2=I^{2p+2}.
\]
Next assume that $k$ is even. Then $(I^{(2p)})^2=(I^{(2)})^{2p}$. Thus
\[
(I^{(2p)})^2=((I^{(2)})^{2})^p\subset (I^3)^p\subset I^{2p+1}.
\]
Here we used that $(I^{(2)})^{2}\subset I^3$.
\end{proof}
Proposition~\ref{onecase} is trivial if $G$ does not contain an odd cycle, in other words, if $G$ is bipartite, because in this case it is known
\cite[Theorem~5.1(b)]{HHT} that the symbolic and ordinary powers of $I$ coincide. The first non-trivial case is that when $G$ is an odd cycle, say on the vertex set $[n]$. In that case it is known, and easy to see, that $I$ is generated by the monomials $u_i=\prod_{j=i}^{(n+1)/2}x_{i+2j}$, where for simplicity of notation $x_i=x_{i-n}$ if $i>n$. It is also known that $I^{(2)}=I^2+(u)$, where $u=\prod_{i=1}^nx_i$, see for example \cite[Proposition~5.3]{HHT}. It follows that $(I^{(2)})^2=I^4+I^2(u)+(u^2)$. Since $u\in I$ we see that $I^2(u)\subset I^3$, and since $u_1u_2u_3$ divides $u^2$ we also have $(u^2)\subset I^3$. So that altogether, $(I^{(2)})^2\subset I^3$ for the vertex cover ideal of any odd cycle. We do not whether the same inclusion holds for any graph containing and odd cycle.
\begin{Example}
{\em There exist squarefree monomial ideals $I$ for which $(I^{(2)})^2$ is not contained in $I^3$. Indeed, let $I_{n,d}$ be the ideal generated by all squarefree monomials of degree $d$ in $n$ variables, and let $2<d<n$. Then $u=\prod_{i=1}^{d+1}x_i\in I_{n,d}^{(2)}$, but $u^2\not \in I_{n,d}^3$, because $I_{n,d}^3$ is generated in degree $3d$, while $\deg u^2=2(d+1)<3d$.
}
\end{Example}
It is immediate that a monomial ideal $I$ is strongly Golod, if for all minimal monomial generators $u,v \in I$ and all integers $i$ and $j$ such that $x_i|u$ and $x_j|v$ it follows that $uv/x_ix_j\in I$. Farkhari and Welker showed \cite{FW} that $IJ$ is Golod for any two proper monomial ideals. However a product of proper monomial ideals need not to be strongly Golod as the following example shows: Let $I=(x,y)$ and $J=(z)$. Then $IJ=(xz,yz)$ and $(xz)(yz)/xy=z^2$ does not belong to $IJ$.
We denote by $\bar{I}$ the integral closure of an ideal, and use the above characterization of strongly Golod monomial ideals to show:
\begin{Proposition}
\label{integral}
Let $I$ be a strongly Golod monomial ideal. Then $\bar{I}$ is strongly Golod. In particular, for any monomial ideal $I$, the ideals $\overline{I^k}$ are strongly Golod for all $k\geq 2$.
\end{Proposition}
\begin{proof}
The integral closure of the monomial ideal $I$ is again a monomial ideal, and a monomial $u\in S$ belongs to $\bar{I}$ if and only if there exists an integer $r>0$ such that $u^r\in I^r$, see \cite[Proposition 1.4.2]{SwaHu} and the remarks preceding this proposition.
Let $u,v\in \bar{I}$ monomials and suppose that $x_i|u$ and $x_j|v$. There exist integers $s,t>0$ such that $u^s\in I^s$ and $v^t\in I^t$. We may assume that both, $s$ and $t$, are even.
We observe that for a monomial $w$ with $x_k|w$ and $w^r\in I^r$ it follows that $(w/x_k)^r\in I^{r/2}$ if $r$ is even. Indeed, $w^r=m_1\cdots m_r$ with $m_i\in I$. Let $d_i$ be the highest exponent that divides $m_i$. We may assume that $d_i=0$ for $i=1,\ldots, a$, $d_i=1$ for $i=a+1,\ldots,b$ and $d_i>1$ for $i>b$. Then
\[
(w/x_k)^r= (m_1\cdots m_a)((m_{a+1}/x_k)\cdots (m_b/x_k))((m_{b+1}\cdots m_r)/x_k^d),
\]
where $d=\sum_{i=b+1}^rd_i$.
Since $I$ is strongly Golod, it follows from this presentation of $(w/x_k)^r$ that
\[
(w/x_k)^r\in I^aI^{\lfloor b/2\rfloor}=I^{\lfloor(2a+b)/2\rfloor},
\]
where $\lfloor c\rfloor$ denotes the lower integer part of the real number $c$.
Note that $r=d+b$ and that $2(r-a-b)\leq d$. This implies that $2a\geq d$, and hence $2a+b\geq d+b=r$, which implies that $(w/x_k)^r\in I^{r/2}$, as desired.
Now it follows that
\[
(uv/x_ix_j)^{s+t}=(u/x_i)^{t+s}(v/x_j)^{t+s}\in I^{s+t}
\]
This shows that $uv/x_ix_j\in\bar{I}$, and proves that $\bar{I}$ is strongly Golod.
\end{proof}
\end{document} |
\begin{document}
\vskip 6cm
\title{\bf On certain topological indices of graphs}
\markright{Abbreviated Article Title}
\author{
Arber Avdullahu\\
University of Primorska, Koper, Slovenia \\
\texttt{[email protected]}
\and
Slobodan Filipovski\footnote{Supported in part by the Slovenian Research Agency (research program P1-0285 and Young Researchers Grant).} \\
University of Primorska, Koper, Slovenia \\
\texttt{[email protected]}
}
\date{}
\maketitle
\begin{abstract} In this paper we give new bounds for a several vertex-based and edge-based topological indices of graphs: Albertson irregularity index, degree variance index, Mostar and the first Zagreb index. Moreover, we give a new upper bound for the energy of graphs through $IRB$-index. Most of our results rely on a well-known characterization of the Laplacian spectral radius.
\end{abstract}
\section{Introduction}
\quad Let $G$ be an undirected graph with $n$ vertices and $m$ edges without loops and multiple edges. The degree of a vertex $v$, denoted $\deg(v)$, is the number of the vertices connected to $v$. Since $\sum_{v\in V(G)} \deg(v) =2m$, average of vertex degrees can be given as $\overline{d}(G)=\frac{2m}{n}.$ A graph is $k$-regular if every degree is equal to $k$. Otherwise, the graph is said to be an irregular graph.
Once we know the average degree of a graph, it is possible to compute more complex measures of the heterogeneity in connectivity across vertices (e.g., the extent to which there is a very big spread between well-connected and not so well-connected vertices in the graph) beyond the simpler measures of range such as the difference between the maximum degree $\Delta$ and the minimum degree $\delta$.
One such measure was proposed by Tom Snijders in \cite{snijders}, called the degree variance of the graph. This vertex-based measure is defined as the average squared deviation between the degree of each vertex and the average degree:
\begin{equation}\label{varr} Var(G)=\frac{\sum_{u\in V(G)}\left(\deg(u)-\overline{d}(G)\right)^{2}}{n}=\frac{\sum_{u\in V(G)}\left(\deg(u)-\frac{2m}{n}\right)^{2}}{n}.
\end{equation}
Note that the degree variance of a regular graph is always zero.\\
Among the oldest and most studied topological indices, there are two classical vertex-degree based topological indices–the first Zagreb index and second
Zagreb index. The Zagreb indices were first introduced by Gutman et al. in \cite{gutman, gutman2}; they present an important molecular descriptor closely correlated with many chemical properties.
The first Zagreb index $M_{1}(G)$ is defined as
\begin{equation} M_{1}(G)=\sum_{v\in V(G)}\deg(v)^{2}.
\end{equation}
A general edge-additive index is defined as the sum, over all edges, of edge effects. These effects can be of various types, but the most common ones are defined in terms of some property of the end-vertices of the considered edge. In many cases the edge contribution represents how similar its end-vertices are.
In 1997, the Albertson irregularity index of a connected graph $G$, introduced by Albertson [1], was defined by
\begin{equation}\label{irr} Irr(G)=\sum_{(u,v)\in E(G)}|\deg(u)-\deg(v)|.
\end{equation}
This index has been of interest to mathematicians, chemists and scientists from related fields due to the
fact that the Albertson irregularity index plays a major role in irregularity measures of graphs \cite{dimitrov,abdo,chen,reti}, predicting the biological activities and properties of chemical compounds in the QSAR/QSPR
modeling and the quantitative characterization of network heterogeneity. Due to their simple computation, the degree-variance $Var(G)$ and the Albertson index $Irr(G)$ belong to the family of the widely used irregularity indices. Another bond-additive index studied in this paper is $IRB$-index, defined as
\begin{equation}\label{irb} IRB(G)=\sum_{e=(u,v)\in E(G)}(\sqrt{\deg(u)}-\sqrt{\deg(v)})^{2}.
\end{equation}
The Mostar index is a recently introduced bond-additive distance-based graph invariant that measures the degree of peripherality of particular edges and of the graph as a whole.
This index was introduced in \cite{mostar}, and independently in \cite{reti}.
Let $e=(u,v)\in E(G).$ Let $n_{e}(u)$ be the number of vertices of $G$ closer to $u$ than to $v$. The Mostar index of $G$ is defined as
\begin{equation}\label{most} Mo(G)=\sum_{e=(u,v)\in E(G)} | n_{e}(u)-n_{e}(v)|.
\end{equation}
In this paper we present an inequality between the degree variance and the Albertson irregularity index, Theorem 2.2.
As a consequence of this result, we improve the well-known lower bound for the first Zagreb index, $M_{1}(G)\geq \frac{4m^{2}}{n}.$
By involving $IRB$ index we provide an upper bound for the energy of graphs, which presents an improvement of the well-known upper bound $\sqrt{2mn},$ Theorem 2.5.
In Section 3 we focus on the Mostar index. In subsection 3.1 we consider bipartite graphs with diameter three; here we derive an upper bound for $Mo(G)$ which depends on the order of the partite sets. We finish our paper with a general upper bound for the Mostar index, in terms of $m$ and $n$, Theorem 3.11.
Almost all results in this paper share the same proving key which comes from a well-known characterization of the Laplacian spectral radius, Lemma 2.1.
\section{Laplacian matrices and their applications in estimating certain graph invariants}
\quad Most of the results in this paper are based on a well-known upper bound for the spectral radius of the laplacian matrix. Let $G=(V, E)$ be a graph whose vertices are labelled $\{1,2,\ldots, n\}$, and let $L$ be the Laplacian matrix of $G$. The Laplacian matrix of the graph $G$ is defined as follows
\begin{equation*} \label{laplacian}
L_{ij} = \left\{
\begin{array}{lc} -1, & \mbox{ if } (i,j)\in E \\
0, & \mbox{ if } (i,j)\notin E \; \mbox {and } i\neq j\\
-\sum_{k\neq i} L_{ik}, & \mbox{ if } i=j.
\end{array}
\right.
\end{equation*}
It is well-known that $L$ is a positive semidefinite matrix. More about the properties
of the Laplacian matrices can be found in \cite{meris, mohar}. In our proofs we use the fact that the largest eigenvalue $\lambda_{max}$ of $L$ satisfies $\lambda_{max}\leq n$, that is, $\frac{\lambda_{\max}}{n}\leq 1.$
\\
For a graph $G$ on $n$ vertices we identify a vector $x\in \mathbb{R}^{n}$ with a function $x: V(G)\rightarrow \mathbb{R}.$ The quadratic form defined by $L$ has the following expression
\begin{equation}\label{quadratic} x^{T}Lx=\sum_{(u,v)\in E(G)}(x(u)-x(v))^{2}.
\end{equation}
We also need Fiedler's \cite{eigenvalue} characterization of $\lambda_{max}.$
\begin{lemma} $$\lambda_{max}=2n\max_{x}\frac{\sum_{(u,v)\in E(G)}(x(u)-x(v))^{2}}{\sum_{u\in V(G)}\sum_{v\in V(G)}(x(u)-x(v))^{2}}$$
where $x$ is a nonconstant vector.
\end{lemma}
\subsection{An inequality between the degree variance and the irregularity index of graphs}
\begin{theorem} Let $G$ be a connected graph on $n$ vertices and $m$ edges. Then
\begin{equation}\label{teorema}Var(G)\geq \frac{Irr^{2}(G)}{mn^{2}}.
\end{equation}
The equality holds if and only if $G$ is a regular graph.
\end{theorem}
\begin{proof}
From the inequality between arithmetic and quadratic mean for the numbers $|\deg(u)-\deg(v)|$, where $(u,v)\in E(G),$ we get
\begin{equation}
\begin{gathered}
Irr(G)=\sum_{(u,v)\in E(G)}|\deg(u)-\deg(v)|\leq m\cdot \sqrt{\frac{\sum_{(u,v)\in E(G)}(\deg(u)-\deg(v))^{2}}{m}}\\
=\sqrt{m}\sqrt{\sum_{(u,v)\in E(G)}(\deg(u)-\deg(v))^{2}}= \sqrt{m}\sqrt{\sum_{(u,v)\in E(G)}((\deg(u)-\frac{2m}{n})-(\deg(v)-\frac{2m}{n}))^{2}}.
\end{gathered}\label{hm}
\end{equation}
For the graph $G$ we define a vector $x \in \mathbb{R}^{n}$ such that $x(u)=\deg(u)-\frac{2m}{n},$ where $u\in V(G).$
The inequality in (\ref{hm}) becomes
\begin{gather*}
Irr(G)\leq \sqrt{m} \sqrt{\sum_{(u,v)\in E(G)}\left((\deg(u)-\frac{2m}{n})-(\deg(v)-\frac{2m}{n})\right)^{2}}=
\sqrt{m}\sqrt{\sum_{(u,v)\in E(G)}(x(u)-x(v))^{2}}.
\end{gather*}
Using arithmetic calculations it is easy to see that:
\begin{equation}\label{eq:1}
\frac{1}{2}\cdot \sum_{u\in V(G)}\sum_{v\in V(G)}(x(u)-x(v))^{2} = n\cdot \sum_{u\in V(G)}x(u)^{2}-(\sum_{v\in V(G)}x(v))^{2}
\end{equation}
Now, applying Lemma 2.1 and equation (\ref{eq:1}) we obtain
\begin{gather*}
Irr(G)\leq \sqrt{m}\sqrt{\frac{\lambda_{\max}}{2n}\cdot \sum_{u\in V(G)}\sum_{v\in V(G)}(x(u)-x(v))^{2}}=
\sqrt{m}\sqrt{\frac{\lambda_{\max}}{n}[n\sum_{u\in V(G)}x(u)^{2}-(\sum_{v\in V(G)}x(v))^{2}]}.
\end{gather*}
From $\sum_{u\in V(G)}x(u)^{2}=\sum_{u\in V(G)} (\deg(u)-\frac{2m}{n})^{2}=n\cdot Var(G)$ and from $\sum_{v\in V(G)}x(v)=\sum_{v\in V(G)}(\deg(v)-\frac{2m}{n})=\sum_{v\in V(G)}(\deg(v)) - 2m=0$ we get
\begin{equation}\label{ravenka} Irr(G)\leq\sqrt{m}\sqrt{\frac{\lambda_{\max}}{n}\cdot n^{2}Var(G)}\leq n\sqrt{m}\sqrt{Var(G)}.
\end{equation}
From (\ref{ravenka}) we get
$$Var(G)\geq \frac{Irr^{2}(G)}{mn^{2}}.$$
Since the equality between arithmetic and quadratic mean holds when $\deg(u)=\deg(v)$ for each $u,v \in V(G)$, we obtain that
the equality in (\ref{teorema}) holds when $G$ is a regular graph. On the other hand, if $G$ is a regular graph we get $Var(G)=Irr(G)=0.$
\qed
\end{proof}
\subsection{A new lower bound for the first Zagreb index}
\quad As a consequence of Theorem 2.2, we improve the well-known lower bound for the first Zagreb index, $M_{1}(G)\geq \frac{4m^{2}}{n}$.
\begin{corollary} Let $G$ be a graph with $m$ edges and $n$ vertices. Then
$$M_{1}(G)\geq \frac{4m^{2}}{n}+\frac{Irr^{2}(G)}{mn}.$$
\end{corollary}
\begin{proof} Recall, the first Zagreb index for the graph $G$ is defined as
\begin{equation}\label{eden} M_{1}(G)=\sum_{u\in V(G)} \deg(u)^{2}.\end{equation}
It is easy to show that
\begin{equation}\label{dva}Var(G)=\frac{M_{1}(G)}{n}-\frac{4m^{2}}{n^{2}}.\end{equation}
From Theorem 2.2 and (\ref{dva}) we obtain the required inequality.
\qed
\end{proof}
\begin{remark} Let $G$ be a graph on $n$ vertices and $m$ edges, and let $d_{1},d_{2},\ldots, d_{n}$ be its vertex-degrees. Corollary 2.3 is equivalent to the inequality
$$d_{1}^{2}+d_{2}^{2}+\ldots+d_{n}^{2}\geq \frac{(d_{1}+d_{2}+\ldots+d_{n})^{2}}{n}+\frac{(\sum_{i\sim j}|d_{i}-d_{j}|)^{2}}{mn},$$
which presents an improvement of the inequality between quadratic and arithmetic means for the positive numbers $d_{1},d_{2},\ldots, d_{n}$.
\end{remark}
\subsection{A new upper bound for the energy of graphs}
\quad An \emph{adjacency matrix} $A=A(G)$ of the graph $G$ is the
$n\times n$ matrix
$[a_{ij}]$ with $a_{ij}=1$ if $v_{i}$ is adjacent to $v_{j},$ and $a_{ij}=0$ otherwise.
The eigenvalues $\lambda_{1}, \lambda_{2}, \ldots , \lambda_{n}$ of the graph $G$ are the eigenvalues of its adjacency
matrix $A$. Since $A$ is a symmetric matrix
with zero trace, these eigenvalues are real and their sum is equal to zero.
The \emph{energy }of $G$, denoted by $E(G)$, was first defined by I. Gutman
in \cite{ivan0} as the sum of the absolute values
of its eigenvalues. Thus,
\begin{equation}\label{energija} E(G)=\sum_{i=1}^{n} |\lambda_{i}|.
\end{equation}
This concept arose in theoretical chemistry, since
it can be used to approximate the total $\pi$-electron energy of a molecule.
The first result relating the
energy of a graph with its order and size is the following upper bound
obtained in 1971 by McClelland \cite{pi}:
\begin{equation}\label{bound2}
E(G)\leq \sqrt{2mn}.
\end{equation}
Since then, numerous other bounds for $E(G)$ were discovered, see \cite{babic, ivan5, jahan1, li}.
In \cite{kulen1} Koolen and Moulton improved the bound (\ref{bound2}) as follows: If $2m>n$ and $G$ is a graph with $n$ vertices and $m$ edges, then
\begin{equation}\label{kulen}
E(G)\leq \frac{2m}{n}+\sqrt{(n-1)\left(2m-\left(\frac{2m}{n}\right)^{2}\right)}.
\end{equation}
We improve the bound in (\ref{bound2}) by using $IRB$ index for a given graph and the technique presented in this paper. Recall, the $IRB$-index for the graph $G$ is defined as \\ $IRB(G)=\sum_{(u,v)\in E(G)}(\sqrt{\deg(u)}-\sqrt{\deg(v)})^2$.
\begin{theorem} Let $G$ be a connected graph on $n$ vertices and $m$ edges. Then
$$E(G) \leq \sqrt{2mn - IRB(G)}.$$
\end{theorem}
\begin{proof}
For the graph $G$ we define a vector $x \in \mathbb{R}^{n}$ such that $x(u)=\sqrt{\deg(u)},$ where $u\in V(G).$ Then we rewrite $IRB(G) = \sum_{(u,v)\in E(G)}(\sqrt{\deg(u)}-\sqrt{\deg(v)})^2=\sum_{(u,v)\in E(G)} (x(u)-x(v))^{2}.$
Applying Lemma 2.1 and equation (8) we obtain
\begin{gather*}\label{energija}
IRB(G)\leq \frac{\lambda_{\max}}{n}\cdot( n\sum_{u\in V(G)}x(u)^{2}-(\sum_{v\in V(G)}x(v))^{2}) \\
\leq n\sum_{u\in V(G)}x(u)^{2}-(\sum_{v\in V(G)}x(v))^{2} = n\sum_{u\in V(G)}\deg(u)-(\sum_{v\in V(G)}\sqrt{\deg(v)})^{2}.
\end{gather*}
In \cite{ariz} was proven that $E(G)\leq \sum_{u\in V(G)} \sqrt{\deg(u)}.$
Using this estimation we get
$$E(G) \leq \sqrt{2mn - IRB(G)}.$$
\qed
\end{proof}
Unfortunately we are not able to compare the bounds in Theorem 2.5 and (\ref{kulen}).
\section{Mostar index}
\quad As we mentioned in the introduction, the Mostar index is a new bond-additive structural invariant which measures the peripherality in graphs. Moreover, the Mostar index can be used to measure how much a given graph $G$ deviates from being distance-balanced. This index was introduced by Došlić et. al in \cite{mostar} and is studied in several publications, for example, see in \cite{mostar,gao,reti}.
In \cite{mostar} was proven that $Mo(P_{n})\leq Mo(T_{n})\leq (n-1)(n-2)=Mo(S_{n}),$ where $P_{n}, T_{n}$ and $S_{n}$ are paths, trees and stars on $n$ vertices, respectively.
We list several known results.
\begin{proposition}\cite{gao} Let $G$ be a graph of diameter $2$. Then $Mo(G)=Irr(G).$
\end{proposition}
\begin{proposition}\cite{gao} Let $T$ be a tree. Then $Mo(T)$ and $Irr(T)$ have the same parity.
\end{proposition}
\begin{theorem}\cite{gao} Let $T_{n}$ be a tree on $n$ vertices. Then
$$Mo(T_{n})\geq Irr(T_{n})$$
with equality if and only if $T_{n}$ is isomorphic with $S_{n}$.
\end{theorem}
\begin{theorem} \label{retii}\cite{reti} If $G$ is a connected graph of order $n\geq 3$ and size $m$, then
$$0\leq Mo(G)\leq m(n-2)$$
with the left equality if and only if $G$ is a distance-balanced graph and with the right equality if and only if $G$ is isomorphic to the star graph $S_{n}.$
\end{theorem}
In this section we derive several new upper bounds for the Mostar index.
\subsection{Bipartite graphs with diameter three}
\quad In \cite{gao} was proven that the graphs with diameter two have equal Mostar and Albertson index, that is, $Mo(G)=Irr(G).$
In this subsection we consider bipartite graphs with diameter three. Let $G=(V, E)$ be a bipartite graph with diameter three, of order $n$ and size $m$. Let $V_{1}$ and $V_{2}$ be the partite sets of $G$, that is, $V=V_{1}\cup V_{2}.$
We suppose that $|V_{1}|=n_{1}$ and $|V_{2}|=n_{2}.$ Let $e=(u,v)\in E(G)$ such that $u\in V_{1}$ and $v\in V_{2}.$ Since the diameter of $G$ is three we easily observe that $n_{e}(u)=n_{1}+\deg(u)-\deg(v)$ and $n_{e}(v)=n_{2}+\deg(v)-\deg(u).$ Thus
\begin{equation} \label{mostar} Mo(G)=\sum_{(u,v)\in E(G)}|n_{e}(u)-n_{e}(v)|=\sum_{(u,v)\in E(G)}|(n_{1}+2\deg(u))-(n_{2}+2\deg(v))|.
\end{equation}
Using Lemma 2.1 we derive the following result.
\begin{theorem} Let $G$ be a bipartite graph on $n$ vertices, $m$ edges and with diameter three. Let $V_{1}$ and $V_{2}$ be the partitive sets of $G$ such that $|V_{1}|=n_{1}$ and $|V_{2}|=n_{2}.$ Then
$$Mo(G)\leq \sqrt{m}\sqrt{\frac{\lambda_{\max}}{n}(n_{1}n_{2}n^{2}-4mn^{2}+4M_{1}(G)n-4n_{1}^{2}n_{2}^{2}-16m^{2}+16n_{1}n_{2}m)}.$$
\end{theorem}
\begin{proof} For the graph $G$ we define a vector $x \in \mathbb{R}^{n}$ as follows:
\begin{equation*} \label{laplacian}
x(w) = \left\{
\begin{array}{lc} n_{1}+2\deg(w), & \mbox{ if } w \in V_{1} \\
n_{2}+2\deg(w), & \mbox{ if } w \in V_{2} .\\
\end{array}
\right.
\end{equation*}
From (\ref{mostar}) and from the inequality between arithmetic and quadratic mean we get
$$Mo(G)\leq \sqrt{m} \sqrt{\sum_{(u,v)\in E(G)}((n_{1}+2\deg(u))-(n_{2}+2\deg(v)))^{2}}=\sqrt{m}\sqrt{\sum_{(u,v)\in E(G)}(x(u)-x(v))^{2}}.$$
Now, applying Lemma 2.1 we obtain
$$Mo(G)\leq \sqrt{m}\sqrt{\frac{\lambda_{\max}}{2n}\cdot \sum_{u\in V(G)}\sum_{v\in V(G)}(x(u)-x(v))^{2}}=\sqrt{m}\sqrt{\frac{\lambda_{\max}}{n}[n\sum_{u\in V(G)}x(u)^{2}-(\sum_{v\in V(G)}x(v))^{2}]}=$$
$$=\sqrt{m}\sqrt{\frac{\lambda_{\max}}{n}\left(n(n_{1}^{3}+n_{2}^{3}+4M_{1}(G)+4n_{1}m+4n_{2}m)-(n^{2}-2n_{1}n_{2}+4m)^{2}\right)}=$$
$$\sqrt{m}\sqrt{\frac{\lambda_{\max}}{n}(n_{1}n_{2}n^{2}-4mn^{2}+4M_{1}(G)n-4n_{1}^{2}n_{2}^{2}-16m^{2}+16n_{1}n_{2}m)}.$$
\qed
\end{proof}
\begin{remark} If $n_{1}=n_{2}$, then $Mo(G)=2\sum_{(u,v)\in E(G)}| \deg(u)-\deg(v)|=2Irr(G).$ Setting $n_{1}=n_{2}=\frac{n}{2}$ in Theorem 3.5 we get
$$Mo(G)=2\cdot Irr(G)\leq 2\sqrt{m(nM_{1}(G)-4m^{2})\frac{\lambda_{\max}}{n}},$$
which matches with the Goldberg's bound given in \cite{gold}.
\end{remark}
\begin{corollary} Let $G$ be a bipartite graph with diameter three such that $|V_{1}|=n_{1}$ and $|V_{2}|=n_{2}$. Then
$$Mo(G)\leq \sqrt{\frac{n_{1}^{2}n_{2}^{2}n^{2}}{3}+\left(\frac{2n_{1}^{2}n_{2}^{2}}{27}+\frac{n_{1}n_{2}n^{2}}{18}\right)\sqrt{4n_{1}^{2}n_{2}^{2}+3n_{1}n_{2}n^{2}}-\frac{4n_{1}^{3}n_{2}^{3}}{27}}.$$
\end{corollary}
\begin{proof} Clearly, $G$ is a triangle-free graph, and for its first Zagreb index holds $M_{1}(G)\leq mn.$ Using this bound in Theorem 3.5 we get
$$Mo(G)\leq \sqrt{m(n_{1}n_{2}n^{2}-4n_{1}^{2}n_{2}^{2}-16m^{2}+16n_{1}n_{2}m)}.$$
Let $f(x)=-16x^{3}+16n_{1}n_{2}x^{2}+(n_{1}n_{2}n^{2}-4n_{1}^{2}n_{2}^{2})x$ be a function defined on the interval $[1, n_{1}n_{2}].$ Clearly $m\in [1, n_{1}n_{2}].$
Since $f^{'}(x)=-48x^{2}+32n_{1}n_{2}x+n_{1}n_{2}n^{2}-4n_{1}^{2}n_{2}^{2}$ we get
\begin{equation} \label{max}f^{'}(x)\geq 0 \Leftrightarrow x\in [\frac{n_{1}n_{2}}{3}-\frac{\sqrt{4n_{1}^{2}n_{2}^{2}+3n_{1}n_{2}n^{2}}}{12}, \frac{n_{1}n_{2}}{3}+\frac{\sqrt{4n_{1}^{2}n_{2}^{2}+3n_{1}n_{2}n^{2}}}{12}].
\end{equation}
From (\ref{max}) we conclude that $f(x)$ achieves its maximum (on $[1,n_{1}n_{2}])$ at $m_{\max}=\frac{n_{1}n_{2}}{3}+\frac{\sqrt{4n_{1}^{2}n_{2}^{2}+3n_{1}n_{2}n^{2}}}{12}.$ Hence
$$Mo(G)\leq \sqrt{f(m)}\leq \sqrt{f(m_{\max})}=\sqrt{m_{\max}(n_{1}n_{2}n^{2}-4n_{1}^{2}n_{2}^{2}-16m_{\max}^{2}+16n_{1}n_{2}m_{\max})}=$$
$$=\sqrt{\frac{n_{1}^{2}n_{2}^{2}n^{2}}{3}+\left(\frac{2n_{1}^{2}n_{2}^{2}}{27}+\frac{n_{1}n_{2}n^{2}}{18}\right)\sqrt{4n_{1}^{2}n_{2}^{2}+3n_{1}n_{2}n^{2}}-\frac{4n_{1}^{3}n_{2}^{3}}{27}}.$$
\qed
\end{proof}
In chemical graph theory, the Szeged index is another edge-based topological index of molecule. This index was introduced by Gutman in \cite{gutman10}. The Szeged index of a connected graph $G$ is defined as
\begin{equation}\label{seged}
Sz(G)=\sum_{e=(u,v)\in E(G)} n_{e}(u)\cdot n_{e}(v).
\end{equation}
In the next result we give a relation between Mostar and Szeged index for bipartite graphs.
\begin{proposition}Let $G$ be a bipartite graph on $n$ vertices and $m$ edges. Then
\begin{equation}\label{bound}Mo(G)\leq \sqrt{m^{2}n^{2}-4mSz(G)}.
\end{equation}
\end{proposition}
\begin{proof} Since $G$ is a bipartite graph, then $n_{e}(u)+n_{e}(v)=n.$ Thus
\begin{equation}\label{haha}
(n_{e}(u)-n_{e}(v))^{2}=n^{2}-4n_{e}(u)n_{e}(v).
\end{equation}
From the inequality between quadratic and arithmetic mean for the numbers $|n_{e}(u)-n_{v}(v)|$ we get
$$\frac{Mo^{2}(G)}{m}=\frac{\left(\sum_{e\in E(G)}|n_{e}(u)-n_{e}(v)|\right)^{2}}{m}\leq \sum_{e\in E(G)}(n_{e}(u)-n_{e}(v))^{2}=mn^{2}-4Sz(G).$$
Thus $Mo^{2}(G)\leq m^{2}n^{2}-4mSz(G).$
\qed
\end{proof}
\begin{remark} When $n_{e}(u)+n_{e}(v)=n$ the parameter $n_{e}(u)n_{e}(v)$ achieves minimum $n-1$ (if one number is $n-1$ and other $1$).
Thus $Sz(G)\geq m(n-1).$ Replacing this in (\ref{bound}) we get
$Mo^{2}(G)\leq m^{2}n^{2}-4mSz(G)\leq m^{2}n^{2}-4m\cdot m\cdot(n-1)=m^{2}(n-2)^{2}.$
Hence
$$Mo(G)\leq m(n-2)$$
which is a trivial upper bound for $Mo(G).$ We note that the equality holds for the complete bipartite graph $K_{1,n-1}.$
\qed
\end{remark}
\subsection{A new upper bound for the Mostar index}
\quad In the last part of the paper we consider non triangle-free graphs. For an edge $(u,v)$ of $G$ we define $n_{uv}=|\{w\;|\; d(w,v)=d(w,u)\}|.$ It can be noted that $0\leq n_{uv}\leq n-2.$ It is easy to see that for any edge $(u,v)$ of $G$, $n_{e}(u)+n_{e}(v)=n-n_{uv}.$
\\Denote by $t(G)$ the number of triangles of a graph $G$.
We will use the following result:
\begin{proposition} For the number of triangles of $G$ it holds
$$\frac{m(4m-n^{2})}{n}\leq 3t(G)\leq \sum_{(u,v)} n_{uv}.$$
\end{proposition}
The first inequality is due to Bollob\' {a}s \cite{bolobas}. Since $n_{uv}$ counts cycles of odd length containing an edge $(u,v)$, the second inequality is obvious.
\begin{theorem} Let $G$ be a graph on $n$ vertices and $m$ edges. Then
$$Mo(G)< \sqrt{m^{2}(n-2)^{2}-\frac{m^{2}(n-2)(4m-n^{2})}{n}}.$$
\end{theorem}
\begin{proof} We already show that $ Mo^{2}(G)\leq m \sum_{(u,v)\in E(G)}(n_{e}(u)-n_{e}(v))^{2}.$ Thus
$$Mo^{2}(G)\leq m\sum_{(u,v)\in E(G)}(n_{e}(u)+n_{e}(v))^{2}-4m\sum_{(u,v)\in E(G)}n_{e}(u)n_{e}(v)\leq$$
$$\leq m\sum _{(u,v)\in E(G)}(n-n_{uv})^{2}-4m \sum_{(u,v)\in E(G)} (n-1-n_{uv})$$
$$=m^{2}n^{2}-4m^{2}(n-1)-m\sum_{(u,v)\in E(G)}n_{uv}(2n-n_{uv}-4)$$
$$\leq m^{2}(n-2)^{2}-m\sum_{(u,v)\in E(G)}n_{uv}(2n-4-(n-2))=$$
$$=m^{2}(n-2)^{2}-m(n-2)\sum_{(u,v)\in E(G)}n_{uv}$$
$$\leq m^{2}(n-2)^{2}-\frac{m^{2}(n-2)(4m-n^{2})}{n}.$$
\qed
\end{proof}
\begin{remark} In the above theorem we can assume that $m > \frac{n{2}}{4}.$ According to the Mantel theorem, it directly implies that $G$ is not a triangle-free graph. This assumption makes our bound better than the existing trivial bound $m(n-2)$ given in Theorem 3.4.
\end{remark}
\section*{Acknowledgment}
The authors would like to thank Dr. Ademir Hujdurović for introducing us with the Mostar index.
\end{document} |
\begin{document}
\title[The linking form and non-negative curvature]{Highly connected $7$-manifolds, the linking form and non-negative curvature}
\author[S.\ Goette]{S.\ Goette}
\address[Goette]{
Mathematisches Institut, Universit\"at Freiburg, Germany.}
\email{[email protected]}
\author[M.\ Kerin]{M.\ Kerin}
\address[Kerin]{School of Mathematics, Statistics and Applied Mathematics, N.U.I. Galway, Ireland.}
\email{[email protected]}
\author[K.\ Shankar]{K.\ Shankar}
\address[Shankar]{Department of Mathematics, University of Oklahoma, U.S.A.}
\email{[email protected]}
\thanks{}
\date{\today}
\subjclass[2010]{primary: 53C20, secondary: 55R55, 57R19, 57R30}
\keywords{highly connected, non-negative curvature, linking form}
\begin{abstract}
In a recent article, the authors constructed a six-parameter family of highly connected $7$-manifolds which admit an $\mathrm{SO}(3)$-invariant metric of non-negative sectional curvature. Each member of this family is the total space of a Seifert fibration with generic fibre $\mathbf{S}^3$ and, in particular, has the cohomology ring of an $\mathbf{S}^3$-bundle over $\mathbf{S}^4$. In the present article, the linking form of these manifolds is computed and used to demonstrate that the family contains infinitely many manifolds which are not even homotopy equivalent to an $\mathbf{S}^3$-bundle over $\mathbf{S}^4$, the first time that any such spaces have been shown to admit non-negative sectional curvature.
\end{abstract}
\maketitle
Closed manifolds admitting non-negative sectional curvature are not very well understood and it is, at present, quite difficult to obtain examples with interesting topology. This is partially explained by the dearth of known constructions, all of which depend in some way on two basic facts: First, compact Lie groups admit a bi-invariant metric (hence, non-negative curvature) and, second, Riemannian submersions do not decrease sectional curvature.
In \cite{GKS1}, a $6$-parameter family of non-negatively curved, $2$-connected $7$-manifolds $M^7_{\ul{a}, \ul{b}}$ was constructed, where the parameters $\underline a = (a_1, a_2, a_3), \underline b = (b_1, b_2, b_3) \in \mathbb{Z}^3$ satisfy $a_i, b_i \equiv 1 \!\! \mod 4$, for all $ i \in \{1,2,3\}$, and
$$
\mf{g}cd(a_1, a_2 \pm a_3) = 1 = \mf{g}cd(b_1, b_2 \pm b_3).
$$
Each of the manifolds $M^7_{\ul{a}, \ul{b}}$ is the total space of a Seifert fibration over an orbifold $\mathbf{S}^4$ with generic fibre $\mathbf{S}^3$ and has the cohomology ring of an $\mathbf{S}^3$-bundle over $\mathbf{S}^4$. In particular, $H^4(M^7_{\ul{a}, \ul{b}}; \mathbb{Z}) = \mathbb{Z}_{|n|}$, where $n = \frac{1}{8} \det \backslashm a_1^2 & b_1^2 \\ a_2^2 - a_3^2 & b_2^2 - b_3^2 \end{smallmatrix}\right)$ and, in the case $n = 0$, the notation $\mathbb{Z}_0$ signifies the integers $\mathbb{Z}$. The manifolds $M^7_{\ul{a}, \ul{b}}$ were shown in \cite{GKS1} to realise all exotic $7$-spheres. To the authors' knowledge, this was the first time that it was observed that all exotic $7$-spheres are Seifert fibred by $\mathbf{S}^3$. The following result is somewhat surprising.
\begin{theorem}
\label{T:thmA}
Infinitely many of the manifolds $M^7_{\ul{a}, \ul{b}}$
are not even homotopy equivalent to an $\mathbf{S}^3$-bundle over $\mathbf{S}^4$.
\end{theorem}
In the context of non-negative curvature, the construction of the manifolds $M^7_{\ul{a}, \ul{b}}$ fits neatly into the general scheme of increasing topological complexity via reducing symmetry assumptions. The standard example of a non-negatively curved manifold is a compact homogeneous space. In \cite{GM}, Gromoll and Meyer discovered the first example of an exotic sphere admitting non-negative curvature by introducing the notion of a \emph{biquotient} $G /\!\!/ H$, that is, the quotient of a compact Lie group $G$ by a closed subgroup $H \subseteq G \times G$ acting freely on $G$ via $(h_1, h_2) \cdot g = h_1 g h_2^{-1}$, $g \in G$, $(h_1, h_2) \in H$. Clearly, the isometry group of a biquotient will, in general, be much smaller than that of a homogeneous space. In contrast to the homogeneous situation, Totaro \cite{To} showed, for example, that there are infinitely many rational homotopy types of (non-negatively curved) biquotients already in dimension $6$.
An alternative approach to reducing symmetry is to assume that the manifold in question has low cohomogeneity, that is, that the quotient by a group of isometries is low dimensional. In particular, when the quotient space is a closed interval, that is, for manifolds of \emph{cohomogeneity one}, Grove and Ziller \cite{GZ} discovered sufficient conditions to ensure the existence of an invariant metric of non-negative curvature, thus generalising earlier work of Cheeger \cite{Ch}, and used this to demonstrate that all $\mathbf{S}^3$-bundles over $\mathbf{S}^4$ admit a metric with non-negative curvature.
A cohomogeneity-one manifold as above naturally admits a codimension-one singular Riemannian foliation whose leaves are the orbits of the action, that is, are homogeneous spaces. It was observed by Wilking in \cite{BWi} that a manifold which admits a codimension-one singular Riemannian foliation with biquotient leaves will also admit non-negative curvature, providing the sufficient conditions of Grove and Ziller \cite{GZ} are satisfied. The
manifolds $M^7_{\ul{a}, \ul{b}}$
fall into this category and can thus be seen as a further success of the strategy of symmetry reduction.
The manifolds mentioned in Theorem \ref{T:thmA} occur in infinitely many cohomology types and are distinguished from $\mathbf{S}^3$-bundles over $\mathbf{S}^4$ by having a non-standard linking form. In particular, these are the first manifolds with non-standard linking form observed to admit non-negative curvature (cf.\ \cite{GoKiSh}), thus implying that the linking form is not an obstruction to non-negative sectional curvature.
\begin{theorem}
\label{T:thmB}
Suppose the manifold $M^7_{\ul{a}, \ul{b}}$ has $H^4(M^7_{\ul{a}, \ul{b}}; \mathbb{Z}) = \mathbb{Z}_{|n|}$, $n \neq 0$. Then there is a generator $\mf{o}ne \in H^4(M^7_{\ul{a}, \ul{b}}; \mathbb{Z})$ such that the linking form of $M^7_{\ul{a}, \ul{b}}$ is given (up to sign) by
\begin{align*}
\lk : H^4(M^7_{\ul{a}, \ul{b}}; \mathbb{Z}) \mf{o}times H^4(M^7_{\ul{a}, \ul{b}}; \mathbb{Z}) \to& \mathbb{Q}/\mathbb{Z} \\
(x \mf{o}ne,y \mf{o}ne) \mapsto&
\pm \left(e_1 \, b_1^2 + e_0 \left(\tfrac{b_2^2 - b_3^2}{8} \right) \right) \frac{xy}{n} \!\! \mod 1
\end{align*}
where $e_0, e_1 \in \mathbb{Z}$ satisfy
$e_1 \, a_1^2 + e_0 \, \frac{1}{8}(a_2^2 - a_3^2) = 1$.
\end{theorem}
Observe that, if $f_0, f_1 \in \mathbb{Z}$ are chosen such that $f_1 \, b_1^2 + f_0 \, \frac{1}{8}(b_2^2 - b_3^2) = 1$, then
$$
\left( f_1 \, a_1^2 + f_0 \left(\tfrac{a_2^2 - a_3^2}{8} \right) \right) \left( e_1 \, b_1^2 + e_0 \left( \tfrac{b_2^2 - b_3^2}{8}\right) \right) \equiv 1 \!\! \mod n.
$$
Therefore, the linking form of $M^7_{\ul{a}, \ul{b}}$ can equivalently be written (up to sign) as
$$
\lk(x \mf{o}ne',y \mf{o}ne') = \pm \left(f_1 \, a_1^2 + f_0 \left( \tfrac{a_2^2 - a_3^2}{8} \right) \right) \frac{xy}{n} \!\! \mod 1
$$
with respect to the generator $\mf{o}ne' := \left(f_1 \, a_1^2 + f_0 \left( \tfrac{a_2^2 - a_3^2}{8} \right) \right) \mf{o}ne \in H^4(M^7_{\ul{a}, \ul{b}}; \mathbb{Z})$.
It will be demonstrated in Lemma \ref{L:stdLF} that $M^7_{\ul{a}, \ul{b}}$ has standard linking form whenever $\mf{g}cd(a_1, b_1) = 1$. In particular, this is the case for all $\mathbf{S}^3$-bundles over $\mathbf{S}^4$. However, it is well known from \cite{DWi} that there exist $2$-connected $7$-manifolds with non-standard linking form which have the same cohomology ring as in the case $\mf{g}cd(a_1, b_1) = 1$: see, for instance, Example \ref{Eg:nonst}.
\begin{obs}
The manifolds $M^7_{\ul{a}, \ul{b}}$ do not realise all $2$-connected $7$-manifolds with the cohomolgy ring of an $\mathbf{S}^3$-bundle over $\mathbf{S}^4$.
\end{obs}
In light of this observation, it is tempting to make the following conjecture.
\begin{conjecture*}
Every $2$-connected $7$-manifold with the cohomology ring of an $\mathbf{S}^3$-bundle over $\mathbf{S}^4$ admits a non-negatively curved, codimension-one singular Riemannian foliation with singular leaves of codimension two, and a Seifert fibration onto an orbifold $\mathbf{S}^4$ with generic fibre $\mathbf{S}^3$.
\end{conjecture*}
The paper is organised as follows. In Section \ref{S:prelim}, the construction and properties of the manifolds $M^7_{\ul{a}, \ul{b}}$ are reviewed and relevant notation introduced, before the linking form is introduced and some important facts recalled. In Section \ref{S:Bockstein}, the structure of the manifolds $M^7_{\ul{a}, \ul{b}}$ is used to obtain an understanding of the Bockstein homomorphism. Theorem \ref{T:thmB} is proved in Section \ref{S:linking}, while Section \ref{S:numth} is dedicated to the elementary number theory necessary to construct explicit manifolds $M^7_{\ul{a}, \ul{b}}$ satisfying the conclusion of Theorem \ref{T:thmA}.
\ack{It is a pleasure to thank Diarmuid Crowley for his interest in this project and for useful conversations about the linking form. Part of this research was performed at the mathematical research institute MATRIX in Australia and the authors wish to thank the institute for its hospitality. S.\ Goette and M.\ Kerin have received support from the DFG Priority Program 2026 \emph{Geometry at Infinity}, while M.\ Kerin and K.\ Shankar received support from SFB 898: \emph{Groups, Geometry \& Actions} at WWU M\"unster. K.\ Shankar received support from the National Science Foundation.\footnote{The views expressed in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation.}}
\section{Preliminaries and notation}
\label{S:prelim}
\subsection{The manifolds $M^7_{\ul{a}, \ul{b}}$\, } \hspace*{1mm}\\
\label{SS:family}
Suppose that a compact Lie group $G$ acts smoothly on a closed, connected, smooth manifold $M$ via $G \times M \to M,\ (g,p) \mapsto g \cdot p$. For each $p \in M$, the \emph{isotropy group} at $p$ is the subgroup $G_p = \{g \in G \mid g \cdot p = p\} \subseteq G$, and the \emph{orbit} through $p$ is the submanifold $G \cdot p = \{g \cdot p \in M \mid g \in G \} \subseteq M$. The manifold $M$ is foliated by $G$-orbits and an orbit $G \cdot p$ is diffeomorphic to the homogeneous space $G/G_p$.
The action $G \times M \to M$ is said to be of \emph{cohomogeneity one} if there is an orbit of codimension one or, equivalently, if $\dim(M/G) = 1$. In such a case, the manifold $M$ is called a \emph{cohomogeneity-one ($G$-)manifold}. If, in addition, $\pi_1(M)$ is assumed to be finite, then the orbit space $M/G$ can be identified with a closed interval. By fixing an appropriately normalised $G$-invariant metric on $M$, it may be assumed that $M/G = [-1,1]$. Let $\pi: M \to M/G = [-1,1]$ denote the quotient map. The orbits $\pi^{-1}(t)$, $t \in (-1,1)$, are called \emph{principal orbits} and the orbits $\pi^{-1}(\pm 1)$ are called \emph{singular orbits}.
Choose a point $p_0 \in \pi^{-1}(0)$ and consider a geodesic $c:\mathbb{R} \to M$ orthogonal to all the orbits, such that $c(0) = p_0$ and $\pi \circ c|_{[-1,1]} = \mathrm{id}_{[-1,1]}$. Then, for every $t \in (-1,1)$, one has $G_{c(t)} = G_{p_0} \subseteq G$, and this \emph{principal isotropy group} will be denoted by $H \subseteq G$. If $p_\pm = c(\pm 1) \in M$, denote the \emph{singular isotropy groups} $G_{p_\pm}$ by $K_\pm$ respectively. In particular, $H \subseteq K_\pm$.
By the slice theorem, $M$ can be decomposed as the union of two disk-bundles, over the singular orbits $G/K_- = \pi^{-1}(- 1)$ and $G/K_+ = \pi^{-1}(+ 1)$ respectively, which are glued along their common boundary $G/H = \pi^{-1}(0)$:
$$
M = (G \times_{K_-} \mathbf{D}^{l_-}) \cup_{G/H} (G \times_{K_+} \mathbf{D}^{l_+}) \, .
$$
Since the principal orbit $G/H$ is the boundary of both disk-bundles, it follows that $K_\pm/H = \mathbf{S}^{l_\pm-1}$, where $l_\pm$ denote the codimensions of $G/K_\pm \subseteq M$.
Conversely, given any chain $H \subseteq K_\pm \subseteq G$, with $K_\pm/H = \mathbf{S}^{d_\pm}$, one can construct a cohomogeneity-one $G$-manifold $M$ with codimension $d_\pm + 1$ singular orbits. For this reason, a cohomogeneity-one manifold is conveniently represented by its group diagram:
$$
\timesymatrix{
& G & \\
K_- \ar@{-}[ur] & & K_+ \ar@{-}[ul] \\
& H \ar@{-}[ur] \ar@{-}[ul] &
}
$$
In \cite{GZ}, the authors determined a sufficient condition for a cohomogeneity-one manifold to admit non-negative curvature.
\begin{thm}[\cite{GZ}]
\label{T:GZ}
Let $G$ be a compact Lie group acting on a manifold $M$ with cohomogeneity one. If the singular orbits are of codimension $2$, then $M$ admits a $G$-invariant metric of non-negative sectional curvature.
\end{thm}
Consider now the subgroups
\begin{align*}
Q &= \{\pm 1, \pm i, \pm j, \pm k\}, \\
\mathrm{Pin}(2) &= \{e^{i \theta} \mid \theta \in \mathbb{R}\} \cup \{e^{i \theta} j \mid \theta \in \mathbb{R}\}, \\
\mathrm{Pjn}(2) &= \{e^{j \theta} \mid \theta \in \mathbb{R}\} \cup \{i \, e^{j \theta} \mid \theta \in \mathbb{R}\}
\end{align*}
of the group $\mathbf{S}^3$ of unit quaternions, where the notation $\mathrm{Pjn}(2)$ is intended to be suggestive since, clearly, the groups $\mathrm{Pin}(2)$ and $\mathrm{Pjn}(2)$ are isomorphic, the only difference being that the roles of $i$ and $j$ are switched.
For $\underline a = (1, a_2, a_3), \underline b = (1, b_2, b_3) \in \mathbb{Z}^3$, with $a_i, b_i \equiv 1$ mod $4$ for all $i \in \{1,2,3\}$ and $\mf{g}cd(a_1, a_2, a_3) = \mf{g}cd(b_1, b_2, b_3) = 1$, a family of cohomogeneity-one ($\mathbf{S}^3 \times \mathbf{S}^3 \times \mathbf{S}^3$)-manifolds $P^{10}_{\ul{a}, \ul{b}}$ was introduced in \cite{GKS1} via the group diagram
\begin{equation}
\label{E:Pab}
\timesymatrix{
& \mathbf{S}^3 \times \mathbf{S}^3 \times \mathbf{S}^3 & \\
\mathrm{Pin}a \ar@{-}[ur] & & \mathrm{Pjn}b \ar@{-}[ul] \\
& \mathfrak{D}elta Q \ar@{-}[ur] \ar@{-}[ul] &
}
\end{equation}
where the principal isotropy group $\mathfrak{D}elta Q$ denotes the diagonal embedding of $Q$ into $\mathbf{S}^3 \times \mathbf{S}^3 \times \mathbf{S}^3$, and the singular isotropy groups are given by
\begin{align*}
\mathrm{Pin}a &= \{(e^{i a_1 \theta}, e^{i a_2 \theta}, e^{i a_3 \theta}) \mid \theta \in \mathbb{R}\} \cup \{(e^{i a_1 \theta} j, e^{i a_2 \theta} j, e^{i a_3 \theta} j) \mid \theta \in \mathbb{R}\}, \\
\mathrm{Pjn}b &= \{(e^{j b_1 \theta}, e^{j b_2 \theta}, e^{j b_3 \theta}) \mid \theta \in \mathbb{R}\} \cup \{(i \, e^{j b_1 \theta}, i\, e^{j b_2 \theta}, i\, e^{j b_3 \theta}) \mid \theta \in \mathbb{R}\}.
\end{align*}
Note that the restriction $a_i, b_i \equiv 1$ mod $4$ is to ensure only that $\mathfrak{D}elta Q$ is a subgroup of both $\mathrm{Pin}a$ and $\mathrm{Pjn}b$. The subfamily consisting of those $P^{10}_{\ul{a}, \ul{b}}$ having $a_1 = b_1 = 1$ describes all principal ($\mathbf{S}^3 \times \mathbf{S}^3$)-bundles over $\mathbf{S}^4$; see \cite{GZ}.
For the sake of notation, let $G = \mathbf{S}^3 \times \mathbf{S}^3 \times \mathbf{S}^3$ from now on. It was proven in \cite[Lemma 1.2]{GKS1} that the subgroup $\{1\} \times \mathfrak{D}elta \mathbf{S}^3 \subseteq \{1\} \times \mathbf{S}^3 \times \mathbf{S}^3 \subseteq G$ acts freely on $P^{10}_{\ul{a}, \ul{b}}$ if and only if
\begin{equation}
\label{E:free}
\mf{g}cd(a_1, a_2 \pm a_3) = 1 \ \textrm{ and } \ \mf{g}cd(b_1, b_2 \pm b_3) = 1.
\end{equation}
Therefore, given a cohomogeneity-one $G$-manifold $P^{10}_{\ul{a}, \ul{b}}$ determined by a group diagram \eqref{E:Pab} satisfying the conditions \eqref{E:free}, one obtains a smooth, $7$-dimensional manifold $M^7_{\ul{a}, \ul{b}}$ defined via
$$
M^7_{\ul{a}, \ul{b}} = (\{1\} \times \mathfrak{D}elta \mathbf{S}^3) \backslash P^{10}_{\ul{a}, \ul{b}} \, .
$$
Since the singular orbits of the cohomogeneity-one $G$-action on $P^{10}_{\ul{a}, \ul{b}}$ are of codimension $2$, it follows from Theorem \ref{T:GZ} that each $P^{10}_{\ul{a}, \ul{b}}$ admits a $G$-invariant metric of non-negative sectional curvature. As the free action of $\{1\} \times \mathfrak{D}elta \mathbf{S}^3$ is by isometries, there is an induced metric of non-negative curvature on $M^7_{\ul{a}, \ul{b}}$.
By construction, there is a codimension-one singular Riemannian foliation of $M^7_{\ul{a}, \ul{b}}$ by biquotients, such that the leaf space is $[-1,1]$ and $M^7_{\ul{a}, \ul{b}}$ decomposes as a union of two-dimensional disk-bundles over the two singular leaves which are glued along their common boundary, a regular leaf. This follows easily from the Slice Theorem applied to $P^{10}_{\ul{a}, \ul{b}}$. Indeed, the action of $\{1\} \times \mathfrak{D}elta \mathbf{S}^3$ preserves the $G$-orbits of $P^{10}_{\ul{a}, \ul{b}}$, and the image of an orbit $G/U$ is a leaf given by
\begin{equation}
\label{E:BiqDiff}
(\{1\} \times \mathfrak{D}elta \mathbf{S}^3) \backslash G / U \cong (\mathbf{S}^3 \times \mathbf{S}^3) /\!\!/ U \, ,
\end{equation}
where this diffeomorphism is induced by
$$
(q_1 \, u_1, q_2 \, u_2, q_3 \, u_3) \mapsto (q_1 \, u_1, u_2^{-1} q_2^{-1} q_3 \, u_3),
$$
for $(q_1, q_2, q_3) \in G$ and $(u_1, u_2, u_3) \in U \subseteq G$. Viewing $M^7_{\ul{a}, \ul{b}}$ in this way, the $\mf{g}cd$ conditions \eqref{E:free} required in the definition are simply the conditions ensuring that each of the biquotient actions on $\mathbf{S}^3 \times \mathbf{S}^3$ is free.
If $\varepsilon \in (-1,1)$ and if $\tau : M^7_{\ul{a}, \ul{b}} \to [-1,1]$ denotes the projection onto the leaf space of the codimension-one foliation of $M^7_{\ul{a}, \ul{b}}$ by biquotients, define
$$
M_- = \tau^{-1}([-1,\varepsilon)), \
M_+ = \tau^{-1}((-\varepsilon, 1]) \
\text{ and } \
M_0 = \tau^{-1}(-\varepsilon, \varepsilon).
$$
The preimages $M_\pm$ are two-dimensional disk-bundles over the singular leaves $(\mathbf{S}^3 \times \mathbf{S}^3) /\!\!/ \mathrm{Pin}a$ and $(\mathbf{S}^3 \times \mathbf{S}^3) /\!\!/ \mathrm{Pjn}b$, while $M_0 = M_- \cap M_+ \cong (\mathbf{S}^3 \times \mathbf{S}^3) /\!\!/ \mathfrak{D}elta Q \times (-\varepsilon,\varepsilon) $. Clearly $M^7_{\ul{a}, \ul{b}} = M_- \cup M_+$.
It was shown in \cite{GKS1} that the manifolds $M^7_{\ul{a}, \ul{b}}$ are $2$-connected and that
\begin{equation}
\label{E:cohom}
H^4(M^7_{\ul{a}, \ul{b}}; \mathbb{Z}) = \mathbb{Z}_{|n|},
\text{ where }
n = \frac{1}{8} \det \begin{pmatrix} a_1^2 & b_1^2 \\ a_2^2 - a_3^2 & b_2^2 - b_3^2 \end{pmatrix}.
\end{equation}
The notation $\mathbb{Z}_0$ signifies the integers $\mathbb{Z}$, in the case $n = 0$. From Lemmas 2.6 and 2.7 of \cite{GKS1} it follows that
\begin{equation}
\label{E:leafcohom}
\begin{split}
H^j(M_\pm ; \mathbb{Z}) &=
\begin{cases}
\mathbb{Z}, & j = 0,3, \\
\mathbb{Z}_2, & j = 2, 5, \\
0, & \text{otherwise,}
\end{cases} \\
H^j(M_0 ; \mathbb{Z}) &=
\begin{cases}
\mathbb{Z}, & j = 0,6, \\
\mathbb{Z}_2 \mf{o}plus \mathbb{Z}_2, & j = 2, 5,\\
\mathbb{Z} \mf{o}plus \mathbb{Z}, & j = 3,\\
0, & \text{otherwise.}
\end{cases}
\end{split}
\end{equation}
Denote by
\begin{equation}
\label{E:inclusions}
i_\pm : M_\pm \hookrightarrow M^7_{\ul{a}, \ul{b}}
\ \text{ and } \
j_\pm : M_0 \hookrightarrow M_\pm
\end{equation}
the respective inclusion maps, and by
\begin{equation}
\label{E:pairmaps}
q_\pm : (M^7_{\ul{a}, \ul{b}}, \emptyset) \to (M^7_{\ul{a}, \ul{b}}, M_\pm)
\ \text{ and } \
f_\pm : (M_\mp, M_0) \to (M^7_{\ul{a}, \ul{b}}, M_\pm)
\end{equation}
the maps of pairs induced by the identity map on $M^7_{\ul{a}, \ul{b}}$ and by $i_\pm$ respectively. Note, furthermore, that the maps on cohomology induced by the inclusions $j_\pm$ are determined by the projection maps $\pi_\pm$ in the circle-bundles
\begin{align*}
\mathbf{S}^1 = \mathrm{Pin}a/\mathfrak{D}elta Q &\longrightarrow (\mathbf{S}^3 \times \mathbf{S}^3) /\!\!/ \mathfrak{D}elta Q \stackrel{\pi_-}{\longrightarrow} (\mathbf{S}^3 \times \mathbf{S}^3) /\!\!/ \mathrm{Pin}a \,, \\[1mm]
\mathbf{S}^1 = \mathrm{Pjn}b/\mathfrak{D}elta Q &\longrightarrow (\mathbf{S}^3 \times \mathbf{S}^3) /\!\!/ \mathfrak{D}elta Q \stackrel{\pi_+}{\longrightarrow} (\mathbf{S}^3 \times \mathbf{S}^3) /\!\!/ \mathrm{Pjn}b \, ,
\end{align*}
since $\pi_\pm$ respect deformation retractions of $M_-$, $M_+$ and $M_0$ onto the respective leaves. In particular, the maps $\pi_\pm^*$ been computed in degree three in \cite[Equation (2.16)]{GKS1} and, with respect to fixed bases $\{x_\pm\}$ of $H^3(M_\pm ; \mathbb{Z}) = \mathbb{Z}$ and $\{v_1, v_2\}$ of $H^3(M_0; \mathbb{Z}) = \mathbb{Z} \mf{o}plus \mathbb{Z} $, yield
\begin{equation}
\label{E:maps}
\begin{split}
j_-^*(x_-) &= \frac{1}{8}(a_2^2 - a_3^2) \, v_1 + a_1^2 \, v_2, \\
j_+^*(x_+) &= -\frac{1}{8}(b_2^2 - b_3^2) \, v_1 - b_1^2 \, v_2,
\end{split}
\end{equation}
which, by the gcd conditions \eqref{E:free}, are each generators of $H^3(M_0; \mathbb{Z})$. Finally, by excision, the induced homomorphisms $f_\pm^* : H^j(M^7_{\ul{a}, \ul{b}}, M_\pm; R) \to H^j(M_\mp, M_0; R)$ are isomorphisms in all degrees, for any choice of coefficient ring $R$.
\subsection{The linking form} \hspace*{1mm}\\
\label{SS:link}
If $M$ is an $(s-1)$-connected, closed, oriented, smooth, $(2s+1)$-dimensional manifold, let $TH_s(M)$ and $TH^{s+1}(M;\mathbb{Z})$ denote the torsion subgroups of respective integral homology and cohomology groups. Let $a \in \mathcal C_{s}(M)$ be a chain representing a homology class $[a] \in TH_{s}(M)$. Then there is some $n_a \in \mathbb{Z}$ such that $n_a \cdot [a] = 0$ and, hence, some $c_a \in \mathcal C_{s+1} (M)$ such that $n_a \cdot a$ is the boundary of $c_a$, that is, $n_a \cdot a = \partial c_a$. The \emph{linking form} is a non-degenerate, bilinear pairing defined by
\begin{equation}
\label{E:LFhom}
\begin{split}
\lk : TH_{s}(N) \mf{o}times TH_{s}(M) &\to \mathbb{Q} / \mathbb{Z} \\
([a], [b]) &\mapsto \frac{\subseteqt(c_a, b)}{n_a} \mod 1,
\end{split}
\end{equation}
where $\subseteqt: \mathcal C_{s+1}(M) \times \mathcal C_{s}(M) \to \mathbb{Z}$ yields the signed count of intersections of its arguments with respect to the orientation of $M$. The linking form is symmetric (respectively, skew-symmetric) for $s$ odd (respectively, $s$ even). It was introduced in \cite{Br} and \cite{ST}.
Consider now the short exact sequence
$$
0 \longrightarrow \mathbb{Z} \stackrel{m}{\longrightarrow} \mathbb{Q} \stackrel{r}{\longrightarrow} \mathbb{Q} / \mathbb{Z} \longrightarrow 0.
$$
The boundary homomorphism $\beta : H^j(M; \mathbb{Q}/\mathbb{Z}) \to H^{j+1}(M; \mathbb{Z})$ in the associated long exact sequence
$$
\dots \longrightarrow H^{j}(M; \mathbb{Z})
\stackrel{m}{\longrightarrow} H^{j}(M; \mathbb{Q})
\stackrel{r}{\longrightarrow} H^{j}(M; \mathbb{Q}/\mathbb{Z})
\stackrel{\beta}{\longrightarrow}H^{j+1}(M; \mathbb{Z})
\longrightarrow \dots
$$
is called the \emph{Bockstein homomorphism}. Observe that $TH^j(M; \mathbb{Z}) \subseteq \im(\beta)$, since $TH^j(M; \mathbb{Z})$ lies in the kernel of $m : H^{j}(M; \mathbb{Z}) \to H^{j}(M; \mathbb{Q}) $.
Now, if $D : H_j(M) \to H^{2s + 1 - j}(M; \mathbb{Z})$ denotes the inverse of Poincar\'e duality, $[M] \in H_{2s+1}(M)$ the fundamental class of $M$ and $\langle \, ,\rangle : H^j(M;R) \mf{o}times H_j(M) \to R$ the $R$-valued Kronecker pairing, the right-hand side of \eqref{E:LFhom} is given, modulo the integers, by
$$
\frac{\subseteqt(c_a,b)}{n_a} = \langle w_a \smile D([b]), [M] \rangle,
$$
where $w_a \in H^{s}(M; \mathbb{Q}/\mathbb{Z})$ is such that $\beta(w_a) = D([a])$. That is, the linking form can be rewritten as a non-degenerate, bilinear form
\begin{equation}
\label{E:LFcohom}
\begin{split}
\lk : TH^{s+1}(M; \mathbb{Z}) \mf{o}times TH^{s+1}(M; \mathbb{Z}) &\to \mathbb{Q}/\mathbb{Z} \\
(x,y) &\mapsto \langle w \smile y, [M] \rangle \mod 1,
\end{split}
\end{equation}
where $\beta(w) = x \in TH^{s+1}(M; \mathbb{Z})$. Note, in particular, that the sign of the linking form depends on the choice of orientation on $M$. Furthermore, if $H^{s+1}(M; \mathbb{Z})$ is torsion, that is, $TH^{s+1}(M; \mathbb{Z}) = H^{s+1}(M; \mathbb{Z})$, then $M$ being $(s-1)$-connected implies that the Bockstein homomorphism is an isomorphism and it follows from \eqref{E:LFcohom} that
\begin{equation}
\label{E:LF}
\lk(x,y) = \langle\beta^{-1}(x) \smile y, [M]\rangle,
\end{equation}
for all $x,y \in H^{s+1}(M; \mathbb{Z})$.
Suppose now that $TH^{s+1}(M; \mathbb{Z})$ is cyclic of order $n$. In this case, bilinearity ensures that the linking form is completely determined by $\lk(\mf{o}ne, \mf{o}ne)$, where $\mf{o}ne$ is some generator of $TH^{s+1}(M; \mathbb{Z}) = \mathbb{Z}_n$. The linking form is said to be \emph{standard} if there exists an isomorphism $\theta : TH^{s+1}(M; \mathbb{Z}) \to TH^{s+1}(M; \mathbb{Z})$ such that
$$
\lk(\theta(\mf{o}ne), \theta(\mf{o}ne)) = \frac{1}{n} \in \mathbb{Q}/\mathbb{Z}.
$$
Recall, however, that the group of isomorphisms of $\mathbb{Z}_n$ is isomorphic to the group of units $\mathbb{Z}_n^* \subseteq \mathbb{Z}_n$. Therefore, the linking form is standard if and only if there is some unit $\lambda \in \mathbb{Z}_n^*$ such that
$$
\lk(\mf{o}ne, \mf{o}ne) = \frac{\lambda^2}{n} \mod 1.
$$
For $2$-connected $7$-manifolds, the linking form being standard imposes topological restrictions on the manifold.
\begin{thm}[{\cite[Corollary 2]{KiSh}}]
\label{T:KS}
A closed, smooth, $2$-connected $7$-manifold $M$, with $H^4(M; \mathbb{Z})$ finite cyclic, is homotopy equivalent to an $\mathbf{S}^3$-bundle over $\mathbf{S}^4$ if and only if its linking form is standard for some choice of orientation on $M$.
\end{thm}
By work of Crowley and Escher \cite{CE} and Kitchloo and Shankar \cite{KiSh} (cf.\ \cite{DWi}), the homotopy equivalence in Theorem \ref{T:KS} can, in fact, be strengthened to equivalence under a PL-homeomorphism.
The strategy for proving Theorem \ref{T:thmA} is now clear. One must identify manifolds $M^7_{\ul{a}, \ul{b}}$ which have a non-standard linking form, regardless of the choice of orientation. A simple example might shed some light on the number theoretic side of the problem, even though, by Lemma \ref{L:stdLF}, this particular example cannot occur among the manifolds $M^7_{\ul{a}, \ul{b}}$.
\begin{eg}
\label{Eg:nonst}
Suppose $M$ is a closed, smooth, $2$-connected $7$-manifold with $H^4(M; \mathbb{Z}) = \mathbb{Z}_5$ and $\lk(\mf{o}ne, \mf{o}ne) = \frac{2}{5} \in \mathbb{Q}/\mathbb{Z}$, for some generator $\mf{o}ne \in H^4(M; \mathbb{Z}) $. Since $\pm 2 \in \mathbb{Z}_5$ is not the square of a unit in $\mathbb{Z}_5^* = \{1,2,3,4\}$, it follows from Theorem \ref{T:KS} that $M$ is not homotopy equivalent to an $\mathbf{S}^3$-bundle over $\mathbf{S}^4$.
\end{eg}
A well-known fact from the study of quadratic reciprocities is being exploited in Example \ref{Eg:nonst} and plays an important role in finding further examples; namely, for an odd prime $p$, the unit $-1 \in \mathbb{Z}_p^*$ is a square if and only if $p \equiv 1$ mod $4$. Since the squares make up only half of all units in $\mathbb{Z}_p$, this implies that multiplying a non-square by $-1$ will not turn it into a square whenever $p \equiv 1$ mod $4$. This observation will yield non-standard linking forms, even up to a change of sign.
\section{The Bockstein homomorphism}
\label{S:Bockstein}
Since the formula \eqref{E:LF} describes the linking form of the manifolds $M^7_{\ul{a}, \ul{b}}$, it will be important in what follows to have a good understanding of the Bockstein homomorphism for these manifolds.
Recall that $M^7_{\ul{a}, \ul{b}} = M_- \cup M_+$ and $M_- \cap M_+ = M_0$ and suppose from now on that
\begin{equation}
\label{E:finite}
H^4(M^7_{\ul{a}, \ul{b}}; \mathbb{Z}) = \mathbb{Z}_{|n|},
\text{ where }
n = \frac{1}{8} \det \begin{pmatrix} a_1^2 & b_1^2 \\ a_2^2 - a_3^2 & b_2^2 - b_3^2 \end{pmatrix} \neq 0.
\end{equation}
It follows from \eqref{E:leafcohom} and the long exact cohomology sequence for the pair $(M^7_{\ul{a}, \ul{b}}, M_\pm)$ that
$$
H^3(M^7_{\ul{a}, \ul{b}}, M_\pm ; \mathbb{Z}) = \mathbb{Z}_2
\ \text{ and } \
H^5(M^7_{\ul{a}, \ul{b}}, M_\pm ; \mathbb{Z}) = 0.
$$
On the other hand, the long exact sequence for the pair $(M_\mp, M_0)$ yields a short exact sequence
$$
0 \longrightarrow H^3(M_\mp; \mathbb{Z}) \stackrel{j_\mp^*}{\longrightarrow }
H^3(M_0; \mathbb{Z}) \longrightarrow
H^4(M_\mp, M_0; \mathbb{Z}) \longrightarrow 0.
$$
By \eqref{E:leafcohom} and \eqref{E:maps}, it now follows that $H^4(M_\mp, M_0; \mathbb{Z}) = \mathbb{Z}$. However, excision implies that the map $f_\pm^* : H^4(M^7_{\ul{a}, \ul{b}}, M_\pm; \mathbb{Z}) \to H^4(M_\mp, M_0; \mathbb{Z})$ is an isomorphism, from which it may be concluded that
$$
H^4(M^7_{\ul{a}, \ul{b}}, M_\pm; \mathbb{Z}) = \mathbb{Z}.
$$
These considerations, together with the Universal Coefficient Theorem for cohomology \cite[Chap.\ 5, Theorem 10]{Sp}, now easily yield the cohomology groups listed in Table \ref{table:cohomgps}.
\begin{table}
\begin{tabular}[h]{|Sc||Sc|c|Sc|}
\cline{2-4}
\multicolumn{1}{c||}{} & $M^7_{\ul{a}, \ul{b}}$ & $M_\pm$ & $(M^7_{\ul{a}, \ul{b}}, M_\pm)$
\\ \hline \hline
$H^3(- , ; \mathbb{Z})$ & $0$ & $\mathbb{Z}$ & $\mathbb{Z}_2$ \\ \hline
$H^3(-; \mathbb{Q})$ & $0$ & $\mathbb{Q}$ & $0$ \\ \hline
$H^3(-; \mathbb{Q}/\mathbb{Z})$ & $\mathbb{Z}_{|n|}$ & $\mathbb{Q}/\mathbb{Z}$ & $0$
\\ \hline\hline
$H^4(-; \mathbb{Z})$ & $\mathbb{Z}_{|n|}$ & $0$ & $\mathbb{Z}$ \\ \hline
$H^4(-; \mathbb{Q})$ & $0$ & $0$ & $\mathbb{Q}$ \\ \hline
$H^4(-; \mathbb{Q}/\mathbb{Z})$ & $0$ & $\mathbb{Z}_2$ & $\mathbb{Q}/\mathbb{Z}$ \\ \hline
\end{tabular}
\caption{Important cohomology groups in $\mathbb{Z}$, $\mathbb{Q}$ and $\mathbb{Q}/\mathbb{Z}$ coefficients.}
\label{table:cohomgps}
\end{table}
From the short exact coefficient sequence
$$
0 \longrightarrow \mathbb{Z} \stackrel{m}{\longrightarrow} \mathbb{Q} \stackrel{r}{\longrightarrow} \mathbb{Q} / \mathbb{Z} \longrightarrow 0,
$$
together with the maps $i_\pm : M_\pm \hookrightarrow M^7_{\ul{a}, \ul{b}}$ and $q_\pm : (M^7_{\ul{a}, \ul{b}}, \emptyset) \to (M^7_{\ul{a}, \ul{b}}, M_\pm)$, one obtains a commutative diagram
\begin{equation}
\label{E:bigdiag}
\timesymatrix@C=0.54cm{
& 0 \ar[d] & 0 \ar[d] & 0 \ar[d] & \\
0 \ar[r] & \mathcal C^*(M^7_{\ul{a}, \ul{b}}, M_\pm; \mathbb{Z}) \ar[r]^(0.55){q_\pm^*} \ar[d]^m &
\mathcal C^*(M^7_{\ul{a}, \ul{b}}; \mathbb{Z}) \ar[r]^{i_\pm^*} \ar[d]^m &
\mathcal C^*(M_\pm; \mathbb{Z}) \ar[r] \ar[d]^m & 0 \\
0 \ar[r] & \mathcal C^*(M^7_{\ul{a}, \ul{b}}, M_\pm; \mathbb{Q}) \ar[r]^(0.55){q_\pm^*} \ar[d]^r &
\mathcal C^*(M^7_{\ul{a}, \ul{b}}; \mathbb{Q}) \ar[r]^{i_\pm^*} \ar[d]^r &
\mathcal C^*(M_\pm; \mathbb{Q}) \ar[r] \ar[d]^r & 0 \\
0 \ar[r] & \mathcal C^*(M^7_{\ul{a}, \ul{b}}, M_\pm; \mathbb{Q}/\mathbb{Z}) \ar[r]^(0.55){q_\pm^*} \ar[d] &
\mathcal C^*(M^7_{\ul{a}, \ul{b}}; \mathbb{Q}/\mathbb{Z}) \ar[r]^{i_\pm^*} \ar[d] &
\mathcal C^*(M_\pm; \mathbb{Q}/\mathbb{Z}) \ar[r] \ar[d] & 0 \\
& 0 & 0 & 0 &
}
\end{equation}
of cochain complexes. This induces a commutative diagram
\begin{equation}
\label{E:cohomdiag}
\resizebox{\displaywidth}{!}{\timesymatrix@C=0.4cm{
& & H^3(M^7_{\ul{a}, \ul{b}}, M_\pm; \mathbb{Q}/\mathbb{Z}) \ar[d] \ar[r]^(0.55){q_\pm^*} & H^3(M^7_{\ul{a}, \ul{b}}; \mathbb{Q}/\mathbb{Z}) \ar[d]^\beta \\
H^3(M^7_{\ul{a}, \ul{b}}; \mathbb{Z}) \ar[r]^{i_\pm^*} \ar[d]^m &
H^3(M_\pm; \mathbb{Z}) \ar[r]^(0.45){\delta_\pm} \ar[d]^m &
H^4(M^7_{\ul{a}, \ul{b}}, M_\pm; \mathbb{Z}) \ar[r]^(0.55){q_\pm^*} \ar[d]^m &
H^4(M^7_{\ul{a}, \ul{b}}; \mathbb{Z}) \ar[d]^m \\
H^3(M^7_{\ul{a}, \ul{b}}; \mathbb{Q}) \ar[r]^{i_\pm^*} \ar[d]^r &
H^3(M_\pm; \mathbb{Q}) \ar[r]^(0.45){\delta_\pm} \ar[d]^r &
H^4(M^7_{\ul{a}, \ul{b}}, M_\pm; \mathbb{Q}) \ar[r]^(0.55){q_\pm^*} \ar[d]^r &
H^4(M^7_{\ul{a}, \ul{b}}; \mathbb{Q}) \ar[d]^r \\
H^3(M^7_{\ul{a}, \ul{b}}; \mathbb{Q}/\mathbb{Z}) \ar[r]^{i_\pm^*} \ar[d]^\beta &
H^3(M_\pm; \mathbb{Q}/\mathbb{Z}) \ar[r]^(0.45){\delta_\pm} \ar[d] &
H^4(M^7_{\ul{a}, \ul{b}}, M_\pm; \mathbb{Q}/\mathbb{Z}) \ar[r]^(0.55){q_\pm^*} &
H^4(M^7_{\ul{a}, \ul{b}}; \mathbb{Q}/\mathbb{Z}) \\
H^4(M^7_{\ul{a}, \ul{b}}; \mathbb{Z}) \ar[r]^{i_\pm^*} &
H^4(M_\pm; \mathbb{Z}) & &
}}
\end{equation}
of long exact sequences for the pair $(M^7_{\ul{a}, \ul{b}}, M_\pm)$, where $\delta_\pm : H^j(M_\pm; R) \to H^{j+1}(M^7_{\ul{a}, \ul{b}}, M_\pm; R)$, $R \in \{\mathbb{Z}, \mathbb{Q}, \mathbb{Q}/\mathbb{Z}\}$, denotes the coboundary homomorphism.
By exactness and by Table \ref{table:cohomgps}, it can immediately be deduced from diagram \eqref{E:cohomdiag}: both the Bockstein homomorphism $\beta : H^3(M^7_{\ul{a}, \ul{b}}; \mathbb{Q}/\mathbb{Z}) \to H^4(M^7_{\ul{a}, \ul{b}};\mathbb{Z})$ and $\delta_\pm : H^3(M_\pm; \mathbb{Q}) \to H^{4}(M^7_{\ul{a}, \ul{b}}, M_\pm; \mathbb{Q})$ are isomorphisms; the homomorphisms $m : H^4(M^7_{\ul{a}, \ul{b}}, M_\pm; \mathbb{Z}) \to H^4(M^7_{\ul{a}, \ul{b}}, M_\pm; \mathbb{Q})$ and $i_\pm^* : H^3(M^7_{\ul{a}, \ul{b}}; \mathbb{Q}/ \mathbb{Z}) \to H^3(M_\pm; \mathbb{Q}/\mathbb{Z})$ are injective; and the homomorphims $r : H^3(M_\pm; \mathbb{Q}) \to H^3(M_\pm; \mathbb{Q}/\mathbb{Z})$ and $q_\pm^* : H^4(M^7_{\ul{a}, \ul{b}}, M_\pm; \mathbb{Z}) \to H^4(M^7_{\ul{a}, \ul{b}}; \mathbb{Z})$ are surjective. From these observations, it is now possible to gain some further understanding of the Bockstein homomorphism.
\begin{prop}
\label{P:Bockstein}
Suppose $M^7_{\ul{a}, \ul{b}}$ satisfies \eqref{E:finite}. Then, with the notation above, the Bockstein homomorphism satisfies
$$
i_\pm^* \circ \beta^{-1} \circ q_\pm^* = r \circ \delta_\pm^{-1} \circ m : H^4(M^7_{\ul{a}, \ul{b}}, M_\pm; \mathbb{Z}) \to H^3(M_\pm; \mathbb{Q}/\mathbb{Z}).
$$
\end{prop}
\begin{proof}
The proof will be on the level of cochains. If $w_\pm \in \mathcal C^4(M^7_{\ul{a}, \ul{b}}, M_\pm; Z)$ represents a cohomology class $[w_\pm] \in H^4(M^7_{\ul{a}, \ul{b}}, M_\pm; \mathbb{Z})$, then, since the Bockstein homomorphism $\beta : H^3(M^7_{\ul{a}, \ul{b}}; \mathbb{Q}/\mathbb{Z}) \to H^4(M^7_{\ul{a}, \ul{b}};\mathbb{Z})$ is an isomorphism, there is a unique class $[w] \in H^3(M^7_{\ul{a}, \ul{b}}; \mathbb{Q}/\mathbb{Z})$ such that $\beta([w]) = q_\pm^*([w_\pm])$. Let $w \in \mathcal C^3(M^7_{\ul{a}, \ul{b}}; \mathbb{Q}/\mathbb{Z})$ represent $[w] \in H^3(M^7_{\ul{a}, \ul{b}}; \mathbb{Q}/\mathbb{Z})$ and $q_\pm^*(w_\pm) \in \mathcal C^4(M^7_{\ul{a}, \ul{b}}; \mathbb{Z})$ represent $\beta([w]) = q_\pm^* ([w_\pm]) \in H^4(M^7_{\ul{a}, \ul{b}}; \mathbb{Z})$.
Recall that $\beta : H^3(M^7_{\ul{a}, \ul{b}}; \mathbb{Q}/\mathbb{Z}) \to H^4(M^7_{\ul{a}, \ul{b}};\mathbb{Z})$ arises by applying the Snake Lemma to the commutative diagram
\begin{equation}
\label{E:Bock}
\timesymatrix{
0 \ar[r]
& \mathcal C^3(M^7_{\ul{a}, \ul{b}}; \mathbb{Z}) \ar[r]^m \ar[d]^\delta
& \mathcal C^3(M^7_{\ul{a}, \ul{b}}; \mathbb{Q}) \ar[r]^r \ar[d]^\delta
& \mathcal C^3(M^7_{\ul{a}, \ul{b}}; \mathbb{Q}/\mathbb{Z}) \ar[r] \ar[d]^\delta
& 0 \\
0 \ar[r]
& \mathcal C^4(M^7_{\ul{a}, \ul{b}}; \mathbb{Z}) \ar[r]^m
& \mathcal C^4(M^7_{\ul{a}, \ul{b}}; \mathbb{Q}) \ar[r]^r
& \mathcal C^4(M^7_{\ul{a}, \ul{b}}; \mathbb{Q}/\mathbb{Z}) \ar[r]
& 0
}
\end{equation}
of exact sequences of cochain groups, where $\delta : \mathcal C^3(M^7_{\ul{a}, \ul{b}}; R) \to \mathcal C^4(M^7_{\ul{a}, \ul{b}}; R)$ is the coboundary map for coefficients in $R$. Since $\beta([w]) = q_\pm^* ([w_\pm])$, a cochain $u \in \mathcal C^3(M^7_{\ul{a}, \ul{b}}; \mathbb{Q})$ may thus be chosen such that
\begin{equation}
\label{E:ruw}
r(u) = w
\ \ \text{ and } \ \
\delta u = m(q_\pm^*(w_\pm)).
\end{equation}
Notice, however, that the middle vertical map in \eqref{E:Bock} also appears in the same position in the commutative diagram
\begin{equation}
\label{E:pairdiag}
\timesymatrix{
0 \ar[r]
& \mathcal C^3(M^7_{\ul{a}, \ul{b}}, M_\pm; \mathbb{Q}) \ar[r]^(0.55){q_\pm^*} \ar[d]^\delta
& \mathcal C^3(M^7_{\ul{a}, \ul{b}}; \mathbb{Q}) \ar[r]^{i_\pm^*} \ar[d]^\delta
& \mathcal C^3(M_\pm; \mathbb{Q}) \ar[r] \ar[d]^\delta
& 0 \\
0 \ar[r]
& \mathcal C^4(M^7_{\ul{a}, \ul{b}}, M_\pm; \mathbb{Q}) \ar[r]^(0.55){q_\pm^*}
& \mathcal C^4(M^7_{\ul{a}, \ul{b}}; \mathbb{Q}) \ar[r]^{i_\pm^*}
& \mathcal C^4(M_\pm; \mathbb{Q}) \ar[r]
& 0
}
\end{equation}
for the pair$(M^7_{\ul{a}, \ul{b}}, M_\pm)$. Observe that, although $u \in \mathcal C^3(M^7_{\ul{a}, \ul{b}}; \mathbb{Q})$ is only a cochain, its image $i_\pm^*(u)$ under $i_\pm^* : \mathcal C^3(M^7_{\ul{a}, \ul{b}}; \mathbb{Q}) \to \mathcal C^3(M_\pm; \mathbb{Q})$ is a cocycle. Indeed, from \eqref{E:ruw} and the diagram \eqref{E:bigdiag} it may be deduced that
$$
\delta(i_\pm^*(u)) = i_\pm^*(\delta u) = i_\pm^*(m(q_\pm^*(w_\pm))) = m(i_\pm^*(q_\pm^*(w_\pm))) = 0.
$$
Therefore, by applying the Snake Lemma to \eqref{E:pairdiag}, the image $\delta_\pm ([i_\pm^*(u)])$ of the class $[i_\pm^*(u)] \in H^3(M_\pm; \mathbb{Q})$ under the boundary homomorphism $\delta_\pm : H^3(M_\pm; \mathbb{Q}) \to H^4(M^7_{\ul{a}, \ul{b}}, M_\pm; \mathbb{Q})$ can be represented by a cocycle $c_\pm \in \mathcal C^4(M^7_{\ul{a}, \ul{b}}, M_\pm; \mathbb{Q})$ such that, by \eqref{E:ruw} and \eqref{E:bigdiag},
$$
q_\pm^*(c_\pm) = \delta u = m(q_\pm^*(w_\pm)) = q_\pm^*(m(w_\pm)).
$$
However, by \eqref{E:bigdiag}, the cochain map $q_\pm^* : \mathcal C^4(M^7_{\ul{a}, \ul{b}}, M_\pm; \mathbb{Z}) \to \mathcal C^4(M^7_{\ul{a}, \ul{b}}; \mathbb{Z})$ is injective, implying that $c_\pm = m(w_\pm) \in \mathcal C^4(M^7_{\ul{a}, \ul{b}}, M_\pm; \mathbb{Q})$ and, hence, that $\delta_\pm([i_\pm^*(u)]) = [m(w_\pm)] = m([w_\pm])$.
Since $\delta_\pm : H^3(M_\pm; \mathbb{Q}) \to H^4(M^7_{\ul{a}, \ul{b}}, M_\pm; \mathbb{Q})$ is an isomorphism, it thus follows from \eqref{E:bigdiag} and \eqref{E:ruw} that
\begin{align*}
r \circ \delta_\pm^{-1} \circ m ([w_\pm])
&= r([i_\pm^*(u)]) \\
&= i_\pm^*([r(u)]) \\
&= i_\pm^*([w]) \\
&= i_\pm^* \circ \beta^{-1} \circ q_\pm^* ([w_\pm]),
\end{align*}
as desired, where the final equality is a consequence of $\beta([w]) = q_\pm^*([w_\pm])$ and the fact that $\beta : H^3(M^7_{\ul{a}, \ul{b}}; \mathbb{Q}/\mathbb{Z}) \to H^4(M^7_{\ul{a}, \ul{b}}; \mathbb{Z})$ is an isomorphism.
\end{proof}
\section{The linking form}
\label{S:linking}
Associated to the decomposition $M^7_{\ul{a}, \ul{b}} = M_-\cup M_+$ of each $M^7_{\ul{a}, \ul{b}}$ into the union of two disk-bundles with $M_- \cap M_+ = M_0$, there is a commutative braid diagram
\begin{equation}
\label{E:braid}
\resizebox{\displaywidth}{!}{
\timesymatrix@=0.4cm{
{\phantom{0}} \ar[dr]^(0.4){q_-^*} \ar@(ur,ul)[rr] &
& H^3(M_+; R) \ar[dr]^{j_+^*} \ar@(ur,ul)[rr]^{\delta_+} &
& H^4(M^7_{\ul{a}, \ul{b}}, M_+; R) \ar[dr]^(0.55){q_+^*} \ar@(ur,ul)[rr] &
& {\phantom{0}} \\
& H^3(M^7_{\ul{a}, \ul{b}}; R) \ar[ur]^{i_+^*} \ar[dr]^(0.55){i_-^*} &
& H^3(M_0; R) \ar[ur]^(0.45){\partial_-} \ar[dr]^(0.5){\partial_+} &
& H^4(M^7_{\ul{a}, \ul{b}}; R) \ar[ur]^(0.55){i_-^*} \ar[dr]^(0.6){i_+^*} & \\
{\phantom{0}} \ar[ur]^(0.4){q_+^*} \ar@(dr,dl)[rr]&
& H^3(M_-; R) \ar[ur]^{j_-^*} \ar@(dr,dl)[rr]_{\delta_-} &
& H^4(M^7_{\ul{a}, \ul{b}}, M_-; R) \ar[ur]^(0.5){q_-^*} \ar@(dr,dl)[rr]&
& {\phantom{0}}
}}
\end{equation}
with coefficients in $R \in \{\mathbb{Z}, \mathbb{Q}, \mathbb{Q}/\mathbb{Z}\}$, where each braid is the long exact sequence of a pair. In particular, the isomorphisms $f_\pm^* : H^j(M^7_{\ul{a}, \ul{b}}, M_\pm; R) \to H^j(M_\mp, M_0; R)$ given by excision are being used implicitly and the homomorphism $\partial_\pm : H^3(M_0;R) \to H^4(M^7_{\ul{a}, \ul{b}}, M_\mp; R)$ corresponds to the boundary homomorphism in the long exact sequence for the pair $(M_\pm, M_0)$.
Furthermore, given the projection $\tau : M^7_{\ul{a}, \ul{b}} \to [-1,1]$ discussed in Section \ref{SS:family}, observe that the inclusion of the submanifold $\tau^{-1}[0,1] \subseteq M^7_{\ul{a}, \ul{b}}$ with boundary $\tau^{-1}\{0\}$ into the disk-bundle $M_+$ induces a homotopy equivalence $(\tau^{-1}[0,1], \tau^{-1}\{0\}) \to (M_+, M_0)$. Therefore, Poincar\'e duality holds for $(M_+, M_0)$ just as for compact, orientable manifolds with boundary. In particular, if $[M_+] \in H_7(M_+, M_0;\mathbb{Z})$ is a fundamental class, then
$$
\frown [M_+] : H^k(M_+, M_0; R) \to H_{7-k}(M_+;R) \,;\, \alpha \mapsto \alpha \frown [M_+]
$$
is an isomorphism for all $k$. An analogous argument works for $(M_-, M_0)$.
Let $[M] \in H_7(M^7_{\ul{a}, \ul{b}})$ be a fundamental class of $M^7_{\ul{a}, \ul{b}}$. Then $(q_-)_* [M]$ is a fundamental class for the pair $(M^7_{\ul{a}, \ul{b}}, M_-)$ and, by excision, there is a fundamental class $[M_+] \in H_7(M_+, M_0)$ for the pair $(M_+, M_0)$ such that $(f_-)_*[M_+] = (q_-)_* [M]$.
Let $x_\pm \in H^3(M_\pm; \mathbb{Z}) = \mathbb{Z}$ be the generators used in \eqref{E:maps}. By the Universal Coefficient Theorem, together with Table \ref{table:cohomgps}, $H^3(M_+; \mathbb{Z})$ is naturally isomorphic to $\mc{H}om(H_3(M_+), \mathbb{Z})$. Therefore, by Poincar\'e duality, a generator $\mf{g}amma_- \in H^4(M^7_{\ul{a}, \ul{b}}, M_-; \mathbb{Z}) = \mathbb{Z}$ may be chosen such that $f_-^*(\mf{g}amma_-) \in H^4(M_+, M_0; \mathbb{Z})$ is a generator and the generator $(f_-^*(\mf{g}amma_-)) \!\frown\! [M_+] \in H_3(M_+)$ is dual to $x_+$.
By exactness and since $H^4(M^7_{\ul{a}, \ul{b}}; \mathbb{Z}) = \mathbb{Z}_{|n|}$, the boundary homomorphisms $\delta_\pm : H^3(M_\pm; \mathbb{Z}) = \mathbb{Z} \to H^4(M^7_{\ul{a}, \ul{b}}, M_\pm; \mathbb{Z}) = \mathbb{Z}$ are given, up to sign, by multiplication by $n$. Therefore, a generator $\mf{g}amma_+ \in H^4(M^7_{\ul{a}, \ul{b}}, M_+; \mathbb{Z})$ can be chosen such that $\delta_+(x_+) = n \mf{g}amma_+$. Moreover, since $m : H^4(M^7_{\ul{a}, \ul{b}}, M_+; \mathbb{Z}) \to H^4(M^7_{\ul{a}, \ul{b}}, M_+; \mathbb{Q})$ is injective and $\delta_+ : H^3(M_+; \mathbb{Q}) \to H^{4}(M^7_{\ul{a}, \ul{b}}, M_+; \mathbb{Q})$ is an isomorphism, it follows from \eqref{E:cohomdiag} that
\begin{equation}
\label{E:gens}
\delta_+^{-1} \circ m (\mf{g}amma_+) = \frac{1}{n} \, m(x_+) \in H^3(M_+; \mathbb{Q}).
\end{equation}
Now, since $q_\pm^* : H^4(M^7_{\ul{a}, \ul{b}}, M_\pm; \mathbb{Z}) \to H^4(M^7_{\ul{a}, \ul{b}}; \mathbb{Z})$ are surjective, a generator of $H^4(M^7_{\ul{a}, \ul{b}}; \mathbb{Z})$ can be defined by $\mf{o}ne := q_+^*(\mf{g}amma_+)$. This is the generator mentioned in Theorem \ref{T:thmB} and, furthermore, there is some $\lambda \in \mathbb{Z}$ such that $\lambda$ mod $|n|$ is a unit in $\mathbb{Z}_{|n|}$ and such that $q_-^*(\lambda \mf{g}amma_-) = \mf{o}ne$.
\begin{prop}
\label{P:LF}
With the notation above, the linking form
$$
\lk : H^4(M^7_{\ul{a}, \ul{b}}; \mathbb{Z}) \mf{o}times H^4(M^7_{\ul{a}, \ul{b}}; \mathbb{Z}) \to \mathbb{Q}/\mathbb{Z}
$$
is given by $\lk(x \mf{o}ne, y \mf{o}ne) = \frac{\lambda xy}{n} \!\! \mod 1$.
\end{prop}
\begin{proof}
By bilinearity, only $\lk(\mf{o}ne, \mf{o}ne)$ needs to be computed. By \eqref{E:LF},
\begin{align*}
\lk(\mf{o}ne, \mf{o}ne) &= \langle \beta^{-1}(\mf{o}ne) \smile \mf{o}ne, [M]\rangle \\
&= \langle \lambda \, \beta^{-1}(\mf{o}ne) \smile (q_-^*(\mf{g}amma_-)), [M] \rangle \\
&= \langle \lambda \, q_-^*(\beta^{-1}(\mf{o}ne) \smile \mf{g}amma_-), [M] \rangle,
\end{align*}
where the last equality follows from \cite[page 251]{Sp}, since $q_- : (M^7_{\ul{a}, \ul{b}}, \emptyset) \to (M^7_{\ul{a}, \ul{b}}, M_\pm)$ is induced by the identity map on $M^7_{\ul{a}, \ul{b}}$. By naturality of the Kronecker pairing, it now follows that
\begin{align*}
\lk(\mf{o}ne, \mf{o}ne) &= \langle \lambda \, \beta^{-1}(\mf{o}ne) \smile \mf{g}amma_-, (q_-)_* [M] \rangle \\
&= \langle \lambda \, \beta^{-1}(\mf{o}ne) \smile \mf{g}amma_-, (f_-)_* [M_+] \rangle \\
&= \langle \lambda \, f_-^*(\beta^{-1}(\mf{o}ne) \smile \mf{g}amma_-), [M_+] \rangle \\
&= \langle \lambda \, i_+^*(\beta^{-1}(\mf{o}ne)) \smile f_-^*(\mf{g}amma_-), [M_+] \rangle,
\end{align*}
where the last equality again follows from \cite[page 251]{Sp}, since the map $f_- : (M_+, M_0) \to (M^7_{\ul{a}, \ul{b}}, M_-)$ is induced by the inclusion $i_+ : M_+ \to M^7_{\ul{a}, \ul{b}}$. Now, by Proposition \ref{P:Bockstein} and \eqref{E:gens},
\begin{align*}
i_+^*(\beta^{-1}(\mf{o}ne)) &= i_+^* \circ \beta^{-1} \circ q_+^*(\mf{g}amma_+) \\
&= r \circ \delta_+^{-1} \circ m(\mf{g}amma_+) \\
&= r \left(\frac{1}{n} \, m(x_+) \right).
\end{align*}
Therefore, by naturality with respect to the inclusion $m : \mathbb{Z} \to \mathbb{Q}$ and the reduction $r : \mathbb{Q} \to \mathbb{Q}/\mathbb{Z}$, it follows that
\begin{align*}
\lk(\mf{o}ne, \mf{o}ne) &= \left\langle\lambda \, r \left(\frac{1}{n} \, m(x_+) \right) \smile f_-^*(\mf{g}amma_-), [M_+] \right\rangle \\
&= r \left( \frac{\lambda}{n} \, m \left( \langle x_+ \smile f_-^*(\mf{g}amma_-), [M_+] \rangle \right) \right) \\
&= r \left( \frac{\lambda}{n} \, m \left( \langle x_+, f_-^*(\mf{g}amma_-) \frown [M_+] \rangle \right) \right) \\
&= r \left( \frac{\lambda}{n} \right) \\
&= \frac{\lambda}{n} \!\! \mod 1,
\end{align*}
as desired, where the second-last equality follows since $(f_-^*(\mf{g}amma_-)) \frown [M_+] \in H_3(M_+)$ is dual to $x_+ \in H^3(M_+; \mathbb{Z})$.
\end{proof}
Therefore, in order to prove Theorem \ref{T:thmB}, it remains only to determine the value of $\lambda \in \mathbb{Z}$ in the formula for the linking form given in Proposition \ref{P:LF}. To this end, it is necessary to first introduce two further bases, $\{u_1, u_2\}$ and $\{w_1, w_2\}$, for $H^3(M_0; \mathbb{Z}) = \mathbb{Z} \mf{o}plus \mathbb{Z}$, in addition to the basis $\{v_1, v_2\}$ used in \eqref{E:maps}. Recall from \eqref{E:free} that
$$
\mf{g}cd(a_1, a_2 \pm a_3) = 1 = \mf{g}cd(b_1, b_2 \pm b_3).
$$
Hence, there exist $e_0, e_1, f_0, f_1 \in \mathbb{Z}$ such that
$$
e_1 \, a_1^2 + e_0 \left(\tfrac{a_2^2 - a_3^2}{8} \right) = 1
\ \ \text{ and } \ \
f_1 \, b_1^2 + f_0 \left(\tfrac{b_2^2 - b_3^2}{8} \right) = 1.
$$
Therefore, as each of the elements $j_-^*(x_-) = \frac{1}{8}(a_2^2 - a_3^2) \, v_1 + a_1^2 \, v_2$ and $j_+^*(x_+) = -\frac{1}{8}(b_2^2 - b_3^2) \, v_1 - b_1^2 \, v_2$ is a generator of $H^3(M_0;\mathbb{Z})$, the two new bases can be defined via
$$
u_1 := j_-^*(x_-),
\mathsf{q}quad
u_2 := - e_1 \, v_1 + e_0 \, v_2
$$
and
$$
w_1 := j_+^*(x_+),
\mathsf{q}quad
w_2 := \varepsilon (- f_1 \, v_1 + f_0 \, v_2),
$$
where $\varepsilon \in \{\pm 1\}$ is such that $\delta_-(x_-) = \varepsilon n \mf{g}amma_-$. Define, in addition, the integers
$$
\kappa := f_1 \, a_1^2 + f_0 \left(\tfrac{a_2^2 - a_3^2}{8} \right)
\ \ \text{ and } \ \
\rho := e_1 \, b_1^2 + e_0 \left(\tfrac{b_2^2 - b_3^2}{8} \right),
$$
for which the following congruence identities hold:
\begin{equation}
\begin{split}
\label{E:cong}
a_1^2 \, \rho &\equiv b_1^2 \!\! \mod n, \\
b_1^2 \, \kappa &\equiv a_1^2 \!\! \mod n, \\
\frac{1}{8}(a_2^2 - a_3^2) \, \rho &\equiv \frac{1}{8}(b_2^2 - b_3^2) \!\! \mod n, \\
\frac{1}{8}(b_2^2 - b_3^2) \, \kappa &\equiv \frac{1}{8}(a_2^2 - a_3^2) \!\! \mod n.
\end{split}
\end{equation}
Observe, finally, that the basis element $u_2$ can be written in terms of the basis $\{w_1, w_2\}$ as
\begin{equation}
\label{E:u2}
u_2 = (e_1 \, f_0 - e_0 \, f_1)\, w_1 + \varepsilon \rho \, w_2.
\end{equation}
It is now possible to complete the proof of Theorem \ref{T:thmB}.
\begin{thm}
\label{T:PfthmB}
With the notation above, the linking form $\lk : H^4(M^7_{\ul{a}, \ul{b}}; \mathbb{Z}) \mf{o}times H^4(M^7_{\ul{a}, \ul{b}}; \mathbb{Z}) \to \mathbb{Q}/\mathbb{Z}$ is given by
$$
\lk(x \mf{o}ne, y \mf{o}ne) = \pm \frac{\rho \, xy}{n} \!\! \mod 1.
$$
Alternatively, with respect to the generator $\mf{o}ne' := \kappa \, \mf{o}ne$, the linking form is given by $\lk(x \mf{o}ne', y \mf{o}ne') = \pm \frac{\kappa \, xy}{n} \!\! \mod 1$.
\end{thm}
\begin{proof}
From exactness and commutativity in the braid diagram \eqref{E:braid}, the following identities hold:
$$
\partial_\pm \circ j_\pm^* = 0
\ \ \text{ and } \ \
\partial_\mp \circ j_\pm^* = \delta_\pm.
$$
Now, recall that $\delta_+(x_+) = n \, \mf{g}amma_+$ and $\delta_-(x_-) = \varepsilon n \mf{g}amma_-$ for some $\varepsilon \in \{\pm 1\}$. Therefore, it is a simple calculation to show that the homomorphisms $\partial_\pm : H^3(M_0; \mathbb{Z}) \to H^4(M^7_{\ul{a}, \ul{b}}, M_\mp; \mathbb{Z})$ are given by
$$
\partial_-(v_1) = -a_1^2 \, \mf{g}amma_+,
\mathsf{q}quad
\partial_-(v_2) = \frac{a_2^2 - a_2^3}{8} \, \mf{g}amma_+
$$
and
$$
\partial_+(v_1) = - \varepsilon b_1^2 \, \mf{g}amma_-,
\mathsf{q}quad
\partial_+(v_2) = \varepsilon \frac{b_2^2 - b_2^3}{8} \, \mf{g}amma_-,
$$
respectively. From the definition of the bases $\{u_1, u_2\}$ and $\{w_1, w_2\}$, it now follows that
$$
\partial_-(u_1) = 0,
\mathsf{q}quad
\partial_-(u_2) = \mf{g}amma_+,
\ \ \text{ and } \ \
\partial_-(w_2) = \varepsilon \kappa \, \mf{g}amma_+,
$$
while
$$
\partial_+(w_1) = 0,
\mathsf{q}quad
\partial_+(w_2) = \mf{g}amma_-,
\ \ \text{ and } \ \
\partial_+(u_2) = \varepsilon \rho \, \mf{g}amma_-.
$$
Therefore, by \eqref{E:braid} and \eqref{E:u2},
\begin{align*}
\lambda \, q_-^* (\mf{g}amma_-) &= \mf{o}ne \\
&= q_+^*(\mf{g}amma_+) \\
&= q_+^*(\partial_-(u_2)) \\
&= q_-^*(\partial_+(u_2)) \\
&= q_-^*(\partial_+((e_1 \, f_0 - e_0 \, f_1)\, w_1 + \varepsilon \rho \, w_2)) \\
&= \varepsilon \rho \, q_-^*(\partial_+(w_2)) \\
&= \varepsilon \rho \, q_-^*(\mf{g}amma_-) \in H^4(M^7_{\ul{a}, \ul{b}}; \mathbb{Z}) = Z_{|n|},
\end{align*}
from which it immediately follows that $\lambda \equiv \varepsilon \rho$ mod $n$, as desired. The final statement in the theorem follows from a direct calculation showing that $\kappa \, \rho \equiv 1$ mod $n$, since this implies that $\mf{o}ne' = \kappa \mf{o}ne$ is a generator of $H^4(M^7_{\ul{a}, \ul{b}}; \mathbb{Z}) = \mathbb{Z}_{|n|}$.
\end{proof}
\section{Some elementary number theory}
\label{S:numth}
As a simple corollary of Theorem \ref{T:PfthmB}, it turns out that any $M^7_{\ul{a}, \ul{b}}$ with $\mf{g}cd(a_1, b_1) = 1$ and satisfying \eqref{E:finite} has standard linking form. Such manifolds include, of course, all $\mathbf{S}^3$-bundles over $\mathbf{S}^4$ with non-trivial $H^4$, as described by Grove and Ziller \cite{GZ}, which are well known to have standard linking form \cite{CE}.
\begin{lem}
\label{L:stdLF}
Every $M^7_{\ul{a}, \ul{b}}$ with $\mf{g}cd(a_1, b_1) = 1$ and satisfying \eqref{E:finite} is homotopy equivalent, hence PL-homeomorphic, to an $\mathbf{S}^3$-bundle over $\mathbf{S}^4$.
\end{lem}
\begin{proof}
Suppose that $M^7_{\ul{a}, \ul{b}}$ has $\mf{g}cd(a_1, b_1) = 1$. Then, by the definition of $n$,
$$
\mf{g}cd(a_1, n) = 1 = \mf{g}cd(b_1, n),
$$
that is, $a_1$ mod $n$ and $b_1$ mod $n$ are units in $\mathbb{Z}_{|n|}$. Therefore, $a_1 \mf{o}ne$ and $b_1 \mf{o}ne$ are generators of $H^4(M^7_{\ul{a}, \ul{b}}; \mathbb{Z}) = \mathbb{Z}_{|n|}$. In particular, by \eqref{E:cong} and Theorem \ref{T:PfthmB},
\begin{align*}
\lk(a_1 \mf{o}ne, a_1 \mf{o}ne) &= \pm \frac{a_1^2 \, \rho}{n} \!\! \mod 1 \\
&= \pm \frac{b_1^2}{n} \!\! \mod 1.
\end{align*}
Now, by the definition of standard linking form in Section \ref{SS:link} and Theorem \ref{T:KS}, the result follows.
\end{proof}
As a consequence of Lemma \ref{L:stdLF}, to have any hope of obtaining manifolds $M^7_{\ul{a}, \ul{b}}$ with non-standard linking form, it is necessary to assume that $\mf{g}cd(a_1, b_1) \neq 1$. In particular, this implies that there is some prime $p$ dividing $n$ such that $p^2$ also divides $n$. Therefore, as in Example \ref{Eg:nonst}, whenever $n$ is not divisible by $p^2$ for all prime divisors $p$ of $n$, there is the possibility of finding manifolds with non-standard linking form which cannot be described as a manifold $M^7_{\ul{a}, \ul{b}}$. Hence, the manifolds $M^7_{\ul{a}, \ul{b}}$ do not realise all $2$-connected $7$-manifolds with $H^4$ finite cyclic.
Returning to the search for manifolds $M^7_{\ul{a}, \ul{b}}$ with non-standard linking form, the following simple observation will prove useful.
\begin{lem}
\label{L:primes}
Suppose $d \in \mathbb{N}$ divides $n \in \mathbb{N}$ and that $k \in \mathbb{Z}$ is not a square mod $d$. Then $k \in \mathbb{Z}$ is not a square mod $n$.
\end{lem}
\begin{proof}
Suppose that there is some $l \in \mathbb{Z}$ such that $k \equiv l^2$ mod $n$. Then it is clear that $k \equiv l^2$ mod $d$, a contradiction.
\end{proof}
It now turns out that it is reasonably straightforward to find examples of manifolds $M^7_{\ul{a}, \ul{b}}$ with non-standard linking form. To avoid that the computations to follow become unnecessarily complicated, let
$$
a_0 := \frac{a_2^2 - a_3^2}{8},
\mathsf{q}quad
b_0 := \frac{b_2^2 - b_3^2}{8}.
$$
With this notation,
\begin{equation}
\label{E:conds}
\begin{split}
n &= a_1^2 \, b_0 - a_0\, b_1^2, \\
1 &= e_1 \, a_1^2 + e_0 \, a_0, \\
1 &= f_1 \, b_1^2 + f_0 \, b_0.
\end{split}
\end{equation}
Recall that, for $p$ an odd prime and $x \in \mathbb{Z}$, the \emph{Legendre symbol} $\bigl(\frac{x}{p} \bigr)$ is defined via
$$
\mathfrak{B}igl(\frac{x}{p} \mathfrak{B}igr) =
\begin{cases}
\phantom{-}1, & \text{ if $x$ is a square mod $p$ and $x \not\equiv 0$ mod $p$},\\
-1, & \text{ if $x$ is not a square mod $p$},\\
\phantom{-}0, & \text{ if $x \equiv 0$ mod $p$}.
\end{cases}
$$
The Legendre symbol has the following properties:
\begin{equation}
\label{E:legendre}
\begin{split}
\mathfrak{B}igl(\frac{x}{p} \mathfrak{B}igr) &= \mathfrak{B}igl(\frac{y}{p} \mathfrak{B}igr), \ \ \text{ if } x \equiv y \!\! \mod p \,; \\
\mathfrak{B}igl(\frac{xy}{p} \mathfrak{B}igr) &= \mathfrak{B}igl(\frac{x}{p} \mathfrak{B}igr) \mathfrak{B}igl(\frac{y}{p} \mathfrak{B}igr).
\end{split}
\end{equation}
The first supplement to the law of quadratic reciprocity states that
\begin{equation}
\label{E:-1}
\mathfrak{B}igl(\frac{-1}{p} \mathfrak{B}igr) = 1 \ \text{ if and only if } p \equiv 1 \!\! \mod 4,
\end{equation}
that is, $-1$ is a square if and only if $p \equiv 1$ mod $4$.
\begin{thm}
\label{T:egs}
Suppose $M^7_{\ul{a}, \ul{b}}$ satisfies \eqref{E:finite} and that there is a prime $p \equiv 1$ mod $4$ such that $p$ divides $\mf{g}cd(a_1, b_1)$. If $a_0$ is not a square mod $p$ and $b_0$ is a square mod $p$, then $M^7_{\ul{a}, \ul{b}}$ has non-standard linking form and, hence, is not even homotopy equivalent to an $\mathbf{S}^3$-bundle over $\mathbf{S}^4$.
\end{thm}
\begin{proof}
By Theorem \ref{T:PfthmB}, there is a generator $\mf{o}ne \in H^4(M^7_{\ul{a}, \ul{b}}; \mathbb{Z}) = \mathbb{Z}_{|n|}$ such that $\lk(\mf{o}ne, \mf{o}ne) = \pm \frac{\rho}{n}$ mod $1$, with $\rho = e_1 \, b_1^2 + e_0 \, b_0$. On the other hand, by \eqref{E:conds},
$e_0 \, a_0 \equiv 1 \!\! \mod p$. Since $1$ is obviously a square mod $p$, \eqref{E:legendre} implies that
$$
1 = \mathfrak{B}igl(\frac{1}{p} \mathfrak{B}igr)
= \mathfrak{B}igl(\frac{e_0 \, a_0}{p} \mathfrak{B}igr)
= \mathfrak{B}igl(\frac{e_0}{p} \mathfrak{B}igr) \mathfrak{B}igl(\frac{a_0}{p} \mathfrak{B}igr)
= - \mathfrak{B}igl(\frac{e_0}{p} \mathfrak{B}igr),
$$
because $\bigl(\frac{a_0}{p} \bigr) = -1$, by hypothesis. That is, $\bigl(\frac{e_0}{p} \bigr) = -1$.
Therefore, since $p \equiv 1$ mod $4$ was assumed to divide $b_1$,
\begin{align*}
\mathfrak{B}igl(\frac{\pm \rho}{p} \mathfrak{B}igr)
&= \mathfrak{B}igl(\frac{\pm (e_1 \, b_1^2 + e_0 \, b_0)}{p} \mathfrak{B}igr) \\
&= \mathfrak{B}igl(\frac{ \pm e_0 \, b_0}{p} \mathfrak{B}igr) \\
&= \mathfrak{B}igl(\frac{\pm 1}{p} \mathfrak{B}igr) \mathfrak{B}igl(\frac{e_0}{p} \mathfrak{B}igr) \mathfrak{B}igl(\frac{b_0}{p} \mathfrak{B}igr) \\
&= -1,
\end{align*}
where the final equality follows from \eqref{E:-1}, $\bigl(\frac{e_0}{p} \bigr) = -1$ and the hypothesis that $b_0$ is a square mod $p$.
Hence, $\pm \rho$ is not a square mod $p$ and, by Lemma \ref{L:primes}, it follows that $\pm \rho$ is not a square mod $n$. However, since $\pm \rho$ is a unit in $\mathbb{Z}_{|n|}$ (by the proof of Theorem \ref{T:PfthmB}), this implies that $M^7_{\ul{a}, \ul{b}}$ has a non-standard linking form, as desired.
\end{proof}
Explicit examples satisfying the hypotheses of Theorem \ref{T:egs} are plentiful. Indeed, note that, by a simply counting argument, for any prime $p \equiv 1$ mod $4$ there must be a pair $m$, $m+1$, $m \in \{1, \dots, p-2\}$, of consecutive integers such that $\bigl( \frac{m}{p} \bigr) = -1$ and $\bigl( \frac{m+1}{p} \bigr) = 1$.
\begin{cor}
Let $p \equiv 1$ mod $4$ be an odd prime. If $m \in \{1, \dots, p-2\}$ is such that $\bigl( \frac{m}{p} \bigr) = -1$ and $\bigl( \frac{m+1}{p} \bigr) = 1$, then $a_1 = b_1 = p$, $|a_2| = 2m - 1$, $|a_3| = |b_2| = 2m +1$ and $|b_3| = 2m+3$ define a manifold $M^7_{\ul{a}, \ul{b}}$ with non-standard linking form.
\end{cor}
\begin{proof}
Observe first that some choice of signs for $\pm(2m-1)$, $\pm(2m+1)$ and $\pm(2m+3)$ yields integers $\equiv 1$ mod $4$. Furthermore, $a_2^2 - a_3^2 = -8m$ and $b_2^2 - b_3^2 = -8(m+1)$ are, by definition, prime to $p = a_1 = b_1$. Therefore, the freeness conditions \eqref{E:free} are satisified and $\underline a$, $\underline b$ define a manifold $M^7_{\ul{a}, \ul{b}}$. Moreover, $n = -p^2$, so that $H^4(M^7_{\ul{a}, \ul{b}}; \mathbb{Z}) = \mathbb{Z}_{p^2}$.
Now $a_0 = -m$ and $b_0 = -(m+1)$. Thus, by the hypotheses on $m \in \{1, \dots, p-2\}$, Theorem \ref{T:egs} implies that $M^7_{\ul{a}, \ul{b}}$ has non-standard linking form.
\end{proof}
\end{document} |
\betagin{document}
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\betagin{abstract}This article describes a structure that metric spaces can be equipped with so that they resemble normed vector spaces and examines necessary and sufficient conditions for the existence of such a structure on a general metric space.\end{abstract}
\section{Introduction}
Many ideas and structures in mathematics emerge as a way of filling the space 'between' concepts that have already been established and studied thoroughly. A classic example of this is the concept of uniform spaces, that came about as an attempt to generalize notions of 'uniformity' to a class of topological spaces beyond just metric spaces.\\
The concept of a metric space with a dilation family constitutes a similar sort of bridge between already established concepts. Specifically, it is a structure that is stronger than a metric space but weaker than a normed real vector space. Given a metric space (not necessarily equipped with any inner algebraic operation), a dilation family on the space is like an action by the nonnegative reals on the space that satisfies an analog of the homogeneity property of a vector norm (i.e. that $\boldsymbol{||}a \vec{v}\boldsymbol{||}=|a|\;\boldsymbol{||}\vec{v}\boldsymbol{||}$ always). Similar concepts have been explored already in papers by Marius Buliga, such as that of a 'dilatation structure' which is somewhere "between a group and a differential structure" \cite{dilatations}. One structure already defined by M. Buliga that is very similar to that of a metric space with a dilation family is a 'normed conical group' \cite{intrinsic}. A normed conical group is essentially a normed vector space where the vector addition operation is allowed to be noncommutative. In this sense, it is a stronger structure than a metric space with a dilation family. We will show that an exact condition for a metric group with a dilation family can be formulated to determine whether or not a norm exists that makes it a normed conical group.\\
As for determining whether a metric space can be equipped with a dilation family, we will demonstrate that any metrizable cone space can be equipped with a metric that preserves its topology and under which a dilation family exists about the cone's center. We will then go on to show that the existence of a special kind of null-homotopy on a locally compact metric space guarantees that the space is equivalent to a metrizable cone, meaning that it can be equipped with a dilation family (there are two cases here, depending on the size of the dilation family). Lastly, we will provide one result that gives a necessary and sufficient characterization for metric spaces to possess dilation families - this characterization relies on the existence of a special family of bi-Lipschitz maps on a metric space.\\
\textbf{Acknowledgement:} I would like to thank Dr. Marius Buliga for pointing me to research relevant to this topic and my undergraduate Senior Project supervisor, Dr. Haniya Azam, for encouraging me to pursue this topic and giving me important advice during my first time working on an independent project.\\
\section{Dilation Families, Properties, and Characterization Results}
This section looks at some properties of dilation families and demonstrates some conditions on a metric space necessary and/ or sufficient for the existence of an associated dilation family.\\
\betagin{defi} Let $(X,d_1)$ and $(Y,d_2)$ be a pair of metric spaces. Let $\alpha$ be a nonnegative real number. If $h:X\rightarrow Y$ is a map such that $d_2(h(x_1),h(x_2))=\alpha d_1(x_1,x_2), \;\forall\; x_1,x_2\in X$, then $h$ is said to be a dilation from $(X,d_1)$ to $(Y,d_2)$ of scale $\alpha$.\\ \end{defi}
It is trivial to see that, just like isometries, all dilations are continuous and all dilations of positive scale are injective.\\
The central object of study here will be something I refer to as a 'dilation family'. Given a metric space $(X,d)$ and an index set $I\subset \mathbb R_{\geq 0}$ this is an indexed set of functions $\{T_{\alpha}\}_{\alpha\in I}\subset X^X$ that 'scale distances exactly' about a fixed point $x_0\in X$, and satisfy the property that $T_{\alpha}\circ T_{\beta}=T_{\alpha\beta}$ always.\\
More precisely, we have the following definition:\\
\betagin{defi} Let $(X,d)$ be a metric space. Let $x_0$ be a point in $X$. Given any index set $I$ closed under multiplication and any collection $\mathcal{A}=\{T_{\alpha}\}_{\alpha\in I}\subset X^X$, we call $\mathcal{A}$ a dilation family in $X$ about $x_0$ if:\\
$1)\; T_{\alpha}$ is a dilation under $d$ of scale $\alpha, \;\forall\; \alpha \in I$\\
$2)\; T_{\alpha}(x_0)=x_0, \;\forall\; \alpha\in I$;\\
$3)\; T_{\alpha}\circ T_{\beta}=T_{\alpha\beta}, \;\forall\; \alpha,\beta\in I$; and\\
$4)\; \lim_{\alpha\rightarrow 1} T_{\alpha}(x)$ exists $\;\forall\; x\in X$\\ \end{defi}
We are mainly concerned with dilation families where the index set is a 'pure' index set, as described below:
\betagin{defi} Let $I$ be a subset of $\mathbb R_{\geq 0}$ closed under multiplication containing $1$. If - $1)\; I\subset [1,\infty)$ and for any $\alpha<\beta\in I, \frac{\beta}{\alpha}\in I$; or $2)\; I\subset [0,1]$ and for any $\alpha<\beta\in I, \frac{\alpha}{\beta}\in I$; or $3)\; I$ is closed under nonzero division - then $I$ is called a pure set.\\ \end{defi}
Regarding the $4$th condition in the definition of a dilation family — this is a, seemingly weak, kind of continuity requirement on the dilation family. As we will see, these structures represent a strengthening of classical contractibility conditions that appear in general topology. However, they may also be relevant to the study of certain kinds of metric measures, such as many Hausdorff measures, in which a dilation of a measurable set by scale $\alpha$ yields a measurable set with a measure scaled by a factor of $\alpha^d$ ($d$ being the dimension of the measure). It also allows one to define (once suitably extended) differentiability of a function at the 'center' of the dilation family.\\
We will start off with some results that just establish basic properties of dilation families and necessary conditions for their existence.\\
\betagin{prop}
\lambdabel{prop: if 1 or 0 is in I} Let $(X,d)$ be a metric space. Let $\{T_{\alpha}\}_{\alpha\in I}$ be a dilation family on $(X,d,x_0)$. Then $T_1=\textrm{Id}_X$. If $0\in I$, $T_0=O_{x_0}$.\end{prop}
\betagin{proof} For any $x\in X$, we have $T_1(T_1(x))=T_{1^2}(x)=T_1(x)$, so $d(x,T_1(x))=1\cdot d(x,T_1(x))=d(T_1(x),T_1(T_1(x)))=0$, i.e. $x=T_1(x), \;\forall\; x\in X$. Also, we must have $T_0(x_0)=x_0$, but for any $x\in X$, we have $d(T_0(x),T_0(x_0))=0\cdot d(x,x_0)=0$, so $T_0(x)=x_0,\;\forall\; x\in X$.\\ \end{proof}
\betagin{prop}
\lambdabel{prop: limit is identity} Let $(X,d)$ be a metric space. Let $\{T_{\alpha}\}_{\alpha\in I}$ be a dilation family on $(X,d,x_0)$. For any $x\in X$, we have $\lim_{\alpha\rightarrow 1} T_{\alpha}(x)=x$.\end{prop}
\betagin{proof} Let $x$ be any point of $X$. Let $y$ denote $\lim_{\alpha \rightarrow 1} T_{\alpha}(x)$. Then we also have that $y=\lim_{\alpha\rightarrow 1} T_{\alpha^2}(x)$.\\
This means
$$0=d(y,y)=d(\lim_{\alpha\rightarrow 1} T_{\alpha^2}(x),\lim_{\alpha\rightarrow 1} T_{\alpha}(x))$$
$$=\lim_{\alpha\rightarrow 1} d(T_{\alpha^2}(x),T_{\alpha}(x))=\lim_{\alpha\rightarrow 1} d(T_{\alpha}(T_{\alpha}(x)),T_{\alpha}(x))$$
$$=\lim_{\alpha\rightarrow 1} \alpha d(T_{\alpha}(x),x)=\lim_{\alpha \rightarrow 1} d(T_{\alpha}(x),x)$$
$$=d(\lim_{\alpha \rightarrow 1} T_{\alpha}(x),\lim_{\alpha \rightarrow 1} x)=d(y,x)$$
It follows that $y=x$.\\ \end{proof}
\betagin{prop}
\lambdabel{prop: x-function is cont} Let $(X,d)$ be a metric space. Let $\{T_{\alpha}\}_{\alpha\in I}$ be a dilation family on $(X,d,x_0)$. Then, for any $x\in X$, the function $T^x: I\rightarrow (X,d)$ defined as $\alpha \mathbbpsto T_{\alpha}(x)$ is continuous and uniformly continuous on any set $[a,b]\cap I$ where $a>0$.\end{prop}
\betagin{proof} Let any $\alpha\in I$ be given. We show continuity of $T^x$ at $\alpha$. There are three cases depending on the type of pure set $I$ is - these break down into three cases for $\alpha$, whether it's in $(0,1]$, $[1,\infty)$, or equal to $0$.\\
Suppose $\alpha$ is positive. Now, let any $\epsilon\in\mathbb R^+$ be given. Since $\lim_{\beta \rightarrow 1} T_{\beta}(x)$ exists and equals $x$, there exists $0<\delta<1$ such that whenever $\beta \in B(1;\delta)\cap I$ we have $d(T_{\beta}(x),x)<\frac{\epsilon}{2\alpha}$. Now suppose $\beta$ is any member of the set $B(\alpha; \frac{\alpha \delta}{2})\cap I$. If $\alpha \in (0,1]$ and $\beta<\alpha$, then we have:
$$d(T^x(\alpha),T^x(\beta))=d(T_{\alpha}(x),T_{\beta}(x))$$
$$=d(T_{\alpha}(x),T_{\alpha}(T_{\frac{\beta}{\alpha}}(x)))$$
$$=\alpha d(x,T_{\frac{\beta}{\alpha}}(x))$$
As $|\beta-\alpha|<\frac{\alpha \delta}{2}$, $|\frac{\beta}{\alpha}-1|<\frac{\delta}{2}<\delta$, so we have $d(T^x(\alpha),T^x(\beta))<\alpha \cdot \frac{\epsilon}{2\alpha}<\epsilon$.\\
If $\alpha\in (0,1]$ and $\beta>\alpha$, then we have:
$$d(T^x(\alpha),T^x(\beta))=d(T_{\alpha}(x),T_{\beta}(x))$$
$$=d(T_{\beta}(T_{\frac{\alpha}{\beta}}(x)),T_{\beta}(x))$$
$$=\beta d(T_{\frac{\alpha}{\beta}}(x),x)$$
As $|\beta-\alpha|<\frac{\alpha \delta}{2}$, $|1-\frac{\alpha}{\beta}|<\frac{\alpha}{\beta}\frac{\delta}{2}<\frac{\delta}{2}<\delta$, so we have $d(T^x(\alpha),T^x(\beta))<\beta \cdot \frac{\epsilon}{2\alpha}$. As $0<\delta<1$, $0<\beta<\frac{3\alpha}{2}<2\alpha$, so this means $d(T^x(\alpha),T^x(\beta))<\epsilon$.\\
If $\alpha\in [1,\infty)$ and $\beta<\alpha$, then we have:
$$d(T^x(\alpha),T^x(\beta))=d(T_{\alpha}(x),T_{\beta}(x))$$
$$=d(T_{\beta}(T_{\frac{\alpha}{\beta}}(x)),T_{\beta}(x))$$
$$=\beta d(T_{\frac{\alpha}{\beta}}(x),x)$$
As $|\beta-\alpha|<\frac{\alpha \delta}{2}$, $|1-\frac{\alpha}{\beta}|<\frac{\alpha}{\beta}\frac{\delta}{2}$. As $\delta<1$, $\beta>\frac{\alpha}{2}$, so $|1-\frac{\alpha}{\beta}|<\delta$, giving us $d(T^x(\alpha),T^x(\beta))<\beta \cdot \frac{\epsilon}{2\alpha}$. As $0<\delta<1$, $0<\beta<\frac{3\alpha}{2}<2\alpha$, so this means $d(T^x(\alpha),T^x(\beta))<\epsilon$.\\
And if $\alpha\in [1,\infty)$ and $\beta>\alpha$, then we have:
$$d(T^x(\alpha),T^x(\beta))=d(T_{\alpha}(x),T_{\beta}(x))$$
$$=d(T_{\alpha}(x),T_{\alpha}(T_{\frac{\beta}{\alpha}}(x)))$$
$$=\alpha d(x,T_{\frac{\beta}{\alpha}}(x))$$
As $|\beta-\alpha|<\frac{\alpha \delta}{2}$, $|\frac{\beta}{\alpha}-1|<\frac{\delta}{2}<\delta$, so we have $d(T^x(\alpha),T^x(\beta))<\alpha \cdot \frac{\epsilon}{2\alpha}<\epsilon$.\\
So for any $\alpha>0$, we have $T^x(B(\alpha;\frac{\alpha\delta}{2})\cap I)\subset B(T^x(\alpha);\epsilon)$. But this means that for any set $[a,b]\cap I$ with $a>0$ and any $\alpha\in [a,b]\cap I$, we have $T^x(B(\alpha;\frac{a\delta}{2})\cap I)\subset T^x(B(\alpha;\frac{\alpha\delta}{2})\cap I)\subset B(T^x(\alpha);\epsilon)$. As $\epsilon$ was an arbitrary positive real, we have that $T^x$ is uniformly continuous on any set $[a,b]\cap I$, and so is also continuous on $(0,\infty)\cap I$.\\
Now, suppose $0\in I$. To prove continuity of $T^x$ at $0$, again let any $\epsilon>0$ be given. If $x=x_0$, then $T^x$ is the constant function $x_0$, and so $T^x$ is trivially continuous at $0$. Assuming $x\neq x_0$, then given any $\beta\in B(0;\frac{\epsilon}{d(x_0,x)})$, we have:
$$d(T^x(0),T^x(\beta))=d(T_0(x),T_{\beta}(x))$$
$$=d(x_0,T_{\beta}(x))=d(T_{\beta}(x_0),T_{\beta}(x))$$
$$=\beta d(x_0,x)<\frac{\epsilon}{d(x_0,x)}\cdot d(x_0,x)=\epsilon$$
making $T^x$ continuous at $0$ as well.\\ \end{proof}
\betagin{prop}
\lambdabel{prop: extend dilation family} Let $(X,d)$ be a Cauchy metric space. Let $\{T_{\alpha}\}_{\alpha\in I}$ be a dilation family on $(X,d,x_0)$. Then, setting $J=\overline{I}$ (where the closure is taken in $\mathbb R_{\geq 0}$), $J$ is also a pure set and there exists a dilation family about $x_0$, $\{T_{\alpha}\}_{\alpha\in J}$, that extends $\{T_{\alpha}\}_{\alpha\in I}$.\end{prop}
\betagin{proof} First, I show that $J=\overline{I}$ is pure. As $I$ contains $1$, $J$ does too.\\
If $I\subset [0,1]$, then $J\subset [0,1]$ as well. Given any $c<d\in J$, we can choose sequences of elements, $\{\alpha_i\}_{i=1}^{\infty}\subset I$ converging to $c$ and $\{\beta_i\}_{i=1}^{\infty}\subset I$ converging to $d$, such that $\alpha_i<\beta_i, \;\forall\; i\in \mathbb Z^+$. Then $\frac{\alpha_i}{\beta_i}\in I, \;\forall\; i\in\mathbb Z^+$, and $\lim_{i\rightarrow \infty} \frac{\alpha_i}{\beta_i}=\frac{c}{d}$, so $\frac{c}{d}\in J$, making $J$ pure.\\
If $I\subset [1,\infty)$, then $J\subset [1,\infty)$ as well. Given any $c<d\in J$, we can choose sequences of elements, $\{\alpha_i\}_{i=1}^{\infty}\subset I$ converging to $c$ and $\{\beta_i\}_{i=1}^{\infty}\subset I$ converging to $d$, such that $\alpha_i<\beta_i, \;\forall\; i\in \mathbb Z^+$. Then $\frac{\beta_i}{\alpha_i}\in I, \;\forall\; i\in\mathbb Z^+$, and $\lim_{i\rightarrow \infty} \frac{\beta_i}{\alpha_i}=\frac{d}{c}$, so $\frac{d}{c}\in J$, making $J$ pure.\\
If $I$ intersects both $[0,1)$ and $[1,\infty)$, then $I$ closed under nonzero division, so $J$ is as well, making $J$ a pure set as well.\\
Now, using Proposition \ref{prop: x-function is cont} (in particular, the fact that $T^x$ is uniformly continuous on $[a,b]$ for $a>0$), for any $\alpha\in J$, define a function $T_{\alpha}\in X^X$ such that $T_{\alpha}(x)=\lim_{\substack{\beta\rightarrow \alpha,\\ \beta\in I}} T^x(\beta)$. The extension, $T^x:J\rightarrow X$, defined as $T^x(\alpha)=T_{\alpha}(x)$ is continuous.\\
Now, for any $\alpha\in J$, we have
$$T_{\alpha}(x_0)=\lim_{\substack{\beta\rightarrow \alpha,\\ \beta\in I}} T^x(\beta)$$
$$=\lim_{\substack{\beta\rightarrow \alpha,\\ \beta\in I}} T_{\beta}(x_0)=\lim_{\substack{\beta\rightarrow \alpha,\\ \beta\in I}} x_0=x_0$$
Also $T_1(x_0)=x_0$.\\
Next, for any $\alpha\in \overline{I}$ and any $x,y\in X$, we have
$$d(T_{\alpha}(x),T_{\alpha}(y))=d(\lim_{\substack{\beta\rightarrow \alpha,\\ \beta\in I}} T^x(\beta),\lim_{\substack{\beta\rightarrow \alpha,\\ \beta\in I}} T^y(\beta))$$
$$=\lim_{\substack{\beta\rightarrow \alpha,\\ \beta\in I}} d( T_{\beta}(x),T_{\beta}(y))$$
$$=\lim_{\substack{\beta\rightarrow \alpha,\\ \beta\in I}} \beta d(x,y)$$
$$=\alpha d(x,y)$$
so $T_{\alpha}$ is a dilation of scale $\alpha, \;\forall\; \alpha\in J$.\\
Given any $\alpha,\beta\in \overline{I}$ and any $x\in X$ (appealing to the continuity of $T_{\alpha}$), we have:
$$(T_{\alpha}\circ T_{\beta})(x)=T_{\alpha}(T_{\beta}(x))=T_{\alpha}(\lim_{\substack{\gamma\rightarrow \beta,\\ \gamma\in I}} T^x(\gamma))$$
$$=\lim_{\substack{\gamma\rightarrow \beta,\\ \gamma\in I}} T_{\alpha}(T^x(\gamma))=\lim_{\substack{\gamma\rightarrow \beta,\\ \gamma\in I}} T_{\alpha}(T_{\gamma}(x))$$
$$=\lim_{\substack{\gamma\rightarrow \beta,\\ \gamma\in I}} \lim_{\substack{\delta\rightarrow \alpha,\\ \delta\in I}} T^{T_{\gamma}(x)}(\delta)=\lim_{\substack{\gamma\rightarrow \beta,\\ \gamma\in I}} \lim_{\substack{\delta\rightarrow \alpha,\\ \delta\in I}} T_{\delta}(T_{\gamma}(x))$$
$$=\lim_{\substack{\gamma\rightarrow \beta,\\ \gamma\in I}} \lim_{\substack{\delta\rightarrow \alpha,\\ \delta\in I}} T_{\delta\gamma}(x)=\lim_{\substack{\delta\gamma\rightarrow \alpha\beta,\\ \delta,\gamma\in I}} T^x(\delta\gamma)$$
$$=T_{\alpha\beta}(x)$$
where the last two equalities hold because $T^x: J \rightarrow X$ is continuous.\\
As $x$ was an arbitrary point of $X$, this means $T_{\alpha}\circ T_{\beta}=T_{\alpha\beta}, \;\forall\; \alpha,\beta\in J$.\\
Lastly, as $T^x:J\rightarrow X$ is continuous $\;\forall\; x\in X$, we have for any $x\in X$ that $\lim_{\alpha\rightarrow 1} T_{\alpha}(x)$ exists. So $\{T_{\alpha}\}_{\alpha\in J}$ is a dilation family about $x_0$ extending $\{T_{\alpha}\}_{\alpha\in I}$, as desired.\\ \end{proof}
\betagin{prop}
\lambdabel{prop: adding 0} Let $(X,d)$ be a metric space. Let $\{T_{\alpha}\}_{\alpha\in I}$ be a dilation family on $(X,d,x_0)$. Suppose $I\cap [0,1)\neq \emptyset$. Then $J=I\cup \{0\}$ is pure and, setting $T_0=O_{x_0}$, the collection of functions $\{T_{\alpha}\}_{\alpha\in J}$ is also a dilation family on $(X,d,x_0)$.\end{prop}
\betagin{proof} Supposing $I\cap [0,1)\neq \emptyset$, let $J=I\cup\{0\}$.\\
If $I\subset [0,1]$, then $J\subset [0,1]$ as well. Given any $\alpha<\beta\in K$, either $\alpha,\beta\in I$, in which case (as $I$ as pure), $\frac{\alpha}{\beta}\in J$. If $0<\alpha$, then $\frac{0}{\alpha}=0\in K$. So $J$ is pure.\\
If $I\not\subset [0,1]$, then $I$ is closed under nonzero division. $J$ will then also be closed under nonzero division. And given any $\alpha\neq 0\in K$, we have $\frac{0}{\alpha}=0\in J$, so it follows that $J$ is closed under nonzero division as well, i.e. $J$ is pure too.\\
Finally, we set $T_0$ equal the constant map $O_{x_0}$. By Proposition \ref{prop: if 1 or 0 is in I}, this definition produces no inconsistency. Given any $\alpha\in K$, we have $T_{\alpha}\circ T_0=T_{\alpha}\circ O_{x_0}=O_{x_0}=O_{x_0}\circ T_{\alpha}=T_0\circ T_{\alpha}$. So $\{T_{\alpha}\}_{\alpha\in K}$ is also a dilation family on $(X,d,x_0)$, completing the proof.\\ \end{proof}
We now establish one of the central claims made earlier about dilation families - that they are a 'a strengthening of classical contractibility conditions' (specifically, dilation families with index sets that cover at least $[0,1]$):
\betagin{prop}
\lambdabel{prop: continuous action} Let $(X,d)$ be a metric space and $x_0$ be a point in $X$. If $\{T_{\alpha}\}_{\alpha\in I}$ is a dilation family on $(X,d,x_0)$, then the function $F:I\times X\rightarrow X$ defined as $F(\alpha,x)=T_{\alpha}(x), \;\forall\; (\alpha,x)\in I\times X$ is continuous.\end{prop}
\betagin{proof} Let $(\alpha,x)$ be any point of $I\times X$ and let $\epsilon$ be any positive real number. By Proposition \ref{prop: x-function is cont}, we can choose some $\delta\in (0,\alpha)$ such that whenever $\beta\in B(\alpha;\delta)$, $d(T^x(\alpha),T^x(\beta))<\frac{\epsilon}{2}$. Now, if $\alpha\geq 1$, for any $(\beta,y)\in B(\alpha;\delta)\times B_d(x;\frac{\epsilon}{4\alpha})$, we have:
$$d(F(\alpha,x),F(\beta,y))=d(T_{\alpha}(x),T_{\beta}(y))$$
$$=d(T_{\alpha}(x),T_{\beta}(x))+d(T_{\beta}(x),T_{\beta}(y))$$
$$=d(T^x(\alpha),T^x(\beta))+\beta d(x,y)$$
$$<\frac{\epsilon}{2}+\beta\cdot \frac{\epsilon}{4\alpha}=\frac{\epsilon}{2}+\frac{\epsilon}{2}=\epsilon$$
where the second-last equality follows from the fact that (as $\beta\in B(\alpha;\delta)$ and $0<\delta<\alpha$) $\beta<2\alpha$.\\
If $\alpha<1$, then for any $(\beta,y)\in B(\alpha;\delta)\times B_d(x;\frac{\epsilon}{4})$, we have:
$$d(F(\alpha,x),F(\beta,y))=d(T_{\alpha}(x),T_{\beta}(y))$$
$$=d(T_{\alpha}(x),T_{\beta}(x))+d(T_{\beta}(x),T_{\beta}(y))$$
$$=d(T^x(\alpha),T^x(\beta))+\beta d(x,y)$$
$$<\frac{\epsilon}{2}+\beta\cdot \frac{\epsilon}{4}<\frac{\epsilon}{2}+2\alpha\cdot \frac{\epsilon}{4}$$
$$<\frac{\epsilon}{2}+\frac{\epsilon}{2}=\epsilon$$
As $(\alpha,x)$ was an arbitrary point of $I\times X$ and $\epsilon$ was an arbitrary positive real, $F$ is continuous everywhere, as desired.\\ \end{proof}
\betagin{co}
\lambdabel{co: continuous map into function space} Let $(X,d)$ be a metric space and $x_0$ be a point in $X$. If $\{T_{\alpha}\}_{\alpha\in I}$ is a dilation family on $(X,d,x_0)$, then the function $F:I\rightarrow C(X)$ defined as $F(\alpha)(x)=T_{\alpha}(x), \;\forall\; (\alpha,x)\in I\times X$ is continuous ($C(X)$ has the compact-open topology). \end{co}
\betagin{proof} This follows directly from Proposition \ref{prop: continuous action} and the fact that the induced function of a continuous function produces a continuous function into the function space (provided the function space is equipped with the compact-open topology).\\ \end{proof}
\betagin{defi} Let $X$ be a topological space. The symbol $C_{x_0}(X)$ denotes the set $\{f\in C(X) | f(x_0)=x_0\}$. \\ \end{defi}
We mostly work with the compact-open topology (on $C_{x_0}(X)$, we usually apply the subspace topology induced by the compact-open topology).\\
The rest of this paper will be devoted to finding sufficient conditions for the existence of dilation families. The way we will do this is by first demonstrating that all metrizable cone spaces can be given a metric under which they possess a very special kind of dilation family and then describing conditions under which topological or metric spaces may be converted into such metrizable cone spaces. These 'special kind' of dilation families are the following:
\betagin{defi} Let $(X,d)$ be a metric space and $x_0$ be a point in $X$. If $\{T_{\alpha}\}_{\alpha\in I}$ is a dilation family on $(X,d,x_0)$, and we have for any $\alpha\leq \beta\leq \gamma\in I$, $d(T_{\alpha}(x),T_{\gamma}(x))=d(T_{\alpha}(x),T_{\beta}(x))+d(T_{\beta}(x),T_{\gamma}(x)), \;\forall\; x\in X$, then we say that $\{T_{\alpha}\}_{\alpha\in I}$ is a linear dilation family on $(X,d,x_0)$.\\ \end{defi}
\betagin{defi} Let $C$ be a topological space, $I$ be a subset of $\mathbb R$ containing $0$, and $p$ be any point. $\mathcal{C}_p(I,C)$ denotes the cone formed from $I\times C$ by identifying $\{0\}\times C$ with $p$.\\ \end{defi}
The following result provides us with a general method of producing metrics for metrizable cones under which they have linear dilation families:\\
\betagin{prop}
\lambdabel{prop: metrizable cones} Let $(C,d)$ be a metric space with diameter at most $2$. Letting $X$ denote $\mathcal{C}_{x_0}([0,1],C)$, the function $D:X\times X\rightarrow \mathbb R_{\geq 0}$ defined as $D(x_0,x_0)=0, D((a,c_1),x_0)=D(x_0,(a,c_1))=a$ and $D((a,c_1),(b,c_2))=|a-b|+\min(a,b)d(c_1,c_2), \;\forall\; a,b\in (0,1],\;\forall\; c_1,c_2\in C$ is a metric on $X$ preserving the topology of $X$. Also, the family of functions $\{F(\alpha)\}_{\alpha\in [0,1]}$ defined as $F(\alpha)(x_0)=x_0, F(\alpha)(a,x)=(a\alpha,x), \;\forall\; \alpha\in (0,1]$, and $F(0)=O_{x_0}$ is a linear dilation family on $(X,D,x_0)$. Furthermore, $\textrm{Bd}(B_D(x_0;1))=\{1\}\times C$.\end{prop}
\betagin{proof}
\textbf{\underline{Part 1:}}\\
Clearly $D$ is a well-defined nonnegative symmetric function.\\
We have $D(x_0,x_0)=0$ and for any $(a,c)\in X-\{x_0\}$, $D((a,c),(a,c))=|a-a|+\min(a,a)d(c,c)=0+0=0$.\\
Conversely, if $D(z_1,z_2)=0$, there are two cases, the first of which is where $z_1,z_2\neq x_0$ in which case $z_1=(a,c_1), z_2=(b,c_2)$ for some $a,b\in (0,1]$, and some $c_1,c_2\in C$. This means $|a-b|+\min(a,b)d(c_1,c_2)=0\Rightarrow |a-b|=0, \min(a,b)d(c_1,c_2)=0$. As $a,b>0$, $\min(a,b)>0$, so we get $d(c_1,c_2)=0$ as well, i.e. $a=b$ and $c_1=c_2$, giving us that $z_1=(a,c_1)=(b,c_2)=z_2$. The other case (WLOG) is where $z_1=x_0$. If $z_2\neq x_0$, then $z_2=(a,c)$ for some $a\in (0,1]$ and some $c\in C$, so $0=D(z_1,z_2)=D(x_0,(a,c))=a$, a contradiction, so we must have $z_2=x_0=z_1$. By the symmetry of $D$, the same occurs if $z_2=x_0$. So $D$ satisfies the identity of indiscernibles.\\
Now we must show that $D$ satisfies the triangle inequality, i.e. $D(x,z)\leq D(x,y)+D(y,z), \;\forall\; x,y,z\in X$. The first case is where $x,y,$ and $z$ are all not equal to $x_0$. We can write $x=(a,c_1), y=(b,c_2),$ and $z=(c,c_3)$ for some $a,b,c\in (0,1]$ and some $c_1,c_2,c_3\in C$. We assume WLOG that $a\geq c$. There are two subcases - the first is where $b\geq c$ and the second is where $b\leq c$. If $b\geq c$, then:
$$D(x,z)=|a-c|+\min(a,c)d(c_1,c_3)$$
$$=|a-c|+c d(c_1,c_3)$$
$$\leq |a-b|+|b-c|+c d(c_1,c_3)$$
$$\leq |a-b|+|b-c|+c d(c_1,c_2)+c d(c_2,c_3)$$
$$\leq |a-b|+|b-c|+\min(a,b) d(c_1,c_2)+\min(b,c) d(c_2,c_3)$$
$$=D(x,y)+D(y,z)$$
If $b\leq c$, then
$$D(x,z)=|a-c|+\min(a,c)d(c_1,c_3)$$
$$=a-c+c d(c_1,c_3)$$
Now
$$[a+c-2b+b d(c_1,c_3)]-[a-c+c d(c_1,c_3)]$$
$$=2c-2b+(b-c)d(c_1,c_3)$$
$$=(c-b)(2-d(c_1,c_3))$$
$$\geq 0$$
where the final inequality follows as $d$ is bounded above by $2$. So:
$$D(x,z)=a-c+c d(c_1,c_3)$$
$$\leq a+c-2b+b d(c_1,c_3)$$
$$=(a-b)+(c-b)+b d(c_1,c_3)$$
$$=|a-b|+|b-c|+b d(c_1,c_3)$$
$$\leq |a-b|+|b-c|+b d(c_1,c_2)+b d(c_2,c_3)$$
$$= |a-b|+|b-c|+\min(a,b) d(c_1,c_2)+\min(b,c) d(c_2,c_3)$$
$$= D(x,y)+D(y,z)$$
The next case is where $z=x_0$, and $x,y\neq x_0$. We have $x=(a,c_1), y=(b,c_2)$ for some $a,b\in (0,1]$, and some $c_1,c_2\in C$. There are again two subcases here - where $a\geq b$ and $a\leq b$.\\
If $\alpha\geq \beta$, then:
$$D(x,z)=D(x,x_0)=a$$
$$=b+(a-b)$$
$$\leq b+|a-b|+\min(a,b)d(c_1,c_2)$$
$$=D(x,y)+D(y,x_0)=D(x,y)+D(y,z)$$
If $a\leq b$, then:
$$D(x,z)=D(x,x_0)=a\leq b$$
$$\leq b +|a-b|+\min(a,b)d(c_1,c_2)$$
$$=D(x,y)+D(y,x_0)=D(x,y)+D(y,z)$$
And the next case (and the last nontrivial case) is where $y=x_0$ but $x,z\neq x_0$. We can write $x=(a,c_1), z=(c,c_3)$ for some $a,c\in (0,1]$, and some $c_1,c_3\in C$. Here (again, assuming WLOG that $a\geq c$), we have:
$$D(x,z)=|a-c|+\min(a,c) d(c_1,c_3)$$
$$=a-c + c d(c_1,c_3)$$
$$\leq a-c+2c=a+c$$
$$=D(x,x_0)+D(x_0,z)=D(x,y)+D(y,z)$$
These cover all nontrivial cases (by appropriately also applying the symmetry property of $D$). The remaining cases are just those where two or more of $x,y,$ and $z$ equal $x_0$, but these are all trivial.\\
If $y=x_0=z$, then $D(y,z)=0$, so $D(x,z)\leq D(x,z)+0=D(x,z)+D(y,z)=D(x,y)+D(y,z)$. Similarly, if $x=z_0=z$, then $D(x,z)=0$ which, by nonnegativity of $D$, is automatically $\leq D(x,y)+D(y,z)$. So $D$ satisfies the triangle inequality, making it a metric extending $d$.\\
\textbf{\underline{Part 2:}}\\
Next, we show that the topology induced by $D$ is that of $\mathcal{C}_{x_0}([0,1],C)$.\\
First, let $O$ be any open set of $\mathcal{C}_{x_0}([0,1],C)$ and $x$ be any member of $O$. If $x$ equals some $(a,c)\in (0,1]\times C$, then we may choose some $\epsilon>0$ such that $x=(a,c_1)\in B(a;\epsilon)\times B_d(c_1;\epsilon)\subset O$ (where $B(a;\epsilon)$ denotes the open ball of radius $\epsilon$ about $a$ in $(0,1]$). We may assume WLOG that $\epsilon<\frac{a}{2}$. Now let $y$ be any member of $B_D\left(x;\frac{a\epsilon}{2}\rightarrowght)$. If $y=x_0$, then $a=D(x_0,(a,c_1))=D(y,x)<\frac{a\epsilon}{2}<\frac{a^2}{2}<a$, a contradiction, so $y$ equals some $(b,c_2)\in (0,1]\times C$. Since $y\in B_D\left(x;\frac{a\epsilon}{2}\rightarrowght)$, we have:
$$|a-b|\leq |a-b|+\min(a,b)d(c_1,c_2)$$
$$=D(x,y)<\frac{a\epsilon}{2}<\epsilon$$
so $b\in B(a;\epsilon)$. Also, as $\epsilon<\frac{a}{2}$, $b\in B\left(a;\frac{a}{2}\rightarrowght)$, which means $b>\frac{a}{2}$. This means:
$$\frac{a}{2}d(c_1,c_2)<\min(a,b)d(c_1,c_2)$$
$$\leq |a-b|+\min(a,b)d(c_1,c_2)$$
$$=D(x,y)<\frac{a\epsilon}{2}$$
Cancelling $\frac{a}{2}$ from the first and last expressions, we get that $d(c_1,c_2)<\epsilon$, i.e. $c_2\in B_d(c_1;\epsilon)$. So $y=(b,c_2)\in B(a;\epsilon)\times B_d(c_1;\epsilon)\subset O$. As $y$ was an arbitrary point of $B_D\left(x;\frac{a\epsilon}{2}\rightarrowght)$, we have $x\in B_D\left(x;\frac{a\epsilon}{2}\rightarrowght)\subset O$.\\
If instead we have $x=x_0$, then for some $a\in (0,1]$, we have that $\{x_0\}\cup (0,a)\times C\subset O$. But this former set is easily seen to just be $B_D(x_0;a)$, so $x_0\in B_D(x_0;a)\subset O$. It follows that the topology induced by $D$ is at least as fine as that of $\mathcal{C}_{x_0}([0,1],C)$.\\
Conversely, let $O$ be any open set of $X$ under $D$ and $x$ be any member of $O$. Then we can choose some $\epsilon\in \mathbb R^+$ such that $B_D(x;\epsilon)\subset O$. If $x$ equals some $(a,c_1)\in (0,1]\times C$, then let $y=(b,c_2)$ be any element of $B\left(a;\frac{\epsilon}{2}\rightarrowght)\times B_d\left(c_1;\frac{\epsilon}{2a}\rightarrowght)$ (where the first open ball is taken in $(0,1]$).\\
$$D(x,y)=|a-b|+\min(a,b)d(c_1,c_2)$$
$$<\frac{\epsilon}{2}+\min(a,b)\frac{\epsilon}{2a}$$
$$\leq \frac{\epsilon}{2}+a\times \frac{\epsilon}{2a}=\epsilon$$
so $y\in B_D(x;\epsilon)$, i.e. $x\in B\left(a;\frac{\epsilon}{2}\rightarrowght)\times B_d\left(c_1;\frac{\epsilon}{2a}\rightarrowght)\subset B_D(x;\epsilon)$.\\
If instead $x=x_0$, then $B_D(x;\epsilon)=\{x_0\}\cup (0,\epsilon)\times C$. It follows that the topology of $\mathcal{C}_{x_0}([0,1],C)$ is at least as fine as that induced by $D$ as well, so that the two topologies are the same.\\
\textbf{\underline{Part 3:}}\\
Now, for any $\alpha\in (0,1]$ and any $x,y\in X-\{x_0\}$ we can write $x=(a,c_1), y=(b,c_2)$ for some $a,b\in (0,1]$ and some $c_1,c_2\in C$ we have:
$$D(F(\alpha)(x),F(\alpha)(y))=D(F(\alpha)(a,c_1),F(\alpha)(b,c_2))$$
$$=D((\alpha a,c_1),(\alpha b,c_2))$$
$$=|\alpha a-\alpha b|+\min(\alpha a,\alpha b)d(c_1,c_2)$$
$$=\alpha(|a-b|+\min(a,b)d(c_1,c_2))$$
$$=\alpha D((a,c_1),(b,c_2))=\alpha D(x,y)$$
and
$$=D(F(\alpha)(x),F(\alpha)(x_0))=D(F(\alpha)(a,c_1),x_0)$$
$$=D((\alpha a,c_1),x_0)$$
$$=\alpha a=\alpha D((a,c_1),x_0)$$
$$=\alpha D(x,x_0)$$
so $F(\alpha)$ is a dilation of scale $\alpha$, $\;\forall\; \alpha\in (0,1]$. Since $F(0)$ is a constant map, it follows that $F(\alpha)$ is dilation of scale $\alpha$, $\;\forall\; \alpha\in [0,1]$.\\
Also, given any $\alpha,\beta\in (0,1]$ and any $(a,c)\in (0,1]\times C$ we have:
$$(F(\alpha)\circ F(\beta))(a,c)=F(\alpha)(\beta a,c)$$
$$=((\alpha\beta)a,c)$$
$$=F(\alpha\beta)(a,c)$$
and $(F(\alpha)\circ F(\beta))(x_0)=F(\alpha)(x_0)=x_0=F(\alpha\beta)(x_0)$, so $F(\alpha)\circ F(\beta)=F(\alpha\beta), \;\forall\; \alpha,\beta\in (0,1]$. Also, $F(\alpha)$ fixes $x_0, \;\forall\; \alpha\in [0,1]$, so:
$$F(\alpha)\circ F(0)=F(\alpha)\circ O_{x_0}$$
$$=O_{x_0}=F(\alpha\times 0)=F(0\times\alpha)$$
$$=O_{x_0}=O_{x_0}\circ F(\alpha)$$
$$=F(0)\circ F(\alpha)$$
so $F(\alpha)\circ F(\beta)=F(\alpha\beta), \;\forall\; \alpha,\beta \in [0,1]$, meaning $\{F(\alpha)\}_{\alpha\in [0,1]}$ is a dilation family on $(X,D,x_0)$.\\
Next, let any $\alpha\leq \beta\leq \gamma\in [0,1]$ be given. For any $(a,c)\in (0,1]\times C$, if $\alpha>0$, we have:
$$D(F(\alpha)(a,c),F(\gamma)(a,c))=D((\alpha a,c),(\gamma a, c))$$
$$=|\alpha a-\gamma a|+d(c,c)$$
$$=(\gamma-\alpha)a=(\gamma-\beta+\beta-\alpha)a$$
$$=|\alpha-\beta|a+|\beta-\gamma|a$$
$$=|\alpha a-\beta a|+|\beta a-\gamma a|$$
$$=(|\alpha a-\beta a|+d(c,c))+(|\beta a-\gamma a|+d(c,c))$$
$$=D((\alpha a,c),(\beta a,c))+D((\beta a,c),(\gamma a,c))$$
$$=D(F(\alpha)(a,c),F(\beta)(a,c))+D(F(\beta)(a,c),F(\gamma)(a,c))$$
and if $\alpha=0$:
$$D(F(\alpha)(a,c),F(\gamma)(a,c))=D(x_0,(\gamma a, c))$$
$$=\gamma a=\beta a+|\beta a-\gamma a|$$
$$=D(x_0,(\beta a,c))+(|\beta a-\gamma a|+d(c,c))$$
$$=D(F(\alpha)(a,c),F(\beta)(a,c))+D((\beta a,c),(\gamma a,c))$$
$$=D(F(\alpha)(a,c),F(\beta)(a,c))+D(F(\beta)(a,c),F(\gamma)(a,c))$$
(the cases where $\beta$ or $\gamma$ are $0$ are trivial). Also, for any $\alpha\leq \beta\leq \gamma\in [0,1]$, we have:
$$D(F(\alpha)(x_0),F(\gamma)(x_0))=D(x_0,x_0)$$
$$=0=0+0=D(x_0,x_0)+D(x_0,x_0)$$
$$=D(F(\alpha)(x_0),F(\beta)(x_0))+D(F(\beta)(x_0),F(\gamma)(x_0))$$
So $\{F(\alpha)\}_{\alpha\in [0,1]}$ is in fact a linear dilation family on on $(X,D,x_0)$.\\
Lastly, for any $x\in \mathcal{C}_{x_0}([0,1],C)$, we have:
$$x\in \textrm{Bd}(B_D(x_0;1))\iff D(x,x_0)=1$$
$$\iff x=(1,c), c\in C\iff x\in \{1\}\times C$$
so $\textrm{Bd}(B_D(x_0;1))=\{1\}\times C$, completing the proof.\\ \end{proof}
Now, we demonstrate a condition for compact spaces that resembles that of the existence of dilation families that 'contract' and is strong enough to ensure they are metrizable cones (note that dilation families that 'expand' are not possible over compact spaces).\\
\betagin{prop}
\lambdabel{prop: compact space cone condition} Let $(X,\tau)$ be a compact Hausdorff topological space. Suppose for some $x_0\in X$, there exists a continuous monoidal monomomorphism $F: ([0,1],\times) \rightarrow (C_{x_0}(X),\circ)$ such that: $1) F(0)=O_{x_0}$, $2) F(1)=\textrm{Id}_X$, and $3) F(\alpha)$ is injective $\;\forall\; \alpha\in (0,1]$. Then, with $C=X-F[0,1)(X)$, $\{F(\alpha)(C)\}_{\alpha\in [0,1]}$ is a partition of $X$. If $F[0,1)(X)$ is an open subset of $X$, then the function $f':\mathcal{C}_{x_0}([0,1],C)\rightarrow X$ defined so that $x_0\mathbbpsto x_0$ and $(\alpha,c)\mathbbpsto F(\alpha)(c)$ is a homeomorphism.\\
\end{prop}
\betagin{proof}
As $F$ is a monomorphism, $F(0)\neq F(1)$, i.e. $O_{x_0}\neq \textrm{Id}_X$. This means $X-\{x_0\}$ is non-empty.\\
As $X$ is compact and $F: ([0,1],\times)\rightarrow (C_{x_0}(X),\circ)$ is continuous, the function $f: [0,1]\times X\rightarrow X$ defined as $f(\alpha,x)=F(\alpha)(x), \;\forall\; (\alpha,x)\in [0,1]\times X$ is continuous as well. For any $x\in X$ then (as $X$ is Hausdorff), $f^{-1}(\{x\})$ is a closed subset of $[0,1]\times X$. Letting $\pi_1:[0,1]\times X\rightarrow [0,1]$ denote the projection map onto the first component, as $X$ is compact, $\pi_1$ is a closed map, so $\pi_1(f^{-1}(\{x\}))$ is a closed subset of $[0,1], \;\forall\; x\in X$. As $[0,1]$ is compact, this means that $\pi_1(f^{-1}(\{x\}))$ always has a least element - let $\Gamma: X\rightarrow [0,1]$ be the function that gives this least element for each $x$. As $F(0)=O_{x_0}$, $\Gamma(x_0)=0$ and $\Gamma(x)>0, \;\forall\; x\in X-\{x_0\}$.\\
Now, by the definition of $\Gamma$, for any $x\in X-\{x_0\}$, we have:
$$x\in F(\Gamma(x))(X)-\cup_{\alpha\in [0,\Gamma(x))} F(\alpha)(X)$$
$$=F(\Gamma(x))(X)-\cup_{\beta\in [0,1)} F(\Gamma(x)\beta)(X)$$
$$=F(\Gamma(x))(X)-\cup_{\beta\in [0,1)} (F(\Gamma(x))\circ F(\beta))(X)$$
$$=F(\Gamma(x))(X-\cup_{\beta\in [0,1)} F(\beta)(X))$$
$$=F(\Gamma(x))(X-F[0,1)(X))=F(\Gamma(x))(C)$$
where the third equality holds as $\Gamma(x)>0$ and $F(\alpha)$ is injective $\;\forall\; \alpha\in (0,1]$.\\
As $x$ was an arbitrary point of $X-\{x_0\}$, the collection $\{F(\alpha)(C)\}_{\alpha\in (0,1]}$ covers $X-\{x_0\}$. If $F(\alpha)(C)$ is empty for any $\alpha\in (0,1]$, then $C$ itself will be empty so that $X-\{x_0\}=\cup_{\alpha\in (0,1]} F(\alpha)(C)=\cup_{\alpha\in (0,1]} F(\alpha)(\emptyset)=\cup_{\alpha\in (0,1]} \emptyset=\emptyset$, a contradiction. So each element of the collection $\{F(\alpha)(C)\}_{\alpha\in (0,1]}$ is nonempty. In particular, $C=F(1)(C)$ itself is nonempty.\\
As $F(0)=O_{x_0}$ and $C$ is nonempty, $F(0)(C)=\{x_0\}$, so $\{F(\alpha)(C)\}_{\alpha\in [0,1]}$ is a covering of $X$ by nonempty sets. Lastly, for any $\alpha>\beta\in [0,1]$, if $x\in F(\alpha)(C)\cap F(\beta)(C)$, then:
$$x\in F(\alpha)(C)=F(\alpha)(X-F[0,1)(X))$$
$$=F(\alpha)(X)-F(\alpha)(F[0,1)(X))$$
$$=F(\alpha)(X)-F[0,\alpha)(X)$$
$$\subset F(\alpha)(X)-F(\beta)(X)$$
$$\subset F(\alpha)(X)-F(\beta)(C)$$
a contradiction. It follows that all sets in $\{F(\alpha)(C)\}_{\alpha\in [0,1]}$ are pairwise disjoint. Hence, $\{F(\alpha)(C)\}_{\alpha\in [0,1]}$ is a partition of $X$.\\
Now, suppose $F[0,1)(X)$ is open in $X$. Then $C$ is closed in $X$, and hence compact, making $\mathcal{C}_{x_0}([0,1],C)$ compact as well. As $f$ is continuous, so is its restriction to $[0,1]\times C$. By what we have already shown, this restriction has image $X$ and has fibres coinciding exactly with the elements (as equivalence classes) of $\mathcal{C}_{x_0}([0,1],C)$. This means $f':\mathcal{C}_{x_0}([0,1],C)\rightarrow X$ is a continuous bijection. As $\mathcal{C}_{x_0}([0,1],C)$ is compact and $X$ is Hausdorff, it follows that $f'$ is a homeomorphism.\\ \end{proof}
\betagin{ex} The condition that $F[0,1)(X)$ be open in $X$ is necessary. Consider the family of functions $\{F_{\alpha}\}_{\alpha\in [0,1]}$ on $[0,1]^{\omega}$ defined so that $F_{\alpha}((x_i)_{i\in\mathbb N})=(\alpha x_i)_{i\in\mathbb N}$ and equip $[0,1]^{\omega}$ with the product topology. $[0,1]^{\omega}$ is compact and the family $\{F_{\alpha}\}_{\alpha\in [0,1]}$ is easily seen to be a dilation family in $[0,1]^{\omega}$ about $(0)_{i\in \mathbb N}$. However, the set $C=[0,1]^{\omega}-\cup_{\alpha\in [0,1)}F_{\alpha}([0,1]^{\omega})$ consists of exactly those elements $(a_i)_{i\in\mathbb N}\in [0,1]^{\omega}$ such that $\sup\{a_i|i\in \mathbb N\}=1$. $C$ in this case, is hence not only not a closed subset of $[0,1]^{\omega}$, it is a proper dense subset of the ambient space, $[0,1]^{\omega}$.\\ \end{ex}
We can apply Proposition \ref{prop: compact space cone condition} to a special case for compact metric spaces:\\
\betagin{co}
\lambdabel{co: compact metrizable space cone condition} Let $(X,d)$ be a compact metric space. Suppose for some $x_0$ in $X$, there exists a continuous monoidal monomomorphism $F: ([0,1],\times) \rightarrow (C_{x_0}(X),\circ)$ such that $F(0)=O_{x_0}$ and $F(1)=\textrm{Id}_X$. If $d(F(\alpha)(x),x_0)< d(x,x_0), \;\forall\; \alpha\in [0,1), x\in X-\{x_0\}$ and $x_0$ is not a limit point of $C=X-F[0,1)(X)$ , then for any sufficiently small $\epsilon\in\mathbb R^+$, the function $f':\mathcal{C}_{x_0}([0,1],\textrm{Bd}(B_D(x_0;\epsilon)))\rightarrow \overline{B_D(x_0;\epsilon)}$ defined so that $x_0\mathbbpsto x_0$ and $(\alpha,c)\mathbbpsto F(\alpha)(c)$ is a homeomorphism. \end{co}
\betagin{proof} As $x_0$ is not a limit point of $C$, we can choose some $\delta>0$ such that the open ball $B_D(x_0;\delta)$ is disjoint from $C$. Because $d(F(\alpha)(x),x_0)<d(x,x_0), \;\forall\; \alpha\in [0,1)$, $F[0,1)(\overline{B_D(x_0;\epsilon)})\subset B_D(x_0;\epsilon)$. This means $\overline{B_D(x_0;\epsilon)}-F[0,1)(\overline{B_D(x_0;\epsilon)})\supset \overline{B_D(x_0;\epsilon)}-B_D(x_0;\epsilon)=\textrm{Bd}(B_D(x_0;\epsilon)))$.\\
By the reasoning explained in the proof of Proposition \ref{prop: compact space cone condition}, for each $x\in B_D(x_0;\epsilon)$, there exists some $c\in C$ and some $\alpha\in [0,1)$ such that $x=F(\alpha)(c)$. Now define $\Gamma:[\alpha,1]\rightarrow \mathbb R$ as $\Gamma(\delta)=d(F(\delta)(c),x_0), \;\forall\; \delta\in [\alpha,1]$. This is a continuous real-valued function on a connected interval. We have $\Gamma(1)=d(F(1)(c),x_0)=d(c,x_0)\geq\epsilon$ (this, by the choice of $\epsilon$) and $\Gamma(\alpha)=d(F(\alpha)(c),x_0)=d(x,x_0)<\epsilon$, so there exists some $\beta\in (\alpha,1]$ such that $\Gamma(\beta)=\epsilon$, i.e. $d(F(\beta)(c),x_0)=\epsilon$. Setting $y=F(\beta)(x)$, we have $y\in \overline{B_D(x_0;\epsilon)}$. As $\beta>\alpha$, we have that
$$x=F(\alpha)(c)=F\left(\frac{\alpha}{\beta}\rightarrowght)(F(\beta)(c))$$
$$=F\left(\frac{\alpha}{\beta}\rightarrowght)(y)$$
$$\in F[0,1)(\overline{B_D(x_0;\epsilon)})$$
This holds $\;\forall\; x\in X$, hence it follows directly that $B_D(x_0;\epsilon)\subset F[0,1)(\overline{B_D(x_0;\epsilon)})\Rightarrow \overline{B_D(x_0;\epsilon)}-F[0,1)(\overline{B_D(x_0;\epsilon)})\subset \textrm{Bd}(B_D(x_0;1)))$, so $\overline{B_D(x_0;\epsilon)}-F[0,1)(\overline{B_D(x_0;\epsilon)})= \textrm{Bd}(B_D(x_0;\epsilon)))$. Since $F(1)=\textrm{Id}_X$, $F[0,1](\overline{B_D(x_0;\epsilon)})\subset \overline{B_D(x_0;\epsilon)}$, so the result now follows directly from Proposition \ref{prop: compact space cone condition}.\\ \end{proof}
Now, we look at a similar condition as in the statement of Proposition \ref{prop: compact space cone condition} but modified for locally compact spaces with dilation families that both 'contract' and 'expand':\\
\betagin{prop}
\lambdabel{prop: dilation family existence - locally compact case} Let $(X,\tau)$ be a locally compact Hausdorff space. Suppose for some $x_0$ in $X$ there exists a continuous monoidal monomorphism, $F: ([0,\infty),\times) \rightarrow (C_{x_0}(X),\circ)$, such that $F(0)=O_{x_0}$ and $F(1)=\textrm{Id}_X$. If there is a compact set, $D$, such that $x_0\in \textrm{Int}(D)$ and $F[0,1)(D)\subset \textrm{Int}(D)$, then $F[0,1)(D)=\textrm{Int}(D)$ and the function $f':\mathcal{C}_{x_0}([0,\infty),\textrm{Bd}(D))\rightarrow X$ defined so that $x_0\mathbbpsto x_0$ and $(\alpha,c)\mathbbpsto F(\alpha)(c)$ is a homeomorphism.\end{prop}
\betagin{proof} First, we show that $X$ cannot be compact. Suppose by way of contradiction that $X$ is compact. As $F$ is a monomorphism and $F(0)=O_{x_0}, F(1)=\textrm{Id}_X$, we must have that $X\neq\{x_0\}$, i.e. we can choose some point $y\in X$ so that $X-\{y\}$ is an open neighborhood of $x_0$. We also have $F(0)(X)=O_{x_0}(X)=\{x_0\}$, so $F(0)$ belongs to the subbasis element $S(X,X-\{y\})$. As $F$ is continuous, this means we can choose some $\alpha>0$, such that $F(\beta)\in S(X,X-\{y\}), \;\forall\; \beta\in [0,\alpha)$. In particular:
$$F\left(\frac{\alpha}{2}\rightarrowght)\in S(X,X-\{y\})$$
$$\Rightarrow F\left(\frac{\alpha}{2}\rightarrowght)(X)\subset X-\{y\}\subsetneq X$$
However, we also have that $F\left(\frac{\alpha}{2}\rightarrowght)\circ F\left(\frac{2}{\alpha}\rightarrowght)=F(1)=\textrm{Id}_X$, so we must have $F\left(\frac{\alpha}{2}\rightarrowght)(X)=X$ as well, a contradiction. So $X$ cannot be compact.\\
Now, let any point of $X-\{x_0\}$, $x$, be given. I claim that $F[0,\infty)(x)\not\subset D$. Suppose by way of contradiction that $F[0,\infty)(x)\subset D$. This means that $\overline{F[0,\infty)(x)}$ is compact. Also, given any $\alpha\in [0,\infty)$, we have (by continuity) that
$$F(\alpha)(\overline{F[0,\infty)(x)})\subset \overline{F(\alpha)(F[0,\infty)(x))}$$
$$=\overline{(F(\alpha)\circ F[0,\infty))(x)}\subset \overline{F[0,\infty)(x)}$$
so each $F(\alpha)$ maps $\overline{F[0,\infty)(x)}$ into itself. Lastly, if $F(\alpha)(x)=F(\beta)(x)$ for any $\alpha\neq \beta\in [0,\infty)$, then (supposing WLOG that $\alpha<\beta$) $F\left(\frac{\alpha}{\beta}\rightarrowght)(x)=x$, so $F\left(\left(\frac{\alpha}{\beta}\rightarrowght)^n\rightarrowght)(x)=x, \;\forall\; n\in \mathbb Z^+$, but, as $x\neq x_0$, this contradicts the fact that $\lim_{\gamma\rightarrow 0} F(\gamma)(x)=x_0$, so $F(\alpha)(x)\neq F(\beta)(x), \;\forall\; \alpha\neq \beta\in [0,\infty)$. This means the restricted map, $F': ([0,\infty),\times) \rightarrow (C_{x_0}(\overline{F[0,\infty)(x)}),\circ)$ is also a continuous monoidal monomorphism. However, this contradicts the result proven in the first paragraph. So we must have $F[0,\infty)(x)\not\subset D$.\\
Next, I claim that $F[0,\infty)(x)\cap \textrm{Bd}(D)\neq \emptyset$. For otherwise, $X-D$ and $\textrm{Int}(D)$ are a pair of disjoint open sets both of which intersect $F[0,\infty)(x)$ (by the last paragraph), contradicting the connectedness of $F[0,\infty)(x)$. So $F(\alpha)(x)\in \textrm{Bd}(D)$ for some $\alpha\in (0,\infty)$, i.e. $F(\beta)(c)=x$ for some $c\in \textrm{Bd}(D)$ with $\beta=\frac{1}{\alpha}$. I claim that $F(\alpha)(c_1)=F(\beta)(c_2)$ for any $c_1,c_2\in \textrm{Bd}(D)$ only if $\alpha=\beta$ and $c_1=c_2$ or simply just $\alpha=\beta=0$. As if one of $\alpha$ and $\beta$ are $0$, then so must the other be (as neither of $c_1$ and $c_2$ can equal $x_0$). And if both $\alpha$ and $\beta$ are positive, then (assuming WLOG that $\alpha<\beta$) we have:
$$c_2=F(1)(c_2)=F\left(\frac{1}{\beta}\rightarrowght)(F(\beta)(c_2))$$
$$=F\left(\frac{1}{\beta}\rightarrowght)(F(\alpha)(c_1))=F\left(\frac{\alpha}{\beta}\rightarrowght)(c_1)$$
but as $c_1\in \textrm{Bd}(D)\subset D$ and $\frac{\alpha}{\beta}\in (0,1)$, this means $c_2\in F[0,1)(D)\subset \textrm{Int}(D)$, contradicting the assumption that $c_2\in \textrm{Bd}(D)$.\\
I also claim that if $x\in \textrm{Int}(D)-\{x_0\}$, then $\beta$ must lie in $(0,1)$ for otherwise, $\frac{1}{\beta}\in (0,1)$ (if $\beta=1$, then $x=F(1)(c)=c$) and
$$c=F(1)(c)=F\left(\frac{1}{\beta}\rightarrowght)(F(\beta)(c))=F\left(\frac{1}{\beta}\rightarrowght)(x)$$
$$\in F\left(\frac{1}{\beta}\rightarrowght)(\textrm{Int}(D))\subset F\left(\frac{1}{\beta}\rightarrowght)(D) \subset \textrm{Int}(D)$$
a contradiction. It also follows from this that $\textrm{Int}(D)\subset F[0,1)(\textrm{Bd}(D))\subset F[0,1)(D)$, meaning $\textrm{Int}(D)=F[0,1)(D)$.\\
This holds $\;\forall\; x\in F[0,1)(D)-\{x_0\}$ and $x_0=F(0)(c)$ for any $c\in \textrm{Bd}(D)$, hence it follows that $F[0,1)(D)\subset F[0,1)(\textrm{Bd}(D))\Rightarrow F[0,1)(D)=F[0,1)(\textrm{Bd}(D))$. Since $F(1)=\textrm{Id}_X$, $F[0,1](D)\subset D$. Additionally, as $\textrm{Bd}(D)\neq \emptyset$ and $d(F(\alpha)(x),x_0)<d(x,x_0), \;\forall\; \alpha\in [0,1), \;\forall\; x\in U-\{x_0\}$, the restricted map, $F': ([0,\infty),\times) \rightarrow (C_{x_0}(D),\circ)$ is a continuous monoidal monomorphism as well. So by Proposition \ref{prop: compact space cone condition}, the restricted map, $f'_{[0,1]}:\mathcal{C}_{x_0}([0,1],\textrm{Bd}(D))\rightarrow D$, is a homeomorphism.\\
We now need to show that the map $f':\mathcal{C}_{x_0}([0,\infty),\textrm{Bd}(D))\rightarrow X$ is a homeomorphism. As $X$ is locally compact and Hausdorff, the restricted evaluation map, $f: [0,\infty)\times \textrm{Bd}(D)\rightarrow X$ is continuous. By what we have already shown, it is also surjective, and its fibres coincide with the members of $\mathcal{C}_{x_0}([0,\infty),\textrm{Bd}(D))$, so $f'$ is a continuous bijection. Let $O$ be any open basis element in $\mathcal{C}_{x_0}([0,\infty),\textrm{Bd}(D))$. There exists some $\alpha>0$ such that $O\subset \mathcal{C}_{x_0}([0,\alpha),\textrm{Bd}(D))$. Let $G_{\alpha}$ denote the map $:\mathcal{C}_{x_0}([0,\infty),\textrm{Bd}(D))\rightarrow \mathcal{C}_{x_0}([0,\infty),\textrm{Bd}(D))$ that fixes $x_0$ and sends any point $(a,c)$ to $\left(\frac{a}{\alpha},c\rightarrowght)$. This map is clearly a homeomorphism. It is now trivial to check that $f'(O)=(F(\alpha)\circ f'_{[0,1]}\circ G_{\alpha})(O)$. $G_{\alpha}$ is a homeomorphism. As $O$ is an open subset of $\mathcal{C}_{x_0}([0,\alpha),\textrm{Bd}(D))$, $G_{\alpha}(O)$ is an open subset of $\mathcal{C}_{x_0}([0,1),\textrm{Bd}(D))$. As $f_{[0,1]}'$ is a homeomorphism, $f_{[0,1]}'(G_{\alpha}(O))$ is an open subset of $F[0,1)(D)$, which is open in $X$. Finally, as $F(\alpha)$ is a homeomorphism, we have that $f'(O)$ is open in $X$, making $f'$ a homeomorphism.\\ \end{proof}
There is a similar convenient Corollary to Proposition \ref{prop: dilation family existence - locally compact case} in the case where the locally compact space is also a metric space:
\betagin{co}
\lambdabel{co: dilation family existence - locally compact metric case} Let $(X,d)$ be a locally compact metric space. Suppose for some $x_0$ in $X$ there exists a continuous monoidal monomorphism, $F: ([0,\infty),\times) \rightarrow (C_{x_0}(X),\circ)$, such that $F(0)=O_{x_0}$ and $F(1)=\textrm{Id}_X$. If $d(F(\alpha)(x),x_0)< d(x,x_0), \;\forall\; \alpha\in [0,1), x\in X-\{x_0\}$, then for all sufficiently small $\epsilon>0$, the function $f':\mathcal{C}_{x_0}([0,\infty),\textrm{Bd}(B(x_0;\epsilon))\rightarrow X$ defined so that $x_0\mathbbpsto x_0$ and $(\alpha,c)\mathbbpsto F(\alpha)(c)$ is a homeomorphism.\end{co}
\betagin{proof} Choose any $\epsilon$ small enough that $B(x_0;\epsilon)$ has compact closure. Then, by the shrinking condition, $F[0,1)(\overline{B(x_0;\epsilon)})\subset B(x_0;\epsilon)\subset \textrm{Int}(\overline{B(x_0;\epsilon)})$. The result now follows directly from Proposition \ref{prop: dilation family existence - locally compact case}.\\ \end{proof}
We observe that, if we use the result in Proposition \ref{prop: metrizable cones} in combination with either Corollary \ref{co: compact metrizable space cone condition} or Corollary \ref{co: dilation family existence - locally compact metric case}, we get resulting dilation families on spaces satisfying the Corollary statement conditions that are also linear dilation families. So it turns out that in many situations, the existence of a dilation family on a space can imply a 'flatness' about the center of dilation.\\
We now look at some additional characterizations of dilation families and similar structures that do not involve the same reliance on converting spaces into metrizable cones.\\
\betagin{prop}
\lambdabel{prop: general space condition} Let $(X,d)$ be a metric space. Let $I$ be a pure set containing $0$ and $F:(I,\times)\rightarrow (X^X,\circ)$ be a monoidal homomorphism with $F(0)=O_{x_0}$. If for each $x\in X$, there exists some $\alpha_x, A_x>0$ such that $\frac{\beta d(x,y)}{A_x}\leq d(F(\beta)(x),F(\beta)(y))\leq A_x\beta d(x,y), \;\forall\; y\in X, \;\forall\; \beta\in (0,\alpha_x)$ and $\lim_{\substack{\alpha\rightarrow 1\\ \alpha\in I}} F(\alpha)(x)$ exists, then the function $D:X\times X\rightarrow \mathbb R_{\geq 0}$ defined as $(x,y)\rightarrow \lim_{\alpha\rightarrow 0^+}\sup\left\{\frac{d(F(\beta)(x),F(\beta)(y))}{\beta}\Big|\beta\in I\cap (0,\alpha]\rightarrowght\}$ defines a metric on $X$ that induces the same topology as $d$ and under which the collection $\{F(\alpha)\}_{\alpha\in I}$ forms a dilation family on $(X,d,x_0)$.\end{prop}
\betagin{proof} First, we show that $D$ is a metric on $X$.
Given the inequality condition in the proposition statement, $D$ is clearly a well-defined nonnegative real-valued function.\\
If $x=y$, then
$$\frac{d(F(\beta)(x),F(\beta)(y))}{\beta}=\frac{d(F(\beta)(x),F(\beta)(x))}{\beta}=\frac{0}{\beta}=0$$
so
$$D(x,y)=\lim_{\alpha\rightarrow 0^+} \sup\{0\}=\lim_{\alpha\rightarrow 0^+} 0=0$$
Conversely, if $D(x,y)=0$, there exists a sequence of positive real numbers, $\{a_n\}_{n\in\mathbb Z^+}$ such that $\lim_{n\rightarrow \infty} a_n=0$ and
$$\sup\left\{\frac{d(F(\beta)(x),F(\beta)(y))}{\beta}\Big|\beta\in I\cap (0,a_n]\rightarrowght\}< \frac{1}{n} \;\forall\; n\in \mathbb Z^+$$
This means
$$\frac{d(F(a_n)(x),F(a_n)(y))}{a_n}<\frac{1}{n}, \;\forall\; n\in \mathbb Z^+\Rightarrow\lim_{n\rightarrow \infty}\frac{d(F(a_n)(x),F(a_n)(y))}{a_n}=0$$
As $\lim_{n\rightarrow \infty} a_n=0$, there exists some $M$ such that $\;\forall\; n\geq M, a_n<\alpha_x$. Since
$$\frac{d(x,y)}{A_x}\leq \frac{d(F(\beta)(x),F(\beta)(y))}{\beta}, \;\forall\; \beta<\alpha_x$$
we have:
$$\frac{d(x,y)}{A_x}\leq \lim_{\substack{n\rightarrow \infty\\ n\geq M}} \frac{d(F(a_n)(x),F(a_n)(y))}{a_n}$$
$$=\lim_{n\rightarrow \infty}\frac{d(F(a_n)(x),F(a_n)(y))}{a_n}=0$$
It follows that $d(x,y)=0\Rightarrow x=y$, so $D$ satisfies the identity of indiscernibles.\\
$D$ is also trivially seen to be symmetric.\\
And given any $x,y,z\in X$ and any $\beta\in \mathbb R^+$, we have
$$\frac{d(F(\beta)(x),F(\beta)(z))}{\beta}\leq \frac{d(F(\beta)(x),F(\beta)(y))}{\beta}+\frac{d(F(\beta)(y),F(\beta)(z))}{\beta}$$
So for any $\alpha\in \mathbb R^+$, we have:
$$\sup\left\{\frac{d(F(\beta)(x),F(\beta)(z))}{\beta}\Big|\beta\in I\cap (0,\alpha]\rightarrowght\}$$
$$\leq \sup\left\{\frac{d(F(\beta)(x),F(\beta)(y))}{\beta}+\frac{d(F(\beta)(y),F(\beta)(z))}{\beta}\Big|\beta\in I\cap (0,\alpha]\rightarrowght\}$$
$$\leq \sup\left\{\frac{d(F(\beta)(x),F(\beta)(y))}{\beta}\Big|\beta\leq \alpha\rightarrowght\}+\sup\left\{\frac{d(F(\beta)(y),F(\beta)(z))}{\beta}\Big|\beta\in I\cap (0,\alpha]\rightarrowght\}$$
Taking limits as $\alpha\rightarrow 0$ on both sides gives us $D(x,z)\leq D(x,y)+D(y,z)$, making $D$ a metric on $X$.\\
Now we show that $D$ induces the same topology on $X$ as $d$. Let any $x\in X$ and any $\epsilon>0$ be given. If $y$ is any point in $B_d\left(x;\frac{\epsilon}{2A_x}\rightarrowght)$ then for any $\beta<\alpha_x$, we have:
$$\frac{d(F(\beta)(x),F(\beta)(y))}{\beta}\leq A_xd(x,y)<\frac{\epsilon}{2}$$
So:
$$D(x,y)=\lim_{\alpha\rightarrow 0^+}\sup\left\{\frac{d(F(\beta)(x),F(\beta)(y))}{\beta}\Big|\beta\in I\cap (0,\alpha]\rightarrowght\}$$
$$\leq \sup\left\{\frac{d(F(\beta)(x),F(\beta)(y))}{\beta}\Big|\beta\in I\cap (0,\alpha_x]\rightarrowght\}\leq \frac{\epsilon}{2}<\epsilon$$
meaning $B_d\left(x;\frac{\epsilon}{2A_x}\rightarrowght)\subset B_D(x;\epsilon)$.\\
Similarly, let $y$ be any point in $B_D\left(x;\frac{\epsilon}{A_x}\rightarrowght)$. We have $D(x,y)<\frac{\epsilon}{A_x}\Rightarrow \lim_{\alpha\rightarrow 0^+}\sup\left\{\frac{d(F(\beta)(x),F(\beta)(y))}{\beta}\Big|\beta\in I\cap (0,\alpha]\rightarrowght\}<\frac{\epsilon}{A_x}$. This means we can choose some $\gamma>0$ such that $\sup\left\{\frac{d(F(\beta)(x),F(\beta)(y))}{\beta}\Big|\beta\leq \gamma\rightarrowght\}<\frac{\epsilon}{A_x}$, so if $\delta$ is any positive real that is less than both $\gamma$ and $\alpha_x$, we have:
$$\frac{d(x,y)}{A_x}\leq \frac{d(F(\delta)(x),F(\delta)(y))}{\delta}<\frac{\epsilon}{A_x}$$
$$\Rightarrow d(x,y)<\epsilon$$
so $y\in B_d(x;\epsilon)$, meaning $B_D\left(x;\frac{\epsilon}{A_x}\rightarrowght)\subset B_d(x;\epsilon)$. So $d$ and $D$ both induce the same topology on $X$.\\
Lastly, given any $\gamma$ in $I$ and any $x,y\in I$, we have:
$$D(F(\gamma)(x),F(\gamma)(y))=\lim_{\alpha\rightarrow 0^+}\sup\left\{\frac{d(F(\beta)(F(\gamma)(x)),F(\beta)(F(\gamma)(y)))}{\beta}\Big|\beta\in I\cap (0,\alpha]\rightarrowght\}$$
$$=\gamma\lim_{\alpha\rightarrow 0^+}\sup\left\{\frac{d(F(\beta\gamma)(x),F(\beta\gamma)(y))}{\beta\gamma}\Big|\beta\in I\cap (0,\alpha]\rightarrowght\}$$
$$=\gamma\lim_{\alpha\rightarrow 0^+}\sup\left\{\frac{d(F(\delta)(x)),F(\delta)(y))}{\delta}\Big|\delta\in I\cap (0,\alpha]\rightarrowght\}$$
$$=\gamma D(x,y)$$
where $\delta$ in the second-last expression is a substitution for $\beta\gamma$ (this works because $I$ is a pure set). So $F(\alpha)$ is a dilation of scale $\alpha$ under $D$ for each $\alpha\in I$, completing the proof.\\ \end{proof}
\betagin{prop}
\lambdabel{prop: translation-invariant metric} Let $(X,*)$ be a topological group with identity $x_0$. Let $d$ be a metric on $X$ that induces said topology. Suppose that there is a dilation family, $\{T_{\alpha}\}_{\alpha\in [0,\infty)}$, on $(X,d,x_0)$ consisting of group homomorphisms on $X$ with respect to $*$. If the collection of translation maps $\{\rho_c\}_{c\in X}$ (where each $\rho_c$ is the map $x\mathbbpsto c*x$) is equicontinuous with respect to $d$, then the function $D:X\times X\rightarrow \mathbb R_{\geq 0}$ defined as $D(x,y)=\sup\{d(c*x,c*y)| \;\forall\; c\in X\}$ is a metric on $X$ that induces the same topology on $X$ as $d$ does, and $\{T_{\alpha}\}_{\alpha\in [0,\infty)}$ is a dilation family on $(X,D,x_0)$. Furthermore, the norm on $X$ defined as $||x||=D(x,x_0), \;\forall\; x\in X$ is a homogeneous norm on $X$.\end{prop}
\betagin{proof}
It is easy to see that, as long as we can show that $\{d(c*x,c*y)|\;\forall\; c\in X\}$ is bounded for each $(x,y)\in X\times X$, then $D$ is well-defined.\\
Let any $x$ in $X$ be given. As the collection $\{\rho_x\}_{x\in X}$ is equicontinuous with respect to $d$, there exists some open neighborhood, $U$ of $x_0$, such that $\{d(c*x_0,c*z)|\;\forall\; c\in X\}$ is bounded for each $z\in U$. Now by Proposition \ref{prop: x-function is cont}, there exists some $\gamma>0$ such that $T_{\alpha}(x)\in U$, so $\{d(c*x_0,c*T_{\gamma}(x))|\;\forall\; c\in X\}$ is bounded. But this set can be reexpressed as:
$$=\{d(c*T_{\gamma}(x_0),c*T_{\gamma}(x))|\;\forall\; c\in X\}$$
$$=\{d(T_{\gamma}(T_{\gamma}^{-1}(c)*x_0),T_{\gamma}(T_{\gamma}^{-1}(c)*x)|\;\forall\; c\in X\}$$
As $\gamma>0$, $T_{\gamma}$ is a bijection on $X$, so (with the substitution $e=T_{\gamma}^{-1}(c)$), this set is just:
$$=\{d(T_{\gamma}(e*x_0),T_{\gamma}(e*x)|\;\forall\; e\in X\}$$
$$=\{\gamma d(e*x_0,e*x)| \;\forall\; e\in X\}$$
So the set $\{\gamma d(e*x_0,e*x)| \;\forall\; e\in X\}$ must be bounded. As $\gamma>0$, this implies that $\{d(e*x_0,e*x)| \;\forall\; e\in X\}$ must be bounded, $\;\forall\; x\in X$. Now, let any $x,y\in X$ be given. By what we have just shown, $\{d(e*x_0,e*(x^{-1}*y))|\;\forall\; e\in X\}$ is bounded. Multiplication by any fixed element of $X$ is a bijection, so, letting $c$ denote $e*x^{-1}$, we have that
$$\{d(e*x_0,e*(x^{-1}*y))|\;\forall\; e\in X\}$$
$$=\{d((c*x)*x_0,(c*x)*(x^{-1}*y)| \;\forall\; c\in X\}$$
$$=\{d(c*x,c*y)| \;\forall\; c\in X\}$$
and hence this final set is also bounded, $\;\forall\; (x,y)\in X\times X$, meaning $D$ is well-defined.\\
That $D$ is symmetric follows from the fact that $d$ is symmetric.\\
If $x=y$, then $D(x,y)=\sup\{d(c*x,c*y)| \;\forall\; c\in X\}=\sup\{d(c*x,c*x)| \;\forall\; c\in X\}=\sup\{0\}=0$. Similarly, if $D(x,y)=0$, then $0\leq d(x,y)=d(x_0*x,x_0*y)\leq \sup\{d(c*x,c*y)| \;\forall\; c\in X\}=0$, so $d(x,y)=0\Rightarrow x=y$, meaning $D$ satisfies the identity of indiscernables as well.\\
And for any $x,y,z\in X$, we have $d(c*x,c*z)\leq d(c*x,c*y)+d(c*y,c*z), \;\forall\; c\in X$, so:
$$D(x,z)=\sup\{d(c*x,c*y)| \;\forall\; c\in X\}\leq \sup\{d(c*x,c*y)+d(c*y,c*z)| \;\forall\; c\in X\}$$
$$\leq \sup\{d(e*x,e*y)| \;\forall\; e\in X\}+\sup\{d(f*y,f*z)| \;\forall\; f\in X\}=D(x,y)+D(x,z)$$
meaning $D$ satisfies the triangle inequality as well. Hence, $D$ is a metric on $X$.\\
To show that $D$ induces the same topology on $X$ as $d$, let $x$ be any point in $X$ and $\epsilon$ be any positive real. As $D(x,y)<\epsilon\Rightarrow d(x,y)=d(x_0*x,x_0*y)\leq D(x,y)<\epsilon$, we have $x\in B_D(x;\epsilon)\subset B_d(x;\epsilon)$, meaning $D$ induces a topology finer than that induced by $d$. At the same time, as the collection $\{\rho_x\}_{x\in X}$ is equicontinuous with respect to $d$, there exists some $\delta>0$ such that $d(c*x,c*y)<\frac{\epsilon}{2}, \;\forall\; c\in X$ and for all $y\in B_d(x;\delta)$, meaning $D(x,y)=\sup\{d(c*x,c*y)| \;\forall\; c\in X\}\leq \frac{\epsilon}{2}<\epsilon$. So $x\in B_d(x;\delta)\subset B_D(x;\epsilon)$, meaning $d$ induces a topology finer than that induced by $D$. So $D$ and $d$ induce equivalent topologies on $X$.\\
Next, for any $x,y\in X$ and any $\alpha\in (0,\infty)$, we have:
$$D(T_{\alpha}(x),T_{\alpha}(y))=\{d(c*T_{\alpha}(x),c*T_{\alpha}(y))|\;\forall\; c\in X\}$$
$$=\sup\{d(T_{\alpha}(T_{\alpha}^{-1}(c)*x),T_{\alpha}(T_{\alpha}^{-1}(c)*y)|\;\forall\; c\in X\}$$
$$=\sup\{d(T_{\alpha}(e*x),T_{\alpha}(e*y)|\;\forall\; e\in X\}$$
$$=\sup\{\alpha d(e*x,e*y)|\;\forall\; e\in X\}$$
$$=\alpha D(x,y)$$
where in the $3$rd equality, $e$ represents $T_{\alpha}^{-1}(c)$. It follows directly that $\{T_{\alpha}\}_{\alpha\in [0,\infty)}$ is a dilation family on $(X,D,x_0)$.\\
Finally, we show that $D$ induces a group norm on $X$. Given any $x,y\in X$ and any $z\in X$, we have:
$$D(z*x,z*y)=\sup\{d(c*(z*x),c*(z*y))| \;\forall\; c\in X\}$$
$$=\sup\{d((c*z)*x,(c*z)*y)| \;\forall\; c\in X\}$$
$$=\sup\{d(e*x,e*y)|\;\forall\; e\in X\}$$
$$=D(x,y)$$
so $D$ is (left) translation-invariant. For any $x\in X$, we now have:
$$||x||=D(x,x_0)=D(x_0,x)$$
$$=D(x^{-1}*x_0,x^{-1}*x)=D(x^{-1},x_0)$$
$$=||x^{-1}||$$
and for any $x,y\in X$, we have:
$$||x*y||=D(x*y,x_0)=D(x^{-1}*(x*y),x^{-1}*x_0)$$
$$=D(y,x^{-1})\leq D(y,x_0)+D(x_0,x^{-1})$$
$$=D(y,x_0)+D(x^{-1},x_0)=||y||+||x^{-1}||$$
$$=||x||+||y||$$
making the induced norm a group norm as well.\\ \end{proof}
\section{Links with Marius Buliga's work}
First, some results here may be phrased as telling us when metric spaces are 'metric cones'. As defined by M. Buliga (see \cite{intrinsic}, page $10$, Def $2.17$), a metric cone is basically a pointed locally compact metric space $(X,d,x_0)$ with an open ball about $x_0$ that admits dilations into itself about $x_0$ of all scales in $(0,1]$ \cite{intrinsic}. This is somewhat weaker than the condition of a space possessing a dilation family.\\
So we have:
\betagin{prop}
\lambdabel{prop: metric cone existence - locally compact metric case} Let $(X,d)$ be a locally compact metric space. Suppose for some $x_0$ in $X$ there exists a continuous action $F:[0,\infty)\times X\rightarrow X$ with $F(0,x)=x_0, \;\forall\; x\in X$ ($F(1,x)=x, \;\forall\; x\in X$, as well). If $d(F(\alpha,x),x_0)< d(x,x_0), \;\forall\; \alpha\in [0,1), x\in X-\{x_0\}$, then there exists a metric, $D$ on $X$, such that $D$ and $d$ induce the same topologies on $X$ and $(X,D,x_0)$ is a metric cone.\end{prop}
There is similarly the following result relating to 'normed local groups with dilations' (from \cite{intrinsic}, page $13$, Def $3.2$):
\betagin{prop}
\lambdabel{prop: local group dilations - locally compact metric case} Let $(X,d)$ be a locally compact metric space such that $X$ is a local group with identity $x_0$. Suppose there exists a continuous action $F:[0,\infty)\times X\rightarrow X$ with $F(0,x)=x_0, \;\forall\; x\in X$ ($F(1,x)=x, \;\forall\; x\in X$, as well). If $d(F(\alpha,x),x_0)< d(x,x_0), \;\forall\; \alpha\in [0,1), x\in X-\{x_0\}$, then there exists a metric, $D$ on $X$, such that $D$ and $d$ induce the same topologies on $X$ and $(X,||\cdot ||, \delta)$ is a normed local group with dilations with a homogeneous norm where $||x||$ is defined as $D(x,x_0), \;\forall\; x\in X$ and $\delta_{\epsilon}(x)=F(\epsilon,x), \;\forall\; (\epsilon,x)\in [0,\infty)\times X$.\end{prop}
There is also the following result for 'normed conical groups' (these are normed groups equipped with dilation families such that the dilations are group homomorphisms - see \cite{intrinsic}, page $14$, Def $3.3$):
\betagin{prop}
\lambdabel{prop: normed group dilations - metric case} Let $(X,d)$ be a metric space such that $X$ is a group with the operation $*$ and identity $x_0$. Suppose also that there is a family of dilations about $x_0$ in $(X,d)$, $\{T_{\alpha}\}_{\alpha\in [0,\infty)}$ where each $T_{\alpha}$ is a group homomorphism. If the family of translations $\{\rho_c\}_{c\in X}$ where $\rho_c(x)=c*x, \;\forall\; c,x\in X$ is equicontinuous at $x_0$, then the function $D:X\times X\rightarrow \mathbb R$ defined as $$D(x,y)=\sup\{d(c*x,c*y)|c\in X\}$$ is a translation-invariant metric on $X$ that induces the same topology on $X$ as $d$ and the norm induced by $D$ makes $X$ a normed conical group.\end{prop}
\end{document} |
\begin{document}
\title[Equivariant-constructible Koszul duality]
{Equivariant-constructible Koszul duality for dual toric varieties}
\date{\today}
\author{Tom Braden}
\operatorname{add}ress{Dept.\ of Mathematics and Statistics\\
University of Massachusetts, Amherst}
\email{[email protected]}
\author{Valery A.~Lunts}
\operatorname{add}ress{Department of Mathematics, Indiana University,
Bloomington, IN 47405, USA}
\email{[email protected]}
\thanks{The first named author was partially supported by NSF grant
DMS-0201823.\\
The second named author was partially supported by NSA grant MDA904-01-1-0020
and CRDF grant RM1-2405-MO-02}
\begin{abstract} For affine toric varieties $X$ and $\check X$
defined by dual cones, we define an equivalence of categories between
mixed versions of the equivariant derived category $D^b_T(X)$ and
the derived category of sheaves on $\check X$ which are locally constant
with unipotent monodromy on each orbit. This equivalence satisfies the
Koszul duality formalism of Beilinson, Ginzburg, and Soergel.
\end{abstract}
\subjclass{14M25; 16S37, 55N33, 18F20}
\mathfrak maketitle
\setcounter{tocdepth}{1}
\tableofcontents
\section{Introduction}
\subsection{}
Let $T$ and $\check T$ be dual complex tori. The ring ${\mathfrak mathcal O} (T)$ of
regular functions on $T$ is canonically isomorphic to the group ring
${\mathfrak mathbb C} [\pi _1(\check T)]$ of the fundamental group $\pi _1(\check T)$.
Thus the category of quasi-coherent sheaves on $T$ is identified with
the category of local systems on $\check T$. Let $\mathfrak mathfrak{t}$ be the
Lie algebra of $T$. The (evenly graded) algebra $A$ of polynomial
functions on $\mathfrak mathfrak{t}$ is canonically isomorphic to the
equivariant cohomology ring $H^*_T(pt)$.
The exponential map
$$exp: \mathfrak mathfrak{t}\to T$$ identifies
$A$-modules supported at the origin with quasi-coherent sheaves on
$T$ supported at the identity and hence with unipotent local systems
on $\check T$.
Notice that under this correspondence the logarithm of
the monodromy operator becomes the multiplication by the first Chern
class of a line bundle. This phenomenon also occurs in
mirror symmetry, where the logarithm of the monodromy of
the Gauss-Manin local system around certain loops in the moduli space
of complex structures of a Calabi-Yau manifold is identified with
multiplication by a Chern class on the cohomology of
the mirror Calabi-Yau. This is worked out in more generality for
the case of complete intersections in toric varieties in
\cite{Ho}.
\subsection{} \langlebel{intro to main results}
In this paper we extend the above idea to affine toric varieties.
Let $X$ be an affine $T$-toric variety whose fan
consists of a cone ${\sigma}$ and all its faces,
and let $\check X$ be the toric variety whose fan
consists of the dual cone $\check {\sigma}$ and all its faces.
The $T$-orbits of $X$ are indexed by the faces $\tau$ of the
cone ${\sigma}$. The $T$-orbit $O_\tau \subset X$ is
identified with a quotient $T/T_\tau$ by a subtorus $T_\tau$.
The dual variety has a corresponding orbit $O_{\tau^\bot}$ which is
isomorphic as a $\check T$-space to $\check T/T_\tau^\bot$,
where $T_\tau^\bot$ is the ``perpendicular'' subtorus
whose Lie algebra is the annihilator of
$\mathfrak mathfrak{t}_\tau = \mathfrak mathop{\rm Lie} T_\tau$.
The ring of regular functions on $T_\tau$ is canonically
isomorphic to the group ring of $\pi_1(\check T/T_\tau^\bot)$.
Let $A_\tau$ be the algebra of polynomial functions on
$\mathfrak mathfrak{t}_\tau$; it is canonically isomorphic to
the equivariant cohomology $H^*_T(T/T_\tau)$.
In the same way as before, the exponential map identifies
$A_\tau$-modules supported at the origin with unipotent
local systems on $O_{\tau^\bot}$.
We use this idea (together with a more combinatorial
duality, see \S\ref{combinatorial Koszul}) to relate
$T$-equivariant sheaves on $X$ to complexes of sheaves
with unipotent monodromy on $\check X$. We define an
equivalence of triangulated categories $K$, which fits
into the following diagram of categories and functors.
\begin{equation} \langlebel{main diagram}
\xymatrix{
D^b({\mathfrak mathcal A}\text{\rm -mod}_{\text{\it f}}) \ar[r]^K_\sim\ar[d]^{F_T} & D^b(LC_{\mathfrak mathcal F}(X^\vee))\ar[d]^{F_{cf}} \\
D^b_T(X) & D^b(LC_{cf}(X^\vee))
}\end{equation}
The categories on the bottom are topological categories of sheaves
on the dual toric varieties $X$ and $X^\vee$. The categories
above them are ``mixed'' versions of these categories, where
objects have been given an (extra) grading,
and the vertical functors forget the grading.
On the left hand side we have $T$-equivariant sheaves on $X$,
while on the right we have complexes of orbit-constructible sheaves
on $\check X$ with unipotent monodromy.
Let us describe these categories in more detail.
$D^b_T(X)$ is the (bounded, constructible)
$T$-equivariant derived category of sheaves on
$X$ defined in \cite{BL}. By results of \cite{L} it is equivalent
to a full subcategory of
the category of differential graded modules (DG-modules) over
a sheaf ${\mathfrak mathcal A} = {\mathfrak mathcal A}_{[{\sigma}]}$ of rings on the finite poset
$[{\sigma}]$ of faces of ${\sigma}$. Sections of this sheaf on a
face $\tau\prec {\sigma}$ are complex valued polynomial functions
on $\tau$, graded so linear functions have degree $2$.
Our mixed version of this category is $D^b({\mathfrak mathcal A}\text{\rm -mod}_{\text{\it f}})$, the
derived category of finitely generated graded ${\mathfrak mathcal A}$-modules; it
has two gradings -- the module grading, and the grading in the complex.
The forgetful functor $F_T$ combines these two gradings into
the single grading on DG-modules. There is a ``twist''
automorphism of $D^b({\mathfrak mathcal A}\text{\rm -mod}_{\text{\it f}})$, denoted $\langle 1\rangle$,
which shifts both the complex and module gradings so that
$F_T\langle 1\rangle = F_T$.
The right side of \eqref{main diagram} involves sheaves on
$\check X$. $LC_{cf}(\check X)$
denotes the category of sheaves of ${\mathfrak mathbb C}$-vector spaces
on $\check X$ which are
\begin{enumerate}
\item[(1)] ``locally constant'' -- the restriction to each
$T$-orbit $O \subset \check X$ is a local system,
\item[(2)] ``unipotent'' -- for any orbit $O$ and any
$\gamma\in \pi_1(O)$, the action of $\gamma-1$ on
the stalk at a point $p \in O$ is locally nilpotent.
\item[(3)] ``cofinite'' -- the $\pi_1(O)$ invariants
of the stalk at $p$ is finite dimensional, for all
orbits $O$ and $p\in O$.
\end{enumerate}
$D^b(LC_{cf}(\check X))$ is then the derived category of
this abelian category. Note that although
objects of $D^b(LC_{cf}(\check X))$ are locally constant on orbits,
they are not constructible in the usual sense, since the stalks
need not be finite-dimensional. As we will see, though, conditions
(1)--(3) imply that these objects are
still well-behaved. In particular, the full subcategory of
objects all of whose stalk cohomology groups are
finite-dimensional is equivalent to a full subcategory of
the usual constructible derived category.
See Proposition \ref{constructible = finite length}.
To define the mixed version $LC_{\mathfrak mathcal F}(\check X)$ of
$LC_{cf}(X^\vee)$, we use a self-map ${\mathfrak mathcal F}\text{\rm co-}lon
\check X \to \check X$ which is a lift of the Frobenius
map to characteristic zero defined for toric varieties.
An object in $LC_{\mathfrak mathcal F}(\check X)$ is an object $S \in LC_{cf}(\check X)$
together with an isomorphism $\theta \text{\rm co-}lon {\mathfrak mathcal F} ^{-1}F\to F$
whose eigenvalues on the stalks at points of $(\check X)^{\mathfrak mathcal F}$
are powers of $2^{1/2}$. We again have a ``shift of grading''
functor $\langle 1\rangle$, which multiplies $\theta$ by $2^{1/2}$.
It is clear that $F_{cf}\langle 1 \rangle = F_{cf}$.
\subsection{$t$-structures}
All four categories in \eqref{main diagram}
come equipped with natural perverse
$t$-structures, whose abelian cores we denote by $P_T(X)$, $P({\mathfrak mathcal A})$,
$P_{cf}(\check X)$, and $P_{\mathfrak mathcal F}(\check X)$. These $t$-structures are
particularly nice, in that each triangulated category is equivalent to
the bounded derived category of its core: $D^b_T(X) \text{\rm co-}ng
D^b(P_T(X))$, etc.
The forgetful functors $F_T$, $F_{cf}$ are $t$-exact, so
they restrict to functors $P({\mathfrak mathcal A})\to P_T(X)$ and
$P_{\mathfrak mathcal F}(\check X) \to P_{cf}(\check X)$.
The twist functors $\langle 1\rangle$ are also $t$-exact, and so they give
automorphisms of $P({\mathfrak mathcal A})$ and $P_{\mathfrak mathcal F}(X^\vee)$. This induces
bijections between isomorphism classes of ``ungraded'' simple objects
and ``graded'' simples up to twists:
\begin{eqnarray*}
Irr(P_T(X)) & \leftrightarrow & Irr(P({\mathfrak mathcal A}))/{\mathfrak mathbb Z},\\
Irr(P_{cf}(\check X)) & \leftrightarrow & Irr(P_{\mathfrak mathcal F}(\check X))/{\mathfrak mathbb Z}.
\end{eqnarray*}
The set $Irr(P_T(X))$ consists of all equivariant IC-sheaves supported
on the closures of $T$-orbits. For each IC-sheaf $IC_T(\overline{O_\tau})$
we will fix a certain lift to $P({\mathfrak mathcal A})$, which we denote ${\mathfrak mathcal L}^\tau$;
it is a complex of ${\mathfrak mathcal A}$-modules with nonzero cohomology in a single degree.
Up to a grading shift, it is the combinatorial equivariant
intersection cohomology sheaf studied in \cite{BBFK, BrLu, Ka}.
\subsection{} \langlebel{injectives and projectives}
The abelian category $P({\mathfrak mathcal A})$ has enough projective objects. Let ${\mathfrak mathcal P}\in
P({\mathfrak mathcal A})$ be a projective cover of $\oplus_{\tau\in [{\sigma}]}{\mathfrak mathcal L}^\tau$,
and let $R$ be the opposite ring to the graded ring
\[\tmop{end}({\mathfrak mathcal P}) := \oplus_{i\ge 0} \operatorname{Hom}_{P({\mathfrak mathcal A})}({\mathfrak mathcal P},{\mathfrak mathcal P}\langle i\rangle).\]
In the usual way we see that $P({\mathfrak mathcal A})$ is equivalent to $R\text{\rm -mod}_{\text{\it f}}$, the
category of finitely generated $R$-modules. Furthermore, $P_T(X)$ is
equivalent to $R\text{\rm -Mod}_{\text{\it f}}$, the category of finitely generated ungraded
$R$-modules. It follows that we have equivalences $D^b({\mathfrak mathcal A}\text{\rm -mod}_{\text{\it f}})\text{\rm co-}ng
D^b(R\text{\rm -mod}_{\text{\it f}})$, $D^b_T(X)\text{\rm co-}ng D^b(R\text{\rm -Mod}_{\text{\it f}})$. With respect to these
equivalences, $F_T$ is the functor of forgetting the grading.
A similar story holds on the right-hand side of (\ref{main diagram}).
The simple objects in $P_{cf}(\check X)$ are the intersection
cohomology complexes $IC^{\scriptscriptstyle \bullet}(\overline{O_\alpha})$ for $O_\alpha \subset
\check X$ a $T$-orbit. We will single out a distinguished
lift $L^{\scriptscriptstyle \bullet}_\alpha\in P_{\mathfrak mathcal F}(\check X)$ of $IC^{\scriptscriptstyle \bullet}(\overline{O_\alpha})$.
There are enough injectives in $P_{\mathfrak mathcal F}(\check X)$; let $I$ denote the
injective hull of $\oplus L^{\scriptscriptstyle \bullet}_\alpha$, and put $R^\vee =
\tmop{end}(I)^{opp}$. The functor $\oplus_{i\ge 0} \operatorname{Hom}_{P_{\mathfrak mathcal F}(\check
X)}(-,I\langle -i\rangle)^*$ gives an equivalence between $P_{\mathfrak mathcal F}(\check X)$
and $\check R\text{\rm -mod}_{\text{\it cf}}$, the category of ``co-finite'' graded $\check
R$-modules (see \S\ref{module conventions}).
Similarly $P_{cf}(\check X)$ is equivalent to the
category $\check R\text{\rm -Mod}_{\text{\it cf}}$ of ungraded co-finite modules. With respect
to these equivalences, $F_{cf}$ is the functor of forgetting the
grading.
The full subcategory $P_{u,fl}(\check X)$ of $P_{cf}(\check X)$
consisting of objects of finite length is the full subcategory of the
usual topological category of orbit-constructible perverse sheaves
consisting of objects with unipotent monodromy. Further, the category
$P_{{\mathfrak mathcal F},fl}(\check X)$ of finite length objects in $P_{\mathfrak mathcal F}(\check X)$
is a mixed version of $P_{u,fl}(\check X)$. These categories are
equivalent to finite-dimensional ungraded and graded $\check
R$-modules, respectively.
\subsection{}
The functor $K$ which relates the two sides of
(\ref{main diagram}) is {\em not\/} $t$-exact, but it
does have an interesting relationship with the
$t$-structures: it is a Koszul equivalence.
Roughly this means that $K$ takes simples of weight $0$ in
$P({\mathfrak mathcal A})$ to indecomposable injectives in $P_{\mathfrak mathcal F}(\check X)$
and indecomposable projectives in $P({\mathfrak mathcal A})$ to simples
in $P_{\mathfrak mathcal F}(\check X)$. It also implies that
$R$ and $\check R$ are Koszul graded rings, and are naturally
Koszul dual to each other, in the sense of \cite{BGS}.
We explain this in more detail in \S\ref{def of Koszul functor} below.
\subsection{} A similar Koszul functor was constructed
for the varieties $X$ and $\check X$ in \cite{Br}. In that
construction the source and target categories were
combinatorially defined triangulated categories ${\mathfrak mathbf D}_\Phi(X)$
and ${\mathfrak mathbf D}_{\check \Phi}(\check X)$, which model mixed sheaves
on $X$ and $X^\vee$ with ``conditions at infinity'' described
by auxiliary data $\Phi$, $\Phi^\vee$. This auxiliary choice
(essentially the choice of a toric normal slice to each stratum)
is somewhat artificial, and as a result it is not clear how
to relate these categories directly to a topological category,
although the corresponding abelian category of perverse
objects in ${\mathfrak mathbf D}_\Phi(X)$ is a mixed version of a category
${\mathfrak mathcal P}_\Phi(X)$ of perverse sheaves on $X$.
The construction in this paper
removes these defects, and it is our hope that this more
canonical approach will in turn inspire a more intrinsic point
of view on this phenomenon, in which the duality functor is
defined directly, perhaps in terms of filtered $D$-modules.
\subsection{Ideas for future work}
We expect that our work can be extended in several directions.
a) We hope that a particular instance of our constructible-equivariant
correspondence can be considered as a ``limit case'' of the mirror
symmetry between dual families of Calabi-Yau hypersurfaces in dual toric
varieties.
b) We believe that the same constructible-equivariant component should be
present in the Koszul duality on flag manifolds constructed in
[BGS]. This should be in agreement with Soergel's conjectures [S].
c) One should be able to extend to toric varieties the full
correspondence (local systems on
$T$) $\leftrightarrow$ (quasi-coherent sheaves on $\check T$). In
our work we restricted ourselves to unipotent local systems and
quasi-coherent sheaves supported at the identity. The language of
configuration schemes [L2] may be appropriate in this problem.
\subsection{} We briefly describe the structure of the paper.
Section 2 contains some basic background from homological
algebra, including definitions on graded modules,
discussion of Koszul equivalences, and mixed categories and
gradings. Section 3 introduces the formalism of sheaves on
fans considered as finite partially ordered sets, and defines
three sheaves of rings on fans which are important later.
In Section 4 we consider sheaves on toric varieties which are
locally constant on orbits. We prove that the derived category
of these sheaves is the same as the category of complexes of
sheaves with locally constant cohomology; this means that locally
constant sheaves have enough flexibility for our homological
calculations. We also show that the category $LC(X)$ of
locally constant sheaves is equivalent to comodules over a sheaf
of rings ${\mathfrak mathcal B}$ defined in section 3.
In section 5 we define our ``mixed'' version
$LC_{\mathfrak mathcal F}(X)$ of locally constant sheaves, and show that they are equivalent
to graded comodules over ${\mathfrak mathcal B}$. We define a perverse $t$-structure
on $D^b(LC_{\mathfrak mathcal F}(X))$ and prove some basic properties of perverse
objects, including the local purity of simple objects. This
purity allows us to define a mixed structure on the category
of perverse objects, which is the first step to proving that
they are equivalent to modules over some graded algebra.
Section 6 turns to the equivariant side of our picture. We first
describe a topological realization functor from
complexes of ${\mathfrak mathcal A}$-modules on a fan ${\Sigma}$ to equivariant complexes on
the corresponding toric variety $X$; this was originally defined in \cite{L}.
We next study the homological algebra of ${\mathfrak mathcal A}$-modules; the main result is
that $D^b({\mathfrak mathcal A}\text{\rm -mod}_{\text{\it f}})$ is equivalent to the
homotopy category of complexes of ``pure'' ${\mathfrak mathcal A}$-modules, which
are direct sums of shifts of the combinatorial
intersection cohomology sheaves ${\mathfrak mathcal L}^\tau$ studied in \cite{BBFK,BrLu}.
In Section 7 we finally define our toric Koszul functor $K$
and prove that it has the asserted properties. In fact
the existence of enough projectives in $P({\mathfrak mathcal A})$ and
enough injectives in $P(LC_{\mathfrak mathcal F}(\check X))$ is deduced
by applying $K$ and $K^{-1}$ to the appropriate simple
perverse objects, and all the assertions of
\S\ref{injectives and projectives} are proved here.
We banish a few technical proofs to Section 8.
\subsection{Acknowledgments}
The first author would like to thank David Cox for suggesting
that the results of \cite{Br} should be reformulated equivariantly.
The second author would like to thank V.\ Golyshev for stimulating
discussions.
\section{Ideas from homological algebra}
\subsection{Conventions on graded rings and modules} \langlebel{module conventions}
Fix a field $k$, and let $R = \oplus_{n\ge 0} R_n$ be a positively
graded $k$-algebra whose zeroth graded piece $R_0$ is isomorphic to
$k^{\oplus s}$ for some $s \ge 1$. Suppose that all graded pieces
$R_n$ are finite-dimensional (this holds if $R$ is either left or
right Noetherian, for instance).
Let $R\mathfrak mod$, $R\text{\rm -Mod}$ denote the abelian categories of graded (resp.\
ungraded) left $R$-modules. Let $R\text{\rm -mod}_{\text{\it f}}$, $R\text{\rm -Mod}_{\text{\it f}}$ be their respective
full subcategories of finitely generated modules; they are abelian
subcategories if and only if $R$ is left Noetherian.
The shift of grading functors $\langle j\rangle, j\in {\mathfrak mathbb Z}$ act on $R\mathfrak mod$ by
$(M\langle j\rangle)_n = M_{n+j}$ (note that this is the opposite convention
to \cite{BGS}). They preserve the subcategory $R\text{\rm -mod}_{\text{\it f}}$.
Given $M$ in $R\mathfrak mod$, define the ``graded dual'' $M^*$ of $M$ by
$(M^*)_n = \operatorname{Hom}_k(M_{-n}, k)$. Then $M\mathfrak mapsto M^*$ is a functor
$R\mathfrak mod \to (R^{opp})\mathfrak mod^{opp}$. Put $R^\circledast := (R^{opp})^*$;
it is an injective object of $R\mathfrak mod$.
We will also need ``cofinite'' graded $R$-modules, the dual notion to
finitely generated modules. If $n \ge 0$, put $R_{> n} = \oplus_{k>
n}R_k$.
\begin{prop} Let $M \in R\mathfrak mod$ or $R\text{\rm -Mod}$. The following are equivalent:
\begin{enumerate}
\item $\dim_k \{m\in M \mathfrak mid R_{>0}\cdot m = 0\}<\infty$, and every
$m\in M$ is annihilated by some $R_{> n}$.
\item $M^*$ is contained in a finite direct sum of shifted copies of
$R^{\circledast}$.
\end{enumerate}
\end{prop}
Such modules are called $R_{>0}$-cofinite, or simply ``cofinite''.
Let $R\text{\rm -mod}_{\text{\it cf}}$, $R\text{\rm -Mod}_{\text{\it cf}}$ denote the category of cofinite graded and ungraded
$R$-modules, respectively.
The graded dual gives an equivalence of categories $R\text{\rm -mod}_{\text{\it f}}
\to (R^{opp})\text{\rm -mod}_{\text{\it cf}}^{opp}$. It follows that $R\text{\rm -mod}_{\text{\it f}}$ is an abelian
subcategory of $R\mathfrak mod$ if and only if $R$ is right Noetherian.
\subsection{Koszul functors and Koszul duality} \langlebel{def of Koszul functor}
We present the ideas of Koszul duality on derived categories at a
level of generality appropriate for our purposes. For a more general
discussion, see \cite{BGS}.
Fix a field $k$. Let $R$ and $R^\vee$ be algebras of the type
considered in the previous section.
\begin{defn} \langlebel{Koszul functor definition}
A covariant functor
\[K\text{\rm co-}lon D^b(R\text{\rm -mod}_{\text{\it f}}) \to D^b(R^\vee\text{\rm -mod}_{\text{\it cf}})\]
is a {\em Koszul equivalence} if the following are satisfied:
\begin{enumerate}
\item $K$ is a triangulated equivalence of categories. In particular
$K(M[1]) = (KM)[1]$ for all $M\in D^b(R\text{\rm -mod}_{\text{\it f}})$.
\item For all $M \in D^b(R\text{\rm -mod}_{\text{\it f}})$, we have
\[K(M\langle 1\rangle) = (KM)\langle -1 \rangle[1].\]
\item $KR_0 \text{\rm co-}ng (R^\vee)^\circledast$.
\item $KR \text{\rm co-}ng (R^\vee_0)^\circledast$.
\end{enumerate}
\end{defn}
Since $R_0$, $R^\vee_0$ are semisimple, $R$ is projective and \
$(R^\vee)^\circledast$ is injective, the conditions (3) and (4) can be
replaced by the following.
\begin{enumerate}
\item[($3'$)] $K$ sends simple objects of grading degree $0$ to
injective hulls of simples of degree $0$.
\item[($4'$)] $K$ sends projective covers of simples of grading degree
$0$ to simples of degree $0$.
\end{enumerate}
Here we use the standard embeddings of $R\text{\rm -mod}_{\text{\it f}}$, $\check R\text{\rm -mod}_{\text{\it cf}}$ into
their derived categories as complexes with cohomology only in degree
$0$.
\begin{thm} \langlebel{Koszul functors and rings}
If a Koszul functor $K$ exists as in Definition \ref{Koszul functor definition},
then
\begin{enumerate}
\item[(a)] $R$, $R^\vee$ are Koszul graded rings, i.e.\
\[\operatorname{Ext}^i_{R}(R_0,R_0\langle j\rangle) = 0\;\text{for}\; i\ne -j,\]
and similarly for $R^\vee$.
\item[(b)] $R^\vee$ is the Koszul dual ring to $R$, i.e.\ we have an
isomorphism of rings
\[\langlebel{Koszul dual ring}
R^\vee \text{\rm co-}ng \oplus_{i\ge 0} \operatorname{Ext}^i_R(R_0,R_0\langle -i\rangle).\]
\end{enumerate}
\end{thm}
The proof is immediate from the definition \ref{Koszul functor
definition}.
The conclusion (a) implies in particular that $R$ is a quadratic algebra, with
generators in degree $1$ and relations in degree $2$. (b)
implies that $R$ and $R^\vee$ are quadratic dual rings: $R_0 =
R^\vee_0$ canonically, $R_1$ and $R_1^\vee$ are dual $R_0$-modules,
and the relations for $R$ and $R^\vee$ are orthogonal. See \cite{BGS}
for more precise statements and a proof.
\begin{rmk}
The original example of a Koszul equivalence was defined by
Bernstein, Gelfand, and Gelfand \cite{BGG} for $R$ a polynomial ring
and $R^\vee$ the dual exterior algebra. This example is related to
a duality between equivariant and ordinary cohomology, and underlies
the ``local'', one-orbit case of our toric Koszul duality.
In \cite{BGS} it is shown that under mild finiteness conditions
(e.g.\ if $\dim_k R^\vee <\infty$, so $R^\vee\text{\rm -mod}_{\text{\it cf}} = R^\vee\text{\rm -mod}_{\text{\it f}}$),
then a Koszul dual pair of rings $(R, R^\vee)$ gives rise to a
Koszul equivalence $D^b(R\text{\rm -mod}_{\text{\it f}})\to D^b(R^\vee\text{\rm -mod}_{\text{\it f}})$. We are taking
the opposite point of view and considering the functor $K$ as
the primary object.
\end{rmk}
\subsection{Mixed categories} \langlebel{mixed categories}
We will need the notion of a ``mixed'' abelian category. This
generalizes the category of finitely generated graded modules over a
finitely generated positively graded ring. We mostly follow
\cite{BGS}, but we do not wish to assume our algebras are
finite-dimensional, so our abelian categories are not assumed to be
Artinian.
Fix a field $k$. Consider triples $({\mathfrak mathbf M}, W_{\scriptscriptstyle \bullet}, \langle 1\rangle)$, where
\begin{itemize}
\item ${\mathfrak mathbf M}$ is an abelian $k$-category.
\item $\langle 1\rangle$ is an automorphism of ${\mathfrak mathbf M}$, and
\item For each $M \in {\mathfrak mathbf M}$, $\{W_jM\}_{j\in {\mathfrak mathbb Z}}$ is a functorial
increasing filtration of $M$.
\end{itemize}
We call such a triple a {\em mixed} category if the following are
satisfied:
\begin{enumerate}
\item The filtration $W$ is strictly compatible with morphisms, so
${\mathfrak mathbb G}r^W_j = W_j/W_{j-1}$ is an exact functor.
\item For any $M \in {\mathfrak mathbf M}$, we have $W_j(M\langle 1\rangle) = W_{j-1}(M)\langle
1\rangle$.
\item If ${\mathfrak mathbb G}r^W_jM = 0$ for $j\ne w$ (we call such an object {\em pure
of weight $w$}), then $M$ is a finite direct sum of simple
objects.
\item There are only finitely many isomorphism classes of
simples of weight $0$.
\end{enumerate}
Define automorphisms $\langle n\rangle$, $n\in {\mathfrak mathbb Z}$ of ${\mathfrak mathbf M}$ by taking powers:
$\langle n\rangle = \langle 1\rangle^n$.
We say an object $M$ of ${\mathfrak mathbf M}$ has weights $\le j$ (resp.\ has weights
$\ge j$) if $W_jM = M$, (resp.\ $W_{j-1}M = 0$). If both hold, we say
$M$ is pure of weight $j$; such an object is semisimple of finite
length by (3).
Given a mixed category $({\mathfrak mathbf M},W_{\scriptscriptstyle \bullet},\langle 1\rangle)$, and objects $X,Y \in
{\mathfrak mathbf M}$, define the graded hom and graded ext by
\begin{align*}
\mathop{\rm hom}\nolimits(X,Y)_n & = \operatorname{Hom}_{\mathfrak mathbf M}(X,Y\langle n\rangle) \\
\mathfrak mathop{\rm ext}\nolimits^i(X,Y)_n & = \operatorname{Ext}^i_{\mathfrak mathbf M}(X,Y\langle n\rangle).
\end{align*}
The graded vector space $\tmop{end}(X):= \mathop{\rm hom}\nolimits(X,X)$ naturally has the
structure of a graded ring.
Let $L \in {\mathfrak mathbf M}$ be the direct sum of one object from each
isomorphism class of weight $0$ simples. We call a projective object
$M\in {\mathfrak mathbf M}$ a {\em mixed projective generator (resp.\ a mixed injective
generator)} if
\begin{enumerate}
\item $M$ is projective (resp.\ injective),
\item $M/W_{-1}M \text{\rm co-}ng L$ (resp.\ $W_0M \text{\rm co-}ng L$), and
\item for any $X \in {\mathfrak mathbf M}$ there exist $r_k\in {\mathfrak mathbb Z}$ and a surjection
$\oplus_{k=1}^n M\langle r_k\rangle \to X$ (resp.\ an injection $X \to
\oplus_{k=1}^n M\langle r_k\rangle$).
\end{enumerate} If $P$ is a mixed projective generator,
then it is is a projective cover of $L$, and the graded endomorphism
ring $\tmop{end}(P)$ is positively graded. If in addition the endomorphisms
of simple objects in ${\mathfrak mathbf M}$ are reduced to scalars, then
$\tmop{end}(P)_0$ is isomorphic to $k^r$, where $r$ is the number of
isomorphism classes of weight $0$ simples in ${\mathfrak mathbf M}$. Similar
statements hold for mixed injective generators.
The main examples of mixed categories are categories of graded modules
over graded rings.
Given a positively graded ring $R$ with $R_0$ semisimple, we have a
mixed category $({\mathfrak mathbf M}, W_{\scriptscriptstyle \bullet}, \langle 1\rangle)$ where ${\mathfrak mathbf M}$ consists of
graded $R$-modules $M$ with $\dim_k M_j < \infty$ for all $j$,
$\langle 1\rangle$ is the degree shift as defined previously, and
$W_jM = \oplus_{i\ge -j} M_i$. If $R$
is left (resp.\ right) Noetherian, then this restricts to a mixed
structure on $R\text{\rm -mod}_{\text{\it f}}$ (resp.\ $R\text{\rm -mod}_{\text{\it cf}}$).
We want sufficient conditions for a mixed category to be of the form
$R\text{\rm -mod}_{\text{\it f}}$ or $R\text{\rm -mod}_{\text{\it cf}}$.
\begin{prop} \langlebel{mixed categories are modules}
Let $({\mathfrak mathbf M},W_{\scriptscriptstyle \bullet},\langle 1\rangle)$ be a mixed category.
\begin{itemize}
\item[(a)] If ${\mathfrak mathbf M}$ has a mixed projective generator $P$ and
$\tmop{end}(P)$ is Noetherian, then $\mathop{\rm hom}\nolimits(P,-)$ defines an equivalence of
categories ${\mathfrak mathbf M}\to R\text{\rm -mod}_{\text{\it f}}$, where $R = \tmop{end}(P)^{opp}$.
\item[(b)] If ${\mathfrak mathbf M}$ has a mixed injective generator $I$ and $\tmop{end}(I)$
is Noetherian, then $\mathop{\rm hom}\nolimits(-,I)^*$ defines an equivalence of
categories ${\mathfrak mathbf M} \to R\text{\rm -mod}_{\text{\it cf}}$, where $R = \tmop{end}(I)^{opp}$.
\end{itemize}
In either case the mixed structure on ${\mathfrak mathbf M}$ agrees with the one on
graded modules.
\end{prop}
\subsubsection{Gradings on abelian categories} \langlebel{grading section}
Let ${\mathfrak mathbf C}$ be an abelian category. By a {\em pre-grading} on ${\mathfrak mathbf C}$ we
mean a collection $({\mathfrak mathbf M},W_{\scriptscriptstyle \bullet},\langle 1\rangle, v, \epsilon)$, where
$({\mathfrak mathbf M},W_{\scriptscriptstyle \bullet},\langle 1\rangle)$ is a mixed category, $v\text{\rm co-}lon {\mathfrak mathbf M} \to {\mathfrak mathbf C}$
is an exact functor,
and $\epsilon$ is a natural isomorphism $v \to v\circ \langle 1\rangle$,
satisfying: (1) $v$ sends simples to simples and (2)
for any $X,Y$ in ${\mathfrak mathbf M}$, the map \[\mathop{\rm hom}\nolimits_{\mathfrak mathbf M}(X,Y) \to
\operatorname{Hom}_{\mathfrak mathbf C}(vX,vY)\] induced by $v, \epsilon$ is bijective.
\begin{prop} \langlebel{ungraded modules}
Let $({\mathfrak mathbf M},W_{\scriptscriptstyle \bullet},\langle 1\rangle, v, \epsilon)$ be a pre-grading on ${\mathfrak mathbf C}$.
\begin{enumerate}
\item[(a$'$)] If part (a) of Proposition \ref{mixed categories are
modules} holds, and in addition $vP$ is a projective generator of
${\mathfrak mathbf C}$, then ${\mathfrak mathbf C}$ is equivalent to $R^{opp}\text{\rm -Mod}_{\text{\it f}}$.
\item[(b$'$)] If part (b) of Proposition \ref{mixed categories are
modules} holds, and in addition $vI$ is an injective generator of
${\mathfrak mathbf C}$, then ${\mathfrak mathbf C}$ is equivalent to $R^{opp}\text{\rm -Mod}_{\text{\it cf}}$.
\end{enumerate}
in either case $v$ is the functor of forgetting the grading.
\end{prop}
In either situation (a$'$) or (b$'$) it follows that for any $X,Y$ in
${\mathfrak mathbf M}$ and $i\ge 0$, the induced map
\[\mathfrak mathop{\rm ext}\nolimits^i_{\mathfrak mathbf M}(X,Y) \to \operatorname{Ext}^i_{\mathfrak mathbf C}(X,Y)\]
is bijective. Thus our pre-grading is what in \cite{BGS} was termed a
``grading'' on ${\mathfrak mathbf C}$.
\subsection{Triangulated gradings} \langlebel{triangulated gradings}
Let $D$ be a triangulated category. A triangulated grading
on $D$ is defined to be a tuple $(D_m, \langle 1\rangle, v,\epsilon)$, where
$D_m$ is a triangulated category, $\langle 1\rangle$ is a triangulated automorphism
of $D_m$, $v\text{\rm co-}lon D_m\to D$ is a triangulated functor, and
$\epsilon\text{\rm co-}lon v \to v \circ \langle 1\rangle$ is a natural isomorphism,
subject to the condition that the induced map
\[\mathop{\rm hom}\nolimits_{D_m}(X,Y) \to \operatorname{Hom}_D(vX,vY)\]
is an isomorphism for any $X,Y\in D_m$, where
$\mathop{\rm hom}\nolimits_{D_m}$ is defined as in the previous section.
If we have a grading on an abelian category as in the previous
section, we get a triangulated grading by letting
$D_m = D^b({\mathfrak mathbf M})$, $D = D^b({\mathfrak mathbf C})$. One can also go in
the other direction, starting from a triangulated grading
as defined above, and endowing $D_m$ and $D$
with $t$-structures for which $\langle 1\rangle$ and $v$ are $t$-exact.
Letting ${\mathfrak mathbf M}$ and ${\mathfrak mathbf C}$ be the abelian cores of $D_m$ and $D$,
respectively, we get functors $\langle 1 \rangle\text{\rm co-}lon {\mathfrak mathbf M}\to {\mathfrak mathbf M}$ and
$v\text{\rm co-}lon {\mathfrak mathbf M}\to {\mathfrak mathbf C}$ as above. Endowing ${\mathfrak mathbf M}$ with a suitable
mixed structure, we get a pre-grading. If Proposition
\ref{ungraded modules} applies, it is a grading. In
\S\ref{main proofs}) we use this approach to
prove the fact stated in the introduction
that the functors $F_T$ and $F_{cf}$ are gradings on the
appropriate perverse abelian categories.
\section{Sheaves of rings associated to toric varieties}
\subsection{Ringed quivers}
Let ${\mathfrak mathbb G}amma$ be a finite partially ordered set which we consider as a
category: for any $\alpha ,\beta \in {\mathfrak mathbb G}amma$ the set of morphisms
$\operatorname{Hom}(\beta ,\alpha)$ contains a single element if $\beta \geq \alpha$
and is empty otherwise. A (covariant) functor from ${\mathfrak mathbb G}amma $ to the
category of rings is called a sheaf of rings on ${\mathfrak mathbb G}amma$. Let ${\mathfrak mathcal A}={\mathfrak mathcal A}
_{{\mathfrak mathbb G}amma}$ be such a sheaf of rings, i.e. ${\mathfrak mathcal A}$ is a collection of
rings $\{{\mathfrak mathcal A} _{\alpha}\}_{\alpha \in {\mathfrak mathbb G}amma}$ with ring homomorphisms
$\phi _{\beta \alpha}\text{\rm co-}lon {\mathfrak mathcal A} _{\beta}\to {\mathfrak mathcal A} _{\alpha}$, if $\beta
\geq \alpha$, satisfying $\phi _{\gamma \beta}\phi _{\beta
\alpha}=\phi _ {\gamma \alpha}$ if $\gamma \geq \beta \geq \alpha$.
We call the pair $({\mathfrak mathbb G}amma ,{\mathfrak mathcal A} _{{\mathfrak mathbb G}amma})$ a {\it ringed quiver}.
Assume that for every $\alpha \in {\mathfrak mathbb G}amma$ there is given a ${\mathfrak mathcal A}
_{\alpha}$-module ${\mathfrak mathcal M}_{\alpha}$ with a morphism $\psi _{\beta
\alpha}\text{\rm co-}lon {\mathfrak mathcal M}_{\beta}\to {\mathfrak mathcal M}_{\alpha}$ of ${\mathfrak mathcal A} _{\beta}$ modules
(for $\beta \geq \alpha$), such that $\psi _{\gamma \beta}\psi _{\beta
\alpha}=\psi _{\gamma \alpha}$ if $\gamma \geq \beta \geq \alpha$.
This data will be called an ${\mathfrak mathcal A} $-module. If each ${\mathfrak mathcal M}_\alpha$ is
finitely generated over ${\mathfrak mathcal A}_\alpha$, we call the resulting
${\mathfrak mathcal A}$-module locally finitely generated.
${\mathfrak mathcal A}$-modules (resp.\ locally finitely generated ${\mathfrak mathcal A}$-modules) form
an abelian category which we denote ${\mathfrak mathcal A} \text{\rm -Mod}$ (resp. ${\mathfrak mathcal A}\text{\rm -Mod}_f$).
Notice that ${\mathfrak mathbb G}amma$ can be viewed as a topological space where $\beta
$ is in the closure of $\alpha $ iff $\beta \geq \alpha$. So the
subsets $[\beta]= \{ \alpha \mathfrak mid \alpha \leq \beta\}$ are the
irreducible open subsets in ${\mathfrak mathbb G}amma$. Then ${\mathfrak mathcal A}$ induces a sheaf of
rings on this topological space, so that the corresponding category of
sheaves of modules is equivalent to ${\mathfrak mathcal A} \text{\rm -Mod}$.
We call $({\mathfrak mathbb G}amma ,{\mathfrak mathcal A} )$ a graded ringed quiver if rings ${\mathfrak mathcal A}
_{\alpha}$ are graded and $\phi _{\beta \alpha}$ are morphisms of
graded rings. In this case let ${\mathfrak mathcal A} \mathfrak mod$ (resp. ${\mathfrak mathcal A}\text{\rm -mod}_{\text{\it f}}$) denote
the abelian category of graded ${\mathfrak mathcal A}$-modules (resp. locally finitely
generated graded ${\mathfrak mathcal A}$-modules) with morphisms of degree zero.
\begin{rmk} The category ${\mathfrak mathcal A}\mathfrak mod$ can also be described as graded modules over the
quiver algebra $R = R_{{\mathfrak mathbb G}amma,{\mathfrak mathcal A}}$ generated by idempotents
$e_\alpha$, $\alpha\in {\mathfrak mathbb G}amma$ in degree $0$, maps
$\psi_{\beta,\alpha}$, $\beta \ge \alpha$ in degree $1$, together
with all the elements of the rings ${\mathfrak mathcal A}_\alpha$, $\alpha\in {\mathfrak mathbb G}amma$,
and satisfying obvious relations (for instance, for $a\in
{\mathfrak mathcal A}_\gamma$, $\psi_{\beta,\alpha}a = \phi_{\beta,\alpha}(a)$ if
$\gamma = \alpha$, and is zero otherwise). ${\mathfrak mathcal A}\text{\rm -mod}_{\text{\it f}}$ is then the
category of finitely generated $R$-modules.
\end{rmk}
\subsubsection{Co-sheaves of modules on a ringed quiver}
Given a ringed quiver $({\mathfrak mathbb G}amma ,{\mathfrak mathcal A} _{{\mathfrak mathbb G}amma})$, by a {\it co-sheaf}
of ${\mathfrak mathcal A} _{{\mathfrak mathbb G}amma}$-modules we mean the following data: for every
$\alpha \in {\mathfrak mathbb G}amma$ there is given a ${\mathfrak mathcal A} _{\alpha}$-module
${\mathfrak mathcal M}_{\alpha}$ with a morphism $\varphi _{\alpha \beta}\text{\rm co-}lon
{\mathfrak mathcal M}_{\alpha}\to {\mathfrak mathcal M}_{\beta}$ of ${\mathfrak mathcal A} _{\beta}$ modules if $\beta \geq
\alpha$, so that $\varphi _{\beta \gamma}\varphi _{\alpha
\beta}=\varphi _{\alpha \gamma}$ for $\gamma \geq \beta \geq
\alpha$. We call co-sheaves of ${\mathfrak mathcal A} $-modules co-${\mathfrak mathcal A} $-modules and
denote this abelian category by co-${\mathfrak mathcal A}\text{\rm -Mod}$ (resp. co-${\mathfrak mathcal A}\mathfrak mod$ in
the graded case).
\subsection{DG ringed quiver} \langlebel{DG sheaves}
A {\it sheaf of DG algebras} ${\mathfrak mathcal C}={\mathfrak mathcal C} _{{\mathfrak mathbb G}amma}$
on ${\mathfrak mathbb G}amma$ is defined in the same way as a sheaf of rings, except the
stalks ${\mathfrak mathcal C} _{\alpha}$ are DG algebras and morphisms $\phi _{\beta
\alpha}\text{\rm co-}lon {\mathfrak mathcal C} _{\beta}\to {\mathfrak mathcal C} _{\alpha}$ are homomorphisms of DG
algebras. Similarly, a DG ${\mathfrak mathcal C}$-module ${\mathfrak mathcal N}$ is a collection $\{
{\mathfrak mathcal N}_{\alpha}, \psi _{\beta \alpha}\} _{\alpha \leq \beta }$, where
${\mathfrak mathcal N}_{\alpha}$ is a DG module over ${\mathfrak mathcal C} _{\alpha}$ and $\psi _{\beta
\alpha}\text{\rm co-}lon {\mathfrak mathcal N}_{\beta}\to {\mathfrak mathcal N}_{\alpha}$ is a homomorphism of DG
modules over ${\mathfrak mathcal C} _{\beta}$. Denote by ${\mathfrak mathcal C} \text{-DG-Mod}$ the
abelian category of DG ${\mathfrak mathcal C}$-modules. One can define a natural
triangulated category $D(\text{DG-${\mathfrak mathcal C}$})$ which is called the derived
category of DG ${\mathfrak mathcal C}$-modules (see \cite{L}).
As mentioned before it is sometimes convenient to consider ${\mathfrak mathbb G}amma $
as a topological space. Then ${\mathfrak mathcal C}$ induces a sheaf of DG algebras on
this space and DG ${\mathfrak mathcal C}$-modules become sheaves of DG modules over that
sheaf of DG algebras.
\begin{defn} Let ${\mathfrak mathcal M}$ be a sheaf (or a co-sheaf, or a DG-module) on a quiver ${\mathfrak mathbb G}amma$.
a) If $\Phi\subset {\mathfrak mathbb G}amma$ is a locally closed subset (i.e.\ the
difference of two open sets), denote by ${\mathfrak mathcal M} _{\Phi}$ the extension
by zero to ${\mathfrak mathbb G}amma$ of the restriction ${\mathfrak mathcal M} \vert _{\Phi}$. Thus
$({\mathfrak mathcal M} _{\Phi})_\alpha = {\mathfrak mathcal M}_\alpha$ if $\alpha \in \Phi$, and $0$
otherwise. The restriction map $({\mathfrak mathcal M} _{\Phi})_\alpha \to ({\mathfrak mathcal M}
_{\Phi})_\beta$ is the one from ${\mathfrak mathcal M}$ if $\alpha$, $\beta\in \Phi$,
and is zero otherwise.
b) In case ${\mathfrak mathcal M}$ is graded and $k\in {\mathfrak mathbb Z}$ denote by ${\mathfrak mathcal M}\lb k\rb$
the same object
shifted ``down" by $k$, i.e. ${\mathfrak mathcal M}\lb k\rb_i={\mathfrak mathcal M} _{k+1}$.
\end{defn}
\begin{ex} \langlebel{forget the grading}
Let $({\mathfrak mathbb G}amma ,{\mathfrak mathcal A} )$ be a graded ringed quiver. Assume that the
algebras ${\mathfrak mathcal A}_\alpha$ are {\em evenly} graded. We may consider
${\mathfrak mathcal A}$ as a sheaf of DG algebras with zero differential. There is a
natural ``forgetful" exact functor between the corresponding derived
categories
$$\nu \text{\rm co-}lon D({\mathfrak mathcal A}\mathfrak mod)\to D(\text{DG-${\mathfrak mathcal A}$}).$$
Namely, an object
of $D({\mathfrak mathcal A}\mathfrak mod)$ is a complex ${\mathfrak mathcal M}^{\scriptscriptstyle \bullet}$ of graded ${\mathfrak mathcal A}$-modules
(thus it has a double grading), whereas an object of
$D(\text{DG-${\mathfrak mathcal A}$})$ is a single DG ${\mathfrak mathcal A}$-module. We put $\nu
({\mathfrak mathcal M}^{\scriptscriptstyle \bullet})=\oplus _i{\mathfrak mathcal M}^i\lb -i\rb$ as ${\mathfrak mathcal A}$-modules,
with the obvious induced differential. It is easy to see
that $\nu$ is a triangulated grading (\S\ref{triangulated gradings}).
\end{ex}
\subsection{Generalities on tori and toric varieties}
Fix a complex torus $T\simeq ({\mathfrak mathbb C} ^*)^n$ with Lie algebra
$\mathfrak mathfrak{t}$.
The lattice $N=N_T = \operatorname{Hom}({\mathfrak mathbb C}^*, T)$ of
co-characters embeds naturally into $\mathfrak mathfrak{t}$.
Namely, given a group homomorphism $\phi \text{\rm co-}lon {\mathfrak mathbb C}^*\to T$,
the corresponding point in $\mathfrak mathfrak{t}$ is $d\phi
_*(1)$, where $d\phi _* \text{\rm co-}lon {\mathfrak mathbb C} \to \mathfrak mathfrak{t}$ is the induced
map of Lie algebras. Then $\mathfrak mathfrak{t} \text{\rm co-}ng N_{\mathfrak mathbb C} := N \otimes_Z {\mathfrak mathbb C}$.
Note that $N$ is naturally isomorphic to the fundamental
group $\pi _1(T)$: given an element $n\in N$ the corresponding map
$f\text{\rm co-}lon [0,1]\to T$ is defined by the formula
$$f(t)=e^{2\pi i tn}.$$
Clearly this correspondence is functorial with
respect to homomorphisms of tori.
There is also a natural lattice of characters
$M=M_T\subset \mathfrak mathfrak{t}^*$ defined similarly. The abelian groups
$M$ and $N$ are dual to each other: $M = \operatorname{Hom}(N,{\mathfrak mathbb Z})$.
The dual torus $\check T$ is the torus for which
$M_{\check T}=N_T$ and $N_{\check T}=M_T$; it
is isomorphic to $T$, but not canonically.
\subsection{Toric varieties and fans}
Let $X$ be a normal $T$-toric variety. The $T$-orbits $\{O_{\alpha}\}$
in $X$ are
indexed by the cones $\alpha$ in a finite polyhedral fan ${\Sigma}$ in
the vector space $N_{\mathfrak mathbb R} := N \otimes_{\mathfrak mathbb Z} {\mathfrak mathbb R}$ which is rational with respect to the
lattice $N$. If $\alpha \in {\Sigma}$, then
$\operatorname{Span}_{\mathfrak mathbb C}(\alpha) \subset \mathfrak mathfrak{t}$ is
the Lie algebra of the stabilizer $T_\alpha$ of the corresponding
orbit $O_\alpha$ (since $T$ is abelian, the stabilizer can be taken
at any point).
We put the
natural inclusion order on ${\Sigma}$, where
$\alpha \le \beta$ if and only if $\alpha$ is a face of $\beta$.
Then $\alpha \le \beta$ if and only if $O_\beta \subset \overline{O_\alpha}$.
Thus open unions of orbits correspond to subfans of ${\Sigma}$.
More generally we will want to consider locally closed unions of
orbits in $X$. Such a subvariety $Y$ corresponds to a locally closed
subset $\Lambda \subset {\Sigma}$, which satisfies
all the fan properties, except that it need not be closed under
taking faces. Instead if $\alpha \le \beta$ are cones in
$\Lambda$, then $\Lambda$ must contain all faces of $\beta$ which
contain $\alpha$. We call such subsets {\em quasifans}.
\subsection{} \langlebel{toric projections}
Let $X$ be a $T$-toric variety.
For each orbit $O_\alpha$ in $X$ denote by
$$\operatorname{St}(O_\alpha)=\bigcup_{O_\alpha \subset \overline{O_\beta}}O_\beta$$
its star in $X$. Consider the stabilizer $T_\alpha \subset T$ of the
orbit $O_\alpha$. Since $X$ is normal, the group $T_\alpha $ is
connected, and hence is a torus. There exists a non-canonical
homeomorphism
$$\operatorname{St}(O_\alpha)\simeq X^\prime \times (T/T_\alpha) \simeq X' \times
O_\alpha,$$
where $X^\prime $ is an affine $T_\alpha$-toric variety
with a single fixed point.
The action of $T_\alpha $ on $\operatorname{St} (O_\alpha)$ defines a canonical
projection
$$p_\alpha\text{\rm co-}lon \operatorname{St}(O_\alpha)\to O_\alpha,$$
which is compatible with
the product decomposition above.
If $O_\beta \subset \operatorname{St}(O_\alpha)$ we denote by $p_{\beta \alpha}$ the
restriction of $p_{\alpha}$ to $O_\beta$. The collection of
projections $\{p_{\beta \alpha}\}$ is compatible in the sense that
$p_{\beta \alpha}p_{\gamma \beta}=p_{\gamma \alpha}$ wherever all maps
are defined.
\begin{lemma} \langlebel{fund nbds}
Each point in $O_\alpha$ has a fundamental system of distinguished
contractible neighborhoods $U\subset X$, such that $U\cap
\operatorname{St}(O_\alpha)\subset p_\alpha ^{-1}(U\cap O_\alpha)$ and for each
orbit $O_\beta \subset \operatorname{St}(O_\alpha)$ the inclusion $U\cap
O_\beta\hookrightarrow p^{-1}_{\alpha \beta}(U\cap O_\alpha)$ is a
homotopy equivalence.
\end{lemma}
\subsection{The ringed quiver $({\Sigma} ,{\mathfrak mathcal A} _{\Sigma})$}
\langlebel{the sheaf cA}
Let ${\Sigma}$ be a finite polyhedral fan, rational or not, with the
inclusion partial order on faces.
There is a natural graded ringed quiver ${\mathfrak mathcal A} ={\mathfrak mathcal A}
_{\Sigma}$ on ${\Sigma}$: for $\tau \in {\Sigma}$ the stalk ${\mathfrak mathcal A}
_{\tau}$ is the graded ring of complex-valued polynomial functions on
the span of $\tau$. The structure homomorphisms $\phi $ are the
restrictions of functions. We consider linear functions as having
degree $2$, so that ${\mathfrak mathcal A} $ is evenly graded.
Note that the ringed quiver ${\mathfrak mathcal A}$ makes sense even for non-rational
fans (which do not correspond to toric varieties).
\begin{rmk} Notice that the topological space associated to the
partially ordered set ${\Sigma}$ is homeomorphic to the quotient space
$\overline{X}=X/T$. If $Y\subset X$ is $T$-invariant and locally
closed, the space of sections of ${\mathfrak mathcal A}$ on $\overline{Y}$ is canonically
identified with the equivariant cohomology $H^*_T(Y;{\mathfrak mathbb C})$.
This is why this sheaf is useful for studying the equivariant
topology of $X$. Later in \S\ref{equivariant sheaves} we will use the
categories ${\mathfrak mathcal A}_X\text{\rm -mod}_{\text{\it f}}$, DG-${\mathfrak mathcal A}_X$, and their derived categories to
model equivariant sheaves and complexes on $X$.
\end{rmk}
\subsection{The ringed quiver $({\Sigma} ^\circ,{\mathfrak mathcal B} _{\Sigma})$}
Now consider the partially ordered set ${\Sigma} ^\circ$ which is
${\Sigma}$ with the opposite ordering. One may think about ${\Sigma}
^\circ$ as the partially ordered set of orbits of $X$, where $O_\alpha
\leq O_\beta$ iff $O_\alpha\subset \overline{O_\beta}$. There is a
natural sheaf of rings ${\mathfrak mathcal B} _X={\mathfrak mathcal B}$ on ${\Sigma} ^\circ$: for an orbit
$O_\alpha$ take ${\mathfrak mathcal B} _{\alpha}$ to be the group ring ${\mathfrak mathbb C} [\pi
_1(O_\alpha)]$. If $O_\alpha \leq O_\beta$ the canonical projections
$p_{\alpha \beta}\text{\rm co-}lon O_\beta \to O_\alpha$ induce homomorphisms
${\mathfrak mathcal B} _\beta \to {\mathfrak mathcal B} _\alpha$. Thus we obtain a ringed quiver $({\Sigma}
^\circ,{\mathfrak mathcal B})$.
As with the ringed quiver ${\mathfrak mathcal A}$, ${\mathfrak mathcal B}$ can be described entirely
in terms of the fan ${\Sigma}$, without reference to the
toric variety.
For any orbit $O_\alpha$, there is a canonical identification
$\pi_1(O_\alpha) = N_\alpha$, where $N_\alpha$ is the lattice
$N/(N\cap \operatorname{Span}(\alpha))$.
If $\alpha$ is a face of $\beta$, so $\beta \le \alpha$
in ${\Sigma}^\circ$, the homomorphism $\pi_1(O_\alpha) \to \pi_1(O_\beta)$
comes from the natural map $N_\alpha \to N_\beta$.
\subsection{The ringed quiver $({\Sigma} ^\circ, {\mathfrak mathcal T} _{\Sigma})$}
We can define another ringed quiver on ${\Sigma}^\circ$ as
follows. For $\alpha\in {\Sigma}$, let ${\mathfrak mathcal T}_\alpha=\operatorname{Sym}(N_{\alpha,{\mathfrak mathbb C}})$.
We consider it as a graded
polynomial algebra, where elements of $N_{\alpha,{\mathfrak mathbb C}}$ have
degree $2$. If $\alpha$ is a face of $\beta$,
the homomorphism ${\mathfrak mathcal T}_\alpha \to {\mathfrak mathcal T}_\beta$
comes from the natural map $N_\alpha \to N_\beta$.
Notice that $N_{\alpha,{\mathfrak mathbb C}}$
is canonically isomorphic to the Lie algebra of $T/T_\alpha$,
which can be canonically identified with $O_\alpha$.
Thus our graded ringed quiver can be described
more geometrically: if $O_\alpha \leq O_\beta$,
the canonical projections
$p_{\alpha \beta}\text{\rm co-}lon O_\beta \to O_\alpha$ induce morphisms of tori
$T/T_\beta \to T/T_\alpha$, hence they induce homomorphisms of graded
polynomial algebras ${\mathfrak mathcal T} _\beta \to {\mathfrak mathcal T} _\alpha$.
\subsection{Dual affine toric varieties} \langlebel{dual toric varieties}
Let $X$ be an affine $T$-toric variety with a single fixed point. The
corresponding fan ${\Sigma} = {\Sigma} _X$ consists of a single full-dimensional cone
${\sigma}ma = {\sigma}ma_X\subset N_{T,{\mathfrak mathbb R}}$ together with its faces.
We have the dual cone $\check{\sigma}ma$ in the dual vector
space $M_{T,{\mathfrak mathbb R}} = (N_{T,{\mathfrak mathbb R}})^*$, defined by
\[\check{\sigma}ma = \{y \in M_{T,{\mathfrak mathbb R}} \mathfrak mid \langle x, y\rangle \ge 0\;\text{for all}\;x\in {\sigma}.\]
Let $\check T$ be the dual torus to $T$; then
$N_{\check T, {\mathfrak mathbb R}} = (N_{T,{\mathfrak mathbb R}})^*$ canonically.
\begin{defn}
The dual toric variety $\check X$ to $X$ is the
affine $\check T$-toric variety defined by the fan
${\Sigma}^\vee$ consisting of $\check {\sigma}$ and all its
faces, with respect to the lattice $M_T = N_{\check T}$.
In other words, we have $\check {\sigma}ma _X={\sigma}ma_{\check X}$.
\end{defn}
There is an order-reversing isomorphism $\alpha \mathfrak mapsto \alpha^\bot$
between ${\Sigma}$ and ${\Sigma}^\vee$, defined by $\alpha^\bot =
{\sigma}^\vee \cap \operatorname{Span}(\alpha)^\bot$. In particular we have
$\operatorname{Span}(\alpha^\bot) = \operatorname{Span}(\alpha)^\bot$.
This map gives
an identification ${\Sigma}^\vee = {\Sigma}^\circ$ of partially ordered sets.
With respect to this identification, the
ringed quivers $({\Sigma},{\mathfrak mathcal A}_{\Sigma})$ and
$((\check{\Sigma})^\circ,{\mathfrak mathcal T} _{\check{\Sigma}})$ are identical.
This will be important
for the definition of our equivariant-constructible
duality in \S\ref{main proofs}.
\section{Locally constant sheaves on toric varieties}
\subsection{Some lemmas about sheaves on toric varieties}
For a topological space $Y$ denote by $\operatorname{Sh}(Y)$ the abelian category of
sheaves of complex vector spaces on $Y$.
Let $X$ be a normal toric variety. We consider $X$ as a topological
space in the classical topology. Let $Z$ be a $T$-invariant subspace
of $X$. Denote by $LC(Z)\subset \operatorname{Sh}(Z)$ the full subcategory of
sheaves which are locally constant on each orbit.
Consider the full subcategory $D^b_{LC}(\operatorname{Sh}(Z))$ of the bounded
derived category $D^b(\operatorname{Sh}(Z))$, consisting of complexes with
cohomologies in $LC(Z)$.
Fix an orbit $O_\alpha \subset X$ and choose a locally closed
$T$-invariant subset $W\subset \operatorname{St}(O_\alpha)$, which contains
$O_\alpha$. Denote by $i\text{\rm co-}lon O_\alpha \hookrightarrow W$ the
corresponding closed embedding. Let $j\text{\rm co-}lon U\hookrightarrow W$ be
the complementary open embedding of $U=W-O_\alpha$. Denote by $q\text{\rm co-}lon
W\to O_\alpha$, $p\text{\rm co-}lon U\to O_\alpha$ the restrictions of the
projection $p_\alpha$ to $W$ and $U$ respectively.
\begin{lemma} \langlebel{a lemma}
In the above notation the functors ${\bf R} q_*$ and $i^*$ from
$D^b_{LC}(W)$ to $D^b_{LC}(O_\alpha)$ are naturally isomorphic. In
particular, the functors ${\bf R} p_*$ and $i^*{\bf R} j_*$ from
$D^b_{LC}(U)$ to $D^b_{LC}(O_\alpha)$ are naturally isomorphic.
Hence, the functors $p_*$ and $i^*j_*$ from $LC(U)$ to
$LC(O_\alpha)$ are naturally isomorphic.
\end{lemma}
\begin{proof} Using distinguished neighborhoods of points in
$O_\alpha$ (Lemma \ref{fund nbds}) we see that there exists a
natural morphism of functors ${\bf R} q_*\to i^*$. Let us show that
it is an isomorphism.
The category $D^b_{LC}(W)$ is the triangulated envelope of objects
${\bf R} j_{\beta *}L$, where $j_\beta \text{\rm co-}lon O_\beta
\hookrightarrow W$ is the embedding of an orbit $O_\beta$ and $L$ is
an object in $LC(O_\beta)$. So we may assume that $U=O_\beta$ and
it suffices to show that $i^*{\bf R} j_*L={\bf R}p_*L$ (the case
$\alpha =\beta$ is clear).
Choose a distinguished neighborhood $V\subset X$ of a point in
$O_\alpha$. Then the complex ${\mathfrak mathbb G}amma (V\cap O_\alpha, i^*{\bf R}
j_*L)$ is quasi-isomorphic to the complex ${\bf R} {\mathfrak mathbb G}amma (V\cap
O_\beta,L)$. But the inclusion $V\cap O_\beta \subset p^{-1}(V\cap
O_\alpha)$ is a homotopy equivalence, hence it induces a
quasi-isomorphism ${\bf R} {\mathfrak mathbb G}amma (p^{-1}(V\cap O_\alpha),L)\simeq
{\bf R} {\mathfrak mathbb G}amma (V\cap O_\beta,L)$. This proves that $i^*{\bf R}
j_*L={\bf R}p_*L$.
The last statement now follows by taking $H^0$.
\end{proof}
\subsection{Equivalence of derived categories}
\begin{thm} \langlebel{D^b(LC)}
The natural functor $D^b(LC(X))\to D^b_{LC}(\operatorname{Sh}(X))$ is an
equivalence.
\end{thm}
\begin{proof} Let $i=i_O\text{\rm co-}lon O\hookrightarrow X$ be the
(locally closed) embedding of an orbit $O$. For $F\in LC(O)$ we may
consider two different (derived) direct images of $F$ under the
embedding $i$: one in the category $D^b_{LC}(\operatorname{Sh}(X))$, denoted as
usual by ${\bf R}i_*F$, and the other in the category $D^b(LC(X))$,
which we denote by ${\bf R}_{LC}i_*F$. It is clear that the category
$D^b_{LC}(\operatorname{Sh}(X))$ (resp. $D^b(LC(X))$) is the triangulated envelope
of the objects ${\bf R}i_*F$ (resp. ${\bf R}_{LC}i_*F$) for various
orbits $O$ and locally constant sheaves $F$ on them. So it suffices
to prove the following two claims.
\mathfrak medskip
\noindent{\it Claim 1.} The complexes ${\bf R}i_*F$ and ${\bf R}_{LC}i_*F$ are
quasi-isomorphic.
\mathfrak medskip
\noindent{\it Claim 2.} Let $i$ and $F$ be as above, $j\text{\rm co-}lon O^\prime \hookrightarrow X$
be the embedding of an orbit and $G\in LC(O^\prime)$. Then
$$\operatorname{Ext} ^{\scriptscriptstyle \bullet}_{D^b(LC(X))}({\bf R}_{LC}j_*G,{\bf R}_{LC}i_*F)= \operatorname{Ext}
^{\scriptscriptstyle \bullet}_{D^b_{LC}(\operatorname{Sh}(X))}({\bf R}_{LC}j_*G,{\bf R}_{LC}i_*F).$$
\mathfrak medskip
Let us prove the second claim first. Using the adjunction we need to
prove that
$$\operatorname{Ext} ^{\scriptscriptstyle \bullet}_{D^b(LC(O))}(i^*{\bf R}_{LC}j_*G,F)= \operatorname{Ext}
^{\scriptscriptstyle \bullet}_{D^b_{LC}(\operatorname{Sh}(O))}(i^*{\bf R}_{LC}j_*G,F).$$
By devissage this
is a consequence of the following lemma.
\begin{lemma} Let $Y$ be a $K(\pi ,1)$-space, $LC(Y)$ -- the category of locally constant
sheaves on $Y$. Then for any $A,B\in LC(Y)$
$$\operatorname{Ext} ^{\scriptscriptstyle \bullet}_{LC(Y)}(A,B)=\operatorname{Ext} ^{\scriptscriptstyle \bullet} _{\operatorname{Sh}(Y)}(A,B).$$
\end{lemma}
\begin{proof} Let $f\text{\rm co-}lon \tilde{Y}\to Y$ be the universal covering map. Then the functor $f^*$
establishes an equivalence of abelian categories
$$f^*\text{\rm co-}lon \operatorname{Sh}(Y)\rightarrow \operatorname{Sh} _{\pi}(\tilde{Y}),$$
where
$\operatorname{Sh}_{\pi}(\tilde{Y})$ is the category of $\pi$-equivariant sheaves
on $\tilde{Y}$ \cite{Gr}. Clearly $f^*$ preserves locally constant
sheaves.
\begin{rmk}
It is well known that the category $\operatorname{Sh}_{\pi ,LC}(\tilde{Y})$ of
locally constant (=constant) $\pi $-equivariant sheaves on
$\tilde{Y}$ is equivalent to the category of $\pi$-modules. The
equivalence is provided by the functor of global sections ${\mathfrak mathbb G}amma$.
\end{rmk}
Put $\tilde{A}=f^*A$, $\tilde{B}=f^*B$. We will prove that
$\operatorname{Ext}^{\scriptscriptstyle \bullet}_{\operatorname{Sh}_{\pi, LC}(\tilde{Y})}(\tilde{A},\tilde{B})= \operatorname{Ext}
^{\scriptscriptstyle \bullet}_{\operatorname{Sh}_{\pi}(\tilde{Y})}(\tilde{A},\tilde{B}).$ Choose an
injective resolution
$$\tilde{B}\to I^0\to I^1\to ...$$
in the category
$\operatorname{Sh}_{\pi,LC}(\tilde{Y})$. It suffices to prove that
$\operatorname{Ext}^k_{\operatorname{Sh}_{\pi}(\tilde{Y})}(\tilde{A},I^t)=0$ for any $t$ and
$k>0$. Put $I=I^t$ and choose a resolution $0\to I\to J^0\to
J^1\to...$, where $J$'s are injective objects in
$\operatorname{Sh}_{\pi}(\tilde{Y})$. So
$$\operatorname{Ext} ^k_{\operatorname{Sh}_{\pi}(\tilde{Y})}(\tilde{A},I)= H^k(\operatorname{Hom}
^{\scriptscriptstyle \bullet}_{\operatorname{Sh}_{\pi}(\tilde{Y})}(\tilde{A},J^{\scriptscriptstyle \bullet})).$$
Notice that $I$, as a sheaf, is constant and each $J^s$, as a sheaf,
is injective \cite{Gr}. Hence the complex of global sections
$$0\to {\mathfrak mathbb G}amma (I)\to {\mathfrak mathbb G}amma (J^0)\to {\mathfrak mathbb G}amma (J^1)\to ...$$
is exact
($\tilde{Y}$ is contractible). Since $\tilde{A}$, as a sheaf, is also
constant, the complex
$$0\to \operatorname{Hom} _{\operatorname{Sh}_{\pi}(\tilde{Y})}(\tilde{A},I)\to
\operatorname{Hom}_{\operatorname{Sh}_{\pi}(\tilde{Y})}(\tilde{A},J^0)\to...$$
is isomorphic to the complex
$$0\to \operatorname{Hom} _{\pi}({\mathfrak mathbb G}amma (\tilde{A}),{\mathfrak mathbb G}amma (I))\to \operatorname{Hom}
_{\pi}({\mathfrak mathbb G}amma (\tilde{A}),{\mathfrak mathbb G}amma (J^0))\to ...$$
The $\pi$-module
${\mathfrak mathbb G}amma (I)$ is injective and so are the $\pi$-modules ${\mathfrak mathbb G}amma (J^s)$
for all $s\geq 0$. Hence the last complex is exact. This proves the
lemma and Claim 2.
\end{proof}
Let us prove Claim 1. Choose a locally constant sheaf $I$ on $O$ which
is injective in the category $LC(O)$. It suffices to prove that the
complex of sheaves ${\bf R}i_*I$ is acyclic except in degree zero.
Choose an orbit $O_\alpha \subset \overline{O}$ and let $p\text{\rm co-}lon O\to
O_\alpha$ be the canonical projection. Fix a distinguished
neighborhood $U\subset X$ of a point in $O_\alpha$ (Remark \ref{fund
nbds}). By Lemma \ref{a lemma} above the complex ${\mathfrak mathbb G}amma (U\cap
O_\alpha,{\bf R}i_*I)$ is quasi-isomorphic to the complex
$${\bf R}{\mathfrak mathbb G}amma ( p^{-1}(U\cap O_\alpha), I)= {\bf R}\operatorname{Hom}
^{{\scriptscriptstyle \bullet}}({\mathfrak mathbb C} _{p^{-1}(U\cap O_\alpha)}, I\vert _{p^{-1}(U\cap
O_\alpha)}).$$
Since the space $p^{-1}(U\cap O_\alpha)$ is $K(\pi
,1)$, and the restriction of the local system $I$ to $p^{-1}(U\cap
O_\alpha)$ remains injective, it follows from the above lemma that the
complex ${\bf R}\operatorname{Hom} ^{{\scriptscriptstyle \bullet}}({\mathfrak mathbb C} _{p^{-1}(U\cap O_\alpha)}, I\vert
_{p^{-1}(U\cap O_\alpha)})$ is acyclic in positive degrees. This
proves Claim 1 and the theorem.
\end{proof}
\begin{rmk} The key property of toric varieties which is used in the proof of the above
theorem is that the star of an orbit is homotopy equivalent to the
orbit itself. For example, the analogue of the above theorem does
not hold for ${\mathfrak mathbb P} ^1$ which is stratified by two cells: ${\mathfrak mathbb C} $ and
a point.
\end{rmk}
The category $LC(X)$ has enough injectives: injective objects are sums
of objects of the form $i_*I$, where $i\text{\rm co-}lon O \hookrightarrow X$ is
an embedding of an orbit and $I\in LC(O)$ is an injective local
system. Furthermore, $LC(X)$ has finite cohomological dimension, so
objects in $D^b(LC(X))$ can be represented by bounded complexes of
injectives.
Thus if $j\text{\rm co-}lon Y\hookrightarrow X$ is an embedding of a locally
closed $T$-invariant subspace, we can take derived functors of $j_*$
and ${\mathfrak mathbb G}amma_Y$ (sections with support in $Y$), giving functors
${{\mathfrak mathbf R}}_{LC}j_*\text{\rm co-}lon D^b(LC(Y)) \to D^b(LC(X))$ and
${{\mathfrak mathbf R}}_{LC}{\mathfrak mathbb G}amma_Y\text{\rm co-}lon D^b(LC(X)) \to D^b(LC(X))$. Define
$j^!_{LC} = j^*{{\mathfrak mathbf R}}_{LC}{\mathfrak mathbb G}amma_Y$.
On the other hand, the usual derived functors restrict to functors
${{\mathfrak mathbf R}}j_*\text{\rm co-}lon D^b_{LC}(Y) \to D^b_{LC}(X)$ and
${{\mathfrak mathbf R}}{\mathfrak mathbb G}amma_Y\text{\rm co-}lon D^b_{LC}(X) \to D^b_{LC}(X)$, and we have
$j^! = j^*{\mathfrak mathbf R}{\mathfrak mathbb G}amma_Y$. The following corollary to the proof of Theorem
\ref{D^b(LC)} will be used later when we discuss the intersection
cohomology sheaves.
\begin{cor} \langlebel{R vs R_LC}
The functors ${{\mathfrak mathbf R}}_{LC}j_*$ and ${\mathfrak mathbf R} j_*$ are isomorphic under
the equivalence of Theorem \ref{D^b(LC)}, as are $j^!_{LC}$ and
$j^!$.
\end{cor}
\begin{proof}
Since $D^b(LC(X))$ is generated by injective objects of $LC(X)$, for
the first claim it will be enough to consider $i_*I$, where $i\text{\rm co-}lon
O \hookrightarrow Y$ is the inclusion of an orbit and $I$ is
injective in $LC(O)$. Then ${\mathfrak mathbf R}_{LC}j_*(i_*I) = j_*i_*I = (j\circ
i)_*I\simeq R_{LC}(j\circ i)_*I$, since $i_*I$ is injective. On the
other hand, we have
\[Rj_*(i_*I) = Rj_*R_{LC}i_*I \simeq Rj_*Ri_*I \simeq R(j\circ i)_*I ,\]
using Claim 1 from the proof of Theorem \ref{D^b(LC)}. Applying Claim
1 once more gives ${\mathfrak mathbf R}_{LC}j_*(i_*I) \simeq Rj_*(i_*I)$.
For the second part, let $i\text{\rm co-}lon O\hookrightarrow X$ be the inclusion
of an orbit, and let $I\in LC(O)$ be injective; we will show that
$j^!(i_*I)$ and $j^!_{LC}(i_*I)$ are quasi-isomorphic. The inclusion
$j$ can be factored as the composition of an open embedding and a
closed embedding. The required isomorphism is obvious when $Y$ is
open, so we can assume that $Y$ is closed. If $Y$ contains
$\overline{O}$, then $j^!i_*I = j^*i_*I = j^!_{LC}i_*I$. Otherwise,
we have $j^!_{LC}(i_*I) = j^*{\mathfrak mathbb G}amma_Y(i_*I) = 0$, since $i_*I$ is
injective and all nonzero sections of $i_*I$ must contain points of
$O$ in their support. On the other hand, $i_*I \simeq {\mathfrak mathbf R} i_*I$, so
$j^!(i_*I) = 0$ as well. This completes the proof.
\end{proof}
\subsection{Quiver description of the category $LC(X)$}
Recall the ringed quiver $({\Sigma} ^\circ,{\mathfrak mathcal B})$ associated with the toric
variety $X$. We are going to define a functor
$$\eta \text{\rm co-}lon LC(X)\to \text{\rm co-}{\mathfrak mathcal B}\text{\rm -Mod}.$$
For this we need to recall how to
glue sheaves on topological spaces. Surely this construction is well
known, but we do not know a reference.
Let $Y$ be a topological space, $i\text{\rm co-}lon Z\hookrightarrow Y$ the
embedding of a closed subset and $j\text{\rm co-}lon U=Y-Z\hookrightarrow Y$ the
complementary open embedding. Consider the abelian category $\operatorname{Sh}(Y,Z)$
consisting of triples $(G,H,\xi)$, where $G\in \operatorname{Sh}(Z)$, $H\in \operatorname{Sh}(U)$
and $\xi $ is a morphism of sheaves in $\operatorname{Sh}(Z)$ $\xi\text{\rm co-}lon G\to
i^*j_*H$. We have a natural functor $\tau \text{\rm co-}lon \operatorname{Sh}(Y)\to \operatorname{Sh}(Y,Z)$
which associates to a sheaf $F\in \operatorname{Sh}(Y)$ its restrictions $i^*F\in
\operatorname{Sh}(Z)$, $j^*F\in \operatorname{Sh}(U)$ and the pullback under $i^*$ of the
adjunction morphism $F\to j_*j^*F$.
\begin{lemma} The functor $\tau$ is an equivalence.
\end{lemma}
\begin{proof} Let us define the inverse functor $\eta \text{\rm co-}lon \operatorname{Sh}(Y,Z)\to
\operatorname{Sh}(Y)$. Given $(G,H,\xi)\in \operatorname{Sh}(Y,Z)$ define a presheaf
$\overline{F}$ on $Y$ as follows. For an open subset $V\subset Y$
put $\overline{F}(V)=H(V)$ if $V\subset U$. Otherwise set
$$\overline{F}(V)=\{(g,h)\in G(V\cap Z)\times H(V\cap U)\mathfrak mid \xi
(g)=h^\prime \},$$
where $h^\prime $ is the image of $h$ in
$i^*j_*H(V\cap Z)$. Then let $\eta (G,H,\xi)\in \operatorname{Sh}(Y)$ be the
sheafification of $\overline{F}$.
\end{proof}
Let $F\in LC(X)$. Denote the stalk of $F$ at the distinguished point
in an orbit $O_\alpha$ by $F_\alpha$. Then $F_\alpha $ is a ${\mathfrak mathcal B}
_\alpha $-module. Given two orbits $O_\alpha \subset
\overline{O_\beta}$ consider the canonical projection $p = p_{\beta
\alpha}\text{\rm co-}lon O_\beta \to O_\alpha$. By Lemma 6.7 and Lemma 6.1 the
restriction of the sheaf $F$ to the union of the two orbits defines a
morphism of sheaves
$$F\vert _{O_\alpha}\to p_*(F\vert _{O_\beta}),$$
or equivalently a
morphism of sheaves
$$p^{-1}(F\vert _{O_\alpha})\to (F\vert _{O_\beta}).$$
Such a morphism
is equivalent to a homomorphism of ${\mathfrak mathcal B} _\beta$-modules $F_\alpha\to
F_\beta$. So the sheaf $F$ defines a co-${\mathfrak mathcal B}$-module. This is our
functor
$$\eta \text{\rm co-}lon LC(X)\to \text{\rm co-}{\mathfrak mathcal B}\text{\rm -Mod}.$$
\begin{thm} The functor $\eta$ is an equivalence.
\end{thm}
For example, in case the toric variety $X$ is the affine line with two
orbits, ${\mathfrak mathbb C} ^*$ and the origin, an object in $LC(X)$ is the same as
a vector space $P$, a ${\mathfrak mathbb C} [\pi _1 ({\mathfrak mathbb C} ^*)]$-module $Q$ and a linear
map $P\to Q^{\pi_1({\mathfrak mathbb C}^*)}$.
\begin{proof} We will prove the theorem by induction on the number of
orbits in $X$. For one orbit the statement of the theorem is a well
known equivalence between the category of locally constant sheaves
and that of $\pi _1$-modules.
Now we proceed with the induction step. Pick an orbit $O_\alpha
\subset X$ of smallest dimension. We may assume that
$X=\operatorname{St}(O_\alpha)$. Indeed, otherwise $X$ may be covered by open
$T$-invariant subsets $V$, which are strictly smaller than $X$. By
induction, the theorem is true for each $V$ and so we obtain the
equivalence for $X$ by gluing the corresponding equivalences
$LC(V)\simeq \text{\rm co-}{\mathfrak mathcal B} _V\text{\rm -Mod}$.
Put $U=X-O_\alpha$ and let $j\text{\rm co-}lon U\hookrightarrow X$ and $i\text{\rm co-}lon
O_\alpha \hookrightarrow X$ be the open and closed embeddings
respectively. By Lemma 6.7 a sheaf $F\in LC(X)$ is the same as a
triple $(G,H,\xi)$, where $G\in LC(O_\alpha)$, $H\in LC(U)$ and $\xi
\text{\rm co-}lon G\to i^*j_*H$. By Lemma 6.1 $i^*j_*H=p_{\alpha *}H$. Thus by
adjunction the morphism $\xi$ is the same as a morphism $\iota
\text{\rm co-}lon p_\alpha^{-1}G\to H$. Let $G_\alpha$ be the ${\mathfrak mathcal B}
_\alpha$-module corresponding to $G$. It is easy to see that the
sheaf $p_\alpha^{-1}G$ considered as a co-${\mathfrak mathcal B} _U$-module is the
constant one equal to $G_\alpha$. Thus the triple $(G,H,\iota)$ is
the same as a co-${\mathfrak mathcal B}$-module. This proves the theorem.
\end{proof}
\subsection{Unipotent sheaves}
\begin{defn} A sheaf $F\in LC(X)$ is called {\it unipotent} if for each
orbit $O_\alpha$ and $x\in \pi_1(O_\alpha)$ the action of the
operator $x-1$ on the stalk $F_\alpha$ of $F$ at a point of
$O_\alpha$ is locally nilpotent. It is called {\it co-finite} if in
addition the space of invariants $F_\alpha ^{\pi _1(O_\alpha)}$ is
finite-dimensional for all $\alpha$. Let $LC_{u}(X)$ and
$LC_{cf}(X)$ be the full subcategories of $LC(X)$ consisting of
unipotent (resp.\ co-finite) sheaves.
The next result describes the corresponding subcategories of
$\text{\rm co-}{\mathfrak mathcal T}\text{\rm -Mod}$ under the equivalence $\eta$. Let
$\text{\rm co-}{\mathfrak mathcal T}\text{\rm -Mod}_{n}$ be the full subcategory of co-${\mathfrak mathcal T}$-modules
${\mathfrak mathcal M}$ which are ``supported at the origin'', i.e.\ for which every
$m\in {\mathfrak mathcal M}_\alpha$ is annihilated by some power of the homogeneous
maximal ideal $\mathfrak m_\alpha\subset {\mathfrak mathcal T} _\alpha$. Let $\text{\rm co-}{\mathfrak mathcal T}\text{\rm -Mod}_{cf}$
be the further full subcategory of modules ${\mathfrak mathcal M}$ for which each
${\mathfrak mathcal M}_\alpha$ is a cofinite ${\mathfrak mathcal T}_\alpha$-module (\S\ref{module conventions}).
In other words, in addition to being supported at the origin,
for each $\alpha$ the space
$\{m\in {\mathfrak mathcal M}_\alpha\mathfrak mid \mathfrak m_\alpha \cdot m = 0\}$ should be finite dimensional.
\end{defn}
\begin{thm} \langlebel{cofinite LC sheaves}
The functor $\eta$ restricts to give equivalences of full
abelian subcategories
$$LC_u(X)\simeq \text{\rm co-}{\mathfrak mathcal T}\text{\rm -Mod} _n,\;\text{and}$$
$$LC_{cf}(X)\simeq \text{\rm co-}{\mathfrak mathcal T}\text{\rm -Mod}_{\text{\it cf}}.$$
\end{thm}
\begin{proof} Take a sheaf $F \in LC_u(X)$, and let
${\mathfrak mathcal M} = \eta(F) \in \text{\rm co-}{\mathfrak mathcal B}\text{\rm -Mod}$ be the corresponding $\text{\rm co-}{\mathfrak mathcal B}$-module.
Since the action of any
$x\in \pi _1(O_\alpha)\text{\rm co-}ng N_\alpha$ on ${\mathfrak mathcal M}_\alpha$ is
unipotent, the action of the power series
$\frac{1}{2\pi i}\ln x$ is well-defined.
We can extend this uniquely to a map
$v \mathfrak mapsto \frac{1}{2\pi i}\ln v$
from $N_{\alpha} \otimes {\mathfrak mathbb C}$ to $\mathfrak mathop{\rm End}\nolimits({\mathfrak mathcal M}_\alpha)$.
Any two of these operators commute, since $\pi_1(O_\alpha)$ is abelian.
This gives ${\mathfrak mathcal M}_\alpha$ the structure of a $\operatorname{Sym}(N_{\alpha}\otimes{\mathfrak mathbb C}) =
{\mathfrak mathcal T}_\alpha$-module, and in fact makes ${\mathfrak mathcal M}$ into a co-${\mathfrak mathcal T}$-module.
The resulting co-${\mathfrak mathcal T}$-module is clearly supported at the origin.
Conversely, given a co-${\mathfrak mathcal T}$-module ${\mathfrak mathcal M}$ supported at the origin,
we can exponentiate the action of elements of $N_\alpha$
to get an action of $\pi_1(O_\alpha)$ on ${\mathfrak mathcal M}_\alpha$. These
actions combine to give the structure of a $\text{\rm co-}{\mathfrak mathcal B}$-module
on ${\mathfrak mathcal M}$, and then applying $\eta^{-1}$ gives the required object
in $LC_u(X)$.
The second equivalance follows immediately, since the maximal
ideal $\mathfrak m_\alpha \subset {\mathfrak mathcal T}_\alpha$ is generated by
$\frac{1}{2\pi i}\ln x, x\in \pi_1(O_\alpha)$.
\end{proof}
\section{Mixed locally constant sheaves}
\subsection{pre-${\mathfrak mathcal F}$-sheaves}
For toric varieties the Frobenius endomorphism has a natural lift to
characteristic zero -- see \cite{We}. We will use it to define a
mixed version of the
category $LC(X)$.
Consider the group homomorphism $\phi \text{\rm co-}lon T\to T$, $a\mathfrak mapsto a^2$.
For any toric variety $X$ the homomorphism $\phi$ extends uniquely to
a morphism ${\mathfrak mathcal F}={\mathfrak mathcal F}_X\text{\rm co-}lon X\to X$. Namely, recall that each orbit
$O_\alpha$ is identified with the quotient torus $T/T_\alpha$; then
the map ${\mathfrak mathcal F}\text{\rm co-}lon O_\alpha \to O_\alpha$ is again squaring. The maps
$\phi$ and ${\mathfrak mathcal F}$ have degree $2^n$.
\begin{defn} A pre-${\mathfrak mathcal F}$-sheaf is a pair
$(F,\theta)$, where $F\in LC(X)$ and $\theta $
is an isomorphism
$$\theta \text{\rm co-}lon {\mathfrak mathcal F} ^{-1}F\to F .$$
\end{defn}
Let us describe the inverse image functor ${\mathfrak mathcal F} ^{-1}\text{\rm co-}lon LC(X)\to
LC(X)$ in terms of co-${\mathfrak mathcal B}$-modules. The map ${\mathfrak mathcal F}$ induces the
endomorphism of the sheaf ${\mathfrak mathcal B}$, where each element $x\in
\pi_1(O_\alpha)$ maps to $x^2$. Denote this endomorphism $\psi \text{\rm co-}lon
{\mathfrak mathcal B} \to {\mathfrak mathcal B}$. Fix $F\in LC(X)$ and let ${\mathfrak mathcal M}$ be the corresponding
co-${\mathfrak mathcal B}$-module. Then the sheaf ${\mathfrak mathcal F} ^{-1}F$ corresponds to the
co-${\mathfrak mathcal B}$-module $\psi _*{\mathfrak mathcal M}$, i.e. it is obtained from ${\mathfrak mathcal M}$ by
restriction of scalars via $\psi$. So the isomorphism $\theta \text{\rm co-}lon
\psi_*{\mathfrak mathcal M}\to {\mathfrak mathcal M} $ corresponding to the isomorphism $\theta \text{\rm co-}lon {\mathfrak mathcal F}
^{-1}F\to F$ amounts to a compatible system of linear maps $\theta
_\alpha \text{\rm co-}lon {\mathfrak mathcal M}_\alpha \to {\mathfrak mathcal M}_\alpha $ such that for $x\in \pi
_1(O_\alpha)$, $m\in {\mathfrak mathcal M}_\alpha$
$$\theta _\alpha(x^2m)=x\theta _\alpha (m).$$
\subsection{${\mathfrak mathcal F}$-sheaves and graded co-finite co-${\mathfrak mathcal T}$-modules}
\langlebel{F-sheaves}
\begin{defn} A pre-${\mathfrak mathcal F}$-sheaf $(F,\theta)$
is called an ${\mathfrak mathcal F}$-{\it sheaf} if $F$ is co-finite (unipotent) and
for each $\alpha$ the endomorphism $\theta _\alpha \text{\rm co-}lon F_\alpha
\to F_\alpha $ is diagonalizable with eigenvalues $2^{n/2}$, $n\in
{\mathfrak mathbb Z}$. We will refer to ${\mathfrak mathcal F}$-sheaves
on a single orbit as ``${\mathfrak mathcal F}$-local systems''.
Denote by $LC_{\mathfrak mathcal F}(X)$ the category of ${\mathfrak mathcal F}$-sheaves on $X$.
\end{defn}
For any $n\in {\mathfrak mathbb Z}$ define an automorphism $\langle n \rangle$ of $LC_{\mathfrak mathcal F}(X)$
by $(F,\theta) \mathfrak mapsto (F,2^{n/2}\theta)$.
Let $(F,\theta)$ be an ${\mathfrak mathcal F}$-sheaf, and take $x\in \pi_1(O_\alpha)$.
The relation
$$\theta \cdot x^2=x\cdot \theta$$
in $\mathfrak mathop{\rm End}\nolimits(F_\alpha)$ is equivalent
to
$$\theta \cdot 2(\frac{1}{2\pi i}\ln x)=(\frac{1}{2\pi i}\ln x)\cdot
\theta.$$
Thus $F$ considered as the co-${\mathfrak mathcal T}$-module ${\mathfrak mathcal M}$ via Theorem
\ref{cofinite LC sheaves} is graded:
$${\mathfrak mathcal M}_k=\{ a\in {\mathfrak mathcal M}\mathfrak mid \theta (a)=2^{-k/2}a\},$$
and the operators
$\frac{1}{2\pi i}\ln x$ map ${\mathfrak mathcal M}_k$ to ${\mathfrak mathcal M}_{k+2}$.
\begin{defn}
Let co-${\mathfrak mathcal T}\mathfrak mod _n$ denote the category of {\it graded}
co-${\mathfrak mathcal T}$-modules which are supported at the origin, and let
co-${\mathfrak mathcal T}\text{\rm -Mod}_{\text{\it cf}}$ denote the full subcategory of objects
${\mathfrak mathcal M}$ for which ${\mathfrak mathcal M}_\alpha$ is a co-finite graded
${\mathfrak mathcal T}_\alpha$-module for all $\alpha$.
\end{defn}
The following result is an immediate consequence of Theorem
\ref{cofinite LC sheaves} and the above discussion.
\begin{thm} \langlebel{combinatorial F-sheaves}
There is a natural equivalence of abelian
categories
$$LC_{\mathfrak mathcal F}(X)\simeq \text{\rm co-}{\mathfrak mathcal T}\text{\rm -mod}_{\text{\it cf}},$$
and hence an equivalence
$$D^b(LC_{\mathfrak mathcal F}(X))\simeq D^b(\text{\rm co-}{\mathfrak mathcal T}\text{\rm -mod}_{\text{\it cf}}).$$\
Under these equivalences the twist operator $\langle n\rangle$ goes to
the grading shift $\lb n\rb$.
\end{thm}
Note that these isomorphisms and the isomorphisms of Theorem
\ref{cofinite LC sheaves} are compatible with the forgetful functors
$LC_{\mathfrak mathcal F}(X) \to LC_{cf}(X)$ and co-${\mathfrak mathcal T}\text{\rm -mod}_{\text{\it cf}} \to $ co-${\mathfrak mathcal T}\text{\rm -Mod}_{\text{\it cf}}$. This
means that $D^b(LC_{\mathfrak mathcal F}(X)) \to D^b(LC_{cf}(X))$ is a
triangulated grading in the sense of
\S\ref{triangulated gradings}.
\subsection{Simple and injective mixed sheaves} \langlebel{injective mixed sheaves}
Since the category co-${\mathfrak mathcal T}\text{\rm -mod}_{\text{\it cf}}$ has enough injectives, so does
$LC_{\mathfrak mathcal F}(X)$. It will be helpful to have a concrete description of
simple and injective objects in this category.
First consider the case of a single $T$-orbit $O = O_\alpha$.
We have an equivalence
$LC_{\mathfrak mathcal F}(O_\alpha) \simeq \text{\rm co-}{\mathfrak mathcal T}_\alpha\text{\rm -mod}_{\text{\it cf}}$. Up to degree
shifts there is one simple object of $\text{\rm co-}{\mathfrak mathcal T}_\alpha\text{\rm -mod}_{\text{\it cf}}$, namely
$({\mathfrak mathcal T}_\alpha/\mathfrak m_\alpha{\mathfrak mathcal T}_\alpha)^*$. The corresponding object in
$LC_{\mathfrak mathcal F}(O_\alpha)$ is the constant local system ${\mathfrak mathbb C}_{O_\alpha}$, with
${\mathfrak mathcal F}$-structure given by $\theta_\alpha = 1$. We will denote this
${\mathfrak mathcal F}$-local system by ${\mathfrak mathbb C}_\alpha$.
The injective hull of $({\mathfrak mathcal T}_\alpha/\mathfrak m_\alpha{\mathfrak mathcal T}_\alpha)^*$ is
${\mathfrak mathcal T}_\alpha^*$; let $\Theta_\alpha$ denote the corresponding injective
object in $LC_{\mathfrak mathcal F}(O_\alpha)$. It has the following topological
description. Let $q_\alpha\text{\rm co-}lon \widetilde{O}_\alpha \to O_\alpha$ be the
universal cover of $O_\alpha$. Then $\Theta_\alpha$ is the largest
subsheaf of the local system $q_{\alpha
*}{\mathfrak mathbb C}_{\widetilde{O}_\alpha}$ on which all the monodromy operators $x\in
\pi_1(O_\alpha)$ act (locally) unipotently.
Let $b \in O_\alpha$ be the distinguished point. Since there is a canonical
identification $\pi_1(O_\alpha) \text{\rm co-}ng N_\alpha$, we can identify the
stalk $(q_{\alpha *}{\mathfrak mathbb C}_{\widetilde{O}_\alpha})_b$ with the space of functions
$N_\alpha \to {\mathfrak mathbb C}$, at the price of choosing a point $\tilde b \in
q^{-1}(b)$. The action of $x \in \pi_1(O_\alpha)$ on this stalk is
identified with the pushforward by
the translation $\tau_x\text{\rm co-}lon n\mathfrak mapsto n + x$ of the lattice
$N_\alpha$.
The stalk $(\Theta_\alpha)_b$ is thus the space of all functions
$N_\alpha \to {\mathfrak mathbb C}$ which are annihilated by some power of $x - 1$
for every $x \in N_\alpha$. This is the space of polynomial
functions $N_\alpha \to {\mathfrak mathbb C}$. The logarithm of $x$ acts on these
functions as the differential operator $\partial_x$.
We can now make $\Theta_\alpha$ into an ${\mathfrak mathcal F}$-sheaf by letting the
operator $\theta_\alpha$ be the pullback by $N_\alpha \to N_\alpha$,
$x\mathfrak mapsto 2x$. The corresponding grading is just our usual even
grading on polynomial functions. The resulting ${\mathfrak mathcal F}$-sheaf is the
injective hull of ${\mathfrak mathbb C}_\alpha$.
If $X$ has more than one orbit, then up to grading shift the injective
objects of co-${\mathfrak mathcal T}\text{\rm -mod}_{\text{\it cf}}$ are the sheaves ${\mathfrak mathcal T}^*_{\overline \alpha}$ for
$\alpha\in {\Sigma}_X$, where the closure is taken in the fan topology, so
$\overline{\alpha} = \{\beta\in {\Sigma}_X \mathfrak mid \alpha \le \beta\}$.
The corresponding injective objects in $LC_{\mathfrak mathcal F}(X)$ are (up to twists
$\langle n\rangle$) the sheaves $j_{\alpha *} \Theta_\alpha, \alpha\in {\Sigma}_X$
where $j_\alpha\text{\rm co-}lon O_\alpha \to X$ is the inclusion. $j_{\alpha *}
\Theta_\alpha$ is the injective hull of the extension by zero
$j_{\alpha!}{\mathfrak mathbb C}_\alpha$ of ${\mathfrak mathbb C}_\alpha$.
Note that the forgetful functor $F_{cf}\text{\rm co-}lon LC_{\mathfrak mathcal F}(X) \to
LC_{cf}(X)$ preserves injectivity, as does the inclusion
$LC_{cf}(X) \subset LC(X)$. In particular, this implies that
$D^b(LC_{cf}(X))$ is a full subcategory of $D^b(LC(X))$.
\subsection{Extension and restriction functors}
Let $j\text{\rm co-}lon Y \hookrightarrow X$ be the inclusion of a $T$-invariant
locally closed subset of $X$. Since $LC_{\mathfrak mathcal F}(X)$, $LC_{\mathfrak mathcal F}(Y)$ have
enough injectives, we can take derived functors of the left exact
functors $j_*$ and $j^*{\mathfrak mathbb G}amma_Y$ to get functors
${\mathfrak mathbf R} j_* \text{\rm co-}lon D^b(LC_{\mathfrak mathcal F}(Y)) \to D^b(LC_{\mathfrak mathcal F}(X))$ and
$j^!\text{\rm co-}lon D^b(LC_{\mathfrak mathcal F}(X)) \to D^b(LC_{\mathfrak mathcal F}(Y))$. The restriction
and extension by zero functors $j_!$ and $j^*$ are already exact,
so they do not need to be derived.
In the same way we get derived functors between
$D^b(LC(X))$ and $D^b(LC(Y))$. We will denote them by the same symbols
${\mathfrak mathbf R} j_*$, $j_!$, $j^*$, $j^!$; context will make clear which functor is
meant. These functors correspond to the ones on ${\mathfrak mathcal F}$-sheaves:
${\mathfrak mathbf R} j_* F_{cf} = F_{cf} {\mathfrak mathbf R} j_*$, etc. Furthermore, by Corollary
\ref{R vs R_LC}, these functors agree with the usual topological versions.
\subsection{Perverse $t$-structure} \langlebel{perverse t-structure}
These functors satisfy the usual adjuntions and distinguished
triangles which allow one to define perverse $t$-structures; see
\cite{GM} or \cite{Br}.
To do this, define $c(\alpha) = \ranglenk N - \dim \alpha = \dim_{\mathfrak mathbb C}
O_\alpha$ for any $\alpha\in {\Sigma}$, and define full subcategories of
$D = D^b(LC_{\mathfrak mathcal F}(X))$ by
\[D^{\le 0}_{\mathfrak mathcal F}(X) = \{F^{\scriptscriptstyle \bullet} \in D \mathfrak mid H^i(j_\alpha^* F^{\scriptscriptstyle \bullet}) = 0
\;\text{for}\; i > - c(\alpha)\},\]
\[D^{\ge 0}_{\mathfrak mathcal F}(X) = \{F^{\scriptscriptstyle \bullet} \in D \mathfrak mid
H^i({j^!_\alpha} F^{\scriptscriptstyle \bullet}) = 0 \;\text{for}\; i < -c(\alpha)\}.\] The
core $P_{\mathfrak mathcal F}(X) = D^{\le 0}_{\mathfrak mathcal F}(X) \cap D^{\ge 0}_{\mathfrak mathcal F}(X)$ is an abelian
category whose objects will be called perverse ${\mathfrak mathcal F}$-sheaves.
The same formulas define a perverse $t$-structure $(D^{\le 0}_{cf}(X),
D^{\ge 0}_{cf}(X))$ on $D^b(LC_{cf}(X))$. The resulting core of
perverse objects will be denoted $P_{cf}(X)$. The forgetful functor
$F_{cf}\text{\rm co-}lon D^b(LC_{\mathfrak mathcal F}(X))\to D^b(LC_{cf}(X))$ is $t$-exact, so it
restricts to an exact functor $P_{\mathfrak mathcal F}(X)\to P_{cf}(X)$.
Simple objects in $P_{\mathfrak mathcal F}(X)$ and $P_{cf}(X)$ are obtained as usual by
applying the Deligne-Goresky-MacPherson middle extension
$j_{\alpha!*}$ to a simple local system on an orbit $O_\alpha$,
shifted so as to be perverse. In particular, \[L^{\scriptscriptstyle \bullet}_\alpha :=
j_{\alpha!*} {\mathfrak mathbb C}_\alpha[c(\alpha)]\langle -c(\alpha)\rangle\] is simple in
$P_{\mathfrak mathcal F}(X)$, and all simple objects are isomorphic to
$L^{\scriptscriptstyle \bullet}_\alpha\langle n\rangle$ for some $\alpha \in {\Sigma}$, $n\in {\mathfrak mathbb Z}$ (we add
the twist by $-c(\alpha)$ so $L^{\scriptscriptstyle \bullet}_\alpha$ will have weight $0$ in
the mixed structure we define below). Applying the forgetful functor
$F_{cf}$ to $L^{\scriptscriptstyle \bullet}_\alpha$ gives the usual intersection cohomology
sheaf $IC^{\scriptscriptstyle \bullet}(\overline{O_\alpha};{\mathfrak mathbb C})$; these give all the simple objects
of $P_{cf}(X)$.
Note that unlike the usual category of constructible perverse sheaves,
$P_{\mathfrak mathcal F}(X)$ is not artinian, since even if $X$ has only one stratum,
objects like $\Theta_\alpha$ have infinite length. However, Homs are
finite-dimensional in $D^b(LC_{\mathfrak mathcal F}(X))$, and hence in $P_{\mathfrak mathcal F}(X)$.
\begin{prop} \langlebel{lower star is exact}
Let $i\text{\rm co-}lon O \hookrightarrow X$ be the inclusion of an
orbit. Then the functor ${\mathfrak mathbf R} i_* \text{\rm co-}lon D^b(LC_\square(O)) \to
D^b(LC_\square(X))$ is $t$-exact, $\square = {\mathfrak mathcal F},cf$.
\end{prop}
\begin{proof} Suppose that $O=O_\alpha$.
We have ${\mathfrak mathbf R} i_*(D^{\ge 0}_{\mathfrak mathcal F}(O)) \subset D^{\ge 0}_{\mathfrak mathcal F}(X)$
automatically, since $j^!_\beta{\mathfrak mathbf R} i_* = 0$ for any $\beta \ne
\alpha$.
Suppose that $S^{\scriptscriptstyle \bullet} \in D^{\le 0}(LC_{\mathfrak mathcal F}(O))$, and let $M^{\scriptscriptstyle \bullet}
\in D^b({\mathfrak mathcal T}_\alpha\text{\rm -mod}_{\text{\it cf}})$ be the corresponding complex of
${\mathfrak mathcal T}_\alpha$-modules; we have $H^d(M^{\scriptscriptstyle \bullet}) = 0$ for $d >
-c(\alpha)$. Using the description of injective ${\mathfrak mathcal F}$-sheaves from
\S\ref{injective mixed sheaves} we see that the complex in
$D^b({\mathfrak mathcal T}_\beta\text{\rm -mod}_{\text{\it cf}})$ corresponding to $j^*_\beta{\mathfrak mathbf R} i_*S^{\scriptscriptstyle \bullet}$ is
$M_\beta^{\scriptscriptstyle \bullet} = {\mathfrak mathbf R}\mathop{\rm hom}\nolimits_{{\mathfrak mathcal T}_\alpha}({\mathfrak mathcal T}_\beta, M^{\scriptscriptstyle \bullet})$. The
functor ${\mathfrak mathbf R}\mathop{\rm hom}\nolimits$ can be defined by deriving either the first or the
second variable, so the fact that $H^d(M_\beta^{\scriptscriptstyle \bullet}) = 0$ \ for $d
> -c(\beta)$ follows from the fact that ${\mathfrak mathcal T}_\beta$ has a resolution
of length $c(\alpha) - c(\beta)$ by free ${\mathfrak mathcal T}_\alpha$-modules.
\end{proof}
Thus for any $\alpha \in {\Sigma}$ we can define an object in $P_{\mathfrak mathcal F}(X)$
by
\[\nabla^{\scriptscriptstyle \bullet}_\alpha = {\mathfrak mathbf R} j_{\alpha *}
\Theta_\alpha[c(\alpha)]\langle -c(\alpha)\rangle.\] Note that since
$\Theta_\alpha$ is an injective ${\mathfrak mathcal F}$-local system, taking $j_{\alpha
*}$ instead of ${\mathfrak mathbf R} j_{\alpha *}$ defines the same object. Under
the isomorphism of Theorem \ref{combinatorial F-sheaves},
$\nabla^{\scriptscriptstyle \bullet}_\alpha$ corresponds to
${\mathfrak mathcal T}^*_{\overline\alpha}[c(\alpha)]\lb -c(\alpha)\rb$.
This object will be important in the proof of the main properties of
our Koszul duality functor in \S\ref{main proofs} below.
\subsection{Constructible ${\mathfrak mathcal F}$-sheaves}
We can also consider the full subcategories $D^b_c(LC_{\mathfrak mathcal F}(X)) \subset
D^b(LC_{\mathfrak mathcal F}(X))$ and $D^b_c(LC_{cf}(X)) \subset D^b(LC_{cf}(X))$
consisting of complexes $S^{\scriptscriptstyle \bullet}$
whose cohomology sheaves have finite
dimensional stalks on each orbit $O_\alpha$. We call
such objects ``constructible''. Note that by Theorem \ref{D^b(LC)},
$D^b_c(LC_{cf}(X))$ is equivalent to a full subcategory of the usual
constructible derived category of $D^b_c(X)$:
namely the category of objects whose cohomology sheaves are orbit-constructible
(and have finite-dimensional stalks), with
unipotent monodromy on each orbit.
The $t$-structures we defined in the previous section restrict to
$t$-structures on these subcategories, giving abelian cores
$P_{{\mathfrak mathcal F},c}(X) \subset P_{\mathfrak mathcal F}(X)$ and $P_{cf,c}(X) \subset P_{cf}(X)$.
\begin{prop} \langlebel{constructible = finite length}
$P_{{\mathfrak mathcal F},c}(X)$, (resp. $P_{cf,c}(X)$) is the full subcategory of
objects in $P_{\mathfrak mathcal F}(X)$ (resp. $P_{cf}(X)$) consisting of all objects
of finite length. In particular, $P_{cf,c}(X)$ is equivalent to the
full subcategory of the category of constructible perverse sheaves
on $X$ consisting of objects all of whose simple constituents are of
the form $IC^{\scriptscriptstyle \bullet}(\overline{O_\alpha};{\mathfrak mathbb C})$, $\alpha \in {\Sigma}$.
\end{prop}
\begin{rmk} In \cite{Br} a triangulated category ${\mathfrak mathbf D}({\Sigma})$
was defined for any fan ${\Sigma}$ to model mixed $T$-constructible
complexes on the toric variety $X_{\Sigma}$ (in the case ${\Sigma}$ is
rational). It can be shown that $D^b_c(LC_{\mathfrak mathcal F}(X))$ is equivalent to
${\mathfrak mathbf D}({\Sigma})$; under this equivalence the $t$-structure here is the
same as the $t$-structure in \cite{Br}.
\end{rmk}
\subsection{Mixed structure and pure ${\mathfrak mathcal F}$-sheaves}
In the categories of mixed $l$-adic sheaves or mixed Hodge modules,
simple perverse objects are pure. We need the following analog of
this fact in our combinatorial setting. We call an object $S^{\scriptscriptstyle \bullet}
\in D^b(LC_{\mathfrak mathcal F}(X))$ {\em pure of weight $0$} if for any orbit
$O_\beta$ and any $i \in {\mathfrak mathbb Z}$, the ${\mathfrak mathcal F}$-local systems
$H^i(j_\beta^*S^{\scriptscriptstyle \bullet})$ and $H^i(j_\beta^!S^{\scriptscriptstyle \bullet})$ are direct sums of
finitely many copies of ${\mathfrak mathbb C}_{\beta}\langle i\rangle$. More generally we say
$F^{\scriptscriptstyle \bullet}$ is pure of weight $k$ if $S^{\scriptscriptstyle \bullet}[-k]$ is pure of weight
$0$.
\begin{prop} \langlebel{unipotent Koszul}
If $S_1^{\scriptscriptstyle \bullet}, S_2^{\scriptscriptstyle \bullet}\in P_{{\mathfrak mathcal F},c}(X)$ are pure of weights $r_1$
and $r_2$, respectively, then $\operatorname{Hom}_{D^b(LC_{\mathfrak mathcal F}(X))}(S_1^{\scriptscriptstyle \bullet},
S_2^{\scriptscriptstyle \bullet}[k]) = 0$ unless $r_2 = r_1 - k$. In particular,
$\operatorname{Ext}^1_{P_{\mathfrak mathcal F}(X)}(S_1^{\scriptscriptstyle \bullet}, S_2^{\scriptscriptstyle \bullet}) = 0$ unless $r_2 = r_1 -
1$.
\end{prop}
\begin{proof} There is a spectral sequence with $E_1$ term
\[E_1^{p,q} = \bigoplus_{\dim\alpha = p} \operatorname{Hom}_{D^b(LC_{\mathfrak mathcal F}(O_\alpha))}
(j_\alpha^*S_1^{\scriptscriptstyle \bullet},j_\alpha^!S_2^{\scriptscriptstyle \bullet}),\] which converges to
$\operatorname{Hom}_{D^b(LC_{\mathfrak mathcal F}(X))}(S_1^{\scriptscriptstyle \bullet}, S_2^{\scriptscriptstyle \bullet}[p+q])$. Theorem
\ref{purity of simples} implies that $E_1^{p,q} = 0$ unless $p + q =
r_1 - r_2$, which implies the result.
\end{proof}
\begin{thm} \langlebel{purity of simples}\langlebel{mixed F-sheaves}
The simple perverse sheaf $L^{\scriptscriptstyle \bullet}_\alpha$ is pure of weight $0$.
\end{thm}
The proof will be given in \S\ref{appendix}.
\begin{rmk} Purity of IC sheaves enters our main argument twice,
once via Theorem \ref{purity of simples}, and once in the next
section, where equivariant IC sheaves are used. In that setting
the purity follows from a proof of Karu \cite{Ka}, which makes sense
even for non-rational fans. In fact Theorem \ref{purity of simples}
can also be stated and proved
for non-rational fans, without reference to a toric variety.
Although the category $LC_{\mathfrak mathcal F}(X)$ doesn't make sense, $\text{\rm co-}{\mathfrak mathcal T}\text{\rm -mod}_{\text{\it cf}}$
still does, and one
can define a functor from the combinatorial equivariant sheaves
(${\mathfrak mathcal A}$-modules) considered in the next section to $D^b(\text{\rm co-}{\mathfrak mathcal T}\text{\rm -mod}_{\text{\it cf}})$,
which sends the equivariant IC sheaves to the $L^{\scriptscriptstyle \bullet}_\alpha$'s.
The required purity can then be deduced from Karu's result.
\end{rmk}
We now define a mixed structure on $P_{\mathfrak mathcal F}(X)$. We have already defined
the twist functor $\langle 1\rangle$. What remains is to construct the
filtration $W_{\scriptscriptstyle \bullet}$.
\begin{thm} \langlebel{weight filtration}
There exists a unique
functorial increasing filtration $W_{\scriptscriptstyle \bullet}$ on objects of $P_{\mathfrak mathcal F}(X)$ satisfying
the following:
\begin{enumerate}
\item[(a)] For any $S^{\scriptscriptstyle \bullet} \in P_{\mathfrak mathcal F}(X)$ there exists $n\in {\mathfrak mathbb Z}$ depending on $S^{\scriptscriptstyle \bullet}$
so that $W_{n}S^{\scriptscriptstyle \bullet} = 0$,
\item[(b)] For all $i$ and all $S^{\scriptscriptstyle \bullet}\in P_{\mathfrak mathcal F}(X)$, ${\mathfrak mathbb G}r_i^WS^{\scriptscriptstyle \bullet} = W_iS^{\scriptscriptstyle \bullet}/W_{i-1}S^{\scriptscriptstyle \bullet}$
is isomorphic to a finite direct sum
of objects $L^{\scriptscriptstyle \bullet}_\alpha\langle i\rangle$
(thus ${\mathfrak mathbb G}r_i^WS^{\scriptscriptstyle \bullet}$ is pure of weight $i$), and
\item[(c)] $(P_{\mathfrak mathcal F}(X),W_{\scriptscriptstyle \bullet},\langle 1\rangle)$ is a
mixed category (\S\ref{mixed categories}).
\end{enumerate}
\end{thm}
For finite length objects, i.e.\ objects in $P_{{\mathfrak mathcal F},c}(X)$, this
follows in a standard way from Theorem \ref{purity of simples} and
Proposition \ref{unipotent Koszul}.
We give the complete proof in \S\ref{appendix}.
\begin{cor} \langlebel{weights of nabla}
For any $\alpha \in {\Sigma}$, we have $W_0\nabla^{\scriptscriptstyle \bullet}_\alpha \text{\rm co-}ng
L^{\scriptscriptstyle \bullet}_\alpha$. In particular, $\nabla^{\scriptscriptstyle \bullet}_\alpha$ has weights
$\ge 0$.
\end{cor}
\begin{proof} If $m$ is the minimum weight in $\nabla^{\scriptscriptstyle \bullet}_\alpha$, then
$W_m\nabla^{\scriptscriptstyle \bullet}_\alpha$ is a semisimple subobject of
$\nabla^{\scriptscriptstyle \bullet}_\alpha$. But by adjunction $\operatorname{Hom}(L_\beta^{\scriptscriptstyle \bullet}\langle
k\rangle, \nabla^{\scriptscriptstyle \bullet}_\alpha)$ is one-dimensional if $\alpha = \beta$
and $k = 0$, and vanishes otherwise.
\end{proof}
\section{Equivariant sheaves} \langlebel{equivariant sheaves}
\subsection{} \langlebel{equivariant sheaves and DG modules}
Let us very briefly recall the notion of the bounded, constructible
equivariant derived category $D_{T}^b(X)$ \cite{BL} (note that in
\cite{BL} this category was denoted $D^b_{T,c}(X)$). Let $E$ be a
contructible space with a free $T$-action, and put $X_T=(X\times
E)/T$. Then $E/T=BT$ is the classifying space for $T$ and $X_T\to BT$
is a locally trivial fibration with the fiber $X$. Similarly, a
$T$-invariant subspace $U\subset X$ induces the corresponding subspace
$U_T\subset X_T$. The triangulated category $D_{T}^b(X)$ can be
canonically identified as a full triangulated subcategory of the
bounded derived categories of sheaves on $X_T$. For example, it can be
defined as the triangulated envelope of the collection of all sheaves
$\{ {\mathfrak mathbb C} _{U_T}\}$ (${\mathfrak mathbb C} _{U_T}$ is the extension by zero to $X_T$ of
the constant sheaf ${\mathfrak mathbb C}$ on $U_T$), where $U\subset X$ is a star of
an orbit. The following theorem is one of the main results in
\cite{L}.
\begin{thm} \langlebel{equivariant-DG equivalence}
There exists a natural equivalence of triangulated categories
$$\epsilon\text{\rm co-}lon D_{T}^b(X)\to D_f(\text{DG-${\mathfrak mathcal A}_X$}).$$
\end{thm}
The equivalence of categories in the above theorem is of ``local
nature" and actually comes from a continuous map of topological
spaces. Namely there exists a natural map
$$\mathfrak mu \text{\rm co-}lon X_T\to X/T,$$
and the functor $\epsilon$ is essentially
the derived direct image functor ${\bf R}\mathfrak mu _*$. In particular, if
$U\subset X$ is the star of an orbit in $X$ and $\tau \in {\Sigma}$ is the
cone corresponding to that orbit, then $\epsilon({\mathfrak mathbb C} _{U_T})={\mathfrak mathcal A}
_{[\tau]}$. Also $\epsilon $ takes the constant sheaf ${\mathfrak mathbb C} _{X_T}$ on
$X_T$ to the sheaf ${\mathfrak mathcal A}$. (In \cite{L} the sheaf ${\mathfrak mathcal A}$ is denoted by
${\mathfrak mathcal H}$).
This allows us to define the functor $F_T\text{\rm co-}lon D^b({\mathfrak mathcal A}) \to D^b_T(X)$
described in the introduction (\S\ref{intro to main results}). It is
obtained by composing the functor of Example \ref{forget the grading}
with $\epsilon^{-1}$.
\subsection{Combinatorial equivariant complexes}
Let ${\Sigma}ma$ be a fan in the vector space $V$, and let ${\mathfrak mathcal A} =
{\mathfrak mathcal A}_{\Sigma}$ be the sheaf of conewise polynomial functions introduced in
\S\ref{the sheaf cA}. For the remainder of this section all our
${\mathfrak mathcal A}$-modules will be assumed to be locally finite, so to simplify
notation we put $D^b({\mathfrak mathcal A}\text{\rm -mod}_{\text{\it f}}) = D^b({\mathfrak mathcal A})$.
If $\operatorname{D}elta \subset {\Sigma}$ is a subfan or more generally a difference of
subfans, the space of sections of a sheaf ${\mathfrak mathcal M}$ on $\operatorname{D}elta$ will be
denoted ${\mathfrak mathcal M}(\operatorname{D}elta)$. If $\Lambda \subset \operatorname{D}elta$ is another subfan,
we put ${\mathfrak mathcal M}(\operatorname{D}elta,\Lambda) = \mathop{\rm ker}\nolimits({\mathfrak mathcal M}(\operatorname{D}elta) \to {\mathfrak mathcal M}(\Lambda))$.
If ${\sigma} \in {\Sigma}ma$, recall that $[{\sigma}]$ is the fan of all faces of
${\sigma}$. It follows that ${\mathfrak mathcal M}({[{\sigma}]})$ is isomorphic to the stalk
${\mathfrak mathcal M}_{{\sigma}}$. Define ${\partial}{\sigma} = [{\sigma}] \setminus \{{\sigma}\}$. To
simplify notation, we write ${\mathfrak mathcal M}({\sigma},{\partial}{\sigma}) =
{\mathfrak mathcal M}([{\sigma}],{\partial}{\sigma})$.
Note that a map of ${\mathfrak mathcal A}$-modules ${\mathfrak mathcal M} \to {\mathfrak mathcal N}$ is determined by the
collection of induced maps ${\mathfrak mathcal M}({\sigma})\to {\mathfrak mathcal N}({\sigma})$ over all cones
${\sigma}\in {\Sigma}$. A sequence ${\mathfrak mathcal E}\to{\mathfrak mathcal M}\to {\mathfrak mathcal N}$ is exact if and only if
${\mathfrak mathcal E}({\sigma}) \to {\mathfrak mathcal M}({\sigma}) \to {\mathfrak mathcal N}({\sigma})$ is exact for all ${\sigma}\in
{\Sigma}$.
\begin{defn} Let ${\mathfrak mathcal M}$ be an ${\mathfrak mathcal A}$-module. If
the stalk ${\mathfrak mathcal M}({\sigma})$ is a free ${\mathfrak mathcal A}_{\sigma}$-module for every ${\sigma}
\in {\Sigma}$, we say that ${\mathfrak mathcal M}$ is {\em locally free}. If the
restriction ${\mathfrak mathcal M}({\sigma}) \to {\mathfrak mathcal M}({{\partial} {\sigma}})$ is surjective for every
${\sigma} \in {\Sigma}$, we say that ${\mathfrak mathcal M}$ is {\em flabby}. If both
conditions hold, we say ${\mathfrak mathcal M}$ is {\em combinatorially pure}
(``pure'' for short).
\end{defn}
Note that flabbiness of ${\mathfrak mathcal M}$ even implies that ${\mathfrak mathcal M}(\operatorname{D}elta) \to
{\mathfrak mathcal M}(\Lambda)$ is surjective for any subfans $\Lambda \subset
\operatorname{D}elta\subset {\Sigma}$.
Let $\mathfrak mathop{\rm Pure}({\mathfrak mathcal A}_{\Sigma})$ denote the full subcategory of ${\mathfrak mathcal A}\text{\rm -mod}_{\text{\it f}}$
consisting of all pure sheaves, and let $K^b(\mathfrak mathop{\rm Pure}({\mathfrak mathcal A}))$ be the
category of bounded complexes of pure sheaves, with morphisms taken up
to chain homotopy. Our main theorem is the following.
\begin{thm} \langlebel{homotopy of pure sheaves}
The natural functor $K^b(\mathfrak mathop{\rm Pure}({\mathfrak mathcal A})) \to D^b({\mathfrak mathcal A})$ is an equivalence
of categories.
\end{thm}
This sort of theorem is familiar when instead of pure objects we have
complexes of projective or injective objects in an abelian category.
The idea of the theorem is that a pure sheaf ${\mathfrak mathcal M}$ is half injective
and half projective: the locally free condition says that the stalk
${\mathfrak mathcal M}({\sigma})$ is a projective ${\mathfrak mathcal A}_{\sigma}$-module, and flabbiness means
that ${\mathfrak mathcal M}$ is injective as a sheaf of vector spaces.
\subsection{Flabby sheaves}
Let us make more precise in what sense flabby sheaves are injective.
We first need the following definition.
\begin{defn} \langlebel{strongly injective}
An injective map ${\mathfrak mathcal M}\to {\mathfrak mathcal N}$ is called {\em strongly injective} if
the inclusion of stalks ${\mathfrak mathcal M}({\sigma})\to {\mathfrak mathcal N}({\sigma})$ splits for every
${\sigma}\in {\Sigma}$.
\end{defn}
\begin{prop} \langlebel{quasi-injectivity}
Suppose that ${\mathfrak mathcal I}$ is a flabby ${\mathfrak mathcal A}$-module. If $\eta\text{\rm co-}lon{\mathfrak mathcal M} \to {\mathfrak mathcal N}$ is a
strongly injective map, and ${\mathfrak mathcal N}$ is locally free, then the induced
homomorphism $\operatorname{Hom}_{{\mathfrak mathcal A}}({\mathfrak mathcal N}, {\mathfrak mathcal I}) \to \operatorname{Hom}_{{\mathfrak mathcal A}}({\mathfrak mathcal M}, {\mathfrak mathcal I})$ is
surjective.
\end{prop}
\begin{proof} Take a map $\phi\text{\rm co-}lon{\mathfrak mathcal M} \to {\mathfrak mathcal I}$.
We define a lift $\psi\text{\rm co-}lon {\mathfrak mathcal N}\to {\mathfrak mathcal I}$ of $\phi$ inductively on an
increasing sequence of subfans. Defining $\psi$ on the zero cone is
trivial. So suppose $\operatorname{D}elta\subset {\Sigma}$ is a subfan with more than
one cone, that $\tau\in \operatorname{D}elta$ is a maximal cone, and that
$\psi|_{\operatorname{D}elta\setminus\{\tau\}}$ has been defined already.
Since $\eta$ is strongly injective, we can choose a
splitting of ${\mathfrak mathcal A}_\tau$-modules
${\mathfrak mathcal N}(\tau) = \eta({\mathfrak mathcal M}(\tau)) \oplus M$;
since ${\mathfrak mathcal N}(\tau)$ is a free ${\mathfrak mathcal A}_\tau$-module, so are ${\mathfrak mathcal M}(\tau)$
and $M$. Define the restriction of $\psi$ to $\eta({\mathfrak mathcal M}(\tau))$
to be $\phi \eta^{-1}$. To define $\psi$ on
$M$, we need a map $M \to {\mathfrak mathcal I}(\tau)$ making the square
\[\xymatrix{
M \ar[r]\ar[d] & {\mathfrak mathcal I}(\tau)\ar[d] \\
{\mathfrak mathcal N}({{\partial}\tau})\ar[r] & {\mathfrak mathcal I}({{\partial}\tau}) }\] commute. The right-hand
vertical map is surjective, since ${\mathfrak mathcal I}$ is flabby, and since $M$ is free, the
required map exists.
\end{proof}
\begin{prop} \langlebel{fully faithful}
If $\phi\text{\rm co-}lon {\mathfrak mathcal M}^{\scriptscriptstyle \bullet}\to {\mathfrak mathcal N}^{\scriptscriptstyle \bullet}$ is a quasi-isomorphism of
pure complexes, then it has a homotopy inverse.
\end{prop}
\begin{lemma}
\langlebel{map from acyclic}
If ${\cal Z}^{\scriptscriptstyle \bullet}$ and ${\mathfrak mathcal M}^{\scriptscriptstyle \bullet}$ are bounded complexes of pure sheaves,
and ${\cal Z}^{\scriptscriptstyle \bullet}$ is acyclic, then any map ${\cal Z}^{\scriptscriptstyle \bullet} \to {\mathfrak mathcal M}^{\scriptscriptstyle \bullet}$ is
chain-homotopic to zero.
\end{lemma}
\begin{proof}
Proposition \ref{quasi-injectivity} allows us to copy the standard
argument used when the objects ${\cal Z}^i$ are injective, provided that
we know that each $\mathfrak mathop{\rm coker}\nolimits d_{\cal Z}^i \to {\cal Z}^{i+2}$ is strongly
injective. In other words, we need to show that $\mathfrak mathop{\rm coker}\nolimits
(d_{{\cal Z}^i({\sigma})}) \to {\cal Z}^{i+2}({\sigma})$ is a split injection for all
${\sigma}\in{\Sigma}$. This follows from the fact that ${\cal Z}^{\scriptscriptstyle \bullet}({\sigma})$ is
an acyclic complex of free ${\mathfrak mathcal A}_{\sigma}$-modules.
\end{proof}
\begin{proof}[Proof of Proposition \ref{fully faithful}]
Let ${\cal Z}^{\scriptscriptstyle \bullet}$ be the mapping cone of $\phi$. Applying the lemma
to the connecting map ${\cal Z}^{\scriptscriptstyle \bullet}\to {\mathfrak mathcal M}^{\scriptscriptstyle \bullet}[1]$ gives a chain
homotopy whose components are maps $h^i\text{\rm co-}lon {\cal Z}^i \to {\mathfrak mathcal M}[1]^{i-1}
= {\mathfrak mathcal M}^i$. But ${\cal Z}^i = {\mathfrak mathcal N}^i \oplus {\mathfrak mathcal M}^{i+1}$, so the first
component of $h^i$ gives a map $\psi^i\text{\rm co-}lon {\mathfrak mathcal N}^i \to {\mathfrak mathcal M}^i$. The
resulting map of complexes $\psi\text{\rm co-}lon {\mathfrak mathcal N}^{\scriptscriptstyle \bullet} \to {\mathfrak mathcal M}^{\scriptscriptstyle \bullet}$ is a
homotopy inverse of $\phi$.
\end{proof}
\subsection{Locally free sheaves}
Locally free sheaves act like projective objects in a very similar
way. We do not need the following result, but we include it to
illustrate the parallels with the situation for flabby sheaves.
\begin{defn} A surjective morphism ${\mathfrak mathcal M} \to {\mathfrak mathcal N}$ between
${\mathfrak mathcal A}_{\Sigma}$-modules is called {\em strongly surjective} if the
induced homomorphism ${\mathfrak mathcal M}({\sigma}, {\partial}{\sigma}) \to {\mathfrak mathcal N}({\sigma},{\partial}{\sigma})$ is
surjective for every ${\sigma}\in {\Sigma}$.
\end{defn}
\begin{prop} \langlebel{quasi-projectivity} Let ${\mathfrak mathcal P}$ be a locally free sheaf. If
${\mathfrak mathcal M} \to {\mathfrak mathcal N}$ is strongly surjective and ${\mathfrak mathcal M}$ is flabby, then the
homomorphism $\operatorname{Hom}_{{\mathfrak mathcal A}}({\mathfrak mathcal P},{\mathfrak mathcal M}) \to \operatorname{Hom}_{{\mathfrak mathcal A}}({\mathfrak mathcal P},{\mathfrak mathcal N})$ is
surjective.
\end{prop}
The proof is left to the reader.
\subsection{There are enough pure sheaves}
To finish the proof of Theorem \ref{homotopy of pure sheaves} we need
to show that there are ``enough'' pure sheaves to represent any
complex in $D^b({\mathfrak mathcal A})$. This follows from a two-step resolution
process, using the following result.
\begin{prop} \langlebel{enough pures} Take any object ${\mathfrak mathcal M}\in {\mathfrak mathcal A}\text{\rm -mod}_{\text{\it f}}$.
\begin{enumerate}
\item[(a)] There exists a locally free ${\mathfrak mathcal A}$-module ${\mathfrak mathcal P}$ and a strong
surjection ${\mathfrak mathcal P} \to {\mathfrak mathcal M}$. If ${\mathfrak mathcal M}$ is flabby, then ${\mathfrak mathcal P}$ can be
chosen to be pure.
\item[(b)] There exists a flabby ${\mathfrak mathcal A}$-module ${\mathfrak mathcal I}$ and a strong
injection ${\mathfrak mathcal M} \to {\mathfrak mathcal I}$. If ${\mathfrak mathcal M}$ is locally free, then ${\mathfrak mathcal I}$ can
be chosen to be pure.
\item[(c)] If ${\mathfrak mathcal M}$ is zero on the subfan ${\partial}{\sigma}$ for some
${\sigma}\in{\Sigma}$, then the maps in {\rm (a)} and {\rm (b)} can be
chosen to be isomorphisms on all of $[{\sigma}]$.
\end{enumerate}
\end{prop}
Assuming this result, we can now prove Theorem \ref{homotopy of pure
sheaves}. Consider any complex ${\mathfrak mathcal M}^{\scriptscriptstyle \bullet} \in D^b({\mathfrak mathcal A})$. Part (a)
of the proposition allows us to find a complex ${\mathfrak mathcal P}^{\scriptscriptstyle \bullet}$ of locally
free sheaves and a quasi-isomorphism ${\mathfrak mathcal P}^{\scriptscriptstyle \bullet} \mathfrak mathop{\rm st}\nolimitsackrel{\sim}{\to}
{\mathfrak mathcal M}^{\scriptscriptstyle \bullet}$. Note that (c) implies that ${\mathfrak mathcal P}^{\scriptscriptstyle \bullet}$ can be chosen to
be a bounded complex.
Part (b) then implies that there is a complex ${\mathfrak mathcal I}^{\scriptscriptstyle \bullet}$ of pure
sheaves and a quasi-isomorphism ${\mathfrak mathcal P}^{\scriptscriptstyle \bullet} \mathfrak mathop{\rm st}\nolimitsackrel\sim\to
{\mathfrak mathcal I}^{\scriptscriptstyle \bullet}$. Note that here it is crucial that (b) gives strong
injections: to construct ${\mathfrak mathcal I}^j$ we apply (b) to ${\mathfrak mathcal M} =
\mathfrak mathop{\rm coker}\nolimits({\mathfrak mathcal P}^{j-1} \to {\mathfrak mathcal P}^j \oplus {\mathfrak mathcal I}^{j-1})$, which is locally free
since ${\mathfrak mathcal P}^j$ and ${\mathfrak mathcal I}^{j-1}$ are and ${\mathfrak mathcal P}^{j-1} \to {\mathfrak mathcal I}^{j-1}$ is a
strong injection. Using (c) again, we see that ${\mathfrak mathcal I}^{\scriptscriptstyle \bullet}$ can be
chosen to be a bounded complex.
Thus the functor $K^b(\mathfrak mathop{\rm Pure}({\mathfrak mathcal A})) \to D^b({\mathfrak mathcal A})$ is essentially
surjective. The same two-step resolution process also shows it is
fully faithful, by the following well-known result. Let ${\mathfrak mathcal C}$ be an
abelian category, $K({\mathfrak mathcal C})$ the homotopy category of complexes in
${\mathfrak mathcal C}$.
\begin{lemma} Let $K_1 \subset K_2$ be full triangulated subcategories
of $K({\mathfrak mathcal C})$, and let $D_1$, $D_2$ be the corresponding derived
categories. If either of the following conditions holds, then the
functor $D_1 \to D_2$ is fully faithful.
\begin{enumerate}
\item For any quasi-isomorphism $X^{\scriptscriptstyle \bullet} \to Y^{\scriptscriptstyle \bullet}$ in $K_2$, with
$X^{\scriptscriptstyle \bullet}$ in $K_1$, there exists a quasi-isomorphism $A^{\scriptscriptstyle \bullet} \to
X^{\scriptscriptstyle \bullet}$ with $A^{\scriptscriptstyle \bullet} \in K_1$.
\item For any quasi-isomorphism $X^{\scriptscriptstyle \bullet} \to Y^{\scriptscriptstyle \bullet}$ in $K_2$, with
$Y^{\scriptscriptstyle \bullet}$ in $K_1$, there exists a quasi-isomorphism $Y^{\scriptscriptstyle \bullet} \to
B^{\scriptscriptstyle \bullet}$ with $B^{\scriptscriptstyle \bullet} \in K_1$.
\end{enumerate}
\end{lemma}
\begin{cor} \langlebel{loc free --> pure}
Suppose ${\mathfrak mathcal M}$, ${\mathfrak mathcal P}$ are ${\mathfrak mathcal A}$-modules, ${\mathfrak mathcal M}$ is locally free, and
${\mathfrak mathcal P}$ is pure. Then $\operatorname{Hom}_{D^b({\mathfrak mathcal A})}({\mathfrak mathcal M},{\mathfrak mathcal P}[i]) = 0$ unless $i =
0$.
\end{cor}
\begin{proof} We can replace ${\mathfrak mathcal M}$ by a
quasi-isomorphic complex ${\mathfrak mathcal I}^{\scriptscriptstyle \bullet}$ of pure sheaves, with ${\mathfrak mathcal I}^j =
0$ for $j < 0$ and $\mathfrak mathop{\rm coker}\nolimits(\partial^{j-1}) \to {\mathfrak mathcal I}^{j+1}$ strongly
injective for all $j \ge 0$. Now apply Proposition
\ref{quasi-injectivity}.
\end{proof}
\begin{proof}[Proof of Proposition \ref{enough pures}]
To prove (a), we construct the object ${\mathfrak mathcal P}$ and the map $\phi\text{\rm co-}lon
{\mathfrak mathcal P} \to {\mathfrak mathcal M}$ simultaneously, by induction on subfans. For the base
case when $\operatorname{D}elta = \{o\}$, we set ${\mathfrak mathcal I}(o) = {\mathfrak mathcal M}(o)$ and let
$\phi|_o$ be the identity map.
Now suppose $\operatorname{D}elta$ is a fan with at least two cones, $\tau$ is a
maximal cone, and the restrictions of ${\mathfrak mathcal P}$ and $\phi$ to $\operatorname{D}elta
\setminus \{\tau\}$ have already been defined. To define them on
all of $\operatorname{D}elta$ it is enough to define them on $[\tau]$, since the
resulting sheaves and maps can be glued.
This amounts to choosing a free ${\mathfrak mathcal A}_\tau$-module ${\mathfrak mathcal P}(\tau)$ and
homomorphisms ${\partial}_{\mathfrak mathcal P}\text{\rm co-}lon {\mathfrak mathcal P}(\tau) \to {\mathfrak mathcal P}({\partial}\tau)$ and
$\phi_\tau \text{\rm co-}lon {\mathfrak mathcal P}(\tau) \to {\mathfrak mathcal M}(\tau)$ so that $\phi_\tau$ and
the induced map $\mathop{\rm ker}\nolimits {\partial}_{\mathfrak mathcal P} \to \mathop{\rm ker}\nolimits {\partial}_{\mathfrak mathcal M}$ are surjective and
the square
\[\xymatrix{
{\mathfrak mathcal P}(\tau) \ar[r]^{\phi_\tau} \ar[d]_{{\partial}_{\mathfrak mathcal P}} & {\mathfrak mathcal M}(\tau)\ar[d]^{{\partial}_{\mathfrak mathcal M}}\\
{\mathfrak mathcal P}({{\partial}\tau}) \ar[r]_{\phi|_{{\partial}\tau}} & {\mathfrak mathcal M}({{\partial}\tau}) }\]
commutes. To do this, find free ${\mathfrak mathcal A}_\tau$-modules $M_1$,$M_2$ and
maps $p_1\text{\rm co-}lon M_1 \to {\mathfrak mathcal P}({{\partial}\tau})$ and $p_2\text{\rm co-}lon M_2 \to
{\mathfrak mathcal M}(\tau)$ so that $\operatorname{Im} p_1 = (\phi|_{{\partial}\tau})^{-1}(\operatorname{Im} {\partial}_{\mathfrak mathcal M})$
and $\operatorname{Im} p_2 = \mathop{\rm ker}\nolimits({\partial}_{\mathfrak mathcal M})$. Then let ${\mathfrak mathcal P}({\sigma}) = M_1 \oplus
M_2$, and ${\partial}_{\mathfrak mathcal P} = p_1\oplus 0$. To define $\phi_\tau$, let
$\phi_\tau|_{M_2} = p_2$, and for each $a$ in a basis of $M_1$, define
$\phi_\tau(a)$ to satisfy $(\phi|_{{\partial}\tau})(p_1(a)) =
{\partial}_{\mathfrak mathcal M}\phi_\tau(a)$.
To prove (b), we again proceed by induction. The base case is again
trivial, and we are reduced to the problem of extending the sheaf
${\mathfrak mathcal I}$ and morphism $\phi\text{\rm co-}lon {\mathfrak mathcal M}\to {\mathfrak mathcal I}$ from ${\partial}\tau$ to $[\tau]$
as before. This, in turn, amounts to finding an ${\mathfrak mathcal A}_\tau$-module
${\mathfrak mathcal I}(\tau)$, a surjective restriction homomorphism ${\partial}_{\mathfrak mathcal I}\text{\rm co-}lon
{\mathfrak mathcal I}(\tau) \to {\mathfrak mathcal I}({{\partial}\tau})$, and a split injection $\phi_\tau\text{\rm co-}lon
{\mathfrak mathcal M}(\tau) \to {\mathfrak mathcal I}(\tau)$, such that the square
\[\xymatrix{
{\mathfrak mathcal M}(\tau) \ar[r]^{\phi_\tau} \ar[d]_{{\partial}_{\mathfrak mathcal M}} & {\mathfrak mathcal I}(\tau)\ar[d]^{{\partial}_{\mathfrak mathcal I}}\\
{\mathfrak mathcal M}({{\partial}\tau}) \ar[r]_{\phi|_{{\partial}\tau}} & {\mathfrak mathcal I}({{\partial}\tau}) }\]
commutes. This can be done by letting ${\mathfrak mathcal I}(\tau) = {\mathfrak mathcal I}({{\partial}\tau})
\oplus {\mathfrak mathcal M}(\tau)$, and letting $\phi_\tau = (0, {\rm id}_{\mathfrak mathcal M})$ and
${\partial}_{\mathfrak mathcal I} = {\rm id}_{{\mathfrak mathcal I}({{\partial}\tau})} \oplus (\phi|_{{\partial}\tau} \circ
{\partial}_{\mathfrak mathcal M})$.
For the second statement of (b), we make a different choice at the
inductive step: take a free ${\mathfrak mathcal A}_\tau$-module $M$ and a
surjective homomorphism $p\text{\rm co-}lon M\to {\mathfrak mathcal I}({{\partial}\tau})$. We then
define ${\mathfrak mathcal I}(\tau) = M \oplus {\mathfrak mathcal M}(\tau)$ and ${\partial}_{\mathfrak mathcal I} = p \oplus 0$.
Since ${\mathfrak mathcal M}(\tau)$ is free by assumption, ${\mathfrak mathcal I}(\tau)$ is free as well.
The required map $\phi_\tau$ now exists because ${\partial}_{\mathfrak mathcal I}$ is
surjective and ${\mathfrak mathcal M}(\tau)$ is free.
Checking that these constructions satisfy (c) is easy.
\end{proof}
\subsection{Indecomposible pure sheaves}
The notion of pure ${\mathfrak mathcal A}$-module was first used in \cite{BrLu,BBFK} to
model direct sums of (shifted) intersection cohomology sheaves. The
indecomposible pure objects are models of single intersection
cohomology sheaves. We recall here their basic properties.
For a cone ${\sigma}\in{\Sigma}$, let $c({\sigma})$ denote the codimension of
${\sigma}$ in the ambient vector space.
For $n\in {\mathfrak mathbb Z}$, recall that $\lb n\rb \text{\rm co-}lon D^b({\mathfrak mathcal A}) \to D^b({\mathfrak mathcal A})$ is the
functor which shifts the degree {\em down} by $n$.
\begin{thm} \langlebel{pure simples}
For every ${\sigma}\in{\Sigma}$, there is an indecomposible pure
${\mathfrak mathcal A}$-module ${\mathfrak mathcal L}^{\sigma}$, unique up to a scalar isomorphism, for
which (1) ${\mathfrak mathcal L}^{\sigma}(\tau) = 0$ unless ${\sigma} \prec\tau$, and (2)
${\mathfrak mathcal L}^{\sigma}({\sigma}) = {\mathfrak mathcal A}_{\sigma}\lb c({\sigma})\rb $.
These objects satisfy the following:
\begin{enumerate}
\item Every pure sheaf is isomorphic to a finite direct sum $\oplus_i
{\mathfrak mathcal L}^{{\sigma}_i}\lb n_i\rb $ with ${\sigma}_i\in {\Sigma}$ and $n_i \in {\mathfrak mathbb Z}$.
\item For all $\tau \in {\Sigma}\setminus\{{\sigma}\}$, ${\mathfrak mathcal L}^{\sigma}(\tau)$ is
generated in degrees $< -c(\tau)$.
\item For all $\tau \in {\Sigma}$, ${\mathfrak mathcal L}^{\sigma}({\tau,{\partial}\tau})$ is a free
${\mathfrak mathcal A}_\tau$-module; it is generated in degrees $> -c(\tau)$, unless
${\sigma}ma = \tau$.
\end{enumerate}
\end{thm}
\begin{rmk}
We put the generator of ${\mathfrak mathcal L}^{\sigma}({\sigma})$ in degree $-c({\sigma})$
(rather than degree $0$ as in \cite{BrLu,BBFK}) so that the
resulting object will be perverse, i.e.\ in the core of the
$t$-structure which we define in the next section.
\end{rmk}
A proof of (1) can be found in \cite{BrLu,BBFK}, while (2) and (3)
follow from work of Karu \cite{Ka}.
Next we look more carefully at homomorphisms between pure sheaves.
Because of Theorem \ref{pure simples}, it is enough to look at the
objects ${\mathfrak mathcal L}^{\sigma}\lb n\rb $.
\begin{thm} \langlebel{Homs between simples}
Take ${\sigma},\tau\in {\Sigma}$, $n\in {\mathfrak mathbb Z}$. Let $H =
\operatorname{Hom}_{{\mathfrak mathcal A}}({\mathfrak mathcal L}^{\sigma},{\mathfrak mathcal L}^\tau\lb n\rb )$.
\begin{enumerate}
\item[(a)] If $n < 0$, then $H=0$.
\item[(b)] If $n=0$, then $H=0$ unless ${\sigma} = \tau$, in which case
$\dim_{\mathfrak mathbb R} H = 1$, with a basis given by the identity map.
\item[(c)] If $n = 1$ and ${\sigma}\prec\tau$, then restricting to $\tau$
gives an isomorphism
\[ H \text{\rm co-}ng
\operatorname{Hom}_{{\mathfrak mathcal A}_\tau}({\mathfrak mathcal L}^{\sigma}(\tau),{\mathfrak mathcal L}^\tau({\tau,{\partial}\tau}) ) =
{\mathfrak mathcal L}^{\sigma}(\tau)^*_{c(\tau)-1},\] while if $\tau\prec{\sigma}$, restricting
to ${\sigma}$ gives an isomorphism
\[H \text{\rm co-}ng
\operatorname{Hom}_{{\mathfrak mathcal A}_{\sigma}}({\mathfrak mathcal L}^{\sigma}({\sigma}),{\mathfrak mathcal L}^\tau({\sigma},{\partial}{\sigma})) =
{\mathfrak mathcal L}^\tau({\sigma},\partial{\sigma})_{-c({\sigma})+1}.\] If $\tau$ and ${\sigma}$ are
not comparable, then $\operatorname{Hom}_{{\mathfrak mathcal A}}({\mathfrak mathcal L}^{\sigma},{\mathfrak mathcal L}^\tau\lb 1\rb ) = 0$.
\end{enumerate}
\end{thm}
The proof is by induction, using Theorem \ref{pure simples} and the
following lemma.
\begin{lemma} Suppose ${\mathfrak mathcal M},{\mathfrak mathcal N} $ are ${\mathfrak mathcal A}$-modules, ${\mathfrak mathcal M}$
is locally free, and ${\mathfrak mathcal N}$ is flabby. If ${\sigma}\in {\Sigma}$ is a
maximal cone, then there is a short exact sequence
\[0 \to \operatorname{Hom}_{{\mathfrak mathcal A}_{\sigma}}({\mathfrak mathcal M}({\sigma}),{\mathfrak mathcal N}({\sigma},{\partial}{\sigma}))
\to \operatorname{Hom}_{{\mathfrak mathcal A}}({\mathfrak mathcal M},{\mathfrak mathcal N})\]\[ \to
\operatorname{Hom}_{{\mathfrak mathcal A}|_{{\Sigma}\setminus\{{\sigma}\}}}({\mathfrak mathcal M}|_{{\Sigma}\setminus\{{\sigma}\}},
{\mathfrak mathcal N}|_{{\Sigma}\setminus\{{\sigma}\}}) \to 0.\]
\end{lemma}
The following corollary of Theorem \ref{Homs between simples}(b) is
useful.
\begin{cor} \langlebel{Cor to Homs between simples}
If ${\mathfrak mathcal M} \to {\mathfrak mathcal N}$ is a morphism between two pure sheaves, each of
which is a direct sum of various ${\mathfrak mathcal L}^{\sigma}$ (without degree shifts),
then the kernel and cokernel are both pure.
\end{cor}
We will also need the following technical lemma.
\begin{lemma} \langlebel{Noetherian}
Suppose that ${\mathfrak mathcal M}$ is a pure ${\mathfrak mathcal A}$-module. Then the graded
endomorphism ring
\[R = \mathfrak mathop{\rm End}\nolimits_{{\mathfrak mathcal A}\text{\rm -Mod}}({\mathfrak mathcal M}) = \oplus_{n\in {\mathfrak mathbb Z}}\operatorname{Hom}_{{\mathfrak mathcal A}\mathfrak mod}({\mathfrak mathcal M},{\mathfrak mathcal M}\lb n\rb )\]
is Noetherian.
\end{lemma}
\begin{proof}
There is a homomorphism from ${\mathfrak mathcal A}_o$ (polynomial functions on
$N\otimes {\mathfrak mathbb C}$) to $R$ given by pointwise multiplication. The ring
$R$ is contained in $\oplus_{\tau \in {\Sigma}}
\mathfrak mathop{\rm End}\nolimits_{{\mathfrak mathcal A}_o\text{\rm -Mod}}({\mathfrak mathcal M}(\tau))$, which is a finitely generated
${\mathfrak mathcal A}_o$-module.
\end{proof}
\subsection{Perverse $t$-structure} \langlebel{t-structure on cA-complexes}
We define a $t$-structure on the triangulated category $D^b({\mathfrak mathcal A})$,
analogous to the usual one on the equivariant derived category
$D^b_T(X)$ whose core consists of equivariant perverse sheaves.
Let $K^{\ge 0}$ (respectively $K^{\le 0}$) be the full subcategory of
$K^b(\mathfrak mathop{\rm Pure}({\mathfrak mathcal A}))$ consisting of complexes which are quasi-isomorphic
to a complex ${\mathfrak mathcal M}^{\scriptscriptstyle \bullet}$, where ${\mathfrak mathcal M}^i \text{\rm co-}ng \bigoplus_k
{\mathfrak mathcal L}^{{\sigma}_k}\lb n_k\rb $, ${\sigma}_k\in {\Sigma}$, $n_k \le i$ (respectively $n_k
\ge i$).
\begin{thm} \langlebel{t-structure thm}
This defines a $t$-structure on $K^b(\mathfrak mathop{\rm Pure}({\mathfrak mathcal A}))$. The heart
$P({\Sigma}) = K^{\ge 0}\cap K^{\le 0}$ is equivalent to the full
subcategory of $P({\Sigma})$ consisting of bounded complexes ${\mathfrak mathcal P}^{\scriptscriptstyle \bullet}$
so that for any $i$ we have
\begin{equation*} \tag{*} {\mathfrak mathcal P}^i \text{\rm co-}ng
\bigoplus_{k=1}^l {\mathfrak mathcal L}^{{\sigma}_k}\lb i\rb
\end{equation*}
with ${\sigma}_1,\dots,{\sigma}_l \in {\Sigma}$.
\end{thm}
Since $D^b({\mathfrak mathcal A})$ and $K^b(\mathfrak mathop{\rm Pure}({\mathfrak mathcal A}))$ are equivalent categories, this
defines a $t$-structure $(D^{\le 0}, D^{\ge 0})$ on $D^b({\mathfrak mathcal A})$ as
well.
\begin{rmk} Note that all chain homotopies between complexes satisfying
(*) automatically vanish, by Theorem \ref{Homs between simples}.
Thus if objects in $P({\Sigma})$ are represented by such complexes,
morphisms are just morphisms of complexes.
The resulting category of mixed equivariant perverse sheaves is similar
to a construction of Vybornov \cite{V}.
\end{rmk}
\begin{proof}
There are four things to check to show that $(K^{\ge 0}, K^{\le 0})$
forms a $t$-structure. It is clear that $K^{\ge 0} \subset K^{\ge
0}[1]$ and $K^{\le 0}[1] \subset K^{\le 0}$. If ${\mathfrak mathcal M}^{\scriptscriptstyle \bullet} \in
K^{\le 0}$ and ${\mathfrak mathcal N}^{\scriptscriptstyle \bullet} \in K^{\ge 1} = K^{\ge 0}[-1]$, we have
$\operatorname{Hom}({\mathfrak mathcal M}^{\scriptscriptstyle \bullet},{\mathfrak mathcal N}^{\scriptscriptstyle \bullet}) = 0$, by Theorem \ref{Homs between
simples}(a). Given a distinguished triangle
\begin{equation*}
{\mathfrak mathcal E}^{\scriptscriptstyle \bullet} \to {\mathfrak mathcal M}^{\scriptscriptstyle \bullet} \to {\mathfrak mathcal N}^{\scriptscriptstyle \bullet} \mathfrak mathop{\rm st}\nolimitsackrel{[1]}{\to},
\end{equation*}
where ${\mathfrak mathcal E}^{\scriptscriptstyle \bullet}$ and ${\mathfrak mathcal N}^{\scriptscriptstyle \bullet}$ are both in $K^{\le 0}$ (resp.\
$K^{\ge 0}$), then ${\mathfrak mathcal M}^{\scriptscriptstyle \bullet} \in K^{\le 0}$ (resp. ${\mathfrak mathcal M}^{\scriptscriptstyle \bullet}\in
K^{\ge 0}$), since the triangle comes from a short exact sequence $0
\to {\mathfrak mathcal E}^{\scriptscriptstyle \bullet} \to\widetilde{\mathfrak mathcal M}^{\scriptscriptstyle \bullet} \to {\mathfrak mathcal N}^{\scriptscriptstyle \bullet} \to 0$, with $\widetilde{\mathfrak mathcal M}^{\scriptscriptstyle \bullet}
\text{\rm co-}ng {\mathfrak mathcal M}^{\scriptscriptstyle \bullet}$.
Finally, we need to show that for any ${\mathfrak mathcal M}^{\scriptscriptstyle \bullet}\in K^b(\mathfrak mathop{\rm Pure}({\mathfrak mathcal A}_{\Sigma}))$
there exists a triangle ${\mathfrak mathcal E}^{\scriptscriptstyle \bullet} \to {\mathfrak mathcal M}^{\scriptscriptstyle \bullet} \to {\mathfrak mathcal N}^{\scriptscriptstyle \bullet}
\mathfrak mathop{\rm st}\nolimitsackrel{[1]}{\to}$ with ${\mathfrak mathcal E}^{\scriptscriptstyle \bullet} \in K^{\le 0}$ and ${\mathfrak mathcal N}^{\scriptscriptstyle \bullet} \in
K^{\ge 1}$. To do this, we write each ${\mathfrak mathcal M}^i$ as a direct sum
$\oplus_j {\mathfrak mathcal M}^i_j$, where ${\mathfrak mathcal M}^i_j$ is isomorphic to a sum of various
${\mathfrak mathcal L}^{\sigma}\lb j\rb $. Then we can write the differential $d^i\text{\rm co-}lon {\mathfrak mathcal M}^i
\to {\mathfrak mathcal M}^{i+1}$ as a sum $\sum\limits_{\mathfrak mathop{\rm st}\nolimitsackrel{\scriptstyle j,k\in
{\mathfrak mathbb Z}}{k \ge 0}}d^i_{jk}$, where $d^i_{jk}\text{\rm co-}lon {\mathfrak mathcal M}^i_j \to
{\mathfrak mathcal M}^{i+1}_{j+k}$.
We then let
\[{\mathfrak mathcal E}^i = \mathop{\rm ker}\nolimits d^i_{i,0} \oplus\bigoplus_{i-j < 0} {\mathfrak mathcal M}^i_j ,\]
which is pure by Corollary \ref{Cor to Homs between simples}. It is a
subcomplex of ${\mathfrak mathcal M}^{\scriptscriptstyle \bullet}$, and it clearly lies in $K^{\le 0}$.
Moreover, it is compatible with the decomposition ${\mathfrak mathcal M}^i = \oplus_j
{\mathfrak mathcal M}^i_j$, so if we let ${\mathfrak mathcal N}^{\scriptscriptstyle \bullet} = {\mathfrak mathcal M}^{\scriptscriptstyle \bullet}/{\mathfrak mathcal E}^{\scriptscriptstyle \bullet}$, we have a
decomposition ${\mathfrak mathcal N}^i = \oplus {\mathfrak mathcal N}^i_j$ compatible with the quotient
map. Let $\tilde d$ be the differential of ${\mathfrak mathcal N}^{\scriptscriptstyle \bullet}$; it can be
decomposed $\tilde d^i = \oplus \tilde d^i_{jk}$ as before.
We must show that ${\mathfrak mathcal N}^{\scriptscriptstyle \bullet}$ lies in $K^{\ge 1}$. Note that letting
\[{\mathfrak mathcal N}^i = \widetilde{\mathfrak mathcal N}^i_{i} \oplus \operatorname{Im}(\tilde{d}^i_{i,0})\]
defines a subcomplex ${\mathfrak mathcal N}^{\scriptscriptstyle \bullet}$ of ${\mathfrak mathcal N}^{\scriptscriptstyle \bullet}$ which is
quasi-isomorphic to $0$. Since ${\mathfrak mathcal N}^{\scriptscriptstyle \bullet}/{\mathfrak mathcal N}^{\scriptscriptstyle \bullet}$ is clearly in
$K^{\ge 1}$, so is ${\mathfrak mathcal N}^{\scriptscriptstyle \bullet}$.
For the second statement, suppose ${\mathfrak mathcal M}^{\scriptscriptstyle \bullet}$ is in $K^{\le 0}\cap
K^{\ge 0}$. Then $\widetilde{{\mathfrak mathcal M}}^i = (\mathop{\rm ker}\nolimits d^i_{i,0})/(\operatorname{Im}
d^{i-1}_{i-1,0})$ gives a quasi-isomorphic complex which satisfies
(*).
\end{proof}
\subsection{$t$-exactness}
In this section, we prove
\begin{thm} \langlebel{t-exact}
The functor $F_T\text{\rm co-}lon D^b({\mathfrak mathcal A}) \to D^b_T(X_{\Sigma})$
defined in \S\ref{equivariant sheaves and DG modules} is $t$-exact.
\end{thm}
Here $D^b({\mathfrak mathcal A})$ has the
$t$-structure just defined, and $D^b_T(X)$ has the perverse
$t$-structure from \cite{BL}.
In terms of the presentation we use of $D^b_T(X)$ as a full
subcategory of $D^b(X_T)$, we define this second $t$-structure in
terms of the usual perverse $t$-structure $(D^{\le 0}(X), D^{\ge
0}(X))$ on $D^b(X)$. Since $X_T$ is a fiber bundle over $BT$ with
fiber $X$, we get an embedding of $X$ into $X_T$ by choosing a
basepoint in $BT$. This gives rise to a ``forgetful functor''
\[{\mathfrak mathbb F}or\text{\rm co-}lon D^b_T(X) \to D^b(X)\]
which is simply restriction to $X$. Then our perverse $t$-structure
is $({\mathfrak mathbb F}or^{-1}D^{\le 0}(X),{\mathfrak mathbb F}or^{-1}D^{\ge 0}(X))$.
For a face ${\sigma}\in {\Sigma}$ let $j_{\sigma}$ be the
inclusion of $O_{\sigma} \hookrightarrow X$. Let $F_{T,{\Sigma}}\text{\rm co-}lon
D^b({\mathfrak mathcal A}) \to D^b_T(X)$ and $F_{T,{\sigma}}\text{\rm co-}lon D^b({\mathfrak mathcal A}_{\sigma}\text{\rm -mod}_{\text{\it f}})\to
D^b_T(O_{\sigma})$ be the realization functors. We have restriction
functors
\[j_{\sigma}^*,j^!_{\sigma}\text{\rm co-}lon D^b_T(X) \to D^b_T(O_{\sigma})\]
which are simply restriction and corestriction to $O_{{\sigma},T}$ (note
that although the space $X_T$ is not locally compact, the
corestriction can be defined using the derived ``restriction with
supports'' functor ${\mathfrak mathbf R}{\mathfrak mathbb G}amma_{O_{{\sigma},T}}$; all the usual adjunction
and base change properties still apply).
Theorem \ref{t-exact} follows from Theorem \ref{pure simples} and the
following result, which describes the stalk and costalk functors
in terms of ${\mathfrak mathcal A}$-modules. The proof will be given in \S\ref{appendix}.
\begin{thm} If ${\mathfrak mathcal M}^{\scriptscriptstyle \bullet} \in K^b(\mathfrak mathop{\rm Pure}({\mathfrak mathcal A}))$, there are natural
isomorphisms in $D^b_T(O_{\sigma})$:
\begin{itemize}
\item[(a)] $j_{\sigma}^*F_{T,{\Sigma}} {\mathfrak mathcal M}^{\scriptscriptstyle \bullet} \simeq
F_{T,{\sigma}}({\mathfrak mathcal M}^{\scriptscriptstyle \bullet}({\sigma}))$,
\item[(b)] $j_{\sigma}^!F_{T,{\Sigma}} {\mathfrak mathcal M}^{\scriptscriptstyle \bullet} \simeq
F_{T,{\sigma}}({\mathfrak mathcal M}^{\scriptscriptstyle \bullet}({\sigma},\partial {\sigma}))$.
\end{itemize}
Let $i_x$ be the inclusion of a point $x$ into $ O_{{\sigma},T}$. Then
there are natural isomorphisms in $D^b({\mathfrak mathbb C}\text{\rm -mod}_{\text{\it f}}) = D^b(pt)$
\begin{itemize}
\item[(c)] $i_x^*j_{\sigma}^*F_{T,{\Sigma}} {\mathfrak mathcal M}^{\scriptscriptstyle \bullet} \simeq {\mathfrak mathbb C}
\otimes_{{\mathfrak mathcal A}_{\sigma}} \nu{\mathfrak mathcal M}^{\scriptscriptstyle \bullet}({\sigma})$,
\item[(d)] $i_x^*j_{\sigma}^!F_{T,{\Sigma}} {\mathfrak mathcal M}^{\scriptscriptstyle \bullet} \simeq {\mathfrak mathbb C}
\otimes_{{\mathfrak mathcal A}_{\sigma}} \nu{\mathfrak mathcal M}^{\scriptscriptstyle \bullet}({\sigma},\partial {\sigma})$.
\end{itemize}
\end{thm}
Here $\nu$ is the functor $D({\mathfrak mathcal A}_{\sigma}\mathfrak mod) \to
D(\text{DG-${\mathfrak mathcal A}_{\sigma}$})$ of Example \ref{forget the grading}.
We can make a similar statement for general complexes ${\mathfrak mathcal M}^{\scriptscriptstyle \bullet}
\in D^b({\mathfrak mathcal A})$ if we replace the functors on the right hand
sides of (a), (b), (c), and (d) with the appropriate derived
functors: replace $\otimes_{{\mathfrak mathcal A}_{\sigma}}$ by $\mathfrak mathop{\rm st}\nolimitsackrel{L}{\otimes}_{{\mathfrak mathcal A}_{\sigma}}$
and $-({\sigma},\partial {\sigma}) = {\mathfrak mathbb G}amma_{\sigma}(-)|_{\{{\sigma}\}}$ by
${\mathfrak mathbf R}{\mathfrak mathbb G}amma_{\sigma}(-)|_{\{{\sigma}\}}$.
\subsection{Mixed structure} \langlebel{mixed equivariant sheaves}
For any $n\in {\mathfrak mathbb Z}$,
define an automorphism of $D^b({\mathfrak mathcal A})$ by $\langle n\rangle = [n]\lb -n\rb $.
It is obviously $t$-exact, so it induces an automorphism of
$P({\Sigma})$. Define a functorial filtration on objects of
$P({\Sigma})$ by $W_j{\mathfrak mathcal P}^{\scriptscriptstyle \bullet} = \oplus_{i\ge -j} {\mathfrak mathcal P}^i$, assuming
${\mathfrak mathcal P}^{\scriptscriptstyle \bullet}$ is a complex satisfying condition (*) of Theorem
\ref{t-structure thm}. It is easy to see this defines a mixed structure on
$P({\Sigma})$. We will show in the next section that the functor
$F_T\text{\rm co-}lon P({\Sigma})\to P_T(X_{\Sigma})$ is a grading on $P_T(X_{\Sigma})$
in the sense of \S\ref{grading section}.
\section{The toric Koszul functor} \langlebel{main proofs}
\subsection{}
Let $X = X_{\sigma}$ be a normal affine $T$-toric variety with a single
fixed point defined by a
cone ${\sigma}ma \subset N_T\otimes {\mathfrak mathbb R}$,
with $\dim {\sigma} = \ranglenk N_T = n$. Let $\check X$ be
the dual $\check T$-toric variety defined by the dual cone $\check
{\sigma}ma \subset N_{\check T}\otimes {\mathfrak mathbb R}$.
Put ${\Sigma} = {\Sigma} _X =[{\sigma}ma]$,
$\check {\Sigma} = {\Sigma} _{\check X}=[\check {\sigma}ma]$, ${\mathfrak mathcal A} = {\mathfrak mathcal A}_{\Sigma}$,
and ${\mathfrak mathcal T} = {\mathfrak mathcal T}_{\check {\Sigma}}$. Recall the identification
of ringed quivers
\begin{equation}\langlebel{aaa}
({\Sigma},{\mathfrak mathcal A}) = ((\check{\Sigma})^\circ,{\mathfrak mathcal T})
\end{equation}
from \S\ref{dual toric varieties}.
In this section we define our Koszul equivalence
\[K\text{\rm co-}lon D^b({\mathfrak mathcal A}\text{\rm -mod}_{\text{\it f}})\to D^b(LC_{\mathfrak mathcal F}(X^\vee)).\]
It will be a composition of three equivalences:
\[D^b({\mathfrak mathcal A}\text{\rm -mod}_{\text{\it f}}) \mathfrak mathop{\rm st}\nolimitsackrel{\kappa}{\to} D^b(\text{\rm co-}{\mathfrak mathcal A}\text{\rm -mod}_{\text{\it cf}})
\to D^b(LC_{\mathfrak mathcal F}(X^\vee)) \mathfrak mathop{\rm st}\nolimitsackrel{\langle -n\rangle}\to D^b(LC_{\mathfrak mathcal F}(X^\vee));\]
The last functor $\langle -n \rangle$ is the twist defined in \S\ref{F-sheaves}.
The middle functor is the equivalence
of Theorem \ref{combinatorial F-sheaves} combined with
\eqref{aaa}. The functor $\kappa$ is a combinatorial form of Koszul duality
which makes sense for any fan, rational or not. We define it in the
next section.
Our main results can be summarized as follows.
\begin{thm} \langlebel{K is Koszul}
$K$ is a Koszul equivalence in the sense of Definition
\ref{Koszul functor definition}. Here we use the $t$-structures
and mixed structure on
$D^b({\mathfrak mathcal A}\text{\rm -mod}_{\text{\it f}})$ and $D^b(LC_{\mathfrak mathcal F}(X^\vee))$ defined in \S\ref{mixed F-sheaves} and
\S\ref{mixed equivariant sheaves}, and the ring $R$, resp.\ $R^\vee$, is
the opposed ring of the
graded endomorphism ring of a mixed projective generator of $P({\mathfrak mathcal A}_{\Sigma})$ (resp.\
a mixed injective generator of $P_{\mathfrak mathcal F}(X^\vee)$).
\end{thm}
\subsection{Combinatorial Koszul functor} \langlebel{combinatorial Koszul}
Fix a fan ${\Sigma}$ (rational or not) in ${\mathfrak mathbb R} ^n$ with
the corresponding ``structure sheaf" ${\mathfrak mathcal A} ={\mathfrak mathcal A} _{\Sigma}$.
We define the functor $\kappa\text{\rm co-}lon D^b({\mathfrak mathcal A}\text{\rm -mod}_{\text{\it f}}) {\to} D^b(\text{\rm co-}{\mathfrak mathcal A}\text{\rm -mod}_{\text{\it cf}})$
as follows.
\begin{defn}
Fix an abelian category ${\mathfrak mathcal C}$.
\begin{enumerate}
\item[a)] A ${\Sigma}$-{\it diagram} in ${\mathfrak mathcal C}$ is a collection $\{ {\mathfrak mathcal M}
_\tau\} _{\tau\in{\Sigma}}$ of objects of ${\mathfrak mathcal C}$ together with
with morphisms $p_{\tau \xi}\text{\rm co-}lon {\mathfrak mathcal M} _\tau \to {\mathfrak mathcal M} _\xi$ for $\tau
\geq \xi$, satisfying $p_{\rho\xi}p_{\tau\rho} = p_{\tau\xi}$
whenever $\tau \geq \rho \geq \xi$.
\item[b)] Fix an orientation of each cone in ${\Sigma}$. Then every ${\Sigma}
$-diagram ${\mathfrak mathcal M}=\{ {\mathfrak mathcal M} _\tau \}$ gives rise to the corresponding {\it
cellular complex} in ${\mathfrak mathcal C}$:
$$C^\bullet({\mathfrak mathcal M})=\ \bigoplus_{\dim (\tau)=n}{\mathfrak mathcal M} _\tau \to
\bigoplus_{\dim (\xi)=n-1}{\mathfrak mathcal M} _\xi \to ...$$
where the terms
${\mathfrak mathcal M}_\rho$ appear in degree $- \dim \rho$, and the differential is
the sum of the maps $p_{\tau \xi}$ with $\pm$ sign depending on
whether the orientations of $\tau $ and $\xi$ agree or not.
\end{enumerate}
\end{defn}
\begin{lemma} \langlebel{Cech complex of constant sheaf}
Let ${\mathfrak mathcal M}=\{{\mathfrak mathcal M} _\tau \}$ be a constant ${\Sigma}$-diagram supported
between cones $\eta $ and $\xi$ in ${\Sigma}$. That is ${\mathfrak mathcal M} _\tau =M$
for a fixed $M$ if $\eta \geq \tau \geq \xi$, and ${\mathfrak mathcal M}_\tau =0$
otherwise; for $\eta \geq \tau _1 \geq \tau _2 \geq \xi$ the maps
$p_{\tau _1\tau _2}$ are the identity. If $\eta\ne \xi$, then the
cellular complex $C^\bullet ({\mathfrak mathcal M})$ is acyclic.
\end{lemma}
\begin{proof} The complex $C^\bullet ({\mathfrak mathcal M})$ is isomorphic to an
augmented cellular
chain complex of a closed ball of dimension $\dim (\eta )-\dim
(\xi)-1$.
\end{proof}
Recall that the sheaves ${\mathfrak mathcal A} ^*_{[\tau ]}$ are injective objects of
co-${\mathfrak mathcal A} \text{-Mod}$ for every $\tau \in {\Sigma}$. Consider the
${\Sigma}$-diagram ${\mathfrak mathcal K}=\{ {\mathfrak mathcal K} _\tau \}$ in co-${\mathfrak mathcal A} \text{-Mod}$, where
${\mathfrak mathcal K} _\tau ={\mathfrak mathcal A} ^*_{[\tau ]}$ and the maps $p_{\tau \xi}$ are the
projections. This diagram ${\mathfrak mathcal K}$ defines a covariant functor
$$\kappa \text{\rm co-}lon D^b({\mathfrak mathcal A}\text{\rm -mod}_{\text{\it f}})\to D^b(\text{\rm co-}{\mathfrak mathcal A}\text{\rm -mod}_{\text{\it cf}})$$
in the following
way. If ${\mathfrak mathcal N} \in {\mathfrak mathcal A}\text{\rm -mod}_{\text{\it f}}$ is locally free, the collection \[{\mathfrak mathcal K}
\otimes {\mathfrak mathcal N}=\{ {\mathfrak mathcal K} _\tau \otimes _{{\mathfrak mathcal A} _\tau}{\mathfrak mathcal N} _\tau \}\] is a ${\Sigma}
$-diagram of co-${\mathfrak mathcal A}$-modules. Thus its cellular complex $C^\bullet
({\mathfrak mathcal K} \otimes {\mathfrak mathcal N})$ is a complex of co-${\mathfrak mathcal A}$-modules. By Theorem
\ref{homotopy of pure sheaves}, every
object in $D^b({\mathfrak mathcal A}\text{\rm -mod}_{\text{\it f}})$ is quasi-isomorphic to a complex of locally free
(in fact, pure) ${\mathfrak mathcal A}$-modules, so we obtain a derived functor
$$\kappa(\cdot )=C^\bullet({\mathfrak mathcal K} \mathfrak mathop{\rm st}\nolimitsackrel{{\mathfrak mathbb L}}{\otimes}\cdot)\text{\rm co-}lon
D^b({\mathfrak mathcal A}\text{\rm -mod}_{\text{\it f}})\to D^b(\text{\rm co-}{\mathfrak mathcal A}\text{\rm -mod}_{\text{\it cf}}).$$
We call it the combinatorial Koszul
functor.
\begin{lemma} \langlebel{calculations of kappa}
For any cone $\tau\in {\Sigma}$, there are isomorphisms $\kappa
({\mathfrak mathcal A}_{\{\tau \}})\text{\rm co-}ng {\mathfrak mathcal A}^*_{[\tau]}[\dim\tau]$ and $\kappa
({\mathfrak mathcal A}_{[\tau ]})\text{\rm co-}ng{\mathfrak mathcal A}^*_{\{ \tau\}}[\dim\tau]$.
\end{lemma}
The first isomorphism is obvious; the second follows from Lemma
\ref{Cech complex of constant sheaf}.
\begin{prop} The functor $\kappa$ is an equivalence of triangulated categories.
\end{prop}
\begin{proof}
The category $D^b({\mathfrak mathcal A} _f\text{-Mod})$ is the triangulated envelope of
either all objects of the form ${\mathfrak mathcal A}_{[\tau]}\lb k\rb$
or all objects of the form ${\mathfrak mathcal A} _{\{\tau\}} \lb k\rb$, in either case taken over all
$\tau \in {\Sigma}$ and $k\in {\mathfrak mathbb Z}$. Similarly, $D^b(\text{\rm co-}{\mathfrak mathcal A}_{cf}\text{-Mod})$) is the
triangulated envelope of all objects of the form ${\mathfrak mathcal A}^*_{[\tau]} \lb k\rb$ or
all objects of the form ${\mathfrak mathcal A}^*_{\{\tau \}} \lb k\rb$.
So it suffices to show that for any $k,l,\tau , \xi$ the functor
$\kappa$ induces an isomorphism
$$\kappa \text{\rm co-}lon \operatorname{Hom} _{{\mathfrak mathcal A} _f\text{-Mod}}({\mathfrak mathcal A} _{[\tau]}\lb k\rb , {\mathfrak mathcal A}
_{\{\xi \}}\lb l\rb )\to \operatorname{Hom} _{\text{\rm co-}{\mathfrak mathcal A} _{cf}\text{-Mod}}({\mathfrak mathcal A} ^*_{\{\tau
\}}\lb k\rb ,{\mathfrak mathcal A} ^*_{[\xi]}\lb l\rb ).$$
Both sides are equal to the $l-k$
graded part of ${\mathfrak mathcal A} _{\tau}$ if $\tau =\xi$ and vanish otherwise.
\end{proof}
Thus $K$ satisfies property (1) of the definition of
a Koszul equivalence (Definition
\ref{Koszul functor definition}). Property (2) follows immediately
from the definition of the twist functors $\langle n\rangle$ in the
categories $D^b({\mathfrak mathcal A})$ and $D^b(LC_{\mathfrak mathcal F}(\check X))$.
Showing that $K$ sends simples to injectives and indecomposable
projectives to simples (properties (3) and (4)) will take up the
remainder of the paper.
\begin{rmk} We can think of $\kappa$ as convolution with
kernel ${\mathfrak mathcal K}$. This is more enlightening if one considers ${\mathfrak mathcal K}$
as a sheaf on ${\Sigma} \times (\check {\Sigma})^\circ$ by the natural
identification ${\Sigma} = (\check {\Sigma})^\circ$.
The support of ${\mathfrak mathcal K}$ is then the ``combinatorial conormal variety''
\[\Lambda = \{(\tau,\alpha)\in {\Sigma} \times (\check {\Sigma})^\circ \mathfrak mid
\tau^\bot \le \alpha\}.\] If $p_1\text{\rm co-}lon \Lambda \to {\Sigma}$, $p_2\text{\rm co-}lon \Lambda \to
(\check {\Sigma})^\circ$ are the projections,
then ${\mathfrak mathcal K} = p_2^{-1}{\mathfrak mathcal T}^*$, using the identification \eqref{aaa}.
Note that ${\mathfrak mathcal K}$ has a natural action of $p_1^{-1}{\mathfrak mathcal A}$ which
commutes with the action of $p_2^{-1}{\mathfrak mathcal T}$. In fact,
$\Lambda$ is the largest subset of ${\Sigma} \times (\check {\Sigma})^\circ$
for which this is true.
\end{rmk}
\subsection{Proof of Theorem \ref{K is Koszul}, part I:
$K(\text{simple})$ is injective}
Let us examine what the functor $K$ does to indecomposable pure
sheaves. For a face $\alpha \in [\check{\sigma}]$, define $I^{\scriptscriptstyle \bullet}_\alpha
= K({\mathfrak mathcal L}^{\alpha^\bot})$. It is perverse, as follows from the
following more general statement. Recall the objects
$\nabla_\alpha^{\scriptscriptstyle \bullet} \in P_{\mathfrak mathcal F}(\check X)$ from \S\ref{perverse
t-structure}.
\begin{prop} \langlebel{locally free to perverse}
If ${\mathfrak mathcal M}^{\scriptscriptstyle \bullet}\in D^b({\mathfrak mathcal A}\text{\rm -mod}_{\text{\it f}})$ is given by placing a locally free
${\mathfrak mathcal A}$-module in degree $0$, then $K({\mathfrak mathcal M}^{\scriptscriptstyle \bullet})$ is perverse, and has
a filtration whose graded pieces are objects $\nabla_\alpha^{\scriptscriptstyle \bullet}\langle
k\rangle$, $\alpha \in \check {\Sigma}$, $k \in {\mathfrak mathbb Z}$.
\end{prop}
\begin{proof} A locally free ${\mathfrak mathcal A}$-module has a filtration whose
subquotients are sheaves ${\mathfrak mathcal A}_{\{\tau\}}\{ k\}$, $\tau\in {\Sigma}$,
$k\in {\mathfrak mathbb Z}$. By Lemma \ref{calculations of kappa}, we have
$K({\mathfrak mathcal A}_{\{\tau\}}) \text{\rm co-}ng \nabla_{\tau^\bot}^{\scriptscriptstyle \bullet}\langle \dim \tau - n
\rangle$. The result follows.
\end{proof}
\begin{thm} \langlebel{K(simple) is injective}
$I^{\scriptscriptstyle \bullet}_\alpha$ is an injective object in $P_{\mathfrak mathcal F}(X)$, and its
image $F_{cf}(I^{\scriptscriptstyle \bullet}_\alpha)$ is injective in $P_{cf}(X)$.
With respect
to the mixed structure defined in \S\ref{mixed F-sheaves},
we have $W_0I^{\scriptscriptstyle \bullet}_\alpha \text{\rm co-}ng L^{\scriptscriptstyle \bullet}_\alpha$.
\end{thm}
\begin{proof}
To show that $I^{\scriptscriptstyle \bullet}_\alpha$ is injective, we will show that the
following statement holds for any $S^{\scriptscriptstyle \bullet} \in P_{\mathfrak mathcal F}(\check X)$:
\begin{equation}
\tag{*} \operatorname{Hom}_{D^b(LC_{\mathfrak mathcal F}(\check X))}(S^{\scriptscriptstyle \bullet}, I^{\scriptscriptstyle \bullet}_\alpha[k]) =
0 \;\text{for all}\; k > 0.
\end{equation}
First note that if
\[0\to S_1^{\scriptscriptstyle \bullet} \to S_2^{\scriptscriptstyle \bullet} \to S_3^{\scriptscriptstyle \bullet} \to 0\]
is a short exact sequence in $P_{\mathfrak mathcal F}(\check X)$ and (*) holds for
$S_1^{\scriptscriptstyle \bullet}$ and $S_3^{\scriptscriptstyle \bullet}$ or for $S_2^{\scriptscriptstyle \bullet}$ and $S_3^{\scriptscriptstyle \bullet}$, then
it also holds for all three objects.
Note that (*) holds for $S^{\scriptscriptstyle \bullet} = \nabla_\beta^{\scriptscriptstyle \bullet}\langle k\rangle$ for any
$\beta\in [\check{\sigma}]$ and $k\in {\mathfrak mathbb Z}$, by applying $K$ to Corollary
\ref{loc free --> pure}. Thus (*) holds for any object of the form
$S^{\scriptscriptstyle \bullet} = {\mathfrak mathbf R} j_{\beta *}E_\beta[\dim O_\beta]$, where $E_\beta$ is
any object in $LC_{\mathfrak mathcal F}(O_\beta)$, since ${\mathfrak mathbf R} j_{\beta *}$ is $t$-exact
(Proposition \ref{lower star is exact}),
and $E_\beta$ can be resolved by a finite complex of injective
${\mathfrak mathcal F}$-local systems, i.e.\ by direct sums of copies of objects
$\Theta_\beta\langle k\rangle$, $k\in {\mathfrak mathbb Z}$.
Now we prove (*) for general $S^{\scriptscriptstyle \bullet}$, by induction on the number of
orbits in the support. If $O_\beta$ is an open orbit contained in
$\operatorname{Supp} S^{\scriptscriptstyle \bullet}$, then $S^{\scriptscriptstyle \bullet}|_{O_\beta}$ is a ${\mathfrak mathcal F}$-local system
placed in degree $-\dim O_\beta$. Consider the adjunction map
$\phi\text{\rm co-}lon S^{\scriptscriptstyle \bullet} \to {\mathfrak mathbf R} j_{\beta *}(S^{\scriptscriptstyle \bullet}|_{O_{\beta}})$, and
note that (*) holds for the target of $\phi$ by the previous
paragraph. If $\operatorname{Supp} S^{\scriptscriptstyle \bullet}$ consists of the unique closed orbit
$O_{{\sigma}^\vee}$, so $\beta = \check {\sigma}$, then $\phi$ is an
isomorphism, and we are done. Otherwise note that (*) holds by
induction for the kernel and cokernel of $\phi$, since they have
strictly smaller support, and thus it holds for $S^{\scriptscriptstyle \bullet}$.
Thus
$I^{\scriptscriptstyle \bullet}_\alpha$ is injective. The same argument shows that
$F_{cf}(I^{\scriptscriptstyle \bullet}_\alpha)$ is injective in $P_{cf}(X)$.
Apply Proposition \ref{locally free to perverse} to obtain a
filtration
\[M^{\scriptscriptstyle \bullet}_0 \subset \dots \subset M^{\scriptscriptstyle \bullet}_l = I^{\scriptscriptstyle \bullet}_\alpha\]
with $M_0^{\scriptscriptstyle \bullet} = \nabla^{\scriptscriptstyle \bullet}_\alpha$ , and where the
$M^{\scriptscriptstyle \bullet}_i/M^{\scriptscriptstyle \bullet}_{i-1}$ for $i >0$ are sums of objects
$\nabla^{\scriptscriptstyle \bullet}_\beta\langle k\rangle$ with $\beta \in [\alpha]\setminus
\{\alpha\}$ and $k \in {\mathfrak mathbb Z}$. In fact, using Lemma \ref{calculations of
kappa} and property (2) of Theorem \ref{pure simples}, we see that
only twists $k > 0$ can occur. The remaining statements of the
theorem follow using Corollary \ref{weights of nabla}.
\end{proof}
Let $I^{\scriptscriptstyle \bullet} = \oplus_{\alpha \in [{\sigma}^\vee]} I^{\scriptscriptstyle \bullet}_\alpha$, so
$I^{\scriptscriptstyle \bullet} = K({\mathfrak mathcal L})$, where ${\mathfrak mathcal L} = \oplus_{\tau\in [{\sigma}]} {\mathfrak mathcal L}^\tau$.
\begin{prop} $I^{\scriptscriptstyle \bullet}$ is a mixed injective generator (\S\ref{mixed categories})
of $P_{\mathfrak mathcal F}(\check X)$; $F_{cf}(I^{\scriptscriptstyle \bullet})$ is an
injective generator of $P_{cf}(\check X)$.
\end{prop}
\begin{proof}
Let $Inj$ be the category of all finite direct sums of objects
of the form $I^{\scriptscriptstyle \bullet}_\alpha\langle k\rangle$, $\alpha \in [\check {\sigma}]$, $k\in
{\mathfrak mathbb Z}$. We need to show that any object of $P_{\mathfrak mathcal F}(\check X)$ embeds into
an object of $Inj$.
We showed in the previous proof that $\nabla^{\scriptscriptstyle \bullet} _\alpha\langle k\rangle =
j_{\alpha *}\Theta_\alpha[\dim O_\alpha]\langle k\rangle$ embeds into
$I^{\scriptscriptstyle \bullet}_\alpha\langle k\rangle$.
Let $E_\alpha$ be a ${\mathfrak mathcal F}$-local system on $O_\alpha$. It
embeds into a finite direct sum of injective ${\mathfrak mathcal F}$-local systems
$\Theta_\alpha\langle n\rangle$, so $j_{\alpha *}E_\alpha[\dim O_\alpha]$
embeds into a finite sum of $I^{\scriptscriptstyle \bullet}_\alpha\langle k\rangle$.
We prove that an arbitrary $S^{\scriptscriptstyle \bullet} \in P_{\mathfrak mathcal F}(\check X)$ embeds into
an object of $Inj$ by induction on the number of orbits in $\operatorname{Supp}
S^{\scriptscriptstyle \bullet}$. The case when $\operatorname{Supp} S^{\scriptscriptstyle \bullet}$ is a single orbit
follows from the previous paragraph. Otherwise,
take an open orbit $O_\beta$ in $\operatorname{Supp} S^{\scriptscriptstyle \bullet}$, and let
$\phi\text{\rm co-}lon S^{\scriptscriptstyle \bullet} \to {\mathfrak mathbf R} j_{\beta *}(S^{\scriptscriptstyle \bullet}|_{O_{\beta}})$ be
the adjunction map. We have seen that the target of $\phi$
embeds into an object $I^{\scriptscriptstyle \bullet}_1 \in Inj$, so the image of $\phi$
does as well. Since $\mathop{\rm ker}\nolimits \phi$ has support strictly smaller than
$S^{\scriptscriptstyle \bullet}$, by induction it embeds into an object $I^{\scriptscriptstyle \bullet}_2\in Inj$.
Since $I^{\scriptscriptstyle \bullet}_2$ is injective, this embedding extends to a map
$S^{\scriptscriptstyle \bullet} \to I^{\scriptscriptstyle \bullet}_2$. Thus we get an embedding of $S^{\scriptscriptstyle \bullet}$
into $I^{\scriptscriptstyle \bullet}_1 \oplus I^{\scriptscriptstyle \bullet}_2 \in Inj$.
The argument for $P_{cf}(\check X)$ is essentially the same.
\end{proof}
Now define
\[R = \tmop{end}(I^{\scriptscriptstyle \bullet})^{opp} \text{\rm co-}ng
\oplus_{n\in {\mathfrak mathbb Z}}\operatorname{Hom}_{{\mathfrak mathcal A}\mathfrak mod}({\mathfrak mathcal L},{\mathfrak mathcal L}\{n\})^{opp}.\]
By Lemma \ref{Noetherian}, $R$ is (left and right) Noetherian.
Then applying Propositions
\ref{mixed categories are modules} and \ref{ungraded modules}, we
conclude that $P_{\mathfrak mathcal F}(\check X)$ is equivalent to $R\text{\rm -mod}_{\text{\it cf}}$,
$P_{cf}(\check X)$ is equivalent to $R\text{\rm -Mod}_{\text{\it cf}}$, and $F_{cf}$ is the
functor of forgetting the grading.
\begin{cor} \langlebel{unipotent equivalence}
There are equivalences of triangulated categories:
$D^b(LC_{\mathfrak mathcal F}(\check X))\simeq D^b(P_{\mathfrak mathcal F}(\check X))$ and
$D^b(LC_{cf}(\check X))\simeq D^b(P_{cf}(\check X))$.
\end{cor}
The argument for the two equivalences is the same, so we concentrate
on the first one. Both $D^b(LC_{\mathfrak mathcal F}(\check X))$ and $D^b(P_{\mathfrak mathcal F}(\check
X))$ are generated by the injectives in $P_{\mathfrak mathcal F}(\check X)$; note that
any complex has a bounded injective resolution, since any object of
$D^b({\mathfrak mathcal A})$ can be represented by a bounded complex of pure sheaves.
We thus need to show that for any injectives $I^{\scriptscriptstyle \bullet}_1$, $I^{\scriptscriptstyle \bullet}_2
\in P_{\mathfrak mathcal F}(\check X)$
and any $d \in {\mathfrak mathbb Z}$ there is an isomorphism
\[\operatorname{Ext}^d_{P_{\mathfrak mathcal F}(\check X)}(I^{\scriptscriptstyle \bullet}_1, I^{\scriptscriptstyle \bullet}_2) \mathfrak mathop{\rm st}\nolimitsackrel{\sim}{\to}
\operatorname{Hom}_{D^b(LC_{\mathfrak mathcal F}(\check X))}(I^{\scriptscriptstyle \bullet}_1, I^{\scriptscriptstyle \bullet}_2[d]).\] Both sides
are automatically isomorphic for $d \le 0$, while for $d > 0$ the left
side vanishes by the injectivity of $I^{\scriptscriptstyle \bullet}_2$. The vanishing of the
right side is just (*) from the proof of Theorem \ref{K(simple) is
injective}.
\subsection{Proof of Theorem \ref{K is Koszul}, part II:
$K^{-1}(\text{simple})$ is projective}
For any $\tau \in [{\sigma}]$, let ${\mathfrak mathcal P}^{\scriptscriptstyle \bullet}_\tau = K^{-1}(L^{\scriptscriptstyle \bullet}_{\tau^\bot})$.
Analogously to Theorem \ref{K(simple) is injective},
we have
\begin{thm} \langlebel{K(projective) is simple}
${\mathfrak mathcal P}^{\scriptscriptstyle \bullet}_\tau$ lies in the core of the perverse $t$-structure on
$D^b({\mathfrak mathcal A})$ defined in \S\ref{t-structure on cA-complexes}.
In the abelian category $P({\mathfrak mathcal A})$, it is the projective cover
of ${\mathfrak mathcal L}^\tau$.
\end{thm}
\begin{proof} Let $L^{\scriptscriptstyle \bullet} = L^{\scriptscriptstyle \bullet}_{\tau^\bot}$.
Consider an injective resolution of $L^{\scriptscriptstyle \bullet}$ (as remarked before, it
can be chosen to be bounded):
\[L^{\scriptscriptstyle \bullet} \mathfrak mathop{\rm st}\nolimitsackrel\sim\to (J^{\scriptscriptstyle \bullet}_0 \to J^{\scriptscriptstyle \bullet}_1 \to \dots \to J^{\scriptscriptstyle \bullet}_k).\]
Taking $K^{-1}$ gives a complex ${\mathfrak mathcal M}_0\to {\mathfrak mathcal M}_1 \to\dots\to {\mathfrak mathcal M}_k$ of pure
${\mathfrak mathcal A}$-modules which represents the object ${\mathfrak mathcal P}^{\scriptscriptstyle \bullet}_\tau$.
This complex will be perverse if ${\mathfrak mathcal M}_j[-j]$ is perverse
for $j = 0,\dots,k$, or in other words, if each $J^{\scriptscriptstyle \bullet}_l$ is a direct sum
of objects $I^{\scriptscriptstyle \bullet}_{\alpha}\langle l\rangle$, $\alpha \in \check {\Sigma}$.
The existence such a resolution follows from Proposition
\ref{unipotent Koszul} and Corollary \ref{unipotent equivalence}.
Since objects of $P({\mathfrak mathcal A})$ have finite length, to show that ${\mathfrak mathcal P}^{\scriptscriptstyle \bullet}_\tau$
is the projective cover of ${\mathfrak mathcal L}^\tau$ it will be enough to show that for any
$\rho \in {\Sigma}$, $k,l \in {\mathfrak mathbb Z}$ we have
$\operatorname{Hom}_{D^b({\mathfrak mathcal A})}({\mathfrak mathcal P}^{\scriptscriptstyle \bullet}_\tau, {\mathfrak mathcal L}^\rho[k]\langle l\rangle)$ is one-dimensional
if $k = l = 0$ and $\rho = \tau$, and vanishes otherwise.
By applying $K$, this follows from Theorem \ref{K(simple) is injective}.
\end{proof}
Define ${\mathfrak mathcal P}^{\scriptscriptstyle \bullet} := \oplus_{\tau\in[{\sigma}]} {\mathfrak mathcal P}^{\scriptscriptstyle \bullet}_\tau$, and let
\[\check R = \tmop{end}_{P({\mathfrak mathcal A})}({\mathfrak mathcal P}^{\scriptscriptstyle \bullet})^{opp}.\]
\begin{cor} ${\mathfrak mathcal P}^{\scriptscriptstyle \bullet}$ is a graded projective
generator of $P({\mathfrak mathcal A})$; $F_T({\mathfrak mathcal P}^{\scriptscriptstyle \bullet})$ is a projective generator of $P_T(X)$.
There are equivalences of abelian categories $P({\mathfrak mathcal A}) \simeq \check R\text{\rm -mod}_{\text{\it f}}$ and
$P_T(X) \simeq \check R\text{\rm -Mod}_{\text{\it f}}$; with respect to these equivalences $F_T$ is
the functor of forgetting the grading.
\end{cor}
\begin{cor} There are equivalences of triangulated categories: $D^b({\mathfrak mathcal A})\simeq D^b(P({\mathfrak mathcal A}))$
and $D^b_T(X) \simeq D^b(P_T(X))$.
\end{cor}
The proofs are the same as in the previous section; note that since objects of
$P({\mathfrak mathcal A})$ have finite length the ring $\check R$ is automatically Noetherian.
\section{Some proofs} \langlebel{appendix}
\subsection{Proof of Theorem \ref{purity of simples}}
The functor $[k]\langle -k\rangle$ on $D^b(LC_{\mathfrak mathcal F}(X))$ preserves the
property of being pure of weight $0$, so we can instead prove that
\[S^{\scriptscriptstyle \bullet} := L^{\scriptscriptstyle \bullet}_\alpha[-c(\alpha)]\langle c(\alpha)\rangle
= j_{\alpha !*}{\mathfrak mathbb C}_\alpha\] is pure of weight $0$. The support of
$S^{\scriptscriptstyle \bullet}$ is $\overline{O_\alpha}$, which is itself a toric variety (for a
smaller torus). Thus we can assume that $\alpha = o$ is the zero cone
and the support of $S^{\scriptscriptstyle \bullet}$ is all of $X$.
Let ${\mathfrak mathbf S}^{\scriptscriptstyle \bullet} = F_{cf}S^{\scriptscriptstyle \bullet}$; it is an
intersection cohomology sheaf shifted so that the restriction
to the open orbit $O_o$ is a local
system in degree $0$. The ${\mathfrak mathcal F}$-structure on $S^{\scriptscriptstyle \bullet}$
defines an isomorphism $\theta\text{\rm co-}lon {\mathfrak mathcal F}^{-1}{\mathfrak mathbf S}^{\scriptscriptstyle \bullet}
\mathfrak mathop{\rm st}\nolimitsackrel{\sim}\to {\mathfrak mathbf S}^{\scriptscriptstyle \bullet}$. It induces an action on the stalk of
the cohomology sheaves $H^i(j_\beta^*{\mathfrak mathbf S}^{\scriptscriptstyle \bullet})$ and
$H^i(j_\beta^!{\mathfrak mathbf S}^{\scriptscriptstyle \bullet})$; we need to show this action is
multiplication by $2^{i/2}$.
Note that if $O_\beta$ has positive dimension, there is an
${\mathfrak mathcal F}$-stable normal slice to $O_\beta$ at a point of $(O_\beta)^{\mathfrak mathcal F}$
which is itself an affine toric variety. By restricting to this slice
we can restrict to the case when $O_\beta = \{b\}$ is a single point.
Note that ${\mathfrak mathcal F}^{-1}{\mathfrak mathbf S}^{\scriptscriptstyle \bullet} \text{\rm co-}ng {\mathfrak mathbf S}^{\scriptscriptstyle \bullet}$ (see \cite{BM}), and all
automorphisms of ${\mathfrak mathbf S}^{\scriptscriptstyle \bullet}$ are multiplication by scalars. Therefore
$\theta$ is uniquely determined by its action on the stalk at an
${\mathfrak mathcal F}$-fixed point of the open orbit $O_o$, where it acts as the
identity.
Let $\pi \text{\rm co-}lon \widetilde{X} \to X$ be a toric resolution of singularities,
and let $\widetilde{{\mathfrak mathcal F}}$ be our geometric Frobenius map on $\widetilde{X}$; we have
$\widetilde{\mathfrak mathcal F}\pi = \pi{\mathfrak mathcal F}$.
Since $\widetilde{\mathfrak mathcal F}^*{\mathfrak mathbb C}_{\widetilde X} \text{\rm co-}ng {\mathfrak mathbb C}_{\widetilde X}$, we can put a
$\widetilde{\mathfrak mathcal F}$-structure on the constant sheaf ${\mathfrak mathbb C}_{\widetilde{X}}$ by letting
$\tilde\theta\text{\rm co-}lon \widetilde{{\mathfrak mathcal F}}^*{\mathfrak mathbb C}_{\widetilde{X}} \to {\mathfrak mathbb C}_{\widetilde{X}}$ act as the
identity on the stalk at a point of $(O_o)^{\mathfrak mathcal F}$. By adjunction
$\tilde \theta$ induces a map ${\mathfrak mathbb C}_{\widetilde{X}} \to
{\mathfrak mathbf R}\widetilde{{\mathfrak mathcal F}}_*{\mathfrak mathbb C}_{\widetilde{X}}$, and applying ${\mathfrak mathbf R}\pi_*$ and adjunction
again gives
$\theta'\text{\rm co-}lon {\mathfrak mathcal F}^*{\mathfrak mathbf R}\pi_*{\mathfrak mathbb C}_{\widetilde X}\to {\mathfrak mathbf R}\pi_*{\mathfrak mathbb C}_{\widetilde X}$.
By the decomposition theorem \cite{BBD}, ${\mathfrak mathbf S}^{\scriptscriptstyle \bullet}$ is a direct
summand of ${\mathfrak mathbf R}\pi_*{\mathfrak mathbb C}_{\widetilde X}$ and of ${\mathfrak mathcal F}^*{\mathfrak mathbf R}\pi_*{\mathfrak mathbb C}_{\widetilde X}$.
Composing $\theta'$ with the inclusion and projection gives a map
$S^{\scriptscriptstyle \bullet} \to S^{\scriptscriptstyle \bullet}$; it is easy to see that it agrees with $\theta$
on the open orbit, so it must equal $\theta$ on all of $X$.
The cohomology groups of $j_\beta^*{\mathfrak mathbf R}\pi_*{\mathfrak mathbb C}_{\widetilde X}$ and
$j_\beta^!{\mathfrak mathbf R}\pi_*{\mathfrak mathbb C}_{\widetilde X}$ are $H^{\scriptscriptstyle \bullet}(\pi^{-1}(b))$ and
$H^{\scriptscriptstyle \bullet}(\widetilde{X},\widetilde{X}\setminus \pi^{-1}(b))$, respectively, and the
action of $\theta'$ is the action of the pullback $\widetilde{{\mathfrak mathcal F}}^*$. Thus
we have reduced the proof of the theorem to showing that this action
is multiplication by $2^{i/2}$ on the cohomology in degree $i$.
Here is one way to see this: $\widetilde{X}$ has a completion to a smooth
complete toric variety $Y$. There is a homomorphism ${\mathfrak mathbb C}^* \to T$ so
that the induced action of ${\mathfrak mathbb C}^*$ on $X$ is ``attractive'':
$\lim_{t\to 0} t\cdot x = b$ for all $x\in X$, and the induced action
on $Y$ has isolated fixed points. Then by Bia\l ynicki-Birula
$\pi^{-1}(b)$ has a decomposition into $\bigcup_x {C_x}$ into affine
cells, so $H^{\scriptscriptstyle \bullet}(\pi^{-1}(b)) \text{\rm co-}ng \oplus_x H^{\scriptscriptstyle \bullet}_c(C_x)$. The
cells are $T$- and ${\mathfrak mathcal F}$-invariant, and $\widetilde{\mathfrak mathcal F}$ acts on each
$k$-dimensional cell as the map ${\mathfrak mathbb C}^k \to {\mathfrak mathbb C}^k$,
$(x_1,\dots,x_k)\mathfrak mapsto (x_1^2,\dots,x_k^2)$. The result for
$H^{\scriptscriptstyle \bullet}(\pi^{-1}(b))$ follows immediately.
For $H^{\scriptscriptstyle \bullet}(\widetilde{X},\widetilde{X}\setminus \pi^{-1}(b))$, we use the Bia\l
ynicki-Birula cells for the opposite character ${\mathfrak mathbb C}^* \to T$. Then
$\widetilde{X}$ is an open union of these cells which deformation retracts
onto $\pi^{-1}(b)$ by our action. Therefore we have
$H^{\scriptscriptstyle \bullet}(\widetilde{X},\widetilde{X}\setminus \pi^{-1}(b)) \text{\rm co-}ng H^{\scriptscriptstyle \bullet}_c(\widetilde{X})$, and
can use the argument of the previous paragraph.
\subsection{Proof of Theorem \ref{weight filtration}}
We begin by defining the filtration $W_{\scriptscriptstyle \bullet} S^{\scriptscriptstyle \bullet}$
when $S^{\scriptscriptstyle \bullet} \in P_{{\mathfrak mathcal F},c}(X)$, i.e.\
when $S^{\scriptscriptstyle \bullet}$ has finite length. We proceed by induction on the length
of $S^{\scriptscriptstyle \bullet}$. If $S^{\scriptscriptstyle \bullet}$ has length $1$, it is simple, say of weight
$m$, and we can let $W_k S^{\scriptscriptstyle \bullet} = 0$ if $k < m$, $W_k S^{\scriptscriptstyle \bullet} = S^{\scriptscriptstyle \bullet}$
for $k \ge m$.
Otherwise suppose the filtration has already been defined for objects of
smaller length.
Find a simple subobject $L^{\scriptscriptstyle \bullet}$ of $S^{\scriptscriptstyle \bullet}$,
and suppose it is pure of weight $m$. Let $\phi \text{\rm co-}lon
S^{\scriptscriptstyle \bullet} \to C^{\scriptscriptstyle \bullet} = S^{\scriptscriptstyle \bullet}/L^{\scriptscriptstyle \bullet}$ be the corresponding
quotient map. By induction we can assume
we have already defined our filtration on $C^{\scriptscriptstyle \bullet}$.
For any $k < m$ consider the
exact sequence
\[0 \to L^{\scriptscriptstyle \bullet} \to \phi^{-1} W_k C^{\scriptscriptstyle \bullet} \to W_k C^{\scriptscriptstyle \bullet} \to 0.\]
Since the simple constituents of $W_k C^{\scriptscriptstyle \bullet}$ are all pure of weights
$< m$, Proposition \ref{unipotent Koszul} implies
\[\operatorname{Hom}(L^{\scriptscriptstyle \bullet}, W_k C^{\scriptscriptstyle \bullet}) = \operatorname{Ext}^1(L^{\scriptscriptstyle \bullet}, W_k C^{\scriptscriptstyle \bullet}) = 0,\]
and so the exact sequence splits canonically. We then define
$W_kS^{\scriptscriptstyle \bullet}$ to be the image of $W_kC^{\scriptscriptstyle \bullet} \to \phi^{-1} W_k C^{\scriptscriptstyle \bullet}
\to S^{\scriptscriptstyle \bullet}$, where the first map is the splitting map.
For $k \ge m$ we let $W_k S^{\scriptscriptstyle \bullet} = \phi^{-1}(W_k C^{\scriptscriptstyle \bullet})$.
Then ${\mathfrak mathbb G}r^W_m S^{\scriptscriptstyle \bullet} \text{\rm co-}ng {\mathfrak mathbb G}r^W_m C^{\scriptscriptstyle \bullet} \oplus L^{\scriptscriptstyle \bullet}$, while
${\mathfrak mathbb G}r^W_k S^{\scriptscriptstyle \bullet} \text{\rm co-}ng {\mathfrak mathbb G}r^W_k C^{\scriptscriptstyle \bullet}$ if $k \ne m$, so
${\mathfrak mathbb G}r^W_k S^{\scriptscriptstyle \bullet}$ is pure of weight $k$ for all $k\in{\mathfrak mathbb Z}$, since
the same was true for $C^{\scriptscriptstyle \bullet}$ by induction.
Next we extend this filtration to arbitrary objects of $P_{\mathfrak mathcal F}(X)$,
which may not have finite length. In order to do this, we need to
show that for any object $S^{\scriptscriptstyle \bullet}\in P_{\mathfrak mathcal F}(X)$ there is a lower bound
on the weights of the simple constituents of $S^{\scriptscriptstyle \bullet}$. To see this,
use induction on the number of orbits in the support of $S^{\scriptscriptstyle \bullet}$. If
the support is a single orbit $O_\alpha$, the result follows from the
equivalence $P_{\mathfrak mathcal F}(O_\alpha) \simeq LC_{\mathfrak mathcal F}(O_\alpha) \simeq
\text{\rm co-}{\mathfrak mathcal T}_\alpha\text{\rm -mod}_{\text{\it cf}}$.
Otherwise, let $O_\alpha$ be an open orbit in the support of
$S^{\scriptscriptstyle \bullet}$. Let $\phi$ denote the natural adjunction morphism $S^{\scriptscriptstyle \bullet}
\to {\mathfrak mathbf R} j_{\alpha*}(S^{\scriptscriptstyle \bullet}|_{O_\alpha})$, and consider the short
exact sequence
\[0 \to \mathop{\rm ker}\nolimits \phi \to S^{\scriptscriptstyle \bullet} \to \operatorname{Im} \phi \to 0.\]
The support of $\mathop{\rm ker}\nolimits \phi$ is strictly smaller, so its weights are
bounded below by the inductive hypothesis. Thus it will suffice to
show the weights of $\operatorname{Im} \phi$ are bounded below as well. But by the
preceding paragraph the weights in $S^{\scriptscriptstyle \bullet}|_{O_\alpha}$ are bounded
below, say by $w$. Since ${\mathfrak mathbf R} j_{\alpha *}$ is a $t$-exact functor,
this implies that the weights of ${\mathfrak mathbf R}
j_{\alpha*}(S^{\scriptscriptstyle \bullet}|_{O_\alpha})$, and hence of $\operatorname{Im} \phi$, are
bounded below by $w+w'$, where $w'$ is a lower bound for the weights
of ${\mathfrak mathbf R} j_{\alpha*}{\mathfrak mathbb C}_\alpha[\dim O_\alpha]$. This lower bound exists
because ${\mathfrak mathbf R} j_{\alpha*}{\mathfrak mathbb C}_\alpha[\dim O_\alpha]$ has finite length by
Proposition \ref{constructible = finite length} (in fact, $w' = 0$
works).
The existence of the filtration $W_{\scriptscriptstyle \bullet}$ follows immediately: if
$S^{\scriptscriptstyle \bullet} \in P_{\mathfrak mathcal F}(X)$ and $k\in{\mathfrak mathbb Z}$, the collection $W_k\hat
S^{\scriptscriptstyle \bullet}$ forms a directed system over all finite-length subobjects
$\hat S^{\scriptscriptstyle \bullet}$ contained in $S^{\scriptscriptstyle \bullet}$. It vanishes identically for
$k \ll 0$, so we can proceed by induction on $k$: assume that
$W_{k-1}S^{\scriptscriptstyle \bullet}$ has been defined in such a way that $S_+^{\scriptscriptstyle \bullet} =
S^{\scriptscriptstyle \bullet}/W_{k-1}S^{\scriptscriptstyle \bullet}$ has only simple constituents of weights $\ge
k$. The family $\{W_kS^{\scriptscriptstyle \bullet}_+\}$ must be eventually constant, since
$\operatorname{Hom}(L^{\scriptscriptstyle \bullet}_\alpha\langle k\rangle, S^{\scriptscriptstyle \bullet}_+)$ is finite-dimensional for
all $\alpha$. Thus $\{W_kS^{\scriptscriptstyle \bullet}\}$ also stabilizes, so the limit is
a finite-length subobject.
The properties of a mixed category are easy to verify.
\subsection{Proof of Theorem \ref{t-exact}}
Let us briefly recall how the functor $\epsilon\text{\rm co-}lon
D^b_T(X) \to D^b({\mathfrak mathcal A}_X\text{\rm -mod}_{\text{\it f}})$ of \cite{L} is defined. Since $X$
is affine, we can choose a $T$-equivariant embedding $X \hookrightarrow
{\mathfrak mathbb P}^n$, where the action of $T$ on ${\mathfrak mathbb P}^n$ is linear. We can choose
a representative for the classifying space $BT$ so that ${\mathfrak mathbb P}^n_T$ is
an infinite dimensional manifold in the sense of \cite{BL} --
essentially this means it is a limit of finite-dimensional manifolds
by closed embeddings. ${\mathfrak mathbb P}^n_T$ has a ``de Rham complex''
$\Omega^{\scriptscriptstyle \bullet}_{{\mathfrak mathbb P}^n_T}$ which is a resolution of the constant sheaf
${\mathfrak mathbb R}_{{\mathfrak mathbb P}^n_T}$ by soft sheaves. It is also naturally a
supercommutative sheaf of DG-algebras. We then let
$\Omega^{\scriptscriptstyle \bullet}_{X_T} = \Omega^{\scriptscriptstyle \bullet}_{{\mathfrak mathbb P}^n_T}|_{X_T}.$
Let $\pi\text{\rm co-}lon X_T\to X/T$ be the map sending $O_T$ to $O/T$ for any
$T$-orbit $O$. Given $S^{\scriptscriptstyle \bullet} \in D^b_T(X)$,
the complex $M^{\scriptscriptstyle \bullet} = \pi_*(\Omega^{\scriptscriptstyle \bullet}_{X_T} \otimes S^{\scriptscriptstyle \bullet})$ is
naturally a DG-module over the DG-sheaf $\widetilde{\mathfrak mathcal A}:=
\pi_*(\Omega^{\scriptscriptstyle \bullet}_{X_T})$. Here $\otimes$ is tensoring over ${\mathfrak mathbb R}$.
Since all ${\mathfrak mathbb R}$ sheaves are flat, $\Omega^{\scriptscriptstyle \bullet}_{X_T} \otimes
S^{\scriptscriptstyle \bullet}$ is quasi-isomorphic to $S^{\scriptscriptstyle \bullet}$.
The DG-sheaf $\widetilde{\mathfrak mathcal A}$ is formal, i.e.\ it is quasi-isomorphic to its
cohomology $H(\widetilde{\mathfrak mathcal A})$. Under the natural identification of $X/T$
with the fan ${\Sigma}$ defining $X$, $H(\widetilde{\mathfrak mathcal A})$ is canonically
isomorphic to our sheaf of rings ${\mathfrak mathcal A}_{\Sigma}$. This gives an
equivalence of categories
\begin{equation}\langlebel{xxx}
D(\text{DG-$\widetilde{\mathfrak mathcal A}$})\simeq D(\text{DG-${\mathfrak mathcal A}$})
\end{equation}
which commutes with restriction and corestriction. The functor
$\epsilon$ is the composition of this equivalence with
$\pi_*(\Omega^{\scriptscriptstyle \bullet}_{X_T} \otimes {\scriptscriptstyle \bullet})$.
\newcommand{\hat\jmath}{\hat\jmath}
Let us prove (a). Since the functor $F_{T,{\Sigma}}$ is defined locally,
we can assume that ${\Sigma} = [{\sigma}]$, so $O_{\sigma}$ is the unique closed
orbit in $X$. Let $S^{\scriptscriptstyle \bullet} = F_{T,{\Sigma}}{\mathfrak mathcal M}^{\scriptscriptstyle \bullet}$. Let $j\text{\rm co-}lon
O_{{\sigma},T} \to X_T$, $\hat\jmath\text{\rm co-}lon \{{\sigma}\}\to {\Sigma}$ denote the
inclusions, and let $\pi_{\sigma}$ be the constant map $O_{\sigma} \to
\{{\sigma}\}$, so $\pi\circ j = \hat\jmath\circ \pi_{\sigma}$. We will show that
there are quasi-isomorphisms
\[\hat\jmath^*\pi_*(\Omega^{\scriptscriptstyle \bullet}_{X_T} \otimes S^{\scriptscriptstyle \bullet}) \simeq
\pi_{{\sigma} *}j^*(\Omega^{\scriptscriptstyle \bullet}_{X_T} \otimes S^{\scriptscriptstyle \bullet}) \simeq \pi_{{\sigma}
*}(\Omega^{\scriptscriptstyle \bullet}_{O_{{\sigma},T}}\otimes j^*S^{\scriptscriptstyle \bullet}).\] This will imply
our result -- the equivalence \eqref{xxx} commutes with taking stalks,
so the left hand side is $\hat\jmath^*\nu{\mathfrak mathcal M}^{\scriptscriptstyle \bullet} = \nu({\mathfrak mathcal M}({\sigma}))$, while
the right hand side is $\epsilon(j^*F_{T,{\Sigma}}{\mathfrak mathcal M}^{\scriptscriptstyle \bullet})$. Applying
$\epsilon^{-1}$ gives (a).
The second isomorphism is standard; see \cite[Proposition 2.3.5]{KS},
for instance. For the first isomorphism, note that since the smallest
open subset of the fan $[{\sigma}]$ containing ${\sigma}$ is $[{\sigma}]$ itself,
the functor $\hat\jmath^*$ is naturally isomorphic to $\hat p_*$, where
$\hat p\text{\rm co-}lon [{\sigma}]\to \{{\sigma}\}$ is the constant map. Therefore it
will be enough to construct a quasi-isomorphism
\[\hat p_*\pi_*(\Omega^{\scriptscriptstyle \bullet}_{X_T} \otimes S^{\scriptscriptstyle \bullet}) =
\pi_{{\sigma} *}p_*(\Omega^{\scriptscriptstyle \bullet}_{X_T} \otimes S^{\scriptscriptstyle \bullet})
\mathfrak mathop{\rm st}\nolimitsackrel{\sim}{\to} \pi_{{\sigma} *}j^*(\Omega^{\scriptscriptstyle \bullet}_{X_T} \otimes
S^{\scriptscriptstyle \bullet}),\] where
\[p = p_{{\sigma},T} \text{\rm co-}lon X_T \to O_{{\sigma},T}\] is the map induced by
the projection map $p_{\sigma}$ defined in \S\ref{toric projections}.
Since $p\circ j$ is the identity on $O_{{\sigma},T}$, adjunction gives a
natural transformation $p_* \to p_*j_*j^* = j^*$. We will show that applying
it to $\widetilde S^{\scriptscriptstyle \bullet} = \Omega^{\scriptscriptstyle \bullet}_{X_T} \otimes S^{\scriptscriptstyle \bullet}$ gives a
quasi-isomorphism. Without loss of generality we can assume that
$S^{\scriptscriptstyle \bullet} = {\mathfrak mathbf R} i_{\tau *}{\mathfrak mathbb R}_{O_{\tau,T}}$, where $\tau \in [{\sigma}]$ and
$i_\tau\text{\rm co-}lon O_{\tau,T} \to X_T$ is the inclusion. Note that $\widetilde
S^{\scriptscriptstyle \bullet}$ is a complex of soft sheaves, so applying $p_*$ to it is the
same as applying ${\mathfrak mathbf R} p_*$. The stalk cohomology of ${\mathfrak mathbf R} p_*\widetilde S^{\scriptscriptstyle \bullet}$
and $j^*\widetilde S^{\scriptscriptstyle \bullet}$ at a point $x\in O_{{\sigma},T}$ are both isomorphic
to the cohomology of the torus $p^{-1}(x) \cap O_{\tau,T}$, which
implies the claim.
For (b), consider the chain of maps
\[{\mathfrak mathbf R} \pi_{{\sigma} *}(j^*\Omega^{\scriptscriptstyle \bullet}_{X_T} \otimes j^!S^{\scriptscriptstyle \bullet}) \to
{\mathfrak mathbf R} \pi_{{\sigma} *}j^!(\Omega^{\scriptscriptstyle \bullet}_{X_T}\otimes S^{\scriptscriptstyle \bullet})
\mathfrak mathop{\rm st}\nolimitsackrel{\sim}{\to} \hat\jmath^! {\mathfrak mathbf R}\pi_*(\Omega^{\scriptscriptstyle \bullet}_{X_T}\otimes
S^{\scriptscriptstyle \bullet}).\] For the first map see \cite[Proposition 3.1.11]{KS}; the
second map is the usual base change. To check this is an isomorphism
it is enough to consider the case $S^{\scriptscriptstyle \bullet} = {\mathfrak mathbf R} i_{\tau *}
{\mathfrak mathbb R}_{O_{\tau,T}}$ as before. If $\tau = {\sigma}$ the isomorphism is
clear, while if $\tau \ne {\sigma}$ both sides vanish. (b) then follows.
The isomorphisms (c) and (d) follow from these statements using
results of \cite{BL}. In the case of a single orbit $O_{\sigma}$,
the equivalence
$\epsilon\text{\rm co-}lon D^b_T(O_{\sigma}) \to D_f(\text{DG-}{\mathfrak mathcal A}_{\sigma})$
can be factored as an equivalence
\begin{equation} \langlebel{yyy}
D^b_T(O_{\sigma})
\mathfrak mathop{\rm st}\nolimitsackrel\sim\to D^b_{T/T_{\sigma}}(pt)
\end{equation}
\cite[Theorem 2.6.2]{BL} (here $T_{\sigma}$ is the stabilizer of any
point of $O_{\sigma}$) followed by an equivalence
$D^b_{T/T_{\sigma}}(pt) \mathfrak mathop{\rm st}\nolimitsackrel\sim\to D_f(\text{DG-}{\mathfrak mathcal A}_{\sigma})$
\cite[Theorem 12.7.2(ii)]{BL}. The pullback functor $i^*_x$
is $Q^*_f$, where $f\text{\rm co-}lon \{y\} \to O_{\sigma}$
is the inclusion of a point $\{y\}$, which is a $\phi$-map for
the homomorphism $\phi\text{\rm co-}lon \{1\} \to T$ (for the definition and
properties of $Q^*_f$, see \cite[\S3.6]{BL}). The equivalence
\eqref{yyy} is $Q^*_g$, where $g\text{\rm co-}lon O_{\sigma}\to pt$ is the quotient
map and $pt$ carries a $T/T_{\sigma}$-action. Thus $i^*_x = Q^*_fQ^*_g =
Q^*_{gf}$. Applying \cite[Theorem 12.7.2(iii)]{BL} completes the proof.
\end{document} |
\begin{document}
\title{The spectral function of a first order elliptic system}
\maketitle
\begin{abstract}
We consider an elliptic self-adjoint first order pseudodifferential
operator acting on columns of complex-valued half-densities over
a connected compact manifold without boundary. The eigenvalues of the principal
symbol are assumed to be simple but no assumptions are made on their
sign, so the operator is not necessarily semi-bounded.
We study the following objects:
the propagator (time-dependent operator which solves
the Cauchy problem for the dynamic equation),
the spectral function (sum of squares of Euclidean norms
of eigenfunctions evaluated at a given point of the manifold,
with summation carried out over all eigenvalues between zero
and a positive~$\lambda$) and
the counting function (number of eigenvalues between zero
and a positive~$\lambda$).
We derive explicit two-term asymptotic formulae for all three.
For the propagator ``asymptotic'' is understood as asymptotic
in terms of smoothness, whereas for the spectral and counting
functions ``asymptotic'' is understood as asymptotic with respect
to $\lambda\to+\infty$.
\end{abstract}
\textbf{Mathematics Subject Classification (2010).}
Primary 35P20; Secondary 35J46, 35R01.
\
\textbf{Keywords.}
Spectral theory, asymptotic distribution of eigenvalues.
\section{Main results}
\label{Main results}
The aim of the paper is to extend the classical results of \cite{DuiGui}
to systems. We are motivated by the observation that,
to our knowledge, all previous publications on systems give
formulae for the second asymptotic coefficient that are either
incorrect or incomplete (i.e.~an algorithm for the calculation
of the second asymptotic coefficient rather than an actual formula).
The appropriate bibliographic review is presented in
Section~\ref{Bibliographic review}.
Consider a first order classical pseudodifferential
operator $A$ acting on columns
$v=\begin{pmatrix}v_1&\ldots&v_m\end{pmatrix}^T$
of complex-valued half-densities
over a connected compact $n$-dimensional manifold $M$ without boundary.
Throughout this paper we assume that $m,n\ge2\,$.
We assume the coefficients of the operator $A$ to be infinitely smooth. We also
assume that the operator $A$ is formally self-adjoint (symmetric):
$\int_Mw^*Av\,dx=\int_M(Aw)^*v\,dx$ for all infinitely smooth
$v,w:M\to\mathbb{C}^m$. Here and further on
the superscript $\,{}^*\,$ in matrices, rows and columns
indicates Hermitian conjugation in $\mathbb{C}^m$
and $dx:=dx^1\ldots dx^n$, where $x=(x^1,\ldots,x^n)$ are local
coordinates on $M$.
Let $A_1(x,\xi)$ be the principal symbol of the operator $A$.
Here $\xi=(\xi_1,\ldots,\xi_n)$ is the variable dual to the position
variable $x$; in physics literature the $\xi$ would be referred to
as \emph{momentum}. Our principal symbol $A_1$ is an $m\times m$
Hermitian matrix-function on $T'M:=T^*M\setminus\{\xi=0\}$,
i.e.~on the cotangent bundle with the zero section removed.
Let $h^{(j)}(x,\xi)$ be the eigenvalues of the principal symbol. We
assume these eigenvalues to be nonzero (this is a version of the
ellipticity condition) but do not make any assumptions on their
sign. We also assume that the eigenvalues $h^{(j)}(x,\xi)$ are
simple for all $(x,\xi)\in T'M$. The techniques developed in our
paper do not work in the case when eigenvalues of the principal
symbol have variable multiplicity, though they could probably be
adapted to the case of constant multiplicity different from
multiplicity 1. The use of the letter ``$h$'' for an eigenvalue of
the principal symbol is motivated by the fact that later it will
take on the role of a Hamiltonian, see formula (\ref{Hamiltonian system of equations}).
We enumerate the eigenvalues of the principal symbol
$h^{(j)}(x,\xi)$ in increasing order, using a positive index
$j=1,\ldots,m^+$ for positive $h^{(j)}(x,\xi)$ and a negative index
$j=-1,\ldots,-m^-$ for negative $h^{(j)}(x,\xi)$. Here $m^+$ is the
number of positive eigenvalues of the principal symbol and $m^-$ is
the number of negative ones. Of course, $m^++m^-=m$.
Under the above assumptions $A$ is a self-adjoint operator, in the
full functional analytic sense, in the Hilbert space
$L^2(M;\mathbb{C}^m)$ (Hilbert space of square integrable
complex-valued column ``functions'') with domain $H^1(M;\mathbb{C}^m)$
(Sobolev space of complex-valued column ``functions'' which are
square integrable together with their first partial derivatives) and
the spectrum of $A$ is discrete. These facts are easily established
by constructing the parametrix (approximate inverse) of the operator
$A+iI$.
Let $\lambda_k$ and
$v_k=\begin{pmatrix}v_{k1}(x)&\ldots&v_{km}(x)\end{pmatrix}^T$ be
the eigenvalues and eigenfunctions of the operator $A$. The
eigenvalues $\lambda_k$ are enumerated in increasing order with
account of multiplicity,
using a positive index $k=1,2,\ldots$ for positive $\lambda_k$
and a nonpositive index $k=0,-1,-2,\ldots$ for nonpositive $\lambda_k$.
If the operator $A$ is bounded from below (i.e.~if $m^-=0$)
then the index $k$ runs from some integer value to $+\infty$;
if the operator $A$ is bounded from above (i.e.~if $m^+=0$)
then the index $k$ runs from $-\infty$ to some integer value;
and if the operator $A$ is unbounded from above
and from below (i.e.~if $m^+\ne0$ and $m^-\ne0$)
then the index $k$ runs from $-\infty$ to $+\infty$.
\
We will be studying the following three objects.
\
\textbf{Object 1.}
Our first object of study is the \emph{propagator},
which is the one-parameter family of operators defined as
\begin{equation}
\label{definition of wave group}
U(t):=e^{-itA}
=\sum_k e^{-it\lambda_k}v_k(x)\int_M[v_k(y)]^*(\,\cdot\,)\,dy\,,
\end{equation}
$t\in\mathbb{R}$.
The propagator provides a solution to the Cauchy problem
\begin{equation}
\label{initial condition most basic}
\left.w\right|_{t=0}=v
\end{equation}
for the dynamic equation
\begin{equation}
\label{dynamic equation most basic}
D_tw+Aw=0\,,
\end{equation}
where $D_t:=-i\partial/\partial t$.
Namely, it is easy to see that if the column of half-densities $v=v(x)$
is infinitely smooth,
then, setting
$\,w:=U(t)\,v$, we get a time-dependent column of half-densities $w(t,x)$
which is also infinitely smooth
and which satisfies the equation
(\ref{dynamic equation most basic})
and the initial condition
(\ref{initial condition most basic}).
The use of the letter ``$U$'' for the propagator is motivated by the
fact that for each $t$ the operator $U(t)$ is unitary.
\
\textbf{Object 2.}
Our second object of study is the \emph{spectral function},
which is the real density defined as
\begin{equation}
\label{definition of spectral function}
e(\lambda,x,x):=\sum_{0<\lambda_k<\lambda}\|v_k(x)\|^2,
\end{equation}
where $\|v_k(x)\|^2:=[v_k(x)]^*v_k(x)$ is the square of the
Euclidean norm of the eigenfunction $v_k$ evaluated at the point
$x\in M$ and $\lambda$ is a positive parameter (spectral parameter).
\
\textbf{Object 3.}
Our third and final object of study is the \emph{counting function}
\begin{equation}
\label{definition of counting function}
N(\lambda):=\,\sum_{0<\lambda_k<\lambda}1\ =\int_Me(\lambda,x,x)\,dx\,.
\end{equation}
In other words, $N(\lambda)$ is the number of eigenvalues $\lambda_k$
between zero and $\lambda$.
\
It is natural to ask the question: why, in defining the spectral function
(\ref{definition of spectral function})
and the counting function
(\ref{definition of counting function}),
did we choose to perform summation
over all \emph{positive} eigenvalues up to a given positive $\lambda$
rather than
over all \emph{negative} eigenvalues up to a given negative $\lambda$?
There is no particular reason. One case reduces to the other by the change
of operator $A\mapsto-A$. This issue will be revisited in
Section~\ref{Spectral asymmetry}.
Further on we assume that $m^+>0$, i.e.~that the operator
$A$ is unbounded from above.
\
Our objectives are as follows.
\
\textbf{Objective 1.}
We aim to construct the propagator
(\ref{definition of wave group})
explicitly in terms of
oscillatory integrals, modulo an integral operator with an
infinitely smooth, in the variables $t$, $x$ and $y$, integral kernel.
\
\textbf{Objectives 2 and 3.}
We aim to derive, under appropriate assumptions on Hamiltonian
trajectories, two-term asymptotics for the spectral function
(\ref{definition of spectral function})
and the counting function
(\ref{definition of counting function}),
i.e.~formulae of the type
\begin{equation}
\label{two-term asymptotic formula for spectral function}
e(\lambda,x,x)=a(x)\,\lambda^n+b(x)\,\lambda^{n-1}+o(\lambda^{n-1}),
\end{equation}
\begin{equation}
\label{two-term asymptotic formula for counting function}
N(\lambda)=a\lambda^n+b\lambda^{n-1}+o(\lambda^{n-1})
\end{equation}
as $\lambda\to+\infty$.
Obviously, here we expect the real constants $a$, $b$ and real densities
$a(x)$, $b(x)$ to be related in accordance with
\begin{equation}
\label{a via a(x)}
a=\int_Ma(x)\,dx,
\end{equation}
\begin{equation}
\label{b via b(x)}
b=\int_Mb(x)\,dx.
\end{equation}
\
It is well known that the above three objectives are closely
related: if one achieves Objective 1, then Objectives 2 and 3 follow via
Fourier Tauberian theorems \cite{DuiGui,mybook,ivrii_book,Safarov_Tauberian_Theorems}.
\
We are now in a position to state our main results.
\
\textbf{Result 1.}
We construct the propagator as a sum of $m$ oscillatory integrals
\begin{equation}
\label{wave group as a sum of oscillatory integrals}
U(t)\overset{\operatorname{mod}C^\infty}=\sum_j
U^{(j)}(t)\,,
\end{equation}
where the phase function of each oscillatory integral
$U^{(j)}(t)$ is associated with the corresponding
Hamiltonian $h^{(j)}(x,\xi)$. The symbol of the oscillatory integral
$U^{(j)}(t)$ is a complex-valued $m\times m$ matrix-function
$u^{(j)}(t;y,\eta)$, where $y=(y^1,\ldots,y^n)$ is the position of the
source of the wave (i.e.~this is the same $y$ that appears in formula
(\ref{definition of wave group})) and
$\eta=(\eta_1,\ldots,\eta_n)$ is the corresponding dual variable
(covector at the point $y$).
When $|\eta|\to+\infty$, the symbol admits an asymptotic expansion
\begin{equation}
\label{decomposition of symbol of OI into homogeneous components}
u^{(j)}(t;y,\eta)=u^{(j)}_0(t;y,\eta)+u^{(j)}_{-1}(t;y,\eta)+\ldots
\end{equation}
into components positively homogeneous in $\eta$, with the subscript
indicating degree of homogeneity.
The formula for the principal symbol of the oscillatory integral
$U^{(j)}(t)$ is known
\cite{SafarovDSc,NicollPhD}
and reads as follows:
\begin{multline}
\label{formula for principal symbol of oscillatory integral}
u^{(j)}_0(t;y,\eta)=
[v^{(j)}(x^{(j)}(t;y,\eta),\xi^{(j)}(t;y,\eta))]
\,[v^{(j)}(y,\eta)]^*
\\
\times\exp
\left(
-i\int_0^tq^{(j)}(x^{(j)}(\tau;y,\eta),\xi^{(j)}(\tau;y,\eta))\,d\tau
\right),
\end{multline}
where $v^{(j)}(z,\zeta)$ is the normalised eigenvector of the principal
symbol $A_1(z,\zeta)$ corresponding to the eigenvalue (Hamiltonian)
$h^{(j)}(z,\zeta)$,
\ $(x^{(j)}(t;y,\eta),\xi^{(j)}(t;y,\eta))$ is the Hamiltonian trajectory
originating from the point $(y,\eta)$, i.e.~solution of the system of
ordinary differential equations (the dot denotes differentiation in $t$)
\begin{equation}
\label{Hamiltonian system of equations}
\dot x^{(j)}=h^{(j)}_\xi(x^{(j)},\xi^{(j)}),
\qquad
\dot\xi^{(j)}=-h^{(j)}_x(x^{(j)},\xi^{(j)})
\end{equation}
subject to the initial condition $\left.(x^{(j)},\xi^{(j)})\right|_{t=0}=(y,\eta)$,
\ $q^{(j)}:T'M\to\mathbb{R}$ is the function
\begin{equation}
\label{phase appearing in principal symbol}
q^{(j)}:=[v^{(j)}]^*A_\mathrm{sub}v^{(j)}
-\frac i2
\{
[v^{(j)}]^*,A_1-h^{(j)},v^{(j)}
\}
-i[v^{(j)}]^*\{v^{(j)},h^{(j)}\}
\end{equation}
and
\begin{equation}
\label{definition of subprincipal symbol}
A_\mathrm{sub}(z,\zeta):=
A_0(z,\zeta)+\frac i2
(A_1)_{z^\alpha\zeta_\alpha}(z,\zeta)
\end{equation}
is the subprincipal symbol of the operator $A$,
with the subscripts $z^\alpha$ and $\zeta_\alpha$
indicating partial derivatives and
the repeated index $\alpha$ indicating summation over $\alpha=1,\ldots,n$.
Curly brackets in formula
(\ref{phase appearing in principal symbol})
denote the Poisson bracket on matrix-functions
\begin{equation}
\label{Poisson bracket on matrix-functions}
\{P,R\}:=P_{z^\alpha}R_{\zeta_\alpha}-P_{\zeta_\alpha}R_{z^\alpha}
\end{equation}
and its further generalisation
\begin{equation}
\label{generalised Poisson bracket on matrix-functions}
\{P,Q,R\}:=P_{z^\alpha}QR_{\zeta_\alpha}-P_{\zeta_\alpha}QR_{z^\alpha}\,.
\end{equation}
As the derivation of formula
(\ref{formula for principal symbol of oscillatory integral})
was previously performed only in theses \cite{SafarovDSc,NicollPhD},
we repeat it in Sections
\ref{Algorithm for the construction of the wave group}
and
\ref{Leading transport equations}
of our paper.
Our derivation differs slightly from that in \cite{SafarovDSc} and \cite{NicollPhD}.
Formula (\ref{formula for principal symbol of oscillatory integral})
is invariant under changes of local coordinates on the manifold $M$,
i.e.~elements of the $m\times m$ matrix-function
$u^{(j)}_0(t;y,\eta)$ are scalars on
$\mathbb{R}\times T'M$.
Moreover, formula (\ref{formula for principal symbol of oscillatory integral})
is invariant under the transformation of the eigenvector of
the principal symbol
\begin{equation}
\label{gauge transformation of the eigenvector}
v^{(j)}\mapsto e^{i\phi^{(j)}}v^{(j)},
\end{equation}
where
\begin{equation}
\label{phase appearing in gauge transformation}
\phi^{(j)}:T'M\to\mathbb{R}
\end{equation}
is an arbitrary smooth function.
When some quantity is defined up to the action of a certain
transformation, theoretical physicists refer to such a
transformation as a \emph{gauge transformation}. We follow this
tradition.
Note that our particular gauge
transformation
(\ref{gauge transformation of the eigenvector}),
(\ref{phase appearing in gauge transformation})
is quite common in quantum mechanics:
when $\phi^{(j)}$ is a function of the position variable $x$ only
(i.e.~when $\phi^{(j)}:M\to\mathbb{R}$) this gauge transformation
is associated with electromagnetism.
Both Y.~Safarov \cite{SafarovDSc} and W.J.~Nicoll \cite{NicollPhD} assumed
that the operator $A$ is semi-bounded from below but this assumption
is not essential and their formula
(\ref{formula for principal symbol of oscillatory integral})
remains true in the more general case that we are dealing with.
However, knowing the principal symbol
(\ref{formula for principal symbol of oscillatory integral})
of the oscillatory integral $U^{(j)}(t)$
is not enough if one wants to derive two-term
asymptotics
(\ref{two-term asymptotic formula for spectral function})
and
(\ref{two-term asymptotic formula for counting function}).
One needs information about $u^{(j)}_{-1}(t;y,\eta)$,
the component of the symbol of the
oscillatory integral $U^{(j)}(t)$
which is positively homogeneous in $\eta$ of degree~-1,
see formula (\ref{decomposition of symbol of OI into homogeneous components}),
but here the problem is that $u^{(j)}_{-1}(t;y,\eta)$
is not a true invariant in the sense that it depends on the choice of
phase function in the oscillatory integral. We overcome this difficulty
by observing that $U^{(j)}(0)$ is a pseudodifferential operator, hence,
it has a well-defined subprincipal symbol
$[U^{(j)}(0)]_\mathrm{sub}$. We prove that
\begin{equation}
\label{subprincipal symbol of OI at time zero}
\operatorname{tr}[U^{(j)}(0)]_\mathrm{sub}
=-i\{[v^{(j)}]^*,v^{(j)}\}
\end{equation}
and subsequently show that information contained in formulae
(\ref{formula for principal symbol of oscillatory integral})
and
(\ref{subprincipal symbol of OI at time zero})
is sufficient for the derivation of two-term
asymptotics
(\ref{two-term asymptotic formula for spectral function})
and
(\ref{two-term asymptotic formula for counting function}).
Note that the RHS of formula (\ref{subprincipal symbol of OI at time zero})
is invariant under the gauge transformation
(\ref{gauge transformation of the eigenvector}),
(\ref{phase appearing in gauge transformation}).
Formula~(\ref{subprincipal symbol of OI at time zero})
plays a central role in our paper.
Sections~\ref{Algorithm for the construction of the wave group}
and~\ref{Leading transport equations}
provide auxiliary material needed for the proof of
formula~(\ref{subprincipal symbol of OI at time zero}),
whereas the actual proof
of formula~(\ref{subprincipal symbol of OI at time zero})
is given in Section~\ref{Proof of formula}.
Let us elaborate briefly on the geometric meaning of the
RHS of (\ref{subprincipal symbol of OI at time zero})
(a more detailed exposition is presented in Section~\ref{U(1) connection}).
The eigenvector of the principal
symbol is defined up to a gauge transformation
(\ref{gauge transformation of the eigenvector}),
(\ref{phase appearing in gauge transformation})
so it is natural to introduce a $\mathrm{U}(1)$ connection on $T'M$ as
follows: when parallel transporting an eigenvector of the principal
symbol along a curve in $T'M$ we require that the derivative of the
eigenvector along the curve be orthogonal to the eigenvector itself.
This is equivalent to the introduction of an (intrinsic) electromagnetic
field on $T'M$, with the $2n$-component real quantity
\begin{equation}
\label{electromagnetic covector potential}
i\,(\,[v^{(j)}]^*v^{(j)}_{x^\alpha}\,,\,[v^{(j)}]^*v^{(j)}_{\xi_\gamma}\,)
\end{equation}
playing the role of the electromagnetic covector potential. Our
quantity (\ref{electromagnetic covector potential}) is a 1-form on
$T'M$, rather than on $M$ itself as is the case in ``traditional''
electromagnetism. The above $\mathrm{U}(1)$ connection generates
curvature which is a 2-form on $T'M$, an analogue of the
electromagnetic tensor. Out of this curvature 2-form one can
construct, by contraction of indices, a real scalar. This scalar
curvature is the expression appearing in the RHS of
formula (\ref{subprincipal symbol of OI at time zero}).
Observe now that $\sum_jU^{(j)}(0)$ is the identity operator
on half-densities. The subprincipal symbol of the identity operator
is zero, so formula (\ref{subprincipal symbol of OI at time zero})
implies
\begin{equation}
\label{sum of curvatures is zero}
\sum_j\{[v^{(j)}]^*,v^{(j)}\}=0.
\end{equation}
One can check the identity (\ref{sum of curvatures is zero})
directly, without constructing the oscillatory integrals
$U^{(j)}(t)$: it follows from the fact that the $v^{(j)}(x,\xi)$ form an
orthonormal basis, see end of Section \ref{U(1) connection} for details.
We mentioned the identity
(\ref{sum of curvatures is zero})
in order to highlight, once again, the fact that the curvature effects
we have identified are specific to systems and do not have an
analogue in the scalar case.
\
\textbf{Results 2 and 3.}
We prove, under appropriate assumptions on Hamiltonian trajectories
(see Theorems~\ref{theorem spectral function unmollified two term}
and \ref{theorem counting function unmollified two term}),
asymptotic formulae
(\ref{two-term asymptotic formula for spectral function})
and
(\ref{two-term asymptotic formula for counting function})
with
\begin{equation}
\label{formula for a(x)}
a(x)=\sum_{j=1}^{m^+}
\ \int\limits_{h^{(j)}(x,\xi)<1}{d{\hskip-1pt\bar{}}\hskip1pt}\xi\,,
\end{equation}
\begin{multline}
\label{formula for b(x)}
b(x)=-n\sum_{j=1}^{m^+}
\ \int\limits_{h^{(j)}(x,\xi)<1}
\Bigl(
[v^{(j)}]^*A_\mathrm{sub}v^{(j)}
\\
-\frac i2
\{
[v^{(j)}]^*,A_1-h^{(j)},v^{(j)}
\}
+\frac i{n-1}h^{(j)}\{[v^{(j)}]^*,v^{(j)}\}
\Bigr)(x,\xi)\,
{d{\hskip-1pt\bar{}}\hskip1pt}\xi\,,
\end{multline}
and $a$ and $b$ expressed via the above densities
(\ref{formula for a(x)})
and
(\ref{formula for b(x)})
as
(\ref{a via a(x)})
and
(\ref{b via b(x)}).
In
(\ref{formula for a(x)})
and
(\ref{formula for b(x)})
\,${d{\hskip-1pt\bar{}}\hskip1pt}\xi$ is shorthand for
${d{\hskip-1pt\bar{}}\hskip1pt}\xi:=(2\pi)^{-n}\,d\xi
=(2\pi)^{-n}\,d\xi_1\ldots d\xi_n$,
and the Poisson bracket on matrix-functions
$\{\,\cdot\,,\,\cdot\,\}$
and its further generalisation
$\{\,\cdot\,,\,\cdot\,,\,\cdot\,\}$
are defined by formulae
(\ref{Poisson bracket on matrix-functions})
and
(\ref{generalised Poisson bracket on matrix-functions})
respectively.
To our knowledge, formula (\ref{formula for b(x)}) is a new result.
Note that in \cite{SafarovDSc} this formula
(more precisely, its integrated over $M$ version (\ref{b via b(x)}))
was written incorrectly,
without the curvature terms
$\,-\frac{ni}{n-1}\int h^{(j)}\{[v^{(j)}]^*,v^{(j)}\}$.
See also Section~\ref{Bibliographic review}
where we give a more detailed bibliographic review.
It is easy to see that the right-hand sides of
(\ref{formula for a(x)})
and
(\ref{formula for b(x)})
behave as densities under changes of local coordinates
on the manifold $M$ and that these expressions are invariant
under gauge transformations
(\ref{gauge transformation of the eigenvector}),
(\ref{phase appearing in gauge transformation})
of the eigenvectors of the principal symbol.
Moreover, the right-hand sides of
(\ref{formula for a(x)})
and
(\ref{formula for b(x)})
are unitarily invariant,
i.e.~invariant under the transformation of the operator
\begin{equation}
\label{unitary transformation of operator A}
A\mapsto RAR^*,
\end{equation}
where
\begin{equation}
\label{matrix appearing in unitary transformation of operator}
R:M\to\mathrm{U}(m)
\end{equation}
is an arbitrary smooth unitary matrix-function.
The fact that the RHS of
(\ref{formula for b(x)})
is unitarily invariant is non-trivial: the appropriate calculations
are presented in Section~\ref{U(m) invariance}.
The observation that without the curvature terms
$\,-\frac{ni}{n-1}\int h^{(j)}\{[v^{(j)}]^*,v^{(j)}\}$
(as in \cite{SafarovDSc}) the RHS of
(\ref{formula for b(x)}) is not unitarily invariant
was a major motivating factor in the writing of this paper.
\
Formula (\ref{formula for b(x)}) is the main result of our paper.
Note that even though the two-term asymptotic
expansion (\ref{two-term asymptotic formula for spectral function})
holds only under certain assumptions
on Hamiltonian trajectories (loops),
the second asymptotic coefficient (\ref{formula for b(x)})
is, in itself,
well-defined irrespective of how many loops we have. If one wishes
to reformulate the asymptotic expansion
(\ref{two-term asymptotic formula for spectral function})
in such a way that it remains valid without assumptions on the
number of loops, this can easily be achieved, say, by
taking a convolution with a function
from Schwartz space $\mathcal{S}(\mathbb{R})$:
see Theorem~\ref{theorem spectral function mollified}.
\section{Algorithm for the construction of the propagator}
\label{Algorithm for the construction of the wave group}
We construct the propagator as a sum of $m$ oscillatory integrals
(\ref{wave group as a sum of oscillatory integrals}) where each
integral is of the form
\begin{equation}
\label{algorithm equation 1}
U^{(j)}(t)
=
\int e^{i\varphi^{(j)}(t,x;y,\eta)}
\,u^{(j)}(t;y,\eta)
\,\varsigma^{(j)}(t,x;y,\eta)\,d_{\varphi^{(j)}}(t,x;y,\eta)\,
(\ \cdot\ )\,dy\,{d{\hskip-1pt\bar{}}\hskip1pt}\eta\,.
\end{equation}
Here we use notation from the book \cite{mybook}, only adapted to
systems. Namely, the expressions appearing in formula
(\ref{algorithm equation 1}) have the following meaning.
\begin{itemize}
\item
The function
$\varphi^{(j)}$ is a phase function, i.e.~a function
$\mathbb{R}\times M\times T'M\to\mathbb{C}$
positively homogeneous in $\eta$ of degree 1 and satisfying
the conditions
\begin{equation}
\label{algorithm equation 2}
\varphi^{(j)}(t,x;y,\eta)
=(x-x^{(j)}(t;y,\eta))^\alpha\,\xi^{(j)}_\alpha(t;y,\eta)
+O(|x-x^{(j)}(t;y,\eta)|^2),
\end{equation}
\begin{equation}
\label{algorithm equation 3}
\operatorname{Im}\varphi^{(j)}(t,x;y,\eta)\ge0,
\end{equation}
\begin{equation}
\label{algorithm equation 4}
\det\varphi^{(j)}_{x^\alpha\eta_\beta}(t,x^{(j)}(t;y,\eta);y,\eta)\ne0.
\end{equation}
Recall that according to Corollary 2.4.5 from
\cite{mybook} we are guaranteed to have
(\ref{algorithm equation 4}) if we choose a phase function
\begin{multline}
\label{algorithm equation 5}
\varphi^{(j)}(t,x;y,\eta)
=(x-x^{(j)}(t;y,\eta))^\alpha\,\xi^{(j)}_\alpha(t;y,\eta)
\\
+\frac12C^{(j)}_{\alpha\beta}(t;y,\eta)
\,(x-x^{(j)}(t;y,\eta))^\alpha\,(x-x^{(j)}(t;y,\eta))^\beta
\\
+O(|x-x^{(j)}(t;y,\eta)|^3)
\end{multline}
with complex-valued symmetric matrix-function $C^{(j)}_{\alpha\beta}$ satisfying the strict inequality
$\operatorname{Im}C^{(j)}>0$
(our original requirement (\ref{algorithm equation 3}) implies only the
non-strict inequality $\operatorname{Im}C^{(j)}\ge0$).
Note that even though the matrix-function $C^{(j)}_{\alpha\beta}$ is
not a tensor, the inequalities $\operatorname{Im}C^{(j)}\ge0$ and
$\operatorname{Im}C^{(j)}>0$ are invariant under transformations of
local coordinates $x$; see Remark 2.4.9 in \cite{mybook} for details.
\item
The quantity $u^{(j)}$ is the symbol of our oscillatory integral,
i.e.~a complex-valued $m\times m$ matrix-function
$\mathbb{R}\times T'M\to\mathbb{C}^{m^2}$
which admits the asymptotic expansion
(\ref{decomposition of symbol of OI into homogeneous components}).
The symbol is the unknown quantity in our construction.
\item
The quantity $d_{\varphi^{(j)}}$ is defined in accordance with
formula (2.2.4) from \cite{mybook} as
\begin{equation}
\label{algorithm equation 6}
d_{\varphi^{(j)}}(t,x;y,\eta)
:=({\det}^2\varphi^{(j)}_{x^\alpha\eta_\beta})^{1/4}
=|\det\varphi^{(j)}_{x^\alpha\eta_\beta}|^{1/2}
\,e^{\,i\arg({\det}^2\varphi^{(j)}_{x^\alpha\eta_\beta})/4}.
\end{equation}
Note that in view of (\ref{algorithm equation 4}) our $d_{\varphi^{(j)}}$
is well-defined and smooth for $x$ close to $x^{(j)}(t;y,\eta)$. It is known
\cite{mybook} that under coordinate transformations
$d_{\varphi^{(j)}}$ behaves as a half-density in $x$ and
as a half-density to the power $-1$ in $y$.
In formula (\ref{algorithm equation 6}) we wrote
$({\det}^2\varphi^{(j)}_{x^\alpha\eta_\beta})^{1/4}$
rather than
$(\det\varphi^{(j)}_{x^\alpha\eta_\beta})^{1/2}$
in order to make this expression truly invariant under coordinate transformations.
Recall that local coordinates $x$ and $y$ are chosen independently
and that $\eta$ is a covector based at the point $y$.
Consequently,
$\det\varphi^{(j)}_{x^\alpha\eta_\beta}$ changes sign under inversion
of one of the local coordinates $\,x^\alpha$, $\alpha=1,\ldots,n$,
or $\,y^\beta$, $\beta=1,\ldots,n$,
whereas ${\det}^2\varphi^{(j)}_{x^\alpha\eta_\beta}$ retains sign under
inversion.
The choice of (smooth) branch of $\arg({\det}^2\varphi^{(j)}_{x^\alpha\eta_\beta})$ is assumed
to be fixed. Thus, for a given phase function
$\varphi^{(j)}$
formula (\ref{algorithm equation 6}) defines the quantity
$d_{\varphi^{(j)}}$
uniquely up to a factor $e^{ik\pi/2}$, $k=0,1,2,3$.
Observe now that if we set $t=0$ and choose the same local coordinates
for $x$ and $y$, we get $\varphi^{(j)}_{x^\alpha\eta_\beta}(0,y;y,\eta)=I$.
This implies that we can fully specify the choice of branch of
$\arg({\det}^2\varphi^{(j)}_{x^\alpha\eta_\beta})$
by requiring that
$d_{\varphi^{(j)}}(0,y;y,\eta)=1$.
The purpose of the introduction of the factor $d_{\varphi^{(j)}}$
in (\ref{algorithm equation 1}) is twofold.
\begin{itemize}
\item[(a)]
It ensures that the symbol $u^{(j)}$ is a function on
$\mathbb{R}\times T'M$ in the full differential geometric sense of the word,
i.e.~that it is invariant under transformations of local coordinates $x$ and $y$.
\item[(b)]
It ensures that the principal symbol $u^{(j)}_0$ does not depend
on the choice of phase function $\varphi^{(j)}$.
See Remark 2.2.8 in \cite{mybook} for more details.
\end{itemize}
\item
The quantity $\varsigma^{(j)}$ is a smooth cut-off function
$\mathbb{R}\times M\times T'M\to\mathbb{R}$
satisfying the following conditions.
\begin{itemize}
\item[(a)]
$\varsigma^{(j)}(t,x;y,\eta)=0$ on the set
$\{(t,x;y,\eta):\ |h^{(j)}(y,\eta)|\le1/2\}$.
\item[(b)]
$\varsigma^{(j)}(t,x;y,\eta)=1$ on the intersection
of a small conic neighbourhood of the set
\begin{equation}
\label{algorithm equation 7.1}
\{(t,x;y,\eta):\ x=x^{(j)}(t;y,\eta)\}
\end{equation}
with the set $\{(t,x;y,\eta):\ |h^{(j)}(y,\eta)|\ge1\}$.
\item[(c)]
$\varsigma^{(j)}(t,x;y,\lambda\eta)=\varsigma^{(j)}(t,x;y,\eta)$
for $\,|h^{(j)}(y,\eta)|\ge1$, $\,\lambda\ge1$.
\end{itemize}
\item
It is known (see Section 2.3 in \cite{mybook} for details)
that Hamiltonian trajectories generated by a Hamiltonian
$h^{(j)}(x,\xi)$ positively homogeneous in $\xi$ of degree~1
satisfy the identity
\begin{equation}
\label{algorithm equation 7.1.5}
(x^{(j)}_\eta)^{\alpha\beta}\xi^{(j)}_\alpha=0,
\end{equation}
where $(x^{(j)}_\eta)^{\alpha\beta}:=\partial(x^{(j)})^\alpha/\partial\eta_\beta$.
Formulae (\ref{algorithm equation 2}) and (\ref{algorithm equation 7.1.5})
imply
\begin{equation}
\label{algorithm equation 7.2}
\varphi^{(j)}_\eta(t,x^{(j)}(t;y,\eta);y,\eta)=0.
\end{equation}
This allows us to apply the stationary phase method in the neighbourhood
of the set (\ref{algorithm equation 7.1}) and disregard what happens
away from it.
\end{itemize}
\
Our task now is to construct the symbols $u^{(j)}_0(t;y,\eta)$, $j=1,\ldots,m$,
so that our oscillatory integrals $U^{(j)}(t)$, $j=1,\ldots,m$,
satisfy the dynamic equations
\begin{equation}
\label{algorithm equation 8}
(D_t+A(x,D_x))\,U^{(j)}(t)\overset{\operatorname{mod}C^\infty}=0
\end{equation}
and initial condition
\begin{equation}
\label{algorithm equation 9}
\sum_jU^{(j)}(0)\overset{\operatorname{mod}C^\infty}=I\,,
\end{equation}
where $I$ is the identity operator on half-densities;
compare with formulae
(\ref{dynamic equation most basic}),
(\ref{initial condition most basic})
and (\ref{wave group as a sum of oscillatory integrals}).
Note that the pseudodifferential operator $A$ in formula
(\ref{algorithm equation 8}) acts on the oscillatory integral
$U(t)$ in the variable $x$; say, if $A$ is a differential
operator this means that in order to evaluate $A\,U^{(j)}(t)$
one has to perform the appropriate
differentiations of the oscillatory integral
(\ref{algorithm equation 1})
in the variable $x$.
Following the conventions of Section 3.3 of \cite{mybook},
we emphasise the fact that the pseudodifferential operator $A$ in formula
(\ref{algorithm equation 8}) acts on the oscillatory integral
$U(t)$ in the variable $x$ by writing this pseudodifferential operator
as $A(x,D_x)$, where
$D_{x^\alpha}:=-i\partial/\partial x^\alpha$.
We examine first the dynamic equation (\ref{algorithm equation 8}).
We have
\[
(D_t+A(x,D_x))\,U^{(j)}(t)=F^{(j)}(t)\,,
\]
where $F^{(j)}(t)$ is the oscillatory integral
\[
F^{(j)}(t)
=
\int e^{i\varphi^{(j)}(t,x;y,\eta)}
\,f^{(j)}(t,x;y,\eta)
\,\varsigma^{(j)}(t,x;y,\eta)\,d_{\varphi^{(j)}}(t,x;y,\eta)\,
(\ \cdot\ )\,dy\,{d{\hskip-1pt\bar{}}\hskip1pt}\eta
\]
whose matrix-valued amplitude $f^{(j)}$ is given by the formula
\begin{equation}
\label{algorithm equation 12}
f^{(j)}=D_tu^{(j)}+
\bigl(
\varphi^{(j)}_t+(d_{\varphi^{(j)}})^{-1}(D_t d_{\varphi^{(j)}})+s^{(j)}
\bigr)
\,u^{(j)},
\end{equation}
where the matrix-function $s^{(j)}(t,x;y,\eta)$ is defined as
\begin{equation}
\label{algorithm equation 13}
s^{(j)}=e^{-i\varphi^{(j)}}(d_{\varphi^{(j)}})^{-1}\,A(x,D_x)\,(e^{i\varphi^{(j)}}d_{\varphi^{(j)}})\,.
\end{equation}
Theorem 18.1 from \cite{shubin} gives us the following explicit asymptotic
(in inverse powers of $\eta$) formula for the
matrix-function (\ref{algorithm equation 13}):
\begin{equation}
\label{algorithm equation 14}
s^{(j)}=(d_{\varphi^{(j)}})^{-1}\sum_{\bm\alpha}
\frac1{{\bm\alpha}!}
\,A^{({\bm\alpha})}(x,\varphi^{(j)}_x)\,(D_z^{\bm\alpha}\chi^{(j)})\bigr|_{z=x}\ ,
\end{equation}
where
\begin{equation}
\label{algorithm equation 15}
\chi^{(j)}(t,z,x;y,\eta)
=e^{i\psi^{(j)}(t,z,x;y,\eta)}d_{\varphi^{(j)}}(t,z;y,\eta),
\end{equation}
\begin{equation}
\label{algorithm equation 16}
\psi^{(j)}(t,z,x;y,\eta)
=\varphi^{(j)}(t,z;y,\eta)
-\varphi^{(j)}(t,x;y,\eta)
-\varphi^{(j)}_{x^\beta}(t,x;y,\eta)\,(z-x)^\beta.
\end{equation}
In formula (\ref{algorithm equation 14})
\begin{itemize}
\item
${\bm\alpha}:=(\alpha_1,\ldots,\alpha_n)$ is a multi-index
(note the bold font which we use to distinguish
multi-indices and individual indices),
${\bm\alpha}!:=\alpha_1!\cdots\alpha_n!\,$,
$D_z^{\bm\alpha}:=D_{z^1}^{\alpha_1}\cdots D_{z^n}^{\alpha_n}$,
$D_{z^\beta}:=-i\partial/\partial z^\beta$,
\item
$A(x,\xi)$ is the full symbol of the pseudodifferential operator $A$
written in local coordinates~$x$,
\item
$A^{({\bm\alpha})}(x,\xi):=\partial_\xi^{\bm\alpha}A(x,\xi)$,
$\partial_\xi^{\bm\alpha}:=\partial_{\xi_1}^{\alpha_1}\cdots\partial_{\xi_n}^{\alpha_n}$
and $\partial_{\xi_\beta}:=\partial/\partial\xi_\beta\,$.
\end{itemize}
When $|\eta|\to+\infty$
the matrix-valued amplitude $f^{(j)}(t,x;y,\eta)$ defined by formula
(\ref{algorithm equation 12}) admits an asymptotic expansion
\begin{equation}
\label{algorithm equation 17}
f^{(j)}(t,x;y,\eta)=f^{(j)}_1(t,x;y,\eta)+f^{(j)}_0(t,x;y,\eta)+f^{(j)}_{-1}(t,x;y,\eta)+\ldots
\end{equation}
into components positively homogeneous in $\eta$, with the subscript
indicating degree of homogeneity. Note the following differences between formulae
(\ref{decomposition of symbol of OI into homogeneous components})
and (\ref{algorithm equation 17}).
\begin{itemize}
\item
The leading term in
(\ref{algorithm equation 17})
has degree of homogeneity 1, rather than 0 as in
(\ref{decomposition of symbol of OI into homogeneous components}).
In fact, the leading term in
(\ref{algorithm equation 17})
can be easily written out explicitly
\begin{equation}
\label{algorithm equation 18}
f^{(j)}_1(t,x;y,\eta)=
(\varphi^{(j)}_t(t,x;y,\eta)+A_1(x,\varphi^{(j)}_x(t,x;y,\eta)))\,u^{(j)}_0(t;y,\eta)\,,
\end{equation}
where $A_1(x,\xi)$ is the (matrix-valued) principal symbol of the pseudodifferential
operator $A$.
\item
Unlike the symbol $u^{(j)}(t;y,\eta)$, the amplitude
$f^{(j)}(t,x;y,\eta)$ depends on $x$.
\end{itemize}
We now need to exclude the dependence on $x$ from the amplitude
$f^{(j)}(t,x;y,\eta)$. This can be done by means of the algorithm
described in subsection 2.7.3 of \cite{mybook}.
We outline this algorithm below.
Working in local coordinates, define the matrix-function
$\varphi^{(j)}_{x\eta}$ in accordance with
$(\varphi^{(j)}_{x\eta})_\alpha{}^\beta:=\varphi^{(j)}_{x^\alpha\eta_\beta}$
and then define its inverse $(\varphi^{(j)}_{x\eta})^{-1}$ from the identity
$(\varphi^{(j)})_\alpha{}^\beta[(\varphi^{(j)}_{x\eta})^{-1}]_\beta{}^\gamma:=\delta_\alpha{}^\gamma$.
Define the ``scalar'' first order linear differential operators
\begin{equation}
\label{algorithm equation 19}
L^{(j)}_\alpha:=[(\varphi^{(j)}_{x\eta})^{-1}]_\alpha{}^\beta\,(\partial/\partial x^\beta),
\qquad\alpha=1,\ldots,n.
\end{equation}
Note that the coefficients of these differential operators are functions of the position
variable $x$ and the dual variable $\xi$. It is known, see part 2 of Appendix E in \cite{mybook},
that the operators (\ref{algorithm equation 19}) commute:
$\ L^{(j)}_\alpha L^{(j)}_\beta=L^{(j)}_\beta L^{(j)}_\alpha$,
$\ \alpha,\beta=1,\ldots,n$.
Denote
$\ L^{(j)}_{\bm\alpha}:=(L^{(j)}_1)^{\alpha_1}\cdots(L^{(j)}_n)^{\alpha_n}$,
$\ (-\varphi^{(j)}_\eta)^{\bm\alpha}:=(-\varphi^{(j)}_{\eta_1})^{\alpha_1}\cdots(-\varphi^{(j)}_{\eta_n})^{\alpha_n}$,
and, given an $r\in\mathbb{N}$, define the ``scalar'' linear differential operator
\begin{equation}
\label{algorithm equation 21}
\mathfrak{P}^{(j)}_{-1,r}:=
i(d_{\varphi^{(j)}})^{-1}
\,
\frac\partial{\partial\eta_\beta}
\,d_{\varphi^{(j)}}
\left(1+
\sum_{1\le|{\bm\alpha}|\le2r-1}
\frac{(-\varphi^{(j)}_\eta)^{\bm\alpha}}{{\bm\alpha}!\,(|{\bm\alpha}|+1)}
\,L^{(j)}_{\bm\alpha}
\right)
L^{(j)}_\beta\,,
\end{equation}
where $|{\bm\alpha}|:=\alpha_1+\ldots+\alpha_n$ and the repeated index $\beta$ indicates
summation over $\beta=1,\ldots,n$.
Recall Definition 2.7.8 from \cite{mybook}:
the linear operator $L$ is said to be
positively homogeneous in $\eta$ of degree $p\in\mathbb{R}$
if for any $q\in\mathbb{R}$ and any function $f$
positively homogeneous in $\eta$ of degree $q$
the function $Lf$ is
positively homogeneous in $\eta$ of degree $p+q$.
It is easy to see that the operator (\ref{algorithm equation 21}) is
positively homogeneous in $\eta$ of degree $-1$
and the first subscript in $\mathfrak{P}^{(j)}_{-1,r}$ emphasises this fact.
Let $\mathfrak{S}^{(j)}_0$ be the (linear) operator of restriction to $x=x^{(j)}(t;y,\eta)$,
\begin{equation}
\label{algorithm equation 22}
\mathfrak{S}^{(j)}_0:=\left.(\,\cdot\,)\right|_{x=x^{(j)}(t;y,\eta)}\,,
\end{equation}
and let
\begin{equation}
\label{algorithm equation 23}
\mathfrak{S}^{(j)}_{-r}:=\mathfrak{S}^{(j)}_0(\mathfrak{P}^{(j)}_{-1,r})^r
\end{equation}
for $r=1,2,\ldots$. Observe that our linear operators
$\mathfrak{S}^{(j)}_{-r}$, $r=0,1,2,\ldots$, are
positively homogeneous in $\eta$ of degree $-r$.
This observation allows us to define the linear operator
\begin{equation}
\label{algorithm equation 24}
\mathfrak{S}^{(j)}:=\sum_{r=0}^{+\infty}\mathfrak{S}^{(j)}_{-r}\ ,
\end{equation}
where the series is understood as an asymptotic series in inverse powers of $\eta$.
According to subsection 2.7.3 of \cite{mybook},
the dynamic equation (\ref{algorithm equation 8}) can now be rewritten in the equivalent form
\begin{equation}
\label{algorithm equation 25}
\mathfrak{S}^{(j)}f^{(j)}=0\,,
\end{equation}
where the equality is understood in the asymptotic sense, as
an asymptotic expansion in inverse powers of $\eta$.
Recall that the matrix-valued amplitude $f^{(j)}(t,x;y,\eta)$
appearing in (\ref{algorithm equation 25}) is defined
by formulae (\ref{algorithm equation 12})--(\ref{algorithm equation 16}).
Substituting (\ref{algorithm equation 24}) and (\ref{algorithm equation 17})
into (\ref{algorithm equation 25}) we obtain a hierarchy of equations
\begin{equation}
\label{algorithm equation 26}
\mathfrak{S}^{(j)}_0f^{(j)}_1=0,
\end{equation}
\begin{equation}
\label{algorithm equation 27}
\mathfrak{S}^{(j)}_{-1}f^{(j)}_1+\mathfrak{S}^{(j)}_0f^{(j)}_0=0,
\end{equation}
\[
\mathfrak{S}^{(j)}_{-2}f^{(j)}_1+\mathfrak{S}^{(j)}_{-1}f^{(j)}_0+\mathfrak{S}^{(j)}_0f^{(j)}_{-1}=0,
\]
\[
\ldots
\]
positively homogeneous in $\eta$ of degree 1, 0, $-1$, $\ldots$.
These are the \emph{transport} equations for the determination of the unknown
homogeneous components $u^{(j)}_0(t;y,\eta)$, $u^{(j)}_{-1}(t;y,\eta)$, $u^{(j)}_{-2}(t;y,\eta)$, $\ldots$,
of the symbol of the oscillatory integral (\ref{algorithm equation 1}).
Let us now examine the initial condition (\ref{algorithm equation 9}).
Each operator $U^{(j)}(0)$ is a pseudodifferential operator, only
written in a slightly nonstandard form. The issues here are as follows.
\begin{itemize}
\item
We use the invariantly defined phase function
$
\varphi^{(j)}(0,x;y,\eta)
=(x-y)^\alpha\,\eta_\alpha
+O(|x-y|^2)
$
rather than the linear phase function $(x-y)^\alpha\,\eta_\alpha$
written in local coordinates.
\item
When defining the (full) symbol of the operator $U^{(j)}(t)$ we excluded the variable
$x$ from the amplitude rather than the variable $y$. Note that when dealing
with pseudodifferential operators it is customary to exclude the variable $y$
from the amplitude; exclusion of the variable $x$ gives the dual symbol of
a pseudodifferential operator, see subsection 2.1.3 in \cite{mybook}.
Thus, at $t=0$, our symbol $u^{(j)}(0;y,\eta)$ resembles
the dual symbol of a pseudodifferential operator rather
than the ``normal'' symbol.
\item
We have the extra factor $d_{\varphi^{(j)}}(0,x;y,\eta)$ in our representation
of the operator $U^{(j)}(0)$ as an oscillatory integral.
\end{itemize}
The (full) dual symbol
of the pseudodifferential operator $U^{(j)}(0)$
can be calculated in local coordinates in accordance with the following
formula which addresses the issues highlighted above:
\begin{equation}
\label{algorithm equation 30}
\sum_{\bm\alpha}
\frac{(-1)^{|{\bm\alpha}|}}{{\bm\alpha}!}\,
\bigl(
D_x^{\bm\alpha}\,\partial_\eta^{\bm\alpha}\,
u^{(j)}(0;y,\eta)\,
e^{i\omega^{(j)}(x;y,\eta)}\,d_{\varphi^{(j)}}(0,x;y,\eta)
\bigr)
\bigr|_{x=y}\ ,
\end{equation}
where
$\omega^{(j)}(x;y,\eta)=\varphi^{(j)}(0,x;y,\eta)-(x-y)^\beta\,\eta_\beta\,$.
Formula (\ref{algorithm equation 30})
is a version of the formula from subsection 2.1.3 of \cite{mybook}, only with
the extra factor $(-1)^{|{\bm\alpha}|}$. The latter is needed because we are writing
down the dual symbol of the pseudodifferential operator $U^{(j)}(0)$ (no dependence on $x$)
rather than its ``normal'' symbol (no dependence on $y$).
The initial condition (\ref{algorithm equation 9}) can now be rewritten in explicit form as
\begin{equation}
\label{algorithm equation 32}
\sum_j
\sum_{\bm\alpha}
\frac{(-1)^{|{\bm\alpha}|}}{{\bm\alpha}!}\,
\bigl(
D_x^{\bm\alpha}\,\partial_\eta^{\bm\alpha}\,
u^{(j)}(0;y,\eta)\,
e^{i\omega^{(j)}(x;y,\eta)}\,d_{\varphi^{(j)}}(0,x;y,\eta)
\bigr)
\bigr|_{x=y}=I\,,
\end{equation}
where $I$ is the $m\times m$ identity matrix.
Condition (\ref{algorithm equation 32})
can be decomposed into components positively homogeneous in $\eta$
of degree $0,-1,-2,\ldots$, giving us a hierarchy of initial conditions.
The leading (of degree of homogeneity 0) initial condition reads
\begin{equation}
\label{algorithm equation 33}
\sum_j
u^{(j)}_0(0;y,\eta)=I\,,
\end{equation}
whereas lower order initial conditions are more complicated
and depend on the choice of our phase functions $\varphi^{(j)}$.
\section{Leading transport equations}
\label{Leading transport equations}
Formulae
(\ref{algorithm equation 22}),
(\ref{algorithm equation 18}),
(\ref{algorithm equation 2}),
(\ref{Hamiltonian system of equations})
and the identity $\xi_\alpha h^{(j)}_{\xi_\alpha}(x,\xi)=h^{(j)}(x,\xi)$
(consequence of the fact that $h^{(j)}(x,\xi)$ is positively homogeneous in $\xi$ of degree~1)
give us the following explicit representation
for the leading transport equation
(\ref{algorithm equation 26}):
\begin{equation}
\label{Leading transport equations equation 1}
\!\!
\bigl[
A_1\bigl(x^{(j)}(t;y,\eta),\xi^{(j)}(t;y,\eta)\bigr)
-
h^{(j)}\bigl(x^{(j)}(t;y,\eta),\xi^{(j)}(t;y,\eta)\bigr)
\bigr]
\,u^{(j)}_0(t;y,\eta)=0.
\end{equation}
Here, of course,
$h^{(j)}\bigl(x^{(j)}(t;y,\eta),\xi^{(j)}(t;y,\eta)\bigr)=h^{(j)}(y,\eta)$.
Equation (\ref{Leading transport equations equation 1}) implies that
\begin{equation}
\label{Leading transport equations equation 2}
u^{(j)}_0(t;y,\eta)=v^{(j)}(x^{(j)}(t;y,\eta),\xi^{(j)}(t;y,\eta))
\,[w^{(j)}(t;y,\eta)]^T,
\end{equation}
where $v^{(j)}(z,\zeta)$ is the normalised eigenvector of the principal
symbol $A_1(z,\zeta)$ corresponding to the eigenvalue $h^{(j)}(z,\zeta)$
and $w^{(j)}:\mathbb{R}\times T'M\to\mathbb{C}^m$ is a column-function,
positively homogeneous in $\eta$ of degree 0, that remains to be found.
Formulae
(\ref{algorithm equation 33})
and
(\ref{Leading transport equations equation 2})
imply the following initial condition for the unknown column-function $w^{(j)}$:
\begin{equation}
\label{Leading transport equations equation 3}
w^{(j)}(0;y,\eta)=\overline{v^{(j)}(y,\eta)}.
\end{equation}
We now consider the
next transport equation in our hierarchy,
equation (\ref{algorithm equation 27}).
We will write down the two terms appearing in
(\ref{algorithm equation 27}) separately.
In view of formulae
(\ref{algorithm equation 18})
and
(\ref{algorithm equation 21})--(\ref{algorithm equation 23}),
the first term in (\ref{algorithm equation 27}) reads
\begin{multline}
\label{Leading transport equations equation 4}
\mathfrak{S}^{(j)}_{-1}f^{(j)}_1=
\\
i
\left.
\left[
(d_{\varphi^{(j)}})^{-1}
\frac\partial{\partial\eta_\beta}
d_{\varphi^{(j)}}
\left(1-
\frac12
\varphi^{(j)}_{\eta_\alpha}
L^{(j)}_\alpha
\right)
\left(
L^{(j)}_\beta
\bigl(\varphi^{(j)}_t+A_1(x,\varphi^{(j)}_x)\bigr)
\right)
u^{(j)}_0
\right]
\right|_{x=x^{(j)}}\,,
\end{multline}
where we dropped, for the sake of brevity,
the arguments $(t;y,\eta)$ in $u^{(j)}_0$ and $x^{(j)}$,
and the arguments $(t,x;y,\eta)$
in $\varphi^{(j)}_t$, $\varphi^{(j)}_x$, $\varphi^{(j)}_\eta$ and $d_{\varphi^{(j)}}\,$.
Recall that the differential operators $L^{(j)}_\alpha$ are defined in accordance with
formula (\ref{algorithm equation 19})
and the coefficients of these operators depend on $(t,x;y,\eta)$.
In view of formulae
(\ref{algorithm equation 12})--(\ref{algorithm equation 17})
and
(\ref{algorithm equation 22}),
the second term in (\ref{algorithm equation 27}) reads
\begin{multline}
\label{Leading transport equations equation 5}
\mathfrak{S}^{(j)}_0f^{(j)}_0=
D_tu^{(j)}_0
\\
+\left.\left[
(d_{\varphi^{(j)}})^{-1}
\left(D_t+(A_1)_{\xi_\alpha}D_{x^\alpha}\right)
d_{\varphi^{(j)}}
+A_0
-\frac i2(A_1)_{\xi_\alpha\xi_\beta}C^{(j)}_{\alpha\beta}
\right]
\right|_{x=x^{(j)}}u^{(j)}_0
\\
+\bigl[A_1-h^{(j)}\bigr]u^{(j)}_{-1}\,,
\end{multline}
where
\begin{equation}
\label{Leading transport equations equation 6}
C^{(j)}_{\alpha\beta}:=\left.\varphi^{(j)}_{x^\alpha x^\beta}\right|_{x=x^{(j)}}
\end{equation}
is the matrix-function from
(\ref{algorithm equation 5}).
In formulae
(\ref{Leading transport equations equation 5})
and
(\ref{Leading transport equations equation 6})
we dropped, for the sake of brevity,
the arguments $(t;y,\eta)$ in $u^{(j)}_0$, $u^{(j)}_{-1}$, $C^{(j)}_{\alpha\beta}$ and $x^{(j)}$,
the arguments
$(x^{(j)}(t;y,\eta),\xi^{(j)}(t;y,\eta))$
in $A_0$, $A_1$, $(A_1)_{\xi_\alpha}$, $(A_1)_{\xi_\alpha\xi_\beta}$ and $h^{(j)}$,
and the arguments $(t,x;y,\eta)$
in $d_{\varphi^{(j)}}$ and $\varphi^{(j)}_{x^\alpha x^\beta}\,$.
Looking at
(\ref{Leading transport equations equation 4})
and
(\ref{Leading transport equations equation 5})
we see that the transport equation (\ref{algorithm equation 27}) has a complicated
structure.
Hence, in this section we choose not to perform the analysis
of the full equation (\ref{algorithm equation 27})
and analyse only one particular subequation of this equation.
Namely, observe that equation (\ref{algorithm equation 27})
is equivalent to $m$ subequations
\begin{equation}
\label{Leading transport equations equation 7}
\bigl[v^{(j)}\bigr]^*
\,
\bigl[\mathfrak{S}^{(j)}_{-1}f^{(j)}_1+\mathfrak{S}^{(j)}_0f^{(j)}_0\bigr]
=0,
\end{equation}
\begin{equation}
\label{Leading transport equations equation 8}
\bigl[v^{(l)}\bigr]^*
\,
\bigl[\mathfrak{S}^{(j)}_{-1}f^{(j)}_1+\mathfrak{S}^{(j)}_0f^{(j)}_0\bigr]
=0,
\qquad l\ne j,
\end{equation}
where we dropped, for the sake of brevity, the arguments
$(x^{(j)}(t;y,\eta),\xi^{(j)}(t;y,\eta))$
in $\bigl[v^{(j)}\bigr]^*$ and $\bigl[v^{(l)}\bigr]^*$.
In the remainder of this section we analyse (sub)equation
(\ref{Leading transport equations equation 7}) only.
Equation (\ref{Leading transport equations equation 7})
is simpler than each of the $m-1$ equations
(\ref{Leading transport equations equation 8})
for the following two reasons.
\begin{itemize}
\item
Firstly, the term
$\bigl[A_1-h^{(j)}\bigr]u^{(j)}_{-1}$
from
(\ref{Leading transport equations equation 5})
vanishes after multiplication by
$\bigl[v^{(j)}\bigr]^*$
from the left.
Hence, equation
(\ref{Leading transport equations equation 7})
does not contain $u^{(j)}_{-1}$.
\item
Secondly, if we substitute
(\ref{Leading transport equations equation 2})
into
(\ref{Leading transport equations equation 7}),
then the term with
\[
\partial[d_{\varphi^{(j)}}w^{(j)}(t;y,\eta)]^T/\partial\eta_\beta
\]
vanishes.
This follows from the fact that the scalar function
\[
\bigl[v^{(j)}\bigr]^*
\bigl(\varphi^{(j)}_t+A_1(x,\varphi^{(j)}_x)\bigr)
v^{(j)}
\]
has a second order zero, in the variable $x$, at $x=x^{(j)}(t;y,\eta)$.
Indeed, we have
\begin{multline*}
\left.
\left[
\frac\partial{\partial x^\alpha}
\bigl[v^{(j)}\bigr]^*
\bigl(\varphi^{(j)}_t+A_1(x,\varphi^{(j)}_x)\bigr)
v^{(j)}
\right]
\right|_{x=x^{(j)}}
\\
=
\bigl[v^{(j)}\bigr]^*
\left.
\left[
\bigl(\varphi^{(j)}_t+A_1(x,\varphi^{(j)}_x)\bigr)_{x^\alpha}
\right]
\right|_{x=x^{(j)}}
v^{(j)}
\\
=
\bigl[v^{(j)}\bigr]^*
\bigl(
-h^{(j)}_{x^\alpha}-C^{(j)}_{\alpha\beta}h^{(j)}_{\xi_\beta}
+(A_1)_{x^\alpha}+C^{(j)}_{\alpha\beta}(A_1)_{\xi_\beta}
\bigr)
v^{(j)}
\\
=
\bigl[v^{(j)}\bigr]^*(A_1)_{x^\alpha}v^{(j)}-h^{(j)}_{x^\alpha}
+
C^{(j)}_{\alpha\beta}
\bigl(
\bigl[v^{(j)}\bigr]^*(A_1)_{\xi_\beta}v^{(j)}-h^{(j)}_{\xi_\beta}
\bigr)
=0\,,
\end{multline*}
where in the last two lines we dropped,
for the sake of brevity,
the arguments
$(x^{(j)}(t;y,\eta),\xi^{(j)}(t;y,\eta))$
in $(A_1)_{x^\alpha}$, $(A_1)_{\xi_\beta}$,
$h^{(j)}_{x^\alpha}$, $h^{(j)}_{\xi_\beta}$,
and the argument $(t;y,\eta)$ in
$C^{(j)}_{\alpha\beta}$
(the latter is the
matrix-function
from
formulae (\ref{algorithm equation 5}) and (\ref{Leading transport equations equation 6})).
Throughout the above argument we used the fact that our
$\bigl[v^{(j)}\bigr]^*$ and $v^{(j)}$ do not depend on $x$:
their argument is
$(x^{(j)}(t;y,\eta),\xi^{(j)}(t;y,\eta))$.
\end{itemize}
Substituting
(\ref{Leading transport equations equation 4}),
(\ref{Leading transport equations equation 5})
and
(\ref{Leading transport equations equation 2})
into
(\ref{Leading transport equations equation 7})
we get
\begin{equation}
\label{Leading transport equations equation 9}
(D_t+p^{(j)}(t;y,\eta))\,[w^{(j)}(t;y,\eta)]^T=0\,,
\end{equation}
where
\begin{multline}
\label{Leading transport equations equation 10}
p^{(j)}=
i
\left.
[v^{(j)}]^*
\left[
\frac\partial{\partial\eta_\beta}
\left(1-
\frac12
\varphi^{(j)}_{\eta_\alpha}
L^{(j)}_\alpha
\right)
\left(
L^{(j)}_\beta
\bigl(\varphi^{(j)}_t+A_1(x,\varphi^{(j)}_x)\bigr)
\right)
v^{(j)}
\right]
\right|_{x=x^{(j)}}
\\
-i[v^{(j)}]^*\{v^{(j)},h^{(j)}\}
+\left.\left[
(d_{\varphi^{(j)}})^{-1}
\left(D_t+h^{(j)}_{\xi_\alpha}D_{x^\alpha}\right)
d_{\varphi^{(j)}}
\right]
\right|_{x=x^{(j)}}
\\
+[v^{(j)}]^*
\left(
A_0
-\frac i2(A_1)_{\xi_\alpha\xi_\beta}C^{(j)}_{\alpha\beta}
\right)
v^{(j)}.
\end{multline}
Note that the ordinary differential operator in the LHS of formula
(\ref{Leading transport equations equation 9}) is a scalar one, i.e.
it does not mix up the different components of the column-function
$w^{(j)}(t;y,\eta)$. The solution of the ordinary
differential equation
(\ref{Leading transport equations equation 9})
subject to the initial condition
(\ref{Leading transport equations equation 3})
is
\begin{equation}
\label{Leading transport equations equation 11}
w^{(j)}(t;y,\eta)=\overline{v^{(j)}(y,\eta)}
\exp\left(-i\int_0^tp^{(j)}(\tau;y,\eta)\,d\tau\right).
\end{equation}
Comparing formulae
(\ref{Leading transport equations equation 2}),
(\ref{Leading transport equations equation 11})
with formula
(\ref{formula for principal symbol of oscillatory integral})
we see that in order to prove the latter we need only to establish the
scalar identity
\begin{equation}
\label{Leading transport equations equation 12}
p^{(j)}(t;y,\eta)=q^{(j)}(x^{(j)}(t;y,\eta),\xi^{(j)}(t;y,\eta))\,,
\end{equation}
where $q^{(j)}$ is the function
(\ref{phase appearing in principal symbol}).
In view of the definitions of the quantities
$p^{(j)}$ and $q^{(j)}$,
see formulae
(\ref{Leading transport equations equation 10})
and
(\ref{phase appearing in principal symbol}),
and the definition of the subprincipal symbol
(\ref{definition of subprincipal symbol}),
proving the identity
(\ref{Leading transport equations equation 12})
reduces to proving the identity
\begin{multline}
\label{Leading transport equations equation 13}
\{
[v^{(j)}]^*,A_1-h^{(j)},v^{(j)}
\}
(x^{(j)},\xi^{(j)})
=
\\
-2
\left.
[v^{(j)}(x^{(j)},\xi^{(j)})]^*
\left[
\frac\partial{\partial\eta_\beta}
\left(1-
\frac12
\varphi^{(j)}_{\eta_\alpha}
L^{(j)}_\alpha
\right)
\left(
L^{(j)}_\beta
\bigl(\varphi^{(j)}_t+A_1(x,\varphi^{(j)}_x)\bigr)
\right)
v^{(j)}(x^{(j)},\xi^{(j)})
\right]
\right|_{x=x^{(j)}}
\\
+2\left.\left[
(d_{\varphi^{(j)}})^{-1}
\left(\partial_t+h^{(j)}_{\xi_\alpha}\partial_{x^\alpha}\right)
d_{\varphi^{(j)}}
\right]
\right|_{x=x^{(j)}}
\\
+[v^{(j)}(x^{(j)},\xi^{(j)})]^*
\left(
(A_1)_{x^\alpha\xi_\alpha}+(A_1)_{\xi_\alpha\xi_\beta}C^{(j)}_{\alpha\beta}
\right)
v^{(j)}(x^{(j)},\xi^{(j)}).
\end{multline}
Note that the expressions in the LHS and RHS of
(\ref{Leading transport equations equation 13}) have different
structure. The LHS of
(\ref{Leading transport equations equation 13})
is the generalised Poisson bracket
$\{
[v^{(j)}]^*,A_1-h^{(j)},v^{(j)}
\}$,
see (\ref{generalised Poisson bracket on matrix-functions}),
evaluated at $z=x^{(j)}(t;y,\eta)$, $\zeta=\xi^{(j)}(t;y,\eta)$,
whereas the RHS of
(\ref{Leading transport equations equation 13})
involves partial derivatives (in $\eta$) of
$v^{(j)}(x^{(j)}(t;y,\eta),\xi^{(j)}(t;y,\eta))$
(Chain Rule).
In writing
(\ref{Leading transport equations equation 13}) we also
dropped, for the sake of brevity,
the arguments $(t,x;y,\eta)$ in
$\varphi^{(j)}_t$, $\varphi^{(j)}_x$,
$\varphi^{(j)}_\eta$, $d_{\varphi^{(j)}}\,$
and the coefficients of the differential operators
$L^{(j)}_\alpha$ and $L^{(j)}_\beta$,
the arguments $(x^{(j)},\xi^{(j)})$
in $h^{(j)}_{\xi_\alpha}$,
$(A_1)_{x^\alpha\xi_\alpha}$ and $(A_1)_{\xi_\alpha\xi_\beta}$,
and the arguments $(t;y,\eta)$
in $x^{(j)}$, $\xi^{(j)}$ and $C^{(j)}_{\alpha\beta}$.
Before performing the calculations that will establish the identity
(\ref{Leading transport equations equation 13}) we make several
observations that will allow us to simplify these calculations
considerably.
Firstly, our function $p^{(j)}(t;y,\eta)$ does not depend on the choice
of the phase function $\varphi^{(j)}(t,x;y,\eta)$. Indeed, if
$p^{(j)}(t;y,\eta)$ did depend on the choice of phase function, then,
in view of formulae
(\ref{Leading transport equations equation 2})
and
(\ref{Leading transport equations equation 11})
the principal symbol of our oscillatory integral $U^{(j)}(t)$ would depend
on the choice of phase function, which would contradict Theorem
2.7.11 from \cite{mybook}. Here we use the fact that operators
$U^{(j)}(t)$ with different $j$ cannot compensate each other to give
an integral operator whose integral kernel is infinitely
smooth in $t$, $x$ and $y$ because all our $U^{(j)}(t)$ oscillate in
$t$ in a different way:
$\varphi^{(j)}_t(t,x^{(j)}(t;y,\eta);y,\eta)=-h^{(j)}(y,\eta)$
and we assumed the eigenvalues $h^{(j)}(y,\eta)$ of our principal
symbol $A_1(y,\eta)$ to be simple.
Secondly, the arguments (free variables) in
(\ref{Leading transport equations equation 13}) are
$(t;y,\eta)$. We fix an arbitrary point
$(\tilde t;\tilde y,\tilde\eta)\in\mathbb{R}\times T'M$
and prove formula
(\ref{Leading transport equations equation 13})
at this point.
Put
$(\xi^{(j)}_\eta)_\alpha{}^\beta:=\partial(\xi^{(j)})_\alpha/\partial\eta_\beta$.
According to Lemma 2.3.2 from \cite{mybook}
there exists a local coordinate system $x$ such that
$\det(\xi^{(j)}_\eta)_\alpha{}^\beta\ne0$.
This opens the way to the use of the linear phase function
\begin{equation}
\label{Leading transport equations equation 14}
\varphi^{(j)}(t,x;y,\eta)
=(x-x^{(j)}(t;y,\eta))^\alpha\,\xi^{(j)}_\alpha(t;y,\eta)
\end{equation}
which will simplify calculations to a great extent.
Moreover, we can choose a local coordinate system $y$ such that
\begin{equation}
\label{Leading transport equations equation 15}
(\xi^{(j)}_\eta)_\alpha{}^\beta(\tilde t;\tilde y,\tilde\eta)=\delta_\alpha{}^\beta
\end{equation}
which will simplify calculations even further.
The calculations we are about to perform will make use of the symmetry
\begin{equation}
\label{Leading transport equations equation 16}
(x^{(j)}_\eta)^{\gamma\alpha}(\xi^{(j)}_\eta)_\gamma{}^\beta
=
(x^{(j)}_\eta)^{\gamma\beta}(\xi^{(j)}_\eta)_\gamma{}^\alpha
\end{equation}
which is an immediate
consequence of formula (\ref{algorithm equation 7.1.5}).
Formula (\ref{Leading transport equations equation 16}) appears
as formula (2.3.3) in \cite{mybook} and the accompanying text
explains its geometric meaning. Note that at the point
$(\tilde t;\tilde y,\tilde\eta)$ formula
(\ref{Leading transport equations equation 16}) takes the
especially simple form
\begin{equation}
\label{Leading transport equations equation 17}
(x^{(j)}_\eta)^{\alpha\beta}(\tilde t;\tilde y,\tilde\eta)
=
(x^{(j)}_\eta)^{\beta\alpha}(\tilde t;\tilde y,\tilde\eta).
\end{equation}
Our calculations will also involve the quantity
$\varphi^{(j)}_{\eta_\alpha\eta_\beta}(\tilde t,\tilde x;\tilde y,\tilde\eta)$
where $\tilde x:=x^{(j)}(\tilde t;\tilde y,\tilde\eta)$.
Formulae
(\ref{Leading transport equations equation 14}),
(\ref{algorithm equation 7.1.5}),
(\ref{Leading transport equations equation 15})
and
(\ref{Leading transport equations equation 17})
imply
\begin{equation}
\label{Leading transport equations equation 18}
\varphi^{(j)}_{\eta_\alpha\eta_\beta}(\tilde t,\tilde x;\tilde y,\tilde\eta)
=
-(x^{(j)}_\eta)^{\alpha\beta}(\tilde t;\tilde y,\tilde\eta).
\end{equation}
Further on we denote $\tilde\xi:=\xi^{(j)}(\tilde t;\tilde y,\tilde\eta)$.
With account of all the simplifications listed above, we can rewrite
formula
(\ref{Leading transport equations equation 13}),
which is the identity that we are proving, as
\begin{multline}
\label{Leading transport equations equation 19}
\{
[v^{(j)}]^*,A_1-h^{(j)},v^{(j)}
\}
(\tilde x,\tilde\xi)
=
\\
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
-2
[\tilde v^{(j)}]^*
\Bigl[
\frac{\partial^2}{\partial x^\alpha\partial\eta_\alpha}
\bigl(A_1(x,\xi^{(j)})-h^{(j)}(\tilde y,\eta)
\\
\qquad\qquad\qquad\qquad\qquad\qquad
-(x-x^{(j)})^\gamma h^{(j)}_{x^\gamma}(x^{(j)},\xi^{(j)})\bigr)
\,v^{(j)}(x^{(j)},\xi^{(j)})
\Bigr]
\Bigr|_{(x,\eta)=(\tilde x,\tilde\eta)}
\\
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
-(\tilde x^{(j)}_\eta)^{\alpha\beta}\,
[\tilde v^{(j)}]^*
\Bigl[
\frac{\partial^2}{\partial x^\alpha\partial x^\beta}
\bigl(A_1(x,\xi^{(j)})-h^{(j)}(\tilde y,\eta)
\\
\qquad\qquad\qquad\qquad\qquad\qquad
-(x-x^{(j)})^\gamma h^{(j)}_{x^\gamma}(x^{(j)},\xi^{(j)})\bigr)
\,v^{(j)}(x^{(j)},\xi^{(j)})
\Bigr]
\Bigr|_{(x,\eta)=(\tilde x,\tilde\eta)}
\\
+[\tilde v^{(j)}]^*
(\tilde A_1)_{x^\alpha\xi_\alpha}
\tilde v^{(j)}
-\tilde h^{(j)}_{x^\alpha\xi_\alpha}
-\tilde h^{(j)}_{x^\alpha x^\beta}(\tilde
x^{(j)}_\eta)^{\alpha\beta}\,,
\qquad
\end{multline}
\noindent
where
$\tilde v^{(j)}=v^{(j)}(\tilde x,\tilde\xi)$,
$\tilde x^{(j)}_\eta=x^{(j)}_\eta(\tilde t;\tilde y,\tilde\eta)$,
$(\tilde A_1)_{x^\alpha\xi_\alpha}=(A_1)_{x^\alpha\xi_\alpha}(\tilde x,\tilde\xi)$,
$\tilde h^{(j)}_{x^\alpha\xi_\alpha}=h^{(j)}_{x^\alpha\xi_\alpha}(\tilde x,\tilde\xi)$,
$\tilde h^{(j)}_{x^\alpha x^\beta}=h^{(j)}_{x^\alpha x^\beta}(\tilde x,\tilde\xi)$,
$x^{(j)}=x^{(j)}(\tilde t;\tilde y,\eta)$
and
$\xi^{(j)}=\xi^{(j)}(\tilde t;\tilde y,\eta)$.
Note that the last two terms in the RHS of
(\ref{Leading transport equations equation 19})
originate from the term with
$d_{\varphi^{(j)}}$ in
(\ref{Leading transport equations equation 13}):
we used the fact that $d_{\varphi^{(j)}}$ does not depend on
$x$ and that
\begin{equation}
\label{Leading transport equations equation 20}
\left.
\left[
(d_{\varphi^{(j)}})^{-1}
\partial_t
d_{\varphi^{(j)}}
\right]
\right|_{(t,x;y,\eta)=(\tilde t,\tilde x;\tilde y,\tilde\eta)}
=-\frac12
\bigl(
\tilde h^{(j)}_{x^\alpha\xi_\alpha}
+\tilde h^{(j)}_{x^\alpha x^\beta}(\tilde x^{(j)}_\eta)^{\alpha\beta}
\bigr).
\end{equation}
Formula (\ref{Leading transport equations equation 20})
is a special case of formula (3.3.21) from \cite{mybook}.
Note also that the term $-h^{(j)}(\tilde y,\eta)$
appearing (twice) in the RHS of
(\ref{Leading transport equations equation 19})
will vanish after being acted upon with
the differential operators
$\frac{\partial^2}{\partial x^\alpha\partial\eta_\alpha}$
and
$\frac{\partial^2}{\partial x^\alpha\partial x^\beta}$
because it does not depend on $x$.
We have
\begin{multline}
\label{Leading transport equations equation 21}
[\tilde v^{(j)}]^*
\left.
\left[
\frac{\partial^2}{\partial x^\alpha\partial\eta_\alpha}
\bigl(A_1(x,\xi^{(j)})-(x-x^{(j)})^\gamma h^{(j)}_{x^\gamma}(x^{(j)},\xi^{(j)})\bigr)
\,v^{(j)}(x^{(j)},\xi^{(j)})
\right]
\right|_{(x,\eta)=(\tilde x,\tilde\eta)}
\\
=
[\tilde v^{(j)}]^*
(\tilde A_1)_{x^\alpha\xi_\alpha}
\tilde v^{(j)}
-\tilde h^{(j)}_{x^\alpha\xi_\alpha}
-\tilde h^{(j)}_{x^\alpha x^\beta}(\tilde x^{(j)}_\eta)^{\alpha\beta}
\\
+
[\tilde v^{(j)}]^*
\bigl(
(\tilde A_1)_{x^\alpha}
-\tilde h^{(j)}_{x^\alpha}
\bigr)
\bigl(
\tilde v^{(j)}_{\xi_\alpha}
+\tilde v^{(j)}_{x^\beta}(\tilde x^{(j)}_\eta)^{\alpha\beta}
\bigr),
\end{multline}
\begin{multline}
\label{Leading transport equations equation 22}
[\tilde v^{(j)}]^*
\left.
\left[
\frac{\partial^2}{\partial x^\alpha\partial x^\beta}
\bigl(A_1(x,\xi^{(j)})-(x-x^{(j)})^\gamma h^{(j)}_{x^\gamma}(x^{(j)},\xi^{(j)})\bigr)
\,v^{(j)}(x^{(j)},\xi^{(j)})
\right]
\right|_{(x,\eta)=(\tilde x,\tilde\eta)}
\\
=
[\tilde v^{(j)}]^*
(\tilde A_1)_{x^\alpha x^\beta}
\tilde v^{(j)}\,,
\end{multline}
where
$(\tilde A_1)_{x^\alpha}=(A_1)_{x^\alpha}(\tilde x,\tilde\xi)$,
$\tilde h^{(j)}_{x^\alpha}=h^{(j)}_{x^\alpha}(\tilde x,\tilde\xi)$,
$\tilde v^{(j)}_{\xi_\alpha}=v^{(j)}_{\xi_\alpha}(\tilde x,\tilde\xi)$
and
$\tilde v^{(j)}_{x^\beta}=v^{(j)}_{x^\beta}(\tilde x,\tilde\xi)$.
We also have
\begin{multline}
\label{Leading transport equations equation 23}
[\tilde v^{(j)}]^*
\bigl(
(\tilde A_1)_{x^\alpha}
-\tilde h^{(j)}_{x^\alpha}
\bigr)
\tilde v^{(j)}_{x^\beta}
+
[\tilde v^{(j)}]^*
\bigl(
(\tilde A_1)_{x^\beta}
-\tilde h^{(j)}_{x^\beta}
\bigr)
\tilde v^{(j)}_{x^\alpha}
\\
=
\tilde h^{(j)}_{x^\alpha x^\beta}
-
[\tilde v^{(j)}]^*
(\tilde A_1)_{x^\alpha x^\beta}
\tilde v^{(j)}.
\end{multline}
Using formulae
(\ref{Leading transport equations equation 23})
and
(\ref{Leading transport equations equation 17})
we can rewrite formula
(\ref{Leading transport equations equation 21})
as
\begin{multline}
\label{Leading transport equations equation 24}
[\tilde v^{(j)}]^*
\left.
\left[
\frac{\partial^2}{\partial x^\alpha\partial\eta_\alpha}
\bigl(A_1(x,\xi^{(j)})-(x-x^{(j)})^\gamma h^{(j)}_{x^\gamma}(x^{(j)},\xi^{(j)})\bigr)
\,v^{(j)}(x^{(j)},\xi^{(j)})
\right]
\right|_{(x,\eta)=(\tilde x,\tilde\eta)}
\\
=
[\tilde v^{(j)}]^*
(\tilde A_1)_{x^\alpha\xi_\alpha}
\tilde v^{(j)}
-\tilde h^{(j)}_{x^\alpha\xi_\alpha}
+
[\tilde v^{(j)}]^*
\bigl(
(\tilde A_1)_{x^\alpha}
-\tilde h^{(j)}_{x^\alpha}
\bigr)
\tilde v^{(j)}_{\xi_\alpha}
\\
-\frac12
\bigl(
[\tilde v^{(j)}]^*
(\tilde A_1)_{x^\alpha x^\beta}
\tilde v^{(j)}
+
\tilde h^{(j)}_{x^\alpha x^\beta}
\bigr)
(\tilde x^{(j)}_\eta)^{\alpha\beta}.
\end{multline}
Substituting
(\ref{Leading transport equations equation 24})
and
(\ref{Leading transport equations equation 22})
into
(\ref{Leading transport equations equation 19})
we see that all the terms with $(\tilde x^{(j)}_\eta)^{\alpha\beta}$ cancel out
and we get
\begin{multline}
\label{Leading transport equations equation 25}
\{
[v^{(j)}]^*,A_1-h^{(j)},v^{(j)}
\}
(\tilde x,\tilde\xi)
=
\\
-[\tilde v^{(j)}]^*
\bigl(
(\tilde A_1)_{x^\alpha\xi_\alpha}
-
\tilde h^{(j)}_{x^\alpha\xi_\alpha}
\bigr)
\tilde v^{(j)}
-2
[\tilde v^{(j)}]^*
\bigl(
(\tilde A_1)_{x^\alpha}
-\tilde h^{(j)}_{x^\alpha}
\bigr)
\tilde v^{(j)}_{\xi_\alpha}.
\end{multline}
Thus, the proof of the identity
(\ref{Leading transport equations equation 13})
has been reduced to the proof of the identity~(\ref{Leading transport equations equation 25}).
Observe now that formula
(\ref{Leading transport equations equation 25})
no longer has Hamiltonian trajectories present in it.
This means that we can drop all the tildes and
rewrite
(\ref{Leading transport equations equation 25})
as
\begin{multline}
\label{Leading transport equations equation 26}
\{
[v^{(j)}]^*,A_1-h^{(j)},v^{(j)}
\}
=
\\
-[v^{(j)}]^*
\bigl(
A_1
-h^{(j)}
\bigr)_{x^\alpha\xi_\alpha}
v^{(j)}
-2
[v^{(j)}]^*
\bigl(
A_1
-h^{(j)}
\bigr)_{x^\alpha}
v^{(j)}_{\xi_\alpha}\,,
\end{multline}
where the arguments are $(x,\xi)$.
We no longer need to restrict our consideration to the particular
point $(x,\xi)=(\tilde x,\tilde\xi)$:
if we prove
(\ref{Leading transport equations equation 26})
for an arbitrary $(x,\xi)\in T'M$
we will prove it for a particular
$(\tilde x,\tilde\xi)\in T'M$.
The proof of the identity
(\ref{Leading transport equations equation 26})
is straightforward. We note that
\begin{multline}
\label{Leading transport equations equation 27}
[v^{(j)}]^*
(A_1-h^{(j)})_{x^\alpha\xi_\alpha}
v^{(j)}=
\\
-
[v^{(j)}]^*
(A_1-h^{(j)})_{x^\alpha}
v^{(j)}_{\xi_\alpha}
-
[v^{(j)}]^*
(A_1-h^{(j)})_{\xi_\alpha}
v^{(j)}_{x^\alpha}
\end{multline}
and substituting
(\ref{Leading transport equations equation 27})
into
(\ref{Leading transport equations equation 26})
reduce the latter to the form
\begin{multline}
\label{Leading transport equations equation 28}
\{
[v^{(j)}]^*,A_1-h^{(j)},v^{(j)}
\}
=
\\
[v^{(j)}]^*
\bigl(
A_1
-h^{(j)}
\bigr)_{\xi_\alpha}
v^{(j)}_{x^\alpha}
-
[v^{(j)}]^*
\bigl(
A_1
-h^{(j)}
\bigr)_{x^\alpha}
v^{(j)}_{\xi_\alpha}.
\end{multline}
But
\begin{equation}
\label{Leading transport equations equation 29}
[v^{(j)}]^*
\bigl(
A_1
-h^{(j)}
\bigr)_{x^\alpha}
=
-
[v^{(j)}_{x^\alpha}]^*
\bigl(
A_1
-h^{(j)}
\bigr),
\end{equation}
\begin{equation}
\label{Leading transport equations equation 30}
[v^{(j)}]^*
\bigl(
A_1
-h^{(j)}
\bigr)_{\xi_\alpha}
=
-
[v^{(j)}_{\xi_\alpha}]^*
\bigl(
A_1
-h^{(j)}
\bigr).
\end{equation}
Substituting
(\ref{Leading transport equations equation 29})
and
(\ref{Leading transport equations equation 30})
into
(\ref{Leading transport equations equation 28})
we get
\[
\{
[v^{(j)}]^*,A_1-h^{(j)},v^{(j)}
\}
=
[v^{(j)}_{x^\alpha}]^*
\bigl(
A_1
-h^{(j)}
\bigr)
v^{(j)}_{\xi_\alpha}
-
[v^{(j)}_{\xi_\alpha}]^*
\bigl(
A_1
-h^{(j)}
\bigr)
v^{(j)}_{x^\alpha}
\]
which agrees with the definition of the generalised Poisson bracket
(\ref{generalised Poisson bracket on matrix-functions}).
\section{Proof of formula (\ref{subprincipal symbol of OI at time zero})}
\label{Proof of formula}
In this section we prove formula
(\ref{subprincipal symbol of OI at time zero}).
Our approach is as follows.
We write down explicitly the transport equations
(\ref{Leading transport equations equation 8}) at $t=0$,
i.e.
\begin{equation}
\label{Proof of formula equation 1}
\bigl[v^{(l)}\bigr]^*
\,
\left.
\bigl[\mathfrak{S}^{(j)}_{-1}f^{(j)}_1+\mathfrak{S}^{(j)}_0f^{(j)}_0\bigr]
\right|_{t=0}
=0,
\qquad l\ne j.
\end{equation}
We use the same local coordinates for $x$ and $y$ and we
assume all our phase functions to be linear, i.e.~we assume
that for each $j$ we have
(\ref{Leading transport equations equation 14}).
Using linear phase functions is justified for small $t$ because we
have
$(\xi^{(j)}_\eta)_\alpha{}^\beta(0;y,\eta)=\delta_\alpha{}^\beta$
and, hence, $\det\varphi^{(j)}_{x^\alpha\eta_\beta}(t,x;y,\eta)\ne0$
for small $t$. Writing down equations
(\ref{Proof of formula equation 1}) for linear phase functions
is much easier than for general phase functions
(\ref{algorithm equation 2}).
Using linear phase functions has the additional
advantage that the initial condition
(\ref{algorithm equation 32}) simplifies and reads now
$\sum_ju^{(j)}(0;y,\eta)=I$.
In view of
(\ref{decomposition of symbol of OI into homogeneous components}),
this implies, in particular, that
\begin{equation}
\label{Proof of formula equation 2}
\sum_j
u^{(j)}_{-1}(0)=0.
\end{equation}
Here and further on in this section we drop,
for the sake of brevity, the arguments
$(y,\eta)$ in $u^{(j)}_{-1}$.
Of course, the formula we are proving,
formula (\ref{subprincipal symbol of OI at time zero}),
does not depend
on our choice of phase functions. It is just easier to carry
out calculations for linear phase functions.
We will show that
(\ref{Proof of formula equation 1})
is a system of complex linear algebraic equations for the unknowns
$u^{(j)}_{-1}(0)$. The total number of equations
(\ref{Proof of formula equation 1}) is $m^2-m$. However, for each
$j$ and $l$
the LHS of (\ref{Proof of formula equation 1}) is a row of $m$
elements, so (\ref{Proof of formula equation 1}) is, effectively,
a system of $m(m^2-m)$ scalar equations.
Equation
(\ref{Proof of formula equation 2})
is a single matrix equation, so it is,
effectively,
a system of $m^2$ scalar equations.
Consequently, the system
(\ref{Proof of formula equation 1}),
(\ref{Proof of formula equation 2}) is, effectively,
a system of $m^3$ scalar equations.
This is exactly the number of unknown scalar elements
in the $m$ matrices $u^{(j)}_{-1}(0)$.
In the remainder of this section we write down explicitly
the LHS of (\ref{Proof of formula equation 1})
and solve the linear algebraic system
(\ref{Proof of formula equation 1}),
(\ref{Proof of formula equation 2})
for the unknowns
$u^{(j)}_{-1}(0)$.
This will allow us to prove formula
(\ref{subprincipal symbol of OI at time zero}).
Before starting explicit calculations we observe that
equations (\ref{Proof of formula equation 1}) can be equivalently rewritten as
\begin{equation}
\label{Proof of formula equation 3}
P^{(l)}
\,
\left.
\bigl[\mathfrak{S}^{(j)}_{-1}f^{(j)}_1+\mathfrak{S}^{(j)}_0f^{(j)}_0\bigr]
\right|_{t=0}
=0,
\qquad l\ne j,
\end{equation}
where $P^{(l)}:=[v^{(l)}(y,\eta)]\,[v^{(l)}(y,\eta)]^*$
is the orthogonal projection onto the eigenspace corresponding to
the (normalised) eigenvector $v^{(l)}(y,\eta)$ of the principal
symbol.
We will deal with
(\ref{Proof of formula equation 3})
rather than with
(\ref{Proof of formula equation 1}).
This is simply a matter of convenience.
\subsection{Part 1 of the proof of formula (\ref{subprincipal symbol of OI at time zero})}
\label{Part 1}
Our task in this subsection is to calculate
the LHS of (\ref{Proof of formula equation 3}).
In our calculations we use the explicit formula
(\ref{formula for principal symbol of oscillatory integral})
for the principal symbol $u^{(j)}_0(t;y,\eta)$
which was proved in Section~\ref{Leading transport equations}.
At $t=0$ formula (\ref{Leading transport equations equation 4}) reads
\[
\left.
\bigl[\mathfrak{S}^{(j)}_{-1}f^{(j)}_1\bigr]
\right|_{t=0}
=
i
\left.
\left[
\frac{\partial^2}{\partial x^\alpha\eta_\alpha}
\bigl(
A_1(x,\eta)
-h^{(j)}(y,\eta)
-(x-y)^\gamma h^{(j)}_{y^\gamma}(y,\eta)
\bigr)
P^{(j)}(y,\eta)
\right]
\right|_{x=y}
\]
which gives us
\begin{equation}
\label{Proof of formula equation 4}
\left.
\bigl[\mathfrak{S}^{(j)}_{-1}f^{(j)}_1\bigr]
\right|_{t=0}
=
i
\left[
(A_1-h^{(j)})_{y^\alpha\eta_\alpha}P^{(j)}
+
(A_1-h^{(j)})_{y^\alpha}P^{(j)}_{\eta_\alpha}
\right].
\end{equation}
In the latter formula we dropped, for the sake of brevity,
the arguments $(y,\eta)$.
At $t=0$ formula (\ref{Leading transport equations equation 5}) reads
\begin{multline}
\label{Proof of formula equation 5}
\left.
\bigl[\mathfrak{S}^{(j)}_0f^{(j)}_0\bigr]
\right|_{t=0}
=
-i\{v^{(j)},h^{(j)}\}[v^{(j)}]^*
+
\left(
A_0
-
q^{(j)}
+
\frac i2h^{(j)}_{y^\alpha\eta_\alpha}
\right)
P^{(j)}
\\
+[A_1-h^{(j)}]u^{(j)}_{-1}(0)\,,
\end{multline}
where $q^{(j)}$ is the function
(\ref{phase appearing in principal symbol})
and we dropped, for the sake of brevity,
the arguments $(y,\eta)$.
Note that in writing down
(\ref{Proof of formula equation 5})
we used the fact that
\[
\left.
\left[
(d_{\varphi^{(j)}})^{-1}
\partial_t
d_{\varphi^{(j)}}
\right]
\right|_{(t,x;y,\eta)=(0,y;y,\eta)}
=-\frac12
h^{(j)}_{y^\alpha\eta_\alpha}(y,\eta)\,,
\]
compare with formula
(\ref{Leading transport equations equation 20}).
Substituting formulae
(\ref{Proof of formula equation 4})
and
(\ref{Proof of formula equation 5})
into
(\ref{Proof of formula equation 3})
we get
\begin{equation}
\label{Proof of formula equation 6}
(h^{(l)}-h^{(j)})P^{(l)}u^{(j)}_{-1}(0)+P^{(l)}B^{(j)}_0=0,
\qquad l\ne j,
\end{equation}
where
\begin{equation}
\label{Part 1 result}
B^{(j)}_0=
\left(
A_0-q^{(j)}-\frac i2h^{(j)}_{y^\alpha\eta_\alpha}+i(A_1)_{y^\alpha\eta_\alpha}
\right)
P^{(j)}
-i
h^{(j)}_{\eta_\alpha}P^{(j)}_{y^\alpha}
+i(A_1)_{y^\alpha}P^{(j)}_{\eta_\alpha}.
\end{equation}
The subscript in $B^{(j)}_0$ indicates the degree of homogeneity in $\eta$.
\subsection{Part 2 of the proof of formula (\ref{subprincipal symbol of OI at time zero})}
\label{Part 2}
Our task in this subsection is to
solve the linear algebraic system
(\ref{Proof of formula equation 6}),
(\ref{Proof of formula equation 2})
for the unknowns
$u^{(j)}_{-1}(0)$.
It is easy to see that
the unique solution to the system
(\ref{Proof of formula equation 6}),
(\ref{Proof of formula equation 2})
is
\begin{equation}
\label{Part 2 result}
u^{(j)}_{-1}(0)
=\sum_{l\ne j}
\frac
{P^{(l)}B^{(j)}_0+P^{(j)}B^{(l)}_0}
{h^{(j)}-h^{(l)}}\,.
\end{equation}
Summation in (\ref{Part 2 result}) is carried out over all $l$
different from $j$.
\subsection{Part 3 of the proof of formula (\ref{subprincipal symbol of OI at time zero})}
\label{Part 3}
Our task in this subsection is to calculate $[U^{(j)}(0)]_\mathrm{sub}$.
We have
\begin{equation}
\label{subprincipal symbol of Uj0 equation 1}
[U^{(j)}(0)]_\mathrm{sub}
=u^{(j)}_{-1}(0)-\frac i2P^{(j)}_{y^\alpha\eta_\alpha}.
\end{equation}
Here the sign in front of $\frac i2$ is opposite to that in
(\ref{definition of subprincipal symbol})
because the way we write $U^{(j)}(0)$ is using the dual symbol.
Substituting
(\ref{Part 2 result})
and
(\ref{Part 1 result})
into (\ref{subprincipal symbol of Uj0 equation 1})
we get
\begin{multline}
\label{subprincipal symbol of Uj0 equation 2}
[U^{(j)}(0)]_\mathrm{sub}
=
-\frac i2P^{(j)}_{y^\alpha\eta_\alpha}
+\sum_{l\ne j}\frac1{h^{(j)}-h^{(l)}}
\\
\times
\bigl(
P^{(l)}
[
(A_0+i(A_1)_{y^\alpha\eta_\alpha})P^{(j)}
-ih^{(j)}_{\eta_\alpha}P^{(j)}_{y^\alpha}
+i(A_1)_{y^\alpha}P^{(j)}_{\eta_\alpha}
]
\\
\qquad\qquad+
P^{(j)}
[
(A_0+i(A_1)_{y^\alpha\eta_\alpha})P^{(l)}
-ih^{(l)}_{\eta_\alpha}P^{(l)}_{y^\alpha}
+i(A_1)_{y^\alpha}P^{(l)}_{\eta_\alpha}
]
\bigr)
\\
=
\sum_{l\ne j}
\frac
{
P^{(l)}A_\mathrm{sub}P^{(j)}
+
P^{(j)}A_\mathrm{sub}P^{(l)}
}
{
h^{(j)}-h^{(l)}
}
+\frac i2
\Bigl(
-P^{(j)}_{y^\alpha\eta_\alpha}
+
\sum_{l\ne j}
\frac
{
G_{jl}
}
{
h^{(j)}-h^{(l)}
}
\Bigr)\,,
\end{multline}
where
\begin{multline*}
G_{jl}:=
P^{(l)}
[
(A_1)_{y^\alpha\eta_\alpha}P^{(j)}
-2h^{(j)}_{\eta_\alpha}P^{(j)}_{y^\alpha}
+2(A_1)_{y^\alpha}P^{(j)}_{\eta_\alpha}
]
\\
+
P^{(j)}
[
(A_1)_{y^\alpha\eta_\alpha}P^{(l)}
-2h^{(l)}_{\eta_\alpha}P^{(l)}_{y^\alpha}
+2(A_1)_{y^\alpha}P^{(l)}_{\eta_\alpha}
]
\,.
\end{multline*}
We have
\begin{multline*}
G_{jl}
=
2P^{(l)}\{A_1,P^{(j)}\}
+
2P^{(j)}\{A_1,P^{(l)}\}
\\
+
P^{(l)}
[
(A_1-h^{(j)})_{y^\alpha\eta_\alpha}P^{(j)}
+2(A_1-h^{(j)})_{\eta_\alpha}P^{(j)}_{y^\alpha}
]
\\
+
P^{(j)}
[
(A_1-h^{(l)})_{y^\alpha\eta_\alpha}P^{(l)}
+2(A_1-h^{(l)})_{\eta_\alpha}P^{(l)}_{y^\alpha}
]
\\
=
2P^{(l)}\{A_1,P^{(j)}\}
+
2P^{(j)}\{A_1,P^{(l)}\}
-
P^{(l)}\{A_1-h^{(j)},P^{(j)}\}
-
P^{(j)}\{A_1-h^{(l)},P^{(l)}\}
\\
+
P^{(l)}
[
(A_1-h^{(j)})_{y^\alpha\eta_\alpha}P^{(j)}
+(A_1-h^{(j)})_{\eta_\alpha}P^{(j)}_{y^\alpha}
+(A_1-h^{(j)})_{y^\alpha}P^{(j)}_{\eta_\alpha}
]
\\
+
P^{(j)}
[
(A_1-h^{(l)})_{y^\alpha\eta_\alpha}P^{(l)}
+(A_1-h^{(l)})_{\eta_\alpha}P^{(l)}_{y^\alpha}
+(A_1-h^{(l)})_{y^\alpha}P^{(l)}_{\eta_\alpha}
]
\\
=
P^{(l)}\{A_1+h^{(j)},P^{(j)}\}
+
P^{(j)}\{A_1+h^{(l)},P^{(l)}\}
\\
-
P^{(l)}
(A_1-h^{(j)})
P^{(j)}_{y^\alpha\eta_\alpha}
-
P^{(j)}
(A_1-h^{(l)})
P^{(l)}_{y^\alpha\eta_\alpha}
\\
=
P^{(l)}\{A_1+h^{(j)},P^{(j)}\}
+
P^{(j)}\{A_1+h^{(l)},P^{(l)}\}
\\
-
P^{(l)}
(h^{(l)}-h^{(j)})
P^{(j)}_{y^\alpha\eta_\alpha}
-
P^{(j)}
(h^{(j)}-h^{(l)})
P^{(l)}_{y^\alpha\eta_\alpha}
\\
=
P^{(l)}\{A_1+h^{(j)},P^{(j)}\}
+
P^{(j)}\{A_1+h^{(l)},P^{(l)}\}
+(h^{(j)}-h^{(l)})
(
P^{(l)}
P^{(j)}_{y^\alpha\eta_\alpha}
-
P^{(j)}
P^{(l)}_{y^\alpha\eta_\alpha}
)\,,
\end{multline*}
so formula (\ref{subprincipal symbol of Uj0 equation 2}) can be rewritten as
\begin{multline}
\label{subprincipal symbol of Uj0 equation 3}
[U^{(j)}(0)]_\mathrm{sub}
=
\frac i2
\Bigl(
-P^{(j)}_{y^\alpha\eta_\alpha}
+
\sum_{l\ne j}
(
P^{(l)}
P^{(j)}_{y^\alpha\eta_\alpha}
-
P^{(j)}
P^{(l)}_{y^\alpha\eta_\alpha}
)
\Bigr)
\\
+
\frac12
\sum_{l\ne j}
\frac
{
P^{(l)}(2A_\mathrm{sub}P^{(j)}+i\{A_1+h^{(j)},P^{(j)}\})
+
P^{(j)}(2A_\mathrm{sub}P^{(l)}+i\{A_1+h^{(l)},P^{(l)}\})
}
{
h^{(j)}-h^{(l)}
}\,.
\end{multline}
But
\begin{multline*}
\sum_{l\ne j}
(
P^{(l)}
P^{(j)}_{y^\alpha\eta_\alpha}
-
P^{(j)}
P^{(l)}_{y^\alpha\eta_\alpha}
)
=
\Bigl(\,
\sum_{l\ne j}
P^{(l)}
\Bigr)
P^{(j)}_{y^\alpha\eta_\alpha}
-
P^{(j)}
\Bigl(\,
\sum_{l\ne j}
P^{(l)}
\Bigr)_{y^\alpha\eta_\alpha}
\\
=(I-P^{(j)})P^{(j)}_{y^\alpha\eta_\alpha}
-P^{(j)}(I-P^{(j)})_{y^\alpha\eta_\alpha}
=P^{(j)}_{y^\alpha\eta_\alpha},
\end{multline*}
so formula (\ref{subprincipal symbol of Uj0 equation 3}) can be simplified to read
\begin{multline}
\label{Part 3 result}
[U^{(j)}(0)]_\mathrm{sub}
\\
=
\frac12
\sum_{l\ne j}
\frac
{
P^{(l)}(2A_\mathrm{sub}P^{(j)}+i\{A_1+h^{(j)},P^{(j)}\})
+
P^{(j)}(2A_\mathrm{sub}P^{(l)}+i\{A_1+h^{(l)},P^{(l)}\})
}
{
h^{(j)}-h^{(l)}
}
\,.
\end{multline}
\subsection{Part 4 of the proof of formula (\ref{subprincipal symbol of OI at time zero})}
\label{Part 4}
Our task in this subsection is to calculate $\operatorname{tr}[U^{(j)}(0)]_\mathrm{sub}$.
Formula (\ref{Part 3 result}) implies
\begin{equation}
\label{trace of subprincipal symbol of Uj0 equation 1}
\operatorname{tr}[U^{(j)}(0)]_\mathrm{sub}
=
\frac i2\operatorname{tr}\sum_{l\ne j}
\frac
{
P^{(l)}\{A_1,P^{(j)}\}
+
P^{(j)}\{A_1,P^{(l)}\}
}
{
h^{(j)}-h^{(l)}
}
\,.
\end{equation}
Put $A_1=\sum_kh^{(k)}P^{(k)}$ and observe that
\begin{itemize}
\item
terms with the derivatives of $h$ vanish and
\item
the only $k$ which may give nonzero contributions are $k=j$ and $k=l$.
\end{itemize}
Thus, formula (\ref{trace of subprincipal symbol of Uj0 equation 1}) becomes
\begin{multline}
\label{trace of subprincipal symbol of Uj0 equation 2}
\operatorname{tr}[U^{(j)}(0)]_\mathrm{sub}
=
\frac i2\operatorname{tr}\sum_{l\ne j}
\frac1
{
h^{(j)}-h^{(l)}
}
\\
\times\bigl(
h^{(j)}
[
P^{(l)}\{P^{(j)},P^{(j)}\}
+
P^{(j)}\{P^{(j)},P^{(l)}\}
]
+
h^{(l)}
[
P^{(l)}\{P^{(l)},P^{(j)}\}
+
P^{(j)}\{P^{(l)},P^{(l)}\}
]
\bigr).
\end{multline}
We claim that
\begin{multline}
\label{Part 4 auxiliary equation 1}
\operatorname{tr}(P^{(l)}\{P^{(j)},P^{(j)}\})
=
\operatorname{tr}(P^{(j)}\{P^{(j)},P^{(l)}\})
\\
=
-\operatorname{tr}(P^{(l)}\{P^{(l)},P^{(j)}\})
=
-\operatorname{tr}(P^{(j)}\{P^{(l)},P^{(l)}\})
=[v^{(l)}]^*\{v^{(j)},[v^{(j)}]^*\}v^{(l)}
\\
=([v^{(l)}]^*v^{(j)}_{y^\alpha})([v^{(j)}_{\eta_\alpha}]^*v^{(l)})
-([v^{(l)}]^*v^{(j)}_{\eta_\alpha})([v^{(j)}_{y^\alpha}]^*v^{(l)}).
\end{multline}
These facts are established by writing the orthogonal projections
in terms of the eigenvectors and using, if required,
the identities
\[
[v^{(l)}_{y^\alpha}]^*v^{(j)}+[v^{(l)}]^*v^{(j)}_{y^\alpha}=0,
\qquad
[v^{(l)}_{\eta_\alpha}]^*v^{(j)}+[v^{(l)}]^*v^{(j)}_{\eta_\alpha}=0,
\]
\[
[v^{(j)}_{y^\alpha}]^*v^{(l)}+[v^{(j)}]^*v^{(l)}_{y^\alpha}=0,
\qquad
[v^{(j)}_{\eta_\alpha}]^*v^{(l)}+[v^{(j)}]^*v^{(l)}_{\eta_\alpha}=0.
\]
In view of the identities (\ref{Part 4 auxiliary equation 1})
formula (\ref{trace of subprincipal symbol of Uj0 equation 2})
can be rewritten as
\begin{multline}
\label{Part 4 auxiliary equation 2}
\operatorname{tr}[U^{(j)}(0)]_\mathrm{sub}
=
i\operatorname{tr}
\sum_{l\ne j}
P^{(l)}\{P^{(j)},P^{(j)}\}
\\
=
i\operatorname{tr}
(\{P^{(j)},P^{(j)}\}-P^{(j)}\{P^{(j)},P^{(j)}\})
=
-i\operatorname{tr}
(P^{(j)}\{P^{(j)},P^{(j)}\}).
\end{multline}
It remains only to simplify the expression in the RHS of (\ref{Part 4 auxiliary equation 2}).
We have
\begin{multline}
\label{Part 4 auxiliary equation 3}
\operatorname{tr}
(P^{(j)}\{P^{(j)},P^{(j)}\})
=
\{[v^{(j)}]^*,v^{(j)}\}
\\
+[([v^{(j)}]^*v^{(j)}_{y^\alpha})([v^{(j)}]^*v^{(j)}_{\eta_\alpha})-([v^{(j)}]^*v^{(j)}_{\eta_\alpha})([v^{(j)}]^*v^{(j)}_{y^\alpha})]
\\
+[([v^{(j)}_{y^\alpha}]^*v^{(j)})([v^{(j)}_{\eta_\alpha}]^*v^{(j)})-([v^{(j)}_{\eta_\alpha}]^*v^{(j)})([v^{(j)}_{y^\alpha}]^*v^{(j)})]
\\
+[([v^{(j)}]^*v^{(j)}_{y^\alpha})([v^{(j)}_{\eta_\alpha}]^*v^{(j)})-([v^{(j)}]^*v^{(j)}_{\eta_\alpha})([v^{(j)}_{y^\alpha}]^*v^{(j)})]
\\
=
\{[v^{(j)}]^*,v^{(j)}\}
+[([v^{(j)}]^*v^{(j)}_{y^\alpha})([v^{(j)}_{\eta_\alpha}]^*v^{(j)})-([v^{(j)}]^*v^{(j)}_{\eta_\alpha})([v^{(j)}_{y^\alpha}]^*v^{(j)})]
\\
=
\{[v^{(j)}]^*,v^{(j)}\}
-[([v^{(j)}]^*v^{(j)}_{y^\alpha})([v^{(j)}]^*v^{(j)}_{\eta_\alpha})-([v^{(j)}]^*v^{(j)}_{\eta_\alpha})([v^{(j)}]^*v^{(j)}_{y^\alpha})]
\\
=
\{[v^{(j)}]^*,v^{(j)}\}.
\end{multline}
Formulae
(\ref{Part 4 auxiliary equation 2})
and
(\ref{Part 4 auxiliary equation 3})
imply formula (\ref{subprincipal symbol of OI at time zero}).
\section{$\mathrm{U}(1)$ connection}
\label{U(1) connection}
In the preceding Sections
\ref{Algorithm for the construction of the wave group}--\ref{Proof of formula}
we presented technical details
of the construction of the propagator. We saw that
the eigenvectors of the principal symbol, $v^{(j)}(x,\xi)$, play a major role
in this construction. As pointed out in Section~\ref{Main results},
each of these eigenvectors is
defined up to a $\mathrm{U}(1)$ gauge transformation
(\ref{gauge transformation of the eigenvector}),
(\ref{phase appearing in gauge transformation}).
In the end, the full symbols
(\ref{decomposition of symbol of OI into homogeneous components})
of our oscillatory integrals $U^{(j)}(t)$
do not depend on the choice of gauge for the eigenvectors $v^{(j)}(x,\xi)$.
However, the effect of the gauge transformation
(\ref{gauge transformation of the eigenvector}),
(\ref{phase appearing in gauge transformation})
is not as trivial as it may appear at first sight.
We will demonstrate in this section that the gauge transformation
(\ref{gauge transformation of the eigenvector}),
(\ref{phase appearing in gauge transformation})
shows up, in the form of invariantly defined curvature, in the lower
order terms $u^{(j)}_{-1}(t;y,\eta)$ of the symbols of our oscillatory integrals $U^{(j)}(t)$.
More precisely, we will show that the RHS of
formula~(\ref{subprincipal symbol of OI at time zero})
is the scalar curvature of a connection associated with the gauge transformation
(\ref{gauge transformation of the eigenvector}),
(\ref{phase appearing in gauge transformation}).
Further on in this section, until the very last paragraph, the index $j$ enumerating eigenvalues and
eigenvectors of the principal symbol is assumed to be fixed.
Consider a smooth curve $\Gamma\subset T'M$ connecting points $(y,\eta)$ and $(x,\xi)$.
We write this curve in parametric form as $(z(t),\zeta(t))$, $t\in[0,1]$,
so that $(z(0),\zeta(0))=(y,\eta)$ and $(z(1),\zeta(1))=(x,\xi)$.
Put
\begin{equation}
\label{derivative of eigenvector is orthogonal to eigenvector auxiliary}
w(t):=e^{i\phi(t)}v^{(j)}(z(t),\zeta(t))\,,
\end{equation}
where
$\phi:[0,1]\to\mathbb{R}$
is an unknown function which is to be determined from the condition
\begin{equation}
\label{derivative of eigenvector is orthogonal to eigenvector}
iw^*\dot w=0
\end{equation}
with the dot indicating the derivative with respect to the parameter $t$.
Substituting (\ref{derivative of eigenvector is orthogonal to eigenvector auxiliary})
into
(\ref{derivative of eigenvector is orthogonal to eigenvector})
we get an ordinary differential equation for $\phi$ which
is easily solved, giving
\begin{multline}
\label{formula for phi(1)}
\phi(1)
=\phi(0)+\int_0^1(\dot z^\alpha(t)\,P_\alpha(z(t),\zeta(t))+\dot\zeta_\gamma(t)\,Q^\gamma(z(t),\zeta(t)))\,dt
\\
=\phi(0)+\int_\Gamma(P_\alpha dz^\alpha+Q^\gamma d\zeta_\gamma)\,,
\end{multline}
where
\begin{equation}
\label{formula for P and Q}
P_\alpha:=i[v^{(j)}]^*v^{(j)}_{z^\alpha},
\qquad
Q^\gamma:=i[v^{(j)}]^*v^{(j)}_{\zeta_\gamma}.
\end{equation}
Note that the $2n$-component real quantity $(P_\alpha,Q^\gamma)$
is a covector field (1-form) on $T'M$. This quantity already appeared
in Section~\ref{Main results} as formula (\ref{electromagnetic covector potential}).
Put $f(y,\eta):=e^{i\phi(0)}$, $f(x,\xi):=e^{i\phi(1)}$
and rewrite formula (\ref{formula for phi(1)}) as
\begin{equation}
\label{formula for a(1)}
f(x,\xi)
=f(y,\eta)\,e^{i\int_\Gamma(P_\alpha dz^\alpha+Q^\gamma d\zeta_\gamma)}.
\end{equation}
Let us identify the group $\mathrm{U}(1)$ with the unit circle in the complex
plane, i.e. with $f\in\mathbb{C}$, $|f|=1$.
We see that formulae (\ref{formula for a(1)}) and (\ref{formula for P and Q})
give us a rule for the parallel transport of elements
of the group $\mathrm{U}(1)$ along curves in $T'M$. This is the natural
$\mathrm{U}(1)$ connection generated by the normalised field of columns of
complex-valued scalars
\begin{equation}
\label{jth eigenvector of the principal symbol}
v^{(j)}(z,\zeta)=
\bigl(
\begin{matrix}v^{(j)}_1(z,\zeta)&\ldots&v^{(j)}_m(z,\zeta)\end{matrix}
\bigr)^T.
\end{equation}
Recall that the $\Gamma$ appearing in formula (\ref{formula for a(1)}) is a curve
connecting points $(y,\eta)$ and $(x,\xi)$, whereas
the $v^{(j)}(z,\zeta)$ appearing in formulae
(\ref{formula for P and Q}) and (\ref{jth eigenvector of the principal symbol})
enters our construction as
an eigenvector of the principal symbol of our $m\times m$ matrix pseudo\-differential
operator $A$.
In practice, dealing with a connection is not as convenient as dealing with
the covariant derivative $\nabla$. The covariant derivative
corresponding to the connection (\ref{formula for a(1)}) is determined as follows.
Let us view the $(x,\xi)$ appearing in formula (\ref{formula for a(1)})
as a variable which takes values close to $(y,\eta)$,
and suppose that the curve $\Gamma$ is a short straight (in local coordinates)
line segment connecting the point $(y,\eta)$ with the point $(x,\xi)$.
We want the covariant derivative of our function
$f(x,\xi)$, evaluated at $(y,\eta)$, to be zero.
Examination of formula (\ref{formula for a(1)}) shows that
the unique covariant derivative satisfying this condition is
\begin{equation}
\label{formula for U(1) covariant derivative}
\nabla_\alpha:=\partial/\partial x^\alpha-iP_\alpha(x,\xi),
\qquad
\nabla^\gamma:=\partial/\partial\xi_\gamma-iQ^\gamma(x,\xi).
\end{equation}
We define the curvature of our $\mathrm{U}(1)$ connection as
\begin{equation}
\label{definition of U(1) curvature}
R:=
-i
\begin{pmatrix}
\nabla_\alpha\nabla_\beta-\nabla_\beta\nabla_\alpha&
\nabla_\alpha\nabla^\delta-\nabla^\delta\nabla_\alpha
\\
\nabla^\gamma\nabla_\beta-\nabla_\beta\nabla^\gamma&
\nabla^\gamma\nabla^\delta-\nabla^\delta\nabla^\gamma
\end{pmatrix}.
\end{equation}
It may seem that the entries of the $(2n)\times(2n)$ matrix (\ref{definition of U(1) curvature})
are differential operators. They are, in fact, operators of multiplication
by ``scalar functions''. Namely, the more explicit form of (\ref{definition of U(1) curvature}) is
\begin{equation}
\label{explicit formula for U(1) curvature}
R=
\begin{pmatrix}
\frac{\partial P_\alpha}{\partial x^\beta}-\frac{\partial P_\beta}{\partial x^\alpha}&
\frac{\partial P_\alpha}{\partial\xi_\delta}-\frac{\partial Q^\delta}{\partial x^\alpha}
\\
\frac{\partial Q^\gamma}{\partial x^\beta}-\frac{\partial P_\beta}{\partial\xi_\gamma}&
\frac{\partial Q^\gamma}{\partial\xi_\delta}-\frac{\partial Q^\delta}{\partial\xi_\gamma}
\end{pmatrix}.
\end{equation}
The $(2n)\times(2n)$\,-\,component real quantity (\ref{explicit formula for U(1) curvature})
is a rank 2 covariant antisymmetric tensor (2-form) on $T'M$.
It is an analogue of the electromagnetic tensor.
Substituting (\ref{formula for P and Q}) into
(\ref{explicit formula for U(1) curvature})
we get an expression for curvature in terms of the eigenvector
of the principal symbol
\begin{equation}
\label{more explicit formula for U(1) curvature}
R=i
\begin{pmatrix}
[v^{(j)}_{x^\beta}]^*v^{(j)}_{x^\alpha}-[v^{(j)}_{x^\alpha}]^*v^{(j)}_{x^\beta}&
[v^{(j)}_{\xi_\delta}]^*v^{(j)}_{x^\alpha}-[v^{(j)}_{x^\alpha}]^*v^{(j)}_{\xi_\delta}
\\
[v^{(j)}_{x^\beta}]^*v^{(j)}_{\xi_\gamma}-[v^{(j)}_{\xi_\gamma}]^*v^{(j)}_{x^\beta}&
[v^{(j)}_{\xi_\delta}]^*v^{(j)}_{\xi_\gamma}-[v^{(j)}_{\xi_\gamma}]^*v^{(j)}_{\xi_\delta}
\end{pmatrix}.
\end{equation}
Examination of formula (\ref{more explicit formula for U(1) curvature}) shows that,
as expected, curvature is invariant under the gauge transformation
(\ref{gauge transformation of the eigenvector}),
(\ref{phase appearing in gauge transformation}).
It is natural to take the trace of the upper right block
in (\ref{definition of U(1) curvature}) which,
in the notation (\ref{Poisson bracket on matrix-functions}), gives us
\begin{equation}
\label{scalar curvature of U(1) connection}
-i(\nabla_\alpha\nabla^\alpha-\nabla^\alpha\nabla_\alpha)
=-i\{[v^{(j)}]^*,v^{(j)}\}.
\end{equation}
Thus, we have shown that the RHS of
formula~(\ref{subprincipal symbol of OI at time zero})
is the scalar curvature of our $\mathrm{U}(1)$ connection.
\
We end this section by proving, as promised in Section~\ref{Main results},
formula (\ref{sum of curvatures is zero}) without referring to microlocal analysis.
In the following arguments we use our standard notation for the orthogonal
projections onto the eigenspaces of the principal symbol,
i.e.~we write $P^{(k)}:=v^{(k)}[v^{(k)}]^*$.
We have $\operatorname{tr}\{P^{(j)},P^{(j)}\}=0$
and $\sum_lP^{(l)}=I$
which implies
\begin{multline}
\label{sum of curvatures is zero proof equation 1}
0=\sum_{l,j}\operatorname{tr}(P^{(l)}\{P^{(j)},P^{(j)}\})
\\
=\sum_j\operatorname{tr}(P^{(j)}\{P^{(j)},P^{(j)}\})
+\sum_{l,j:\ l\ne j}\operatorname{tr}(P^{(l)}\{P^{(j)},P^{(j)}\}).
\end{multline}
But, according to formula (\ref{Part 4 auxiliary equation 1}),
for $l\ne j$ we have
\[
\operatorname{tr}(P^{(l)}\{P^{(j)},P^{(j)}\})
=-\operatorname{tr}(P^{(j)}\{P^{(l)},P^{(l)}\}),
\]
so the expression in the last sum in the RHS of (\ref{sum of curvatures is zero proof equation 1})
is antisymmetric in the indices $l,j$, which implies that this sum is zero.
Hence, formula
(\ref{sum of curvatures is zero proof equation 1}) can be rewritten as
$\sum\limits_j\operatorname{tr}(P^{(j)}\{P^{(j)},P^{(j)}\})=0$.
It remains only to note that,
according to formula (\ref{Part 4 auxiliary equation 3}),
$\operatorname{tr}(P^{(j)}\{P^{(j)},P^{(j)}\})=\{[v^{(j)}]^*,v^{(j)}\}$.
\section{Singularity of the propagator at $t=0$}
\label{Singularity of the wave group at time zero}
Following the notation of \cite{mybook}, we denote by
\[
\mathcal{F}_{\lambda\to t}[f(\lambda)]=\hat f(t)=\int e^{-it\lambda}f(\lambda)\,d\lambda
\]
the one-dimensional Fourier transform and by
\[
\mathcal{F}^{-1}_{t\to\lambda}[\hat f(t)]=f(\lambda)=(2\pi)^{-1}\int e^{it\lambda}\hat f(t)\,dt
\]
its inverse.
Suppose that we have a Hamiltonian trajectory
$(x^{(j)}(t;y,\eta),\xi^{(j)}(t;y,\eta))$
and a real number $T>0$ such that
$x^{(j)}(T;y,\eta)=y$. We will say in this case
that we have a loop of length $T$ originating
from the point $y\in M$.
\begin{remark}
\label{remark on reversibility}
There is no need to consider loops of negative length $T$ because,
given a $T>0$, we have
$x^{(j)}(T;y,\eta^+)=y$
for some $\eta^+\in T'_yM$ if and only if we have
$x^{(j)}(-T;y,\eta^-)=y$
for some $\eta^-\in T'_yM$. Indeed,
it suffices to relate the
$\eta^\pm$ in accordance with
$\eta^\mp=\xi^{(j)}(\pm T;y,\eta^\pm)$.
\end{remark}
Denote by $\mathcal{T}^{(j)}\subset\mathbb{R}$ the set of lengths $T>0$
of all possible loops generated by the Hamiltonian $h^{(j)}$.
Here ``all possible'' refers to all possible starting points
$(y,\eta)\in T'M$ of Hamiltonian trajectories.
It is easy to see that $0\not\in\overline{\mathcal{T}^{(j)}}$.
We put
\[
\mathbf{T}^{(j)}:=
\begin{cases}
\inf\mathcal{T}^{(j)}\quad&\text{if}\quad\mathcal{T}^{(j)}\ne\emptyset,
\\
+\infty\quad&\text{if}\quad\mathcal{T}^{(j)}=\emptyset.
\end{cases}
\]
In the Riemannian case (i.e.~the case when the Hamiltonian
is a square root of a quadratic polynomial in $\xi$) it is known \cite{sabourau,rotman}
that there is a loop originating from every point of the
manifold $M$ and, moreover, there is an explicit estimate from above for
the number $\mathbf{T}^{(j)}$.
We are not aware of similar results for general Hamiltonians.
We also define
$\mathbf{T}:=\min\limits_{j=1,\ldots,m^+}\mathbf{T}^{(j)}$.
\begin{remark}
\label{remark on negative Hamiltonians}
Note that negative eigenvalues of the principal symbol,
i.e.~Hamiltonians $h^{(j)}(x,\xi)$ with negative index
$j=-1,\ldots,-m^-$,
do not affect the asymptotic formulae
we are about to derive. This is because we are dealing
with the case $\lambda\to+\infty$ rather than $\lambda\to-\infty$.
\end{remark}
Denote by
\begin{equation}
\label{definition of integral kernel of wave group}
u(t,x,y):=
\sum_k e^{-it\lambda_k}v_k(x)[v_k(y)]^*
\end{equation}
the integral kernel of the propagator (\ref{definition of wave group}).
The quantity (\ref{definition of integral kernel of wave group})
can be understood as a distribution in the variable
$t\in\mathbb{R}$ depending on the parameters $x,y\in M$.
The main result of this section is the following
\begin{lemma}
\label{Singularity of the wave group at time zero lemma}
Let $\hat\rho:\mathbb{R}\to\mathbb{C}$ be an infinitely smooth function such that
\begin{equation}
\label{condition on hat rho 1}
\operatorname{supp}\hat\rho\subset(-\mathbf{T},\mathbf{T}),
\end{equation}
\begin{equation}
\label{condition on hat rho 2}
\hat\rho(0)=1,
\end{equation}
\begin{equation}
\label{condition on hat rho 3}
\hat\rho'(0)=0.
\end{equation}
Then, uniformly over $y\in M$, we have
\begin{equation}
\label{Singularity of the wave group at time zero lemma formula}
\mathcal{F}^{-1}_{t\to\lambda}[\hat\rho(t)\operatorname{tr}u(t,y,y)]=
n\,a(y)\,\lambda^{n-1}+(n-1)\,b(y)\,\lambda^{n-2}+O(\lambda^{n-3})
\end{equation}
as $\lambda\to+\infty$.
The densities $a(y)$ and $b(y)$ appearing in the RHS of formula
(\ref{Singularity of the wave group at time zero lemma formula})
are defined in accordance with formulae
(\ref{formula for a(x)}) and (\ref{formula for b(x)}).
\end{lemma}
\emph{Proof\ }
Denote by $(S^*_yM)^{(j)}$ the $(n-1)$-dimensional unit cosphere in the cotangent fibre
defined by the equation $h^{(j)}(y,\eta)=1$
and denote by $d(S^*_yM)^{(j)}$ the
surface area element on $(S^*_yM)^{(j)}$
defined by the condition
$d\eta=d(S^*_yM)^{(j)}\,dh^{(j)}$.
The latter means that we use spherical coordinates in the cotangent fibre
with the Hamiltonian $h^{(j)}$
playing the role of the radial coordinate, see subsection 1.1.10 of \cite{mybook} for details.
In particular, as explained in subsection 1.1.10 of \cite{mybook},
our surface area element $d(S^*_yM)^{(j)}$ is expressed via the Euclidean surface area element as
\[
d(S^*_yM)^{(j)}=
\biggl(\,\sum_{\alpha=1}^n\bigl(h^{(j)}_{\eta_\alpha}(y,\eta)\bigr)^2\biggr)^{-1/2}
\times\,
\text{Euclidean surface area element}
\,.
\]
Denote also
$\,{d{\hskip-1pt\bar{}}\hskip1pt}(S^*_yM)^{(j)}:=(2\pi)^{-n}\,d(S^*_yM)^{(j)}\,$.
According to Corollary 4.1.5 from \cite{mybook} we have
uniformly over $y\in M$
\begin{multline}
\label{Singularity of the wave group at time zero lemma equation 1}
\mathcal{F}^{-1}_{t\to\lambda}[\hat\rho(t)\operatorname{tr}u(t,y,y)]=
\\
\sum_{j=1}^{m^+}
\left(c^{(j)}(y)\,\lambda^{n-1}+d^{(j)}(y)\,\lambda^{n-2}+e^{(j)}(y)\,\lambda^{n-2}\right)
+O(\lambda^{n-3})\,,
\end{multline}
where
\begin{equation}
\label{Singularity of the wave group at time zero lemma equation 2}
c^{(j)}(y)=\int\limits_{(S^*_yM)^{(j)}}
\operatorname{tr}u^{(j)}_0(0;y,\eta)
\,{d{\hskip-1pt\bar{}}\hskip1pt}(S^*_yM)^{(j)}\,,
\end{equation}
\begin{multline}
\label{Singularity of the wave group at time zero lemma equation 3}
d^{(j)}(y)=
\\
(n-1)\int\limits_{(S^*_yM)^{(j)}}
\operatorname{tr}
\left(
-\,i\,\dot u^{(j)}_0(0;y,\eta)
+\frac i2\bigl\{u^{(j)}_0\bigr|_{t=0}\,,h^{(j)}\bigr\}(y,\eta)
\right)
{d{\hskip-1pt\bar{}}\hskip1pt}(S^*_yM)^{(j)}\,,
\end{multline}
\begin{equation}
\label{Singularity of the wave group at time zero lemma equation 4}
e^{(j)}(y)=\int\limits_{(S^*_yM)^{(j)}}
\operatorname{tr}[U^{(j)}(0)]_\mathrm{sub}(y,\eta)
\,{d{\hskip-1pt\bar{}}\hskip1pt}(S^*_yM)^{(j)}\,.
\end{equation}
Here $u^{(j)}_0(t;y,\eta)$ is the principal symbol of the oscillatory integral
(\ref{algorithm equation 1}) and $\dot u^{(j)}_0(t;y,\eta)$ is its time derivative.
Note that in writing the term with the Poisson bracket in
(\ref{Singularity of the wave group at time zero lemma equation 3})
we took account of the fact that Poisson brackets in \cite{mybook}
and in the current paper have opposite signs.
Observe that the integrands in formulae
(\ref{Singularity of the wave group at time zero lemma equation 2})
and
(\ref{Singularity of the wave group at time zero lemma equation 3})
are positively homogeneous in $\eta$ of degree 0,
whereas the integrand in formula
(\ref{Singularity of the wave group at time zero lemma equation 4})
is positively homogeneous in $\eta$ of degree $-1$.
In order to have the same degree of homogeneity, we rewrite
formula
(\ref{Singularity of the wave group at time zero lemma equation 4})
in equivalent form
\begin{equation}
\label{Singularity of the wave group at time zero lemma equation 5}
e^{(j)}(y)=\int\limits_{(S^*_yM)^{(j)}}
\bigl(
h^{(j)}\operatorname{tr}[U^{(j)}(0)]_\mathrm{sub}
\bigr)
(y,\eta)
\,{d{\hskip-1pt\bar{}}\hskip1pt}(S^*_yM)^{(j)}\,.
\end{equation}
Switching from surface integrals to volume integrals with the help of formula (1.1.15) from \cite{mybook},
we rewrite formulae
(\ref{Singularity of the wave group at time zero lemma equation 2}),
(\ref{Singularity of the wave group at time zero lemma equation 3})
and
(\ref{Singularity of the wave group at time zero lemma equation 5})
as
\begin{equation}
\label{Singularity of the wave group at time zero lemma equation 6}
c^{(j)}(y)=n\int\limits_{h^{(j)}(y,\eta)<1}
\operatorname{tr}u^{(j)}_0(0;y,\eta)
\,{d{\hskip-1pt\bar{}}\hskip1pt}\eta\,,
\end{equation}
\begin{multline}
\label{Singularity of the wave group at time zero lemma equation 7}
d^{(j)}(y)=n(n-1)\times
\\
\int\limits_{h^{(j)}(y,\eta)<1}
\operatorname{tr}
\left(
-\,i\,\dot u^{(j)}_0(0;y,\eta)
+\frac i2\bigl\{u^{(j)}_0\bigr|_{t=0}\,,h^{(j)}\bigr\}(y,\eta)
\right)
{d{\hskip-1pt\bar{}}\hskip1pt}\eta\,,
\end{multline}
\begin{equation}
\label{Singularity of the wave group at time zero lemma equation 8}
e^{(j)}(y)=n\int\limits_{h^{(j)}(y,\eta)<1}
\bigl(
h^{(j)}\operatorname{tr}[U^{(j)}(0)]_\mathrm{sub}
\bigr)
(y,\eta)
\,{d{\hskip-1pt\bar{}}\hskip1pt}\eta\,.
\end{equation}
Substituting formulae
(\ref{formula for principal symbol of oscillatory integral})
and
(\ref{phase appearing in principal symbol})
into formulae
(\ref{Singularity of the wave group at time zero lemma equation 6})
and
(\ref{Singularity of the wave group at time zero lemma equation 7})
we get
\begin{equation}
\label{Singularity of the wave group at time zero lemma equation 9}
c^{(j)}(y)=n\int\limits_{h^{(j)}(y,\eta)<1}
{d{\hskip-1pt\bar{}}\hskip1pt}\eta\,,
\end{equation}
\begin{multline}
\label{Singularity of the wave group at time zero lemma equation 10}
d^{(j)}(y)=-n(n-1)\times
\\
\int\limits_{h^{(j)}(y,\eta)<1}
\left(
[v^{(j)}]^*A_\mathrm{sub}v^{(j)}
-\frac i2
\{
[v^{(j)}]^*,A_1-h^{(j)},v^{(j)}
\}
\right)(y,\eta)
\,{d{\hskip-1pt\bar{}}\hskip1pt}\eta\,.
\end{multline}
Substituting formula
(\ref{subprincipal symbol of OI at time zero})
into formula
(\ref{Singularity of the wave group at time zero lemma equation 8})
we get
\begin{equation}
\label{Singularity of the wave group at time zero lemma equation 11}
e^{(j)}(y)=-n\,i\int\limits_{h^{(j)}(y,\eta)<1}
\bigl(
h^{(j)}\{[v^{(j)}]^*,v^{(j)}\}
\bigr)
(y,\eta)
\,{d{\hskip-1pt\bar{}}\hskip1pt}\eta\,.
\end{equation}
Substituting formulae
(\ref{Singularity of the wave group at time zero lemma equation 9})--(\ref{Singularity of the wave group at time zero lemma equation 11})
into formula
(\ref{Singularity of the wave group at time zero lemma equation 1})
we arrive
at (\ref{Singularity of the wave group at time zero lemma formula}).~$\square$
\begin{remark}
The proof of Lemma~\ref{Singularity of the wave group at time zero lemma}
given above was based on the use of Corollary 4.1.5 from \cite{mybook}.
In the actual statement of Corollary 4.1.5 in \cite{mybook}
uniformity in $y\in M$ was not mentioned because the authors were dealing with
a manifold with a boundary. Uniformity reappeared in the subsequent
Theorem 4.2.1 which involved pseudodifferential cut-offs
separating the point $\,y\,$ from the boundary.
\end{remark}
\section{Mollified spectral asymptotics}
\label{Mollified spectral asymptotics}
\begin{theorem}
\label{theorem spectral function mollified}
Let $\rho:\mathbb{R}\to\mathbb{C}$ be a function from Schwartz space $\mathcal{S}(\mathbb{R})$
whose Fourier transform $\hat\rho$ satisfies conditions
(\ref{condition on hat rho 1})--(\ref{condition on hat rho 3}).
Then, uniformly over $x\in M$, we have
\begin{equation}
\label{theorem spectral function mollified formula}
\int e(\lambda-\mu,x,x)\,\rho(\mu)\,d\mu=
a(x)\,\lambda^n+b(x)\,\lambda^{n-1}+
\begin{cases}
O(\lambda^{n-2})\quad&\text{if}\quad{n\ge3},
\\
O(\ln\lambda)\quad&\text{if}\quad{n=2},
\end{cases}
\end{equation}
as $\lambda\to+\infty$.
The densities $a(x)$ and $b(x)$ appearing in the RHS of formula
(\ref{theorem spectral function mollified formula})
are defined in accordance with formulae
(\ref{formula for a(x)}) and (\ref{formula for b(x)}).
\end{theorem}
\emph{Proof\ }
Our spectral function $e(\lambda,x,x)$ was initially defined only for $\lambda>0$,
see formula (\ref{definition of spectral function}). We extend the definition
to the whole real line by setting
\[
e(\lambda,x,x):=0\quad\text{for}\quad\lambda\le0.
\]
Denote by $e'(\lambda,x,x)$ the derivative, with respect to the spectral
parameter, of the spectral function. Here ``derivative'' is understood in the
sense of distributions. The explicit formula for $e'(\lambda,x,x)$ is
\begin{equation}
\label{theorem spectral function mollified equation 2}
e'(\lambda,x,x):=\sum_{k=1}^{+\infty}\|v_k(x)\|^2\,\delta(\lambda-\lambda_k).
\end{equation}
Formula (\ref{theorem spectral function mollified equation 2}) gives us
\begin{equation}
\label{theorem spectral function mollified equation 3}
\int e'(\lambda-\mu,x,x)\,\rho(\mu)\,d\mu=
\sum_{k=1}^{+\infty}\|v_k(x)\|^2\,\rho(\lambda-\lambda_k).
\end{equation}
Formula (\ref{theorem spectral function mollified equation 3})
implies, in particular, that, uniformly over $x\in M$, we have
\begin{equation}
\label{theorem spectral function mollified equation 4}
\int e'(\lambda-\mu,x,x)\,\rho(\mu)\,d\mu=O(|\lambda|^{-\infty})
\quad\text{as}\quad\lambda\to-\infty\,,
\end{equation}
where $O(|\lambda|^{-\infty})$ is shorthand for ``tends to zero faster
than any given inverse power of $|\lambda|$''.
Formula (\ref{theorem spectral function mollified equation 3})
can also be rewritten as
\begin{equation}
\label{theorem spectral function mollified equation 5}
\int e'(\lambda-\mu,x,x)\,\rho(\mu)\,d\mu=
\mathcal{F}^{-1}_{t\to\lambda}[\hat\rho(t)\operatorname{tr}u(t,x,x)]
-\sum_{k\le0}\|v_k(x)\|^2\,\rho(\lambda-\lambda_k)\,,
\end{equation}
where the distribution $u(t,x,y)$ is defined in accordance with
formula (\ref{definition of integral kernel of wave group}).
Clearly, we have
\begin{equation}
\label{theorem spectral function mollified equation 6}
\sum_{k\le0}\|v_k(x)\|^2\,\rho(\lambda-\lambda_k)=O(\lambda^{-\infty})
\quad\text{as}\quad\lambda\to+\infty\,.
\end{equation}
Formulae
(\ref{theorem spectral function mollified equation 5}),
(\ref{theorem spectral function mollified equation 6})
and Lemma~\ref{Singularity of the wave group at time zero lemma}
imply that, uniformly over $x\in M$, we have
\begin{multline}
\label{theorem spectral function mollified equation 7}
\int e'(\lambda-\mu,x,x)\,\rho(\mu)\,d\mu=
\\
n\,a(x)\,\lambda^{n-1}+(n-1)\,b(x)\,\lambda^{n-2}+O(\lambda^{n-3})
\quad\text{as}\quad\lambda\to+\infty\,.
\end{multline}
It remains to note that
\begin{equation}
\label{theorem spectral function mollified equation 8}
\frac d{d\lambda}\int e(\lambda-\mu,x,x)\,\rho(\mu)\,d\mu
=
\int e'(\lambda-\mu,x,x)\,\rho(\mu)\,d\mu\,.
\end{equation}
Formulae
(\ref{theorem spectral function mollified equation 8}),
(\ref{theorem spectral function mollified equation 4})
and
(\ref{theorem spectral function mollified equation 7})
imply
(\ref{theorem spectral function mollified formula}).~$\square$
\begin{theorem}
\label{theorem counting function mollified}
Let $\rho:\mathbb{R}\to\mathbb{C}$ be a function from Schwartz space $\mathcal{S}(\mathbb{R})$
whose Fourier transform $\hat\rho$ satisfies conditions
(\ref{condition on hat rho 1})--(\ref{condition on hat rho 3}).
Then we have
\begin{equation}
\label{theorem counting function mollified formula}
\int N(\lambda-\mu)\,\rho(\mu)\,d\mu=
a\,\lambda^n+b\,\lambda^{n-1}+
\begin{cases}
O(\lambda^{n-2})\quad&\text{if}\quad{n\ge3},
\\
O(\ln\lambda)\quad&\text{if}\quad{n=2},
\end{cases}
\end{equation}
as $\lambda\to+\infty$.
The constants $a$ and $b$ appearing in the RHS of formula
(\ref{theorem counting function mollified formula})
are defined in accordance with formulae
(\ref{a via a(x)}),
(\ref{formula for a(x)}),
(\ref{b via b(x)})
and
(\ref{formula for b(x)}).
\end{theorem}
\emph{Proof\ }
Formula
(\ref{theorem counting function mollified formula})
follows from formula
(\ref{theorem spectral function mollified formula})
by integration over $M$,
see also formula (\ref{definition of counting function}).~$\square$
\
In stating Theorems
\ref{theorem spectral function mollified}
and
\ref{theorem counting function mollified}
we assumed the mollifier $\rho$ to be complex-valued.
This was done for the sake of generality but may seem
unnatural when mollifying real-valued functions
$e(\lambda,x,x)$ and $N(\lambda)$. One can make our
construction look more natural by dealing only with
real-valued mollifiers $\rho$. Note that if the function $\rho$
is real-valued and even then its Fourier transform
$\hat\rho$ is also real-valued and even and, moreover,
condition (\ref{condition on hat rho 3}) is automatically satisfied.
\section{Unmollified spectral asymptotics}
\label{Unmollified spectral asymptotics}
In this section we derive asymptotic formulae for
the spectral function $e(\lambda,x,x)$ and the
counting function $N(\lambda)$ without mollification.
The section is split into two subsections: in the first
we derive one-term asymptotic formulae and
in the second --- two-term asymptotic formulae.
\subsection{One-term spectral asymptotics}
\label{One-term spectral asymptotics}
\begin{theorem}
\label{theorem spectral function unmollified one term}
We have, uniformly over $x\in M$,
\begin{equation}
\label{theorem spectral function unmollified one term formula}
e(\lambda,x,x)=a(x)\,\lambda^n+O(\lambda^{n-1})
\end{equation}
as $\lambda\to+\infty$.
\end{theorem}
\emph{Proof\ }
The result in question is an immediate consequence of
formulae
(\ref{theorem spectral function mollified equation 8}),
(\ref{theorem spectral function mollified equation 7})
and
Theorem~\ref{theorem spectral function mollified}
from the current paper
and Corollary~B.2.2 from \cite{mybook}.~$\square$
\begin{theorem}
\label{theorem counting function unmollified one term}
We have
\begin{equation}
\label{theorem counting function unmollified one term formula}
N(\lambda)=a\lambda^n+O(\lambda^{n-1})
\end{equation}
as $\lambda\to+\infty$.
\end{theorem}
\emph{Proof\ }
Formula
(\ref{theorem counting function unmollified one term formula})
follows from formula
(\ref{theorem spectral function unmollified one term formula})
by integration over $M$,
see also formula (\ref{definition of counting function}).~$\square$
\subsection{Two-term spectral asymptotics}
\label{Two-term spectral asymptotics}
Up till now, in Section~\ref{Mollified spectral asymptotics}
and subsection~\ref{One-term spectral asymptotics},
our logic was to derive asymptotic formulae for the spectral
function $e(\lambda,x,x)$ first and then obtain corresponding
asymptotic formulae for the counting function $N(\lambda)$
by integration over $M$. Such an approach will not work
for two-term asymptotics because
the geometric conditions required for the existence of
two-term asymptotics of $e(\lambda,x,x)$ and $N(\lambda)$
will be different:
for $e(\lambda,x,x)$ the appropriate geometric conditions
will be formulated in terms of \emph{loops},
whereas
for $N(\lambda)$ the appropriate geometric conditions
will be formulated in terms of \emph{periodic trajectories}.
Hence, in this subsection we deal with
the spectral function $e(\lambda,x,x)$
and the counting function $N(\lambda)$ separately.
In what follows the point $y\in M$ is assumed to be fixed.
Denote by $\Pi_y^{(j)}$ the set of normalised ($h^{(j)}(y,\eta)=1$)
covectors $\eta$ which serve as starting points for loops generated by the Hamiltonian
$h^{(j)}$. Here ``starting point'' refers to the starting point
of a Hamiltonian trajectory
$(x^{(j)}(t;y,\eta),\xi^{(j)}(t;y,\eta))$
moving forward in time ($t>0$),
see also Remark~\ref{remark on reversibility}.
The reason we are not interested in large negative $t$ is that the refined
Fourier Tauberian theorem we will be applying,
Theorem~B.5.1 from \cite{mybook},
does not require information regarding large negative $t$.
And the underlying reason for the latter is the fact that the function
we are studying, $e(\lambda,x,x)$ (and, later, $N(\lambda)$), is real-valued.
The real-valuedness of the function $e(\lambda,x,x)$ implies that its
Fourier transform, $\hat e(t,x,x)$, possesses the symmetry
$\hat e(-t,x,x)=\overline{\hat e(t,x,x)}$.
The set $\Pi_y^{(j)}$ is a subset of the $(n-1)$-dimensional unit cosphere
$(S^*_yM)^{(j)}$ and the latter is equipped with
a natural Lebesgue measure, see proof of
Lemma~\ref{Singularity of the wave group at time zero lemma}.
It is known, see Lemma 1.8.2 in \cite{mybook}, that the
set $\Pi_y^{(j)}$ is measurable.
\begin{definition}
\label{definition of nonfocal point 1}
A point $y\in M$ is said to be \emph{nonfocal} if for each
$j=1,\ldots,m^+$ the set $\Pi_y^{(j)}$ has measure zero.
\end{definition}
With regards to the range of the index $j$ in
Definition~\ref{definition of nonfocal point 1},
as well as in sub\-sequent
Definitions~\ref{definition of nonfocal point 2}--\ref{definition of nonperiodicity condition 2},
see Remark~\ref{remark on negative Hamiltonians}.
We call a loop of length $T>0$ \emph{absolutely focused} if
the function
\[
|x^{(j)}(T;y,\eta)-y|^2
\]
has an infinite order zero in the variable $\eta$, and we denote
by $(\Pi_y^a)^{(j)}$ the set of normalised ($h^{(j)}(y,\eta)=1$)
covectors $\eta$ which serve as starting points for absolutely focused loops
generated by the Hamiltonian $h^{(j)}$.
It is known, see Lemma 1.8.3 in \cite{mybook}, that the
set $(\Pi_y^a)^{(j)}$ is measurable and,
moreover, the set $\Pi_y^{(j)}\setminus(\Pi_y^a)^{(j)}$ has measure zero.
This allows us to reformulate Definition~\ref{definition of nonfocal point 1}
as follows.
\begin{definition}
\label{definition of nonfocal point 2}
A point $y\in M$ is said to be \emph{nonfocal} if for each
$j=1,\ldots,m^+$ the set $(\Pi_y^a)^{(j)}$ has measure zero.
\end{definition}
In practical applications it is easier to work with
Definition~\ref{definition of nonfocal point 2}
because the set $(\Pi_y^a)^{(j)}$ is usually much
thinner than the set $\Pi_y^{(j)}$.
In order to derive a two-term asymptotic formula for the
spectral function $e(\lambda,x,x)$ we need the following
lemma (compare with Lemma~\ref{Singularity of the wave group at time zero lemma}).
\begin{lemma}
\label{Singularity of the wave group at time nonzero lemma pointwise}
Suppose that the point $y\in M$ is nonfocal.
Then for any complex-valued function $\hat\gamma\in C_0^\infty(\mathbb{R})$
with $\operatorname{supp}\hat\gamma\subset(0,+\infty)$ we have
\begin{equation}
\label{Singularity of the wave group at time nonzero lemma pointwise formula}
\mathcal{F}^{-1}_{t\to\lambda}[\hat\gamma(t)\operatorname{tr}u(t,y,y)]=
o(\lambda^{n-1})
\end{equation}
as $\lambda\to+\infty$.
\end{lemma}
\emph{Proof\ }
The result in question is a special case of
Theorem~4.4.9 from \cite{mybook}.~$\square$
\
The following theorem is our main result regarding the spectral function $e(\lambda,x,x)$.
\begin{theorem}
\label{theorem spectral function unmollified two term}
If the point $x\in M$ is nonfocal then
the spectral function $e(\lambda,x,x)$ admits
the two-term asymptotic expansion
(\ref{two-term asymptotic formula for spectral function})
as $\lambda\to+\infty$.
\end{theorem}
\emph{Proof\ }
The result in question is an immediate consequence of
formulae
(\ref{theorem spectral function mollified equation 7}),
Theorem~\ref{theorem spectral function mollified}
and
Lemma~\ref{Singularity of the wave group at time nonzero lemma pointwise}
from the current paper
and Theorem~B.5.1 from \cite{mybook}.~$\square$
\
We now deal with the counting function $N(\lambda)$.
Suppose that we have a Hamiltonian trajectory
$(x^{(j)}(t;y,\eta),\xi^{(j)}(t;y,\eta))$
and a real number $T>0$ such that
$(x^{(j)}(T;y,\eta),\xi^{(j)}(T;y,\eta))=(y,\eta)$.
We will say in this case
that we have a $T$-periodic trajectory originating
from the point $(y,\eta)\in T'M$.
Denote by $(S^*M)^{(j)}$ the unit cosphere bundle,
i.e.~the $(2n-1)$-dimensional surface in the cotangent
bundle defined by the equation $h^{(j)}(y,\eta)=1$.
The unit cosphere bundle is equipped with a natural Lebesgue measure:
the $(2n-1)$-dimensional surface area element on $(S^*M)^{(j)}$ is
$dy\,d(S^*_yM)^{(j)}$ where $d(S^*_yM)^{(j)}$ is the
$(n-1)$-dimensional surface area
element on the unit cosphere $(S^*_yM)^{(j)}$, see proof of
Lemma~\ref{Singularity of the wave group at time zero lemma}.
Denote by $\Pi^{(j)}$ the set of points in $(S^*M)^{(j)}$
which serve as starting points for periodic trajectories generated by the Hamiltonian
$h^{(j)}$.
It is known, see Lemma 1.3.4 in \cite{mybook}, that the
set $\Pi^{(j)}$ is measurable.
\begin{definition}
\label{definition of nonperiodicity condition 1}
We say that the nonperiodicity condition is fulfilled
if for each
$j=1,\ldots,m^+$ the set $\Pi^{(j)}$ has measure zero.
\end{definition}
We call a $T$-periodic trajectory \emph{absolutely periodic} if
the function
\[
|x^{(j)}(T;y,\eta)-y|^2+|\xi^{(j)}(T;y,\eta)-\eta|^2
\]
has an infinite order zero in the variables $(y,\eta)$, and we denote
by $(\Pi^a)^{(j)}$ the set of points in $(S^*M)^{(j)}$
which serve as starting points for absolutely periodic trajectories
generated by the Hamiltonian $h^{(j)}$.
It is known, see Corollary 1.3.6 in \cite{mybook}, that the
set $(\Pi^a)^{(j)}$ is measurable and,
moreover, the set $\Pi^{(j)}\setminus(\Pi^a)^{(j)}$ has measure zero.
This allows us to reformulate Definition~\ref{definition of nonperiodicity condition 1}
as follows.
\begin{definition}
\label{definition of nonperiodicity condition 2}
We say that the nonperiodicity condition is fulfilled
if for each
$j=1,\ldots,m^+$ the set $(\Pi^a)^{(j)}$ has measure zero.
\end{definition}
In practical applications it is easier to work with
Definition~\ref{definition of nonperiodicity condition 2}
because the set $(\Pi^a)^{(j)}$ is usually much
thinner than the set $\Pi^{(j)}$.
In order to derive a two-term asymptotic formula for the
counting function $N(\lambda)$ we need the following
lemma.
\begin{lemma}
\label{Singularity of the wave group at time nonzero lemma integrated}
Suppose that the nonperiodicity condition is fulfilled.
Then for any complex-valued function $\hat\gamma\in C_0^\infty(\mathbb{R})$
with $\operatorname{supp}\hat\gamma\subset(0,+\infty)$ we have
\begin{equation}
\label{Singularity of the wave group at time nonzero lemma integrated formula}
\int_M
\mathcal{F}^{-1}_{t\to\lambda}[\hat\gamma(t)\operatorname{tr}u(t,y,y)]\,dy=
o(\lambda^{n-1})
\end{equation}
as $\lambda\to+\infty$.
\end{lemma}
\emph{Proof\ }
The result in question is a special case of
Theorem~4.4.1 from \cite{mybook}.~$\square$
\
The following theorem is our main result regarding the counting function $N(\lambda)$.
\begin{theorem}
\label{theorem counting function unmollified two term}
If the nonperiodicity condition is fulfilled then
the counting function $N(\lambda)$ admits
the two-term asymptotic expansion
(\ref{two-term asymptotic formula for counting function})
as $\lambda\to+\infty$.
\end{theorem}
\emph{Proof\ }
The result in question is an immediate consequence of
formulae
(\ref{definition of counting function}),
(\ref{theorem spectral function mollified equation 7}),
Theorem~\ref{theorem spectral function mollified}
and
Lemma~\ref{Singularity of the wave group at time nonzero lemma integrated}
from the current paper
and Theorem~B.5.1 from \cite{mybook}.~$\square$
\section{$\mathrm{U}(m)$ invariance}
\label{U(m) invariance}
We prove in this section that the RHS of formula
(\ref{formula for b(x)})
is invariant under unitary transformations
(\ref{unitary transformation of operator A}),
(\ref{matrix appearing in unitary transformation of operator})
of our operator $A$.
The arguments presented in this section bear some
similarity to those from Section~\ref{U(1) connection},
the main difference being that the unitary matrix-function in question
is now a function on the base manifold~$M$ rather than on $T'M$.
Fix a point $x\in M$ and an index $j$ (index enumerating the eigenvalues
and eigenvectors of the principal symbol) and consider the expression
\begin{multline}
\label{formula for bj(x)}
\int\limits_{h^{(j)}(x,\xi)<1}
\biggl(
[v^{(j)}]^*A_\mathrm{sub}v^{(j)}
\\
-\frac i2
\bigl\{
[v^{(j)}]^*,A_1-h^{(j)},v^{(j)}
\bigr\}
+\frac i{n-1}h^{(j)}\bigl\{[v^{(j)}]^*,v^{(j)}\bigr\}
\biggr)(x,\xi)\,
d\xi\,,
\end{multline}
compare with (\ref{formula for b(x)}).
We will show that this expression
is invariant under the transformation
(\ref{unitary transformation of operator A}),
(\ref{matrix appearing in unitary transformation of operator}).
The transformation
(\ref{unitary transformation of operator A}),
(\ref{matrix appearing in unitary transformation of operator})
induces the following transformation of the principal
and subprincipal symbols of the operator $A$:
\begin{equation}
\label{transformation of the principal symbol}
A_1\mapsto RA_1R^*,
\end{equation}
\begin{equation}
\label{transformation of the subprincipal symbol}
A_\mathrm{sub}\mapsto
RA_\mathrm{sub}R^*
+\frac i2
\left(
R_{x^\alpha}(A_1)_{\xi_\alpha}R^*
-
R(A_1)_{\xi_\alpha}R^*_{x^\alpha}
\right).
\end{equation}
The eigenvalues of the principal symbol remain unchanged,
whereas the eigen\-vectors transform as
\begin{equation}
\label{transformation of the eigenvectors of the principal symbol}
v^{(j)}\mapsto Rv^{(j)}.
\end{equation}
Substituting formulae
(\ref{transformation of the principal symbol})--(\ref{transformation of the eigenvectors of the principal symbol})
into the RHS of
(\ref{formula for bj(x)})
we conclude that the increment of
the expression (\ref{formula for bj(x)}) is
\begin{multline*}
\int\limits_{h^{(j)}(x,\xi)<1}
\biggl(\,
\frac i2[v^{(j)}]^*
\left(
R^*R_{x^\alpha}(A_1)_{\xi_\alpha}-(A_1)_{\xi_\alpha}R^*_{x^\alpha}R
\right)
v^{(j)}
\\
-
\frac i2
\left(
[v^{(j)}]^*R^*_{x^\alpha}R(A_1-h^{(j)})v^{(j)}_{\xi_\alpha}
-
[v^{(j)}_{\xi_\alpha}]^*(A_1-h^{(j)})R^*R_{x^\alpha}v^{(j)}
\right)
\\
+
\frac i{n-1}h^{(j)}
\left(
[v^{(j)}]^*R^*_{x^\alpha}Rv^{(j)}_{\xi_\alpha}
-
[v^{(j)}_{\xi_\alpha}]^*R^*R_{x^\alpha}v^{(j)}
\right)
\biggr)(x,\xi)\,
d\xi\,,
\end{multline*}
which can be rewritten as
\begin{multline*}
-\frac i2\int\limits_{h^{(j)}(x,\xi)<1}
\biggl(
h^{(j)}_{\xi_\alpha}
\left(
[v^{(j)}]^*R^*_{x^\alpha}Rv^{(j)}
-
[v^{(j)}]^*R^*R_{x^\alpha}v^{(j)}
\right)
\\
-\frac 2{n-1}h^{(j)}
\left(
[v^{(j)}]^*R^*_{x^\alpha}Rv^{(j)}_{\xi_\alpha}
-
[v^{(j)}_{\xi_\alpha}]^*R^*R_{x^\alpha}v^{(j)}
\right)
\biggr)(x,\xi)\,
d\xi\,.
\end{multline*}
In view of the identity $R^*R=I$ the above expression can be further simplified,
so that it reads now
\begin{multline}
\label{increment of bj(x) first iteration}
i\int\limits_{h^{(j)}(x,\xi)<1}
\biggl(
h^{(j)}_{\xi_\alpha}[v^{(j)}]^*R^*R_{x^\alpha}v^{(j)}
\\
-\frac1{n-1}h^{(j)}
\left(
[v^{(j)}]^*R^*R_{x^\alpha}v^{(j)}_{\xi_\alpha}
+
[v^{(j)}_{\xi_\alpha}]^*R^*R_{x^\alpha}v^{(j)}
\right)
\biggr)(x,\xi)\,
d\xi\,.
\end{multline}
Denote
$B_\alpha(x):=-iR^*R_{x^\alpha}$
and observe that this set of matrices,
enumerated by the tensor index $\alpha$ running through the values $1,\ldots,n$,
is Hermitian.
Denote also $b_\alpha(x,\xi):=[v^{(j)}]^*B_\alpha v^{(j)}$
and observe that these $b_\alpha$ are positively homogeneous in $\xi$ of degree 0.
Then the expression
(\ref{increment of bj(x) first iteration})
can be rewritten as
\begin{equation*}
\label{increment of bj(x) second iteration}
-
\int\limits_{h^{(j)}(x,\xi)<1}
\left(
h^{(j)}_{\xi_\alpha}\,b_\alpha
-\frac 1{n-1}\,h^{(j)}\,\frac{\partial b_\alpha}{\partial\xi_\alpha}
\right)\!(x,\xi)\,
d\xi\,.
\end{equation*}
Lemma 4.1.4 and formula (1.1.15) from \cite{mybook} tell us
that this expression is zero.
\section{Spectral asymmetry}
\label{Spectral asymmetry}
In this section we deal with the special case when the operator
$A$ is differential (as opposed to pseudodifferential).
Our aim is to examine what happens when we change the sign of the operator.
In other words, we compare the original operator $A$ with the operator
$\tilde A:=-A$. In theoretical physics the transformation
$A\mapsto-A$ would be interpreted as time reversal,
see equation (\ref{dynamic equation most basic}).
It is easy to see that for a differential operator the number $m$
(number of equations in our system) has to be even and that the
principal symbol has to have the same number of positive and negative
eigenvalues.
In the notation of Section~\ref{Main results}
this fact can be expressed as $m=2m^+=2m^-$.
It is also easy to see that the
principal symbols of the two operators, $A$ and $\tilde A$,
and the eigenvalues and eigenvectors of the principal symbols
are related as
\begin{equation}
\label{Spectral asymmetry equation 1}
A_1(x,\xi)=\tilde A_1(x,-\xi),
\end{equation}
\begin{equation}
\label{Spectral asymmetry equation 2}
h^{(j)}(x,\xi)=\tilde h^{(j)}(x,-\xi),
\end{equation}
\begin{equation}
\label{Spectral asymmetry equation 3}
v^{(j)}(x,\xi)=\tilde v^{(j)}(x,-\xi),
\end{equation}
whereas the subprincipal symbols are related as
\begin{equation}
\label{Spectral asymmetry equation 4}
A_\mathrm{sub}(x)=-\tilde A_\mathrm{sub}(x).
\end{equation}
Formulae
(\ref{formula for a(x)}),
(\ref{formula for b(x)}),
(\ref{generalised Poisson bracket on matrix-functions}),
(\ref{Poisson bracket on matrix-functions})
and
(\ref{Spectral asymmetry equation 1})--(\ref{Spectral asymmetry equation 4})
imply
\begin{equation}
\label{Spectral asymmetry equation 5}
a(x)=\tilde a(x),
\qquad
b(x)=-\tilde b(x).
\end{equation}
Substituting (\ref{Spectral asymmetry equation 5}) into
(\ref{a via a(x)}) and (\ref{b via b(x)}) we get
\begin{equation}
\label{Spectral asymmetry equation 6}
a=\tilde a,
\qquad
b=-\tilde b.
\end{equation}
Formulae (\ref{two-term asymptotic formula for counting function})
and (\ref{Spectral asymmetry equation 6}) imply that the spectrum
of a generic first order differential operator is asymmetric about $\lambda=0$.
This phenomenon is known as
\emph{spectral asymmetry}
\cite{atiyah_short_paper,atiyah_part_1,atiyah_part_2,atiyah_part_3}.
If we square our operator $A$ and consider the spectral problem
$A^2v=\lambda^2v$,
then the terms $\pm b\lambda^{n-1}$ cancel
out and the second asymptotic coefficient of the counting function
(as well as the spectral function) of the operator $A^2$ turns to zero.
This is in agreement with the known fact that for an even order semi-bounded
matrix differential operator acting on a manifold without boundary
the second asymptotic coefficient of the counting function is zero, see
Section 6 of \cite{VassilievFuncAn1984} and \cite{SafarovIzv1989}.
\section{Bibliographic review}
\label{Bibliographic review}
To our knowledge, the first publication on the subject
of two-term spectral asymptotics for systems
was Ivrii's 1980 paper \cite{IvriiDoklady1980}
in Section 2 of
which the author stated, without proof, a formula for the second
asymptotic coefficient of the counting function.
In a subsequent 1982 paper \cite{IvriiFuncAn1982}
Ivrii acknowledged that the formula from
\cite{IvriiDoklady1980} was incorrect and gave a new formula, labelled (0.6), followed by a ``proof''.
In his 1984 Springer Lecture Notes \cite{ivrii_springer_lecture_notes}
Ivrii acknowledged on page 226 that both his
previous formulae for the second asymptotic coefficient were
incorrect and stated, without proof, yet another formula.
Roughly at the same time Rozenblyum \cite{grisha} also stated
a formula for the second asymptotic coefficient of the counting function
of a first order system.
The formulae from \cite{IvriiDoklady1980}, \cite{IvriiFuncAn1982} and \cite{grisha}
are fundamentally flawed because they are proportional to the subprincipal
symbol. As our formulae
(\ref{b via b(x)}) and (\ref{formula for b(x)})
show, the second
asymptotic coefficient of the counting function
may be nonzero even when the subprincipal symbol is zero.
This illustrates, yet again, the difference between scalar
operators and systems.
The formula on page 226 of \cite{ivrii_springer_lecture_notes}
gives an algorithm for the calculation of the correction term
designed to take account of the effect
described in the previous paragraph. This algorithm
requires the evaluation of a limit of a complicated expression
involving the integral, over the cotangent bundle,
of the trace of the symbol of the resolvent of the operator $A$
constructed by means of pseudodifferential calculus. This algorithm
was revisited in Ivrii's 1998 book, see formulae (4.3.39) and (4.2.25)
in \cite{ivrii_book}.
The next contributor to the subject was Safarov
who, in his 1989 DSc Thesis~\cite{SafarovDSc}, wrote down a formula
for the second asymptotic coefficient of the counting function
which was ``almost'' correct.
This formula appears in \cite{SafarovDSc} as formula (2.4).
As explained in Section~\ref{Main results},
Safarov lost only the curvature terms
$\,-\frac{ni}{n-1}\int h^{(j)}\{[v^{(j)}]^*,v^{(j)}\}$.
Safarov's DSc Thesis \cite{SafarovDSc} provides arguments which are sufficiently
detailed and we were able to identify the precise point
(page 163) at which the mistake occurred.
In 1998 Nicoll rederived \cite{NicollPhD} Safarov's formula
(\ref{formula for principal symbol of oscillatory integral})
for the principal symbols of the propagator, using a method
slightly different from \cite{SafarovDSc}, but stopped short
of calculating the second asymptotic coefficient
of the counting function.
In 2007 Kamotski and Ruzhansky \cite{kamotski}
performed an analysis of the
propagator of a first order elliptic system based on the
approach of Rozenblyum \cite{grisha}, but stopped short
of calculating the second asymptotic coefficient
of the counting function.
One of the authors of this paper, Vassiliev, considered systems in Section 6 of
his 1984 paper \cite{VassilievFuncAn1984}. However, that paper dealt with systems of a very special type:
differential (as opposed to pseudodifferential) and of even (as opposed to odd) order.
In this case the second asymptotic coefficients
of the counting function and the spectral function vanish, provided the
manifold does not have a boundary.
\end{document} |
\begin{document}
\title{On Poisoned Wardrop Equilibrium in Congestion Games }
\author{Yunian Pan\inst{1}\orcidID{0000-0002-7277-3657}
\and Quanyan Zhu \inst{1}\orcidID{0000-0002-0008-2953}}
\authorrunning{Pan and Zhu}
\institute{New York University, Brooklyn, NY, USA; E-mail: \email{\{yp1170, qz494\}@nyu.edu}
}
\maketitle
\begin{abstract}
Recent years have witnessed a growing number of attack vectors against increasingly interconnected traffic networks. Informational attacks have emerged as the prominent ones that aim to poison traffic data, misguide users, and manipulate traffic patterns. To study the impact of this class of attacks, we propose a game-theoretic framework where the attacker, as a Stackelberg leader, falsifies the traffic conditions to change the traffic pattern predicted by the Wardrop traffic equilibrium, achieved by the users, or the followers.
The intended shift of the Wardrop equilibrium is a consequence of strategic informational poisoning. Leveraging game-theoretic and sensitivity analysis, we quantify the system-level impact of the attack by characterizing the concept of poisoned Price of Anarchy, which compares the poisoned Wardrop equilibrium and its non-poisoned system optimal counterpart.
We use an evacuation case study to show that the Stackelberg equilibrium can be found through a two-time scale zeroth-order learning process and demonstrate the disruptive effects of informational poisoning, indicating a compelling need for defense policies to mitigate such security threats.
\keywords{Congestion Games \and Adversarial Attack \and Stackelberg Game \and Sensitivity Analysis}
\end{abstract}
\section{Introduction}\label{intro}
With the rapid growth of the Internet-of-Things (IoT), there has been a significant number of vulnerable devices in the past decade, widening the cyber-physical attack surface of modern Intelligent Transportation Systems (ITS).
For example, the adoption of IoT technologies for Vehicle-to-Vehicle (V2V), Vehicle-to-Infrastructure (V2I), and Infrastructure-to-Infrastructure (I2I) communications has enabled automated toll collection, traffic cameras and signals, road sensors, barriers, and Online Navigation Platforms (ONP) \cite{huq2017cyberattacks}.
It, however, creates opportunities for attackers to disrupt the infrastructure by exploiting cyber vulnerabilities.
A quintessential example of such attacks is the hijacking of traffic lights and smart signs.
The recent work \cite{cerrudo2014hacking} demonstrates that due to lack of authentication, the wireless sensors and repeaters of the lighting control system can be accessed and manipulated through antenna, exposing serious vulnerabilities of the traffic infrastructure.
The impact of a local attack on the traffic systems propagates and creates a global disruption of the infrastructure.
System-level modeling of cyber threats in traffic systems is crucial to understanding and assessing the consequences of cyber threats and the associated defense policies.
One significant system-level impact is on the traffic conditions, including delays and disruptions. Attackers can launch a man-in-the-middle (MITM) on ONP systems to mislead the population to choose routes that are favored by the attackers.
For instance, in 2014, two Israel students hacked the Google-owned Waze GPS app, causing the platform to report fake traffic conditions to its users; they used bot users to crowdsource false location information to the app, causing congestion \cite{popularnavihack2014}.
A similar recent case happened in Berlin \cite{hackerjam2013}, where an artist loaded $99$ smartphones in the street, causing Google-Map to mark that street as having bad traffic.
It has been reported in \cite{google2020} that real-time traffic systems can be deceived by malicious attacks such as modified cookie replays and simulated delusional traffic flows.
This class of attacks is referred to as {\it informational attacks} on traffic systems. They aim to exploit the vulnerabilities in the data and information infrastructures and strategically craft information to misguide users and achieve a target traffic condition.
The advent of information infrastructures and ONP has made user decisions more reliant on services offered by Google and Apple.
This reliance has made the attack easily influence the populational behaviors in a much faster and more direct way. Fig. \ref{anexample} illustrates an example attack scenario.
The attack manipulates the information collected by an ONP, including traffic demand and travel latency, and misleads it to make false traffic prediction and path recommendations.
\begin{figure}
\caption{
An example attack scenario:
a radio transmitter interferes the GPS communication channel, falsifying the user location information received by an ONP. ITS components, such as smart traffic signal and road cameras, can be hijacked to achieve the same goal.
}
\label{anexample}
\end{figure}
{\it Wardrop Equilibrium} (WE) \cite{wardrop1952road} has been widely used to predict the long-term behavioral patterns of the users and the equilibrium outcome of traffic conditions.
It is a natural system-level metric for the impact assessment of informational attacks.
Based on WE, we formulate a Stackelberg game as our attack model.
In this model, the attacker, or the leader, aims to disrupt the traffic system by poisoning the traffic conditions in a stealthy manner with bounded capabilities.
To capture this strategic behavior, we let the attacker's utility consist of the cost of modifying the traffic conditions and the payoff of disruption outcome. In addition, stealthy information falsification attacks seek to satisfy flow conservation constraints to evade inconsistency check.
The best response of the users, or the followers, to such informational attacks is the path-routing equilibrium outcome subject to falsified traffic conditions, which are encapsulated by the poisoned traffic latency function and demand vector.
We refer to the resulting behavioral pattern as the {\it Poisoned Wardrop Equilibrium} (PWE).
The disruptive effects of such attack is measured by the {\it Poisoned Price of Anarchy} (PPoA), which is the ratio of the aggregated latency under PWE to its non-poisoned system-optimal counterpart.
The local first-order stationary point is called differential Stackelberg equilibrium.
The sensitivity analysis of the PWE and PPoA shows that the attacker's utility function is sufficiently smooth under regularity assumptions of the latency functions. We characterize the implicit relation between the PWE and the attack parameters, based on which we give an explicit expression for the gradient of attack utility.
By analyzing the attack gradient, we find that the existence of a differential Stackelberg equilibrium is determined by the weighting coefficient of attack payoff that captures the tradeoff between ``disruption'' and ``stealthiness''.
We also uniformly characterize the locally Lipschitz parameters for both the attack utility and its gradient, which scale with a set of parameters, including the network size and topology, total traffic demand, and the smoothness level for the latency functions.
We propose a zeroth order two-time scale learning algorithm to find the differential Stackelberg equilibrium and study the iterative adversarial behavior.
We approximate the attack gradient by sampling the aggregated latency outcome of PWE and give a polynomial
sample efficient guarantee for gradient approximation.
We test our algorithm using an evacuation case study on a Sioux Falls network, where the attacker consistently learns to manipulate the information during the evacuation process through bandit feedback.
We show that after several iterations, the PPoA of the entire traffic network converges to a PWE where the traffic flow concentrates on several particular edges, causing congestion and low road utilization rates. As
congestion games are ubiquitous not only in transportation networks but also in applications related to smart grid, distributed control, and wireless spectrum sharing, it is anticipated that similar attacks can occur in a broader range of scenarios, and there is a need for the development of secure and resilient mechanisms as future work.
{\bf Content organization:} We briefly introduce WE and some related works in Section \ref{rw}. In Section \ref{pf}, we present the model for WE and introduce two problem formulations corresponding to two fundamental principles, following which the attacker's problem is discussed. Section \ref{wediff} provides several theoretical aspects for attack objective function. In Section \ref{ag}, we explore the algorithmic development of the Stackelberg learning framework. We demonstrate the attack effects in Section \ref{sfdemo}.
\section{Related Work}\label{rw}
WE was introduced in 1952 \cite{wardrop1952road,beckmann1956studies} as an equilibrium model to predict the traffic patterns in transportation networks.
The equilibrium concept is related to the notion of Nash equilibrium in game theory that was developed separately.
Rosenthal in \cite{rosenthal1973class} introduced the class of congestion games and showed its existence of a pure-strategy Nash equilibrium.
There have been an extensive and growing literature that studies congestion games and their variants, and they have been used to model and understand the various technological impact on the transportation networks, including speed limits \cite{speedexogenous}, road pricing \cite{groot2014toward} or direct ONP assignment \cite{stackelbergweighted}.
In these works, congestion games are subsumed as a building block to formulate Stackelberg games \cite{rockafellar2002} to design incentives, pricing, and policies.
This work leverages the approach to create a formal framework to quantify and analyze the impact of the worst-case attack strategies on the transportation networks.
PoA is commonly used as a metric and analytical tool for congestion games.
Cominetti et al. in \cite{DBLP:journals/corr/abs-1907-10101} have shown that PoA is a $C^1$ function of demand under certain conditions, which coincides to our results showing the smoothness of attack utility w.r.t. the demand poisoning parameter.
Aligned with our discussions on the latency corruption, the effects of biased cost function have been investigated in \cite{meir2015playing}, their results are based on the notion of $(\lambda, \mu)$-smoothness \cite{roughgarden2010algorithmic}, which differs from our methods.
In general, PoA is sharply bounded by the condition number of the set of latency functions \cite{roughgarden2003price}, called the Pigou-bound. We refer the readers to \cite{Correa05onthe} for tighter analysis.
Specifically, for affine cost functions, this bound becomes $4/3$. Our numerical study shows that the inefficiency can be worse than the established results under informational attacks.
\section{Problem Formulation}\label{pf}
\subsection{Preliminary Background: Congestion Game and Wardrop Equilibrium}
Consider the traffic network as a directed graph $\mathcal{G} = (\mathcal{V}, \mathcal{E})$,
with the vertices $\mathcal{V}$ representing road junctions, and edges $\mathcal{E}$ representing road segments.
We assume that $\mathcal{G}$ is finite, connected without buckles, i.e., the edges that connect a vertex to itself. The network contains the following elements:
\begin{itemize}
\item $\mathcal{W} \subseteq \mathcal{V} \times \mathcal{V}$ is the set of distinct origin-destination (OD) pairs in the network; for $w \in \mathcal{W}$, $(o_w, d_w) \in \mathcal{V} \times \mathcal{V}$ is the OD pair;
\item $\mathcal{P}_w \subseteq \mathcal{P}(\mathcal{E})$ is the set of all directed paths from $o_w$ to $d_w$;
\item $\mathcal{P} = \bigcup_{w\in\mathcal{W}}\mathcal{P}_w$ is the set of paths in a network, each $\mathcal{P}_w$ is disjoint;
\item $Q \in \R^{|\mathcal{W}|}_{\geq 0}$ is the OD demand vector ,
$Q_w$ represents the traffic demand between OD pair $w \in \mathcal{W}$;
\item $q \in \R^{|\mathcal{E}|}_{\geq 0}$ is the edge flow vector, $q_e$ is the amount of traffic flow that goes through edge $e \in \mathcal{E}$
\item $\mu \in \R^{|\mathcal{P}|}_+$ is the path flow vector, $\mu_p$ is the amount of traffic flow that goes through path $p \in \mathcal{P}$.
\item $\ell_e: \R_{\geq 0} \to \R_+ \ \ e \in \mathcal{E}$ is the cost/latency functions, determined by the edge flow. Let $\ell : \R^{|\mathcal{E}|}_{\geq 0} \to \R^{|\mathcal{E}|}_+$ denote the vector-valued latency function.
\end{itemize}
We assume that there is a set of infinite, infinitesimal players over this graph $\mathcal{G}$, denoted by a measurable space $(\mathcal{X}, \mathcal{M}, m)$.
The players are non-atomic, i.e., $m(x) = 0 \ \ \forall x \in \mathcal{X}$; they are split into distinct populations indexed by the OD pairs, i.e., $\mathcal{X} = \bigcup_{w\in\mathcal{W}} \mathcal{X}_w$ and $\mathcal{X}_w \bigcap \mathcal{X}_{w^{\prime}} = \empty \ \ \forall w, w^{\prime} \in \mathcal{W}$.
For each player $x \in \mathcal{X}_w$, we assume that the path is fixed at the beginning, and thus the action of player $x$ is $A(x) \in \mathcal{P}_w$, which is $\mathcal{M}$-measurable.
The action profile of all the players $\mathcal{X}$ induces the edge flows $q_e := \int_{\mathcal{X}} \mathds{1}_{\{e \in A(x)\}} m(dx) \ \ e \in \mathcal{E}$, and a path flow $\mu_p := {\int_{\mathcal{X}_w} \mathds{1}_{\{A(x) = p\}} m(dx)} \ \ p \in \mathcal{P}_w$, which are the fraction of players using edge $e$, and the fraction of players using $p \in \mathcal{P}_w$, respectively.
The path flow can also be interpreted as a mixed strategy played by a single centralized planner.
By definition, a feasible flow pattern $(q, \mu) \in \R^{|\mathcal{E}|} \times \R^{|\mathcal{P}|}$ is constrained by \eqref{feasibleconstraint}:
\begin{equation}
\label{feasibleconstraint}
\begin{aligned}
\Lambda \mu - Q & = 0 \\
\Delta \mu - q & = 0 \\
-\mu & \preceq 0 .
\end{aligned}
\end{equation}
where $\Lambda \in \R^{|\mathcal{W}| \times |\mathcal{P}|}$, $\Delta \in \R^{|\mathcal{E}| \times |\mathcal{P}|}$ are the path-demand incidence matrix and the path-edge incidence matrix, respectively, which are defined in \eqref{pathdemandedge}.
The two matrices only depend on the topology of network $\mathcal{G}$.
\begin{equation}
\label{pathdemandedge}
\Lambda_{w p}=\left\{\begin{array}{ll}1 & \text { if } p \in \mathcal{P}_{w} \\ 0 & \text { otherwise }\end{array} \quad \text { and } \quad \Delta_{e p}=\left\{\begin{array}{ll}1 & \text { if } e \in p \\ 0 & \text { otherwise }\end{array}\right.\right. .
\end{equation}
The utility function for a single player is the aggregated cost for the path she selects, $\ell_p (\mu) = \sum_{e \in p} \ell_e(q_e)$. Note that the path latency is a function of the path flow vector $\mu$.
We hereby impose the first assumption about the edge latency functions.
\begin{assumption}[($\ell$-Regularity)]
\label{latencyassumption}
For all $e \in \mathcal{E}$, the latency functions $\ell_e$ are $l_0$-Lipschitz continuous, twice differentiable with $\ell_e^{\prime} (q_e) > 0 $, and $\ell_e^{\prime\prime}(q_e) \geq 0$ for $q_e \geq 0$. In addition, $\ell^{\prime}_e$ are $l_1$-Lipschitz continuous and $\ell^{\prime\prime}_e$ is bounded by $l_1$.
\end{assumption}
The path latency $\ell_p$ can be bounded by $D(\mathcal{G})c_0$, where $D(\mathcal{G})$ the diameter of the graph $\mathcal{G}$, and $c_0 : = \| \ell\|_{\infty} = \max_{e \in \mathcal{E}} \ell_e (D)$.
This congestion game $\mathcal{G}_c$ is thus encapsulated by the triplet $(\mathcal{X}, \ell, \mathcal{P})$.
\subsection{System Optimum and Wardrop Equilibrium}
In the seminar work \cite{wardrop1952road}, Wardrop proposed two different principles, leading to two solution concepts.
\begin{itemize}
\item \textit{Wardrop's first principle (Nash equilibrium principle)}: Players aim to minimize their own travel cost, i.e., for a mixed strategy $\mu$ to be a Nash equilibrium, whenever a path $ \mu_p > 0$ is chosen for the OD pair $w$, it holds that $\ell_p(\mu) \leq \ell_{p'}(\mu)$ $\forall p' \in \mathcal{P}_w$, implying that every flow has the same latency.
\item \textit{Wardrop's second principle (social optimality principle)}: Players pick routes cooperatively such that the overall latency is minimized. The coordinated behaviors minimize the aggregated system performance $\sum_{e \in \mathcal{E}} q_e \ell_e(q_e)$ under proper constraints.
\end{itemize}
We hereby formalize the notion of { \it System Optimum} (SO) and WE.
Definition \ref{wardropsecond} follows Wardrop's second principle, characterizing the cooperative behaviors of individuals that minimize the aggregated latency.
\begin{definition}[System Optimum (SO)]\label{wardropsecond}
The socially optimal routing $(q^{\star}, \mu^{\star})$ is a feasible flow pattern that optimizes the social welfare by minimizing the aggregated latency $S(q) = \sum_{e \in \mathcal{E}} q_e \ell_e(q_e)$, obtained from the optimization problem \eqref{socialopt}
\begin{equation} \label{socialopt}
\begin{aligned}
\min_{q, \mu } \quad &\sum_{e \in \mathcal{E}} q_e \ell_e(q_e) \\
\text{s.t.} \quad & (q, \mu) \in F_Q
\end{aligned}
\end{equation}
where $F_Q := \{(q, \mu) \in \R^{|\mathcal{E}|} \times \R^{|\mathcal{P}|} | (q, \mu) \text{ satisfies \eqref{feasibleconstraint}.} \}$
\end{definition}
By assumption \ref{latencyassumption}, problem \eqref{socialopt} is strictly convex in $\R^{|\mathcal{E}|}$, admitting a strict global minimum edge flow $q^{\star}$, the corresponding path flow set $\pmb{\mu}^{\star}$ is generally the non-unique solution to the linear equation $\Delta \mu = q^{\star}$, satisfying \eqref{feasibleconstraint}.
The optimal aggregated latency is denoted by $S^{\star} := S(q^{\star})$.
The Nash equilibrium, on the other hand, exploits the self-interest nature of the individuals in a transportation network.
Definition \ref{wardropfirst} follows Wardrop's first principle, characterizing the non-cooperative behaviors of individuals that minimize their own latency.
\begin{definition}[Wardrop Equilibrium (WE)] \label{wardropfirst}
A flow pattern $(q, \mu)$ is said to be a Wardrop Equilibrium (WE), if it satisfies $(q, \mu) \in F_Q $, and for all $w \in \mathcal{W}$:
\begin{itemize}
\item $\ell_p (\mu) = \ell_{p'}(\mu)$ for all $p, p' \in \mathcal{P}_w$ with $ \mu_p, \mu_{p'} >0$;
\item $\ell_p(\mu) \geq \ell_p(\mu)$ for all $p, p' \in \mathcal{P}_w$ with $ \mu_p >0$ and $\mu_{p'} = 0$.
\end{itemize}
Equivalently, WE can be characterized as the minimizer of the following convex program:
\begin{equation} \label{beckmanopt}
\begin{aligned}
\min_{ q, \mu } \quad & \sum_{e \in \mathcal{E}} \int_{0}^{q_e} \ell_e(z) dz\\
\text{s.t.} \quad & (q, \mu ) \in F_Q,
\end{aligned}
\end{equation}
where $\sum_{e \in \mathcal{E}} \int_{0}^{q_e} \ell_e(z) dz = : J(q)$ is called the Beckman potential.
\end{definition}
Since, by assumption \ref{latencyassumption}, $\ell_e$ is strictly increasing, the equilibrium edge flow $q^*$ is uniquely defined; the corresponding equilibrium path flow set $\pmb{\mu}^*$ is generally the non-unique solution to the linear equation $\Delta \mu = q^*$, satisfying \eqref{feasibleconstraint}.
\subsection{Stackelberg Congestion Security Game}
This section formulates a Stackelberg congestion security game. We consider
an attacker who manipulates latency and demand data to mislead the ONP and its users.
To capture this malicious behavior, we introduce a pair of attack parameters $ (\theta, d) \in (\Theta \times \mathcal{D})$ as the attack action, which parameterize two global traffic condition operators, $\Phi_{\theta}: \Theta \times \R^{|\mathcal{E}|} \times \R^{|\mathcal{P}|} \to \R^{|\mathcal{E}|} \times \R^{|\mathcal{P}|} $ and $\Phi_d: \mathcal{D} \times \R^{|\mathcal{W}|} \to \R^{| \mathcal{W}|}$.
The flow operator $\Phi_{\theta}$ modifies the real-time traffic flow to poison the latency function;
the demand operator $\Phi_d$ poisons the traffic demand prediction.
After the poisoning, the demand prediction and latency function are corrupted to be $\tilde{Q} : = \Phi_d \cdot Q$ and $\tilde{\ell} = \ell \circ \Phi_{\theta} $, respectively.
We hereby introduce the {\it $(\theta,d)$-Poisoned Wardrop Equilibrium} ($(\theta,d)$-PWE) as described in \ref{pwe}.
\begin{definition}[$(\theta,d)$-PWE]\label{pwe}
A flow pattern $(q, \mu)$ is said to be a $(\theta,d)$-PWE, if it is a solution to the problem \eqref{beckmanopt}, with the latency function being $\tilde{\ell} = \Phi_{\theta} \circ \ell$ and the OD demand vector being $\tilde{Q} = \Phi_d \cdot Q$.
The equilibrium edge flow and path flow set are denoted by $q^*(\theta, d)$, and $ \pmb{\mu}^*(\theta, d)$, respectively.
\end{definition}
As illustrated in Fig. \ref{attackfeedback}, the corruption of real-time traffic conditions, the poisoned path recommendation by ONP, and the user path selection (the formation of PWE) form a closed-loop system that is interfered by the attacker.
\begin{figure}
\caption{An illustration of the $(\theta,d)$-PWE-formation loop: the attacker stealthily intercepts the communication channel that collects traffic conditions, forcing the formation of traffic flow that is favored by the attacker.}
\label{attackfeedback}
\end{figure}
To quantify the disruption caused by such informational attack, we introduce the notion of $(\theta,d)$-{\it Poisoned Price of Anarchy} ($(\theta,d)$-PPoA).
\begin{definition}\label{ppoadef}
The ratio of aggregated latency at $(\theta,d)$-PWE
to the aggregated latency at non-poisoned SO is called $(\theta,d)$-PPoA , i.e.:
\begin{equation}\label{ppoa}
\text{$(\theta,d)$-PPoA} = \frac{\sum_{e \in \mathcal{E}} q^*_e(\theta,d) \ell_e(q^*_e(\theta,d))}{\sum_{e \in \mathcal{E}} q^{\star}_e \ell_e (q^{\star}_e)}.
\end{equation}
\end{definition}
Now, we are ready to define attacker's cost function and complete the attack model.
We give two formulations in the sequel, based on the malicious manipulation of edge flow and path flow, respectively.
\subsubsection{Edge Flow Poisoning}
In this case, the attacker corrupts the latency function through a global edge flow operator $\Phi_{\theta}: \Theta \times \R^{|\mathcal{E}|} \mapsto \R^{|\mathcal{E}|}$. For simplicity, we consider the attack operators to be matrices of proper dimensions, i.e., $\Phi_{\theta} \in \R^{|\mathcal{E}| \times |\mathcal{E}|}$, and $ \Phi_d \in \R^{|\mathcal{W}| \times |\mathcal{W}|}$.
The operators $\Phi_{\theta}$ and $\Phi_d$ have the following interpretation.
Through data manipulation, the fraction $\Phi_{\theta; i, j}$ of traffic flow in edge $i$ is redistributed to edge $j$; the fraction $\Phi_{d, i, j}$ of demand between OD pair $i$ is redirected to OD pair $j$.
It is reasonable to let $\|\Phi_{\theta}\|_{op}$ and $\|\Phi_d\|_{op}$ be $1$ such that the flow and demand corruption cannot be identified by checking the norm of the flow and demand vectors. The set of column-stochastic matrix satisfies such a constraint.
The problem \eqref{edgeflowattackopt} is to optimize the attack utility $\mathcal{L}: \Theta \times \mathcal{Q} \times \R^{|\mathcal{E}|} \mapsto \R$.
The utility $\mathcal{L}$ contains two terms.
The attack cost term is measured by the $\|\cdot\|_F$ norm of deviation from ``no-attack'' to ``attack'';
the attack payoff term is the $(\theta,d)$-PPoA weighted by parameter $\gamma$, which measures the disruption of the transportation network.
\begin{equation} \label{edgeflowattackopt}
\begin{aligned}
\min_{ (\theta, d) \in \Theta \times \mathcal{Q} , \ q = q^*(\theta, d)} \ \ &\mathcal{L}\left((\theta, d) , q\right) : = \frac{1}{2} (\| \Phi_{\theta} - I \|^2_F + \| \Phi_d - I\|^2_F) - \gamma \frac{\sum_{e \in \mathcal{E}} q_e \ell_e(q_e)}{\sum_{e \in \mathcal{E}} q^{\star}_e \ell_e (q^{\star}_e)} \\
\text{s.t. } & \quad \Phi_{\theta}^{\top} \mathds{1} = \mathds{1} ,
\quad \\
& \quad \Phi_{\theta; i,j} \geq 0 \quad \forall i, j \in 1, \ldots, |\mathcal{E}|, \\
& \quad \Phi_d^{\top} \mathds{1} = \mathds{1} , \\
&\quad \Phi_{d; i,j} \geq 0 \quad \forall i, j \in 1, \ldots, |\mathcal{W}|.
\end{aligned}
\end{equation}
\subsubsection{Path Flow Poisoning}
In this case, the attacker corrupts the latency function through a global path flow operator $\Phi_{\theta}: \Theta \times \R^{|\mathcal{P}|} \mapsto \R^{|\mathcal{P}|}$.
Let $\Phi_{\theta} \in \R^{|\mathcal{P}|\times |\mathcal{P}|}$ and $\Phi_d \in \R^{|\mathcal{W}| \times |\mathcal{W}|}$, with similar path flow and demand operating interpretation.
Writing the $(\theta,d)$-PPoA term with respect to the path flow, we can restate the problem as in \eqref{pathflowattackopt}:
\begin{equation} \label{pathflowattackopt}
\begin{aligned}
\min_{ (\theta, d) \in \Theta \times \mathcal{Q} } \sup_{ \mu \in \pmb{\mu}^*(\theta, d)} \ \ &\mathcal{L}\left((\theta, d) , \mu \right) : = \frac{1}{2} (\| \Phi_{\theta} - I \|^2_F + \| \Phi_d - I\|^2_F) - \gamma \frac{\sum_{p \in \mathcal{P}} \mu_p \ell_p(\mu) }{\sum_{p \in \mathcal{P}} \mu^{\star}_p \ell_p (\mu^{\star})}\\
\text{s.t. } & \quad \Phi_{\theta}^{\top} \mathds{1} = \mathds{1} ,
\quad \\
& \quad \Phi_{\theta; i,j} \geq 0 \quad \forall i, j \in 1, \ldots, |\mathcal{P}|, \\
& \quad \Phi_d^{\top} \mathds{1} = \mathds{1}, \\
&\quad \Phi_{d; i,j} \geq 0 \quad \forall i, j \in 1, \ldots, |\mathcal{W}| ,
\end{aligned}
\end{equation}
where we take the supremum over the path flow set of $(\theta,d)$-PWE.
One can verify that the normalizing denominator $\sum_{p \in \mathcal{P}} \mu^{\star}_p \ell_p (\mu^{\star}) = \sum_{e \in \mathcal{E}} q^{\star}_e\ell_e(q^{\star}_e) $, i.e., while the one edge flow may correspond to multiple path flows, the aggregated latency remains the same.
Since in general the optimal path flow $\pmb{\mu}^*(\theta,d)$ is a set-valued mapping, we focus on problem \eqref{edgeflowattackopt} for analytical convenience in the sequel.
For the Stackelberg game defined in \eqref{edgeflowattackopt}, we refer to the constraint set as $\mathcal{C}$.
The convexity of the mathematical program \eqref{edgeflowattackopt} can not be determined due to the implicity of parameterization $(\theta,d)$-PWE. Assuming that the parameterization yields sufficient smoothness conditions, we adopt the first-order local stationary point as the solution concept, called {\it Differential Stackelberg Equilibrium} (DSE), as described in Definition \ref{dse}.
\begin{definition}[Differential Stackelberg Equilibrium (DSE)] \label{dse}
A pair $\left((\theta^*,d^*), (q, \pmb{\mu})\right)$ with $(\theta^*, d^*)\in\mathcal{C}$, $(q,\pmb{\mu}) = (q^*(\theta^*, d^*), \pmb{\mu}^*(\theta^*,d^*))$ being the $(\theta^*,d^*)$-PWE, is said to be a Differential Stackelberg Equilibrium (DSE) for the Stackelberg game defined in \eqref{edgeflowattackopt}, if
$\nabla_{\theta,d} \mathcal{L} = 0$, and $\nabla^2_{\theta,d} \mathcal{L}$ is positive definite.
\end{definition}
In practice, we consider the explicit case where $\Phi_{\theta}$ is a matrix in $\R^{|\mathcal{E}|\times |\mathcal{E}|}$ (or $\R^{|\mathcal{P}|\times |\mathcal{P}|}$) and is parameterized by $\theta \in \Theta = \R^{|\mathcal{E}|^2 }$ (or $\R^{|\mathcal{P}|^2 }$) such that $vec(\Phi_{\theta}) = \theta$, and $\Phi_d$ is parameterized by $d \in \mathcal{D} = \R^{|\mathcal{W}|^2 }$ such that $vec(\Phi_d) = d$. In this case, $\mathcal{C}$ is a compact and convex set. Later on, we use the operator $\operatorname{Proj}_{\mathcal{C}}( \theta, d)$ to represent the Euclidean projection onto $\mathcal{C}$, i.e., $\operatorname{Proj}_{\mathcal{C}}( \theta, d) = \arg \min_{z \in \mathcal{C}} \| z - ( \theta, d)\|^2$.
\section{Sensitivity Analysis} \label{wediff}
\subsection{Smoothness of $(\theta, d)$-PWE}
Let $\Theta, \mathcal{D}$ be open sets, for some fixed parameter $(\theta, d) \in \Theta \times \mathcal{D}$, a unique minimizer $q^*(\theta, d)$ of the parameterized Beckman program \eqref{parabeck} is uniquely determined.
\begin{equation}\label{parabeck}
\begin{aligned}
\min_{q, \mu} \ \ & J( (q, \mu) |\theta, d) := \sum_{e \in \mathcal{E}} \int_{0}^{q}(\ell \circ \Phi_{\theta})_e(z) d\\
\text{s.t.} \quad & (\Phi_{\theta} q, \mu ) \in F_{\Phi_d Q}.
\end{aligned}
\end{equation}
To study the sensitivity of $\mathcal{L}$ and $q^{\star}(\theta,d)$ to the perturbations of $\theta$ and $d$, we reduce the feasibility set for the parameterized version of Beckman program \eqref{beckmanopt} to the $q$ variable first.
In doing so, we give Lemma \ref{charfeasibility}.
\begin{lemma}\label{charfeasibility}
~Given attack parameter $\theta, d$, define the feasible set of edge flow
\begin{equation*}
\pmb{q}_{\theta, d} := \{ q \in \mathbb{R}^{|\mathcal{E}|} \ \big\vert \ \exists \mu \text{ such that } (\Phi_{\theta}q ,\mu) \in F_{\Phi_d Q} \},
\end{equation*} which has the following properties:
\begin{itemize}
\item[(a)] There exists $A \in \R^{r \times |\mathcal{E}|}$ and $B \in \R^{r \times |\mathcal{W}|}$ of proper dimensions, with $r$ depending only on $ \mathcal{G}$, such that
\begin{equation*}
\pmb{q}_{\theta,d} = \{ q \in \mathbb{R}^{| \mathcal{E}|} \ \big\vert \ A \Phi_{\theta}q \leq B\Phi_d Q\} .
\end{equation*}
\item[(b)] Any $q \in \pmb{q}_{\theta,d}$ is bounded by
\begin{equation*}
\| q\| \leq D\sqrt{|\mathcal{E}|}.
\end{equation*}
\item[(c)] There exists a constant $l_d$ such that for any $d^{\prime}, d \in \mathcal{D}$ and $ q \in \pmb{q}_{\theta,d}$ there exists $q^{\prime} \in \pmb{q}_{\theta, d^{\prime}}$ satisfying
\begin{equation*}
\|q^{\prime} - q \| \leq l_d \|d^{\prime} - d \|.
\end{equation*}
\end{itemize}
\end{lemma}
By lemma \ref{charfeasibility}, the feasibility set $\pmb{q}_{\theta,d}$ can be projected onto $q$-space as a linear inequality constraint on $q$-variable, which is bounded and local Lipschitz smooth w.r.t. $d$.
\begin{lemma}\label{continuity}
Let $q^*(\theta, d)$ be the unique minimizer of \eqref{parabeck}. Then, at each $(\bar{\theta}, \bar{d}) \in \Theta \times \mathcal{D}$, there exists $\varepsilon$ such that for all $(\theta, d) \in B_{\varepsilon}(\bar{\theta}, \bar{d})$:
\begin{itemize}
\item[(a)] The edge flow at $(\theta,d)$-PWE, $q^*(\theta, d)$ is continuous, i.e., for any sequence $ (\theta_n, d_n) \to (\bar{\theta}, \bar{d}), n \in \mathbb{N}$, we have $q^*(\theta_n, d_n) \to q^*(\theta, d)$. In addition, there exists a Lipschitz constant $ l_q > 0$ that is related to $\|B \Phi_d \|$ such that
\begin{equation}
\label{lipqthetad}
\|q^*(\theta, d) - q^*(\bar{\theta}, \bar{d})\| \leq l_q \|(\theta,d ) - (\bar{\theta}, \bar{d}) \|
\end{equation}
\item[(b)] The poisoned aggregated latency function $ S(q^*(\theta,d))$ is locally Lipschitz continuous, i.e.,
\begin{equation*}
\| S(q^*(\theta, d)) - S(q^*(\bar{\theta}, \bar{d})) \| \leq (c_0 + l_0 D) l_q \sqrt{|\mathcal{E}|}\| (\theta, d) - (\bar{\theta}, \bar{d})\| .
\end{equation*}
\end{itemize}
\end{lemma}
The Lipschitz constant in \ref{continuity} (b) has the following interpretation.
The smoothness level of the poisoned aggregated latency function scales with three factors: the upper estimate scale of latency ($\|\ell\|_{\infty} \text{ and } l_0 D$), the network size ($\sqrt{|\mathcal{E}|}$), and the smoothness level of $(\theta,d)$-PWE ($l_q$).
This Lipschitz constant directly implies the smoothness level of $(\theta,d)$-PPoA.
\subsection{Differentiability of $(\theta, d)$-PWE}
By lemma \ref{charfeasibility}, the feasibility set can be reduced to a linear inequality constraint.
Define the $(\theta, d)$-poisoned Lagrangian:
\begin{equation}\label{plagrangian}
L(q, \lambda, \theta, d) = \sum_{e\in \mathcal{E}} \int_{0}^{(\Phi_{\theta} q)_e} \ell_e (z) dz + \lambda^{\top}(A \Phi_{\theta}q - B\Phi_d Q).
\end{equation}
The KKT condition states that a vector $\tilde{q} \in \R^{|\mathcal{E}|}$ is the solution $q^*(\theta,d)$ if and only if there exists $\tilde{\lambda} \in \R^{r}$ such that:
\begin{equation*}
\begin{aligned}
A \Phi_{\theta}\tilde{q} - B\Phi_d Q & \preceq 0 \\
\tilde{\lambda}_i & \geq 0, \quad i = 1, \ldots, r \\
\tilde{\lambda}_i (A \Phi_{\theta} \tilde{q} - B \Phi_d Q)_i & = 0, \quad i = 1, \ldots, r \\
\sum_{e^{\prime} \in \mathcal{E}} \Phi_{\theta; e, e^{\prime}}^{\top} \ell_{e^{\prime}}( (\Phi_{\theta} \tilde{q} )_{e^{\prime}}) + (\Phi_{\theta}^{\top } A^{\top} \tilde{\lambda} )_e & = 0 , \quad e = 1, \ldots, |\mathcal{E}| ,
\end{aligned}
\end{equation*}
To apply Implicit Function Theorem (IFT) to the poisoned Beckman program \eqref{beckmanopt}, we define the vector-valued function $g = \nabla_{(q, \lambda)} L$,
\begin{equation}\label{ifth}
g( \tilde{q}, \tilde{\lambda}, \theta, d) = \begin{bmatrix}
\sum_{e^{\prime} \in \mathcal{E}} \Phi_{\theta; e^{\prime}, 1} \ell_{e^{\prime}}( (\Phi_{\theta} \tilde{q} )_{e^{\prime}}) + (\Phi_{\theta}^{\top } A^{\top} \tilde{\lambda} )_1 \\
\ldots\\
\sum_{e^{\prime} \in \mathcal{E}} \Phi_{\theta; e^{\prime}, |\mathcal{E}|} \ell_{e^{\prime}}( (\Phi_{\theta} \tilde{q} )_{e^{\prime}}) + (\Phi_{\theta}^{\top } A^{\top} \tilde{\lambda} )_{|\mathcal{E}|}\\
\operatorname{diag}(\lambda) (A \Phi_{\theta} \tilde{q} - B\Phi_d Q) \end{bmatrix},
\end{equation}
where $\operatorname{diag}(\cdot)$ transforms the vector $\lambda$ into the matrix with $\lambda_i$ being the diagonal entries. For a candidate WE solution $(\tilde{q}, \tilde{\lambda})$ such that $g(\tilde{q}, \tilde{\lambda}, \theta, d ) = 0$, we define the partial Jacobian w.r.t. variable $(q, \lambda)$:
\begin{equation}\label{dqlambda}
\mathrm{D}_{(q, \lambda)} g(\tilde{q}, \tilde{\lambda}, \theta, d)=\begin{bmatrix}
\mathrm{D}_{q} \nabla_{q} L(\tilde{q}, \tilde{\lambda}, \theta, d) & \Phi_{\theta}^{\top} A^{\top} \\
\operatorname{diag}(\tilde{\lambda}) A \Phi_{\theta} & \operatorname{diag}(A \Phi_{\theta} \tilde{q} - B\Phi_d Q)
\end{bmatrix} ,
\end{equation}
where the first diagonal term $$
\mathrm{D}_{q} \nabla_{q} L(\tilde{q}, \tilde{\lambda}, \theta, d) = [\sum_{e^{\prime} \in [\mathcal{E}]} \Phi_{\theta; e^{\prime}, i} \Phi_{\theta; e^{\prime}, j} \ell^{\prime}_{e^{\prime}}( (\Phi_{\theta} \tilde{q} )_{e^{\prime}})]_{i, j \in [\mathcal{E}]} = \Phi_{\theta}^{\top} \nabla_q \tilde{\ell},$$ is positive definite according to assumption \ref{latencyassumption}. By Shur's complement, one can verify that if $\{ i \ \big \vert \tilde{\lambda}_i = 0 \text{ and } (A \Phi_{\theta} \tilde{q} - \Phi_d Q)_i = 0\} =\emptyset$, the partial Jacobian is non-singular.
The partial Jacobian w.r.t. variable $(\theta, d)$ is
\begin{equation}\label{jacobianthetad}
\mathrm{D}_{(\theta, d)} g(\tilde{q}, \tilde{\lambda}, \theta, d)
=\begin{bmatrix}
\mathrm{D}_{\theta} \nabla_{q} L(\tilde{q}, \tilde{\lambda}, \theta, d) & \mathrm{D}_{d} \nabla_{q} L(\tilde{q}, \tilde{\lambda}, \theta, d)\\
\operatorname{diag}(\tilde{\lambda}) \mathrm{D}_{\theta} (A \Phi_{\theta} \tilde{q}) & - \operatorname{diag}(\tilde{\lambda}) \mathrm{D}_d(B\Phi_dQ)
\end{bmatrix}.
\end{equation}
Lemma \ref{iftbeckman} gives the local differentiability result for $(\theta,d)$-PWE.
\begin{lemma}[IFT for $(\theta,d)$-PWE] \label{iftbeckman}
Let $g(\tilde{q}, \tilde{\lambda}, \theta, d) = 0$, if the set $\{ i \ \big \vert \tilde{\lambda}_i = 0 \text{ and } (A \Phi_{\theta} \tilde{q} - \Phi_d Q)_i = 0\} =\emptyset$, then $\mathrm{D}_{(q, \lambda)} g(\tilde{q}, \tilde{\lambda}, \theta, d )$ is non-singular, then the solution mapping for WE \eqref{beckmanopt} has a single-value localization $q^*(\theta, d)$ around $(\tilde{q}, \tilde{\lambda})$, which is continuously differentiable in the neighbor of $(\theta,d)$ with partial Jacobian satisfying:
\begin{equation}
\mathrm{D}_{\theta} q^*(\theta, d)=-\mathrm{D}_{(q, \lambda)} g(\tilde{q}, \tilde{\lambda}, \theta, d)^{-1} \mathrm{D}_{\theta} g(\tilde{q}, \tilde{\lambda}, \theta, d) \quad\quad \forall \theta \in \Theta,
\label{qwrttheta}
\end{equation}
and
\begin{equation}
\mathrm{D}_{d} q^*(\theta, d)=-\mathrm{D}_{(q, \lambda)} g(\tilde{q}, \tilde{\lambda}, \theta , d)^{-1} \mathrm{D}_{d} g(\tilde{q}, \tilde{\lambda}, \theta, d) \quad\quad \forall d \in \mathcal{D} ,
\label{qwrtd}
\end{equation}
where $\mathrm{D}_{(q, \lambda)} g(\tilde{q}, \tilde{\lambda}, \theta, d)$ is defined in \eqref{dqlambda}, and $[\mathrm{D}_{\theta} g(\tilde{q}, \tilde{\lambda}, \theta, d), \mathrm{D}_{d} g(\tilde{q}, \tilde{\lambda}, \theta, d)]$ is defined in \eqref{jacobianthetad}.
\end{lemma}
A similar derivation for the path flow case is given in Appendix \ref{appendixa}.
\subsection{Characterizing Attacker Objective}
Equipped with Lemma \ref{iftbeckman}, we arrive at the explicit expression for $\nabla \mathcal{L}$ in Theorem \ref{attackgrad}.
\begin{theorem}\label{attackgrad}
For problem \eqref{edgeflowattackopt}, the gradient of $\mathcal{L}$ w.r.t. $\theta$ is:
\begin{equation}\label{attackedgegradtheta}
\nabla_{\theta} \mathcal{L} = \theta - vec(I_{|\mathcal{E}|}) - \frac{\gamma}{S^{\star}} \sum_{e \in \mathcal{E}} \left( q_e^*(\theta, d) \frac{d \ell_e(z)}{d z}\big\vert_{ q_e^*(\theta,d)} + \ell_e(q_e^*(\theta,d)) \right) \nabla_{\theta}q_e^*(\theta,d),
\end{equation}
where $\nabla_{\theta}q_e^*(\theta,d)$ is the transpose of $\mathrm{D}_{\theta}q_e^*(\theta,d)$ defined in \eqref{qwrttheta}.
The gradient of $\mathcal{L}$ w.r.t. $d$ is:
\begin{equation}\label{attackedgegradd}
\nabla_{d} \mathcal{L} = d - vec(I_{|\mathcal{W}|}) - \frac{\gamma}{S^{\star}} \sum_{e \in \mathcal{E}} \left( q_e^*(\theta, d) \frac{d \ell_e(z)}{d z}\big\vert_{q_e^*(\theta,d)} + \ell_e(q_e^*(\theta,d)) \right) \nabla_{d}q_e^*(\theta,d),
\end{equation}
where $\nabla_{d}q_e^*(\theta,d)$ is the transpose of $\mathrm{D}_{d}q_e^*(\theta,d)$ defined in \eqref{qwrtd}.
\end{theorem}
Theorem \ref{attackgrad} also indicates that the existence of a DSE can be controlled by the weighting factor $\gamma$.
To see this, we first notice that the first-order condition $\nabla \mathcal{L}$ may not be achievable within $\mathcal{C}$ when $\gamma$ is too large.
For the second-order condition, observe that the Hessian $\nabla^2_{\theta}\mathcal{L}$ takes the form similar to an $M$-matrix, i.e., $\nabla^2_{\theta} \mathcal{L} = I - \gamma H$, where $H$ is:
\begin{equation*}
M = \frac{1}{S^{\star}}\nabla_{\theta} \sum_{e \in \mathcal{E}} \left( q_e^*(\theta, d) \frac{d \ell_e(z)}{d z}\big\vert_{ q_e^*(\theta,d)} + \ell_e(q_e^*(\theta,d)) \right) \mathrm{D}_{\theta}q_e^*(\theta,d).
\end{equation*}
Under proper scaling of $\gamma$, the positive definiteness of $\nabla^2_{\theta}\mathcal{L}$ can be guaranteed, given the spectral radius of $M$ is strictly less than $\frac{1}{\gamma}$ everywhere in $\Theta \times \mathcal{D}$. The same analysis can be applied to $\nabla^2_d \mathcal{L}$.
The weighting parameter $\gamma$ also plays a role in balancing the local sensitivities of attack cost and payoff, as described in Theorem \ref{attackobjlip}.
\begin{theorem}\label{attackobjlip}
The attacker objective function $\mathcal{L}$ is $L_0$-locally Lipshcitz continuous w.r.t. its argument $\theta$ and $d$, where $L_0$ is:
\begin{equation}\label{lipconst}
L_0 = (\sqrt{2} + \gamma \frac{(c_0 + l_0 D) l_q}{S^{\star}} )\sqrt{|\mathcal{E}|} .
\end{equation}
\end{theorem}
$L_0$ consists of two terms:
one is the smoothness level of quadratic cost that scales with the network size factor $\sqrt{|\mathcal{E}|}$;
one is the smoothness level of $(\theta, d)$-PPoA that scales with not only $\sqrt{|\mathcal{E}|}$, but also the ratio between Lipschitz constants of $S(q^*(\theta,d))$ and $S^{\star}$.
It can be computed that $S^{\star}$ roughly scales with $c_0 D \sqrt{|\mathcal{E}|}$,
hence $\gamma l_q$ must scale with $\sqrt{|\mathcal{E}|}$ to match the sensitivities of attack cost and payoff.
This again indicates that the selection of weighting factor $\gamma$ is non-trivial.
The gradient smoothness is an important condition for the convergence analysis of gradient-based algorithms.
Determining the Lipschitz constant of $\nabla \mathcal{L}$ requires the upper estimates of $\|\nabla_{\theta, d} q_e^{*}(\theta,d)\|_{op}$, which in turn requires the lower eigenvalue estimates $ \lambda_{\min} (\mathrm{D}_{(q, \lambda)} g)$ and upper eigenvalue estimates $\lambda_{\max}(\mathrm{D}_{\theta,d} g)$.
Intuitively, the boundedness of the partial Jacobians of $g$ can be guaranteed by the regularity assumption of $\ell$ and $\Phi_{\theta}$, which is already made in our context.
We end this section with Lemma \ref{gradientlip}, which characterize the gradient smoothness under the regularity assumptions of $\|\nabla_{\theta, d} q_e^{*}(\theta,d)\|_{op}$.
\begin{lemma}\label{gradientlip}
Given $\|\nabla_{\theta, d} q_e^{*}(\theta,d)\|_{op}$ is bounded by $C_0$ and $C_1$-locally Lipschitz continuous, the attacker objective gradient $\nabla_{\theta} \mathcal{L}$ is $L_1$-locally Lipschitz continuous w.r.t. its argument $\theta$, where $L_1$ is:
\begin{equation}
L_1 = 1 + \frac{\gamma}{S^{\star}}\left(C_0 lq (l_0 + \ell^{\prime}(D))+ C_1 c_0 + D \sqrt{|\mathcal{E}|} (C_0 l_1 l_q + C_1 \ell^{\prime}(D) )\right)\sqrt{|\mathcal{E}|} .
\end{equation}
\end{lemma}
\section{Algorithmic Development} \label{ag}
\subsection{Consistent Attack as a Stackelberg Learning Process}
Projected gradient-based method is a standard approach to find a first-order stationary point or a DSE.
As shown in algorithm \ref{firstorderlearn}, the two-time scale Stackelberg learning procedure requires the attacker to have access to the first-order oracle, which gives the zeroth and first-order information of the edge latencies, the traffic flow at PWE, and the partial Jacobians of $g$.
\begin{algorithm}[htbp]
\SetKwInOut{Input}{Input}
\SetAlgoLined
\Input{Admissible initial parameter $\theta, d$, learning rate $\eta$;}
\While{not done}{
Attacker initiates attack $\Phi_{\theta}, \Phi_d$\;
\While{Attacking}{
Players form $(\theta,d)$-PWE according to $\ell\circ \Phi_{\theta}$ and demand $\Phi_d Q$\;
Attacker observes $(q^*, \mu^*)(\theta,d)$ and performs projected gradient updates\;
\begin{equation}
\begin{aligned}
\theta \leftarrow \mathrm{Proj}_{\mathcal{C}}[\theta - \eta \nabla_{\theta} \mathcal{L}] \quad
d \leftarrow \mathrm{Proj}_{\mathcal{C}}[d - \eta \nabla_{d} \mathcal{L}]
\end{aligned}
\end{equation}
}
}
\caption{First-Order Poisoning}
\label{firstorderlearn}
\end{algorithm}
This framework can be viewed as a closed-loop feedback learning process. Every attack iteration is a period of $(\theta,d)$-PWE formation, given the poisoning configuration as input; the first-order oracle reveals the result for the attacker to consistently adjust the poisoning strategy.
The first-order oracle is oftentimes unavailable in practice. WHAT QUESTION IS HERE? The question is whether the attacker is able to approximately find the Stackelberg differential equilibria through bandit-feedback, i.e., the aggregated latency results of $(\theta,d)$-PWE.
To this end, we define two smoothed versions of attacker utility,
\begin{equation}
\begin{aligned}
\mathcal{L}^{\theta}_r((\theta,d), q^* ) = \E_{ u \sim \mathbb{B}^{\theta}_r} [\mathcal{L}((\theta + u, d), q^*) ], \\ \mathcal{L}^{d}_r((\theta,d), q^* ) = \E_{ v \sim \mathbb{B}^d_r} [\mathcal{L}((\theta , d + v), q^*) ],
\end{aligned}
\end{equation}
where $u,v$ are uniformly sampled from $r$-radius Frobenius norm balls $\mathbb{B}^{\theta}_r, \mathbb{B}^{d}_r$ with proper dimensions.
As smoothed functions, $\mathcal{L}^{\theta}_r, \mathcal{L}^d_r$ have Lipschitz constants no worse than $L$ for all $r > 0$, and their gradients, by standard volume argument from \cite{flaxman2004online} Lemma 2.1,
\begin{equation}\label{zerograd}
\begin{aligned}
\nabla_{\theta} \mathcal{L}^{\theta}_r((\theta,d), q^* ) = \frac{dim(\Theta)}{r^2}\E_{ u \sim \mathbb{S}^{\theta}_r} [\mathcal{L}((\theta + u, d), q^*) u], \\
\nabla_{d} \mathcal{L}^{d}_r((\theta,d), q^* ) = \frac{dim(\mathcal{D})}{r^2}\E_{ v \sim \mathbb{S}^{\theta}_r} [\mathcal{L}((\theta , d + v), q^*) v],
\end{aligned}
\end{equation}
where $\mathbb{S}^{\theta}_r, \mathbb{S}^{d}_r$ are $r$-radius spheres of proper dimensions.
Equipped with the smoothness results and \eqref{zerograd}, by standard concentration inequalities, we show that it suffices to use polynomial numbers of samples to approximate the gradients.
\begin{prop}[Gradient Approximation Efficiency] \label{gradefficiency}
Given a small $\epsilon > 0$, one can find fixed polynomials $h_r(1/ \epsilon)$, $h_{sample}(dim(\Theta), 1/ \epsilon)$, $h_{sample}(dim(\mathcal{D}), 1/ \epsilon)$, for $r \leq h_r(1/\epsilon)$, with $m \geq \max \{h_{sample}(dim(\Theta), 1/ \epsilon), h_{sample}(dim(\mathcal{D}), 1/ \epsilon) \}$ samples of $u_1$, $\ldots$, $u_m$ and $v_1, \ldots, v_m$, with probability at least $1 - (d/\epsilon)^{-d}$ the sample averages
\begin{equation} \label{sampleavg}
\frac{dim(\Theta)}{mr^2}\sum_{i=1}^m\mathcal{L}((\theta + u_i,d), q^*) u_i , \quad \frac{dim(\mathcal{D})}{mr^2}\sum_{i=1}^m\mathcal{L}((\theta,d + v_i), q^*) v_i
\end{equation}
are $\epsilon$ close to $\nabla_{\theta}\mathcal{L}$ and $\nabla_d \mathcal{L}$, respectively.
\end{prop}
Leveraging the one-point gradient approximation technique, we propose the derivative-free Algorithm \ref{zeroorderlearn} as an alternative to Algorithm \ref{firstorderlearn}.
This algorithm asynchronously perturbs the parameters $\theta$ and $d$ to obtain the one-point gradient estimates.
\begin{algorithm}[htbp]
\SetKwInOut{Input}{Input}
\SetAlgoLined
\Input{Admissible initial parameter $\theta, d$, learning rate $\eta$, sample size $m$, radius $r$;}
\While{not done}{
Attacker initiates attack $\Phi_{\theta}, \Phi_d$\;
\While{Attacking}{
\For{$i = 1,\ldots, m$}{
Sample $(\theta,d)$-PWE for searching directions $u_i \sim \mathbb{S}^{\theta}_r$, $v_i \sim \mathbb{S}^d_r$, obtain:
\begin{equation*}
\mathcal{L}^{\theta}_i = \mathcal{L} (\operatorname{Proj}_{\mathcal{C}}(\theta + u_i, d ), q^*) , \quad \mathcal{L}^{d}_i = \mathcal{L} (\operatorname{Proj}_{\mathcal{C}}(\theta, d + v_i), q^*).
\end{equation*}
}
Projected gradient updates:
\begin{equation}
\theta \leftarrow \mathrm{Proj}_{\mathcal{C}}[\theta - \eta \frac{dim(\Theta)}{mr^2}\sum_{i=1}^m \mathcal{L}_{\theta}^i u_i] \quad
d \leftarrow \mathrm{Proj}_{\mathcal{C}}[d - \eta \frac{dim(\mathcal{D})}{mr^2}\sum_{i=1}^m \mathcal{L}_d^i v_i] .
\end{equation}
}
}
\caption{Zeroth-Order Poisoning}
\label{zeroorderlearn}
\end{algorithm}
By projecting perturbed $\theta, d$ to the constraint set $\mathcal{C}$, we ensure the feasibility of the perturbed attack strategies when sampling $\mathcal{L}^{\theta}_i$ and $\mathcal{L}^d_i$.
Algorithm \ref{zeroorderlearn} can proceed without the aid of first-order oracle, but it requires the number of samples to be polynomial w.r.t. the smoothness level of attack utility.
\section{Case Study} \label{sfdemo}
Through an emergency evacuation case study \cite{ng2010hybrid}, over the classical Sioux Falls, South Dakota Transportation Network \cite{siouxfalls} (Fig. \ref{sf} (b)), we test our Stackelberg learning algorithm and demonstrate the attack effects.
In our example, the evacuation lasts for one month.
During each day, a total of 34200 individuals are transported from emergency locations (the red nodes) (14), (15), (22), and (23), to shelter places (the green nodes) (4), (5), (6), (8), (9), (10), (11), (16), (17), and (18).
The transportation network data, including node attributes, free travel time, and road capacity, etc., are available at \cite{github}.
\begin{figure}
\caption{The topological (left) and geographical layout \cite{chakirov2014enriched}
\label{sfsnetworktopo}
\label{sf}
\end{figure}
The edge latency is given by the standard Bureau of Public Roads (BPR) function:
\begin{equation}\label{bprfunc}
\ell_{e}\left(q_{e}\right)=t_{e}^{f}\left(1+\alpha\left(\frac{q_{e}}{\mathrm{C}_{e}}\right)^{\beta}\right),
\end{equation}
where $t_e^f$ is the free time for edge $e$, $\mathrm{C}_e$ is the road capacity for edge $e$, and $\alpha, \beta$ are some parameters.
The attacker's goal is to slow down the evacuation process through latency and demand poisoning.
The attacker can launch multiple attacks during one day, for each attack, the aggregated latency at the corresponding PWE is revealed as an observation to the attacker.
These observations are then used to update the attack strategy.
The weighting factor $\gamma$ and sample size $m$ are both picked to scale with $\sqrt{|\mathcal{E}|}$, where $|\mathcal{E}| = 76$ is the total edge number.
An annealing factor of $0.95$ is used for the learning rate.
We sample perturbations $u_i$ and $v_i$ from a standard normal distribution for practical purposes.
The PPoA evolution curve is shown in Fig. \ref{evopoa}.
\begin{figure}
\caption{The evolution of PPoA: after $15$ days, the PPoA of this attack scenario reaches above $4$; and the process is stabilized at day-$20$ and attains the PPoA around $3.6$.}
\label{evopoa}
\end{figure}
Fig. \ref{evopoa} shows that a stealthy attacker can decrease the efficiency of WE, pushing it far away from the SO.
The convergence of the Stackelberg learning process implies the finding of a DSE.
We compare the edge efficiencies of SO and PWE in Fig. \ref{edgeefficiency}. Fig. \ref{edgetime} shows the comparison of latency function values for each edge, given by \eqref{bprfunc} and the assigned edge flow; Fig. \ref{edgeutilization} shows the comparison of utilization ratio between the actual flow on that edge and its road capacity, $q_e / \mathrm{C}_e$.
\begin{figure}
\caption{This bar chart compares the edge travel time caused by SO and PWE of the last day.
Edges indexed by 49, 51, 60, and 63 are experiencing significant traffic delays.}
\caption{This bar chart compares the edge utilization rate caused by SO and PWE of the last day.
Corresponding to Fig. \ref{edgetime}
\label{edgetime}
\label{edgeutilization}
\label{edgeefficiency}
\end{figure}
Fig. \ref{edgetime} shows that at the end of the iteration, PWE assigns overwhelming traffic flow on several high-capacity edges, causing edge latencies to be higher than those of SO.
It can be inferred that the congestion is likely to occur on those edges with the overwhelming flow.
Fig. \ref{edgeutilization} shows that those edges with significantly high traffic latencies are severely overloaded, which indicates that the evacuation process is highly disrupted.
\section{Conclusion}
In this paper, we have formulated a Stackelberg game framework to quantify and analyze the impact of informational attacks that aim to manipulate the traffic data to mislead the Online Navigation Platforms (ONP) to provide users with falsified route recommendations.
Through sensitivity analysis, we have shown the continuity and differentiability of the attack utility function and characterized its smoothness level.
The result has shown that the PPoA is a $C^1$-function with respect to the poisoning attack parameters, and an optimal strategy of the attack model can be achieved by a consistent Stackelberg learning process.
It reveals the vulnerabilities of the Wardrop Equilibrium (WE)-based flow planning systems and showcases the disruptive effects that an attacker can inflict on the entire traffic network.
Future research directions would include the investigation of the poisoning of transient equilibrium formation behavior and the development of effective defensive and detective strategies against this class of attacks.
\appendix
\section{Path Flow IFT} \label{appendixa}
In the path flow poisoning scenario, the parameterized Lagrangian can be written as:
\begin{equation*}
L(\mu, \lambda, \nu, \theta, d) = \sum_{e \in \mathcal{E}} \int_{0}^{(\Delta \Phi_{\theta} \mu)_e} \ell_e (z)dz - \lambda^{\top} \Phi_{\theta} \mu + \nu^{\top} ( \Lambda \Phi_{\theta} \mu - \Phi_dQ) .
\end{equation*}
Similarly for a candidate solution $\tilde{\mu}, \tilde{\lambda}, \tilde{\nu}$, write down the KKT conditions as:
\begin{equation*}
\begin{aligned}
- \Phi_{\theta} \mu & \preceq 0 \\
\Delta \Phi_{\theta} \mu - \Phi_dQ & = 0 \\
\tilde{\lambda}_p & \geq 0, \quad p = 1, \ldots, | \mathcal{P}| \\
\tilde{\lambda}_p (\Phi_{\theta} \mu)_p & = 0, \quad p = 1, \ldots, | \mathcal{P}| \\
\sum_{e \in \mathcal{E}} (\Delta\Phi_{\theta})^{\top}_{pe} \ell_e((\Delta \Phi_{\theta} \mu)_e) - (\Phi_{\theta}^{\top} \lambda)_p + (\Phi_{\theta}^{\top} \Lambda^{\top} \nu)_p & = 0 , \quad p = 1, \ldots, | \mathcal{P}|
\end{aligned}
\end{equation*}
Again, we define the parameterized function $g(\tilde{\mu}, \tilde{\lambda}, \tilde{\nu}, \theta, d )$
\begin{equation}
\label{gfuncforpath}
g(\tilde{\mu}, \tilde{\lambda}, \tilde{\nu}, \theta, d ) =
\begin{bmatrix}
\nabla_{\mu} L(\mu, \lambda, \nu, \theta,d) \\
- \operatorname{diag}(\lambda) \Phi_{\theta} \mu \\
\Delta \Phi_{\theta} \mu - \Phi_d Q
\end{bmatrix}
\end{equation}
The partial Jacobian of variable $(\tilde{\mu}, \tilde{\lambda}, \tilde{\nu})$ is
\begin{equation*}
\mathrm{D}_{(\tilde{\mu}, \tilde{\lambda}, \tilde{\nu})} g(\tilde{\mu}, \tilde{\lambda}, \tilde{\nu}, \theta, d)=
\begin{bmatrix}
\mathrm{D}_{\mu} \nabla_{\mu} L(\tilde{\mu}, \tilde{\lambda}, \tilde{\nu}, \theta, d) & \Phi_{\theta}^{\top} & (\Delta \Phi_{\theta})^{\top} \\
\operatorname{diag}(\tilde{\lambda}) \Phi_{\theta} & \operatorname{diag}(- \Phi_{\theta}\mu) & 0 \\
\Delta \Phi_{\theta} & 0 & 0\end{bmatrix}
\end{equation*}
And the partial Jacobian for $\theta$ and $d$ is
\begin{equation}
\mathrm{D}_{(\theta, d)} g(\tilde{\mu}, \tilde{\lambda}, \tilde{\nu}, \theta, d)=
\begin{bmatrix}
\mathrm{D}_{\mu} \nabla_{\mu} L(\tilde{\mu}, \tilde{\lambda}, \tilde{\nu}, \theta, d) & \Phi_{\theta}^{\top} & (\Delta \Phi_{\theta})^{\top} \\
\operatorname{diag}(\tilde{\lambda}) \Phi_{\theta} & \operatorname{diag}(- \Phi_{\theta}\mu) & 0 \\
\Delta \Phi_{\theta} & 0 & 0
\end{bmatrix} .
\end{equation}
We omit the explicit gradient calculation as there are diverse possibilities of parameterization.
Note that in this formulation, the conditions for $\mathrm{D}_{(\tilde{\mu}, \tilde{\lambda}, \tilde{\nu})}g$ to be non-singular becomes $ \operatorname{diag}(-\Phi_{\theta} \mu)$ and $(\Delta \Phi_{\theta})$ being invertible. A result like Theorem \ref{iftbeckman} can be derived using a similar analysis.
\section{Sketch of Proofs for Sensitivity Analysis} \label{appendixb}
We omit the proofs of Lemma \ref{charfeasibility} and \ref{continuity} (a), which is adapted from Lemma 8.3 and Corollary 8.1 of \cite{still2018lectures} by inserting $\Phi_{\theta}$ and $\Phi_d$. The following proofs are based on these preliminary results.
\begin{proof}[Lemma \ref{continuity} (b)]
It suffices to show the smoothness of $\langle q^*(\theta, d) \ell (q^* (\theta, d)) \rangle$, for $(\theta_1, d_1), (\theta_2, d_2) \in \Theta \times \mathcal{D}$, denote variable $z_1 =(\theta_1, d_1)$ and $z_2 =(\theta_2, d_2)$, by triangular inequality and Cauchy-Schwarz inequality,
\begin{equation*}
\begin{aligned}
& \| \langle q^*(z_1 ) \ell (q^* (z_1)) \rangle - \langle q^*(z_2) \ell (q^* (z_2)) \rangle \| \\ \leq & \| \langle q^*(z_1 ) \ell (q^* (z_1)) \rangle - \langle q^*(z_1) \ell (q^* (z_2)) \rangle \| + \| \langle q^*(z_1 ) \ell (q^* (z_2)) \rangle - \langle q^*(z_2) \ell (q^* (z_2)) \rangle \| \\
\leq & \|q^*(z_1) \| \| \ell(q^* (z_1)) - \ell(q^* (z_2)) \| + \|\ell (q^* (z_2))\|\|q^*(z_1) - q^*(z_2) \| \\
\leq & \sqrt{|\mathcal{E}|}D l_q l_0 \|z_1 - z_2 \| + \sqrt{|\mathcal{E}|} l_q c_0 \| z_1 - z_2\| .
\end{aligned}
\end{equation*}
\qed
\end{proof}
\begin{proof}[Lemma \ref{iftbeckman}]
Immediately follows substituting the condition $H(x, t) = 0$ in general IFT with Stationarity KKT condition $g(q, \lambda, \theta, d) = 0$, and checking the Shur complement of partial Jacobian $\mathrm{D}_{(q, \lambda)} g(\tilde{q}, \tilde{\lambda}, \theta, d)$.
\qed
\end{proof}
\begin{proof}[Theorem \ref{attackgrad}]
It suffices to show for variable $\theta$. Taking derivative gives:
$$
\nabla_{\theta} \mathcal{L} = \theta - vec(I_{|\mathcal{E}|}) - \frac{\gamma}{S^{\star} } \langle \nabla_{\theta}q^*, \ell (q^* )\rangle + \langle q^* , \nabla_{\theta} q^* \mathrm{D} \ell(q^*)\rangle
$$
Rearranging the terms yields the results.
\qed
\end{proof}
\begin{proof}[Theorem \ref{attackobjlip}]
For the attack cost term, we can compute the Lipschitz constant with respect to the two variables as $ \sqrt{| \mathcal{E}|} \|\theta_1 - \theta_2 \|$ and $\sqrt{| \mathcal{W}|} \|d_1 - d_2 \|$, respectively. Thus the first part for the constant should be $\sqrt{2}\max\{\sqrt{| \mathcal{E}|}, \sqrt{| \mathcal{W}|} \} = \sqrt{2}\sqrt{| \mathcal{E}|}$. For the second part, multiplying the constant in Lemma \ref{continuity} (b) with $S^{\star}$ and $\gamma$ yields the result.
\qed
\end{proof}
\begin{proof}[Lemma \ref{gradientlip}]
We proceed under the assumption of boundedness and Lipschitz smoothness of $\|\nabla_{\theta} q^*\|_{op}$. We analyze two terms, $ \langle \nabla_{\theta}q^*, \ell (q^* )\rangle$ and $\langle q^* , \nabla_{\theta} q^* \mathrm{D} \ell(q^*)\rangle$. Write $q^*(\theta_1, d)$ and $q^*(\theta_2, d)$ as $q^*_1$ and $q^*_2$, respectively.
For the first term, we have
\begin{align*}
& \|\langle \nabla_{\theta}q^*_1, \ell (q^*_1 )\rangle - \nabla_{\theta}q^*_2, \ell (q^*_2 )\rangle\| \\
\leq & \| \langle\nabla_{\theta}q^*_1, \ell (q^*_1 )\rangle - \nabla_{\theta}q^*_1, \ell (q^*_2 )\rangle\| + \| \langle \nabla_{\theta}q^*_1, \ell (q^*_2 )\rangle - \nabla_{\theta}q^*_2, \ell (q^*_2 )\rangle\| \\
\leq & (C_0 l_0 l_q \sqrt{|\mathcal{E}|} + C_1 c_0 \sqrt{|\mathcal{E}|} ) \| \theta_1 - \theta_2\|.
\end{align*}
For the second term, by the monotonicity of $\ell$ and the sensitivity results,
\begin{align*}
& \| \langle q^*_1 , \nabla_{\theta} q^*_1 \mathrm{D} \ell(q^*_1)\rangle - \langle q^*_2 , \nabla_{\theta} q^*_2 \mathrm{D} \ell(q^*_2)\rangle \| \\
\leq & \| \langle q^*_1 , \nabla_{\theta} q^*_1 \mathrm{D} \ell(q^*_1)\rangle - \langle q^*_1 , \nabla_{\theta} q^*_2 \mathrm{D} \ell(q^*_2)\rangle\| + \|\langle q^*_1 , \nabla_{\theta} q^*_2 \mathrm{D} \ell(q^*_2)\rangle - \langle q^*_2 , \nabla_{\theta} q^*_2 \mathrm{D} \ell(q^*_2)\rangle\| \\
\leq & \left(\sqrt{|\mathcal{E}|} D (C_0\sqrt{|\mathcal{E}|}l_1 l_q + C_1 \ell^{\prime}(D) \sqrt{|\mathcal{E}|} ) + l_q C_0 \sqrt{|\mathcal{E}|} \ell^{\prime}(D) \right) \| \theta_1 - \theta_2\|
\end{align*}
Summing the two terms together yields the result.
\qed
\end{proof}
\section{Sketch Proof of Proposition \ref{gradefficiency}}\label{propproof}
\begin{proof}
We show the sample bound for $\nabla_{\theta}\mathcal{L}$ approximation, the proof of sample bound for $\nabla_d\mathcal{L}$ follows the similar procedure.
Let $\hat{\nabla}$ denote the sample average in \eqref{sampleavg}, the approximation error can be broken into two terms:
\begin{equation*}
\begin{aligned}
\hat{\nabla}-\nabla_{\theta} \mathcal{L}((\theta, d), q^*)=\nabla_{\theta} \mathcal{L}^{\theta}_r((\theta, d), q^*)-\nabla_{\theta} \mathcal{L}((\theta, d), q^*) + \hat{\nabla}-\nabla_{\theta}\mathcal{L}^{\theta}_r((\theta, d), q^*)
\end{aligned}
\end{equation*}
For the first term, choose $h_{r}(1 / \epsilon)=\min \left\{1 / r_{0}, 2 L_1 / \epsilon\right\}$, by Lemma \ref{gradientlip} when $r < 1/h_r(1/\epsilon) = \epsilon / 2 L_1$, $\| \nabla_{\theta} \mathcal{L} ((\theta + u, d), q^{*}) - \nabla_{\theta}\mathcal{L} ((\theta, d), q^{*})\| \leq \epsilon /4$. Since $$\nabla_{\theta} \mathcal{L}_r^{\theta} ((\theta, d), q^*) = \nabla_{\theta} \mathcal{L}^{\theta}_r((\theta,d), q^* ) = \frac{dim(\Theta)}{r^2}\E_{ u \sim \mathbb{S}^{\theta}_r} [\mathcal{L}((\theta + u, d), q^*) u],$$
by triangular inequality, $\| \nabla_{\theta} \mathcal{L}^{\theta}_r ((\theta, d), q^{*}) - \nabla_{\theta} \mathcal{L} ((\theta, d), q^{*})\| \leq \epsilon /2$.
Select $r_0$ such that for any $u \sim \mathbb{S}_r$, it holds that $ \mathcal{L}((\theta + u, d), q^*)$. By Theorem \ref{attackobjlip}, one can select such a $r_0$ by examining related constants.
Since $ \E[\hat{\nabla}] = \nabla_{\theta} \mathcal{L}^{\theta}_r((\theta,d), q^* )$, and each sampled norm is bounded by $ 2 dim(\Theta) \mathcal{L} /r$,
by vector Bernstein's inequality, when $ m \geq h_{sample}(d, 1/\epsilon) \propto d (\frac{d \mathcal{L}^2}{\epsilon r}) \log d / \epsilon$, with probability at least $ 1 - (d/\epsilon)^{-d}$, we have
\begin{equation*}
\| \hat{\nabla} - \nabla_{\theta}\mathcal{L}^{\theta}_r((\theta, d), q^*)\| \leq \epsilon/2,
\end{equation*}
hence proving the claim.
\qed
\end{proof}
\end{document} |
\begin{document}
\begin{titlepage}
\title{Optimal Delegation in a Multidimensional World
}
\author{
Andreas Kleiner\thanks{Department of Economics, Arizona State University. Email: \mailto{[email protected]}.}
}
\maketitle
We study a model of delegation in which a principal takes a multidimensional action and an agent has private information about a multidimensional state of the world. The principal can design any direct mechanism, including stochastic ones. We provide necessary and sufficient conditions for an arbitrary mechanism to maximize the principal's expected payoff. We also discuss simple conditions which ensure that some \textit{convex delegation set} is optimal.
A key step of our analysis shows that a mechanism is incentive compatible if and only if its induced indirect utility is convex and lies below the agent's first-best payoff.
\thispagestyle{empty}
\end{titlepage}
\section{Introduction}
In many economic and political environments, a principal delegates decisions to a better-informed agent: a firm appoints a manager to choose investment levels in different projects;
US Congress delegates power to federal agencies;
a legislative forms a committee to draft bills; a regulator lets a monopolist choose prices.
Following \citet{Holmstrom:77}, an extensive literature models such delegation problems by assuming that both the action and the state of the world lie in a one-dimensional space. A main result of this literature characterizes when it is optimal for the principal to constrain the agent's choice to lie in an interval, and this conclusion has been used to explain why managers face spending caps, regulators impose price ceilings, and trade agreements specify maximum tariff levels.
The assumption that the action and state space are one-dimensional is made for tractability. In many applications, the underlying states and actions are more complex and more realistically modeled as multidimensional: managers invest in several projects, Congress delegates many decisions to the EPA, and committees draft multiple bills. What mechanisms are optimal in such multidimensional settings? How robust are conclusions obtained for one-dimensional models? And can we still expect that relatively simple mechanisms are often optimal?
To study these questions, we consider a principal that takes a multidimensional action and faces an agent with private information about a multidimensional state of the world (the agent's \textit{type}). Payoffs depend on the action and the state of the world, and transfers are infeasible. The principal can design arbitrary mechanisms, including stochastic ones, to maximize her expected payoff. Our main result characterizes, for an arbitrary mechanism, when this mechanism is optimal. Often, it is optimal to delegate the decision to the agent but to constrain the agent by requiring that her action lies in some set. For convex delegation sets,
we provide a simple characterization, which is a direct analog of conditions characterizing when interval delegation is optimal in one-dimensional models.
Our results illustrate how a principal can benefit from optimally bundling independent decisions.
Even for one-dimensional models, our approach provides new insights: Our main result characterizes for arbitrary mechanisms---not just interval delegation sets---when this mechanism is optimal. And as corollaries, we obtain a novel condition under which some interval delegation set will be optimal and a comparative statics result showing when the agent will get more discretion.\todo{This result still needs to be added to the file.}
A key step to deriving our results lies in obtaining a simple characterization of the set of feasible mechanisms. Given a mechanism, the corresponding \textit{indirect utility} assigns to any type the payoff this type would get by choosing his report optimally. This payoff must be less than the \textit{first-best payoff}, i.e., the payoff this type would get if he could choose the action without any constraints. Moreover, the indirect utility must be a convex function since it is the maximum of a family of affine functions. \autoref{lemma:feasible_set} shows that any function satisfying these two properties is the indirect utility of an incentive-compatible mechanism.
This characterization is easy to use and already helpful for one-dimensional delegation models. Our formulation differs from the previous literature, which often considered only deterministic mechanisms. Since the convex combination of two incentive-compatible mechanisms is not necessarily incentive compatible, the set of deterministic mechanisms is not even convex.\footnote{Some earlier papers also consider stochastic mechanisms (or allow for money burning/restricted transfer) and obtain a convex set of mechanisms; see, for example, \citet{AB:13,KM:09,AE:17,amador20,kartik21,kleiner21}.} Moreover, a common approach is to first treat the model as one with transfers and then impose that these transfers are zero (or negative). Compared to this approach, formulating the problem via indirect utilities is more direct and provides valuable geometric insights into which mechanisms can be optimal. For the multidimensional problem, the approach via indirect utilities provides additional benefits because it circumvents intricate characterizations of incentive compatibility \citep[see][]{rochet87}.
To find the optimal mechanism, we formulate the principal's problem in terms of indirect utilities. In this formulation, the problem becomes a linear program, and we use linear programming duality to derive necessary and sufficient conditions for a given mechanism to be optimal. Typically, optimal mechanisms pool certain types, and our main result shows that a mechanism is optimal if conditional on any pooling region, a stochastic dominance condition (using the convex order) is satisfied. Intuitively, this condition requires that restricted to the pooling region (where the indirect utility function is affine), any convex indirect utility yields a lower payoff.
If the pooling regions are at most one-dimensional, the stochastic dominance condition has a simple formulation in terms of majorization. Using this observation, we provide necessary and sufficient conditions for a convex delegation set with a smooth boundary to be optimal. These conditions are easy to check and are straightforward extensions of conditions that ensure the optimality of interval delegation sets in one-dimensional models \citep[see][]{AB:13}.
\paragraph{Related Literature}
The literature on delegation has focused mainly on problems in which the principal delegates a single one-dimensional decision and therefore assumed that both the action and state spaces are one-dimensional; see, for example, \citet{Holmstrom:77,Holmstrom:84,MS:91,AM:08,AB:13,KZ:19}.
A few delegation papers do consider richer action and/or type spaces. \citet{Armstrong95} considers an agent with two-dimensional private information and discusses several applications. Since the principal's action is assumed to be one-dimensional (and only interval delegation sets are considered), there is only limited scope to screen two-dimensional types in his analysis.
\citet{koessler2012optimal} characterize the optimal mechanism in a setting where two decisions depend on a single-dimensional underlying state.
\citet{galperti2019theory} considers multidimensional information and actions but restricts the principal's choice to a particular class of ``budgeting mechanisms''. The closest paper to ours is \citet{frankel2016delegating}, which studies the delegation of several independent decisions, which yield multidimensional action and state spaces. For quadratic preferences with a constant bias, he shows that if the states are independently and identically distributed according to normal distributions then it is optimal to delegate a `half space'. Without the normality assumption, he shows that the principal's payoff from such a mechanism converges to the first-best as the number of independent decision problems grows.
\citet{frankel2014aligned} also considers multidimensional delegation problems and characterizes the max-min optimal mechanism, which maximizes the principal's payoff against the worst-case preference type of the agent.
The elicitation of information about multiple independent decisions from a biased agent has been studied in general mechanism design \citep[e.g.,][]{jackson07} and cheap talk environments \citep{chakraborty2007comparative,lipnowski2020cheap}. \citet{jackson07} show that by linking independent decisions, the principal's payoff converges to the first-best as the number of decisions grows. Our results can be used to show how the principal should optimally link decisions, which can be important if there is a limited number of decisions.
On a methodological level, our work is related to the literature on multidimensional mechanism design, and in particular on multiproduct monopolists \citep[see, e.g.,][]{rochet87,manelli06,manelli07,daskalakis-etal2017,haghpanah21}.
\section{Model}
A principal chooses an action $a\in \mathbb{R}^n$. An agent is privately informed about the state of the world $s\in S$, where $S\subseteq \mathbb{R}^n$ is compact and convex. The agent's and principal's payoffs depend on both the action and the state of the world, and are given by
\begin{align*}
u_A(a,s)&:= a\cdot s + b(a)\\
u_P(a,s)&:=a\cdot g(s) + \kappa b(a),
\end{align*}
respectively, where $b:\mathbb{R}^n\rightarrow \mathbb{R}$ is strictly concave, differentiable with a Lipschitz-continuous gradient mapping, and satisfies $\lim_{\hat{\mathbf{n}}orm{a}\rightarrow \infty}\frac{b(a)}{\hat{\mathbf{n}}orm{a}}=-\infty$,\footnote{Since we do not artificially constrain the set of actions, some assumptions are needed to ensure that for every type $s\in S$ there is an optimal action. The current assumptions ensure this and simplify some arguments, but weaker conditions on $b$ could be imposed.} $g: S\rightarrow \mathbb{R}^n$ is continuous, and $\kappa>0$.
We assume that the state $s$ is distributed according to a probability distribution $F$ with differentiable density $f$ and support $S$.
The principal aims to maximize her expected payoff and can design arbitrary mechanisms.
The revelation principle applies and we define a \emph{mechanism} to be a function $m:S\rightarrow \Delta(\mathbb{R}^n)$ such that expected payoffs are integrable.\footnote{We denote by $\Delta(\mathbb{R}^n)$ the Borel $\sigma$-algebra on $\mathbb{R}^n$.} To simplify notation, we extend the domain of $b(\cdot)$ and $u_i(\cdot,s)$ linearly to include probability distributions over $\mathbb{R}^n$, so that $b(m(s))=\mathbb{E}_{m(s)}[b(a)]$ and analogously for $u_i(\cdot,s)$. A mechanism is \emph{incentive compatible} if for all $s$ and $s'$ in $S$,
\begin{align*}
u_A(m(s),s)\ge u_A(m(s'),s).
\end{align*}
\section{Characterizing incentive-compatible mechanisms}
We characterize the set of incentive-compatible mechanisms in terms of their indirect utilities.
To any incentive-compatible mechanism $m$ corresponds an \emph{indirect utility} $U:\mathbb{R}^n\rightarrow\mathbb{R}$ defined by
\[U(s):=\sup_{s'\in S} \mathbb{E}[m(s')]\cdot s + b(m(s')).\]
Which indirect utilities correspond to some incentive-compatible mechanism? First, any indirect utility is convex as the supremum of a family of functions that are affine in the state $s$. Second, in the absence of transfers the agent's utility cannot be higher than if he was free to choose his action. Defining the \textit{first-best payoff} $h:\mathbb{R}^n\rightarrow \mathbb{R}$ by
\[h(s):=\sup_{a\in \mathbb{R}^n} a\cdot s + b(a),\]
$U\le h$ is clearly necessary.\footnote{We denote the pointwise order by $\le$, so $U\le h$ means $U(s)\le h(s)$ for all $s$ in the common domain of $U$ and $h$.} The following result shows that these two conditions characterize the set of feasible indirect utilities.
\begin{lemma}\label{lemma:feasible_set}
An indirect utility $U$ corresponds to an incentive-compatible mechanism if and only if $U$ is convex and lies below the first-best payoff: $U\le h$.
\end{lemma}
\begin{figure}
\caption{The function $U$ satisfies $U(s)\le h(s)$ for all $s\in S$ but does not correspond to a feasible mechanism. To see this, note that there is no convex extension of $U$ to $\mathbb{R}
\label{fig:domain_indirect_util}
\end{figure}
Intuitively, if $U$ is convex then it would correspond to an incentive-compatible mechanism if transfers were available and the agent had quasi-linear preferences. If the required transfers are all negative then we can use the agent's risk aversion (coming from the strict concavity of $b$) to simulate these transfers via stochastic actions. One can show that $U\le h$ implies that the required transfers are negative. This last step relies on the domain of $U$ and $h$ being large enough and it would not suffice to require only that $U(s)\le h(s)$ for all $s\in S$. \autoref{fig:domain_indirect_util} illustrates a convex function $U$ which lies below $h$ on all of $S$, but which does not correspond to a mechanism because the lotteries assigned to low types would yield a strictly higher payoff than the first-best payoffs for some hypothetical types, an impossibility.
\begin{proof}
Let us first state three basic observations from convex analysis. The convex conjugate of a function $U$ is denoted by $U^*$ and defined by $U^*(a):=\sup_{s\in\mathbb{R}^n} a\cdot s - U(s)$. We will use the following facts, which follow immediately from this definition: (i) $h=(-b)^*$, (ii) $U\le h$ implies $h^*\le U^*$, and (iii) $a\in\partial U(s)$ implies $U^*(a)=a\cdot s -U(s)$.\footnote{Here, $\partial U(s)$ denotes the subdifferential of $U$ at $s$. To see (iii), note that the definition of $U^*$ implies $U^*(a)\ge a\cdot s -U(s)$. Conversely, convexity of $U$ implies that for all $s'$,
$a\cdot s-U(s)\ge a\cdot s'-U(s')$. Taking the supremum of the right-hand side with respect to $s'$ yields $a\cdot s-U(s) \ge U^*(a)$.}
Suppose $U$ is convex and satisfies $U\le h$. Let the mechanism $m$ assign to any type $s\in S$ a lottery with expected value $a\in\partial U(s)$ that yields the payoff $a\cdot s + b(a) - U^*(a)+h^*(a)$. Such a lottery exists because $a\cdot s + b(a)$ would be the payoff for type $s$ from always getting action $a$, because fact (ii) implies that the agents payoff is lower, and because $b$ is strictly concave.\footnote{More formally, strict concavity of $b$ implies that for any $a\in\mathbb{R}^n$ and nonzero $d\in\mathbb{R}^n$ there is $\underline{v}arepsilon>0$ such that $1/2[b(a+d)+b(a-d)]< b(a)-\underline{v}arepsilon$. It follows that for any $\lambda>1$, $1/2 [b(a+ \lambda d)+b(a- \lambda d)] \le b(a)-\lambda \underline{v}arepsilon.$
Therefore, by choosing $\lambda$ arbitrarily large, one can design lotteries with expected value $a$ that yield arbitrarily low payoff to the agent.
}
Then facts (i) and (iii) imply that the payoff of a truthful type $s$ is $U(s)$:
\begin{align*}
u_A(m(s),s) = s\cdot a + b(a) - U^*(a)+h^*(a) = U(s).
\end{align*}
It remains to show that $m$ is incentive compatible. For all $s$ and $s'$,
\begin{align*}
&u_A(m(s),s) = U(s) \ge U(s') + \mathbb{E}[m(s')]\cdot (s-s') \\
= &\mathbb{E}[m(s')]\cdot s' + b(m(s')) + \mathbb{E}[m(s')]\cdot (s-s') = u_A(m(s'),s),
\end{align*}
where the first inequality follows since $E[m(s)]\in \partial U(s')$.
\end{proof}
\autoref{fig:example_feasible_mechanisms} illustrates the result for one-dimensional types and quadratic payoffs. It shows four indirect utilities that, according to \autoref{lemma:feasible_set}, correspond to incentive-compatible mechanisms. In \autoref{fig:first}, all types between $s_1$ and $s_2$ obtain their first-best utility and $U$ is affine below $s_1$ and above $s_2$. This indirect utility can be obtained by letting types choose their preferred action from the interval of deterministic actions $[s_1,s_2]$. In \autoref{fig:second}, the menu of actions from which the agent can choose contains an additional deterministic action above $s_2$. The indirect utility in \autoref{fig:third} contains an affine piece that lies strictly below the graph of $h$. This part of the indirect utility corresponds to types that obtain a (nondegenerate) stochastic action, which yields no type its first-best payoff. Finally, \autoref{fig:fourth} illustrates an indirect utility corresponding to a mechanism in which types in two adjacent regions obtain a stochastic action.
\begin{figure}
\caption{Interval delegation}
\label{fig:first}
\caption{A deterministic mechanism}
\label{fig:second}
\caption{A stochastic mechanism}
\label{fig:third}
\caption{A stochastic mechanism with two adjacent stochastic actions}
\label{fig:fourth}
\caption{Examples of indirect utilities. The blue curves show the function $h$ for one-dimensional types and quadratic payoffs (i.e., assuming $b(a)=-\frac{a^2}
\label{fig:example_feasible_mechanisms}
\end{figure}
\section{Characterizing optimal mechanisms}
We characterize the optimal mechanisms in this section. To do so, we first formulate the principal's problem in terms of indirect utilities (\autoref{sec:formulating_problem}). We then state the main characterization of optimal mechanisms in \autoref{sec:optimal_mechanisms} and illustrate the result for particular mechanisms. Finally, we outline the proof of the main result in \autoref{sec:proof_sketch}.
\subsection{Formulating the principal's problem}
\label{sec:formulating_problem}
Consider an indirect utility $U$ that corresponds to some incentive-compatible mechanism. In general, there are many incentive-compatible mechanisms that induce the same indirect utility; however, all such mechanism induce the same payoff for the principal. To see this, let $m$ be an incentive-compatible mechanism with corresponding indirect utility $U$. Using $\hat{\mathbf{n}}abla U(s)=\mathbb{E}[m(s)]$ (by an Envelope theorem) and $U(s) = \hat{\mathbf{n}}abla U(s)\cdot s + b(m(s))$, the principal's payoff from mechanism $m$ in state $s$ is completely determined by $U$:
\begin{align*}
\mathbb{E}[m(s)]\cdot g(s) + \kappa b(m(s)) = \hat{\mathbf{n}}abla U(s) \cdot [g(s)- \kappa s]+ \kappa U(s).
\end{align*}
This observation implies that the principal's payoff is a linear function of $U$. Therefore, a solution to the principal's problem can be found at an extreme point of the feasible set.
Returning to \autoref{fig:example_feasible_mechanisms}, it is easy to see that the indirect utilities in Figures \ref{fig:first}--\ref{fig:third} are extremal in that they cannot be written as a nontrivial convex combination of two feasible indirect utilities. In contrast, the indirect utility in \autoref{fig:fourth} can be written as such a convex combination. This implies that whenever this mechanism is optimal, there is another (and simpler) mechanism which is also optimal. Intuively, one can write this indirect utility as a convex combination because two adjacent regions obtain distinct stochastic actions. This insight shows how, without loss of optimality, one can restrict attention to a smaller class of mechanism.\footnote{\cite{kleiner21} develop this point more formally in the context of one-dimensional types/actions and quadratic utilities and characterize the set of extremal mechanisms. Formulating the problem in terms of indirect utilities and using our \autoref{lemma:feasible_set}, one can obtain this characterization more directly. It would be interesting to extend this characterization of extremal mechanisms to the multidimensional setting.} For multidimensional settings, analogous arguments show that many complicated mechanisms are not extremal and therefore the principal need not consider these mechanisms.
As is standard in multidimensional mechanism design \parencite[see, for example,][]{rochet1998ironing}, we can use the divergence theorem to reformulate the objective function as follows:
\begin{align*}
&\int \big[ \kappa U(s)+ \hat{\mathbf{n}}abla U(s)\cdot [g(s)-\kappa s] \big] \,\mathrm dF(s)\\
=& \int U(s) \big[\kappa f(s) - \diver[(g(s)-\kappa s)f(s)] \big] \,\mathrm ds + \int_{\bd S} U(s)[g(s)-\kappa s]f(s) \cdot \hat{\mathbf{n}}_S(s) \,\mathrm d\mathcal{H}(s),
\end{align*}
where $\div$ denotes the divergence of a function, for any set $A$, $\bd A$ denotes the boundary of $A$, $\mathcal{H}$ denotes the $n-1$-dimensional Hausdorff measure on the boundary of $S$, and $\hat{\mathbf{n}}_S(s)$ denotes the outward normal vector to the convex set $S$ at $s\in \bd S$.
This allows us to write the principal's problem as\footnote{The existence of a maximizer follows from standard arguments.}
\begin{align*}
&\max_{U \text{ convex}} \int U(s) \,\mathrm d\mu(s)\\
& \text{s.\,t. } U\le h,
\end{align*}
where the measure $\mu$ is defined by
\begin{align*}
\mu(E) = \int_E \hat{\mathbf{n}}u(s) \,\mathrm d\lambda(s) ,
\end{align*}
$\lambda$ is the Lebesgue measure on $S$ plus the Hausdorff measure on the boundary of $S$, and
\begin{align*}
\hat{\mathbf{n}}u(s):= \begin{cases}
\kappa f(s) - \diver[(g(s)-\kappa s)f(s)] &\text{ if } s\in\interior S \\
[g(s)-\kappa s]f(s) \cdot \hat{\mathbf{n}}_S(s) &\text{ if } s\in\bd S.
\end{cases}
\end{align*}
Cearly, $\hat{\mathbf{n}}u$ plays an important role in determining which mechanisms are optimal. Heuristically, $\hat{\mathbf{n}}u(s)$ measures how much the prinicipal's payoff increases if the indirect utility of type $s$ is increased, but where types on the boundary get extra weight.
To illustrate $\hat{\mathbf{n}}u$ and for later use, let use compute $\hat{\mathbf{n}}u$ for a one-dimensional type space $S=[\underline{s},\overline{v}erline{s}]$:
\begin{align}
\hat{\mathbf{n}}u(s):= \begin{cases}
\kappa f(s) - [g'(s)-\kappa] f(s) -[g(s)- \kappa s]f'(s) &\text{ if } s\in (\underline{s},\overline{v}erline{s}) \\
[g(\overline{v}erline{s})-\kappa \overline{v}erline{s}]f(\overline{v}erline{s}) &\text{ if } s=\overline{v}erline{s}\\
[\kappa \underline{s} - g(\underline{s})]f(\underline{s}) &\text{ if } s=\underline{s}.
\end{cases}\label{eq:nu_onedimensional}
\end{align}
\begin{example}
\label{ex:uniform}
Suppose $S=[-\frac{1}{2},\frac{1}{2}]^n$ and $F$ is the uniform distribution on $S$. Let us assume payoffs are quadratic and that $g(s)=\alpha s$ for some $\alpha\in (0,\kappa]$; this implies that the principal is biased towards the ex-ante optimal action $0$. In that case, $\hat{\mathbf{n}}u$ simplifies to
\begin{align}
\hat{\mathbf{n}}u(s):= \begin{cases}
\kappa + (\kappa - \alpha)n &\text{ if } s\in\interior S \\
(\alpha-\kappa)s \cdot \hat{\mathbf{n}}_S(s) &\text{ if } s\in\bd S. \label{eq:nu_uniform}
\end{cases}
\end{align}
\end{example}
\subsection{Optimal mechanisms}
\label{sec:optimal_mechanisms}
Given an indirect utility $U$, we let $\mathcal{Q}$ denote a coarsest partition of $\mathbb{R}^n$ such that $U$ is affine on each partition element. We denote by $\{\mu|_Q\}_{Q\in \mathcal{Q}}$ a conditional measure of $\mu$ given $Q$. \todo{Check $X$ versus $\mathbb{R}^n$.}
\begin{theorem}\label{thm:main_result}
Let $U$ be a feasible indirect utility. Then $U$ is optimal if for all $Q\in \mathcal{Q}$, $\mu|_Q(Q)\ge 0$ and $\mu|_Q\le_{cx} \delta_Q$, where $\delta_Q$ is a point mass of mass $\mu|_Q(Q)$ at $s$ if there is $s\in Q$ satisfying $U(s)=h(s)$ and $\delta_Q$ is the zero-measure otherwise.
Moreover, this condition is necessary for $U$ to be optimal if $U$ is differentiable $|\mu|$-almost everywhere.
\end{theorem}
Two comments on the conditions in \autoref{thm:main_result} are in order. First, if there is $s\in Q$ such that $U(s)=h(s)$ then it is unique because $U$ is affine on $Q$ and $h$ is strictly convex.
Second, for the necessity result, observe that $U$ is differentiable Lebesgue-almost everywhere since it is a convex function. Since $|\mu|$ is absolutely continuous with respect to the Lebesgue measure on the interior of $S$, $U$ is differentiable $|\mu|$-almost everywhere if, for example, the density $f$ is zero on the boundary of $S$ or if $U$ is differentiable $\mathcal{H}$-almost everywhere on the boundary of $S$. In the one-dimensional case, this last condition can always be satisfied.
Why are the conditions in \autoref{thm:main_result} sufficient for $U$ to be optimal? Consider a partition element $Q\in \mathcal{Q}$ and suppose $\delta_Q$ is a point mass at $s^*$. Then any feasible indirect utility $V$ will be convex and lie below $U$ at $s^*$. Also, $\mu|_Q\le_{cx} \delta_Q$ implies $\int V(s) \,\mathrm d\mu|_Q(s) \le \int V(s) \,\mathrm d\delta_Q(s)$. Moreover, if $a$ is an affine function that coincides with $V$ at the barycenter of $\mu|_Q$ then we get $\int V(s) \,\mathrm d\mu|_Q(s) \le \int a(s) \,\mathrm d\mu|_Q(s)$.
Since $U$ restricted to $Q$ is affine, lies above $V$ at $s^*$, and $\mu|Q(Q)\ge 0$, this implies that conditional on the type belonging to $Q$, the principal's expected payoff under $U$ is higher than under $V$. And if $\delta_Q$ is the zero measure then $V$ might lie above $U$ but the same conclusion follows since $\mu|_Q(Q)=0$ and conditional on $Q$, adding a constant to the indirect utility does not change the principal's payoff. The conclusion that these conditions are also essentially necessary shows that the problem can in some sense be decomposed: whenever the principal can improve $U$ conditional on $Q$, she can extend this improved version to a feasible indirect utility that yields unconditionally a higher payoff.
A particularly simple mechanism is if the principal delegates the decision to the agent, potentially restricting the agents action to belong to some set $A$. Note that any deterministic mechanism can be implemented as an indirect mechanism in this way.
For a closed set $A\subseteq S$, we say that \emph{delegating to $A$} is optimal if an optimal mechanism takes the form that any type in $A$ gets her first-best action, and any other type gets her most preferred action among the first-best actions of types in $A$. For example, if $n=1$ and $A=[s_1,s_2]$ then delegating to $A$ is optimal if there is an optimal mechanism in which any type below $s_1$ gets the first-best action of type $s_1$, any type in $[s_1,s_2]$ gets her first best action, and any type above $s_2$ gets the first-best action of type $s_2$. In the following we will specialize \autoref{thm:main_result} and discuss under what conditions such a mechanism is optimal.
We can simplify the conditions in \autoref{thm:main_result} by recalling that the convex order has a simple structure for one-dimensional spaces. A cdf $H_1$ on a one-dimensional interval $[x,y]$ dominates a cdf $H_2$ in the convex order if and only if $H_2$ \emph{majorizes} $H_1$:
\[ \int_s^y H_1(z) \,\mathrm dz \le \int_s^y H_2(z) \,\mathrm dz \]
for all $s\in [x,y]$ with equality for $s=x$ \parencite[][Theorem 3.A.1]{shaked2007stochastic}. This observation simplifies the characterization in \autoref{thm:main_result} whenever $U$ is affine on at most one-dimensional sets. As we will see, this is useful even if the type space is multidimensional. To illustrate the simpler conditions, we first consider when interval delegation is optimal with one-dimensional types \parencite[for earlier characterizations, see][]{AM:08,AB:13}.
\begin{corollary}\label{cor:onedimension}
Suppose $n=1$ and $s_1,s_2\in S$ with $s_1<s_2$. Delegating to the interval $[s_1,s_2]$ is optimal if and only if
\begin{enumerate}[label=(\roman*)]
\item $\hat{\mathbf{n}}u(s)\ge 0$ for all $s\in[s_1,s_2]$,
\item $\int_s^{\overline{v}erline{s}} (x-s)\hat{\mathbf{n}}u(x) \,\mathrm d \lambda(x|x\ge s_2)\le 0$ for all $s\ge s_2$ with equality for $s=s_2$, and
\item $\int_{\underline{s}}^{{s}} (s-x)\hat{\mathbf{n}}u(x) \,\mathrm d \lambda(x|x\le s_1)\le 0$ for all $s\le s_1$ with equality for $s=s_1$.
\end{enumerate}
\end{corollary}
\begin{proof}
Note that the partition $\mathcal{Q}$ induced by $U$ has elements $[\underline{s},s_1]$, $[s_2,\overline{v}erline{s}]$, and $\{s\}$ for all $s\in(s_1,s_2)$. For all $s\in(s_1,s_2)$, $\hat{\mathbf{n}}u(s)\ge 0$ is equivalent to $\mu|_Q(Q)\ge 0$ and $\mu|_Q\le_{cx} \delta_Q$ for $Q=\{s\}$.\footnote{For $s\in\{s_1,s_2\}$, if $s\in\interior S$ then $\hat{\mathbf{n}}u(s)\ge 0$ follows because $\hat{\mathbf{n}}u$ is continuous on the interior of $S$. And if $s\in\bd S$, there is $Q\in\mathcal{Q}$ with $Q\cap S=\{s\}$ and hence $\mu|_Q(Q)\ge 0$ implies $\hat{\mathbf{n}}u(s)\ge0$.} Now consider $Q=[s_2,\overline{v}erline{s}]$ and let $\lambda(x|x\ge s_2)$ denote the conditional distribution of $\lambda$ conditional on $x\ge s_2$. Since $\delta_Q$ is a point mass of mass $\mu|_Q(Q)$ at $s_2$, we can use majorization to rewrite $\mu|_Q\le_{cx} \delta_Q$ as
\[ \int_s^{\overline{v}erline{s}} \int_x^{\overline{v}erline{s}} \hat{\mathbf{n}}u(z) \,\mathrm d \lambda(z|z\ge s_2) \, \mathrm dx \le 0 \]
for all $s\ge s_2$ with equality for $s=s_2$. Integrating by parts, this becomes condition (ii). Moreover, since the derivative with respect to $s$ of the left-hand side evaluated at $s_2$ is negative, we obtain $\mu|_Q(Q)\ge 0$. The argument for $Q=[\underline{s},s_1]$ is analogous.
\end{proof}
\begin{figure}
\caption{Optimality of interval delegation}
\label{fig:economic_iterpretation_interval}
\end{figure}
\autoref{fig:economic_iterpretation_interval} illustrates condition (ii). Suppose that starting with interval delegation (represented by the solid indirect utility), the principal changes the mechanism and assigns a lottery with expected value strictly above $\hat{\mathbf{n}}abla U(s_2)$ to all types above $s$. This tilts the indirect utility starting at $s$ upwards (see the dashed indirect utility) and therefore increases the indirect utility for every type $x\ge s$ in proportion to $x-s$. The change in the principal's expected payoff is therefore proportional to $\int_s^{\overline{v}erline{s}} (x-s)\hat{\mathbf{n}}u(x) \,\mathrm d \lambda(x|x\ge s_2)$. Consequently, condition (ii) ensures that such changes are not profitable. Equality for $s=s_2$ implies, in addition, that it would not be profitable to marginally reduce the action for all types above $s_2$ either.
Interestingly, the conditions identified in \autoref{cor:onedimension} are in our setting equivalent to the ones obtained in \citet[Proposition 2a]{AB:13}. This might initially be surprising since we characterize optimality of interval delegation in the class of stochastic mechanisms and \citeauthor{AB:13} characterize optimality in the class of deterministic mechanisms (and stochastic mechanisms can do strictly better in general). \autoref{fig:economic_iterpretation_interval} illustrates why the conditions are the same: Suppose the principal strictly benefits from deviating to the dashed indirect utility, which represents a stochastic mechanism. Since her payoff is linear in $U$, the arguments in the previous paragraph imply that she also benefits from deviating to the dotted indirect utility. Since the dotted linear utility corresponds to a deterministic mechanism, we conclude that conditions (ii) in (iii) in \autoref{cor:onedimension} are necessary for interval delegation to be optimal in the class of deterministic mechanisms (and necessity of condition (i) can be shown easily). Later, it will become clear that this equivalence is specific to the one-dimensional setting.
\begin{corollary}\label{cor:suff_one_dim}
If $n=1$ and $\{s\in S: \hat{\mathbf{n}}u(s)\ge 0\}$ is an interval, then delegating to an interval is optimal.
\end{corollary}
The key insight for this result is that any pooling region (i.e., any $Q$ such that $Q\cap S$ is not a singleton) must contain types $s$ with $\hat{\mathbf{n}}u(s)\ge 0$ (since $\mu|_Q(Q)\ge 0$) and types $s$ with $\hat{\mathbf{n}}u(s)\le 0$ (since no point measure $\delta_Q$ can dominate a distinct positive measure in the convex order). If $\hat{\mathbf{n}}u$ is positive on an interval, it follows that there can be at most two pooling regions. A simple argument then shows that delegating to an interval is an optimal mechanism.
\autoref{cor:suff_one_dim} extends Proposition 2(a) in \citet{ABF:18}, which in our notation requires $\hat{\mathbf{n}}u$ to be positive on $(\underline{s},\overline{v}erline{s})$. An simple implication of our result is the following, which can be useful for applications.
\begin{corollary}\label{cor:log_concave}
Suppose the type space is one-dimensional (i.e., $n=1$), $\kappa=1$, and the agent has a constant bias (i.e., $g(s)=s+\beta$ for some $\beta\in\mathbb{R}$). If $f$ is logconcave then delegating to an interval is optimal.
\end{corollary}\todo{Mention comparative statics result implied by this result in combination with Holmstrom's?}
As another illustration, let us return to \autoref{ex:uniform} specializing to a one-dimensional type space.
\addtocounter{example}{-1}
\begin{example}[continued]
For $n=1$ and $\kappa=1$, the objective function simplifies to $\hat{\mathbf{n}}u(s) =2\kappa - \alpha$ for $ s\in (\underline{s},\overline{v}erline{s})$, $\hat{\mathbf{n}}u(\underline{s})=(\kappa - \alpha)\underline{s}$, and $\hat{\mathbf{n}}u(\overline{v}erline{s})=(\alpha-\kappa)\overline{v}erline{s}$. Since $\hat{\mathbf{n}}u$ is positive on an interval, \autoref{cor:suff_one_dim} implies that delegating to an interval is optimal, and it only remains to find the best interval.
The optimal interval must satisfy Condition (ii) as an equality for $s=s_2$, which requires
\[ (2- \alpha) \left[\frac{1}{8} - \frac{1}{2} s_2^2 - s_2 \left(\frac{1}{2}-s_2\right)\right]=0, \]
and simple algebra yields $s_2=\frac{\alpha}{2- \alpha}$. Using symmetry, it follows that it is optimal to delegate to the interval $\left[-\frac{\alpha}{2- \alpha}, \frac{\alpha}{2- \alpha} \right]$.
\end{example}
For a one-dimensional type space, the approach used in \autoref{cor:onedimension} can be used to simplify the conditions in \autoref{thm:main_result} for any mechanism, not just interval delegation. More generally, this approach is useful even with multidimensional types. To see this, let $A$ be a closed and convex set and, for $s\in\bd A$, let $N_A(s)$ denote the normal cone to $A$ at $s$. With quadratic payoffs, if the principal delegates $A$ and $s\in \bd A$, then all types in $s+N_A(s)$ will choose action $s$. Moreover, if the boundary of $A$ is differentiable then $N_A(s)$ is a (one-dimensional) ray and we can again use majorization to simplify the convex dominance conditions in \autoref{thm:main_result}.
\begin{figure}
\caption{Indirect utility for delegation to a convex set. }
\label{fig:convex_delegation}
\end{figure}
\begin{corollary}\label{cor:convex_delegation}
Suppose payoffs are quadratic and $A\subseteq S$ is closed, convex, has nonempty interior and a differentiable boundary. Delegating to $A$ is optimal if and only if
\begin{enumerate}[label=(\roman*)]
\item $\hat{\mathbf{n}}u(s)\ge 0$ for all $s\in A$ and
\item for all $s\in \bd A$ and $z>0$,
\[\int_z^{\infty} (x-z) \hat{\mathbf{n}}u(s+x \hat{\mathbf{n}}_A(s)) \,\mathrm d\lambda(s+x \hat{\mathbf{n}}_A(s)|s+N_A(s)) \le 0\]
with equality for $z=0$.
\end{enumerate}
\end{corollary}
The conditions in \autoref{cor:convex_delegation} closely resemble those in \autoref{cor:onedimension}. Indeed, Condition (i) in either case requires that $\hat{\mathbf{n}}u$ is positive on the set of types that obtain their first-best payoffs, and Condition (ii) (and Conditions (ii) and (iii), respectively) impose that for each point on the boundary the analogous stochastic dominance condition holds.
The economic interpretation of Condition (ii) is analogous to how we interpreted Condition (ii) in \autoref{cor:onedimension}. This condition ensures that the principal does not benefit from marginally tilting the indirect utility along line segments that are orthogonal to the boundary of $A$, e.g., the solid line segment in \autoref{fig:convex_delegation}. Observe that there is a stochastic mechanism in which the indirect utility is increased only in a small neighborhood of the solid line segment (by \autoref{lemma:feasible_set}). On the other hand, there is no deterministic mechanism achieving this because for any deterministic action the indirect utility would have to increase significantly along the solid line segment (in order to reach the first-best payoff for some type) and convexity then requires that all types in a neighborhood of the line segment obtain higher indirect utilities. This indicates that our characterization relies in the multidimensional setting on stochastic mechanisms being feasible.
\addtocounter{example}{-1}
\begin{example}[continued]
Consider a two-dimensional example and recall that $F$ is the uniform distribution and $g(s)=\alpha s$ for some $\alpha\in[0,\kappa)$. We assume quadratic payoffs; in that case, the problem is separable across dimensions: the principal's optimal action in dimension 1 depends only on the first component of the state and is independent of the action in dimension 2.
Suppose first that there are two agents: For $i=1,2$, agent $i$ has private information about $s_i$ (but not $s_j$ for $j\hat{\mathbf{n}}eq i$) and cares only about the action and state in dimension $i$. It follows that the principal faces two independent delegation problems, and our earlier analysis implies that it is optimal to let each agent choose any action in $\left[ -\frac{\alpha}{2- \alpha},\frac{\alpha}{2- \alpha} \right]$. In effect, the agents' choice will be the action in the red square in \autoref{fig:optimal_bundling} that is closed to the realized state.
Now compare this to the situation where there is only one agent. This agent has private information about both dimensions of the state and cares about both dimensions of the action. How can the principal improve her expected payoff? Intuitively, she could offer the agent to take more extreme actions in one dimension if he moderates his action in the other dimension. How can the principal optimally bundle the two decision problems?
\autoref{cor:convex_delegation} provides insights into how to solve the problem: if one can find a $A$ satisfying the conditions stated there, delegating to this set will be an optimal mechanism. Since $\hat{\mathbf{n}}u$ is positive on the interior of $S$ and strictly negative on the boundary of $S$, Condition (i) will be satisfied if $A\subseteq \interior S$ and Condition (ii) will be satisfied if, for every $s\in\bd A$, equality holds in Condition (ii) for $z=0$. This yields a second-order differential equation, whose solution describes the boundary of the optimal delegation set, see the blue curve in \autoref{fig:optimal_bundling} for an illustration.
\end{example}
\begin{figure}
\caption{Optimal bundling}
\label{fig:optimal_bundling}
\end{figure}
\subsection{Proof Sketch}
\label{sec:proof_sketch}
To proof \autoref{thm:main_result}, we use duality in linear programming. To formulate the dual program, it is more convenient to work with indirect utilities that are defined on a compact domain. But recall that it is not enough to only require that $U(s)\le h(s)$ for all $s\in S$.
The following technical result ensures that we can restrict the indirect utilities to have a compact domain as long as this domain is chosen large enough.
\begin{lemma}\label{lemma:compact_domain}
There is a compact $X\subseteq\mathbb{R}^n$ such that the principal's problem can be written as $\max \{\int U \,\mathrm d\mu| U:X\rightarrow\mathbb{R},\ U \text{ convex},\ U\le h\}$.
\end{lemma}
Formally, we show that if $X$ is chosen large enough then for any solution to the above problem there is a corresponding solution to the original problem. For a convex function $U$ defined on $S$, we consider the smallest convex function defined on $\mathbb{R}^n$ that extends $U$. If this extension lies below $h$ on a large set $X$ then $h(y)<U(y)$ for some $y$ is possible only if $\hat{\mathbf{n}}orm{\hat{\mathbf{n}}abla U(s)}$ is large for some $s\in S$, i.e., the expected action for some type is large. We show that this implies that the principal's expected payoff is low, contradicting that $U$ is a solution.
Now let $X$ be as in the above lemma and denote by $\mathcal{U}$ the set of convex continuous functions that map $X$ to $\mathbb{R}$ and by $\mathcal{M}_+$ the set of positive measures on $X$. We can formulate the principal's problem as follows (and call this formulation the primal problem):
\begin{align*}\label{eq:primal_problem}
&\max_{U \in\mathcal{U}} \int U(s) \,\mathrm d\mu(s) \tag{P}\\
& \text{s.\,t. } U\le h
\end{align*}
\paragraph{The dual problem}
We will show that the following problem is the dual problem:
\begin{align}\label{eq:dual_problem}
&\inf_{\gamma \in \mathcal{M}_+} \int h(s) \,\mathrm d\gamma(s) \tag{D}\\
& \text{s.\,t. } \gamma \ge_{cx} \mu,\hat{\mathbf{n}}onumber
\end{align}
where $\ge_{cx}$ denotes the convex order on the space of measures.
Note that $h$ is a convex function; therefore, if $\mu$ was a positive measure, this would be a trivial problem with solution $\gamma= \mu$. However, since $\mu$ is a signed measure and $\gamma$ has to be a positive measure, $\mu$ is not feasible in general.
It is easy to see that weak duality holds, that is, the value of the primal problem \eqref{eq:primal_problem} is always below the value of the dual problem \eqref{eq:dual_problem}. Indeed, for any feasible $U$ and $\gamma$,
\begin{align}\label{eq:weak_duality}
\int U(s) \,\mathrm d\mu(s)\underbrace{\le}_{(i)} \int U(s)\,\mathrm d \gamma(s)\underbrace{\le}_{(ii)} \int h(s) \,\mathrm d \gamma(s)
\end{align}
since (i) $U$ is convex and $\mu\le_{cx} \gamma$ and (ii) $\gamma$ is a positive measure and $U\le h$. The following result shows that strong duality holds, that is the optimal values of both problems are equal and the dual problem has a solution.
\begin{lemma}[Strong duality]\label{l:strong_duality}
A feasible mechanism $U$ is optimal if and only if there exists a positive measure $\gamma\ge_{cx}\mu$ such that
\begin{align}
U(s)&=h(s) \text{ for $\gamma$-almost every\ $v$ } \label{eq:cs1} \\
\int U(s) \,\mathrm d\mu(s) &= \int U(s) \,\mathrm d\gamma(s). \label{eq:cs2}
\end{align}
\end{lemma}
This result is an analogue of a result in the revenue-maximization problem of a multiproduct monopolist \parencite[see Theorem 2 in][]{daskalakis-etal2017}. Our formulation of the delegation problem allows us to easily deduce strong duality. Note that there is a convex function $U$ such that $h(x)-U(x)>0$ for all $x\in X$. Therefore, Slater's condition is satisfied and standard results from linear programming imply that the dual problem has a solution and that the optimal solutions of the primal and dual problems achieve the same value. Since both inequalities in \eqref{eq:weak_duality} have to hold as equalities, \autoref{l:strong_duality} follows.
\paragraph{Proof idea for \autoref{thm:main_result}.} It is easy to show that the conditions in \autoref{thm:main_result} imply that $U$ is optimal: by aggregating the measures $\delta_Q$, one obtains a positive measure $\gamma$ satisfying the complementary slackness conditions \eqref{eq:cs1} and \eqref{eq:cs2} and $\gamma\ge_{cx}\mu$. \autoref{l:strong_duality} then implies that $U$ is optimal.
For the converse direction, suppose $U$ is optimal. By \autoref{l:strong_duality}, there is a positive measure $\gamma$ such that the complementary slackness conditions \eqref{eq:cs1} and \eqref{eq:cs2} hold and $\gamma \ge_{cx} \mu$. Letting $\mu^+$ ($\mu^-$) denote the positive (negative) part of $\mu$, this last condition is equivalent to $\gamma + \mu^- \ge_{cx} \mu^+$. Strassen's theorem then implies that $\gamma + \mu^-$ is a mean-preserving spread of $\mu^+$: one can obtain the measure $\gamma+\mu^-$ by taking, for every $s$, the mass $\mu^+$ puts on $s$ and spreading it according to a probability measure $D_s$ with expected value $s$. Since $U$ is convex, Jensen's inequality implies that $U(s)\le \int U(x)\,\mathrm dD_s(x)$ and equality holds only if $U$ is affine on the convex hull of the support of $D_s$. Since equality must hold by \eqref{eq:cs2}, we obtain that for all $Q\in\mathcal{Q}$ and $s\in Q$, the support of $D_s$ is contained in the closure of $Q$.
To simplify this informal discussion, suppose that for all $Q\in\mathcal{Q}$ and $s\in Q$, the support of $D_s$ is actually contained in $Q$ (and not just the closure of $Q$) and consider a partition element $Q$ of positive measure. Then the conditional measure $\gamma|_Q$ is positive (since $\gamma$ is positive) and satisfies $\gamma|_Q+\mu^-|_Q\ge_{cx} \mu^+|_Q$ (since the left-hand side is a mean-preserving spread of the right-hand side). Moreover, by \eqref{eq:cs2} we get $U(s)=h(s)$ for every $s$ in the support of $\gamma|_Q$. Since $h$ is strictly convex and $U$ is affine on $Q$, there is at most one $s\in Q$ with $U(s)=h(s)$ and therefore $\gamma|_Q$ is a point mass at this $s$ or the zero measure. It follows that $\mu|_Q\le_{cx} \delta_Q$, where $\delta_Q$ is a point mass at $s$ or is the zero measure. The proof in the Appendix follows this sketch but uses the additional assumption in \autoref{thm:main_result} and additional arguments to deal with the case where the support of $D_s$ is a subset of the closure of $Q$ but not a subset of $Q$.\footnote{If $s$ lies in the closure of $Q$ and $Q'\hat{\mathbf{n}}eq Q$, then $U$ is not differentiable at $s$ and, therefore, $U(s)<h(s)$. If follows from \eqref{eq:cs1} that such points have measure zero under $\gamma$. The additional assumption ensures that such points also have measure zero under $\mu^+$ and $\mu^-$, and hence play no role.}
\begin{comment}
\section{Extensions}
\subsection{More general payoffs for the principal}
Suppose the principal's payoff from action $a$ in state $s$ s $u_P(a,s)$, where $u_P$ is continuous, concave in $a$, and the maximizing $a$ exists and is continuous in $s$. Let $\kappa:=\sup\{\kappa'\in \mathbb{R}: u_P(a,s) -\kappa' b(a) \text{ is concave}\}$.
Consider an incentive-compatible mechanism $m$ with corresponding indirect utility $U$. Then
\[ \mathbb{E}[u_P(m(s),s)] \le u_P(\hat{\mathbf{n}}abla U(s),s) - \kappa [U(s)-\hat{\mathbf{n}}abla U(s) \cdot s + b(\hat{\mathbf{n}}abla U(s))] \]
(since ?).
Moreover, equality holds whenever $m$ is a deterministic mechanism.
\begin{align*}
&\max_{U\in\mathcal{U}} \int u_P(\hat{\mathbf{n}}abla U(s),s) - \kappa [U(s)-\hat{\mathbf{n}}abla U(s) \cdot s + b(\hat{\mathbf{n}}abla U(s))] \,\mathrm dF(s) \\
&\text{s.t. } U\le h
\end{align*}
Suppose there is $\gamma\in\mathcal{M}_+$ such that $\int U\,\mathrm d \gamma=\int h\,\mathrm d \gamma$
and directional derivatives in all feasible directions are negative. Then $U$ is optimal.
\begin{align*}
1/\underline{v}arepsilon \int u_P(\hat{\mathbf{n}}abla U(s)+\underline{v}arepsilon \hat{\mathbf{n}}abla V(s),s) -u_P(\hat{\mathbf{n}}abla U(s),s) - \kappa [\underline{v}arepsilon V(s)-(\underline{v}arepsilon \hat{\mathbf{n}}abla V(s) )\cdot s + b(\hat{\mathbf{n}}abla U(s)+\underline{v}arepsilon \hat{\mathbf{n}}abla V(s))] - b(\hat{\mathbf{n}}abla U(s)) \,\mathrm dF(s) +\int V(s) \,\mathrm d\gamma(s)
\end{align*}
\begin{align*}
\int D_a u_P(\hat{\mathbf{n}}abla U(s),s) \cdot \hat{\mathbf{n}}abla V(s) - \kappa [ V(s)- \hat{\mathbf{n}}abla V(s) \cdot s + \hat{\mathbf{n}}abla b(\hat{\mathbf{n}}abla U(s))\cdot \hat{\mathbf{n}}abla V(s)] \,\mathrm dF(s) +\int V(s) \,\mathrm d\gamma(s)\le 0
\end{align*}
for all $V\in\mathcal{U}$.
\end{comment}
\begin{singlespace}
\addcontentsline{toc}{section}{References}
\printbibliography
\end{singlespace}
\appendix
\hat{\mathbf{n}}ewpage
\section{Omitted Proofs}
\hat{\mathbf{n}}ewcommand{\tilde{U}}{\tilde{U}}
\begin{proof}[Proof of \autoref{lemma:compact_domain}]
Let $B_r$ denote a ball of radius $r$ around $0$ and let $U$ be a solution to $\max \{\int U \,\mathrm d\mu| U:B_r\rightarrow\mathbb{R},\ U \text{ convex},\ U\le h\}$. We will show that $U$ can be extended to a solution to the principal's original problem. Let $\tilde{U}$ denote the smallest convex extension to $\mathbb{R}^n$ of the restriction of $U$ to $S$ \parencite[see][]{dragomirescu92}.
If $\tilde{U}$ is not feasible for the original problem then there is $y\hat{\mathbf{n}}ot\in B_r$ such that $\tilde{U}(y)>h(y)$ and there is $s\in S$ such that $\tilde{U}(y)=U(s)+\hat{\mathbf{n}}abla U(s)\cdot (y-s)$ (since $\tilde{U}$ is the smallest convex extension).
Using strong convexity of $h$ (which follows since $b$ has Lipschitz-continuous gradients, see Theorem E.4.2.2 in \citet{hiriart2004}), one can show that $U(s)< h(s) - z(r)$, where $z(r)\rightarrow \infty$ as $r\rightarrow \infty$.\footnote{ Let $c'$ denote modulus of convexity of $h$. Then, for all $y\in B_r$ that lie on the line segment from $s$ to x, and all $t\in\partial (h-\tilde{U})(y)$, $h(s)-\tilde{U}(s) \ge [h(y)-\tilde{U}(y)] + t\cdot (s-y) + \frac{c'}{2} \hat{\mathbf{n}}orm{y-s}^2. $ Since $\tilde{U}(s)=U(s)$ and the first two terms of the RHS are positive, the claim follows.}
Then either $U(s')\le h(s)-z(r)/2$ for all $s'\in S$ or, on a set of positive Lebesgue-measure, $\hat{\mathbf{n}}abla U(s')\hat{\mathbf{n}}ot\in B_{r/c}$ for some constant $c>0$ independent of $r,s$ and $s'$. Since $\lim_{\hat{\mathbf{n}}orm{a}\rightarrow \infty} b(a)=-\infty$ by assumption, this implies that in either case for $r$ large enough, the principals payoff from $U$ will be less than her payoff from taking the ex-ante optimal action. This contradicts our assumption that $U$ was optimal. Hence, any solution can be extended to a solution of the original problem.
\end{proof}
\begin{proof}[Proof of \autoref{l:strong_duality}]
Let $\mathcal{C}(X)$ denote the vector space of continuous functions on $X$ with the supremum norm and recall that its dual space is the space of (Radon) measures on $X$, which we denote by $\mathcal{M}(X)$. Let $\mathcal{V} :=\{ g\in \mathcal{C}(X): \forall x\in V, g(x)\ge 0 \}$; the polar cones of $\mathcal{U}$ and $\mathcal{V}$ are defined by
\begin{align*}
\mathcal{U}^* &:=\{ \gamma \in \mathcal{M}(X): \forall U\in\mathcal{U}, \int U \,\mathrm d\gamma \ge 0 \}\\
\mathcal{V}^* &:=\{ \gamma \in \mathcal{M}(X): \forall g\in\mathcal{V}, \int g \,\mathrm d\gamma \ge 0 \}.
\end{align*}
The principal's problem can be written as $\max_{U \in \mathcal{U}} \int U\,\mathrm d\mu $ subject to $h-U\in \mathcal{V}$.
This is a conical linear program and its dual is $\inf_{\gamma \in \mathcal{V}^*} \int h\,\mathrm d\gamma $ subject to $\mu-\gamma\in U^*$ \parencite[e.g.,][]{shapiro10}.
Since $\mathcal{V}^*=\mathcal{M}_+(X)$ by the Riesz representation theorem \parencite[][p.\,65]{dunford-schwartz88a} and $\mu-\gamma\in U^*$ is equivalent to $\mu\ge_{cx} \gamma$, \eqref{eq:dual_problem} is the dual problem.
Since there is $U\in \mathcal{U}$ such that $h-U$ is in the interior of $\mathcal{V}$, Slater's condition is satisfied and standard results imply that strong duality holds \parencite[e.g.,][Proposition 2.8]{shapiro10}. Therefore, $U$ is optimal if and only if there is a positive measure $\gamma\ge_{cx} \mu$ such that $\int U \,\mathrm d\mu = \int h\, \mathrm d\gamma$, which implies the result.
\end{proof}
\begin{proof}[Proof of \autoref{thm:main_result}]
Given $s\in X$, we denote by $Q(s)$ the partition element of $\mathcal{Q}$ that contains $s$.
``$\mathcal{L}eftarrow$'': Let $\gamma := \int \delta_{Q(s)} \,\mathrm d |\mu|(s)$. Given the properties of $\mu|_Q$, we conclude that $\gamma\in\mathcal{M}_+$ and $\supp \gamma \subseteq \{s:U(s)=h(s)\}$. Moreover, for all $c\in\mathcal{U}$,
\begin{align*}
\int c(x)\, \mathrm d \gamma(x) = \int \int c(x)\, \mathrm d \delta_{Q(s)}(x) \, \mathrm d|\mu|(s) \ge \int \int c(x) \, \mathrm d\mu|_{Q(s)} \, \mathrm d|\mu|(s) = \int c(x)\, \mathrm d\mu(x).
\end{align*}
and equality holds for $c\equiv U$ because (i) $U$ is affine on each $Q\in\mathcal{Q}$ and (ii) $\delta_Q\ge_{cx} \mu|_Q$ implies $\int a(x) \, \mathrm d\delta_Q = \int a(x) \, \mathrm d\mu|_Q$ for any affine function $a\in \mathcal{C}(X)$.
Therefore, $\gamma$ is feasible for the dual problem and satisfies the complementary slackness conditions \eqref{eq:cs1} and \eqref{eq:cs2}. We conclude that $U$ is optimal.
``$\mathbb{R}ightarrow$'':
By Lemma \ref{l:strong_duality}, $U$ is optimal if and only if there is $\gamma\in \mathcal{M}_+$ satisfying \eqref{eq:cs1}, \eqref{eq:cs2}, and $\gamma \ge_{cx}\mu$. Letting $\mu^+$ and $\mu^-$ denote the positive and negative parts of $\mu$, respectively, the last condition becomes $\gamma + \mu^- \ge_{cx} \mu^+$. Since both sides of the inequality are positive measures, Strassen's theorem \parencite[see, for example,][p. 93-94]{phelps01} implies that there is a dilation $D_s$ (that is, for each $s$, $D_s$ is a probability measure with barycenter $s$) satisfying $\gamma+\mu^- = \int D_s \,\mathrm d\mu^+(s)$.
Let $\mu|_Q$ be a (regular, proper) system of conditional measures (such conditional measures exist by Example 10.4.11 in \cite{bogachev07b}), which by definition satisfies
\[ \int_X c(s) \,\mathrm d\mu(s) = \int_X \int_X c(y) \,\mathrm d\mu|_{Q(s)}(y) \,\mathrm d |\mu|(s) \]
for all $c\in\mathcal{C}(X)$. Letting $\alpha_Q := \int D_s \,\mathrm d\mu|_Q^+ (s) - \mu|_Q^-$, we claim that there is $\mathcal{Q}'\subseteq \mathcal{Q}$ such that $\mathcal{Q}'$ has $|\mu|$-measure 0
and, for all $Q\in\mathcal{Q}\setminus \mathcal{Q}'$, $\alpha_Q$ is a positive measure that has support on $Q\cap\{s:U(s)=h(s)\}$.
Before proving this claim, we show that it implies the result: From the definition of $\alpha_Q$ it follows that if $\alpha_Q$ is a positive measure then $\mu|_Q(Q)\ge \alpha_Q(Q)\ge 0$. Also, $\alpha_Q\ge_{cx} \mu|_Q$ since $D_s$ is a dilation; therefore, if $\alpha_Q$ has support in $Q\cap \{s:U(s)=h(s)\}$ then $\delta_Q\ge_{cx} \mu|_Q$, where $\delta_Q$ is a point mass at $Q\cap \{s:U(s)=h(s)\}$ or the zero measure if $U(s)<h(s)$ for all $s\in Q$. For each $Q\in \mathcal{Q}'$, we let $\mu|_Q$ be the zero measure and observe that $\mu|_Q$ is still a conditional measure for $\mu$ since $\mathcal{Q}'$ has measure 0. This proves the result.
First, suppose there is $\mathcal{Q}'\subseteq \mathcal{Q}$ with strictly positive $|\mu|$-measure such that, for all $Q'\in\mathcal{Q}'$, the support of $\alpha_{Q'}$ is not a subset of the closure of $Q$. Fix arbitrary $Q'\in\mathcal{Q}'$. Since the support of $\alpha_{Q'}$ is not contained in the closure of $Q'$, there is a set $A\subseteq Q'$ of strictly positive $\mu|_{Q'}^+$-measure such that, for all $x\in A$, the support of $D_x$ is not contained in the closure of $Q'$. Since Jensen's inequality is strict whenever the convex function is not affine on the convex hull of the support \parencite[][Proposition 16.C.1]{marshall79}, we obtain
\[ \int U(s) \,\mathrm d\mu|_{Q'}^+(s)< \int \left[\int U(x) \,\mathrm dD_s(x)\right] \,\mathrm d\mu|_{Q'}^+(s). \]
This yields
\begin{align*}
\int U(s)\,\mathrm d \mu(s) &= \int \left[\int U(x) \,\mathrm d\mu|_{Q(s)}^+(x) - \int U(x) \,\mathrm d\mu|_{Q(s)}^-(x)\right] \,\mathrm d |\mu|(s) \\
&< \int \left[\int \left( \int U(y) \,\mathrm d D_x(y)\right) \,\mathrm d\mu|_{Q(s)}^+(x) - \int U(x) \,\mathrm d\mu|_{Q(s)}^-(x)\right] \,\mathrm d |\mu|(s) \\
&= \int \int U(y) \,\mathrm d D_s(y) \,\mathrm d \mu^+(s) - \int U(x) \,\mathrm d \mu^-(s) \\
&=\int U(s) \,\mathrm d \gamma(s),
\end{align*}
which contradicts \eqref{eq:cs2}. We conclude that, except possibly on a $|\mu|$-Null set, the support of $\alpha_{Q}$ is a subset of the closure of $Q$.
Second, we show that $\alpha_{Q(s)}$ is a positive measure for $|\mu|$-almost every $s$:
Let
\[ B := \{s\in X: s\in \closure Q\cap \closure Q' \text{ for }Q\hat{\mathbf{n}}eq Q'\}, \]
and note that for any $s\in B$, $U$ is not differentiable at $s$ and therefore $U(s)<h(s)$. Since $\supp \gamma\subseteq \{s: U(s)=h(s)\}$ by \eqref{eq:cs1}, $\gamma(B)=0$. Moreover, $\mu^-(B)=0$ because $U$ is continuously differentiable $|\mu|$-almost everywhere by assumption.
Let $\mathcal{G}$ denote the $\sigma$-algebra generated by $\mathcal{Q}$ and note that the Borel $\sigma$-algebra on $X$ is generated by some countable algebra $\{A_1,A_2,...\}$ \parencite[][Propositions 3.1 and 3.3]{preston08}.
For each $n$ and $G\in \mathcal{G}$,
\begin{align*}
\int_G \alpha_{Q(s)}(A_n) \,\mathrm d|\mu|(s) = &\int_G \int_X D_{s'}(A_n)\,\mathrm d \mu|_{Q(s)}^+(s') - \mu|_{Q(s)}^-(A_n) \,\mathrm d|\mu|(s) \\
= &\int_G D_s(A_n)\,\mathrm d\mu^+(s)-\mu^-(A_n\cap G) \\
\ge &\left[\int_X D_s(A_n\cap G)\,\mathrm d\mu^+(s)-\mu^-(A_n\cap G)\right] - \int_{X\setminus G} D_s(A_n\cap G)\,\mathrm d\mu^+(s).
\end{align*}
The bracketed term equals $\gamma(A_n\cap G)$ and is therefore positive. The last term is zero since
\[\int_{X\setminus G} D_s(A_n\cap G)\,\mathrm d\mu^+(s) \le \int D_s(A_n\cap G\cap B)\,\mathrm d\mu^+(s) - \mu^-(A_n\cap G\cap B) = \gamma(A_n\cap G\cap B)=0\]
(recall that $\gamma(B)=\mu^-(B)=0$).
Since $\alpha_{Q(s)}(A_n)$ is $\mathcal{G}$-measurable in $s$, it follows that there is a $|\mu|$-Null set $Z_n$ such that $\alpha_{Q(s)}(A_n)\ge 0$ for all $s\in X\setminus Z_{n}$. Letting $Z:=\bigcup_{n=1}^{\infty} Z_n$, for all $s\in X\setminus Z$ and Borel sets $A$, $\alpha_{Q(s)}(A)\ge 0$ by Caratheodory's extension theorem \parencite[see][Theorem 1.5.6 and the comment afterward]{bogachev07a}.
Finally, if it is not true that for $|\mu|$-almost every $s$, the support of $\alpha_{Q(s)}$ is a subset of $\{s:U(s)=h(s)\}$, then $\int U \,\mathrm d \gamma < \int h \, \mathrm d \gamma$, contradicting \eqref{eq:cs1}.
Moreover, for any $s\in \closure Q\setminus Q$, $U$ is not differentiable at $s$ and therefore $U(s)<h(s)$ (because $U\le h$ and $h$ is differentiable). We conclude that there is a collection $\mathcal{Q}'\subset \mathcal{Q}$ with $|\mu|$-measure 0 such that, for all $Q\in \mathcal{Q}\setminus \mathcal{Q}'$, $\alpha_Q$ is a positive measure that has support on $Q\cap\{s:U(s)=h(s)\}$.
\end{proof}
\begin{proof}[Proof of \autoref{cor:suff_one_dim}]
Let $U$ be an optimal indirect utility and $\mathcal{Q}$ a corresponding partition. Since $\mu|_Q(Q)\ge 0$ and $\mu|_Q\le_{cx} \delta_Q$, any pooling region\footnote{That is, any $Q$ such that $Q\cap S$ contains strictly more than one element.} $Q\in \mathcal{Q}$ must contain types with $\hat{\mathbf{n}}u(s)\ge 0$ and types with $\hat{\mathbf{n}}u(s)\le s$.
If $\hat{\mathbf{n}}u(\underline{s})\ge 0$ and $\hat{\mathbf{n}}u(\overline{v}erline{s})\ge 0$, $\hat{\mathbf{n}}u$ is positive everywhere and the claim follows. So suppose $\hat{\mathbf{n}}u(\underline{s})< 0$; then there is a pooling region $Q:=[x,y]\in\mathcal{Q}$ which contains $\underline{s}$ and some $s$ with $\hat{\mathbf{n}}u(s)>0$. If $\hat{\mathbf{n}}u(y)<0$, then $[x,y]\subseteq Q$ must hold and the claim follows. Therefore, assume $\hat{\mathbf{n}}u(y)\ge 0$. The measure $\delta_Q$ from \autoref{thm:main_result} must be a point mass at some $z\in Q$ with $\hat{\mathbf{n}}u(z)\ge 0$ (if $\delta_Q$ were the zero measure or a point mass at $z'$ with $\hat{\mathbf{n}}u(z')<0$, then $\int x-x^* \,\mathrm d\mu|_Q > \int x-x^* \,\mathrm d \delta_Q$ whenever $x^*=\inf \{x:\hat{\mathbf{n}}u(x)\ge 0\}$, which contradicts $\mu|_Q\le_{cx} \delta_Q$).
It follows that $U(z)=h(z)$.
If $\hat{\mathbf{n}}u(\overline{v}erline{s})\ge 0$ then $\hat{\mathbf{n}}u(s)\ge 0$ for all $s\in[z,\overline{v}erline{s}]$ and delegating to $[z,\overline{v}erline{s}]$ is optimal. If $\hat{\mathbf{n}}u(\overline{v}erline{s})<0$, repeating our previous argument implies that there is an interval $[x',y']\in\mathcal{Q}$ which contains $\overline{v}erline{s}$ and some $z'$ with $\hat{\mathbf{n}}u(z')\ge 0$ and $U(z')=h(z')$. Since $\hat{\mathbf{n}}u(s)\ge 0$ for all $s\in[z,z']$, delegating to $[z,z']$ is optimal. If $\hat{\mathbf{n}}u(\underline{s})\ge 0$ and $\hat{\mathbf{n}}u(\overline{v}erline{s})<0$, a symmetric argument applies.
\end{proof}
\begin{proof}[Proof of \autoref{cor:log_concave}]
It follows from \eqref{eq:nu_onedimensional} that $\hat{\mathbf{n}}u(s)= f(s)\left[ 1- \beta \frac{f'(s)}{f(s)} \right]$ for $s\in(\underline{s},\overline{v}erline{s})$. If $\beta\ge 0$ then $\hat{\mathbf{n}}u$ is singlecrossing from below on $(\underline{s},\overline{v}erline{s})$ (since $f$ is logconcave) and $\hat{\mathbf{n}}u(\underline{s})\le 0$. The claim then follows from \autoref{cor:suff_one_dim}.
\end{proof}
\begin{proof}[Proof of \autoref{cor:convex_delegation}]
The corresponding indirect utility induces the partition with the following elements: for any $a$ in the interior of $A$, $\{a\}$, and for any $a\in \bd A$, the normal cone $N_A(a)$, which is a ray through $a$ and orthogonal to $\bd A$.
For any such normal ray $Q$, condition (ii) is equivalent to $\mu|_Q\ge_{cx} \delta_Q$ by the same argument as in \autoref{cor:onedimension}.
``$\mathcal{L}eftarrow$'': Condition (i) ensures that $\mu|_{\{a\}}$ is positive for all $a$ in the interior of $A$. Since it has singleton support, $\mu|_{\{a\}}\ge_{cx} \delta_{\{a\}}$. For any normal ray $Q$, $\mu|_Q\ge_{cx} \delta_Q$ by condition (ii) and $\mu|_Q(Q)\ge 0$ since $\int_0^{\infty} \hat{\mathbf{n}}u(s+x \hat{\mathbf{n}}_A(s)) \,\mathrm d\lambda(s+x \hat{\mathbf{n}}_A(s)|ray)\ge 0$ follows from condition (ii). It follows from \autoref{thm:main_result} that $U$ is optimal.
``$\mathbb{R}ightarrow$'':
If $\hat{\mathbf{n}}u(a)<0$ for some $a$ in the interior of $A$ then there is a subset of $A$ with positive $|\mu|$-measure on which $\hat{\mathbf{n}}u$ is strictly negative, which implies $\mu|_Q(Q)<0$ on a set of positive measure, which contradicts optimality of $U$. Similarly, if $\hat{\mathbf{n}}u(a)<0$ for some $a\in \bd A$ then it can be shown that $\mu|_Q(Q)<0$ on a set of positive measure, which contradicts optimality of $U$ by \autoref{thm:main_result}.
If condition (ii) is violated, $\mu|_Q \hat{\mathbf{n}}ot\ge_{cx} \delta_Q$ on a set of positive measure, which again contradicts optimality of $U$ by \autoref{thm:main_result}.
\end{proof}
\end{document} |
\begin{document}
\abovedisplayskip=6pt plus 1pt minus 1pt \belowdisplayskip=6pt
plus 1pt minus 1pt
\thispagestyle{empty} \vskip {1mm}ace*{-1.0truecm} \noindentindent
\vskip 10mm
\begin{center}{\large\bf When is region crossing change an unknotting operation?\\[2mm]
\footnotetext{\footnotesize The authors are supported by NSF 11171025 and Scientific Research Foundation of Beijng Normal University}} \end{center}
\vskip 5mm
\begin{center}{\bf Cheng Zhiyun\\
{\small School of Mathematical Sciences, Beijing Normal University
\\Laboratory of Mathematics and Complex Systems, Ministry of
Education, Beijing 100875, China
\\(email: [email protected])}}\end{center}
\vskip 1 mm
\noindentindent{\small {\small\bf Abstract} In this paper, we prove that region crossing change on a link diagram is an unknotting operation if and only if the link is proper. A description of the behavior of region crossing change on link diagrams is given. Furthermore we also discuss the relation between region crossing change and the Arf invariant of proper links.
\ \
\vskip {1mm}ace{1mm}\baselineskip 12pt
\noindent{\small\bf Keywords} region crossing change; unknotting operation \ \
\noindent{\small\bf MR(2000) Subject Classification} 57M25\ \ {\rm }}
\vskip 1 mm
\vskip {1mm}ace{1mm}\baselineskip 12pt
\section{Introduction}
In this paper. we consider some local transformations on link diagrams. In [4], H. Murakami defined $\sharp$-operation and showed that $\sharp$-operation is an unknotting operation. In [5], $\triangle$-unknotting operation was defined by H. Murakami and Y. Nakanishi. At a later time, Y. Nakanishi [6] proved that a $\triangle$-unknotting operation can be obtained from a finite sequence of 3-gon moves. Hence 3-gon move is also an unknotting operation. In [1], Haruko Aida generalized 3-gon moves to $n$-gon moves, which was also proved to be an unknotting operation, see the figure below.
\begin{center}
\includegraphics{figure1.eps} \centerline{\small Figure
1\quad}
\end{center}
Recently, a new local transformation on link diagram was introduced in [9], named as region crossing change. Here a \textit{region crossing change} at a region of $R^2$ divided by a link diagram is defined to be the crossing changes at all the crossing points on the boundary of the region. For example, the figure below shows the effect of taking region crossing change on the region with capital letter R:
\begin{center}
\includegraphics{figure2.eps} \centerline{\small Figure
2\quad}
\end{center}
Evidently, $\sharp$-operation and $n$-gon move mentioned above are both special cases of region crossing changes. Therefore we say region crossing change is an \textit{unknotting operation} on a link diagram if there exist some regions of $R^2$ divided by the link diagram such that if we apply region crossing changes on these regions the new diagram represents a trivial link. We remark that during the process Reidemeister moves are forbidden, i.e. the diagram are kept if we regard it as a 4-valent graph and ignore the information of the crossings. For the case of knots, the theorem below was proved in [9].
\begin{theorem}$^{[9]}$
Let $D$ be a knot diagram and $p$ a crossing point of $D$, then there exist some regions such that if one takes region crossing changes on these regions, $D$ will be transformed into a new knot diagram $D'$, here $D'$ is obtained from $D$ by a crossing change at $p$.
\end{theorem}
Obviously it follows that region crossing change is an unknotting operation on knot diagrams. In general, region crossing change is not always an unknotting operation for link diagrams. For instance, the standard diagram of Hopf link can not be transformed into a diagram of trivial link by region crossing changes, since the two crossing points are both on the boundary of each region of the diagram. Hence a natural question is on which kind of link diagrams, region crossing change is an unknotting operation. In [2], we give an answer to this question for 2-component links.
\begin{theorem}$^{[2]}$
Region crossing change is an unknotting operation on a diagram of $L=K_1\cup K_2$ if and only if $lk(K_1, K_2)$ is even.
\end{theorem}
In this paper, we will prove the following theorem, which can be regarded as a generalization of the theorem above.
\begin{theorem}
Region crossing change is an unknotting operation on a diagram of $L=K_1\cup K_2\cup \cdots\cup K_n$ if and only if
\begin{center}
$\sum\limits_{j\neq i} lk(K_i, K_j)=0$ $(mod$ $2)$
\end{center}
for all $1\leq i\leq n$.
\end{theorem}
We say a link is \textit{proper} if it satisfies the condition in Theorem 1.3. In [5], H. Murakami and Y. Nakanishi proved that $L=K_1\cup K_2\cup \cdots\cup K_n$ can be obtained from $L'=K'_1\cup K'_2\cup \cdots\cup K'_n$ by a finite sequence of $\sharp$-operations (some Reidemeister moves may be needed) if and only if
\begin{center}
$\sum\limits_{j\neq i} lk(K_i, K_j)=\sum\limits_{j\neq i} lk(K'_i, K'_j)$ $(mod$ $2)$
\end{center}
for all $1\leq i\leq n$. Similarly, in [1], the author proved that $L=K_1\cup K_2\cup \cdots\cup K_n$ can be deformed into $L'=K'_1\cup K'_2\cup \cdots\cup K'_n$ by a finite sequence of $n$-gon moves (some Reidemeister moves may be needed) if and only if \begin{center}
$\sum\limits_{j\neq i} lk(K_i, K_j)=\sum\limits_{j\neq i} lk(K'_i, K'_j)$ $(mod$ $2)$
\end{center}
for all $1\leq i\leq n$. Since when we talk about the equivalence generated by region crossing changes, Reidemeister moves are forbidden, hence Theorem 1.3 can not be obtained from the two results above evidently. For the same reason, we can only discuss the necessary condition of the equivalence up to region crossing changes, see Proposition 4.1. The sufficient condition does not make sense unless we are given two link diagrams which are isotopic as 4-valent graphs.
The rest of the paper are arranged as follows: in Section 2 we will take a brief review of the incidence matrix defined in [2] and some related results of it. In Section 3 we will prove Theorem 1.3 for 3-component links, which is the initial step of the induction used in Section 4. In Section 4, we will give the proof of Theorem 1.3 and offer a complete solution to detect whether some given crossing points of a link diagram can be switched by region crossing changes. Hence the behavior of region crossing change on link diagrams are well understood. Finally the relation between region crossing change and the Arf invariant is discussed.
\section{Incidence matrix of a link diagram}
In this section we will take a quick review of incidence matrix which was defined in [2]. Given a link diagram $D$, let $G$ and $G'$ be the Tait graph of $D$ and the dual graph respectively. In graph theory [10], the incidence matrix of a graph is defined as below
\begin{center}
$M(G)=(m_x(y)),\quad x\in V(G)$ and $y\in E(G)$
\end{center}
and
\begin{center}
$m_x(y)=
\begin{cases}
1& \text{if $y$ is incident with $x$;}\\
0& \text{otherwise.}
\end{cases}$
\end{center}
If we use $M(G)$ $(M(G'))$ to denote the incidence matrix of $G$ $(G')$, since $G$ and $G'$ have the same size, we can obtain a new $(c+2)\times c$ matrix from $M(G)$ and $M(G')$ as below
\begin{center}
$M(D)=\begin{bmatrix}
M(G) \\
M(G') \\
\end{bmatrix},$
\end{center}
here $c$ denotes the number of crossing points of $D$. If we work with $\mathbb{Z}_2$ coefficients, it is not difficult to find that the incidence matrix $M(D)$ is closely related to region crossing changes. In fact each row of $M(D)$ corresponds to a region of $D$, and the positions of 1's of one row tell us which crossings will be changed if we take region crossing change on the corresponding region. Moreover, given a set of regions, in order to understand the effect of region crossing changes on these regions, one just need to read the positions of 1's on the sum of the corresponding rows. The following theorem was proved in [2], here the rank means the $\mathbb{Z}_2$-rank.
\begin{theorem}$^{[2]}$
Let $L$ denote a $n$-component link, and $D$ a diagram of $L$, then the rank of $M(D)$ equals to $c-n+1$, here $c$ denotes the crossing number of $D$.
\end{theorem}
Before ending this section, we want to fix two conventions we will use throughout and mention two useful propositions about region crossing changes. First all diagrams mentioned in this paper are non-split. Besides, sometimes we will abuse our notation, letting $L=K_1\cup K_2\cup \cdots\cup K_n$ refer both to a link diagram and the link itself, so is each component $K_i$ of $L$. It is not difficult to determine the precise meaning from context. Given a diagram $D$ of $L=K_1\cup K_2\cup \cdots\cup K_n$, we define a set of crossing points of $D$, say $P$, are \textit{region crossing change admissible} if we can obtain a new link diagram $D'$ from $D$ by a sequence of region crossing changes, here $D'$ is obtained from $D$ by taking crossing changes on every crossing point of $P$. Then we have
\begin{proposition}
Let $L$ be a link diagram, and $L_1$ is a sub-link of $L$. Choose a set of crossing points of $L_1$, say $P$, if $P$ is region crossing change admissible on the diagram of $L_1$, then it is also region crossing change admissible on the diagram of $L$.
\end{proposition}
\begin{proof}
Notice that any region of $L_1$ is the union of some regions of $L$, if there is no nugatory crossing, then the effect of region crossing changes on the union of these regions is equivalent to the effect on the original one. If there exist some nugatory crossings, with the skill of handling nugatory crossings, see Proposition 2.1 in [2], we can still suitably choose some regions of $L$ which satisfy our requirement. This finishs the proof.
\end{proof}
Finally we want to recall a result in [2].
\begin{proposition}$^{[2]}$
Given an $n$-component link diagram $L=K_1\cup \cdots \cup K_n$, each crossing point of $K_i{\scr A }p K_i$ $(1\leq i\leq n)$ is region crossing change admissible, and each pair of crossing points of $K_i{\scr A }p K_j$ $(1\leq i< j\leq n)$ are region crossing change admissible.
\end{proposition}
\section{The case of 3-component links}
In this section we will prove Theorem 1.3 for the case of 3-component links.
\begin{proposition}
Region crossing change is an unknotting operation on a diagram of $L=K_1\cup K_2\cup K_3$ if and only if $L$ is a proper link.
\end{proposition}
\begin{proof}
Let us consider the sufficient part first. Now $L$ is a proper link and assume $D$ is a diagram of $L$, with crossing number $c$. Let $P$ denote an unknotting set of crossing points, i.e. if one takes crossing changes on all points of $P$ then the new diagram represents a trivial link. Obviously if $lk(K_i, K_j)$ is odd (even), then $(K_i{\scr A }p K_j){\scr A }p P$ contains odd (even) crossing points. Since $L$ is proper, we can divide our discussion in three cases:
\begin{itemize}
\item $K_1{\scr A }p K_3=\varnothing$.
Since $D$ is non-split and $L$ is proper, it follows that $lk(K_1, K_2)=lk(K_2, K_3)=0$ $($mod 2$)$. Then according to Proposition 2.3, we conclude that $P$ is region crossing change admissible.
\item $K_1{\scr A }p K_2\neq\varnothing$, $K_2{\scr A }p K_3\neq\varnothing$, $K_3{\scr A }p
K_1\neq\varnothing$ and $lk(K_1, K_2)=lk(K_2, K_3)=lk(K_3, K_1)=0$ $($mod 2$)$.
As above, $P$ is also region crossing change admissible in this case.
\item $K_1{\scr A }p K_2\neq\varnothing$, $K_2{\scr A }p K_3\neq\varnothing$, $K_3{\scr A }p K_1\neq\varnothing$ and $lk(K_1, K_2)=lk(K_2, K_3)=lk(K_3, K_1)=1$ $($mod 2$)$.
In this case we claim that $\forall p_1\in K_1{\scr A }p K_2$, $\forall p_2\in K_2{\scr A }p K_3$ and $\forall p_3\in K_3{\scr A }p K_1$, $\{p_1, p_2, p_3\}$ are region crossing change admissible. Then combining Proposition 2.3, the conclusion follows. According to the orientation of $K_1$ and $K_2$, we smooth $p_1$ such that $K_1$ and $K_2$ become one component, say $K'$. Now there are only two components in the new diagram, $K'$ and $K_3$. Therefore it follows from Proposition 2.3, $\{p_2, p_3\}$ are region crossing change admissible on $K'\cup K_3$. In other words, there exist some regions of $D$ such that taking region crossing changes on them, $\{p_2, p_3\}$ will be changed. If these region crossing changes also changes $p_1$, then these regions satisfy our requirement. Otherwise we smooth $p_2$ and $p_3$ respectively. If all these three cases can not offer some regions as required, then together with Proposition 2.3 it means that for any pair points of $(K_1{\scr A }p K_2)\cup(K_2{\scr A }p K_3)\cup(K_3{\scr A }p K_1)$, they are region crossing changes admissible. Hence with those rows of $M(D)$, we can construct a matrix as below
\begin{center}
$\begin{bmatrix}
1& &&&&&& \\
& \ddots &&&&&&\\
& & 1& &&&& \\
&&&1 & 1 & & & \\
&&& & 1 & 1 & & \\
&&& & &\ddots&\ddots& \\
&&& & & & 1 & 1 \\
&&&1 & & & & 1\\
\end{bmatrix}$,
\end{center}
where the top left identity submatrix corresponds to those self-crossing points, i.e. the crossing points of $K_i{\scr A }p K_i$, and the right bottom submatrix is referred to those crossing points between different components. It is obvious that the rank of this matrix is $c-1$, hence the rank of $M(D)$ is at least $c-1$, which contradicts with Theorem 2.1. Hence we finish the proof of the sufficient part.
\end{itemize}
Now we turn to the proof of the necessary part. Assume $L$ is not a proper link, there are two possibilities:
\begin{itemize}
\item $K_1{\scr A }p K_3=\varnothing$. It follows that $lk(K_1, K_2)$ and $lk(K_2, K_3)$ can not be both even. Without loss of generality, we suppose $lk(K_1, K_2)$ is odd.
If $lk(K_2, K_3)$ is odd. Since region crossing change is an unknotting operation, we conclude that for any $p_1\in K_1{\scr A }p K_2$ and $p_2\in K_2{\scr A }p K_3$, $\{p_1, p_2\}$ are region crossing change admissible. It means that any pair of non-self-crossing points are region crossing change admissible. Then we can construct a matrix with those rows of $M(D)$ as above. The contradiction follows.
If $lk(K_2, K_3)$ is even. Since region crossing change is an unknotting operation, therefore any crossing point of $K_1{\scr A }p K_2$ is region crossing change admissible. One can also construct a matrix as above, which also leads to a contradiction.
\item $K_1{\scr A }p K_2\neq\varnothing$, $K_2{\scr A }p K_3\neq\varnothing$, $K_3{\scr A }p K_1\neq\varnothing$. Without loss of generality, we assume $lk(K_1, K_2)$ is odd, and $lk(K_2, K_3)$ is even. We continue our discussion in two cases.
If $lk(K_3, K_1)$ is odd. Since for any $p_1\in K_1{\scr A }p K_2$, $p_2\in K_2{\scr A }p K_3$ and $p_3\in K_3{\scr A }p K_1$, $\{p_1, p_2, p_3\}$ are region crossing change admissible. If region crossing change is an unknotting operation then any crossing point of $K_2{\scr A }p K_3$ is region crossing change admissible. It follows that any pair crossing points of $(K_1{\scr A }p K_2)\cup(K_3{\scr A }p K_1)$ are region crossing change admissible. Similarly we can obtain a contradiction as above.
If $lk(K_3, K_1)$ is even. In this case any crossing point of $K_1{\scr A }p K_2$ is region crossing change admissible. The contradiction follows similarly.
\end{itemize}
In conclusion, if $L$ is not proper then region crossing change is impossible to be an unknotting operation. The proof is finished.
\end{proof}
It is worth noting that the proof of the necessary part is direct, all the possible cases of a 3-component link are discussed. In Section 4, Proposition 4.1 will offer a solution of it with another viewpoint.
\section{The proof of the main theorem}
Before giving the proof of Theorem 1.3, we need some preliminary results.
\begin{proposition}
If a link diagram $L=K_1\cup K_2\cup \cdots\cup K_n$ can be obtained from another link diagram $L'=K'_1\cup K'_2\cup \cdots\cup K'_n$ by a sequence of region crossing changes, then $($after suitably ordered if necessary$)$
\begin{center}
$\sum\limits_{j\neq i} lk(K_i, K_j)=\sum\limits_{j\neq i} lk(K'_i, K'_j)$ $(mod$ $2)$
\end{center}
for all $1\leq i\leq n$.
\end{proposition}
\begin{proof}
It suffices to show that for any $1\leq i\leq n$, $\sum\limits_{j\neq i} lk(K_i, K_j)$ $($mod 2$)$ is unaffected by one region crossing change. In fact, it is easy to observe that given a region of the diagram, there are even crossing points on the boundary that are generated by $L-K_i$ and $K_i$. Consequently $\sum\limits_{j\neq i} lk(K_i, K_j)$ $($mod 2$)$ is invariant, then the result follows.
\end{proof}
The next proposition plays an important role in the proof of the main theorem, it can be regarded as a generalization of Proposition 2.3.
\begin{proposition}
Given a link diagram $L$, $\{K_1, \cdots, K_n\}$ are some components of it. If $K_i{\scr A }p K_{j}\neq\varnothing$ for all $\{i, j\}$ which satisfy $|i-j|=1$ or $|i-j|=n-1$, then for any crossing point $p_1\in K_1{\scr A }p K_2, \cdots, p_{n-1}\in K_{n-1}{\scr A }p K_n, p_n\in K_n{\scr A }p K_1$, $\{p_1, p_2, \cdots, p_n\}$ are region crossing change admissible.
\end{proposition}
\begin{proof}
When $n=1$ or 2, the statement follows from Proposition 2.3, the case $n=3$ follows from Proposition 2.2 and the claim in Proposition 3.1. Now we assume the statement is correct for $n\leq k$, it suffices to show it is also correct for $n=k+1$.
If there exist a pair of integers $\{i, j\}$ with $1<j-i<k$, such that $K_i{\scr A }p K_j\neq \varnothing$, then we can choose a crossing point $q$ from $K_i{\scr A }p K_j$. According to the assumption, $\{p_1, \cdots, p_{i-1}, q, p_{j}, \cdots, p_{k+1}\}$ and $\{p_i, \cdots, p_{j-1}, q\}$ are both region crossing change admissible. As a result, $\{p_1, p_2, \cdots, p_{k+1}\}$ are region crossing change admissible.
If for any $\{i, j\}$ which satisfy $1<j-i<k$, $K_i$ and $K_j$ have no intersection, let us consider the diagram of the sub-link $L'=K_1\cup \cdots \cup K_{k+1}$. According to Proposition 2.2, it is sufficient to prove $\{p_1, p_2, \cdots, p_{k+1}\}$ are region crossing change admissible on the diagram of $L'$. Similar to the proof of Proposition 3.1, we can smooth $p_{k+1}$ to obtain a $k$-component link diagram $L''$. By induction, $\{p_1, p_2, \cdots, p_k\}$ are region crossing change admissible on the diagram of $L''$. If the corresponding region crossing changes of $L'$ will affect $p_{k+1}$, then the result follows. Consequently we only need to consider the case whichever crossing point of $\{p_1, p_2, \cdots, p_{k+1}\}$ is chosen to smooth, the corresponding region crossing changes will not affect itself. In other words, any $k$ points of $\{p_1, p_2, \cdots, p_{k+1}\}$ are region crossing change admissible on the diagram of $L'$. Due to these facts, we can use the rows of $M(L')$ to construct a $c\times c$ matrix $M$ as below, here $c$ denotes the crossing number of $L'$.
\begin{center}
$M=\begin{bmatrix}
I& &&&& \\
& A &&&&\\
& & \ddots& && \\
&&&A & & \\
&B & \cdots & B & I & \\
&B&\cdots& B & & I\\
\end{bmatrix}$,
where $I=\begin{bmatrix}
1& && &\\
& 1 &&&\\
& & \ddots& & \\
&&&1& \\
&&&&1\\
\end{bmatrix}$, $A=\begin{bmatrix}
1 & 1 & & & \\
& 1 & 1 & & \\
& &\ddots&\ddots& \\
& & & 1 & 1 \\
1 & & & & 1\\
\end{bmatrix}$ and $B=\begin{bmatrix}
1 & & & & \\
1 & & & & \\
\vdots & &&& \\
1 & & & & \\
1 & & & & \\
\end{bmatrix}$.
\end{center}
As a result, $rank M(L')\geq rank M=c-k+1$, which contradicts the fact $rank M(L')=c-(k+1)+1=c-k$. The proof is finished.
\end{proof}
Now we are going to turn to the proof of Theorem 1.3.
\begin{proof}
The necessary part directly follows from Proposition 4.1, therefore it suffices to prove the sufficient part of the theorem. If $lk(K_i, K_j)$ is even for all $1\leq i<j\leq n$, the result follows from Proposition 2.3.
Therefore we assume there exist some pairs of components with odd linking number. Let us construct a graph $G$ which contains $n$ vertices, each vertex $v_i$ corresponds to a component $K_i$ of $L$. If $lk(K_i, K_j)$ is odd, then we add an edge between $v_i$ and $v_j$. Let $P$ be a unknotting set of $L$. By Proposition 2.3, we can change all the self-intersections and each pair of crossing points between two components in $P$. Thus for any pair of components with odd linking number, there exists one crossing point between them in the remainder of $P$. Hence what we want to do is to remove all the edges from $G$ by region crossing changes. Notice that Proposition 4.2 tells us that a loop of $G$ can be removed by region crossing changes, therefore we can remove loops one by one. Because $L$ is proper, then for any vertex $v_i$ there are even vertices which are adjacent to $v_i$. Hence the process will continue until all the edges have been removed. The proof is finished.
\end{proof}
As we mentioned in Section 1, we can talk about two link diagrams being related by finite region crossing changes only if these two link diagrams are isotopic as 4-valent graphs. Under this condition, it can be proved analogously that $L=K_1\cup K_2\cup \cdots\cup K_n$ and $L'=K'_1\cup K'_2\cup \cdots\cup K'_n$ are related by a sequence of region crossing changes if and only if
\begin{center}
$\sum\limits_{j\neq i} lk(K_i, K_j)=\sum\limits_{j\neq i} lk(K'_i, K'_j)$ $(mod$ $2)$
\end{center}
for all $1\leq i\leq n$.
Given a link diagram $L=K_1\cup K_2\cup \cdots\cup K_n$ and some crossing points of it, say $Q$, a natural question is whether $Q$ is region crossing change admissible? In order to answer this question, we just need to switch all the crossing points of $Q$, then $Q$ is region crossing change admissible if and only if $L$ and the new link $L'$ satisfy the condition above. Or we can construct a graph with vertices $\{v_1, \cdots, v_n\}$ which correspond to the components $\{K_1, \cdots, K_n\}$ respectively. For each non-self-intersection point of $Q$, we add an edge between the corresponding two vertices. Finally we obtain a graph $G(L; Q)$, then it is evident that $Q$ is region crossing change admissible if and only if each vertex of $G(L; Q)$ has even valency.
\section{Region crossing change and Arf invariant}
The aim of this section is to study the relation between region crossing change and Arf invariant. According to Theorem 1.3, region crossing change is an unknotting operation on $L$ if and only if $L$ is proper. Proper links are very special since Arf invariant is well defined on them. Hence a natural question arises: is there any relations between region crossing change and Arf invariant? Before discussing this question, we take a short review of the proper link and its Arf invariant.
Recall that we say a link $L$ is a \emph{proper link} if for any component of $L$, the sum of the linking numbers between this component and the rests is an even integer. According to [7], we can define the Arf invariant of a proper link in this way: let $M=S^3\times [0, 1]$, then $\partial M=S^3\times \{0\}\cup S^3\times \{1\}=\partial M_+\cup \partial M_-$. Given a proper link $L$ and a knot $K$ which are embedded in $\partial M_+$ and $\partial M_-$ respectively, if there exists a regularly embedded 2-manifold $N$ of genus zero such that $\partial N{\scr A }p \partial M_+=L$ and $\partial N{\scr A }p \partial M_-=K$, then we say $K$ is \emph{related} to $L$. It was proved in [7] that if $K$ and $K'$ are two knots related to the same proper link $L$, then Arf$(K)=$Arf$(K')$. Therefore we can define Arf$(L)\triangleq $Arf$(K)$ where $K$ is a knot related to $L$.
In practice, given a proper link $L=K_1\cup K_2\cup \cdots\cup K_s$ $($without loss of generality, we assume that $L$ is non-split$)$, in order to calculate the Arf invariant of $L$, we can handle it as follows. First choose a crossing point between $K_i$ and $K_j$, then smooth it according to the orientations of $K_i$ and $K_j$$($see the figure below$)$. Now we obtain a proper link with $s-1$ components. Repeating this process until we get one component, i.e. a knot $K$. By the definition above, we have Arf$(L)=$Arf$(K)$.
\begin{center}
\includegraphics{figure3.eps} \centerline{\small Figure
3\quad}
\end{center}
In [4], it was shown that one $\sharp$-operation changes the Arf invariant of the knot, i.e. if $K$ and $K'$ are related by one $\sharp$-operation, then Arf$(K)+$Arf$(K')=1$. In [5], a similar result was given for $\triangle$-operation, i.e. if $K$ and $K'$ are related by one $\triangle$-operation, we also have Arf$(K)$+Arf$(K')=1$.
Let $L$ be a diagram of a proper link, and $R$ a region of it. After taking region crossing change on $R$, one obtain a new proper link $($Proposition 4.1$)$, say $L'$. Now we want to investigate the relation between Arf$(L)$ and Arf$(L')$.
Consider the region $R$, we denote the crossing points on the boundary of $R$ by $\{c_1, \cdots, c_n\}$. Color the regions of $L$ in checkerboard fashion, such that $R$ is colored white. For each crossing $c_i$, we assign two integers $a(c_i)$ and $w(c_i)$, according to the figure below.
\begin{center}
\includegraphics{figure4.eps} \centerline{\small Figure
4\quad}
\end{center}
The main theorem of this section can be described as below:
\begin{theorem}
Let $L$ be a diagram of a proper link, $L'$ is obtained by taking region crossing change on region $R$ of $L$, then
\begin{center}
\emph{Arf}$(L)+$\emph{Arf}$(L')=$
$\begin{cases}
0$ $ ($\emph{mod 2}$)& \text{\emph{if} $\frac{1}{2}\sum\limits_{i=1}^n(a(c_i)-w(c_i))=0$ $($\emph{mod}$ $ $4 )$;}\\
1$ $ ($\emph{mod 2}$)& \text{\emph{if} $\frac{1}{2}\sum\limits_{i=1}^n(a(c_i)-w(c_i))=2$ $($\emph{mod}$ $ $4 )$.}
\end{cases}$
\end{center}
Here $\{c_1, \cdots, c_n\}$ denote the crossing points on the boundary of $R$.
\end{theorem}
If we denote $\frac{1}{2}\sum\limits_{i=1}^n(a(c_i)-w(c_i))$ by $A(R)$, in fact $A(R)$ is an even integer $($see the proof below$)$, and now the equality above can be written as
\begin{center}
Arf$(L)+$Arf$(L')=$
$\begin{cases}
0$ $ ($mod 2$)& \text{if $A(R)=0$ $(${mod} 4$ )$;}\\
1$ $ ($mod 2$)& \text{if $A(R)=2$ $(${mod} 4$ )$.}
\end{cases}$
\end{center}
We remark that when $R$ is the changed region in $\sharp$-operation, it is easy to find that $A(R)=2$, therefore after one $\sharp$-operation the Arf invariant will change. Note that the equality above is also valid for $n$-gon move defined in [1].
Next we give the proof of Theorem 5.1.
\begin{proof}
The idea of the proof basically comes from the related result in [4] and [5]. The key point is that with the given region $R$, there exists an $n$-component proper link $L_R$ which is completed determined by $\{a(c_1), \cdots, a(c_n)\}$ and $\{w(c_1), \cdots, w(c_n)\}$, such that Arf$(L)+$Arf$(L')=$ Arf$(L_R)$ $($mod 2$)$. The figure below shows how to find this proper link $L_R$.
\begin{center}
\includegraphics{figure5.eps} \centerline{\small Figure
5\quad}
\end{center}
Since $L_R$ is an $n$-component link, we can suppose $L_R=K_{R1}\cup K_{R2}\cup \cdots\cup K_{Rn}$ as above, then $lk(K_{R1}, K_{R2})=\pm1, \cdots, lk(K_{R(n-1)}, K_{Rn})=\pm1, lk(K_{Rn}, K_{R1})=\pm1$. Note that after taking $n$ connected sum operations between $L$ and $L_R$, we obtain $L'$. See the figure below. According to [7], it follows that Arf$(L)+$Arf$(L')=$ Arf$(L_R)$ $($mod 2$)$. Hence it suffices to find out the Arf invariant of $L_R$.
\begin{center}
\includegraphics{figure6.eps} \centerline{\small Figure
6\quad}
\end{center}
As we mentioned before, it order to calculate Arf$(L_R)$, we just need to smooth $n-1$ crossing points from $K_{R1}{\scr A }p K_{R2}, \cdots, K_{R(n-1)}{\scr A }p K_{Rn}$ according to their orientations, then we will obtain a knot $K_R$ which has the same Arf invariant with $L_R$. Assign each $c_i$ with a pair of integers $(a(c_i), w(c_i))$, there are totally four cases for all $\{c_1, \cdots, c_n\}$, i.e. $(-1, +1), (+1, -1), (+1, +1), (-1, -1)$. Let $m_{-+}, m_{+-}, m_{++}$ and $m_{--}$ denote the number of the crossing points of these four types respectively. We claim that $K_R$ can be described as one of the four cases $($or their inverses$)$ below:
\begin{center}
\includegraphics{figure7.eps} \centerline{\small Figure
7\quad}
\end{center}
In order to see this, it suffices to notice that for a crossing $c_i$ of type $(+1,+1)$ or $(-1,-1)$, smoothing one crossing point between $K_{Ri}$ and $K_{R(i+1)}$ will provide no twist. However if $c_i$ is of type $(-1,+1)$ or $(+1,-1)$, the same operation will increase one positive half-twist or one negative half-twist respectively. Since $K_R$ is a knot, from Figure 7 it is obvious that $m_{-+}+m_{+-}$ is an even integer, it follows that $A(R)=\frac{1}{2}\sum\limits_{i=1}^n(a(c_i)-w(c_i))=m_{+-}-m_{-+}$ is even. Because two full-twists preserve the Arf invariant, it follows that
\begin{center}
Arf$(K_R)=$
$\begin{cases}
0& \text{if $m_{-+}-m_{+-}=0$ $(${mod} 4$ )$;}\\
1& \text{if $m_{-+}-m_{+-}=2$ $(${mod} 4$ )$.}
\end{cases}$
\end{center}
The proof is complete.
\end{proof}
As a corollary, we have
\begin{corollary}
Let $L$ be a diagram of a proper link, $\{R_1, \cdots, R_n\}$ some regions of $L$, such that taking region crossing changes on $\{R_1, \cdots, R_n\}$ will turn $L$ to be trivial. Then
\begin{center}
\emph{Arf}$(L)=$
$\begin{cases}
0& \text{\emph{if} $\sum\limits_{i=1}^nA(R_i)=0$ $($\emph{{mod} 4}$ )$;}\\
1& \text{\emph{if} $\sum\limits_{i=1}^nA(R_i)=2$ $($\emph{{mod} 4}$ )$.}
\end{cases}$
\end{center}
\end{corollary}
\textbf{Acknowledgement} The authors wish to thank Professor Gao Hongzhu, Professor Lorenzo Traldi and Ayaka Shimizu for their useful suggestions and comments.
\noindent
\vskip0.2in
\noindent {\bf References}
\vskip0.1in
\footnotesize
{\Bbb R}EF{[1]}Haruko Aida, {\it Unknotting operation for Polygonal type}. Tokyo J. Math. Vol. 15, No. 1, 111-121, 1992
{\Bbb R}EF{[2]}Cheng Zhiyun, Gao Hongzhu, {\it On region crossing change and incidence matrix}. math.GT/1101.1129v2, 2011. To appear in Science China Mathematics.
{\Bbb R}EF{[3]}Hoste, J., Nakanishi, Y. and Taniyama, K., {\it Unknotting operations involving trivial tangles}. Osaka J. Math. 27, 555-566, 1990
{\Bbb R}EF{[4]}H. Murakami, {\it Some metrics on classical knots}. Math. Ann. 270, 35-45, 1985
{\Bbb R}EF{[5]}H. Murakami, Y. Nakanishi, {\it On a certain move generating link-homology}. Math. Ann. 284, 75-89, 1989
{\Bbb R}EF{[6]}Y. Nakanishi, {\it Replacements in the Conway third identity}. Tokyo J. Math. 14, 197-203, 1991
{\Bbb R}EF{[7]}R. Robertello, {\it An invariant of knot cobordism}. Commun. Pure Appl. Math. 18, 543-555, 1965
{\Bbb R}EF{[8]}D. Rolfsen, {\it Knots and links}. Publish or Perish, Inc. 1976
{\Bbb R}EF{[9]}Ayaka Shimizu, {\it Region crossing change is an unknotting operation}. math.GT/1011.6304v2, 2010
{\Bbb R}EF{[10]}Junming Xu, {\it Theory and Application of Graphs}. Kluwer Academic Publishers, 2003
\end{document} |
\begin{document}
\title{Nearest Neighbor Search for Hyperbolic Embeddings}
\begin{abstract}
Embedding into hyperbolic space is emerging as an effective representation technique for datasets that exhibit hierarchical structure. This development motivates the need for algorithms that are able to effectively extract knowledge and insights from datapoints embedded in negatively curved spaces. We focus on the problem of nearest neighbor search, a fundamental problem in data analysis. We present efficient algorithmic solutions that build upon established methods for nearest neighbor search in Euclidean space, allowing for easy adoption and integration with existing systems. We prove theoretical guarantees for our techniques and our experiments demonstrate the effectiveness of our approach on real datasets over competing algorithms.
\end{abstract}
\section{Introduction}
We study the nearest neighbor problem for vector representations in hyperbolic space: given a dataset $\mathcal{D}$ of vectors and a query $q$, find the nearest neighbor of $q$ among the elements of $\mathcal{D}$ according to the hyperbolic distance metric. Nearest neighbor search is an important building block in many applications, including classification, recommendation systems, DNA sequencing, web search, and near duplicate detection. Yet for embeddings into negatively curved spaces, we still lack simple, practical, experimentally verified and theoretically justified solutions to tackle this question.
Hyperbolic embeddings have emerged as a useful way of representing data that exhibit hierarchical structure. \cite{nickel2017poincare} studies the representation and generalization performance of hyperbolic embeddings in comparison to Euclidean and translational embeddings and shows that hyperbolic embeddings outperforms with just a few dimensions. Later work focuses on techniques to produce even higher quality hyperbolic embeddings, including different training algorithms in different models of hyperbolic space \cite{nickel2018learning} and combinatorial embedding algorithms \cite{de2018representation}, and hybrid training models \cite{le2019inferring}. These developments motivate the need for algorithms that are able to effectively extract knowledge and insights from hyperbolic data representations, for example neural networks that can work with hyperbolic embeddings as feature vectors \cite{ganea2018hyperbolic}. We focus on the problem of nearest neighbor search.
Despite the extensive literature on nearest neighbor search, most focus on the Euclidean setting and very few existing algorithms can be applied to hyperbolic embeddings. One relevant work for hyperbolic space is \cite{krauthgamer2006algorithms}, which proposes an approximate nearest neighbor search scheme that involves iteratively partitioning the space using special separator points.
They prove the existence of such points, but do not give an efficient algorithm to find them. Moreover, their solution requires precise a-priori knowledge of intrinsic parameters, such as the hyperbolicity of the dataset, that are computationally very difficult to compute exactly, \cite{borassi2015computing}. Their approximation guarantees are in terms of these parameters, so using upper bounds could lead to poor performance.
There are also nearest neighbor graph methods \cite{malkov2018efficient} \cite{naidan2015permutation} \cite{fu2019fast} \cite{subramanya2019rand} that create a search graph for a dataset by linking elements are close together in a generic distance metric, and hyperbolic distance applies. The drawback is that they do not come with any theoretical guarantees and require a lot of hyperparameter tuning and high indexing costs.
Our focus is on developing efficient nearest neighbor algorithms for hyperbolic space with provable guarantees that also work well in practice. We leverage solutions for provably efficient nearest neighbor search in Euclidean space and show how those algorithms can be used in a black box fashion to find nearest neighbors in hyperbolic space with minimal additional cost in query time and storage. Our solution is simple, intuitive, and easy to adopt by practitioners. We experiment on real datasets and show that our technique compares favorably against benchmark methods. Our theoretical analysis develops a rigorous understanding of our techniques and our ideas offer insights on key properties of negatively curved spaces that we hope will benefit future algorithmic work on hyperbolic space.
\section{Related Work}
Our work adds to a fast-growing collection of exciting progress on hyperbolic representation learning, recently popularized by the work of \cite{nickel2017poincare} and \cite{nickel2018learning}. \cite{ganea2018hyperbolic} \cite{gu2018learning} \cite{law2019lorentzian} \cite{de2018representation} \cite{tifrea2018poincar} study techniques for learning more effective hyperbolic embeddings from hierarchical data, including both neural network and combinatorial based approaches. Works such as \cite{cho2019large} \cite{davidson2018hyperspherical} \cite{tran2020hyperml} develop techniques for performing downstream tasks such as classification and recommendation given pretrained embeddings. \cite{dhingra2018embedding} and \cite{tay2018hyperbolic} work in the NLP domain and train hyperbolic word embeddings and use them for downstream tasks such a Question Answering. \cite{chamberlain2017neural} embeds graphs into hyperbolic space. \cite{gulcehre2018hyperbolic} \cite{ganea2018hyperbolic} develop neural network architectures for transformers and recurrent neural networks that use hyperbolic geometry to learn from datasets with hierachical structure.
Nearest neighbor methods in Euclidean space are well studied, see \cite{reza2014survey} for a general survey. There are many different techniques that come with provable guarantees, including Locality Sensitive Hashing \cite{wang2014hashing}, KD trees \cite{bentley1975multidimensional}, and many others, see \cite{reza2014survey} and references therein . On the empirical side, https://github.com/erikbern/ann-benchmarks compares performance of common nearest neighbor algorithms for benchmark datasets. However, these techniques and analyses are focused on Euclidean space, and do not apply immediately to hyperbolic space. To our knowledge, we are the first to present a theoretically justified and empirically validated solution for nearest neighbors in hyperbolic space.
\section{Problem formulation and approach overview}
We are given a dataset $\mathcal{D}$ of $n$ points and a query $q$ in hyperbolic space and want to find the nearest neighbor or approximate nearest neighbor to $q$ from $p \in \mathcal{D}$. We call a point $p$ a $c$-approximate nearest neighbor for $c > 1$ if $d_H(p, q) \leq c \cdot d_H(p^*, q)$, where $p^*$ is the nearest neighbor to $q$ in the hyperbolic metric, and $d_H$ is the hyperbolic distance function. There are several models of hyperbolic space and we focus on the popular and intuitive Poincar\'e ball model in $r$ dimensions, which we denote $\mathbb{H}_r$. The different models are isometric, so one can apply our techniques to points embedded into other models by translating them to the Poincar\'e ball, see \cite{cannon1997hyperbolic} for details.
\subsection{Preliminaries}
In $\mathbb{H}_r$, all points are inside the $r$-dimensional unit ball, and distance between points $x$ and $y$ is defined by
\begin{equation}
\label{eq:poincare_distance}
d_H(x, y) = \arccosh \left(1 + \frac{2\|x-y\|^2}{(1-\|x\|^2) (1-\|y\|^2)}\right) ~,
\end{equation}
where $\|\cdot\|$ denotes Euclidean norm or Euclidean distance.
We denote $\mathcal{B}_H(q, d)$ the hyperbolic ball around center $q$ with hyperbolic radius $d$. We denote $\mathcal{B}_E(q, d)$ as the Euclidean ball around center $q$ with Euclidean radius $d$. One useful fact is that for every $q, d$, $\mathcal{B}_H(q, d) = \mathcal{B}_E(q', d')$ for some $q', d'$ that can be solved via simple calculations (ie, hyperbolic balls in Poincar\'e space are Euclidean balls with different centers and radii) \cite{cannon1997hyperbolic}.
\subsection{Overall approach}
Our overall approach is to leverage existing Euclidean nearest neighbor methods to find near exact hyperbolic nearest neighbors. {\emph{Our first class of algorithms use the key fact that hyperbolic balls in $\mathbb{H}_r$ are Euclidean balls with different centers of gravity.}} For query $q$, if we had $p \in \mathcal{D}$ such that $p \in \mathcal{B}_H(q, d_H(p, q)) = \mathcal{B}_E(q', d')$, then we can find a better neighbor by doing Euclidean nearest neighbor search on $q'$.
{\emph{Our second main class of algorithms uses the insight that when $p \in \mathcal{D}$ have similar Euclidean norms, the denominator term $(1-\|p\|^2) (1-\|q\|^2)$ in Eq. \ref{eq:poincare_distance} is similar for different $p$, so the problem reduces to minimizing $2\|p-q\|^2$, which is a Euclidean nearest neighbor problem.}} We first partition our dataset so that elements in one partition have similar Euclidean norms, perform Euclidean nearest neighbor search in these partitions separately, and then aggregate results. For massive datasets, this idea also provides a way to shard the database that maintains efficient search and indexing.
We abstract our use of Euclidean nearest neighbor algorithms into black box oracles; our algorithms are compatible with any implementation of Euclidean nearest neighbor search, however performance varies depending on the underlying algorithm. We use the following classes of oracles:
\begin{definition}[Exact Euclidean Nearest Neighbor Oracle $\O$]
The exact Euclidean nearest neighbor oracle, $\O$ takes as input a query $q$ and a dataset $\mathcal{D}$ and returns $O(q, \mathcal{D})$, which is an element $d$ in $\mathcal{D}$ that minimizes Euclidean distance to $q$ in query time $\mathcal{T}$ and space $\mathcal{S}$.
\end{definition}
\begin{definition}[$(1+\epsilon)$-approximate Euclidean Nearest Neighbor Oracle $\tO$]
For $\epsilon > 0$, a $(1+\epsilon)$-approximate Euclidean nearest neighbor oracle, $\tO$ takes as input a query $q$ and a dataset $\mathcal{D}$, and returns $\tO(q, \mathcal{D})$, which is some $d \in \mathcal{D}$ such that $\|d - q\| \leq (1+\epsilon) \|q-n_E\|$ in query time $\mathcal{T}$ and space $\mathcal{S}$, where $n_E$ is the Euclidean nearest neighbor to $q$ in $\mathcal{D}$.
\end{definition}
We do not include failure probability into our definition of $\tO$ even though many of them give high probability guarantees, because this can be resolved using independent trials. Examples of common oracles and their performance are in \cite{wang2014hashing} and references therein.
To summarize, our main contributions are:
\begin{itemize}
\item \emph{Recentering-HyperbolicNN }NS, an exact hyperbolic nearest neighbor algorithm that uses an exact Euclidean nearest neighbor oracle.
\item \emph{Binary-Search-HyperbolicNN }NS, a $c$-approximate hyperbolic nearest neighbor algorithm that uses an exact Euclidean nearest neighbor oracle.
\item \emph{Spherical-Shell-HyperbolicNN }NS, a $c$-approximate hyperbolic nearest neighbor algorithm that uses a $(1+\epsilon)$-approximate Euclidean nearest neighbor oracle.
\end{itemize}
\section{Recentering algorithms using exact Euclidean oracles}
In each iteration of \emph{Recentering-HyperbolicNN }NS, Algorithm \ref{alg:nn_recentering_alg}, we take the current best hyperbolic nearest neighbor $n_H$ (initially set to be the Euclidean nearest neighbor of $q$) and attempt to find a closer point in hyperbolic distance. We exploit the fact that the hyperbolic ball around $q$ that has $n_H$ on its boundary is a Euclidean ball around a different point $q_{new}$ \cite{cannon1997hyperbolic}. Performing Euclidean nearest neighbor search around $q_{new}$ either finds a point strictly inside this ball (which is closer to q than $n_H$ in hyperbolic distance), or establishes that $n_H$ indeed is the hyperbolic nearest neighbor of $q$.
\emph{Recentering-HyperbolicNN } uses \emph{Euclidean-Center-of-Hyperbolic-Ball }NS, an elementary subroutine that performs the recentering. Details can be found in \cite{cannon1997hyperbolic} and in the appendix.
\begin{algorithm}[ht]
\caption{\emph{Recentering-HyperbolicNN }}
\label{alg:nn_recentering_alg}
\begin{algorithmic}[1]
\mathbb{R}EQUIRE{query $q$, dataset $\mathcal{D}$, exact Euclidean nearest neighbor oracle $\O$}
\mathcal{S}TATE $n_H \leftarrow q$
\mathcal{S}TATE $n_E \leftarrow \O(q, \mathcal{D})$.
\WHILE{$d_H(n_E, q) \neq d_H(n_H, q)$}
\mathcal{S}TATE $n_H \leftarrow n_E$
\mathcal{S}TATE $q_{new} = \emph{Euclidean-Center-of-Hyperbolic-Ball }(q, d_H(q,n_H))$
\mathcal{S}TATE $n_E \leftarrow \O(q_{new}, \mathcal{D})$
\ENDWHILE
\mathcal{S}TATE \textbf{Return} $n_H$
\end{algorithmic}
\end{algorithm}
\begin{theorem}
\label{thm:nn_recentering_alg}
Suppose that the Euclidean nearest neighbor to $q$, is the $k$-th nearest hyperbolic neighbor to $q$. Then Algorithm \ref{alg:nn_recentering_alg} returns the hyperbolic nearest neighbor $n_H$ after at most $k+1$ invocations of the exact Euclidean nearest neighbor oracle $\O$. The runtime of this algorithm is at most $(k+1) \mathcal{T}$, where $\mathcal{T}$ is the runtime for one invocation of $\O$. The storage of this algorithm is $\mathcal{S}$, where $\mathcal{S}$ is the storage requirement of $\O$.
\end{theorem}
\begin{proof}
If there is an exact match, we would invoke $\O$ once. If there is no exact match, the first invocation of $\O$ returns the Euclidean nearest neighbor to the query, $n_E$. We can draw a hyperbolic ball around $q$ with radius $d_H(q, n_E)$. Clearly, any point that is closer in hyperbolic distance to $q$ must lie inside this ball. So we will find these points by calling $\O$ on the Euclidean center of this ball, $q_{new}$, which guarantees an improvement over $n_E$. We recurse on this logic. If at round $r$, we do not get an improvement, then we terminate, as there cannot be a point that is a nearer neighbor.
Since each round results in a strict improvement or a termination, if the Euclidean nearest neighbor of $q$ is the $k$-th nearest hyperbolic neighbor to $q$, then \emph{Recentering-HyperbolicNN } terminates in at most $k+1$ invocations of $\O$. The runtime guarantee follows trivially.
\end{proof}
\emph{Recentering-HyperbolicNN } generalizes to provably return $K$ nearest neighbors using an oracle that finds $K$ Euclidean nearest neighbors when the recentering and termination criterion use the $K$-th nearest neighbor found so far.
Theorem \ref{thm:nn_recentering_alg} provides a worst case guarantee in terms $k$, the ranking of the Euclidean nearest neighbor with respect to the hyperbolic metric. Our algorithm doesn't need to know $k$; moreover, in the best case, the datapoints could be such that the Euclidean nearest neighbor of $q$ is the $k$-th hyperbolic nearest neighbor to $q$ for arbitrarily high $k$ but \emph{Recentering-HyperbolicNN } returns the hyperbolic nearest neighbor in 3 invocations to $\O$.
However, in the worst case, \emph{Recentering-HyperbolicNN } returns the hyperbolic nearest neighbor in exactly $k+1$ invocations of $\O$ for arbitrary $k$. We give the construction below.
\begin{lemma}
Let $q$ be our query in 1 dimension, and $\|q\|$ is close to 1. Suppose for arbitrary $k \in \mathbb{N}$, we have data points $q+z, q-z, p_1, \ldots p_{k-2}$, where $p_i = \frac{2^i -1}{2^i}$, and $z$ is very small and satisfies $q - z \geq \frac{2^{k} -1}{2^{k}}$ and $q + z < 1$, and $d_H(q, 0) = d_H(q, q+z)$. Then \emph{Recentering-HyperbolicNN } returns $q-z$, hyperbolic nearest neighbor in exactly $k+1$ invocations to $\O$.
\end{lemma}
\begin{proof}
\emph{Recentering-HyperbolicNN } first returns $n_E = q + z$, the $k$-th hyperbolic nearest neighbor to $q$. $n_E$ is close to the edge of the disk whereas all the other points are closer to the origin, so $d_H(q, n_E)$ is high even though the Euclidean distance is small. The new center from the first recentering is near the point $\frac{1}{2}$, so the next call to $\O$ returns $p_1 = \frac{1}{2}$. Subsequent calls to $\O$ will return $\frac{3}{4} = p_2$, and then $p_3, \ldots p_{k-2}$ until we finally find $q-z$.
\end{proof}
\subsection{$k$-Independent approximate hyperbolic nearest neighbor algorithm}
\emph{Binary-Search-HyperbolicNN } is an approximate hyperbolic nearest neighbor algorithm that aims to approximate the smallest possible radius $r$ around the query such that $B_H(q, r)$ is non-empty, which essentially isolates the nearest neighbor. It performs binary search on $r$, starting from the upper bound $r = d_H(q, n_E)$, and continues until it finds a small enough non-empty radius that satisfies the desired approximation guarantee.
Using the same recentering idea, we can use $\O$ to determine whether $B_H(q, r)$ is non-empty for any $r$. The nearest neighbor that $\O$ outputs is the certificate that indicates whether to recurse on the left or right side of the binary search. The algorithm maintains upper and lower bounds $R_i$ and $L_i$ on $r$ in each round $i$, ensuring that $\frac{R_{i+1}}{L_{i+1}}\leq\sqrt{R_i/L_i}$.
\begin{algorithm}[ht]
\caption{\emph{Binary-Search-HyperbolicNN }}
\label{alg:binary_search}
\begin{algorithmic}[1]
\mathbb{R}EQUIRE{query $q$, exact Euclidean nearest neighbor oracle $\O$, approximation guarantee $c > 1$}
\mathcal{S}TATE $n_E \leftarrow \O(q, \mathcal{D})$
\mathcal{S}TATE $n_H \leftarrow n_E$
\IF {$n_E = q$}
\mathcal{S}TATE \textbf{Return} $n_H$
\ENDIF
\mathcal{S}TATE $L = d_H\left(q, \left(1-\frac{\|d_E -q\|}{\|q\|}\right)\cdot q\right)$
\mathcal{S}TATE $R = d_H(n_E, q)$
\WHILE{$R > cL$}
\mathcal{S}TATE $q_{new} = \emph{Euclidean-Center-of-Hyperbolic-Ball }(q, \sqrt{RL})$
\mathcal{S}TATE $n_E \leftarrow \O(q_{new}, \mathcal{D})$
\IF {$d_H(n_E, q) > \sqrt{RL}$}
\mathcal{S}TATE $L \leftarrow \sqrt{RL}$
\ELSE
\mathcal{S}TATE $n_H \leftarrow n_E$
\mathcal{S}TATE $R \leftarrow d_H(n_H, q)$
\ENDIF
\ENDWHILE
\mathcal{S}TATE \textbf{Return} $n_H$
\end{algorithmic}
\end{algorithm}
\begin{theorem}
\label{thm:binary_search}
Given query $q$, and approximation constant $c > 1$, and letting $R_{initial} = d_H(q, n_E), L_{initial}$ be initial non-zero upper and lower bounds on the distance of the hyperbolic nearest neighbor to $q$, \emph{Binary-Search-HyperbolicNN } returns a $c$-approximate hyperbolic nearest neighbor in at most $\log_2 \left( \frac{\log \left( \frac{R_{initial}}{L_{initial}}\right)}{\log(c)}\right)$ rounds. The total runtime is $\mathcal{T} \cdot \log_2 \left( \frac{\log \left( \frac{R_{initial}}{L_{initial}}\right)}{\log(c)}\right)$, where $\mathcal{T}$ is the runtime for one invocation of $\O$. The storage of this algorithm is $\mathcal{S}$, where $\mathcal{S}$ is the storage requirement of $\O$.
\end{theorem}
\begin{proof}
$n^*$, the hyperbolic nearest neighbor to $q$, is always within hyperbolic distance $L$ and $R$ to $q$ in every iteration. This is true at the beginning of the algorithm: $d_H(q, n^*) \leq d_H(q, n_E) = R$. $\O$ produces a Euclidean nearest neighbor $n_E$, which means that the interior of the Euclidean ball around $q$ with radius $n_E$ is empty. Therefore $d_H(q, n^*)$ must be at least as far away from $q$ as the closest point on this ball to $q$ in hyperbolic distance. The closest. point can be expressed as $t\cdot q$ for $0< t < 1$, and satisfies $\|q - t\cdot q\| = \|d_E, q\|$. Therefore, this point is $\left(1-\frac{\|d_E -q\|}{\|q\|}\right)\cdot q$, and so $L$ as initialized in the algorithm is a valid lower bound. In the first iteration of the algorithm, we search within the hyperbolic ball around $q$ with hyperbolic radius $\sqrt{RL}$ by finding the Euclidean center to this ball and searching for the Euclidean nearest neighbor. If we find $n_E$ such that $d_H(q, n_E) \leq \sqrt{RL}$, this means that this ball is nonempty and so $n^*$ must be within hyperbolic distance $L$ and $\sqrt{RL}$. Furthermore, we have a point $n_E$ such that $L \leq d_H(q, n_E) \leq \sqrt{RL}$. Otherwise if this ball is empty then the nearest neighbor must have hyperbolic distance at least $\sqrt{RL}$ and so we update the lower threshold, $L$. Therefore, at any point in the algorithm, $L$ and $R$ represent valid upper and lower bounds for $d_H(q, n^*)$. Note also that the current $n_H$ is always a point such that $L \leq d_H(q, n_H) \leq R$. At each iteration, the square root the ratio $\frac{R}{L}$ from the previous round until we hit the termination condition that $\frac{R}{L} \leq c$, so that $n_H$ is a $c$-approximate nearest neighbor.
Let $R_i$ and $L_i$ be the upper and lower thresholds at round $i$. Then in the next round, $\frac{R_{i+1}}{L_{i+1}} \leq \sqrt{\frac{R_i}{L_i}}$. Suppose the algorithm starts off with $R_{initial}$ and $L_{initial}$. Then \emph{Binary-Search-HyperbolicNN } terminates in $\delta$ rounds, where $\left( \frac{R_{initial}}{L_{initial}} \right)^{\frac{1}{2^\delta}}\leq c$. Solving for $\delta$ yields $\delta \geq \log_2 \left( \frac{\log \left( \frac{R_{initial}}{L_{initial}}\right)}{\log(c)}\right)$.
\end{proof}
In the worst case, we establish in Lemma \ref{lem:R_L_high} that $\frac{R_{initial}}{L_{initial}}$ can be arbitrarily high. The construction is simple -- we choose (Euclidean) co-linear $n_E, n_H, q$ where $\|n_E - q\| = \|n_H - q\|$, and show that the ratio can be arbitrary bad as the points approach the edge of the disk.
Even though the ratio can become arbitrarily high as points approach the edge of the disk, for finite datasets, we prove the upper bound $\frac{R_{initial}}{L_{initial}} \leq O(\ln(\frac{1}{1-\|q\|^2}) + \ln(\frac{1}{1-\|n_E\|^2}))$. Practitioners can understand how long \emph{Binary-Search-HyperbolicNN } might take in the worst case with some prior knowledge on the largest $\|x\|^2$ for $x \in \mathcal{D}$ in their dataset. We formalize this in Lemma \ref{lem:precision}.
\begin{lemma}
\label{lem:R_L_high}
Fix large $s$, and let $\gamma, \delta$ be such that $0 < \gamma < \delta < 1$, $\delta^{s+1} < \gamma < \delta^s$, and $\frac{\delta - 2\delta^s}{\delta + \delta^s} \geq \frac{1}{2}$. Further let the query $q = (0, 1- \frac{\gamma + \delta}{2})$, $n_E = (0, 1-\gamma)$, and $n_H = (0, 1-\delta)$. Then $\frac{R_{initial}}{L_{initial}} = \Omega (s)$.
\end{lemma}
\begin{proof}
We start with the following 3 points: $n_E = (0, 1-\gamma)$, $q = \left(0, 1- \left(\frac{\gamma + \delta}{2}\right)\right)$, $n_H = (0, 1-\delta)$, where $0<\gamma <\delta < 1$. $q$ is exactly the midpoint between $n_E$ and $n_H$ in the Euclidean metric.
Now fix some very large constant $s$ where $s > 1$. Suppose that $\gamma$ is small enough that $\delta^{s+1} < \gamma < \delta^{s}$. Further suppose $s$ is large enough that $\frac{\left( \delta - 2\delta^s\right)}{\delta + \delta^s} \geq \frac{1}{2}$. We will show that when $\gamma$ and $\delta$ satisfy this regime, $d_H(n_E, q) / d_H(n_H, q) = \Omega(s)$, so to make this ratio very high, one can use a very large $s$.
From \eqref{eq:poincare_distance}, we have $d_H(n_E, q) \geq \arccosh \left(1 + \frac{2 \left( \frac{\delta - \delta^s}{2}\right)^2}{(2\delta^s)(\delta + \delta^s)} \right)$. Note that:
\begin{align*}
\frac{\left(\frac{\delta - \delta^s}{2} \right)^2}{\delta + \delta^s}&= \frac{ \left( \frac{\delta}{2}\right)^2 - 2 \left( \frac{\delta}{2}\right) \left( \frac{\delta^s}{2}\right) + \left( \frac{\delta^s}{2}\right)^2}{\delta + \delta^s}\\
&\geq \frac{\frac{\delta^2}{4} - \frac{\delta^{s+1}}{2}}{\delta + \delta^s} = \frac{\frac{\delta}{4} \left( \delta - 2\delta^s\right)}{\delta + \delta^s} \geq \frac{\delta}{8}
\end{align*}
Therefore,
$$d_H(n_E, q) \geq \arccosh \left(1 + \frac{\delta}{8\delta^s} \right) = \arccosh \left(1 + \frac{1}{8\delta^{s-1}} \right)$$
$$d_H(n_H, q) \leq \arccosh \left(1 + \frac{2 \left( \frac{\delta - \delta^{s+1}}{2}\right)^2}{(\delta^2)\left(\frac{\delta + \delta^{s+1}}{2}\right)^2} \right) \leq \arccosh \left(1+ \frac{2}{\delta^2} \right)$$
Using the identity $\arccosh(x) = \ln (x + \sqrt{x^2-1})$, we have:
$$d_H(n_E, q) \geq \ln \left(1 + \frac{1}{8\delta^{s-1}} \right) \geq \ln \left(\frac{1}{8\delta^{s-1}} \right)$$
$$d_H(n_H, q) \leq \ln \left( 2+ \frac{4}{\delta^2} \right) \leq \ln \left(\frac{6}{\delta^2} \right)$$
To conclude, we have:
$$ \frac{d_H(n_E, q)}{d_H(n_H, q)} \geq \frac{\ln \left( \frac{1}{8\delta^{s-1}}\right)}{\ln \left(\frac{6}{\delta^2} \right)} = \frac{\ln \left( \frac{1}{\delta^{s-1}}\right) + \ln \left(\frac{1}{8}\right)}{\ln \left(\frac{1}{\delta^2} \right) + \ln(6)} = \frac{s-1}{2} + o(1)$$
Therefore, $ \frac{d_H(n_E, q)}{d_H(n_H, q)} =\frac{R_{initial}}{L_{initial}} = \Omega(s)$.
\end{proof}
\begin{lemma}
\label{lem:precision}
\emph{Binary-Search-HyperbolicNN } returns a c-approximate hyperbolic nearest neighbor in at most $\log_2 ((\log_2 b)/(\log c)) + O(1)$ rounds, where $b$ is the number of bits used to represent an arbitrary $x \in \mathcal{D}$.
\end{lemma}
\begin{proof}
We show $\frac{R_{initial}}{L_{initial}} \leq O\left(\ln(\frac{1}{1-\|q\|^2}) + \ln(\frac{1}{1-\|n_E\|^2}) \right)$.
Let $q$ be the query, $n_E$ be the Euclidean nearest neighbor to $q$, and $n_H$ be the point such that $\|q - n_H\| = \|q - n_E\|$ and $d_H(q, n_H)$ is minimized. This maximizes $\frac{R_{initial}}{L_{initial}}$. Let $\epsilon = \|q - n_H\| = \|q - n_E\|$, let $\delta = 1-\|n_E\|^2$.
First we assume the case that $\|n_E\| \geq 2\epsilon$. We have: $\|q\| \geq \|n_E\| - \epsilon$, so that $1 - \|q\|^2 \leq 1-\|n_e\|^2 + 2\epsilon\|n_E\| - \epsilon^2 \leq 1 - \|n_E\|^2 + 2\epsilon = \delta + 2\epsilon$. We also have that $\|n_H\| \geq \|n_E\| - 2\epsilon$, therefore $1-\|n_H\|^2 \leq 1 - \|n_E\|^2 + 4\epsilon \|n_E\| - 4\epsilon^2 \leq \delta + 4\epsilon$. We can write $d_H(q, n_H) \geq \arccosh \left( 1 + \frac{2\epsilon^2}{(\delta + 2\epsilon)(\delta + 4\epsilon)}\right)$. Therefore, if $\delta < \epsilon$, then $d_H(q, n_H) = \Omega (1)$, and $d_H(q, n_E) = O\left(\ln(\frac{1}{1-\|q\|^2}) + \ln(\frac{1}{1-\|n_E\|^2}) \right)$. If $\delta \geq \epsilon$, then $\frac{d_H(q, n_E)}{d_H(q, n_H)} = \frac{\arccosh(1 + f_1)}{\arccosh(1+f_2)}$, where $\frac{f_1}{f_2} = \frac{\delta + 4\epsilon}{\delta} \leq 5$, so we conclude that $\frac{d_H(q, n_E)}{d_H(q, n_H)} = O(1)$ in this case.
We consider the case that $\|n_E\| < 2\epsilon$. Suppose $1-\|n_E\|^2 = \delta > \frac{1}{2}$. Then $\frac{d_H(q, n_E)}{d_H(q, n_H)} = \frac{\arccosh(1 + f_1)}{\arccosh(1+f_2)}$, where $\frac{f_1}{f_2} = \frac{1-\|n_H\|^2}{1-\|n_E\|^2} \leq 2$, so $\frac{d_H(q, n_E)}{d_H(q, n_H)} = O(1)$ in this case. When $1-\|n_E\|^2 = \delta \leq \frac{1}{2}$, it follows that $\frac{1}{\sqrt{2}} \leq \|n_E\| < 2\epsilon$, so $\epsilon > \frac{1}{2\sqrt{2}}$. Therefore, $d_H(q, n_H) \geq \arccosh(1 + \frac{1}{4}) = \Omega (1)$. Therefore \\ $\frac{d_H(q, n_E)}{d_H(q, n_H)} = O\left(\ln(\frac{1}{1-\|q\|^2}) + \ln(\frac{1}{1-\|n_E\|^2}) \right)$ in this case.
\end{proof}
\subsubsection{Integration with approximate Euclidean nearest neighbor oracles}
Since approximate nearest neighbor algorithms are heavily used, we consider \emph{Recentering-HyperbolicNN } and \emph{Binary-Search-HyperbolicNN } when powered by approximate Euclidean nearest neighbor oracles $\tO$. We show, somewhat surprisingly, that replacing the exact Euclidean nearest neighbor oracle by an approximate oracle can cause both algorithms to return points with arbitrarily bad approximation ratios. The next section shows how approximate oracles can be used to derive approximate hyperbolic nearest neighbor algorithms.
\begin{lemma}
For any $\epsilon > 0$, \emph{Recentering-HyperbolicNN } using a $(1+\epsilon)$-approximate Euclidean nearest neighbor oracle $\tO$ can return an approximate hyperbolic nearest neighbor with an arbitrarily bad approximation ratio.
\end{lemma}
\begin{proof}
Suppose $q = (0, y)$, and $n_E = (0, y+r)$ and $n_H = (0, y-r)$ for $r > 0$.
Then we have:
\[
d_H(q, n_E) = \arccosh \left( 1 + \frac{2r^2}{(1-y^2)(1-(y+r)^2)} \right)
\]
The bottom of the hyperbolic circle with radius $d_H(q, n_E)$ is a point $B = (0, y-b)$ for $0<b$ that satisfies:
\[
d_H(q, B) = d_H(q, n_E) = \arccosh \left( 1 + \frac{2b^2}{(1-y^2)(1-(y-b)^2)} \right)
\]
The Euclidean center, denoted $n_c$ is $\frac{y+r+y-b}{2} = \frac{2y+r-b}{2} = y + \frac{r-b}{2}$.
In order for \emph{Recentering-HyperbolicNN } to fail with a $(1+\epsilon)$-Euclidean oracle, we want $y-r > y + \frac{r-b}{2} + \frac{d_E(n_c, n_E)}{1+\epsilon}$, where $d_E(n_c, n_E) = \frac{r+b}{2}$. This means that we want $b > \frac{r(4+3\epsilon)}{\epsilon}$.
We want $b$ such that
\[
\frac{r^2}{1-(y+r)^2} = \frac{b^2}{1-(y-b)^2}
\]
This implies that $b = \frac{r-ry^2}{1-y^2-2ry}$. Combined with the condition that $b > \frac{r(4+3\epsilon)}{\epsilon}$, we want:
\[
\frac{1-y^2}{1-y^2-2ry} > \frac{4}{\epsilon} + 3
\]
Now we substitute in $y = 1 - \frac{\delta + \gamma}{2}$ and $r = \frac{\delta - \gamma}{2}$, and we maintain the condition that $\delta^{s+1} < \gamma < \delta^{s}$.
This implies:
\begin{align*}
\frac{1-y^2}{1-y^2-2ry} &= \frac{\delta + \gamma - \left( \frac{\gamma + \delta}{2}\right)^2}{\delta + \gamma - \left( \frac{\gamma + \delta}{2}\right)^2 - 2 \left( \frac{\delta - \gamma}{2}\right)\left(1 - \frac{\delta + \gamma}{2} \right)} \\
& \geq \frac{\delta + \delta^{s+1} - \left( \frac{\delta^{s} + \delta}{2}\right)^2}{\delta + \gamma - \left( \frac{\gamma + \delta}{2}\right)^2 - 2 \left( \frac{\delta - \gamma}{2}\right) + 2\left(\frac{\delta - \gamma}{2} \right)\left(\frac{\delta + \gamma}{2} \right)} \\
& = \frac{\delta + \delta^{s+1} - \left( \frac{\delta^{s} + \delta}{2}\right)^2}{2\gamma - \left( \frac{\gamma + \delta}{2}\right)^2 + 2\left(\frac{\delta - \gamma}{2} \right)\left(\frac{\delta + \gamma}{2} \right)} \\
& \geq \frac{\delta + \delta^{s+1} - \left( \frac{\delta^{s} + \delta}{2}\right)^2}{2\gamma + \left( \frac{\gamma + \delta}{2}\right)^2}
\end{align*}
Note that since $\left(\frac{\delta + \gamma}{2}\right)^2 = \left( \frac{\delta}{2}\right)^2 + \frac{\delta \cdot \gamma}{2} + \left( \frac{\gamma}{2}\right)^2 \leq \left( \frac{\delta}{2}\right)^2 + \frac{\delta^{s+1}}{2} + \left( \frac{\delta^s}{2}\right)^2$
We therefore have,
\[
\frac{1-y^2}{1-y^2-2ry} \geq \frac{\delta + \delta^{s+1} - \left( \frac{\delta^{s} + \delta}{2}\right)^2}{2\delta^s + \left( \frac{\delta}{2}\right)^2 + \frac{\delta^{s+1}}{2} + \left( \frac{\delta^s}{2}\right)^2} = \frac{\delta + o(\delta)}{\frac{\delta^2}{4} + o(\delta^2)} = \Theta \left(\frac{1}{\delta}\right)
\]
Suppose that $\frac{1-y^2}{1-y^2-2ry} \geq \frac{k_1}{\delta}$ for some $k_1>0$. Then we need $\delta$ such that
\[
\frac{k_1}{\delta} > \frac{4}{\epsilon} + 3
\]
This implies that $\delta < \frac{k_1 \cdot \epsilon}{4 + 3\epsilon}$. So for sufficiently small $\delta$, \emph{Recentering-HyperbolicNN } will fail to find $n_H$ during the recentering phase. Moreover, for sufficiently small $\epsilon$, given $\delta$, the ratio $d_H(q, n_E) / d_H(q, n_H)$ can be arbitrarily high. Therefore we conclude that \emph{Recentering-HyperbolicNN } with a $(1+\epsilon)$ approximate Euclidean oracle can return an answer with arbitrarily high approximation ratio.
\end{proof}
\begin{lemma}
For any $\epsilon > 0$, \emph{Binary-Search-HyperbolicNN } using a $(1+\epsilon)$-approximate Euclidean nearest neighbor oracle $\tO$ can return an approximate hyperbolic nearest neighbor with an arbitrarily bad approximation ratio.
\end{lemma}
The proof is similar and is in the appendix.
\section{Approximate Near Neighbors}
The previous section shows that \emph{Recentering-HyperbolicNN } and \emph{Binary-Search-HyperbolicNN } cannot guarantee a close hyperbolic nearest neighbor when using $\tO$, an approximate Euclidean nearest neighbor oracle. We now develop \emph{Spherical-Shell-HyperbolicNN }NS, which uses $\tO$ to return neighbors with provable guarantees on the hyperbolic approximation ratio.
Our idea is inspired by the formula for hyperbolic distance in $\mathbb{H}_r$.
\[
d_H(q, x) = \arccosh\left(1 + \frac{\|q - x\|^2}{(1-\|q\|^2)(1-\|x\|^2)}\right) \eqref{eq:poincare_distance}
\]
If 2 points $x_1, x_2$ are such that $\|x_1\| \approx \|x_2\|$, then finding the nearer neighbor to $q$ reduces to minimizing $\|q - x\|$, which is a Euclidean nearest neighbor problem. Our overall scheme divides the dataset based on their Euclidean squared distance to the origin. Each batch of points in an annulus is organized into its own data structure that $\tO$ accesses. We probe relevant batches and return the best approximate nearest neighbor that we find from the different partitions.
In the preprocessing to divide $\mathcal{D}$, we take the multiplicative width of each annulus $w > 1$, and put into the $i$-th annulus, or partition, all data points $x$ such that $w^{i-1} \leq \frac{1}{1-\|x\|^2} \leq w^{i}$. The width $w$ controls the granularity of the $\ell_2$ norm at which we divide the dataset.
The nearest neighbor algorithm, \emph{Spherical-Shell-HyperbolicNN } probes different annuli using $\tO$ and returns the nearest hyperbolic neighbor from among $n_F$ returned by $\tO$ applied to each partition. One important detail is which partitions to probe and in which order. Algorithm \ref{alg:nn_banded_alg} offers one strategy. We first probe the band that the query falls into, $i$. Then we maintain two lists. The first list contains the indices higher than $i$ in sorted order, the other list contains the lowest. We choose from the top of the two lists, based on which choice maximizes the radius of the hyperbolic ball around $q$ that is completely covered by the union of bands probed so far as well as the new band under consideration. This is implemented in \emph{Check-Intersection }MaxRadiusNS. We terminate based on \emph{Check-Intersection }NS, which takes $n_H$, the best hyperbolic nearest neighbor found so far, and checks if there exists $x \in \mathbb{H}_r$ such that $w^{b-1} \leq \frac{1}{1-\|x\|^2}\leq w^b$ and also belongs to $\mathcal{B}_H(q, d_H(q, n_H))$.
\begin{algorithm}[!htp]
\caption{\emph{Spherical-Shell-HyperbolicNN }}
\label{alg:nn_banded_alg}
\begin{algorithmic}[1]
\mathbb{R}EQUIRE{query $q$, approximate Euclidean nearest neighbor oracle $\tO$, partitions $B$ with width $w$}
\mathcal{S}TATE $i \leftarrow \left\lceil\frac{-\log (1-\|q\|^2)}{\log w}\right\rceil$ \text{ // the annulus that the query belongs to}
\mathcal{S}TATE $ProbingListTop \leftarrow [i+1, i+2, \ldots]$
\mathcal{S}TATE \text{ // the list of partitions arranged in probing order}
\mathcal{S}TATE $ProbingListBottom \leftarrow [i-1, i-2, \ldots]$
\mathcal{S}TATE \text{ // the list of partitions arranged in probing order}
\mathcal{S}TATE $n_H \leftarrow \tO(q, B[i])$ \text{ // current best nearest neighbor candidate}
\mathcal{S}TATE $dist_H \leftarrow d_H(q, n_H)$
\mathcal{S}TATE \text{ // hyperbolic distance of current best nearest neighbor candidate}
\WHILE {$\emph{Check-Intersection }(q, n_H, w, ProbingListTop[0])$ or $\emph{Check-Intersection }(q, n_H, w, ProbingListBottom[0])$}
\mathcal{S}TATE ${ band = \emph{Check-Intersection }MaxRadius(q, n_H, w, ProbingListTop[0], ProbingListBottom[0])}$
\mathcal{S}TATE $n_F \leftarrow \tO(q, B[band])$
\IF {$d_H(q, n_F) < dist_H$}
\mathcal{S}TATE $dist_H \leftarrow d_H(q, n_F)$
\mathcal{S}TATE $n_H \leftarrow n_F$
\ENDIF
\ENDWHILE
\mathcal{S}TATE \textbf{Return} $n_H$
\end{algorithmic}
\end{algorithm}
The routine calculations for $\emph{Check-Intersection }MaxRadius$ and $\emph{Check-Intersection }$ use elementary properties of hyperbolic geometry and are in the appendix.
\emph{Spherical-Shell-HyperbolicNN }NS, Algorithm \ref{alg:nn_banded_alg} provides the following approximation guarantee:
\begin{theorem}
Using a $(1+\epsilon)$-Euclidean nearest neighbor oracle $\tO$ and a dataset split with a multiplicative width of $w$, \emph{Spherical-Shell-HyperbolicNN } returns a hyperbolic approximate nearest neighbor $n_H$ to any query $q$ such that $d_H(q, n_H) \leq \sqrt{w}(1+\epsilon) d_H(q, n^*)$, where $n^*$ is the exact hyperbolic nearest neighbor.
\end{theorem}
\begin{proof}
The true hyperbolic nearest neighbor, $n^*$ is organized into a bucket $j$ that \emph{Spherical-Shell-HyperbolicNN } is guaranteed to probe. Suppose that instead of finding $n^*$, the algorithm finds $n_H$ in bucket $j$. The hyperbolic distance between the query $q$ and $n_H$, $D$, is upper bounded by
\[
D \leq \arccosh \left(1 + \frac{2\|q - n_H\|^2 \cdot w^j}{(1-\|q\|^2)} \right)~,
\]
where the inequality comes from the guarantee that all elements $y$ in bucket $j$ satisfy $w^{j-1} \leq \frac{1}{1-\|y\|^2} \leq w^j$. This also implies that $\frac{\|q-n_H\|^2}{1-\|q\|^2} \geq \frac{\cosh(D) - 1}{2w^j}$.
In the worst case, the true nearest neighbor $n^*$ is such that $\|q-n^*\|$ is much smaller than $\|q-n_H\|$ and also $\frac{1}{1-\|n^*\|^2}$ is much smaller than $\frac{1}{1-\|n_H\|^2}$. To make $\|q-n^*\|$ small, the worst case is that $n^*$ is actually the nearest neighbor in bucket $j$ to $q$. However, the guarantee of the approximate Euclidean oracle is that $\|q-n_H\| \leq (1+\epsilon) \|q-n^*\|$, so that $\|q-n^*\|^2 \geq \frac{\|q-n_H\|^2}{(1+\epsilon)^2}$. We also have that $d_H(q, n^*) = \arccosh\left(1 + \frac{2\|q - n^*\|^2}{(1-\|q\|^2)(1-\|n^*\|^2)}\right) \geq \arccosh\left(1 + \frac{\cosh(D)-1}{w(1+\epsilon)^2}\right)$.
Now we want to analyze $\frac{D}{\arccosh\left(1 + \frac{\cosh(D)-1}{w(1+\epsilon)^2}\right)}$. \\
\begin{align*}
\arccosh\left(1 + \frac{\cosh(D)-1}{w(1+\epsilon)^2}\right) &= \arccosh \left(1 + \frac{\frac{e^D + e^{-D}}{2}-1}{w(1+\epsilon)^2} \right) \\
& = \arccosh \left(1 + \frac{\sum\limits_{i=1}^\infty \frac{D^{2i}}{(2i)!}}{w(1+\epsilon)^2} \right) \geq \left( 1 + \sum\limits_{i=1}^\infty \frac{\left(\frac{D}{\sqrt{w}(1+\epsilon)} \right)^{2i}}{(2i)!}\right)\\
& = \arccosh(1 + \cosh(\frac{D}{\sqrt{w}(1+\epsilon)} -1) = \frac{D}{\sqrt{w}(1+\epsilon)}
\end{align*}
Therefore, we conclude that $d_H(q, n_H) \leq \sqrt{w}(1+\epsilon)d_H(q, n^*)$.
\end{proof}
The runtime of \emph{Spherical-Shell-HyperbolicNN } depends on the number of partitions that are probed, which we now analyze.
We first define $b_1, b_2, i_q$. Let $x = \argmax\limits_{z \in \mathcal{B}_H(q, d_H(q, n_H))} \|z\|$, and let $b_1 = \left\lceil\frac{-\log(1-\|x\|^2)}{\log(w)}\right\rceil$ denote the index of the partition that $x$ falls into, which is also the largest index that intersects this hyperbolic ball. Let $y = \argmin\limits_{z \in \mathcal{B}_H(q, d_H(q, n_H))} \|z\|$, and let $b_2 = \left\lfloor\frac{-\log(1-\|y\|^2)}{\log(w)}\right\rfloor$ be the index of the partition that $y$ falls into, which is also the smallest index possible that intersects the hyperbolic ball when 0 is not contained in this hyperbolic ball. When 0 is contained in the hyperbolic ball, the smallest partition index that intersects the hyperbolic ball is 1.
\begin{lemma}
For a query $q$, suppose that $n_H$ is the approximate hyperbolic nearest neighbor returned by \emph{Spherical-Shell-HyperbolicNN }. Further suppose that $d_H(0, q) > d_H(q, n_H)$. Then the number of partitions probed is $b_1 - b_2 + 1$.
\end{lemma}
\begin{lemma}
For a query $q$, suppose that $n_H$ is the approximate hyperbolic nearest neighbor output of \emph{Spherical-Shell-HyperbolicNN }. Further suppose that $d_H(0, q) \leq d_H(q, n_H)$. Then the number of partitions probed is $b_1$.
\end{lemma}
\emph{Spherical-Shell-HyperbolicNN } generalizes to return $K$ nearest neighbors with the worst case approximation guarantee for each neighbor if $\emph{Check-Intersection }MaxRadius$ and $\emph{Check-Intersection }$ use the distance of the $K$-th best nearest neighbor found so far.
One can design variants of \emph{Spherical-Shell-HyperbolicNN } that differ in the probing sequence and probing criteria. We explore a randomized probing order in the appendix. This scheme uses a $(1+\epsilonilon, R)$-approximate Euclidean Near Neighbor Decision Oracle that gives a Yes/No answer for whether there is an element within distance $R$ to any point. We only probe a partition if the Decision Oracle says there is definitely a nearer neighbor in that partition than the current best.
We show this variant has the same $\sqrt{w}(1+\epsilon)$-approximation guarantee as in \emph{Spherical-Shell-HyperbolicNN }NS, and will fully search (using $\tilde{O}$) $\log(B)$ partitions in expectation, though the Decision Oracle could be applied to all partitions.
Lastly, we show in the appendix that even with an exact Euclidean oracle $\O$, \emph{Spherical-Shell-HyperbolicNN } is not guaranteed to return an exact hyperbolic nearest neighbor.
\section{Evaluation}
We compare the techniques we develop to existing solutions. \cite{krauthgamer2006algorithms} presents an idea for hyperbolic nearest neighbor search but omits key implementation details (and we were unable to extract an efficient implementation from their proof). To our knowledge, the only other practical algorithms for this problem are nearest neighbor graph methods \cite{subramanya2019rand} \cite{malkov2018efficient} \cite{fu2019fast}, where the graph is constructed using hyperbolic distance. We compare the effectiveness of our technique against Vamana, a graph method that exhibited superior performance against the other in-class methods in the evaluation in \cite{subramanya2019rand}. As this family of algorithms does not come with any guarantee on the search quality, our experiments use a fixed sampling budget and compare the nearest neighbor found by the different algorithms under this budget. For the algorithms developed in this paper, if during the search the algorithm terminates before hitting this budget, we stop early. For the graph-based method, if the graph search terminates before hitting the budget, we initialize another round of search by starting at a different random initial point and search until we hit the budget.
We use a low-dimensional and a high dimensional dataset. Our queries are points that we withheld from the dataset. We solve the $K$-nearest neighbor ($K$-NN) problem for $K = 1,5$. We report the average recall for our batch of queries, defined as \# of the $K$ true nearest neighbors found / K. We also report the average approximation ratios and the max approximation ratio, where for $K > 1$, the ratio is computed pointwise: $d_H(q, n_k) / d_H(q, n^*_k)$. for each $k \in [K]$. For Vamana, we experimented with a range of hyperparameters and report the most favorable results. Our results largely show that our simple algorithms perform very well against Vamana. We find more exact nearest neighbors and we report better approximation ratios on average when we do not find the exact nearest neighbor. We also report the CPU running time of each our algorithms.
\subsection{Low dimensional hyperbolic embeddings}
For the low-dimensional regime, we embed into 10 dimensions a dataset of 82,115 words from the WordNet noun hierarchy using the source code in \cite{nickel2017poincare}. As a sanity check, our trained embeddings achieve a rank of 4.739 and a MAP score of 0.811 in the reconstruction evaluation criteria as described in \cite{nickel2017poincare}, which is close to their reported results.
Since we have exact oracles in this regime, we first consider whether it is efficient to use \emph{Recentering-HyperbolicNN } for real-world datasets by evaluating the number of calls to $\O$. We use a standard kd-tree\footnote{Source code for the kd-tree can be found at https://github.com/stefankoegl/kdtree.} as the underlying Euclidean oracle. To further optimize, we make the minor modification to the classic kd-tree -- whenever the algorithm solves for the Euclidean distance between a data point and $q$, we also solve for the hyperbolic distance. The traversal and termination criteria are all based on Euclidean distance; our modification also keeps track of the closest hyperbolic neighbor seen so far and returns that point. The analysis that we develop in this paper assuming a black box oracle still holds in this modified setting.
In 2 independent trials, we withhold 800 queries from the dataset and record the number of calls to $\O$ that \emph{Recentering-HyperbolicNN } uses to find the exact nearest neighbor. We see that the number is low (Table \ref{table_kd_recenter}). Therefore, we use \emph{Recentering-HyperbolicNN } in our subsequent experiments to evaluate against Vamana.
\begin{table}[h]
\caption{Statistics of number of calls to $\O$ in \emph{Recentering-HyperbolicNN } for sets of 800 queries}
\label{table_kd_recenter}
\centering
\begin{tabular}{ |p{0.8cm}||p{3cm}|p{0.8cm}|p{0.8cm}|p{0.8cm}| }
\hline
Trial & Average \# of calls to $\O$ & SD &Min &Max \\
\hline
1 & 2.36 &0.51& 2 &4\\
2 & 2.3 & 0.49 &2 & 4\\
\hline
\end{tabular}
\end{table}
To compare against Vamana, we withhold 50 queries from the dataset. We report the results for the $1$-NN problem in Table~\ref{table:kd_tree_vs_graph}, and $5$-NN problem in Table ~\ref{table:kd_tree_vs_graph_5_NN}. We report a second trial with the same experimental setup in Tables \ref{table:kd_tree_vs_graph_2_trial} and \ref{table:kd_tree_vs_graph_5_NN_trial_2} We vary the budget of datapoints that the algorithm is able to search: 100, 500, 1000. After some hyperparameter tuning for Vamana, we use $L = 10$ and $R = 10$ and $\alpha = 1.5$, see \cite{subramanya2019rand} for more details on these hyperparameters. Our results show that \emph{Recentering-HyperbolicNN } with kd-tree generally finds more exact nearest neighbors than Vamana and approximate near neighbors with lower approximation ratios. For $K > 1$, we use the KD tree to return $K$ nearest Euclidean neighbors, and we first recenter based on the nearest neighbor. When that termination criteria is hit, then we recenter based on the $K$-th nearest neighbor.
\begin{table}[h]
\caption{Trial 1. \emph{Recentering-HyperbolicNN } vs Vamana for $1$-NN search in the 10-dimensional noun hierarchy dataset}
\centering
\scalebox{0.8}{
\begin{tabular}{|c|ccc|ccc|}
\hline
& \multicolumn{3}{c|} {\emph{Recentering-HyperbolicNN }} & \multicolumn{3}{c|}{Vamana} \\
\hline
\# Samples & Recall & Avg Ratio & Avg Max & Recall & Avg Ratio & Avg Max\\
\hline
100 &0.46&1.10 &1.66 &0.46 & 1.21 & 2.61 \\
\hline
500 & 0.7&1.036 & 1.37 &0.52 & 1.18 & 2.61 \\
\hline
1000 &0.84& 1.017& 1.37 &0.52 & 1.19 & 2.61 \\
\hline
\end{tabular}}
\label{table:kd_tree_vs_graph}
\end{table}
\begin{table}[h]
\caption{Trial 2. \emph{Recentering-HyperbolicNN } vs Vamana for $1$-NN search in the 10-dimensional noun hierarchy dataset}
\centering
\scalebox{0.8}{
\begin{tabular}{|c|ccc|ccc|}
\hline
& \multicolumn{3}{c|} {\emph{Recentering-HyperbolicNN }} & \multicolumn{3}{c|}{Vamana} \\
\hline
\# Samples & Recall & Avg Ratio & Avg Max & Recall & Avg Ratio & Avg Max\\
\hline
100 &0.56&1.10 &1.69 &0.6 & 1.38 & 6.42 \\
\hline
500 & 0.78&1.035 & 1.59 &0.6 & 1.372 & 6.42 \\
\hline
1000 & 0.9 & 1.018& 1.27 &0.64& 1.29 & 6.40 \\
\hline
\end{tabular}
}
\label{table:kd_tree_vs_graph_2_trial}
\end{table}
\begin{table}[h]
\caption{Trial 1. \emph{Recentering-HyperbolicNN } vs Vamana for $5$-NN search in the 10-dimensional noun hierarchy dataset}
\centering
\scalebox{0.8}{
\begin{tabular}{|c|ccc|ccc|}
\hline
& \multicolumn{3}{c|} {\emph{Recentering-HyperbolicNN }} & \multicolumn{3}{c|}{Vamana} \\
\hline
\# Samples & Recall & Avg Ratio & Avg Max & Recall & Avg Ratio & Avg Max\\
\hline
100 & 0.23 & 1.15&1.24 & 0.420 &1.132 &1.24 \\
\hline
500 &0.48 &1.07 &1.12 &0.424 & 1.12 &1.23 \\
\hline
1000 & 0.59&1.04 &1.09 &0.452 &1.10 &1.20 \\
\hline
\end{tabular}}
\label{table:kd_tree_vs_graph_5_NN}
\end{table}
\begin{table}[h]
\caption{Trial 2. \emph{Recentering-HyperbolicNN } vs Vamana for $5$-NN search in the 10-dimensional noun hierarchy dataset}
\centering
\scalebox{0.8}{
\begin{tabular}{|c|ccc|ccc|}
\hline
& \multicolumn{3}{c|} {\emph{Recentering-HyperbolicNN }} & \multicolumn{3}{c|}{Vamana} \\
\hline
\# Samples & Recall & Avg Ratio & Avg Max & Recall & Avg Ratio & Avg Max\\
\hline
100 &0.312 &1.173 &1.25 & 0.576 &1.21 &1.39\\
\hline
500 &0.584 & 1.07&1.12 &0.596 &1.20 &1.38\\
\hline
1000 &0.687 &1.044 &1.07 &0.576 &1.20 &1.39\\
\hline
\end{tabular}}
\label{table:kd_tree_vs_graph_5_NN_trial_2}
\end{table}
\subsection{Approximate nearest neighbors for high dimensional hyperbolic embeddings}
For the high dimensional regime, we use provided embeddings from \cite{de2018representation} constructed using a higher dimensional extension of Sarkar's embedding algorithm \cite{sarkar2011low}. We use a dataset of 63,000 embeddings in 100 dimensions from the WordNet Hypernym noun hierarchy. Our Euclidean approximate nearest neighbor algorithm is the random hyperplane based scheme in \cite{datar2004locality}. We draw random hyperplanes uniformly from the unit sphere. For a random normal hyperplane $r$, the hash value of an element $x$ is $\frac{r \cdot x}{g}$, where $g$ is a granularity constant that determines how many equi-width segments we want to split the line segment $(-1, 1)$ into. As described in \cite{datar2004locality}, points that are close together tend to fall into the same segment.
We use \emph{Spherical-Shell-HyperbolicNN } with width $w = 3$, and 25 bands for extra tolerance. Each band $i$ containing normalized elements $x$ such that $3^{i-1} \leq \frac{1}{1-\|x\|^2} \leq 3^i$ is organized into an LSH data structure that uses 5 tables, with 15 random normalized hyperplanes per table, and with granularity $g=\min\{3^i, 10000\}$. We choose granularities based on data characteristics; in locality sensitive hashing, bucket widths are proportional to typical Euclidean nearest neighbor distances scaled by an appropriate function of the dimension and the number of random hyperplanes. The hyperplanes used for the LSH tables of each partition are the same. We probe buckets within distance 1 of the query bucket.
Tables \ref{table:lsh_graph_experiments} and \ref{table:lsh_graph_experiments_5_NN} give the results for 49 queries withheld for the $1$-KNN and the $5$-KNN problems respectively. Tables \ref{table:lsh_graph_experiments_trial_2} and \ref{table:lsh_graph_experiments_5_NN_trial_2} give the results for a second trial of 38 queries withheld.
After tuning for Vamana, we use $L = 40, R = 20, \alpha = 1.5$. Our results show that with \emph{Spherical-Shell-HyperbolicNN } with LSH generally finds much more exact nearest neighbors than Vamana. Interestingly, for this type of very structured Sarkar embeddings, we outperform Vamana by a larger margin than for the trained embeddings in the previous experiment.
\begin{table}[h]
\centering
\caption{Trial 1. \emph{Spherical-Shell-HyperbolicNN } vs Vamana for $1$-NN in the 100-dimensional noun hierarchy dataset}
\scalebox{0.8}{
\centering
\begin{tabular}{|c|ccc|ccc|}
\hline
& \multicolumn{3}{c|} {\emph{Spherical-Shell-HyperbolicNN }} & \multicolumn{3}{c|}{Vamana} \\
\hline
\# Samples & Recall & Avg Ratio & Avg Max & Recall & Avg Ratio& Avg Max \\
\hline
100 & 0.43 & 2.01 &8.99 &0.04 & 3.66 & 8.68 \\
\hline
500 &0.71 &1.39 & 5.91 &0.18 & 2.001 & 4.49 \\
\hline
1000 & 0.90& 1.053&1.81 &0.39 & 1.52 & 3.50 \\
\hline
\end{tabular}}
\label{table:lsh_graph_experiments}
\end{table}
\begin{table}[h]
\caption{Trial 2. \emph{Spherical-Shell-HyperbolicNN } vs Vamana for $1$-NN in the 100-dimensional noun hierarchy dataset}
\centering
\scalebox{0.8}{
\begin{tabular}{|c|ccc|ccc|}
\hline
& \multicolumn{3}{c|} {\emph{Spherical-Shell-HyperbolicNN }} & \multicolumn{3}{c|}{Vamana} \\
\hline
\# Samples & Recall & Avg Ratio & Avg Max & Recall & Avg Ratio& Avg Max \\
\hline
100 &0.58 &1.63 & 6.05 &0.05 & 3.413 & 8.53 \\
\hline
500 &0.71 &1.28 & 4.74 &0.16 & 1.798& 5.06 \\
\hline
1000 &0.92 &1.08 &3.68&0.32 & 1.55 & 3.98 \\
\hline
\end{tabular}
}
\label{table:lsh_graph_experiments_trial_2}
\end{table}
\begin{table}[h]
\centering
\caption{Trial 1. \emph{Spherical-Shell-HyperbolicNN } vs Vamana for $5$-NN in the 100-dimensional noun hierarchy dataset}
\scalebox{0.8}{
\centering
\begin{tabular}{|c|ccc|ccc|}
\hline
& \multicolumn{3}{c|} {\emph{Spherical-Shell-HyperbolicNN }} & \multicolumn{3}{c|}{Vamana} \\
\hline
\# Samples & Recall & Avg Ratio & Avg Max & Recall & Avg Ratio& Avg Max \\
\hline
100 & 0.32 &1.57 &2.19 &0.016 &2.76 & 3.76 \\
\hline
500 &0.65 & 1.20& 1.41 & 0.09&1.73 & 2.35 \\
\hline
1000 &0.81 &1.052 & 1.12&0.187 & 1.40 & 1.85 \\
\hline
\end{tabular}}
\label{table:lsh_graph_experiments_5_NN}
\end{table}
\begin{table}[h]
\centering
\caption{Trial 2. \emph{Spherical-Shell-HyperbolicNN } vs Vamana for $5$-NN in the 100-dimensional noun hierarchy dataset}
\scalebox{0.8}{
\centering
\begin{tabular}{|c|ccc|ccc|}
\hline
& \multicolumn{3}{c|} {\emph{Spherical-Shell-HyperbolicNN }} & \multicolumn{3}{c|}{Vamana} \\
\hline
\# Samples & Recall & Avg Ratio & Avg Max & Recall & Avg Ratio& Avg Max \\
\hline
100 & 0.410&1.40& 1.75&0.063 & 2.79 &3.66 \\
\hline
500 &0.668 &1.22&1.36 &0.147 & 1.65 & 2.13 \\
\hline
1000 &0.784 &1.14&1.244 &0.236 & 1.521 & 1.97 \\
\hline
\end{tabular}}
\label{table:lsh_graph_experiments_5_NN_trial_2}
\end{table}
\subsection{Running Time}
We report the running time ratio for the $5$-KNN problem in Table \ref{table:running_time_5nn}, where the ratio is defined as the time for our techniques / Vamana's running time (so lower is better). Overall, our methods are faster than Vamana. This difference is likely because our algorithms have termination criteria that may not exhaust the given sampling budget, and so we stop early, whereas for the graph based Vamana, we maximize the budget. In the latter case, we do so because Vamana (and other in-class graph algorithms \cite{malkov2018efficient} \cite{fu2019fast}) perform better when the graph is searched multiple times using different initial points (even so, there are no theoretical guarantees).
\begin{table}[h]
\caption{Running time Ratios}
\centering
\scalebox{1}{
\begin{tabular}{|c|ccc|ccc|}
\hline
& \multicolumn{3}{|c|} {Low dimensional} & \multicolumn{3}{c|}{High dimensional} \\
\hline
\# samples&100& 500 &1000 & 100& 500 &1000\\
\hline
Ratio &0.07 & 0.03 & 0.02 & 0.017 & 0.006 &0.0018 \\
\hline
\end{tabular}}
\label{table:running_time_5nn}
\end{table}
\section{Conclusion}
We consider the problem of nearest neighbor search for hyperbolic embeddings. We give theoretical guarantees and hardness results for our techniques. Experimental validation shows the effectiveness of our techniques against baseline methods.
\appendix
\section{Details for \emph{Euclidean-Center-of-Hyperbolic-Ball }NS}
We now provide the helper routine to recenter the hyperbolic ball to its Euclidean center, \emph{Euclidean-Center-of-Hyperbolic-Ball }NS. The reasoning that \emph{Euclidean-Center-of-Hyperbolic-Ball } will return the Euclidean center of the hyperbolic ball is as follows:
\begin{itemize}
\item Hyperbolic distance is additive on the line.
\item $t_1c_H, t_2c_H$ and $c_H$ are collinear. Moreover, $d_H(t_1c_H, c_H) = d_H(t_2c_H, c_H) = r$ and therefore, $d_H(t_1c_H, t_2c_H) = 2r$ and so $t_1c_H$ and $t_2c_H$ are points on the sphere whose distance achieves the largest possible according to the hyperbolic metric, and so they form the endpoints of a line segment that passes through the center of the Euclidean circle. So we can take their average to find the center.
\end{itemize}
\begin{algorithm}[h]
\caption{\emph{Euclidean-Center-of-Hyperbolic-Ball }}
\begin{algorithmic}[1]
\mathbb{R}EQUIRE{hyperbolic center $c_H$, radius of hyperbolic ball $r$}
\IF {$c_H = \vec{0}$}
\mathcal{S}TATE \textbf{Return} $\vec{0}$
\ENDIF
\mathcal{S}TATE Find scalar $t_1$ such that $t_1 \|c_H\|_2 = \tanh \left( \frac{d_H(0, c_H) + r}{2} \right)$
\mathcal{S}TATE Find scalar $t_2$ such that $t_2 \|c_H\|_2 = \tanh \left( \frac{d_H(0, c_H) - r}{2} \right)$
\mathcal{S}TATE \textbf{Return} $\frac{t_1c_H + t_2c_H}{2}$
\end{algorithmic}
\end{algorithm}
\subsubsection{Best Case Configuration for \emph{Recentering-HyperbolicNN }}
A best case configuration is the following. The query, $q$, is point $(0, 0.99)$. Suppose now that the true hyperbolic nearest neighbor, $n_H$, is at point $(0,.981)$ and there is another point $n_E$, at $(0, 0.998)$, which is the Euclidean nearest neighbor to $q$.
At the first iteration, the Euclidean nearest neighbor oracle $\O$ returns $n_E$. Then the hyperbolic circle radius is:
\[d_H(q, n_E) = \arccosh \left(1 + \frac{2 \|q-n_E\|^2}{(1-\|q\|^2)(1-\|n_E\|^2)} \right)\]
The other boundary of the hyperbolic ball in the direction of the query $q$, denoted $n_B$ is a point of the form $(0, b)$. We solve for $b$ by noticing that $n_B$ satisfies:
\begin{align*} d_H(q, n_B) &= \arccosh \left(1 + \frac{2 \|q-n_B\|^2}{(1-\|q\|^2)(1-\|n_B\|^2)} \right) \\
& = \arccosh \left(1 + \frac{2 (0.99 - b)^2}{(1-(0.99)^2)(1-b^2)} \right)
\end{align*}
Equating the expression to $d_H(q, n_E)$ gives us that $b \approx .912252$.
\\\\
Therefore, the Euclidean center of this hyperbolic circle, denoted $q_{new}$, is $(0, 0.9551260)$.
\\\\
Now suppose additionally we have $k-2$ points on the $y$-axis between $.912252$ and $.928$, for arbitrary $k$. Clearly then $n_E$ is the $k$-th hyperbolic nearest neighbor of $q$ but \emph{Recentering-HyperbolicNN } will return the hyperbolic nearest neighbor in 3 rounds.
\section{Integration with approximate Euclidean nearest neighbor oracles}
\begin{lemma}
For any $\epsilon > 0$, \emph{Binary-Search-HyperbolicNN } using a $(1+\epsilon)$-approximate Euclidean nearest neighbor oracle $\tO$ can return an approximate hyperbolic nearest neighbor with an arbitrarily bad approximation ratio.
\end{lemma}
\begin{proof}
As before, let
$$n_E = (0, 1-\gamma)$$
$$q = \left(0, 1- \left(\frac{\gamma + \delta}{2}\right)\right)$$
$$n_H = (0, 1-\delta)$$
Suppose that $\frac{d_H(q, n_E)}{d_H(q, n_H)} = S$ for some very high $S$. Then we want to show that if $\delta$ is sufficiently high, \emph{Binary-Search-HyperbolicNN } will return $n_E$ and fail to find $n_H$, leading to a bad approximation ratio of $S$.
Clearly, $RL = S(d_H(q, n_H))^2$ in this case, so $\sqrt{RL} = \sqrt{S}d_H(q, n_H)$. We want to find $T_1 = (0, t_1)$ and $T_2 = (0, t_2)$ such that $d_H(q, T_1) = d_H(q, T_2) = \sqrt{RL}$. \emph{Binary-Search-HyperbolicNN } will call the $(1+\epsilon)$-Euclidean oracle to find the nearest neighbor of $n_c = \frac{T_1 + T_2}{2}$.
For clarity, let's say that $q = (0, y), n_E = (0, y+r), n_H = (0, y-r)$, where $y>0, r>0$.
For \emph{Binary-Search-HyperbolicNN } to fail, the condition we want is:
\[
y - r > \frac{t_1 + t_2}{2} + \frac{d_E(n_E, n_c)}{1+\epsilon} = \frac{t_1 + t_2}{2} + \frac{y + r - \frac{t_1 + t_2}{2}}{1+\epsilon}
\]
This condition is equivalent to:
\[
\frac{t_1 + t_2}{2} < \frac{\epsilon(y-r)-2r}{\epsilon}
\]
Let $D = \sqrt{RL}$. One can calculate that
\[
t_1 = \frac{\sinh\left(\frac{D}{2}\right) - y \cosh \left( \frac{D}{2}\right)}{y\sinh\left(\frac{D}{2}\right) - \cosh \left( \frac{D}{2}\right)}
\]
\[
t_2 = \frac{\sinh\left(\frac{D}{2}\right) + y \cosh \left( \frac{D}{2}\right)}{y\sinh\left(\frac{D}{2}\right) + \cosh \left( \frac{D}{2}\right)}
\]
Therefore, we have:
\begin{align*}
\frac{t_1 + t_2}{2} &= \frac{y \left( \sinh^2 \left( \frac{D}{2}\right) - \cosh^2 \left( \frac{D}{2} \right) \right)}{y^2 \sinh^2 \left( \frac{D}{2}\right) - \cosh^2 \left( \frac{D}{2}\right)} \\
&= \frac{y}{\cosh^2 \left( \frac{D}{2}\right) - y^2 \sinh^2 \left( \frac{D}{2}\right)} \\
& = \frac{y}{\cosh^2 \left( \frac{D}{2}\right) - y^2 \left( \cosh^2 \left( \frac{D}{2}\right) - 1\right)} \\
& = \frac{y}{(1-y^2)\cosh^2 \left( \frac{D}{2}\right) + y^2 } \\
& = \frac{2y}{(1-y^2)\left( 1+ \cosh(D)\right) + 2y^2 } \\
& = \frac{2y}{1-y^2 + \cosh(D)(1-y^2) + 2y^2} \\
& = \frac{2y}{1+y^2 + \cosh(D)(1-y^2)} \\
& \leq \frac{2y}{1+y^2 + (1-y^2)\frac{e^D}{2}} \\
& \leq \frac{2y}{(1-y^2)(1 + \frac{e^D}{2})} \\
& \leq \frac{4y}{(1-y^2)(e^D)}
\end{align*}
Note that $D = \sqrt{RL} = \sqrt{S} d_H(q, n_H)$.
Remember that we have:
\begin{align*}
d_H(n_H, q) &= \arccosh \left(1+ \frac{2 \left(\frac{\delta - \gamma}{2}\right)^2}{\left(2\delta - \delta^2\right)\left(\gamma + \delta - \left(\frac{\gamma + \delta}{2}\right)^2\right)} \right) \\
& \geq \arccosh \left(1+ \frac{2 \left(\frac{\delta - \gamma}{2}\right)^2}{\left(2\delta\right)\left(\gamma + \delta\right)} \right) \\
& \geq \arccosh \left( 1 + \frac{1}{\delta} \cdot \frac{\delta}{8}\right) \\
& \geq \arccosh \left( 1 + \frac{1}{8} \right) \geq 0.49
\end{align*}
where we again use that $\gamma$ is sufficiently small that $\frac{\left( \delta - 2\gamma\right)}{\delta +\gamma} \geq \frac{1}{2}$.
This implies that $D \geq 0.49\sqrt{S}$, so $e^D \geq e^{ 0.49\sqrt{S}}$.
So we want:
\[
\frac{4y}{(1-y^2)(e^D)} \leq \frac{\epsilon(y-r)-2r}{\epsilon} = y-r - \frac{2}{\epsilon}r
\]
This is equivalent to:
\[
r\left(1+ \frac{2}{\epsilon}\right) \leq y\left(1-\frac{4}{(1-y^2)(e^D)}\right)
\]
Remember that $r = \frac{\delta - \gamma}{2} < \frac{\delta}{2}$, so we have:
\[
r\left(1+ \frac{2}{\epsilon}\right) \leq \frac{3\delta}{2\epsilon}
\]
Now to focus on the right hand side, if we have:
\[
\frac{4}{e^D} < \frac{1}{2} (1-y^2) ~,
\]
then we have
\[
y\left(1-\frac{4}{(1-y^2)(e^D)}\right) \geq \frac{y}{2}
\]
Also we can say that $y = 1- \left(\frac{\gamma + \delta}{2}\right) > \frac{1}{2}$, so that $y\left(1-\frac{4}{(1-y^2)(e^D)}\right) > \frac{1}{4}$.
Then for a given $(1+\epsilon)$-approximate Euclidean oracle, as long as $\delta$ is small enough that $\frac{3\delta}{2\epsilon} \leq \frac{1}{4}$ or $\delta < \frac{\epsilon}{6}$, then \emph{Binary-Search-HyperbolicNN } will fail.
Now to see how to satisfy the constraint that $\frac{4}{e^D} < \frac{1}{2} (1-y^2)$.
Note that
\begin{align*}
\frac{1}{2} (1-y^2) &= \frac{1}{2} \left( \delta + \gamma - \left( \frac{\gamma + \delta}{2}\right)^2\right) \\
&\geq \frac{1}{2}\left( \frac{\gamma + \delta}{2}\right)^2 \\
& \geq \frac{\delta^2}{8}
\end{align*}
From before, we had that $\frac{4}{e^D} \leq \frac{4}{e^{ 0.49\sqrt{S}}}$.
Then a sufficient condition is that $S$ is large enough that $\frac{4}{e^{ 0.49\sqrt{S}}} \leq \frac{\delta^2}{8}$.
\end{proof}
\section{\emph{Spherical-Shell-HyperbolicNN }}
\subsection{Details for \emph{Spherical-Shell-Partition }NS}
We first describe the partitioning algorithm to divide the dataset into bands based on Euclidean norm. \emph{Spherical-Shell-Partition }NS, Algorithm \ref{alg:nn_make_bands} is the formal pre-processing algorithm to divide the dataset. The algorithm works by taking in the largest possible norm that one wishes to support; for a given dataset, this could be the norm of the largest data point or a norm slightly higher than that for extra tolerance, as well as the multiplicative width of each annulus $w$, for $(w > 1)$. The width $w$ controls the granularity at which we divide the dataset based on $1 - \|x\|^2$. The $i$-th annulus, or partition, contains all data points $x$ such that $w^{i-1} \leq \frac{1}{1-\|x\|^2} \leq w^{i}$.
\begin{algorithm}[ht]
\caption{\emph{Spherical-Shell-Partition }}
\label{alg:nn_make_bands}
\begin{algorithmic}[1]
\mathbb{R}EQUIRE{dataset $\mathcal{D}$, multiplicative width of annulus, $w$, largest possible norm to support, $L$}
\mathcal{S}TATE \text{num\_bands} $\leftarrow \left\lceil\frac{-\log (1-\|L\|^2)}{\log w}\right\rceil$
\mathcal{S}TATE \text{Initialize (num\_bands -1) partitions to organize datasets into, denote $B[i]$ as the $i$-th partition}.
\FORALL{$x \in \mathcal{D} $}
\mathcal{S}TATE $i = \left\lceil\frac{-\log (1-\|x\|^2)}{\log w}\right\rceil$
\mathcal{S}TATE \text{Insert $x$ into $B[i]$}
\ENDFOR
\mathcal{S}TATE \textbf{Return} $B$
\end{algorithmic}
\end{algorithm}
\subsection{Details for \emph{Check-Intersection }}
We now describe the helper routine for \emph{Spherical-Shell-HyperbolicNN } that determines whether to probe a band (Algorithm \ref{alg:probe_bucket}). The idea behind \emph{Check-Intersection } is very simple. It takes in the center, and a point on the intended hyperbolic ball, which in our case is the query $q$ and the current best nearest neighbor, $n_H$, respectively, as well as the multiplicative width of the buckets and the bucket index to evaluate. The point $x$ with the largest possible Euclidean norm of any of the points in this ball satisfies $d_H(x, 0) = d_H(q, n_H) + d_H(0, q)$. Moreover, if $x$ were of the form $t_1c_H$ for some scalar $t_1$, since hyperbolic distance is additive on the line, we also satisfy that $t_1c_H$ is on the boundary of the ball. Therefore, we just have to solve for this $t_1$ and calculate the bucket index $j$ that $t_1c_H$ would ordinarily partition to. If the bucket index under consideration $b$ is greater than $i$ (the bucket index that the query partitions to), we should probe $b$ if $b < j$. If $b < i$, then we do the same calculation but for the reverse situation where we analyze the smallest possible Euclidean norm of any point in the hyperbolic ball. One small difference is that the origin might be contained in this ball, in which case the $t_2$ might be negative. In that case, we should search all buckets with indices smaller than $i$.
\begin{algorithm}[ht]
\caption{\emph{Check-Intersection }}
\label{alg:probe_bucket}
\begin{algorithmic}[1]
\mathbb{R}EQUIRE{hyperbolic center $c_H$, point on the boundary of hyperbolic ball $p$, multiplicative width of annulus $w$, bucket index to evaluate $b$}
\mathcal{S}TATE $i \leftarrow \left\lceil\frac{-\log (1-\|c_H\|^2)}{\log w}\right\rceil$
\IF {$p = NULL$}
\mathcal{S}TATE \textbf{Return} \text{True}
\ELSIF {$b \geq i$}
\mathcal{S}TATE Find scalar $t_1$ such that $t_1 \|c_H\|_2 = \tanh \left( \frac{d_H(0, c_H) + d_H(c_H, p)}{2} \right)$
\mathcal{S}TATE $j \leftarrow \left\lceil\frac{-\log(1-\|t_1c_H\|^2)}{\log(w)}\right\rceil$
\IF {$b \leq j$}
\mathcal{S}TATE \textbf{Return} \text{True}
\ENDIF
\ELSE
\mathcal{S}TATE Find scalar $t_2$ such that $t_2 \|c_H\|_2 = \tanh \left( \frac{d_H(0, c_H) - d_H(c_H, p)}{2} \right)$
\IF {$t_2 \leq 0$}
\mathcal{S}TATE \textbf{Return} \text{True}
\ELSE
\mathcal{S}TATE $j \leftarrow \left\lfloor\frac{-\log(1-\|t_2c_H\|^2)}{\log(w)}\right\rfloor$
\IF {$b \geq j$}
\mathcal{S}TATE \textbf{Return} \text{True}
\ENDIF
\ENDIF
\ENDIF
\mathcal{S}TATE \textbf{Return} \text{False}
\end{algorithmic}
\end{algorithm}
\subsection{Details for \emph{Check-Intersection }MaxRadius}
We describe the helper routine that decides whether the algorithm should search in band $b_1$ or $b_2$, when the algorithm is guaranteed to have already searched in bands $i, i+1 \ldots b_1 - 1$, and $i-1, i-2 \ldots b_2+1$, where $i$ is the band index that the query falls into. The overall idea is that when deciding which next band to probe, we choose the band which maximizes the radius of the hyperbolic ball around $q$ that is completely covered by the union of bands probed so far as well as the new band under consideration.
\begin{algorithm}[ht]
\caption{\emph{Check-Intersection }MaxRadius}
\label{alg:choose_bucket}
\begin{algorithmic}[1]
\mathbb{R}EQUIRE{hyperbolic center $c_H$, current best neighbor $n_H$, multiplicative width of annulus $w$, bucket index to evaluate $b_1$, $b_2$, wlog $b_1 > b_2$}
\mathcal{S}TATE $d_1 \leftarrow -\infty$
\mathcal{S}TATE $d_2 \leftarrow -\infty$
\IF {$\emph{Check-Intersection }(c_H, n_H, w, b_1)$}
\mathcal{S}TATE Find scalar $t_1$ such that $\frac{1}{1-\|t_1 c_H\|^2} = w^{b_1}$
\mathcal{S}TATE Find scalar $t_2$ such that $\frac{1}{1-\|t_2 c_H\|^2} = w^{b_2}$
\mathcal{S}TATE $d_1 \leftarrow \min \{ d_H(c_H, t_1c_H), d_H(c_H, t_2 c_H)\}$
\ENDIF
\IF {$\emph{Check-Intersection }(c_H, n_H, w, b_2)$}
\mathcal{S}TATE Find scalar $t_3$ such that $\frac{1}{1-\|t_3 c_H\|^2} = w^{b_1-1}$
\mathcal{S}TATE Find scalar $t_4$ such that $\frac{1}{1-\|t_4 c_H\|^2} = w^{b_2-1}$
\mathcal{S}TATE $d_2 \leftarrow \min \{ d_H(c_H, t_3c_H), d_H(c_H, t_4 c_H)\}$
\ENDIF
\IF {$d_1 \geq b_2$}
\mathcal{S}TATE \textbf{Return} \text{$b_1$}
\ENDIF
\mathcal{S}TATE \textbf{Return} \text{$b_2$}
\end{algorithmic}
\end{algorithm}
\subsection{\emph{Randomized-Spherical-Shell-HyperbolicNN }}
The probing strategy for \emph{Spherical-Shell-HyperbolicNN } in the worst case (for large hyperbolic distances between $q$ and $n_H$) would probe many buckets, possibly all the buckets. To reduce the number of buckets probed, we introduce a randomized algorithm that orders the buckets uniformly at random among all possible permutations, and calls the Euclidean nearest neighbor oracle $\tO$ on the first bucket on the list to find a starting nearest neighbor candidate with hyperbolic radius $r$ to the query. On subsequent buckets, we first use a decision oracle to determine whether that bucket will definitely contain an element closer to $q$ than the current best. If the decision oracle says yes, then we do a full probe on that bucket. Otherwise we move onto the next bucket on the list. The advantage here is that a query to the decision oracle can be very fast, so if \emph{Spherical-Shell-HyperbolicNN } would do a full probe on all the buckets, this randomized algorithm would in expectation do a full probe on a small number of buckets. However, this algorithm uses a decision oracle, which is not always available, or efficient. We first define the decision oracle.
\begin{definition}[$(1+\epsilon, R)$-approximate Euclidean Near Neighbor Decision Oracle, $\tDO$]
The $(1+\epsilon, R)$-approximate Euclidean Near Neighbor Oracle, $\tDO$ takes as input a query $q$, radius of interest $R$, approximation factor $\epsilon > 0$, and a dataset of elements $\mathcal{D}$. If the Euclidean nearest neighbor to $q$, denoted $n_E$, satisfies $\|q-n_E\| \leq R$, this oracle returns a certificate element $x'$ such that $\|x' - q\| \leq (1+\epsilon) R$.
\end{definition}
It is actually possible to build a $(1+\epsilon)$-approximate Euclidean nearest neighbor oracle by calling on the $(1+\epsilon, R)$-approximate Euclidean Near Neighbor Decision Oracle multiple times using successively smaller values of $R$ in a binary search fashion. The query times for the decision oracle are typically smaller than for the approximate near neighbor oracles (since we are not searching for the nearest, just for something nearer than $R$), the saving is about a factor logarithmic in $n$.
\begin{algorithm}[!htp]
\caption{\emph{Randomized-Spherical-Shell-HyperbolicNN }}
\label{alg:nn_banded_rand_alg}
\begin{algorithmic}[1]
\mathbb{R}EQUIRE{query $q$, $(1+\epsilon)$-approximate Euclidean NN oracle $\tO$, $(1+\epsilon, R)$-approximate decision oracle $\tDO$, buckets $B$ with width $w$}
\mathcal{S}TATE $ProbingList \leftarrow Unif(B)$ \text{ // the list of buckets arranged in a random order}
\mathcal{S}TATE $n_H \leftarrow \text{NULL}$ \text{ // current best nearest neighbor candidate}
\mathcal{S}TATE $dist_H = \infty$ \text{ // hyperbolic distance of current best nearest neighbor candidate}
\FOR{buckets $b$ in ProbingList}
\mathcal{S}TATE $R \leftarrow \sqrt{\frac{\left( \frac{\cosh(dist_H)-1}{2}\right) (1-\|q\|^2)}{w^i (1+\epsilon)^2}}$
\IF {$dist_H = \infty$ or $\tDO(q, R, B[b]) = YES$}
\mathcal{S}TATE $n_F \leftarrow \tO(q, B[b])$
\IF {$d_H(q, n_F) < dist_H$}
\mathcal{S}TATE $dist_H \leftarrow d_H(q, n_F)$
\mathcal{S}TATE $n_H \leftarrow n_F$
\ENDIF
\ENDIF
\ENDFOR
\mathcal{S}TATE \textbf{Return} $n_H$
\end{algorithmic}
\end{algorithm}
We first analyze the approximation guarantee of this \emph{Randomized-Spherical-Shell-HyperbolicNN }. Then we give the analysis for the expected number of full probes made by the approximate nearest neighbor oracle $\tO$.
\begin{theorem}
Using a $(1+\epsilon)$-Euclidean nearest neighbor oracle $\tO$, a $(1+\epsilon, R)$-Euclidean near neighbor decision oracle and a dataset split with a multiplicative width of $w$, \emph{Randomized-Spherical-Shell-HyperbolicNN } returns a hyperbolic approximate nearest neighbor $n_H$ to any query $q$ such that $d_H(q, n_H) \leq \sqrt{w}(1+\epsilon) d_H(q, n^*)$.
\end{theorem}
\begin{proof}
Suppose that the current best nearest neighbor candidate, $n_H$ has hyperbolic distance $D$ to the query. Further suppose we are looking at the $i$-th bucket. This bucket contains elements $y$ such that $\frac{1}{w^i} \leq 1-\|y\|^2 \leq \frac{1}{w^{i-1}}$. We want to ask this bucket if it contains an element $x$ such that $d_H(q, x) < D$.
So we want:
\[
d_H(q, x) = \arccosh \left(1 + \frac{2\|q-x\|^2}{(1-\|q\|^2)(1-\|x\|^2} \right) \leq D
\]
This implies that
\begin{align*}
\|q-x\|^2 &\leq \left( \frac{\cosh(D)-1}{2}\right) (1-\|q\|^2) (1-\|x\|^2) \\
&\leq \frac{\left( \frac{\cosh(D)-1}{2}\right) (1-\|q\|^2)}{w^i}
\end{align*}
So if bucket $i$ contains an element $x$ such that
\[
\|q-x\| \leq \sqrt{\frac{\left( \frac{\cosh(D)-1}{2}\right) (1-\|q\|^2)}{w^i}}
\]
then $x$ is definitely a nearer neighbor to $q$ than $n_H$.
But since we are using a $(1+\epsilon)$-approximate nearest neighbor oracle, to guarantee that the oracle only returns an element if bucket $i$ is guaranteed to contain a nearer neighbor, we let $R = \sqrt{\frac{\left( \frac{\cosh(D)-1}{2}\right) (1-\|q\|^2)}{w^i (1+\epsilon)^2}}$.
Now to analyze the approximation factor. Some error could be introduced in the fact that the decision oracle could have said ``NO" but the bucket actually contained a closer element, but this closer element was just slightly closer to $q$ than $n_H$. Say that this happened and we just missed $n^*$. Then clearly,
\[
\|q-n^*\| \geq \sqrt{\frac{\left( \frac{\cosh(D)-1}{2}\right) (1-\|q\|^2)}{w^i(1+\epsilon)^2}}
\]
Moreover, $\frac{1}{1-\|n^*\|} \geq w^{i-1}$
Therefore,
\begin{align*}
d_H(q, n^*) &= \arccosh \left(1 + \frac{2\|q-n^*\|^2}{(1-\|q\|^2)(1-\|n^*\|^2} \right) \\
& \geq \arccosh \left(1 + \frac{2 w^{i-1} \|q-n^*\|^2}{(1-\|q\|^2)} \right) \\
& \geq \arccosh \left( 1 + \frac{\cosh(D) -1}{w(1+\epsilon)^2} \right)
\end{align*}
The rest of the proof follows similarly to the proof for \emph{Spherical-Shell-HyperbolicNN }NS.
\end{proof}
Now we want to provide an analysis on the expected number of invocations of the approximate nearest neighbor oracle $\tO$. We have the following theorem:
\begin{lemma}
Suppose that there are $N$ buckets in total, and the probing order is selected uniformly at random among all the possible permutations of the $N$ buckets. Then the expected number of invocations of the approximate nearest neighbor oracle $\tO$ is $O(\ln N)$.
\end{lemma}
\begin{proof}
We proceed with a proof by induction. The base case when $N=1$ holds. Now suppose that for $k= 2, 3, \ldots N-1$ buckets, the expected number of invocations is $\sum\limits_{n=1}^{k} \frac{1}{n}$.
Now let us consider the case when we have $N$ buckets. First of all, we always probe the first bucket. Now suppose the hyperbolic nearest neighbor to $q$ in the first bucket is the $k$-th hyperbolic nearest neighbor to $q$ among the entire dataset. Then we subsequently have to probe at most $k-1$ buckets, so the problem has been reduced to the subproblem of solving for the number of expected probes where the total number of buckets is $k-1$, which by our inductive assumption is $\sum\limits_{j=1}^{k-1} \frac{1}{j}$. This event happens with probability $\frac{1}{N}$. Now, summing over all possible values of $k$ gives us the following expression:
\[
1 + \frac{1}{N} \sum\limits^{N-1}_{k=1} \sum\limits^k_{j=1} \frac{1}{j} ~.
\]
Also note that by this reasoning combined with the inductive hypothesis gives that
\[
1 + \frac{1}{N-1}\sum\limits^{N-2}_{k=1} \sum\limits^k_{j=1} \frac{1}{j} = \sum\limits_{n=1}^{N-1} \frac{1}{n} ~.
\]
Now to evaluate:
\begin{align*}
1 + \frac{1}{N} \sum\limits^{N-1}_{k=1} \sum\limits^k_{j=1} \frac{1}{j} &= 1 + \frac{N-1}{N} \cdot \frac{1}{N-1}\sum\limits^{N-2}_{k=1} \sum\limits^k_{j=1} \frac{1}{j} + \frac{1}{N} \sum\limits^{N-1}_{j=1} \frac{1}{j} \\
&=1 + \frac{N-1}{N} \cdot \sum\limits_{n=2}^{N-1} \frac{1}{n} + \frac{1}{N} \sum\limits^{N-1}_{j=1} \frac{1}{j} \\
&=1 + \sum\limits_{n=2}^{N-1} \frac{1}{n} + \frac{1}{N} \\
&=\sum\limits_{n=1}^N \frac{1}{n}
\end{align*}
\end{proof}
Then we come to the final runtime guarantee of \emph{Randomized-Spherical-Shell-HyperbolicNN }NS.
\begin{theorem}[Runtime of \emph{Randomized-Spherical-Shell-HyperbolicNN }NS]
The expected runtime of \emph{Randomized-Spherical-Shell-HyperbolicNN } is $O(\mathcal{T} \cdot \ln N + \mathcal{T}_\mathcal{D} \cdot N)$,
where $\mathcal{T}$ is the runtime for one invocation of $\tO$ and $\mathcal{T}_\mathcal{D}$ is the runtime for one invocation of the decision oracle $\tDO$ and $N$ is the total number of buckets.
\end{theorem}
\subsection{\emph{Spherical-Shell-HyperbolicNN } cannot return an exact hyperbolic nearest neighbor with an exact Euclidean oracle $\O$}
We provide a simple example demonstrating that even with an exact Euclidean oracle $\O$, \emph{Spherical-Shell-HyperbolicNN } is not guaranteed to return an exact hyperbolic nearest neighbor. However, if the dataset has already been divided into buckets according to \emph{Spherical-Shell-Partition }NS, one can additionally leverage the recentering idea that forms the core of \emph{Recentering-HyperbolicNN } to return an exact hyperbolic nearest neighbor. We leave the implementation details to the reader.
The example is as follows. Suppose we have a dataset of two points, $n^* = (0, 0.5)$ and $n_E = (0.15, 0.55)$ and the query $q$ is $(0,0.99)$. Straightforward calculation shows that:
\[
\frac{1}{1-\|n^*\|^2} \approx 1.33
\]
and
\[
\frac{1}{1-\|n_E\|^2} \approx 1.48 ~.
\]
Therefore for $w \geq 1.5$, \emph{Spherical-Shell-Partition } will designate them into the same bucket.
We also remark that the hyperbolic nearest neighbor is $n^*$, since $d_H(q, n^*) \approx 4.19$ and $d_H(q, n_E) \approx 4.384$.
However, the Euclidean nearest neighbor of $q$ is $n_E$, not $n^*$, with $\|q - n_E\| \approx .464 $ and $\|q - n^*\| = .49$. Therefore, \emph{Spherical-Shell-HyperbolicNN } will return $n_E$ when using an exact Euclidean nearest neighbor oracle, which is an approximate nearest neighbor to the query.
This example relies crucially on the fact that depending on the placement of the hyperbolic nearest neighbor $n^*$ on the Poincare disk, the hyperbolic ball around the query $q$ with radius $d_H(q, n^*)$, call it $B_H(q, d_H(q, n^*))$, can be completely contained in $B_E(q, d_E(q, n^*))$, the Euclidean ball around $q$ with radius $d_E(q, n^*)$. When this is true, for any predetermined value of $c$, one can find a set of $q, n^*, n_E$ where \emph{Spherical-Shell-HyperbolicNN } cannot guarantee an exact nearest neighbor even when using an exact Euclidean nearest neighbor oracle.
\end{document} |
\begin{document}
\title[]{On trivialities of Chern classes}
\author{Aniruddha C. Naolekar}
\address{Indian Statistical Institute, 8th Mile, Mysore Road, RVCE Post, Bangalore 560059, INDIA.}
\email{[email protected]}
\author{ Ajay Singh Thakur}
\address{Department of Mathematics, University of Haifa, Mount Carmel, Haifa 31905, ISRAEL.}
\email{[email protected]}
\keywords{$C$-trivial, $W$-trivial, Chern class, Stiefel-Whitney class, stunted projective space.}
\begin{abstract} A finite $CW$-complex $X$ is $C$-trivial if for every complex vector bundle $\xi$ over $X$, the total Chern class $c(\xi)=1$. In this note we completely determine when each of the following spaces are $C$-trivial: suspensions of stunted real projective spaces, suspensions of stunted complex projective spaces and suspensions of stunted quaternionic projective spaces.
\end{abstract}
\subjclass[2010] {57R20 (55R50, 57R22).}
\email{}
\date{}
\maketitle
\section{Introduction}
A $CW$-complex $X$ is said to be $C$-trivial if for any complex vector bundle $\xi$ over $X$, the total Chern class $c(\xi)=1$.
A related notion is that of $W$-triviality. A $CW$-complex $X$ is said to be $W$-trivial if for any real vector bundle $\eta$ over $X$, the total Stiefel-Whitney class $w(\eta)=1$. A central result in this direction is a theorem of Atiyah-Hirzebruch (\cite{atiyahirz}, Theorem\,2) which states that for any finite $CW$-complex $X$, the $9$-fold suspension $\Sigma^9X$ of $X$ is always $W$-trivial. In particular, the spheres $S^k$ are all $W$-trivial for $k\geq 9$. In fact, a sphere $S^k$ is $W$-trivial if and only if $k\neq 1,2,4,8$ (see \cite{atiyahirz}, Theorem\,1).
Understanding which spaces
are $W$-trivial has been of some interest in recent times (see \cite{aniajay}, \cite{tanaka}, \cite{ajay} and the references therein).
In \cite{tanaka}, the author has completely determined which suspensions $\Sigma^k \mathbb F\mathbb P^n$ are $W$-trivial. Here $\mathbb F$ denotes either the field $\mathbb R$ of real number, the field $\mathbb C$ of complex numbers or the skew-field $\mathbb H$ of quaternions and $\mathbb F\mathbb P^n$ denotes the appropriate projective space. In \cite{ajay}, the second named author has determined, in most cases, which suspensions of the Dold manifolds are $W$-trivial. In \cite{aniajay}, the authors have completely determined which suspensions of the stunted real projective spaces are $W$-trivial.
In this note we study the notion of $C$-triviality and determine whether some of the familiar spaces and their suspensions are $C$-trivial.
To begin with it is well known that there is no analogue of the Atiyah-Hirzebruch theorem for Chern classes. Indeed, by the Bott integrality theorem (see Theorem\,\ref{bott} below for the precise statement) the even dimensional spheres are not $C$-trivial.
Thus if $d>4$, then $S^{2d}$ is $W$-trivial but not $C$-trivial. The circle $S^1$ is $C$-trivial but not $W$-trivial.
We give other examples in the sequel.
However, there are sufficient conditions under which one implies the other. We point out conditions under which this happens.
In this note we completely determine when the suspension of a stunted real projective space is $C$-trivial (see Theorem\,\ref{secondtheorem} and Theorem\,\ref{thirdtheorem} below). We also
completely determine which suspensions of the stunted complex and quaternionic projective spaces are $C$-trivial (see Corollary\,\ref{complex} below).
We now state the main results of this paper.
The following theorem completely describes which suspensions $\Sigma^k\mathbb R\mathbb P^n$ of the real projective spaces are $C$-trivial. Since $S^1 = \mathbb R \mathbb P^1$ is $C$-trivial and $\mathbb R \mathbb P^n$ is not $C$-trivial for $n >1$, we shall assume $k>0$.
\begin{Thm} \label{secondtheorem} Let $X^k_n=\Sigma^k \mathbb R\mathbb P^n$ with $k,n>0$. Then $X^k_n$ is not $C$-trivial if and only if one of the following conditions is satisfied.
\begin{enumerate}
\item $k,n$ are both odd.
\item $k=2,4$ and $n\geq k$.
\end{enumerate}
\end{Thm}
Next we look at the suspensions of the stunted real projective spaces. To state the results we introduce the following notations.
Let $X_{m,n}$ denote the stunted real projective space
$$X_{m,n}=\mathbb R\mathbb P^m/\mathbb R\mathbb P^n$$
and $X^k_{m,n}$ the $k$-fold suspension
$$X^k_{m,n}=\Sigma^k\left(\mathbb R\mathbb P^m/\mathbb R\mathbb P^n\right).$$
\begin{Thm}\label{thirdtheorem}
Let $X^k_{m,n}$ be as above with $k\geq 0$ and $0<n<m$.
\begin{enumerate}
\item If $k$ is odd and $m$ is even, then $X^k_{m,n}$ is $C$-trivial.
\item If $k,m$ are odd, then $X^k_{m,n}$ is not $C$-trivial.
\item If $ n= 2t$, then $X_{m,n}$ is $C$-trivial if and only if $m< 2^{t+1}$.
\item If $k,n$ are even and $k\geq 2$, then $X^k_{m,n}$ is $C$-trivial.
\item If $k$ is even and $n$ is odd, then $X^k_{m,n}$ is not $C$-trivial.
\end{enumerate}
\end{Thm}
The paper is organized as follows. In Section 2 we prove some general facts about $C$-triviality. In Section 3 we determine which suspensions of stunted complex and quaternionic projective spaces are $C$-trivial and prove the main theorems.
{\em Conventions.} By a space we mean a finite connected $CW$-complex. Given a map $\alpha:X\longrightarrow Y$ between spaces the induced homomorphism in $K$-theory and singular cohomology will again be denoted by $\alpha$.
\section{Generalities}
In this section we prove some general facts about $C$-triviality. We give sufficient conditions under which $W$-triviality implies $C$-triviality and
$C$-triviality implies $W$-triviality.
To begin, note that a finite $CW$-complex $X$ is $C$-trivial if the reduced complex $\widetilde{K}$-group $\widetilde{K}(X)=0$.
\begin{Lem} \label{prelim} For any space $X$, we have the following.
\begin{enumerate}
\item If $H^{2s}(X;\mathbb Z)=0$ for all $s> 0$, then $X$ is $C$-trivial.
\item If $X$ has cells only in odd dimensions, then $X$ is $C$-trivial.
\item If $X$ is $C$-trivial, then $H^2(X;\mathbb Z)=0$
\item If $X$ is $W$-trivial and the mod-$2$ reduction homomorphism $\rho_2:H^{2i}(X;\mathbb Z)\longrightarrow H^{2i}(X;\mathbb Z_2)$ is monomorphic for all $i>0$, then $X$ is $C$-trivial.
\end{enumerate}
\end{Lem}
{\bf Proof.} We omit the easy proofs of (1)-(3). The claim (4) follows from the fact that for a complex bundle $\xi$ we have $\rho_2(c_i(\xi))=w_{2i}(\xi_{\mathbb R})$. Here $\xi_{\mathbb R}$ denotes the underlying real bundle of $\xi$. \qed
Thus the real and complex projective spaces $\mathbb R\mathbb P^n$ ($n>1$) and $\mathbb C\mathbb P^n$ are not $C$-trivial as their second integral cohomology group is non-zero.
The following observations are straightforward and we omit their easy proofs.
\begin{Lem}\label{prelim1}
Let $f:X\longrightarrow Y$ be a map between spaces.
\begin{enumerate}
\item If $f:\widetilde{K}(Y)\longrightarrow \widetilde{K}(X)$ is onto and $Y$ is $C$-trivial, then $X$ is $C$-trivial.
\item Suppose $f:H^{2i}(Y;\mathbb Z)\longrightarrow H^{2i}(X;\mathbb Z)$ is a monomorphism for all $i>0$. Then if $X$ is $C$-trivial so is $Y$. \qed
\end{enumerate}
\end{Lem}
We shall use the above observations in the sequel sometimes without an explicit reference. We have already noted that the even dimensional spheres $S^{2d}$ with $d>4$ are examples of spaces that are $W$-trivial but not $C$-trivial and also that $S^1$ is $C$-trivial but not $W$-trivial.
We give some more examples below.
\begin{Exm}
Let $X=\Sigma\mathbb R\mathbb P^2$ be the suspension of the real projective space $\mathbb R\mathbb P^2$. Then $X$ has non-zero integral cohomology only in degree $3$ and hence $X$ must be $C$-trivial. It is known that
$X$ is not $W$-trivial (\cite{tanaka}, Theorem 1.4).
\end{Exm}
\begin{Exm}
Let $X=M(\mathbb Z_3,1)$ be the Moore space of type $(\mathbb Z_3,1)$. This is a $2$-dimensional $CW$-complex. Since the second (integral) cohomology of $X$ is non-zero, $X$ is not $C$-trivial. However, as $H^i(X;\mathbb Z_2)=0$
for $i>0$, $X$ is $W$-trivial.
\end{Exm}
We now state a necessary condition for a space to be $C$-trivial.
\begin{Lem}
Let $X$ be a $C$-trivial space. Then for any real bundle $\xi$ over $X$ we have $w_i^2(\xi)=0$ for all $i>0$.
\end{Lem}
{\bf Proof.} Let $\eta$ denote the underlying real bundle of the complexification $\xi\otimes \mathbb C$ of $\xi$. Then the total Stiefel-Whitney class
of $\eta$ is given by
$$w(\eta)=1+w_1^2+w_2^2+\cdots$$
where $w_i=w_i(\xi)$. If $w(\eta)\neq 1$, then clearly $c(\xi\otimes \mathbb C)\neq 1$. This completes the proof. \qed
In particular, if there exists a real bundle $\xi$ over $X$ with $w_i^2(\xi)\neq 0$ for some $i>0$, then $X$ cannot be $C$-trivial.
The above lemma, in particular, implies that the quaternionic projective space $\mathbb H\mathbb P^n$ is not $C$-trivial for $n>1$. This is because for the canonical line bundle $\xi$ over $\mathbb H\mathbb P^n$ we have $w_4^2(\xi)\neq 0$. That $\mathbb H\mathbb P^1=S^4$ is not $C$-trivial is clear. The (even dimensional) spheres show that the converse of the above lemma is not true.
For a space $X$, we have the realification homomorphism $r:\widetilde{K}(X)\longrightarrow \widetilde{KO}(X)$.
The following lemma gives a sufficient condition for a $C$-trivial space to be $W$-trivial.
\begin{Lem}
Suppose that $r:\widetilde{K}(X)\longrightarrow \widetilde{KO}(X)$ is onto. If $X$ is $C$-trivial, then $X$ is $W$-trivial.
\end{Lem}
{\bf Proof.} The surjectivity of $r$ implies that every real vector bundle $\xi$ over $X$ is stably equivalent to the underlying real bundle
$\eta_{\mathbb R}$ of a complex vector bundle $\eta$ over $X$. Now $w(\xi) =w(\eta_{\mathbb R})=1$. This completes the proof. \qed
We mention that the converse to the above lemma is not true. Indeed, if $X=S^6$, then as $\widetilde{KO}(S^6)=0$, $X$ is $W$-trivial but $X$ is not
$C$-trivial.
Some of our proofs depend upon the following important observation which gives
a sufficient condition for a $W$-trivial space to be $C$-trivial.
\begin{Prop}\label{onetytwo}
Let $X$ be a space. Assume that $H_*(X;\mathbb Z)$ is concentrated in odd degrees and is direct sum of copies of $\mathbb Z _2$. If $X$ is $W$-trivial, then $X$ is $C$-trivial.
\end{Prop}
{\bf Proof.} Given the assumptions on the integral homology of $X$ it follows from the universal coefficients theorem that the integral cohomology of $X$ is concentrated in even degrees and is a direct sum of copies of $\mathbb Z_2$. Again, by the universal coefficients theorem, it follows that $H^{2i}(X;\mathbb Z_2)$ is a direct sum of copies of $\mathbb Z_2$ with the same number of $\mathbb Z_2$ summands as in the integral cohomology.
As $X$ has no integral cohomology in odd degrees, the mod-$2$ reduction map
$$\rho_2:H^{2i}(X;\mathbb Z)\longrightarrow H^{2i}(X;\mathbb Z_2)$$
is surjective and hence an isomorphism as both the groups are a direct sum of equal number of copies of $\mathbb Z_2$. The proposition now follows from Lemma\,\ref{prelim} (4). This completes the proof. \qed
\begin{Rem}
We remark that the converse of the above proposition is not true. For consider the space $X^6_2=\Sigma^6\mathbb R\mathbb P^2$. Then $X^6_2$ satisfies the conditions of the above proposition. By Theorem\,\ref{secondtheorem}, $X^6_2$ is $C$-trivial. However $X^6_2$ is not $W$-trivial, by Theorem\,1.4 (3) of \cite{tanaka}.
\end{Rem}
We end this section by noting that second suspension of a $C$-trivial space is $C$-trivial. The idea of the proof is same
as that of Theorem\,1 in \cite{atiyahirz} and Theorem\,1.1 of \cite{tanaka}.
\begin{Thm}\label{hahaha}
Let $X$ be a $C$-trivial space. Then the second suspension $\Sigma^2X$ of $X$ is $C$-trivial.
\end{Thm}
{\bf Proof.} Let $\pi_1,\pi_2$ denote the projection maps of $S^2\times X$ to the first and second factor respectively and $p:S^2\times X\longrightarrow \Sigma^2X$ the quotient map. Let $\nu$ denote the Hopf bundle over $S^2$. Given a bundle $\xi$ over $\Sigma^2X$ there exists, by Bott periodicity, a bundle $\theta$ of rank $n$ (say) over $X$ such that $p^*\xi$ is stably isomorphic to the tensor product
$$(\pi_1^*\nu-1)\otimes (\pi_2^*\theta-n).$$
Thus
$$c(p^*\xi)=c(\pi_1^*\nu\otimes\pi_2^*\theta)c(\pi_1^*\nu)^{-n}c(\pi_2^*\theta)^{-2}.$$
Since $X$ is $C$-trivial we have $c(\pi_2^*\theta)=1$. It is clear that $c(\pi_1^*\nu)=1+t\times 1$ where $t\in H^2(S^2;\mathbb Z)$ is a generator.
Finally, one checks that
$$c(\pi_1^*\nu\otimes\pi_2^*\theta)=(1+t\times 1)^n$$
so that
$$c(p^*\xi)=1.$$
But as $f:H^*(\Sigma^2X;\mathbb Z)\longrightarrow H^*(S^2\times X;\mathbb Z)$ is a monomorphism it follows that $c(\xi)=1$. This completes the proof. \qed
\section{Proof of Theorems\,\ref{secondtheorem} and \ref{thirdtheorem}}
In this section we prove the main theorems and derive some consequences. We first state the Bott integrality theorem which we shall use in the sequel.
\begin{Thm} {\rm(Bott integrality theorem)}\label{bott} {\rm(\cite{hus}, Chapter 20, Corollary\,$9.8$)} Let $a\in H^{2n}(S^{2n};\mathbb Z)$ be a generator. For any complex vector bundle $\xi$ over $S^{2n}$, the Chern class $c_n(\xi)$ is divisible by $(n-1)!a$. For each $m$ divisible by $(n-1)!$, there exists a unique $\xi\in\widetilde{K}(S^{2n})$ with $c_n(\xi)=ma$. \qed
\end{Thm}
The Bott integrality theorem implies that if $\xi$ is a complex vector bundle over the even dimensional sphere $S^{2n}$ with $n \geq 3$, then $c_n(\xi) \in H^{2n}(S^{2n};\mathbb Z)$ is an even multiple of a generator.
The following observation is now immediate from the Bott integrality theorem.
\begin{Prop}\label{firstcor}
Let $X$ be a finite $CW$-complex. Assume that there is a map $\alpha:X\longrightarrow S^d$ such that the homomorphism
$\alpha:H^d(S^d;\mathbb Z)\longrightarrow H^d(X;\mathbb Z)$ is injective. Then $\Sigma^kX$ is not $C$-trivial whenever $k+d$ even.
\end{Prop}
{\bf Proof.} The map $\Sigma^kf$ induces a monomorphism in cohomology in degree $(k+d)$ for all $k$. If $(k+d)$ is even, then as there exists a vector bundle $\xi$ over $S^{k+d}$ with $c(\xi)\neq 1$ we must have $c(f^*\xi)\neq 1$. \qed
If $X$ is a connected closed orientable manifold we have a degree one map $f:X\longrightarrow S^{\mathrm{dim}(X)}$. This induces an isomorphism in top integral cohomology. Thus we have the following.
\begin{Cor}\label{secondcor}
Suppose $X$ is a connected closed orientable manifold. Then $\Sigma^k X$ is not $C$-trivial whenever $\mathrm{dim}(X)+k$ is even. \qed
\end{Cor}
If $k$ is odd then the suspensions $\Sigma^k (\mathbb C\mathbb P^m/\mathbb C\mathbb P^n)$ and $\Sigma^k (\mathbb H\mathbb P^m/\mathbb H\mathbb P^n)$ have cells only in odd dimensions and hence are $C$-trivial. The following observation is now immediate from the the above noted facts.
\begin{Cor}\label{complex}
Let $\mathbb F= \mathbb C$ or $\mathbb H$. Let $0\leq n<m$. Then $\Sigma^k(\mathbb F\mathbb P^m/\mathbb F\mathbb P^n)$ is $C$-trivial if and only if $k$ is odd. \qed
\end{Cor}
\begin{Cor}
The product of two connected closed orientable manifolds is not $C$-trivial.
\end{Cor}
{\bf Proof.} In the case the product $M\times N$ is even dimensional, the claim follows from Corollary\,\ref{secondcor}. In the case that $M\times N$ is odd dimensional, assume that $M$ is even dimensional. Then $M$ is not $C$-trivial and since the composition
$$M\stackrel{i}\longrightarrow M\times N\stackrel{\pi_1}{\longrightarrow} M$$
where $i(x)=(x,y)$ for a fixed $y\in N$ and $\pi_1$ the projection to the first factor, is identity it follows that $M\times N$ is not $C$-trivial. This completes the proof. \qed
In particular, a product of spheres is not $C$-trivial. We make some more observations before proving the main theorems.
\begin{Lem}\label{howdoyoudo} Suppose $k$ is even.
\begin{enumerate}
\item If $X^k_n$ is $W$-trivial, then $X^k_n$ is $C$-trivial.
\item If $n$ is even and $X^k_{m,n}$ is $W$-trivial, then $X^k_{m,n}$ is $C$-trivial.
\end{enumerate}
\end{Lem}
{\bf Proof.} We prove (1), the proof of (2) is similar. As $k$ is even, the integral cohomology is zero in odd degrees except in degree $n$ when $n$ is odd in which case it is infinite cyclic.
The integral cohomology in even degrees is cyclic of order two. The mod-$2$ reduction map in even degrees is readily seen to be an isomorphism.
By Lemma\,\ref{prelim} (4), the proof is complete. \qed
\begin{Lem}\label{newlemma}
Let $k,m,n$ be even. If $X^k_{m+1,n}$ is not $C$-trivial, then $X^k_{m,n}$ is not $C$-trivial.
\end{Lem}
{\bf Proof.} The lemma follows from the fact that the obvious map
$$j:X_{m,n}\longrightarrow X_{m+1,n}$$
induces isomorphism in integral cohomology in even degrees. Hence so does the map $\Sigma^kj$. \qed
{\em Proof of Theorem\,\ref{secondtheorem}.}
If both $k$ and $n$ are odd, then as $\mathbb R\mathbb P^n$ is orientable, it follows from
Corollary\,\ref{secondcor} that $X^k_n$ is not $C$-trivial. Next assume that
$k$ is odd and $n$ is even. Then as $H^i(X^k_n;\mathbb Z)=0$ if $i>0$ is even it follows that $X^k_n$ is $C$-trivial in this case. This proves the theorem when $k$ is odd.
Next we assume that $k$ is even. We first show that $X^4_n$ is not $C$-trivial if and only if $n\geq 4$.
We know that $X^4_n$ for $n \leq 3$ is $W$-trivial (\cite{tanaka}, Theorem\,1.4).
Hence it follows from Lemma\,\ref{howdoyoudo} that $X^4_n$ is $C$-trivial for $n \leq 3$. So assume that $n\geq 4$. Let $\xi$ be a complex $2$-plane bundle over $S^4$ with $c_2(\xi)$ a generator and let $\eta$ be a complex line bundle over $\mathbb R\mathbb P^n$ with
$c_1(\eta)=t\in H^2(\mathbb R\mathbb P^n;\mathbb Z)\cong \mathbb Z_2$ the non-zero element.
The cofiber sequence
$$S^4\vee \mathbb R\mathbb P^n\stackrel{j}\longrightarrow S^4\times \mathbb R\mathbb P^n\stackrel{\alpha}\longrightarrow\Sigma^4\mathbb R\mathbb P^n$$
gives rise to an exact sequence
$$0\rightarrow \widetilde{K}(\Sigma^4\mathbb R\mathbb P^n)\stackrel{\alpha}\longrightarrow\widetilde{K}(S^4\times\mathbb R\mathbb P^n)
\stackrel{j}\longrightarrow\widetilde{K}(S^4\vee \mathbb R\mathbb P^n)\rightarrow 0.$$
We compute (see, for example, Lemma\,2.1, \cite{tanaka})
$$\begin{array}{rcl}
c((\pi_1^*\xi-2)\otimes (\pi_2^*\eta-1)) & = & 1+ c_2(\xi)\times ((1+t)^{-2}-1)\\
& = & 1+ c_2(\xi)\times (-2t+3t^2-4t^3+\cdots)\\
& \neq & 1
\end{array}$$
as $c_2(\xi)\neq 0$ is a generator and $3t^2\neq 0$.
Now as $j((\pi_1^*\xi-2)\otimes (\pi_2^*\eta-1))=0$, there exists a bundle $\theta\in \widetilde{K}(\Sigma^k\mathbb R\mathbb P^n)$ with
$$\alpha(\theta)=(\pi_1^*\xi-2)\otimes (\pi_2^*\eta-1).$$
Clearly, $c(\theta)\neq 1$. This completes the proof that $X^4_n$ is not $C$-trivial if and only if $n\geq 4$.
We now look at the case $k = 2$. Note that $X^2_1=S^3$ is $C$-trivial. We now check that $X^2_n$ is not $C$-trivial if $n \geq 2$.
Let $\xi$ be a complex bundle over $S^{2}$ with $c(\xi)\neq 1$ and $c_1(\xi)$ a generator. Let $\eta$ denote the non-trivial complex line bundle over $\mathbb R\mathbb P^n$. Let $\pi_1,\pi_2$ be the two projections of $S^{2}\times\mathbb R\mathbb P^n$ onto the the first and the second factor respectively. Then,
$$\begin{array}{rcl}
c((\pi_1^*\xi-1)\otimes (\pi_2^*\eta-1)) & = & 1+ c_1(\xi)\times ((1+t)^{-1}-1)\\
& = & 1+ c_1(\xi)\times (-t+t^2-t^3+\cdots)\\
& \neq & 1
\end{array}$$
as $c_1(\xi)$ is a generator.
Here $t\in H^2(\mathbb R\mathbb P^n;\mathbb Z)=\mathbb Z_2$ is the unique non-zero element. Then, arguing as in the above case, it follows that
there must exist a bundle $\theta\in \widetilde{K}(\Sigma^k\mathbb R\mathbb P^n)$ with $c(\theta)\neq 1$. Thus $X^2_n$ is not $C$-trivial if and only if $n\geq 2$.
To complete the proof of the theorem we finally show that $X^6_n$ is $C$-trivial for all $n>0$. This will imply, by Theorem\,\ref{hahaha}, that $X^k_n$ is $C$-trivial for all $k\geq 6$ and $k$ even.
First note that $X^6_1=S^7$ is $C$-trivial. That $X^6_2$ is $C$-trivial follows from the fact that $X^4_2$ is $C$-trivial and by Theorem\, \ref{hahaha}.
By Theorem\,1.4 of \cite{tanaka}, $X^6_n$ is $W$-trivial whenever $n>3$.
Hence by Lemma\,\ref{howdoyoudo} $X^6_n$ is $C$-trivial when $n>3$. Finally, we look at $X^6_3$. The long exact sequence of the pair $(\mathbb R\mathbb P^3,\mathbb R\mathbb P^2)$ shows that
the inclusion map $i:\mathbb R\mathbb P^2\longrightarrow \mathbb R\mathbb P^3$ induces an isomorphism $i:H^2(\mathbb R\mathbb P^3;\mathbb Z)\longrightarrow H^2(\mathbb R\mathbb P^2;\mathbb Z)$. Hence the map $\Sigma^6i:\Sigma^6\mathbb R\mathbb P^2\longrightarrow \Sigma^6\mathbb R\mathbb P^3$ induces isomorphism in integral cohomology in degree $8$. Since the only non-zero cohomology in even degree (for both $X^6_2$ and $X^6_3$) is in
degree $8$ and $X^6_2$ is $C$-trivial, it follows by Lemma\,\ref{prelim1} (2) that $X^6_3$ is $C$-trivial. This completes the proof that $X^6_n$ is $C$-trivial.
This takes care of all the cases and completes the proof of the theorem. \qed
We now come to the proof of Theorem\,\ref{thirdtheorem}. First note that if $m$ is odd then the stunted real projective space $X_{m,m-2}$
admits a splitting
$$X_{m,m-2}=S^m\vee S^{m-1}$$
and if $m$ is even then
$$X_{m,m-2}=\Sigma^{m-2}\mathbb R\mathbb P^2.$$
We now prove Theorem\,\ref{thirdtheorem}.
{\em Proof of Theorem\,\ref{thirdtheorem}.} We first prove (1). If $k$ is odd and $m$ is even, then the integral cohomology of $X^k_{m,n}$ is trivial in even degrees and hence $X^k_{m,n}$ is $C$-trivial in this case proving (1).
Next, if $k,m$ are both odd there exists a map $\alpha:X^k_{m,n}\longrightarrow S^{k+m}$ inducing isomorphism in top integral cohomology. By Corollary\,\ref{secondcor}, $X^k_{m,n}$ is not $C$-trivial. This proves (2).
Next we prove (3). By Theorem 7.3 \cite{adams}, the projection $j:\mathbb R\mathbb P^m \rightarrow X_{m,n}$ maps $\widetilde{K}(X_{m,n})$ isomorphically into the subgroup of $\widetilde{K}(\mathbb R\mathbb P^m)$ generated by class of $2^t\nu$, where $\nu = \xi \otimes \mathbb C$ is the complexification of the canonical line bundle $\xi$ over $\mathbb R \mathbb P^m$. Let $\alpha \in \widetilde{K}(X_{m,n})$ be the generator such that $j^*(\alpha) = 2^t\nu$.
If $z \in H^2(\mathbb R \mathbb P^m;\mathbb Z)$ is the unique non-zero element, then the total Chern class
$$c(2^t\nu) = c(\nu)^{2^t} = (1+z)^{2^t} = (1+z^{2^t}).$$
Since $j^*:H^k(X_{m,n};\mathbb Z) \rightarrow H^k(\mathbb R \mathbb P^m;\mathbb Z)$ is injective for $0 \leq k \leq m$, we have $c(\alpha) = 1$ if and only if $m < 2^{t+1}$. As $\alpha$ is a generator, we conclude that $X_{m,n}$ is $C$-trivial if and only of $m< 2^{t+1}$.
We now prove (4).
We shall only prove the $C$-triviality for the case $k=2$. Then in view of Theorem\,\ref{hahaha}, $X^k_{m,n}$ will be $C$-trivial for all $k\geq 4$ and $k$ is even. By Theorem\, 1.3 of \cite{aniajay}, $X^2_{m,n}$ is $W$-trivial if $m \neq 6,7$. By Lemma\,\ref{howdoyoudo} we see that for $n$ even, $X^2_{m,n}$ is $C$-trivial if $m \neq 6,7$. We are now left to prove $C$-triviality of the following cases: $X^2_{6,n}$ and $X^2_{7,n}$ for $n$ even. We prove these as follows.
We first prove that $X^2_{6,n}$ is $C$-trivial for $n$ even. As $X^2_{6,4}=\Sigma^6\mathbb R\mathbb P^2$, it follows from Theorem\,\ref{secondtheorem} that
$X^2_{6,4}$ is $C$-trivial. We next look at the case $X^2_{6,2}$ and
let $\xi$ be a complex vector bundle over $X^2_{6,2}$. We shall show that $c_3(\xi)=0=c_4(\xi)$.
We first claim that $c_3(\xi)=0$. For if $c_3(\xi)\neq 0$, then $w_6(\xi_{\mathbb R})\neq 0$ as the mod-$2$ reduction homomorphism is an isomorphism.
Now observe that $w_i(\xi_{\mathbb R})=0$ for $1\leq i\leq 5$. This contradicts the well-known fact that for a real bundle the first non-zero Stiefel-Whitney class appears in degree a power of two. Thus $c_3(\xi)=0$. Assume now that $c_4(\xi)\neq 0$.
consider the exact sequence
$$\cdots\rightarrow \widetilde{K}^{-2}(X_{8,6})\stackrel{\alpha}\longrightarrow\widetilde{K}^{-2}(X_{8,2})\stackrel{j}\longrightarrow
\widetilde{K}^{-2}(X_{6,2})\longrightarrow\widetilde{K}^{-1}(X_{8,6})\rightarrow \cdots.$$
Using the Atiyah-Hirzebruch spectral sequence in complex $K$-theory it is easy to see that if $m$ is even, then
$$\widetilde{K}^{-1}(X_{m,n})=0.$$
Thus the homomorphism $j$ is epimorphic. Hence, $X^2_{6,2}$ is $C$-trivial as $X^2_{8,2}$ is $C$-trivial.
Next we prove that $X^2_{7,n}$ is $C$-trivial for $n$ even. Clearly, $X^2_{7,6}=S^9$ is $C$-trivial.
That $X^2_{7,2}$ and $X^2_{7,4}$ is $C$-trivial follows from $C$-triviaility of $X^2_{6,2}$ and $X^2_{6,4}$ and by Lemma\,\ref{newlemma}. This completes the proof of (4).
We now prove (5). Here $k$ is even and $n$ is odd. First we assume $m$ is even. We look at the cofiber sequence
$$X_{n+1,n}\stackrel{j}\longrightarrow X_{m,n}\stackrel{\alpha}\longrightarrow X_{m,n+1}$$
and the associated exact sequence
$$\cdots\rightarrow \widetilde{K}^{-k}(X_{m,n+1})\stackrel{\alpha}\longrightarrow\widetilde{K}^{-k}(X_{m,n})\stackrel{j}\longrightarrow
\widetilde{K}^{-k}(X_{n+1,n})\longrightarrow\widetilde{K}^{-k+1}(X_{m,n+1})\rightarrow\cdots.$$
As noted above, since the last group in the above exact sequence is zero the homomorphism $j$ is epimorphic. Since $\Sigma^kX_{n+1,n}$ is an even dimensional sphere, and therefore not $C$-trivial, we conclude that $X^k_{m,n}$ is not $C$-trivial. Note that, in particular, there exists a complex vector bundle $\xi$ over $X^k_{m,n}$ with $$c_{\frac{k+n+1}{2}}(\xi)\neq 0.$$
Next we assume $m$ is odd. Consider the obvious map
$$j: X_{m,n}\longrightarrow X_{m+1,n}.$$
The homomorphism $j$ in integral cohomology is an isomorphism in degree $(n+1)$. The homomorphism $\Sigma^kj$ is an isomorphism in integral cohomology in degree $(k+n+1)$. Now if $\xi$ is a complex vector bundle over $X^k_{m+1,n}$ with $$c_{\frac{k+n+1}{2}}(\xi)\neq 0,$$ we have that $$c_{\frac{k+n+1}{2}}(j^*\xi)\neq 0.$$
Thus $X^k_{m,n}$ is not $C$-trivial when $k$ is even and $m,n$ are odd. This completes the proof of (5) and the theorem.
\qed
We remark that the fact that
$$\widetilde{K}^{-k}(X_{m,n})=0$$
whenever $k$ is odd and $m$ is even also gives another proof of Theorem\,\ref{thirdtheorem} (1).
\end{document} |
\begin{document}
\title{Ultrafilters maximal for finite embeddability}
\author{Lorenzo Luperi Baglini\thanks{University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, 1090 Vienna, AUSTRIA, e-mail: \texttt{[email protected]}, supported by grant P25311-N25 of the Austrian Science Fund FWF.}}
\maketitle
\date{}
\begin{abstract}
In \cite{fe} the authors showed some basic properties of a pre-order that arose in combinatorial number theory, namely the finite embeddability between sets of natural numbers, and they presented its generalization to ultrafilters, which is related to the algebraical and topological structure of the Stone-\v{C}ech compactification of the discrete space of natural numbers. In this present paper we continue the study of these pre-orders. In particular, we prove that there exist ultrafilters maximal for finite embeddability, and we show that the set of such ultrafilters is the closure of the minimal bilateral ideal in the semigroup $(\beta\mathbb{N},\oplus)$, namely $\overline{K(\beta\mathbb{N},\oplus)}$. As a consequence, we easily derive many combinatorial properties of ultrafilters in $\overline{K(\beta\mathbb{N},\oplus)}$. We also give an alternative proof of our main result based on nonstandard models of arithmetic.
\end{abstract}
\section{Introduction}
This paper is a planned sequel of the paper \cite{fe} written by Andreas Blass and Mauro Di Nasso. Both in \cite{fe} and in this present paper it is studied a notion that arose in combinatorial number theory (see \cite{DN} and \cite{ruzsa}, where this notion was implicitly used), the finite embeddability between sets of natural numbers. We recall its definition:
\begin{defn}[\cite{fe}, Definition 1] For $A,B$ subsets of $\mathbb{N}$, we say that $A$ is finitely embeddable in $B$ and we write $A\leq_{fe} B$ if each finite subset $F$ of $A$ has a rightward translate $F+k$ included in $B$. \end{defn}
We use the standard notation $n+F=\{n+a\mid a\in F\}$ and we use the standard convention that $\mathbb{N}=\{0,1,2,...\}$. In \cite{fe} the authors also considered the generalization of $\leq_{fe}$ to ultrafilters:
\begin{defn}[\cite{fe}, Definition 2] For ultrafilters $\mathcal{U}, \mathcal{V}$ on $\mathbb{N}$, we say that $\mathcal{U}$ is finitely embeddable in $\mathcal{V}$ and we write $\mathcal{U}\leq_{fe}\mathcal{V}$ if, for each set $B\in\mathcal{V}$, there is some $A\in\mathcal{U}$ such that $A\leq_{fe} B$. \end{defn}
It is easy to prove (see \cite{fe}, \cite{Tesi}) that both $(\mathcal{P}(\mathbb{N}),\leq_{fe})$ and $(\beta\mathbb{N},\leq_{fe})$ are preorders. In \cite{fe} the authors studied some properties of $\leq_{fe}$, giving in particular many equivalent characterization of the relations $A\leq_{fe} B$ and $\mathcal{U}\leq_{fe}\mathcal{V}$ using standard and nonstandard techniques; in this present paper we use similar techniques to continue the study of these pre-orders. Our main result is that there exist ultrafilters maximal for finite embeddability and that the set of such maximal ultrafilters is the closure of the minimal bilateral ideal in $(\beta\mathbb{N},\oplus)$, namely $\overline{K(\beta\mathbb{N},\oplus)}$. This result allows to easily deduce many combinatorial properties of ultrafilters in $\overline{K(\beta\mathbb{N},\oplus)}$, e.g. that for every ultrafilter $\mathcal{U}\in\overline{K(\beta\mathbb{N},\oplus)}$, for every $A\in\mathcal{U}$, $A$ has positive upper Banach density, it contains arbitrarily long arithmetic progressions and it is piecewise syndetic\footnote{Let us note that many of these combinatorial properties of ultrafilters in $\overline{K(\beta\mathbb{N},\oplus)}$ where already known.}. We will also show that there do not exist minimal sets in $(\mathcal{P}_{\aleph_{0}}(\mathbb{N}),\leq_{fe})$ or minimal ultrafilters in $(\beta\mathbb{N}\setminus\mathbb{N},\leq_{fe})$, where $\mathcal{P}_{\aleph_{0}}(\mathbb{N})$ is the set of infinite subsets of $\mathbb{N}$ and $\beta\mathbb{N}\setminus\mathbb{N}$ is the set of nonprincipal ultrafilters. These topics are studied in sections \ref{propy} and \ref{extreme}. In section \ref{NS23} we reprove our main result by nonstandard methods; nevertheless, this is the only section in which nonstandard methods are used, so the rest of the paper is accessible also to readers unfamiliar with nonstandard methods.\par
We refer to \cite{rif12} for all the notions about combinatorics and ultrafilters that we will use, to \cite{rif5}, §4.4 for the foundational aspects of nonstandard analysis and to \cite{davis} for all the nonstandard notions and definitions. Finally, we refer the interested reader to \cite{Tesi}, Chapter 4 for other properties and characterizations of the finite embeddability.
\section{Some basic properties of $(\mathcal{P}(\mathbb{N}),\leq_{fe})$}\label{propy}
Let $n$ be a natural number. Throughout this section we will denote by $\mathcal{P}_{\geq n}(\mathbb{N})$ the set
\begin{equation*} \mathcal{P}_{\geq n}(\mathbb{N})=\{A\subseteq\mathbb{N}\mid |A|\geq n\}; \end{equation*}
similarly, we will denote by $\mathcal{P}_{\aleph_{0}}(\mathbb{N})$ the set
\begin{equation*} \mathcal{P}_{\aleph_{0}}(\mathbb{N})=\{A\subseteq\mathbb{N}\mid |A|=\aleph_{0}\}. \end{equation*}
Moreover, we will denote by $\equiv_{fe}$ the equivalence relation such that, for every $A,B\subseteq\mathbb{N}$,
\begin{equation*} A\equiv_{fe} B\Leftrightarrow A\leq_{fe} B \wedge B\leq_{fe} A \end{equation*}
and, for every set $A$, we will denote by $[A]$ its equivalence class. Finally we will denote by $\leq_{fe}$ the ordering induced on the space of equivalence classes defined by setting, for every $A,B\subseteq\mathbb{N}$,
\begin{equation*} [A]\leq_{fe} [B]\Leftrightarrow A\leq_{fe} B.\end{equation*}
It is immediate to see that the relation $\leq_{fe}$ on $\mathcal{P}(\mathbb{N})$ is not antysimmetric (e.g., $\{2n\mid n\in\mathbb{N}\}\equiv_{fe}\{2n+1\mid n\in\mathbb{N}\}$), so to search for maximal and minimal sets we will actually work in $(\mathcal{P}(\mathbb{N})/\mathord\equiv_{fe},\leq_{fe})$.\par
In \cite{fe} the authors proved that the finite embeddability has the following properties (for the relevant definitions, see \cite{rif12}):
\begin{prop}[\cite{fe}, Proposition 6]\label{trs} Let $A,B$ be sets of natural numbers.
\begin{enumerate}
[leftmargin=*,label=(\roman*),align=left ]
\item $A$ is maximal with respect to $\leq_{fe}$ if and only if it is thick;
\item if $A\leq_{fe} B$ and $A$ is piecewise syndetic then $B$ is also piecewise syndetc;
\item if $A\leq_{fe} B$ and $A$ contains a $k$-term arithmetic progression then also $B$ contains a $k$-term arithmetic progression;
\item if $A\leq_{fe} B$ then the upper Banach densities satisfy $BD(A)\leq BD(B)$;
\item if $A\leq_{fe} B$ then $A-A\subseteq B-B$;
\item if $A\leq_{fe} B$ then $\bigcap\limits_{t\in G} (A-t)\leq_{fe} \bigcap\limits_{t\in G}(B-t)$ for every finite $G\subseteq\mathbb{N}$.
\end{enumerate}
\end{prop}
We will use Proposition \ref{trs} to (re)prove some combinatorial properties of ultrafilters in $\overline{K(\beta\mathbb{N},\oplus)}$ in Section \ref{extreme}. In this present section we want to study the existence of minimal elements with respect to $\leq_{fe}$ in various subsets of $\mathcal{P}(\mathbb{N})$, and a nice property of the ordering $\leq_{fe}$ on the set of equivalence classes, namely that for every set $A$ there does not exist a set $B$ such that $[A]<_{fe} [B] <_{fe} [A+1]$. To prove this result we need the following lemma:
\begin{lem}\label{basico} For every $A,B\subseteq\mathbb{N}$ the following two properties hold:
\begin{enumerate}
[leftmargin=*,label=(\roman*),align=left ]
\item\label{a1} if $B\nleq_{fe} A$ and $B\leq_{fe} A+1$ then $B\subseteq A+1$;
\item\label{a2} if $A\leq B$ and $A+1\nleq B$ then $A\subseteq B$.
\end{enumerate}\end{lem}
\begin{proof} We prove only \ref{a1}, since \ref{a2} can be proved similarly. Let $F\subseteq B$ be a finite subset of $B$ such that $F+n\nsubseteq A$ for every $n\in\mathbb{N}$. In particular, for every finite $H\subseteq B$ such that $F\subseteq H$ and for every $n\in\mathbb{N}$ we have that $n+H\nsubseteq A$. But, by hypothesis, there exists $n\in\mathbb{N}$ such that $n+H\subseteq A+1$. If $n\geq 1$ we have a contradition, so it must be $n=0$, i.e $H\subseteq A+1$. Since this holds for every finite $H\subseteq B$ (with $F\subseteq H$) we deduce that $B\subseteq A+1$. \end{proof}
\begin{thm}\label{ledzeppelin} Let $A,B\subseteq\mathbb{N}$. If $A\leq_{fe} B\leq_{fe} A+1$ then $[A]=[B]$ or $[A+1]=[B]$. \end{thm}
\begin{proof} Let us suppose that $A+1\nleq_{fe} B\nleq_{fe} A$. Then, since $A\leq_{fe} B\leq_{fe} A+1$, by Lemma \ref{basico} we deduce that $A\subseteq B\subseteq A+1$, so $A\subseteq A+1$. This is absurd since $A\setminus (A+1)\supseteq\{\min A\}\neq\emptyset$. \end{proof}
We now turn the attention to the existence of minimal elements in various subsets of $\mathcal{P}(\mathbb{N})$. Two immediate observations are that the empty set is the minimum in $(\mathcal{P}(\mathbb{N}),\leq_{fe})$ and that $\{0\}$ is the minimum in $(\mathcal{P}(\mathbb{N})_{\geq 1}\equiv_{fe},\leq_{fe})$. Moreover, if we identify each natural number $n$ with the singleton $\{n\}$, it is immediate to see that $(\mathbb{N},\leq)$ forms an initial segment of $(\mathcal{P}_{\geq 1}(\mathbb{N}),\leq_{fe})$ and that, more in general, the following easy result holds:
\begin{prop} A set $A$ is minimal in $(\mathcal{P}_{\geq n}(\mathbb{N}),\leq_{fe})$ if and only if $0\in A$ and $|A|=n$. \end{prop}
The proof follows easily from the definitions. Let us note that, in particular, the following facts follow:
\begin{enumerate}
[leftmargin=*,label=(\roman*),align=left ]
\item for every natural number $m\geq n-1$ there are $\binom{m}{n-1}$ inequivalent minimal elements in $(\mathcal{P}_{\geq n}(\mathbb{N}),\leq_{fe})$ that are subsets of $\{0,...,m\}$;
\item if $n\geq 2$ then $(\mathcal{P}_{\geq n}(\mathbb{N}),\leq_{fe})$ does not have a minimum element.
\end{enumerate}
If we consider only infinite subsets of $\mathbb{N}$ the situation is different: there are no minimal elements in $(\mathcal{P}_{\aleph_{0}}(\mathbb{N})/\mathord\equiv_{fe},\leq_{fe})$, as we are now going to show.
\begin{defn} Let $A,B\subseteq\mathbb{N}$. We say that $A$ is strongly non f.e. in $B$ (notation: $A\nleq_{fe}^{S} B$) if for every set $C\subseteq A$ with $|C|= 2$ we have that $C\nleq_{fe}B$. If both $A\nleq_{fe}^{S} B$ and $B\nleq_{fe}^{S} A$ we say that $A,B$ are strongly mutually unembeddable (notation: $A\not\equiv_{S} B$). \end{defn}
Let us observe that, in the previous definition, we can equivalenty substitute the condition "$|C|=2$" with "$|C|\geq 2$".
\begin{prop}\label{incomparabili1} Let $X$ be an infinite subset of $\mathbb{N}$. Then there are $A,B\subseteq X$, $A,B$ infinite, such that $A\cap B=\emptyset$ and $A\not\equiv_{S} B$. \end{prop}
\begin{proof} To prove the thesis we construct $A,B\subseteq X$ such that, for any $C\subseteq A$, $D\subseteq B$ with $|C|=|D|=2$, we have $C\nleq_{fe} B$ and $D\nleq_{fe} B$.\par
Let $X=\{x_{n}\mid n\in\mathbb{N}\}$, with $x_{n}<x_{n+1}$ for every $n\in\mathbb{N}$. We set
\begin{equation*} a_{0}=x_{0}, b_{0}=x_{1} \end{equation*}
and, recursively, we set
\begin{equation*} a_{n+1}=\min\{x\in X\mid x>a_{n}+b_{n}+1\}, \ b_{n+1}=\min\{x\in X\mid x>b_{n}+a_{n+1}+1\}. \end{equation*}
Finally, we set $A=\{a_{n}\mid n\in\mathbb{N}\}$ and $B=\{b_{n}\mid n\in\mathbb{N}\}$. Clearly $A\cap B=\emptyset$, and both $A,B$ are infinite subsets of $X$. Now we let $a_{n_{1}}<a_{n_{2}}$ be any elements in $A$. Let us suppose that there are $b_{m_{1}}<b_{m_{2}}$ in $B$ with $a_{n_{2}}-a_{n_{1}}=b_{m_{2}}-b_{m_{1}}$ and let us assume that $b_{n_{2}}>a_{n_{2}}$ (if the converse hold, we can just exchange the roles of $a_{n_{1}},a_{n_{2}},b_{m_{1}},b_{m_{2}}$). By construction, since $b_{m_{2}}>a_{n_{2}}$, we have $b_{m_{2}}-b_{m_{1}}\geq a_{n_{2}}+1 >a_{n_{2}}$, while $a_{n_{2}}-a_{n_{1}}\leq a_{n_{2}}$. So $A\not\equiv_{S} B$.\end{proof}
Three corollaries follow immediatly by Proposition \ref{incomparabili1}:
\begin{cor}\label{nitrogeno} For every infinite set $X\subseteq\mathbb{N}$ there is an infinite set $A\subseteq X$ such that $X\nleq_{fe} A$. \end{cor}
\begin{proof} Let $A,B$ be infinite subsets of $X$ such that $A\not\equiv_{S} B$. Then $X$ cannot be finitely embeddable in both $A$ and $B$ otherwise, since clearly $A,B\leq_{fe} X$, we would have that $[A]=[X]=[B]$, which is absurd. \end{proof}
\begin{cor}\label{popporoppo} For every infinite set $X\subseteq\mathbb{N}$ there is an infinite descending chain $X=X_{0}\supset X_{1}\supset X_{2}...$ in $\mathcal{P}_{\aleph_{0}}(\mathbb{N})$ such that $X_{i+1}\nleq_{fe} X_{i}$ for every $i\in\mathbb{N}$. \end{cor}
\begin{proof} The result follows immediatly by Corollary \ref{nitrogeno}.\end{proof}
\begin{cor}\label{gugugaga} There are no minimal elements in $(\mathcal{P}_{\aleph_{0}}(\mathbb{N})/\mathord\equiv_{fe},\leq_{fe})$. \end{cor}
\begin{proof} The result follows immediatly by Corollary \ref{popporoppo}. \end{proof}
\section{Properties of $(\beta\mathbb{N},\leq_{fe})$}\label{extreme}
In this section we want to prove some basic properties of $(\beta\mathbb{N},\leq_{fe})$, in particular the generalization of Theorem \ref{ledzeppelin} to ultrafilters, and to characterize the maximal ultrafilters with respect to $\leq_{fe}$. We fix some notations: we will denote by $\equiv_{fe}$ the equivalence relation such that, for every $\mathcal{U},\mathcal{V}$ ultrafilters on $\mathbb{N}$,
\begin{equation*} \mathcal{U}\equiv_{fe} \mathcal{V}\Leftrightarrow \mathcal{U}\leq_{fe} \mathcal{V} \wedge \mathcal{U}\leq_{fe} \mathcal{V} \end{equation*}
and, for every ultrafilter $\mathcal{U}$, we will denote by $[\mathcal{U}]$ its equivalence class. Finally we will denote by $\leq_{fe}$ the ordering induced on the space of equivalence classes defined by setting, for every $\mathcal{U},\mathcal{V}\in\beta\mathbb{N}$,
\begin{equation*} [\mathcal{U}]\leq_{fe} [\mathcal{V}]\Leftrightarrow \mathcal{U}\leq_{fe} \mathcal{V}.\end{equation*}
\subsection{Some basic properties of $(\beta\mathbb{N},\leq_{fe})$}
The first result that we prove is that Theorem \ref{ledzeppelin} can be generalized to ultrafilters:
\begin{thm} For every $\mathcal{U},\mathcal{V}\in\beta\mathbb{N}$ if $\mathcal{U}\leq_{fe} \mathcal{V}\leq_{fe} \mathcal{U}\oplus 1$ then $[\mathcal{U}]=[\mathcal{V}]$ or $[\mathcal{U}\oplus 1]=[\mathcal{V}]$. \end{thm}
\begin{proof} Let us suppose that $\mathcal{U}\oplus 1\nleq_{fe} \mathcal{V}\nleq_{fe} \mathcal{U}$. In particular, $\mathcal{U}\oplus 1\neq \mathcal{V}$, so there exists $A\in\mathcal{U}$ such that $A+1\notin \mathcal{V}$. Since $\mathcal{V}\nleq_{fe} \mathcal{U}$ there exists $B\in\mathcal{U}$ such that $K\nleq_{fe} B$ for every $K\in\mathcal{V}$. In particular, $K\nleq A\cap B$ for every $K\in\mathcal{V}$.\par
Moreover, since $(A\cap B)+1\in\mathcal{U}\oplus 1$ we derive that there exists $C\in\mathcal{V}$ such that $C\leq_{fe} (A\cap B)+1$. So we have that
\begin{equation*} C\nleq_{fe} (A\cap B) \ \mbox{and} \ C\leq_{fe} (A\cap B)+1; \end{equation*}
by Lemma \ref{basico} we conclude that $C\subseteq (A\cap B)+1$. But $C\in\mathcal{V}$, so $(A\cap B)+1\in\mathcal{V}$ and, since $(A\cap B)+1\subseteq A+1$, this entails that $A+1\in\mathcal{V}$, which is absurd. \end{proof}
Another result that we want to prove is that $(\beta\mathbb{N},\leq_{fe})$ is not a total preorder:
\begin{prop} There are nonprincipal ultrafilters $\mathcal{U},\mathcal{V}$ such that $\mathcal{U}$ is not finitely embeddable in $\mathcal{V}$ and $\mathcal{V}$ is not finitely embeddable in $\mathcal{U}$. \end{prop}
\begin{proof} Let $A,B$ be strongly mutually unembeddable infinite sets (which existence is a consequence of Proposition \ref{incomparabili1}). Let $\mathcal{U},\mathcal{V}$ be nonprincipal ultrafilters such that $A\in\mathcal{U}, B\in\mathcal{V}$ and let us suppose that $\mathcal{U}\leq_{fe}\mathcal{V}$. Let $C\in \mathcal{U}$ be such that $C\leq_{fe} B$. Since $C\in\mathcal{U}$, $A\cap C$ is in $\mathcal{U}$ and it is infinite (since $\mathcal{U}$ is nonprincipal). So we have that
\begin{itemize}
\item $A\cap C\leq_{fe} B$, since $A\cap C\subseteq C$;
\item $A\cap C\nleq_{fe} B$, since $A\not\equiv_{S} B$.
\end{itemize}
This is absurd, so $\mathcal{U}$ is not finitely embeddable in $\mathcal{V}$. In the same way we can prove that $\mathcal{V}$ is not finitely embeddable in $\mathcal{U}$.\end{proof}
It is easy to show that, if we identity each natural number $n$ with the principal ultrafilter $\mathcal{U}_{n}=\{A\in\mathcal{P}(\mathbb{N})\mid n\in A\}$, then $(\mathbb{N},\leq)$ is an initial segment in $(\beta\mathbb{N},\leq_{fe})$. In particular, $\mathcal{U}_{0}$ is the minimum element in $\beta\mathbb{N}$. One may wonder if there is a minimum element in $(\beta\mathbb{N}\setminus\mathbb{N},\leq_{fe}),$ and the answer is no. In the following proposition, by $\Theta_{X}$ we mean the clopen set
\begin{equation*} \Theta_{X}=\{\mathcal{U}\in\beta\mathbb{N}\mid X\in\mathcal{U}\}. \end{equation*}
\begin{prop} For every infinite set $X\subseteq\mathbb{N}$ there is not a minimum in $((\Theta_{X}\setminus\mathbb{N})/\mathord\equiv_{fe},\leq_{fe})$. \end{prop}
\begin{proof} Let us suppose that such a minimum $M$ exists, and let $\mathcal{U}\in\Theta_{X}$ be such that $M=[\mathcal{U}]$. Let $A,B\subseteq X$ be mutually unembeddable subsets of $X$ and let $\mathcal{V}_{1},\mathcal{V}_{2}$ be nonprincipal ultrafilters such that $A\in \mathcal{V}_{1}$ and $B\in \mathcal{V}_{2}$ (in particular, $\mathcal{V}_{1},\mathcal{V}_{2}\in\Theta_{X}$). Since, by hypothesis, $[\mathcal{U}]$ is the minumum in $((\Theta_{X}\setminus\mathbb{N})/\mathord\equiv_{fe},\leq_{fe})$, there are $C_{1},C_{2}\in \mathcal{U}$ such that $C_{1}\leq_{fe} A$ and $C_{2}\leq_{fe} B$. Let us consider $C_{1}\cap C_{2}\in\mathcal{U}$. By construction, $C_{1}\cap C_{2}$ is finitely embeddable in $A$ and in $B$. But this is absurd: in fact, let $c_{1}<c_{2}$ be any two elements in $C_{1}\cap C_{2}$. Then there are $n,m$ such that $n+\{c_{1},c_{2}\}=\{a_{1},a_{2}\}\subset A$ and $m+\{c_{1},c_{2}\}=\{b_{1},b_{2}\}\subset B$, and this cannot happen, because in this case we would have $b_{2}-b_{1}=c_{2}-c_{1}=a_{2}-a_{1}$, while $A\not\equiv_{S} B$. \end{proof}
In particular, by taking $X=\mathbb{N}$, we prove that:
\begin{cor} There is not a minimum in $((\beta\mathbb{N}\setminus\mathbb{N})/\mathord\equiv_{fe},\leq_{fe})$.\end{cor}
\subsection{Maximal Ultrafilters}
To study maximal ultrafilters in $(\beta\mathbb{N},\leq_{fe})$ we need to recall three results that have been proved in \cite{fe}:
\begin{thm}[\cite{fe}, Theorem 10]\label{fondamentali} Let $\mathcal{U},\mathcal{V}$ be ultrafilters on $\mathbb{N}$. Then $\mathcal{U}\leq_{fe}\mathcal{V}$ if and only if $\mathcal{V}\in\overline{\{\mathcal{U}\oplus\mathcal{W}\mid \mathcal{W}\in\beta\mathbb{N}\}}$. \end{thm}
\begin{cor}[\cite{fe}, Corollary 12]\label{updir} The ordering $\leq_{fe}$ on ultrafilters on $\mathbb{N}$ is upward directed.\end{cor}
We also recall that, actually, Corollary \ref{updir} can be improved: in fact, for every $\mathcal{U},\mathcal{V}\in\beta\mathbb{N}$ we have
\begin{equation*} \mathcal{U},\mathcal{V}\leq_{fe}\mathcal{U}\oplus\mathcal{V}. \end{equation*}
Let us introduce the following definition:
\begin{defn} For any $\mathcal{U}\in\beta\mathbb{N}$ the upward cone generated by $\mathcal{U}$ is the set
\begin{equation*} \mathcal{C}(\mathcal{U})=\{\mathcal{V}\in\beta\mathbb{N}\mid \mathcal{U}\leq_{fe}\mathcal{V}\}. \end{equation*}
\end{defn}
\begin{cor}[\cite{fe}, Corollary 13]\label{hui} For any $\mathcal{U}\in\beta\mathbb{N}$, the upward cone $\mathcal{C}(\mathcal{U})$ is a closed, two-sided ideal in $\beta\mathbb{N}$. It is the smallest closed right ideal containing $\mathcal{U}$ and therefore it is also the smallest two-sided ideal containing $\mathcal{U}$.\end{cor}
Let us note that from Theorem \ref{fondamentali} it easily follows that the relation $\leq_{fe}$ is not antisymmetric: in fact, if $R$ is a minimal right ideal in $(\beta\mathbb{N},\oplus)$ and $\mathcal{U}\in R$ then $\mathcal{C}(\mathcal{U})=\mathcal{C}(\mathcal{U}\oplus 1)$, so $\mathcal{U}\leq_{fe}\mathcal{U}\oplus 1$ and $\mathcal{U}\oplus 1\leq_{fe} \mathcal{U}$.\par
We want to prove that there is a maximum in $(\beta\mathbb{N}/\mathord\equiv_{fe},\leq_{fe})$. Due to Corollary \ref{hui}, since $(\beta\mathbb{N}/\mathord\equiv_{fe},\leq_{fe})$ is an order then to prove that it has a maximum if is enough\footnote{An upward directed ordered set $(A,\leq)$ has at most one maximal element which, if it exists, is the greatest element of the order.} to prove that it has maximal elements.\par
To prove the existence of maximal elements we use Zorn's Lemma. A technical lemma that we need is the following:
\begin{lem}\label{ultraordine} Let $I$ be a totally ordered set. Then there is an ultrafilter $\mathcal{V}$ on $I$ such that, for every element $i\in I$, the set
\begin{equation*} G_{i}=\{j\in I\mid j\geq i\}.\end{equation*}
is included in $\mathcal{V}$.
\end{lem}
\begin{proof} We have just to observe that $\{G_{i}\}_{i\in I}$ is a filter and to recall that every filter can be extended to an ultrafilter.\end{proof}
The key property of these ultrafilters is the following:
\begin{prop}\label{ultraordine2} Let $I$ be a totally ordered set and let $\mathcal{V}$ be given as in Lemma \ref{ultraordine}. Then for every $A\in \mathcal{V}$ and $i\in I$ there exists $j\in A$ such that $i\leq j$. \end{prop}
We omit the straightforward proof.\par
In the next Theorem we use the notion of limit ultrafilter. We recall that, given an ordered set $I$, an ultrafilter $\mathcal{V}$ on $I$ and a family $\mathcal{U}_{i}$ of ultrafilters on $\mathbb{N}$, the $\mathcal{V}$-limit of the family $\langle \mathcal{U}_{i}\mid i\in I\rangle$ (denoted by $\mathcal{V}-\lim\limits_{i\in I}\mathcal{U}_{i}$) is the ultrafilter such that, for every $A\subseteq\mathbb{N}$,
\begin{equation*} A\in\mathcal{V}-\lim\limits_{i\in I}\mathcal{U}_{i}\Leftrightarrow\{i\in I\mid A\in\mathcal{U}_{i}\}\in\mathcal{V}. \end{equation*}
Let us introduce the notion of $\leq_{fe}$-chain:
\begin{defn} Let $(I,<)$ be an ordered set. We say that the family $\langle \mathcal{U}_{i}\mid i\in I\rangle$ is an $\leq_{fe}$-chain if for every $i<j\in I$ we have $\mathcal{U}_{i}\leq_{fe}\mathcal{U}_{j}$.\end{defn}
\begin{thm} Every $\leq_{fe}$-chain $\langle \mathcal{U}_{i}\mid i\in I\rangle$ has an $\leq_{fe}$-upper bound $\mathcal{U}$. \end{thm}
\begin{proof} Let $\mathcal{V}$ be an ultrafilter on $I$ with the property expressed in Lemma \ref{ultraordine}. We claim that the ultrafilter
\begin{equation*} \mathcal{U}=\mathcal{V}-\lim\limits_{i\in I} \mathcal{U}_{i} \end{equation*}
is an $\leq_{fe}$-upper bound for the $\leq_{fe}$-chain $\langle \mathcal{U}_{i}\mid i\in I\rangle$. We have to prove that $\mathcal{U}_{i}\leq_{fe} \mathcal{U}$ for every index $i$; let $A$ be an element of $\mathcal{U}$. By definition,
\begin{equation*} A\in \mathcal{U}\Leftrightarrow I_{A}=\{i\in I\mid A\in \mathcal{U}_{i}\} \in \mathcal{V}. \end{equation*}
$I_{A}$ is a set in $\mathcal{V}$ so, by Proposition \ref{ultraordine2}, there is an element $j>i$ in $I_{A}$. Therefore $A\in\mathcal{U}_{j}$ and, since $\mathcal{U}_{i}\leq_{fe} \mathcal{U}_{j}$, there exists an element $B$ in $\mathcal{U}_{i}$ with $B\leq_{fe} A$. Hence $\mathcal{U}_{i}\leq_{fe}\mathcal{U}$, and the thesis is proved.\end{proof}
As an immediate consequence we have that:
\begin{cor} Every $\leq_{fe}$-chain $\langle [\mathcal{U}_{i}]\mid i\in I\rangle$ has an upper bound $[\mathcal{U}]$. \end{cor}
Being an upward directed set with maximal elements, $(\beta\mathbb{N}/\mathord{\equiv_{fe}},\leq_{fe})$ has a maximum, that we denote by $M$.
\begin{defn} We say that an ultrafilter $\mathcal{U}$ on $\mathbb{N}$ is maximal if $[\mathcal{U}]=M$. We denote by $\mathcal{M}$ the set of maximal ultrafilters.\end{defn}
By definition, for every ultrafilter $\mathcal{U}$ we have the following equivalences:
\begin{equation*} [\mathcal{U}]=M\Leftrightarrow \mathcal{U}\in\mathcal{M}\Leftrightarrow \mathcal{V}\leq_{fe}\mathcal{U} \ \mbox{for every} \ \mathcal{V}\in\beta\mathbb{N}. \end{equation*}
In particular, we can characterize $\mathcal{M}$ in terms of the $\leq_{fe}$-cones:
\begin{cor}\label{zumpa} $\mathcal{M}=\bigcap\limits_{\mathcal{U}\in\beta\mathbb{N}} \mathcal{C}(\mathcal{U}).$\end{cor}
\begin{proof} We have just to observe that $\mathcal{M}\subseteq\mathcal{C}(\mathcal{U})$ for every ultrafilter $\mathcal{U}$ and that, if $\mathcal{U}$ is a maximal ultrafilter, then $\mathcal{C}(\mathcal{U})=\mathcal{M}$.\end{proof}
We can now prove our main result:
\begin{thm}\label{eccolo} $\mathcal{M}=\overline{K(\beta\mathbb{N},\oplus)}$.
\end{thm}
\begin{proof} Given any ultrafilter $\mathcal{U}$, by Proposition \ref{fondamentali} we know that $\mathcal{C}(\mathcal{U})$ is the minimal closed bilateral ideal containing $\mathcal{U}$. By Corollary $\ref{zumpa}$ we know that $\mathcal{M}=\bigcap\limits_{\mathcal{U}\in\beta\mathbb{N}}\mathcal{C}(\mathcal{U})$ so, in particular, being the intersection of a family of closed bilateral ideal $\mathcal{M}$ itself is a closed bilater ideal. So if $\mathcal{U}$ is any ultrafilter in $K(\beta\mathbb{N},\oplus)$, we know that:
\begin{enumerate}
[leftmargin=*,label=(\roman*),align=left ]
\item $\mathcal{M}\subseteq\mathcal{C}(\mathcal{U})$;
\item $\mathcal{C}(\mathcal{U})=\overline{K(\beta\mathbb{N},\oplus)}$.
\end{enumerate}
So $\mathcal{M}$ is a closed bilateral ideal included in $\overline{K(\beta\mathbb{N},\oplus)}$, and the only such ideal is $\overline{K(\beta\mathbb{N},\oplus)}$ itself.\end{proof}
This result has a few interesting consequences:
\begin{cor}\label{uno} An ultrafilter $\mathcal{U}$ is maximal if and only if every element $A$ of $\mathcal{U}$ is piecewise syndetic. \end{cor}
\begin{proof} This follows from this well-known characterization of $\overline{K(\beta\mathbb{N},\oplus)}$: an ultrafilter $\mathcal{U}$ is in $\overline{K(\beta\mathbb{N},\oplus)}$ if and only if every element $A$ of $\mathcal{U}$ is piecewise syndetic (see, e.g., \cite{rif12}). \end{proof}
As mentioned in the introduction, the notion of finite embeddability is related with some properties that arose in combinatorial number theory. A particularity of maximal ultrafilters is that every set in a maximal ultrafilter satisfies many of these combinatorial properties:
\begin{defn} We say that a property $P$ is $\leq_{fe}$-upward invariant if the following holds: for every $A,B\subseteq \mathbb{N}$, if $P(A)$ holds and $A\leq_{fe} B$ then $P(B)$ holds.\par
We way that $P$ is partition regular if the family $S_{P}=\{A\subseteq\mathbb{N}\mid P(A)$ holds$\}$ contains an ultrafilter (i.e., if for every finite partition $\mathbb{N}=A_{1}\cup... \cup A_{n}$ there exists at least one index $i\leq n$ such that $A_{i}\in S_{P})$.\end{defn}
By Proposition \ref{trs} it follows that the following properties are $\leq_{fe}$-upward invariant:
\begin{enumerate}
[leftmargin=*,label=(\roman*),align=left ]
\item $A$ is thick;
\item\label{ps} $A$ is piecewyse syndetic;
\item\label{alap} $A$ contains arbitrarily long arithmetic progressions;
\item\label{bd} $BD(A)>0$, where $BD(A)$ is the upper Banach density of $A$.
\end{enumerate}
In particular, properties \ref{ps}, \ref{alap}, \ref{bd} are also partition regular. These kind of properties are important in relation with maximal ultrafilters:
\begin{prop}\label{consequence} Let $P$ be a partition regular $\leq_{fe}$-upward invariant property of sets. Then for every maximal ultrafilter $\mathcal{U}$, for every $A\in\mathcal{U}$, $P(A)$ holds.\end{prop}
\begin{proof} Let $P$ be given, let $S_{P}=\{A\subseteq\mathbb{N}\mid P(A)$ holds$\}$ and let $\mathcal{V}\subseteq S_{P}$ (such an ultrafilter exists because $P$ is partition regular). Let $B\in\mathcal{U}$. Since $\mathcal{U}$ is maximal, $\mathcal{V}\leq_{fe}\mathcal{U}$. Let $A\in\mathcal{V}$ be such that $A\leq_{fe} B$. Since $P$ is $\leq_{fe}$-upward invariant and $P(A)$ holds, we obtain that $P(B)$ holds, hence we have the thesis. \end{proof}
E.g., as a consequence of Proposition \ref{consequence} we can prove the following:
\begin{cor}\label{due} Let $\mathcal{U}\in\overline{K(\beta\mathbb{N},\oplus)}$. Then:
\begin{enumerate}
[leftmargin=*,label=(\roman*),align=left ]
\item each set $A$ in $\mathcal{U}$ has positive Banach density;
\item each set $A$ in $\mathcal{U}$ contains arbitrarily long arithmetic progressions;
\item each set $A$ in $\mathcal{U}$ is piecewise syndetic.
\end{enumerate}
\end{cor}
In particular, by combining Corollaries \ref{uno} and \ref{due} we obtain an alternative proof of the following known results:
\begin{itemize}
\item every piecewise syndetic set contains arbitrarily long arithmetic progressions;
\item every piecewise syndetic set has positive upper Banach density.
\end{itemize}
In the forthcoming paper \cite{functions} we will show how, actually, similar arguments can be used to prove combinatorial properties of other families of ultrafilters, e.g. to prove that for every ultrafilter $\mathcal{U}\in\overline{K(\beta\mathbb{N},\odot)}$, for every $A\in\mathcal{U}$, $A$ contains arbitrarily long arithmetic progression and it contains a solution to every partition regular homogeneous equation\footnote{An equation $P(x_{1},...,x_{n})=0$ is partition regular if and only if for every finite coloration $\mathbb{N}=C_{1}\cup...\cup C_{n}$ of $\mathbb{N}$ there exists an index $i$ and monocromatic elements $a_{1},...,a_{n}\in C_{i}$ such that $P(a_{1},...,a_{n})=0$.}.
\section{A Direct Nonstandard Proof that $M_{fe}=\overline{K(\beta\mathbb{N},\oplus)}$}\label{NS23}
In this section we assume the reader to be familiar with the basics of nonstandard analysis. In particular, we will use the notions of nonstandard extension of subsets of $\mathbb{N}$ and the transfer principle. We refer to \cite{rif5} and \cite{davis} for an introduction to the foundations of nonstandard analysis and to the nonstandard tools that we are going to use.\par
Both in \cite{fe} and in \cite{Tesi} it has been shown that the relation of finite embeddability between sets has a very nice characterization in terms of nonstandard analysis, which allows to study some of its properties in a quite simple, and elegant, way. We recall the characterization (in the following proposition, it is assumed for technical reasons that the nonstandard extension that we consider satisfies at least the $\mathfrak{c}^{+}$-enlarging property\footnote{We recall that a nonstandard extension $^{*}\mathbb{N}$ of $\mathbb{N}$ has the $\mathfrak{c}^{+}$ enlarging property if, for every family $\mathcal{F}$ of subsets of $\mathbb{N}$ with the finite intersection property, the intersection $\bigcap\limits_{A\in\mathcal{F}}$$^{*}A$ is nonempty.}, where $\mathfrak{c}$ is the cardinality of $\mathcal{P}(\mathbb{N})$):
\begin{prop}[\cite{fe}, Proposition 15]\label{NSCAR} Let $A,B$ be subsets of $\mathbb{N}$. The following two conditions are equivalent:
\begin{enumerate}
[leftmargin=*,label=(\roman*),align=left ]
\item $A$ is finitely embeddable in $B$;
\item there is an hypernatural number $\alpha$ in $^{*}\mathbb{N}$ such that $\alpha+A\subseteq$$^{*}B$.
\end{enumerate}
\end{prop}
We use Proposition \ref{NSCAR} to reprove directly, with nonstandard methods, Theorem \ref{eccolo}:
\begin{proof}[Theorem \ref{eccolo}] Let $A$ be a set in $\mathcal{U}$, and let $\mathcal{V}$ be an ultrafilter on $\mathbb{N}$. Since $A$ is piecewise syndetic there is a natural number $n$ such that
\begin{equation*} T=\bigcup_{i=1}^{n} (A+i) \end{equation*}
is thick. By transfer\footnote{Thick set can be characterized by mean of nonstandard analysis as follows (see e.g. \cite{Tesi}): a set $T\subseteq\mathbb{N}$ is thick if and only if $T^{*}$ contains an interval of infinite lenght.} it follows that there are hypernatural numbers $\alpha\in$$^{*}\mathbb{N}$ and $\eta\in$$^{*}\mathbb{N}\setminus\mathbb{N}$ such that the interval $[\alpha,\alpha+\eta]$ is included in $^{*}T$. In particular, since $\eta$ is infinite, $\alpha+\mathbb{N}\subseteq$$^{*}T$.\par
For every $i\leq n$ we consider
\begin{equation*} B_{i}=\{n\in\mathbb{N}\mid \alpha+n\in\mbox{}^{*}(A+i)\}. \end{equation*}
Since $\bigcup_{i=1}^{n} B_{i}=\mathbb{N}$, there is an index $i$ such that $B_{i}\in\mathcal{V}$. We claim that $B_{i}\leq_{fe} A$. In fact, by construction $\alpha+B_{i}\subseteq$$^{*}A+i$, so
\begin{equation*} (\alpha-i)+B_{i}\subseteq\mbox{}^{*}A. \end{equation*}
By Proposition \ref{NSCAR}, this entails that $B_{i}\leq_{fe} A$, and this proves that $\mathcal{V}\leq_{fe}\mathcal{U}$ for every ultrafilter $\mathcal{V}$. Hence $\mathcal{U}$ is maximal.\end{proof}
In bibliografia devo aggiungere un lavoro di Beiglbock ed uno di Krautzberger
\end{document} |
\begin{document}
\section{Introduction}
One of the main tasks of statistical modeling is to exploit the association between a response variable and multiple predictors. Linear model (LM), as a simple parametric regression model, is often used to capture linear dependence between response and predictors. The other two common models: generalized linear model (GLM) and Cox's proportional hazards (CoxPH) model, can be considered as the extensions of linear model, depending on the types of responses. Parameter estimation in these models can be computationally intensive when the number of predictors is large. Meanwhile, Occam's razor is widely accepted as a heuristic rule for statistical modeling, which balances goodness of fit and model complexity. This rule leads to a relative small subset of important predictors.
The canonical approach to subset selection problem is to choose $k$ out of $p$ predictors for each $k \in \{0,1,2,\dots,p\}$. This involves exhaustive search over all possible $2^p$ subsets of predictors, which is an NP-hard combinatorial optimization problem.
To speed up, \cite{furnival1974regressions} introduced a well-known branch-and-bound algorithm
with an efficient updating strategy for LMs, which was later implemented by \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}roglang{R} packages such as the \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}kg{leaps} \citep{lumley2017leaps} and the \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}kg{bestglm} \citep{mcleod2010bestglm}. Yet for GLMs, a simple exhaustive screen is undertaken in \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}kg{bestglm}. When the exhaustive screening is not feasible for GLMs, fast approximating approaches have been proposed based on a genetic algorithm. For instance, \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}kg{kofnGA}\citep{wolters2015a} implemented a genetic algorithm to search for a best subset of a pre-specified model size $k$, while \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}kg{glmuti} \citep{calcagno2010glmulti} implemented a genetic algorithm to automatically select the best model for GLMs with no more than 32 covariates.
These packages can only deal with dozens of predictors but not high-dimensional data arising in modern statistics. Recently, \cite{bertsimas2016best} proposed a mixed integer optimization approach to find feasible best subset solutions for LMs with relatively larger $p$, which relies on certain third-party integer optimization solvers. Alternatively, regularization strategy is widely used to transform the subset selection problem into computational feasible problem. For example, \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}kg{glmnet} \citep{friedman2010regularization, simon2011regularization} implemented a coordinate descent algorithm to solve the LASSO problem, which is a convex relaxation by replacing the cardinality constraint in best subset selection problem by the $L_1$ norm.
In this paper, we consider a primal-dual active set (PDAS) approach to solve the best subset selection problem for LM, GLM and CoxPH models. The PDAS algorithm for linear least squares problems was first introduced by \cite{ito2013variational} and later discussed by \cite{jiao2015primal}, \cite{Huang-et-al-2017} and \cite{ghilli2017monotone}. It utilizes an active set updating strategy and fits the sub-models through use of complementary primal and dual variables. We generalize the PDAS algorithm for general convex loss functions with the best subset constraint, and further extend it to support both sequential and golden section search strategies for optimal $k$ determination. We develop a new package \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}kg{BeSS} (BEst Subset Selection, \cite{wen2017bess}) in the \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}roglang{R} programming system \citep{R} with \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}roglang{C++} implementation of PDAS algorithms and memory optimized for sparse matrix output. This package is publicly available from the Comprehensive \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}roglang{R} Archive Network (CRAN) at \bfm{u}} \def\v{\bfm{v}} \def\w{\bfm{w}} \def\x{\bfm{x}} \def\y{\bfm{y}rl{https://cran.r-project.org/package=BeSS}.
We demonstrate through enormous datasets that \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}kg{BeSS} is efficient and stable for high dimensional data, and may solve best subset problems with $n$ in 1000s and $p$ in 10000s in just seconds on a single personal computer.
The article is organized as follows. In Section 2, we provide a general primal-dual formulation for the best subset problem that includes linear, logistic and CoxPH models as special cases. Section 3 presents the PDAS algorithms and related technical details. Numerical experiments with enormous simulations and real datasets are conducted in Section 4. We conclude with a short discussion in Section 5.
\section{Primal-dual formulation}\label{sec:2}
The best subset selection problem with the subset size $k$ is given by the following optimization problem:
\begin{equation}\label{eqn:best}
\min_{\beta \in \mathbb{R}^p} \ l(\bbeta) \quad \text{ s.t. } \quad \|\bbeta\|_0 = k,
\end{equation}
where $l(\bbeta)$ is a convex loss function of the model parameters $\bbeta \in \mathbb{R}^p$
and $k$ is an unknown positive integer. The $L_0$ norm $\|\bbeta\|_0 =\sum_{j=1}^p |\beta_j|_0 = \sum_{j=1}^p 1_{\beta_j\neq0}$ counts the number of nonzeros in $\bbeta$.
It is known that the problem (\ref{eqn:best}) admits non-unique local optimal solutions, among which the coordinate-wise minimizers possess promising properties. For a coordinate-wise minimizer $\bbeta^\diamond$, denote the vectors of gradient and Hessian diagonal by
\begin{equation}\label{gh}
\g^\diamond = \nabla l (\bbeta^\diamond), \quad
\h^\diamond = \mbox{\rm diag}(\nabla^2 l (\bbeta^\diamond)),
\end{equation}
respectively. For each coordinate $j=1,\ldots,p$, write $l_j(t) = l(\beta_1^\diamond, \dots, \beta_{j-1}^\diamond, t, \beta_{j+1}^\diamond, \dots, \beta_p^\diamond)$
while fixing the other coordinates.
Then the local quadratic approximation of $l_j(t)$ around $\beta_j^\diamond$ is given by
\begin{equation}\label{eqn:quadapprox}
\bfm{a}} \def\b{\bfm{b}} \def\c{\bgk{c}} \def\d{\bfm{d}} \def\e{\bfm{e}rraycolsep=1.4pt\def2.0{2.0}
\begin{array}{l l}
l_j^Q(t) & = l_j(\beta_j^\diamond) + g_j^\diamond (t-\beta_j^\diamond) + \bfm{f}} \def\g{\bfm{g}} \def\h{\bfm{h}} \def\i2{\bfm{i}} \def\j{\bfm{j}rac{1}{2} h_j^\diamond (t - \beta_j^\diamond)^2\\
&= \dfrac{1}{2} h_j^\diamond \left(t - \beta_j^\diamond + g_j^\diamond/h_j^\diamond \right)^2 + l_j(\beta_j^\diamond) - \dfrac{1}{2}[g_j^\diamond] ^2/h_j^\diamond \\
&= \dfrac{1}{2} h_j^\diamond \left(t - (\beta_j^\diamond + \gamma_j^\diamond) \right)^2 + l_j(\beta_j^\diamond) - \dfrac{1}{2}[g_j^\diamond] ^2/h_j^\diamond,
\end{array}
\end{equation}
which gives rise of an important quantity $\gamma_j^\diamond$ of the following scaled gradient form
\begin{equation}\label{gamma}
\gamma_j^\diamond = - g_j^\diamond/h_j^\diamond.
\end{equation}
Minimizing the objective function $l_j^Q(t)$ yields $t_j^*= \beta_j^\diamond +\gamma_j^\diamond$ for $j=1,\ldots,p$.
The constraint in (\ref{eqn:best}) says that there are $p-k$ components of $\{t_j^*, j=1,\ldots,p\}$ that would be enforced to be zero. To determine which $p-k$ components, we consider the sacrifices of $l_j^Q(t)$ when switching each $t_j^*$ from $ \beta_j^\diamond +\gamma_j^\diamond$ to $0$, which are given by
\begin{equation}\label{eqn:Delta}
\Delta_j^\diamond =\bfm{f}} \def\g{\bfm{g}} \def\h{\bfm{h}} \def\i2{\bfm{i}} \def\j{\bfm{j}rac{1}{2} h_j^\diamond (\beta_j^\diamond +\gamma_j^\diamond)^2, \quad j=1,\ldots,p.
\end{equation}
Among all the candidates, we may enforce those $t_j^*$'s to zero
if they contribute the {\em least total sacrifice} to the overall loss.
To realize this, let $\Delta_{[1]}^\diamond\geq \cdots \geq \Delta_{[p]}^\diamond$ denote the decreasing rearrangement of $\Delta_j^\diamond$ for $j=1,\ldots,p$, then truncate the ordered sacrifice vector at position $k$.
Combining the analytical result by (\ref{eqn:quadapprox}), we obtain that
\begin{equation}\label{eqn:hard}
\beta_j^\diamond = \left\{
\begin{array}{l l}
\beta_j^\diamond + \gamma_j^\diamond , & \text{ if } \Delta_j^\diamond\geq \Delta_{[k]}^\diamond \\
0, & \text{ otherwise}.
\end{array} \right.
\end{equation}
In (\ref{eqn:hard}), we treat $\bbeta^\diamond = (\beta_1^\diamond, \ldots, \beta_p^\diamond)$ as primal variables, $\ggamma^\diamond = (\gamma_1^\diamond, \ldots, \gamma_p^\diamond)$ as dual variables, and $\bm\Delta^\diamond = (\Delta_1^\diamond, \ldots, \Delta_p^\diamond)$ as reference sacrifices.
Next we provide their explicit expressions for three important statistical models.
\noindent \textbf{Case 1: Linear regression}.
Consider the LM $\y=\X\bbeta + \veve$ with design matrix $\X\in\RRR^{n\times p}$ and i.i.d. errors.
Here $\X$ and $\y$ are standardized such that the intercept term is removed from the model and each column of $\X$ has $\sqrt{n}$ norm.
Take the loss function $l(\bbeta)=\bfm{f}} \def\g{\bfm{g}} \def\h{\bfm{h}} \def\i2{\bfm{i}} \def\j{\bfm{j}rac{1}{2n}\|\y-\X\bbeta\|^2$. For $j=1,\ldots,p$, it is easy to obtain
\begin{equation}\label{eqn:gh:lm}
g_j^\diamond = \bfm{f}} \def\g{\bfm{g}} \def\h{\bfm{h}} \def\i2{\bfm{i}} \def\j{\bfm{j}rac{1}{n} \X_{(j)}^T(\X\bbeta-\y), \quad h_j^\diamond= 1,
\end{equation}
where
$\X_{(j)}$ denotes the $j$th column of $\X$, so
\begin{equation}\label{eqn:delta:lm}
\gamma_j^\diamond = \bfm{f}} \def\g{\bfm{g}} \def\h{\bfm{h}} \def\i2{\bfm{i}} \def\j{\bfm{j}rac{1}{n} \X_{(j)}^\top(\y - \X\bbeta), \quad \Delta_j^\diamond =\bfm{f}} \def\g{\bfm{g}} \def\h{\bfm{h}} \def\i2{\bfm{i}} \def\j{\bfm{j}rac{1}{2}(\beta_j^\diamond + \gamma_j^\diamond)^2.
\end{equation}
\noindent \textbf{Case 2: Logistic regression}.
Consider the GLM
$$
\log(p(\x)/(1-p(\x))) = \beta_0 + \x^\top\bbeta, \quad \x\in\RRR^p
$$
with $p(\x) = \mbox{\rm Prob}(Y=1|\x)$. Given the data $\big\{(\x_i, y_i)\big\}_{i=1}^n$ with binary responses $y_i\in\{0,1\}$, the negative log-likelihood function is given by
\begin{equation}\label{eqn:loglike:glm}
l(\beta_0, \bbeta) = - \sum_{i=1}^n \Big\{ y_i(\beta_0 + \x_i^\top\bbeta) - \log(1+\exp(\beta_0 + \x_i^\top\bbeta)) \Big\}.
\end{equation}
We give only the primal-dual quantities for $\bbeta\in\RRR^p$ according to the $L_0$ constraint in (\ref{eqn:best}), while leaving $\beta_0$ to be estimated by unconstrained maximum likelihood method. For $j=1,\ldots,p$,
\begin{equation}\label{eqn:gh:glm}
g_j^\diamond = -\sum_{i=1}^nx_{ij}(y_i - p_i^\diamond), \quad h_j^\diamond = \sum_{i=1}^n x_{ij}^2p_i^\diamond (1-p_i^\diamond)
\end{equation}
where $p_i^\diamond ={\exp(\beta_0 + \x_i^\top\bbeta^\diamond)}/{(1 + \exp(\beta_0 + \x_i^\top\bbeta^\diamond))}$ denotes the $i$-th predicted probability. Then,
\begin{equation}\label{eqn:delta:glm}
\gamma_j^\diamond = \dfrac{\sum_{i=1}^n x_{ij} (y_i - p_i^\diamond) }{ \sum_{i=1}^n x_{ij}^2 p_i^\diamond(1-p_i^\diamond)}, \quad\Delta_j^\diamond = \bfm{f}} \def\g{\bfm{g}} \def\h{\bfm{h}} \def\i2{\bfm{i}} \def\j{\bfm{j}rac{1}{2}{\sum_{i=1}^n x_{ij}^2 p_i^\diamond(1-p_i^\diamond)} (\beta_j^\diamond+\gamma_j^\diamond)^2.
\end{equation}
\noindent \textbf{Case 3: CoxPH regression.} Consider the CoxPH model
$$
\lambda(t|\x) = \lambda_0(t) \exp(\x^\top\bbeta), \quad \x\in\RRR^p
$$
with an unspecified baseline hazard $\lambda_0(t)$. Given the data $\{(T_i,\delta_i, \x_i): i=1,\dots,n\}$ with observations of survival time $T_i$ and censoring indicator $\delta_i$,
by the method of partial likelihood \citep{cox1972regression}, the parameter $\bbeta$ can be obtained by minimizing the following convex loss
\begin{equation}\label{eqn:cox:like}
l(\bbeta) = - \sum_{i: \delta_i=1}\bigg(\x_i^\top\bbeta - \log\bigg(\sum_{i': T_{i'}\geq T_i} \exp(\x_{i'}^\top\beta) \bigg) \bigg).
\end{equation}
By writing $\omega_{i,i'}^\diamond = \exp(\x_{i'}^\top\bbeta^\diamond)/\sum_{i': T_{i'}\geq T_i} \exp(\x_{i'}^\top\bbeta^\diamond)$, it can be verified that
\begin{align}\label{eqn:gh:cox}
g_j^\diamond & = - \sum_{i: \delta_i=1}\bigg(x_{ij} - \sum_{i': T_{i'}\geq T_i} \omega_{i,i'}^\diamond x_{i'j}\bigg)\\
h_j^\diamond & = \sum_{i: \delta_i=1} \sum_{i': T_{i'}\geq T_i} \omega_{i,i'}^\diamond\bigg(x_{i'j} - \sum_{i': T_{i'}\geq T_i}
\omega_{i,i'}^\diamond x_{i'j}\bigg)^2
\end{align}
so that $\gamma_j^\diamond = -g_j^\diamond/h_j^\diamond$ and $\Delta_j^\diamond = \bfm{f}} \def\g{\bfm{g}} \def\h{\bfm{h}} \def\i2{\bfm{i}} \def\j{\bfm{j}rac{1}{2}h_j^\diamond(\beta_j^\diamond + \gamma_j^\diamond)^2$ for $j=1,\ldots,p$.
\section{Active set algorithm}
For the best subset problem (\ref{eqn:best}), define the active set $\calA = \{j: \beta_j \neq 0\}$ with cardinality $k$ and the inactive set $\calI = \{j: \beta_j = 0\}$ with cardinality $p-k$.
For the coordinate-wise minimizer $\bbeta^\diamond$ satisfying (\ref{eqn:hard}), we have that
\begin{enumerate}[(C1)]
\item \label{itm:C1} $\beta_j^\diamond = 0$ when $j\in \calI$;
\item $\gamma_j^\diamond= 0$ when $j\in \calA $;\label{con:C2}
\item $\Delta_j^\diamond \geq \Delta_{j'}^\diamond$ whenever $j\in\calA$ and $j' \in\calI$.\label{con:C3}
\end{enumerate}
By (C\ref{itm:C1}) and (C2), the primal variables $\beta_j^\diamond$'s and the dual variables $\gamma_j^\diamond$'s have complementary supports. (C3) can be viewed as a local stationary condition.
These three conditions lay the foundation for the primal-dual active set algorithm we develop in this section.
Let $\calA$ be a candidate active set. By (C1), we may estimate the $k$-nonzero primal variables by standard convex optimization:
\begin{equation}\label{eqn:exactmin}
\bm{\hat\beta} =
\bfm{a}} \def\b{\bfm{b}} \def\c{\bgk{c}} \def\d{\bfm{d}} \def\e{\bfm{e}rgmin_{\bbeta_{\calI}= \bfm{z}ero} l(\bbeta), \quad\mbox{where } \calI = \calA^c.
\end{equation}
Given $\bm{\hat\beta}$, the $\g$ and $\h$ vectors (\ref{gh}) can be computed, with their explicit expressions derived for linear, logistic and CoxPH models in the previous section. The $\ggamma$ and $\bm\Delta$ vectors are readily obtainable by (\ref{gamma}), (\ref{eqn:Delta}) and (C2). Then we may check if (C3) is satisfied; otherwise, update the active and inactive sets by
\begin{equation}\label{eqn:act}
\calA\leftarrow\left\{j: \Delta_j \geq \Delta_{[k]}\right\} , \qquad \calI\leftarrow \left\{j: \Delta_j < \Delta_{[k]} \right\}.
\end{equation}
This leads to the following iterative algorithm.
\noindent\rule{\textwidth}{1.5pt}
\textbf{Algorithm 1} Primal-dual active set (PDAS) algorithm\\
\noindent\rule{\textwidth}{0.8pt}
\begin{enumerate}
\item Specify the cardinality $k$ of the active set and the maximum number of iterations $m_{\max}$.
Initialize $\calA$ to be a random $k$-subset of $\{1,\ldots,p\}$ and $\calI = \calA^c$.
\item For $m = 1,2,\dots, m_{\max}$, do
\begin{enumerate}[(2.a)]
\item Estimate $\bm{\hat\beta}$ by (\ref{eqn:exactmin});
\item Compute $\g, \h, \ggamma, \bm\Delta$;
\item Update $\calA, \calI$ by (\ref{eqn:act});
\item If $A$ is invariant, stop.
\end{enumerate}
\item Output $\{\calA, \bm{\hat\beta}, \bm\Delta\}$.
\end{enumerate}
\noindent\rule{\textwidth}{1.5pt}
\begin{remark}
The proposed PDAS algorithm is close to the primal-dual active set strategy first developed by \cite{ito2013variational}, but different from their original algorithm in two main aspects. First, our PDAS algorithm is derived from the quadratic argument (\ref{eqn:quadapprox}) and it involves the second-order partial derivatives (i.e. Hessian diagonal $\h$). Second, our algorithm extends the original linear model setting to the general setting with convex loss functions.
\end{remark}
\subsection{Determination of optimal $k$}\label{sec:tuning}
The subset size $k$ is usually unknown in practice, thus one has to determine it in a data-driven manner. A heuristic way is using the cross-validation technique to achieve the best prediction performance. Yet it is time consuming to conduct the cross-validation method especially for high-dimensional data. An alternative way is to run the PDAS algorithm from small to large $k$ values, then identify an optimal choice according to some criteria, e.g., Akaike information criterion (\cite{akaike1974new}, AIC) and Bayesian information criterion (\cite{schwarz1978estimating}, BIC) and extended BIC (\cite{chen2008extended, chen2012extended}, EBIC) for small-$n$-large-$p$ scenarios.
This leads to the sequential PDAS algorithm.
\noindent\rule{\textwidth}{1.5pt}
\textbf{Algorithm 2} Sequential primal-dual active set (SPDAS) algorithm\\
\noindent\rule{\textwidth}{0.8pt}
\begin{enumerate}
\item Specify the maximum size $k_{\max}$ of the active set, and initialize $\calA^0=\emptyset$.
\item For $k = 1,2,\dots, k_{\max}$, do
\hspace{0.5cm} \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}arbox{0.9\textwidth}{
Run \textbf{PDAS} with initial $\calA^{k-1}\cup \{j \in\calI^{k-1}: j \in \mbox{arg}\max \Delta_j^{k-1}\} $. Denote the output by $\{\calA^{k}, \bbeta^k, \bm\Delta^k\}$.
}
\item Output the optimal choice $\{\calA^*, \bbeta^*, \bm\Delta^*\}$ that attains the minimum AIC, BIC or EBIC.
\end{enumerate}
\noindent\rule{\textwidth}{1.5pt}
\begin{figure}
\caption{Plot of the loss function against the model complexity $k$ and solution path for each coefficients. The orange vertical dash line indicates number of true nonzero coefficients. }
\label{fig:loss}
\end{figure}
To alleviate the computational burden of determining $k$ as in SPDAS, here we provide an alternative method: the golden section search algorithm. We begin by plotting the loss function $l(\bbeta)$ as a function of $k$ for a simulated data from linear model with standard Gaussian error. The true coefficient $\bbeta=(3, 1.5, 0,0,-2,0,0,0,-1,0,\dots,0)$ and the design matrix $\X$ is generated as in Section \ref{sec:simulate} with $\rho=0.2$. From Figure~\ref{fig:loss}, it can be seen that the slope of the loss plot goes from steep to flat and there is an `elbow' exists near the true number of active set, i.e., $k=4$.
The solution path for the same data is presented at the bottom of Figure~\ref{fig:loss} for a better visualization on the relationship between loss function and coefficient estimation. When a true active predictor is included in the model, the loss function drops dramatically and the predictors already in the model adjust their estimates to be close to the true values. When all the active predictors are included in the model, their estimates would not change much as $k$ becomes larger.
Motivated by this interesting phenomenon, we develop a search algorithm based on the golden section method to determine the location of such an {\em elbow} in the loss function. In this way, we can avoid to run the PDAS algorithm extensively for a whole sequential list. The golden section primal-dual active set (GPDAS) algorithm is summarized as follows.
\noindent\rule{\textwidth}{1.5pt}
\textbf{Algorithm 3} Golden section primal-dual active set (GPDAS) algorithm \\
\noindent\rule{\textwidth}{0.8pt}
\begin{enumerate}
\item Specify the number of maximum iterations $m_{\max}$, the maximum size $k_{\max}$ of the active set and the tolerance $\eta \in (0,1)$. Initialize $k_L=1$, and $k_R=k_{\max}$.
\item For $m = 1,2,\dots, m_{\max}$, do
\begin{enumerate}[(2.a)]
\item Run {PDAS} with $k = k_L$ and initial $\calA^{m-1}_L \cup \{j \in\calI^{m-1}_L: j \in \mbox{arg}\max (\Delta_L^{m-1})_{j}\} $. Output $\{\calA^{m}_L, \bbeta^m_L, \bm\Delta^m_L\}$.
\item Run {PDAS} with $k = k_R$ and initial $\calA^{m-1}_R \cup \{j \in\calI^{m-1}_R: j \in \mbox{arg}\max (\Delta_R^{m-1})_{j}\} $. Output $\{\calA^{m}_R, \bbeta^m_R, \bm\Delta^m_R\}$.
\item Calculate $k_M = k_L + 0.618\times (k_R -k_L)$. Run {PDAS} with $k = k_M$ and initial $\calA^{m-1}_M \cup \{j \in\calI^{m-1}_M: j \in \mbox{arg}\max (\Delta_M^{m-1})_{j}\} $. Output $\{\calA^{m}_M, \bbeta^m_M, \bm\Delta^m_M\}$.
\item Determine whether $k_M$ is an elbow point:
\begin{itemize}
\item Run {PDAS} with $k = k_M-1$ and initial $\calA^{m}_M $. Output $\{\calA^{m}_{M-}, \bbeta^m_{M-}, \bm\Delta^m_{M-}\}$.
\item Run {PDAS} with $k = k_M+1$ and initial $\calA^{m}_M $. Output $\{\calA^{m}_{M+}, \bbeta^m_{M+}, \bm\Delta^m_{M+}\}$.
\item
If ${|l(\bbeta^{m}_M) - l(\bbeta^{m}_{M_-})|}>\eta{ | l(\bbeta^{m}_M)|}$ and ${|l(\bbeta^{m}_M) - l(\bbeta^{m}_{M_+})|}< \eta { | l(\bbeta^{m}_M)|}/2$, then stop and denote $k_M$ as an elbow point, otherwise go ahead.
\end{itemize}
\item Update $k_L, k_R$ and $\calA ^{m}_L, \calA ^{m}_R$:
\begin{itemize}
\item If ${|l(\bbeta^{m}_M) - l(\bbeta^{m}_{L})|}>\eta{ | l(\bbeta^{m}_M)|}>{|l(\bbeta^{m}_R) - l(\bbeta^{m}_{L})|}$, then $k_R = k_M$, $\calA ^{m}_R = \calA ^{m}_M$;
\item If $\min \left\{{|l(\bbeta^{m}_M) - l(\bbeta^{m}_{L})|},{|l(\bbeta^{m}_R) - l(\bbeta^{m}_{L})|}\right\} > \eta{ | l(\bbeta^{m}_M)|}$, then $k_L = k_M$, $\calA ^{m}_L = \calA ^{m}_M$;
\item Otherwise, $k_R = k_M, \calA ^{m}_R = \calA ^{m}_M$ and $k_L = 1, \calA ^{m}_L = \emptyset $.
\end{itemize}
\item If $k_L= k_R - 1$, then stop, otherwise $m=m+1$.
\end{enumerate}
\item Output $\{\calA^m_M, \bbeta^{m}_M, \bm\Delta^m_M\}$.
\end{enumerate}
\noindent\rule{\textwidth}{1.5pt}
\subsection{Computational details}
The proposed PDAS, SPDAS and SPDAS algorithms are much faster than existing methods reviewed in Section 1. For the exhaustive methods like \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}kg{leaps} and \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}kg{bestglm}, they essentially deal with $\sum_{k=1}^{k_{\max}} C(p,k)$ sub-models in order to search for the best subset with size no more than $k_{\max}$.
It is infeasible even when $k_{\max}$ is moderate. That is why the greedy methods (e.g., \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}kg{glmuti}) and the relaxed methods (e.g., \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}kg{glmnet}) become popular.
Our proposed algorithms belong to the greedy methods and their computational complexity is discussed below.
In general, consider one iteration in step (2) of the PDAS algorithm with a pre-specified $k$.
Denote by $N_{l}$ the computational complexity for solving $\bbeta$ on the active set; and
denote by $N_g$ and $N_h$ the computational complexity for calculating $\g$ and $\h$, respectively. The calculation of $\ggamma$ in steps (2.b)-(2.c) costs $O((p-k)\max(N_h, N_g) )$, and the calculation of $\bm\Delta$ in steps (2.b)-(2.c) costs $O(pN_h)$. Then the overall cost of one iteration is $O(\max(N_{l}, pN_h, (p-k)N_g))$.
The total number of iterations of the PDAS algorithm could depend on the signal-to-noise ratio, the dimensionality $p$, and the sparsity level $k$. The algorithm may usually converge in finite steps (otherwise capped by $m_{\max}$). Denote by $N_{\text{P}}$ the complexity for each run of the PDAS algorithm, then the total complexity of the SPDAS and GPDAS algorithms are $O(k_{\max} \times N_{\text{P}})$ and $O(\log(k_{\max}) \times N_{\text{P}})$, respectively.
\noindent \textbf{Case 1: Linear regression.} Since $\h=\one$, $N_h = 0$. The matrix vector product in the computation of $\h$ takes $O(n)$ flops. For the least squares problem on the active set, we use Cholesky factorization to obtain the estimate, which leads to $N_{l}= O(\max(nk^2, k^3))$. Thus the total cost of one iteration in step (2) is ${O}(\max(n k^2, k^3, n(p-k)))$, and the overall cost of the PDAS algorithm is the same since the number of iterations is often finite.
In particular, if the true coefficient vector is sparse with $ k \ll p$ and $n =O(\log(p))$, the cost of the PDAS algorithm is $O(np)$, a linear time with respective to the size $p$.
With an unknown $k$, we can choose an appropriate $k_{\max}$ value, e.g., $k_{\max}=n/\log(n)$, to speed up the SPDAS and GPDAS algorithms. Their costs become $O(n^2p/\log(n))$ and $O(np\log(n/\log(n)))$, respectively. These rates are comparable with the sure independence screening procedure \citep{fan2008sure} in handling ultrahigh-dimensional data. In fact, even if the true coefficient vector is not sparse, we could use a conjugate gradient \citep{golub2012matrix} algorithm with a preconditioning matrix to achieve a similar computational rate.
\noindent \textbf{Case 2: Logistic regression.} It costs $O(p)$ flop to compute the predicted probabilities $p_i$'s. Thus $N_g=O(np)$ and $N_h=O(np)$. We use the iteratively reweighted least squares (IRLS)
for parameter estimation on the active set. The complexity of each IRLS step is the same as that of the least squares, so $N_{l}= O(N_I \max(nk^2, k^3))$ with $N_I$ denoting the finite number of IRLS iterations. The total cost of one iteration in step (2) is ${O}(\max(np^2, nk^2N_I, k^3N_I))$.
\noindent \textbf{Case 3: CoxPH regression.} It costs $O(np)$ flops to compute $\omega_{i,i'}$'s. Assume the censoring rate is $c$, then $N_g=O(n^3p(1-c))$ and $N_h=O(n^3p(1-c))$.
Like the \code{coxph} command from the \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}kg{survival} package, we adopt the standard Newton-Raphson algorithm for the maximum partial likelihood estimation on the active set. Its difficulty arises in the computation of the inverse of the Hessian matrix, which is full and dense. The Hessian matrix has $k^2$ entries and it requires $O(n^3 k(1-c))$ flops for the computation of each entry. The matrix inversion costs $O(k^3)$ via Gauss-Jordan elimination or Cholesky decomposition. Hence, for each Newton-Raphson iteration, the updating equation requires $O(\max(n^3 k^3(1-c), k^3))$ flops. We may speed up the algorithm by replacing the Hessian matrix with its diagonal, which reduces the computational complexity per updating to $O(\max(n^3 k^2(1-c), k^3))$. Denote by $N_{nr}$ the number of Newton-Raphson iterations, then $N_{l}= O(N_{nr} \max(n^3 k^2(1-c), k^3))$ and the total cost of one iteration in step (2) is ${O}(\max(n^3p^2(1-c), n^3 k^2(1-c)N_{nr}, k^3N_{nr})). $
\subsection{R package}
We have implemented the active set algorithms described above into an \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}roglang{R} package called \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}kg{BeSS} (BEst Subset Selection), which is publicly available from the CRAN at \bfm{u}} \def\v{\bfm{v}} \def\w{\bfm{w}} \def\x{\bfm{x}} \def\y{\bfm{y}rl{https://cran.r-project.org/package=BeSS}. The package is implemented in \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}roglang{C++} with memory optimized using sparse matrix output
and it can be called from \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}roglang{R} by a user-friendly interface.
The package contains two main functions, i.e., \code{bess.one} and \code{bess}, for solving the best subset selection problem with or without specification of $k$. In \code{bess}, two options are provided to determine the optimal $k$: one is based on the SPDAS algorithm with criteria including AIC, BIC and EBIC; the other is based on the GPDAS algorithm. The function \code{plot.bess} generates plots of loss functions for the best sub-models for each candidate $k$, together with solution paths for each predictor. We also include functions \code{predict.bess} and \code{predict.bess.one} to make prediction on the new data.
\section{Numerical examples}
In this section we compare the performance of our new \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}kg{BeSS} package to other well-known packages for best subset selection: \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}kg{leaps}, \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}kg{bestglm} and \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}kg{glmulti}. We also include \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}kg{glmnet} as an approximate subset selection method and use the default cross-validation method to determine an optimal tuning parameter. All parameters use the default values of the corresponding main functions in those packages unless otherwise stated. In presenting the results of \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}kg{BeSS}, \code{bess.seq} represents \code{bess} with argument \code{method = "sequential"} and \code{bess.gs} represents \code{bess} with argument \code{method = "gsection"}, two different ways to determine the optimal parameter $k$. In \code{bess.seq}, we use AIC for examples with $n\geq p$ and EBIC
for examples with $n<p$.
We choose $k_{\max} = \min(n/2, p)$ for linear models and $k_{\max} = \min(n/\log(n), p)$ for logistic and CoxPH models.
All the \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}roglang{R} codes are demonstrated in Section 4.3. All computations are carried out on a 64-bit Intel machine with a single 3.30 GHz CPU and 4 GB of RAM.
\subsection{Simulation data}\label{sec:simulate}
We demonstrate the practical application of our new \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}kg{BeSS} package on synthetical data under both low and high dimensional settings. For the low-dimensional data, \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}kg{BeSS} has comparable performance with other state-of-the-art methods. For the high-dimensional data, while most state-of-the-art methods become incapable to deal with them, \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}kg{BeSS} still performs fairly well. For an instance, \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}kg{BeSS} is scalable enough to identify the best sub-model over all candidates efficiently in seconds or a few minutes when the dimension $p = 10000$.
We compare the performances of different methods in three aspects. The first aspect is the run time in seconds (Time). The second aspect is the selection performance in terms of true positive (TP) and false positive (FP) numbers, which are defined by the numbers of true relevant and true irrelevant variables among the selective predictors. The third aspect is the predictive performance on a held out test data of size 1000. For linear regression, we use the relative mean squares error (MSE) as defined by $\|\X\hat\bbeta - \X\bbeta^*\|_2/\|\X\bbeta^*\|_2$. For logistic regression, we calculate the classification accuracy by the average number of observations being correctly classified. For CoxPH regression, we compute the median time on the test data, then derive the area under the receiver operator characteristic curve (i.e., AUC) using nearest neighbor estimation method as in \cite{heagerty2000time}.
We generate the design matrix $\X$ and the underlying coefficients $\bbeta$ as follows. The design matrix $\X$ is generated with $\X_{(j)} = \bfm{Z}_j + 0.5\times(\bfm{Z}_{j-1} + \bfm{Z}_{j+1}), ~j=1,\dots, p$, where $\bfm{Z}_0=\bfm{z}ero, \bfm{Z}_{p+1}=\bfm{z}ero$ and $\{\bfm{Z}_j, j=1,\dots,p\}$ were i.i.d. random samples drawn from standard Gaussian distribution and subsequently normalized to have $\sqrt{n}$ norm. The true coefficient $\bbeta^*$ is a vector with $q$ nonzero entries uniformly distributed in $[b, B]$, with $b$ and $B$ to be specified. In the simulation study, the sample size is fixed to be $n=1000$. For each scenario, 100 replications are conducted .
\textbf{Case 1: Linear regression.} For each $\X$ and $\bbeta^*$, we generate the response vector $\y = \X\bbeta^* + \sigma \epsilon$, with $\epsilon \sim \mathcal{N}(0,1)$. We set $b=5\sigma \sqrt{2\log(p)/n}, ~B = 100b$ and $\sigma=3$.
Different choices of $(p, q)$ are taken to cover both the low-dimensional cases $(p = 20, 30, \text{or }40, \ q=4)$ and the high-dimensional cases ($p = 100, 1000, \text{or }10000,\ q= 40$). For \code{glmulti}, we only present the result for $p=20$ and $p=30$ since it can only deal with at most 32 predictors. Since \code{leaps} and \code{bestglm} cannot deal with high-dimensional case, we only report the results of \code{glmnet}, \code{bess.seq} and \code{bess.gs}. The results are summarized in Table~\ref{tab:sim:lm}.
In the low-dimensional cases, the performances of all best subset selection methods are comparable in terms of prediction accuracy and selection consistency. However, the regularization method \code{glmnet} has much higher MSE and lower FP, which suggests that LASSO incurs bias in the coefficient estimation. In terms of computational times, both \code{bess.seq} and \code{bess.gs} have comparable performance with \code{glmnet}, which cost much less run times than the state-of-the-art methods. Unlike \code{leaps}, \code{bestglm} and \code{glmulti}, the run times of \code{bess.seq} and \code{bess.gs} remain fairly stable across different dimensionality.
In the high-dimensional cases, both \code{bess.seq} and \code{bess.gs} work quite well and they have similar performance in prediction and variable selection. Furthermore, their performances become better as $p$ and $q$ increase (from left to right in Table~\ref{tab:sim:lm}). On the other hand, \code{glmnet} has higher FP as $p$ increases. In particular, when $p=10000$ and only $40$ nonzero coefficients are involved, the average TP equals $40$ and the average FP is less than 3.06. In contrast, the average FP of \code{glmnet} increases to 30. As for the computational issues, both \code{bess.seq} and \code{bess.gs} seem to grow at a linear rate of $p$, but \code{bess.gs} offers speedups by factors of 2 up to 10 and more.
\begin{table}[!h]
\centering
\begin{tabular}{lccccc}
\toprule
Low-dimensional & & Method & $p=20$ & $p=30$ & $p=40$ \\
\cmidrule(r){2-6}
& Time & \code{leaps} & 0.00(0.01) & 0.39(0.13) & 58.79(28.78) \\
&& \code{bestglm}& 0.02(0.01) & 0.51(0.15) & 69.39(32.27) \\
&& \code{glmulti}& 11.91(2.60) & 18.41(4.13) & --- \\
&& \code{glmnet} & 0.08(0.02) & 0.09(0.02) & 0.08(0.01) \\
&& \code{bess.seq}& 0.18(0.01) & 0.23(0.02) & 0.25(0.03) \\
&& \code{bess.gs} & 0.16(0.01) & 0.18(0.02) & 0.17(0.02) \\
\cmidrule(r){2-6}
&MSE & \code{leaps} & 1.91(0.83) & 2.18(0.81) & 2.44(1.15) \\
&($\times 10^{-2}$) & \code{bestglm} & 1.91(0.83) & 2.18(0.81) & 2.44(1.15) \\
&& \code{glmulti}& 1.87(0.72) & 2.16(0.79) & --- \\
&& \code{glmnet} & 3.90(1.30) & 3.51(1.23) & 3.51(1.37) \\
&& \code{bess.seq} & 1.93(0.82) & 2.12(0.76) & 2.43(1.21) \\
&& \code{bess.gs} & 2.14(2.45) & 2.06(1.78) & 2.80(3.37) \\
\cmidrule(r){2-6}
&TP & \code{leaps}& 3.97(0.17) & 3.99(0.10) & 3.97(0.17) \\
&& \code{bestglm} & 3.97(0.17) & 3.99(0.10) & 3.97(0.17) \\
&& \code{glmulti}& 3.99(0.10) & 4.00(0.00) & --- \\
&& \code{glmnet} & 3.96(0.20) & 3.97(0.17) & 3.95(0.22) \\
&& \code{bess.seq} & 3.96(0.20) & 3.91(0.35) & 3.84(0.44) \\
&& \code{bess.gs} & 3.78(0.42) & 3.73(0.51) & 3.63(0.61) \\
\cmidrule(r){2-6}
&FP & \code{leaps} & 2.37(1.83) & 3.92(2.39) & 5.53(2.66) \\
&& \code{bestglm} & 2.37(1.83) & 3.92(2.39) & 5.53(2.66) \\
&& \code{glmulti}& 2.29(1.63) & 4.15(2.29) & --- \\
&& \code{glmnet} & 0.73(0.80) & 0.82(0.83) & 0.78(1.10) \\
&& \code{bess.seq} & 3.75(4.25) & 4.98(5.80) & 7.59(8.64) \\
&& \code{bess.gs} & 1.35(2.94) & 4.31(6.93) & 5.42(8.74) \\
\midrule
High-dimensional & & Method & $p=100$ & $p=1000$ & $p=10000$ \\
\cmidrule(r){2-6}
& Time & \code{glmnet} & 0.16(0.03) & 1.77(0.09) & 14.82(1.73) \\
& & \code{bess.seq} & 1.29(0.09) & 74.54(1.33) & 137.04(13.80) \\
& & \code{bess.gs} & 0.53(0.12) & 3.72(0.41) & 12.87(2.89) \\
\cmidrule(r){2-6}
& MSE & \code{glmnet}& 1.42(0.18) & 2.51(0.28) & 2.47(0.22) \\
& ($\times 10^{-2}$) & \code{bess.seq} & 1.65(0.41) & 1.20(0.62) & 0.70(0.23) \\\
&& \code{bess.gs} & 1.33(0.29) & 0.98(0.37) & 1.00(0.35) \\
\cmidrule(r){2-6}
&TP & \code{glmnet} & 39.74(0.54) & 39.80(0.45) & 39.75(0.46) \\
&& \code{bess.seq} & 35.30(2.17) & 38.72(1.29) & 39.53(0.70) \\
&& \code{bess.gs} & 35.78(2.12) & 39.43(0.88) & 39.58(0.71) \\
\cmidrule(r){2-6}
&FP & \code{glmnet} & 15.45(3.65) & 12.73(5.50) & 29.82(11.91) \\
&& \code{bess.seq} & 27.15(10.66) & 4.92(6.99) & 0.32(1.92) \\
&& \code{bess.gs} & 28.86(8.90) & 1.51(2.53) & 3.06(3.84) \\
\bottomrule
\end{tabular}\label{tab:sim:lm}
\caption{Simulation results for linear regression. Time stands for run time (CPU seconds), MSE stands for Mean Squared Error, TP stands for true positive number and FP stands for false positive number. The number of true nonzero coefficients is $q=4$ for low-dimensional cases and $q=40$ for high-dimensional cases. }
\end{table}
\textbf{Case 2: Logistic regression.} For each $\x$ and $\bbeta^*$, the binary response is generated by $y = \text{Bernoulli}(\mbox{\rm Prob}(Y=1))$, where $\mbox{\rm Prob}(Y=1) = \exp(\x^\top\bbeta^*)/(1+\exp(\x^\top\bbeta^*))$. The range of nonzero coefficients are set as $b=10 \sqrt{2\log(p)/n},\ B = 5b$. Different choices of $p$ are taken to cover both the low-dimensional cases $(p = 8, 10, \text{ or }12)$ and the high-dimensional cases $(p = 100, 1000, \text{ or } 10000)$. The number of true nonzero coefficients is chosen to be $q=4$ for low-dimensional cases and $q=20$ for high-dimensional cases. Since \code{bestglm} is based on complete enumeration, it may be used for low-dimensional cases yet it becomes computationally infeasible for high dimensional cases.
The simulation results are summarized in Table~\ref{tab:sim:logistic}. When $p$ is small, both \code{bess.seq} and \code{bess.gs} have comparable performance with \code{bestglm}, \code{glmulti} and \code{glmnet}, but have considerably faster speed in computation than \code{bestglm} and \code{glmulti}. In the high-dimensional cases, we see that all three methods perform very well in terms of accuracy and TP. Yet both \code{bess.seq} and \code{bess.gs} have much smaller FP than \code{glmnet}. Among them, the run time for \code{bess.gs} is around a quarter of that for \code{bess.seq} and is similar to that for \code{glmnet}.
\begin{table}[!h]
\centering
\begin{tabular}{lccccc}
\toprule
Low-dimensional & & Method& $p=8$ & $p=10$ & $p=12$ \\
\cmidrule(r){2-6}
&Time & \code{bestglm} & 1.83(0.15) & 7.55(0.26) & 28.35(1.93) \\
&& \code{glmulti} & 2.08(0.11) & 13.91(2.43) & 21.61(4.54) \\
&& \code{glmnet} & 0.49(0.07) & 0.56(0.09) & 0.63(0.17) \\
&& \code{bess.seq} & 0.70(0.33) & 0.79(0.35) & 0.78(0.52) \\
&& \code{bess.gs} & 0.52(0.20) & 0.78(1.14) & 0.65(0.23) \\
\cmidrule(r){2-6}
&Acc & \code{bestglm} & 0.949(0.012) & 0.950(0.013) & 0.950(0.011) \\
&& \code{glmulti} & 0.949(0.012) & 0.950(0.013) & 0.950(0.011) \\
&& \code{glmnet} & 0.949(0.013) & 0.951(0.013) & 0.950(0.011) \\
&& \code{bess.seq} & 0.949(0.012) & 0.950(0.013) & 0.950(0.011) \\
&& \code{bess.gs} & 0.948(0.013) & 0.951(0.012) & 0.949(0.013) \\
\cmidrule(r){2-6}
&TP & \code{bestglm} & 3.99(0.10) & 4.00(0.00) & 3.99(0.10) \\
&& \code{glmulti} & 3.99(0.10) & 4.00(0.00) & 4.00(0.00) \\
& & \code{glmnet}& 4.00(0.00) & 4.00(0.00) & 4.00(0.00) \\
& & \code{bess.seq}& 3.96(0.20) & 3.95(0.30) & 3.91(0.32) \\
& & \code{bess.gs}& 3.87(0.37) & 3.87(0.42) & 3.89(0.40) \\
\cmidrule(r){2-6}
&FP & \code{bestglm} & 0.73(0.85) & 1.02(1.05) & 1.41(1.44) \\
&& \code{glmulti} & 0.73(0.85) & 1.02(1.05) & 1.37(1.20) \\
&& \code{glmnet} & 1.62(0.96) & 2.07(1.16) & 2.83(1.44) \\
&& \code{bess.seq} & 1.77(1.59) & 2.19(2.20) & 2.39(2.40) \\
&& \code{bess.gs} & 0.15(0.41) & 0.31(0.93) & 0.64(1.57) \\
[1ex]
\midrule
[0.1ex]
High-dimensional & & Method& $p=100$ & $p=1000$ & $p=10000$ \\
\cmidrule(r){2-6}
&Time & \code{glmnet} & 4.75(0.89) & 4.38(0.49) & 17.01(0.24) \\
&& \code{bess.seq} & 43.99(7.42) & 54.85(4.46) & 108.66(2.47) \\
&& \code{bess.gs} & 7.34(2.10) & 11.46(1.81) & 22.43(2.16) \\
\cmidrule(r){2-6}
&Acc & \code{glmnet} & 0.969(0.006) & 0.945(0.009) & 0.922(0.011) \\
&& \code{bess.seq} & 0.963(0.012) & 0.972(0.011) & 0.979(0.006) \\
&& \code{bess.gs} & 0.970(0.010) & 0.976(0.008) & 0.978(0.009) \\
\cmidrule(r){2-6}
&TP & \code{glmnet} & 19.96(0.20) & 19.97(0.17) & 19.79(0.52) \\
&& \code{bess.seq} & 16.50(2.38) & 19.34(1.23) & 19.92(0.34) \\
&& \code{bess.gs} & 18.62(1.15) & 19.81(0.49) & 19.82(0.61) \\
\cmidrule(r){2-6}
&FP & \code{glmnet} & 34.59(4.74) & 122.82(19.80) & 222.77(43.63) \\
&& \code{bess.seq} & 5.61(3.37) & 1.82(2.03) & 0.49(0.67) \\
&& \code{bess.gs} & 3.16(2.46) & 0.95(1.34) & 0.54(0.92) \\
\bottomrule
\end{tabular}\label{tab:sim:logistic}
\caption{Simulation results for logistic regression. Time stands for run time (CPU seconds), Acc stands for classification accuracy, TP stands for true positive number and FP stands for false positive number. The number of true nonzero coefficients is $q=4$ for low-dimensional cases and $q=20$ for high-dimensional cases. }
\end{table}
\textbf{Case 3: CoxPH regression.} For each $\x$ and $\bbeta^*$, we generate data from the CoxPH model with hazard rate $\lambda(t|\x) = \exp(\x^\top\bbeta^*)$. The ranges of nonzero coefficients are set same as those in logistic regression, i.e., $b=10 \sqrt{2\log(p)/n}, ~B = 5b$. Different choices of $p$ are taken to cover both the low-dimensional cases $(p = 8, 10, \text{ or }12)$ and the high-dimensional cases $(p = 100, 1000, \text{ or } 10000)$. The number of true nonzero coefficients is chosen to be $q=4$ for low-dimensional cases and $q=20$ for high-dimensional cases. Since \code{glmulti} cannot handle more than 32 predictors, we only report the low dimensional result for \code{glmulti}.
The simulation results are summarized in Table~\ref{tab:sim:cox}. Our findings about \code{bess.seq} and \code{bess.gs} are similar to those for the logistic regression.
\begin{table}[!h]
\centering
\begin{tabular}{lccccc}
\toprule
Low-dimensional & & Method& $p=8$ & $p=10$ & $p=12$ \\
\cmidrule(r){2-6}
&Time & \code{glmulti} & 1.53(0.06)& 10.11(1.75) & 15.20(2.86) \\
&& \code{glmnet} & 1.07(0.20) & 1.09(0.20) & 1.16(0.23) \\
&& \code{bess.seq} & 0.42(0.20) & 0.49(0.23) & 0.52(0.22) \\
&& \code{bess.gs} & 0.35(0.15) & 0.46(0.19) & 0.51(0.18) \\
\cmidrule(r){2-6}
&AUC & \code{glmulti} & 0.973(0.012) & 0.972(0.010) & 0.974(0.010) \\
&& \code{glmnet} & 0.973(0.012) & 0.972(0.010) & 0.974(0.010) \\
&& \code{bess.seq} & 0.973(0.012) & 0.972(0.010) & 0.974(0.010) \\
&& \code{bess.gs} & 0.972(0.012) & 0.972(0.010) & 0.974(0.011) \\
\cmidrule(r){2-6}
&TP& \code{glmulti} & 4.00(0.00) & 3.99(0.10) & 4.00(0.00) \\
& & \code{glmnet}& 4.00(0.00) & 4.00(0.00) & 4.00(0.00) \\
& & \code{bess.seq}& 4.00(0.00) & 4.00(0.00) & 4.00(0.00) \\
& & \code{bess.gs}& 3.89(0.35) & 3.96(0.20) & 3.99(0.10) \\
\cmidrule(r){2-6}
&FP& \code{glmulti} & 0.60(0.77) & 1.06(1.17) & 1.14(1.21) \\
&& \code{glmnet} & 1.17(1.01) & 1.56(1.04) & 1.82(1.14) \\
&& \code{bess.seq} & 1.62(1.69) & 1.98(2.25) & 2.38(2.69) \\
&& \code{bess.gs} & 0.11(0.35) & 0.04(0.20) & 0.06(0.37) \\
[1ex]
\midrule
[0.1ex]
High-dimensional & & Method& $p=100$ & $p=1000$ & $p=10000$ \\
\cmidrule(r){2-6}
&Time & \code{glmnet} & 16.61(1.90) & 297.01(62.83) & 832.69(73.26) \\
& & \code{bess.seq}& 20.57(1.77) & 72.53(2.58) & 233.53(11.94) \\
& & \code{bess.gs}& 4.86(1.59) & 15.36(1.69) & 63.23(7.21) \\
\cmidrule(r){2-6}
&AUC & \code{glmnet} & 0.993(0.005) & 0.992(0.006) & 0.991(0.007) \\
& & \code{bess.seq} & 0.993(0.005) & 0.992(0.006) & 0.991(0.007) \\
& & \code{bess.gs} & 0.990(0.008) & 0.992(0.006) & 0.991(0.007) \\
\cmidrule(r){2-6}
&TP & \code{glmnet} & 20.00(0.00) & 20.00(0.00) & 20.00(0.00) \\
& & \code{bess.seq} & 18.06(1.67) & 19.70(0.70) & 20.00(0.00) \\
& & \code{bess.gs} & 17.09(2.03) & 19.93(0.33) & 19.99(0.10) \\
\cmidrule(r){2-6}
&FP & \code{glmnet} & 41.26(4.10) & 245.82(19.41) & 541.13(34.33) \\
& & \code{bess.seq} & 11.80(9.25) & 1.64(3.78) & 0.02(0.14) \\
& & \code{bess.gs} & 13.65(11.84) & 0.19(0.60) & 0.05(0.22) \\
\bottomrule
\end{tabular}\label{tab:sim:cox}
\caption{Simulation results for CoxPH regression. Time stands for run time (CPU seconds), AUC stands for the integrated time-dependent area under the curve, TP stands for true positive number and FP stands for false positive number. The number of true nonzero coefficients is $q=4$ for low-dimensional cases and $q=20$ for high-dimensional cases. }
\end{table}
\subsection{Real data}\label{sec:real}
We also evaluate the performance of the \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}kg{BeSS} package in modeling several real data sets. Table~\ref{tab:data} lists these instances and their descriptions. All datasets
are saved as \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}roglang{R} data objects and available online with this publication.
\begin{table}[!h]
\centering
\renewcommand{2.0}{1.2}
\begin{tabular}{lccll}
\toprule
Dataset & $n$ & $p$ & Type & Data source \\
\midrule
\code{prostate} & 97 & 9 & Continuous & \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}roglang{R} package \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}kg{ElemStatLearn}\\
\code{SAheart} & 462 & 8 & Binary & \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}roglang{R} package \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}kg{ElemStatLearn} \\
\code{trim32} & 120 & 18975 & Continuous &\cite{scheetz2006regulation} \\
\code{leukemia} & 72 & 3571 & Binary & \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}roglang{R} package \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}kg{spikeslab} \\
{\code{gravier}} & {168} &{2905} & {Binary} & \href{https://github.com/ramhiser/datamicroarray/wiki/Gravier-(2010)}{https://github.com/ramhiser/}\\
\code{er0} & 609 & 22285 & Survival & \href{https://www.ncbi.nlm.nih.gov/geo/}{https://www.ncbi.nlm.nih.gov/geo/}\\
\bottomrule
\end{tabular}\label{tab:data}
\caption{Description for the real data sets. Here $n$ denotes the number of observations, $p$ denotes the number of predictors, and `Type' denotes the type of response.}
\end{table}
We randomly split the data into a training set with two-thirds observations and a test set with remaining observations. Different best subset selection methods are used to identify the best sub-model. For each method, the run time in seconds (Time) and the size of selected model (MS) are recorded. We also include measurements of the predictive performance on test data according to the metrics as in Section~\ref{sec:simulate}. For reliable evaluation, the aforementioned procedure is replicated for 100 times.
The modeling results are displayed in Table~\ref{tab:real}. Again in low-dimensional cases, \code{bess} has comparable performance with the state-of-art algorithms (branch-and-bound algorithm for linear models and complete enumeration algorithm and genetic algorithm for GLMs). Besides, \code{bess.gs} has comparable run time with \code{glmnet} and is considerably faster than \code{bess.seq} especially in high-dimensional cases.
\begin{table}[!h]
\centering
\renewcommand{0.1cm}{0.1cm}
\begin{tabular}{lccccccc}
\toprule
Data &Method & \code{leaps} & \code{bestglm}& \code{glmulti} & \code{glmnet} &\code{bess.seq} & \code{bess.gs}\\
\midrule
\code{prostate} & Time & 0.00(0.01) & 0.01(0.01)&0.61(0.05) & 0.07(0.01) & 0.22(0.01) & 0.22(0.01) \\
& PE & 0.61(0.14) & 0.61(0.14)& 0.61(0.14) & 0.65(0.19) & 0.60(0.13) & 0.60(0.14) \\
& MS & 4.27(1.11) & 4.25(1.12)& 4.25(1.12)& 3.58(0.87) & 4.29(1.17) & 6.11(0.87) \\
\cmidrule(r){1-8}
\code{SAheart} & Time & --- & 1.58(0.07)& 4.03(0.53)& 0.13(0.01) & 0.27(0.04) & 0.26(0.04) \\
& Acc & --- & 0.72(0.03)&0.72(0.03) & 0.70(0.04)& 0.72(0.03) & 0.72(0.03) \\
& MS & --- & 5.68(0.98)& 5.68(0.98)& 4.61(0.84)& 5.68(0.99) & 6.29(1.09) \\
\cmidrule(r){1-8}
\code{trim32} & Time & --- & --- & --- & 3.23(0.15)& 1.95(0.53) & 1.08(0.19) \\
& PE & --- & --- & --- & 0.01(0.01)& 0.01(0.01) & 0.01(0.00) \\
& MS & --- & --- & --- & 24.89(11.79)& 1.60(0.62) & 7.82(2.26) \\
\cmidrule(r){1-8}
\code{leukemia} & Time & --- & --- & --- & 0.38(0.01)& 1.74(0.77) & 1.14(0.53) \\
& Acc & --- & --- & --- & 0.93(0.05)& 0.90(0.06) & 0.91(0.06) \\
& MS & --- & --- & --- & 11.76(4.40)& 1.54(0.77) & 2.00(0.00) \\
\cmidrule(r){1-8}
\code{gravier} & Time & --- & --- & --- & 0.68(0.03)& 6.64(4.09) & 2.93(2.50) \\
& Acc & --- & --- & --- & 0.71(0.07)& 0.72(0.06) & 0.72(0.06) \\
& MS & --- & --- & --- & 10.83(7.39)& 9.23(1.05) & 10.80(2.47) \\
\cmidrule(r){1-8}
\code{er0} & Time & --- & --- & --- & 154.97(15.75)& 184.51(86.15) & 55.20(22.07) \\
& AUC & --- & --- & --- & 0.52(0.04)& 0.53(0.05) & 0.60(0.05) \\
& MS & --- & --- & --- & 3.06(7.35)& 1.02(0.14) & 56.85(6.90) \\
\bottomrule
\end{tabular}\label{tab:real}
\caption{Results for the real data sets. Time stands for run time (CPU seconds), MS stands for the size of selected model. PE stands for mean prediction error in linear model; Acc stands for classification accuracy in logistic regression model; AUC stands for the integrated time-dependent area under the curve in CoxPH regression model. }
\end{table}
\subsection{Code demonstration}
We demonstrate how to use the package \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}kg{BeSS} on a synthesis data as discussed in Section~\ref{sec:tuning} and a real data in Section~\ref{sec:real}. Firstly, load \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}kg{BeSS} and generate data with the \code{gen.data} function.
\begin{Sinput}
R> require("BeSS")
R> set.seed(123)
R> Tbeta <- rep(0, 20)
R> Tbeta[c(1, 2, 5, 9)] <- c(3, 1.5, -2, -1)
R> data <- gen.data(n = 200, p = 20, family = "gaussian", beta = Tbeta,
+ rho = 0.2, sigma = 1)
\end{Sinput}
We may call the \code{bess.one} function to solve the best subset selection problem with a specified cardinality. Then we can \code{print} or \code{summary} the \code{bess.one} object. While the \code{print} method allows users to obtain a brief summary of the fitted model, the \code{summary} method presents a much more detailed description.
\begin{Sinput}
R> fit.one <- bess.one(data$x, data$y, s = 4, family = "gaussian")
R> print(fit.one)
\end{Sinput}
\begin{Soutput}
Df MSE AIC BIC EBIC
4.0000000 0.8501053 -24.4790117 -11.2857422 12.6801159
\end{Soutput}
\begin{Sinput}
R> summary(fit.one)
\end{Sinput}
\begin{Soutput}
----------------------------------------------------------------------
Primal-dual active algorithm with maximum iteration being 15
Best model with k = 4 includes predictors:
X1 X2 X5 X9
3.019296 1.679419 -2.021521 -1.038276
log-likelihood: 16.23951
deviance: -32.47901
AIC: -24.47901
BIC: -11.28574
EBIC: 12.68012
----------------------------------------------------------------------
\end{Soutput}
The estimated coefficients of the fitted model can be extracted by using the \code{coef} function, which provides a sparse output with the control of argument \code{sparse = TRUE}. It is recommended to output a non-sparse vector when \code{bess.one} is used, and to output a sparse matrix when \code{bess} is used.
\begin{Sinput}
R> coef(fit.one, sparse = FALSE)
\end{Sinput}
\begin{Soutput}
(intercept) X1 X2 X3 X4 X5
-0.07506287 3.01929556 1.67941924 0.00000000 0.00000000 -2.02152109
X6 X7 X8 X9 X10 X11
0.00000000 0.00000000 0.00000000 -1.03827568 0.00000000 0.00000000
X12 X13 X14 X15 X16 X17
0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000
X18 X19 X20
0.00000000 0.00000000 0.00000000
\end{Soutput}
To make prediction on new data, the \code{predict} function can be used as follows.
\begin{Sinput}
R> pred.one <- predict(fit.one, newdata = data$x)
\end{Sinput}
To extract the selected best model, we provide the \code{lm}, \code{glm}, or \code{coxph} type of object named
the \code{bestmodel} in the fitted \code{bess.one} object.
Users could \code{print}, \code{summary} or \code{predict} this \code{bestmodel} object just like working with classical regression modeling. This would be helpful for statistical analysts who are familiar with \code{lm}, \code{glm}, or \code{coxph} functions.
\begin{Sinput}
R> bm.one <- fit.one$bestmodel
R> summary(bm.one)
\end{Sinput}
\begin{Soutput}
Call:
lm(formula = ys ~ xbest)
Residuals:
Min 1Q Median 3Q Max
-2.54220 -0.63600 -0.04702 0.64100 3.11518
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.07506 0.06603 -1.137 0.257
xbestX1 3.01930 0.06715 44.962 <2e-16 ***
xbestX2 1.67942 0.06577 25.535 <2e-16 ***
xbestX5 -2.02152 0.06577 -30.735 <2e-16 ***
xbestX9 -1.03828 0.06313 -16.446 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.9338 on 195 degrees of freedom
Multiple R-squared: 0.9566, Adjusted R-squared: 0.9557
F-statistic: 1075 on 4 and 195 DF, p-value: < 2.2e-16
\end{Soutput}
In practice when the best subset size is unknown, we have to determine the optimal choice of such sub-model size. The function \code{bess} provides two options: \code{method = "sequential"} corresponds to the SPDAS algorithm, and \code{method = "gsection"} corresponds to the GPDAS algorithm. Next we illustrate the usage of \code{bess} in the \code{trim32} data. We first load the data into the environment and show that it has 18975 variables, a much larger number compared with the sample size 120.
\begin{Sinput}
R> load("trim32.RData")
R> dim(X)
\end{Sinput}
\begin{Soutput}
[1] 120 18975
\end{Soutput}
Below is an example of running \code{bess} with argument \code{method = "sequential", epsilon = 0} and other argument being default values. We use the \code{summary} function to give a summary of the fitted \code{bess} object.
\begin{Sinput}
R> fit.seq <- bess(X, Y, method="sequential", epsilon = 0)
R> summary(fit.seq)
\end{Sinput}
\begin{Soutput}
----------------------------------------------------------------------------
Primal-dual active algorithm with tuning parameter determined by
sequential method
Best model determined by AIC includes 25 predictors with AIC = -890.9282
Best model determined by BIC includes 25 predictors with BIC = -821.2409
Best model determined by EBIC includes 2 predictors with EBIC = -561.2689
-----------------------------------------------------------------------------
\end{Soutput}
As in the \code{bess.one}, the \code{bess} function outputs an \code{lm} type of object \code{bestmodel} associated with the selected best model. Here the \code{bestmodel} component outputs the largest fitted model since we did not use any early stopping rule as shown in the argument \code{epsilon = 0}.
\begin{Sinput}
R> bm.seq <- fit.seq$bestmodel
R> summary(bm.seq)
\end{Sinput}
\begin{Soutput}
Call:
lm(formula = ys ~ xbest)
Residuals:
Min 1Q Median 3Q Max
-0.039952 -0.012366 -0.001078 0.011401 0.075677
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 5.618703 0.407769 13.779 < 2e-16 ***
xbest1368348_at -0.089394 0.014563 -6.139 1.97e-08 ***
xbest1370558_a_at -0.122228 0.010712 -11.410 < 2e-16 ***
xbest1372548_at -0.179410 0.012085 -14.846 < 2e-16 ***
xbest1377032_at -0.062936 0.016733 -3.761 0.000294 ***
xbest1382223_at 0.497858 0.023655 21.047 < 2e-16 ***
xbest1388491_at 0.266606 0.021538 12.378 < 2e-16 ***
xbest1388657_at -0.085292 0.015030 -5.675 1.53e-07 ***
xbest1389122_at -0.101926 0.015317 -6.655 1.88e-09 ***
xbest1390269_at 0.106434 0.012130 8.774 7.40e-14 ***
xbest1378024_at -0.123666 0.017614 -7.021 3.40e-10 ***
xbest1378552_at -0.049578 0.010397 -4.768 6.77e-06 ***
xbest1379586_at -0.066086 0.013526 -4.886 4.22e-06 ***
xbest1379772_at -0.096651 0.010166 -9.507 2.05e-15 ***
xbest1379933_at 0.186271 0.015806 11.785 < 2e-16 ***
xbest1380696_at 0.028347 0.006882 4.119 8.19e-05 ***
xbest1380977_at 0.104704 0.018148 5.769 1.01e-07 ***
xbest1382392_at -0.033764 0.005830 -5.791 9.21e-08 ***
xbest1384690_at -0.083789 0.013985 -5.991 3.80e-08 ***
xbest1385015_at 0.131036 0.011803 11.102 < 2e-16 ***
xbest1385032_at 0.100631 0.012171 8.268 8.73e-13 ***
xbest1385395_at -0.139164 0.010919 -12.745 < 2e-16 ***
xbest1385673_at 0.071119 0.011828 6.013 3.46e-08 ***
xbest1392605_at -0.051400 0.008229 -6.246 1.21e-08 ***
xbest1394502_at 0.020363 0.006134 3.320 0.001283 **
xbest1398128_at -0.084070 0.012728 -6.605 2.36e-09 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.02241 on 94 degrees of freedom
Multiple R-squared: 0.981, Adjusted R-squared: 0.976
F-statistic: 194.5 on 25 and 94 DF, p-value: < 2.2e-16
\end{Soutput}
Alternatively, we might use criteria like AIC to select the best model among a sequential list of candidate models. As shown above, the output of the \code{bess} function includes the AIC, BIC and EBIC values for best subset selection. Since the \code{trim32} data is high dimensional, we opt to use the EBIC criterion to determine the optimal model size. Then we run the \code{coef} function to extract the coefficients in the \code{bess} object and output the nonzero coefficients of the selected model.
\begin{Sinput}
R> K.opt.ebic <- which.min(fit.seq$EBIC)
R> coef(fit.seq)[, K.opt.ebic][which(coef(fit.seq)[, K.opt.ebic]!=0)]
\end{Sinput}
\begin{Soutput}
(intercept) 1382223_at 1388491_at
0.8054785 0.5715478 0.3555834
\end{Soutput}
We can also run the \code{predict} function for a given \code{newdata}. The argument \code{type} specifies which criteria is used to select the best fitted model.
\begin{Sinput}
R> pred.seq <- predict(fit.seq, newdata = data$x, type="EBIC")
\end{Sinput}
The \code{plot} routine provides the loss function plot for the sub-models with different $k$ values, as well as solution paths for each predictor. It also adds a vertical dashed line to indicate the optimal $k$ value as determined by EBIC. Figure~\ref{fig:seq} shows the result from the following \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}roglang{R} code.
\begin{Sinput}
R> plot(fit.seq, type = "both", breaks = TRUE, K = K.opt.ebic)
\end{Sinput}
\begin{figure}
\caption{Best subset selection results for the \code{trim32}
\label{fig:seq}
\end{figure}
Next we call the function \code{bess} with argument \code{method = "gsection"} to perform the GPDAS algorithm. At each iteration, it outputs the split information.
\begin{Sinput}
R> fit.gs <- bess(X, Y, family = "gaussian", method = "gsection",
R+ epsilon = 1e-2)
\end{Sinput}
\begin{Soutput}
1-th iteration s.left:1 s.split:16 s.right:25
2-th iteration s.left:1 s.split:10 s.right:16
3-th iteration s.left:1 s.split:7 s.right:10
4-th iteration s.left:1 s.split:5 s.right:7
5-th iteration s.left:5 s.split:6 s.right:7
\end{Soutput}
From the above code, we know that the best selected model has 6 predictors and the algorithm ends at the 5{\it th} iteration. To show more information about the best selected model, we may extract \code{fit.gs$bestmodel} and present its summary information via the S3 method \code{summary}.
\begin{Sinput}
R> bm.gs <- fit.gs$bestmodel
R> summary(bm.gs)
\end{Sinput}
\begin{Soutput}
Call:
lm(formula = ys ~ xbest)
Residuals:
Min 1Q Median 3Q Max
-0.114598 -0.036829 -0.007365 0.041804 0.161688
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.41219 0.50792 2.780 0.006363 **
xbest1368316_at -0.16680 0.03479 -4.794 5.02e-06 ***
xbest1372248_at 0.22120 0.05579 3.965 0.000129 ***
xbest1373887_at 0.27947 0.05707 4.897 3.27e-06 ***
xbest1387160_at -0.12456 0.02989 -4.168 6.05e-05 ***
xbest1389910_at 0.54459 0.07220 7.543 1.25e-11 ***
xbest1381978_a_at -0.16091 0.03454 -4.658 8.77e-06 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.05926 on 113 degrees of freedom
Multiple R-squared: 0.8405, Adjusted R-squared: 0.8321
F-statistic: 99.28 on 6 and 113 DF, p-value: < 2.2e-16
\end{Soutput}
Running the \code{coef} function directly on the \code{bess} object returns a sparse matrix as shown below. The last column corresponds to the best fitted coefficients.
\begin{Sinput}
R> beta <- coef(fit.gs, sparse = TRUE)
R> class(beta)
\end{Sinput}
\begin{Soutput}
[1] "dgCMatrix"
attr(,"package")
[1] "Matrix"
\end{Soutput}
\begin{Sinput}
R> beta[, ncol(beta)][which(beta[, ncol(beta)]!=0)]
\end{Sinput}
\begin{Soutput}
(intercept) 1368316_at 1372248_at 1373887_at 1387160_at
1.4121869 -0.1668030 0.2211982 0.2794672 -0.1245576
1389910_at 1381978_a_at
0.5445936 -0.1609133
\end{Soutput}
Based on the \code{fit.gs}, we can predict for the new data via the \code{predict} function as follows.
\begin{Sinput}
R> pred.gs <- predict(fit.gs, newdata = X)
\end{Sinput}
\section{Discussion}
In this paper, we introduce a primal dual active set (PDAS) algorithm for solving the best subset selection problem under the general convex loss setting. The PDAS algorithm identifies the best sub-model with a pre-specified model size via a primal-dual formulation on feasible solutions. To determine the best sub-model over different model sizes, both sequential search and golden section search are proposed, i.e., SPDAS and GPDAS algorithms. We find that the GPDAS algorithm is especially efficient and accurate in selecting variables for high-dimensional and sparse data.
The proposed algorithms are implemented with \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}roglang{C++} through the new \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}kg{BeSS} package in the \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}roglang{R} statistical environment. Package \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}kg{BeSS} provides \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}roglang{R} users with a new and flexible way to carry out best subset selection for LM, GLM and CoxPH models. It allows us to identify the best sub model efficiently (usually in seconds or a few minutes) even when the number of predictors is extremely large, say $p \bfm{a}} \def\b{\bfm{b}} \def\c{\bgk{c}} \def\d{\bfm{d}} \def\e{\bfm{e}pprox 10000$, based on a standard personal computer. In both simulation and real data examples, it was shown that the \bfm{p}} \def\q{\bfm{q}} \def\r{\bfm{r}} \def\s{\bfm{s}} \def\t{\bfm{t}kg{BeSS} package is highly efficient compared to other state-of-the-art methods.
\section*{Acknowledgments}
We are grateful to the anonymous referees for valuable comments that lead to the improvement of the current paper. Wen's research is partially supported by NSFC(11801540), the Natural Science Foundation of Guangdong (2017A030310572), the Fundamental Research Funds for the Central Universities (WK2040170015, WK2040000016). Zhang's research is partially supported by Basic Research Seed Fund (201611159250) and Big Data Project Fund of The University of Hong Kong. Wang's research is partially supported by NSFC(11771462), The National Key Research and Development Program of China(2018YFC1315400), and The Key Research and Development Program of Guangdong, China(2019B020228001).
\end{document} |
\begin{document}
\title{Localization of the Kobayashi distance for any visibility domain}
\author{Amar Deep Sarkar}
\address{ADS: Indian Institute of Science Education and Research Kolkata, India}
\email{[email protected]}
\keywords{Visibility, weak visibility, Kobayashi distance, localization}
\subjclass{Primary: 32F45}
\thanks{The author is supported by the postdoctoral fellowship of Indian Institute of Science Education and Research Kolkata.}
\begin{abstract}
In this article, we prove localization results for the Kobayashi distance of Kobayashi hyperbolic domains with local visibility property in $\mathbb{C}^d$, $d \geq 1$. This is done by proving a localization result for the Kobayashi-Royden pseudometric, along with some other results for domains satisfying local weak visibility.
\end{abstract}
\maketitle
\section{Introduction}
Let $\Omega \subset \mathbb{C}^d$, $d \geq 1$, be a domain and $\mathsf{k}_{\Omega}: \Omega \times \Omega \longrightarrow \mathbb{R}_{\geq 0}$ and $\kappa: \Omega \times \mathbb{C}^d \longrightarrow \mathbb{R}_{\geq 0}$ denote the Kobayashi pseudodistance and Kobayashi-Royden pseudometric respectively. When $\mathsf{k}_{\Omega}$ is indeed a distance, we say that $\Omega$ is a Kobayashi hyperbolic domain. The aim of this paper is to obtain localization results for Kobayashi distance under the assumption of weak visibility property (see Definition~\ref{D:Weak_vis}). These localization results could be used to infer about the global geometry from the local geometry and vice versa.
This kind of localization result (see Theorem~\ref{T:Main_Thm_1}) is well-known for bounded strongly convex and strongly pseudoconvex domains (cf. \cite{Abate, Balogh_Bonk, Jarnicki_Pflug}); from the work of Zimmer \cite{Zimmer_1, Zimmer_2} it follows that such kind of localization is also true for bounded convexifiable domains of finite type, $\mathbb{C}$-strictly convex domains with $C^{1, \alpha}$ boundary. Recently, Liu--Wang obtained such kind of localization result under the assumption that locally the domains are log-type convex domains, see \cite{Liu_et_al}, and this was substantially generalized by Bracci--Nikolov--Thomas under the assumption that the domains are locally convex and locally has weak visibility property, see \cite{BNT}. In all the above results mentioned, the local convexity or convexifiability assumption is present along with the weak visibility property. However, there exists a large class of domains in $\mathbb{C}^d$ which are not locally convex or locally convexifiable. For example, not all pseudoconvex domains of finite type near a boundary point is locally convexifiable; however, these domains satisfy local visibility property, this follows from the work of Bharali--Zimmer \cite{Bharali_Zimmer_1}, also see \cite{BM}, \cite{BNT}, \cite{CMS}.
Now we present our first main result which shows that no assumption of local convexity is required; only local weak visibility assumption is enough to get such localization result. This generalizes \cite[Theorem~1.1]{BNT}. The idea of the proof is inspired by the proof of Theorem~1.1 in \cite{BNT} along with a few important new observations.
\begin{theorem}\label{T:Main_Thm_1}
Suppose $\Omega \subset \mathbb{C}^d$ is a Kobayashi hyperbolic domain and $U$ is an open subset of $\mathbb{C}^d$ such that $U \cap \partial \Omega \neq \emptyset$ and $U \cap \Omega$ is connected. Suppose $( U \cap \Omega, \mathsf{k}_{U \cap \Omega} )$ has weak visibility property for every pair of distinct points in $ U \cap \partial \Omega$. Then for every open sets $W, W_0$ with $W\cap \Omega \neq \emptyset$, $W \subset \subset W_0 \subset \subset U$ and $\mathsf{k}_{\Omega}(W_0 \cap \Omega, \Omega \setminus U) > 0$, there exists a constant $C> 0$ which depends only on $U, W, W_0$ such that for every $z, w \in W \cap \Omega$,
\[
\mathsf{k}_{U \cap \Omega} (z, w) \leq \mathsf{k}_{\Omega} (z, w) + C.
\]
\end{theorem}
\begin{remark}
If $\Omega$ is a bounded domain, then for every pair $W_0, U$ as in the above theorem, it follows $\mathsf{k}_{\Omega}(W_0 \cap \Omega, \Omega \setminus U) > 0$. Moreover, {\it (3)} of Proposition~\ref{P:Visibility_Outside_Point} gives a sufficient condition for $\mathsf{k}_{\Omega}(W_0 \cap \Omega, \Omega \setminus U) > 0$ when $\Omega$ possibly an unbounded domain. This condition is mentioned in the previous version of this paper without a proof, and later a proof of this also appears in \cite{NOT}.
\end{remark}
Our next main result is the following
\begin{theorem}\label{T:Main_Thm_2}
Suppose $\Omega \subset \mathbb{C}^d$ is a Kobayashi hyperbolic domain and $U$ is an open subset of $\mathbb{C}^d$ such that $U \cap \partial \Omega \neq \emptyset$. Suppose $( \Omega, \mathsf{k}_{\Omega} )$ has weak visibility property for every pair of distinct points in $ U \cap \partial \Omega$. Then for every open set $W$ with $W \cap \Omega \neq \emptyset$ and $W \subset \subset U$, there exists a constant $C> 0$ which depends only on $U, W$ such that for every $z, w \in W \cap \Omega$,
\[
\mathsf{k}_{U \cap \Omega} (z, w) \leq \mathsf{k}_{\Omega} (z, w) + C.
\]
\end{theorem}
\begin{remark}
Note that in Theorem~\ref{T:Main_Thm_2} the weak visibility assumption is taken with respect to $(\Omega, \mathsf{k}_{\Omega} )$ whereas in Theorem~\ref{T:Main_Thm_1} the weak visibility is taken with respect to $(U \cap \Omega, \mathsf{k}_{U \cap \Omega} )$. The second assumption in Theorem~\ref{T:Main_Thm_1} is that $\mathsf{k}_{\Omega}(W_0 \cap \Omega, \Omega \setminus U) > 0$, however there is no such assumption in Theorem~\ref{T:Main_Thm_2} --- although it is necessary in the proof --- is because of {\it (3)} of Proposition~\ref{P:Visibility_Outside_Point}.
\end{remark}
As a consequence of the above statement, we obtain the following
\begin{corollary}\label{C:Pseudo_finte_type}
Suppose $\Omega \subset \mathbb{C}^d$ is a bounded domain with $C^{\infty}$-smooth boundary and of finite D'Angelo type. Let $p \in \partial \Omega$. Then for every neighbourhood $ U$ and $W$ of $p$ with $W \subset \subset U$ there exists a constant $C> 0$ such that for every $z, w \in W \cap \Omega$,
\[
\mathsf{k}_{U \cap \Omega} (z, w) \leq \mathsf{k}_{\Omega} (z, w) + C.
\]
\end{corollary}
\begin{remark}
To the best of our knowledge, such kind of additive localization result for the Kobayashi distance has not been obtained before in this generality. The above corollary could be mentioned for a larger class of domains. We refer the reader to the following papers for such examples of domains: \cite{Bharali_Zimmer_1, Bharali_Zimmer_2, BM, BNT, CMS}. These domains are not necessarily of finite type near every boundary point and in many cases, the
type may not even be defined because of the low regularity of the boundary of the domains. We also emphasize that these domains need not be Cauchy-complete with respect to the Kobayashi distance.
\end{remark}
\subsection*{An application of additive localization}
Next, we show that for bounded domains, multiplicative localization (given below) as a consequence of the additive localization. This shows that additive localization is indeed stronger than multiplicative localization, at least in this case.
\begin{theorem}\label{T:Main_Thm_3}
Suppose $\Omega \subset \mathbb{C}^d$ is a bounded domain. Suppose $U$ and $W$ are open subsets of $\mathbb{C}^d$ with $W \subset \subset U$, $W \cap \Omega \neq \emptyset$ and $U \cap \Omega$ is connected, and there exists a constant $C_0> 0$ such that for all $z, w \in W \cap \Omega$
\[
\mathsf{k}_{U \cap \Omega}(z, w) \leq \mathsf{k}_{\Omega}(z, w) + C_0.
\]
Then there exists a constant $ C \geq 1$ such that for all $z, w \in W \cap \Omega$,
\[
\mathsf{k}_{U \cap \Omega}(z, w) \leq C\mathsf{k}_{\Omega}(z, w).
\]
\end{theorem}
The above theorem follows from a more general result with no boundedness assumption on $\Omega$ but with some other conditions.
\begin{theorem}\label{T:Main_Thm_4}
Suppose $\Omega \subset \mathbb{C}^d$ is a Kobayashi hyperbolic domain and $W_1, W_2$ are open subsets of $\mathbb{C}^d$ with $W_1 \subset \subset W_2$, $W_1 \cap \Omega \neq \emptyset$ and $\mathsf{k}_{\Omega}(W_1 \cap \Omega, \Omega \setminus W_2) > 0$. Suppose $U$ and $W$ are open subsets of with $W \subset \subset W_1 \subset \subset W_2 \subset \subset U$, $W \cap \Omega \neq \emptyset$, $U \cap \Omega$ is connected, and there exists a constant $C_0> 0$ such that for all $z, w \in W \cap \Omega$
\[
\mathsf{k}_{U \cap \Omega}(z, w) \leq \mathsf{k}_{\Omega}(z, w) + C_0.
\]
Then there exists a constant $ C \geq 1$ such that for all $z, w \in W \cap \Omega$,
\[
\mathsf{k}_{U \cap \Omega}(z, w) \leq C\mathsf{k}_{\Omega}(z, w).
\]
\end{theorem}
\begin{remark}
The above result is proved using a weaker assumption than that of Theorem~1.4 in \cite{NOT} although the proofs are similar. It can be applied in situations where no form of visibility is present, but additive localization is present. We do not know any non-trivial examples of such domains.
\end{remark}
Very recently, similar localization results are obtained by Nikolov--\"Okten--Thomas \cite{NOT} under some weaker conditions along with results on local and global visibility.
\section{Preliminaries}
\subsection{Notations and conventions}
Let $\Omega \subset \mathbb{C}^d$, $d \geq 1$, be a domain, $U \subset \mathbb{C}^d$ an open set, and $X, Y \subset \mathbb{C}^d $. We denote $\Omega \setminus X$ as the complement of $X \cap \Omega$ in $\Omega$.
\begin{itemize}
\item $X \subset \subset U$ means that $X$ is a relatively compact subset of $U$, i.e., $\overline{X} \subset U$ and $\overline{X}$ is a compact set.
\item For $z \in \Omega$, $\delta_{\Omega}(z) = \inf_{w \in \mathbb{C}^d \setminus \Omega}||z - w||$, where $||z - w||$ denotes the Euclidean distance between $z$ and $w$.
\item For $z \in \Omega$, $\mathsf{k}_{\Omega}(z, X) \defeq \inf_{x \in X}\mathsf{k}_{\Omega}(z, x)$.
\item $\mathsf{k}_{\Omega}(X, Y) \defeq \inf_{x \in X, y \in Y}\mathsf{k}_{\Omega}(x, y)$.
\item Throughout this paper, whenever for an open set $U \subset \mathbb{C}^d$ if $U \cap \partial \Omega \neq \emptyset$, we assume the cardinality of $U \cap \partial \Omega$ is greater than one, i.e., $\#(U \cap \partial \Omega) > 1$, because when $\#(U \cap \partial \Omega) = 1$ the main results of this paper follows trivially. Here $\partial \Omega$ denotes the boundary of the domain $\Omega$.
\item For a curve $\gamma: I \longrightarrow \Omega$ of an interval $I \subset \mathbb{R}$, by a slight abuse of notation we denote the range of the curve $\gamma$ by $\gamma$ itself.
\end{itemize}
\begin{definition}
For $ \lambda \geq 1$ and $\kappa \geq 0$, a curve $\gamma: I \longrightarrow \Omega$ of an interval $I \subset \mathbb{R}$ is said to be $(\lambda, \kappa)${\bf -quasi-geodesic} if for all $s, t \in I$
\[
\frac{1}{\lambda}|t - s| - \kappa\leq \mathsf{k}_{\Omega}(\gamma(s), \gamma(t) ) \leq \lambda |s -t| + \kappa,
\]
and when $\lambda =1$ and $\kappa = 0$ $\gamma$ is called a geodesic.
Furthermore, for $ \lambda \geq 1$ and $\kappa \geq 0$, a $(\lambda, \kappa)$-quasi-geodesic $\gamma: I \longrightarrow \Omega$ is said to be $(\lambda, \kappa)${\bf-almost-geodesic} of $( \Omega, \mathsf{k}_{\Omega} )$ if,
\begin{itemize}
\item $\gamma$ is an absolutely continuous curve, and
\item $\kappa_{\Omega}(\gamma(t); \gamma^{\prime}(t) ) \leq \lambda$ for almost every $t \in I$, ($\gamma^{\prime}$ exists almost everywhere on $I$ since $\gamma$ is absolutely continuous).
\end{itemize}
\end{definition}
Given any $ \kappa > 0$ and $\Omega \subset \mathbb{C}^d$ a Kobayashi hyperbolic domain (need not be bounded), it is shown in \cite[Proposition~5.3]{Bharali_Zimmer_2} that given any two points $z, w \in \Omega$ there exists a $(1, \kappa)$-almost-geodesic $
\gamma: [0, a] \longrightarrow \Omega$ joining $z$ and $w$, that is, $\gamma(0) = z$ and $\gamma(a) = w$.
Now, we state the definition of visibility and weak visibility.
\begin{definition}\label{D:Weak_vis}
Let $\Omega \subset \mathbb{C}^d$ be a Kobayashi hyperbolic domain. For $\lambda \geq 1$ and $\kappa \geq 0$, we say a pair of points $p, q \in \partial \Omega$, $p \neq q$ has visibility property with respect to $( \Omega, \mathsf{k}_{\Omega} )$ if there exist neighbourhoods $U$ of $p$ and $V$ of $q$ in $\mathbb{C}^d$ such that $\overline U \cap \overline V \neq \emptyset$ and a compact set $K \subset \Omega$ such that for any $(\lambda, \kappa)$-almost-geodesic $\gamma: [0, a] \longrightarrow \Omega$ with $\gamma(0) \in U \cap \Omega$ and $\gamma(a) \in V \cap \Omega$, $\gamma \cap K \neq \emptyset$. When the above condition is only required for $\lambda = 1$ and $\kappa \geq 0$, we say that the distinct pair $p, q$ has {\bf weak visibility} property with respect to $(\Omega, \mathsf{k}_{\Omega} )$.
\end{definition}
\section{Proofs of the main results}
The following localization lemma, by L. H. Royden \cite[Lemma~2]{Royden}, whose proof can be found in \cite[Lemma~4]{Ian_Gramham}, is used to prove localization of the Kobayashi distance and to study the relation between local and global visibility and Gromov hyperbolicity in \cite{BNT} and \cite{BGNT} respectively.
\noindent {\bf Royden's Localization Lemma.}
Suppose $\Omega \subset \mathbb{C}^d $ is a Kobayashi hyperbolic domain and $U $ is an open subset of $\mathbb{C}^d$ such that $U \cap \Omega \neq \emptyset$ and $U \cap \Omega $ is connected.
Then for all $z \in U \cap \Omega$ and $v \in \mathbb{C}^d$,
\begin{equation}
\kappa_{\Omega}(z; v) \leq \kappa_{U \cap \Omega}(z; v) \leq \coth(\mathsf{k}_{\Omega}(z, \Omega \setminus U)) \kappa_{\Omega}(z; v).
\end{equation}
Next, we prove a lemma that is used to prove the localization of the Kobayashi distance for visibility domains stated above.
\begin{lemma}\label{L:Varied_Loc_Kob_metric}
Suppose $\Omega \subset \mathbb{C}^d $ is a Kobayashi hyperbolic domain and $U $ is an open subset of $\mathbb{C}^d$ such that $U \cap \Omega \neq \emptyset$ and connected.
Then for every $W \subset \subset U$ with $W \cap \Omega \neq \emptyset$ and $\mathsf{k}_{\Omega}(W \cap \Omega, \Omega \setminus U) > 0$, there exists $L > 0$ such that for all $z \in W \cap \Omega$ and $v \in \mathbb{C}^d$,
\begin{equation}
\kappa_{U \cap \Omega}(z; v) \leq \left( 1 + L e^{-\mathsf{k}_{\Omega} (z, \Omega \setminus U) } \right)\kappa_{\Omega}(z; v).
\end{equation}
\end{lemma}
\begin{proof}
We first note the inequality,
\begin{equation}\label{E:Ineq_tanh}
\tanh(x) \geq 1 - e^{-x}\,\, \forall \,\, x \geq 0.
\end{equation}
This follows from the following formula obtained by algebraic manipulations: for all $x \in \mathbb{R}$,
\[
\tanh(x) - (1 - e^{-x}) = \frac{(1 -e^{-x})^{2}}{e^{x} + e^{-x}} \geq 0.
\]
Now note that, for all $z \in W \cap \Omega$, we have $\coth(\mathsf{k}_{\Omega}(z, \Omega \setminus U)) \leq \coth(\mathsf{k}_{\Omega}(W \cap \Omega, \Omega \setminus U)) =: L < +\infty$. By the Royden's Localization Lemma, for all $z \in U\cap \Omega$ and $v \in \mathbb{C}^d$, we have
\[
\tanh(\mathsf{k}_{\Omega}(z, \Omega \setminus U))\kappa_{U \cap \Omega}(z; v) \leq \kappa_{\Omega}(z; v).
\]
This gives, after applying $\kappa_{ U \cap \Omega}(z; v) \leq \coth(\mathsf{k}_{\Omega}(z, \Omega \setminus U)) \kappa_{\Omega}(z; v) \leq L \kappa_{\Omega}(z; v) $ for every $z \in W \cap \Omega$ and (\ref{E:Ineq_tanh}), for all $z \in W \cap \Omega$
\begin{equation*}
\begin{split}
\kappa_{U \cap \Omega}(z; v)
&\leq (1 - \tanh(\mathsf{k}_{\Omega}(z, \Omega \setminus U)) ) \kappa_{U \cap \Omega}(z; v) +\kappa_{\Omega}(z; v)\\
&\leq \left( 1 + L\left(1 - \tanh(\mathsf{k}_{\Omega}(z, \Omega \setminus U)) \right) \right) \kappa_{ \Omega}(z; v) \\
&\leq \left(1 + Le^{- \mathsf{k}_{\Omega}(z, \Omega \setminus U) } \right)\kappa_{ \Omega}(z; v) \quad [\text{by applying (\ref{E:Ineq_tanh})}].
\end{split}
\end{equation*}
This proves the lemma.
\end{proof}
\begin{proposition}\label{P:Visibility_Outside_Point}
Suppose $\Omega \subset \mathbb{C}^d$ is a Kobayashi hyperbolic domain and $U$ is an open subset of $\mathbb{C}^d$, and $U \cap \partial \Omega \neq \emptyset$. Suppose $(\Omega, \mathsf{k}_{\Omega} )$ has weak visibility property for every distinct pair $\{p, q\}$, $ p,q \in U \cap \partial \Omega$. Let $W \subset \subset U$ be an open set with $W \cap \Omega \neq \emptyset$.
Then
\begin{enumerate}
\item for any $\xi_2 \in \partial \Omega \setminus U$ (whenever $\partial \Omega \setminus U \neq \emptyset$) and $\xi_1 \in U \cap \partial \Omega$, the distinct pair $\{\xi_1, \xi_2\}$ has weak visibility property with respect to $(\Omega, \mathsf{k}_{\Omega} )$. Consequently, every pair $\{\xi_1, \xi_2\}$, with $\xi_1 \in U \cap \partial \Omega$, $\xi_2 \in \partial \Omega$ and $\xi_1 \neq \xi_2$, has weak visibility property with respect to $(\Omega, \mathsf{k}_{\Omega} )$.
\item Let $V$ is a subset of $\mathbb{C}^d$ with $V \cap \Omega \neq \emptyset$ and $\overline W \cap \overline V = \emptyset$. Then for $\kappa \geq 0$ there is a compact set $K \subset \Omega$ such for every $z \in W \cap \Omega$ and $w \in V \cap \Omega $ and $\gamma$ is a $(1, \kappa)$-almost-geodesic joining $z$ and $w$, $\gamma \cap K \neq \emptyset$.
\item For every open sets $W_1, W_2$ with $W \subset \subset W_1 \subset \subset W_2 \subset \subset U$, we have $\mathsf{k}_{\Omega}(W_1 \cap \Omega, \Omega \setminus W_2) > 0$. Consequently, $\mathsf{k}_{\Omega}(W \cap \Omega, \Omega \setminus U) > 0$.
\end{enumerate}
\end{proposition}
\begin{proof}
If possible, suppose that {\it (1)} is not true. Then we get a sequence of $(1, \kappa)$-almost-geodesics $\gamma_n: [0, a_n] \longrightarrow \Omega$ such that $\gamma_n(0) \to \xi_1$ and $\gamma_n(a_n) \to \xi_2$ and $ \lim_{n \to \infty} \sup_{z \in \gamma_n \cap B(0, R)}\delta_{\Omega}(z) = 0 $, for all $R> R_0$, where we choose $R_0 > 0$ such that $\gamma_n \cap B(0, R_0) \neq \emptyset $ for all $n \in \mathbb{Z}_{+}$. Since $U$ is an open subset of $\mathbb{C}^d$ and $\xi_1 \in U$, there exists $0 < r < ||\xi_1 - \xi_2||/4$ such that the open ball of radius $2r$ centred at $\xi_1$, $B(\xi_1, 2r) \subset U$. Without loss of generality (by passing to a subsequence if necessary), we may assume that $\gamma_n(0) \in B(\xi_1, r/2)$ and $\gamma_n(a_n) \in B(\xi_2, r/2)$ for all $n \in \mathbb{Z}_{+}$. Now, let
\[
t_n \defeq \inf\{t \in [0, a_n] : \gamma_n(t) \in \partial B(\xi_1, r) \cap \Omega \}.
\]
Again, by passing to a subsequence, we may assume that $ \gamma_n(t_n) \to \xi_0 \in (\partial B(\xi_1, r) \cap \partial \Omega) \subset U \cap \partial \Omega$. Now, note that the sequence of $(1, \kappa)$-almost-geodesics defined as
\[
\sigma_n \defeq \gamma_n|_{[0, t_n]}
\]
shows that the distinct pair $\{ \xi_1, \xi_0 \} \subset U \cap \partial \Omega $ does not have weak visibility property with respect to $(\Omega, \mathsf{k}_{\Omega} )$. This gives a contradiction to our assumption. This proves {\it (1)}.
Next, we shall prove {\it (2)}. If possible, to get a contradiction, suppose that {\it (2)} does not hold. Then there exist $\kappa \geq 0$, sequences of points $z_n \in W \cap \Omega$ and $w_n \in V \cap \Omega$ and $(1, \kappa)-$almost-geodesic $\gamma_n$ of $(\Omega, \mathsf{k}_{\Omega} )$ joining $z_n$ and $w_n$ for all $n \in \mathbb{Z}_{+}$, such that $\lim_{n \to \infty} \sup_{z \in \gamma_n \cap B(0, R)}\delta_{\Omega}(z) = 0 $, for all $R> R_1$, where we choose $R_1 > 0$ such that $ \overline W \subset B(0, R_1) $. By passing to a subsequence, we may assume that $z_n \to \xi'_1 \in \overline W \cap \partial \Omega \subset U \cap \partial \Omega $ as $n \to \infty$.
Since $\overline {W \cap \Omega} \cap \overline{V} = \emptyset$, let $0 < r_1 < \frac{||x - y||}{4}$ for all $x \in W \cap \Omega $ and $y \in V$. Assume, by passing to a subsequence, $z_n \in B(\xi'_1, r_1/2) $ for all $n \in \mathbb{Z}_{+}$, and define
\[
t'_n \defeq \inf\{t \in [0, a_n] : \gamma_n(t) \in \partial B(\xi'_1, r_1) \cap \Omega \}.
\]
Without loss of generality, we may assume that $\gamma_n(t'_n) \to \xi'_2 \in \partial B(\xi'_1, r_1) \cap \partial \Omega$ as $n \to \infty$. By construction,
$\xi'_1 \neq \xi'_2$. Now, note that the sequence of $(1, \kappa)$-almost-geodesics defined as
\[
\sigma'_n \defeq \gamma_n|_{[0, t'_n]}
\]
shows that the distinct pair $\{ \xi'_1, \xi'_2 \} \subset \partial \Omega $ with $\xi'_1 \in U \cap \partial \Omega$ does not have weak visibility property with respect to $(\Omega, \mathsf{k}_{\Omega} )$. This gives a contradiction to {\it (1)}. Hence, {\it (2)} holds true.
To prove {\it (3)} by contradiction if possible let $\mathsf{k}_{\Omega}(W_1 \cap \Omega, \Omega \setminus W_2) = 0$. Then there exist sequences of points $z_n \in W_1 \cap \Omega$ and $w_n \in \Omega \setminus W_2$ such that $\mathsf{k}_{\Omega}(z_n, w_n) \to 0$ as $n \to \infty$. Let $\gamma_n: [0, a_n] \longrightarrow \Omega$ is a sequence of $(1, 1/n)$-almost-geodesic such that $\gamma_n(0) = z_n \in W_1 \cap \Omega$ and $\gamma_n(a_n) = w_n \in \Omega \setminus W_2$ for all $n \in \mathbb{Z}_{+}$. Since $W_1$ is a relatively compact subset of U, and $\overline {W_1} \cap \overline{\Omega \setminus W_2} = \emptyset$, by {\it (2)}, for $\kappa \geq 1 $ there exists a compact set $K \subset \Omega$ such that for every $z \in W_1 \cap \Omega$ and $w \in \Omega \setminus W_2 $ if $\gamma$ is a $(1, \kappa)$-almost-geodesic joining $z$ and $w$, then $\gamma \cap K \neq \emptyset$. Hence,
$\gamma_n \cap K \neq \emptyset$ for all $n \in \mathbb{Z}_{+}$. Let $o_n \in \gamma_n \cap K$ and if necessary by passing to a subsequence, assume that $o_n \to o \in K$. This gives, using the fact that $\gamma_n$ is a $(1, 1/n)$-almost-geodesic passing through $o_n$,
\[
\mathsf{k}_{\Omega}(z_n, o_n) + \mathsf{k}_{\Omega}(o_n, w_n) - 3/n \leq \mathsf{k}_{\Omega}(z_n, w_n).
\]
This implies as $n \to \infty$, $\mathsf{k}_{\Omega}(z_n, o_n) \to 0$ and $\mathsf{k}_{\Omega}( o_n, w_n) \to 0$. Since $\Omega$ is a Kobayashi hyperbolic, it follows $z_n \to o$ and $w_n \to o$. This is a contradiction because $z_n \in W_1 \cap \Omega$, $w_n \in \Omega \setminus W_2$ for all $n \in \mathbb{Z}_{+}$, and $\overline{W_1 \cap \Omega} \cap \overline{\Omega \setminus W_2} = \emptyset$. Hence $\mathsf{k}_{\Omega}(W_1 \cap \Omega, \Omega \setminus W_2) > 0$.
Moreover, using the fact that if $X, Y \subset \Omega$ and $A \subset X$ and $B \subset Y$, then by definition $\mathsf{k}_{\Omega}( A, B) \geq \mathsf{k}_{\Omega}(X, Y)$, we have $\mathsf{k}_{\Omega}(W \cap \Omega, \Omega \setminus U) \geq \mathsf{k}_{\Omega}(W_1 \cap \Omega, \Omega \setminus W_2) > 0$. This completes the proof of {\it (3)}.
\end{proof}
\begin{lemma}\label{L:Comple_dist_Compar}
Suppose $\Omega \subset \mathbb{C}^d$ is a Kobayashi hyperbolic domain and $U$ is an open subset of $\mathbb{C}^d$ such that $U \cap \partial \Omega \neq \emptyset$. Suppose $( \Omega, \mathsf{k}_{\Omega} )$ has weak visibility property for every pair of distinct points $\{p, q\}$, $p, q \in U \cap \partial \Omega$.
Then for every open sets $W_1, W_2$ with $W_1 \subset \subset W_2 \subset \subset U$, $W_1 \cap \Omega \neq \emptyset$ and $o \in \Omega$ there exists $L > 0$ such that for all $z \in W_1 \cap \Omega$
\[
\mathsf{k}_{\Omega} ( z, o) \leq \mathsf{k}_{\Omega} (z, \Omega \setminus W_2) + L.
\]
\end{lemma}
\begin{proof}
Let $z \in W_1 \cap \Omega$ and $w \in \Omega \setminus W_2$.
Let $\kappa > 0$ and $\gamma$ is a $(1, \kappa)$-almost-geodesic joining $z$ and $w$. Then, by {\it (2)} of Proposition~\ref{P:Visibility_Outside_Point}, there exists a compact subset $K \subset \Omega$ such that $\gamma \cap K \neq \emptyset$. Let $o_w \in \gamma \cap K$, then we have (using the fact that $\gamma$ is $(1, \kappa)$-almost-geodesic)
\begin{equation*}
\begin{split}
\mathsf{k}_{\Omega}(z, o)
&\leq \mathsf{k}_{\Omega}(z, o_w) + \mathsf{k}_{\Omega}(o_w, o) \\
& \leq \mathsf{k}_{\Omega}(z, w) + 2\kappa + \sup_{y \in K}\mathsf{k}_{\Omega}(y, o).
\end{split}
\end{equation*}
Since $w \in \Omega \setminus W_2$ is arbitrary, by taking $L \defeq 2\kappa + \sup_{y \in K}\mathsf{k}_{\Omega}(y, o) $, we have,
\[
\mathsf{k}_{\Omega} ( z, o) \leq \mathsf{k}_{\Omega} (z, \Omega \setminus W_2) + L.
\]
\end{proof}
As a corollary of the above lemma, we have a similar local result.
\begin{lemma}\label{L:Comple_dist_Compar_local}
Suppose $\Omega \subset \mathbb{C}^d$ is a Kobayashi hyperbolic domain and $U$ is an open subset of $\mathbb{C}^d$ such that $U \cap \partial \Omega \neq \emptyset$ and $U \cap \Omega $ is connected. Suppose $( U \cap \Omega, \mathsf{k}_{U \cap \Omega} )$ has weak visibility property for every pair of distinct points $\{p, q\}$, $p, q \in U \cap \partial \Omega$.
Then for every open sets $W_1, W_2$ with $W_1 \subset \subset W_2 \subset \subset U$, $W_1 \cap \Omega \neq \emptyset$ and $o \in U \cap\Omega$ there exists $L > 0$ such that for all $z \in W_1 \cap \Omega$
\[
\mathsf{k}_{U \cap \Omega} ( z, o) \leq \mathsf{k}_{U \cap \Omega} (z, U \cap \Omega \setminus W_2) + L.
\]
\end{lemma}
\begin{proof}
The proof follows replacing $\Omega$ by $U \cap \Omega$ in Lemma~\ref{L:Comple_dist_Compar}.
\end{proof}
\begin{lemma}\label{L:One_kappa_geodesic}
Suppose $\Omega \subset \mathbb{C}^d$ is a Kobayashi hyperbolic domain and $U$ is an open subset of $\mathbb{C}^d$ such that $U \cap \partial \Omega \neq \emptyset$ and $U \cap \Omega $ is connected. Suppose $( U \cap \Omega, \mathsf{k}_{U \cap \Omega} )$ has weak visibility property for every pair of distinct points $\{p, q\}$, $p, q \in U \cap \partial \Omega$. Let $W, W_1$ be open sets with $W \subset \subset W_1 \subset \subset U $, $W \cap \Omega \neq \emptyset$ and $\mathsf{k}_{\Omega}(W_1 \cap \Omega, \Omega \setminus U) > 0$, and suppose that $z, w \in W \cap \Omega $ and $\gamma$ is a $(1, \kappa)$-almost-geodesic of $(\Omega, \mathsf{k}_{\Omega})$ joining $z$ and $w$ such that $\gamma \subset W$. Then there exists a $\kappa_0 > 0$ such that $\gamma$ is $(1, \kappa_0)$-quasi-geodesic of $(U \cap \Omega, \mathsf{k}_{U \cap \Omega} )$.
\end{lemma}
\begin{proof}
By Lemma~\ref{L:Varied_Loc_Kob_metric}, there exists $L > 0$ which depends on $U, W_1$ such that for all $z \in W_1 \cap \Omega$ and $v \in \mathbb{C}^d$,
\begin{equation*}
\kappa_{U \cap \Omega}(z; v) \leq \left( 1 + L e^{-\mathsf{k}_{\Omega} (z, \Omega \setminus U) } \right)\kappa_{\Omega}(z; v).
\end{equation*}
Since $\mathsf{k}_{\Omega} (z, \Omega \setminus W_1) \leq \mathsf{k}_{\Omega} (z, \Omega \setminus U)$, we have
\begin{equation}\label{E:Lemma_use_Roy_loc}
\kappa_{U \cap \Omega}(z; v) \leq \left( 1 + L e^{-\mathsf{k}_{\Omega} (z, \Omega \setminus W_1) } \right)\kappa_{\Omega}(z; v).
\end{equation}
Suppose that $\gamma: [0, a] \longrightarrow \Omega$, $a \geq 0$.
Since $\gamma \subset W$, from above inequality and by \cite[Theorem~3.1]{Venturini}, for every $s_1, s_2 \in [0, a]$, we have
\begin{equation}\label{E:One_kappa_geo_ineq}
\begin{split}
\mathsf{k}_{U \cap \Omega}(\gamma(s_1), \gamma(s_2))
& \leq \int_{s_1}^{s_2} \kappa_{U \cap \Omega}(\gamma(t); \gamma^{\prime}(t)) dt\\
& \leq \int_{s_1}^{s_2} \left( 1 + L e^{-\mathsf{k}_{\Omega} (\gamma(t), \Omega \setminus W_1) } \right)\kappa_{\Omega}(\gamma(t); \gamma^{\prime}(t)) dt\\
&\leq |s_2 - s_1| + L \int_{s_1}^{s_2} e^{-\mathsf{k}_{\Omega} (\gamma(t), \Omega \setminus W_1) } dt \\
& \leq \mathsf{k}_{\Omega}(\gamma(s_1), \gamma(s_2)) + \kappa + L \int_{0}^{a} e^{-\mathsf{k}_{\Omega} (\gamma(t), \Omega \setminus W_1) } dt.
\end{split}
\end{equation}
{\bf Claim.}
There exist $C_0, C_1 > 0$ such that for all $z \in W \cap \Omega$
\[
\mathsf{k}_{U \cap \Omega}(z, U \cap \Omega \setminus W_1) \leq C_0 \mathsf{k}_{\Omega}(z, \Omega \setminus W_1) + C_1.
\]
Assuming the claim and deferring the proof, first note that, for $o \in U \cap \Omega$, by Lemma~\ref{L:Comple_dist_Compar_local}, there exists $L_0 > 0$ which depends on $o, W, W_1, U$ such that for all $z \in W \cap \Omega$,
\[
\mathsf{k}_{U \cap \Omega} ( z, o) \leq \mathsf{k}_{U \cap \Omega} (z, U \cap \Omega \setminus W_1) + L_0.
\]
Next, let $t_0 \in [0, a]$ such that for all $t \in [0, a]$,
\[
\mathsf{k}_{\Omega}(\gamma(t_0), \Omega \setminus W_1) \leq \mathsf{k}_{\Omega}(\gamma(t), \Omega \setminus W_1).
\]
Now, applying the above inequalities in the following computation, we obtain, for all $t \in [0, a]$
\begin{equation*}
\begin{split}
|t - t_0| - \kappa
&\leq \mathsf{k}_{\Omega}(\gamma(t), \gamma(t_0))\\
& \leq \mathsf{k}_{\Omega}(\gamma(t), o) + \mathsf{k}_{\Omega}(\gamma(t_0), o)\\
& \leq \mathsf{k}_{U \cap \Omega}(\gamma(t), o) + \mathsf{k}_{U \cap \Omega}(\gamma(t_0), o)\\
&\leq \mathsf{k}_{U \cap \Omega} (\gamma(t), U \cap \Omega \setminus W_1) + \mathsf{k}_{U \cap \Omega} (\gamma(t_0), U \cap \Omega \setminus W_1) + 2L_0\\
&\leq 2\mathsf{k}_{U \cap \Omega} (\gamma(t), U \cap \Omega \setminus W_1) + 2L_0\\
\end{split}
\end{equation*}
If we apply the claim to the above inequality, we get
\begin{equation*}
\begin{split}
|t - t_0| - \kappa
&\leq 2\mathsf{k}_{U \cap \Omega} (\gamma(t), U \cap \Omega \setminus W_1) + 2L_0\\
& \leq 2C_0 \mathsf{k}_{\Omega}(\gamma(t), \Omega \setminus W_1) + 2C_1 + 2 L_0.
\end{split}
\end{equation*}
This gives, for all $ t \in [0, a]$
\[
-\mathsf{k}_{\Omega}(\gamma(t), \Omega \setminus W_1) \leq \frac{ -|t - t_0|}{2C_0} + \frac{2C_1 + 2L_0 + \kappa}{2C_0}.
\]
After using the above inequality in inequality~(\ref{E:One_kappa_geo_ineq}), we obtain for every $s_1, s_2 \in [0, a]$,
\begin{equation}
\begin{split}
\mathsf{k}_{U \cap \Omega}(\gamma(s_1), \gamma(s_2))
& \leq \mathsf{k}_{\Omega}(\gamma(s_1), \gamma(s_2)) + \kappa + LC_3 \int_{0}^{a} e^{-\frac{|t - t_0|}{2C_0} } dt\\
&\leq \mathsf{k}_{\Omega}(\gamma(s_1), \gamma(s_2)) + \kappa + LC_3 \int_{0}^{+\infty} e^{-\frac{|t - t_0|}{2C_0} } dt\\
&\leq \mathsf{k}_{\Omega}(\gamma(s_1), \gamma(s_2)) + \kappa + 4LC_3 C_0
\end{split}
\end{equation}
where $C_3 = e^{\frac{2C_1 + 2L_0 + \kappa}{2C_0}}$, and the last inequality follows from the following bound:
\begin{equation}
\int_{0}^{+\infty} e^{-\frac{|t - t_0|}{2C_0} } dt \leq 4C_0.
\end{equation}
This shows that $\gamma$ is a $(1, \kappa_0)$-quasi-geodesic for $\kappa_0 = \kappa + 4LC_3C_0$, given that the claim is true (note the fact that $\kappa_0$ depends only on $W,W_1,U$ and $\kappa$). Now, we give a proof of the claim.
\begin{proof}[Proof of the claim.]
By the Royden's Localization Lemma and setting $ \coth{\mathsf{k}_{\Omega}(W_1 \cap \Omega, \Omega \setminus U)} := C_0 < +\infty$, for all $z \in W_1 \cap \Omega $ and $v \in \mathbb{C}^d$,
\[
\kappa_{U \cap \Omega}(z; v) \leq \coth{\mathsf{k}_{\Omega}(z, \Omega \setminus U)} \kappa_{\Omega}(z;v) \leq \coth{\mathsf{k}_{\Omega}(W_1 \cap \Omega, \Omega \setminus U)} \kappa_{\Omega}(z;v) = C_0 \kappa_{\Omega}(z;v).
\]
Now, suppose that $w_n \in \Omega \setminus W_1$ such that $\lim_{n \to \infty} \mathsf{k}_{\Omega}(z, w_n) = \mathsf{k}_{\Omega}(z, \Omega \setminus W_1) $ and $\gamma_n : [0, a_n] \longrightarrow \Omega$ is sequence of $(1, \kappa)$-almost-geodesic of $(\Omega, \mathsf{k}_{\Omega} )$ joining $z$ and $w_n$, for some $\kappa > 0$.
\[
t_n = \sup \{t : t \in [0, a_n]\,\, \text{such that}\,\, \gamma_n(s) \in W_1 \cap \Omega \,\forall s \in [0, t]\},
\]
By definition, $\gamma_n([0, t_n)) \subset W_1 \cap \Omega$. Then,
\begin{equation*}
\begin{split}
\mathsf{k}_{U \cap \Omega}(z, U \cap \Omega \setminus W_1)
&\leq \mathsf{k}_{U \cap \Omega}(z, \gamma_n(t_n) )\\
& \leq \int_{0}^{t_n} \kappa_{U \cap \Omega}(\gamma_n(t); \gamma_n^{\prime}(t)) dt\\
& \leq C_0 \int_{0}^{t_n} \kappa_{ \Omega}(\gamma_n(t); \gamma_n^{\prime}(t)) dt\\
&\leq C_0 \int_{0}^{a_n} \kappa_{ \Omega}(\gamma_n(t); \gamma_n^{\prime}(t)) dt\\
&\leq C_0 \mathsf{k}_{\Omega}(z, w_n) + C_0 \kappa.\\
\end{split}
\end{equation*}
The third inequality follows because $\kappa_{U \cap \Omega}(\gamma_n(t); \gamma_n^{\prime}(t)) \leq C_0 \kappa_{\Omega}(\gamma_n(t); \gamma_n^{\prime}(t)) $ a.e. $t \in [0, a_n] $.
Taking limit as $n \to \infty$, we get
\[
\mathsf{k}_{U \cap \Omega}(z, U \cap \Omega \setminus W_1) \leq C_0\mathsf{k}_{\Omega}(z, \Omega \setminus W_1) + C_0 \kappa.
\]
This proves the claim. Hence the proof is complete.
\end{proof}
\end{proof}
\begin{lemma}\label{L:One_kappa_geodesic_Omega}
Suppose $\Omega \subset \mathbb{C}^d$ is a domain and $U$ is an open subset of $\mathbb{C}^d$ such that $U \cap \partial \Omega \neq \emptyset$. Suppose $(\Omega, \mathsf{k}_{\Omega} )$ has weak visibility property for every pair of distinct points $\{p, q\}$, $p, q \in U \cap \partial \Omega$. Let $W \subset \subset U $ be an open set with $W \cap \Omega \neq \emptyset$, and suppose that $z, w \in W \cap \Omega $ and $\gamma$ is a $(1, \kappa)$-almost-geodesic of $(\Omega, \mathsf{k}_{\Omega})$ joining $z$ and $w$ such that $\gamma \subset W$. Then there exists a $\kappa_0 > 0$ such that $\gamma$ is $(1, \kappa_0)$-quasi-geodesic of $(U \cap \Omega, \mathsf{k}_{U \cap \Omega} )$.
\end{lemma}
\begin{proof}
Let $W_1 \subset \mathbb{C}^d$ be an open set such that $W \subset \subset W_1 \subset \subset U$. By {\it (3)} of Proposition~\ref{P:Visibility_Outside_Point}, $\mathsf{k}_{\Omega}(W_1 \cap \Omega, \Omega \setminus U) >0 $.
Therefore, (as observed at the start of the proof of Lemma~\ref{L:One_kappa_geodesic}),
by Lemma~\ref{L:Varied_Loc_Kob_metric}, there exists $L > 0$ which depends on $U, W_1$ such that for all $z \in W_1 \cap \Omega$ and $v \in \mathbb{C}^d$,
\begin{equation*}
\kappa_{U \cap \Omega}(z; v) \leq \left( 1 + L e^{-\mathsf{k}_{\Omega} (z, \Omega \setminus W_1) } \right)\kappa_{\Omega}(z; v).
\end{equation*}
Suppose that $\gamma: [0, a] \longrightarrow \Omega$. Then by doing similar computations as of (\ref{E:One_kappa_geo_ineq}) of Lemma~ \ref{L:One_kappa_geodesic}, for every $s_1, s_2 \in [0, a]$, we have
\begin{equation}\label{E:One_kappa_geo_ineq_2}
\begin{split}
\mathsf{k}_{U \cap \Omega}(\gamma(s_1), \gamma(s_2))
& \leq \mathsf{k}_{\Omega}(\gamma(s_1), \gamma(s_2)) + \kappa + L \int_{0}^{a} e^{-\mathsf{k}_{\Omega} (\gamma(t), \Omega \setminus W_1) } dt.
\end{split}
\end{equation}
Now, by Lemma~\ref{L:Comple_dist_Compar}, there exists $L_0 > 0$ which depends on $o, W, W_1, U$ such that for all $z \in W \cap \Omega$,
\[
\mathsf{k}_{\Omega} ( z, o) \leq \mathsf{k}_{\Omega} (z, \Omega \setminus W_1) + L_0.
\]
Next, let $t_0 \in [0, a]$ such that for all $t \in [0, a]$,
\[
\mathsf{k}_{\Omega}(\gamma(t_0), \Omega \setminus W_1) \leq \mathsf{k}_{\Omega}(\gamma(t), \Omega \setminus W_1).
\]
Since $\gamma$ is a $(1, \kappa)$-almost-geodesic of $(\Omega, \mathsf{k}_{\Omega} )$, we have by a simple computation (using the above inequalities and the triangle inequality), for all $t \in [0, a]$
\begin{equation*}
\begin{split}
|t - t_0| - \kappa
&\leq 2\mathsf{k}_{\Omega} (\gamma(t), \Omega \setminus W_1) + 2L_0.
\end{split}
\end{equation*}
This gives, for all $ t \in [0, a]$
\[
-\mathsf{k}_{\Omega}(\gamma(t), \Omega \setminus W_1) \leq \frac{ -|t - t_0|}{2} + \frac{2L_0 + \kappa}{2}.
\]
After using the above inequality in inequality~(\ref{E:One_kappa_geo_ineq_2}), we obtain for every $s_1, s_2 \in [0, a]$,
\begin{equation}
\begin{split}
\mathsf{k}_{U \cap \Omega}(\gamma(s_1), \gamma(s_2))
& \leq \mathsf{k}_{\Omega}(\gamma(s_1), \gamma(s_2)) + \kappa + LC_3 \int_{0}^{a} e^{-\frac{|t - t_0|}{2} } dt\\
&\leq \mathsf{k}_{\Omega}(\gamma(s_1), \gamma(s_2)) + \kappa + 4LC_3 C_0
\end{split}
\end{equation}
where $C_3 = e^{\frac{2L_0 + \kappa}{2}}$, and the last inequality follows from the following bound:
\begin{equation}
\int_{0}^{+\infty} e^{-\frac{|t - t_0|}{2} } dt \leq 4.
\end{equation}
This shows that $\gamma$ is a $(1, \kappa_0)$-quasi-geodesic for $\kappa_0 = \kappa + 4LC_3$ (note the fact that $\kappa_0$ depends only on $W,W_1,U$ and $\kappa$). Hence the proof is complete.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{T:Main_Thm_1}]
Let $W_1, W_2$ be open sets such that $W \subset \subset W_1 \subset \subset W_2 \subset \subset W_0$. Let $z, w \in W \cap \Omega$. Let $ \kappa > 0$ and $\gamma : [0, a] \longrightarrow \Omega$ be a $(1, \kappa)$-almost-geodesic of $(\Omega, \mathsf{k}_{\Omega} )$ joining $z$ and $w$.
\noindent
{\bf Case 1.} $\gamma \subset W_2 \cap \Omega $.
Since $\mathsf{k}_{\Omega}(W_0 \cap \Omega, \Omega \setminus U) > 0$, by replacing $W$ and $W_1$ by $W_2$ and $W_0$ respectively in Lemma~\ref{L:One_kappa_geodesic}, there exists $\kappa_0 > 0$ which depends on $W_2, W_0, U$ and $\kappa$ such that $\gamma$ is a $(1, \kappa_0)$-quasi-geodesic of $(U \cap \Omega, \mathsf{k}_{U \cap \Omega} )$.
This gives, using the fact that $\gamma$ is a $(1, \kappa_0)$-quasi-geodesic of $(U \cap \Omega, \mathsf{k}_{U \cap \Omega} )$ and $\gamma$ is a $(1, \kappa)$-almost-geodesic of $(\Omega, \mathsf{k}_{\Omega} )$,
\[
\mathsf{k}_{U \cap \Omega }(z, w) \leq a + \kappa_0 \leq \mathsf{k}_{ \Omega }(z, w) + \kappa + \kappa_0.
\]
\noindent
{\bf Case 2.} $\gamma \nsubseteq W_2 \cap \Omega $.
Now, set
\[
s_0 = \sup \{t : t \in [0, a]\,\, \text{such that}\,\, \gamma(s) \in W_1 \,\forall s \in [0, t]\},
\]
\[
t_0= \inf \{t : t \in [0, a]\,\, \text{such that}\,\, \gamma(s) \in W_1 \,\forall s \in [t, a]\}.
\]
Note that by definition, $\gamma([0, s_0]) \subset W_2$ and $\gamma([t_0, a]) \subset W_2$. Hence $\gamma|_{[0, s_0]}$ and $ \gamma|_{[t_0, a]}$ are $(1, \kappa_0)$-quasi-geodesics of $(U \cap \Omega, \mathsf{k}_{U \cap \Omega} )$ (as observed in Case 1).
Now, let $\sigma^1 : [0, b^1] \longrightarrow U \cap \Omega $ and $\sigma^2: [0, b^2] \longrightarrow U \cap \Omega $ be $(1, \kappa)$-almost-geodesics of $(U \cap \Omega, \mathsf{k}_{U \cap \Omega} )$ joining $z$ and $\gamma(s_0)$, and $w$ and $\gamma(t_0)$ respectively. Let $U' \subset \subset U$ such that $\#(U' \cap \partial \Omega) > 1$ and $W_0 \subset \subset U'$. Since $U' \cap \partial (U \cap \Omega) = U'\cap \partial \Omega$, every pair of distinct boundary points in $U'\cap \partial \Omega$ has weak visibility property with respect to $(U \cap \Omega, \mathsf{k}_{\Omega} )$.
Then, by {\it (2)} of Proposition ~\ref{P:Visibility_Outside_Point} by replacing the domain $\Omega$ by $ U\cap \Omega$, $U$ by $U'$, $W$ by $W$ and $V$ by $ (U \cap \Omega) \setminus W_1$, there exists a compact set $K \subset U \cap \Omega$ such that $\sigma^1 \cap K$ and $\sigma^2 \cap K$ are non-empty sets. Let $ o^1 \in \sigma^1 \cap K$ and $o^2 \in \sigma^2 \cap K$.
Now, using the fact that $\sigma^1 $ and $\sigma^2 $ are $(1, \kappa)$-almost-geodesics of $(U \cap \Omega, \mathsf{k}_{U \cap \Omega} )$ and the triangle inequality, we have the following inequality
\begin{equation}\label{E:C_inequality_1}
\begin{split}
\mathsf{k}_{U \cap \Omega}(z, \gamma(s_0))
&\geq \mathsf{k}_{U \cap \Omega}(z, o^1) + \mathsf{k}_{U \cap \Omega}(o^1, \gamma(s_0)) - 3\kappa\\
&\geq \mathsf{k}_{U \cap \Omega}(z, o) - \mathsf{k}_{U \cap \Omega}(o, o^1) + \mathsf{k}_{U \cap \Omega}(o, \gamma(s_0)) - \mathsf{k}_{U \cap \Omega}(o, o^1) - 3\kappa\\
&\geq \mathsf{k}_{U \cap \Omega}(z, o) + \mathsf{k}_{U \cap \Omega}(o, \gamma(s_0)) - 2 \sup_{y \in K }\mathsf{k}_{U \cap \Omega}(o, y) - 3\kappa,
\end{split}
\end{equation}
and similarly
\begin{equation}\label{E:C_inequality_2}
\begin{split}
\mathsf{k}_{U \cap \Omega}(w, \gamma(t_0))
&\geq \mathsf{k}_{U \cap \Omega}(w, o) + \mathsf{k}_{U \cap \Omega}(o, \gamma(t_0)) - 2 \sup_{y \in K }\mathsf{k}_{U \cap \Omega}(o, y) - 3\kappa.
\end{split}
\end{equation}
Let $C_0 \defeq 2 \sup_{y \in K }\mathsf{k}_{U \cap \Omega}(o, y) + 3\kappa$.
Now, using the fact that $\gamma$ is a $(1, \kappa)$-almost-geodesic of $ (\Omega, \mathsf{k}_{\Omega} )$, and $\gamma|_{[0, s_0]}$ and $ \gamma|_{[t_0, a]}$ are $(1, \kappa_0)$-quasi-geodesics of $ (U \cap\Omega, \mathsf{k}_{U \cap \Omega} )$, we have
\begin{equation*}
\begin{split}
\mathsf{k}_{\Omega}(z, w)
&\geq a - \kappa\\
&\geq a - t_0 + s_0 -\kappa \quad [\text{since $s_0 \leq t_0$ }]\\
&\geq \mathsf{k}_{U \cap \Omega}(w, \gamma(t_0) ) - \kappa_0 + \mathsf{k}_{U \cap \Omega}(\gamma(s_0), z) - \kappa_0 - \kappa\\
&\geq \mathsf{k}_{U \cap \Omega} (w, o) + \mathsf{k}_{U \cap \Omega}(o, \gamma(t_0) ) - C_0 + \mathsf{k}_{U \cap \Omega} (z, o) + \mathsf{k}_{U \cap \Omega}(o, \gamma(s_0) ) - C_0 - 2 \kappa_0- \kappa\\
&\geq \mathsf{k}_{U \cap \Omega} (z, o) + \mathsf{k}_{U \cap \Omega} (w, o) - 2C_0- 2 \kappa_0- \kappa\\
&\geq \mathsf{k}_{U \cap \Omega} (z, w) - 2C_0- 2 \kappa_0 - \kappa.
\end{split}
\end{equation*}
The fourth inequality follows from (\ref{E:C_inequality_1}) and (\ref{E:C_inequality_2}).
This proves the theorem by taking $C \defeq 2C_0 + 2 \kappa_0 + \kappa$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{T:Main_Thm_2}]
Let $W_1, W_2$ be open sets such that $W \subset \subset W_1 \subset \subset W_2 \subset \subset U$. Let $z, w \in W \cap \Omega$. Let $ \kappa > 0$ and $\gamma : [0, a] \longrightarrow \Omega$ be a $(1, \kappa)$-almost-geodesic of $(\Omega, \mathsf{k}_{\Omega} )$ joining $z$ and $w$.
\noindent
{\bf Case 1.} $\gamma \subset W_2 \cap \Omega $.
Now, replacing $W$ by $W_2$ in Lemma~\ref{L:One_kappa_geodesic_Omega}, there exists $\kappa_0 > 0$ which depends on $W_2, U$ and $\kappa$ such that $\gamma$ is a $(1, \kappa_0)$-quasi-geodesic of $(U \cap \Omega, \mathsf{k}_{U \cap \Omega} )$.
This gives, using the fact that $\gamma$ is a $(1, \kappa_0)$-quasi-geodesic of $(U \cap \Omega, \mathsf{k}_{U \cap \Omega} )$ and $\gamma$ is a $(1, \kappa)$-almost-geodesic of $(\Omega, \mathsf{k}_{\Omega} )$,
\[
\mathsf{k}_{U \cap \Omega }(z, w) \leq a + \kappa_0 \leq \mathsf{k}_{ \Omega }(z, w) + \kappa + \kappa_0.
\]
\noindent
{\bf Case 2.} $\gamma \nsubseteq W_0 \cap \Omega $.
Now, set
\[
s_0 = \sup \{t : t \in [0, a]\,\, \text{such that}\,\, \gamma(s) \in W_1 \,\forall s \in [0, t]\},
\]
\[
t_0= \inf \{t : t \in [0, a]\,\, \text{such that}\,\, \gamma(s) \in W_1 \,\forall s \in [t, a]\}.
\]
Note that by definition, $\gamma([0, s_0]) \subset W_2$ and $\gamma([t_0, a]) \subset W_2$. Hence $\sigma^1 \defeq \gamma|_{[0, s_0]}$ and $\sigma^2 \defeq \gamma|_{[t_0, a]}$ are $(1, \kappa_0)$-quasi-geodesic of $(U \cap \Omega, \mathsf{k}_{U \cap \Omega} )$ (as observed in Case 1). Also note that $\sigma_1: [0, s_0] \longrightarrow \Omega$ and $\sigma_2: [t_0, a] \longrightarrow \Omega$ are $(1, \kappa)$-almost-geodesics of $(\Omega, \mathsf{k}_{\Omega} )$ such that $\sigma_1(0), \sigma_2(a) \in W \cap \Omega$ and $\sigma_1(s_0), \sigma_2(t_0) \in \Omega \setminus W_1 $. Furthermore, $\overline W \cap \overline{ \Omega \setminus W_1} = \emptyset$. Now, since $(\Omega, \mathsf{k}_{\Omega} )$ has weak visibility for every pair of distinct points in $U \cap \partial \Omega$, by {\it (2)} of Proposition ~\ref{P:Visibility_Outside_Point} after replacing $W$ and $V$ by $W$ and $\Omega \setminus W_1$ respectively, for $\kappa $ as above, there exists a compact set $K \subset \Omega$ such that $\sigma^1 \cap K \neq \emptyset$ and $\sigma^2 \cap K \neq \emptyset$. Let $o^1 \in \sigma^1 \cap K \subset \overline W_1 \cap \Omega$ and $o^2 \in \sigma^2 \cap K \subset \overline W_1 \cap \Omega$. Now, applying the fact that $\gamma$ is a $(1, \kappa)$-almost-geodesic of $(\Omega, \mathsf{k}_{\Omega} )$ and $\sigma_1$ and $\sigma_2$ are $(1, \kappa_0)$-quasi-geodesic of $(U \cap \Omega, \mathsf{k}_{U \cap \Omega} )$ and by the triangle inequality, we obtain the following sequence of inequalities:
\begin{equation*}
\begin{split}
\mathsf{k}_{\Omega}(z, w)
&\geq a - \kappa\\
&\geq a - t_0 + s_0 -\kappa \quad [\text{since $s_0 \leq t_0$ }]\\
&\geq \mathsf{k}_{U \cap \Omega}(w, \gamma(t_0) ) - \kappa_0 + \mathsf{k}_{U \cap \Omega}(\gamma(s_0), z) - \kappa_0 - \kappa\\
&\geq \mathsf{k}_{U \cap \Omega}(w, o^2) + \mathsf{k}_{U \cap \Omega}(o^2, \gamma(t_0) ) - 3\kappa_0 + \mathsf{k}_{U \cap \Omega}(z, o^1) + \mathsf{k}_{U \cap \Omega}(o^1, \gamma(s_0) ) - 3\kappa_0 - 2 \kappa_0 - \kappa\\
&\geq \mathsf{k}_{U \cap \Omega}(w, o^2) + \mathsf{k}_{U \cap \Omega}(z, o^1) - 8 \kappa_0 - \kappa\\
&= \mathsf{k}_{U \cap \Omega}(w, o^2) + \mathsf{k}_{U \cap \Omega}(o^2, o^1) + \mathsf{k}_{U \cap \Omega}(z, o^1) - \mathsf{k}_{U \cap \Omega}(o^2, o^1) - 8 \kappa_0- \kappa\\
&\geq \mathsf{k}_{U \cap \Omega}(z, w) - \sup_{x, y \in K \cap \overline W_1} \mathsf{k}_{U \cap \Omega}(x, y) - 8\kappa_0 - \kappa.
\end{split}
\end{equation*}
Note that $K \cap \overline W_1$ is a relatively compact subset of $U \cap \Omega$, and hence $\sup_{x, y \in K \cap \overline W_1} \mathsf{k}_{U \cap \Omega}(x, y) < + \infty$. This proves the theorem by taking $C \defeq \sup_{x, y \in K \cap \overline W_1} \mathsf{k}_{U \cap \Omega}(x, y) + 8\kappa_0 + \kappa$.
\end{proof}
\begin{proof}[Proof of Corollary~\ref{C:Pseudo_finte_type}]
This follows from Theorem~\ref{T:Main_Thm_2} by noting that $\Omega$ is also a Goldilocks domain, by \cite{Bharali_Zimmer_1}. Hence $(\Omega, \mathsf{k}_{\Omega} )$ has weak visibility property for every pair of distinct boundary points in $\partial \Omega$.
\end{proof}
Next, we give a proof of Theorem~\ref{T:Main_Thm_4} which is similar to the proof of \cite[Theorem]{NOT} by Nikolov-\"Okten-Thomas although our result holds in a more general setting.
\begin{proof}[Proof of Theorem~\ref{T:Main_Thm_4}]
We break the problem in two cases. Set $c\defeq \mathsf{k}_{\Omega}(W_1 \cap \Omega, \Omega \setminus W_2) > 0$.
{\bf Case 1.} $z, w \in W \cap \Omega$ such that $\mathsf{k}_{\Omega} (z, w) \geq \frac{1}{4} \mathsf{k}_{\Omega}(W_1 \cap \Omega, \Omega \setminus W_2) = \frac{c}{4}$. In this case, using the following (additive localization):
\[
\mathsf{k}_{U \cap \Omega}(z, w) \leq \mathsf{k}_{\Omega}(z, w) + C_0,
\]
we get
\[
\frac{\mathsf{k}_{U \cap \Omega}(z, w)}{\mathsf{k}_{\Omega}(z, w)} \leq 1 + \frac{C_0}{\mathsf{k}_{\Omega}(z, w)} \leq 1 + \frac{4C_0}{c}.
\]
{\bf Case 2.} $z, w \in W \cap \Omega$ such that $\mathsf{k}_{\Omega} (z, w) < c/4 $.
Suppose that $\gamma_n : [0, a_n] \longrightarrow \Omega$ is a sequence of $(1, c/4n)$-almost-geodesics such that $\gamma_n(0) = z$ and $\gamma_n (a_n) = w$ for all $n \in \mathbb{Z}_{+}$ (existence follows from \cite[Proposition~5.3]{Bharali_Zimmer_2}).
\noindent {\bf Claim.} $\gamma_n \subset W_2$ for all $n \in \mathbb{Z}_{+}$.
\begin{proof}[Proof of the claim]
If not then there exists $N \in \mathbb{Z}_{+}$ such that $\gamma_N \cap \Omega \setminus W_2 \neq \emptyset$. Let $z_0 \in \gamma_N \cap \Omega \setminus W_2 $. Since $z, w \in W \cap \Omega$ and $z_0 \in \Omega \setminus W_2$, it follows by definition that
\[
\mathsf{k}_{\Omega}(z,z_0) \geq \mathsf{k}_{\Omega}(W_1 \cap \Omega, \Omega \setminus W_2)= c, \quad \mathsf{k}_{\Omega}(w,z_0) \geq \mathsf{k}_{\Omega}(W_1 \cap \Omega, \Omega \setminus W_2)= c.
\]
Since $\gamma_N$ is a $(1, c/4N)$-almost-geodesic of $(\Omega, \mathsf{k}_{\Omega})$, using the inequalities above, we have
\[
\mathsf{k}_{\Omega}(z,w) \geq \mathsf{k}_{\Omega}(z,z_0) + \mathsf{k}_{\Omega}(z_0,w) - 3c/4N \geq c + c - 3c/4N
> c.
\]
This is a contradiction because by assumption $\mathsf{k}_{\Omega}(z,w) < c/4$. Hence the claim is true.
\end{proof}
Now, using the Royden's Localization Lemma, it follows for all $z \in W_1 \cap \Omega$ and $v \in \mathbb{C}^d$,
\[
\kappa_{U \cap \Omega}(z;v) \leq c \kappa_{\Omega}(z;v).
\]
This gives (by definition $\gamma_n$ is is absolutely continuous, hence differentiable almost everywhere on $[0, a_n]$ for all $n \in \mathbb{Z}_{+}$) by \cite[Theorem~3.1]{Venturini}, for all $n \in \mathbb{Z}_{+}$,
\begin{equation}
\begin{split}
\mathsf{k}_{U \cap \Omega}(z, w)
& \leq \int_{0}^{a_n} \kappa_{U \cap \Omega}(\gamma_n(t); \gamma_n^{\prime}(t)) dt\\
& \leq c\int_{0}^{a_n} \kappa_{ \Omega}(\gamma_n(t); \gamma_n^{\prime}(t)) dt\\
&\leq c (a_n - c/4n) +c^2/4n\\
& \leq c\mathsf{k}_{\Omega}(z, w) + c^2/4n.
\end{split}
\end{equation}
By letting $n \to \infty$, we get
\[
\mathsf{k}_{U \cap \Omega}(z, w) \leq c\mathsf{k}_{\Omega}(z, w).
\]
By combining both cases, we obtain the required result.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{T:Main_Thm_3}]
Since for the bounded domain $\Omega$, $\mathsf{k}_{\Omega}(W_1 \cap U, \Omega \setminus W_2) > 0$ for every open sets $W_1, W_2$ with $W_1 \subset \subset W_2$ and $W_1 \cap \Omega \neq \emptyset$. This can be seen from \cite[Proposition 3.5.(1)]{Bharali_Zimmer_1}. Hence the result follows from Theorem~\ref{T:Main_Thm_4}.
\end{proof}
\noindent \textbf{Acknowledgements.} The author would like to thank Anwoy Maitra, Vikramjeet Singh Chandel and Sushil Gorai for useful discussions and Gautam Bharali for useful suggestions and corrections. The author also would like to thank the referee for the corrections and many useful suggestions which improved the exposition of the previous version of this paper, and also for suggesting short and transparent proof of Lemma~\ref{L:Comple_dist_Compar}, and consequently, the proofs of the main theorems are simplified.
\end{document} |
\begin{document}
\title{A Three-Operator Splitting Perspective of a Three-Block ADMM for Convex Quadratic Semidefinite Programming and Extensions
}
\titlerunning{A Three-Operator Splitting Perspective of a Three-Block ADMM}
\author{X. K. Chang \and L. Chen \and S. Y. Liu}
\institute{
Xiaokai Chang
\at School of Mathematics and Statistics, Xidian University, Xi’an 710071, P.R. China.\\
\email{[email protected]}
\and
Liang Chen
\at College of Mathematics and Econometrics, Hunan University, Changsha 4100082, P.R. China
\\
\email{[email protected]}
\and
Sanyang Liu
\at
School of Mathematics and Statistics, Xidian University, Xi'an 710071, P.R. China.
\\
\email{[email protected]}
}
\date{2017/10/23, revised on 2018/07/05}
\maketitle
\begin{abstract}
In recent years, several convergent multi-block variants of the alternating direction method of multipliers (ADMM) have been proposed for solving the convex quadratic semidefinite programming via its dual, which is naturally a $3$-block separable convex optimization problem with one coupled linear equality constraint. Among of these ADMM-type algorithms, the modified $3$-block ADMM in [Chang et al., Neurocomput. 214: 575--586 (2016)] bears a peculiar feature that the augmented Lagrangian function is not necessarily to be minimized with respect to the block-variable corresponding to the quadratic term of the objective function. In this paper, we lay the theoretical foundation of this phenomena by interpreting this modified $3$-block ADMM as a realization of a $3$-operator splitting framework.
Based on this perspective, we are able to extend this modified $3$-block ADMM to a generalized $3$-block ADMM, which not only applies to the more general convex composite quadratic programming setting but also admits the potential of achieving even a better numerical performance.
\keywords{Convex composite quadratic programming\and Convex quadratic semidefinite programming\quad Multi-Block\and Alternating direction method of multipliers (ADMM)\and Operator splitting\and Generalized ADMM}
\subclass{90C25\and 90C22 \and 65K05 \and 47H05 }
\end{abstract}
\section{Introduction}
\label{sec_Introduction}
The convex quadratic semidefinite programming (CQSDP) has found many concrete applications in economics and engineering, and captures many well-studied problems, including the linearly constrained semidefinite least squares problem, the nearest Euclidean distance matrix (EDM) problem and the nearest correlation matrix problem.
To solve CQSDP problems, several algorithms have been proposed from different angles and here we only mention a few typical and relevant approaches.
Based on certain perturbations of the Kuhn-Krash-Tucker (KKT) system of the CQSDP problem and its dual, Toh and his coauthors have proposed several inexact interior-point methods for solving them \cite{A polynomial-time,Inexact primal dual,An inexact primal dual path}.
By using the generalized Newton method together with the conjugate gradient method, many efficient methods were proposed for solving CQSDP problems \cite{semis-mooth Newton-CG,augmented Lagrangian dual-H}.
For an important class of the CQSDP problem, i.e. the nearest correlation matrix problem, the quadratic convergence of the Newton-CG method has been obtained by Qi and Sun \cite{semis-mooth Newton-CG}.
For general CQSDP problems, the most recently developed solver QSDPNAL in Li, Sun and Toh \cite{qsdpnal} has demonstrated that a two-phase augmented Lagrangian method, which properly combines both first-order and second-order algorithms, possesses a pretty promising numerical performance.
A closer look of this solver shows that the inexact multi-block proximal ADMM studied in \cite{chenine}, in which the inexact block symmetric Gauss Seidel iteration technique elaborated by Li, Sun and Toh \cite{lisgs} was tightly incorporated, has been utilized to generate an approximate solution with a low to medium accuracy to warm-start an augmented Lagrangian method, whose subproblems are solved by a semismooth Newton method.
Just as in QSDPNAL, first-order ADMM-type algorithms are of their own importance for solving CQSDP problems. In fact, many extensions and modifications of the classic ADMM of Glowinski and Marroco \cite{GLOWINSKI75} and Gabay and Mercier \cite{GABAY76} have been considered in recent years for solving the CQSDP problem via its dual, which is innately a $3$-block separable convex optimization problem with one coupled linear constraint.
Indeed, the most intuitive idea is to directly extend the classic ADMM to $3$-block problems and the corresponding numerical performance is pretty good for many instances of problems \cite{chang1,chenine,Schur Complement,admm3c}.
However, the direct extension of ADMM (ADMMe) to problems with more than two blocks of variables is not guaranteed to be convergent (c.f. \cite{direct extension} for a concrete example).
Therefore, attentions have been paid to the design of multi-block ADMM-type algorithms and, fortunately, several algorithms of this type have been successfully applied to solving the CQSDP problem via its dual with a satisfactory numerical efficiency and a theoretical guarantee of convergence \cite{modified admm,chenine,Schur Complement}.
Among of these ADMM-type algorithms for CQSDP problems, the modified $3$-block ADMM by Chang et al. \cite{modified admm} has a distinct feature that one of the subproblems, i.e., the minimization of the augmented Lagrangian function with respect to a certain block of variables, can always be skipped.
This saves both the computational cost and the memory for variable storage, and, more importantly, the convergence is guaranteed under only one extra condition on the penalty parameter $\sigma$, while the proof for the convergence is much more involved.
The peculiar feature of this method inspired us to get a further understanding of its theoretical foundation.
Moreover, we are also concerned with the question that to what extent can this algorithm being improved or generalized, as well as whether this ADMM-type algorithm can be applied to more general problems.
In order to conduct the theoretical analysis to address the concerns mentioned above, an indispensable tool is the concept of the maximal monotone operator as well as the corresponding operator splitting methods \cite{DRS,forward-backward-forward splitting} which are designed to find its roots.
The interconnection between the operator splitting methods and ADMM-type algorithms was first established by Gabay \cite{GABAY83}, who showed that the classic ADMM with unit step-length can be explained as the well-known Douglas-Rachford operator splitting method.
Based on this observation, Eckstein and Bertsekas \cite{ECKSTEIN92} presented a generalized ADMM for the purpose of improving the performance of the classic ADMM (with unit step-length) via an over-relaxation step.
We mention that for a recent survey on this topic one may refer to \cite{eck12}, and one also can refer to \cite{xiao} and references therein for more details and recent progresses on generalized ADMM.
Consequently, we are interested if one can also interpret the modified $3$-block ADMM in \cite{modified admm} via a certain operator splitting scheme and get further improvements on this algorithm via certain over-relaxation steps.
In this paper, we fulfil our objective by showing that the modified $3$-block ADMM in \cite{modified admm} can be explained as an application of the $3$-operator splitting framework studied in Davis and Yin \cite{Three-Operator}.
We conduct our analysis in a much general setting in which the model that we will consider contains the CQSDP problem as a special case.
Moreover, based on this operator splitting perspective, we present a generalized $3$-block ADMM, in the sense of Eckstein and Bertsekas \cite{ECKSTEIN92},
in which an over-relaxation step is incorporated.
We mention that, such as in Xiao et al. \cite{xiao}, this kind of over-relaxation can lead to an obvious improvement on the numerical efficiency of ADMM-type algorithms.
The remaining parts of this paper are organized as follows.
In Section \ref{prel}, we give a quick review of the CQSDP problem and the modified 3-block ADMM algorithm proposed by Chang et al. \cite{modified admm}.
Section \ref{sec_Three-Operator} is devoted to the operator-splitting perspective of this modified $3$-block ADMM for the CQSDP problem.
In Section \ref{main}, we introduce the convex composite quadratic optimization model and present a generalized version of the modified $3$-block ADMM in \cite{modified admm} for solving this problem.
With the result established in Section \ref{sec_Three-Operator}, the convergence analysis of the proposed algorithm can be conducted in a very concise manner.
We conclude this paper in Section \ref{conclusion}.
\section{Preliminaries}
\label{prel}
\subsection{Basic Concepts}
Let ${\mathcal H}$ be an arbitrary finite dimensional real Hilbert space endowed with an inner product denoted by $\langle \cdot, \cdot\rangle$ and its induced norm $\|\cdot\|$.
Let ${\mathcal F}:{\mathcal H}\to{\mathcal H}$ be an arbitrary set-valued mapping.
If ${\mathcal F}$ is single-valued, it is called $\beta$-cocoercive (or $\beta$-inverse-strongly monotone) for a certain constant $\beta>0$, if
$$\langle {\mathcal{F}}(x)-{\mathcal{F}}(x'), x-x'\rangle\geq \beta \|{\mathcal{F}}x- {\mathcal{F}}x'\|^2,\quad
\forall x, x'\in {\mathcal H};
$$
If ${\mathcal F}$ is a self-adjoint positive semidefinite linear operator, we use $\lambda_{\max}({\mathcal F})$ to denote its largest eigenvalue, i.e. $\lambda_{\max}({\mathcal F}):=\max_{\|x\|=1}\langle x, {\mathcal F} x\rangle$. In this case, it is easy to verify that
${\mathcal F}$ is $\frac{1}{\lambda_{\max}({\mathcal F})}$-coercive.
If ${\mathcal F}$ is a multi-valued maximal monotone operator and $\sigma>0$ is a constant,
the mapping ${\mathcal J}_{\sigma{\mathcal F}}:=({\mathcal I}+\sigma{\mathcal F})^{-1}$, which is called the Minty resolvant of ${\mathcal F}$, is a single valued mapping, and this mapping is also nonexpansive \cite[Theorem 12.12]{rocva}. Here, ${\mathcal I}$ denotes the identity operator from ${\mathcal H}$ to itself and it will be kept as the notation for the identity operator from any space to itself, if no ambiguity is caused.
Let $f:{\mathcal H}\to(-\infty,\infty]$ be a closed proper convex function. The subdifferential mapping $\partial f$ of $f$ is then a maximal monotone operator and in this case
\begin{eqnarray*}
{\mathcal{J}}_{\sigma \partial f}(x)
= ({\mathcal I}+\sigma\partial f)^{-1}(x)
= \argmin_z\left\{ f(z)+\frac{1}{2\sigma}\|x-z\|^2\right\},\quad\forall\, x\in{\mathcal H}.
\end{eqnarray*}
For any set ${\mathcal C}\subset {\mathcal H}$, the indicator function $\delta_{\mathcal C}:{\mathcal H}\to(-\infty,\infty]$ is defined by $\delta_{\mathcal C}(x)=0$ if $x\in{\mathcal C}$ and $\delta_{\mathcal C}(x)=+\infty$ otherwise.
If ${\mathcal C}$ is a closed convex set, $\delta_{\mathcal C}$ is therefore a closed proper convex function and, in this case,
$
{\mathcal{J}}_{\sigma \partial \delta_{\mathcal C}}(x)=\Pi_{\mathcal C}(x)
$, i.e., the metric projection of $x$ onto ${\mathcal C}$,
and
$$
\partial\delta_{\mathcal C}(x)={\mathcal N}_{{\mathcal C}}(x):=\{z\,|\, \langle z, x'-x\rangle\le 0,\ \forall x'\in{\mathcal H}\}.
$$
Here, the mapping ${\mathcal N}_{\mathcal C}$ is called the normal cone mapping of the set ${\mathcal C}$ and ${\mathcal N}_c(x)$ is called the normal cone of ${\mathcal C}$ at $x$, which is a closed convex cone.
Let ${\mathcal{S}}^n$ be the space of $n\times n$ real symmetric matrices endowed with the standard trace inner product $\langle \cdot,\cdot \rangle$ and the Frobenius norm $\|\cdot\|$. We use ${\mathcal{S}}_{+}^{n}$ and ${\mathcal{S}}_{++}^{n}$ to denote the sets of symmetric positive semidefinite and positive definite matrices in ${\mathcal{S}}^{n}$, respectively.
\subsection{The CQSDP Problem}
The CQSDP problem takes the following standard form:
\begin{eqnarray}
\label{CQSDP}
\min_X \;\; && \frac{1}{2} \left\langle X,\varphi (X)\right\rangle +\langle C , X\rangle \nonumber\\
\mbox{s.t.} \;\;&& {\mathcal{A}}(X)= b ,\ X \in {\mathcal{S}}^n_+,
\end{eqnarray}
where $\varphi: {\mathcal{S}}^n \rightarrow
{\mathcal{S}}^n$ is a given self-adjoint positive semidefinite linear operator,
${\mathcal{A}}: {\mathcal{S}}^n \rightarrow {\mathbb{R}}^m $ is a linear map defined by
\\
$$
{\mathcal{A}}(X):=\left(
\begin{array}{c}
\langle A_1 ,X\rangle \\
\vdots \\
\langle A_m ,X\rangle
\end{array}
\right)\in {\mathbb{R}}^m,
\quad\forall X\in{\mathcal S}^n
$$
with the given $A_i \in {\mathcal{S}}^n$, $i=1,\ldots,m$, and $ b \in {\mathbb{R}}^m$.
The adjoint of ${\mathcal{A}}$, with respect to the standard inner products in ${\mathcal{S}}^n$ and $ {\mathbb{R}}^m$, is denoted by ${\mathcal{A}}^*$.
Note that ${\mathcal{S}}_{+}^{n}$ is a closed convex self-dual cone. Then, the dual of problem (\ref{CQSDP}) can be equivalently formulated in minimization form as follows
\begin{eqnarray}
\label{CQSDPD}
\min_{W,y,Z} \;\;&&\frac{1}{2} \langle W,\varphi (W)\rangle -b^Ty+\delta_{{\mathcal{S}}^n_+}(Z) \nonumber\\
\mbox{s.t.} \;\; && -\varphi (W)+{\mathcal{A}}^*(y)+Z=C,
\end{eqnarray}
where $\delta_{{\mathcal S}_+^n}$ is the indicator function of ${\mathcal S}_+^n$, $W\in{\mathcal S}^n$, $Z\in{\mathcal S}^n$ and $y\in{\mathbb R}^m$.
The Lagrangian function of problem \eqref{CQSDPD} is defined by
$$
\begin{array}{r}
l(W,y,Z;X):=\frac{1}{2}\langle W,\varphi (W)\rangle -b^Ty+\delta_{{\mathcal{S}}^n_+}(Z)+\langle -\varphi (W)+{\mathcal{A}}^*(y)+Z-C, X\rangle,\quad
\\begin{equation}1mm]
\forall(W,y,Z,X)\in{\mathcal S}^n\times{\mathbb R}^m\times{\mathcal S}^n\times{\mathcal S}^n.
\end{array}
$$
Therefore, the KKT system of problem \eqref{CQSDPD} is given by
\begin{eqnarray}\label{KKT}
\left\{
\begin{array}{l}
\varphi(W)=\varphi(X),\
{\mathcal{A}}(X)=b,\\begin{equation}1mm]
~-\varphi (W)+{\mathcal{A}}^*(y)+Z=C,\\begin{equation}1mm]
~X\in {\mathcal{S}}^n_+,~Z\in {\mathcal{S}}^n_+,~\langle Z,X\rangle=0.
\end{array}\right.
\end{eqnarray}
For any $(W,y,Z,X)\in{\mathcal S}^n\times{\mathbb R}^m\times{\mathcal S}^n\times{\mathcal S}^n$ satisfying the KKT system \eqref{KKT}, $(W,y,Z)$ is a solution to problem \eqref{CQSDPD} while $X$ is a solution to problem \eqref{CQSDP}.
\subsection{A Modified $3$-Block ADMM for Problem (\ref{CQSDPD})}
\label{sec_Modified ADMM}
Let $\sigma>0$ be the penalty parameter. The augmented Lagrangian function of problem (\ref{CQSDPD}) can be defined by
\begin{equation}
\label{auglag}
\begin{array}{l}
{\mathcal L}_\sigma(W,y,Z;X):=l(W,y,Z;X)+\frac{\sigma}{2}\|-\varphi (W)+{\mathcal{A}}^*(y)+Z-C\|^2,
\\begin{equation}1mm]
\hspace{5cm}\forall(W,y,Z,X)\in{\mathcal S}^n\times{\mathbb R}^m\times{\mathcal S}^n\times{\mathcal S}^n.
\end{array}
\end{equation}
Choose an initial points $(y^0,Z^0,X^0)\in{\mathbb{R}}^m\times{\mathcal{S}}^n_+\times\in {\mathcal{S}}^n$.
A direct extension of the classic ADMM to the $3$-block problem (\ref{CQSDPD}) consists of the following steps, for $k=0,1,\ldots$,
\begin{equation}
\left\{
\label{ADPsi}
\begin{array}{lcl}
W^{k+1}&:=&\displaystyle\argmin_{W } {\mathcal{L}}_\sigma(W,y^{k},Z^{k};X^k), \\begin{equation}1mm]
y^{k+1}&:=&\displaystyle\argmin_{y} {\mathcal{L}}_\sigma(W^{k+1},y,Z^k;X^k), \\begin{equation}1mm]
Z^{k+1}&:=&\displaystyle\argmin_{Z } {\mathcal{L}}_\sigma(W^{k+1},y^{k+1},Z;X^k), \\begin{equation}1mm]
X^{k+1}&:=&X^k+\tau\sigma\left({\mathcal{A}}^*(y^{k+1})+Z^{k+1}-\varphi (W^{k+1})- C\right),
\end{array}
\right.
\end{equation}
where $\tau> 0$ is the step-length. Generally, the convergence of the iteration scheme (\ref{ADPsi}) can not be guaranteed.
In Chang et al. \cite{modified admm}, the authors have proposed the following algorithm to solve problem \eqref{CQSDPD}, by modifying the iteration scheme (\ref{ADPsi}).
\centerline{\fbox{\parbox{0.97\textwidth}{
\begin{algo}[A Modified ADMM for the CQSDP problem (\ref{CQSDPD})]
\label{algo1}
\end{algo}
Let $\sigma>0$ be the given parameter. Choose $Z^0\in {\mathcal{S}}^n_+$ and $X^0\in {\mathcal{S}}^n$. For $k=0,1,\ldots,$
\begin{description}
\item[{\bf Step 1.}] Set $W^{k+1}:=X^{k}$;
\item[{\bf Step 2.}] Compute $y^{k+1}:=\argmin\limits_{y} {\mathcal{L}}_\sigma(W^{k+1},y,Z^k;X^k)$;
\item[{\bf Step 3.}] Compute $Z^{k+1}:=\argmin\limits_{Z} {\mathcal{L}}_\sigma(W^{k+1},y^{k+1},Z;X^k)$;
\item[{\bf Step 4.}] Update $X^{k+1}:=X^k+\sigma\left({\mathcal{A}}^*(y^{k+1})+Z^{k+1}-\varphi (W^{k+1})- C\right)$.
\end{description}
}}}
Compared with the directly extended $3$-block ADMM scheme \eqref{ADPsi}, Algorithm \ref{algo1} always set $W^{k+1}$ as $X^k$ instead of minimizing the augmented Lagrangian function with respect to the block-variable $W$.
The original idea of this $3$-block ADMM is quite intuitive since that for any $(W,y,Z,X)\in{\mathcal S}^n\times{\mathbb R}^m\times{\mathcal S}^n\times{\mathcal S}^n$ being a solution to the KKT system of the CQSDP problem, $(X,y,Z,X)$ is also a solution to it.
Therefore, whenever dealing with a subproblem with respect to $W$, one may directly use the value of $X$ to substitute $W$ instead of solving this subproblem. However, the convergence analysis of Algorithm \ref{algo1} in \cite{modified admm} is very complicated.
The following Assumption was used
in \cite{modified admm} for
analyzing the convergence of Algorithm \ref{algo1}.
\begin{asmp}
\label{assum1}
The linear operator ${\mathcal A}$ is surjective
and the Slater's constraint qualification holds for problem \eqref{CQSDP}, i.e. there exists a matrix $\tilde{X} \in {\mathcal{S}}^n_{++}$ satisfying ${\mathcal{A}}( \tilde{X} )= b$.
\end{asmp}
\begin{rem}
The first part of Assumption \ref{assum1} implies that the linear operator ${\mathcal{A}}{\mathcal{A}}^*$ is nonsingular.
Consequently, Step 2 of Algorithm \ref{algo1} is well-defined.
Moreover, under Assumption \ref{assum1}, we know from \cite[Corollaries 28.2.2 \& 28.3.1]{rocbook} that $X\in{\mathcal S}^n$ is a solution to problem \eqref{CQSDP} if and only if there exists a vector $(W,y,Z)\in{\mathcal S}^n\times{\mathbb R}^m\times{\mathcal S}^n$ such that $(W,y,Z,X)$ is a solution to the KKT system \eqref{KKT}.
Additionally, for any $(W,y,Z,X)$ that satisfies \eqref{KKT}, from \cite[Corollary 30.5.1]{rocbook} we know that $X$ is an optimal solution to problem \eqref{CQSDP} and $(W,y,Z)$ is an optimal solution to problem \eqref{CQSDPD}.
\end{rem}
The convergence properties of Algorithm \ref{algo1} have been given in \cite[Theorem 1]{modified admm}. We summarize these results as follows.
\begin{prop}
\label{propconv}
Suppose that the solutions set to problem \eqref{CQSDP} is nonempty, Assumption \ref{assum1} holds and $\sigma\in \left(0,\frac{1}{\lambda_{\max}(\varphi)}\right]$. Then, the sequence $\{(W^k,y^k,Z^k, X^k)\}$ generated by Algorithm \ref{algo1} converges to a point which is a solution to the KKT system \eqref{KKT}.
\end{prop}
\begin{rem}
If $\varphi=0$, problem (\ref{CQSDP}) is then a standard linear semidefinite programming problem and its dual will be a 2-block convex optimization problem.
In this case, the requirement that $\sigma\in\left(0,\frac{1}{\lambda_{\max}(\varphi)}\right]$ is no longer necessary and Algorithm \ref{algo1} is then automatically the classic 2-block ADMM with the unit step-length.
\end{rem}
\section{A Three-Operator Splitting Perspective}
\label{sec_Three-Operator}
In this section, we narrate Algorithm \ref{algo1} from a $3$-operator splitting perspective. Note that the problem \eqref{CQSDP} can be written as
$$
\min_{X\in{\mathcal S}^n} f(X) + g(X) + h(X),
$$
where
$$
\left\{
\begin{array}{l}
\displaystyle
f(X):= \delta_{K}(X)\quad\mbox{with}\quad K:=\{X\in{\mathcal S}^n |{\mathcal A}(X)=b\},
\\begin{equation}2mm]
\displaystyle
g(X):=\delta_{{\mathcal S}^n_+}(X),
\\begin{equation}2mm]
\displaystyle
h(X):=\frac{1}{2}\langle X,\varphi(X)\rangle+\langle C, X\rangle.
\end{array}
\right.
$$
Under Assumption \ref{assum1}, we know from \cite[Theorem 23.8]{rocbook} that $X$ is a solution to problem \eqref{CQSDP} if and only if
$$
0\in{\mathcal N}_{K}(X)+{\mathcal N}_{{\mathcal S}_+}(X) +\left( \varphi(X)+C\right) = \partial f(X) + \partial g(X) +\nabla h (X).
$$
Therefore, one can try to solve problem \eqref{CQSDP} via solving the above inclusion problem.
In fact, Algorithm \ref{algo1} can be interpreted as an operator splitting algorithm applied to solve this inclusion problem.
This will be explained in details as follows.
Let $\{X^k\}$, $\{y^k\}$ and $\{Z^k\}$ be the sequences generated by Algorithm \ref{algo1}. We define for $k\ge 0$,
\begin{equation}
\label{defuk}
U^{k}:=X^k-\sigma(\varphi(X^k)+C-{\mathcal A}^*y^{k+1}).
\end{equation}
Moreover, just as \cite[Equation (1.2)]{Three-Operator}, we define the mapping ${\mathcal T}:{\mathcal S}^n\to{\mathcal S}^n$ by
$$
{\mathcal T}:={\mathcal I}-{\mathcal J}_{\sigma{\mathcal N}_{{\mathcal S}_+^n}}
+{\mathcal J}_{\sigma{\mathcal N}_{K}}\circ\big(2{\mathcal J}_{\sigma{\mathcal N}_{{\mathcal S}_+^n}}-{\mathcal I}-\sigma\nabla h ({\mathcal J}_{\sigma{\mathcal N}_{{\mathcal S}_+^n}})\big).
$$
Then, we have the following result.
\begin{thm}
\label{thmmain}
Let $\{X^k\}$, $\{y^k\}$ and $\{Z^k\}$ be the sequences generated by Algorithm \ref{algo1}. Then the sequence $\{U^k\}$ defined in \eqref{defuk} satisfies
$$
U^{k+1}={\mathcal T}(U^k).
$$
\end{thm}
\begin{proof}
Note that for any $k\ge 0$, steps $1$ to $4$ of Algorithm \ref{algo1} can be reorganized as follows
\begin{equation}
\left\{
\label{yk}
\begin{array}{l}
y^{k+1}:=-({\mathcal A}{\mathcal A}^*)^{-1}(({\mathcal A} X^k-b)/\sigma+{\mathcal A}(Z^k-\varphi(X^k)-C)),
\\begin{equation}2mm]
X^{k+1}:=\Pi_{{\mathcal S}^n_+}(U^{k}),
\\begin{equation}2mm]
Z^{k+1}:=(X^{k+1}-U^{k})/\sigma.
\end{array}
\right.
\end{equation}
Therefore, it holds that
\begin{equation}
\label{xplus}
X^{k+1}=({\mathcal I}+\sigma {\mathcal N}_{{\mathcal S}^n_+})^{-1}(U^k)={\mathcal J}_{\sigma{\mathcal N}_{{\mathcal S}_+^n}}(U^k).
\end{equation}
Moreover, one can readily obtain that
$$
\begin{array}{ll}
y^{k+2}&=-\frac{1}{\sigma}({\mathcal A}{\mathcal A}^*)^{-1}( {\mathcal A} X^{k+1}-b +\sigma{\mathcal A}(Z^{k+1}-\varphi(X^{k+1})-C))
\\begin{equation}2mm]
&=-\frac{1}{\sigma}({\mathcal A}{\mathcal A}^*)^{-1}\left( {\mathcal A} X^{k+1}-b +\sigma{\mathcal A}\left[\frac{1}{\sigma}(X^{k+1}-U^k)-\varphi(X^{k+1})-C\right]\right)
\\begin{equation}2mm]
&=-\frac{1}{\sigma}({\mathcal A}{\mathcal A}^*)^{-1}\left( {\mathcal A} X^{k+1}-b +{\mathcal A} (X^{k+1}-U^k)-\sigma{\mathcal A}\left[\varphi(X^{k+1})+C\right]\right)
\\begin{equation}2mm]
&
=-\frac{1}{\sigma}({\mathcal A}{\mathcal A}^*)^{-1}\left( {\mathcal A} \left(2X^{k+1}- U^k-\sigma\big[\varphi(X^{k+1})+C\big]\right)-b\right).
\end{array}
$$
Note that
$$\Pi_{K}(X)=X-{\mathcal A}^*({\mathcal A}{\mathcal A}^*)^{-1}({\mathcal A} X-b),
\quad\forall X\in{\mathcal S}^n. $$
Then, by using \eqref{xplus} we can get
$$
\begin{array}{l}
\left({\mathcal J}_{\sigma{\mathcal N}_{K}}\circ\big(2{\mathcal J}_{\sigma{\mathcal N}_{{\mathcal S}_+^n}}-{\mathcal I}-\sigma\nabla h ({\mathcal J}_{\sigma{\mathcal N}_{{\mathcal S}_+^n}})\big)\right)(U^k)
\\begin{equation}2mm]
=
\Pi_{K}\left(2X^{k+1}- U^k-\sigma\big[\varphi(X^{k+1})+C\big]\right)
\\begin{equation}2mm]
=\left(2X^{k+1}- U^k-\sigma\big[\varphi(X^{k+1})+C\big]\right)+\sigma{\mathcal A}^*y^{k+2}.
\end{array}
$$
Moreover, it is easy to see from \eqref{defuk} that
$$
\begin{array}{ll}
U^{k+1}
&
=X^{k+1}-\sigma(\varphi(X^{k+1})+C-{\mathcal A}^*y^{k+2})
\\begin{equation}2mm]
&
=X^{k+1}+\sigma{\mathcal A}^*y^{k+2}-\sigma\big[\varphi(X^{k+1})+C\big ]
\\begin{equation}2mm]
&=U^{k}+\left(X^{k+1}- U^k-\sigma\big[\varphi(X^{k+1})+C\big ]\right)+\sigma{\mathcal A}^*y^{k+2}
\\begin{equation}2mm]
&=U^{k}-X^{k+1}+\left(2X^{k+1}- U^k-\sigma\big[\varphi(X^{k+1})+C\big ]\right)+\sigma{\mathcal A}^*y^{k+2}
\\begin{equation}2mm]
&=U^{k}-{\mathcal J}_{\sigma{\mathcal N}_{{\mathcal S}_+^n}}(U^k)+
\left({\mathcal J}_{\sigma{\mathcal N}_{K}}\circ\big(2{\mathcal J}_{\sigma{\mathcal N}_{{\mathcal S}_+^n}}-{\mathcal I}-\sigma\nabla h ({\mathcal J}_{\sigma{\mathcal N}_{{\mathcal S}_+^n}})\big)\right)(U^k),
\end{array}
$$
which, together with the definition of ${\mathcal T}$, completes the proof.
\qed
\end{proof}
\begin{rem}
The definition of the operator ${\mathcal T}$ was introduced by Davis and Yin \cite[Equation (1.2)]{Three-Operator}. This operator was regarded as a combination of the well-known Douglas-Rachford splitting and the forward-backward splitting.
As will be seen in the next section, based on the properties of ${\mathcal T}$, the global convergence of Algorithm \ref{algo1}, with the looser requirement $\sigma\in (0, \frac{2}{\lambda_{\max}(\varphi)})$, can be alternatively proved by using \cite[Theorem 2.1]{Three-Operator} together with Theorem \ref{thmmain}.
\end{rem}
\section{Generalizations and Extensions}
\label{main}
The successful application of Algorithm \ref{algo1} to the CQSDP problem \eqref{CQSDP} via its dual \eqref{CQSDPD} and the explanation from the operator splitting perspective made in Section
\ref{sec_Three-Operator} inspired us to consider extending this algorithm to much general problems.
In this section, we consider the following convex composite quadratic programming \cite{Schur Complement} problem
\begin{eqnarray}
\label{cqp}
\min_{x\in{\mathcal X}} \;\; && \theta^*(x)+\frac{1}{2} \left\langle x,{\mathcal Q} (x)\right\rangle +\langle c , x\rangle \nonumber\\
\mbox{s.t.} \;\;&& {\mathcal{A}} x = b,
\end{eqnarray}
where ${\mathcal A}:{\mathcal X}\to{\mathcal Y}$ is a linear map, ${\mathcal X}$ and ${\mathcal Y}$ are finite dimensional Euclidean spaces each endowed with a inner product $\langle\cdot,\cdot\rangle$ and its induced norm $\|\cdot\|$. $\theta^*$ is the Fenchel conjugate function of the closed proper convex (possibly nonsmooth) function $\theta:{\mathcal X}\to(-\infty,\infty]$, ${\mathcal Q}:{\mathcal X}\to{\mathcal X}$ is a self-adjoint positive semidefinite linear operator, and $c\in{\mathcal X}$ and $b\in{\mathcal Y}$ are the given data.
Obviously, problem (\ref{CQSDPD}) is an instance of problem (\ref{cqpd})
in which ${\mathcal X}={\mathcal S}^n$, ${\mathcal Y}=\mathbb{R}^m$ and $\theta^*$ being the indicator function of ${\mathcal S}^n_+$.
We make the following assumption on problem \eqref{cqp}.
\begin{asmp}
\label{ass2}
The linear operator ${\mathcal A}$ is surjective and there exists a point $x\in\ri(\dom\theta^*)$ such that ${\mathcal A} x=b$.
\end{asmp}
Under Assumption \ref{ass2} we know that $x$ is a solution to problem \eqref{cqp} if and only if there exists a vector $(w,y,z)\in{\mathcal X}\times{\mathcal Y}\times{\mathcal X}$ such that $(x,w,y,z)$ solves the following KKT system of problem \eqref{cqpd}
\begin{eqnarray}
\label{KKT2}
\left\{
\begin{array}{l}
{\mathcal Q} w={\mathcal Q} x,\\begin{equation}1mm]
{\mathcal A} x-b=0,\\begin{equation}1mm]
0\in x-\partial \theta(-z),\\begin{equation}1mm]
{\mathcal A}^*y+z-{\mathcal Q} w-c=0.
\end{array}\right.
\end{eqnarray}
Moreover, such a vector $(w,y,z)$ is a solution to the dual of problem \eqref{cqp}, which can equivalently be recast in minimization form as
\begin{equation}
\label{cqpd}
\min_{w,y,z}\left\{ \frac{1}{2}\langle w,{\mathcal Q} w\rangle -\langle b,y\rangle+\theta(-z)
\ |\
-{\mathcal Q} w+{\mathcal A}^*y+z=c
\right\},
\end{equation}
where the decision variables $w\in{\mathcal X},y\in{\mathcal Y}$ and $z\in{\mathcal X}$.
Let $\sigma>0$ be the penalty parameter. The augmented Lagrangian function of problem \eqref{cqpd} is defined by
$$
\begin{array}{ll}
\displaystyle
{\mathcal L}_{\sigma}(w,y,z;x):=&\frac{1}{2}\langle w,{\mathcal Q} w\rangle -\langle b,y\rangle+\theta(-z)
+\langle {\mathcal A}^*y+z-{\mathcal Q} w-c,x\rangle+\frac{\sigma}{2}\|{\mathcal A}^*y+z-{\mathcal Q} w-c\|^2,\\begin{equation}2mm]
&
\forall(w,y,z;x)\in{\mathcal X}\times{\mathcal Y}\times{\mathcal X}\times{\mathcal X}.
\end{array}
$$
In sequel, we will extend and generalize Algorithm \ref{algo1} to problem (\ref{cqpd}), and prove its convergence via the existing convergence theorem of the 3-operator splitting method in \cite{Three-Operator}.
\centerline{\fbox{\parbox{0.97\textwidth}{
\begin{algo}[A Generalized Modified ADMM for problem (\ref{cqpd})]\label{algo2}
\end{algo}
Let $\sigma>0$ and $\rho\in(0,2)$. Choose initial variables $z^0$ such that $-z^0\in\dom\theta$ and $x^0\in{\mathcal X}$. For $k=0,1,\ldots,$
\begin{description}
\item[{\bf Step 1.}] Set $w^{k+1}:=x^{k}$;
\item[{\bf Step 2.}] Compute $y^{k+1}:=\argmin\limits_{y} {\mathcal{L}}_\sigma(w^{k+1},y,z^k;x^k)$;
\item[{\bf Step 3.}]
Compute
$$
\begin{array}{rl}
\displaystyle z^{k+1}:=\argmin_{z}&\left\{
\theta(-z)+\langle z,x^k\rangle
+\frac{\sigma}{2}\|\rho{\mathcal A}^*y^{k+1}-(1-\rho)z^k+z-\rho{\mathcal Q} w^{k+1}-\rho c\|^2\right\};\\
\end{array}
$$
\item[{\bf Step 4.}] Update $x^{k+1}:=x^k+
\sigma (\rho{\mathcal A}^*y^{k+1}-(1-\rho)z^k+z^{k+1}-\rho{\mathcal Q} w^{k+1}-\rho c).$
\end{description}
}}}
\begin{rem}
The above algorithm is called a generalized modified ADMM since that it can viewed as a direct extension of the generalized $2$-block ADMM \cite{ECKSTEIN92} to problem \eqref{cqpd}. Moreover, we should mention that, generally, since that ${\mathcal A}$ is surjective, all the subproblems are well-defined and admit unique solutions.
The well-definedness of subproblems is very essential for ADMM-type algorithm. On this part, one may refer to a counterexample by Chen et al. \cite[Section 3]{admmnote}.
\end{rem}
\begin{rem}
For the case that $\rho=1$, step $3$ of Algorithm \ref{algo2} turns to
$$
z^{k+1}=\argmin_z{\mathcal L}_{\sigma}(w^{k+1},y^{k+1},z;x^k).
$$
In this case, the direct extension of the generalized ADMM is then a direct extension of the classic ADMM with unit step-length, whose $k$-th step takes the following form
\begin{equation}
\label{admm3e}
\left
\{\begin{array}{l}
w^{k+1}\in\argmin_{w} {\mathcal{L}}_\sigma(w,y^{k},z^{k};x^k),\\begin{equation}2mm]
y^{k+1}\in\argmin_{y} {\mathcal{L}}_\sigma(w^{k+1},y,z^k;x^k),\\begin{equation}2mm]
z^{k+1}\in\argmin_{z} {\mathcal{L}}_\sigma(w^{k+1},y^{k+1},z;x^k),\\begin{equation}2mm]
x^{k+1}=x^k+\sigma ( {\mathcal A}^*y^{k+1}+z^{k+1}-{\mathcal Q} w^{k+1}- c).
\end{array}
\right.
\end{equation}
If the order of solving the subproblems is further changed as follows
$$
\left
\{\begin{array}{l}
y^{k+1}\in\argmin_{y} {\mathcal{L}}_\sigma(w^{k},y,z^k;x^k),\\begin{equation}2mm]
w^{k+1}\in\argmin_{w} {\mathcal{L}}_\sigma(w,y^{k+1},z^{k};x^k),\\begin{equation}2mm]
z^{k+1}\in\argmin_{z} {\mathcal{L}}_\sigma(w^{k+1},y^{k+1},z;x^k),\\begin{equation}2mm]
\end{array}
\right.
$$
the convergence of this direct extension of the class ADMM has been established in Li et al. \cite[Theorem 2.1]{limin} under certain conditions\footnote{
In \cite{limin}, the authors also have considered adding proximal terms to subproblems and using a dual step-size which can be chosen in $\big(0,(1+\sqrt{5})/2\big)$.
Since that one can restrict $w$ always in the range space of the linear operator ${\mathcal Q}$ so that $f_1(w)=\frac{1}{2}\langle w,{\mathcal Q} w\rangle$ is a strongly convex function. Hence, the results in \cite{limin} are applicable.}.
In \cite[Section 4.2]{Three-Operator}, the authors have considered another $3$-block extension of the classic ADMM, i.e., \cite[Algorithm 7]{Three-Operator}.
The difference of this extension from \eqref{admm3e} is that the subproblem for computing $w^{k+1}$ does not contain the penalty term $\frac{\sigma}{2}\|{\mathcal A}^*y^k+z^k-{\mathcal Q} w-c\|$.
Moreover, in the corresponding convergence analysis, it requires $\sigma\in\left(0,\frac{2\lambda^+_{\min}({\mathcal Q})}{(\lambda_{\max} ({\mathcal Q}))^2}\right)$, where $\lambda^+_{\min}({\mathcal Q})$ denotes the smallest positive eigenvalue of ${\mathcal Q}$.
This requirement of $\sigma$ is obviously stronger than the condition that $\sigma\in \left(0,\frac{2}{\lambda_{\max}({\mathcal Q})}\right)$, which will be used in the forthcoming convergence analysis of Algorithm \ref{algo2}.
\end{rem}
Next, we analyze the convergence properties of Algorithm \ref{algo2}.
Suppose that $\{w^k\}$, $\{y^k\}$, $\{z^k\}$ and $\{x^k\}$ be the infinite sequences generated by Algorithm \ref{algo2}. Define for $k\ge 0$
\begin{equation}
\label{sequk}
u^{k}:=x^k+
\sigma (\rho{\mathcal A}^*y^{k+1}-(1-\rho)z^k -\rho{\mathcal Q} w^{k+1}-\rho c)
=x^{k+1}-\sigma z^{k+1}.
\end{equation}
For convenience, we define the convex set
$$
{\mathcal K}:=\{x\in{\mathcal X}|{\mathcal A} x=b\},
$$
and the
quadratic function $q:{\mathcal X}\to(-\infty,\infty)$ by
$$
q(x):=\frac{1}{2}\langle x,{\mathcal Q} x\rangle+\langle c,x\rangle,\quad \forall x\in{\mathcal X}.
$$
Then, the gradient of the function $q$ is given by
$\nabla q(x)={\mathcal Q} x+c$, $\forall x\in{\mathcal X}$.
Moreover, we define a single-valued mapping $\Gamma:{\mathcal X}\to{\mathcal X}$ by
\begin{equation}
\label{mainoper}
\Gamma:={\mathcal I}-{\mathcal J}_{\sigma\partial \theta^*}+
{\mathcal J}_{\sigma{\mathcal N}_{\mathcal K}}\circ\left(2 {\mathcal J}_{\sigma\partial\theta^*}-{\mathcal I}-\sigma \nabla q\circ {\mathcal J}_{\sigma\partial\theta^*}\right) .
\end{equation}
Based on the above definitions we have the following result.
\begin{prop}
\label{prop2}
Suppose that $\{w^k\}$, $\{y^k\}$ and $\{z^k\}$ are the infinite sequences generated by Algorithm \ref{algo2}, and $\{u^k\}$ is the sequence defined by \eqref{sequk}.
Then, one has that
$$
u^{k+1}=(1-\rho)u^k+\rho\Gamma(u^{k}).
$$
\end{prop}
\begin{proof}
Note that for any $k\ge 0$
\begin{equation}
\label{optcond1}
\begin{array}{rl}
0&\in-\partial \theta(-z^{k+1})+ x^k+\sigma (\rho{\mathcal A}^*y^{k+1}-(1-\rho)z^k+z^{k+1}-\rho{\mathcal Q} w^{k+1}-\rho c)
\\begin{equation}2mm]
&=-\partial \theta(-z^{k+1})+ x^{k+1}.
\end{array}
\end{equation}
Since that $\theta$ is a closed proper convex function, by using \cite[Theorem 23.5]{rocbook} we have that
$
x^{k+1}\in \partial \theta(-z^{k+1})
$
so that
$
-z^{k+1}\in\partial \theta^*( x^{k+1})
$.
Therefore, it holds that
$$
\begin{array}{ll}
0\in
& \partial \theta^*( x^{k+1})+ z^{k+1}
= \partial \theta^*( x^{k+1})
+\frac{1}{\sigma}\left(x^{k+1}-(x^{k+1}-\sigma z^{k+1})\right)
\\begin{equation}2mm]
&= \partial \theta^*( x^{k+1})
+\frac{1}{\sigma} (x^{k+1}-u^k),
\end{array}
$$
where we have used the fact that $u^k=x^{k+1}-\sigma z^{k+1}$ from \eqref{sequk}.
Thus, by using \cite[Theorem 23.8 \& 23.9]{rocbook} and the above inclusion one can get that
\begin{equation}
\label{pf21}
x^{k+1}=
\argmin_{x}\left\{ \theta^*( x)+\frac{1}{2\sigma}\|x-u^k\|^2\right\}
=\left({\mathcal I}+\sigma\partial \theta^*\right)^{-1}(u^k)
={\mathcal J}_{\sigma\partial\theta^*}(u^k).
\end{equation}
On the other hand, one can readily obtain that
\begin{equation}
\label{optt}
0=-b+{\mathcal A} x^{k+1}
+\sigma{\mathcal A}({\mathcal A}^*y^{k+2}+z^{k+1}-{\mathcal Q} w^{k+2}-c).
\end{equation}
Therefore,
\begin{equation}
\label{ykp2}
\begin{array}{ll}
y^{k+2}&=-[\sigma{\mathcal A}{\mathcal A}^*]^{-1}\left(({\mathcal A} x^{k+1}-b)
+\sigma {\mathcal A}(z^{k+1}-{\mathcal Q} x^{k+1}-c)\right)
\\begin{equation}2mm]
&=-[\sigma{\mathcal A}{\mathcal A}^*]^{-1}\left( {\mathcal A} (2x^{k+1}-u^k-\sigma({\mathcal Q} x^{k+1}+c) )-b\right).
\end{array}
\end{equation}
Note that for any $\xi\in{\mathcal X}$ one has $\Pi_{{\mathcal K}}(\xi)=\xi-{\mathcal A}^*({\mathcal A}{\mathcal A}^*)^{-1}({\mathcal A} \xi-b)$.
Consequently, by using \eqref{pf21} and \eqref{ykp2} we can get that
$$
\begin{array}{l}
{\mathcal J}_{\sigma{\mathcal N}_{{\mathcal K}}}\left( \big(2{\mathcal J}_{\sigma\partial\theta^*}-{\mathcal I}-\sigma\nabla f\circ {\mathcal J}_{\sigma\partial\theta^*}\big) (u^k)\right)
\\begin{equation}2mm]
=
\Pi_{{\mathcal K}}\left(2x^{k+1}-u^k-\sigma({\mathcal Q} x^{k+1}+c )
\right)
\\begin{equation}2mm]
=2x^{k+1}-u^k-\sigma({\mathcal Q} x^{k+1}+c)
+\sigma{\mathcal A}^*y^{k+2}.
\end{array}
$$
From \eqref{sequk} and the fact that $w^{k+1}=x^{k}$ one has that
$$
\begin{array}{ll}
u^{k+1}
&
=x^{k+1}+\sigma(\rho{\mathcal A}^*y^{k+2}-(1-\rho)z^{k+1} -\rho{\mathcal Q} x^{k+1}-\rho c)
\\begin{equation}2mm]
&
=u^k-\rho x^{k+1}+(1+\rho)x^{k+1}-u^k+\sigma\rho{\mathcal A}^*y^{k+2}
\\begin{equation}1mm]
&\quad\quad
-\sigma(1-\rho)z^{k+1}
-\sigma\rho({\mathcal Q} x^{k+1}+ c)
\\begin{equation}2mm]
&
=u^k-\rho x^{k+1}
+\rho(
2x^{k+1}
- u^k
-\sigma ({\mathcal Q} x^{k+1}+ c)+\sigma {\mathcal A}^*y^{k+2})
\\begin{equation}1mm]
&\quad\quad
+(1-\rho)(x^{k+1}
-u^k
-\sigma z^{k+1}).
\end{array}
$$
Note that \eqref{sequk} tells that $u^{k}=x^{k+1}-\sigma z^{k+1}$.
Therefore, we can readily get
$$
\begin{array}{ll}
u^{k+1}
&=u^k-\rho x^{k+1}
+\rho(
2x^{k+1}
- u^k
-\sigma ({\mathcal Q} x^{k+1}+ c)
+\sigma {\mathcal A}^*y^{k+2})
\\begin{equation}2mm]
&=(1-\rho)u^{k}+\rho
\left(u^k-
x^{k+1}
+
2x^{k+1}
- u^k
-\sigma ({\mathcal Q} x^{k+1}+ c)
+\sigma {\mathcal A}^*y^{k+2}\right)\\begin{equation}2mm]
&=(1-\rho)u^{k}+\rho\Gamma(u^k),
\end{array}
$$
which completes the proof.
\qed
\end{proof}
According to Proposition \ref{prop2}, Algorithm \ref{algo2} can also be viewed as a realization of the 3-operator splitting scheme proposed in \cite{Three-Operator} applied to the following problem
\begin{equation}
\label{3blkdual}
\min_x~~\left\{\theta^*(x)+\delta_{\mathcal K}(x)+\frac{1}{2} \langle x,{\mathcal Q} x\rangle+\langle c,x\rangle\right\}.
\end{equation}
Therefore, by using Proposition \ref{prop2}, a part of the convergence properties of Algorithm \ref{algo2} can be deduced directly from \cite[Theorem 1.1]{Three-Operator}. We summarize it as follows.
\begin{prop}
\label{thm:conv1}
Suppose that the solution set to problem \eqref{3blkdual} is nonempty and Assumption \ref{ass2} holds.
Let the infinite sequences
$\{w^k\}$, $\{y^k\}$, $\{z^k\}$ and $\{x^k\}$ be generated by Algorithm \ref{algo2} with
$\sigma\in \left(0,\frac{2}{\lambda_{\max}({\mathcal Q})}\right)$
and
$\rho\in\left(0,\frac{4-\sigma\lambda_{\max}({\mathcal Q})}{2}\right)$
.
Then, $\{x^k\}$ converges to a solution to problem \eqref{3blkdual}. Moreover, the sequence $\{u^k\}$ defined by \eqref{sequk} converges to a unique point, say $u^{\infty}$, such that $0\in\Gamma(u^{\infty})$.
\end{prop}
Since that Algorithm \ref{algo2} is intentionally designed for problem \eqref{cqpd}, Proposition \ref{thm:conv1} is still not enough for this algorithm.
Therefore, we need to further analyze its convergence properties.
The following theorem fulfils this objective.
\begin{thm}
Suppose that the solution set to problem \eqref{3blkdual} is nonempty and Assumption \ref{ass2} holds.
Let
$\sigma\in \left(0,\frac{2}{\lambda_{\max}({\mathcal Q})}\right)$
and
$\rho\in\left(0,\frac{4-\sigma\lambda_{\max}({\mathcal Q})}{2}\right)$.
Then, the infinite sequences $\{w^k\}$, $\{y^k\}$, $\{z^k\}$ and $\{x^k\}$ can be generated by Algorithm \ref{algo2}, and the sequence $\{(w^k,y^k,z^k)\}$ converges to a solution to problem \eqref{cqpd} while the sequence $\{x^k\}$ converges to a solution to problem \eqref{cqp}.
\end{thm}
\begin{proof}
According to Proposition \ref{thm:conv1} we know that both sequences $\{x^k\}$ and $\{u^k\}$ are convergent, especially that
$\{x^k\}$ converges to a solution of problem \eqref{cqp}.
Define $x^{\infty}:=\lim_{k\to\infty}x^k$ and
$u^{\infty}:=\lim_{k\to\infty}u^k$.
Then, by \eqref{ykp2} we know that the sequence $\{y^k\}$ is convergent. Moreover, by \eqref{sequk} we know that $\{z^k\}$ is also convergent.
We define $y^{\infty}:=\lim_{k\to\infty}y^k$
and $z^{\infty}:=\lim_{k\to\infty}z^k$.
Note that
$$
\lim_{k\to\infty} (\rho{\mathcal A}^*y^{k+1}-(1-\rho)z^k+z^{k+1}-\rho{\mathcal Q} w^{k+1}-\rho c)
=0,
$$
which implies that ${\mathcal A}^*y^{\infty}+z^{\infty}-{\mathcal Q} x^{\infty}-c=0$.
Then, by taking limits on both sides of \eqref{optt}, one has that ${\mathcal A} x^{\infty}-b=0$.
Also, one can take limits in \eqref{optcond1} and obtains that
$0\in-\partial \theta(-z^{\infty})+ x^{\infty}$.
Therefore, by denoting $w^{\infty}=x^{\infty}$ we can conclude that $(w^\infty,y^{\infty},z^{\infty},x^{\infty})$ is a solution to the KKT system \eqref{KKT2}, so that
$\{(w^k,y^k,z^k)\}$ converges to a solution to problem \eqref{cqpd}.
This completes the proof.
\qed
\end{proof}
\section{Conclusions}
\label{conclusion}
In this paper, we have shown that the modified $3$-block ADMM in Chang et al. \cite{modified admm} is an instance of the $3$-operator splitting scheme in \cite{Three-Operator}. Based on this observation, we considered a generalized modified $3$-block ADMM applied to the more general convex composite quadratic programming model, and derived its convergence via a very concise approach.
The obtained results paved the way for further study of the proposed generalized modified ADMM for convex composite quadratic programming such as the iteration complexity and the local or global convergence rate, which we leave as our future work.
\end{document} |
\begin{document}
\title
{A twisted link invariant derived from a virtual link invariant}
\author{Naoko Kamada
\thanks{This work was supported by JSPS KAKENHI Grant Number 15K04879.}
\\ Graduate School of Natural Sciences, Nagoya City University\\
1 Yamanohata, Mizuho-cho, Mizuho-ku, Nagoya, Aichi 467-8501 Japan\\
}
\date{}
\maketitle
\begin{abstract}
Virtual knot theory is a generalization of knot theory which is based on Gauss chord diagrams and link diagrams on closed oriented surfaces.
A twisted knot is a generalization of a virtual knot, which corresponds to a link diagram on a possibly non-orientable surface.
In this paper, we discuss an invariant of twisted links which is obtained from the JKSS invariant of virtual links by use of double coverings. We also discuss some properties of double covering diagrams.
\end{abstract}
\section{Introduction}
L. H. Kauffman introduced virtual knot theory, which is a generalization of knot theory based on Gauss chord diagrams and link diagrams in closed oriented surfaces \cite{rkauD}.
Twisted knot theory was introduced by Bourgoin. It is an extension of virtual knot theory. Twisted links correspond to stable equivalence classes of links in oriented 3-manifolds which are line bundles over (possibly non-orientable) closed surfaces \cite{rBor}, \cite{rCKS}.
F. Jaeger, L. H. Kauffman and H. Saleur defined an invariant of links in thickened surfaces, where the surfaces are oriented \cite{rJKS}. J. Sawollek \cite{rSaw} applied it to virtual links, which is so-called the {\it JKSS invariant}.
In this paper, we introduce an invariant of twisted links obtained from the JKSS invariant by use of double coverings.
We also discuss some properties of double coverings of twisted link diagrams.
A {\it virtual link diagram\/} is a generically immersed loops whose double points have information of positive, negative or virtual crossing. A {\it virtual crossing\/} is an encircled double point without over-under information. A {\it twisted link diagram\/} is a virtual link diagram, possibly with {\it bars\/} on arcs. Examples of twisted link diagrams are depicted in Figure~\ref{fig:extwtdiag}.
\begin{figure}
\caption{Examples of twisted link diagrams}
\label{fig:extwtdiag}
\end{figure}
A {\it twisted link\/} (resp. a {\it virtual link}) is an equivalence class of a twisted link diagram (resp. virtual link diagram) under Reidemeister moves, virtual Reidemeister moves and twisted Reidemeister moves (resp. Reidemeister moves and virtual Reidemeister moves) depicted in Figures~\ref{fig:movesR}, \ref{fig:movesV} and \ref{fig:movesT}.
\begin{figure}
\caption{Reisdemeister moves}
\label{fig:movesR}
\end{figure}
\begin{figure}
\caption{Virtual Reidemeister moves}
\label{fig:movesV}
\end{figure}
\begin{figure}
\caption{Twisted Reidemeister moves}
\label{fig:movesT}
\end{figure}
\section{The JKSS invariant}
We recall the definition of the JKSS invariant of a virtual link.
Let $D$ be a virtual link diagram with $n$ real crossings.
Let $c_1, \dots, c_n$ be the real crossings of $D$.
We define a $2n\times 2n$ matrix, $M=\mathrm{diag}(M_1,\dots, M_n)$, where
$M_i=M_+$ (or $M_-$) if $c_i$ is positive (or negative) crossing. Here $M_+$ and $M_-$
are $2\times 2$ matrices:
$M_{+}=\begin{pmatrix}
1-x&-y&\\
-xy^{-1}&0
\end{pmatrix}$
and
$M_{-}=\begin{pmatrix}
0&-x^{-1}y\\
-y^{-1}&1-x^{-1}\\
\end{pmatrix}$.
Let $|D|$ be the 4-valent graph obtained from $D$ by regarding each real crossing as a vertex of $|D|$. We
denote by the same symbols $c_1, \dots, c_n$ the vertices of $|D|$.
The graph $|D|$ is immersed in $\mathbb{R}^2$ and the multiple points of $|D|$ are virtual crossings of $D$.
For each vertex $c_i$ of $|D|$, consider an open regular neighborhood $N(c_i, |D|)$ of $c_i$ in $|D|$. Then $N(c_i, |D|)-\{c_i\}$ is the union of four open arcs, which we call the {\it short edges around} $c_i$. According to the position, we denote by $i_0^-, i_1^-, i_0^+, i_1^+$ the short edges as in
Figure \ref{fig:labeledge}.
\begin{figure}
\caption{Labels of four edges}
\label{fig:labeledge}
\end{figure}
We define a $2n\times 2n$ matrix, $P=(p_{kl})$ as follows. For each $i, j\in \{1, \dots, n\}$,
$$p_{(2i-1+\epsilon)(2j-1+\lambda)}=\left\{
\begin{array}{ll}
1&\left(\parbox{10cm}{if two short edges $i^{-}_{\epsilon}$ and $ j^{+}_{\lambda}$ are on the same edge of $|D|$}\right)\\
0&\parbox{3cm}{(otherwise)}
\end{array}\right. ,$$
where $\epsilon, \lambda\in \{0,1\}$.
\begin{thm}[Jaeger, Kauffman and Saleur \cite{rJKS}, Sawollek \cite{rSaw}]
For a virtual link diagram $D$, $Z_{D}(x, y) = (-1)^{w(D)}\mathrm{det}(M-P )$ is an invariant of the virtual link up to multiplication by powers of $x^{\pm 1}$, i.e., for any virtual link diagram $D'$ representing the same virtual link with $D$, we have $Z_{D'}(x, y)=x^mZ_D(x, y)$ for some $m\in \mathbb{Z}$.
\end{thm}
The {\it JKSS invariant} of a virtual link $L$ is defined by $Z_L(x, y)=Z_D(x,y)$ for a diagram $D$ of $L$.
For example, the JKSS invariant of a virtual link depicted in Figure~\ref{fig:exJKSSv1} is $(x-1)(y+1)(x+y)y^{-1}$.
\begin{figure}
\caption{A virtual link diagram}
\label{fig:exJKSSv1}
\end{figure}
We define an invariant of twisted links which is related to the JKSS invariant.
Let $D$ be a {twisted} link diagram with $n$ real crossings $c_1,\dots, c_n$.
We define a $4n\times 4n$ matrix $\widetilde{M}$, by $\widetilde{M}=\mathrm{diag}(\widetilde{M}_1,\dots, \widetilde{M}_n)$, where
$\widetilde{M}_i=\widetilde{M}_+$ (or $\widetilde{M}_-$) if the crossing $c_i$ is positive (or negative). Here
$\widetilde{M}_+$ and $\widetilde{M}_-$ are $4\times 4$ matrices $\mathrm{diag}(M_+, M_+)$ and $\mathrm{diag}(M_-, M_-)$ respectively, i.e.,
$$\widetilde{M}_+=
\begin{pmatrix}
1-x&-y&0&0\\
-xy^{-1}&0&0&0\\
0&0&1-x&-y\\
0&0&-xy^{-1}&0
\end{pmatrix}
\text{ and }
\widetilde{M}_-=
\begin{pmatrix}
0&-x^{-1}y&0&0\\
-y^{-1}&1-x^{-1}&0&0\\
0&0&0&-x^{-1}y\\
0&0&-y^{-1}&1-x^{-1}
\end{pmatrix}
.$$
For a twisted link diagram $D$, the graph $|D|$ is defined by the same way before. Each edge of $|D|$ may have bars on it. For each vertex $c_i$ of $|D|$, we denote by $i_0^-, i_1^-, i_0^+$, and $i_1^+$ the short edge around $c_i$ as before.
We denote
$i_{\epsilon} \overset{\mathrm{e}}{\leftarrow} j_{\lambda}$ (or $i_{\epsilon} \overset{\mathrm{o}}{\leftarrow} j_{\lambda}$),
if two short edges $i^{-}_{\epsilon}$ and $ j^{+}_{\lambda}$ are on the same edge of $|D|$ and there are an even (or odd) number of bars on the edge.
We defined $4n\times 4n$ matrix, $\widetilde{P}=(\tilde{p}_{kl})$ as follows. For each $i, j\in\{1,\dots, n\}$,
$$\tilde{p}_{(4i-3+a)(4j-3+b)}=\left\{
\begin{array}{ll}
1&\left(\parbox{9cm}{$i_a \overset{\mathrm{e}}{\leftarrow} j_b$ or $i_{3-a} \overset{\mathrm{e}}{\leftarrow} j_{3-b}$ or $i_a \overset{\mathrm{o}}{\leftarrow} j_{3-b}$ or $i_{3-a} \overset{\mathrm{o}}{\leftarrow} j_{b}$}\right)\\
0&\parbox{3cm}{(otherwise)}
\end{array}\right. , $$
where $a, b \in \{0, 1, 2, 3 \}$
Note that $i_k^-$ and $j_k^-$ arc not defined for $k\in\{2, 3\}$. We assume that $i_k \overset{\mathrm{e}}{\leftarrow} j_l$ and $i_k \overset{\mathrm{o}}{\leftarrow} j_l$ are false when $k\in\{2, 3\}$ or $l\in\{2, 3\}$.
\begin{thm}\label{thm1}
For a twisted link diagram $D$, $\widetilde{Z}_{D}(x, y) = \mbox{det}(\widetilde{M}-\widetilde{P })$ is an invariant of the twisted link up to multiplication by powers of $x^{\pm 1}$, i.e., for any twisted link diagram $D'$ representing the same twisted link with $D$, we have $Z_{D'}(x, y)=x^mZ_D(x, y)$ for some $m\in \mathbb{Z}$.
\end{thm}
For a twisted link $L$, we define the {\it twisted JKSS invariant} of $L$, denoted by $\widetilde{Z}_L(x, y)$, by $\widetilde{Z}_D(x, y)$ for a diagram $D$ of $L$.
\begin{figure}
\caption{The twisted JKSS invariants of twisted links}
\label{fig:exJKSScalt1}
\end{figure}
The twisted JKSS invariant of the diagram in Figure \ref{fig:exJKSScalt1} (a)
is $0$ and that of the diagram in Figure \ref{fig:exJKSScalt1} (b) is $y^{-2}\left(x^2-1\right) \left(y^2-1\right) \left(x^2-y^2\right)$. We see that they are not equivalent.
Note that they are not distinguished by the twisted Jones polynomial defined in \cite{rBor}. The twisted Jones polynomials of the twisted links in Figure \ref{fig:exJKSScalt1} are $-A^{-6}(A^{4}+A^{-4})M$.
\section{Proof of Theorem \ref{thm1}}
Let $D$ be a twisted link diagram with bars $b_1,\dots , b_k$.
Assume that $D$ is on the left of the $y$-axis and all bars are parallel to the $x$-axis with disjoint $y$-coordinates. Let $s(D)$ be the twisted link diagram which is obtained from $D$ by reflection with respect to the $y$-axis and switching all real crossings of $D$.
See Figure~\ref{fig:doublecv1}.
We construct the double covering of $D$ as follows:
\begin{figure}
\caption{A twisted link diagram $D$ and $s(D)$}
\label{fig:doublecv1}
\end{figure}
For horizontal lines $l _1, \dots , l_k$ such that $l_i$ contains $b_i$ and $s(b_i)$,
we replace each part of $D\amalg s(D)$ in a neighborhood of $N(l_i)$ as in Figure~\ref{fig:doublecv2}. We call this diagram $\widetilde{D}$ the {\it double covering diagram} of $D$.
\begin{figure}
\caption{Construct double covering }
\label{fig:doublecv2}
\end{figure}
The double covering diagram of the twisted link diagram $D$ in Figure~\ref{fig:doublecv1} is shown in Figure~\ref{fig:doublecv3}.
\begin{figure}
\caption{The double covering diagram of $D$}
\label{fig:doublecv3}
\end{figure}
\begin{thm}[\cite{rkk7}]\label{thm2}
Let $D_1$ and $D_2$ be twisted link diagrams and
$\widetilde{D_1}$ and $\widetilde{D_2}$ their double coverings diagrams of $D_1$ and $D_2$.
If $D_1$ and $D_2$ are equivalent as twisted links, then $\widetilde{D_1}$ and $\widetilde{D_2}$ are equivalent
as virtual links.
\end{thm}
\begin{thm}\label{thm0}
For a twisted link diagram $D$, $\widetilde{Z}_{D}(x,y)$ coincides to ${Z}_{\widetilde{D}}(x,y)$, where $\widetilde{D}$ is the double covering of $D$.
\end{thm}
\noindent
{\bf Proof}\quad
Let $D$ be a twisted link diagram with $n$ real crossings $c_1, \cdots , c_n$.
Let $s(c_1), \dots, s(c_n)$ be the real crossings of $s(D)$ corresponding to $c_1, \dots, c_n$.
We regard $c_1, \cdots , c_n, s(c_1), \dots, s(c_n)$ as the real crossings of the double covering diagram $\widetilde{D}$.
We rename the real crossings of $\widetilde{D}$ by $\tilde{c}_1, \cdots , \tilde{c}_{2n}$ such that $c_i=\tilde{c}_{2i-1}$ and $s(c_i)=\tilde{c}_{2i}$ for $i\in \{1,\dots, n\}$.
Then the matrix $\widetilde{M}$ for $D$ coincides with matrix $M$ for $\widetilde{D}$. We show that the matrix $\widetilde{P}$ for $D$ coincides with the matrix $P$ for $\widetilde{D}$.
Suppose that real crossings $c_i$ and $c_j$ of $D$ are the boundary points of an edge of $|D|$, i.e., $i_{\epsilon} \overset{\mathrm{e}}{\leftarrow} j_{\lambda}$ or $i_{\epsilon} \overset{\mathrm{o}}{\leftarrow} j_{\lambda}$ holds for $D$.
\begin{enumerate}
\item
If $i_{\epsilon} \overset{\mathrm{e}}{\leftarrow} j_{\lambda}$ for $D$,
we see that ${c_i}$ and ${c_j}$ (or $s(c_i)$ and ${s(c_j)}$ ) are the boundary points of an edge of $|\widetilde{D}|$.
The short edges labeled by $(2i-1)_{\epsilon}^-$ and $(2j-1)_{\lambda}^+$ (or $(2i)_{1-\epsilon}^-$ and $(2j)_{1-\lambda}^+$) are on the same edge of $|\widetilde{D}|$. Thus we have
$$
\tilde{p}_{(4i-3+a)(4j-3+b)}=
\left\{\begin{array}{ll}
1&(a=\epsilon \text{ and } b=\lambda)\\
1&(a=2+1-\epsilon \text{ and } b=2+1-\lambda)\\
0&(\text{otherwise}).
\end{array}\right.
$$
\item
If $i_{\epsilon} \overset{\mathrm{o}}{\leftarrow} j_{\lambda}$ for $D$,
we see that ${c_i}$ and ${s(c_j)}$ (or ${s(c_i)}$ and ${c_j}$ ) are the boundary points of an edge of $|\widetilde{D}|$.
Namely, the short edges labeled by $(2i-1)_{\epsilon}^-$ and $(2j)_{1-\lambda}^+$ (or $(2i)_{1-\epsilon}^-$ and $(2j-1)_{\lambda}^+$) are on the same edge of $|\widetilde{D}|$. Thus we have
$$
\tilde{p}_{(4i-3+a)(4j-3+b)}=
\left\{\begin{array}{ll}
1&(a=\epsilon \text{ and } b=2+1-\lambda)\\
1&(a=2+1-\epsilon \text{ and } b=\lambda)\\
0&(\text{otherwise}).
\end{array}\right.
$$
\end{enumerate}
Thus we have the $\widetilde{P}=P$. Since $w(\widetilde{D})$ is $2w(D)$, $Z_{\widetilde{D}}(x, y)=\mathrm{det}(M-P)=\mathrm{det}(\widetilde{M}-\widetilde{P})
=\widetilde{Z}_D(x, y)$.
$\square$
\section{Properties of double covering diagrams}
For a twisted (virtual or classical) link diagram $D$, we denote by $r(D)$ the number of real crossings of $D$.
For a twisted link (or a virtual link) $L$, we denote by $r(L)$ (or $r_v(L)$) the minimal number of real crossings of all of diagrams of $L$.
\begin{thm}\label{prop1}
Let $L$ be a twisted link presented by a twisted link diagram $D$ and let $\widetilde{L}$ be the virtual link presented by the double covering diagram $\widetilde{D}$ of $D$.
If $r(\widetilde{D})=r_v(\widetilde{L})$, then $r(D)=r(L)$.
\end{thm}
\noindent
{\bf Proof}\quad
Note that $r(\widetilde{D})=2r(D)$.
If $r(D)\ne r(L)$, then there is a diagram of $L$, $D_0$ such that $r(D_0)=r(L)<r(D)$.
For the double covering diagram $\widetilde{D}_0$ of $D_0$, we have $r(\widetilde{D}_0)=2r(D_0)<2r(D)$,
which is a contradiction of the assumption since $\widetilde{D}_0$ is equivalent to $\widetilde{D}$ as a virtual link.
$\square$
\begin{prop}\label{prop2}
Let $D$ be a twisted link diagram obtained from a virtual link diagram ${D_0}$ by adding a bar on an arc. Then $\widetilde{D}$ is equivalent to a connected sum of ${D_0}$ and $s({D_0})$ modulo virtual Reidemeister moves.
\end{prop}
\noindent
{\bf Proof}\quad
For the double covering $\widetilde{D}$ of $D$, there are two kinds of sets of virtual crossings. One is the set of virtual crossings which correspond to virtual crossings of $D$ and $s(D)$, and the other
is the set of virtual crossings which occur when we construct $\widetilde{D}$ from $D\amalg s(D)$ by replacement as in Figure~\ref{fig:doublecv2}. The second set of virtual crossings of $\widetilde{D}$ look as in Figure~\ref{fgprop2} (i).
As in Figure~\ref{fgprop2} (ii), such virtual crossings are eliminated by some virtual Reidemeister moves.
The virtual link diagram shown in Figure~\ref{fgprop2} (ii) is a connected sum of two virtual link diagrams which are equivalent to ${D_0}$ and $s({D_0})$ modulo virtual Reidemeister moves.
$\square$
\begin{figure}
\caption{Double covering of a twisted link diagram with a bar}
\label{fgprop2}
\end{figure}
{\bf Acknowledgements }
The author would like to thank Seiichi Kamada for his useful suggestion.
\end{document} |
\hbox{\bf B}egin{document}
\hbox{\hbox{\bf B}f T}itle{On slant helices in Minkowski space $\hbox{\hbox{\bf B}f E}_1^3$}
\author{ Ahmad T. Ali\\Mathematics Department\\
Faculty of Science, Al-Azhar University\\
Nasr City, 11448, Cairo, Egypt\\
email: [email protected]\\
\vspace*{1cm}\\
Rafael L\'opez\footnote{Partially
supported by MEC-FEDER
grant no. MTM2007-61775.}\\
Departamento de Geometr\'{\i}a y Topolog\'{\i}a\\
Universidad de Granada\\
18071 Granada, Spain\\
email: [email protected]}
\date{}
\title{On slant helices in Minkowski space $\e_1^3$}
\hbox{\bf B}egin{abstract} We consider a curve $\alpha=\alpha(s)$ in Minkowski 3-space $\hbox{\hbox{\bf B}f E}_1^3$ and denote by $\{\hbox{\hbox{\bf B}f T},\hbox{\hbox{\bf B}f N},\hbox{\bf B}\}$ the Frenet frame of $\alpha$. We say that $\alpha$ is a slant helix if there exists a fixed direction $U$ of $\hbox{\hbox{\bf B}f E}_1^3$ such that the function $\langle \hbox{\hbox{\bf B}f N}(s),U\hbox{\hbox{\bf B}b R}angle$ is constant. In this work we give characterizations of slant helices in terms of the curvature and torsion of $\alpha$.
\hbox{\hbox{\bf B}f E}nd{abstract}
\hbox{\hbox{\bf B}f E}mph{MSC:} 53C40, 53C50
\hbox{\hbox{\bf B}f E}mph{Keywords}: Minkowski 3-space; Frenet equations; Slant helix.
\hbox{\hbox{\bf B}b S}ection{Introduction and statement of results}
Let $\hbox{\hbox{\bf B}f E}_1^3$ be the Minkowski 3-space, that is, $\hbox{\hbox{\bf B}f E}_1^3$ is the real vector space $\hbox{\hbox{\bf B}b R}^3$ endowed with the standard flat metric
$$\langle,\hbox{\hbox{\bf B}b R}angle=dx_1^2+dx_2^2-dx_3^2,$$
where $(x_1,x_2,x_3)$ is a rectangular coordinate system of $\hbox{\hbox{\bf B}f E}_1^3$. An arbitrary vector $v\in\hbox{\hbox{\bf B}f E}_1^3$ is said spacelike if $\langle v,v\hbox{\hbox{\bf B}b R}angle>0$ or $v=0$, timelike if $\langle v,v\hbox{\hbox{\bf B}b R}angle<0$, and lightlike (or null) if $\langle v,v\hbox{\hbox{\bf B}b R}angle =0$ and $v\hbox{\hbox{\bf B}f N}eq0$. The norm (length) of a vector $v$ is given by $\parallel v\parallel=\hbox{\hbox{\bf B}b S}qrt{|\langle v,v\hbox{\hbox{\bf B}b R}angle|}$.
Given a regular (smooth) curve $\alpha:I\hbox{\hbox{\bf B}b S}ubset\hbox{\hbox{\bf B}b R}\hbox{\hbox{\bf B}b R}ightarrow\hbox{\hbox{\bf B}f E}_1^3$, we say that $\alpha$ is spacelike (resp. timelike, lightlike) if all of its velocity vectors $\alpha'(t)$ are spacelike (resp. timelike, lightlike). If $\alpha$ is spacelike or timelike we say that $\alpha$ is a non-null curve. In such case, there exists a change of the parameter $t$, namely, $s=s(t)$, such that $\parallel\alpha'(s)\parallel=1$. We say then that $\alpha$ is parametrized by the arc-length parameter. If the curve $\alpha$ is lightlike, the acceleration vector $\alpha''(t)$ must be spacelike for all $t$. Then we change the parameter $t$ by $s=s(t)$ in such way that $\parallel \alpha''(s)\parallel=1$ and we say that $\alpha$ is parameterized by the pseudo arc-length parameter. In any of the above cases, we say that $\alpha$ is a unit speed curve.
Given a unit speed curve $\alpha$ in Minkowski space $\hbox{\hbox{\bf B}f E}_1^3$ it is possible to define a Frenet frame $\{\hbox{\hbox{\bf B}f T}(s),\hbox{\hbox{\bf B}f N}(s),\hbox{\bf B}(s)\}$ associated for each point $s$ \cite{ku,wa}. Here $\hbox{\hbox{\bf B}f T}$, $\hbox{\hbox{\bf B}f N}$ and $\hbox{\bf B}$ are the tangent, normal and binormal vector field, respectively. The geometry of the curve $\alpha$ can be describe by the differentiation of the Frenet frame, which leads to the corresponding Frenet equations. Although different expressions of the Frenet equations appear depending of the causal character of the Frenet trihedron (see the next sections below), we have the concepts of curvature $\kappa$ and torsion $\hbox{\hbox{\bf B}f T}au$ of the curve. With this preparatory introduction, we give the following
\hbox{\bf B}egin{definition} A unit speed curve $\alpha$ is called a slant helix if there exists a constant vector field $U$ in $\hbox{\hbox{\bf B}f E}_1^3$ such that the function $\langle \hbox{\hbox{\bf B}f N}(s),U\hbox{\hbox{\bf B}b R}angle$ is constant.
\hbox{\hbox{\bf B}f E}nd{definition}
This definition is motivated by what happens in Euclidean ambient space $\hbox{\hbox{\bf B}f E}^3$. In this setting, we recall that a helix is a curve where the tangent lines make a constant angle with a fixed direction. Helices are characterized by the fact that the ratio $\hbox{\hbox{\bf B}f T}au/\kappa$ is constant along the curve \cite{dc}. Helices in Minkowski space have been studied depending on the causal character of the curve $\alpha$: see for example \cite{fgl,ko,ps}.
Recently, Izumiya and Takeuchi have introduced the concept of slant helix in Euclidean space by saying that the normal lines make a constant angle with a fixed direction \cite{it}. They characterize a slant helix if and only if the function
\hbox{\bf B}egin{equation}\label{slant}
\dfrac{\kappa^2}{(\kappa^2+\hbox{\hbox{\bf B}f T}au^2)^{3/2}}\Big(\dfrac{\hbox{\hbox{\bf B}f T}au}{\kappa}\Big)'
\hbox{\hbox{\bf B}f E}nd{equation}
is constant. See also \cite{ky, okkk}. Thus, our definition of slant helix is the Lorentzian version of the Euclidean one. Only it is important to point out that, in contrast to what happens in Euclidean space, in Minkowski ambient space we can not define the angle between two vectors (except that both vectors are of timelike type). For this reason, we avoid to say about the angle between the vector fields $\hbox{\hbox{\bf B}f N}(s)$ and $U$.
Our main result in this work is the following characterization of slant helices in the spirit of the one given in equation (\hbox{\hbox{\bf B}b R}ef{slant}). We will assume throughout this work that the curvature and torsion functions do not equal zero. Exactly, we prove
\hbox{\bf B}egin{theorem}\label{t1} Let $\alpha$ be a unit speed timelike curve in $\hbox{\hbox{\bf B}f E}_1^3$. Then $\alpha$ is a slant helix if and only if either one the next two functions
\hbox{\bf B}egin{equation}\label{slant2}
\frac{\kappa^2}{(\hbox{\hbox{\bf B}f T}au^2-\kappa^2)^{3/2}}\Big(\dfrac{\hbox{\hbox{\bf B}f T}au}{\kappa}\Big)'\hspace*{1cm}\mbox{or}\hspace*{1cm}
\frac{\kappa^2}{(\kappa^2-\hbox{\hbox{\bf B}f T}au^2)^{3/2}}\Big(\dfrac{\hbox{\hbox{\bf B}f T}au}{\kappa}\Big)'
\hbox{\hbox{\bf B}f E}nd{equation}
is constant everywhere $\hbox{\hbox{\bf B}f T}au^2-\kappa^2$ does not vanish.
\hbox{\hbox{\bf B}f E}nd{theorem}
\hbox{\bf B}egin{theorem}\label{t2} Let $\alpha$ be a unit speed spacelike curve in $\hbox{\hbox{\bf B}f E}_1^3$.
\hbox{\bf B}egin{enumerate}
\item If the normal vector of $\alpha$ is spacelike, then $\alpha$ is a slant helix if and only if either one the next two functions
\hbox{\bf B}egin{equation}\label{slant3}
\frac{\kappa^2}{(\hbox{\hbox{\bf B}f T}au^2-\kappa^2)^{3/2}}\Big(\dfrac{\hbox{\hbox{\bf B}f T}au}{\kappa}\Big)'\hspace*{1cm}\mbox{or}\hspace*{1cm}
\frac{\kappa^2}{(\kappa^2-\hbox{\hbox{\bf B}f T}au^2)^{3/2}}\Big(\dfrac{\hbox{\hbox{\bf B}f T}au}{\kappa}\Big)'
\hbox{\hbox{\bf B}f E}nd{equation}
is constant everywhere $\hbox{\hbox{\bf B}f T}au^2-\kappa^2$ does not vanish.
\item If the normal vector of $\alpha$ is timelike, then $\alpha$ is a slant helix if and only if the function
\hbox{\bf B}egin{equation}\label{slant4}
\frac{\kappa^2}{(\hbox{\hbox{\bf B}f T}au^2+\kappa^2)^{3/2}}\Big(\dfrac{\hbox{\hbox{\bf B}f T}au}{\kappa}\Big)'
\hbox{\hbox{\bf B}f E}nd{equation}
is constant.
\item Any spacelike curve with lightlike normal vector is a slant curve.
\hbox{\hbox{\bf B}f E}nd{enumerate}
\hbox{\hbox{\bf B}f E}nd{theorem}
In the case that $\alpha$ is a lightlike curve, we have
\hbox{\bf B}egin{theorem}\label{t3} Let $\alpha$ be a unit speed lightlike curve in $\hbox{\hbox{\bf B}f E}_1^3$. Then $\alpha$ is a slant helix if and only if the torsion is
\hbox{\bf B}egin{equation}\label{slant5}
\hbox{\hbox{\bf B}f T}au(s)=\frac{a}{(bs+c)^2},
\hbox{\hbox{\bf B}f E}nd{equation}
where $a$, $b$ and $c$ are constant.
\hbox{\hbox{\bf B}f E}nd{theorem}
The proof of Theorems \hbox{\hbox{\bf B}b R}ef{t1}, \hbox{\hbox{\bf B}b R}ef{t2} and \hbox{\hbox{\bf B}b R}ef{t3} is carried in the successive sections.
\hbox{\hbox{\bf B}b S}ection{Timelike slant helices}
Let $\alpha$ be a unit speed timelike curve in $\hbox{\hbox{\bf B}f E}_1^3$. The Frenet frame $\{\hbox{\hbox{\bf B}f T},\hbox{\hbox{\bf B}f N},\hbox{\bf B}\}$ of $\alpha$ is given by
$$\hbox{\hbox{\bf B}f T}(s)=\alpha'(s),\ \ \hbox{\hbox{\bf B}f N}(s)=\dfrac{\alpha''(s)}{\parallel\alpha''(s)\parallel},\ \ \hbox{\bf B}(s)=\hbox{\hbox{\bf B}f T}(s)\hbox{\hbox{\bf B}f T}imes\hbox{\hbox{\bf B}f N}(s).$$
The Frenet equations are
\hbox{\bf B}egin{equation}\label{equi1}
\left[
\hbox{\bf B}egin{array}{c}
\hbox{\hbox{\bf B}f T}'(s) \\
\hbox{\hbox{\bf B}f N}'(s) \\
\hbox{\bf B}'(s)
\hbox{\hbox{\bf B}f E}nd{array}
\hbox{\hbox{\bf B}b R}ight]=\left[
\hbox{\bf B}egin{array}{ccc}
0 & \kappa(s) & 0 \\
\kappa(s) & 0 &\hbox{\hbox{\bf B}f T}au(s)\\
0 &-\hbox{\hbox{\bf B}f T}au(s) & 0\\
\hbox{\hbox{\bf B}f E}nd{array}
\hbox{\hbox{\bf B}b R}ight]\left[
\hbox{\bf B}egin{array}{c}
\hbox{\hbox{\bf B}f T}(s) \\
\hbox{\hbox{\bf B}f N}(s) \\
\hbox{\bf B}(s) \\
\hbox{\hbox{\bf B}f E}nd{array}
\hbox{\hbox{\bf B}b R}ight].
\hbox{\hbox{\bf B}f E}nd{equation}
In order to prove Theorem \hbox{\hbox{\bf B}b R}ef{t1}, we first assume that $\alpha$ is a slant helix. Let $U$ be the vector field such that the function $\langle \hbox{\hbox{\bf B}f N}(s),U\hbox{\hbox{\bf B}b R}angle:=c$ is constant. There exist smooth functions $a_1$ and $a_3$ such that
\hbox{\bf B}egin{equation}\label{u1}
U=a_1(s)\hbox{\hbox{\bf B}f T}(s)+c \hbox{\hbox{\bf B}f N}(s)+a_3(s) \hbox{\bf B}(s),\ \ s\in I.
\hbox{\hbox{\bf B}f E}nd{equation}
As $U$ is constant, a differentiation in (\hbox{\hbox{\bf B}b R}ef{u1}) together (\hbox{\hbox{\bf B}b R}ef{equi1}) gives
\hbox{\bf B}egin{equation}\label{u2}
\left.\hbox{\bf B}egin{array}{ll}
a_1'-c\kappa&=0\\
\kappa a_1-\hbox{\hbox{\bf B}f T}au a_3&=0\\
a_3'+c \hbox{\hbox{\bf B}f T}au &=0
\hbox{\hbox{\bf B}f E}nd{array}\hbox{\hbox{\bf B}b R}ight\}
\hbox{\hbox{\bf B}f E}nd{equation}
From the second equation in (\hbox{\hbox{\bf B}b R}ef{u2}) we have
\hbox{\bf B}egin{equation}\label{u5}
a_1=a_3\hbox{\bf B}ig(\dfrac{\hbox{\hbox{\bf B}f T}au}{\kappa}\hbox{\bf B}ig).
\hbox{\hbox{\bf B}f E}nd{equation}
Moreover
\hbox{\bf B}egin{equation}\label{u3}
\langle U,U\hbox{\hbox{\bf B}b R}angle=-a_1^2+c^2+a_3^2=\mbox{constant}.
\hbox{\hbox{\bf B}f E}nd{equation}
We point out that this constraint, together the second and third equation of (\hbox{\hbox{\bf B}b R}ef{u2}) is equivalent to the very system (\hbox{\hbox{\bf B}b R}ef{u2}). From (\hbox{\hbox{\bf B}b R}ef{u5}) and (\hbox{\hbox{\bf B}b R}ef{u3}), set
$$a_3^2\Big(\hbox{\bf B}ig(\frac{\hbox{\hbox{\bf B}f T}au}{\kappa}\hbox{\bf B}ig)^2-1\Big)=\hbox{\hbox{\bf B}f E}psilon m^2,\ \ m>0,\hbox{\hbox{\bf B}f E}psilon\in\{-1,0,1\}.$$
If $\hbox{\hbox{\bf B}f E}psilon=0$, then $a_3=0$ and from (\hbox{\hbox{\bf B}b R}ef{u2}) we have $a_1=c=0$. This means that $U=0$: contradiction. Thus $\hbox{\hbox{\bf B}f E}psilon=1$ or $\hbox{\hbox{\bf B}f E}psilon=-1$ which gives
$$a_3=\pm\dfrac{m}{\hbox{\hbox{\bf B}b S}qrt{\hbox{\bf B}ig(\dfrac{\hbox{\hbox{\bf B}f T}au}{\kappa}\hbox{\bf B}ig)^2-1}}\hspace*{1cm}\mbox{or}\hspace*{1cm}a_3=\pm\dfrac{m}{\hbox{\hbox{\bf B}b S}qrt{1-\hbox{\bf B}ig(\dfrac{\hbox{\hbox{\bf B}f T}au}{\kappa}\hbox{\bf B}ig)^2}}$$
on $I$. The third equation in (\hbox{\hbox{\bf B}b R}ef{u2}) yields
$$\dfrac{d}{ds}\Big[\pm\dfrac{m}{\hbox{\hbox{\bf B}b S}qrt{\hbox{\bf B}ig(\dfrac{\hbox{\hbox{\bf B}f T}au}{\kappa}\hbox{\bf B}ig)^2-1}}\Big]=-c \hbox{\hbox{\bf B}f T}au \hspace*{1cm} \mbox{or}\hspace*{1cm}\dfrac{d}{ds}\Big[\pm\dfrac{m}{\hbox{\hbox{\bf B}b S}qrt{1-\hbox{\bf B}ig(\dfrac{\hbox{\hbox{\bf B}f T}au}{\kappa}\hbox{\bf B}ig)^2}}\Big]=c \hbox{\hbox{\bf B}f T}au$$
on $I$. This can be written as
$$\frac{\kappa^2}{(\hbox{\hbox{\bf B}f T}au^2-\kappa^2)^{3/2}}\Big(\dfrac{\hbox{\hbox{\bf B}f T}au}{\kappa}\Big)'=\mp\dfrac{c}{m}\hspace*{1cm}\mbox{or}\hspace*{1cm}
\frac{\kappa^2}{(\kappa^2-\hbox{\hbox{\bf B}f T}au^2)^{3/2}}\Big(\dfrac{\hbox{\hbox{\bf B}f T}au}{\kappa}\Big)'=\pm\dfrac{c}{m}$$
This shows a part of Theorem \hbox{\hbox{\bf B}b R}ef{t1}. Conversely, assume that the condition (\hbox{\hbox{\bf B}b R}ef{slant2}) is satisfied. In order to simplify the computations, we assume that the first function in (\hbox{\hbox{\bf B}b R}ef{slant2}) is a constant, namely, $c$ (the other case is analogous). We define
\hbox{\bf B}egin{equation}\label{u9}
U=\dfrac{\hbox{\hbox{\bf B}f T}au}{\hbox{\hbox{\bf B}b S}qrt{\hbox{\hbox{\bf B}f T}au^2-\kappa^2}}\hbox{\hbox{\bf B}f T}+
c\hbox{\hbox{\bf B}f N}+\dfrac{\kappa}{\hbox{\hbox{\bf B}b S}qrt{\hbox{\hbox{\bf B}f T}au^2-\kappa^2}}\hbox{\bf B}\Big.
\hbox{\hbox{\bf B}f E}nd{equation}
A differentiation of (\hbox{\hbox{\bf B}b R}ef{u9}) together the Frenet equations gives $\dfrac{dU}{ds}=0$, that is, $U$ is a constant vector.
On the other hand, $\langle\hbox{\hbox{\bf B}f N}(s),U\hbox{\hbox{\bf B}b R}angle=1$ and this means that $\alpha$ is a slant helix.
\hbox{\bf B}egin{remark} In Theorem \hbox{\hbox{\bf B}b R}ef{t1} we need to assure that the function $\hbox{\hbox{\bf B}f T}au^2-\kappa^2$ does not vanish everywhere. We do not know that happens if it vanishes at some points. On the other hand, any timelike curve that satisfies $\hbox{\hbox{\bf B}f T}au(s)^2-\kappa(s)^2=0$ is a slant curve. The reasoning is the following. For simplicity, we only consider the case that $\hbox{\hbox{\bf B}f T}au=\kappa$. We define $U=\hbox{\hbox{\bf B}f T}(s)+\hbox{\bf B}(s)$, which is constant using the Frenet equations (\hbox{\hbox{\bf B}b R}ef{equi1}). Moreover,
$\langle \hbox{\hbox{\bf B}f N},U\hbox{\hbox{\bf B}b R}angle=0$, that is, $\alpha$ is a slant curve. Finally, we point that there exist curves in $\hbox{\hbox{\bf B}f E}_1^3$ that satisfies the relation
$\hbox{\hbox{\bf B}f T}au=\kappa$: it suffices to put $\hbox{\hbox{\bf B}f T}au=\kappa:=c=\mbox{constant}$ and the fundamental theorem of the theory of curves assures the existence of a timelike curve $\alpha$ with curvature and torsion $c$.
\hbox{\hbox{\bf B}f E}nd{remark}
\hbox{\hbox{\bf B}b S}ection{Spacelike slant helices}
Let $\alpha$ be a unit speed spacelike curve in $\hbox{\hbox{\bf B}f E}_1^3$. In the case that
the normal vector $\hbox{\hbox{\bf B}f N}(s)$ of $\alpha$ is spacelike or timelike, the proof of Theorem \hbox{\hbox{\bf B}b R}ef{t2} is similar to the one given for Theorem \hbox{\hbox{\bf B}b R}ef{t1}. We omit the details.
The case that remains to study is that the normal vector $\hbox{\hbox{\bf B}f N}(s)$ of the curve is a lightlike vector for any $s\in I$. Now the Frenet trihedron is
$\hbox{\hbox{\bf B}f T}(s)=\alpha'(s)$, $\hbox{\hbox{\bf B}f N}(s)=\hbox{\hbox{\bf B}f T}'(s)$ and $\hbox{\bf B}(s)$ is the unique lightlike vector orthogonal to $\hbox{\hbox{\bf B}f T}(s)$ such that $\langle\hbox{\hbox{\bf B}f N}(s),\hbox{\bf B}(s)\hbox{\hbox{\bf B}b R}angle=1$. Then the Frenet equations as
\hbox{\bf B}egin{equation}\label{u11}
\left[
\hbox{\bf B}egin{array}{c}
\hbox{\hbox{\bf B}f T}' \\
\hbox{\hbox{\bf B}f N}' \\
\hbox{\bf B}'
\hbox{\hbox{\bf B}f E}nd{array}
\hbox{\hbox{\bf B}b R}ight]=\left[
\hbox{\bf B}egin{array}{ccc}
0 & 1 & 0 \\
0 & \hbox{\hbox{\bf B}f T}au & 0 \\
-1 & 0 & \hbox{\hbox{\bf B}f T}au\\
\hbox{\hbox{\bf B}f E}nd{array}
\hbox{\hbox{\bf B}b R}ight]\left[
\hbox{\bf B}egin{array}{c}
\hbox{\hbox{\bf B}f T} \\
\hbox{\hbox{\bf B}f N} \\
\hbox{\bf B} \\
\hbox{\hbox{\bf B}f E}nd{array}
\hbox{\hbox{\bf B}b R}ight].
\hbox{\hbox{\bf B}f E}nd{equation}
Here $\hbox{\hbox{\bf B}f T}au$ is the torsion of the curve (recall that $\hbox{\hbox{\bf B}f T}au(s)\hbox{\hbox{\bf B}f N}ot=0$ for any $s\in I$). We show that \hbox{\hbox{\bf B}f E}mph{any} such curve is a slant helix. Let $a_2(s)$ any non-trivial solution of the O.D.E. $y'(s)+\hbox{\hbox{\bf B}f T}au(s)y(s)=0$ and define $U=a_2(s)\hbox{\hbox{\bf B}f N}(s)$. By using (\hbox{\hbox{\bf B}b R}ef{u11}),
$dU(s)/ds=0$, that is, $U$ is a (non-zero) constant vector field of $\hbox{\hbox{\bf B}f E}_1^3$ and, obviously, the function $\langle\hbox{\hbox{\bf B}f N}(s),U\hbox{\hbox{\bf B}b R}angle$ in constant (and equal to $0$).
\hbox{\hbox{\bf B}b S}ection{Lightlike slant helices }
In this section we show Theorem \hbox{\hbox{\bf B}b R}ef{t3}. Let $\alpha$ be a unit lightlike in $\hbox{\hbox{\bf B}f E}_1^3$. The Frenet frame of $\alpha$ is
$\hbox{\hbox{\bf B}f T}(s)=\alpha'(s)$, $\hbox{\hbox{\bf B}f N}(s)=\hbox{\hbox{\bf B}f T}'(s)$ and $\hbox{\bf B}(s)$ the unique lightlike vector orthogonal to $\hbox{\hbox{\bf B}f N}(s)$ such that
$\langle\hbox{\hbox{\bf B}f T}(s),\hbox{\bf B}(s)\hbox{\hbox{\bf B}b R}angle=1$. The Frenet equations are
\hbox{\bf B}egin{equation}\label{u21}
\left[
\hbox{\bf B}egin{array}{c}
\hbox{\hbox{\bf B}f T}' \\
\hbox{\hbox{\bf B}f N}' \\
\hbox{\bf B}'
\hbox{\hbox{\bf B}f E}nd{array}
\hbox{\hbox{\bf B}b R}ight]=\left[
\hbox{\bf B}egin{array}{ccc}
0 & 1 & 0 \\
\hbox{\hbox{\bf B}f T}au & 0 & -1 \\
0 & -\hbox{\hbox{\bf B}f T}au & 0\\
\hbox{\hbox{\bf B}f E}nd{array}
\hbox{\hbox{\bf B}b R}ight]\left[
\hbox{\bf B}egin{array}{c}
\hbox{\hbox{\bf B}f T} \\
\hbox{\hbox{\bf B}f N} \\
\hbox{\bf B} \\
\hbox{\hbox{\bf B}f E}nd{array}
\hbox{\hbox{\bf B}b R}ight].
\hbox{\hbox{\bf B}f E}nd{equation}
Here $\hbox{\hbox{\bf B}f T}au(s)$ is the torsion of $\alpha$, which is assumed with the property $\hbox{\hbox{\bf B}f T}au(s)\hbox{\hbox{\bf B}f N}ot=0$, for any $s\in I$.
Assume that $\alpha$ is a slant helix. Let $U$ be the constant vector field such that the function $\langle \hbox{\hbox{\bf B}f N}(s),U\hbox{\hbox{\bf B}b R}angle$ is constant.
As in the above cases
$$U=a_1(s)\hbox{\hbox{\bf B}f T}(s)+c \hbox{\hbox{\bf B}f N}(s)+a_3(s) \hbox{\bf B}(s),\ \ s\in I,$$
where $c$ is a constant and
\hbox{\bf B}egin{equation}\label{u23}
\left.\hbox{\bf B}egin{array}{ll}
a_1'+c \hbox{\hbox{\bf B}f T}au&=0\\
a_1-\hbox{\hbox{\bf B}f T}au a_3&=0\\
a_3'-c &=0
\hbox{\hbox{\bf B}f E}nd{array}\hbox{\hbox{\bf B}b R}ight\}
\hbox{\hbox{\bf B}f E}nd{equation}
Then $a_3(s)=cs+m$, $m\in \hbox{\hbox{\bf B}b R}$ and $a_1=(cs+m)\hbox{\hbox{\bf B}f T}au$. Using the first equation of (\hbox{\hbox{\bf B}b R}ef{u23}), we have
$(cs+m)\hbox{\hbox{\bf B}f T}au'+2c\hbox{\hbox{\bf B}f T}au=0$. The solution of this equation is
$$\hbox{\hbox{\bf B}f T}au(s)=\frac{n}{(cs+m)^2},$$
where $m$ and $n$ are constant. This proves (\hbox{\hbox{\bf B}b R}ef{slant5}) in Theorem \hbox{\hbox{\bf B}b R}ef{t3}. Conversely, if the condition (\hbox{\hbox{\bf B}b R}ef{slant5}) is satisfied,
we define
$$U=\frac{a}{bs+c}\hbox{\hbox{\bf B}f T}(s)+b\hbox{\hbox{\bf B}f N}(s)+(bs+c)\hbox{\bf B}(s).$$
Using the Frenet equations (\hbox{\hbox{\bf B}b R}ef{u21}) we obtain that $dU(s)/ds=0$, that is, $U$ is a constant vector field of $\hbox{\hbox{\bf B}f E}_1^3$. Finally, $\langle \hbox{\hbox{\bf B}f N}(s),U\hbox{\hbox{\bf B}b R}angle=b$ and this proves that $\alpha$ is a slant helix.
\hbox{\bf B}egin{thebibliography}{99}
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\hbox{\bf B}ibitem{fgl} A. Ferrandez, A. Gimenez, P. Lucas,
Null helices in Lorentzian space forms,
Int. J. Mod. Phys. A. 16 (2001), 4845--4863.
\hbox{\bf B}ibitem{it} S. Izumiya , N. Takeuchi,
New special curves and developable surfaces,
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\hbox{\bf B}ibitem{ko} H. Kocayi\v{g}it H, M. \"{O}nder, Timelike curves of constant slope in Minkowski space $\hbox{\hbox{\bf B}f E}_1^4$, J. Science Techn. Beykent Univ. 1 (2007), 311--318.
\hbox{\bf B}ibitem{ku} W. Kuhnel, Differential geometry: Curves, Surfaces, Manifolds. Weisbaden: Braunschweig 1999.
\hbox{\bf B}ibitem{ky} L. Kula, Y. Yayli, On slant helix and its spherical indicatrix,
Appl. Math. Comp. 169 (2005), 600--607.
\hbox{\bf B}ibitem{okkk} M. \"{O}nder, M. Kazaz, H. Kocayi\v{g}it, O. Kilic,
$B_2$-slant helix in Euclidean 4-space $E^4$,
Int. J. Cont. Math. Sci. vol. 3, no. 29 (2008), 1433--1440.
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\hbox{\hbox{\bf B}f E}nd{thebibliography}
\hbox{\hbox{\bf B}f E}nd{document} |
\begin{document}
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\parindent=20pt
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\begin{center}
{\bf Kostka functions associated to complex reflection groups}
\end{center}
\par
\begin{center}
Toshiaki Shoji
\\
\end{center}
\title{}
\begin{abstract}
Kostka functions $K^{\pm}_{\Bla, \Bmu}(t)$ associated to complex reflection groups are
a generalization of Kostka polynomials, which are indexed by a pair $\Bla, \Bmu$
of $r$-partitions of $n$ (and by the sign $+, -$). It is expected that there exists a close relationship
between those Kostka functions and the intersection cohomology associated
to the enhanced variety $\SX$ of level $r$.
In this paper, we study combinatorial properties of $K^{\pm}_{\Bla,\Bmu}(t)$ based on
the geometry of $\SX$.
In paticular, we show that in the case where $\Bmu = (-,\dots, -,\mu^{(r)})$
(and for arbitrary $\Bla$), $K^-_{\Bla, \Bmu}(t)$ has a Lascoux-Sch\"utzenberger
type combinatorial description.
\end{abstract}
\maketitle
\pagestyle{myheadings}
\begin{center}
{\sc Introduction}
\end{center}
\par
In 1981, Lusztig gave a geometric interpretation of Kostka polynomials
in the following sense; let $V$ be an $n$-dimensional vector space
over an algebraically closed field, and put $G = GL(V)$. Let $\SP_n$
be the set of partitions of $n$. Let $\SO_{\la}$ be the
unipotent class in $G$ labelled by $\la \in \SP_n$, and $K = \IC(\ol \SO_{\la}, \Ql)$
the intersection cohomology associated to the closure $\ol\SO_{\la}$ of $\SO_{\la}$.
Let $K_{\la,\mu}(t)$ be the Kostka polynomial indexed by $\la, \mu \in \SP_n$, and
$\wt K_{\la,\mu}(t) = t^{n(\mu)}K_{\la,\mu}(t\iv)$ the modified Kostka polynomial
(see 1.1 for the definition $n(\mu)$). Lusztig proved that
\begin{equation*}
\tag{0.1}
\wt K_{\la,\mu}(t) = t^{n(\la)}\sum_{i \ge 0}\dim (\SH^{2i}_xK)t^i
\end{equation*}
for $x \in \SO_{\mu} \subset \ol\SO_{\la}$, where $\SH^{2i}_xK$ is the stalk at
$x$ of the $2i$-th cohomology sheaf $\SH^{2i}K$ of $K$.
(0.1) implies that $K_{\la,\mu}(t) \in \BZ_{\ge 0}[t]$.
\par
Let $\SP_{n,r}$ be the set of $r$-tuple of partitions
$\Bla = (\la^{(1)}, \dots, \la^{(r)})$ such that $\sum_{i=1}^r |\la^{(i)}| = n$
(we write $|\la^{(i)}| = m$ if $\la^{(i)} \in \SP_m$).
In [S1], [S2], Kostka functions $K^{\pm}_{\Bla,\Bmu}(t)$
associated to complex reflections groups (depending on the signs $+, -$) are introduced,
which are apriori rational functions in $t$ indexed by $\Bla, \Bmu \in \SP_{n,r}$.
In the case where $r = 2$ (in this case $K^-_{\Bla,\Bmu}(t) = K^+_{\Bla, \Bmu}(t)$),
it is proved in [S2] that $K^{\pm}_{\Bla, \Bmu}(t) \in \BZ[t]$.
In this case, Achar-Henderson [AH] proved that those (generalized) Kostka polynomials
have a geometric interpretation in the following sense; under the previous notation,
consider the variety $\SX = G \times V$ on which $G$ acts naturally. Put
$\SX\uni = G\uni \times V$, where $G\uni$ is the set of unipotent elements in $G$.
$\SX\uni$ is a $G$-stable subset of $\SX$, and is isomorphic
to the enhanced nilpotent cone introduced by [AH].
It is known by [AH], [T] that $\SX\uni$ has finitely many $G$-orbits, which are
naturally parametrized by $\SP_{n,2}$.
They proved in [AH] that the modified Kostka polynomial $\wt K^{\pm}_{\Bla, \Bmu}(t)$
$(\Bla, \Bmu \in \SP_{n,2}$), defined
in a similar way as in the original case,
can be written as in (0.1) in terms of the intersection cohomology
associated to the closure $\ol\SO_{\Bla}$ of the $G$-orbit $\SO_{\Bla} \subset \SX\uni$.
\par
In the case where $r = 2$, the interaction of geometric properties and combinatorial
properties of Kostka polynomials was studied in [LS].
In particular, it was proved that in the special case where $\Bmu = (-,\mu^{(2)})$
(and for arbitrary $\Bla \in \SP_{n,2}$), $K_{\Bla,\Bmu}(t)$ has a combinatorial description
analogous to Lascoux-Sch\"utzenberger theorem for the original Kostka polynomials
([M, III, (6.5)]).
\par
We now consider the variety $\SX = G \times V^{r-1}$ for an integer $r \ge 1$, on which $G$
acts diagonally, and let $\SX\uni = G\uni \times V^{r-1}$ be the $G$-stable subset of $\SX$.
The variety $\SX$ is called the enhanced variety of level $r$.
In [S4], the relationship between Kostka functions
$K^{\pm}_{\Bla, \Bmu}(t)$ indexed by
$\Bla, \Bmu \in \SP_{n,r}$ and the geometry of $\SX\uni$ was studied.
In contrast to the case where $r = 1,2$, $\SX\uni$ has infinitely many $G$-orbits if
$r \ge 3$. A partition $\SX\uni = \coprod_{\Bla \in \SP_{n,r}}X_{\Bla}$ into
$G$-stable pieces $X_{\Bla}$ was constructed in [S3],
and some formulas expressing the Kostka functions
in terms of the intersection cohomology associated to the closure of $X_{\Bla}$
were obtained in [S4],
though it is a partial generalization of the result of Achar-Henderson for the case $r = 2$.
\par
In this paper, we prove a formula (Theorem 2.6) which is a generalization
of the formula in [AH, Theorem 4.5] (and also in [FGT (11)]) to arbitrary $r$.
Combined this formula with the results in [S4],
we extend some results in [LS] to arbitrary $r$. In particular, we show
in the special case where $\Bmu = (-,\dots,-,\mu^{(r)}) \in \SP_{n,r}$
(and for arbitrary $\Bla \in \SP_{n,r}$) that $K^{-}_{\Bla,\Bmu}(t)$ has
a Lasacoux-Sch\"utzenberger type combinatorial description.
\par
\par
\section{Review on Kostka functions}
\para{1.1.}
First we recall basic properties of Hall-Littlewood functions and Kostka polynomials
in the original setting, following [M].
Let $\vL = \vL(y) = \bigoplus_{n \ge 0}\vL^n$ be the ring of symmetric functions
over $\BZ$ with respect to the variables $y = (y_1, y_2, \dots)$, where $\vL^n$ denotes the
free $\BZ$-module of symmetirc functions of degree $n$.
We put $\vL_{\BQ} = \BQ\otimes_{\BZ}\vL$, $\vL^n_{\BQ} = \BQ\otimes_{\BZ}\vL^n$.
Let $s_{\la}$ be the Schur function associated to $\la \in \SP_n$. Then
$\{ s_{\la} \mid \la \in \SP_n \}$ gives a $\BZ$-baisis of $\vL^n$.
Let $p_{\la} \in \vL^n$ be the power sum symmetric function associated to
$\la \in \SP_n$,
\begin{equation*}
p_{\la} = \prod_{i = 1}^kp_{\la_i},
\end{equation*}
where $p_m$ denotes the $m$-th power sum symmetric function for each integer $m > 0$.
Then $\{ p_{\la} \mid \la \in \SP_n \}$ gives a $\BQ$-basis of
$\vL^n_{\BQ}$.
For $\la = (1^{m_1}, 2^{m_2}, \dots) \in \SP_n$, define an integer $z_{\la}$ by
\begin{equation*}
\tag{1.1.1}
z_{\la} = \prod_{i \ge 1}i^{m_i}m_i!.
\end{equation*}
Following [M, I], we introduce a scalar product on $\vL_{\BQ}$ by
$\lp p_{\la}, p_{\mu} \rp = \d_{\la\mu}z_{\la}$.
It is known that $\{s_{\la}\}$ form an orthonormal basis of $\vL$.
\par
Let $P_{\la}(y;t)$ be the Hall-Littlewood function associated to a partition $\la$.
Then $\{ P_{\la} \mid \la \in \SP_n \}$ gives a $\BZ[t]$-basis of
$\vL^n[t] = \BZ[t]\otimes_{\BZ}\vL^n$, where $t$ is an indeterminate.
Kostka polynomials $K_{\la, \mu}(t) \in \BZ[t]$ ($\la, \mu \in \SP_n$) are defined by the formula
\begin{equation*}
\tag{1.1.2}
s_{\la}(y) = \sum_{\mu \in \CP_n}K_{\la,\mu}(t)P_{\mu}(y;t).
\end{equation*}
\par
Recall the dominance order $\la \ge \mu$ in $\SP_n$, which is defined by the condition
$\sum_{j= 1}^i\la_j \ge \sum_{j = 1}^i\mu_j$ for
each $i \ge 1$.
For each partition $\la = (\la_1, \dots, \la_k)$, we define an integer
$n(\la)$ by $n(\la) = \sum_{i=1}^k(i-1)\la_i$.
It is known that $K_{\la,\mu}(t) = 0$ unless $\la \ge \mu$, and that
$K_{\la,\mu}(t)$ is a monic of degree $n(\mu) - n(\la)$ if $\la \ge \mu$
([M, III, (6.5)]).
Put $\wt K_{\la,\mu}(t) = t^{n(\mu)}K_{\la, \mu}(t\iv)$. Then
$\wt K_{\la, \mu}(t) \in \BZ[t]$, which we call the modified Kostka polynomial.
\par
For $\la = (\la_1, \dots, \la_k) \in \SP_n$ with $\la_k > 0$,
we define $z_{\la}(t) \in \BQ(t)$ by
\begin{equation*}
\tag{1.1.3}
z_{\la}(t) = z_{\la}\prod_{i \ge 1}(1 - t^{\la_i})\iv,
\end{equation*}
where $z_{\la}$ is as in (1.1.1).
Following [M, III], we introduce a scalar product on $\vL_{\BQ}(t) = \BQ(t)\otimes_{\BZ}\vL$ by
$\lp p_{\la}, p_{\mu} \rp = z_{\la}(t)\d_{\la,\mu}$.
Then $P_{\la}(y;t)$ form an orthogonal basis of $\vL[t] = \BZ[t]\otimes_{\BZ}\vL$.
In fact, they are characterized by the following two properties
([M, III, (2.6) and (4.9)]);
\begin{equation*}
\tag{1.1.4}
P_{\la}(y;t) = s_{\la}(y) + \sum_{\mu < \la}w_{\la\mu}(t)s_{\mu}(y)
\end{equation*}
with $w_{\la\mu}(t) \in \BZ[t]$ , and
\begin{equation*}
\tag{1.1.5}
\lp P_{\la}, P_{\mu} \rp = 0 \text{ unless $\la = \mu$. }
\end{equation*}
\para{1.2.}
We fix a positive integer $r$.
Let
$\Xi = \Xi(x) \simeq \vL(x^{(1)})\otimes\cdots\otimes\vL(x^{(r)})$ be
the ring of symmetric functions over $\BZ$
with respect to variables $x = (x^{(1)}, \dots, x^{(r)})$, where
$x^{(i)} = (x^{(i)}_1, x^{(i)}_2, \dots)$.
We denote it as $\Xi = \bigoplus_{n \ge 0}\Xi^n$, similarly to the case of $\vL$.
Let $\SP_{n,r}$ be as in Introduction.
For $\Bla \in \SP_{n,r}$, we define a Schur function
$s_{\Bla}(x) \in \Xi^n$ by
\begin{equation*}
\tag{1.2.1}
s_{\Bla}(x) = s_{\la^{(1)}}(x^{(1)})\cdots s_{\la^{(r)}}(x^{(r)}).
\end{equation*}
Then
$\{ s_{\Bla} \mid \Bla \in \SP_{n,r} \}$ gives a $\BZ$-basis of $\Xi^n$.
Let $\z$ be a primitive $r$-th root of unity in $\BC$.
For an integer $m \ge 1$ and $k$ such that $1 \le k \le r$, put
\begin{equation*}
p_m^{(k)}(x) = \sum_{j = 1}^{r}\z^{(k-1)(j-1)}p_m(x^{(j)}),
\end{equation*}
where $p_m(x^{(j)})$ denotes the $m$-th power sum symmetric function with respect to
the variables $x^{(j)}$.
For $\Bla \in \SP_{n,r}$, we define $p_{\Bla}(x) \in \Xi^n_{\BC} = \Xi^n \otimes_{\BZ}\BC$ by
\begin{equation*}
\tag{1.2.2}
p_{\Bla}(x) = \prod_{k = 1}^r\prod_{j= 1}^{m_k}p^{(k)}_{\la^{(k)}_j}(x),
\end{equation*}
where $\la^{(k)} = (\la^{(k)}_1, \dots, \la^{(k)}_{m_k})$ with $\la^{(k)}_{m_k} > 0$.
Then $\{ p_{\Bla} \mid \Bla \in \SP_{n,r} \}$ gives a $\BC$-basis of
$\Xi^n_{\BC}$.
For a partition $\la^{(k)}$ as above,
we define a function
$z_{\la^{(k)}}(t) \in \BC(t)$ by
\begin{equation*}
z_{\la^{(k)}}(t) = \prod_{j = 1}^{m_k}(1 - \z^{k-1}t^{\la_j^{(k)}})\iv.
\end{equation*}
For $\Bla \in \SP_{n,r}$,
we define an integer $z_{\Bla}$ by $z_{\Bla} = \prod_{k=1}^rr^{m_k}z_{\la^{(k)}}$,
wher $z_{\la^{(k)}}$ is as in (1.1.1).
We now define a function $z_{\Bla}(t) \in \BC(t)$ by
\begin{equation*}
\tag{1.2.3}
z_{\Bla}(t) = z_{\Bla}\prod_{k=1}^rz_{{\la}^{(k)}}(t).
\end{equation*}
Let $\Xi[t] = \BZ[t]\otimes_{\BZ}\Xi$ be the free $\BZ[t]$-module,
and $\Xi_{\BC}(t) = \BC(t)\otimes_{\BZ}\Xi$ be the $\BC(t)$-space.
Then $\{ p_{\Bla}(x) \mid \Bla \in \SP_{n,r} \}$ gives a basis of $\Xi^n_{\BC}(t)$.
We define a sesquilinear form on $\Xi_{\BC}(t)$ by
\begin{equation*}
\tag{1.2.4}
\lp p_{\Bla}, p_{\Bmu} \rp = \d_{\Bla,\Bmu}z_{\Bla}(t).
\end{equation*}
\par
We express an $r$-partition $\Bla = (\la^{(1)}, \dots, \la^{(r)})$ as
$\la^{(k)} = (\la^{(k)}_1, \dots, \la^{(k)}_m)$ with a common $m$,
by allowing zero on parts $\la^{(i)}_j$, and define a composition
$c(\Bla)$ of $n$ by
\begin{equation*}
c(\Bla) = (\la^{(1)}_1, \dots, \la^{(r)}_1, \la^{(1)}_2, \dots, \la^{(r)}_2,
\dots, \la^{(1)}_m, \dots, \la^{(r)}_m).
\end{equation*}
We define a partial order $\Bla \ge \Bmu$ on $\SP_{n,r}$ by the condition
$c(\Bla) \ge c(\Bmu)$, where $\ge $ is the dominance order on the set of
compositions of $n$ defined in a similar way as in the case of partitions.
We fix a total order $\Bla \gv \Bmu$ on $\SP_{n,r}$ compatible with the
partial order $\Bla > \Bmu$.
\par
The following result was proved in Theorem 4.4 and Proposition 4.8 in [S1],
combined with [S2, \S 3].
\begin{prop}
For each $\Bla \in \SP_{n,r}$, there exist unique functions
$P^{\pm}_{\Bla}(x;t) \in \Xi^n_{\BQ}(t)$ $($depending on the signs $+$, $-$ $)$
satisfying the following properties.
\begin{enumerate}
\item
$P^{\pm}_{\Bla}(x;t)$ can be written as
\begin{equation*}
P^{\pm}_{\Bla}(x;t) = s_{\Bla}(x) + \sum_{\Bmu \lv \Bla}u^{\pm}_{\Bla,\Bmu}(t)s_{\Bmu}(x)
\end{equation*}
with $u^{\pm}_{\Bla,\Bmu}(t) \in \BQ(t)$.
\item
$\lp P_{\Bla}^-, P^+_{\Bmu} \rp = 0$ unless $\Bla = \Bmu$.
\end{enumerate}
\end{prop}
\para{1.4.}
$P^{\pm}_{\Bla}(x;t)$ are called Hall-Littlewood functions associated to
$\Bla \in \SP_{n,r}$.
By Proposition 1.3, for $\ve \in \{ +,-\}$, $\{ P^{\ve}_{\Bla} \mid \Bla \in \SP_{n,r}\}$
gives
a $\BQ(t)$-basis for $\Xi_{\BQ}(t)$.
For $\Bla, \Bmu \in \SP_{n,r}$, we define functions $K^{\pm}_{\Bla, \Bmu}(t) \in \BQ(t)$ by
\begin{equation*}
\tag{1.4.1}
s_{\Bla}(x) = \sum_{\Bmu \in \SP_{n,r}}K^{\pm}_{\Bla,\Bmu}(t)P^{\pm}_{\Bmu}(x;t).
\end{equation*}
\par
$K^{\pm}_{\Bla, \Bmu}(t)$ are called Kostka functions associated to complex
reflection groups since they are closely related to the complex reflection group
$S_n\ltimes (\BZ/r\BZ)^n$ (see [S1, Theorem 5,4]).
For each $\Bla \in \SP_{n,r}$, by putting
$n(\Bla) = n(\la^{(1)}) + \cdots + n(\la^{(r)})$, we define an $a$-function
$a(\Bla)$ on $\SP_{n,r}$ by
\begin{equation*}
\tag{1.4.2}
a(\Bla) = r\cdot n(\Bla) + |\la^{(2)}| + 2|\la^{(3)}| + \cdots + (r-1)|\la^{(r)}|.
\end{equation*}
We define modifed Kostka functions
$\wt K^{\pm}_{\Bla, \Bmu}(t)$ by
\begin{equation*}
\tag{1.4.3}
\wt K^{\pm}_{\Bla, \Bmu}(t) = t^{a(\Bmu)}K^{\pm}_{\Bla, \Bmu}(t\iv).
\end{equation*}
\remark{1.5.}
In the case where $r = 1$, $P^{\pm}_{\Bla}(x;t)$ coincides with the original
Hall-Littlewood function given in 1.1. In the case where $r = 2$,
it is proved by [S2, Prop. 3.3] that $P^-_{\Bla}(x;t) = P^+_{\Bla}(x;t) \in \Xi[t]$,
hence $K^-_{\Bla,\Bmu}(t) = K^+_{\Bla, \Bmu}(t) \in \BZ[t]$.
Moreover it is shown that $K^{\pm}_{\Bla, \Bmu}(t) \in \BZ[t]$,
which is a monic of degree $a(\Bmu) - a(\Bla)$. Thus $\wt K^{\pm}_{\Bla, \Bmu}(t) \in \BZ[t]$.
As mentioned in Introduction $\wt K^{\pm}_{\Bla, \Bmu}(t)$ has a geometric
interpretation, which imples that $K^{\pm}_{\Bla, \Bmu}(t)$, and so
$P^{\pm}_{\Bla}(x;t)$ are independent of the choice of the total order $\lv$ on $\SP_{n,r}$.
In the case where $r \ge 3$, it is not known whether Hall-Littlewood functions do not
depend on the choice of the total order $\lv$,
whether $K^{\pm}_{\Bla, \Bmu}(t)$ are polynomials in $t$.
\par
\section{Enhanced variety of level $r$ }
\para{2.1.}
Let $V$ be an $n$-dimensional vector space over an algebraic closure $\Bk$ of a finite
field $\Fq$,
and $G = GL(V) \simeq GL_n$.
Let $B = TU$ be a Borel subgroup of $G$, $T$ a maximal torus and $U$ the unipotent radical
of $B$. Let $W = N_G(T)/T$ be the Weyl group of $G$, which is isomorphic to the symmetric
group $S_n$.
By fixing an integer $r \ge 1$, put $\SX = G \times V^{r-1}$
and $\SX\uni = G\uni \times V^{r-1}$, where $G\uni$ is the set of unipotent elements in $G$.
The variety $\SX$ is called the enhanced variety of level $r$.
We consider the diagonal action of $G$ on $\SX$.
Put $\SQ_{n,r} = \{ \Bm = (m_1, \dots, m_r) \in \BZ^r_{\ge 0} \mid \sum m_i = n\}$.
For each $\Bm \in \SQ_{n,r}$,
we define integers $p_i = p_i(\Bm)$ by $p_i = m_1 + \cdots + m_i$ for $i = 1, \dots, r$.
Let $(M_i)_{1 \le i \le n}$ be the total flag in $V$ whose stabilizer in $G$
coincides with $B$. We define varieties
\begin{align*}
\wt\SX_{\Bm} &= \{ (x, \Bv, gB) \in G \times V^{r-1} \times G/B \mid g\iv xg \in B,
g\iv \Bv \in \prod_{i=1}^{r-1}M_{p_i} \}, \\
\SX_{\Bm} &= \bigcup_{g \in G}g(B \times \prod_{i=1}^{r-1}M_{p_i}),
\end{align*}
and the map $\pi_{\Bm} : \wt \SX_{\Bm} \to \SX_{\Bm}$ by $(x,\Bv, gB) \mapsto (x,\Bv)$.
We also define the varieties
\begin{align*}
\wt\SX_{\Bm, \unip} &= \{ (x, \Bv, gB) \in G\uni \times V^{r-1} \times G/B \mid g\iv xg \in U,
g\iv \Bv \in \prod_{i=1}^{r-1}M_{p_i} \}, \\
\SX_{\Bm} &= \bigcup_{g \in G}g(U \times \prod_{i=1}^{r-1}M_{p_i}),
\end{align*}
and the map $\pi_{\Bm,1}: \wt\SX_{\Bm,\unip} \to \SX_{\Bm,\unip}$, similarly.
Note that in the case where $\Bm = (n,0,\dots, 0)$, $\SX_{\Bm}$ (resp. $\SX_{\Bm,\unip}$)
coincides with $\SX$ (resp. $\SX\uni$).
In that case, we denote $\wt\SX_{\Bm}, \pi_{\Bm}$, etc.
by $\wt\SX, \pi$, etc. by omitting the symbol $\Bm$.
(Note: here we follow the notation in [S4], but, in part, it differs from [S3].
In [S3], our $\pi_{\Bm}, \pi_{\Bm,1}$ are denoted by $\pi^{(\Bm)}, \pi^{(\Bm)}_1$
for the consistency with the exotic case).
\para{2.2.}
In [S3, 5.3], a partition of $\SX\uni$ into pieces $X_{\Bla}$ is defined
\begin{equation*}
\SX\uni = \coprod_{\Bla \in \SP_{n,r}}X_{\Bla},
\end{equation*}
where $X_{\Bla}$ is a locally closed, smooth irreducible, $G$-stable subvariety
of $\SX\uni$. If $r = 1$ or 2, $X_{\Bla}$ is a single $G$-orbit. However, if
$r \ge 3$, $X_{\Bla}$ is in general a union of infinitely many $G$-orbits.
\par
For $\Bm \in \SQ_{n,r}$, let $W_{\Bm} = S_{m_1} \times \cdots \times S_{m_r}$
be the Young subgroup of $W = S_n$.
For $\Bm \in \SQ_{n,r}$, we denote by $\SP(\Bm)$ the set of $\Bla \in \SP_{n,r}$
such that $|\la^{(i)}| = m_i$.
The (isomorphism classes of)
irreducible representations (over $\Ql$) of $W_{\Bm}$ are parametrized by $\SP(\Bm)$.
We denote by $V_{\Bla}$ an irreducible representation of $W_{\Bm}$ corresponding to
$\Bla$, namley $V_{\Bla} = V_{\la^{(1)}}\otimes\cdots\otimes V_{\la^{(r)}}$,
where $V_{\mu}$ denotes the irreducible representation of $S_n$ corresponding to
the partition $\mu$ of $n$. (Here we use the parametrization such that $V_{(n)}$ is the
trivial representation of $S_n$).
The following results were proved in [S3].
\begin{thm}[{[S3, Thm. 4.5]}]
Put $d_{\Bm} = \dim \SX_{\Bm}$. Then $(\pi_{\Bm})_*\Ql[d_{\Bm}]$ is a
semsimple perverse sheaf equipped with the action of $W_{\Bm}$, and is decomposed as
\begin{equation*}
(\pi_{\Bm})_*\Ql[d_{\Bm}] \simeq \bigoplus_{\Bla \in \SP(\Bm)}
V_{\Bla} \otimes \IC(\SX_{\Bm}, \SL_{\Bla})[d_{\Bm}],
\end{equation*}
where $\SL_{\Bla}$ is a simple local system on a certain open dense subvariety of $\SX_{\Bm}$.
\end{thm}
\begin{thm}[{[S3, Thm. 8.13, Thm. 7.12]}]
Put $d'_{\Bm} = \dim \SX_{\Bm,\unip}$.
\begin{enumerate}
\item
$(\pi_{\Bm,1})_*\Ql[d'_{\Bm}]$
is a semisimple perverse sheaf equipped with the action of $W_{\Bm}$, and
is decomposed as
\begin{equation*}
(\pi_{\Bm,1})_*\Ql[d'_{\Bm}] \simeq \bigoplus_{\Bla \in \SP(\Bm)}
V_{\Bla} \otimes \IC(\ol X_{\Bla}, \Ql)[\dim X_{\Bla}].
\end{equation*}
\item
We have
$\IC(\SX_{\Bm}, \SL_{\la})|_{\SX_{\Bm, \unip}} \simeq
\IC(\ol X_{\Bla}, \Ql)[\dim X_{\Bla} - d'_{\Bm}]$.
\end{enumerate}
\end{thm}
\para{2.5.}
For a partition $\la$, we denote by $\la^t$ the dual partition of $\la$.
For $\Bla = (\la^{(1)}, \dots, \la^{(r)}) \in \SP(\Bm)$, we define
$\Bla^t \in \SP(\Bm)$ by
$\Bla^t = ((\la^{(1)})^t, \dots, (\la^{(r)})^t)$.
Assume that $\Bla \in \CP(\Bm)$.
We write $(\la^{(i)})^t$
as ($\mu^{(i)}_1 \le \mu^{(i)}_2 \le \cdots \le \mu^{(i)}_{\ell_i})$,
in the increasing order, where $\ell_i = \la^{(i)}_1$.
For each $1 \le i \le r, 1\le j < \ell_i$, we define an integer $n(i,j)$ by
\begin{equation*}
n(i,j) = (|\la^{(1)}| + \cdots + |\la^{(i-1)}|) + \mu_1^{(i)} + \cdots + \mu_j^{(i)}.
\end{equation*}
Let $Q = Q_{\Bla}$ be the stabilizer of the partial flag $(M_{n(i,j)})$ in $G$, and $U_Q$
the unipotent radical of $Q$.
In particular, $Q$ stabilizes the subspaces $M_{p_i}$.
Let us define a variety $\wt X_{\Bla}$ by
\begin{equation*}
\begin{split}
\wt X_{\Bla} = \{ (x, \Bv, gQ) \in G\uni \times V^{r-1} \times G/Q
\mid g\iv xg \in U_Q, g\iv \Bv \in \prod_{i=1}^{r-1}M_{p_i} \}.
\end{split}
\end{equation*}
We define a map $\pi_{\Bla} : \wt X_{\Bla} \to \CX\uni$ by
$(x,\Bv, gQ) \mapsto (x,\Bv)$. Then $\pi_{\Bla}$ is a proper map.
Since
$\wt X_{\Bla} \simeq G\times^{Q}(U_{Q}
\times \prod_i M_{p_i})$,
$\wt X_{\Bla}$ is smooth and irreducible.
It is known by [S3, Lemma 5.6] that $\dim \wt X_{\Bla} = \dim X_{\la}$
and that $\Im \pi_{\Bla}$ coincides with $\ol X_{\Bla}$, the closure of
$X_{\Bla}$ in $\SX\uni$.
\par
For $\la, \mu \in \SP_n$, let $K_{\la, \mu} = K_{\la,\mu}(1)$ be the Kostka number.
We have $K_{\la,\mu} = 0$ unless $\la \ge \mu$.
For $\Bla = (\la^{(1)}, \dots, \la^{(r)})$,
$\Bmu = (\mu^{(1)}, \dots, \mu^{(r)}) \in \SP(\Bm)$, we define an integer
$K_{\Bla, \Bmu}$ by
\begin{equation*}
K_{\Bla,\Bmu} = K_{\la^{(1)}, \mu^{(1)}}K_{\la^{(2)},\mu^{(2)}}
\cdots K_{\la^{(r)}, \mu^{(r)}}.
\end{equation*}
We define a partial order $\Bla \trreq \Bmu$ on $\SP_{n,r}$ by the condition
$\la^{(i)} \ge \mu^{(i)}$ for $i = 1, \dots, r$.
Hence $\Bla \trreq \Bmu$ implies that $\Bla, \Bmu \in \SP(\Bm)$ for a commom $\Bm$.
We have $K_{\Bla, \Bmu} = 0$
unless $\Bla \trreq \Bmu$.
Note that $\Bla \trreq \Bmu$ implies that $\Bmu^t \trreq \Bla^t$.
We show the following theorem. In the case where
$r = 2$, this result was proved by [AH, Thm. 4.5].
\begin{thm}
Assume that $\Bla \in \SP_{n,r}$. Then $(\pi_{\Bla})_*\Ql[\dim X_{\Bla}]$
is a semisimple perverse sheaf on $\ol X_{\Bla}$, and is decomposed as
\begin{equation*}
\tag{2.6.1}
(\pi_{\Bla})_*\Ql[\dim X_{\Bla}] \simeq \bigoplus_{\Bmu \trleq \Bla}
\Ql^{K_{\Bmu^t, \Bla^t}}\otimes \IC(\ol X_{\Bmu}, \Ql)[\dim X_{\Bmu}].
\end{equation*}
\end{thm}
\para{2.7.}
The rest of this section is devoted to the proof of Theorem 2.6.
First we consider the case where $r = 1$. Actually, the result in this case
is contained in [AH]. Their proof (for $r = 2$) depends on the result of Spaltenstein [Sp]
concerning the ``Springer fibre'' $(\pi_{\Bla})\iv(z)$ for $z \in \ol X_{\Bla}$
in the case $r = 1$. In the following, we give an alternate proof independent of [Sp]
for the later use.
Let $Q$ be a parabolic subgroup of $G$ containing $B$, $M$ the Levi
subgroup of $Q$ containing $T$ and $U_Q$ the unipotent radical of $Q$.
(In this stage, this $Q$ is independent of $Q$ in 2.5.)
Let $W_Q$ be the Weyl subgroup of
$W$ corresponding to $Q$.
Let $G\reg$ be the set of regular semisimple elements in $G$, and put $T\reg = G\reg \cap T$.
Consider the map $\psi: \wt G\reg \to G\reg$, where
\begin{align*}
\wt G\reg = \{ (x, gT) \in G\reg \times G/T \mid g\iv xg \in T\reg \} \\
\end{align*}
and $\psi : (x, gT) \mapsto x$.
Then $\psi$ is a finite Galois covering with group $W $.
We also consider a variety
\begin{align*}
\wt G\reg^M &= \{ (x, gM) \in G\reg \times G/M \mid g\iv xg \in M\reg \},
\end{align*}
where $M\reg = G\reg \cap M$.
The map $\psi$ is decomposed as
\begin{equation*}
\begin{CD}
\psi : \wt G\reg @>\psi' >> \wt G\reg^M @> \psi''>> G\reg,
\end{CD}
\end{equation*}
\par\noindent
where $\psi': (x, gT) \mapsto (x, gM)$, $\psi'': (x, gM) \mapsto x$.
Here $\psi'$ is a finite Galois covering with group $W_Q$.
Now $\psi_*\Ql$ is a semisimple local system on $G\reg$ such that
$\End (\psi_*\Ql) \simeq \Ql[W]$, and
is decomposed as
\begin{equation*}
\tag{2.7.1}
\psi_*\Ql \simeq \bigoplus_{\r \in W\wg} \r \otimes \SL_{\r},
\end{equation*}
where $\SL_{\r} = \Hom_W(\r, \psi_*\Ql)$ is a simple local system on $G\reg$.
We also have
\begin{equation*}
\tag{2.7.2}
\psi'_*\Ql \simeq \bigoplus_{\r' \in W_Q\wg}\r' \otimes \SL'_{\r'},
\end{equation*}
where $\SL'_{\r'}$ is a simple local system on $\wt G\reg^M$.
Hence
\begin{equation*}
\tag{2.7.3}
\psi_*\Ql \simeq \psi_*''\psi_*'\Ql \simeq \bigoplus_{\r' \in W_Q\wg}
\r'\otimes \psi''_*\SL'_{\r'}.
\end{equation*}
(2.7.3) gives a decompostion of $\psi_*\Ql$ with respect to the action of $W_Q$.
Comparing (2.7.1) and (2.7.3), we have
\begin{equation*}
\tag{2.7.4}
\psi''_*\SL'_{\r'} \simeq \bigoplus_{\r \in W\wg}\Ql^{(\r: \r')}\otimes \SL_{\r},
\end{equation*}
where $(\r: \r')$ is the multiplicity of $\r'$ in the restricted
$W_Q$-module $\r$.
\par
We consider the map $\pi : \wt G \to G$, where
\begin{equation*}
\wt G = \{ (x, gB) \in G \times G/B \mid g\iv xg \in B \} \simeq G \times^BB ,
\end{equation*}
and $\pi: (x, gB) \mapsto x$. We also consider
\begin{align*}
\wt G^Q = \{ (x, gQ) \in G \times G/Q \mid g\iv xg \in Q \} \simeq G \times^QQ.
\end{align*}
The map $\pi$ is decomposed as
\begin{equation*}
\begin{CD}
\pi: \wt G @>\pi'>> \wt G^Q @>\pi''>> G,
\end{CD}
\end{equation*}
where $\pi': (x, gB) \mapsto (x, gQ)$, $\pi'': (x, gQ) \mapsto x$.
It is well-known ([L1]) that
\begin{equation*}
\tag{2.7.5}
\pi_*\Ql \simeq \bigoplus_{\r \in W\wg} \r \otimes \IC(G, \SL_{\r}).
\end{equation*}
Let $B_M = B \cap M$ be the Borel subgroup of $M$ containing $T$.
We consider the following commutative diagram
\begin{equation*}
\tag{2.7.6}
\begin{CD}
G \times ^BB @<\wt p<< G \times (Q\times^BB) @>\wt q>> M \times^{B_M}B_M \\
@V\pi'VV @VVr V @VV\pi^M V \\
G \times^QQ @<p<< G \times Q @>q>> M ,
\end{CD}
\end{equation*}
where under the identification $G \times^BB \simeq G \times^Q(Q \times^BB)$,
the maps $p,\wt p$ are defined by the quotient by $Q$. The map $q$ is
a projection to the $M$-factor of $Q$, and $\wt q$ is the map induced
from the projection $Q \times B \to M \times B_M$.
$\pi^M$ is defined similarly to $\pi$ replacing $G$ by $M$. The map $r$
is defined by $(g, h*x) \mapsto (g, hxh\iv)$.
(We use the notation $h*x \in Q\times^BB$ to denote the $B$-orbit in $Q \times B$
containing $(h,x)$.)
Here all the squares are cartesian squares. Moreover,
\par
(a) $p$ is a principal $Q$-bundle.
\par
(b) $q$ is a locally trivial fibration with fibre isomorphic to $G \times U_Q$.
\par
\noindent
Thus as in [S4, (1.5.2)], for any $M$-equivariant simple pervere sheaf $A_1$
on $M$, there exists a unique (up to isomorphism) simple perverse sheaf $A_2$
on $\wt G^Q$ such that $p^*A_2[a] \simeq q^*A_1[b]$, where
$a = \dim Q$ and $b = \dim G + \dim U_Q$.
\par
By using the cartesian squares in (2.7.6), and by (2.7.2),
we see that $\pi'_*\Ql \simeq \IC(\wt G^Q, \psi'_*\Ql)$, and $\pi'_*\Ql$ is decomposed as
\begin{equation*}
\tag{2.7.7}
\pi'_*\Ql \simeq \bigoplus_{\r' \in W_Q\wg}\r' \otimes \IC(\wt G^Q, \SL'_{\r'}).
\end{equation*}
By comparing (2.7.4) and (2.7.7), we have
\begin{equation*}
\tag{2.7.8}
\pi''_*\IC(\wt G^Q, \SL'_{\r'}) \simeq \bigoplus_{\r \in W\wg}\Ql^{(\r:\r')}\otimes
\IC(G,\SL_{\r}).
\end{equation*}
Note that if $\r = V_{\la}$ for $\la \in \SP_n$, we have
\begin{equation*}
\tag{2.7.9}
\IC(G, \SL_{\r})|_{G\uni} \simeq \IC(\ol\SO_{\la}, \Ql)[\dim \SO_{\la} - 2\nu_G]
\end{equation*}
by [BM], where $\nu_G = \dim U$.
Hence by restricting on $G\uni$, we have
\begin{equation*}
\tag{2.7.10}
\pi''_*\IC(\wt G^Q, \SL'_{\r'})[2\nu_G]|_{G\uni}
\simeq \bigoplus_{\la \in \SP_n}\Ql^{(V_{\la} : \r')}\otimes
\IC(\ol\SO_{\la}, \Ql)[\dim \SO_{\la}].
\end{equation*}
\para{2.8.}
Now assume that
$W_Q \simeq S_{\mu}$ for a partition $\mu$, where we put
$S_{\mu} = S_{\mu_1} \times \cdots \times S_{\mu_k}$ if
$\mu = (\mu_1, \dots, \mu_k) \in \SP_n$.
Take $\r' = \ve$ the sign representation of $W_Q$.
We have
\begin{equation*}
\tag{2.8.1}
(V_{\la} : \ve) = (V_{\la^t} : 1_{W_Q}) = K_{\la^t,\mu},
\end{equation*}
where $1_{W_Q}$ is the trivial representation of $W_Q$.
\par
The restriction of the diagram (2.7.6) to the ``unipotent parts'' makes sense, and
we have the commutative diagram
\begin{equation*}
\tag{2.8.2}
\begin{CD}
G \times^BU @<<< G \times^Q(Q \times^BU) @>>> M \times^{B_M}U_M \\
@VVV @VVV @VVV \\
G \times^QQ\uni @<p_1<< G \times Q\uni @>q_1>> M\uni,
\end{CD}
\end{equation*}
where $U_M$ is the unipotent radical of $B_M$, and $Q\uni, M\uni$ are the set of
unipotent elements in $Q, M$, respectively.
$p_1, q_1$ have similar properties as (a), (b) in 2.7.
We consider $\IC(M, \SL^M_{\ve})$ on $M$, where $\SL^M_{\ve}$ is the simple local
system on $M\reg$ corresponding to
$\ve \in W\wg_Q$. Then by (2.7.6), we see that
\begin{equation*}
p^*\IC(\wt G^Q, \SL'_{\ve}) \simeq q^*\IC(M, \SL^M_{\ve}).
\end{equation*}
By applying (2.7.9) to $M$,
$\IC(M, \SL^M_{\ve})|_{M\uni} \simeq \IC(\ol\SO'_{\ve}, \Ql)[\dim \SO'_{\ve} - 2\nu_M]$,
where $\SO'_{\ve}$ is the orbit in $M\uni$ corresponding to
$\ve$ under the Springer correspondence, and $\nu_M$ is defined similarly to $\nu_G$.
It is known that $\SO'_{\ve}$ is the orbit $\{ e \} \subset M\uni$, where $e$ is the identity element
in $M$. Hence $\IC(M, \SL^M_{\ve})|_{M\uni}$ coincides with $\Ql[-2\nu_M]$ supported on
$\{e\}$.
It follows, by (2.8.2)
\par
\noindent
(2.8.3) \ The restriction of $\IC(\wt G^Q, \SL'_{\ve})$ on $G\times^QQ\uni$
coincides with $i_*\Ql[-2\nu_M]$, where $i: G \times^QU_Q \hra G \times^QQ\uni$ is
the closed embedding.
\par
We deifne a map $\pi_Q : G\times^QU_Q \to G\uni$ by $g*x \mapsto gxg\iv$.
Put $\wt G^Q_1 = G\times^QU_Q$.
\begin{prop}
Under the notation as above,
\begin{enumerate}
\item
$\pi''_*\IC(\wt G^Q, \SL'_{\ve})[2\nu_G]|_{G\uni} \simeq (\pi_Q)_*\Ql[\dim \wt G^Q_1]$.
\item
We have
\begin{equation*}
(\pi_Q)_*\Ql[\dim \wt G^Q_1] \simeq \bigoplus_{\substack{\mu \in \SP_n \\ \mu \le {}^t\la}}
\Ql^{K_{{}^t\la, \mu}}\otimes \IC(\ol\SO_{\la}, \Ql)
[\dim \SO_{\la}].
\end{equation*}
\end{enumerate}
\end{prop}
\begin{proof}
Note that $2\nu_G - 2\nu_M = 2\dim U_Q = \dim \wt G^Q_1$.
Thus by (2.8.3),
\begin{equation*}
\tag{2.9.1}
\IC(\wt G^Q, \SL'_{\ve})[2\nu_G]|_{G \times^QQ\uni} \simeq i_*\Ql[\dim \wt G^Q_1].
\end{equation*}
By applying the base change theorem to the cartesian square
\begin{equation*}
\begin{CD}
G \times^QQ\uni @>>> G \times^QQ \\
@V\pi_1''VV @VV\pi''V \\
G\uni @>>> G,
\end{CD}
\end{equation*}
we obtain (i) from (2.9.1) since $\pi_Q = \pi''_1\circ i$.
Then (ii) follows from (i)
by using (2.7.10) and (2.8.1).
\end{proof}
\para{2.10.}
Returning to the setting in 2.5, we consider the case where $r$ is arbitrary.
We fix $\Bm \in \SQ_{n,r}$, and let $P = P_{\Bm}$ be the parabolic subgroup of
$G$ containing $B$ which is the stabilizer of the partial flag $(M_{p_i})_{1 \le i \le r}$.
Let $L$ be the Levi subgroup of $P$ containing $T$, and $B_L = B \cap L$ the Borel
subgroup of $L$ containing $T$. Let $U_L$ be the unipotent radical of $B_L$.
Put $\ol M_{p_i} = M_{p_i}/M_{p_{i-1}}$ for each $i$, under the convention $M_{p_0} = 0$.
Then $L$ acts naturally on $\ol M_{p_i}$, and by applying the definition of
$\pi_{\Bm,1} : \wt\SX_{\Bm,\unip} \to \SX_{\Bm, \unip}$ to $L$, we can define
\begin{align*}
\wt\SX^L_{\Bm, \unip} &\simeq L \times^{B_L}(U_L \times \prod_{i=1}^{r-1}\ol M_{p_i}), \\
\SX^L_{\Bm,\unip} &= \bigcup_{g \in L}g(U_L \times \prod_{i = 1}^{r-1}\ol M_{p_i})
= L\uni \times \prod_{i=1}^{r-1} \ol M_{p_i}
\end{align*}
and the map $\pi^L_{\Bm,1} : \wt\SX^L_{\Bm, \unip} \to \SX^L_{\Bm, \unip}$ similarly.
Let $Q = Q_{\Bla}$ be as in 2.5 for $\Bla \in \SP(\Bm)$.
Thus we have $B \subset Q \subset P$, and
$Q_L = Q \cap L$ is a parabolic subgroup of
$L$ containing $B_L$.
We consider the following commutative diagram
\begin{equation*}
\tag{2.10.1}
\begin{CD}
\wt\SX_{\Bm,\unip} @<\wt p_1<< G \times \wt\SX^P_{\Bm,\unip} @>\wt q_1>> \wt\SX^L_{\Bm,\unip} \\
@V\a'_1VV @VV r'_1 V @VV\b'_1 V \\
\wh \SX^Q_{\Bm,\unip} @<\wh p_1 << G \times \wt\SX_{\Bm,\unip}^{P,Q} @>\wh q_1>>
\wt\SX^{L, Q_L}_{\Bm,\unip} \\
@V\a''_1VV @VVr_1''V @VV\b''_1V \\
\wh \SX^P_{\Bm, \unip} @<p_1<< G \times \SX_{\Bm, \unip}^P @>q_1>> \SX^L_{\Bm,\unip} \\
@V\pi''_1 VV \\
\SX_{\Bm,\unip},
\end{CD}
\end{equation*}
where, by putting $P\uni = L\uni U_P$ (the set of unipotent elements in $P$),
\begin{align*}
\SX_{\Bm,\unip}^P &= \bigcup_{g \in P}g(U \times \prod_i M_{p_i}) = P\uni \times \prod_i M_{p_i}, \\
\wh \SX^P_{\Bm,\unip} &= G \times^P\SX_{\Bm,\unip}^P = G \times^P(P\uni \times \prod_iM_{p_i}), \\
\wt \SX^P_{\Bm,\unip} &= P \times^{B}(U \times \prod_iM_{p_i}), \\
\wh \SX^Q_{\Bm,\unip} &= G \times^Q(Q\uni \times \prod_i M_{p_i}), \\
\wt\SX^{P,Q}_{\Bm, \unip} &= P \times^Q(Q\uni \times \prod_i M_{p_i}).
\end{align*}
$\wt\SX^{L,Q_L}_{\Bm \unip}$ is a similar variety as $\wh\SX^P_{\Bm, \unip}$
defined with respecto to $(L, Q_L)$, namely,
\begin{equation*}
\wt\SX^{L, Q_L}_{\Bm, \unip} = L \times^{Q_L}((Q_L)\uni \times \prod_i\ol M_{p_i}).
\end{equation*}
The maps are defined as follows;
under the identification $\wt\SX_{\Bm, \unip} \simeq G \times^B(U \times \prod_iM_{p_i})$,
$\a'_1, \a''_1$ are the natural maps induced from the inclusions
$G \times (U \times \prod M_{p_i}) \to G \times (Q\uni \times \prod M_{p_i})
\to G \times (P\uni \times \prod M_{p_i})$.
$\pi_1'': g*(x,\Bv) \mapsto (gxg\iv, g\Bv)$.
$q_1$ is defined by $(g,x,\Bv) \mapsto (\ol x, \ol \Bv)$, where
$x \to \ol x$, $\Bv \mapsto \ol \Bv$ are natural maps
$P \to L, \prod_iM_{p_i} \to \prod_i\ol M_{p_i}$.
$\wt q_1$ is the composite of the projection
$G \times \wt\SX^P_{\Bm,\unip} \to \wt\SX^P_{\Bm, \unip}$
and the map $\wt\SX^P_{\Bm, \unip} \to \wt\SX^L_{\Bm,\unip}$ induced from
the projection $P \times (U \times \prod M_{p_i}) \to L \times (U_L \times \prod \ol M_{p_i})$.
$\wh q_1$ is defined similarly by using the map
$\wt\SX^{P,Q}_{\Bm,\unip} \to \wh\SX^{L,Q_L}_{\Bm,\unip}$
induced from the projection $P \times (Q\uni \times \prod M_{p_i})
\to L \times ((Q_L)\uni \times \prod \ol M_{p_i})$.
$p_1$ is the quotient by $P$. $\wt p_1$ and $\wh p_1$ are also quotient by $P$
under the identifications $\wt\SX_{\Bm,\unip} \simeq G \times^P \wt\SX^P_{\Bm,\unip}$,
$\wh\SX^Q_{\Bm\unip} \simeq G \times^P\wt\SX^{P,Q}_{\Bm, \unip}$.
$\b_1'$ is defined similarly to $\a_1'$ and $\b_1''$ is defined similarly to $\pi_1''$.
$r'_1$ is the natural map induced from the injection
$P \times (U \times \prod M_{p_i}) \to P \times (Q\uni \times \prod M_{p_i})$,
and $r_1''$ is the natural map induced from the map
$P \times^Q(Q\uni \times \prod M_{p_i}) \to P\uni \times \prod M_{p_i}$,
$g*(x,\Bv) \mapsto (gxg\iv, g\Bv)$.
\par
Put $\pi'_1 = \a_1''\circ \a_1': \wt\SX_{\Bm, \unip} \to \wh\SX^P_{\Bm, \unip}$.
We have $\b_1''\circ \b_1' = \pi^L_{\Bm,1}$, and the diagram (2.10.1) is the refinement of
the diagram (6.3.2) in [S4] (see also the diagram (1.5.1) in [S4]).
In particular, the map $p_1$ is a principal $P$-bundle, and the map $q_1$ is a locally
trivial fibration with fibre isomorphic to $G \times U_P \times \prod_{i=1}^{r-2}M_{p_i}$.
Moreover, all the squares appearing in (2.10.1) are caetesian squares.
Hence the diagram (2.10.1) satisfies similar properties as in the diagram (2.8.2).
\par
Note that $L \simeq G_1 \times \cdots \times G_r$ with $G_i = GL(\ol M_{p_i})$.
Then $Q_L$ can be written as $Q_L \simeq Q_1 \times \cdots \times Q_r$, where
$Q_i$ is a parabloic subgroup of $G_i$.
We have
\begin{align*}
\wt\SX^L_{\Bm, \unip} &\simeq \prod_{i=1}^r(\wt G_i)\uni \times V, \\
\wh \SX^{L,Q_L}_{\Bm, \unip} &\simeq \prod_{i=1}^r (\wt G_i^{Q_i})\uni \times V, \\
\SX^L_{\Bm, \unip} &\simeq \prod_{i=1}^r (G_i)\uni \times V,
\end{align*}
where $(\wt G_i)\uni, (\wt G_i^{Q_i})\uni$, etc. denote the unipotent parts of
$\wt G_i, \wt G_i^{Q_i}$, etc. as in (2.8.2). The maps $\b_1', \b_1''$ are induced from
the maps $(\wt G_i)\uni \to (\wt G_i^{Q_i})$, $(\wt G_i^{Q_i})\uni \to (G_i)\uni$, and those
maps coincide with the maps $\pi', \pi''$ in 2.7 defined with respect to $G_i$.
Note that $W_{Q_i} \simeq S_{(\la^{(i)})^t}$ for each $i$ by
the construction of $Q = Q_{\Bla}$ in 2.5.
Put
\begin{equation*}
\wh \SX^Q_1 = G \times^Q(U_Q \times \prod M_{p_i}), \quad
\wt\SX^{L,Q_L}_1 = L \times^{Q_L}(U_{Q_L} \times \prod \ol M_{p_i}),
\end{equation*}
and let $i_Q : \wh\SX_1^Q \hra \wh \SX^Q_{\Bm, \unip},
i_{Q_L} : \wt \SX^{L, Q_L}_1 \hra \wt \SX^{L,Q_L}_{\Bm \unip}$ be
the closed embeddings.
Let $\pi^L_{Q_L} : \wt \SX^{L,Q_L}_1 \to \SX^L_{\Bm, \unip}$ be
the restriction of $\b_1''$.
Let $\SO^L_{\Bmu} \simeq \SO'_{\mu^{(1)}} \times \cdots \times \SO'_{\mu^{(r)}}$ be
the $L$-orbit in $\SX^L_{\Bm, \unip}$,
where $\SO'_{\mu^{(i)}}$ is the $G_i$-oribt in $(G_i)\uni \times \ol M_{p_i}$ of type
$(\mu^{(i)}, \emptyset)$. Note that if we denote by $\SO_{\mu^{(i)}}$ the $G_i$-orbit
in $(G_i)\uni$ of type $\mu^{(i)}$, we have
$\IC(\ol\SO'_{\mu^{(i)}}, \Ql) \simeq \IC(\ol\SO_{\mu^{(i)}}, \Ql) \boxtimes \Ql$
(the latter term $\Ql$ denotes the constatn sheaf on $\ol M_{p_i}$).
Hence the decompostion of $\pi^L_{Q_L}$ into simple components is described by
considering the factors
$\IC(\ol \SO_{\mu^{(i)}}, \Ql)$.
In particular, by Proposition 2.9, we have
\begin{equation*}
\tag{2.10.2}
(\pi^L_{Q_L})_*\Ql[\dim \wt\SX_1^{L,Q_L}] \simeq
\bigoplus_{\Bmu \trleq \Bla}
\Ql^{K_{\Bmu^t, \Bla^t}}\otimes \IC(\ol \SO^L_{\Bmu}, \Ql)[\dim \SO^L_{\Bmu}].
\end{equation*}
\par
By using the diagram (2.10.1), we see that
\begin{equation*}
\wh q_1^*(i_{Q_L})_*\Ql[\dim \wt\SX^{L, Q_L}_1] \simeq \wh p_1^*(i_Q)_*\Ql[\dim \wt X_{\Bla}].
\end{equation*}
It follows, again by using the diagram (2.10.1), we have
\begin{equation*}
\tag{2.10.3}
(\a_1'')_*(i_Q)_*\Ql[\dim \wt X_{\Bla}] \simeq \bigoplus_{\Bmu \trleq \Bla}
\Ql^{K_{\Bmu^t, \Bla^t}}\otimes B_{\Bmu},
\end{equation*}
where $B_{\Bmu}$ is the simple perverse sheaf on $\wh\SX^P_{\Bm, \unip}$
characterized by the property that
\begin{equation*}
p_1^*B_{\Bmu}[a'] \simeq q_1^*\IC(\ol\SO^L_{\Bmu}, \Ql)[b' + \dim \SO^L_{\Bmu}]
\end{equation*}
with $a' = \dim P$, $b' = \dim G + \dim U_P + \dim \prod_{i=1}^{r-2}M_{p_i}$.
\par
On the other hand, by Proposition 1.6 in [S4], we have
\begin{equation*}
\pi''_*A_{\Bmu} \simeq \IC(\SX_{\Bm}, \SL_{\Bmu})[d_{\Bm}],
\end{equation*}
where $\pi'': \wh\SX^P_{\Bm} = G \times^P(P \times \prod_iM_{p_i})
\to \SX_m $
is an analogous map to $\pi''_1$, and $A_{\Bmu}$ is a simple perverse sheaf on $\wh \SX^P_{\Bm}$
such that the restriction of $A_{\Bmu}$ on $\wh\SX^P_{\Bm, \unip}$ coincides with $B_{\Bmu}$,
up to shift.
Thus by Theorem 2.4 (ii), we have
\begin{equation*}
\tag{2.10.4}
(\pi''_1)_*B_{\Bmu} \simeq \IC(\ol X_{\Bmu}, \Ql)[\dim X_{\Bmu}].
\end{equation*}
Since $\pi_{\Bla} = \pi_1''\circ \a_1''\circ i_Q$, by applying $(\pi''_1)_*$ on both sides of
(2.10.3), we obtain the formula (2.6.1). This completes the proof of Theorem 2.6.
\par
\section{$G^F$-invariant functions on the enhanced variety \\
and Kostka functions}
\para{3.1.}
We now assume that $G$ and $V$ are defined over $\Fq$, and let
$F: G \to G, F: V \to V$ be the corresponding Frobenius maps.
Assume that $B$ and $T$ are $F$-stable.
Then $X_{\Bla}$ and $\wt X_{\Bla}$ have natrual $\Fq$-structures, and the map
$\pi_{\Bla}: \wt X_{\Bla} \to \ol X_{\Bla}$ is $F$-equivariant.
Thus one can define a canonical isomorphsim $\vf : F^*K_{\Bla} \,\raise2pt\hbox{$\underrightarrow{\sim}$}\, K_{\Bla}$
for $K_{\Bla} = (\pi_{\Bla})_*\Ql$.
By using the decomposition in Theorem 2.6,
$\vf$ can be written as $\vf = \sum_{\Bmu}\s_{\Bmu} \otimes \vf_{\Bmu}$,
where $\s_{\Bmu}$ is the identity map on $\Ql^{K_{\Bmu^t, \Bla^t}}$ and
$\vf_{\Bmu} : F^*L_{\Bmu} \,\raise2pt\hbox{$\underrightarrow{\sim}$}\, L_{\Bmu}$ is the isomorphism induced from $\vf$
for $L_{\Bmu} = \IC(\ol X_{\Bmu}, \Ql)$.
(Note that $\dim X_{\Bla} - \dim X_{\Bmu}$ is even if $\Bmu \trleq \Bla$ by
[S4, Prop. 4.3], so the degree shift is negligible).
We also consider the natural isomorphism $\f_{\Bmu} : F^*L_{\Bmu} \,\raise2pt\hbox{$\underrightarrow{\sim}$}\, L_{\Bmu}$
induced from the $\Fq$-strucutre of $X_{\Bmu}$.
By using a similar argument as in [S4, (6.1.1)], we see that
\begin{equation*}
\tag{3.1.1}
\vf_{\Bmu} = q^{d_{\Bmu}}\f_{\Bmu},
\end{equation*}
where $d_{\Bmu} = n(\Bmu)$.
We consider the characteristic function $\x_{L_{\Bmu}}$ of $L_{\Bmu}$
with respect to $\f_{\Bmu}$, which is a $G^F$-invariant function on $\ol X_{\Bmu}^F$.
\para{3.2.}
Take $\Bmu, \Bnu \in \SP_{n,r}$, and assume that $\Bnu \in \SP(\Bm)$.
For each $z = (x, \Bv) \in X_{\Bmu}$ with $\Bv = (v_1, \dots, v_{r-1})$, we define a
variety $\SG_{\Bnu,z}$ by
\begin{equation*}
\tag{3.2.1}
\begin{split}
\SG_{\Bnu,z} = \{ (W _{p_i}) &\text{ : $x$-stable flag } \mid v_i \in W_{p_i}
\ (1 \le i \le r-1), \\
&x|_{W_{p_i}/W_{p_{i-1}}}
\text{: type $\nu^{(i)}$ } \ (1 \le i \le r) \}.
\end{split}
\end{equation*}
If $z \in X_{\Bmu}^F$, the variety $\SG_{\Bnu,z}$ is defined over $\Fq$.
Put $g_{\Bnu,z}(q) = |\SG_{\Bnu,z}^F|$.
Let $\wt K_{\la,\mu}(t)$ be the modified Kostka polynomial indexed by partitions
$\la, \mu$.
The following result is a generalization of Proposiition 5.8 in [AH].
\begin{prop}
Assume that $\Bla, \Bmu \in \SP_{n,r}$. For each $z \in X_{\Bmu}^F$, we have
\begin{equation*}
\x_{L_{\Bla}}(z) = q^{-n(\Bla)}\sum_{\Bnu \trleq \Bla}g_{\Bnu, z}(q)
\wt K_{\la^{(1)},\nu^{(1)}}(q)\cdots \wt K_{\la^{(r)},\nu^{(r)}}(q).
\end{equation*}
\end{prop}
\begin{proof}
Let $\x_{K_{\Bla}, \vf}$ be the characteristic function of $K_{\Bla}$ with respect to
$\vf$.
By Theorem 2.6 together with (3.1.1), we have
\begin{equation*}
\tag{3.3.1}
\x_{K_{\Bla}, \vf} = \sum_{\Bxi \trleq \Bla}K_{\Bxi^t, \Bla^t}q^{n(\Bxi)}\x_{L_{\Bxi}}.
\end{equation*}
On the other hand, by the Grothendieck's fixed point formula, we have
$\x_{K_{\Bla}, \vf}(z) = |\pi_{\Bla}\iv(z)^F|$ for $z \in \ol X_{\Bla}^F$.
Then if $z = (x, \Bv) \in X_{\Bmu}^F$,
\begin{equation*}
\tag{3.3.2}
|\pi_{\Bla}\iv(z)^F| = \sum_{\Bnu \in \SP_{n,r}}|\SG_{\Bnu,z}^F|\prod_i|\pi_{\la^{(i)}}\iv(x_i)^F|,
\end{equation*}
where $\pi_{\la^{(i)}} : \wt\SO_{\la^{(i)}} \to \ol\SO_{\la^{(i)}}$ is a similar map
as $\pi_{\Bla}$ applied to the case $r = 1$, by replacing $G$ by $G_i = GL(\ol M_{p_i})$,
and $x_i = x|_{\ol M_{p_i}}$ has Jordan type $\nu^{(i)}$.
It is known by [L1] that
$q^{n(\xi^{(i)})}\x_{L_{\xi^{(i)}}}(x_i) = \wt K_{\xi^{(i)}, \nu^{(i)}}(q)$
for a partition $\xi^{(i)}$ of $m_i$.
It follows, by applying (3.3.1) to the case where $r = 1$,
and by the Grothendieck's fixed point formula,
we have
\begin{equation*}
|\pi_{\la^{(i)}}\iv(x_i)^F| = \sum_{\xi^{(i)} \le \la^{(i)}}
K_{\xi^{(i)t}, \la^{(i)t}}\wt K_{\xi^{(i)}, \nu^{(i)}}(q).
\end{equation*}
Then (3.3.2) implies that
\begin{equation*}
\tag{3.3.3}
\x_{K_{\Bla}, \vf} =
|\pi_{\Bla}\iv(z)^F| = \sum_{\Bnu \in \SP_{n,r}}g_{\Bnu,z}(q)\sum_{\Bxi \trleq \Bla}
K_{\Bxi^t, \Bla^t}\wt K_{\xi^{(1)}, \nu^{(1)}}(q)\cdots
\wt K_{\xi^{(r)}, \nu^{(r)}}(q).
\end{equation*}
Since $(K_{\Bxi^t, \Bla^t})_{\Bla, \Bxi}$ is a unitriangular matrix with respect
to the partial order $\Bxi \trleq \Bla$, by comparing (3.3.1) and (3.3.3),
we obtain the required formula.
\end{proof}
\remark{3.4.}
In general, $X_{\Bmu}$ consists of infinitely many $G$-orbits. Hence the value
$g_{\Bnu, z}(q)$ may depend on the choice of $z \in X_{\Bmu}^F$. However, if
$X_{\Bmu}$ is a single $G$-orbit, then $X_{\Bmu}^F$ is also a single $G^F$-orbit, and
$g_{\Bnu,z}(q)$ is constant for $z \in X_{\Bmu}^F$, in which case, we denote
$g_{\Bnu,z}(q)$ by $g_{\Bnu}^{\Bmu}(q)$. In what follows, we show in some special
cases that there exists a polynomial $g_{\Bnu}^{\Bmu}(t) \in \BZ[t]$ such that
$g_{\Bnu}^{\Bmu}(q)$ coincides with the value at $t = q$ of $g_{\Bnu}^{\Bmu}(t)$.
\para{3.5.}
We consider the special case where $\Bmu \in \SP(\Bm')$ is such that
$m_i' = 0$ for $i = 1, \dots, r-2$. In this case, $X_{\Bmu}$ consists of a single
$G$-orbit.
In particular, for $\Bla \in \SP_{n,r}$,
$\dim \SH^i_z\IC(\ol X_{\Bla}, \Ql)$ does not depend on the chocie
of $z \in X_{\Bmu}$.
We define a polynomial $\IC^-_{\Bla, \Bmu}(t) \in \BZ[t]$ by
\begin{equation*}
\IC^-_{\Bla,\Bmu}(t) = \sum_{i \ge 0}\dim \SH^{2i}_z\IC(\ol X_{\Bla}, \Ql)t^i.
\end{equation*}
The following result was proved in [S4].
\begin{prop}[{[S4, Prop. 6.8]}]
Let $\Bla, \Bmu \in \SP_{n,r}$, and assume that $\Bmu$ is as in 3.5.
\begin{enumerate}
\item
Assume that $z \in X^F_{\Bmu}$. Then $\SH^i_z\IC(\ol X_{\Bla}, \Ql) = 0$ if $i$ is odd,
and the eigenvalues of $\f_{\Bla}$ on $\SH^{2i}_z\IC(\ol X_{\Bla}, \Ql)$ are $q^i$.
In particular, $\x_{L_{\Bla}}(z) = \IC^-_{\Bla, \Bmu}(q)$.
\item
$\wt K^-_{\Bla, \Bmu}(t) = t^{a(\Bla)}\IC^-_{\Bla, \Bmu}(t^r)$.
\end{enumerate}
\end{prop}
As a corollary, we have the following result, which is a
generalization of [AH, Prop. 5.8] (see also [LS, Prop. 3.2]).
\begin{cor}
Assume that $\Bmu$ is as in 3.5.
\begin{enumerate}
\item
There exists a polynomial $g_{\Bnu}^{\Bmu}(t) \in \BZ[t]$
such that $g_{\Bnu}^{\Bmu}(q)$ coincides with the value at
$t = q$ of $g^{\Bmu}_{\Bnu}(t)$.
\item
We have
\begin{equation*}
\tag{3.7.1}
\wt K^-_{\Bla, \Bmu}(t) = t^{a(\Bla)-rn(\Bla)}
\sum_{\Bnu \trleq \Bla}g^{\Bmu}_{\Bnu}(t^r)
\wt K_{\la^{(1)}, \nu^{(1)}}(t^r)\cdots \wt K_{\Bla^{(r)}, \Bnu^{(r)}}(t^r).
\end{equation*}
\end{enumerate}
\end{cor}
\begin{proof}
By Proposition 3.6 (i) and Proposition 3.3, we have
\begin{equation*}
\tag{3.7.2}
\IC^-_{\Bla, \Bmu}(q) = q^{-n(\Bla)}\sum_{\Bnu \trleq \Bla}g_{\Bnu}^{\Bmu}(q)
\wt K_{\la^{(1)},\nu^{(1)}}(q)\cdots \wt K_{\la^{(r)}, \nu^{(r)}}(q)
\end{equation*}
By fixing $\Bmu$, we consider two sets of functions
$\{ \IC^-_{\Bla \Bmu}(q) \mid \Bla \in \SP_{n,r} \}$ and
$\{ g^{\Bmu}_{\Bnu}(q) \mid \Bnu \in \SP_{n,r} \}$.
If we notice that $\wt K_{\la^{(1)}, \nu^{(1)}}(q)\cdots \wt K_{\la^{(r)}, \nu^{(r)}}(q)
= q^{n(\Bla)}$ for $\Bnu = \Bla$, (3.7.2) shows that the transition matrix
between those two sets is unitriangular.
Hence $g^{\Bmu}_{\Bnu}(q)$ is determined from $\IC^-_{\Bla, \Bmu}(q)$, and a similar
formula makes sense if we replace $q$ by $t$. This implies (i).
(ii) now follows from (3.7.2) by replacing $q$ by $t$.
\end{proof}
\para{3.8.}
In what follows, we assume that $\Bmu$ is of the form
$\Bmu = (-, \dots, -, \xi)$ with $\xi \in \SP_n$.
In this case, $g^{\Bmu}_{\Bnu}(t)$ coincides with the polynomial
$g^{\xi}_{\nu^{(1)}, \dots, \nu^{(r)}}(t)$
obtained from
$G^{\xi}_{\nu^{(1)}, \dots, \nu^{(r)}}(\Fo)$ discussed in [M, II, 2].
On the other hand, we define a polynomial $f^{\xi}_{\nu^{(1)}, \dots, \nu^{(r)}}(t)$ by
\begin{equation*}
\tag{3.8.1}
P_{\nu^{(1)}}(y;t)\cdots P_{\nu^{(r)}}(y;t) =
\sum_{\xi \in \SP_n}f^{\xi}_{\nu^{(1)}, \dots, \nu^{(r)}}(t)P_{\xi}(y;t).
\end{equation*}
In the case where $r = 2$, $g^{\xi}_{\nu^{(1)}, \nu^{(2)}}(t)$ coincides with the Hall polynomial,
and a simple formula relating it with $f^{\xi}_{\nu^{(1)}, \nu^{(2)}}(t)$ is konwn
([M, III (3.6)]). In the general case, we also have a formula
\begin{equation*}
\tag{3.8.2}
g^{\xi}_{\nu^{(1)}, \dots, \nu^{(r)}}(t) = t^{n(\xi)- n(\Bnu)}
f^{\xi}_{\nu^{(1)}, \dots, \nu^{(r)}}(t\iv).
\end{equation*}
The proof is easily reduced to [M, III (3.6)].
\par
For partitions $\la, \nu^{(1)}, \dots, \nu^{(r)}$, we
define an integer $c^{\la}_{\nu^{(1)}, \dots, \nu^{(r)}}$ by
\begin{equation*}
s_{\nu^{(1)}}\cdots s_{\nu^{(r)}} = \sum_{\la}c^{\la}_{\nu^{(1)},\dots, \nu^{(r)}}s_{\la}.
\end{equation*}
In the case where $r = 2$, $c^{\la}_{\nu^{(1)},\nu^{(2)}}$ coincides with the Littlewood-Richardson
coefficient.
\par
For $\Bla \in \SP_{n,r}$, put
\begin{equation*}
\tag{3.8.3}
b(\Bla) = a(\Bla) - r\cdot n(\Bla) = |\la^{(2)}| + 2|\la^{(3)}| + \cdots + (r-1)|\la^{(r)}|.
\end{equation*}
The following lemma is a generalization of [LS, Lemma 3.4].
\begin{lem}
Let $\Bla, \Bmu \in \SP_{n,r}$, and assume that
$\Bmu = (-, \dots, -, \xi)$. Then we have
\begin{align*}
\tag{3.9.1}
K^-_{\Bla, \Bmu}(t) &= t^{b(\Bmu) - b(\Bla)}
\sum_{\Bnu \trleq \Bla}
f^{\xi}_{\nu^{(1)}, \dots, \nu^{(r)}}(t^{r})
K_{\la^{(1)}, \nu^{(1)}}(t^r)\cdots K_{\la^{(r)}, \nu^{(r)}}(t^r), \\
\tag{3.9.2}
K^-_{\Bla, \Bmu}(t) &=
t^{b(\Bmu) - b(\Bla)}
\sum_{\e \in \SP_n}c^{\eta}_{\la^{(1)}, \dots, \la^{(r)}}K_{\eta, \xi}(t^r).
\end{align*}
\end{lem}
\begin{proof}
The formula (3.7.1) can be rewritten as
\begin{equation*}
\tag{3.9.3}
K^-_{\Bla, \Bmu}(t) = t^{a(\Bmu) - a(\Bla) + rn(\Bla)}\sum_{\Bnu \trleq \Bla}t^{-rn(\Bnu)}
g^{\xi}_{\nu^{(1)}, \dots, \nu^{(r)}}(t^{-r})
K_{\la^{(1)}, \nu^{(1)}}(t^r)\cdots K_{\la^{(r)}, \nu^{(r)}}(t^r).
\end{equation*}
Substituting (3.8.2) into (3.9.3), we obtain (3.9.1).
Next we show (3.9.2).
One can write as
\begin{equation*}
s_{\la^{(i)}}(y) = \sum_{\nu^{(i)}}K_{\la^{(i)}, \nu^{(i)}}(t)P_{\nu^{(i)}}(y;t).
\end{equation*}
Hence
\begin{align*}
\tag{3.9.4}
s_{\la^{(1)}}(y)\cdots s_{\la^{(r)}}(y) &= \sum_{\Bnu \in \SP_{n,r}}
K_{\la^{(1)}, \nu^{(1)}}(t)\cdots K_{\la^{(r)}, \nu^{(r)}}(t)
P_{\nu^{(1)}}(y;t)\cdots P_{\nu^{(r)}}(y;t) \\
&= \sum_{\Bnu \in \SP_{n,r}}\sum_{\xi \in \SP_n}f^{\xi}_{\nu^{(1)}, \dots, \nu^{(r)}}(t)
K_{\la^{(1)},\nu^{(1)}}(t)\cdots K_{\la^{(r)},\nu^{(r)}}(t)P_{\xi}(y;t).
\end{align*}
On the other hand,
\begin{align*}
\tag{3.9.5}
s_{\la^{(1)}}(y)\cdots s_{\la^{(r)}}(y) &= \sum_{\e \in \SP_n}
c^{\eta}_{\la^{(1)}, \dots, \la^{(r)}}s_{\eta}(y) \\
&= \sum_{\eta \in \SP_n}c^{\eta}_{\la^{(1)}, \dots, \la^{(r)}}
\sum_{\xi \in \SP_n} K_{\eta, \xi}(t)P_{\xi}(y;t).
\end{align*}
By comparing (3.9.4) and (3.9.5), we have an equality for each $\xi \in \SP_n$,
\begin{equation*}
\sum_{\e \in \SP_n}c^{\eta}_{\la^{(1)}, \dots, \la^{(r)}}K_{\eta, \xi}(t)
= \sum_{\Bnu \in \SP_{n,r}}f^{\xi}_{\nu^{(1)}, \dots, \nu^{(r)}}(t)
K_{\la^{(1)}, \nu^{(1)}}(t), \dots K_{\la^{(r)}, \nu^{(r)}}(t).
\end{equation*}
Combining this with (3.9.1), we obtain (3.9.2). The lemma is proved.
\end{proof}
\para{3.10.}
Let $\e' = \la' - \th', \e'' = \la'' - \th''$ be skew diagrams, where
$\th' \subset \la', \th'' \subset \la''$ are partitions.
We define a new skew diagram $\e'*\e'' = \la - \th$ as follows;
write the partitions $\la', \la''$ as
$\la' = (\la'_1, \dots, \la'_{k'}), \la'' = (\la''_1, \dots, \la''_{k''})$
with $\la'_{k'} > 0, \la''_{k''} > 0$.
Put $a = \la_1''$. We define a partition $\la = (\la_1, \dots, \la_{k' + k''})$
by
\begin{equation*}
\la_i = \begin{cases}
\la'_i + a &\quad\text{ for } 1 \le i \le k', \\
\la''_{i-k'} &\quad\text{ for } k' + 1 \le i \le k' + k''.
\end{cases}
\end{equation*}
Write partitions $\th', \th''$ as $\th' = (\th'_1, \dots, \th'_{k'}),
\th'' = (\th''_1, \dots, \th''_{k''})$ with $\th'_{k'} \ge 0$,
$\th''_{k''} \ge 0$.
We define a partition $\th = (\th_1, \dots, \th_{k' + k''})$, in a similar
way as above, by
\begin{equation*}
\th_i = \begin{cases}
\th'_i + a &\quad\text{ for } 1 \le i \le k', \\
\th''_{i-k'} &\quad\text{ for } k' + 1 \le i \le k' + k''.
\end{cases}
\end{equation*}
We have $\th \subset \la$, and the skew diagram $\e'*\e'' = \la - \th$
can be defined.
\par
For $\la, \mu \in \SP_n$, let $SST(\la, \mu)$ be the set of semistandard tableaux of
shape $\la$ and weight $\mu$.
Let $\Bla \in \SP_{n,r}$. An $r$-tuple $T = (T^{(1)}, \dots, T^{(r)})$
is called a semistandard tableau of shape $\Bla$ if $T^{(i)}$ is a semistandard
tableau of shape $\la^{(i)}$ with respect to the letters $\{ 1, \dots, n\}$.
We denote by $SST(\Bla)$ the set of semistandard tableaux of shape $\Bla$.
For $\Bla \in \SP_{n,r}$, let $\wt\Bla$ be the skew diagram
$\la^{(1)}*\la^{(2)}*\cdots *\la^{(r)}$.
Then $T \in SST(\Bla)$ is regarded as a usual semistandard tableau $\wt T$
associated to the skew diagram $\wt\Bla$.
Assume $\pi \in \SP_n$. We say that $T \in SST(\Bla)$ has weight $\pi$ if the
corresponidng tableau $\wt T$ has shape $\wt\Bla$ and weight $\pi$.
We denote by $SST(\Bla, \pi)$ the set of semistandard tableaux of shape $\Bla$ and
weight $\pi$.
\para{3.11.}
In [M, I, (9.4)], a bijective map $\varTheta$
\begin{equation*}
\tag{3.11.1}
\varTheta : SST(\wt\Bla, \pi) \,\raise2pt\hbox{$\underrightarrow{\sim}$}\, \coprod_{\nu \in \SP_n}(SST^0(\wt\Bla, \nu) \times SST(\nu, \pi))
\end{equation*}
was constructed, where $SST^0(\wt\Bla, \nu)$ is the set of tableau $T$ such that
the associated word $w(T)$ is a lattice permutation (see [M, I, 9] for the definition).
Under the identification $SST(\wt\Bla, \pi) \simeq SST(\Bla, \pi)$,
the subset $SST^0(\Bla,\nu)$ of $SST(\Bla, \nu)$ is also defined.
Then we can regard $\varTheta$ as a bijection with respect to the set $SST(\Bla, \pi)$
(and $SST^0(\Bla, \nu)$).
\par
In the case where $r = 2$, it is shown in [LS, Cor. 3.9] that $|SST^0(\Bla, \nu)|$
coincides with the Littlewood-Richardson coefficient $c^{\nu}_{\la^{(1)}, \la^{(2)}}$.
A similar argument can be applied also to the general case, and we have
\begin{cor}
Assume that $\Bla \in \SP_{n,r}, \nu \in \SP_n$. Then we have
\begin{equation*}
|SST^0(\Bla, \nu)| = c^{\nu}_{\la^{(1)}, \dots, \la^{(r)}}.
\end{equation*}
\end{cor}
\para{3.13.}
For a semistandard tableau $S$, the charge $c(S)$ is defined as in [M, III, 6].
It is known that Lascoux-Sch\"utzenberger Theorem
([M, III, (6.5)]) gives a combinatorial description of
Koskta polynomials $K_{\la,\mu}(t)$ in terms of sesmistandard tableaux,
\begin{equation*}
\tag{3.13.1}
K_{\la,\mu}(t) = \sum_{S \in SST(\la, \mu)}t^{c(S)}.
\end{equation*}
In the case where $r = 2$, a similar formula was proved for $K_{\Bla, \Bmu}(t)$
in [LS, Thm. 3.12], in the special case where $\Bmu = (-,\mu'')$.
Here we consider $K_{\Bla, \Bmu}(t)$ for general $r$.
Assume that $\Bla \in \SP_{n,r}$ and $\xi \in \SP_n$. For $T \in SST(\Bla, \xi)$,
we write $\varTheta(T) = (D, S)$ with $S \in SST(\nu, \xi)$ for some $\nu$.
we define a charge $c(T)$ of $T$ by $c(T) = c(S)$.
We have the following theorem. Note that the proof is quite similar to
[LS].
\begin{thm}
Let $\Bla, \Bmu \in \SP_{n,r}$, and assume that $\Bmu = (-, \dots, -, \xi)$. Then
\begin{equation*}
K^-_{\Bla,\Bmu}(t) = t^{b(\Bmu) - b(\Bla)}\sum_{T \in SST(\Bla, \xi)}t^{r\cdot c(T)}.
\end{equation*}
\end{thm}
\begin{proof}
We define a map $\Psi : SST(\Bla, \xi) \to \coprod_{\nu \in \SP_n}SST(\nu,\xi)$
by $T \mapsto S$, where $\varTheta(T) = (D, S)$. Then by Corollary 3.12, for each
$S \in SST(\nu, \xi)$, the set $\Psi\iv(S)$ has the cardinality
$c^{\xi}_{\la^{(1)}, \dots, \la^{(r)}}$, and by definition, any $T \in \Psi\iv(S)$ has the
charge $c(T) = c(S)$. Hence
\begin{align*}
\sum_{T \in SST(\Bla, \xi)}t^{c(T)} &= \sum_{\nu \in \SP_n}
\sum_{S \in SST(\nu, \xi)}
c^{\nu}_{\la^{(1)}, \dots, \la^{(r)}}t^{c(S)} \\
&= \sum_{\nu \in \SP_n}c^{\nu}_{\la^{(1)}, \dots, \la^{(r)}}K_{\nu, \xi}(t).
\end{align*}
The last equality follows from (3.13.1).
The theorem now follows from (3.9.2).
\end{proof}
\begin{cor}
Under the assumption of Theorem 3.14, we have
\begin{equation*}
K^-_{\Bla, \Bmu}(1) = |SST(\Bla, \xi)|.
\end{equation*}
\end{cor}
\para{3.16.}
In the rest of this section, we shall give an alternate description
of the polynomial $g^{\Bmu}_{\Bnu}(t)$ in the case where
$\Bmu = (-, \dots, -, \xi)$.
For $\Bnu \in \SP_{n,r}$, put
$R_{\Bnu}(x;t) = P_{\nu^{(1)}}(x^{(1)};t^r)\cdots P_{\nu^{(r)}}(x^{(r)};t^r)$.
Then $\{ R_{\Bnu} \mid \Bnu \in \SP_{n,r} \}$ gives a basis of $\Xi^n[t]$.
We define funtions $h^{\Bmu}_{\Bnu}(t) \in \BQ(t)$ by the condition that
\begin{equation*}
\tag{3.16.1}
R_{\Bnu}(x;t) = \sum_{\Bmu \in \SP_{n,r}}h^{\Bmu}_{\Bnu}(t)P^-_{\Bmu}(x;t).
\end{equation*}
The following formula is a generalization of Proposition 4.2 in [LS].
\begin{prop}
Assume that $\Bmu = (-,\dots,-,\xi)$. Then
\begin{equation*}
h^{\Bmu}_{\Bnu}(t) = t^{a(\Bmu) - a(\Bnu)}g^{\Bmu}_{\Bnu}(t^{-r}).
\end{equation*}
\end{prop}
\begin{proof}
The proof is quite similar to that of [LS, Prop. 4.2].
For $\Bla \in \SP_{n,r}$, we have
\begin{align*}
s_{\Bla}(x) &= s_{\la^{(1)}}(x^{(1)})\cdots s_{\la^{(r)}}(x^{(r)}) \\
&= \prod_{i=1}^r\sum_{\nu^{(i)}}K_{\la^{(i)},\nu^{(i)}}(t^r)P_{\nu^{(i)}}(x^{(i)};t^r) \\
&= \sum_{\Bnu}K_{\la^{(1)},\nu^{(1)}}(t^r)\cdots K_{\la^{(r)},\nu^{(r)}}(t^r)
\sum_{\Bmu \in \SP_{n,r}}h^{\Bmu}_{\Bnu}(t)P^-_{\Bmu}(x;t) \\
&= \sum_{\Bmu \in \SP_{n,r}}\biggl(
\sum_{\Bnu}K_{\la^{(1)},\nu^{(1)}}(t^r)\cdots K_{\la^{(r)}, \nu^{(r)}}(t^r)
h^{\Bmu}_{\Bnu}(t)\biggr)P^-_{\Bmu}(x;t).
\end{align*}
Since $s_{\Bla}(x) = \sum_{\Bmu \in \SP_{n,r}}K^-_{\Bla,\Bmu}(t)P_{\Bmu}^-(x;t)$,
by comparing the coefficients of $P^-_{\Bmu}(x;t)$, we have
\begin{equation*}
\tag{3.17.1}
K^-_{\Bla,\Bmu}(t) = \sum_{\Bnu \in \SP_{n,r}}h^{\Bmu}_{\Bnu}(t)
K_{\la^{(1)}, \nu^{(1)}}(t^r)\cdots K_{\la^{(r)}, \nu^{(r)}}(t^r).
\end{equation*}
Now assume that $\Bmu = (-,\dots,-,\xi)$. If we notice that
$K_{\la^{(i)},\nu^{(i)}}(t^r) \ne 0$ only when $|\la^{(i)}| = |\nu^{(i)}|$,
(3.9.3) implies that
\begin{equation*}
\tag{3.17.2}
K^-_{\Bla,\Bmu}(t) = \sum_{\Bnu \in \SP_{n,r}}t^{a(\Bmu) - a(\Bnu)}g^{\Bmu}_{\Bnu}(t^{-r})
K_{\la^{(1)}, \nu^{(1)}}(t^r)\cdots K_{\la^{(r)},\nu^{(r)}}(t^r).
\end{equation*}
Since $(K_{\la^{(1)},\nu^{(1)}}(t^r)\cdots K_{\la^{(r)},\nu^{(r)}}(t^r))_{\Bla, \Bnu \in \SP_{n,r}}$
is a unitriangular matrix, the proposition follows by comparing (3.17.1) and (3.17.2).
\end{proof}
\par
\noindent
T. Shoji \\
Department of Mathematics, Tongji University \\
1239 Siping Road, Shanghai 200092, P. R. China \\
E-mail: \verb|[email protected]|
\end{document} |
\begin{document}
\maketitle
\defAbstract{Abstract}
\begin{abstract}
We consider a multivariate non-linear Hawkes process in a multi-class setup where particles are organised within two populations of possibly different sizes, such that one of the populations acts excitatory on the system while the other population acts inhibitory on the system. The goal of this note is to present a class of Hawkes Processes with stable dynamics without assumptions on the spectral radius of the associated weight function matrix. This illustrates how inhibition in a Hawkes system significantly affects the stability properties of the system.
\end{abstract}
{\it Key words} : Multivariate nonlinear Hawkes processes, Stability, Piecewise deterministic Markov processes, Lyapunov functions.
\\
{\it MSC 2000} : 60G55; 60G57; 60J25; 60Fxx
\section{Introduction and main result}\label{sec:1}
We consider a system of interacting Hawkes processes structured within two populations. We shall label the two populations with ``$+$'' or ``$-$'' signaling that the population acts excitatory or inhibitory on the system, respectively. Let $N_+,N_{-} \in \mathbb {N}$ be the number of units in each population. Introduce weight functions given by
\begin{eqnarray}\label{eq:weightfunctions}
h_{++} ( t)& =&\frac{c_{++}}{N_{+}} e^{ -\nu_{+}t}, \quad h_{+-}( t)=\frac{c_{+-}}{N_{+}}e^{ -\nu_{+}t},\\
h_{-+}( t)&=&\frac{c_{-+}}{N_{-}} e^{ -\nu_{-}t},\quad h_{--}( t)=\frac{c_{--}}{N_{-}} e^{ -\nu_{-}t},
\end{eqnarray}
for $ t \geq 0.$
In the above formula, $h_{+-}$ indicates the weight function from a unit in the excitatory group ``$+$'' to a unit in the inhibitory group ``$-$'', and so on. The coefficients of the system of interacting Hawkes processes are the exponential leakage terms $ \nu_+ > 0 , \nu_- > 0 $ and the weights $ c_{++} , c_{+-} , c_{-+}, c_{--} $ satisfying that
\begin{equation}
c_{++} \geq 0, \; c_{+-} \geq 0, \; c_{--} \leq 0, \; c_{-+} \leq 0.
\end{equation}
The multivariate linear Hawkes process with these parameters is given as
\begin{eqnarray}\label{eq:dyn}
Z_{+}^i (t) &= & \int_0^t \int_0^\infty
{\bf 1}_{ \{ z \leq \psi^{i}_+ ( X_{+} ({s-})) \}} \pi_{+}^i (ds, dz) , 1 \le i\leq N_{+}, \\
Z_{-}^j (t) &= & \int_0^t \int_0^\infty
{\bf 1}_{ \{ z \leq \psi^{j}_- ( X_{-} ({s-})) \}} \pi_{-}^j (ds, dz) ,1 \le j\leq N_{-}, \\
X_+ (t) &= & e^{- \nu_+ t } X_+ (0) + \frac{c_{++}}{N_+} \sum_{i=1}^{N_{+}}\int_0^t e^{- \nu_+ ( t- s) } Z_{+}^i (d s) + \frac{c_{-+}}{N_-} \sum_{j=1}^{N_{-}}\int_0^t e^{- \nu_+ ( t- s) } Z_{-}^j (ds) ,\\
X_{-} (t) &= & e^{- \nu_- t } X_{-} (0 )+ \frac{c_{+-}}{N_+} \sum_{i=1}^{N_{+}}\int_0^t e^{- \nu_- ( t- s) } Z_{+}^i (ds) +\frac{ c_{--}}{N_-} \sum_{j=1}^{N_{-}}\int_0^t e^{- \nu_- ( t- s) } Z_{-}^j (ds) ,
\end{eqnarray}
where the jump rate functions $\psi^i_+ : \mathbb {R} \to \mathbb {R}_+ , \psi^i_- : \mathbb {R} \to \mathbb {R}_+ $ are given by
\begin{equation}\label{eq:aplus}
\psi^i_{\pm} (x) = a^i_{\pm} + \max (x, 0), \; \mbox{ where } a^i_{\pm } > 0 ,
\end{equation}
and where the $ \pi^i_{\pm} , i \geq 1, $ are i.i.d. Poisson random measures on $ \mathbb {R}_+ \times \mathbb {R}_+ $ having intensity $ dt dz.$
Notice that the process $ ( X_+, X_{-} ) $ is a piecewise deterministic Markov process having generator
\begin{multline}
A g (x, y ) = - \nu_+ x \partial_x g (x,y ) - \nu_- y \partial_y g (x, y ) + \sum_{i=1}^{N_+} \psi^i_+ ( x) [ g ( x + \frac{c_{++}}{N_+} , y+ \frac{c_{+-}}{N_+} )- g(x,y ) ] \\
+\sum_{j=1}^{N_-} \psi_-^j (y ) [ g ( x + \frac{c_{-+}}{N_-} , y + \frac{c_{--}}{N_-} ) - g(x, y ) ] ,
\end{multline}
for sufficiently smooth test functions $g.$
Classical stability results for multivariate nonlinear Hawkes processes found e.g. in \cite{bm} or in the recent paper \cite{manonetal}, which is devoted to the study of the stabilising effect of inhibitions, are stated in terms of an associated weight function matrix $\Lambda,$ imposing that the spectral radius of $\Lambda$ is strictly smaller than one. In this case the process is termed to be {\it subcritical}. This spectral radius stability condition has a natural interpretation in terms of a multitype branching process with immigration which is spatially structured and where each jump of a given type ($+ $ or $-$) gives rise to future jumps of the same or of the opposite type, see \cite{ho}. The subcriticality condition ensures the recurrence of this process (see \cite{kaplan}). In our system, the weight function matrix is given by
\begin{equation}\label{eq:Lambda}
\Lambda = \left( \begin{array}{cc}
\frac{c_{++}}{\nu_+} & \frac{|c_{-+}|}{\nu_+} \\
\frac{c_{+-}}{\nu_-} & \frac{|c_{--}|}{\nu_-}
\end{array}
\right) .
\end{equation}
Notice that in \eqref{eq:Lambda}, negative synaptic weights do only appear through their absolute values. This is due to the fact that using the Lipschitz continuity of the rate functions leads automatically to considering absolute values and does not enable us to make profit from the inhibitory action of $c_{-+} $ and $ c_{--}. $ Obviously, having sufficiently fast decay, that is, $ \min (\nu_+ , \nu_-) >> 1, $ is a sufficient condition fo subcriticality.
The purpose of this note is to show how the presence of sufficiently high (in absolute value) negative weights helps stabilising the process without imposing such a subcriticality condition, in particular, without imposing $ \nu_+, \nu_- $ being large. To the best of our knowledge, only few results have been obtained on this natural question in the literature. \cite{bm} gives an attempt in this direction but does only deal with the case when $ c_{+- } $ and $ c_{-+}$ are of the same sign (see Theorem 6 in \cite{bm}), and \cite{manonetal} do only work with the positive part of the weight functions, without profiting from the explicit inhibitory part within the system.
Our approach is based on the construction of a convenient Lyapunov function using the inhibitory part of the dynamics. As such, this approach is limited to the present Markovian framework where the weight functions are decreasing exponentials.
In the following, we shall write
$$ c_{++}^* := c_{++} - \nu_+ , \; c_{-- }^* := c_{--} - \nu_{-} .$$
Notice that $ c_{++}^* $ could be interpreted as the net increase of $ X_+ $ due to self-interactions of $X_+ $ with itself. $ c_{--}^* $ is always negative.
\begin{ass}\label{aslol}
We assume the following inequalities.
\begin{eqnarray}\label{eq:stab}
c_{++}^* +c_{--}^* &<& 0 ,\\
( c_{++}^*- c_{--}^* )^2 &< &4 c_{+- } | c_{-+}| ,\\
c_{++}^* - c_{--}^* &>& 0.
\end{eqnarray}
\end{ass}
This assumption ensures that the system is balanced. Notice that Assumption \ref{aslol} does not imply - nor is implied by - that the spectral radius of $\Lambda$ is strictly smaller than $1$. For example, if Assumption \ref{aslol} is satisfied for some parameters $ ( c_{++},c_{+-},c_{-+},c_{--},\nu,\nu ) , $ i.e., $\nu_+ = \nu_- = \nu, $ such that additionally $ c_{++} + c_{--} < 0, $ then for all $C>1 $ and all $\varepsilon>0,$ the set of parameters $( Cc_{++},Cc_{+-},Cc_{-+},Cc_{--},\varepsilon\nu,\varepsilon\nu ) $ satisfies Assumption \ref{aslol} as well. But the associated offspring matrix $\Lambda_{C,\varepsilon}$ of the scaled parameters is equal to $(C/\varepsilon) \Lambda , $ and thus the spectral radius is also scaled by $C/\varepsilon$.
\begin{ass}\label{ass:2d}
We assume that either $ \nu_+ \neq \nu_- $ or $ \nu_+ = \nu_- $ and $( c_{++},c_{+-} ) ,( c_{-+},c_{--}) $ are linearly independent.
\end{ass}
We are now able to state our main result. It states that under Assumptions \ref{aslol} and \ref{ass:2d}, the process $ X = (X_+, X_{-} ) $ is positive Harris recurrent, together with a strong mixing result. To state our result, for any $ t > 0 $ and for $ z = (x,y ) \in \mathbb {R}^2 ,$ we write $ P_t ( z, \cdot )$ for the transition semigroup of the process, defined through $P_t (z, A) = E_z ( 1_A (X (t)) ) .$ Moreover, for any pair of probability measures $\mu_1, \mu_2 $ on $ {\mathcal B} (\mathbb {R}^2)$ and for any function $ V : \mathbb {R}^2 \to [1, \infty [, $ we put
$$ \| \mu_1- \mu_2 \|_{ V} := \sup_{ g : |g| \le V } | \mu_1 ( g) - \mu_2 (g) | .$$
\begin{theo}\label{theo:harris}
Grant Assumptions \ref{aslol} and \ref{ass:2d}. \\
1) Then the process $ X = (X_+, X_- ) $ is positive recurrent in the sense of Harris, and its unique invariant probability measure $ \mu $ possesses a Lebesgue continuous part. \\
2) There exists a function $V (x, y ) : \mathbb {R}^2 \to [1, \infty [ $ such that $\lim_{ |x| + |y| \to \infty } V ( x,y ) = \infty $ and there exist $ c_1, c_2 > 0 $ such that for all $z \in \mathbb {R}^2$ and all $ t \geq 0, $
\begin{equation}\label{eq:last}
\| P_t(z , \cdot ) - \mu\|_{ V} \le c_1 V (z) e^{ - c_2 t} .
\end{equation}
\end{theo}
\begin{rem}
Notice that if Assumption \ref{ass:2d} is not satisfied, that is, if $ \nu_+ = \nu_- $ and if
$$\left( \begin{array}{c} c_{-+}\\
c_{--}
\end{array}\right) \in H:= \mathbb {R} \left( \begin{array}{c} c_{++}\\
c_{+-}
\end{array}\right),$$
then it is easily shown that almost surely, $ dist ( X (t) , H) \to 0 $ as $t \to \infty $ and that $H$ is invariant under the dynamics. Moreover, the restriction of the dynamics to $H$ is Harris recurrent, having a unique invariant measure $ \mu $ which is absolutely continuous with respect to the Lebesgue measure on $H.$ However, it is easy to show that the original process $X,$ defined on $ \mathbb {R}^2, $ is not Harris in this case, since it is not $ \mu-$irreducible.
\end{rem}
\section{Proof of Theorem \ref{theo:harris}}
This section is devoted to the proof of Theorem \ref{theo:harris}.
\subsection{A Lyapunov function for $X$}
We start this section with the following useful property.
\begin{prop}\label{prop:Feller}
The process $ X$ is a Feller process, that is, for any $f : \mathbb {R}^2 \to \mathbb {R}$ which is bounded and continuous, we have that $\mathbb {R}^2 \ni (x, y ) = z \mapsto E_{z} f (X (t) ) = P_t f (z) $ is continuous.
\end{prop}
The proof of this result follows from classical arguments, see e.g.\ the proof of Proposition 4.8 in \cite{evaflow}, or \cite{ikeda1966}.
The next result shows that if the cross-interactions, that is, influence from $ X_+ $ to $X_{-} $ and vice versa, are sufficiently strong, then -- under mild additional assumptions -- it is possible to construct a Lyapunov function for the system that does mainly profit from the inhibitory part of the jumps.
\begin{prop}\label{prop:lyapunov}
Grant Assumption \ref{aslol} and put
$$ V ( x, y ) := \left\{
\begin{array}{ll}
V_{++} ( x, y ) : = c_{+- } x^2 -c_{-+}y^2 - (c_{++}^* - c_{--}^*) xy & x \in \mathbb {R}_+ , y\in \mathbb {R}_+ \\
V_{+-} (x,y ) := c_{+- } x^2 + q y^2 - (c_{++}^* - c_{--}^*) xy & x\in \mathbb {R}_+ , y \in \mathbb {R}_- \\
V_{-+} (x,y ) := px^2 -c_{-+}y^2 - (c_{++}^* - c_{--}^*) xy & x \in \mathbb {R}_- , y\in \mathbb {R}_+ \\
V_{--} (x,y ) := p x^2 + qy^2 - (c_{++}^* - c_{--}^*) xy & x \in \mathbb {R}_- , y\in \mathbb {R}_-
\end{array}
\right\} , $$
with $p$ so small such that
$$ - (c_{++}^* - c_{--}^* ) (c_{--} - \nu_+ - \nu_- ) + 2 p c_{-+} > 0 $$
and
$q$ so large such that
$$ (c_{++}^* - c_{--}^* ) [ \nu_+ + \nu_- - c_{++} ] + 2 q c_{+- } > 0 \mbox{ and } 4 pq > (c_{++}^* - c_{--}^*)^2 .$$
Then $\lim_{ |x| + |y| \to \infty } V ( x,y ) = \infty $ and there exist $ \kappa, c, K > 0 $ such that
\begin{equation}
A V (x,y ) \le - \kappa V ( x, y ) + c 1_{\{ | x| + |y| \geq K\}} .
\end{equation}
\end{prop}
\begin{proof}
We calculate $ A V ( x, y ) = A^1 V (x, y ) + A^2 V (x, y ) , $ with
$$ A^1 V ( x, y ) = - \nu_+ \partial_x V (x,y ) - \nu_- \partial_y V (x, y ) $$
and $ A^2 $ the jump part of the generator.
{\bf Part 1.1} Suppose first that $ x \geq |c_{-+}|/ N_- , y \geq | c_{--} |/N_- . $ Then
$$ A V (x, y ) = A^1 V_{++} (x, y) + A^2 V_{++} (x,y ) = a_{++} x^2 + b_{++} xy + d_{++} y^2 + L_{++} (x,y ) ,$$
where $L_{++} $ is a polynomial of degree $1.$ A straightforward calculus shows that
\begin{eqnarray*}
a_{++} &=& c_{+-} (c_{++}^* + c_{--}^* ) , \\
b_{++} &=& - (c_{++}^* - c_{--}^* ) (c_{++}^* + c_{--}^* )\\
d_{++} & =& - c_{-+} (c_{++}^* + c_{--}^* ) ,
\end{eqnarray*}
proving that
$$ A V( x, y ) = (c_{++}^* + c_{--}^* ) V (x,y) + L_{++} (x,y ) .$$
This implies that there exist $K, \kappa > 0 $ such that
$$ A V ( x, y ) \le - \kappa V ( x,y ) $$
for all $ x > K , y > K,$ since $ c_{++}^* + c_{--}^* < 0 $ by assumption.
{\bf Part 1.2} Suppose now that $0 \le x < |c_{-+} |/N_- $ and $y \geq | c_{--} |/N_- .$ Then a jump of one of the inhibitory neurons will lead to a change $ x \mapsto x + c_{-+}/N_- < 0 .$ In this case we obtain
$$ A V (x,y) = A V_{++} ( x,y ) + \sum_{j=1}^{N_-}(a^{j}_- + y ) ( V_{-+} ( x + \frac{c_{-+}}{N_-}, y + \frac{c_{--}}{N_-} ) - V_{++} ( x + \frac{c_{-+}}{N_-}, y + \frac{c_{-+}}{N_-} )) .$$
But
$$ |V_{-+} ( x + \frac{c_{-+}}{N_-}, y + \frac{c_{--}}{N_-} ) - V_{++} ( x + \frac{c_{-+}}{N_-}, y + \frac{c_{-+}}{N_-} )| \le C ,$$
since $ | x| < |c_{-+} |,$ and therefore
$$ A V (x,y) \le A V_{++} ( x,y ) + L (y) , $$
where $ L(y) $ is a monomial in $ y.$
The other case $0 \le y < |c_{--} |/N_- $ and $x \geq | c_{-+} |/N_- $ is treated analogously.
{\bf Part 2.1} Suppose now that $ x \geq |c_{-+}|/N_- , y \leq - c_{+-} /N_+ . $
Then
$$ A V (x, y ) = A^1 V_{+-} (x, y) + A^2 V_{+-} (x,y ) = a_{+-} x^2 + b_{+-} xy + d_{+-} y^2 + L_{+-} (x,y ) ,$$
where $L_{+-} $ is a polynomial of degree $1.$ We obtain
\begin{eqnarray*}
a_{+-} &=& c_{+-} (c_{++}^* + c_{--}^* ) , \\
b_{+-} &=& (c_{++}^* - c_{--}^* ) (\nu_+ + \nu_- - c_{++} ) + 2 q c_{+- } \\
d_{+- } & =& - 2 \nu_- q .
\end{eqnarray*}
Since $ b_{+-} > 0 $ by choice of $q,$ this implies that for a suitable positive constant $ \kappa > 0 ,$
$$ A V (x,y) \le - \kappa V(x,y) + L_{+-} (x,y) , $$
which allows to conclude as before.
{\bf Part 2.2} The cases $ x \geq |c_{-+}|/N_- , 0 \geq y > - c_{+-}/N_+ $ or $ 0 \le x < |c_{-+}|/N_- , y \leq - c_{+-}/N_+ $ are treated analogously to Part 1.2.
{\bf Part 3} Suppose now that $ x \le - c_{++} /N_+ , y \geq - c_{- -} /N_- . $ Then
$$ A V (x, y ) = A^1 V_{-+} (x, y) + A^2 V_{-+} (x,y ) = a_{-+} x^2 + b_{-+} xy + d_{-+} y^2 + L_{-+} (x,y ) ,$$
where $L_{-+} $ is a polynomial of degree $1$ and where
\begin{eqnarray*}
a_{-+} &=& -2 \nu_+ p , \\
b_{-+} &=& (c_{++}^* - c_{--}^* ) ( \nu_+ + \nu_- - c_{--} ) + 2 p c_{-+} \\
d_{-+} & =& - c_{-+} (c_{++}^* + c_{--}^* ) .
\end{eqnarray*}
Notice that by choice of $p,$ $ b_{-+} > 0 .$ The conclusion of this part follows analogously to the previous parts 1.1 and 2.1.
{\bf Part 4} Suppose finally that $ x \le - c_{++}/N_+ , y \le - c_{+ -} /N_+ . $ Then
$$ A V (x, y ) = A^1 V_{--} (x, y) + A^2 V_{--} (x,y ) = a_{--} x^2 + b_{--} xy + d_{--} y^2 + L_{--} (x,y ) ,$$
where $L_{--} $ is a polynomial of degree $1$ and where
\begin{eqnarray*}
a_{--} &=& -2 \nu_+ p , \\
b_{--} &=& (c_{++}^* - c_{--}^* ) (\nu_+ + \nu_- ) \\
d_{--} & =& - 2 \nu_- q ,
\end{eqnarray*}
leading to the same conclusion as in the previous parts.
\end{proof}
As a consequence of Proposition \ref{prop:lyapunov}, the process $X$ is stable in the sense that it necessarily possesses invariant probability measures, maybe several of them. The uniqueness of the invariant probability measure together with the Harris recurrence will follow from the following local Doeblin type lower bound.
\begin{prop}\label{thm:Doeblin}
For all $ T> 0 $ and for all $z_* = (x_*, y_*) \in \mathbb {R}^2 $ the following holds. There exist $R > 0 , $ an open set $ I \subset \mathbb {R}^2 $ with strictly positive Lebesgue measure and a constant $\beta \in (0, 1), $ depending on $I , R$ and the coefficients of the system with
\begin{equation}\label{doblinminorization}
P_{T} (z , dz' ) \geq \beta 1_C (z) \nu ( dz') ,
\end{equation}
where $ C = B_R ( z_* ) $ is the (open) ball of radius $R$ centred at $z_* ,$ and where $ \nu $ is the uniform probability measure on $ I.$
\end{prop}
\begin{proof}
We start with the case $ \nu_+ \neq \nu_- ,$ under the assumption that $ c_{++}, c_{--}, c_{+-}, c_{-+} \neq 0 . $ In this case, \cite{Clinet} in the proof of their Lemma 6.4 establish the lower bound \eqref{doblinminorization} for the four-dimensional Markov process
$ \bar X = (X_{++}, X_{+-}, X_{-+}, X_{--} ) $ given by
$$
X_{++} (t) = e^{- \nu_+ t } X_{++} (0) + \frac{c_{++}}{N_+} \sum_{i=1}^{N_{+}}\int_0^t e^{- \nu_+ ( t- s) } Z_{+}^i (d s) , $$
$$ X_{-+} (t) = e^{- \nu_+ t } X_{-+} (0)+ \frac{c_{-+}}{N_-} \sum_{j=1}^{N_{-}}\int_0^t e^{- \nu_+ ( t- s) } Z_{-}^j (ds),$$
$$X_{+-} (t) = e^{- \nu_- t } X_{+-} (0) + \frac{c_{+-}}{N_+} \sum_{i=1}^{N_{+}}\int_0^t e^{- \nu_- ( t- s) } Z_{+}^i (ds) ,$$
$$ X_{--} (t) = e^{- \nu_- t } X_{--} (0)+ \frac{ c_{--}}{N_-} \sum_{j=1}^{N_{-}}\int_0^t e^{- \nu_- ( t- s) } Z_{-}^j (ds) ,
$$
where $ X_{++} (0) + X_{-+} (0) = X_+ (0) , X_{--} (0) + X_{+-} (0) = X_- (0) .$
More precisely, they show that for any $ \bar z_* \in \mathbb {R}^4 ,$ there exist $\bar R > 0 , $ an open rectangle $ \bar I \subset \mathbb {R}^4 $ with strictly positive Lebesgue measure and a constant $\bar \beta \in (0, 1), $ such that
$$\bar P_{T} (\bar z , d\bar z' ) \geq \bar \beta 1_{\bar C} (\bar z) \bar \nu ( d \bar z') ,
$$
where $ \bar C = B_R ( \bar z_* ) $ is the (open) ball of radius $\bar R$ centred at $\bar z_* ,$ and where $ \bar \nu $ is the uniform probability measure on $ \bar I.$ The above formula can be interpreted in the following way: For any $ \bar z \in \bar C, $ with probability $\bar \beta, $ the law of $ \bar X (T) $ is equal to the law of $ U = (U_1, U_2, U_3, U_4) $ where $ U $ is a uniform random vector on $ \bar I .$ Since $ \bar I $ is supposed to be a rectangle, this implies in particular the independence of its coordinates $ U_1, \ldots , U_4.$
Notice that we have $ X (T) = A \bar X (T) ,$ where
$$ A = \left( \begin{array}{cccc}
1&0&1&0\\
0&1&0&1
\end{array}
\right) .$$
We now show how the above result implies the local lower bound for the original process $X.$ For that sake
let $ z_* \in \mathbb {R}^2 $ be arbitrary and fix any $ \bar z_* \in \mathbb {R}^4 $ such that $A \bar z_* = z_* .$ Let $\bar R$ be the associated radius and choose $R$ such that $ B_R ( z_* ) \subset A B_{\bar R} ( \bar z_*) .$ Then for all $ z \in B_R (z_* ) $ and $ \bar z \in B_{\bar R} ( z_* ) $ with $ A \bar z = z, $
$$ P_z ( X (T) \in \cdot ) = P_{\bar z} (A \bar X (T) \in \cdot ) \geq \bar \beta \P ( A U \in \cdot ) .$$
Since
$$ A U = \left( \begin{array}{c}
U_1 + U_3 \\
U_2 + U_4
\end{array}
\right) ,$$
by independence of the coordinates $ U_1, \ldots , U_4,$ this implies the desired result for the two-dimensional Markov process $X $ as well.
We finally deal with the case $ \nu_+ = \nu_- $ and $( c_{++},c_{+-} ) ,( c_{-+},c_{--}) $ linearly independent. Fix $ z_* = (x_*, y_* ) $ and $ M> |x_*|+ |y_*| $ arbitrarily and let $ H := \{ z = (x, y ) : |x| \le M, |y|\le M \} .$ Recall \eqref{eq:aplus} and introduce finally the event $E$ given by
\begin{itemize}
\item $\pi^1_{+} ( [ 0,T] \times [ 0,a^1_{+}]) =1,$
\item $\pi_{+}^1 ( [ 0,T] \times ]a^1_{+}, a^1_{+}+c_{++} + M ) =0,$
\item $\pi_{+}^i ( [ 0,T] \times [ 0, a^i_{+}+c_{++} + M ) =0$ for all $ 2 \le i \le N_+,$
\item $\pi^1_{-} ( [ 0,T] \times [ 0,a^1_{-}]) =1,$
\item $\pi_{-}^1 ( [ 0,T] \times ] a^1_{-}, a^1_{-}+c_{+-} + M ) =0,$
\item $\pi_{-}^j ( [ 0,T] \times [ 0, a^j_{-}+c_{+-} + M ) =0$ for all $ 2 \le j \le N_- .$
\end{itemize}
Define the substochastic kernel
$$
Q^T_{z} ( A) =P_z( E\cap \{ X ({T}) \in A\} )=P( E) P_z( X ({T}) \in A | E) .
$$
The conditional law of $ X ({T}) $ given $ E,$ under $P_z,$ is equal to the law of
$$
Y_z ({T}) =ze^{-\nu_+ T}+e^{-\nu_+ U_{+}}\left(\begin{array}{c}c_{++}/N_+ \\c_{+-}/N_+ \end{array}\right)+ e^{-\nu_+ U_{-}}\left( \begin{array}{c}c_{-+}/N_-\\c_{--}/N_- .\end{array}\right) ,
$$
where the two jump-times $U_{+},U_{-}$ are independent uniform variables on $[ 0,T] .$ Since $$C=\left( \begin{array}{cc} c_{++}/N_+ & c_{-+}/N_-\\c_{+-}/N_+& c_{--}/N_- \end{array}
\right) $$
is invertible and the law of $( e^{-\nu_+ U_{+}},e^{-\nu_+ U_{-}})$ is equivalent with the Lebesgue measure on $[ e^{-\nu_+ T},1] ^2, $ the law of $ Y_z ({T})$ has density
$$
f_{z}:v\mapsto | det\; C|^{-1} f\circ C^{-1}( v-ze^{-\nu_+ T}),
$$
where $f$ is the density of $( e^{-\nu_+ U_{+}}, e^{-\nu_+ U_{-}})$. The density is positive on the interior of its support
$$
supp( Y_z ({T}))= e^{-\nu_+ T}z +C [ e^{-\nu_+ T},1]^2 .
$$
Since $C$ is a homeomorphism, it is an open mapping. Thus we can find balls $B_r ( v_{0})\subset B_{2r} ( v_{0}) \subset C [ e^{- \nu_+ T},1]^2 $ for all $T>1.$ Take now $T$ so large that $e^{-\nu_+ T}\sup_{v \in H} \| v \| <r.$ For such $T$ and all $ z \in H$ we have
$$
\overline{B}_r ( v_{0}) \subset e^{-\nu_+ T} z + B_{2r} ( v_{0})\subset supp( Y_z ({T})) .
$$
Note now that $H\times \overline{B}_r ( v_{0}) \ni (z,v) \mapsto f_{z}( v) $ is continuous, so the positivity of the density gives $\inf_{z \in H,v\in \overline{B}_r ( v_{0}) } f_{z}( v):=\alpha>0.$ We therefore conclude that
$$
Q^T_{z}( A)\geq P( E) \cdot \alpha \cdot \lambda ( A\cap B_r ( v_{0}) ),
$$
for all $z \in H,$ where $ \lambda $ denotes the Lebesgue measure on $\mathbb {R}^2.$ This proves the desired result.
\end{proof}
We do now dispose of all ingredients to conclude the proof of Theorem \ref{theo:harris}.
\begin{proof}[Proof of Theorem \ref{theo:harris}]
1) We apply Proposition \ref{thm:Doeblin} with $z_* = 0 .$ Let $R$ be the associated radius.
By Proposition \ref{prop:lyapunov}, we know that for a suitable compact set $K \subset \mathbb {R}^2, $ $X $ comes back to $K $ infinitely often almost surely. For $ z = ( x, y ), $ write
\begin{equation}\label{eq:flowy}
\varphi_t (z) = ( \varphi^{(1)}_t ( x) ,\varphi^{(2)}_t (y) ) = (e^{ - \nu_+ t}x , e^{- \nu_- t} y)
\end{equation}
for the flow of the process in between successiv jumps and let $ \| z\|_1 := |x| + |y|.$ Then
\begin{equation}
\sup_{z \in K, t \geq 0} \| \varphi_t (z) \|_1 := F < \infty \; \; \mbox{ and } \; \; \sup_{z \in K} \| \varphi_t (z) \|_1 \to 0
\end{equation}
as $t \to \infty .$ Therefore there exists $t_* $ such that $\varphi_t (z) \in B_{R } ( 0) $ for all $t \geq t_* , $ for all $ z \in K .$
Hence,
$$ \inf_{z\in K} P_z ( X ({t_* + s }) \in B_R ( 0 ), 0 \le s \le 2T ) > 0 . $$
Consequently, the Markov chain $(X ({kT}))_{k \in \mathbb {N}} $ visits $ B_{R } ( 0 )$ infinitely often almost surely. \\
The standard regeneration technique (see e.g. \cite{dashaeva}) allows to conclude that $(X ({kT}))_{k \in \mathbb {N}} $ and therefore $(X(t))_t $ are Harris recurrent. This concludes the proof of the Harris recurrence of the process.
2) The sampled chain $ (X ({kT }))_{k \geq 0 }$ is Feller according to Proposition \ref{prop:Feller}. Moreover it is $ \nu-$irreducible, where $ \nu $ is the measure introduced in Proposition \ref{thm:Doeblin}, associated with the point $z_* = (0,0) .$ Since $\nu$ is the uniform measure on some open set of strictly positive Lebesgue measure, the support of $\nu $ has non-empty interior. Theorem 3.4 of \cite{MT1992} implies that all compact sets are `petite' sets of the sampled chain. The Lyapunov condition established in Proposition \ref{prop:lyapunov} allows to apply Theorem 6.1 of \cite{MT1993} which implies the second assertion of the theorem.
\end{proof}
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\end{document}
\section{A priori estimates}
\begin{prop}\label{prop:2}
Grant Assumption \ref{ass:1}.
Any solution $(X^N_t)_{t\geq 0}$ to \eqref{eq:dyn} satisfies that
\begin{equation}\label{eq:nice}
\frac1N \sum_{i=1}^N \E \int_0^t f ( X^{N, i }_s ) ( (X^{N, i }_s)^2 + \sigma^2 ) ds \le \frac3N \sum_{i=1}^N \E ( (X^{N, i }_0)^2) + 4 \sigma^2 f ( \sqrt{2} \sigma ) t
\end{equation}
and
\begin{equation}
\frac1N \sum_{i=1}^N \E (X^{N, i }_t)^2 \le \frac1N \sum_{i=1}^N \E ( (X^{N, i }_0)^2) + \frac{4 \sigma^2}{3 } f( \sqrt{2} \sigma ) t.
\end{equation}
\end{prop}
\begin{proof}
For $ x = ( x^1, \ldots , x^N), $ put $ V(x) := \frac1N \sum_{i=1}^N (x^i)^2 $ and let $ V_t := V( X^N_t) .$ Then
\begin{multline*}
d V_t = \frac2N \sum_{i=1}^N b ( x^i ) x^i dt - \frac1N \sum_{i=1}^N (X^{N, i }_{t-})^2 \int_\mathbb {R} \int_0^\infty
{\bf 1}_{ \{ z \le f ( X^{N, i}_{s-}) \}} {\mathbf{N}}^i (ds,du, dz) \\
+ \frac{1}{N} \sum_{i=1}^N \sum_{ j \neq i } \int_0^t \int_\mathbb {R} \int_0^\infty [ \frac{2 u X^{N, i}_{s-}}{\sqrt{N}} + \frac{u^2 }{N} ] {\bf 1}_{ \{ z \le f ( X^{N, j}_{s-}) \}} {\mathbf{N}}^j (ds,du, dz) .
\end{multline*}
Taking expectation and writing $ v_t= \E V_t, $ this yields
\begin{equation}\label{eq:vt}
d v_t \le - \frac1N \sum_{i=1}^N E [ f ( X^{N, i }_t ) ( (X^{N, i }_t)^2 - \sigma^2 ) ] dt.
\end{equation}
Now we use that $ x^2 - \sigma^2 \geq \frac{x^2 + \sigma^2}{3} - \frac{4 \sigma^2}{3} {\bf 1}_{ x^2 \le 2 \sigma^2 } $ and the fact that $f$ is non-decreasing to deduce from this that
$$
\frac1N \sum_{i=1}^N \E \int_0^t f ( X^{N, i }_s ) ( (X^{N, i }_s)^2 + \sigma^2 ) ds \le 3 v_0 + 4 \sigma^2 f ( \sqrt{2} \sigma ) t,
$$
which is the first assertion, and that
$$ d v_t \le \frac{4 \sigma^2 }{3} f( \sqrt{2} \sigma ) ,$$
implying the second assertion.
\end{proof}
WE SHOULD ALSO SAY HERE THAT THIS IMPLIES THE EXISTENCE OF A SOLUTION !
\section{Convergence of the associated empirical measures}
We endow the set ${\mathbb D}(\mathbb {R}_+, \mathbb {R} )$ of c\`adl\`ag functions on $\mathbb {R}_+$ taking values in $\mathbb {R} $ with the topology of the Skorokhod convergence on compact time intervals,
see Jacod and Shiryaev \cite{js}.
\begin{theo}\label{theo:6}
Grant Assumption \ref{ass:1}. Consider a probability distribution $g_0$
on $\mathbb {R}$ such that $\int_\mathbb {R} y^2 g_0(dy) = v_0 <\infty$. For each $N\geq 1$,
consider the unique solution $(X^N_t)_{t\geq 0}$ to \eqref{eq:dyn} starting from
some i.i.d. $g_0$-distributed initial conditions $X^{N,i}_0$.
\vskip0.2cm
(i) The sequence of processes $(X^{N,1}_t)_{t\geq 0}$ is tight in ${\mathbb D}(\mathbb {R})$.
\vskip0.2cm
(ii) The sequence of empirical measures $ \mu_N=N^{-1}\sum_{i=1}^N \delta_{(X^{N,i}_t)_{t\geq 0}}$
is tight in ${\mathcal P}({\mathbb D}(\mathbb {R}))$.
\end{theo}
\begin{proof}
First, it is well-known that point (ii) follows from point (i) and the exchangeability
of the system, see Sznitman \cite[Proposition 2.2-(ii)]{s}. We thus
only prove (i). We consider a probability distribution $g_0$
on $\mathbb {R}_+$ such that $\int_0^\infty x g_0(dx)<\infty$ and, for each $N\geq 1$,
the unique solution $(X^N_t)_{t\geq 0}$ to \eqref{eq:dyn} starting from
some i.i.d. $g_0$-distributed initial conditions $X^{N,i}_0$.
To show that the family $((X^{N,1}_t)_{t\geq 0})_{N\geq 1}$ is tight ${\mathbb D}(\mathbb {R}_+)$,
we use the criterion of Aldous, see Jacod and Shiryaev \cite[Theorem 4.5 page 356]{js}. It is sufficient to prove that
\vskip0.2cm
(a) for all $ T> 0$, all $\varepsilon >0$,
$ \lim_{ \delta \downarrow 0} \limsup_{N \to \infty } \sup_{ (S,S') \in A_{\delta,T}}
\P ( |X_{S'}^{N, 1 } - X_S^{N , 1 } | > \varepsilon ) = 0$,
where $A_{\delta,T}$ is the set of all pairs of stopping times $(S,S')$ such that
$0\leq S \leq S'\leq S+\delta\leq T$ a.s.,
\vskip0.2cm
(b) for all $ T> 0$, $\lim_{ K \uparrow \infty } \sup_N
\P ( \sup_{ t \in [0, T ] } |X_t^{N, 1 }| \geq K ) = 0$.
\vskip0.2cm
To check (a), consider $(S,S')\in A_{\delta,T}$ and write
\begin{multline*}
X_{S'}^{N, 1 } - X_S^{N , 1 } =
- \int_S^{S'} \int_\mathbb {R} \int_0^\infty X^{ N, 1 }_{s- } {\bf 1}_{\{ z \le f ( X_{s- }^{N, 1} ) \}} {\mathbf{N}}^1 (ds, du, dz ) + \int_S^{S'} b(X^{N, 1 }_s) ds
\\
+ \frac{1}{ \sqrt{N} } \sum_{j=2}^N \int_S^{S'} \int_\mathbb {R} \int_0^\infty u {\bf 1}_{\{ z \le f ( X_{s- }^{N, j} ) \}}
{\mathbf{N}}^j (ds, du, dz ) ,
\end{multline*}
implying that
\begin{multline*}
|X_{S'}^{N, 1 } - X_S^{N , 1 }| \le | \int_S^{S'} \int_\mathbb {R} \int_0^\infty X^{ N, 1 }_{s- } {\bf 1}_{\{ z \le f ( X_{s- }^{N, 1} ) \}} {\mathbf{N}}^1 (ds, du, dz ) | \\
+ \delta + | \frac{1}{ \sqrt{N} } \sum_{j=2}^N \int_S^{S'} \int_\mathbb {R} \int_0^\infty u {\bf 1}_{\{ z \le f ( X_{s- }^{N, j} ) \}}
{\mathbf{N}}^j (ds, du, dz ) | =: I_{S, S'} + |J_{S, S'} |,
\end{multline*}
since $b$ is bounded.
We first note that $I_{S,S'}>0$ implies that
$\tilde I_{S,S'}:=
\int_S^{S'} \int_\mathbb {R} \int_0^\infty {\bf 1}_{\{ z \le f ( X_{s- }^{N, 1} ) \}} {\mathbf{N}}^i (ds, du, dz)\geq 1$, whence
$$
\P ( I_{S, S'} > 0 )\leq \P (\tilde I_{S,S'}\geq 1)\leq \E[\tilde I_{S,S'}]\le
\E\mathcal{B}ig[ \int_S^{S+\delta} f( X_s^{N, 1 } ) ds \mathcal{B}ig] \le \delta,
$$
since $ f$ is bounded.
We proceed similarly to check that
$$
\P ( |J_{S, S'}| \geq \varepsilon ) \le \frac{1}{\varepsilon^2} \E[(J_{S,S'})^2 ]\leq \frac{\sigma^2}{N\varepsilon^2 } \sum_{j=2}^N\E\mathcal{B}ig[ \int_S^{S+\delta} f( X_s^{N, j} ) ds\mathcal{B}ig]
\le \frac{\sigma^2}{\varepsilon^2} \| f \|_\infty \delta .
$$
To check (b), we write, using the same notation as above,
$$ \sup_{s \le T} | X_s^{N, 1}| \le \int_0^T \int_\mathbb {R} \int_0^\infty |X^{ N, 1 }_{s- } | {\bf 1}_{\{ z \le f ( X_{s- }^{N, 1} ) \}} {\mathbf{N}}^1 (ds, du, dz ) + \| b\|_\infty T + \sup_{s \le T }|J_ {0, s}| ,$$
where
\begin{multline*}
\E \int_0^T \int_\mathbb {R} \int_0^\infty |X^{ N, 1 }_{s- } | {\bf 1}_{\{ z \le f ( X_{s- }^{N, 1} ) \}} {\mathbf{N}}^1 (ds, du, dz ) = \E \int_0^T |X^{ N, 1 }_{s } | f ( X_{s- }^{N, 1} ) ds\\
\le \E \int_0^T [ (X^{ N, 1 }_{s } )^2 + 1 ] f ( X_{s- }^{N, 1} ) ds \le C_T ,
\end{multline*}
where $C_T$ does not depend on $N, $ which follows from \eqref{eq:nice}. Moreover, since $ J (0, t ) $ is a square integrable local martingale,
$$ \E \sup_{ s \le T} | J_{0, s }| \le C \E [J_{0, T }^2] \le C \sigma^2 \int_0^T \E [ f( X_s^{N, 1} ) ds \le C \sigma^2 \|f\|_\infty T ,$$
concluding the proof.
\end{proof}
\begin{rem}
We can probably get better results if we suppose that
$$ x b(x) \le - C x^2 $$
for all $|x| \geq K $ for a suitable $K.$
\end{rem}
\section{The limit process}
We start with some informal discussion of how the limit process of the particle system $X^N $ should a priori look like, if it exists. So we suppose that there exists a process $ (Y^1, Y^2 , Y^3, \ldots ) \in {\mathbb D} ( \mathbb {R}_+, \mathbb {R})^\mathbb {N} $ such that for all $ K > 0, $ we have weak convergence $ {\mathcal L }(X^{N, 1, }, \ldots , X^{N, K} ) \to {\mathcal L} ( Y^1, \ldots, Y^K) $ in ${\mathbb D} (\mathbb {R}_+, \mathbb {R} ) .$ Since the law of the $N-$particle system $ (X^{N, 1}, \ldots, X^{N, N} ) $ is symmetric, the law of $ Y $ must be exchangeable, that is, for all finite permutations $\pi, $ we have that
$ {\mathcal L} ( Y^{\pi ( 1) }, Y^{\pi ( 2) } , \ldots ) = {\mathcal L} (Y).$ In particular, the random limit
$$ \mu := \lim_{N\to \infty}\frac1N \sum_{i=1}^N \delta_{Y^i } $$
exists.
Supposing that $ \mu_N$ converges, it necessarily converges towards $\mu. $ Therefore, $Y_t$ should solve the limit system \begin{equation}\label{eq:dynlimit}
Y^i_t = Y^i_0 + \int_0^t b(Y^i_s) ds
- \int_0^t \int_\mathbb {R} \int_0^\infty Y^i_{s- } {\bf 1}_{ \{ z \le f ( Y^i_{s-}) \}} {\mathbf{N}}^i (ds,du, dz) + \sigma \int_0^t \sqrt{ \mu_t ( f) } d B_t , i \in \mathbb {N},
\end{equation}
where $(B_t)_{t\geq 0}$ is a standard one-dimensional Brownian motion which is independent of the Poisson random measures.
{\it Discussion of $ \mu.$}
The presence of the common Brownian motion $ B$ implies that even in the large population limit, particles do not become independent. However, they are conditionally independent given the Brownian motion path.
Therefore, $\mu $ will be the conditional law of the solution given the Brownian path, that is, $P-$almost surely
$$ \mu ( \cdot ) = P ( Y^i \in \cdot | (B_t)_{ t \geq 0 } ) = P( Y^i \in \cdot | B ) ,$$
for any $ i \in \mathbb {N} .$ The conditionning on $B$ reflects the correlations between the particles.
We are now going to give a precise mathematical definition of what we call a {\it strong solution of the non-linear limit process}.
\begin{defin}
Fix some $T > 0 $ and let $ (\Omega, ({\mathcal F}_t)_{ t \in [0, T ] }, P) $ be a filtered probability space on which are defined an $ ({\mathcal F}_t)_{ t \in [0, T ] }-$Poisson random measure $ {\mathbf{N}} ( ds, du, dz ) $ and
an $({\mathcal F}_t)_{ t \in [0, T ] }-$Brownian motion $ B$ in dimension one. We say that an $({\mathcal F}_t)_{ t \in [0, T ] }-$adapted process $ (Y_t)_{ t \in [0, T ] } $ is a strong solution of the non-linear limit problem if
\begin{equation}\label{eq:dynlimit}
Y_t = Y_0 + \int_0^t b(Y_s) ds
- \int_0^t \int_\mathbb {R} \int_0^\infty Y_{s- } {\bf 1}_{ \{ z \le f ( Y_{s-}) \}} {\mathbf{N}} (ds,du, dz) + \sigma \int_0^t \sqrt{ \mu_t ( f) } d B_t ,
\end{equation}
where the process $ \mu_t $ is $({\mathcal F}_t)_{ t \in [0, T ] }-$adapted such that $P-$almost surely,
$$ \mu ( \cdot ) = P ( Y \in \cdot | B ) .$$
\end{defin}
To prove the well-posedness of this limit equation \eqref{eq:dynlimit} is not evident. The common jumps of the particles, due to their scaling with $ 1/\sqrt{N} $ and the fact that they are centred, by the Central Limit Theorem,
create the single Brownian motion
$ B_t $ which is underlying each particle's motion and which induces a common noise factor for all particles. To prove the trajectorial uniqueness of \eqref{eq:dynlimit}, due to the presence of jumps and of the diffusive term at the same time demands actually some non-trivial work. Roughly speaking, the jump terms demand to work in an $L^1 - $framework, and the diffusive terms to work in an $L^2-$framework. Carl Graham \cite{carl} in his important paper of 1992 proposes a unified approach to deal both with jump and with diffusion terms in a non-linear framework, and we shall rely on his ideas in the sequel. The presence of the random volatility term $ \mu_t ( f) $ which involves conditional expectation causes however additional technical difficulties in our present frame, due to the fact that conditional expectation does not behave in a continuous way.
Another difficulty comes from the fact that the jumps do behave in a
``non-Lipschitz"-way comparable to the TCP process - indeed, even if two particles have been close by just before jumping, and if one of the particles jumps but not the other, the distance between the two right after jumping migth be very big. For this reason, a classical Wasserstein-$1-$coupling is difficult for the jump terms.
In order to overcome these difficulties, we need to work under the following additional assumption on the jump rate of each particle.
\begin{ass}\label{ass:2}
$f \in C^3(\mathbb {R} , \mathbb {R}_+ )$ is strictly increasing, bounded and lowerbounded. Moreover, $\sup_{x} [f'(x)/f(x)+|f''(x)|/f'(x) + |f''' (x)| /| f'' (x)| + |b' (x)| / f'(x) ]<\infty$.
\end{ass}
As a consequence, we have that for a suitable constant $C,$
$$ |f'' ( x) - f'' (y) | + |f'(x) - f' (y) | + |b(x) - b(y) | \le C | f(x) - f(y) |.$$
\begin{theo}\label{prop:42}
Suppose that $f$ satisfies Assumptions \ref{ass:1} and \ref{ass:2}.
Then pathwise uniqueness holds for the nonlinear SDE \eqref{eq:dynlimit}.
\end{theo}
\begin{proof}
Consider two solutions $ (Y_t)_{t \geq 0}$ and $ (\tilde Y_t)_{t \geq 0 } , $ defined on the same probability space and driven by the same Poisson random measure ${\mathbf{N}} $ and the same Brownian motion $ B,$ and with $ Y_0 = \tilde Y_0.$ We consider $ Z_t := f(Y_t) - f( \tilde Y_t) ,$ for all $ t \le T.$ Then
\begin{multline*}
Z_t = \int_0^t \left( b ( Y_s ) f' ( Y_s ) - b ( \tilde Y_s) f' ( \tilde Y_s) \right) ds +\frac12 \int_0^t ( f'' ( Y_s) \mu_s ( f) - f'' ( \tilde Y_s ) \tilde \mu_s ( f) ) \sigma^2 ds \\
+ \int_0^t ( f' ( Y_s) \sqrt{\mu_s (f)} +f' (\tilde Y_s ) \sqrt{ \tilde \mu_s (f)} ) \sigma d B_s \\
- \int_0^t \int_\mathbb {R} \int_0^\infty \left( f (Y_{s- }) - f( \tilde Y_{s-}) \right) {\bf 1}_{ \{ z \le f ( Y_{s-}) \wedge f ( \tilde Y_{s-}) \}} {\mathbf{N}} (ds, du, dz)\\
+ \int_0^t \int_\mathbb {R} \int_0^\infty [f(0)- f( Y_{s-} )] {\bf 1}_{ \{ f ( \tilde Y_{s-} ) < z \le f ( Y_{s-} ) \}} {\mathbf{N}} (ds, du, dz) \\
+ \int_0^t \int_\mathbb {R} \int_0^\infty [ f( \tilde Y_{s-} ) - f(0) ] {\bf 1}_{ \{ f ( Y_{s-} ) < z \le f ( \tilde Y_{s-} ) \}} {\mathbf{N}} (ds,du, dz) = : A_t + M_t +{\mathbb D}elta_t ,
\end{multline*}
where $ A_t $ denotes the bounded variation part of the evolution, $M_t$ the martingale part and $ {\mathbb D}elta_t$ the sum of the three jump terms. Notice that
$$M_t= \int_0^t ( f' ( Y_s) \sqrt{\mu_s (f)} -f' (\tilde Y_s ) \sqrt{ \tilde \mu_s (f)} ) \sigma d B_s$$
is a square integrable martingale since $ f$ is bounded.
We wish to obtain a control on $ |Z^* _t | := \sup_{ s\le t } |Z_s | .$ We first take care of the jumps of $ |Z_t|.$ Notice first that, since $f$ is bounded,
\begin{multline*}
{\mathbb D}elta (x,y):= (f(x) \wedge f(y)) | f (x) - f(y ) | + | f (x ) - f( y ) | \; \mathcal{B}ig| | f ( x \wedge y ) - f(0) | - | f (x) - f(y ) | \mathcal{B}ig| \\
\le C | f (x) - f( y ) |
\end{multline*}
implying that
$$ \E \sup_{s \le t } | {\mathbb D}elta_s | \le C \E \int_0^t | f(Y_s^i) - f(\tilde Y_s^i ) | ds \le C t \, \E |Z_t^* | . $$
Moreover, for a constant $C$ depending on $\sigma^2 ,$ $\| f \|_\infty , \| f'\|_\infty, \| f'' \|_\infty $ and $ \| b \|_\infty , $
\begin{multline*}
\E \sup_{ s \le t } | A_s | \le C \int_0^t \E |b ( Y_s ^i ) - b ( \tilde Y_s^i ) | ds + C \int_0^t \E |f' ( Y_s ^i ) - f' ( \tilde Y_s^i ) | ds
\\
+ C \left[ \int_0^t | f'' ( Y_s ^i ) -f '' ( \tilde Y_s^i ) | ds + \int_0^t | \mu_s ( f) - \tilde \mu_s ( f) | ds \right] .
\end{multline*}
We know that $ |b ( Y_s ^i ) - b ( \tilde Y_s^i ) | + |f' ( Y_s ^i ) - f' ( \tilde Y_s^i ) | + |f'' ( Y_s ^i ) - f'' ( \tilde Y_s ) | \le C |f ( Y_s ) - f ( \tilde Y_s ) |= C | Z_s| .$ Therefore,
$$ \E \sup_{ s \le t } | A_s | \le C \E \left[ \int_0^t | Z_s | ds + \int_0^t | \mu_s ( f) - \tilde \mu_s ( f) | ds \right].$$
Moreover,
$$ |\mu_s (f)- \tilde \mu_s (f) | = \mathcal{B}ig| \E \left( f ( Y_s ) - f ( \tilde Y_s ) | B \right) \mathcal{B}ig| \le \E \left( | f ( Y_s ) - f ( \tilde Y_s^i )| | B \right) = \E ( |Z_s| | B) ,$$
and thus,
$$ \E \int_0^t | \mu_s ( f) - \tilde \mu_s ( f) | ds \le \E \int_0^t |Z_s| ds \le t \E | Z^*_t| .$$
Putting all these upper bounds together we conclude that for a constant $C$ not depending on $t,$
$$ \E \sup_{s \le t} |A_s| \le C t \E |Z_t^*| .$$
Finally, we treat the martingale part using the Burkholder-Davis-Gundy inequality, and we obtain
$$
\E \sup_{s \le t} |M_s| \le C \E \left[ \left( \int_0^t (f' (Y_s ) \sqrt{ \mu_s ( f) } - f' (\tilde Y_s ) \sqrt{ \tilde \mu_s ( f) })^2 ds \right)^{1/2}\right].$$
But
\begin{multline}\label{eq:varquadratique}
(f' (Y_s ) \sqrt{ \mu_s ( f) } - f' (\tilde Y_s ) \sqrt{ \tilde \mu_s ( f) })^2 \le
C \left[ ( (f' (Y_s ) - f' (\tilde Y_s ))^2 + (\sqrt{ \mu_s ( f) } - \sqrt{ \tilde \mu_s ( f) })^2 \right] \\
\le C | Z_t^*|^2 + C (\sqrt{ \mu_s ( f) } - \sqrt{ \tilde \mu_s ( f) })^2 ,
\end{multline}
where we have used once more that $ | f' (x) - f' (y) | \le C | f(x) - f(y) | $ and that $f$ and $f'$ are bounded.
Finally, since $ f$ is lowerbounded,
$$| \sqrt{ \mu_s ( f) } - \sqrt{ \tilde \mu_s ( f) }|^2 \le C | \mu_s ( f) - \tilde \mu_s ( f) |^2 \le C \left( \E ( |Z_s^*| | B ) \right)^2 \le C \left( \E ( |Z_t^*| | B ) \right)^2,$$
since $ |Z_s^* | \le | Z_t^*| ,$ which implies the control of
$$ \E \sup_{s \le t} |M_s| \le C \sqrt{t} \E | Z_t^* | .$$
The above upper bounds imply that, for a constant $C$ not depending on $t, $
$$ \E |Z_t^*| \le C (t + \sqrt{t} ) \E | Z_t^* | ,$$
and therefore, for $ t $ sufficiently small, $ \E |Z_t^*| = 0,$ which implies the assertion.
\end{proof}
The same ideas now allow us to prove that
\begin{theo}
Suppose that $f$ satisfies Assumptions \ref{ass:1} and \ref{ass:2}.
Then there exists a strong solution of the nonlinear SDE \eqref{eq:dynlimit}.
\end{theo}
\begin{proof}
The proof is done using a classical Picard-iteration. Therefore we introduce the sequence of processes $ Y_t^{[0] } \equiv Y_0 , $ and
$$ Y^{[n+1]}_t := Y_0 + \int_0^t b( Y_s^{[n]} ) ds - \int_\mathbb {R} \int_0^\infty Y^{[n+1]}_{s- } {\bf 1}_{ \{ z \le f ( Y^{[n]}_{s-}) \}} {\mathbf{N}} (ds,du, dz) + \sigma \int_0^t \sqrt{ \mu^n_t ( f) } d B_t ,$$
where
$$ \mu^n = P ( Y^{[n]} \in \cdot | B ) .$$
Then the same strategy as the one of the proof of Proposition \ref{prop:42} allows to show that $$\delta_t^n := \E \sup_{s \le t } | f ( Y_s^{[n]} ) - f( Y_s^{[n-1]} ) | $$ satisfies
$$ \delta_t^n \le C (t + \sqrt{t} ) \delta_t^{n-1} ,$$
for all $ n \geq 1 , $ for a constant $C$ only depending on the parameters of the model, but not on $ n, $ neither on $t. $ Choose $t_1 $ such that
$$ C (t_1 + \sqrt{t_1} ) \le \frac13.$$
Since $ \sup_{s \le t_1 } | f ( Y_s^{[0]} ) | = f ( Y_0) \le \| f \|_\infty, $ we deduce from this
that
$$ \delta_{t_1}^n \le 3^{- n } \| f \|_\infty .$$
We want to deduce from this that the sequence of processes $ (f(Y^{[n]} ))_n $ converges in the Skorokhod space $D( [0, t_1], \mathbb {R} ) .$ For that sake, for c\`adl\`ad functions $\eta, \xi\in D([0,t_1 ],\mathbb {R})$ we consider the distance
$d_S(\eta,\xi)$ defined by
\begin{equation}
\label{def:skorohod_like_dist}
d_S(\eta,\xi)=\inf_{\phi\in I}\left\{\|\phi\|_{[0,t_1],*}\varepsilone \|\eta-\xi(\phi)\|_{[0,t_1],\infty}\right\},
\end{equation}
where $I$ is the set of non-decreasing functions $\phi:[0,t_1]\to [0,t_1]$ satisfying $\phi(0)=0$ and $\phi(t_1)=t_1$ and where for any function $\phi\in I$ the norm $\|\phi\|_{[0,t_1],*}$ is defined as
$$
\|\phi\|_{[0,t_1],*}=\sup_{0\leq s<t\leq t_1}\log\left(\frac{\phi(t)-\phi(s)}{t-s}\right).
$$
The metric $d_S(\cdot,\cdot)$ is equivalent to the classical Skorokhod distance. More importantly the metric space $(D([0,t_1],\mathbb {R}),d_S)$ is Polish, see for instance \cite{Billingsley:68}.
Clearly,
$$ d_S ( \eta, \xi ) \le \sup_{s \le t_1} | \eta ( s) - \xi ( s) | .$$
Therefore,
$$ \sum_{n \geq 1} \P ( d_S ( f ( Y^{[n]} ), f ( Y ^{[n-1]} )) > 2^{-n} ) \le \sum_{n \geq 1} \P ( \sup_{s \le t_1 } | f ( Y_s^{[n]} ) - f( Y_s^{[n-1]} ) | > 2^{-n} ) \le \sum_{n } 2^n \delta_{t_1}^n < \infty ,$$
implying that almost surely,
\begin{equation}\label{eq:fort}
d_S ( f ( Y^{[n]} ), f ( Y ^{[n-1]} )) \le \sup_{s \le t_1 } | f ( Y_s^{[n]} ) - f( Y_s^{[n-1]} ) | \le 2^{-n } ,
\end{equation}
for sufficiently large $n.$ This implies that almost surely, the sequence of processes $ (f(Y^{[n]} ))_n $ is a Cauchy sequence, hence it converges in the Skorokhod space $D( [0, t_1], \mathbb {R} ) $ to a limit process $ f ( Y^\infty ) .$ $f$ being continuous and
strictly increasing, this implies the almost sure convergence of $ Y^{[n]} \to Y $ in $D( [0, t_1], \mathbb {R} ) .$ Finally, \eqref{eq:fort} also implies that
$$ \sup_{s \le t_1} | f ( Y_s^{[n]} ) - f (Y_s) | \to 0 $$
almost surely, as $n \to \infty .$
It remains to prove that $Y$ is solution of the limit equation which follows by standard arguments (note that the jump term does not cause troubles because it is of finite activity).
The most important point is to notice that
$$ \mu_t^n ( f) = E ( f ( Y_t^{[n]} | B ) \to E ( f (Y_t) | B ) $$
almost surely, which follows from the almost sure convergence of $ f ( Y_t^{[n]} ) \to f (Y_t ) ,$ using dominated convergence.
Finally, once the convergence is proven on the time interval $ [0, t_1 ], $ we can proceed iteratively over successive intervals $ [ k t_1, (k+1) t_1] $ to conclude the proof.
\end{proof}
\begin{rem}
So the above result implies trajectoral uniqueness. Does this also imply uniqueness in law of the limit process? What about Yamada-Watanabe theorem???
\end{rem}
\section{Convergence to the limit system}
\subsection{An auxiliary particle system and coupling using KMT}
Let $N_t$ be a standard Poisson process of rate $1,$ independent of ${\mathbf{N}}^1 .$ Let moreover $ U_n , n \geq 1, $ be i.i.d. variables, distributed according to $ \mu, $ independent of
everything else. Put finally $ Z_t := \sum_{n = 1}^{N_t} U_n , $ which is a centered compound Poisson process.
Then we can rewrite the dynamics of $ X_t^{N, 1 } $ as
$$ X_t^{N, 1 } = X_0^{N, 1 } + \int_0^t b( X^{N, 1}_s ) ds - \int_0^t \int_\mathbb {R} \int_0^\infty
X^{N, 1}_{s-} {\bf 1}_{ \{ z \le f ( \tilde X^{N, 1}_{s-}) \}} {\mathbf{N}}^1 (ds,du, dz) + \frac{1}{\sqrt{N} } Z_{A_t^{N, X} } , $$
where
$$ A_t^{N, X} = \sum_{ j = 2}^N \int_0^t f ( X_s^{N, j } ) ds . $$
The important point is that we can couple the centered compound Poisson process $ Z$ with a Brownian motion. Indeed,
\begin{lem}\label{lem:KMT}
The centered compound Poisson process can be constructed on the same sample space as $ \sigma B_t , $ $ \sigma^2 = Var (U_1) , $ such that
$$ \sup_{t \geq 0} \frac{ | Z_t - \sigma B_t|}{ \log t \varepsilone 2 } \le K < \infty $$
almost surely, where $K$ is a random variable having exponential moments.
\end{lem}
Applying the above result, we will show that $X^{N, 1 }$ behaves, for large $N,$ as the following process
\begin{eqnarray}\label{eq:dynapprox}
Y^{N, 1}_t &= & X^{N,1}_0 + \int_0^t b(Y^{N, 1}_s ) ds - \int_0^t \int_\mathbb {R} \int_0^\infty
Y^{N, 1}_{s-} {\bf 1}_{ \{ z \le f ( Y^{N, 1}_{s-}) \}} {\mathbf{N}}^1 (ds,du, dz) \\
&&+\frac{\sigma}{\sqrt{N} } B_{A_t^{N, Y}} ,
\nonumber
\end{eqnarray}
where $B$ is the standard one-dimensional Brownian motion of Lemma \ref{lem:KMT}, where the time change is given by
$$ A_t^{N, Y}= \sum_{j=2}^N \int_0^t f ( Y_s^{N, j} ) ds ,$$
and where the $ Y^{N, j } $ follow the same dynamics as $ Y^{N, 1 },$ see \eqref{eq:dynapproxbis} below.
Let us now describe more in detail the coupling we are going to construct. Based on Lemma \ref{lem:KMT}, we rewrite
\begin{equation}\label{eq:dec}
X^{N, 1}_t = \tilde X^{N, 1 }_t + R_t^N
\end{equation}
where
\begin{equation}\label{eq:rewrite}
\tilde X^{N, 1 }_t = X_0^{N, 1 } + \int_0^t b( X^{N, 1}_s ) ds
- \int_0^t \int_\mathbb {R} \int_0^\infty
X^{N, 1}_{s-} {\bf 1}_{ \{ z \le f ( \tilde X^{N, 1}_{s-}) \}} {\mathbf{N}}^1 (ds,du, dz) + \frac{\sigma}{\sqrt{N} } B_{A_t^{N, X} }
\end{equation}
and
$$ R_t^N = \frac{1}{\sqrt{N} } ( Z_{A_t^{N, X} } -\sigma B_{A_t^{N, X} } ) \mbox{ is such that } | R_t^N | \le \frac{1}{\sqrt{N} } \log ( \|f\|_\infty N ) K \le C N^{-1 /2 } \log N K ,$$
with $K$ is the random variable of Lemma \ref{lem:KMT}. Notice that the martingale part in \eqref{eq:rewrite} can be written as
$$\frac{1}{\sqrt{N} } B_{A_t^{N, X}} = \int_0^t \sqrt{ \frac1N \sum_{j=1, j \neq 1 }^N f ( X_s^{N, j} ) } d W_s , $$
for some standard one-dimensional Brownian $W.$
Let us now precise the dynamics of \eqref{eq:dynapprox}. We take $\tilde {\mathbf{N}}^i (ds,du, dz) , 2 \le i \le N, $ i.i.d. Poisson random measures, having intensity measure $ ds \mu ( du ) dz $ each, which are independent of $ {\mathbf{N}}^1 , $ of the compound Poisson process $Z $ and of the Brownian motion $W.$ We let $ Y^{N, 1 } $ be the strong solution of the stochastic differential equation driven by $ W, {\mathbf{N}}^1, \tilde {\mathbf{N}}^i, 2 \le i \le N , $ given by
\begin{eqnarray}\label{eq:dynapproxbisbis}
Y^{N, 1}_t &= & X^{N,1}_0 + \int_0^t b(Y^{N, 1}_s ) ds - \int_0^t \int_\mathbb {R} \int_0^\infty
Y^{N, 1}_{s-} {\bf 1}_{ \{ z \le f ( Y^{N, 1}_{s-}) \}} {\mathbf{N}}^1 (ds,du, dz) \\
&&+ \sigma \int_0^t \sqrt{ \frac1N \sum_{j=1, j \neq 1 }^N f ( Y_s^{N, j} ) } d W_s,
\nonumber
\end{eqnarray}
and we complete \eqref{eq:dynapproxbisbis} by
\begin{eqnarray}\label{eq:dynapproxbis}
Y^{N, i}_t &= & X^{N,i}_0 + \int_0^t b(Y^{N, i}_s ) ds - \int_0^t \int_\mathbb {R} \int_0^\infty
Y^{N, i}_{s-} {\bf 1}_{ \{ z \le f ( Y^{N, i}_{s-}) \}} \tilde {\mathbf{N}}^i (ds,du, dz) \\
&&+ \sigma \int_0^t \sqrt{ \frac1N \sum_{j=1, j \neq i }^N f ( Y_s^{N, j} ) } d W_s, \; 2 \le i \le N .
\nonumber
\end{eqnarray}
\begin{rem}
We stress that the above coupling is constructed for the evolution of the first particle $ X^{N, 1 }$ only : we express the small jumps of $ X^{N, 1}$ by the means of a compound Poisson process which is then approximated by a Brownian motion. And then we use this same Brownian motion to construct the first component $ Y^{N, 1 } $ of the auxiliary particle system $ Y^N.$ Notice that $ Y^{N, 1 } $ is coupled to $ X^N$ since it uses the same Poisson random measure $ {\mathbf{N}}^1 $ and the same Brownian motion $ W.$ It is this way that $ X^N $ and $Y^N $ are coupled. Of course, by exchangeability of the particles, it is not important which particle we take as a representative one - but it is important to notice that the coupling does indeed depend on this choice.
\end{rem}
In the sequel, we shall consider $ Z^{N, 1} _t := f(Y^{N, 1}_t) - f( X^{N, 1}_t) ,$ for all $ t \le T.$
Using the decomposition of $ X^{N, 1} = \tilde X^{N, 1 } + R^N $ of \eqref{eq:dec}, we have
$$ | f( X_t^{N, 1 } ) - f ( \tilde X^{N, 1 }_t ) | \le \| f' \|_\infty C N^{-1 /2 } (\log N) K .$$
As a consequence, it remains to control
$$ \tilde Z_t^{N, 1 } := f ( Y^{N , 1 }_t ) - f ( \tilde X^{N, 1 }_t ) ,$$ which is done using Ito's formula. The same arguments as those given in the proof of Theorem \ref{prop:42} allow to conclude. We recall them briefly in what follows.
We have
$$ \tilde Z^{N, 1}_t = A^N_t + M^N_t +{\mathbb D}elta^N_t ,$$
where
\begin{multline*}
{\mathbb D}elta^N_t =
- \int_0^t \int_\mathbb {R} \int_0^\infty \left( f (Y^{N, 1}_{s- }) - f( X^{N, 1}_{s-}) \right) {\bf 1}_{ \{ z \le f ( Y^{N, 1}_{s-}) \wedge f ( X^{N, 1}_{s-}) \}} {\mathbf{N}} ^1 (ds, du, dz)\\
+ \int_0^t \int_\mathbb {R} \int_0^\infty [f(0)- f( Y^{N, 1}_{s-} )] {\bf 1}_{ \{ f ( X^{N, 1}_{s-} ) < z \le f ( Y^{N, 1}_{s-} ) \}} {\mathbf{N}}^1 (ds, du, dz) \\
+ \int_0^t \int_\mathbb {R} \int_0^\infty [ f( X^{N, 1}_{s-} ) - f(0) ] {\bf 1}_{ \{ f ( Y^{N, 1}_{s-} ) < z \le f ( X^{N, 1}_{s-} ) \}} {\mathbf{N}}^1 (ds,du, dz) .
\end{multline*}
This term is treated as in the proof of Theorem \ref{prop:42}. Moreover, with $ \mu^{N, X, 1} _t := \frac1N \sum_{i=2}^N \delta_{X^{N, i }_t } , $ and $ \mu^{N, Y, 1} _t := \frac1N \sum_{i=2}^N \delta_{Y^{N, i }_t } ,$
\begin{multline*}
A_t^N = \int_0^t (f' (Y^{N , 1 }_t ) b ( Y^{N , 1 }_t ) - f' ( X^{N , 1 }_t ) b ( X^{N , 1 }_t )) + \\
\frac12 \int_0^t ( f'' ( Y^{N, 1}_s) \mu^{N, Y, 1 }_s ( f) - f'' ( X^{N ,1 }_s ) \mu^{N, X, 1 }_s ( f)) \sigma^2 ds
\end{multline*}
which is handled thanks to Assumption \ref{ass:2}, following the lines of the proof of Theorem \ref{prop:42}.
Finally,
$$M^N_t= \sigma \int_0^t ( f' ( Y^{N, 1 }_s) \sqrt{\mu^{N, Y, 1}_s (f)} -f' (X^{N, 1}_s ) \sqrt{ \tilde \mu^{N, X, 1}_s (f)} ) d W_s $$
is controlled using the Burkholder-Davis-Gundy inequality again, which gives
$$
\E \sup_{s \le t} |M^N_s|\le C \E \left[ \left( \int_0^t \left(f' ( Y^{N, 1 }_s) \sqrt{\mu^{N, Y, 1}_s (f)} - f' (X^{N, 1}_s ) \sqrt{ \tilde \mu^{N, X, 1}_s (f)}\right)^2 ds \right)^{1/2}\right]
$$
and which is controlled as in \eqref{eq:varquadratique}, leading to
$$ \E \sup_{s \le t} |M^N_s| \le C \sqrt{t} \left( \frac1N \sum_{j=2}^N \E \sup_{s \le t} | Z_s^{N, j } | + \E \sup_{s \le t } | Z_s^{N, 1 } | \right) .$$
By exchangeability of $ (Y^{N, 1 }, \ldots, Y^{N, N} ),$ we end up with the upper bound
$$ \E \sup_{s \le t} |M^N_s| \le C \sqrt{t} \E \sup_{s \le t} | Z_s^{N, 1 } | .$$
Resuming the above steps,
$$ \E \sup_{ s \le t } |Z^{N, 1}_s| \le C (t + \sqrt{t} ) \E \sup_{ s \le t } | Z^{N, 1}_s | + C \frac{\log N}{\sqrt{N}} ,$$
and concluding as in the proof of Theorem \ref{prop:42}, we deduce the following
\begin{theo}
Grant Assumptions \ref{ass:1} and \ref{ass:2}. Then for any $ T < \infty $ there exists a constant $C_T$ only depending on $ T$ and on the parameters of the system, but not on $N, $ such that for all $ t \le T,$
$$ \E | f(X_t^{N, 1 }) - f(Y_t^{N, 1 } ) | \le C_T \frac{\log N}{\sqrt{N}}.$$
\end{theo}
\subsection{Convergence of $ Y^N $ to the limit equation}
We now introduce the usual coupling of $ Y^N $ and the limit process $ Y$ using the same Brownian motion and the same Poisson random measures for the two processes.
Then we have
\begin{theo}
Grant Assumptions \ref{ass:1} and \ref{ass:2}. Consider a probability distribution $g_0$ on $\mathbb {R}$ such that $\int_{\mathbb {R}} x^2 g_0(dx)<\infty$ and, for each $N\geq 1$,
the unique solution $(Y^N_t)_{t\geq 0}$ to \eqref{eq:dynapprox} starting from
some i.i.d. $g_0$-distributed initial conditions $Y^{N,i}_0 = X^{N, i }_0 = Y^{i}_0 .$
Then for all $s \le t, $
$$ \E \sup_{s \le t } | f ( Y^{N, 1 }_t - f ( Y_t) | \le C_T N^{-1/2}.$$
\end{theo}
\begin{proof}
The proof is done by decomposing the evolution of the limit process in the following way.
\begin{eqnarray}\label{eq:dynapproxbisbis}
Y^1_t &= & Y_0 + \int_0^t b(Y^1_s ) ds - \int_0^t \int_\mathbb {R} \int_0^\infty
Y^1_{s-} {\bf 1}_{ \{ z \le f ( Y^1_{s-}) \}} {\mathbf{N}}^1 (ds,du, dz) \\
&&+ \sigma \int_0^t \sqrt{ \frac1N \sum_{j=1}^N f ( Y_s^{ j} ) } d B_s + M_t^N ,
\nonumber
\end{eqnarray}
where
$$ M_t^N = \sigma \int_0^t \left( \sqrt{ \frac1N \sum_{j=1}^N f ( Y_s^{ j} ) } - \sqrt{\E ( f ( Y_s^{ 1} ) | B) }\right) d B_s$$
is such that
$$ <M^N>_t \le \sigma^2 \int_0^t \left( \sqrt{ \frac1N \sum_{j=1}^N f ( Y_s^{ j} ) } - \sqrt{\E ( f ( Y_s^{ 1} ) | B) }\right)^2 ds. $$
Taking conditional expectation $\E ( \cdot | B ) $ implies that
$$ \E <M^N>_t \le C_T N^{-1} , $$
and this implies the result.
\end{proof}
\section{OLD Stuff}
In what follows, we shall write $\omega = (\omega_t)_{t \geq 0 } $ for the canonical process on ${\mathbb D}(\mathbb {R}_+, \mathbb {R} ) ,$ and we endow ${\mathbb D}(\mathbb {R}_+, \mathbb {R} )$ with the usual
filtration $( {\mathcal F}_t)_{t \geq 0 } , $ where
$$ {\mathcal F}_t = \sigma \{ \omega_s , s \le t \} .$$
\begin{defin}
We say that $ \mu \in {\mathcal P} ({\mathbb D}(\mathbb {R}_+, \mathbb {R} )) $ is solution of ${\mathcal M} $ if for all $ \varphi \in C^\infty_0 ( \mathbb {R} ) , $
$$ \varphi( \omega_t ) - \varphi ( \omega_0 ) - \int_0^t \cL_s \varphi ( \omega_s) ds $$
is a $ ( \mu , ( {\mathcal F}_t)_{t \geq 0 }) -$martingale, where
$$ \cL_s \varphi ( x) = b(x) \varphi ' ( x) + f( x) [ \varphi ( 0 ) - \varphi ( x) ] + \frac{\sigma^2 }{2} \left( \int f ( \omega_s ) \mu ( d \omega ) \right) \varphi '' ( x) .$$
\end{defin}
\begin{theo}
Grant Assumption \ref{ass:1}. Consider a probability distribution $g_0$ on $\mathbb {R}$ such that $\int_{\mathbb {R}} x^2 g_0(dx)<\infty$ and, for each $N\geq 1$,
the unique solution $(X^N_t)_{t\geq 0}$ to \eqref{eq:dyn} starting from
some i.i.d. $g_0$-distributed initial conditions $X^{N,i}_0$. Write $\mu_N=N^{-1}\sum_{i=1}^N \delta_{(X^{N,i}_t)_{t\geq 0}}.$ Then any limit point $\mu$ of $ \mu_N $
almost surely belongs to $\cS := \{ \mu \in {\mathcal P} ({\mathbb D}(\mathbb {R}_+, \mathbb {R} )) : \mu \mbox{ solution of } {\mathcal M} \} .$
\end{theo}
\begin{proof}
By Theorem \ref{theo:6}-(ii), we know that the sequence $ \mu_N$ is tight. We thus consider a (not relabeled) subsequence $\mu_N$ going
in law to some ${\mathcal P} ({\mathbb D}(\mathbb {R}_+, \mathbb {R} ))$-valued random variable $\mu$. We want to show that
$\mu$ a.s.\ belongs to $\cS .$
\vskip0.2cm
{\it Step 1.} For $t\geq 0$, we introduce $\pi_t: D(\mathbb {R}_+, \mathbb {R} )\mapsto \mathbb {R}_+$
defined by $\pi_t(\omega)=\omega_t$. We claim that
$Q\in{\mathcal P}( D(\mathbb {R}_+, \mathbb {R} ))$ belongs to $\cS$ if the following conditions are satisfied:
\vskip0.2cm
(a) $Q\circ \pi_0^{-1}=g_0$;
\vskip0.2cm
(b) for all $t\geq 0$, $\E \int_{{\mathbb D}(\mathbb {R}_+, \mathbb {R} )}\int_0^t (\omega_s)^2 ds Q(d\omega)<\infty$;
\vskip0.2cm
(c) for any $ 0 \le s_1 < \ldots < s_k < s < t$, any $\varphi_1,\dots,\varphi_k
\in C_b ( \mathbb {R})$, any $\varphi \in C^3_b (\mathbb {R} )$,
\begin{multline*}
F(Q):=\int_{{ D} ( \mathbb {R}_+, \mathbb {R} )} \int_{{ D} (\mathbb {R}_+ , \mathbb {R} )}
Q ( d \omega ) Q ( d \tilde \omega ) \; \varphi_1 ( \omega_{s_1} ) \ldots
\varphi_{k} (\omega_{s_k} ) \\
\mathcal{B}ig[ \varphi ( \omega_t) - \varphi ( \omega_s) - \int_s^t f( \omega_u)
( \varphi ( 0) - \varphi (\omega_u ) ) du - \int_s^t b( \omega_u) \varphi' ( \omega_u) du- \frac12 \sigma^2 \int_s^t \varphi'' ( \omega_u )
f (\tilde \omega_u ) du\mathcal{B}ig]=0 .
\end{multline*}
{\it Step 2.} Here we check that for any $t\geq 0$, a.s., $\mu(\{\omega \, : \, {\mathbb D}elta\omega(t)\ne 0\})=0$.
We assume by contradiction that there exists $t > 0 $
such that $\mu ( \{ \omega : {\mathbb D}elta \omega (t) \neq 0 \} ) > 0 $
with positive probability. Hence there are $a,b>0$ such that the event
$E:=\{\mu ( \{ \omega : |{\mathbb D}elta \omega (t) | > a \} ) > b\}$ has a
positive probability. For every $\varepsilon > 0$, we have
$E\subset \{ \mu ( \cB^\varepsilon_a ) > b\}$, where
$\cB^\varepsilon_a := \{ \omega : \sup_{ s \in (t- \varepsilon , t + \varepsilon)}| {\mathbb D}elta \omega ( s) | > a \}$, which
is an open subset of $D ( \mathbb {R}_+ , \mathbb {R} )$. Thus ${\mathcal P}^{\varepsilon}_{a,b} :=
\{ Q \in {{\mathcal P}} ( {D} ( \mathbb {R}_+, \mathbb {R} ) ) : Q ( \cB^\varepsilon_a ) > b \}$
is an open subset of $ {{\mathcal P}} ( {{\mathbb D}} ( \mathbb {R}_+) )$.
The Portmanteau theorem implies then that for any $\varepsilon>0$,
$$
\liminf_{N \to \infty } \P ( \mu_N \in {\mathcal P}^{\varepsilon}_{a,b} ) \geq \P ( \mu \in {\mathcal P}^{\varepsilon}_{a,b} )
\geq \P ( E) > 0.
$$
But
\begin{multline*}
\{\mu_N \in {\mathcal P}^{\varepsilon}_{a,b}\} \subset \mathcal{B}ig\{\frac1N \sum_{ i= 1 }^N
{\bf 1}_{\{ \int_{t- \varepsilon}^{t + \varepsilon} \int_{\mathbb {R} \times \mathbb {R}_+ } {\bf 1}_{ \{ z \le f( X^{N, i }_{v- } ) \}} {\mathbf{N}}^i (dv, du, dz) \geq 1\}} \geq b/2 \mathcal{B}ig\} \\
\cup \mathcal{B}ig\{\frac1N \sum_{ i= 1 }^N \sum_{j \neq i }
{\bf 1}_{\{ \int_{t- \varepsilon}^{t + \varepsilon}\int_{\mathbb {R} \times \mathbb {R}_+ } u {\bf 1}_{ \{ z \le f( X^{N, j }_{v- } ) \}} {\mathbf{N}}^j (dv, du, dz) \geq 1\}} \geq \sqrt{N} b/2 \mathcal{B}ig\}.
\end{multline*}
Using exchangeability, we obtain
\begin{multline*}
\P ( \mu_N \in {\mathcal P}^{\varepsilon}_{a,b} )
\le \frac{2}{b N} \sum_{i=1}^N \E \mathcal{B}ig(\int_{t- \varepsilon}^{t + \varepsilon} \int_{\mathbb {R} \times \mathbb {R}_+ } {\bf 1}_{ \{ z \le f( X^{N, i }_{v- } ) \}} {\mathbf{N}}^i (dv, du, dz)
\mathcal{B}ig ) +\\
\frac{4}{b^2 N } \E \left[ \big( \frac1N \sum_{ i= 1 }^N \sum_{j \neq i }
{\bf 1}_{\{ \int_{t- \varepsilon}^{t + \varepsilon}\int_{\mathbb {R} \times \mathbb {R}_+ } u {\bf 1}_{ \{ z \le f( X^{N, j }_{v- } ) \}} {\mathbf{N}}^j (dv, du, dz) \geq 1\}} \big)^2 \right] \\
\le \frac{4}{b} \| f\|_\infty \varepsilon + \frac{8 }{b^2 } \sigma^2 \| f \|_\infty \varepsilon ,
\end{multline*}
which does not depend on $N$ and tends to $0$ as $\varepsilon \to 0$. We thus have the contradiction
$$
0 < \P ( E) \le \liminf_{\varepsilon \to 0 } \liminf_{N \to \infty } \P ( \mu_N \in {\mathcal P}^{\varepsilon}_{a, b}) =0.
$$
{\it Step 3.} Our limit $\mu$ a.s. satisfies (a), because $\mu \circ \pi_0^{-1}$
is the limit in law of $\mu^N \circ \pi_0^{-1}=N^{-1}\sum_{i=1}^N \delta_{X^{N,i}_0}$,
which goes to $g_0$ because the $X^{N,i}_0$ are i.i.d. with common law
$g_0$. It also a.s. satisfies (b) since for all $t\geq 0$ and $K > 0,$ using the Fatou Lemma
and \eqref{eq:nice},
\begin{align*}
\E\mathcal{B}ig[\int_{D(\mathbb {R}_+, \mathbb {R} )}\intot [(\omega_s )^2 \wedge K ]ds \mu(d\omega)\mathcal{B}ig]
\leq& \liminf_N \E\mathcal{B}ig[\int_{{\mathbb D}(\mathbb {R}_+)}\intot [(\omega_s)^2 \wedge K] ds \mu_N(d\omega)\mathcal{B}ig]\\
= & \liminf_N N^{-1} \sum_{i=1}^N\intot \E[(X^{N,i}_s)^2 ]ds < \infty.
\end{align*}
The conclusion follows by letting $K \to \infty .$
\vskip0.2cm
{\it Step 4.} It remains to check that $\mu$ a.s. satisfies (c). We thus consider
$F:D(\mathbb {R}_+, \mathbb {R} )\mapsto \mathbb {R}$ as in (c).
\vskip0.2cm
{\it Step 4.1.} Here we prove that $\lim_N\E[|F(\mu_N)|]=0$.
We have
\begin{align*}
F( \mu_N) =& \frac1N \sum_{i= 1}^N \varphi_1 ( X^{N, i }_{s_1} )
\ldots \varphi_k ( X^{N, i }_{s_k} ) \\
&\mathcal{B}igg[ \varphi (X^{N, i }_{t}) - \varphi (X^{N, i }_{s}) - \int_s^t f( X^{N, i }_{u})
[\varphi (0) - \varphi (X^{N, i }_{u}) ] du
- \int_s^t b(X^{N, i }_{u}) \varphi' (X^{N, i }_{u}) du\\
& \hskip5cm - \frac{\sigma^2}{2} \int_s^t \varphi'' (X^{N, i }_{u}) \frac1N \sum_{j=1}^N f ( X_u^{ N, j}) du
\mathcal{B}igg] .
\end{align*}
But recalling \eqref{eq:dyn} and using the It\^o formula for jump processes,
\begin{align*}
\varphi ( X_t^{N, i } ) =& \varphi (X_0^{N, i } ) +
\int_0^t \int_{\mathbb {R}}\int_0^\infty \!
[ \varphi ( 0 ) - \varphi( X^{N, i }_{v-} ) ] {\bf 1}_{ \{ z \le f( X^{N, i }_{v- } ) \}}
{\mathbf{N}}^{i} (dv, du, dz)
+ \intot b(X^{N,i}_v) \varphi'( X^{N, i }_v) dv \\
&+ \sum_{ j \neq i } \int_0^t \int_{\mathbb {R}} \int_0^\infty \mathcal{B}ig( \varphi ( X^{N, i }_{ v - } + \frac{u}{\sqrt{N}} )
- \varphi ( X_{v-}^{N, i } ) \mathcal{B}ig) {\bf 1}_{ \{ z \le f( X_{v-}^{N, j } ) \}} {\mathbf{N}}^j (dv, du, dz).
\end{align*}
Consequently, using the notation $\tilde {\mathbf{N}}^i (dv, du, dz ) = {\mathbf{N}}^i (dv, du, dz ) - dv \mu (du) dz$ and setting
$$
M_t^{N, i } := \int_0^t \int_\mathbb {R} \int_0^\infty
[ \varphi ( 0 ) - \varphi( X^{N, i }_{v-} ) ] {\bf 1}_{ \{ z \le f( X^{N, i }_{v- } ) \}}
\tilde {\mathbf{N}}^{i} (dv, du, dz)
$$
and
\begin{multline*}
{\mathbb D}elta_t^{N, i } := \sum_{ j \neq i } \! \int_0^t \!\int_{\mathbb {R}} \int_0^\infty \!\!\! \mathcal{B}ig( \varphi ( X^{N, i }_{ v - } +
\frac{u}{\sqrt{N}} ) - \varphi ( X_{v-}^{N, i } ) \mathcal{B}ig) {\bf 1}_{ \{ z \le f( X_{u-}^{N, j } ) \}}{\mathbf{N}}^j (du, dz)
- \\
- \frac{\sigma^2}{2} \int_s^t \varphi'' (X^{N, i }_{u}) \frac1N \sum_{j=1}^N f ( X_u^{ N, j}) du ,
\end{multline*}
we see that
$$
F(\mu_N) = \frac1N \sum_{i= 1}^N \varphi_1 ( X^{N, i }_{s_1} ) \ldots
\varphi_k ( X^{N, i }_{s_k} ) \big[ ( M_t^{N, i } - M_s^{N, i } )
+ ( {\mathbb D}elta_t^{N, i } - {\mathbb D}elta_s^{N, i } ) \big] .
$$
Since the Poisson measures ${\mathbf{N}}^i$ are i.i.d., the martingales $M^{N, i }$
are orthogonal.
Using exchangeability and the boundedness of the $\varphi_k$, we thus find that
\begin{equation}\label{eq:318}
\E [ |F ( \mu_N) | ] \le C_F \frac{1}{\sqrt{N}}
\E [ ( M_t^{N, 1} - M_s^{N, 1 } )^2]^{1/2} + C_F \E[ | {\mathbb D}elta_t^{N, 1}| +|{\mathbb D}elta_s^{N, 1 }|].
\end{equation}
First, since $\varphi$ and $ f$ are bounded,
$$
\E[( M_t^{N, 1} - M_s^{N, 1 } )^2]=\int_s^t \E[(\varphi ( 0 ) - \varphi( X^{N, 1}_{u} ))^2
f( X^{N, 1}_{u} )] du \leq C_F.
$$
Next,
\begin{align*}
| {\mathbb D}elta_t^{N, 1 }| \le & \int_0^t \int_{\mathbb {R}} \int_0^\infty \mathcal{B}ig|\varphi ( X^{N, 1 }_{ v - } + \frac{u}{\sqrt{N}} ) -
\varphi ( X_{v-}^{N, 1 } )\mathcal{B}ig| {\bf 1}_{ \{ z \le f( X_{v-}^{N, 1 } ) \}}{\mathbf{N}}^1 (dv, du, dz) \\
& + \mathcal{B}ig| \sum_{j=1 }^N \int_0^t \int_{\mathbb {R}} \int_0^\infty \big( \varphi ( X^{N, 1 }_{ v - } + \frac{u}{\sqrt{N}} ) -
\varphi ( X_{v-}^{N, 1 } ) \big) {\bf 1}_{ \{ z \le f( X_{v-}^{N, j} ) \}}\tilde {\mathbf{N}}^j (dv, du, dz)\mathcal{B}ig|
\\
& + \sum_{j=1}^N \int_0^t \int_{\mathbb {R}} \mathcal{B}ig| \varphi ( X^{N, 1 }_{ v} + \frac{u}{\sqrt{N}} ) -
\varphi ( X_{v}^{N, 1 } ) - \frac1N \varphi' (X_v^{N, 1 } )\mathcal{B}ig| f( X_v^{N, j} ) dv \mu ( du) \\
& =: I^N_t+J^N_t+K^N_t.
\end{align*}
Using that $\varphi'$ and $f$ are bounded, we find
$$
\E [ I^N_t ] \le \frac{C_F}{\sqrt{N}} \int |u| \mu ( du ) \int_0^t \E[f ( X_u^{N, 1})] du
\le \frac{C_F}{\sqrt{N}}.
$$
Moreover, since $\varphi'''$ is bounded and by \eqref{ethop} again,
$$
\E [ K^N_t] \le \frac{C_F}{N^2}\sum_{j=1}^N \int_0^t \E[f ( X_u^{N, j} )] du \le \frac{C_F}{N}.
$$
{\bf The problem is actually the martingale term in the middle.}
It is of the kind
$$ \sum_{j=1 }^N \int_0^t \int_{\mathbb {R}} \int_0^\infty \big( \varphi' ( X^{N, 1 }_{ v - } ) \frac{u}{\sqrt{N}} \big) {\bf 1}_{ \{ z \le f( X_{v-}^{N, j} ) \}}\tilde {\mathbf{N}}^j (dv, du, dz).$$
And this should behave as
$$ \sigma \int_0^t \varphi' ( X^{N, 1 }_{ v } ) \sqrt{ \mu^N_v ( f)} d B_v .$$
Which would be a common martingale part for all the particles.
Idea would be to change the original dynamics and to introduce an approximating system
{\bf We have to treat the smoothness of limit semigroup !}
\vskip0.2cm
{\it Step 4.2.} Clearly, $F$ is continuous at any point $Q\in {\mathcal P}({\mathbb D}(\mathbb {R}_+))$
such that $Q(\omega\, : \, {\mathbb D}elta\omega(s_1)=\dots={\mathbb D}elta\omega(s_k)={\mathbb D}elta\omega(s)
={\mathbb D}elta\omega(t)=0)=1$ and such that $\int_{{\mathbb D}(\mathbb {R}_+)}\intot [\omega_u+f(\omega_u)]du Q(d\omega)<\infty$.
Our limit point $\mu$ a.s. satisfies these two conditions by Steps 2 and 3 (because
$x+f(x)\leq C(1+xf(x))$).
Since $\mu$ is the limit in law of $\mu_N$ and since $F$ is a.s. continuous at $\mu$,
we thus deduce that for any $K>0$, $\E[|F(\mu)|\land K]=\lim_N \E[|F(\mu_N)|\land K]$.
Consequently, $\E[|F(\mu)|\land K] \leq \limsup_N \E[|F(\mu_N)|]$ for all $K>0$.
Using Step 4.1, we deduce that $\E[|F(\mu)|\land K] =0$ for any $K>0$.
By the monotone convergence theorem, we conclude that $\E[|F(\mu)|]=0$, whence
$F(\mu)=0$ a.s.
\end{proof}
From the above, we obtain the weak convergence of $\hat \mu_N $ along a subsequence to a limit law ${\mathcal L} ( \mu | P^\infty ) .$
{\bf Question :} Does the limit law $P^\infty $ satisfy the following ? $P^\infty -$almost surely, $ \mu -$ which is a random law on $ D ( \mathbb {R} ) $ satisfies : for all $ \varphi \in C_0^\infty ( \mathbb {R} ) , $ we have that
$$ \varphi ( \omega_t ) - \varphi ( \omega_0) - \int_0^t L_s \varphi ( \omega_s ) ds $$
is a $ \mu-$martingale. Here, $ \omega $ is the canonical process and
$$ L_t \varphi ( x) = b(x) \varphi ' (x) + f(x) [ \varphi( 0) - \varphi( x) ] + \frac12 \left( \int f ( \omega_t ) \mu ( d \omega ) \right) \varphi'' ( x) .$$
Should be, no?
Do we have uniqueness of the solution of this martingale problem?
{\bf Are we sure that this limit law is random ??? }
{\bf Do we have a density for $ Y_t^{\infty, 1 } $ ???}
\section{Auxiliary process}
We cut time into intervals of length $ \delta > 0 $ and we consider an approximation of our process which has constant jump rate over such intervals.
In other words we consider an approximation $ X^{N, \delta, i }_{n \delta }, n \geq 0, $ such that
\begin{equation}\label{eq:XNdelta}
X^{N, \delta , i}_{(n+1) \delta } = X^{N, \delta ,i}_{n \delta } + \int_{ n \delta}^{(n+1) \delta} b( X^{N,\delta, i}_s) ds - \int_0^t \int_\mathbb {R} \int_0^\infty
X^{N,\delta, i}_{s-} {\bf 1}_{ \{ z \le f ( X^{N,\delta, i}_{n \delta}) \}} {\mathbf{N}}^i (ds,du, dz) + {\mathbb D}elta M_{n \delta}^{N,\delta, i }
\end{equation}
where
$${\mathbb D}elta M_{n \delta}^{N,\delta, i }
= \frac{1}{\sqrt{N}}\sum_{ j \neq i } \int_{n \delta}^{(n+1) \delta} \int_\mathbb {R} \int_0^\infty u {\bf 1}_{ \{ z \le f ( X^{N,\delta, j}_{n \delta }) \}} {\mathbf{N}}^j (ds,du, dz) .$$
{\bf Maybe we do also have to discretise the first line, I do not know for the moment! }
\begin{prop}
1) The random variables $ (X^{N, \delta , i}_{n \delta } , 1 \le i \le N ) $ are exchangeable for all $n.$
2) The associated empirical measures
$$ \hat \mu_{n\delta}^{N, \delta} := \frac1N \sum_{i=1}^N \delta_{X^{N, \delta , i}_{n \delta } } $$ converge to a limit measure that will be denoted $ \mu^\delta_{n \delta } .$ (Or at least, they are tight!)
\end{prop}
As a consequence, conditionally on ${\mathcal F}_{n \delta}, $ we have that
$${\mathbb D}elta [M^{N,\delta, i }] _{n \delta} = \frac{\sigma^2 \delta }{N} \sum_{j \neq i } f ( X^{N,\delta, j }_{n \delta} ) \sim \sigma^2 \delta \; \hat \mu_{n \delta }^{N, \delta} ( f) \to \sigma^2 \delta \; \mu^\delta_{n \delta} ( f) ,$$
from which we deduce the weak convergence of
$$ {\mathbb D}elta M_{n \delta}^{N,\delta, i } \to \sigma \sqrt{ \mu^\delta_{n \delta} ( f) } B_\delta .$$
\section{Auxiliary process-bis}
We cut time into intervals of length $ \delta > 0 $ and we consider an approximation of our process such that at most one single jump may happen during any such interval, per particle.
Our goal is to prove that for all $n, $ $Y^N( n \delta) $ is an exchangeable random vector. This should be proved inductively over $n.$
Suppose at time $n \delta $ we have configuration $ y = (y_1, \ldots, y_N) . $ Choose independent exponential random variables $ \tau_1 , \ldots, \tau_N $ such that $ \tau_i \sim \exp ( f ( y_i) ) .$ Put $ \Phi_i (n) = 1_{\{ \tau_i \le \delta \} } $ and let $ U_i (n ) $ be i.i.d. $\sim \mu .$ Finally we have
$$ q_N := \frac{1}{\sqrt{N}} \sum_{i=1}^N \Phi_i ( n) U_i (n) .$$
Then for all $ i $ such that $\Phi_i (n ) = 0 $ we put
$$ Y_i ((n+1)\delta ) := e^{- \lambda \delta } y_i + q_N .$$
Take now those particles that jump. We have
$$ N(n) := \sum_{i=1}^N \Phi_i (n) $$ such particles that we number according to increasing order of jumps
$$ j_1 , \ldots, j_{N(n) } .$$
Then we put for any $ 1 \le i \le N(n) , $
$$ Y_{j_i } ( (n+1) \delta ) := \frac{1}{\sqrt{N}} \sum_{k=i+1}^{N(n) } U_{j_k ( n) } .$$
In particular, $ Y_{j_{N(n) } }( (n+1) \delta ) = 0.$
It should be clear that the coupling of the true process with $ Y^\delta $ is ok.
And we should also have that the fact that
$$ \hat \mu^N ( n \delta ) := \frac{1}{N} \sum_{i=1}^N
\delta_{ Y_i (n\delta ) } \to \mu_{n \delta } $$
(which has to proven by induction in $n$) as $ N \to \infty $ implies weak convergence
$$ q_N \stackrel{\mathcal L}{\to } \sigma \sqrt{ \mu_{n \delta} ( f) } \int_{n \delta}^{(n+1) \delta } d B_s .$$
Then a simple re-ordering of the $U_i (n) $ gives
$$ \hat \mu^N ( (n+1) \delta ) = \frac{1}{N} \sum_{i=1}^N 1_{ \{ \tau_i > \delta \} } \delta_{ e^{- \lambda \delta } y_i + q_n } +
\frac{1}{N} \sum_{i=1}^{N(n) } \delta_{ N^{- 1/2} \sum_{ j =i+1}^{N(n) } U_j (n ) } .$$
Consider for instance a test function $ \Psi , $ then we have that
\begin{equation}
\hat \mu^N ( (n+1) \delta ) ( \Psi ) = \frac{1}{N} \sum_{i=1}^N 1_{ \{ \tau_i > \delta \} } \Psi \left( e^{- \lambda \delta } y_i + q_N \right) + \frac{1}{N} \sum_{i=1}^{N(n) } \Psi \left( N^{- 1/2} \sum_{ j =i+1}^{N(n) } U_j (n ) \right) .
\end{equation}
Question : what is the limit of (5.8)?
Q1 : If we know that $ \hat \mu^N (n \delta ) \to \mu ( n \delta ) , $ does that imply that - in a certain sense that has to be made precise - we have that $ (Y^N_1, \ldots , Y^N_N) \to ( Y_1, \ldots, Y_N, \ldots ) $ where the limit sequence is necessarily exchangeable?
\end{document} |
\begin{document}
\title[Expansion of a simplicial complex]{Expansion of a simplicial complex}
\author[S. Moradi and F. Khosh-Ahang]{Somayeh Moradi and Fahimeh Khosh-Ahang}
\address{Somayeh Moradi, Department of Mathematics,
Ilam University, P.O.Box 69315-516, Ilam, Iran and School of Mathematics, Institute
for Research in Fundamental Sciences (IPM), P.O.Box: 19395-5746, Tehran, Iran.} \email{[email protected]}
\address{Fahimeh Khosh-Ahang, Department of Mathematics,
Ilam University, P.O.Box 69315-516, Ilam, Iran.}
\email{fahime$_{-}[email protected]}
\keywords{Cohen-Macaulay, edge ideal, expansion, projective dimension, regularity, shellable, vertex
decomposable.\\
}
\subseteqbjclass[2010]{Primary 13D02, 13P10; Secondary 16E05}
\begin{abstract}
\noindent
For a simplicial complex $\displaystyleelta$, we introduce a simplicial complex attached to $\displaystyleelta$, called the expansion of $\displaystyleelta$, which is a natural generalization of the notion of expansion in graph theory. We are interested in knowing how the properties of a
simplicial complex and its Stanley-Reisner ring relate to those of its expansions. It is shown that taking expansion preserves vertex decomposable and
shellable properties and in some cases Cohen-Macaulayness. Also it is proved that some homological invariants of Stanley-Reisner ring of a simplicial complex relate to those invariants in the
Stanley-Reisner ring of its expansions.
\end{abstract}
\maketitle
\section*{Introduction}
Simplicial complexes are widely used structures which have many applications in algebraic topology and commutative algebra. In particular, in order to characterize monomial quotient rings with a desired property, simplicial complex is a very strong tool considering the Stanley-Reisner correspondence between simplicial complexes and monomial ideals.
Characterizing simplicial complexes which have properties like vertex decomposability, shellability and Cohen-Macaulayness are some main problems in combinatorial commutative algebra. It is rather hopeless to give a full classification of simplicial complexes with each of these properties.
In this regard, finding classes of simplicial complexes, especially independence complexes of graphs with a desired property have been considered by many researchers (cf. \cite{F,FV,HMV,VVi,W,W1}). Constructing new simplicial complexes from the existing ones satisfying a desired property is another way to know more about the characterization. In the works \cite{CN,DE,FH,Moha,V}, the idea of making modifications to a graph like adding whiskers and ears to the graph in order to obtain sequentially
Cohen-Macaulay, Cohen-Macaulay and vertex decomposable graphs is investigated. In \cite{BV}, the authors developed a construction similar to whiskers to build a vertex decomposable simplicial complex $\displaystyleelta_{\chi}$ from a coloring $\chi$ of
the vertices of a simplicial complex $\displaystyleelta$, and in \cite{BFHV} for colorings of subsets of the vertices, necessary and sufficient conditions are given for this
construction to produce vertex decomposable simplicial complexes.
Motivated by the above works and the concept of expansion of a graph in graph theory, in this paper, we
introduce the concept of expansion of simplicial complexes which is a natural generalization of expansion of graphs. Also, we study some properties of this expansion to see how they are related to corresponding properties of the initial simplicial complex. This tool allows us construct
new vertex decomposable and shellable simplicial complexes from vertex decomposable and shellable ones. Moreover, some families of Cohen-Macaulay simplicial complexes are introduced. We are also interested in knowing how the homological invariants of the Stanley-Reisner ring of a simplicial complex and its expansions are related.
The paper is organized as follows. In the first section, we review some preliminaries from the literature.
In Section 2, first in Theorem \ref{evd} we show that for a simplicial complex $\displaystyleelta$, vertex decomposability of $\displaystyleelta$ is equivalent to vertex decomposability of an expansion of $\displaystyleelta$. Also it is proved that expansions of a shellable simplicial complex are again shellable (see Theorem \ref{vI}). Moreover, it is shown that under some conditions, expansions of a simplicial complex inherit Cohen-Macaulayness (see Corollaries \ref{cor2}, \ref{cor3}, \ref{cor1} and \ref{CM}). Finally, in Section 3, for a shellable simplicial complex, the projective dimension and the regularity of its Stanley-Reisner ring are compared with the corresponding ones in an expansion of $\displaystyleelta$ (see Propositions \ref{pd} and \ref{shreg}).
\section{Preliminaries}
Throughout this paper, we assume that $\displaystyleelta$ is a simplicial complex on the vertex set $V(\displaystyleelta)=\{x_1, \dots, x_n\}$. The set of facets (maximal faces) of $\displaystyleelta$
is denoted by $\mathcal{F}(\displaystyleelta)$.
In this section, we recall some preliminaries which are needed in the sequel. We begin with definition of a vertex decomposable simplicial complex. To this aim, we need to recall definitions of the link and the deletion of a face in $\displaystyleelta$.
For a simplicial complex $\displaystyleelta$ and $F\in \displaystyleelta$, the \textbf{link} of $F$ in
$\displaystyleelta$ is defined as $$\mathrm{lk}_{\displaystyleelta}(F)=\{G\in \displaystyleelta: G\cap
F=\emptyset, G\cup F\in \displaystyleelta\},$$ and the \textbf{deletion} of $F$ is the
simplicial complex $$\mathrm{del}_{\displaystyleelta}(F)=\{G\in \displaystyleelta: G\cap
F=\emptyset\}.$$
\begin{defn}\label{1.1}
{\rm A simplicial complex $\displaystyleelta$ is called \textbf{vertex decomposable} if
$\displaystyleelta$ is a simplex, or $\displaystyleelta$ contains a vertex $x$ such that
\begin{itemize}
\item[(i)] both $\mathrm{del}_{\displaystyleelta}(x)$ and $\mathrm{lk}_{\displaystyleelta}(x)$ are vertex decomposable, and
\item[(ii)] every facet of $\mathrm{del}_{\displaystyleelta}(x)$ is a facet of $\displaystyleelta$.
\end{itemize}
A vertex $x$ which satisfies condition (ii) is called a
\textbf{shedding vertex} of $\displaystyleelta$.}
\end{defn}
\begin{rem}\label{remark1}
{\rm It is easily seen that $x$ is a shedding vertex of $\displaystyleelta$ if and only if no facet of $\mathrm{lk}_{\displaystyleelta}(x)$ is a facet of $\mathrm{del}_{\displaystyleelta}(x)$.}
\end{rem}
\begin{defn}
{\rm A simplicial complex $\displaystyleelta$ is called \textbf{shellable} if there exists an ordering $F_1<\cdots<F_m$ on the
facets of $\displaystyleelta$
such that for any $i<j$, there exists a vertex
$v\in F_j\mbox{set}\,minus F_i$ and $\ell<j$ with
$F_j\mbox{set}\,minus F_\ell=\{v\}$. We call $F_1,\ldots,F_m$ a \textbf{shelling} for
$\displaystyleelta$.}
\end{defn}
The above definition is referred to as non-pure shellable and
is due to Bj\"{o}rner and Wachs \cite{BW}. In this paper we will
drop the adjective ``non-pure".
\begin{defn}
{\rm A graded $R$-module $M$ is called
\textbf{sequentially Cohen--Macaulay} (over a field $K$) if there exists a
finite filtration of graded $R$-modules $$0=M_0\subseteqbset M_1\subseteqbset
\cdots \subseteqbset M_r=M$$ such that each $M_i/M_{i-1}$ is
Cohen--Macaulay and
$$\mbox{dim}\,(M_1/M_0)<\mbox{dim}\,(M_2/M_1)<\cdots<\mbox{dim}\,(M_r/M_{r-1}).$$}
\end{defn}
For a $\mathbb{Z}$-graded $R$-module $M$, the \textbf{Castelnuovo-Mumford regularity} (or briefly regularity)
of $M$ is defined as
$$\mathrm{reg}(M) = \max\{j-i: \ \beta_{i,j}(M)\neq 0\},$$
and the \textbf{projective dimension} of $M$ is defined as
$$\mathrm{pd}(M) = \max\{i:\ \beta_{i,j}(M)\neq 0 \ \text{for some}\ j\},$$
where $\beta_{i,j}(M)$ is the $(i,j)$th graded Betti number of $M$.
Let $V = \{x_1,\ldots, x_n\}$ be a finite set, and let $\mathcal{E} = \{E_1,\ldots,E_s\}$ be a family of nonempty subsets
of $V$. The pair $\mathcal{H} = (V, \mathcal{E})$ is called a \textbf{simple hypergraph} if for each $i$, $|E_i| \geq 2$ and whenever
$E_i,E_j\in \mathcal{E}$ and $E_i \subseteqbseteq E_j$, then $i =j$. The elements of $V$ are
called the vertices and the elements of $\mathcal{E}$ are called the edges of $\mathcal{H}$. For a hypergraph $\mathcal{H}$, the \textbf{independence complex} of $\mathcal{H}$ is defined as $$\displaystyleelta_{\mathcal{H}}=\{F\subseteqbseteq V(\mathcal{H}):\ E\nsubseteq F, \text{ for each } E\in \mathcal{E}(\mathcal{H})\}.$$
A simple graph $G=(V(G), E(G))$ is a simple hypergraph with the vertices $V(G)$ and the edges $E(G)$, where each of its edges has cardinality exactly two.
For a simple graph $G$, the \textbf{edge ideal} of $G$ is defined as the ideal $I(G)=(x_ix_j:\ \{x_i,x_j\}\in E(G))$. It is easy to see that $I(G)$ can be viewed as the Stanley-Reisner ideal of the simplicial complex
$\displaystyleelta_{G}$ i.e., $I(G)=I_{\displaystyleelta_G}$. Also, the \textbf{big height} of $I(G)$, denoted by $\mathrm{bight}(I(G))$, is
defined as the maximum height among the minimal prime divisors of $I(G)$.
A graph $G$ is called vertex decomposable, shellable, sequentially Cohen-Macaulay or Cohen-Macaulay if
the independence complex $\displaystyleelta_G$ is vertex decomposable, shellable, sequentially Cohen-Macaulay or Cohen-Macaulay.
A graph $G$ is called \textbf{chordal}, if it contains no induced cycle of length $4$ or greater.
\begin{defn}\label{1.2}
{\rm
A monomial ideal $I$ in the ring $R=K[x_1,\ldots,x_n]$ has \textbf{linear quotients} if there exists an ordering $f_1, \dots, f_m$ on the minimal generators of $I$ such that the colon ideal $(f_1,\ldots,f_{i-1}):_R(f_i)$ is generated by a subset of $\{x_1,\ldots,x_n\}$ for all $2\leq i\leq m$. We show this ordering by $f_1<\dots <f_m$ and we call it \textbf{an order of linear quotients} on $\mathcal{G}(I)$.
Also for any $1\leq i\leq m$, $\mbox{set}\,_I(f_i)$ is defined as
$$\mbox{set}\,_I(f_i)=\{x_k:\ x_k\in (f_1,\ldots, f_{i-1}) :_R (f_i)\}.$$
We denote $\mbox{set}\,_I(f_i)$ by $\mbox{set}\, (f_i)$ if there is no ambiguity about the ideal $I$.
}
\end{defn}
A monomial ideal $I$ generated by monomials of degree $d$ has a \textbf{linear resolution} if $\beta _{i,j}(I)=0$ for all $j\neq i+d$. Having linear quotients is a strong tool to determine some classes of ideals with linear resolution. The main tool in this way is the following lemma.
\begin{lem}(See \cite[Lemma 5.2]{F}.)\label{Faridi}
Let $I=(f_1, \dots, f_m)$ be a monomial ideal with linear quotients such that all $f_i$s are of the same degree. Then $I$ has a linear resolution.
\end{lem}
For a squarefree monomial ideal $I=( x_{11}\cdots
x_{1n_1},\ldots,x_{t1}\cdots x_{tn_t})$, the \textbf{Alexander dual ideal} of $I$, denoted by
$I^{\vee}$, is defined as
$$I^{\vee}:=(x_{11},\ldots, x_{1n_1})\cap \cdots \cap (x_{t1},\ldots, x_{tn_t}).$$
For a simplicial complex $\displaystyleelta$ with the vertex set $X=\{x_1, \dots, x_n\}$, the \textbf{Alexander dual simplicial complex} associated to $\displaystyleelta$ is defined as
$$\displaystyleelta^{\vee}=\{X\mbox{set}\,minus F:\ F\notin \displaystyleelta\}.$$
For a subset $C\subseteqbseteq X$, by $x^C$ we mean the monomial $\prod_{x\in C} x$ in the ring $K[x_1, \dots, x_n]$.
One can see that
$(I_{\displaystyleelta})^{\vee}=(x^{F^c} \ : \ F\in \mathcal{F}(\displaystyleelta)),$
where $I_{\displaystyleelta}$ is the Stanley-Reisner ideal associated to $\displaystyleelta$ and $F^c=X\mbox{set}\,minus F$.
Moreover, one can see that $(I_{\displaystyleelta})^{\vee}=I_{\displaystyleelta^{\vee}}$.
The following theorem which was proved in \cite{T}, relates projective dimension and regularity of a
squarefree monomial ideal to its Alexander dual. It is one of our
tools in the study of the projective dimension and regularity of the ring $R/I_{\displaystyleelta}$.
\begin{thm}(See \cite[Theorem 2.1]{T}.) \label{1.3}
Let $I$ be a squarefree monomial ideal. Then
$\mathrm{pd}(I^{\vee})=\mathrm{reg}(R/I)$.
\end{thm}
\section{Expansions of a simplicial complex and their algebraic properties}
In this section, expansions of a simplicial complex and their Stanley-Reisner rings are studied. The main goal is to explore
how the combinatorial and algebraic properties of a simplicial complex $\displaystyleelta$ and its Stanley-Reisner ring affects on the expansions.
\begin{defn}\label{2.1}
{\rm
Let $\displaystyleelta=\langle F_1,\ldots,F_m\rangle$ be a simplicial complex with the vertex set $V(\displaystyleelta)=\{x_1,\ldots,x_n\}$ and $s_1,\ldots,s_n\in \mathbb{N}$ be arbitrary integers.
For any $F_i=\{x_{i_1},\ldots,x_{i_{k_i}}\}\in \mathcal{F}(\displaystyleelta)$, where $1\leq i_1<\cdots<i_{k_i}\leq n$ and any $1\leq r_1\leq s_{i_1},\ldots, 1\leq r_{k_i}\leq s_{i_{k_i}}$, set
$$F_i^{r_1,\ldots, r_{k_i}}=\{x_{i_1r_1},\ldots,x_{i_{k_i}r_{k_i}}\}.$$
We define the
$(s_1,\ldots,s_n)$-expansion of $\displaystyleelta$ to be a simplicial complex with the vertex set $\{\{x_{11},\ldots,x_{1s_1},x_{21},\ldots,x_{2s_2},\ldots,x_{n1},\ldots,x_{ns_n}\}$ and the facets $$\{x_{i_1r_1},\ldots,x_{i_{k_i}r_{k_i}}\} \ :\ \{x_{i_1},\ldots,x_{i_{k_i}}\}\in \mathcal{F}(\displaystyleelta), \ (r_1,\ldots,r_{k_i})\in [s_{i_1}]\times \cdots \times [s_{i_{k_i}}]\}.$$
We denote this simplicial complex by $\displaystyleelta^{(s_1,\ldots,s_n)}$}.
\end{defn}
\begin{exam}
{\rm Consider the simplicial complex $\displaystyleelta=\langle\{x_1,x_2,x_3\},\{x_1,x_2,x_4\},\{x_4,x_5\}\rangle$ depicted in Figure $1$. Then $$\displaystyleelta^{(1,2,1,1,2)}=\langle\{x_{11},x_{21},x_{31}\},\{x_{11},x_{22},x_{31}\},\{x_{11},x_{21},x_{41}\},\{x_{11},x_{22},
x_{41}\},\{x_{41},x_{51}\},\{x_{41},x_{52}\}\rangle.$$
\begin{figure}
\caption{The simplicial complex $\displaystyleelta$ and the $(1,2,1,1,2)$-expansion of $\displaystyleelta$}
\label{fig:graph}
\label{Fig1}
\end{figure}
}
\end{exam}
The following definition, gives an analogous concept for the expansion of a hypergraph, which is also a generalization of \cite[Definition 4.2]{FHV}.
\begin{defn}\label{2.3}
{\rm
For a hypergraph $\mathcal{H}$ with the vertex set $V(\mathcal{H})=\{x_1,\ldots,x_n\}$ and the edge set $\mathcal{E}(\mathcal{H})$, we define the
$(s_1,\ldots,s_n)$-expansion of $\mathcal{H}$ to be a hypergraph with the vertex set $\{x_{11},\ldots,x_{1s_1},x_{21},\ldots,x_{2s_2},\ldots,x_{n1},\ldots,x_{ns_n}\}$ and the edge set
\begin{align*}
\{\{x_{i_1r_1},\ldots, x_{i_tr_t}\}:\ \{x_{i_1},\ldots, x_{i_t}\}\in \mathcal{E}(\mathcal{H}),\
(r_1,\ldots,r_{t})\in [s_{i_1}]\times \cdots \times [s_{i_{t}}]\}\cup\\
\{\{x_{ij},x_{ik}\}: \ 1\leq i\leq n, \ j\neq k\}.
\end{align*}
We denote this hypergraph by $\mathcal{H}^{(s_1,\ldots,s_n)}$.
}
\end{defn}
\begin{rem}
{\rm From Definitions \ref{2.1} and \ref{2.3} one can see that for a hypergraph $\mathcal{H}$ and integers $s_1,\ldots,s_n\in \mathbb{N}$,
$\displaystyleelta_{\mathcal{H}^{(s_1,\ldots,s_n)}}=\displaystyleelta_{\mathcal{H}}^{(s_1,\ldots,s_n)}.$
Thus the expansion of a simplicial complex is the natural generalization of the concept of expansion in graph theory.
}
\end{rem}
\begin{exam}
{\rm
Let $G$ be the following graph.
\begin{figure}\label{fig5}
\label{fig:graph}
\end{figure}
The graph $G^{(1,1,2,1,2)}$ and the independence complexes $\displaystyleelta_G$ and $\displaystyleelta_{G^{(1,1,2,1,2)}}$ are
shown in Figure $2$.
\begin{figure}
\caption{The graph $G^{(1,1,2,1,2)}
\label{fig6}
\label{fig:graph2}
\end{figure}
}
\end{exam}
In the following proposition, it is shown that a graph is chordal if and only if some of its expansions is chordal.
\begin{prop}\label{cl}
For any $s_1,\ldots,s_n\in \mathbb{N}$, $G$ is a chordal graph if and only if $G^{(s_1,\ldots,s_n)}$ is chordal.
\end{prop}
\begin{proof}
If $G^{(s_1,\ldots,s_n)}$ is chordal, then clearly $G$ is also chordal, since it can be considered as an induced subgraph of
$G^{(s_1,\ldots,s_n)}$. Now, let $G$ be chordal, $V(G)=\{x_1,\ldots,x_n\}$ and consider a cycle $C_m: x_{i_1j_1},\ldots, x_{i_mj_m}$ in $G^{(s_1,\ldots,s_n)}$, where $m\geq 4$ and $1\leq j_k\leq s_{i_k}$ for all $1\leq k\leq m$. We consider two cases.
Case 1. $i_k=i_\ell$ for some distinct integers $k$ and $\ell$ with $1\leq k<\ell\leq m$. Then by the definition of expansion,
$x_{i_kj_k}x_{i_\ell j_\ell}\in E(G^{(s_1,\ldots,s_n)})$. Thus if $x_{i_kj_k}x_{i_\ell j_\ell}$ is not an edge of $C_m$, then it is a chord in $C_m$.
Now, assume that $x_{i_kj_k}x_{i_\ell j_\ell}$ is an edge of $C_m$. Note that since
$x_{i_\ell j_\ell}x_{i_{\ell+1}j_{\ell+1}}\in E(C_m)$, either $i_\ell=i_{\ell+1}$ or $x_{i_\ell}x_{i_{\ell+1}}\in E(G)$ (if $\ell=m$, then set $\ell+1:=1$). Thus $x_{i_kj_k}x_{i_{\ell+1}j_{\ell+1}}\in E(G^{(s_1,\ldots,s_n)})$ is a chord in $C_m$.
Case 2. $i_k\neq i_\ell$ for any distinct integers $1\leq k,\ell\leq m$. By the definition of expansion, one can see that
$x_{i_1},\ldots, x_{i_m}$ forms a cycle of length $m$ in $G$. So it has a chord. Let $x_{i_k}x_{i_\ell}\in E(G)$ be a chord in this cycle. Then
$x_{i_kj_k}x_{i_\ell j_\ell}\in E(G^{(s_1,\ldots,s_n)})$ is a chord in $C_m$.
Thus $G^{(s_1,\ldots,s_n)}$ is also chordal.
\end{proof}
The following theorem illustrates that the vertex decomposability of a simplicial complex is equivalent to the vertex decomposability of its expansions.
\begin{thm}\label{evd}
Assume that $s_1, \dots, s_n$ are positive integers. Then $\displaystyleelta$ is vertex decomposable if and only if $\displaystyleelta^{(s_1,\ldots,s_n)}$ is vertex decomposable.
\end{thm}
\begin{proof}
Assume that $\displaystyleelta$ is a simplicial complex with the vertex set $V(\displaystyleelta)=\{x_1,\dots, x_n\}$ and $s_1, \dots, s_n$ are positive integers. To prove the `only if' part, we use generalized induction on $|V(\displaystyleelta^{(s_1,\ldots,s_n)})|$ (note that $|V(\displaystyleelta^{(s_1,\ldots,s_n)})|\geq |V(\displaystyleelta)|$). If $|V(\displaystyleelta^{(s_1,\ldots,s_n)})|=|V(\displaystyleelta)|$, then $\displaystyleelta=\displaystyleelta^{(s_1, \dots, s_n)}$ and so there is nothing to prove in this case. Assume inductively that for all vertex decomposable simplicial complexes $\displaystyleelta'$ and all positive integers $s'_1, \dots, s'_n$ with $|V(\displaystyleelta'^{(s'_1,\dots,s'_n)})|< t$, $\displaystyleelta'^{(s'_1,\ldots,s'_n)}$ is vertex decomposable. Now, we are going to prove the result when $t=|V(\displaystyleelta^{(s_1,\ldots,s_n)})|>|V(\displaystyleelta)|$. Since $|V(\displaystyleelta^{(s_1,\ldots,s_n)})|>|V(\displaystyleelta)|$, there exists an integer $1\leq i\leq n$ such that $s_i>1$. If $\displaystyleelta=\langle F\rangle$ is a simplex, we claim that $x_{i1}$ is a shedding vertex of $\displaystyleelta^{(s_1,\ldots,s_n)}$. It can be easily checked that
$$\mathrm{lk}_{\displaystyleelta^{(s_1,\ldots,s_n)}}(x_{i1})=\langle F\mbox{set}\,minus \{x_i\}\rangle^{(s_1, \dots, s_{i-1},s_{i+1}, \dots, s_n)}$$
and
$$\mathrm{del}_{\displaystyleelta^{(s_1,\ldots,s_n)}}(x_{i1})=\displaystyleelta^{(s_1, \dots, s_{i-1}, s_i-1, s_{i+1}, \dots, s_n)}.$$
So, inductive hypothesis ensures that $\mathrm{lk}_{\displaystyleelta^{(s_1,\ldots,s_n)}}(x_{i1})$ and $\mathrm{del}_{\displaystyleelta^{(s_1,\ldots,s_n)}}(x_{i1})$ are vertex decomposable. Also, it can be seen that every facet of $\mathrm{del}_{\displaystyleelta^{(s_1,\ldots,s_n)}}(x_{i1})$ is a facet of $\displaystyleelta^{(s_1,\ldots,s_n)}$. This shows that $\displaystyleelta^{(s_1,\ldots,s_n)}$ is vertex decomposable in this case. Now, if $\displaystyleelta$ is not a simplex, it has a shedding vertex, say $x_1$. We claim that $x_{11}$ is a shedding vertex of $\displaystyleelta^{(s_1,\ldots,s_n)}$. To this end, it can be seen that
$$\mathrm{lk}_{\displaystyleelta^{(s_1,\ldots,s_n)}}(x_{11})={\mathrm{lk}_\displaystyleelta (x_1)}^{(s_2,\ldots,s_n)}$$
and
$$\mathrm{del}_{\displaystyleelta^{(s_1,\ldots,s_n)}}(x_{11})=\left\lbrace
\begin{array}{c l}
\displaystyleelta^{(s_1-1, s_2,\ldots,s_n)} & \text{if $s_1>1$;}\\
{\mathrm{del}_\displaystyleelta (x_1)}^{(s_2,\ldots,s_n)} & \text{if $s_1=1$.}
\end{array}
\right.$$
Hence, inductive hypothesis deduces that $\mathrm{lk}_{\displaystyleelta^{(s_1,\ldots,s_n)}}(x_{11})$ and $\mathrm{del}_{\displaystyleelta^{(s_1,\ldots,s_n)}}(x_{11})$ are vertex decomposable simplicial complexes. Now, suppose that $F^{j_1, \dots, j_k}=\{x_{i_1j_1}, \dots, x_{i_kj_k} \}$ is a facet of $\mathrm{lk}_{\displaystyleelta^{(s_1,\ldots,s_n)}}(x_{11})$, where $F=\{x_{i_1}, \dots, x_{i_k}\}$ is a face of $\displaystyleelta$. Then since $\mathrm{lk}_{\displaystyleelta^{(s_1,\ldots,s_n)}}(x_{11})={\mathrm{lk}_\displaystyleelta (x_1)}^{(s_2,\ldots,s_n)}$,
$F$ is a facet of $\mathrm{lk}_\displaystyleelta (x_1)$. So, there is a vertex $x_{i_{k+1}}\in V(\displaystyleelta)$ such that $\{x_{i_1}, \dots,x_{i_k}, x_{i_{k+1}}\}$ is a face of $\mathrm{del}_\displaystyleelta (x_1)$ (see Remark \ref{remark1}). Hence $\{x_{i_1j_1}, \dots, x_{i_kj_k}, x_{i_{k+1}1} \}$ is a face of $\mathrm{del}_{\displaystyleelta^{(s_1,\ldots,s_n)}}(x_{11})$. This completes the proof of the first part.
To prove the `if' part, we also use generalized induction on $|V(\displaystyleelta^{(s_1,\ldots,s_n)})|$. If $|V(\displaystyleelta^{(s_1,\ldots,s_n)})|=|V(\displaystyleelta)|$, then $\displaystyleelta=\displaystyleelta^{(s_1, \dots, s_n)}$ and so there is nothing to prove in this case. Assume inductively that for all simplicial complexes $\displaystyleelta'$ and all positive integers $s'_1, \dots, s'_n$ with $|V(\displaystyleelta'^{(s'_1,\dots,s'_n)})|< t$ such that $\displaystyleelta'^{(s'_1,\dots,s'_n)}$ is vertex decomposable, we have proved that $\displaystyleelta'$ is also vertex decomposable. Now, we are going to prove the result when $t=|V(\displaystyleelta^{(s_1,\ldots,s_n)})|>|V(\displaystyleelta)|$. Now, since $|V(\displaystyleelta^{(s_1,\ldots,s_n)})|>|V(\displaystyleelta)|$ and $\displaystyleelta^{(s_1,\ldots,s_n)}$ is vertex decomposable, it has a shedding vertex, say $x_{11}$. If $s_1>1$, then
$$\mathrm{del}_{\displaystyleelta^{(s_1,\ldots,s_n)}}(x_{11})=\displaystyleelta^{(s_1-1, s_2,\ldots,s_n)},$$
and the inductive hypothesis ensures that $\displaystyleelta$ is vertex decomposable as desired. Else, we should have $s_1=1$,
$$\mathrm{lk}_{\displaystyleelta^{(s_1,\ldots,s_n)}}(x_{11})={\mathrm{lk}_\displaystyleelta (x_1)}^{(s_2,\ldots,s_n)}$$
and
$$\mathrm{del}_{\displaystyleelta^{(s_1,\ldots,s_n)}}(x_{11})={\mathrm{del}_\displaystyleelta (x_1)}^{(s_2,\ldots,s_n)}.$$
So, inductive hypothesis implies that $\mathrm{lk}_\displaystyleelta (x_1)$ and $\mathrm{del}_\displaystyleelta (x_1)$ are vertex decomposable simplicial complexes. Now, assume that $F=\{x_{i_1}, \dots, x_{i_k}\}$ is a facet of $\mathrm{del}_\displaystyleelta (x_1)$. Then $\{x_{i_11}, \dots, x_{i_k1}\}$ is a facet of $\mathrm{del}_{\displaystyleelta^{(s_1,\ldots,s_n)}}(x_{11})$. Since $x_{11}$ is a shedding vertex of $\displaystyleelta^{(s_1,\ldots,s_n)}$, $\{x_{i_11}, \dots, x_{i_k1}\}$ is a facet of $\displaystyleelta^{(s_1,\ldots,s_n)}$. Hence, $F$ is a facet of $\displaystyleelta$ and the proof is complete.
\end{proof}
\begin{rem}\label{pure}
{\rm By the notations as in Definition \ref{2.1}, $\displaystyleelta$ is pure if and only if $\displaystyleelta^{(s_1,\ldots,s_n)}$ is pure, since any facet $F_i^{r_1,\ldots, r_{k_i}}$ of $\displaystyleelta^{(s_1,\ldots,s_n)}$ has the same cardinality as $F_i$.}
\end{rem}
The following theorem together with Theorem \ref{evd} help us to see how the Cohen-Macaulayness propery in a vertex decomposable simplicial complex and its expansions are related.
\begin{thm}\label{vc}
A vertex decomposable simplicial complex $\displaystyleelta$ is Cohen-Macaulay if and only if $\displaystyleelta$ is pure.
\end{thm}
\begin{proof}
See \cite[Theorem 11.3]{BW2} and \cite[Theorem 5.3.18]{VIL}.
\end{proof}
\begin{cor}\label{cor2}
Let $\displaystyleelta$ be a vertex decomposable simplicial complex and $s_1, \dots, s_n$ be positive integers. Then $\displaystyleelta$ is Cohen-Macaulay if and only if $\displaystyleelta^{(s_1,\ldots,s_n)}$ is Cohen-Macaulay.
\end{cor}
\begin{proof}
By Theorem \ref{evd}, $\displaystyleelta^{(s_1,\ldots,s_n)}$ is also vertex decomposable. Also, by Theorem \ref{vc}, $\displaystyleelta$, respectively $\displaystyleelta^{(s_1,\ldots,s_n)}$, is Cohen-Macaulay if and only if $\displaystyleelta$, respectively $\displaystyleelta^{(s_1,\ldots,s_n)}$, is pure. Now, by Remark \ref{pure}, the result is clear.
\end{proof}
\begin{cor}\label{cor3}
Let $G$ be a Cohen-Macaulay chordal graph or a Cohen-Macaulay bipartite graph. Then $G^{(s_1,\ldots,s_n)}$ is Cohen-Macaulay.
\end{cor}
\begin{proof}
By \cite[Corollary 7]{W} and \cite[Corollary 2.12]{VT} chordal graphs and Cohen-Macaulay bipartite graphs are vertex decomposable. The result now follows from Corollary \ref{cor2}.
\end{proof}
In the following theorem, it is shown that shellability is preserved under expansion and from a shelling for $\displaystyleelta$, a shelling for its expansion is constructed.
\begin{thm}\label{vI}
Let $\displaystyleelta$ be a shellable simplicial complex with $n$ vertices. Then $\displaystyleelta^{(s_1,\ldots,s_n)}$ is shellable for any $s_1,\ldots,s_n\in \mathbb{N}$.
\end{thm}
\begin{proof}
Use the notations as in Definition \ref{2.1}. Let $\displaystyleelta$ be a shellable simplicial complex with the shelling order $F_1<\cdots<F_m$ on the facets of $\displaystyleelta$. Consider an order on $\mathcal{F}(\displaystyleelta^{(s_1,\ldots,s_n)})$ as follows.
For two facets $F_i^{r_1,\ldots, r_{k_i}}$ and $F_j^{r'_1,\ldots, r'_{k_j}}$ of $\displaystyleelta^{(s_1,\ldots,s_n)}$
\begin{itemize}
\item[(i)] if $i<j$, set $F_i^{r_1,\ldots, r_{k_i}}<F_j^{r'_1,\ldots, r'_{k_j}}$,
\item[(ii)] if $i=j$, set $F_i^{r_1,\ldots, r_{k_i}}<F_i^{r'_1,\ldots, r'_{k_i}}$, when $(r_1,\ldots, r_{k_i})<_{lex} (r'_1,\ldots, r'_{k_i})$.
\end{itemize}
We show that this ordering forms a shelling order. Consider two facets $F_i^{r_1,\ldots, r_{k_i}}$ and $F_j^{r'_1,\ldots, r'_{k_j}}$ with $i<j$.
Since $F_i<F_j$, there exists an integer $\ell<j$ and $x_{j_t}\in F_j\mbox{set}\,minus F_i$ such that $F_j\mbox{set}\,minus F_\ell=\{x_{j_t}\}$. So
$x_{j_tr'_t}\in F_j^{r'_1,\ldots, r'_{k_j}}\mbox{set}\,minus F_i^{r_1,\ldots, r_{k_i}}$. Let
$F_\ell=\{x_{\ell_1},\ldots,x_{\ell_{k_\ell}}\}$, where $\ell_1<\cdots<\ell_{k_\ell}$. Then there exist indices $h_1,\ldots,h_{t-1},h_{t+1},\ldots,h_{k_j}$ such that
$j_1=\ell_{h_1},\ldots, j_{t-1}=\ell_{h_{t-1}},j_{t+1}=\ell_{h_{t+1}},\ldots,j_{k_j}=\ell_{h_{k_j}}$.
Thus
$$F_j^{r'_1,\ldots, r'_{k_j}}\mbox{set}\,minus F_\ell^{r''_1,\ldots,r''_{k_\ell}}=\{x_{j_tr'_t}\},$$ where
$r''_{h_1}=r'_1,\ldots,r''_{h_{t-1}}=r'_{t-1},r''_{h_{t+1}}=r'_{t+1},\ldots,r''_{h_{k_j}}=r'_{k_j}$ and $r''_{\lambda}=1$ for other indices $\lambda$. Since $\ell<j$, we have $F_\ell^{r''_1,\ldots,r''_{k_\ell}}<F_j^{r'_1,\ldots, r'_{k_j}}$.
Now assume that $i=j$ and $F_i^{r_1,\ldots, r_{k_i}}<F_i^{r'_1,\ldots, r'_{k_i}}$. Thus $$(r_1,\ldots, r_{k_i})<_{lex} (r'_1,\ldots, r'_{k_i}).$$ Let $1\leq t\leq k_i$ be an integer with $r_t<r'_t$. Then $x_{i_tr'_t}\in F_i^{r'_1,\ldots, r'_{k_i}}\mbox{set}\,minus
F_i^{r_1,\ldots, r_{k_i}}$, $$F_i^{r'_1,\ldots,r'_{k_i}}\mbox{set}\,minus F_i^{r'_1,\ldots,r'_{t-1},r_t,r'_{t+1},\ldots, r'_{k_i}}=\{x_{i_tr'_t}\}$$
and $$(r'_1,\ldots,r'_{t-1},r_t,r'_{t+1},\ldots, r'_{k_i})<_{lex} (r'_1,\ldots,r'_{k_i}).$$ Thus $F_i^{r'_1,\ldots,r'_{t-1},r_t,r'_{t+1},\ldots, r'_{k_i}}<F_i^{r'_1,\ldots,r'_{k_i}}$. The proof is complete.
\end{proof}
The following corollary is an immediate consequence of Theorem \ref{vI}, Remark \ref{pure} and \cite[Theorem 5.3.18]{VIL}.
\begin{cor}\label{cor1}
Let $\displaystyleelta$ be a pure shellable simplicial complex. Then $\displaystyleelta^{(s_1,\ldots,s_n)}$ is Cohen-Macaulay for any $s_1,\ldots,s_n\in \mathbb{N}$.
\end{cor}
\begin{thm}\label{one dimension}
Let $\displaystyleelta$ be a pure one dimensional simplicial complex. Then the following statements are equivalent.
\begin{itemize}
\item[(i)] $\displaystyleelta$ is connected.
\item[(ii)] $\displaystyleelta$ is vertex decomposable.
\item[(iii)] $\displaystyleelta$ is shellable.
\item[(iv)] $\displaystyleelta$ is sequantially Cohen-Macaulay.
\item[(v)] $\displaystyleelta$ is Cohen-Macaulay.
\end{itemize}
\end{thm}
\begin{proof}
\begin{itemize}
\item[$(i\mathbb{R}ightarrow ii)$] Suppose that $\displaystyleelta=\langle F_1, \dots, F_m\rangle$. We use induction on $m$. If $m=1$, $\displaystyleelta$ is clearly vertex decomposable. Suppose inductively that the result has been proved for smaller values of $m$. We consider two cases. If $\displaystyleelta$ has a free vertex (a vertex which belongs to only one facet), then there is a facet, say $F_m=\{x,y\}$, of $\displaystyleelta$ such that $x\not\in \bigcup_{i=1}^{m-1}F_i$. In this case
$\mathrm{lk}_\displaystyleelta(x)=\langle \{ y\}\rangle,$
which is clearly vertex decomposable.
Also, since $\displaystyleelta$ is connected,
$$\mathrm{del}_\displaystyleelta(x)=\langle F_1,\dots, F_{m-1}\rangle$$ is a pure one dimensional connected simplicial complex. So, by inductive hypothesis $\mathrm{del}_\displaystyleelta(x)$ is also vertex decomposable. Moreover each facet of $\mathrm{del}_\displaystyleelta(x)$ is a facet of $\displaystyleelta$. This shows that $\displaystyleelta$ is vertex decomposable. Now, suppose that $\displaystyleelta$ doesn't have any free vertex. So, each vertex belongs to at least two facets. Hence, there is a vertex $x$ such that $\mathrm{del}_\displaystyleelta(x)$ is also connected and one dimensional. (Note that since $\displaystyleelta$ is connected and one dimensional, it may be illustrated as a connected graph. Also, from graph theory, we know that every connected graph has at least two vertices such that by deleting them, we still have a connected graph). Now, by induction hypothesis we have that $\mathrm{del}_\displaystyleelta(x)$ is vertex decomposable. Also, $\mathrm{lk}_\displaystyleelta(x)$ is a discrete set and so vertex decomposable. Furthermore, in view of the choice of $x$, it is clear that every facet of $\mathrm{del}_\displaystyleelta(x)$ is a facet of $\displaystyleelta$. Hence, $\displaystyleelta$ is vertex decomposable as desired.
\item[$(ii\mathbb{R}ightarrow iii)$] follows from \cite[Theorem 11.3]{BW2}.
\item[$(iii\mathbb{R}ightarrow iv)$] is firstly shown by Stanley in \cite{Stanley}.
\item[$(iv\mathbb{R}ightarrow v)$] The result follows from the fact that every pure sequantially Cohen-Macaulay simplicial complex is Cohen-Macaulay.
\item[$(v\mathbb{R}ightarrow i)$] follows from \cite[Corollary 5.3.7]{VIL}.
\end{itemize}
\end{proof}
\begin{cor}\label{CM}
Let $\displaystyleelta$ be a Cohen-Macaulay simplicial complex of dimension one. Then $\displaystyleelta^{(s_1,\ldots,s_n)}$ is Cohen-Macaulay for any $s_1,\ldots,s_n\in \mathbb{N}$.
\end{cor}
\begin{proof}
Since $\displaystyleelta$ is Cohen-Macaulay of dimension one, Theorem \ref{one dimension} implies that $\displaystyleelta$ is pure shellable. Hence, Corollary \ref{cor1} yields the result.
\end{proof}
The evidence suggests when $\displaystyleelta$ is Cohen-Macaulay, its expansions are also Cohen-Macaulay. Corollaries \ref{cor2}, \ref{cor3}, \ref{cor1} and \ref{CM} are some results in this regard. But in general, we did not get to a proof or a counter example for this statement. So, we just state it as a conjecture as follows.
\textbf{Conjecture.} If $\displaystyleelta$ is a Cohen-Macaulay simplicial complex, then $\displaystyleelta^{(s_1,\ldots,s_n)}$ is Cohen-Macaulay for any $s_1,\ldots,s_n\in \mathbb{N}$.
\section{Homological invariants of expansions of a simplicial complex}
We begin this section with the next theorem which presents formulas for the projective dimension and depth of the Stanley-Reisner ring of an expansion of a shellable simplicial complex in terms of the corresponding invariants of the Stanley-Reisner ring of the simplicial complex.
\begin{thm}\label{pd}
Let $\displaystyleelta$ be a shellable simplicial complex with the vertex set $\{x_1,\ldots,x_n\}$, $s_1,\ldots,s_n\in \mathbb{N}$ and $R=K[x_1,\ldots,x_n]$ and $R'=K[x_{11},\ldots,x_{1s_1},\ldots,x_{n1},\ldots,x_{ns_n}]$ be polynomial rings over a field $K$. Then $$\mathrm{pd}(R'/I_{\displaystyleelta^{(s_1,\ldots,s_n)}})=\mathrm{pd}(R/I_{\displaystyleelta})+s_1+\cdots+s_n-n$$
and
$$\mathrm{depth}(R'/I_{\displaystyleelta^{(s_1,\ldots,s_n)}})=\mathrm{depth}(R/I_{\displaystyleelta}).$$
\end{thm}
\begin{proof}
Let $\displaystyleelta$ be a shellable simplicial complex. Then it is sequentially Cohen-Macaulay. By Theorem \ref{vI}, $\displaystyleelta^{(s_1,\ldots,s_n)}$ is also shellable and then
sequentially Cohen-Macaulay. Thus by \cite[Corollary 3.33]{MVi}, $$\mathrm{pd}(R'/I_{\displaystyleelta^{(s_1,\ldots,s_n)}})=\mbox{bight}\,(I_{\displaystyleelta^{(s_1,\ldots,s_n)}})$$
and $$\mathrm{pd}(R/I_{\displaystyleelta})=\mbox{bight}\,(I_{\displaystyleelta}).$$ Let $k=\min\{|F|:\ F\in \mathcal{F}(\displaystyleelta)\}$. It is easy to see that
$\min\{|F|:\ F\in \mathcal{F}(\displaystyleelta^{(s_1,\ldots,s_n)})\}=k$. Then $\mbox{bight}\,(I_{\displaystyleelta})=n-k$ and
$$\mbox{bight}\,(I_{\displaystyleelta^{(s_1,\ldots,s_n)}})=|V(\displaystyleelta^{(s_1,\ldots,s_n)})|-k=s_1+\cdots+s_n-k=s_1+\cdots+s_n+\mathrm{pd}(R/I_{\displaystyleelta})-n.$$
The second equality holds by Auslander-Buchsbaum formula, since $\mathrm{depth}(R')=s_1+\cdots+s_n$.
\end{proof}
In the following example, we compute the invariants in Theorem \ref{pd} and illustrate the equalities.
\begin{exam}
{\rm Let $\displaystyleelta=\langle \{x_1,x_2,x_3\}, \{x_1,x_2,x_4\},\{x_4,x_5\}\rangle$. Then $\displaystyleelta$ is shellable with the order as listed in $\displaystyleelta$. Then
$$\displaystyleelta^{(1,1,2,1,2)}=\langle \{x_{11},x_{21},x_{31}\},\{x_{11},x_{21},x_{32}\},\{x_{11},x_{21},x_{41}\},\{x_{41},x_{51}\},\{x_{41},x_{52}\}\rangle.$$
computations by Macaulay2 \cite{GS}, show that $\mathrm{pd}(R/I_{\displaystyleelta})=3$ and $\mathrm{pd}(R'/I_{\displaystyleelta^{(s_1,\ldots,s_n)}})=5=\mathrm{pd}(R/I_{\displaystyleelta})+s_1+\cdots+s_n-n=3+1+1+2+1+2-5$.
Also $\mathrm{depth}(R'/I_{\displaystyleelta^{(s_1,\ldots,s_n)}})=\mathrm{depth}(R/I_{\displaystyleelta})=2$. }
\end{exam}
The following result, which is a special case of \cite[Corollary 2.7]{Leila}, is our main tool to prove Proposition \ref{shreg}.
\begin{thm}(See \cite[Corollary 2.7]{Leila}.)\label{Leila}
Let $I$ be a monomial ideal with linear quotients with the ordering $f_1<\cdots<f_m$ on the minimal generators of $I$.
Then $$\beta_{i,j}(I)=\subseteqm_{\deg(f_t)=j-i} {|\mbox{set}\,_I(f_t)|\choose i}.$$
\end{thm}
\begin{prop}\label{shreg}
Let $\displaystyleelta=\langle F_1,\ldots,F_m\rangle$ be a shellable simplicial complex with the vertex set $\{x_1,\ldots,x_n\}$, $s_1,\ldots,s_n\in \mathbb{N}$ and $R=K[x_1,\ldots,x_n]$ and $R'=K[x_{11},\ldots,x_{1s_1},\\ \ldots,x_{n1},\ldots,x_{ns_n}]$ be polynomial rings over a field $K$. Then
\begin{itemize}
\item[(i)] if $s_1,\ldots,s_n>1$, then
$\mathrm{reg}(R'/I_{\displaystyleelta^{(s_1,\ldots,s_n)}})=\mathrm{dim}(\displaystyleelta)+1=\mathrm{dim}(R/I_{\displaystyleelta});$
\item[(ii)] if for each $1\leq i\leq m$, $\lambda_i=|\{x_\ell\in F_i:\ s_\ell>1\}|$, then $$\mathrm{reg}(R'/I_{\displaystyleelta^{(s_1,\ldots,s_n)}})\leq \mathrm{reg}(R/I_{\displaystyleelta})+\max \{\lambda_i:\ 1\leq i\leq m\}.$$
\end{itemize}
\end{prop}
\begin{proof}
Without loss of generality assume that $F_1<\cdots<F_m$ is a shelling for $\displaystyleelta$. We know that $I_{\displaystyleelta^{\vee}}$ has linear quotients with the ordering $x^{F_1^c}<\cdots<x^{F_m^c}$ on its minimal generators (see \cite[Theorem 1.4]{HD}). Moreover by Theorem \ref{1.3}, $\mathrm{reg}(R'/I_{\displaystyleelta^{(s_1,\ldots,s_n)}})=\mathrm{pd}(I_{\displaystyleelta^{(s_1,\ldots,s_n)^{\vee}}})$ and by \cite[Theorem 5.1.4]{BH} we have $\mathrm{dim}(R/I_\displaystyleelta)=\mathrm{dim}(\displaystyleelta)+1$. Thus, to prove (i), it is enough to show that $\mathrm{pd}(I_{\displaystyleelta^{(s_1,\ldots,s_n)^{\vee}}})=\mbox{dim}\,(\displaystyleelta)+1$. By Theorem \ref{Leila}, $\mathrm{pd}(I_{\displaystyleelta^{\vee}})=\max\{|\mbox{set}\,(x^{F_i^c})|:\ 1\leq i\leq m\}$.
For any $1\leq i\leq m$, $\mbox{set}\,(x^{F_i^c})\subseteqbseteq F_i$, since any element $x_\ell\in \mbox{set}\,(x^{F_i^c})$ belongs to
$(x^{F_j^c}):_R(x^{F_i^c})$ for some $1\leq j<i$. Thus $x_\ell=x^{F_j^c}/\gcd(x^{F_j^c},x^{F_i^c})=x^{F_i\mbox{set}\,minus F_j}$. Let $F_i=\{x_{i_1},\ldots,x_{i_{k_i}}\}$ and $\mbox{set}\,(x^{F_i^c})=\{x_{i_\ell}:\ \ell\in L_i\}$, where $L_i\subseteqbseteq \{1,\ldots,k_i\}$.
Consider the shelling for $\displaystyleelta^{(s_1,\ldots,s_n)}$ constructed in the proof of Theorem \ref{vI}. Using again of \cite[Theorem 1.4]{HD} shows that this shelling induces an order of linear quotients on the minimal generators of $I_{\displaystyleelta^{(s_1,\ldots,s_n)^{\vee}}}$. With this order
\begin{equation}\label{tasavi}
\mbox{set}\,(x^{(F_i^{r_1,\ldots, r_{k_i}})^c})=\{x_{i_\ell r_\ell}:\ \ell\in L_i\}\cup \{x_{i_tr_t}:\ r_t>1\}.
\end{equation}
More precisely, if $r_t>1$ for some $1\leq t\leq k_i$, then
\begin{align*}
x_{i_tr_t} & =x^{(F_i^{r_1,\ldots, r_{k_i}}\mbox{set}\,minus F_i^{r_1,\ldots,r_{t-1},r_t-1,r_{t+1},\ldots, r_{k_i}})}\\
& \in (x^{(F_i^{r_1,\ldots,r_{t-1},r_t-1,r_{t+1},\ldots, r_{k_i}})^c}):_{R'}(x^{(F_i^{r_1,\ldots, r_{k_i}})^c}).
\end{align*}
Hence, $x_{i_tr_t}\in \mbox{set}\,(x^{(F_i^{r_1,\ldots, r_{k_i}})^c})$.
Also for any $x_{i_\ell}\in \mbox{set}\,(x^{F_i^c})$, there exists $1\leq j<i$ such that $x_{i_\ell}=x^{F_i\mbox{set}\,minus F_j}\in (x^{F_j^c}):_R(x^{F_i^c})$.
Thus there exist positive integers $r''_1,\ldots,r''_j$ such that
\begin{align*}
x_{i_\ell r_\ell} & =x^{(F_i^{r_1,\ldots,r_{k_i}}\mbox{set}\,minus F_j^{r''_1,\ldots,r''_{k_j}})}\\
& \in (x^{(F_j^{r''_1,\ldots, r''_{k_j}})^c}):_{R'}(x^{(F_i^{r_1,\ldots,r_{k_i}})^c}).
\end{align*}
Hence, $x_{i_\ell r_\ell}\in \mbox{set}\,(x^{(F_i^{r_1,\ldots, r_{k_i}})^c})$.
Now, if $s_1,\ldots,s_n>1$, then
$\mbox{set}\,(x^{(F_i^{s_{i_1},\ldots, s_{i_{k_i}}})^c})=\{x_{i_1s_{i_1}},\ldots,x_{i_{k_i}s_{i_{k_i}}}\}.$ Thus
$$\mathrm{pd}(I_{{\displaystyleelta^{(s_1,\ldots,s_n)}}^\vee})=\max\{|\mbox{set}\,(x^{(F_i^{s_{i_1},\ldots, s_{i_{k_i}}})^c})|:\ 1\leq i\leq m\}=\max\{|F_i|:\ 1\leq i\leq m \}=\mathrm{dim}(\displaystyleelta)+1.$$ To prove (ii), notice that by equality \ref{tasavi}, $|\mbox{set}\,(x^{(F_i^{r_1,\ldots, r_{k_i}})^c})|\leq |\mbox{set}\,(x^{F_i^c})|+ \lambda_i$. Therefore $\mathrm{pd}(I_{{\displaystyleelta^{(s_1,\ldots,s_n)}}^\vee})\leq \mathrm{pd}(I_{\displaystyleelta^{\vee}})+\max \{\lambda_i:\ 1\leq i\leq m\}.$
Now, by Theorem \ref{1.3}, the result holds.
\end{proof}
\begin{exam}
{\rm
Consider the chordal graph $G$ depicted in Figure $3$ and its $(2,2,3,2,3)$-expansion which is a graph with $12$ vertices. Then $\displaystyleelta_G=\langle\{x_1,x_3\},\{x_3,x_5\},\{x_4,x_5\},\{x_2\}\rangle$.
Since $G$ is shellable, by Proposition \ref{shreg}, $\mathrm{reg}(R'/I(G^{(2,2,3,2,3)}))=\mbox{dim}\,(\displaystyleelta_G)+1=2$.
\begin{figure}
\caption{The graph $G$ and the $(2,2,3,2,3)$-expansion of $G$}
\label{fig7}
\end{figure}
}
\end{exam}
\providecommand{\bysame}{\leavevmode\mbox{ht}\,box
to3em{\mbox{ht}\,rulefill}\thinspace}
\end{document} |
\begin{document}
\title{Projected Dynamical Systems on Irregular, Non-Euclidean Domains for Nonlinear Optimization hanks{Submitted to the editors DATE.
unding{This work was supported by ETH Zurich and the SNF AP Energy Grant \#160573.}
\begin{abstract}
Continuous-time projected dynamical systems are an elementary class of discontinuous dynamical systems with trajectories that remain in a feasible domain by means of projecting outward-pointing vector fields. They are essential when modeling physical saturation in control systems, constraints of motion, as well as studying projection-based numerical optimization algorithms. Motivated by the emerging application of feedback-based continuous-time optimization schemes that rely on the physical system to enforce nonlinear hard constraints, we study the fundamental properties of these dynamics on general locally-Euclidean sets. Among others, we propose the use of Krasovskii solutions, show their existence on nonconvex, irregular subsets of low-regularity Riemannian manifolds, and investigate how they relate to conventional Carath\'eodory solutions. Furthermore, we establish conditions for uniqueness, thereby introducing a generalized definition of prox-regularity which is suitable for non-flat domains. Finally, we use these results to study the stability and convergence of projected gradient flows as an illustrative application of our framework. We provide simple counter-examples for our main results to illustrate the necessity of our already weak assumptions.
\end{abstract}
\section{Introduction}
Projected dynamical systems form an important class of discontinuous dynamical systems whose trajectories remain in a domain $\mathcal{X}$. This invariance (or \emph{viability}) of $\mathcal{X}$ is achieved by projecting a vector field $f$ on the tangent cone of $\mathcal{X}$. More specifically, in the interior of $\mathcal{X}$, trajectories follow the vector field $f$. At the boundary, instead of leaving $\mathcal{X}$, trajectories ``slide'' along the boundary of $\mathcal{X}$ in the feasible direction that is closest to the direction imposed by $f$. This qualitative behavior is illustrated in \cref{fig:qual_pds1}.
Even though projected dynamical systems have a long history in different contexts such as the study of variational inequalities or differential inclusions, new compelling applications in the context of real-time optimization require a different, more general approach. Hence, this paper is primarily motivated by the renewed interest in dynamical systems that solve optimization problems. Early works in this spirit such as~\cite{brockettDynamicalSystemsThat1988,helmkeOptimizationDynamicalSystems1996} have designed continuous-time systems to solve computational problems such as diagonalizing matrices or solving linear programs. This has further resulted in the study of optimization algorithms over manifolds~\cite{absilOptimizationAlgorithmsMatrix2008}. Recently, interest has shifted towards analyzing existing iterative schemes with tools from dynamical systems including Lyapunov theory~\cite{wilsonLyapunovAnalysisMomentum2016} and integral quadratic constraints~\cite{lessardAnalysisDesignOptimization2016,fazlyabAnalysisOptimizationAlgorithms2017}. Most of these have considered unconstrained optimization problems~\cite{suDifferentialEquationModeling2014} and algorithms that can be modelled with a standard ODE~\cite{kricheneAcceleratedMirrorDescent2015} or with variational tools~\cite{wibisonoVariationalPerspectiveAccelerated2016}. With this paper we hope to pave the way for the analysis of algorithms for constrained optimization whose continuous-time limits are discontinuous.
Recently, this idea of studying the dynamical aspects of optimization algorithms has given rise to a new type of feedback control design that aims at steering a physical system in real time to the solution of an optimization problem~\cite{nelsonIntegralQuadraticConstraint2018,zhangDistributedControlReaching2018,mentaStabilityDynamicFeedback2018,colombinoOnlineOptimizationFeedback2019,bernsteinOnlinePrimalDualMethods2019} without external inputs.
Precursors of this idea have been used in the analysis of congestion control in communication networks~\cite{kellyRateControlCommunication1998,lowInternetCongestionControl2002}. More recently, the concept has been widely applied to power systems~\cite{hauswirthOnlineOptimizationClosed2017,dallaneseOptimalPowerFlow2018,molzahnSurveyDistributedOptimization2017,tangRealTimeOptimalPower2017}. This context is particularly challenging, because the physical laws of power flow, saturating components, and other constraints define a highly non-linear, nonconvex feasible domain.
Projected dynamical systems provide a particularly useful framework to model actuation constraints and physical saturation in this context, but existing results are of limited applicability for complicated problems. Hence, in this paper, we consider new, generalized features for projected dynamical systems. We consider for example \emph{irregular} feasible domains (\cref{fig:qual_pds2}) for which traditional \emph{Carath\'eodory solutions} can fail to exist or may not be unique. Furthermore, \emph{non-orthogonal projections} occur in non-Euclidean spaces and may alter the dynamics.
Finally, coordinate-free definitions are required to study projected dynamical systems on subsets of manifolds (\cref{fig:qual_pds4}).
\begin{figure}
\caption{Qualitative behavior of projected dynamical systems: (a) projected gradient flow on a convex polyhedron, (b) flow on an irregular set with non-unique trajectory, (c) periodic projected trajectory on a subset of a sphere.}
\label{fig:qual_pds}
\end{figure}
\subsection*{Literature review}
Different approaches have been reviewed and explored to establish the results in this paper.
One of the earliest formulations of projected dynamical systems goes back to~\cite{henryExistenceTheoremClass1973} which establishes the existence of Carath\'eodory solutions on closed convex domains.
In~\cite{cornetExistenceSlowSolutions1983} this requirement is relaxed to $\mathcal{X}$ being Clarke regular (for existence) and prox-regular (for uniqueness).
In the larger context of differential inclusions and viability theory~\cite{aubinViabilityTheory1991,aubinDifferentialInclusionsSetValued1984}, projected dynamical systems are often presented as specific examples of more general differential inclusions, but without substantially generalizing the results of~\cite{henryExistenceTheoremClass1973,cornetExistenceSlowSolutions1983}.
In the context of variational equalities,~\cite{nagurneyProjectedDynamicalSystems1996} provides alternative proofs of existence and uniqueness of Carath\'eodory solutions when the domain $\mathcal{X}$ is a convex polyhedron by using techniques from stochastic analysis.
In~\cite{brogliatoEquivalenceComplementaritySystems2006}, various equivalence results between the different formulations are established for convex $\mathcal{X}$.
Finally, projected dynamical systems have been defined and studied in the more general context of Hilbert~\cite{cojocaruExistenceSolutionsProjected2004} and Banach spaces~\cite{cojocaruNonpivotImplicitProjected2012,giuffreClassesProjectedDynamical2008}. The latter, in particular, is complicated by the lack of an inner product and consequently more involved projection operators~\cite{alberGeneralizedProjectionOperators1996}.
The behavior of projected dynamical systems as illustrated in \cref{fig:qual_pds} suggests the presence of switching mechanics that result in different vector fields being active in different parts of the domain and its boundary in particular. This idea is further supported by the fact that in the study of optimization problems with a feasible domain delimited by explicit constraints, it is often useful to define the (finite) \emph{set of active constraints} at a given point. This suggests that projected dynamical systems should be modeled as switched~\cite{liberzonSwitchingSystemsControl2003} or even hybrid systems~\cite{goebelHybridDynamicalSystems2012} or hybrid automata~\cite{lygerosDynamicalPropertiesHybrid2003,simicGeometricTheoryHybrid2000}. However, projected dynamical systems are much more easily (and generally) modeled as differential inclusions without explicitly considering any type of switching.
A special case of projected dynamical systems are subgradient and saddle-point flows arising in non-smooth and constrained optimization. Whereas projection-based algorithms and subgradients are ubiquitous in the analysis of iterative algorithms, work on their continuous-time counterparts is far less prominent has only been studied with limited generality~\cite{arrowStudiesLinearNonlinear1958,cherukuriAsymptoticConvergenceConstrained2016,cortesDiscontinuousDynamicalSystems2008,hauswirthProjectedGradientDescent2016}, e.g., restricted to convex problems.
\subsection*{Contributions}
In this paper, we study a generalized class of projected dynamical systems in finite dimensions that allows for oblique projection directions. These variable projection directions are described by means of a (possibly non-differentiable) metric $g$ and are essential in providing a coordinate-free definition of projected dynamical systems on low-regularity Riemannian manifolds. Compared to previous work, we do not make a-priori assumptions on the regularity (or convexity) of the feasible domain $\mathcal{X}$ or the vector field $f$. Instead, we strive to illustrate the necessity of those assumptions that we require by a series of (non-)examples.
Our main contribution is the development of a self-contained and comprehensive theory for this general setup. Namely, we provide weak requirements on the feasible set $\mathcal{X}$, the vector field $f$, the metric $g$ and the differentiable structure of the underlying manifold that guarantee existence and uniqueness of trajectories, as well as other properties. \cref{tab:summary} at the end of the paper concisely summarizes these results.
To be able to work with projected dynamical systems on irregular domains and with discontinuous vector fields, we resort to so-called \emph{Krasovskii solutions} that are a weaker notion than the classical \emph{Carath\'eodory solutions} and are commonly used in the study of differential inclusions because their existence is guaranteed under minimal requirements. We derive this set of regularity conditions in the specific context of projected dynamical system.
Under slightly stronger assumptions involving continuity and Clarke regularity, we show that Krasovskii solutions coincide with the classical Carath\'eodory solutions, thus recovering (in case of the Euclidean metric) known requirements for the existence of the latter. Finally, we lay out the requirements for uniqueness of solutions which are based on Lipschitz-continuity and a new, generalized definition of prox-regularity which is suitable for low-regularity Riemannian manifolds, i.e., manifolds that do not necessarily have a $C^\infty$ structure~\cite{hosseiniMetricProjectionProxregular2013,bernicotSweepingProcessProxregular2015}. Our already weak regularity conditions are sharp in the sense that counter-examples can be constructed to show that requirements cannot be violated individually without the respective result failing to hold.
A major appeal of our analysis framework is its geometric nature: All of our notions are preserved by sufficiently regular coordinate transformations, which allows us to extend all of our results to constrained subsets of differential manifolds. A noteworthy by-product of this analysis is the fact that our generalized definition of prox-regularity is an intrinsic property of subsets of $C^{1,1}$ manifolds, i.e., independent of the metric, even though the traditional definition (on $\mathbb{R}^n$) suggests that prox-regularity depends on the choice of metric.
Through a series of examples, we demonstrate the application of our framework to general (nonlinear and nonconvex) optimization problems and study the stability and convergence of projected gradient dynamics under very weak regularity assumptions.
Thus, we believe that our results are not only of interest in the context of discontinuous dynamical systems, but we also envision their use in the analysis of algorithms for nonlinear, nonconvex optimization problems, possibly on manifolds.
The properties developed in the present paper also form a solid foundation for constrained feedback control and online optimization in various contexts. Some preliminary results for online optimization in power systems can be found in~\cite{hauswirthProjectedGradientDescent2016,hauswirthOnlineOptimizationClosed2017}.
\subsection*{Paper organization}
After introducing notation and preliminary definitions in \cref{sec:preliminaries,sec:pds}, we establish the existence of Krasovskii solutions to projected dynamical systems on $\mathbb{R}^n$ in \cref{sec:exist}. \Cref{sec:equiv} establishes equivalence of Krasovskii and Carath\'eodory solutions under Clarke regularity and we point out the connection to related work. In \cref{sec:uniq}, we elaborate on the requirements for uniqueness and in \cref{sec:mfd} we define projected dynamical systems on low-regularity Riemannian manifolds and establish the requirements on the differentiable structure that guarantee existence and uniqueness. As an illustration of optimization applications, in \cref{sec:stab} we consider Krasovskii solutions of projected gradient systems on irregular domains, we study their convergence and stability and revisit the connection to subgradient flows.
Throughout the paper, we illustrate our theoretical developments with insightful examples. Finally, \cref{sec:conclusion} concisely summarizes our results in the form of \cref{tab:summary} and concludes the paper.
The appendix includes technical definitions and results that are used in proofs but are not required to understand the main results of the paper.
\unless\ifARXIV
Some lengthy algebraic manipulations and technical, though standard, proofs are only available online, in the extended version of this paper~\cite{hauswirthProjectedDynamicalSystems2018a}.
\fi
\section{Preliminaries}\label{sec:preliminaries}
\subsection{Notation}
We only consider finite-dimensional spaces. Unless explicitly noted otherwise, we will work in the usual Euclidean setup for $\mathbb{R}^n$ with inner product $\left\langle \cdot, \cdot \right\rangle$ and 2-norm $\| \cdot \|$. Whenever it is informative, we make a formal distinction between $\mathbb{R}^n$ and its tangent space $T_x \mathbb{R}^n$ at $x \in \mathbb{R}$, even though they are isomorphic.
For a set $A \subset \mathbb{R}^n$ we use the notation $\| A \| := \sup_{v \in A} \| v \|$. The closure, convex hull and closed convex hull of $A$ are denoted by $\cl A$, $\co A$, and $\cocl A$, respectively.
The set $A$ is \emph{locally compact} if it is the intersection of a closed and an open set. A neighborhood $U\subset A$ of $x \in A$ is understood to be a \emph{relative neighborhood}, i.e., with respect to the subspace topology on $A$.
Given a convergent sequence $\{x_k\}$, the notation $x_k \underset{A} \rightarrow x$ implies that $x_k \in A$ for all $k$. If $x_k \in \mathbb{R}$, the notation $x \rightarrow 0^+$ means $x_k > 0$ for all~$k$ and $x_k$ converges to 0.
Let $V$ and $W$ be vector spaces endowed with norms $\| \cdot \|_V$ and $\| \cdot \|_W$, respectively, and let $A \subset V$. Continuous maps $\Phi: A \rightarrow W$ are denoted by $C^0$. The map $\Phi$ is \emph{(locally) Lipschitz} (denoted by $C^{0,1}$) if for every $x \in A$ there exists $L>0$ such that for all $z, y \in A$ in a neighborhood of $x$ it holds that
\begin{equation}\label{eq:def_lipschitz}
\| \Phi(z) - \Phi(y) \|_W \leq L \| z - y \|_V \, .
\end{equation}
The map $\Phi$ is \emph{globally Lipschitz} if~\eqref{eq:def_lipschitz} holds for the same $L$ for all $z,y$.
Differentiability is understood in the sense of Fr\'echet. Namely, if $A$ is open, then the map $\Phi$ is \emph{differentiable at $x$} if there is a linear map $D_x \Phi: V \rightarrow W$ such that
\begin{equation*}
\underset{y \rightarrow x}{\lim} \, \frac{\| \Phi(y) - \Phi(x) - D_x \Phi (y - x) \|_W}{\| y - x \|_V} = 0 \, .
\end{equation*}
The map $\Phi$ is \emph{differentiable} ($C^1$) if it is differentiable at every $x \in A$. It is $C^{1,1}$ if it is $C^1$ and $D_x \Phi$ is $C^{0,1}$ (as function of $x$). Finally, given bases for $V$ ($\dim V= m$) and $W$ ($\dim W = n$), the \emph{Jacobian of $\Phi$ at $x$} is denoted by the $n\times m$-matrix $\nabla \Phi(x)$.
In our context, a \emph{set-valued map} $F: A \rightrightarrows \mathbb{R}^n$ where $A \subset \mathbb{R}^n$ is a map that assigns to every point $x \in A$ a set $F(x) \subset T_x\mathbb{R}^n$.
The set-valued map $F$ is \emph{non-empty}, \emph{closed}, \emph{convex}, or \emph{compact} if for every $x\in A$ the set $F(x)$ is non-empty, closed, convex, or compact, respectively.
It is \emph{locally bounded} if for every $x \in A$ there exists $L > 0$ such that $\| F(y) \| \leq L$ for all $y \in A$ in a neighborhood of $x$. The same definition also applies to single-valued functions. The map $F$ is \emph{bounded} if there exists $L > 0$ such that $\| F(y) \| \leq L$ for all $x \in A$. The \emph{inner} and \emph{outer limits} of $F$ at $x$ are denoted by $\lim \inf_{y \rightarrow x} F(y)$ and $\lim \sup_{y \rightarrow x} F(y)$ respectively (see appendix for a formal definition and summary of continuity concepts which are required for certain proofs only).
\subsection{Tangent and Clarke Cones}
The ensuing definitions follow~\cite[Chap.~6]{rockafellarVariationalAnalysis2009}.
\begin{definition}\label{def:tgt_cone}
Given a set $\mathcal{X} \subset \mathbb{R}^n$ and $x \in \mathcal{X}$, a vector $v \in T_x\mathbb{R}^n$ is a \emph{tangent vector of $\mathcal{X}$ at $x$} if there exist sequences $x_k \underset{\mathcal{X}}{\rightarrow} x$ and $\delta_k \rightarrow 0^+$ such that $\tfrac{x_k - x}{\delta_k} \rightarrow v$. The set of all tangent vectors is the \emph{tangent cone of $\mathcal{X}$ at $x$} and denoted by $T_x \mathcal{X}$.
\end{definition}
The tangent cone $T_x \mathcal{X}$ (also known as \emph{(Bouligand's) contingent cone~\cite{clarkeNonsmoothAnalysisControl1998}}) is closed and non-empty (namely, $0 \in T_x \mathcal{X}$) for any $x \in \mathcal{X}$.
In the following definition of Clarke regularity and in most of paper we limit ourselves to locally compact subsets of $\mathbb{R}^n$. In our context, a more general definition of Clarke regularity does not improve our results and only adds to the technicalities.
\begin{definition}\label{def:clarke_tgt}
For a locally compact set $\mathcal{X} \subset \mathbb{R}^n$ the \emph{Clarke tangent cone at $x \in \mathcal{X}$} is defined as the inner limit of the tangent cones, i.e., $T^C_x \mathcal{X} := \underset{y \rightarrow x}{\lim \inf} \, T_y \mathcal{X}$.
\end{definition}
By definition of the inner limit, we have $T^C_x \mathcal{X} \subseteq T_x \mathcal{X}$.
Furthermore, $T^C_x \mathcal{X}$ is closed, convex and non-empty for all $x \in \mathcal{X}$~\cite[Thm.~6.26]{rockafellarVariationalAnalysis2009}.
\begin{definition}\label{def:clarke_reg} We call a set $\mathcal{X} \subset \mathbb{R}^n$ \emph{Clarke regular at $x$} if it is locally compact and $T_x \mathcal{X} = T_x^C \mathcal{X}$. The set $\mathcal{X}$ is \emph{Clarke regular} if it is Clarke regular for all $x \in \mathcal{X}$.
\end{definition}
\cref{fig:tgt_cone} illustrates the definition of a tangent vector by a sequence $\{x_k \}$ that approaches $x$ in a tangent direction. \cref{fig:ctgt_cone} shows a set that is not Clarke regular.
The following example illustrates that, under standard constraint qualifications as used in optimization theory, sets defined by $C^1$ inequality constraints are Clarke regular. Such sets are generally encountered in nonlinear programming.
\begin{figure}
\caption{Tangent cone construction (a), Clarke tangent cone at an irregular point (b), and oblique normal cones induced by a non-Euclidean metric (c).}
\label{fig:tgt_cone}
\label{fig:ctgt_cone}
\label{fig:norm_cone}
\end{figure}
\begin{example}[sets defined by inequality constraints]\label{ex:clarke_reg_constraint_set}
Let $h: \mathbb{R}^n \rightarrow \mathbb{R}^m$ be $C^{1}$ such that $\nabla h(x)$ has full rank for all $x$.\footnote{This rank condition is a standard \emph{constraint qualification} in nonlinear programming~\cite{bazaraaNonlinearProgrammingTheory2006}. In general, instead of $\nabla h(x)$ having full rank for all $x$, it suffices that for a given $x$ only the active constraints (i.e., $\nabla h_{I(x)}(x)$) have full rank. Furthermore, equality constraints can be easily incorporated.}
Then, the set $\mathcal{X} := \{ x \, | \, h(x) \leq 0 \}$ is Clarke regular~\cite[Thm.~6.31]{rockafellarVariationalAnalysis2009}. In particular, let $h$ be expressed componentwise as $h(x) = {\left[ h_1(x), \ldots, h_m(x) \right]}^T$, let $I(x) := \{ i \, | \, h_i(x) = 0 \}$ denote the set of active constraints at $x \in \mathcal{X}$ and define $h_{I(x)} := {[ h_i(x) ]}_{i\in I(x)}$ as the function obtained from stacking the active constraint functions. Then, the (Clarke) tangent cone at $x$ in the canonical basis is given by
$T^C_ x \mathcal{X} = T_x \mathcal{X} = \{ v \, | \, \nabla h_{I(x)}(x) v \leq 0 \}$.
\end{example}
\subsection{Low-regularity Riemannian metrics}
A natural extension for projected dynamical systems are oblique projection directions. These are conveniently defined via a (Riemannian) metric which defines a variable inner product on $T_x \mathbb{R}^n$ as function of $x$. Furthermore, the notion of a Riemannian metric is essential to define projected dynamical systems in a coordinate-free setup on manifolds.
We quickly review the definition of bilinear forms and inner products. Let $L_2^n$ denote the space of bilinear forms on $\mathbb{R}^n$, i.e., every $g \in L_2^n$ is a map $g: \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$ such that for every $u, v, w \in \mathbb{R}^n$ and $\lambda \in \mathbb{R}$ it holds that $g(u + v, w) = g(u, w) + g(v, w)$ and $g(u, v + w) = g(u, v) + g(u, w)$ as well as $g(\lambda v, w) = \lambda g(v, w) = g(v, \lambda w)$. Given the canonical basis of $\mathbb{R}^n$, $g$ can be written in matrix form as $g(u, v) := u^T G v$ where $G \in \mathbb{R}^{n \times n}$. In particular, $L_2^n$ is itself a $n^2$-dimensional space isomorphic to $\mathbb{R}^{n \times n}$.
An \emph{inner product} $g \in L_2^n$ is a symmetric, positive-definite bilinear form, that is, for all $u, v \in \mathbb{R}^n$ we have $g(u, v) = g(v,u)$. Further, $g(u,u) \geq 0$, and $g(u,u) = 0$ holds if and only if $u = 0$. If $g$ is an inner product we use the notation $\left\langle u, v \right\rangle_g := g(u,v)$. In matrix form, we can write $\left\langle u, v \right\rangle_g := u^T G v$ where $G$ is symmetric positive definite.
We write $\|\cdot \|_{g}$ given by $\| v \|_{g} := \sqrt{\left\langle v, v \right\rangle_{g}}$ to denote the 2-norm induced by $g$. The \emph{maximum} and \emph{minimum eigenvalues} of $g$ are denoted by $\maxEig{g} : = \max \{ \| v \|_{g} \, |\, \| v \| = 1 \}$ and $\minEig{g} = \min \{ \| v \|_{g} \, |\, \| v \| = 1 \}$ respectively, and the \emph{condition number} is defined as $\condN{g} := \maxEig{g} / \minEig{g}$.
In this context, also recall that the 2-norms induced by any two inner products on a finite-dimensional vector space are equivalent, that is, for a vector space $V$ with norms $\| \cdot \|_a$ and $\| \cdot \|_b$ there are constants $\ell>0$ and $L>0$ such that for every $v \in V$ it holds that $\ell \| v \|_a \leq \| v \|_b \leq L \| v \|_a $. For instance, $\ell = \minEig{b} /\maxEig{a}$ and $L = \maxEig{b} / \minEig{a}$.
Hence, we can define a metric as a variable inner product over a given set.
\begin{definition}\label{def:metric}
Given a set $\mathcal{X} \subset \mathbb{R}^n$, a \emph{(Riemannian) metric} is a map $g: \mathcal{X} \rightarrow L_2^n$ that assigns to every point $x \in \mathcal{X}$ an inner product $\left\langle \cdot , \cdot \right\rangle_{g(x)}$. A metric is (Lipschitz) continuous if is (Lipschitz) continuous as a map from $\mathcal{X}$ to $L^n_2$.
\end{definition}
If clear from the context at which point $x$ the metric $g$ is applied, we drop the argument in the subscript and write $\left\langle \cdot, \cdot \right\rangle_g$ or $\| \cdot \|_g$. We always retain the subscript $g$, in order to draw a distinction between the Euclidean norm $\| \cdot \|$.
Since $g$ is positive definite for all $x$ by definition, it follows that $\maxEig{g(x)}, \minEig{g(x)}$ and $\condN{g(x)}$ are well-defined for all $x$.
However, $\condN{g(x)}$ is not necessarily locally bounded (even if $g$ is bounded as a map). In particular, $\minEig{g(x)}$ might not be bounded below, away from 0. Hence, for metrics we require the following definition of local boundedness.
\begin{definition}
A metric $g$ on $\mathcal{X}$ is \emph{locally weakly bounded} if for every $x \in \mathcal{X}$ there exist $\ell, L > 0$ such that $\ell \leq \condN{g(y)} \leq L$ holds for all $y \in \mathcal{X}$ in a neighborhood of~$x$. It is \emph{weakly bounded} if $\ell \leq \condN{g(x)} \leq L$ holds for all $x \in \mathcal{X}$.
\end{definition}
A metric $g$ can be locally weakly bounded even if its not locally bounded as a map $\mathcal{X} \rightarrow L^n_2$. Furthermore, since maximum and minimum eigenvalues (and hence the condition number) are continuous functions of a metric (or the representing matrix) it follows that a continuous metric is always locally weakly bounded.
\begin{remark}
In the following, we will continue to use the Euclidean norm as a distance function on $\mathbb{R}^n$ and use any Riemannian metric only in the context of projection directions.
Thereby, we avoid the notational complexity introduced by Riemannian geometry, and more importantly we do not need to make an a priori assumption on the differentiability on the metric $g$ (which is a prerequisite for many Riemannian constructs to exist), thus preserving a high degree of generality.
\end{remark}
\subsection{Normal Cones} Given a metric $g$, we can define (oblique) normal cones induced by $g$ (see \cref{fig:norm_cone}).
\begin{definition}\label{def:norm_cone} Let $\mathcal{X} \subset \mathbb{R}^n$ be Clarke regular and let $g$ be a metric on $\mathcal{X}$, then the \emph{normal cone at $x \in \mathcal{X}$ with respect to $g$} is defined as the polar cone of $T^C_x \mathcal{X}$ with respect to the metric $g$, i.e.,
\begin{equation}\label{eq:norm_cone}
N^g_x \mathcal{X} := {\left(T_x^C \mathcal{X} \right)}^* = \left\lbrace \eta \, \middle|\, \forall v \in T^C_x \mathcal{X}: \, \left\langle v, \eta \right\rangle_{g(x)} \leq 0 \right\rbrace \, .
\end{equation}
The normal cone with respect to the Euclidean metric is simply denoted by $N_x \mathcal{X}$.
\end{definition}
\begin{remark}\label{rem:normal_cones} For simplicity, we will use the notion of normal cone only in the context of Clarke regular sets. If $\mathcal{X}$ is not Clarke regular, one needs to distinguish between the \emph{regular}, \emph{general} and \emph{Clarke normal cones}~\cite{rockafellarVariationalAnalysis2009}.
\end{remark}
\begin{example}[normal cone to constraint-defined sets]\label{ex:clarke_reg_normal_cone}
As in \cref{ex:clarke_reg_constraint_set} consider $\mathcal{X} := \{ x \, | \, h(x) \leq 0\}$ where $h: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is $C^1$ and $\nabla h(x)$ has full rank for all~$x$. Further, let $g$ denote a metric on $\mathcal{X}$ represented by $G(x)\in \mathbb{R}^{n \times n}$. Then, the normal cone of $\mathcal{X}$ at $x$ is given by
\begin{align*}
N^g_x \mathcal{X} = \left\lbrace \eta \, \middle| \, \eta = \sum\nolimits_{i \in I(x)} \alpha_i G^{-1}(x) {\nabla h_i(x)}^T, \, \alpha_i \geq 0 \right\rbrace
\end{align*}
which can be derived by inserting any $\eta$ into~\eqref{eq:norm_cone} and using $T_x \mathcal{X}$ in \cref{ex:clarke_reg_constraint_set}.
\end{example}
\section{Projected Dynamical Systems}\label{sec:pds}
With the above notions we can now formally define our main object of study.
\begin{definition}\label{def:proj_vf}
Given a set $\mathcal X \subset \mathbb{R}^n$, a metric $g$ on $\mathcal{X}$, and a vector field $f: \mathcal{X} \rightarrow \mathbb{R}^n$, the \emph{projected vector field} of $f$ is defined as the set-valued map
\begin{align}\label{eq:def_proj_vf}
\tproj{\mathcal{X}}{g}{f}: \mathcal{X} \rightrightarrows \mathbb{R}^n \qquad
x \mapsto \underset{v \in T_x \mathcal{X}}{\arg \min} \| v - f(x) \|^2_{g(x)}
\end{align}
\end{definition}
For simplicity, we call $\tproj{\mathcal{X}}{g}{f}$ a \emph{vector field} even though $\tproj{\mathcal{X}}{g}{f}(x)$ might not be a singleton.
We will write $\tproj{}{}{f}$ whenever $\mathcal{X}$ and $g$ are clear from the context.
\begin{example}[pointwise evaluation of a projected vector field] As in \cref{ex:clarke_reg_constraint_set,ex:clarke_reg_normal_cone} let $\mathcal{X} := \{ x \, | \, h(x) \leq 0\}$ where $h: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is $C^1$ and $\nabla h(x)$ has full rank for all~$x$ and let $g$ denote a metric on $\mathcal{X}$ represented by $G(x)\in \mathbb{R}^{n \times n}$. Furthermore, consider a vector field $f: \mathcal{X} \rightarrow \mathbb{R}^n$. Then, the projected vector field $\tproj{\mathcal{X}}{g}{f}(x)$ at $x \in \mathcal{X}$ is given as the solution of the convex quadratic program
\begin{align*}
\underset{v \in \mathbb{R}^n}{\minimize} \quad {(f(x) - v)}^T G(x) (f(x) - v) \qquad
\subjto \quad \nabla h_{I(x)}(x) v \leq 0 \, .
\end{align*}
Note that $x$ is not an optimization variable. Hence, the properties of $f$ and $g$ as function of $x$ are irrelevant when doing a pointwise evaluation of $\tproj{\mathcal{X}}{g}{f}(x)$.
\end{example}
Since $T_x \mathcal{X}$ is non-empty and closed, a minimum norm projection exists, and therefore $\tproj{\mathcal{X}}{g}{f}(x)$ is non-empty for all $x \in \mathcal{X}$.\footnote{See, e.g., the first part of the proof of Hilbert's projection theorem~\cite[Prop.~1.37]{peypouquetConvexOptimizationNormed2015}.}
Hence, a \emph{projected dynamical system} is described by the initial value problem
\begin{equation}\label{eq:pds_ivp}
\dot x \in \tproj{\mathcal{X}}{g}{f}(x) \,, \qquad x(0) = x_0 \,,
\end{equation}
where $x_0 \in \mathcal{X}$.
If $T_x \mathcal{X}$ is convex for all $x$ then $\tproj{\mathcal{X}}{g}{f}(x)$ is a singleton for all $x \in \mathcal{X}$ (note that $ \| v - f(x) \|^2_{g(x)}$ is always strictly convex as function of $v$). In this case we will slightly abuse notation and not distinguish between the set-valued map and its induced vector field, i.e., instead of~\eqref{eq:pds_ivp} we simply write $\dot x = \tproj{\mathcal{X}}{g}{f}(x)$, $x(0) = x_0$.
An absolutely continuous function $x: [0, T) \rightarrow \mathcal{X}$ with $T>0$ and $x(0) = x_0$ that satisfies $\dot x \in \tproj{\mathcal{X}}{g}{f}(x)$ almost everywhere (i.e., for all $t \in [0, T)$ except on a subset of Lebesgue measure zero) is called a \emph{Carath\'eodory solution} to~\eqref{eq:pds_ivp}.
\begin{remark}
The class of systems~\eqref{eq:pds_ivp} can be generalized to $f$ being set-valued, i.e., $f:\mathbb{R}^n \rightrightarrows \mathbb{R}^n$. This avenue has been explored in~\cite{henryExistenceTheoremClass1973,cornetExistenceSlowSolutions1983,aubinViabilityTheory1991,aubinDifferentialInclusionsSetValued1984}, albeit only for $g$ Euclidean and $\mathcal{X}$ Clarke regular. In order not to overload our contributions with technicalities we assume that $f$ is single-valued, although an extension is possible.
\end{remark}
As the following example shows, Carath\'eodory solutions to~\eqref{eq:pds_ivp} can fail to exist unless various regularity assumptions $\mathcal X$, $f$ and $g$ hold. Hence, in the next section we propose the use of \emph{Krasovskii solutions} which exist in more general settings. Furthermore, we will show that the Krasovskii solutions reduce to Carath\'eodory solutions under the same assumptions that guarantee the existence of the latter.
\begin{example}[non-existence of Carath\'eodory solution]\label{ex:marble_run1}
Consider $\mathbb{R}^2$ with the Euclidean metric, the uniform ``vertical'' vector field $f = (0,1)$, and the self-similar closed set $\mathcal{X}$ illustrated in \cref{fig:marble_run} and defined by
\begin{equation}\label{eq:marble_run_def}
\mathcal{X} = \left\lbrace (x_1, x_2) \, \middle|\, \forall k \in \mathbb{Z}: \, x_2 = \pm 2x_1 - \frac{2}{9^k}, |x_2| \leq |x_1| \right\rbrace \cup \{0 \} \, .
\end{equation}
\begin{figure}
\caption{(a) Tangent cone and projected vector field at 0, (b) local equilibria for \cref{ex:marble_run1}
\label{fig:marble_run_proj}
\label{fig:marble_run_equi}
\label{fig:marble_run_krasovskii}
\label{fig:marble_run}
\end{figure}
The tangent cone at $0$ is given by $T_0 \mathcal{X} = \{ (v_1, v_2) \, | \, | v_2 | \leq | v_1 | \}$. It is not ``derivable'', that is, there are no differentiable curves leaving 0 in a tangent direction and remaining in $\mathcal{X}$. However, by definition there is a sequence of points in $\mathcal{X}$ approaching~$0$ in the direction of any tangent vector.
At 0 the projection of $f$ on the tangent cone is not unique as seen in \cref{fig:marble_run_proj}, namely $\tproj{}{}{f}(0) = \left \{\left(\frac{1}{2}, \frac{1}{2}\right), \left(-\frac{1}{2}, \frac{1}{2}\right)\right \}$.
Furthermore, there is no Carath\'eodory solution to $\dot x \in \tproj{}{}{f}(x)$ for $x(0) = 0$. To see this, we can argue that any solution starting at~0 can neither stay at~0 nor leave~0.
More precisely, on one hand the constant curve $x(t) = 0$ for $t \in [0, T)$ with $T > 0$ cannot be a solution since it does not satisfy $\dot x \in \tproj{}{}{f}(0)$.
On the other hand, the points $p_k = \left(\pm \frac{2}{3^{1+2k}}, \frac{2}{3^{1+2k}}\right)$ illustrated in \cref{fig:marble_run_equi} are locally asymptotically stable equilibria of the system. Namely there is an equilibrium point arbitrarily close to $0$. Thus, loosely speaking, any solution leaving $0$ would need to converge to an equilibrium arbitrarily close to~$0$.
\end{example}
\section{Existence of Krasovskii solutions}\label{sec:exist}
The pathology in \cref{ex:marble_run1} can be resolved either by placing additional assumptions on the feasible set $\mathcal{X}$ or by relaxing the notion of a solution. In this section we focus on the latter.
\begin{definition}\label{def:krasovskii_reg}
Given a set-valued map $F: \mathcal{X} \rightrightarrows \mathbb{R}^n$, its \emph{Krasovskii regularization} is defined as the set-valued map given by
\begin{align*}
\Kras{F}: \mathcal{X} \rightrightarrows \mathbb{R}^n \qquad
x \mapsto \cocl \underset{y \rightarrow x}{\lim \sup}\, F(y) \, .
\end{align*}
\end{definition}
Given a set-valued map $F: \mathcal{X} \rightrightarrows \mathbb{R}^n$, an absolutely continuous function $x:[0, T) \rightarrow \mathcal{X}$ with $T>0$ and $x(0) = x_0$ is a \emph{Krasovskii solution} of the inclusion
\begin{equation*}
\dot x \in F(x)\, , \qquad x(0) = x_0
\end{equation*}
if it satisfies $\dot x \in \Kras{F}(x)$ almost everywhere. In other words, a Carath\'eodory solution to the regularized set-valued map $\Kras{F}$ is a Krasovskii solution of the original problem.
Hence we can state the following existence result about Krasovskii solutions.
\begin{theorem}[existence of Krasovskii solutions]\label{thm:main_exist}
Let $\mathcal{X} \subset \mathbb{R}^n$ be a locally compact set, $f:\mathcal{X} \rightarrow \mathbb{R}^n$ a locally bounded vector field and $g$ a locally weakly bounded metric defined on~$\mathcal{X}$. Then, for any $x_0 \in \mathcal{X}$ there exists a Krasovskii solution $x:[0, T)\rightarrow \mathcal{X}$ for some $T>0$ to
\begin{equation}\label{eq:main_krasovskii_system}
\dot x \in \tproj{\mathcal{X}}{g}{f}(x) \qquad x(0) = x_0 \, .
\end{equation}
In addition, for $r>0$ such that $U_r := \{ x \in \mathcal{X} \, | \, \| x - x_0 \| \leq r \}$ is closed and $L = \max_{y \in U_r} \| \Kras{\tproj{\mathcal{X}}{g}{f}}(y) \|$ exists, the solution is $C^{0,1}$ and exists for $T > r/ L$.
\end{theorem}
\ifARXIV
\begin{proof}
We show that the general existence result~\cite[Cor.~1.1]{haddadMonotoneTrajectoriesDifferential1981} (\cref{prop:haddad}) is applicable to Krasovskii regularized projected vector fields. Namely, we need to verify that $\Kras{\tproj{\mathcal{X}}{g}{f}}$ is convex, compact, non-empty, upper semicontinuous (usc), and
\begin{equation}\label{eq:haddad_cond}
\Kras{\tproj{\mathcal{X}}{g}{f}}(x) \cap T_x \mathcal{X} \neq \emptyset \qquad \forall x \in \mathcal{X} \, .
\end{equation}
The fact that $\Kras{\tproj{\mathcal{X}}{g}{f}}$ is closed and convex is immediate from its definition. It is non-empty since $\tproj{\mathcal{X}}{g}{f}(x)$ is non-empty and $\tproj{\mathcal{X}}{g}{f}(x) \subset \Kras{\tproj{\mathcal{X}}{g}{f}}(x)$ for all $x \in \mathcal{X}$.
Further, we have $\tproj{\mathcal{X}}{g}{f}(x) \subset T_x \mathcal{X}$ by definition for all $x \in \mathcal{X}$ and therefore~\eqref{eq:haddad_cond} holds.
For the rest of the proof let $F(x) := {\lim \sup}_{y \rightarrow x} \tproj{\mathcal{X}}{g}{f}(y)$ (hence, $\Kras{\tproj{\mathcal{X}}{g}{f}} = \cocl F$).
Next, we show that $\Kras{\tproj{\mathcal{X}}{g}{f}}(x)$ is compact for all $x \in \mathcal{X}$.
For this, we first introduce an auxiliary metric $\hat g$ defined as $\hat g(x) := g(x) / \maxEig{g(x)}$,
that is, we scale the metric at every $x \in \mathcal{X}$ by dividing it by its maximum eigenvalue at that point.
This implies that $\| f(x) \|_{\hat{g}(x)} \leq \| f(x) \|$ for all $x \in \mathcal{X}$.
Note that the projected vector field is unchanged, i.e., $\tproj{\mathcal{X}}{\hat{g}}{f} = \tproj{\mathcal{X}}{g}{f}$, since in~\eqref{eq:def_proj_vf} only the objective function is scaled.
Furthermore, $\condN{g(x)} = \condN{\hat{g}(x)}$ for all $x \in \mathcal{X}$, and consequently $\hat{g}$ is locally weakly bounded since $g$ is locally weakly bounded.
Given any $x \in \mathcal{X}$, since $0 \in T_x \mathcal{X}$ it follows that $\|v \|_{\hat{g}(x)} \leq \| f(x) - 0 \|_{\hat{g}(x)}$ for every $v \in \tproj{\mathcal{X}}{\hat{g}}{f}(x)$. Consequently, by local boundedness of $f$ there exists $L''>0$ such that $\| \tproj{\mathcal{X}}{\hat{g}}{f}(y) \|_{\hat{g}(y)} \leq L''$ for every $y \in \mathcal{X}$ in a neighborhood of $x$.
Furthermore, by weak local boundedness of $\hat{g}$ there exists $L' > 0$ such that $\condN{\hat{g}(x)} \leq L'$ in a neighborhood of $x$. Since $\maxEig{\hat{g}(x)} = 1$, it follows that $\minEig{g(x)} \geq 1/L'$ and therefore $ \| v \| \leq L' \| v \|_{g(y)}$ for all $v \in T_y \mathbb{R}^n$ and all $y \in \mathcal{X}$ in a neighborhood of $x$.
Combining these arguments, there exist $L', L'' > 0$ such that for every $y \in \mathcal{X}$ in a neighborhood of $x$ it holds that
\begin{align}\label{eq:proj_bounded}
\tfrac{1}{L'} \| \tproj{\mathcal{X}}{\hat{g}}{f}(y) \| \leq \| \tproj{\mathcal{X}}{\hat{g}}{f}(y) \|_{\hat{g}(y)} \leq \| f(y) \|_{\hat{g}(y)} \leq \| f(y) \| \leq L'' \, .
\end{align}
Hence, since $\tproj{\mathcal{X}}{\hat{g}}{f} = \tproj{\mathcal{X}}{g}{f}$, it follows that $\tproj{\mathcal{X}}{g}{f}$ is locally bounded.
Let $U \subset \mathcal{X}$ be a compact neighborhood of $x$ such that~\eqref{eq:proj_bounded} holds. Consider the graph of $\tproj{\mathcal{X}}{g}{f}$ restricted to $U$ given by $\gph \tproj{\mathcal{X}}{g}{f}|_U := \{ (x, v) \, | \, x \in U, v \in \tproj{\mathcal{X}}{g}{f}(x) \}$. By definition of the outer limit we have $\cl \gph \tproj{\mathcal{X}}{g}{f}|_U = \gph F|_U$, i.e., $F$ is the so-called \emph{closure} of $\tproj{\mathcal{X}}{g}{f}|_U$~\cite[p. 154]{rockafellarVariationalAnalysis2009}. Thus, since $\gph \tproj{\mathcal{X}}{g}{f}|_U$ is bounded, $\gph F|_U$ is compact, and consequently $F(y)$ is locally bounded for every $y \in U$. In particular, since $F(x)$ is compact, and the closed convex hull of a bounded set is compact~\cite[Thm.~1.4.3]{hiriart-urrutyFundamentalsConvexAnalysis2012}, it follows that $\cocl F(x) = \Kras{\tproj{\mathcal{X}}{g}{f}}(x)$ is compact for all $x \in \mathcal{X}$.
Finally, we need to show that $\Kras{\tproj{\mathcal{X}}{g}{f}}$ is usc. For this, note that the map $F$ is outer semicontinuous (osc) and closed by definition. Furthermore, it is locally bounded (as shown above). Consequently, by \cref{lem:outer_sem_closedgraph}, $F$ is also usc. Hence, \cref{lem:filippov_convex} states that $\co F$ is usc as well. Since $F(x)$ is compact for all $x \in \mathcal{X}$, it follows that $\co F(x) = \cocl F(x)$~\cite[Thm.~1.4.3]{hiriart-urrutyFundamentalsConvexAnalysis2012},
and therefore $\Kras{\tproj{\mathcal{X}}{g}{f}} = \cocl F$ is usc.
Thus, $\Kras{\tproj{\mathcal{X}}{g}{f}}$ satisfies the conditions for \cref{prop:haddad} to be applicable, and therefore the existence of Krasovskii solution to~\eqref{eq:main_krasovskii_system} is guaranteed for all $x_0 \in \mathcal{X}$.
\end{proof}
\else
\cref{thm:main_exist} can be derived from standard viability results, e.g.,~\cite{aubinViabilityTheory1991,goebelHybridDynamicalSystems2012}. The primary technicality is to show that a locally weakly bounded metric results in $\Kras{\tproj{\mathcal{X}}{g}{f}}$ being locally bounded. For completeness, a self-contained proof can be found in~\cite{hauswirthProjectedDynamicalSystems2018a}.
\fi
Besides weaker requirements for existence, the choice to consider Krasovskii solutions is also motivated by their inherent ``robustness'' towards perturbations, i.e., solutions to a perturbed system still approximate the solutions of the nominal systems~\cite[Chap.~4]{goebelHybridDynamicalSystems2012}. In the same spirit, one can also establish results about the continuous dependence of solutions on initial values and problem parameters~\cite{filippovDifferentialEquationsDiscontinuous1988}.
The existence of solutions for $t\rightarrow \infty$ is guaranteed under the following conditions.
\begin{corollary}[existence of complete solutions]\label{cor:max_sol}
Consider the same setup as in \cref{thm:main_exist}. If either
\begin{enumerate}[label = (\roman*)]
\item\label{enum:glob_exist_1} $\mathcal{X}$ is closed, $f$ is bounded, and $g$ is weakly bounded, or
\item\label{enum:glob_exist_2} $\mathcal{X}$ is compact, $f$ and $g$ are continuous, or
\item\label{enum:glob_exist_3} $\mathcal{X}$ is closed, $f$ is globally Lipschitz
and $g$ is weakly bounded,
\end{enumerate}
then for every $x_0 \in \mathcal{X}$ every Krasovskii solution to \eqref{eq:main_krasovskii_system} can be extended to $T \rightarrow \infty$.
\end{corollary}
\ifARXIV
\begin{proof}
\ref{enum:glob_exist_1} If $f$ is bounded and $g$ is weakly bounded, then the local boundedness argument of the proof of \cref{thm:main_exist} can be applied globally, i.e.,~\eqref{eq:proj_bounded} holds for all $y \in \mathcal{X}$ for the same $L', L''$ and hence $\Kras{\tproj{\mathcal{X}}{g}{f}}$ is bounded. Hence, in \cref{thm:main_exist} the constant $L > 0$ exists for $r \rightarrow \infty$ and consequently $T \rightarrow \infty$.
\ref{enum:glob_exist_2} Since $f$ is continuous it only takes bounded values on a compact set. Furthermore, continuity of $g$ implies local weak boundedness, i.e., for every $x \in \mathcal{X}$ there exist $\ell_x, L_x > 0$ such that $\ell_x < \condN{g(y)} < L_x$ for all $y \in \mathcal{X}$ in a neighborhood of $x$. Since $\mathcal{X}$ is compact, there exist $\ell := \min_{x \in \mathcal{X}} \ell_x$ and $L := \max_{x \in \mathcal{X}} L_x$ and~\eqref{eq:proj_bounded} holds for all $y \in \mathcal{X}$. Hence, $g$ is weakly bounded. Then, the same arguments as for \cref{enum:glob_exist_1} apply.
\ref{enum:glob_exist_3} Assume without loss of generality that $0 \in \mathcal{X}$ (possibly after a linear translation). Global Lipschitz continuity of $f$ implies the existence of $L'' > 0$ such that $\| f(x) \| \leq L'' ( \| x \| + 1)$ for all $x \in \mathcal{X}$ (\emph{linear growth} property~\cite{aubinViabilityTheory1991}). To see this, recall that by the reverse triangle inequality and the definition of Lipschitz continuity there exists $L' > $ such that $| \| f(x) \| - \| f(0) \| | \leq \| f(x) - f(0) \| \leq L' \| x \|$ for all $x, y \in \mathcal{X}$. It follows that $\| f(x) \| \leq L' \| x \| + \|f(0)\|$ and hence $L''$ can be chosen as the maximum of $L'$ and $\|f(0)\|$ to yield the linear growth property.
Since $g$ is weakly bounded, the same arguments used for~\eqref{eq:proj_bounded} can be used to establish that there exists $L'''> 0$ such that for all $x \in \mathcal{X}$ it holds that
\begin{align*}\label{eq:proj_bounded2}
L''' \| \tproj{\mathcal{X}}{g}{f}(x) \| < \| \tproj{\mathcal{X}}{g}{f}(x) \|_{g(x)} \leq \| f(x) \|_{g(x)} \leq \| f(x) \| < L'' (\|x \| + 1) \, .
\end{align*}
It follows by the same arguments as in the proof of \cref{thm:main_exist} that $\| \Kras{\tproj{\mathcal{X}}{g}{f}}(x) \| \leq L (\|x \| + 1)$ where $L = L''/L'''$, i.e., the linear growth condition applies to $\Kras{\tproj{\mathcal{X}}{g}{f}}$.
Hence using standard bounds~\cite[p. 100]{aubinViabilityTheory1991}, one can conclude that any Krasovskii solution to~\eqref{eq:main_krasovskii_system} satisfies $\| x (t) \| \leq (\|x_0 \| + 1) e^{L t}$. Namely, define $u(t) := L( \| x(t) \| + 1)$ and note that $\dot u(t) = L \frac{d}{dt} \| x(t) \| = L \langle x(t) / \|x(t) \|, \dot x(t) \rangle \leq L \| \dot x(t) \| \leq L^2 ( \| x(t) \| + 1) = L u(t)$ holds for all $t$ where $\dot x(t)$ exists. Hence, Gronwall's inequality (for discontinuous ODEs) implies the desired bound. It immediately follows that $x(t)$ cannot have finite escape time and therefore can be extended to $t \rightarrow \infty$, completing the proof of \cref{enum:glob_exist_3}.
\end{proof}
\else
The proof of \cref{cor:max_sol} is standard and can be found in~\cite{hauswirthProjectedDynamicalSystems2018a}. For instance, \cref{enum:glob_exist_3} requires a Gronwall-argument to preclude finite escape times.
\fi
\begin{example}[existence of Krasovskii solutions]\label{ex:marble_run_krasovskii}
Consider again the setup of \cref{ex:marble_run1}. The Krasovskii regularization at $0$ of the projected vector field $\tproj{}{}{f}$ is shown in \cref{fig:marble_run_krasovskii}. It is the convex hull of five limiting vectors: the two vectors in $\tproj{}{}{f}(0)$, the projected vector field at the arbitrarily close-by equilibria $p_k$ which is $\tproj{}{}{f}(p_k) = 0$ and the projected vectors at the ascending and descending slopes.
Note that the map $x(t) = 0$ for all $t \geq 0$ is a valid solution to the differential inclusion $\dot x \in \Kras{\tproj{}{}{f}}(x)$ with initial point $0$ and hence a Krasovskii solution to the projected dynamical system, but not a Carath\'eodory solution.
\end{example}
\subsection{Additional Lemmas}
For future reference we state the following two key lemmas about projected vector fields and their Krasovskii regularizations.
\unless\ifARXIV
Proofs for both results are simple but tedious and can be found in~\cite{hauswirthProjectedDynamicalSystems2018a}. They both rely on Moreau's Decomposition Theorem~\cite[Thm.~3.2.5]{hiriart-urrutyFundamentalsConvexAnalysis2012} and generalize results in~\cite{cornetExistenceSlowSolutions1983} to the case of a variable metric and Krasovskii-regularized maps.
\fi
\begin{lemma}\label{lem:moreau_gen}
Given $\mathcal X$, $g$, and $f$ as in \cref{def:proj_vf}, for any $v \in \tproj{\mathcal{X}}{g}{f}(x)$ one has $\left\langle f(x), v \right\rangle_{g(x)} = \| v \|^2_{g(x)}$. If in addition $\mathcal{X}$ is Clarke regular at $x$, then $\tproj{\mathcal{X}}{g}{f}(x)$ is a singleton and there is $\hat{\eta} \in N_x^g \mathcal{X}$ such that the following equivalent statements hold:
\begin{enumerate} [label = (\roman*)]
\item\label{moreau1} $\tproj{\mathcal{X}}{g}{f}(x) = f(x) - \hat{\eta}$,
\item\label{moreau2} $\arg {\min}_{\eta \in N^g_x \mathcal{X}} \| \eta - f(x)\|_{g(x)} = \hat{\eta}$,
\item $f(x) - \hat{\eta} \in T_x \mathcal{X}$ and $\left\langle x - \hat{\eta}, \hat{\eta} \right\rangle_{g(x)} = 0$.
\end{enumerate}
\end{lemma}
\ifARXIV
\begin{proof}
Let $v \in \tproj{\mathcal{X}}{g}{f}(x)$. As $T_x \mathcal{X}$ is a cone we have $\lambda v \in T_x \mathcal{X}$ for all $\lambda \geq 0$. Since $v$ (locally) minimizes $\| v - f(x) \|_{g(x)}^2$ over $T_x \mathcal{X}$, it follows that $\lambda = 1$ minimizes $M(\lambda) := \tfrac{1}{2}\| \lambda v - f(x) \|_{g(x)}^2$ for $v$ fixed. Hence, for $\lambda = 1$ the optimality condition $\tfrac{dM}{d\lambda}(\lambda) = \lambda \left\langle v - f(x), v \right\rangle_{g(x)} = 0$ holds. This proves the first part.
The second part follows from Moreau's Theorem~\cite[Thm.~3.2.5]{hiriart-urrutyFundamentalsConvexAnalysis2012} since $T_x \mathcal{X}$ is convex by Clarke regularity.
\end{proof}
\else
\fi
\begin{lemma}\label{lem:kras_normal} Consider $\mathcal X \subset \mathbb{R}^n$, let $g$ be a continuous metric on $\mathcal{X}$ and $f$ a continuous vector field on $\mathcal{X}$. Then, for every $v \in \Kras{\tproj{\mathcal{X}}{g}{f}}(x)$, one has $\left\langle f(x), v \right\rangle_{g(x)} \geq \| v \|_{g(x)}^2$. If in addition $\mathcal{X}$ is Clarke regular, then for $\hat{\eta} := f(x) - v$ we have $\hat{\eta} \in N^g_x \mathcal{X}$.
\end{lemma}
\ifARXIV
\begin{proof} Let $F(x) := {\lim \sup}_{y \rightarrow x} \tproj{\mathcal{X}}{g}{f}(y)$. By definition of the outer limit, there exist sequences $x_k \rightarrow x$ with $x_k \in \mathcal{X}$ and $v_k \rightarrow v$ with $v_k \in \tproj{\mathcal{X}}{g}{f}(x_k)$ for every $v \in F(x)$ and every $x \in \mathcal{X}$.
In particular, $\left\langle f(x_k), v_k \right\rangle_{g(x_k)}= \| v_k \|_{g(x_k)}^2$ holds for every $k$ by \cref{lem:moreau_gen}. Since $f$ and $g$ are continuous the equality holds in the limit, i.e., $\left\langle f(x), v \right\rangle_{g(x)} = \| v \|^2_{g(x)}$ for every $v \in F(x)$. Taking any convex combination $v = \sum_i \alpha_i v_i$ with $v_i \in F(x)$ and $\alpha_i \geq 0$ and $\sum_i \alpha_i = 1$, we have
\begin{equation*}
\sum\nolimits_i \left\langle f(x), \alpha_i v_i \right\rangle_{g(x)} =
\sum\nolimits_i \alpha_i \| v_i \|^2_{g(x)} \geq
{\left \| \sum \nolimits_i \alpha_i v_i \right \|}_{g(x)}^2 = \left \| v \right \|^2_{g(x)} \, ,
\end{equation*}
and therefore $\left\langle f(x), v \right\rangle_{g(x)} \geq \| v \|^2_{g(x)}$ for every $v \in \cocl F(x) = \Kras{\tproj{\mathcal{X}}{g}{f}}(x)$.
According to \cref{lem:moreau_gen}, if $\mathcal{X}$ is Clarke regular, given a sequence $x_k \rightarrow x$, the sequences $v_k = \tproj{\mathcal{X}}{g}{f}(x_k)$ and $\hat{\eta}_k \in N_{x_k}^g \mathcal{X}$ for which $\hat{\eta}_k = f(x_k) - \tproj{\mathcal{X}}{g}{f}(x_k)$ are uniquely defined. Since $g$ is continuous, the mapping $x \mapsto N^g_x \mathcal{X}$ is outer semi-continuous (\cref{lem:normal_outer_semi}) and therefore $\lim_{k \rightarrow \infty} \hat{\eta}_k \in N^g_x \mathcal{X}$. In other words, for every $v \in F(x)$ it holds that $f(x) - v \in N_x^g \mathcal{X}$. Since by Clarke regularity $N_x^g\mathcal{X}$ is convex, it follows that, for any convex combination $\eta = \sum_i \alpha_i (f(x) - v_i)$ with $v_i \in F(x)$ and $\alpha_i \geq 0$ and $\sum_i \alpha_i = 1$, it must hold that $\eta \in N_x^g \mathcal{X}$, which completes the proof.
\end{proof}
\else
\fi
\section{Equivalence of Krasovskii and Carath\'eodory Solutions}\label{sec:equiv}
In this section we study the relation between Carath\'eodory and Krasovskii solutions. In particular, we show that the solutions are equivalent if the metric is continuous and the feasible domain is Clarke regular, thus recovering (for the Euclidean metric) known existence conditions for Carath\'eodory solutions. Further, we establish the connection to related work~\cite{aubinViabilityTheory1991,aubinDifferentialInclusionsSetValued1984,cornetExistenceSlowSolutions1983}.
\begin{definition} Consider a set $\mathcal{X} \subset \mathbb{R}^n$, a metric $g$ and a vector field $f$, both defined on $\mathcal{X}$. The \emph{sets of Carath\'eodory} and \emph{Krasovskii solutions} of~\eqref{eq:pds_ivp} with initial condition $x_0 \in \mathcal{X}$ are respectively given by
\begin{align*}
\mathcal{S}_C(x_0) & : = \left\lbrace x \middle| x: [0, T) \rightarrow \mathcal{X}, \, T > 0, \, x \in C^A, \, x(0) = x_0, \, \dot x(t) \in \tproj{\mathcal{X}}{g}{f}(x(t))\text{a.e.} \right\rbrace \\
\mathcal{S}_K(x_0) & : = \left\lbrace x \middle| x: [0, T) \rightarrow \mathcal{X}, \, T > 0, \, x \in C^A, \, x(0) = x_0, \, \dot x(t) \in \Kras{\tproj{\mathcal{X}}{g}{f}}(x(t)) \mbox{a.e.} \right\rbrace
\end{align*}
where \emph{a.e.} means \emph{almost everywhere} and $C^A$ denotes absolutely continuous functions.
\end{definition}
Since $\tproj{\mathcal{X}}{g}{f}(x) \subset \Kras{\tproj{\mathcal{X}}{g}{f}}(x)$, it is clear that every Carath\'eodory solution of~\eqref{eq:pds_ivp} is also a Krasovskii solution, i.e., $\mathcal{S}_C(x_0) \subset \mathcal{S}_K(x_0)$ for all $x_0 \in \mathcal{X}$. A pointwise condition for the equivalence of the solution sets is given as follows:
\begin{lemma}\label{lem:equiv}
Given any set $\mathcal{X}$, metric $g$ and vector field $f$, if $\Kras{\tproj{\mathcal{X}}{g}{f}}(x) \cap T_x \mathcal{X} = \tproj{\mathcal{X}}{g}{f}(x)$ holds for all $x \in \mathcal{X}$, then $\mathcal{S}_C(x_0) = \mathcal{S}_K(x_0)$ for all $x_0 \in \mathcal{X}$.
\end{lemma}
\begin{proof}
Since, $\mathcal{S}_C(x_0)~\subset~\mathcal{S}_K(x_0)$, we only need to consider $x \in \mathcal{S}_K(x_0)$ and show that $x \in \mathcal{S}_C(x_0)$. By \cref{lem:tgt_deriv}, $\dot x(t) \in T_{x(t)}\mathcal{X}$ holds for $x(t)$ almost everywhere. Consequently, $\dot x(t) \in \Kras{\tproj{\mathcal{X}}{g}{f}(x}(t)) \cap T_{x(t)} \mathcal{X}$ almost everywhere, and therefore, by assumption, $\dot x(t) \in \tproj{\mathcal{X}}{g}{ f(x}(t))$.
\end{proof}
The proof of the next result follows ideas from~\cite{cornetExistenceSlowSolutions1983}. The requirement that $g$ and $f$ need to be continuous deserves particular attention.
\begin{theorem}[equivalence of solution sets]\label{thm:main_equiv}
If $\mathcal{X}$ is Clarke regular, $g$ is a continuous metric on $\mathcal{X}$, and $f$ is continuous on $\mathcal{X} $, then $ \mathcal{S}_C (x_0) = \mathcal{S}_K (x_0)$ for all $x_0 \in \mathcal{X}$.
\end{theorem}
\begin{proof} It suffices to show that under the proposed assumptions \cref{lem:equiv} is applicable. By definition of $\tproj{\mathcal{X}}{g}{f}(x)$ we have $\tproj{\mathcal{X}}{g}{f}(x) \subset \Kras{\tproj{\mathcal{X}}{g}{f}}(x) \cap T_x \mathcal{X} $. For the converse, let $ v \in \Kras{\tproj{\mathcal{X}}{g}{f}}(x) \cap T_x \mathcal{X}$.
By \cref{lem:kras_normal}, $v = f(x) - \hat{\eta}$ for some $\hat{\eta} \in N^g_x \mathcal{X}$ and $\| v \|^2_{g(x)} \leq \left\langle v, f(x) \right\rangle_{g(x)}$.
Since $\left\langle v, \eta \right\rangle_{g(x)} \leq 0$ for all $\eta \in N^g_x \mathcal{X}$ we have
\begin{equation*}
\| v \|_{g(x)}^2 \leq \left\langle v, f(x) \right\rangle_{g(x)} - \left\langle v, \eta \right\rangle_{g(x)} \leq \| v \|_{g(x)} \| f(x) - \eta \|_{g(x)} \qquad \forall \eta \in N^g_x \mathcal{X} \, ,
\end{equation*}
where the second inequality is due to Cauchy-Schwarz, and therefore $\| v - \hat{\eta}\|_{g(x)} \leq \| f(x) - \eta \|_{g(x)}$ holds for all $\eta \in N^g_x \mathcal{X}$.
However, according to \cref{lem:moreau_gen} the fact that $\hat{\eta} = \arg \underset{\eta \in N^g_x \mathcal{X}}{\min} \| f(x) - \eta \|_{g(x)}$ is equivalent to $v \in \tproj{\mathcal{X}}{g}{f }(x)$.
\end{proof}
Note that \cref{ex:marble_run1,ex:marble_run_krasovskii} show a case where the conclusion of \cref{thm:main_equiv} fails to hold because $\mathcal{X}$ is not Clarke regular at the origin. Hence, our sufficient characterization in terms of Clarke regularity is also a sharp one.
\cref{thm:main_equiv} also serves as an existence result of Carath\'eodory solutions, that recovers the conditions derived in~\cite{cornetExistenceSlowSolutions1983}, but for a general metric.
\begin{corollary}[Existence of Carath\'eodory solutions]\label{cor:cara_exist}
If $\mathcal{X}$ is Clarke regular, and $g$ and $f$ are continuous on $\mathcal{X}$, then there exists a Carath\'eodory solution $x:[0, T) \rightarrow \mathcal{X}$ of~\eqref{eq:pds_ivp} with $x(0) = x_0$ for some $T> 0$, and every $x_0 \in \mathcal{X}$.
\end{corollary}
Uniqueness, however, requires additional assumptions as will be shown in \cref{sec:uniq}. In particular, uniqueness of the projection $\tproj{\mathcal{X}}{g}{f}(x)$ does not imply uniqueness of the trajectory (see forthcoming \cref{rem:unique_proj}).
\subsection{Related work and alternative formulations}
With the statements of \cref{sec:equiv} at hand, we discuss their connection to related literature.
As discussed in the introduction, projected dynamical system have been studied from different perspectives and with various applications in mind. In particular, a number of alternative, but equivalent formulations do exist~\cite{brogliatoEquivalenceComplementaritySystems2006,heemelsProjectedDynamicalSystems2000}, but none considers the case of a variable metric. In the following, we discuss a well-established formulation~\cite{aubinDifferentialInclusionsSetValued1984,aubinViabilityTheory1991,cornetExistenceSlowSolutions1983} that has a number of insightful properties.
Namely, under Clarke regularity of the feasible set $\mathcal{X}$ we may define an alternative differential inclusion given by the initial value problem
\begin{equation}\label{eq:norm_incl}
\dot x \in f(x) - N^g_x \mathcal{X} \,, \qquad x(0) = x_0 \in \mathcal{X}
\end{equation}
and define the solution set as
\begin{equation*}
\mathcal{S}_N(x_0) : = \left\lbrace x \,\middle|\, x: [0, T) \rightarrow \mathcal{X}, \, T > 0, \, x \in C^A, \, x(0) = x_0, \, \dot x \in f(x) - N^g_x \mathcal{X} \text{ a.e.}\right\rbrace \, .
\end{equation*}
The next result is an adaptation of~\cite[Thm.~2.3]{cornetExistenceSlowSolutions1983} to arbitrary metrics. We provide a self-contained proof for completeness.
\begin{corollary}\label{cor:equiv_normal}
Consider a Clarke regular set $\mathcal{X} \subset \mathbb{R}^n$, a continuous vector field $f$, and a continuous metric $g$, both defined on $\mathcal{X}$. Then, $\mathcal{S}_N(x_0) = \mathcal{S}_C(x_0)$ holds for systems of the form~\eqref{eq:pds_ivp} and \eqref{eq:norm_incl}, and for all $ x_0 \in \mathcal{X}$.
\end{corollary}
In short, any solution to~\eqref{eq:norm_incl} is a Carath\'eodory solution of~\eqref{eq:pds_ivp} and vice versa.
However, \cref{cor:equiv_normal} makes no statement about existence of solutions. In fact, the non-compactness of $N_x^g \mathcal{X}$ prevents us from applying the same viability result as for \cref{thm:main_exist}.
\begin{proof}
We first note that $\mathcal{S}_C(x_0) \subset \mathcal{S}_N(x_0)$ since $\tproj{\mathcal{X}}{g}{f}(x) \subset f(x) - N^g_x \mathcal{X}$ for all $x \in \mathcal{X}$ by virtue of \cref{lem:kras_normal} and since $\mathcal{X}$ is Clarke regular.
Conversely, let $x \in \mathcal{S}_N(x_0)$ be defined for $t \in [0, T)$ for $T > 0$. Then for almost all $t$, we have $\dot x(t) \in f(x(t)) - N^g_{x(t)}\mathcal{X}$ and $\dot x(t) \in T_{x(t)} \mathcal{X} \cap -T_{x(t)} \mathcal{X}$ by \cref{lem:tgt_deriv}. Thus, for $\dot x(t) = f(x(t)) - \eta(x(t))$ with $\eta(x(t)) \in N^g_{x(t)}\mathcal{X}$ it must hold that
\begin{equation*}
\left\langle f(x(t)) - \eta(x(t)), \eta(x(t)) \right\rangle_{g(x(t))} \leq 0 \quad \mbox{and} \quad \left\langle f(x(t)) - \eta(x(t)), - \eta(x(t)) \right\rangle_{g(x(t))} \leq 0 \, .
\end{equation*}
Consequently, $\left\langle f(x(t)) - \eta(x(t)), \eta(x(t)) \right\rangle_{g(x(t))} = 0$, and using \cref{lem:moreau_gen} it follows that $\dot x(t) = \tproj{\mathcal{X}}{g}{ f(x}(t))$.
\end{proof}
\begin{remark}
Defining inclusions of the form~\eqref{eq:norm_incl} for a set $\mathcal{X}$ that is not Clarke regular is possible but technical since one would need to distinguish between different types of normal cones (\cref{rem:normal_cones}). Furthermore, depending on the choice of normal cone the resulting set of solutions can be overly relaxed or too restrictive.
\end{remark}
\begin{remark}
Using \cref{moreau2} in \cref{lem:moreau_gen} it follows that whenever $\dot x$ exists, we have $\dot x = \arg \min_{v \in f(x) - N_x^g \mathcal{X}} \| v \|_{g(x)}$. When $g$ is the Euclidean metric, this \emph{minimum norm} property gives rise to so-called \emph{slow} solutions of~\eqref{eq:norm_incl}~\cite[Chap.~10.1]{aubinViabilityTheory1991}. For a general metric, the definition of a slow solution generalizes accordingly. However, the property of being ``slow'' depends on the metric.
\end{remark}
\section{Prox-regularity and Uniqueness of Solutions}\label{sec:uniq}
Next, we introduce a generalized definition of \emph{prox-regular} sets on non-Euclidean spaces with a variable metric and show their significance for the uniqueness for solutions of projected dynamical systems. In the Euclidean setting prox-regularity is well-known to be a sufficient condition on the feasible domain $\mathcal{X}$ for uniqueness~\cite{cornetExistenceSlowSolutions1983}.
The key issue of this section is thus to generalize the definition of prox-regular sets that can be used on low-regularity Riemannian manifolds. Previously, prox-regularity has been defined and studied on \emph{smooth} (i.e., $C^\infty$) Riemannian manifolds in~\cite{hosseiniMetricProjectionProxregular2013,bernicotSweepingProcessProxregular2015} using standard \emph{geodesic} notions from Riemannian geometry. In this paper, we weaken the smoothness assumption but, consequently, we cannot apply to the same toolset that requires the existence of unique geodesics (which is only guaranteed on sufficiently smooth manifolds~\cite{hartmanLocalUniquenessGeodesics1950}). Instead we pursue a more low-level approach which the novel insights prox-regularity of a set is independent of the choice of metric (and, more precisely, preserved under $C^{1,1}$ coordinate transformations). This feature is particularly important for envisioned applications in optimization where the feasible domain is given, but choice of metric is often a design parameter of an algorithm.
\subsection{Prox-regularity on non-Euclidean spaces}
For illustration, we first recall and discuss the definition of prox-regularity in Euclidean space. Our treatment of the topic is deliberately kept limited. For a more general overview see~\cite{adlyPreservationProxRegularitySets2016,poliquinProxregularFunctionsVariational1996}.
\begin{definition}\label{def:prox_reg_trad} A Clarke regular set $\mathcal{X} \subset \mathbb{R}^n$ is \emph{prox-regular at $x \in \mathcal{X}$} if there is $L > 0$ such that for every $z,y \in \mathcal{X}$ in a neighborhood of $x$ and $\eta \in N_y \mathcal{X}$ we have
\begin{align}\label{eq:prox_reg_trad}
\left\langle \eta, z - y \right\rangle \leq L \| \eta \| \| z - y \|^2 \, .
\end{align}
The set $\mathcal{X}$ is \emph{prox-regular} if it is prox-regular at every $x \in \mathcal{X}$.
\end{definition}
One of the key features of a prox-regular set $\mathcal{X}$ is that for every point in a neighborhood of $\mathcal{X}$ there exists a unique projection on the set~\cite[Def.~2.1, Thm.~2.2]{adlyPreservationProxRegularitySets2016}.
\begin{example}[Prox-regularity in Euclidean spaces]\label{ex:prox_reg_basic}
Consider the parametric set
\begin{equation}
\mathcal{X}_\alpha := \left\lbrace (x_1, x_2) \, \middle | \, |x_2| \geq {\max \{0, x_1 \}}^\alpha\right\rbrace\label{eq:prox_reg_example}
\end{equation}
where $0 < \alpha < 1$ and which is illustrated in \cref{fig:prox_reg}. For $\alpha \leq 0.5$ the set is prox-regular everywhere. In particular for the origin, a ball with non-zero radius can be placed tangentially such that it only intersects the set at 0. For $\alpha > 0.5$ on the other hand the set is not prox-regular at the origin. In fact, all points on the positive axis have a non-unique projection on $\mathcal{X}_\alpha$ as illustrated in \cref{fig:non-uniq_proj}.
\begin{figure}
\caption{Set $\mathcal{X}
\label{fig:non-uniq_proj}
\label{fig:prox_reg}
\end{figure}
\end{example}
\cref{def:prox_reg_trad} cannot be directly generalized to non-Euclidean spaces since it requires the distance $\| y - x \|$ between two points in $\mathcal{X}$. Hence, in~\cite{hosseiniMetricProjectionProxregular2013,bernicotSweepingProcessProxregular2015} prox-regularity is defined on smooth (i.e., $C^{\infty}$) Riemannian manifolds resorting to geodesic distances.
For our purposes we can avoid the notational complexity of Riemannian geometry, yet preserve a higher degree of generality. Thus, we introduce the following definitions.
\begin{definition}\label{def:prox_normal}
Given a Clarke regular set $\mathcal{X} \subset \mathbb{R}^n$ and a metric $g$, a normal vector $\eta \in N^g_x \mathcal{X}$ at $ x \in \mathcal{X}$ is \emph{$L$-proximal with respect to $g$} for $L \geq 0$ if for all $y \in \mathcal{X}$ in a neighborhood of $x$ we have
\begin{align}\label{eq:prox_reg_new}
\left\langle \eta, y - x \right\rangle_{g(x)} \leq L \| \eta \|_{g(x)} \| y - x \|^2_{g(x)} \, .
\end{align}
The cone of all $L$-proximal normal vectors at $x$ with respect to $g$ is denoted by $\bar{N}^{g,L}_x \mathcal{X}$.
\end{definition}
A crucial detail in~\eqref{eq:prox_reg_new} is the fact that $g$ is evaluated at $x$ and is used as an inner product on $\mathbb{R}^n$ (which is a slight abuse of notation). In other words, we exploit the canonical isomorphism between $\mathbb{R}^n$ and $T_x \mathbb{R}^n$ to use $g(x)$ as an inner product on $\mathbb{R}^n$.
\begin{definition}\label{def:prox_reg_new}
A Clarke regular set $\mathcal{X} \subset \mathbb{R}^n$ with a metric $g$ is \emph{$L$-prox-regular at $x \in \mathcal{X}$ with respect to $g$} if $\bar{N}_y^{g,L} \mathcal{X} = N^g_y \mathcal{X}$ for all $y \in \mathcal{X}$ in a neighborhood of $x$.
The set $\mathcal{X}$ is \emph{prox-regular with respect to $g$} if for every $x \in \mathcal{X}$ there exists $L > 0$ such that $\mathcal{X}$ is $L$-prox-regular at $x$ with respect to $g$.
\end{definition}
\begin{remark}
Note that if $g$ is the Euclidean metric, \cref{def:prox_reg_new} reduces to \cref{def:prox_reg_trad}.
Moreover, when applied to a smooth Riemannian manifold, \cref{def:prox_reg_new} reduces to the definition of prox-regularity given in \cite{hosseiniMetricProjectionProxregular2013,bernicotSweepingProcessProxregular2015}.
To see this, consider a closed subset $\mathcal{X}$ of a (geodesically complete) smooth Riemannian manifold $\mathcal{M}$ with metric $g$. In \cite{hosseiniMetricProjectionProxregular2013,bernicotSweepingProcessProxregular2015}, the $L$-proximal normal cone of $\mathcal{X}$ at $x \in \mathcal{X}$ is defined as the set of all $\eta \in T_x \mathcal{M}$ such that
\begin{align*}
\left\langle \eta, \exp^{-1}_x(y) \right\rangle_{g(x)} \leq L \| \eta \| \left \| \exp^{-1}_x(y) \right\|^2_{g(x)}
\end{align*}
holds for all $y \in \mathcal{M}$ in a neighborhood of $x$ and $\exp^{-1}_x(y)$ is the inverse of the exponential map. Namely, $\exp^{-1}_{x}(y)$ maps $y$ to a tangent vector $w \in T_x \mathcal{M}$ at $x$ such that the geodesic segment between $x$ and $y$ starting from $x$ in the direction $w$ has length $\| w \|_{g(x)}$. With this local bijection between $T_x \mathcal{M}$ and $\mathcal{M}$, prox-regularity of $\mathcal{X}$ can be defined similarly to \cref{def:prox_reg_new}, albeit smoothness and geodesic completeness of $\mathcal{M}$ (as well as other technical assumptions, e.g., \cite[Ass. 2.9]{bernicotSweepingProcessProxregular2015}) are a prerequisite.
\end{remark}
The following result shows that prox-regularity is in fact independent of the metric. This is the first step towards a coordinate-free definition of prox-regularity.
\begin{proposition}\label{prop:prox_invariance}
Let $\mathcal{X} \subset \mathbb{R}^n$ be Clarke regular. If $\mathcal{X}$ is prox-regular with respect to a $C^{0}$ metric $g$, then it is prox-regular with respect to any other $C^0$ metric.
\end{proposition}
In particular if $\mathcal{X}$ is prox-regular with respect to the Euclidean metric, i.e., according to \cref{def:prox_reg_trad}, then it is prox-regular in any other continuous metric on $\mathbb{R}^n$. For the proof of \cref{prop:prox_invariance} we require the following lemma.
\begin{lemma}\label{lem:prox_invar_local}
Let $\mathcal{X} \subset \mathbb{R}^n$ be Clarke regular and consider to metrics $g, g'$ defined on $\mathcal{X}$. If for $x \in \mathcal{X}$ there is $L > 0$ such that $\bar{N}_x^{g,L} \mathcal{X} = N^g_x \mathcal{X}$ then $\bar{N}_x^{g',L'} \mathcal{X} = N^{g'}_x \mathcal{X}$ holds for $L' \geq \condN{g(x)} \condN{g'(x)} L$.
\end{lemma}
\begin{proof}
First note that for every $x \in \mathcal{X}$ the two metrics $g$ and $g'$ induce a bijection between $N^g_x \mathcal{X}$ and $N^{g'}_x \mathcal{X}$. Namely, we define $q: T_x \mathbb{R}^n \rightarrow T_x\mathbb{R}^n$ as the unique element $q(v)$ that satisfies by $\left\langle v, w \right\rangle_{g(x)} = \left\langle q(v), w \right\rangle_{g'(x)}$ for all $w \in T_x\mathbb{R}^n$.
To clarify, in matrix notation we can write $ v^T G(x) w = {q(v)}^T G'(x) w$ and since $G(x), G'(x)$ are symmetric positive definite we have $q(v) := {G'(x)}^{-1} G(x) v$.
It follows that if $\eta \in N_x^g \mathcal{X}$ (hence, by definition $\left\langle \eta, w \right\rangle_{g(x)} \leq 0$ for all $w \in T_x \mathbb{R}^n$), then $q(\eta) \in N_x^{g'}\mathcal{X}$.
Furthermore, omitting the argument $x$, we have $\| q(\eta)\|_{g'} = \eta^T G {G'}^{-1} G \eta \geq 1/\maxEig{g'} \| G \eta \| $ and $\| \eta\|_{g} = \eta^T G {G}^{-1} G \eta \leq 1/\minEig{g} \| G \eta \|$, and therefore $\| q(\eta)\|_{g'(x)} \geq \minEig{g(x)} /\maxEig{g'(x)} \| \eta \|_{g(x)}$.
Hence, let $\eta \in N^g_x \mathcal{X} \setminus \{ 0 \}$ be a $L$-proximal normal vector, then
\begin{equation*}
\left\langle \tfrac{q(\eta)}{\| q(\eta) \|_{g'(x)}}, y - x \right\rangle_{g'(x)}
\leq \tfrac{\maxEig{g'(x)}}{\minEig{g(x)}} \left\langle \tfrac{\eta}{\| \eta \|_{g(x)}}, y - x \right\rangle_{g(x)}
\leq \tfrac{\maxEig{g'(x)}}{\minEig{g(x)}}L \| y - x \|^2_{g(x)} \, .
\end{equation*}
Finally, using the equivalence of norms, we have
\begin{equation}\label{eq:prox_reg_local}
\tfrac{\maxEig{g'(x)}}{\minEig{g(x)}}L \| y - x \|^2_{g(x)}
\leq \tfrac{\maxEig{g'(x)}}{\minEig{g(x)}} \tfrac{\maxEig{g(x)}}{\minEig{g'(x)}} L \| y - x \|^2_{g'(x)}
\leq L' \| y - x\|^2_{g'(x)} \, ,
\end{equation}
where $L' \geq \condN{g(x)} \condN{g'(x)} L$. Thus, we have shown that if $v \in \bar{N}_x^{g,L} \mathcal{X} = N^{g}_x \mathcal{X}$ then $q(v) \in \bar{N}_x^{g',L'} = N^{g'}_x \mathcal{X}$ which completes the proof.
\end{proof}
\begin{proof}[Proof of \cref{prop:prox_invariance}]
Since $g$ and $g'$ are continuous it follows that $\condN{g(x)}$ and $\condN{g'(x)}$ are continuous in $x$ and therefore locally bounded. Given any $x \in \mathcal{X}$ and using the pointwise result in \cref{lem:prox_invar_local}, we can choose $L'> 0$ such that~\eqref{eq:prox_reg_local} is satisfied for all $y \in \mathcal{X}$ in a neighborhood of $x$.
\end{proof}
We conclude this section by showing that feasible domains defined by $C^{1,1}$ constraint functions are prox-regular under the usual constraint qualifications.
\begin{example}[prox-regularity of constraint-defined sets]\label{ex:clarke_reg_prox_cone}
As in \cref{ex:clarke_reg_constraint_set,ex:clarke_reg_normal_cone} let $h: \mathbb{R}^n \rightarrow \mathbb{R}^m$ be $C^1$ and $\nabla h(x)$ have full rank for all $x$ and consider $\mathcal{X} := \{ x \, | \, h(x) \leq 0 \}$. If in addition, $h$ is a $C^{1,1}$ map, then $\mathcal{X} := \{ x \, | \, h(x) \leq 0 \}$ is prox-regular with respect to any $C^0$ metric $g$ on $\mathbb{R}^n$.
To see this, we consider the Euclidean case without loss of generality as a consequence of \cref{prop:prox_invariance}. We first analyze the sets $\mathcal{X}_i := \{ x \, | \, h_i(x) \leq 0 \}$ and then show prox-regularity of their intersection. For this, we only need to consider points $x \in \partial \mathcal{X}_i$ on the boundary of $\mathcal{X}_i$ since for all $\bar{x} \notin \partial \mathcal{X}_i$ we have $N_{\bar{x}} \mathcal{X}_i = \{ 0 \}$ and prox-regularity is trivially satisfied. Hence, using the Descent \cref{lem:c11_lipschitz}, for all $z,y \in \mathbb{R}^n$ in a neighborhood of $x$ and all $i=1, \ldots, m$ there exists $L_i > 0$ such that
\begin{equation*}
- L_i \| z - y\|^2 \leq h_i(z) - h_i(y) - \left \langle \nabla h_i^T(z), z - y \right \rangle \, .
\end{equation*}
In particular, for $z \in \mathcal{X}_i$ (i.e., $h_i(z) \leq 0$) and $y \in \partial \mathcal{X}_i$ (i.e., $h_i(y) = 0$) in a neighborhood of $x$ we have
\begin{equation}\label{eq:pre_prox}
\left\langle \nabla h_i^T(y), z-y \right\rangle \leq h_i(z)+ L_i \| z - y\|^2 \leq L_i \| z - y \|^2 \, .
\end{equation}
For the set $\mathcal{X} = \bigcap_{i=1}^m \mathcal{X}_i$ recall from \cref{ex:clarke_reg_normal_cone} that for $x \in \mathcal{X}$ we have
\begin{align*}
N_x \mathcal{X} = \left\lbrace \eta \, \middle| \, \eta = \sum\nolimits_{i \in I(x)} \alpha_i {\nabla h^T_i(x)}, \, \alpha_i \geq 0 \right\rbrace \, .
\end{align*}
Consider $z \in \mathcal{X}$ and $y \in \partial \mathcal{X}$ in a small enough neighborhood of $x$. Note that $y \in \partial \mathcal{X}$ implies that $y \in \partial \mathcal{X}_i$ for all $i \in I(y)$. Using~\eqref{eq:pre_prox}, for all $\eta \in N_y\mathcal{X}$ with $\eta = \sum_{i \in I(y)} \alpha_i {\nabla h^T_i(y)}/ \| \nabla h_i(y) \|$ we have
\begin{equation*}
\left\langle \eta, z- y \right\rangle
= \left\langle
\sum\nolimits_{i \in I(y)} \alpha_i {\nabla h_i(y)}^T, z - y \right\rangle
\leq
\left(
\sum\nolimits_{i \in I(y)} \alpha_i L_i \, ,
\right) \| z - y\|^2 \,
\end{equation*}
and therefore $
\langle \eta, z - y \rangle
\leq L(y) \| \eta \| \| z - y\|^2$, where
\begin{align*}
L(y) := \tfrac{\sum\nolimits_{i \in I(y)} \alpha_i L_i}{\| \eta \|} = \tfrac{\sum\nolimits_{i \in I(y)} \alpha_i L_i}{\left \| \sum\nolimits_{i \in I(y)} \alpha_i \nabla h_i(y) \right\|}
\leq \max_{i \in I(y)} \tfrac{\alpha_i \nabla L_i}{\alpha_i \| \nabla h_i(y) \|} \leq \max_{i=1, \ldots m} \tfrac{L_i}{\| \nabla h_i(y) \|} \, .
\end{align*}
The first inequality can be shown by taking the square and proceeding by induction. Since the final bound is with respect to all $h_i$, it is continuous in $y$ in a neighbhorhood of $x$. Consequently, we can choose $\bar{L}$ such that $\bar{L} \geq L(y)$ for all $y \in \mathcal{X}$ in a neighborhood of $x$, and therefore $\langle \eta, z - y \rangle
\leq \bar{L} \| \eta \| \| z - y\|^2 $ for $z \in \mathcal{X}$ in a neighborhood of $y$. This proves $\bar{L}$-prox-regularity at $x$ and prox-regularity follows accordingly.
\end{example}
\subsection{Uniqueness of solutions to projected dynamical systems}
Before formulating our main uniqueness result, we present an example that illustrates the impact of prox-regularity on the uniqueness of solutions.
\begin{example}[prox-regularity and uniqueness of solutions]\label{ex:prox_uniq}
We consider the set $\mathcal{X}_\alpha := \left\lbrace (x_1, x_2) \, \middle | \, |x_2| \geq {\max \{0, x_1\}}^\alpha\right\rbrace$ for $0 < \alpha < 1$, as in \cref{ex:prox_reg_basic}.
We study how the value of $\alpha$ affects the uniqueness of solutions of the projected dynamical system defined by the uniform ``horizontal'' vector field $f(x) = (1, 0)$ for all $x \in \mathcal{X}$ and the initial condition $x(0) = 0$ as illustrated in \cref{fig:prox_uniq}.
\begin{figure}
\caption{Projected vector field on $\mathcal{X}
\label{fig:prox_uniq}
\end{figure}
Since $\mathcal{X}_\alpha$ is Clarke regular and closed, since the vector field is uniform, and since we use the Euclidean metric, the existence of Krasovskii solutions and the equivalence of Carath\'eodory solutions is guaranteed for $t\rightarrow \infty$ by \cref{cor:max_sol} and \cref{thm:main_equiv}, respectively.
The prox-regularity of $\mathcal{X}_\alpha$ at the origin is however only guaranteed for $0 < \alpha \leq \frac{1}{2}$ (\cref{ex:prox_reg_basic}).
A formal analysis reveals that for $0 < \alpha \leq \frac{1}{2}$ the origin is a strong equilibrium, i.e., the constant solution $x(t) = 0$ is the unique solution to the projected dynamical system. For $\frac{1}{2} < \alpha < 1$, however, the origin is only a weak equilibrium point. Namely, a solution may remain at $0$ for an arbitrary amount of time before leaving $0$ on either upper or lower halfplane, and thus uniqueness is not guaranteed.
\end{example}
\begin{remark}\label{rem:unique_proj}
Whether $\tproj{}{}{f}(x_0)$ is a singleton or not is generally unrelated to the uniqueness of solutions starting from $x_0$. For instance, in \cref{ex:prox_uniq}, if $\alpha > 0$ multiple solutions exists even though $\tproj{}{}{f}(x)$ is a singleton at $x = 0$. Conversely, \cref{ex:marble_run_krasovskii} shows that even if $\tproj{}{}{f}(x_0)$ is not unique, the (Krasovskii) solution starting from~$x_0$ is unique.
\end{remark}
\ifARXIV
For the proof of uniqueness under prox-regularity, we require the following lemma.
\else
For the proof of uniqueness under prox-regularity, we require the following technical lemma whose proof is purely technical and can be found in~\cite{hauswirthProjectedDynamicalSystems2018a}.
\fi
\begin{lemma}\label{lem:hypomonotone} Let $\mathcal{X}$ be $L$-prox-regular at $x$ with respect to a $C^{0,1}$ metric $g$. Then, there exist $\bar{L} > 0$ such that for all $y \in \mathcal{X}$ in a neighborhood of $x$ and all $\eta \in N_y^{g,L}$ with $\| \eta \|_{g(y)} = 1$ we have $\left\langle \eta, x - y \right\rangle_{g(x)} \leq \bar{L} \| y - x \|^2_{g(x)}$.
\end{lemma}
\ifARXIV
\begin{proof} We know that $ \left\langle \eta, y - x \right\rangle_{g(y)} \leq L \| y - x \|_{g(y)}^2 $ for $y$ close enough to $x$ because $\eta$ is a $L$-proximal normal vector at $y$ with respect to $g$. Furthermore, by the equivalence of norms there exists $L' > 0$ sucht that $ \left\langle \eta, y - x \right\rangle_{g(y)} \leq L' \| y - x \|_{g(x)}^2$.
Next, we show that $| \left\langle \eta, x - y \right\rangle_{g(y)} - \left\langle \eta, x -y \right\rangle_{g(x)} | \leq M \| y - x \|^2_{g(x)}$ for some $M > 0$. Since $L^n_2$ is a vector space, we may write
\begin{equation*}
\left\langle \eta, x-y \right\rangle_{g(y)} - \left\langle \eta, x-y \right\rangle_{g(x)} = \left\langle \eta, x-y \right\rangle_{g(y) - g(x)}
\end{equation*}
which is a slight abuse of notation since $\left\langle \cdot, \cdot \right\rangle_{g(y) - g(x)}$ is not necessarily positive definite and therefore not a metric. Nevertheless, any map of the form $(u,v,g) \mapsto \left\langle u, w \right\rangle_{g}$ where $g \in L^n_2$ is linear in $u,v$ and in $g$ (e.g., $(u,v,g) \mapsto \left\langle u, w \right\rangle_{\lambda g} = \lambda \left\langle u, w \right\rangle_{g}$ for any $\lambda \in \mathbb{R}$). Therefore, there exist $M', M > 0$ such that
\begin{equation*}
\left| \left\langle \eta, x-y \right\rangle_{g(y) - g(x)} \right | \leq M' \| g(y) - g(x) \|_{L^n_2} \| x - y \|_{g(x)} \leq M \| x - y \|^2_{g(x)} \, ,
\end{equation*}
where $\| \cdot \|_{L^n_2}$ denotes any norm on the vector space $L^n_2$, and the second inequality follows directly from the Lipschitz continuity of $g$. Hence, we can conclude that that
\begin{align*}
\left\langle \eta, x -y \right\rangle_{g(x)}
\leq \left\langle \eta, x -y \right\rangle_{g(y)}
+ | \left\langle \eta, x - y \right\rangle_{g(y) - g(x)} |
\leq (L' + M) \| y - x \|^2_{g(x)} \, .
\end{align*}
\end{proof}
\fi
Next, we can show the following Lipschitz-type property of projected vector fields.
\begin{proposition}\label{prop:lipschitz_flat} Let $f$ be a $C^{0,1}$ field on $\mathcal{X}$. If $g$ is a $C^{0,1}$ metric and $\mathcal{X}$ is prox-regular, then for every $x \in \mathcal{X}$ there exists $L > 0$ such that for all $y \in \mathcal{X}$ in a neighborhood of $x$ we have
\begin{equation*}
\left\langle \tproj{\mathcal{X}}{g}{f}(y) - \tproj{\mathcal{X}}{g}{f}(x), y - x \right\rangle_{g(x)} \leq L \| y - x \|^2_{g(x)} \, .
\end{equation*}
\end{proposition}
\begin{proof} As a consequence of \cref{lem:moreau_gen}, we can write
\begin{multline}\label{eq:onesided_lip}
\left\langle \tproj{\mathcal{X}}{g}{f}(y) - \tproj{\mathcal{X}}{g}{f}(x), y - x \right\rangle_{g(x)} \\
= \left\langle f(y) - f(x) , y - x \right\rangle_{g(x)}
+ \left\langle \eta_y, x - y \right\rangle_{g(x)}
+ \left\langle \eta_x, y - x \right\rangle_{g(x)} \, .
\end{multline}
where $\eta_y \in N^g_y \mathcal{X} = \bar{N}_y^{g, L} \mathcal{X}$ and $\eta_x \in N^g_x \mathcal{X} = \bar{N}_x^{g, L}$ for some $L> 0$.
For the first term, we get
$\left\langle f(y) - f(x) , y - x \right\rangle_{g(x)} \leq \| f(y) - f(x) \|_{g(x)} \| y - x\|_{g(x)}$.
by applying Cauchy-Schwarz.
Since $f$ is Lipschitz and using the equivalence of norms there exists $L_a > 0$ such that
$\| f(y) - f(x) \|_{g(x)} \leq L_a \| y - x\|_{g(x)}$
for all $y \in \mathcal{X}$ in a neighborhood of $x$. Thus, we have
$\left\langle f(y) - f(x) , y - x \right\rangle_{g(x)} \leq L_a \| y - x\|^2_{g(x)}$.
For the second and third term in~\eqref{eq:onesided_lip} we have
\begin{align*}
\left\langle \eta_y, x - y \right\rangle_{g(x)} & \leq L' \| y - x \|_{g(x)}^2 \| \eta_y \|_{g(y)} \\
\left\langle \eta_x, y - x \right\rangle_{g(x)} & \leq L \| y - x \|_{g(x)}^2 \| \eta_x \|_{g(x)}
\end{align*}
by \cref{lem:hypomonotone} and the definition of a $L$-proximal normal vector, respectively.
By \cref{lem:moreau_gen} we know that $\| \eta_y \|_{g(y)} \leq \| f(y) \|_{g(y)}$ and $\| \eta_x \|_{g(x)} \leq \| f(x) \|_{g(x)}$. Since $g$ and $f$ are continuous we can choose $M > 0$ such that $\| f(z) \|_{g(z)} \leq M$ for all $z \in \mathcal{X}$ in a neighborhood of $x$. Therefore,~\eqref{eq:onesided_lip} can be bounded by
\begin{align*}
\left\langle \tproj{\mathcal{X}}{g}{f}(y) - \tproj{\mathcal{X}}{g}{f}(x), y - x \right\rangle_{g(x)}
\leq (L_a + L' M + L M)\| y - x \|^2_{g(x)}
\end{align*}
which completes the proof.
\end{proof}
Hence, we can state our main result on the uniqueness of solutions which complements results in~\cite{cornetExistenceSlowSolutions1983} by considering a variable (but non-differentiable) metric and using our general definition of prox-regularity. In this context, uniqueness is understood in the sense that any two solutions are equal on the interval on which they are both defined.
\begin{theorem}[uniqueness of solutions]\label{thm:main_uniq} Let $f$ be a $C^{0,1}$ vector field on $\mathcal{X}$. If $g$ is a $C^{0,1}$ metric and $\mathcal{X}$ is prox-regular, then for every $x_0 \in \mathcal{X}$ there exists $T> 0$ such that the initial value problem
$\dot x \in \tproj{\mathcal{X}}{g}{f}(x)$ with $x(0) = x_0$
has a unique Carath\'eodory solution $x:[0, T) \rightarrow \mathcal{X}$ (which is also the unique Krasovskii solution).
\end{theorem}
\begin{proof}[Proof of \cref{thm:main_uniq}] The proof follows standard contraction ideas~\cite{filippovDifferentialEquationsDiscontinuous1988}. Let $x(t)$ and $y(t)$ be two solutions solving the same initial value problem $\dot x \in \tproj{\mathcal{X}}{g}{f}(x)$ with $x(0) = x_0 \in \mathcal{X}$, both defined on a non-empty interval $[0, T)$.
Using \cref{prop:lipschitz_flat}, there exists $M > 0$ and a neighborhood $V$ of $x_0$ such that
\begin{equation}\label{eq:lipschitz_uniq}
\begin{split}
\tfrac{d}{dt} \left( \tfrac{1}{2}\| y(t) - x(t) \|^2_{g(x_0)} \right) & = \left\langle \tproj{\mathcal{X}}{g}{ f(y}(t)) - \tproj{\mathcal{X}}{g}{ f(x}(t)), y(t) - x(t) \right\rangle_{g(x_0)} \\ & \leq M || y(t) - x(t) ||^2_{g(x_0)}
\end{split}
\end{equation}
for all $t$ in some non-empty subinterval $[0, T') \subset [0, T)$ for which $x(t)$ and $y(t)$ remain in $V$.
Next, consider the non-negative, absolutely continuous function $q: [0, T') \rightarrow \mathbb{R}$ defined as $q(t) := \frac{1}{2} \| y(t) - x(t) \|^2_{g(x_0)} e^{-2M t}$. Note that $q(0) = 0$. Furthermore, using~\eqref{eq:lipschitz_uniq} and applying the product rule we have
\begin{align*}
\tfrac{d}{dt} q(t) & = ( \left\langle \tproj{\mathcal{X}}{g}{ f(y}(t)) - \tproj{\mathcal{X}}{g}{ f(x}(t)), y(t) - x(t) \right\rangle_{g(x_0)} - M || y(t) - x(t) ||^2_{g(x_0)} ) e^{-2M t}
\end{align*}
and since $y(0) = x(0)$ it follows that $\frac{d}{dt} q(t) \leq 0$ for $t \geq 0$. However, since $q$ is non-negative and absolutely continuous, we conclude that $x(t) = y(t)$ for all $t \in [0, T')$ thus finishing the proof of uniqueness.
\end{proof}
Combining all the insights so far, we arrive at the following ready-to-use result:
\begin{example}[Existence and uniqueness on constraint-defined sets] As in \cref{ex:clarke_reg_prox_cone} consider a set $\mathcal{X} := \{ x \in \mathbb{R}^n \, | \, h(x) \leq 0 \}$ where $h: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is of class $C^{1,1}$ and has full rank for all $x \in \mathbb{R}^n$. Further, consider a globally Lipschitz continuous vector field $f: \mathbb{R}^n \rightarrow \mathbb{R}$.
Then, for every $x_0 \in \mathcal{X}$ there exists a unique and complete Carath\'eodory solution $x: [0, \infty) \rightarrow \mathcal{X}$ to the initial value problem $\dot x = \tproj{\mathcal{X}}{g}{f}(x)$ with $x(0) = x_0$ where $g$ is any weakly bounded $C^{0,1}$ metric on $\mathcal{X}$.
\end{example}
\section{Existence and Uniqueness on low-regularity Riemannian Manifolds}\label{sec:mfd}
The major appeal of \cref{thm:main_exist,thm:main_equiv,thm:main_uniq} is their geometric nature. Namely, as we will show next, their assumptions are preserved by sufficiently regular coordinate transformations which allows us to give a coordinate-free definition of projected dynamical system on manifolds with minimal degree of differentiability.
Recall that for open sets $V, W \subset \mathbb{R}^n$ a map $\Phi: V \rightarrow W$ is a \emph{$C^k$ diffeomorphism} if it is a $C^k$ bijection with a $C^k$ inverse where, for our purposes, $C^k$ stands for either $C^1$ or $C^{1,1}$.
We employ the usual definition of a \emph{$C^k$ manifold} as locally Euclidean, second countable Hausdorff space endowed with a $C^k$ differentiable structure. In particular, for a point $p$ on a $n$-dimensional manifold $\mathcal{M}$ there exists a chart $(U, \phi)$ where $U \subset \mathcal{M}$ is open and $\phi: U \rightarrow \mathbb{R}^n$ is a homeomorphism onto its image. For any two charts $(U, \phi), (V, \psi)$ for which $U \cap V \neq \emptyset$, the map $\phi \circ \psi^{-1}: \psi(U \cap V) \rightarrow \phi(U \cap V)$ is a $C^k$ diffeomorphism.
A \emph{$C^k$ (Riemannian) metric $g$} is a map that assigns to every point $p \in \mathcal{M}$ an inner product on the \emph{tangent space}\footnote{Note that the definition (and hence the notation) of the tangent space $T_x \mathcal{M}$ of a manifold $\mathcal{M}$ is consistent with the definition of the tangent cone $T_x \mathcal{X}$ of an arbitrary set $\mathcal{X}$~\cite[Ex.~6.8]{rockafellarVariationalAnalysis2009}.}
$T_p \mathcal{M}$ such that in local coordinates $(U, \phi)$ the metric $g(\phi^{-1}(x))$ is a $C^k$ metric for $x \in \phi(U)$ according to \cref{def:metric}.
A vector field defined on $\mathcal{M}$ is \emph{locally bounded at $x$} if it is locally bounded in any local coordinate domain for $x$. Similarly, a metric is \emph{locally weakly bounded at $x$} if its locally weakly bounded in local coordinates.
Given a $C^k$ manifold $\mathcal{M}$ with $k\geq 1$, a curve $\gamma:[0, T) \rightarrow \mathcal{M}$ is \emph{absolutely continuous} if it is absolutely continuous in any chart domain where it is defined.\footnote{Note that local (weak) boundedness of a vector field or metric are properties that are preserved by $C^1$ diffeomorphisms. Similarly, absolute continuity is preserved by $C^1$ maps~\cite[Ex.~6.44]{roydenRealAnalysis1988}. Hence, it is sufficient if these properties hold in any local coordinate domain.}
The next lemma shows that a $C^1$ diffeomorphism maps (Clarke) tangent cones to (Clarke) tangent cones. Hence, Clarke regularity is preserved by $C^1$ diffeomorpisms.
\unless\ifARXIV
The proof simple but technical and can be found in~\cite{hauswirthProjectedDynamicalSystems2018a}.
\fi
\begin{lemma}\label{lem:clarke_reg_trans} Let $V, W \subset \mathbb{R}^n$ be open and consider a $C^1$ diffeomorphism $\Phi: V \rightarrow W$. Given $\mathcal{X} \subset \mathbb{R}^n$ and $\tilde{\mathcal{X}} := \mathcal{X} \cap V$, for every $x \in \tilde{\mathcal{X}}$ it holds that
\begin{align}\label{eq:tgt_equiv}
T_{\Phi(x)}\Phi(\tilde{\mathcal{X}}) & = D_x\Phi (T_x \tilde{\mathcal{X}}) \\\label{eq:ctgt_equiv}
T^C_{\Phi(x)}\Phi(\tilde{\mathcal{X}}) & = D_x\Phi (T^C_x \tilde{\mathcal{X}}) \, .
\end{align}
Hence, $\Phi(\tilde{\mathcal{X}})$ is Clarke regular at $\Phi(x)$ if and only if $\tilde{\mathcal{X}}$ is Clarke regular at $x \in \tilde{\mathcal{X}}$.
\end{lemma}
\ifARXIV
\begin{proof}
We only need to show that
$T_{\Phi(x)}\Phi(\tilde{\mathcal{X}}) \subset D_x\Phi (T_x \tilde{\mathcal{X}})$.
Since $\Phi$ is a $C^1$ diffeomorphism the other direction follows by applying the same arguments to $\Phi^{-1}$.
Let $v \in T_x \tilde{\mathcal{X}}$. Then, by definition there exist $x_k \rightarrow x$ with $x_k \in \tilde{\mathcal{X}}$ and $\delta_k \rightarrow 0^+$ such that $(x_k - x)/ \delta_k \rightarrow v$. Furthermore, $\| x_k - x \| / \delta_k$ converges to $\| v \|$. According to the definition of the derivative of $\Phi$, for the same sequence $\{ x_k \}$ we have
$\underset{k \rightarrow \infty}{\lim} \| \Phi(x_k) - \Phi(x) - D_x \Phi(x_k - x) \| / \| x_k - x \| = 0 $.
Since the limit of the element-wise product of convergent sequences equals the product of its limits we can write
\begin{equation*}
\underset{k \rightarrow \infty}{\lim} \tfrac{\left \| \Phi (x_k) - \Phi(x) - D_x \Phi(x_k - x) \right \|}{\| x_k - x \|} \tfrac{\|x_k - x \|}{\delta_k} = 0
\end{equation*}
which, using the fact that $D_x \Phi$ is linear, simplifies to
\begin{equation*}
\underset{k \rightarrow \infty}{\lim} \left \| \tfrac{\Phi(x_k) - \Phi(x)}{\delta_k} - D_x \Phi\left(\tfrac{x_k - x}{\delta_k} \right) \right \| = 0 \, .
\end{equation*}
This implies that $(\Phi(x_k) - \Phi(x))/\delta_k \rightarrow D_x \Phi (v)$, and hence $D_x \Phi(v)$ is a tangent vector of $\Phi(\tilde{\mathcal{X}})$ at $\Phi(x)$. This proves~\eqref{eq:tgt_equiv}.
To show~\eqref{eq:ctgt_equiv} we use~\eqref{eq:tgt_equiv} together with the definition of the Clarke tangent cone as the inner limit of the surrounding tangent cones (\cref{def:clarke_tgt}). We can write
\begin{equation*}
T_{\Phi(x)}^C \Phi(\tilde{\mathcal{X}}) =
\underset{\hat{y} \rightarrow \Phi(x)}{\lim \inf} \,
T_{\hat{y}} \Phi(\tilde{\mathcal{X}})
= \underset{y \rightarrow x}{\lim \inf} \,
D_{y} \Phi \left( T_{y} \tilde{\mathcal{X}} \right) \, .
\end{equation*}
Since $D_x \Phi$ is continuous in $x$, we have
$\underset{y \rightarrow x}{\lim \inf} \,
D_{y} \Phi ( T_{y} \tilde{\mathcal{X}} ) = \underset{y \rightarrow x}{\lim \inf} \,
D_{x} \Phi ( T_{y} \tilde{\mathcal{X}} )$.
Further, \cref{lem:continuity_set} implies that
$\underset{y \rightarrow x}{\lim \inf} \,
D_{x} \Phi ( T_{y} \tilde{\mathcal{X}} ) \supset
D_{x} \Phi ( \underset{y \rightarrow x}{\lim \inf} \, T_{y} \tilde{\mathcal{X}})
= D_x\Phi (T^C_x \tilde{\mathcal{X}} ) $
and therefore we have $ T_{\Phi(x)}^C \Phi(\tilde{\mathcal{X}}) \supset D_x\Phi (T^C_x \tilde{\mathcal{X}})$. Again, since $\Phi$ is a diffeomorphism, the opposite inclusion holds by applying the same argument to $\Phi^{-1}$. This shows~\eqref{eq:ctgt_equiv} and completes the proof.
\end{proof}
\fi
Hence, the notions of (Clarke) tangent cone and Clarke regularity are independent of the coordinate representation on a $C^1$ manifold.
\begin{definition} Let $\mathcal{M}$ be a $C^1$ manifold with a metric $g$ and consider a subset $\mathcal{X} \subset \mathcal{M}$. The (Clarke) tangent cone $T_x \mathcal{X}$ $(T^C_x \mathcal{X})$ is a subset of $T_x \mathcal{M}$ such that $D_x \phi(T_x \mathcal{X})$ $(D_x \phi(T^C_x \mathcal{X}))$ is the (Clarke) tangent cone of $\phi(\mathcal{X} \cap U)$ for any coordinate chart $(U, \phi)$ defined at $x$. The set $\mathcal{X}$ is Clarke regular at $x \in \mathcal{X}$ if it is Clarke regular in any local coordinate domain defined at $x$.
\end{definition}
The next key result establishes that solutions of projected dynamical systems remain solutions of projected dynamical systems under $C^{1}$ coordinate transformations.
\begin{proposition}\label{prop:inv_pds} Let $V, W \subset \mathbb{R}^n$ be open and consider a $C^{1}$ diffeomorphism $\Phi: V \rightarrow W$. Let $\mathcal{X} \subset \mathbb{R}^n$ be locally compact and $\tilde{\mathcal{X}} := \mathcal{X} \cap V$. Further, let $g$ be a locally weakly bounded metric on $W$ and let $\Phi^* g$ denote the \emph{pull-back metric along $\Phi$}, i.e.,
\begin{equation}\label{eq:pullback_metric}
\left\langle v, w \right\rangle_{\Phi^* g(x)} := \left\langle D_{x} \Phi (v), D_{x} \Phi (w) \right\rangle_{g(\Phi(x))}
\end{equation}
for all $x \in V$ and $v,w \in T_x \mathbb{R}^n$. Further, let $f: \tilde{\mathcal{X}} \rightarrow \mathbb{R}^n$ be a locally bounded vector field. If $x:[0, T) \rightarrow \tilde{\mathcal{X}}$ for some $T> 0$ is a Krasovskii (respectively, Carath\'eodory) solution to the initial value problem
\begin{equation}\label{eq:non_trans_prob}
\dot x \in \tproj{\tilde{\mathcal{X}}}{\Phi^*g}{f}(x) \, , \quad x(0) = x_0 \, ,
\end{equation}
then $\Phi \circ x: [0, T) \rightarrow \Phi(\tilde{\mathcal{X}})$ is a Krasovskii (respectively, Carath\'eodory) solution to
\begin{equation}\label{eq:transf_prob}
\dot{y} \in \tproj{{g}}{{\Phi(\tilde{\mathcal{X}})}}{\hat{f}}(y) \, , \quad y(0) = y_0 \,,
\end{equation}
where $y_0 := \Phi(x_0)$ and $\hat{f}(y) := D_{\Phi^{-1}(y)} \Phi (f ( \Phi^{-1}(y)))$ is the \emph{pushforward vector field of $f$ along $\Phi^{-1}$}.
\end{proposition}
\begin{proof} First, note that since $x$ is absolutely continuous and $\Phi$ is differentiable, $\Phi\circ x$ is absolutely continuous~\cite[Ex.~6.44]{roydenRealAnalysis1988}. Second, it holds that $y(t) \in \Phi(\tilde{\mathcal{X}})$ for all $t \in [0, T)$. Third, using~\eqref{eq:tgt_equiv} we can write for every $x \in \tilde{\mathcal{X}}$ and $y := \Phi(x)$ that
\begin{align*}
\tproj{{g}}{{\Phi(\tilde{\mathcal{X}})}}{\hat{f}}(y)
& = \underset{w \in T_{y} \Phi(\tilde{\mathcal{X}})}{\arg \min}
\left \|w - D_{x} \Phi(f(x)) \right \|_{g}
= \underset{w \in D_{x} \Phi \left(T_{x} \tilde\mathcal{X}\right)}{\arg \min}
\left \|w - D_{x} \Phi(f(x)) \right \|_{g} \\
& = D_{x}\Phi \left(\underset{v \in T_{x}\tilde{\mathcal{X}}}{\arg \min}
\left \| D_{x}\Phi(v) - D_{x} \Phi(f(x)) \right \|_{g} \right) \, ,
\end{align*}
where for the last equality we introduce the transformation $w := D_x \Phi(v)$ for $v \in T_{x}\tilde\mathcal{X}$. Hence, using the definition of the pullback metric~\eqref{eq:pullback_metric} we continue with
\begin{align*}
\tproj{{g}}{{\Phi(\tilde{\mathcal{X}})}}{\hat{f}}(y)
= D_{x}\Phi \left( \underset{ v \in T_{x}{\tilde{\mathcal{X}}}}{\arg \min}
\left \|v - f(x) \right \|_{\Phi^* g} \right) = D_{x} \Phi \left(\tproj{\tilde{\mathcal{X}}}{\Phi^*g}{f}(x)\right) \, .
\end{align*}
Thus, if $x(\cdot)$ is a Carath\'eodory solution of~\eqref{eq:non_trans_prob} and hence
$\dot x(t) \in \tproj{\tilde{\mathcal{X}}}{\Phi^*g}{f}(x(t))$
holds almost everywhere, then $\Phi \circ x(\cdot)$ satisfies
\begin{equation*}
\frac{d}{dt}\left(\Phi \circ x \right) \in D_{x} \Phi \left(\tproj{\tilde{\mathcal{X}}}{\Phi^*g}{f}(x)\right) = \tproj{\Phi(\tilde{\mathcal{X}})}{g}{\hat{f}}(\Phi \circ x(t))
\end{equation*}
almost everywhere and hence $\Phi \circ x(\cdot)$ is a Carath\'eodory solution to~\eqref{eq:transf_prob}.
It remains to prove the statement is also true for Krasovskii solutions. For this, we need to show that
$\Kras{ \tproj{{g}}{{\Phi(\tilde{\mathcal{X}})}}{\hat{f}}}(y)
\supset D_{x} \Phi (\Kras{\tproj{\tilde{\mathcal{X}}}{\Phi^*g}{f } }(y) )$.
Expanding the definition of the Krasovskii regularization we get
\begin{align*}
\Kras{ \tproj{{g}}{{\Phi(\tilde{\mathcal{X}})}}{\hat{f}}}(y)
& =
\cocl \, \underset{\tilde{y} \rightarrow y}{\lim \sup} \, \tproj{\Phi(\tilde{\mathcal{X}})}{g}{\hat{f}}(\tilde{y}) \\
& =
\cocl \, \underset{\tilde{x} \rightarrow x}{\lim \sup} \,
D_{\tilde{x}} \Phi \left( \tproj{\tilde{\mathcal{X}}}{\Phi^*g}{f }(\tilde{x}) \right) \\
& =
\cocl \, \underset{\tilde{x} \rightarrow x}{\lim \sup} \,
D_{x} \Phi \left( \tproj{\tilde{\mathcal{X}}}{\Phi^*g}{f }(x_k) \right) \, ,
\end{align*}
where the last equation is due to the fact that $D_x \Phi$ is continuous in $x$. Next, with \cref{lem:continuity_set} we can write
\begin{align*}
\Kras{ \tproj{{g}}{{\Phi(\tilde{\mathcal{X}})}}{\hat{f}}}(y)
& \supset
\cocl \, D_{x} \Phi \left(\underset{\tilde{x} \rightarrow x}{\lim \sup} \,
\tproj{\tilde{\mathcal{X}}}{\Phi^*g}{f }(x_k) \right)
= D_{x} \Phi \left(\Kras{\tproj{\tilde{\mathcal{X}}}{\Phi^*g}{f } }(x) \right)
\end{align*}
where the equation follows from the fact that $D_x \Phi$ is a linear map and hence commutes with taking the convex closure.
To conclude we can proceed similar to the case of Carath\'eodory solutions. Let $x(\cdot)$ be a Krasovskii solution to~\eqref{eq:non_trans_prob} and $y(\cdot) := \Phi \circ x(\cdot)$. Then, $\dot y(t) = \frac{d}{dt}(\Phi \circ x)(t) = D_{x(t)} \Phi( \dot x(t))$ for almost all $t \in [0, T)$ and we have that
\begin{equation*}
\dot y(t) \in
D_{x(t)} \Phi \left(\Kras{\tproj{\tilde{\mathcal{X}}}{\Phi^*g}{f }( x}(t) ) \right) \subset
\Kras{ \tproj{\Phi(\tilde{\mathcal{X}})}{g}{\hat{f}}}(y(t))
\end{equation*}
for almost all $t \in [0, T)$, and thus $y$ is a Krasovskii solution of~\eqref{eq:transf_prob}.
\end{proof}
Hence, \cref{thm:main_exist,thm:main_equiv} combined with \cref{prop:inv_pds} give rise to our main result on the existence of Krasovskii (Careth\'eodory) solutions to on manifolds.
\begin{theorem}[existence on manifolds]\label{thm:main_mfd}
Let $\mathcal{M}$ be $C^1$ manifold, $g$ a locally weakly bounded Riemannian metric, $\mathcal{X} \subset \mathcal{M}$ locally compact, and $f$ a locally bounded vector field on $\mathcal{X}$. Then for every $x_0 \in \mathcal{X}$ there exists a Krasovskii solution $x: [0, T) \rightarrow \mathcal{X}$ for some $T> 0$ that solves
$\dot x(t) \in \tproj{\mathcal{X}}{g}{f}(x(t))$ with $x(0) = x_0$.
Furthermore, if $\mathcal{X}$ is Clarke regular, and if $f$ and $g$ are continuous, then every Krasovskii solution is a Carath\'eodory solution and vice versa.
\end{theorem}
Similarly, \cref{prop:inv_pds} directly implies that other results such as \cref{cor:max_sol} extend to $C^1$ manifolds. For instance, if $\mathcal{M}$ is compact and $f$ and $g$ are continuous, every initial condition admits a complete trajectory. However, to extend our uniqueness results, we require stronger conditions.
\begin{proposition}\label{prop:c11_prox} Let $V,W \subset \mathbb{R}^n$ be open and $\Phi: V \rightarrow W$ a $C^{1,1}$ diffeo\-morphism. Let $\mathcal{X} \subset \mathbb{R}^n$ be locally compact and consider $\tilde{\mathcal{X}} := \mathcal{X} \cap V$. If $\tilde{\mathcal{X}}$ is prox-regular then $\Phi(\tilde{\mathcal{X}})$ is prox-regular.
\end{proposition}
\begin{proof} By \cref{prop:prox_invariance} it suffices to show prox-regularity with respect to a single metric on $V$ and $W$ respectively. Hence, let $W$ be endowed with the Euclidean metric, and let $e^*$ denote its pullback metric on $V$ along $\Phi$, i.e., $\left\langle v, w \right\rangle_{e^*(x)} := \left\langle D_x\Phi(v), D_x\Phi(w) \right\rangle$.
Similarly to \cref{lem:clarke_reg_trans}, we show that (proximal) normal cones are preserved by $C^1$ coordinate transformations, i.e.,
\begin{align}
\eta \in N^{e^*}_x \tilde{\mathcal{X}} & \quad \Longleftrightarrow \quad D_x\Phi(\eta) \in N_{\Phi(x)} \Phi(\tilde{\mathcal{X}}) \qquad \forall x \in \tilde{\mathcal{X}}\label{eq:norm_impl} \\
\eta \in \bar{N}^{e^*, L}_y \tilde{\mathcal{X}} & \quad \Longleftrightarrow \quad D_y\Phi(\eta) \in \bar{N}^{L'}_{\Phi(y)} \Phi(\tilde{\mathcal{X}}) \qquad \forall y \in \mathcal{N}_x\label{eq:norm_impl_prox}
\end{align}
for some $L', L > 0$ where $\mathcal{N}_x \subset \tilde{\mathcal{X}}$ is a neighborhood of $x$. Since $\Phi$ is a diffeomorphism it suffices to show one direction only.
Hence, consider $\eta \in N^{e^*}_x \tilde{\mathcal{X}}$. By \cref{def:norm_cone} and using~\eqref{eq:tgt_equiv} we have
\begin{align*}
\eta \in N^{e^*}_x \tilde{\mathcal{X}} \quad & \Leftrightarrow \quad \left\langle \eta, w \right\rangle_{e^*(x)} = \left\langle D_x\Phi(\eta), D_x\Phi(w) \right\rangle \leq 0 \quad \forall w \in T_x \tilde{\mathcal{X}} \\
& \Leftrightarrow \quad \left\langle D_x\Phi(\eta), w \right\rangle \leq 0 \quad \forall w \in D_{x} \Phi (T_x \tilde{\mathcal{X}}) = T_{\Phi(x)} \Phi(\tilde{\mathcal{X}}) \, .
\end{align*}
We conclude that $D_x\Phi(\eta) \in N_{\Phi(x)} \Phi(\tilde{\mathcal{X}})$ and~\eqref{eq:norm_impl} holds.
For~\eqref{eq:norm_impl_prox} we consider $y \in \tilde{\mathcal{X}}$ in a neighborhood of $x$ and $\eta \in \bar{N}_y^{e^*, L} \tilde{\mathcal{X}}$ such that
\begin{align*}
\left\langle \eta, z - y \right\rangle_{e^*(y)}
=
\left\langle D_y\Phi(\eta), D_y\Phi (z - y) \right\rangle
\leq L \| z - y \|^2_{e^* g(y)}
\end{align*}
holds for all $z \in \tilde{\mathcal{X}}$ in a neighborhood of $y$. However, we need to show that for some $L' > 0$ we have
\begin{align}\label{eq:prox_goal}
\left\langle D_y\Phi(\eta), \Phi(z) - \Phi(y) \right\rangle
\leq L' \| \Phi(z) - \Phi(y) \|^2 \, .
\end{align}
Hence, we define the $C^{1,1}$ function $\psi(z) := \left\langle D_y\Phi(\eta), \Phi(z) \right\rangle$ and note that by linearity we have $D_z \psi (v) := \left\langle D_y\Phi(\eta), D_z\Phi(v) \right\rangle$. This enables us to apply the Desent \cref{lem:c11_lipschitz} and state that for some $M > 0$ it holds that
\begin{align*}
| \psi(z) - \psi(y) - D_y \psi(z - y) | =
\underbrace{| \left\langle D_y\Phi(\eta), \Phi(z) - \Phi(y) - D_y \Phi(z - y) \right\rangle |}_{ =: \gamma(z)}
\leq M \| z - y \|^2 \, .
\end{align*}
This bound can be used to establish
\begin{align*}
\left\langle D_y\Phi(\eta), \Phi(z) - \Phi(y) \right\rangle \leq
\left\langle D_y\Phi(\eta), D_y\Phi (z - y) \right\rangle
+ \gamma(z) \leq (L + M) \| z - y \|^2 \, .
\end{align*}
Finally note that $\| z - y \|^2 \leq L' \| \Phi(z) - \Phi(y) \|^2$ for some $L'$ since $\Phi^{-1}$ is Lipschitz continuous. Hence,~\eqref{eq:prox_goal} and therefore~\eqref{eq:norm_impl_prox} holds for $L' = L'' ( L + M)$.
\end{proof}
Apart from \cref{prop:c11_prox}, we note that Lipschitz continuity of a metric and of vector fields is preserved under $C^{1,1}$ coordinate transformations. This allows us to generalize \cref{thm:main_uniq} to the following uniqueness result on manifolds.
\begin{theorem}[uniqueness on manifolds]\label{thm:main_mfd_uniq}
Let $\mathcal{M}$ be $C^{1,1}$ manifold, $g$ a $C^{0,1}$ Riemannian metric, $\mathcal{X} \subset \mathcal{M}$ is prox-regular, and $f$ a $C^{0,1}$ vector field on $\mathcal{X}$. Then, for every $x_0 \in \mathcal{X}$ there exists a unique Carath\'eodory solution $x: [0, T) \rightarrow \mathcal{X}$ for some $T> 0$ that solves $\dot x(t) \in \tproj{\mathcal{X}}{g}{f}(x(t)))$ with $x(0) = x_0$.
\end{theorem}
In conclusion, thanks to our coordinate-free definition of projected dynamical systems, our existence and uniqueness results seamlessly extend to systems defined on abstract manifolds.
\section{Stability of Projected Gradient Flows}\label{sec:stab}
To illustrate how established stability concepts easily apply to Krasovskii solutions of projected dynamical systems, we consider projected gradient systems, i.e., projected dynamical systems for which the vector field is the gradient of a function. Naturally, these systems are of prime interest for constrained optimization. Similar techniques can also be used to assess the stability of equilibria of other vector fields ranging from saddle-point flows~\cite{cherukuriAsymptoticConvergenceConstrained2016} to momentum methods~\cite{wilsonLyapunovAnalysisMomentum2016}. In what follows, we will establish convergence and stability results that generalize the work in~\cite{hauswirthProjectedGradientDescent2016}.
For simplicity, we consider systems defined on $\mathbb{R}^n$. Extensions to subsets of manifolds are possible using the results from \cref{sec:mfd} (see \cref{rem:grad_mfd} below).
\subsection{Preliminaries and LaSalle Invariance}
We quickly review some basic terminology for continuous-time systems defined by a constrained differential inclusion
\begin{align}\label{eq:gen_inclusion}
\dot x \in F(x) \quad x \in \mathcal{X} \, ,
\end{align}
where $\mathcal{X} \subset \mathbb{R}^n$ is closed and $F: \mathcal{X} \rightrightarrows \mathbb{R}^n$ is non-empty, closed, convex, locally bounded, and outer semicontinuous. In the following, a \emph{solution of \eqref{eq:gen_inclusion}} refers to a Carath\'eodory solution of \eqref{eq:gen_inclusion}, whereas a \emph{Krasovskii solution} of \eqref{eq:gen_inclusion} is a (Carath\'eodory) solution of the inclusion obtained from regularizing \eqref{eq:gen_inclusion}.
The \emph{$\omega$-limit set} of a complete solution $x$ of \eqref{eq:gen_inclusion} is the set of all points $\hat{x}$ for which there exists a sequence $\{t_k \}$ with $\lim_{k \rightarrow \infty} t_k = \infty$ and $\lim_{k \rightarrow \infty} x(t_k) = \hat{x}$.
A set $\mathcal{A} \subset \mathcal{X}$ is \emph{weakly invariant}, if for every initial condition $x_0 \in \mathcal{A}$, there exists a complete solution starting at $x_0$ that remains in $\mathcal{A}$ for all $t \geq 0$. The union of any weakly invariant subsets is weakly invariant, hence the notion of \emph{largest weakly invariant set} is well-defined.
A set $\mathcal{A} \subset \mathcal{X}$ is \emph{invariant}, if for every initial condition $x_0 \in \mathcal{A}$, every complete solution starting at $x_0$ remains in $\mathcal{A}$ for all $t \geq 0$.
Also recall that $\hat{x} \in \mathcal{X}$ is a \emph{weak equilibrium} for \eqref{eq:gen_inclusion} if and only if $x(t) = \hat{x}$ for all $t \geq 0$ is a solution. Namely, $\hat{x}$ is a weak equilibrium if and only if $0 \in F(\hat{x})$.
A \emph{strong equilibrium} is a point $\hat{x}$ such that $x(t) = \hat{x}$ for all $t \geq 0$ is the only solution starting at $\hat{x}$.
A compact set $\mathcal{A} \subset \mathcal{X}$ is \emph{stable} for \eqref{eq:gen_inclusion} if for every (relative) neighborhood $\mathcal{V}$ of $\mathcal{A}$ there exists a neighborhood $\mathcal{W}$ of $\mathcal{A}$ such that every complete solution of \eqref{eq:gen_inclusion} starting in $\mathcal{W}$ satisfies $x(t) \in \mathcal{V} $ for all $t \geq 0$. The set $\mathcal{A}$ is \emph{locally asymptotically stable}, if it is stable and there exists $\delta > 0$ such that every solution $x$ with $d_\mathcal{A}(x(0)) \leq \delta$ converges to $\mathcal{A}$, i.e., $\lim_{t \rightarrow \infty} d_\mathcal{A}(x(t)) = 0$.
We will make use of the following invariance principle for differential inclusions. The result is a special case of \cite[Thm.~8.2]{goebelHybridDynamicalSystems2012} which applies to hybrid systems. For similar results for differential inclusions see also \cite{aubinViabilityTheory1991,ryanIntegralInvariancePrinciple1998}.
\begin{theorem}\label{thm:invar}
Consider a continuous function $V: \mathbb{R}^n \rightarrow \mathbb{R}$, any function $u: \mathbb{R}^n \rightarrow [- \infty, \infty]$, and a set $\mathcal{U} \subset \mathbb{R}^n$ such that $u(x) \leq 0$ for every $x \in \mathcal{U}$ and such that the growth of $V$ along solutions of \eqref{eq:gen_inclusion} is bounded by $u$ on $\mathcal{U}$. In other words, any solution $x: [0, T) \rightarrow \mathcal{U}$ of~\eqref{eq:gen_inclusion} satisfies $V(x(t_1)) - V(x(t_0)) \leq \int_{t_0}^{t_1} u ( x(\tau)) d\tau$ for any $t_0, t_1 \in [0, T)$ and $t_0 < t_1$.
Let $x$ be a complete and bounded solution of~\eqref{eq:gen_inclusion} such that $x(t) \in \mathcal{U}$ for all $t \geq 0$. Then, for some $r \in V(\mathcal{U})$, $x$ approaches the nonempty set that is the largest weakly invariant subset of $V^{-1}(r) \cap \mathcal{U} \cap \cl u^{-1}(0)$.
\end{theorem}
\subsection{Convergence of Projected Gradient Flows}
In the following, we consider projected gradient flows of the form
\begin{align}\label{eq:proj_grad_intro}
\dot x \in \tproj{\mathcal{X}}{g}{- \grad_g \Phi}(x)
\end{align}
where $\mathcal{X} \subset \mathbb{R}^n$ is closed, and $g$ is a locally weakly bounded metric on $\mathcal{X}$. Further, $\Phi: \mathbb{R}^n \rightarrow \mathbb{R}$ is an objective function, continuously differentiable in a neighborhood of $\mathcal{X}$.
The \emph{gradient} $\grad_g \Phi(x)$ of $\Phi$ with respect to $g$ at $x \in \mathcal{X}$ is the unique vector that satisfies $\left \langle \grad_g \Phi(x), w \right \rangle_{g(x)} = D_x \Phi (w)$ for all $w \in T_x \mathbb{R}^n$. In matrix notation we have
\begin{align*}
\grad_g \Phi(x) = G^{-1}(x) \nabla \Phi(x)^T \, .
\end{align*}
The results of the previous sections can be used to guarantee the existence and uniqueness of (Carath\'eodory or Krasovskii) solutions of \eqref{eq:proj_grad_intro} under appropriate condtions on $\mathcal{X}, g$ and $\Phi$. In fact, \eqref{eq:proj_grad_intro} is well-defined on subsets of abstract $C^{1}$-manifolds.
Dynamics of the form \eqref{eq:proj_grad_intro} serve to find local solutions of the constrained problem
\begin{equation}\label{eq:min_prob_intro}
\text{minimize } \, \Phi(x) \quad \text{subject to } \, x \in \mathcal{X} \,.
\end{equation}
A \emph{(strict) local minimizer of \eqref{eq:min_prob_intro}} is a point $x^\star \in \mathcal{X}$ such that there exists a relative neighborhood $\mathcal{N} \subset \mathcal{X}$ of $x^\star$ and $\Phi(y) \geq (>) \Phi(x^\star)$ holds for all $y \in \mathcal{N} \setminus \{ x^\star\}$.
A \emph{critical point} of \eqref{eq:min_prob_intro} is a point $x^\star \in \mathcal{X}$ satisfying
\begin{align}\label{eq:equil_crit_pt}
D_{x^\star} \Phi ( w ) = \nabla \Phi(x^\star) w \geq 0 \qquad \text{for all} \qquad w \in T_{x^\star} \mathcal{X} \,.
\end{align}
Every local minimizer of \eqref{eq:min_prob_intro} is a critical point \cite[Thm.~6.12]{rockafellarVariationalAnalysis2009}. Further, if $\mathcal{X}$ is Clarke regular and of the same form as in \cref{ex:clarke_reg_constraint_set}, \eqref{eq:equil_crit_pt} is equivalent to the well-known Karush-Kuhn-Tucker (KKT) conditions~\cite[Chap. 4]{bazaraaNonlinearProgrammingTheory2006}.
The metric $g$ is a property of the system~\eqref{eq:proj_grad_intro} only and does not affect the optimizers of~\eqref{eq:min_prob_intro}.
Furthermore, it is reasonable (but important to note) that, in general, the metric that defines the gradient has to be the same metric that defines the projection.
A particular choice of $g$ is, for example, induced by the Hessian of $\Phi$ if $\Phi$ is twice continuously differentiable and strongly convex. This leads to Newton-type dynamics (\cref{ex:newton} below).
When considering the projected gradient flow \eqref{eq:proj_grad_intro} we need to distinguish between equilibrium points for Carath\'eodory and Krasovskii solutions. In particular, we say that $x^\star$ is a \emph{weak (strong) K-equilibrium}, if it is a weak (strong) equilibrium of the Krasovskii-regularized inclusion. Analogously, $x^\star$ is a \emph{weak (strong) C-equilibrium} if it is an equilibrium for Carath\'eodory solutions (i.e., solutions of the unregularized inclusion).
Since every Carath\'eodory solution of \eqref{eq:proj_grad_intro} is also a Krasovskii solutions, it follows that every strong K-equilibrium is also a strong C-equilibrium. On the other hand, a weak C-equilibrium is a weak K-equilibrium.
We can now establish the relation between critical points and minimizers of \eqref{eq:min_prob_intro}, and the different types of equilibria of \eqref{eq:proj_grad_intro}.
\begin{lemma}\label{lem:weak_k_equil}
Every critical point of \eqref{eq:min_prob_intro} is a weak K-equilibrium of \eqref{eq:proj_grad_intro}, and every weak C-equilibrium of \eqref{eq:proj_grad_intro} is a critical point of \eqref{eq:min_prob_intro}.
\end{lemma}
\begin{proof}
Let $x^\star$ be a critical point of \eqref{eq:min_prob_intro}. By definition of $\grad_g \Phi$, we can reformulate \eqref{eq:equil_crit_pt} as $\left\langle - \grad_g \Phi(x^\star), w \right\rangle_{g(x^\star)} \leq 0$ for all $w \in T_{x^\star} \mathcal{X}$. Furthermore, by \cref{lem:kras_normal}, we have, for all $x \in \mathcal{X}$,
\begin{align*}
\left\langle - \grad_g \Phi(x), w \right\rangle_{g(x)} \geq \| w \|^2_{g(x)} \quad \forall w\in \Kras{\tproj{\mathcal{X}}{g}{-\grad_g \Phi}}(x).
\end{align*}
Combining these two statements we get
\begin{align*}
0 \geq \left\langle - \grad_g \Phi(x^\star), w \right\rangle_{g(x^\star)} \geq \| w \|^2_{g(x^\star)} \quad \forall w \in T_{x^\star} \mathcal{X} \cap \Kras{\tproj{\mathcal{X}}{g}{-\grad_g \Phi}}(x^\star).
\end{align*}
We know that $T_{x} \mathcal{X} \cap \Kras{\tproj{\mathcal{X}}{g}{-\grad_g \Phi}}(x) \neq \emptyset$ holds for all $x \in \mathcal{X}$ by viability of $\tproj{\mathcal{X}}{g}{-\grad_g \Phi}$. Therefore, we conclude that $w = 0 \in \Kras{\tproj{\mathcal{X}}{g}{-\grad_g \Phi}}(x^\star)$ and $x^\star$ is a weak K-equilibrium.
Next, assume that $x^\star \in \mathcal{X}$ is a weak C-equilibrium, i.e., $0 \in \tproj{\mathcal{X}}{g}{- \grad_g \Phi}(x^\star)$. If $x^\star$ were not a critical point of \eqref{eq:min_prob_intro}, then $\left\langle - \grad_g \Phi(x^\star), v \right\rangle_{g(x^\star)} > 0$ holds for some $v \in T_{x^\star} \mathcal{X}$. This, however, means that $0 \notin \tproj{\mathcal{X}}{g}{- \grad_g \Phi}(x^\star)$.
To see this, note that the projection of $u := - \grad_g \Phi(x^\star)$ onto the ray/cone spanned by $v$ is given by $w := ({\left\langle u, v\right\rangle_{g(x^\star)}}/{\| v\|^2_{g(x^\star)}}) v$ (note that $\left\langle u, v\right\rangle_{g(x^\star)} \geq 0$). Applying the Pythagorean theorem to the right triangle $\{0, u, w\}$, we have $\| u - w \| < \| u - 0 \|$. Hence, $0$ cannot be a projection of $u$ onto $T_{x^\star} \mathcal{X}$ since it does not achieve the minimal distance to $T_{x^\star} \mathcal{X}$ which contradicts the fact that $x^\star$ is a C-equilibrium.
\end{proof}
\begin{lemma}\label{lem:invar_k_grad}
Along Krasovskii solutions of \eqref{eq:proj_grad_intro}, $\Phi$ is nonincreasing and, consequently, the sublevel sets $S_\ell := \{ x\, | \, \Phi(x) \leq \ell \} \cap \mathcal{X}$ for $\ell \in \mathbb{R}$ are invariant.
\end{lemma}
\begin{proof}
Given any Krasovskii solution $x: [0, T) \rightarrow \mathcal{X}$ of \eqref{eq:proj_grad_intro}, for almost all $t \in [0, T)$ there exists $w(t) \in \Kras{\tproj{\mathcal{X}}{g}{- \grad_g \Phi } }(x(t))$ such that
\begin{align*}
\tfrac{d}{dt} \Phi(x(t)) = D_{x(t)} \Phi(w(t)) = \left\langle \grad_g \Phi(x), w(t) \right\rangle_{g(x(t))} \, .
\end{align*}
Using \cref{lem:kras_normal} on \cpageref{lem:kras_normal}, we then have
\begin{align}\label{eq:grad_liederiv}
\tfrac{d}{dt} \Phi(x(t)) = - \left\langle -\grad_g \Phi(x(t)), w(t) \right\rangle_{g(x(t))}
\leq - \| w(t) \|^2_{g(x(t))} \leq 0 \, .
\end{align}
Thus $\Phi$ is non-increasing along Krasovskii solutions of \eqref{eq:proj_grad_intro} and hence $\mathcal{S}_\ell$ is invariant.
\end{proof}
\begin{lemma}\label{lem:strong_k_equil}
Every local minimizer of \eqref{eq:min_prob_intro} is a strong K-equilibrium of \eqref{eq:proj_grad_intro}.
\end{lemma}
\begin{proof}
By \cref{thm:main_exist}, there exists a Krasovskii solution $x: [0, T) \rightarrow \mathcal{X}$ of \eqref{eq:proj_grad_intro} starting at the local minimizer $x^\star \in \mathcal{X}$ of \eqref{eq:min_prob_intro}. Assume for the sake of contradiction that $x(0) = x^\star$ but $x(T) \neq x^\star$.
The sublevel set $\mathcal{S}_{\ell^\star}$ with $\ell^\star := \Phi(x^\star)$ is invariant and $x(t) \in \mathcal{S}_{\ell^\star}$ for all $t \in [0, T)$, by \cref{lem:invar_k_grad}. Since $x^\star \in \mathcal{X}$ is a local minimizer there exists a neighborhood $\mathcal{N} \subset \mathcal{X}$ of $x^\star$ such that $\Phi(x') \geq \Phi(x^\star)$ for all $x' \in \mathcal{N}$. If necessary, restrict the solution $x$ such that $x : [0, T) \rightarrow \mathcal{N}$. We have $\Phi(x(t)) = \Phi(x^\star)$ and $\tfrac{d}{dt} \Phi(x(t)) = 0$ for all $t \in [0, T)$, and therefore, for almost all $t \in [0, T)$, we have
\begin{align*}
0 = \tfrac{d}{dt} \Phi(x(t)) = - \left\langle -\grad_g \Phi(x(t)), \dot{x}(t) \right\rangle_{g(x)} \leq - \| \dot{x}(t) \|^2 \, ,
\end{align*}
where the inequality follows from \cref{lem:kras_normal}. Consequently, we have $\dot{x}(t) = 0$ for almost all $t \in [0, T)$ and thus $x(T) = \int_0^T \dot{x}(t) dt = x^\star$, establishing the contradiction.
\end{proof}
\Cref{lem:weak_k_equil,lem:strong_k_equil,thm:main_equiv} can be summarized as follows:
\begin{proposition}[connection between equilibria]\label{prop:proj_grad_cd}
Consider the projected gradient flow \eqref{eq:proj_grad_intro} and the problem \eqref{eq:min_prob_intro}. The following inclusions hold:
\begin{multline*}
\text{local minimizer} \, \, \subset \, \,
\text{strong K-eq.} \, \, \subset \, \,
\text{strong C-eq.} \, \, \\ \subset \, \,
\text{weak C-eq.} \, \, \subset \, \,
\text{critical pt.} \, \, \subset \, \,
\text{weak K-eq.}
\end{multline*}
If, in addition, $\mathcal{X}$ is Clarke regular and $g$ is continuous, then we have
\begin{align*}
\text{local minimizer} \, \, \subset \, \,
\text{strong eq.} \, \, \subset \, \,
\text{weak eq.} \, \, = \, \,
\text{critical pt.}
\end{align*}
\end{proposition}
If solutions of \eqref{eq:proj_grad_intro} are unique we do not distinguish between weak and strong equilibria and \cref{prop:proj_grad_cd} simplifies to equivalence of critical points and equilibria.
As an example of a critical point that is a weak (C-)equilibrium, but not a strong (C-)equilibrium we refer back to \cref{ex:prox_uniq} which illustrates this case for $\Phi(x) := x_1$. In that example, non-unique solutions may leave the critical point at arbitrary times, but the constant function is nevertheless a solution.
Unfortunately, convergence is generally guaranteed only to the set of weak K-equilibria as the following application of the invariance principle \cref{thm:invar} shows.
\begin{proposition}\label{prop:pgs_stab}
Consider \eqref{eq:proj_grad_intro} and let $\Phi: \mathbb{R}^n \rightarrow \mathbb{R}$ have compact sublevel sets on $\mathcal{X}$, i.e., for every $\ell \in \mathbb{R}$ the set $\mathcal{S}_\ell := \{ x\, | \, \Phi(x) \leq \ell \} \cap \mathcal{X}$ is compact. Then, \eqref{eq:proj_grad_intro} admits a complete Krasovskii solution $x: [0, \infty) \rightarrow \mathcal{X}$ for every initial condition $x(0) \in \mathcal{X}$. Furthermore, for some $r \in \Phi(\mathcal{S}_{\ell})$, $x$ converges to set of weak K-equilibrium points in $\Phi^{-1}(r) \cap \mathcal{X}$.
If, in addition, $\mathcal{X}$ is Clarke regular and $g$ is continuous, then convergence is to the set of critical points of \eqref{eq:min_prob_intro}.
\end{proposition}
\begin{proof}
We consider the Krasovskii regularization of \eqref{eq:proj_grad_intro} which is non-empty, closed, convex, locally bounded, and outer semicontinuous. As before, the compactness and invariance of the sublevel sets $\mathcal{S}_\ell$ of $\Phi$ on $\mathcal{X}$ implies that (Krasovskii) solutions cannot escape to the horizon in finite time and therefore must be complete.
Hence, \cref{thm:invar} guarantees convergence to the largest weakly invariant subset for which $\tfrac{d}{dt} \Phi(x(t)) = 0$ (and which lies on a level set of $\Phi$ relative to $\mathcal{X}$). Using \cref{eq:grad_liederiv}, we know that every limit point $\hat{x}$ of $x$ satisfies $0 \in \Kras{\tproj{\mathcal{X}}{g}{- \grad_g \Phi}}(\hat{x})$, i.e., $\hat{x}$ is a weak K-equilibrium of~\eqref{eq:proj_grad_intro}. Finally, under Clarke regularity of $\mathcal{X}$ and continuity of $g$, \cref{prop:proj_grad_cd} implies that every weak equilibrium is a critical point.
\end{proof}
Although convergence is generally only to weak equilibria, the following theorem, inspired by \cite{absilStableEquilibriumPoints2006}, establishes the connection between stability and optimality.
\begin{theorem}[stability \& optimality]\label{thm:stab}
Consider \eqref{eq:proj_grad_intro} and let $\Phi$ have compact sublevel sets on $\mathcal{X}$ as in \cref{prop:pgs_stab}. For some $r$, let $\hat{\mathcal{X}} \subset \{x \in \mathcal{X} \, | \, \Phi(x) = r \}$ be a connected set of weak K-equilibria. Then, the following statements hold:
\begin{enumerate}[label = (\roman*)]
\item\label{enum:stab1} If $\hat{\mathcal{X}}$ is locally asymptotically stable for~\eqref{eq:proj_grad_intro} then it is a \emph{strict set of minimizer of~\eqref{eq:min_prob_intro}}, i.e., $\Phi(y) > r$ for all $y \in \mathcal{N} \setminus \hat{\mathcal{X}}$ where $\mathcal{N}$ is a neighborhood of $\hat{\mathcal{X}}$.
\item\label{enum:stab2} If $\hat{\mathcal{X}}$ is a strict set of minimizers of \eqref{eq:min_prob_intro} then it is stable for~\eqref{eq:proj_grad_intro}.
\end{enumerate}
\end{theorem}
\begin{proof}
Recall from \cref{prop:pgs_stab} that the compactness of the sublevel sets of $\Phi$ guarantees the existence of complete solutions.
To show~\ref{enum:stab1}, let $\mathcal{V} \subset \mathcal{X}$ be a neighborhood of $\hat{\mathcal{X}}$ such that any solution $x: [0, \infty) \rightarrow \mathcal{X}$ of~\eqref{eq:proj_grad_intro} with $x(0) \in \mathcal{V}$ converges to $\hat{\mathcal{X}}$. Since $\Phi$ is $C^1$ and $x$ is absolutely continuous, $\Phi \circ x$ is absolutely continuous, and we may write
\begin{equation*}
\underset{t \rightarrow +\infty}{\lim}(\Phi \circ x)(t) = \Phi( x(0)) + \int\nolimits_{0}^{+\infty} D_x \Phi(\dot x(t)) dt = r \, .
\end{equation*}
Since $D_x \Phi(\dot{x}(t)) \leq 0$ holds for almost all $t \geq 0$, it follows that $\int_{0}^{+\infty} D_x \Phi(\dot x(t)) dt \leq 0$, and hence $r \leq \Phi(x(0))$ for all $t \geq 0$. Because this reasoning applies to all $x(0)$ in the region of attraction of $\hat{\mathcal{X}}$, it follows that $\hat{\mathcal{X}}$ is a local minimizer of $\Phi$.
To see that $\hat{\mathcal{X}}$ is a strict local minimizer, assume for the sake of contradiction that for some $\widetilde{x}$ in the region of attraction of $\hat{\mathcal{X}}$ it holds that $\Phi(\widetilde{x}) \leq r$.
Every solution $y$ to~\eqref{eq:proj_grad_intro} with $y(0) = \widetilde{x}$ nevertheless converges to $\hat{\mathcal{X}}$ by assumption. Therefore, it must hold that $\int_{0}^{+\infty} D_y \Phi(\dot{y}(t)) = 0$ and since $D_y \Phi(\dot{y}(t)) \leq 0$, it follows that $D_y \Phi(\dot{y}(t)) = 0$ for almost all $t \geq 0$.
But as a consequence of \cref{prop:pgs_stab}, all points in the $\omega$-limit set are weak K-equilibrium points, this holds in particular for $\widetilde{x}$ and therefore $\hat{\mathcal{X}}$ cannot be locally asymptotically stable.
For~\ref{enum:stab2}, assume that $\hat{\mathcal{X}} \neq \mathcal{X}$ (otherwise stability is trivial). Hence, consider a bounded (relative) neighborhood $\mathcal{W} \subset \mathcal{X}$ of $\hat{\mathcal{X}}$ in which $\hat{\mathcal{X}}$ is a strict local minimizer.
Next, we construct a neighborhood $\mathcal{V} \subset \mathcal{W}$ such that all trajectories starting in $\mathcal{V}$ remain in $\mathcal{W}$. Namely, let $\alpha$ be such that $r < \alpha < {\min}_{x \in \partial \mathcal{W}} \Phi(x)$ where $\partial \mathcal{W}$ is the boundary of~$\mathcal{W}$ relative to $\mathcal{X}$. Define $\mathcal{V} := \{ x \in \mathcal{W}\, | \, \Phi(x) \leq \alpha \} \subseteq \mathcal{W}$ which has a non-empty interior because $r < \alpha$. Since for any trajectory, we have $D_x \Phi(\dot x(\tau)) \leq 0$ we conclude that $\mathcal{V}$ is strongly invariant and remains in $\mathcal{V}$, thus establishing stability.
\end{proof}
It is not possible to draw stronger conclusions (e.g., that strict minimizers are always locally asymptotically stable) than in \cref{thm:stab}, unless additional assumptions are satisfied. A counter-example for an unconstrained gradient flow (which is, technically, a special case of a projected gradient flow) is documented in \cite{absilStableEquilibriumPoints2006}.
\begin{remark}\label{rem:grad_mfd}
The results of this section can be generalized to projected gradient flows on $C^1$ manifolds.
For instance, since any (equilibrium or critical) point under consideration can be locally mapped into $\mathbb{R}^n$, \cref{prop:proj_grad_cd} applies directly to projected gradient flows on manifolds. Similarly, the statements of \cref{thm:stab} about the relation between stability and optimality hold true on manifolds, especially if the set $\hat{\mathcal{X}}$ of weak K-equilibria is contained in a single chart domain.
On the other hand, because \cref{prop:pgs_stab} is a global statement, for it to generalize to manifolds an invariance principle akin to \cref{thm:invar} but for differential inclusion on manifolds is required. Such a generalization is plausible, but has not yet been documented.
\end{remark}
\begin{remark}
Projected gradient flows like \eqref{eq:proj_grad_intro} can be approximated (or implemented) in different ways. On one hand, standard numerical integration schemes can be adapted for (Euclidean) projected dynamical systems on convex domains as documented in \cite{nagurneyProjectedDynamicalSystems1996}, yielding well-known numerical optimization algorithms. In the non-Euclidean, non-convex setting, oblique projected gradient flows can be implemented, e.g., as in \cite{haberleNonconvexFeedbackOptimization2020} by linearizing constraints around the current state. This leads to algorithms similar to sequential quadratic programming schemes~\cite{nocedalNumericalOptimization2006}.
Another possibility are \emph{anti-windup approximations} \cite{hauswirthAntiWindupApproximationsOblique2020a,hauswirthDifferentiabilityProjectedTrajectories2020a} which serve to implement projected dynamical systems as the closed-loop behavior of feedback control loops that are subject to input saturation in \emph{feedback-based optimization} \cite{bernsteinOnlinePrimalDualMethods2019,colombinoOnlineOptimizationFeedback2019,hauswirthProjectedGradientDescent2016}.
\end{remark}
As a specific example of a projected gradient flow, we consider the metric $g$ to be the Hessian of the objective function, resulting in a \emph{projected Newton flow}:
\begin{example}\label{ex:newton}
Let $\mathcal{X} \subset \mathbb{R}^n$ be closed, and let $\Psi: \mathbb{R}^n \rightarrow \mathbb{R}$ be strongly convex and globally Lipschitz continuous and twice differentiable. In particular, the Hessian of $\Psi$ (denoted by $\nabla^2 \Psi$) is continuous and has lower and upper bounded eigenvalues. Therefore, we may use $\nabla^2 \Psi$ to define the weakly bounded metric $\left\langle u , v \right\rangle_{g(x)} := u^T \nabla^2 \Psi(x) v$ for $u, v \in T_x \mathbb{R}^n$. Hence, the projected gradient flow
\begin{align}\label{eq:newton_flow}
\dot{x} \in \tproj{\mathcal{X}}{g}{\left ( - \grad_g \Psi \right)}(x) \, , \quad x(0) = x_0 \in \mathcal{X}
\end{align}
where $\grad_g \Psi(x) = {(\nabla^2 \Psi(x))}^{-1} \nabla \Psi(x)^T$ is a constrained form of a \emph{Newton flow}, i.e., the continuous-time limit of the well-known \emph{Newton method} for optimization.
If $\mathcal{X}$ is convex, one can recover a a \emph{proximal Newton-type method} \cite{leeProximalNewtonTypeMethods2014} for solving \eqref{eq:min_prob_intro} as a projected forward Euler discretization of \eqref{eq:newton_flow} (possibly with variable step size).
\end{example}
\subsection{Connection to Subgradient Flows}
Assuming that $f$ is the gradient field of an objective function and $\mathcal{X}$ is Clarke regular, we can establish the connection between oblique projected gradients and subgradients. This fact is well-known for convex functions (and lesser known for regular functions~\cite{clarkeNonsmoothAnalysisControl1998,cortesDiscontinuousDynamicalSystems2008}) in the Euclidean metric, but, as we show next, generalizes to a variable metric.
Recall that $\Psi: \mathcal{V} \rightarrow \overline{\mathbb{R}}$, where $\mathcal{V} \subset \mathbb{R}^n$ is open and $\overline{\mathbb{R}} := \mathbb{R} \cup \{ \infty \}$, is \emph{(subdifferentially) regular} if its epigraph $\epi \Psi := \{ (x, y) \, | \, x\in \mathcal{V}, \, y \geq \Psi(x) \}$ is non-empty and Clarke regular.
\begin{definition}
Given a metric $g$ on an open set $\mathcal{V} \subset \mathbb{R}^n$ and a regular function $\Psi: \mathcal{V} \rightarrow \overline{\mathbb{R}}$, $v$ is a \emph{subgradient of $\Psi$ with respect to $g$ at $x$}, denoted by $v \in \partial\Psi(x)$, if
\begin{align*}
\underset{y \rightarrow x}{\lim \inf} \, \tfrac{\Psi(y) - \Psi(x) - \left\langle v, y - x \right\rangle_{g(x)}}{\| y - x \|} \geq 0 \, .
\end{align*}
\end{definition}
Namely, if $\Psi$ is differentiable at $x$, then $\partial \Psi(x) = \{ \grad_g \Psi(x) \}$. Further, if $\mathcal{X} \subset \mathcal{V}$ is Clarke regular and $I_\mathcal{X}: \mathcal{V} \rightarrow \overline{\mathbb{R}}$ denotes its indicator function, then $\partial I_\mathcal{X}(x) = N^g_x \mathcal{X}$.
The next result is a direct combination of~\cite[Ex.~8.14]{rockafellarVariationalAnalysis2009} and~\cite[Cor.~10.9]{rockafellarVariationalAnalysis2009}.
\begin{proposition}
Let $\hat \Psi := \Psi + I_\mathcal{X}$ where $\Psi: \mathcal{V} \rightarrow \mathbb{R}$ is a $C^1$ function and $I_\mathcal{X}$ is the indicator function of a Clarke regular set $\mathcal{X} \subset \mathcal{V}$ where $\mathcal{V} \subset \mathbb{R}^n$ is open. Then, for all $x \in \mathcal{X}$ one has
\begin{align*}
\partial \hat \Psi(x) = \grad_g \Psi(x) + N^g_x \mathcal{X} \, .
\end{align*}
\end{proposition}
It follows immediately from \cref{cor:equiv_normal} that under the appropriate assumptions trajectories of projected gradient flows are also solutions to subgradient flows.
\begin{corollary}[equivalence with subgradient flows]\label{cor:subgrad_equiv}
Let $\mathcal{X}$ be Clarke regular, let $g$ be a continuous metric on $\mathcal{X}$, and let $\Psi$ be a $C^1$ objective function on an open neighborhood of $\mathcal{X}$. Then, for any $x_0 \in \mathcal{X}$ there exists a Carath\'eodory solution $x: [0, T) \rightarrow \mathcal{X}$ to the subgradient flow
\begin{align*}
\dot x \in - \partial( \Psi + I_\mathcal{X})(x)\,, \quad x(0) \in \mathcal{X} \, .
\end{align*}
Furthermore, $x$ is a solution if and only if it is a Carath\'eodory (and Krasovskii) solution to the projected gradient flow \eqref{eq:proj_grad_intro}.
\end{corollary}
In summary, we have seen that projected gradient flows are well-defined in very general settings if one considers Krasovskii solutions. The convergence behavior is more fine-grained than for special cases (e.g., convex optimization problems) since the notion of equilibrium depends on the definition of the solution concept. Further, projected gradient flows exhibit the same connection between stability and optimality of equilibria as unconstrained gradient flows. Finally, \emph{oblique} projected gradient flows on Clarke regular sets can be interpreted subgradient flows of a composite function that is the sum of a smooth objective and the indicator function of the feasible set.
\section{Conclusion}\label{sec:conclusion}
\begin{table}[bt]
\makegapedcells
\centering
{\footnotesize
\begin{tabular}{lrrrrl}
& $f$ & $g$ & $\mathcal{X}$ & $\mathcal{M}$ & \\
\toprule
\makecell[cl]{Local Existence of Krasovskii \\ solutions} &
LB &
LWB & loc.\ compact & $C^1$ &
\makecell[cl]{Thm.~\ref{thm:main_exist} \\ Thm.~\ref{thm:main_mfd}} \\ \midrule
\makecell[cl]{Global Existence of Krasovskii \\ solutions (multiple possibilities)} &
$C^0$ & $C^0$ & compact & $C^1$ &
Cor.~\ref{cor:max_sol} \\ \midrule
\makecell[cl]{Equivalence of Krasovskii \\ and Carath\'eodory solutions} &
$C^0$ & $C^0$ & Clarke regular & $C^1$ &
\makecell[cl]{Thm.~\ref{thm:main_equiv} \\ Thm.~\ref{thm:main_mfd}} \\ \midrule
\makecell[cl]{Equivalence of projected gradient \\ and subgradient flows} &
$C^0$ & $C^0$ & Clarke regular & $C^1$ & Cor.~\ref{cor:subgrad_equiv} \\ \midrule
\makecell[cl]{Uniqueness of (Krasovskii \& \\ Carath\'eodory) solutions} &
$C^{0,1}$ & $C^{0,1}$ & prox-regular & $C^{1,1}$ &
\makecell[cl]{Thm.~\ref{thm:main_uniq} \\ Thm.~\ref{thm:main_mfd_uniq}}\\ \bottomrule
\end{tabular}
}
\caption{Summary of results: regularity requirements for projected dynamical systems for a vector field $f$, metric $g$, feasible domain $\mathcal{X}$ and regularity of the manifold $\mathcal{M}$. (LB\@: locally bounded; LWB\@: locally weakly bounded)}\label{tab:summary}
\end{table}
We have provided an extensive study of projected dynamical systems on irregular subset on manifolds, including the model of oblique projection directions. We have carved out sharp regularity requirements on the feasible domain, vector field, metric and differentiable structure that are required for the existence, uniqueness and other properties of solution trajectories. \cref{tab:summary} summarizes these results. In the process, we have established auxiliary findings, such as the fact that prox-regularity is an intrinsic property of subset of $C^{1,1}$ manifolds and independent of the choice of Riemannian metric.
While we believe these results are of general interest in the context of discontinuous dynamical systems, they particularly provide a solid foundation for the study of continuous-time constrained optimization algorithms for nonlinear, nonconvex problems. To illustrate this point, we have included a study the stability and convergence of Krasovskii solutions to projected gradient descent---arguably the most prototypical continuous-time constrained optimization algorithm.
\section*{Acknowledgments}
We would like to thank Gabriela Hug and Matthias Rungger for their support in putting together this paper.
\appendix
\section{Technical definitions and results}\label{app:basic_notions}
\unless\ifARXIV
The following technical lemmas are required for different results in the current paper. More background material can be found in the appendix of~\cite{hauswirthProjectedDynamicalSystems2018a}.
\fi
\begin{lemma}\label{lem:tgt_deriv} Given a set $\mathcal{X} \subset \mathbb{R}^n$, for any absolutely continuous function $x: [0, T) \rightarrow \mathcal{X}$ with $T>0$ it holds that $\dot x(t) \in T_{x(t)} \mathcal{X} \cap -T_{x(t)} \mathcal{X}$ almost everywhere on $[0, T)$, where $-T_{x(t)} := \{ v | -v \in T_{x(t)} \}$.
\end{lemma}
\begin{proof} Let $t \in [0, T)$ be such that $\dot x(t)$ exists. This implies that by definition
\begin{equation*}
\dot x(t) = \underset{\tau \rightarrow 0^+}{\lim} \tfrac{x(t+ \tau) - x(t)}{\tau} = \underset{\tau \rightarrow 0^+}{\lim} \tfrac{x(t) - x(t-\tau) }{\tau},
\quad
\end{equation*}
Thus, by choosing any sequence $\tau_k \rightarrow 0$ with $\tau_k > 0$, the sequence $\frac{x(t+ \tau_k) - x(t)}{\tau_k}$ converges to a tangent vector and $\frac{ - x(t-\tau_k) + x(t)}{\tau_k}$ converges to a vector in $-T_{x(t)} \mathcal{X}$ by definition of $T_{x(t)} \mathcal{X}$ and the fact that $x(t) \in \mathcal{X}$ for all $t \in [0, T)$.
\end{proof}
The following is a local version of~\cite[Lem.~1.30]{peypouquetConvexOptimizationNormed2015}.
\begin{lemma}[Descent Lemma]\label{lem:c11_lipschitz}
Let $\Phi: V \rightarrow \mathbb{R}$ be a $C^{1,1}$ map where $V \subset \mathbb{R}^n$ is open. Given $x \in V$ there exists $L > 0$ such that for all $z,y \in V$ in a neighborhood of $x$ it holds that
\begin{align*}
| \Phi(z) - \Phi(y) - D_y \Phi(z - y) | \leq L \| z - y \|^2
\end{align*}
\end{lemma}
\ifARXIV
For a comprehensive treatment of the following definitions and results see~\cite{rockafellarVariationalAnalysis2009,aubinDifferentialInclusionsSetValued1984,peypouquetConvexOptimizationNormed2015,hiriart-urrutyFundamentalsConvexAnalysis2012}.
Given a sequence $\{x_{k}\}$ and a set $\mathcal X$, the notation $x_k \overset{sub}{\underset{\mathcal{X}} \longrightarrow} x$ denotes the existence of a subsequence $\{x_{k'}\}$ that converges to $x$ and $x_{k'} \in \mathcal{X}$ for all $k'$. Similarly, $x_k \overset{ev}{\underset{\mathcal{X}} \longrightarrow} x$ implies that $x_k \in \mathcal{X}$ holds \emph{eventually}, i.e., for all $k$ larger than some $K$, and that $\{x_k\}$ converges to $x$.
Given a sequence of sets $\{C_k\}$ in $\mathbb{R}^n$, its \emph{outer limit} and \emph{inner limit} are given as
\begin{align*}
\underset{k \rightarrow \infty}{\lim \sup} \, C_k
& := \left\lbrace x \, \middle|\,
\exists \{x_i\}: x_i \underset{C_i}{\overset{sub}{\longrightarrow}} x
\right\rbrace \quad \mbox{and} \quad
\underset{k \rightarrow \infty}{\lim \inf} \, C_k
& := \left\lbrace x \, \middle|\,
\exists \{x_i\}: x_i \underset{C_i}{\overset{ev}{\longrightarrow}} x
\right\rbrace \
\end{align*}
respectively. As a pedagogical example to distinguish between inner and outer limits, consider an alternating sequence of sets given by
$C_{2m} := A$ and $C_{2m+1} := B$. Then, we have ${\lim \sup}_{k \rightarrow \infty} \, C_k = A \cup B$ and $ {\lim \inf}_{k \rightarrow \infty} \, C_k = A \cap B$. On the one hand any constant sequence $\{x_k\}$ with $x_k = c \in A \cap B$ for all $k$ satisfies the requirement such that $c \in {\lim \inf}_{k \rightarrow \infty} \, C_k$. On the other hand, any sequence $\{x_k\}$ with $x_{2m} = a \in A$ for $m \in \mathbb{N}$ has a trivial (constant) subsequence converging to $a \in A$ and hence $a \in {\lim \sup}_{k \rightarrow \infty} \, C_k $. The following result relates the image of an outer (inner) limit to the outer (inner) limit of images of a map $f$.
\fi
\begin{lemma}{\cite[Thm.~4.26]{rockafellarVariationalAnalysis2009}}\label{lem:continuity_set}
For a sequence of sets $\{C_k \}$ in $V \subset \mathbb{R}^n$ and a continuous map $f: V \rightarrow \mathbb{R}^m$, one has
\begin{equation*}
f\left(\underset{k \rightarrow \infty}{\lim \inf} \, C_k \right)
\subset \underset{k \rightarrow \infty}{\lim \inf} \, f(C_k) \, , \qquad f\left(\underset{k \rightarrow \infty}{\lim \sup} \, C_k \right)
\subset \underset{k \rightarrow \infty}{\lim \sup} \, f(C_k) \, .
\end{equation*}
\end{lemma}
For a set-valued map $F: V \rightrightarrows W$ with $V \subset \mathbb{R}^n$ and $W \subset \mathbb{R}^m$ its \emph{outer limit} and \emph{inner limit} at $x$ are defined respectively as
\begin{equation*}
\underset{y \rightarrow x}{\lim \sup} \, F(y) := \bigcup_{x_k \underset{V}{\longrightarrow} x} \underset{k \rightarrow \infty}{\lim \sup} \, F(x_k) \quad \mbox{and} \quad
\underset{y \rightarrow x}{\lim \inf} \, F(y) := \bigcap_{x_k \underset{V}{\longrightarrow} x} \underset{k \rightarrow \infty}{\lim \inf} \, F(x_k) \, .
\end{equation*}
\ifARXIV
A set-valued map $F : V \rightrightarrows \mathbb{R}^m$ for $V \subset \mathbb{R}^n$ is \emph{outer semicontinuous (osc) at $x \in V$} if ${\lim \sup}_{y \rightarrow x} \, F(y) \subset F(x)$~\cite[Def.~5.4]{rockafellarVariationalAnalysis2009}.
The map $F$ is \emph{upper semicontinuous (usc) at $x$} if for any open neighborhood $A \subset V$ of $ F(x)$ there exists a neighborhood $B \subset V$ of $x$ such that for all $ y\in B$ one has $F(y) \subset A$~\cite[Def.~2.1.2]{aubinViabilityTheory1991}. The map $F$ is \emph{outer (upper) semi-continuous} if and only if it is osc (usc) at every $x \in V$.
For locally bounded, closed set-valued maps outer and upper semicontinuity are equivalent.
\begin{lemma}{\cite[Lem.~5.15]{goebelHybridDynamicalSystems2012}}\label{lem:outer_sem_closedgraph}
Let $F : V \rightrightarrows \mathbb{R}^m$ be closed and locally bounded for $V \subset \mathbb{R}^n$. Then, $F$ is osc at $x \in V$ if and only if it is usc at $x$.
Furthermore, $F$ is osc/usc at $x$ if and only if
$\gph F := \left\lbrace (x,v) \, \middle|\, x\in V, v \in F(x) \right\rbrace$
locally closed at $x$.
\end{lemma}
The next result states that upper semicontinuity is preserved by convexification.
\begin{lemma}{\cite[Lem.~16, \S 5]{filippovDifferentialEquationsDiscontinuous1988}}\label{lem:filippov_convex}
Given a set-valued map $F: V \rightrightarrows \mathbb{R}^m$ with $V \subset \mathbb{R}^n$, if $F$ is usc and $F(x)$ is non-empty and compact for each $x \in V$, then the map $\co F: V \rightrightarrows \mathbb{R}^m$ defined as $x \mapsto \co F(x)$ is usc.
\end{lemma}
The following result is a generalization of~\cite[Prop.~6.5]{rockafellarVariationalAnalysis2009} to the case of a continuous metric instead of the standard Euclidean metric:
\begin{lemma}\label{lem:normal_outer_semi}
Let $\mathcal{X}$ be Clarke regular. If the metric $g$ on $\mathcal{X}$ is continuous, then the set-valued map $\mathcal{X} \mapsto N^g_x \mathcal{X}$ is outer semi-continuous.
\end{lemma}
\begin{proof} Consider any two sequences $x_k \rightarrow x$ with $x_k \in \mathcal{X}$ and $\eta_k \rightarrow \eta$ with $\eta_k \in N^g_{x_k} \mathcal{X}$. To complete the proof we need to show that $\eta \in N^g_x \mathcal{X}$.
By definition of $N^g_{x_k} \mathcal{X}$ we have $\left\langle v, \eta_k \right\rangle_{g(x_k)} \leq 0$ for all $v \in T^C_x \mathcal{X}$. Furthermore, by continuity of $g$ we have $\left\langle v, \eta \right\rangle_{g(x)} \leq 0$ for all $v \in {\lim \sup}_{x_k \rightarrow x} \, T_{x_k}^C \mathcal{X}$. (Namely, we must have $\left\langle v_k, \eta_k \right\rangle_{g(x_k)} \leq 0$ for every sequence $v_k \rightarrow v$ with $v_k \in T^C_{x_k} \mathcal{X}$, hence the use of $\lim \sup$.) By definition of the Clarke tangent cone, we note that $\left\langle v, \eta \right\rangle_{g(x)} \leq 0$ holds for all
\begin{equation*}
v \in T_x^C \mathcal{X} = \underset{x_k \rightarrow x}{\lim \inf} \, T_{x_k} \mathcal{X} =
\underset{x_k \rightarrow x}{\lim \inf} \, T_{x_k}^C \mathcal{X} \subset
\underset{x_k \rightarrow x}{\lim \sup} \, T_{x_k}^C \mathcal{X} \, ,
\end{equation*}
and therefore $\eta \in N^g_x \mathcal{X}$.
\end{proof}
The following general existence and viability theorem goes back to~\cite{haddadMonotoneTrajectoriesDifferential1981}. Similar results can also be found in~\cite{aubinViabilityTheory1991,clarkeOptimizationNonsmoothAnalysis1990,goebelHybridDynamicalSystems2012}.
\begin{proposition}[{\cite[Cor.~1.1, Rem 3]{haddadMonotoneTrajectoriesDifferential1981}}]\label{prop:haddad}
Let $\mathcal{X}$ be a locally compact subset of $\mathbb{R}^n$ and $F: \mathcal{X} \rightrightarrows \mathbb{R}^n$ an usc, non-empty, convex and compact set-valued map. Then, for any $x_0 \in \mathcal{X}$ there exists $T > 0$ and a Lipschitz continuous function $x: [0, T) \rightarrow \mathcal{X}$ such that $x(0) = x_0$ and $\dot x(t) \in F(x(t))$ almost everywhere in [0, T) if and only if the condition $F(x) \cap T_x \mathcal{X} \neq \emptyset$ holds for all $x \in \mathcal{X}$.
Furthermore, for $r>0$ such that $U_r := \{ x \in \mathcal{X} \, | \, \| x - x_0 \| \leq r \}$ is closed and $L = \max_{y \in U_r} \| F(y) \|$ exists, the solution is Lipschitz and exists for $T > r/ L$.
\end{proposition}
\fi
\end{document} |
\begin{document}
\title{Breaking a quantum key distribution system through a timing side channel}
\author{Ant\'{\i}a Lamas-Linares and Christian Kurtsiefer}
\address{Department of Physics, National University of
Singapore\\2 Science Drive 3, Singapore 117542}
\email{antia\[email protected]}
\homepage{http://qoptics.quantumlah.org/lah/}
\begin{abstract}
The security of quantum key distribution relies on the validity of
quantum mechanics as a description of nature and on the
non-existence of leaky degrees of freedom in the practical
implementations. We experimentally demonstrate how, in some
implementations, timing information revealed during public
discussion between the communicating parties can be used by an
eavesdropper to undetectably access a significant portion of the
``secret'' key.
\end{abstract}
\ocis{(030.5260) Coherence and statistical optics: photon counting; (270.5290) Quantum optics: photon statistics; (999.9999) Quantum cryptography}
\begin{thebibliography}{10}
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\expandafter\ifx\csname url\endcsname\relax
\def\url#1{\texttt{#1}}\fi
\expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL
}\fi \providecommand{\eprint}[2][]{\url{#2}}
\bibitem{dusek:06}
M.~D{\^u}sek, N.~L{\"u}tkenhaus, and M.~Hendrych, \enquote{Quantum
Cryptography,} Prog. in Opt. \textbf{49}, 381--454 (2006).
\bibitem{bennett:84}
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\section{Introduction}
Theoretical proofs of the security of quantum key distribution
(QKD), are a well developed subfield in quantum communication
research (see~\cite{dusek:06}), both in highly
idealized~\cite{bennett:84, ekert:91} and more realistic
scenarios~\cite{gisin:07}. By construction, these proofs assume that
the legitimate parties measurement results are isolated from the
environment and thus from an eavesdropper. Comparatively little work
has been done studying the possible physical side channels
associated with particularities of the physical devices
used~\cite{kurtsiefer:02, kurtsiefer:01a} or possible attacks based
on the external manipulation of the expected response of the
apparatus~\cite{makarov:05, makarov:06, gisin:06, zhao:07}.
All photon-counting implementations of QKD identify a signal photon
from background by measurement of the arrival time at detectors.
In an ideal scenario, there can be no correlation between the measurement
outcome on the quantum variable (e.g. polarization in the original
BB84 proposal), and this publicly exchanged timing information.
However, in a recent entanglement based QKD implementation, a pulsed
down-conversion source provided photon pairs with a well-defined
timing signature~\cite{ursin:06}. For photon identification, timing
information was recorded with a high resolution and communicated to
the other side (similar scheme as
in~\cite{kurtsiefer:02,poppe:04,resch:05,marcikic:06}). We show that
there may be an exploitable correlation between the exchanged timing
information and the measurement results in the quantum channel.
\section{Time response analysis}
A configuration implementing the detection scheme just described is
shown in Fig.~\ref{fig:topology}. An incoming photon is randomly
directed by a beam splitter towards two possible polarizing beam
splitters each of which performs a measurement in one basis ($H/V$
or $45^\circ/-45^\circ$). Finally, there are four possible outcomes
of the measurement (two bits of information) of which one bit will
be made public. The remaining bit is the raw material for generating
the secret key and must be kept secret. Although the optical
distance from the entrance of the module to the four detectors
differs by less than 1\,mm, there is a measurable difference in the
timing of the electronic signal from the different possibilities.
\begin{figure}
\caption{A typical configuration of photocounting detectors for
quantum key distribution. A beam splitter (BS), polarizing beam
splitters (PBS) and a half wave plate ($\lambda/2$), divert incoming
photons onto a set of detectors, which generate a macroscopic timing
signal. This timing information and e.g. a projection basis is
revealed publicly, while information on which detector out of two
absorbed a photon is the secret used to subsequently generate a
key.}
\label{fig:topology}
\end{figure}
In order to determine the timing differences between the four single
photon detectors, we used an attenuated fraction of a pulse train
emitted by a Ti:Sapphire femtosecond laser as a light source (see
Fig.~\ref{fig:expsetup}). Single photon detectors consisted of
Silicon Avalanche Photodiodes (type C30902S, Perkin-Elmer), operated
in a passively quenched configuration. The breakdown of the
avalanche region was converted into a digital pulse signal by a high
speed comparator, registering a voltage drop over the measurement
resistor $R_M=100\,\Omega$ of 150\,mV, which has to be compared to a
maximal voltage drop across $R_M$ of about 700\,mV. The distribution
of peak amplitudes for the breakdown signal exhibits a spread below
10\% for photodetector event rates of 5000--6000\,s$^{-1}$, and the
pulse duration before the comparator is on the order of 2\,ns.
We obtained the timing distribution with an oscilloscope sampling at
20\,GS/s, by interpolating the time when the comparator output
passed through the 50\% value between the two logical levels. Time
reference is a trigger signal supplied by a MSM Schottky reference
photodiode (G7096-03, Hamamatsu) looking at another fraction of the
optical pulse train. The timing jitter of 10\,ps (FWHM) we observe
between consecutive pulses from the mode-locked laser gives an upper
bound for the total timing uncertainty.
\begin{figure}
\caption{Experimental set-up to characterize the timing jitter of a
single
photon detector. A train of ultrashort light pulses from a mode-locked
Ti:Sapphire laser is sent with strong attenuation into
a passively quenched Si avalanche photodiode (APD). A histogram of timing differences (TDH) with respect to the signal of a
trigger photodidode (TD) is recorded.
}
\label{fig:expsetup}
\end{figure}
The resulting timing histograms of the different detectors
(Fig.~\ref{fig:histograms}) show a clearly different centroid
location with respect to the trigger pulse. We model the observed
distribution with a convolution product of an exponential decay and
a Gaussian distribution,
\begin{equation}\label{eq:modelfunction}
d_i(t)={1\over2\tau_e} e^{-{\tau_G^2\over4\tau_e^2}}\cdot
e^{t-t_0\over\tau_e}{\rm erfc}\left({t-t_0\over\tau_G}\right)
\end{equation}
\begin{figure}
\caption{Photoevent timing histograms for the four detectors
involved in a
quantum key distribution receiver. While the general shape of the
distributions is similar, there is a distinction in the response time
visible for detectors 1 and 4 with respect to detectors 2 and 3, which, if
not compensated, can be exploited by an eavesdropper to gain knowledge
about the measurement result. The solid lines represent a fit to the model
in equation \ref{eq:modelfunction}
\label{fig:histograms}
\end{figure}
The fit values for the temporal offset $t_0$ and the exponential and
Gaussian decay constants $\tau_e, \tau_G$ for the four detectors
$i=1,2,3,4$ are summarized in table~\ref{tab:fitresults}. While the
difference between $\tau_e$ and $\tau_G$ differ maximally by 38\,ps
and 20\,ps, respectively, the time offsets $t_0$ can differ up to
240\,ps between detectors 2 and 4. The physical origin of this
difference could be attributed to differences in the electrical
delays for the different detectors on the order of a few cm on the
circuit board layouts, and to different absolute pulse heights of
the detected breakdown currents due to different parasitic
capacities for the different diodes.
\begin{table}
\caption{Extracted model parameters for the time distributions of
the different photodetectors with their statistical uncertainties.}
\begin{center}
\begin{tabular}{c||c|c|c}
Detector $i$&$t_0$\,(ps)&$\tau_e$\,(ps)&$\tau_G$\,(ps)\\
\hline\hline 1&$1138\pm7$&$395\pm7$&$288\pm4$\\ \hline
2&$1356\pm6$&$433\pm7$&$279\pm4$\\ \hline
3&$1248\pm4$&$409\pm5$&$292\pm3$\\ \hline
4&$1117\pm7$&$415\pm7$&$302\pm4$
\end{tabular}
\label{tab:fitresults}
\end{center}
\end{table}
\section{Information extraction}
An eavesdropper can exploit these differences in the detector
responses $d_i$, and obtain information on the secret key by
listening in the publicly communicated detection times. The
knowledge in principle attainable by the eavesdropper is quantified
by the mutual information $I(X;T)$ between the time distribution of
detector clicks (publicly revealed) and the bits composing the
secret key:
\begin{equation}
I(X;T)=H(X)+H(T)-H(X,T) \label{eq:mutinfo}
\end{equation}
There, $X$ represents the distribution of logical 0 and 1, and $T$
is the distribution of detection times. The entropies and joint
entropies of the distributions are given by
\begin{eqnarray*}
H(T)&=&-\int \bar{d}(t)\log_2[\bar{d}(t)]\,{\rm d}t
\\
H(X)&=&-\sum_x p^0(x)\log_2[p^0(x)]\\
H(X,T)&=&-\sum_x\int p(x,t)\log_2[p(x,t)]\,{\rm d}t\\
&=&-\sum_x\int p^0(x)d_x(t)\log_2[p^0(x)d_x(t)]\,{\rm d}t
\end{eqnarray*}
where $\bar{d}(t)=\sum_xp^0(x)d_x(t)$ is the probability of a click
occurring at time $t$ for the ensemble of detectors, and $d_x(t)$
the probabilities of a click at a particular time $t$ for a detector
corresponding to logical value $x\in \{0,1\}$. In most protocols,
the prior distribution of logical values is balanced such that
$p^0(0)=p^0(1)=0.5$.
If we bin the detector results in the manner most favorable to the
eavesdropper by assigning detectors (1,2) to one basis, (3,4) to the
other basis, and taking detectors groups (1, 3) and (2, 4) to
represent 0 and 1, the average extractable information is
$3.8\pm0.38\%$.
\begin{figure}
\caption{Eavesdropper's information on the secret bit as a function
of delay
$\Delta t_0$ between detector timing distributions with identical
shapes. The three curves represent different levels of
discretization of the data. The top curve corresponds to the
continuous distribution and the subsequent are for 0.5 ns and
1 ns time bins. As expected, with an increasing time bin there is less information available for the eavesdropper. For
$\Delta t_0$ as small as 0.5\,ns the eavesdropper will gain
access to more than a quarter of the ``secret key''.}
\label{fig:timeshift}
\end{figure}
It is worth considering in detail how the distinguishability of the
distributions comes about, and how quickly the eavesdropper
knowledge of the key changes. Figure~\ref{fig:timeshift} shows the
eavesdropper's knowledge of the secret bit for two distributions
$d_0(t),d_1(t)$ with the same $\tau_e=400$\,ps, $\tau_G=290\,$ps,
but with different relative delays $\Delta t_0$. Detectors that are
uncompensated by as little as $\Delta t_0=500\,{\rm ps}$ will give
the eavesdropper access to more than 25\% of the ``secret'' key.
Since a small relative delay is not visible in the usual
experimental setups which employ coincidence windows between 1 and
20\,ns~\cite{poppe:04,resch:05,peng:05,marcikic:06}, it requires an
additional effort to make sure that this leakage channel is closed.
The solution to this particular side channel is not complex, the
timing information should be characterized and the delays equalized,
randomized or the precision truncated such that the potential
information leakage is below a certain threshold. Quantum
cryptography protocols can then deal with this in the same way they
deal with errors, by applying an appropriate amount of privacy
amplification~\cite{bennett:88}. In every real experiment the timing
information is communicated with a finite precision that could be
adjusted for this purpose. Figure 4 shows the effect of discretizing
the time information into 0.5 ns and 1 ns time bins (a typical
experimental value of $\approx150$~ps gives a negligible difference
with the continuous distribution). As expected, the eavesdropper's
information is reduced as the bin width increases. Somewhat
counterintuitively there is still a strong leakage even at bin sizes
comparable to the width of the distribution $d(t)$; furthermore
there is a penalty in the form of increased background. For our
particular device, the main distinguishing feature is the time
offset. If this is compensated for (i.e. made identical for all
detectors), and applying the same procedure as before to obtain the
leakage to an eavesdropper given the probability distributions, we
find the leakage to be around 0.3\%.
It is reasonable to ask whether this problem affects ``prepare and
measure'' protocols as well. A typical BB84 QKD system based on weak
coherent pulses has a synchronous operation, and the detector side
will locally determine whether the detected event falls in the right
part of the timing frame to be counted as genuine. This binary
decision will not provide information to the eavesdropper from the
detector side. However, the problem has just been displaced from the
detectors to the emitters: if the states to be sent are prepared by
different physical devices, their temporal response needs to be
charaterized, and the possible information leakage should be
evaluated with a similar analysis.
\section{Conclusions}
Quantum cryptography is slowly leaving the purely academic
environment and starting to appear in commercial
products~\cite{magiqidq}. The theoretical aspects of its security
are a very active research area but comparatively little has been
done in terms of scrutinizing the practical systems. However, there
is increasing interest in looking at the side channels arising from
the physical realization in practical systems (see recent work by
Zhao et al.~\cite{zhao:07} for an attack on a commercial product
based on a proposal by Makarov et al.~\cite{makarov:06}). We have
shown here how some of the information publicly revealed by the
communicating parties in reasonable mature implementations, may lead
to a large proportion of the key becoming insecure.
\end{document} |
\begin{document}
\date{}
\title{ \huge Greening Multi-Tenant Data Center Demand Response}
\author{Niangjun Chen, Xiaoqi Ren, Shaolei Ren, Adam Wierman}
\maketitle
\abstract
Data centers have emerged as promising resources for demand response, particularly for emergency demand response (EDR\xspace), which saves the power grid from incurring blackouts during emergency situations. However, currently, data centers typically participate in EDR\xspace by turning on backup (diesel) generators, which
is both expensive and environmentally unfriendly.
In this paper, we focus on ``greening'' demand response in multi-tenant data centers, i.e., colocation data centers, by designing a pricing mechanism through which the data center operator can efficiently extract load reductions from tenants during emergency periods to fulfill energy reduction requirement for EDR\xspace. In particular, we propose a pricing mechanism for both mandatory and voluntary EDR\xspace programs, {\sf ColoEDR}\xspace, that is based on parameterized supply function bidding and provides provably near-optimal efficiency guarantees, both when tenants are price-taking and when they are price-anticipating. In addition to analytic results, we extend the literature on supply function mechanism design, and evaluate {\sf ColoEDR}\xspace using trace-based simulation studies. These validate the efficiency analysis and conclude that the pricing mechanism is both beneficial to the environment and to the data center operator (by decreasing the need for backup diesel generation), while also aiding tenants (by providing payments for load reductions).
\section{Introduction}\label{sec:introduction}
Data centers have emerged as a promising demand response opportunity. However, data center demand response today is not environmentally friendly since data centers typically participate by turning on backup (diesel) generators. In this paper, we focus on designing a pricing mechanism for a crucial class of data centers for demand response -- multi-tenant colocation data centers -- that allows the data center operator to encourage load shedding among tenants in response to demand response signals; thus greening data center demand response by reducing the need for use of backup (diesel) generation.
\textbf{Data center demand response.}
Power-hungry data centers have been quickly expanding in both number and scale to support the exploding IT demand, consuming 91 billion kilowatt-hour (kWh) electricity in 2013 in the U.S. alone \cite{NRDC_Colocation_2014}.
While traditionally viewed purely as a negative, the massive energy usage of data centers has recently begun to be recognized as an opportunity. In particular, because the energy usage of data centers tends to be flexible, they are promising candidates for \emph{demand response}, which is a crucial tool for improving grid reliability and incorporating renewable energy into the power grid. From the grid operator's perspective, a data center's flexible power demand serves as a valuable energy buffer, helping balance grid power's supply and demand at runtime \cite{AdamWierman_DataCenterDemandResponse_Survey_IGCC_2014}.
To this point, data center is a promising, but still largely under-utilized opportunity for demand response. However, this is quickly changing as data centers play an increasing role in emergency demand response (EDR\xspace) programs. EDR\xspace is the most widely-adopted demand response program in the U.S.,
representing 87\% of demand reduction capabilities across all reliability regions \cite{EDR_Market_Overview}.
Specifically,
during emergency events (e.g., extreme weather or natural disasters),
EDR\xspace coordinates many large energy consumers, including data
centers, to shed their power loads, serving as the last protection
against cascading blackouts that could potentially result in economic losses of billions of dollars \cite{pjm_emergency_demand_response_Performance,Demand_response_Evidence_Blackouts_Canada_US}.
The U.S. EPA has identified data centers as critical resources for EDR\xspace \cite{Demand_response_US_EPA_EnerNOC}, which was
attested to by the following example: on July 22,
2011, hundreds of data centers participated in EDR\xspace by cutting their electricity
usage before a large-scale blackout would have occurred \cite{Demand_response_Evidence_Blackouts_Canada_US}.
While data centers are increasingly contributing to EDR\xspace, they typically participate by turning on
their on-site backup diesel generators, which is neither cost effective
nor environmentally friendly. For example, in California (a major
data center market), a standby diesel generator often produces 50-60 times more nitrogen oxides (a smog-forming pollutant) compared to a typical power plant for each kWh of electricity, and diesel particulate represents
the state's most significant toxic air pollution problem \cite{Diesel_Pollution_Reference}.
In addition, relying on diesel generation for EDR\xspace
presents emerging challenges which, if left unaddressed, may forfeit data center's EDR\xspace capability.
{First}, as EDR\xspace is becoming more frequent, the
current financial compensation offered by power grid to data centers (for committed
energy reduction during EDR\xspace) may not be enough to cover
the growing cost of diesel generation.
{Second}, data center operators are aggressively cutting the huge capital
investment in their power infrastructure (e.g., 10-15\$/watt \cite{Hoelzle_datacenter_book_2013,LimKansalLiu_ATC2011}),
by down-sizing the capacity
of diesel generator and uninterrupted power supply (UPS) system \cite{Wang:2014:UBP:2541940.2541966}.
Such under-provisioning of diesel generator may compromise data center's EDR\xspace capability.
Therefore, to retain and encourage data center's participation in EDR\xspace without contaminating the environment, it is critical and urgent that data centers seek alternative
ways to shed load.
Consequently, modulating
server energy for green EDR\xspace (as well as other demand response programs such as regulation service \cite{Xiaorui_2013data_frequency_regulation})
has received an increasing amount
of attention in recent years, e.g., \cite{AksanliRosing14_ProvidingRegulationServicesManagingDataCenterPeakPower,
Chen_PowerControl_Regulation_CDC_2013,aikema2012data_ancillary_IGCC_2012,
Liu:2014:PDC:2591971.2592004,hamed_datacenter_ancillary_smartgridcom_2012,Xiaorui_2013data_frequency_regulation,
DataCenterDemandResponsePreliminary_Feedback_2012,AdamWierman_DataCenterDemandResponse_Survey_IGCC_2014}.
These studies leverage various widely-available IT computing knobs
(e.g., server turning on/off and workload migration)
in data centers and provide algorithms to optimize them for participation in demand response markets. Importantly, these are not simply theoretical studies. For example, a field study by Lawrence Berkeley National Laboratory (LNBL) has illustrated that data centers can reduce energy consumption by 10-25\% in response to demand response signals, without noticeably impacting data center's normal operation \cite{DataCenterDemandResponse_Report_Berkeley}.
\textbf{Demand response in collocation data centers.}
While existing studies on data center demand response show promising progress, they are primarily focused on owner-operated data centers (e.g., Google) whose operators have full control over both servers and facilities.
Unfortunately, such companies may actually be the least likely to participate in demand response programs, because many of their workloads are extremely delay sensitive and their data centers have been optimized for delay.
In this paper, we focus on another type of data centers ---
multi-tenant colocation data centers (e.g., Equinix). These have been investigated much less frequently, but are actually better targets for demand response then owner-operated data centers. In a colocation data center (simply called ``colocation'' or ``colo''),
multiple tenants deploy and keep full control of their own physical servers in a shared space, while the colo operator only provides facility support (e.g., high-availability power and cooling). Colos are less studied than owner-operated data centers, but they are actually more common in practice. Colos offer
data center solutions to many industry sectors, and serve as physical home to many private clouds,
medium-scale public clouds (e.g., VMware) \cite{colocation_cloud_in_SuperNAP_Switch_LasVegas_2014},
and content delivery providers (e.g., Akamai). Further, a recent study shows that colos consume nearly 40\% data center energy in the U.S.,
while Google-type data centers collectively account for less than 8\%, with the remaining
going to enterprise in-house data centers \cite{NRDC_Colocation_2014}.
In addition to consuming a significant amount of energy (more than Google-type data centers), colos are often located in places more useful for demand response. While many
mega-scale owner-operated data centers are built in rural areas,
colos are mostly located in metropolitan areas (e.g.,
Los Angeles, New York) \cite{colocation_usa_datacentermap},
which are the very places where EDR\xspace
is most needed. Further, workloads in colos are highly heterogenous, and
many tenants run non-mission-critical workloads (e.g., lab computing \cite{Colocation_Symantec_Megawwat_Lease_2015}) that have very high scheduling flexibilities, different delay sensitivities, peak load periods, etc., which is ideal for demand response participation.
For all these reasons, colos are key participants in EDR\xspace programs.
Compared to owner-operated data centers that can leverage various computing knobs, however,
greening colos' participation in EDR\xspace by reducing reliance on diesel generator is significantly more challenging,
because of colo operators' lack of control over their tenants' servers.
On the other hand, many tenants in colos run servers hosting highly-flexible
and non-critical workloads with a great potential for shedding loads when called upon \cite{Colocation_Symantec_Megawwat_Lease_2015}.
Thus, tenants' load shedding potentials, if appropriately exploited,
can altogether form a green alternative to diesel generation for colo EDR\xspace.
Nonetheless, tenants manage their own servers independently and may not have
incentive to cooperate with the operator for EDR\xspace, thus raising
the research question: how can a colo operator \emph{efficiently} incentivize its tenants' load shedding for EDR\xspace?\footnote{Tenants receive UPS-protected power from colo operator and share
cooling systems. In other words, tenants' total energy consumption is not directly
provided by grid and includes non-separable cooling
energy, which makes tenants ineligible for direct participation in EDR\xspace \cite{pjm_emergency_demand_response_Performance}.}
\textbf{Contributions of this paper.} In this paper, we focus on ``greening'' colocation
demand response by extracting load reduction from tenants instead of relying on backup diesel generation.
We study both
\emph{mandatory} EDR\xspace, a type of EDR\xspace program in which participants
sign contracts and are obliged to reduce loads when requested \cite{pjm_emergency_demand_response_Performance}, and \emph{voluntary} EDR\xspace, where participants voluntarily reduce loads for financial compensation upon grid request.
In both cases, we propose a new pricing mechanism with which colo operators can extract load shedding from tenants. In particular, our proposed approach, called {\sf ColoEDR}\xspace, can
effectively provide incentives for tenants to reduce energy consumption during EDR\xspace events, complementing (and even substituting for) the high-cost and environmentally-unfriendly diesel generation.
{\sf ColoEDR}\xspace works as follows. After an EDR\xspace signal arrives at the colo operator, tenants bid using a parameterized supply function, and then the colo operator announces a market clearing price which, when plugged into the bids, specifies how much energy tenants will reduce and how much they will be paid. Participation by the tenants is straightforward, since they are required to bid only one parameter, which can be viewed as a proxy of how much flexibility in energy usage they have at that moment. This participation can be automated and so can be easily incorporated into current practice, and mimics the way generation resources participate in electricity markets more broadly. For example, colo operators at Verizon Terremark already communicate with tenants in preparation for an EDR\xspace event.
The main technical contribution of the paper is the analysis of the efficiency of the supply function mechanism proposed in {\sf ColoEDR}\xspace. In particular, while there is a large literature studying supply function bidding \cite{johari2011,day2002, baldick2004, green1992, green1996}, our setting here is novel and different because the colo operator can either satisfy the EDR\xspace request using flexibility from the tenants (as in prior supply funding literature) or through its backup diesel generator. Thus, the diesel generator is an outside option that allows for \emph{elasticity} in the amount of response extracted from the tenants. Further, the colo operator can combine and balance between its two options (i.e., tenant load shedding and backup generator) in order to minimize costs. This creates a multi-stage game and adds a considerable complexity as compared to the standard setting without an outside option, e.g., \cite{johari2011}.
Despite the added complexity, our analysis precisely characterizes the equilibrium outcome, both when tenants are price-taking and when they are price-anticipating. In both cases, our results highlight that {\sf ColoEDR}\xspace suffers little performance loss compared to the socially optimal outcome, both from the operator's and the tenants' perspectives. However, our analysis does highlight one possible drawback of {\sf ColoEDR}\xspace. In the worst case, it is possible that {\sf ColoEDR}\xspace may result in
using significantly more on-site diesel generation than would the socially optimal. However, this bad event occurs only in cases where one tenant has an overwhelmingly dominant amount of servers and has a unit cost (for energy reduction) just below that of on-site diesel generation. Such an exploitation of market power is unlikely to be possible in practical multi-tenant colocation data centers.
In addition to our theoretical analysis, we investigate a case study of colocation demand response in \xref{sec:simulation} using trace-based experiments. The results further validate the design of {\sf ColoEDR}\xspace, and show that it achieves the mandatory energy reduction for EDR\xspace while benefiting tenants through financial incentives and decreasing the operator's cost. Moreover, our simulation study shows that the efficiency loss in practical settings is even lower than what is suggested by the analytic bounds. This is especially true for the amount of on-site generation, which the analytic results suggest can (in the worst-case) be significantly larger than socially optimal but in realistic settings is very close to the social optimal.
\section{Problem Formulation}\label{sec:model}
Our focus is to design a mechanism for a colo operator to extract tenant load reductions in response to to an EDR\xspace signal. Thus, we need to begin by describing a model for a colo operator.
Recall that the colo operator is responsible for non-IT facility support (e.g., high-availability power, cooling). We capture the non-IT energy consumption using Power Usage Effectiveness (PUE) $\gamma$, which is
the ratio of the total colocation energy consumption to the IT energy consumption. Typically, $\gamma$ ranges from 1.1 to 2.0, depending on factors such as outside temperature.
When the operator receives an EDR\xspace signal from the LSE\xspace, it has two options for satisfying the load reduction. First, without involving the tenants, the colo operator can use its on-site backup diesel generator.\footnote{Other alternatives, e.g., battery \cite{Wang:2014:UBP:2541940.2541966}, usually
only last for $<5$ minutes. So, diesel
generation is the typical method \cite{Demand_response_US_EPA_EnerNOC}.}
We denote the amount of energy reduction by diesel generation by $y$ and
the cost per kWh of diesel generation (e.g., for fuels) by $\alpha$.
Alternatively, the colo operator could try to extract IT load reductions from the tenants. We consider a setting where there are $N$ tenants, $i \in \mathcal{N}=\{1,2, \cdots,N\}$.
When shedding energy consumption, a tenant $i$ will incur some
costs and we denote the cost from shedding $s_i$ by a function $c_i(s_i)$. These costs could be due to wear-and-tear, performance degradation, workload shifting, etc. For the purposes of our model, we do not specify which technique reduces the IT load, only its cost. For details on how one might model such costs, see \cite{ong2010impacts, fan2007power, andrew2010optimality, wierman2009power}.
A standard, natural assumption on the costs is the following.
\begin{assumption} For each $n$, the cost function $c_n(s_n)$ is continuous, with $c_n(s_n) = 0$ if $s_n \le 0$. Over the domain $s_n \ge 0$, the cost function $c_n$ is convex and strictly increasing.
\label{asn: cost_convexity}
\end{assumption}
Intuitively, convexity follows from the conventional assumption that the unit cost increases as tenants reduce more energy (e.g., utilization becomes higher when servers are off, leading to a faster increase in response time of tenants' workloads).
\section{Pricing Tenant Load Shedding in Mandatory EDR\xspace}
EDR\xspace is the last line of protection against cascading power failures, and represents 87\% of demand reduction capabilities across all the U.S. reliability regions \cite{EDR_Market_Overview}. In general, there are two types of EDR\xspace programs: mandatory and voluntary (also called economic) \cite{pjm_emergency_demand_response_Performance}. We focus on mandatory EDR\xspace first, and return to voluntary EDR\xspace in Section \ref{sec: voluntary_EDR}.
For mandatory EDR\xspace, participants typically sign contracts with a load serving entity (LSE\xspace) in advance (e.g., 3 years ahead in PJM \cite{pjm_emergency_demand_response_Performance}) and receive financial rebates for their committed energy reduction even if no EDR\xspace signals are triggered during the participation year, whereas non-compliance (i.e., failure to cut load as required during EDR\xspace) incurs heavy penalty \cite{pjm_emergency_demand_response_Performance}. If an LSE\xspace anticipates that an emergency will occur, participants are notified, usually at least 10 minutes in advance, and obliged to fulfill their contracted amounts of energy reduction for the length of the event, which may span a few minutes to a few hours.
In mandatory EDR\xspace, the colo operator has two options for obtaining load reductions in response to an EDR\xspace signal
that specifies the reduction amount -- tenants or on-site generation.
Thus, it must balance between paying tenants for reduction and using on-site generation in order to minimize cost. Note that tenants' load
reduction can also reduce the usage of diesel generator, mitigating environmental impacts. Nonetheless,
the challenge is that the operator does not know the tenant cost functions, and so cannot determine the cost-minimizing price.
Consequently, the operator has two options: (i) predict the tenant supply function and compute prices based on the predictions, or (ii) allow tenants to supply some information about their cost functions through bids. Clearly, there is a tradeoff here between the accuracy of predictions and the manipulation possible in the bids.
Both of these approaches have been looked at in the literature \cite{mohsenian2010, Liu:2014:PDC:2591971.2592004,johari2011,day2002, anderson2008}, though not in the context of colo demand response. In general, the broad conclusion is that approach (i) is appropriate when predictions are accurate and one bidder has market power (e.g., is significantly larger than other bidders). While market power is a considerable issue for the participation of owner-operated data centers in demand response programs due to their large size compared
to other participants, it is not an issue within a specific colo that houses
multiple tenants (typically of comparable sizes), and so we adopt approach (ii) in this paper.
Specifically, we design a mechanism, named {\sf ColoEDR}\xspace, where tenants bid using parameterized supply functions and then, given the bids, the operator decides how much load to shed via tenants and how much to shed via on-site generation. In the following, we describe the mechanism and then contrast our approach with other potential alternatives.
Note that, throughout this paper, we focus on one EDR\xspace event, and thus we omit the time index. In the case of multiple consecutive EDR\xspace events,
{\sf ColoEDR}\xspace will be executed once at the beginning of each event, as is standard in the literature \cite{Liu:2014:PDC:2591971.2592004,Shaolei_Colocation_ICAC_2014}.
\subsection{An overview of ColoDR}
The operation of {\sf ColoEDR}\xspace is summarized below, and then discussed in detail in the text that follows.
\begin{enumerate}
\item The colo operator receives an EDR\xspace reduction target $\delta$ and broadcasts the supply function $S(b_n, p)$ to tenants according to \eqref{eqn: supply_function};
\item Participating tenants respond by placing their bids $b_n$;
\item The colo operator decides the amount of on-site generation $y$ and market clearing price $p$ to minimize its cost, using equations \eqref{eqn: price} and \eqref{eqn: local_gen1};
\item EDR\xspace is exercised. $\forall n\in\mathcal{N}$, tenant $n$ sheds $S(b_n, p)$, and receives $pS(b_n, p)$ reward.
\end{enumerate}
Given the overview above, we now discuss each step in more detail.
\emph{Step 1.} Upon receiving an EDR\xspace notification of an energy reduction target $\delta$, the colo operator broadcasts a parameterized supply function $S(b,p)$ to tenants (by, e.g., signalling
to the tenants' server control interfaces, which are widely existing
today). The form of $S(b, p)$ is the following parameterized family\footnote{The supply function allows tenants to have negative supply, i.e., tenants consume more energy intentionally, which is neither profit maximizing nor practical. We show in \xref{sec: eff_analysis} that energy reduction of each tenant is always nonnegative in both equilibrium and social optimal outcomes.}:
\begin{equation}
S(b_n, p) = \delta - \frac{b_n}{p}.
\label{eqn: supply_function}
\end{equation}
where $p$ is offered reward for each kWh of energy reduction and $b_n$ is the bidding values that can be chosen by tenant $n$. This form is inspired by \cite{johari2011}, where it is shown that by restricting the supply function to this parameterized family, the mechanism can guide the firms in the market reach to an equilibrium with desirable properties.\footnote{\cite{johari2011} studies the case where firms bid to supply an inelastic demand, which is equivalent to fixing the diesel generation $y=0$ in our case. Allowing the operator to choose $y$ in a cost-minimizing manner leads to significantly different results, as will be shown in \xref{sec: price-taking} and \xref{sec: price-anticipating}.} Note that, to be consistent with the supply function literature, we exchangeably use ``price'' and ``reward rate'' wherever applicable.
\emph{Step 2.} Next, according to the supply function, each participating tenant submits its
bid $b_n$ to the colo operator. This bid specifies that, at each price $p$, it is willing to reduce $S(b_n, p)$ unit of energy. The bid is chosen by tenants individually to maximize their own utility and can be interpreted as the amount of IT service revenue that tenant $n$ is willing to forgo. Note that $b_n$ can be chosen to ensure that tenant $n$ will not be required to reduce more energy than its capacity. To see this, note that since the operator is cost-minimizing, $p(\mathbf{b}, y) \le \alpha$ always holds, i.e., the market clearing price is lower than the unit cost of diesel generation. Hence, if $K_n$ is the capacity of reduction for tenant $n$, as long as $b_n \ge \alpha (\delta - K_n)$, then
\begin{equation*}
S(b_n, p) = \delta - \frac{b_n}{p} \le \delta - \frac{b_n}{\alpha} \le K_n.
\end{equation*}
An important note about the tenant bids is that the supply function is likely of a different form than the true cost function $c_n$, and so it is unlikely for the tenants to reveal their cost functions truthfully. This is necessary in order to provide a simple form for tenant bids. Bidding their true cost functions is too complex
and intrusive. However, a consequence of this is that one must carefully analyze the emergent equilibrium to understand the efficiency of the pricing mechanism. We study both the cases of price-taking and price-anticipating equilibrium in \xref{sec: eff_analysis}.
\emph{Step 3.} After tenants have submitted their bids, the colo operator
decides the amount of energy $y$ to produce via on-site generation and the clearing price $p$. Given $y$, the market clearing price has to satisfy~ $\Sigma_n S(p(\mathbf{b}), b_n) + y = \delta$, thus
\begin{equation}
p(\mathbf{b}, y) = \frac{\sum_n b_n}{(N-1)\delta + y}.
\label{eqn: price}
\end{equation}
To determine the amount of local generation $y$, the operator minimizes the cost of the two
load-reduction options, i.e.,
\begin{align}
y = \argmin_{0 \le y \le \delta} (\delta - y)\cdot p(\mathbf{b}, y) + \alpha y.
\label{eqn: local_gen1}
\end{align}
\emph{Step 4.} Finally, EDR\xspace is exercised and tenants receive financial compensation from
the colo operator via the realized price in \eqref{eqn: price}, shed load $S(p,b_n)$, and on-site generation produces \eqref{eqn: local_gen1}.
\subsection{Discussion}
To the best of our knowledge, this paper represents the first attempt to design a supply function bidding mechanism for colocation demand response. Although alternative
mechanisms may be applicable,
there are compelling advantages to the supply function approach.
First, bidding for the tenants is simple -- they only need to communicate one number, and it is already common practice for operators to communicate with tenants before EDR\xspace events, so the overhead is small. Second, the colo operator collects
just enough information (i.e., how much energy reduction
each tenant will contribute to EDR\xspace), while
tenants' private information (i.e., how
much performance penalty/cost each for energy reduction)
is masked by the form of the supply function and hence not solicited.
Third, {\sf ColoEDR}\xspace guarantees that the colo operator
will not incur a higher cost than the case where only
diesel generator is used. Further, {\sf ColoEDR}\xspace pays
a uniform price to all participating tenants and hence ensures fairness.
The most natural alternative design to supply function bidding is a VCG-based mechanism, as is suggested in \cite{Shaolei_Colo_TruthDR_Tech}. While VCG-based mechanisms have the benefits of incentive compatibility, however, these mechanisms violate all the four properties discussed above. Under such approaches, tenants must submit very complex bids describing their precise cost functions, the true private cost of tenants is disclosed, payment made to tenants may be unbounded, and prices to different tenants are differentiated and thus raises unfairness issues.
Due to these shortcomings, VCG-based mechanisms are typically not adopted in complex resource allocation settings such as power markets, where supply-function based designs are common \cite{johari2011}. In fact, nearly all generation markets use a variation of supply function bidding.
\newcommand{\yterm}{\frac{\alpha}{2N\delta}(y+(N-1)\delta)^2}
\section{Efficiency Analysis of {\sf ColoEDR}\xspace for Mandatory EDR}\label{sec: eff_analysis}
Given the {\sf ColoEDR}\xspace mechanism described above, our task now is to characterize its efficiency. There are two potential causes of inefficiency in the mechanism: the cost minimizing behavior of the operator and the strategic behavior (bidding) of the tenants. In particular, since the forms of the tenant's cost functions are likely more complex than the supply function bids, tenants cannot bid their true cost function even if they wanted to. This means that evaluating the equilibrium outcome is crucial to understanding the efficiency of the mechanism.
Further, the equilibrium outcome that emerges depends highly on the behavior of the tenants -- whether they are \emph{price-taking}, i.e., they passively accept the offered market price $p$ as given when deciding their own bids; or \emph{price-anticipating}, i.e., they anticipate how the price $p$ will be impacted by
their own bids. We investigate both models, in \xref{sec: price-taking} and \xref{sec: price-anticipating}, respectively.
In both cases, the goal of our analysis is to assess the efficiency of {\sf ColoEDR}\xspace. To this end, we adopt a notion of a (socially) optimal outcome, and focus on the following social
cost minimization problem.
\begin{subequations}
\label{eqn: edr1}
\begin{eqnarray}
\label{eqn:obj_edr}
\mathsf{SCM}:&& \min \;\; \alpha y + \sum_{i\in\mathcal{N}}c_i(s_i)\\
\label{eqn:constraint_edr}
\text{s.t.} && y + \gamma\cdot\sum_{i\in\mathcal{N}}s_i = \delta\\
\label{eqn:nonnegative} \label{eqn:nonnegative_y}
&& s_i \ge 0,\;\forall i\in\mathcal{N} \text{, } \quad y \geq 0.
\end{eqnarray}
\end{subequations}
where $s_i$ and $c_i$ are tenant $i$'s energy reduction and corresponding
cost, respectively.
The objective in $\mathsf{SCM}$\xspace can be interpreted as the tenants' cost plus the colo operator's cost. Note that the internal payment transfer between the colo operator and tenants cancels, and does not impact the social cost.
Also, note that payment from the LSE\xspace to the colo operator is not included
in the social cost objective, since it is independent of how the operator obtains the amount of $\delta$ load reduction. Additionally, we do not include the option of ignoring the event and taking the penalty, since the penalties for lack of participation are typically extreme. Finally, the Lagrangian multiplier of \eqref{eqn:constraint_edr} can be interpreted as the social optimal price $p^*$, i.e., given this price as reward for energy reduction, each tenant will individually reduce their energy by $s_n$ that corresponds to the social cost minimization solution in \eqref{eqn: edr1}.
Before moving to the analysis, in order to simplify notation, we suppress the PUE $\gamma$ by, without loss of generality, setting $\gamma=1$. This is equivalent to a change of notation $y' = y / \gamma$, $\delta' = \delta / \gamma$, and $\alpha' = \alpha \gamma$, i.e., translating the diesel generation, unit cost of diesel generation, and EDR\xspace energy reduction target into their respective equivalent amounts in terms of server energy.
\subsection{Price-Taking Tenants}
\label{sec: price-taking}
When tenants are price-taking, they maximize their net utility, which is the difference between the payment they receive and the cost of energy reduction, given the assumption that they consider their action does not impact the price.
\begin{subequations}
\label{eqn: cost_taking}
\begin{eqnarray}
P_n(b_n, p) &= pS_n(b_n, p) - c_n(S_n(b_n, p)) \\
&= p\delta - b_n - c_n\left(\delta - \frac{b_n}{p}\right).
\end{eqnarray}
\end{subequations}
Here, the price-taking assumption implies that the variable $p$ is considered to be as is. The market equilibrium for price-taking tenants is thus defined as follows.
\begin{definition}
A triple $(\mathbf{b}, p, y)$ is a (price-taking) market equilibrium if each tenant maximizes its payoff defined in \eqref{eqn: cost_taking}, market is cleared by setting price $p$ according to \eqref{eqn: price}, and the amount of on-site generation is decided by \eqref{eqn: local_gen1}, i.e.,
\begin{align} \label{def: ne1_taking}
P_n(b_n; p) &\ge P_n(\bar{b}_n; p)\quad \forall \bar{b}_n \ge 0, \quad n=1, \ldots, N .\quad \\
\label{def: price_ne1}
p &= \frac{ \sum_{i\in\mathcal{N}}b_i}{(N-1)\delta+y}.\\
\label{def: local_gen_ne1}
y &= \argmin_{0 \le y \le \delta} (\delta - y)\cdot p(\mathbf{b}, y) + \alpha y.
\end{align}
\end{definition}
\subsubsection{Market Equilibrium Characterization}
The key to our analysis is the observation that the equilibrium can be characterized by an optimization problem. Once we have this optimization, we can use it to characterize the efficiency of the equilibrium outcome. This approach parallels that used in \cite{johari2011}; however, the optimization obtained has a different structure due to local diesel generation. Additionally, though we use an optimization to characterize the equilibrium, the game is not a potential game.
Our first result highlights that, given any choice for on-site generation, a unique market equilibrium exists for the tenants, and can be characterized via a simple optimization.
\begin{prop}
Under Assumption \ref{asn: cost_convexity}, when tenants are price-taking, for any on-site generation level $0 \le y < \delta$, there exists a market equilibrium, i.e., a vector $\mathbf{b}^t = (b_1^t, \ldots, b_N^t)\ge 0$ and a scalar $p > 0$ that satisfies \eqref{eqn: price}, and the resulting allocation $s_n = S(b_n, p)$ is the optimal solution of the following
\begin{subequations}
\begin{align}
\label{eqn: p1-1}
\min_{\mathbf{s}} & \quad\sum_{i\in \mathcal{N}} c_i(s_i) \\
\label{eqn: p1-2}
s.t. & \quad \sum_{i\in \mathcal{N}} s_i = (\delta - y), \\
\label{eqn: p1-3}
&\quad s_i \ge 0,\ \forall i \in \mathcal{N}.
\end{align}
\label{eqn: p1}
\end{subequations}
\label{prop: ec1}
\end{prop}
This result is a key tool for understanding the overall market outcome.
Intuitively, the operator running {\sf ColoEDR}\xspace is more likely (than the social optimal) to use on-site generation, since this reduces the price paid to tenants. The following proposition quantifies this statement.
\begin{prop}
Under Assumption \ref{asn: cost_convexity}, it is optimal for price-taking tenants to use on-site generation if and only if
\begin{equation}
\alpha < \frac{(\Sigma_n b_n)}{(N-1)\delta},\footnote{We adopt the convention that $\frac{0}{0} = 0$ and $\frac{x}{0} = +\infty$ when $x>0$. Therefore, when $N=1,$ unless the bid is 0, the condition is always satisfied.}
\label{eqn: cheap_on-site}
\end{equation}
However, when the operator is profit maximizing, it will turn on on-site generation if and only if
\begin{equation}
\alpha < \frac{N}{N -1} \frac{(\Sigma_n b_n)}{(N-1)\delta}.
\label{eqn: cheap_on-site2}
\end{equation}
\label{prop: on-site_gen}
\end{prop}
This proposition is an important building block because the most interesting case to consider is when it is optimal to use some on-site generation and some tenant load shedding, i.e., $\delta>y^* >0 $. Otherwise the EDR demand should be entirely fulfilled by tenants, and the analysis reduces to the case of an inelastic demand, as studied in \cite{johari2011}. Thus, subsequently, we make the following assumption, which ensures that on-site generation is valuable.
\begin{assumption}
The unit cost of on-site generation is cheap enough that the optimal on-site generation is non-zero, i.e., $\alpha$ satisfies \eqref{eqn: cheap_on-site}.
\label{asn: cheap_on-site}
\end{assumption}
Note that, when Assumption~\ref{asn: cheap_on-site} holds, by first-order optimality condition of \eqref{eqn: local_gen1} we have
\begin{equation}
y = \sqrt{\frac{(\Sigma_{i\in \mathcal{N}}b_i) N\delta}{\alpha }} - (N-1)\delta,
\label{eqn: on-site_gen}
\end{equation}
and so the market clearing price for the tenants given on-site generation is
\begin{equation}
p =\frac{ \sum_{i\in\mathcal{N}}b_i}{(N-1)\delta+y} = \sqrt{\frac{(\Sigma_{i\in \mathcal{N}}b_i)\alpha}{N\delta}}.
\label{eqn: price2}
\end{equation}
Using these allows us to prove a complete characterization of the market equilibrium under price-taking tenants.
This theorem is the key to our analysis of market efficiency.
\begin{theorem}
When Assumptions \ref{asn: cost_convexity} and \ref{asn: cheap_on-site} hold there is a unique {market equilibrium}, i.e., a vector $\mathbf{b}^t = (b_1^t, \ldots, b_N^t) \ge 0$, $y^t > 0$ and a scalar $p^t>0$ that satisfies \eqref{def: ne1_taking}-\eqref{def: local_gen_ne1}, and the resulting allocation $(\mathbf{s}^t, y^t)$ where $s_n^t = S(b_n^t, p^t)$ is the optimal solution of the following problem
\begin{subequations}
\label{eqn: price-taking0}
\begin{align}
\label{eqn: price-taking1}
\min_{\mathbf{s}, y}\quad& \sum_n c_n(s_n) + \yterm \\
\label{eqn: price-taking2}
s.t. \quad& \sum_n s_n = \delta - y, \\
\label{eqn: price-taking3}
& s_n \ge 0, \;\forall n, \quad y \ge 0.
\end{align}
\end{subequations}
\label{thm: price-taking-characterization}
\end{theorem}
\subsubsection{Bounding Efficiency Loss}
We now use Theorem~\ref{thm: price-taking-characterization} to bound the efficiency loss due to strategic behavior in the market. Denote the socially optimal on-site generation by $y^*$, the optimal price that leads to the optimal allocation $s_i, \forall i \in \mathcal{N}$ by $p^*$, and let $y^t$ and $p^t$ be the allocation under the price-taking assumption.
Our first result highlights that, due to the cost-minimizing behavior of the operator, the equilibrium outcome uses more on-site generation and pays a lower price to the tenants than the social optimal.
\begin{prop}
Suppose that Assumptions~\ref{asn: cost_convexity} and~\ref{asn: cheap_on-site} hold. When tenants are price-taking, the operator running {\sf ColoEDR}\xspace uses more on-site generation and pays a lower price for power reduction to its tenants than the social optimal. Specifically,
$y^t \ge y^*$ and~ $\frac{N-1}{N}p^* \le p^t \le p^*.$
\label{prop: ec2}
\end{prop}
Now, we move to more detailed comparisons. There are three components of market efficiency that we consider: social welfare, operator cost, and tenant cost.
First, let us consider the social cost.
\begin{theorem}
Suppose that Assumptions \ref{asn: cost_convexity} and \ref{asn: cheap_on-site} hold. Let $(\mathbf{s}^t, y^t)$ be the allocation when tenants are price-taking, and $(\mathbf{s}^*, y^*)$ be the optimal allocation. Then the welfare loss is bounded by:
$\sum_n c_n(s_n^t) + \alpha y^t \le \sum_n c_n(s_n^*) + \alpha y^* + \alpha\delta/2N.$
\label{thm: welfare_loss_taking}
\end{theorem}
Importantly, this theorem highlights that the market equilibrium is quite efficient, especially if the number of tenants is large (the efficiency loss decays to zero as $O(1/N)$). However, the market could maintain good overall social welfare at the expense of either the operator or the tenants. The following results show this is not true.
Let $\mathrm{cost}_o(p, y)$ be the operator's cost, i.e.,
\begin{align}
\mathrm{cost}_o(p, y) &= p(\delta - y) + \alpha y.
\label{eqn: operator_cost}
\end{align}
Then, we have the following results.
\begin{theorem}
Suppose that Assumptions \ref{asn: cost_convexity} and \ref{asn: cheap_on-site} are satisfied. The cost of colo operator with price-taking tenants is smaller than the cost in the socially optimal case. Further, we have
$\mathrm{cost}_o(p^*, y^*) -\alpha\delta/N\le\mathrm{cost}_o(p^t, y^t) \le \mathrm{cost}_o(p^*, y^*).$
\label{thm: colo_cost1_taking}
\end{theorem}
\subsection{Price-Anticipating Tenants}
\label{sec: price-anticipating}
In contrast to the price-taking model, price-anticipating tenants realize that they can change the market price by their bids, i.e., that $p$ is set according to \eqref{eqn: price2}, and adjust their bids accordingly. Clearly, this additional strategic behavior can lead to larger efficiency loss. But, in this section, we show that the extra loss is surprisingly small, especially when a large number of tenants participate in {\sf ColoEDR}\xspace.
Given bids from the other tenants, each price-anticipating tenant $n$ optimizes the following cost over bidding value $b_n$
\begin{align*}
Q_n(b_n, \mathbf{b}_{-n}) &= p(\mathbf{b}) S_n(b_n, p) - c_n(S_n(b_n, p))
\end{align*}
where we use $\mathbf{b}_{-n}$ to denote the vector of bids of tenants other than $n$; i.e., $\mathbf{b}_{-n} = (b_1, \ldots, b_{n-1}, b_{n+1}, \ldots, b_N)$. Thus, substituting \eqref{eqn: supply_function} and \eqref{eqn: price2}, we have
\begin{equation}
Q_n(b_n; \mathbf{b}_{-n}) = \sqrt{\frac{(\Sigma_n b_n)\alpha\delta}{N}} - b_n - c_n\left(\delta - \frac{b_n}{\sqrt{\Sigma_m b_m}}\sqrt{\frac{N\delta}{\alpha}} \right).
\label{eqn: cost_anticipating}
\end{equation}
Note that the payoff function $Q_n$ is similar to the payoff function $P_n$ in the price-taking case, except that the tenants anticipate that the colo operator will set the price $p$ according to $p = p(\mathbf{b}, y)$ from \eqref{eqn: price2}.
\begin{definition}
A triple $(\mathbf{b}, p, y)$ is a (price-anticipating) market equilibrium if each tenant maximizes its payoff defined in \eqref{eqn: cost_anticipating}, the market is cleared by setting the price $p$ according to \eqref{eqn: price} and the amount of on-site generation is decided by \eqref{eqn: local_gen1}, i.e.,
\begin{align}
\label{def: ne1}
Q_n(b_n; \mathbf{b}_n) &\ge Q_n(\bar{b}_n; \mathbf{b}_n)\quad \forall \bar{b}_n \ge 0, \quad n=1, \ldots, N \\
\label{def: ne2}
p &= \frac{ \sum_n b_n}{(N-1)\delta + y}.\\
\label{def: ne3_price_anticipating}
y &= \argmin_{0 \le y \le \delta} (\delta - y)\cdot p(\mathbf{b}, y) + \alpha y.
\end{align}
\end{definition}
Note that our analysis in this section requires one additional technical assumption about the tenant cost functions.
\begin{assumption}
The marginal cost of all the tenants at 0 is greater than $\frac{\alpha}{2N}$, i.e.,
$\frac{\partial^+ c_n(0)}{\partial s_n} \ge \frac{\alpha}{2N}, \ \forall n.$
\label{asn: mc_lowerbound}
\end{assumption}
This assumption is quite mild, especially if the number of tenants $N$ is large. Intuitively, it says that the unit cost of on-site generation is competitive with the cost of tenants reducing their server energy.
\subsubsection{Market Equilibrium Characterization}
Our analysis of market equilibria proceeds along parallel lines to the price-taking case. We again show that there exists a unique equilibrium and, furthermore, that the tenants and operator behave in equilibrium as if they were solving an optimization problem of the same form as the aggregate cost minimization \eqref{eqn: edr1}, but with ``modified'' cost functions.
\begin{theorem}
Suppose that Assumption \ref{asn: cost_convexity}-\ref{asn: mc_lowerbound} are satisfied, then there exists a unique equilibrium of the game defined by \\$(Q_1, \ldots, Q_n)$ satisfying \eqref{def: ne1}-\eqref{def: ne3_price_anticipating}. For such an equilibrium, the vector $\mathbf{s}^a$ defined by $s_n^a = S(p(\mathbf{b}^a), b_n^a)$ is the unique optimal solution to the following optimization:
\begin{subequations}
\label{eqn: mc0}
\begin{align}
\label{eqn: mc1}
\min \quad & \sum_n \hat{c}_n(s_n) + \frac{\alpha}{2N\delta}(y+(N-1)\delta)^2\\
\label{eqn: mc2}
\text{s.t.} \quad & \sum_n s_n = \delta - y \\
\label{eqn: mc3}
& y \ge 0,\ s_n \ge 0, \quad n = 1, \ldots, N,
\end{align}
\end{subequations}
where, for $s_n \ge 0$,
\begin{align}
\hat{c}_n(s_n) = &\frac{1}{2}\left(c_n(s_n) + s_n \frac{\alpha}{2N} \right)
+ \frac{1}{2}\int^{s_n}_0 \sqrt{ \left( \mz - \frac{\alpha}{2N} \right)^2 + 2 \mz \frac{ z\alpha}{N\delta}} dz,
\label{eqn: modified_cost}
\end{align}
and for $s_n <0, \quad \hat{c}_n(s_n) = 0.$
\label{thm: modified_cost}
\end{theorem}
Although the form of $\hat{c}_n(s_n)$ looks complicated, there is a simple linear approximation that gives useful intuition.
\begin{lem}
Suppose that Assumption \ref{asn: cost_convexity}-\ref{asn: mc_lowerbound} are satisfied. For all modified cost $\hat{c}_n, n \in 1, \ldots, N$, for any $0 \le s_n \le \delta$,
\[ c_n(s_n) \le \hat{c}_n(s_n) \le c_n(s_n) + s_n\frac{\alpha}{2N}, \]
Furthermore, when the left or right derivatives of $\hat{c}(\cdot)$ is defined, it can be bounded by
\[ \lmc \le \frac{\partial^-\hat{c}(s_n)}{\partial s_n} \le \frac{\partial^+\hat{c}(s_n)}{\partial s_n} \le \rmc + \frac{\alpha}{2N}.\]
\label{lem: mc_bound}
\end{lem}
The form of Lemma~\ref{lem: mc_bound} shows that the difference between the modified cost function in \eqref{eqn: modified_cost} and the true cost diminishes as $N$ increases, and this is the key observation that underlies our subsequent results upper bounding the efficiency loss of {\sf ColoEDR}\xspace.
\subsubsection{Bounding Efficiency Loss}
We now use Theorem \ref{thm: modified_cost} to bound the efficiency loss due to strategic behavior. Note that, by comparing to both the socially optimal and the price-taking outcomes, we can understand the impact of both strategic behavior by the operator and the tenants.
Our first result focuses on comparing the price-anticipating and price-taking equilibrium outcomes. It highlights that price-anticipating behavior leads to tenants receiving higher price while providing less load shedding.
\begin{theorem}
Suppose Assumption \ref{asn: cost_convexity}-\ref{asn: mc_lowerbound} hold. Let $(p^t, y^t)$ be the equilibrium price and on-site generation when tenants are price-taking, and $(p^a, y^a)$ be those when tenants are price-anticipating, then we have,
$y^t \le y^a \le y^t + \delta/2$ and~ $p^t \le p^a \le p^t + \alpha/2N.$
\label{thm: diff-price-anticipate}
\end{theorem}
Next, combining Theorem \ref{thm: diff-price-anticipate} and Proposition \ref{prop: ec2} yields the following comparison between the price-anticipating and socially optimal outcomes.
\begin{corollary}
Suppose Assumption \ref{asn: cost_convexity}-\ref{asn: mc_lowerbound} hold. When tenants are price-anticipating, an operator running {\sf ColoEDR}\xspace uses more on-site generation and pays lower market price than in the socially optimal case, i.e.,
$y^a \ge y^*$ and~$\frac{N-1}{N}p^* \le p^a \le p^*. $
\label{cor: diff-price-anticipate2}
\end{corollary}
Now, we move to more detailed comparisons. There are three components of market efficiency that we consider: social welfare, operator cost, and tenant cost.
First, let us consider the social cost.
\begin{theorem} \label{thm: welfare_loss2}
Suppose that Assumption \ref{asn: cost_convexity}-\ref{asn: mc_lowerbound} hold. Let $(\mathbf{s}^a, y^a)$ be the allocation when tenants are price-anticipating, and $(\mathbf{s}^*, y^*)$ be the optimal allocation. The welfare loss is bounded by:
$\sum_n c_n(s^a_n) + \alpha y^a \le \sum_n c_n(s^*_n) + \alpha y^* + \alpha\delta/N.$
\end{theorem}
Similarly to the price-taking case, the efficiency loss in the price-anticipating case decays to zero as $O(1/N)$, only with a larger constant. Also, as in the case of price-taking tenants, we again see that neither the tenants nor the operator suffers significant efficiency loss.
\begin{theorem}
\label{thm: diff-payment}
Suppose that Assumption \ref{asn: cost_convexity}-\ref{asn: mc_lowerbound} hold. The cost of colo operator for price-anticipating tenants is smaller than the cost in the socially optimal case. Further, we have
\begin{align*}
\mathrm{cost}_o(p^*, y^*) -\frac{\alpha\delta}{N}\le\mathrm{cost}_o(p^a, y^a)\le \mathrm{cost}_o(p^*, y^*), \\
\mathrm{cost}_o(p^a, y^a) -\frac{\alpha\delta}{N}\le\mathrm{cost}_o(p^t, y^t)\le \mathrm{cost}_o(p^a, y^a)
\end{align*}
\end{theorem}
Finally, let us end by considering the amount of on-site generation used in equilibrium.
Here, in the worst-case, the equilibrium on-site generation for price-anticipating tenants can be arbitrarily worse than the socially optimal, i.e., the socially optimal can use no on-site generation while the equilibrium outcome uses only on-site generation.
\begin{theorem}
Suppose that Assumption \ref{asn: cost_convexity}-\ref{asn: mc_lowerbound} hold. For any $\varepsilon >0$, $N \ge 1$, there exist cost functions $c_1, \ldots, c_N$, such that the on-site generation in the market equilibrium compared to the optimal is given by $y^a - y^* \ge \delta - \varepsilon.$
\label{thm: on-site_gap}
\end{theorem}
This is a particularly disappointing result since a key goal of the mechanism is to obtain load shedding from the tenants. However, the proof emphasizes that this is unlikely to occur in practice. In particular, the worst-case scenario is that there exists a dominant (monopoly) tenant, which is unlikely in a multi-tenant
colo, that has a cost function asymptotically linear with unit cost roughly matching the on-site generation price $\alpha$. We confirm this in a case study in Section \ref{sec:simulation}.
\subsection{Discussion}
\label{sec: compare_models}
The main results for the price-taking and price-anticipating analyses are summarized in Table \ref{table: performance_guarantee}. Note that simplified bounds are presented in the table, to ease interpretation, and the interested reader should refer to the theorems in \xref{sec: price-taking} and \xref{sec: price-anticipating} for the tightest bounds. Also, note that the benchmark for social cost we consider is an ideal, but not achievable, mechanism.
\begin{table}[h]
{\mathrm{c}\mkern-6.5mu{\mid}}ering
\begin{tabular}{|c|c|c|c|}\hline
Tenants & Price Ratio& Colo Saving & Welfare Loss \\ \hline
Price-taking & $[\frac{N-1}{N},~ 1 ]$ & $[0, ~ \alpha\delta / N ]$ & $[0, ~ \alpha\delta/ 2N]$ \\ \hline
Price-anticipating & $[\frac{N-1}{N},~ 1 ]$ & $[0, ~ \alpha\delta / N]$ & $[0, ~ \alpha\delta /N]$\\ \hline
\end{tabular}
\caption{Performance guarantee of {\sf ColoEDR}\xspace compared to the social optimal allocation.}
\label{table: performance_guarantee}
\end{table}
To summarize the results in Table \ref{table: performance_guarantee} briefly, note first that {\sf ColoEDR}\xspace always benefits the operator, since the price paid to tenants to reduce energy is always less than the socially optimal price, and the total cost incurred by operator for energy reduction is also less than that of the social optimal. Secondly, {\sf ColoEDR}\xspace also gives the tenants approximately the social optimal payment, since the operator's additional benefit is bounded above by $\alpha\delta / N$. This naturally means that the loss in payment for tenants compared to the social optimal is also $\alpha\delta / N$, which approaches 0 as $N$ grows. Third, regardless of tenants being price-taking or price-anticipating, {\sf ColoEDR}\xspace is approximately socially cost-minimizing as the number of tenants grows.
However, while {\sf ColoEDR}\xspace is good in terms of operator, tenant, and social cost, it may not use the most environmentally friendly form of load reduction: in the worst case, the upper bound on the extra on-site generation that {\sf ColoEDR}\xspace uses is not decreasing with $N$. However, the analysis highlights that this worst-case occurs when there exists a dominant tenant with unit cost of energy reduction that is consistently just below the cost of diesel over a large range of energy reduction. As our case study in \xref{sec:simulation} shows, this is unlikely to occur in practice. So, {\sf ColoEDR}\xspace can be expected to use an environmentally friendly mix in most realistic situations.
\newcommand{\sum_{i=1}^n}{\sum_{i=1}^n}
\newcommand{\gamma_n}{\gamma_n}
\newcommand{\half}{\frac{1}{2}}
\section{Pricing Tenant Load Shedding in Voluntary EDR\xspace}\label{sec: voluntary_EDR}
We now turn from mandatory EDR\xspace to voluntary EDR\xspace and show how the analysis and design of {\sf ColoEDR}\xspace can be extended. Under voluntary EDR\xspace, a colo operator is offered a certain compensation rate for load reduction and can cut any amounts of energy \emph{at will} without any obligation. Voluntary EDR\xspace often supplements mandatory EDR\xspace, and both are widely adopted in practice \cite{pjm_emergency_demand_response_Performance,EDR_Market_Overview}.
Since the colo operator can freely decide on the amount of energy to cut based on the compensation rate
\cite{pjm_emergency_demand_response_Performance}, the amount of energy reduction responses from tenants is \emph{fully} elastic, differing
from mandatory EDR\xspace where the total energy reduction (including diesel generation if necessary)
needs to satisfy a constraint $\delta$.
In the following, we formulate the problem and generalize {\sf ColoEDR}\xspace for the voluntary EDR\xspace setting. Furthermore, we illustrate that the efficiency
analysis, though more complicated, parallels that of mandatory EDR\xspace.
\subsection{Problem Formulation}
During a voluntary EDR\xspace event, the LSE\xspace offers a reward of $u$ for each unit of
energy reduction (or diesel generation if applicable). In our setting, the colo operator aims at maximizing its profit through extracting loads from tenants using parameterized supply function bidding, as considered for mandatory EDR\xspace.
A key difference with the case of mandatory EDR\xspace is that, since the reduction is voluntary, diesel generation need not be considered. In particular, if the reward offered the the LSE\xspace for reduction is larger than the cost of diesel, then the operator can contribute its whole diesel capacity and, if the reward is smaller than the cost of diesel, no diesel need be used. Compared to the mandatory EDR\xspace setting, operator need to use more diesel generation when tenants' bids are high in order to meet the fixed reduction target $\delta$; in the voluntary EDR\xspace case, the operator can simply reduce the DR contribution by tenants when their bids are high. Thus, the optimization of diesel generation by the operator is separable from the optimization of tenant reduction.
This yields a situation where the net profit (from tenant reduction) received by the colo operator is:
\begin{equation}
u\cdot d - p\cdot d \label{eqn: vdr-net-benefit}
\end{equation}
where $p$ is the unit price the colo operator pays to the tenants to solicit $d$ units of reduction in aggregate, which arises from $N$ tenants where tenant $i$ has reduction capacity $D_i$.
\paragraph{An overview of {\sf ColoEDR}\xspace}
It is straightforward to adapt {\sf ColoEDR}\xspace to this setting. We outline its operation in four steps below, which parallel the steps in the case of mandatory EDR\xspace.
\begin{enumerate}
\item The colo operator receives the voluntary EDR\xspace reduction price $u$ and broadcasts the supply function $S(b_n, p)$ to tenants according to \begin{equation}
S_i(b_i, p) = D_i - \frac{b_i}{p}, \label{eqn: vdr-supply-function}
\end{equation}
where $D_i$ is the capacity of tenant $i$ for reduction determined exogenously.
\item Participating tenants respond by placing their bids $b_n$ in order to maximize their own payoff;
\item The colo operator decides the total amount of reduction from tenants $d$ and market clearing price $p$ to maximize its utility. Given the bids $\mathbf{b} = (b_1, \ldots, b_n)$, if the operator decides to offer $d$ amount of energy reduction to the utility, then the market clearing price $p$ will be
\begin{equation}
p = \frac{\sum_{i=1}^n b_i}{\sum_{i=1}^n D_i - d}. \label{eqn: vdr-price1}
\end{equation}
Hence to maximize the operator's profit, the operator will chooose $d$ such that
\begin{equation}
d = \argmax_{0 \le d \le \sum_{i=1}^n D_i } (u - p) d = \left(u - \frac{\sum_{i=1}^n b_i}{\sum_{i=1}^n D_i - d}\right) d.
\label{eqn: profit-maximize-d}
\end{equation}
It follows from the first order optimality of \eqref{eqn: profit-maximize-d} that
\begin{equation}
d = \sum_{i=1}^n D_i - \sqrt{\frac{ (\sum_{i=1}^n b_i) (\sum_{i=1}^n D_i)}{u} }, \label{eqn: vdr-quantity}
\end{equation}
which gives that the price set by a profit maximizing operator will be
\begin{equation}
p = \sqrt{\frac{u\sum_{i=1}^n b_i}{\sum_{i=1}^n D_i }}. \label{eqn: vdr-price2}
\end{equation}
\item Voluntary EDR\xspace is exercised. $\forall n\in\mathcal{N}$, tenant $n$ sheds $S(b_n, p)$, and receives $pS(b_n, p)$ reward.
\end{enumerate}
\paragraph{Discussion}
The key difference in the operation of {\sf ColoEDR}\xspace for mandatory EDR\xspace and voluntary EDR\xspace is in the form of the supply function used. In particular, we allow heterogeneity in the supply function for tenants in terms of their capacity $D_n$. Recall, that in the case of mandatory EDR\xspace the desired reduction capacity $\delta$ was used. This difference stems from the fact that the reduction target is flexible for voluntary demand response and creates significant challenges -- both in terms of efficiency, since it allows the chance of market power to emerge because of capacity differences, and for analysis, since it adds considerable complexity.
\subsection{Efficiency Analysis of {\sf ColoEDR}\xspace for Voluntary EDR\xspace}
Given the adaptation of {\sf ColoEDR}\xspace to the voluntary EDR\xspace setting, it is natural to ask how the efficiency of the mechanism changes when the operator has flexibility in the amount of response to provide to an EDR\xspace signal. Intuitively, the increased flexibility leads to the possibility of more inefficiency, but how large is this effect?
We again quantify efficiency through a comparison with the (socially) optimal outcome. Assuming that each tenant has a cost $c_i(\cdot)$ associated with energy reduction that is convex, increasing, and $c_i(x) = 0, \forall x \le 0$ (Assumption \ref{asn: cost_convexity}). Then the allocation that maximizes social utility (the sum of operator's and tenants' utility) solves the following problem
\begin{subequations}
\label{eqn: vdr-utility-maximization}
\begin{align}
\max_{d, \mathbf{s}} \quad& ud - \sum_{i=1}^n c_i(s_i) \\
\text{subject to} \quad & \sum_{i=1}^n s_i = d \\
\quad & 0 \le s_i \le D_i .
\end{align}
\end{subequations}
Finally, note that our analysis makes the following natural assumptions on the unit price $u$ and the marginal cost of each tenants. Note that they are analogous to Assumption \ref{asn: cheap_on-site} and Assumption \ref{asn: mc_lowerbound}.
\begin{assumption}
The market clearing price $p$ is lower than the price offered by the utility for any $d>0$, i.e., $u \ge \frac{\sum_{i=1}^n b_i}{\sum_{i=1}^n D_i}$.
\label{asn: vdr-u-condition}
\end{assumption}
\begin{assumption}
The marginal cost of each tenants satisfies $ \mz\Big|_{z=0} \ge \frac{\gamma_n u}{2}, \forall n.$ \label{asn: vdr-mc-lbound}
\end{assumption}
Before moving to the main results, let us first define some notation. Let $\gamma_n = \frac{D_n}{\sum_{i=1}^n D_i}$, we have $\sum_n \gamma_n = 1.$ Here $\gamma_n$ behaves like ``market share'' of tenant $n$ in the voluntary DR market. In the EDR case, $\gamma_n = 1/N $ for all $n$. Furthermore, define $\gamma = \max_n \gamma_n$, as the ``dominant share'' in load reduction among the tenants, and $D = \max_n D_n$.
\subsection{Market Equilibrium Characterization}
As in the case of mandatory EDR\xspace, we consider both the cases price-taking and and price-anticipating tenants.
\subsubsection{Price-taking Tenants}
Given other tenants, each price-taking tenant $n$ optimizes the following cost over bidding value $b_n$,
\begin{align*}
P_n(b_n, \mathbf{b}_{-n}) = pS_n(b_n, p) - c_n(S_n(b_n, p))
= pD_n - b_n - c_n(D_n - \frac{b_n}{p})
\end{align*}
So, in a price-taking equilibrium $(\mathbf{b}, d, p)$, we must have $P_n(b_n; \mathbf{b}_{-n}) \ge P_n(\bar{b}_n; \mathbf{b}_{-n})$ hold for each tenant $n$ over all $\bar{b}_n \ge 0$. Also, the market clearing price must satisfy \eqref{eqn: vdr-price1} and the total reduction must satisfy \eqref{eqn: profit-maximize-d}. Using techniques similar to the proof of Theorem \ref{thm: price-taking-characterization}, we can completely characterize the the price-taking equilibrium of {\sf ColoEDR}\xspacevdr in voluntary EDR\xspace as follows:
\begin{theorem}
There exists a unique equilibrium of the game defined by $(P_1, \ldots, P_N)$ for {\sf ColoEDR}\xspacevdr. For such an equilibrium, the vector $\mathbf{s}^t$ defined by $s_n^t = S(p(\mathbf{b}^t), b_n^t)$ is the unique optimal solution to the following optimization:
\begin{subequations}
\label{eqn: mc0_taking}
\begin{align}
\label{eqn: mc1_taking}
\max \quad & ud - \frac{ud^2}{2\sum_n D_n} - \sum_n c_n(s_n) \\
\label{eqn: mc2_taking}
\text{s.t.} \quad & \sum_n s_n = d \\
\label{eqn: mc3_taking}
& d \ge 0,\ 0 \le s_n \le D_n, \quad n = 1, \ldots, N,
\end{align}
\end{subequations}
\label{thm: vdr_ne_taking}
\end{theorem}
\subsubsection{Price-anticipating Tenants} Given other tenants, each price-anticipating tenant $n$ optimizes the following cost over bidding value $b_n$,
\begin{align*} Q_n(b_n, \mathbf{b}_{-n}) = p(\mathbf{b}) S_n(b_n, p) - c_n(S_n(b_n, p))
= \gamma_n \sqrt{\Sigma_m b_m}\sqrt{u \sum_{i=1}^n D_i} - b_n - c_n(D_n - \frac{b_n}{\Sigma_m b_m} \sqrt{\frac{\sum_{i=1}^n D_i }{u}}),
\end{align*}
So, in a price-anticipating equilibrium $(\mathbf{b}, d, p)$, we must have $Q_n(b_n; \mathbf{b}_{-n}) \ge Q_n(\bar{b}_n; \mathbf{b}_{-n})$ for all $n$ over all $\bar{b}_{n}$. Also, the market clearing price must satisfy \eqref{eqn: vdr-price1} and the total reduction $d$ must satisfy \eqref{eqn: profit-maximize-d}.
Using techniques similar to the proof of Theorem \ref{thm: modified_cost}, we can completely characterize the the price-anticipating equilibrium of {\sf ColoEDR}\xspacevdr in voluntary EDR\xspace as follows.
\begin{theorem}
There exists a unique equilibrium of the game defined by $(Q_1, \ldots, Q_N)$ for {\sf ColoEDR}\xspacevdr. For such an equilibrium, the vector $\mathbf{s}^a$ defined by $s_n^a = S(p(\mathbf{b}^a), b_n^a)$ is the unique optimal solution to the following optimization:
\begin{subequations}
\label{eqn: mc0_anticipating}
\begin{align}
\label{eqn: mc1_anticipating}
\max \quad & ud - \frac{ud^2}{2\sum_n D_n} - \sum_n \hat{c}_n(s_n) \\
\label{eqn: mc2_anticipating}
\text{s.t.} \quad & \sum_n s_n = d \\
\label{eqn: mc3_anticipating}
& d \ge 0,\ 0 \le s_n \le D_n, \quad n = 1, \ldots, N,
\end{align}
\end{subequations}
where, for $s_n \ge 0$,
\begin{align}
\hat{c}_n(s_n) = &\frac{1}{2}\left(s_n \frac{\gamma_n u}{2} + c_n(s_n)\right)
+ \frac{1}{2}\int^{s_n}_0 \sqrt{ \left(\frac{\gamma_n u}{2} - \mz \right)^2
+ 2 \mz \frac{ zu}{\Sigma_i D_i}} dz ,
\label{eqn: vdr_modified_cost}
\end{align}
and for $s_n <0, \quad \hat{c}_n(s_n) = 0.$
\label{thm: vdr_ne_anticipating}
\end{theorem}
Like in the case of mandatory EDR\xspace, the above characterization can be approximated using a modified cost function when $\gamma_n$ is small, i.e., when there are a large number of firms and all firms have similar market shares.
\begin{lem}
For $0 \le s_n \le D_n$, the modified cost in \eqref{eqn: vdr_modified_cost} can be upper and lower bounded by,
\[
c_n(s_n) \le \hat{c}_n (s_n)\le c_n(s_n) + s_n \frac{\gamma_n u}{2},
\]
Furthermore, where the left or right derivatives are defined, we have
\begin{subequations}
\begin{align}
&\frac{\partial^- c_n(s_n)}{\partial s_n} \le \frac{\partial^-\hat{c}_n(s_n)}{\partial s_n}
\le \frac{\partial^+\hat{c}_n(s_n)}{\partial s_n} \le \frac{\partial^+ c_n(s_n)}{\partial s_n} + \frac{\gamma_n u}{2}.
\end{align}
\end{subequations}
\label{lemma: vdr_mc_bound}
\end{lem}
\subsection{Bounding Efficiency Loss}
We now use the characterization results of Theorem \ref{thm: vdr_ne_taking} and Theorem \ref{thm: vdr_ne_anticipating} to analyze the social efficiency of {\sf ColoEDR}\xspacevdr in the voluntary EDR\xspace setting for both price-taking and price-anticipating tenants.
\begin{theorem}
For price taking tenants, the welfare loss of {\sf ColoEDR}\xspacevdr is bounded by
$ud^t - \sum_n c_n(s_n^t) \ge ud^* - \sum_n c_n(s_n^*) - \frac{u {d^*}^2 }{2\sum_n D_n}.$
Moreover, the bound is tight.
\label{thm: vdr_loss_taking}
\end{theorem}
\begin{theorem}
For price anticipating tenants, the welfare loss of {\sf ColoEDR}\xspacevdr is bounded by
$ ud^a - \sum_n c_n(s_n^a) \ge ud^* - \sum_n c_n(s_n^*) - \frac{u}{2} \left(\Sigma_n D_n \gamma_n + \frac{{d^*}^2}{\Sigma_n D_n }\right).$
\label{thm: vdr_loss_anticipating}
\end{theorem}
Theorem \ref{thm: vdr_loss_taking} highlights that the price-taking market equilibrium is efficient when the optimal energy reduction $d^*$ is small. This is due to the profit maximizing behavior of the operator: when the social optimal $d^*$ is large, the operator has greater opportunity to raise his profit by lowering the market price.
Comparing Theorem \ref{thm: vdr_loss_anticipating} with Theorem \ref{thm: vdr_loss_taking}, we can see that when tenants are price-anticipating, the additional welfare loss due to the price-anticipating behavior of tenants is a function of $\gamma_n$, the market share of the tenants. It is easy to see the additional loss of social utility is minimized when $\gamma_n = 1/N$ for all $n$, i.e., when the reduction capacity of each tenant is equal.
Additionally, we can obtain tight bounds on the market clearing price, energy reduction quantity, and operator's profit in a similar fashion as our analysis done for the mandatory EDR\xspace case using Theorem \ref{thm: vdr_ne_taking} and Theorem \ref{thm: vdr_ne_anticipating}. The results are summarized in Table \ref{table: vdr_comp_optimal} and Table \ref{table: vdr_comp_taking}.
\subsection{Market Clearing Price}
\begin{prop}
When tenants are price-taking, the operator running {\sf ColoEDR}\xspacevdr uses more on-site generation and pays a lower price for power reduction to its tenants than the social optimal. Specifically,
$d^t \le d^*$ and~ $(1 - \frac{d^*}{\sum_n D_n}) p^* \le p^t \le p^*.$
\label{prop: vdr_price_taking}
\end{prop}
By Lemma \ref{lemma: vdr_mc_bound}, we can characterize the the price markup under the supply function bidding mechanism:
\begin{theorem}
Let $(p^t, d^t)$ be the equilibrium price and total tenant energy reduction when tenants are price-taking, and $(p^a, d^a)$ be those when tenants are price-anticipating, then let $\gamma = \max_n \gamma_n$, $D = \max_n D_n$, we have,
$d^t \ge d^a \ge d^t - D/2$ and~ $p^t \le p^a \le \min( p^*, p^t + u\gamma/2).$
\label{thm: vdr_price_diff}
\end{theorem}
\subsection{Operator's profit}
Let $U_o(p, d) = (u - p) d$ be the operator's when the market clearing price is $p$ and the total demand response from tenants are $d$. From the price and vdr-quantity bounds provided in the previous sections, we can give bound on the utility of {\sf ColoEDR}\xspacevdr.
\begin{theorem}
\label{thm: vdr_operator_profit}
Suppose that Assumptions \ref{asn: cost_convexity}, \ref{asn: vdr-u-condition}, \ref{asn: vdr-mc-lbound} hold. The net utility for the colo operator of {\sf ColoEDR}\xspacevdr can be characterized by $0= U_o(p^*, d^*) \le U_o(p^a, d^a) \le U_o(p^t, d^t) \le \frac{u{d^*}^2}{\Sigma_n D_n}$, and furthermore, $U_o(p^t, d^t) \le U_o(p^a, d^a) + uD.$
\end{theorem}
Table \ref{table: vdr_comp_optimal} shows that as the optimal reduction $d^*$ increases, there is more opportunity for the operator to profitably reduce market price and increase his own profit. Table \ref{table: vdr_comp_taking} shows further that, when tenants are price-anticipating, they will drive the market clearing price up, provide less energy reduction and reduce the operator's profit. However, all these additional losses can be bounded by linear functions of $\gamma$, the dominant share of the energy reduction capacity. Hence the loss due to price-anticipating behavior of tenants are minimized $D_1 = D_2 = \cdots = D_N$.
\begin{table}
{\mathrm{c}\mkern-6.5mu{\mid}}ering
{\small
\begin{tabular}{|c|c|c|c|}\hline
Tenants & Price Ratio& Colo Extra Profit & Welfare Loss \\ \hline
Price-taking & $[1 - \frac{d^*}{\Sigma_n D_n},~ 1 ]$ & $[0, ~ u {d^*}^2/ \Sigma_n D_n ]$ & $[0, ~ u{d^*}^2 / 2 \Sigma_n D_n]$ \\ \hline
Price-anticipating & $[1 - \frac{d^*}{\Sigma_n D_n},~ 1 ]$ & $[0, ~ u {d^*}^2/ \Sigma_n D_n ]$ & $[0, ~ u (\Sigma_n D_n \gamma_n + {d^*}^2 / \Sigma_n D_n ) /2 ]$\\ \hline
\end{tabular}
}
\caption{Performance guarantee of {\sf ColoEDR}\xspacevdr compared to the social optimal allocation.}
\label{table: vdr_comp_optimal}
\end{table}
\begin{table}
{\mathrm{c}\mkern-6.5mu{\mid}}ering
{\small
\begin{tabular}{|c|c|c|c|}\hline
Price Markup & Load Reduction & Operator's cost \\ \hline
$[0, ~ u\gamma/2]$ & $[-D / 2,~ 0 ]$ & $[0, ~ uD]$ \\ \hline
\end{tabular}
}
\caption{Performance guarantee of {\sf ColoEDR}\xspacevdr when tenants are price-anticipating compared to them being price-taking.}
\label{table: vdr_comp_taking}
\end{table}
\newcommand{{\mathrm{c}\mkern-6.5mu{\mid}}}{{\mathrm{c}\mkern-6.5mu{\mid}}}
\section{Case Study}\label{sec:simulation}
Our goal in this section is to investigate {\sf ColoEDR}\xspace in a realistic scenario. Given the theoretical results in the prior sections, we know that {\sf ColoEDR}\xspace is efficient for both the operator and tenants when the number of tenants is large, but that it may use excessive on-site generation (in the worst case). Thus, two important issues to address in the case study are: \textit{How efficient is the pricing mechanism in small markets, i.e., when $N$ is small? What is the impact of the pricing mechanism on on-site generation in realistic scenarios?} Additionally, the case study allows us to better understand when it is feasible to obtain load shedding from tenants, i.e., \emph{how flexible must tenants be in order to actively participate in a load shedding program?}
We discuss only on mandatory EDR\xspace in this section. The results in the case of voluntary EDR\xspace are parallel.
\subsection{Simulation Settings}
We use trace-based simulations in our case study. Our simulator takes the tenants' workload trace and a trace of mandatory EDR\xspace signals from PJM as its inputs. It then executes {\sf ColoEDR}\xspace (by emulating the bidding process and tenants' energy reduction for EDR\xspace),
and outputs the resulting equilibrium. The settings we use for modeling the colocation data center and the tenant costs follow.
\textbf{Colocation data center setup.}
We consider a colocation data center located in Ashburn, VA, which is a major data center market served by PJM Interconnection \cite{pjm}.
By default, there are three participating tenants interested in EDR\xspace, though we vary the number of participating tenants during the experiments.
Each participating tenant has 2,000 servers, and each server has an idle and peak power of $150$W and $250$W, respectively.
The default PUE of the colo is set to $1.5$ (typical for colo), and hence, whenever a tenant reduces 1kWh energy, the corresponding energy reduction at the colo level amounts to 1.5kWh. Thus, the maximum possible power reduction is 2.25MW (i.e., 1.5MW IT plus 0.75 non-IT).
We assume that the colo operator counts the extra energy reduction at the colo level as part of the tenants' contributions, and rewards the tenants accordingly.
The colo has an on-site diesel generator, which has cost $0.3$\$/kWh estimated based on typical fuel efficiency \cite{wiki_diesel_cost}.
For setting the energy reduction target received by the colo, we follow the EDR\xspacep signals issued by PJM Interconnection on January $7$, $2014$, when
many states in eastern U.S. experienced an extremely cold weather and faced electricity production shortage \cite{pjm}. Fig.~\ref{fig:pjm_edr} shows the total energy reduction requirement by PJM, which we further normalize and scale down such
that maximum energy reduction target for our considered colo is 900kWh.
\textbf{Tenant workloads characteristics.} We choose three representative types of workloads for participating tenants: tenant 1 is running delay-sensitive workloads (e.g., user-facing web service), tenant 2 is running delay-moderate workloads (e.g., enterprise's internal services), and tenant 3 is running delay-tolerant workload (e.g., back-end processing).
The workload traces for the three participating tenants were collected from logs of MSR \cite{thereska2009sierra}, Wiki \cite{urdaneta2009wikipedia}, and a public university (anonymous for review), respectively. Fig.~\ref{fig:workload} illustrates a snapshot of the traces, where the workloads are normalized with respect to each tenant's maximum service capacity.
The illustrated results us an average utilization for each tenant of 30\%, consistent with reported values from real systems \cite{Hoelzle_datacenter_book_2013}. Our results are not particularly sensitive to this choice.
\begin{figure}
\caption{\textbf{(a)}
\label{fig:workload}
\label{fig:pjm_edr}
\label{fig:trace}
\end{figure}
\begin{figure*}
\caption{Performance comparison under default settings. Throughout this and later plots, the bars in each cluster are the price-taking, price-anticipating, socially optimal, and diesel only (if applicable) outcomes. }
\label{fig:default:welfareLoss}
\label{fig:default:energy}
\label{fig:default:netUtilityTenant}
\label{fig:default:netUtilityDC}
\label{fig:default:price}
\label{fig:default:u1}
\label{fig:default:u2}
\label{fig:default:u3}
\label{fig:default}
\end{figure*}
There are various power management techniques, e.g., load migration/scheduling, that can be used for reducing tenants' server energy consumption. Here, as a concrete example, we consider that tenants dynamically turn on/off servers according to workloads for energy saving subject to SLA \cite{LinWiermanAndrewThereska}. This power-saving technique has been widely studied \cite{LinWiermanAndrewThereska,Gandhi:2012:ADR:2382553.2382556} and also recently applied in real systems (e.g., Facebook's AutoScale \cite{Facebook_AutoScale_Energy_2014}).
When tenants save energy for EDR\xspace by turning off some unused servers, their application performance might be affected. We adopt a simple model based on an M/G/1/Processor-Sharing queueing model, as follows. For a tenant with $M$ servers each with a service rate of $\mu$, denote the workload arrival rate by $\lambda$. When $m$ servers are shut down, we model the
total delay cost as
$\bar{c}(m) =\lambda \cdot \beta\cdot T \cdot \text{delay}(m)=\frac{\beta T}{\frac{1}{uM}-\frac{1}{M-m}}$,
where $u=\frac{\lambda}{\mu M}$ denotes the normalized workload arrival (i.e.,
utilization without turning off servers),
$T$ is the duration of an EDR\xspace event, and $\beta$ is a cost parameter ({\$/time unit/job}). In our simulations, we set the cost parameter for tenant 1, tenant 2 and tenant 3 as 0.1, 0.03, 0.006, respectively, which
are already higher than those considered in the prior context of turning off servers for energy saving \cite{LinWiermanAndrewThereska}. Note that we have experimented with a variety of other models as well and the results do not qualitatively change.
We use a standard model for energy usage \cite{Hoelzle_datacenter_book_2013} and take the energy reduction $s$ as linear in the number of servers shut down, i.e., $s = \theta\cdot m$, where $\theta$ is a constant decided by server's idle power and $T$.
Then, it yields the following cost function for tenants's energy reduction $c(s) =\bar{c}(\frac{s}{\theta})-\bar{c}(0)$, where $\bar{c}(\,\cdot\,)$ is defined in the above paragraph.
Note that we have experimented with a variety of other forms, and our results are not sensitive to the details of this cost function.
Finally, note that tenants typically have delay performance requirement which, based on the above queueing model, is translated as an utilization upper bound. Such
translation is also common in real systems (e.g., default policy for auto-scaling virtual machines \cite{Azure_AutoScaling_Rule}). In our simulation, we capture the performance constraint by setting utilization upper bounds for tenant 1, tenant 2, and tenant 3 as 0.5, 0.6, and 0.8, respectively.
\textbf{Efficiency benchmarks.} Throughout our experiments, we consider the price-taking, price-anticipating, and social optimal outcomes. Additionally, we consider one other benchmark, \emph{diesel only}, which is meant to capture common practice today. Under diesel only, the full EDR\xspace response is provided by the on-site diesel generator. \textit{Throughout, our results are presented in grouped bar plots with the bars representing (from left to right) the price-taking, price-anticipating, social optimal, and diesel only (if applicable) outcomes. }
While other mechanisms (e.g., direct pricing \cite{Liu:2014:PDC:2591971.2592004},
auction \cite{Shaolei_Colo_TruthDR_Tech}) have been introduced in recent papers, we do not compare {\sf ColoEDR}\xspace with them here because {\sf ColoEDR}\xspace is already typically indistinguishable from the social optimal cost.
\subsection{Performance Evaluation}
We now discuss our main results, shown in Fig.~\ref{fig:default}.
\textbf{Social cost.} We first compare in
Fig.~\ref{fig:default:welfareLoss} the social costs incurred by different algorithms. Note that {\sf ColoEDR}\xspace is close to the social cost optimal under both price-taking
and price-anticipating cases even though there are only three participating
tenants. Further, the resulting social costs in both the price-taking and price-anticipating scenarios are significantly lower than that of the diesel only outcome. This shows a great potential of tenants' IT power reduction
for EDR\xspace, which is consistent with the prior literature on owner-operated
data center demand response \cite{aikema2012data_ancillary_IGCC_2012,Liu:2014:PDC:2591971.2592004,AdamWierman_DataCenterDemandResponse_Survey_IGCC_2014}.
\textbf{Energy reduction contributions.}
Fig.~\ref{fig:default:energy} plots EDR\xspace energy reduction contributions from tenants and the diesel generator. As expected from analytic results, both price-taking and price-anticipating tenants tend to
contribute less to EDR\xspace (compared to the social optimal) because of their self-interested decisions.
In other words, given self-interested tenants, the colo operator needs more diesel generation than the social optimal. Nonetheless, the difference is fairly small, much smaller than predicted by the worst-case analytic results. This highlights that worst-case results were too pessimistic in this case. Of course, one must remember that all tenant reduction extracted is in-place of diesel generation, and so serves to make the demand response more environmentally friendly.
\textbf{Benefits for tenants and colocation operator.}
We show in Fig.~\ref{fig:default:netUtilityTenant} and
Fig.~\ref{fig:default:netUtilityDC} that both the tenants and the colo operator can benefit from {\sf ColoEDR}\xspace.
Specifically, Fig.~\ref{fig:default:netUtilityTenant} presents net profit (i.e., payment made by colo operator minus
performance cost) received by tenants, showing that all participating tenants receive positive net rewards.
While price-anticipating tenants can receive higher net rewards than when they are price-taking, the extra reward
gained is quite small. Similarly, Fig.~\ref{fig:default:netUtilityDC} shows cost saving for the colo operator, compared to the ``diesel only'' case .
\textbf{Market clearing price.}
Fig.~\ref{fig:default:price} shows the market clearing price. Naturally, when using {\sf ColoEDR}\xspace to incentivize tenants for EDR\xspace while
minimizing the total cost, the colo operator will not pay the tenants
at price higher than its diesel price (shown via the red horizontal line). We also note that the price under the price-anticipating case is higher than that under the price-taking case, because the price-anticipating tenants are more strategic. However, the price difference between price-anticipating and price-taking cases is quite small, which again confirms our analytic results.
\textbf{Tenant' server utilization.}
Tenants' server utilizations are shown in
Figs.~\ref{fig:default:u1}, \ref{fig:default:u2} and \ref{fig:default:u3}, respectively. These illustrate that, while tenants reduce energy for EDR\xspace, their server utilizations still stay within their respective limits (shown via the red horizontal lines), satisfying performance constraints. This is because
tenants typically provision their servers based on the maximum possible workloads (plus a certain margin), while in practice their workloads are usually quite low, resulting
in a ``slackness'' that allows for saving energy while still meeting their performance requirements.
\subsection{Sensitivity Analysis}
To complete our case study, we investigate the sensitivity of the conclusions discussed above to the settings used. For each study, we only show results that are significantly different than those in Fig.~\ref{fig:default}.
\begin{figure}
\caption{Impact of number of tenants.}
\label{fig:diffN:welfareLoss}
\label{fig:diffN:price}
\label{fig:diffN:netUtilityTenant}
\label{fig:diffN:netUtilityDC}
\label{fig:N}
\end{figure}
\textbf{Impact of the number of tenants.}
First, we vary the number of participating tenants and show the results in Fig.~\ref{fig:N}.
To make results comparable, we fix the EDR\xspace energy reduction requirement as well as total number of servers: tenant 1, tenant 2 and tenant 3 are each equally split into multiple smaller tenants, each having fewer servers.
We then aggregate replicas of the same tenant together for an easy viewing in the figures, e.g., ``tenant 1'' in the figures represent the whole group of tenants that are obtained by splitting the original tenant 1.
One interesting observation is that as more tenants participate in EDR\xspace, the market becomes more ``competitive''.
Hence, each individual
tenant can only gain less net reward, but both the price and the aggregate net reward become
higher (see Figs.~\ref{fig:diffN:price} and~\ref{fig:diffN:netUtilityTenant}).
Motivated by this, one might suggest a possible trick: a tenant may gain more utility by splitting its servers and pretending as multiple tenants. In practice, however, each tenant has only one account (for billing, etc.) which requires contracts and base fees, and thus pretending as multiple tenants is not possible in a colo.
\begin{figure}
\caption{Impact of diesel price.}
\label{fig:diffAlpha:welfareLoss}
\label{fig:diffAlpha:energy}
\label{fig:diffAlpha:price}
\label{fig:diffAlpha:netUtilityTenant}
\label{fig:alpha}
\end{figure}
\textbf{Impact of the price of diesel.}
Fig.~\ref{fig:alpha} illustrates how our result changes as the diesel price varies. Intuitively, as
shown in Fig.~\ref{fig:diffAlpha:welfareLoss}, the social cost (which includes diesel cost as a key component) increases with the diesel price. We see from Figs.~\ref{fig:diffAlpha:energy} and \ref{fig:diffAlpha:price} that, when diesel price is very low (e.g., $0.1$\$/kWh), the colo operator is willing to use more diesel and offers a lower price to tenants.
As a result, tenants contribute less to EDR\xspace.
As the diesel price increases (e.g., from $0.2$\$/kWh to $0.3$\$/kWh), the colo operator increases the market price (but still below the diesel price) to encourage tenants to cut more energy for EDR\xspace. Nonetheless,
tenants' energy reduction contribution cannot increase arbitrarily due to their performance constraints.
Specifically, after the diesel price exceeds 0.4\$/kWh, tenants will not contribute more to EDR\xspace (i.e., almost all their IT energy reduction capabilities have been exploited), even
though the colo operator increases the reward. In this case, tenants simply receive higher net
rewards without further contributing to EDR\xspace, as shown in Fig.~\ref{fig:diffAlpha:netUtilityTenant}.
\begin{figure}
\caption{Impact of EDR\xspace energy reduction target.}
\label{fig:diffDelta:energy}
\label{fig:diffDelta:price}
\label{fig:delta}
\end{figure}
\textbf{Impact of EDR\xspace requirement.}
Fig.~\ref{fig:delta} varies the EDR\xspace energy reduction target, with the maximum reduction ranging from $20\%$ to $120\%$ of the colo's peak IT power consumption.
As the EDR\xspace energy reduction target increases, tenants' energy reduction for EDR\xspace also increases;
after a certain threshold, diesel generation becomes the main approach to EDR\xspace, while the increase in tenant's contribution is diminishing (even though
the colo operator increases the market price), because of tenants' performance requirements
that limit their energy reduction capabilities.
\begin{figure}
\caption{Impact of tenants' workloads and the workload prediction errors.}
\label{fig:diffU:welfareLoss}
\label{fig:diffU:energy}
\label{fig:preError:welfareLoss}
\label{fig:preError:price}
\label{fig:u}
\end{figure}
\textbf{Impact of tenants' workloads.}
In Fig. \ref{fig:diffU:welfareLoss}-\ref{fig:diffU:energy}, we vary the tenants' workload intensity (measured in terms of the average server utilization
when all servers are active) from $10\%$ to $50\%$, while
still keeping the maximum utilization bounds to $50\%$, $60\%$ and $80\%$ as
the performance requirements for the three tenants, respectively.
While it is straightforward that when tenants have more workloads, they tend
to contribute less to EDR\xspace, because they need to keep more servers active
to deliver a good performance. Nonetheless, even when their average
utilization without turning off servers is as high as 50\% (which is quite high in real systems,
considering that the average utilization is only around 10-30\% \cite{Hoelzle_datacenter_book_2013}),
tenants can still contribute more than 20\% of EDR\xspace energy reduction under {\sf ColoEDR}\xspace,
showing again the potential of IT power management for EDR\xspace.
\textbf{Impact of workload prediction error.}
In practice, tenants may not perfectly estimate their own workload arrival rates. To cope with possible traffic spikes, tenants can either keep more servers active as a backup or deliberately overestimate the workload arrival rate by a certain overestimation factor. We choose the later approach in our simulation.
Fig.~\ref{fig:preError:welfareLoss}-\ref{fig:preError:price} shows the result under workload prediction errors. We see that
both the social cost and market price are fairly robust against tenants' workload over-predictions.
For example, the social cost
increases by less than $10\%$, even when tenants overestimate their workloads
by $20\%$ (which is already sufficiently high in practice, as shown in \cite{Gandhi:2012:ADR:2382553.2382556}).
Other results (e.g., tenants' net reward, colo operator's total cost)
are also only minimally affected,
thereby demonstrating the robustness of {\sf ColoEDR}\xspace against
tenants' workload over-predictions.
\section{Related Work}\label{sec:related_work}
Our work contributes both to the growing literature on data center demand response, and to the literature studying supply function equilibria. We discuss each in turn below.
Recently, data center demand response has received a growing amount of attention. A variety of approaches have been considered, such as optimizing grid operator's pricing strategies for data centers \cite{Liu:2014:PDC:2591971.2592004}
and tuning computing (e.g., server control and scheduling) and/or non-computing knobs (e.g., cooling system) in data centers for various types of demand response programs
\cite{aikema2012data_ancillary_IGCC_2012,Wang:2014:ESG:2567529.2567556,DataCenterDemandResponse_CollaborativeOPtimization_10.1109/ICDEW.2013.6547462,AksanliRosing14_ProvidingRegulationServicesManagingDataCenterPeakPower,Chen_PowerControl_Regulation_CDC_2013}.
Field tests by LBNL also verify the practical feasibility of
data center demand response using a combination of existing power management techniques
(e.g., load migration) \cite{DataCenterDemandResponse_Report_Berkeley}.
These studies, however, have all focused on large owner-operated data centers.
In contrast, to our best knowledge, colocation demand response has been investigated by only a few previous works. The first is \cite{Shaolei_Colocation_ICAC_2014}, which proposes a simple mechanism, called iCODE, to
incentivize tenants' load reduction. But, iCODE is purely based on ``best effort'' and does not include any energy reduction target (needed for EDR\xspace). More importantly, iCODE is designed without considering strategic behavior by tenants, and can be compromised by price-anticipating tenants \cite{Shaolei_Colocation_ICAC_2014}. More relevant to the current work is \cite{Shaolei_Colo_TruthDR_Tech},
which proposes a VCG-type auction mechanism where colocation participation in EDR\xspace programs. While the mechanism is approximately truthful, it
asks participating tenants to reveal their private cost information through complex bidding functions. Further, the colocation operator may be forced to make arbitrarily high payments to tenants. In contrast, our proposed solution provides a simple bidding space, protects tenants' private valuation,
and ensures that the colocation operator does not incur a higher cost for EDR\xspace than
the case tenant contributions. Thus, unlike \cite{Shaolei_Colo_TruthDR_Tech},
{\sf ColoEDR}\xspace benefits both colocation operator and tenants, giving both parties incentives to cooperate for EDR\xspace.
Finally, it is important to note that our approach builds on, and adds to, the supply function mechanism literature. Supply function bidding (c.f. the seminal work by \cite{klemperer1989}) is frequently used in electricity markets due to its simple bidding language and the avoidance of the unbounded payments typical in VCG-like mechanisms. Supply function bidding mechanisms have been extensively studied, e.g., \cite{day2002, baldick2004, green1992, green1996, anderson2008, vives2011}. The literature primarily focuses on existence and computation of supply function equilibrium, sometimes additionally proving bounds on efficiency loss. Our work is most related to \cite{johari2011}, which considers an inelastic demand $\delta$ that must be satisfied via extracting load shedding from consumers and proves efficient bounds on supply function equilibrium. In contrast, our work assumes that the operator has an outside option (diesel) that can be used to satisfy the inelastic demand. This leads to a multistage game between the tenants and the profit-maximizing operator, a dynamic which has not been studied previously in the supply function literature.
\section{Conclusion}
In this paper, we focused on ``greening'' colocation demand response by designing a pricing mechanism that can extract load reductions from tenants during EDR\xspace events. Our mechanism, {\sf ColoEDR}\xspace, can be used in both mandatory and voluntary EDR\xspace programs and is easy put in place given systems available in colos today. The main technical contribution of the work is the analysis of the {\sf ColoEDR}\xspace mechanism, which is a supply function mechanism for an elastic setting, a setting for which efficiency results have not previously been attained in the supply function literature. Our results highlight that {\sf ColoEDR}\xspace provides provably near-optimal efficiency guarantees, both when tenants are price-taking and when they are price-anticipating. We also evaluate {\sf ColoEDR}\xspace using trace-based simulation studies and validate
that {\sf ColoEDR}\xspace is both beneficial to the colo operator (by reducing costs), to the environment (by reducing diesel usage), and to the tenants (by providing payments for reductions).
{\scriptsize
}
\appendix
\newcommand{\lmmc}{\frac{
\partial^-\hat{c}_n(s_n)}{
\partial s_n}}
\newcommand{\rmmc}{\frac{
\partial^+\hat{c}_n(s_n)}{
\partial s_n}}
\newcommand{section}{section}
\section{Price taking tenants}
\subsection{Proof of Proposition \ref{prop: ec1}} When tenants are price takers, they maximize the payout $P_n(b_n, p) = pS_n(b_n, p) - c_n(s_n)$ over the bid $b_n$. Note that $b_n \in [0, p\delta]$ as no tenant will bid beyond $p\delta$ otherwise the payout $P_n<0$. Hence $\mathrm{b} = (b_1, \ldots, b_n)$ is an equilibrium if and only if the following condition is satisfied
{\small \begin{subequations}
\begin{align}
\label{eqn: ne1} \frac{
\partial^-c_n(s_n)}{
\partial s_n} \le p, \quad & 0 \le b_n < p\delta, \\
\label{eqn: ne2} \frac{
\partial^+c_n(s_n)}{
\partial s_n} \ge p, \quad & 0 < b_n \le p\delta.
\end{align}
\end{subequations}
}
At least one feasible solution to \eqref{eqn: p1} exists because it is minimizing a continuous function over a compact set. Furthermore, \eqref{eqn: p1-2} - \eqref{eqn: p1-3} satisfy standard constraint qualification, hence for the Lagrangian
{\small\[L(\mathrm{s}, \mu) = \sum_{n} c_n(s_n) + \mu((\delta-y) - \sum_n s_n), \]}
there exists optimal primal dual pair $(\mathrm{s}, \mu)$, such that \eqref{eqn: p1-2} and \eqref{eqn: p1-3} are satisfied, and
{\small\begin{subequations}
\begin{align}
\label{eqn: opt1} \frac{
\partial^-c_n(s_n)}{
\partial s_n} \le {\mu}, \quad & s_n > 0, \\
\label{eqn: opt2} \frac{
\partial^+c_n(s_n)}{
\partial s_n} \ge {\mu}, \quad & s_n \ge 0.
\end{align}
\end{subequations}}
Given the optimal $(\mathrm{s}, \mu)$, let $p = {\mu}$, and $b_n = p(\delta - s_n)$, then \eqref{eqn: p1-2} implies $p$ satisfies \eqref{eqn: price}, and \eqref{eqn: opt1}-\eqref{eqn: opt2} implies \eqref{eqn: ne1} - \eqref{eqn: ne2}, hence an equilibrium exists.
Conversely, if $(\mathrm{b}, p)$ is an equilibrium and $p$ satisfies \eqref{eqn: price}, the resulting allocation $\mathrm{s}$ is optimal to \eqref{eqn: p1}. To see this, if $0 \le s_n < \delta - y$ for all $n$, \eqref{eqn: ne1}-\eqref{eqn: ne2} is equivalent to \eqref{eqn: opt1}-\eqref{eqn: opt2} if we set $\mu = p$, hence $(\mathrm{s}, \mu)$ is primal dual optimal pair for \eqref{eqn: p1}. If $s_n = (\delta - y)$, then $s_m = 0, \forall m \ne n$. In this case, we set $\bar{\mu} = \min\{ p,
\partial^+c_n(s_n)/
\partial s_n\}$, and we can check that $(\mathrm{s}, \bar{\mu})$ is the primal dual optimal solution for \eqref{eqn: p1}.
\subsection{Proof of Theorem \ref{thm: price-taking-characterization}} By Proposition \ref{prop: ec1}, when tenants are price-taking, for any $y$, the there is always an equilibrium, and the resulting $\mathbf{s}$ is always the optimal allocation to provide $(\delta - y)$ energy reduction.
Hence we only need to verify that the on-site generation level $y$ is the solution to \eqref{eqn: price-taking1}-\eqref{eqn: price-taking3}. Similar to the proof of Proposition \ref{prop: ec1}, by Assumption \ref{asn: cheap_on-site}, the first order optimality condition for the $y$ in \eqref{eqn: price-taking1}-\eqref{eqn: price-taking3} is $\frac{\alpha}{N\delta}(y+(N-1)\delta) = p.$
By Proposition \ref{prop: ec1}, $p$ satisfies the relation \eqref{eqn: price}, substitute the left-hand-side into \eqref{eqn: price} and solve for $y$, we have
$y = \sqrt{\frac{\Sigma_n b_n N\delta}{\alpha}} - (N-1)\delta.$
This is exactly the on-site generation $y$ that minimizes $\mathrm{cost}_o(\mathbf{b}, y)$ given in \eqref{eqn: on-site_gen}. Hence the datacenter will always pick $y$ that is optimal for \eqref{eqn: price-taking1}-\eqref{eqn: price-taking3}, together with Proposition \ref{prop: ec1}, an equilibrium exists, and the resulting allocation $(\mathbf{s}, y)$ is optimal for \eqref{eqn: price-taking1}-\eqref{eqn: price-taking3}.
\subsection{Proof of Proposition \ref{prop: ec2}} Since $y \ge 0$, it suffices to prove that whenever the optimal on-site generation is non-zero, $y^* >0$, $y^t \ge y^*$. From \eqref{eqn: edr1}, the Lagrangian of $\mathsf{SCM}$\xspace is
\[ L(\mathrm{s}, y, \mu^*, \lambda^*) = \sum_n c_n(s_n) + \alpha y + \mu^*((\delta - y) - \sum_n s_n) - \lambda^* y. \]
By constraint qualification and the KKT conditions,
assuming $y^* >0$, then $\lambda = 0$, $\mu^* = \alpha$, hence the market clearing price in the optimal allocation should be $p^* = \alpha $.
Next, consider the market price for price taking tenants. From \eqref{eqn: price2},
\begin{equation}
\label{eqn: price_t1} p^t = \frac{ \sum_{i\in\mathcal{N}}b^t_i}{(N-1)\delta+y^t} = \sqrt{\frac{(\Sigma_{i\in \mathcal{N}}b^t_i)\alpha}{N\delta}}.
\end{equation}
The second equality yields $ \sum_{i\in\mathcal{N}}b^t_i = \frac{((N-1)\delta+y^t)^2}{N\delta}\alpha$. Substitute this back to \eqref{eqn: price_t1},
\begin{align}
\label{eqn: price_t2} p^t &= \frac{ \sum_{i\in\mathcal{N}}b^t_i}{(N-1)\delta+y^t}= \frac{(N-1)\delta+y^t}{N\delta}\alpha.
\end{align}
And note that $y_t \in [0, \delta]$ and $p^* = \alpha$, thus \eqref{eqn: price_t2} yields
$\frac{N-1}{N}p^* \le p^t \le p^*.$
Finally, from \eqref{eqn: price-taking0}, the Lagrangian of the price-taking characterization optimization is,
\begin{align*}
L(\mathrm{s}, y, \mu^t, \lambda^t) = &\sum_n c_n(s_n) + \yterm
+ \mu^t((\delta - y) - \sum_n s_n) - \lambda^t y.
\end{align*}
By examining the KKT condition and using a similar argument to the proof of Proposition \ref{prop: ec1}, we have $p^t = \mu^t$, also,
$\frac{\partial^{-}c_n(s^t_n)}{\partial s^t_n} \le p^t \le p^* \le \frac{\partial^{+}c_n(s^*_n)}{\partial s^*_n}.$
Thus, $\forall n, s^t_n \le s^*_n$. Since $y = \delta - \sum s_n$, $y^t \ge y^*$.
\subsection{Proof of Proposition \ref{prop: on-site_gen}} From the proof of Proposition \ref{prop: ec2}, we see that when $y^*>0$, $\lambda^* = 0$, and $\mu^* = \alpha$. Furthermore, we have $ \sum_n s_n < \delta$, but $s_n = \delta - \frac{b_n}{\mu^*}$. Hence
$ (N\delta - \frac{\Sigma_n b_n}{\alpha}) < \delta. $
Conversely, if \eqref{eqn: cheap_on-site} holds, then
$ \alpha (N-1)\delta < \sum_n b_n.$
But by Proposition \ref{prop: ec1} and \eqref{eqn: price}, we have
$ \sum_n b_n = (p^* (N-1)\delta + y). $
By combining the two equations above:
$ \alpha(N-1)\delta < p^*((N-1)\delta+y^*).$
However, from the proof in Proposition \ref{prop: ec1}, we have $p^* \le \alpha$, hence we must have $y^*>0$.
On the other hand, when the data center operator is profit maximizing, the cost to the operator
$\mathrm{cost}_o(\mathbf{b}, y) = \frac{(\Sigma_n b_n)(\delta-y)}{(N-1)\delta + y} + \alpha y$
is a convex function in $y$ over the domain $y \ge 0$. By first order condition, the cost is minimized when
\begin{align}
y' = \sqrt{\frac{N\ \delta \Sigma_n b_n}{\alpha}} - (N-1)\delta, \label{eqn: yunc}
\end{align}
then $y = y'$ if and only if $y' \in [0, \delta]$. However,
$\Sigma_n b_n = \Sigma_n p(\delta - s_n)={p}((N-1)\delta + y) \le \alpha (N\delta), $
where the last inequality is because $y \le \delta$, and $p \le \alpha$, since operator always has the option to use on-site generation to get unit cost of energy reduction at $\alpha$. Hence we always have $y' \le \delta$. So, if $y >0$, by \eqref{eqn: yunc}, \eqref{eqn: cheap_on-site2} must hold, conversely, if \eqref{eqn: cheap_on-site2} holds, then by \eqref{eqn: yunc}, $y'>0$, so operator will use $y = y'$.
\subsection{Proof of Theorem \ref{thm: welfare_loss_taking}} Note that $(\mathbf{s}^*, y^*)$ is a feasible solution to \eqref{eqn: price-taking0}. By Theorem \ref{thm: price-taking-characterization}, we have
$\sum_n c_n(s^t_n) + \frac{\alpha}{2N\delta} (y^t + (N-1)\delta)^2 \le \sum_n c_n(s^*_n) + \frac{\alpha}{2N\delta} (y^* + (N-1)\delta)^2.$
Rearranging, we have
\begin{align*}
&\sum_n c_n(s^t_n) +\alpha y^t - \left(\sum_n c_n(s^*) + \alpha y^*\right)
\le \frac{\alpha}{2N\delta}(y^t - y^*)\left(2\delta - (y^t + y^*)\right)\\
=& \frac{\alpha}{2N\delta}[-(y^t-y^*)^2+2(\delta-y^*)(y^t-y^*)]
\le \frac{\alpha}{2N\delta}[-(y^t-y^*-(\delta-y^*))^2 + (\delta-y^*)^2]\\
= &\frac{\alpha}{2N\delta}(\delta-y^*)^2\le \frac{\alpha\delta}{2N}.
\end{align*}
\subsection{Proof of Theorem \ref{thm: colo_cost1_taking}} From Proposition \ref{prop: ec2}, we have $\frac{N-1}{N}\alpha \le p^t\le p^* = \alpha$, and $0 \le y^t \le \delta$, which yields
\begin{align*}
&\mathrm{cost}_o^*(p^*, y^*)-\mathrm{cost}_o(p^t, y^t)
= p^*(\delta-y^*)+\alpha y^* - \left(p^t(\delta-y^t)+\alpha y^t\right)=(\alpha-p^t)(\delta-y^t)
\end{align*}
Substituting the above bounds for $p^t$ and $y^t$ gives
$
0\le \mathrm{cost}_o^*(p^*, y^*)-\mathrm{cost}_o(p^t, y^t) \le \frac{\alpha \delta }{N}.
$
\section{price-anticipating tenants}
\subsection{Proof of Theorem \ref{thm: modified_cost}} The proof proceeds in a number of steps. We first show that the payoff function $Q_n$ is a concave and continuous function for each firm $n$. We then establish necessary and sufficient conditions for $\mathbf{b}$ to be an equilibrium; these conditions look similar to the optimality conditions \eqref{eqn: ne1}-\eqref{eqn: ne2} in the proof of Proposition \ref{prop: ec1}, but for a ``modified'' cost function defined according to \eqref{eqn: modified_cost}. We then show the correspondence between these conditions and the optimality conditions for the problem \eqref{eqn: mc1}-\eqref{eqn: mc3}. This correspondence establishes existence of an equilibrium, and uniqueness of the resulting allocation.
\begin{enumerate}
[label= Step \arabic*:, leftmargin=0cm,itemindent=.5cm,labelwidth=
\itemindent,labelsep=0cm,align=left]
\item \emph{ If $\mathbf{b}$ is an equilibrium, and Assumption \ref{asn: cheap_on-site} is satisfied, at least one coordinate of $\mathbf{b}$ is positive.}
By Assumption \ref{asn: cheap_on-site}, $0 < \alpha < \frac{\Sigma_n b_n}{(N-1)\delta}$, hence at least one coordinate of $\mathbf{b}$ must be positive.
\item \emph{ The function $Q_n(\bar{b}_n; \mathbf{b}_{-n})$ is concave and continuous in $\bar{b}_n$, for $\bar{b}_n \ge 0$. } From \eqref{eqn: cost_anticipating} and by plugging $p(\mathbf{b})$ into $s_n$ in \eqref{eqn: supply_function}, we have \label{step: concave}
\begin{align*}
Q_n(\bar{b}_n; \mathbf{b}_{-n}) &= \sqrt{\frac{(\Sigma_{m \ne n} b_m + \bar{b}_n)\alpha\delta}{N}} - \bar{b}_n - c_n\left(\delta - \frac{\bar{b}_n}{\sqrt{\Sigma_{m \ne n} b_m + \bar{b}_n}} \sqrt{\frac{N\delta}{\alpha}}\right).
\end{align*}
When $\Sigma_{m\ne n} b_m+ \bar{b}_n >0$, the function $\bar{b}_n / \sqrt{\Sigma_{m \ne n} b_m + \bar{b}_n}$ is a strictly concave function of $\bar{b}_n$ (for $\bar{b}_n \ge 0$). Since $c_n$ is assumed to be convex and nondecreasing (and hence continuous), it follows that $Q_n(\bar{b}_n, \mathbf{b}_{-n})$ is concave and continuous in $\bar{b}_n$, for $\bar{b}_n \ge 0$.
It is easy to show that for $s_n$ to be positive, we need $b_n \le \overline{b_n}$ where
$\overline{b_n} = \brange.$
\item \emph{In an equilibrium, $0 \le b_n \le \overline{b_n}, \forall n$.}
Tenant $n$ would never bid more than $\bar{b}_n$ given $\mathbf{b}_{-n}$. If $b_n > \overline{b_n}$, then
$S(p(\mathbf{b}), b_n) = \delta - \frac{b_n}{\sqrt{b_n + \Sigma_{m \ne n} b_m}}\frac{N\delta}{\alpha} <0.$
so the payoff $Q_n(b_n; \mathbf{b}_{-n})$ becomes negative; on the other hand, $Q_n(\overline{b_n}; \mathbf{b}_{-n}) = 0$.
We specify the following condition when marginal cost of production is not less than the price:
\begin{equation}
\forall n, \quad \lmc \le p(\mathbf{b}), \quad s_n >0. \label{eqn: p>mc}
\end{equation}
This condition is satisfied when tenants are price-taking, in the next step, we show that \eqref{eqn: p>mc} also holds in an equilibrium outcome when tenants are price-anticipating.
\item \emph{ The vector $b$ is an equilibrium if and only if \eqref{eqn: p>mc} is satisfied, at least one component of $\mathbf{b}$ is positive, and for each $n$, $b_n \in [0, \overline{b_n}],$ and the following conditions hold:
{ \begin{subequations}
\begin{align}
\label{eqn: ne3}
&\text{if }\ 0<b_n \le \overline{b_n}, \quad \frac{1}{2} \left( \rmc + \frac{\alpha}{2N} \right)
+ \frac{1}{2}\sqrt{ \left(\rmc - \frac{ \alpha}{2N} \right)^2 + \rmc \frac{2 s_n\alpha}{N\delta}} \ge p(\mathbf{b}), \\
\label{eqn: ne4}
&\text{if }\ 0 \le b_n < \overline{b_n}, \quad \frac{1}{2} \left( \lmc + \frac{\alpha}{2N} \right)
+ \frac{1}{2}\sqrt{ \left(\lmc - \frac{ \alpha}{2N} \right)^2 + \lmc \frac{2 s_n\alpha}{N\delta}} \le p(\mathbf{b}).
\end{align}
\end{subequations}}
} By Step 2, $Q_n(b_n; \mathbf{b}_{-n})$ is concave and continuous for $b_n \ge 0$. By Step 3, $b_n \in [0, \overline{b_n}]$. $b_n$ must maximize $Q_n(b_n; \mathbf{b}_{-n})$ over $0 \le b_n \le \overline{b_n}$, and satisfy the following first order optimality conditions:
{\begin{align*}
\frac{
\partial^+Q_n(b_n; \mathbf{b}_{-n})}{
\partial b_n} &\le 0 , \quad \text{if } \ 0 < b_n \le \overline{b_n}; \\
\frac{
\partial^-Q_n(b_n; \mathbf{b}_{-n})}{
\partial b_n} &\ge 0 , \quad \text{if } \ 0 \le b_n < \overline{b_n};
\end{align*}}
Recalling the expression for $p(\mathbf{b})$ given in \eqref{eqn: price2}, we have
{ \begin{align*}
\frac{1}{2\sqrt{\Sigma_m b_m}} \sqrt{\frac{\alpha\delta}{N}} - 1 + \lmc \frac{1}{p(\mathbf{b})} (1 - \frac{b_n}{2 \Sigma_m b_m}) \le 0,
\quad \text{if }\ 0 \le b_n < \overline{b_n}; \\
\frac{1}{2\sqrt{\Sigma_m b_m}} \sqrt{\frac{\alpha\delta}{N}} - 1 + \rmc \frac{1}{p(\mathbf{b})} (1 - \frac{b_n}{2 \Sigma_m b_m}) \ge 0,
\quad \text{if }\ 0 < b_n \le \overline{b_n}.
\end{align*}}
We now note that by \eqref{eqn: price2} and \eqref{eqn: supply_function}, we have :
$\frac{1}{\sqrt{\Sigma_m b_m}} = \frac{1}{p(\mathbf{b})} \sqrt{\frac{\alpha}{N\delta}},$ and $\frac{b_n}{\sqrt{\Sigma_m b_m}} = (\delta - s_n) \sqrt{\frac{\alpha }{N\delta}}. $
Substituting these two equations into the above, we have
\begin{subequations}
\begin{align}
\label{eqn: p>mc1} \frac{1}{2p(\mathbf{b})} \frac{\alpha}{N} -1 + \lmc \frac{1}{p(\mathbf{b})}\left(1 - \frac{1}{2p(\mathbf{b})} \frac{\alpha}{N} \frac{\delta - s_n}{\delta} \right) \le 0. \\
\label{eqn: p>mc2} \frac{1}{2p(\mathbf{b})} \frac{\alpha}{N} -1 + \rmc \frac{1}{p(\mathbf{b})}\left(1 - \frac{1}{2p(\mathbf{b})} \frac{\alpha}{N} \frac{\delta - s_n}{\delta} \right) \ge 0.
\end{align}
\end{subequations}
To show \eqref{eqn: p>mc} holds, we divide into two cases, when $N \ge 2$, by rearranging \eqref{eqn: p>mc1}, we have
\begin{align*}
\lmc \frac{1}{p(\mathbf{b})} &\le \frac{2Np(\mathbf{b}) - \alpha }{ 2Np(\mathbf{b}) - \alpha \frac{\delta - s_n}{\delta} } \le 1. \\
\end{align*}
This is because by Assumption \ref{asn: cheap_on-site}, $2Np(\mathbf{b}) - \alpha >0$ when $N\ge 2$. Also, we have
$2Np(\mathbf{b}) - \alpha\frac{\delta-s_n}{\delta} \ge 2Np(\mathbf{b}) - \alpha .$
Hence \eqref{eqn: p>mc} holds for $N \ge 2$.
When $N = 1$, we can simplify \eqref{eqn: p>mc1} further to
\begin{align*}
\frac{1}{2p(\mathbf{b})} {\alpha} -1 + \lmc \frac{1}{2p(\mathbf{b})} \le 0, \
\Rightarrow p(\mathbf{b}) \ge \frac{1}{2}\left(\alpha + \lmc\right) \ge \lmc.
\end{align*}
The last inequality is because $\alpha \ge \lmc$, otherwise $p(\mathbf{b}) > \alpha$, but profit maximizing operator will not pay for price more than $\alpha$, contradiction. Hence \eqref{eqn: p>mc} must hold for all $N$. After multiplying through \eqref{eqn: p>mc1}-\eqref{eqn: p>mc2} by $p(\mathbf{b})$ and rearranging, we have two quadratic inequalities in terms of $p(\mathbf{b})$.
Solving the inequalities lead to two sets of conditions of $p(\mathbf{b})$ that satisfy the first order optimality conditions, they are:
\begin{subequations}
\begin{align}
\label{eqn: p=mc2} & \text{if }\ 0 \le b_n < \overline{b_n}, \quad \frac{1}{2} \left( \lmc + \frac{\alpha}{2N} \right) \pm \half \sqrt{ \left(\lmc - \frac{ \alpha}{2N} \right)^2 + 4 \lmc \frac{ s_n\alpha}{2N\delta}} &\le p(\mathbf{b})\\
\label{eqn: p=mc1} & \text{if }\ 0 < b_n \le \overline{b_n}, \quad \frac{1}{2} \left( \rmc + \frac{\alpha}{2N} \right) \pm \half \sqrt{ \left(\rmc - \frac{ \alpha}{2N} \right)^2 + 4 \rmc \frac{ s_n\alpha}{2N\delta}} &\ge p(\mathbf{b})
\end{align}
\end{subequations}
However, only the conditions with plus sign satisfies \eqref{eqn: p>mc}, the conditions with minus sign violates \eqref{eqn: p>mc} because since
\[\forall s_n >0, \quad p(\mathbf{b}) \le \frac{\alpha}{2N} \le \frac{
\partial^+c_n(0)}{
\partial s_n} < \frac{
\partial^-c_n(s_n)}{
\partial s_n}.\]
Hence we discard the conditions with minus sign and note that \eqref{eqn: p=mc1}-\eqref{eqn: p=mc2} corresponds to \eqref{eqn: ne3}-\eqref{eqn: ne4}.
Conversely, suppose that $\mathbf{b}$ has at least one strictly positive component, that $0 \le b_n \le \overline{b_n}$, and that $\mathbf{b}$ satisfies \eqref{eqn: p>mc} and \eqref{eqn: ne3}-\eqref{eqn: ne4}. Then we may simply reverse the argument: by Step 2, $Q_n(b_n; \mathbf{b}_{-n})$ is concave and continuous in $b_n \ge 0$, and in this case the conditions \eqref{eqn: ne3}-\eqref{eqn: ne4} imply that $b_n$ maximizes $Q_n(b_n; \mathbf{b}_{-n})$ over $0 \le b_n \le \overline{b_n}$. Since we have already shown that choosing $b_n > \overline{b_n}$ is never optimal for firm $n$, we conclude that $\mathbf{b}$ is an equilibrium, and it is easy to check that in this case condition \eqref{eqn: p>mc} is satisfied.
\item \emph{ If Assumption \ref{asn: cheap_on-site} holds, then the function $\hat{c}_n(s_n)$ defined in \eqref{eqn: modified_cost} is continuous, and strictly convex and strictly increasing over $s_n \ge 0$, with $\hat{c}(s_n) = 0$ \ for $s_n \le 0$.}
$\hat{c}_n(s_n)$ is continuous on $s_n >0$ by continuity of $c_n$ and on $s_n <0$ by definition. We only need to show that $\hat{c}_n(0) = 0$, this is because when $s_n=0$, $c_n(s_n) = 0, s_n\frac{\alpha}{2N} = 0$, and integrating from 0 to $s_n$ is 0. Hence $\hat{c}_n(s_n) = 0$ for $s_n \le 0$.
For $s_n \ge 0$, we simply compute the directional derivatives of $\hat{c}_n$:
\begin{align*}
\frac{
\partial^+\hat{c}_n(s_n)}{
\partial s_n} &= \frac{1}{2} \left(\frac{\alpha}{2N} + \rmc \right)
+ \frac{1}{2}\sqrt{\left(\frac{\alpha}{2N} - \rmc\right)^2 + 2\rmc\frac{s_n\alpha}{N\delta}}, \\
\frac{
\partial^-\hat{c}_n(s_n)}{
\partial s_n} &= \frac{1}{2} \left(\frac{\alpha}{2N} + \lmc \right)
+ \frac{1}{2}\sqrt{\left(\frac{\alpha}{2N} - \rmc\right)^2 + 2\rmc\frac{s_n\alpha}{N\delta}}.
\end{align*}
Since $c_n$ is strictly increasing and convex, for $0 \le s_n < \bar{s}_n$, we will have
\[ 0 \le \frac{
\partial^+\hat{c}(s_n)}{
\partial s_n} < \frac{
\partial^-\hat{c}(\bar{s}_n)}{
\partial s_n} \le \frac{
\partial^+\hat{c}(\bar{s}_n)}{
\partial s_n}.\]
This guarantees that $\hat{c}_n$ is strictly increasing and strictly convex over $s_n \ge 0$.
\item \emph{ There exists a unique vector $\mathbf{s} \ge 0, y\ge 0$ and at least one scalar $\rho > 0$ such that:
\begin{subequations}
\begin{align}
\label{eqn: mopt1} & \frac{1}{2} \left( \rmc + \frac{\alpha}{2N} \right)
+ \frac{1}{2}\sqrt{ \left(\rmc - \frac{ \alpha}{2N} \right)^2 + \rmc \frac{2 s_n\alpha}{N\delta}} \ge \rho, \quad \text{if }\ s_n \ge 0; \\
\label{eqn: mopt2} & \frac{1}{2} \left( \lmc + \frac{\alpha}{2N} \right)
+ \frac{1}{2}\sqrt{ \left(\rmc - \frac{ \alpha}{2N} \right)^2 + \rmc \frac{2 s_n\alpha}{N\delta}} \le \rho, \quad \text{if }\ s_n >0; \\
\label{eqn: mopt3} & \frac{\alpha}{N\delta}\left(y + (N - 1)\delta\right) = \rho; \\
\label{eqn: mopt4} &\sum_n s_n = (\delta - y).
\end{align}
\end{subequations}
The vector $\mathbf{s}$ and $y$ is then the unique optimal solution to \eqref{eqn: mc1}-\eqref{eqn: mc3}. }
By Step 5, since $\hat{c}_n$ is continuous and strictly over the convex, compact feasible region for each $n$, we know that \eqref{eqn: mc1}-\eqref{eqn: mc3} have a unique optimal solution $\mathbf{s}, y$. As in the proof of Proposition \ref{prop: ec1}, form the Lagrangian
\begin{align*}
L(\mathbf{s}, y; \rho) &= \sum_n \hat{c}_n(s_n) + \yterm
+ \rho((\delta-y) -\sum_n s_n).
\end{align*}
By assumption \ref{asn: cheap_on-site}, $y>0$, and by the fact that $\hat{c}_n(s_n) = 0$ for $s_n \le 0$, $s_n \ge 0$. there exists a Lagrange multiplier $\rho$ such that $(\mathbf{s}, y, \rho)$ satisfy the stationarity conditions which corresponds to
\eqref{eqn: mopt1}-\eqref{eqn: mopt3} when we expand the definition of $\hat{c}_n(s_n)$, together with the constraint \eqref{eqn: mopt4}. The fact that $\rho>0$ follows by \eqref{eqn: mopt3} as $y >0$.
\item \emph{ If $\mathbf{s} \ge 0, y \ge 0$ and $\rho > 0$ satisfy \eqref{eqn: mopt1}-\eqref{eqn: mopt4}, then the triple $(\mathbf{b}, \rho, y)$ defined by $b_n = (\delta - s_n)\rho$ is an equilibrium as defined in \eqref{def: ne1} and \eqref{def: ne2}.}
First observe that with this definition, together with \eqref{eqn: mopt4} and the fact that $s_n \ge 0$, we have $b_n \ge 0$ for all $n$. Furthermore, we can show $b_n \le \overline{b_n}$, since $s_n \ge 0$, $b_n \le \rho \delta$, but by \eqref{eqn: mopt3}-\eqref{eqn: mopt4}, we have
\begin{align}
\label{eqn: rho} \rho = \frac{\alpha}{N\delta}(y+(N-1)\delta)
=\frac{\alpha}{N\delta} (N\delta - \sum_n s_n)
\end{align}
Substitute the definition $s_n = \delta - \frac{b_n}{\rho}$ into \eqref{eqn: rho}, we have
\begin{equation}
\rho = \frac{\alpha}{N\delta} \frac{\Sigma_n b_n}{\rho} \Rightarrow \rho = \sqrt{\frac{\Sigma_n b_n \alpha}{N\delta}}. \label{eqn: rho2}
\end{equation}
Substituting \eqref{eqn: rho2} into $b_n \le \rho\delta$, we have
$b_n \le \sqrt{\frac{(\Sigma_{m\ne n} b_m + b_n) \alpha\delta}{N}},$
Solving this inequality we have $b_n \le \overline{b_n}$.
Finally, at least one component of $\mathbf{b}$ is strictly positive, since otherwise we have $s_{n1} = s_{n2}=\delta$ for some $n1\ne n2$, in which case $\Sigma_n s_n >\delta,$ which contradicts \eqref{eqn: mopt4}. (or $s_n=\delta$, $y=0$, contradicting our assumption that $y>0$.)
By Step 4, to check that $\mathbf{b}$ is an equilibrium, we must only check the stationarity conditions \eqref{eqn: ne3}-\eqref{eqn: ne4}. We simply note that under the identification $b_n = \rho(\delta - s_n)$, using \eqref{eqn: rho2} and \eqref{eqn: mopt3}, we have
\begin{align*}
y = \sqrt{\frac{\Sigma_n b_n N\delta}{\alpha}} - (N-1)\delta ; \quad
\rho =\frac{ \Sigma_n b_n}{(N-1)\delta + y} = p(\mathbf{b}).
\end{align*}
Substitute $p(\mathbf{b})$ into \eqref{eqn: mopt1} will correspond to \eqref{eqn: ne3}, and \eqref{eqn: mopt2} implies \eqref{eqn: ne4} and \eqref{eqn: p>mc} because $\lmc \le \rmc$. Thus $(\mathbf{b}, \rho, y)$ is an equilibrium.
\item \emph{ If ($\mathbf{b}, p(\mathbf{b}), y)$ is an equilibrium, then there exists a scalar $\rho \ge 0$ such that the vector $\mathbf{b}$ defined by $s_n = S(p(\mathbf{b}), b_n)$ satisfies \eqref{eqn: mopt1}-\eqref{eqn: mopt4}.}
We simply reverse the argument of Step 7. Since $\mathbf{b}$ is an equilibrium bids, by \eqref{def: ne2} and $s_n = S(p(\mathbf{b}), b_n)$, we have $\sum_n s_n = (\delta - y)$, i.e., \eqref{eqn: mopt4} is satisfied. By Step 4, $\mathbf{b}$ satisfies \eqref{eqn: ne3}-\eqref{eqn: ne4}. Since $y>0$ by Assumption \ref{asn: cheap_on-site}, $0\le s_n < \delta$ for all $n$, let
\begin{align*}
\rho = \max \Big\{ p(\mathbf{b}), &\half\left(\lmc + \frac{\alpha}{2N}\right) + \half\sqrt{(\rmc-\frac{\alpha}{2N})^2 + \rmc\frac{2s_n\alpha}{N\delta} } \Big\}.
\end{align*}
In this case $\rho >0$ and $0 \le b_n \le \overline{b_n}$ for all $n$, so \eqref{eqn: ne4} implies \eqref{eqn: mopt2} by definition of $\rho$, and \eqref{eqn: mopt1} holds by \eqref{eqn: ne3} and the fact that $
\partial^-c_n(s_n) \le
\partial^+ c_n(s_n)$ (by convexity).
\item \emph{ There exists an equilibrium $\mathbf{b}$, and for any equilibrium that price is greater than marginal cost, the vector $\mathbf{s}$ defined by $s_n = S(p(\mathbf{b}), b_n)$ is the unique optimal solution of \eqref{eqn: mopt1}-\eqref{eqn: mopt4}. }
The conclusion is now straightforward. Existence follows from Steps 6 and 7. Uniqueness of the resulting production vector $\mathbf{s}$, and the fact that $\mathbf{s}$ is an optimal solution to \eqref{eqn: mc1}-\eqref{eqn: mc3}, follows by Steps 6 and 8.
\end{enumerate}
\subsection{Proof of Lemma \ref{lem: mc_bound}} We exploit the structure of the modified cost $\hat{c}_n$ to prove the result. Note that, for all $n$, $ s_n \ge 0$, if we define $G_n(s_n) = \int^{s_n}_0 \sqrt{(\frac{
\partial^+c_n(z)}{
\partial z} - \frac{\alpha}{2N})^2 + \frac{
\partial^+c_n(z)}{
\partial z} \frac{2z\alpha}{N\delta}} dz$, then
\begin{align}
G_n(s_n)
&\ge \int^{s_n}_0 \sqrt{\left(\frac{
\partial^+c_n(z)}{
\partial z} - \frac{\alpha}{2N}\right)^2} dz =c_n(s_n) - s_n\frac{\alpha}{2N} \notag.
\end{align}
First inequality is because $z\ge 0$, last equality is because by convexity and Assumption \ref{asn: mc_lowerbound}, we have
$\mz \ge \frac{
\partial^+c_n(0) }{
\partial s_n} \ge \frac{\alpha}{2N}.$
Hence we have
$\hat{c}_n(s_n) = \frac{1}{2}\left(c_n(s_n) + s_n\frac{\alpha}{2N}\right) + \frac{1}{2}G_n(s_n) \ge c_n(s_n)$.
On the other hand, notice that $s_n \le \delta$, we have:
\begin{align}
G_n(s_n)
&\le \int^{s_n}_0 \sqrt{\left(\frac{
\partial^+c_n(z)}{
\partial z} - \frac{\alpha}{2N}\right)^2 + \frac{
\partial^+c_n(z)}{
\partial z} \frac{2\delta\alpha}{N\delta}} dz \notag \\
&= \int^{s_n}_0 \sqrt{\left(\frac{
\partial^+c_n(z)}{
\partial z} + \frac{\alpha}{2N}\right)^2 } dz = c_n(s_n) + s_n \frac{\alpha}{2N} \notag.
\end{align}
Hence we have
$\hat{c}_n(s_n) = \frac{1}{2}\left(c_n(s_n) + s_n\frac{\alpha}{2N}\right) + \frac{1}{2}G_n(s_n) \le c_n(s_n) + s_n\frac{\alpha}{2N}.$ The bounds for the left and right derivatives can be obtained from taking the left (or right) derivatives at the bounds of $G_n(s_n)$.
\subsection{Proof of Theorem \ref{thm: diff-price-anticipate}} Firstly we will prove one side of the inequality $p^t \le p^a, y^t \le y^a.$ Recall that by the examinging the Lagrangians of the optimizations in Proposition \ref{prop: ec2} in and Theorem \ref{thm: modified_cost}, we have $p^t \ge \partial^-c_n(s_n^t)/\partial s_n$, $p^t \le \partial^+c_n(s_n^t)/\partial s_n$, $p^a \ge \partial^-\hat{c}_n(s_n^a)/\partial s_n$, $p^a \le \partial^+\hat{c}_n(s_n^a)/\partial s_n$, at the domain where the left or right derivative is defined, and $ p^t = \frac{\alpha}{N\delta}(y^t + (N-1)\delta), p^a = \frac{\alpha}{N\delta}(y^a + (N-1)\delta).$
If $y^t > y^a$, then
$p^t > p^a$. Also, because the total energy reduction $\delta$ is constant, we have $\sum_n s_n^t < \sum_n s_n^a $.
Hence there exist $s_r > 0$ such that $s_r^a > s_r^t$ for some $r \in \{1, \ldots, N\}$. Therefore, by strict convexity of $c_n$ (Assumption \ref{asn: cost_convexity}):
{ \begin{equation}
p^t \le \frac{
\partial^+ c_r(s_r^t)}{\partial s_r} < \frac{\partial^- c_r(s_r^a)}{\partial s_r}. \label{eqn: pt_ub}
\end{equation}}
However, by Lemma \ref{lem: mc_bound} we have
$ \frac{
\partial^- \hat{c}_r(s_r)}{
\partial s_r}
\ge \frac{
\partial^-c_r(s_r)}{
\partial s_r}.
$ Hence, we have
{ \begin{align}
p^a \ge \frac{\partial^- \hat{c}_r(s^a_r)}{\partial s_r} \ge \frac{\partial^- c_r(s_r^a)}{\partial s_r}. \label{eqn: pa_lb}
\end{align}}
Combining \eqref{eqn: pt_ub} and \eqref{eqn: pa_lb}, we have $p^t < p^a$, contradiction. Hence we have $y^t \le y^a$, and $p^t \le p^a$.
Next we show the other side of the inequality $p^a \le p^t + \frac{\alpha}{2N}, y^a \le y^t + \frac{\delta}{2},$ by the previous part, we have $\sum_n s_n^a \le \sum_n s^t_n$.
Let $n = \argmax_m (s^t_m - s_m^a)$, clearly $s^t_n \ge s^a_n$, otherwise $\sum_n s^t_n < \sum_n s^a_n$, contradiction.
If $s^t_n = s^a_n,$ then $\forall m, s^t_m = s^a_m$, and $y^t = y^a$, then $p^t = p^a$.
If $s^t_n > s_n^a$, then by strict convexity of $c_n$ (assumption \ref{asn: cost_convexity}), and the fact that $s_n^a \ge 0, s^t_n > 0$, we have
\begin{equation}
\frac{\partial^+ \hat{c}_n(s^a_n)}{s_n} < \frac{\partial^- c_n(s^t_n)}{s_n} \le p^a. \label{eqn: pt_lb}
\end{equation}
Also, by Lemma \ref{lem: mc_bound}, we have
$
\frac{\partial^+ \hat{c}_n(s_n)}{\partial s_n}
\le \frac{\partial^+c_n(s_n)}{
\partial s_n} + \frac{\alpha}{2N},
$
this gives us
\begin{equation}
p^a \le \frac{\partial^+ \hat{c}_n(s^a_n)}{\partial s_n} \le \frac{\partial^+ c_n(s^a_n)}{\partial s_n} + \frac{\alpha}{2N}. \label{eqn: pa_ub}
\end{equation}
Combining \eqref{eqn: pt_lb} and \eqref{eqn: pa_ub}, we have
$p^a < p^t + \frac{\alpha}{2N}.$
Hence we have
\begin{align*}
\frac{\alpha}{N\delta}(y^a + (N -1)\delta) < \frac{\alpha}{N\delta}(y^t + (N-1)\delta) + \frac{\alpha}{2N}, \Rightarrow
y^a < y^t + \frac{\delta}{2}.
\end{align*}
\subsection{Proof of Theorem \ref{thm: on-site_gap}} Given any $\varepsilon >0$, let $\varepsilon' = \frac{1}{2}\varepsilon$. Consider the following set of cost function:
{\[ c_1(s_1) =
\begin{cases}
\frac{\alpha}{2N}s_1 , & \text{ if } s_1 < \varepsilon' ; \\
\alpha(1- \frac{3\varepsilon'}{2N\delta})s_1+C_1 , & \varepsilon' \le s_1 \le \delta - \varepsilon'; \\
2\alpha s_1 + C_2, & s_1 > \delta - \varepsilon'
\end{cases}
\]}
where $C_1, C_2$ are constants that make $c_1$ continuous\footnote{$C_1 = -\alpha\varepsilon'(\frac{ (2N-1)\delta - 3\varepsilon'}{2N\delta} )$, and $C_2 = -\frac{\alpha}{N\delta}(N\delta^2 +\delta\varepsilon' - 3\varepsilon')$}, then $c_1$ is piece-wise linear and convex. Also, $\forall m \ne 1, c_m(s_m) = 2\alpha s_m$. It is easy to see that $s_1^* = \delta - \varepsilon'$ and $y^* = \varepsilon'$ is the optimal allocation.
Let $s^a_1 = \varepsilon', y^a = \delta - \varepsilon',$ and $\forall m\ne 1, s^a_m = 0$, we claim that $(\mathbf{s}^a, y^a)$ is the unique optimal solution to \eqref{eqn: mc1}-\eqref{eqn: mc3}. To see this, let $\rho = \alpha(1 - \varepsilon / (N\delta))$, then,
\begin{subequations}
\begin{align}
\frac{\alpha}{N\delta}(y^a + (N-1)\delta) &= \rho; \quad \sum_n s^a_n = \delta - y^a; \\
\frac{\partial^- \hat{c}_1(s^a_1)}{\partial s_1} &\le \rho ; \quad
\frac{\partial^+ \hat{c}_1(s^a_1)}{\partial s_1} \ge \rho ; \quad
\frac{\partial^+\hat{c}_m(0)}{\partial s_m} \ge \rho , \quad \forall m \ne 1.
\end{align}
\end{subequations}
where the second inequality is because if we let $H_n$ be the term under square root for $\rmmc$, then
\begin{align*}
H_n &= \sqrt{\left(\rmc - (\frac{\alpha}{2N} - \frac{\alpha}{N}\frac{s_n}{\delta})\right)^2 + (\frac{\alpha^2}{N^2}\frac{(\delta+s_n)(\delta-s_n)}{\delta^2})} \\
&\ge \rmc - (\frac{\alpha}{2N} - \frac{\alpha}{N}\frac{s_n}{\delta}).
\end{align*}
Note that $\mrmc = \frac{1}{2} (\rmc + \frac{\alpha}{2N}) + \frac{1}{2}H_n$. Hence we have
$\frac{\partial^+ \hat{c}_1(s^a_1)}{\partial s_1} \ge \frac{\partial^+c_1(s_1^a)}{\partial s_1} + \frac{\alpha s_1}{2N\delta} = \rho.$
These conditions correspond to \eqref{eqn: mopt1}-\eqref{eqn: mopt4}, so we conclude that $(\mathbf{s^a}, y^a)$ is the unique optimal solution to \eqref{eqn: mc1}-\eqref{eqn: mc3}. Hence $y^a - y^* = \delta - 2\varepsilon' = \delta - \varepsilon.$
\subsection{Proof of Theorem \ref{thm: welfare_loss2}} As $(\mathbf{s}^*, y^*)$ is a feasible solution to \eqref{eqn: mc0}, by Theorem \ref{thm: modified_cost}, we have
\begin{align}
\label{eqn: cost_compare1} \sum_n \hat{c}_n(s_n^a) + \frac{\alpha}{2N\delta}(y^a + (N -1)\delta)^2
\le \sum_n \hat{c}_n(s^*_n) + \frac{\alpha}{2N\delta}(y^* + (N -1)\delta)^2 .
\end{align}
Rearranging, we have
$\sum_n \hat{c}_n(s^a_n) + \alpha y^a - \left(\sum_n \hat{c}_n(s_n^*) + \alpha y^*\right)
\le \frac{\alpha}{N}\left( (y^a - y^*) (1 - \frac{y^a+y^*}{2\delta})\right). $
By Corollary \ref{cor: diff-price-anticipate2} and the fact that $y^* \le \delta, y^a \le \delta,$ both terms in the brackets are positive, hence right-hand-side expression is maximized when $y^* \rightarrow 0^+$ and $y^a = \delta$, hence
\begin{align}
\label{eqn: cost_compare2} \left( \sum_n \hat{c}_n(s^a_n)+ \alpha y^a\right) - \left(\sum_n \hat{c}_n(s_n^*) + \alpha y^*\right) \le \frac{\alpha\delta}{2N}.
\end{align}
However, by Lemma \ref{lem: mc_bound}, we have
$\sum_n \hat{c}_n(s^*_n) \le \sum_n c_n(s^*_n) + \frac{\alpha}{2N}(\sum_n s_n) \le \sum_n c_n(s^*_n) + \frac{\alpha\delta}{2N}$; and
$\sum_n \hat{c}_n(s^a_n) \ge \sum_n c_n(s^a_n). $
Substituting the above relations into \eqref{eqn: cost_compare2} and rearranging, we have the desired result.
\subsection{Proof of Theorem \ref{thm: diff-payment}} First, we compare the cost by operator between the price-taking and price anticipating cases, by definition \eqref{eqn: operator_cost} and rearranging, we have
$ \mathrm{cost}_o(p^a, y^a) - \mathrm{cost}_o(p^t, y^t) = (p^a - p^t) \left({\delta - y^t} \right) + \left(\alpha - {p^a}\right) (y^a - y^t).$
By the fact that $p^a=\frac{\alpha}{N\delta}(y^a + (N-1)\delta)$ (shown in Theorem \ref{thm: diff-price-anticipate}) and the fact that $0 \le y^a \le \delta$, we have
\begin{equation}
\alpha\left(\frac{N-1}{N}\right) \le p^a \le \alpha . \label{eqn: pa_range}
\end{equation}
By the upper bound of $p^a$ in \eqref{eqn: pa_range} and the upper bounds of $p^t, y^t$ in Theorem \ref{thm: diff-price-anticipate}, we have
\begin{equation}
\mathrm{cost}_o(p^a, y^a) - \mathrm{cost}_o(p^t, y^t) \ge 0. \label{eqn: anticipating_v_taking}
\end{equation}
Similarly, using the lower bound of $p^a$ in \eqref{eqn: pa_range} and the upper bounds of $p^a, y^a$ in Theorem \ref{thm: diff-price-anticipate}, we have
\begin{align*}
\mathrm{cost}_o(p^a, y^a) - \mathrm{cost}_o(p^t, y^t)
\le \left(\frac{\alpha}{2N} \right)\cdot \left({\delta}\right) + \left(\alpha \cdot \frac{1}{N}\right)\left(\frac{\delta}{2} \right)= \frac{\alpha\delta}{N}.
\end{align*}
Second, we compare the cost by the operator to the social optimal. Since the energy reduction goal $\delta$ is the same, by Proposition \ref{prop: ec2} and Corollary \ref{cor: diff-price-anticipate2}, we have $p^t \le p^*$ and $p^a \le p^*$. Hence we have $\mathrm{cost}_o(p^t, y^t) \le \mathrm{cost}_o(p^a, y^a) \le \mathrm{cost}_o(p^*, y^*).$
Furthermore,
\begin{align}
&\cost_o(p^*, y^*) - \cost_o(p^t, y^t) = \alpha\delta - (p^t(\delta - y^t) + \alpha y^t)\notag \\
= &(\alpha - p^t)(\delta - y^t) = \alpha \left( \frac{\delta - y^t}{N\delta} \right)(\delta - y^t) \le \frac{\alpha\delta}{N}. \label{eqn: taking_v_optimal}
\end{align}
Lastly by \eqref{eqn: anticipating_v_taking} and \eqref{eqn: taking_v_optimal}, we have
$\cost(p^*, y^*) -\cost(p^a, y^a) \le \cost(p^*, y^*) - \cost(p^t , y^t) \le \frac{\alpha\delta}{N}.$
\section{Results for {\sf ColoEDR}\xspacevdr}
\subsection{Proof Sketch of Theorem \ref{thm: vdr_ne_taking}}
Theo proof is similar to that of Theorem \ref{thm: price-taking-characterization}, which uses Proposition \ref{prop: ec1}, note that in the VDR case, we can change $N\delta$ in the proof of Theorem \ref{thm: price-taking-characterization} to $\Sigma_n D_n$, and interpret the variable $y$ as $\Sigma_n D_n - d $, $\alpha$ as $u$ and $\gamma_n$ as $1/N$ in the proof of Theorem \ref{thm: price-taking-characterization}.
\subsection{Proof Sketch of Theorem \ref{thm: vdr_ne_anticipating}}
Theo proof is similar to that of Theorem \ref{thm: modified_cost}. note that in the VDR case, we can change $N\delta$ in the proof of Theorem \ref{thm: price-taking-characterization} to $\Sigma_n D_n$, and interpret the variable $y$ as $\Sigma_n D_n - d $, $\alpha$ as $u$ and $\gamma_n$ as $1/N$ in the proof of Theorem \ref{thm: modified_cost}.
\subsection{Proof of Lemma \ref{lemma: vdr_mc_bound}}
For the bound on the magnitude of the modified cost, we exploit the structure of the modified cost $\hat{c}_n$ to prove the result. Note that, for all $n$, $ s_n \ge 0$, if we define $G_n(s_n) = \int_{0}^{s_n}\sqrt{\left( \frac{\partial^+c_n(z)}{\partial z} - \frac{\gamma_n u}{2} \right)^2 + 2\frac{\partial^+c_n(z)}{\partial z} \frac{zu}{\Sigma_i D_i}}$, then
\begin{align}
G_n(s_n)
&\ge \int^{s_n}_0 \sqrt{\left( \frac{\partial^+c_n(z)}{\partial z} - \frac{\gamma_n u}{2} \right)^2} dz =c_n(s_n) - s_n\frac{u\gamma_n}{2} \notag.
\end{align}
First inequality is because $z\ge 0$, last equality is because by convexity and Assumption \ref{asn: vdr-mc-lbound}, we have
$\mz \ge \frac{
\partial^+c_n(0) }{
\partial s_n} \ge \frac{u\gamma_n}{2}.$
Hence we have
$\hat{c}_n(s_n) = \frac{1}{2}\left(c_n(s_n) + s_n\frac{u\gamma}{2}\right) + \frac{1}{2}G_n(s_n) \ge c_n(s_n)$.
On the other hand, notice that $s_n \le D_n$, we have:
\begin{align}
G_n(s_n)
&\le \int^{s_n}_0 \sqrt{\left(\frac{\partial^+c_r(s_n)}{\partial s_n} - \frac{u \gamma_n}{2}\right)^2 + 2\frac{\partial^+c_n(s_n)}{\partial s_n}\gamma_n u} dz = c_n(s_n) + s_n \frac{\alpha}{2N} \notag.
\end{align}
Hence we have
$\hat{c}_n(s_n) = \frac{1}{2}\left(c_n(s_n) + s_n\frac{\alpha}{2N}\right) + \frac{1}{2}G_n(s_n) \le c_n(s_n) + s_n\frac{u\gamma_n}{2}.$ The bounds for the left and right derivatives can be obtained from taking the left (or right) derivatives at the bounds of $G_n(s_n)$.
\subsection{Proof of Theorem \ref{thm: vdr_loss_taking}}
We can combine \eqref{eqn: vdr-quantity} with \eqref{eqn: vdr-price2} to eliminate the $\sqrt{\sum_{i=1}^n b_i}$ term to get a relation between market price and the vdr-quantity decided by the profit maximizing operator:
\begin{equation}
p = \frac{u}{\sum_{i=1}^n D_i} (\sum_{i=1}^n D_i - d)
\label{eqn: price_d}
\end{equation}
By the characterization theorem, we have
$ ud^* - \frac{u{d^*}^2}{2\sum_n D_n} - \sum_n c_n(s_n^*) \le ud^t - \frac{u{d^t}^2}{2\sum_n D_n} - \sum_n c_n(s_n^t). $
Rearranging, we have
\begin{align*}
ud^* - \sum_n c_n(s_n^*) &\le ud^t - \sum_n c_n(s_n^t) + \frac{u ({d^*}^2 - {d^t}^2)}{2\sum_n D_n} \\
&\le ud^t - \sum_n c_n(s_n^t) + \frac{u\sum_n {d^*}^2}{2\sum_n D_n}
\end{align*}
where the last inequality is due to the fact that $d^t \ge 0.$
\subsection{Proof of Theorem \ref{thm: vdr_loss_anticipating}}
By Theorem \ref{thm: vdr_ne_anticipating}, we have
$ud^a - \frac{u{d^a}^2}{2} - \sum_n \hat{c}_n(s_n^a) \ge ud^a - \frac{u{d^*}^2}{2} - \sum_n \hat{c}_n(s_n^*)$
Using Lemma \ref{lemma: vdr_mc_bound}, and rearranging, we have
\begin{align*}
& ud^a - \sum_n c_n(s_n^a) \\
\ge & ud^* - \sum_n c_n(s_n^*) - \frac{u({d^*}^2 - {d^a}^2)}{2\sum_n D_n} - \sum_n s_n^*\frac{\gamma_n u}{2} \\
\ge & ud^* - \sum_n c_n(s_n^*) - \frac{u \sum_n D_n }{2} - \sum_n D_n \frac{\gamma_n u}{2}\\
= & ud^* - \sum_n c_n(s_n^*) - \frac{u}{2}\sum_n D_n(1 + \gamma_n).
\end{align*}
where the first inequality is because $c_n(s_n^a) \le \hat{c}_n(s_n^a)$, and $\hat{c}_n(s_n^*) \le c_n(s_n^*) + s_n^* \frac{\gamma_n u}{2}$, and the second inequality is becuase $s_n^* \le D_n$.
\subsection{Proof of Proposition \ref{prop: vdr_price_taking}}
The Lagrangian of the welfare maximization problem \eqref{eqn: vdr-utility-maximization} is
\[ L(\mathbf{s}, d; \mu, \bar{\lambda}, \underline{\lambda}) = ud - \sum_{i=1}^n c_i(s_i) + \mu(\sum_{i=1}^n s_i - d) + \sum_{i=1}^n \underline{\lambda}_is_i + \sum_{i=1}^n \bar{\lambda}_i(D_i - s_i). \]
By constraint qualification, the optimal primal dual solutions $(\mathbf{s}, y; \mu)$ satisfies the KKT conditions
\begin{align*}
& \mu^* = u, \\
& \lmc \le \mu^*, \text{ if } 0 < s_n \le D_n \\
& \rmc \ge \mu^*, \text{ if } 0 \le s_n < D_n.
\end{align*}
Hence the market clearing price in the optimal allocation should be $p^* = u$.
Now consider the market clearing price for price taking tenants, from \eqref{eqn: price_d}, we know that
$p^t = u - \frac{u d^t}{\sum_n D_n} \le u = p^*$. Similarly, by Theorem \ref{thm: vdr_ne_taking} and looking at the Lagrangian of \eqref{eqn: mc0_taking}, we have $ \frac{\partial^- c_n(s_n^t)}{\partial s_n} \le p^t$ for all $0< s_n^t \le D_n$, hence for all $n$, such that $s_n^t >0$ and $s_n^* < D_n$, we have
\[ \frac{\partial^- c_n(s_n^t)}{\partial s_n} \le p^t \le p^* \le \frac{\partial^+ c_n(s_n^*)}{\partial s_n}, \]
hence $s_n^t \le s_n^*$ for all such $n$, on the other hand, if $s_n^t = 0$ or $s_n^* = D_n$, we also have $s_n^t \le s_n^*$, hence $d^t = \sum_n s_n^t \le \sum_n s_n^* = d^*.$ Finally, by the fact that $d^t \le d^*$ and \eqref{eqn: price_d}, we have
\[ p^t = u - \frac{u d^t}{\sum_n D_n} \ge u - \frac{u d^*}{\sum_n D_n} = \left(1 - \frac{d^*}{\sum_n D_n}\right)p^*. \]
\subsection{Proof of Theorem \ref{thm: vdr_price_diff}}
Firstly we will prove one side of the inequality $p^t \le p^a, d^t \ge d^a.$
We can prove this by contradiction. Suppose $d^t < d^a$, then by \eqref{eqn: price_d}, $p^t > p^a$. Also, $\sum_n s_n^t < \sum_n s_n^a $.
Hence there exist $s_r^a > 0$ such that $s_r^a > s_r^t$ for some $r \in \{1, \ldots, N\}$. Therefore, by the stationarity of the Lagrangian of \eqref{eqn: mc0_taking} and strict convexity of $c_n$ (Assumption \ref{asn: cost_convexity}):
\begin{equation}
p^t \le \frac{
\partial^+ c_r(s_r^t)}{\partial s_r} < \frac{\partial^- c_r(s_r^a)}{\partial s_r} \label{eqn: vdr_pt_ub}
\end{equation}
However, by the stationarity of the Lagrangian of \eqref{eqn: mc0_anticipating} and Lemma \ref{lemma: vdr_mc_bound}, we have
\begin{align}
p^a \ge \frac{\partial^- \hat{c}_r(s_r^a)}{\partial s_r} \ge \frac{\partial^- c_r(s_r^a)}{\partial s_r}.
\label{eqn: vdr_pa_lb}
\end{align}
Combining \eqref{eqn: pt_ub} and \eqref{eqn: pa_lb}, we have $p^t < p^a$, contradiction. Hence we have $y^t \le y^a$, and $p^t \le p^a$.
Next we show the other side of the inequality $p^a \le p^t + \frac{u \gamma }{2}, d^a \le d^t - \frac{D}{2},$ by the previous part, we have $\sum_n s_n^a \le \sum_n s^t_n$.
Let $n = \argmax_m (s^t_m - s^a_m)$, clearly $s^t_n \ge s^a_n$, otherwise $\sum_n s^t_n < \sum_n s^a_n$, contradiction.
If $s^t_n = s^a_n,$ then $\forall m, s^t_m = s^a_m$, and $d^t = d^a$. By \eqref{eqn: price_d}, $p^t = p^a$.
If $s^t_n > s^a_n$, then by stationary condition of the Lagrangian of \eqref{eqn: mc0_taking} and strict convexity of $c_n$ (assumption \ref{asn: cost_convexity}), and the fact that $s^a_n \ge 0, s^t_n > 0$, we have
\begin{equation}
\frac{\partial^+ c_n(s_n^a)}{\partial s_n} < \frac{\partial^- c_n(s^t_n)}{s_n} \le p^t. \label{eqn: pt_lb}
\end{equation}
Also, by Lemma \ref{lemma: vdr_mc_bound} and stationary condition of Lagrangian of \eqref{eqn: mc0_anticipating}, we have definition of \begin{equation}
p^a \le \frac{\partial^+\hat{c}_n(s_n^a)}{\partial s_n} \le \frac{\partial^+\hat{c}_n(s_n^a)}{\partial s_n} + \frac{\gamma_n u}{2}. \label{eqn: pa_ub}
\end{equation}
Combining \eqref{eqn: pt_lb} and \eqref{eqn: pa_ub}, we have
$p^a < p^t + \frac{\gamma_n u}{2} \le p^t + \frac{u\gamma}{2}.$
Substitute the above relation into \eqref{eqn: price_d}, we have
\begin{align*}
u - \frac{ud^a}{\sum_n D_n} &< u - \frac{ud^t}{\sum_n D_n} + \frac{u\gamma}{2}, \\
d^a & > d^t - \frac{D}{2},
\end{align*}
the last inequality is because $D = \max_n D_n = (\sum_n D_n) \gamma.$
\subsection{Proof of Theorem \ref{thm: vdr_operator_profit}}
Firstly, by theorem \ref{thm: vdr_price_diff}, $d^a \le d^t$, and $p^a \ge p^t$, hence $U_o(p^a, d^a) \le U_o(p^t, d^t)$. Furthermore,
\begin{align}
&U_o(p^t, d^t) - U_o(p^a, d^a) = (u-p^t)d^t - (u-p^a)d^a \notag \\
\label{eqn: cost_o1} = & (u-p^t)(d^t - d^a) + d^a(p^a - p^t).
\end{align}
By theorem \ref{thm: vdr_price_diff}, we have $d^t \le d^a + D/2$, $p^a \le p^t + u\gamma/2$, and by the fact that $d^a \le \sum_n D_n$, we have
\begin{equation}
U_o(p^t, d^t) - U_o(p^a, d^a) \le u\cdot\frac{D}{2} + (\sum_n D_n)\frac{u\gamma}{2} = uD.
\end{equation}
\end{document} |
\begin{document}
\begin{center}
\emph{Dedicado a Biblioco 34}
\end{center}
\begin{abstract}
This is a survey on the state-of-the-art of the classification of finite-dimensio\-nal complex Hopf algebras.
This general question is addressed through the consideration of different classes of such Hopf algebras.
Pointed Hopf algebras constitute the class best understood; the classification of those with abelian group is expected to be
completed soon and there is substantial progress in the non-abelian case.
\end{abstract}
\begin{classification}
16T05, 16T20, 17B37, 16T25, 20G42.
\end{classification}
\begin{keywords}
Hopf algebras, quantum groups, Nichols algebras.
\end{keywords}
\maketitle
\section{Introduction}
Hopf algebras were introduced in the 1950's from three different perspectives: algebraic groups in positive characteristic,
cohomology rings of Lie groups, and group objects in the category of von Neumann algebras.
The study of non-commutative non-cocommutative Hopf algebras started in the 1960's.
The fundamental breakthrough is Drinfeld's report \cite{Dr1}. Among many contributions and ideas,
a systematic construction of solutions of the quantum Yang-Baxter equation (qYBE) was presented. Let $V$ be a vector space.
The qYBE is equivalent to the braid equation:
\begin{align}\label{eq:braid}
(c\otimes \operatorname{id}) (\operatorname{id}\otimes c) (c\otimes \operatorname{id}) &= (\operatorname{id}\otimes c) (c\otimes \operatorname{id})
(\operatorname{id}\otimes c),&
c&\in GL(V\otimes V).
\end{align}
If $c$ satisfies \eqref{eq:braid}, then $(V,c)$ is called a braided vector space; this is a down-to-the-earth version
of a braided tensor category \cite{JS}.
Drinfeld introduced the notion of quasi-triangular Hopf algebra, meaning a pair $(H, R)$ where $H$ is a Hopf algebra and
$R\in H\otimes H$ is invertible and satisfies the approppriate conditions, so that every $H$-module $V$ becomes a braided
vector space, with $c$ given by the action of $R$ composed with the usual flip.
Furthermore, every finite-dimensional Hopf algebra $H$ gives rise to a quasi-triangular Hopf algebra, namely the Drinfeld double $D(H) = H \otimes H^*$
as vector space. If $H$ is not finite-dimensional, some precautions have to be taken to construct $D(H)$, or else one considers
Yetter-Drinfeld modules, see \S \ref{subsec:repns}. In conclusion, every Hopf algebra is a source of solutions of the braid equation.
Essential examples of quasi-triangular Hopf algebras are the quantum groups $U_q(\mathfrak g)$ \cite{Dr1, J} and the
finite-dimensional variations $\mathfrak{u}_q(\mathfrak g)$ \cite{L1, L2}.
In the approach to the classification of Hopf algebras exposed in this report, braided vector spaces and braided tensor categories play a decisive role;
and the finite quantum groups are the main actors in one of the classes that splits off.
By space limitations, there is a selection of the topics and references included.
Particularly, we deal with finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero with special emphasis on
description of examples and classifications.
Interesting results on Hopf algebras either infinite-dimensional, or over other fields, unfortunately can not be reported.
There is no account of the many deep results on tensor categories, see \cite{EGNO}.
Various basic fundamental results are not explicitly cited, we refer to \cite{bariloche, Ma3, Mo1, Rad, Sch2, Sw} for them; classifications of Hopf algebras
of fixed dimensions are not evoked, see \cite{BG, Ng, Z}.
\smallbreak
\section{Preliminaries}
Let $\theta\in\mathbb N$ and $\I= \I_\theta = \{1,2,\dots,\theta\}$. The base field is $\mathbb C$. If $X$ is a set, then
$\vert X\vert$ is its cardinal and $\mathbb C X$ is the vector space with basis $(x_i)_{i\in X}$.
Let $G$ be a group: we denote by $\operatorname{Irr} G$ the set of isomorphism classes of irreducible
representations of $G$ and by $\widehat{G}$ the subset of those of dimension 1; by $G^x$ the centralizer of $x\in G$; and by ${\mathcal O}_x^G$ its conjugacy class.
More generally we denote by $\operatorname{Irr} {\mathfrak C}$ the set of isomorphism classes of simple objects in an abelian category ${\mathfrak C}$.
The group of $n$-th roots of 1 in $\mathbb C$ is denoted $\G_n$; also $\G_{\infty} = \bigcup_{n\mathfrak{g}e1} \G_n$.
The group presented by $(x_i)_{i\in I}$ with relations $(r_j)_{j\in J}$ is denoted $\langle (x_i)_{i\in I} \vert (r_j)_{j\in J} \rangle$.
The notation for Hopf algebras is standard: $\Delta$, $\varep$, $\Ss$, denote
respectively the comultiplication, the counit, the antipode (always assumed bijective, what happens in the finite-dimensional case). We use Sweedler's
notation: $\Delta(x) = x_{(1)} \otimes x_{(2)}$.
Similarly, if $C$ is a coalgebra and $V$ is a left comodule with structure map
$\delta: V \to C \otimes V$, then $\delta(v) = v_{(-1)} \otimes v_{(0)}$.
If $D, E$ are subspaces of $C$, then $D \wedge E = \{c\in C: \Delta(c) \in D\otimes C + C \otimes E\}$; also
$\wedge^{0} D = D$ and $\wedge^{n+1} D = (\wedge^{n} D) \wedge D$ for $n>0$.
\smallbreak\subsection{Basic constructions and results}
The first examples of finite-dimensional Hopf algebras are the group algebra $\mathbb C G$ of a finite group $G$
and its dual, the algebra of functions $\mathbb C^G$. Indeed,
the dual of a finite-dimensional Hopf algebra is again a Hopf algebra by transposing operations.
By analogy with groups, several authors explored the notion of extension of Hopf algebras at various
levels of generality; in the finite-dimensional context, every extension
$\mathbb C \to A \rightarrow C \rightarrow B \to \mathbb C$ can be described as $C$ with underlying vector space
$A\otimes B$, via a heavy machinery of actions,
coactions and non-abelian cocycles, but actual examples are rarely found in this way (extensions from a different perspective
are in \cite{AG-compo}).
Relevant exceptions are the so-called \emph{abelian extensions} \cite{K-ext} (rediscovered by Takeuchi and Majid):
here the input is a matched pair of groups $(F, G)$
with mutual actions $\operatorname{tr}iangleright$, $\operatorname{tr}iangleleft$ (or equivalently, an exact factorization of a finite group).
The actions give rise to a Hopf algebra $\mathbb C^G \# \mathbb C F$.
The multiplication and comultiplication can be further modified by compatible cocycles
$(\sigma, \kappa)$, producing to the abelian extension
$\mathbb C \to \mathbb C^G \rightarrow \mathbb C^G {}^\kappa\mathfrak{h}space{-3pt}\#_\sigma \mathbb C F \rightarrow \mathbb C F \to \mathbb C$.
Here $(\sigma, \kappa)$ turns out to be a 2-cocycle in the total complex associated to a double complex built
from the matched pair; the relevant $H^2$ is computed via the so-called Kac exact sequence.
\smallbreak
It is natural to approach Hopf algebras by considering algebra or coalgebra invariants.
There is no preference in the finite-dimensional setting but coalgebras and comodules are locally finite, so we privilege the coalgebra ones
to lay down general methods. The basic coalgebra invariants of a Hopf algebra $H$ are:
\smallbreak \mathbf{n}oindent $\circ$ The group $G(H) =\{g\in H - 0: \Delta(g) = x\otimes g\}$ of group-like elements of $H$.
\smallbreak \mathbf{n}oindent $\circ$ The space of skew-primitive elements $\mathcal{P}_{g,h}(H)$, $g, h\in G(H)$; $\mathcal{P}(H) := \mathcal{P}_{1,1}(H)$.
\smallbreak \mathbf{n}oindent $\circ$ The coradical $H_0$, that is the sum of all simple subcoalgebras.
\smallbreak \mathbf{n}oindent $\circ$ The coradical filtration $H_0 \subset H_1 \subset \dots$, where $ H_n= \wedge^{n} H_0$; then $H = \bigcup_{n \mathfrak{g}eq 0} H_n$.
\smallbreak\subsection{Modules}\label{subsec:repns}
The category $\mc{H}{}$ of left modules over a Hopf algebra $H$ is monoidal with tensor product defined by the comultiplication;
ditto for the category $\mc{}{H}$ of left comodules, with tensor product defined by the multiplication. Here are two ways
to deform Hopf algebras without altering one of these categories.
\smallbreak \mathbf{n}oindent $\bullet$ Let $F\in H\otimes H$ be invertible such that $(1\otimes F) (\operatorname{id} \otimes \Delta)(F) =
(F\otimes 1) (\Delta \otimes \operatorname{id})(F)$ and $(\operatorname{id} \otimes \varepsilon)(F) =
(\varepsilon \otimes \operatorname{id})(F) = 1$. Then $H^{F}$ (the same
algebra with comultiplication $\Delta^{F}:= F \Delta F^{-1}$) is again a Hopf algebra,
named the \emph{twisting} of $H$ by $F$ \cite{Dr-quasi}. The monoidal categories $\mc{H}{}$ and $\mc{H^F}{}$ are equivalent.
If $H$ and $K$ are finite-dimensional Hopf algebras with $\mc{H}{}$ and $\mc{K}{}$ equivalent as monoidal categories,
then there exists $F$ with $K\simeq H^F$ as Hopf algebras (Schauenburg, Etingof-Gelaki).
Examples of twistings not mentioned elsewhere in this report are in \cite{EN, Momb}.
\smallbreak \mathbf{n}oindent $\bullet$ Given a linear map $\sigma: H\otimes H \to \mathbb C$ with analogous conditions, there is a Hopf algebra
$H_{\sigma}$ (same coalgebra, multiplication twisted by $\sigma$) such that the monoidal categories $\mc{}{H}$ and $\mc{}{H_\sigma}$ are equivalent \cite{DT}.
\smallbreak
A Yetter-Drinfeld module $M$ over $H$ is left $H$-module and left $H$-comodule with the compatibility
$\delta(h.m) = h_{(1)} m_{(-1)}\Ss(h_{(3)}) \otimes h_{(2)} \cdot m_{(0)}$, for all $m\in M$ and $h\in H$.
The category ${}_{H}^{H}\mathcal{YD}$ of Yetter-Drinfeld modules is braided monoidal.\label{page:yd}
That is, for every $M,N \in {}_{H}^{H}\mathcal{YD}$, there is a natural isomorphism $c: M\otimes N \to N\otimes
M$ given by $c(m\otimes n) = m_{(-1)}\cdot n \otimes m_{(0)}$, $m\in M$, $n\in N$.
When $H$ is finite-dimensional, the category ${}_{H}^{H}\mathcal{YD}$ is equivalent, as a braided monoidal category, to $\mc{D(H)}{}$.
\smallbreak
The definition of Hopf algebra makes sense in any braided monoidal category. Hopf algebras in ${}_{H}^{H}\mathcal{YD}$ are interesting because of the following
facts--discovered by Radford and interpreted categorically by Majid, see \cite{Ma3, Rad}:
\smallbreak \mathbf{n}oindent $\diamond$ If $R$ is a Hopf algebra in ${}_{H}^{H}\mathcal{YD}$, then $R\# H := R \otimes H$ with semidirect product and coproduct is a Hopf algebra,
named the \emph{bosonization} of $R$ by $H$.
\smallbreak \mathbf{n}oindent $\diamond$ Let $\pi, \iota$ be Hopf algebra maps as in
$\mathtt{x}ymatrix{K \ar@{->>}_{\pi}[r] & \ar@/^-1pc/_{\iota}[l] H}$ with $\pi\iota = \operatorname{id}_H$.
Then $R = H^{\text{co} \pi} := \{x\in K: (\operatorname{id} \otimes \pi)\Delta(x) = 1\otimes x\}$ is a Hopf algebra in ${}_{H}^{H}\mathcal{YD}$ and $K \simeq R\# H$.
\smallbreak
For instance, if $V\in {}_{H}^{H}\mathcal{YD}$, then the tensor algebra $T(V)$ is a Hopf algebra in ${}_{H}^{H}\mathcal{YD}$, by requiring
$V \mathfrak{h}ookrightarrow \mathcal{P}(T(V))$. If $c:V\otimes V \to V\otimes V$ satisfies $c = - \tau$, $\tau$ the usual flip, then the exterior algebra $\Lambda(V)$
is a Hopf algebra in ${}_{H}^{H}\mathcal{YD}$.
There is a braided adjoint action of a Hopf algebra $R$ in ${}_{H}^{H}\mathcal{YD}$ on itself, see e.g. \cite[(1.26)]{AHS}. If $x\in \mathcal{P}(R)$ and $y \in R$,
then $\ad_c(x)(y) = xy - \text{mult } c(x\otimes y)$.
\smallbreak\subsubsection{Triangular Hopf algebras}\label{subsec:triangular}
A quasitriangular Hopf algebra $(H,R)$ is \emph{triangular} if the braiding induced by $R$ is a symmetry:
$c_{V\otimes W}c_{W\otimes V} = \operatorname{id}_{W\otimes V}$ for all $V, W \in \mc{H}{}$. A finite-dimensional
triangular Hopf algebra is a twisting of a bosonization $\Lambda(V)\# \mathbb C G$, where $G$ is a finite group and $V\in {}_{\Gamma}^{\Gamma}\mathcal{YD}g$ has $c = -\tau$ \cite{AEG}.
This lead eventually to the classification of triangular finite-dimensional Hopf algebras \cite{EG-clas-triang}; previous work on the semisimple
case culminated in \cite{EG-clas-triang-ss}.
\smallbreak\subsection{Semisimple Hopf algebras}
The algebra of functions $\mathbb C^G$ on a finite group $G$ admits a Haar measure,
i.e., a linear function $\smallint: \mathbb C^G \to \mathbb C$
invariant under left and right translations, namely $\smallint = $
sum of all elements in the standard basis of $\mathbb C G$.
This is adapted as follows: a right integral \emph{on} a Hopf algebra $H$ is
a linear function $\smallint: H \to \mathbb C$
which is invariant under the left regular coaction: analogously there is the notion of left integral.
The notion has various applications. Assume that $H$ is finite-dimensional.
Then an integral \emph{in} $H$ is an integral on $H^*$; the subspace of left integrals in $H$ has dimension one, and there is a generalization of
Maschke's theorem for finite groups: $H$ is semisimple if and only if $\varep(\Lambda)\mathbf{n}eq 0$ for any integral $0\mathbf{n}eq \Lambda\in H$.
This characterization of semisimple Hopf algebras, valid in any characteristic, is one of several, some valid only in characteristic 0.
See \cite{Sch2}. Semisimple Hopf algebras can be obtained as follows:
\smallbreak \mathbf{n}oindent $\diamond$ A finite-dimensional Hopf algebra $H$ is semisimple if and only if it is cosemisimple (that is, $H^*$ is semisimple).
\smallbreak \mathbf{n}oindent $\diamond$ Given an extension $\mathbb C \to K \rightarrow H \rightarrow L \to \mathbb C$, $H$ is semisimple iff $K$ and $L$ are.
Notice that there are semisimple extensions that are not abelian \cite{GNN, Nat-ext, Nik}.
\smallbreak \mathbf{n}oindent $\diamond$ If $H$ is semisimple, then so are $H^{F}$ and $H_{\sigma}$, for any twist $F$ and cocycle $\sigma$.
If $G$ is a finite simple group, then any twisting of $\mathbb C G$ is a simple Hopf algebra (i.e., not a non-trivial extension) \cite{Nik-tw},
but the converse is not true \cite{GN}.
\smallbreak \mathbf{n}oindent $\diamond$ A bosonization $R\# H$ is semisimple iff $R$ and $H$ are.
To my knowledge, all examples of semisimple Hopf algebras arise from group algebras by the preceding constructions; this was proved in
\cite{N4-mem, N4-60} in low dimensions and in \cite{ENO} for dimensions $p^aq^b$, $pqr$, where $p$, $q$ and $r$ are primes.
See \cite[Question 2.6]{bariloche}. An analogous question in terms of fusion categories: is any semisimple Hopf algebra weakly group-theoretical? See
\cite[Question 2]{ENO}.
There are only finitely many isomorphism classes of \emph{semisimple} Hopf algebras in each dimension \cite{St-finite-ss}, but this fails in general
\cite{AS-jalg, BDG}.
\begin{conjecture} (Kaplansky). Let $H$ be a semisimple Hopf algebra. The dimension of every $V\in \operatorname{Irr} \mc{H}{}$ divides the dimension of $H$.
\end{conjecture}
The answer is affirmative for iterated extensions of group algebras and duals of group algebras \cite{MW}
and notably for semisimple quasitriangular Hopf algebras \cite{EG-frob-prop}.
\section{Lifting methods}\label{sec:methods}
\smallbreak\subsection{Nichols algebras}\label{subsec:nichols}
Nichols algebras are a special kind of Hopf algebras in braided tensor categories.
We are mainly interested in Nichols algebras in the braided category ${}_{H}^{H}\mathcal{YD}$, where $H$ is a
Hopf algebra, see page \pageref{page:yd}. In fact, there is a functor $V\mapsto {\mathfrak B}(V)$ from ${}_{H}^{H}\mathcal{YD}$ to the category of
Hopf algebras in ${}_{H}^{H}\mathcal{YD}$. Their first appearence is in the precursor \cite{N}; they were
rediscovered in \cite{W} as part of a ``quantum differential calculus", and in \cite{Lu}
to present the positive part of $U_q(\mathfrak g)$.
See also \cite{Ro2, Sbg}.
There are several, unrelated at the first glance, alternative definitions.
Let $V\in {}_{H}^{H}\mathcal{YD}$. The first definition uses the representation of the braid group $\mathbb B_n$ in $n$ strands on $V^{\otimes n}$,
given by $\varsigma_i \mapsto \operatorname{id}\otimes c\otimes \operatorname{id}$, $c$ in $(i, i+1)$ tensorands; here recall that
$\mathbb B_n = \langle\varsigma_1, \dots, \varsigma_{n-1}\vert \varsigma_i\varsigma_j = \varsigma_j\varsigma_i, \vert i-j\vert >1,
\varsigma_i\varsigma_j\varsigma_i = \varsigma_j\varsigma_i\varsigma_j, \vert i-j\vert = 1\rangle$. Let
$M: \mathbb S_n \to\mathbb B_n$ be the Matsumoto section and let $\mathcal{Q}_n :V^{\otimes n} \to V^{\otimes n}$ be the
quantum symmetrizer, $\mathcal{Q}_n = \sum_{s\in \mathbb S_n} M(s) :V^{\otimes n} \to V^{\otimes n}$. Then define
\begin{align}\label{eq:nichols-first-def}
\mathfrak{J}^n(V) &= \ker \mathcal{Q}_n, & \mathfrak{J}(V) &= \oplus_{n\mathfrak{g}eq 2} \mathfrak{J}^n(V),& {\mathfrak B}(V) &= T(V) / \mathfrak{J}(V).
\end{align}
Hence ${\mathfrak B}(V)= \oplus_{n\mathfrak{g}eq 0}{\mathfrak B}^n(V)$ is a graded Hopf algebra in ${}_{H}^{H}\mathcal{YD}$ with ${\mathfrak B}^0(V) = \mathbb C$, ${\mathfrak B}^1(V) \simeq V$;
by \eqref{eq:nichols-first-def} the algebra structure depends only on $c$. To explain the second definition, let us observe that the tensor
algebra $T(V)$ is a Hopf algebra in ${}_{H}^{H}\mathcal{YD}$ with comultiplication determined by $\Delta(v) = v\otimes 1 + 1 \otimes v$ for $v\in V$.
Then $\mathfrak{J}(V)$ coincides with the largest homogeneous ideal of $T(V)$ generated by elements of degree $\mathfrak{g}e 2$ that is also a coideal.
Let now
$T= \oplus_{n\mathfrak{g}eq 0}T^n$ be a graded Hopf algebra in ${}_{H}^{H}\mathcal{YD}$ with $T^0 = \mathbb C$. Consider the conditions
\begin{align}\label{eq:nichols-charac-alg}
T^1 & \text{ generates $T$ as an algebra}, \\ \label{eq:nichols-charac-coalg}
T^1 &= \mathcal{P} (T).
\end{align}
These requirements are dual to each other: if $T$ has finite-dimensional homogeneous components and $R = \oplus_{n\mathfrak{g}eq 0}R^n$ is the graded
dual of $T$, i.e., $R^n = (T^n)^*$, then $T$ satisfies \eqref{eq:nichols-charac-alg} if and only if
$R$ satisfies \eqref{eq:nichols-charac-coalg}. These conditions determine ${\mathfrak B}(V)$ up to isomorphisms,
as the unique graded connected Hopf algebra $T$ in ${}_{H}^{H}\mathcal{YD}$ that satisfies $T^1 \simeq V$, \eqref{eq:nichols-charac-alg} and \eqref{eq:nichols-charac-coalg}.
There are still other characterizations of $\mathfrak{J}(V)$, e.g. as the radical of a suitable homogeneous bilinear form on $T(V)$,
or as the common kernel of some suitable skew-derivations. See \cite{AS-cambr} for more details.
Despite all these different definitions, Nichols algebras are extremely difficult to deal with, e.g. to present by generators and relations,
or to determine when a Nichols algebra has finite dimension or finite Gelfand-Kirillov dimension.
It is not even known a priori whether the ideal $\mathfrak{J}(V)$ is finitely generated, except in a few specific cases.
For instance, if $c$ is a symmetry, that is $c^2 =\operatorname{id}$, or satisfies a Hecke condition with generic parameter, then ${\mathfrak B}(V)$ is quadratic.
By the efforts of various authors, we have some understanding
of finite-dimensional Nichols algebras of braided vector spaces either of diagonal or of rack type, see \S \ref{subsec:nichols-diagonal},
\ref{subsec:nichols-rack}.
\smallbreak\subsection{Hopf algebras with the (dual) Chevalley property}\label{subsec:chevalley}
We now explain how Nichols algebras enter into our approach to the classification of Hopf algebras. Recall that
a Hopf algebra has the \emph{dual Chevalley property} if the tensor product of two simple comodules is semisimple, or
equivalently if its coradical is a (cosemisimple) Hopf subalgebra. For instance, a \emph{pointed} Hopf algebra,
one whose simple comodules have all dimension one, has the dual Chevalley property and its coradical is a group algebra.
Also, a \emph{copointed} Hopf algebra (one whose coradical is the algebra of functions on a finite group)
has the dual Chevalley property.
The Lifting Method is formulated in this context \cite{AS-jalg}. Let $H$ be a Hopf
algebra with the dual Chevalley property and set $K := H_0$. Under this assumption, the graded coalgebra
$\operatorname{gr} H = \oplus_{n\in \mathbb N_0}\operatorname{gr}^n H$ associated to the coradical filtration becomes a Hopf algebra and
considering the homogeneous projection $\pi$ as in
$\mathtt{x}ymatrix{R = H^{\text{co}\, \pi} \ar@{^{(}->}[r] & \operatorname{gr} H \ar@<-0.5ex>@{->}_{\quad \pi}[r] & \ar@<-0.5ex>[l] K}
$
we see that $\operatorname{gr} H \simeq R \# K$. The subalgebra of coinvariants $R$ is a graded Hopf algebra in ${}_{K}^{K}\mathcal{YD}$
that inherits the grading with $R^0 = \mathbb C$; it satisfies \eqref{eq:nichols-charac-coalg} since the grading comes from the coradical filtration.
Let $R'$ be the subalgebra of $R$ generated by $ V :=R^1$; then $R' \simeq {\mathfrak B} (V)$. The braided vector space $V$ is a basic invariant
of $H$ called its \emph{infinitesimal braiding}.
Let us fix then a semisimple Hopf algebra $K$.
To classify all finite-dimensional Hopf algebras $H$ with $H_0 \simeq K$ as Hopf algebras, we have to address the
following questions.
\begin{enumerate}\renewcommand{(\alph{enumi})}{(\alph{enumi})}\itemsep1pt \parskip0pt \parsep0pt
\item\label{item:method-nichols} Determine those $V \in {}_{K}^{K}\mathcal{YD}$ such that $\mathfrak B(V)$ is finite-dimensional,
and give an efficient defining set of relations of these.
\item\label{item:method-gen-deg-one} Investigate whether any {\it finite-dimensional} graded Hopf
algebra $R$ in ${}_{K}^{K}\mathcal{YD}$
satisfying $R^0 = \mathbb C$ and $P(R) = R^1$, is a Nichols algebra.
\item\label{item:method-lifting} Compute all Hopf algebras $H$ such that $\operatorname{gr} H \simeq \mathfrak B(V) \# K$, $V$ as in \ref{item:method-nichols}.
\end{enumerate}
Since the Nichols algebra ${\mathfrak B}(V)$ depends as an algebra (and as a coalgebra) only on the braiding $c$,
it is convenient to restate Question \ref{item:method-nichols} as follows:
\begin{enumerate}\itemsep1pt \parskip0pt \parsep0pt
\item[(a$_1$).] Determine those braided vector spaces $(V, c)$ in a suitable class
such that $\dim \mathfrak B(V) < \infty$, and give an efficient defining set of relations of these.
\item[(a$_2$).] For those $V$ as in (a$_1$), find in how many ways, if any, they can be realized as Yetter-Drinfeld modules over $K$.
\end{enumerate}
For instance, if $K = \mathbb C\Gamma$, $\Gamma$ a finite abelian group, then the suitable class is that of braided vector spaces of diagonal type.
In this context, Question (a$_2$) amounts to solve systems of equations in $\Gamma$.
The answer to \ref{item:method-nichols} is instrumental to attack \ref{item:method-gen-deg-one} and \ref{item:method-lifting}.
Question \ref{item:method-gen-deg-one} can be rephrased in two equivalent statements:
\begin{enumerate}\itemsep1pt \parskip0pt \parsep0pt
\item[(b$_1$).] Investigate whether any {\it finite-dimensional} graded Hopf
algebra $T$ in ${}_{K}^{K}\mathcal{YD}$ with $T^0 = \mathbb C$ and generated as algebra by $T^1$, is a Nichols algebra.
\item[(b$_2$).] Investigate whether any {\it finite-dimensional} Hopf
algebra $H$ with $H_0 = K$ is generated as algebra by $H_1$.
\end{enumerate}
We believe that the answer to \ref{item:method-gen-deg-one} is affirmative at least when $K$ is a group algebra.
In other words, by the reformulation (b$_2$):
\begin{conjecture}\label{conj:gendegone} \emph{\cite{AS-adv}}
Every {\it finite-dimensional} pointed Hopf algebra is generated by group-like and skew-primitive elements.
\end{conjecture}
As we shall see in \S \ref{subsec:gen-deg-1},
the complete answer to \ref{item:method-nichols} is needed in the approach proposed in \cite{AS-adv} to attack Conjecture \ref{conj:gendegone}.
It is plausible that the answer of (b$_2$) is affirmative for every semisimple Hopf algebra $K$.
Question \ref{item:method-lifting}, known as \emph{lifting} of the relations, also
requires the knowledge of the generators of $\mathfrak{J}(V)$, see \S \ref{subsec:deformations}.
\smallbreak\subsection{Generalized Lifting method}\label{subsec:generalized-lifting-method}
Before starting with the analysis of the various questions in \S \ref{subsec:chevalley}, we discuss a possible approach to more general Hopf algebras \cite{AC}.
Let $H$ be a Hopf algebra; we consider the following invariants of $H$:
\smallbreak \mathbf{n}oindent $\circ$ The \emph{Hopf coradical} $H_{[0]}$ is the subalgebra generated by $H_0$.
\smallbreak \mathbf{n}oindent $\circ$ The \emph{standard filtration} $H_{[0]} \subset H_{[1]} \subset \dots$, $H_{[n]}=\wedge^{n+1} H_{[0]}$;
then $H = \bigcup_{n \mathfrak{g}eq 0} H_{[n]}$.
\smallbreak
If $H$ has the dual Chevalley property, then $H_{[n]} = H_n$ for all $n\in \mathbb N_0$.
In general, $H_{[0]}$ is a Hopf subalgebra of $H$ with coradical $H_0$ and we may consider the graded Hopf
algebra $\operatorname{gr} H=\oplus_{n \mathfrak{g}eq 0} H_{[n]}/H_{[n-1]}$.
As before, if $\pi:\operatorname{gr} H \rightarrow H_{[0]}$ is the homogeneous projection, then $R = (\operatorname{gr} H)^{\text{co}\, \pi}$ is a
Hopf algebra in ${}_{H}^{H}\mathcal{YD}o$ and $\operatorname{gr} H \cong R \# H_{[0]}$.
Furthermore, $R= \oplus_{n \mathfrak{g}eq 0} R^n$ with grading inherited from $\operatorname{gr} H$.
This discussion raises the following questions.
\begin{enumerate}\renewcommand{(\alph{enumi})}{(\uglph{enumi})}\itemsep1pt \parskip0pt \parsep0pt
\item\label{item:gen-method-coalgebras}
Let $C$ be a finite-dimensional cosemisimple coalgebra and $S: C\to C$ a bijective
anti-coalgebra map.
Classify all finite-dimensional Hopf algebras $L$ generated by $C$, such that $\Ss {\vert_C} = S$.
\item\label{item:gen-method-R} Given $L$ as in the previous item, classify all finite-dimensional connected graded Hopf algebras $R$ in ${}_{L}^{L}\mathcal{YD}$.
\item\label{item:gen-method-lifting} Given $L$ and $R$ as in previous items, classify all deformations or liftings, that is, classify all Hopf algebras $H$
such that $\operatorname{gr} H \cong R\# L$.
\end{enumerate}
Question \ref{item:gen-method-coalgebras} is largely open, except for the remarkable \cite[Theorem 1.5]{St}:
if $H$ is a Hopf algebra generated by an $\Ss$-invariant 4-dimensional simple subcoalgebra $C$, such that
$1 < \operatorname{ord} (\Ss^{2}\mathfrak{h}space{-1pt}\vert_{_C}) < \infty$,
then $H$ is a Hopf algebra quotient of the quantized algebra of functions on $SL_2$ at a root of unity $\omega$.
Nichols algebras enter into the picture in Question \ref{item:gen-method-R}; if $V = R^1$, then ${\mathfrak B}(V)$ is a subquotient of $R$.
Question \ref{item:gen-method-lifting} is completely open, as it depends on the previous Questions.
\smallbreak\subsection{Generalized root systems and Weyl groupoids}\label{subsec:grs}
Here we expose two important notions introduced in \cite{HY}.
Let $\theta \in \N$ and $\I = \I_{\theta}$.
A \emph{basic datum} of type $\I$ is a pair $(\mathcal{X}, \rho)$, where $\mathcal{X} \mathbf{n}eq \emptyset$ is a set and
$\rho: \I \to \Sb_{\mathcal{X}}$ is a map such that $\rho_i^2 = \operatorname{id}$ for all $i\in \I$.
Let $\mathcal{Q}_{\rho}$ be the quiver $\{\sigma_i^x := (x, i, \rho_i(x)): i\in \I, x\in \mathcal{X}\}$ over $\mathcal{X}$, with
$t(\sigma_i^x) = x$, $s(\sigma_i^x) = \rho_i(x)$ (here $t$ means target, $s$ means source). Let $F(\mathcal{Q}_{\rho})$ be the free groupoid over $\mathcal{Q}_{\rho}$;
in any quotient of $F(\mathcal{Q}_{\rho})$, we denote
\begin{align}\label{eq:conventio}
\sigma_{i_1}^x\sigma_{i_2}\cdots \sigma_{i_t} =
\sigma_{i_1}^x\sigma_{i_2}^{\rho_{i_1}(x)}\cdots \sigma_{i_t}^{\rho_{i_{t-1}} \cdots \rho_{i_1}(x)};
\end{align}
i.e., the implicit superscripts are those allowing compositions.
\smallbreak\subsubsection{Coxeter groupoids}\label{subsubsec:cox-grpds}
A \emph{Coxeter datum} is a triple $(\mathcal{X}, \rho, \mathbf{M} )$, where $(\mathcal{X}, \rho)$ is a basic datum of type $\I$
and $\mathbf{M} = (\mathbf{m} ^x)_{x\in \mathcal{X}}$ is a family of Coxeter matrices $\mathbf{m} ^x = (m^x_{ij})_{i,j\in \I}$ with
\begin{align}\label{eq:coxeter-datum}
s((\sigma^x_i\sigma_j)^{m^x_{ij}}) &= x,& i,j&\in \I,& x&\in \mathcal{X}.
\end{align}
The \emph{Coxeter groupoid} $\mathcal{W}(\mathcal{X}, \rho, \mathbf{M} )$
associated to $(\mathcal{X}, \rho, \mathbf{M} )$ \cite[Definition 1]{HY}
is the groupoid presented by generators $\mathcal{Q}_{\rho}$ with relations
\begin{align}\label{eq:def-coxeter-gpd}
(\sigma_i^x\sigma_j)^{m^x_{ij}} &= \operatorname{id}_x, &i, j\in \I,\, &x\in \mathcal{X}.
\end{align}
\smallbreak\subsubsection{Generalized root system}\label{subsubsec:grs}
A generalized root system (GRS for short) is a collection $\mathcal{R}:= (\mathcal{X}, \rho, \mathcal{C}, \Delta)$, where
$\mathcal{C} = (C^x)_{x\in \mathcal{X}}$ is a family of generalized Cartan matrices $C^x = (c^x_{ij})_{i,j \in \I}$, cf. \cite{K-libro},
and $\Delta = (\Delta^x)_{x\in \mathcal{X}}$ is a family of subsets $\Delta^x \subset \mathbb Z^{\I}$. We need the following notation:
Let $\{\alpha_i\}_{i\in\I}$ be the canonical basis of $\mathbb Z^{\I}$ and
define $s_i^x\in GL(\mathbb Z^{\I})$ by
$s_i^x(\alpha_j) = \alpha_j-c_{ij}^x\alpha_i$, $i, j \in \I$, $x \in \mathcal{X}$.
The collection should satisfy the following axioms:
\begin{align}
c^x_{ij}&=c^{\rho_i(x)}_{ij} & \mbox{for all }&x \in \mathcal{X}, \, i,j \in \I. \\
\label{eq:def root system 1}
\Delta^x &= \Delta^x_+ \cup \Delta^x_-, & \Delta^x_{\pm} &:= \pm(\Delta^x \cap \mathbb N_0^{\I}) \subset \pm\mathbb N_0^{\I};
\\ \label{eq:def root system 2}
\Delta^x \cap \mathbb Z \alpha_i &= \{\pm \alpha_i \};& &
\\\label{eq:def root system 3}
s_i^x(\Delta^x)&=\Delta^{\rho_i(x)};&
\\ \label{eq:def root system 4}
(\rho_i\rho_j)^{m_{ij}^x}(x)&=(x), & m_{ij}^x &:=|\Delta^x \cap (\mathbb N_0\alpha_i+\mathbb N_0 \alpha_j)|,
\end{align}
for all $x \in \mathcal{X}$, $i \mathbf{n}eq j \in \I$.
We call $\Delta^x_+$, respectively $ \Delta^x_-$, the set of \emph{positive}, respectively \emph{negative}, roots.
Let $\mathcal{G} = \mathcal{X} \times GL_{\theta}(\mathbb Z) \times \mathcal{X}$, $\varsigma_i^x = (x, s_i^x,\rho_i(x))$, $i \in \I$, $x \in \mathcal{X}$,
and $\mathcal{W} = \mathcal{W}(\mathcal{X}, \rho, \mathcal{C})$ the subgroupoid of $\mathcal{G}$ generated by all the $\varsigma_i^x$,
i.e., by the image of the morphism of quivers $\mathcal{Q}_{\rho} \to \mathcal{G}$, $\sigma_i^x \mapsto \varsigma_i^x$.
There is a Coxeter matrix $\mathbf{m} ^x = (m^x_{ij})_{i,j\in \I}$, where $m^x_{ij}$ is the smallest natural number such that
$(\varsigma_i^x\varsigma_j)^{m^x_{ij}} = \operatorname{id} x$. Then
$\mathbf{M} = (\mathbf{m} ^x)_{x\in \mathcal{X}}$ fits into a Coxeter datum $(\mathcal{X}, \rho, \mathbf{M} )$, and
there is an isomorphism of groupoids
$\mathtt{x}ymatrix{ \mathcal{W}(\mathcal{X}, \rho, \mathbf{M} ) \ar@{->>}[r] &\mathcal{W} = \mathcal{W}(\mathcal{X}, \rho, \mathcal{C})}
$ \cite{HY}; this is called the \emph{Weyl groupoid} of $\mathcal{R}$.
If $w \in \mathcal{W}(x, y)$, then $w(\Delta^x)= \Delta^y$, by \eqref{eq:def root system 3}.
The sets of \emph{real} roots at $x \in \mathcal{X}$ are
$(\Delta^{\re})^x = \bigcup_{y\in \mathcal{X}}\{ w(\alpha_i): \ i \in \I, \ w \in \mathcal{W}(y,x) \}$; correspondingly the \emph{imaginary} roots are
$(\Delta^{\im})^x = \Delta^{x} - (\Delta^{\re})^x$.
Assume that $\mathcal{W}$ is connected.
Then the following conditions are equivalent
\cite[Lemma 2.11]{CH-at most 3}:
\begin{itemize} \itemsep1pt \parskip0pt \parsep0pt
\item $\Delta^x$ is finite for some $x\in \mathcal{X}$,
\item $\Delta^x$ is finite for all $x\in \mathcal{X}$,
\item $(\Delta^{\re})^x$ is finite for all $x\in \mathcal{X}$,
\item $\mathcal{W}$ is finite.
\end{itemize}
If these hold, then all roots are real \cite{CH-at most 3}; we say that $\mathcal{R}$ is \emph{finite}.
We now discuss two examples of GRS, central for the subsequent discussion.
\begin{example}\label{exa:grs-lie-super} \cite{AA}
Let $\ku$ be a field of characteristic $\ell \mathfrak{g}eq 0$, $\theta\in\N$, $\mathbf{p} \in\G_2^\theta$ and $A =(a_{ij})\in\ku^{\theta\times\theta}$.
We assume $\ell \mathbf{n}eq 2$ for simplicity. Let $\mathfrak{h} = \ku^{2\theta-\rk A}$.
Let $\mathfrak{g}(A,\mathbf{p} )$ be the Kac-Moody Lie superalgebra over $\ku$ defined as in \cite{K-libro};
it is generated by $\mathfrak{h}$, $e_i$ and $f_i$, $i\in \I$, and the parity is given by
$|e_i|=|f_i|=p_i$, $i\in\I$, $|h|=0$, $h\in\mathfrak{h}$. Let $\Delta^{A, \mathbf{p} }$ be the root system of $\mathfrak{g}(A,\mathbf{p} )$.
We make the following technical assumptions:
\begin{align}\label{eq:symmetrizable}
a_{jk} &=0 \implies a_{kj}=0, & j&\mathbf{n}eq k; \\
\label{eq:f-nilpotent}
\ad f_i &\text{ is locally nilpotent in } \mathfrak{g}(A,\mathbf{p} ), & i&\in\I.
\end{align}
The matrix $A$ is \emph{admissible} if \eqref{eq:f-nilpotent} holds \cite{Ser-superKM}.
Let $C^{A,\mathbf{p} }= (c_{ij}^{A,\mathbf{p} })_{i,j\in\I}$ be given by
\begin{align}\label{eq:cryst-g(A)}
c_{ij}^{A,\mathbf{p} }&:=-\min\{m\in\N_0:(\ad f_i)^{m+1} f_j= 0 \}, i\mathbf{n}eq j \in \I, & c_{ii}^{A,\mathbf{p} }&:= 2.
\end{align}
We need the following elements of $\ku$:
\begin{align}
\label{eq:def dn, i par} \mbox{if } p_i &= 0, & d_m&= m\, a_{ij}+\binom{m}{2} a_{ii}; \\
\label{eq:def dn, i impar} \mbox{if }p_i &= 1; &
d_m&= \left\{ \begin{array}{ll} k\, a_{ii}, & m=2k, \\ k\, a_{ii}+a_{ij},& m=2k+1; \end{array}\right.
\\
\label{eq:formula nu}
\mathbf{n}u_{j,0}&=1, & \mathbf{n}u_{j,n}&= \prod_{t=1}^n (-1)^{p_i((t-1)p_i+p_j)}d_t; \\
\label{eq:formula mu}
\mu_{j,0}&=0, & \mu_{j,n}&= (-1)^{p_ip_j}n\left(\prod_{t=2}^n (-1)^{p_i((t-1)p_i+p_j)}d_t\right)a_{ji} .
\end{align}
With the help of these scalars, we define a reflection $r_i(A,\mathbf{p} ) = (r_iA, r_i\mathbf{p} )$, where
$r_i\mathbf{p} =(\overline{p}_j)_{j\in\I}$, with $\overline{p}_j=p_j-c_{ij}^{A,\mathbf{p} }p_i$, and $r_i A=(\overline{a}_{jk})_{j,k\in\I}$, with
\begin{align}\label{eq:formula matriz si(A)}
\overline{a}_{jk}=\left\{\begin{array}{ll} -c_{ik}^{A,\mathbf{p} }\mu_{j,-c_{ij}^{A,\mathbf{p} }}a_{ii}+\mu_{j,-c_{ij}^{A,\mathbf{p} }}a_{ik} & \\
\qquad -c_{ik}^{A,\mathbf{p} }\mathbf{n}u_{j,-c_{ij}^{A,\mathbf{p} }}a_{ji}+\mathbf{n}u_{j,-c_{ij}^{A,\mathbf{p} }}a_{jk}, & j,k\mathbf{n}eq i; \\
c_{ik}^{A,\mathbf{p} }a_{ii}-a_{ik}, & j=i\mathbf{n}eq k; \\ -\mu_{j,-c_{ij}^{A,\mathbf{p} }}a_{ii}-\mathbf{n}u_{j,-c_{ij}^{A,\mathbf{p} }}a_{ji}, & j\mathbf{n}eq k=i; \\ a_{ii}, & j=k=i.\end{array} \right.
\end{align}
\begin{theorem}\label{thm:isomorfismo Ti} There is an isomorphism $T_i^{A,\mathbf{p} }:\mathfrak{g}(r_i (A,\mathbf{p} ))\to\mathfrak{g}(A,\mathbf{p} )$
of Lie superalgebras given (for an approppriate basis $(h_i)$ of $\mathfrak{h}$) by
\end{theorem}
\begin{equation}\label{eq:def Ti}
\begin{split}
T_i^{A,\mathbf{p} }(e_j)&= \begin{cases}
(\ad e_i)^{-c_{ij}^{A,\mathbf{p} }} (e_j), & i\mathbf{n}eq j\in\I,
\\ f_i, & j=i
\end{cases}
\\
T_i^{A,\mathbf{p} }(f_j)&= \begin{cases} (\ad f_i)^{-c_{ij}^{A,\mathbf{p} }}f_j, & j\in\I,j\mathbf{n}eq i, \\
(-1)^{p_i}e_i, & j=i,\end{cases}
\\
T_i^{A,\mathbf{p} }(h_j)&= \begin{cases} \mu_{j,-c_{ij}^{A,\mathbf{p} }} h_i+\mathbf{n}u_{j,-c_{ij}^{A,\mathbf{p} }} h_j,& i\mathbf{n}eq j\in\I \\
-h_i, & j=i,\\ h_j, & \theta + 1 \leq j \leq 2\theta-\rk A.\end{cases}
\end{split}
\end{equation}
Assume that $\dim \mathfrak{g}(A,\mathbf{p} ) < \infty$; then \eqref{eq:symmetrizable} and \eqref{eq:f-nilpotent} hold. Let
\begin{align*}
\mathcal{X} =&\{ r _{i_1}\cdots r _{i_n}(A,\mathbf{p} )\,|\, n\in \N _0, i_1,\ldots ,i_n\in \I\}.
\end{align*}
Then $(\mathcal{X}, r, \mathcal{C}, \Delta)$, where $\mathcal{C} = (C^{(B, \mathbf{q} )})_{(B, \mathbf{q} )\in \mathcal{X}}$ and $\Delta = (\Delta^{(B, \mathbf{q} )})_{(B, \mathbf{q} )\in \mathcal{X}}$,
is a finite GRS, an invariant of
$\mathfrak{g}(A,\mathbf{p} )$. \end{example}
\begin{example}\label{exa:grs-nichols-ss-yd} Let $H$ be a Hopf algebra, assumed semisimple for easiness.
Let $M\in {}_{H}^{H}\mathcal{YD}$ be finite-dimensional,
with a fixed decomposition $M = M_1\oplus \dots \oplus M_{\theta}$, where
$M_1,\dots,M_{\theta} \in \operatorname{Irr} {}_{H}^{H}\mathcal{YD}$. Then $T(M)$ and ${\mathfrak B}(M)$ are $\zt$-graded, by $\deg x =\alpha_i$
for all $x\in M_i$, $i\in \I_\theta$. Recall that $\zt_{\mathfrak{g}e 0} = \sum_{i\in \I_\theta}\mathbb Z_{\mathfrak{g}e 0}\alpha_i$.
\begin{theorem}\label{th:grs-semisimple} \emph{\cite{HeckSch, HS-israel}}
If $\dim {\mathfrak B}(M) < \infty$, then $M$ has a finite GRS.
\end{theorem}
We discuss the main ideas of the proof. Let $i\in \I = \I_\theta$. We define
$M'_i=V_i^*$,
\begin{align*}
c_{ij}^M &= - \sup \{h\in \N_0: \ad_c^h (M_i)(M_j)\mathbf{n}eq 0 \text{ in }{\mathfrak B}(M)\},& i&\mathbf{n}eq j, \quad c^M_{ii} = 2;
\\ M'_j&= \ad_c^{-c_{ij}} (M_i)(M_j), & \rho_i(M) &= M'_1 \oplus \dots \oplus M'_\theta.
\end{align*}
Then $\dim {\mathfrak B}(M) = \dim {\mathfrak B}(\rho_i(M))$ and $C^M = (c^M_{ij})_{i,j\in\I}$ is a
generalized Cartan matrix \cite{AHS}.
Also, $M'_j$ is irreducible \cite[3.8]{AHS}, \cite[7.2]{HeckSch}.
Let $\mathcal{X}$ be the set of objects in ${}_{H}^{H}\mathcal{YD}$ with fixed decomposition (up to isomorphism) of the form
\begin{align*}
\{ \rho _{i_1}\cdots \rho _{i_n}(M)\,|\, n\in \N _0, i_1,\ldots ,i_n\in \I\}.
\end{align*}
Then $(\mathcal{X}, \rho, \mathcal{C})$, where $\mathcal{C} = (C^N)_{N\in \mathcal{X}}$, is a crystallographic datum. Next we need:
\smallbreak\mathbf{n}oindent $\bullet$ \cite[Theorem 4.5]{HeckSch}; \cite{HeckGran07} There exists a totally ordered index set $(L,\le)$ and families
$(W_l)_{l\in L}$ in $\operatorname{Irr}{}_{H}^{H}\mathcal{YD}$, $(\beta_l)_{l\in L}$ such that ${\mathfrak B} (M)\simeq \otimes _{l\in L}{\mathfrak B} (W_l)$ as
$\zt$-graded objects in ${}_{H}^{H}\mathcal{YD}$, where $\deg x =\beta_l$ for all $x\in W_l$, $l\in L$.
\smallbreak
Let $\Delta_{\pm}^M = \{\pm\beta_l: l\in L\}$, $\Delta^M = \Delta _+^M \cup \Delta_-^M$, $\Delta = (\Delta^N)_{N\in \mathcal{X}(M)}$.
Then $\mathcal{R} = (\mathcal{X}, \rho, \mathcal{C}, \Delta)$ is a finite GRS.
\end{example}
\begin{theorem}\label{th:classif-GRS} \emph{\cite{CH}}
The classification of all finite GRS is known.
\end{theorem}
The proof is a combinatorial tour-de-force and requires computer calculations.
It is possible to recover from this result the classification of the finite-dimensional contragredient Lie superalgebras
in arbitrary characteristic \cite{AA}. However, the list of \cite{CH} is substantially larger than the classifications of
the alluded Lie superalgebras or the braidings of diagonal type with finite-dimensional Nichols algebra.
\smallbreak\subsection{Nichols algebras of diagonal type}\label{subsec:nichols-diagonal}
Let $G$ be a finite group. We denote ${}_{\Gamma}^{\Gamma}\mathcal{YD}g = {}_{H}^{H}\mathcal{YD}$ for $H = \mathbb C G$. So
$M\in {}_{\Gamma}^{\Gamma}\mathcal{YD}g$ is a left $G$-module with a $G$-grading $M = \oplus_{g\in G}M_g$ such that
$t\cdot M_g = M_{tgt^{-1}}$, for all $g,t\in G$.
If $M,N \in {}_{\Gamma}^{\Gamma}\mathcal{YD}g$, then the braiding $c: M\otimes N \to N\otimes
M$ is given by $c(m\otimes n) = g\cdot n \otimes m$, $m\in M_g$, $n\in N$, $g\in G$.
Now assume that $G = \Gamma$ is a finite abelian group. Then every $M\in {}_{\Gamma}^{\Gamma}\mathcal{YD}$ is a $\Gamma$-graded $\Gamma$-module, hence of the form
$M = \oplus _{g\in \Gamma, \chi\in \widehat{\Gamma}} M_g^{\chi}$, where $M_g^{\chi}$ is the $\chi$-isotypic component of $M_g$.
So ${}_{\Gamma}^{\Gamma}\mathcal{YD}$ is just the category
of $\Gamma \times \widehat{\Gamma}$-graded modules, with the braiding $c: M\otimes N \to N\otimes
M$ given by $c(m\otimes n) = \chi(g) n \otimes m$, $m\in M^\eta_g$, $n\in N_t^\chi$, $g, t\in G$, $\chi, \eta\in \widehat{\Gamma}$.
Let $\theta\in \N$, $\I = \I_{\theta}$.
\begin{definition}\label{def:diagonal} Let $\mathbf{q} = (q_{ij})_{i, j\in \I}$ be a matrix with entries in $\mathbb C^{\times}$.
A braided vector space $(V,c)$ is of \emph{diagonal type} with matrix $\mathbf{q} $ if $V$ has a basis $(x_i)_{i\in \I}$
with
\begin{align}\label{eq:diagonal}
c(x_i \otimes x_j) &= q_{ij} x_j \otimes x_i, & i, j &\in \I.
\end{align}
\end{definition}
Thus, every finite-dimensional $V\in {}_{\Gamma}^{\Gamma}\mathcal{YD}$ is a braided vector space of diagonal type.
Question \ref{item:method-nichols}, more precisely (a$_1$), has a complete answer in this setting.
First we can assume that $q_{ii} \mathbf{n}eq 1$ for $i\in \I$, as otherwise $\dim {\mathfrak B}(V) = \infty$.
Also, let $\mathbf{q} ' = (q'_{ij})_{i, j\in \I} \in (\mathbb C^{\times})^{\I \times \I}$ and $V'$ a braided vector space with matrix $\mathbf{q} '$.
If $q_{ii} = q'_{ii}$ and $q_{ij}q_{ji} = q'_{ij}q'_{ji}$ for all $j\mathbf{n}eq i\in \I_\theta$, then ${\mathfrak B}(V)\simeq {\mathfrak B}(V')$
as braided vector spaces.
\begin{theorem}\label{th:classif-diagonal} \emph{\cite{H-classif}}
The classification of all braided vector spaces of diagonal type with finite-dimensional Nichols algebra is known.
\end{theorem}
The proof relies on the Weyl groupoid introduced in \cite{H-inv}, a particular case of Theorem \ref{th:grs-semisimple}.
Another fundamental ingredient is the following result, generalized at various levels in \cite{HeckGran07, HeckSch, HS-israel}.
\begin{theorem}\label{th:pbw} \emph{\cite{Khar99}}
Let $V$ be a braided vector space of diagonal type. Every Hopf algebra quotient of $T(V)$ has a PBW basis.
\end{theorem}
The classification in Theorem \ref{th:classif-diagonal} can be organized as follows:
\smallbreak \mathbf{n}oindent $\diamond$ For most of the matrices $\mathbf{q} = (q_{ij})_{i, j\in \I_\theta}$ in the list of \cite{H-classif}
there is a field $\ku$ and a pair $(A, \mathbf{p} )$ as in Example \ref{exa:grs-lie-super} such that $\dim \mathfrak{g}(A, \mathbf{p} ) < \infty$, and $\mathfrak{g}(A, \mathbf{p} )$
has the same GRS as the Nichols algebra corresponding to $\mathbf{q} $ \cite{AA}.
\smallbreak \mathbf{n}oindent $\diamond$ Besides these, there are 12 (yet) unidentified examples.
\mathbf{n}oindent We believe that Theorem \ref{th:classif-diagonal} can be proved from Theorem \ref{th:classif-GRS}, via Example \ref{exa:grs-lie-super}.
\begin{theorem}\label{th:presentation-diagonal} \emph{\cite{A-jems, A-crelle}}
An efficient set of defining relations of each finite-dimen\-sional Nichols algebra of a braided vector space of diagonal type is known.
\end{theorem}
The proof uses most technical tools available in the theory of Nichols algebras; of interest in its own is the introduction of the notion of convex order
in Weyl groupoids. As for other classifications above, it is not possible to state precisely the list of relations.
We just mention different types of relations that appear.
\begin{itemize}\renewcommand{$\circ$}{$\circ$}\itemsep1pt \parskip0pt \parsep0pt
\item Quantum Serre relations, i.e., $\ad_c(x_i)^{1- a_{ij}} (x_j)$ for suitable $i\mathbf{n}eq j$.
\item Powers of root vectors, i.e., $x_\beta^{N_\beta}$, where the $x_\beta$'s are part of the PBW basis.
\item More exotic relations; they involve 2, 3, or at most 4 $i$'s in $\I$.
\end{itemize}
\smallbreak\subsection{Nichols algebras of rack type}\label{subsec:nichols-rack}
We now consider Nichols algebras of objects in ${}_{\Gamma}^{\Gamma}\mathcal{YD}g$, where $G$ is a finite not necessarily abelian group.
The category ${}_{\Gamma}^{\Gamma}\mathcal{YD}g$ is semisimple and the simple objects are parametrized by pairs $({\mathcal O}, \rho)$, where ${\mathcal O}$ is a conjugacy class
in $G$ and $\rho\in \operatorname{Irr} G^x$, for a fixed $x\in {\mathcal O}$; the corresponding simple Yetter-Drinfeld module
$M({\mathcal O}, \rho)$ is $\operatorname{Ind}_{G^x}^G \rho$ as a module. The braiding $c$ is described in terms of the conjugation in ${\mathcal O}$. To describe the related suitable class,
we recall that a rack is a set $X \mathbf{n}eq \emptyset$ with a map $\operatorname{tr}id: X \times X \to X$ satisfying
\begin{itemize}\renewcommand{$\circ$}{$\circ$}\itemsep1pt \parskip0pt \parsep0pt
\item $\varphi_x := x\operatorname{tr}id \underline{\quad}$ is a
bijection for every $x \in X$.
\item $x\operatorname{tr}id (y\operatorname{tr}id z) = (x\operatorname{tr}id y) \operatorname{tr}id (x\operatorname{tr}id z)$ for all $x,y,z \in X$ (self-distributivity).
\end{itemize}
For instance, a conjugacy class ${\mathcal O}$ in $G$ with the operation $x\operatorname{tr}id y = xyx^{-1}$, $x, y \in {\mathcal O}$
is a rack; actually we only consider racks realizable as conjugacy classes.
Let $X$ be a rack and $\mathfrak X = (X_k)_{k\in I}$ a decomposition of $X$, i.e., a disjoint family of subracks with $X_l\operatorname{tr}id X_k = X_k$ for
all $k,l\in I$.
\begin{definition}\label{def:braiding-rack}\cite{AG-adv} A \emph{2-cocycle of degree $\mathbf{n} = (n_k)_{k\in I}$},
associated to $\mathfrak X$, is a family $\mathbf{q}
= (q_k)_{k\in I}$ of maps $q_k: X \times X_k \to \mathbf{GL}(n_k,\mathbb C)$
such that
\begin{align}\label{eqn:non-ppal-cocycle-braiding}
q_k(i,j\operatorname{tr}id h)q_k(j,h)&= q_k(i\operatorname{tr}id j,i\operatorname{tr}id h)q_k(i,h),& i,j &\in X, \, h\in X_k, \, k\in I.
\end{align}
Given such $\mathbf{q} $, let $V = \oplus_{k\in I} \mathbb C X_k\otimes\mathbb C^{n_k}$ and let $c^{\mathbf{q} }\in \mathbf{GL}(V\otimes V)$ be given by
\begin{align*}
c^{\mathbf{q} }(x_i v\otimes x_jw) &= x_{i\operatorname{tr}id j}q_k(i,j)(w)\otimes x_iv,& i&\in X_l, \, j\in X_k, \, v\in\mathbb C^{n_l}, \, w\in\mathbb C^{n_k}.
\end{align*}
Then $(V, c^{\mathbf{q} })$ is a braided vector space called of \emph{rack type}; its Nichols algebra is denoted ${\mathfrak B}(X, \mathbf{q} )$.
If $\mathfrak X = (X)$, then we say that $\mathbf{q} $ is \emph{principal}.
\end{definition}
Every finite-dimensional $V\in {}_{\Gamma}^{\Gamma}\mathcal{YD}g$ is a braided vector space of rack type \cite[Theorem 4.14]{AG-adv}.
Question (a$_1$) in this setting has partial answers in three different lines: computation of some finite-dimensional Nichols
algebras, Nichols algebras of reducible Yetter-Drinfeld modules and collapsing of racks.
\smallbreak\subsubsection{Finite-dimensional Nichols algebras of rack type}\label{subsubsec:nichols-rack-findim}
The algorithm to compute a Nichols algebra ${\mathfrak B}(V)$ is as follows:
compute the space $\mathfrak{J}^i(V) = \ker \mathcal{Q}_i$ of relations of degree $i$, for $i = 2, 3, \dots, m$; then compute
the $m$-th partial Nichols algebra $\widehat{{\mathfrak B}_m}(V) = T(V)/ \langle \oplus_{2\le i \le m} \mathfrak{J}^i(V)\rangle$,
say with a computer program. If lucky enough to get $\dim \widehat{{\mathfrak B}_m}(V) < \infty$, then check whether it is a Nichols algebra, e.g. via skew-derivations;
otherwise go to $m+1$. The description of $\mathfrak{J}^2(V) = \ker (\operatorname{id} + c)$ is not difficult \cite{GG} but for higher degrees it turns out to be very complicated.
We list all known examples of finite-dimensional Nichols algebras ${\mathfrak B}(X, \mathbf{q} )$ with $X$ indecomposable and $\mathbf{q} $ principal
and abelian ($n_1 = 1$).
\begin{example}\label{exa:nichols-sim} Let ${\mathcal O}^m_d$ be the conjugacy class of $d$-cycles in
$\sm$, $m \mathfrak{g}e 3$. We start with the rack of transpositions in $\sm$ and the cocycles $-1$, $\chi$ that arise from the $\rho \in \operatorname{Irr} \sm^{(12)}$
with $\rho(12) = -1$, see \cite[(5.5), (5.9)]{MS}.
Let $V$ be a vector space with basis $(x_{ij})_{(ij)\in {\mathcal O}^m_2}$ and consider the relations
\begin{align}
\label{eq:trasp-quadratic}
x_{ij}^2 &= 0, & &(ij)\in {\mathcal O}^m_2;
\\ \label{eq:trasp-far}
x_{ij}x_{kl} + x_{kl}x_{ij} &= 0, & &(ij), (kl)\in {\mathcal O}^m_2, \, \vert \{i,j,k,l\}\vert = 4;
\\ \label{eq:trasp-far-sign}
x_{ij}x_{kl} - x_{kl}x_{ij} &= 0, & &(ij), (kl)\in {\mathcal O}^m_2, \, \vert \{i,j,k,l\}\vert = 4;
\\ \label{eq:trasp-braided}
x_{ij}x_{ik} + x_{jk}x_{ij}+ x_{ik}x_{jk} &= 0, & &(ij), (ik), (jk)\in {\mathcal O}^m_2, \, \vert \{i,j,k\}\vert = 3;
\\ \label{eq:trasp-braided-sgn}
x_{ij}x_{ik} - x_{jk}x_{ij} - x_{ik}x_{jk} &= 0, & &(ij), (ik), (jk)\in {\mathcal O}^m_2, \, \vert \{i,j,k\}\vert = 3.
\end{align}
The quadratic algebras ${\mathfrak B}_m := \widehat{{\mathfrak B}_2}( {\mathcal O}^m_2, -1) = T(V)/\langle \text{\eqref{eq:trasp-quadratic}, \eqref{eq:trasp-far}, \eqref{eq:trasp-braided}}\rangle$
and
${\mathcal E}_m:= \widehat{{\mathfrak B}_2}( {\mathcal O}^m_2, \chi) = T(V)/\langle \text{\eqref{eq:trasp-quadratic}, \eqref{eq:trasp-far-sign}, \eqref{eq:trasp-braided-sgn}}\rangle$
were considered in \cite{MS}, \cite{FK} respectively; ${\mathcal E}_m$ are named the \emph{Fomin-Kirillov algebras}. It is known that
\smallbreak \mathbf{n}oindent $\circ$ The Nichols algebras ${\mathfrak B}( {\mathcal O}^m_2, -1)$ and ${\mathfrak B}( {\mathcal O}^m_2, \chi)$ are twist-equivalent, hence have the same Hilbert series.
Ditto for the algebras ${\mathfrak B}_m$ and ${\mathcal E}_m$ \cite{V}.
\smallbreak \mathbf{n}oindent $\circ$ If $3 \le m \le 5$, then ${\mathfrak B}_m = {\mathfrak B}( {\mathcal O}^m_2, -1)$ and ${\mathcal E}_m ={\mathfrak B}( {\mathcal O}^m_2, \chi)$ are finite-dimensional
\cite{FK, MS, GG} (for $m=5$ part of this was done by Gra\~na). In fact
\begin{align*}
\dim {\mathfrak B}_3 &= 12, &\dim {\mathfrak B}_4 &= 576, &\dim {\mathfrak B}_5 &= 8294400.
\end{align*}
But for $m\mathfrak{g}eq 6$, it is not known whether the Nichols algebras ${\mathfrak B}( {\mathcal O}^m_2, -1)$ and ${\mathfrak B}( {\mathcal O}^m_2, \chi)$ have finite
dimension or are quadratic.
\end{example}
\begin{example}\label{exa:nichols-cube} \cite{AG-adv}
The Nichols algebra ${\mathfrak B}( {\mathcal O}^4_4, -1)$ is quadratic, has the same Hilbert series as ${\mathfrak B}( {\mathcal O}^4_2, -1)$
and is generated by $(x_\sigma)_{\sigma\in {\mathcal O}_4^4}$ with defining relations
\begin{align}
x_\sigma^2 &= 0, & &\\
x_\sigma x_{\sigma^{-1}}+x_{\sigma^{-1}} x_\sigma &= 0, & & \\
x_\sigma x_\kappa+x_\mathbf{n}u x_\sigma+x_\kappa x_\mathbf{n}u&= 0,&
\sigma\kappa &=\mathbf{n}u\sigma, \ \kappa\mathbf{n}eq \sigma\mathbf{n}eq \mathbf{n}u\in {\mathcal O}_4^4.
\end{align}
\end{example}
\begin{example}\label{exa:nichols-affines} \cite{grania1} Let $A$ be a finite abelian group
and $g\in \operatorname{Aut} A$. The \emph{affine} rack $(A, g)$ is the set $A$ with product $a\operatorname{tr}id b = g(b) + (\operatorname{id} - g) (a)$, $a, b \in A$.
Let $p\in \N$ be a prime, $q =p^{v(q)}$ a power of $p$, $A = \F_q$ and $g$ the multiplication by $N \in \F_q^{\times}$;
let $X_{q, N} = (A, g)$. Assume that $q = 3$, 4, 5, or 7, with
$N = 2$, $\omega\in \F_4 - \F_2$, 2 or 3, respectively. Then
$\dim {\mathfrak B}(X_{q, N}, -1) = q \varphi(q)(q-1)^{q-2}$, $\varphi$ being the Euler function, and
$\mathfrak{J}(X_{q, N}, -1) = \langle\mathfrak{J}^2 + \mathfrak{J}^{v(q)(q-1)}\rangle$, where $\mathfrak{J}^2$ is generated by
\begin{align}
&x_i^2, &&\text{always}\\
&x_ix_j+x_{-i+2j}x_i+x_jx_{-i+2j}, &&\text{for $q=3$,}\\
&x_ix_j+x_{(\omega+1)i+\omega j}x_i+x_jx_{(\omega+1)i+\omega j,} &&\text{for $q=4$,}\\
&x_ix_j+x_{-i+2j}x_i+x_{3i-2j}x_{-i+2j}+x_jx_{3i-2j}, &&\text{for $q=5$,}\\
&x_ix_j+x_{-2i+3j}x_i+x_jx_{-2i+3j}, &&\text{for $q=7$,}
\end{align}
with $i, j \in \F_q$; and $\mathfrak{J}^{v(q)(q-1)}$ is generated by $\displaystyle\sum_{h}T^h(V)\mathfrak{J}^2T^{v(q)(q-1) -h -2}(V)$ and
\begin{align}
&(x_\omega x_1x_0)^2 + (x_1x_0x_\omega)^2 + (x_0x_\omega x_1)^2, &&\text{for $q=4$,} \\
&(x_1x_0)^2 + (x_0x_1)^2, &&\text{for $q=5$,} \\
&(x_2x_1x_0)^2 + (x_1x_0x_2)^2 + (x_0x_2x_1)^2, &&\text{for $q=7$.}
\end{align}
Of course $X_{3, 2} = {\mathcal O}_2^3$; also $\dim {\mathfrak B}(X_{4, \omega}, -1) = 72$. By duality, we get
\begin{align*}
\dim {\mathfrak B}(X_{5, 3}, -1) = \dim {\mathfrak B}(X_{5, 2}, -1) &= 1280, \\\dim {\mathfrak B}(X_{7, 5}, -1) = \dim {\mathfrak B}(X_{7, 3}, -1) &= 326592.
\end{align*}
\end{example}
\begin{example}\label{exa:nichols-tetra}\cite{HLV} There is another finite-dimensional Nichols algebra associated to $X_{4, \omega}$
with a cocycle $\mathbf{q} $ with values $\pm \mathtt{x}i$, where $1\mathbf{n}eq \mathtt{x}i\in\G_3$. Concretely, $\dim {\mathfrak B}(X_{4, \omega}, \mathbf{q} ) = 5184$ and
${\mathfrak B}(X_{4, \omega}, \mathbf{q} )$ can be presented by generators $(x_i)_{i\in \F_4}$ with defining relations
\begin{gather*}
\begin{aligned}
x_{0}^{3}=x_{1}^{3}=x_{\omega}^{3}=x_{\omega^2}^{3}&=0, && \\
\mathtt{x}i ^2x_{0}x_{1} + \mathtt{x}i x_{1}x_{\omega} - x_{\omega}x_{0} &=0,& \mathtt{x}i ^2x_{0}x_{\omega} + \mathtt{x}i x_{\omega}x_{\omega^2} - x_{\omega^2}x_{0} &= 0,\\
\mathtt{x}i x_{0}x_{\omega^2} -\mathtt{x}i ^2x_{1}x_{0} + x_{\omega^2}x_{1} &=0, & \mathtt{x}i x_{1}x_{\omega^2} + \mathtt{x}i ^2x_{\omega}x_{1} + x_{\omega^2}x_{\omega} &=0,
\end{aligned}
\\
x_{0}^2x_{1}x_{\omega}x_{1}^2 + x_{0}x_{1}x_{\omega}x_{1}^2x_{0} + x_{1}x_{\omega}x_{1}^2x_{0}^2 + x_{\omega}x_{1}^2x_{0}^2x_{1} + x_{1}^2x_{0}^2x_{1}x_{\omega} + x_{1}x_{0}^2x_{1}x_{\omega}x_{1} \\
+ x_{1}x_{\omega}x_{1}x_{0}^2x_{\omega} + x_{\omega}x_{1}x_{0}x_{1}x_{0}x_{\omega} + x_{\omega}x_{1}^2x_{0}x_{\omega}x_{0} =0.
\end{gather*}
\end{example}
\smallbreak\subsubsection{Nichols algebras of decomposable Yetter-Drinfeld modules over groups}\label{subsubsec:nichols-rack-decomposable}
The ideas of Example \ref{exa:grs-nichols-ss-yd} in the context of decomposable Yetter-Drinfeld modules over groups were pushed further
in a series of papers culminating with a remarkable classification result \cite{HV2}. Consider the groups
\begin{align}
\Gamma_n &= \langle a,b,\mathbf{n}u \vert ba=\mathbf{n}u ab,\quad \mathbf{n}u a=a\mathbf{n}u^{-1},\quad \mathbf{n}u b=b\mathbf{n}u,\quad \mathbf{n}u ^n=1\rangle,\quad n\mathfrak{g}e 2;
\\ T &= \langle\zeta ,\chi _1,\chi _2\vert \zeta \chi_1=\chi_1\zeta ,\quad
\zeta \chi_2=\chi_2\zeta ,\quad \chi_1\chi_2\chi_1=\chi_2\chi_1\chi_2,\quad
\chi_1^3=\chi_2^3\rangle.
\end{align}
\smallbreak \mathbf{n}oindent $\circ$ \cite{HS2} Let $G$ be a quotient of $\Gamma_{2}$. Then there exist $V_1$, $W_1 \in \operatorname{Irr}{}_{\Gamma}^{\Gamma}\mathcal{YD}g$ such that
$\dim V_1 = \dim W_1 = 2$ and $\dim{\mathfrak B}(V_1\oplus W_1) = 64 = 2^6$.
\smallbreak \mathbf{n}oindent $\circ$ \cite{HV2} Let $G$ be a quotient of $\Gamma_{3}$. Then there exist $V_2$, $V_3$, $V_4$, $W_2, W_3, W_4 \in \operatorname{Irr}{}_{\Gamma}^{\Gamma}\mathcal{YD}g$ such that
$\dim V_2 = 1$, $\dim V_3 = \dim V_4 = 2$, $\dim W_2 = \dim W_3 = \dim W_4 = 3$ and $\dim{\mathfrak B}(V_2\oplus W_2) = \dim{\mathfrak B}(V_3\oplus W_3) = 10368 = 2^73^4$,
$\dim{\mathfrak B}(V_4\oplus W_4) = 2304 = 2^{18}$.
\smallbreak \mathbf{n}oindent $\circ$ \cite{HV1} Let $G$ be a quotient of $\Gamma_{4}$. Then there exist $V_5$, $W_5 \in \operatorname{Irr} {}_{\Gamma}^{\Gamma}\mathcal{YD}g$ such that
$\dim V_5 = 2$, $\dim W_5 = 4$ and $\dim{\mathfrak B}(V_5\oplus W_5) = 262144 = 2^{18}$.
\smallbreak \mathbf{n}oindent $\circ$ \cite{HV1} Let $G$ be a quotient of $T$. Then there exist $V_6$, $W_6 \in \operatorname{Irr}{}_{\Gamma}^{\Gamma}\mathcal{YD}g$ such that
$\dim V_6 = 1$, $\dim W_6 = 4$ and $\dim{\mathfrak B}(V_6\oplus W_6) = 80621568 = 2^{12}3^9$.
\begin{theorem} \emph{\cite{HV2}}
Let $G$ be a non-abelian group and $V, W\in \operatorname{Irr} {}_{\Gamma}^{\Gamma}\mathcal{YD}g$ such that $G$ is generated by the support of $V\oplus W$.
Assume that $c^2_{\vert V\otimes W} \mathbf{n}eq \operatorname{id}$ and that $\dim {\mathfrak B} (V\oplus W) < \infty$.
Then $V\oplus W$ is one of $V_i\oplus W_i$, $i\in \I_6$, above, and correspondingly $G$ is a quotient of
either $\Gamma_n$, $2\le n\le 4$, or $T$.
\end{theorem}
\smallbreak\subsubsection{Collapsing racks}\label{subsubsec:nichols-rack-collapsing}
Implicit in Question (a$_1$) in the setting of racks is the need to compute all non-principal 2-cocycles for a fixed rack $X$.
Notably, there exist criteria that dispense of this computation. To state them and explain their significance, we need some terminology.
All racks below are finite.
\smallbreak \mathbf{n}oindent $\circ$ A rack $X$ is \emph{abelian} when $x\operatorname{tr}id y = y$, for all $x, y \in X$.
\smallbreak \mathbf{n}oindent $\circ$
A rack is \emph{indecomposable} when it is not a disjoint union
of two proper subracks.
\smallbreak \mathbf{n}oindent $\circ$ A rack $X$ with $\vert X\vert > 1$ is \emph{simple} when for any projection of racks
$\pi: X\to Y$, either $\pi$ is an isomorphism or $Y$ has only one element.
\begin{theorem} \emph{\cite[3.9, 3.12]{AG-adv}, \cite{jo}}
Every simple rack is isomorphic to one of:
\begin{enumerate}\renewcommand{(\alph{enumi})}{(\alph{enumi})}\itemsep1pt \parskip0pt \parsep0pt
\item Affine racks $(\F_p^t, T)$, where $p$ is a prime, $t\in \N$, and
$T$ is the companion matrix of a monic irreducible polynomial $f\in \F_p[\mathtt X]$ of degree $t$,
$f\mathbf{n}eq \mathtt X, \mathtt X-1$.
\item\label{item:twisted-conjugacy} Non-trivial (twisted) conjugacy classes in simple groups.
\item\label{item:twisted-homogeneous} Twisted conjugacy classes of type $(G,u)$, where $G = L^t$, with $L$ a simple non-abelian group and $1 < t\in \N$;
and $u\in \operatorname{Aut} (L^t)$ acts by $u(\ell_1,\ell_2, \dots,\ell_t)=(\theta(\ell_t),\ell_1,\ell_2, \dots,\ell_{t-1})$,
where $\theta\in\operatorname{Aut}(L)$.
\end{enumerate}
\end{theorem}
\begin{definition}\label{def:rack-typeD}
\cite[3.5]{AFGV-ampa} We say that a finite rack $X$ is \emph{of type D}
when there are a decomposable subrack
$Y = R\coprod S$, $r\in R$ and $s\in S$ such that
$r\operatorname{tr}id(s\operatorname{tr}id(r\operatorname{tr}id s)) \mathbf{n}eq s$.
Also,
$X$ is \emph{of type F} \cite{ACG-I} if there are a disjoint family of subracks $(R_a)_{a \in \I_4}$
and a family $(r_a)_{a \in \I_4}$ with $r_a\in R_a$, such that
$R_a \operatorname{tr}iangleright R_b = R_b$,
$r_a\operatorname{tr}iangleright r_b \mathbf{n}eq r_b$, for all $a\mathbf{n}eq b \in \I_4$.
\end{definition}
An indecomposable rack $X$ \emph{collapses} when $\dim {\mathfrak B}(X, \mathbf{q} ) = \infty$
for every finite \emph{faithful} 2-cocycle $\mathbf{q} $ (see \cite{AFGV-ampa} for the definition of faithful).
\begin{theorem}\label{th:type4} \emph{\cite[3.6]{AFGV-ampa}; \cite[2.8]{ACG-I}} If a rack is of type D or F, then it collapses.
\end{theorem}
The proofs use results on Nichols algebras from \cite{AHS, HeckSch, CH}.
If a rack projects onto a rack of type D (or F), then it is also
of type D (or F), hence it collapses by Theorem \ref{th:type4}.
Since every indecomposable rack $X$, $\vert X\vert > 1$, projects onto a simple rack,
it is natural to ask for the determination of all \emph{simple} racks of type D or F.
A rack is \emph{cthulhu}
if it is neither of type D nor F;
it is \emph{sober} if every subrack is either abelian or indecomposable \cite{ACG-I}. Sober implies cthulhu.
\smallbreak \mathbf{n}oindent $\circ$ Let $m\mathfrak{g}eq 5$. Let ${\mathcal O}$ be either ${\mathcal O}^{\sm}_{\sigma}$, if
$\sigma\in \sm -\am$, or else ${\mathcal O}^{\am}_{\sigma}$ if $\sigma\in \am$. The type of $\sigma$ is formed by the lengths of the cycles in its decomposition.
\smallbreak \mathbf{n}oindent $\diamond$ \cite[4.2]{AFGV-ampa} If the type of $\sigma$ is $(3^2)$, $(2^2,3)$, $(1^n,3)$, $(2^4)$, $(1^2,2^2)$, $(2,3)$, $(2^3)$, or
$(1^n,2)$, then ${\mathcal O}$ is cthulhu.
If the type of $\sigma$ is $(1,2^2)$, then ${\mathcal O}$ is sober.
\smallbreak \mathbf{n}oindent $\diamond$ \cite{F} Let $p\in \N$ be a prime. Assume the type of $\sigma$ is $(p)$.
If $p = 5, 7$ or not of the form $(r^k - 1)/(r -1)$, $r$ a prime power, then ${\mathcal O}$ is sober; otherwise ${\mathcal O}$ is of type D.
Assume the type of $\sigma$ is $(1, p)$. If $p = 5$ or not of the form $(r^k - 1)/(r -1)$, $r$ a prime power,
then ${\mathcal O}$ is sober; otherwise ${\mathcal O}$ is of type D.
\smallbreak \mathbf{n}oindent $\diamond$ \cite[4.1]{AFGV-ampa} For all other types, ${\mathcal O}$ is of type D, hence it collapses.
\smallbreak \mathbf{n}oindent $\circ$ \cite{ACG-I} Let $n \mathfrak{g}e 2$ and $q$ be a prime power. Let $x\in \mathbf{PSL}_n(q)$ not semisimple and ${\mathcal O} = {\mathcal O}^{\mathbf{PSL}_n(q)}_{x}$.
The type of a unipotent element are the sizes of its Jordan blocks.
\smallbreak \mathbf{n}oindent $\diamond$ Assume $x$ is unipotent. If $x$ is either of type $(2)$ and $q$ is even or not a square, or
of type $(3)$ and $q =2$, then ${\mathcal O}$ is sober. If $x$ is either of type $(2,1)$ and $q$ is even, or of type $(2,1,1)$ and $q=2$
then ${\mathcal O}$ is cthulhu. If $x$ is of type $(2,1, 1)$ and $q > 2$ is even, then ${\mathcal O}$ is not of type D, but it is open if it is of type F.
\smallbreak \mathbf{n}oindent $\diamond$ Otherwise, ${\mathcal O}$ is either of type D or of type F, hence it collapses.
\smallbreak \mathbf{n}oindent $\circ$ \cite{AFGV-espo, FV} Let ${\mathcal O}$ be a conjugacy class in a sporadic simple group $G$.
If ${\mathcal O}$ appears in Table \ref{tab:notD}, then ${\mathcal O}$ is not of type D.
If $G = M$ is the Monster and ${\mathcal O}$ is one of \textup{32A, 32B, 41A, 46A, 46B, 47A, 47B, 59A, 59B, 69A, 69B, 71A, 71B,
87A, 87B, 92A, 92B, 94A, 94B}, then it is open whether ${\mathcal O}$ is of type D.
Otherwise, ${\mathcal O}$ is of type D.
\begin{table}[t]
\caption{Classes in sporadic simple groups not of type D}\label{tab:notD}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\mathfrak{h}line Group & Classes & Group & Classes\\
\mathfrak{h}line $T$ & \textup{2A} & $Co_{3}$ & \textup{23A, 23B}\\
\mathfrak{h}line $M_{11}$ & \textup{8A, 8B, 11A, 11B} & $J_{1}$ & \textup{15A, 15B, 19A, 19B, 19C}\\
\mathfrak{h}line $M_{12}$ & \textup{11A, 11B} & $J_{2}$ & \textup{2A, 3A}\\
\mathfrak{h}line $M_{22}$ & \textup{11A, 11B} & $J_{3}$ & \textup{5A, 5B, 19A, 19B}\\
\mathfrak{h}line $M_{23}$ & \textup{23A, 23B} & $J_4$ & \textup{29A, 43A, 43B, 43C}\\
\mathfrak{h}line $M_{24}$ & \textup{23A, 23B}& $Ly$ & \textup{37A, 37B, 67A, 67B, 67C}\\
\mathfrak{h}line $Ru$ & \textup{29A, 29B}& $O'N$ & \textup{31A, 31B}\\
\mathfrak{h}line $Suz$ & \textup{3A} & $Fi_{23}$ & \textup{2A}\\
\mathfrak{h}line $HS$ & \textup{11A, 11B} & $Fi_{22}$ & \textup{2A, 22A, 22B}\\
\mathfrak{h}line $McL$ & \textup{11A, 11B} & $Fi'_{24}$ & \textup{29A, 29B}\\
\mathfrak{h}line $Co_{1}$ & \textup{3A} & $B$ & \textup{2A, 46A, 46B, 47A, 47B}\\
\mathfrak{h}line $Co_{2}$ & \textup{2A, 23A, 23B} \\
\cline{1-2}
\end{tabular}
\end{center}
\end{table}
\subsection{Generation in degree one}\label{subsec:gen-deg-1}
Here is the scheme of proof proposed in \cite{AS-adv} to attack Conjecture \ref{conj:gendegone}:
Let $T$ be a finite-dimensional graded Hopf algebra in ${}_{K}^{K}\mathcal{YD}$ with $T^0 = \mathbb C$ and generated as algebra by $T^1$. We have a commutative diagram of Hopf algebra maps
$\mathtt{x}ymatrix@R=2pt@C=10pt{T \ar@{->>}^{\pi}[rr] & & {\mathfrak B}(V) \\ & T(V)\ar@{->}[ul]^p \ar@{->}[ur] &}
$. To show that $\pi$ is injective, take a generator $r$ (or a family of generators) of $\mathfrak{J}(V)$ such that $r\in \mathcal{P}(T(V))$
and consider the Yetter-Drinfeld submodule $U = \mathbb C r\oplus V$ of $T(V)$; if $\dim {\mathfrak B}(U) = \infty$, then $p(r) = 0$.
Then $p$ factorizes through $T(V)/\mathfrak{J}_1(V)$, where $\mathfrak{J}_1(V)$ is the ideal generated by primitive generators of $\mathfrak{J}(V)$, and so on.
The Conjecture has been verified in all known examples in characteristic 0 (it is false in positive characteristic or for infinite-dimensional Hopf algebras).
\begin{theorem}
A finite-dimensional pointed Hopf algebra $H$ is generated by group-like and skew-primitive elements if either of the following holds:
\smallbreak \mathbf{n}oindent $\diamond$ \emph{\cite{A-crelle}} The infinitesimal braiding is of diagonal type, e.~g. $G(H)$ is abelian.
\smallbreak \mathbf{n}oindent $\diamond$ \emph{\cite{AG-ama, GG}.} The infinitesimal braiding of $H$ is any of $({\mathcal O}^m_2, -1)$, $({\mathcal O}^m_2, \chi)$ ($m= 3,4, 5$), $(X_{4, \omega}, -1)$,
$(X_{5, 2}, -1)$, $(X_{5, 3}, -1)$, $(X_{7, 3}, -1)$, $(X_{7, 5}, -1)$.
\end{theorem}
\smallbreak\subsection{Liftings}\label{subsec:deformations}
We address here Question \ref{item:method-lifting} in \S \ref{subsec:chevalley}. Let $X$ be a finite rack and $q: X\times X \to \G_{\infty}$
a 2-cocycle. A Hopf algebra $H$ is a \emph{lifting} of $(X, q)$ if $H_0$ is a Hopf subalgebra, $H$ is generated by $H_1$ and
its infinitesimal braiding is a realization of $(\mathbb C X,c^{q})$. See \cite{GIV} for liftings in the setting of copointed Hopf algebras.
We start discussing realizations of braided vector spaces as Yetter-Drinfeld modules. Let $\theta\in \N$ and $\I = \I_\theta$.
First, a \emph{YD-datum of diagonal type} is a collection
\begin{align}\label{eq:YD-datum-diagonal}
\D = ((q_{ij})_{i, j\in \I}, G, (g_i)_{i\in \I}, (\chi_i)_{i\in \I}),
\end{align}
where $q_{ij} \in \G_{\infty}$, $q_{ii}\mathbf{n}eq 1$, $i, j\in \I$; $G$ is a finite group; ${g_i} \in Z(G)$; ${\chi_i} \in \widehat{G}$, $i\in \I$;
such that $q_{ij} := \chi_j(g_i)$, $i, j\in \I$.
Let $(V,c)$ be the braided vector space of diagonal type with matrix $(q_{ij})$
in the basis $(x_i)_{i\in \I_\theta}$. Then $V\in {}_{\Gamma}^{\Gamma}\mathcal{YD}g$ by declaring $x_i \in V_{g_i}^{\chi_i}$, $i\in \I$. More generally,
a \emph{YD-datum of rack type} \cite{AG-ama, MS} is a collection
\begin{align}\label{eq:YD-datum-rack}
\D = (X, q, G, \cdot, g, \chi),
\end{align}
where $X$ is a finite rack; $q: X\times X \to \G_{\infty}$ is a $2$-cocycle; $G$ is a finite group;
$\cdot$ is an action of $G$ on $X$; $g: X \to G$ is equivariant
with respect to the conjugation in $G$; and $\chi = (\chi_i)_{i\in X}$ is a family of 1-cocycles
$\chi_i: G \to \mathbb C^{\times}$ (that is, $\chi_i(ht) =\chi_i(t) \chi_{t \cdot i}(h)$, for
all $i\in X$, $h, t\in G$) such that $g_i \cdot j = i\operatorname{tr}id j$ and
$\chi_i(g_j) = q_{ij}$ for all $i, j\in X$.
Let $(V, c) = (\mathbb C X, c^q)$ be the associated braided vector space.
Then $V$ becomes an object in ${}_{\Gamma}^{\Gamma}\mathcal{YD}g$ by $\delta(x_i) = g_i \otimes x_i$ and
$t\cdot x_{i} = \chi_i(t) x_{t\cdot i}$, $t \in G$, $i\in X$.
Second, let $\D$ be a YD-datum of either diagonal or rack type and $V \in {}_{\Gamma}^{\Gamma}\mathcal{YD}g$ as above; let $\T(V) := T(V)\# \mathbb C G$.
The desired liftings are quotients of $\T(V)$; write $a_i$ in these quotients
instead of $x_i$ to distinguish them from the elements in ${\mathfrak B}(V)\# \mathbb C G$.
Let $\mathcal{G}$ be a minimal set of generators of $\mathfrak{J}(V)$, assumed homogeneous both for the $\N$- and the $G$-grading.
Roughly speaking, the deformations will be defined by replacing the relations $r = 0$ by $r = \phi_r$, $r\in \mathcal{G}$,
where $\phi_r\in \T(V)$ belongs to a lower term of the coradical filtration, and the ideal $\mathfrak{J}_{\phi}(V)$ generated by $\phi_r$, $r\in \mathcal{G}$,
is a Hopf ideal. The problem is to describe the $\phi_r$'s and to check that $\T(V)/\mathfrak{J}_{\phi}(V)$ has the right dimension.
If $r\in \mathcal{P}(T(V))$ has $G$-degree $g$, then $\phi_r = \lambda(1 - g)$ for some $\lambda\in \mathbb C$; depending on the action of $G$ on $r$,
it may happen that $\lambda$ should be 0. In some cases, all $r\in \mathcal{G}$ are primitive, so all deformations can be described; see \cite{AS-jalg}
for quantum linear spaces (their liftings can also be presented as Ore extensions \cite{BDG})
and the Examples \ref{exa:lifting-transpositions-s3} and \ref{exa:lifting-transpositions-s4}.
But in most cases, not all $r\in \mathcal{G}$ are primitive and some recursive construction of the deformations is needed. This was achieved in
\cite{AS-cambr} for diagonal braidings of Cartan type $A_n$, with explicit formulae, and in \cite{AS-ann} for diagonal braidings of finite Cartan type,
with recursive formulae. Later it was observed that the so obtained liftings are cocycle deformations of ${\mathfrak B}(V)\# \mathbb C G$, see e.g. \cite{M4}.
This led to the strategy in \cite{AAGMV}: pick an adapted stratification $\mathcal{G} = \mathcal{G}_0 \cup \mathcal{G}_1 \cup\dots \cup \mathcal{G}_{N}$ \cite[5.1]{AAGMV};
then construct recursively the deformations of $T(V) / \langle \mathcal{G}_0 \cup \mathcal{G}_1 \cup\dots \cup \mathcal{G}_{k-1}\rangle$
by determining the cleft extensions of the deformations in the previous step and applying the theory of Hopf bi-Galois extensions \cite{S-bigal}.
In the Examples below, $\chi_i = \chi \in \widehat{G}$ for all $i\in X$ by \cite[3.3 (d)]{AG-ama}.
\begin{example}\label{exa:lifting-transpositions-s3} \cite{AG-ama, GIV}
Let $\D = ({\mathcal O}^3_2, -1, G, \cdot, g, \chi)$ be a YD-datum. Let $\lambda \in \mathbb C^2$ be such that
\begin{align}
\label{norm3} \lambda_1 &= \lambda_2 = 0,& &\text{ if }\chi^2\mathbf{n}eq \varepsilon;
\\
\label{norm1}
\lambda_1 &= 0, & &\text{ if } g_{12}^2 = 1; &
\lambda_2 &= 0, & &\text{ if } g_{12}g_{13} = 1.
\end{align}
Let $\ug = \ug(\D, \lambda)$ be the quotient of $\T(V)$ by the relations
\begin{align}
\label{relsym31}
a_{12}^2 &= \lambda_1(1-g_{12}^2),
\\\label{relsym33}
a_{12}a_{13} + a_{23}a_{12}+ a_{13}a_{23} &= \lambda_2 (1-g_{12} g_{13}).
\end{align}
Then $\ug$ is a pointed Hopf algebra, a cocycle deformation of $\operatorname{gr} \ug\simeq {\mathfrak B}(V)\#\mathbb C G$ and $\dim \ug = 12 |G|$;
$\ug(\D, \lambda) \simeq \ug(\D, \lambda')$ iff $\lambda = c\lambda'$ for some $c\in \mathbb C^{\times}$.
Conversely, any lifting of $({\mathcal O}^3_2,-1)$ is isomorphic to $\ug(\D, \lambda)$ for some YD-datum
$\D = ({\mathcal O}^3_2, -1, G, \cdot, g, \chi)$ and $\lambda \in \mathbb C^2$ satisfying \eqref{norm3}, \eqref{norm1}.
\end{example}
\begin{example}\label{exa:lifting-transpositions-s4} \cite{AG-ama}
Let $\D = ({\mathcal O}^4_2, -1, G, \cdot, g, \chi)$ be a YD-datum.
Let $\lambda \in \mathbb C^3$ be such that
\begin{align}\label{norm3bis}
\lambda_i &= 0,& &i\in \I_3,& \text{ if }\chi^2&\mathbf{n}eq \varepsilon;
\\\label{norm14}
\lambda_1 &= 0, & &\text{ if } g_{12}^2 = 1; &
\lambda_2 &= 0, & \text{ if } g_{12}g_{34} &= 1; &
\lambda_3 &= 0, & \text{ if } g_{12}g_{13} &= 1.
\end{align}
Let $\ug = \ug(\D, \lambda)$ be the quotient of $\T(V)$ by the relations
\begin{align}
\label{relsym41}
a_{12}^2 &= \lambda_1(1-g_{12}^2),
\\\label{relsym42}
a_{12} a_{34} +a_{34} a_{12} &= \lambda_2 (1-g_{12} g_{34}),
\\\label{relsym43}
a_{12}a_{13} + a_{23}a_{12}+ a_{13}a_{23} &= \lambda_3 (1- g_{12} g_{13}).
\end{align}
Then $\ug$ is a pointed Hopf algebra, a cocycle deformation of ${\mathfrak B}(V)\#\mathbb C G$ and $\dim \ug = 576 |G|$;
$\ug(\D, \lambda) \simeq \ug(\D, \lambda')$ iff $\lambda = c\lambda'$ for some $c\in \mathbb C^{\times}$.
Conversely, any lifting of $({\mathcal O}^4_2, -1)$ is isomorphic to $\ug(\D, \lambda)$ for some YD-datum
$\D = ({\mathcal O}^4_2, -1, G, \cdot, g, \chi)$ and $\lambda \in \mathbb C^3$ satisfying \eqref{norm3bis}, \eqref{norm14}.
\end{example}
\begin{example}\label{exa:lifting-affine-f4} \cite{GIV}
Let $\D = (X_{4,\omega}, -1, G, \cdot, g, \chi)$ be a YD-datum. Let $\lambda \in \mathbb C^3$ be such that
\begin{align}
\label{eqn:cond1-A} \lambda_1 &= \lambda_2 = 0,& \text{ if }\chi^2&\mathbf{n}eq \varepsilon;& \lambda_3 &= 0, \text{ if }\chi^6\mathbf{n}eq \varepsilon;\\
\label{eqn:cond1-L} \lambda_1 &=0, \text{ if }\, g_0^2=1, & \lambda_2 &=0, \text{ if } \, g_0g_1 =1, & \lambda_3&=0, \text{ if }\, g_0^3g_1^3=1.
\end{align}
Let $\ug = \ug(\D, \lambda)$ be the quotient of $\T(V)$ by the relations
\begin{align}
x_0^2 &= \lambda_1(1- g_0^2), \\
x_0x_1 + x_{\omega} x_0 + x_1x_{\omega} &= \lambda_2(1 -g_0g_1) \\
(x_{\omega}x_1x_0)^2 + (x_1x_0x_{\omega})^2 + (x_0x_{\omega}x_1)^2 &= \zeta_6 - \lambda_3(1-g_0^3g_1^3),& &\text{where}
\end{align}
\begin{align*}&\zeta_6=\lambda_2(x_{\omega}x_1x_0x_{\omega}+ x_1x_0x_{\omega}x_1
+x_0x_{\omega}x_1x_0)-\lambda_2^3(g_0g_1-g_0^3g_1^3)
\\\mathbf{n}otag
+ & \lambda_1^2g_0^2\big(g_{1 + \omega}^2(x_{\omega}x_3 +x_0x_{\omega}) +g_1g_{1 + \omega}(x_{\omega}x_1+
x_1x_3) +g_1^2(x_1x_0+x_0x_3)\big)
\\\mathbf{n}otag
- & 2\lambda_1^2g_0^2(x_0x_3- x_{\omega}x_3 -x_1x_{\omega} + x_1x_0) - 2\lambda_1^2g_{\omega}^2(x_{\omega}x_3
-x_1x_3+x_0x_{\omega} - x_0x_1 )
\\\mathbf{n}otag
- & 2\lambda_1^2g_1^2(x_{\omega}x_1 + x_1x_3 + x_1x_{\omega} - x_0x_3 + x_0x_1)
\\\mathbf{n}otag
+ & \lambda_2\lambda_1(g_{\omega}^2x_0x_3+ g_1^2x_{\omega}x_3+ g_0^2x_1x_3)
+\lambda_2^2g_0g_1(x_{\omega}x_1 + x_1x_0 + x_0x_{\omega}-\lambda_1)
\\ \mathbf{n}otag
- & \lambda_2\lambda_1^2(3g_0^3g_{1 + \omega}-2 g_0g_1^3- g_0^2g_{\omega}^ 2
-2 g_0^3g_1+g_{\omega}^2 - g_1^2+g_0^2)
\\\mathbf{n}otag
- &\lambda_2(\lambda_1-\lambda_2)\big(\lambda_1\,g_0^2(g_{1 + \omega}^2+g_1g_{1 + \omega} +g_1^2+2g_0g_1^3)
+x_{\omega}x_1 + x_1x_0 + x_0x_{\omega}\big).
\end{align*}
Then $\ug$ is a pointed Hopf algebra, a cocycle deformation of $\operatorname{gr} \ug\simeq {\mathfrak B}(V)\#\mathbb C G$ and $\dim \ug = 72 |G|$;
$\ug(\D, \lambda) \simeq \ug(\D, \lambda')$ iff $\lambda = c\lambda'$ for some $c\in \mathbb C^{\times}$.
Conversely, any lifting of $(X_{4,\omega}, -1)$ is isomorphic to $\ug(\D, \lambda)$ for some YD-datum
$\D = (X_{4,\omega}, -1, G, \cdot, g, \chi)$ and $\lambda \in \mathbb C^3$ satisfying \eqref{eqn:cond1-A}, \eqref{eqn:cond1-L}.
\end{example}
\begin{example}\label{exa:lifting-affine-f5} \cite{GIV}
Let $\D = (X_{5,2}, -1, G, \cdot, g, \chi)$ be a YD-datum. Let $\lambda \in \mathbb C^3$ be such that
\begin{align}
\label{eqn:cond1-Aa} \lambda_1 &= \lambda_2 = 0,& \text{ if }\chi^2&\mathbf{n}eq \varepsilon;& \lambda_3 &= 0, \text{ if }\chi^4\mathbf{n}eq \varepsilon;\\
\label{eqn:cond1-La} \lambda_1 &=0, \text{ if }\, g_0^2=1, & \lambda_2 &=0, \text{ if } \, g_0g_1 =1, & \lambda_3&=0, \text{ if }\,g_0^2g_1g_2 = 1.
\end{align}
Let $\ug = \ug(\D, \lambda)$ be the quotient of $\T(V)$ by the relations
\begin{align}
&x_0^2 = \lambda_1(1- g_0^2), \\
&x_0x_1 + x_2x_0 + x_3x_2 + x_1x_3 = \lambda_2(1 -g_0g_1), \\
&(x_1x_0)^2 + (x_0x_1)^2 = \zeta_4 - \lambda_3(1-g_0^2g_1g_2),& &\text{where}
\end{align}
$\zeta_4= \lambda_2\, (x_1x_0 +
x_0x_1)+\lambda_1\, g_1^2(x_3x_0+ x_2x_3) - \lambda_1\, g_0^2(x_2x_4+ x_1x_2) +
\lambda_2\lambda_1\,g_0^2(1- g_1g_2)$.
Then $\ug$ is a pointed Hopf algebra, a cocycle deformation of $\operatorname{gr} \ug\simeq {\mathfrak B}(V)\#\mathbb C G$ and $\dim \ug = 1280 |G|$;
$\ug(\D, \lambda) \simeq \ug(\D, \lambda')$ iff $\lambda = c\lambda'$ for some $c\in \mathbb C^{\times}$.
Conversely, any lifting of $(X_{5,2}, -1)$ is isomorphic to $\ug(\D, \lambda)$ for some YD-datum
$\D = (X_{5,2}, -1, G, \cdot, g, \chi)$ and $\lambda \in \mathbb C^3$ satisfying \eqref{eqn:cond1-Aa}, \eqref{eqn:cond1-La}.
\end{example}
\begin{example}\label{exa:lifting-affine-f5-3} \cite{GIV}
Let $\D = (X_{5,3}, -1, G, \cdot, g, \chi)$ be a YD-datum. Let $\lambda \in \mathbb C^3$ be such that
\begin{align}
\label{eqn:cond1-Ab} \lambda_1 &= \lambda_2 = 0,& \text{ if }\chi^2&\mathbf{n}eq \varepsilon;& \lambda_3 &= 0, \text{ if }\chi^4\mathbf{n}eq \varepsilon;\\
\label{eqn:cond1-Lb} \lambda_1 &=0, \text{ if }\, g_0^2=1, & \lambda_2 &=0, \text{ if } \, g_1g_0 =1, & \lambda_3&=0, \text{ if }\,g_0^2g_1g_3 = 1.
\end{align}
Let $\ug = \ug(\D, \lambda)$ be the quotient of $\T(V)$ by the relations
\begin{align}
&x_0^2 = \lambda_1(1- g_0^2), \\
&x_1x_0 + x_0x_2 + x_2x_3 + x_3x_1 = \lambda_2(1 -g_1g_0) \\
&x_0x_2x_3x_1 + x_1x_4x_3x_0 = \zeta'_4 - \lambda_3(1-g_0^2g_1g_3),
\end{align}
where
$\zeta'_4= \lambda_2\, (x_0x_1 + x_1x_0) -\lambda_1\, g_1^2(x_3x_2 +x_0x_3) - \lambda_1\,
g_0^2(x_3x_4 +x_1x_3) + \lambda_1\lambda_2( g_1^2 +
g_0^2- 2 g_0^2g_1g_3)$.
Then $\ug$ is a pointed Hopf algebra, a cocycle deformation of $\operatorname{gr} \ug\simeq {\mathfrak B}(V)\#\mathbb C G$ and $\dim \ug = 1280 |G|$;
$\ug(\D, \lambda) \simeq \ug(\D, \lambda')$ iff $\lambda = c\lambda'$ for some $c\in \mathbb C^{\times}$.
Conversely, any lifting of $(X_{5,3}, -1)$ is isomorphic to $\ug(\D, \lambda)$ for some YD-datum
$\D = (X_{5,3}, -1, G, \cdot, g, \chi)$ and $\lambda \in \mathbb C^3$ satisfying \eqref{eqn:cond1-Ab}, \eqref{eqn:cond1-Lb}.
\end{example}
\section{Pointed Hopf algebras}\label{sec:pointed}
\smallbreak\subsection{Pointed Hopf algebras with abelian group}\label{subsec:pointed-ab}
Here is a classification from \cite{AS-ann}.
Let $\D = ((q_{ij})_{i, j\in \I_{\theta}}, \Gamma, (g_i)_{i\in \I_\theta}, (\chi_i)_{i\in \I_{\theta}})$ be a YD-datum of diagonal type
as in \eqref{eq:YD-datum-diagonal} with $\Gamma$ a finite abelian group and let $V\in {}_{\Gamma}^{\Gamma}\mathcal{YD}$ be the corresponding realization.
We say that $\D$ is a \emph{Cartan datum} if
there is a Cartan matrix (of finite type) $\mathbf{a} = (a_{ij})_{i, j\in \I_{\theta}}$
such that $q_{ij} q_{ji} = q_{ii}^{a_{ij}}$, $i\mathbf{n}eq j \in \I_{\theta}$.
\smallbreak
Let $\Phi$ be the root system associated to $\mathbf{a} $, $\alpha_1,\dots,\alpha_{\theta}$ a choice of simple roots,
$\mathcal{X}$ the set of connected components of the Dynkin diagram of $\Phi$ and set
$i \sim j$ whenever $\alpha_i, \alpha_j$ belong to the same $J\in \mathcal X$. We consider two classes of parameters:
\begin{itemize}\renewcommand{$\circ$}{$\circ$}\itemsep1pt \parskip0pt \parsep0pt
\item $ \lambda =(\lambda_{ij})_{\substack{i < j \in \I_\theta,\\ i\mathbf{n}ot\sim j}}$ is a family in $\{0,1\}$ with $\lambda_{ij} =0$ when
$g_ig_j =1$ or $\chi_i \chi_j \mathbf{n}eq \varepsilon$.
\item
$\mu=(\mu_{\alpha})_{\alpha \in \Phi^+}$ is a family in $\mathbb C$
with $\mu_{\alpha} =0$ when $\supp \alpha \subset J$, $J \in \mathcal{X}$, and $g_{\alpha}^{N_J} =1$ or $\chi_{\alpha}^{N_J} \mathbf{n}eq\varepsilon$.
Here $N_J =\operatorname{ord} q_{ii}$ for an arbitrary $i\in J$.
\end{itemize}
We attach a family ($u_{\alpha}(\mu))_{\alpha \in \Phi^+}$ in $\mathbb C\Gamma$ to the parameter $\mu$ , defined
recursively on the length of $\alpha$, starting by $u_{\alpha_i}(\mu) = \mu_{\alpha_i}(1 - g_i^{N_i})$.
From all these data we define a Hopf algebra $\mathfrak{u}(\D,\lambda,\mu)$ as the quotient of
$\T(V) = T(V) \#\mathbb C \Gamma$ by the relations
\begin{align}
ga_i g^{-1} &= \chi_i(g)a_i, & & \\
\ad_c(a_i)^{1 - a_{ij}}(a_j) &= 0,& &i \mathbf{n}eq j, i\sim j,\\
\ad_c(a_i)(a_j) &= \lambda_{ij}(1 - g_ig_j),& & i<j ,i \mathbf{n}sim j,\\
a_{\alpha}^{N_J} &= u_{\alpha}(\mu).
\end{align}
\begin{theorem} The Hopf algebra ${\mathfrak{u}(\D,\lambda,\mu)}$ is pointed, $G({\mathfrak{u}(\D,\lambda,\mu)}) \simeq \Gamma$ and
$\dim {\mathfrak{u}(\D,\lambda,\mu)} = \prod_{J \in \mathcal{X}} N_J^{|\Phi_J^+|} |\Gamma|$.
Let $H$ be a pointed finite-dimensional Hopf algebra and set $\Gamma = G(H)$.
Assume that the prime divisors of $\vert\Gamma\vert$ are $>7$. Then there exists a Cartan datum
$\D$ and parameters $\lambda$ and $\mu$ such that $H \simeq \mathfrak{u}(\D,\lambda,\mu)$.
It is known when two Hopf algebras ${\mathfrak{u}(\D,\lambda,\mu)}$ and ${\mathfrak{u}(\D',\lambda',\mu')}$ are isomorphic.
\end{theorem}
The proof offered in \cite{AS-ann} relies on \cite{AS-adv, H-inv, L1, L2}.
Some comments: the hypothesis on $\vert \Gamma\vert$ forces the infinitesimal braiding $V$ of $H$ to be of Cartan type,
and the relations of ${\mathfrak B}(V)$ to be just quantum Serre and powers of root vectors. The quantum Serre relations
are not deformed in the liftings, except those linking different components of the Dynkin diagram; the powers of the root vectors
are deformed to the $u_{\alpha}(\mu)$ that belong to the coradical. All this can fail without the hypothesis, see \cite{helbig} for examples
in rank 2.
\smallbreak\subsection{Pointed Hopf algebras with non-abelian group}\label{subsec:pointed-non-ab}
We present some classification results of pointed Hopf algebras with non-abelian group.
We say that a finite group $G$ \emph{collapses} whenever any finite-dimensional pointed Hopf algebra $H$ with $G(H)\simeq G$ is isomorphic to $\mathbb C G$.
\smallbreak \mathbf{n}oindent $\bullet$ \cite{AFGV-ampa, AFGV-espo} Let $G$ be either $\am$, $m\mathfrak{g}eq 5$, or a
sporadic simple group, different from $Fi_{22}$, the Baby Monster $B$ or the Monster $M$.
Then $G$ collapses.
The proof uses \S \ref{subsubsec:nichols-rack-collapsing}; the remaining Yetter-Drinfeld modules are discarded
considering abelian subracks of the supporting conjugacy class and the list in \cite{H-classif}.
\smallbreak \mathbf{n}oindent $\bullet$ \cite{AHS} Let $V = M({\mathcal O}^3_2, \sgn)$ and let $\D$ be the corresponding YD-datum.
Let $H$ be a finite-dimensional pointed Hopf algebra with $G(H)\simeq \st$.
Then $H$ is isomorphic either to $\mathbb C\st$, or to $\ug(\D, 0) = {\mathfrak B}(V)\# \mathbb C\st$, or to $\ug(\D, (0,1))$, cf. Example \ref{exa:lifting-transpositions-s3}.
\smallbreak \mathbf{n}oindent $\bullet$ \cite{GG} Let $H\mathbf{n}ot\simeq \mathbb C\sk$ be a finite-dimensional pointed Hopf algebra with $G(H)\simeq \sk$.
Let $V_1 = M({\mathcal O}^4_2, \sgn \otimes \operatorname{id})$, $V_2 = M({\mathcal O}^4_2, \sgn \otimes \sgn)$, $W = M({\mathcal O}^4_4, \sgn \otimes \operatorname{id})$, with corresponding data $\D_1$,
$\D_2$ and $\D_3$.
Then $H$ is isomorphic to one of
\begin{align*}
&\ug(\D_1, (0,\mu)),\quad \mu\in \mathbb C^2;&
&\ug(\D_2, t), \quad t\in \{0,1\};& &\ug(\D_3, \lambda),\quad \lambda\in\mathbb C^2.
\end{align*}
Here $\ug(\D_1, (0,\mu))$ is as in Example \ref{exa:lifting-transpositions-s4};
$\ug(\D_2, t)$ is the quotient of $\T(V_2)$ by the relations
$ a_{12}^2 = 0$, $a_{12}a_{34} - a_{34}a_{12} = 0$,
$a_{12}a_{23} - a_{13}a_{12} - a_{23}a_{13} = t(1- g_{(12)}g_{(23)})$
and
$\ug(\D_3, \lambda)$ is the quotient of $\T(W)$ by the relations
\begin{align*}
& a_{(1234)}^2= \lambda_1(1-g_{(13)}g_{(24)}); \qquad a_{(1234)}a_{(1432)}+a_{(1432)}a_{(1234)}=0;\\
&a_{(1234)}a_{(1243)}+a_{(1243)}a_{(1423)}+a_{(1423)}a_{(1234)}=\lambda_2(1-g_{(12)}g_{(13)}).
\end{align*}
Clearly $\ug(\D_1, 0) = {\mathfrak B}(V_1)\# \mathbb C\sk$, $\ug(\D_2, 0) = {\mathfrak B}(V_2)\# \mathbb C\sk$, $\ug(\D_3, 0) = {\mathfrak B}(W)\# \mathbb C\sk$.
Also $\ug(\D_1, (0,\mu)) \simeq \ug(\D_1, (0,\mathbf{n}u))$ iff $\mu = c\mathbf{n}u$ for some $c \in \mathbb C^{\times}$, and $\ug(\D_3, \lambda)
\simeq \ug(\D_3, \kappa)$ iff $\lambda = c\kappa$ for some $c \in \mathbb C^{\times}$.
\smallbreak \mathbf{n}oindent $\bullet$ \cite{AFGV-ampa, GG} Let $H$ be a finite-dimensional pointed Hopf algebra with $G(H)\simeq \sco$, but $H\mathbf{n}ot\simeq \mathbb C\sco$.
It is not known whether $\dim {\mathfrak B}({\mathcal O}_{2,3}^5,\sgn\otimes\varepsilon) <\infty$.
Let $\D_1$, $\D_2$ be the data corresponding to $V_1 = M({\mathcal O}^5_2, \sgn \otimes \operatorname{id})$, $V_2 = M({\mathcal O}^5_2, \sgn \otimes \sgn)$.
If the infinitesimal braiding of
$H$ is not $M({\mathcal O}_{2,3}^5,\sgn\otimes\varepsilon)$, then $H$ is isomorphic to one of
$\ug(\D_1, (0,\mu))$, $\mu\in \mathbb C^2$ (defined as in Example \ref{exa:lifting-transpositions-s4}),
or ${\mathfrak B}(V_2)\# \mathbb C\sco$, or $\ug(\D_2, 1)$ (defined as above).
\smallbreak \mathbf{n}oindent $\bullet$ \cite{AFGV-ampa} Let $m > 6$.
Let $H\mathbf{n}ot\simeq \mathbb C\sm$ be a finite-dimensional pointed Hopf algebra with $G(H)\simeq \sm$.
Then the infinitesimal braiding of $H$ is $V = M({\mathcal O}, \rho)$, where the type of $\sigma$ is $(1^{m-2}, 2)$ and $\rho = \rho_1 \otimes \sgn$,
$\rho_1 =\sgn$ or $\varepsilon$; it is an open question whether $\dim {\mathfrak B}(V) < \infty$, see Example \ref{exa:nichols-sim}. If $m = 6$, there
are two more Nichols algebras with unknown dimension corresponding to the class of type $(2^3)$, but they are conjugated to those
of type $(1^4, 2)$ by the outer automorphism of $\sei$.
\smallbreak \mathbf{n}oindent $\bullet$ \cite{FG} Let $m \mathfrak{g}e 12$, $m = 4h$ with $h\in \N$.
Let $G = \dm$ be the dihedral group of order $2m$. Then there are infinitely many finite-dimensional Nichols algebras
in ${}_{\Gamma}^{\Gamma}\mathcal{YD}g$; all of them are exterior algebras as braided Hopf algebras.
Let $H$ be a finite-dimensional pointed Hopf algebra with $G(H)\simeq \dm$, but $H\mathbf{n}ot\simeq \mathbb C\dm$.
Then $H$ is a lifting of an exterior algebra, and there infinitely many such liftings.
\smallbreak\subsubsection{Copointed Hopf algebras}\label{subsubsec:co-pointed}
We say that a semisimple Hopf algebra $K$ \emph{collapses} if any finite-dimensional Hopf algebra $H$ with $H_0\simeq K$ is isomorphic to $K$.
Thus, if $G$ collapses, then $\mathbb C^G$ and $(\mathbb C G)^F$ collapse, for any twist $F$.
Next we state the classification of the finite-dimensional copointed Hopf algebras over $\st$ \cite{AV}.
Let $V = M({\mathcal O}^3_2, \sgn)$ as a Yetter-Drinfeld module over $\mathbb C^{\st}$.
Let $\lambda \in\mathbb C^{{\mathcal O}_2^3}$ be such that $\displaystyle\sum_{(ij)\in{\mathcal O}_2^3}\lambda_{ij}=0$.
Let $\mathfrak{v} = \mathfrak{v}(V, \lambda)$ be the quotient of $T(V)\#\mathbb C^{\st}$ by the relations
$\mathtt{x}ij{13}\mathtt{x}ij{23}+\mathtt{x}ij{12}\mathtt{x}ij{13} + \mathtt{x}ij{23}\mathtt{x}ij{12} = 0$, $\mathtt{x}ij{23}\mathtt{x}ij{13}+\mathtt{x}ij{13}\mathtt{x}ij{12}+\mathtt{x}ij{12}\mathtt{x}ij{23} =0$,
\mathbf{n}ewline $\mathtt{x}ij{ij}^2 = \sum_{g\in\st}(\lambda_{ij} - \lambda_{g^{-1}(ij)g})\delta_g$, for $(ij)\in {\mathcal O}_2^3$.
Then $\mathfrak{v}$ is a Hopf algebra of dimension 72 and $\operatorname{gr} \mathfrak{v} \simeq {\mathfrak B}(V)\# \mathbb C^{\st}$.
Any finite-dimensional copointed Hopf algebra $H$ with $H_0 \simeq \mathbb C^{\st}$ is isomorphic to
$\mathfrak{v}(V, \lambda)$ for some $\lambda$ as above; $\mathfrak{v}(V, \lambda)\simeq \mathfrak{v}(V, \lambda')$ iff $\lambda$
and $\lambda'$ are conjugated under $\mathbb C^{\times}\times\operatorname{Aut} \st$.
\end{document} |
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\src@spec\l@icbegin{abstract}
Assuming Generalized Riemann's Hypothesis, Bach proved that the class group $\mathcal C\!\mathbf ell_\mathbf K$ of a number field
$\mathbf K$ may be generated using prime ideals whose norm is bounded by $12\log\DKsq$, and by $(4+o(1))\log\DKsq$
asymptotically, where $\Delta_\mathbf K$ is the absolute value of the discriminant of $\mathbf K$. Under the same assumption,
Belabas, Diaz y Diaz and Friedman showed a way to determine a set of prime ideals that generates $\mathcal C\!\mathbf ell_\mathbf K$
and which performs better than Bach's bound in computations, but which is asymptotically worse. In this
paper we show that $\mathcal C\!\mathbf ell_\mathbf K$ is generated by prime ideals whose norm is bounded by the minimum of
$4.01\log\DKsq$, $4\big(1+\big(2\pi e^{\gamma})^{-n_\K}\big)^2\log\DKsq$ and $4\big(\log\DK+\log\lDK-(\gamma+\log
2\pi)n_\K+1+(n_\K+1)\frac{\log(7\log\DK)}{\log\DK}\big)^2$. Moreover, we prove explicit upper bounds for the size
of the set determined by Belabas, Diaz y Diaz and Friedman's algorithms, confirming that it has size
$\asymp (\log\DK\log\lDK)^2$. In addition, we propose a different algorithm which produces a set of generators
which satisfies the above mentioned bounds and in explicit computations turns out to be smaller than
$\log\DKsq$ except for $7$ out of the $31292$ fields we tested.
\mathbf end{abstract}
\title{Explicit bounds for generators of the class group}
\section{Introduction}
Let $\mathbf K$ be a number field of degree $n_\K\gammaeq 2$, with $r_1$ (resp. $r_2$) real (resp. pair of complex)
embeddings and denote $\Delta_\mathbf K$ the absolute value of its discriminant. Throughout the paper $\mathfrak p$ will
always denote a non-zero prime ideal, $\rho$ a non-trivial zero of ${\zeta_\K}$, $\gamma$ the imaginary part of such
a $\rho$ and, since we are assuming Generalized Riemann's Hypothesis, $\rho=\frac{1}{2}+i\gamma$. The
Euler--Mascheroni constant will also be denoted $\gamma$, but the context will make it clear.
Buchmann's algorithm is an efficient method to compute both the class group $\mathcal C\!\mathbf ell_\mathbf K$ and a basis for a
maximal lattice of the unity group of $\mathbf K$. However, it needs as input a set of generators for $\mathcal C\!\mathbf ell_\mathbf K$.
Minkowski's bound shows that ideals having a norm (essentially) below $\sqrt\Delta_\mathbf K$ may be used, but it
works just for a few fields since the discriminant increases very quickly. Assuming Generalized
Riemann's Hypothesis, Eric Bach proved in~\cite{Bach:explicit} that ideals with a norm below $12\log\DKsq$
suffice, and that the bound improves up to $(4+o(1))\log\DKsq$ as $\Delta_\mathbf K$ diverges, where the function in
$o(1)$ is not made explicit in that paper, but has order at least $\log^{-2/3}\Delta_\mathbf K$.
This is a remarkable improvement, but for certain applications it is still too large. A different method
to find a good bound $T$ for norms of ideal generating $\mathcal C\!\mathbf ell_\mathbf K$ has been proposed by Karim Belabas,
Francisco Diaz y Diaz and Eduardo Friedman~\cite{small-generators}. In all tests their method behaves
very well, producing a good bound $T(\mathbf K)$ (see Section~3 of~\cite{small-generators}) which is much lower
than $4\log\DKsq$. However, the authors prove \cite[Theorem~4.3]{small-generators} that $T(\mathbf K)\gammaeq
\big(\big(\frac1{4n_\K}+o(1)\big)\log\DK\log\lDK\big)^2$, and advance the conjecture that
$T(\mathbf K)\sim\big(\frac14\log\DK\log\lDK\big)^2$. Thus, its impressive performance is the combined effect of
relatively large/small constants in front of these bounds and of the present computational power, but
which will disappear for large $\Delta_\mathbf K$.
In this paper we first prove in Theorem~\ref{theo:Teasynt} an explicit, easy and better version of Bach's
$(4+o(1))\log\DKsq$ bound, in Corollary~\ref{coro:2} that $4\log\DKsq$ is sufficient for a wide range of fields
and in Corollary~\ref{coro:Bach4.01} that $4.01\,\log\DKsq$ is sufficient for all fields.
Corollary~\ref{coro:2} also contains an explicit bound showing that the universal constant $4.01$ actually
decays exponentially to $4$ with the degree of the field.\\
Secondly, in Theorem~\ref{estim-T(K)} we prove that $T(\mathbf K)\leq \big(\frac{1+o(1)}4\log\DK\log\lDK\big)^2$, and
that $T(\mathbf K)\leq 3.9(\log\DK\log\lDK)^2$ with only three exceptions which are explicit. In a private
communication K.~Belabas told us that he also has a proof for the first part of this claim. Together with
the lower bound in~\cite{small-generators} it shows in particular that $T(\mathbf K)\asymp(\log\DK\log\lDK)^2$ for
fixed $n_\K$.
The weight function of~\cite{small-generators} can be seen as the convolution square of a characteristic
function, i.e. of a one step function. Using the convolution square of a two (resp. three) steps
function, we show in Corollary~\ref{coro:boundT1cst} that the bound already improve to
$(6.04+o(1))\log\DKsq$ (resp. $(4.81+o(1))\log\DKsq$),
where moreover in both cases $o(1)<0$ for fields of degree $n_\K\gammaeq 3$. To further improve the result we
propose in Subsections~\ref{subsec:algo}--\ref{subsec:algoend} a different algorithm producing a new
bound $T_1(\mathbf K)$, and which is essentially a multistep version of Belabas, Diaz y Diaz and Friedman's
algorithm, where the number of steps is not set in advance. By design, it performs better than $T(\mathbf K)$
and is lower than all the bounds we have proved in the first part of the paper.
In Subsection~\ref{subsec:tests} we report the conclusions about extensive tests we have conducted on a
few thousands of pure and biquadratic fields: in all cases the algorithm produces $T_1(\mathbf K)$ lower
than $\log\DKsq$ except for some biquadratic fields where it is anyway $\leq 1.004\log\DKsq$.\\
All `little-$o$' terms in these formulas are explicit, simple, of order $\frac{\log\lDK}{\log\DK}$ and with
small coefficients.
We have made two sample implementations of our algorithms. The first one is a script for
PARI/GP~\cite{PARI2} which can be found at the following address:\\
\url{http://users.mat.unimi.it/users/molteni/research/generators/bounds.gp}\\
The other is the branch \texttt{loic-bnf-optim} of the \texttt{git} tree of PARI/GP, available at\\
\url{http://pari.math.u-bordeaux.fr/git/pari.git}
\src@spec\l@icbegin{acknowledgements}
We wish to thank Giacomo Gigante for his comments and interesting discussions, and the referee for
her/his useful comments. We also thank the referee of an early version of the second part of this paper
for the suggestion to use Cholesky's decomposition. The authors are members of the INdAM group GNSAGA.
\mathbf end{acknowledgements}
\mathbf enlargethispage{\baselineskip}
\section{Preliminary}
\src@spec\l@icbegin{defi}
Let $\mathcal W$ be the set of functions $F\colon [0,{+\infty})\to\mathbf R$ such that
\src@spec\l@icbegin{itemize}
\item $F$ is continuous;
\item $\mathbf exists\varepsilon>0$ such that the function
$F(x)e^{(\frac12+\varepsilon)x}$ is integrable and of bounded variation;
\item $F(0)>0$;
\item $(F(0)-F(x))/x$ is of bounded variation.
\mathbf end{itemize}
Let then, for $T>1$, $\mathcal W(T)$ be the subset of $\mathcal W$ such that
\src@spec\l@icbegin{itemize}
\item $F$ has support in $[0,\log T]$;
\item the Fourier cosine transform of $F$ is non-negative.
\mathbf end{itemize}
\mathbf end{defi}
\src@spec\l@icbegin{defi}
For any compactly supported function $F$ on $[0,\infty)$, we set
\[
\Iint(F) := \int_{0}^{+\infty}\frac{F(0)-F(x)}{2\sinh(x/2)}\,\mathrm d x
\quad\text{and}\quad
\Jint(F) := \int_{0}^{+\infty}\frac{F(x)}{2\cosh(x/2)}\,\mathrm d x.
\]
\mathbf end{defi}
\src@spec\l@icbegin{defi}
Let $T_{\mathcal{C}}K$ be the lowest $T$ such that the set $\{\mathfrak p\colon\mathrm N\mathfrak p\leq T\}$ generates $\mathcal C\!\mathbf ell_\mathbf K$.
\mathbf end{defi}
The main result of \cite[Th.~2.1]{small-generators} can be reformulated as follows.
\src@spec\l@icbegin{theo}[\textbf{Belabas, Diaz y Diaz, Friedman}]\label{theoKB}
Let $\mathbf K$ be a number field satisfying the Riemann Hypothesis for all $\mathrm L$-functions attached to
non-trivial characters of its ideal class group $\mathcal C\!\mathbf ell_\mathbf K$, and suppose that there exists, for some $T>1$,
an $F\in\mathcal W(T)$ such that
\src@spec\l@icbegin{equation}\label{theoeq}
2\sum_\mathfrak p\log\mathrm N\mathfrak p\sum_{m=1}^{+\infty}\frac{F(m\log\mathrm N\mathfrak p)}{\mathrm N\mathfrak p^{m/2}}
>
F(0)(\log\DK - (\gamma + \log 8\pi)n_\K)
+ \Iint(F) n_\K
- \Jint(F) r_1.
\mathbf end{equation}
Then $T_{\mathcal{C}}K<T$.
\mathbf end{theo}
Assuming \textrm{\upshape GRH}, Weil's Explicit Formula (see~\cite[Ch.~XVII, Th.~3.1]{Lang:algnumtheory}), as simplified by
Poitou in~\cite{Poitou:petits-discs}, can be written for $F\in\mathcal W$ as
\src@spec\l@icbegin{multline}\label{WEF}
2\sum_\gamma \int_{0}^{+\infty}F(x)\cos(x\gamma) \,\mathrm d x
=
4\int_0^{+\infty}F(x)\cosh\Big(\frac{x}{2}\Big)\,\mathrm d x \\
- 2\sum_\mathfrak p\log\mathrm N\mathfrak p\sum_{m=1}^{+\infty}\frac{F(m\log\mathrm N\mathfrak p)}{\mathrm N\mathfrak p^{m/2}}
+ F(0)(\log\DK - (\gamma + \log 8\pi)n_\K)
+ \Iint(F) n_\K
- \Jint(F) r_1.
\mathbf end{multline}
Hence~\mathbf eqref{theoeq} can be stated as
\src@spec\l@icbegin{equation}\label{equivtheo}
2\int_{0}^{+\infty}F(x)\cosh\Big(\frac{x}{2}\Big)\,\mathrm d x
>
\sum_\gamma \int_{0}^{+\infty}F(x)\cos(x\gamma) \,\mathrm d x.
\mathbf end{equation}
Let $\Phi$ be an even, integrable and compactly supported function, and let $F=\Phi\ast\Phi$. Then
\src@spec\l@icbegin{align*}
\int_{0}^{+\infty} F(x)\cosh\Big(\frac{x}{2}\Big)\,\mathrm d x
&= 2\Big(\int_{0}^{+\infty}\Phi(x)\cosh\Big(\frac{x}{2}\Big)\,\mathrm d x\Big)^2, \\
\int_{0}^{+\infty} F(x)\cos(xt)\,\mathrm d x
&= 2\Big(\int_{0}^{+\infty}\Phi(x)\cos(xt)\,\mathrm d x\Big)^2,
\mathbf end{align*}
and $F$ satisfies~\mathbf eqref{equivtheo} and hence~\mathbf eqref{theoeq} if and only if
\src@spec\l@icbegin{equation}\label{eqzero}
8\Big(\int_{0}^{+\infty}\Phi(x)\cosh\Big(\frac{x}{2}\Big)\,\mathrm d x\Big)^2
>
\sum_\gamma\widehat\Phi(\gamma)^2,
\mathbf end{equation}
where we have set $\widehat\Phi(t):=\int_{\mathbf R}\Phi(x)e^{ixt}\,\mathrm d x=2\int_{0}^{+\infty}\Phi(x)\cos(xt)\,\mathrm d x$.
\section{Bounds for class group generators}
Assume $T>1$ and let $L:=\log T$. Let $\Phi^+$ be a real, non-negative, piecewise continuous function with
positively measured support in $[0,L]$, and let
\src@spec\l@icbegin{equation}\label{eq:setup}
\src@spec\l@icbegin{split}
\Phi^-(x) &:= \Phi^+(-x), \\
\Phi^\circ(x) &:= \Phi^+(L/2+x), \\
\Phi(x) &:= \Phi^\circ(x) + \Phi^\circ(-x), \\
F &:= \Phi\ast\Phi.
\mathbf end{split}
\mathbf end{equation}
These choices ensure that $F\in\mathcal W(T)$.
\src@spec\l@icbegin{prop}\label{prop:eqzero-simplifiee}
Assume \textrm{\upshape GRH}\ and let $F$ as in~\mathbf eqref{eq:setup}. Then~\mathbf eqref{theoeq} is satisfied by $F$ if
\src@spec\l@icbegin{multline}\label{equiveqzero-Phi-simplifiee}
\sqrt{T}
\gammaeq
\frac{2T\int_{0}^{L}\big(\Phi^+(x)\big)^2\,\mathrm d x}{\big(\int_{0}^L\Phi^+(x)e^{x/2}\,\mathrm d x\big)^2}
( \log\DK
- (\gamma + \log 8\pi)n_\K
) \\
- \frac{4T\sum_{\mathfrak p,m}\log\mathrm N\mathfrak p\frac{(\Phi^+\ast\Phi^-)(m\log\mathrm N\mathfrak p)}{\mathrm N\mathfrak p^{m/2}}}{\big(\int_{0}^L\Phi^+(x)e^{x/2}\,\mathrm d x\big)^2}
+ \frac{2T\,\Iint(\Phi^+\ast\Phi^-)n_\K}{\big(\int_{0}^L\Phi^+(x)e^{x/2}\,\mathrm d x\big)^2}
+ \frac{2T\int_{0}^L\Phi^+(x)e^{-x/2}\,\mathrm d x}{\int_{0}^L\Phi^+(x)e^{x/2}\,\mathrm d x}.
\mathbf end{multline}
\mathbf end{prop}
\src@spec\l@icbegin{proof}
We have
\src@spec\l@icbegin{align*}
2\int_{0}^{+\infty}\Phi(x)\cosh\Big(\frac{x}{2}\Big)\,\mathrm d x
= \frac{1}{T^{1/4}}\int_{0}^L\Phi^+(x)e^{x/2}\,\mathrm d x
+ T^{1/4} \int_{0}^L\Phi^+(x)e^{-x/2}\,\mathrm d x.
\mathbf end{align*}
Hence
\src@spec\l@icbegin{multline*}
8\Big(\int_{0}^{+\infty}\Phi(x)\cosh\Big(\frac{x}{2}\Big)\,\mathrm d x\Big)^2\\
> \frac{2}{\sqrt{T}} \Big(\int_{0}^L\Phi^+(x)e^{x/2}\,\mathrm d x\Big)^2
+ 4\int_{0}^L\Phi^+(x)e^{x/2}\,\mathrm d x\int_{0}^L\Phi^+(x)e^{-x/2}\,\mathrm d x.
\mathbf end{multline*}
Moreover
\src@spec\l@icbegin{equation}\label{eq:1c}
\widehat{\Phi}(t)
= 2\Ree\int_{\mathbf R}\Phi^\circ(x)e^{ixt}\,\mathrm d x
= 2\Ree\big[e^{-i\frac{Lt}{2}}\widehat{\Phi^+}(t)\big].
\mathbf end{equation}
Hence
\src@spec\l@icbegin{equation}\label{eq:2c}
|\widehat{\Phi}(t)|^2
\leq 4 \big|e^{-i\frac{Lt}{2}}\widehat{\Phi^+}(t)\big|^2
= 4 \big|\widehat{\Phi^+}(t)\big|^2.
\mathbf end{equation}
We have $\big|\widehat{\Phi^+}(t)\big|^2 = \widehat{\Phi^+}(t) \overline{\widehat{\Phi^+}(t)} =
\widehat{\Phi^+}(t) \widehat{\Phi^-}(t) = \widehat{\Phi^+\ast\Phi^-}(t)$. Thus to
satisfy~\mathbf eqref{eqzero} it is sufficient that
\src@spec\l@icbegin{equation}\label{equiveqzero-Phi}
\frac{2}{\sqrt{T}} \Big(\int_{0}^L\Phi^+(x)e^{x/2}\,\mathrm d x\Big)^2
+ 4\int_{0}^L\Phi^+(x)e^{x/2}\,\mathrm d x\int_{0}^L\Phi^+(x)e^{-x/2}\,\mathrm d x
\gammaeq \sum_\gamma \widehat{\,\mathrm digamma}(\gamma),
\mathbf end{equation}
where
\[
\,\mathrm digamma := 4\Phi^+\ast\Phi^-.
\]
By Weil's Explicit Formula~\mathbf eqref{WEF},
\src@spec\l@icbegin{multline}\label{Weil-Phi}
\sum_\gamma \widehat{\,\mathrm digamma}(\gamma)
\leq \,\mathrm digamma(0)\big(\log\DK - (\gamma + \log 8\pi)n_\K\big)
- 2\sum_{\mathfrak p,m}\log\mathrm N\mathfrak p\frac{\,\mathrm digamma\big(m\log\mathrm N\mathfrak p\big)}{\mathrm N\mathfrak p^{m/2}} \\
+ \Iint(\,\mathrm digamma)n_\K
+ 4\int_{0}^{+\infty} \,\mathrm digamma(x)\cosh\Big(\frac{x}{2}\Big)\,\mathrm d x
\mathbf end{multline}
where we cancelled the term $-\Jint(\,\mathrm digamma)r_1$ because $\,\mathrm digamma\gammaeq 0$. Notice that
\mathbf enlargethispage{2\baselineskip}
\src@spec\l@icbegin{align}
\,\mathrm digamma(0)
&=4\int_{0}^{L}\big(\Phi^+(x)\big)^2\,\mathrm d x \label{bho}
\intertext{and}
\int_{0}^{+\infty} \,\mathrm digamma(x)\cosh\Big(\frac{x}{2}\Big)\,\mathrm d x
&= \frac{1}{2}\int_{\mathbf R}\,\mathrm digamma(x)e^{x/2}\,\mathrm d x \notag\\
&= 2\int_{0}^{L} \Phi^+(x) e^{x/2}\,\mathrm d x \int_{0}^{L} \Phi^+(x) e^{-x/2}\,\mathrm d x. \label{ribho}
\mathbf end{align}
The claim follows combining~\mathbf eqref{equiveqzero-Phi}, \mathbf eqref{Weil-Phi}, \mathbf eqref{bho} and~\mathbf eqref{ribho}.
\mathbf end{proof}
\subsection{Upper bound for $T_\eK$}
The coefficient of $\log\DK$ in Proposition~\ref{prop:eqzero-simplifiee} is
\[
\frac{2T\int_{0}^{L}\big(\Phi^+(x)\big)^2\,\mathrm d x}{\big(\int_{0}^L\Phi^+(x)e^{x/2}\,\mathrm d x\big)^2}.
\]
Cauchy--Schwarz's inequality shows that its minimum value is $\frac{2T}{T-1}$ and it is attained only for
$\Phi^+(x)=e^{x/2}$ on $[0,L]$. We are interested into small values for this coefficient, hence this is
the best choice we can do. However, this function produces in~\mathbf eqref{equiveqzero-Phi-simplifiee} an
inequality for $T$ that cannot be solved easily and, moreover, this choice does not give the best possible
results for secondary coefficients. To overcome this problem in the next theorem we consider the
functions $e^{x/2}\chi_{[L-a,L]}(x)$, where $a$ is a parameter which is fixed in $(0,L]$. This is a
suboptimal choice for the coefficient of $\log\DK$ if $a \neq L$, but every value of $a$ independent of $T$
produces an inequality which can be solved easily, still having the correct order for the main term.
Furthermore, acting on $a$ we can minimize also the total contribution coming from the other terms
in~\mathbf eqref{equiveqzero-Phi-simplifiee}. Theorem~\ref{theo:Teasynt} is proved using several values of $a$,
and would not be accessible using only the conclusions coming from the choice $a=L$.
\src@spec\l@icbegin{defi}
Assume $a\in(0,L]$. Let $\Phi_\e^+(x):=e^{x/2}{\raise0.2ex\hbox{$\chi$}}_{[L-a,L]}(x)$ and let $F_{\mathbf e}$ be the $F$ defined
in~\mathbf eqref{eq:setup} when $\Phi^+=\Phi_\e^+$.
\mathbf end{defi}
\src@spec\l@icbegin{rem}\label{rem:Fe}
We recall that $F_{\mathbf e}$ is even with support in $[-L,L]$. Moreover, we find that for every $x\in [0,L]$,
\src@spec\l@icbegin{align*}
F_{\mathbf e}(x)&= \,\mathrm delta_1(x)(2a-L+x)e^{x/2}\sqrt{T}
+ \,\mathrm delta_2(x)(L-x)e^{x/2}\sqrt{T} \\
&\quad
+ 2\,\mathrm delta_3(x)\big(e^{-x/2}-e^{x/2-a}\big)T
+ \,\mathrm delta_4(x)(2a-L-x)e^{-x/2}\sqrt{T}
\mathbf end{align*}
where $\,\mathrm delta_1:={\raise0.2ex\hbox{$\chi$}}_{[L-2a,L-a)}$, $\,\mathrm delta_2:={\raise0.2ex\hbox{$\chi$}}_{[L-a,L]}$, $\,\mathrm delta_3:={\raise0.2ex\hbox{$\chi$}}_{[0,a]}$ and
$\,\mathrm delta_4:={\raise0.2ex\hbox{$\chi$}}_{[0,2a-L]}$.
\mathbf end{rem}
\src@spec\l@icbegin{defi}
Let $T_\eK$ be the minimal $T$ such that the function $F_{\mathbf e}$ satisfies~\mathbf eqref{theoeq} for some $a$.
\mathbf end{defi}
Note that $T_{\mathcal{C}}K\leq T_\eK$ so that we will state most results about $T_\eK$ below as results on $T_{\mathcal{C}}K$ as
well.
\src@spec\l@icbegin{theo}\label{theo:Phie}
Assume \textrm{\upshape GRH}. Fix $T_0>1$. We then have
\src@spec\l@icbegin{align}\label{eq:Phielong}
\sqrt{T_\eK}
&\leq \max\Big(\sqrt{T_0},
r(\log\DK,n_\K,T_0)
- \frac{4}{\big(1-T_0^{-1}\big)^2}\sum_{\mathrm N\mathfrak p^m\leq T_0}\Big(\frac{1}{\mathrm N\mathfrak p^m}-\frac{1}{T_0}\Big)\log\mathrm N\mathfrak p
\Big),
\intertext{in particular}
\label{eq:Phie}
\sqrt{T_\eK}
&\leq \max\big(\sqrt{T_0},r(\log\DK,n_\K,T_0)\big),
\mathbf end{align}
where
\[
r(\mathcal{L},n,t) := \frac{2}{1-t^{-1}}\Big(\mathcal{L}
+ \log t
- \Big(\gamma
+ \log 2\pi
- \frac{\log t}{t-1}
+ \log\big(1-t^{-1}\big)\Big)n\Big).
\]
\mathbf end{theo}
\src@spec\l@icbegin{proof}
We are assuming $\Phi^+=\Phi_\e^+$ for some $a\leq L$. In this case we have
\src@spec\l@icbegin{align*}
\int_{0}^{L}\Phi_\e^+(x)e^{x/2}\,\mathrm d x
&= \big(1-e^{-a}\big)T
= \int_{0}^{L}\big(\Phi_\e^+(x))^2\,\mathrm d x, \\
\int_{0}^{L}\Phi_\e^+(x)e^{-x/2}\,\mathrm d x
&= a.
\mathbf end{align*}
Moreover, $\forall x\in[0,a]$,
\src@spec\l@icbegin{align*}
\Phi_\e^+\ast\Phi_\e^-(x)
&= \big(e^{-x/2}-e^{x/2-a}\big)T.
\mathbf end{align*}
In addition $\forall x>a$, $\Phi_\e^+\ast\Phi_\e^-(x)=0$. This means that
\src@spec\l@icbegin{align*}
\Iint(\Phi_\e^+\ast\Phi_\e^-)
& = \int_{0}^{+\infty} \frac{\Phi_\e^+\ast\Phi_\e^-(0)-\Phi_\e^+\ast\Phi_\e^-(x)}{2\sinh(x/2)}\,\mathrm d x \\
& = (\log 4)\big(1-e^{-a}\big)T
+ aT
- \big(1-e^{-a}\big)\log\big(e^{a}-1\big)T.
\mathbf end{align*}
We now set $a=:\log T_0$ for some $T_0>1$. Since we need to have $L=\log T\gammaeq a$, we get
that~\mathbf eqref{equiveqzero-Phi-simplifiee} is satisfied for any $T\gammaeq T_0$ such that
{\small
\src@spec\l@icbegin{multline*}
\sqrt{T}
\gammaeq \\
\frac{2}{1-T_0^{-1}}\Big(\log\DK
- \frac{2}{1-T_0^{-1}}\sum_{\mathrm N\mathfrak p^m\leq T_0}\Big(\frac{\log\mathrm N\mathfrak p}{\mathrm N\mathfrak p^m}-\frac{\log\mathrm N\mathfrak p}{T_0}\Big)
- \Big(\gamma + \log 2\pi
- \frac{\log T_0}{T_0-1}
+ \log\big(1-T_0^{-1}\big)\Big)n_\K
+ \log T_0\Big).
\mathbf end{multline*}
}
Since the right-hand side does not depend on $T$, this proves the first claim. The second is an obvious
consequence, because the sum on prime ideals is non-negative.
\mathbf end{proof}
\subsection{Upper bounds for class group generators}
Theorem~\ref{theo:Teasynt} below gives an upper bound for $T_\eK$, and hence for $T_{\mathcal{C}}K$. It is
essentially the best result we can deduce from Theorem~\ref{theo:Phie} (see the remark immediately
following the proof). The theorem has Corollaries~\ref{coro:2} and~\ref{coro:Bach4.01} as easy
consequences.
\src@spec\l@icbegin{theo}\label{theo:Teasynt}
We have
\src@spec\l@icbegin{align*}
\sqrt{T_{\mathcal{C}}K}\leq
\sqrt{T_\eK} &\leq 2\Big(\log\DK
+ \log\lDK
- (\gamma+\log 2\pi)n_\K
+ 1
+ (n_\K+1)\tfrac{\log(7\log\DK)}{\log\DK}
\Big).
\intertext{Moreover, if $\log\DK\gammaeq n_\K2^{n_\K}$, we have}
\sqrt{T_{\mathcal{C}}K}\leq
\sqrt{T_\eK} &\leq 2(\log\DK
+ \log\lDK
- (\gamma+\log 2\pi)n_\K
+ 1).
\mathbf end{align*}
\mathbf end{theo}
\src@spec\l@icbegin{proof}
We use~\mathbf eqref{eq:Phie} with $T_0=\log\DK+1$. We have
\src@spec\l@icbegin{align*}
\frac{1}{2}r(\mathcal{L},n,\mathcal{L}+1) &= \mathcal{L}
+ \log\mathcal{L}
- (\gamma+\log 2\pi)n
+ 1
+ (n+1)\frac{\log\mathcal{L}}{\mathcal{L}}
- f(\mathcal{L})n
+ g(\mathcal{L}),
\mathbf end{align*}
where
\src@spec\l@icbegin{align*}
f(\mathcal{L}) &:= (\gamma + \log 2\pi)\mathcal{L}^{-1}-(1+\mathcal{L}^{-1})^2-\mathcal{L}^{-2}\log\mathcal{L}, \\
g(\mathcal{L}) &:= (1+\mathcal{L}^{-1})\log(1+\mathcal{L}^{-1}).
\mathbf end{align*}
We have $f(\mathcal{L})\gammaeq 0$ and $g(\mathcal{L})-2f(\mathcal{L})\leq 0$ for any $\mathcal{L}\gammaeq 4$. This proves that
\src@spec\l@icbegin{multline}\label{eq:rlDK+1}
\frac{1}{2}r(\log\DK,n_\K,\log\DK+1) \\
\leq \log\DK
+ \log\lDK
- (\gamma+\log 2\pi)n_\K
+ 1
+ (n_\K+1)\frac{\log\lDK}{\log\DK}
\mathbf end{multline}
for any $\mathbf K$ such that $\log\DK\gammaeq 4$.
We look under which condition $\frac{1}{2}\sqrt{T_0}$ satisfies the same bound, i.e. when
\src@spec\l@icbegin{equation}\label{eq:sqrtlDK+1}
\frac{1}{2}\sqrt{\log\DK+1} \leq \log\DK
+ \log\lDK
- (\gamma+\log 2\pi)n_\K
+ 1
+ (n_\K+1)\frac{\log\lDK}{\log\DK}.
\mathbf end{equation}
Let, for $n\gammaeq 2$ and $\mathcal{L}\gammaeq 1$
\[
h(\mathcal{L},n):= \mathcal{L}
+ \log\mathcal{L}
- (\gamma+\log 2\pi)n
+ 1
+ (n+1)\frac{\log\mathcal{L}}{\mathcal{L}}
- \frac{1}{2}\sqrt{\mathcal{L}+1},
\]
so that~\mathbf eqref{eq:sqrtlDK+1} holds true if $h(\log\DK,n_\K)\gammaeq 0$. We have $\frac{\partial h}{\partial \mathcal{L}}
\gammaeq 0$ if $\mathcal{L}\gammaeq 0.2n-1$,
which is true each time $\mathcal{L}=\log\DK$ and $n=n_\K$. This allows to prove that if $\mathcal{L}_0\gammaeq 0.2n-1$
satisfies $h(\mathcal{L}_0,n)\gammaeq 0$, then $h(\mathcal{L},n)\gammaeq 0$ if $\mathcal{L}\gammaeq \mathcal{L}_0$. Case $b=2.3$ in Table~3
of~\cite{Odlyzko:tables} shows that $\log\DK\gammaeq 2.8n_\K-9.6$:
using this inequality we have $h(\log\DK,n_\K)\gammaeq 0$ if $n_\K\gammaeq 17$.
For $2\leq n_\K\leq 16$, we still have $h(\log\DK,n_\K)\gammaeq 0$ for $\log\DK\gammaeq \mathcal{L}_0(n_\K)$ where $\mathcal{L}_0(n_\K)$ is
indicated in the table below. The table also gives the minimum possible $\log\DK$ for the given $n_\K$,
computed either with ``megrez'' number field table or with Odlyzko's Table~3.
\[
\src@spec\l@icbegin{array}{rcr|rcr|rcr}
n_\K & \min\log\DK & \mathcal{L}_0(n_\K) &
n_\K & \min\log\DK & \mathcal{L}_0(n_\K) &
n_\K & \min\log\DK & \mathcal{L}_0(n_\K) \\
\hline
2 & 1.098 & 2.697 &
7 & 12.125 & 13.676 &
12 & 24.336 & 25.675 \\
3 & 3.135 & 4.576 &
8 & 13.972 & 16.053 &
13 & 27.749 & 28.096 \\
4 & 4.762 & 6.728 &
9 & 17.118 & 18.446 &
14 & 29.748 & 30.520 \\
5 & 7.383 & 8.995 &
10 & 19.060 & 20.849 &
15 & 33.256 & 32.948 \\
6 & 9.184 & 11.319 &
11 & 22.359 & 23.259 &
16 & 35.277 & 35.378
\mathbf end{array}
\]
We note that~\mathbf eqref{eq:sqrtlDK+1} holds also for $n_\K=15$. By~\mathbf eqref{eq:Phie}, \mathbf eqref{eq:rlDK+1}
and~\mathbf eqref{eq:sqrtlDK+1}, the first claim is proved for $n_\K\gammaeq 17$ or $\log\DK\gammaeq \max(4,\mathcal{L}_0(n_\K))$ (in
an even stronger form, because now we have $\log \log\DK$ instead of $\log(7\log\DK)$).\\
To complete the proof and to extend the claim to $2\leq n_\K\leq 16$ and $\log\DK\leq \max(4,\mathcal{L}_0(n_\K))$ we
use a different strategy. Let
\[
\ell(n,t) := \frac{1}{2}(t^{1/2} - t^{-1/2})
- \log t
+ \Big(\gamma
+ \log 2\pi
- \frac{\log t}{t-1}
+ \log\big(1-t^{-1}\big)\Big)n,
\]
for $n>0$ and $t>1$. It is the function such that
\[
r(\ell(n,t),n,t) = \sqrt{t},
\]
hence, if $\log\DK = \ell(n_\K,T_0)$, then by~\mathbf eqref{eq:Phie} $T_0$ is an upper-bound for $T_\eK$; note that
this corresponds to the case $a=L=\log T$ in Theorem~\ref{theo:Phie}. Observe that $\ell$ is increasing as
a function of $t$, and that it diverges to $-\infty$ and to $+\infty$ for $t\to 1^-$ and $t\to +\infty$,
respectively, for every fixed $n$. As a consequence, for given $n_\K$ and $\log\DK$ there is a unique $T_0$
such that $\ell(n_\K,T_0)=\log\DK$, and this $T_0$ is also an upper-bound for $T_\eK$.\\
Thus, for $2\leq n_\K\leq 16$ (only a finite set of cases) and $\log\DK\leq \max(4,\mathcal{L}_0(n_\K))$ (a bounded
range for $\log\DK$) we set $T_0$ such that $\log\DK=\ell(n_\K,T_0)$ and we directly check that
\[
\frac{1}{2\sqrt{T_0}}
+ \Big(\frac{\log T_0}{T_0-1} - \log(1-T_0^{-1})\Big)n_\K
\leq 1
+ \log\Big(\frac{\ell(n_\K,T_0)}{T_0}\Big)
+ (n_\K+1)\frac{\log(7\ell(n_\K,T_0))}{\ell(n_\K,T_0)}
\]
which is equivalent to
\[
\frac{1}{2}
\sqrt{T_0} \leq \log\DK
+ \log\lDK
- (\gamma+\log 2\pi)n_\K
+ 1
+ (n_\K+1)\frac{\log(7\log\DK)}{\log\DK}.
\]
(Note that now $\log(7\log\DK)$ appears, as in the claim.)
For the second claim of the theorem, we use~\mathbf eqref{eq:Phielong} still with $T_0=\log\DK+1$. We compute a
lower bound for the sum of prime ideals choosing two prime ideals $\mathfrak p_0$ and $\mathfrak p_1$ above respectively
$2$ and $3$. We get
\src@spec\l@icbegin{align}
\sum_{\mathrm N\mathfrak p^m\leq T_0}\Big(\frac{1}{\mathrm N\mathfrak p^m}-\frac{1}{T_0}\Big)\log\mathrm N\mathfrak p
&\gammaeq \Big(\frac{1}{\mathrm N\mathfrak p_0}-\frac{1}{T_0}\Big)\log\mathrm N\mathfrak p_0
+ \,\mathrm delta_{n_\K,2}\Big(\frac{1}{\mathrm N\mathfrak p_1}-\frac{1}{T_0}\Big)\log\mathrm N\mathfrak p_1
\notag
\intertext{where $\,\mathrm delta_{n_\K,2}$ is $1$ if $n_\K=2$ and $0$ otherwise. Note that this holds in any case
because if $\mathfrak p_0$ or $\mathfrak p_1$ does not appear in the original sum, then the chosen lower bound is
negative. In its turn this is}
&\gammaeq n_\K(\log 2)\Big(\frac{1}{2^{n_\K}}-\frac{1}{T_0}\Big)
+ n_\K(\log 3)\,\mathrm delta_{n_\K,2}\Big(\frac{1}{3^{n_\K}}-\frac{1}{T_0}\Big),\label{eq:lowerboundsum}
\mathbf end{align}
because the inert case gives the least contribution. Since $\max(4,\mathcal{L}_0(n_\K))\leq n_\K2^{n_\K}$ for all
$n_\K\leq 16$, \mathbf eqref{eq:rlDK+1} holds if $\log\DK\gammaeq n_\K2^{n_\K}$. Hence to prove the second claim it is
sufficient to prove that if $\log\DK\gammaeq n_\K2^{n_\K}$, then
\src@spec\l@icbegin{multline}\label{eq:restorlDK+1}
(n_\K+1)\frac{\log\log\DK}{\log\DK} \\
\leq 2n_\K(\log 2)\Big(\frac{1}{2^{n_\K}}-\frac{1}{\log\DK+1}\Big)
+ n_\K(\log 3)\,\mathrm delta_{n_\K,2}\Big(\frac{1}{3^{n_\K}}-\frac{1}{\log\DK+1}\Big)
\mathbf end{multline}
and
\[
\sqrt{\log\DK+1} \leq 2(\log\DK
+ \log\lDK
- (\gamma+\log 2\pi)n_\K
+ 1).
\]
The second statement is elementary and is true for any $n_\K\gammaeq 2$.
For~\mathbf eqref{eq:restorlDK+1}, we observe that the left-hand side is decreasing in $\log\DK$ while the
right-hand side is increasing, hence it is sufficient to verify it with $\log\DK$ substituted by
$n_\K2^{n_\K}$. One can see that it is true for $n_\K\gammaeq 7$ and for $2\leq n_\K\leq 6$ and $\log\DK\gammaeq
\mathcal{L}_1(n_\K)$ as indicated in table below
\[
\src@spec\l@icbegin{array}{rrr}
n_\K & n_\K2^{n_\K} & \mathcal{L}_1(n_\K) \\
\hline
2 & 8 & 15.670 \\
3 & 24 & 35.173 \\
4 & 64 & 78.801 \\
5 & 160 & 174.859 \\
6 & 384 & 384.395
\mathbf end{array}
\]
To fill the gap, we use~\mathbf eqref{eq:Phielong}, \mathbf eqref{eq:lowerboundsum} and $T_0=\log\DK+7$.
\mathbf end{proof}
\src@spec\l@icbegin{rem*}
The first claim is somehow the best we can hope from~\mathbf eqref{eq:Phie}. Indeed the optimal $T_0$
for~\mathbf eqref{eq:Phie} is such that
\[
\log\DK = T_0 - (n_\K+1)\log T_0 + \Big(\gamma+\log 2\pi - 2\frac{\log T_0}{T_0-1} + \log(1-T_0^{-1})\Big)n_\K - 1
\]
for all but a finite number of fields of degree $n_\K\leq 22$. Using this formula, one checks that the best
bound we can get from~\mathbf eqref{eq:Phie} is
\[
2(\log\DK + \log\lDK - (\gamma+\log 2\pi)n_\K + 1 + \mathbf epsilon(\log\DK))
\]
where $\mathbf epsilon(\log\DK)\sim (n_\K+1)\log\lDK/\log\DK$ for $\log\DK\to\infty$ and fixed $n_\K$.
\mathbf end{rem*}
\src@spec\l@icbegin{rem*}
Using the full strength of~\mathbf eqref{eq:Phielong}, one can prove that for quadratic fields the second claim
of Theorem~\ref{theo:Teasynt} is true for $T_{\mathcal{C}}K$ also for $\log\DK\leq n_\K2^{n_\K}=8$ with only the four
exceptions $\mathbf Q[\sqrt{-15}]$, $\mathbf Q[\sqrt{-5}]$, $\mathbf Q[\sqrt{-23}]$ and $\mathbf Q[\sqrt{-6}]$ (for which
$T_{\mathcal{C}}K=2$) and the ten fields of discriminant in $[-11,13]\cup\{-19\}$ (for which the class group is
trivial).
\mathbf end{rem*}
We now prove that, for fixed $n_\K$, the absolute upper bound for $T_{\mathcal{C}}K/\log\DKsq$ is near $4$ and that the
asymptotic limit $4\log\DKsq$ is true for a very large set of fields.
\src@spec\l@icbegin{coro}\label{coro:2}
We have
\[
T_{\mathcal{C}}K\leq T_\eK \leq 4\big(1+\big(2\pi e^{\gamma}\big)^{-n_\K}\big)^2\log\DKsq.
\]
Moreover,
\[
\text{if}\quad
\log\DK\leq \frac{1}{e}\big(2\pi e^{\gamma}\big)^{n_\K}\quad\text{then}\quad
T_{\mathcal{C}}K\leq T_\eK\leq 4\log\DKsq.
\]
\mathbf end{coro}
Notice that $2\pi e^\gammaamma> 11.19$.
\src@spec\l@icbegin{proof}
It is sufficient to prove that $T_\eK\leq 4\log\DKsq$ if $\log\DK\leq n_\K2^{n_\K}$, because the second statement
of Theorem~\ref{theo:Teasynt} already proves both statements in the remaining ranges.
The right-hand side of the first statement of Theorem~\ref{theo:Teasynt} is $2\log\DK+2f(\log\DK,n_\K)$ with
\[
f(\mathcal{L},n) := \log \mathcal{L}
+ 1
- (\gamma+\log 2\pi)n
+ (n+1)\frac{\log(7\mathcal{L})}{\mathcal{L}}.
\]
We just need to check that $f(\log\DK,n_\K)\leq 0$ if $\log\DK\leq n_\K2^{n_\K}$. As a function of $\mathcal{L}\gammaeq
1$, for fixed $n\gammaeq 2$, $\frac{\partial f}{\partial \mathcal{L}}=\frac{n+1}{\mathcal{L}}\big(\frac{1}{n+1}
-\frac{\log(7\mathcal{L})-1}{\mathcal{L}}\big)$ is negative then positive,
hence to check that $f$ is negative, it is sufficient to check its value for the minimum and the maximum
$\mathcal{L}$ we are interested in. We have $f(n 2^n,n)<0$ for any $n\gammaeq 2$ and $f(\log 23,n)<0$ for any
$n\gammaeq 3$.
Thus $f(\log\DK,n_\K)<0$ for any field of degree $n_\K\gammaeq 3$. For quadratic fields, we come back
to~\mathbf eqref{eq:Phie} with $T_0=2\log\DK$ to directly check that $T_\eK\leq 4\log\DKsq$ if $\log\DK\leq
n_\K 2^{n_\K}=8$.
\mathbf end{proof}
\src@spec\l@icbegin{rem*}
The upper bound for the quotient $T_\eK/\log\DKsq$ tends obviously very fast to $4$. For instance, for
$n_\K\gammaeq 10$, we have $T_{\mathcal{C}}K\leq T_\eK\leq (4+2.6\cdot 10^{-10})\log\DKsq$
-- but notice that the second claim in Corollary~\ref{coro:2} shows that $T_{\mathcal{C}}K\leq T_\eK\leq 4\log\DKsq$ if
$\Delta_\mathbf K\leq \mathbf exp(10^{10})$.
\mathbf end{rem*}
For a much tighter range of discriminants one can prove bounds of the form $T_\eK\leq c\log\DKsq$ with
$c<4$. For instance, we have the psychologically important bound $T_{\mathcal{C}}K\leq T_\eK\leq \log\DKsq$ as soon as
$$\log\DK+2\log\lDK+2+2(n_\K+1)\frac{\log(7\log\DK)}{\log\DK}\leq 2(\gamma+\log 2\pi)n_\K.$$
For a given degree $n_\K\gammaeq 4$, this happens for $\Delta_\mathbf K$ lower than a certain limit. As $n_\K$ goes to
infinity, the limit corresponds to a root-discriminant tending to $(2\pi e^{\gamma})^2=125.23\,\mathrm dots$.
There are infinitely many fields satisfying this condition. Indeed, consider the field
$F=\mathbf Q[\cos(2\pi/11),\sqrt{2},\sqrt{-23}]$. Martinet~\cite{Martinet:tours} proved that the Hilbert
class field tower of $F$ is infinite because $F$ satisfies Golod--Shafarevich's condition. Since
$n_F=20$ and $\log\Delta_F\leq 90.6$, this shows that there is an infinite number of fields $\mathbf K$ such that
$\log\DK\leq 4.53n_\K$.
For one of those fields, we have $T_{\mathcal{C}}K\leq T_\eK\leq \log\DKsq$ if $n_\K\gammaeq 47$ and the quotient improves
when the degree increases, with $\limsup\{T_\eK/\log\DKsq\colon\allowbreak \log\DK\leq 4.53n_\K\}\leq 0.88$.\\
As a second example, consider $F=\mathbf Q[x]/(f)$, where $f=x^{10} + 223x^8 + 18336x^6 + 10907521x^4 +
930369979x^2 + 18559139599$. Hajir and Maire~\cite{HajirMaire:asymptotically-good} proved that the Hilbert
class field tower of $F$ is infinite because $F$ satisfies Golod--Shafarevich's condition. Since
$\log\Delta_F\leq 44.4$, this shows that there is an infinite number of fields $\mathbf K$ such that $\log\DK\leq
4.44n_\K$.
For one of those fields, we have $T_{\mathcal{C}}K\leq T_\eK\leq \log\DKsq$ if $n_\K\gammaeq 34$ with $\limsup\{T_\eK/\log\DKsq
\colon \log\DK\leq 4.44n_\K\}\leq 0.84$.\\
\mathbf enlargethispage{-\baselineskip}
As a third example, consider the field $F=\mathbf Q[x]/(f)$, where $f=x^{12} + 339x^{10} - 19752x^8 -
2188735x^6 + 284236829x^4 + 4401349506x^2 + 15622982921$. In~\cite{HajirMaire:TamelyRamifiedTowers}, the
authors proved that $F$ admits an infinite tower of extensions ramified at most above a single prime ideal
of $F$ of norm $9$. Since $\log(9\Delta_F)/12\leq 4.41$, there is an infinite number of fields $\mathbf K$ such
that $\log\DK\leq 4.41n_\K$.
For one of those fields, we have $T_{\mathcal{C}}K\leq T_\eK\leq \log\DKsq$ if $n_\K\gammaeq 32$ with $\limsup\{T_\eK/\log\DKsq
\colon \log\DK\leq 4.41n_\K\}\leq 0.82$.\\
Assuming \textrm{\upshape GRH}, Serre~\cite{Serre:discriminants} proved that there are only finitely many
fields such that $\log\DK\leq cn_\K$ for every $c<\gamma+\log 8\pi$. Suppose that $\log\DK\leq (\gamma+\log 8\pi)n_\K$,
then $T_{\mathcal{C}}K\leq T_\eK\leq \log\DKsq$ if $n_\K\gammaeq 11$. Serre's result does not rule out the possibility that
there are infinitely many such fields; in this case $\limsup \{T_\eK/\log\DKsq\colon \log\DK\leq
(\gamma+\log 8\pi)n_\K\} \leq \big(\frac{4\log 2}{\gamma+\log 8\pi}\big)^2 \leq 0.54$.
\src@spec\l@icbegin{coro}\label{coro:Bach4.01}
Assume \textrm{\upshape GRH}. Then
\[
T_{\mathcal{C}}K\leq T_\eK\leq 4.01\log\DKsq.
\]
\mathbf end{coro}
\src@spec\l@icbegin{proof}
For $n_\K\gammaeq 3$ or $n_\K=2$ and $\log\DK\leq (2\pi e^{\gamma})^{n_\K}/e$, the claim follows from
Corollary~\ref{coro:2}.\\
For $n_\K=2$ and $\log\DK\gammaeq (2\pi e^{\gamma})^{n_\K}/e$ we apply a different argument. Let
\[
f_\mathbf K(n,t) = \log t
- \Big(\gamma
+ \log 2\pi
- \frac{\log t}{t-1}
+ \log\big(1-t^{-1}\big)\!\Big)n
- \frac{2}{1-t^{-1}}\sum_{\mathrm N\mathfrak p^m\leq t}\Big(\frac{1}{\mathrm N\mathfrak p^m}-\frac{1}{t}\Big)\log\mathrm N\mathfrak p,
\]
so that~\mathbf eqref{eq:Phielong} can be written as
\[
\sqrt{T_\eK} \leq \max\Big(\sqrt{T_0},
\frac{2}{1-T_0^{-1}}(\log\DK + f_\mathbf K(n_\K,T_0))
\Big).
\]
Suppose we have a $T_0$ such that $f_\mathbf K(n_\K,T_0)\leq 0$, then we have
\[
\sqrt{T_\eK} \leq \max\Big(\sqrt{T_0},
\frac{2}{1-T_0^{-1}}\log\DK
\Big),
\]
and hence
\[
T_\eK \leq \frac{4}{(1-T_0^{-1})^2}\log\DKsq
\]
if $\log\DK\gammaeq\frac{1}{2}(T_0^{1/2}-T_0^{-1/2})$. Recalling that $n_\K=2$, we choose $T_0=935$:
in this case
\[
f_\mathbf K(2,935)\leq 2 - \frac{935}{467}\sum_{\mathrm N\mathfrak p^m\leq 935}\Big(\frac{1}{\mathrm N\mathfrak p^m}-\frac{1}{935}\Big)\log\mathrm N\mathfrak p.
\]
The value of the sum on prime ideals depends on $\mathbf K$, but it is always larger than what we get
assuming that all primes are inert. This gives
\[
f_\mathbf K(2,935)\leq 2 - \frac{935}{467}\sum_{p^{2m}\leq 935}\Big(\frac{1}{p^{2m}}-\frac{1}{935}\Big)\log(p^2)
\leq -0.02
\]
which therefore produces $T_\eK\leq 4.0086\log\DKsq$ for $\log\DK\gammaeq 15.3$. The proof is complete because
$(2\pi e^{\gamma})^2/e\gammaeq 46$.
\mathbf end{proof}
\subsection{Lower bound for $T_\e(\mathbf K)$}
\src@spec\l@icbegin{prop}\label{prop:3.9}
Assume \textrm{\upshape GRH}. Then
\[
\sqrt{T_\eK} \gammaeq (1 + o(1))\frac{\log\DK}{n_\K}.
\]
\mathbf end{prop}
\src@spec\l@icbegin{proof}
Let $S(T)$ denote the Dirichlet series appearing on the left-hand side of~\mathbf eqref{theoeq}. Then,
introducing the generalized von Mangoldt function $\tilde{\Lambda}_\mathbf K(n) := \sum_{\mathrm N\mathfrak p^m=n}\log\mathrm N\mathfrak p$ we get
\src@spec\l@icbegin{equation}\label{eq:1a}
S(T)
= \sum_{n} \frac{2F_{\mathbf e}(\log n)}{\sqrt{n}}\tilde{\Lambda}_\mathbf K(n).
\mathbf end{equation}
Using $\tilde{\Lambda}_\mathbf K(n)\leq n_\K\Lambda(n)$ in~\mathbf eqref{eq:1a} and introducing Stieltjes' integral
notation, we have
\src@spec\l@icbegin{equation*}
\frac{S(T)}{n_\K}
\leq \int_{2^-}^{+\infty} \frac{2F_{\mathbf e}(\log x)}{\sqrt{x}}\,\mathrm d\psi(x).
\mathbf end{equation*}
Let $g(x)=\frac{2F_{\mathbf e}(\log x)}{\sqrt{x}}$ and notice that it is a continuous function which is derivable
except at most in $T^{\pm1}$ and $(Te^{-2a})^{\pm1}$ with a derivative which is continuous where it exists
and bounded. Thus, with a partial integration we get:
\src@spec\l@icbegin{align*}
\frac{S(T)}{n_\K} &
\leq -\int_{2}^{+\infty} g'(x)\psi(x)\,\mathrm d x
\leq -\int_{2}^{+\infty} g'(x)x\,\mathrm d x
+ \int_{2}^{+\infty} |g'(x)||\psi(x)-x|\,\mathrm d x \\
& = \int_{2}^{+\infty} g(x)\,\mathrm d x
+ \int_{2}^{+\infty} |g'(x)||\psi(x)-x|\,\mathrm d x
+ 2g(2).
\intertext{Since under RH $|\psi(x) - x|\leq 2\sqrt{x}\log^2x$ for every $x\gammaeq 2$ (Schoenfeld proved that
RH implies $|\psi(x) -x|\leq \frac{1}{8\pi}\sqrt{x}\log^2x$ as soon as $x\gammaeq 74$, a direct computation
shows that inequality for the intermediate range $x\in[2,74]$) we get}
&\leq \int_{2}^{+\infty} g(x)\,\mathrm d x
+ 2\int_{2}^{+\infty} |g'(x)|\sqrt{x}\,\log^2 x\,\mathrm d x
+ 3F_{\mathbf e}(\log 2) \\
& = 2\int_{2}^{+\infty} \frac{F_{\mathbf e}(\log x)}{\sqrt{x}}\,\mathrm d x
+ 2\int_{2}^{+\infty}\Big|\Big(\frac{F_{\mathbf e}(\log x)}{\sqrt{x}}\Big)'\Big|\sqrt{x}\,\log^2x\,\mathrm d x
+ 3F_{\mathbf e}(\log 2) \\
& = 2\int_{\log 2}^{+\infty} F_{\mathbf e}(x)e^{x/2}\,\mathrm d x
+ \int_{\log 2}^{+\infty} |2F_{\mathbf e}'(x)-F_{\mathbf e}(x)|x^2\,\mathrm d x
+ 3F_{\mathbf e}(\log 2).
\mathbf end{align*}
We extend the range of the integrals, getting
\src@spec\l@icbegin{equation}\label{eq:1b}
\frac{S(T)}{n_\K}
\leq 2\int_{0}^{+\infty} F_{\mathbf e}(x)e^{x/2}\,\mathrm d x
+ \int_{0}^{+\infty} |2F_{\mathbf e}'(x)-F_{\mathbf e}(x)|x^2\,\mathrm d x
+ 3F_{\mathbf e}(\log 2).
\mathbf end{equation}
We notice that
\src@spec\l@icbegin{equation}
\int_{0}^{+\infty} F_{\mathbf e}(x)e^{x/2}\,\mathrm d x
\leq \int_{\mathbf R} F_{\mathbf e}(x)e^{x/2}\,\mathrm d x
= \Big(\int_{\mathbf R} \Phi_\e(x)e^{x/2}\,\mathrm d x\Big)^2. \label{eq:2b}
\mathbf end{equation}
We observe that, since $\Phi_\e\gammaeq 0$, $\max F_{\mathbf e} = F_{\mathbf e}(0)$ and that the non-negative part of the
support of $F_{\mathbf e}$ is included in $[0,a]\cup[L-2a,L]$ (the intervals may overlap) hence
\src@spec\l@icbegin{align}
\int_{0}^{+\infty}|F_{\mathbf e}(x)|x^2\,\mathrm d x
&= \int_{0}^{L} F_{\mathbf e}(x)x^2\,\mathrm d x
\leq F_{\mathbf e}(0)\Big(\int_{0}^a x^2\,\mathrm d x + \int_{L-2a}^{L} x^2\,\mathrm d x\Big) \notag\\
& = F_{\mathbf e}(0)\big(2aL^2-4a^2L + 3a^3\big)
\leq 2aF_{\mathbf e}(0)L^2. \label{eq:3b}
\mathbf end{align}
Moreover, from Remark~\ref{rem:Fe}, we see that the function is piecewise of the form
$(ax+b)e^{x/2}+(cx+d)e^{-x/2}$, with $a$, $b$, $c$ and $d$ constants, with at most four pieces. Deriving
the expression we find that it can have at most three variations in each piece. The total variation of
$F_{\mathbf e}$ on $[0,L]$ is thus at most $12\max F_{\mathbf e}=12F_{\mathbf e}(0)$. It follows that
\src@spec\l@icbegin{equation}\label{eq:4b}
\int_{0}^{+\infty}|F_{\mathbf e}'(x)|x^2\,\mathrm d x
\leq 12F_{\mathbf e}(0)L^2.
\mathbf end{equation}
Plugging~\mathbf eqref{eq:2b},~\mathbf eqref{eq:3b} and \mathbf eqref{eq:4b} into~\mathbf eqref{eq:1b} we get
\src@spec\l@icbegin{align*}
\frac{S(T)}{n_\K}
&\leq 2\Big(\int_{\mathbf R} \Phi_\e(x)e^{x/2}\,\mathrm d x\Big)^2
+ 2(a+12)F_{\mathbf e}(0)L^2
+ 3F_{\mathbf e}(0) \\
& = 2\Big(e^{-a}T^{3/4}\int_{0}^a e^{x}\,\mathrm d x
+ T^{-1/4}\int_{0}^a \,\mathrm d x\Big)^2
+ 2(a+12)F_{\mathbf e}(0)L^2
+ 3F_{\mathbf e}(0) \\
& = 2\Big(\big(1-e^{-a}\big)T^{3/4} + aT^{-1/4}\Big)^2
+ 2(a+12)F_{\mathbf e}(0)L^2
+ 3F_{\mathbf e}(0) \\
& = 2\big(1-e^{-a}\big)^2T^{3/2}
+ 4a\big(1-e^{-a}\big)T^{1/2}
+ a^2T^{-1/2}
+ 2(a+12)F_{\mathbf e}(0)L^2
+ 3F_{\mathbf e}(0).
\mathbf end{align*}
We have $\Jint(F_{\mathbf e})\leq\frac{\pi}{2}F_{\mathbf e}(0)$ hence in order to satisfy~\mathbf eqref{theoeq} we must have
\src@spec\l@icbegin{multline*}
2\big(1-e^{-a}\big)^2T^{3/2}
+ 4a\big(1-e^{-a}\big)T^{1/2}
+ a^2T^{-1/2}
+ 2(a+12)F_{\mathbf e}(0)L^2
+ 3F_{\mathbf e}(0) \\
\gammaeq \frac{S(T)}{n_\K}
\gammaeq F_{\mathbf e}(0)\Big(\frac{\log\DK}{n_\K} - \Big(\gamma + \log8\pi + \frac{\pi}{2}\Big)\Big)
\mathbf end{multline*}
that we simplify to
\[
(1+o(1))\sqrt{T}
\gammaeq
\frac{F_{\mathbf e}(0)}{2\big(1-e^{-a}\big)^2T}
\Big(\frac{\log\DK}{n_\K}
- 2(a+12)L^2
+ O(1)
\Big).
\]
Since $F_{\mathbf e}(0)=\int_{\mathbf R}(\Phi_\e(y))^2\,\mathrm d y\gammaeq 2\int_{L/2-a}^{L/2}e^{L/2+y}\,\mathrm d y =2\big(1-e^{-a}\big)T$,
this requires
\[
\big(1+o(1)\big)^2\sqrt{T}
\gammaeq \frac{1}{1-e^{-a}}
\Big(\frac{\log\DK}{n_\K}
- 2(a+12)L^2
+ O(1)
\Big).
\]
In the given range for $a$, we can assume that the right-hand side is positive otherwise the claim is
evident. In that case the minimum for the main term is obviously $a=\log T$, hence we can assume that
\[
(1+o(1))\sqrt{T}
\gammaeq \frac{1}{1-T^{-1}}
\Big(\frac{\log\DK}{n_\K}
+ O(\log^3\log\DK)
\Big).
\]
The claim follows.
\mathbf end{proof}
\section{Upper bound for $T(\mathbf K)$}
Belabas, Diaz y Diaz and Friedman~\cite[Section~3]{small-generators} applied Theorem~\ref{theoKB} with
$F(x)=F_L(x):=(L-x)\chi_{[-L,L]}(x)=(\Phi\ast\Phi)(x)$, where $\Phi$ is the characteristic function of
$[-L/2,L/2]$, with $L=\log T$, $T>1$. (Actually they chose $F=\frac1L\Phi\ast\Phi$, but the difference
does not matter since Equation~\mathbf eqref{theoeq} is homogeneous). For this weight function, \mathbf eqref{theoeq}
reads
\src@spec\l@icbegin{equation}\label{biribiri}
2\sum_{\substack{\mathfrak p,m\\\mathrm N\mathfrak p^m<T}}\frac{\log\mathrm N\mathfrak p}{\mathrm N\mathfrak p^{m/2}}\Big(1-\frac{\log\mathrm N\mathfrak p^m}{L}\Big)
>
\log\DK
- (\gamma+\log 8\pi)n_\K
+ \frac{\Iint(F_L)}{L}n_\K
- \frac{\Jint(F_L)}{L}r_1.
\mathbf end{equation}
with
\src@spec\l@icbegin{align*}
\Iint(F_L) &= \frac{\pi^2}2-4\,\mathrm dilog\left(\frac1{\sqrt T}\right)+\,\mathrm dilog\left(\frac1T\right)
\leq \frac{\pi^2}2
\intertext{and}
\Jint(F_L) &= \frac{\pi L}{2}-4C+4\Imm\,\mathrm dilog\left(\frac i{\sqrt T}\right)
\gammaeq \frac{\pi L}{2}-4C,
\mathbf end{align*}
where $\,\mathrm dilog x=-\int_0^x\frac{\log(1-u)}{u}\,\mathrm d u$ and $C=\sum_{k\gammaeq 0}(-1)^k(2k+1)^{-2}=0.9159\ldots$ is
Catalan's constant. Note that Belabas, Diaz y Diaz and Friedman use the estimated values for $I(F_L)$
and $J(F_L)$ instead of their exact values: this is a legitimate simplification which affects the
conclusions only by very small quantities.
In this way they produce a quick algorithm giving a bound $T(\mathbf K)$ for $T_{\mathcal{C}}K$ which, in explicit
computations, is very small. Unfortunately they also prove that it is $\gammaeq ((\frac{1}{4n_\K} +
o(1))\log\DK\log\lDK)^2$, and that therefore it is asymptotically worse than Bach's bound. In the same paper
they advance the conjecture that $(\log\DK\log\lDK)^2$ is the correct size of $T(\mathbf K)$, guessing that
$T(\mathbf K)=((\frac{1}{4} + o(1))\log\DK\log\lDK)^2$. In this section we prove for $T(\mathbf K)$ the corresponding upper
bound and some explicit bounds.
\subsection{Estimation of the number of zeros}
We first prove an estimation of the number of zeros of Dedekind's zeta function that we will use to prove
the main result of this section.
\src@spec\l@icbegin{defi}
Let, for $t\in\mathbf R$,
\[
\mathrm NK(t):=\#\{\rho\colon |\gamma|\leq t\},
\]
where the number is intended including the multiplicity.
\mathbf end{defi}
Trudgian~\cite{Trudgian:zero-counting} gives an estimation of $\mathrm NK(t)$ for $t\gammaeq 1$. The bound depends
on a parameter $\mathbf eta$ which we take equal to $0.05$. In that case the formula is:
\src@spec\l@icbegin{align*}
\forall t\gammaeq 1,\quad
\mathrm NK(t) & =\frac{t}{\pi}\log\Big(\Delta_\mathbf K\Big(\frac{t}{2\pi e}\Big)^{n_\K}\Big)+R_\mathbf K(t),
\intertext{with}
|R_\mathbf K(t)| &\leq 0.247(\log\DK+n_\K\log t)+8.851n_\K+3.024.
\mathbf end{align*}
For $t\in(0,1]$, we use a different strategy.
\src@spec\l@icbegin{prop}\label{N(T)-T<1}
Assume \textrm{\upshape GRH}. We have
\[
\forall t\in(0,1],\quad
\mathrm NK(t) \leq 0.637t\Big(\log\DK-2.45n_\K+S\Big(\frac{3.03}t\Big)\Big)
\]
where
\[
S(U)
:= 960\frac{\big((U-4)e^{\frac{U}{4}}+(U+4)e^{-\frac{U}{4}}\big)^2}{U^5}.
\]
\mathbf end{prop}
\src@spec\l@icbegin{proof}
We use an analog of \cite[Proposition~1]{Omar}, but with a different weight: the function
$F:=30\phi\ast\phi$ with $\phi(x):=\big(\frac14-x^2\big){\raise0.2ex\hbox{$\chi$}}_{[-\frac12,\frac12]}(x)$.
We then have
\[
F(x) = \src@spec\l@icbegin{array}\{{l@{}>{\quad}l}.
-|x|^5 + 5|x|^3 - 5x^2 + 1 &\text{if } |x|\leq 1\\
0 &\text{if } 1<|x|
\mathbf end{array}
\]
and
\src@spec\l@icbegin{align*}
\widehat F(t)
&= 30\big(\widehat\phi(t)\big)^2
= 120\frac{\big(2\sin\big(\frac{t}{2}\big)-t\cos\big(\frac{t}{2}\big)\big)^2}{t^6}.
\mathbf end{align*}
Observe that $\widehat F$ is decreasing on $[0,8.98]$. Consider, for $U>0$,
\[
F_U(x):=F\Big(\frac{x}{U}\Big).
\]
We then have
\[
\widehat{F_U}(t)=U\widehat F(Ut)
=120U\frac{\big(2\sin\big(\frac{Ut}2\big)-Ut\cos\big(\frac{Ut}2\big)\big)^2}{(Ut)^6}.
\]
Using Weil's Explicit Formula~\mathbf eqref{WEF} we have
\src@spec\l@icbegin{multline}\label{eqfromWEF}
U\sum_\gamma\widehat F(U\gamma) = 4\int_{0}^{+\infty} F_U(x)\cosh\Big(\frac{x}{2}\Big)\,\mathrm d x
- 2\sum_{\mathfrak p,m}\frac{\log\mathrm N\mathfrak p}{\mathrm N\mathfrak p^{\frac{m}{2}}}F_U(m\log\mathrm N\mathfrak p) \\
+ \log\DK
- (\gamma + \log8\pi)n_\K
+ \Iint(F_U) n_\K
- \Jint(F_U) r_1.
\mathbf end{multline}
We have
\src@spec\l@icbegin{align*}
4\int_{0}^{+\infty} F_U(x)\cosh\Big(\frac{x}{2}\Big)\,\mathrm d x
&= 4\int_{0}^{+\infty} F\Big(\frac{x}{U}\Big)\cosh\Big(\frac{x}{2}\Big)\,\mathrm d x \\
&= 4U\int_{0}^1(1-5x^2+5x^3-x^5)\cosh\Big(\frac{Ux}{2}\Big)\,\mathrm d x \\
&= 960\frac{\big((U-4)e^{\frac U4}+(U+4)e^{-\frac U4}\big)^2}{U^5}
= S(U).
\mathbf end{align*}
Moreover,
\src@spec\l@icbegin{align*}
\Iint(F_U) & = \int_{0}^{+\infty}\frac{1-F_U(x)}{2\sinh(x/2)}\,\mathrm d x
= U\int_{0}^{+\infty}(1-F(x))\frac{e^{-Ux/2}}{1-e^{-Ux}}\,\mathrm d x.
\intertext{Integrating by parts, which is possible because $F$ is $C^2$, it becomes}
& = 5\int_{0}^1(2x-3x^2+x^4)\log\Big(\frac{1+e^{-Ux/2}}{1-e^{-Ux/2}}\Big)\,\mathrm d x
\mathbf end{align*}
from which we readily see that $\Iint(F_U)$ is decreasing. Removing the positive terms $\sum_{\mathfrak p,m}$ and
$\Jint(F_U)r_1$ from~\mathbf eqref{eqfromWEF}, we get
\[
\forall U>0,\quad U\sum_\gamma\widehat F(U\gamma) \leq \log\DK-(\gamma+\log8\pi-\Iint(F_U))n_\K+S(U).
\]
Let $t\in(0,1]$ and $c$ such that $0<c\leq 8.98$, then setting $U=\frac{c}{t}$ and using
$\Iint(F_{c/t})\leq \Iint(F_c)$ we have
\[
\sum_\gamma\widehat F\Big(\frac{c\gamma}{t}\Big)
\leq
\frac{t}{c}\Big(\log\DK-(\gamma+\log8\pi-\Iint(F_c))n_\K+S\Big(\frac{c}{t}\Big)\Big)
\]
and
\[
\widehat F(c)\mathrm NK(t)
\leq
\sum_{|\gamma|\leq t}\widehat F\Big(\frac{c\gamma}{t}\Big)
\leq
\sum_\gamma\widehat F\Big(\frac{c\gamma}{t}\Big)
\]
so that $\forall t\in(0,1]$, $\forall c\in(0,8.98]$ we have
\[
\mathrm NK(t)
\leq
\frac{c^5 t}{120(2\sin(c/2)-c\cos(c/2))^2}\Big(\log\DK-\big(\gamma + \log8\pi - \Iint(F_c)\big)n_\K + S\Big(\frac{c}{t}\Big)\Big).
\]
The value of $c$ minimizing the coefficient of $t\log\DK$ is $3.051\,\mathrm dots$.
The claim follows setting $c=3.03$.
\mathbf end{proof}
\src@spec\l@icbegin{defi}
Let ${M_\K}(t)$ be the function
\[
{M_\K}(t) :=
\src@spec\l@icbegin{cases}
0.637t\big(\log\DK-2.45n_\K+S\big(\frac{3.03}{t}\big)\big) &\text{if }0< t<1 \\
\frac{t}{\pi}\log\big(\Delta_\mathbf K\big(\frac{t}{2\pi e}\big)^{n_\K}\big)
+ 0.247(\log\DK+n_\K\log t)+8.851n_\K+3.024 &\text{if }1\leq t.
\mathbf end{cases}
\]
\mathbf end{defi}
Recalling the previous proposition we thus have
\[
\forall t> 0,\quad\mathrm NK(t)\leq {M_\K}(t).
\]
\subsection{The bound}
Now we are in position to prove the announced upper bounds for $T(\mathbf K)$.
\src@spec\l@icbegin{theo}\label{estim-T(K)}
Assume \textrm{\upshape GRH}. We have for any fixed $n_\K$
\[
\limsup_{\Delta_\mathbf K\to\infty} \frac{T(\mathbf K)}{\big(\log\DK\log\lDK\big)^2}\leq \frac{1}{16}.
\]
Moreover, for any field
$\mathbf K\not\in\big\{\mathbf Q[\sqrt{-1}],\mathbf Q[\sqrt{-3}],\mathbf Q[\sqrt5]\big\}$
we have
\[
T(\mathbf K)\leq 3.9\big(\log\DK\log\lDK\big)^2.
\]
\mathbf end{theo}
\src@spec\l@icbegin{rem*}
Computing $T(\mathbf K)$ for the whole ``megrez'' number field table~\cite{MegrezTables}, we find that the
quotient $T(\mathbf K)/(\log\DK\log\lDK)^2$ is mostly $\leq0.27$ for them. In fact, apart the fields appearing
as exceptions in the theorem and for which the quotient is $\gammaeq10$, there are only six more fields for
which it is $\gammaeq1$: the quadratic fields of discriminant in $\{-11,-8,-7,8,12,13\}$.
\mathbf end{rem*}
\src@spec\l@icbegin{proof}
Assume $T>e$, $L=\log T$ and $F:=F_L=\Phi\ast\Phi$ with $\Phi:={\raise0.2ex\hbox{$\chi$}}_{[-L/2,L/2]}$. Then~\mathbf eqref{eqzero}
becomes
\src@spec\l@icbegin{equation*}
4\Big(\sqrt{T} - 2 + \frac1{\sqrt{T}}\Big)
\gammaeq \sum_\gamma\frac{1-\cos(L\gamma)}{\gamma^2},
\mathbf end{equation*}
but to estimate $T(\mathbf K)$ we have to add the function
\[
G(T) = \Big(4\,\mathrm dilog\Big(\frac{1}{\sqrt{T}}\Big) - \,\mathrm dilog\Big(\frac{1}{T}\Big)\Big)n_\K
+ 4\Imm\,\mathrm dilog\Big(\frac{i}{\sqrt{T}}\Big)r_1
\]
to the right-hand side, as a consequence of the approximations used in~\cite{small-generators} for
$\Iint(F_L)$ and $\Jint(F_L)$. Thus, introducing $f(t):=\frac{1-\cos t}{t^2}$, the condition becomes
\src@spec\l@icbegin{equation}\label{condition-equivalente-2-BDyDF}
4\Big(\sqrt{T} - 2 + \frac1{\sqrt{T}}\Big) \gammaeq L^2\sum_\gamma f(L\gamma) + G(T),
\mathbf end{equation}
and we need an upper bound for the sum appearing on the right-hand side.
Function $G(T)$ may be easily estimated, for $T\gammaeq e$, as
\src@spec\l@icbegin{equation}\label{eq:G}
G(T) \leq \frac{8n_\K}{\sqrt{T}}\sum_{k=0}^\infty \frac{T^{-2k}}{(4k+1)^2}
\leq \frac{8.05n_\K}{\sqrt{T}}.
\mathbf end{equation}
The main contribution to the sum on zeros comes from those which are close to $0$, the remaining ones
being easily and quite well estimated via the partial summation formula. Thus, we consider first the
range $|t|\leq 1$. The best absolute bound for $f(Lt)$ in this range is $\tfrac12 L^2$, and if we bound
the sum $\sum_{|\gamma|\leq}f(L\gamma)$ simply as $\sup_{t\in[0,1]}|f(Lt)|\mathrm NK(1)$ then we get a term of size
$\frac12 L^2\log\DK$. With this bound~\mathbf eqref{condition-equivalente-2-BDyDF} would become
\[
(4 + o(1))\sqrt{T}> \Big(\frac{1}{2} + o(1)\Big) L^2\log\DK
\]
forcing $T$ to $\gammaeq (\frac{1}{4} +o(1))(\log\DK)^2(\log\log\DK)^4$, which is much larger than what we want to
prove.\\
We overcome this problem using the conclusion of Proposition~\ref{N(T)-T<1} to bound $\mathrm NK(t)$ for $t\leq
1$, but in itself this is not yet sufficient since that bound diverges as $t$ goes to $0$. Hence, for
very small $\gamma$ we apply a more involved argument. In a way similar to the proof of
Proposition~\ref{N(T)-T<1} but using $F:=\Phi\ast\Phi$ with $\Phi:={\raise0.2ex\hbox{$\chi$}}_{[-1/2,1/2]}$, we have,
\src@spec\l@icbegin{align}
\forall U&>0,&
2U\sum_\gamma f(U\gamma)
&\leq \log\DK
- (\gamma + \log8\pi - {\mathcal I}(U))n_\K
+ \frac{8}{U}\big(e^{U/4}-e^{-U/4}\big)^2, \notag
\intertext{where ${\mathcal I}(U) := \Iint(F_U) = \frac{1}{U}\big(\frac{\pi^2}{2}+4\,\mathrm dilog(e^{-U/2})
- \,\mathrm dilog(e^{-U})\big)$. Hence}
\forall U&\gammaeq 3.545,&
2U\sum_\gamma f(U\gamma)
&\leq \log\DK - 2.6016\,n_\K + \frac{8}{U}e^{\frac{U}{2}}. \label{majoration-en-U}
\mathbf end{align}
Setting $U=L$, this bound gives immediately a bound for $L^2\sum_\gamma f(L\gamma)$ of the right order
$\frac{1}{2}L\log\DK$ for the part depending on the discriminant. Unfortunately, it also contains the term
$4e^{\frac{L}{2}} = 4\sqrt{T}$ which makes the bound completely useless when inserted
in~\mathbf eqref{condition-equivalente-2-BDyDF}. As a consequence we have to modify a bit this approach, and we
use~\mathbf eqref{majoration-en-U} with $U:=L-2\log L$. In fact, we notice that $\forall t\gammaeq 0$, $f'(t)\leq
0.014$.
This means that
\src@spec\l@icbegin{equation}\label{eq:delta-bound}
\forall t\gammaeq 0,\quad
f(Lt) \leq f((L-2\log L)t) + 0.028t\log L.
\mathbf end{equation}
Below $\gammaimel:=\big(0.014L^2\log L\big)^{-1/3}$ we bound $f(Lt)$ using~\mathbf eqref{eq:delta-bound} otherwise we
use the trivial bound $f(Lt)\leq \frac2{(Lt)^2}$. We thus define
\[
\forall t\gammaeq 0,
\quad
g(t) :=
\src@spec\l@icbegin{cases}
0.028t L^2\log L &\text{if } 0\leq t\leq \gammaimel \\
\frac2{t^2} &\text{if } \gammaimel<t.
\mathbf end{cases}
\]
Note that we have chosen $\gammaimel$ in such a way that $g$ is continuous. With this definition of $g$, we
thus have $L^2f(Lt)\leq L^2f((L-2\log L)t){\raise0.2ex\hbox{$\chi$}}_{[0,\gammaimel]}(t)+g(t)$.\\
Since we use~\mathbf eqref{majoration-en-U} with $U=L-2\log L$, we need that $L-2\log L\gammaeq 3.545$ so that we
suppose $T\gammaeq 2000$. This, in turn, means that $\gammaimel\leq 1$.
In this way we get
\src@spec\l@icbegin{equation}\label{partiel}
L^2\sum_\gamma f(L\gamma)
\leq
L^2\sum_{|\gamma|\leq \gammaimel}f((L-2\log L)\gamma)
+ \sum_\gamma g(\gamma).
\mathbf end{equation}
By~\mathbf eqref{majoration-en-U}, for the first part we have
\src@spec\l@icbegin{align}
L^2\sum_{|\gamma|\leq \gammaimel}f((L-2\log L)\gamma)
\leq
\frac{L^2}{2(L-2\log L)}\Big(\log\DK-2.6016n_\K+\frac{8\sqrt{T}}{L(L-2\log L)}\Big). \label{prim-part}
\mathbf end{align}
Notice that in this way the term containing $\sqrt{T}$ is actually $O(\sqrt{T}/\log T)$
and does not interfere any more.\\
We now estimate the second part of~\mathbf eqref{partiel}. We have
\src@spec\l@icbegin{align}
\sum_\gamma g(\gamma)
& = \sum_{|\gamma|\leq \gammaimel}g(\gamma)
+ \sum_{\gammaimel<|\gamma|}g(\gamma)
\leq g(\gammaimel)\mathrm NK(\gammaimel)
+ \int_{\gammaimel^+}^{+\infty} g(t)\,\mathrm d\mathrm NK(t)
= -\int_\gammaimel^{+\infty} g'(t)\mathrm NK(t)\,\mathrm d t. \notag
\intertext{Since $g'$ is non-positive on $[\gammaimel,\infty)$ we can estimate $\mathrm NK$ by
${M_\K}$, getting}
&\leq -\int_\gammaimel^{+\infty} g'(t){M_\K}(t)\,\mathrm d t
= -\int_\gammaimel^1g'(t){M_\K}(t)\,\mathrm d t
- \int_{1}^{+\infty} g'(t){M_\K}(t)\,\mathrm d t. \notag
\intertext{The first integral can be estimated by noticing that $S$ is increasing so that
${M_\K}(t)\leq \frac{t}{\gammaimel}{M_\K}(\gammaimel)$ in $[\gammaimel,1]$. The second integral can be
computed. In this way we have}
\sum_\gamma g(\gamma)
&\leq 2.55\Big(\log\DK-2.45n_\K+S\Big(\frac{3.03}\gammaimel\Big)\Big)\int_\gammaimel^1\frac{\!\,\mathrm d t}{t^2}
+ g(1){M_\K}(1)
+ \int_{1}^{+\infty} g(t)F'_\mathbf K(t)\,\mathrm d t \notag\\
& = 2.55\Big(\log\DK-2.45n_\K+S\Big(\frac{3.03}\gammaimel\Big)\Big)\Big(\frac1\gammaimel-1\Big) \notag\\
&\quad
+ \frac2\pi(\log\DK-n_\K\log(2\pi e))
+ 2\cdot 0.247\log\DK
+ 2(8.851n_\K+3.024) \notag\\
&\quad
+ 2\int_{1}^{+\infty}\Big(\frac1\pi\Big(\log\DK+n_\K\log\big(\frac{t}{2\pi}\big)\Big)+\frac{0.247n_\K}t\Big)\frac{\,\mathrm d t}{t^2} \notag\\
& = 2.55\Big(\log\DK-2.45n_\K+S\Big(\frac{3.03}\gammaimel\Big)\Big)\Big(\frac1\gammaimel-1\Big) \label{second-part}\\
&\quad
+ \frac4\pi(\log\DK-n_\K\log2\pi)
+ 0.247(2\log\DK+n_\K)
+ 2(8.851n_\K+3.024). \notag
\mathbf end{align}
Inserting~\mathbf eqref{prim-part} and~\mathbf eqref{second-part} in~\mathbf eqref{partiel} we finally obtain
\src@spec\l@icbegin{align*}
L^2\sum_\gamma f(L\gamma)
&\leq
\frac{L^2}{2(L-2\log L)}\Big(\log\DK-2.6016n_\K+\frac{8\sqrt{T}}{L(L-2\log L)}\Big) \\
&\quad
+ 2.55\Big[\log\DK{-}2.45n_\K{+}S\Big(3.03\big(0.014L^2\log L\big)^{1/3}\Big)\Big]
\Big[\big(0.014L^2\log L\big)^{1/3}{-}1\Big] \\
&\quad
+ \frac4\pi(\log\DK-n_\K\log2\pi)
+ 0.247(2\log\DK+n_\K)
+ 2(8.851n_\K+3.024) \\
&\leq
\Big(\frac{L^2}{2(L-2\log L)}
+ 0.62\big(L^2\log L\big)^{1/3}
- 0.78\Big)\log\DK
+ \frac{4L}{(L-2\log L)^2}\sqrt{T} \\
&\quad
+ \Big(-\frac{1.3008L^2}{L-2\log L}
- 1.5057\big(L^2\log L\big)^{1/3}
+ 21.857\Big)n_\K \\
&\quad
+ 2.55\,
S\Big(3.03\big(0.014L^2\log L\big)^{1/3}\Big)\Big(\big(0.014L^2\log L\big)^{1/3} - 1\Big)
+ 6.05.
\mathbf end{align*}
We are assuming $T\gammaeq 2000$, thus the coefficient of $n_\K$ is smaller than $3.17-1.3L$
and we get
\src@spec\l@icbegin{multline}\label{derniere}
L^2\sum_\gamma f(L\gamma)
\leq
\Big(\frac{L^2}{2(L{-}2\log L)}
{+} 0.62\big(L^2\log L\big)^{\frac13}
{-} 0.78\Big)\log\DK
{+} \frac{4L}{(L{-}2\log L)^2}\sqrt{T} \\
{+} 2.55\,
S\Big(3.03\big(0.014L^2\log L\big)^{\frac13}\Big)\Big(\big(0.014L^2\log L\big)^{1/3} {-} 1\Big)
{+} (3.17 {-} 1.3L)n_\K
{+} 6.05.
\mathbf end{multline}
We can now deal with the first part of the proposition, that is
\[
\limsup_{\Delta_\mathbf K\to\infty} \frac{T(\mathbf K)}{\big(\log\DK\log\lDK\big)^2}\leq \frac1{16}
\]
inserting~\mathbf eqref{derniere} in~\mathbf eqref{condition-equivalente-2-BDyDF}, with the bound~\mathbf eqref{eq:G}.
Obviously, $T$ diverges as $\Delta_\mathbf K$ goes to infinity; in particular, the restriction $T\gammaeq 2000$ does not
matter. Moreover, we observe that the second term in the first line of~\mathbf eqref{derniere} is $o(\log
T)\log\DK$ and that the second line is $O(\sqrt{T}/\log T)$, hence the first claim is proved.
We now study the second part of the proposition, which means the claimed inequality
\[
T(\mathbf K)\leq 3.9\big(\log\DK\log\lDK\big)^2.
\]
We note that, to have $T<2000$ in~\mathbf eqref{condition-equivalente-2-BDyDF} with~\mathbf eqref{eq:G}
and~\mathbf eqref{derniere}, we need $\log\DK<5.15 + 0.63n_\K$.
According to Table~3 in~\cite{Odlyzko:tables} (entry $b=1.3$), this may happen only if $n_\K\leq 5$
and also in this case, only when $\Delta_\mathbf K\leq 607$ (resp. $1141$, $2143$, $4023$) for fields of degree $2$
(resp. $3$, $4$, $5$).\\
For fields of degree $n_\K\gammaeq 6$ and $\Delta_\mathbf K\gammaeq 1.7\cdot10^5$, by elementary arguments one sees
from~\mathbf eqref{condition-equivalente-2-BDyDF}, \mathbf eqref{eq:G} and~\mathbf eqref{derniere} that
\[
T(\mathbf K)\leq 3.6\big(\log\DK\log\lDK\big)^2.
\]
According to Table~3 in~\cite{Odlyzko:tables}, this covers in particular all fields with degree $n_\K\gammaeq
7$.\\
For fields of degree $n_\K\leq 5$, we see that
\src@spec\l@icbegin{equation}\label{eq:beppe}
T(\mathbf K)\leq 3.9\big(\log\DK\log\lDK\big)^2
\mathbf end{equation}
as soon as $\Delta_\mathbf K\gammaeq 3\cdot10^6$ for quadratic fields,
or $\Delta_\mathbf K\gammaeq 10^6$ for $3\leq n_\K\leq 5$.
There remains a finite number of fields of degree $2\leq n_\K\leq 6$. All those with $n_\K\gammaeq 3$ appear in
``megrez'' number field table~\cite{MegrezTables} and for all fields, including the quadratic ones,
we use the algorithm indicated in \cite{small-generators} as implemented in~\cite{PARI2}. For
$\mathbf K\in\big\{\mathbf Q[\sqrt{-3}],\mathbf Q[\sqrt{-1}]\big\}$ we find $T(\mathbf K)=5$ and for $\mathbf K=\mathbf Q[\sqrt5]$, $T(\mathbf K)=7$.
All other fields satisfy~\mathbf eqref{eq:beppe}.
\mathbf end{proof}
\section{Multi-step}
\subsection{Bounds for two and three steps}
The original choice $F=\Phi\ast\Phi$ with $\Phi$ the characteristic function of $[-L/2,L/2]$ is of the
type considered in~\mathbf eqref{eq:setup} with $\Phi^+(x)=\chi_{[0,L]}(x)$, i.e. a function assuming only one
value in $[0,L]$. We call this choice the \mathbf emph{one-step} case. The following corollary shows that the
performance of the algorithm significantly improves already when $\Phi^+(x)$ is allowed to assume two or
three values in the $[0,L]$ interval, as long as it is zero when $x$ is close to $L/2$ (so that
$\Phi(x)=0$ if $x$ is close to $0$). In particular, the extra factor $\log\lDK$ disappears. We call these
choices \mathbf emph{two-} and \mathbf emph{three-steps}.
\src@spec\l@icbegin{coro}\label{coro:boundT1cst}
Let $\Phi^+(x)=b{\raise0.2ex\hbox{$\chi$}}_{[L-2a,L]}(x)+ {\raise0.2ex\hbox{$\chi$}}_{[L-a,L]}(x)$ and define $F$ as in~\mathbf eqref{eq:setup}. Denote
respectively $T_{\mathrm{c}2}(\mathbf K)$ (`two steps') and $T_{\mathrm{c}3}(\mathbf K)$ (`three steps') the lowest $T$
such that~\mathbf eqref{theoeq} is satisfied by such an $F$, with respectively $b=0$ and $b\neq 0$.
We have
\[
\sqrt{T_{\mathrm{c}2}(\mathbf K)}\leq \max\big(2.456\log\DK - 5.623n_\K + 14,\sqrt{13}\big),
\]
where the result is obtained with $a=2.5$, and
\[
\sqrt{T_{\mathrm{c}3}(\mathbf K)}\leq \max\big(2.193\log\DK - 6.19n_\K + 16,\sqrt{32}\big),
\]
which is obtained with $a=1.722$ and $b=\frac{e^{-a/2}}{1-e^{-a/2}}$.
\mathbf end{coro}
\src@spec\l@icbegin{proof}
The claim is proved directly applying Proposition~\ref{prop:eqzero-simplifiee} with
$\Phi^+(x)=b{\raise0.2ex\hbox{$\chi$}}_{[L-2a,L]}(x)+ {\raise0.2ex\hbox{$\chi$}}_{[L-a,L]}(x)$. Conditions $\sqrt{T_{c2}}\gammaeq \sqrt{13}$ and
$\sqrt{T_{c3}}\gammaeq \sqrt{32}$ come from the need to ensure that the support of $\Phi^+$ be in $[0,L]$, so
that we need to assume $L>a$ for the two-steps and $L>2a$ for the three-steps, respectively.
\mathbf end{proof}
\subsection{The algorithm}\label{subsec:algo}
Our aim is to find a good $T$ for the number field $\mathbf K$ as fast as possible exploiting the bilinearity of
the convolution product. We introduce some definitions to make the discussion easier.
\src@spec\l@icbegin{defi}
Let $\mathcal{S}$ be the real vector space of even and compactly supported step functions and, for $T>1$, let
$\mathcal{S}(T)$ be the subspace of $\mathcal{S}$ of functions supported in $[-L/2,L/2]$, with $L=\log T$.
\mathbf end{defi}
\src@spec\l@icbegin{defi}
For any integer $N\gammaeq 1$ and positive real $\,\mathrm delta$ we define the subspace
$\mathcal{S}_d(N,\,\mathrm delta)$ of $\mathcal{S}(e^{2N\,\mathrm delta})$ made of functions which are constant
$\forall k\in\mathrm NM$ on $[k\,\mathrm delta,(k+1)\,\mathrm delta)$.
\mathbf end{defi}
The elements of $\mathcal{S}_d(N,\,\mathrm delta)$ are thus step functions with fixed step width
$\,\mathrm delta$. If $N\gammaeq 1$, $\,\mathrm delta>0$ and $T=e^{2N\,\mathrm delta}$ we have
\src@spec\l@icbegin{subequations}
\src@spec\l@icbegin{alignat}{1}
\mathcal{S}_d(N,\,\mathrm delta) & \subset\mathcal{S}(T)\subset\mathcal{S} \\
\forall\Phi\in\mathcal{S}(T),\qquad\Phi\ast\Phi & \in\mathcal W(T) \\
\mathcal{S}_d(N,\,\mathrm delta) & \subset\mathcal{S}_d(N+1,\,\mathrm delta)\label{incnext} \\
\forall k\gammaeq 1,\quad\mathcal{S}_d\Big(kN,\frac\,\mathrm delta k\Big) & \subseteq\mathcal{S}_d(N,\,\mathrm delta)\label{incmul}.
\mathbf end{alignat}
\mathbf end{subequations}
If, for some $T>1$, $\Phi\in\mathcal{S}(T)$ and $F=\Phi\ast\Phi$ satisfies \mathbf eqref{theoeq} then, according to
Theorem~\ref{theoKB}, $T_{\mathcal{C}}K<T$. This leads us to define the linear form $\mathbf ell_\mathbf K$ on
$\bigcup_{T>1}\mathcal W(T)$ by
\[
\mathbf ell_\mathbf K(F) = -2\sum_\mathfrak p\log\mathrm N\mathfrak p\sum_{m=1}^{+\infty}\frac{F(m\log\mathrm N\mathfrak p)}{\mathrm N\mathfrak p^{m/2}}
+ F(0)(\log\DK - (\gamma + \log 8\pi)n_\K)
+ \Iint(F) n_\K
- \Jint(F) r_1
\]
and the quadratic form $q_\mathbf K$ on $\mathcal{S}$ by $q_\mathbf K(\Phi)=\mathbf ell_\mathbf K(\Phi\ast\Phi)$. From Theorem~\ref{theoKB}
we deduce the following consequence.
\src@spec\l@icbegin{coro}\label{coroKBL}
Let $\mathbf K$ be a number field satisfying \textrm{\upshape GRH}\ and $T>1$. If the restriction of $q_\mathbf K$ to $\mathcal{S}(T)$ has a
negative eigenvalue then $T_{\mathcal{C}}K<T$.
\mathbf end{coro}
\src@spec\l@icbegin{defi}
A \mathbf emph{bound} for $\mathbf K$ is an $L=\log T$ with $T$ as in Theorem~\ref{theoKB}.
\mathbf end{defi}
Note that $q_\mathbf K$ is a continuous function as a function from $(\mathcal{S}(T),\|.\|_1)$ to $\mathbf R$. Therefore if $L$
is a bound for $\mathbf K$ then there exists an $L'<L$ such that $L'$ is a bound for $\mathbf K$. Note also that, in
terms of $T$, only the norms of prime ideals are relevant, which means that we do not need the smallest
possible $T$ to get the best result.
\src@spec\l@icbegin{rem}\label{rem:varepsilon}
If $T>1$ and $\Phi\in\mathcal{S}(T)$, then for any $\varepsilon>0$ there exists $N\gammaeq 1$, $\,\mathrm delta>0$ and
$\Phi_\,\mathrm delta\in\mathcal{S}_d(N,\,\mathrm delta)$ such that $\|\Phi\ast\Phi-\Phi_\,\mathrm delta\ast\Phi_\,\mathrm delta\|_\infty\leq
\varepsilon$ and $e^{2N\,\mathrm delta}\leq T$. Hence we do not loose anything in terms of bounds for $\mathbf K$ if we
consider only the subspaces of the form $\mathcal{S}_d(N,\,\mathrm delta)$.
\mathbf end{rem}
As we will see later, we can compute $q_\mathbf K(\Phi)$ for a generic $\Phi\in \mathcal{S}(T)$ combining its values for
$\Phi=\chi_{[-L/2,L/2]}$ at different $L$'s. Thus, let~$\textrm{\upshape GRH}check(\mathbf K,L)$ be the function that returns the
right-hand side of~\mathbf eqref{biribiri} minus its left-hand side (without the approximations for $\Iint(F_L)$
and $\Jint(F_L)$), and $\BDyDF(\mathbf K)$ be the function which implements the algorithm
of~\cite[Section~3]{small-generators}. The computation of $\BDyDF(\mathbf K)$ is very fast because the only
arithmetic information we need on $\mathbf K\simeq\mathbf Q[x]/(P)$ is the splitting information for primes $p<T$ and
is determined easily for nearly all $p$. Indeed if $p$ does not divide the index of $\mathbf Z[x]/(P)$ in $\OC_{\K}$,
then the splitting of $p$ in $\mathbf K$ is determined by the factorization of $P\mod p$. We can also store such
splitting information for all $p$ that we consider and do not recompute it each time we test whether a
given $L$ is a bound for $\mathbf K$.
We denote $q_{\mathbf K,N,\,\mathrm delta}$ the restriction of $q_\mathbf K$ to $\mathcal{S}_d(N,\,\mathrm delta)$. According to
Corollary~\ref{coroKBL}, if $q_{\mathbf K,N,\,\mathrm delta}$ has a negative eigenvalue then $2N\,\mathrm delta$ is a bound for
$\mathbf K$. This justifies the following definition.
\src@spec\l@icbegin{defi}
The pair $(N,\,\mathrm delta)$ is \gammaood{} when $q_{\mathbf K,N,\,\mathrm delta}$ has a negative eigenvalue.
\mathbf end{defi}
We can reinterpret Functions~\textrm{\upshape GRH}check and \BDyDF saying that if $\textrm{\upshape GRH}check(\mathbf K,2\,\mathrm delta)$ is negative then
$(1,\,\mathrm delta)$ is \gammaood{} and that $\big(1,\frac12\log\BDyDF(\mathbf K)\big)$ is \gammaood.
The fundamental step for our algorithm is the following: given $\,\mathrm delta>0$ we look for the smallest $N$
such that $(N,\,\mathrm delta)$ is \gammaood. Looking for such an $N$ can be done fairly easily with this setup. For
any $i\gammaeq 1$, let $\Phi_i$ be the characteristic function of $({-i\,\mathrm delta},i\,\mathrm delta)$. Then
$(\Phi_i)_{1\leq i\leq N}$ is a basis of $\mathcal{S}_d(N,\,\mathrm delta)$. We have
$\Phi_i\ast\Phi_i=F_{2i\,\mathrm delta}=(2i\,\mathrm delta-|x|)\chi_{[-2i\,\mathrm delta,2i\,\mathrm delta]}(x)$. We observe that
$$\Phi_i\ast\Phi_j=F_{(i+j)\,\mathrm delta}-F_{|i-j|\,\mathrm delta}\ .$$
This means that the matrix $A_N$ of $q_{\mathbf K,N,\,\mathrm delta}$ can be computed by computing only the values of
$\mathbf ell_\mathbf K(F_{i\,\mathrm delta})$ for $1\leq i\leq 2N$ and subtracting those values.
We then stop when the determinant of $A_N$ is negative or when $2N\,\mathrm delta\gammaeq \BDyDF(\mathbf K)$. This does not
guarantee that we stop as soon as there is a negative eigenvalue. Indeed, consider the following sequence
of signatures:
$$(0,p,0)\to(1,p,0)\to(1,p,1)\to(0,p+1,2)\to\cdots$$
here a signature is $(z,p,m)$ where $z$ is the dimension of the kernel and $p$ (resp. $m$) the dimension
of a maximal subspace where $q_\mathbf K$ is positive (resp. negative) definite. We should have stopped when the
signature was $(1,p,1)$ however the determinant was zero there. Our algorithm will stop as soon as there
is an odd number of negative eigenvalues (and no zero) or we go above $\BDyDF(\mathbf K)$. Such unfavorable
sequence of signatures is however very unlikely and does not happen in practice.
The corresponding algorithm is presented in Function~\mathrm NDelta. We have added a limit $N_{\max}$ for $N$
which is not needed right now but will be used later. Note that $(\Phi_i)$ is a basis adapted to the
inclusion~\mathbf eqref{incnext} so that we only need to compute the edges of the matrix $A_N$ at each step. The
test $\,\mathrm det A<0$ in line~\ref{line:detA<0} can be efficiently implemented using Cholesky $LDL^*$
decomposition because it is incremental; moreover, if the last coefficient of $D$ is negative, the last
line of $L^{-1}$ is a vector $v$ such that $vA{}^tv<0$ so that we can check the result.
One way to use this function is to compute $T=\BDyDF(\mathbf K)$ and for some $N_{\max}\gammaeq 2$, let
$\,\mathrm delta=\frac{L}{2N_{\max}}$ and $N=\mathrm NDelta(\mathbf K,\,\mathrm delta,N_{\max})$. Using the inclusion~\mathbf eqref{incmul}, we
see that $(N,\,\mathrm delta)$ is \gammaood{} and that $N\leq N_{\max}$, so that we have improved the bound.
\mathbf enlargethispage{\baselineskip}
\subsection{Adaptive steps}
Unfortunately Function~\mathrm NDelta is not very efficient mostly for two reasons. To explain them and to
improve the function we introduce some extra notations.\\ For any $\,\mathrm delta>0$, let $N_\,\mathrm delta$ be the
minimal $N$ such that $(N,\,\mathrm delta)$ is \gammaood. Observe that Function~\mathrm NDelta computes $N_\,\mathrm delta$, as long as
$N_\,\mathrm delta\leq N_{\max}$ and no zero eigenvalue prevents success. Obviously, using~\mathbf eqref{incnext}, we see
that for any $N\gammaeq N_\,\mathrm delta$, $(N,\,\mathrm delta)$ is \gammaood. We have observed numerically that the sequence
$N\,\mathrm delta_N$ is roughly decreasing, i.e. for most values of $N$ we have $N\,\mathrm delta_N\gammaeq
(N+1)\,\mathrm delta_{N+1}$.\\
For any $N\gammaeq 1$, let $\,\mathrm delta_N$ be the infimum of the $\,\mathrm delta$'s such that $(N,\,\mathrm delta)$ is \gammaood. It is
not necessarily true that if $\,\mathrm delta\gammaeq \,\mathrm delta_N$ then $(N,\,\mathrm delta)$ is \gammaood, however we have never found
a counterexample. The function $\,\mathrm delta\mapsto\,\mathrm delta N_\,\mathrm delta$ is piecewise linear with discontinuities at
points where $N_\,\mathrm delta$ changes; the function is increasing in the linear pieces and decreasing at the
discontinuities. This means that if we take $0<\,\mathrm delta_2<\,\mathrm delta_1$ but we have $N_{\,\mathrm delta_2}>N_{\,\mathrm delta_1}$
then we may have $\,\mathrm delta_2N_{\,\mathrm delta_2}>\,\mathrm delta_1N_{\,\mathrm delta_1}$ so the bound we get for $\,\mathrm delta_2$ is not
necessarily as good as the one for $\,\mathrm delta_1$.\\
The resolution of Function~\mathrm NDelta is not very good: going from $N-1$ to $N$ the bound for the norm of the
prime ideals is multiplied by $e^{2\,\mathrm delta}$. This is the first reason reducing the efficiency of the
function. The second one is that if $N_{\max}$ is above $20$ or so, the number
$\,\mathrm delta=\frac{\log\BDyDF(\mathbf K)}{2N_{\max}}$ has no specific reason to be near $\,\mathrm delta_{N_\,\mathrm delta}$; as
discussed above, this means that we can get a better bound for $\mathbf K$ by choosing $\,\mathrm delta$ to be just above
either $\,\mathrm delta_{N_\,\mathrm delta}$ or $\,\mathrm delta_{1+N_\,\mathrm delta}$. Both reasons derive from the same facts and give a
bound for $\mathbf K$ that can be overestimated by at most $2\,\mathrm delta$ for the considered
$N=\mathrm NDelta(\mathbf K,\,\mathrm delta,N_{\max})$.
To improve the result, we can use once again inclusion~\mathbf eqref{incmul} and determine a good approximation
of $\,\mathrm delta_N$ for $N=2^n$. We determine first by dichotomy a $\,\mathrm delta_0$ such that $(N_0,\,\mathrm delta_0)$ is
\gammaood{} for some $N_0\gammaeq 1$. For any $k\gammaeq 0$, we take $N_{k+1}=2N_k$ and determine by dichotomy a
$\,\mathrm delta_{k+1}$ such that $(N_{k+1},\,\mathrm delta_{k+1})$ is \gammaood; we already know that $\frac{\,\mathrm delta_k}2$ is an
upper bound for $\,\mathrm delta_{k+1}$ and we can either use $0$ as a lower bound or try to find a lower bound not
too far from the upper bound because the upper bound is probably not too bad. The algorithm is described
in Function~\Bound. It uses a subfunction $\OptimalT(\mathbf K,N,T_\mathbf ell,T_h)$ which returns the smallest integer
$T\in[T_\mathbf ell,T_h]$ such that $\mathrm NDelta(\mathbf K,L/(2N),N)>0$. The algorithm does not return a bound below those
proved in~\ref{theo:Teasynt} and~\ref{coro:Bach4.01}.
\subsection{Further refinements}\label{subsec:algoend}
To improve the speed of the algorithm, we decided to make the dichotomy in~$\OptimalT(\mathbf K,N,T_\mathbf ell,T_h)$,
not on all value of $T$ but only on the norms of the prime ideals in $[T_\mathbf ell,T_h]$.\\
To reduce the time used to compute the determinants, we tried to use steps of width $4\,\mathrm delta$ in
$[-L/2,L/2]$ and of width $2\,\mathrm delta$ in the rest of $[-3L/4,3L/4]$, to halve the dimension of
$\mathcal{S}_d(N,\,\mathrm delta)$. It worked in the sense that we found substantially the same $T$ faster. However we
decided that the total time of the algorithm is not high enough to justify the increase in code
complexity.
\src@spec\l@icbegin{function}
\relax
\mathbf KwIn{a number field $\mathbf K$}
\mathbf KwIn{a positive real $\,\mathrm delta$}
\mathbf KwIn{a positive integer $N_{\max}$}
\mathbf KwOut{an $N\leqslant N_{\max}$ such that $(N,\,\mathrm delta)$ is \gammaood{} or $0$}
$tab\leftarrow\text{$(2N_{\max}+1)$-dimensional array}$\;
$tab[0]\leftarrow0$\;
$A\leftarrow\text{$N_{\max}\times N_{\max}$ identity matrix}$\;
$N\leftarrow 0$\;
\While{$N<N_{\max}$}{
$N\leftarrow N+1$\;
$tab[2N-1]\leftarrow(2N-1)\textrm{\upshape GRH}check(\mathbf K,(2N-1)\,\mathrm delta)$\;
$tab[2N]\leftarrow2N\textrm{\upshape GRH}check(\mathbf K,2N\,\mathrm delta)$\;
\For{$i\leftarrow1$ \mathbf KwTo $N$}{
$A[N,i]\leftarrow tab[N+i]-tab[N-i]$\;
$A[i,N]\leftarrow A[N,i]$\;
}
\If{$\,\mathrm det A<0$}{\label{line:detA<0}
\mathbf KwRet{$N$}\;
}
}
\mathbf KwRet{$0$}\;
\caption{NDelta($\mathbf K$,$\,\mathrm delta$,$N_{\max}$)}
\mathbf end{function}
\src@spec\l@icbegin{function}
\relax
\mathbf KwIn{a number field $\mathbf K$}
\mathbf KwOut{a bound for the norm of a system of generators of $\mathcal C\!\mathbf ell_\mathbf K$}
\uIf{$\log\DK< n_\K 2^{n_\K}$}{
$T_0\leftarrow 4\Big(\log\DK+\log\lDK-(\gamma+\log 2\pi)n_\K+1+(n_\K+1)\frac{\log(7\log\DK)}\log\DK\Big)^2$\;
}
\Else{
$T_0\leftarrow 4(\log\DK+\log\lDK-(\gamma+\log 2\pi)n_\K+1)^2$\;
}
$T_0\leftarrow \min\big(T_0,4.01\log\DKsq\big)$\;
$N\leftarrow8$;
$\,\mathrm delta\leftarrow0.0625$\;
\While{$\mathrm NDelta(\mathbf K,\,\mathrm delta,N)=0$}{
$\,\mathrm delta\leftarrow\,\mathrm delta+0.0625$\;
}
$T_h\leftarrow \OptimalT(\mathbf K,N,e^{2N\,(\,\mathrm delta-0.0625)},e^{2N\,\,\mathrm delta})$\;
$T\leftarrow T_h+1$\;
\While{$T_h<T\mathop{||}T>T_0$}{
$T\leftarrow T_h$;
$N\leftarrow2N$\;
$T_h\leftarrow \OptimalT(\mathbf K,N,1,T_h)$\;
}
\mathbf KwRet{$T$}\;
\caption{Bound($\mathbf K$)}
\mathbf end{function}
\subsection{Theoretical performance}
We denote $T_1(\mathbf K)$ the result of Function~\Bound. The algorithm reaches bounds of the same quality as
those of $T_\eK$.
\src@spec\l@icbegin{coro}\label{coro:boundT1}
Assume \textrm{\upshape GRH}. Then Function~\Bound terminates. Moreover we have
\src@spec\l@icbegin{align*}
\sqrt{T_1(\mathbf K)}&\leq 2\log\DK
+ 2\log\lDK
+ 2
- 2(\gamma+\log 2\pi)n_\K
+ 2(n_\K+1)\frac{\log(7\log\DK)}{\log\DK}, \\
\sqrt{T_1(\mathbf K)}&\leq 2\log\DK
+ 2\log\lDK
+ 2
- 2(\gamma+\log 2\pi)n_\K\qquad\text{if }\log\DK\gammaeq n_\K 2^{n_\K}, \\
T_1(\mathbf K) &\leq 4\big(1+\big(2\pi e^{\gamma}\big)^{-n_\K}\big)^2\log\DKsq, \\
T_1(\mathbf K) &\leq 4\log\DKsq\qquad\text{if }\log\DK\leq \frac{1}{e}\big(2\pi e^{\gamma}\big)^{n_\K}, \\
T_1(\mathbf K) &\leq 4.01\log\DKsq.
\mathbf end{align*}
\mathbf end{coro}
\src@spec\l@icbegin{proof}
Consider one of the bounds of Theorem~\ref{theo:Teasynt} or of Corollary~\ref{coro:Bach4.01}. It is
associated to a certain $F_{\mathbf e}=\Phi_\e\ast\Phi_\e$ with a certain $a=\log T_0$ and $T$ given by the bound.
As in Remark~\ref{rem:varepsilon}, For any $\varepsilon>0$, there exists a step function $\Phi$ with
support in $[-L/2,L/2]$ and $2^N$ steps, for $N$ large enough, such that
$\|F-\Phi\ast\Phi\|_\infty<\varepsilon$. Since the inequality in~\mathbf eqref{theoeq} is strict, we can take
$\varepsilon$ small enough so that $\Phi\ast\Phi$ satisfies~\mathbf eqref{theoeq}. Hence, for $N$ large enough,
the algorithm will find a negative eigenvalue in $\mathcal{S}_d(2^N,2^{-N-1}L)$ and hence it will terminate. The
bound $T$ it will give satisfies obviously the first two and last inequality of the statement of the
corollary. Since the intermediate inequalities are consequences of the first two, $T$ will also satisfy
the intermediates inequalities.
\mathbf end{proof}
\subsection{Effective performance}\label{subsec:tests}
\subsubsection{Various fields}
We tested the algorithm on several fields. Let first $\mathbf K=\mathbf Q[x]/(P)$ where
$$\catcode`\*=\active\,\mathrm def*{}
P=x^3 + 559752270111028720*x + 55137512477462689.
$$
The polynomial $P$ has been chosen so that for all primes $2\leq p\leq 53$ there are two prime ideals of
norms $p$ and $p^2$. This ensures that there are lots of small norms of prime ideals. We have
$T(\mathbf K)=19162$. There are $2148$ non-zero prime ideals with norms up to $T(\mathbf K)$. We found that
$T_1(\mathbf K)=11071$ and that there are $1343$ non-zero prime ideals of norms up to $T_1(\mathbf K)$.
We tested also the algorithm on the set of $4686$ fields of degree $2$ to $27$ and small discriminant
coming from a benchmark of~\cite{PARI2}. The mean value of $T_1(\mathbf K)/T(\mathbf K)$ for those fields is lower than
$1/2$.
We have tested the cyclotomic fields $\mathbf K=\mathbf Q[\zeta_n]$ for $n\leq 250$. For them we have found that the
quotient $T_1(\mathbf K)/T(\mathbf K)$ becomes smaller and smaller as the degree increases, reaching the value $1/2$ for
the higher order cyclotomic fields. However, we have observed that the fraction is generally higher than
what we get for the generic fields with comparable degree and discriminant. Certainly cyclotomic fields
are not typical fields: for instance for them the class number grows more than exponentially as a
function of the order~\cite[Th.~4.20]{Washington1}. The weight function that we observe in the tests for
generic fields contains several parameters and therefore could have generic profiles but actually always
shows two bumps, one centered at the origin and one near the end of the support. On the contrary, the
weight producing $T(\mathbf K)$ has a unique bump in the origin, by design. The fact that the original algorithm
already produces a good bound for cyclotomic of small order in some sense means that the second bump is
not necessary, and this is probably due to the existence of a lots of ideals of small norms. However, we
admit we do not have any convincing explanation of this phenomenon.
\subsubsection{Pure fields, small discriminants}
We computed $T(\mathbf K)$ and $T_1(\mathbf K)$ for fields of the form $\mathbf Q[x]/(P)$ with $P=x^n\pm p$ and $p$ is the
first prime after $2^a$ for a certain family of integers $n$ and $a$ such that $\log\DK\leq 250$. We limited
the discriminant because, while at the time of writing the record for which the Buchmann algorithm has
been successfully completed has $\log\DK\gammaeq 646$, this has been done for only very few fields with $\log\DK\gammaeq
100\log 10\simeq 230$. We computed the family of $\frac{T_1(\mathbf K)}{T(\mathbf K)}$ for each fixed degree. The
results are presented in Figure~\ref{fig1}. We can see that in the right-half of the graph, the fields
adopt the asymptotical behavior where $\frac{T_1(\mathbf K)}{T(\mathbf K)}\asymp(\log\lDK)^{-2}$.\\
Let $t(\mathbf K)$ denote the time needed to compute $T(\mathbf K)$ and $t_1(\mathbf K)$ the additional time needed to compute
$T_1(\mathbf K)$. In Figure~\ref{fig2}, we have drawn the points $\frac{t_1(\mathbf K)}{t(\mathbf K)}$ for the four families of
fields we have tested. We have removed three points with $\frac{t_1(\mathbf K)}{t(\mathbf K)}\gammaeq 25$ (in details:
$30.17$, $30.31$ and $47.19$) whose $\log\DK$ is respectively $160.81$, $162.20$ and $167.74$.
In spite of the relatively large value for the quotient, the value of $t_1(\mathbf K)$ in all cases has been
lower than $2$s
(and actually larger than $0.35$s in only $22$ out of the $8308$ fields, including the three fields for
which $t_1(\mathbf K)/t(\mathbf K)\gammaeq 25$, all having $\log\DK\gammaeq 153$).
This shows that the time needed to compute $T_1(\mathbf K)$ is limited anyway with respect to the time needed for
the full Buchmann algorithm.
\subsubsection{Pure fields}
We once again computed $T(\mathbf K)$ and $T_1(\mathbf K)$ for fields of the form $\mathbf Q[x]/(P)$ with $P=x^n\pm p$ and $p$
is the first prime after $10^a$ for a certain family of integers $n$ and $a$.
The graph of $\frac{T_1(\mathbf K)}{T(\mathbf K)}$ looks like a continuation of the right-half of Figure~\ref{fig1} so
that we do not draw it once again. The graph of $\frac{T_1(\mathbf K)}{T(\mathbf K)}(\log\lDK)^2$ is much more regular and
looks to have a non-zero limit, see Figure~\ref{fig3} below. We plotted the graph of
$\frac{T_1(\mathbf K)}{\log\DKsq}$ for the same fields in Figure~\ref{fig4} as well. We computed the mean of
$\frac{T_1(\mathbf K)}{T(\mathbf K)}(\log\lDK)^2$ and the maximum of $\frac{T_1(\mathbf K)}{\log\DKsq}$ for each fixed degree. The
results are summarized below:
\[
\src@spec\l@icbegin{array}{l|r|r|c|c}
P & a\leq & \log\DK\leq & \text{mean of }\frac{T_1(\mathbf K)}{T(\mathbf K)}(\log\lDK)^2 & 1-\max\big(\frac{T_1(\mathbf K)}{\log\DKsq}\big) \\
\hline
x^2-p & 3999 & 9212 & 13.19 & 2\cdot 10^{-5} \\
x^6+p & 1199 & 13818 & 13.38 & 9\cdot 10^{-6} \\
x^{21}-p & 328 & 15169 & 13.68 & 4\cdot 10^{-5}
\mathbf end{array}
\]
The small discriminants are (obviously) much less sensitive to the new algorithm. We reduced the range for
each series to have $\log\DK\leq 500$. The results are as follows:
\[
\src@spec\l@icbegin{array}{l|r|c|c}
P & a\leq & \text{mean of }\frac{T_1(\mathbf K)}{T(\mathbf K)}(\log\lDK)^2 & 1-\max\big(\frac{T_1(\mathbf K)}{\log\DKsq}\big) \\
\hline
x^2-p & 218 & 12.35 & 0.018 \\
x^6+p & 43 & 13.66 & 0.073 \\
x^{21}-p & 10 & 17.19 & 0.279
\mathbf end{array}
\]
\subsubsection{Biquadratic fields}
We repeated the computations above also for biquadratic fields $\mathbf Q[\sqrt{p_1},\sqrt{p_2}]$ where each
$p_i$ is the first prime after $2^{a_i}$ (respectively $10^{a_i}$) for certain families of integers $a_i$
and included them in Figures~\ref{fig1}--\ref{fig4}.\\
In the case where $p_i$ is the first prime after $10^{a_i}$, we found that the mean of
$\frac{T_1(\mathbf K)}{T(\mathbf K)}(\log\lDK)^2$ is $13.63$ for the $7119$ fields computed, and $13.88$ if we restrict the
family to the $1537$ ones with $\log\DK\leq 500$, while the maximum of $\frac{T_1(\mathbf K)}{\log\DKsq}$ is lower than
$1.0038$ for all fields and $0.957$ for the fields with $\log\DK\leq 500$.\\
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\caption{$\frac{T_1(\mathbf K)}{T(\mathbf K)}$ for some fields of small discriminant; in abscissa $\log\DK$.}
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\caption{$\frac{t_1(\mathbf K)}{t(\mathbf K)}$ for some fields of small discriminant; in abscissa $\log\DK$.}
\label{fig2}
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Package color not loaded in conjunction with
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\put(7577,2813){\makebox(0,0)[l]{\strut{}$0.8$}}
\put(7577,3110){\makebox(0,0)[l]{\strut{}$0.9$}}
\put(7577,3406){\makebox(0,0)[l]{\strut{}$1$}}
\put(7577,3703){\makebox(0,0)[l]{\strut{}$1.1$}}
}
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\\
\caption{$\frac{T_1(\mathbf K)}{\log\DKsq}$ for some fields; in abscissa $\log\DK$.}
\label{fig4}
\mathbf end{fixedfig}
\subsection{A simplified algorithm}
Since the expression of $F_{\mathbf e}$ as given in Remark~\ref{rem:Fe} is simpler for $a=L/2$, we can implement a
variant of the algorithm of Belabas, Diaz y Diaz and Friedman with that weight. We have for $x\gammaeq 0$,
\[
F_{\mathbf e}(x) = \Big(\big(x-2+2e^{-x}\sqrt{T}\big)\chi_{[0,L/2)}(x)+(L-x)\chi_{[L/2,L]}(x)\Big)e^{x/2}\sqrt{T}
\]
so that~\mathbf eqref{theoeq} becomes
\src@spec\l@icbegin{multline*}
\sum_{\mathrm N\mathfrak p^m<\sqrt{T}}\Big(m\log\mathrm N\mathfrak p-2+2\frac{\sqrt{T}}{\mathrm N\mathfrak p^m}\Big)\log\mathrm N\mathfrak p
+ \sum_{\sqrt{T}\leq \mathrm N\mathfrak p^m<T}(L-m\log\mathrm N\mathfrak p)\log\mathrm N\mathfrak p \\
>
\big(\sqrt{T}-1\big)(\log\DK - (\gamma + \log 8\pi)n_\K)
+ \frac{\Iint(F_{\mathbf e})}{2\sqrt{T}} n_\K
- \frac{\Jint(F_{\mathbf e})}{2\sqrt{T}} r_1
\mathbf end{multline*}
where
\src@spec\l@icbegin{align*}
\frac{\Iint(F_{\mathbf e})}{2\sqrt{T}}
&= \big(\sqrt{T}-1\big)\log\Big(\frac{4}{1-T^{-1/2}}\Big)-\frac{L^2}{8}+\frac{L}{2}
- \frac{\pi^2}{12}-\,\mathrm dilog(-T^{-1/2}) \\
\frac{\Jint(F_{\mathbf e})}{2\sqrt{T}}
&= \big(\sqrt{T}+1\big)\log\Big(\frac{2}{1+T^{-1/2}}\Big)+\frac{L^2}{8}-\frac{L}{2}
- \frac{\pi^2}{24}-\,\mathrm dilog(-T^{-1/2})+\frac{1}{2}\,\mathrm dilog(-T^{-1}).
\mathbf end{align*}
Applying Theorem~\ref{theo:Phie} with $T_0=\sqrt{T}$, we can see that the result $T_2(\mathbf K)$ of this
algorithm satisfies
\[
\sqrt{T_2(\mathbf K)}\leq 2\log\DK + 2\log\lDK - (\gamma+\log 2\pi)n_\K + 1 + 2\log 2 + cn_\K\frac{\log\lDK}{\log\DK}
\]
for some absolute constant $c$. This means that the asymptotical expansion is nearly optimal: the first
term that changes with respect to Theorem~\ref{theo:Teasynt} is the constant term which increases from $2$
to $1+2\log 2\simeq 2.38$. For small discriminants this algorithm is sometimes worse than~\BDyDF and
always significantly worse than~\Bound, given in Subsections~\ref{subsec:algo}--\ref{subsec:algoend}. For
larger discriminants, it gives only sightly bigger results than~\Bound but is always faster. Numerically,
in our experiments with the above mentioned fields, $T_2(\mathbf K)\gammaeq T(\mathbf K)$ if $\log\DK\leq 48$ (resp. $142$,
$83$, $162$) for $n_\K=2$ (resp. $4$, $6$, $21$) for a total of $401$ fields out the $12648$ tested.
\section{A final comment}
We have proved that
\[
(1+o(1))\frac{\log\DK}{n_\K}\leq \sqrt{T_\eK}\leq (2+o(1))\log\DK.
\]
We have three reasons to believe that the ``true'' behavior is
\[
\sqrt{T_\eK}\sim \log\DK
\]
as $\Delta_\mathbf K\to\infty$, for fixed $n_\K$.
The first one is computational. We have observed that $\sqrt{T_1(\mathbf K)}/\log\DK$ seems to tend to $1$ (from
below, see Figure~\ref{fig4}) for several series of pure fields and one series of biquadratic fields. We
also tested some restricted cases with $\Phi_\e^+$ and $a=L$ and the $\Phi^+$ of the form indicated in
Corollary~\ref{coro:boundT1cst} with $a=\log 4$ and $b=0$: in all cases we observed the same phenomenon,
which is that the experimental result seems to be half the one we can prove.
The second one is related to the upper bound. The function $\widehat F(t)=4(\Ree[e^{-iLt/2}
\widehat{\Phi^+}(t)])^2$ in~\mathbf eqref{eq:1c} is estimated with
$\widehat\,\mathrm digamma(t)=4|\widehat{\Phi^+}(t)|^2$ in~\mathbf eqref{eq:2c}. This step removes the quick oscillations
of $e^{-iLt/2}$ and allows the conclusion of the argument, but it overestimates the contribution of this
object, which would be of this size only in the case where the $\gamma$'s were placed very close to the
maxima of $\cos^2(Lt)$ and which would be smaller by a factor $1/2$, in mean, for uniformly spaced zeros.
Unfortunately, the actual information we have for the vertical distribution of zeros is not strong enough
to distinguish between these two behaviors.
The third one is related to the lower bound in Proposition~\ref{prop:3.9}. For its computation we have
considered each prime integer as totally split. This allows an explicit bound, but it should be corrected
by a factor $1/n_\K$, because this is the density of the totally split primes, and the contribution of the
other primes should be negligible.
All three reasons indicate that $1\cdot \log\DK$ should be the correct asymptotic. We are unable to prove it,
though.
\,\mathrm defPARI}\def\megrez{megrez{PARI}\,\mathrm def\megrez{megrez}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
\providecommand{\MRhref}[2]{
\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}
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\mathbf end{thebibliography}
\mathbf end{document} |
\begin{document}
\begin{abstract}
We introduce \emph{global model categories} as a general framework to capture several phenomena in global equivariant homotopy theory. We then construct \emph{genuine stabilizations} of these, generalizing the usual passage from unstable to stable global homotopy theory. Finally, we define the \emph{global topological André-Quillen cohomology} of an ultra-commutative ring spectrum and express it in terms of a genuine stabilization in our framework in analogy with the classical non-equivariant description obtained by Basterra and Mandell.
\end{abstract}
\maketitle
\setcounter{tocdepth}{1}
\tableofcontents
\section*{Introduction}
Cohomology theories like topological $K$-theory and the various flavors of cobordism are among the most fundamental tools of algebraic topology. Many of the examples one encounters in practice come with additional structure in the form of multiplications and power operations, and these can often be exploited fruitfully, as for example in the classical proof that the Hopf maps define non-trivial elements in the stable homotopy groups of spheres via the Steenrod operations on singular cohomology. In terms of the representing spectra, these extra algebraic data are encoded in a highly structured multiplication, making the representing spectra so-called \emph{$E_\infty$-ring spectra}. In good model categories of diagram spectra like symmetric or orthogonal spectra \cite{hss, mmss}, such $E_\infty$-ring spectra are represented by strictly commutative algebras, and they can also be conveniently described in the language of $\infty$-categories.
\emph{$G$-equivariant cohomology theories}, as modelled by \emph{genuine $G$-spectra} in the sense of equivariant stable homotopy theory, are a refinement of cohomology theories to the context of objects endowed with extra symmetries in the form of an action of a (finite) group $G$. Many classical cohomology theories have equivariant analogues, leading for example to $G$-equivariant topological $K$-theory and cobordism.
When we study highly structured multiplications on these, interesting new structure arises in the form of \emph{norm maps}, which can be thought of as twisted multiplications. As a motivating algebraic example, consider an ordinary commutative ring $R$ with an action by a finite group $G$ and an $H$-fixed point $r\in R^H$ for a subgroup $H\subset G$. Then the \emph{norm} of $r$ is the product
\[ N_H^G(r) = {\textup{pr}}od_{gH\in G/H} g. r,\]
which is now a \emph{$G$-fixed} point. This yields a multiplicative map $N_H^G\colon R^H\to R^G$, and together with inclusions of fixed points as restrictions and similarly defined additive \emph{transfers}, this gives rise to the structure of a \emph{Tambara functor} \cite{tambara} on the collection $\{ R^H\}_{H\subset G}$ of fixed points of $R$. More generally, the zeroth homotopy groups of any \emph{genuine $G$-$E_\infty$-ring spectrum} naturally admit the structure of a Tambara functor and in particular come with norm maps. These norms have been famously exploited in the solution of the Kervaire-invariant-one problem by Hill, Hopkins, and Ravenel \cite{hhr}, renewing interest in this rich structure.
It turns out that many important equivariant cohomology theories like cobordism and $K$-theory exist in a uniform fashion for large classes of groups, like all finite or all compact Lie groups. This is the perspective taken by \emph{global homotopy theory} \cite{schwede-book, hausmann-global}. Such global objects come with extra structure in the form of \emph{inflations} (restrictions along surjective group homomorphisms), and incorporating this additional information can allow for easier analysis of equivariant phenomena, the most prominent example being the recent proof of an equivariant Quillen theorem linking equivariant bordism to formal group laws by Hausmann \cite{hausmann-formal}, which crucially relies on the global perspective.
The correct notion of `multiplicative global cohomology theories' are the \emph{ultra-commutative ring spectra}, which are commutative algebras in a suitable model category of global spectra. The zeroth homotopy groups of such an ultra-commutative ring spectrum form a so-called \emph{global power functor}, the global analogue of a $G$-Tambara functor for fixed $G$, in particular coming with norm maps for any inclusion $H\subset G$ of finite groups. Currently, no purely $\infty$-categorical description of ultra-commutative ring spectra is known.
\subsection*{Obstruction theory and topological André-Quillen cohomology}
As commutative ring spectra encode such a rich additional structure in their homotopy groups and represented cohomology, their study has received much attention. However, the existence of structured multiplications on spectra is a very subtle question: while many important spectra do come with the structure of an $E_\infty$-ring spectrum, such as the sphere spectrum, Eilenberg-MacLane spectra, $K$-theory and Thom spectra, or spectra of topological modular forms, some other naturally defined spectra do not admit such a structured multiplication. The prime examples of this are Moore spectra, where it has been long known that $\mathbb S/2$ does not admit a unital multiplication in the homotopy category and no $\mathbb S/p$ can have a structured associative \textup{h}box{(i.e.~$A_\infty$-)}\kern0ptmultiplication, see \cite[Theorem 1.1]{araki-toda} and \cite[Example 3.3]{angeltveit-thh}. Similarly, the Brown-Peterson spectrum $BP$ (for any prime $p$) does not support an $E_\infty$-multiplication \cite{lawson-BP, senger-BP}.
In order to derive positive results, in some cases obstruction theoretic methods can be applied, with famous examples being the result of Goerss and Hopkins \cite{goerss-hopkins} that Morava $E$-theory admits a unique $E_\infty$-ring structure, or the usage of Postnikov towers of commutative ring spectra in order to construct an $E_4$-multiplication on $BP$ by Basterra and Mandell \cite{basterra-mandell-BP}. Both of these results rely on \emph{topological André-Quillen cohomology}, a cohomology theory for commutative ring spectra introduced by Basterra \cite{basterra-TAQ} as an adaptation of a cohomology theory originally defined by André and Quillen for ordinary commutative rings \cite{andre-homology, quillen-cohomology-algebras}. The latter is defined as a derived functor of Kähler differentials, and a similar approach is used by Basterra for the topological version, using a model category of spectra in which commutative ring spectra model $E_\infty$-ring spectra.
A crucial observation in the construction of classical \emph{algebraic} André-Quillen cohomology is that for a fixed commutative ring $R$, one can identify the category of $R$-modules with the category of abelian group objects in augmented $R$-algebras, which allows to interpret André-Quillen homology and cohomology as a derived abelianization procedure. In the topological case, a similar result was obtained by Basterra and Mandell \cite{basterra-mandell-stab}, who showed that for a commutative ring spectrum $R$, topological André-Quillen cohomology exhibits the category of $R$-modules as a stabilization of the category of augmented $R$-algebras.
\subsection*{Global topological André-Quillen cohomology}
Given the wealth of structure encoded in the multiplication on an ultra-commutative ring spectrum (even compared to a non-equivariant $E_\infty$-ring spectrum), a \emph{global obstruction theory} would be particularly desirable. As the first step towards this, we adapt the theory of topological André-Quillen cohomology to the context of global homotopy theory in this article. In particular, we define the global topological André-Quillen homology and cohomology of ultra-commutative ring spectra, prove a Hurewicz theorem in this setting, and construct Postnikov towers for ultra-commutative ring spectra. As the main result of this paper (see Theorem~\ref{introthm:global-stab-calg} below) we then identify global topological André-Quillen cohomology as a suitable \emph{global stabilization} in analogy with the non-equivariant result of Basterra--Mandell, justifying that this is indeed the `correct' global analogue of classical topological André-Quillen cohomology.
To put this into perspective, recall that any suitably nice model or $\infty$-category admits a \emph{stabilization} \cite{schwede-stab,higher-algebra}, obtained by inverting the suspension-loop adjunction. However, the passage between unstable and stable equivariant or global homotopy theory is more subtle, related to the existence of transfer maps between equivariant stable homotopy groups mentioned above. In the equivariant setting, we can instead concisely express it as a \emph{genuine stabilization} \cite[Appendix~C]{gepner-meier}: the passage from unstable to stable $G$-equivariant homotopy theory is given by universally inverting the $1$-point compactification $S^G$ of the regular real representation, instead of just the usual sphere $S^1$ (i.e.~the $1$-point compactification of the trivial $1$-dimensional representation). Sadly, however, this approach can not be immediately adapted to the global world---for example, a computation by Schwede shows that only the ordinary non-equivariant spheres are inverted when passing from unstable to stable global homotopy theory.
Similarly to an idea described by Gepner and Nikolaus \cite{gepner-nikolaus}, we solve this issue in the present paper by looking more generally at \emph{$G$-global homotopy theory} in the sense of \cite{g-global} for all finite groups $G$, which for the trivial group $G=1$ recovers usual global homotopy theory. In this setting, we then describe the passage from unstable to stable $G$-global homotopy theory \emph{for all finite groups $G$ simultaneously}: it is given by inverting $S^G$ in $G$-global homotopy theory for all $G$ in a compatible way (see Theorem~\ref{introthm:global-stab-s} below).
\subsection*{Global model categories}
To make this precise, we introduce the notion of a \emph{global model category} (Definition~\ref{defi:global-model-cat}). Roughly speaking, such a global model category $\ul{\mathscr C}$ consists of a category $\mathscr C$ together with two (suitably nice) Quillen equivalent model structures on the category $G\text-\mathscr C$ of $G$-objects in $\mathscr C$, called the \emph{projective} and \emph{flat} model structures, that interact in a prescribed way with restriction along group homomorphisms, formalizing the behaviour established for unstable and stable $G$-global homotopy theory in \cite{g-global}. As our main examples, we introduce and study global model categories $\glo{GlobalSpaces}$ of global spaces, $\glo{GlobalSpectra}$ of global spectra, $\glo{Mod}_R$ of $R$-modules, and $\glo{Comm}_R/R$ of augmented $R$-algebras for an ultra-commutative ring spectrum $R$.
If $\ul{\mathscr C}$ is a \emph{pointed} global model category (i.e.~$\mathscr C$ has a zero object) and $G$ is finite, the homotopy category ${\textup{Ho}}(G\text-\mathscr C)$ comes with an equivariant suspension-loop adjunction
\begin{equation*}
S^G\smashp^{\cat L}{\textup{--}}\colon {\textup{Ho}}(G\text-\mathscr C)\rightleftarrows{\textup{Ho}}(G\text-\mathscr C) :\!\cat{R}\Omega^G
\end{equation*}
(recovering the usual one for pointed global spaces), and we call $\ul{\mathscr C}$ \emph{(genuinely) stable} if this adjunction is an adjoint equivalence for every $G$.
With this definition, both the global model category of global spectra and of $R$-modules are stable. For a general global model category $\ul{\mathscr C}$ on the other hand, we can construct a \emph{global stabilization} in the form of a homotopy universal map to a stable global model category by considering suitable $G$-global spectrum objects in $\mathscr C$, refining and generalizing the non-equivariant construction of \cite{schwede-stab}. We then compute this global stabilization in two key cases:
First, we show that the global stabilization of global spaces is indeed given by global spectra in this setting:
\begin{introthm}[see \Cref{thm:stabilization-global-spaces}]\label{introthm:global-stab-s}
The suspension spectrum-loop space adjunction
\begin{equation*}
\ul{\Sigma^\bullet_+}\colon\glo{GlobalSpaces}\rightleftarrows\glo{GlobalSpectra} :\!\ul{\Omega^\bullet}
\end{equation*}
exhibits the global model category $\glo{GlobalSpectra}$ of global spectra as global stabilization of the global model category $\glo{GlobalSpaces}$ of global spaces.
\end{introthm}
This in particular serves as a sanity check for our framework, but it also allows us to provide a description of the previously elusive passage between unstable and stable global homotopy theory in terms of a universal property.
The main application of this theory, however, lies in the calculation of the global stabilization of the category of augmented $R$-algebras for an ultra-commutative ring spectrum $R$:
\begin{introthm}[see \Cref{thm:stabilization-aug-algebras} for a precise statement]\label{introthm:global-stab-calg}
Let $R$ be a flat ultra-commutative ring spectrum. Then the global model category $\glo{Mod}_R$ of $R$-modules is the global stabilization of the global model category $\glo{Comm}_R/R$ of augmented $R$-algebras, and the universal map reduces on homotopy categories to the `global abelianization' functor from the construction of global topological André-Quillen cohomology.
\end{introthm}
\subsection*{Related work}
Another perspective on the relation between unstable and stable equivariant homotopy theory is provided by \emph{parameterized higher category theory} in the sense of Barwick, Dotto, Glasman, Nardin, and Shah \cite{elements-param,nardin-orbi}, which emphasizes the extra algebraic structure encoded in the additive transfers (or more precisely the so-called \emph{Wirthmüller isomorphisms} underlying them) as a form of `genuine semiadditivity.'
Using this language, Cnossen, Linskens, and the first author \cite{global-param} have concurrently introduced the concept of \emph{global $\infty$-categories}. These again come with a notion of \emph{genuine stability} (now defined via the aforementioned Wirthmüller isomorphisms), and the main result of \emph{op.~cit.} describes a certain explicit global $\infty$-category of global spectra (again built from $G$-global spectra for all finite $G$) as the genuine stabilization in this sense of an analogous global $\infty$-category of global spaces---in fact, global spaces and global spectra admit universal descriptions in this framework as the free presentable and free presentable genuinely stable global $\infty$-category, respectively, which immediately implies the above description.
It is not hard to show that any global model category gives rise to a (presentable) global $\infty$-category via Dwyer-Kan localization, which for $\glo{GlobalSpaces}$ and $\glo{GlobalSpectra}$ precisely recovers the aforementioned global $\infty$-categories of global spaces and global spectra; in particular, the two notions of global stabilization agree in this case. However, while it is natural to expect them to also agree in general, this is not clear a priori; the first author plans to come back to this question in future work.
\subsection*{Outline}
In Section~\ref{sec:g-global} we give a recollection of $G$-equivariant and $G$-global homotopy theory, in particular describing the rich `change of group'-calculus present in the latter. We then formalize this calculus in Section~\ref{sec:global-model-cat} in the notion of a \emph{global model category}.
Section~\ref{sec:global-stab} is devoted to the notion of \emph{stability} for global model categories and the general construction of global stabilizations. In Section~\ref{sec:global-spectra} we compute this stabilization in the case of global spaces, proving Theorem~\ref{introthm:global-stab-s}.
In Section~\ref{section:brave-new-algebra} we develop the theory of modules and algebras in stable $G$-global homotopy theory, and in particular define corresponding global model categories. Afterwards, we introduce ($G$-)global topological André-Quillen cohomology in Section~\ref{sec:gtaq} and express it via the global stabilization of augmented commutative algebras, proving Theorem~\ref{introthm:global-stab-calg}. This proof in turn relies on a hard technical result about the stabilization of the free-forgetful adjunction between modules and non-unital commutative algebras (so-called `NUCAs'), to which all of Section~\ref{section:stab-nucas} is devoted.
\section{A reminder on $G$-equivariant and $G$-global homotopy theory}\label{sec:g-global}
Throughout, let $G$ be a finite group. To set the stage, we recall several model categorical aspects of \emph{$G$-equivariant} and \emph{$G$-global homotopy theory}.
\subsection{\texorpdfstring{$\bm G$}{G}-equivariant homotopy theory} We begin with the classical unstable equivariant story:
\begin{prop}
Let $\mathcal F$ be a family of subgroups of $G$, i.e.~a non-empty collection of subgroups that is closed under subconjugates. Then there is a unique model structure on $\cat{$\bm G$-SSet}$ in which a map $f$ is a weak equivalence or fibration if and only if $f^H$ is a weak homotopy equivalence or Kan fibration, respectively, of simplicial sets for all $H\in\mathcal F$. We call this the \emph{$\mathcal F$-equivariant model structure} and its weak equivalences the \emph{$\mathcal F$-weak equivalences}. It is simplicial, proper, and combinatorial with generating cofibrations
\begin{equation*}
\{ G/H\times(\del\Delta^n\textup{h}ookrightarrow\Delta^n) : H\in\mathcal F,n\ge0\}
\end{equation*}
and generating acyclic cofibrations
\begin{equation*}
\{ G/H\times(\Lambda^n_k\textup{h}ookrightarrow\Delta^n) : H\in\mathcal F,0\le k\le n\}.
\end{equation*}
Moreover, a map is a cofibration in it if and only if it is levelwise injective and every simplex not in its image has isotropy in $\mathcal F$. Finally, filtered colimits in it are homotopical.
\begin{proof}
By \cite[Example~2.14]{cellular} we get a cofibrantly generated (hence combinatorial) model structure with the above weak equivalences, fibrations, and generating (acyclic) cofibrations, while Proposition~2.16 of \emph{op.~cit.} provides the characterization of the cofibrations. The remaining properties are easy to check, also see \cite[Proposition~1.1.2]{g-global} for a complete proof.
\end{proof}
\end{prop}
\begin{ex}
If we take $\mathcal F=\mathcal A\ell\ell$ to be the collection of all subgroups of $G$, then we get a model structure with cofibrations the underlying cofibrations of simplicial sets, while weak equivalences and fibrations are those maps $f$ such that $f^H$ is a weak equivalence or fibration, respectively, for every subgroup $H\subset G$.
We will refer to this model structure simply as the \emph{$G$-equivariant model structure} and to its weak equivalences as \emph{$G$-equivariant weak equivalences}.
\end{ex}
\begin{ex}\label{ex:graph-model-structure}
Let $G,H$ be finite groups. We write $\mathcal G_{G,H}$ for the family of \emph{graph subgroups} of $G\times H$, i.e.~groups of the form $\Gamma_{K,\phi}\mathrel{:=}\{(k,\phi(k)) : k\in K\}$ for a subgroup $K\subset G$ and a homomorphism $\phi\colon K\to H$. Note that $K$ and $\phi$ are actually uniquely determined for any graph subgroup, and a subgroup $L\subset G\times H$ is a graph subgroup if and only if it intersects $H$ trivially, i.e.~$L\cap (1\times H)=1$.
We now apply the proposition for $\mathcal F=\mathcal G_{G,H}$ to get a \emph{graph model structure} on $\cat{$\bm{(G\times H)}$-SSet}$, which in the case $H=1$ recovers the previous model structure. For general $H$, a map is a cofibration in this model structure if and only if it is levelwise injective and $H$ acts freely outside the image.
\end{ex}
We will also need a variant of the above model structure with more cofibrations:
\begin{prop}
Let $G$ be a finite group and let $\mathcal F$ be a family of subgroups. Then there is a unique model structure on $\cat{$\bm G$-SSet}$ with weak equivalences the $\mathcal F$-equivariant weak equivalences and cofibrations the injective cofibrations, i.e.~the underlying cofibrations of simplicial sets. We call this the \emph{injective $\mathcal F$-equivariant model structure}. It is simplicial, proper, combinatorial with generating cofibrations
\begin{equation*}
\{G/H\times(\del\Delta^n\textup{h}ookrightarrow\Delta^n) : H\subset G,n\ge0\},
\end{equation*}
and filtered colimits in it are homotopical.
\end{prop}
Observe that for $\mathcal F=\mathcal A\ell\ell$ this recovers the $G$-equivariant model structure again.
\begin{proof}
This is a folklore result; the earliest appearance in the literature (with $\cat{SSet}_*$ instead of $\cat{SSet}$) we are aware of is \cite[Proposition~1.3]{shipley-mixed} where this is already referred to as a `well-known' model structure. A full proof in our setting can be found as \cite[Proposition~1.1.15]{g-global}.
\end{proof}
We will frequently use the following well-known `change of group' properties of the above model structures, all of which can also be found in \cite[1.1.4]{g-global}:
\begin{lemma}\label{lemma:graph-target}
Let $G$ be a finite group and let $\alpha\colon H\to H'$ be a homomorphism of finite groups. Then the adjunction
\begin{equation*}
\alpha_!\mathrel{:=}(G\times\alpha)_!\colon \cat{$\bm{(G\times H)}$-SSet}_{\mathcal G_{G,H}}\rightleftarrows \cat{$\bm{(G\times H')}$-SSet}_{\mathcal G_{G,H
}} :(G\times\alpha)^*\mathrel{=:}\alpha^*
\end{equation*}
is a Quillen adjunction. If $\alpha$ is injective, then also
\begin{equation*}
\alpha^*\mathrel{:=}(G\times\alpha)^*\colon \cat{$\bm{(G\times H')}$-SSet}_{\mathcal G_{G,H'}}\rightleftarrows \cat{$\bm{(G\times H)}$-SSet}_{\mathcal G_{G,H}} :(G\times\alpha)_*\mathrel{=:}\alpha_*
\end{equation*}
is a Quillen adjunction.\qed
\end{lemma}
\begin{lemma}\label{lemma:graph-source}
Let $\alpha\colon G\to G'$ be an injective homomorphism of finite groups and let $H$ be a finite group. Then
\begin{equation*}
\alpha^*\mathrel{:=}(\alpha\times H)^*\colon\cat{$\bm{(G'\times H)}$-SSet}_{\mathcal G_{G',H}}\to \cat{$\bm{(G\times H)}$-SSet}_{\mathcal G_{G,H}}
\end{equation*}
is both left and right Quillen.\qed
\end{lemma}
Next, we come to the stable situation, where we will use Hausmann's model \cite{hausmann-equivariant} based on symmetric spectra, which we briefly recall:
\begin{constr}
We write $\bm\Sigma$ for the $\cat{SSet}_*$-enriched category whose objects are finite sets and with morphism spaces
\begin{equation*}
\mathord{\textup{maps}}_{\bm\Sigma}(A,B)\mathrel{:=}\bigvee_{i\colon A\to B\text{ injective}} S^{B\setminus i(A)}.
\end{equation*}
Composition is given by smashing, i.e.~if $C$ is yet another object, then the composition $\mathord{\textup{maps}}_{\bm\Sigma}(A,B)\smashp\mathord{\textup{maps}}_{\bm\Sigma}(B,C)\to\mathord{\textup{maps}}_{\bm\Sigma}(A,C)$ is given on the wedge summands corresponding to $i\colon A\to B$ and $j\colon B\to C$ as
\begin{equation*}
S^{B\setminus i(A)}\smashp S^{C\setminus j(B)}\cong S^{j(B)\setminus ji(A)}\smashp S^{C\setminus j(B)}\cong S^{C\setminus ji(A)}\textup{h}ookrightarrow \bigvee_{k\colon A\to C\text{ injective}} S^{C\setminus k(A)}
\end{equation*}
where the first isomorphism is induced by $j$, the second one is the canonical isomorphism, and the final map is the inclusion of the summand indexed by $ji$.
\end{constr}
\begin{defi}
A \emph{symmetric spectrum} (in simplicial sets) is an $\cat{SSet}_*$-enriched functor $\bm\Sigma\to\cat{SSet}_*$. We write $\cat{Spectra}$ for the corresponding category of enriched functors and enriched natural transformations, which is itself enriched, tensored, and cotensored over $\cat{SSet}_*$ with (co)tensoring defined levelwise.
If $G$ is any group, then we write $\cat{$\bm G$-Spectra}$ for the category of $G$-objects in $\cat{Spectra}$ and call its objects \emph{$G$-symmetric spectra} or simply (by slight abuse of language) \emph{$G$-spectra}.
\end{defi}
\begin{rk}
Symmetric spectra are often instead defined in a `coordinatized' fashion as sequences $(X_n)_{n\ge0}$ of based $\Sigma_n$-simplicial sets together with maps $S^1\smashp X_n\to X_{n+1}$ that interact suitably with the actions. For the equivalence to the above definition we refer the reader to \cite[2.4]{hausmann-equivariant}.
\end{rk}
We will now construct equivariant model structures on the category of $G$-spectra. Just like non-equivariantly, these come in a \emph{projective} and a \emph{flat} version, and will be obtained by Bousfield localizing suitable level model structures:
\begin{prop}
There is a unique model structure on $\cat{$\bm G$-Spectra}$ in which a map $f$ is a weak equivalence or fibration if and only if $f(A)$ is a $\mathcal G_{G,\Sigma_A}$-weak equivalence or fibration, respectively, for every finite set $A$. We call this the \emph{$G$-equivariant projective level model structure} and its weak equivalences the \emph{$G$-equivariant level weak equivalences}. It is combinatorial with generating cofibrations
\begin{equation*}
\big\{\big(G_+\smashp\bm\Sigma(A,{\textup{--}})\big)/H\smashp(\del\Delta^n\textup{h}ookrightarrow\Delta^n)_+ : \text{$A$ finite set}, H\in\mathcal G_{G,\Sigma_A}, n\ge0\big\}.
\end{equation*}
\end{prop}
More precisely, for the above to be a set (as opposed to a proper class), we should restrict to a set of finite sets hitting all isomorphism classes; in all what follows we will ignore this and similar technicalities.
\begin{proof}
See \cite[Corollary~2.26 and discussion afterwards]{hausmann-equivariant}.
\end{proof}
\begin{prop}
There is a unique model structure on $\cat{$\bm G$-Spectra}$ in which a map $f$ is a weak equivalence or fibration if and only if $f(A)$ is a weak equivalence or fibration, respectively, in the \emph{injective} $\mathcal G_{G,\Sigma_A}$-model structure for every finite set $A$. We call this the \emph{$G$-equivariant flat level model structure}. Its weak equivalences are precisely the $G$-equivariant level weak equivalence; moreover, a map is a cofibration in the flat level model structure on $\cat{$\bm G$-Spectra}$ if and only if it is a cofibration in the flat level model structure on $\cat{Spectra}$ (i.e.~for $G=1$); we will refer to these maps as \emph{flat cofibrations}.
Finally, the $G$-equivariant flat level model structure is combinatorial with generating cofibrations
\begin{equation*}
\big\{\big(G_+\smashp\bm\Sigma(A,{\textup{--}})\big)/H\smashp(\del\Delta^n\textup{h}ookrightarrow\Delta^n)_+ : \text{$A$ finite set}, H\subset G\times\Sigma_A, n\ge0\big\}.
\end{equation*}
\begin{proof}
The construction of the model structure and the identification of the generating cofibrations is \cite[Corollary~2.25 and discussion afterwards]{hausmann-equivariant}, while the characterization of the flat cofibrations is Remark~2.20 of \emph{op.~cit.}
\end{proof}
\end{prop}
\begin{rk}
We will never need to know how the generating \emph{acyclic} cofibrations of the above two model structures look like; the curious reader can find them in Hausmann's treatment referred to above.
\end{rk}
\begin{defi}
A $G$-spectrum $X$ is called a \emph{$G$-$\Omega$-spectrum} if for every $H\subset G$ and all finite $H$-sets $A\subset B$ the derived adjoint structure map $X(A)\to \cat{R}\Omega^{B\setminus A} X(B)$ is an $H$-equivariant weak equivalence, where $H$ acts on $X$, $A$, and $B$.
\end{defi}
Here we are deriving $\Omega^{B\setminus A}$ with respect to the $H$-equivariant model structure on $\cat{$\bm H$-SSet}_*$; in particular, if $X$ is fibrant in either of the above level model structures, then the above is already represented by the ordinary adjoint structure map.
We will also frequently reexpress the above condition as saying that for all finite $H$-sets $A,C$ the map $X(A)\to\cat{R}\Omega^C X(A\amalg C)$ is a weak equivalence.
\begin{thm}
The projective $G$-equivariant level model structure on $\cat{$\bm G$-Spectra}$ admits a Bousfield localization with fibrant objects precisely the projectively level fibrant $G$-$\Omega$-spectra. Similarly, the flat $G$-equivariant level model structure admits a Bousfield localization with fibrant objects the flatly level fibrant $G$-$\Omega$-spectra. Both of these model structures are combinatorial, and they have the same weak equivalences, which we call the \emph{$G$-equivariant weak equivalences}.
\begin{proof}
See \cite[Theorems~4.7 and~4.8]{hausmann-equivariant}.
\end{proof}
\end{thm}
\begin{rk}\label{rk:equivariant-proj-gen-cof}
Let us say something about the generating acyclic cofibrations of the above model structure, see \cite[Example~2.37 and discussion after Theorem~4.8]{hausmann-equivariant}: for any finite set $A$, the spectrum $\bm\Sigma(A,{\textup{--}})$ corepresents evaluation at $A$ by the enriched Yoneda lemma; similarly $S^B\smashp\bm\Sigma(A\amalg B,{\textup{--}})$ corepresents $X\mathord{\textup{maps}}to\Omega^B X(A\amalg B)$. By another application of the Yoneda Lemma we therefore get a map $\lambda_{A,B}\colon S^B\smashp\bm\Sigma(A\amalg B,{\textup{--}})\to\bm\Sigma(A,{\textup{--}})$ such that $\mathord{\textup{maps}}(\lambda_{A,B},X)$ agrees up to conjugation by natural isomorphisms with the adjoint structure map $X(A)\to\Omega^B X(A\amalg B)$ for any symmetric spectrum $X$.
If now $H\subset G$ acts on $A,B$, then $\lambda_{A,B}$ becomes a map of $H$-spectra (denoted $\lambda_{H,A,B}$) with respect to the induced actions, and we factor it as a projective cofibration $\kappa_{H,A,B}$ followed by a level weak equivalence $\rho_{H,A,B}$. Then a set of generating acyclic cofibrations is given by taking a set of generating acyclic \emph{level} cofibrations and adding the pushout product maps $(G_+\smashp_H\kappa_{H,A,B})\ppo i$ for all $H\subset G$, all finite $H$-sets $A,B$ (up to isomorphism), and all generating cofibrations $i$.
\end{rk}
\begin{rk}\label{rk:equivariant-stabilization}
As a teaser for the things to come, we recall that any combinatorial simplicial (left) proper model category admits a \emph{stabilization} \cite{schwede-stab}. We remark without proof that on associated $\infty$-categories this models the universal stabilization in the sense of \cite[Corollary~1.4.4.5]{higher-algebra}, i.e.~the initial example of an adjunction to a presentable stable $\infty$-category, or equivalently the result of universally inverting $\Sigma\mathrel{:=} S^1\smashp{\textup{--}}$ in the presentable world.
However, while we have a Quillen adjunction $\Sigma^\infty_+\colon\cat{$\bm G$-SSet}\rightleftarrows\cat{$\bm G$-Spectra}:\!\Omega^\infty$ for either of the above model structures, as already mentioned in the introduction this does \emph{not} model the stabilization in the above na\"ive sense. Rather, this defines a `genuine stabilization' universally inverting the functor $S^G\smashp{\textup{--}}$ (in presentable $\infty$-categories), where $G$ acts via permuting the smash functors of $S^G=\bigwedge_GS^1$, or equivalently the functors $S^A\smashp{\textup{--}}$ for all finite $G$-sets $A$, see~\cite[Appendix C]{gepner-meier} or \cite[Theorem~A.2]{clausen-mathew-naumann-noel}.
\end{rk}
The main disadvantage of the approach via symmetric spectra is that the weak equivalences are only indirectly defined in terms of specifying the local objects. However, there is a notion of \emph{$\ul\pi_*$-isomorphism}, which we will now introduce, that while not accounting for all $G$-equivariant weak equivalences is at least coarse enough for many purposes:
\begin{constr}
Let $H\subset G$ and let $\mathcal U_H$ be a complete $H$-set universe, i.e.~a countable $H$-set into which any other countable $H$-set embeds equivariantly, and write $s(\mathcal U_H)$ for the poset of finite $H$-subsets of $\mathcal U_H$.
For every $G$-spectrum $X$ and every $k\ge0$ we then define
\begin{equation*}
\pi_k^H(X)=\mathop{\textup{colim}}\nolimits_{A\in s(\mathcal U_H)} [S^{A\amalg\{1,\dots,k\}}, |X(A)|]^H_*
\end{equation*}
where $[\,{,}\,]^H_*$ denotes the set of $H$-equivariant based homotopy classes (for maps of $H$-topological spaces) and the transition maps are given by
\begin{align*}
[S^{A\amalg\{1,\dots,k\}}, |X(A)|]^H_*&\xrightarrow{S^{B\setminus A}\smashp{\textup{--}}}
[S^{B\setminus A}\smashp S^{A\amalg\{1,\dots,k\}}, S^{B\setminus A}\smashp|X(A)|]^H_*\\
&\cong[S^{B\amalg\{1,\dots,k\}},S^{B\setminus A}\smashp |X(A)|]^H_*\\
&\xrightarrow{\sigma} [S^{B\amalg\{1,\dots,k\}},|X(B)|]^H_*
\end{align*}
for all $A\subset B$, where $\sigma$ denotes the structure map of the symmetric spectrum and the unlabelled isomorphism is the canonical one. Similarly, for $k<0$ we define
\begin{equation*}
\pi_k^H(X)=\mathop{\textup{colim}}\nolimits_{A\in s(\mathcal U_H)}[S^A, |X(A\amalg\{1,\dots,-k\})|]^H_*
\end{equation*}
with the analogously defined transition maps.
For every $k\in\mathbb Z$ and $H\subset G$, $\pi_k^HX$ is naturally an abelian group \cite[Definition~3.1]{hausmann-equivariant}; however, we will not need this group structure below.
\end{constr}
\begin{defi}
A map $f\colon X\to Y$ of $G$-spectra is called a \emph{($G$-equivariant) $\ul\pi_*$-isomorphism} if $\pi_k^Hf$ is an isomorphism for all $H\subset G$ and all $k\in\mathbb Z$.
\end{defi}
\begin{rk}
The above homotopy groups are independent of the choice of $\mathcal U_H$ up to natural, but in general non-canonical isomorphism \cite[3.3]{hausmann-equivariant}. In particular, the notion of $\ul\pi_*$-isomorphism is independent of any choices.
\end{rk}
\begin{thm}
Every $\ul\pi_*$-isomorphism of $G$-spectra is a $G$-equivariant weak equivalence.
\begin{proof}
See \cite[Theorem~3.36]{hausmann-equivariant}
\end{proof}
\end{thm}
\begin{warn}
While restriction along \emph{injective} homomorphisms preserves all of the above structure \cite[5.2]{hausmann-equivariant}, the equivariant model structures do not interact reasonably with restrictions along \emph{general} homomorphisms, unlike their unstable siblings. In particular, if $\alpha\colon G\to G'$ is not injective, then $\alpha^*$ will typically \emph{not} send $G'$-equivariant weak equivalences to $G$-equivariant ones.
One nice property of the $G$-global theory we will introduce in the following two subsections is that it comes with homotopically meaningful `change of group' adjunctions (which we will later formalize in the notion of a \emph{global model category}), and in particular that restriction along arbitrary homomorphisms will indeed be homotopical.
\end{warn}
\subsection{Unstable \texorpdfstring{$\bm G$}{G}-global homotopy theory}
Let $G$ continue to denote a finite group. We will now recall \emph{$G$-global homotopy theory} in the sense of \cite{g-global}; this generalizes global homotopy theory (for finite groups) in the sense of Schwede \cite{schwede-book}, and will be the key tool in this article to express and prove properties of the latter. Again, we begin with the unstable story:
\begin{constr}\label{constr:indiscrete}
The forgetful functor $\cat{SSet}\to\cat{Set}$ sending a simplicial set $X$ to its set of vertices admits a right adjoint $E$, given explicitly by $(EX)_n=X^{1+n}$ with the evident functoriality in $X$ and with structure maps induced by the canonical identification $X^{1+n}\cong{\textup{Hom}}_{\cat{Set}}([n], X)$.
\end{constr}
\begin{defi}
We write $I$ for the category of finite sets and injective maps, and we let $\mathcal I$ denote the simplicial category obtained by applying $E$ to all hom sets. We write $\cat{$\bm{\mathcal I}$-SSet}$ for the enriched category of enriched functors $\mathcal I\to\cat{SSet}$ and call its objects \emph{$\mathcal I$-spaces} or \emph{global spaces}. More generally, we write $\cat{$\bm G$-$\bm{\mathcal I}$-SSet}$ for the category of $G$-objects in $\cat{$\bm{\mathcal I}$-SSet}$ and call its objects \emph{$G$-$\mathcal I$-spaces} or \emph{$G$-global spaces}.
\end{defi}
\begin{prop}\label{prop:I-g-glob-level}
There is a unique model structure on $\cat{$\bm G$-$\bm{\mathcal I}$-SSet}$ in which a map is a weak equivalence or fibration if and only if $f(A)$ is a weak equivalence or fibration, respectively, in the $\mathcal G_{\Sigma_A,G}$-equivariant model structure for every finite set $A$. We call this the \emph{$G$-global level model structure} and its weak equivalences the \emph{$G$-global level weak equivalences}. This model structure is proper, simplicial, and combinatorial with generating cofibrations
\begin{equation*}
\{ \mathcal I(A,{\textup{--}})\times_\phi G\times (\del\Delta^n\textup{h}ookrightarrow\Delta^n) : H\subset\Sigma_A, \phi\colon H\to G,n\ge0\},
\end{equation*}
where $\times_\phi$ denotes the quotient of the ordinary product by the diagonal of the right action of $H$ on $\mathcal I(A,{\textup{--}})$ via its tautological action on $A$ and the right action on $G$ via $g.h=g\phi(h)$.
Finally, filtered colimits of $G$-global level weak equivalences are again $G$-global level weak equivalences.
\begin{proof}
See \cite[Proposition~1.4.3]{g-global}.
\end{proof}
\end{prop}
Just like for $G$-symmetric spectra, we will now Bousfield localize this to get the model structure we are actually after. However, unlike for $G$-symmetric spectra, we can actually explicitly describe both the weak equivalences and the local objects. We start with the former:
\begin{constr}
Let $A$ be any set, possibly infinite, and let $X$ be an $\mathcal I$-simplicial set. Then we define
\begin{equation*}
X(A)\mathrel{:=}\mathop{\textup{colim}}\nolimits\limits_{B\subset A\text{ finite}} X(B)
\end{equation*}
with transition maps induced by functoriality of $X$. This becomes a functor in $X$ in the obvious way; in particular, if $G$ acts on $X$, then $X(A)$ becomes naturally a $G$-simplicial set.
In addition, the monoid $\textup{End}(A)$ of self-maps of $A$ acts naturally on the above by permuting the terms of the colimit. Thus, if $A$ is an $H$-set and $X$ is a $G$-$\mathcal I$-simplicial set, then $X(A)$ becomes an $(H\times G)$-simplicial set.
\end{constr}
\begin{defi}
A map $f\colon X\to Y$ of $G$-$\mathcal I$-simplicial sets is called a \emph{$G$-global weak equivalence} if for every finite group $H$ and some (hence any) complete $H$-set universe $\mathcal U_H$ the induced map $f(\mathcal U_H)$ is a $\mathcal G_{H,G}$-equivariant weak equivalence, or equivalently (replacing $H$ by a subgroup if necessary) for every $\phi\colon H\to G$ the map $(\phi^* f)(\mathcal U_H)$ is an $H$-equivariant weak equivalence.
\end{defi}
Next, we come to the analogue of the notion of an $\Omega$-spectrum in this setting:
\begin{defi}
A $G$-$\mathcal I$-simplicial set is called \emph{static} if for every finite group $H$ and all finite \emph{faithful} $H$-sets $A\subset B$ the map $X(A)\to X(B)$ induced by the inclusion is a $\mathcal G_{H,G}$-weak equivalence.
\end{defi}
\begin{thm}\label{thm:I-G-glob}
The $G$-global level model structure on $\cat{$\bm G$-$\bm{\mathcal I}$-SSet}$ admits a Bousfield localization with weak equivalences the $G$-global weak equivalences. Its fibrant objects are precisely the level fibrant \emph{static} $G$-$\mathcal I$-simplicial sets.
This model structure is again combinatorial (with the same generating cofibrations), simplicial, proper, and filtered colimits in it are homotopical.
\begin{proof}
See \cite[Theorem~1.4.30]{g-global}.
\end{proof}
\end{thm}
We will also need the following injective variant of the above model structure:
\begin{thm}\label{thm:I-inj-G-glob}
There is a unique model structure on $\cat{$\bm G$-$\bm{\mathcal I}$-SSet}$ with weak equivalences the $G$-global weak equivalences and cofibrations the injective cofibrations. We call this the \emph{injective $G$-global model structure}. It is combinatorial, simplicial, proper, and filtered colimits in it are homotopical.
\begin{proof}
See~\cite[Theorem~1.4.37]{g-global}.
\end{proof}
\end{thm}
\begin{rk}
The category $\cat{$\bm G$-$\bm I$-SSet}$ of $G$-objects in $\textup{Fun}(I,\cat{SSet})$ also carries a $G$-global level model structure analogous to Proposition~\ref{prop:I-g-glob-level} and this admits a Bousfield localization with the static objects as local objects such that the resulting model category is Quillen equivalent to $\cat{$\bm G$-$\bm{\mathcal I}$-SSet}$, see~\cite[Theorem~1.4.31]{g-global}. However, the weak equivalences of this model structure are somewhat complicated (similarly to the situation for symmetric spectra), and in particular they cannot just be checked by evaluation at complete $H$-set universes. The passage from $I$ to $\mathcal I$ is precisely what eliminates this subtlety, which is why the above model will be more convenient for us.
In addition to these, \cite[Chapter~1]{g-global} also studies various models of $G$-global homotopy theory based on a certain monoid $\mathcal M$ and the simplicial monoid $E\mathcal M$ obtained via Construction~\ref{constr:indiscrete} from this, that are related to the above via (zig-zags of) Quillen equivalences.
\end{rk}
As promised, the above $G$-global model structures support a rich `change of group' calculus:
\begin{prop}\label{prop:I-functoriality-general}
Let $\alpha\colon G\to G'$ be any group homomorphism. Then the restriction $\alpha^*\colon\cat{$\bm{G'}$-$\bm{\mathcal I}$-SSet}\to\cat{$\bm G$-$\bm{\mathcal I}$-SSet}$ is homotopical and we have Quillen adjunctions
\begin{align*}
\alpha_!\colon\cat{$\bm{G}$-$\bm{\mathcal I}$-SSet}_\textup{$G$-global}&\rightleftarrows\cat{$\bm{G'}$-$\bm{\mathcal I}$-SSet}_\textup{$G'$-global} :\!\alpha^*\\
\alpha^*\colon\cat{$\bm{G'}$-$\bm{\mathcal I}$-SSet}_\textup{injective $G'$-global}&\rightleftarrows\cat{$\bm{G}$-$\bm{\mathcal I}$-SSet}_\textup{injective $G$-global} :\!\alpha_*.
\end{align*}
\begin{proof}
See \cite[Lemma~1.4.40 and Corollary~1.4.41]{g-global}.
\end{proof}
\end{prop}
\begin{prop}\label{prop:I-functoriality-injective}
Let $\alpha\colon G\to G'$ be an \emph{injective} homomorphism. Then also
\begin{align*}
\alpha_!\colon\cat{$\bm{G}$-$\bm{\mathcal I}$-SSet}_\textup{injective $G$-global}&\rightleftarrows\cat{$\bm{G'}$-$\bm{\mathcal I}$-SSet}_\textup{injective $G'$-global} :\!\alpha^*\\
\alpha^*\colon\cat{$\bm{G'}$-$\bm{\mathcal I}$-SSet}_\textup{$G'$-global}&\rightleftarrows\cat{$\bm{G}$-$\bm{\mathcal I}$-SSet}_\textup{$G$-global} :\!\alpha_*.
\end{align*}
are Quillen adjunctions.
\begin{proof}
See \cite[Lemmas~1.4.42 and~1.4.43]{g-global}.
\end{proof}
\end{prop}
As every $G$-$\mathcal I$-simplicial set is injectively cofibrant, the above implies via Ken Brown's Lemma that $\alpha_!$ is homotopical for \emph{injective} $\alpha$. The following generalization of this (which makes precise that `free quotients are homotopical') will be a key input in many arguments, see in particular~Example~\ref{ex:S-Sp-global}.
\begin{prop}\label{prop:I-free-quotients}
Let $\alpha\colon G\to G'$ be any homomorphism and let $f\colon X\to Y$ be a $G$-global weak equivalence such that $\ker(\alpha)$ acts levelwise freely on $X$ and $Y$. Then $\alpha_!f$ is a $G'$-global level weak equivalence.
\begin{proof}
We factor $f$ in the $G$-global model structure as an acyclic cofibration $j\colon X\to Z$ followed by a fibration $p$ (automatically acyclic). Then $\alpha_!j$ is a $G'$-global weak equivalence by Proposition~\ref{prop:I-functoriality-general}, so it suffices to show that also $\alpha_!p$ is a weak equivalence; we will show that it is even a $G'$-global level weak equivalence.
To this end, we observe that for any generating cofibration $i$ and every finite set $A$ the map $i(A)$ is a cofibration in the $\mathcal G_{\Sigma_A,G}$-equivariant model structure since $G$ acts freely on $\mathcal I(B,A)\times_\phi G$ for every finite faithful $H$-set $B$ and homomorphism $\phi\colon H\to G$. As evaluation at $A$ is cocontinuous, we see that the claim holds more generally for all cofibrations, and in particular for the above map $j$. Thus, $\ker(\alpha)$ also acts levelwise freely on $Z$; the claim therefore follows by applying \cite[Proposition~1.1.22]{g-global} levelwise (with $M=\Sigma_A$ and $\mathcal E=\mathcal A\ell\ell$).
\end{proof}
\end{prop}
\subsection{Stable \texorpdfstring{$\bm G$}{G}-global homotopy theory} Finally, we come to models of \emph{stable} $G$-global homotopy theory \cite[Chapter~3]{g-global}; we restrict ourselves to the basics here and will recall further constructions and results (in particular monoidal properties and the tensoring over $G$-global spaces) later when needed.
\subsubsection{Model structures} On the pointset level, our models will again be simply given by symmetric spectra with a $G$-action, and we once more start with suitable level model structures \cite[Propositions~3.1.20 and~3.1.23]{g-global}:
\begin{prop}
There is a unique model structure on $\cat{$\bm G$-Spectra}$ in which a map is a weak equivalence or fibration if and only if $f(A)$ is a $\mathcal G_{\Sigma_A,G}$-weak equivalence or fibration, respectively, for every $A\in\bm\Sigma$. We call this the \emph{$G$-global projective level model structure} and its weak equivalences the \emph{$G$-global level weak equivalences}. It is proper, simplicial, combinatorial with generating cofibrations
\begin{equation*}
\{\bm\Sigma(A,{\textup{--}})\smashp_H G_+\smashp(\del\Delta^n\textup{h}ookrightarrow\Delta^n)_+ : A\in\bm\Sigma, H\in\mathcal G_{\Sigma_A,G},n\ge0\},
\end{equation*}
and filtered colimits in it are homotopical.\qed
\end{prop}
\begin{prop}
There is a unique model structure on $\cat{$\bm G$-Spectra}$ in which a map is a weak equivalence or fibration if and only if $f(A)$ is a weak equivalence or fibration, respectively, in the \emph{injective} $\mathcal G_{\Sigma_A,G}$-model structure. We call this the \emph{$G$-global flat level model structure}; its weak equivalences are precisely the $G$-global level weak equivalences and its cofibrations are the flat cofibrations. This model structure is proper, simplicial, combinatorial with generating cofibrations
\begin{equation*}
\{\bm\Sigma(A,{\textup{--}})\smashp_H G_+\smashp(\del\Delta^n\textup{h}ookrightarrow\Delta^n)_+ : A\in\bm\Sigma, H\subset \Sigma_A\times G,n\ge0\},
\end{equation*}
and filtered colimits in it are homotopical.\qed
\end{prop}
\begin{warn}
Beware that the notion of \emph{$G$-global level weak equivalence} differs from the \emph{$G$-equivariant level weak equivalences}: the former is a condition on the $H$-fixed points for $H\in\mathcal G_{\Sigma_A,G}$ for varying $A$, while the latter is a condition for $H\in\mathcal G_{G,\Sigma_A}$.
\end{warn}
\begin{defi}
A $G$-spectrum $X$ is called a \emph{$G$-global $\Omega$-spectrum} if for all finite groups $H$ and all finite \emph{faithful} $H$-sets $A\subset B$ the derived adjoint structure map
\begin{equation*}
X(A)\to\cat{R}\Omega^{B\setminus A}X(B)
\end{equation*}
is a $\mathcal G_{H,G}$-weak equivalence.
\end{defi}
Again, for a $G$-spectrum that is fibrant in either of the above level model structures, the derived adjoint structure map is already modelled by the ordinary one.
\begin{defi}
A map $f$ in $\cat{$\bm G$-Spectra}$ is called a \emph{$G$-global weak equivalence} if $\phi^*f$ is an $H$-equivariant weak equivalence for every finite group $H$ and every homomorphism $\phi\colon H\to G$.
\end{defi}
\begin{thm}
The $G$-global projective level model structure admits a Bousfield localization with weak equivalences the $G$-global weak equivalences. We call this the \emph{$G$-global projective model structure}; its fibrant objects are precisely those $G$-global $\Omega$-spectra that are fibrant in the $G$-global projective level model structure.
This model structure is again combinatorial (with the same generating cofibrations as before), simplicial, proper, and filtered colimits in it are homotopical.
\begin{proof}
See \cite[Theorem~3.1.41 and Proposition~3.1.47]{g-global}.
\end{proof}
\end{thm}
\begin{thm}
The $G$-global flat level model structure admits a Bousfield localization with weak equivalences the $G$-global weak equivalences. We call this the \emph{$G$-global flat model structure}; its fibrant objects are precisely those $G$-global $\Omega$-spectra that are fibrant in the $G$-global flat level model structure.
This model structure is again combinatorial (with the same generating cofibrations as before), simplicial, proper, and filtered colimits in it are homotopical.
\begin{proof}
See \cite[Theorem~3.1.40 and Proposition~3.1.47]{g-global}.
\end{proof}
\end{thm}
\begin{rk}
For $G=1$ the above two model structures agree and recover Hausmann's \emph{global model structure} \cite[Theorem~2.18]{hausmann-global}.
\end{rk}
Again, there is also an injective version of the above model structures:
\begin{thm}
There is a unique model structure on $\cat{$\bm G$-Spectra}$ with weak equivalences the $G$-global weak equivalences and cofibrations the injective cofibrations. We call this the \emph{$G$-global injective model structure}. It is combinatorial, simplicial, proper, and filtered colimits in it are homotopical.
\begin{proof}
See \cite[Corollary~3.1.46]{g-global}.
\end{proof}
\end{thm}
\subsubsection{Change of group adjunctions} As promised (and unlike their equivariant counterparts), these model structures behave nicely under changing the group:
\begin{prop}\label{prop:sp-functoriality-general}
Let $\alpha\colon G\to G'$ be any homomorphism. Then the restriction $\alpha^*\colon\cat{$\bm{G'}$-Spectra}\to\cat{$\bm G$-Spectra}$ is homotopical and we have Quillen adjunctions
\begin{align*}
\alpha_!\colon\cat{$\bm{G}$-Spectra}_\textup{$G$-gl.~proj.}&\rightleftarrows\cat{$\bm{G'}$-Spectra}_\textup{$G'$-gl.~proj.} :\!\alpha^*\\
\alpha^*\colon\cat{$\bm{G'}$-Spectra}_\textup{$G'$-gl.~flat} &\rightleftarrows\cat{$\bm{G}$-Spectra}_\textup{$G$-gl.~flat} :\!\alpha_*\\
\alpha^*\colon\cat{$\bm{G'}$-Spectra}_\textup{$G'$-gl.~inj.}&\rightleftarrows\cat{$\bm{G}$-Spectra}_\textup{$G$-gl.~inj.} :\!\alpha_*.
\end{align*}
\begin{proof}
Everything except for the statement about the injective model structures appears in \cite[Lemmas~3.1.49 and~3.1.50]{g-global}. For the final statement, it then only remains to show that $\alpha^*$ preserves injective cofibrations, which is immediate from the definition.
\end{proof}
\end{prop}
\begin{prop}\label{prop:sp-functoriality-injective}
Let $\alpha\colon G\to G'$ be an \emph{injective} homomorphism. Then we also have Quillen adjunctions
\begin{align*}
\alpha^*\colon\cat{$\bm{G'}$-Spectra}_\textup{$G'$-gl.~proj.}&\rightleftarrows\cat{$\bm{G}$-Spectra}_\textup{$G$-gl.~proj.} :\!\alpha_*\\
\alpha_!\colon\cat{$\bm{G}$-Spectra}_\textup{$G$-gl.~flat} &\rightleftarrows\cat{$\bm{G'}$-Spectra}_\textup{$G'$-gl.~flat} :\!\alpha^*\\
\alpha_!\colon\cat{$\bm{G}$-Spectra}_\textup{$G$-gl.~inj.}&\rightleftarrows\cat{$\bm{G'}$-Spectra}_\textup{$G'$-gl.~inj.} :\!\alpha^*.
\end{align*}
\begin{proof}
The latter two statements are \cite[Propositions~3.1.52 and~3.1.53]{g-global}. For the first statement it then only remains (as $\alpha^*$ is homotopical) that $\alpha_*$ sends acyclic fibrations of the $G$-global projective (level) model structure to acyclic fibrations in the $G$-global projective (level) model structure. Using that acyclic fibrations are defined levelwise and adjoining, this amounts to saying that
\begin{equation*}
(\Sigma_A\times\alpha)^*\colon\cat{$\bm{(\Sigma_A\times G')}$-SSet}_{\mathcal G_{\Sigma_A,G'}}\to \cat{$\bm{(\Sigma_A\times G)}$-SSet}_{\mathcal G_{\Sigma_A,G}}
\end{equation*}
preserves cofibrations for every finite set $A$. This is immediate from Lemma~\ref{lemma:graph-target}.
\end{proof}
\end{prop}
Again, suitably free quotients are homotopical in our setting:
\begin{prop}\label{prop:free-quotient-spectra}
Let $\alpha\colon G\to G'$ be any homomorphism, and let $f\colon X\to Y$ be a $G$-global weak equivalence in $\cat{$\bm G$-Spectra}$ such that $\ker(\alpha)$ acts levelwise freely on $X$ and $Y$ outside the basepoint. Then $\alpha_!f$ is a $G'$-global weak equivalence.
\begin{proof}
See \cite[Proposition~3.1.54]{g-global}.
\end{proof}
\end{prop}
\subsubsection{The smash product} The usual smash product of symmetric spectra gives us a smash product on $\cat{$\bm G$-Spectra}$ by pulling through the $G$-actions. This is compatible with the above model structures:
\begin{thm}\label{thm:smash-g-global}
The smash product defines left Quillen bifunctors
\begin{align*}
\cat{$\bm G$-Spectra}_\textup{$G$-global flat}\times\cat{$\bm G$-Spectra}_\textup{$G$-global flat}&\to\cat{$\bm G$-Spectra}_\textup{$G$-global flat}\\
\cat{$\bm G$-Spectra}_\textup{$G$-global proj.}\times\cat{$\bm G$-Spectra}_\textup{$G$-global flat}&\to\cat{$\bm G$-Spectra}_\textup{$G$-global proj.}
\end{align*}
\end{thm}
Note that in the second adjunction, indeed only \emph{one} of the input factors is equipped with the projective model structure (the corresponding statement where both factors are equipped with the projective model structures follows immediately).
\begin{proof}
See \cite[Propositions~3.1.63 and~3.1.64]{g-global}.
\end{proof}
In particular, smashing with a fixed flat $G$-spectrum is left Quillen for either of the above model structures, so it preserves weak equivalences between cofibrant objects by Ken Brown's Lemma. The following (easy) $G$-global analogue of the equivariant \emph{Flatness Theorem} \cite[Proposition~6.2]{hausmann-equivariant} strengthens this result:
\begin{prop}\label{prop:flatness-theorem}
\begin{enumerate}
\item Let $X$ be any $G$-spectrum. Then $X\smashp{\textup{--}}$ preserves $G$-global weak equivalences between \emph{flat} $G$-spectra.
\item Let $X$ be a \emph{flat} $G$-spectrum. Then $X\smashp{\textup{--}}$ preserves $G$-global weak equivalences.
\end{enumerate}
\begin{proof}
See~\cite[Proposition~3.1.62]{g-global}.
\end{proof}
\end{prop}
\subsubsection{Relation to stable equivariant homotopy theory} On the pointset level, $G$-global and $G$-equivariant spectra are the same objects, and every $G$-global weak equivalence is in particular a $G$-equivariant weak equivalence. Thus, the identity of $\cat{$\bm G$-Spectra}$ descends to exhibit the $G$-equivariant stable homotopy category as a localization of the $G$-global one. This localization admits both adjoints (fully faithful for formal reasons), which has the following model categorical manifestation:
\begin{prop}\label{prop:equivariant-vs-global}
The adjunctions
\begin{align*}
{\textup{id}}\colon\cat{$\bm G$-Spectra}_\textup{$G$-equivariant projective}&\rightleftarrows\cat{$\bm G$-Spectra}_\textup{$G$-global flat} :\!{\textup{id}}\\
{\textup{id}}\colon\cat{$\bm G$-Spectra}_\textup{$G$-global flat}&\rightleftarrows\cat{$\bm G$-Spectra}_\textup{$G$-equivariant flat} :\!{\textup{id}}
\end{align*}
are Quillen adjunctions.
\begin{proof}
See \cite[Proposition~3.3.1 and Corollary~3.3.3]{g-global}.
\end{proof}
\end{prop}
\subsubsection{Suspension spectra} Finally, we come to the relation between unstable and stable $G$-global homotopy theory:
\begin{constr}
Let $X$ be an $\mathcal I$-simplicial set (or an $I$-simplicial set). We define a symmetric spectrum $\Sigma^\bullet_+X$ via $(\Sigma^\bullet_+X)(A) = S^A\smashp X(A)_+$; if $i\colon A\to B$ is an injection of finite sets, then the structure map is given by
\begin{equation*}
S^{B\setminus i(A)}\smashp (S^A\smashp X(A)_+)\cong S^B\smashp X(A)_+\xrightarrow{S^B\smashp X(i)_+} S^B\smashp X(B)_+
\end{equation*}
where the unlabelled isomorphism is induced by $i$. This becomes an enriched functor in $X$ in the obvious way, which we then lift to a functor
\begin{equation}\label{eq:suspension}
\Sigma^\bullet_+\colon\cat{$\bm G$-$\bm{\mathcal I}$-SSet}\to\cat{$\bm G$-Spectra}
\end{equation}
by pulling through the $G$-actions.
\end{constr}
\begin{prop}\label{prop:suspension-loop-G-gl}
The functor $(\ref{eq:suspension})$ admits a simplicial right adjoint $\Omega^\bullet$. We have Quillen adjunctions
\begin{align*}
\Sigma^\bullet_+\colon\cat{$\bm G$-$\bm{\mathcal I}$-SSet}_\textup{$G$-gl.~proj.}&\rightleftarrows\cat{$\bm G$-Spectra}_\textup{$G$-gl.~proj.} :\!\Omega^\bullet\\
\Sigma^\bullet_+\colon\cat{$\bm G$-$\bm{\mathcal I}$-SSet}_\textup{$G$-gl.~inj.}&\rightleftarrows\cat{$\bm G$-Spectra}_\textup{$G$-gl.~inj.} :\!\Omega^\bullet,
\end{align*}
and in particular $\Sigma^\bullet_+$ is homotopical.
\end{prop}
Beware that \cite{g-global} uses $\Omega^\bullet$ for the corresponding right adjoint in $I$-simplicial sets instead and introduces more complicated notation for the above right adjoint in $\mathcal I$-simplicial sets. As we will only need the latter, we have decided to change notation here.
\begin{proof}
See \cite[Corollary~3.2.6 and Remark~3.2.7]{g-global}.
\end{proof}
\begin{rk}\label{rk:Omega-bullet-on-fibrant}
We briefly remark on the above right adjoint. As a functor from $\cat{$\bm I$-SSet}$, $\Sigma^\bullet_+$ has a left adjoint $\omega^\bullet$ defined via $(\omega^\bullet X)(A)=\Omega^AX(A)$ with the evident functoriality in each variable. The functor $\Omega^\bullet$ is accordingly obtained by postcomposing $\omega^\bullet$ with the right adjoint $\cat{$\bm I$-SSet}\to\cat{$\bm{\mathcal I}$-SSet}$ of the forgetful functor, and this as usual gives the right adjoint for general $G$ by pulling through the action. As the forgetful functor $\cat{$\bm G$-$\bm{\mathcal I}$-SSet}\to\cat{$\bm G$-$\bm I$-SSet}$ is the left half of a Quillen equivalence for the projective $G$-global model structures \cite[Theorem~1.4.49]{g-global}, we obtain a natural $G$-global level weak equivalence between the restriction of
\begin{equation*}
\cat{$\bm G$-Spectra}\xrightarrow{\Omega^\bullet}\cat{$\bm{G}$-$\bm{\mathcal I}$-SSet}\xrightarrow{\textup{forget}}\cat{$\bm G$-$\bm I$-SSet}
\end{equation*}
to projectively fibrant objects and the corresponding restriction of $\omega^\bullet$; this is all we will need about $\Omega^\bullet$ below.
\end{rk}
\section{Global model categories}\label{sec:global-model-cat}
In this section, we introduce the framework of \emph{global model categories} which will then in particular allow us later to express the universal property of the passage from global spaces to global spectra.
\subsection{Preglobal model categories} We begin by describing a slightly more general notion:
\begin{defi}
A \emph{preglobal model category} consists of a locally presentable category $\mathscr C$, which is enriched, tensored, and cotensored over $\cat{SSet}$, together with two model structures on the category $G\text{-}\mathscr C$ of $G$-objects in $\mathscr C$ for each finite group $G$, called the \emph{projective} and \emph{flat $G$-global model structures}, such that the following conditions are satisfied:
\begin{enumerate}
\item The projective and the flat $G$-global model structure have the same weak equivalences (which we call the \emph{$G$-global weak equivalences}) and the adjunction ${\textup{id}}\colon G\text-\mathscr C_{\text{proj}}\rightleftarrows G\text{-}\mathscr C_\text{flat} :\!{\textup{id}}$ is a Quillen adjunction (i.e.~every projective cofibration is also a flat cofibration, or equivalently every fibration of the flat model structure is also a fibration in the projective one).
\item Both the projective and the flat model structure on $G\text{-}\mathscr C$ are left proper, combinatorial, and simplicial.
\item For every homomorphism $\alpha\colon H\to G$ of finite groups the restriction functor $\alpha^*\colon G\text{-}\mathscr C\to H\text-\mathscr C$ preserves weak equivalences, flat cofibrations, and projective fibrations. Moreover, if $\alpha$ is injective, then $\alpha^*$ also preserves projective cofibrations and flat fibrations.
In particular, $\alpha^*$ is always left Quillen for the flat model structures and right Quillen for the projective ones; if $\alpha$ is injective, then $\alpha^*$ is also right Quillen for the flat model structures and left Quillen for the projective ones.
\end{enumerate}
\end{defi}
\begin{ex}
Let $\mathscr C=\cat{$\bm{\mathcal I}$-SSet}$. For every finite group $G$, we can equip $\cat{$\bm G$-$\bm{\mathcal I}$-SSet}$ with the \emph{$G$-global model structure} and the \emph{injective $G$-global model structure} as projective and flat model structures, respectively, and these are left proper, combinatorial, and simplicial (Theorems~\ref{thm:I-G-glob} and~\ref{thm:I-inj-G-glob}). Propositions~\ref{prop:I-functoriality-general} and~\ref{prop:I-functoriality-injective} then show that this yields a preglobal model category, which we denote by $\glo{GlobalSpaces}$ and call the \emph{preglobal model category of global spaces}.
\end{ex}
\begin{ex}\label{ex:global-spectra}
Let $\mathscr C=\cat{Spectra}$ be the category of symmetric spectra. For every finite $G$, we can equip $\cat{$\bm G$-Spectra}$ with the projective and injective $G$-global model structure. Propositions~\ref{prop:sp-functoriality-general} and~\ref{prop:sp-functoriality-injective} then show that this yields a preglobal model category $\glo{GlobalSpectra}$, which we call the \emph{preglobal model category of global spectra}.
\end{ex}
\begin{rk}
By the same argument we could have used the \emph{flat} instead of the \emph{injective} model structures to yield another preglobal model category (which is the motivation for the above terminology). However, for now the above choices will be more convenient; later, when we consider algebraic structures on global spectra, we will see a return of the flat model structures (or more precisely their `positive' cousins).
\end{rk}
\begin{ex}\label{ex:shifts}
Let $\underline{\mathscr C}$ be a preglobal model category and let $G$ be a finite group. Then we have a preglobal model category $\underline{G\text{-}\mathscr C}$ with underlying category $G\text{-}\mathscr C$ and with the model structures on $G'\text{-}(G\text{-}\mathscr C)$ transported from $(G'\times G)\text{-}\mathscr C$ along the evident isomorphism of categories. In particular, specializing to the previous examples we get preglobal model categories of $G$-global spaces and of $G$-global spectra.
\end{ex}
\begin{ex}\label{ex:exotic}
Finally, we introduce a more exotic example. Let $\mathcal F$ be a \emph{global family}, i.e.~a non-empty collection of finite groups closed under subquotients (hence in particular under isomorphisms). We make $\mathscr E\mathrel{:=}\cat{SSet}$ into a preglobal model category as follows: for every finite group $G$, we write $G\cap\mathcal F$ for the family of subgroups of $G$ that belong to $\mathcal F$, and we equip $\cat{$\bm G$-SSet}$ with the $(G\cap\mathcal F)$-model structure and the injective $(G\cap\mathcal F)$-model structure, respectively.
If now $\alpha\colon G\to G'$ is any group homomorphism, then $\alpha^*$ clearly preserves injective cofibrations, and it preserves weak equivalences and projective fibrations as $(\alpha^*X)^H=X^{\alpha(H)}$ and $\alpha(H)$ is a quotient of $H$. Moreover, if $\alpha$ is injective, then $\alpha^*$ also preserves projective cofibrations as for any $G'$-simplicial set $X$ and any simplex $x$ the isotropy groups satisfy $\textup{Iso}_{\alpha^*X}(x)=\alpha^{-1}(\textup{Iso}_X(x))$, which is isomorphic to a subgroup of $\textup{Iso}_X(x)$. Finally, $\alpha_!$ always preserves injective cofibrations (as quotients by group actions preserve injections of sets), and if $\alpha$ is injective, then this moreover sends $(G\cap\mathcal F)$-weak equivalences to $(G'\cap\mathcal F)$-weak equivalences by \cite[Proposition~1.1.18]{g-global} (for $M=1$), so it is left Quillen for the injective equivariant model structures as claimed.
We call the resulting preglobal model category $\ul{\mathscr E}_{\mathcal F}$ the \emph{preglobal model category of $\mathcal F$-equivariant spaces} (the reader is invited to choose for themselves whether $\mathscr E$ is an abbreviation for `equivariant' or `exotic').
\end{ex}
The simplicial enrichment, tensoring, and cotensoring of $\mathscr C$ make $G\text{-}\mathscr C$ not only enriched, tensored, and cotensored over $\cat{SSet}$ but over all of $\cat{$\bm G$-SSet}$: the tensoring and cotensoring are just given by equipping the non-equivariant tensoring or cotensoring with the diagonal and conjugation $G$-action, and the enrichment is given by taking the mapping space in $\mathscr C$ (i.e.~without regards to $G$-actions) and equipping it with the conjugation $G$-action. In the same way, we obtain for any finite group $H$ a functor ${\textup{--}}\times{\textup{--}}\colon\cat{$\bm{(G\times H)}$-SSet}\times G\text-\mathscr C\to (G\times H)\text-\mathscr C$ generalizing the tensoring.
\begin{lemma}\label{lemma:g-sset-enriched}\label{lemma:tensoring-generalized}
Let $G,H$ be finite groups. Then the above functors
\begin{align*}
\cat{$\bm{(G\times H)}$-SSet}_{\mathcal G_{G,H}}\times G\text-\mathscr C_\textup{proj}&\to (G\times H)\text-\mathscr C_\textup{proj}\\
\cat{$\bm{(G\times H)}$-SSet}\times G\text-\mathscr C_\textup{flat}&\to (G\times H)\text-\mathscr C_\textup{flat}
\end{align*}
are left Quillen bifunctors.
In particular, specializing to $H=1$, both the projective and flat model structure on $G\text{-}\mathscr C$ are enriched in the model categorical sense over $\cat{$\bm G$-SSet}$ with the usual equivariant model structure.
\begin{proof}
Let us consider the case of the projective model structures first. By adjointness, it is enough to show that
\begin{align*}
G\text-\mathscr C^{\textup{op}}_\textup{proj}\times (G\times H)\text-\mathscr C_\textup{proj}&\to\cat{$\bm{(G\times H)}$-SSet}_{\mathcal G_{G,H}}\\
(X,Y)&\mathord{\textup{maps}}to \mathord{\textup{maps}}(\textup{triv}_H X,Y)
\end{align*}
is a right Quillen bifunctor. By definition of the model structure on the right this amounts to saying that for every $K\subset G$ and $\phi\colon K\to H$ the functor $(X,Y)\mathord{\textup{maps}}to \mathord{\textup{maps}}^K(i^*X,(i,\phi)^*Y)$ to $\cat{SSet}$ is a right Quillen bifunctor, where $i\colon K\textup{h}ookrightarrow G$ is the inclusion. However, as $(i,\phi)^*$ is right Quillen and $i^*$ is left Quillen for the projective model structures, this follows at once from the fact that $K\text{-}\mathscr C_\textup{proj}$ is a simplicial model category.
For the flat model structures, we observe that $\text{triv}_H\colon G\text-\mathscr C_\textup{flat}\to (G\times H)\text-\mathscr C_\textup{flat}$ is left Quillen. Replacing $G$ by $G\times H$ if necessary, we may therefore assume without loss of generality that $H=1$. However, in this case we are similarly reduced by adjointness to proving that $X,Y\mathord{\textup{maps}}to\mathord{\textup{maps}}^K(X,Y)$ is a right Quillen bifunctor for any subgroup $K\subset G$. This in turns follows again from the fact that the restriction $G\text-\mathscr C_\textup{flat}\to K\text-\mathscr C_\text{flat}$ is both left and right Quillen and that $K\text-\mathscr C_\textup{flat}$ is simplicial.
\end{proof}
\end{lemma}
\subsection{Global model categories} In order to support a good theory of \emph{genuine stabilizations} we will need one extra condition in addition to the axioms of a preglobal model category. To motivate this we first recall:
\begin{lemma}\label{lemma:Beck-Chevalley}
Let
\begin{equation}\label{diag:pb-surj}
\begin{tikzcd}
A\arrow[d, "q"']\arrow[r, "g"] & B\arrow[d, "p"]\\
C\arrow[r, "f"'] & D
\end{tikzcd}
\end{equation}
be a pullback square of groups such that $p$ (whence also $q$) is surjective, and let $\mathscr C$ be a complete category. Then the Beck-Chevalley transformation
\begin{equation*}
\begin{tikzcd}
A\text-\mathscr C\arrow[d, "q_*"']\twocell[from=dr] & \arrow[l, "g^*"'] B\text-\mathscr C\arrow[d, "p_*"]\\
C\text-\mathscr C & \arrow[l, "f^*"] D\text-\mathscr C
\end{tikzcd}
\end{equation*}
(i.e.~the canonical mate
\begin{equation*}
f^*p_*\xrightarrow{\eta} q_*q^*f^*p_*=q_*g^*p^*p_*\xrightarrow{\epsilon} q_*g^*
\end{equation*}
of the identity transformation) is an isomorphism.
\begin{proof}
This is well-known (see e.g.~\cite[Proposition~11.6]{joyal-book} for a result in much greater generality), but also easy enough to prove directly: if $X\in B\text-\mathscr C$, then $f^*p_*X=X^{\ker p}$ as objects of $\mathscr C$, while $q_*g^*X=(g^*X)^{\ker q}$, and the Beck-Chevalley transformation is the unique map under $X$. The claim follows as $g(\ker q)=\ker p$ by virtue of $(\ref{diag:pb-surj})$ being a pullback.
\end{proof}
\end{lemma}
\begin{prop}\label{prop:global-tfae}
Let $\underline{\mathscr C}$ be a preglobal model category. Then the following are equivalent:
\begin{enumerate}
\item For every pullback square $(\ref{diag:pb-surj})$ of finite groups in which all maps are surjections, the Beck-Chevalley transformation $f^*\circ\cat{R}p_*\Rightarrow \cat{R}q_*\circ g^*$ is an isomorphism in ${\textup{Ho}}(C\text-\mathscr C)$.
\item For every pullback square $(\ref{diag:pb-surj})$ of finite groups and surjections, the Beck-Chevalley transformation $\cat{L}g_!\circ q^*\Rightarrow p^*\circ\cat{L}f_!$ is an isomorphism in ${\textup{Ho}}(B\text-\mathscr C)$.
\item \label{item:gtfae-cofree-fixed-points} For every diagram
\begin{equation}\label{diag:span-global}
\begin{tikzcd}
C & \arrow[l, "q"'] A\arrow[r, "g"] & B
\end{tikzcd}
\end{equation}
of finite groups such that $\ker q\cap\ker g=1$, every fibrant $X\in B\text-\mathscr C_\textup{flat}$, and some (hence any) fibrant replacement $g^*X\to Y$ in $A\text-\mathscr C_\textup{flat}$ the induced map $q_*g^*X\to q_*Y$ is a $C$-global weak equivalence.
\item \label{item:gtfae-free-quotient} For every diagram $(\ref{diag:span-global})$, every cofibrant $X\in C\text{-}\mathscr C_\textup{proj}$, and some (hence any) cofibrant replacement $Y\to q^*X$ in $A\text-\mathscr C_\textup{proj}$ the induced map $g_!Y\to g_!q^*X$ is a $B$-global weak equivalence.
\end{enumerate}
\end{prop}
\begin{defi}\label{defi:global-model-cat}
A preglobal model category satisfying the equivalent conditions of Proposition~\ref{prop:global-tfae} is called a \emph{global model category}.
\end{defi}
\begin{proof}[Proof of Proposition~\ref{prop:global-tfae}]
The equivalence $(1)\Leftrightarrow(2)$ follows at once from the fact that the two maps in question are total mates of each other. We will now show that $(1)\Leftrightarrow(3)$; the argument that $(2)\Leftrightarrow(4)$ is then analogous.
For the proof of $(3)\Rightarrow(1)$ we consider a pullback as in $(\ref{diag:pb-surj})$ and we fix a fibrant object $X\in B\text-{\mathscr C}_\text{flat}$ and a fibrant replacement functor $\iota\colon{\textup{id}}\Rightarrow P$ for $A\text-\mathscr C_\text{flat}$. By naturality of $\iota$, we then have a commutative diagram
\begin{equation*}
\begin{tikzcd}
f^*p_*X\arrow[r,"\eta"] & q_*q^*f^*p_*X\arrow[d, "q_*\iota"'] \arrow[r,equal] & q_*g^*p^*p_*X\arrow[r, "q_*g^*\epsilon"]\arrow[d, "q_*\iota"] &[1em] q_*g^*X\arrow[d, "q_*\iota"]\\
& q_*Pq^*f^*p_*X\arrow[r,equal] & q_*Pg^*p^*p_*X\arrow[r, "q_*Pg^*\epsilon"'] & q_*Pg^*X
\end{tikzcd}
\end{equation*}
in $C\text{-}\mathscr C$. Here the top horizontal composite is an isomorphism by the previous lemma; moreover, the bottom composite $f^*p_*X\to q_*Pg^*X$ represents the Beck-Chevalley transformation $f^*\cat{R}p_*\Rightarrow \cat{R}q_*g^*$, so this is the map we want to be a $C$-global weak equivalence. However, the right hand vertical map is a $C$-global weak equivalence by assumption, so the claim follows by $2$-out-of-$3$.
Conversely, for the proof of $(1)\Rightarrow(3)$ we first observe that the above shows that for any pullback square $(\ref{diag:pb-surj})$ of surjections and any fibrant $X\in B\text{-}\mathscr C_\text{flat}$ the canonical map $q_*g^*X\to\cat{R}q_*g^*X$ is a $C$-global weak equivalence.
We will now use this to prove the special case of $(3)$ in which both $q$ and $g$ are surjective: namely, we set $D=A/\ker(g)\ker(q)$ and we define $p\colon B\to D$ as the composite
\begin{equation*}
B\xrightarrow[\cong]{g^{-1}} A/\ker(q)\twoheadrightarrow A/\ker(g)\ker(q)=D
\end{equation*}
and analogously we define a surjective homomorphism $f\colon C\to D$. We now claim that the commutative square
\begin{equation*}
\begin{tikzcd}
A\arrow[d, "q"']\arrow[r, "g"] & B\arrow[d, "p"]\\
C\arrow[r, "f"'] & D
\end{tikzcd}
\end{equation*}
of surjections is a pullback square; with this established, the above observation will then complete the proof of the special case.
To prove that this is indeed a pullback, first note that $(g,q)\colon A\to B\times_DC\subset B\times C$ is injective as $\ker(g,q)=\ker(g)\cap\ker(q)=1$. Thus, it only remains to prove surjectivity. For this we let $(b,c)\in B\times C$ with $p(b)=f(c)$; thus, if $a\in A$ with $g(a)=b$, then $fq(a)=pg(a)=p(b)=f(c)$, so $c=q(a)\bar x$ for some $\bar x\in\ker(f)$. By surjectivity of $q$, we can then write $c=q(ax)$ for some $x\in q^{-1}(\ker(f))=\ker(g)$; but we also have $g(ax)=g(a)g(x)=b$, so $ax$ is the desired preimage of $(b,c)\in B\times_DC$.
Now we can prove the general case of $(3)$: we factor $q$ as a surjection $q'\colon A\to C'$ followed by an injection $i\colon C'\to C$, and we factor $g$ as a surjection $g'\colon A\to B'$ followed by an injection $j\colon B'\to B$. If now $X\in B\text{-}\mathscr C_\text{flat}$ is fibrant, then also $j^*X\in B'\text-\mathscr C_{\text{flat}}$ is fibrant as $j$ is injective. Thus, if $g^*X=(g')^*j^*X\to Y$ is any fibrant replacement in $A\text{-}\mathscr C_\text{flat}$, then applying the above special case to $j^*X$ shows that $q'_*g^*X\to q'_*Y$ is a $C'$-global weak equivalence. The right hand side is fibrant in $C'\text-\mathscr C_\text{flat}$, and we claim that the left hand side is at least fibrant in $C'\text-\mathscr C_\text{proj}$; as $i_*$ is right Quillen for the projective model structures by injectivity, Ken Brown's Lemma will then show that also the induced map $i_*q'_*g^*X\to i_*q'_*Y$ is a $C$-global weak equivalence, and as this agrees with $q_*g^*X\to q_*Y$ up to conjugation by isomorphisms, this will then complete the proof of the proposition.
To prove the claim, we note that $\ker(g')\cap\ker(q')=\ker(g)\cap\ker(q)=1$, so the above argument yields a pullback square
\begin{equation*}
\begin{tikzcd}
A\arrow[r, "g'"]\arrow[d,"q'"'] & B'\arrow[d, "f'"]\\
C'\arrow[r, "p'"'] & D'
\end{tikzcd}
\end{equation*}
of finite groups and surjections. Thus, another application of Lemma~\ref{lemma:Beck-Chevalley} shows $q'_*g^*X=q'_*(g')^*i^*X\cong (p')^*f'_*i^*X$. However, $f'_*i^*X$ is fibrant in $D'\text-\mathscr C_\text{flat}$, hence in particular in $D'\text-\mathscr C_\text{proj}$, so $(p')^*f'_*i^*X$ is fibrant in $C'\text-\mathscr C_\text{proj}$ as desired.
\end{proof}
\begin{rk}
The above argument more generally shows that the Beck-Chevalley transformation $f^*\cat{R}p_*\Rightarrow\cat{R}q_*g^*$ is an isomorphism as soon as $p$ (whence also $q$) is surjective, and similarly $\cat{L}g_!q^*\Rightarrow p^*\cat{L}f_!$ is an isomorphism as soon as $f$ is surjective. Conversely, in the latter two conditions we could equivalently restrict to the case that $g$ and $q$ are surjective.
\end{rk}
\begin{rk}
The first two formulations above are the `morally correct ones,' and they correspond to the notions of \emph{global continuity} and \emph{global cocontinuity} in parameterized higher category theory \cite{elements-param, martini-wolf}. In contrast to that, Condition~$(\ref{item:gtfae-cofree-fixed-points})$ is the statement we will actually use later (namely, in the proof of Theorem~\ref{thm:sp-global-model-cat}), while the final formulation is the one that is easiest to check in our examples below. Intuitively, we can think of a projectively cofibrant object in $G\text-\mathscr C$ as a flat one for which the $G$-action is `free' in some sense (cf.~for example Lemma~\ref{lemma:tensoring-generalized}); the final formulation of the above axiom can then be viewed as a very abstract incarnation of the slogan that `free quotients are homotopical.'
\end{rk}
\begin{ex}\label{ex:S-Sp-global}
Both $\glo{GlobalSpaces}$ and $\glo{GlobalSpectra}$ are global model categories: this follows from Propositions~\ref{prop:I-free-quotients} and~\ref{prop:free-quotient-spectra}, respectively: if $X\in B\text-\mathscr C$ is projectively cofibrant, then $B$ acts freely on it, hence $\ker(q)$ acts freely on $g^*X$ as $g$ is injective when restricted to $\ker(q)$.
\end{ex}
\begin{ex}
Any shift of a global model category is again a global model category. This follows immediately from the third formulation.
\end{ex}
\begin{nex}
Let $p$ be a prime and let $\mathcal F$ be a global family of groups containing $\mathbb Z/p$ but not $\mathbb Z/p\times\mathbb Z/p$, for example the global family of groups of order at most $p$. We claim that the preglobal model category $\ul{\mathscr E}_{\mathcal F}$ from Example~\ref{ex:exotic} is \emph{not} a global model category; in particular, the conditions of the above proposition are not vacuous.
To this end, we will show that Condition~$(\ref{item:gtfae-free-quotient})$ is not satisfied. We set $B=C=\mathbb Z/p$ and $A=B\times C$ with $q$ and $g$ the respective projections. Then the $(B\cap\mathcal F)$-model structure is just the $\mathcal A\ell\ell$-model structure, and in particular $X=*$ is cofibrant in it. We now let $Y\to *=g^*(*)$ be a cofibrant replacement in the $(A\cap\mathcal F)$-model structure, and we claim that $q_!(Y)=Y/B$ has non-connected $C$-fixed points, so it is in particularly not weakly equivalent to $q_!(*)=*$.
For this, let us observe that the isotropy $I_\sigma$ of a simplex $\sigma$ of $Y$ only depends on the class $[\sigma]$ in $Y/B$ (as $A$ is abelian). Moreover, $[\sigma]$ is $C$-fixed if and only if $I_\sigma$ contains an element of the form $(b,1)$ with $b\in B$ arbitrary. Conversely, for any $b\in B$ there exists a vertex $y_b$ of $Y$ with $I_{y_b}\ni (b,1)$: namely, $(b,1)$ generates a subgroup $K\subset A$ isomorphic to $\mathbb Z/p$, so $Y^{K}$ is weakly contractible, hence in particular non-empty.
We now claim that $[y_0]$ and $[y_1]$ cannot be joined by a sequence of $C$-fixed edges in $Y/B$. For this, let us consider any $C$-fixed edge $[e]$ with vertices $[x],[y]$. Then $I_x\supset I_e\subset I_y$. On the other hand, $I_e\not=1$ as $[e]$ is $C$-fixed, while $I_x,I_y\not=A$ as $Y^A=\varnothing$ since $A\notin\mathcal F$. Thus, it follows for cardinality reasons that all of the above inclusions are equalities, and in particular $I_x=I_y$.
Now assume $[y_0]$ and $[y_1]$ were actually connected by a sequence of $C$-fixed edges. Then it would follow from the above by induction that $I_{y_0}=I_{y_1}$, whence in particular $(0,1),(1,1)\in I_{y_0}$. However, then $I_{y_0}=A$ which is impossible by the same argument as above.
\end{nex}
\subsection{Global Quillen adjunctions} Finally, let us discuss the appropriate notion of Quillen adjunctions in this context:
\begin{defi}
Let $\ul{\mathscr C},\ul{\mathscr D}$ be preglobal model categories. A \emph{global Quillen adjunction} $\underline F\colon\underline{\mathscr C}\rightleftarrows\underline{\mathscr D} :\!\underline U$ is a simplicially enriched adjunction $F\colon\mathscr C\rightleftarrows\mathscr D:\!U$ of the underlying categories such that for every finite group $G$ the induced adjunction $G\text{-}\mathscr C\rightleftarrows G\text{-}\mathscr D$ is a Quillen adjunction for both model structures.
We call $\underline{F}\dashv\underline{G}$ a \emph{global Quillen equivalence} if in addition each $G\text{-}\mathscr C\rightleftarrows G\text{-}\mathscr D$ is a Quillen equivalence (for the projective or, equivalently, for the flat model structures).
\end{defi}
\begin{ex}\label{ex:S-Sp-adjunction}
The suspension-loop adjunction $\Sigma^\bullet_+\colon\cat{$\bm{\mathcal I}$-SSet}\rightleftarrows \cat{Spectra}:\!\Omega^\bullet$ defines a global Quillen adjunction $\glo{GlobalSpaces}\rightleftarrows\glo{GlobalSpectra}$, see Proposition~\ref{prop:suspension-loop-G-gl}.
\end{ex}
\begin{lemma}
Let $\underline F\colon\underline{\mathscr C}\rightleftarrows\underline{\mathscr D} :\!\underline U$ be a global Quillen equivalence of preglobal model categories. Then $\underline{\mathscr C}$ is a global model category if and only if $\underline{\mathscr D}$ is so.
\begin{proof}
Associated to any pullback square $(\ref{diag:pb-surj})$ of finite groups and surjective maps, we obtain a coherent cube
\begin{equation*}
\begin{tikzcd}[row sep=small, column sep=small]
& {\textup{Ho}}(A\text-\mathscr C)\arrow[from=dd, "q^*"{near start}] &&\arrow[ll, "g^*"'] {\textup{Ho}}(B\text-\mathscr C)\arrow[from=dd,"p^*"']\\
{\textup{Ho}}(A\text-\mathscr D)\arrow[ur] &&\arrow[ll, "g^*"'{near start}, crossing over] {\textup{Ho}}(B\text-\mathscr D)\arrow[ur]\\
& {\textup{Ho}}(C\text-\mathscr C) &&\arrow[ll, "f^*"'{near end}] {\textup{Ho}}(D\text-\mathscr C)\\
{\textup{Ho}}(C\text-\mathscr D)\arrow[ur]\arrow[uu, "q^*"] && \arrow[ll, "f^*"] {\textup{Ho}}(D\text-\mathscr D)\arrow[ur]\arrow[uu, "p^*"'{near start}, crossing over]
\end{tikzcd}
\end{equation*}
where all front-to-back maps are given by $\cat{R}U$, the front and back face commute strictly, and the remaining squares are filled by the natural isomorphisms coming from the fact that all functors are right Quillen for the projective model structures and commute strictly on the pointset level.
Passing to canonical mates with respect to the adjunctions $q^*\dashv\cat{R}q_*$ and $p^*\dashv\cat{R}p_*$ we then get a coherent cube
\begin{equation*}
\begin{tikzcd}[row sep=small, column sep=small]
& {\textup{Ho}}(A\text-\mathscr C)\arrow[dd, "\cat{R}q_*"'{near end}] &&\arrow[ll, "g^*"'] {\textup{Ho}}(B\text-\mathscr C)\arrow[dd,"\cat{R}p_*"]\\
{\textup{Ho}}(A\text-\mathscr D)\arrow[ur]\arrow[dd, "\cat{R}q_*"'] &&\arrow[ll, "g^*"'{near start}, crossing over] {\textup{Ho}}(B\text-\mathscr D)\arrow[ur]\\
& {\textup{Ho}}(C\text-\mathscr C) &&\arrow[ll, "f^*"'{near end}] {\textup{Ho}}(D\text-\mathscr C)\\
{\textup{Ho}}(C\text-\mathscr D)\arrow[ur] && \arrow[ll, "f^*"] {\textup{Ho}}(D\text-\mathscr D)\arrow[ur]\arrow[from=uu, "\cat{R}p_*"{near end}, crossing over]
\end{tikzcd}
\end{equation*}
in which the transformations in the top, bottom, left, and right face are isomorphisms (the latter two use that $\cat{R}U$ is an equivalence). Using again that all front-to-back maps are equivalences, it follows that the natural transformation filling the front square is an isomorphism if and only if the one filling the back square is so, which immediately yields the claim.
\end{proof}
\end{lemma}
\section{Global stability}\label{sec:global-stab}
In this section we will introduce a notion of \emph{genuine stability} for global model categories and construct universal stabilizations in this setting.
\subsection{Pointed (pre)global model categories} As usual, in order to talk about stability we first have to talk about pointedness:
\begin{defi}
A preglobal model category $\mathscr C$ is called \emph{pointed} if the underlying category $\mathscr C$ is pointed in the usual sense, i.e.~has a zero object.
\end{defi}
\begin{rk}\label{rk:basepoint}
Recall \cite{hirschhorn, hirschhorn-slice} that for a model category $\mathscr C$ the category $\mathscr C_*$ of pointed objects (i.e.~the slice $*/\mathscr C$ under our favourite terminal object) carries a model structure in which a map is a weak equivalence, fibration, or cofibration, if and only if it is so in $\mathscr C$. If $\mathscr C$ is left proper, right proper, or combinatorial, then so is $\mathscr C_*$, with generating (acyclic) cofibrations in the latter case being given by applying the left adjoint $({\textup{--}})_+=({\textup{--}})\amalg*\colon\mathscr C\to\mathscr C_*$ to a set of generating (acyclic) cofibrations of $\mathscr C$. Moreover, if $\mathscr C$ is simplicial, then $\mathscr C_*$ is enriched as a model category over $\cat{SSet}_*$ (hence in particular over $\cat{SSet}$) by taking the basepoints of the mapping spaces to be the zero maps, while the tensoring $K\smashp X$ is induced by the tensoring $K\times X$ in $\mathscr C$ over $\cat{SSet}$ by collapsing $*\times X\amalg K\times *$. Analogously, we again get ${\textup{--}}\smashp{\textup{--}}\colon\cat{$\bm{(G\times H)}$-SSet}_*\times G\text-\mathscr C_*\to (G\times H)\text-\mathscr C_*$.
If now $F\colon \mathscr C\rightleftarrows\mathscr D :\!U$ is a Quillen adjunction, then $U$ lifts to $U_*\colon\mathscr D_*\to \mathscr C_*$ (as it preserves terminal objects); $F$ does not necessarily lift directly, but we get $\mathscr C_*\to F(*)/\mathscr D$, which we can postcompose with pushforward along the unique map $F(*)\to *$ to get a functor $F_*\colon\mathscr C_*\to\mathscr D_*$ left adjoint to $U_*$. As fibrations and weak equivalences are defined in the underlying categories, we immediately see that $F_*\dashv U_*$ is a Quillen adjunction again.
In particular, given a preglobal model category $\ul{\mathscr C}$ we can make the category $\mathscr C_*$ into a preglobal model category $\ul{\mathscr C}_*$ this way, coming with a global Quillen adjunction $\ul{({\textup{--}})_+}\colon\ul{\mathscr C}\rightleftarrows\ul{\mathscr C}_* :\!\ul{\text{forget}}$. If $\ul{\mathscr C}$ is actually a global model category, then so is $\ul{\mathscr C}_*$ (see the third formulation in Proposition~\ref{prop:global-tfae}). Moreover, if
\begin{equation}\label{eq:gQa-pointed}
\ul F\colon\ul{\mathscr C}\rightleftarrows\ul{\mathscr D}:\!\ul{U}
\end{equation}
is a global Quillen adjunction, then $F_*\dashv U_*$ defines a global Quillen adjunction $\ul{F}_*\dashv\ul{U}_*$.
\end{rk}
\begin{lemma}\label{lemma:pointed-equivalence}
If $(\ref{eq:gQa-pointed})$ is a global Quillen \emph{equivalence}, then so is the induced global Quillen adjunction $\ul{F}_*\dashv\ul{U}_*$.
\begin{proof}
We prove more generally that for any Quillen equivalence \textup{h}box{$L\colon\mathscr A\rightleftarrows\mathscr B :\!R$} of \emph{left proper} model categories the induced Quillen adjunction $L_*\dashv R_*$ is a Quillen equivalence. This is well-known, and can also be deduced with a bit of work from \cite[Proposition~2.7]{rezk-proper}, but we do not know of an explicit reference in the literature, so we provide a direct argument.
For this, we first observe that $R_*$ still reflects weak equivalences between fibrant objects (as everything is defined in underlying categories), i.e.~its right derived functor is conservative. To complete the proof it suffices that for every cofibration $*\to X$ and some (hence any) fibrant replacement $L_*(*\to X)\to Z$ the induced map $X\to R_*L_*(*\to X)\to RZ$ in $\mathscr C$ is a weak equivalence.
For this, we pick a cofibrant replacement $Q\to *$. Factoring the induced map $Q\to *\to X$ as a cofibration followed by a weak equivalence we then get a commutative diagram
\begin{equation}\label{diag:pseudo-po}
\begin{tikzcd}
Q\arrow[d]\arrow[r] & Y\arrow[d]\\
*\arrow[r] & X
\end{tikzcd}
\end{equation}
in which both vertical maps are weak equivalences. In particular, this is a homotopy pushout, i.e.~(as $Q\to Y$ is a cofibration), the induced map $Y/Q\to X$ is a weak equivalence of cofibrant objects in $\mathscr A_*$. It therefore suffices to prove the claim for $Y/Q$ instead of $X$, i.e.~we may assume without loss of generality that $(\ref{diag:pseudo-po})$ is an \emph{honest} pushout. We then consider the diagram
\begin{equation*}
\begin{tikzcd}
LQ\arrow[d]\arrow[r] & LY\arrow[d]\\
L*\arrow[r]\arrow[d] & LX\arrow[d]\\
*\arrow[r] & L_*X
\end{tikzcd}
\end{equation*}
where the upper half is the image of $(\ref{diag:pseudo-po})$ under $L$ (whence a pushout) and the lower half is the pushout defining $L_*(*\to X)$; in particular, also the total rectangle is a pushout. As $L(Q\to Y)$ is a cofibration ($L$ being left Quillen) and the left hand vertical map is a weak equivalence, we conclude that the right hand vertical composite $LY\to L_*(*\to X)$ is a weak equivalence because $\mathscr D$ is left proper. Thus, if $L_*(*\to X)\to Z$ is any fibrant replacement (in $\mathscr C_*$), then the composite $LY\to Z$ is a fibrant replacement (in $\mathscr C$). We conclude that the composite $Y\to RLY\to RZ$ represents the derived unit of $L\dashv R$ (as $Y$ is cofibrant), so it is a weak equivalence. The claim follows by $2$-out-of-$3$ as the composite $X\to Y\to RZ$ represents the derived unit for $L_*\dashv R_*$.
\end{proof}
\end{lemma}
Finally, arguing as in Remark~\ref{rk:basepoint}, we deduce from Lemma~\ref{lemma:g-sset-enriched}:
\begin{cor}\label{cor:pointed-g-sset-enriched}
Let $G,H$ be finite groups and let $\ul{\mathscr C}$ be a pointed preglobal model category. Then the smash product defines left Quillen bifunctors
\begin{align*}
\big(\cat{$\bm{(G\times H)}$-SSet}_*\big)_{\mathcal G_{G,H}}\times G\text-\mathscr C_\textup{proj}&\to (G\times H)\text-\mathscr C_\textup{proj}\\
\cat{$\bm{(G\times H)}$-SSet}_*\times G\text-\mathscr C_\textup{flat}&\to (G\times H)\text-\mathscr C_\textup{flat}.
\end{align*}
In particular, both the projective and flat model structure on $G\text{-}\mathscr C$ are enriched as model categories over $\cat{$\bm G$-SSet}_*$.\qed
\end{cor}
\subsection{Genuine stability} Recall from Remark~\ref{rk:equivariant-stabilization} that the passage from $G$-spaces to $G$-spectra can be understood as inverting smashing with the spheres $S^A$ for all finite $G$-sets $A$. While we cannot apply this directly to the passage from global spaces to global spectra in that way (as smashing with $S^A$ for a non-trivial $G$-set $A$ does not make sense directly), smashing with $S^A$ makes sense for \emph{$G$-global} spaces, and following an idea of \cite{gepner-nikolaus} (or more generally the philosophy of parameterized higher category theory) we can then try to characterize the passage from unstable to stable global homotopy theory by looking more generally at what happens in $G$-global homotopy theory for all $G$ simultaneously:
\begin{defi}\label{def:global-stability}
A global model category $\ul{\mathscr C}$ is called \emph{genuinely stable} (or simply \emph{stable}) if it is pointed and the Quillen adjunction $S^A\smashp{\textup{--}}\colon G\text{-}\mathscr C\rightleftarrows G\text{-}\mathscr C :\!\Omega^A$ is a Quillen equivalence for every finite group $G$ and every finite $G$-set $A$ (for the flat or, equivalently, the projective model structures).
\end{defi}
Note that specializing to $A=*$ with trivial $G$-action shows that each $G\text{-}\mathscr C$ is in particular stable in the usual sense.
\begin{prop}\label{prop:GlobalSpectra-stable}
The global model category $\glo{GlobalSpectra}$ from Example~\ref{ex:global-spectra} is stable.
\begin{proof}
The usual smash product of symmetric spectra defines a left Quillen bifunctor
\begin{equation}\label{eq:equivariant-global-bifunctor}
\cat{$\bm G$-Spectra}_\text{$G$-equiv.~proj.}\times \cat{$\bm G$-Spectra}_\text{$G$-gl.~flat}\to \cat{$\bm G$-Spectra}_\text{$G$-gl.~flat}
\end{equation}
by Theorem~\ref{thm:smash-g-global} together with Proposition~\ref{prop:equivariant-vs-global}, making the $G$-global flat model structure tensored over the $G$-equivariant projective one.
If now $A$ is any finite $G$-set, then $\Sigma^\infty S^A$ is cofibrant in the $G$-equivariant projective model structure, so $\Sigma^\infty S^A\smashp^{\cat L}{\textup{--}}$ agrees with the left derived functor of $S^A\smashp{\textup{--}}$; it therefore suffices to show that $\Sigma^\infty S^A\smashp^{\cat L}{\textup{--}}$ is an autoequivalence of the $G$-global stable homotopy category. But by \cite[Proposition~4.9]{hausmann-equivariant} the analogous functor is an autoequivalence of the $G$-\emph{equivariant} stable homotopy category, so we get a projectively cofibrant $G$-equivariant spectrum $D$ together with a zig-zag of weak equivalences of projectively cofibrant objects between $D\smashp \Sigma^\infty S^A$ and $\mathbb S$. As $(\ref{eq:equivariant-global-bifunctor})$ is a left Quillen bifunctor, this then shows that $D\smashp^{\cat L}{\textup{--}}$ is the desired quasi-inverse.
\end{proof}
\end{prop}
\begin{rk}
In the above argument we crucially use that $\Sigma^\infty S^A$ is actually cofibrant in the equivariant \emph{projective} model structure (and not just flat); in particular, $(\ref{eq:equivariant-global-bifunctor})$ is not left Quillen for the $G$-equivariant \emph{flat} model structure. On the other hand, we could have just as well used the $G$-global projective model structure instead of the flat one everywhere.
\end{rk}
\begin{lemma}\label{lemma:stability-stable}
Let $\ul{F}\colon \ul{\mathscr C}\rightleftarrows\ul{\mathscr D} :\!\ul{U}$ be a global Quillen equivalence of pointed global model categories. Then $\ul{\mathscr C}$ is stable if and only if $\ul{\mathscr D}$ is so.
\begin{proof}
Let $G$ be a finite group and $A$ a finite $G$-set. As $F$ is a simplicial and hence also $\cat{SSet}_*$-enriched left adjoint, there is a (canonical) isomorphism filling
\begin{equation*}
\begin{tikzcd}
G\text-\mathscr C\arrow[d, "F"']\arrow[r, "S^A\smashp{\textup{--}}"] &[1em] G\text-\mathscr C\arrow[d, "F"]\\
G\text-\mathscr D\arrow[r, "S^A\smashp{\textup{--}}"'] & G\text-\mathscr D,
\end{tikzcd}
\end{equation*}
and as all participating functors are left Quillen (say, for the projective model structures), this induces an isomorphism $\cat{L}(S^A\smashp{\textup{--}})\circ\cat{L}F\cong\cat{L}F\circ\cat{L}(S^A\smashp{\textup{--}})$ of left derived functors. The claim follows by $2$-out-of-$3$.
\end{proof}
\end{lemma}
\begin{rk}
Let $\ul{\mathscr C}$ be a stable global model category and let $G$ be a finite group. Then the shift $\ul{G\text-\mathscr C}$ is again stable: if $G'$ is any finite group and $A$ is a finite $G'$-set, then viewing it as a $(G'\times G)$-set with trivial $G$-action shows that the left Quillen functor $S^A\smashp{\textup{--}}\colon G'\text-G\text-\mathscr C\to G'\text-G\text-\mathscr C$ is a Quillen equivalence.
\end{rk}
\subsection{Spectrification}\label{subsec:spectrification}
Non-equivariantly, the universal stabilization of a suitably nice simplicial model category $\mathscr C$ can be computed by spectrum objects in $\mathscr C$. In this subsection, we introduce a refinement of this construction to our framework; the universal property will then be established in the next subsection.
\begin{constr}
Let $\underline{\mathscr C}$ be a pointed preglobal model category. We write $\textup{Sp}(\underline{\mathscr C})$ for the category of $\cat{SSet}_*$-enriched functors $\bm\Sigma\to\mathscr C$. A map $f$ in $\textup{Sp}(\underline{\mathscr C})$ is called a \emph{global level weak equivalence} if $f(A)$ is a weak equivalence in $\Sigma_A\text{-}\mathscr C$ for every finite set $A$. Moreover, $f$ is called a \emph{projective global level fibration} or \emph{flat global level fibration} if each $f(A)$ is a fibration in the projective $\Sigma_A$-global model structure or flat $\Sigma_A$-global model structure, respectively.
More generally, if $\ul{\mathscr C}$ is an arbitrary preglobal model category, then we define $\textup{Sp}(\ul{\mathscr C})\mathrel{:=}\textup{Sp}(\ul{\mathscr C}_*$).
\end{constr}
\begin{prop}\label{prop:level-model-structures}
Assume $\ul{\mathscr C}$ is pointed. The global level weak equivalences and global projective (flat) level fibrations are part of a simplicial, combinatorial, and left proper model structure on $\textup{Sp}(\underline{\mathscr C})$. We call this the \emph{global projective (flat) level model structure}. A possible set of generating cofibrations is given by
\begin{equation*}
\{\bm\Sigma(A,{\textup{--}})\smashp_{\Sigma_A}i : \text{$A\in\bm\Sigma$, $i\in I_{\Sigma_A}$}\}
\end{equation*}
where $I$ denotes a set of generating cofibrations of the projective (flat) model structure on $\Sigma_A\text{-}\mathscr C$, and similarly a set of generating acyclic cofibrations is given by
\begin{equation*}
\{\bm\Sigma(A,{\textup{--}})\smashp_{\Sigma_A}j : \text{$A\in\bm\Sigma$, $j\in J_{\Sigma_A}$}\}
\end{equation*}
for sets $J_{\Sigma_A}$ of generating acyclic cofibrations of the respective model structure on $\Sigma_A\text-\mathscr C$.
\end{prop}
Here $\bm\Sigma(A,{\textup{--}})\smashp_{\Sigma_A} f$ is the map obtained from the levelwise tensoring over $\cat{SSet}_*$ by dividing out the diagonal $\Sigma_A$-action.
\begin{rk}
Replacing $\ul{\mathscr C}$ by $\ul{\mathscr C}_*$ we immediately get an analogous statement for unpointed $\mathscr C$, with generating (acyclic) cofibrations now of the form $\bm\Sigma(A,{\textup{--}})\smashp_{\Sigma_A} f_+$ for generating (acyclic) cofibrations $f$ of $\Sigma_A\text-\mathscr C$.
\end{rk}
For the proof of the proposition we will use the following easy criterion, cf.~\cite[Proposition~C.23]{schwede-book} or \cite[Definition~2.21]{hausmann-equivariant}:
\begin{lemma}
Let $\mathbb I$ be a small $\cat{SSet}_*$-enriched category, and let $\mathscr C$ be a locally presentable category enriched over $\cat{SSet}_*$. Assume we are given for each $X\in\mathbb I$ a combinatorial model structure on ${\mathbb E\textup{nd}}(X)\text-\mathscr C$ (the category of enriched functors from the full $\cat{SSet}_*$-subcategory spanned by $X$ to $\mathscr C$) with generating cofibrations $I_X$ and generating acyclic cofibrations $J_X$, such that the following `consistency condition' is satisfied: for every $Y\in\mathbb I$, any relative $\{{\mathbb I}(X,Y)\smashp_{{\mathbb E\textup{nd}}(X)} j : X\in\mathbb I,j\in J_X\}$-cell complex is a weak equivalence in ${\mathbb E\textup{nd}}(Y)\text-\mathscr C$.
Then there is a unique model structure on the category $\mathbb I\text-\mathscr C$ of enriched functors $\mathbb I\to\mathscr C$ in which a map $f$ is a weak equivalence if and only if $f(X)$ is a weak equivalence or fibration, respectively, in the given model structure on ${\mathbb E\textup{nd}}(X)\text-\mathscr C$ for all $X\in\mathbb I$. This model structure is combinatorial with generating cofibrations
\begin{equation*}
\{\mathbb I(X,{\textup{--}})\smashp_{{\mathbb E\textup{nd}}(X)} i : X\in\mathbb I, i\in I_X\}
\end{equation*}
and generating acyclic cofibrations
\begin{equation*}
\{\mathbb I(X,{\textup{--}})\smashp_{{\mathbb E\textup{nd}}(X)} j : X\in\mathbb I, j\in J_X\}.
\end{equation*}
\begin{proof}
The forgetful functor $\mathbb I\text-\mathscr C\to{\textup{pr}}od_{X\in\mathbb I}{\mathbb E\textup{nd}}(X)\text-\mathscr C$ has a left adjoint given by $F\mathrel{:=}\coprod_{X\in\mathbb I}\mathbb I(X,{\textup{--}})\smashp_{{\mathbb E\textup{nd}}(X)}{\textup{pr}}_{X}$. We will verify the conditions of the Crans-Kan Transfer Criterion \cite[Theorem~11.3.2]{hirschhorn} for this adjunction, which will then provide the desired model structure and show that it is cofibrantly generated (hence combinatorial) with the above sets of generating (acyclic) cofibrations. By local presentability, every set permits the small object argument, so we only have to show that every relative $F(J)$-cell complex is a weak equivalence where $J$ is a set of generating acyclic cofibrations of the right hand side. But for the standard choice of generating acyclic cofibrations this precisely amounts to the above consistency condition.
\end{proof}
\end{lemma}
\begin{proof}[Proof of Proposition~\ref{prop:level-model-structures}]
Let us consider the case of the flat model structures first. To verify the above consistency condition, it suffices to show that for all finite sets $A\subset B$ the map $\bm\Sigma(A,B)\smashp_{\Sigma_A}{\textup{--}}\colon\Sigma_A\text{-}\mathscr C_\text{flat}\to\Sigma_B\text{-}\mathscr C_\text{flat}$ is left Quillen. To this end, we observe that we can identify $\bm\Sigma(A,B)$ with $(\Sigma_B)_+\smashp_{\Sigma_{B\setminus A}}S^{B\setminus A}$ as simplicial sets with left $\Sigma_B$- and right $\Sigma_A$-action, see \cite[proof of Proposition~2.24]{hausmann-equivariant}, so $\bm\Sigma(A,B)\smashp_{\Sigma_A}{\textup{--}}$ factors as the composite
\begin{equation*}
\Sigma_A\text{-}\mathscr C_\text{flat}\xrightarrow{S^{B\setminus A}\smashp{\textup{--}}}(\Sigma_A\times\Sigma_{B\setminus A})\text{-}\mathscr C_\text{flat}\xrightarrow{k_!}\Sigma_{B}\text{-}\mathscr C_\text{flat}
\end{equation*}
where $k\colon\Sigma_A\times\Sigma_{B\setminus A}\to\Sigma_B$ is the evident embedding. As $k$ is injective, the second arrow is left Quillen, and so is the first one by Corollary~\ref{cor:pointed-g-sset-enriched}.
The consistency condition for the projective model structure follows immediately from the one for the flat model structure as it has fewer cofibrations and the same weak equivalences, proving the existence of the projective level model structure.
As (acyclic) fibrations and the cotensoring on $\textup{Sp}(\ul{\mathscr C})$ are simply defined levelwise, we immediately see that these model structures are again simplicial. Similarly, for left properness it is enough to observe that all generating cofibrations of either model structure are levelwise flat cofibrations (as $\bm\Sigma(A,B)\smashp_{\Sigma_A}{\textup{--}}$ is left Quillen for the flat model structures by the above), so that every cofibration in $\textup{Sp}(\ul{\mathscr C})$ is in particular a levelwise flat cofibration.
\end{proof}
Applying this to the shifts $\underline{G\text-\mathscr C}$ of $\underline{\mathscr C}$ (Example~\ref{ex:shifts}) gives us two left proper, simplicial, and combinatorial model structures on $G\text{-}\textup{Sp}(\underline{\mathscr C})$ that we call the \emph{$G$-global projective level model structure} and \emph{$G$-global flat level model structure}. Their weak equivalences and fibrations are those maps $f$ such that $f(A)$ is a weak equivalence or fibration, respectively, in the corresponding model structure on $(G\times\Sigma_A)\text{-}\mathscr C$ for every $A\in\bm\Sigma$.
In contrast to this, the cofibrations are not just simply defined levelwise, but rather in terms of a left lifting property. Nevertheless we can say something about the individual levels of the above cofibrations; we begin with the flat case where we have already noticed in the above proof:
\begin{cor}\label{cor:flat-implies-level-flat}
Let $f$ be a cofibration in $G\text-\textup{Sp}(\ul{\mathscr C})_\textup{flat}$ and let $B$ be a finite set. Then $f(B)$ is a cofibration in $(G\times\Sigma_B)\text-\mathscr C_\textup{flat}$.\qed
\end{cor}
In the projective case we get a slightly weaker statement:
\begin{lemma}\label{lemma:projective-cofibrations-levelwise}
Let $i$ be a projective cofibration in $G\text-\textup{Sp}(\ul{\mathscr C})$. Then $i(B)$ is a cofibration in $G\text-\mathscr C_\textup{proj}$ for every finite $G$-set $B$.
\begin{proof}
It is enough to prove this for the generating cofibrations. But for any finite set $A$ and any $(G\times\Sigma_A)$-global projective cofibration $i$
\begin{equation*}
(\bm\Sigma(A,{\textup{--}})\smashp_{\Sigma_A} i)(B)= (\bm\Sigma(A,B)\smashp i)/\Sigma_A
\end{equation*}
and $\bm\Sigma(A,B)\smashp i$ is a $(G\times\Sigma_A)$-global projective cofibration by Corollary~\ref{cor:pointed-g-sset-enriched}; the claim follows as quotients preserve projective cofibrations.
\end{proof}
\end{lemma}
\begin{lemma}\label{lemma:level-model-structures-functoriality}
The projective and flat $G$-global level model structures define a preglobal model category $\underline{\textup{Sp}}(\underline{\mathscr C})_\textup{level}$. If $\ul{\mathscr C}$ is actually a global model category, then so is $\underline{\textup{Sp}}(\underline{\mathscr C})_\textup{level}$.
\begin{proof}
We may assume without loss of generality that $\ul{\mathscr C}$ is pointed.
By the above, both model structures are simplicial, combinatorial, and left proper. Moreover, it is clear that they have the same weak equivalences and that every (generating) projective cofibration is also a flat cofibration. Thus, it only remains to verify the change-of-group properties.
For this let $\alpha\colon G\to G'$ be any group homomorphism. Then the restriction $(\alpha\times{\textup{id}})^*\colon (G'\times\Sigma_A)\text-\mathscr C\to (G\times\Sigma_A)\text-\mathscr C$ preserves weak equivalences and projective fibrations. Thus, $\alpha^*\colon G'\text-\textup{Sp}(\underline{\mathscr C})\to G\text-\textup{Sp}(\underline{\mathscr C})$ preserves level weak equivalences as well as projective level fibrations. Moreover, if $\alpha$ is injective, then each $(\alpha\times{\textup{id}})^*\colon(G'\times\Sigma_A)\text-\mathscr C\to (G\times\Sigma_A)\text-\mathscr C$ also preserves flat fibrations, so that $\alpha^*$ preserves flat level fibrations.
Similarly, one shows that $\alpha_*\colon G\text-\textup{Sp}(\underline{\mathscr C})\to G'\text-\textup{Sp}(\underline{\mathscr C})$ preserves acyclic flat level fibrations for any $\alpha$, so that $\alpha^*$ preserves flat level cofibrations, and that $\alpha_*$ also preserves acyclic projective level fibrations if $\alpha$ is injective, so that $\alpha^*$ preserves projective level cofibrations in this case.
Altogether we have shown that $\underline{\textup{Sp}}(\underline{\mathscr C})$ is a preglobal model category. Now assume $\ul{\mathscr C}$ is actually a global model category, let $g\colon A\to B$ and $q\colon A\to C$ be homomorphisms with $\ker g\cap\ker q = 1$, let $X\in B\text-\textup{Sp}(\underline{\mathscr C})_\text{flat level}$ be fibrant, and let $i\colon g^*X\to Y$ be a fibrant replacement in $A\text-\textup{Sp}(\underline{\mathscr C})_\text{flat level}$; we have to show that $q_*(i)$ is a $C$-global level weak equivalence. But for any finite set $D$, $X(D)$ is fibrant in $(B\times\Sigma_D)\text-\mathscr C_\text{flat}$ and $i(D)$ is a fibrant replacement in $(A\times\Sigma_D)\text-\mathscr C_\text{flat}$. Thus, $(g_*i)(D)=(g\times\Sigma_D)_*(i(D))$ is a $(C\times\Sigma_D)$-global weak equivalence as $\underline{\mathscr C}$ is a global model category and $\ker(g\times{\textup{id}})\cap \ker(q\times{\textup{id}})=1$, whence $g_*i$ is a $G$-global level weak equivalence as desired.
\end{proof}
\end{lemma}
\begin{lemma}\label{lemma:sp-level-induced-adjunction}
Let $\ul F\colon\ul{\mathscr C}\rightleftarrows\ul{\mathscr D} :\!\ul U$ be a global Quillen adjunction. Then also
\begin{equation*}
\ul\textup{Sp}(\ul{F})\colon \ul\textup{Sp}(\ul{\mathscr C})_\textup{level}\rightleftarrows\textup{Sp}(\ul{\mathscr D})_\textup{level} :\!\ul\textup{Sp}(\ul U)
\end{equation*}
is a global Quillen adjunction. If $\ul F\dashv\ul G$ is a global Quillen equivalence, then so is $\ul\textup{Sp}(\ul F)\dashv\ul\textup{Sp}(\ul U)$.
\begin{proof}
By Lemma~\ref{lemma:pointed-equivalence}, we may assume without loss of generality that $\mathscr C$ and $\mathscr D$ are pointed. Passing to shifts it further suffices to prove that $\textup{Sp}(\ul F)\dashv\textup{Sp}(\ul U)$ is a Quillen adjunction for both level model structures and a Quillen equivalence if $\ul F\dashv\ul U$ is a global Quillen equivalence.
For the first statement it is enough to observe that $\textup{Sp}(\ul U)$ preserves (acyclic) fibrations in either model structure as they are simply defined levelwise.
For the second statement, it suffices to prove that this is a Quillen equivalence for the flat model structures. But indeed, if $X\in\textup{Sp}(\ul{\mathscr C})_\text{flat}$ is flat, then $X(A)$ is flat for every $A$ by Corollary~\ref{cor:flat-implies-level-flat}; similarly, if $FX\to Y$ is a fibrant replacement, then each $FX(A)\to Y(A)$ is a fibrant replacement in $\Sigma_A\text-\mathscr C_\text{flat}$. Now the composite $X\to UFX\to UY$ represents the derived unit for $X$ by definition, but at the same time each $X(A)\to UY(A)$ represents the derived unit for $\Sigma_A\text-F\dashv \Sigma_A\text-U$ by the above, so it is a $\Sigma_A$-global weak equivalence by assumption. Analogously one shows that also the derived \emph{co}unit is a global level weak equivalence, finishing the proof of the lemma.
\end{proof}
\end{lemma}
As before, we now want to restrict to a suitable notion of {$\Omega$-spectra} via Bousfield localization.
\begin{defi}
Let $\underline{\mathscr C}$ be a preglobal model category. An object $X\in\textup{Sp}(\underline{\mathscr C})$ is called a \emph{global $\Omega$-spectrum} if for every finite group $H$, every finite $H$-set $A$ and every finite $H$-set $B$ the derived adjoint structure map
\begin{equation}\label{eq:der-adj-str-global}
X(A)\to\cat{R}\Omega^BX(A\amalg B)
\end{equation}
is an $H$-global weak equivalence.
\end{defi}
Here we are deriving $\Omega^B$ with respect to the $H$-global projective model structure; in particular, if $X$ is fibrant in either of the above level model structures, then $(\ref{eq:der-adj-str-global})$ is already represented by the ordinary adjoint structure map. Note that there are \emph{no} faithfulness assumptions on $A$ and $B$ here.
Again we can apply this to the shifts of $\mathscr C$ by $G$; we will refer to the corresponding objects of $G\text{-}\textup{Sp}(\underline{\mathscr C})$ as \textit{$G$-global $\Omega$-spectra}.
\begin{rk}\label{rk:g-global-omega-g-h-sets}
In the definition of a $G$-global $\Omega$-spectrum we can equivalently ask for the adjoint structure map $X(A)\to\cat{R}\Omega^BX(A\amalg B)$ to be a $(G\times H)$-global weak equivalence for all finite $(G\times H)$-sets $A, B$ (instead of just for $H$-sets). Namely, we can simply replace $H$ by $G\times H$ in the above and then restrict along the diagonal $G\times H\to G\times H\times G$.
\end{rk}
In order to construct the corresponding Bousfield localizations we need some additional notation:
\begin{constr}
As already used in the description of the generating (acyclic) cofibrations, performing the tensoring over $\cat{SSet}_*$ levelwise gives us a bifunctor
\begin{equation}\label{eq:tensoring-spectra}
{\textup{--}}\smashp{\textup{--}}\colon\cat{Spectra}\times\mathscr C\to\textup{Sp}(\underline{\mathscr C}),
\end{equation}
which preserves colimits in each variable separately. In particular, $X\smashp{\textup{--}}$ has a right adjoint $F(X,{\textup{--}})\colon\textup{Sp}(\underline{\mathscr C})\to\mathscr C$ for every spectrum $X$. From this, we get bifunctors
\begin{align}
{\textup{--}}\smashp{\textup{--}}\colon \cat{$\bm G$-Spectra}\times G\text-{\mathscr C} &\to G\text-\textup{Sp}(\underline{\mathscr C})\label{eq:tensoring-g-spectra}\\
F\colon\cat{$\bm G$-Spectra}^{\textup{op}}\times G\text-\textup{Sp}(\underline{\mathscr C})&\to G\text-\mathscr C\label{eq:cotensoring-g-spectra}
\end{align}
by pulling through the $G$-actions everywhere.
\end{constr}
\begin{lemma}\label{lemma:tensoring-g-global-level}
The smash product $(\ref{eq:tensoring-g-spectra})$ is a left Quillen bifunctor with respect to the $G$-\emph{equivariant} projective level model structure on $\cat{$\bm G$-Spectra}$ and the projective $G$-global (level) model structures elsewhere. Dually, $(\ref{eq:cotensoring-g-spectra})$ is a right Quillen bifunctor for these model structures.
\begin{proof}
Again, we may assume $\ul{\mathscr C}$ to be pointed. By adjunction, it will be enough to show that the functor
\begin{equation*}
G\text-\mathscr C_\text{proj.}^{\textup{op}}\times G\text-\textup{Sp}(\underline{\mathscr C})_\text{proj.~level}\to \cat{$\bm G$-Spectra}_\text{$G$-equiv.~proj.~level}
\end{equation*}
given by taking mapping spaces levelwise is a right Quillen bifunctor. As also the weak equivalences and fibrations are defined levelwise, it suffices to show that
\begin{equation*}
\mathord{\textup{maps}}\colon G\text-\mathscr C_\text{proj.}^{\textup{op}}\times (G\times \Sigma_A)\text-\mathscr C_\text{proj.}\to \big(\cat{$\bm{(G\times \Sigma_A)}$-SSet}_*\big)_\text{$\mathcal G_{G,\Sigma_A}$-equiv.}
\end{equation*}
is a right Quillen bifunctor for every finite set $A$. By another adjointness argument, this then follows from Corollary~\ref{cor:pointed-g-sset-enriched}.
\end{proof}
\end{lemma}
\begin{prop}\label{prop:projective-model-structure-G-gl}
The $G$-global projective level model structure on $G\text{-}\textup{Sp}(\underline{\mathscr C})$ admits a Bousfield localization whose fibrant objects are precisely the $G$-globally projectively level fibrant $G$-global $\Omega$-spectra. We call the resulting model structure the \emph{$G$-global projective model structure}. It is again left proper, combinatorial, and simplicial.
\begin{proof}
As before we may assume that $\ul{\mathscr C}$ is pointed, and replacing $\ul{\mathscr C}$ by $\underline{G\text{-}\mathscr C}$ if necessary, it suffices to prove this for $G=1$.
As $\textup{Sp}(\underline{\mathscr C})_\text{proj.~level}$ is left proper, combinatorial, and simplicial, it will be enough by the localization machinery of \cite[Proposition A.3.7.3]{htt} to give a set $S$ of cofibrations such that a fibrant $X\in\textup{Sp}(\underline{\mathscr C})_\text{proj.~level}$ is a global $\Omega$-spectrum if and only if for every $f\in S$ the restriction $\mathord{\textup{maps}}(f,X)$ is an acyclic Kan fibration.
For this we recall from Remark~\ref{rk:equivariant-proj-gen-cof} for any finite group $H$ and any finite $H$-sets $A,B$ the map $\lambda_{H,A,B}\colon S^B\smashp\bm\Sigma(A\amalg B,{\textup{--}})\to\bm\Sigma(A,{\textup{--}})$ corepresenting $X(A)^H\to(\Omega^B X(A\amalg B))^H$ and its factorization $\lambda_{H,A,B}=\rho_{H,A,B}\kappa_{H,A,B}$ into an $H$-equivariant projective cofibration followed by a level weak equivalence. Moreover, let us pick for each finite group $H$ a set $I_H$ of generating cofibrations of $H\text-\mathscr C_\text{proj}$. We now claim that the set
\begin{equation}\label{eq:defining-weak-equivalences}
S\mathrel{:=}\{\kappa_{H,A,B}\ppo_H i : H,A,B,i\in I_H\}
\end{equation}
of (balanced) pushout products, where $H$ runs through all finite groups and $A$ and $B$ through finite $H$-sets, has the desired properties.
To this end we first observe that each ordinary pushout product $\kappa_{H,A,B}\ppo i$ is a cofibration in $H\text-\textup{Sp}(\ul{\mathscr C})_\text{proj.~level}$ by Lemma~\ref{lemma:tensoring-g-global-level}, so that $\kappa_{H,A,B}\ppo_Hi$ is indeed a projective cofibration in $\textup{Sp}(\ul{\mathscr C})$ as $\ul{\textup{Sp}}(\ul{\mathscr C})_\textup{level}$ is a preglobal model category. Now by adjointness $\mathord{\textup{maps}}(\kappa_{H,A,B}\ppo_H i, X)$ is an acyclic Kan fibration if and only if the map $F(\kappa_{H,A,B},\mathop{\textup{tr}}\nolimitsiv_HX)$ has the right lifting property in $H\text{-}\mathscr C$ against all maps of the form $i\ppo(\del\Delta^n\textup{h}ookrightarrow\Delta^n)_+$. Letting $i$ vary, we conclude (as on the one hand $H\text{-}\mathscr C$ is tensored over $\cat{SSet}$ and as on the other the pushout product with $(\del\Delta^0\to\Delta^0)_+$ gives back the original map up to isomorphism) that $\mathord{\textup{maps}}(\kappa_{H,A,B}\ppo_Hi,X)$ is an acyclic Kan fibration for all $i\in I_H$ if and only if $F(\kappa_{H,A,B},\mathop{\textup{tr}}\nolimitsiv_HX)$ is an acyclic fibration in $H\text{-}\mathscr C_\text{proj}$. On the other hand, $F(\kappa_{H,A,B},\mathop{\textup{tr}}\nolimitsiv_HX)$ is always a fibration because $F$ is a right Quillen bifunctor (Lemma~\ref{lemma:tensoring-g-global-level} again) and $X$ was assumed to be projectively level fibrant (so that $\mathop{\textup{tr}}\nolimitsiv_HX$ is $H$-globally projectively level fibrant). Thus, $\mathord{\textup{maps}}(\kappa_{H,A,B}\ppo_H i, X)$ is an acyclic fibration for all $i$ if and only if $F(\kappa_{H,A,B},\mathop{\textup{tr}}\nolimitsiv_HX)$ is a weak equivalence in $H\text{-}\mathscr C$. However, $\rho_{H,A,B}$ is an $H$-equivariant level weak equivalence between projectively cofibrant $H$-equivariant spectra (as $\kappa_{H,A,B}$ is a cofibration and the source and target of $\lambda_{H,A,B}$ were cofibrant), so Ken Brown's Lemma shows that $F(\rho_{H,A,B},\mathop{\textup{tr}}\nolimitsiv_HX)$ is an $H$-global weak equivalence. Thus, $F(\lambda_{H,A,B},\mathop{\textup{tr}}\nolimitsiv_HX)$ is an $H$-global weak equivalence if and only if $F(\kappa_{H,A,B},\mathop{\textup{tr}}\nolimitsiv_HX)$ is so. Since $F(\lambda_{H,A,B},\mathop{\textup{tr}}\nolimitsiv_HX)$ is conjugate to the adjoint structure map $X(A)\to\Omega^B X(A\amalg B)$, the claim follows.
\end{proof}
\end{prop}
\begin{rk}
We also explicitly describe the set of maps used to obtain the $G$-global projective model structure by localization in the above proof. Considering the shift $\ul{G\text-\mathscr C}$, we see that this is given as
\begin{equation}\label{eq:localizing-set-G-gl}
S_G\mathrel{:=}\{\kappa_{H,A,B}\ppo_H i : H\text{ finite group}, A,B \text{ finite $H$-sets}, i\in I_{G\times H}\}.
\end{equation}
\end{rk}
\begin{cor}
For any finite group $G$, there is a unique model structure on $G\text-\textup{Sp}(\underline{\mathscr C})$ whose cofibrations are the $G$-global flat level cofibrations and whose weak equivalences are the $G$-global weak equivalences. This model structure is simplicial, combinatorial, and left proper. Moreover, its fibrant objects are precisely the $G$-global $\Omega$-spectra that are fibrant in the $G$-global flat level model structure.
\begin{proof}
Again, we may assume that $\ul{\mathscr C}$ is pointed and $G=1$. As every global projective level cofibration is also a global flat level cofibration, we can localize the global flat level model structure at the maps $(\ref{eq:defining-weak-equivalences})$; as every fibration in the global flat level model structure is also a fibration in the global projective level model structure, the above argument then shows that a level fibrant spectrum is fibrant in this new model structure if and only if it is a global $\Omega$-spectrum. It only remains to show that the weak equivalences of this model structure are again the global weak equivalences. But by abstract nonsense about Bousfield localizations, a map $f$ in $\textup{Sp}(\underline{\mathscr C})$ is a weak equivalence in this global flat model structure if and only if $[f,X]$ is bijective for every global $\Omega$-spectrum, where $[\,{,}\,]$ denotes the hom set in the localization of $\textup{Sp}(\underline{\mathscr C})$ at the global level weak equivalences. As the same characterization applies to the global projective model structure, the claim follows immediately.
\end{proof}
\end{cor}
\begin{warn}
It might be tempting to assume that the generating acyclic cofibrations of the above $G$-global model structures are given by adding the set $S_G$ from $(\ref{eq:localizing-set-G-gl})$ to the generating acyclic cofibrations of the level model structure, analogously to the equivariant situation (Remark~\ref{rk:equivariant-proj-gen-cof}). We warn the reader however that while all of these are clearly acyclic cofibrations, it is not clear whether they generate, and more generally the localization machinery employed above does \emph{not} provide any control about the generating acyclic cofibrations. In fact, the explicit identification of the generating acyclic cofibrations in the equivariant case referred to above crucially relies on \emph{right} properness of the level model structure, which is not assumed in our setting.
\end{warn}
Despite these words of warning, the following result will often allow us to \emph{pretend} that the generating acyclic cofibrations are of the above form:
\begin{prop}\label{prop:proj-level-to-global}
Let $G$ be a finite group and let $\mathscr D$ be a left proper simplicial model category with a simplicial Quillen adjunction $F\colon G\text-\textup{Sp}(\ul{\mathscr C})_\textup{proj.~level}\rightleftarrows\mathscr D :\!U$. Then the following are equivalent:
\begin{enumerate}
\item $F$ is left Quillen as a functor $G\text-\textup{Sp}(\ul{\mathscr C})_\textup{proj}\to\mathscr D$.
\item $F$ sends the maps in $S_G$ to weak equivalences.
\item $U$ sends fibrant objects to $G$-global $\Omega$-spectra.
\end{enumerate}
The analogous statement for the flat $G$-global (level) model structure also holds.
\end{prop}
The proof will in turn rely on the following general result:
\begin{lemma}\label{lemma:check-QA-fibrant}
Let $F\colon\mathscr C\rightleftarrows\mathscr D :\!U$ be a simplicial adjunction of left proper simplicial model categories. Then $F\dashv U$ is a Quillen adjunction if and only if $F$ preserves cofibrations and $U$ preserves fibrant objects.
\begin{proof}
See \cite[Corollary~A.3.7.2]{htt}.
\end{proof}
\end{lemma}
\begin{proof}[Proof of Proposition~\ref{prop:proj-level-to-global}]
The equivalence $(1)\Leftrightarrow(3)$ is an instance of the above lemma, while for $(1)\Rightarrow(2)$ it suffices to observe that the maps in $S_G$ are acyclic cofibrations. Thus, it only remains to prove $(2)\Rightarrow(3)$.
For this, let $X$ be fibrant and let $f\in S_G$. Then $\mathord{\textup{maps}}(f,UX)$ agrees by enriched adjointness up to conjugation with $\mathord{\textup{maps}}(Ff,X)$, so it is an acyclic fibration as $Ff$ is an acyclic cofibration and $X$ was assumed to be fibrant. But $UX$ is level fibrant, so letting $f$ vary this implies by the proof of Proposition~\ref{prop:projective-model-structure-G-gl} that $UX$ is a $G$-global $\Omega$-spectrum as desired.
\end{proof}
\begin{thm}\label{thm:sp-global-model-cat}
Let $\ul{\mathscr C}$ be a global model category. Then the projective and flat $G$-global model structures on $G\text{-}\textup{Sp}(\underline{\mathscr C})$ assemble into a global model category $\glo{Sp}(\underline{\mathscr C})$.
\begin{proof}
Again, we may assume that $\ul{\mathscr C}$ is pointed. We have already seen that all of these model structures are left proper and combinatorial. Moreover, the projective and the flat $G$-global model structure have the same weak equivalences (namely, the $G$-global weak equivalences), and every projective cofibration is also a flat one by Lemma~\ref{lemma:level-model-structures-functoriality}.
For the functoriality properties, let $\alpha\colon G\to G'$ be any homomorphism. Then $\alpha_!\dashv\alpha^*$ is a Quillen adjunction for the projective \emph{level} model structures, hence in particular with respect to the projective \emph{level} model structure on $G\text-\textup{Sp}(\ul{\mathscr C})$ and the actual projective model structure on $G'\text-\textup{Sp}(\ul{\mathscr C})$. By Proposition~\ref{prop:proj-level-to-global} it will therefore suffice to prove that $\alpha^*$ sends any fibrant $X\in G'\text-\textup{Sp}(\underline{\mathscr C})_\text{proj}$ to a $G$-global $\Omega$-spectrum. Indeed, as we already know that $\alpha^*X$ is projectively level fibrant, this amount to saying that the adjoint structure map
\begin{equation*}
(\alpha^*X)(A)\to\Omega^B(\alpha^*X)(A\amalg B)
\end{equation*}
is a $(G\times H)$-global weak equivalence for all finite $H$-sets $A,B$. However, this map agrees with the restriction of the adjoint structure map along $\alpha\times{\textup{id}}\colon G\times H\to G'\times H$; as $(\alpha\times {\textup{id}})^*$ is homotopical, the claim follows.
Next, we will show that $\alpha^*\dashv\alpha_*$ is a Quillen adjunction for the flat model structures; arguing as above, this amounts to saying that if $X\in G\text-\textup{Sp}(\underline{\mathscr C})_\text{flat}$ is fibrant, then $\alpha_*(X)(A)\to\Omega^B(\alpha_*X)(A\amalg B)$ is a $(G'\times H)$-global weak equivalence for all finite $H$-sets $A,B$. This is where we will need that $\underline{\mathscr C}$ is a \emph{global} (and not just a preglobal) model category: namely, we pick functorial fibrant replacements in $(G\times H)\text-\mathscr C_\text{flat}$ to get a commutative diagram
\begin{equation*}
\begin{tikzcd}
X(A)\arrow[d, "\sim"'] \arrow[r, "\sim"] & \Omega^B X(A\amalg B)\arrow[d, "\sim"]\\
Y_1\arrow[r] & Y_2\rlap.
\end{tikzcd}
\end{equation*}
Here the top arrow is a $(G\times H)$-global weak equivalence by assumption on $X$, while the vertical arrows are so by construction; thus, also the lower horizontal arrow is a weak equivalence by $2$-out-of-$3$. We want to show that applying $(\alpha\times{\textup{id}})_*$ sends the top arrow to a $(G'\times H)$-global weak equivalence, for which it is then enough to show by another application of $2$-out-of-$3$ that it sends all the remaining arrows to $(G'\times H)$-global weak equivalences. For the lower horizontal arrow this is simply an instance of Ken Brown's Lemma. We will now show that also the left hand vertical arrow is sent to a $(G'\times H)$-global weak equivalence; the argument for the right hand vertical arrow will then be analogous.
For this, we observe that $X(A)$ is fibrant in $(G\times\Sigma_A)\text-\mathscr C_\text{flat}$ by definition of the $G$-global flat model structure. Thus, if $\rho\colon H\to\Sigma_A$ classifies the $H$-action on $A$ and we define $g\mathrel{:=}({\textup{id}}\times\rho)\colon G\times H\to G\times\Sigma_A, q\mathrel{:=}(\alpha\times {\textup{id}})\colon G\times H\to G'\times H$, then we precisely want to show that $q_*$ sends the fibrant replacement $g^*X(A)\to Y_1$ in $(G\times H)\text-{\mathscr C}_\text{flat}$ to a weak equivalence. However, as $\ker g=1\times(\ker\rho)$ has trivial intersection with $(\ker\alpha)\times1=\ker q$, this is simply an instance of what it means for a preglobal model category to be global.
As $\alpha^*$ is left Quillen for the flat global model structures, it in particular sends acyclic cofibrations to $G$-global weak equivalences. However, any acyclic \emph{fibration} in the $G'$-global flat model structure is in particular a $G'$-global level weak equivalence, and as $\underline\textup{Sp}(\underline{\mathscr C})_\text{level}$ is a preglobal model category, it follows that $\alpha^*$ sends these to $G$-global (level) weak equivalences. Thus, $\alpha^*$ is actually homotopical.
Now assume that $\alpha$ is injective. Then $\alpha^*$ is left Quillen for the projective model structures as it preserves projective (level) cofibrations by Lemma~\ref{lemma:level-model-structures-functoriality} and is homotopical by the above. Moreover, $\alpha_!\dashv\alpha^*$ is a Quillen adjunction for the flat level model structures and $\alpha^*$ sends fibrant objects to $G$-global $\Omega$-spectra by the above, so $\alpha_!\dashv\alpha^*$ is also a Quillen adjunction for the flat model structures.
Finally, let $g\colon A\to C$ and $q\colon A\to B$ be homomorphisms with $\ker(g)\cap\ker(q)=1$, let $X$ be fibrant in $C\text-\textup{Sp}(\underline{\mathscr C})_\text{flat}$, and let $\iota\colon g^*X\to Y$ be a fibrant replacement in $A\text-\textup{Sp}(\underline{\mathscr C})_\text{flat}$; we have to show that $q_*\iota$ is a $B$-global weak equivalence. However, by the above $g^*X$ is an $A$-global $\Omega$-spectrum, so $\iota$ is actually an $A$-global \emph{level} weak equivalence. The claim therefore follows from $\underline{\textup{Sp}}(\underline{\mathscr C})_\textup{level}$ being a global model category.
\end{proof}
\end{thm}
The above spectrification construction is compatible with global Quillen adjunctions and global Quillen equivalences:
\begin{prop}\label{prop:sp-induced-adjunction}
Let $\underline{F}\colon\underline{\mathscr C}\rightleftarrows\underline{\mathscr D}:\!\underline{U}$ be a global Quillen adjunction of global model categories. Then $\glo{Sp}(\underline{F})\colon\glo{Sp}(\underline{\mathscr C})\rightleftarrows\glo{Sp}(\underline{\mathscr D}):\!\glo{Sp}(\underline{U})$ is a global Quillen adjunction. If $\underline{F}\dashv\underline{U}$ is a global Quillen equivalence, then so is $\glo{Sp}(\underline{F})\dashv\glo{Sp}(\underline{U})$.
\begin{proof}
By Lemma~\ref{lemma:pointed-equivalence} we may assume that $\ul{\mathscr C}$ and $\ul{\mathscr D}$ are pointed. Moreover, it suffices as usual to show that $\textup{Sp}(\ul{F})\colon\textup{Sp}(\ul{\mathscr C})\rightleftarrows\textup{Sp}(\ul{\mathscr D}):\!\textup{Sp}(\ul{U})$ is a Quillen adjunction in the usual sense for the projective and flat model structures, and a Quillen equivalence (say, for the flat ones) if $\ul{F}\dashv\ul{U}$ is a global Quillen equivalence.
For the first statement, we observe that this holds for the level model structures by Lemma~\ref{lemma:sp-level-induced-adjunction}, so that it suffices that the right adjoint sends projectively fibrant objects to global $\Omega$-spectra, which is a direct consequence of Ken Brown's Lemma.
For the second statement, we let $Y\in\textup{Sp}(\ul{\mathscr D})_\text{flat}$ fibrant and $X\to GY$ a cofibrant replacement in the global flat level model structure. Then $FX\to FUY\to Y$ represents the derived counit $\cat{L}F\cat{R}UY\to Y$ for the global flat model structure, but also for the global flat \emph{level} model structure; thus, it is a global weak equivalence by Lemma~\ref{lemma:sp-level-induced-adjunction}.
The proof that also the derived unit is an isomorphism is more involved. The crucial observation for this is the following:
\begin{claim*}
Let $W\in\textup{Sp}(\ul{\mathscr C})$ be a flat global $\Omega$-spectrum. Then $\textup{Sp}(\ul F)(W)$ is again a global $\Omega$-spectrum.
\begin{proof}
Let $H$ be a finite group. As $F\dashv U$ is an $\cat{SSet}_*$-enriched adjunction, there are natural comparison isomorphisms $K\smashp F(X)\to F(K\smashp X)$ for all $K\in\cat{$\bm H$-SSet}_*, X\in H\text-\mathscr C$. Specializing to $K=S^B$ for some finite $H$-set $B$ and using that all functors in sight are left Quillen for the flat model structures, these derive to isomorphisms $S^B\smashp^{\cat L}\cat{L}F(X)\to \cat{L}F(S^B\smashp^{\cat L}X)$ in the homotopy category. Passing to canonical mates gives us a natural comparison map $\cat{L}F\cat{R}\Omega^B X\to \cat{R}\Omega^B\cat{L}F(X)$ which is again an isomorphism as $\cat{L}F$ is assumed to be an equivalence. Plugging in the definitions, this comparison map is represented for an $X$ that is cofibrant in the \emph{flat} model structure and fibrant in the \emph{projective} one by
\begin{equation}\label{eq:comparison-Omega-LF}
F(Y)\xrightarrow{F(\pi)} F(\Omega^BX) \to \Omega^B F(X) \xrightarrow{\Omega^B\iota} \Omega^B Z
\end{equation}
where $\pi\colon Y\to\Omega^BX$ is a cofibrant replacement in the flat model structure, the unlabelled arrow is the comparison map coming from the enrichment, and $\iota\colon F(X)\to Z$ is a fibrant replacement (say, in the flat model structure); in particular, the composite $(\ref{eq:comparison-Omega-LF})$ is an $H$-global weak equivalence.
With this at hand, we can now prove the claim. As $\textup{Sp}(\ul{F})$ is left Quillen with respect to the flat level model structures, we may assume without loss of generality that $W$ is also fibrant in the flat level model structure. Let now $H$ be a finite group, and let $A,B$ be finite $H$-sets; we have to show that the composite
\begin{equation}\label{eq:derived-adjunct-struc-map-LF}
F(W(A))\xrightarrow{\tilde\sigma}\Omega^B F(W(A\amalg B))\xrightarrow{\Omega^B\iota} \Omega^B Z
\end{equation}
is an $H$-global weak equivalence, where the first map is the ordinary adjoint structure map and $\iota$ is a fibrant replacement $F(W(A\amalg B))\to Z$ in $H\text-\mathscr C_\text{flat}$. For this we pick a factorization
\begin{equation*}
\begin{tikzcd}[cramped]
W(A)\arrow[r, tail, "\kappa"] & Y\arrow[r, "\pi", "\sim"'] & \Omega^B W(A\amalg B)
\end{tikzcd}
\end{equation*}
in $H\text-\mathscr C_\text{flat}$ of $\tilde\sigma$ into a cofibration $\kappa$ followed by a weak equivalence $\pi$; note that $\kappa$ is actually an acyclic cofibration by $2$-out-of-$3$ and that $Y$ is cofibrant as $W(A)$ is, so that $\pi$ is a cofibrant replacement. Then we have a commutative diagram
\begin{equation*}
\begin{tikzcd}
&[1.5em] \Omega^B F(W(A\amalg B))\arrow[r, "\Omega^B\iota"] & \Omega^B Z\\
F(W(A))\arrow[r, "F(\tilde\sigma)"{description}]\arrow[dr, "F(\kappa)"', bend right=17pt]\arrow[ur, "\tilde\sigma", bend left=15pt] & F(\Omega^B W(A\amalg B))\arrow[u] & \\
& F(Y)\arrow[u, "F(\pi)"']
\end{tikzcd}
\end{equation*}
in which $F(\kappa)$ is a weak equivalence as $F$ is left Quillen, while the composite $F(Y)\to\Omega^BZ$ agrees with $(\ref{eq:comparison-Omega-LF})$ for $X\mathrel{:=}W(A\amalg B)\in H\text-\mathscr C$, so it is a weak equivalence by the above. Thus, also the top composite $(\ref{eq:derived-adjunct-struc-map-LF})$ is a weak equivalence by $2$-out-of-$3$, finishing the proof of the claim.
\end{proof}
\end{claim*}
It suffices now to show that the derived unit $X\to\cat{R}U\cat{L}FX$ is an isomorphism whenever $X$ is a flat \emph{global $\Omega$-spectrum}. But indeed, this is represented by the composite $X\to UFX\to UY$ where $FX\to Y$ is a fibrant replacement in the global projective model structure. By the claim, $FX$ is a global $\Omega$-spectrum, so this is actually a level fibrant replacement; thus, this composite also represents the derived unit for the corresponding level model structures and the claim follows again from Lemma~\ref{lemma:sp-level-induced-adjunction}.
\end{proof}
\end{prop}
Finally, let us show that the above construction actually fulfills its purpose:
\begin{thm}\label{thm:sp-is-stable}
Let $\ul{\mathscr C}$ be a global model category. Then $\glo{Sp}(\underline{\mathscr C})$ is stable.
\end{thm}
For the proof of the theorem, we may as usual assume without loss of generality that $\ul{\mathscr C}$ is pointed. We now want to proceed in the same way as for $\glo{GlobalSpectra}$ (see Proposition~\ref{prop:GlobalSpectra-stable} above), so we first need to introduce a suitable smash product with $G$-equivariant symmetric spectra:
\begin{constr}
Let us write $\smashp$ for the essentially unique $\cat{SSet}_*$-enriched functor $\cat{Spectra}\times\textup{Sp}(\underline{\mathscr C})\to\textup{Sp}(\underline{\mathscr C})$
that preserves tensors and colimits in each variable and such that $\bm\Sigma(A,{\textup{--}})\smashp (\bm\Sigma(B,{\textup{--}})\smashp X)=\bm\Sigma(A\amalg B,{\textup{--}})\smashp X$ with the evident functoriality in $A,B\in\bm\Sigma$ and $X\in\mathscr C$. For any finite group $G$, we then obtain a pairing
\begin{equation}\label{eq:tensoring-g-global}
{\textup{--}}\smashp{\textup{--}}\colon\cat{$\bm G$-Spectra}\times G\text{-}\textup{Sp}(\underline{\mathscr C})\to G\text{-}\textup{Sp}(\underline{\mathscr C})
\end{equation}
by pulling through the $G$-actions.
\end{constr}
There is then a unique way to extend the evident isomorphisms
\begin{align*}
\bm\Sigma(\varnothing,{\textup{--}})\smashp (\bm\Sigma(B,{\textup{--}})\smashp X)&= \bm\Sigma(\varnothing\amalg B,{\textup{--}})\smashp X\cong\bm\Sigma(B,{\textup{--}})\smashp X
\intertext{and}
(\bm\Sigma(A,{\textup{--}})\smashp\bm\Sigma(B,{\textup{--}}))\smashp(\bm\Sigma(C,{\textup{--}})\smashp X)&=\bm\Sigma((A\amalg B)\amalg C,{\textup{--}})\smashp X\\&\cong \bm\Sigma(A\amalg(B\amalg C),{\textup{--}})\smashp X\\&=\bm\Sigma(A,{\textup{--}})\smashp(\bm\Sigma(B,{\textup{--}})\smashp (\bm\Sigma(C,{\textup{--}})\smashp X))
\end{align*}
to make $G\text-\textup{Sp}(\ul{\mathscr C})$ tensored over $\cat{$\bm G$-Spectra}$. Since all adjoints exist by local presentability, $G\text-\textup{Sp}(\ul{\mathscr C})$ is then also enriched and cotensored over $\cat{$\bm G$-Spectra}$; the next proposition can therefore be reformulated as saying that $G\text-\textup{Sp}(\ul{\mathscr C})_\text{proj}$ is enriched in the model categorical sense over $\cat{$\bm G$-Spectra}_\text{$G$-equiv.~proj.}$.
\begin{prop}\label{prop:tensoring-omega}
The pairing $(\ref{eq:tensoring-g-global})$ is a left Quillen bifunctor with respect to the $G$-equivariant projective model structure on $\cat{$\bm G$-Spectra}$ and the $G$-global projective model structure on $G\text-\textup{Sp}(\ul{\mathscr C})$.
\end{prop}
For the proof we will need:
\begin{lemma}\label{lemma:shift-right-Quillen}
Let $A$ be any finite $G$-set. Then the shift $\mathop{\textup{sh}}\nolimits^A\colon G\text-\textup{Sp}(\underline{\mathscr C})\to G\text-\textup{Sp}(\underline{\mathscr C})$ is right Quillen for the $G$-global projective model structures.
\begin{proof}
As before, it suffices to show that $\mathop{\textup{sh}}\nolimits^A$ is right Quillen for the corresponding level model structures and that it sends fibrant objects to $G$-global $\Omega$-spectra.
For the first statement, we simply note that ${\textup{ev}}_B\circ\mathop{\textup{sh}}\nolimits^A\colon G\text{-}\textup{Sp}(\mathscr C)\to (G\times\Sigma_B)\text-\mathscr C$ factors as the composition
\begin{equation*}
G\text{-}\textup{Sp}(\underline{\mathscr C})_\text{proj.~lev.}\textup{h}skip0pt minus 2pt\xrightarrow{\textup{h}skip0pt minus 1pt{\textup{ev}}_{A\amalg B}}\textup{h}skip0pt minus 2pt(G\times\Sigma_{A\amalg B})\text{-}\mathscr C_\text{proj}\textup{h}skip0pt minus 2pt\xrightarrow{\textup{res}}\textup{h}skip0pt minus 2pt(G\times\Sigma_A\times\Sigma_B)\text{-}\mathscr C_\text{proj}\textup{h}skip0pt minus 2pt\xrightarrow{\phi^*\textup{h}skip0pt minus 3pt}\textup{h}skip0pt minus 2pt (G\times\Sigma_B)\text-\mathscr C_\text{proj}
\end{equation*}
where $\phi$ is defined via $\phi(g,\sigma)=(g, g.{\textup{--}},\sigma)$, and each of these is right Quillen by definition.
Finally, if $X$ is fibrant, then $\mathop{\textup{sh}}\nolimits^A X$ is level fibrant by the above, hence $\mathop{\textup{sh}}\nolimits^AX$ will be a $G$-global $\Omega$-spectrum if and only if $X(A\amalg B)\to\Omega^{C} X(A\amalg B\amalg C)$ is a $(G\times H)$-global weak equivalence for all finite $H$-sets $B,C$. This follows easily from Remark~\ref{rk:g-global-omega-g-h-sets} (letting $H$ act trivially on $A$ and $G$ act trivially on $B,C$).
\end{proof}
\end{lemma}
\begin{proof}[Proof of Proposition~\ref{prop:tensoring-omega}]
Let us first prove the analogous statement where we equip $\cat{$\bm G$-Spectra}$ with the $G$-equivariant projective \emph{level} model structure. Adjoining over, it will be enough here to show that
\begin{align*}
G\text{-}\textup{Sp}(\underline{\mathscr C})^{\textup{op}}\times G\text{-}\textup{Sp}(\underline{\mathscr C})&\to \cat{$\bm G$-Spectra}\\
X, Y&\mathord{\textup{maps}}to \big(A\mathord{\textup{maps}}to \mathord{\textup{maps}}(X, \mathop{\textup{sh}}\nolimits^A Y)\big)
\end{align*}
(with the evident functoriality in $A,X,Y$) is a right Quillen bifunctor. As all structure in sight is defined levelwise, this amounts to saying that $(X,Y)\mathord{\textup{maps}}to \mathord{\textup{maps}}(X,\mathop{\textup{sh}}\nolimits^AY)$ is a right Quillen bifunctor to $\cat{$\bm{(G\times\Sigma_A)}$-SSet}_{*}$ with the $\mathcal G_{G,\Sigma_A}$-model structure. However, as $\underline{\textup{Sp}}(\underline{\mathscr C})$ is a global model category, $\text{triv}_{\Sigma_A}\colon G\text-\textup{Sp}(\underline{\mathscr C})\to(G\times\Sigma_A)\text-\textup{Sp}(\underline{\mathscr C})$ is right Quillen for the projective model structures, and so is the endofunctor $\mathop{\textup{sh}}\nolimits^A$ of $(G\times\Sigma_A)\text-\textup{Sp}(\underline{\mathscr C})$ (where $G$ acts trivially on $A$ and $\Sigma_A$ acts tautologically) by the previous lemma. Thus, the claim follows from Corollary~\ref{cor:pointed-g-sset-enriched} applied to the global model category $\ul{\textup{Sp}}(\ul{\mathscr C})$.
For the original statement, all that remains now is to show that the pushout product of any standard generating acyclic cofibration $j$ of $\cat{$\bm G$-Spectra}_\text{$G$-equiv.~proj.}$ with any generating cofibration $i$ of $G\text{-}\textup{Sp}(\ul{\mathscr C})$ is again a $G$-global weak equivalence. However, if $j$ is even a level weak equivalence, this follows from the above, while the other generating acyclic cofibrations are precisely the maps $(G_+\smashp_H\kappa_{H,A,B})\ppo k$ for subgroups $H\subset G$, finite $H$-sets $A,B$, and (generating) cofibrations $k$ of $G\text-\textup{Sp}(\cat{SSet})_\text{$G$-equiv.~proj.}$, see Remark~\ref{rk:equivariant-proj-gen-cof}. The pushout product $(G_+\smashp_H\kappa_{H,A,B}\ppo k)\ppo i$ can then be identified with $\kappa_{H,A,B}\ppo_H(G_+\smashp (k\ppo i))$ where $H$ acts on the second factor via its right action on $G$. By the above, $k\ppo i$ is a (level) cofibration; by construction of the $G$-global projective model structure it therefore only remains to show that
\begin{equation*}
G_+\smashp{\textup{--}}\colon G\text-\textup{Sp}(\underline{\mathscr C})\to (G\times H)\text-\textup{Sp}(\underline{\mathscr C})
\end{equation*}
(where $G$ acts diagonally and $H$ acts from the right) is left Quillen for the projective (level) model structures. However, as $\ul{\textup{Sp}}(\ul{\mathscr C})$ is a (pre)global model category and $G$ is cofibrant in $\cat{$\bm{(G\times H)}$-SSet}_{\mathcal G_{G,H}}$ ($H$ acting freely from the right), this is simply an instance of Corollary~\ref{cor:pointed-g-sset-enriched} again.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:sp-is-stable}]
This follows from Proposition~\ref{prop:tensoring-omega} in the same way as in the proof of Proposition~\ref{prop:GlobalSpectra-stable}.
\end{proof}
\subsection{The universal property of spectrification} Throughout, let $\ul{\mathscr C}$ and $\ul{\mathscr D}$ be global model categories. We begin by explaining in which sense the above construction $\ul{\textup{Sp}}$ is idempotent:
\begin{lemma}
Assume $\ul{\mathscr C}$ is pointed. Then we have a global Quillen adjunction
\begin{equation}\label{eq:stabilization-adjunction}
\ul{\Sigma^\infty}\mathrel{:=}\ul{\mathbb S\smashp{\textup{--}}}\colon\underline{\mathscr C}\rightleftarrows\glo{Sp}(\underline{\mathscr C}) :\!\ul{\Omega^\infty}\mathrel{:=}\ul{{\textup{ev}}_\varnothing}.
\end{equation}
\begin{proof}
It is clear from the definitions that $\ul{\Omega^\infty}$ is right Quillen (already for the level model structures).
\end{proof}
\end{lemma}
\begin{cor}
Let $\ul{\mathscr C}$ be any global model category. Then we have a global Quillen adjunction
\begin{equation*}
\ul{\Sigma^\infty_+}\mathrel{:=}\ul{\mathbb S\smashp({\textup{--}})_+}\colon\ul{\mathscr C}\rightleftarrows\ul{\textup{Sp}}(\ul{\mathscr C}) :\!\ul{\Omega^\infty}\mathrel{:=}\ul{{\textup{ev}}_\varnothing}.\pushQED{\qed}\qedhere\popQED
\end{equation*}
\end{cor}
\begin{thm}\label{thm:stabilization-preserves-stable}
Let $\underline{\mathscr C}$ be a global model category. Then $\underline{\mathscr C}$ is stable if and only if it is pointed and the global Quillen adjunction $(\ref{eq:stabilization-adjunction})$ is a global Quillen equivalence.
\begin{proof}
Stability is preserved under global Quillen equivalences (Lemma~\ref{lemma:stability-stable}), so `$\Leftarrow$' follows from Theorem~\ref{thm:sp-is-stable}. By the usual shifting argument it now suffices to show that $\Sigma^\infty\colon\mathscr C\rightleftarrows\textup{Sp}(\underline{\mathscr C}) :\!\Omega^\infty$ is a Quillen equivalence whenever $\ul{\mathscr C}$ is stable.
For this we first observe that $\cat{L}\Sigma^\infty$ takes values in global $\Omega$-spectra. Namely, if $X\in \mathscr C_\text{flat}$ is cofibrant, then the ordinary structure maps $S^B\smashp (\Sigma^\infty X)(A)\to(\Sigma^\infty X)(A\amalg B)$ are isomorphisms for all finite groups $H$ and finite $H$-sets $A$ and $B$. However, $(\Sigma^\infty X)(A)=S^A\smashp X$ is cofibrant in the flat $H$-global model structure, so we can identify this with the derived map $S^B\smashp^{\cat L}(\Sigma^\infty X)(A)\to (\Sigma^\infty X)(A\amalg B)$. By stability, the adjoint map $(\Sigma^\infty X)(A)\to \cat{R}\Omega^B(\Sigma^\infty X)(A\amalg B)$ is then also an $H$-global weak equivalence as desired.
It follows immediately that for every cofibrant $X\in \mathscr C_\text{flat}$ the ordinary unit $X\to\Omega^\infty\Sigma^\infty X$ already represents the derived unit. As the former is an isomorphism, the derived unit is a global weak equivalence.
To complete the proof, it is now enough to show that $\Omega^\infty$ reflects weak equivalences between global $\Omega$-spectra. But indeed, if $f\colon X\to Y$ is a map of global $\Omega$-spectra such that $f(\varnothing)$ is a global weak equivalence, then $\cat{R}\Omega^A f(A)$ is a $\Sigma_A$-global weak equivalence as it is conjugate to $f(\varnothing)$ (equipped with the trivial $\Sigma_A$-action). Thus, $f(A)$ must be a $\Sigma_A$-global weak equivalence as $\Omega^A\colon \Sigma_A\text-\mathscr C\to\Sigma_A\text-\mathscr C$ is part of a Quillen equivalence (say, for the flat model structures).
\end{proof}
\end{thm}
\begin{defi}\label{defi:global-stabilization}
A global Quillen adjunction $\underline{\mathscr C}\rightleftarrows\underline{\mathscr D}$ is called a \emph{global stabilization} if $\underline{\mathscr D}$ is stable and the induced global Quillen adjunction $\glo{Sp}(\underline{\mathscr C})\rightleftarrows\glo{Sp}(\underline{\mathscr D})$ is a global Quillen equivalence.
\end{defi}
\begin{thm}\label{thm:sp-is-stabilization}
Let $\underline{\mathscr C}$ be any global model category. Then $\ul{\Sigma^\infty_+}\colon\underline{\mathscr C}\rightleftarrows\glo{Sp}(\underline{\mathscr C}):\!\ul{\Omega^\infty}$ is a global stabilization.
\begin{proof}
We have already shown in Theorem~\ref{thm:sp-is-stable} that $\glo{Sp}(\underline{\mathscr C})$ is stable. Replacing $\underline{\mathscr C}$ by $\underline{G\text{-}\mathscr C}$ as usual, it then only remains that $\textup{Sp}(\underline{\Sigma^\infty_+})\colon\textup{Sp}(\underline{\mathscr C})\rightleftarrows\textup{Sp}(\glo{Sp}(\underline{\mathscr C})):\textup{Sp}(\underline{\Omega^\infty})$ is a Quillen equivalence. Comparing the right adjoints we immediately see that this agrees with $\textup{Sp}(\ul{\Sigma^\infty}):\textup{Sp}(\ul{\mathscr C}_*)\rightleftarrows\textup{Sp}(\textup{Sp}(\ul{\mathscr C}_*)):\textup{Sp}(\ul{\Omega^\infty})$, so we may further assume that $\ul{\mathscr C}$ is pointed.
For the proof, we identify the $\cat{SSet}_*$-category $\textup{Sp}(\glo{Sp}(\underline{\mathscr C}))$ with the category of $\cat{SSet}_*$-enriched functors $\bm\Sigma\otimes\bm\Sigma\to\mathscr C$ where $\bm\Sigma\otimes\bm\Sigma$ denotes the $\cat{SSet}_*$-category with objects the pairs $(A,B)$ of finite sets and with mapping spaces
\begin{equation*}
\mathord{\textup{maps}}_{\bm\Sigma\otimes\bm\Sigma}((A,B),(A',B'))=\mathord{\textup{maps}}_{\bm\Sigma}(A,A')\smashp\mathord{\textup{maps}}_{\bm\Sigma}(B,B')
\end{equation*}
with the evident composition. We take the convention that the first factor of $\bm\Sigma\otimes\bm\Sigma$ corresponds to the `outer $\textup{Sp}$', so $\textup{Sp}(\underline{\Omega^\infty})$ corresponds to restriction along the inclusion $i_1\colon\bm\Sigma\to \bm\Sigma\otimes\bm\Sigma,A\mathord{\textup{maps}}to(A,\varnothing)$. On the other hand, restricting along the other inclusion $i_2\colon\bm\Sigma\to\bm\Sigma\otimes\bm\Sigma$ corresponds to the Quillen equivalence $\Omega^\infty$.
We now write $\Pi\colon\bm\Sigma\otimes\bm\Sigma\to\bm\Sigma$ for the usual symmetric monoidal structure (used to construct the smash product of spectra), given on objects by $(A,B)\mathord{\textup{maps}}to A\amalg B$. By restricting, this gives rise to a simplicial functor $\Pi^*\colon\textup{Sp}(\underline{\mathscr C})\to\textup{Sp}(\glo{Sp}(\underline{\mathscr C}))$, which admits a simplicial left adjoint $\Pi_!$ by enriched Kan extension.
\begin{claim*}
The simplicial adjunction $\Pi_!\dashv\Pi^*$ is a Quillen adjunction with respect to the projective model structures.
\begin{proof}
Let us consider the case of the level model structures first, which amounts to saying that $\textup{Sp}(\ul{\mathscr C})\to \Sigma_A\text-\textup{Sp}(\ul{\mathscr C}), X\mathord{\textup{maps}}to (\Pi^*X)(A)$ is right Quillen for the projective model structures everywhere. However, this can be identified with the composition
\begin{equation*}
\textup{Sp}(\underline{\mathscr C})_\text{proj}\xrightarrow{\text{triv}_{\Sigma_A}}\Sigma_A\text-\textup{Sp}(\underline{\mathscr C})_\text{proj}\xrightarrow{\mathop{\textup{sh}}\nolimits^A}\Sigma_A\text-\textup{Sp}(\underline{\mathscr C})_\text{proj}
\end{equation*}
of which the first functor is right Quillen as $\underline{\textup{Sp}}(\underline{\mathscr C})$ is a global model category while the second one is so by Lemma~\ref{lemma:shift-right-Quillen}.
It then only remains to show that if $X$ is fibrant in the global projective model structure on $\textup{Sp}(\underline{\mathscr C})$, then $\Pi^*X$ is a global $\Omega$-spectrum, i.e.~for every finite group $H$ and every finite $H$-sets $A,B$, the map $\mathop{\textup{sh}}\nolimits^A X\to\Omega^B\mathop{\textup{sh}}\nolimits^{A\amalg B}X$ is an $H$-global weak equivalence. But indeed, this is even an $H$-global level weak equivalence: if $C$ is any finite set, then after evaluating at $C$ this is simply the adjoint structure map $X(C\amalg A)\to \Omega^B(C\amalg A\amalg B)$, which is an $(H\times\Sigma_C)$-global weak equivalence since $X$ was assumed to be a global $\Omega$-spectrum (letting $\Sigma_C$ act trivially on $A$ and $B$ while $H$ acts trivially on $C$).
\end{proof}
\end{claim*}
The unitality isomorphisms of the symmetric monoidal structure on $\bm\Sigma$ now give us isomorphisms $\Omega^\infty\circ\Pi^*\cong{\textup{id}}\cong \textup{Sp}(\underline{\Omega^\infty})\circ\Pi^*$. As all functors are right Quillen, this induces isomorphisms $\cat{R}\Omega^\infty\circ\cat{R}\Pi^*\cong{\textup{id}}\cong \cat{R}\textup{Sp}(\ul{\Omega^\infty})\circ\cat{R}\Pi^*$ of derived functors. In particular, as $\cat{R}\Omega^\infty$ is an equivalence (Theorems~\ref{thm:sp-is-stable} and~\ref{thm:stabilization-preserves-stable}), also $\cat{R}\Pi^*$ is an equivalence by $2$-out-of-$3$, and hence so is $\cat{R}\textup{Sp}(\underline{\Omega^\infty})$ by the same argument.
\end{proof}
\end{thm}
\begin{rk}
Write $\cat{GLOBMOD}$ for the opposite of the large category of global model categories and global right Quillen functors, localized with respect to the global Quillen equivalences. Then Proposition~\ref{prop:sp-induced-adjunction} implies that $\ul\textup{Sp}$ descends to an endofunctor $Q$ of $\cat{GLOBMOD}$. Moreover the global Quillen adjunctions $\smash{\ul{\Sigma^\infty_+}\dashv \ul{\Omega^\infty}}$ induce a natural transformation $\eta\colon{\textup{id}}\Rightarrow Q$.\footnote{Here it comes in handy that we defined the underlying $1$-category to consist of right Quillen functors: $\ul{\Omega^\infty}=\ul{{\textup{ev}}_\varnothing}$ is strictly natural while $\ul{\Sigma^\infty_+}$ is only pseudonatural. However, the approach via left Quillen functors or by encoding both adjoints at the same time could also be made to work, as it is not hard to show using a cocylinder argument that isomorphic functors become \emph{equal} after localizing at the global Quillen equivalences in either case.}
By Theorems~\ref{thm:sp-is-stable} and~\ref{thm:stabilization-preserves-stable}, the map $\eta_{Q\ul{\mathscr C}}$ is an isomorphism in $\cat{GLOBMOD}$ for every $\ul{\mathscr C}$; similarly, Theorem~\ref{thm:sp-is-stabilization} above shows that $Q\eta_{\ul{\mathscr C}}$ is an isomorphism. Thus, it follows by abstract nonsense that $Q$ is a Bousfield localization onto its essential image (i.e.~the stable global model categories) with unit given by $\eta$. Put differently, for any global model category $\ul{\mathscr C}$, $\ul{\Omega^\infty}\colon\textup{Sp}(\ul{\mathscr C})\to \ul{\mathscr C}$ (or more generally the right adjoint in any global stabilization in the sense of Definition~\ref{defi:global-stabilization}) is the homotopy universal example of a global right Quillen functor from a \emph{stable} global model category, and dually for the left adjoints.
\end{rk}
\section{The stabilization of global spaces}\label{sec:global-spectra}
In this section we will prove:
\begin{thm}\label{thm:stabilization-global-spaces}
The global Quillen adjunction
\begin{equation}\label{eq:global-stabilization-S-Sp}
\ul{\Sigma^\bullet_+}\colon\glo{GlobalSpaces}\rightleftarrows\glo{GlobalSpectra} :\!\ul{\Omega^\bullet}
\end{equation}
from Example~\ref{ex:S-Sp-adjunction} is a global stabilization.
\end{thm}
This can be seen as a sanity check for our framework, but it also allows us to describe the passage from unstable to stable global homotopy theory via a universal property: $(\ref{eq:global-stabilization-S-Sp})$ is the homotopy-universal example of a global Quillen adjunction from $\glo{GlobalSpaces}$ to a stable global model category.
For the proof, we will compare $\glo{GlobalSpectra}$ to the global stabilization constructed in the previous section. This will require some preparations:
\begin{constr}
We define a functor $\Delta^*\colon\textup{Sp}(\glo{GlobalSpaces})\to\cat{Spectra}$ as follows: if $X$ is a spectrum in pointed $\mathcal I$-simplicial sets, then $\Delta^*X(A)=X(A)(A)$ with structure maps
\begin{equation*}
S^{B\setminus i(A)} \smashp X(A)(A) \xrightarrow{\sigma} X(B)(A) \xrightarrow{X(B)(i)} X(B)(B)
\end{equation*}
for every injection $i\colon A\to B$ and with the evident enriched functoriality in $X$.
By the enriched Yoneda Lemma, $\Delta^*$ admits an enriched left adjoint $\Delta_!$ which is characterized up to unique enriched isomorphism by the condition that it preserves colimits and tensors and satisfies $\Delta_!\bm\Sigma(A,{\textup{--}})=\bm\Sigma(A,{\textup{--}})\smashp \mathcal I(A,{\textup{--}})_+$ with the evident functoriality in $A$.
Writing simplices of $\bm\Sigma(A,B)$ for $B\in\bm\Sigma$ as equivalence classes $[i,\sigma]$ of an injection $i\colon A\to B$ and a simplex $\sigma$ of $S^{B\setminus i(A)}$, the unit of $\bm\Sigma(A,{\textup{--}})$ is then given in degree $B$ by $\bm\Sigma(A,B)\to\bm\Sigma(A,B)\smashp\mathcal I(A,B)_+, [i,\sigma]\mathord{\textup{maps}}to [i,\sigma]\smashp i$.
By enriched Kan extension, $\Delta^*$ moreover admits a simplicial right adjoint $\Delta_*$.
\end{constr}
Our actual goal now is to prove:
\begin{thm}\label{thm:delta-quillen-equivalence}
For any finite group $G$ the simplicial adjunction
\begin{equation}\label{eq:delta-shriek-star}
\Delta_!\colon \cat{$\bm G$-Spectra}_\textup{$G$-global projective}\rightleftarrows G\text-\textup{Sp}(\glo{GlobalSpaces})_\textup{$G$-global projective} :\!\Delta^*
\end{equation}
is a Quillen equivalence.
\end{thm}
The proof will occupy the remainder of this section; for now, let us already remark how it implies Theorem~\ref{thm:stabilization-global-spaces}:
\begin{proof}[Proof of Theorem~\ref{thm:stabilization-global-spaces}]
We already know that $\ul{\Sigma^\bullet_+}\dashv\ul{\Omega^\bullet}$ is a global Quillen adjunction and that $\glo{GlobalSpectra}$ is stable (Proposition~\ref{prop:GlobalSpectra-stable}). Thus, it only remains to show that $G\text-\textup{Sp}(\ul{\Omega^\bullet})$ derives to an equivalence of homotopy categories for every finite $G$. To this end, we consider the diagram
\begin{equation}\label{diag:omega-delta}
\begin{tikzcd}[column sep=tiny]
G\text-\textup{Sp}(\glo{GlobalSpectra})_\text{proj.~$G$-gl.}\arrow[dr, bend right=10pt, "G\text-\textup{Sp}(\underline{\Omega^\bullet})"']\arrow[rr, "\Omega^\infty"] && \cat{$\bm G$-Spectra}_\text{proj.~$G$-gl.}\\
& G\text-\textup{Sp}(\glo{GlobalSpaces})_\text{proj.~$G$-gl.}\arrow[ur, bend right=10pt, "\Delta^*"']
\end{tikzcd}
\end{equation}
of right Quillen functors. This does not commute strictly, but after restricting to fibrant objects, Remark~\ref{rk:Omega-bullet-on-fibrant} shows that the lower composite is given up to natural $G$-global (level) weak equivalence by sending a $G$-bispectrum $X$ to the spectrum $\delta(X)$ given by $A\mathord{\textup{maps}}to\Omega^A X(A)(A)$ with the evident functoriality. We moreover have a natural map $\sigma\colon\Omega^\infty X\to \delta(X)$ given in degree $A$ by the map $X(\varnothing)(A)\to \Omega^AX(A)(A)$ induced by the adjunct structure map. If $X$ is fibrant, then each $X(\varnothing)\to\Omega^AX(A)$ is a $(G\times\Sigma_A)$-global weak equivalence of fibrant objects in the projective $(G\times\Sigma_A)$-global model structure, hence a $(G\times\Sigma_A)$-global level weak equivalence. In particular, after evaluating at $A$, this is a $\mathcal G_{\Sigma_A,G\times\Sigma_A}$-equivariant weak equivalence, hence a $\mathcal G_{\Sigma_A,G}$-equivariant weak equivalence with respect to the diagonal $\Sigma_A$-action. Thus, $\sigma$ is a $G$-global (level) weak equivalence for every fibrant $X$. Altogether, we therefore have an isomorphism $\cat{R}\Omega^\infty\cong\cat{R}(G\text-\textup{Sp}(\ul{\Omega^\bullet}))\circ\cat{R}\Delta^*$ of right derived functors.
However, the top arrow in $(\ref{diag:omega-delta})$ induces an equivalence of homotopy categories by stability of $\glo{GlobalSpectra}$ (Proposition~\ref{prop:GlobalSpectra-stable}), and so does $\Delta^*$ by the previous theorem. The claim now follows from $2$-out-of-$3$.
\end{proof}
It remains to prove Theorem~\ref{thm:delta-quillen-equivalence}.
\begin{lemma}
The simplicial adjunction $(\ref{eq:delta-shriek-star})$ is a Quillen adjunction.
\begin{proof}
We will first show that $\Delta^*$ preserves (acyclic) level fibrations. Indeed, if $f\colon X\to Y$ is any map in $G\text-\textup{Sp}(\glo{GlobalSpaces})$, then $(\Delta^*f)(A)=f(A)(A)$. If now $f$ is a $G$-global (acyclic) level fibration, then $f(A)$ is a $(G\times\Sigma_A)$-global (acyclic) fibration, hence in particular a $(G\times\Sigma_A)$-global (acyclic) level fibration, so that $f(A)(A)$ is an (acyclic) $\mathcal G_{\Sigma_A,G\times\Sigma_A}$-equivariant fibration. As before, it is then in particular a $\mathcal G_{\Sigma_A,G}$-equivariant (acyclic) fibration with respect to the diagonal action, i.e.~$\Delta^*f$ is an (acyclic) level fibration as desired.
To complete the proof, it now only remains to show that $\Delta^*$ sends fibrant objects to $G$-global $\Omega$-spectra. But indeed, if $X$ is fibrant and $A,B$ are finite $H$-sets, then $X(A)\to\Omega^BX(A\amalg B)$ is a $(G\times H)$-global weak equivalence between projectively fibrant $(G\times H)$-global spaces, hence a $(G\times H)$-global level weak equivalence. In particular, $(\Delta^*X)(A)=X(A)(A)\to\Omega^BX(A\amalg B)(A)$ is a $\mathcal G_{\Sigma_A,G\times H}$-equivariant weak equivalence. Thus, if $A$ is \emph{faithful}, then this is a $\mathcal G_{H,G\times H}$-weak equivalence by Lemma~\ref{lemma:graph-source} and whence a $\mathcal G_{H,G}$-equivariant weak equivalence with respect to the diagonal action.
Now the adjoint structure map of $\Delta^*X$ factors as
\begin{equation*}
X(A)(A)\xrightarrow{\tilde\sigma(A)}\Omega^BX(A\amalg B)(A)\xrightarrow{\Omega^BX(A\amalg B)(\text{incl})} \Omega^BX(A\amalg B)(A\amalg B);
\end{equation*}
if $H$ acts faithfully on $A$, then the first map is a $\mathcal G_{H,G}$-equivariant weak equivalence by the above, and so is the second one by the same computation as $X(A\amalg B)$ is a fibrant $(G\times H)$-$\mathcal I$-space.
\end{proof}
\end{lemma}
\begin{lemma}\label{lemma:Delta-star-conservative}
The functor $\Delta^*\colon G\text-\textup{Sp}(\glo{GlobalSpaces})_\textup{$G$-global proj.}\to\cat{$\bm G$-Spectra}_\textup{$G$-global proj.}$ reflects $G$-global weak equivalences between fibrant objects, i.e.~its right derived functor $\cat{R}\Delta^*$ is conservative.
\begin{proof}
Let $f\colon X\to Y$ be a map of fibrant objects such that $\Delta^*f$ is a $G$-global weak equivalence. As $\Delta^*$ is right Quillen, $\Delta^*X$ and $\Delta^*Y$ are fibrant, so $\Delta^*f$ is even a $G$-global level weak equivalence, i.e.~$f(A)(A)$ is a $\mathcal G_{H,G}$-equivariant weak equivalence of $(G\times H)$-simplicial sets for every finite group $H$ and any finite faithful $H$-set $A$. We want to show that $f(A)$ is already a $(G\times\Sigma_A)$-global level weak equivalence, i.e.~that for every finite set $B$ the map $f(A)(B)$ is a $\mathcal G_{\Sigma_B,G\times\Sigma_A}$-equivariant weak equivalence, or put differently that for every finite group $H$ acting arbitrarily on $A$ and faithfully on $B$ the map $f(A)(B)$ is a $\mathcal G_{H,G}$-equivariant weak equivalence.
For this we consider the commutative diagram
\begin{equation*}
\begin{tikzcd}
X(A)(B)\arrow[r,"\tilde\sigma"]\arrow[d,"f(A)(B)"'] & \Omega^B X(A\amalg B)(B)\arrow[d, "\Omega^B f(A\amalg B)(B)"]\arrow[r, "\Omega^B X(A\amalg B)(\textup{incl})"] &[5em] \Omega^B(\Delta^*X)(A\amalg B)\arrow[d, "\Omega^B(\Delta^*f)(A\amalg B)"]\\
Y(A)(B)\arrow[r,"\tilde\sigma"'] & \Omega^B Y(A\amalg B)(B)\arrow[r, "\Omega^B X(A\amalg B)(\textup{incl})"'] & \Omega^B(\Delta^*Y)(A\amalg B)
\end{tikzcd}
\end{equation*}
As $H$ acts faitfully on $A\amalg B$, the right hand vertical map is a $\mathcal G_{H,G}$-equivariant weak equivalence by the above. Moreover, as $X(A)\to\Omega^BX(A\amalg B)$ is a $(G\times H)$-global weak equivalence of fibrant objects and $H$ acts faithfully on $B$, the top left horizontal map is a $\mathcal G_{H,G}$-weak equivalence, and so is the lower left horizontal map by the same argument. Finally, $X(A\amalg B)$ is a fibrant $(G\times H)$-$\mathcal I$-space, so that the top right horizontal map is a $\mathcal G_{H,G}$-equivariant weak equivalence, and likewise for the lower right horizontal map. The claim now follows by $2$-out-of-$3$.
\end{proof}
\end{lemma}
We now want to prove the following strengthening of the above lemma:
\begin{prop}\label{prop:Delta-star-homotopical}
The functor $\Delta^*$ creates $G$-global weak equivalences.
\end{prop}
For the proof, it will be crucial to understand the behaviour of $\Delta^*$ on generating (acyclic) \emph{cofibrations}, and more generally on maps of the form $\bm\Sigma(A,{\textup{--}})\smashp_{\Sigma_A}f$ for maps $f$ in $\cat{$\bm{(G\times\Sigma_A)}$-$\bm{\mathcal I}$-SSet}_*$. For this we recall the following standard construction:
\begin{constr}\label{constr:tensor-Sp-I}
Let $X$ be a spectrum and let $Y$ be a pointed $\mathcal I$-simplicial set. Then we write $X\otimes Y$ for the spectrum with $(X\otimes Y)(A)=X(A)\smashp Y(A)$ and structure maps
\begin{equation*}
S^{B\setminus i(A)}\smashp (X\otimes Y)(A)=S^{B\setminus A}\smashp X(A)\smashp Y(A)\xrightarrow{\sigma\smashp Y(i)} X(B)\smashp Y(B)=(X\otimes Y)(B)
\end{equation*}
for every injection $i\colon A\to B$. This becomes a functor $\cat{Spectra}\times\cat{$\bm{\mathcal I}$-SSet}_*\to\cat{Spectra}$ in the evident way, which we as usual promote to
\begin{equation}\label{eq:tensor-product}
{\textup{--}}\otimes{\textup{--}}\colon\cat{$\bm G$-Spectra}\times\cat{$\bm G$-$\bm{\mathcal I}$-SSet}_*\to\cat{$\bm G$-Spectra}.
\end{equation}
\end{constr}
Put differently we have for any $G$-spectrum $X$ and any $G$-$\mathcal I$-simplicial set $Y$ an equality $X\otimes Y=\Delta^*(X\smashp Y)$ for the levelwise smash product $\cat{$\bm G$-Spectra}\times\cat{$\bm G$-$\bm{\mathcal I}$-SSet}_*\to G\text-\textup{Sp}(\glo{GlobalSpaces}_*)=G\text-\textup{Sp}(\glo{GlobalSpaces})$, and likewise for maps.
\begin{prop}\label{prop:tensor-homotopical}
The tensor product $(\ref{eq:tensor-product})$ preserves $G$-global weak equivalences in each variable.
\begin{proof}
Let $X$ be a $G$-spectrum and let $g\colon Y\to Y'$ be a $G$-global weak equivalence in $\cat{$\bm G$-$\bm{\mathcal I}$-SSet}_*$. We will show that $X\otimes g$ is even a $\ul\pi_*$-isomorphism, for which we let $\phi\colon H\to G$ be any homomorphism. Then the effect of $\phi^*(X\otimes g)=(\phi^*X)\otimes(\phi^*g)$ on $\pi^H_*$ agrees up to conjugation by isomorphisms with the one of $\phi^*X\smashp (\phi^*g)(\mathcal U_H)$, see \cite[Lemma~3.2.11]{g-global}. But $(\phi^*g)(\mathcal U_H)$ is an $H$-equivariant weak equivalence, so $\phi^*X\smashp (\phi^*g)(\mathcal U_H)$ is even an $H$-equivariant level weak equivalence, in particular a $\ul\pi_*$-isomorphism. This completes the proof that the tensor product preserves $G$-global weak equivalences in the second variable.
On the other hand, let $f\colon X\to X'$ be a $G$-global weak equivalence of $G$-global spectra and let $Y$ be any $G$-global space; we want to show that $f\otimes Y$ is a $G$-global weak equivalence. Arguing precisely as above, we see that ${\textup{--}}\otimes Y$ preserves $\ul\pi_*$-isomorphisms, so we may assume without loss of generality that $f$ is a map between flat $G$-spectra. Under this assumption, \cite[Proposition~3.2.14]{g-global} provides us with a commutative diagram
\begin{equation*}
\begin{tikzcd}
X\smashp\Sigma^\bullet Y\arrow[r, "\psi", "\sim"']\arrow[d, "f\smashp\Sigma^\bullet Y"'] & X\otimes Y \arrow[d, "f\otimes Y"]\\
X'\smashp\Sigma^\bullet Y\arrow[r, "\psi"', "\sim"] & X'\otimes Y
\end{tikzcd}
\end{equation*}
in which the horizontal maps are $\ul\pi_*$-isomorphisms (in particular $G$-global weak equivalences). By Proposition~\ref{prop:flatness-theorem} the left hand vertical arrow is a $G$-global weak equivalence, so the claim follows by $2$-out-of-$3$.
\end{proof}
\end{prop}
\begin{proof}[Proof of Proposition~\ref{prop:Delta-star-homotopical}]
By Lemma~\ref{lemma:Delta-star-conservative}, it suffices to prove that $\Delta^*$ is homotopical. We already know that $\Delta^*$ preserves acyclic fibrations, so it only remains to show that it also sends acyclic cofibrations to weak equivalences. To this end, we will show that $\Delta^*$ is in fact also \emph{left} Quillen as a functor to $\cat{$\bm G$-Spectra}_\textup{injective $G$-global}$.
Let us first prove that $\Delta^*$ is left Quillen as a functor $G\text-\textup{Sp}(\glo{GlobalSpaces})_\text{proj.~$G$-global level}\to\cat{$\bm G$-Spectra}_\text{inj.~$G$-global}$. Indeed, it clearly sends generating cofibrations to injective cofibrations, so we only need to show that it sends generating acyclic cofibrations to weak equivalences. Such a generating acyclic cofibration is now of the form $\bm\Sigma(A,{\textup{--}})\smashp_{\Sigma_A} j_+$ for some $A\in\bm\Sigma$ and $j$ a (generating) acyclic cofibration in $\cat{$\bm{(G\times\Sigma_A)}$-$\bm{\mathcal I}$-SSet}$. Then
\begin{equation}\label{eq:Delta-star-gen-acyc-level}
\Delta^*(\bm\Sigma(A,{\textup{--}})\smashp_{\Sigma_A}j_+)=(\bm\Sigma(A,{\textup{--}})\otimes j_+)/\Sigma_A;
\end{equation}
but $\bm\Sigma(A,{\textup{--}})\otimes j_+$ is a $(G\times\Sigma_A)$-global weak equivalence by Proposition~\ref{prop:tensor-homotopical}, and $\Sigma_A$ acts freely on $\bm\Sigma(A,{\textup{--}})$, so Proposition~\ref{prop:free-quotient-spectra} shows that the quotient $(\ref{eq:Delta-star-gen-acyc-level})$ is a $G$-global weak equivalence. This completes the argument for the level model structure.
For the actual claim, it suffices now by Proposition~\ref{prop:proj-level-to-global} to show that $\Delta^*$ sends the maps in the set $S_G$ from $(\ref{eq:localizing-set-G-gl})$ to weak equivalences, i.e.~the maps $\kappa_{H,A,B}\ppo_H i_+$ for finite groups $H$, finite $H$-sets $A,B$, and generating cofibrations $i\colon X\to Y$ of the projective model structure on $\cat{$\bm{(G\times H)}$-$\bm{\mathcal I}$-SSet}_*$.
For this, we will pick the generating cofibrations as in Theorem~\ref{thm:I-G-glob} (so that they are maps between cofibrant objects), and we will show that $f\mathrel{:=}\Delta^*(\kappa_{H,A,B}\ppo i_+)$ is a $(G\times H)$-global weak equivalence; the claim will then follow from Proposition~\ref{prop:free-quotient-spectra} again as $f$ is an injective cofibration and $H$ acts levelwise freely on projectively cofibrant $(G\times H)$-global spaces.
To show that $f$ is a $(G\times H)$-global weak equivalence, write $S,T$ for the source and target of $\kappa_{H,A,B}$ and consider the image
\begin{equation*}
\begin{tikzcd}
S\otimes X_+\arrow[dr,phantom, "\ulcorner"{very near end}]\arrow[d, "\kappa_{H,A,B}\otimes X_+"']\arrow[r, "S\otimes i_+"] & S\otimes Y_+\arrow[d]\arrow[ddr, bend left=15pt, "\kappa_{H,A,B}\otimes Y_+"]\\
T\otimes X_+\arrow[r]\arrow[drr, bend right=15pt, "T\otimes i_+"'] & P\arrow[dr, "f"{description}]\\[-1em]
&&[-1em] T\otimes Y_+
\end{tikzcd}
\end{equation*}
under $\Delta^*$ of the diagram defining $\kappa_{H,A,B}\ppo i_+$. Now $\kappa_{H,A,B}$ is an $H$-equivariant weak equivalence of $H$-equivariantly \emph{projectively} cofibrant $H$-spectra, whence an $H$-global weak equivalence by Proposition~\ref{prop:equivariant-vs-global} and thus a $(G\times H)$-global weak equivalence with respect to the trivial $G$-actions. Proposition~\ref{prop:tensor-homotopical} therefore shows that $\kappa_{H,A,B}\otimes X_+$ and $\kappa_{H,A,B}\otimes Y_+$ are $(G\times H)$-global weak equivalences. But on the other hand $\kappa_{H,A,B}\otimes X_+$ is also an injective cofibration by direct inspection, so the pushout $S\otimes Y_+\to P$ is again a $(G\times H)$-global weak equivalence. We conclude by $2$-out-of-$3$ that also $f$ is a $(G\times H)$-global weak equivalence, which then completes the proof that $\Delta^*(\kappa_{H,A,B}\ppo_H i_+)$ is a $G$-global weak equivalence and hence the proof of the proposition.
\end{proof}
\begin{prop}
Let $H$ be a finite group, let $A$ be a finite faithful $H$-set, and let $\phi\colon H\to G$ be any homomorphism. Then the unit
\begin{equation*}
\eta\colon\bm\Sigma(A,{\textup{--}})\smashp_\phi G_+ \to \Delta^*\Delta_!(\bm\Sigma(A,{\textup{--}})\smashp_\phi G_+)
\end{equation*}
is a $G$-global weak equivalence.
\begin{proof}
By design, $\eta$ is induced by the `diagonal' map $\bm\Sigma(A,{\textup{--}})\to \bm\Sigma(A,{\textup{--}})\otimes \mathcal I(A,{\textup{--}})_+$; in particular, it has a left inverse induced by the unique map $p\colon\mathcal I(A,{\textup{--}})\to *$. By $2$-out-of-$3$ it therefore suffices to show that $(\bm\Sigma(A,{\textup{--}})\otimes p_+)\smashp_\phi G_+=\phi_!(\bm\Sigma(A,{\textup{--}})\otimes p_+)$ is a $G$-global weak equivalence.
For this we note that $p$ is an $H$-global weak equivalence, whence so is $\bm\Sigma(A,{\textup{--}})\otimes p_+$ by Proposition~\ref{prop:tensor-homotopical}. By faithfulness, $H$ acts freely on $\bm\Sigma(A,{\textup{--}})$ outside the basepoint, so the claim follows again from Proposition~\ref{prop:free-quotient-spectra}.
\end{proof}
\end{prop}
\begin{proof}[Proof of Theorem~\ref{thm:delta-quillen-equivalence}]
As $\cat{R}\Delta^*$ is conservative (Lemma~\ref{lemma:Delta-star-conservative}), it is enough to show that the derived unit is an isomorphism in the homotopy category. As $\Delta^*$ is homotopical (Proposition~\ref{prop:Delta-star-homotopical}), this amounts to saying that the ordinary unit $X\to\Delta^*\Delta_!X$ is a $G$-global weak equivalence for every projectively cofibrant $G$-global spectrum $X$.
This is a standard cell induction argument: namely, $\Delta_!$ is left Quillen while $\Delta^*$ is a left adjoint sending cofibrations to \emph{injective} cofibrations, so that it is enough to prove the claim for the sources and targets of the standard generating cofibrations, see e.g.~\cite[Lemma~1.2.64]{g-global}. However, these are of the form $\bm\Sigma(A,{\textup{--}})\smashp_\phi G_+\smashp K_+$ for some simplicial set $K$; as the tensoring over $\cat{SSet}$ is homotopical and the unit is compatible with the tensoring ($\Delta_!\dashv\Delta^*$ being a simplicial adjunction), the claim therefore follows from the previous proposition.
\end{proof}
\section{Global brave new algebra}\label{section:brave-new-algebra}
We now turn our attention to multiplicative structures in stable $G$-global homotopy theory, generalizing results for $G=1$ by Schwede \cite[Chapter~5]{schwede-book} and Hausmann \cite[Section~3]{hausmann-global}.
\subsection{Positive model structures} Already in the non-equivariant setting, the study of commutative ring spectra from a model categorical perspective requires one to introduce suitable `positive' model structures. This subsection is devoted to the construction of positive flat and projective $G$-global model structures; however, as most of this is entirely parallel to the construction of the usual $G$-global model structures \cite[3.1]{g-global}, we will be somewhat terse here.
\begin{prop}
There is a unique model structure on $\cat{$\bm G$-Spectra}$ in which a map $f$ is a weak equivalence or fibration if and only if $f(A)$ is a $\mathcal G_{\Sigma_A,G}$-weak equivalence or fibration, respectively, for every \emph{non-empty} finite set $A$. We call this the \emph{positive $G$-global projective level model structure} and its weak equivalences the \emph{positive $G$-global level weak equivalences}. It is combinatorial with generating cofibrations
\begin{equation}\label{eq:gen-cof-pos-proj}
\{\bm\Sigma(A,{\textup{--}})\smashp_\phi G_+\smashp(\del\Delta^n\textup{h}ookrightarrow\Delta^n)_+ : A\not=\varnothing, H\subset\Sigma_A,\phi\colon H\to G,n\ge0\},
\end{equation}
simplicial, proper, and filtered colimits in it are homotopical.
\begin{proof}
To see that the model structure exists and is cofibrantly generated (hence combinatorial) with the above generating cofibrations it suffices to check the `consistency condition' of \cite[Proposition~C.23]{schwede-book} for the $\mathcal G_{\Sigma_A,G}$-model structure on $\cat{$\bm{(G\times\Sigma_A)}$-SSet}_*$ for $A\not=\varnothing$ and the model structure on $\cat{$\bm{(G\times\Sigma_\varnothing)}$-SSet}_*$ in which only isomorphisms are cofibrations. However, it is clear that for $A\not=\varnothing$ each $\bm\Sigma(A,B)\smashp_{\Sigma_A}{\textup{--}}$ sends the usual generating acyclic cofibrations of $\cat{$\bm{(G\times\Sigma_A)}$-SSet}_*$ to injective cofibrations and weak equivalences (also see \cite[proof of Proposition~3.1.20]{g-global}) and even to isomorphisms for $A=\varnothing$.
Right properness, the statement about filtered colimits, and the Pullback Power Axiom for simplicial model categories follow immediately as all relevant constructions and notions are levelwise. Finally, all (generating) cofibrations are injective cofibrations, so left properness follows in the same way.
\end{proof}
\end{prop}
Similarly one gets:
\begin{prop}
There is a unique model structure on $\cat{$\bm G$-Spectra}$ in which a map $f$ is a weak equivalence or fibration if and only if $f(A)$ is a weak equivalence or fibration, respectively, in the \emph{injective} $\mathcal G_{\Sigma_A,G}$-model structure for every $A\not=\varnothing$. We call this the \emph{positive flat $G$-global level model structure}; its weak equivalences are again the $G$-global positive level weak equivalences. Moreover, it is combinatorial with generating cofibrations
\begin{equation}\label{eq:gen-cof-pos-flat}
\{(\bm\Sigma(A,{\textup{--}})\smashp G_+)/H\smashp(\del\Delta^n\textup{h}ookrightarrow\Delta^n)_+ : A\not=\varnothing, H\subset\Sigma_A\times G,n\ge0\},
\end{equation}
simplicial, proper, and filtered colimits in it are homotopical.\qed
\end{prop}
\begin{rk}
The above generating cofibrations agree with the generating cofibrations of Hausmann's \emph{$G$-equivariant positive flat model structure} from \cite[discussion after Proposition~2.28]{hausmann-equivariant}. In particular, the above cofibrations are independent of the group $G$ and agree with what Hausmann calls \emph{positive flat cofibrations}, cf.~Remark~2.20 of \emph{op. cit.}
\end{rk}
For later use we record the following relationship to the usual projective and flat cofibrations:
\begin{lemma}\label{lemma:positive-vs-absolute}
Let $f\colon X\to Y$ be a map in $\cat{$\bm G$-Spectra}$. Then:
\begin{enumerate}
\item $f$ is a positive flat cofibration if and only if it is a flat cofibration and $f(\varnothing)$ is an isomorphism.
\item $f$ is a positive $G$-global projective cofibration if and only if it is a $G$-global projective cofibration and $f(\varnothing)$ is an isomorphism.
\end{enumerate}
\begin{proof}
This is immediate from the characterization of cofibrations in terms of latching maps given in \cite[Proposition~C.23]{schwede-book}.
\end{proof}
\end{lemma}
\begin{defi}
A $G$-spectrum $X$ is called a \emph{positive $G$-global $\Omega$-spectrum} if for every finite group $H$, any \emph{non-empty} finite faithful $H$-set $A$, and every finite $H$-set $B$ the adjoint structure map
\begin{equation*}
X(A)\to\cat{R}\Omega^{B}X(A\amalg B)
\end{equation*}
is a $\mathcal G_{H,G}$-weak equivalence.
\end{defi}
As before, if $X$ is fibrant in either of the above positive level model structures, then this is represented by the ordinary adjoint structure map.
\begin{prop}\label{prop:pos-proj}
There is a unique model structure on $\cat{$\bm G$-Spectra}$ whose cofibrations are the positive $G$-global projective cofibrations and whose weak equivalences are the usual $G$-global weak equivalences. We call this the \emph{positive $G$-global projective model structure}. Its fibrant objects are precisely the positively projectively level fibrant positive $G$-global $\Omega$-spectra.
Moreover, it is again combinatorial with generating cofibrations $(\ref{eq:gen-cof-pos-proj})$, simplicial, proper, and filtered colimits in it are homotopical.
\begin{proof}
We will first construct a Bousfield localization with the above fibrant objects. For this we recall the maps $\lambda_{H,A,B}\colon S^B\smashp\bm\Sigma(A\amalg B,{\textup{--}})\to\bm\Sigma(A,{\textup{--}})$ from Remark~\ref{rk:equivariant-proj-gen-cof} for finite groups $H$ and finite $H$-sets $A,B$. Varying over all homomorphisms $\phi\colon H\to G$ from finite groups to $H$ and restricting $A$ to \emph{non-empty faithful} $H$-sets, the maps $\phi_!\lambda_{H,A,B}$ are then maps between cofibrant objects of the positive $G$-global projective level model structure corepresenting $\phi$-fixed points of the adjoint structure maps. Factoring each of them as a cofibration $\kappa_{\phi,A,B}$ followed by a $G$-global positive level weak equivalence, \cite[Proposition~A.3.7.3]{htt} applied to the set of all $\kappa_{\phi,A,B}$'s then gives a Bousfield localization with the above fibrant objects, and this is automatically again combinatorial, left proper, simplicial, and filtered colimits in it are homotopical (see \cite[Lemma~A.2.4]{g-global} for the final statement).
We now claim that the weak equivalences agree with the $G$-global weak equivalences. For this we first observe that the identity constitutes a Quillen adjunction $\cat{$\bm G$-Spectra}_\text{positive $G$-global proj.}\rightleftarrows\cat{$\bm G$-Spectra}_\text{$G$-global proj.}$ by Lemma~\ref{lemma:check-QA-fibrant} as the left adjoint preserves cofibrations and the right adjoint preserves fibrant objects. On the other hand, a simple cofinality argument shows that positive $G$-global \emph{level} weak equivalences are $\ul\pi_*$-isomorphisms, hence in particular $G$-global weak equivalences, so every weak equivalence in the above model structure is a $G$-global weak equivalence.
Conversely, let $f\colon X\to Y$ be a $G$-global weak equivalence; we want to show that it is a weak equivalence in the above model structure. Using the previous direction and functorial factorizations in the above model structure, we reduce by $2$-out-of-$3$ to the case that $X$ and $Y$ are fibrant in the above sense, i.e.~they are positive $G$-global $\Omega$-spectra and positively projectively level fibrant. Then the natural maps $X\to\Omega\mathop{\textup{sh}}\nolimits X, Y\to\Omega\mathop{\textup{sh}}\nolimits Y$ are positive $G$-global level weak equivalences, and $\Omega\mathop{\textup{sh}}\nolimits f$ is a $G$-global weak equivalence of $G$-global $\Omega$-spectra, hence in particular a (positive) $G$-global level weak equivalence. Thus, another application of $2$-out-of-$3$ shows that also $f$ is a positive $G$-global level weak equivalence, hence in particular a weak equivalence in the above model structure as claimed.
Finally, we observe that despite its definition right properness is independent of the class of fibrations \cite[Proposition~2.5]{rezk-proper}, so right properness of the positive $G$-global projective model structure follows from right properness of the usual $G$-global projective model structure.
\end{proof}
\end{prop}
\begin{prop}\label{prop:pos-flat}
There is a unique model structure on $\cat{$\bm G$-Spectra}$ whose cofibrations are the positive flat cofibrations and whose weak equivalences are the usual $G$-global weak equivalences. We call this the \emph{positive flat $G$-global model structure}. Its fibrant objects are precisely those positive $G$-global $\Omega$-spectra that are fibrant in the positive flat $G$-global level model structure.
Moreover, this model structure is again combinatorial with generating cofibrations $(\ref{eq:gen-cof-pos-flat})$, simplicial, proper, and filtered colimits in it are homotopical.
\begin{proof}
Arguing precisely as before we get a model structure with the desired cofibrations and fibrant objects. By abstract nonsense about Bousfield localizations, a map $f$ is a weak equivalence in this model structure or the one from the previous proposition if and only if $[f,T]$ is bijective for every positive $G$-global $\Omega$-spectrum $T$, where $[\,{,}\,]$ denotes hom sets in the localization at the positive $G$-global level weak equivalences. In particular, its weak equivalences agree with the ones from the previous proposition, i.e.~with the $G$-global weak equivalences.
Finally, all the remaining properties are established precisely as in the previous proposition.
\end{proof}
\end{prop}
\begin{rk}
For $G=1$ the above two model structures again agree, and they recover Hausmann's \emph{positive global model structure} \cite[Theorem~2.18]{hausmann-global}.
\end{rk}
\begin{rk}\label{rk:gen-acyclic-pos-flat}
We will never need to know explicitly how the generating acyclic cofibrations in the above model structures look like. However, we record for later use that the generating cofibrations $(\ref{eq:gen-cof-pos-flat})$ are maps between cofibrant objects, so \cite[Corollary~2.7]{barwick-tractable} shows that we can also find a set of generating acyclic cofibrations for the positive $G$-global flat model structure consisting of maps between cofibrant objects.
\end{rk}
Next, we come to functoriality properties:
\begin{lemma}
Let $\alpha\colon H\to G$ be any homomorphism. Then we have Quillen adjunctions
\begin{align*}
\alpha_!\colon\cat{$\bm H$-Spectra}_\textup{pos.~$H$-global proj.}&\rightleftarrows\cat{$\bm G$-Spectra}_\textup{pos.~$G$-global proj.} :\!\alpha^*\\
\alpha^*\colon\cat{$\bm G$-Spectra}_\textup{pos.~$G$-global flat}&\rightleftarrows
\cat{$\bm H$-Spectra}_\textup{pos.~$H$-global flat} :\!\alpha_*.
\end{align*}
\begin{proof}
For the first statement, we first observe that this holds for the corresponding level model structures as a consequence of Lemma~\ref{lemma:graph-target}. For the actual model structures at hand it suffices then to observe that $\alpha^*$ sends fibrant objects to positive $H$-global $\Omega$-spectrum by direct inspection.
For the second statement, it is clear that $\alpha^*$ preserves positive flat cofibrations and sends $G$-global weak equivalences to $H$-global weak equivalences.
\end{proof}
\end{lemma}
\begin{lemma}
Let $\alpha\colon H\to G$ be \emph{injective}. Then we also have Quillen adjunctions
\begin{align*}
\alpha_!\colon\cat{$\bm H$-Spectra}_\textup{pos.~$H$-global flat}&\rightleftarrows\cat{$\bm G$-Spectra}_\textup{pos.~$G$-global flat} :\!\alpha^*\\
\alpha^*\colon\cat{$\bm G$-Spectra}_\textup{pos.~$G$-global proj.}&\rightleftarrows
\cat{$\bm H$-Spectra}_\textup{pos.~$H$-global proj.} :\!\alpha_*.
\end{align*}
\begin{proof}
For the first statement we observe that $\alpha_!$ sends generating cofibrations to generating cofibrations by direct inspection and that it is homotopical by Proposition~\ref{prop:sp-functoriality-injective}.
For the second statement, the corresponding statement for level model structures follows from the fact that $\alpha^*$ sends $\mathcal G_{\Sigma_A,G}$-cofibrations to $\mathcal G_{\Sigma_A,H}$-cofibrations by Lemma~\ref{lemma:graph-target}. The actual claim then follows as $\alpha^*$ is homotopical.
\end{proof}
\end{lemma}
Arguing as for the usual model structures we then conclude from the above:
\begin{cor}
The positive $G$-global projective and flat model structures for varying $G$ make $\cat{Spectra}$ into a global model category $\glo{GlobalSpectra}^{+}$.\qed
\end{cor}
On the other hand we have straight from the definition of cofibrations and weak equivalences:
\begin{cor}
The identity defines a global Quillen equivalence $\glo{GlobalSpectra}^{+}\rightleftarrows\glo{GlobalSpectra}$.\qed
\end{cor}
Finally let us record how the smash product behaves with respect to the above model structures:
\begin{cor}\label{cor:smash-left-Quillen}
The smash product $\cat{$\bm G$-Spectra}\times\cat{$\bm G$-Spectra}\to\cat{$\bm G$-Spectra}$ is a left Quillen bifunctor in each of the following cases:
\begin{enumerate}
\item the $G$-global positive flat model structure and the $G$-global flat model structure on the source, and the $G$-global positive flat model structure on the target
\item the $G$-global positive flat model structure and the $G$-global projective model structure on the source, and the $G$-global positive projective model structure on the target
\item the $G$-global positive projective model structure and the $G$-global flat model structure on the source, and the $G$-global positive projective model structure on the target.
\end{enumerate}
\begin{proof}
For the ordinary projective and flat model structures this is Theorem~\ref{thm:smash-g-global}. The claims now follow from this via Lemma~\ref{lemma:positive-vs-absolute} and the natural isomorphism $(X\smashp Y)(\varnothing)\cong X(\varnothing)\smashp Y(\varnothing)$ for all symmetric spectra $X,Y$.
\end{proof}
\end{cor}
\subsection{Smash powers and norms} Next, we come to homotopical properties of smash powers.
\begin{constr}
Let $X$ be a $G$-spectrum and let $n\ge1$. Then the $n$-fold smash power $X^{\smashp n}$ carries $n$ commuting $G$-actions as well as a $\Sigma_n$-action (by permuting the factors). Together these assemble into a natural action of the \emph{wreath product} $\Sigma_n\wr G=\Sigma_n\ltimes G^n$, lifting $({\textup{--}})^{\smashp n}$ to a functor $\cat{$\bm G$-Spectra}\to\cat{$\bm{(\Sigma_n\wr G)}$-Spectra}$.
\end{constr}
\begin{constr}
Let $H\subset G$ be finite groups and set $n\mathrel{:=} |G/H|$. Then any choice of right $H$-coset representatives $g_1,\dots,g_n$ defines an injective homomorphism $\iota\colon G\to\Sigma_n\wr H$ as follows: $\iota(g)=(\sigma(g);h_1(g),\dots,h_n(g))$ with $gg_i=g_{\sigma(g)(i)}h_i(g)$. The composite
\begin{equation}\label{eq:hhr-norm}
N^G_H\colon \cat{$\bm H$-Spectra}\xrightarrow{({\textup{--}})^{\smashp n}} \cat{$\bm{(\Sigma_n\wr H)}$-Spectra}\xrightarrow{\iota^*}\cat{$\bm G$-Spectra}
\end{equation}
is then called the (Hill-Hopkins-Ravenel) \emph{norm}.
\end{constr}
For our purposes, the key result on the equivariant behaviour of the norm will be the following:
\begin{thm}\label{thm:norm-equivariant}
The composite $(\ref{eq:hhr-norm})$ sends $H$-equivariant weak equivalences between flat $H$-spectra to $G$-equivariant weak equivalences (of flat $G$-spectra).
\begin{proof}
This is the special case $N=1$ of \cite[Theorem~6.8]{hausmann-equivariant}.
\end{proof}
\end{thm}
We now want to consider the smash powers and norms from a $G$-global perspective. Here we will prove the following stronger result:
\begin{thm}\label{thm:smash-power-wreath}
Let $f\colon X\to Y$ be a $G$-global weak equivalence and assume $X$ and $Y$ are flat. Then $f^{\smashp n}$ is a $(\Sigma_n\wr G)$-global weak equivalence.
\end{thm}
Restricting along the above homomorphism $\iota$ this immediately implies:
\begin{cor}
Let $H\subset G$ be finite groups. Then $N^G_H\colon\cat{$\bm H$-Spectra}\to\cat{$\bm G$-Spectra}$ sends $H$-global weak equivalences of flat $H$-spectra to $G$-global weak equivalences (of flat $G$-spectra).\qed
\end{cor}
\begin{proof}[Proof of Theorem~\ref{thm:smash-power-wreath}]
Let $f\colon X\to Y$ be a $G$-global weak equivalence of flat $G$-spectra. As a first step, we will show that $f^{\smashp n}$ is a $(\Sigma_n\wr G)$-\emph{equivariant} weak equivalence. For this, the key observation will be that while the norm is defined in terms of the smash power, in the global setting we can also go the other way round, cf.~\cite[Remark~5.1.7-(iv)]{schwede-book}. Namely, write $K\subset\Sigma_n\wr G$ for the subgroup of those $(\sigma;g_1,\dots,g_n)$ with $\sigma(1)=1$, which comes with a projection homomorphism $\pi\colon K\to G,\pi(\sigma;g_1,\dots,g_n)=g_1$. If we now fix for each $i=1,\dots,n$ a permutation $\sigma_i$ with $\sigma_i(1)=i$, then the $(\sigma_i;1,\dots,1)$ form a system of coset representatives of $(\Sigma_n\wr G)/K$, and one easily checks from the definitions that the resulting homomorphism $\iota\colon\Sigma_n\wr G\to \Sigma_n\wr K$ is of the form
\begin{equation*}
(\sigma;g_1,\dots,g_n)\mathord{\textup{maps}}to (\sigma; (?;g_1,?,\dots), (?;g_2,?,\dots),\dots)
\end{equation*}
where `$?$' denotes an entry we don't care about.
Now $\pi^*f$ is a $K$-equivariant weak equivalence by assumption on $f$, so $N^{\Sigma_n\wr G}_K(\pi^*f)$ is a $(\Sigma_n\wr G)$-equivariant weak equivalence by Theorem~\ref{thm:norm-equivariant}. But by the above description of $\iota$, this agrees with $f^{\smashp n}$ as map of $(\Sigma_n\wr G)$-spectra, completing the proof of the claim.
Now let $\phi\colon H\to \Sigma_n\wr G$ be any map. We have to show that $\phi^*(f^{\smashp n})$ is an $H$-equivariant weak equivalence. For this we view $f$ as a map of $(G\times H)$-spectra via the trivial $H$-action. Applying the above with $G$ replaced by $G\times H$ then shows that $f^{\smashp n}$ is a $\Sigma_n\wr (G\times H)$-equivariant weak equivalence where all copies of $H$ act trivially. The claim now follows by restricting along the \emph{injective} homomorphism
\begin{equation*}
H\xrightarrow{(\phi,{\textup{id}})} (\Sigma_n\wr G)\times H\xrightarrow\delta \Sigma_n\wr (G\times H)
\end{equation*}
with $\delta$ given by $\delta(\sigma;g_1,\dots,g_n;h)=(\sigma;(g_1,h),\dots,(g_n,h))$.
\end{proof}
Finally, we come to the key property of the \emph{positive} model structures that will allow us to establish model structures on commutative algebras below:
\begin{lemma}\label{lemma:smash-power-free}
Let $X$ be a \emph{positive} flat spectrum and let $n\ge1$. Then the $\Sigma_n$-action on $X^{\smashp n}$ is levelwise free outside the basepoint.
\begin{proof}
This is a special case of \cite[Proposition~7.7*-(a)]{harper-corrigendum}.
\end{proof}
\end{lemma}
\begin{cor}\label{cor:sym-powers-pos-flat}
Let $f\colon X\to Y$ be a $G$-global weak equivalence and assume $X$ and $Y$ are \emph{positively} flat. Then $f^{\smashp n}/\Sigma_n$ is a $G$-global weak equivalence again.
\begin{proof}
By Theorem~\ref{thm:smash-power-wreath}, $f^{\smashp n}$ is a $(\Sigma_n\wr G)$-global weak equivalence, hence in particular a $(\Sigma_n\times G)$-global weak equivalence. The claim follows as $\Sigma_n$ acts freely on both source and target by the previous lemma.
\end{proof}
\end{cor}
\subsection{Global model categories of modules}
Throughout, let $G$ be a finite group. We now introduce the key objects of study for the rest of this paper:
\begin{defi}
A \emph{$G$-global ultra-commutative ring spectrum} is a commutative monoid $R$ (for the smash product) in $\cat{$\bm G$-Spectra}$. We write:
\begin{enumerate}
\item $\cat{UCom}^G=\cat{$\bm G$-UCom}$ for the category of commutative monoids
\item $\cat{Mod}_R^G$ (or simply $\cat{Mod}_R$ if $G$ is clear from the context) for the category of modules in $\cat{$\bm G$-Spectra}$ over the commutative monoid $R$.
\item $\cat{UCom}^G_R$ for the category of $R$-algebras (i.e.~commutative monoids in $\cat{Mod}_R^G$ for the relative smash product, which is canonically isomorphic to the slice $R/\cat{UCom}_G$).
\end{enumerate}
\end{defi}
\begin{prop}
Let $R$ be a $G$-global ultra-commutative ring spectrum. Then the positive projective and positive flat $G$-global model structure on $\cat{$\bm G$-Spectra}$ transfer along the free-forgetful adjunction
\begin{equation*}
R\smashp{\textup{--}}\colon\cat{$\bm G$-Spectra}\rightleftarrows\cat{Mod}_R :\!\mathbb U.
\end{equation*}
This model structure is proper, simplicial, and combinatorial with generating cofibrations $R\smashp I$ and generating acyclic cofibrations $R\smashp J$ for sets of generating (acyclic) cofibrations $I,J$ of the positive projective/flat $G$-global model structure. Moreover, filtered colimits in it are homotopical.
\begin{proof}
For the existence of the model structure, we verify the assumptions of the Crans-Kan Transfer Criterion. By local presentability, every set admits the small object argument, so it only remains to show that for some (hence any) set $J$ of generating acyclic cofibrations of $\cat{$\bm G$-Spectra}_\textup{$G$-global flat}$ the forgetful functor $\mathbb U$ sends relative $(R\smashp J)$-cell complexes to weak equivalences. However, $\mathbb U$ is also a left adjoint, so it suffices to show that it sends maps in $R\smashp J$ to acyclic cofibrations in the \emph{injective $G$-global model structure}. Taking $J$ to consist of maps between cofibrant objects (see~Remark~\ref{rk:gen-acyclic-pos-flat}), this is immediate from Proposition~\ref{prop:flatness-theorem}.
The model structure is clearly combinatorial with the above generating (acyclic) cofibrations, right proper, and simplicial, and filtered colimits in it are homotopical. To see that it is also left proper it suffices to observe that $\mathbb U$ is also left Quillen as a functor into the \emph{injective $G$-global model structure} on $\cat{$\bm G$-Spectra}$ by the above, so that the claim follows from left properness of $\cat{$\bm G$-Spectra}_\textup{$G$-global injective}$ via \cite[Lemma~A.2.15]{g-global}.
\end{proof}
\end{prop}
\begin{rk}
The corresponding statement for the usual projective and flat model structures hold as well (by the same argument); however, for us the above version will be more convenient as we later want to relate the above to categories of commutative algebras.
\end{rk}
\begin{constr}
Let $H$ be a finite group. Identifying $\cat{$\bm H$-Mod}_R^G$ with the category of modules in $\cat{$\bm{(H\times G)}$-Spectra}$ (with $H$ acting trivially on $R$), we get positive projective and flat model structures on $\cat{$\bm H$-Mod}_R^G$, which are equivalently transferred from $\cat{$\bm{(H\times G)}$-Spectra}$ along the evident forgetful functor.
\end{constr}
\begin{lemma}\label{lemma:module-global}
The above make $\cat{Mod}_R^G$ into a global model category $\glo{Mod}_R^G$.
\begin{proof}
We already observed that all of these model structures are combinatorial, simplicial, and (left) proper. To complete the proof that this defines a \emph{pre}global model category it is then enough to note that $\alpha^*$ commutes with both $\mathbb U$ and $R\smashp{\textup{--}}$ for any $\alpha\colon H\to H'$, so that $\alpha_*$ commutes with $\mathbb U$ by passing to mates; thus, all the functoriality properties follow from the corresponding statements for $\glo{GlobalSpectra}^+$.
Similarly, the fact that $\glo{Mod}_R^G$ is a \emph{global} model category follows from the corresponding result for $G\text-\glo{GlobalSpectra}^+$ using the third formulation in Proposition~\ref{prop:global-tfae}.
\end{proof}
\end{lemma}
\begin{cor}\label{cor:modules-stable}
The global model category $\glo{Mod}_R^G$ is stable.
\begin{proof}
Let $H$ be a finite group and $A$ a finite $H$-set. Then $S^A\smashp{\textup{--}}$ commutes with the forgetful functor $\mathbb U$, so it is homotopical. Thus, also the (total or equivalently left) derived functors commute. Similarly, $\Omega^A$ commutes with $\mathbb U$, and as both are right Quillen their right derived functors also commute (via the mate of the above equivalence). The claim now follows immediately from the corresponding statement for $G\text-\glo{GlobalSpectra}$ (Proposition~\ref{prop:GlobalSpectra-stable}) as ${\textup{Ho}}(\mathbb U)$ is conservative.
\end{proof}
\end{cor}
\begin{lemma}\label{lemma:module-adjunctions}
Let $G$ be a finite group and let $R$ be a \emph{flat} $G$-global ultra-commutative ring spectrum. Then we have global Quillen adjunctions
\begin{equation*}
\ul{R\smashp{\textup{--}}}\colon G\text-\glo{GlobalSpectra}^+\rightleftarrows\glo{Mod}^G_R :\!\ul{\mathbb U}
\qquad\text{and}\qquad
\ul{\mathbb U}\colon \glo{Mod}^G_R\rightleftarrows G\text-\glo{GlobalSpectra}^+ :\!\ul{F(R,{\textup{--}})}
\end{equation*}
Moreover, both $\ul{R\smashp{\textup{--}}}$ and $\ul{\mathbb U}$ are homotopical.
\begin{proof}
By definition, $\mathbb U$ preserves weak equivalences as well as fibrations in either model structure; in particular, it is right Quillen. Moreover, it sends generating cofibrations to cofibrations by Proposition~\ref{cor:smash-left-Quillen}, so it is also left Quillen. Finally, also $R\smashp{\textup{--}}$ is homotopical by the Flatness Theorem (Proposition~\ref{prop:flatness-theorem}).
\end{proof}
\end{lemma}
As $\cat{Mod}^R_G$ is a stable model category, its homotopy category is naturally a triangulated category. For later use, we record a t-structure on this as well as its interaction with the smash product. To define this, we have to recall the \emph{true} homotopy groups (as opposed to the na\"ive ones considered before) of a $G$-global spectrum:
\begin{constr}
Let $X$ be a $G$-global spectrum, let $\phi\colon H\to G$ be a homomorphism from a finite group $H$, and let $k\in\mathbb Z$. We define the \emph{true $k$-th $\phi$-equivariant homotopy group} as
\begin{equation*}
\textup{h}at\pi_k^\phi(X)\mathrel{:=}[\Sigma^{\bullet+k}_+ I(H,{\textup{--}})\times_\phi G,X],
\end{equation*}
where $[\,{,}\,]$ denotes the hom set in the $G$-global stable homotopy category; note that $\textup{h}at\pi_k^\phi(X)$ carries a natural abelian group structure by additivity of the latter, and that distinguished triangles induce long exact sequences in $\textup{h}at\pi_*^\phi$ by general nonsense about triangulated categories.
We moreover write $\underline{\textup{h}at\pi}_k(X)$ for the collection of all $\phi$-equivariant homotopy groups for varying $\phi$, together with all natural maps between them. For $G=1$, this structure can be explicitly described as a \emph{global (Mackey) functor} \cite[Remark~6.5]{schwede-k-theory}; we will never need an explicit description for non-trivial $G$.
\end{constr}
\begin{rk}
The true $\phi$-equivariant homotopy groups of $X$ can be equivalently described as the true $H$-equivariant homotopy groups of the $H$-equivariant spectrum $\phi^*X$, see \cite[Corollary~3.3.4]{g-global}.
\end{rk}
\begin{prop}\label{prop:t-structure}
Let $R$ be a $G$-global ultra-commutative ring spectrum.
\begin{enumerate}
\item The triangulated category ${\textup{Ho}}(\cat{Mod}^G_R)$ is compactly generated with generators $R\smashp\Sigma^\bullet_+\big(I(A,{\textup{--}})\times_\phi G\big)$ for finite sets $A$, subgroups $H\subset\Sigma_A$, and homomorphisms $\phi\colon H\to G$.
\item Assume that $R$ is \emph{connective}, i.e.~for every $k<0$ the true homotopy groups $\underline{\textup{h}at\pi}_k R$ (of the underlying $G$-global spectrum) are trivial. Then ${\textup{Ho}}(\cat{Mod}^G_R)$ carries a t-structure with
\begin{itemize}
\item connective part the connective $R$-modules
\item coconnective part those $R$-modules $X$ that are \emph{coconnective} in the sense that $\underline{\textup{h}at\pi}_kX=0$ for all $k>0$.
\end{itemize}
\item In this case, the derived smash product restricts to functors
\begin{equation*}
{\textup{--}}\smashp_R^{\cat L}{\textup{--}}\colon{\textup{Ho}}(\cat{Mod}^G_R)_{\ge m}\times{\textup{Ho}}(\cat{Mod}^G_R)_{\ge n}\to{\textup{Ho}}(\cat{Mod}^G_R)_{\ge m+n}
\end{equation*}
for all $m,n\in\mathbb Z$.
\end{enumerate}
\begin{proof}
We first consider the special case that $R=\mathbb S$, i.e.~the case of $G$-global spectra. In this setting, the first two statements are part of \cite[Theorem~7.1.12]{global-param}. For the final statement we reduce (as ${\textup{--}}\smashp^{\cat L}{\textup{--}}$ is naturally exact in each variable and in particular preserves suspension) first to the case $m=n=0$ and then to the case of the above generators. By flatness we can then simply compute the derived smash product in terms of the non-derived one. A simple Yoneda argument now shows $\Sigma^\bullet_+I(A,{\textup{--}})\smashp\Sigma^\bullet_+ I(B,{\textup{--}})\cong\Sigma^\bullet_+ I(A\amalg B,{\textup{--}})$ naturally in $A$ and $B$, so
\begin{equation*}
\big(\Sigma^\bullet_+ I(A,{\textup{--}})\times_\phi G\big)\smashp\big(\Sigma^\bullet_+ I(B,{\textup{--}})\times_\psi G\big)\cong \Sigma^\bullet_+\big(I(A\amalg B)\times (G\times G)\big)/(H\times K)
\end{equation*}
(where $H\subset\Sigma_A,K\subset\Sigma_B$ are the sources of $\phi$ and $\psi$, respectively), which is clearly connective again.
Now we treat the general case. For this we first note that we have an exact adjunction
\begin{equation}\label{eq:exact-adj}
R\smashp^{\cat L}{\textup{--}}\colon{\textup{Ho}}(\cat{$\bm G$-Spectra}_\textup{$G$-global})\rightleftarrows {\textup{Ho}}(\cat{Mod}^G_R) :\!\mathbb U
\end{equation}
with conservative right adjoint by construction, so the $R\smashp\Sigma^\bullet_+I(A,{\textup{--}})\times_\phi G$ are a set of generators (here we secretly used Proposition~\ref{prop:flatness-theorem} to identify $R\smashp^{\cat L}\Sigma^\bullet_+I(A,{\textup{--}})\times_\phi G$ with the underived smash product). We now observe that coproducts of $G$-global spectra are homotopical \cite[Lemma~3.1.43]{g-global} and that $\mathbb U$ preserves them on the pointset level; it follows that also coproducts in $\cat{Mod}^G_R$ are homotopical, and that the right adjoint in $(\ref{eq:exact-adj})$ preserves coproducts, so that its left adjoint preserves compact objects. This completes the proof of the first statement.
For the second statement, we use the above together with \cite[Theorem~A.1]{t-struc-compact} to obtain a t-structure on ${\textup{Ho}}(\cat{Mod}^G_R)$ whose connective part is the smallest subcategory closed under coproducts, suspensions, and extensions containing the $R\smashp\Sigma_+^\bullet I(A,{\textup{--}})\times_\phi G$'s; we claim that this is the t-structure described above. To see this, we first observe that an $R$-module $X$ is coconnective in this t-structure if and only if $[\Sigma T,X]=0$ for all $T\in{\textup{Ho}}(\cat{Mod}^G_R)_{\ge 0}$. Specializing to $T=R\smashp\Sigma^{\bullet+k}_+ I(A,{\textup{--}})\times_\phi G$ and using the adjunction isomorphisms shows that $X$ is then coconnective in the above sense. Conversely, for fixed $X$ the class of objects $T$ such that $[\Sigma^kT,X]=0$ for all $k>0$ is easily seen to be closed under coproducts, suspension, and extensions (using the long exact sequence for the last statement), so if $X$ has trivial positive homotopy groups, then $X\in{\textup{Ho}}(\cat{Mod}^G_R)_{\le0}$.
To identify the connective part, we first observe that the exact coproduct preserving functor $\mathbb U\colon{\textup{Ho}}(\cat{Mod}^G_R)\to{\textup{Ho}}(\cat{$\bm G$-Spectra})$ sends the chosen generators to connective objects (as $R$ is connective and the smash product of connective spectra is connective by the above); thus, ${\textup{Ho}}(\cat{Mod}^G_R)_{\ge0}$ is contained in the preimage under $\mathbb U$ of ${\textup{Ho}}(\cat{$\bm G$-Spectra})_{\ge0}$, i.e.~all objects of ${\textup{Ho}}(\cat{Mod}^G_R)_{\ge0}$ are connective in the above sense. Conversely, if $X$ is connective then we have a distinguished triangle
\begin{equation*}
X_{\ge 0}\to X\to X_{\le -1}\to \Sigma X_{\ge 0}
\end{equation*}
with $X_{\ge0}\in{\textup{Ho}}(\cat{Mod}^G_R)_{\ge0}$ and $X_{\le -1}\in{\textup{Ho}}(\cat{Mod}^G_R)_{\le-1}$ by the axioms of a t-structure. Then $X_{\ge0}$ is connective by the above and hence so is $X_{\le -1}$ by the long exact sequence. But on the other hand, $X_{\le -1}$ has vanishing non-negative homotopy groups by the above identification of the coconnective part, so $X_{\le -1}=0$ and hence $X\cong X_{\ge0}\in{\textup{Ho}}(\cat{Mod}^G_R)_{\ge0}$ as claimed.
Finally, for the compatibility of the smash product with the t-structure we reduce as above to proving that $(R\smashp\Sigma^\bullet_+ I(A,{\textup{--}})\times_\phi G)\smashp_R(R\smashp\Sigma^\bullet_+ I(B,{\textup{--}})\times_\phi G)$ is connective. But this is isomorphic to $R\smashp(\Sigma^\bullet_+ I(A,{\textup{--}})\times_\phi G)\smashp (\Sigma^\bullet_+ I(A,{\textup{--}})\times_\psi G)$, so the claim follows from the special case $R=\mathbb S$ considered above (and Proposition~\ref{prop:flatness-theorem}).
\end{proof}
\end{prop}
As a consequence of Corollary~\ref{cor:modules-stable}, the suspension spectrum-loop space adjunction defines a global Quillen equivalence between $\glo{Mod}_R^G$ and its stabilization $\ul\textup{Sp}(\glo{Mod}_R^G)$, so the latter contains no new homotopy theoretic information. Nevertheless, the concrete model will be useful later for comparisons to other stabilizations, and we close this discussion by understanding its weak equivalences a bit better.
\begin{lemma}\label{lemma:g-global-sp-sp-colim}
Let $R$ be a $G$-global ultra-commutative ring spectrum. Then the $G$-global weak equivalences in $\textup{Sp}(\glo{Mod}^G_R)$ are closed under pushouts along injective cofibrations (i.e.~levelwise injections) as well as under arbitrary filtered colimits.
\begin{proof}
For the first statement, let $i\colon X\to Y$ be an injective cofibration, and let $f\colon X\to Z$ be a $G$-global weak equivalence. We factor $f$ (say, in the projective model structure) as an acyclic cofibration $k$ followed by an acyclic fibration $p$; in particular, $p$ is a $G$-global \emph{level} weak equivalence. Then we have an iterated pushout square
\begin{equation*}
\begin{tikzcd}
X \arrow[d, "i"']\arrow[dr, phantom, "\ulcorner"{very near end}]\arrow[rr, "f", yshift=5pt, bend left=15pt] \arrow[r, "k"', tail, "\sim"] & H\arrow[dr, phantom, "\ulcorner"{very near end}] \arrow[r, "p"', "\sim", two heads]\arrow[d, "j"] & Z\arrow[d]\\
Y\arrow[r, "\ell"'] & K\arrow[r, "q"'] & P
\end{tikzcd}
\end{equation*}
in which $\ell$ is an acyclic cofibration (as a pushout of an acyclic cofibration) and $j$ is again an injective cofibration. Thus, applying left properness of the injective model structure levelwise we see that $q$ is a $G$-global (level) weak equivalence; the claim follows as $q\ell$ is a pushout of $f=pk$ along $i$.
The second statement follows similarly from the corresponding statement for $\cat{Mod}_R^G$, also see \cite[Lemma~A.2.4]{g-global}.
\end{proof}
\end{lemma}
\begin{lemma}\label{lemma:module-transferred}
Let $R$ be a flat $G$-global ultra-commutative ring spectrum. Then both adjoints in the Quillen adjunction
\begin{equation*}
\textup{Sp}(\ul{R\smashp{\textup{--}}})\colon\textup{Sp}(G\text-\glo{GlobalSpectra}^+)\rightleftarrows\textup{Sp}(\glo{Mod}^G_R):\!\textup{Sp}(\ul{\mathbb U})
\end{equation*}
are homotopical. Moreover, the projective model structure on $\textup{Sp}(\glo{Mod}^R_G)$ is transferred from the projective model structure on $\textup{Sp}(G\text-\glo{GlobalSpectra}^+)$ along the above adjunction, and likewise for the flat model structures.
\end{lemma}
Beware that in general stabilization does \emph{not} commute with transferring model structures!
\begin{proof}
By Lemma~\ref{lemma:module-adjunctions} \emph{both} functors are left Quillen (for either model structure), in particular they send flat acyclic cofibrations to weak equivalences. Moreover, both $\textup{Sp}(\ul{\mathbb U})$ and $\textup{Sp}(\ul{R\smashp{\textup{--}}})$ preserve level weak equivalences as $\ul{\mathbb U}$ and $\ul{R\smashp{\textup{--}}}$ are homotopical by the aforementioned lemma. Factoring an arbitrary weak equivalence into a flat cofibration followed by a level weak equivalence we see that both functors are in fact homotopical.
Now let $f\colon X\to Y$ be a map in $\textup{Sp}(\glo{Mod}^G_R)$ such that $\textup{Sp}(\ul{\mathbb U})(f)$ is a weak equivalence; we have to show that $f$ is a weak equivalence. As $\textup{Sp}(\ul{\mathbb U})$ is homotopical, we may assume by $2$-out-of-$3$ that $f$ is a map of fibrant objects. But then also $\textup{Sp}(\ul{\mathbb U})(f)$ is a map of fibrant objects, whence a level weak equivalence, so that also $f$ is a (level) weak equivalence.
Finally, to prove that the projective and flat model structures on $\textup{Sp}(\glo{Mod}^G_R)$ are transferred along the above adjunction, we will first show that the transferred model structures exist, and then prove that they agree with the given ones.
For the existence, it suffices by local presentability that $\textup{Sp}({R\smashp{\textup{--}}})(J)$-cell complexes are sent by $\textup{Sp}(\ul{\mathbb U})$ to weak equivalences, where $J$ is our favourite set of generating acyclic cofibrations. But as $\textup{Sp}(\ul{\mathbb U})$ is left Quillen we only have to show that $\textup{Sp}(\ul{\mathbb U})$ sends maps in $\textup{Sp}(\ul{R\smashp{\textup{--}}})(J)$ to acyclic cofibrations, which follows at once since also $\textup{Sp}(\ul{R\smashp{\textup{--}}})$ is left Quillen.
Now we simply observe that the transferred model structure has the correct acyclic fibrations (obviously) as well as weak equivalences (by the above), so it agrees with the given model structure as claimed.
\end{proof}
\begin{rk}
The second half of the lemma holds true more generally for all (not necessarily flat) $G$-global ultra-commutative $R$; however, we will only need the above version, which is slightly easier to prove.
\end{rk}
\subsection{Global model categories of algebras}\label{subsection:model-structures-algebras}
Next, we turn to model structures on $\cat{UCom}^G$ and more generally on categories of $R$-algebras for $G$-global ultra-commutative ring spectra $R$. These model structures can be constructed using the monoid axiom of \cite{schwede-shipley-monoidal} and the commutative monoid axiom of \cite{white-cmon}. We recall the relevant results and then show that these axioms are satisfied for the projective and flat $G$-global model structures.
\begin{constr}\label{constr:iterated-pushout-product}
Let $(\mathscr C, \otimes, \bm 1)$ be a cocomplete closed symmetric monoidal category. For any object $X$ in $\mathscr C$, we denote by $\power{m} X = X^{\otimes m}/\Sigma_m$ the $m$-th symmetric power of $X$. We consider a generalization of the pushout product
\[ f\ppo g\colon A\otimes Y \amalg_{A\otimes B} X\otimes B\to X\otimes Y \]
of two morphisms $f\colon A\to X$ and $g\colon B\to Y$:
Let $f\colon A\to X$ be a morphism in $\mathscr C$, and $m\geq 1$. We consider the poset category $\mathscr P(\{1, \ldots, m\})$ and the hypercube-shaped diagram
\begin{align*}
W(f)\colon \mathscr P(\{1, \ldots, m\}) &\to \mathscr C, \\
S & \mathord{\textup{maps}}to W_1(f,S)\otimes \ldots \otimes W_m(f,S),
\end{align*}
where
\[ W_k(f,S) = \begin{cases}
A & \textup{if } k\notin S,\\
X & \textup{if } k \in S.
\end{cases}\]
To an inclusion $i\colon S\textup{h}ookrightarrow T$, the functor $W(f)$ assigns the morphism $W(f, i) = W_1(f, i)\otimes \ldots \otimes W_m(f,i)$, where
\[ W_k(f,i) = \begin{cases}
{\textup{id}} & \textup{if } k\notin T\setminus S,\\
f & \textup{if } k \in T\setminus S.
\end{cases}\]
This diagram defines a morphism
\[ f^{\ppo m}\colon Q^m := \mathop{\textup{colim}}\nolimits_{\mathscr P(\{1, \ldots, m\})\setminus \{1, \ldots, m\}} W(f) \to X^{\otimes m} = W(f, \{1, \ldots, m\}).\]
This morphism is $\Sigma_m$-equivariant for the actions induced by the permutation action of $\Sigma_m$ on $\{1, \ldots, m\}$, and hence induces a map
\begin{equation}\label{eq:symmetric-iterated-pushout-product}
f^{\ppo m}/\Sigma_m \colon Q^m/\Sigma_m \to \power{m} X = X^{\otimes m}/\Sigma_m.
\end{equation}
on coinvariants.
\end{constr}
\begin{defi}\label{def:monoid-axioms}
Let $(\mathscr C, \otimes, \bm 1)$ be a symmetric monoidal model category. We say that $\mathscr C$ satisfies the
\begin{enumerate}
\item \emph{monoid axiom} if every $\big(\{\textup{acyclic cofibrations}\}\otimes \mathscr C\big)$-cellular map is a weak equivalence.
\item \emph{commutative monoid axiom} if for any acyclic cofibration $f\colon X\to Y$ and any $n\geq 0$, the map $f^{\ppo n}/\Sigma_n$ is an acyclic cofibration.
\item \emph{strong commutative monoid axiom} if for any cofibration or acyclic cofibration $f\colon X\to Y$ and any $n\geq 0$, the map $f^{\ppo n}/\Sigma_n$ is a cofibration or an acyclic cofibration, respectively.
\end{enumerate}
\end{defi}
\begin{prop}\label{prop:model-str-on-algebras}
Let $(\mathscr C, \otimes, \bm 1)$ be a combinatorial symmetric monoidal model category satisfying the monoid axiom and commutative monoid axiom, and let $R$ be a commutative monoid in $\mathscr C$. Then:
\begin{enumerate}
\item the category $\Mod_R$ of $R$-modules inherits a model structure from $\mathscr C$, transferred along the adjunction
\[ \begin{tikzcd}[column sep = large]
\mathscr C \arrow[r, shift left, "R\otimes \_"] & \Mod_R \arrow[l, shift left, "\mathbb U"].
\end{tikzcd}\]
\item the category $\CAlg_R$ of commutative $R$-algebras inherits a model structure from $\mathscr C$, transferred along the free-forgetful adjunction
\[ \begin{tikzcd}[column sep = large]
\mathscr C \arrow[r, shift left, "{R\otimes\power{} \mathrel{:=}R\otimes\coprod_{n\ge0}\mathbb P^n}"] &[4.2em] \CAlg_R \arrow[l, shift left, "\mathbb U"].
\end{tikzcd}\]
\item the category $\NUCA_R$ of non-unital commutative $R$-algebras (i.e.~$R$-modules $M$ equipped with an associative and commutative multiplication $M\smashp_RM\to M$) inherits a model structure from $\mathscr C$, transferred along the free-forgetful adjunction
\[ \begin{tikzcd}[column sep = large]
\mathscr C \arrow[r, shift left, "{R\otimes\power{>0}\mathrel{:=}R\otimes\coprod_{n>0}\mathbb P^n}"]&[4.2em] \NUCA_R \arrow[l, shift left, "\mathbb U"].
\end{tikzcd}\]
\end{enumerate}
\begin{proof}
The first assertion is part of \cite[Theorem 4.1]{schwede-shipley-monoidal}, the second of \cite[Theorem 3.2, Remark 3.3]{white-cmon} and the last of \cite[Theorem B.2.6]{stahlhauer}.
\end{proof}
\end{prop}
\begin{rk}\label{rk:pushout-analysis-nucas}
We comment on the proof of the result for non-unital commutative algebras in \cite{stahlhauer}. As for the other results, the main step is an analysis of certain pushouts in the category of non-unital commutative algebras. For simplicity, we restrict to $R=\bm 1$. Explicitly, for a symmetric monoidal model category $(\mathscr C, \otimes, \bm 1)$, we consider morphisms $h\colon K\to L$ and $p\colon K\to X$ in $\mathscr C$, where $X$ is a non-unital commutative algebra, and we analyze the pushout
\begin{equation}\label{diag:pushout-free-nucas}
\begin{tikzcd}
\power{>0} K \arrow[r, "\power{>0}h"] \arrow[d, "\tilde p", swap] & \power{>0} L \arrow[d]\\
X \arrow[r, "f"] & P
\end{tikzcd}
\end{equation}
in the category $\NUCA$ of non-unital commutative algebras. For the proof of \Cref{prop:model-str-on-algebras}, we need to check that if $h$ is an acyclic cofibration, then the pushout morphism $f\colon X\to P$ is a weak equivalence. We show this by defining a filtration
\[ X = P_0\xrightarrow{f_1} P_1\xrightarrow{f_2} \ldots \to \mathop{\textup{colim}}\nolimits_{n\geq 0} P_n = P. \]
Here, the $P_n$ are iteratively defined via pushouts
\[\textup{h}skip-3.76pt\textup{h}fuzz=4pt\begin{tikzcd}[cramped,column sep = 12em]
(Q^{n+1}/\Sigma_{n+1}) \amalg X\otimes (Q^{n+1}/\Sigma_{n+1}) \arrow[r, "h^{\ppo n+1}/\Sigma_{n+1} \amalg X\otimes h^{\ppo n+1}/\Sigma_{n+1}"] \arrow[d, "t_{n+1}", swap]
& \power{n+1} L \amalg X\otimes \power{n+1} L \arrow[d, "T_{n+1}"]\\
P_n \arrow[r, "f_{n+1}", swap]
& P_{n+1}.
\end{tikzcd} \]
We will never need to know how the vertical maps are defined precisely; informally, $t_n$ is given by mapping all entries in $K$ to $X$ via $p$ and multiplying all resulting terms from $X$ together, whereas all entries in $L$ are collected into $\power{0<\ast\leq n} L$.
\end{rk}
We will also need the following criterion for left properness:
\begin{prop}\label{prop:alg-left-proper}
Let $\mathscr C$ be a combinatorial symmetric monoidal model category satisfying the monoid axiom and strong commutative monoid axiom. Then the transferred model structure on $\CAlg$ is left proper provided all of the following conditions are satisfied:
\begin{enumerate}
\item $\mathscr C$ is left proper and filtered colimits in it are homotopical.
\item There exists a set of generating cofibrations for $\mathscr C$ consisting of maps between \emph{cofibrant} objects.
\item For any $X\in\mathscr C$ and any cofibration $i$, pushouts along $X\otimes i$ are homotopy pushouts.
\item For any cofibrant $X$, the functor $X\otimes{\textup{--}}$ is homotopical.
\end{enumerate}
\begin{proof}
This is a special case of \cite[Theorem~4.17]{white-cmon}, also see \cite[Theorem~2.1.34]{g-global} for the reduction to White's result.
\end{proof}
\end{prop}
Let us now specialize this to algebraic structures on $G$-global spectra:
\begin{prop}\label{prop:white-assumptions}
Let $G$ be a finite group. Then the positive projective and positive flat $G$-global model structures on $\cat{$\bm G$-Spectra}$ are symmetric monoidal. Moreover, they satisfy both the monoid axiom as well as the strong commutative monoid axiom.
\begin{proof}
The pushout product axiom is immediate from Corollary~\ref{cor:smash-left-Quillen}. Moreover, if $p\colon\mathbb S^+\to\mathbb S$ is a cofibrant replacement in either of these model structures, and $X$ is any $G$-spectrum, then $X\smashp p$ is a $G$-global weak equivalence by flatness of $\mathbb S$ and Proposition~\ref{prop:flatness-theorem}. Thus, both model structures are symmetric monoidal.
For the monoid axiom, we simply observe once more that for any $G$-global spectrum $R$ and any acyclic cofibration (say, in the positive flat model structure) $j$ the map $R\smashp j$ is a $G$-global weak equivalence and an injective cofibration.
If now $f$ is a positive flat cofibration then $f^{\ppo n}$ is a positive flat cofibration simply by monoidality, also see \cite[Remark~6.9]{hausmann-equivariant}; the strong commutative monoid axiom for positive flat cofibrations follows as $({\textup{--}})/\Sigma_n$ preserves positive flat cofibrations (since $\mathop{\textup{tr}}\nolimitsiv_{\Sigma_n}$ clearly preserves acyclic fibrations).
In the projective case we explicitly compute that for a generating cofibration $f=\bm\Sigma(A,{\textup{--}})\smashp_{H} G_+\smashp (\del\Delta^m\textup{h}ookrightarrow\Delta^m)$ with $H$ acting faithfully on $A\not=\varnothing$ the map $f^{\ppo n}/\Sigma_n$ agrees up to isomorphism with
\begin{equation}\label{eq:com-mon-proj-cof}
\bm\Sigma(\bm n\times A,{\textup{--}})\smashp_{\Sigma_n\wr H} G^n_+\smashp(\del\Delta^m\textup{h}ookrightarrow\Delta^m)^{\ppo n}_+
\end{equation}
where $\Sigma_n\wr H$ acts on $\bm n\times A$ via $(\sigma,h_\bullet).(i,a)\mathord{\textup{maps}}to (\sigma(i), h_i.a)$, which is faithful as $A\not=\varnothing$ and $H$ acts faithfully. Now $(\del\Delta^m\textup{h}ookrightarrow\Delta^m)^{\ppo n}_+$ is a cofibration of simplicial sets and $G$ acts freely on $G^n$, so $G^n_+\smashp(\del\Delta^m\textup{h}ookrightarrow\Delta^m)^{\ppo n}_+$ is a $\mathcal G_{\Sigma_n\wr H,G}$-cofibration, whence $(\ref{eq:com-mon-proj-cof})$ is a projective cofibration again.
Finally, for the (strong) commutative monoid axiom for acyclic cofibrations, we note that we can as before pick sets of generating acyclic cofibrations consisting of maps between positively flat $G$-spectra. It therefore suffices by \cite[Corollaries~10 and~23]{sym-powers} that for any $G$-global weak equivalence $f$ of positively flat $G$-spectra also $f^{\smashp n}/\Sigma_n$ is a weak equivalence, which is precisely the content of Corollary~\ref{cor:sym-powers-pos-flat}.
\end{proof}
\end{prop}
\begin{cor}\label{cor:ucomRG}
Let $G$ be a finite group and let $R$ be a $G$-global ultra-commutative ring spectrum. Then the positive projective and flat $G$-global model structures on $\cat{Mod}_R^G$ transfer along the free-forgetful adjunction $\mathbb{P}\colon\cat{Mod}_R^G\rightleftarrows\cat{UCom}_R^G :\!\mathbb U$. The resulting model structures are again combinatorial, proper, simplicial, and filtered colimits in it are homotopical.
\begin{proof}
Identifying $\cat{UCom}_R^G$ with the slice $R/\cat{UCom}^G$, it suffices to consider the case $R=\mathbb S$. Thus, the existence of the model structure follows from White's criterion (Proposition~\ref{prop:model-str-on-algebras}) and the previous proposition.
As the model structure is transferred from $\cat{$\bm G$-Spectra}$, we immediately see that it is combinatorial, right proper, simplicial, and that filtered colimits in it are homotopical. It remains to check left properness, for which we will verify the assumptions of Proposition~\ref{prop:alg-left-proper}:
We know from Propositions~\ref{prop:pos-proj} and~\ref{prop:pos-flat} that both positive model structures on $\cat{$\bm G$-Spectra}$ are left proper, that filtered colimits in them are homotpical, and moreover the standard generating cofibrations are maps of cofibrant objects. As already observed above, for any $X$ and any (generating) cofibration $i$, $X\smashp i$ is an injective cofibration, so pushouts along it are homotopy pushouts. Finally, if $X$ is cofibrant in either of the two model structures, then $X\smashp{\textup{--}}$ is homotopical by Proposition~\ref{prop:flatness-theorem}.
\end{proof}
\end{cor}
Again, we more generally get positive projective and flat model structures on $\cat{$\bm H$-UCom}^G_R$ for all finite groups $H$.
\begin{lemma}
These make $\cat{UCom}^G_R$ into a global model category $\glo{Comm}_R^G$, and the free-forgetful adjunction defines a global Quillen adjunction $\ul{\mathbb P}\colon\glo{Mod}^G_R\rightleftarrows\glo{Comm}_R^G :\!\ul{\mathbb U}$.
\begin{proof}
For the first statement one argues precisely as for $\glo{Mod}_R^G$ (Lemma~\ref{lemma:module-global}). The second statement is clear.
\end{proof}
\end{lemma}
\begin{defi}
Let $R$ be a $G$-global ultra-commutative ring spectrum, considered as a commutative algebra over itself in the obvious way. An \emph{augmented commutative $R$-algebra} is an object of the slice $\cat{UCom}^G_R/R$, i.e.~a commutative $R$-algebra $A$ together with an $R$-algebra homomorphism $\epsilon\colon A\to R$ (note that $\epsilon$ is automatically a retraction of the unit $R\to A$ by unitality).
\end{defi}
\begin{rk}
Arguing as in Remark~\ref{rk:basepoint}, $\cat{UCom}^G_R/R$ carries a left proper simplicial combinatorial model structure in which a map is a weak equivalence, fibration, or cofibration, if and only if it is so in the positive projective model structure on $\cat{UCom}^G_R$. Analogously, we get a positive flat model structure and via the usual identifications these make $\cat{UCom}^G_R/R$ into a global model category $\glo{Comm}_R^G/R$.
\end{rk}
\begin{defi}\label{defi:nuca}
Let $R$ be a $G$-global ultra-commutative ring spectrum. We denote the category of non-unital commutative $R$-algebras by $\cat{NUCA}^G_R$.
\end{defi}
\begin{cor}
Let $G$ be a finite group and let $R$ be a $G$-global ultra-commutative ring spectrum. Then the positive projective and flat $G$-global model structures on $\cat{Mod}_R^G$ transfer along the free-forgetful adjunction $\mathbb{P}^{>0}\colon\cat{Mod}_R^G\rightleftarrows\cat{NUCA}_R^G :\!\mathbb U$. The resulting model structures are again combinatorial, proper, simplicial, and filtered colimits in them are homotopical.
\begin{proof}
All statements except for left properness follow as in the case of ordinary commutative algebras (Corollary~\ref{cor:ucomRG}). For left properness, on the other hand, we observe that the (non-full) inclusion $\cat{UCom}^G_R\textup{h}ookrightarrow\cat{NUCA}^G_R$ admits a left adjoint $K$ given by $X\mathord{\textup{maps}}to (R\textup{h}ookrightarrow R\vee X)$ with the unique multiplication extending the one on $X$. Then $K$ preserves (generating) cofibrations by direct inspection, and it is homotopical by \cite[Lemma~3.1.43]{g-global}. Finally, it also reflects weak equivalences as on the level of underlying $G$-spectra $X$ is naturally a retract of $R\vee X=K(X)$. Thus, left properness of $\cat{NUCA}^G_R$ follows from left properness of $\cat{UCom}^G_R$ by \cite[Lemma~A.2.15]{g-global}.
\end{proof}
\end{cor}
Again, we get more generally positive projective and flat model structures on $\cat{$\bm H$-NUCA}^G_R$ for all finite groups $H$.
\begin{lemma}
These make $\cat{NUCA}^G_R$ into a global model category $\glo{NUCA}_R^G$, and the free-forgetful adjunction defines a global Quillen adjunction $\ul{\mathbb P^{>0}}\colon\glo{Mod}^G_R\rightleftarrows\glo{NUCA}_R^G :\!\ul{\mathbb U}$.
\begin{proof}
For the first statement one argues precisely as for $\glo{Mod}_R^G$, while the second statement is clear from the definitions.
\end{proof}
\end{lemma}
\section{Global topological André-Quillen cohomology\texorpdfstring{\except{toc}{{\textup{pr}}otect\footnote{The contents of the first two subsections of this section are an adapted version of part of the thesis of the second author \cite[Chapter 2]{stahlhauer}.}}}{}}\label{sec:gtaq}
In this section, we introduce $G$-global versions of topological André-Quillen cohomology. The classical non-equivariant cohomology theory was introduced by Basterra in \cite{basterra-TAQ} as a cohomology theory for (augmented) commutative $S$-algebras. The algebraic predecessor homology theory, as defined by André and Quillen \cite{andre-homology, quillen-cohomology-algebras}, is defined as a derived functor of Kähler differentials which in turn may be constructed as the module of indecomposables of the augmentation ideal of an augmented algebra. Both Basterra's construction of topological André-Quillen (co)homology as well as our $G$-global version below mimic this construction.
\subsection{The cotangent complex in general model categories}
Our construction works in a general abstract setting: throughout, we fix a combinatorial and stable symmetric monoidal model category $\mathscr C$ satisfying the monoid axiom and commutative monoid axiom, such that finite coproducts in $\mathscr C$ are homotopical; the examples the reader should keep in mind and to which we will later specialize are the positive $G$-global model structures on $\cat{$\bm G$-Spectra}$ constructed in \Cref{section:brave-new-algebra}.
\begin{defi}\label{def:augmentation-ideal}
Let $R$ be a commutative monoid in $\mathscr C$, and let $S$ be an augmented $R$-algebra, with augmentation $\epsilon\colon S\to R$. Then we define the \emph{augmentation ideal} $I(S)$ of $S$ as the strict pullback
\[ \begin{tikzcd}
I(S) \arrow[r]\arrow[dr, phantom, "\lrcorner"{very near start}] \arrow[d] & S \arrow[d, "\epsilon"] \\
\ast \arrow[r] & R
\end{tikzcd}\]
in the category $\Mod_R$ of $R$-modules.
\end{defi}
This inherits the structure of a non-unital $R$-algebra from $S$, yielding a functor $I\colon \CAlg_R/R \to \NUCA_R$. There also is a \emph{unitalization} functor in the other direction, given as
\[ K\colon \NUCA_R \to \CAlg_R/R, \quad J\mathord{\textup{maps}}to R\amalg J. \]
This object is equipped with a multiplication using the multiplications on $R$ and $J$ as well as the $R$-module structure on $J$.
Additionally, we define a functor taking a non-unital commutative algebra to its module of indecomposables.
\begin{defi}\label{def:indecomposables}
Let $R$ be a commutative monoid in $\mathscr C$, and $J$ be a non-unital commutative $R$-algebra with multiplication map $\mu\colon J\otimes_R J\to J$. Then we define the \emph{module of indecomposables} $Q(J)$ of $J$ as the strict pushout
\[ \begin{tikzcd}
J\otimes_R J \arrow[r, "\mu"]\arrow[dr, phantom, "\ulcorner"{very near end}] \arrow[d] & J \arrow[d]\\
\ast \arrow[r] & Q(J)
\end{tikzcd}\]
in $\Mod_R$.
\end{defi}
This defines a functor $Q\colon \NUCA_R \to \Mod_R$. In the other direction, we can equip an $R$-module with the zero multiplication to obtain a functor
\[ Z\colon \Mod_R \to \NUCA_R.\]
\begin{prop}\label{prop:aug-ideal-Quillen-equivalence}
The functors
\[ \begin{tikzcd}
\CAlg_R/R \arrow[r, "I", swap, shift right] & \NUCA_R \arrow[l, "K", swap, shift right] \arrow[r, "Q", shift left] & \Mod_R \arrow[l, "Z", shift left]
\end{tikzcd}\]
define two Quillen adjunctions for the induced model structures from \Cref{subsection:model-structures-algebras}, with left adjoints the top arrows. The Quillen adjunction $K \dashv I$ is a Quillen equivalence. Moreover, the functors $K$ and $Z$ are homotopical.
\end{prop}
\begin{proof}
The fact that the two pairs of functors are adjoint can be seen from explicitly considering the unit and counit. For the first adjunction, the unit is given by the square
\[ \begin{tikzcd}
J \arrow[r, "\textup{incl}"] \arrow[d] & R\amalg J \arrow[d, "{\textup{pr}}"] \\
\ast \arrow[r] & R
\end{tikzcd}\]
and the counit by $(\eta,\text{incl})\colon R\amalg I(A) \to A$, where $\eta\colon R\to A$ is the unit map for the algebra $A$. For the second adjunction, the unit is given by the projection $J\to Q(J)$ from the defining pushout, considered as a map of non-unital commutative monoids for the zero multiplication on $Q(J)$. The counit is defined by the square
\[ \begin{tikzcd}
M\otimes_R M \arrow[r, "\ast"] \arrow[d] & M \arrow[d, "{\textup{id}}"] \\
\ast \arrow[r] & M
\end{tikzcd}\]
for any $R$-module $M$, where the top morphism is the zero map.
The first adjunction then is a Quillen adjunction, since $K$ preserves both cofibrations and acyclic cofibrations of non-unital commutative algebras. For this, it suffices to check the generating (acyclic) cofibrations, which are of the form $R\otimes \power{>0} f$ for $f$ a generating (acyclic) cofibration of $\mathscr C$. These morphisms are sent by $K$ to $R\otimes \power{} f$, which are precisely the generating (acyclic) cofibrations for the model structure on the category of augmented algebras. We conclude that $K$ is left Quillen and hence the first adjunction is a Quillen adjunction. Furthermore, since coproducts are homotopical in $\mathscr C$, also $K$ is homotopical.
For the second adjunction, we observe that the right adjoint $Z$ is the identity on underlying objects and morphisms. Since for both the model category of non-unital algebras and modules, the fibrations and weak equivalences are defined on underlying objects, we see that $Z$ is right Quillen and homotopical.
Finally, we need to check that $K\dashv I$ is a Quillen equivalence. We check the following criterion from \cite[Definition 1.3.12]{hovey}: Let $J\in \NUCA_R$ be a cofibrant non-unital commutative $R$-algebra and $S\in \CAlg_R/R$ be a fibrant augmented algebra. Then we have to show that a morphism $f\colon KJ\to S$ is a weak equivalence if and only if its adjoint $\tilde{f} \colon J\to IS$ is a weak equivalence. Since $S$ is fibrant, the augmentation $\epsilon \colon S\to R$ is a fibration in $\mathscr C$, and hence the defining diagram
\[ \begin{tikzcd}
I(S) \arrow[r, "\text{incl}"] \arrow[d] & S \arrow[d, "\epsilon"] \\
\ast \arrow[r] & R
\end{tikzcd}\]
is a homotopy pullback square in $\mathscr C$. The map $\eta\colon R\to A$ defines a section to $\epsilon$, so the distinguished triangle
\[
I(S)\xrightarrow{\text{incl}} S\xrightarrow{\;\epsilon\;} R\to \Sigma I(S)
\]
in ${\textup{Ho}}(\mathscr C)$ splits and the counit $(\eta,\text{incl})$ is a weak equivalence. Moreover, since $\tilde{f}$ is a retract of $R\amalg \tilde{f}$ and $R\amalg (\_)$ is homotopical, one is a weak equivalence if and only if the other is one. In total, since we have the relation $f = (\eta, i) \circ (R\amalg \tilde{f})$, we see that $K\dashv I$ is a Quillen equivalence.
\end{proof}
We also have a base-change adjunction
\begin{equation}\label{eq:base-change-adj}
\begin{tikzcd}[column sep = large]
\CAlg_R /S \arrow[r, shift left, "S\otimes_R \_"] & \CAlg_S /S \arrow[l, shift left, "\textup{forget}"]
\end{tikzcd}
\end{equation}
for any $R$-algebra $S$. Since the forgetful functor is the identity on underlying objects and morphisms, it is right Quillen, so also this adjunction is a Quillen adjunction.
\begin{defi}
Let $R$ be a commutative monoid in $\mathscr C$ and let $S$ be a commutative $R$-algebra. Then we define the \emph{abelianization} functor as the composite
\[ \Ab_{S/R} \colon {\textup{Ho}}(\CAlg_R /S) \xrightarrow{S\otimes_R^{\cat{L}} \_ } {\textup{Ho}}(\CAlg_S /S) \xrightarrow{\cat{R}I} {\textup{Ho}}(\NUCA_S) \xrightarrow{\cat{L}Q} {\textup{Ho}}(\Mod_S). \]
\end{defi}
\begin{rk}
The name \emph{abelianization} for the above construction originates in algebra, where it is an observation going back to Beck \cite{beck-cohomology} that abelian group objects in the category of augmented algebras over a commutative ring $R$ are equivalently $R$-modules, with the equivalence given by square-zero extensions in one direction and the augmentation ideal in the other. One then observes that Kähler differentials define a left adjoint functor to the inclusion of abelian group objects, interpreted as modules, into all augmented algebras.
In our context of $G$-global homotopy theory, we show in Theorem~\ref{thm:stabilization-aug-algebras} that $\Ab_{R/R}=\cat{L}Q\cat{R}I$ can be interpreted as a global stabilization.
\end{rk}
\begin{prop}\label{prop:abelianization-adj}
Let $R$ be a commutative monoid in $\mathscr C$ and $S$ be a commutative $R$-algebra. The functors
\[ \begin{tikzcd}[column sep = large]
{\textup{Ho}}(\CAlg_R/S) \arrow[r, shift left, "\Ab_{S/R}"] & {\textup{Ho}}(\Mod_S) \arrow[l, shift left, "\cat{L}K\circ \cat{R}Z"]
\end{tikzcd}\]
are adjoint, with abelianization being the left adjoint.
\begin{proof}
We use the Quillen adjunctions from \Cref{prop:aug-ideal-Quillen-equivalence} and \eqref{eq:base-change-adj} to calculate for an $S$-module $M$ and an $R$-algebra $T$ with augmentation to $S$:\phantom{\qedhere}
\begin{align*}
{\textup{Ho}}(\Mod_S)(\Ab_{S/R}(T), M) & \cong {\textup{Ho}}(\NUCA_S) ((\cat{R}I)(S\otimes_R^{\cat{L}} T), (\cat{R}Z)(M))\\
& \cong {\textup{Ho}}(\CAlg_S/S) ((\cat{L}K)(\cat{R}I)(S\otimes_R^{\cat{L}} T), (\cat{L}K)(\cat{R}Z)(M))\\
& \cong {\textup{Ho}}(\CAlg_R/S) (T, (\cat{L}K)(\cat{R}Z)(M)).\pushQED{\qed}\qedhere\popQED
\end{align*}
\end{proof}
\end{prop}
Using these adjunctions, we now define the cotangent complex:
\begin{defi}\label{def:cotangent-complex}
Let $R$ be a commutative monoid in $\mathscr C$ and $S$ be an $R$-algebra. The cotangent complex of $S$ over $R$ is defined as the object
\[ \Omega_{S/R} = \Ab_{S/R} (S) =(\cat{L}Q) (\cat{R}I)(S \otimes_R^{\cat{L}} S)\]
in ${\textup{Ho}}(\Mod_S)$.
\end{defi}
For any square
\[\begin{tikzcd}
R\arrow[r]\arrow[d] & R^{\textup{pr}}ime\arrow[d]\\
S \arrow[r] & S^{\textup{pr}}ime
\end{tikzcd}\]
in the model category $\CAlg$, we obtain an induced morphism on cotangent complexes $\Omega_{S/R}\to \Omega_{S^{\textup{pr}}ime/R^{\textup{pr}}ime}$ as $S$-modules. This arises from the universal property of pushouts and pullbacks.
For any commutative $R$-algebra $S$, we also obtain a \emph{universal derivation}
\begin{equation}\label{eq:universal-derivation}
d_{S/R}\colon S\to \Omega_{S/R}
\end{equation}
by composing the unit $S\to S\amalg \Omega_{S/R} = (KZ)(\Ab_{S/R}(S))$ of the adjunction in \Cref{prop:abelianization-adj} with the projection to $\Omega_{S/R}$.
This cotangent complex behaves just as in the classical case of commutative algebras, in that is comes with a transitivity exact sequence and a base-change formula. These properties allow us to consider the cotangent complex as defining a cohomology theory for augmented commutative algebras. In order to establish these properties, we first need to consider how the augmentation ideal and indecomposables behave under base changes.
\begin{lemma}\label{lemma:aug-ideal-base-change}
Let $R$ be a commutative monoid in $\mathscr C$, let $S$ be a commutative $R$-algebra, and let $T$ be an augmented commutative $R$-algebra. Then the commutative square
\begin{equation}\label{diag:aug-ideal-base-change}
\begin{tikzcd}
I_R(T) \otimes_R S \arrow[r]\arrow[d] & T\otimes_R S \arrow[d]\\
\ast \arrow[r] & S
\end{tikzcd}
\end{equation}
induces a morphism
\[ \cat{R}I_R (T) \otimes_R^{\cat{L}} S \to \cat{R}I_S(T\otimes_R^{\cat{L}} S), \]
which is an isomorphism in the homotopy category of non-unital commutative $S$-algebras.
\end{lemma}
\begin{proof}
The square \eqref{diag:aug-ideal-base-change} induces a natural transformation $I_R(\_) \otimes_R S \Rightarrow I_S(\_\otimes_R S)$ by considering pullbacks. Here, the functor $(\_)\otimes_R S$ can be left derived, and the functors $I$ can be right derived. We thus consider the double category of model categories and left and right derivable functors. In this context, \cite[Theorem 7.6]{shulman-left-right-derived} shows that taking homotopy categories and derived functors is a double pseudofunctor. In particular, the natural transformation above induces a transformation
\[ \cat{R}I_R(\_) \otimes_R^{\cat{L}} S \Rightarrow \cat{R}I_S(\_\otimes_R^{\cat{L}} S)\]
as desired.
Since $I$ is a Quillen equivalence with inverse $K$, we can show that this transformation is a natural isomorphism in the homotopy category by considering its mate
\[ \cat{L}K_S(\_\otimes_R^{\cat{L}} S) \Rightarrow \cat{L}K_R(\_)\otimes_R^{\cat{L}} S. \]
This transformation is induced from the natural isomorphism $S\amalg (\_\otimes_R S) \cong (R\amalg \_) \otimes_R S$ of left Quillen functors, and hence is a natural isomorphism.
\end{proof}
\begin{lemma}\label{lemma:indec-base-change}
Let $R$ be a commutative monoid in $\mathscr C$, let $S$ be a commutative $R$-algebra and $J$ be a non-unital commutative $R$-algebra. Then the commutative square
\begin{equation}\label{diag:indec-base-change}
\begin{tikzcd}
(J\otimes_R J) \otimes_R S \arrow[r]\arrow[d] & J\otimes_R S \arrow[d]\\
\ast \arrow[r] & Q_R(J) \otimes_R S
\end{tikzcd}
\end{equation}
induces a morphism
\[ \cat{L}Q_S (J\otimes_R^{\cat{L}} S) \to \cat{L}Q_R(T) \otimes_R^{\cat{L}} S, \]
which is an isomorphism in the homotopy category of $S$-modules.
\end{lemma}
\begin{proof}
As both the functors labelled $Q$ and the base change $(\_)\otimes_R S$ are left Quillen functors, the desired morphism can be obtained from the natural morphism $Q_S(J \otimes_R^{\cat{L}} S) \to Q_R(J) \otimes_R^{\cat{L}} S$ induced on pushouts by the diagram \eqref{diag:indec-base-change}, by considering cofibrant replacements. Moreover, as $(\_)\otimes_R S$ is a left adjoint, it preserves pushouts, which implies that this morphism is an isomorphism.
\end{proof}
\begin{lemma}\label{lemma:cofib-base-change}
Let $R$ be a commutative monoid in $\mathscr C$ and let $S$ and $T$ be commutative $R$-algebras with a map $S\to T$ of commutative $R$-algebras. Assume moreover that $\mathscr C$ is left proper or that $S$ is cofibrant as a commutative monoid. Then a homotopy cofiber of $S\otimes_R^{\cat{L}} T\to T\otimes_R^{\cat{L}} T$ in $\CAlg_T/T$ is given by $T\otimes_S^{\cat{L}} T$.
\end{lemma}
\begin{proof}
This statement may be shown by considering cofibrant replacements. In order to calculate $S\otimes_R^{\cat{L}} T$, we consider a cofibrant replacement $\Gamma_RS\to S$ of $S$ as a commutative $R$-algebra. Moreover, in order to calculate $T \otimes_R^{\cat{L}} T$, we decompose the morphism $\Gamma_R S\to S\to T$ into a cofibration followed by a weak equivalence as $\Gamma_R S\to \Gamma_R T\to T$.
We now take the homotopy cofiber of
\[ \Gamma_R S\otimes_R T\to \Gamma_R T\otimes_R T.\]
This morphism arises from applying the left Quillen functor $\_\otimes_R T\colon \CAlg_R/T \to \CAlg_T/T$ to the cofibration $\Gamma_RS\to \Gamma_RT$ between cofibrant objects, so it is itself a cofibration between cofibrant objects. Hence, its homotopy cofiber is represented by the $1$-categorical cofiber, which can be calculated as
\[ (\Gamma_R T\otimes_R T)\otimes_{\Gamma_R S\otimes_R T} T \cong (\Gamma_R T\otimes_{\Gamma_R S} \Gamma_R S\otimes_R T)\otimes_{\Gamma_R S\otimes_R T} T \cong \Gamma_R T\otimes_{\Gamma_R S} T. \]
We now need to compare this with $\Gamma_S T\otimes_S T$, where $\Gamma_ST$ is a cofibrant replacement of $T$ as a commutative $S$-algebra. For this, it suffices to establish a weak equivalence $\Gamma_RT \otimes_{\Gamma_RS} S \to \Gamma_ST$ of commutative $S$-algebras. As both $\Gamma_RT \otimes_{\Gamma_RS} S$ and $\Gamma_ST$ are cofibrant $S$-algebras (since $\_\otimes_{\Gamma_RS}S$ is left Quillen and by definition, respectively), the left Quillen functor $\_\otimes_ST$ then preserves this weak equivalence.
In order to establish this weak equivalence, we consider the diagram
\[\begin{tikzcd}[column sep = large]
\Gamma_RS \arrow[r, "\simeq"] \arrow[dd] & S \arrow[r] \arrow[d] & \Gamma_S T\arrow[dd, "\simeq"]\\
& \Gamma_RT\otimes_{\Gamma_RS} S \arrow[ru, dashed] \\
\Gamma_R T \arrow[rr, "\simeq", swap] \arrow[ru] \arrow[rruu, dashed, bend right=20] && T.
\end{tikzcd}\]
Here, $S\to \Gamma_ST \to T$ is the factorization of $S\to T$ into a cofibration followed by an acyclic fibration provided by the cofibrant replacement $\Gamma_S T$. As $\Gamma_R S\to \Gamma_R T$ is a cofibration, so is $S\to \Gamma_RT\otimes_{\Gamma_RS} S$ and the diagonal dashed morphisms exist by the lifting property. Moreover, since either the categories in question are left proper or $S$ is cofibrant, we conclude that the morphism $\Gamma_R T\to \Gamma_RT\otimes_{\Gamma_RS} S$ is a weak equivalence. Hence we indeed have a weak equivalence $\Gamma_RT \otimes_{\Gamma_RS} S \to \Gamma_ST$ as desired by the 2-out-of-3 property.
\end{proof}
From this cofiber calculation, we deduce the transitivity sequence for the cotangent complex.
\begin{thm}\label{thm:transitivity_sequence_topological_cotangent_complex}
Let $R\to S\to T$ be a sequence of commutative monoids in $\mathscr C$. Moreover, assume that either $\mathscr C$ is left proper or that $S$ is cofibrant. Then the sequence
\[ \Omega_{S/R}\otimes_S^{\cat{L}} T \to \Omega_{T/R} \to \Omega_{T/S}, \]
induced from functoriality of $\Omega$, admits the structure of a homotopy cofiber sequence of $T$-modules.
\end{thm}
\begin{proof}
We consider the morphism $S\otimes_R^{\cat{L}} T \to T\otimes_R^{\cat{L}} T$ of $T$-algebras augmented to $T$. By \Cref{lemma:cofib-base-change}, the homotopy cofiber of this morphism in $\CAlg_T/T$ is given by $T\otimes_S^{\cat{L}} T$. Since $I$ is a Quillen equivalence and $Q$ is left Quillen, applying $\cat{L} Q_T\circ \cat{R} I_T$ to the resulting homotopy cofiber sequence yields a homotopy cofiber sequence of $T$-modules. This takes the form
\[ \cat{L} Q_T(\cat{R} I_T(S\otimes_R^{\cat{L}} T)) \to \cat{L} Q_T(\cat{R} I_T(T\otimes_R^{\cat{L}} T)) \to \cat{L} Q_T(\cat{R} I_T(T\otimes_S^{\cat{L}} T)). \]
The last two terms are by definition the cotangent complexes $\Omega_{T/R}$ and $\Omega_{T/S}$. We thus only have to identify the first term as $\Omega_{S/R} \otimes_S^{\cat{L}} T$.
For this, we observe that $S\otimes_R^{\cat{L}} T \cong (S\otimes_R^{\cat{L}} S) \otimes_S^{\cat{L}} T$ and use \Cref{lemma:aug-ideal-base-change,lemma:indec-base-change} to calculate
\begin{align*}
\cat{L} Q_T(\cat{R} I_T(S\otimes_R^{\cat{L}} T)) & \cong \cat{L} Q_T(\cat{R} I_T((S\otimes_R^{\cat{L}} S)\otimes_S^{\cat{L}} T))\\
& \cong \cat{L} Q_T( \cat{R} I_S(S\otimes_R^{\cat{L}} S) \otimes_S^{\cat{L}} T)\\
& \cong \cat{L} Q_S(\cat{R} I_S(S\otimes_R^{\cat{L}} S)) \otimes_S^{\cat{L}} T\\
& \cong \Omega_{S/R} \otimes_S^{\cat{L}} T.
\end{align*}
In total, we obtain the desired homotopy cofiber sequence.
\end{proof}
Moreover, we also get base change and additivity results.
\begin{prop}\label{prop:base_change-cotangent-complex}
Let $R$ be a commutative monoid in $\mathscr C$ and $S$ and $T$ be two commutative $R$-algebras. Then there are natural isomorphisms
\begin{align*}
\Omega_{S\otimes_R^{\cat{L}} T/T} &\cong \Omega_{S/R} \otimes_R^{\cat{L}} T \qquad \textrm{and}\\
\Omega_{S\otimes_R^{\cat{L}} T/R} &\cong (\Omega_{S/R} \otimes_R^{\cat{L}} T) \amalg (S\otimes_R^{\cat{L}}\Omega_{T/R}).
\end{align*}
\end{prop}
\begin{proof}
For the first assertion, we calculate
\begin{align*}
\Omega_{S\otimes_R^{\cat{L}} T/T} & \cong \cat{L} Q_{S\otimes_R^{\cat{L}} T}(\cat{R} I_{S\otimes_R^{\cat{L}} T}((S\otimes_R^{\cat{L}} T)\otimes_T^{\cat{L}} (S\otimes_R^{\cat{L}} T)))\\
& \cong \cat{L} Q_{S\otimes_R^{\cat{L}} T}(\cat{R} I_{S\otimes_R^{\cat{L}} T}((S\otimes_R^{\cat{L}} S)\otimes_S^{\cat{L}} (S\otimes_R^{\cat{L}} T)))\\
& \cong \cat{L} Q_{S\otimes_R^{\cat{L}} T}(\cat{R} I_S(S\otimes_R^{\cat{L}} S) \otimes_S^{\cat{L}} (S\otimes_R^{\cat{L}} T)) \\
& \cong \cat{L} Q_S(\cat{R} I_S(S\otimes_R^{\cat{L}} S)) \otimes_S^{\cat{L}} (S\otimes_R^{\cat{L}} T)\\
& \cong \Omega_{S/R}\otimes_R^{\cat{L}} T.
\end{align*}
The second assertion follows from observing that the transitivity cofiber sequences for $R\to S\to S\otimes_R^{\cat{L}} T$ and $R\to T\to S\otimes_R^{\cat{L}} T$ fit together to define splittings for each other, and applying the first assertion to the resulting cotangent complexes.
\end{proof}
\subsection{Topological André-Quillen cohomology of \texorpdfstring{$\bm G$}{G}-global ring spectra}
By Proposition~\ref{prop:white-assumptions}, the positive $G$-global model structures on $\cat{$\bm G$-Spectra}$ are symmetric monoidal and satisfy the monoid and strong commutative monoid axiom; moreover, coproducts in them are homotopical by \cite[Lemma~3.1.43]{g-global}. We can therefore specialize the above discussion to this setting, yielding:
\begin{defi}\label{def:TAQ}
Let $R$ be a $G$-global ultra-commutative ring spectrum and $S$ be an $R$-algebra. The cotangent complex of $S$ over $R$ is defined as
\[ \Omega_{S/R} = (\cat{L}Q) (\cat{R}I)(S \smashp_R^{\cat{L}} S).\]
This is an $S$-module, and the homology and cohomology theories represented by it are called \emph{($G$-global) topological André-Quillen (co)homology}. Explicitly, for an $S$-module $M$, topological André-Quillen (co)homology of $S$ over $R$ with coefficients in $M$ is defined as
\begin{align*}
\ul{\TAQ}_\ast (S, R;M) & = \ul{\textup{h}at\pi}_\ast (\Omega_{S/R}\smashp_S^{\cat{L}} M) \\
\ul{\TAQ}^\ast (S, R;M) & = \ul{\textup{h}at\pi}_{-\ast} (\cat{R} F(\Omega_{S/R}, M)).
\end{align*}
Here, $F$ denotes the usual function spectrum, carrying the structure of an $S$-module.
\end{defi}
As a consequence of \Cref{thm:transitivity_sequence_topological_cotangent_complex} and \Cref{prop:base_change-cotangent-complex}, we obtain a transitivity sequence and base change for these. We explicitly state the transitivity sequence in homotopy groups.
\begin{cor}\label{cor:transitivity-seq-homotopy-groups}
Let $R\to S\to T$ be morphisms of $G$-global ultra-commutative ring spectra, and let $M$ be a $T$-module. Then there are long exact sequences
\[\textup{h}skip-11.67pt\textup{h}fuzz=12pt \cdots \to \ul{\TAQ}_{n+1}(T, S; M) \to \ul{\TAQ}_n(S,R; M)\to \ul{\TAQ}_n(T,R; M) \to \ul{\TAQ}_{n}(T,S;M) \to \cdots \]
and
\[\textup{h}skip-11.67pt\textup{h}fuzz=12pt \cdots \to \ul{\TAQ}^n(T, S; M) \to \ul{\TAQ}^n(T,R; M)\to \ul{\TAQ}^n(S,R; M) \to \ul{\TAQ}^{n+1}(T,S;M) \to \cdots \]
induced from the cofibre sequence from \Cref{thm:transitivity_sequence_topological_cotangent_complex}. \qed
\end{cor}
We now present two applications of the above theory to the study of $G$-global ultra-commutative ring spectra. The first is a Hurewicz theorem, which says that vanishing of the (relative) André-Quillen homology detects equivalences of $G$-global ultra-commutative ring spectra. The other application is a construction of Postnikov towers for global ultra-commutative ring spectra, with $k$-invariants in global topological André-Quillen cohomology. These applications are analogous to the usage of topological André-Quillen homology in \cite[Chapter 8]{basterra-TAQ}, but the proofs are simplified by the use of t-structures.
Both of these results need a connectivity hypothesis, and the first step is to consider how the indecomposables functor $Q$ interacts with connectivity.
\begin{lemma}\label{lemma:Q-connectivity}
Let $R$ be a connective $G$-global ultra-commutative ring spectrum and $J$ be a cofibrant non-unital commutative $R$-algebra. Suppose moreover that $J$ is $n$-connected for $n\geq 0$. Then also $Q(J)$ is $n$-connected, and the adjunction unit $q\colon J\to Q(J)$ induces an isomorphism on $\ul{\textup{h}at\pi}_k$ for $n+1\leq k< 2n$.
\end{lemma}
\begin{proof}
The module of indecomposables is defined via the cofiber sequence
\[ J\smashp_R J \to J\xrightarrow{\eta} Q(J) \]
of $R$-modules. Here, $J$ is $n$-connected by assumption, while $J\smashp_RJ$ is $(2n-1)$-connected by Proposition~\ref{prop:t-structure}; the claim follows from the long exact sequence.
\end{proof}
\begin{thm}[Hurewicz theorem]\label{thm:Hurewicz}
Let $R$ be a connective $G$-global ultra-commuta\-tive ring spectrum and let $S$ be a connective commutative $R$-algebra such that the unit map $\eta\colon R\to S$ is an $n$-equivalence for $n\geq 1$. Then $\Omega_{S/R}$ is $n$-connected, and the universal derivation $d_{S/R}\colon S\to \Omega_{S/R}$ factors through the cone $C(\eta)$, where it induces an isomorphism $\ul{\textup{h}at\pi}_{n+1} (C(\eta)) \cong \ul{\textup{h}at\pi}_{n+1}(\Omega_{S/R})$.
\end{thm}
\begin{proof}
After cofibrant replacement, we may assume that $S$ is a cofibrant commutative $R$-algebra. We consider the diagram
\[ \begin{tikzcd}
R \arrow[r, "\eta"] \arrow[d, "\eta", swap] & S \arrow[r] \arrow[d] & C(\eta)\arrow[d, "\iota"]\\
S \arrow[r] & S\smashp_R S \arrow[r] & S\smashp_R C(\eta).
\end{tikzcd}\]
The first line of the diagram is a cofiber sequence by definition, and the second line arises from it by applying $S\smashp_R \_$. Since $S$ is cofibrant, this again is a cofiber sequence. The vertical morphisms are the inclusions as the right factors. By \Cref{prop:aug-ideal-Quillen-equivalence}, the counit
\[ S\vee \cat{R}I(S\smashp_R S) \to S\smashp_R S\]
is an equivalence, and thus by comparing cofibers we obtain $S\smashp_R C(\eta) \cong \cat{R}I (S\smashp_R S)$. We now consider the composition
\[ C(\eta)\xrightarrow{\iota} S\smashp_R C(\eta) \cong \cat{R}I(S\smashp_R S) \xrightarrow{q} \Omega_{S/R},\]
where the last map is an instance of the unit map $J\to \cat{L}Q(J)$ considered in \Cref{lemma:Q-connectivity}, for the non-unital algebra $J= S\smashp_R C(\eta)\cong \cat{R}I(S\smashp_R S) $. Since $\eta$ is $n$-connected, so is $C(\eta)$. As $S$ is connective, also $S\smashp_R C(\eta)$ is $n$-connected, and the morphism $\iota$ is an isomorphism on $\ul{\textup{h}at\pi}_k$ for $k\leq {n+1}$. By \Cref{lemma:Q-connectivity}, also $q$ induces an isomorphism on $\ul{\textup{h}at\pi}_k$ for $k\leq n+1$, and this finishes the proof.\\
Unravelling the definition of the morphism $C(\eta)\to \Omega_{S/R}$ considered here, we observe that it is indeed induced by the universal derivation $d_{S/R}\colon S\to \Omega_{S/R}$, using that this map vanishes on the image of $R$ in $S$.
\end{proof}
\begin{cor}\label{cor:TAQ-detects-equiv}
Let $R$ be a connective $G$-global ultra-commutative ring spectrum and $S$ be a connective $R$-algebra such that $\Omega_{S/R}\simeq0$. Then the unit map $\eta\colon R\to S$ is an equivalence.
\end{cor}
\begin{proof}
Suppose $\eta$ is not an equivalence, and let $\ul{\textup{h}at\pi}_k(C(\eta))$ be the first non-trivial homotopy group of the cone. Then also $\ul{\textup{h}at\pi}_k(\Omega_{S/R})\neq 0$ by \Cref{thm:Hurewicz}, in contradiction to triviality of the cotangent complex.
\end{proof}
Hence, we see that topological André-Quillen cohomology detects equivalences of connective $G$-global ultra-commutative ring spectra.
Next, we construct Postnikov towers for global ultra-commutative ring spectra. For this, we explain how elements of the topological André-Quillen cohomology parametrize extensions of algebras.
\begin{constr}\label{constr:extension-k-inv}
Let $R$ be a $G$-global ultra-commutative ring spectrum, $S$ be a commutative $R$-algebra and $M$ be an $S$-module. By the definition of topological André-Quillen cohomology, we have the identification
\[ \TAQ_e^n(S, R; M) \cong {\textup{Ho}}(\cat{Mod}^G_S)(\Omega_{S/R}, \Sigma^n M) \cong {\textup{Ho}}(\cat{UCom}^G_R/S) (S, S\vee \Sigma^nM)\]
where $e\colon1\to G$ is the trivial homomorphism.
In particular, for a given class $k\in \TAQ_e^n(S, R; M)$, we may interpret it as a morphism $S\to S\vee \Sigma^nM$ of commutative $R$-algebras over $S$. We form the homotopy pullback
\[ \begin{tikzcd}
S[k] \arrow[r] \arrow[d] & S \arrow[d, "\iota"]\\
S \arrow[r, "k", swap] & S\vee \Sigma^nM
\end{tikzcd}\]
in $R$-algebras and call it the extension of $S$ by $k$. Here, the right vertical map is the inclusion of $S$ as the first wedge summand.
Topological André-Quillen cohomology can be related to usual cohomology represented by a $G$-global spectrum by the universal derivation $d_{S/R}\colon S\to \Omega_{S/R}$ constructed in \eqref{eq:universal-derivation}. Precomposition with this derivation takes a map $\tilde{k}\colon \Omega_{S/R}\to \Sigma^n M$ representing an element in $\TAQ_e^n(S, R; M)$ to the map $\tilde{k}\circ d_{S/R}\colon S\to \Sigma^n M$. By the definition of the universal derivation, this agrees (in the homotopy category) with the composition
\[ {\textup{pr}}\circ k\colon S\to S\vee \Sigma^n M \to \Sigma^n M,\]
where $k$ is the adjoint to $\tilde{k}$ under the adjunction between square-zero extensions and Kähler differentials.
Using this translation, we observe that $S[k]$ is also the homotopy pullback in the total square
\[ \begin{tikzcd}
S[k] \arrow[r] \arrow[d] & S \arrow[d, "\iota"] \arrow[r] & \ast \arrow[d]\\
S \arrow[r, "k", swap] & S\vee \Sigma^nM \arrow[r, "{\textup{pr}}", swap] & \Sigma^n M
\end{tikzcd}\]
and hence the homotopy fiber of $\tilde{k}\circ d_{S/R}$ in the category of $R$-modules.
\end{constr}
If $X$ is any ultra-commutative ring spectrum, then its zeroth homotopy groups $\ul{\textup{h}at\pi}_0X$ come with additional norm maps, defined via the multiplication on smash powers, giving them the structure of a so-called \emph{global power functor} \cite[Definition~5.1.6]{schwede-book}. In the next theorem, we will need that conversely any global power functor $F$ gives rise to an \emph{ultra-commutative} Eilenberg-MacLane spectrum $HF$, i.e.~an ultra-commutative global ring spectrum with $\ul{\textup{h}at\pi}_0(HF)\cong F$ as global power functors and $\ul{\textup{h}at\pi}_k(HF)=0$ for $k\not=0$, see \cite[Theorem~5.4.14]{schwede-book}. As the corresponding result in $G$-global homotopy theory for general $G$ has not been established yet, this means we have to restrict to $G=1$ here; however, once the corresponding theory of Eilenberg-MacLane spectra is set up, the same result will hold for arbitrary $G$, with the same proof.
\begin{thm}\label{thm:Postnikov-towers}
Let $R$ be a connective global ultra-commutative ring spectrum. Then there is a sequence $R_0, R_1, \ldots$ of commutative $R$-algebras together with maps $R_{n+1}\to R_n$ of $R$-algebras and classes $k_n\in \TAQ_e^{n+1}(R_n, R; H\ul{\textup{h}at\pi}_{n+1}(R))$, such that the following properties are satisfied:
\begin{enumerate}
\item $R_0\cong H\ul{\textup{h}at\pi}_0(R)$ and $R_{n+1} \cong R_{n}[k_n]$,
\item $\ul{\textup{h}at\pi}_k(R_n)=0$ for $k>n$,
\item the unit maps $\eta_n\colon R\to R_n$ are $(n+1)$-equivalences.
\end{enumerate}
\end{thm}
\begin{proof}
We define $R_0= H\ul{\textup{h}at\pi}_0(R)$ as an Eilenberg-MacLane spectrum for the global power functor $\ul{\textup{h}at\pi}_0(R)$. This is a global ultra-commutative ring spectrum, and it comes with a morphism $\eta_0\colon R\to H\ul{\textup{h}at\pi}_0(R)$ of ultra-commutative ring spectra inducing an isomorphism on $\ul{\textup{h}at\pi}_0$ (see the remark before \cite[Theorem 5.4.14]{schwede-book}). Hence $R_0$ is a possible first stage of the Postnikov tower.
Now suppose that we have constructed the tower up to level $n$. In particular, we have a morphism $\eta_n\colon R\to R_n$ of ultra-commutative ring spectra that is an $(n+1)$-equivalence, and $\ul{\textup{h}at\pi}_{n+2}(R_n)= \ul{\textup{h}at\pi}_{n+1}(R_n) = 0$. The Hurewicz theorem \ref{thm:Hurewicz} shows that thus $\Omega_{R_n/R}$ is $(n+1)$-connected and $\ul{\textup{h}at\pi}_{n+2}(\Omega_{R_n/R})\cong \ul{\textup{h}at\pi}_{n+1}(R)$. This isomorphism defines a morphism $\tilde{k}_n\colon \Omega_{R_n/R}\to \Sigma^{n+2} H\ul{\textup{h}at\pi}_{n+1}(R)$ of $R$-modules, which corresponds to an element $k_n\in \TAQ_e^{n+1}(R_n, R; H\ul{\textup{h}at\pi}_{n+1}(R))$.
Using this element and \Cref{constr:extension-k-inv}, we define $R_{n+1}= R_n[k_n]$. This comes with a map $R_{n+1}\to R_n$ of $R$-algebras. Furthermore, as an $R$-module, the algebra $R_{n+1}$ is the homotopy fiber of the map \[R_n\xrightarrow{d_{R_n/R}} \Omega_{R_n/R} \xrightarrow{\tilde{k}_n} \Sigma^{n+2} H\ul{\textup{h}at\pi}_{n+1}(R).\] Hence the morphism $\eta_{n+1}\colon R\to R_{n+1}$ is indeed an $(n+2)$-equivalence and all higher homotopy groups of $R_{n+1}$ vanish. Thus, the theorem follows by induction.
\end{proof}
\subsection{(Co)Homology as a stabilization}
We now explain how $G$-global topological André-Quillen cohomology can be interpreted as a global stabilization:
\begin{thm}\label{thm:stabilization-aug-algebras}
Let $R$ be a flat $G$-global ultra-commutative ring spectrum. Then all the global Quillen adjunctions
\begin{equation}
\begin{tikzcd}
\glo{Mod}_R\arrow[r, "\Sigma^\infty", shift left] &\arrow[l, "\Omega^\infty", shift left] \glo{Sp}(\glo{Mod}_R) \arrow[r, "\glo{Sp}(\ul{Z})", shift right, swap] &[2em]\arrow[l, "\glo{Sp}(\ul{Q})", shift right, swap] \glo{Sp}(\glo{NUCA}_R) \arrow[r, "\glo{Sp}(\ul{K})", shift left] &[2em] \arrow[l, "\glo{Sp}(\ul{I})", shift left] \glo{Sp}(\glo{Comm}_R/R)
\end{tikzcd}
\end{equation}
are global Quillen equivalences. In particular, the composite
\begin{equation*}
\glo{Mod}_R\xrightarrow{\;Z\;}\glo{NUCA}_R\xrightarrow{I^{-1}}\glo{Comm}_R/R
\end{equation*}
in $\cat{GLOBMOD}$ (which on homotopy categories gives the right adjoint to $\Ab_{R/R}$) is the universal map from a stable global model category to $\glo{Comm}_R/R$.
\end{thm}
For the proof, the main step is to show that another stabilization of a global Quillen adjunction is an equivalence, namely the free-forgetful adjunction for non-unital commutative $R$-algebras:
\begin{thm}\label{thm:stabilization-nucas}
In the above situation, the global Quillen adjunction
\begin{equation}\label{eq:stab-free-forgetful-nuca}
\begin{tikzcd}
\glo{Sp}(\glo{Mod}_R) \arrow[r, "\glo{Sp}(\ul{\mathbb P^{>0}})", yshift=3pt] &[2em]\arrow[l, "\glo{Sp}(\ul{\mathbb U})", yshift=-3pt] \glo{Sp}(\glo{NUCA}_R)
\end{tikzcd}
\end{equation}
is a global Quillen equivalence.
\end{thm}
The proof of this requires a substantial amount of work, and we devote all of \Cref{section:stab-nucas} to it. For now, let us use it to deduce \Cref{thm:stabilization-aug-algebras}:
\begin{proof}[Proof of \Cref{thm:stabilization-aug-algebras}]
Since $\ul K\dashv\ul I$ is a global Quillen equivalence, so is the globally stabilized adjunction $\ul{\textup{Sp}}(\ul K)\dashv\ul{\textup{Sp}}(\ul I)$ by \Cref{prop:sp-induced-adjunction}. Moreover, the global model category $\glo{Mod}_R$ is globally stable (Corollary~\ref{cor:modules-stable}), so the stabilization adjunction
\begin{equation*}
\begin{tikzcd}
\glo{Mod}_R\arrow[r, "\Sigma^\infty", shift left] &\arrow[l, "\Omega^\infty", shift left] \glo{Sp}(\glo{Mod}_R)
\end{tikzcd}
\end{equation*}
is a global Quillen equivalence by \Cref{thm:stabilization-preserves-stable}.
Finally, we have to consider how the adjunction $\ul Q\dashv\ul Z$ behaves upon stabilization. For this, we use that by \Cref{thm:stabilization-nucas} the adjunction $(\ref{eq:stab-free-forgetful-nuca})$ is a global Quillen equivalence. We now observe that the composite $\ul Q\circ \ul{\power{>0}}\colon \glo{Mod}_R\to \glo{NUCA}_R \to \glo{Mod}_R$ of left Quillen functors is naturally isomorphic to the identity. Thus, the same is true after stabilization, and also the associated composite of left derived functors is naturally isomorphic to the identity. Hence, $\ul Q\dashv\ul Z$ induces a global Quillen equivalence after stabilization by $2$-out-of-$3$, finishing the proof.
\end{proof}
\section{The stabilization of global NUCAs}\label{section:stab-nucas}
\subsection{Computing the stabilization} This section is devoted to the proof of Theorem~\ref{thm:stabilization-nucas}. We begin by recasting the left adjoint $\ul{\textup{Sp}}(\ul{\mathbb P^{>0}})$ into a more convenient form:
\begin{constr}\label{constr:diamond-sp-sp}
Let $X,Y\in \textup{Sp}(\glo{GlobalSpectra})$. We define $X\diamond Y$ as the bispectrum with $(X\diamond Y)(B)=X(B)\smashp Y(B)$ (where the right hand side is the usual Day convolution smash product on $\cat{Spectra}$) and structure maps
\begin{align*}
S^A\smashp (X\diamond Y)(B)&=S^A\smashp X(B)\smashp Y(B)\xrightarrow{\delta} S^A\smashp X(B)\smashp S^A \smashp Y(B)\\&\xrightarrow{\sigma\smashp\sigma} X(A\amalg B)\smashp Y(A\amalg B)=(X\diamond Y)(A\amalg B)
\end{align*}
where the first map is induced by the diagonal $S^A\to S^A\smashp S^A$. This becomes a functor in the obvious way.
If $G$ acts on $X$ and $H$ acts on $Y$, then $G\times H$ acts on $X\diamond Y$ via functoriality; if $G=H$, we will typically equip $X\diamond Y$ with the diagonal $G$-action, yielding a bifunctor on $G\text-\textup{Sp}(\glo{GlobalSpectra})=\textup{Sp}(G\text-\glo{GlobalSpectra})$.
Finally, let $X\in G\text-\textup{Sp}(\glo{GlobalSpectra})$ and let $n\ge 1$. Then we obtain $n$ commuting $G$-actions on the $n$-fold $\diamond$-product $X^{\diamond n}$, which together with the $\Sigma_n$-action via permuting the factors assemble into a $(\Sigma_n\wr G)$-action. We will also frequently view $X^{\diamond n}$ as an object in $(\Sigma_n\times G)\text-\textup{Sp}(\glo{GlobalSpectra})$ via the diagonal $G$-action.
More generally, if $R$ is a ($G$-)global ultra-commutative ring spectrum, then we define $\diamond_R$ as the pointwise smash product over $R$, yielding functors $\textup{Sp}(\glo{Mod}^G_R)\times\textup{Sp}(\glo{Mod}^G_R)\to\textup{Sp}(\glo{Mod}^G_R)$.
\end{constr}
\begin{rk}
For later use we record that the same construction can be applied more generally in the categories $\cat{Fun}(\bm\Sigma,\cat{Spectra})$ and $\cat{Fun}(\bm\Sigma,\cat{Mod}_R^G)$ of all $\cat{SSet}$-enriched (as opposed to $\cat{SSet}_*$-enriched) functors $\bm\Sigma\to\cat{Spectra}$ or $\bm\Sigma\to\cat{Mod}_R^G$, respectively. For these larger categories, $\diamond$ and $\diamond_R$ are the tensor product of a symmetric monoidal structure with unit the constant functor at $\mathbb S$ or $R$, respectively (while unitality fails in $\textup{Sp}(\glo{GlobalSpectra})$ and $\textup{Sp}(\glo{Mod}^G_R)$).
\end{rk}
If now $X\in \textup{Sp}(\glo{GlobalSpectra})$, then comparing right adjoints yields a natural isomorphism $\textup{Sp}(\mathbb P_R^{>0})\textup{Sp}(R\smashp{\textup{--}})(X)\cong\textup{Sp}(R\smashp{\textup{--}})\textup{Sp}(\mathbb P_{\mathbb S}^{>0})(X) =\textup{Sp}(R\smashp{\textup{--}})\big(\bigvee_{n\ge1} X^{\diamond n}/\Sigma_n\big)$. Non-equivariantly, Basterra and Mandell \cite[Theorem~2.8]{basterra-mandell-stab} proved that the summands indexed by $n>1$ vanish (for suitably cofibrant $X$), and this is the key non-formal ingredient used to compare spectra of $R$-NUCAs with $R$-modules (Theorem~3.7 of \emph{op.~cit.}). Similarly, the following $G$-global comparison will be the key computational ingredient to the proof of our Theorem~\ref{thm:stabilization-nucas}:
\begin{thm}\label{thm:diamond-sp-sp-trivial}
\begin{enumerate}
\item Let $X,Y\in G\text{-}\textup{Sp}(\glo{GlobalSpectra})$ levelwise flat. Then $X\diamond Y$ is $G$-globally weakly equivalent to $0$.
\item Let $X\in G\text{-}\textup{Sp}(\glo{GlobalSpectra})$ levelwise \emph{positively} flat. Then $X^{\diamond n}/\Sigma_n$ is $G$-globally weakly equivalent to $0$ for every $n>1$.
\item If $i\colon X\to Y$ is a levelwise positive flat cofibration, then $i^{\ppo n}/\Sigma_n\colon Q^n/\Sigma_n\to Y^{\diamond n}/\Sigma_n$ is a levelwise positive flat cofibration and a {$G$-global weak equivalence} for every $n>1$. In particular, if $Y$ is levelwise positively flat, then $Q^n/\Sigma_n$ is $G$-globally weakly equivalent to $0$.
\end{enumerate}
\end{thm}
The proof will be given below after some preparations; for now, let us use it to deduce the theorem:
\begin{proof}[Proof of Theorem~\ref{thm:stabilization-nucas}]
Fix a finite group $H$. We have to show that $H\text-\textup{Sp}(\ul{\mathbb P^{>0}})\dashv H\text-\textup{Sp}(\ul{\mathbb U})$ is a Quillen equivalence for the flat model structures. Replacing $G$ by $G\times H$ (and letting $H$ act trivially on $R$), we may assume without loss of generality that $H=1$.
\begin{claim*}
The positive flat global model structure on $\textup{Sp}(\glo{Mod}_R^G)$ transfers to $\textup{Sp}(\glo{NUCA}_R^G)$ along $\textup{Sp}(\ul{\mathbb P}^{>0})\dashv \textup{Sp}(\ul{\mathbb U})$.
\begin{proof}
As the positive flat global model structure on $\textup{Sp}(\glo{Mod}_R^G)$ is in turn transferred from $\textup{Sp}(G\text-\glo{GlobalSpectra}^+)$ (Lemma~\ref{lemma:module-transferred}) it suffices to consider the composite adjunction
\begin{equation*}
\textup{Sp}(\ul{R\smashp\mathbb P^{>0}_\mathbb S})\colon\textup{Sp}(G\text-\glo{GlobalSpectra}^+)\rightleftarrows\textup{Sp}(\glo{NUCA}_R^G) :\!\textup{Sp}(\ul{\mathbb U}).
\end{equation*}
By local presentability, we only have to show that every $\textup{Sp}(\ul{R\smashp\mathbb P^{>0}_\mathbb S})(J)$-cell complex is sent by $\textup{Sp}(\ul{\mathbb U})$ to a $G$-global weak equivalence, where $J$ is our favourite set of generating acyclic cofibrations of $G\text-\textup{Sp}(\glo{GlobalSpectra})$.
For this, we consider a pushout square
\begin{equation*}
\begin{tikzcd}
\textup{Sp}(\ul{R\smashp\mathbb P}^{>0})(A)\arrow[r, "{\textup{Sp}(\ul{R\smashp\mathbb P}^{>0})(j)}"]\arrow[d] &[3em] \textup{Sp}(\ul{R\smashp\mathbb P}^{>0})(B)\arrow[d]\\
X \arrow[r, "k"'] & Y
\end{tikzcd}
\end{equation*}
in $\textup{Sp}(\glo{NUCA}_R^G)$ with $j\in J$. We want to express $\textup{Sp}(\ul{\mathbb U})(k)$ as a transfinite composition of weak equivalences, for which we note that we can identify the $1$-category $\textup{Sp}(\glo{NUCA}_R^G)$ over $\textup{Sp}(\glo{GlobalSpectra})$ with non-unital commutative monoids in $\textup{Sp}(\glo{Mod}_R^G)$ with respect to the non-unital symmetric product $\diamond_R$. While we cannot literally apply Remark~\ref{rk:pushout-analysis-nucas} in this setting (for lack of unitality), we can do this inside the larger category $\cat{Fun}(\bm\Sigma,\cat{Mod}^G_R)$, which (as spectrum objects are closed inside $\cat{SSet}$-enriched functors under all colimits) then factors $k$ as a map in $G\text-\textup{Sp}(\glo{GlobalSpectra})$ into a transfinite composition of maps $k_n$ fitting into pushout squares
\begin{equation*}\textup{h}skip-30.71pt\textup{h}fuzz=31pt
\begin{tikzcd}
\textup{Sp}(\ul{R\smashp{\textup{--}}})(Q^n/\Sigma_n \vee X\diamond Q^{n}/\Sigma_{n})\arrow[r, "\textup{Sp}(\ul{R\smashp{\textup{--}}})(j^{\ppo n}/\Sigma_n\vee X\diamond j^{\ppo n}/\Sigma_{n})"]\arrow[d] &[9em] \textup{Sp}(\ul{R\smashp{\textup{--}}})(L^{\diamond n}/\Sigma_n\vee X\diamond L^{\diamond n}/\Sigma_n)\arrow[d]\\
\cdot\arrow[r, "k_n"'] & \cdot
\end{tikzcd}
\end{equation*}
in $G\text-\textup{Sp}(\glo{GlobalSpectra})$. Appealing to Theorem~\ref{thm:diamond-sp-sp-trivial} above, the top map is a map between weakly contractible objects, hence a weak equivalence, and moreover a levelwise positive flat cofibration (in particular an injective cofibration). Thus, also $k_n$ is a weak equivalence, and hence so is $\textup{Sp}(\ul{\mathbb U})(k)$ since filtered colimits in $\textup{Sp}(\glo{Mod}_R^G)$ are homotopical (Lemma~\ref{lemma:g-global-sp-sp-colim}). As $\textup{Sp}(\ul{\mathbb U})$ preserves filtered colimits, it then follows by the same argument that also any transfinite compositions of maps of the above form are sent to $G$-global weak equivalences as desired.
\end{proof}
\end{claim*}
We now observe that the transferred model structure and the global model structure on $\textup{Sp}(\glo{NUCA}_R^G)$ have the same acyclic fibrations (namely, those maps $f$ for which each $f(A)$ is a fibration in the positive flat $(G\times\Sigma_A)$-global model structure on $(G\times\Sigma_A)\text-\textup{Sp}(\glo{GlobalSpectra}^+)$) and the same fibrant objects (namely, those $X$ for which each $X(A)$ is $(G\times\Sigma_A)$-globally positively flatly fibrant and for which $X(A)\to\Omega^BX(A\amalg B)$ is a $(G\times H)$-global weak equivalence of NUCAs or equivalently of spectra for all $H$-sets $A,B$). Thus, the two model structures actually agree, and in particular we see that $\textup{Sp}(\mathbb U)\colon \textup{Sp}(\glo{NUCA}_R^G)\to \textup{Sp}(\glo{Mod}_R^G)$ preserves and reflects weak equivalences. To complete the proof, it suffices now to show that the ordinary unit $X\to\textup{Sp}(\ul{\mathbb U\mathbb P^{>0}})(X)$ is a weak equivalence for every cofibrant $X$.
This is again a standard cell induction argument \cite[Corollary~1.2.65]{g-global} using \Cref{thm:diamond-sp-sp-trivial}: we let $\mathscr H$ denote the class of all objects for which the above is a weak equivalence. If $Y$ is a levelwise positively flat $G$-bispectrum and $X=\textup{Sp}(\ul{R\smashp{\textup{--}}})(Y)$, then the unit for $X$ agrees up to isomorphism with the inclusion of the first summand of $\bigvee_{n\ge 1} \textup{Sp}(\ul{R\smashp{\textup{--}}})(Y^{\diamond n}/\Sigma_n)$, so it is a weak equivalence by Theorem~\ref{thm:diamond-sp-sp-trivial}, i.e.~$\textup{Sp}(\ul{R\smashp{\textup{--}}})(Y)\in\mathscr H$; in particular, $\mathscr H$ contains the initial object $0$ as well as all sources and targets of the usual generating cofibrations $I$. However, $\mathscr H$ is closed under pushouts along cofibrations (as the right Quillen functor $\textup{Sp}(\mathbb U)$ sends these to homotopy pushouts by stability and left properness) as well as filtered colimits (as these are homotopical by Lemma~\ref{lemma:g-global-sp-sp-colim} and preserved by $\textup{Sp}(\mathbb U))$, so $\mathscr H$ more generally contains all $I$-cell complexes, whence all cofibrant objects by Quillen's Retract Argument.
\end{proof}
\subsection{The levelwise smash product is trivial} In this subsection we will complete the proof of Theorem~\ref{thm:stabilization-nucas} by proving Theorem~\ref{thm:diamond-sp-sp-trivial}.
\subsubsection{The case of ordinary spectra} For this, we first consider the analogues of Construction~\ref{constr:diamond-sp-sp} and Theorem~\ref{thm:diamond-sp-sp-trivial} for ordinary $G$-spectra:
\begin{constr}
Let $X,Y$ be spectra. We define $X\diamond Y$ as the spectrum with $(X\diamond Y)(B)=X(B)\smashp Y(B)$ (smash product of pointed simplicial sets) and structure maps
\begin{align*}
S^A\smashp (X\diamond Y)(B)&=S^A\smashp X(B)\smashp Y(B)\xrightarrow{\delta} S^A\smashp X(B)\smashp S^A \smashp Y(B)\\&\xrightarrow{\sigma\smashp\sigma} X(A\amalg B)\smashp Y(A\amalg B)=(X\diamond Y)(A\amalg B)
\end{align*}
where the first map is induced by the diagonal $S^A\to S^A\smashp S^A$. This becomes a functor in the obvious way.
If $X$ and $Y$ are $G$-spectra, then we equip $X\diamond Y$ with the induced action. Given any $n\ge1$, we view $X^{\diamond n}$ as a $(\Sigma_n\wr G)$- or $(\Sigma_n\times G)$-spectrum.
\end{constr}
\begin{prop}\label{prop:diamond-sp-trivial}
Let $X,Y$ be $G$-spectra.
\begin{enumerate}
\item $X\diamond Y$ is $G$-globally weakly contractible.
\item Let $n>1$. Then $X^{\diamond n}$ is $(\Sigma_n\wr G)$-globally weakly contractible.\label{item:dst-iterated}
\end{enumerate}
\begin{proof}
We will prove the second statement, the proof of the first one being similar but easier. We will show that it is even $\ul\pi_*$-isomorphic to $0$, for which we will need the following easy geometric input:
\begin{claim*}
Let $H$ be a finite group and let $F,A$ be finite $H$-sets such that $F$ is non-empty and free while $|A| > 1$. Then the diagonal embedding $\delta\colon S^F\to S^{A\times F}$ of $H$-\emph{topological} spaces is $H$-equivariantly based nullhomotopic.
\begin{proof}
The map $\delta$ is the $1$-point compactification of the diagonal map $\delta\colon\mathbb R[F]\to \mathbb R[A\times F]$. Now the $H$-fixed points of the source have dimension $|F/H|$ while the $H$-fixed points of the target have dimension $|(A\times F)/H| = |A|\cdot|F/H| > |F/H|$ where the first equality uses freeness of $F$. In particular, there exists an $H$-fixed point $p\in \mathbb R[A\times F]$ outside the diagonal. But then
\begin{align*}
(0,1]\times \mathbb R[F]&\to \mathbb R[A\times F]\\
(t,x)&\mathord{\textup{maps}}to p + t^{-1}(\delta(x)-p)
\end{align*}
is $H$-equivariant and one easily checks that this extends to an equivariant based homotopy from the map constant at $\infty$ to the diagonal embedding $S^F\to S^{A\times F}$.
\end{proof}
\end{claim*}
Fix now a finite group $H$ and a homomorphism $\phi\colon H\to\Sigma_n\wr G$; we will show that $\pi_0^\phi(X^{\diamond n})=0$, the argument in other dimensions being analogous but requiring slightly more notation. For this we pick an exhaustive sequence $B_0\subset B_1\subset\cdots\subset\mathcal U_H$ of our favourite complete $H$-set universe $\mathcal U_H$ such that each $B_{k+1}\setminus B_k$ contains a free $H$-orbit. Then
\begin{equation*}
\pi_0^\phi(X^{\diamond n})=\mathop{\textup{colim}}\nolimits_{B\in s(\mathcal U_H)}[S^B, |\phi^*(X^{\diamond n})(B)|]^H_*\cong \mathop{\textup{colim}}\nolimits_k [S^{B_k},|\phi^*(X^{\diamond n})(B_k)|]^H_*
\end{equation*}
by cofinality. But on the other hand, if $F\subset B_{k+1}\setminus B_k$ is a free $H$-orbit, then the transition map $[S^{B_k},|\phi^*(X^{\diamond n})(B_k)|]^H_*\to [S^{B_{k+1}},|\phi^*(X^{\diamond n})(B_{k+1})|]^H_*$ factors by definition through
\begin{equation*}\textup{h}skip-20.384pt\textup{h}fuzz=21pt
[S^F\smashp S^{B_k}, S^F\smashp |\phi^*(X^{\diamond n})(B_{k})|]^H_*\xrightarrow{\delta\smashp |\phi^*(X^{\diamond n})(B_k)|} [S^F\smashp S^{B_k}, S^{({\textup{pr}}_{\Sigma_n}\circ\phi)^*\bm{n}\times F}\smashp |\phi^*(X^{\diamond n})(B_{k})|]^H_*
\end{equation*}
which is null by the claim.
\end{proof}
\end{prop}
Beware that Theorem~\ref{thm:diamond-sp-sp-trivial} does not follow simply by applying the proposition levelwise; in particular, taking $X=Y=\Sigma^\infty\mathbb S$ in the first item we have $(X\diamond Y)(A)\cong\Sigma^{2\cdot A}\mathbb S$, so $X\diamond Y$ is not globally \emph{level} equivalent to $0$. Instead, the reduction will require further preparation.
\subsubsection{The external smash product}
We begin by introducing yet another smash product:
\begin{constr}
Let $X,Y\in\cat{Spectra}$. Then we define the \emph{external smash product} $X\mathbin{\widehat\smashp} Y\in\textup{Sp}(\cat{Spectra})$ as the bispectrum with $(X\mathbin{\widehat\smashp} Y)(A)(B)= X(A)\smashp Y(B)$ and with the evident structure maps, i.e.~$(X\mathbin{\widehat\smashp} Y)(A)=X(A)\smashp Y$ and $(X\mathbin{\widehat\smashp} Y)({\textup{--}})(B)=X\smashp Y(B)$ as spectra. Again, we extend this to $G$-spectra by pulling through the $G$-action.
\end{constr}
Note that the above agrees with the functor ${\textup{--}}\smashp{\textup{--}}\colon\cat{Spectra}\times\mathscr C\to\textup{Sp}(\ul{\mathscr C})$ considered in Subsection~\ref{subsec:spectrification}, specialized to $\mathscr C=\cat{Spectra}$. However, in our setting using `$\smashp$' again would be highly ambiguous, which is why we introduced the above notation.
\begin{thm}\label{thm:ext-smash}
Let $G$ be any finite group. Then the external smash product
\begin{equation*}
{\textup{--}}\mathbin{\widehat\smashp}{\textup{--}}\colon\cat{$\bm G$-Spectra}_\textup{$G$-global}\times\cat{$\bm G$-Spectra}_\textup{$G$-global}\to G\text-\textup{Sp}(\glo{GlobalSpectra})_\textup{$G$-global}
\end{equation*}
is homotopical in both variables.
\end{thm}
The statement for the second variable is actually quite easy:
\begin{lemma}\label{lemma:ext-smash-easy}
Let $X$ be a $G$-spectrum and let $f$ be a $G$-global weak equivalence of $G$-spectra. Then $X\mathbin{\widehat\smashp} f$ is a $G$-global weak equivalence in $G\text-\textup{Sp}(\glo{GlobalSpectra})$.
\begin{proof}
If $A$ is any finite set, then $f$ is a $(G\times\Sigma_A)$-global weak equivalence (with respect to the trivial $\Sigma_A$-actions), and hence so is $X(A)\smashp f$. Thus, $X\mathbin{\widehat\smashp} f$ is even a $G$-global level weak equivalence.
\end{proof}
\end{lemma}
The proof that the external smash product is also homotopical in the first variable is much harder. We begin with some further closure properties of the $G$-global weak equivalences of $G\text-\textup{Sp}(\glo{GlobalSpectra})$ similar to Lemma~\ref{lemma:g-global-sp-sp-colim}:
\begin{lemma}\label{lemma:g-global-sp-sp-quotient}
Let $f\colon X\to Y$ be a $G$-global weak equivalence in $G\text-\textup{Sp}(\glo{GlobalSpectra})$ and let $\alpha\colon G\to G'$ be a homomorphism of finite groups. Assume that $\ker(\alpha)$ acts levelwise freely outside the basepoint on $X$ and $Y$. Then $\alpha_!f$ is a $G'$-global weak equivalence.
\begin{proof}
Employing functorial factorizations in the $G$-global projective \emph{level} model structure we obtain a commutative diagram
\begin{equation*}
\begin{tikzcd}
X'\arrow[d, "\sim"']\arrow[r, "f'"] & Y'\arrow[d, "\sim"]\\
X\arrow[r, "f"'] & Y
\end{tikzcd}
\end{equation*}
such that the vertical maps are $G$-global level weak equivalences and $X',Y'$ are projectively cofibrant. Then $f'$ is a $G$-global weak equivalence by $2$-out-of-$3$, and hence $\alpha_!f'$ is a $G'$-global weak equivalence by Ken Brown's Lemma. By another application of $2$-out-of-$3$, it will then be enough to show that $\alpha_!$ sends the vertical maps to $G'$-global weak equivalences. However, $G$ (and hence in particular $\ker\alpha$) acts levelwise freely outside the basepoint on $X'$ and $Y'$ by \cite[Remark~3.1.22]{g-global} together with Lemma~\ref{lemma:projective-cofibrations-levelwise} (for the trivial $G$-action on the indexing set), whence the claim follows from Proposition~\ref{prop:free-quotient-spectra}.
\end{proof}
\end{lemma}
Moreover, one proves in the same way:
\begin{lemma}\label{lemma:G-sset-bispectra-homotopical}
The levelwise smash product
\begin{equation*}
(\cat{$\bm G$-SSet}_*)_\textup{$G$-equivariant}\times G\text-\textup{Sp}(\glo{GlobalSpectra})_\textup{$G$-global}\to G\text-\textup{Sp}(\glo{GlobalSpectra})_\textup{$G$-global}
\end{equation*}
is homotopical in each variable.\qed
\end{lemma}
\begin{proof}[Proof of Theorem~\ref{thm:ext-smash}]
Write $\mathscr H$ for the class of $G$-global spectra $X$ such that ${\textup{--}}\mathbin{\widehat\smashp} X$ is homotopical. Our goal is to show that $\mathscr H$ consists of all objects, which will be done in several steps.
\noindent\textit{Step 1. For every $Y\in\cat{$\bm G$-$\bm{\mathcal I}$-SSet}_*$ we have $\Sigma^\bullet Y\in\mathscr H$.}\\
Plugging in the definitions, we have for every $G$-global spectrum $T$ and all finite sets $A,B$ a natural isomorphism
\begin{align*}
(T\mathbin{\widehat\smashp} \Sigma^\bullet Y)(A)(B)=T(A)\smashp S^B\smashp Y(B)&\cong S^B\smashp T(A)\smashp Y(B)\\&= \big(G\text-\textup{Sp}(\ul{\Sigma^\bullet})(T\mathbin{\widehat\otimes} Y)\big)(A)(B)
\end{align*}
where $\mathbin{\widehat\otimes}$ denotes the `external tensor product' of a $G$-spectrum with a $G$-$\mathcal I$-simplicial set, i.e.~the $G$-spectrum object in $\cat{$\bm{\mathcal I}$-SSet}_*$ given by $(T\mathbin{\widehat\otimes} Y)(A)(B)=T(A)\smashp Y(B)$ with the obvious functoriality. Letting $A$ and $B$ vary, one then easily checks that the above induces a natural isomorphism $T\mathbin{\widehat\smashp} \Sigma^\bullet Y\cong G\text-\textup{Sp}(\ul{\Sigma^\bullet})(T\mathbin{\widehat\otimes} Y)$. However, $G\text-\textup{Sp}(\ul{\Sigma^\bullet})$ is left Quillen (say, for the projective model structures) and preserves $G$-global level weak equivalences (Proposition~\ref{prop:suspension-loop-G-gl}), so it is in fact fully homotopical. Thus, it will be enough to show that ${\textup{--}}\mathbin{\widehat\otimes} Y$ sends $G$-global weak equivalences of $G$-spectra to $G$-global weak equivalences in $G\text-\textup{Sp}(\glo{GlobalSpaces})$. However, by Proposition~\ref{prop:Delta-star-homotopical} the latter are detected by the diagonal restriction $\Delta^*$, and $\Delta^*({\textup{--}}\mathbin{\widehat\otimes} Y)={\textup{--}}\otimes Y$ preserves $G$-global weak equivalences of $G$-spectra by Proposition~\ref{prop:tensor-homotopical}.
\noindent\textit{Step 2. For every $Y\in\cat{$\bm G$-$\bm{I}$-SSet}_*$ we have $\Sigma^\bullet Y\in\mathscr H$.}\\
By Lemma~\ref{lemma:ext-smash-easy} and $2$-out-of-$3$, $\mathscr H$ is closed under $G$-global weak equivalences. But \cite[Theorem~1.4.31 and~Proposition~3.2.2]{g-global} yield a $G$-global weak equivalence $\Sigma^\bullet Y\simeq \Sigma^\bullet(\mathcal I\times_IY)$ for some pointed $G$-$\mathcal I$-simplicial set $\mathcal I\times_I Y$, so the claim follows from the previous step.
\noindent\textit{Step 3. For every finite $G$-set $A$ and every $K\in\cat{$\bm G$-SSet}_*$, $\bm\Sigma(A,{\textup{--}})\smashp K\in\mathscr H$.}\\
The endofunctor $S^A\smashp{\textup{--}}$ of $G\text-\textup{Sp}(\glo{GlobalSpectra})$ is homotopical (by the previous lemma) and part of a Quillen equivalence (by stability), so it reflects weak equivalences. Thus, it suffices to show that $S^A\smashp (f\mathbin{\widehat\smashp} \bm\Sigma(A,{\textup{--}})\smashp K)$ is a $G$-global weak equivalence for every $G$-global weak equivalence $f$. However, this is conjugate to $f\mathbin{\widehat\smashp} \big(S^A\smashp\bm\Sigma(A,{\textup{--}})\big)\smashp K$, and by a simple Yoneda argument $S^A\smashp\bm\Sigma(A,{\textup{--}})\cong\Sigma^\bullet_+ I(A,{\textup{--}})$ naturally in $A$ (hence $G$-equivariantly). Thus, the claim follows from the previous step.
\noindent\textit{Step 4. Let $H$ be a finite group, $\phi\colon H\to G$ a homomorphism, and $A$ a finite faithful $H$-set. Then $\bm\Sigma(A,{\textup{--}})\smashp_\phi G_+\smashp K\in\mathscr H$ for every pointed $G$-simplicial set $K$.}\\
Applying the previous step with $G$ replaced by $G\times H$ shows that the functor ${\textup{--}}\mathbin{\widehat\smashp}\bm\Sigma(A,{\textup{--}})\smashp G_+\smashp K$ sends $G$-global weak equivalences to $(G\times H)$-global weak equivalences, where $H$ acts on $A$ in the given way, on $G$ from the right via $\phi$, and trivially everywhere else. However, as $A$ is faithful, $H$ acts freely on $\bm\Sigma(A,B)$ outside the basepoint for every finite set $B$. Thus, Lemma~\ref{lemma:g-global-sp-sp-quotient} immediately implies that $({\textup{--}}\mathbin{\widehat\smashp}\bm\Sigma(A,{\textup{--}})\smashp G_+\smashp K)/H\cong {\textup{--}}\mathbin{\widehat\smashp}\bm\Sigma(A,{\textup{--}})\smashp_\phi G_+\smashp K$ is homotopical.
\noindent\textit{Step 5. Every projectively cofibrant $G$-global spectrum is contained in $\mathscr H$.}\\
Fix a $G$-global weak equivalence $f\colon T\to U$, which induces a natural transformation $T\mathbin{\widehat\smashp}{\textup{--}}\Rightarrow U\mathbin{\widehat\smashp}{\textup{--}}$. We want to show that this is a weak equivalence on all projectively cofibrant objects. This is again a standard cell induction argument \cite[Lemma~1.2.64]{g-global}: by the previous step the claim is true for the sources and targets of the standard generating cofibrations, and moreover for any $G$-spectrum $V$ the functor $V\mathbin{\widehat\smashp} {\textup{--}}$ preserves colimits as well as injective cofibrations; the claim therefore follows from Lemma~\ref{lemma:g-global-sp-sp-colim}.
\noindent\textit{Step 6. All $G$-global spectra belong to $\mathscr H$.}\\
Every $G$-global spectrum is weakly equivalent to a projectively cofibrant one. The claim therefore follows from the previous step together with Lemma~\ref{lemma:ext-smash-easy}.
\end{proof}
\subsubsection{Proof of triviality} Using this, we can prove a key special case of Theorem~\ref{thm:diamond-sp-sp-trivial}:
\begin{prop}\label{prop:diamond-power-corep}
Let $A$ be any finite set, let $X$ be a positively flat $(G\times\Sigma_A)$-spectrum, and let $n>1$. Then $(\bm\Sigma(A,{\textup{--}})\mathbin{\widehat\smashp}_{\Sigma_A} X)^{\diamond n}/\Sigma_n$ is $G$-globally weakly contractible.
\begin{proof}
Reordering factors we have
\begin{equation*}
(\bm\Sigma(A,{\textup{--}})\mathbin{\widehat\smashp} X)^{\diamond n}(B)(C)\cong\bm\Sigma(A,B)^{\smashp n}\smashp {X^{\smashp n}}(C)
\end{equation*}
inducing a $G\times(\Sigma_n\wr\Sigma_A)$-equivariant isomorphism of bispectra $(\bm\Sigma(A,{\textup{--}})\mathbin{\widehat\smashp} X)^{\diamond n}\cong \bm\Sigma(A,{\textup{--}})^{\diamond n}\mathbin{\widehat\smashp} X^{\smashp n}$. By Proposition~\ref{prop:diamond-sp-trivial}, $\bm\Sigma(A,{\textup{--}})^{\diamond n}$ is $\Sigma_n\wr(G\times\Sigma_A)$- and hence also $G\times(\Sigma_n\wr\Sigma_A)$-globally weakly contractible, so Theorem~\ref{thm:ext-smash} shows that $\bm\Sigma(A,{\textup{--}})^{\diamond n}\mathbin{\widehat\smashp} X^{\smashp n}$ is $G\times(\Sigma_n\wr\Sigma_A)$-globally weakly contractible. Now $\Sigma_A^n$ acts levelwise freely on this (as it already does so on $\bm\Sigma(A,{\textup{--}})^{\smashp n}$), whence Lemma~\ref{lemma:g-global-sp-sp-quotient} shows that $(\bm\Sigma(A,{\textup{--}})\mathbin{\widehat\smashp}_{\Sigma_A}X)^{\diamond n}\cong \bm\Sigma(A,{\textup{--}})^{\diamond n}\mathbin{\widehat\smashp}_{\Sigma_A^n} X^{\smashp n}$ is $(G\times\Sigma_n)$-globally weakly contractible. But $\bm\Sigma(A,{\textup{--}})\mathbin{\widehat\smashp}_{\Sigma_A} X$ is levelwise positively flat, so the $\Sigma_n$-action on its $n$-th $\diamond$-power is free by Lemma~\ref{lemma:smash-power-free}, and the proposition follows by another application of Lemma~\ref{lemma:g-global-sp-sp-quotient}.
\end{proof}
\end{prop}
In the same way one shows:
\begin{prop}
Let $A,B$ be finite sets, let $X$ be a flat $(G\times\Sigma_A)$-spectrum, and $Y$ a flat $(G\times\Sigma_B)$-spectrum. Then $(\bm\Sigma(A,{\textup{--}})\mathbin{\widehat\smashp}_{\Sigma_A}X)\diamond (\bm\Sigma(B,{\textup{--}})\mathbin{\widehat\smashp}_{\Sigma_B} Y)$ is $G$-globally weakly contractible.\qed
\end{prop}
From this we can immediately easily deduce the following slight strenghtening of the first part of Theorem~\ref{thm:diamond-sp-sp-trivial}:
\begin{prop}\label{prop:diamond-sp-sp-trivial-first-half}
Let $X,Y\in G\text-\textup{Sp}(\glo{GlobalSpectra})$ and assume at least one of them is levelwise flat. Then $X\diamond Y$ is $G$-globally weakly contractible.
\begin{proof}
We first observe the following closure properties:
\begin{claim*}
For any (levelwise) flat $X\in G\text-\textup{Sp}(\glo{GlobalSpectra})$ the class of objects $Y$ for which $X\diamond Y$ is weakly contractible is closed under (a) filtered colimits \emph{and} (b) pushouts along levelwise flat cofibrations.
\begin{proof}
We will prove the second statement, the argument for the first one being similar. Consider a pushout in $G\text-\textup{Sp}(\glo{GlobalSpectra})$ as on the left
\begin{equation*}
\begin{tikzcd}
A\arrow[d]\arrow[dr, phantom, "\ulcorner"{very near end}]\arrow[r, "i"] & B\arrow[d]\\
C\arrow[r]&D
\end{tikzcd}
\qquad\qquad
\begin{tikzcd}
X\diamond A\arrow[dr, phantom, "\ulcorner"{very near end}]\arrow[d]\arrow[r, "X\diamond i"] & X\diamond B\arrow[d]\\
X\diamond C\arrow[r]& X\diamond D
\end{tikzcd}
\end{equation*}
such that $i$ is levelwise flat. Applying $X\diamond{\textup{--}}$ to this yield a pushout as on the right (as $X\diamond{\textup{--}}$ is cocontinuous), and $X\diamond i$ is a levelwise flat cofibration (in particular an injective cofibration). Thus, if $X\diamond A,X\diamond B,X\diamond D$ are weakly contractible then so is $X\diamond D$ by Lemma~\ref{lemma:g-global-sp-sp-colim}.
\end{proof}
\end{claim*}
Using this, the previous proposition immediately proves the special case that $Y$ is flat and $X=\bm\Sigma(A,{\textup{--}})\mathbin{\widehat\smashp}_{\Sigma_A}Z$ for some flat $Z$. But for any flat $Y$, the class of $X$ for which $X\diamond Y$ is weakly contractible is again closed under filtered colimits and pushouts along flat cofibrations (by symmetry), proving the case that both $X$ and $Y$ are flat. As for any levelwise flat $Z$ both ${\textup{--}}\diamond Z$ and $Z\diamond{\textup{--}}$ preserve $G$-global \emph{level} weak equivalences, the claim now follows by cofibrant replacement.
\end{proof}
\end{prop}
\begin{proof}[Proof of Theorem~\ref{thm:diamond-sp-sp-trivial}]
The first statement is a special case of the previous proposition. For the second statement, we observe that the class of \emph{levelwise positively flat} $X\in G\text-\textup{Sp}(\glo{GlobalSpectra})$ such that $X^{\diamond n}/\Sigma_n$ is $G$-globally weakly contractible contains $0$ and is closed under filtered colimits. To complete the proof it is then enough to show that it is also closed under pushouts along generating cofibrations, for which we more generally consider any pushout
\begin{equation}\label{diag:po-sp-sp-trivial}
\begin{tikzcd}
\bm\Sigma(A,{\textup{--}})\mathbin{\widehat\smashp}_{\Sigma_A} X\arrow[r, "{\bm\Sigma(A,{\textup{--}})\mathbin{\widehat\smashp}_{\Sigma_A}i}"]\arrow[d] &[3em] \bm\Sigma(A,{\textup{--}})\mathbin{\widehat\smashp}_{\Sigma_A} Y\arrow[d]\\
Z\arrow[r, "j"'] & P\arrow[from=ul,phantom, "\ulcorner"{very near end,xshift=.75em,yshift=2pt}]
\end{tikzcd}
\end{equation}
such that $i\colon X\to Y$ is a positive flat cofibration of $(G\times\Sigma_A)$-spectra and $Z$ is levelwise positively flat with $Z^{\diamond n}/\Sigma_n\simeq0$ for all $n>1$; we will prove that also $P^{\diamond n}/\Sigma_n\simeq0$ for all $n>1$. For this, we apply \cite[Theorem~22]{sym-powers} (to the larger category of all $\cat{SSet}$-enriched functors) yielding factorizations
\begin{equation*}
Z^{\diamond n}/\Sigma_n=Q^n_0\to Q^n_1\to\cdots\to Q^n_{n-1}\to Q^n_n=P^{\diamond n}/\Sigma_n
\end{equation*}
of $j^{\diamond n}/\Sigma_n$ for all $n$ such that we have for $k<n-1$ a pushout
\begin{equation}\label{diag:qkn-po}
\begin{tikzcd}
Z^{\diamond (n-k)}/\Sigma_{n-k}\diamond Q^k_{k-1} \arrow[r, "Z^{\diamond(n-k)}/\Sigma_{n-k}\diamond j^{\ppo k}/\Sigma_k"]\arrow[d] &[6.6em] Z^{\diamond (n-k)}/\Sigma_{n-k}\diamond P^{\diamond k}/\Sigma_k\arrow[d]\\
Q^n_k\arrow[r] & Q^n_{k+1}\arrow[from=ul,phantom, "\ulcorner"{very near end, xshift=1.7em,yshift=2pt}]
\end{tikzcd}
\end{equation}
(where for $k=0$ the empty $\diamond$-power has to be interpreted as a formal unit) while $Q^n_{n-1}\to Q^n_n$ recovers $j^{\ppo n}/\Sigma_n$.
\begin{claim*}
For all $0\le k<n$ the map $Q^n_k\to Q^{n}_{k+1}$ is a levelwise flat cofibration.
\begin{proof}
For $k=n-1$ this is an instance of the strong commutative monoid axiom for the (say, non-equivariant) positive flat model structure on symmetric spectra.
For $k<n-1$, on the other hand, we observe that the top arrow in the pushout square $(\ref{diag:qkn-po})$ is a positive levelwise flat cofibration by the above special case and levelwise flatness of $Z^{\diamond(n-k)}/\Sigma_{n-k}$. The claim follows immediately.
\end{proof}
\end{claim*}
By (ordinary) stability, it therefore suffices to show that the $1$-categorical cofiber of $Q^n_k\to Q^n_{k+1}$ is trivial for all $0\le k<n$. However by the aforementioned \cite[Theorem~22]{sym-powers}, this cofiber is isomorphic to $Z^{\diamond(n-k)}/\Sigma_{n-k}\diamond {\textup{cofib}}(j)^{\diamond k}/\Sigma_{k}$. For $0<k<n$ this is $G$-globally weakly contractible by Proposition~\ref{prop:diamond-sp-sp-trivial-first-half}, while for $k=0$ this is $G$-globally weakly contractible by assumption on $Z$. Finally, if $k=n$, then we observe that ${\textup{cofib}}(j)\cong \bm\Sigma(A,{\textup{--}})\mathbin{\widehat\smashp}_{\Sigma_A}{\textup{cofib}}(i)$ because of the pushout $(\ref{diag:po-sp-sp-trivial})$; as ${\textup{cofib}}(i)$ is levelwise positive flat, the claim now follows from Proposition~\ref{prop:diamond-power-corep}.
Finally, for the third statement, we simply argue as before to see that for any levelwise positive flat cofibration $i$ the pushout product $i^{\ppo n}/\Sigma_n$ is again a levelwise positive flat cofibration, and that its ($1$-categorical) cofiber agrees with ${\textup{cofib}}(i)^{\diamond n}/\Sigma_n$, which is $G$-globally weakly contractible by the second statement.
\end{proof}
\frenchspacing
\end{document} |
\begin{document}
\title{Feasibility of continuous-variable quantum key distribution with noisy coherent states}
\author{Vladyslav C. Usenko}
\email{[email protected]}
\affiliation{Bogolyubov Institute for Theoretical Physics of National Academy of Sciences,
Metrolohichna st. 14-b, 03680, Kiev, Ukraine}
\affiliation{Department of Optics, Palack\' y University, 17. listopadu 50, 772~07 Olomouc, Czech Republic}
\author{Radim Filip}
\email{[email protected]}
\affiliation{Department of Optics, Palack\' y University, 17. listopadu 50, 772~07 Olomouc, Czech Republic}
\date{\today}
\begin{abstract}
We address security of the quantum key distribution scheme based on the noisy modulation of coherent states and investigate how it is robust against noise in the modulation regardless of the particular technical implementation. As the trusted preparation noise is shown to be security breaking even for purely lossy channels, we reveal the essential difference between two types of trusted noise, namely sender-side preparation noise and receiver-side detection noise, the latter being security preserving. We consider the method of sender-side state purification to compensate the preparation noise and show its applicability in the realistic conditions of channel loss, untrusted channel excess noise, and trusted detection noise. We show that purification makes the scheme robust to the preparation noise (i.e., even the arbitrary noisy coherent states can in principle be used for the purpose of quantum key distribution). We also take into account the effect of realistic reconciliation and show that the purification method is still efficient in this case up to a limited value of preparation noise.
\end{abstract}
\pacs{03.67.Hk, 03.67.Dd}
\maketitle
\section{Introduction}
Secure key distribution \cite{cr} with continuous variables \cite{cm} is a recent practical example of the usefulness of quantum properties exhibited by coherent states generated with ordinary laser systems. The pure coherent states were surprisingly found to be the sufficient resource to distribute a secret key, even through a strongly attenuating channel \cite{RR,exp,coll,Gauss}, while the inefficiency and electronic noise of a homodyne detector in a trusted receiver station does not break security of the key distribution \cite{Lod}. Recently, security analysis of excess noise in the channel and detector was discussed in \cite{Lod2}; moreover, the noise at the remote side was shown to be useful for the scheme security \cite{Cerf}. However, it was also recently shown that excess noise in trusted state preparation can break the security even for a purely lossy channel and ideal homodyne detector \cite{Filip08}. To eliminate this security break, an optimized purification of the prepared noisy coherent states in the trusted sender station was proposed. Interestingly, it was shown that secure communication through any purely lossy channel is possible for arbitrary noisy coherent states, assuming ideal homodyne detection \cite{Filip08}. From this follows that even a noisy coherent state can be a useful quantum resource for secure key distribution in the perfect experimental conditions of pure channel loss. Still an open question is how generally valid is this result for an imperfect homodyne detector, with remaining excess noise in the channel and realistic key reconciliation.
In this article, we discuss security of the continuous-variable (CV) quantum key distribution (QKD) with noisy coherent states through a lossy and noisy channel. The preparation noise is due to the imperfect modulation, which was shown to be destructive for the security of the scheme. We reveal the fundamental difference between various types of trusted noise and show the possibility to establish the secure key transmission with noisy coherent states upon a high level of preparation noise using the local state manipulations in the presence of either trusted detection noise or untrusted channel noise. We also investigate the limits of high modulation variance and show that in this case we can practically eliminate the negative effect of the preparation noise using optimal purification. In addition, we show that the positive purification effect is present also upon limited reconciliation efficiency.
The article is structured as follows: in Sec. \ref{protocol} we briefly recall the QKD with noisy coherent states and mention the effect of the purely lossy channel; in Sec. \ref{noisedet} we add detection noise on the trusted remote party side; in Sec. \ref{channoise} we consider untrusted channel noise; in Sec. \ref{recon} we take into account the effect of the realistic error correction with limited efficiency; and then we finish the article with some concluding remarks.
\section{Coherent states protocol}
\label{protocol}
CV QKD with coherent states is based on the Gaussian modulation of the states produced by a laser in such a way
that their mixture constitutes a thermal state centered in an origin of phase space and characterized by a source variance $V=1+\sigma$, where $\sigma$ is the modulation variance. Such a setup is referred to as the prepare and measure ($P \& M$) as one of the trusted parties (Alice) for each next
key bit applies modulation to the coherent states, generating them centered at some two Gaussian-distributed values of
quadratures (Fig. \ref{scheme}). The prepared state travels though a quantum channel to the remote trusted party (Bob),
which performs the homodyne measurement of a randomly selected quadrature and stores the result.
Lately, as the sufficient number of preparation and measurement events is carried out, the parties perform key sifting when Bob, using a classical channel,
reveals which of two quadratures he was measuring for each bit and Alice keeps the corresponding bit value, such a scheme
being named reverse reconciliation \cite{RR}. From the obtained classically correlated data, if they they are correlated enough
(which is checked by comparison of the randomly selected subsets),
trusted parties can distill the secure cryptographic key. Physically, the excess noise arises mainly in the
imperfect optical modulator so that each coherent state has the variance $1+\Delta V$, where $\Delta V$ is the preparation noise.
Besides preparation noise, another main source of imperfection is the quantum channel, which has nonunit transmittivity
$\eta$, thus suppressing the signal with losses and adding excess noise $\eta$ to the signal modulation. Also imperfect detection
at Bob's side is assumed, as his homodyne detector has some nonunit efficiency and the excess electronic noise;
we consider imperfect detection as adding excess detection noise $\chi$. Here, for the trusted detector we treat loss and noise in the detection jointly, as the output of the detector is typically classically amplified; thus the detector can be purely noisy so that $\chi$ is the total additive noise of the detector.
At that, the state preparation is assumed to be completely secure (i.e., no attack of an eavesdropper in the sender station is allowed and no side signal leaves the station). Also, the receiver station is trusted, which means that no information is leaking to a potential eavesdropper, whereas the quantum channel is untrusted as it is out of control of the trusted parties.
It already is well known that untrusted channel noise is limiting the security of CV QKD, whereas trusted detection noise is not contributing to the the knowledge of Eve and is just quantitatively reducing the key rate \cite{RR}. At the same time, it was recently shown that the preparation noise, although being also trusted, is breaking the security of the CV QKD \cite{Filip08}. Thus, in this article we stress out the essential difference between two types of trusted noise, namely preparation and detection, in their effect on the CV QKD security and study the method to compensate the negative effect of trusted preparation noise by applying state purification.
In our theoretical analysis, we model the CV QKD coherent states-based scheme in order to estimate its security, taking realistic values of the parameters of the scheme. In particular, we suppose that the quantum channel has low transmittance $\eta\in (0.01,0.1)$, which corresponds to long-distance optical fibers; the source variance $V$ is mostly taken within the range $V\in (10,100)$, which well corresponds to recent experiments in CV QKD, where the source variance reached approximately $20$ \cite{Lod2} or $40$ \cite{RR} shot-noise units. The typical values of the homodyne detector electonic noise in the mentioned experiments were $0.041$ and $0.33$, respectively, but the limited detection efficiency makes the overall homodyne detection noise higher.
At the same time, it is hard to assess the possible values of the preparation noise, as the sources of this noise may be essentially different; it can be either imperfect modulation of a shot-noise limited coherent source or the increased variance of a laser, which is used to produce the exact state, displaced according to a next bit value. Besides, the values of such noise were not discussed in experimental CV QKD as this noise is difficult to estimate and calibrate \cite{Lod2}. Hence all the noise in the receiving station was considered to be the impact of Eve and the sources were usually assumed as ideal, focusing on other possible imperfections. However, the simplest sources for a classical coherent optical communication link are cheap laser diodes together with the usual low-frequency integrated amplitude and phase modulators \cite{Betti}. While such a source still has enough temporal coherence to build the coherent communication link, its intensity and phase noise added at the lower modulation frequencies makes the source to be not shot-noise limited, exhibiting additional excess noise \cite{Bachor}. For the classical coherent communication, this is not a serious problem, and these issues can be considered to be just technical, but it appears to be essential for QKD, as the preparation noise can quickly lead to a security break. Hence, we propose to distinguish between trusted preparation noise and the rest of the noise and show that this may improve the applicability of CV QKD as we can compensate security-breaking preparation noise. Note, that in the case of the discrete-variable coding the mixed signal states were considered already in \cite{Koashi}.
It is worth mentioning, that the method of sender-side attenuation, which we propose and investigate in this article in order to purify the states and make CV QKD more robust to the preparation noise, is most likely just one of the possible ways to reduce the negative effect of the state preparation noise. Yet we propose it as the simplest method to purify the states by using just a linear optical device, the beam-splitter, in case the noisy source is given and cannot be otherwise improved. In addition, by introducing attenuation $T$, we can clearly show the effect analytically and optimize it as well as estimate the limitations, imposed on the purification method by the reconciliation efficiency.
As a matter of fact, the attenuation of the signal on the trusted source side was already applied in \cite{Lod2} without the analysis of the impact. Its effect is actually twofold: to purify the noisy states by suppressing the noise and to optimize the signal-to-noise ratio for the given reconciliation efficiency. Still, the purifying effect was not theoretically analyzed. Moreover, the attenuation of laser diodes was proposed by decreasing their power \cite{Lod}, but this method is not applicable for the case, when modulation is performed directly in the laser, not by a standalone modulator. Also, the feedback systems were applied \cite{RR,Lod2} to control the level of modulation and they can be used to reduce the noise, but as their effect is nonlinear, it is not considered in this article.
\begin{figure}
\caption{Continuous-variables quantum key distribution scheme based on the noisy coherent states: Laser beam is modulated to encode information in modulator M producing coherent states with superposed additive phase-insensitive noise $\Delta V$. It passes through a variable attenuator with transmittance $T$ (with a second port in vacuum mode $C$) followed by the channel described by excess noise $\epsilon$ and channel transmittivity $\eta$ toward Bob's homodyne detector with additive noise $\chi$. Inlay: equivalent model of Alice noisy state preparation based on the entangled signal source (modes $A,B$) of variance $V$ and entangled noise source (modes $F,G$) of variance $\Delta V$, coupled to signal mode, Alice performing heterodyne measurement on mode B; rest of the scheme is the same as for coherent state modulation.}
\label{scheme}
\end{figure}
For the security analysis, it is assumed that Eve couples signal pulses to her ancillary states and stores them in a quantum memory. Then she can perform either an individual attack after the key sifting procedure, when she measures her ancillary states individually, or a collective attack during the key distillation, when she applies the optimal collective measurement on the stored ancillary states. We limit our study of security of the scheme by Gaussian collective attacks as they were shown to be optimal \cite{Gauss} and keep to the the recent proof, that, similar to the case of discrete-variable QKD, the protocol is secure against the most general attacks, if it is secure against the collective ones \cite{Renner}. For individual attacks, in case of reverse reconciliation, the security analysis is based on the fact that maximum information available to Eve is bounded by the classical (Shannon) mutual information, characterizing her knowledge of Bob's data, while in the case of collective attacks Eve is possessing the quantum (von Neumann) information on Bob's measurement results, which is limited by the Holevo bound; in all cases, the criterion for security is formulated as the exceeding of mutual information between Alice and Bob over the information available to Eve on Bob's data; under this condition the secure key can be distilled by the standard procedures. Thus, for the collective attack scenario and one-way
classical postprocessing, there is always a key distillation
protocol generating secure key with minimal rate
\begin{equation}\label{gen}
I=I_{AB}-\chi_{BE},
\end{equation}
where $I_{AB}$ is Shannon mutual information from data obtained by Alice and Bob,
$\chi_{BE}$ is the Holevo information quantity between receiver $B$ and
potential eavesdropper Eve \cite{DV}. The Holevo quantity can be
written as $\chi_{BE}=S_E-\int P(B)S_{E|B}dB$, where $S_E$ is von Neumann
entropy of the eavesdropper's state $\rho_E$ \cite{Gauss}. The quantity
$S_{E|B}$ is the von Neumann entropy of the eavesdropper state
$\rho_{E|B}$ conditioned by the receiver measurement result $B$, and $P(B)$ is the distribution of the measured results.
If $I>0$, then information shared by Alice and Bob is larger than information accessible to an eavesdropper which is performing collective attacks. Then optimal distillation procedure can generate a secure key between trusted parties.
A weaker condition is for security against only the individual attacks. In this case,
the lower bound on secure key rate is given $I_i=I_{AB}-I_{BE}$,
where $I_{BE}$ is Shannon mutual information between the
eavesdropper and the receiver, if the eavesdropper applies an optimal strategy.
Remarkably, for ideal coherent-state preparation without any excess
noise, it is always possible to achieve $I^{max}=-\log_2(1-\eta)/2>0$ for an arbitrary lossy channel
(but without excess noise) with $\eta>0$ and for arbitrary modulation
variance $\sigma > 0$ \cite{coll}.
While we consider the security of the scheme for collective attacks, we will use the individual attack case to estimate the regions of parameters, where the scheme already becomes insecure for any attacks, because insecurity against individual attacks automatically means insecurity against the collective, as the latter are more effective. Furthermore, the individual attack case enables us to analytically show the effect of state purification as well as constitute the limits to which collective attacks tend in the high modulation regime.
So, we investigate the security of the scheme based on the noisy coherent states in different realistic conditions. The situation, when only the state preparation noise is present in the CV QKD based on the coherent states and the quantum channel is purely lossy, was already addressed in \cite{Filip08}. It was shown that the preparation noise is strongly destructive for security of the scheme, but the purification on the side of the trusted source can provide the possibility to establish secure key distribution upon any value of the preparation noise, and that security is guaranteed against both individual and collective attacks. Although seeming contradictory at first glance, this result can be clearly explained from the physical considerations. As the state coherence is preserved upon any attenuation, by suppressing the noisy coherent state we suppress the noise, but the state coherence remains, so that while the intensity becomes lower and the key rate decreases, the state becomes pure enough to provide the security of the key distribution, which without purification could not be possible. The earlier work provided the proof of principle for suppressing preparation noise with purification, while in this article we perform the feasibility check of the method in the realistic conditions of trusted detection noise, untrusted channel noise, and imperfect key reconciliation.
Generally we investigate whether a noisy source can be used for a CV coherent states-based QKD given that the source cannot be replaced with a better one. While the exact level of preparation noise is the question of cost and complexity of the state preparation equipment, in this article we rather generally discuss the impact of the preparation noise on the QKD security, than refer to a specific experimental arrangement. In this sense, our work is a step from perfect laboratory conditions toward the realistic CV QKD, based on the affordable signal sources.
\section{Noiseless channel and trusted detection noise}
\label{noisedet}
First we generalize the result presented in \cite{Filip08} that is valid for the lossy channel and ideal homodyne detector. Keeping the noiseless channel in mind, we assume a realistic lossy and noisy trusted homodyne detector. This represents an optimistic scenario of realistic experimental setup, taking into account the minimal impact of natural channel background noise into narrow-band homodyne detection.
First we calculate the impact of noise on the security against individual attacks to estimate the insecurity region. As we perform the calculations in the equivalent entangled-based source setup \cite{Lod2}, the overall expression for the lower bound on the key rate, which is secure against individual attacks using the reverse reconciliation, is \cite{RR}
\begin{equation}
\label{indkeyrgen}
I_i=\frac{1}{2}\log_2{\frac{V_{B|E}}{V_{B|A^M}}},
\end{equation}
where $V_{B|E}=V_B-\frac{C_{BE}^2}{V_E}$ and $V_{B|A^M}=V_B-\frac{C_{AB}^2}{V_A+1}$ are the relevant conditional variances. The one unit of noise added to $V_A$ appears from the heterodyne measurement on Alice's side. In our case, the variances are $V_A=V$, $V_B=\eta(V+\Delta V)+1-\eta+\chi$ and $V_E=(1-\eta)(V+\Delta V)+\eta$ and the mode correlations are $C_{BE}=\sqrt{\eta(1-\eta)}(1-V-\Delta V)$ whereas $C_{AB}=\sqrt{\eta (V^2-1)}$. Detection noise $\chi$, being out of control by the eavesdropper, is not involved in the correlation $C_{BE}$; hence, it does not break the security but only lowers the key rate. The conditional variances are $V_{B|A^M}=1+\eta\Delta V+\chi$ and
\begin{equation}
V_{B|E}=\frac{1}{\frac{\eta}{V+\Delta V}+1-\eta}+\chi;
\end{equation}
then from the explicit expression (\ref{indkeyrgen}) for the lower bound on key rate in this case we can derive the security constraint on the level of the preparation noise:
\begin{equation}
\label{maxdvind}
\Delta V < \frac{1}{2}-\frac{V}{2}+\sqrt{(V-1)\bigg(V-1+\frac{4}{1-\eta}\bigg)},
\end{equation}
which does not depend on the detection noise, added on the remote receiving side.
In the limit of arbitrary large source variance (i.e., arbitrary high modulation), this constraint turns to
\begin{equation}
\label{maxdvindlim}
\Delta V < \frac{1}{1-\eta},
\end{equation}
which means that for strongly attenuating channels $\eta << 1$, even if the modulation is arbitrarily high, the preparation noise should not exceed one shot noise unit, while this condition becomes more restrictive for the realistic cases of finite modulation.
In the case of collective attacks, as mentioned, Eve's information on the key is limited by the Holevo quantity.
Since we are considering a purely attenuating channel, we substitute it by a beam splitter with the transmittance equivalent to the channel transmittivity $\eta$. We perform the straightforward calculations of the Holevo quantity $\chi_{BE}=S_E-\int P(B)S_{E|B}dB$, bounding Eve's quantum information from a state going from the beam splitter to Eve. Since we are working with Gaussian states, entropy $S_{E|B}$ does not depend on the Bob's measurement result and we can simply use $\chi_{BE}=S_E-S_{E|B}$. Using the expression for von Neumann entropies \cite{entropy}, the Holevo quantity is calculated from
\begin{equation}\label{holevo1}
\chi_{BE}=G(\frac{\lambda_1-1}{2})-G(\frac{\lambda_2-1}{2}),
\end{equation}
where $G(x)=(x+1)\log (x+1)-x\log x$, $\lambda_1$ is the symplectic eigenvalue of Eve's mode covariance matrix $\gamma_E=V_E\mathbb{I}$, in fact $\lambda_1=\sqrt{\mbox{Det}\gamma_E}$ and $\lambda_2$ is the symplectic eigenvalue of the covariance matrix $\gamma_E^{x_B}$ characterizing the state of Eve's mode after Bob's projective measurement:
\begin{equation}
\gamma_E^{x_B}=\gamma_E-\sigma_{BE}(X \gamma_B X)^{MP}\sigma_{BE}^T,
\end{equation}
where $\gamma_B=V_B\mathbb{I}$ is the covariance matrix of Bob's mode, $\sigma_{BE}$ characterizes correlation between Bob's and Eve's modes, MP stands for Moore Penrose inverse of a matrix (also known as pseudoinverse) and
\begin{equation}
X =
\left( \begin{array}{cc}
1 & 0 \\
0 & 0
\end{array} \right).
\end{equation}
Finally, the mutual information $I_{AB}$ is calculated using Shannon entropies as
\begin{equation}
I_{AB}=\frac{1}{2}\log_2{\frac{V_{A}+1}{V_{A|B}+1}}.
\end{equation}
The explicit expression for the key rate is obtainable analytically, but it is too lengthly. However, similarly to the case of individual attacks, the security of the scheme is limited by the preparation noise and the boundary for collective attacks is close to that of the individual attacks (\ref{maxdvind}), while in the limit of arbitrary high modulation it exactly coincides with (\ref{maxdvindlim}). The effect of the preparation noise in comparison to the effect of the trusted detection noise in case of collective attacks is shown in Fig. \ref{trusted_noise} for the lossy channel with $\eta=0.01$ and realistic source variance $V=20$. One may see, that despite the fact, that both kinds of noise are trusted, the preparation noise is security breaking, unlike the detection noise, which is only quantitatively reducing the key rate -- the effect, which we observed above for the individual attacks case.
\begin{figure}
\caption{The effect of trusted detection noise $\chi$ and trusted preparation noise $\Delta V$ on the secret key in the case of no purification. Security against collective attacks is considered, source variance $V=20$, and channel transmittivity $\eta=0.01$.}
\label{trusted_noise}
\end{figure}
Now let us assume that purifying attenuation is applied to the signal states prior to sending them through the quantum channel. By supposing the attenuator transmittivity is $T$, we can see its positive affect already for individual attacks, as the lower bound on the key rate in this case is
\begin{eqnarray}
\label{indkr}
I_i=\frac{1}{2}\log_2{\bigg(
\frac{1+T(V+\Delta V -1)}{1+T(V+\Delta V -1)(1-\eta)}+\chi\bigg)}-{}
\nonumber \\
{}-\frac{1}{2}\log_2{(1+T\eta\Delta V+\chi)} &
\end{eqnarray}
and from this expression it is clear that attenuation $T$ is suppressing the preparation noise $\Delta V$. If the modulation variance is arbitrary high, this effect is even more evident, as the key rate turns to
\begin{eqnarray}
\label{indkrlim}
I_i\big|_{V \to \infty}=\frac{1}{2}\log_2{\bigg(\frac{1}{1-\eta}+\chi\bigg)}-{}
\nonumber \\
{}-\frac{1}{2}\log_2{(1+T\eta\Delta V+\chi)}, &
\end{eqnarray}
while the preparation noise $\Delta V$ threshold becomes
\begin{equation}
\label{inddvmax}
\Delta V_{i,max}\big|_{V \to \infty}=\frac{1}{T(1-\eta)}
\end{equation}
and can be made arbitrary high by purification.
For the finite $V$, if we take the derivative of the key rate (\ref{indkr}) by purifying attenuation $T$ in the point $T=0$, we obtain
\begin{equation}
\label{indder}
\frac{dI_{i}}{dT}\bigg\arrowvert_{T=0}=\frac{1}{\log{4}}\frac{\eta(V-1)}{1+\chi},
\end{equation}
which is always positive as $V \ge 1$. This means that close to $T=0$ there is always some $T>0$ that is sufficient to provide with the secure key rate because at point $T=0$, the key rate is equal to 0.
From the given expressions it is evident that detection noise, being not under the control of an eavesdropper, only decreases the key rate but does not destroy the transmission, even for finite source variance $V$, the detection noise $\chi$ does not change the security bounds, whereas preparation noise $\Delta V$ plays crucial role.
It is also worth mentioning that while secure key transmission is possible for arbitrarily high preparation noise upon arbitrary strong purification, for the given parameters, there always exists an optimal purification level, which maximizes the secure key rate. We skip the corresponding lengthly equations here, but in the further exposition we calculate the maximal key rate values in various conditions.
If the optimal purification is applied on the noisy signal and the modulation is extremely intense so that the source variance $V \to \infty$, then the expression for the bound on the key rate for individual attacks (\ref{indkr}) analytically turns to the expression for the case when the preparation is pure, i.e. $\Delta V=0$ and the modulation is infinitely high:
\begin{equation}
I_i\big|_{V \to \infty}^{T=T_{opt}}=
\frac{1}{2}\log{\frac{1+\chi(1-\eta)}{(1+\chi)(1-\eta)}}=I_i\big|_{V \to \infty}^{\Delta V=0},
\end{equation}
which means that by combining optimal purification with large modulation variance we can completely compensate the negative effect of preparation noise and achieve the values of key rate corresponding to the generation of pure states. Furthermore we will obtain a similar result in other cases.
In the case of collective attacks, similarly to the individual attacks, we can calculate the derivative of the key rate by purifying attenuation $T$ and evaluate it in the limit of $T\to 0$, arriving at the same result as (\ref{indder}) for individual attacks, proving that for finite modulation variance $\sigma$, we can always find some $T$ close to 0, which will provide secure transmission for any preparation noise $\Delta V$ as $I_{c}\arrowvert_{T\to 0}=0$. This result was also confirmed numerically by estimation of the maximal key rate secure against collective attacks upon optimal attenuator setting for given preparation noise and channel loss level. The optimized key rate versus preparation and detection noise is presented in Fig. \ref{trusted_noise_opt}. It is evident from the plot that by applying optimal purification to the noisy states we can turn the negative security-breaking effect of the preparation noise to the same as that of the detection noise, which is only reducing the key rate, but not turning it to zero.
\begin{figure}
\caption{The effect of trusted detection noise $\chi$ and trusted preparation noise $\Delta V$ on the secret key if the optimal purification is applied. Security against collective attacks is considered, source variance $V=20$, and channel transmittivity $\eta=0.01$.}
\label{trusted_noise_opt}
\end{figure}
In the limiting case of infinitely high modulation (i.e., source variance $V \to \infty$), the expressions for key rates and preparation noise thresholds are the same as for individual attacks, namely (\ref{indkrlim},\ref{inddvmax}), which means that by combining optimal attenuation and arbitrary high modulation, we can completely remove the negative effect of the preparation noise.
\section{Channel noise and untrusted detection noise}
\label{channoise}
While state purification was shown to be effective in case when trusted noise is present in the system, let us estimate the applicability of the method in the worst-case scenario, considering that noise, observable in the receiving station, is not trusted. This can be channel background noise in the case of broad-band detectors, untrusted detection noise or the influence of Eve, but as channel is the only untrusted part of our scheme, we refer to this noise as to the channel one.
Let us consider excess noise $\epsilon$, which is out of control of trusted parties. In the case of individual attacks, we use the "entangling cloner" scheme \cite{RR}, which allows Eve to perform optimal eavesdropping by substituting the channel of loss $\eta$ and excess noise $\epsilon$ with her apparatus. The optimality of the cloner attack is due to the fact that it enables Eve to reach the limit of her knowledge on Bob's data, described by the Heisenberg uncertainty principle, with the statement being valid under the assumptions that the used states are Gaussian and Eve knows the parameters of the channel, which well holds in our case. The attack uses pure two-mode squeezed vacuum state having local variance $N$; one mode is reflected to Bob's mode through the beam splitter with reflectivity $1-\eta$, whereas the transmitted part is kept by Eve. The second mode from the two-mode squeezed vacuum state is purely in the hands of Eve. Eve is fixing $N$ in order to satisfy $N=1+\frac{\eta \epsilon}{1-\eta}$, this way emulating the channel by her apparatus. She can store both of her modes in quantum memories and measure the appropriate quadratures of both modes in order to decrease the noise added to the measured part of the signal mode upon eavesdropping.
If Eve's modes variances before interaction with the signal are $V_{E^0_1}=V_{E^0_2}=N$ and Bob's mode is $V_{B_0}=T(V+\Delta V)+1-T$, then after interaction between modes $B$ and $E_1$ their variances are $V_B=\eta V_{B_0}+1-\eta+\eta\epsilon$ and $V_{E_1}=\eta V_{E^0_1}+(1-\eta)V_{B_0}$, while correlation between the modes is $C_{BE_1}=\sqrt{\eta(1-\eta)}(V_{E^0_1}-V_{B_0})$.
By calculating conditional variance as $V_{B|E_1}=V_B-C^2_{BE_1}/V_{E_1}$ and taking into account that after Eve's measurement on mode ${E_2}$ the conditional variance of measurement on mode ${E_1}$ becomes $V_{E_1|E_2}=1/N$, we obtain the expression for conditional variance of Eve's results on Bob's measurement results, which is
\begin{equation}
V_{B|E}=V_{B|E_1E_2}=\frac{1}{\eta\big(\frac{1}{T(V+\Delta V)+1-T}-1+\epsilon\big)+1}
\end{equation}
whereas Alice's uncertainty from the measured results is
\begin{equation}
V_{B|A^M}=1+\eta T \Delta V+\eta\epsilon.
\end{equation}
From the expression (\ref{indkr}) we can explicitly obtain the lower bound on the key rate in case of the untrusted channel noise:
\begin{eqnarray}
\label{indkrcn}
I_i=\frac{1}{2}\log_2{\bigg(
\frac{1}{\eta\big(\frac{1}{T(V+\Delta V)+1-T}-1+\epsilon\big)+1}\bigg)}-{}
\nonumber \\
{}-\frac{1}{2}\log_2{(1+T\eta\Delta V+\epsilon\eta)},
\end{eqnarray}
The derivative of the key rate by $T$ in the point $T=0$ is again always positive with $\frac{1}{\log_{10}{4}}\frac{\eta(V-1)}{1+\eta \epsilon}$, but as the key rate at $T=0$ is equal to $0$ only when $\epsilon=0$, being negative at any other $\epsilon>0$, the security upon channel noise is no longer guaranteed only by purification.
However, we may estimate the region of channel loss in which the purification can provide security for the given noise levels. As the channel excess noise can be hardly detected by trusted parties, in the entanglement cloner scenario, we take Eve's Einstein-Podolsky-Rosen (EPR) source variance $N$ as the parameter, which is independent on $\eta$. Within this approach we may estimate the bound on $\eta$ which restricts the security of key distribution for the given noise levels using the condition for optimal purification $T_{opt}<1$, this bound on security can be expressed as
\begin{equation}
\label{etamax}
\eta < \frac{1}{1+\frac{V-1}{N(1-V-\Delta V+\Delta V(V+\Delta V)^2)}},
\end{equation}
it can be compared to the similar bound on $\eta$, restricting the security of the key upon no purification, $T=1$. We skip the lengthy expression for the latter bound, but the calculations show that the region (\ref{etamax}) is no larger than the similar region for no trusted side attenuation, i.e. purification provides security of the key for any $\Delta V$ in the same region of parameters, where the security was provided for no purification and no preparation noise.
The optimal purification, which maximizes the secure key rate upon given parameters can be expressed as
\begin{eqnarray}
\label{topt}
\lefteqn{T_{opt}=\frac{1}{V+\Delta V-1}\times{}}
\nonumber \\
&&{}\times\Bigg(\sqrt{\frac{(V+\Delta V-1)(\eta\epsilon+1)-\eta\Delta V}{\Delta V(\eta\epsilon+1-\eta)}}-1\Bigg)
\end{eqnarray}
In the limit of infinitely high modulation (i.e., source variance $V \to \infty$), the bound on the key rate turns to
\begin{eqnarray}
\label{indkrlim2}
I_i\big|_{V \to \infty}=\frac{1}{2}\log_2{\frac{1}{\eta\epsilon+1-\eta}}-{}
\nonumber \\
{}-\frac{1}{2}\log_2{(1+T\eta\Delta V+\eta\epsilon)}, &
\end{eqnarray}
while the threshold on the preparation noise involves channel noise, which is additionally limiting the tolerable preparation noise:
\begin{equation}
\Delta V_{i,max}=\frac{1-\epsilon}{T(1-\eta+\eta\epsilon)}-\frac{\epsilon}{T}.
\end{equation}
However, if the optimal purification (\ref{topt}) is applied in this case, the expression for the secure key rate turns to the one for the case of infinite modulation and no preparation noise:
\begin{eqnarray}
I_i\big|_{V \to \infty}^{T=T_{opt}}=\frac{1}{2}\log_2{\frac{1}{\eta\epsilon+1-\eta}}-{}
\nonumber \\
{}-\frac{1}{2}\log_2{(1+\eta\epsilon)}=I_i\big|_{V \to \infty}^{\Delta V=0}, &
\end{eqnarray}
which means that the combination of optimal purification and sufficiently large modulation variance can completely eliminate the negative effect of the preparation noise. Still, for the limited source variance, the security can be provided for any given preparation noise within the limited range of noise introduced by an eavesdropper as described by (\ref{etamax}).
In the case of collective attacks performed by an eavesdropper we can no longer use direct calculations of the Holevo quantity used in the previous section. Rather, we assume that Eve can purify the complete state shared among all the parties. In order to perform calculations for this case we must switch to the entangled-based scheme, which was shown to be equivalent to the prepare-and-measure one \cite{Lod2} and take into account all the trusted modes in Alice's station.
Similarly, we will assume the worst-case detection scenario, that all the noise and loss in the Bob's station is untrusted.
In the entanglement-based scheme, the entanglement is generated in the source $EPR:V$ by mixing two pure orthogonally squeezed states with the squeezing variance $V_S$ at a balanced beam splitter producing two entangled modes A and B. The mode A in the thermal state of variance $V=(V_S+V_S^{-1})/2$ is measured by Alice simultaneously in both quadratures by the heterodyne detector, while mode B is coupled to the noise mode G from the similar entangled source $EPR:\Delta V_0$ of variance $\Delta V_0$, emulating preparation noise in modulator (see Fig.~\ref{scheme}, inlay).
To emulate the additive noise with noise variance $\Delta V$, the coupling between $B$ and $G$ is strongly asymmetrical, with almost unit transmittivity $T_N\approx 1$. This setup corresponds to the preparation noise of variance $\Delta V=(1-T_N)\Delta V_0$.
Now the state of ABCFG is pure in the absence of channel noise and we can use the fact that Eve purifies this state so that von Neumann entropy $S_E=S_{ABCFG}$. Then, after Bob's measurement the system ACFG is pure and the conditional entropy $S_{E|B}=S_{ACFG|B}$. Thus, the Holevo quantity becomes $\chi_{BE}=S_{ABCFG}-S_{ACFG|B}$. We perform calculation of these values similarly to the case of individual attacks, by obtaining symplectic eigenvalues of the five-mode covariance matrix $ABCFG$ and four-mode matrix $ACFG|B$, which can be done purely numerically. As a result, we get the method allowing us to perform calculations of the lower bound of the key rate upon any conditions in the presence of both preparation and channel excess noise and we investigate the effect of purification on the applicability of the scheme. Moreover, we have numerically confirmed that in the absence of channel noise this method, based on the state purification, is equivalent to the method based on Eve's mode calculation, which was used in the previous section.
Like in the individual attacks scenario, in the case of the collective attacks, security is now limited by both preparation and channel excess noise. We calculated the maximal tolerable channel excess noise as the function of preparation noise both for optimal purification and no purification on Alice's side; the results are given in Fig. \ref{channel_noise} for typical source variance of $V=10$ and for almost ideal case of extremely large source variance $V=10^5$. It was checked numerically and can be seen from the graphs that, similarly to the case of individual attacks, when the purification is optimized, the transmission is now possible upon arbitrarly high preparation noise if the channel itself was not destructive, and the scheme is not becoming more sensitive to the channel noise than without purification. Also for large source variance $V$ the dependence of maximal tolerable excess noise on given state preparation noise approaches saturation, starting to be very slow.
Thus, we need to increase source variance $V$ (by increasing modulation variance $\sigma$) in order to weaken the dependence of key rate on the preparation noise $\Delta V$. However, for small source variance $V \to 1$, the dependence on preparation noise is qualitatively the same, just the scheme quickly becomes sensitive to channel noise.
\begin{figure}
\caption{Maximal tolerable channel excess noise versus given preparation noise for no Alice-side purification (dashed line) and with optimal attenuator setting (solid line), source variance $V=10$ (lower lines) and $V=10^5$ (upper lines), and channel loss $\eta=0.01$}
\label{channel_noise}
\end{figure}
For the realistic cases, in order to perform the qualitative comparison, the maximal key rate was calculated as the function of preparation noise and channel excess noise and is given in Fig. \ref{KR_3D_10} for $V=10$ and in Fig. \ref{KR_3D_100} for $V=100$. It is evident from the graph, that for relatively small channel noise the security of the scheme can be provided upon any preparation noise level, while as the channel noise increases, the security is not guaranteed anymore.
\begin{figure}
\caption{Maximal key rate as the function of channel excess noise $\epsilon$ and preparation noise $\Delta V$, source variance $V=10$, and channel loss $\eta=0.01$}
\label{KR_3D_10}
\end{figure}
\begin{figure}
\caption{Maximal key rate as the function of channel excess noise $\epsilon$ and preparation noise $\Delta V$, source variance $V=100$, and channel loss $\eta=0.01$}
\label{KR_3D_100}
\end{figure}
The region of parameters for which the security is guaranteed is plotted for source variance $V=100$ at Fig. \ref{security_surface-dB} and one can see that over the broad values of channel loss $\eta$ and channel noise $\epsilon$, the security is guaranteed for any preparation noise $\Delta V$, while dependence on $\Delta V$ upon its high values becomes more linear and flat, approaching the horizontal surface for extremely high source varance $V\to \infty$.
\begin{figure}
\caption{The region of parameters for which the security is guaranteed for source variance $V=100$}
\label{security_surface-dB}
\end{figure}
Let us now consider the relatively simple case of no preparation noise (i.e., $\Delta V=0$). The explicit expressions for the key rate even in this case are lengthy [although analytically obtainable through the symplectic eigenvalues of matrices $AB$ and $A|x_B$ and extended expression (\ref{holevo1})]. We are interested in extremely lossy channels, so we perform a series expansion of the key rate with respect to source variance $V$ around infinity and to channel loss $\eta$ around zero. Such expansion up to the first order has the form
\begin{eqnarray}
I_c\big|^{\Delta V=0}\approx\frac{\eta}{\log{4}}
\Big(1-\epsilon+\epsilon\log{\frac{\eta\epsilon}{2}}\Big)+O[\eta]^{3/2}+{}
\nonumber \\
{}+\frac{1}{V}\Big(-\frac{2\eta}{\log{8}}+O[\eta]^{3/2}\Big)+O\Big[\frac{1}{V}\Big]^{3/2}, &
\end{eqnarray}
while for $V \to \infty$ the approximate normalized expression for the key rate turns to
\begin{equation}
\label{approxseries}
I_c\big|_{V \to \infty}^{\Delta V=0}\approx 0.721\eta-1.221\eta\epsilon+0.721\eta\epsilon\log{\eta\epsilon}.
\end{equation}
Now we perform the analytical least-squares fit of the numerical data \cite{hudson}, obtained for the optimal purification when preparation noise is present:
\begin{equation}
\label{approxnum}
I_c\big|_{V \to \infty}^{T=T_{opt}}\approx 0.722\eta-1.237\eta\epsilon+0.731\eta\epsilon\log{\eta\epsilon}.
\end{equation}
which very well fits the numerical data and at the same time is close to the approximate expression for the key rate in the case of no preparation noise (\ref{approxseries}). We also performed a comparison between a numerically calculated maximized key rate upon optimal purification, high source variance $V=10^5$, preparation noise $\Delta V \in (0,5)$ and the analytically obtained values of the key rate upon the same source variance, no purification and no preparation noise $\Delta V=0$ in the region of loss $\eta \in (0.01,0.1)$ and channel noise $\epsilon \in (0.01,0.1)$. The standard (root-mean-square) deviation of the maximized key from the analytical key rate in case of pure coherent states is approximately $6\times 10^{-5}$, which is calculated as $s=\sqrt{\frac{1}{n-1}\sum_n{\big(I_c\big|_{V =10^5}^{\Delta V=0}(\eta,\epsilon)-I_c\big|_{V = 10^5}^{T=T_{opt}}(\eta,\Delta V,\epsilon)\big)^2}}$, where $n=10^3$ is the number of points $(\eta,\Delta V,\epsilon)$ taken within the regions of parameters \cite{korn}. The average relative deviation, being the ratio of the standard deviation to the average value of the key rate for pure states $\bar{I_c}\big|_{V =10^5}^{\Delta V=0}(\eta,\epsilon)\approx 0.026$, is around $2\%$. This divergence can be made even smaller upon further increase of source variance; for example, at $V=10^6$ it already becomes less than $1\%$.
This way we claim that by combination of optimal purification and extremely large modulation variance the key rate for collective attacks can reach the values as for the case of no preparation noise. Thus, with the optimal purification upon high modulation variance we can completely eliminate preparation noise for both types of attacks, which in the absence of purification would otherwise be destructive for the key transmission.
\section{Realistic reconciliation}
\label{recon}
In previous sections we calculated the lower bound on the secure key rate assuming that the classical postprocessing algorithms, which are aimed at deriving the secure key from the raw key, obtained by Alice and Bob, are absolutely efficient and do not reduce the key length. In practice, though the realistic error correction, which is also referred to as key reconciliation \cite{Lutk1}, has limited efficiency and is being done by the cost of the key length, thus decreasing the overall secure key rate. The negative effect of the realistic key reconciliation was shown to essentially limit the applicability of the secure key distribution with binary modulation of coherent states, especially in case of the reverse reconciliation \cite{Lutk2}. The similar effect is observed in the case of continuous-variables coding; thus, we must investigate whether the state purification is still efficient against the preparation noise if the error correction has limited efficiency.
In practical CV QKD the influence of the realistic key reconciliation is quantitatively characterized by the efficiency $\beta$, which is reducing the mutual information, available to Alice and Bob and stands for the fraction of the data, discarded in the process of error correction. Thus, in the case of collective attacks the efficient lower bound on the secure key is given by \cite{Lod2}:
\begin{equation}\label{geneff}
I_{eff}=\beta I_{AB}-\chi_{BE}
\end{equation}
Reconciliation effectiveness $\beta$ is a function of the signal-to-noise ratio (SNR), but it also depends on the algorithm, being used for the reconciliation and on the computational power, which is available for the trusted parties. Generally, the lower the SNR, the stronger the negative impact of the inefficient error correction, although for a given $\beta$, there is always some optimal value of SNR (i.e., of the modulation, applied to the coherent signal \cite{Lod2} if other parameters are fixed). So, as reconciliation efficiency $\beta$ depends on the particular technical parameters, we do not use the explicit function describing the dependence of $\beta$ on SNR in our theoretical analysis. Instead we assume that the dependence of $\beta$ on SNR is unknown, so we must keep both as the independent parameters and estimate how tolerable is the CV QKD to the preparation noise upon realistic reconciliation when we perform state purification by attenuating the signal or when the purification is not applied. As a result, we describe the area of values of SNR and $\beta$, where the method gives positive effect. Then in any particular case, when the dependence of $\beta$ on SNR is known, it can be easily found, whether given reconciliation efficiency allows improvement by the proposed purification method.
Let us consider the purely lossy channel, as the effects of both preparation noise and inefficient reconciliation are significant already in this case. In the previous sections the lower bound on the key rate $I$ was the function $I(V,\Delta V,\eta,T)$ of source variance, preparation noise, channel loss and trusted-side attenuation $T$. At the same time, SNR $\Sigma$, being the ratio of the signal variance to the overall excess noise variance at the channel output is also the function $\Sigma(V,\Delta V,\eta,T)$ of the same parameters:
\begin{equation}
\Sigma=\frac{T\eta(V-1)}{1+T\eta\Delta V}
\end{equation}
So, in order to describe the security region in terms of the preparation noise, taking into account the inefficient reconciliation, when the purification is absent i.e. $T=1$, we perform the transition $V \to V(\Sigma,\Delta V,\eta)$, recalculate the lower bound on the key rate in new terms as $I_{eff}(\Sigma,\Delta V,\eta,\beta)$, taking into account (\ref{geneff}) and can estimate the threshold values of preparation noise $\Delta V_{max}$, which turn this lower bound to zero. The threshold is presented graphically as the three-dimensional (3D) plot versus reconciliation efficiency $\beta$ and SNR for typical channel loss $\eta=0.1$ in Fig. \ref{3D-dv_max_no_t}, note that it is independent of source variance $V$, which is given by the value of SNR.
\begin{figure}
\caption{Threshold on the preparation noise versus reconciliation efficiency $\beta$ and SNR for any source variance $V$ and typical channel loss $\eta=0.1$ when no trusted-side purification is applied ($T=1$)}
\label{3D-dv_max_no_t}
\end{figure}
Now let us suppose we plug in the attenuation at the trusted side, which is aimed at the purification of the signal states. Then SNR puts the constraint on attenuation $T$ as all other parameters are given and we can express $T$ as $T(\Sigma,V,\Delta V,\eta)$. Similarly, we reculaculate the lower bound on the secure key as $I_{eff}(\Sigma,V,\Delta V,\eta,\beta)$ and estimate the threshold on preparation noise $\Delta V$, when the attenuation $T$, unlike in the previous sections, is no more optimized, but is explicitly given by the SNR and other parameters. In Fig. \ref{3D-dv_max_t} we present the results as the 3D plot of the security region, describing the maximal tolerable preparation noise $\Delta V_{max}$ versus reconciliation efficiency $\beta$ and SNR, realistic source variance $V=20$ and typical channel loss $\eta=0.1$.
\begin{figure}
\caption{Threshold on the preparation noise versus reconciliation efficiency $\beta$ and SNR for realistic source variance $V=20$ and typical channel loss $\eta=0.1$ when the states are purified by attenuation $T$; values of $\Delta V_{max}
\label{3D-dv_max_t}
\end{figure}
Any practical scheme has its own dependence of reconciliation efficiency on SNR, which is related to the intristic parameters of the scheme, which can be graphically described by a curve in the $\beta,SNR$ plain. Thus, in order to estimate the tolerance of the particular scheme to the preparation noise and the effectiveness of state purification, one should project this curve on surfaces at Figs. \ref{3D-dv_max_no_t} and \ref{3D-dv_max_t} respectively. Nevertheless, in the general case, it is evident from both plots, that the preparation noise is significantly limiting the securirty of the scheme in the large region of SNR and reconciliation efficiency $\beta$. At the same time, the applicable purification on the trusted side can essentially improve the tolerance to the preparation noise, not reducing the applicability area in terms of SNR and $\beta$. Interestingly, this area is even increased by purification, giving a high threshold on tolerable preparation noise for low SNR and $\beta$, although the key rate in this area is very low and can be thus drastically affected by other practical imperfections.
\section{Conclusions}
We have investigated the influence of the noisy modulation on the security of the quantum key distribution scheme based on the coherent states upon realistic conditions of channel loss and detection (trusted) or channel (untrusted) excess noise. While the preparation noise was shown to be destructive to secure transmission, which reveals an essential difference between two types of trusted noise, we investigate the possibility of suppressing the preparation noise. It is shown that optimal purification on the trusted sender side drastically increases the security region in terms of the tolerable preparation noise if the channel itself is not security breaking. Thus, noisy coherent states are as useful for the secure key distribution as the pure coherent states. In the case of the high source variance, the optimal purification is shown to practically compensate the negative effect of preparation noise, which otherwise would be destructive for the transmission. The positive effect of purification is preserved also in the conditions of realistic key reconciliation with nonunity efficiency. While purification by attenuation is already being used to reduce the modulation noise and optimize the modulation for the given reconciliation efficiency, the more advanced methods out of the scope of linear operations, in particular, feedback control of the modulated signals, can be even more effective and may be the subject for further research.
\noindent {\bf Acknowledgments} The research has been supported by
Projects No. MSM 6198959213 and No. LC06007 of the Czech Ministry
of Education and Project No. 202/07/J040 of GACR.
R.F. also acknowledges support by the Alexander von Humboldt
Foundation, V.U. thanks the Ukrainain State Fundamental Research Foundation.
\end{document} |
\begin{equation}gin{document}
\title{On the Bang-Bang Type Nash Equilibrium Point for Markovian Nonzero-sum Stochastic Differential Game}
\slugger{mms}{xxxx}{xx}{x}{x--x}
\begin{equation}gin{abstract}
In this paper, we study a nonzero-sum stochastic differential game of bang-bang type in the Markovian framework. We show the existence of a Nash equilibrium point for this game. The main tool is the notion of backward stochastic differential equations which, in our case, are multidimensional with discontinuous generators with respect to $z$ component.
\end{abstract}
\begin{equation}gin{keywords}
Nonzero-sum Stochastic Differential Games; Nash Equilibrium Point; Backward Stochastic Differential Equations ; Bang-bang type control.
\end{keywords}
\begin{equation}gin{AMS}
49N70; 49N90; 91A15.
\end{AMS}
\pagestyle{myheadings}
\thispagestyle{plain}
\markboth{TEX PRODUCTION}{USING SIAM'S \LaTeX\ MACROS}
\section{Introduction}\label{se:intro}
This paper is related to nonzero-sum stochastic differential games (NZSDG, for short) of bang-bang type in the Markovian framework which we describe below.
We consider the case of two players $\pi_1$ and $\pi_2$. If there are more than two players, the adaptation is straightforward. The two players $\pi_1$ and $\pi_2$ act on a system through two admissible controls $u:=(u_s)_{s\leq T}$ and $v:=(v_s)_{s\leq T}$ which are adapted stochastic processes, with values in compact metric spaces $U$ and $V$, respectively. The dynamics of the controlled system is given by a stochastic process $(X^{u,v}_s)_{s\leq T}$, solution of the following standard stochastic differential equation (SDE, for short):
\begin{equation}gin{equation}\label{eq: intro x}
dX_s^{u,v}=\Gamma(s,X_s^{u,v},u_s,v_s)ds+\sigma(s,X_s^{u,v})dB_s\text{ for }s\leq T, \text{ and } X_0=x_0
\end{equation}
where $B:=(B_s)_{s\leq T}$ is a Brownian motion. Next with each player $\pi_i$, $i=1,2$, is associated a payoff $J_i(u,v)$, $i=1,2$, given by:
\begin{equation}gin{equation}\label{eq: payoff intro}
J_i(u,v)=\textbf{E}[g_i(X_T^{u,v})].
\end{equation}
The objective is to find a pair $(u^*,v^*)$ which satisfy
\begin{equation}gin{equation}\label{eq: obj intro}
J_1(u^*, v^*)\geq J_1(u, v^*) \text{ and } J_2(u^*, v^*)\geq J_2(u^*, v)\mbox{ for any pair }(u,v).
\end{equation}
As we can see, the payoff function of the player $\pi_1$ (resp. $\pi_2$) depends not only on its own control $u$ (resp. $v$) but also on the control used by the other player $\pi_2$ (resp. $\pi_1$). Therefore there is a game between the two players which is of cooperative relationship. This kind of game is known as the nonzero-sum stochastic differential game. The pair $(u^*,v^*)$ of (\ref{eq: obj intro}) is called a Nash equilibrium point (NEP, for short) of the game. Its meaning is that when it is implemented by both players, then if $\pi_1$ (resp. $\pi_2$) decides unilaterally to change a control while $\pi_2$ (resp. $\pi_1$) keeps $v^*$ (resp. $u^*$) then her payoff $J_1(u, v^*)$ (resp. $J_2(u^*, v)$) is lesser than
$J_1(u^*, v^*)$ (resp. $J_2(u^*, v^*)$), i.e., her action of unilateral deviation induces a penalty.
In the case when $J_1(u,v)+J_2(u,v)=0$, the game turns into the well-known zero-sum differential game which is widely studied in the literature (see e.g. \cite{zs3, zs4, zs5, zs1, zs2}, etc. and the references therein). On the other hand, if $X^{u,v}$ does not depend on $v$ then the problem turns merely into a control problem. In this specific case, we know that an optimal control exists and is of bang-bang type since it takes values only on the boundaries of $U$ according to the derivative of the value function of the control problem. This is the consequence of the fact that the instantaneous reward in (\ref{eq: payoff intro}) is null. So one would expect the same features of the NEP of this game if it exists.
\ms
The nonzero-sum differential game is also considered by several authors in the literature, see eg. \cite{bensoussan1, bensoussan2,buckdahn2004, carmona2013control, rainer, friedman2, H1997, hamadene1999, hamadene1998, qianlin, mannucci, carmona2013probabilistic}, to name a few. There are typically two approaches. One method is related to partial differential equation (PDE) theory. Some of the results show that the payoff function of the game is the unique viscosity solution of a related Hamilton-Jacobi-Bellman-Isaacs equation, e.g., \cite{qianlin}. Other works make use of the Sobolev theory of PDEs (see \cite{bensoussan1, bensoussan2, friedman2, mannucci}, etc.) to deal with NZSDGs. Comparatively, another popular way to deal with stochastic differential game is the backward stochastic differential equation (BSDE) approach \cite{H1997, hamadene1999, hamadene1998, qianlin}, which characterizes the payoffs of the game through solutions of associated BSDEs. However those BSDEs are of multidimensional type and usually their generators are non-Lipschitz. Therefore proving that they have solutions is not an easy task.
In the present article, we study the above NZSDG via the BSDE arguments in the Markovian framework. For clarity reasons we consider a special game model in assuming that:
(i) the process $X$ of \eqref{eq: intro x} is $\textbf{R}$-valued, and $U=[0,1]$, $V=[-1,1]$ ;
(ii) the drift $\Gamma$ of \eqref{eq: intro x} has the following structure:
$$
\Gamma(t,x,u,v)=f(t,x)+u+v.
$$
The conditions on the functions $f$ and $g_i$, $i=1,2$, are rather weak since they are related to measurability and growth conditions.
At the end of the paper we give hints which allow to generalize this setting in several directions, especially if the dynamics contains a diffusion term and the multidimensional case for the process $X^{u,v}$.
\ms
In this problem the main difficulty is lodged at the level of the main BSDE (\ref{eq:main bsde}) associated with this NZSDG. Once the existence of its solution is stated, it provides the NEP of the game. As pointed out previously, this BSDE is of multidimensional type (here of dimension two since there are two players) and whose generator is discontinuous in $z$. The main challenge we overcome is to show that this BSDE has a solution and then we constructed a NEP for the game defined by (\ref{eq: intro x}), (\ref{eq: payoff intro}) and (i)-(ii) above. Like in the control framework, this NEP is of bang-bang type
since the payoffs have no instantaneous payoffs included. This is the main novelty of this article. The closest work to ours is the one by P.Manucci \cite{mannucci}, but this latter concerns only diffusions in bounded domains and the requirements on the regularity of the data are stronger than ours due to the method she employed based on PDEs in Sobolev spaces.
\ms
This paper is organized as follows:
In Subsection \ref{subsec:statement 1-d}, we introduce the game problem and some preliminaries. The formulation we adopt is of weak type. Besides, for intuitive understanding, we work with a particular setting of controls and state process $X^{u,v}$. The extension to the multidimensional situation obviously holds following the same ideas. The explicit form of discontinuous controls, namely, bang-bang controls are presented in Subsection \ref{subsec: bang-bang}. In Subsection \ref{subsec: main result}, we give the main result (Theorem \ref{th: nash}) of this work and some other related important results. We first provide a link between the game problem and Backward SDEs (Proposition \ref{prop: yuv}). The payoff of the game can be characterized by the initial value of the solution for an associated BSDE. Then, by Proposition \ref{prop: equilibrium}, we prove that the existence of a NEP for the game is equivalent to the existence of a solution of a BSDE which is of multidimensional and with discontinuous generator with respect to $z$. Finally, under some reasonable assumption, we provide the solution of this special BSDE (Theorem \ref{th: main bsde}). All the proofs are stated in Subsection \ref{subsec:proofs}. The proofs of Propositions \ref{prop: yuv} and \ref{prop: equilibrium} are standard. For Theorem \ref{th: main bsde}, the method is mainly based on an approximating scheme. In Section \ref{sec:gener}, we investigate some possible generalizations. The idea is the same with a bit modification which is indicated.
\section{Bang-bang type NZSDG and multidimensional BSDEs with discontinuous generators}\label{sec: 1-d}
In this section, we first deal with the bang-bang type nonzero-sum stochastic differential game problem in 1-dimensional framework. A more general setting will be given in the next section.
u^*bsection{Statement of the problem}\label{subsec:statement 1-d}
Let $T>0$ be fixed and let $(\Omega, \mathcal{F}, \textbf{P})$ be a probability space, on which is defined a 1-dimensional standard Brownian motion $B:=(B_t)_{t\leq T}$. For $t\leq T$, let us set $F_t:=\sigma(B_u,u\leq t)$ and denote by $(\mathcal{F}_t)_{t\leq T}$ the completion of $(F_t)_{t\leq T}$ with the $\textbf{P}$-null sets of $\mathcal{F}$. Next let $\mathcal{P}$ be the $\sigma$-algebra on $[0,T]\times \Omega$ of ${\mathcal{F}}_t$-progressively measurable sets. For a real constant $p\geq 1$, we introduce the following spaces:
\noindent \begin{equation}gin{itemize}
\item $L^p=\{\xi:\ \mathcal{F}_T$-measurable and $\textbf{R}$-valued random variable s.t.$\textbf{E}[|\xi|^p]<\infty\}$;
\item $\mathcal{S}_T^p(\textbf{R})=\{Y=(Y_s)_{s\in[0,T]}:\ \mathcal{P}$-measurable, continuous and $\textbf{R}$-valued stochastic process s.t. $\textbf{E}[u^*p_{s\in [0,T]}|Y_s|^p]<\infty\}$;
\item $\mathcal{H}_T^p(\textbf{R})=\{Z=(Z_s)_{s\in[0,T]}:\ \mathcal{P}$-measurable and $\textbf{R}$-valued stochastic process s.t. $\textbf{E}[(\int_0^T|Z_s|^2ds)^{p/2}]<\infty\}$.
\end{itemize}
We consider, in this article, the 2-player case which we describe accurately below. The general multiple players case is a straightforward adaptation.
Let $(t,x)\in [0,T]\times \textbf{R}$ and $X^{t,x}$ be the stochastic process defined as follows:
\begin{equation}gin{equation}\label{eq: SDE}
\forall s\leq T,\ X_s^{t,x}=x+(B_{s\vee t}-B_t).
\end{equation}
\begin{equation}gin{remark}\label{re:sigma} Note that we consider a trivial situation for SDE \eqref{eq: SDE} with an identity diffusion process, just for easy understanding. The trick of the technique in this article still valid for general diffusion process with appropriate properties. We will introduce this point in Section \ref{sec:gener}.
\end{remark}
Each player $\pi_i$, $i=1, 2$, has her own control. Let us denote next by $U=[0,1]$, $V=[-1,1]$ those two compact subsets of $\textbf{R}$ and $\mathcal{M}_1$ (resp. $\mathcal{M}_2$) the set of $\mathcal{P}$-measurable process $u=(u_t)_{t\leq T}$ (resp. $v=(v_t)_{t\leq T}$) on $[0,T]\times \Omega$ with value in $U$ (resp. $V$). Hereafter, we call $\mathcal{M}:=\mathcal{M}_1\times \mathcal{M}_2$ (resp. $\mathcal{M}_1$, resp. $\mathcal{M}_2$) the set of admissible controls for the two players (resp. first player ; resp. second player).
Let $f: (t,x)\in [0,T]\times \textbf{R}\rightarrow \textbf{R}$ be a Borelian function.
We will say that $f$ is of linear growth if there exists a constant $C\geq 0$ such that for any $\tx$,
\begin{equation}gin{equation}\lb {lg}|f(t,x)|\leq C(1+|x|).\end{equation}
Next let $\Gamma$ be the function such that for any $(t,x,u,v)\in [0,T]\times \textbf{R}\times U\times V$ associates $\Gamma(t,x,u,v)=f(t,x)+u+v \in \textbf{R}$. The function $\Gamma$ stands for the drift of the dynamics of the system when controlled by the two players $\pi_i$, $i=1, 2$. When $f$ is of linear growth, the function $\G$ is so since $U$ and $V$ are bounded sets.
\ms
Next for $(u.,v.)\in \mathcal{M}$, let $\textbf{P}^{u,v}_{t,x}$ be the positive measure on $(\Omega,\mathcal{F})$ defined as follows:
\begin{equation}gin{equation}\label{eq:density fun}\begin{equation}gin{array}{c}
d\textbf{P}^{u,v}_{t,x}=\zeta_T(\Gamma(.,X.^{t,x},u.,v.))d\textbf{P}\ \text{with}\ \zeta_s(\Theta):=1+\int_0^s \Theta_r \zeta_r dB_r,\ \sT,\end{array}
\end{equation}
for any $\mathcal{P}$-measurable $\textbf{R}$-valued process $\Theta:=(\Theta_s)_{s\leq T}$. It follows from the uniform linear growth property of $\Gamma$ that $\textbf{P}^{u,v}_{t,x}$ is a probability on $(\Omega, \mathcal{F})$ (see Appendix A of \cite{karoui and hamadene} or \cite{karatzas}, pp.200). Then, by Girsanov's Theorem (\cite{girsanov}), the process $B^{u,v}=(B_s-\int_0^s\Gamma(r, X_r^{t,x},u_r,v_r)dr)_{s\leq T}$ is a $(\mathcal{F}_s, \textbf{P}^{u,v}_{t,x})$-Brownian motion and $(X_s^{t,x})_{s\leq T}$ satisfies the following SDE
\begin{equation}gin{equation}\label{eq: SDE with Gamma}\begin{equation}gin{array}{c}
dX_s^{t,x}=\Gamma(s, X_s^{t,x},u_s,v_s)ds+dB_s^{u,v},\ \forall s\in[t,T]\ \text{and}\ X_s^{t,x}=x,\ s\in [0,t].\end{array}
\end{equation}
As a matter of fact, the process $X^{t,x}$ is not adapted with respect to the filtration generated by the Brownian motion $B^{u,v}$. Thereby, $X^{t,x}$ is a weak solution for the SDE \eqref{eq: SDE with Gamma}. If the system starts from $x_0\in \textbf{R}$ at $t=0$ and is controlled by player $\pi_1$ (resp. $\pi_2$) with $u.$ (resp. $v.$), the law of its dynamics is the same as the one
of $X^{0,x_0}$ under $\textbf{P}^{u,v}_{0,x_0}$.
Once more let $x_0\in \textbf{R}$ fixed. We will precise the payoffs of the players when they implement the pair of strategies $(u.,v.)$. It is of terminal type and given, for player $\pi_1$ (resp. $\pi_2$), by
\begin{equation}gin{eqnarray}\label{eq: payoffs}
J_1(u,v):=\textbf{E}^{u,v}_{0,x_0}[g_1(X_T^{0,x_0})]
\mbox{ (resp. }J_2(u,v):=\textbf{E}^{u,v}_{0,x_0}[g_2(X_T^{0,x_0})]),
\end{eqnarray}
where:
\noindent (i) $g_1$ and $g_2$ are two Borel measurable functions from $\textbf{R}$ to $\textbf{R}$ which are of polynomial growth, \textit{i.e.}, there exist non-negative constants $C$ and $\gamma\geq 1$ such that for any $x\in \textbf{R}$,
\begin{equation}gin{eqnarray}\label{(A2)}|g_1(x)|+|g_2(x)|\leq C(1+|x|^{\gamma})\end{eqnarray}
(ii) For any fixed $(t,x)\in \sp$, $\textbf{E}^{u,v}_{t,x}$ is the expectation under the probability $\textbf{P}^{u,v}_{t,x}$ ; hereafter $\textbf{E}^{u,v}_{0,x}(.)$ (resp. $\textbf{P}^{u,v}_{0,x}$) will be simply denoted by $\textbf{E}^{u,v}_x(.)$
(resp. $\textbf{P}^{u,v}_x$).
As we can see from \eqref{eq: SDE with Gamma} and \eqref{eq: payoffs}, the choice of control of each player has influence on the other one's payoff through the state process $X^{0,x_0}$ under $\textbf{P}^{u,v}_{x_0}$. What we discussed here is a nonzero-sum stochastic differential game which means that the two players are of cooperate relationship. Both of them want to reach the maximum payoff. Therefore, naturally, we are concerned with the existence of a \textit{Nash equilibrium point}, which is a couple of controls $(u^*,v^*)\in \mathcal{M}$, such that, for all $(u,v)\in \mathcal{M}$,
\[
J_1(u^*,v^*)\geq J_1(u, v^*)\ \text{and}\ J_2(u^*,v^*)\geq J_2(u^*,v).
\]
This means that when the strategy $(u^*,v^*)$ is implemented by the players, one who makes unilaterally the decision to deviate or to change a strategy, while the other one keeps its own choice, is penalized.\qedh
u^*bsection{Bang-bang type control}\label{subsec: bang-bang}
As pointed out in \eqref{eq: payoffs}, there are no instantaneous payoffs in $J_1$ and $J_2$. Therefore, in comparison with optimal control which is a particular case of our problem (see e.g. \cite{balak, benes, christopeit, maurerosmo}), the equilibrium point of this game, if exists, should be of bang-bang type, \textit{i.e.}, the optimal control $u^*$ (resp.$v^*$) will jump between the two bounds of the value set $U$ (resp.$V$).
To proceed, let $H_1$ and $H_2$ be the \textit{Hamiltonian functions} of this game problem, \textit{i.e.}, the functions (which do not depend on $\omega$) defined from $[0,T]\times \textbf{R}\times \textbf{R}\times U\times V$ into $\textbf{R}$ by:
\begin{equation}gin{eqnarray*}
H_1(t,x,p,u,v)&:=p\Gamma(t,x,u,v)=p(f(t,x)+u+v);\\
H_2(t,x,q,u,v)&:=q\Gamma(t,x,u,v)=q(f(t,x)+u+v).
\end{eqnarray*}
Next let $\epsilon_1$ and $\epsilon_2$ be two arbitrary elements of $U$ and $V$ respectively. Let $\begin{array}r{u}$ and $\begin{array}r{v}$ be two functions defined on $\textbf{R}\times U$ and $\textbf{R}\times V$, valued on $U$ and $V$ respectively, as follows: $\forall p,q\in \textbf{R}$,
\begin{equation}gin{equation}
\begin{array}r{u}(p,\epsilon_1)=
\left\{
\begin{equation}gin{aligned}
1, \ p>0,\\
\epsilon_1,\ p=0,\\
0,\ p<0,
\end{aligned}
\right.
\quad
\text{and}
\quad
\begin{array}r{v}(q,\epsilon_2)=
\left\{
\begin{equation}gin{aligned}
1,\ q>0,\\
\epsilon_2,\ q=0,\\
-1,\ q<0.
\end{aligned}
\right.
\end{equation}
Then, we can easily check that $\begin{array}r{u}$ and $\begin{array}r{v}$ satisfy the \textit{generalized Isaacs' condition} which reads as follows: $\forall \,\,(t,x,p,q,u,v)\in [0,T]\times \textbf{R}\times \textbf{R}\times U\times V$,
\begin{equation}gin{equation}\label{eq:Isaacs}
\left\{
\begin{equation}gin{aligned}
H_1^*(t,x,p,q,\epsilon_2)&:=H_1(t,x,p,\begin{array}r{u}(p,\epsilon_1),\begin{array}r{v}(q,\epsilon_2))\geq H_1(t,x,p,u,\begin{array}r{v}(q,\epsilon_2))\mbox{ and }\\
H_2^*(t,x,p,q,\epsilon_1)&:=H_2(t,x,q,\begin{array}r{u}(p,\epsilon_1),\begin{array}r{v}(q,\epsilon_2))\geq H_2(t,x,q,\begin{array}r{u}(p,\epsilon_1),v).
\end{aligned}
\right.
\end{equation}
\begin{equation}gin{remark}
Let us notice that the function $H_1^*$ (resp. $H_2^*$) does not depend on $\epsilon_1$ (resp. $\epsilon_2$) since, $p\begin{array}r{u}(p, \epsilon_1)=p\vee 0$ (resp. $q\begin{array}r{v}(q,\epsilon_2)=|q|$) does not depend on $\epsilon_1$ (resp. $\epsilon_2$). Besides, they are discontinuous w.r.t. $(p,q)$ since $\begin{array}r{v}$ and $\begin{array}r{u}$ are so.
\end{remark}
We next give the main result of this article without proofs for intuitive understanding. All the proofs are given in Subsection \ref{subsec:proofs}.
u^*bsection{Main result}\label{subsec: main result}
As in several papers on the same subject (\cite{karoui and hamadene, H1997, H2014}, etc.), we will adopt the BSDE approach in order to show that this
particular nonzero-sum stochastic differential game has a Nash equilibrium point. For sake of clarity, in this subsection we give the main result and the intermediary ones which we need. We collect all their proofs in the next subsection. To begin with, the following result characterizes the payoffs (\ref{eq: payoffs}) through a solution of a multidimensional BSDE.
\begin{equation}gin{proposition}\label{prop: yuv}
Assume that (\ref{lg}) and (\ref{(A2)}) are satisfied. Then for all $(u,v)\in \mathcal{M}$ and $i=1,2$, there exists a unique pair of $\mathcal{P}$-measurable processes\\ $(Y^{i;x_0;u,v}, Z^{i;x_0;u,v})$, with values in $\textbf{R}\times \textbf{R}$, such that: For $i=1,2$,
\begin{equation}gin{description}
\item[(i)] for all constant $q\geq 1$,
\begin{equation}gin{equation}\label{eq:esti yuv and zuv}
\textbf{E}_{x_0}^{u,v}\Big[u^*p_{s\in [0,T]}|Y_s^{i;x_0;u,v}|^q+(\int_0^T |Z_s^{i;x_0;u,v}|^2ds)^{\frac{q}{2}}\Big]<\infty.
\end{equation}
\item[(ii)]
\begin{equation}gin{equation}\label{eq:BSDE yiuv}\left\{\begin{equation}gin{array}{l}
-dY_t^{i;x_0;u,v}=H_i(s, X_s^{0,x_0},Z_s^{i;x_0;u,v},u_s,v_s)ds - Z_s^{i;x_0;u,v}dB_s,\,\,s\leq T; \\ Y_T^{i;x_0;u,v}=g_i(X_T^{0,x_0}).\end{array}\right.
\end{equation}
\item[]Moreover $Y_0^{i;x_0;u,v}=J_i(u,v)$.\qedh
\end{description}
\end{proposition}
The following result is a verification theorem for the existence of NEP of the game of bang-bang type.
\begin{equation}gin{proposition}\label{prop: equilibrium} Assume that (\ref{lg}) and (\ref{(A2)}) are satisfied.
Besides, suppose that there exist two deterministic functions $\eta^1$, $\eta^2$ and stochastic processes $(Y^1, Z^1)$, $(Y^2, Z^2)$ and $\theta$, $\vartheta$ such that:
\begin{equation}gin{description}
\item[(i) (a)] $\theta$ (resp. $\vartheta$) is a $\mathcal{P}$-measurable process with values in $U$ (resp. $V$) and $(Y^1, Z^1)$ and $(Y^2, Z^2)$ are two couples of $\mathcal{P}$-measurable processes $\textbf{R}^{1+1}$-valued which satisfy:
\item[(b)] for $i=1,2$ $\textbf{P}$-a.s. $(Z^i_s(\omega))_{s\leq T}$ is $ds$-square integrable and for all $s\leq T$,
\begin{equation}gin{equation}\label{eq:main bsde}
\left\{
\begin{equation}gin{aligned}
-dY_s^1&=H_1^*(s, X_s^{0,x_0},Z_s^1, Z_s^2,\vartheta_s)ds- Z_s^1dB_s,\ Y_T^1=g_1(X_T^{0,x_0});\\
-dY_s^2&=H_2^*(s, X_s^{0,x_0},Z_s^1, Z_s^2,\theta_s)ds- Z_s^2dB_s,\ Y_T^2=g_2(X_T^{0,x_0}).
\end{aligned}
\right.
\end{equation}
\item[(ii)] $\eta^1$ and $\eta^2$ are two deterministic measurable functions with polynomial growth from $[0,T]\times \textbf{R}$ to $\textbf{R}$ such that
$\textbf{P}$-a.s., $\forall s\leq T$, $Y_s^i=\eta^i(s, X_s^{0,x_0})$.
\end{description}
Then, the pair of controls $(\begin{array}r{u}(Z_s^1, \theta_s), \begin{array}r{v}(Z_s^2, \vartheta_s))_{s\leq T}$ is a bang-bang type Nash equilibrium point of the nonzero-sum stochastic differential game.\qedh
\end{proposition}
Finally since the diffusion coefficient in equation (\ref{eq: SDE}) is equal to the identity and in using a result by El-Karoui et al. \cite{el karoui} which allows the representation of solutions of BSDEs through deterministic functions, in the markovian case of randomness, we prove the existence of processes and deterministic functions which satisfy the requirements of Proposition \ref{prop: equilibrium}. The main difficulty relies on the discontinuity of the generator $H_1^*$ (resp. $H_2^*$) w.r.t. $(p,q)$ which comes from the discontinuity of $\begin{array}r{v}$ (resp. $\begin{array}r{u}$) on $q=0$ (resp. $p=0$). However we can overcome this difficulty and we have:
\begin{equation}gin{theorem}\label{th: main bsde} Assume that
$f$ and $g_i$, $i=1,2$,
satisfy to (\ref{lg}) and (\ref{(A2)}) respectively. Then there exist $\eta^1$, $\eta^2$, $(Y^1\!,Z^1\!)$, $(Y^2,Z^2)$ and $\theta$, $\vartheta$ which satisfy (i),(a)-(b) and (ii) of Proposition \ref{prop: equilibrium}.
\end{theorem}
As a consequence of Theorem \ref{th: main bsde} and Proposition \ref{prop: equilibrium}, we obtain the main result of this article.
\begin{equation}gin{theorem}\label{th: nash} The pair $(\begin{array}r{u}(Z_s^1, \theta_s), \begin{array}r{v}(Z_s^2, \vartheta_s))_{s\leq T}$ of $\mathcal{M}$ is a bang-bang type Nash equilibrium point for the nonzero-sum stochastic differential game defined by (\ref{eq: SDE}), (\ref{eq:density fun}) and (\ref{eq: payoffs}).
\end{theorem}
u^*bsection{Proofs}\label{subsec:proofs}
u^*bsubsection{Pre-results}
We would like to introduce first two estimates about the process $X^{t,x}$ defined in \eqref{eq: SDE} which will be used in order to prove the above results. They are related to moments of $X^{t,x}$ under the probabilities $\textbf{P}$ and $\textbf{P}^{u,v}$, $(u,v)\in \mathcal{M}$ (see. Karatzas, I.1991 \cite{karatzas}, pp.306). Indeed we have:
\begin{equation}gin{equation}\label{eq: est x}
\forall q\in [1,\infty),\quad \textbf{E}\big[(u^*p_{s\leq T}|X_s^{t,x}|)^{q}\big]\leq C(1+|x|^{q})
\end{equation}
and for any $(u,v)\in \mathcal{M}$
\begin{equation}gin{equation}\label{eq: est x weak}
\forall q\in [1, \infty),\quad \textbf{E}^{u,v}_{t,x}\big[(u^*p_{s\leq T}|X_s^{t,x}|)^{q}\big]\leq C(1+|x|^{q}).
\end{equation}
Finally let us recall the following important result by U.G.Hausmann (see Theorem 2.2, pp.14 \cite{haussmann}) related to the integrability of the exponential local martingale defined by \eqref{eq:density fun}.
\begin{equation}gin{lemma}\label{lemma: haussmann}({\cite{haussmann}, pp.14}) Let
$\Theta$ be a $\mathcal{P}\otimes \cal{B} (\textbf{R})$-measurable application from $[0,T]\times \Omega\times \textbf{R}$ to $\textbf{R}$ which is of uniformly linear growth, that is, $\textbf{P}$-a.s. $\forall(s,x)\in [0,T]\times \textbf{R}$, $|\Theta(s,\omega,x)|\leq C_0(1+|x|)$. Then, there exist constants $p\in (1,2)$ and $C$, where $p$ depends only on $C_0$ while the constant $C$, depends only on $p$, but not on $\Theta$, such that:
\[
\textbf{E}\Big[\Big(\zeta_T\{\Theta(s, X_s^{t,x})\}\Big)^p\Big]\leq C,
\]
where the process $\zeta_T(.)$ is the density function defined by \eqref{eq:density fun}.
\end{lemma}
As a by-product we have:
\begin{equation}gin{corollary} \label{coro: Haussmann}
For any admissible control $(u,v)\in \mathcal{M}$ and $(t,x)\in [0,T]\times \textbf{R}$, there exists a constant $p\in (1,2)$ such that:
\[
\textbf{E}\Big[\Big(\zeta_T\{\Gamma(s, X_s^{t,x},u_s, v_s)\}\Big)^p\Big]\leq C.
\]
\end{corollary}
u^*bsubsection{Proof of Proposition \ref{prop: yuv}}\label{subsubsec: proof prop 1}
We will prove this Proposition by constructing the candidate solution of BSDE \eqref{eq:BSDE yiuv} directly. Then we check by It\^o's formula that, the process defined is exactly the solution we anticipate. In this proof, Corollary \ref{coro: Haussmann} plays an important role. Let us illustrate it for player $\pi_1$. The same can be done for
player $\pi_2$.
For simplicity, only in this proof, we use the notation $(Y^{u,v}, Z^{u,v})$ instead of $(Y^{1;x_0;u,v}, Z^{1;x_0;u,v})$.
For any $(u,v)\in \mathcal{M}$, let us define the process $(Y^{u,v}_s)_{s\leq T}$ as follows:
\begin{equation}gin{align}\label{eq:def yuv}
Y^{u,v}_s:=\textbf{E}^{u,v}_{x_0}[g_1(X_T^{0,x_0})|\mathcal{F}_s],\quad \forall s\leq T.
\end{align}
This process is well defined by noticing that, for any constant $q\geq 1$, we have $\textbf{E}^{u,v}_{x_0}[|g_1(X_T^{0,x_0})|^{q}]\leq C\textbf{E}^{u,v}_{x_0}[C(1+u^*p_{s\leq T}|X_s^{0,x_0}|^{q\gamma})]<\infty$ which is obtained by (\ref{(A2)}) and \eqref{eq: est x weak}. For writing convenience, we denote by $\zeta_s$, the function\\ $\zeta_s(\Gamma(., X_.^{0,x_0},u_.,v_.))$ as mentioned in \eqref{eq:density fun}. Therefore \eqref{eq:def yuv} can be transformed into:
\begin{equation}gin{flushleft}
$\qquad \qquad
Y^{u,v}_s=\zeta_s^{-1}\textbf{E}[\zeta_T\cdot g_1(X_T^{0,x_0})|\mathcal{F}_s],\quad \forall s\leq T.
$
\end{flushleft}
In the following, we show that $\zeta_T\cdot g_1(X_T^{0,x_0})\in L^{\begin{array}r{q}}$ for some $\begin{array}r{q}\in (1,2)$. Indeed, according to Corollary \ref{coro: Haussmann}, there exists some $p_0\in (1,2)$, such that $\zeta_T\in L^{p_0}(d\textbf{P})$. Therefore, for any $\begin{array}r{q}\in (1,p_0)$, Young's inequality leads to:
\begin{equation}gin{flushleft}
$\qquad \qquad
\textbf{E}[|\zeta_T\cdot g_1(X_T^{0,x_0})|^{\begin{array}r{q}}]\leq \frac{\begin{array}r{q}}{p_0}\textbf{E}[|\zeta_T|^{p_0}]+\frac{p_0-\begin{array}r{q}}{p_0}\textbf{E}[|g_1(X_T^{0,x_0})|^{\begin{array}r{q}\cdot \frac{p_0}{p_0-\begin{array}r{q}}}],
$
\end{flushleft}
which is obviously finite by the polynomial growth of $g_1$ and \eqref{eq: est x}. Therefore the process $Y^{u,v}$ is defined. On the other hand, by Doob's inequality, (\ref{(A2)}) and estimate (\ref{eq: est x weak}) we have:
\begin{equation}\label{estimyuv}
\forall q>1, \textbf{E}^{u,v}_{x_0}[u^*p_{s\leq T}|Y^{u,v}_s|^q]\leq C\textbf{E}^{u,v}_{x_0}[|g_1(X_T^{0,x_0})|^q]\leq C(1+|x_0|^q).
\end{equation}
Next thanks to representation Theorem of martingales (\cite{revuzyor}, pp.199) applied to the process $(\textbf{E}[\zeta_T\cdot g_1(X_T^{0,x_0})|\mathcal{F}_s])_{s\le T}$, there exists a $\mathcal{P}$-measurable and $\textbf{R}$-valued process $(\Delta_s)_{s\leq T}$ which satisfies
$
\textbf{E}[(\int_0^T|\Delta_s|^2ds)^{\frac{\begin{array}r{q}}{2}}]<\infty
$
and
\begin{equation}gin{flushleft}
$\qquad \qquad
Y^{u,v}_s=\zeta_s^{-1}\{\textbf{E}[\zeta_T\cdot g_1(X_T^{0,x_0})]+\int_0^s \Delta_rdB_r\}:=\zeta_s^{-1}R_s, \quad \forall s\leq T,
$
\end{flushleft}
with $R_s:=\textbf{E}[\zeta_T\cdot g_1(X_T^{0,x_0})|\mathcal{F}_s]=\textbf{E}[\zeta_T\cdot g_1(X_T^{0,x_0})]+\int_0^s \Delta_rdB_r$. Next let us set
\begin{equation}gin{equation}\label{eq:def zuv}
Z^{u,v}_s:=-\zeta_s^{-1}\big\{R_s\Gamma(s, X_s^{0,x_0},u_s,v_s)-\Delta_s\big\}, \,s\leq T.
\end{equation}
As for any $s\in [0,T]$, $$
d\zeta_s=\zeta_s\cdot \Gamma(s, X_s^{0,x_0},u_s,v_s)dB_s$$ then by It\^o's formula applied to $\zeta^{-1}R$, one can easily check that $(Y^{u,v},Z^{u,v})$ verifies (\ref{eq:BSDE yiuv}) for $i=1$. The same happens for $i=2$.
\ms
Now for any $(u,v)\in \mathcal{M}$ and $s\leq T$, if $(B^{u,v}_s)_{s\leq T}$ is the Brownian motion under $\textbf{P}^{u,v}_{x_0}$, we then deduce from (\ref{eq:BSDE yiuv}) that
$$
-dY^{i;x_0;u,v}_s=-Z^{i;x_0;u,v}_sdB^{u,v}_s,\,\;Y^{i;x_0;u,v}_T=g_i(X^{0,x_0}_T)
$$ and $Y^{i;x_0;u,v}_0=\textbf{E}^{u,v}_{x_0}[g_i(X^{0,x_0}_T]=J_i(u,v)$ since $\mathcal{F}_0$ is the trivial $\sigma$-algebra completed with the $\textbf{P}$-null sets of $\mathcal{F}$ and taking into account that $\textbf{P}$ and $\textbf{P}^{u,v}_{x_0}$ are equivalent probabilities.
Therefore taking into account of (\ref{estimyuv}) and using the Burkholder-Davis-Gundy inequality we have
$$
\forall q>1, \textbf{E}^{u,v}_{x_0}[(\int_0^T|Z^{i;{x_0};u,v}_r|^2dr)^{\frac{q}{2}}]<\infty.$$
This and (\ref{estimyuv}) imply the estimate (\ref{eq:esti yuv and zuv}) of Proposition \ref{prop: yuv} for $q>1$. Finally for $q=1$, (\ref{eq:esti yuv and zuv}) is obviously true since it is valid for any $q>1$.
The proof of the Proposition \ref{prop: yuv} is completed.\qedh
u^*bsubsection{Proof of Proposition \ref{prop: equilibrium}}
For $s\leq T$, let us set $\begin{array}r{u}_s=\begin{array}r{u}(Z_s^1,\theta_s)$ and $\begin{array}r{v}_s=\begin{array}r{v}(Z_s^2,\vartheta_s)$, then $(\begin{array}r{u},\begin{array}r{v})\in \mathcal{M}$. On the other hand, thanks to Proposition \ref{prop: yuv}, we obviously have $Y_0^1=J_1(\begin{array}r{u},\begin{array}r{v})$.
Next let $u$ be an arbitrary element of $\mathcal{M}_1$ and let us show that $Y^1\geq Y^{1;x_0;u,\begin{array}r{v}}$, which yields $Y^1_0=J_1(\begin{array}r{u},\begin{array}r{v})\geq Y^{1;x_0;u, \begin{array}r{v}}_0=J^1(u, \begin{array}r{v})$.
The control $(u, \begin{array}r{v})$ is admissible and thanks to Proposition \ref{prop: yuv}, there exists a pair of $\mathcal{P}$-measurable processes $(Y^{1;x_0;u,\begin{array}r{v}},Z^{1;x_0;u,\begin{array}r{v}})$ such that for any $q>1$,
\begin{equation}gin{equation}\label{eq:bsde yuvb}
\left\{
\begin{equation}gin{aligned}
&\textbf{E}^{u,\begin{array}r{v}}_{x_0}\Big[u^*p_{0\leq s\leq T}|Y_s^{1;x_0;u,\begin{array}r{v}}|^q+\big(\int_0^T|Z_s^{1;x_0;u,\begin{array}r{v}}|^2ds\big)^{\frac{q}{2}}\Big]<\infty\,;\\
&Y_s^{1;x_0;u,\begin{array}r{v}}=g_1(X_T^{0,x_0})+\int_s^T\! H_1(r, X^{0,x_0}_r, Z_r^{1;x_0;u,\begin{array}r{v}}, u_r, \begin{array}r{v}_r)dr-\int_s^T\! Z_r^{1;x_0;u,\begin{array}r{v}}dB_r,\\
&\qquad \qquad \qquad \forall s\leq T.
\end{aligned}
\right.
\end{equation}
\noindentindent Afterwards, we aim to compare $Y^1$ in \eqref{eq:main bsde} and $Y^{1;x_0;u,\begin{array}r{v}}$ in \eqref{eq:bsde yuvb}. So let us denote by
\begin{equation}gin{equation*}
\triangle Y= Y^{1;x_0;u, \begin{array}r{v}}-Y^1 \mbox{ and }\triangle Z= Z^{1;x_0;u,\begin{array}r{v}}-Z^1.
\end{equation*}
\noindentindent For $k\geq 0$, we define the stopping time $\tau_k$ as follows:
\begin{equation}gin{center}
$
\tau_k:= \inf\{s\geq 0, |\triangle
Y_s|+\int_0^s|\triangle Z_r|^2dr\geq k\}\wedge T.
$
\end{center}
\noindent The sequence of stopping times $(\tau_k)_{k\geq 0}$ is of stationary type and converges to $T$. Next applying It\^{o}-Meyer formula to $|(\triangle Y)^+|^q$ $(q>1)$ (see Theorem 71, P. Protter, \cite{protter}, pp.221), between $s\wedge \tau_k$ and $\tau_k$, we obtain: $\forall s\leq T$,
\begin{equation}gin{equation}\label{eq:tem 3.17}
\begin{equation}gin{aligned}
|(\triangle &Y_{s\wedge \tau_k})^+|^q+c(q)\int_{s\wedge \tau_k}^{\tau_k}|(\triangle Y_r)^+|^{q-2}1_{\triangle Y_r>0}|\triangle Z_r|^2dr\\
&= |(\triangle Y_{\tau_k})^+|^q+ q\int_{s\wedge
\tau_k}^{\tau_k}|(\triangle Y_{r})^+|^{q-1}
1_{\triangle Y_r>0} \big[H_1(r, X_r^{0,x_0}, Z_r^{1;x_0;u, \begin{array}r{v}}, u_r, \begin{array}r{v}_r)-\\
&\qquad H_1(r, X_r^{0,x_0}, Z_r^1, \begin{array}r{u}_r,
\begin{array}r{v}_r)\big]dr -
q\int_{s\wedge \tau_k}^{\tau_k}|(\triangle
Y_{r})^+|^{q-1}1_{\triangle Y_r>0}\triangle Z_rdB_r
\end{aligned}
\end{equation}
where $c(q)=\frac{1}{2}q(q-1)$. Besides for any $s\le T$,
\begin{equation}gin{align*}
H_1(s, X_s^{0,x_0}, Z_s^{1;x_0;u, \begin{array}r{v}}, u_s, \begin{array}r{v}_s)&- H_1(s, X_s^{0,x}, Z_s^1, \begin{array}r{u}_s,
\begin{array}r{v}_s)=\\
&H_1(s, X_s^{0,x_0}, Z_s^{1;x_0;u, \begin{array}r{v}}, u_s, \begin{array}r{v}_s)- H_1(s,X_s^{0,x_0},Z_s^1, u_s, \begin{array}r{v}_s) \\
&+H_1(s, X_s^{0,x_0}, Z_s^1, u_s, \begin{array}r{v}_s)-H_1(s, X_s^{0,x_0}, Z_s^1, \begin{array}r{u}_s, \begin{array}r{v}_s)
\end{align*}
\noindent Considering now the generalized Isaacs' condition \eqref{eq:Isaacs}, we have $$H_1(s, X_s^{0,x_0}, Z_s^1, u_s, \begin{array}r{v}_s)- H_1(s,X_s^{0,x_0},Z_s^1, \begin{array}r{u}_s, \begin{array}r{v}_s)\leq 0,\quad \forall s\leq T.$$Additionally
$$H_1(s, X_s^{0,x_0}, Z_s^{1;x_0;u, \begin{array}r{v}}, u_s, \begin{array}r{v}_s)-H_1(s, X_s^{0,x_0}, Z_s^1, u_s, \begin{array}r{v}_s)=\triangle Z_s\Gamma(s,X_s^{0,x_0},u_s,\begin{array}r{v}_s).$$Thus equation \eqref{eq:tem 3.17} can be simplified into: $\forall s\in [0,T]$,
\begin{equation}gin{equation}
\begin{equation}gin{aligned}
|(\triangle &Y_{s\wedge \tau_k})^+|^q+c(q)\int_{s\wedge \tau_k}^{\tau_k}|(\triangle Y_r)^+|^{q-2}1_{\triangle Y_r>0}|\triangle Z_r|^2dr\\
&\leq |(\triangle Y_{\tau_k})^+|^q + q\int_{s\wedge
\tau_k}^{\tau_k}|(\triangle Y_{r})^+|^{q-1}
\mathbbm{1}_{\triangle Y_r>0} \triangle Z_r \Gamma(r, X_r^{0,x_0}, u_r, \begin{array}r{v}_r)dr\\
&\quad - q\int_{s\wedge \tau_k}^{\tau_k}|(\triangle Y_{r})^+|^{q-1}1_{\triangle Y_r>0}\triangle Z_rdB_r\noindentnumber\\
&=|(\triangle Y_{\tau_k})^+|^q-q\int_{s\wedge \tau_k}^{\tau_k}|(\triangle
Y_{r})^+|^{q-1}1_{\triangle Y_r>0}\triangle Z_rdB_r^{u,\begin{array}r{v}},
\end{aligned}
\end{equation}
where $B^{u, \begin{array}r{v}}= (B_s- \int_0^{s} \Gamma(r, X_r^{0,x_0}, u_r, \begin{array}r{v}_r)dr)_{s\leq T}$ is an $(\mathcal{F}_s, \textbf{P}^{u, \begin{array}r{v}}_{x_0})$-Brownian motion. Then for any $s\leq T$,
\begin{equation}gin{equation*}
\begin{equation}gin{aligned}
|(\triangle Y_{s\wedge \tau_k})^+|^q\leq |(\triangle Y_{\tau_k})^+|^q- q\int_{s\wedge \tau_k}^{\tau_k}|(\triangle Y_{r})^+|^{q-1}1_{\triangle Y_r>0}\triangle Z_rdB_r^{u, \begin{array}r{v}}.
\end{aligned}
\end{equation*}
\noindentindent By definition of the stopping time $\tau_k$, we have
\begin{equation}gin{center}
$
\textbf{E}^{u, \begin{array}r{v}}_{x_0}\big[\int_{s\wedge \tau_k}^{\tau_k}|(\triangle Y_r)^+|^{q-1}1_{\triangle Y_r>0}\triangle Z_rdB_r^{u,\begin{array}r{v}}\big]=0.
$
\end{center}
Thus for any $s\le T$,
\begin{equation}gin{align}\label{eq:tem 3.18}
\textbf{E}^{u, \begin{array}r{v}}_{x_0}\left[|(\triangle Y_{s\wedge \tau_k})^+|^q\right] &\leq \textbf{E}_{x_0}^{u, \begin{array}r{v}}\big[|(Y^{1;x_0;u,\begin{array}r{v}}_{\tau_k}-Y_{\tau_k}^1)^+|^q\big].
\end{align}
Next taking into account \eqref{eq: est x weak} and the fact that $Y^1$ has a representation through $X^{0,x_0}$ and $\eta^1(s,y)$, $(s,y)\in \sp$, a deterministic function with polynomial growth, we have
\begin{equation}gin{equation}\label{eq:uni int}
\textbf{E}^{u, \begin{array}r{v}}_{x_0}\Big[u^*p_{s\leq T}(|Y^1_s|+|Y^{1;x_0;u,\begin{array}r{v}}_s|)^q\Big]<\infty\end{equation}
As the sequence $((Y_{\tau_k}^{1;x_0;u,\begin{array}r{v}}-Y^1_{\tau_k})^+)_{k\geq 0}$
converges to $0$ when $k\rightarrow \infty$, $\textbf{P}_{x_0}^{u,\begin{array}r{v}}$-a.s., since $\lim_{k\rightarrow \infty} Y_{\tau_k}^{1;x_0;u,\begin{array}r{v}}=\lim_{k\rightarrow \infty}Y_{\tau_k}^1=g_1(X_T^{0,x})\ \textbf{P}_{x_0}^{u,\begin{array}r{v}}$-a.s.. Then it converges also to $0$ in $L^1(d\textbf{P}_{x_0}^{u,\begin{array}r{v}})$ thanks to \eqref{eq:uni int}. Take now $k\rightarrow \infty$ on \eqref{eq:tem 3.18}, it follows from Fatou's Lemma that:
$$
\textbf{E}^{u, \begin{array}r{v}}_{x_0}\left[(\triangle Y_{s})^+\right]=0,\,\,\forall s\leq T,
$$
which implies that $Y^1\geq Y^{1;x_0;u, \begin{array}r{v}}$, $\textbf{P}$-a.s., since the probabilities $\textbf{P}_{x_0}^{u,\begin{array}r{v}}$ and $\textbf{P}$ are equivalent. Thus
$Y^1_0=J^1(\begin{array}r{u}, \begin{array}r{v})\geq Y^{1;x_0;u, \begin{array}r{v}}_0=J^1(u, \begin{array}r{v})$.
Similarly, we can show that, $Y^2_0=J^2(\begin{array}r{u}, \begin{array}r{v})\geq Y^{2;x_0;\begin{array}r{u}, v}_0=J^2(\begin{array}r{u},v)$ for arbitrary $v\in \mathcal{M}_2$. Henceforth $(\begin{array}r{u},\begin{array}r{v})$ is a Nash equilibrium point for the NZSDG. \qedh
u^*bsubsection{Proof of Theorem \ref{th: main bsde}}\label{subsubsec: proof th}
The proof will be split into several steps. Firstly, we construct an approximating sequence of BSDEs with continuous Lipschitz generators by smoothing the functions $\begin{array}r{u}$ and $\begin{array}r{v}$. Next we provide appropriate uniform estimates of the solutions of the approximating scheme. Finally we show that the approximating scheme contains at least a convergent subsequence which provides the stochastic processes and deterministic functions verifying the requirements of Proposition \ref{prop: equilibrium}.
\noindent \underline{\textit{Step 1}}: Approximating scheme.
\ms
At the beginning of this proof, we would like to clarify that the functions $p\in \textbf{R}\mapsto p\begin{array}r{u}(p,\epsilon_1)$ and $q\in \textbf{R}\mapsto q\begin{array}r{v}(q,\epsilon_2)$ are uniformly Lipschitz for any $\epsilon_1$ and $\epsilon_2$, since $p\begin{array}r{u}(p,\epsilon_1)=p\begin{array}r{u}(p,0)=u^*p_{u\in U}pu$ and $q\begin{array}r{v}(q,\epsilon_2)=q\begin{array}r{v}(q,0)=u^*p_{v\in V}qv$. Hereafter $\begin{array}r{u}(p,0)$ (resp. $\begin{array}r{v}(q,0)$) will be simply denoted by $\begin{array}r{u}(p)$ (resp. $\begin{array}r{v}(q)$).
\noindent Next for integer $n\ge 1$, let $\begin{array}r{u}^n$ and $\begin{array}r{v}^n$ be the functions defined as follows:
\begin{equation}gin{equation*}
\begin{array}r{u}^{n}(p)=\left\{
\begin{equation}gin{aligned}
0&\mbox{ if}\ \ p\leq -1/n,\\
1&\mbox{ if}\ \ p\geq 0,\\
np+1&\mbox{ if}\ \ p\in (-1/n,0),
\end{aligned}
\right. \mbox{ and }
\quad
\begin{array}r{v}^{n}(q)=\left\{
\begin{equation}gin{aligned}
-1&\mbox{ if}\ \ q\leq -1/n,\\
1&\mbox{ if}\ \ q\geq 1/n,\\
nq&\mbox{ if}\ \ q\in (-1/n,1/n).
\end{aligned}
\right.
\end{equation*}
Note that $\begin{array}r{u}^n$ and $\begin{array}r{v}^n$ are Lipschitz in $p$ and $q$ respectively. Roughly speaking, they are the approximations of $\begin{array}r{u}$ and $\begin{array}r{v}$. Below, let $\Phi_n$ be the truncation function $x\in \textbf{R}\mapsto \Phi_n(x)=(x\wedge n)\vee(-n) \in \textbf{R}$, which is bounded by $n$. Now for $(t,x)\in \sp$ and $n\geq 1$, we consider the following BSDE of dimension two, with Lipschitz generator: For any $s\leq T$,
\begin{equation}gin{equation}\label{eq:bsdeapprox}
\left\{
\begin{equation}gin{aligned}
-dY_s^{1,n;t,x}&= \{\Phi_n(Z^{1,n;t,x}_r)\Phi_n(f(r,X_r^{t,x}))+
\Phi_n(Z^{1,n;t,x}_r\begin{array}r{u}(Z^{1,n;t,x}_r))+\\
&\Phi_n(Z^{1,n;t,x}_r)\begin{array}r{v}^n(Z^{2,n;t,x}_r)\}dr-Z_r^{1,n;t,x}dB_r ,\ Y_T^{1,n;t,x}=g_1(X_T^{t,x});\\
-dY_s^{2,n;t,x}&=\{\Phi_n(Z^{2,n;t,x}_r)\Phi_n(f(r,X_r^{t,x}))+
\Phi_n(Z^{2,n;t,x}_r\begin{array}r{v}(Z^{2,n;t,x}_r))+\\
& \Phi_n(Z^{2,n;t,x}_r)\begin{array}r{u}^n(Z^{1,n;t,x}_r)\}dr-Z_r^{2,n;t,x}dB_r,\ Y_T^{2,n;t,x}=g_2(X_T^{t,x}).
\end{aligned}
\right.
\end{equation}
From Pardoux-Peng's result (\cite{pardoux peng}), for any $n\ge 1$, this equation has a unique solution $(Y^{i,n;t,x}, Z^{i,n;t,x})\in \mathcal{S}_T^2(\textbf{R})\times \mathcal{H}_T^2(\textbf{R})$, $i=1,2$. Taking account of the result by El-Karoui et al.(\cite{el karoui}, pp.46, Theorem 4.1), there exist measurable deterministic functions $\eta^{i,n}$ and $\varsigma^{i,n}$, $i=1,2$ and $n\ge 1$, defined on $\sp$ and $\textbf{R}$-valued such that:
\begin{equation}gin{equation}\label{eq:eta in varsigma in}
\forall s\in [t,T],\,\,Y_s^{i,n;t,x}=\eta^{i,n}(s, X_s^{t,x})\quad \text{and}\quad Z_s^{i,n;t,x}=\varsigma^{i,n}(s, X_s^{t,x}).
\end{equation}
Moreover, for $n\geq 1$ and $i=1,2,$ the functions $\eta^{i,n}$ verify:
\begin{equation}gin{equation}\label{eq:eta n}\forall (t,x) \in [0,T]\times \textbf{R},\,\,
\eta^{i,n}(t,x)=\textbf{E}[g_i(X_T^{t,x})]+\int_t^T H_i^n(r, X_r^{t,x})dr]
\end{equation}
with, for any $(s,x)\in [0,T]\times \textbf{R}$,
\begin{equation}gin{equation}\label{eq:Hn}
\left\{
\begin{equation}gin{aligned}
H_1^n(s,x)&=\Phi_n(\varsigma^{1,n}(s,x))\Phi_n(f(s,x))+
\Phi_n\big(\varsigma^{1,n}(s,x)\begin{array}r{u}(\varsigma^{1,n}(s,x))\big)+\\
&\quad + \Phi_n(\varsigma^{1,n}(s,x))\begin{array}r{v}^n(\varsigma^{2,n}(s,x));\\
H_2^n(s,x)&=\Phi_n(\varsigma^{2,n}(s,x))\Phi_n(f(s,x))+
\Phi_n\big(\varsigma^{2,n}(s,x)\begin{array}r{v}(\varsigma^{2,n}(s,x))\big)+\\
&\quad +\Phi_n(\varsigma^{2,n}(s,x))\begin{array}r{u}^n(\varsigma^{1,n}(s,x)).
\end{aligned}
\right.
\end{equation}
\noindent \underline{\textit{Step 2}}: Estimates for processes $(Y^{i,n;t,x}, Z^{i,n;t,x}),i=1,2$.
\ms
In order to show the needed uniform estimates for $Y^{i,n;t,x}$ of BSDE \eqref{eq:bsdeapprox}, we use comparison. For that let us consider the following BSDE: For $i=1,2$ and any $s\in[0,T]$,
\begin{equation}gin{equation}\label{eq: bar yin}
\begin{array}r{Y}_s^{i,n}=g_i(X_T^{t,x})+\int_s^T \{\Phi_n(C(1+|X_r^{t,x}|))|\begin{array}r{Z}_r^{i,n}|+C|\begin{array}r{Z}_r^{i,n}|\}dr-\int_s^T \begin{array}r{Z}_r^{i,n}dB_r,
\end{equation}
\noindent where the constant $C$ is the one such that the generators
$(H_i^n(s, X_s^{t,x}))_{s\le T}$ satisfy
\begin{equation}\label{compourestim}\forall s\le T,\,\,|H_i^n(s, X_s^{t,x})|\leq \Phi_n(C(1+|X_s^{t,x}|))|Z_s^{i,n;t,x}|+C|Z_s^{i,n;t,x}|.\end{equation}
This constant exists since $f$ is of linear growth and $\begin{array}r{u}$, $\begin{array}r{v}$, $\begin{array}r{u}^n$ and $\begin{array}r{v}^n$ are uniformly bounded. Observing now that the application $z\in \textbf{R}\mapsto \Phi_n(C(1+|X_r^{t,x}|))|z|+C|z|$ is Lipschitz continuous, therefore the solution $(\begin{array}r{Y}^{i,n},\begin{array}r{Z}^{i,n})$ of the above BSDE (\ref{eq: bar yin}) exists in the space $\mathcal{S}_T^2(\textbf{R})\times \mathcal{H}_T^2(\textbf{R})$ and is unique. Note that we have omitted the dependence w.r.t $(t,x)$ of $(\begin{array}r{Y}^{i,n},\begin{array}r{Z}^{i,n})$ in order to alleviate notations as there is no possible confusion. On the other hand by the standard comparison theorem of solutions of BSDEs
(\cite{el karoui}, pp.46, Theorem 4.1) one has
\begin{equation}\label{comp2}
\begin{array}r Y^{i,n}\geq Y^{i,n;t,x}, \textbf{P}-a.s.\end{equation}
Next provided that we show uniform estimates, w.r.t. $n$, for $\begin{array}r{Y}^{i,n}$, then estimates for $Y^{i,n;t,x}$ will be an immediate consequence. Below, we will focus on the properties of $\begin{array}r{Y}^{i,n}$.
Using again the result by El Karoui et al. yields that, there exist deterministic measurable functions $\begin{array}r{\eta}^{i,n}:[0,T]\times \textbf{R}\rightarrow \textbf{R}$ such that, for any $s\in [t,T]$,
\begin{equation}gin{equation}\label{eq: bar eta in}
\begin{array}r{Y}_s^{i,n}=\begin{array}r{\eta}^{i,n}(s, X_s^{t,x}),\ i=1,2.
\end{equation}
Next let us consider the process\\ $B^{i,n}=(B_s-\int_0^s[\Phi_n(C(1+|X_r^{t,x}|))+C]\text{sign}(\begin{array}r Z_r^{i,n})dr)_{s\leq T}$, $i=1,2$, which is, thanks to Girsanov's Theorem, a Brownian motion under the probability $\textbf{P}^{i,n}$ on $(\Omega, \mathcal{F})$ whose density with respect to $\textbf{P}$ is $\zeta_T\{[\Phi_n(C(1+|X_s^{t,x}|))+C]\text{sign}(\begin{array}r Z_s^{i,n})\}$ where for any $z\in \textbf{R}, \text{sign}(z)=1_{\{|z|\neq 0\}}\frac{z}{|z|}$ and $\zeta_T(.)$ is defined by \eqref{eq:density fun}. Then the BSDE \eqref{eq: bar yin} will be simplified into,
\[
\begin{array}r{Y}_s^{i,n}=g_i(X_T^{t,x})-\int_s^T \begin{array}r{Z}_r^{i,n}dB_r^{i,n},\ s\le T,\ i=1,2.
\]
\noindent In view of \eqref{eq: bar eta in}, we obtain,
\[
\begin{array}r{\eta}^{i,n}(t,x)=\textbf{E}^{i,n}[g_i(X_T^{t,x})|\mathcal{F}_t],\ i=1,2,
\]
\noindent where $\textbf{E}^{i,n}$ is the expectation under probability $\textbf{P}^{i,n}$. By taking the expectation on both sides of the above equation under the probability $\textbf{P}^{i,n}$ and considering $\begin{array}r{\eta}^{i,n}(t,x)$ is deterministic, we arrive at,
\[
\begin{array}r{\eta}^{i,n}(t,x)=\textbf{E}^{i,n}[g_i(X_T^{t,x})],\ i=1,2.
\]
As the functions $g_i$, $i=1,2$, verify the polynomial growth condition (\ref{(A2)})
and for any $s\leq T$,
$$
|\{\Phi_n(C(1+|X_s^{t,x}|))+C\}\text{sign}(\begin{array}r Z_s^{i,n})|\le \begin{array}r C(1+|X^{t,x}_s|$$
then by estimate \eqref{eq: est x weak} we have,
for some constant $\lambda \geq 0$ and $C$ a constant which does not depend on $n$,
$$
|\begin{array}r{\eta}^{i,n}(t,x)|\leq C(1+|x|^{\lambda}).
$$ Therefore by (\ref{comp2}) we obtain $\eta^{i,n}(t,x)\le C(1+|x|^\lambda)$, for any $(t,x)\in \sp$.
In a similar way, we can show that $\eta^{i,n}(t,x)\geq -C(1+|x|^{\lambda})$, $(t,x)\in[0,T]\times \textbf{R}$. Therefore, $\eta^{i,n},\ i=1,2$ are of polynomial growth with respect to $(t,x)$ uniformly in $n$.
To conclude this step, we have the following results: There exists a constant $C$ independent of $n$ and $t,x$ such that, for $(t,x)\in [0,T]\times \textbf{R}$, $i=1,2,$
\begin{equation}gin{equation}\label{eq:rst step 2}
\left\{
\begin{equation}gin{aligned}
&\text{(a) } |\eta^{i,n}(t,x)|\leq C(1+|x|^{\lambda}),\text{ for any }\lambda\geq 0;\\
&\text{(b) by the combination of (a), \eqref{eq: est x} with \eqref{eq:eta in varsigma in}, it holds: }\\&\qquad \forall \alpha\geq 1,\,\,
\textbf{E}[u^*p_{s\in[t,T]}|Y^{i,n;t,x}_s|^{\alpha}]\leq C\,;\\
&\text{(c) } \text{for any } (t,x)\in \sp,\ \textbf{E}[\int_t^T |Z_s^{i,n;t,x}|^2ds]\leq C \text{ which is a }\\
&\qquad \text{straightforward result by using It\^o's formula with } (Y^{i,n;t,x})^2 \mbox{ and using (b)}.
\end{aligned}
\right.
\end{equation}
\noindent \underline{\textit{Step 3}}: Convergence of a subsequence of $(Y^{i,n;0,x},Z^{i,n;0,x})_{n\geq 1}$, $i=1,2$.
\ms
First let us define on $\textbf{R}$ the measure $\mu(0,x_0;s,dy)$ as the law under $\textbf{P}$ of $X_s^{0,x_0}$, i.e., $\mu(0,x_0;s,dy):=\textbf{P}(X^{0,x_0}_s\in dy)
=\frac{1}{\sqrt{2\pi s}}e^{-\frac{(y-x_0)^2}{2s}}dy$ for any $s\in (0,T]$.
Let $q\in (1,2)$ be fixed. We are going to show that the sequence $(H_i^n(s, y))_{n\ge 1}$ belongs to $L^q([0,T]\times \textbf{R}; \mu(0,x_0;s,dy)ds)$, $i=1,2$. Actually,
\begin{equation}gin{align}\label{eq: Hin in lq}
\textbf{E}[\int_0^T|H_i^n(s,X_s^{0,x_0})|^q ds]
&=\int_{[0,T]\times \textbf{R}}|H_i^n(s,y)|^q\mu(0,x_0;s,dy)ds\noindentnumber\\
&\leq C\textbf{E}[\int_0^T|Z^{i,n;0,x_0}_s|^q(1+|X_s^{0,x}|^q)ds]\noindentnumber\\
&\leq C\{\textbf{E}[\int_0^T|Z_s^{i,n;0,x_0}|^2ds]+\textbf{E}[1+u^*p_{s\in[0,T]}|X_s^{0,x_0}|^{\frac{2q}{2-q}}]\}\le C\noindentnumber\\
\end{align}
($C$ is a generic constant whose value may change from line to line). The last inequality is obtained from the fact that $\textbf{E}[\int_0^T |Z_s^{i,n;0,x_0}|^2 ds]\leq C$ and estimate \eqref{eq: est x}. As a result, there exists a subsequence $\{n_k\}$ (still denoted by $\{n\}$ for simplification) and two $\mathcal{B}([0,T]\times \textbf{R})$-measurable deterministic functions $H_i(s,y)$, $i=1,2$, such that,
\begin{equation}gin{equation}\label{eq:Hin weak conv}
H_i^n\rightharpoonup H_i \text{ weakly in } L^q([0,T]\times \textbf{R}; \mu(0,x_0;s,dy)ds).
\end{equation}
Next we focus on passing from the weak convergence to strong sense convergence by proving that $(\eta^{i,n}(t,x))_{n\geq 1}$ defined in \eqref{eq:eta n} is a Cauchy sequence for each $(t,x)\in [0,T]\times \textbf{R}$, $i=1,2.$ Let $(t,x)$ be fixed in $[0,T)\times \textbf{R}$ (w.l.o.g we assume $t<T$), $\delta>0, k,n$ and $m\geq 1$ be integers. From \eqref{eq:eta n}, we have,
\begin{equation}gin{align}\label{eq:eta in - eta im}
|\eta^{i,n}(t,x)-\eta^{i,m}(t,x)|&= \big|\textbf{E}[\int_t^T H_i^n(s, X_s^{t,x})-H_i^m(s,X_s^{t,x})ds]\big|\noindentnumber\\
&\leq \big|\textbf{E}[\int_t^{t+\delta} H_i^n(s, X_s^{t,x})-H_i^m(s,X_s^{t,x})ds]\big|\noindentnumber\\
&\quad +\big|\textbf{E}[\int_{t+\delta}^T (H_i^n(s, X_s^{t,x})-H_i^m(s,X_s^{t,x}))\cdot 1_{\{|X_s^{t,x}|\leq k\}}ds]\big|\noindentnumber\\
&\quad +\big|\textbf{E}[\int_{t+\delta}^T (H_i^n(s, X_s^{t,x})-H_i^m(s,X_s^{t,x}))\cdot 1_{\{|X_s^{t,x}|> k\}}ds]\big|.\noindentnumber\\
&
\end{align}
We first deal with the first term of the right-hand side of \eqref{eq:eta in - eta im}. By H\"older's inequality, definition of $H_i^n(s, X_s^{t,x})$ and (\ref{eq:rst step 2}) we have$$\begin{equation}gin{array}{ll}
\big|\textbf{E}[\int_t^{t+\delta}\!\!\! H_i^n(s, X_s^{t,x})-\!\!\!{}&\!\!\!H_i^m(s,X_s^{t,x})ds]\big|\\ &\leq\!
\textbf{E}[\int_t^{t+\delta}| H_i^n(s, X_s^{t,x})-H_i^m(s,X_s^{t,x})|ds] \\{}&\leq
C\textbf{E}[\int_t^{t+\delta}|Z^{i,n;t,x}_s|(1+|X^{t,x}_s|)ds]\\{}&\leq C\delta^{\frac{1}{4}}\{\textbf{E}[\int_t^T\!\! |Z^{i,n;t,x}_s|^2ds]\}^{\frac{1}{2}}\{\textbf{E}[\int_t^T\!\! (1+|X^{t,x}_s|)^4ds]\}^{\frac{1}{4}} \\{}& \leq C\delta^{\frac{1}{4}}.
\end{array}
$$
We now focus on the third term of the right hand-side of inequality \eqref{eq:eta in - eta im}. In the same way as previous we have:
$$\begin{equation}gin{array}{ll}
\big|\textbf{E}&[\int_{t+\delta}^T (H_i^n(s, X_s^{t,x})-H_i^m(s,X_s^{t,x}))\cdot 1_{\{|X_s^{t,x}|> k\}}ds]\big|\\ &\leq \textbf{E}[\int_{t+\delta}^T |H_i^n(s, X_s^{t,x})-H_i^m(s,X_s^{t,x})|\cdot 1_{\{|X_s^{t,x}|> k\}}ds]\\
&\leq C\{\textbf{E}[\int_{t+\delta}^T1_{\{|X_s^{t,x}|>k\}}ds]\}^{\frac{1}{4}}
\{\textbf{E}[\int_t^T\!\! |Z^{i,n;t,x}_s|^2ds]\}^{\frac{1}{2}}\{\textbf{E}[\int_t^T\!\! (1+|X^{t,x}_s|)^4ds]\}^{\frac{1}{4}} \\
&\leq Ck^{-\frac{1}{4}}
\end{array}
$$
by using Markov inequality. Finally we deal with the second term of the right-hand side of \eqref{eq:eta in - eta im}. By using the law of $X^{t,x}_s$ we have
\begin{equation}gin{align}\label{eq:sec term}
& \textbf{E}[\int_{t+\delta}^T (H_i^n(s, X_s^{t,x})-H_i^m(s,X_s^{t,x}))\cdot 1_{\{|X_s^{t,x}|\leq k\}}ds]\noindentnumber\\&=
\int_{\textbf{R}}\int_{t+\delta}^T (H_i^n(s,y)-H_i^m(s,y))1_{\{|y|\leq k\}}\frac{1}{\sqrt{2\pi(s-t)}}e^{-\frac{(y-x)^2}{2(s-t)}}dsdy\noindentnumber\\
&=\int_{\textbf{R}}\int_{t+\delta}^T (H_i^n(s,y)-H_i^m(s,y))1_{\{|y|\leq k\}}\Phi_{t,x,x_0}(s,y)\frac{1}{\sqrt{2\pi s}}e^{-\frac{(y-x_0)^2}{2s}}dy
\end{align}
where
$$
\Phi_{t,x,x_0}(s,y)=\frac{\sqrt{s}}{\sqrt{s-t}}e^{-\frac{(y-x)^2t}{2(s-t)}+\frac{(y-x_0)^2}{2s}}
$$which is bounded when $s\in [t+\delta,T]$ and $|y|\leq k$ and then \\
$\Phi_{t,x,x_0}(s,y)1_{[t+\delta,T]\times [-k,k]}(s,y)$ belongs
$L^{\begin{array}r q}([0,T]\times \textbf{R}; \mu(0,x_0;s,dy)ds)$ with $\begin{array}r q$ is the conjugate of $q$. Now as the sequence of functions $(H_i^n)_{n\geq 1}$ converges weakly in $
L^q([0,T]\times \textbf{R}; \mu(0,x_0;s,dy)ds)$ then $$\begin{equation}gin{array}{ll}
&\textbf{E}[\int_{t+\delta}^T (H_i^n(s, X_s^{t,x})-H_i^m(s,X_s^{t,x}))\cdot 1_{\{|X_s^{t,x}|\leq k\}}ds]\\&\quad =
\int_{\textbf{R}}\int_{t+\delta}^T (H_i^n(s,y)-H_i^m(s,y))1_{\{|y|\leq k\}}\Phi_{t,x,x_0}(s,y)\frac{1}{\sqrt{2\pi s}}e^{-\frac{(y-x_0)^2}{2s}}dy\\&\quad \rightarrow 0 \mbox{ as }n,m\rightarrow \infty.\end{array}
$$
Henceforth for $i=1,2$, and any $(t,x)\in \sp$, the sequences $(\eta^{i,n}(t,x))_{n\geq 1}$ is of Cauchy type and then converges to a deterministic measurable function $\eta^i(t,x)$. Additionally from the uniform polynomial growth of $\eta^{i,n}$ (see (a) of Step 2) we deduce that $\eta^i$ is also of polynomial growth, i.e., \begin{equation}\label{polygrowthetai}
\forall (t,x)\in \sp, \,\,|\eta^i(t,x)|\le C(1+|x|^\lambda) \,\,(\lg \ge 0).\end{equation}
Next it turns out that, for $i=1,2$ and any $s\in [0,T]$,
$$
\lim_{n\rightarrow \infty} Y_s^{i,n;0,x_0}(\omega)=\eta^i(s,X_s^{0,x_0}(\omega)) \mbox{ and }|Y_s^{i,n;0,x_0}(\omega)|\leq C(1+|X_s^{0,x_0}(\omega)|^{\lambda}),\ \textbf{P}-a.s.
$$
Setting $Y^i:=(\eta^i(s,X_s^{0,x_0}))_{s\leq T}$, then by Lebesgue's dominated convergence theorem it holds \begin{equation}gin{align}\label{eq:yin conv l alpha}
\textbf{E}[\int_0^T |Y_s^{i,n;0,x_0}-Y_s^i|^{\alpha}ds]\rightarrow 0, \quad \text{as}\ n \rightarrow \infty \quad \text{for any}\ \alpha\geq 1\,\, (i=1,2).
\end{align}
It remains to show the convergence of sequences $((Z_s^{i,n;0,x_0})_{s\leq T})_{n\geq 1}$, $i=1,2$. Taking It\^o's formula with $(Y^{i,n;0,x_0}-Y^{i,m;0,x_0})^2$ and considering (\ref{lg}), we get:
\begin{equation}gin{align}\label{eq:2.31}
&|Y_s^{i,n;0,x_0}-Y_s^{i,m;0,x_0}|^2+\int_s^T|Z_r^{i,n;0,x_0}-Z_r^{i,m;0,x_0}|^2ds\noindentnumber\\
&\leq 2\int_s^TC|Y_r^{i,n;0,x_0}-Y_r^{i,m;0,x_0}|(1+|X_r^{0,x_0}|)(|Z_r^{i,n;0,x_0}|+|Z_r^{i,m;0,x_0}|)dr\noindentnumber\\
&\quad -2\int_s^T(Y_r^{i,n;0,x_0}-Y_r^{i,m;0,x_0})(Z_r^{i,n;0,x_0}-Z_r^{i,m;0,x_0})dB_r,\,\,\forall s\le T.
\end{align}
\noindentindent Since for any $a,b,c\in \textbf{R}$ and for any $\epsilon>0$, $|abc|\leq \frac{\epsilon^2}{2}a^2+\frac{\epsilon^4}{4}b^4+\frac{1}{4\epsilon^8}c^4$, we then have: $\forall s\le T$,
\begin{equation}gin{align}\label{eq:2.32}
&|Y_s^{i,n;0,x_0}-Y_s^{i,m;0,x_0}|^2+\int_s^T|Z_r^{i,n;0,x_0}-Z_r^{i,m;0,x_0}|^2dr\noindentnumber\\
&\leq C\big\{\frac{\epsilon^2}{2}\int_s^T(|Z_r^{i,n;0,x_0}|+|Z_r^{i,m;0,x_0}|)^2dr+\frac{\epsilon^4}{4}\int_s^T(1+|X_r^{0,x_0}|)^4dr\noindentnumber\\
&\qquad+\frac{1}{4\epsilon^8}\int_s^T|Y_r^{i,n;0,x_0}-Y_r^{i,m;0,x_0}|^4dr\big\}\noindentnumber\\
&\quad-2\int_s^T(Y_r^{i,n;0,x_0}-Y_r^{i,m;0,x_0})(Z_r^{i,n;0,x_0}-Z_r^{i,m;0,x_0})dB_r.
\end{align}
Since $\epsilon$ is arbitrary, taking now $s=0$, expectation on both hand-sides and the limit w.r.t. $n,m$, combining with \eqref{eq:yin conv l alpha}, \eqref{eq: est x}, \eqref{eq:rst step 2}-(c) yields that,
\begin{equation}gin{equation}\label{eq:2.33}
\limsup_{n,m\rightarrow \infty} \textbf{E}[\int_0^T |Z_r^{i,n;0,x_0}-Z_r^{i,n;0,x_0}|^2dr]\rightarrow 0,\quad i=1,2.
\end{equation}
Consequently, for $i=1,2$, the sequence $(Z^{i,n;0,x_0}=(\varsigma^{i,n}(t, X_t^{0,x}))_{t\leq T})_{n\geq 1}$ is convergent in $\mathcal{H}^2_T(\textbf{R})$ to a process $Z^i$ which belongs also to $\mathcal{H}^2_T(\textbf{R})$.
On the other hand one can substract a subsequence which we still denote by $\{n\}$ such that
$(Z^{i,n;0,x_0}_s)_{n\ge 1}\rw Z^i_s$, $ds\otimes d\textbf{P}$-a.e. and $u^*p_{n\ge 1}|Z^{i,n;0,x_0}_s|$ belongs to $L^2(\sp,ds\otimes d\textbf{P})$.
Next going back to inequality \eqref{eq:2.32}, taking the supremum on interval $[0,T]$ and using BDG's inequality, we deduce that,
\begin{equation}gin{align*}
&\textbf{E}\big[u^*p_{s\in [0,T]}|Y_s^{i,n;0,x_0}-Y_s^{i,m;0,x_0}|^2+\int_0^T|Z_r^{i,n;0,x_0}-Z_r^{i,m;0,x_0}|^2dr\big]\noindentnumber\\
&\leq C\textbf{E}\big\{\frac{\epsilon^2}{2}\int_0^T(|Z_r^{i,n;0,x_0}|+|Z_r^{i,m;0,x_0}|)^2dr+\frac{\epsilon^4}{4}\int_0^T(1+|X_r^{0,x_0}|)^4dr\noindentnumber\\
&\qquad \qquad +\frac{1}{4\epsilon^8}\int_0^T|Y_r^{i,n;0,x_0}-Y_r^{i,m;0,x_0}|^4dr\big\}\noindentnumber\\
&\quad+\frac{1}{2}\textbf{E}\big[u^*p_{r\in[0,T]}|Y_r^{i,n;0,x_0}-Y_r^{i,m;0,x_0}|^2\big]+2\textbf{E}\big[\int_0^T|Z_r^{i,n;0,x_0}-Z_r^{i,m;0,x_0}|^2dr\big],
\end{align*}
\noindentindent which implies,
$$
\limsup_{n,m\rightarrow \infty}\textbf{E}\big[u^*p_{s\in[0,T]}|Y_s^{i,n;0,x_0}-Y_s^{i,m;0,x_0}|^2\big]=0,
$$
since $\epsilon$ is arbitrary and the facts of \eqref{eq: est x}, \eqref{eq:2.33}, \eqref{eq:yin conv l alpha} and \eqref{eq:rst step 2}-(c). Thus the sequence of processes $(Y^{i,n;0,x_0})_{n\geq 1}$ converges in $\mathcal{S}_T^2(\textbf{R})$ to $Y^i$ for $i=1,2$ which are continuous processes.
To summarize this step, we have the following results (at least for a subsequence $\{n\}$): for $i=1,2$,
\begin{equation}gin{equation}\label{eq:rst step3}
\left\{
\begin{equation}gin{aligned}
&\text{(a)}\ H_i^n(s,y)\in L^q([0,T]\times \textbf{R};\ \mu(0,x_0;s,dy)ds)\text{ uniformly w.r.t. }n;\\
&\text{(b)}\ Y^{i,n;0,x_0}\rightarrow_{n\rightarrow \infty} Y^i \text{ in } L^{\alpha}([0,T]\times \textbf{R}, ds\otimes d\textbf{P})\text{ for any } \alpha\geq 1,\text{ besides, }\\
&\qquad Y^{i,n;0,x_0}\rightarrow_{n\rightarrow \infty} Y^i \text{ in } \mathcal{S}_T^2(\textbf{R});\\
&(c)\ Z^{i,n;0,x_0}\rightarrow_{n\rightarrow\infty}Z^i\text{ in } L^2([0,T]\times \textbf{R},ds\otimes d\textbf{P}), \text{ additionally, there exists a}\\
&\qquad \text{subsequence } \{n\} \text{ s.t }Z^{i,n;0,x_0}\rightarrow_{n\rightarrow\infty}Z^i\ ds\otimes d\textbf{P}-a.e.\text{ and }\\
&\qquad u^*p_{n\geq 1}|Z^{i,n;0,x_0}|\in L^2([0,T]\times \textbf{R},ds\otimes d\textbf{P}).
\end{aligned}
\right.
\end{equation}
\noindent \underline{\textit{Step 4}}: Convergence of $(H_i^n)_{n\geq 1},\ i=1,2$.
In this step, we are going to define the processes $(\theta_s)_{s\leq T}$ and $(\vartheta_s)_{s\leq T}$ and verify that $(Y^i,Z^i)$, $i=1,2$, and $\theta$, $\vartheta$ satisfy (\ref{eq:main bsde}). We demonstrate first for $i=1$. Let us consider the subsequence which satisfies (\ref{eq:rst step3}).
Recall \eqref{eq:Hn} for $(t,x)=(0,x_0)$ which reads as: For any $\sT$,
\begin{equation}gin{equation}\label{eq:rec Hn}
\begin{equation}gin{aligned}
H_1^n(s,X^{0,x_0}_s)&=\Phi_n(Z_s^{1,n;0,x_0})\Phi_n(f(s,X^{0,x_0}_s))+
\Phi_n(Z_s^{1,n;0,x_0}\begin{array}r{u}(Z_s^{1,n;0,x_0}))\noindentnumber\\
&\quad+\Phi_n(Z_s^{1,n;0,x_0})\begin{array}r{v}^n(Z_s^{2,n;0,x_0}).
\end{aligned}
\end{equation}
Note that,
\begin{equation}gin{align*}
\Phi_n(Z_s^{1,n;0,x_0})\Phi_n(f(s,X^{0,x_0}_s))&+
\Phi_n(Z_s^{1,n;0,x_0}\begin{array}r{u}(Z_s^{1,n;0,x_0}))\\
&\rightarrow_{n\rightarrow\infty} Z_s^{1}f(s,X^{0,x_0}_s)+Z_s^1\begin{array}r{u}(Z_s^{1}),\
ds\otimes d\textbf{P}\text{-a.e.}
\end{align*}
since $Z^{1,n;0,x_0}\rightarrow_{n\rightarrow \infty}Z^1$, $ds\otimes d\textbf{P}$-a.e. as stated in \eqref{eq:rst step3}-(c), $\Phi_n(x)\rightarrow_{n\rightarrow \infty}x$ and finally by the continuity of $p\in \textbf{R}\mapsto p\begin{array}r{u}(p)$. This convergence holds also in $\mathcal{H}^1_T(\textbf{R})$, by Lebesgue's dominated convergence theorem, since
the process $(u^*p_{n\ge 1}|Z_s^{1,n;0,x_0}|)_{\sT}$ belongs to $\mathcal{H}^2_T(\textbf{R})$, $f$ is of linear growth and $\begin{array}r{u}$ is uniformly bounded. The rest part in \eqref{eq:rec Hn} is
$$\begin{equation}gin{array}{l}
\Phi_n(Z_s^{1,n;0,x_0})\begin{array}r{v}^n(Z_s^{2,n;0,x_0})\\\qquad =\Phi_n(Z_s^{1,n;0,x_0})\begin{array}r{v}^n(Z_s^{2,n;0,x_0})1_{\{Z_s^{2}\neq 0\}}+\Phi_n(Z_s^{1,n;0,x_0})
\begin{array}r{v}^n(Z_s^{2,n;0,x_0})1_{\{Z_s^{2}=0\}}.
\end{array} $$
But $$
\Phi_n(Z_s^{1,n;0,x_0})\begin{array}r{v}^n(Z_s^{2,n;0,x_0})
1_{\{Z_s^{2}\neq 0\}} \rightarrow_{n\rightarrow \infty} Z_s^1\begin{array}r{v}(Z_s^2)1_{\{Z_s^2\neq 0\}} \quad ds\otimes d\textbf{P}\text{-a.e.}
$$
since for any $z\neq 0$, if $n$ is large enough then $\begin{array}r v^n(z')=\begin{array}r v(z')$ if $z'\in (z-a,z+a)u^*bset (-\infty,0)\cup (0,+\infty)$ for small $a>0$ and $\begin{array}r v$ is continuous in $z$. Once more
the convergence holds in $\mathcal{H}^2_T(\textbf{R})$, by Lebesgue's dominated convergence theorem, since
the process $(u^*p_{n\ge 1}|Z_s^{1,n;0,x_0}|)_{\sT}$ belongs to $\mathcal{H}^2_T(\textbf{R})$ and $\begin{array}r{v}^n$ is uniformly bounded.
To proceed let us define a $\mathcal{P}$-measurable process $(\vartheta_s)_{s\leq T}$ valued on $V$ as the weak limit in $\mathcal{H}_T^2(\textbf{R})$ of some subsequence $(\begin{array}r{v}^{n_k}(Z^{2,n_k;0,x_0})1_{\{Z^{2}=0\}})_{k\geq 0}$. The weak limit exists since $(\begin{array}r{v}^{n_k})_{k\geq 0}$ is bounded. Let now $\tau$ be an arbitrary stopping time such that $\tau \in [0,T]$, $ \textbf{P}-a.s.$, then
\begin{equation}gin{align*}
\int_0^{\tau}&\Phi_{n_k}(Z_s^{1,n_k;0,x_0})\begin{array}r{v}^{n_k}(Z_s^{2,n_k;0,x_0})1_{\{Z_s^2=0\}}ds\\
&\rightarrow_{k\rightarrow\infty} \int_0^{\tau}Z_s^1\vartheta_s1_{\{Z_s^2=0\}}ds\text{ weakly in } L^2(\Omega, d\textbf{P}).\end{align*}
Indeed
\begin{equation}gin{align*}
\int_0^{\tau}&\Phi_{n_k}(Z_s^{1,n_k;0,x_0})\begin{array}r{v}^{n_k}(Z_s^{2,n_k;0,x_0})1_{\{Z_s^2=0\}}ds\\
&=\quad\int_0^{\tau}(\Phi_{n_k}(Z_s^{1,n_k;0,x_0})-Z_s^1)\begin{array}r{v}^{n_k}(Z_s^{2,n_k;0,x_0})1_{\{Z_s^2=0\}}ds\\
&\quad+\int_0^{\tau}Z_s^1\begin{array}r{v}^{n_k}(Z_s^{2,n_k;0,x_0})1_{\{Z_s^2=0\}}ds.
\end{align*}
\noindent On the right side, the first integral converges to $0$ in $L^2(d\textbf{P})$ by Lebesgue's dominated convergence theorem since
$\Phi_{n_k}(Z^{1,n_k;0,x_0})\rightarrow Z^1 \ dt\otimes d\textbf{P}$-a.e., $u^*p_{k\geq 0}|Z_t^{1,n_k}|\in L^2([0,T]\times \textbf{R})$ as shown in \eqref{eq:rst step3}-(c), $Z^1\in L^2([0,T]\times \textbf{R})$, the sequence $(\begin{array}r{v}^{n_k})_{k\geq 0}$ is bounded and $|\Phi_{n_k}(x)|\leq |x|$, $\forall x\in \textbf{R}$. Below, we will give the weak convergence in $L^2(\Omega, d\textbf{P})$ of the integral $\int_0^{\tau}Z_s^1\begin{array}r{v}^{n_k}(Z_s^{2,n_k;0,x_0})1_{\{Z_s^2=0\}}ds$ to $\int_0^{\tau}Z_s^1\vartheta_s1_{\{Z_s^2=0\}}ds$. That is, for any random variable $\xi\in L^2(\Omega, \mathcal{F}_T, d\textbf{P})$, we need to show,
\begin{equation}gin{equation}\label{eq:weak cov int xi}
\textbf{E}[\xi\int_0^{\tau}Z_s^1\begin{array}r{v}^{n_k}(Z_s^{2,n_k;0,x_0})1_{\{Z_s^2=0\}}ds]\rightarrow_{k\rightarrow\infty}\textbf{E}[\xi\int_0^{\tau}Z_s^1\vartheta_s 1_{\{Z_s^2=0\}}ds].
\end{equation}
\noindentindent Thanks to martingale representation theorem, there exists a process $( \Lambda_s)_{s\leq T}\in \mathcal{H}_{T}^2(\textbf{R})$ such that,
$
\textbf{E}[\xi|\mathcal{F}_{\tau}]=\textbf{E}[\xi]+\int_0^{\tau}\Lambda_sdB_s.
$
Therefore,
\begin{equation}gin{align*}
\textbf{E}\Big[\xi\int_0^{\tau}&Z_s^1\begin{array}r{v}^{n_k}(Z_s^{2,n_k;0,x_0})1_{\{Z_s^2=0\}}ds\Big]\\
&=\textbf{E}\Big[\textbf{E}\big[\xi\int_0^{\tau}Z_s^1\begin{array}r{v}^{n_k}(Z_s^{2,n_k;0,x_0})1_{\{Z_s^2=0\}}ds|\mathcal{F}_{\tau}]\Big]\\
&=\textbf{E}\Big[\textbf{E}[\xi|\mathcal{F}_{\tau}]\cdot \int_0^{\tau}Z_s^1\begin{array}r{v}^{n_k}(Z_s^{2,n_k;0,x_0})1_{\{Z_s^2=0\}}ds\Big]\\
&=\textbf{E}\Big[\textbf{E}[\xi]\int_0^{\tau}Z_s^1\begin{array}r{v}^{n_k}(Z_s^{2,n_k;0,x_0})1_{\{Z_s^2=0\}}ds\Big]\\
&\qquad+ \textbf{E}\Big[\int_0^{\tau}\Lambda_sdB_s\cdot\int_0^{\tau}Z_s^1\begin{array}r{v}^{n_k}(Z_s^{2,n_k;0,x_0})1_{\{Z_s^2=0\}}ds\Big].
\end{align*}
Notice that $\textbf{E}[\xi]\textbf{E}[\int_0^{\tau}Z_s^1\begin{array}r{v}^{n_k}(Z_s^{2,n_k;0,x_0})1_{\{Z_s^2=0\}}ds] \rightarrow_{k\rightarrow \infty} \textbf{E}[\xi]\textbf{E}[\int_0^{\tau}Z_s^1\vartheta_s1_{\{Z_s^2=0\}}ds]$, since $(Z_s^1)_{s\leq T}\in \mathcal{H}_T^2(\textbf{R})$ and $\begin{array}r{v}^{n_k}(Z_s^{2,n_k;0,x_0})1_{\{Z_s^2=0\}}\rightarrow_{k\rightarrow\infty}\vartheta_s$ weakly in $\mathcal{H}_T^2(\textbf{R})$. Next, by It\^o's formula,
\begin{equation}gin{align*}
\textbf{E}[&\int_0^{\tau}\Lambda_sdB_s\cdot\int_0^{\tau}Z_s^1\begin{array}r{v}^{n_k}(Z_s^{2,n_k;0,x_0})1_{\{Z_s^2=0\}}ds]\\
&=\textbf{E}[\int_0^{\tau}\big(\int_0^s\Lambda_udB_u\big)Z_s^1\begin{array}r{v}^{n_k}(Z_s^{2,n_k;0,x_0})1_{\{Z_s^2=0\}}ds]+\\
&\quad+\textbf{E}[\int_0^{\tau}\big(\int_0^sZ_u^1\begin{array}r{v}^{n_k}(Z_u^{2,n_k;0,x_0})1_{\{Z_u^2=0\}}du\big)\Lambda_sdB_s].
\end{align*}
The latter one on the right side is 0, since $\int_0^{.}(\int_0^sZ_u^1\begin{array}r{v}^{n_k}(Z_u^{2,n_k;0,x_0})1_{\{Z_u^2=0\}}du)\Lambda_sdB_s$ is an $\mathcal{F}_{t}$-martingale which is followed by $(Z_s^1)_{s\leq T}\in \mathcal{H}_T^2(\textbf{R}),\ (\Lambda_s)_{s\leq T}\in \mathcal{H}_T^2(\textbf{R})$ and the boundness of $\begin{array}r{v}^{n_k}$ and then \\
$\textbf{E}[\{\int_0^T|\int_0^sZ_u^1\begin{array}r{v}^{n_k}(Z_u^{2,n_k;0,x_0})1_{\{Z_u^2=0\}}du|^2|\Lambda_s|^2ds\}^{\frac{1}{2}}]<\infty$. For the former part, let us denote $\int_0^s\Lambda_udB_u$ by $\psi_s$ for any $s\in [0,\tau]$. Then for any integer $\kappa>0$, we have,
\begin{equation}gin{align*}
|\textbf{E}&[\int_0^{\tau}\psi_sZ_s^1\Big(\begin{array}r{v}^{n_k}(Z_s^{2,n_k})1_{\{Z_s^2=0\}}-\vartheta_s\Big)ds]|\\
&=|\textbf{E}[\int_0^{\tau}\psi_sZ_s^1\Big(\begin{array}r{v}^{n_k}(Z_s^{2,n_k;0,x_0})1_{\{Z_s^2=0\}}-\vartheta_s\Big)1_{\{|\psi_sZ_s^1|\leq \kappa\}}ds]|+\\%
&\quad+|\textbf{E}[\int_0^{\tau}\psi_sZ_s^1\Big(\begin{array}r{v}^{n_k}(Z_s^{2,n_k;0,x_0})1_{\{Z_s^2=0\}}-\vartheta_s\Big)1_{\{|\psi_sZ_s^1|> \kappa\}}ds]|.
\end{align*}
On the right side of the above equation, the first component converges to 0 which is the consequence of $\begin{array}r{v}^{n_k}(Z_s^{2,n_k;0,x_0})1_{\{Z_s^2=0\}}\rightarrow_{k\rightarrow\infty}\vartheta_s$ weakly in $\mathcal{H}_T^2(\textbf{R})$. For the second term, considering both $(\begin{array}r{v}^{n_k}(Z_s^{2,n_k;0,x_0}))_{s\leq \tau}$ and $(\vartheta_s)_{s\leq \tau}$ are bounded, it is smaller than $ C|\textbf{E}[\int_0^{\tau}|\psi_sZ_s^1|1_{\{|\psi_sZ_s^1|\geq \kappa\}}ds]|$ which obviously converges to 0 as $\kappa\rightarrow \infty$. Thus \eqref{eq:weak cov int xi} holds true.
Finally, we also have
\begin{equation}gin{equation}\label{eq:conv z tau}
\int_0^{\tau}Z_s^{1,n_k;0,x_0}dB_s \rightarrow_{k\rightarrow\infty} \int_0^{\tau}Z_s^1dB_s \text{ in } L^2(\Omega, d\textbf{P}),
\end{equation}
which is obtained from the convergence of $(Z^{1,n_k;0,x_0})_{k\geq 0}$ to $Z^1$ in $\mathcal{H}_T^2(\textbf{R})$.
Then by observing the approximation BSDE \eqref{eq:bsdeapprox} in a forward way, \textit{i.e.} for any stopping time $\tau$,
$$
Y_{\tau}^{1,n_k;0,x_0}=Y_0^{1,n_k;0,x_0}-\int_0^{\tau} H_1^{n_k}(s, X_s^{0,x_0})ds+\int_0^{\tau} Z_s^{1,n_k;0,x_0}dB_s,
$$
\noindent combining with the convergence of $(Y^{1,n_k;0,x_0})_{k\geq 0}$ to $Y^1$ in $\mathcal{S}_T^2(\textbf{R})$ we infer that
\begin{equation}gin{equation*}
\textbf{P}\text{-a.s.},\ Y_{\tau}^1=Y_0^1-\int_0^{\tau}H_1^*(s,X^{0,x_0}_s,Z_s^1,Z_s^2,\vartheta_s)ds+\int_0^{\tau}Z_s^1dB_s
\end{equation*}
since
$$
\int_0^{\tau} H_1^{n_k}(s, X_s^{0,x_0})ds \rightharpoonup_{k\rw \infty}\int_0^{\tau}H_1^*(s,X^{0,x_0}_s,Z_s^1,Z_s^2,\vartheta_s)ds \mbox{ weakly in }L^2(\Omega, d\textbf{P}).
$$
As $\tau$ is arbitrary then the processes $Y^1$ and $Y_0^1-\int_0^\cdot H_1^*(s,X^{0,x_0}_s,Z_s^1,Z_s^2,\vartheta_s)ds+\int_0^\cdot Z_s^1dB_s$ are indistinguishable, \textit{i.e.}, $\textbf{P}$-a.s.
\begin{equation}gin{equation*}
\forall s\leq T,\quad Y_s^1=Y_0^1-\int_0^sH_1^*(r,X^{0,x_0}_r,Z_r^1,Z_r^2,\vartheta_r)dr+\int_0^sZ_r^1dB_r.
\end{equation*}
On the other hand, $Y_T^1=g_1(X^{0,x_0}_T)$, then,
\begin{equation}gin{equation*}\textbf{P}\text{-a.s.},\,\quad
\forall s\leq T,\quad Y_s^1=g_1(X^{0,x_0}_T)+\int_s^TH_1^*(r,X^{0,x_0}_r,Z_r^1,Z_r^2,\vartheta_r)dr-\int_s^TZ_r^1dB_r.
\end{equation*}
Similarly, for player $\pi_2$, there exists a $\mathcal{P}$-measurable process $(\theta_s)_{s\leq T}$ valued on $U$, which is obtained in the same way as previously as a weak limit of
a subsequence of
$({\begin{array}r u}^n(Z^{1,n;0,x_0}_s)1_{\{Z^1_s\neq 0\}})_{s\leq T} \mbox{ in }\mathcal{H}^2_T(\textbf{R})$ such that
\begin{equation}gin{equation*}
\textbf{P}\text{-a.s.},\quad \forall s\leq T,\quad Y_s^2=g_2(X^{0,x_0}_T)+\int_s^T
H_2^*(r,X^{0,x_0}_r,Z_r^1,Z_r^2,\theta_r)dr-\int_s^TZ_r^2dB_r.
\end{equation*}
The proof is completed.\qedh
\section{Generalizations}\label{sec:gener}
In this Section, we are going to deal with some generalizations of Theorem \ref{th: nash} in the following three aspects:
\noindentindent \textbf{(i)} For the drift term $\Gamma$ in SDE \eqref{eq: SDE with Gamma} which reads,
$$
\Gamma(t,x,u,v)=f(t,x)+u+v, \,\,u\in [0,1]\mbox{ and }v\in [-1,1],
$$
one can replace:
(a) $[0,1]$ and $[-1,1]$ with arbitrary closed bounded intervals ;
(b) $u$ (resp. $v$) of $\G$ with $h(u)$ (resp. $\ell(v)$) where $h$ and $\ell$ are continuous functions defined on $U=[a,b]$ and $V=[c,d]$ respectively. In this case $U'=h(U)$ and $V'=\ell(V)$ are also bounded closed intervals. The Nash equilibrium point $(\begin{array}r{u},\begin{array}r{v})$ exists and is still of bang-bang type. The unique difference is that, it will jump between the bound of set $U'$ (resp. $V'$) instead of $U$ (resp. $V$). \qedh
\noindent \textbf{(ii)} As we indicated in Remark \ref{re:sigma}, the dynamics of the process $X^{t,x}$ of \eqref{eq: SDE} may contain a diffusion term $\sigma (t,x)$ (see equation \eqref{eq:sde sigma}) which is a function defined as:
$$
(t,x)\in [0,T]\times \textbf{R} \mapsto \sigma(t,x) \in \textbf{R},
$$
with the following assumption:
\ms
\noindentindent \textbf{Assumption (A1)}: The function $\sigma(t,x)$ is uniformly Lipschitz w.r.t. $x$, it is invertible and bounded and its inverse is bounded.
\noindent Under (A1), for any $\tx$, the following SDE
\begin{equation}gin{equation}\label{eq:sde sigma}\begin{equation}gin{array}{c}
X_s^{t,x}=x+\int_t^s \sigma(r, X_r^{t,x})dB_r,\ \forall s\in [t,T]\text{ and } X_s^{t,x}=x \text{ for } s\in [0,t],\end{array}
\end{equation}
has a unique solution (see e.g. Karatzas and Shreve, pp.289, \cite{karatzas}). Moreover $\sigma$ satisfies the uniform elliptic condition, \textit{i.e.} there exists a constant $\Upsilon>0$ such that for any $(t,x)\in [0,T]\times \textbf{R}$, $\Upsilon\leq \sigma(t,x)^2\leq \Upsilon^{-1}$.
\ms
\noindent In this framework, the Hamiltonian functions associated with the NZSDG of payoffs given by (\ref{eq: payoffs}) are defined from $[0,T]\times \textbf{R}\times \textbf{R} \times U\times V$ into $\textbf{R}$ by:
\begin{equation}gin{eqnarray*}
H_1(t,x,p,u,v)&:=p\sigma^{-1}(t,x)\Gamma(t,x,u,v)=p\sigma^{-1}(t,x)(f(t,x)+u+v);\\
H_2(t,x,q,u,v)&:=q\sigma^{-1}(t,x)\Gamma(t,x,u,v)=q\sigma^{-1}(t,x)(f(t,x)+u+v).
\end{eqnarray*}
Noticing that $\sigma^{-1}$ is bounded, it follows by the generalized Isaacs' condition (\ref{eq:Isaacs}) and the same approach in this article that, the Nash equilibrium point exists and is of bang-bang type.
\noindent Actually we should point out that, all the results in this article will hold by the same techniques in this case (ii) with only some minor adaptions, except the convergence to $0$ of the second term on the right side of inequality \eqref{eq:eta in - eta im} which needs to be checked carefully. Indeed the objective is to show that
\begin{equation}gin{equation}\label{eq:sec term in gen}\begin{array}{l}
|\textbf{E}[\int_{t+\delta}^T (H_i^n(s, X_s^{t,x})-H_i^m(s,X_s^{t,x}))\cdot 1_{\{|X_s^{t,x}|\leq k\}}ds]\big|\rightarrow_{n,m\rightarrow \infty}0.\end{array}
\end{equation}
for fixed $(t,x)$ and $k$. To begin with we give the following result related to domination of laws of the process $X^{t,x}$.
\begin{equation}gin{lemma}\label{lemma:domi}
{\bf ($L^q$-Domination)} Let $t\in [0,T]$ and $x,x_0\in \textbf{R}$. For
$s\in [t,T]$, we denote by $\mu(t,x;s,dy)$ the law of $X_s^{t,x}$. Under Assumption (A1) on $\sigma$, for any $\begin{array}r q\in (1,\infty)$, the family of laws $\{\mu(t,x;s,dy),\ s\in (t,T]\}$ is $L^{\begin{array}r q}$-dominated by $\{\mu(0,x_0;s,dy),\ s\in (t,T]\}$, \textit{i.e.}, for any $\delta\in (0,T-t)$, there exists an application $\phi_{t,x,x_0}^{\delta}:[t+\delta, T]\times \textbf{R}\rightarrow \textbf{R}^+$ such that:
(a) $\mu(t,x;s,dy)ds=\phi_{t,x,x_0}^{\delta}(s,y)\mu(0,x_0;s,dy)ds$ for any $(s,y)\in [t+\delta,T]\times\textbf{R} ;$
(b) $\forall k\geq 1,\ \phi_{t,x,x_0}^{\delta}(s,y)\in L^{ \begin{array}r q}([t+\delta,T]\times [-k,k];\ \mu(0,x_0;s,dy)ds)$.
\end{lemma}
\noindent \textit{Proof.} Readers are referred to \cite{H1997} (Section 28, pp.123) and \cite{H2014} (Lemma 4.3 and Corollary 4.4, pp.14-15) for the proof of this Lemma. However basically it uses the Aronson estimates \cite{aronson} for densities of the laws of the solution of SDE (\ref{eq:sde sigma}) under Assumption (A1).
\noindent {\bf Proof of convergence \eqref{eq:sec term in gen}:} Thanks to Lemma \ref{lemma:domi}, there exists a function $\phi_{t,x,x_0}^{\delta}:[t+\delta,T]\times \textbf{R}\rightarrow \textbf{R}^+$ such that:
\begin{equation}gin{equation}\label{eq: ge phi}
\forall k\geq 1,\ \phi_{t,x,x_0}^{\delta}(s,y)\in L^{\frac{q}{q-1}}([t+\delta, T]\times [-k,k];\ \mu(0,x_0;s,dy)ds).
\end{equation}
Then
$$\begin{array}{ll}
|\textbf{E}[&\!\!\!\!\!\int_{t+\delta}^T (H_i^n(s, X_s^{t,x})-H_i^m(s,X_s^{t,x}))\cdot 1_{\{|X_s^{t,x}|\leq k\}}ds]\big|\\\\
{}&=|\int_{\textbf{R}}\int_{t+\delta}^T (H_i^n(s, y)-H_i^m(s,y))\cdot 1_{\{|y|\leq k\}}\mu(t,x;s,dy)ds|\\\\
{} &=|\int_{\textbf{R}}\int_{t+\delta}^T (H_i^n(s, y)-H_i^m(s,y))\cdot 1_{\{|y|\leq k\}}\phi_{t,x,x_0}^{\delta}(s,y)\mu(0,x_0;s,dy)ds|.\end{array}
$$
The constant $q$ in \eqref{eq: ge phi} is the one which makes that $H_i^n(s,y)\rightarrow_{n\rw \infty} H_i(s,y)$ weakly in $L^q([0,T]\times \textbf{R};\ \mu(0,x_0;s,dy)ds)$ for $i=1,2$ and a fixed $q\in (1,2)$. Then combining this weak convergence result and \eqref{eq: ge phi} yields \eqref{eq:sec term in gen}.\qedh
\noindent \textbf{(iii)} In the same way one can deal with the multi-dimensional case for diffusion processes $X^{t,x}$ satisfying (\ref{eq:sde sigma}) when $\sigma (t,x)$ verifies Assumption (A1). \qedh
\begin{equation}gin{thebibliography}{1}
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\end{document} |
\begin{document}
\title{{$(SGD)^2$}\xspacetext, {$(SGD)^2$}\xspace}
\author{\IEEEauthorblockN{Reyan Ahmed, Felice De Luca, Sabin Devkota, Stephen Kobourov, Mingwei Li}
\thanks{An earlier version of this paper appears in GD'20~\cite{ahmed2020gd}; in this extended version we use stochastic gradient descent which allows for multicriteria optimization on larger graphs.}
\IEEEauthorblockA{\\Department of Computer Science, University of Arizona, USA}}
\IEEEtitleabstractindextext{
\begin{abstract}
Readability criteria, such as distance or neighborhood preservation, are often used to optimize node-link representations of graphs to enable the comprehension of the underlying data.
With few exceptions, graph drawing algorithms typically optimize one such criterion, usually at the expense of others.
We propose a layout approach, {$(SGD)^2$}\xspacetext, {$(SGD)^2$}\xspace, that can handle multiple readability criteria.
\blue{
{$(SGD)^2$}\xspace can optimize any criterion that can be described by a differentiable function.
}
Our approach is flexible and can be used to optimize several criteria that have already been considered earlier (e.g., obtaining ideal edge lengths, stress, neighborhood preservation) as well as other criteria which have not yet been explicitly optimized in such fashion (e.g., node resolution, angular resolution, aspect ratio).
The approach is scalable and can handle large graphs.
A variation of the underlying approach can also be used to optimize many desirable properties in planar graphs, while maintaining planarity.
\blue{
Finally, we provide quantitative and qualitative evidence of the effectiveness of {$(SGD)^2$}\xspace: we analyze the interactions between criteria, measure the quality of layouts generated from {$(SGD)^2$}\xspace as well as the runtime behavior, and analyze the impact of sample sizes.
The source code
is available on github and we
also provide an interactive demo for small graphs.
}
\end{abstract}
\begin{IEEEkeywords}
Graph drawing, gradient descent, quality metrics.
\end{IEEEkeywords}}
\maketitle
\IEEEdisplaynontitleabstractindextext
\IEEEpeerreviewmaketitle
\IEEEraisesectionheading{
\section{Introduction}\label{sec:introduction}}
Graphs represent relationships between entities and visualization of this information is relevant in many domains.
Several criteria have been proposed to evaluate the readability of graph drawings, including the number of edge crossings, distance preservation, and neighborhood preservation.
Such criteria evaluate different aspects of the drawing and different layout algorithms optimize different criteria.
It is challenging to optimize multiple readability criteria at once and there are few approaches that can support this.
Examples of approaches that can handle a small number of related criteria include the stress majorization framework of Wang et al.~\cite{wang2017revisiting}, which optimizes distance preservation via stress as well as ideal edge length preservation.
The Stress Plus X (SPX) framework of Devkota et al.~\cite{devkota2019stress} can minimize the number of crossings, or maximize the minimum angle of edge crossings.
While these frameworks can handle a limited set of related criteria, it is not clear how to extend them to arbitrary optimization goals.
The reason for this limitation is that these frameworks are dependent on a particular mathematical formulation.
For example, the SPX framework
was designed for crossing minimization, which can be easily modified to handle crossing angle maximization (by adding a cosine factor to the optimization function).
This ``trick" can be applied only to a limited set of criteria but not the majority of other criteria that are incompatible with the basic formulation.
\begin{figure}
\caption{\blue{An interactive prototype of {$(SGD)^2$}
\label{fig:tensorflowjs-ui}
\end{figure}
In this paper, we propose a general approach, {$(SGD)^2$}\xspacetext, {$(SGD)^2$}\xspace, that can optimize a large set of drawing criteria, provided that the corresponding metrics that evaluate the criteria are differentiable functions.
\blue{
If the criterion is not naturally differentiable, we design a differentiable surrogate function to approximate and optimize the original criterion.
In {$(SGD)^2$}\xspace, auto-differentiation tools are used for the gradient-based optimization.
}
To demonstrate the flexibility of the approach, we consider a set of nine criteria: minimizing stress, maximizing node resolution, obtaining ideal edge lengths, maximizing neighborhood preservation, maximizing crossing angle, optimizing total angular resolution, minimizing aspect ratio, optimizing the Gabriel graph property, and minimizing edge crossings.
\blue{
We evaluate the effectiveness of our approach quantitatively and qualitatively with evidence drawn from a set of experiments.
To illustrate the effectiveness and efficiency of multicriteria optimization, we evaluate the compatibility of every pair of criteria, measure the quality of each criterion, and demonstrate the distinctive looks of graph layouts under different drawing objectives.
We also evaluate the runtime performance and the impact of sample sizes used in the optimization, and compare our methods with existing ones.
}
We implemented our method with PyTorch.
The code is available at: \url{https://github.com/tiga1231/graph-drawing/tree/sgd}.
For demonstration purposes, we also built an interactive prototype \blue{(that implements full-batch gradient descent on small graphs)} in JavaScript using tensorflow.js and D3.js, which is available on \url{http://hdc.cs.arizona.edu/~mwli/graph-drawing/}.
This interactive prototype allows nodes to be moved manually and combinations of criteria can be optimized by selecting a weight for each;
\blue{see Fig.~\ref{fig:tensorflowjs-ui}.}
\begin{comment}
\begin{figure}
\caption{Generating symmetric layout using {$(SGD)^2$}
\label{fig:inputsym_in}
\label{fig:inputsym_out}
\label{fig:symgdqnfirst}
\end{figure}
\end{comment}
\section{Related Work}
Many criteria associated with the readability of graph drawings have been proposed~\cite{ware2002cognitive}. Most graph layout algorithms are designed to (explicitly or implicitly) optimize a single criterion.
For instance, a classic layout criterion is stress minimization ~\cite{kamada_1989}, where stress is defined by $\sum\limits_{i < j}w_{ij} (|X_i-X_j| - d_{ij})^2$. Here, $X$ is a $n\times2$ matrix containing coordinates for the $n$ nodes, $d_{ij}$ is typically the graph-theoretical distance between two nodes $i$ and $j$ and $w_{ij}=d_{ij}^{-\alpha}$ is a normalization factor with $\alpha$ equal to $0, 1$ or $2$. Thus reducing the stress in a layout corresponds to computing node positions so that the actual distance between pairs of nodes is proportional to the graph theoretic distance between them. Optimizing stress can be accomplished by stress minimization, or stress majorization, which can speed up the computation~\cite{gansner2004graph}. In this paper we only consider drawing in the Euclidean plane, however, stress can be also optimized in other spaces such as the torus~\cite{chen2020doughnets}.
Stress minimization corresponds to optimizing the global structure of the layout, as the stress metric takes into account all pairwise distances in the graph. The t-SNET algorithm of Kruiger et al.~\cite{kruiger2017graph} directly optimizes neighborhood preservation, which captures the local structure of a graph, as the
neighborhood preservation metric only considers distances between pairs of nodes that are close to each other.
Optimizing local or global distance preservation can be seen as special cases of the more general dimensionality reduction approaches such as multi-dimensional scaling~\cite{shepard1962analysis,kruskal1964multidimensional}.
Purchase et al.~\cite{Purchase1997} showed that the readability of graphs increases if a layout has fewer edge crossings. The underlying optimization problem is NP-hard and several graph drawing contests have been organized with the objective of minimizing the number of crossings in the graph drawings~\cite{Abrego12,Buchheim13}. Recently several algorithms that directly minimize crossings have been proposed
~\cite{bennett2010,Radermacher18}.
The negative impact on graph readability due to edge crossings can be mitigated if crossing pairs of edges have a large crossings angle~\cite{Argyriou2010,huang2014,huang2013,didimo2014crossangle}. Formally, the crossing angle of a straight-line drawing of a graph is the minimum angle between two crossing edges in the layout, and optimizing this property is also NP-hard.
Recent graph drawing contests have been organized with the objective of maximizing the crossings angle in graph drawings and this has led to several heuristics for this problem~\cite{Demel2018AGH,Bekos18}.
The algorithms above are very effective at optimizing the specific readability criterion they are designed for, but they cannot be directly used to optimize additional criteria. This is a desirable goal, since optimizing one criterion often leads to poor layouts with respect to one or more other criteria: for example, algorithms that optimize the crossing angle tend to create drawings with high stress and no neighborhood preservation~\cite{devkota2019stress}.
Davidson and Harel~\cite{davidson1996drawing} used simulated annealing to optimize different graph readability criteria (keeping nodes away from other nodes and edges, uniform edge lengths, minimizing edge crossings). \blue{Huang et al.~\cite{huang2013} extended a force-directed algorithm to optimize crossing angle and angular resolution by incorporating two additional angle forces. The authors show that in addition to optimizing crossing angle and angular resolution, the algorithm also improves other desirable properties (average size of crossing angles, standard deviation of crossing angles, standard deviations of angular resolution, etc.). In a force-directed method similar to the algorithm proposed by Huang et al.~\cite{huang2013}, to optimize each criterion one needs to design a new force. The new force can be considered as a gradient update by hand, whereas {$(SGD)^2$}\xspace is a gradient descent based algorithm where the gradients are computed automatically using auto-differentiation tools.}
Recently, several approaches have been proposed to simultaneously improve multiple layout criteria. Wang et al.~\cite{wang2017revisiting} propose a revised formulation of stress that can be used to specify ideal edge direction in addition to ideal edge lengths in a graph drawing. Wang et al.~\cite{wang2018structure} extended that stress formulation to produce structure-aware and smooth fish-eye views of graphs.
Devkota et al.~\cite{devkota2019stress}
also use a stress-based approach to minimize edge crossings and maximize crossing angles.
Eades et al.~\cite{10.1007/978-3-319-27261-0_41} provided a technique to draw large graphs while optimizing different geometric criteria, including the Gabriel graph property. Although the approaches above are designed to optimize multiple criteria, they cannot be naturally extended to handle other optimization goals.
Constraint-based layout algorithms such as COLA~\cite{ipsepcola_2006, scalable_cola_2009}, can be used to enforce separation constraints on pairs of nodes to support properties such as customized node ordering or downward pointing edges. The coordinates of two nodes are related by inequalities in the form of $x_i \geq x_j + gap$ for a node pair $(i,j)$. Dwyer et al.~\cite{dwyer2009constrained} use gradient projection to handle these constraints, be moving nodes as little as needed to satisfy the inequalities/ equalities after each iteration of the layout method. The gradient projection method has been extended to also handle non-linear constraints~\cite{dwyer2009layout}. These hard constraints are powerful but a bit restrictive and are different from \blue{the soft constraints} in our {$(SGD)^2$}\xspace framework.
\begin{comment}
{\reyan{Also there are other readability criteria for which we can apply {$(SGD)^2$}\xspace: crossings, NP, stress, area, aspect ratio,
symmetry,
upwardness drawing, label overlapping removal, \href{https://arxiv.org/pdf/1908.03586.pdf}{edge-length ratio}, \href{https://arxiv.org/pdf/1908.07363.pdf}{node overlap removal}, \href{https://arxiv.org/pdf/1908.06504.pdf}{total angular resolution}, \href{https://link.springer.com/chapter/10.1007/978-3-030-04414-5_20}{crossing-angle maximization}, \href{https://arxiv.org/pdf/1708.09815.pdf}{drawings with few
slopes},
\href{https://pdfs.semanticscholar.org/be7e/4c447ea27e0891397ae36d8957d3cbcea613.pdf}{maximizing the minimum angle between edges leaving a node (angular resolution)},
\href{https://pdfs.semanticscholar.org/be7e/4c447ea27e0891397ae36d8957d3cbcea613.pdf}{maximizing consistent flow direction}, \href{https://arxiv.org/pdf/1908.07792.pdf}{clustering quality}.}}
stress in torus
https://dl.acm.org/doi/pdf/10.1145/3313831.3376180
\end{comment}
\section{The {$(SGD)^2$}\xspace Framework}
The {$(SGD)^2$}\xspace framework is a general optimization approach to generate a layout with any desired set of aesthetic metrics, provided that they can be expressed by a smooth function. The basic principles underlying this framework are simple.
The first step is to select a set of layout readability criteria and loss functions that measure each of them.
Then we define the function to optimize as a linear combination of the loss functions for each individual criterion.
Finally, we iterate the gradient descent steps, from which we obtain a slightly better drawing at each iteration.
Fig.~\ref{fig:gdgdframework} depicts the framework of {$(SGD)^2$}\xspace:
Given any graph with $n$ nodes and a readability criterion $Q$, we design a loss function $L_{Q}: \mathbb{R}^{n \times 2} \to \mathbb{R}$ that maps the current layout $X \in \mathbb{R}^{n \times 2}$ to a measure $L_{Q}(X)$ with respect to the readability criterion.
Then we combine multiple loss functions from different criteria into a single one by taking a weighted sum, $L(X) = \Sigma_{Q}w_Q L_{Q}(X)$, where a lower value is always desirable.
At each iteration, a slightly better layout can be found by taking a small ($\epsilon$) step along the (negative) gradient direction: $X^{(new)} = X - \epsilon \cdot \nabla\; L(X)$.
\blue{Algorithm~\ref{alg:gd2} summarises the {$(SGD)^2$}\xspace optimization procedure.}
\begin{figure}
\caption{The {$(SGD)^2$}
\label{fig:gdgdframework}
\end{figure}
\begin{algorithm}[t]
\DontPrintSemicolon
\caption{\blue{The {$(SGD)^2$}\xspace Algorithm}\label{alg:gd2}}
\KwInput{
\\
$G = (V,E)$ \tcp{graph}
$C = \{\dots c \dots\}$ \tcp{criteria}
$S: c \mapsto s_c$ \tcp{sample sizes for each $c$}
$L_c: \mathbb{R}^{s_c \times 2} \to \mathbb{R}_+$ \tcp{loss functions}
$maxiter \in \mathbb{Z}_+$ \tcp{number of iterations}
$W: c \mapsto w_c$, where $w_c: [1, maxiter] \to \mathbb{R_+}$ \tcp{weight schedules}
$\eta: [1, maxiter] \to \mathbb{R_+}$ \tcp{learning rate}
$q$ \tcp{criterion for safe update}
$Q_{q}$ \tcp{quality measure of $q$}
}
\KwOutput{
$X$ \tcp{layout that optimizes multiple criteria}
}
\Fn{Layout($G; C, S, W, maxiter, \eta$)}{
$X \leftarrow$ InitializeLayout($G$)\;
\If{`crossings' $\in C$}{
$cd \leftarrow$ InitializeCrossingDetector()\;
}
\For {$t = 1, \dots, maxiter$}{
$l \leftarrow 0$\;
\For {$c \in C$ s.t. $w_c(t) > 0$}{
$sample \leftarrow$ Sample($c, s_c$)\;
\If{c == `crossings'}{
UpdateCrossingDetector($cd, sample$)\;
$l_{c} \leftarrow L_{c}(sample; G, cd)$\;
}
\Else{
$l_{c} \leftarrow L_{c}(sample; G)$\;
}
$l \leftarrow l + w_c(t) \cdot l_c$\;
}
\If{`Safe update' is enabled}{
$X_{prev} \leftarrow X$\;
$X_{new} \leftarrow X - \eta(t) \cdot \nabla_{X} l$\;
$X \leftarrow$ SafeUpdate($X_{prev}, X_{new}; G, Q_q$) \tcp{Alg. \ref{alg:safe_update_2}}
}
\Else{
$X \leftarrow X - \eta(t) \cdot \nabla_{X} l$\;
}
}
Return X\;
}
\end{algorithm}
\subsection{Gradient Descent Optimization}
There are different kinds of gradient descent algorithms.
The standard method considers all nodes, computes the gradient of the objective function, and updates node coordinates based on the gradient.
Some objectives may consider all the nodes in every step.
For example, the basic stress formulation~\cite{kamada_1989} falls in this category.
To compute the gradient for optimization, one has to iterate through all the nodes which makes it not scalable to very large graphs.
Fortunately,most of these objectives can be decomposed into optimization over only subsets of nodes.
Consider stress minimization again, if we sample a set of node pairs randomly and minimize the stress between the nodes in each pair, the stress of the whole graph is also minimized~\cite{zheng2018graph}.
\blue{This approach is known as stochastic gradient descent (SGD) and we use this idea extensively.}
In section~\ref{sect:properties-and-measures}, we specify the objective loss functions and sampling methods we used for each readability criterion we consider.
Not all readability criteria come naturally in the form of differentiable functions.
We cannot compute the gradient of or apply SGD on non-differentiable functions.
In cases that the original objective is continuous but not everywhere differentiable e.g., a 'hinge' function f(x)=max(0,x), we can compute the subgradient and update the objective based on the subgradient.
Hence, as long as the function is continuously defined on a connected component in the domain, we can apply the subgradient descent algorithm.
When a function is not defined in a connected domain, we can introduce surrogate loss functions to `connect the pieces'.
For example, when optimizing neighborhood preservation we maximize the Jaccard similarity between graph neighbors and nearest neighbors in graph layout.
However, Jaccard similarity is only defined between two binary vectors.
To solve this problem we extend Jaccard similarity to all real vectors by its Lov\'{a}sz extension~\cite{berman2018lovasz} and apply that to optimize neighborhood preservation.
An essential part of gradient descent based algorithms is to compute the gradient/subgradient of the objective function.
In practice, it is not necessary to write down the gradient analytically as it can be computed automatically via (reverse-mode) automatic differentiation~\cite{griewank2008evaluating}.
Deep learning packages such as Tensorflow~\cite{abadi2016tensorflow} and PyTorch~\cite{paszke2019pytorch} apply automatic differentiation to compute the gradient of complicated functions.
\blue{
Most of the objective functions that we consider here are not convex and do not have unique global minimizers.
Therefore, even though SGD is known to converge (to at least a local optimum) in relatively relaxed settings~\cite{gower2019sgd, bassily2018exponential}, few optimization objectives are guaranteed to find the global optimum.
In particular, unlike methods such as stress majorization~\cite{gansner2004graph}, most of our optimization objectives are not guaranteed to converge to the global optimum.
Meanwhile, most of the objective functions for which SGD works well in practice (e.g., in deep learning) are neither convex nor have unique global minimizers~\cite{li2017visualizing}.
With this in mind, we follow the common practice of applying an annealing process, if necessary, to ensure convergence (to a possibly local minimum).
}
When optimizing multiple criteria simultaneously, we combine them via a weighted sum.
However, choosing a proper weight for each criterion can be tricky.
Consider, for example, maximizing crossing angles and minimizing stress simultaneously with a fixed pair of weights.
At the very early stage, the initial drawing may have many crossings and stress minimization often removes most of the early crossings.
As a result, maximizing crossing angles in the early stage can be harmful as it moves nodes in directions that contradict those that come from stress minimization.
Therefore, a well-tailored \textit{weight scheduling} is needed for a successful outcome.
Continuing with the same example, a better outcome can be achieved by first optimizing stress until it converges, and later adding weights for the crossing angle maximization.
To explore different ways of scheduling, we provide an interface that allows manual tuning of the weights.
\blue{
We consider weight schedules for different criteria sets in Section~\ref{sect:analysis-of-qualities}.
}
\subsection{Implementation}
We implemented the {$(SGD)^2$}\xspace framework in Python.
In particular we used PyTorch~\cite{paszke2019pytorch} automatic differentiation, NetworkX~\cite{networkx} for processing graphs, and matplotlib~\cite{matplotlib} for drawing.
The code is available at \url{https://github.com/tiga1231/graph-drawing/tree/sgd}.
To demonstrate our method with small graphs, we have provided an interactive tool written in JavaScript \footnote{ \url{http://hdc.cs.arizona.edu/~mwli/graph-drawing/}}, where we used the automatic differentiation tools in tensorflow.js~\cite{mlsys2019_154} and the drawing library D3.js~\cite{2011-d3}.
\begin{comment}
{
\begin{figure}
\caption{Layouts computed by {$(SGD)^2$}
\label{fig:blocknp}
\label{fig:blocknpue}
\label{fig:gridst}
\label{fig:treest}
\label{fig:rand_samples}
\end{figure}
}
\end{comment}
\section{Properties and Measures}\label{sect:properties-and-measures}
In this section we specify the aesthetic goals, definitions, quality measures and loss functions for each of the $9$ graph drawing properties we optimized: stress, ideal edge lengths, neighborhood preservation, crossing number, crossing angle, aspect ratio, angular resolution, node resolution and Gabriel graph property.
Other standard graph notation is summarized in Table~\ref{table:notations}.
In each subsection, we first define our loss function for the entire graph.
For small graphs, one can apply (full-batch) gradient descent directly on this loss.
To speed up our method for larger graphs, we sample portions of our loss functions at each iteration and apply (mini-batch) stochastic gradient descent on them.
\blue{
The definition of a sample can be different for each criterion.
For example, for stress minimization we sample pairs of nodes; for ideal edge length, we sample edges.
Hence, the sample sizes of different criteria can be set independently.
Moreover, when the sample size for a certain criterion exceeds the number of possible samples, our method is automatically equivalent to (full-batch) gradient descent for that criterion.
In Section~\ref{sect:analysis-of-sample-size}, we discuss the effect of the sample sizes on the convergence rates.
The analysis has helped us set the default values for each readability criterion.
In general, for each criterion we sample mini-batches from a pool of all sample points (e.g., all pairs of nodes for stress, all edges for ideal edge length) without replacement, and `refill the pool' when all sample points are drawn.
In practice, we shuffle the list of data points, draw mini-batches from the list in consecutive order, and re-shuffle the list once every data point is drawn.
}
\begin{table}[h]
\resizebox{\columnwidth}{!}{
\begin{tabular}{l|l}
\toprule
Notation & Description\\
\midrule
$G$ & Graph\\
$V$ & The set of nodes in $G$, indexed by $i$, $j$ or $k$\\
$E$ & The set of edges in $G$, indexed by a pair of nodes $i,j$ in $V$\\
$n=|V|$ & Number of nodes in $G$\\
$m$ & sample size for a certain criterion in SGD\\
$|E|$ & Number of edges in $G$\\
$Adj$ and $A_{i,j}$ & Adjacency matrix of $G$ and its $(i,j)$-th entry\\
$d_{ij}$ & Graph-theoretic distance between node $i$ and $j$\\
$X_{n \times 2}$ & 2D-coordinates of nodes in the drawing\\
$||X_i - X_j||$ & The Euclidean distance between nodes $i$ and $j$ \\
$\theta_i$ & $i^{th}$ crossing angle\\
$\varphi_{ijk}$ & Angle between incident edges $(i,j)$ and $(j,k)$\\
\bottomrule
\end{tabular}
\caption{Graph notation used in this paper.}
\label{table:notations}
}
\end{table}
\subsection{Stress}
We minimize stress, $L_{ST}$, to draw a graph that matches the Euclidean distances between pairs of nodes in the drawing to their graph theoretic distances.
Following the original definition of stress~\cite{kamada_1989}, we minimize
\begin{align}
L_{ST} = \sum\limits_{i<j}\;w_{ij}(||X_i - X_j||_2 - d_{ij})^2 \label{eq:loss-stress}
\end{align}
Where $d_{ij}$ is the graph-theoretical distance between nodes $i$ and $j$, $X_i$ and $X_j$ are the coordinates of nodes $i$ and $j$ in the layout.
The normalization factor $w_{ij}=d_{ij}^{-2}$ balances the influence of short and long distances: the longer the graph theoretic distance, the more tolerance we give to the discrepancy between two distances.
When comparing two drawings of the same graph with respect to stress, a smaller value (lower bounded by $0$) corresponds to a better drawing.
To work with large graphs, we take the mean loss for any pairs of nodes turn it into the expectation of stress
\begin{align}
\hat{L}_{ST} = \mathbb{E}_{i\neq j}\; [w_{ij}(||X_i - X_j||_2 - d_{ij})^2] \label{eq:loss-stress-expectation}
\end{align}
\blue{The quality measure for stress, $Q_{ST}$, is equal to the loss $\hat{L}_{ST}$ over all pairs of nodes.}
In each SGD iteration we minimize the loss by sampling a number of node pairs.
\blue{
Since the expectation of the gradient of the sample loss equals the true loss, we can use the gradient of the sample loss as an estimate of the true gradient and update the drawing through SGD accordingly.
In each SGD iteration, we sample $m$ pairs of nodes.
By default, we set $m=32$ based on our experiments with different sample sizes in Section~\ref{sect:analysis-of-sample-size}.
Before a round that goes over all
pairs of nodes, we shuffle a list of node-pairs and take mini-batches from the shuffled list.
This guarantees that we process every pair of nodes exactly once per round.
}
\subsection{Ideal Edge Length}
Given a set of ideal edge lengths $\{l_{ij}: (i,j) \in E\}$ we minimize the variance from the ideal lengths:
\begin{align}
L_{IL} &= \sum\limits_{(i,j) \in E}\;
(\frac{||X_i - X_j|| - l_{ij}}{l_{ij}})^2 \label{eq:loss-ideal-edge-length}
\end{align}
For unweighted graphs, by default we use $1$ as the ideal edge length for all edges $(i,j) \in E$.
As with stress minimization, for large graphs we replace the summation by the expectation and estimate it through sampling the edges.
\begin{align}
\hat{L}_{IL} &= \mathbb{E}_{(i,j) \in E}[
(\frac{||X_i - X_j|| - l_{ij}}{l_{ij}})^2
]
\label{eq:loss-ideal-edge-length-expectation}
\end{align}
\blue{
The quality measure $Q_{IL} = \hat{L}_{IL}$ is lower bounded by $0$ and a lower score yields a better layout.
Similar to the sampling strategy for stress, here we keep a list of all edges in random order, draw mini-batches (by default, of size $m=32$) from it, and re-shuffle the list after all edges are processed once.}
\subsection{Neighborhood Preservation}
\label{sec:neighbor}
Neighborhood preservation aims to keep adjacent nodes close to each other in the layout.
Similar to Kruiger et al.~\cite{kruiger2017graph}, the idea is to have the $k$-nearest (Euclidean) neighbors (k-NN) of node $i$ in the drawing to align with the $k$ nearest nodes (in terms of graph distance from $i$).
Here we choose $k$ to be the degree of node - for nodes of different degrees, we consider a different number of neighbors.
A natural quality measure for the alignment is the Jaccard index between the two pieces of information. Let, $Q_{NP} = JaccardIndex(K, Adj) = \frac{|\{(i,j): K_{ij}=1 \text{ and } A_{ij}=1\}|}{|\{(i,j): K_{ij}=1 \text{ or } A_{ij}=1\}|}$, where $Adj$ denotes the adjacency matrix and the $i$-th row in $K$ denotes the $k$-nearest neighborhood information of $i$:
$K_{ij} = 1$ if $j$ is one of the k-nearest neighbors of $i$ and $K_{ij}$ = 0 otherwise.
To express the Jaccard index as a differentiable minimization problem,
first, we express the neighborhood information in the drawing as a smooth function of node positions $X_i$ and store it in a matrix $\hat{K}$.
In $\hat{K}$, a positive entry $\hat{K}_{i,j}$ means node $j$ is one of the k-nearest neighbors of $i$, otherwise the entry is negative.
Next, we take a differentiable surrogate function of the Jaccard index, the Lov\'{a}sz hinge loss (LHL) given by Berman et al.~\cite{berman2018lovasz}, to make the Jaccard loss optimizable via gradient descent.
We minimize
\begin{align}
L_{NP} &= LHL(\hat{K}, Adj)\label{eq:lovasz-hinge}
\end{align}
where $\hat{K}$ denotes the $k$-nearest neighbor estimation.
For simplicity, let $d_{i,j}=||X_i - X_j||$ denote the Euclidean distance between node $i$ and $j$, then we design $\hat{K}$ as:
\begin{align}
\hat{K}_{i,j} &=
\left\{\begin{array}{ll}
-(d_{i,j} - \frac{d_{i,\pi_k} + d_{i,\pi_{k+1}}}{2} ) & \text{ if } i \neq j\\
0 & \text{ if } i=j\\
\end{array}\right.\label{eq:neighbor-pred}
\end{align}
where $\pi_{k}$ denotes the $k^{th}$ nearest neighbor of node $i$.
In other words, for every node $i$, we treat the average distance to its $k^{th} $ and $(k+1)^{th}$ nearest neighbor as a threshold, and use it to measure whether node $j$ is in the neighbor or not.
Note that $d_{i,j}$, $d_{i,\pi_k}$ and $d_{i,\pi_{k+1}}$ are all smooth functions of node positions in the layout, so $\hat{K}_{i,j}$ is also a smooth function of node positions $X$.
Furthermore, $\hat{K}_{i,j}$ is positive if node $j$ is a k-NN of node $i$, otherwise it is negative, as is required by LHL~\cite{berman2018lovasz}.
In order to handle large graphs we sample nodes for stochastic gradient descent.
However, note that the nearest neighbors $\pi_k$ and $\pi_{k+1}$ in $\hat{K}_{i,j}$ depend on distances from all nodes.
To derive a reliable estimation of the Jaccard index, instead of letting $k$ equal to the degree of node $i$ in the full graph, we need $k$ equal to the degree of the subgraph that we sample.
In other words, in every gradient descent iteration we sample a subgraph from the full graph and compute $LHL$ of the subgraph.
In practice, we randomly select a small set of $m$ nodes \blue{(by default, $m=16$)}, along with nodes that are $1$ or $2$ hops away from any of them.
We also include a fraction of nodes that are not already in the sample.
We extract the subgraph induced by this set of nodes and apply stochastic gradient descent.
\subsection{Crossing Number}
Reducing the number of edge crossings is one of the classic optimization goals in graph drawing, known to affect readability~\cite{Purchase1997}.
\blue{
Shabbeer et al.~\cite{bennett2010}, employed an expectation-maximization-like algorithm to minimize the number of crossings.
Since two edges do not cross if and only if there exists a line that separates their extreme points, they trained many support vector machine (SVM) classifiers to separate crossing pairs and use the classifiers as a guide to eliminate crossings.
Since one has to train as many SVM classifiers as the number of crossings in the graph and knowledge learned by one SVM does not naturally transfer to another, we found that this approach does not work well on large graphs.
With this in mind we modified our initial approach to that of Tiezzi et al.\cite{tiezzi2021graph}, which uses Graph Neural Networks to reduce the number of crossings in two steps.
First, they train a generic neural network to predict if any two edges cross each other.
Since neural networks are differentiable, the well-trained edge crossing predictor from this step will serve as a guide to gradient descent steps later on.
In the second step they train a Graph Neural Network and use the edge crossing predictor as a guide to improve the layout.
Our method uses only the first step above and utilizes a different training strategy.
Instead of training the edge crossing predictor using a synthetic dataset before the layout optimization, we train the crossing predictor directly on the current graph layout while simultaneously updating the node coordinates in the same graph, using the crossing predictor as a guide.
}
\blue{
Formally, let $f_\beta$ denote a neural network with trainable parameters $\beta$ that takes the coordinates of the four nodes of any two edges $X^{(i)} \in \mathbb{R}^{4 \times 2}$ and outputs a scalar from the $(0,1)$ interval.
An output close to $0$ means ``no crossing'' and one close to $1$ means ``crossing''.
In practice, $f_{\beta}$ is a simple multi-layer perceptron (MLP) with batch normalization~\cite{ioffe2015batch} and LeakyReLU activation.
To train a neural crossing detector $f_{\beta}$, we feed different edge pairs $X^{(i)} \in \mathbb{R}^{4 \times 2}$ to approximate the ground truth $t^{(i)} \in \{0, 1\}$ where $0$ means ``no crossing'' and $1$ means ``crossing''.
We optimize the parameters $\beta$ to minimize the cross entropy (CE) loss $L_{\beta}$ between the prediction $f_{\beta}(X^{(i)})$ and the ground truth $t^{(i)}$, averaging over a sample of $n$ instances of edge pairs:
$$
L_{\beta}(\beta;X^{(1)} \dots X^{(n)}) = \frac{1}{n}\sum\limits_{i=1}^n
CE(f_\beta (X^{(i)}), t^{(i)})
$$
where
\begin{align}
CE(y, t) := - t \cdot log(y) - (1-t) \cdot log(1-y) \label{eq:cross-entropy}
\end{align}
We use the neural crossing detector $f_\beta$ to construct a differentiable surrogate loss function for crossing minimization.
Specifically, given a well-trained $f_\beta$, we can reduce the number of crossings in a layout by minimizing the cross entropy between the prediction of edge pairs $f_\beta(X^{(i)})$ and the desired target (i.e., no crossing $t=0$):
\begin{align}
L_{CR}(X; \beta) = \frac{1}{n}\sum\limits_{i=1}^n CE(f_\beta (X^{(i)}), 0)
\end{align}
In practice, we minimize $L_{\beta}$ and $L_{CR}$ simultaneously in each {$(SGD)^2$}\xspace iteration.
We first improve the neural crossing predictor using a sample of edge pairs from the graph.
For simplicity, we describe the training by SGD, although in practice one can utilize any SGD variants (e.g. SGD with momentum, ADAM\cite{kingma2014adam} or RMSProp~\cite{Tieleman2012}) to train the predictor more efficiently.
\begin{align}
\beta^{(new)} = \beta - \epsilon' \cdot \nabla L_{\beta}
\end{align}
In the meantime we update the layout in a similar manner:
\begin{align}
X^{(new)} = X - \epsilon \cdot \nabla L_{CR}
\end{align}
Although one could improve the neural crossing predictor by multiple steps in every {$(SGD)^2$}\xspace iteration, we found little difference when varying the number of steps.
Therefore, we only take one step to improve the neural crossing predictor in every {$(SGD)^2$}\xspace iteration.
As with other criteria, we randomly draw mini-batches (by default, of size $m=128$) and iterate through all edge pairs over the course of the SGD iterations.
}
\blue{
When a graph layout does not have many crossings (e.g., a stress-minimized layout of a near-planar graph), this sampling strategy is not efficient.
In that case, we use an efficient algorithm (Bentley-Ottmann~\cite{bentley1979algorithms}) to find all crossing edges in a graph, and sample a mini-batch of crossings.
Since finding all crossings can be slow for large graphs, we only do this once every few iterations and reuse the finding across a few iterations.
Specifically, when we draw mini-batches from the pool of all crossings, we recompute all crossings again once the pool is drained.
To evaluate the quality we simply count the number of crossings.
}
\subsection{Crossing Angle Maximization}
When edge crossings are unavoidable, the graph drawing can still be easier to read when edges cross at angles close to 90 degrees~\cite{ware2002cognitive}.
Heuristics such as those by Demel et al.~\cite{Demel2018AGH} and Bekos et al.~\cite{Bekos18} have been proposed and have been successful in graph drawing challenges~\cite{devanny2017graph}.
We use an approach similar to the force-directed algorithm given by Eades et al.~\cite{eades2010force} and minimize the squared cosine of crossing angles:
\begin{align}
L_{CAM}
= \sum_{\substack{\text{all crossed edge pairs }\\(i,j), (k,l) \in E}}
(\frac{\langle X_{i}-X_{j}, X_{k}-X_{l}\rangle}{|X_{i}-X_{j}|\cdot|X_{k}-X_{l}|})^2
\end{align}
We evaluate quality by measuring the worst (normalized) absolute discrepancy between each crossing angle $\theta$ and the target crossing angle (i.e. 90 degrees):
$
Q_{CAM} = \max_{\theta} |\theta - \frac{\pi}{2}| / \frac{\pi}{2}
$.
As with crossing numbers, for large graphs we sample a subset \blue{(by default, of size $m=16$)} of edge pairs and consider their crossing angles if the edge pair cross each other.
Again, if there are not many crossing pairs we use an efficient algorithm to find all crossings.
When optimizing the number of crossings and crossing angles simultaneously, we sample from the same pool of crossings formed via the Bentley-Ottmann algorithm.
\subsection{Aspect Ratio}
Good use of drawing area is often measured by the aspect ratio~\cite{duncan1998balanced} of the bounding box of the drawing, with 1:1 as the optimum.
\blue{
The idea here is to consider different rotations of the current layout and try to ``squarify" the corresponding bounding boxes.
In practice,
we rely on the singular values of the matrix of node coordinates to approximate the worst aspect ratio.
Formally, assume vertex coordinates are centered with zero mean and let $X$ denote the collection of (centered) vertex coordinates as rows in a matrix.
Since the coordinates are two dimensional, $X$ has only two (non-zero) singular values, denoted by $\sigma_1$ and $\sigma_2$ and each measures the standard deviation of the layout along with two orthogonal directions.
Then we approximate the aspect ratio using the quotient of the two singular values of $X$ and encourage the ratio to be close to the target ratio $r=1$ using the cross entropy (CE) in Eq.~\ref{eq:cross-entropy}:
$$
L_{AR} = CE(\frac{\sigma_2}{\sigma_1}, r)
$$
Note that although we only consider 1:1 ratios, the formulation of cross entropy let us consider arbitrary ratios.
During mini-batch SGD, we simply sample a subset of nodes (by default, of size $m=128$) and use the singular values of the matrix formed by the subset to optimize the aspect ratio.
}
Finally, we evaluate the drawing quality by measuring the worst aspect ratio on a finite set of rotations.
The quality score ranges from 0 to 1.
In our case, 1 is optimal and the minimal ratio among different rotations is the worst:
$
Q_{AR} = \min_{
\theta \in \{
\frac{2\pi k}{N}, \text{ for } k=0, \cdots (N-1)
\}
} \frac{\min(w_{\theta}, h_{\theta})}{\max(w_{\theta}, h_{\theta})}
$,
where $N$ is the number of rotations sampled (e.g., $N=7$), and $w_{\theta}$, $h_{\theta}$ are the width and height of the bounding box when rotating the drawing around its center by an angle $\theta$.
\subsection{Angular Resolution}
Distributing edges adjacent to a node makes it easier to perceive the information presented in a node-link diagram~\cite{huang2013}.
Angular resolution~\cite{Argyriou2010}, defined as the minimum angle between incident edges, is one way to quantify this goal.
Formally,
$
ANR = \min_{j \in V} \min_{(i,j),(j,k) \in E} \varphi_{ijk}
$,
where $\varphi_{ijk}$ is the angle formed by between edges $(i,j)$ and $(j,k)$.
Note that for any given graph, an upper bound of this quantity is $\frac{2\pi}{d_{max}}$ where $d_{max}$ is the maximum degree of nodes in the graph.
Therefore in the evaluation, we will use this upper bound to normalize our quality measure to $[0,1]$, i.e.
$
Q_{ANR} = \frac{ANR}{2\pi / d_{max}}
$.
To achieve a better drawing quality via gradient descent, we define the angular energy of an angle $\varphi$ to be $e^{-s \cdot \varphi}$, where $s$ is a constant controlling the sensitivity of angular energy with respect to the angle (by default $s=1$), and minimize the total angular energy over all incident edges:
\begin{align}
L_{ANR} = \sum_{(i,j),(j,k) \in E} e^{-s \cdot \varphi_{ijk}}
\end{align}
When the graph is large, it is expensive to compute the energy of all pairs of incident edges.
Therefore, in {$(SGD)^2$}\xspace we randomly sample a minibatch of pairs of incident edges \blue{(by default, of size $m=128$)} and minimize the energy of the sample accordingly.
\begin{figure}
\caption{Optimizing Planar Graphs: (a) An initial layout without crossings, (b) A layout after optimizing stress while maintaining planarity.}
\label{fig:maintian_planarity}
\end{figure}
\begin{figure}
\caption{(a) The edge uniformity loss is increasing when we optimize the stress of a nested triangular graph, (b) The loss is decreasing when we update the coordinates carefully.}
\label{fig:maintian_EU_triangular}
\end{figure}
\subsection{Node Resolution}
Good node resolution is associated with the ability to distinguish different nodes by preventing nodes from occluding each other.
Node resolution is typically defined as the minimum Euclidean distance between two nodes in the drawing~\cite{chrobak1996convex,schulz2011drawing}.
However, in order to align with the units in other objectives such as stress, we normalize the minimum Euclidean distance with respect to a reference value.
Hence we define the node resolution to be the ratio between the shortest and longest distances between pairs of nodes in the drawing,
$
VR = \frac{\min_{i \neq j}||X_i - X_j||}{r \cdot d_{max}}
$,
where $d_{max} = \max_{k,l}||X_k - X_l||$.
To achieve a certain target resolution $r \in [0,1]$ by minimizing a loss function, we minimize
\begin{align}
L_{VR} = \sum_{i,j \in V, i \neq j}
max(
0,
( 1 - \frac{||X_i - X_j||}{r \cdot d_{max}})^2
)
\end{align}
In practice, we set the target resolution to be $r=\frac{1}{\sqrt{|V|}}$, where $|V|$ is the number of nodes in the graph.
In this way, an optimal drawing will distribute nodes uniformly in the drawing area.
Each term in the summation vanishes when the distance between two nodes meets the required resolution $r$, otherwise it is greater than zero.
In the evaluation, we report, as a quality measure, the ratio between the actual and target resolution and cap its value between $0$ (worst) and $1$ (best).
$
Q_{VR} = \min(1.0, \frac{\min_{i,j} ||X_i - X_j||}{r \cdot d_{max}})
$
For large graphs, we sample a subset of nodes \blue{(by default, of size $m=256$)} and compute the approximate loss of node resolution on the sample.
\subsection{Gabriel Graph Property}
A graph is a Gabriel graph if it can be drawn in such a way that any disk formed by using an edge in the graph as its diameter contains no other nodes.
Not all graphs are Gabriel graphs, but drawing a graph so that as many of these edge-based disks are empty of other nodes has been associated with good readability~\cite{10.1007/978-3-319-27261-0_41}.
This property can be enforced by a repulsive force around the midpoints of edges.
Formally, we establish a repulsive field with radius
$r_{ij}$ equal to half of the edge length, around the midpoint $c_{ij}$ of each edge $(i,j) \in E$, and we minimize the total potential energy:
\begin{align}
L_{GA} = \sum_{
\substack{
(i,j) \in E,\\
k \in V \setminus \{i,j\}
}}
max(0, r_{ij} - |X_k - c_{ij}|) \; ^ 2
\label{eq:gabriel}
\end{align}
where
$
c_{ij} = \frac{X_i + X_j}{2}
$ and
$
r_{ij} = \frac{|X_i - X_j|}{2}
$.
We use the (normalized) minimum distance from nodes to centers to characterize the quality of a drawing with respect to Gabriel graph property:
$
Q_{GA} = \min (1, \min_{(i,j) \in E, k \in V}\frac{|X_k - c_{ij}|}{r_{ij}})
$.
For large graphs, we sample a mini-batch of node-edge pairs \blue{(by default, of size $m=64$)} and compute the approximate loss from the sample.
\begin{comment}
\subsection{\color{red}Symmetry}
Different metrics have been proposed to measure symmetry~\cite{klapaukh2014empirical,klapaukh2018symmetry,brandes2007eigensolver,deluca2017experimental,dunne2015readability,gansner2005graph,koren2002ace,maaten2008visualizing,ortmann2016sparse}. Recently, Meidiana et al.~\cite{meidiana2020quality} proposed an efficient metric to compute symmetry, however, to use this metric, we need to have prior knowledge about the automorphism of the underlying graph. We use a similar metric proposed in~\cite{purchase2002metrics}. We make some modifications to fit the metric in our optimization framework. We consider a bisector of each line connecting a node pair. These bisectors are considered as symmetric axes. For each edge pair $e_1, e_2$ we take the reflection of $e_1$ with respect to a bisector. Then we compute the distance of the end nodes between $e_2$ and the reflection of $e_1$. If the distance is larger than a threshold we clip the distance to that threshold. Then we sum up the distances. We denote the bisector of node $u, v$ by $B(u, v)$, the distance of edge pair $e_1, e_2$ with respect to $B(u, v)$ by $d_{B(u, v)}(e_1, e_2)$, and the threshold by $t$. Then the loss value of symmetry, $$L_{SY} = \sum_{u, v \in V} \sum_{e_!, e_2 \in E} min(d_{B(u, v)}(e_1, e_2), t)$$. For our experiments we keep $t=1$. We provide an example of a symmetric layout computed using our framework in Figure~\ref{fig:optimize_symmetry_torch}.
\begin{figure}
\caption{Optimizing symmetry: (a) A random initial layout of a symmetric graph, (b) A layout after optimizing symmetry.}
\label{fig:optimize_symmetry_torch}
\end{figure}
\end{comment}
\begin{algorithm}[t]
\DontPrintSemicolon
\caption{\blue{Update coordinates without quality decline}}\label{alg:safe_update_2}
\Fn{SafeUpdate($X_{prev}, X_{new}; G, Q_q$)}{
$X \leftarrow X_{prev}$\;
$q_0 \leftarrow Q_q(X; G)$\;
\For{each node $u \in V$}{
$X[u] \leftarrow X_{new}[u]$\;
$q_{tmp} \leftarrow Q_q(X)$\;
\If{QualityDeclines($q_0, q_{tmp}$)}{
$X[u] \leftarrow X_{prev}[u]$\;
}
}
return X\;
}
\end{algorithm}
\begin{figure*}
\caption{\blue{Compatible pairs}
\caption{\blue{Better pairs}
\caption{\blue{Worse pairs}
\caption{\blue{
We observed three types of interactions between criteria pairs: \textbf{(a)}
\label{fig:three-pairs}
\end{figure*}
\begin{figure*}
\caption{
\blue{
Drawings of optimizing single or pair of criteria on a 6x10 grid with 60 nodes (lower left) or a 5-level balanced binary tree with 63 nodes (upper right). The sample size and weight for each criterion are shown in the diagonal entries of the figure.
}
\label{fig:analysis-pairs-drawings}
\end{figure*}
\subsection{Optimizing Layouts without Quality Decline}
Many optimization criteria tend to provide a drawing that has fewer crossings but cannot guarantee a crossing-free drawing. Stress is one such criterion, whose optimization does not guarantee crossing-free layout when graph is planar (or even a tree). One of the reasons for the popularity of stress-based layout methods is that they capture the underlying graph topology well. On the other hand, algorithms that are guaranteed to produce planar drawing (for planar graphs) are well known to dramatically distort the graph topology.
Our crossing minimization optimization is a soft constraint and does not guarantee a planar drawing when graph is planar. Hence, we provide an extra feature in our system that if we start with a planar drawing we can optimize any of the criteria above, while guaranteeing that no edge crossing will arise at any time. To do this, we add one additional test for every gradient descent step: for each proposed coordinate, we first check whether moving a node to this coordinate will introduce a crossing and if so, we do not update the coordinate. Using this method starting with an initial planar drawing, we can improve it, and provide a layout that is also planar and preserves the topology; see Fig.~\ref{fig:maintian_planarity}.
This technique can be useful in other force-directed algorithms that do not guarantee crossing-free drawing when the initial layout is without crossings~\cite{fowler2012planar}.
We can generalize this idea for any graphs while optimizing any criterion. In the above scenario, we maintained the number of crossings of the layouts equal to zero since we started with a planar layout. If the graph is non-planar then we can update the coordinates in a similar way and the number of crossings in the progressive layouts will be decreasing. Similarly, we can consider other criteria, for example, the edge uniformity loss to decrease in the progressive layouts. If we are optimizing another criterion, for example, stress, there is no guarantee that edge uniformity will improve, see Fig.~\ref{fig:maintian_EU_triangular}.
\blue{The general algorithm for safely update the layout with respect to any quality measure is described in Algorithm~\ref{alg:safe_update_2}.
Note that Algorithm~\ref{alg:safe_update_2} is applied in each SGD iteration of Algorithm \ref{alg:gd2}.
Furthermore, it goes through all nodes in the graph and the quality measure is evaluated every time an intermediate layout is generated by a single node update.
Hence, this approach does not scale well to large graphs when the quality measure requires high computational overhead.
}
\section{Experimental Evaluation}
In this section, we assess the effectiveness and limitations of our approach.
Since multiple criteria are not necessarily compatible with each other during optimization, we first identify all compatible pairs of criteria.
After identifying all compatible pairs, we hand-craft weighting schedules to optimize multiple criteria using {$(SGD)^2$}\xspace and compare our layouts from multicriteria optimization with classic drawing algorithms.
Finally, we analyze the runtime behavior and the impact of sample size in our approach.
\subsection{Interactions between Criteria}\label{sect:analysis-of-criteria-pairs}
\blue{We test the interactions between every pair of criteria using two regular graphs, a 6x10 grid (60 nodes) and a balanced binary tree with depth 5 (63 nodes).
Before dealing with pairs of criteria, we first test every single criterion to establish a lower bound of the quality measure.
In this section, we invert some quality measures (e.g., neighborhood preservation, and angular resolution) such that lower values are always better in all quality measures.
Then, we optimize every pair of criteria, monitor the quality measures of the pair over the course of training, and compare them with the corresponding lower bound found when optimizing each single criterion.}
\blue{As expected, we observe that all but one criterion, when optimized on their own, improve or maintain high quality during improvement iterations. The exception is crossing angle maximization, with a quality measure that depends on the worst crossing in the graph. The initial random layout usually has many crossings and maximizing crossing angles on its own (e.g., without also minimizing the number of crossings) does not necessarily lead to high-quality results.
Later we will see that optimizing other criteria together with crossing angle maximization helps. Further, when weight factors can be adjusted with a schedule, we recommend assigning positive weight to crossing angle maximization only at the later iterations.}
\blue{When optimizing pairs of aesthetic criteria, we see three types of pairs: compatible pairs, better pairs and worse pairs. Fig.~\ref{fig:three-pairs} shows an example for each of the three cases.
Most pairs are compatible pairs.
For example, stress minimization is compatible with most other drawing aesthetics, as the qualities of both optimization goals can improve over time and they both achieve their lower bounds.
Some pairs of criteria even do better together than alone.
For example, crossing angle maximization together with stress minimization leads to better results than just crossing angle maximization, confirming the results of Huang et al.~\cite{huang2013}.
A few pairs of criteria are not fully compatible, leading to worse joint optimization.
For example, when simultaneously optimizing vertex resolution and angular resolution, neither value can reach their corresponding lower bound.}
\begin{figure*}
\caption{
\blue{
Learning curves of optimizing pairs of criteria on a 6x10 grid with 60 nodes (lower left) or a 5-level balanced binary tree with 63 nodes (upper right). Better pairs are highlighted in green; worse pairs are highlighted in red. The sample size and weight for each criterion are shown in the diagonal entries of the figure.
In this figure, some quality measures (neighborhood preservation, aspect ratio, angular resolution, vertex resolution and gabriel) are inverted $Q \mapsto 1-Q$, some (stress and ideal edge length) are normalized to the [0,1] interval so that among all criteria smaller values are always better and the worst value is always 1.
}
\label{fig:analysis-pairs-learning-curves}
\end{figure*}
\blue{Out of all 36 pairs, we find we find 20 compatible pairs, 9 better pairs and 7 worse pairs for the $6 \times 10$ grid; for the binary tree with depth 5, we find
13 compatible pairs, 9 better pairs and 14 worse pairs.
Comparing the compatibility between the two graphs, we note that all worse pairs in the grid are also worse pairs in the tree, and most of the better pairs and compatible pairs are shared between two graphs.
See
the additional figures
(Fig.~\ref{fig:analysis-pairs-drawings} and \ref{fig:analysis-pairs-learning-curves})
for the drawings and quality curves of all criteria pairs and singletons.
It is worth noting that the compatibility of criteria also depends on the specific choice of weight factors.
For example, having a dominating criterion by assigning a large weight to it can deteriorate the quality of the other.
In this analysis, we assign a fixed weight factor (and sample size) to each criterion so that every pair yields a reasonable outcome.}
\begin{figure*}
\caption{
\blue{
Distinctive drawings from different algorithms: we compare the drawings of neato, sfdp and {$(SGD)^2$}
\label{fig:all-drawings-partial}
\end{figure*}
\begin{table*}[thbp]
\begin{tabular}{l|rrrrrrrr}
\hline
methods \textbackslash~graphs & dodecahedron & tree-2-6 & grid-12-24 & spx-teaser & 494-bus & grid1 & dwt-307 & dwt-1005 \\
\hline
neato & 0.081 & \textbf{0.078} & \textbf{0.013} & \textbf{0.027} & 0.076 & \textbf{0.062} & 0.083 & \textbf{0.022} \\
sfdp & 0.080 & 0.133 & 0.024 & 0.052 & 0.099 & 0.071 & \textbf{0.081} & 0.029 \\
{$(SGD)^2$}\xspace (ST) & \textbf{0.079} & \textbf{0.078} & \textbf{0.013} & \textbf{0.027} & \textbf{0.071} & \textbf{0.062} & 0.082 & \textbf{0.022} \\
{$(SGD)^2$}\xspace (NP) & 0.188 & 0.233 & 0.065 & 0.241 & 0.825 & 0.149 & 0.279 & 0.275 \\
{$(SGD)^2$}\xspace (ST+IL) & 0.107 & 0.100 & 0.033 & 0.054 & 0.090 & 0.100 & 0.119 & 0.043 \\
{$(SGD)^2$}\xspace (ST+NP) & 0.188 & 0.106 & \textbf{0.013} & 0.051 & 0.739 & 0.080 & 0.178 & 0.059 \\
{$(SGD)^2$}\xspace (ST+CR) & 0.190 & \textbf{0.078} & \textbf{0.013} & 0.028 & 0.079 & 0.073 & 0.091 & 0.045 \\
{$(SGD)^2$}\xspace (ST+CAM) & 0.099 & \textbf{0.078} & 0.015 & 0.029 & 0.075 & 0.063 & 0.094 & 0.029 \\
{$(SGD)^2$}\xspace (ST+AR) & 0.080 & 0.081 & 0.055 & \textbf{0.027} & 0.075 & 0.067 & 0.084 & 0.023 \\
{$(SGD)^2$}\xspace (ST+VR) & 0.083 & 0.080 & \textbf{0.013} & 0.032 & 0.073 & 0.063 & 0.088 & 0.023 \\
{$(SGD)^2$}\xspace (ST+GB) & 0.080 & \textbf{0.078} & \textbf{0.013} & \textbf{0.027} & \textbf{0.071} & \textbf{0.062} & 0.083 & \textbf{0.022} \\
{$(SGD)^2$}\xspace (ST+IL+ANR) & 0.089 & 0.098 & 0.020 & 0.036 & 0.080 & 0.084 & 0.096 & 0.027 \\
{$(SGD)^2$}\xspace (IL+NP+VR) & 0.083 & 0.119 & 0.061 & 0.134 & 0.208 & 0.068 & 0.110 & 0.205 \\
\hline
\end{tabular}
\caption{\blue{Quality Measures of Stress (ST)}}
\label{tab:quality-table-stress}
\end{table*}
\begin{table*}[thbp]
\begin{tabular}{l|rrrrrrrr}
\hline
methods \textbackslash~graphs & dodecahedron & tree-2-6 & grid-12-24 & spx-teaser & 494-bus & grid1 & dwt-307 & dwt-1005 \\
\hline
neato & 0.028 & 0.005 & \textbf{0.002} & 0.099 & 0.484 & 0.026 & 0.299 & 0.280 \\
sfdp & 0.018 & 0.143 & 0.051 & 0.271 & 0.585 & 0.052 & 0.279 & 0.346 \\
{$(SGD)^2$}\xspace (ST) & 0.062 & 0.103 & 0.053 & 0.114 & 0.526 & 0.111 & 0.320 & 0.293 \\
{$(SGD)^2$}\xspace (NP) & 0.679 & 0.867 & 0.351 & 0.884 & 3.348 & 0.649 & 1.083 & 1.312 \\
{$(SGD)^2$}\xspace (ST+IL) & \textbf{0.004} & \textbf{0.003} & \textbf{0.002} & \textbf{0.009} & \textbf{0.462} & \textbf{0.002} & \textbf{0.249} & \textbf{0.245} \\
{$(SGD)^2$}\xspace (ST+NP) & 0.679 & 0.227 & 0.056 & 0.434 & 1.608 & 0.418 & 0.749 & 0.519 \\
{$(SGD)^2$}\xspace (ST+CR) & 0.517 & 0.103 & 0.053 & 0.139 & 0.593 & 0.205 & 0.394 & 0.482 \\
{$(SGD)^2$}\xspace (ST+CAM) & 0.109 & 0.103 & 0.058 & 0.132 & 0.595 & 0.141 & 0.497 & 0.368 \\
{$(SGD)^2$}\xspace (ST+AR) & 0.061 & 0.112 & 0.203 & 0.114 & 0.533 & 0.148 & 0.334 & 0.302 \\
{$(SGD)^2$}\xspace (ST+VR) & 0.083 & 0.257 & 0.067 & 0.159 & 0.569 & 0.155 & 0.399 & 0.309 \\
{$(SGD)^2$}\xspace (ST+GB) & 0.061 & 0.098 & 0.053 & 0.092 & 0.521 & 0.111 & 0.338 & 0.291 \\
{$(SGD)^2$}\xspace (ST+IL+ANR) & 0.016 & 0.027 & 0.012 & 0.029 & 0.486 & 0.027 & 0.277 & 0.268 \\
{$(SGD)^2$}\xspace (IL+NP+VR) & 0.082 & 0.178 & 0.116 & 0.242 & 0.702 & 0.147 & 0.388 & 0.549 \\
\hline
\end{tabular}
\caption{\blue{Quality Measures of Ideal Edge Length (IL)}}
\label{tab:quality-table-ideal_edge_length}
\end{table*}
\begin{table*}[thbp]
\begin{tabular}{l|rrrrrrrr}
\hline
methods \textbackslash~graphs & dodecahedron & tree-2-6 & grid-12-24 & spx-teaser & 494-bus & grid1 & dwt-307 & dwt-1005 \\
\hline
neato & 0.723 & 0.718 & \textbf{0.000} & 0.474 & 0.846 & 0.558 & 0.699 & 0.545 \\
sfdp & 0.571 & 0.592 & 0.063 & 0.533 & 0.750 & 0.651 & 0.584 & 0.653 \\
{$(SGD)^2$}\xspace (ST) & 0.500 & 0.721 & \textbf{0.000} & 0.503 & 0.832 & 0.553 & 0.657 & 0.548 \\
{$(SGD)^2$}\xspace (NP) & \textbf{0.400} & 0.225 & 0.276 & 0.487 & \textbf{0.659} & 0.480 & \textbf{0.428} & 0.516 \\
{$(SGD)^2$}\xspace (ST+IL) & 0.667 & 0.701 & \textbf{0.000} & 0.518 & 0.867 & 0.674 & 0.766 & 0.665 \\
{$(SGD)^2$}\xspace (ST+NP) & \textbf{0.400} & \textbf{0.194} & \textbf{0.000} & 0.467 & 0.664 & \textbf{0.347} & 0.472 & \textbf{0.441} \\
{$(SGD)^2$}\xspace (ST+CR) & 0.481 & 0.714 & \textbf{0.000} & \textbf{0.420} & 0.765 & 0.532 & 0.582 & 0.567 \\
{$(SGD)^2$}\xspace (ST+CAM) & 0.681 & 0.721 & 0.042 & 0.531 & 0.851 & 0.552 & 0.763 & 0.631 \\
{$(SGD)^2$}\xspace (ST+AR) & 0.500 & 0.727 & 0.418 & 0.492 & 0.842 & 0.560 & 0.757 & 0.584 \\
{$(SGD)^2$}\xspace (ST+VR) & 0.750 & 0.708 & \textbf{0.000} & 0.609 & 0.834 & 0.527 & 0.709 & 0.570 \\
{$(SGD)^2$}\xspace (ST+GB) & 0.696 & 0.727 & \textbf{0.000} & 0.566 & 0.831 & 0.555 & 0.702 & 0.547 \\
{$(SGD)^2$}\xspace (ST+IL+ANR) & 0.500 & 0.749 & \textbf{0.000} & 0.539 & 0.823 & 0.622 & 0.705 & 0.520 \\
{$(SGD)^2$}\xspace (IL+NP+VR) & 0.636 & \textbf{0.194} & 0.165 & 0.472 & 0.713 & 0.349 & 0.497 & 0.717 \\
\hline
\end{tabular}
\caption{\blue{Quality Measures of Neighborhood Preservation (NP)}}
\label{tab:quality-table-neighborhood_preservation}
\end{table*}
\begin{table*}[thbp]
\begin{tabular}{l|rrrrrrrr}
\hline
methods \textbackslash~graphs & dodecahedron & tree-2-6 & grid-12-24 & spx-teaser & 494-bus & grid1 & dwt-307 & dwt-1005 \\
\hline
neato & 8 & 1 & \textbf{0} & 80 & 274 & 134 & 1896 & 8001 \\
sfdp & 9 & 2 & \textbf{0} & \textbf{72} & 141 & 109 & 1651 & \textbf{5447} \\
{$(SGD)^2$}\xspace (ST) & 10 & \textbf{0} & \textbf{0} & 73 & 283 & 133 & 1749 & 8367 \\
{$(SGD)^2$}\xspace (NP) & 10 & 5 & 109 & 155 & 216 & \textbf{80} & 1251 & 10736 \\
{$(SGD)^2$}\xspace (ST+IL) & 10 & \textbf{0} & \textbf{0} & 73 & 325 & 151 & 3924 & 13094 \\
{$(SGD)^2$}\xspace (ST+NP) & 6 & \textbf{0} & \textbf{0} & 111 & \textbf{131} & 94 & \textbf{265} & 6932 \\
{$(SGD)^2$}\xspace (ST+CR) & \textbf{0} & \textbf{0} & \textbf{0} & 73 & 187 & 133 & 347 & 8827 \\
{$(SGD)^2$}\xspace (ST+CAM) & 7 & \textbf{0} & 9 & 84 & 430 & 158 & 4673 & 12972 \\
{$(SGD)^2$}\xspace (ST+AR) & 10 & \textbf{0} & 2 & \textbf{72} & 303 & 142 & 2836 & 8628 \\
{$(SGD)^2$}\xspace (ST+VR) & 7 & \textbf{0} & \textbf{0} & 79 & 282 & 134 & 2983 & 8874 \\
{$(SGD)^2$}\xspace (ST+GB) & 6 & \textbf{0} & \textbf{0} & 75 & 264 & 134 & 2034 & 8305 \\
{$(SGD)^2$}\xspace (ST+IL+ANR) & 10 & 24 & \textbf{0} & \textbf{72} & 302 & 171 & 2793 & 8241 \\
{$(SGD)^2$}\xspace (IL+NP+VR) & 6 & 2 & 96 & 105 & 173 & 103 & 1333 & 18276 \\
\hline
\end{tabular}
\caption{\blue{Quality Measures of Crossings (CR)}}
\label{tab:quality-table-crossings}
\end{table*}
\begin{table*}[thbp]
\begin{tabular}{l|rrrrrrrr}
\hline
methods \textbackslash~graphs & dodecahedron & tree-2-6 & grid-12-24 & spx-teaser & 494-bus & grid1 & dwt-307 & dwt-1005 \\
\hline
neato & 0.254 & 0.419 & \textbf{0.000} & 0.786 & 0.956 & 0.927 & 0.970 & 0.999 \\
sfdp & 0.622 & 0.235 & \textbf{0.000} & 0.806 & 0.972 & 0.970 & 0.963 & 1.000 \\
{$(SGD)^2$}\xspace (ST) & 0.601 & \textbf{0.000} & \textbf{0.000} & 0.975 & 0.923 & 0.946 & 0.983 & 0.990 \\
{$(SGD)^2$}\xspace (NP) & 0.969 & 0.808 & 0.945 & 0.999 & 0.913 & 0.939 & 0.988 & 0.992 \\
{$(SGD)^2$}\xspace (ST+IL) & 0.602 & \textbf{0.000} & \textbf{0.000} & 0.575 & 0.970 & 0.837 & 0.989 & 0.998 \\
{$(SGD)^2$}\xspace (ST+NP) & 0.969 & \textbf{0.000} & \textbf{0.000} & 1.000 & \textbf{0.838} & 0.864 & 0.958 & 0.992 \\
{$(SGD)^2$}\xspace (ST+CR) & \textbf{0.000} & \textbf{0.000} & \textbf{0.000} & 0.940 & 0.885 & 0.889 & 0.955 & \textbf{0.984} \\
{$(SGD)^2$}\xspace (ST+CAM) & \textbf{0.000} & \textbf{0.000} & 0.810 & 0.646 & 0.937 & \textbf{0.622} & 0.973 & 0.993 \\
{$(SGD)^2$}\xspace (ST+AR) & 0.605 & \textbf{0.000} & 0.757 & 0.712 & 0.985 & 0.965 & 0.978 & 0.995 \\
{$(SGD)^2$}\xspace (ST+VR) & 0.164 & \textbf{0.000} & \textbf{0.000} & 0.550 & 0.943 & 0.883 & 0.975 & 0.995 \\
{$(SGD)^2$}\xspace (ST+GB) & 0.524 & \textbf{0.000} & \textbf{0.000} & 0.834 & 0.945 & 0.937 & 0.992 & 0.991 \\
{$(SGD)^2$}\xspace (ST+IL+ANR) & 0.603 & 0.725 & \textbf{0.000} & \textbf{0.421} & 0.935 & 0.961 & 0.973 & 0.997 \\
{$(SGD)^2$}\xspace (IL+NP+VR) & 0.323 & 0.143 & 0.954 & 0.963 & 0.913 & 0.631 & \textbf{0.926} & 0.991 \\
\hline
\end{tabular}
\caption{\blue{Quality Measures of Crossing Angle Maximization (CAM)}}
\label{tab:quality-table-crossing_angle_maximization}
\end{table*}
\begin{table*}[thbp]
\begin{tabular}{l|rrrrrrrr}
\hline
methods \textbackslash~graphs & dodecahedron & tree-2-6 & grid-12-24 & spx-teaser & 494-bus & grid1 & dwt-307 & dwt-1005 \\
\hline
neato & 0.062 & 0.145 & 0.483 & 0.010 & 0.143 & 0.314 & 0.065 & \textbf{0.012} \\
sfdp & 0.068 & 0.084 & 0.536 & 0.010 & 0.192 & 0.452 & 0.095 & 0.018 \\
{$(SGD)^2$}\xspace (ST) & 0.049 & 0.150 & 0.470 & \textbf{0.002} & 0.155 & 0.315 & 0.071 & 0.094 \\
{$(SGD)^2$}\xspace (NP) & \textbf{0.047} & 0.124 & 0.481 & 0.176 & 0.162 & \textbf{0.049} & 0.109 & 0.282 \\
{$(SGD)^2$}\xspace (ST+IL) & \textbf{0.048} & 0.169 & 0.507 & 0.018 & 0.160 & 0.470 & 0.157 & 0.045 \\
{$(SGD)^2$}\xspace (ST+NP) & 0.049 & 0.126 & 0.479 & \textbf{0.002} & 0.192 & 0.209 & 0.077 & 0.101 \\
{$(SGD)^2$}\xspace (ST+CR) & 0.134 & 0.149 & 0.473 & 0.028 & 0.136 & 0.368 & 0.030 & 0.079 \\
{$(SGD)^2$}\xspace (ST+CAM) & 0.068 & 0.149 & 0.449 & 0.009 & 0.188 & 0.288 & \textbf{0.028} & 0.141 \\
{$(SGD)^2$}\xspace (ST+AR) & \textbf{0.047} & \textbf{0.017} & \textbf{0.048} & 0.008 & \textbf{0.118} & 0.154 & 0.043 & 0.057 \\
{$(SGD)^2$}\xspace (ST+VR) & 0.068 & 0.135 & 0.466 & 0.009 & 0.152 & 0.300 & 0.044 & 0.091 \\
{$(SGD)^2$}\xspace (ST+GB) & 0.069 & 0.154 & 0.470 & 0.013 & 0.155 & 0.318 & 0.037 & 0.093 \\
{$(SGD)^2$}\xspace (ST+IL+ANR) & \textbf{0.048} & 0.197 & 0.508 & 0.045 & 0.178 & 0.269 & 0.058 & 0.084 \\
{$(SGD)^2$}\xspace (IL+NP+VR) & 0.101 & 0.231 & 0.455 & 0.242 & 0.359 & 0.282 & 0.074 & 0.106 \\
\hline
\end{tabular}
\caption{\blue{Quality Measures of Aspect Ratio (AR)}}
\label{tab:quality-table-aspect_ratio}
\end{table*}
\begin{table*}[thbp]
\begin{tabular}{l|rrrrrrrr}
\hline
methods \textbackslash~graphs & dodecahedron & tree-2-6 & grid-12-24 & spx-teaser & 494-bus & grid1 & dwt-307 & dwt-1005 \\
\hline
neato & 0.753 & 0.749 & 0.525 & 0.999 & 0.998 & 1.000 & 1.000 & \textbf{1.000} \\
sfdp & 0.459 & 0.908 & 0.547 & 0.996 & 0.999 & 0.996 & 1.000 & \textbf{1.000} \\
{$(SGD)^2$}\xspace (ST) & \textbf{0.401} & 0.750 & 0.524 & 0.989 & \textbf{0.993} & 0.941 & 0.999 & \textbf{1.000} \\
{$(SGD)^2$}\xspace (NP) & 0.928 & 0.976 & 0.994 & 0.996 & 1.000 & 0.984 & 1.000 & \textbf{1.000} \\
{$(SGD)^2$}\xspace (ST+IL) & 0.517 & 0.844 & 0.497 & 0.939 & 0.999 & 0.932 & 0.999 & \textbf{1.000} \\
{$(SGD)^2$}\xspace (ST+NP) & 0.952 & 0.466 & \textbf{0.487} & 1.000 & 0.999 & 0.998 & 0.999 & \textbf{1.000} \\
{$(SGD)^2$}\xspace (ST+CR) & 0.746 & 0.769 & 0.526 & 0.996 & 0.998 & 0.969 & \textbf{0.998} & \textbf{0.999} \\
{$(SGD)^2$}\xspace (ST+CAM) & 1.000 & 0.745 & 0.871 & 0.913 & 1.000 & 0.999 & 1.000 & \textbf{0.999} \\
{$(SGD)^2$}\xspace (ST+AR) & 0.403 & 0.779 & 0.991 & 0.963 & 0.998 & 0.994 & 0.999 & \textbf{1.000} \\
{$(SGD)^2$}\xspace (ST+VR) & 0.760 & 0.979 & 0.540 & 0.936 & 1.000 & 0.914 & 1.000 & \textbf{1.000} \\
{$(SGD)^2$}\xspace (ST+GB) & 0.573 & 0.750 & 0.528 & \textbf{0.868} & 1.000 & 0.903 & 1.000 & \textbf{1.000} \\
{$(SGD)^2$}\xspace (ST+IL+ANR) & \textbf{0.401} & \textbf{0.327} & 0.528 & 0.990 & 1.000 & \textbf{0.740} & \textbf{0.999} & \textbf{0.999} \\
{$(SGD)^2$}\xspace (IL+NP+VR) & 0.677 & 0.418 & 0.974 & 0.995 & 0.999 & 0.823 & \textbf{0.999} & \textbf{1.000} \\
\hline
\end{tabular}
\caption{\blue{Quality Measures of Angular Resolution (ANR)}}
\label{tab:quality-table-angular_resolution}
\end{table*}
\begin{table*}[thbp]
\begin{tabular}{l|rrrrrrrr}
\hline
methods \textbackslash~graphs & dodecahedron & tree-2-6 & grid-12-24 & spx-teaser & 494-bus & grid1 & dwt-307 & dwt-1005 \\
\hline
neato & 0.637 & 0.735 & 0.362 & 0.765 & 0.948 & 0.780 & 0.870 & \textbf{0.931} \\
sfdp & 0.391 & 0.594 & 0.807 & 0.825 & 0.951 & 0.877 & 0.859 & 0.967 \\
{$(SGD)^2$}\xspace (ST) & 0.269 & 0.727 & 0.368 & 0.663 & 0.991 & 0.798 & 0.869 & 0.976 \\
{$(SGD)^2$}\xspace (NP) & 1.000 & 0.875 & 0.902 & 0.992 & 0.986 & 0.814 & 0.827 & 0.987 \\
{$(SGD)^2$}\xspace (ST+IL) & 0.334 & 0.705 & \textbf{0.354} & 0.788 & 0.993 & 0.973 & 0.994 & 0.993 \\
{$(SGD)^2$}\xspace (ST+NP) & 1.000 & 0.688 & 0.454 & 1.000 & 0.989 & 0.791 & 0.862 & 0.977 \\
{$(SGD)^2$}\xspace (ST+CR) & 0.570 & 0.762 & 0.379 & 0.789 & 0.954 & 0.908 & 0.785 & 0.950 \\
{$(SGD)^2$}\xspace (ST+CAM) & 0.996 & 0.715 & 0.839 & 0.789 & 0.974 & 0.856 & 0.967 & 0.959 \\
{$(SGD)^2$}\xspace (ST+AR) & 0.278 & 0.682 & 0.633 & 0.698 & 0.938 & 0.802 & 0.963 & 0.991 \\
{$(SGD)^2$}\xspace (ST+VR) & \textbf{0.165} & \textbf{0.266} & 0.388 & \textbf{0.407} & \textbf{0.754} & \textbf{0.590} & \textbf{0.617} & 0.945 \\
{$(SGD)^2$}\xspace (ST+GB) & 0.529 & 0.737 & 0.365 & 0.630 & 0.895 & 0.858 & 0.913 & 0.986 \\
{$(SGD)^2$}\xspace (ST+IL+ANR) & 0.332 & 0.959 & 0.392 & 0.873 & 0.961 & 0.959 & 0.931 & 0.984 \\
{$(SGD)^2$}\xspace (IL+NP+VR) & 0.355 & 0.615 & 0.795 & 0.744 & 0.831 & \textbf{0.590} & 0.751 & 0.993 \\
\hline
\end{tabular}
\caption{\blue{Quality Measures of Vertex Resolution (VR)}}
\label{tab:quality-table-vertex_resolution}
\end{table*}
\begin{table*}[thbp]
\begin{tabular}{l|rrrrrrrr}
\hline
methods \textbackslash~graphs & dodecahedron & tree-2-6 & grid-12-24 & spx-teaser & 494-bus & grid1 & dwt-307 & dwt-1005 \\
\hline
neato & 0.429 & 0.595 & \textbf{0.000} & 0.933 & \textbf{1.000} & 0.920 & \textbf{1.000} & \textbf{1.000} \\
sfdp & 0.860 & 0.806 & 0.148 & 0.758 & \textbf{1.000} & 0.967 & \textbf{1.000} & \textbf{1.000} \\
{$(SGD)^2$}\xspace (ST) & 0.677 & 0.130 & \textbf{0.000} & 0.795 & \textbf{1.000} & 0.963 & \textbf{1.000} & \textbf{1.000} \\
{$(SGD)^2$}\xspace (NP) & 0.951 & 0.954 & 0.973 & 0.920 & \textbf{1.000} & 0.903 & \textbf{1.000} & \textbf{1.000} \\
{$(SGD)^2$}\xspace (ST+IL) & 0.679 & \textbf{0.000} & \textbf{0.000} & 0.834 & \textbf{1.000} & 0.860 & \textbf{1.000} & \textbf{1.000} \\
{$(SGD)^2$}\xspace (ST+NP) & 0.951 & 0.658 & \textbf{0.000} & \textbf{0.039} & \textbf{1.000} & 0.916 & \textbf{1.000} & \textbf{1.000} \\
{$(SGD)^2$}\xspace (ST+CR) & 0.704 & 0.205 & \textbf{0.000} & 0.967 & \textbf{1.000} & 0.988 & \textbf{1.000} & \textbf{1.000} \\
{$(SGD)^2$}\xspace (ST+CAM) & 0.695 & 0.199 & 0.799 & 0.836 & \textbf{1.000} & 0.905 & \textbf{1.000} & \textbf{1.000} \\
{$(SGD)^2$}\xspace (ST+AR) & 0.678 & 0.396 & 0.887 & 0.903 & \textbf{1.000} & 0.958 & \textbf{1.000} & \textbf{1.000} \\
{$(SGD)^2$}\xspace (ST+VR) & 0.437 & 0.957 & \textbf{0.000} & 0.862 & \textbf{1.000} & 0.987 & \textbf{1.000} & \textbf{1.000} \\
{$(SGD)^2$}\xspace (ST+GB) & \textbf{0.368} & 0.036 & \textbf{0.000} & 0.664 & \textbf{1.000} & \textbf{0.791} & \textbf{1.000} & \textbf{1.000} \\
{$(SGD)^2$}\xspace (ST+IL+ANR) & 0.681 & 0.745 & \textbf{0.000} & 0.630 & \textbf{1.000} & 0.967 & \textbf{1.000} & \textbf{1.000} \\
{$(SGD)^2$}\xspace (IL+NP+VR) & 0.421 & 0.800 & 0.982 & 0.926 & \textbf{1.000} & 0.972 & \textbf{1.000} & \textbf{1.000} \\
\hline
\end{tabular}
\caption{\blue{Quality Measures of Gabriel Property (GB)}}
\label{tab:quality-table-gabriel}
\end{table*}
\subsection{Quality Analysis}\label{sect:analysis-of-qualities}
\blue{We compare layouts obtained with {$(SGD)^2$}\xspace when optimizing different aesthetic goals to layouts obtained by neato~\cite{ellson2001graphviz} and sfdp~\cite{ellson2001graphviz}, which are classic implementations of stress-majorization and scalable force-directed methods.
Fig.~\ref{fig:all-drawings-partial} shows the layouts along with information about each graph.
The graphs are chosen to represent a variety of classes such as trees, grids, regular shapes, and to also include real-world examples.
In particular, the last four graphs in Fig.~\ref{fig:all-drawings-partial} are from the Sparse Matrix Collection~\cite{davis2011university} and are also used to evaluate stress minimization via SGD in~\cite{zheng2018graph}; see the supplementary materials for more layouts.
}
\blue{
Next, we evaluate each layout on 9 readability criteria: stress (\texttt{ST}), node resolution (\texttt{VR}), ideal edge lengths (\texttt{IL}), neighbor preservation (\texttt{NP}), crossings (\texttt{CR}), crossing angle (\texttt{CA}), angular resolution (\texttt{ANR}), aspect ratio (\texttt{AR}), and Gabriel graph property (\texttt{GB}).
Our experiment utilizes 8 graphs and layouts computed by neato, sfdp, and 7 runs of {$(SGD)^2$}\xspace using various combinations of objectives.
\blue{
Tables~\ref{tab:quality-table-stress} to~\ref{tab:quality-table-gabriel} summarize the first 4 of the 9 quality measures for the layouts in Fig.~\ref{fig:all-drawings-partial}.
More combinations of criteria used for {$(SGD)^2$}\xspace and the remaining quality measures are included in the supplementary materials.
The quality measure for crossings is the actual number of edge crossings in the layout. For all other criteria, we use the formulas defined in Section~\ref{sect:properties-and-measures}. All quality measures produce values greater than or equal to zero: the lower the value the better the measure. In each column, the best score is bold.}
When optimizing via multicriteria {$(SGD)^2$}\xspace, we choose compatible pairs, better pairs, or compatible triples among the 9 criteria.
When optimizing incompatible pairs or triples, we fix the number of iterations in {$(SGD)^2$}\xspace, select and prioritize one criterion (or compatible pair) in an early stage of the training and postpone the others to the later stage.
For example, when simultaneously optimizing ideal edge length (IL), neighborhood preservation (NP) and vertex resolution (VR), we assign zero weight to VR and positive weights to IL and NP in the first half of the iterations. Then we gradually decrease the weights of IL and NP to 0 (by a smooth function that interpolates the highest and lowest weights) and increase the weight of VR in the second half of the iterations with a similar smooth growth function.
At each stage, we interpolate the two weight levels of each criterion $w_{start}$ and $w_{stop}$ between the start and stopping iterations $t_{start}$ to $t_{stop}$ by a scaled and translated smooth-step function $g(t)$:
$$
g(t) = (w_{stop}-w_{start}) \cdot f(\frac{t-t_{start}}{t_{stop}-t_{start}}) + w_{start}
$$
where $f(x) = 3x^2-2x^3$ for $x \in [0,1]$ is typically called the (standard) smooth-step function.
}
\blue{
The experimental results confirm that {$(SGD)^2$}\xspace yields better or comparable results for most quality measures and on most graphs.
We do note that some criteria (e.g., CR and GB) are harder to optimize on real-world large graphs; improving the performance on such tough criteria is natural direction for future work.}
\begin{figure*}
\caption{\blue{Optimal sample size for stress minimization depends on the learning rate.
(\textbf{two figures on the left}
\label{fig:analysis-of-sample-size}
\end{figure*}
\subsection{Analysis of Sample Size}\label{sect:analysis-of-sample-size}
\blue{
In this section we analyze the impact of sample size on the convergence rate of {$(SGD)^2$}\xspace.
In deep learning, models trained with different sample sizes can converge to different types of minima; e.g., smaller sample tend to lead to a better generalization\cite{keskar2016large}.
In {$(SGD)^2$}\xspace, smaller sample size usually results in faster run time \textit{per iteration} but does not necessarily yield faster \textit{per-second} convergence.
As described in Section~\ref{sect:properties-and-measures}, we use different sampling strategies and sample sizes for each readability criterion.
Consider, for example, stress minimization and how optimal sample size closely depends on other factors in the optimization.
In other words, there is no ``one size fits all'' sample size.
In particular, for any given graph, the optimal sample size depends on the learning rate of the SGD algorithm.
Fig.~\ref{fig:analysis-of-sample-size} shows the quality (i.e. stress) of layouts for a binary tree with 9 levels (1023 nodes) as a function of total run time of the {$(SGD)^2$}\xspace algorithm.
In each plot, we visualize the convergence of the algorithm under a fixed learning rate with various sample sizes.
When the learning rate is small (the two plots on the left of Fig.~\ref{fig:analysis-of-sample-size}), we observe that smaller sample sizes (e.g., 4 or 8) converge faster.
In contrast, a medium sample size (e.g., 16 or 32) can benefit from larger learning rates (the two plots on the right of Fig.~\ref{fig:analysis-of-sample-size}) and converge faster than any cases that use a smaller learning rate.
Moreover, when using a large learning rate, the training with a smaller sample size becomes less stable (due to the high variance of gradients, see the rightmost plot in Fig.~\ref{fig:analysis-of-sample-size}).
We illustrate this observation on the binary tree graph with 9 levels (1023 nodes) and have the same observation on other trees and grids with various sizes.
Moreover, we observed that this interplay between sample size and learning rate on convergence rate is less obvious in variants of the SGD algorithm.
When replacing the SGD with some of its variants (e.g., AdaDelta~\cite{zeiler2012adadelta}, RMSProp~\cite{Tieleman2012} or ADAM~\cite{kingma2014adam}) that takes adaptive step size based on the gradient of previous steps, the convergence rate of stress minimization becomes less sensitive to sample size or learning rate.
}
\begin{figure}
\caption{
Runtime of balanced trees (top) and 2D grids (bottom).
The plots have log scales on both x and y axes.
}
\label{fig:runtime}
\end{figure}
\subsection{Analysis of Run Time}
To test the scalability of our method, we test the runtime of our method on larger graphs.
We tested our code on a MacBook Pro with a 2.9 GHz Dual-Core Intel Core i5 CPU and 16GB of memory.
We picked two families of graphs: balanced binary trees and 2D grids, and measured the convergence time as the size of the graph grows.
For balanced binary trees, we start with a tree with $4$ levels ($15$ nodes) and gradually increase the depth to $12$ levels ($4095$ nodes).
For grids, we start with a grid of size $16 \times 2$ ($32$ nodes) and double the number of columns until we have a grid of size $16 \times 256$ ($4096$ nodes).
For each criterion, we randomly initialize nodes in the layout from standard Gaussian, optimize the layout with respect to only one criterion using SGD and stop as soon as the layout converges.
We ensure the convergence by gradually decreasing the learning rate of SGD: we decrease the learning rate by a factor of $0.7$ every time the loss has not decreased for a certain number of iterations, often referred to as the ``patience.''
In general, we need more patience for larger graphs and for smaller mini-batch to compensate for the large variance in loss estimation due to random sampling.
Here, we set the patience to $max(100, int(|V|/m)*300)$ iterations when we optimize a graph with $|V|$ nodes and take $m$ samples in every SGD iteration.
\blue{To further improve the robustness of the stopping criteria, we smooth the sample loss by taking the exponential moving average.
That is, on the $i^{th}$ iteration, the smoothed loss $L_i$ is defined as
\begin{align*}
L_i &= \frac{SL_i + s^1 SL_{i-1} + \dots + s^{i-1} SL_1}{1+s+s^2+\dots+ s^{i-1}}
= \frac{\Sigma_{k=1}^i s^{i-k} \, SL_k}{\Sigma_{k=1}^i s^{i-k}}
\end{align*}
where $s$ is a smoothing factor.
We set $s=0.5^{1/100}\approx 0.993$, a rate at which the $100^{th}$ preceding iteration will contribute half as much as that of the latest sample loss.
}
Fig.~\ref{fig:runtime} summarizes the runtime analysis for the two families of graphs (trees and grids) for all 9 criteria.
Note that we are using log-log plots (log scales for both the $x$ and $y$ axes). This experimental analysis shows linear or near-linear time for the underlying algorithms.
This is shown as steeper slopes in the log-log plots.
\section{Conclusions, Limitations, Future Work}
We introduced the graph drawing framework, {$(SGD)^2$}\xspace, for multicriteria graph drawing and
showed how this approach can be used to optimize different graph drawing criteria and combinations thereof.
We showed that multiple readability criteria can be optimized jointly via SGD if each of them can be expressed as a differentiable function.
In cases that some readability criteria are not naturally differentiable (e.g., neighborhood preservation or crossing number), one can find differentiable surrogate functions and optimize the criteria indirectly.
\blue{
We measured the quality of generated layouts
and analyzed interactions between criteria, the runtime behavior, and the impact of sample sizes; all of which provide evidence of the effectiveness of {$(SGD)^2$}\xspace
}
\blue{\textbf{Support for More Constraint Types:}
Although {$(SGD)^2$}\xspace is a flexible framework that can optimize a wide range of criteria, we did not consider the class of constraints where the node coordinates are related by some inequalities (i.e., hard constraints). }
Similarly, in the {$(SGD)^2$}\xspace framework we do not naturally support shape-based drawing constraints such as those in~\cite{ipsepcola_2006, scalable_cola_2009,wang2017revisiting}.
\blue{
Incorporating a wider range of constraint types and studying the interactions between them in the multi-objective setting are natural directions for future work.
}
\textbf{Better Weight Balancing for Multicriteria Objectives:}
The {$(SGD)^2$}\xspace framework is flexible and natural directions for future work include adding further drawing criteria and better ways to combine them.
\blue{
An appropriate balance between weights for the different criteria can be crucial as more and more criteria are incorporated into the optimization.
Currently, we manually choose appropriate weight schedules based on specific combinations of criteria.
In the future, we would like to explore ways to automatically design and balance weight schedules in multicriteria graph drawing.
}
\blue{\textbf{Applications of Different Techniques and Frameworks:}
Besides gradient descent, there are other optimization techniques
that could be deployed to multi-objective problems~\cite{orosz2020robust}.
Similarly, while we used Tensorflow.js and PyTorch to implement {$(SGD)^2$}\xspace, there are other frameworks (e.g., pymoo~\cite{blank2020pymoo}) with support for multi-objective optimization.
The application of different optimization techniques and frameworks to multicriteria network visualization seems like an interesting direction for future work.
}
\blue{
\textbf{Scalability for Larger Graphs:}
Currently, not all criteria are fully optimized for speed.
Alternative objective functions, for example tsNET by Kruiger et al.~\cite{kruiger2017graph} for neighborhood preservation, could be considered in the {$(SGD)^2$}\xspace framework as further runtime scalability and quality improvements are needed for graphs with millions of nodes and edges.
One possible direction for improving scalability is to employ a multi-level algorithmic framework.}
\section*{Acknowledgments}
This work was supported in part by NSF grants CCF-1740858, CCF-1712119, and
DMS-1839274.
\begin{IEEEbiography}[{\vspace*{-.5cm}\includegraphics[width=.8in,height=1.0in,clip,keepaspectratio]{./figures/reyan_cropped.png}}]{Reyan Ahmed} is a Ph.D. student at the Department of Computer Science at the University of Arizona. He received B.Sc. and M.Sc. degree in Computer Science and Engineering from Bangladesh University of Engineering and Technology. His research interests include graph algorithms, network visualization, and data science.
\end{IEEEbiography}
\begin{IEEEbiography}[{\includegraphics[width=.8in,height=1.0in,clip,keepaspectratio]{./figures/felice.jpeg}}]{Felice De Luca} is a postdoctoral researcher at the Department of Computer Science at the University of Arizona. He received an MS degree in 2014 and a PhD in 2018 at the Department of Computer and Automation Engineering at the University of Perugia. His research interests include graph drawing, information visualization, algorithm engineering, and computational geometry.
\end{IEEEbiography}
\begin{IEEEbiography}[{\vspace*{-1cm}\includegraphics[width=.8in,height=1.0in,clip,keepaspectratio]{./figures/devkota_cropped.png}}]{Sabin Devkota} is a Ph.D. student at the Department of Computer Science at the University of Arizona. He received a bachelor's in Electronics and Communication Engineering from Tribhuvan University. His research interests include information visualization.
\end{IEEEbiography}
\begin{IEEEbiography}[{\vspace*{-.5cm}\includegraphics[width=.8in,height=1.0in,clip,keepaspectratio]{./figures/kobourov.jpg}}]{Stephen Kobourov} is a Professor at the Department of Computer Science at the University of Arizona. He received a BS degree in Mathematics and Computer Science from Dartmouth College and MS and PhD degrees from Johns Hopkins University. His research interests include information visualisation, graph theory, and geometric algorithms.
\end{IEEEbiography}
\begin{IEEEbiography}[{\vspace*{-1cm}\includegraphics[width=.8in,height=1.0in,clip,keepaspectratio]{./figures/mingwei_cropped.png}}]{Mingwei Li} is a PhD student in the Department of Computer Science, University of Arizona.
He received the BEng degree in electronics engineering from the Hong Kong University
of Science and Technology.
His research interests include data visualization and machine learning.
\end{IEEEbiography}
\resizebox{\columnwidth}{!}{
\begin{tabular}{l|l}
\toprule
Notation & Description\\
\midrule
$G$ & Graph\\
$V$ & The set of nodes in $G$, indexed by $i$, $j$ or $k$\\
$E$ & The set of edges in $G$, indexed by a pair of nodes $i,j$ in $V$\\
$n=|V|$ & Number of nodes in $G$\\
$m$ & sample size for a certain criterion in SGD\\
$|E|$ & Number of edges in $G$\\
$Adj$ and $A_{i,j}$ & Adjacency matrix of $G$ and its $(i,j)$-th entry\\
$d_{ij}$ & Graph-theoretic distance between node $i$ and $j$\\
$X_{n \times 2}$ & 2D-coordinates of nodes in the drawing\\
$||X_i - X_j||$ & The Euclidean distance between nodes $i$ and $j$ \\
$\theta_i$ & $i^{th}$ crossing angle\\
$\varphi_{ijk}$ & Angle between incident edges $(i,j)$ and $(j,k)$\\
\bottomrule
\end{tabular}
\caption{Graph notation used in this paper.}
\label{table:notations}
}
\end{table}
\resizebox{1.1\textwidth}{!}{
\centering
\begin{tabular}{p{2.0cm}|p{4cm}|p{4cm}|p{4cm}}
\toprule
Property
& Gradient Descent
& Subgradient Descent
& Stochastic Gradient Descent\\
\midrule
Stress
& $\sum\limits_{i<j}\;w_{ij}(|X_i - X_j|_2 - d_{ij})^2$
& $\sum\limits_{i<j}\;w_{ij}(|X_i - X_j|_2 - d_{ij})^2$
& $w_{ij}(|X_i - X_j|_2 - d_{ij})^2$ for a random pair of nodes $i, j \in V$\\
\hline
Ideal \hspace{.8cm}Edge Length
& $\sqrt{ \frac{1}{|E|} \sum\limits_{(i,j) \in E}\;
(\frac{||X_i - X_j|| - l_{ij}}{l_{ij}}})^2$ (Eq.~\ref{eq:loss-ideal-edge-length})
& $\frac{1}{|E|} \sum\limits_{(i,j) \in E}\;
|\frac{||X_i - X_j|| - l_{ij}}{l_{ij}}|$
& $|\frac{||X_i - X_j|| - l_{ij}}{l_{ij}}|$ for a random edge $(i,j) \in E$\\
\hline
Crossing \hspace{.5cm} Angle
& $\sum\limits_i\;cos(\theta_i)^2$
& $\sum\limits_i\; |cos(\theta_i)|$
& $|cos(\theta_i)|$ for a random crossing $i$\\
\hline
Neighborhood Preservation
& Lov\'asz \textbf{softmax}~\cite{berman2018lovasz} between
neighborhood prediction (Eq.\ref{eq:neighbor-pred})
and adjacency matrix $Adj$
& Lov\'asz \textbf{hinge}~\cite{berman2018lovasz} between
neighborhood prediction (Eq.\ref{eq:neighbor-pred})
and adjacency matrix $Adj$
& Lov\'asz \textbf{softmax} or \textbf{hinge}~\cite{berman2018lovasz} on a random node.
(i.e. Jaccard loss between a random \textit{row} of K in Eq. \ref{eq:neighbor-pred}
and the corresponding row in the adjacency matrix $Adj$)\\
\hline
Crossing Number
& Shabbeer et al.~\cite{bennett2010}
& Shabbeer et al.~\cite{bennett2010}
& Shabbeer et al.~\cite{bennett2010}\\
\hline
Angular \hspace{.5cm} Resolution
& $\sum\limits_{(i,j),(j,k) \in E}\; e^{-\varphi_{ijk}}$
& $\sum\limits_{v \in E}\; e^{-\varphi_{ijk}}$
& \makecell[l]{$e^{-\varphi_{ijk}}$ \\for random $(i,j),(j,k) \in E$} \\
\hline
Vertex\hspace{.8cm} Resolution
& $\sum_{i,j \in V, i \neq j}$ ${ReLU( 1 - \frac{||X_i - X_j||}{d_{max} \cdot r}) ^2}$ (Eq. \ref{eq:loss-vertex-resolution})
& $\sum_{i,j \in V, i \neq j}$ ${ReLU( 1 - \frac{||X_i - X_j||}{d_{max} \cdot r})}$
& $ReLU( 1 - \frac{||X_i - X_j||}{d_{max} \cdot r})$ for random $i,j \in V, i \neq j$
\\
\hline
Gabriel Graph
&$\sum_{\substack{(i,j) \in E, k \in V \setminus \{i,j\}}} $ $ReLU(r_{ij} - |X_k - c_{ij}|) \; ^ 2$ (Eq. \ref{eq:gabriel})
&$\sum_{\substack{(i,j) \in E,k \in V \setminus \{i,j\}}} $ $ReLU(r_{ij} - |X_k - c_{ij}|)$
&$ReLU(r_{ij} - |X_k - c_{ij}|)$ for random $(i,j) \in E$ and $k \in V \setminus \{i,j\}$
\\
\hline
Aspect Ratio
& Eq. \ref{eq:aspect-ratio}
& Eq. \ref{eq:aspect-ratio}
& Eq. \ref{eq:aspect-ratio}
\\
\bottomrule\\
\end{tabular}
}
\caption{Summary of the objective functions via different optimization methods.}
\label{table:loss-functions}
\end{table}
\end{document} |
\begin{document}
\title{A Latent space solver for PDE generalization}
\begin{abstract}
In this work we propose a hybrid solver to solve partial differential equation (PDE)s in the latent space. The solver uses an iterative inferencing strategy combined with solution initialization to improve generalization of PDE solutions. The solver is tested on an engineering case and the results show that it can generalize well to several PDE conditions.
\end{abstract}
\section{Introduction}
Simulations are important in engineering applications to explore the underlying physics. But, they can be computationally expensive because they involve numerical methods to solve partial differential equations (PDEs) for various conditions associated with the PDE, such as geometry of computational domain, type of boundary conditions (BC), external source terms etc. Researchers have explored the idea of coupling machine learning with PDEs for several decades \citep{crutchfield1987equations, kevrekidis2003equation}. Recently, there is a tremendous focus on improving the predictive capability and generalizability of ML methods by infusing physics, either during training or prediction. A substantial portion of research has focused on introducing physics-based constraints in neural networks through the computation of PDE derivatives \citep{raissi2019physics, raissi2018hidden, rao2020physics, ranade2020discretizationnet, gao2020phygeonet, wu2018physics, qian2020lift, xue2020amortized}. More recently, there is a huge effort on training neural networks within frameworks of differentiable PDE solvers. These approaches use differentiable solvers to learn and control PDE solutions as well as the dynamics of the system \citep{amos2017optnet, um2020solver, de2018end, toussaint2018differentiable, wang2020differentiable, holl2020learning, portwood2019turbulence, bar2019learning, zhuang2020learned, kochkov2021machine}.
There is a growing need for improving generalizability of ML techniques to a wider range of PDE parameters. The existing ML approaches that learn from PDE conditions, such as geometry, BCs and source terms have to handle several challenges. Firstly, the PDE conditions can be high dimensional and sparse, impacting learning and generalizability. Secondly, the inference procedure is static and there are limited opportunities to alter the trajectory of PDE solutions. Finally, the space of physical solutions is very large.
In this paper, we propose a hybrid solver to learning PDE solutions and address the outlined challenges. We use lower dimensional representations to tackle the issues of high dimensionality and sparsity of PDE conditions and solutions. Further, the inferencing methodology uses an iterative procedure to solve the PDE in a lower dimensional latent space with fixed point iterations. Latent space learning has been explored in several works, \citep{wiewel2020latent, maulik2020reduced, kim2019deep, murata2020nonlinear, fukami2020convolutional, champion2019data, fukami2020sparse, he2020unsupervised}, but our approach to solve PDEs is different and novel. The iterative procedure at inference enables us to alter the solution trajectories using existing PDE solvers. This is useful in ML based techniques to ensure solver robustness, generalizability and accuracy. The hybrid solver combines solution initialization using existing coarse grid PDE solvers with this iterative inferencing procedure and is demonstrated on a practical engineering application.
\section{Solution Methodology}
\subsection{Latent space representation}
\begin{figure}
\caption{Autoencoders for PDE conditions and solution}
\label{fig:1}
\end{figure}
Fig. {\textnormal{e}}f{fig:1} shows the neural network architectures used to determine the compressed latent space vectors of the various PDE conditions, such as geometry of computational domain, BCs and source term distributions, and PDE solutions ($u_1$ and $u_2$). The geometry of the computational domain is represented using a binarized level set representation \citep{osher1988fronts}. On the other hand, boundary conditions and source terms already have spatio-temporal distributions. The PDE conditions, as well as the PDE solutions are compressed into their lower dimensional latent vectors, $\eta_g, \eta_h, \eta_b, \eta$ , using CNN encoder-decoder type networks. The loss functions used to constrain the Autoencoder networks of PDE conditions are purely statistical. Conversely, the PDE solution Autoencoders are augmented by including PDE based loss constraints, similar to \citet{ranade2020discretizationnet}. The PDE constraints are computed using the discretization schemes, available in the Ansys suite of software. Additionally, since the PDE solutions are dependent on other PDE conditions, the solution Autoencoders are also conditioned upon the latent vectors of the PDE conditions.
\subsection{Hybrid latent space solver methodology} \label{hyb_sol}
\begin{figure}
\caption{Hybrid solver: Iterative inferencing strategy}
\label{fig:2}
\end{figure}
Fig. {\textnormal{e}}f{fig:2} shows the hybrid latent space solver methodology proposed in this work for using the Autoencoder networks to infer at unknown and unseen conditions. The iterative procedure based on fixed point iterations and is motivated from our previous work, \citep{ranade2020discretizationnet}. The different steps involved in the solution procedure are outlined below.
\begin{enumerate}
\item Latent vectors, $\eta_g$, $\eta_h$ and $\eta_b$, are computed for a new computational domain geometry, boundary condition and source term distribution using the encoder networks.
\item The initial solution of PDE is computed using an existing PDE solver on a very coarse grid. These solutions are interpolated on the fine grid and encoded to their latent vector form, $\eta$.
\item The initial solution latent vector, $\eta$ combined with $\eta_g$, $\eta_h$ and $\eta_b$ is passed through the decoder to generate solution fields, ${u_1}, {u_2}$.
\item Solution fields are compressed to a new solution latent vector, $\hat{\eta}$, using the the encoder.
\item Steps $3$ and $4$ are repeated with the new solution latent vector, $\hat{\eta}$ until $||\eta-\hat{\eta}||_2 < 1e^{-6}$.
\item At convergence, the PDE solutions are decoded using the most recent $\hat{\eta}$.
\end{enumerate}
The hybrid latent space solver described above has two main implications. Firstly, the iterative procedure used for inferencing allows initialization of solutions using existing PDE solvers and moreover provides an opportunity to intervene the solution process and alter the solution trajectory using PDE solvers. Secondly, the solution procedure is conditioned by richer, lower dimensional representations of PDE conditions in the form of latent vectors, thus enhancing the generalizability.
\section{Results and discussions}
\begin{wrapfigure}{r}{0.47\textwidth}
\centering
{\bm{s}}pace{-1em}
\includegraphics[width=6cm]{Fig3.png}
{\bm{s}}pace{-1em}
\caption{Natural convection of electronic chip}
{\bm{s}}pace{-1em}
\label{fig:3}
\end{wrapfigure}
The hybrid solver is demonstrated for a 3-D, steady-state electronic cooling case with natural convection. There are $5$ solution variables, $3$ components of velocity, pressure and temperature. The geometry of the computational domain may be observed in Fig. {\textnormal{e}}f{fig:3}. The domain consists of a chip-mold assembly held by a PCB and the entire geometry is placed inside a fluid domain. The chip is subjected to heat sources with random spatial distributions due to uncertainty in electrical heating. Fig. {\textnormal{e}}f{fig:4} in Appendix {\textnormal{e}}f{App} shows an example of the $8$ different distributions of heat sources. From a physics standpoint, natural convection results in a two-way coupling between temperature and velocity. The heat source specified results in a temperature increase, which generates fluid velocity because of buoyancy effects and in turn cools the chip. At steady state there is sufficient velocity generation to cancel out the heat generation. The main challenges in this case are to capture the complicated physics resulting from the two-way coupling and to generalize to different spatial distributions of heat sources.
\subsection{Data generation and training}
The Autoencoder for heat source distribution is trained with random spatial distributions generated using a Gaussian mixture model, where the number of Gaussians are varied from $1$ to $20$ for randomized mean and variance. The Autoencoder is set up to achieve a compression ratio of around $12$. The PDE solution Autoencoder is trained with $200$ solution generated using Ansys Fluent fluid flow simulation software on a computational mesh with $128^3$ elements for random heat sources. A compression ratio of $64$ is achieved. The normalized reconstruction mean squared error for unseen test samples is to the order of $1e^{-6}$ for all the solution variables as well as the heat source.
\subsection{Comparisons with Ansys Fluent}
The hybrid solver is compared with Ansys Fluent solution for two heat source distributions, which are generated randomly and thus, never seen by the networks. The initial solution is computed at a mesh resolution of $16$ elements in each spatial direction in Ansys Fluent for 200 steady-state iterations.
\subsubsection{Computation time}
Ansys Fluent takes an average of $200$ minutes on a single CPU to obtain a converged solution on $128^3$ mesh. On the other hand, the hybrid solver converges in $50$ seconds on average, including the time it takes to generate the coarse grid solution. Thus, the hybrid solver results in a 200x speedup over Ansys Fluent in generating solutions on fine girds. The averages are calculated over runs on 100 cases with different heat source distributions.
\subsubsection{Contour \& Line plots}
Fig. {\textnormal{e}}f{fig:5} and Fig. {\textnormal{e}}f{fig:6} in Appendix {\textnormal{e}}f{App} compare the solution contours on the X-Y plane for the two test cases. The hybrid solution agrees well with respect to the Ansys Fluent solutions with small mean squared errors. Although not shown here, the performance is similar for a variety of random distributions of heat source, including the examples shown in Figure {\textnormal{e}}f{fig:4} in the Appendix {\textnormal{e}}f{App}. The line plots in Fig. {\textnormal{e}}f{fig:7} and Fig. {\textnormal{e}}f{fig:8} in Appendix {\textnormal{e}}f{App} are plotted along the Y direction through chip center. It can be observed that the overall trends as well as the peak quantities agree well for both velocity and temperature in both test cases. The contour and line plots show that the hybrid latent space solver captures the two-way coupled physics accurately and more importantly, the methodology generalizes well to a variety of heat source distributions.
\begin{figure}
\caption{Comparisons of velocity and temperature between hybrid solver solution and Ansys Fluent solution for test case 1. The contours are plotted in X-Y plane perpendicular to the chip. The velocity contour represents the entire domain (X$\in$(-0.1525m to 0.1525m), Y$\in$(-0.1525m, 0.1525m)), while the temperature contour zooms on the chip (X$\in$(-0.03m, 0.03m), Y$\in$(-0.03m, 0.03m).}
\label{fig:5}
\end{figure}
\begin{figure}
\caption{Comparisons of velocity and temperature between hybrid solver solution and Ansys Fluent solution for test case 1. The line plots are plotted along Y direction at (X, Z)$\in$(0.0m, 0.0m). The velocity line plot represents the entire domain (Y$\in$(-0.1525m, 0.1525m)), while the temperature line plot zooms on the chip (Y$\in$(-0.03m, 0.03m).}
\label{fig:7}
\end{figure}
\section{Conclusion and Future work}
In this work we have proposed a hybrid solver that combines latent space learning with solution initialization to improve generalization of PDE solutions. The results shows that the methodology is computationally fast, accurate and generalizable. In its current form, the proposed methodology has several limitations. Most notably, the approach is restricted to structured domains with fixed resolutions with relatively simpler computational domain geometries. In future, we would like to extend this to solve on unstructured meshes. Furthermore, the latent space iterative strategy will be integrated with an actual PDE solver to alter solution trajectories and improve convergence.
\marginpar{NEW}page
\appendix
\section{Appendix} \label{App}
\begin{figure}
\caption{Example of randomness in spatial distribution of heat source terms}
\label{fig:4}
\end{figure}
\begin{figure}
\caption{Comparisons of velocity and temperature between hybrid solver solution and Ansys Fluent solution for test case 2. The contours are plotted in X-Y plane perpendicular to the chip. The velocity contour represents the entire domain (X=(-0.1525m to 0.1525m), Y$\in$(-0.1525m, 0.1525m)), while the temperature contour zooms on the chip (X$\in$(-0.03m, 0.03m), Y=(-0.03m, 0.03m).}
\label{fig:6}
\end{figure}
\begin{figure}
\caption{Comparisons of velocity and temperature between hybrid solver solution and Ansys Fluent solution for test case 2. The line plots are plotted along Y direction at (X, Z)$\in$0.0m, 0.0m). The velocity line plot represents the entire domain (Y$\in$(-0.1525m, 0.1525m)), while the temperature line plot zooms on the chip (Y=(-0.03m, 0.03m).}
\label{fig:8}
\end{figure}
\end{document} |
\begin{document}
\selectlanguage{english}
\title{Strongly anisotropic diffusion problems; asymptotic analysis}
\author{Mihai Bostan
\thanks{Laboratoire d'Analyse, Topologie, Probabilit\'es LATP, Centre de Math\'ematiques et Informatique CMI, UMR CNRS 7353, 39 rue Fr\'ed\'eric Joliot Curie, 13453 Marseille Cedex 13
France. E-mail : {\tt [email protected]}}
}
\date{ (\today)}
\maketitle
\begin{abstract}
The subject matter of this paper concerns anisotropic diffusion equations: we consider heat equations whose diffusion matrix have disparate eigenvalues. We determine first and second order approximations, we study the well-posedness of them and establish convergence results. The analysis relies on averaging techniques, which have been used previously for studying transport equations whose advection fields have disparate components.
\end{abstract}
\paragraph{Keywords:}
Anisotropic diffusion, Variational methods, Multiple scales, Average operator.
\paragraph{AMS classification:} 35Q75, 78A35.
\section{Introduction}
\label{Intro}
\indent
Many real life applications lead to highly anisotropic diffusion equations: flows in porous media, quasi-neutral plasmas, microscopic transport in magnetized plasmas \cite{Bra65}, plasma thrusters, image processing \cite{PerMal90}, \cite{Wei98}, thermal properties of crystals \cite{DiaShaYun91}. In this paper we investigate the behavior of the solutions for heat equations whose diffusion becomes very high along some direction. We consider the problem
\begin{equation}
\label{Equ1} \partial _t \ue - \divy ( D(y) \nabla _y \ue ) - \frac{1}{\eps} \divy ( b(y) \otimes b(y) \nabla _y \ue ) = 0, \;\;(t,y) \in \R_+ \times \R ^m
\end{equation}
\begin{equation}
\label{Equ2}
\ue (0,y) = \uein (y), \;\;y \in \R^m
\end{equation}
where $D(y) \in {\cal M}_m (\R)$ and $b(y) \in \R^m$ are smooth given matrix field and vector field on $\R^m$, respectively. For any two vectors $\xi, \eta$, the notation $\xi \otimes \eta$ stands for the matrix whose entry $(i,j)$ is $\xi _i \eta _j$, and for any two matrix $A, B$ the notation $A:B$ stands for $\mathrm{trace}(^t AB) = A_{ij} B_{ij}$ (using Einstein summation convention).
We assume that at any $y \in \R^m$ the matrix $D(y)$ is symmetric and $D(y) + b (y) \otimes b(y)$ is positive definite
\begin{equation}
\label{Equ3}
^t D (y) = D(y),\;\;\exists \;d >0 \;\;\mbox{such that}\;\;D(y)\xi\cdot\xi + (b(y) \cdot \xi)^2 \geq d \;|\xi|^2,\;\;\xi \in \R ^m,\;\;y \in \R ^m.
\end{equation}
The vector field $b(y)$, to which the anisotropy is aligned, is supposed divergence free {\it i.e.,} $\divy b = 0$. We intend to analyse the behavior of \eqref{Equ1}, \eqref{Equ2} for small $\eps$, let us say $0 < \eps \leq 1$. In that cases $D(y) + \frac{1}{\eps} b(y) \otimes b(y)$ remains positive definite and if $(\uein)_\eps$ remain in a bounded set of $\lty$, then $(\ue)_\eps$ remain in a bounded set of $\litlty{}$ since, for any $t \in \R_+$ we have
\begin{align*}
\frac{1}{2}\inty{(\ue (t,y))^2} & + d \intsy{|\nabla _y \ue (s,y)|^2} \leq \frac{1}{2}\inty{(\ue (t,y))^2} \\
& + \intsy{\left \{D(y) + \frac{1}{\eps} b(y) \otimes b(y) \right \} : \nabla _y \ue (s,y) \otimes \nabla _y \ue (s,y)} \\
& = \frac{1}{2}\inty{(\uein (y))^2}.
\end{align*}
In particular, when $\eps \searrow 0$, $(\ue)_\eps$ converges, at least weakly $\star$ in $\litlty{}$ towards some limit $u \in \litlty{}$. Notice that the explicit methods are not well adapted for the numerical approximation of \eqref{Equ1}, \eqref{Equ2} when $\eps \searrow 0$, since the CFL condition leads to severe time step constraints like
\[
\frac{d}{\eps} \frac{\Delta t}{|\Delta y |^2} \leq \frac{1}{2}
\]
where $\Delta t$ is the time step and $\Delta y $ is the grid spacing. In such cases implicit methods are desirable \cite{BalTilHow08}, \cite{ShaHam10}.
Rather than solving \eqref{Equ1}, \eqref{Equ2} for small $\eps >0$, we concentrate on the limit model satisfied by the limit solution $u = \lime \ue$. We will see that the limit model is still a parabolic problem, decreasing the $\lty$ norm and satisfying the maximum principle. At least formally, the limit solution $u$ is the dominant term of the expansion
\begin{equation}
\label{Equ6} \ue = u + \eps u ^1 + \eps ^2 u ^2 + ...
\end{equation}
Plugging the Ansatz \eqref{Equ6} into \eqref{Equ1} leads to
\begin{equation}
\label{Equ7} \divy (b \otimes b \nabla _y u ) = 0,\;\;(t,y) \in \R_+ \times \R ^m
\end{equation}
\begin{equation}
\label{Equ8} \partial _t u - \divy (D \nabla _y u ) - \divy ( b \otimes b \nabla _y u^1) = 0,\;\;(t,y) \in \R_+ \times \R ^m
\end{equation}
\[
\vdots
\]
Clearly, the constraint \eqref{Equ7} says that at any time $t \in \R_+$, $b \cdot \nabla _y u = 0$, or equivalently $u(t,\cdot)$ remains constant along the flow of $b$, see \eqref{EquFlow}
\[
u(t, Y(s;y)) = u(t,y),\;\;s \in \R,\;\;y \in \R^m.
\]
The closure for $u$ comes by eliminating $u^1$ in \eqref{Equ8}, combined with the fact that \eqref{Equ7} holds true at any time $t \in \R_+$. The symmetry of the operator $\divy (b \otimes b \nabla _y)$ implies that $\partial _t u - \divy (D \nabla _y u)$ belongs to $(\ker (b \cdot \nabla _y ))^\perp$ and therefore we obtain the weak formulation
\begin{equation}
\label{Equ9}
\frac{\md}{\md t}\inty{u(t,y) \varphi (y)} + \inty{D \nabla _y u (t,y) \cdot \nabla _y \varphi (y) } = 0,\;\;\varphi \in \hoy \cap \kerbg{}.
\end{equation}
The above formulation is not satisfactory, since the choice of test functions is constrained by \eqref{Equ7}; \eqref{Equ9} is useless for numerical simulation. A more convenient situation is to reduce \eqref{Equ9} to another problem, by removing the constraint \eqref{Equ7}. The method we employ here is related to the averaging technique which has been used to handle transport equations with diparate advection fields \cite{BosAsyAna}, \cite{BosTraSin}, \cite{BosGuidCent3D}, \cite{Bos12}
\begin{equation}
\label{Equ10} \partial _t \ue + a(t,y) \cdot \nabla _y \ue + \frac{1}{\eps} b (y) \cdot \nabla _y \ue = 0,\;\;(t,y) \in \R_+ \times \R^m
\end{equation}
\begin{equation}
\label{Equ11} \ue (0,y) = \uein (y),\;\;y \in \R^m.
\end{equation}
Using the same Ansatz \eqref{Equ6} we obtain as before that $b \cdot \nabla _y u (t,\cdot) = 0, t \in \R_+$ and the closure for $u$ writes
\begin{equation}
\label{Equ12}
\mathrm{Proj}_{\kerbg} \{ \partial _t u + a\cdot \nabla _y u \} = 0
\end{equation}
or equivalently
\begin{equation}
\label{Equ14} \frac{\md }{\md t} \inty{u(t,y) \varphi (y) } - \inty{u(t,y) \;a \cdot \nabla _y \varphi } = 0
\end{equation}
for any smooth function satisfying the constraint $b \cdot \nabla _y \varphi = 0$. The method relies on averaging since the projection on $\kerbg$ coincides with the average along the flow of $b$, cf. Proposition \ref{AverageOperator}. As $u$ satisfies the constraint $b \cdot \nabla _y u = 0$, it is easily seen that $\mathrm{Proj}_{\kerbg} \partial _t u = \partial _t u$. A simple case to start with is when the transport operator $a \cdot \nabla _y$ and $b \cdot \nabla _y$ commute {\it i.e.,} $[b \cdot \nabla _y, a \cdot \nabla _y ] = 0$. In this case $a \cdot \nabla _y$ leaves invariant the subspace of the constraints, implying that $\mathrm{Proj}_{\kerbg} \{a \cdot \nabla _y u \} = a \cdot \nabla _y u$. Therefore \eqref{Equ12} reduces to a transport equation and it is easily seen that this equation propagates the constraint, which allows us to remove it. Things happen similarly when the transport operators $a \cdot \nabla _y, b \cdot \nabla _y$ do not commute, but the transport operator of the limit model may change. In \cite{BosTraSin} we prove that there is a transport operator $A \cdot \nabla _y$, commuting with $b \cdot \nabla _y$, such that for any $u \in \kerbg$ we have
\[
\mathrm{Proj}_{\kerbg} \{a \cdot \nabla _y u \} = A \cdot \nabla _y u.
\]
Once we have determined the field $A$, \eqref{Equ12} can be replaced by $\partial _t u + A \cdot \nabla _y u = 0$, which propagates the constraint $b \cdot \nabla _y u (t) = 0$ as well.
Comming back to the formulation \eqref{Equ9}, we are looking for a matrix field $\tilde{D}(y)$ such that $\divy (\tilde{D} \nabla _y)$ commutes with $b \cdot \nabla _y$ and
\[
\mathrm{Proj}_{\kerbg} \{\divy (D(y) \nabla _y u ) \}= \divy (\tilde{D}(y)\nabla _y u),\;\;u \in \kerbg{}.
\]
We will see that, under suitable hypotheses, it is possible to find such a matrix field $\tilde{D}$, and therefore \eqref{Equ9} reduces to the parabolic model
\begin{equation}
\label{Equ15} \partial _t u - \divy (\tilde{D}(y) \nabla _y u ) = 0,\;\;(t,y) \in \R_+ \times \R^m.
\end{equation}
The matrix field $\tilde{D}$ will appear as the orthogonal projection of the matrix field $D$ (with respect to some scalar product to be determined) on the subspace of matrix fields $A$ satisfying $[b\cdot \nabla _y, \divy(A \nabla _y)] = 0$. The field $\tilde{D}$ inherits the properties of $D$, like symmetry, positivity, etc.
Our paper is organized as follows. The main results are presented in Section \ref{ModMainRes}. Section \ref{AveOpe} is devoted to the interplay between the average operator and first and second order linear differential operators. In particular we justify the existence of the {\it averaged} matrix field $\tilde{D}$ associated to any field $D$ of symmetric, positive matrix. The first order approximation is justified in Section \ref{FirstOrdApp} and the second order approximation is discussed in Section \ref{SecOrdApp}. Several technical proofs are gathered in Appendix \ref{A}.
\section{Presentation of the models and main results}
\label{ModMainRes}
\noindent
We assume that the vector field $b :\R^m \to \R^m$ is smooth and divergence free
\begin{equation}
\label{Equ21} b \in W^{1,\infty}_{\mathrm{loc}} (\R^m),\;\;\divy b = 0
\end{equation}
with linear growth
\begin{equation}
\label{Equ22}
\exists \;C > 0\;\;\mbox{such that}\;\; |b(y)| \leq C (1 + |y|),\;\;y \in \R^m.
\end{equation}
We denote by $Y(s;y)$ the characteristic flow associated to $b$
\begin{equation}
\label{EquFlow} \frac{\md Y}{\md s} = b(Y(s;y)),\;\;Y(s;0) = y,\;\;s \in \R,\;\;y \in \R^m.
\end{equation}
Under the above hypotheses, this flow has the regularity $Y \in W^{1,\infty} _{\mathrm{loc}} (\R \times \R^m)$ and is measure preserving.
We concentrate on matrix fields $A(y) \in \loloc{}$ such that $[b(y) \cdot \nabla _y, \divy ( A(y) \nabla _y)] = 0$, let us say in $\dpri$. We check that the commutator between $b \cdot \nabla _y $ and $\divy (A \nabla _y)$ writes cf. Proposition \ref{ComSecOrd}
\[
[b(y) \cdot \nabla _y, \divy ( A(y) \nabla _y)] = \divy ( [b,A]\nabla _y)\;\;\mbox{in}\;\;\dpri
\]
where the bracket between $b$ and $A$ is given by
\[
[b,A] := (b \cdot \nabla _y) A - \partial _y b A (y) - A(y) \;^t \partial _y b,\;\;y \in \R^m.
\]
Several characterizations for the solutions of $[b,A] = 0$ in $\dpri$ are indicated in the Propositions \ref{MFI}, \ref{WMFI}, among which
\begin{equation}
\label{Equ16} A(Y(s;y)) = \partial _y Y (s;y) A(y) \;{^t \partial _y Y} (s;y),\;\;s\in \R,\;\;y \in \R ^m.
\end{equation}
We assume that there is a matrix field $P(y)$ such that
\begin{equation}
\label{Equ56} ^t P = P,\;\;P(y) \xi \cdot \xi >0,\;\;\xi \in \R^m,\;\; y \in \R^m,\;\;P^{-1}, P \in \ltloc{},\;\;[b,P]= 0 \;\mbox{in}\;\dpri.
\end{equation}
We introduce the set
\[
H_Q = \{ A = A(y)\;:\; \inty{Q(y) A(y) : A(y) Q(y) } < +\infty\}
\]
where $Q = P ^{-1}$, and the scalar product
\[
(A,B)_Q = \inty{QA:BQ},\;\;A, B \in H_Q.
\]
The equality \eqref{Equ16} suggests to introduce the family of applications $G(s): H_Q \to H_Q$, $s \in \R$, $G(s)A = (\partial _y Y )^{-1}(s; \cdot) A(Y(s;\cdot)) \;^t (\partial _y Y )^{-1}(s;\cdot)$ which is a $C^0$-group of unitary operators on $H_Q$ cf. Proposition \ref{Groupe}. This allows us to introduce $L$, the infinitesimal generator of $(G(s))_{s\in \R}$. The operator $L$ is skew-adjoint on $H_Q$ and its kernel coincides with $\{A \in H_Q\subset \loloc{} : [b,A] = 0\;\mbox{in} \; \dpri\}$ cf. Proposition \ref{PropOpeL}. The averaged matrix field denoted $\ave{D}_Q$, associated to any $D \in H_Q$ appears as the long time limit of the solution of
\begin{equation}
\label{Equ67}
\partial _t A - L(L(A)) = 0,\;\;t \in \R_+
\end{equation}
\begin{equation}
\label{Equ68} A(0) = D.
\end{equation}
The notation $\ave{\cdot}$ stands for the orthogonal projection (in $\lty{}$) on $\kerbg{}$.
\begin{thm}
\label{AveMatDif} Assume that \eqref{Equ21}, \eqref{Equ22}, \eqref{Equ56} hold true. Then for any $D \in H_Q \cap \liy{}$ the solution of \eqref{Equ67}, \eqref{Equ68} converges weakly in $H_Q$ as $t \to +\infty$ towards the orthogonal projection of $D$ on $\ker L$
\[
\lim _{t \to +\infty} A(t) = \ave{D}_Q\;\mbox{ weakly in }\;H_Q,\;\;\ave{D}_Q := \mathrm{Proj} _{\ker L } D.
\]
If $D$ is symmetric and positive, then so is the limit $\ave{D}_Q = \lim _{t \to +\infty} A(t)$, and satisfies
\begin{equation}
\label{Equ72} L (\ave{D}_Q) = 0,\;\;\nabla _y u \cdot \ave{D}_Q \nabla _y v = \ave{\nabla _y u \cdot D\nabla _y v},\;\;u, v \in H^1(\R^m) \cap \kerbg{}
\end{equation}
\begin{equation}
\label{Equ72Bis}\ave{\nabla _y u \cdot \ave{D}_Q \nabla _y (b \cdot \nabla _y \psi )} = 0,\;\;u \in H^1(\R^m) \cap \kerbg{},\;\;\psi \in C^2_c (\R^m).
\end{equation}
\end{thm}
The first order approximation (for initial data not necessarily well prepared) is justified by
\begin{thm}
\label{MainResult1} Assume that \eqref{Equ21}, \eqref{Equ22}, \eqref{Equ56}, \eqref{Equ26} hold true and that $D$ is a field of symmetric positive matrix, which belongs to $H_Q$. Consider a family of initial conditions $(\uein)_{\eps } \subset \lty$ such that $(\ave{\uein})_\eps$ converges weakly in $\lty{}$, as $\eps \searrow 0$, towards some function $\uin$. We denote by $\ue$ the solution of \eqref{Equ1}, \eqref{Equ2} and by $u$ the solution of
\begin{equation}
\label{Equ75}
\partial _t u - \divy ( \ave{D}_Q \nabla _y u ) = 0,\;\;t \in \R_+,\;\;y \in \R^m
\end{equation}
\begin{equation}
\label{Equ76}
u(0,y) = \uin (y),\;\;y \in \R^m
\end{equation}
where $\ave{D}_Q$ is associated to $D$, cf. Theorem \ref{AveMatDif}. Then we have the convergences
\[
\lime \ue = u\;\;\mbox{weakly} \star \mbox{ in } \litlty{}
\]
\[
\lime \nabla _y \ue = \nabla _y u\;\;\mbox{weakly} \mbox{ in } \lttlty{}.
\]
\end{thm}
The derivation of the second order approximation is more complicated and requires the computation of some other matrix fields. For simplicity, we content ourselves to formal results. The crucial point is to introduce the decomposition given by
\begin{thm}
\label{Decomposition} Assume that \eqref{Equ21}, \eqref{Equ22}, \eqref{Equ56}, \eqref{Equ26} hold true and that $L$ has closed range. Then, for any field of symmetric matrix $D \in H_Q$, there is a unique field of symmetric matrix $F \in \dom (L^2) \cap (\ker L )^\perp$ such that
\[
- \divy ( D \nabla _y) = - \divy ( \ave{D}_Q \nabla _y ) + \divy (L^2 (F)\nabla _y )
\]
that is
\begin{align*}
& \inty{D \nabla _y u \cdot \nabla _y v } - \inty{\ave{D}_Q\nabla _y u \cdot \nabla _y v } \\
& = \inty{L(F) \nabla _y u \cdot \nabla _y (b \cdot \nabla _y v)} + \inty{L(F) \nabla _y ( b \cdot \nabla _y u ) \cdot \nabla _y v} \\
& = - \inty{F \nabla _y ( b \cdot \nabla _y ( b \cdot \nabla _y u)) \cdot \nabla _y v} - 2 \inty{F \nabla _y ( b \cdot \nabla _y u) \cdot \nabla _y ( b \cdot \nabla _y v)}\\
& - \inty{F \nabla _y u \cdot \nabla _y ( b \cdot \nabla _y ( b \cdot \nabla _y v))}
\end{align*}
for any $u, v \in C^3_c(\R^m)$.
\end{thm}
After some computations we obtain, at least formally, the following model, replacing the hypothesis \eqref{Equ56} by the stronger one: there is a matrix field $R(y)$ such that
\begin{equation}
\label{Equ90}
\det R(y)\neq 0,\;y \in \R^m,\;Q = {^t R} R \mbox{ and }P = Q^{-1} \in \ltloc{},\;b \cdot \nabla _y R + R \partial _y b = 0 \mbox{ in } \dpri.
\end{equation}
\begin{thm}
\label{MainResult2}
Assume that \eqref{Equ21}, \eqref{Equ22}, \eqref{Equ23}, \eqref{Equ26}, \eqref{Equ90} hold true and that $D$ is a field of symmetric positive matrix which belongs to $H_Q \cap \liy{}$. Consider a family of initial conditions $(\uein)_\eps \subset \lty{}$ such that $(\frac{\ave{\uein} - \uin }{\eps} ) _{\eps >0}$ converges weakly in $\lty{}$, as $\eps \searrow 0$, towards a function $\vin{}$, for some function $\uin \in \kerbg{}$. Then, a second order approximation for \eqref{Equ1} is provided by
\begin{equation}
\label{IntroEqu87}\partial _t \tue - \divy ( \ave{D}_Q \nabla _y \tue) + \eps [ \divy ( \ave{D}_Q \nabla _y ), \divy (F \nabla _y ) ]\tue - \eps S(\tue) = 0,\;\;(t,y) \in \R_+ \times \R^m
\end{equation}
\begin{equation}
\label{NewIC} \tue (0,y) = \uin (y) + \eps ( \vin (y) + \win (y)),\;\;\win = \divy ( F \nabla _y \uin),\;\;y \in \R ^m
\end{equation}
for some fourth order linear differential operator $S$, see Proposition \ref{DifOpe}, and the matrix field $F$ given by Theorem \ref{Decomposition}.
\end{thm}
\section{The average operator}
\label{AveOpe}
\noindent
We assume that the vector field $b : \R^m \to \R^m$ satisfies \eqref{Equ21}, \eqref{Equ22}. We consider the linear operator $u \to b \cdot \nabla _y u = \divy(ub)$ in $\lty{}$, whose domain is defined by
\[
\dom (b \cdot \nabla _y ) = \{ u \in \lty{} \;:\; \divy(ub) \in \lty\}.
\]
It is well known that
\[
\kerbg = \{ u \in \lty{}\;:\; u (Y(s;\cdot)) = u (\cdot), \;s \in \R\}.
\]
The orthogonal projection on $\kerbg{}$ (with respect to the scalar product of $\lty{}$), denoted by $\ave{\cdot}$, reduces to average along the characteristic flow $Y$ cf. \cite{BosTraSin} Propositions 2.2, 2.3.
\begin{pro}
\label{AverageOperator} For any function $u \in \lty{}$ the family $\ave{u}_T : = \frac{1}{T} \int _0 ^T u (Y(s;\cdot))\md s, T>0$ converges strongly in $\lty{}$, when $T \to + \infty$, towards the orthogonal projection of $u$ on $\kerbg{}$
\[
\lim _{T \to +\infty} \ave{u}_T = \ave{u},\;\;\ave{u} \in \kerbg{} \;\mbox{and} \; \inty{(u - \ave{u}) \varphi } = 0,\;\forall\; \varphi \in \kerbg{}.
\]
\end{pro}
Since $b \cdot \nabla _y$ is antisymmetric, one gets easily
\begin{equation}
\label{Equ24} \overline{\ran (b \cdot \nabla _y ) } = (\kerbg{} ) ^\perp = \ker ( \mathrm{Proj}_{\kerbg{}} ) = \ker \ave{\cdot}.
\end{equation}
\begin{remark}
\label{DetFun} If $u \in \lty{}$ satisfies $\inty{u(y) b \cdot \nabla _y \psi } = 0, \forall \;\psi \in C^1 _c (\R^m)$ and $\inty{u\varphi }= 0, \forall \; \varphi \in \kerbg{}$, then $u = 0$. Indeed, as $u \in \lty{} \subset \loloc{}$, the first condition says that $b \cdot \nabla _y u = 0$ in $\dpri{}$ and thus $u \in \kerbg{}$. Using now the second condition with $\varphi = u$ one gets $\inty{u^2} = 0$ and thus $u = 0$.
\end{remark}
In the particular case when $\ran (b \cdot \nabla _y)$ is closed, which is equivalent to the Poincar\'e inequality (cf. \cite{Brezis} pp. 29)
\begin{equation}
\label{Equ23} \exists\;C_P >0\;:\; \left ( \inty{(u - \ave{u})^2}\right ) ^{1/2} \leq C_P \left ( \inty{(b \cdot \nabla _y u ) ^2} \right ) ^{1/2},\;\;u \in \dom (b \cdot \nabla _y)
\end{equation}
\eqref{Equ24} implies the solvability condition
\[
\exists \; u \in \dom ( b \cdot \nabla _y ) \;\mbox{ such that }\; b \cdot \nabla _y u = v\;\mbox{ iff } \ave{v} = 0.
\]
If $\|\cdot \|$ stands for the $\lty{}$ norm we have
\begin{pro}
\label{Inverse}
Under the hypothesis \eqref{Equ23}, $b \cdot \nabla _y $ restricted to $\ker \ave{\cdot}$ is one to one map onto $\ker \ave{\cdot}$. Its inverse, denoted $(b \cdot \nabla _y )^{-1}$, belongs to ${\cal L}(\ker \ave{\cdot}, \ker \ave{\cdot})$ and
\[
\|(b \cdot \nabla _y ) ^{-1} \|_{{\cal L}(\ker \ave{\cdot}, \ker \ave{\cdot})} \leq C_P.
\]
\end{pro}
Another operator which will play a crucial role is ${\cal T } = - \divy (b \otimes b \nabla _y)$ whose domain is
\[
\dom ({\cal T}) = \{ u \in \dom (b \cdot \nabla _y)\;:\; b \cdot \nabla _y u \in \dom ( b \cdot \nabla _y )\}.
\]
The operator ${\cal T}$ is self-adjoint and under the previous hypotheses, has the same kernel and range as $b\cdot \nabla _y$.
\begin{pro}
\label{KerRanTau} Under the hypotheses \eqref{Equ21}, \eqref{Equ22}, \eqref{Equ23} the operator ${\cal T}$ satisfies
\[
\ker {\cal T} = \kerbg,\;\;\ran {\cal T} = \ran (b \cdot \nabla _y ) = \ker \ave{\cdot}
\]
and $\| u - \ave{u}\| \leq C_P ^2 \|{\cal T} u \|,u \in \dom ({\cal T})$.
\end{pro}
\begin{proof}
Obviously $\kerbg{} \subset \ker {\cal T}$. Conversely, for any $u \in \ker {\cal T}$ we have $\inty{\;(\bg u )^2} = \inty{\;u {\cal T}u} = 0$ and therefore $ u \in \kerbg{}$.
Clearly $\ran {\cal T} \subset \ran ( \bg{}) = \ker \ave{\cdot}$. Consider now $w \in \ker \ave{\cdot} = \ran ( \bg{})$. By Proposition \ref{Inverse} there is $v \in \ker \ave{\cdot} \cap \dom (\bg)$ such that $\bg v = w$. Applying one more time Proposition \ref{Inverse}, there is $ u \in \ker \ave{\cdot} \cap \dom (\bg)$ such that $\bg u = v$. We deduce that $u \in \dom {\cal T}, w = {\cal T}(-u)$. Finally, for any $u \in \dom {\cal T}$ we apply twice the Poincar\'e inequality, taking into account that $\ave{\bg u } = 0$
\[
\| u - \ave{u}\| \leq C_P \|\bg u \| \leq C_P ^2 \|{\cal T} u \|.
\]
\end{proof}
\begin{remark}
\label{AveLone}
The average along the flow of $b$ can be defined in any Lebesgue space $L^q (\R^m)$, $q \in [1,+\infty]$. We refer to \cite{BosTraSin} for a complete presentation of these results.
\end{remark}
\subsection{Average and first order differential operators}
\label{FirstOrdDiffOpe}
\noindent
We are looking for first order derivations commuting with the average operator. Recall that the commutator $[\xi \cdot \nabla _y, \eta \cdot \nabla _y]$ between two first order differential operators is still a first order differential operator, whose vector field, denoted by $[\xi, \eta]$, is given by the Poisson bracket between $\xi$ and $\eta$
\[
[\xi \cdot \nabla _y, \eta \cdot \nabla _y]:= \xi \cdot \nabla _y ( \eta \cdot \nabla _y ) - \eta \cdot \nabla _y ( \xi \cdot \nabla _y ) = [\xi, \eta] \cdot \nabla _y
\]
where $[\xi, \eta] = (\xi \cdot \nabla _y ) \eta - ( \eta \cdot \nabla _y ) \xi$. The two vector fields $\xi$ and $\eta$ are said in involution iff their Poisson bracket vanishes.
Assume that $c(y)$ is a smooth vector field, satisfying $c(Y(s;y)) = \partial _y Y (s;y) c(y), s\in \R, y \in \R^m$, where $Y$ is the flow of $b$ (not necessarily divergence free here). Taking the derivative with respect to $s$ at $s = 0$ yields $(b \cdot \nabla _y ) c = \partial _y b \;c(y)$, saying that $[b,c] = 0$. Actually the converse implication holds true and we obtain the following characterization for vector fields in involution, which is valid in distributions as well (see Appendix \ref{A} for proof details).
\begin{pro}
\label{VFI} Consider $b \in W^{1,\infty}_{\mathrm{loc}} (\R^m)$ (not necessarily divergence free), with linear growth and $c \in \loloc{}$. Then $(b \cdny) c - \partial _y b \;c = 0$ in $\dpri$ iff
\begin{equation}
\label{Equ34} c (Y(s;y)) = \partial _y Y(s;y) c(y),\;\;s\in \R,\;\;y \in \R^m.
\end{equation}
\end{pro}
We establish also weak formulations characterizing the involution between two fields, in distribution sense (see Appendix \ref{A} for the proof). The notation $w_s$ stands for $w \circ Y(s;\cdot)$.
\begin{pro}
\label{WVFI} Consider $b \in W^{1,\infty}_{\mathrm{loc}} (\R^m)$, with linear growth and zero divergence and $c \in \loloc{}$. Then the following statements are equivalent\\
1.
\[[b,c] = 0 \;\mbox{in}\; \dpri{}
\]
2.
\begin{equation}
\label{Equ41}
\inty{(c \cdny u )v_{-s} } = \inty{(c\cdny u_s) v },\;\;\forall \;u, v \in C^1_c(\R^m)
\end{equation}
3.
\begin{equation}
\label{Equ42} \inty{c \cdny u \;b \cdny v } + \inty{c \cdny (b \cdny u ) v } = 0,\;\;\forall\; u \in C^2 _c (\R^m),\;\;v \in C^1 _c (\R^m).
\end{equation}
\end{pro}
\begin{remark}
\label{VecDiv}
If $[b,c]=0$ in $\dpri{}$, applying \eqref{Equ41} with $v = 1$ on the support of $u_s$ (and therefore $v_{-s} = 1$ on the support of $u$) yields
\[
\inty{c \cdny u} = \inty{c\cdny u_s},\;\;u \in C^1_c(\R^m)
\]
saying that $\divy c$ is constant along the flow of $b$ (in $\dpri{}$).
\end{remark}
We claim that for vector fields $c$ in involution with $b$, the derivation $c \cdny $ commutes with the average operator.
\begin{pro}
\label{AveComFirstOrder} Consider a vector field $c \in \loloc{}$ with bounded divergence, in involution with $b$, that is $[b,c] = 0$ in $\dpri{}$. Then the operators $u \to c \cdny u$, $u \to \divy(uc)$ commute with the average operator {\it i.e.,} for any $u \in \dom ( c\cdny )= \dom (\divy (\cdot \;c))$ we have $\ave{u} \in \dom ( c\cdny )= \dom (\divy (\cdot \;c))$ and
\[
\ave{c \cdny u} = c\cdny \ave{u},\;\;\ave{\divy(uc) } = \divy ( \ave{u}c).
\]
\end{pro}
\begin{proof}
Consider $u \in \dom (c \cdny ), s \in \R$ and $\varphi \in C^1 _c (\R^m)$. We have
\begin{align}
\label{Equ43} \inty{u_s c \cdny \varphi } & = \inty{u (c \cdny \varphi)_{-s}} \\
& = \inty{u (c \cdny ) \varphi _{-s} } \nonumber \\
& = - \inty{\divy (uc) \varphi _{-s}} \nonumber \\
& = - \inty{(\divy (uc))_s \varphi (y)} \nonumber
\end{align}
saying that $u _s \in \dom ( c \cdny ) = \dom ( \divy ( \cdot \;c ))$ and $ \divy (u_s c ) = ( \divy (uc))_s$. We deduce $c \cdny u_s = (c \cdny u )_s$ cf. Remark \ref{VecDiv}. Integrating \eqref{Equ43} with respect to $s$ between $0$ and $T>0$ one gets
\begin{align*}
\inty{\frac{1}{T} \int _0 ^T u_s \md s \;c \cdny \varphi } & = \frac{1}{T} \int _0 ^T \inty{u_s c \cdny \varphi }\md s \\
& = - \frac{1}{T} \int _0 ^T \inty{(\divy (uc ))_s \varphi (y) }\md s \\
& = - \inty{\frac{1}{T}\int _0 ^T ( \divy (uc))_s \md s \;\varphi (y) }.
\end{align*}
By Proposition \ref{AverageOperator} we know that $\frac{1}{T} \int _0 ^T u_s \md s \to \ave{u}$ and $\frac{1}{T}\int _0 ^T (\divy (uc))_s \md s \to \ave{\divy (uc)}$ strongly in $\lty{}$, when $T \to +\infty$, and thus we obtain
\[
\inty{\ave{u} c \cdny \varphi } = - \inty{\ave{\divy(uc)} \varphi (y) }
\]
saying that $\ave{u} \in \dom ( c \cdny )$ and $\divy (\ave{u}c) = \ave{\divy (uc)}$, $c \cdny \ave{u} = \ave{c \cdny u }$.
\end{proof}
\subsection{Average and second order differential operators}
\label{SecondOrdDiffOpe}
\noindent
We investigate the second order differential operators $- \divy (A(y) \nabla _y)$ commuting with the average operator along the flow of $b$, where $A(y)$ is a smooth field of symmetric matrix. Such second order operators leave invariant $\kerbg{}$. Indeed, for any $u \in \dom (- \divy (A(y) \nabla _y)) \cap \kerbg{}$ we have
\[
- \divy (A(y) \nabla _y u ) = - \divy (A(y) \ave{u}) = \ave{ - \divy (A(y) \nabla _y u )} \in \kerbg{}.
\]
For this reason it is worth considering the operators $- \divy (A(y) \nabla _y )$ commuting with $b \cdny$. A straightforward computation shows that
\begin{pro}
\label{ComSecOrd} Consider a divergence free vector field $b \in W^{2,\infty} (\R^m)$ and a matrix field $A \in W^{2,\infty} (\R^m)$. The commutator between $b \cdny $ and $- \divy (A(y) \nabla _y)$ is still a second order differential operator
\[
[b\cdny, - \divy(A\nabla _y )] = - \divy ([b,A] \nabla _y )
\]
whose matrix field, denoted by $[b,A]$, is given by
\[
[b,A] = (b \cdny )A - \dyb A(y) - A(y)\; {^t \dyb},\;\;y \in \R^m.
\]
\end{pro}
\begin{remark}
We have the formula ${^t [b,A]} = [b, {^t A}]$. In particular if $A(y)$ is a field of symmetric (resp. anti-symmetric) matrix, the field $[b,A]$ has also symmetric (resp. anti-symmetric) matrix.
\end{remark}
As for vector fields in involution, we have the following characterization (see Appendix \ref{A} for proof details).
\begin{pro}
\label{MFI} Consider $b \in W^{1,\infty}_{\mathrm{loc}} (\R^m)$ (not necessarily divergence free) with linear growth and $A(y) \in \loloc{}$. Then $[b,A] = 0$ in $\dpri{}$ iff
\begin{equation}
\label{Equ35} A(\ysy) = \dyy A(y) \;{^t \dyy},\;\;s\in \R,\;\;y \in \R^m.
\end{equation}
\end{pro}
For fields of symmetric matrix we have the weak characterization (see Appendix \ref{A} for the proof).
\begin{pro}
\label{WMFI} Consider $b \in W^{1,\infty}_{\mathrm{loc}} (\R^m)$ with linear growth, zero divergence and $A \in \loloc{}$ a field of symmetric matrix. Then the following statements are equivalent\\
1.
\[
[b,A] = 0 \;\mbox{ in } \; \dpri{}.
\]
2.
\[
\inty{A(y) \nabla _y u_s \cdot \nabla _y v_s } = \inty{A(y) \nabla _y u \cdot \nabla _y v }
\]
for any $s \in \R$, $u, v \in C^1 _c ( \R^m)$.\\
3.
\[
\inty{A(y) \nabla _y ( b \cdny u ) \cdot \nabla _y v } + \inty{A(y) \nabla _y u \cdot \nabla _y ( b \cdny v ) } = 0
\]
for any $u, v \in C^2 _c (\R^m)$.
\end{pro}
We consider the (formal) adjoint of the linear operator $A \to [b,A]$, with respect to the scalar product $(U,V) = \inty{U(y) : V(y)}$, given by
\[
Q \to - (b \cdny ) Q - {^t \dyb} Q(y) - Q(y) \dyb
\]
when $\divy b = 0$. The following characterization comes easily and the proof is left to the reader.
\begin{pro}
\label{AdyMatFieInv} Consider $b \in W^{1,\infty}_{\mathrm{loc}} (\R^m)$, with linear growth and $Q \in L^1 _{\mathrm{loc}} (\R^m)$. Then $- (b \cdny ) Q - {^t \dyb} Q(y) - Q(y) \dyb = 0$ in $\dpri {}$ iff
\begin{equation}
\label{Equ36} Q(\ysy) = {^t \partial _y Y }^{-1}(s;y) Q(y) \partial _y Y ^{-1}(s;y),\;\;s\in \R,\;\;y \in \R^m.
\end{equation}
\end{pro}
\begin{remark}
\label{InverseQ}
If $Q(y)$ satisfies \eqref{Equ36} and is invertible for any $y \in \R^m$ with $Q^{-1} \in L^1 _{\mathrm{loc}}(\R^m)$, then $Q^{-1} (\ysy) = \dyy Q^{-1} (y) {^t \dyy}$, $s \in \R, y \in \R^m$ and therefore $[b,Q^{-1}] = 0$ in $\dpri{}$. If $P(y)$ satisfies \eqref{Equ35} and is invertible for any $y \in \R^m$, then
\[
P^{-1} (\ysy) = {^t \partial _y Y} ^{-1} (s;y) P ^{-1} (y) \partial _y Y ^{-1} (s;y),\;\;s \in \R,\;\; y \in \R^m
\]
and therefore $- (b \cdny ) P - {^t \dyb} P(y) - P(y) \dyb = 0$ in $\dpri{}$.
\end{remark}
As for vector fields in involution, the matrix fields in involution with $b$ generate second order differential operators commuting with the average operator.
\begin{pro}
\label{AveComSecondOrder} Consider a matrix field $A \in \loloc{}$ such that $\divy A \in \loloc{}$ and $[b,A] = 0$ in $\dpri{}$. Therefore the operator $u \to - \divy (A \nabla _y u )$ commutes with the average operator {\it i.e.,} for any $u \in \dom ( - \divy (A \nabla _y ))$ we have $\ave{u} \in \dom ( - \divy (A \nabla _y ))$ and
\[
-\ave{\divy (A \nabla _y u )} = - \divy (A \nabla _y \ave{u}).
\]
\end{pro}
\begin{proof}
Consider $u \in \dom( - \divy (A\nabla _y )) = \{w \in \lty{}: -\divy (A\nabla _y w ) \in \lty{}\}$. For any $s \in \R, \varphi \in C^2 _c (\R^m)$ we have
\begin{equation}
\label{Equ51} - \inty{u_s \;\divy ( \;{^t A} \nabla _y \varphi ) } = - \inty{u \;( \divy ( \;{^t A} \nabla _\varphi ))_{-s}}.
\end{equation}
By the implication $1.\implies 2.$ of Proposition \ref{WMFI} (which does not require the symmetry of $A(y)$) we know that
\[
\inty{{^t A } \nabla _y \varphi \cdot \nabla _y \psi _s } = \inty{{^t A} \nabla _y \varphi _{-s} \cdot \nabla _y \psi }
\]
for any $\psi \in C^2 _c (\R^m )$. We deduce that
\[
- \inty{\divy ( {^t A} \nabla _y \varphi ) \psi _s } = - \inty{\divy ( {^t A } \nabla _y \varphi _{-s} ) \psi }
\]
and thus $(\divy ( {^t A} \nabla _y \varphi ))_{-s} = \divy ( {^t A} \nabla _y \varphi _{-s})$. Combining with $\eqref{Equ51}$ yields
\begin{eqnarray}
\label{Equ52} - \inty{\;u_s \divy ( {^t A} \nabla _y \varphi )} & = - \inty{\;u\; \divy ( {^t A} \nabla _y \varphi _{-s})} \\
& = - \inty{\;\divy (A \nabla _y u ) \varphi _{-s}} \nonumber \\
& = - \inty{\;(\divy (A \nabla _y u ))_s \varphi (y)} \nonumber
\end{eqnarray}
saying that $u_s \in \dom ( - \divy (A \nabla _y ))$ and
\[
- \divy ( A \nabla _y u_s) = ( - \divy (A \nabla _y u ))_s.
\]
Integrating \eqref{Equ52} with respect to $s$ between $0$ and $T$ we obtain
\[
\inty{\frac{1}{T} \int _0 ^T u_s \;\md s\; \divy ( {^t A } \nabla _y \varphi )} = \inty{\frac{1}{T} \int _0 ^T (\divy (A \nabla _y u))_s \;\md s \;\varphi (y)}.
\]
Letting $T \to +\infty$ yields
\[
\inty{\ave{u} \divy ( {^t A }\nabla _y \varphi ) } = \inty{\ave{\divy (A \nabla _y u )} \varphi (y)}
\]
and therefore $\ave{u} \in \dom ( \divy (A \nabla _y ))$, $\divy (A \nabla _y \ave{u}) = \ave{\divy (A \nabla _y u )}$.
\end{proof}
\subsection{The averaged diffusion matrix field}
\label{AveDifMatFie}
\noindent
We are looking for the limit, when $\eps \to 0$, of \eqref{Equ1}, \eqref{Equ2}. We expect that the limit $u = \lime \ue $ satisfies \eqref{Equ7}, \eqref{Equ8}. By \eqref{Equ7} we deduce that at any time $t \in \R_+$, $u(t,\cdot) \in \kerbg{}$. Observe also that $\divy(b \otimes b \nabla _y u^1) = b \cdny (b \cdny u^1) \in \ran ( b \cdny ) \subset \ker \ave{\cdot}$ and therefore the closure for $u$ comes by applying the average operator to \eqref{Equ8} and by noticing that $\ave{\partial _t u } = \partial _t \ave{u} = \partial _t u $
\begin{equation}
\label{Equ54} \partial _t u - \ave{\divy ( D \nabla _y u )} = 0,\;\;t\in \R_+,\;\;y \in \R^m.
\end{equation}
At least when $[b,D] = 0$, we know by Proposition \ref{AveComSecondOrder} that
\[
\ave{\divy (D \nabla _y u)} = \divy (D \nabla _y \ave{u}) = \divy (D \nabla _y u)
\]
and \eqref{Equ54} reduces to the diffusion equation associated to the matrix field $D(y)$. Nevertheless, even if $[b,D] \neq 0$, \eqref{Equ54} behaves like a diffusion equation. More exactly the $\lty{}$ norm of the solution decreases with a rate proportional to the $\lty{}$ norm of its gradient under the hypothesis \eqref{Equ3}
\begin{align*}
\frac{1}{2}\frac{\md }{\md t} \inty{(u(t,y))^2} & = \inty{\ave{\divy ( D \nabla _y u )} u (t,y) } \\
& = \inty{\divy ( D \nabla _y u ) u }\\
& = - \inty{D \nabla _y u \cdot \nabla _y u }\\
& = - \inty{(D + b\otimes b ) : \nabla _y u \otimes \nabla _y u } \\
& \leq - d \inty{|\nabla _y u (t,y) |^2 }.
\end{align*}
We expect that, under appropriate hypotheses, \eqref{Equ54} coincides with a diffusion equation, corresponding to some {\it averaged} matrix field ${\cal D}$, that is
\begin{equation}
\label{Equ55}
\exists \; {\cal D} (y)\;:\; [b, {\cal D}] = 0\;\mbox{ and } \; \ave{- \divy ( D \nabla _y u )} = - \divy ( {\cal D} \nabla _y u ),\;\;\forall \;u \in \kerbg{}.
\end{equation}
It is easily seen that in this case the limit model \eqref{Equ54} reduces to
\[
\partial _t u - \divy ( {\cal D}\nabla _y u ) = 0,\;\;t \in \R_+,\;\;y \in \R^m.
\]
In this section we identify sufficient conditions which guarantee the existence of the matrix field ${\cal D}$. We will see that it appears as the long time limit of the solution of another parabolic type problem, whose initial data is $D$, and thus as the orthogonal projection of the field $D(y)$ (with respect to some scalar product to be defined) on a subset of $\{A \in \loloc{}:[b,A] = 0\mbox{ in } \dpri{}\}$. We assume that \eqref{Equ56} holds true. We introduce the set
\[
H_Q = \{A = A(y)\;:\; \inty{Q(y)A(y) : A(y) Q(y)} < +\infty\}
\]
where $Q = P^{-1}$ and the bilinear application
\[
(\cdot, \cdot)_Q : H_Q \times H_Q \to \R,\;\;(A,B)_Q = \inty{Q(y)A(y):B(y)Q(y)}
\]
which is symmetric and positive definite. Indeed, for any $A \in H_Q$ we have
\[
(A,A)_Q = \inty{Q^{1/2}AQ^{1/2} : Q^{1/2}AQ^{1/2}} \geq 0
\]
with equality iff $Q^{1/2}AQ^{1/2}= 0$ and thus iff $A = 0$. The set $H_Q$ endowed with the scalar product $(\cdot, \cdot)_Q$ becomes a Hilbert space, whose norm is denoted by $|A|_Q = (A, A)_Q ^{1/2}, A \in H_Q$. Observe that $H_Q \subset \{A(y):A \in \loloc{}\}$. Indeed, if for any matrix $M$ the notation $|M|$ stands for the norm subordonated to the euclidian norm of $\R^m$
\[
|M| = \sup _{\xi \in \R^m \setminus \{0\}} \frac{|M\xi|}{|\xi|} \leq ( M : M ) ^{1/2}
\]
we have for a.a. $y \in \R^m$
\begin{eqnarray}
\label{Equ57} |A(y)| & = & \sup _{\xi, \eta \neq 0} \displaystyle \frac{A(y) \xi \cdot \eta}{|\xi|\;|\eta|} \\
& = & \sup _{\xi, \eta \neq 0} \displaystyle \frac{Q^{1/2}AQ^{1/2} P ^{1/2}\xi \cdot P^{1/2} \eta}{|P^{1/2} \xi|\;|P^{1/2} \eta|}\;\frac{|P^{1/2}\xi|}{|\xi|}\;\frac{|P^{1/2}\eta|}{|\eta|} \nonumber \\
& \leq & |Q^{1/2}AQ^{1/2} |\;|P^{1/2}|^2 \nonumber \\
& \leq & ( Q^{1/2}AQ^{1/2}:Q^{1/2}AQ^{1/2}) ^{1/2} \;|P|.\nonumber
\end{eqnarray}
We deduce that for any $R>0$
\[
\int_{B_R} |A(y)|\;\md y \leq \int _{B_R} ( Q^{1/2}AQ^{1/2}:Q^{1/2}AQ^{1/2}) ^{1/2} \;|P|\;\md y \leq (A,A)_Q ^{1/2} \left ( \int _{B_R} |P(y)|^2 \;\md y \right ) ^{1/2}.
\]
\begin{remark}
\label{Ortho}
We know by Remark \ref{InverseQ} that $Q_s = {^t \partial _y Y ^{-1} }(s;y) Q(y) \partial _y Y ^{-1}(s;y) $ which writes ${^t {\cal O}}(s;y) {\cal O}(s;y) = I$ where ${\cal O}(s;y) = Q_s ^{1/2} \dyy Q^{-1/2}$. Therefore the matrix ${\cal O}(s;y)$ are orthogonal and we have
\begin{equation}
\label{Equ58}
Q_s ^{1/2} \dyy Q^{-1/2} = {\cal O}(s;y) = {^t {\cal O}}^{-1} (s;y) = Q_s ^{-1/2} \;{^t \partial _y Y }^{-1} Q^{1/2}
\end{equation}
\begin{equation}
\label{Equ59}
Q ^{-1/2} \;{^t \dyy} Q_s ^{1/2} = {^t {\cal O}}(s;y) = { {\cal O}}^{-1} (s;y) = Q ^{1/2} { \partial _y Y }^{-1} Q_s^{-1/2}.
\end{equation}
\end{remark}
\begin{pro}
\label{Groupe} The family of applications $A \to G(s)A : = \partial _y Y ^{-1} (s; \cdot) A_s \; {^t \partial _y Y } ^{-1} (s; \cdot)$ is a $C^0$- group of unitary operators on $H_Q$.
\end{pro}
\begin{proof}
For any $A\in H_Q$ observe, thanks to \eqref{Equ59}, that
\begin{align*}
\left | \partial _y Y ^{-1}(s; \cdot) A_s {^t\partial _y Y ^{-1}(s; \cdot) }\right | ^2 _Q &
= \!\!\inty{Q^{1/2}\partial _y Y ^{-1} A_s {^t \partial _y Y ^{-1}}Q^{1/2}:Q^{1/2}\partial _y Y ^{-1} A_s {^t \partial _y Y ^{-1}}Q^{1/2}}\\
& = \!\!\inty{\!\!\!\!{^t {\cal O}} (s;y) Q_s ^{1/2} A_s Q_s ^{1/2} {\cal O}(s;y) \!:\! {^t {\cal O}} (s;y) Q_s ^{1/2} A_s Q_s ^{1/2} {\cal O}(s;y)}\\
& = \inty{Q_s ^{1/2} A_s Q_s ^{1/2} : Q_s ^ {1/2} A_s Q_s ^{1/2}}\\
& = \inty{Q^{1/2}AQ^{1/2} : Q^{1/2}AQ^{1/2}} \\
& = |A|^2 _Q.
\end{align*}
Clearly $G(0)A = A, A\in H_Q$ and for any $s, t \in \R$ we have
\begin{align*}
G(s) G(t) A & = \partial _y Y ^{-1} (s;\cdot) (G(t)A)_s {^t \partial _y Y ^{-1} (s;\cdot)}\\
& = \partial _y Y ^{-1} (s;\cdot) (\partial _y Y )^{-1} (t; Y(s;\cdot))(A_t)_s {^t (\partial _y Y )^{-1} (t; Y(s;\cdot))}{^t \partial _y Y ^{-1} (s;\cdot)} \\
& = \partial _y Y ^{-1} (t + s;\cdot)A_{t+s} {^t \partial _y Y ^{-1} (t + s;\cdot)} = G(t+s) A,\;\;A \in H_Q.
\end{align*}
It remains to check the continuity of the group, {i.e.,} $\lim _{s \to 0 } G(s)A = A$ strongly in $H_Q$ for any $A \in H_Q$. For any $s \in \R$ we have
\begin{align*}
|G(s) A - A|^2 _Q = |G(s)A|^2 _Q + |A|^2 _Q - 2 ( G(s)A, A)_Q = 2|A|^2 _Q - 2 (G(s)A, A)_Q
\end{align*}
and thus it is enough to prove that $\lim _{s \to 0 } G(s)A = A$ weakly in $H_Q$. As $|G(s)| = 1$ for any $s \in \R$, we are done if we prove that $\lim _{s \to 0} (G(s)A, U)_Q = (A, U)_Q$ for any $U \in C^0 _c (\R^m) \subset H_Q$. But it is easily seen that $\lim _{s\to 0} G(-s)U = U$ strongly in $H_Q$, for $U \in C^0 _c (\R^m) $ and thus
\[
\lim _{s \to 0} ( G(s)A, U)_Q = \lim _{s \to 0} (A, G(-s)U)_Q = (A,U)_Q,\;\;U \in C^0 _c (\R^m).
\]
\end{proof}
We denote by $L$ the infinitesimal generator of the group $G$
\[
L:\dom(L) \subset H_Q \to H_Q,\;\;\dom L = \{ A\in H_Q\;:\; \exists \;\lim _{s \to 0} \frac{G(s)A-A}{s}\;\mbox{ in } \;H_Q\}
\]
and $L(A) = \lim _{s \to 0} \frac{G(s)A-A}{s}$ for any $A \in \dom(L)$.
Notice that $C^1 _c (\R^m) \subset \dom(L)$ and $L(A) = b \cdny A - \dyb A - A \;{^t \dyb}$, $A \in C^1 _c (\R^m)$ (use the hypothesis $Q \in \ltloc{}$ and the dominated convergence theorem). Observe also that the group $G$ commutes with transposition {\it i.e.} $G(s) \;{^t A} = {^t G(s)}A$, $s \in \R, A \in H_Q$ and for any $A \in \dom (L)$ we have
$^t A \in \dom (L)$, $L({^t A}) = {^t L(A)}$.
The main properties of the operator $L$ are summarized below (when $b$ is divergence free).
\begin{pro}
\label{PropOpeL} $\;$\\
1. The domain of $L$ is dense in $H_Q$ and $L$ is closed.\\
2. The matrix field $A \in H_Q$ belongs to $\dom (L)$ iff there is a constant $C >0$ such that
\begin{equation}
\label{Equ61} |G(s)A - A |_Q \leq C |s|,\;\;s \in \R.
\end{equation}
3. The operator $L$ is skew-adjoint.\\
4. For any $A \in \dom (L)$ we have
\[
- \divy (L(A) \nabla _y ) = b\cdny ( - \divy (A \nabla _y)) + \divy (A \nabla _y ( b \cdny ))\;\mbox{ in } \; \dpri{}
\]
that is
\[
\inty{L(A) \nabla _y u \cdot \nabla _y v } = - \inty{A \nabla _y u \cdot \nabla _y ( b \cdny v)} - \inty{A\nabla _y ( b \cdny u ) \cdot \nabla _y v }
\]
for any $u, v \in C^2 _c (\R^m)$.
\end{pro}
\begin{proof}
1. The operator $L$ is the infinitesimal generator of a $C^0$-group, and therefore $\dom(L)$ is dense and $L$ is closed. \\
2.
Assume that $A \in \dom(L)$. We know that $\frac{\md }{\md s} G(s)A = L(G(s)A) = G(s)L(A)$ and thus
\[
|G(s)A - A|_Q = \left | \int _0 ^t G(\tau) L(A)\;\md \tau\right |_Q \leq \left | \int _0 ^s |G(\tau)L(A)|_Q \;\md \tau \right | = |s| \;|L(A)|_Q,\;\;s \in \R.
\]
Conversely, assume that \eqref{Equ61} holds true. Therefore we can extract a sequence $(s_k)_k$ converging to $0$ such that
\[
\limk \frac{G(s_k) A - A}{s_k} = V \;\mbox{ weakly in } \;H_Q.
\]
For any $U \in \dom (L)$ we obtain
\[
\left ( \frac{G(s_k) A - A}{s_k}, U \right ) _Q = \left ( A, \frac{G(-s_k)U - U}{s_k} \right ) _Q
\]
and thus, letting $k \to +\infty$ yields
\begin{equation}
\label{Equ62} (V, U)_Q = - (A, L(U))_Q.
\end{equation}
But since $U \in \dom (L)$, all the trajectory $\{G(\tau)U:\tau \in \R\}$ is contained in $\dom(L)$ and $G(-s_k)U = U + \int _0 ^{-s_k}L(G(\tau)U)\md \tau$. We deduce
\begin{align*}
(G(s_k)A - A, U)_Q & = \left ( A, \int _0 ^{-s_k} L(G(\tau)U ) \;\md \tau \right ) \\
& = \int _0 ^{-s_k}( A, L(G(\tau) U))_Q \;\md \tau \\
& = - \int _0 ^{-s_k}( V, G(\tau) U )_Q \;\md \tau \\
& = - \left ( V, \int _0 ^{-s_k} G(\tau)U\;\md \tau \right ) _Q.
\end{align*}
Taking into account that $\left | \int _0 ^{-s_k} G(\tau ) U \md \tau \right |_Q \leq |s_k| \;|U|_Q$ we obtain
\[
\left | \left ( \frac{G(s_k)A - A}{s_k}, U \right ) _Q \right | \leq |V|_Q|U|_Q,\;\;U \in \dom (L)
\]
and thus, by the density of $\dom (L)$ in $H_Q$ one gets
\[
\left | \frac{G(s_k)A - A}{s_k} \right |_Q \leq |V|_Q,\;\;k \in \N.
\]
Since $V$ is the weak limit in $H_Q$ of $\left ( \frac{G(s_k)A - A}{s_k} \right )_k$, we deduce that $\limk \frac{G(s_k)A - A}{s_k} = V$ strongly in $H_Q$. As the limit $V$ is uniquely determined by \eqref{Equ62}, all the family $\left ( \frac{G(s)A - A}{s} \right )_s$ converges strongly , when $s \to 0$, towards $V$ in $H_Q$ and thus $A \in \dom (L)$.\\
3. For any $U, V \in \dom (L)$ we can write
\[
(G(s)U - U, V)_Q + (U, V - G(-s)V)_Q = 0,\;\;s\in \R.
\]
Taking into account that
\[
\lim _{s \to 0} \frac{G(s)U - U}{s} = L(U),\;\;\lim _{s \to 0} \frac{V - G(-s)V}{s} = L(V)
\]
we obtain $(L(U), V)_Q + (U, L(V))_Q = 0$ saying that $V\in \dom (L^\star)$ and $L^\star (V) = - L(V)$. Therefore $L \subset (-L^\star)$. It remains to establish the converse inclusion. Let $V \in \dom (L^\star)$, {\it i.e.,} $\exists C >0$ such that
\[
|(L(U), V)_Q|\leq C|U|_Q,\;\;U \in \dom (L).
\]
For any $s \in \R$, $U \in \dom (L)$ we have
\[
(G(s)V - V , U)_Q = (V, G(-s)U - U)_Q = (V, \int _0 ^{-s}LG(\tau)U \;\md \tau )_Q = \int _0 ^{-s} (V, LG(\tau)U)_Q \;\md \tau
\]
implying
\[
|(G(s) V - V , U )_Q|\leq C |s| \;|U|_Q,\;\;s\in \R.
\]
Therefore $|G(s)V - V|_Q \leq C |s|, s \in \R$ and by the previous statement $V \in \dom (L)$. Finally $\dom (L) = \dom (L^\star)$ and $L^\star (V) = - L(V), V \in \dom (L) = \dom (L^\star)$.\\
4. As $L$ is skew-adjoint, we obtain
\[
- \inty{L(A)\nabla _y u \cdot \nabla _y v } = - ( L(A), Q^{-1} \nabla _y v \otimes \nabla _y u Q^{-1}\;)_Q = ( A, L ( Q^{-1} \nabla _y v \otimes \nabla _y u Q^{-1})\;)_Q.
\]
Recall that $P = Q^{-1}$ satisfies $L(P) = 0$, that is, $G(s)P = P, s \in \R$ and thus
\begin{align*}
L(Q^{-1} \nabla _y v \otimes & \nabla _y u Q^{-1}) = \lim _{ s \to 0} \frac{G(s)P\nabla _y v \otimes \nabla _y u P - P \nabla _y v \otimes \nabla _y u P}{s} \\
& = \lim _{ s \to 0} \frac{\partial _y Y ^{-1} (s;\cdot) P_s (\nabla _y v )_s \otimes (\nabla _y u )_s P_s {^t \partial _y Y ^{-1}}(s;\cdot) - P \nabla _y v \otimes \nabla _y u P}{s}\\
& = \lim _{ s \to 0} \frac{P{^t \partial _y Y} (s;\cdot) (\nabla _y v )_s \otimes (\nabla _y u )_s \partial _y Y (s;\cdot)P - P \nabla _y v \otimes \nabla _y u P}{s} \\
& = \lim _{ s \to 0} \frac{P \nabla _y v_s \otimes \nabla _y u_s P -P \nabla _y v \otimes \nabla _y u P }{s} \\
& = P \nabla _y ( b \cdny v ) \otimes \nabla _y u P + P \nabla _y v \otimes \nabla _y ( b \cdny u ) P.
\end{align*}
Finally one gets
\begin{align*}
- \inty{L(A) \nabla _y u \cdot \nabla _y v } & = ( A, P \nabla _y ( b \cdny v ) \otimes \nabla _y u P) + P \nabla _y v \otimes \nabla _y ( b \cdny u )P)_Q \\
& = \inty{A\nabla _y u \cdot \nabla _y ( b \cdny v)} + \inty{A\nabla _y ( b \cdny u ) \cdot \nabla _y v }.
\end{align*}
\end{proof}
We claim that $\dom (L)$ is left invariant by some special (weighted with respect to the matrix field $Q$) positive/negative part functions. The notations $A^\pm$ stand for the usual positive/negative parts of a symmetric matrix $A$
\[
A^\pm = S \Lambda ^\pm \;{^t S},\;\;A = S\Lambda \;{^t S}
\]
where $\Lambda, \Lambda ^\pm $ are the diagonal matrix containing the eigenvalues of $A$ and the positive/negative parts of these eigenvalues respectively, and $S$ is the orthogonal matrix whose columns contain a orthonormal basis of eigenvectors for $A$. Notice that
\[
A^+ : A^- = 0,\;\;A^+ - A^- = A,\;\;A^+ : A^+ + A^- : A^- = A: A.
\]
We introduce also the positive/negative part functions which associate to any field of symmetric matrix $A(y)$ the fields of symmetric matrix $A^{Q\pm}(y)$ given by
\[
Q^{1/2} A^{Q\pm} \;Q^{1/2} = (Q^{1/2} AQ^{1/2})^\pm.
\]
Observe that $A^{Q+} - A^{Q-} = A$.
\begin{pro}
\label{InvPosNeg}$\;$\\
1. The applications $A \to A^{Q\pm}$ leave invariant the subset $\{A\in \dom (L): {^t A} = A\}$.\\
2. For any $A \in \dom (L), {^t A } = A$ we have
\[
(A^{Q+}, A^{Q-})_Q = 0,\;\;( L(A^{Q+}), L(A^{Q-}))_Q \leq 0.
\]
\end{pro}
\begin{proof}
1. Consider $A \in \dom (L), {^t A} = A$. It is easily seean that ${^t A^{Q\pm}} = A^{Q\pm}$ and
\begin{align*}
|A^{Q+}|^2 _Q + |A^{Q-}|^2 _Q & = \inty{(Q^{1/2}A Q^{1/2} ) ^ + : ( Q^{1/2} A Q^{1/2})^+} \\
& + \inty{(Q^{1/2}A Q^{1/2} ) ^ - : ( Q^{1/2} A Q^{1/2})^-}\\
& = \inty{Q^{1/2}A Q^{1/2} : Q^{1/2} A Q^{1/2}} = |A|^2 _Q < +\infty
\end{align*}
and therefore $A ^{Q\pm} \in H_Q$. The positive/negative parts $A^{Q\pm}$ are orthogonal in $H_Q$
\[
( A^{Q+} , A^{Q-})_Q = \inty{(Q^{1/2}AQ^{1/2})^+ : (Q^{1/2}AQ^{1/2})^-} = 0.
\]
We claim that $A^{Q\pm}$ satisfies \eqref{Equ61}. Indeed, thanks to \eqref{Equ59} we can write, using the notation $X^{:2} = X : X$
\begin{align}
\label{Equ63} |G(s)A^{Q\pm}- A^{Q\pm}|^2 _Q & =
\inty{\{ Q^{1/2} ( \partial _y Y ^{-1} (A ^{Q\pm})_s {^t \partial _y Y ^{-1}} - A^{Q\pm})Q^{1/2} \} ^{:2}}\\
& = \inty{\{{^t {\cal O}} (s;y) Q_s ^{1/2} (A ^{Q\pm})_sQ_s ^{1/2} {\cal O}(s;y) - Q^{1/2} A^{Q\pm}Q^{1/2} \}^{:2}} \nonumber \\
& = \inty{\{ {^t {\cal O}} (s;y) ( Q_s ^{1/2} A_s Q_s ^{1/2})^{\pm} {\cal O}(s;y) - (Q^{1/2}A Q^{1/2} ) ^{\pm} \} ^{:2}}. \nonumber
\end{align}
Similarly we obtain
\begin{equation}
\label{Equ64} |G(s)A - A|^2 _Q = \inty{\{{^t {\cal O}}(s;y) Q^{1/2}_s A_s Q^{1/2}_s {\cal O}(s;y) - Q^{1/2}AQ^{1/2} \} ^{:2}}.
\end{equation}
We are done if we prove that for any symmetric matrix $U, V$ and any orthogonal matrix $R$ we have the inequality
\begin{equation}
\label{Equ65}
( \;{^t R } U ^{\pm} R - V ^\pm \;) : ( \;{^t R } U ^{\pm} R - V ^\pm \;)\leq ( \;{^t R } U R - V \;):( \;{^t R } U R - V \;).
\end{equation}
For the sake of the presentation, we consider the case of positive parts $U^+, V^+$. The other one comes in a similar way. The above inequality reduces to
\[
2 \;{^t R } U R : V - 2 \;{^t R } U ^+ R : V ^+ \leq {^t R } U ^- R : {^t R } U ^- R + V^- : V^-
\]
or equivalently, replacing $U$ by $ U^+ - U^-$ and $V$ by $V^+ - V^-$, to
\[
- 2 \;{^t R } U ^+ R : V^- - 2 \;{^t R } U ^- R : V^+ + 2 \;{^t R } U ^- R : V ^- \leq {^t R } U ^- R : {^t R } U ^- R + V ^- : V^-.
\]
It is easily seen that the previous inequality holds true, since ${^t R } U ^+ R : V^- \geq 0$, ${^t R } U ^- R : V^+ \geq 0$ and
\[
2 \;{^t R } U ^- R : V^- \leq 2 ( {^t R} U ^- R : {^t R} U ^- R) ^{1/2} ( V^- : V^- ) ^{1/2} \leq {^t R} U ^- R : {^t R} U ^- R + V^- : V^-.
\]
Combining \eqref{Equ63}, \eqref{Equ64} and \eqref{Equ65} with
\[
U = Q^{1/2} _s A_s Q^{1/2}_s, \;\;V = Q^{1/2}AQ^{1/2},\;\;R = {\cal O}
\]
yields
\[
\sup _{s \neq 0} \frac{|G(s)A^{Q\pm} - A^{Q\pm} |_Q}{|s|} \leq \sup _{s \neq 0} \frac{|G(s)A - A|_Q}{|s|} \leq |L(A)|_Q
\]
saying that $A^{Q\pm} \in \dom (L)$. \\
2. For any $A \in \dom (L)$, $^t A = A$ we can write
\begin{align*}
(A^{Q+}, A^{Q-})_Q & = \inty{Q^{1/2}A^{Q+}Q^{1/2}: Q^{1/2}A^{Q-}Q^{1/2}} \\
& = \inty{(Q^{1/2}AQ^{1/2})^+ : (Q^{1/2}AQ^{1/2})^-} = 0.
\end{align*}
Since $A^{Q\pm} \in \dom (L)$ we have
\[
L(A^{Q\pm}) = \lim _{s \to 0} \frac{G(s/2) A^{Q\pm} - G(-s/2) A^{Q\pm}}{s}
\]
and therefore, thanks to \eqref{Equ59}, we obtain
\begin{align*}
& ( L(A^{Q+}), L(A^{Q-}))_Q = \lim _{s \to 0} \left (\frac{G(\frac{s}{2}) A^{Q+} - G(-\frac{s}{2}) A^{Q+}}{s}, \frac{G(\frac{s}{2}) A^{Q-} - G(-\frac{s}{2}) A^{Q-}}{s} \right ) _Q\\
& = \lim _{s \to 0} \inty{\frac{Q^{1/2} (\;G(\frac{s}{2}) A^{Q+} - G(-\frac{s}{2}) A^{Q+} \;) Q^{1/2} }{s} : \frac{Q^{1/2} (\;G(\frac{s}{2}) A^{Q-} - G(-\frac{s}{2}) A^{Q-} \;) Q^{1/2}}{s} }\\
& = \lim _{s \to 0} \inty{ \frac{{^t {\cal O}(\frac{s}{2};y)}( Q^{1/2}_{\frac{s}{2}} A_{\frac{s}{2}} Q^{1/2}_{\frac{s}{2}})^+ {\cal O}(\frac{s}{2};y) - {^t {\cal O}(-\frac{s}{2};y)}( Q^{1/2}_{-\frac{s}{2}} A_{-\frac{s}{2}} Q^{1/2}_{-\frac{s}{2}})^+ {\cal O}(-\frac{s}{2};y)}{s} \\
& : \frac{{^t {\cal O}(\frac{s}{2};y)}( Q^{1/2}_{\frac{s}{2}} A_{\frac{s}{2}} Q^{1/2}_{\frac{s}{2}})^- {\cal O}(\frac{s}{2};y) - {^t {\cal O}(-\frac{s}{2};y)}( Q^{1/2}_{-\frac{s}{2}} A_{-\frac{s}{2}} Q^{1/2}_{-\frac{s}{2}})^- {\cal O}(-\frac{s}{2};y)}{s}}\\
& = - \lim _{s \to 0} \inty{\frac{{^t {\cal O}(\frac{s}{2};y)}( Q^{1/2}_{\frac{s}{2}} A_{\frac{s}{2}} Q^{1/2}_{\frac{s}{2}})^+ {\cal O}(\frac{s}{2};y) : {^t {\cal O}(-\frac{s}{2};y)}( Q^{1/2}_{-\frac{s}{2}} A_{-\frac{s}{2}} Q^{1/2}_{-\frac{s}{2}})^- {\cal O}(-\frac{s}{2};y)}{s^2}} \\
& - \lim _{ s \to 0}\inty{\frac{{^t {\cal O}(-\frac{s}{2};y)}( Q^{1/2}_{-\frac{s}{2}} A_{-\frac{s}{2}} Q^{1/2}_{-\frac{s}{2}})^+ {\cal O}(-\frac{s}{2};y) : {^t {\cal O}(\frac{s}{2};y)}( Q^{1/2}_{\frac{s}{2}} A_{\frac{s}{2}} Q^{1/2}_{\frac{s}{2}})^- {\cal O}(\frac{s}{2};y)}{s^2}} \\
& \leq 0
\end{align*}
since
\[
{^t {\cal O}}(\pm s/2;\cdot) ( Q^{1/2} A Q^{1/2}) _{\pm s/2} ^\pm {\cal O}(\pm s/2;\cdot) \geq 0,\;\;{^t {\cal O}}(\mp s/2;\cdot) ( Q^{1/2} A Q^{1/2}) _{\mp s/2} ^\pm {\cal O}(\mp s/2;\cdot) \geq 0.
\]
\end{proof}
We intend to solve the problem \eqref{Equ67}, \eqref{Equ68} by using variational methods. We introduce the space $V_Q = \dom (L) \subset H_Q$ endowed with the scalar product
\[
((A, B))_Q = (A, B)_Q + (L(A), L(B))_Q,\;\;A, B \in V_Q.
\]
Clearly $(V_Q, ((\cdot, \cdot))_Q)$ is a Hilbert space (use the fact that $L$ is closed) and the inclusion $V_Q \subset H_Q$ is continuous, with dense image. The notation $\|\cdot \|_Q$ stands for the norm associated to the scalar product $((\cdot, \cdot))_Q$
\[
\|A\|^2 _Q = ((A, A))_Q = (A, A)_Q + (L(A), L(A))_Q = |A|^2 _Q + |L(A)|^2_Q,\;\;A\in V_Q.
\]
We introduce the bilinear form $\sigma : V_Q \times V_Q \to \R$
\[
\sigma (A, B) = (L(A), L(B))_Q,\;\;A, B \in V_Q.
\]
Notice that $\sigma$ is coercive on $V_Q$ with respect to $H_Q$
\[
\sigma (A, A) + |A|^2_Q = \|A\|^2_Q,\;\;A \in V_Q.
\]
By Theorems 1,2 pp. 620 \cite{DauLions88} we deduce that for any $D \in H_Q$ there is a unique variational solution for \eqref{Equ67}, \eqref{Equ68} that is $A \in C_b (\R_+; H_Q) \cap L^2 (\R_+; V_Q)$, $\partial _t A \in L^2 (\R_+; V_Q ^\prime)$
\[
A(0) = D,\;\;\frac{\md }{\md t } (A(t), U)_Q + \sigma (A(t), U) = 0,\;\;\mbox{in}\;\;\dpri{},\;\;\forall \;U \in V_Q.
\]
The long time limit of the solution of \eqref{Equ67}, \eqref{Equ68} provides the averaged matrix field in \eqref{Equ55}.
\begin{proof}
(of Theorem \ref{AveMatDif}) The identity
\[
\frac{1}{2}\frac{\md }{\md t} |A(t) |^2 _Q + |L(A(t))|^2 _Q = 0,\;\;t \in \R_+
\]
gives the estimates
\[
|A(t)|_Q \leq |D|_Q,\;\;t \in \R_+,\;\;\int _0 ^{+\infty} |L(A(t))|^2 _Q \;\md t\leq \frac{1}{2}|D|^2 _Q.
\]
Consider $(t_k)_k$ such that $t_k \to +\infty$ as $k \to +\infty$ and $(A(t_k))_k$ converges weakly towards some matrix field $X$ in $H_Q$. For any $U \in \ker L$ we have
\[
\frac{\md }{\md t} (A(t), U)_Q = 0,\;\;t \in \R_+
\]
and therefore
\begin{equation}
\label{Equ70} (\mathrm{Proj}_{\ker L} D, U)_Q = (D,U)_Q = (A(0), U)_Q = (A(t_k), U)_Q = ( X, U)_Q,\;\;U \in \ker L.
\end{equation}
Since $L(A) \in L^2 (R_+;H_Q)$ we deduce that $\limk L(A(t_k)) = 0$ strongly in $H_Q$. For any $V \in V_Q$ we have
\[
(X, L(V))_Q = \limk (A(t_k), L(V))_Q = - \limk (L(A(t_k)), V)_Q = 0.
\]
We deduce that $X \in \dom (L^\star) = \dom (L)$ and $L(X) = 0$, which combined with \eqref{Equ70} says that $X = \mathrm{Proj}_{\ker L} D$, or $X = \ave{D}_Q$. By the uniqueness of the limit we obtain $\lim _{t \to +\infty} A(t) = \mathrm{Proj}_{\ker L} D$ weakly in $H_Q$. Assume now that ${^t D } = D$. As $L$ commutes with transposition, we have $\partial _t {^t A} - L (L({^t A})) = 0$, ${^t A }(0) = D$. By the uniqueness we obtain ${^t A } = A$ and thus
\[
^t \ave{D}_Q = \;^t ( \mbox{w}-\lim _{t \to +\infty} A(t) ) = \mbox{w}-\lim _{t \to +\infty} {^t A(t)} = \mbox{w}-\lim _{t \to +\infty} A(t)= \ave{D}_Q.
\]
Suppose that $D\geq 0$ and let us check that $\ave{D}_Q \geq 0$. By Proposition \ref{InvPosNeg} we know that $A^{Q\pm}(t) \in V_Q$, $t \in \R_+$ and
\[
(A^{Q+}(t), A^{Q-}(t))_Q = 0,\;\;(L(A^{Q+}(t)), L(A^{Q-}(t)))_Q \leq 0,\;\;t \in \R_+.
\]
It is sufficient to consider the case of smooth solutions. Multiplying \eqref{Equ67} by $-A^{Q-}(t)$ one gets
\begin{align}
\label{Equ71} \frac{1}{2}\frac{\md }{\md t} |A^{Q-}(t) |^2 _Q + |L(A^{Q-}(t)|^2 _Q & = ( \partial _t A^{Q+}, A^{Q-}(t))_Q + (L(A^{Q+}(t)),L(A^{Q-}(t)) )_Q \\
& \leq ( \partial _t A^{Q+}, A^{Q-}(t))_Q. \nonumber
\end{align}
But for any $0 < h < t $ we have
\begin{align*}
(A^{Q+}(t) - A^{Q+}(t-h), A^{Q-}(t))_Q = - (A^{Q+}(t-h), A^{Q-}(t))_Q \leq 0
\end{align*}
and therefore $(\partial _t A^{Q+}(t), A^{Q-}(t))_Q \leq 0 $. Observe that $Q^{1/2}A^{Q-}(0)Q^{1/2} = (Q^{1/2} D Q^{1/2})^- = 0$, since $Q^{1/2} D Q^{1/2}$ is symmetric and positive. Thus $A^{Q-}(0) = 0$, and from \eqref{Equ71} we obtain
\[
\frac{1}{2} |A^{Q-}(t) |^2 _Q \leq \frac{1}{2}|A^{Q-}(0)|^2 _Q = 0
\]
implying that $Q^{1/2} A(t) Q^{1/2} \geq 0$ and $A(t) \geq 0$, $t \in \R_+$.
Take now any $U \in H_Q$, ${^t U } = U$, $U \geq 0$. By weak convergence we have
\[
( \ave{D}_Q, U)_Q = \lim _{t \to +\infty} (A(t), U)_Q = \lim _{t \to +\infty} \inty{Q^{1/2} A(t)Q^{1/2} :Q^{1/2} UQ^{1/2} }\geq 0
\]
and thus $\ave{D}_Q \geq 0$. By construction $\ave{D}_Q = \mathrm{Proj}_{\ker L} D \in \ker L$. It remains to justify the second statement in \eqref{Equ72}, and \eqref{Equ72Bis}. Take a bounded function $\varphi \in \liy{}$ which remains constant along the flow of $b$, that is $\varphi _s = \varphi, s \in \R$, and a smooth function $u \in C^1 (\R^m)$ such that $u_s = u, s \in \R$ and
\[
\inty{(\nabla _y u \cdot Q^{-1} \nabla _y u )^2 } < +\infty.
\]
We introduce the matrix field $U$ given by
\[
U(y) = \varphi (y) Q^{-1} (y) \;\nabla _y u \otimes \nabla _y u \; Q^{-1}(y),\;\;y \in \R^m.
\]
By one hand notice that $U \in H_Q$
\begin{align*}
|U|^2_Q & = \inty{Q^{1/2}UQ^{1/2}:Q^{1/2}UQ^{1/2}} = \inty{\varphi ^2|Q^{-1/2} \nabla _y u |^4}\\
& \leq \|\varphi \|_{L^\infty} ^2\inty{(\nabla _y u \cdot Q^{-1} \nabla _y u )^2 }.
\end{align*}
By the other hand, we claim that $U \in \ker L$. Indeed, for any $s \in \R$ we have
\[
\nabla _y u = \nabla _y u_s = {^t \dyy}(\nabla _y u )_s
\]
and thus
\begin{align*}
Q_s U_s Q_s & = \varphi _s (\nabla _y u )_s \otimes ( \nabla _y u )_s \\
& = \varphi \;( {^t \partial _y Y ^{-1}}\nabla _y u ) \otimes ( {^t \partial _y Y ^{-1}}\nabla _y u ) \\
& = \varphi \;{^t \partial _y Y ^{-1}}\;\nabla _y u \otimes \nabla _y u \; \partial _y Y ^{-1}\\
& = {^t \partial _y Y ^{-1}} QUQ \partial _y Y ^{-1}.
\end{align*}
Taking into account that $Q_s = {^t \partial _y Y ^{-1}}Q { \partial _y Y ^{-1}}$ we obtain
\[
{^t \partial _y Y ^{-1}} Q { \partial _y Y ^{-1}}U_s {^t \partial _y Y ^{-1}} Q { \partial _y Y ^{-1}} = {^t \partial _y Y ^{-1}} Q U Q { \partial _y Y ^{-1}}
\]
saying that $U_s (y)= \dyy U(y) {^t \dyy}$. As $\ave{D}_Q = \mathrm{Proj}_{\ker L }D$ one gets
\begin{align*}
0 = (D - \ave{D}_Q, U ) _Q & = \inty{(D - \ave{D}_Q) : QUQ} \\
& = \inty{\varphi (y) (D - \ave{D}_Q) :\nabla _y u \otimes \nabla _y u }\\
& = \inty{\varphi (y) \{ \nabla _y u \cdot D \nabla _y u - \nabla _y u \cdot \ave{D}_Q \nabla _y u \}}.
\end{align*}
In particular, taking $\varphi = 1$ we deduce that $\nabla _y u \cdot \ave{D}_Q \nabla _y u \in \loy{}$ and
\[
\inty{\nabla _y u \cdot \ave{D}_Q \nabla _y u } = \inty{\nabla _y u \cdot D \nabla _y u } = (D, Q^{-1}\;\nabla _y u \otimes \nabla _y u \;Q^{-1} )_Q < +\infty
\]
since $D \in H_Q$, $Q^{-1}\nabla _y u \otimes \nabla _y u Q^{-1} \in H_Q$. Since $\ave{D}_Q \in \ker L$, the function $\nabla _y u \cdot \ave{D}_Q \nabla _y u $ remains constant along the flow of $b$
\[
(\nabla _y u )_s \cdot (\ave{D}_Q)_s (\nabla _y u )_s = (\nabla _y u )_s \cdot \dyy \ave{D}_Q \;{^t \dyy} (\nabla _y u )_s = \nabla _y u \cdot \ave{D}_Q\nabla _y u.
\]
Therefore the function $\nabla _y u \cdot \ave{D}_Q \nabla _y u $ verifies the variational formulation
\begin{equation}
\label{Equ73} \nabla _y u \cdot \ave{D}_Q \nabla _y u \in \loy{},\;\;(\nabla _y u \cdot \ave{D}_Q \nabla _y u)_s = \nabla _y u \cdot \ave{D}_Q \nabla _y u,\;\;s \in \R
\end{equation}
and
\begin{equation}
\label{Equ74}
\inty{\nabla _y u \cdot D \nabla _y u \;\varphi } = \inty{\nabla _y u \cdot \ave{D}_Q \nabla _y u\;\varphi },\;\;\forall \;\varphi \in \liy{},\;\varphi _s = \varphi,\; s \in \R.
\end{equation}
It is easily seen, thanks to the hypothesis $D \in \liy{}$, that \eqref{Equ73}, \eqref{Equ74} also make sense for functions $u \in \hoy{}$ such that $u _s = u$, $s \in \R$. We obtain
\[
\nabla _y u \cdot \ave{D}_Q \nabla _y u = \ave{\nabla _y u \cdot D \nabla _y u },\;\;u \in \hoy{},\;\;u_s = u,\;\;s\in \R
\]
where the average operator in the right hand side should be understood in the $\loy{}$ setting cf. Remark \ref{AveLone}. Moreover, if $u, v \in \hoy{} \cap \kerbg{}$ then $\ave{D}_Q ^{1/2} \nabla _y u, \ave{D}_Q ^{1/2} \nabla _y v$ belong to $\lty{}$ implying that $\nabla _y u \cdot \ave{D}_Q \nabla _y v \in \loy{}$. As before we check that $\nabla _y u \cdot \ave{D}_Q \nabla _y v$ remains constant along the flow of $b$ and for any $\varphi \in \liy{}$, $\varphi _s = \varphi, s \in \R$ we can write
\begin{align*}
2 \inty{\nabla _y u \cdot D \nabla _y v \;\varphi } & = \inty{\nabla _y (u + v) \cdot D \nabla _y (u + v) \;\varphi}\\
& - \inty{\nabla _y u \cdot D \nabla _y u \;\varphi} - \inty{\nabla _y v \cdot D \nabla _y v \;\varphi}\\
& = \inty{\nabla _y (u + v) \cdot \ave{D}_Q \nabla _y (u + v) \;\varphi}\\
& - \inty{\nabla _y u \cdot \ave{D}_Q \nabla _y u \;\varphi} - \inty{\nabla _y v \cdot \ave{D}_Q \nabla _y v \;\varphi}\\
& = 2 \inty{\nabla _y u \cdot \ave{D}_Q \nabla _y v \;\varphi }.
\end{align*}
Finally one gets
\[
\nabla _y u \cdot \ave{D}_Q \nabla _y v = \ave{\nabla _y u \cdot D \nabla _y v},\;\;u, v \in \hoy{} \cap \kerbg{}.
\]
Consider now $u \in \hoy{} \cap \kerbg{}$ and $\psi \in C^2 _c (\R^m)$. In order to prove that $\ave{\nabla _y u \cdot \ave{D}_Q \nabla _y ( b \cdny \psi )} = 0$, where the average is understood in the $\loy{}$ setting, we need to check that
\[
\inty{\varphi (y) \;\nabla _y u \cdot \ave{D}_Q \nabla _y ( b \cdny \psi ) } = 0
\]
for any $\varphi \in \liy{}$, $\varphi _s = \varphi, s \in \R$. Clearly $B(y) := \varphi (y) \ave{D}_Q (y) \in \ker L$ and therefore it is enough to prove that
\[
\inty{\nabla _y u \cdot B \nabla _y ( b \cdny \psi ) }= 0
\]
for any $B \in \ker L$, which comes by the third statement of Proposition \ref{WMFI}.
\end{proof}
\begin{remark}
\label{Parametrization} Assume that there is $u_0$ satisfying $u_0 (\ysy) = u_0 (y) + s$, $s \in \R, y \in \R^m$. Notice that $u_0$ could be multi-valued function (think to angular coordinates) but its gradient satisfies for a.a. $y \in \R^m$ and $ s \in \R$
\[
\nabla _y u_0 = {^t \dyy } (\nabla _y u_0 )_s
\]
exactly as any function $u$ which remains constant along the flow of $b$. For this reason, the last equality in \eqref{Equ72} holds true for any $u, v \in \hoy{} \cap \kerbg{} \cup \{u_0\}$. In the case when $m-1$ independent prime integrals of $b$ are known {\it i.e.,} $\exists u_1, ..., u_{m-1} \in \hoy{}\cap \kerbg{}$, the average of the matrix field $D$ comes by imposing
\[
\nabla _y u_i \cdot \ave{D}_Q \nabla _y u_j = \ave{\nabla _y u_i \cdot D \nabla _y u_j},\;\;i, j \in \{0,...,m-1\}.
\]
\end{remark}
\section{First order approximation}
\label{FirstOrdApp}
\noindent
We assume that the fields $D(y), b(y)$ are bounded on $\R^m$
\begin{equation}
\label{Equ26} D \in \liy{},\;\;b \in \liy{}.
\end{equation}
We solve \eqref{Equ1}, \eqref{Equ2} by using variational methods. We consider the Hilbert spaces $V:= \hoy{} \subset H := \lty{}$ (the injection $V \subset H$ being continuous, with dense image) and the bilinear forms $\aeps : V \times V \to \R$ given by
\[
\aeps (u,v) = \inty{D(y) \nabla _y u \cdot \nabla _y v } + \frac{1}{\eps} \inty{(b \cdny u ) \;(b \cdny v)},\;\;u, v \in V.
\]
Notice that for any $0 < \eps \leq 1$ and $v \in V$ we have
\begin{align*}
\aeps (v, v) + d |v|_H ^2 & \geq \inty{D(y) \nabla _y v \cdot \nabla _y v + (b \cdny v ) \;(b \cdny v) } + d \inty{(v(y))^2} \\
& \geq d \inty{|\nabla _y v |^2} + d \inty{(v(y))^2} \\
& = d |v|_V ^2
\end{align*}
saying that $\aeps$ is coercive on $V$ with respect to $H$. By Theorems 1,2 pp. 620 \cite{DauLions88} we deduce that for any $\uein \in H$, there is a unique variational solution for \eqref{Equ1}, \eqref{Equ2}, that is
$\ue \in C_b (\R_+; H) \cap L^2(\R_+;V)$ and
\[
\ue (0) = \uein,\;\;\frac{\md}{\md t } \inty{\ue (t,y) v(y) } + \aeps (\ue (t), v) = 0,\;\;\mbox{in}\;\dpri{},\;\;\forall \; v \in V.
\]
By standard arguments one gets
\begin{pro}
\label{UnifEstim}
The solutions $(\ue)_\eps$ satisfy the estimates
\[
\|\ue \|_{C_b (\R_+; H)} \leq |\uein|_H,\;\;\int _0 ^{+\infty} \!\!\!\!\inty{|\nabla _y \ue |^2}\md t \leq \frac{|\uein |^2 _H}{2d}
\]
and
\[
\|b \cdny \ue \|_{L^2(\R_+; H)} \leq \left ( \frac{\eps}{2(1 - \eps)}\right ) ^{1/2} |\uein |_H,\;\;\eps \in (0,1).
\]
\end{pro}
We are ready to prove the convergence of the family $(\ue )_\eps$, when $\eps \searrow 0$, towards the solution of the heat equation associated to the averaged diffusion matrix field $\ave{D}_Q$.
\begin{proof} (of Theorem \ref{MainResult1}) Based on the uniform estimates in Proposition \ref{UnifEstim}, there is a sequence $(\eps _k)_k$, converging to $0$, such that
\[
\uek \rightharpoonup u \;\mbox{ weakly } \star \mbox{ in } L^\infty(\R_+; H),\;\;\nabla _y \uek \rightharpoonup \nabla _y u \;\mbox{ weakly in }\;L^2(\R_+;H).
\]
Using the weak formulation of \eqref{Equ1} with test functions $\eta (t) \varphi (y)$, $\eta \in C^1 _c (\R_+), \varphi \in C^1 _c (\R^m)$ yields
\begin{align}
\label{Equ77} - \intty{\eta ^\prime (t) \varphi (y) \uek (t,y) } & - \eta (0) \inty{\varphi \uekin } + \intty{\eta \nabla _y \uek \cdot D \nabla _y \varphi } \nonumber \\
& = - \frac{1}{\eps _k} \intty{\eta (t) ( b \cdny \uek) ( b \cdny \varphi )}.
\end{align}
Multiplying by $\eps _k$ and letting $k \to +\infty$, it is easily seen that
\[
\intty{\eta (b \cdny u ) \;(b \cdny \varphi ) } = 0.
\]
Therefore $u(t,\cdot) \in \ker {\cal T} = \kerbg$, $t \in \R_+$, cf. Proposition \ref{KerRanTau}. Clearly \eqref{Equ77} holds true for any $\varphi \in V$. In particular, for any $\varphi \in V \cap \kerbg{}$ one gets
\begin{align}
\label{Equ78} - \intty{\eta ^\prime \uek \varphi } - \eta (0) \inty{\uekin \varphi } + \intty{\eta \nabla _y \uek \cdot D \nabla _y \varphi } = 0.
\end{align}
Thanks to the average properties we have
\[
\inty{\uekin \varphi } = \inty{\ave{\uekin} \varphi } \to \inty{\uin \varphi}
\]
and thus, letting $k \to +\infty$ in \eqref{Equ78}, leads to
\begin{align}
\label{Equ79} - \intty{\eta ^\prime u \varphi } - \eta (0) \inty{\uin \varphi } + \intty{\eta \nabla _y u \cdot D \nabla _y \varphi } = 0.
\end{align}
Since $u(t, \cdot), \varphi \in V \cap \kerbg{}$ we have cf. Theorem \ref{AveMatDif}
\[
\inty{\nabla _y u \cdot D \nabla _y \varphi } = \inty{\nabla _y u \cdot \ave{D}_Q \nabla _y \varphi}
\]
and \eqref{Equ79} becomes
\begin{align}
\label{Equ80} - \intty{\eta ^\prime u \varphi } - \eta (0) \inty{\uin \varphi } + \intty{\eta \nabla _y u \cdot \ave{D}_Q \nabla _y \varphi } = 0.
\end{align}
But \eqref{Equ80} is still valid for test functions $\varphi = b \cdny \psi$, $\psi \in C^2 _c (\R^m)$ since $u(t,\cdot) \in \kerbg$, $\uin = \mbox{w}-\lime \ave{\uein} \in \kerbg$ and $\ave{D}_Q \in \ker L$
\[
\inty{u(t,y) b \cdny \psi } = 0,\;\;\inty{\uin b \cdny \psi } = 0,\;\;\inty{\nabla _y u \cdot \ave{D}_Q \nabla _y ( b \cdny \psi ) }= 0
\]
cf. Theorem \ref{AveMatDif}. Therefore, for any $v \in V$ one gets
\[
\frac{\md}{\md t} \inty{u (t,y) v(y) } + \inty{\nabla _y u \cdot \ave{D}_Q \nabla _y v } = 0\;\mbox{ in } \dpri{}
\]
with $u(0) = \uin$. By the uniqueness of the solution of \eqref{Equ75}, \eqref{Equ76} we deduce that all the family $(\ue)_\eps$ converges weakly to $u$.
\end{proof}
\begin{remark}
\label{Propagation} Notice that \eqref{Equ75} propagates the constraint $b \cdny u = 0$, if satisfied initially. Indeed, for any $v \in C^1 _c (\R^m)$ we have
\begin{equation}
\label{Equ81}\frac{\md }{\md t } \inty{u (t,y) v (y) } + \inty{ \nabla _y u \cdot \ave{D}_Q \nabla _y v } = 0\;\mbox{ in } \dpri{}.
\end{equation}
Since $\ave{D}_Q \in \ker L$, we know by the second statement of Proposition \ref{WMFI} that
\begin{equation*}
\inty{\nabla _y u_s \cdot \ave{D}_Q \nabla _y v } = \inty{\nabla _y u \cdot \ave{D}_Q \nabla _y v_{-s}}.
\end{equation*}
Replacing $v$ by $v_{-s}$ in \eqref{Equ81} we obtain
\[
\frac{\md }{\md t } \inty{u_s v } + \inty{\nabla _y u_s \cdot \ave{D}_Q \nabla _y v } = 0\;\mbox{ in } \dpri{}
\]
and therefore $u_s$ solves
\[
\partial _t u_s - \divy ( \ave{D}_Q \nabla _y u_s) = 0,\;\;(t, y) \in \R_+ \times \R^m
\]
and $u_s (0,y) = \uin (\ysy) = \uin (y), y \in \R^m$. By the uniqueness of the solution of \eqref{Equ75}, \eqref{Equ76} one gets $u_s = u$ and thus, at any time $t \in \R_+$, $b \cdny u (t,\cdot) = 0$.
\end{remark}
\section{Second order approximation}
\label{SecOrdApp}
\noindent
For the moment we have determined the model satisfied by the dominant term in the expansion \eqref{Equ6}. We focus now on second order approximation, that is, a model which takes into account the first order correction term $\eps u ^1$. Up to now we have used the equations \eqref{Equ7}, \eqref{Equ8}. Finding a closure for $u + \eps u ^1$ will require one more equation
\begin{equation}
\label{Equ83} \partial _t u^1 - \divy ( D \nabla _y u^1 ) - \divy ( b \otimes b \nabla _y u^2) = 0,\;\;(t, y) \in \R_+ \times \R ^m.
\end{equation}
Let us see, at least formally, how to get a second order approximation for $(\ue )_\eps$, when $\eps $ becomes small. The first order approximation {\it i.e.}, the closure for $u$, has been obtained by averaging \eqref{Equ8} and by taking into account that $u \in \kerbg{}$
\[
\partial _t u = \ave{\divy( D \nabla _y u ) } = \divy ( \ave{D}_Q \nabla _y u ).
\]
Thus $u^1$ satisfies
\begin{equation}
\label{Equ84} \divy ( \ave{D}_Q \nabla _y u ) - \divy ( D \nabla _y u ) - \divy ( b \otimes b \nabla _y u^1) = 0
\end{equation}
from which we expect to express $u^1$, up to a function in $\kerbg{}$, in terms of $u$.
\begin{proof} (of Theorem \ref{Decomposition})
We claim that $\ran L^2 = \ran L $ and thus $\ran L^2 $ is closed as well. Clearly $\ran L^2 \subset \ran L$. Consider now $Z = L(Y)$ for some $Y \in \dom (L)$. But $Y - \mathrm{Proj}_{\ker L} Y \in \ker L ^\perp = (\ker L^\star ) ^\perp = \overline{\ran L} = \ran L$ and there is $X \in \dom (L)$ such that $Y - \mathrm{Proj}_{\ker L} Y = L(X)$. Finally $X \in \dom (L^2)$ and
\[
Z = L(Y) = L(Y - \mathrm{Proj} _{\ker L} Y ) = L(L(X)).
\]
By construction we have $D - \ave{D}_Q \in ( \ker L)^\perp = ( \ker L^\star ) ^\perp = \overline{\ran L} = \ran L = \ran L^2$ and thus there is a unique $F \in \dom (L^2) \cap ( \ker L )^\perp $ such that $D = \ave{D}_Q - L(L(F))$. As $F \in ( \ker L )^\perp$, there is $C \in \dom (L)$ such that $F = L(C)$ implying that ${^t F} = {^t L(C)} = L ({^t C})$. Therefore ${^t F } \in \dom (L^2) \cap ( \ker L )^\perp$ and satisfies the same equation as $F$
\[
L(L({^t F})) = {^t L}(L(F)) = \ave{D}_Q - D.
\]
By the uniqueness we deduce that $F$ is a field of symmetric matrix.
By Proposition \ref{PropOpeL} we know that
\[
- \divy(L(F) \nabla _y ) = [b \cdot \nabla _y, - \divy ( F \nabla _y )]\;\mbox{ in }\; \dpri{}
\]
{\it i.e.,}
\[
\inty{L(F) \nabla _y u \cdot \nabla _y v } = - \inty{F \nabla _y u \cdot \nabla _y ( b \cdny v ) } - \inty{F \nabla _y ( b \cdny u ) \cdny v }
\]
for any $u, v \in C^2 _c (\R^m)$. Similarly, $E := L(F)$ satisfies
\[
- \divy ( L^2 (F) \nabla _y ) = - \divy (L(E) \nabla _y ) = [b\cdny, - \divy ( E \nabla _y )]\;\mbox{ in }\;\dpri{}
\]
and thus, for any $u, v \in C^3_c (\R^m)$ one gets
\begin{align*}
& \inty{(\ave{D}_Q - D) \nabla _y u \cdny v } = \inty{L^2(F)\nabla _y u \cdny v } \\
& = - \inty{L(F) \nabla _y u \cdny ( b \cdny v ) }- \inty{L(F) \nabla _y ( b \cdny u ) \cdny v } \\
& = \inty{F \nabla _y u \cdny ( b \cdny ( b \cdny v ))} + \inty{F \nabla _y ( b \cdny u ) \cdny ( b \cdny v)} \\
& + \inty{F \nabla _y ( b \cdny u ) \cdny ( b \cdny v)} + \inty{F \nabla _y ( b \cdny ( b \cdny u )) \cdny v}.
\end{align*}
\end{proof}
\noindent
The matrix fields $F \in \dom (L^2)$ and $E = L(F) \in \dom (L)$ have the following properties.
\begin{pro}
\label{PropOpeF} For any $u, v \in C^1 (\R^m)$ which are constant along the flow of $b$ we have in $\dpri{}$
\[
D \nabla _y u \cdny v - \ave{D}_Q \nabla _y u \cdny v = - b \cdny ( E \nabla _y u \cdny v ) = - \divy ( b \otimes b \nabla _y ( F \nabla _y u \cdny v ))
\]
and
\[
\ave{E \nabla _y u \cdny v } = \ave{ F \nabla _y u \cdny v } = 0.
\]
In particular
\[
\inty{E \nabla _y u \cdny v } = \inty{\ave{E \nabla _y u \cdny v}}= 0
\]
\[
\inty{F \nabla _y u \cdny v } = \inty{\ave{F \nabla _y u \cdny v}}= 0
\]
saying that $\ave{\divy ( E \nabla _y u ) } = \ave{\divy ( F \nabla _y u )} = 0$ in $\dpri{}$.
\end{pro}
\begin{proof}
Consider $\varphi \in C^1 _c (\R^m)$, $u, v \in C^1 (\R^m)$ such that $u_s = u, v_s = v$, $s \in \R$ and the matrix field $U = \varphi Q^{-1} \nabla _y v \otimes \nabla _y u Q^{-1} \in H_Q$. Actually $U \in \dom (L)$ and, as in the proof of the last statement in Proposition \ref{PropOpeL}, one gets
\begin{align*}
L(U) & = (b \cdny \varphi ) Q^{-1} \nabla _y v \otimes \nabla _y u Q^{-1} + \varphi \;L ( Q^{-1} \nabla _y v \otimes \nabla _y u Q^{-1}) \\
& = (b \cdny \varphi) Q^{-1} \nabla _y v \otimes \nabla _y u Q^{-1}
\end{align*}
since $ Q^{-1} \nabla _y v \otimes \nabla _y u Q^{-1} \in \ker (L)$. Multiplying by $U$ the equality $D - \ave{D}_Q = - L(E)$, $E = L(F)$, one gets
\[
\inty{\varphi ( D - \ave{D}_Q)\nabla _y u \cdny v } = - (L(E), U)_Q = (E, L(U))_Q = \inty{(b \cdny \varphi) ( E \nabla _y u \cdny v )}
\]
implying that $D \nabla _y u \cdny v = \ave{D}_Q \nabla _y u \cdny v - b \cdny ( E \nabla _y u \cdny v)$ in $\dpri{}$. Multiplying by $U$ the equality $E = L(F)$ yields
\[
\inty{\varphi E \nabla _y u \cdny v } = (E, U)_Q = (L(F), U)_Q = - (F, L(U))_Q = - \inty{(b \cdny \varphi) F \nabla _y u \cdny v}.
\]
We obtain
\[
E \nabla _y u \cdny v = b \cdny ( F \nabla _y u \cdny v) \;\mbox{ in }\; \dpri{}
\]
and thus
\[
D \nabla _y u \cdny v - \ave{D}_Q \nabla _y u \cdny v = - b \cdny (E \nabla _y u \cdny v ) = - b \cdny ( b \cdny ( F \nabla _y u \cdny v ))
\]
in $\dpri{}$. Consider now $U = \varphi Q^{-1} \nabla _y v \otimes \nabla _y u Q^{-1}$ with $\varphi \in \kerbg{}$. We know that $L(U) = 0$ and since, by construction $F \in (\ker L )^\perp$, we deduce
\[
\inty{\varphi F \nabla _y u \cdny v } = (F, U)_Q = 0
\]
saying that $\ave{F \nabla _y u \cdny v} = 0$. Similarly $E = L(F) \in (\ker L )^\perp$ and $\ave{E \nabla _y u \cdny v } = 0$.
\end{proof}
\begin{remark}
\label{ParametrizationBis}
Assume that there is $u_0$ (eventually multi-valued) satisfying $u_0 (\ysy{}) = u_0 (y) + s$, $s \in \R, y \in \R^m$. Its gradient changes along the flow of $b$ exactly as the gradient of any function which is constant along this flow cf. Remark \ref{Parametrization}. We deduce that $Q^{-1} \nabla _y v \otimes \nabla _y u Q^{-1} \in \ker L$ for any $u, v \in \kerbg \cup \{u_0\}$ and therefore the arguments in the proof of Proposition \ref{PropOpeF} still apply when $u, v \in \kerbg{} \cup \{u_0\}$. In the case when $m-1$ independent prime integrals $\{u_1, ..., u_{m-1}\}$ of $b$ are known, the matrix fields $E, F$ come, by imposing for any $i, j \in \{0,1,...,m-1\}$
\[
- b \cdny (E \nabla _y u_i \cdny u _j) = D \nabla _y u_i \cdny u_j - \ave{D \nabla _y u_i \cdny u_j},\;\;\ave{E \nabla _y u_i \cdny u_j} = 0
\]
and
\[
b \cdny (F \nabla _y u_i \cdny u _j) = E \nabla _y u_i \cdny u _j,\;\;\ave{F \nabla _y u_i \cdny u _j } = 0.
\]
\end{remark}
We indicate now sufficient conditions which guarantee that the range of $L$ is closed.
\begin{pro}
\label{CompleteIntegr} Assume that \eqref{Equ21}, \eqref{Equ22}, \eqref{Equ23} hold true and that there is a matrix field $R(y)$ such that \eqref{Equ90} holds true. Then the range of $L$ is closed.
\end{pro}
\begin{proof}
Observe that \eqref{Equ90} implies \eqref{Equ56}. Indeed, it is easily seen that $b \cdny R + R \partial _y b = 0$ in $\dpri{}$ is equivalent to $R = R_s \partial _y Y ( s; \cdot)$, $s \in \R$. We deduce that $P = R ^{-1} \;{^t R} ^{-1}$ satisfies
\[
G(s)P = \partial _y Y ^{-1} (s; \cdot) P_s {^t \partial _y Y ^{-1} (s; \cdot)} = \partial _y Y ^{-1} (s; \cdot)R_s ^{-1} \;{^t R_s} ^{-1} \;{^t \partial _y Y ^{-1} (s; \cdot)} = R^{-1} \;{^t R}^{-1} = P
\]
saying that $[b,P] = 0$ in $\dpri{}$. Therefore we can define $L$ as before, on $H_Q$, which coincides in this case with $\{A:RA\;{^t R} \in \lty{}\}$. We claim that $i \circ L = ( b \cdny ) \circ i$ where $i : H_Q \to \lty{}$, $i(A) = R A\; {^t R}$, $A \in H_Q$, which comes immediately from the equalities
\[
(i\circ G(s))A = RG(s)A {^t R} = R \partial _y Y ^{-1}( s; \cdot ) A_s {^t \partial _y Y }^{-1} {^t R} = R_s A_s {^t R_s} = (i(A))_s,\;s\in \R, A\in H_Q.
\]
In particular we have
\[
\ker L = \{A \in H_Q\;:\; i(A) \in \kerbg\}
\]
and
\begin{align*}
(\ker L )^\perp & = \{A \in H_Q\;:\; \inty{i(A) : U }
= 0\;\forall\;U \in \kerbg{}\} \\
& = \{A \in H_Q\;:\; i(A) \in ( \kerbg)^\perp \}.
\end{align*}
For any $A \in (\ker L)^\perp$ we can apply the Poincar\'e inequality \eqref{Equ23} to $i(A) \in (\kerbg)^\perp$ and we obtain
\[
|A|_Q = |i(A)|_{L^2} \leq C_P |b \cdny (i(A))|_{L^2} = C_P |i (L(A))|_{L^2} = C_P |L(A)|_Q.
\]
Therefore $L$ satisfies a Poincar\'e inequality as well, and thus the range of $L$ is closed.
\end{proof}
\begin{remark}
\label{ClosedRanL}
The hypothesis $b \cdny R + R \dyb = 0$ in $\dpri{}$ says that the columns of $R^{-1}$ form a family of $m$ independent vector fields in involution with respect to $b$, cf. Proposition \ref{VFI}
\[
R_s ^{-1} (y) = \dyy R ^{-1} (y),\;\;s\in \R,\;\;y \in \R^m.
\]
\end{remark}
\begin{remark}
\label{ExplicitAve} For any $U \in \ker L$, that is $i(U) \in \kerbg{}$, we have
\[
\inty{R ( D - \ave{D}_Q) {^t R } : i(U)} = 0.
\]
As $\ave{D}_Q \in \ker L $, we know that $i(\ave{D}_Q) = R \ave{D}_Q {^t R } \in \kerbg{}$ and thus the matrix field $R \ave{D}_Q {^t R }$ is the average (along the flow of $b$) of the matrix field $RD\;{^t R}$, which allows us to express $\ave{D}_Q$ in terms of $R$ and $D$
\[
R \ave{D}_Q {^t R } = \ave{R D \;{^t R }}.
\]
\end{remark}
From now on we assume that \eqref{Equ90} holds true. Applying the decomposition of Theorem \ref{Decomposition} with the dominant term $u \in \kerbg$ in the expansion \eqref{Equ6} and any $v \in C^3 _c (\R^m)$ yields
\[
\inty{(D - \ave{D}_Q) \nabla _y u \cdny v } = - \inty{F \nabla _y u \cdny ( b \cdny ( b \cdny v ))}.
\]
From \eqref{Equ84} one gets
\[
\inty{(D - \ave{D}_Q ) \nabla _y u \cdny v } - \inty{u^1 b \cdny ( b \cdny v )} = 0
\]
and thus
\begin{equation}
\label{CorrSplit}
u^1 = \divy ( F \nabla _y u ) + v^1,\;\;v^1 \in \ker ( b \cdny ( b \cdny )) = \kerbg.
\end{equation}
Notice that $\ave{u^1} = v^1$, since $\ave{\divy ( F \nabla _y u )} = 0$, cf. Proposition \ref{PropOpeF}. The time evolution for $v^1 = \ave{u^1}$ comes by averaging \eqref{Equ83}
\[
\partial _t v ^1 - \ave{\divy ( D \nabla _y v^1)} - \ave{\divy ( D \nabla _y ( \divy ( F \nabla _y u )))} = 0.
\]
As $v^1 \in \kerbg$ we have
\[
- \ave{\divy ( D \nabla _y v^1)} = - \divy ( \ave{D}_Q \nabla _y v^1)
\]
and we can write, with the notation $w^1 = \divy (F \nabla _y u)$
\begin{align}
\label{Equ86} \partial _t \{u + \eps u^1\} - \divy ( \ave{D}_Q \nabla _y \{u + \eps u^1\}) = \eps \partial _t w^1 - \eps \divy ( \ave{D}_Q \nabla _y w^1 ) + \eps \ave{\divy ( D \nabla _y w^1)}.
\end{align}
But the time derivative of $w^1$ is given by
\[
\partial _t w^1 = \divy ( F \nabla _y \partial _t u ) = \divy ( F \nabla _y ( \divy ( \ave{D}_Q \nabla _y u )))
\]
which implies
\begin{align*}
\partial _t w^1 - \divy ( \ave{D}_Q\nabla _y w^1) & = \divy ( F \nabla _y ( \divy ( \ave{D}_Q \nabla _y u )))- \divy ( \ave{D}_Q \nabla _y ( \divy ( F \nabla _y u ))) \\
& = - [\divy(\ave{D}_Q \nabla _y ), \divy ( F \nabla _y )]u.
\end{align*}
Up to a second order term, the equation \eqref{Equ86} writes
\begin{align}
\label{Equ102} \partial _t \{u + \eps u^1\} - \divy ( \ave{D}_Q \nabla _y \{u + \eps u ^1\}) & + \eps [\divy(\ave{D}_Q \nabla _y ), \divy ( F \nabla _y )]\{u + \eps u^1\} \nonumber \\
& - \eps \ave{\divy ( D \nabla _y ( \divy ( F \nabla _y u )))} = {\cal O}(\eps ^2).
\end{align}
We claim that for any $u \in \kerbg$ we have
\begin{equation}
\label{Equ87} \ave{\divy ( D \nabla _y ( \divy ( F \nabla _y u)))} = \ave{\divy ( E \nabla _y ( \divy ( E \nabla _y u )))}.
\end{equation}
By Proposition \ref{PropOpeF} we know that $\ave{\divy ( F \nabla _y u )} = 0$. As $L(\ave{D}_Q) = 0$ we have
\[
[b \cdny, - \divy ( \ave{D}_Q \nabla _y )] = - \divy ( L ( \ave{D}_Q) \nabla _y ) = 0
\]
and thus $\divy ( \ave{D}_Q\nabla _y)$ leaves invariant the subspace of functions which are constant along the flow of $b$. By the symmetry of the operator $\divy ( \ave{D}_Q \nabla _y )$, we deduce that the subspace of zero average functions is also left invariant by $\divy ( \ave{D}_Q \nabla _y )$. Therefore $\ave{\divy ( \ave{D}_Q \nabla _y ( \divy ( F \nabla _y u )))} = 0$ and
\[
\ave{\divy ( D \nabla _y ( \divy ( F \nabla _y u )))} = \ave{\divy ((D - \ave{D}_Q) \nabla _y ( \divy ( F \nabla _y u )))}.
\]
Thanks to Theorem \ref{Decomposition} we have
\begin{align*}
\divy((D - \ave{D}_Q)\nabla _y ) & = [b \cdny, [b \cdny, - \divy ( F \nabla _y )]\;] \\
& = [b \cdny, - \divy (L(F)\nabla _y )]\\
& = [b \cdny, - \divy (E\nabla _y )]
\end{align*}
which implies that
\begin{align*}
& \ave{\divy ( D \nabla _y ( \divy ( F \nabla _y u )))} = \ave{\divy ( (D - \ave{D}_Q) \nabla _y ( \divy ( F \nabla _y u )))} \\
& = \ave{\divy ( E \nabla _y ( b \cdny ( \divy ( F \nabla _y u )))) - b \cdny ( \divy ( E \nabla _y ( \divy ( F \nabla _y u ))))}\\
& = \ave{\divy ( E \nabla _y ( b \cdny ( \divy ( F \nabla _y u ))))}.
\end{align*}
Finally notice that
\[
- \divy ( E \nabla _y u ) = - \divy ( L(F)\nabla _y u ) = [b \cdny, - \divy ( F \nabla _y u )] = - b \cdny ( \divy ( F \nabla _y u ))
\]
and \eqref{Equ87} follows.
We need to average the differential operator $\divy ( E \nabla _y ( \divy ( E \nabla _y )))$ on functions $u \in \kerbg$. For simplicity we perform these computations at a formal level, assuming that all fields are smooth enough. The idea is to express the above differential operator in terms of the derivations ${^t R }^{-1} \nabla _y $ which commute with the average operator (see Proposition \ref{AveComFirstOrder}), since the columns of $R^{-1}$ contain vector fields in involution with $b(y)$.
\begin{lemma}
\label{ChangeOfCoord} Under the hypothesis \eqref{Equ90}, for any smooth function $u(y)$ and matrix field $E(y)$ we have
\begin{equation}
\label{Equ100} \divy ( E \nabla _y u) = \divy ( R \;{^t E}) \cdot ( {^t R}^{-1} \nabla _y u ) + R E \;{^t R} : ( {^t R } ^{-1} \nabla _y \otimes {^t R }^{-1} \nabla _y ) u.
\end{equation}
\end{lemma}
\begin{proof}
Applying the formula $\divy (A\xi) = \divy {^t A} \cdot \xi + {^t A } : \partial _y \xi$, where $A(y)$ is a matrix field and $\xi (y)$ is a vector field, one gets
\[
\divy ( E \nabla _y u ) = \divy ( E \;{^t R } \;{^t R ^{-1}} \nabla _y u ) = \divy ( R \;{^t E}) \cdot ( {^t R }^{-1} \nabla _y u ) + R \;{^t E} : \partial _y ( {^t R }^{-1} \nabla _y u ).
\]
The last term in the above formula writes
\begin{align*}
R \;{^t E } : \partial _y ( {^t R }^{-1} \nabla _y u ) & = R \;{^t E} \;{^t R}\; {^t R } ^{-1} : \partial _y ( {^t R } ^{-1} \nabla _y u ) \\
& = R \;{^t E } \;{^t R } : \partial _y ( {^t R} ^{-1} \nabla _y u ) R ^{-1} \\
& = R E \;{^t R} : {^t R }^{-1} \;{^t \partial _y } ( {^t R } ^{-1} \nabla _y u ) \\
& = R E \;{^t R} : ( {^t R} ^{-1} \nabla _y \otimes {^t R} ^{-1} \nabla _y ) u
\end{align*}
and \eqref{Equ100} follows.
\end{proof}
Next we claim that the term $\ave{\divy ( E \nabla _y ( \divy ( E \nabla _y u )))}$ reduces to a differential operator, if $u \in \kerbg{}$.
\begin{pro}
\label{DifOpe} Under the hypothesis \eqref{Equ90}, for any smooth matrix field $E$ there is a linear differential operator $S(u)$ of order four, such that, for any smooth $u \in \kerbg{}$
\begin{equation}
\label{Equ101} \ave{\divy ( E \nabla _y ( \divy ( E \nabla _y u )))} = S(u).
\end{equation}
\end{pro}
\begin{proof}
For any smooth functions $u, \varphi \in \kerbg{}$ we have, cf. Lemma \ref{ChangeOfCoord}
\begin{align*}
& \inty{\ave{\divy(E \nabla _y ( \divy ( E \nabla _y u )))}\varphi } = \inty{\divy(E \nabla _y ( \divy ( E \nabla _y u ))) \varphi }\\
& = \inty{\divy ( E \nabla _y u ) \;\divy ( E \nabla _y \varphi )} \\
& = \inty{\{\divy ( R \; ^t E) \cdot ( ^t R ^{-1} \nabla _y u ) + R E \; ^t R : ( ^t R ^{-1} \nabla _y \otimes {^t R } ^{-1} \nabla _y )u \}\\
& \times \{\divy ( R \; ^t E) \cdot ( ^t R ^{-1} \nabla _y \varphi ) + R E \; ^t R : ( ^t R ^{-1} \nabla _y \otimes {^t R } ^{-1} \nabla _y )\varphi \}}\\
& = \inty{[\divy (R \;\;^t E) \otimes \divy ( R \;\;^t E)] : [^t R ^{-1} \nabla _y u \otimes {^t R} ^{-1} \nabla _y \varphi] }\\
& + \inty{[R E \;\;^t R \otimes \divy ( R \;\;^t E)] : [( ^t R ^{-1} \nabla _y \otimes {^t R }^{-1}\nabla _y )u \otimes {^t R } ^{-1} \nabla _y \varphi] }\\
& + \inty{[\divy( R \;\;^t E) \otimes R E \;\;^t R] : [(^t R ^{-1} \nabla _y u ) \otimes ( ^t R ^{-1} \nabla _y \otimes {^t R}^{-1} \nabla _y ) \varphi]}\\
& + \inty{[R E \;\;^t R \otimes R E \;\;^t R]:[ ( ^t R ^{-1} \nabla _y \otimes {^t R }^{-1}\nabla _y )u \otimes ( ^t R ^{-1} \nabla _y \otimes {^t R }^{-1}\nabla _y )\varphi ]}
\end{align*}
Recall that $^tR ^{-1} \nabla _y $ leaves invariant $\kerbg$ and therefore
\[
{^t R }^{-1} \nabla _y u \otimes {^t R } ^{-1} \nabla _y \varphi \in \kerbg{}
\]
implying that
\begin{align*}
& \inty{[\divy (R \;\;^t E) \otimes \divy ( R \;\;^t E)] : [^t R ^{-1} \nabla _y u \otimes {^t R} ^{-1} \nabla _y \varphi] }\\
= &
\inty{\ave{\divy (R \;\;^t E) \otimes \divy ( R \;\;^t E)} : [^t R ^{-1} \nabla _y u \otimes {^t R} ^{-1} \nabla _y \varphi] }.
\end{align*}
Similar transformations apply to the other three integrals above, and finally one gets
\begin{align*}
\inty{\ave{\divy(E \nabla _y ( \divy ( E \nabla _y u )))}\varphi } & = \inty{X : [\nablar u \otimes \nablar \varphi ]} \\
& + \inty{Y : [( \nablar \otimes \nablar )u \otimes \nablar \varphi ]}\\
& + \inty{Z : [\nablar u \otimes ( \nablar \otimes \nablar ) \varphi] } \\
& + \inty{T : [( \nablar \otimes \nablar )u \otimes ( \nablar \otimes \nablar ) \varphi]}\\
& = I_1 (u, \varphi) + I_2 (u, \varphi) + I_3 (u, \varphi) + I_4 (u, \varphi)
\end{align*}
where $\nablar := {^t R} ^{-1} \nabla _y $ and $X, Y, Z, T$ are tensors of order two, three, three and four respectively
\[
X_{ij} = \ave{\divy ( R \;\;^t E) _i \;\divy(R \;\;^t E)_j},\;\;i,j\in \{1,...,m\}
\]
\[
Y_{ijk} = \ave{(R E \;\;^t R) _{ij} \;\divy (R \;\;^t E)_k},\;\;Z_{ijk} = \ave{\divy ( R \;\;^t E)_i \;\;(RE \;\;^t R)_{jk}} ,\;\;i,j, k\in \{1,...,m\}
\]
\[
T_{ijkl} = \ave{(RE \;\;^t R)_{ij} \;\;(RE \;\;^tR)_{kl}},\;\;i,j, k, l\in \{1,...,m\}.
\]
Integrating by parts one gets
\[
I_1 (u, \varphi) = \inty{X \nablar u \cdot \nablar \varphi } = \inty{R^{-1} X \nablar u \cdot \nabla _y \varphi } = \inty{S_1 (u) \varphi}
\]
where $S_1 (u) = - \divy ( R^{-1} X \nablar u)$. Notice that the differential operator
\[
\xi \to \divy ( R^{-1} \xi) = \divy (\;^t R ^{-1}) \cdot \xi + {^t R}^{-1} : \partial _y \xi
\]
maps $(\kerbg{})^m$ to $\kerbg{}$, since the columns of $R^{-1}$ contain fields in involution with $b$, and therefore $S_1$ leaves invariant $\kerbg{}$, that is, for any $u \in \kerbg{}$, $\xi = X \nablar u \in (\kerbg{})^m$ and $S_1 (u) = - \divy ( R^{-1} X \nablar u ) = - \divy ( R^{-1} \xi ) \in \kerbg{}$. Similarly we obtain
\[
I_2 (u, \varphi)
= \inty{S_2 (u) \varphi },\;\;
I_3 (u, \varphi) = \inty{S_3 (u) \varphi },\;\;I_4 (u, \varphi) = \inty{S_4 (u) \varphi }
\]
where $S_2, S_3, S_4$ are differential operators of order three, three and four respectively, which leave invariant $\kerbg{}$. We deduce that
\[
\inty{\ave{\divy (E \nabla _y ( \divy ( E \nabla_y u )))} \varphi } = \inty{S(u) \varphi}
\]
for any $u, \varphi \in \kerbg{}$, with $S = S_1 + S_2 + S_3 + S_4$, saying that
\[
\ave{\divy (E \nabla _y ( \divy ( E \nabla_y u )))} - S(u) \perp \kerbg{}.
\]
But we also know that
\[
\ave{\divy (E \nabla _y ( \divy ( E \nabla_y u )))} - S(u) \in \kerbg{}
\]
and thus \eqref{Equ101} holds true.
\end{proof}
Combining \eqref{Equ102}, \eqref{Equ87}, \eqref{Equ101} we obtain
\begin{align*}
\partial _t \{u + \eps u^1\} - \divy ( \ave{D}_Q \nabla _y \{u + \eps u ^1\}) & + \eps [\divy(\ave{D}_Q \nabla _y ), \divy ( F \nabla _y )]\{u + \eps u^1\} \\
& - \eps S(u + \eps u^1) = {\cal O}(\eps ^2)
\end{align*}
which justifies the equation introduced in \eqref{IntroEqu87}. The initial condition comes formally by averaging the Ansatz \eqref{Equ6}
\[
\ave{\ue} = u + \eps v^1 + {\cal O}(\eps ^2).
\]
One gets
\[
v ^1 (0, \cdot) = \mbox{w-} \lime \frac{\ave{\uein} - \uin }{\eps} = \vin
\]
implying that $u ^1 (0, \cdot) = \vin + \divy(F \nabla _y \uin )$, cf. \eqref{CorrSplit}, which justifies \eqref{NewIC}.
\section{An example}
Let us consider the vector field $b(y) = {^\perp y} := (y_2, - y_1)$, for any $y = (y_1, y_2) \in \R^2$ and the matrix field
\[
D (y) = \left(
\begin{array}{cc}
\lambda _ 1 (y) & 0 \\
0 & \lambda _2 (y)
\end{array}
\right),\;\;y \in \R^2
\]
where $\lambda _1, \lambda _2 $ are given functions, satisfying $\min _{y\in \R^2} \{\lambda _1 (y), \lambda _2 (y)\} \geq d>0$. We intend to determine the first order approximation, when $\eps \searrow 0$, for the heat equation
\begin{equation}
\label{Equ91} \partial _t \ue - \divy ( D(y) \nabla _y \ue ) - \frac{1}{\eps} \divy ( b(y) \otimes b(y) \nabla _y \ue ) = 0,\;\;(t, y ) \in \R_+ \times \R ^2
\end{equation}
with the initial condition
\[
\ue (0, y) = \uin (y),\;\;y \in \R^2.
\]
The flow of $b$ is given by $Y(s;y) = {\cal R}(-s)y$, $s \in \R, y \in \R^2$ where ${\cal R}(\alpha)$ stands for the rotation of angle $\alpha \in \R$. The functions in $\kerbg{}$ are those depending only on $|y|$. Notice that the matrix field
\[
R(y) = \frac{1}{|y|}\left(
\begin{array}{rr}
y_2 & -y_1 \\
y_1 & y_2
\end{array}
\right)
\]
satisfies $b \cdot \nabla _y R + R \partial _y b = 0$ and $Q = {^t R } R = I_2$. The averaged matrix field $\ave{D}_Q$ comes, thanks to Remark \ref{ExplicitAve}, by the formula $R \ave{D}_Q {^t R} = \ave{R D \;{^t R}}$ and thus
\[
\ave{D}_Q = {^t R} \ave{RD\; {^t R}} R,\;\;\ave{RD\; {^t R}} = \left(
\begin{array}{rr}
\ave{\frac{\lambda _1 y _2 ^2 + \lambda _2 y_1 ^2 }{|y|^2}} & \ave{\frac{(\lambda _1 - \lambda _2)y_1 y _2 }{|y|^2}} \\
\ave{\frac{(\lambda _1 - \lambda _2)y_1 y _2 }{|y|^2}} & \ave{\frac{\lambda _1 y _1 ^2 + \lambda _2 y_2 ^2 }{|y|^2}}
\end{array}
\right).
\]
In the case when $\lambda _1, \lambda _2$ are left invariant by the flow of $b$, that is $\lambda _1, \lambda _2$ depend only on $|y|$, it is easily seen that
\[
\ave{\frac{y_1 ^2}{|y|^2}} = \ave{\frac{y_2 ^2}{|y|^2}} = \frac{1}{2},\;\;\ave{\frac{y_1 y_2}{|y|^2}} = 0
\]
and thus
\[
\ave{D}_Q = {^t R } \frac{\lambda _1 + \lambda _2}{2} I_2 R = \frac{\lambda _1 + \lambda _2}{2} I_2.
\]
The first order approximation of \eqref{Equ91} is given by
\[
\left\{
\begin{array}{ll}
\partial _t u - \divy \left ( \frac{\lambda _1 (y) + \lambda _2 (y)}{2} \nabla _y u \right ) = 0,& \;\;(t, y ) \in \R_+ \times \R ^2 \\
u(0,y) = \uin (y),& \;\;y \in \R^2.
\end{array}
\right.
\]
We consider the multi-valued function $u_0 (y) = - \theta (y)$, where $y = |y| ( \cos \theta (y), \sin \theta (y))$, which satisfies $b \cdot \nabla _y u_0 = 1$, or $u_0 (Y(s;y)) = u_0 (y) + s$. Notice that the averaged matrix field $\ave{D}_Q$ satisfies (with $u_1 (y) = |y|^2 /2 \in \kerbg{}$\;)
\[
\nabla _y u_i \cdot \ave{D}_Q \nabla _y u _j = \ave{\nabla _y u _i \cdot D \nabla _y u_j },\;\;i, j \in \{0,1\}
\]
as predicted by Remark \ref{Parametrization}.
\appendix
\section{Proofs of Propositions \ref{VFI}, \ref{WVFI}, \ref{MFI}, \ref{WMFI}}
\label{A}
\begin{proof} (of Proposition \ref{VFI}) For simplicity we assume that $b$ is divergence free. The general case follows similarly. Let $c(y)$ be a vector field satisfying \eqref{Equ34}. For any vector field $\phi \in C^1 _c (\R^m)$ we have, with the notation $u _\tau = u (Y(\tau;\cdot))$
\begin{align*}
\inty{c \cdot ( \phi _{-h} - \phi )} = \inty{(c_h - c) \cdot \phi } = \inty{(\partial _y Y (h;y) - I) c \cdot \phi }.
\end{align*}
Multiplying by $h^{-1}$ and passing to the limit when $h \to 0$ imply
\[
- \inty{c ( b \cdny \phi ) } = \inty{\partial _y b c\cdot \phi }
\]
and therefore $(b \cdny ) c - \partial _y b c = 0$ in $\dpri{}$.
Conversely, assume that $[b,c] = 0$ in $\dpri{}$. We introduce $e(s,y) = c(Y(s;y)) - \partial _y Y(s;y) c(y)$. Notice that $e(s,\cdot) \in \loloc{}, s \in \R$ and $e(0, \cdot) = 0$. For any vector field $\phi \in C^1 _c (\R^m)$ we have
\[
E_\phi (s) : = \inty{e(s,y) \cdot \phi (y)} = \inty{c(y) \cdot \phi _{-s}} - \inty{\partial _y Y(s;y) c(y) \cdot \phi (y) }
\]
and thus
\begin{align*}
\frac{\md }{\md s} E_\phi (s) & = - \inty{c(y) \cdot ((b \cdny )\phi )_{-s}} - \inty{\partial _y (b(\ysy)) \;c(y) \cdot \phi (y) } \\
& = - \inty{c \cdot (b \cdny ) \phi _{-s}} - \inty{\partial _y b (Y(s;y)) \dyy c(y) \cdot \phi (y) }\\
& = \inty{\dyb \;c(y) \cdot \phi _{-s}} - \inty{\dyb (\ysy) \dyy c(y) \cdot \phi (y)}\\
& = \inty{\dyb (\ysy) ( c(\ysy) - \dyy c(y)) \cdot \phi (y)} \\
& = \inty{e(s,y)\cdot {^t \dyb} (\ysy) \phi (y)}.
\end{align*}
In the previous computation we have used the fact that the derivation and tranlation along $b$ commute
\[
((b \cdny ) \phi )_{-s} = (b\cdny )\phi _{-s}.
\]
After integration with respect to $s$ one gets
\[
E_\phi (s) = \int _0 ^s \inty{e(\tau,y) \cdot {^t \dyb} (Y(\tau;y)) \phi (y) } \;\md \tau.
\]
Clearly, the above equality still holds true for any $\phi \in C_c (\R^m)$. Consider $R>0, T>0$ and let $K = \|{^t \dyb} \circ Y\|_{L^\infty([-T,T] \times B_R)}$. Therefore, for any $s \in [-T, T]$ we obtain
\begin{align*}
\|e(s,\cdot)\|_{L^\infty(B_R)} & = \sup \{ |E_\phi (s)|\;:\;\phi \in C_c (B_R),\;\;\|\phi \|_{\loy{}} \leq 1\}\\
& \leq K \left |\int _0 ^s \|e (\tau, \cdot) \|_{L^\infty (B_R)}\md \tau \right |.
\end{align*}
By Gronwall lemma we deduce that $\|e(s,\cdot)\|_{L^\infty(B_R)} = 0$ for $-T \leq s \leq T$ saying that $c(\ysy) - \dyy c(y) = 0, s \in \R, y \in \R^m$.
\end{proof}
\begin{proof} (of Proposition \ref{WVFI})\\
1.$\implies$ 2. By Proposition \ref{VFI} we deduce that $c(\ysy) = \dyy c(y)$ and therefore
\begin{align*}
\inty{(c\cdny u) v_{-s}} & = \inty{c(\ysy) \cdot (\nabla _y u ) (\ysy) v(y)}\\
& = \inty{c(y) \cdot {^t \dyy} (\nabla _y u )(\ysy) v(y) } = \inty{(c(y) \cdot \nabla _y u_s) v(y)}.
\end{align*}
2.$\implies$ 3. Taking the derivative with respect to $s$ of \eqref{Equ41} at $s = 0$, we obtain \eqref{Equ42}.
3.$\implies$ 1. Applying \eqref{Equ42} with $v \in C^1 _c (\R^m)$ and $u _i = y_i \varphi (y)$, $\varphi \in C^2 _c (\R^m)$, $\varphi = 1$ on the support of $v$, yields
\[
\inty{c_i \;b \cdny v } + \inty{c \cdny b_i \;v (y)}= 0
\]
saying that $b \cdny c_i = (\dyb \;c) _i$ in $\dpri{}$, $i \in \{1,...,m\}$ and thus $[b,c] = b \cdny c - \dyb c = 0$ in $\dpri{}$.
\end{proof}
\begin{proof} (of Proposition \ref{MFI}) The arguments are very similar to those in the proof of Proposition \ref{VFI}. Let us give the main lines. We assume that $b$ is divergence free, for simplicity. Let $A(y)$ be a matrix field satisfying \eqref{Equ35}. For any matrix field $U \in C^1 _c (\R^m)$ we have
\begin{align*}
\inty{A(y) & : ( U(Y(-h;y)) - U(y) )} = \inty{(A(Y(h;y)) - A(y)) : U(y) } \\
& = \inty{( \partial _y Y (h;y) A(y) {^t \partial _y Y (h;y)} - A(y)) : U(y)} \\
& = \inty{\{( \partial _y Y (h;y) - I ) A(y) {^t \partial _y Y (h;y)} : U(y) + A(y) {^t (\partial _y Y (h;y) - I)} : U(y)\}}.
\end{align*}
Multiplying by $\frac{1}{h}$ and passing $h \to 0$ we obtain
\[
- \inty{A(y) : ( b \cdny U ) } = \inty{(\dyb A(y) + A(y) {^t \dyb }):U(y) }
\]
saying that $[b,A] = 0$ in $\dpri{}$.
For the converse implication define, as before
\[
f(s,y) = A(\ysy ) - \dyy A(y) {^t \dyy},\;\;s \in \R,\;\;y \in \R^m.
\]
For any $U \in C^1 _c (\R^m)$ we have
\begin{align*}
F_U (s)& := \inty{f(s,y) : U(y)} \\
&= \inty{A(y) : U(Y(-s;y))} - \inty{\dyy A(y) {^t \dyy } : U (y)}
\end{align*}
and thus
\begin{align*}
\frac{\md }{\md s} F_U (s) & = - \inty{A(y) : (\;(b \cdny )U\;)_{-s}} - \inty{\partial _y ( b (\ysy)) A(y) {^t \dyy } : U(y)} \\
& - \inty{\dyy A(y) {^t \partial _y ( b (\ysy ))} : U(y)} \\
& = - \inty{A(y) : (b \cdny ) U_{-s} } - \inty{\dyb (\ysy) \dyy A(y) {^t \dyy} : U}\\
& - \inty{\dyy A(y) {^t \dyy} {^t \dyb (\ysy) } : U(y)}\\
& = \inty{ \{ \dyb (\ysy) f(s,y) + f(s,y) {^t \dyb (\ysy)}\} : U(y)} \\
& = \inty{f(s,y) : \{ {^t \dyb (\ysy)} U(y) + U(y) \dyb (\ysy) \}}.
\end{align*}
The previous equality still holds true for $U \in C_c (\R^m)$, and our conclusion follows as in the proof of Proposition \ref{VFI}, by Gronwall lemma.
\end{proof}
\begin{proof} (of Proposition \ref{WMFI})\\
$1.\implies 2.$ By Proposition \ref{MFI} we deduce that $A(\ysy) = \dyy A(y) {^t \dyy}$. Using the change of variable $y \to \ysy$ one gets
\begin{align*}
\inty{A(y) \nabla _y u \cdot \nabla _y v } & = \inty{A(\ysy) (\nabla _y u )(\ysy) \cdot (\nabla _y v ) (\ysy)} \\
& = \inty{A(y) {^t \dyy }(\nabla _y u ) (\ysy) \cdot {^t \dyy } (\nabla _y v ) (\ysy) } \\
& = \inty{A(y) \nabla _y u_s \cdot \nabla _y v_s }.
\end{align*}
$2.\implies 3.$ Taking the derivative with respect to $s$ at $s = 0$ of the constant function $s \to \inty{A(y) \nabla _y u_s \cdot \nabla _y v_s}$ yields
\[
\inty{A(y) \nabla _y ( b \cdny u ) \cdot \nabla _y v } + \inty{A(y) \nabla _y u \cdot \nabla _y ( b \cdny v) } = 0.
\]
$3.\implies 2.$ For any $u, v \in C^2 _c (\R^m)$ we can write, thanks to 3. applied with the functions $u_s, v_s$
\begin{align*}
\frac{\md }{\md s} \inty{A(y) \nabla _y u_s \cdot \nabla _y v_s} & = \inty{A(y) \nabla _y ( \;( b \cdny u)_s )\cdot \nabla _y v_s} \\
& + \inty{A(y) \nabla _y u_s \cdot \nabla _y ( \; (b \cdny v )_s)} \\
& = \inty{A(y) \nabla _y ( b \cdny u_s) \cdot \nabla _y v_s } \\
& + \inty{A(y) \nabla _y u_s \cdot \nabla _y ( b \cdny v_s ) } = 0.
\end{align*}
Therefore the function $s \to \inty{A(y) \nabla _y u_s \cdot \nabla _y v_s}$ is constant on $\R$ and thus
\[
\inty{A(y) \nabla _y u_s \cdot \nabla _y v_s} = \inty{A(y) \nabla _y u \cdot \nabla _y v},\;\;s\in \R.
\]
Up to now, the symmetry of the matrix $A(y)$ did not play any role. We only need it for the implication $2.\implies 1.$\\
$2.\implies 1.$ We have
\begin{align*}
\inty{A(y) \nabla _y u \cdot \nabla _y v } & = \inty{A(y) \nabla _y u_s \cdot \nabla _y v_s } \\
& = \inty{A(y) {^t \dyy} ( \nabla _y u)_s \cdot {^t \dyy } (\nabla _y v )_s } \\
& = \inty{\dyy A(y) {^t \dyy } (\nabla _y u )_s \cdot ( \nabla _y v )_s } \\
& = \inty{(\partial _y Y A \;{^t \partial _y Y})_{-s} \nabla _y u \cdot \nabla _y v }
\end{align*}
where $(\partial _y Y A {^t \partial _y Y})_{-s} = \partial _y Y (s; Y(-s;y)) A(Y(-s;y)) {^t \partial _y Y (s; Y(-s;y))}$. We deduce that
\[
\inty{(A(y) - (\partial _y Y A \;{^t \partial _y Y })_{-s}) \nabla _y u \cdot \nabla _y v} = 0,\;\;u, v \in C^1 _c (\R^m).
\]
Since $A(y) - (\partial _y Y A \;{^t \partial _y Y })_{-s}$ is symmetric, it is easily seen, cf. Lemma \ref{Divergence} below, that $A(y) - (\partial _y Y A \;{^t \partial _y Y })_{-s}= 0$. Therefore we have $A(\ysy) = \dyy A(y) {^t \dyy}$, $s \in \R,y \in \R^m$ and by Proposition \ref{MFI} we deduce that $[b,A] = 0$ in $\dpri{}$.
\end{proof}
\begin{lemma}
\label{Divergence}
Consider a field $A(y) \in \loloc{}$ of symmetric matrix satisfying
\begin{equation}
\label{Equ38} \inty{A(y) \nabla _y u \cdot \nabla _y v} = 0,\;\;u, v \in C^1 _c (\R^m).
\end{equation}
Therefore $A(y) = 0$ a.a. $y \in \R^m$.
\end{lemma}
\begin{proof}
Applying \eqref{Equ38} with $v_j = y_j v$, $v \in C^1 _c (\R^m)$, $u_i = y_i \varphi (y)$ where $\varphi \in C^1 _c (\R^m)$ and $\varphi = 1$ on the support of $v$, yields
\begin{equation}
\label{Equ39}
\inty{A(y) e_i \cdot ( y_j \nabla _y v + v e_j )} = 0.
\end{equation}
Applying \eqref{Equ38} with $v$ and $u_{ij} = y_i y_j \varphi (y)$ one gets
\begin{equation}
\label{Equ40}
\inty{A(y) ( y_j e_i + y_i e _j ) \cdot \nabla _y v } = 0.
\end{equation}
Combining \eqref{Equ39}, \eqref{Equ40} we obtin for any $i, j \in \{1,...,m\}$
\[
2 \inty{(A(y)e_i \cdot e_j) \;v(y)} = \inty{( A(y) e_i \cdot e_j + A(y) e_j \cdot e_i)v(y)} = 0
\]
saying that $A(y) = 0$, a.a. $y \in \R^m$.
\end{proof}
\end{document} |
\begin{document}
\title{Frog model wakeup time on the complete graph}
\begin{abstract}
The frog model is a system of random walks where active particles set sleeping particles in motion. On the complete graph with $n$ vertices it is equivalent to a well-understood rumor spreading model. We given an alternate and elementary proof that the wake-up time, i.e.\ the expected time for every particle to be activated, is $\Theta(\log n)$. Additionally, we give an explicit distributional equation for the wakeup time as a weighted sum of geometric random variables. This project was part of the University of Washington Research Experience for Undergraduates program.
\end{abstract}
\section{Introduction}
The \emph{frog model} starts with an awake frog at the root of a graph and one sleeping frog at each other vertex. In discrete time, awake frogs perform nearest neighbor simple random walks and wake any sleeping frogs they encounter. When first introduced by K. Ravishankar about twenty years ago, the model was known as the egg-model (see \cite{telcs1999}); R. Durrett is
credited with the zoomorphism of viewing particles as frogs. This
likely comes from the chaotic way particles wake up. We study the model on the complete graph with $n$ vertices, $K_n$. In particular we deduce that the \emph{wakeup time} $T_n$, the time for all frogs to wake up, has expected value on the order of $\log n$.
It was brought to our attention in the final stages of this project
that the wakeup time for the frog model on $K_n$ is equivalent to a rumor spreading model introduced in \cite{original}. This model starts with a town of $n$ people where one knows a rumor. At each time step those who know the rumor call a uniformly random resident and inform them. The frog model can be naturally coupled with the spread of the rumor so that the number of awake frogs is the same as the number of informed residents. Hence the wakeup time is equivalent to all $n$ residents knowing the rumor. We remark that the locations of the frogs are an extra bit of randomness not accounted for in the rumor spreading model. For this reason, the coupling only works on $K_n$.
In \cite{original} they show that $T_n/ \log_2 n \overset{\mathbf P} \to 1 + \log n + O(1)$ which implies our theorem.
The idea of the proof in \cite{original} is to break up the spread of the rumor into five stages (see the appendix for a list of the stages). For example, phase one is the time to wake up $N$ frogs with $N$ some large fixed constant. A finer analysis in \cite{second} shows that $T_n = \log_2 n + \log n + O(1)$ in probability. They show that waking is closely approximated by a deterministic equation (see the appendix). A tight analysis for $\E T_n$ is in \cite{third}, where they use three phases and some sophisticated estimates to show that $\E T_n = c \log n + b + o(1)$ with $c = 1+ (1/ \log(2))\approx 2.44$ and $b= 2.765$. Our result is less precise, but the proof is more elementary. We use two phases and only rely on couplings and Markov's inequality.
Although it is equivalent to the spread of a rumor on $K_n$, the wakeup time for the frog model has otherwise not been studied.
The recent article \cite[Open Question 5]{poisson} introduces the problem on finite $d$-ary trees. A survey article \cite{frogs} asks a similar question for a variant of the frog model where frogs perish after taking $t$ steps. They propose a study of the minimal $t$ that guarantees at least half the frogs on a given graph will be activated with probability greater than $1/2$.
It is claimed (without proof) that this value on the complete graph is $O(\log n)$. Though not equivalent, this is closely related to our result for the frog model cover time.
On finite graphs the frog model is a model for epidemics, or the spread of a rumor. The article \cite{Coop09} describes many related variants. It also appears in physics literature as a model for combustion \cite{combustion} known as $A + B \mapsto 2A$, where we replace awake and sleeping frogs with flames and fuel, respectively. The combustion process is studied on $\mathbb Z^d$. Noteworthy theorems include the fact that the origin is visited infinitely often for all $d \geq 1$ \cite{telcs1999} and a shape theorem that says, when properly scaled, the set of activated vertices converges to a convex region in the unit simplex (\cite{shape, random_shape}).
A model that is in a loose sense the reverse of the frog model is coalescing random walk. Introduced in \cite{erdos1974}, this is the process that starts with a particle at each site and when particles collide they coalesce into one. Like the frog model, this is typically studied in $\mathbb Z^d$. For instance, coalescing random walk is recurrent for all $d \geq 1$. This was first shown in \cite{first}, and refined further in \cite{griffeath, kesten_number, arratia, arratia2}. Computer science literature studies coalescing random walk on finite graphs. Of particular interest is the \emph{coalescance time}; the expected time for all particles to coalesce into a single particle. In \cite{cox} they study this on the torus. The recent paper \cite{cs} gives bounds on rather general graphs.
Our result should be compared with the cover time for multiple random walks on a graph (\cite{Elsässer20112623} and \cite{Alon07})). The basic question is how the cover time is reduced by using the combined ranges of $k$ random walks. This is studied on a variety of different graphs, and the speedup depends on the graph structure. For the complete graph, \cite{Alon07} cites the folklore (we give a proof in \thref{lem:couplings} \ref{loop}) that the speedup is linear, meaning the cover time for a single random walk is $k$ times the cover time for $k$ random walkers. All of the results for speedup of multiple walk take the worst-case scenario across every starting configuration for the $k$ walkers. The frog model is different in that we have only one possible starting configuration, and just one active particle. However, the placement of sleeping frogs is optimal in the sense that activated particles are more likely to be near unexplored sites. We ask a question regarding this in Further Questions (i).
\subsection{Main theorem and overview} \label{sec:main}
Before stating the theorem we review \emph{asymptotic notation}. We say that $f(n) = O(g(n))$ if there exists $c >0$ such that for all sufficiently large $n$ it holds that $f(n) \leq c g(n).$ We write $f(n) = \Theta(g(n))$ if $f(n) = O(g(n))$ and $g(n) = O(f(n))$. The wakeup time, $T_n$, is formally defined in the next section.
\begin{theorem} \thlabel{thm:main}
$\E T_n = \Theta(\log(n))$.
\end{theorem}
Additionally, in Section \ref{sec:formula} we give an explicit recursive formula for the distribution of $T_n$. This is in \thref{prop:formula}. The formula involves some sophisticated combinatorial objects and, combined with the formula in \cite{third} for $\E T_n$, yields a bound on their growth that could be of independent interest.
Most of our work is done on $K_n^\circ$, the complete graph with a self loop at each vertex. In \thref{lem:couplings} \ref{loop} we show the frog model on $K_n^\circ$ has a stochastically larger wakeup time.
We then prove \thref{thm:main} in two phases. First, we show
that it takes logarithmic time to wake the first $n/2$ frogs. This is done by embedding a process that grows slower than the frog model, but still (on average) grows exponentially. The idea is to, when say $k$ frogs are awake, only allow more frogs to wake up if at least $\alpha k$ asleep frogs are visited. This occurs with some probability $q_{k,n}$. We show in \thref{prop:p*} that $$\inf_{n \geq 3}\left( \min_{k < n/2} q_{k,n} \right) \geq p_* >0.$$ This lower bound is obtained by having the $k$ awake frogs jump one at a time, and thinking of the number of single jumps to wake the $(i+1)$st frog as a Geometric with mean $\frac{n-k-i}{n}$ waiting time. An application of Markov's inequality gives a uniform bound in terms of $\alpha$ for all $k< n/2$ and $n\geq 3$.
Next, thinking of each time an $\alpha$ proportion wakes up as a Bernoulli($q_{k,n}$) trial, the number of successes after $t$ steps is stochastically larger than a $\Bin(t,p_*)$ random variable.
Thus, the number of frogs awake at time $t$ is stochastically larger than
$$(1 + \alpha)^{\Bin(t,p_*)}.$$
Moreover, the time for this quantity to exceed $n/2$ is a sum of $O(\log n )$ geometric random variables with mean $p_*$. This is made formal in \thref{lem:couplings} \ref{key}. We can conclude that the expected time it takes to wake the first $n/2$ frogs is $O(\log n)$.
Once half the frogs are awake, we ignore the contribution of any new frogs added and show that $n/2$ frogs cover the remaining vertices in $O(\log n)$ steps. This is made precise in \thref{lem:couplings} \ref{loop} by reducing to the coupon collector problem.
\subsection{Further Questions}
The wakeup time for the frog model is a largely unexplored topic. There are many further questions one could ask. We remark that \cite{frogs} and \cite{poisson} discuss a few other problems on finite graphs.
\begin{enumerate} [label = {(\roman*)}]
\item {It is interesting to compare the wakeup time for the frog model on $G$, a graph with $n$ vertices, to the cover time for $n$ independent random walks on $G$ started in the least optimal starting configuration. Perhaps the frogs being evenly spread might overcome the disadvantage of starting with only one awake particle.} \emph{Are there graphs for which the frog model wakeup time is faster than the cover time for $n$-multiple random walks?} {The full binary tree of height $n$ is a good candidate. The expected cover time with $2^{n}$ particles started at the same leaf is $O(n^2 (2/\sqrt 2)^n)$ (see \cite{bincov}), whereas the wakeup time is conjectured to be polynomial in $n$ (see \cite{poisson}).}
\item Let $G(n,p)$ denote an Erd{\H o}s-R{\'e}nyi graph (i.e. the random graph obtained by keeping each edge in $K_n$ with probability $p$). \emph{What is the wakeup time for the frog model on $G(n,p)$?}{ For fixed $p>0$, this should still with high probability be $O(\log n)$, but for $p_n$ decaying with $n$ the graph is sparser and the wakeup time is possibly larger.}
\item \emph{What is the wakeup time for other graphs?} {For instance, the path, cycle, and grid.}
\end{enumerate}
\section{Formal model and couplings}
Here we give a formal definition of the frog model. Then we describe in \thref{lem:couplings} the five couplings we depend on in proving our main theorem.
\subsection{Formal definition of the frog model}
We borrow much of our notation from \cite{shape}. Let $V$ be the vertex set of $K_n$. Consider the collection, $\{F_v(t) \colon v \in V\}$, of independent random walks on $K_n$ each satisfying $F_v(0) = v$. These random walks correspond to the the trajectory of each frog. We now introduce stopping times to account for the waking up that occurs. Define
$$t(v,u) = \min_t \{F_v(t) = u\},$$
the time that the frog originally at vertex $v$ takes to reach vertex $u$. Also define
$$T(v,u) = \inf \left\{\sum_{i=1}^k t(v_{i-1},v_i) \colon v_0=v, \ldots, v_k = u \text{ for some } k\right\},$$
the first passage time from $v$ to $u$ in the frog model. Then $T(v_0,u)$ gives the time it takes for $u$ to be woken. For each $u \in V$, the position of the frog originally at $u$ at time $t$ is defined to be
$$P_u(t) = \begin{cases} u, &t \leq T(v_0,u) \\ F_u(t-T(v_0,u)), &t > T(v_0,u) \end{cases}.$$
With this we can define $\Lambda(t) = \{u \in V \colon T(v_0,u) \leq t\}$, the set of sites that have been visited by time $t$ or the set of awake frogs at time $t$. Define the number of frogs awake at time $t$ to be $N_t = |\Lambda(t)|$. Thus, the time to wake all of the frogs is $T_n = \inf \{t \colon N_t = n\}.$
\subsection{Couplings and stochastic dominance} \label{sec:dominance}
The frog model only depends on the underlying random walk trajectories. It has the nice feature that restricting the range of frogs, or ignoring woken frogs yields models with monotonically slower waking behavior. This is made formal using couplings and stochastic dominance.
Let $X$ and $Y$ be two random variables. If for each $a >0$ we have $\mathbf{P}[Y \geq a] \geq \mathbf{P}[X \geq a]$ then we say that $Y$ \emph{stochastically dominates} $X$, written $X\mathbf{P}receq Y$. A thorough reference on stochastic domination is \cite{SS}. Note that if $A \succeq B$, then $\mathbf{E} A \geq \mathbf{E}B$. An equivalent condition to stochastic dominance is that
$X\mathbf{P}receq Y$ if and only if there exists a coupling $(X,Y)$ with $X\leq Y$ a.s. Formally, a \emph{coupling} is a probability space with (possibly dependent) random variables $X'$ and $Y'$ that have the same distribution as $X$ and $Y$, respectively. Couplings can often be described intuitively and rigorously in words. In the following lemma we describe all of the couplings used in this paper.
\begin{lemma} \thlabel{lem:couplings} The following stochastic dominance relations hold:
\begin{enumerate}[label = (\Roman*)]
\item \label{binom} Let $\{q_i\}_{i=1}^t$ be a sequence in $[0,1]$ with $q_i > p$ for all $i=1,2,\hdots,, t$. It holds that
$$\sum_{i=1}^t \Ber(q_i) \succeq \Bin(t,p).$$
Here $\Ber(p)$ denotes a Bernoulli-$p$ random variable and $\Bin(t,p)$ is a binomial random variable with $t$ trials.
\item \label{loop} Let $K_n^\circ$ be the complete graph with a self-loop added to each vertex. If $T_n^\circ$ is the wakeup time for the frog model on $K_n^\circ$, then
$$T_n \mathbf{P}receq T_n^\circ.$$
\item \label{breakdown} Let $\tau_{n/2}$ and $C_{n/2}$ be as defined in the proof of \thref{thm:main}. It holds that
$$T_n \mathbf{P}receq \tau_{n/2} + C_{n/2}.$$
\item \label{loop} Let $K_n^\circ$ be the complete graph with a self-loop added to each vertex. Define $C_k^\circ$ to be the time for $k$ random walks to collectively visit every vertex of $K_n^\circ$, and define $C_k$ analogously for $K_n$. It holds that
\begin{align}
C_{k} \mathbf{P}receq C_{k}^\circ \overset{d}= \fracrac {C_1^\circ} k \overset{d}= \fracrac 1 k \sum_{i=1}^n \Geo\left(\fracrac{n-i}{n}\right).\label{eq:Ck}
\end{align}
Where $\Geo(p)$ is the number of Bernoulli-$p$ trials until a success occurs.
\item \label{key} Consider a modified frog model on $K_n^\circ$ where, when $k$ frogs are active, more frogs wake up only if at least $\alpha k$ sleeping frogs are visited at the next step (for some fixed $\alpha >0$). When this occurs we select an arbitrary subset of $\lceil \alpha k\rceil $ of these frogs and allow them to wake up. The others remain asleep. Thus, on this event there are at least $(1+\alpha)k$ frogs awake. \thref{prop:p*} shows that waking at least $\alpha k$ frogs occurs with probability at least $p_* >0$ for any $k$ and and sufficiently large $n$. We then have
$$(1 + \alpha)^{\Bin(t,p_*)} \mathbf{P}receq N_t,$$
and for $\tau_{n/2} := \inf\{t \colon N_t \geq n/2\}$ and $n_*:= \left\lfloor{\fracrac{\log(n/2)}{\log(1+ \alpha)}} \right\rfloor$ we have
\begin{align*}\tau_{n/2} &\mathbf{P}receq \inf\{ t \colon (1 + \alpha)^{\Bin(t,p_*)} \geq n/2 \}
\overset{d} = \sum_1^{n_*} \Geo(p_*).
\end{align*}
\end{enumerate}
\end{lemma}
\begin{proof} All of the proofs establish stochastic dominance via couplings.
\begin{enumerate}[label = (\Roman*)]
\item
Define $X = \sum_{i=1}^t\Ber(q_i)$ and $Y = \sum_{i=1}^t \Ber(p) = \Bin(t,p)$. Let $\{U_i\}_{i=1}^t$ be uniform $[0,1]$ random variables, so that
\begin{align*}
X &\overset{d} = \sum_{i=1}^t 1\{U_i \leq q_i\}, \\
Y & \overset{d} = \sum_{i=1}^t 1\{U_i \leq p\}.
\end{align*}
Our hypothesis $q_i > p$ guarantees that $X\geq Y$ for all realizations of the $U_i$.
\item
Pair the frogs on $K_n$ and $K_n^\circ$ in the natural way. Whenever a frog on $K_n^\circ$ moves to a new vertex, have the corresponding frog on $K_n$ follow it. In this way, the frogs on each graph perform random walks, but those on $K_n^\circ$ possibly spend extra steps traveling self-loops. This coupling ensures that $T_n \leq T_n^\circ$ in every realization of the model.
\item
Run the frog model up to time $\tau_{n/2}$. Of the $N_{\tau_{n/2}}$ frogs awake choose a batch of $n/2$ of them. Now think of this batch as paired to another frog model in the same configuration as ours at time $\tau_{n/2}$. Our $n/2$-batch frogs follow their counterparts. The time, $C_{n/2}$, it takes for the batch to visit all $n$ vertices of $K_n$ is at least as large as the $T_n - \tau_{n/2}$ steps taken by the frog model they are coupled with. In this way, the model restricted to a batch spends $\tau_{n/2} + C_{n/2}$ steps, which is at least the $T_n$ steps taken by the frog model.
\item A similar coupling as in (II) gives $C_k \mathbf{P}receq C_k^\circ$. Observe that on $K_n^\circ$ every site is accessible in one step. Thus, the set of sites visited by $k$ random walks has the same law as the range of a single random walk in $k$ steps. It follows that $kC_k^\circ \overset{d} = C_1^\circ$. The last equality
$$C_1^\circ \overset{d} = \sum_{i=1}^n \Geo\left( \fracrac{ n-i }{n}\right)$$
follows from the observation that the waiting time for a single random walk on $K_n^\circ$ to increase its range from $i$ to $i+1$ is the waiting time to have a success in a sequence of Bernoulli($(n-i)/n$) trials. This is distributed as a $\Geo( (n-i)/n)$. As increases in the range are independent and skip-free, the claimed formula follows.
\item By (II) we preserve the dominance relation by working on $K_n^\circ$. Since each successful increase in the number of frogs is a Bernoulli trial with probability at least $p_*$ it follows from (I) that this further dominates the random quantity $(1+ \alpha)^{\text{Bin}(t ,p_*)}$. The fact that $N_t$ dominates this quantity is a straightforward consequence of the fact that we are only ignoring frogs. And, the fact that $\tau_{n/2}$ is less than the time for $(1+ \alpha)^{\text{Bin}(t ,p_*)}$ to exceed $n/2$ follows immediately from the relationship between $(1+ \alpha)^{\Bin(t,p_*)}$ and $N_t$.
Taking the $\log$ of each side, we have the stopping time $\inf\{ t \colon (1+ \alpha)^{\Bin(t,p_*)} \geq n/2\}$ is equivalent to the time for a $\Bin(t,p_*)$ random variable to exceed $\log(n/2)/\log(1+ \alpha).$ The claimed distributional equality is just the fact that $\Bin(t,p_*)$ is a skip-free process which increases independently at each increment after $\Geo(p_*)$ steps.
\end{enumerate}
\end{proof}
\section{Proving \thref{thm:main}}
We start by elaborating a bit on \thref{lem:couplings} \ref{key}. Let $\alpha <1$ be a yet to be chosen parameter. Define the probabilities $q_{k,n} = q_{k,n}(\alpha)$ that the frog model on $K_n^\circ$ with $k$ frogs awake wakes at least $\alpha k$ frogs in one time step. Let $p_*= p_*(\alpha) = \inf_{n \geq 3}\left( \min_{ k < n/2} q_{k,n}\right)$. Our first goal is to establish that $p_*$ is bounded away from 0.
\begin{proposition} \thlabel{prop:p*}
For $\alpha = 1/10$ it holds that $p_* \geq 1/37$.
\end{proposition}
\begin{proof}
It is useful to decompose one time step in the frog model with $k$ frogs awake on $K_n^\circ$ into $k$ steps by a single random walk. Notice that the number of steps by the single random walk to visit the first sleeping frog is the waiting time for a success in a sequence of $\Ber(\frac{n-k}{n})$ trials (i.e.\ a $\Geo(\frac{n-k}{n})$ random variable). Similarly, once $i$ of the $n-k$ frogs have been woken the waiting time is $\Geo(\frac{n-k-i}{n})$. Let $X = X(k,n,\alpha) = \sum_{i=0}^{\lceil \alpha k \rceil} \Geo\left ( \tfrac{ n-k -i }{n} \right) .$ This represents the number of the $k$ awake frogs that jump in order to wake $\lceil \alpha k \rceil$ more.
It follows that $q_{k,n} = 1- \mathbf P[ X > k].$ And, by Markov's inequality
\begin{align}
q_{k,n} \geq 1 - \frac{ \E X }{k}. \label{eq:lb}
\end{align}
We can use linearity and the fact that the mean of a $\Geo(p)$ random variable is $1/p$ to estimate $\E X$:
\begin{align*}
\E X = \sum_{i=0}^{ \lceil \alpha k \rceil } \frac{ n }{ n - k -i } &= n \sum_{i=n- \lceil \alpha k \rceil }^{ n } \frac {1} { i } \leq n \frac{n - (n- \lceil \alpha k \rceil)}{n - \lceil \alpha k \rceil} = \frac{ n \lceil \alpha k \rceil }{ n- \lceil \alpha k \rceil }.
\end{align*}
Since $\lceil \alpha k \rceil \leq \alpha k +1$ we can bound $\E X /k$ by
\begin{align*} \frac{\E X}{k} &\leq \frac{ \alpha k +1}{k} \frac{ n }{n - \alpha k -1} \\
&= \frac{ \alpha n }{n - \alpha k -1}+\frac 1 k \frac{ n }{n - \alpha k -1}.
\end{align*}
Note that $q_{1,n} =1-\frac 1 n \geq \frac 23$ since we assume $n \geq 3$. Thus, we can work with $k \geq 2$. Also by assumption $k$ is no larger than $n/2$. We then preserve the above bound by setting $k=n/2$ in the negative terms and $k=2$ in the $\frac 1 k$ term. This results in
$$q_{k,n} \geq 1- \frac{ \alpha n }{n - \alpha n/2 -1} - \frac{ n }{2(n - \alpha n/2 -1)} = 1 - \frac{ \alpha + \frac 12} { 1 - \frac \alpha 2 - 1/n}.$$
As $n \geq 3$ we arrive at
$p_* \geq 1 - \dfrac{\alpha + \frac 12 }{ \frac 23 - \frac \alpha 2 }.$
If we evaluate at $\alpha = 1/10$, then $p_* \geq 1/37$, which completes the proof.
\end{proof}
\subsection{Proof of \thref{thm:main}}
With \thref{lem:couplings} and \thref{prop:p*} we can prove our main theorem.
\begin{proof}[Proof of \thref{thm:main}]
Since the number of frogs can at most double at each step we have $T_n \geq \log_2(n)$. The lower bound immediately follows. As for the asymptotic upper bound, let $\tau_{n/2} = \inf\{ t \colon N_t \geq n/2\}$ be the time to wake at least $n/2$ frogs, and let $C_{n/2}$ be the time for $n/2$ walkers to visit every vertex of $K_n$ (taken to be the maximum such time over all possible starting configurations of walkers). \thref{lem:couplings} \ref{breakdown} describes a coupling where we ignore the benefit of waking more frogs after time $\tau_{n/2}$ to conclude
\begin{align}T_n \mathbf{P}receq \tau_{n/2} + C_{n/2} .\label{eq:Tn}
\end{align}
Here `$\mathbf{P}receq$' denotes stochastic domination, see Section \ref{sec:dominance} for the definition.
The couplings in \thref{lem:couplings} \ref{loop} and \ref{key} along with \thref{prop:p*} imply that there exists $\alpha, p_*>0$ such that
\begin{align*}
\tau_{n/2} \mathbf{P}receq \sum_{i=1}^{ \left \lfloor \fracrac{ \log( n/2) }{ \log(1+ \alpha)} \right \rfloor } \Geo(p_*)
\quad \text{ and } \quad
C_{n/2} &\mathbf{P}receq \fracrac{2}{n} \sum_{i=1}^{n} \Geo\left( \fracrac{ n-i} {n} \right).
\end{align*}
Using the fact that the expectation of a Geometric($p$), random variable is $1/p$ we can take the expectation of both sides of \eqref{eq:Tn} to obtain
$$\E T_{n} \leq (1/p_*) \left \lfloor \fracrac{ \log( n/2) }{ \log(1+ \alpha)} \right \rfloor + \fracrac{2} n \sum_{i=1}^{n-1} \fracrac{n}{n-i} = O(\log n) .$$
Note the first summand above is $O(\log n)$ because $p_*$ and $\alpha$ are positive. The second summand is $O(\log n)$ by canceling the factors of $1/n$ and $n$ then comparing to the harmonic numbers $\sum_{i=1}^n \fracrac 1 i \approx \log n$.
\end{proof}
\section{Exact distribution of $T_n$} \label{sec:formula}
Let $\sigma_k = \sigma_k(n)$ be the time to wake up all $n$ frogs on the complete graph given that there are $k$ frogs currently awake. So, $\sigma_1 = T_n$.
\begin{proposition} \thlabel{prop:pformula}
Let $$p_{j,k} = \mathbf P[ \text{$k$ awake frogs visit $j$ sleeping frogs on the next step}].$$ It holds that
$$p_{j,k} = \fracrac{1}{(n -1)^k} \sum_{\ell = j}^k {k \choose \ell} {n - k \choose j} S_{j,k}^{\ell} (k - 1)^{k - \ell},$$
where $S_{j,k}^\ell$ the number of ways to distribute $\ell$ balls into $k$ boxes so that no box is empty. This is given by the formula
$$
S_{j,k}^{\ell} =j! \cdot S(k,l) = \sum_{k = 1}^{j } {j \choose j - k} (-1)^{j - k} k^{\ell},
$$
where $S(k,\ell)$ is Stirling's number of the second kind.
\end{proposition}
\begin{proof}
Observe that to wake up $j$ more frogs, we can use between $j$ and all $k$ frogs, and distribute them onto $j$ unvisited vertices. So let $\ell$ range between $j$ and $k$. There are ${k \choose \ell}$ ways to choose $\ell$ of the already awoken frogs to visit the $j$ new vertices. We can choose $j$ new vertices in ${n - k \choose j}$ ways. Once we have chosen the $\ell$ frogs and $j$ new vertices, we can distribute them in (by definition) $S_{j,k}^{\ell}$ ways. Finally the remaining $k - \ell$ frogs can go to any of the $k -1$ already visited vertices they are adjacent to, so they can move in $(k - 1)^{k - \ell}$ ways. Also, since there are $k$ frogs and we are thinking of them as distinguishable, each one of these events happens with probability $1/(n - 1)^k$. Thus we get, by summing over all $\ell$, that
$$
p_{j,k }=\fracrac{1}{(n -1)^k} \sum_{\ell = j}^k {k \choose \ell} {n - k \choose j} S_{j,k}^{\ell} (k - 1)^{k - \ell}.
$$
We claim that for fixed $\ell$:
$$
S_{j,k}^{\ell} =\sum_{k = 1}^{j } {j \choose j - k} (-1)^{j - k} k^{\ell}
$$
To see this we proceed by inclusion exclusion principle. Observe $j^{\ell}$ counts the number of ways to distribute $\ell$ distinguishable balls to $j$ distinguishable boxes with some boxes left empty possibly. So to count the number of ways with no boxes left empty, we should subtract the number with at least one empty, which is ${j \choose j - 1} (j - 1)^{\ell}$ since we have $ j -1$ boxes that we will possibly place balls in, and have $j - 1$ choices for each of the balls. But now all the ways with exactly two boxes left empty have been added once and subtracted twice (since we counted them ${2 \choose 1}$ times in the subtraction), so we should add ${j \choose j -2}(j - 2)^{\ell}$ to count the number of ways with at least two left empty. Keep going in this fashion to get
$$
S_{j,k}^{\ell} = \sum_{ k =1}^{j} {j \choose k} (-1)^{j - k} (j - k)^{\ell} = \sum_{ k =1}^j {j \choose j - k} (-1)^{j- k} (j - k)^{\ell}.
$$
\end{proof}
An explicit formula for time to wake all $n$ frogs on the complete graph with $n$ vertices is defined recursively as follows: \\*
\begin{proposition} \thlabel{prop:formula}
Let $p_{j,k}$ be as in \thref{prop:pformula} and let $p'_{j,k} = \fracrac {1- p_{0,k} }{p_{0,k}}$. The random variables $\sigma_k$ satisfy the following recursive distributional relationship
\begin{align*}
\sigma_1 &\overset{d}= 1 + \sigma_2,\\
\sigma_k &\overset{d}=
\begin{cases}
\Geo(1-p_{0,k})+\sum \limits_{j=k+1}^{2k} p'_{j,k} \sigma_{k+j}, & 2 \leq k\leq \fracrac{n}{2} \\
\Geo(1-p_{0,k})+\sum \limits_{j=k+1}^{n} p'_{j,k} \sigma_{k+j}, & \fracrac{n}{2} < k \leq n-1
\end{cases}.
\end{align*}
Recall that $T_n = \sigma_1$.
\end{proposition}
\begin{proof}
The expression for $\sigma_1$ is the observation that after one step there will always be two frogs awake. When $k \geq 2$ frogs are awake the time to wake more frogs is a geometric random variable with mean $1- p_{0,k}$. Conditioned that the $k$ frogs wake another frog, we obtain $j$ more awake frogs with probability $p'_{j,k}$. In this situation we now must wait $\sigma_{k+j}$ steps.
\end{proof}
\section{Appendix}
\subsection*{Phases for argument in \cite{original}}
\begin{enumerate}[label = \roman*.]
\item The time inform $N$ residents for some fixed constant $N$ (not growing with $n$).
\item The time to go from $N$ to $\zeta n$ informed residents with $0<\zeta<1$ a fixed constant.
\item The time to go from $\zeta n$ to $(1- \epsilon)n$ with $\epsilon >0$ a small fixed constant.
\item The time to go from $(1-\epsilon)n$ to $n-R$ informed residents where $R$ is a large fixed constant.
\item The time to go from $n-R$ to $n$ informed residents.
\end{enumerate}
\subsection*{Deterministic equation in \cite{second}}
Letting $N(t)$ be the number of informed residents at time $t$:
$$N(t+1) =n- (n-N(t))\exp(-N(t)/n ).$$
\subsection*{Phases for argument in \cite{third}}
\begin{enumerate}[label = \roman*.]
\item The time inform $\sqrt n$ residents.
\item The time to go from $\sqrt n $ to $n/2$ informed residents.
\item The time to go from $n/2$ to $n$ informed residents.
\end{enumerate}
\end{document} |
\begin{document}
\begin{abstract}
We show that if $x$ is a strongly extreme point of a bounded closed convex subset of a Banach space and the identity has a geometrically and topologically good enough local approximation at $x$, then $x$ is already a denting point. It turns out that such an approximation of the identity exists at any strongly extreme point of the unit ball of a Banach space with the unconditional compact approximation property. We also prove that every Banach space with a Schauder basis can be equivalently renormed to satisfy the sufficient conditions mentioned. In contrast to the above results we also construct a non-symmetric norm on $c_0$ for which all points on the unit sphere are strongly extreme, but none of these points are denting.
\end{abstract}
\title{Strongly extreme points and approximation properties}
\section{Introduction}\label{sec:intro}
Let $X$ be a (real) Banach space and denote by $B_X$ its unit ball, $S_X$ its
unit sphere, and $X^*$ its topological dual. Let $A$ be a non-empty
set in $X$. By a slice of $A$ we mean a subset of $A$ of the form
\begin{align*}
S(A, x^*,\varepsilon):=\{x \in A: x^*(x) > M - \varepsilon\}
\end{align*}
where $\varepsilon >0$, $x^* \in X^*$ with $x^* \not=0$, and $M = \sup_{x
\in A} x^*(x)$. We will simply write $S(x^*,\varepsilon)$ for a slice of a
set when it is clear from the setting what set we are considering slices of.
\begin{defn} \label{defn:corner-points} Let $B$ be a non-empty
bounded closed convex set in a Banach space $X$ and
let $x \in B$. Then $x$
\begin{enumerate}
\item [a)] is an extreme point of $B$ if for any $y, z$ in $B$ we
have \[x = \frac{y+z}{2} \Rightarrow y = z = x.\]
\item [b)] is a strongly extreme point of $B$ if for any
sequences $(y_n)_{n=1}^\infty, (z_n)_{n=1}^\infty$ in $B$ we have \[
\lim_n \|x - \frac{y_n + z_n}{2}\| = 0 \Rightarrow \lim_n \|y_n -
z_n\| = 0.\] When B is the unit ball, the above condition can be replaced by
\[\lim_n \|x \pm x_n \| = 1 \Rightarrow \lim_n \| x_n \| = 0.\] In
this case we say that the norm is midpoint locally uniformly rotund (MLUR) at $x$.
\item [c)] is a point of continuity for the map $\Phi:B\to X$ if $\Phi$ is weak to norm continuous at $x$. When $\Phi$ is the identity mapping we just say that $x$ is a point of continuity (PC).
\item [d)] is a denting point of $B$ if for every $\varepsilon > 0$ and
$\delta >0$ there exists a slice $S(x^*,\delta)$ of $B$ with diameter less than $\varepsilon.$
\item [e)] is a locally uniformly rotund (LUR) point of $B_X$ if for any sequence
$(x_n)_{n=1}^\infty$ we
have \[\lim_n \|x + x_n\| = 2\lim_n \|x_n\|=2\|x\|=2 \Rightarrow \lim_n \|x - x_n\| = 0.\]
\end{enumerate}
\end{defn}
It is well known that LUR points are denting points and that denting points
are strongly extreme points \cite{MR661446}. Trivially strongly
extreme points are extreme points.
The importance of denting points became clear in the sixties when the Radon-Nikod\'{y}m Property (RNP) got its geometric description. In particular, it became clear that extreme points in many cases are already denting as every bounded closed convex set in a space with the
RNP has at least one denting point. The ``extra'' an extreme point needs to become a denting points is precisely described in the following
\begin{thm}\cite{LLT}\label{thm:LLT} Let $x$ be an extreme point of continuity of a bounded closed convex set $C$ in $X$. Then $x$ is a denting point of $C$.
\end{thm}
It is well known that all points of the unit sphere of $\ell_1$ are points of continuity for the unit ball $B_{\ell_1}$. So from Theorem \loglike{Re}f{thm:LLT} we get that every extreme point of the unit ball of any subspace of $\ell_1$ automatically gets the ``extra'' to become denting.
However, despite the theoretical elegance of Theorem~\loglike{Re}f{thm:LLT}, it is not always easy to to check whether the identity mapping is weak to norm continuous at a certain point of a bounded closed convex set. For this reason it is natural to look for geometrical conditions which ensure weak to norm continuity of the identity operator at $x$ when we approximate it strongly by maps that are weak to norm continuous at $x$.
One such idea could be to assume that $x$ is strongly extreme (not just extreme as in Theorem~\loglike{Re}f{thm:LLT}) and that the identity map is approximated strongly by finite rank operators. But this is not enough to give the extreme point the ``extra'' needed to be denting: Consider $x=x(t)\equiv 1\in B_{C[0,1]}$. Then $x$ is strongly extreme in $B_{C[0,1]}$, but the identity map $I:B_{C[0,1]}\to B_{C[0,1]}$ is not weak to norm continuous there (see the next paragraph), and $\lim_n\|P_n x-x\|= 0$, where $(P_n)$ are the projections corresponding to the Schauder basis in $C[0,1]$. Clearly $P_n$ is weak to norm continuous at any point of $B_{C[0,1]}$ (as any compact operator is).
Actually $C[0,1]$ belongs to the class LD2 of Banach spaces where all slices of the unit ball have diameter 2. Naturally, in such spaces no point of the unit sphere can be a PC point. See e.g. the references in \cite{MR3499106} for more information about the class LD2.
Assuming $x$ is strongly extreme, we need to make stronger assumptions of the approximating sequence of the identity. One such condition which we impose is related to the behaviour of the approximating mappings close to the point $x$. In particular we obtain as a corollary that in
Banach spaces with the unconditional compact approximation property
(UKAP) (see Definition \loglike{Re}f{def:ucap}), every strongly extreme point in the unit ball is PC and therefore denting. In particular, we have that this conclusion holds for Banach spaces with an unconditional basis with unconditionally basis constant 1. Further we show that every Banach space with a Schauder basis can be renormed to satisfy the conditions of Theorem \loglike{Re}f{thm:se=dent}.
Nevertheless we construct, in Section~\loglike{Re}f{sec:cnullgeo}, a closed convex body in $c_0$ where all boundary points are strongly extreme, but none of them is denting. This body is not symmetric; we refer to its gauge as a non-symmetric equivalent norm on $c_0$. Thus, we construct a non-symmetric equivalent norm on $c_0$ for which all points
on the unit sphere are strongly extreme points, but none of these
points is denting. In fact every slice of the unit ball of $c_0$ with
this non-symmetric norm has diameter at least $1/\sqrt{2}.$
In Section \loglike{Re}f{sec:eqnorms} we investigate when equivalent norms conserve strongly extreme and denting points of the corresponding unit balls.
The notation and conventions we use are standard and follow
\cite{MR1863688}. When considered necessary, notation and concepts are
explained as the text proceeds.
\section{Weak to norm continuity of the identity map}\label{sec:se=dent}
Our most general result on how to force a strongly extreme point $x$ to be denting in terms of approximating the identity map $I:X\to X$ at $x$ is the following
\begin{thm}\label{thm:se=dent}
Let $x$ be a strongly extreme point of a non-empty bounded closed convex set $C$ in a Banach space $X$. Let $x$ be a point of continuity for a sequence $\Phi_n:C\to X, n=1,2,\ldots $ of maps such that
\begin{align}
\label{eq:36}
\lim_{n}\|\Phi_n x - x\| = 0
\end{align}
and
\begin{align}
\label{eq:26}
\lim_n\lim_{\varepsilon \to 0^+} f_n(\varepsilon) = 0,
\end{align}
where
\begin{align*}
f_n(\varepsilon) = \sup\{\mbox{dist}((1+\lambda)\Phi_n y -\lambda y, C): y \in C,
\|\Phi_nx-\Phi_ny\| \le \varepsilon\},
\end{align*}
for some $\lambda \in (0,1].$
Then $x$ is a denting point of $C$.
\end{thm}
The proof follows from Theorem~\loglike{Re}f{thm:LLT} and the next proposition which is an interplay between weak and norm topology. With $B(x,\rho)$ we denote the ball with center at $x$ and radius $\rho$.
\begin{prop}\label{prop:PC} Let $x$ be a strongly extreme point of a convex set $C$ of a normed space $X$ and let $0<\lambda\leq 1$. Assume that for every $\eta>0$ there exist a weak neighbourhood $W$ of $x$ and a map $\Phi:W\cap C\to X$ such that
\begin{align}
\label{eq:sk3}
\Phi(W\cap C)\subset B(x,\eta)
\end{align}
and
\begin{align}
\label{eq:sk4}
\sup_{w\in W\cap C}\mbox{dist}((1+\lambda)\Phi w -\lambda w,C)<\eta.
\end{align}
Then $x$ is PC.
\end{prop}
\begin{proof}
Since $x$ is strongly extreme point, for every $\varepsilon>0$ we can find $\delta>0$ such that
\begin{align}
\label{eq:sk5}
\|x-\frac{u+v}{2}\|<\delta, \quad u,v\in C \Rightarrow \|u-v\|<\lambda\varepsilon.
\end{align}
Set $\eta=\min\{\delta,\lambda\varepsilon\}/2$. There is a weak neighbourhood $W$ of $x$ and a map $\Phi$ satisfying (\loglike{Re}f{eq:sk3}) and (\loglike{Re}f{eq:sk4}). Set $\Psi=I-\Phi$ and pick an arbitrary $w\in W\cap C$. Put $y^+ = (1 - \lambda)x + \lambda w.$ Since $x, w \in C$
we get $y^+ \in C$ by convexity. Since
\begin{align*}
\Phi w + \lambda \Psi w - y^+ = (1 - \lambda)(\Phi w -x),
\end{align*}
we have from (\loglike{Re}f{eq:sk3})
\begin{align}
\label{eq:sk6}
\|\Phi w + \lambda \Psi w - y^+\| \le (1 - \lambda) \eta < \eta.
\end{align}
Having in mind (\loglike{Re}f{eq:sk4}) we can find $y^-\in C$ such that
\begin{align}
\label{eq:sk7}
\|(\Phi w-\lambda \Psi w)-y^-\|<\eta.
\end{align}
This and (\loglike{Re}f{eq:sk6}) imply
\begin{align*}
\|x - \frac{y^+ + y^-}{2}\| & \le \|x - \Phi w\| +
\frac{1}{2}\|(\Phi w + \lambda
\Psi w - y^+)+(\Phi w
-\lambda \Psi w -
y^-)\|\\ &< \|x -
\Phi w\| +
\eta \leq 2\eta.\\
\end{align*}
From (\loglike{Re}f{eq:sk5}) we get
\[\|y^+-y^-\|<\lambda\varepsilon.\]
On the other hand, using again (\loglike{Re}f{eq:sk6}) and (\loglike{Re}f{eq:sk7}), we get
\begin{align*}
\|y^+ - y^-\| & = \|y^+ - (\Phi w + \lambda \Psi w) -
y^- + (\Phi w - \lambda \Psi w) +
2\lambda \Psi w\| \\ & > 2\lambda \|\Psi w\| - 2\eta.
\end{align*}
Hence
\[2\lambda \|\Psi w\|<\|y^+-y^-\|+2\eta<\lambda\varepsilon+2\lambda\varepsilon=3\lambda\varepsilon.\]
This and (\loglike{Re}f{eq:sk3}) imply
\[\|w-x\|\leq\|\Phi w-x\|+\|\Psi w\|<2\varepsilon.\]
Since $w$ is an arbitrary element of $W\cap C$ we get that $W\cap C\subset B(x,2\varepsilon)$.
\end{proof}
\begin{rem}\label{rem:afterPC} If $x$ is PC for $C$ we get that $x$ satisfies the hypotheses of Proposition~\loglike{Re}f{prop:PC} just taking $\Phi=I, \lambda\in (0,1]$.
\end{rem}
\begin{proof}[Proof of Theorem~\loglike{Re}f{thm:se=dent}]
Let $\{\varepsilon_n\}$ be a sequence of positive numbers tending to 0. Since $\Phi_n:C\to X, n=1,2,\ldots$ is weak to norm continuous at $x$ there is weak neighbourhood $V_n$ of $x$ such that
\[\Phi_n(V_n\cap C)\subset B(x,\varepsilon_n),\:\:n=1,2,\ldots\]
Thus the conditions of Theorem~\loglike{Re}f{thm:se=dent} imply that for every $\eta>0$ we can find $n=n(\eta)$ such that (\loglike{Re}f{eq:sk3}) and (\loglike{Re}f{eq:sk4}) hold for $W=V_n$
and $\Phi=\Phi_n$.
\end{proof}
Recall that every linear compact operator is weak to norm continuous on bounded sets. This together with Theorem~\loglike{Re}f{thm:se=dent} gives
\begin{cor}\label{rem:se=dent}
Let $(X, \|\cdot\|)$ be a Banach space and let $\vertiii
\cdot$ be an equivalent (not necessarily symmetric)
norm on $X$ with corresponding unit ball $C.$ Let $x$ be a strongly extreme point of $C$. Let $T_n:X\to X$ be linear compact operators such that
\begin{align}
\label{eq:29a}
\lim_n \|T_nx-x\|=0,
\end{align}
\begin{align}
\label{eq:29}
\lim_n\lim_{\varepsilon\to 0^+} \sup\{\vertiii{(1+\lambda)T_ny - \lambda y}: \vertiii{y}
\le 1, \|T_n(x - y)\| \le \varepsilon\} = 1,
\end{align}
for some $\lambda\in (0,1]$. In particular the above is satisfied if
\begin{align}
\label{eq:29b}
\lim_n \vertiii{(1+\lambda)T_n-\lambda I}=1.
\end{align}
Then $x$ is a denting point of $C$.
\end{cor}
\begin{proof}
It is enough to prove that $(\loglike{Re}f{eq:29})$ implies $(\loglike{Re}f{eq:26})$.
Indeed, since there exists $k >0$ such that
$\|\cdot \| \le k\vertiii{\cdot},$ then for every $u \in X \setminus
C$ we
have
\begin{align*}
\mbox{dist}(u, C) &= \inf \{\|u-v\|: v \in C\} \le k
\inf\{\vertiii{u-v}: v \in C\}\\ & \le k\vertiii{u -
\frac{u}{\vertiii{u}} } =
k(\vertiii{u} - 1).
\end{align*}
\end{proof}
\begin{rem}\label{rem:lim-se=dent}
The functions $f_n$ defined in Theorem~\loglike{Re}f{thm:se=dent} can be discontinuous at
$0$. Indeed, $X$ be a Banach space, $e \in B_X$, and
$e^* \in S_{X^*}$ be such that $e^*(e) = \|e\| = \|e^*\| =1.$ Define a
(norm one) projection $P$ on $X$ by $Px = e^*(x)e$ and put
\begin{align*}
f(\varepsilon) = \sup\{ \|Py - Ry\|: \|y\| \le 1, \|P(e - y)\| \le \varepsilon\},
\end{align*}
where $R=I-P$.
Now, if the norm $\|\cdot\|$ on $X$ is either strictly convex or Gateaux
differentiable at $e^*,$ then $f$ is discontinuous at
$0$. Indeed, let $\varepsilon=0, \|y\| \le 1$, and $P(e - y) = 0$. We get $e^*(e - y)e
= 0$. Hence $e^*(y) = e^*(e) = 1$.
By the strict convexity of the norm or the Gateaux differentiability of the norm
at $e^*,$ we have $y =
e$. This implies $Ry = Re = 0,$ so $f(0) = 1$. In order to prove
that $f$ is discontinuous at $0$ we simply apply Corollary~\loglike{Re}f{rem:se=dent} with
$T_n = P$. Since $e$ is strongly extreme, but not denting, we get
$\lim_{\varepsilon \to 0^+} f(\varepsilon) > 1.$
\end{rem}
Remark \loglike{Re}f{rem:lim-se=dent} shows that one cannot replace the limit
condition
(\loglike{Re}f{eq:29}) of Corollary~\loglike{Re}f{rem:se=dent}, by
\[\lim_n\sup\{\vertiii{(1+\lambda)T_ny-\lambda y}:\vertiii{y}\leq 1, T_nx=T_ny\}=1. \]
The conditions in Corollary~\loglike{Re}f{rem:se=dent} are essential. Let us
illustrate this by examples.
\begin{example}\label{ex:app-ess}
Consider the space $c$ of
convergent sequences endowed with its natural norm. Let $e = (1, 1,
\ldots) \in S_c$ and $P_n$ the projection
on $c$ which projects vectors onto their $n$ first
coordinates. Clearly $e$ is a strongly extreme point of $B_c$
which is not denting. Moreover, it is evident that the condition
$\lim_n\|P_ne - e\| = 0$ fails and that the condition (\loglike{Re}f{eq:29})
(moreover (\loglike{Re}f{eq:29a}))
holds for $\lambda = 1$ (and thus for all $ \lambda \in (0,1]).$
It follows that the approximation condition in Corollary~\loglike{Re}f{rem:se=dent} is essential.
\end{example}
\begin{example}
Consider again $c$ endowed with its natural norm. Let $e \in c$ be is as in Example
\loglike{Re}f{ex:app-ess}. Define a projection $P$ on $c$ by $Px =
\lim_n x(n)e$ and put $P_n = P$ for all $n$. By construction $P_ne =
e.$ For $z = (0, 1, 1, \ldots)$ we have $P_n z = P z = e.$
Now, for any $\lambda \in (0,1]$ we have
\begin{align*}
\|(1+\lambda)P_nz - \lambda z\| &= \|(1 + \lambda)e - \lambda z\|\\
& = 1 + \lambda.
\end{align*}
Thus
\begin{align*}
\lim_n\lim_{\varepsilon \to 0^+}\sup\{\|(1+\lambda)P_ny - \lambda y\|: y \in
B_X, \|P_n(e - y)\| < \varepsilon\|\} \ge 1+ \lambda.
\end{align*}
It follows that the
condition (\loglike{Re}f{eq:26}) in Theorem \loglike{Re}f{thm:se=dent} is essential.
\end{example}
We now present our results in terms of an approximation property introduced and studied by Godefroy, Kalton and Saphar.
\begin{defn}\cite{GKS}\label{def:ucap} A Banach space $X$ is said to have the
\emph{unconditional compact approximation property (UKAP)} if there
exists a sequence $(T_n)$ of linear compact operators on $X$ such that $\lim_n \|T_nx - x\| = 0$ for every $x \in X$ and $\lim_n\|I - 2T_n\| = 1.$
\end{defn}
Clearly Banach spaces $X$ with the UKAP satisfy condition (\loglike{Re}f{eq:29b}) for $\lambda = 1$. Clearly also Banach spaces with an unconditional basis with basis
constant $1$ have the UKAP (simply put $T_n = P_n$ the projection onto
the $n$ first vectors of the basis). Thus we immediately have the
following corollary.
\begin{cor}
If $X$ has the UKAP, in particular if $X$ has an unconditional basis
with unconditional basis constant $1,$ then all strongly extreme points in $B_X$ are
denting points.
\end{cor}
Let us mention that the global condition (\loglike{Re}f{eq:29b}) is much stronger than the local condition
(\loglike{Re}f{eq:29}), even in the case when it holds for all $x$ in $S_X$. This
will be clear from the discussion below and in particular from Example
\loglike{Re}f{ex:gc=/=lc} which shows that the condition (\loglike{Re}f{eq:29b}) is
strictly stronger than (\loglike{Re}f{eq:29}). For that example we will use the
following result.
\begin{prop}\label{prop:lur-cond9}
Let $X$ be a Banach space and $x$ a locally uniformly rotund
(LUR) point in $S_X$. Let $(T_n)$ be a sequence of linear bounded operators on $X$, with $\lim_n \|T_n\| =
1,$ and which satisfies condition (\loglike{Re}f{eq:29a}) in
Corollary~\loglike{Re}f{rem:se=dent} for $x \in S_X$. Then condition (\loglike{Re}f{eq:29}) holds for $\lambda =1$ (and thus for all $\lambda
\in (0, 1]$).
\end{prop}
\begin{proof}
Pick an arbitrary sequence $(\varepsilon_n)$ with $\varepsilon_n > 0$ and $\lim_n
\varepsilon_n = 0.$ First we show that
\begin{align}
\label{eq:31}
\lim_n \mbox{diam} D_n = 0,
\end{align}
where $D_n = \{y \in B_X: \|T_n(x-y)\| < \varepsilon_n\}.$ To this end note
that it suffices to show that
\begin{align}
\label{eq:32}
y_n \in B_X, \hskip 2mm \lim_n \|T_n(x-y_n)\| = 0 \Rightarrow
\lim_n \|x - y_n\| = 0.
\end{align}
Indeed,
\begin{align*}
\|T_n\|\,\|x + y_n\|& \ge \|T_n(x + y_n)\|\\&= \|2T_nx + T_n(y_n-x)\|\\&\ge
2\|T_nx\| - \|T_n(y_n - x)\|.
\end{align*}
Hence $\liminf_n \|x + y_n\| \ge 2.$ Since $\|y_n\| \le 1$ we get
$\lim_n\|x + y_n\| = 2.$ Since $x$ is a LUR point, we get that
$(\loglike{Re}f{eq:32})$ holds, and thus (\loglike{Re}f{eq:31}) holds. In order to
prove (\loglike{Re}f{eq:29}) for $\lambda = 1,$ it is enough to show that
\begin{align*}
\lim_n d_n =1,
\end{align*}
where $d_n = \sup\{\|T_nx - R_ny\|: y \in D_n\}, R_n=I-T_n.$ Since $x\in D_n$, we have $d_n\geq\|T_nx-R_nx\|$. So we get from (\loglike{Re}f{eq:29a})
that $\liminf d_n \ge 1.$ Now, pick arbitrary $y \in D_n.$ Then we have
\begin{align*}
\|T_nx - R_ny\| & \le \|T_nx\| + \|R_ny\|\\
& \le \|T_nx\| + \|R_nx\| + \|R_n(y-x)\|\\
& \le \|T_nx\| + \|R_nx\| + \|R_n\|\|(y-x)\|\\
& \le \|T_nx\| + \|R_nx\| + (\|T_n\| + 1)\mbox{diam} D_n.
\end{align*}
Hence $\limsup d_n \le 1.$
\end{proof}
\begin{prop}\label{prop:lur}
Let $(T_n)$ be a bounded sequence of linear compact operators on $X$, $R_n=I-T_n$, and $(f_n) \subset S_{X^*}$
a separating family for $X$. Then the norm
\begin{align*}
\vertiii{u} = \bigg( \sum_{n=1}^\infty 2^{-n}(\|R_nu\|^2 + f^2_n(u) )\bigg)^{\frac{1}{2}}
\end{align*}
is LUR at $x \in X$ provided $\lim_n\|R_nx\| = 0.$ Moreover, if
the operators $(T_n)$ commute and $\lim_n\|T_n\| = 1,$ then
$\lim_n\vertiii{T_n} = 1.$
\end{prop}
\begin{proof}
Pick a sequence $(x_k) \subset X$ with $\lim_k\vertiii{x_k +
x}=\vertiii{x} = \vertiii{x_k}.$ By convex arguments
(\cite[Fact~2.3 p.~45]{MR1211634}) we have
\begin{align}
\label{eq:33}
\lim_k\|R_n x_k\| &= \|R_n x\|, \hskip 2mm n = 1, 2, \ldots
\end{align}
\begin{align}
\label{eq:34}
\lim_k f_n(x_k) &= f_n(x), \hskip 2mm n=1, 2, \ldots.
\end{align}
First we show that $(x_k)$ is norm compact. Given $\varepsilon > 0$ we can
find $n$ with $\|R_nx\| < \varepsilon.$ Using (\loglike{Re}f{eq:33}) we can find
$k_\varepsilon$ such that $\|R_nx_k\| < \varepsilon$ for $k < k_\varepsilon.$ The set $K
= \{x_1, x_2, \ldots, x_{k_\varepsilon}\} \cup \|x\|T_n(B_X)$ is norm
compact. We show that $K$ is an $\varepsilon$-net for $(x_k).$ Indeed, pick
$x_k, k > k_\varepsilon.$ Then $\|x_k - T_nx_k\| = \|R_nx_k\| < \varepsilon$ and
$T_nx_k \in \|x\| T_n(B_X).$ So $(x_k)$ is norm compact. Since
$(f_n)$ is total, we get from (\loglike{Re}f{eq:34}) that $\lim_k\|x_k -x\| =
0.$ Thus the norm $\vertiii{\cdot}$ is LUR at the point $x \in X$.
Now, let us prove the moreover part. As $(T_n)$
commute, we have
\begin{align*}
\vertiii{T_mu}^2 &= \sum_{n=1}^\infty 2^{-n} (\|R_nT_mu\|^2 +
f_n^2(T_mu))\\
& = \sum_{n=1}^\infty 2^{-n} (\|T_mR_nu\|^2 + (T^*_mf_n(u))^2) \\
& \le \sum_{n=1}^\infty 2^{-n} (\|T_m\|^2\|R_nu\|^2 +
\|T^*_m\|^2f_n^2(u)) \\
& = \|T_m\|^2 \vertiii{u}^2.
\end{align*}
Hence $\vertiii{T_m}^2 \le \|T_m\|^2$ for all $m = 1, 2, \ldots,$ so
$\limsup_m \vertiii{T_m} \le \limsup_m\|T_m\|.$ Since $\lim_m\|T_mx
- x\| = 0,$ we get $\liminf_m \vertiii{T_m} \ge 1,$ and so
$\lim_m\vertiii{T_m} =1$ provided $\lim_m \|T_m\| = 1.$
\end{proof}
It is now easy to give the announced example which shows that condition
(\loglike{Re}f{eq:29b}) can fail as condition
(\loglike{Re}f{eq:29}) holds for every $x$ in $S_X$.
\begin{example}\label{ex:gc=/=lc}
Consider $c_0$ endowed with the norm
$\|\cdot\|$ defined by $\|x\| = \sup_{i,j \ge 1} ( x(i) -
x(j) )$ where $x=(x(k)) \in c_0.$
Clearly $\|\cdot\|$ is equivalent to the canonical norm on $c_0$.
Let $P_n$ be the projection onto the $n$ first vectors in the
canonical basis $(e_k)$ of $c_0$ and let $\vertiii
\cdot$ be the norm on $c_0$ given in Proposition
\loglike{Re}f{prop:lur} where $f_n = 0$ for every $n$. Then $(c_0,
\vertiii \cdot)$ fulfils the conditions of
Proposition \loglike{Re}f{prop:lur-cond9} and thus satisfies condition
(\loglike{Re}f{eq:29}) for every $x$ in $S_{c_0}$. Nevertheless we have
$\vertiii {P_k - \lambda R_k} >
1$ for any $\lambda \in (0,1]$, so condition
(\loglike{Re}f{eq:29b}) fails. For the latter, just consider $(P_k - \lambda
R_k)(\sum^{k+1}_{i=1}e_i).$
\end{example}
From the two preceding propositions we also get
\begin{cor}
Let $X$ be a Banach space with a Schauder basis. Then there exists
an equivalent norm $\|\cdot\|$ on $X$ for which the sequence of projections
$P_n$ onto the first $n$ vectors of the basis, satisfy (\loglike{Re}f{eq:29}) for $\lambda = 1$.
\end{cor}
On the other hand we have
\begin{prop}
There exists an equivalent norm $\|\cdot\|$ on $C[0,1]$ such that (\loglike{Re}f{eq:29}) does not hold for any $\lambda > 0$ and any
sequence $(T_n)$ of compact linear operators on $X$when $x \in C[0,1]$ with
$\|x\| = 1$ and $\lim_n \|T_nx -x\|_{\infty} = 0.$
\end{prop}
\begin{proof}
The norm on $C[0,1]$ constructed in \cite[Theorem~2.4]{MR3499106}
is midpoint locally uniformly rotund and has the diameter two
property, i.e. all non-empty relatively weakly open subsets of the
unit ball have diameter $2$. In particular, in this norm all points
on the unit sphere are strongly extreme, but none are denting. Thus
the conclusion follows from Theorem \loglike{Re}f{thm:se=dent}.
\end{proof}
\section{An MLUR non-symmetric norm on $c_0$ without denting points on the unit sphere}\label{sec:cnullgeo}
We will now construct a non-symmetric equivalent norm on $c_0$ for
which all points on the unit sphere are strongly extreme, but none of
these points are denting.
\begin{prop}
There exists in $c_0$ an equivalent (non-symmetric) MLUR norm
$\vertiii{\cdot}$ with no denting points. Moreover, every open slice
of the unit ball corresponding to $\vertiii{\cdot}$ has diameter
$\ge 1/\sqrt{2}.$
\end{prop}
\begin{proof}
With $J$ we denote the set of all finite strictly increasing
sequences $(j_k)_ {k \ge 0}$ of natural numbers. For $x = (x(k))_{k
\ge 1} \in \ell_\infty$ and $j \in J$ we put
\begin{align*}
Q(x,j) &= |x(j_0)| + \sum_{k \ge 1} 2^{-k} x^+(j_k), \hskip 2mm x^+ = \max\{x,0\}\\
q(x) &= (\sum_{k \ge 1} 2^{-k} x^2(k))^{1/2}, \hskip 2mm
\|x\| = \sup\{Q(x,j): j \in J\},\\
\vertiii{x} &= (\|x\|^2 + q^2(x))^{1/2}.
\end{align*}
Clearly
\begin{align}
\label{eq:41}
\|x\|_{\infty} \le \|x\| \le \vertiii{x} \le (\|x\|^2 +
\|x\|_\infty^2)^{1/2} \le \sqrt{2} \|x\| \le 3\|x\|_\infty.
\end{align}
\begin{claim}\label{claim:1}
For every $x \in c_0$ and every $\varepsilon > 0,$ there exists $m = m(x,
\delta) \in \enn$ and $\delta = \delta(x,\varepsilon) > 0$ such that
\begin{align}
\label{eq:42}
\max\{\|x+y\|, \|x-y\|\} \ge \|x\| + \delta
\end{align}
whenever
\begin{align}
\label{eq:43}
y \in \ell_\infty, \|R_my\|_\infty > \varepsilon,
\end{align}
where $P_m$ is the projection onto the first $m$ vectors of the
canonical basis of $c_0$ and $R_m = I - P_m.$
\end{claim}
\begin{proof}
Pick $m$ such that
\begin{align}
\label{eq:44}
\|R_m x\|_ \infty < \varepsilon/8,
\end{align}
and put $\delta = \varepsilon/2^{m+3}.$ Using the definition of
$\|\cdot\|$ we can find $j = (j_i)_{k=0}^p$ such that
\begin{align}
\label{eq:45}
\|x\| - \delta/2 < Q(x,j).
\end{align}
We may assume that $p \ge m.$ Choose $i \ge 1$ such that
\begin{align}
\label{eq:46}
j_{i-1} \le m < j_i
\end{align}
and put $j^1 = (j_k)_{k=0}^{i-1}.$ Using (\loglike{Re}f{eq:46}) we get
\begin{align}
\label{eq:47}
Q(x,j) \le Q(x,j^1) + 2^{-i + 1} \|R_mx\|_\infty.
\end{align}
Pick $y$ satisfying (\loglike{Re}f{eq:43}). There is $r > m$ with $|y(r)|
> \varepsilon.$ We have
\begin{align}
\label{eq:48}
(x(r) + y(r))^+ + (x(r) - y(r))^+ \ge |y(r)| - |x(r)| > \varepsilon - \|R_mx\|_\infty.
\end{align}
Put $j^2 = (j_0, j_1, \ldots, j_{i-1}, r).$ Since $r > m \ge
j_{i-1}$ we have $j^2 \in J.$ So
\begin{align*}
\|x \pm y\| \ge Q(x \pm y, j^2)
= Q(x \pm , j^1) + 2^{-i}(x(r) \pm y(r))^+.
\end{align*}
Since $i \le j_{i-1}$ we get from (\loglike{Re}f{eq:46}), (\loglike{Re}f{eq:47}),
and (\loglike{Re}f{eq:48})
\begin{align*}
(\|x + y\| + \|x - y\|)/2 & \ge Q(x, j^1) + 2^{-i-1}(\varepsilon -
\|R_m\|_\infty)\\
& \ge Q(x,j) + 2^{-i-1}(\varepsilon - 5\|R_m x\|_\infty) \\
& \ge Q(x,j) + 2^{-m -1}(\varepsilon - 5\|R_mx\|_\infty).
\end{align*}
This and (\loglike{Re}f{eq:44}), (\loglike{Re}f{eq:45}) imply (\loglike{Re}f{eq:42}).
\end{proof}
Let $x \in c_0, x_n \in \ell_\infty,$ and $\lim_n\vertiii{x \pm
x_n} = \vertiii{x}.$ By convex arguments as in
\cite[p.~45]{MR1211634} we have
\begin{align}
\label{eq:49}
\lim_n \|x \pm x_n\| = \|x\|,
\end{align}
and
\begin{align}
\label{eq:50}
\lim_n x_n(k) = 0, \hskip 2mm k = 0, 1, \ldots.
\end{align}
We have to show that $\lim_n\|x_n\|_\infty = 0.$ Assume the
contrary. Then we may assume that $\|x_n\|_\infty \ge 2\varepsilon > 0$
for all $n =1, 2, \ldots.$ From (\loglike{Re}f{eq:50}) it follows that there
exists $m$ such that $\|P_m x_n\|_\infty < \varepsilon$ for all $n$ which
are sufficiently big. But, this contradicts (\loglike{Re}f{eq:49}) and Claim
\loglike{Re}f{claim:1}. Hence $\vertiii{\cdot}$ is MLUR.
Now, put $C = \{u \in c_0: \vertiii{u} \le 1\}.$ We will show that
every non-void slice of $C$ has $\|\cdot\|_\infty$ diameter $\ge
1/\sqrt{2}.$ To this end, let $f \in \ell_1, a \in \err, H = \{u \in
c_0: f(u) > a\},$ and $S = H \cap C \not = \emptyset.$
\begin{claim}\label{claim:2}
For every $x \in H$ with $\vertiii{x} < 1,$ there is $y \in S$
with $\vertiii{x-y} \ge \|x\|.$
\end{claim}
\begin{proof}
Choose $\delta > 0$ such that
\begin{align}
\label{eq:51}
(\|x\| + 2\delta)^2 + q^2(x) + \delta < 1, \hskip 2mm f(x) - \delta
> a.
\end{align}
There exists a natural number $m$ such that
\begin{align}
\label{eq:52}
\|R_m x\|_\infty < \delta,
\end{align}
\begin{align}
\label{eq:53}
0 < \|x\| - x(m) < \sqrt{2^m \delta},
\end{align}
and
\begin{align}
\label{eq:54}
f(e_m) < \delta
\end{align}
where $e_m$ is the standard basis vector number $m$ in $c_0.$ Put
$y = x - \|x\|e_m.$ Pick arbitrary $j \in J.$ We will show that
\begin{align}
\label{eq:55}
Q(y,j) \le \|x\| + 2\delta.
\end{align}
If $m < j_0,$ then $y(j_k) = x(j_k)$ for $k = 0, 1, 2, \ldots.$ So
\begin{align*}
Q(y,j) = Q(x,j) \le \|x\|.
\end{align*}
If $m > j_0,$ we get $Q(y,j) \le Q(x,j) \le \|x\|$ since $y^+(m) =
0.$ If $m =j_0$
\begin{align*}
Q(y,j) \le \|x\| - x(m) + \|R_mx\|_\infty.
\end{align*}
This and (\loglike{Re}f{eq:52}) imply (\loglike{Re}f{eq:55}). From (\loglike{Re}f{eq:53}) we
get $q^2(y) < q^2(x) + \delta.$ From this, (\loglike{Re}f{eq:51}), and
(\loglike{Re}f{eq:55}) we have $\vertiii{y} < 1.$ Since $\|x\| < 1$ we get
from (\loglike{Re}f{eq:54}) that $y \in H.$ Thus $y \in S,$ and $\|x
-y\|_\infty = \|x\|.$ Using (\loglike{Re}f{eq:41}) we get $\vertiii{x-y} \ge
\|x - y\|_\infty = \|x\|,$ and the claim follows.
\end{proof}
Since $x \in S$ can be chosen with $\vertiii{\cdot}$-norm
arbitrarily close to $1$ we get from (\loglike{Re}f{eq:41}) that the diameter
of $S \ge 1/\sqrt{2}.$
\end{proof}
\section{Weak continuous perturbation of the norm}\label{sec:eqnorms}
It is natural to expect that weak continuous perturbations of the norm
would preserve points of continuity. We will show that similar
perturbations of the norm preserve strongly extreme points of the
corresponding ball.
\begin{thm}\label{thm:weak-npert}
Let $X$ be a Banach space and let $B$ be the unit ball corresponding
to an equivalent norm $\vertiii{\cdot}$ on $X.$ Assume
that the restriction $\vertiii{\cdot}_{S_X}$ of $\vertiii{\cdot}$ to $S_X$
is continuous at $x \in S_X$
\begin{enumerate}
\item [a)] with respect to the weak topology. Then $x$ is a point of
continuity for $B_X$ provided $x$ is a point of continuity for $B$.
\item[b)] with respect to a topology $\sigma = \sigma(X,Y), Y
\subset X^*$ with the property that for every bounded sequence in
$X$ there exists a $\sigma$-Cauchy subsequence, and let $x$ be an
extreme point of $B_X.$ Then $x$ is a strongly extreme point of
$B_X$ provided it is a strongly extreme point for $B$.
\end{enumerate}
\end{thm}
\begin{proof}
From the assumption $\vertiii{x} = 1.$
a). Pick $\varepsilon > 0.$ Since $x$ is a point of continuity for
$B$, there is a weakly open set $W$ containing $x$ such that
$\mbox{diam}(W \cap B) < \varepsilon.$ Since $\vertiii{\cdot}$ is
uniformly norm continuous, we can find $\delta > 0$ such that
$\mbox{diam} (W \cap (1 + 2\delta)B) < 2\varepsilon.$ Since
$\vertiii{\cdot}_{S_X}$ is weakly continuous at $x$, we can find
a weakly open set $V$ containing $x$ such that $V \cap S_X \subset
(1 + \delta)B.$ Since $\|\cdot\|$ is uniformly norm continuous, we
can find $\eta \in (0,1)$ such that $V \cap (B_X \setminus \eta
B_X) \subset (1 + 2\delta)B.$ Finally there is a weakly open set
$U$ containing $x$ such that $U \cap \eta B_X = \emptyset.$ Hence
$x \in U \cap V \cap W$ and $\mbox{diam}(U \cap V \cap W \cap
B_X) < 2\varepsilon.$
b). Pick an arbitrary sequence $(u_n) \subset X$ with
$\lim_n\|x \pm u_n\| = 1.$ In order to prove that $\lim\|u_n\|
=0,$ it is enough to show that there exists a subsequence of
$(u_n)$ converging to zero. To this end we assume without loss of
generality that
\begin{align*}
\sigma-\lim_{m, n}(u_m - u_n) = 0.
\end{align*}
Taking into account that
\begin{align*}
\|x \pm \frac{u_m - u_n}{2}\| \le \frac{\|x \pm u_m\| + \|x -u_n\|}{2},
\end{align*}
we get
\begin{align*}
\lim_{m,n}\|x \pm \frac{u_m - u_n}{2}\| = 1.
\end{align*}
Put $u_{m,n} = (u_m -u_n)/2,$ $\alpha_{m,n}^{\pm} = \|x \pm u_{m,n}\|
-1.$ Clearly $\lim_{m,n} \alpha^{\pm}_{m,n} = 0$ and $\sigma
-\lim_{m,n}(x \pm u_{m,n})/(1 + \alpha_{m,n}^\pm) = x,$ and $(x \pm
u_{m,n})/(1 + \alpha_{m,n}^\pm) \in S_X.$ Since
$\vertiii{\cdot}_{S_X}$ is $\sigma$-continuous at $x$, we get
\begin{align*}
\lim_{m,n}\vertiii{\frac{x\pm u_{m,n}}{1+\alpha^{\pm}_{m,n}}} = \vertiii{x} = 1.
\end{align*}
This implies $\lim_{m,n}\vertiii{x \pm u_{m,n}} = 1.$ Having in mind that
$x$ is a strongly extreme point of $B,$ we get
$\lim_{m,n}\vertiii{u_{m,n}} = 0.$ Hence $(u_n)$ is a norm Cauchy
sequence. It follows that there is $u \in X$ with $\lim_n\|u_n -
u\| = 0.$ Thus $\|x \pm u\| = 1,$ which implies that $u = 0$ as
$x$ is an extreme point. Now $\lim_n\|u_n\|
= 0$ which finishes the proof.
\end{proof}
\begin{rem}
The requirement that $x$ is an extreme point of $B_X$ is essential
for part b). Indeed, in finite dimensional Banach spaces, all
norms are weak($=$ norm) continuous. Clearly a point $x$ can be
extreme for the ball $B$ and not extreme for $B_X$.
\end{rem}
We end with a result that follows directly by combining Theorems \loglike{Re}f{thm:LLT} and \loglike{Re}f{thm:weak-npert}.
\begin{cor} Let $\|\cdot\|_W$ be a weakly continuous semi norm on bounded sets in a Banach space $(X,\|\cdot\|)$ and let $|\cdot|$ be a lattice norm on $\mathbb{R}^2$. Let $\vertiii{x}=|(\|x\|,\|x\|_W)|$. Let $B$ be the unit ball of $(X,\vertiii{\cdot})$. Then the strongly extreme points of $B$ and $B_X$ coincide, and the denting points of $B$ and $B_X$ coincide.
\end{cor}
\end{document} |
\bar{b}egin{enumerate}gin{document}
\bar{b}egin{enumerate}gin{abstract}
We present definitions of homology groups $H_n$, $n\ge 0$, associated to a family of ``amalgamation functors''. We show that if the generalized amalgamation properties hold, then the homology groups are trivial. We compute the group $H_2$ for strong types in stable theories and show that any profinite abelian group can occur as the group $H_2$ in the model-theoretic context.
\end{abstract}
\bar{m}aketitle
The work described in this paper was originally inspired by Hrushovski's discovery \bar{c}ite{Hr} of striking connections between amalgamation properties and definable groupoids in models of a stable first-order theory. Hrushovski showed that if the theory fails 3-uniqueness (the model-theoretic definition of this property is in Section~3), then there must exist a groupoid whose sets of objects and morphisms, as well as the composition of morphisms, are definable in models of a first-order theory. In~\bar{c}ite{GK}, an explicit construction of such a groupoid was given and it was shown in~\bar{c}ite{gkk} that the group of automorphisms of each object of such a groupoid must be abelian profinite. The morphisms in the groupoid construction in~\bar{c}ite{GK} arise as equivalence classes of ``paths'', defined in a model-theoretic way. In some sense, the groupoid construction paralleled that of the construction of a fundamental groupoid in a topological space. Thus it seems natural to ask whether it is possible to define the notion of a homology group in model-theoretic context and, if yes, would the homology group be linked to the group described in~\bar{c}ite{GK,gkk}. We find that the answer is ``yes'' to both questions.
In this paper, we describe a way to define homology groups for a wide class of first-order theories. We start by describing the construction using category-theoretic language; the few model-theoretic references in Section~1 are provided only as motivation for the notions introduced there. Our goal is to separate, as much as possible, general arguments that do not require heavy use of model-theoretic context from the arguments that do. It turns out that, even at this level of generality, it is possible to show that the homology groups have a fairly simple structure (see, for example, Theorem~\ref{Hn_shells} or Corollary~\ref{Hn_pockets}).
We then define, in Section 2, several natural homology groups in the model-theoretic context (e.g., the type-homology groups and set-homology groups) and show that such groups must be isomorphic. The latter is, of course, what one would expect to see; but the argument turned out to be fairly involved. We show that the homology groups of a complete type of the theory are related to its amalgamation properties: if a type $p$ has $k$-amalgamation for every $k \leq n$, then $H_{n-2}(p) = 0$ (Corollary~\ref{trivial_homology}), and when $4$-amalgamation fails, $H_2(p)$ can be nontrivial, even in a stable theory (see the examples in Section~5). In particular, we show that \emph{any} profinite abelian group can be the group $H_2(p)$ for a suitable $p$ in a stable theory.
Section 4 is devoted to the proof of an analogue of Hurewicz's theorem: in a stable theory, the group $H_2(p)$ is isomorphic to a certain automorphism group $\bar{m}athcal{G}amma_2(p)$ which is analogous to a fundamental group. It turns out that $\bar{m}athcal{G}amma_2(p)$ is always abelian, so there is no need to take the abelianization as in the usual Hurewicz theorem. But in Section~7, we construct a different canonical ``fundamental" group for the type $p$ which seems to give more information: this new group need not be abelian, and the group $\bar{m}athcal{G}amma_2(p)$ is in the center of the new group.
Section 6 gives examples of homology groups in unstable theories, where it is still unclear what model-theoretic properties are measured by these groups.
Amalgamation properties have already been much studied by researchers in simple theories (for instance, in \bar{c}ite{KKT}), and recently the first and third authors of this note investigated analogies with homotopy theory rather than homology theory in \bar{c}ite{GK}. In some sense, this paper is a companion to \bar{c}ite{GK}.
For general background on simple theories, the reader is referred to the book \bar{c}ite{wagner}, which explains nonforking, hyperimaginaries, and much more.
\section{Simplicial homology in a category}
In this section, we define simplicial homology groups in a more general category-theoretic setting than our intended applications to model theory. We aim to provide a general framework for our homological computations and separate some of the category-theoretic arguments from the model-theoretic ones. This section uses model theory only as a source of examples.
The homology groups defined in this section are not, in general, homology groups associated with the entire category $\bar{c}C$, but rather are homology groups of a particular class of ``amalgamation functors'' $\bar{c}A$ from a class of certain partially ordered sets viewed as categories into the category $\bar{c}C$. The class $\bar{c}A$ is assumed to satisfy certain basic closure properties (see Definition~\ref{amenable} below).
\subsection{Basic definitions and facts.}
Throughout this section, let $\bar{c}C$ be a category. If $s$ is a set, then we consider the power set ${\bar{m}athcal P}(s)$ of $s$ to be a category with a single inclusion map ${\hbox{\boldmath $\bar \textup{im}ath$}}ota_{u,v} : u \rightarrow v$ between any pair of subsets $u$ and $v$ with $u \subseteq v$. A subset $X \subseteq {\bar{m}athcal P}(s)$ is called \emph{downward-closed} if whenever $u \subseteq v {\hbox{\boldmath $\bar \textup{im}ath$}}n X$, then $u {\hbox{\boldmath $\bar \textup{im}ath$}}n X$. In this case we consider $X$ to be a full subcategory of ${\bar{m}athcal P}(s)$. An example of a downward-closed collection that we will use often below is ${\bar{m}athcal P}^-(s) := {\bar{m}athcal P}(s) \setminus \{s\}$.
We are interested in subfamilies of functors $f:X \rightarrow \bar{c}C$ for downward-closed subsets $X \subseteq {\bar{m}athcal P}(s)$ for various finite subset sets $s$ of the set of natural numbers. Before specifying the desirable closure properties of a collection $\bar{c}A$ of such functors, we need some auxiliary definitions.
\bar{b}egin{enumerate}gin{definition}
(1) Let $X$ be a downward closed subset of ${\bar{m}athcal P}(s)$ and let $t{\hbox{\boldmath $\bar \textup{im}ath$}}n X$. The symbol $X|_t$ denotes the set $\{u {\hbox{\boldmath $\bar \textup{im}ath$}}n {\bar{m}athcal P}(s\setminus t)\bar{m}id t\bar{c}up u{\hbox{\boldmath $\bar \textup{im}ath$}}n X\}\subseteq X$.
(2) For $s$, $t$, and $X$ as above, let $f:X\to \bar{c}C$ be a functor. Then \emph{the localization of $f$ at $t$} is the functor $f|_t:X|_t\to \bar{c}C$ such that
$$f |_t (u) = f(t \bar{c}up u)$$ and whenever $u \subseteq v {\hbox{\boldmath $\bar \textup{im}ath$}}n X|_t$, $$f |_t({\hbox{\boldmath $\bar \textup{im}ath$}}ota_{u,v}) = f({\hbox{\boldmath $\bar \textup{im}ath$}}ota_{u \bar{c}up t, v \bar{c}up t}).$$
(3) Let $X\subset {\bar{m}athcal P}(s)$ and $Y\subset {\bar{m}athcal P}(t)$ be downward closed subsets, where $s$ and $t$ are finite sets of natural numbers. Let $f:X\to \bar{c}C$ and $g:Y\to \bar{c}C$ be functors. We say that $f$ and $g$ are \emph{isomorphic} if there is an order-preserving bijection $\sigma: s \rightarrow t$ such that $Y = \{\sigma(u) : u {\hbox{\boldmath $\bar \textup{im}ath$}}n X\}$ and a family of isomorphisms $\langlengle h_u : f(u) \rightarrow g(\sigma(u)) : u {\hbox{\boldmath $\bar \textup{im}ath$}}n X \textup{ran}gle$ in $\bar{c}C$ such that for any $u \subseteq v {\hbox{\boldmath $\bar \textup{im}ath$}}n X$, the following diagram commutes:
$$
\bar{b}egin{enumerate}gin{CD}
f(u) @>h_u>> g(\sigma(u))\\
@VV{f({\hbox{\boldmath $\bar \textup{im}ath$}}ota_{u,v})}V @VV{g({\hbox{\boldmath $\bar \textup{im}ath$}}ota_{\sigma(u),\sigma(v)})}V\\
f(v) @>h_v>> g(\sigma(v))
\end{CD}
$$
\end{definition}
\bar{b}egin{enumerate}gin{remark}
If $X$ is a downward closed subset of ${\bar{m}athcal P}(s)$ and $t{\hbox{\boldmath $\bar \textup{im}ath$}}n X$, then $X|_t$ is a downward closed subset of ${\bar{m}athcal P}(s\setminus t)$. Moreover $X|_t$ does not depend on the choice of $s$.
\end{remark}
\bar{b}egin{enumerate}gin{definition}
\langlebel{amenable}
Let $\bar{c}A$ be a non-empty collection of functors $f: X \rightarrow \bar{c}C$ for various non-empty downward-closed
subsets $X \subseteq {\bar{m}athcal P}(s)$ for all finite sets $s$ of natural numbers. We say that $\bar{c}A$ is \emph{amenable} if it
satisfies all of the following properties:
\bar{b}egin{enumerate}gin{enumerate}
{\hbox{\boldmath $\bar \textup{im}ath$}}tem (Invariance under isomorphisms) Suppose that $f: X \rightarrow \bar{c}C$ is in $\bar{c}A$ and $g:Y\to \bar{c}C$ is isomorphic to $f$. Then $g{\hbox{\boldmath $\bar \textup{im}ath$}}n \bar{c}A$.
{\hbox{\boldmath $\bar \textup{im}ath$}}tem (Closure under restrictions and unions) If $X \subseteq {\bar{m}athcal P}(s)$ is downward-closed and $f: X \rightarrow \bar{c}C$ is a functor, then $f {\hbox{\boldmath $\bar \textup{im}ath$}}n \bar{c}A$ if and only if for every $u {\hbox{\boldmath $\bar \textup{im}ath$}}n X$, we have that $f \upharpoonright {\bar{m}athcal P}(u) {\hbox{\boldmath $\bar \textup{im}ath$}}n \bar{c}A$.
{\hbox{\boldmath $\bar \textup{im}ath$}}tem (Closure under localizations) Suppose that $f : X \rightarrow \bar{c}C$ is in $\bar{c}A$ for some $X \subseteq {\bar{m}athcal P}(s)$ and $t {\hbox{\boldmath $\bar \textup{im}ath$}}n X$. Then $f|_t:X|_t\to \bar{c}C$ is also in $\bar{c}A$.
{\hbox{\boldmath $\bar \textup{im}ath$}}tem (De-localization) Suppose that $f: X \rightarrow \bar{c}C$ is in $\bar{c}A$ and $t {\hbox{\boldmath $\bar \textup{im}ath$}}n X \subseteq {\bar{m}athcal P}(s)$ is such that $X|_t=X \bar{c}ap {\bar{m}athcal P}(s \setminus t)$. Suppose that the localization $f |_t : X \bar{c}ap {\bar{m}athcal P}(s \setminus t) \rightarrow \bar{c}C$ has an extension $g : Z \rightarrow \bar{c}C$ in $\bar{c}A$ for some $Z \subseteq {\bar{m}athcal P}(s \setminus t)$. Then there is a map $g_0 : Z_0 \rightarrow \bar{c}C$ in $\bar{c}A$ such that $Z_0 = \{u \bar{c}up v : u {\hbox{\boldmath $\bar \textup{im}ath$}}n Z, v \subseteq t\}$, $f \subseteq g_0$, and $g_0 |_t = g$.
\end{enumerate}
\end{definition}
\bar{b}egin{enumerate}gin{remark}
For example, we could take $\bar{c}C$ to be all boundedly (or algebraically) closed subsets of the monster model of a first-order theory, and let $\bar{c}A$ be all functors which are ``independence-preserving'' (in Hrushovski's terminology
\bar{c}ite{Hr}) and such that every face $f(u)$ is the bounded (or algebraic) closure of its vertices; then $\bar{c}A$ is amenable.
Other examples of amenable collections come from further restricting $\bar{c}A$ by requiring, for instance, that all the ``vertices'' $f(\{i\})$ of functors $f {\hbox{\boldmath $\bar \textup{im}ath$}}n \bar{c}A$ be of a certain type, or by restricting the possible types of edges, faces, et cetera. These examples will be explained more precisely in Section 2.
\end{remark}
\bar{b}egin{enumerate}gin{remark}
Note that any functor $f: X \rightarrow \bar{c}C$ has a ``base'' $f(\emptyset)$ which is embedded into each $f(u)$ for $u {\hbox{\boldmath $\bar \textup{im}ath$}}n X$. This base does not play an important role in computing the homology groups, but it does have model-theoretic significance. In particular, it is often convenient to fix this base.
\end{remark}
\bar{b}egin{enumerate}gin{definition}
Let $B {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Ob}(\bar{c}C)$ and suppose $f(\emptyset)=B$. We say that $f$ \emph{is over $B$} and we let $\bar{c}A_B$ denote the set of all functors $f {\hbox{\boldmath $\bar \textup{im}ath$}}n \bar{c}A$ that are over $B$.
\end{definition}
In model-theoretic applications, there will always be an initial object of $\bar{c}A$ which will be the natural choice for $B$ (either the ``empty type'' for type homology, or the ``empty tuple'' for set homology).
\bar{b}egin{enumerate}gin{remark}
\langlebel{amenable2}
It is easy to see that condition~(2) in Definition~\ref{amenable} is equivalent to the conjunction of the following two conditions:
(Closure under restrictions) If $f : X \rightarrow \bar{c}C$ is in $\bar{c}A$ and $Y \subseteq X$ with $Y$ downward-closed, then $f \upharpoonright Y$ is also in $\bar{c}A$.
(Closure under unions) Suppose that $f: X \rightarrow \bar{c}C$ and $g: Y \rightarrow \bar{c}C$ are both in $\bar{c}A$ and that
$f\upharpoonright X\bar{c}ap Y=g\upharpoonright X\bar{c}ap Y$.
Then the union $f \bar{c}up g: X \bar{c}up Y \rightarrow \bar{c}C$ is also in $\bar{c}A$.
For instance, if these two conditions are true and $f: X \rightarrow \bar{c}C$ is a functor from a downward-closed set $X$ such that $f \upharpoonright {\bar{m}athcal P}(u) {\hbox{\boldmath $\bar \textup{im}ath$}}n \bar{c}A$ for every $u {\hbox{\boldmath $\bar \textup{im}ath$}}n X$, then if $u_1, \ldots, u_n$ are maximal sets in $X$, we can use closure under unions $(n-1)$ times to see that $f {\hbox{\boldmath $\bar \textup{im}ath$}}n \bar{c}A$ (since it is the union of the functors $f \upharpoonright {\bar{m}athcal P}(u_i)$).
\end{remark}
From now on, we assume that $\bar{c}A$ is a nonempty amenable collection of functors mapping into the category $\bar{c}C$. As we mentioned in the above remark, every functor in $\bar{c}A$ can be described as the union of functors whose domain is ${\bar{m}athcal P}(s)$ for for finite set $s$. Such functors will play a central role in this paper.
\bar{b}egin{enumerate}gin{definition}
Let $n\ge 0$ be a natural number. A \emph{(regular) $n$-simplex in $\bar{c}C$} is a functor $f : {\bar{m}athcal P}(s) \rightarrow \bar{c}C$ for some set $s \subseteq \omega$ with $|s| = n+1$. The set $s$ is called the \emph{support of $f$}, or $\textup{supp}(f)$.
Let $S_n(\bar{c}A; B)$ denote the collection of all regular $n$-simplices in $\bar{c}A_B$.
Then put $S(\bar{c}A;B):=\bar{b}igcup_{n} S_n(\bar{c}A;B)$
and $S(\bar{c}A):=\bar{b}igcup_{B {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Ob}(\bar{c}C)} S(\bar{c}A;B)$.
Let $C_n (\bar{c}A; B)$ denote the free abelian group generated by $S_n(\bar{c}A; B)$; its elements are called \emph{$n$-chains in $\bar{c}A_B$}, or \emph{$n$-chains over $B$}. Similarly, we define $C (\bar{c}A; B):=\bar{b}igcup_{n} C_n(\bar{c}A;B)$ and $C (\bar{c}A):=\bar{b}igcup_{B {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Ob}(\bar{c}C)} C(\bar{c}A;B)$. The \emph{support of a chain $c$} is the union of the supports of all the simplices that appear in $c$ with a nonzero coefficient.
\end{definition}
\bar{b}egin{enumerate}gin{remark}
This is more or less a special case of what are known as ``simplicial objects in the category $\bar{c}C$,'' except that we do not equip our simplices with degeneracy maps.
\end{remark}
The adjective ``regular'' in the definition above is to emphasize that none of our simplices are ``degenerate:'' their domains must be \emph{strictly} linearly ordered. It is more usual to allow for degenerate simplices, but for our purposes, this extra generality does not seem to be useful. Since all of our simplices will be regular, we will omit the word ``regular'' in what follows.
\bar{b}egin{enumerate}gin{definition}
If $n \geq 1$ and $0 \leq i \leq n$, then the \emph{ith boundary operator} $\bar{p}artial^i_n : C_n (\bar{c}A;B) \rightarrow C_{n-1} (\bar{c}A; B)$ is defined so that if $f$ is a regular $n$-simplex with domain ${\bar{m}athcal P}(s)$, where $s = \{s_0, \ldots, s_n\}$ with $s_0 < \ldots < s_n$, then $$\bar{p}artial^i_n(f) = f \upharpoonright {\bar{m}athcal P}(s \setminus \{s_i\})$$ and extended linearly to a group map on all of $C_n (\bar{c}A; B)$.
If $n \geq 1$ and $0 \leq i \leq n$, then the \emph{boundary map} $\bar{p}artial_n : C_n(\bar{c}A; B) \rightarrow C_{n-1}(\bar{c}A; B)$ is defined by the rule
$$\bar{p}artial_n(c) = \bar{m}athcal{S}igma_{0 \leq i \leq n} (-1)^i \bar{p}artial^i_n (c).$$
We write $\bar{p}artial^i$ and $\bar{p}artial$ for $\bar{p}artial^i_n$ and $\bar{p}artial_n$, respectively, if $n$ is clear from context.
\end{definition}
\bar{b}egin{enumerate}gin{definition}
The kernel of $\bar{p}artial_n$ is denoted $Z_n(\bar{c}A; B)$, and its elements are called \emph{($n$-)cycles}. The image of $\bar{p}artial_{n+1}$ in $C_n(\bar{c}A; B)$ is denoted $B_n(\bar{c}A; B)$. The elements of $B_n(\bar{c}A; B)$ are called \emph{($n$-)boundaries}.
\end{definition}
It can be shown (by the usual combinatorial argument) that $B_n(\bar{c}A; B) \subseteq Z_n
(\bar{c}A; B)$, or more briefly, ``$\bar{p}artial_n\bar{c}irc \bar{p}artial _{n+1} = 0$.'' Therefore we can define simplicial homology groups relative to $\bar{c}A$:
\bar{b}egin{enumerate}gin{definition}
The \emph{$n$th (simplicial) homology group of $\bar{c}A$ over $B$} is $$H_n(\bar{c}A; B) = Z_n(\bar{c}A; B) / {B_n(\bar{c}A; B)}.$$
\end{definition}
There are two natural candidates for the definition of the boundary of a 0-simplex. One possibility is to define $\bar{p}artial_0 (f) = 0$ for all $f{\hbox{\boldmath $\bar \textup{im}ath$}}n S_0(\bar{c}A;B)$. Another possibility is to extend the definition of an $n$-simplex to $n=-1$; namely a $(-1)$-simplex $f$ is an object $f(\emptyset)$ in $\bar{c}C$. Then the definition of a boundary operator extends naturally to the operator $\bar{p}artial_0:f{\hbox{\boldmath $\bar \textup{im}ath$}}n S_0(\bar{c}A;B)\bar{m}apsto B$.
As we show in Lemma~\ref{h0}, computing the group $H_0$ in a specific context using the first definition gives $H_0\bar{c}ong \bar{m}athbb{Z}$ while using the second definition we get $H_0= 0$. Thus, the difference between the approaches is parallel to that between the homology and reduced homology groups \bar{c}ite{B}.
Next we define the amalgamation properties. We use the convention that $[n]$ denotes the $(n+1)$-element set $\{0, 1, \ldots, n\}$.
\bar{b}egin{enumerate}gin{definition}
Let $\bar{c}A$ be a non-empty amenable family of functors into a category $\bar{c}C$ and let $n\ge 1$.
\bar{b}egin{enumerate}gin{enumerate}
{\hbox{\boldmath $\bar \textup{im}ath$}}tem $\bar{c}A$ has \emph{$n$-amalgamation} if for any functor $f : {\bar{m}athcal P}^-([n-1]) \rightarrow \bar{c}C$, $f{\hbox{\boldmath $\bar \textup{im}ath$}}n \bar{c}A$, there is an $(n-1)$-simplex $g \supseteq f$ such that $g {\hbox{\boldmath $\bar \textup{im}ath$}}n \bar{c}A$.
{\hbox{\boldmath $\bar \textup{im}ath$}}tem $\bar{c}A$ has \emph{$n$-complete amalgamation} or \emph{$n$-CA} if $\bar{c}A$ has $k$-amalgamation for every $k$ with $1 \leq k \leq n$.
{\hbox{\boldmath $\bar \textup{im}ath$}}tem $\bar{c}A$ has \emph{strong $2$-amalgamation} if whenever $f: {\bar{m}athcal P}(s) \rightarrow \bar{c}C$, $g: {\bar{m}athcal P}(t) \rightarrow \bar{c}C$ are simplices in $\bar{c}A$ and $f \upharpoonright {\bar{m}athcal P}(s\bar{c}ap t) = g \upharpoonright {\bar{m}athcal P}(s \bar{c}ap t)$, then $f \bar{c}up g$ can be extended to a simplex $h: {\bar{m}athcal P}(s\bar{c}up t) \rightarrow \bar{c}C$ in $\bar{c}A$.
{\hbox{\boldmath $\bar \textup{im}ath$}}tem $\bar{c}A$ has \emph{$n$-uniqueness} if for any functor $f: {\bar{m}athcal P}^-([n-1]) \rightarrow \bar{c}A$ and any two $(n-1)$-simplices $g_1$ and $g_2$ in $\bar{c}A$ extending $f$, there is a natural isomorphism $F: g_1 \rightarrow g_2$ such that $F \upharpoonright \operatorname{dom}(f)$ is the identity.
\end{enumerate}
\end{definition}
\bar{b}egin{enumerate}gin{remark}
The definition of $n$-amalgamation can be naturally extended to $n=0$: $\bar{c}A$ has $0$-amalgamation if it contains a functor $f:\{\emptyset\} \to \bar{c}C$. This condition is satisfied for any non-empty amenable family of functors.
\end{remark}
\bar{b}egin{enumerate}gin{definition}
We say that an amenable family of functors $\bar{c}A$ is \emph{non-trivial} if $\bar{c}A$ is non-empty, has 1-amalgamation, and satisfies the strong 2-amalgamation property.
\end{definition}
Everywhere below, we are dealing with {\em non-trivial} amenable families of functors. The following claim is immediate from the definitions; we include the proof because it illustrates a typical use of 1-amalgamation, strong 2-amalgamation, and other properties of the amenable families.
\bar{b}egin{enumerate}gin{claim}\langlebel{adding_vertex}
Let $\bar{c}A$ be a non-trivial amenable family of functors and let $f$ be an $n$-simplex with support $s=\{s_0,\bar{d}ots,s_n\}$. For any $m{\hbox{\boldmath $\bar \textup{im}ath$}}n \omega$ such that $m > s_i$ for $i=0,\bar{d}ots n$, there is an $(n+1)$-simplex $h$ with support $s\bar{c}up \{m\}$ such that $\bar{p}artial _{n+1}^{n+1}(h)=f$.
\end{claim}
\bar{b}egin{enumerate}gin{proof}
Let $\bar{c}A$, $f$, and $m$ satisfy the assumptions of the claim. Since $\bar{c}A$ is closed under restrictions, the functor $f\upharpoonright \{\emptyset\}$ is in $\bar{c}A$. By 1-amalgamation, the functor $f\upharpoonright \{\emptyset\}$ has an extension to a functor $g:{\bar{m}athcal P}([0])\to \bar{c}C$, $g{\hbox{\boldmath $\bar \textup{im}ath$}}n \bar{c}A$. If $\sigma$ is the natural functor from the category ${\bar{m}athcal P}(\{m\})$ to ${\bar{m}athcal P}(\{0\})$, then the functor $g':{\bar{m}athcal P}(\{m\})\to \bar{c}C$ defined by $g'=g\bar{c}irc \sigma$ is isomorphic to $g$. Since $\bar{c}A$ is closed under isomorphisms, we have $g'{\hbox{\boldmath $\bar \textup{im}ath$}}n \bar{c}A$. Finally, using strong 2-amalgamation, we obtain a simplex $h$ with support $\{s_0,\bar{d}ots,s_n,m\}$ that extends $f$ and $g'$. By construction, $h$ is an $(n+1)$-simplex such that $\bar{p}artial _{n+1}^{n+1}(h)=f$.
\end{proof}
In particular, we get the following corollary.
\bar{b}egin{enumerate}gin{corollary}
If $\bar{c}A$ is a non-trivial amenable family of functors, then $\bar{c}A$ contains an $n$-simplex for each $n\ge 1$.
\end{corollary}
\subsection{Computing homology groups}
We now establish facts that will describe the general structure of the homology groups. The goal is to show that, under appropriate assumptions, the homology group $H_n$ is equal to the set of cosets $c+B_n(\bar{c}A;B)$ for a set of very simple $n$-cycles $c$. This, in turn, will help with the calculation of the groups in the model-theoretic examples.
We start by defining special kinds of $n$-chains which are useful for computing homology groups.
\bar{b}egin{enumerate}gin{definition}
\langlebel{shell}
If $n \geq 1$, an \emph{$n$-shell} is an $n$-chain $c$ of the form $$\bar{p}m \sum_{0 \leq i \leq n + 1} (-1)^i f_i,$$ where $f_0, \ldots, f_{n+1}$ are $n$-simplices such that whenever $0 \leq i < j \leq n+1$, we have $\bar{p}artial^i f_j = \bar{p}artial^{j-1} f_i$.
\end{definition}
Note that any $n$-shell is an $n$-cycle.
In addition, if $f$ is any $(n+1)$-simplex, then $\bar{p}artial f$ is an $n$-shell. An $n$-shell should be thought of as an attempt to create an $(n+1)$-simplex by piecing together the simplices $f_0, \ldots, f_{n+1}$ along their faces, and the equation $\bar{p}artial^i f_j = \bar{p}artial^{j-1} f_i$ says that we may make the appropriate identifications between these faces.
\bar{b}egin{enumerate}gin{definition}
\langlebel{fan}
If $n \geq 1$, and \emph{$n$-fan} is
an $n$-chain of the form
$$\bar{p}m\sum_{i{\hbox{\boldmath $\bar \textup{im}ath$}}n\{0,..,\bar{b}ar{w}idehat k,..., n+1\}} (-1)^i f_i$$ for some $k {\hbox{\boldmath $\bar \textup{im}ath$}}n [n+1]$, where the $f_i$ are $n$-simplices such that
whenever $0 \leq i < j \leq n$, we have $\bar{p}artial^i f_j = \bar{p}artial^{j-1} f_i$. In other words,
an $n$-fan is an $n$-shell missing one term.
\end{definition}
\bar{b}egin{enumerate}gin{remark}
\bar{b}egin{enumerate}gin{enumerate}
{\hbox{\boldmath $\bar \textup{im}ath$}}tem If $c$ is an $n$-fan, then $\bar{p}artial c$ is an $(n-1)$-shell.
{\hbox{\boldmath $\bar \textup{im}ath$}}tem $\bar{c}A$ has $n$-amalgamation if and only if every $(n-2)$-shell in $\bar{c}A$ is the boundary of an $(n-1)$-simplex in $\bar{c}A$. And $\bar{c}A$ has $n$-uniqueness if and only if every $(n-2)$-shell in $\bar{c}A$ is the boundary of at most one $(n-1)$-simplex in $\bar{c}A$ up to isomorphism.
\end{enumerate}
\end{remark}
\bar{b}egin{enumerate}gin{lemma}
\langlebel{shellcycle}
If $n \geq 2$ and $\bar{c}A$ has $n$-CA, then every
$(n-1)$-cycle is a sum of $(n-1)$-shells. Namely, for each $c{\hbox{\boldmath $\bar \textup{im}ath$}}n Z_{n-1}(\bar{c}A; B)$, $c = \sum_i \bar{b}ar{a}lpha_i f_i$, there is a
finite collection of $(n-1)$-shells $c_i{\hbox{\boldmath $\bar \textup{im}ath$}}n Z_{n-1}(\bar{c}A; B)$
such that $c=\sum_i (-1)^n \bar{b}ar{a}lpha_i c_i$.
Moreover, if $S$ is the support of the chain $c$ and $m$ is any element not in $S$, then we can choose $\sum_i \bar{b}ar{a}lpha_i c_i$ so that its support is $S \bar{c}up \{m\}$.
\end{lemma}
\bar{b}egin{enumerate}gin{proof}
Suppose that $c = \sum_i \bar{b}ar{a}lpha_i f_i$ is a cycle in $Z_{n-1}(\bar{c}A; B)$, where each $f_i$ is an $(n-1)$-simplex, and $\bar{b}ar{a}lpha_i {\hbox{\boldmath $\bar \textup{im}ath$}}n \bar{m}athbb{Z}$.
Let $I$ be the set of all pairs $(i,j)$ such that $f_i$ appears in $c$ and $0 \leq j \leq n-1$. For each $(i,j) {\hbox{\boldmath $\bar \textup{im}ath$}}n I$, let $g_{ij} = \bar{p}artial^j f_i$
(so $g_{ij}$ is an $(n-2)$-simplex).
\bar{b}egin{enumerate}gin{claim}\langlebel{cancelling}
There are $(n-1)$-simplices $h_{ij} {\hbox{\boldmath $\bar \textup{im}ath$}}n \bar{c}A_B$ for each pair $(i,j){\hbox{\boldmath $\bar \textup{im}ath$}}n I$ such that:
\bar{b}egin{enumerate}gin{enumerate}
{\hbox{\boldmath $\bar \textup{im}ath$}}tem[(a)] If $\textup{supp}(g_{ij}) = s_{ij}$, then $\textup{supp}(h_{ij}) = s_{ij} \bar{c}up \{m\}$;
{\hbox{\boldmath $\bar \textup{im}ath$}}tem[(b)] If $g_{ij}=g_{k\ell}$, then $h_{ij}=h_{k\ell}$;
{\hbox{\boldmath $\bar \textup{im}ath$}}tem[(c)] For each $i$, $c_i:=(\sum_{0\leq j\leq n-1} (-1)^jh_{ij})+(-1)^{n}f_i$ is an $(n-1)$-shell.
(Namely, for $j'<j<n$, we have $\bar{p}artial^{j'} h_{ij}=\bar{p}artial^{j-1}h_{ij'}$ and $\bar{p}artial^{j} f_i=g_{ij}=\bar{p}artial^{n-1}h_{ij}$.)
\end{enumerate}
\end{claim}
\bar{b}egin{enumerate}gin{proof}[Proof of Claim.]
First, pick any $0$-simplex $g^*$ in $\bar{c}A_B$ with support $\{m\}$. We define $h_{ij}$ with ``bottom face'' $g_{ij}$ as follows. First set $h_{ij}\upharpoonright {\bar{m}athcal P}(s_{ij})=g_{ij}$. Then set $h_{ij}(\{m\})=g^*(\{m\})$. For $k{\hbox{\boldmath $\bar \textup{im}ath$}}n s_{ij}$, we use $2$-amalgamation and let $h_{ij}(\{k,m\}) {\hbox{\boldmath $\bar \textup{im}ath$}}n \bar{c}A$ be an amalgam of $g^*$ and $g_{ij}\upharpoonright {\bar{m}athcal P}(\{k\})$. Also, for any $k {\hbox{\boldmath $\bar \textup{im}ath$}}n s_{ij} \bar{c}ap s_{pq}$ such that $g_{ij}(\{k\}) = g_{pq}(\{k\})$, we can assume that $h_{ij}(\{k,m\}) = h_{pq}(\{k,m\})$ (by choosing the same amalgam for each).
If $n=2$, stop here, and we have constructed all $h_{ij}$. Now assume $n>2$.
For $k, \ell {\hbox{\boldmath $\bar \textup{im}ath$}}n s_{ij}$ with $k < \ell$, we can use $3$-amalgamation to pick a $2$-simplex $h_{ij}(\{k, \ell, m\}) {\hbox{\boldmath $\bar \textup{im}ath$}}n \bar{c}A$ such that $$\bar{p}artial(h_{ij}(\{k, \ell, m\}) = h_{ij}(\{\ell,m\}) - h_{ij}(\{k,m\}) + g_{ij}(\{k,\ell\}),$$ and again we can ensure that if $\{k,\ell\} \subseteq s_{ij} \bar{c}ap s_{pq}$ then $h_{ij}(\{k,\ell,m\}) = h_{pq}(\{k,\ell,m\})$.
We can continue this procedure inductively, using $n$-CA
to build the simplices $h_{ij}(t)$ for $t \subseteq s_{ij} \bar{c}up \{m\}$ of size $4, 5, \ldots, n$, and we can ensure that $\bar{p}artial^{n-1} h_{ij} = g_{ij}$ and that conditions (a) and (b) hold. Let $(s_{ij})_k$ denote the $k$th element of $s_{ij}$ in increasing order (where $0 \leq k \leq n-3$). If $j'<j\leq n-1$, due to our construction,
$$(\bar{p}artial^{j'}h_{ij})\upharpoonright {\bar{m}athcal P}(s_{ij} \setminus \{(s_{ij})_{j'}\})=
\bar{p}artial^{j'}g_{ij}=\bar{p}artial^{j-1}g_{ij'}=(\bar{p}artial^{j-1}h_{ij'})\upharpoonright {\bar{m}athcal P}(s_{ij'} \setminus \{(s_{ij'})_{j-1}\}).$$
(so, $t:=s_{ij} \setminus \{(s_{ij})_{j'}\}= s_{ij'} \setminus \{(s_{ij'})_{j-1}\}$). Notice then
$$t\bar{c}up\{m\}=\textup{supp}(\bar{p}artial^{j'}h_{ij})=\textup{supp}(\bar{p}artial^{j-1}h_{ij'}).$$
Then again by our construction, $\bar{p}artial^{j'}h_{ij}=\bar{p}artial^{j-1}h_{ij'}$.
Therefore $h_{ij}$ satisfies (c) as well, so we have proved the Claim.
\end{proof}
Now for the sum of $(n-1)$-shells $d:=\sum_i \bar{b}ar{a}lpha_i c_i$, we have
$$d=\sum_i \bar{b}ar{a}lpha_i\left[\left(\sum_{0\leq j\leq n-1} (-1)^jh_{ij}\right)+(-1)^{n}f_i\right]$$
$$= \left[\sum_i \sum_{0 \leq j \leq n-1} (-1)^j \bar{b}ar{a}lpha_i h_{ij}\right] + (-1)^{n} c.$$
Since $\bar{p}artial c = 0$, and $h_{i j} = h_{k \ell}$ whenever $\bar{p}artial^jf_i = \bar{p}artial^\ell f_k$, the first term above is zero.
We have proved the lemma.
\end{proof}
The above lemma allows to make the following conclusions about the structure of the groups $H_n(\bar{c}A; B)$.
\bar{b}egin{enumerate}gin{corollary}\langlebel{shellgen} Assume $\bar{c}A$ has $n$-CA for some $n\geq 2$.
Then $H_{n-1}(\bar{c}A; B)$ is generated by $$\{[c] \, : \, c\bar{m}box{ is an } (n-1)\bar{m}box{-shell over } B\}.$$
In particular, if any $(n-1)$-shell over $B$ is a boundary, then so is any $(n-1)$-cycle.
\end{corollary}
\bar{b}egin{enumerate}gin{corollary}
\langlebel{trivial_homology}
If $\bar{c}A$ has $n$-CA for some $n \geq 3$, then $H_{n-2} (\bar{c}A; B) = 0$.
\end{corollary}
\bar{b}egin{enumerate}gin{proof}
Since $\bar{c}A$ has $n$-amalgamation, any $(n-2)$-shell is the boundary of
an $(n-1)$-simplex. Then due to $(n-1)$-CA and Corollary~\ref{shellgen},
any $(n-2)$-cycle over $B$ is a boundary.
\end{proof}
In fact, Corollary~\ref{shellgen} can be strengthened. We show in Theorem~\ref{Hn_shells} that if $\bar{c}A$ has $(n+1)$-CA for some $n \geq 1$, then the group $H_n(\bar{c}A; B)$ has the universe $\{ [c] : c \textup{ is an } n\textup{-shell over } B \textup{ with support } [n+1] \}$. That is, we are able to show that a linear combination of $n$-shells with integer coefficients is equal, up to a boundary, to an $n$-shell; and we show that the support can be restricted to a prescribed set. To accomplish this, we need to introduce an additional notion.
\bar{b}egin{enumerate}gin{definition}
If $n\ge 1$, an \emph{$n$-pocket} is an $n$-cycle of the form $f-g$, where $f$ and $g$ are $n$-simplices with support $S$ (where $S$ is an $(n+1)$-element set).
\end{definition}
The condition that the boundary of an $n$-pocket $f-g$ is zero implies that $f\restriction {\bar{m}athcal P}(s) = g\restriction {\bar{m}athcal P}(s)$ for each $n$-element subset $s$ of $S$.
\bar{b}egin{enumerate}gin{lemma}\langlebel{trivial_pocket}
Let $\bar{c}A$ be a non-trivial amenable family of functors and suppose that $f,g{\hbox{\boldmath $\bar \textup{im}ath$}}n S_n(\bar{c}A)$ are \emph{isomorphic} functors such that $\bar{p}artial_n f = \bar{p}artial_n g$. Then the $n$-pocket $f-g$ is a boundary.
\end{lemma}
\bar{b}egin{enumerate}gin{proof}
We may assume that the support of both $f$ and $g$ is $[n]$. Using Claim~\ref{adding_vertex}, we can pick an $(n+1)$-simplex $\bar{b}ar{w}idehat{f}$ with the support $[n+1]$ such that $\bar{p}artial^{n+1}_{n+1}\bar{b}ar{w}idehat{f}=f$. Let $\langlengle \bar{b}ar{a}lpha_u:g(u)\to f(u) \bar{m}id u{\hbox{\boldmath $\bar \textup{im}ath$}}n {\bar{m}athcal P}([n])\textup{ran}gle$ be a family of isomorphisms in $\bar{c}C$ that witness the isomorphism of $f$ and $g$.
Define an $(n+1)$-simplex $\bar{b}ar{w}idehat{g}$ by letting, for $u\subset v{\hbox{\boldmath $\bar \textup{im}ath$}}n {\bar{m}athcal P}([n+1])$,
$$
\bar{b}ar{w}idehat{g}(u):=\bar{b}egin{enumerate}gin{cases} \bar{b}ar{w}idehat{f}(u), & u{\hbox{\boldmath \small $\bar n$}}e [n], \\
g([n]), & u=[n],
\end{cases}
$$
and
$$
\bar{b}ar{w}idehat{g}({\hbox{\boldmath $\bar \textup{im}ath$}}ota_{u,v}):=\bar{b}egin{enumerate}gin{cases} g({\hbox{\boldmath $\bar \textup{im}ath$}}ota_{u,v}), & u,v{\hbox{\boldmath $\bar \textup{im}ath$}}n {\bar{m}athcal P}([n]),\\
\bar{b}ar{w}idehat{f}({\hbox{\boldmath $\bar \textup{im}ath$}}ota_{u,v})\bar{c}irc \bar{b}ar{a}lpha_u, & u{\hbox{\boldmath $\bar \textup{im}ath$}}n {\bar{m}athcal P}([n]), v{\hbox{\boldmath \small $\bar n$}}otin {\bar{m}athcal P}([n]),\\
\bar{b}ar{w}idehat{f}({\hbox{\boldmath $\bar \textup{im}ath$}}ota_{u,v}), & u,v{\hbox{\boldmath \small $\bar n$}}otin {\bar{m}athcal P}([n]).
\end{cases}
$$
It is routine to check that $\bar{b}ar{w}idehat{g}$ is indeed an $(n+1)$-simplex and that $f-g$ is the boundary of the $(n+1)$-chain $(-1)^{n+1}(\bar{b}ar{w}idehat{f}-\bar{b}ar{w}idehat{g})$.
\end{proof}
\bar{b}egin{enumerate}gin{lemma}
\langlebel{fans and shells}
Suppose that $n \ge 1$ and $\bar{c}A$ has $(n+1)$-amalgamation. Then for any $n$-fan
$$g = \bar{p}m\sum_{i{\hbox{\boldmath $\bar \textup{im}ath$}}n\{0, \ldots,\bar{b}ar{w}idehat k, \ldots, n+1\}} (-1)^i f_i$$
there is some $n$-simplex $f_k$ and some $(n+1)$-simplex $f$ such that $g + (-1)^k f_k = \bar{p}artial f$.
\end{lemma}
\bar{b}egin{enumerate}gin{proof}
Without loss of generality, $\textup{supp}(g) = [n+1]$ and $k = n+1$. Because $\bar{c}A$ is closed under unions, the union $\bar{b}ar{w}idetilde{f}$ of all the simplices $f_i$ (for $i {\hbox{\boldmath $\bar \textup{im}ath$}}n [n]$) is also in $\bar{c}A$. Let $h=\bar{b}ar{w}idetilde{f} |_{\{n+1\}}$, which is in $\bar{c}A$ by localization. By $(n+1)$-amalgamation, there is a functor $\hat f {\hbox{\boldmath $\bar \textup{im}ath$}}n \bar{c}A$ with support $[n]$ extending $h$. Finally, applying de-localization to $\hat f$, we obtain a functor $ f : {\bar{m}athcal P}([n+1]) \rightarrow \bar{c}C$ in $\bar{c}A$ such that for every $i$ with $0 \leq i \leq n$,
$$ f \upharpoonright {\bar{m}athcal P}(\{0, \ldots, \hat i, \ldots, n+1 \}) = f_i.$$
Letting $f_{n+1} = f \upharpoonright {\bar{m}athcal P}([n])$, we get the result.
\end{proof}
The next lemma says that $n$-pockets are equal to $n$-shells, ``up to a boundary.''
\bar{b}egin{enumerate}gin{lemma}\langlebel{shells and pockets}
Let $\bar{c}A$ be a non-trivial amenable family of functors that has the $(n+1)$-amalgamation property for some $n\ge 1$. For any $B{\hbox{\boldmath $\bar \textup{im}ath$}}n \bar{c}C$, any $n$-shell in $\bar{c}A_B$ with support $[n+1]$ is equivalent, up to a boundary in $B_n(\bar{c}A; B)$, to an $n$-pocket in $\bar{c}A_B$ with support $[n]$. Conversely, any $n$-pocket with support $[n]$ is equivalent, up to a boundary, to an $n$-shell with support $[n+1]$.
\end{lemma}
\bar{b}egin{enumerate}gin{proof}
Suppose $c$ is an $n$-shell with support $[n+1]$ and $$c = \bar{p}m \sum_{0 \leq i \leq n + 1} (-1)^i f_i$$ where $\textup{supp}(f_i) = [n + 1] \setminus \{i\}$. Applying Lemma~\ref{fans and shells} to the $n$-fan $c - f_{n+1}$, we obtain a second $n$-shell
$$
c' = \left(\bar{p}m \sum_{0 \leq i \leq n} (-1)^i f_i\right) + (-1)^{n+1} f'
$$
such that $c'$ is a boundary. Then $c - c' = \bar{p}m (f_{n+1} - f')$ is an $n$-pocket.
Conversely, let $h-h'$ be an $n$-pocket with support $[n]$. By Claim~\ref{adding_vertex}, there is
$\hat h$ extending $h$ with support $[n+1]$ such that $\bar{p}artial^{n+1}\hat h=h$. Then clearly
$$d:=\bar{p}artial \hat h- (-1)^{n+1}(h-h')= \sum^{n+1}_{i=0}(-1)^i\bar{p}artial^i\hat h -(-1)^{n+1}(h-h')$$
is an $n$-shell equivalent to $\bar{p}m (h-h')$.
\end{proof}
From Corollary~\ref{shellgen} and Lemma~\ref{shells and pockets} we get the following.
\bar{b}egin{enumerate}gin{corollary}
If $\bar{c}A$ has $3$-amalgamation, then $H_2(\bar{c}A; B)$ is generated by equivalence classes of $2$-pockets.
\end{corollary}
\bar{b}egin{enumerate}gin{lemma}[Prism lemma]
\langlebel{prism} Let $n\ge 1$.
Suppose that $\bar{c}A$ is a non-trivial amenable family of functors that has $(n+1)$-amalgamation.
Let $f-f'$ be an $n$-pocket in $\bar{c}A_B$ with support $s$, where $|s|=n+1$. Let $t$ be an $(n+1)$-element set disjoint from $s$.
Then given $n$-simplex $g$ in $\bar{c}A_B$ with the domain ${\bar{m}athcal P}(t)$, there is an $n$-simplex $g'$ such that $g-g'$ forms an $n$-pocket in $\bar{c}A_B$
and is equivalent, modulo $B_n(\bar{c}A; B)$, to the pocket $f-f'$. We may choose $g'$ first and then find $g$ to have the same conclusion.
\end{lemma}
\bar{b}egin{enumerate}gin{proof}
We begin with some auxiliary definitions.
\bar{b}egin{enumerate}gin{definition}\langlebel{n-path}
Let $f:{\bar{m}athcal P}(\{0,\bar{d}ots,n\})\to \bar{c}C$ and $g:{\bar{m}athcal P}(\{n+1,\bar{d}ots,2n+1\})\to \bar{c}C$ be $n$-simplices. An \emph{$(n+1)$-path from $f$ to $g$} is a chain $p$ of $(n+1)$-simplices
$$
p=\sum_{k=0}^n (-1)^{n(k+1)}h_k
$$
such that:
\bar{b}egin{enumerate}gin{enumerate}
{\hbox{\boldmath $\bar \textup{im}ath$}}tem
$\operatorname{dom} (h_k) = {\bar{m}athcal P}(\{k,\bar{d}ots,k+n+1\})$ for each $k=0,\bar{d}ots,n$;
{\hbox{\boldmath $\bar \textup{im}ath$}}tem
$\bar{p}artial^0 h_n = g$ and $\bar{p}artial^{n+1} h_0 = f$; and
{\hbox{\boldmath $\bar \textup{im}ath$}}tem
$\bar{p}artial^0 h_k = \bar{p}artial^{n+1} h_{k+1}$ for each $k=0,\bar{d}ots,n-1$.
\end{enumerate}
\end{definition}
\bar{b}egin{enumerate}gin{definition}
Let $f:{\bar{m}athcal P}(\{0,\bar{d}ots,n\})\to \bar{c}C$ and $g:{\bar{m}athcal P}(\{n+1,\bar{d}ots,2n+1\})\to \bar{c}C$ be $n$-simplices; let $p=\bar{d}isplaystyle \sum_{k=0}^n (-1)^{n(k+1)}h_k$ be an $(n+1)$-path from $f$ to $g$.
(1)\
For each $k=0,\bar{d}ots,n$, the $(n+1)$-simplex $h_k$ will be called the \emph{$k$-th leg of the path}.
(2)\
For each leg $h_k$ of the path, the face $\bar{p}artial^{n+1} h_k$ is the \emph{origin face of the leg}, $\bar{p}artial^{0} h_k$ is the \emph{destination face of the leg}, and the remaining faces of the simplex $h_k$ are the \emph{wall faces}.
\end{definition}
With this terminology, the conditions (2) and (3) of Definition~\ref{n-path} are saying that the destination face of the last leg in the path is the simplex $g$, the origin face of the first leg is $f$, and that the destination face of $k$th leg is the same as the origin face of the $(k+1)$st leg.
\bar{b}egin{enumerate}gin{claim}\langlebel{path_claim}
Let $p = \bar{d}isplaystyle \sum_{k=0}^n (-1)^{n(k+1)} h_k$ be a path from $f$ to $g$. Then
$$
\bar{p}artial p = g-f+\sum_{i=1}^n\sum_{k=0}^n (-1)^{i+n(k+1)} \bar{p}artial^ih_k.
$$
That is, the boundary of the path is $g-f$ plus a linear combination of the wall faces of the path.
\end{claim}
\bar{b}egin{enumerate}gin{proof}
Using the definition of the boundary of a chain and changing the order of summation, we have:
\bar{b}egin{enumerate}gin{multline*}
\bar{p}artial p = \bar{p}artial \left(\sum_{k=0}^n (-1)^{n(k+1)} h_k\right)
\\
=\sum_{k=0}^n\sum_{i=0}^{n+1} (-1)^{i+n(k+1)}\bar{p}artial^ih_k
=\sum_{i=0}^{n+1} \sum_{k=0}^n (-1)^{i+n(k+1)}\bar{p}artial^ih_k
\\
=\sum_{k=0}^n (-1)^{n(k+1)} \bar{p}artial^0h_k +\sum_{i=1}^{n} \sum_{k=0}^n (-1)^{i+n(k+1)}\bar{p}artial^ih_k
+\sum_{k=0}^n (-1)^{n+1+n(k+1)}\bar{p}artial^{n+1}h_k.
\end{multline*}
It remains to show that
$$
\sum_{k=0}^n (-1)^{n(k+1)} \bar{p}artial^0h_k +\sum_{k=0}^n (-1)^{n+1+n(k+1)}\bar{p}artial^{n+1}h_k = g-f.
$$
Indeed, the first sum can be rewritten as follows:
\bar{b}egin{enumerate}gin{multline*}
\sum_{k=0}^n (-1)^{n(k+1)} \bar{p}artial^0h_k
\\
= \sum_{k=0}^{n-1}(-1)^{n(k+1)}\bar{p}artial^0 h_k + (-1)^{n(n+1)}\bar{p}artial^0h_n
=\sum_{k=0}^{n-1}(-1)^{n(k+1)}\bar{p}artial^0 h_k + g.
\end{multline*}
Taking into account that $(-1)^{n+1+n(k+1)} = -(-1)^{nk}$, we rewrite the second sum as
$$
\sum_{k=0}^n -(-1)^{nk}\bar{p}artial^{n+1}h_k = -f - \sum_{k=1}^n (-1)^{nk} \bar{p}artial^{n+1}h_k
=-f -\sum_{k=0}^{n-1} (-1)^{n(k+1)} \bar{p}artial^{n+1}h_{k+1}.
$$
Condition (3) in Definition~\ref{n-path} implies that
$$
\sum_{k=0}^{n-1}(-1)^{n(k+1)}\bar{p}artial^0 h_k = \sum_{k=0}^{n-1} (-1)^{n(k+1)} \bar{p}artial^{n+1}h_{k+1},
$$
so the claim follows.
\end{proof}
Here is the plan for the rest of the proof of Lemma~\ref{prism}. We start with the $n$-pocket $f-f'$ in $\bar{c}A_B$ whose support is $s$ for an $(n+1)$-element set $s$. Then given an arbitrary $n$-simplex $g {\hbox{\boldmath $\bar \textup{im}ath$}}n \bar{c}A_B$ with domain $t$, we build a path $p$ from $f$ to $g$ in $\bar{c}A_B$. This is done in Claim~\ref{second_prism} below.
Next we use the walls of the path $p$ and the the simplex $f'$ to gradually build path $p'$ in $\bar{c}A_B$ to some simplex $g'$ such that the walls of $p$
are the same as the walls of $p'$. It will follow immediately from the Claim~\ref{path_claim} that
$$
f-f' +\bar{p}artial (p-p') = g-g',
$$
since the walls of the paths will cancel.
\bar{b}egin{enumerate}gin{claim}
\langlebel{second_prism}
If $\bar{c}A$ is a non-trivial amenable family of functors, and $f: {\bar{m}athcal P}([n]) \rightarrow \bar{c}C$ and $g: {\bar{m}athcal P}(\{n+1, \ldots, 2n+1\})$ are $n$-simplices in $\bar{c}A$, then there is an $(n+1)$-path from $f$ to $g$.
\end{claim}
\bar{b}egin{enumerate}gin{proof}
For $k=0,\bar{d}ots,n$, let $U_k=\{k,\bar{d}ots,k+n+1\}$. We build $(n+1)$-simplices $h_k : {\bar{m}athcal P}(U_k) \rightarrow \bar{c}C$ in $\bar{c}A_B$ by induction on $k$. For $k=0$, use strong $2$-amalgamation to find an $h_0 {\hbox{\boldmath $\bar \textup{im}ath$}}n \bar{c}A_B$ extending both $f$ and $g \upharpoonright {\bar{m}athcal P}(\{n+1\})$. Given $h_{k-1}$ in $\bar{c}A_B$, we can use strong $2$-amalgamation again to build $h_k : {\bar{m}athcal P}(U_k) \rightarrow \bar{c}C$ in $\bar{c}A_B$ such that $\bar{p}artial^{n+1} h_k = \bar{p}artial^0 h_{k-1}$ and $h_k$ extends $g \upharpoonright {\bar{m}athcal P}(\{n+1, n+2, \ldots, k+n+1 \})$. Then $p=\bar{d}isplaystyle \sum_{k=0}^n (-1)^{n(k+1)}h_k$ is a path from $f$ to $g$.
\end{proof}
Now we construct a path from $f'$ to some $n$-simplex $g'$ using the walls of $p$. The walls of $p$ are the following simplices:
$$
\{\bar{p}artial^i h_k \bar{m}id i=1,\bar{d}ots,n;\ k=0,\bar{d}ots,n\}.
$$
By induction on $k=0,\bar{d}ots,n$, we construct simplices $h'_k$ in $\bar{c}A_B$ such that:
\bar{b}egin{enumerate}gin{enumerate}
{\hbox{\boldmath $\bar \textup{im}ath$}}tem
$\bar{p}artial^{n+1}h'_0 = f'$;
{\hbox{\boldmath $\bar \textup{im}ath$}}tem
for $i=1,\bar{d}ots,n$ and $k=0,\bar{d}ots,n$ we have $\bar{p}artial^i h'_k = \bar{p}artial^i h_k$;
{\hbox{\boldmath $\bar \textup{im}ath$}}tem
$\bar{p}artial^0 h'_k = \bar{p}artial^{n+1} h'_{k+1}$ for each $k=0,\bar{d}ots,n-1$.
\end{enumerate}
For $k=0$, consider the $(n-1)$-simplices $(\bar{p}artial^ih_0)|_{\{0\}}$, for $i=1,\bar{d}ots,n$, and the $(n-1)$-simplex $f'|_{\{0\}}$ (that is, we take the walls of the 0th leg of the path and the starting face, which are all $n$-simplices, and localize them at the corner $\{0\}$).
Using $(n+1)$-amalgamation, we can embed these into an $n$-simplex $h''_0$ in $\bar{c}A_{f'(\{0\})}$. Then we apply de-localization to $h''_0$ to get $h'_0 {\hbox{\boldmath $\bar \textup{im}ath$}}n \bar{c}A_B$ into which the wall faces and the starting face are all embedded. Taking the face $\bar{p}artial^0h'_0$, we now have the starting face for the leg $h'_1$, and so on, until we get to $g'$. This completes the proof of Lemma~\ref{prism}. The same argument works when $g'$ is chosen first.
\end{proof}
\bar{b}egin{enumerate}gin{corollary}
\langlebel{Hn_generators} Let $n\ge 1$.
Suppose $\bar{c}A$ is a non-trivial amenable family that has $(n+1)$-CA.
The group $H_n(\bar{c}A; B)$ is generated by equivalence classes $n$-shells with support $[n+1]$.
\end{corollary}
\bar{b}egin{enumerate}gin{proof}
We know by Corollary~\ref{shellgen} above that $H_n(\bar{c}A; B)$ is generated by the equivalence classes of $n$-shells. By Lemma~\ref{shells and pockets}, each equivalence class containing an $n$-shell also contains an $n$-pocket. Each $n$-pocket is equivalent to an $n$-pocket with support $[n]$, via one or two applications of the prism lemma above. The conclusion now follows from the converse clause of Lemma~\ref{shells and pockets}.
\end{proof}
We have a shell version of the prism lemma as well.
\bar{b}egin{enumerate}gin{lemma}[Prism lemma, shell version]
\langlebel{shellprism}
Let $\bar{c}A$ be a non-trivial amenable family of functors that satisfies $(n+1)$-amalgamation for some $n \geq 1$.
Suppose that an $n$-shell $f:=\sum_{0\leq i\leq n+1} (-1)^{i}f_i$ and an $n$-fan
$g^-:=\sum_{i{\hbox{\boldmath $\bar \textup{im}ath$}}n \{0,...,\hat k,... ,n+1\}} (-1)^{i}g_i$ are given, where $f_i,g_i$ are $n$-simplices over $B$, $\textup{supp}(f)=s$ with $|s|=n+2$, and $\textup{supp}(g^-)=t=\{t_0,...,t_{n+1}\}$, where $t_0<...<t_{n+1}$ and $s\bar{c}ap t=\emptyset$. Then there is an
$n$-simplex $g_k$ over $B$ with support $\bar{p}artial_k t := t \setminus \{t_k\}$ such that $g:=g^-+(-1)^{k}g_{k}$
is an $n$-shell over $B$ and $f-g{\hbox{\boldmath $\bar \textup{im}ath$}}n B_n(\bar{c}A;B)$.
\end{lemma}
\bar{b}egin{enumerate}gin{proof}
Assume $f$ and
$g^-$ are given, as supposed.
Now by Lemma~\ref{fans and shells}, there are an $n$-simplex $f'_k$ with $\operatorname{dom}(f'_{k})=\operatorname{dom}(f_{k})$ and an $(n+1)$-chain (indeed a single $(n+1)$-simplex) $c$, such that $f+(-1)^{k}(f'_{k}-f_{k})=\bar{p}artial c$, i.e. $f$ is equivalent to
an $n$-pocket $(-1)^{k+1}(f'_{k}-f_{k})$. Again by Lemma~\ref{fans and shells},
there are an $(n+1)$-simplex $d$ with $\operatorname{dom}(d)={\bar{m}athcal P}(t)$ and an $n$-simplex $g'_{k}$
such that $\bar{p}artial d=g^-+(-1)^{k}g'_{k}.$ Hence
$$f-g^-=(-1)^{k+1}(f'_{k}-f_{k})+(-1)^{k}g'_{k}+\bar{p}artial(c-d).$$
Then by the prism lemma (Lemma~\ref{prism}), there is an $n$-simplex $g_{k}$ such that
$(-1)^{k+1}(f'_{k}-f_{k})+(-1)^{k}g'_{k}$ is equivalent to $(-1)^{k}g_{k}$ up to
a boundary. Hence $f$ is equivalent to $g=g^-+(-1)^{k}g_{k}$ up to a boundary.
Moreover, since $g-g'$ is a pocket, in particular a cycle, clearly $g$ is an
$n$-shell.
\end{proof}
The next theorem gives an even simpler standard form for elements of $H_n(\bar{c}A; B)$. Note that it is a strengthening of Corollary~\ref{Hn_generators} above, which only says that $H_n(\bar{c}A; B)$ is \emph{generated} by shells with support $[n+1]$.
\bar{b}egin{enumerate}gin{theorem}
\langlebel{Hn_shells}
If $\bar{c}A$ is a non-trivial amenable family of functors with $(n+1)$-CA for some $n \geq 1$, then $$H_n(\bar{c}A; B) = \left\{ \left[c\right] : c \textup{ is an } n\textup{-shell over } B \textup{ with support } [n+1] \right\}.$$
\end{theorem}
\bar{b}egin{enumerate}gin{proof} By Corollary~\ref{Hn_generators}, it suffices to show the following: if $d$ and $e$ are any two $n$-shells in $\bar{c}A_B$ with support $[n+1]$, then $[d + e] = [d']$ for some $n$-shell $d'$ in $\bar{c}A_B$ with the same support.
First, pick any $n$-fan in $\bar{c}A_B$ $$c = f_{\bar{b}ar{w}idehat{0}} - f_{\bar{b}ar{w}idehat{1}} + \ldots + (-1)^n f_{\bar{b}ar{w}idehat{n}}$$ such that the domain of $f_{\bar{b}ar{w}idehat{i}}$ is ${\bar{m}athcal P}(\{0, \ldots, \bar{b}ar{w}idehat{i}, \ldots, n+1\})$. Applying the
shell prism lemma to $d$ and to $e$ separately, we see that we can assume (up to equivalence modulo $B_n(\bar{c}A; B)$) that $d = c + (-1)^{n+1} g$ and $e = c + (-1)^{n+1} h$ for some $n$-simplices $g$ and $h$. By Lemma~\ref{fans and shells}, we can pick another $n$-simplex $f_{\bar{b}ar{w}idehat{n+1}}$ such that $d_0 := c + (-1)^{n+1} f_{\bar{b}ar{w}idehat{n+1}}$ is in $B_n(\bar{c}A; B)$.
Next, use Lemma~\ref{fans and shells} two more times to pick an $n$-simplices $k_0$ and $k_1$ such that $$d_1 := k_0 - f_{\bar{b}ar{w}idehat{1}} + f_{\bar{b}ar{w}idehat{2}} - \ldots + (-1)^{n+1} g$$ and $$d_2 := f_{\bar{b}ar{w}idehat{0}} - k_1 + f_{\bar{b}ar{w}idehat{2}} - \ldots + (-1)^{n+1} h$$ are both in $B_n(\bar{c}A; B)$, where the ``$\ldots$'' is filled in with the appropriate $f_{\bar{b}ar{w}idehat{i}}$'s. Finally, let $$d_3 := -k_0 + k_1 - f_{\bar{b}ar{w}idehat{2}} + \ldots + (-1)^n f_{\bar{b}ar{w}idehat{n+1}}.$$
Then since $d_0, d_1,$ and $d_2$ are in $B_n(\bar{c}A; B)$, $$[d_3] = [d_0 + d_1 + d_2 + d_3].$$ On the other hand, simply by canceling terms, we compute:
$$ d_0 + d_1 + d_2 + d_3 = 2 f_{\bar{b}ar{w}idehat{0}} - 2 f_{\bar{b}ar{w}idehat{1}} + 2 f_{\bar{b}ar{w}idehat{2}} - \ldots + 2(-1)^n f_{\bar{b}ar{w}idehat{n}} + (-1)^{n+1} g + (-1)^{n+1} h$$ $$= d + e.$$
\end{proof}
Now using Theorem~\ref{Hn_shells} and Lemma~\ref{shells and pockets}, we get the following.
\bar{b}egin{enumerate}gin{corollary}\langlebel{Hn_pockets}
If $\bar{c}A$ is a non-trivial amenable family with $(n+1)$-CA (for some $n \geq 1$), then
$$
H_n(\bar{c}A;B ) = \left\{ [c] : c \textup{ is an } n\textup{-pocket in } \bar{c}A \textup{ over $B$ with support } [n] \right\}.
$$
\end{corollary}
\section{Homology groups in model theory}
In this section, we define some amenable classes of functors that arise in model theory. The properties of the classes of functors
were the motivation for Definition~\ref{amenable}. Given either a complete rosy theory $T$ or a complete type $p {\hbox{\boldmath $\bar \textup{im}ath$}}n S(A)$ in a rosy theory, we will define both ``type homology groups'' $H^t_n(T)$ (or $H^t_n(p)$) or the ``set homology groups'' $H^{set}_n(T)$ (or $H^{set}_n(p)$). As we show below, these definitions will lead to isomorphic homology groups.
For the remainder of this paper, we assume that $T$ is a rosy theory having elimination of hyperimaginaries. The reason for this is so that we have a nice independence notion. Throughout, ``independent'' or ``nonforking'' will mean independence with respect to thorn nonforking. But the assumption is for convenience not for full generality.
For example if $T$ is simple, then one may assume the independence is usual nonforking in
$\operatorname{\bar{m}athfrak{C}}^{heq}$ while replacing $\operatorname{acl}$ by $\operatorname{bdd}$ and so on. But due to elimination of hyperimaginaries
thorn forking is equivalent to usual forking in simple $T$ \bar{c}ite{EO}. Moreover there are non-rosy examples having suitable independence notions that fit in our amenable category context
\bar{c}ite{KK}.
\subsection{Type homology}
We will work with $*$-types -- that is, types with possibly infinite sets of variables -- and to avoid some technical issues, we will place an absolute bound on the cardinality of the variable sets of the types we consider. Fix some infinite cardinal $\kappa_0 \geq |T|$. We will assume that every $*$-type has no more than $\kappa_0$ free variables. We also fix a set $\bar{c}V$ of variables such that $|\bar{c}V| > \kappa_0$ and assume that all variables in $*$-types come from the set $\bar{c}V$ (which is a ``master set of variables.'') We work in a monster model $\operatorname{\bar{m}athfrak{C}}=\operatorname{\bar{m}athfrak{C}}^{eq}$ which is saturated in some cardinality greater than $2^{|\bar{c}V|}$. We let $\bar{b}ar \kappa=|\operatorname{\bar{m}athfrak{C}}|$. As we will see in the next section, the precise values of $\kappa_0$ and $|\bar{c}V|$ will not affect the homology groups.
Given a set $A$, strictly speaking we should write ``\textbf{a} complete $*$-type of $A$'' instead of ``\textbf{the} complete $*$-type of $A$'' -- there are different such types corresponding to different choices of which set of variables to use, and this distinction is crucial for our purposes.
If $X$ is any subset of the variable set $\bar{c}V$, $\sigma: X \rightarrow \bar{c}V$ is any injective function, and $p(\overlineerline{x})$ is any $*$-type such that $\overlineerline{x}$ is contained in $X$, then we let $$\sigma_* p = \left\{\bar{v}arphi(\sigma(\overlineerline{x})) : \bar{v}arphi(\overlineerline{x}) {\hbox{\boldmath $\bar \textup{im}ath$}}n p \right\}.$$
\bar{b}egin{enumerate}gin{definition}
If $A$ is a small subset of the monster model, then $\bar{c}T_A$ is the category such that
\bar{b}egin{enumerate}gin{enumerate}
{\hbox{\boldmath $\bar \textup{im}ath$}}tem The objects of $\bar{c}T_A$ are are all the complete $*$-types in $T$ over $A$, including (for convenience) a single ``empty type'' $p_\emptyset$ with no free variables;
{\hbox{\boldmath $\bar \textup{im}ath$}}tem $\textup{Mor}_{\bar{c}T_A}(p(\overlineerline{x}), q(\overlineerline{y}))$ is the set of all injective maps $\sigma : \overlineerline{x} \rightarrow \overlineerline{y}$ such that $\sigma_*(p) \subseteq q$
\end{enumerate}
\end{definition}
Note that this definition gives a notion of two types $p(\overlineerline{x})$ and $q(\overlineerline{y})$ being ``isomorphic:'' namely, that $q$ can be obtained from $p$ by relabeling variables.
\bar{b}egin{enumerate}gin{definition}
\langlebel{type_simplex}
If $A$ is a small subset of the monster model, a \emph{closed independent type-functor based on $A$} is a functor $f:X \to \bar{c}T_A$ such that:
\bar{b}egin{enumerate}gin{enumerate}
{\hbox{\boldmath $\bar \textup{im}ath$}}tem $X$ is a downward-closed subset of ${\bar{m}athcal P}(s)$ for some finite $s \subseteq \omega$.
{\hbox{\boldmath $\bar \textup{im}ath$}}tem
Suppose $w {\hbox{\boldmath $\bar \textup{im}ath$}}n X$ and $u,v \subseteq w$. Let $\overlineerline{x}_t$ be the variable set of $f(t)$ and (whenever $r \subseteq t \subseteq s$) let $(f^r_t)_* : p_r(\overlineerline{x}_r) \rightarrow p_t(\overlineerline{x}_t)$ be the image of the inclusion map under the functor $f$. Then whenever $\overlineerline{a}$ realizes the type $f(w)$ and $\overlineerline{a}_u$, $\overlineerline{a}_v$, and $\overlineerline{a}_{u \bar{c}ap v}$ denote subtuples corresponding to the variable sets $f^u_w (\overlineerline{x}_u)$, $f^v_w (\overlineerline{x}_v)$, and $f^{u \bar{c}ap v}_w (\overlineerline{x}_{u \bar{c}ap v})$, then
$$\overlineerline{a}_u {\hbox{\boldmath \small $\bar n$}}onfork _{A \bar{c}up \overlineerline{a}_{u \bar{c}ap v}} \overlineerline{a}_v.$$
{\hbox{\boldmath $\bar \textup{im}ath$}}tem
For all non-empty $u{\hbox{\boldmath $\bar \textup{im}ath$}}n X$ and any $\overlineerline{a}$ realizing $f(u)$, we have (using the notation above) $\overlineerline{a} = \operatorname{acl}\left(A \bar{c}up \bar{b}igcup_{i{\hbox{\boldmath $\bar \textup{im}ath$}}n u} \overlineerline{a}_{\{i\}}\right)$.
\end{enumerate}
(The adjective ``closed'' in the definition refers to the fact that, by (3), all the types $f(u)$ are $*$-types of algebraically closed tuples.)
Let $\bar{c}A^t(T; A)$ denote all closed independent type-functors based on $A$.
\end{definition}
\bar{b}egin{enumerate}gin{remark}
\langlebel{independence}
It follows from the definition above and the basic properties of nonforking that if $f$ is a closed independent type-functor based on $A$ and $u {\hbox{\boldmath $\bar \textup{im}ath$}}n \operatorname{dom}(f)$ is a set of size $k$, then $f(u)$ is the type of the algebraic closure of an $B$-independent set $\{\overlineerline{a}_1, \ldots, \overlineerline{a}_k\}$, where $B$ is some realization of the type $f^{\emptyset}_u(f(\emptyset))$ -- namely, let $\overlineerline{a}_i$ be the subtuple of some realization $\overlineerline{a}$ of $f(u)$ corresponding to the variables in $f^{\{i\}}_u(\overlineerline{x}_{\{i\}})$.
\end{remark}
\bar{b}egin{enumerate}gin{definition}
If $A = \operatorname{acl}(A)$ is a small subset of the monster model and $p {\hbox{\boldmath $\bar \textup{im}ath$}}n S(A)$, then a \emph{closed independent type-functor in $p$} is a closed independent type-functor $f: X \rightarrow \bar{c}T_A$ based on $A$ such that if $X \subseteq {\bar{m}athcal P}(s)$ and $i {\hbox{\boldmath $\bar \textup{im}ath$}}n s$, then $f(\{i\})$ is the complete $*$-type of $\operatorname{acl}(C \bar{c}up \{b\})$ over $A$, where $C$ is some realization of $f(\emptyset)$ and $b$ is some realization of a nonforking extension of $p$ to $C$.
Let $\bar{c}A^t(p)$ denote all closed independent type-functors in $p$.
\end{definition}
\bar{b}egin{enumerate}gin{proposition}
The sets $\bar{c}A^t(T;A)$ and $\bar{c}A^t(p)$ are non-trivial amenable families of functors.
\end{proposition}
\bar{b}egin{enumerate}gin{proof}
The proofs are essentially the same for the two classes of functors, and we point out the differences below. We will prove that these classes are isomorphism invariant, closed under restrictions and unions (see Remark~\ref{amenable2}), and closed under localization and de-localization.
Isomorphism invariance, closure under restrictions, and closure under unions are simple to check directly from the definitions, and closure under localizations just comes down to the fact that for any independent set of elements $A$ and any $B \subseteq A$, the set $A \setminus B$ is independent over $B$. For closure under de-localization, given the functors $f$ and $g$ as in condition~(3) of Definition~\ref{amenable}, the idea behind constructing the functor $g_0$ is as follows: first, following Remark~\ref{independence}, suppose that $|t| = k$ and $f(t)$ is the type of the $B$-independent set $\{a_1, \ldots, a_k\}$ where $B$ is a realization of $f(\emptyset)$ and, in the case of $\bar{c}A^t(p)$, each $a_i$ realizes a nonforking extension of $p$ over $B$. Suppose that $u \bar{c}up v {\hbox{\boldmath $\bar \textup{im}ath$}}n Z_0$ where $u {\hbox{\boldmath $\bar \textup{im}ath$}}n Z$, $v \subseteq t$, and $|v| = \ell$. Then the type $$\left(g^{\emptyset}_u \bar{c}irc f^v_t \right) (f(v))$$ is the type of $\operatorname{acl}(C \bar{c}up \{b_1, \ldots, b_\ell\})$ for some $C$ and $b_1, \ldots, b_\ell$ such that $C$ is a realization of $f(\emptyset)$, the set $\{b_1, \ldots, b_\ell\}$ is $C$-independent, and, in the case of $\bar{c}A^t(p)$, each $b_i$ realizes a nonforking extension of $p$ to $C$. Now $g(u)$ is the type of $$\operatorname{acl}(D \bar{c}up \{c_1, \ldots c_m\}),$$ where $m = |u|$, $D$ realizes $g(\emptyset)$, the set $\{c_1, \ldots, c_m\}$ is $D$-independent, and in the case of $\bar{c}A^t(p)$, each $c_i$ realizes a nonforking extension of $p$ over $D$. We may assume that $\{b_1, \ldots, b_\ell\} \subseteq D$, and we let $g_0(u \bar{c}up v)$ be the type (over $A$) of the set $$\operatorname{acl}(C \bar{c}up \{b_1, \ldots, b_\ell; c_1, \ldots, c_m\}).$$ Similarly, we can define the maps $g_0({\hbox{\boldmath $\bar \textup{im}ath$}}ota_{x,y})$ by combining the images of inclusions under $f$ and under $g$: if $x = u \bar{c}up v$ and $y = u' \bar{c}up v'$, where $u \subseteq u' {\hbox{\boldmath $\bar \textup{im}ath$}}n Z$ and $v \subseteq v' \subseteq t$, then we define $g_0({\hbox{\boldmath $\bar \textup{im}ath$}}ota_{x,y})$ by combining the maps $f({\hbox{\boldmath $\bar \textup{im}ath$}}ota_{v,v'})$ and $g({\hbox{\boldmath $\bar \textup{im}ath$}}ota_{u,u'})$. Again, it is clear that the resulting functor $g_0$ is a closed independent type-functor since for any $x {\hbox{\boldmath $\bar \textup{im}ath$}}n \operatorname{dom}(g_0)$, $g_0(x)$ is the type over $A$ of the algebraic closure of some $C$-independent set.
Finally, strong $2$-amalgamation for $\bar{c}A^t(T;A)$ and $\bar{c}A^t(p)$ follows from the existence of nonforking extensions.
\end{proof}
\bar{b}egin{enumerate}gin{definition}
If $A$ is a small subset of $\operatorname{\bar{m}athfrak{C}}$, then we write $S_n\bar{c}T_A$ as an abbreviation for $S_n(\bar{c}A^t(T;A); p_\emptyset)$ (the collection of closed $n$-simplices in $\bar{c}A^t(T;A)$ over the empty type $p_\emptyset$), $B_n \bar{c}T_A$ and $Z_n\bar{c}T_A$ for the boundary and cycle groups, and $H^t_n(T;A)$ for the homology group $H_n(\bar{c}A^t(T;A); p_\emptyset)$.
If $A = \operatorname{acl}(A)$ and $p {\hbox{\boldmath $\bar \textup{im}ath$}}n S(A)$, then we use the abbreviation $S_n\bar{c}T(p)$ for the collection of all closed $n$-simplices in $\bar{c}A^t(T;A)$ over $p_\emptyset$ of type $p$, and similarly we use the abbreviations $B_n \bar{c}T(p)$, $Z_n\bar{c}T(p)$, and $H^t_n(p)$.
\end{definition}
\subsection{Set homology}
\bar{b}egin{enumerate}gin{definition}
Let $A$ be a small subset of $\operatorname{\bar{m}athfrak{C}}$. By $\bar{c}C_A$ we denote the category of all subsets (not necessarily algebraically closed) of $\operatorname{\bar{m}athfrak{C}}$ of size no more that $\kappa_0$, where morphisms are partial elementary maps over $A$ (that is, fixing $A$ pointwise).
\end{definition}
\bar{b}egin{enumerate}gin{definition}
A \emph{closed independent set-functor based on $A$} is a functor $f:X\to \bar{c}C_A$ such that:
\bar{b}egin{enumerate}gin{enumerate}
{\hbox{\boldmath $\bar \textup{im}ath$}}tem $X$ is a downward-closed subset of ${\bar{m}athcal P}(s)$ for some finite $s \subseteq \omega$; and
$f(\emptyset) \supseteq A$.
{\hbox{\boldmath $\bar \textup{im}ath$}}tem
If $w {\hbox{\boldmath $\bar \textup{im}ath$}}n X$ and $u,v \subseteq w$, and if $f^x_y$ denotes the image under $f$ of the inclusion map $x \subseteq y$ in ${\bar{m}athcal P}(s)$, then $$f^u_w(u){\hbox{\boldmath \small $\bar n$}}onfork _{A \bar{c}up f^{u \bar{c}ap v}_w(u\bar{c}ap v)} f^v_w(v).$$
{\hbox{\boldmath $\bar \textup{im}ath$}}tem
For all non-empty $u{\hbox{\boldmath $\bar \textup{im}ath$}}n X$, we have $f(u) = \operatorname{acl}(A \bar{c}up \bar{b}igcup_{i{\hbox{\boldmath $\bar \textup{im}ath$}}n u} f^{\{i\}}_u(\{i\}))$.
\end{enumerate}
Let $\bar{c}A^{set}(T; A)$ denote all closed independent set-functors based on $A$.
\end{definition}
\bar{b}egin{enumerate}gin{definition}
If $A = \operatorname{acl}(A)$ is a small subset of the monster model and $p {\hbox{\boldmath $\bar \textup{im}ath$}}n S(A)$, then a \emph{closed independent set-functor in $p$} is a closed independent set-functor $f: X \rightarrow \bar{c}C_A$ based on $A$ such that if $X \subseteq {\bar{m}athcal P}(s)$ and $i {\hbox{\boldmath $\bar \textup{im}ath$}}n s$, then $f(\{i\})$ is a set of the form $\operatorname{acl}(C \bar{c}up \{b\})$ where $C = f^{\emptyset}_{\{i\}}(f(\emptyset))\supseteq A$ and $b$ realizes some nonforking extension of $p$ to $C$.
Let $\bar{c}A^{set}(p)$ denote all closed independent set-functors in $p$.
\end{definition}
Just as in the previous subsection (and by an identical argument), we have:
\bar{b}egin{enumerate}gin{proposition}
The sets $\bar{c}A^{set}(T;A)$ and $\bar{c}A^{set}(p)$ are non-trivial amenable families of functors.
\end{proposition}
\bar{b}egin{enumerate}gin{definition}
If $A$ is a small subset of $\operatorname{\bar{m}athfrak{C}}$, then we write ``$S_n \bar{c}C_A$'' for $S_n(\bar{c}A^{set}(T;A); A)$ (the collection of closed $n$-simplices in $\bar{c}A^{set}(T;A)$ {\em over} $A$), and similarly we write $B_n \bar{c}C_A$ and $Z_n \bar{c}C_A$ for the boundary and cycle groups over $A$, and use the notation $H^{set}_n(T;A)$ for the homology group $H_n(\bar{c}A^{set}(T;A); A)$.
If $A = \operatorname{acl}(A)$ and $p {\hbox{\boldmath $\bar \textup{im}ath$}}n S(A)$, then we use similar abbreviations $S_n \bar{c}C(p):=S_n(\bar{c}A^{set}(p); A)$, $B_n\bar{c}C(p)$, $Z_n\bar{c}C(p)$, and $H^{set}_n(p)$ for the type $p$ versions of the groups {\em over $A$} above.
\end{definition}
\bar{b}egin{enumerate}gin{proposition}
\langlebel{homology_equivalence}
\bar{b}egin{enumerate}gin{enumerate}
{\hbox{\boldmath $\bar \textup{im}ath$}}tem For any $n$ and any $A {\hbox{\boldmath $\bar \textup{im}ath$}}n \bar{c}C$, $H^t_n(T; A) \bar{c}ong H_n^{set}(T; A)$.
{\hbox{\boldmath $\bar \textup{im}ath$}}tem For any $n$ and any complete type $p {\hbox{\boldmath $\bar \textup{im}ath$}}n S(A)$, $H^t_n(p) \bar{c}ong H_n^{set}(p)$.
\end{enumerate}
\end{proposition}
\bar{b}egin{enumerate}gin{proof}
The idea is that we can build a correspondence $F: S \bar{c}C_A \rightarrow S \bar{c}T_A$ which maps each set-simplex $f$ to its ``complete $*$-type'' $F(f)$. Note that this will involve some non-canonical choices: namely, which variables to use for $F(f)$, and in what order to enumerate the various sets in $f$ (since our variable set $\bar{c}V$ is indexed and thus implicitly ordered). We will write out a proof of part~(1) of the proposition, and part~(2) can be proved similarly by relativizing to $p$.
Let $S_{\leq n} \bar{c}C_A$ and $S_{\leq n} \bar{c}T_A$ denote, respectively, $\bar{b}igcup_{i \leq n} S_i \bar{c}C_A$ and $\bar{b}igcup_{i \leq n} S_i \bar{c}T_A$. We will build a sequence of maps $F_n : S_{\leq n} \bar{c}C_A \rightarrow S_{\leq n} \bar{c}T_A$ whose union will be $F$. Given such an $F_n$, let $\overline{F}_n: C_{\leq n} \bar{c}C_A \rightarrow C_{\leq n} \bar{c}T_A$ be its natural extension to the class of all set-$k$-chains over $A$ for $k \leq n$.
\bar{b}egin{enumerate}gin{claim}
There are maps $F_n : S_{\leq n} \bar{c}C_A \rightarrow S_{\leq n} \bar{c}T_A$ such that:
\bar{b}egin{enumerate}gin{enumerate}
{\hbox{\boldmath $\bar \textup{im}ath$}}tem $F_{n+1}$ is an extension of $F_n$;
{\hbox{\boldmath $\bar \textup{im}ath$}}tem If $f {\hbox{\boldmath $\bar \textup{im}ath$}}n S_{\leq n} \bar{c}C_A$ and $\operatorname{dom}(f) = {\bar{m}athcal P}(s)$, then $\operatorname{dom}(F_n(f)) = {\bar{m}athcal P}(s)$ and $\left[F_n(f)\right](s)$ is a complete $*$-type of $f(s)$ over $A$;
{\hbox{\boldmath $\bar \textup{im}ath$}}tem For any $k \leq n$, any $f {\hbox{\boldmath $\bar \textup{im}ath$}}n S_k \bar{c}C_A$, and any $i \leq k$, $F_n(\bar{p}artial^i f) = \bar{p}artial^i \left[F_n(f)\right]$; and
{\hbox{\boldmath $\bar \textup{im}ath$}}tem $F_n$ is surjective, and in fact for every $g {\hbox{\boldmath $\bar \textup{im}ath$}}n S_k\bar{c}T_A$ (where $0 \leq k \leq n$), there are \textbf{more} than $2^{|\bar{c}V|}$ simplices $f {\hbox{\boldmath $\bar \textup{im}ath$}}n S_k\bar{c}C_A$ such that $F_n(f) = g$.
\end{enumerate}
\end{claim}
\bar{b}egin{enumerate}gin{proof}
We prove the claim by induction on $n$. The case where $n=0$ is simple: only conditions (2) and (4) are relevant, and note that we can insure (4) because the monster model $\operatorname{\bar{m}athfrak{C}}$ is $(2^{|\bar{c}V|})^+$-saturated and there are at most $2^{|\bar{c}V|}$ elements of $S_0 \bar{c}T_A$. So suppose that $n >0$ and we have $F_0, \ldots, F_n$ satisfying all these properties, and we want to build $F_{n+1}$. We build $F_{n+1}$ as a union of a chain of partial maps from $S_{\leq n+1} \bar{c}C_A$ to $S_{\leq n+1} \bar{c}T_A$ extending $F_n$ (that is, functions whose domains are subsets of $S_{\leq n+1} \bar{c}C_A$).
\bar{b}egin{enumerate}gin{subclaim}
Suppose that $F: X \rightarrow S_{\leq n+1} \bar{c}T_A$ is a function on a set $X \subseteq S_{\leq n+1} \bar{c}C_A$ of size at most $(2^{|\bar{c}V|})^+$ and that $F$ satisfies (1) through (3). Then for any simplex $g {\hbox{\boldmath $\bar \textup{im}ath$}}n S_{n+1} \bar{c}T_A$, there is an extension $F_0$ of $F$ satisfying (1) through (3) such that $|\operatorname{dom}(F_0)| \leq (2^{|\bar{c}V|})^+$ and:
\bar{b}egin{enumerate}gin{center}
(*) There are $(2^{|\bar{c}V|})^+$ distinct $f {\hbox{\boldmath $\bar \textup{im}ath$}}n S_{n+1} \bar{c}C_A$ such that $F'(f) = g$.
\end{center}
\end{subclaim}
\bar{b}egin{enumerate}gin{proof}
Let $\bar{p}artial g = g_0 - g_1 + \ldots + (-1)^n g_n$ (where $g_i = \bar{p}artial^i g$), and let ${\bar{m}athcal P}(s)$ be the domain of $g$. By induction, each $g_i$ is the image under $F_n$ of $(2^{|\bar{c}V|})^+$ different $n$-simplices in $\bar{c}C_A$; let $\langlengle f^j_i : j < (2^{|\bar{c}V|})^+ \textup{ran}gle$ be a sequence of distinct simplices such that for every $j < (2^{|\bar{c}V|})^+$, $F_n(f^j_i) = g_i$. By saturation of the monster model, for each $j < (2^{|\bar{c}V|})^+$ we can pick an $(n+1)$-simplex $f_j {\hbox{\boldmath $\bar \textup{im}ath$}}n \bar{c}C_A$ with domain ${\bar{m}athcal P}(s)$ such that $\bar{p}artial f_j = f^j_0 - f^j_1 + \ldots + (-1)^n f^j_n$ and $\operatorname{tp}(f_j (s)) = g(s)$. Then the $f_j$ are all distinct, so we can let $F_0 = F \bar{c}up \{(f_j, g) : j < (2^{|\bar{c}V|})^+\}$.
\end{proof}
Now by the subclaim, we can use transfinite induction to build a \textbf{partial} map $F' : S_{\leq n+1} \bar{c}C_A \rightarrow S_{\leq n+1} \bar{c}T_A$ satisfying (1) through (4) (also using the fact that there only (at most) $2^{|\bar{c}V|}$ different simplices in $S_{n+1} \bar{c}T_A$ and the fact that the union of a chain of partial maps from $S_{\leq n+1} \bar{c}C_A$ to $S_{\leq n+1} \bar{c}T_A$ satisfying conditions (2) and (3) will still satisfy these conditions).
Finally, we can extend $F'$ to a function on all of $S_{\leq n+1} \bar{c}C_A$ by a second transfinite induction, extending $F'$ to each $f: {\bar{m}athcal P}(s) \rightarrow \bar{c}C_A$ in $\bar{c}C_A$ one at a time; to ensure that properties (2) and (3) hold, we just have to pick $F_{n+1}(f)$ to be some $(n+1)$-simplex with the same domain ${\bar{m}athcal P}(s)$ whose $n$-faces are as specified by $F_n$ and such that $\left[F_{n+1}(f)\right](s)$ is a complete $*$-type of $f(s)$ over $A$.
\end{proof}
Now let $F = \bar{b}igcup_{n < \omega} F_n$. By property~(3) above, it follows that for any chain $c {\hbox{\boldmath $\bar \textup{im}ath$}}n C \bar{c}C_A$, we have $\overline{F}(\bar{p}artial c) = \bar{p}artial \left[\overline{F}(c) \right]$. Hence $\overline{F}$ maps $Z_n\bar{c}C_A$ into $Z_n\bar{c}T_A$ and $B_n\bar{c}C_A$ into $B_n\bar{c}T_A$, and so $\overline{F}$ induces group homomorphisms $\bar{v}arphi_n : H^{set}_n(T;A) \rightarrow H^t_n(T;A)$. Verifying that $\bar{v}arphi_n$ is injective amounts to checking that whenever $\overline{F}(c) {\hbox{\boldmath $\bar \textup{im}ath$}}n B_n\bar{c}T_A$, the set-chain $c$ is in $B_n\bar{c}C_A$, but this is staightforward: if, say, $\overline{F}(c) = \bar{p}artial c'$, then we can pick a set-simplex $\hat{c}$ ``realizing'' $c'$ such that $\bar{p}artial \hat{c} = c$. Finally, condition~(4) implies that $\bar{v}arphi_n$ is surjective, so $H^{set}_n(T;A) \bar{c}ong H^t_n(T;A)$.
\end{proof}
\bar{b}egin{enumerate}gin{remark}
Since Proposition~\ref{homology_equivalence} is true for any choices of $\kappa_0$, $\bar{c}V$, and the monster model $\operatorname{\bar{m}athfrak{C}}$ as long as $|T| \leq \kappa_0 < |\bar{c}V|$ and $2^{|\bar{c}V|} \leq |\operatorname{\bar{m}athfrak{C}}|$, it follows that our homology groups (with the restriction of the set $A$) do not depend on the choices of $\kappa_0$, $|\bar{c}V|$, or the monster model.
Without specifying a base set $A$, one could also define $C_n(T)$ to be the direct sum $\bar{b}igoplus_{i<\bar{b}ar\kappa} C_n \bar{c}C_{A_i}$ where $\{A_i | i < \bar{b}ar \kappa\}$ is the collection of all small subsets of $\operatorname{\bar{m}athfrak{C}}$, and similarly $Z_n(T)$, $B_n(T)$, and
$H_n(T):=Z_n(T)/B_n(T)$. Then the boundary operator $\bar{p}artial$
sends $n$-chains to $(n-1)$-chains componentwise. Hence it follows
$H_n(T)=\bar{b}igoplus_{i<\bar{b}ar\kappa} H_n(T;A_i).$ This means the homology groups defined without specifying a base set depends on the choice of monster model, and so this approach would not give invariants for the theory $T$.
\end{remark}
\subsection{An alternate definition of the set homology groups}
In our definition of the set homology groups $H^{set}_n(T; A)$ and $H^{set}_n(p)$ (where $p {\hbox{\boldmath $\bar \textup{im}ath$}}n S(A)$), we have been implicitly assuming that the base set $A$ is fixed pointwise by all of the elementary maps in a set-simplex -- this is built into our definition of $\bar{c}C_A$, which says that morphisms are elementary maps \emph{over $A$}. It turns out that we get an equivalent definition of the homology groups if we allow the base set to be ``moved'' by the images of the inclusion maps in a set-simplex, as we will show in this subsection.
\bar{b}egin{enumerate}gin{definition}
\bar{b}egin{enumerate}gin{enumerate}
{\hbox{\boldmath $\bar \textup{im}ath$}}tem A \emph{set-$n$-simplex weakly over $A$} is a set-$n$-simplex $f: {\bar{m}athcal P}(s) \rightarrow \bar{c}C$ over $\emptyset$ such that $f(\emptyset) = A$.
{\hbox{\boldmath $\bar \textup{im}ath$}}tem If $p {\hbox{\boldmath $\bar \textup{im}ath$}}n S(A)$, then a set-$n$-simplex $f: {\bar{m}athcal P}(s) \rightarrow \bar{c}C$ is \emph{weakly of type $p$} if it is a closed simplex, $f(\emptyset) = A$, and for every $i {\hbox{\boldmath $\bar \textup{im}ath$}}n s$, $$f(\{i\}) = \operatorname{acl}\left(f^{\emptyset}_{\{i\}}(A) \bar{c}up \{a_i\} \right)$$ for some $a_i$ such that $\operatorname{tp}(a_i / f^{\emptyset}_{\{i\}}(A))$ is a conjugate of $p$.
\end{enumerate}
\end{definition}
Let $C'_n \bar{c}C_A$ be the collection of all set-$n$-simplices weakly over $A$. Note that the boundary operator $\bar{p}artial$ takes an $n$-simplex weakly over $A$ to a chain of $(n-1)$-simplices weakly over $A$, and so we can define ``weak set homology groups over $A$,'' which we denote $H'_n(T; A)$. Similarly, we can define $H'_n(p)$, the ``weak set homology groups of $p$,'' from chains of set-simplices that are weakly of type $p$.
\bar{b}egin{enumerate}gin{proposition}
\langlebel{homology_equivalence_2}
\bar{b}egin{enumerate}gin{enumerate}
{\hbox{\boldmath $\bar \textup{im}ath$}}tem For any $n$ and any $A {\hbox{\boldmath $\bar \textup{im}ath$}}n \bar{c}C$, $H'_n(T; A) \bar{c}ong H_n^{set}(T; A)$.
{\hbox{\boldmath $\bar \textup{im}ath$}}tem For any $n$ and any complete type $p {\hbox{\boldmath $\bar \textup{im}ath$}}n S(A)$, $H'_n(p) \bar{c}ong H_n^{set}(p)$.
\end{enumerate}
\end{proposition}
\bar{b}egin{enumerate}gin{proof}
As usual, the two parts have identical proofs, and we only prove the second part.
We will identify $S'_0\bar{c}C_A$ as a big single {\em complex} as follows.
Due to our cardinality assumption, for each $n<\omega$, there are $\bar{b}ar \kappa$-many
0-simplices in $S'_0\bar{c}C_A$ having the common domain ${\bar{m}athcal P}(\{ n\})$. Then
we consider the following domain set
$\bar{c}D_0=\{\emptyset\}\bar{c}up\{\{(n,i)\}|\ n<\omega, i<\bar{b}ar\kappa\}$. Now as said
we identify $S'_0\bar{c}C_A$ as a {\em single} functor
$F'_0$ from $\bar{c}D_0$ to $\bar{c}C$ such that
$F'_0(\emptyset)=A$, and $F'_0(\{(n,i)\})=(f')^n_i(\{n\})$ where
$(f')^n_i{\hbox{\boldmath $\bar \textup{im}ath$}}n S'_0\bar{c}C_A$ is the corresponding
0-simplex with $((f')^n_i)^{\emptyset}_{\{n\}}=(F'_0)^{\emptyset}_{\{(n,i)\}}$.
Similarly we consider $S_0\bar{c}C_A$ as a functor $F_0$ from
$\bar{c}D_0$ to $\bar{c}C_A$ such that
$F_0(\emptyset)=A$, and $F_0(\{(n,i)\})=f^n_i(\{n\})\equiv (f')^n_i(\{n\})$
where $f^n_i{\hbox{\boldmath $\bar \textup{im}ath$}}n S_0\bar{c}C_A$ is the corresponding
0-simplex over $A$ with $(f^n_i)^{\emptyset}_{\{n\}}=(F_0)^{\emptyset}_{\{(n,i)\}}$.
Now $F'_0$ and $F_0$ are naturally isomorphic by $\eta^0$ with
$\eta^0_{\emptyset}=$the identity map of $A$, and suitable
$\eta^0_{\{(n,i)\}}$ sending $(f')^n_i(\{n\})$ to $f^n_i(\{n\})$.
Now for $S'_1\bar{c}C_A$, note that for each pair $(f')^{n_0}_{i_0},(f')^{n_1}_{i_1} $ with $n_0<n_1$,
there are $\bar{b}ar \kappa$-many
1-simplices $f'_j$ in $S'_1\bar{c}C_A$ having the common domain ${\bar{m}athcal P}(\{ n_0,n_1\})$
with $\bar{p}artial^0f'_j=(f')^{n_1}_{i_1}$ and $\bar{p}artial^1f'_j=(f')^{n_0}_{i_0}$. Hence we now put the
domain set $\bar{c}D_1=\bar{c}D_0\bar{c}up \{\{(n_0,i_0),(n_1,i_1),{j}\}|\ n_0<n_1<\omega; i_0,i_1,j<\bar{b}ar\kappa\}$.
Then we identify $S'_1\bar{c}C_A$ as a functor
$F'_1$ from $\bar{c}D_1$ to $\bar{c}C$ such that $F'_1\upharpoonright \bar{c}D_0=F'_{01}$, and
$F'_1(\{(n_0,i_0),(n_1,i_1),{j}\})$ corresponds $j$th 1-simplex having
$(f')^{n_0}_{i_0},(f')^{n_1}_{i_1} $ as 0-faces. Similarly we try to identify
$S'_1\bar{c}C_A$ as a functor
$F_1$ from $\bar{c}D_1$ to $\bar{c}C_A$, extending $F_0$. But to make $F'_1$ and $F_1$ isomorphic, we need extra care
when defining $F_1$. For each $j<\bar{b}ar \kappa$ and
a set $a'_j=f'_j(\{n_0,n_1\})$ of corresponding 1-simplex $f'_j$, assign
an embedding $\eta^1_j=\eta^1_{\{(n_0,i_0),(n_1,i_1),{j}\}}$ sending
$a'_j$ to $a_j$, extending the inverse of
$(f'_j)^{\emptyset}_{\{n_0,n_1\}}$. Then we define
$F_1(\{(n_0,i_0),(n_1,i_1),{j}\})=a'_j$, and
$$(F_1)^{\{(n_k,i_k)\}}_{\{(n_0,i_0),(n_1,i_1),{j}\}}=
\eta^1_j\bar{c}irc(f'_j)^{\{n_k\}}_{\{n_0,n_1\}}\bar{c}irc(\eta^0_{\{(n_k,i_k)\}})^{-1}.$$
Now then clearly $\eta^1$ with $\eta^1\upharpoonright \bar{c}D_0=\eta^0$ is an isomorphism
between $F'_1$ and $F_1$.
By iterating this argument we can respectively identify $S'_n\bar{c}C_A$ and $S_n\bar{c}C_A$,
as functors $F'_n$ and $F_n$ having the same domain $\bar{c}D_n$ extending $\bar{c}D_1$.
Moreover we can also construct an isomorphism $\eta^n$, extending $\eta^1$, between $F'_n$ and $F_n$.
Note that each $x{\hbox{\boldmath $\bar \textup{im}ath$}}n \bar{c}D_n-\bar{c}D_{n-1}$ corresponds an $n$-simplex $f'{\hbox{\boldmath $\bar \textup{im}ath$}}n S'_n\bar{c}C$, and $\eta^n_x$
corresponds an $n$-simplex over $A$ $f{\hbox{\boldmath $\bar \textup{im}ath$}}n S_n\bar{c}C$.
This correspondence $f'\bar{m}apsto f$ induces a bijection
from $C_n'\bar{c}C_A$ to $C_n\bar{c}C_A$, mapping $c'\bar{m}apsto c$,
which indeed is an isomorphism of the two groups.
Notice that by the construction, if an $n$-shell $c'$ is the boundary of some
$(n+1)$-simplex $f'$, then $c$ is the boundary of $f$.
In general,
it follows $(\bar{p}artial d)'=\bar{p}artial d'$ \ \ (*).
Thus this correspondence also induces an isomorphism between $Z'_n(T;A)$ and $Z_n(T;A)$.
Moreover it follows from (*) that the correspondence sends $B'_n(T;A)$ to $B_n(T;A)$: Let
$c'=\bar{p}artial d'{\hbox{\boldmath $\bar \textup{im}ath$}}n B'_n(T;A)$. Then by (*), we have $c=\bar{p}artial d{\hbox{\boldmath $\bar \textup{im}ath$}}n B_n(T;A).$
Conversely for $c'{\hbox{\boldmath $\bar \textup{im}ath$}}n Z'_n(T;A)$, assume $c=\bar{p}artial e{\hbox{\boldmath $\bar \textup{im}ath$}}n B_n(T;A).$
Now for $e'$, again by (*),
$\bar{p}artial e'=c'$. Hence we have $c'{\hbox{\boldmath $\bar \textup{im}ath$}}n B'_n(T;A) $.
\end{proof}
\section{Homology groups and model-theoretic amalgamation properties: basic facts}
From now on, we will usually drop the superscripts $t$ and $set$ from $H^t_n(p)$ and $H^{set}_n(p)$, since these groups are isomorphic, and use ``$H_n(p)$'' to refer to the isomorphism class of these two groups. In computing the groups below, we generally use $H^{set}_n(p)$ rather than $H^t_n(p)$.
First, we observe that $H_0$ does not give any information, since it is always isomorphic to $\bar{m}athbb{Z}$, if $\bar{p}artial_0(f)=0$ for any $0$-simplex $f$; or is trivial if $\bar{p}artial_0(f)$ is defined to be $f(\emptyset)$:
\bar{b}egin{enumerate}gin{lemma}\langlebel{h0}
\bar{b}egin{enumerate}gin{enumerate}
{\hbox{\boldmath $\bar \textup{im}ath$}}tem If $\bar{p}artial_0(f)=0$, then for any complete type $p$ over an algebraically closed set $A$, $H_0(p) \bar{c}ong \bar{m}athbb{Z}$ and for any small subset $A$ of $\operatorname{\bar{m}athfrak{C}}$, $H_0(T; A) \bar{c}ong \bar{m}athbb{Z}$.
{\hbox{\boldmath $\bar \textup{im}ath$}}tem If $\bar{p}artial_0(f)=f(\emptyset)$, then both groups in (1) are trivial.
\end{enumerate}
\end{lemma}
\bar{b}egin{enumerate}gin{proof}
Both parts of the lemma can be proved by essentially the same argument, so we only write out the proof for the group $H_0(p)$ in (1).
For the proof we will define an augmentation map $\epsilon$ as in
topology. Since we can add parameters to the language for $A$, we can assume that $A = \emptyset$.
Define $\epsilon: C_0\bar{c}C(p) \to \bar{m}athbb{Z}$ by
$\epsilon(c)=\sum_i n_i$ for a 0-chain $c=\sum_i n_if_i$ of type $p$. Then $\epsilon$ is
a homomorphism such that $\epsilon (b)=0$ for any 0-boundary $b$ (since $\epsilon (\bar{p}artial f) = 0$ for any $1$-simplex $f$). Thus
$\epsilon$ induces a homomorphism $\epsilon_*:H_0(p)\to \bar{m}athbb{Z}$. Note that \emph{any} $0$-chain $c$ is in $Z_0(p)$, so clearly $\epsilon_*$ is onto. We claim that $\epsilon_*$ is one-to-one, i.e. $\ker \epsilon_*= B_0(p)$.
Given a $0$-chain $c=\sum_{i {\hbox{\boldmath $\bar \textup{im}ath$}}n I} n_if_i$ such that $\epsilon_*(c)=\sum_{i {\hbox{\boldmath $\bar \textup{im}ath$}}n I} n_i=0$, we shall show $c$ is a boundary. Pick some natural number $m$ greater than every $k_i$ where $\operatorname{dom} f_i={\bar{m}athcal P}(\{k_i\})$.
Let $a_i=\operatorname{acl}(a_i)=f_i(\{k_i\})$. Then choose $a$ realizing $p$ such that $a \bar{m}athop{\bar{m}athpalette\Ind{}} \{a_i : i {\hbox{\boldmath $\bar \textup{im}ath$}}n I\}$.
Now let $g_i$ be a closed 1-simplex of $p$ such that $\operatorname{dom} g_i={\bar{m}athcal P}(\{ k_i,m\})$, $g_i(\{k_i\})=a_i$, and $g_i(\{m\})=a$. Then $\bar{p}artial g_i=c_m-f_i$, where $c_m$ is the $0$-simplex such that $c_m(\emptyset) = \emptyset$ and $c_m(\{m\}) = a$.
Then $c+\bar{p}artial(\sum_i n_i g_i)=\sum_i n_if_i+\sum_in_i(c_m-f_i)=(\sum_in_i)c_m=0.$ Hence $c$ is a 0-boundary,
and $H_0(p) \bar{c}ong \bar{m}athbb{Z}$.
\end{proof}
For $k > 0$, the homology groups $H_k(T; A)$ and $H_k(p)$ are related to standard amalgamation properties. The $n$-amalgamation and $n$-uniqueness properties for simple theories can be stated in terms of shells, and the following is equivalent to the usual definition.
\bar{b}egin{enumerate}gin{definition}
Assume $T = T^{eq}$.
\bar{b}egin{enumerate}gin{enumerate}
{\hbox{\boldmath $\bar \textup{im}ath$}}tem If $A$ is a small subset of $\operatorname{\bar{m}athfrak{C}}$, then $T$ has \emph{$n$-amalgamation property over $A$} if for every $(n-2)$-shell $c$ over $A$, there is an $(n-1)$-simplex $f$ such that $c = \bar{p}artial f$.
{\hbox{\boldmath $\bar \textup{im}ath$}}tem The complete type $p$ has \emph{$n$-amalgamation} if every $(n-2)$-shell $c$ of type $p$, there is an $(n-1)$-simplex $f$ such that $c = \bar{p}artial f$.
{\hbox{\boldmath $\bar \textup{im}ath$}}tem A theory $T$ has \emph{$n$-uniqueness based on $A$} if for every $(n-2)$-shell $c$ based on $A$ and any two $(n-1)$-simplices $f$ and $g$ such that $\bar{p}artial f = \bar{p}artial g = c$, $f$ and $g$ are isomorphic ``over $c$:'' that is, there is an isomorphism $\bar{v}arphi: f \rightarrow g$ such that $\bar{v}arphi$ induces the identity map between $f(u)$ and $g(u)$ whenever $u \subseteq \operatorname{dom}(f)$ has size less than $n-1$. Similarly, we can define what it means for a type $p {\hbox{\boldmath $\bar \textup{im}ath$}}n S(A)$ to have $n$-uniqueness by considering shells of type $p$.
{\hbox{\boldmath $\bar \textup{im}ath$}}tem A theory $T$ has \emph{$n$-amalgamation} if it has $n$-amalgamation over every small subset $A$ of $\operatorname{\bar{m}athfrak{C}}$.
\end{enumerate}
\end{definition}
\bar{b}egin{enumerate}gin{remark}
Intuitively, the $n$-amalgamation property says that we can find a collection $S$ of $n$ independent points such that the algebraic closure of each proper subset of $S$ satisfies a certain specified type (and these types must satisfy obvious coherence conditions). The mismatch between the ``$n$'' in $n$-amalgamation and the dimension of the simplex comes from the fact that there are $n$ vertices in an $(n-1)$-simplex, and it is the types of the vertices that we are trying to amalgamate.
\end{remark}
\bar{b}egin{enumerate}gin{remark}
If $T$ is simple, then $T$ automatically has $n$-amalgamation for $n = 1, 2,$ or $3$: $1$-amalgamation is vacuous, $2$-amalgamation is equivalent to the existence of nonforking extensions, and $3$-amalgamation is by the Independence Theorem \bar{c}ite{KP}. If $T$ is stable, then $T$ has $2$-uniqueness by stationarity.
As well-known, non-simple rosy theory can not have $3$-amalgamation but it may have $n$-amalgamation for all $n\geq 4$ (e.g. the theory of dense linear ordering).
\end{remark}
\bar{b}egin{enumerate}gin{definition}
A theory $T$ (or a complete type $p$) has \emph{$n$-complete amalgamation} (or ``$n$-CA'') if it has $k$-amalgamation for every $k \leq n$.
\end{definition}
Now we can restate Corollary~\ref{Hn_shells} above as:
\bar{b}egin{enumerate}gin{proposition}\langlebel{shellgennew} Assume $T$ has $n$-CA based on $A=\operatorname{acl}(A)$ for $n\geq 2$.
Then $H_{n-1}(T;A)=\{[c]|\ c\bar{m}box{ is an } (n-1)-\bar{m}box{shell over $A$ with support } [n]\}$.
In particular,
if any $(n-1)$-shell is a boundary
then so is any $(n-1)$-cycle.
\end{proposition}
\bar{b}egin{enumerate}gin{corollary}
\langlebel{trivial_homology}
Suppose $n \geq 3$.
Assume $T$ has $n$-CA based on $A=\operatorname{acl}(A)$. Then $H_{n-2}(p) = 0$ for $p{\hbox{\boldmath $\bar \textup{im}ath$}}n S(A)$,
and $H_{n-2}(T;A)=0$.
\end{corollary}
However, the converse of the above corollary is false in general: the theory of the random tetrahedron-free hypergraph does not have $4$-amalgamation, but all of its homology groups are trivial (Example~\ref{tet.free}).
\bar{b}egin{enumerate}gin{corollary}\langlebel{trivialH1}
If $T$ is simple, then $H_1(T; A)=0$ and $H_1(p) = 0$ for any complete type $p$ in $T$.
\end{corollary}
This result is extended to any 1-type in an $o$-minimal theory in Example~\ref{ominh1} below.
\section{Computing $H_2(p)$ (the ``Hurewicz theorem'')}
We assume throughout this section that $T$ is a stable theory and that $p$ is a strong type (without loss of generality, over the empty set). We will prove that the type homology group $H_2(p)$ is isomorphic to a certain automorphism group $\bar{m}athcal{G}amma_2(p)$ defined below. This can be thought of as an analogue of Hurewicz's theorem in algebraic topology, which says that for a path connected topological space $X$, the first homology group $H_1(X)$ is isomorphic to the abelianization of the homotopy group $\bar{p}i_1(X)$. Just as there is a higher-dimensional version of Hurewicz's theorem for $H_n(X)$ under the hypothesis that $X$ is $(n-1)$-connected, we hope that there is a higher-dimensional generalization of our result under the hypothesis that the theory $T$ has $(n+1)$-complete amalgamation. In other words, maybe $n$-CA is analogous to a topological connectedness property.
Throughout this section, ``$\overline{a}$'' denotes the algebraic closure of an element $a$, considered as a possibly infinite ordered \textbf{tuple}, but the choice of ordering will not play any important role in what follows. Implicit in the argument below is that if $a \equiv a_0,$ then there are orderings $\overline{a}, \overline{a_0}$ of their algebraic closures such that $\overline{a} \equiv \overline{a_0}$.
Moreover, $\operatorname{Aut}(A/B)$ denotes the group of
elementary maps from $A$ {\em onto} $A$ fixing $B$ pointwise.
First, suppose that $C = \{a_i : i {\hbox{\boldmath $\bar \textup{im}ath$}}n s\}$ is an independent set of realizations of the type $p$. Pick some $a$ realizing $p$ such that $a \bar{m}athop{\bar{m}athpalette\Ind{}} C$, and let $$\bar{b}ar{w}idetilde{a_s} := \overlineerline{a_s} \bar{c}ap \operatorname{dcl}\left(\bar{b}igcup_{i {\hbox{\boldmath $\bar \textup{im}ath$}}n s} \overlineerline{a, a_{s \setminus \{i\}}}\right).$$ Note that since $T$ is stable, by stationarity, the set $\bar{b}ar{w}idetilde{a}_s$ does not depend on the particular choice of $a$.
Fix some integer $n \geq 2$, and let $\{a_0, \ldots, a_{n-1}\}$ be an independent set of $n+1$ realizations of $p$. Recall our notation that $[k] = \{0, \ldots, k\}$, so that $\bar{b}ar{w}idetilde{a}_{[n-1]} = \bar{b}ar{w}idetilde{a}_{\{0, \ldots, n-1\}}.$ Let $$B_n = \bar{b}igcup_{0 \leq i \leq n-1} \overlineerline{a_{\{0, \ldots, \bar{b}ar{w}idehat{i}, \ldots, n-1\}}}.$$ Finally, we let $\bar{m}athcal{G}amma_n(p) = \operatorname{Aut}(\bar{b}ar{w}idetilde{a}_{[n-1]} / B_n)$.
Note that $\bar{b}ar{w}idetilde{a}_{[n-1]}$ is a subset of $\operatorname{acl}(a_0, \ldots, a_{n-1})$, so $\bar{m}athcal{G}amma_n(p)$ is a quotient of the full automorphism group $\operatorname{Aut}(\overlineerline{a_{[n-1]}} / B_n)$ (namely, the quotient of the subgroup of all automorphisms fixing $\bar{b}ar{w}idetilde{a}_{[n-1]}$ pointwise).
Now we can state the main result of this section:
\bar{b}egin{enumerate}gin{theorem}
\langlebel{hurewicz}
If $T$ is stable, $p$ is stationary, and $(a,b) \bar{m}odels p^{(2)}$, then $H_2(p) \bar{c}ong \bar{m}athcal{G}amma_2(p)$.
\end{theorem}
An immediate consequence of this theorem plus Corollary~\ref{trivial_homology} above is:
\bar{b}egin{enumerate}gin{corollary}
If $p$ is a strong type in a stable theory, then $p$ has $3$-uniqueness (or equivalently, $p$ has $4$-amalgamation) if and only if $H_2(p) \bar{c}ong 0$.
\end{corollary}
\bar{b}egin{enumerate}gin{question}
\langlebel{hurewicz2}
If $T$ is stable with $(n+1)$-complete amalgamation, then is $H_n(p)$ isomorphic to $\bar{m}athcal{G}amma_n(p)$?
\end{question}
\subsection{Preliminaries on definable groupoids}
Here we review some material from \bar{c}ite{GK} and \bar{c}ite{gkk} on definable groupoids that we need for the proof of Theorem~\ref{hurewicz}. We also make a minor correction to a lemma from \bar{c}ite{GK} and set some notation that will be used later. Recall that we assume $T$ is stable.
We know from \bar{c}ite{GK} that in a stable theory, failures of $3$-uniqueness (or equivalently, of $4$-amalgamation) are linked with type-definable connected groupoids which are not retractable. (See that paper for definitions of these terms.) It turns out that the groupoid $\bar{c}G$ associated to such a failure of $3$-uniqueness can even be assumed to have abelian ``vertex groups'' $\textup{Mor}_{\bar{c}G}(a,a)$ (this is proved in Section 2 of \bar{c}ite{gkk}).
Given an $\operatorname{acl}(\emptyset)$-definable connected groupoid $\bar{c}G'$ such that the groups $\bar{c}G'_a := \textup{Mor}_{\bar{c}G'}(a,a)$ are all finite and \emph{abelian}, we can define canonical isomorphisms between any two groups $\bar{c}G'_a$ and $\bar{c}G'_b$ via conjugation by some (any) $h {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}_{\bar{c}G'}(a,b)$. Therefore we can quotient $\bar{b}igcup_{a {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Ob}(\bar{c}G')} \bar{c}G'_a$ by this system of commuting automorphisms to get a {\em binding} group $G'$, and note that $G'$ can be thought of as a subset of $\operatorname{acl}^{eq}(\emptyset)$. In fact, even if the mentioned groupoid $\bar{c}G$ is only type-definable
(more precisely, relatively definable due to the explanation after Claim~\ref{compost}), we can still associate
the binding group $G$ with a subset of $\operatorname{acl}(\emptyset)$: first find a definable connected extension $\bar{c}G'$ of $\bar{c}G$ in which $\bar{c}G$ is a full faithful subcategory, then apply this argument to $\bar{c}G'$. If $h {\hbox{\boldmath $\bar \textup{im}ath$}}n \bar{c}G_a$, let $\left[h\right]_{G'}$ be the corresponding element of $G$ (so identify $G$ and $G'$).
Next we recall from \bar{c}ite{gkk} the definition of a ``full symmetric witness to the failure of $3$-uniqueness.'' For the present paper, we modify the definition slightly so that a full symmetric witness is a tuple $W$ containing a formula $\theta$ witnessing the key property. (Later we will need to keep track of this formula).
\bar{b}egin{enumerate}gin{definition}\langlebel{full_symm_witness}
A \emph{full symmetric witness to non-$3$-uniqueness} (over the set $A$) is a tuple $(a_0, a_1, a_2, f_{01}, f_{12}, f_{02}, \theta(x,y,z))$ such that $a_0, a_1, a_2$ and $f_{01},f_{12}, f_{02}$ are finite tuples, $\theta(x,y,z)$ is a formula over $A$, and:
\bar{b}egin{enumerate}gin{enumerate}
{\hbox{\boldmath $\bar \textup{im}ath$}}tem
$f_{ij} {\hbox{\boldmath $\bar \textup{im}ath$}}n \overlineerline{a}_{ij}$;
{\hbox{\boldmath $\bar \textup{im}ath$}}tem
$f_{01} {\hbox{\boldmath \small $\bar n$}}otin \operatorname{dcl}(\overlineerline{a}_0 \overlineerline{a}_1)$;
{\hbox{\boldmath $\bar \textup{im}ath$}}tem
$a_0a_1f_{01} \equiv_A a_1a_2f_{12} \equiv_A a_0a_2f_{02}$;
{\hbox{\boldmath $\bar \textup{im}ath$}}tem
$f_{01}$ is the unique realization of
$\theta(x, f_{12}, f_{02})$, the element $f_{12}$ is the unique realization of $\theta(f_{01}, y, f_{02})$, and
$f_{02}$ is the unique realization of $\theta(f_{01}, f_{12}, z)$; and
{\hbox{\boldmath $\bar \textup{im}ath$}}tem
$\operatorname{tp}(f_{01}/\overline{a}_0\overline{a}_1)$ is isolated by $\operatorname{tp}(f_{01}/a_0a_1)$.
\end{enumerate}
\end{definition}
The following (proved in \bar{c}ite{gkk}) is the key technical point saying that we have ``enough'' symmetric witnesses:
\bar{b}egin{enumerate}gin{proposition}\langlebel{full symm}
If $T$ does not have $3$-uniqueness, then there is a set $A$ and a full symmetric witness to non-$3$-uniqueness over $A$.
In fact, if $(a_0, a_1, a_2)$ is the beginning of a Morley sequence and $f$ is any element of $\overline{a_{01}} \bar{c}ap \operatorname{dcl}(\overline{a_{02}}, \overline{a_{12}})$ which is not in $\operatorname{dcl}(\overline{a_0}, \overline{a_1})$, then there is some full symmetric witness $(a'_0, a'_1, a'_2, f', g, h, \theta)$ such that $f {\hbox{\boldmath $\bar \textup{im}ath$}}n \operatorname{dcl}(f')$ and $a_i {\hbox{\boldmath $\bar \textup{im}ath$}}n \operatorname{dcl}(a'_i) \subseteq \overline{a_i}$ for $i = 0, 1, 2$.
\end{proposition}
The next lemma states a crucial point in the construction of type-definable groupoids from witnesses to the failure of $3$-uniqueness. This was not isolated as a lemma in \bar{c}ite{GK}, though the idea was there.
\bar{b}egin{enumerate}gin{lemma}
\langlebel{full_symm2}
If $(a_0, a_1, a_2, f_{01}, f_{12}, f_{02}, \theta(x,y,z))$ is a full symmetric witness, and if $f \equiv_{a_0 a_1} f_{01}$ and $g \equiv_{a_1 a_2} f_{12}$, then $$(f, g, \overline{a_0}, \overline{a_1}, \overline{a_2}) \equiv (f_{01}, f_{12}, \overline{a_0}, \overline{a_1}, \overline{a_2}).$$
\end{lemma}
\bar{b}egin{enumerate}gin{proof}
By clause (5) in the definition of a full symmetric witness, $(f, \overline{a_0}, \overline{a_1}) \equiv (f_{01}, \overline{a_0}, \overline{a_1})$ and $(g, \overline{a_1}, \overline{a_2}) \equiv (f_{12}, \overline{a_1}, \overline{a_2})$. It follows (by the stationarity of types over $\overline{a_1}$) that $$(f, g, \overline{a_0}, \overline{a_1}, \overline{a_2}) \equiv (f_{01}, g, \overline{a_0}, \overline{a_1}, \overline{a_2})$$ and $$(f_{01}, g, \overline{a_0}, \overline{a_1}, \overline{a_2}) \equiv (f_{01}, f_{12}, \overline{a_0}, \overline{a_1}, \overline{a_2}),$$ and the lemma follows.
\end{proof}
Given any full symmetric witness to the failure of $3$-uniqueness, we can construct from it a connected, type-definable groupoid:
\bar{b}egin{enumerate}gin{proposition}
\langlebel{groupoid_construction}
Let $W = (a_0, a_1, a_2, f, g, h, \theta(x,y,z))$ be a full symmetric witness (over $\emptyset$). Then from $W$ we can construct a connected groupoid $\bar{c}G_W$ which is type definable over $\operatorname{acl}(\emptyset)$ and has the following properties:
\bar{b}egin{enumerate}gin{enumerate}
{\hbox{\boldmath $\bar \textup{im}ath$}}tem The objects of $\bar{c}G_W$ are the realizations of the type $p = \operatorname{stp}(a_1)$.
{\hbox{\boldmath $\bar \textup{im}ath$}}tem Let $$SW_{a_0, a_1} := \{f' : f' \equiv_{a_0, a_1} f \},$$
There is a bijection $f \bar{m}apsto [f]^{a_0, a_1}_{\bar{c}G_W}$ from $SW_{a_0, a_1}$ onto $\textup{Mor}_{\bar{c}G_W}(a_0, a_1)$ which is definable over $(a_0, a_1)$.
{\hbox{\boldmath $\bar \textup{im}ath$}}tem If $f_0,f_1 {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}_{\bar{c}G}(a_0, a_1)$, then $f_0 \equiv_{a_0, a_1} f_1$.
{\hbox{\boldmath $\bar \textup{im}ath$}}tem The ``vertex groups'' $\textup{Mor}_{\bar{c}G_W}(a,a)$ are finite and abelian.
\end{enumerate}
\end{proposition}
\bar{b}egin{enumerate}gin{proof}
We build $\bar{c}G_W$ using a slight modification of the construction described in subsection~2.2 of \bar{c}ite{GK}. The problem with the construction in that paper is that Remark~2.8 there is incorrect as stated: in general, just because $(a,b,f) \equiv (a_0, a_1, f_{01}) \equiv (b, c, g)$, it does not follow that $(a,b,c,f,g) \equiv (a_0, a_1, a_2, f_{01}, f_{12})$ (if the tuples $a_i$ are not algebraically closed, $f_{01}$ may contain elements of $\operatorname{acl}(a_0) \setminus \operatorname{dcl}(a_0)$, and this could cause $\operatorname{tp}(a, f,g )$ to differ from $\operatorname{tp}(a_0, f_{01}, f_{12})$). However, Lemma~\ref{full_symm2} and the fact that we are using a \emph{full} symmetric witness eliminates this problem. In particular, if $(a,b,f) \equiv (a_0, a_1, f_{01}) \equiv (b, c, g)$, then there is a unique element ``$g \bar{c}irc f$'' such that $\bar{m}odels \theta(f, g, g \bar{c}irc f)$ and $(a, c, g \bar{c}irc f) \equiv (a_0, a_2, f_{02})$.
From here, everything else in the construction of the type-definable groupoid $\bar{c}G = \bar{c}G_W$ in \bar{c}ite{GK} works. Property (1) of the proposition follows directly from the construction, and property (2) is just like Lemma~2.14 of \bar{c}ite{GK}. Because of the definable bijection in (2), any two morphisms in $\textup{Mor}_{\bar{c}G}(a_0, a_1)$ have the same type, yielding (3). Finally, property (4) is Corollary~2.7 of \bar{c}ite{gkk}.
\end{proof}
Next, here is a more detailed version Proposition~2.15 from \bar{c}ite{gkk}, which we will use later.
\bar{b}egin{enumerate}gin{proposition}
\langlebel{corrected2}
Suppose that $(a_0, a_1, a_2, f_{01}, f_{12}, f_{02}, \theta)$ is a full symmetric witness, and $\bar{c}G$ is the associated type-definable groupoid as in Proposition~\ref{groupoid_construction}. If $SW$ is the set $\{f' : \operatorname{tp}(f'/a_0, a_1) = \operatorname{tp}(f_{01} / a_0, a_1)\}$, then there is a group isomorphism $$\bar{p}si^0_{\bar{c}G}: \textup{Mor}_{G}(a_1,a_1) \rightarrow \operatorname{Aut}(SW / a_0, a_1)$$ defined by the rule: if $g {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}_{\bar{c}G}(a_1, a_1)$, then $\bar{p}si^0_{\bar{c}G}(g)$ is the unique element $\sigma {\hbox{\boldmath $\bar \textup{im}ath$}}n \operatorname{Aut}(SW / a_0, a_1)$ which induces the same left action on $\textup{Mor}_{\bar{c}G}(a_0, a_1)$ as left composition by $g$.
Furthermore, the inclusion map $\operatorname{Aut}(SW/ \overline{a_0}, \overline{a_1}) \rightarrow \operatorname{Aut}(SW / a_0, a_1)$ is surjective, so we actually have an isomorphism
$$\bar{p}si_{\bar{c}G}: \textup{Mor}_{G}(a_1,a_1) \rightarrow \operatorname{Aut}(SW / \overline{a_0}, \overline{a_1}).$$
\end{proposition}
\bar{b}egin{enumerate}gin{proof}
The ``Furthermore ...'' clause was not in Proposition~2.15 of \bar{c}ite{gkk}, but it follows from the fact that the witness is fully symmetric: if $f'$ is any element of $SW$, then clause (5) of the definition of a symmetric witness implies that $\operatorname{tp}(f' / \overline{a_0}, \overline{a_1}) = \operatorname{tp}(f_{01} / \overline{a_0}, \overline{a_1})$, and so there is an element $\sigma {\hbox{\boldmath $\bar \textup{im}ath$}}n \operatorname{Aut}(SW / \overline{a_0}, \overline{a_1})$ such that $\sigma(f_{01}) = f'$. This means that there are at least $| \textup{Mor}_{\bar{c}G}(a_1, a_0) |$ different elements in $\operatorname{Aut}(SW / \overline{a_0}, \overline{a_1})$; but, by the first part of the proposition, there are only $| \textup{Mor}_{\bar{c}G}(a_1, a_0) |$ elements in $\operatorname{Aut}(SW / a_0, a_1)$. Since this is a finite set, the injective inclusion map $\operatorname{Aut}(SW/ \overline{a_0}, \overline{a_1}) \rightarrow \operatorname{Aut}(SW / a_0, a_1)$ is surjective.
\end{proof}
\subsection{Proof of Theorem~\ref{hurewicz}}
We assume throughout the proof that $p {\hbox{\boldmath $\bar \textup{im}ath$}}n S(\emptyset)$ and $\operatorname{acl}(\emptyset) = \operatorname{dcl}(\emptyset)$ (since we can add constants for the parameters of $p$ if necessary). It follows directly from the definitions that if $p = \operatorname{tp}(a)$ and $p' = \operatorname{tp}(a')$ where $a$ and $a'$ are interalgebraic, then $H_n(p) = H_n(p')$. Therefore, by Proposition~\ref{full symm} above, we may assume that there are some $(a_0, a_1, a_2)$ realizing $p^{(3)}$ and a full symmetric witness $(a_0, a_1, a_2, f_{01}, f_{12}, f_{02}, \theta(x,y,z))$ to this failure. We pick one such witness which we fix throughout the proof. Note that we assume the $f_{ij}$'s to be finite tuples, and also that there may be more than one such witness (which is the interesting case). We assume that there is \emph{at least} one such witness, since otherwise $H_2(p)$ and $\bar{m}athcal{G}amma_2(p)$ are both trivial.
As already observed in \bar{c}ite{gkk}, the symmetric witnesses in the type $p$ form a directed system. To make this more precise, pick some $(a_0, a_1, a_2)$ realizing $p^{(3)}$ (which we fix for the remainder of the subsection). Now we build a directed system of full symmetric witnesses as follows:
\bar{b}egin{enumerate}gin{claim}
\langlebel{witness system}
There is a directed partially ordered set $\langlengle I, \leq \textup{ran}gle$ and and $I$-indexed collection of symmetric witnesses $\langlengle W_i : i {\hbox{\boldmath $\bar \textup{im}ath$}}n I \textup{ran}gle$ such that for any $i$ and $j$ in $I$:
\bar{b}egin{enumerate}gin{enumerate}
{\hbox{\boldmath $\bar \textup{im}ath$}}tem $W_i = (a^i_0, a^i_1, a^i_2, f_{01}^i, f_{12}^i, f_{02}^i, \theta^*_i(x_i, y_i, z_i))$ is a full symmetric witness to failure of $3$-uniqueness;
{\hbox{\boldmath $\bar \textup{im}ath$}}tem $a^i_0, a^i_1 {\hbox{\boldmath $\bar \textup{im}ath$}}n \operatorname{dcl}(f_{01}^i)$;
{\hbox{\boldmath $\bar \textup{im}ath$}}tem if $i \leq j$, then $f_{01}^i {\hbox{\boldmath $\bar \textup{im}ath$}}n \operatorname{dcl}(f_{01}^j)$, $a^i_0 {\hbox{\boldmath $\bar \textup{im}ath$}}n \operatorname{dcl}(a^j_0) \subseteq \overline{a_0}$, and $a^i_0 a^j_0 \equiv a^i_1 a^j_1 \equiv a^i_2 a^j_2$;
\end{enumerate}
and satisfying the maximality conditions $$\bar{b}ar{w}idetilde{a_{\{0,1\}}} = \operatorname{dcl}\left(\bar{b}igcup_{i {\hbox{\boldmath $\bar \textup{im}ath$}}n I} f^i_{01} \right)$$ and $$\overline{a_0} = \operatorname{dcl} \left(\bar{b}igcup_{i {\hbox{\boldmath $\bar \textup{im}ath$}}n I} a^i_0 \right) .$$
\end{claim}
\bar{b}egin{enumerate}gin{proof}
We will build the partial ordering $\langlengle I, \leq \textup{ran}gle$ as the union of a countable chain of partial orderings $I_0 \subseteq I_1 \subseteq \ldots$ such that for any $i, j {\hbox{\boldmath $\bar \textup{im}ath$}}n I_n$ there is a $k {\hbox{\boldmath $\bar \textup{im}ath$}}n I_{n+1}$ such that $i \leq k$ and $j \leq k$. Then the partial ordering $I = \bar{b}igcup_{n {\hbox{\boldmath $\bar \textup{im}ath$}}n \omega} I_n$ will be directed.
First, let
$$W^0_i = \langlengle a^i_0, a^i_1, a^i_2, f_{01}^i, f_{12}^i, f_{02}^i, \theta^*_i(x_i, y_i, z_i) : i {\hbox{\boldmath $\bar \textup{im}ath$}}n I_0 \textup{ran}gle$$
be any collection of full symmetric witnesses large enough to satisfy the two maximality conditions in the statement of the Claim, where $I_0$ is a trivial partial ordering in which no two distinct elements are comparable. For the induction step, suppose that we have the partial ordering $I_n$ (for some $n {\hbox{\boldmath $\bar \textup{im}ath$}}n \omega$) and full symmetric witnesses $(a^i_0, \ldots, \theta^*_i(x_i, y_i, z_i))$ for each $i {\hbox{\boldmath $\bar \textup{im}ath$}}n I_n$. First, we can build a partial ordering $I_{n+1}$ by adding one new point immediately above every pair of points in $I_n$ and such that any two new points in $I_{n+1} \setminus I_n$ are incomparable. Then by Proposition~\ref{full symm}, there are corresponding full symmetric witnesses $(a^i_0, \ldots, \theta^*_i)$ for each $i {\hbox{\boldmath $\bar \textup{im}ath$}}n I_{n+1} \setminus I_n$ such that if $j$ and $k$ are less than or equal to $i$, then $f^j_{01}, f^k_{01} {\hbox{\boldmath $\bar \textup{im}ath$}}n \operatorname{dcl}(f^i_{01})$ and $a^j_0, a^k_0 {\hbox{\boldmath $\bar \textup{im}ath$}}n \operatorname{dcl}(a^i_0)$. Similarly, we can ensure condition (2) (that $a^i_0, a^i_1 {\hbox{\boldmath $\bar \textup{im}ath$}}n \operatorname{dcl}(f^i_{01})$) for the new symmetric witnesses.
\end{proof}
Let $p_i = \operatorname{stp}(a^i_0)$ and $\bar{c}G^*_i$ be the type-definable groupoid constructed from the full symmetric witness $W_i$ as in Proposition~\ref{groupoid_construction} above. So $\textup{Ob}(\bar{c}G_i) = p_i(\operatorname{\bar{m}athfrak{C}})$ and the groups $\textup{Mor}_{\bar{c}G^*_i}(a^i_0,a^i_0)$ are finite and abelian, and we have the corresponding finite abelian groups $G^*_i$. As explained above, can (and will) assume that the groups $G^*_i$ are subsets of $\operatorname{acl}(\emptyset)$.
For any $i {\hbox{\boldmath $\bar \textup{im}ath$}}n I$, let $SW_i$ be the set of all realizations of $\operatorname{tp}(f_{01}^i / a^i_0, a^i_1)$ (which is a finite set). If $(a,b) \bar{m}odels p_i^{(2)}$, let $SW(a,b)$ be the image of $SW_i$ under an automorphism of $\operatorname{\bar{m}athfrak{C}}$ that maps $(a^i_0, a^i_1)$ to $(a, b)$. Recall from Proposition~\ref{groupoid_construction} that we have a definable map $f \bar{m}apsto \left[ f \right]^{a,b}_{\bar{c}G_i}$ from $SW(a,b)$ onto $\textup{Mor}_{\bar{c}G_i}(a,b)$, from which we can define an inverse map $g \bar{m}apsto \langlengle g \textup{ran}gle^{a,b}_{\bar{c}G_i}$ from $\textup{Mor}_{\bar{c}G_i}(a,b)$ to $SW(a,b)$. For convenience, we will write these maps as ``$\left[ \bar{c}dot \right]^{a,b}_i$'' and ``$\langlengle \bar{c}dot \textup{ran}gle^{a,b}_i$,'' or even just ``$\left[ \bar{c}dot \right]_i$'' and ``$\langlengle \bar{c}dot \textup{ran}gle_i$'' when $(a,b)$ is clear from context.
\bar{b}egin{enumerate}gin{lemma}
\langlebel{commuting_pi}
There are systems of relatively $\emptyset$-definable functions $\langlengle \bar{p}i_{j,i} : i \leq j, j {\hbox{\boldmath $\bar \textup{im}ath$}}n I \textup{ran}gle$ and $\langlengle \tau_{j,i} : i \leq j, j {\hbox{\boldmath $\bar \textup{im}ath$}}n I \textup{ran}gle $ (that is, they are the intersection of an $\emptyset$-definable relation with the product of their domain and range) such that whenever $i \leq j$,
\bar{b}egin{enumerate}gin{enumerate}
{\hbox{\boldmath $\bar \textup{im}ath$}}tem $\tau_{j,i} : p_j(\operatorname{\bar{m}athfrak{C}}) \rightarrow p_i(\operatorname{\bar{m}athfrak{C}})$,
{\hbox{\boldmath $\bar \textup{im}ath$}}tem $\bar{p}i_{j,i} : \bar{b}igcup_{(a,b) \bar{m}odels p_j^{(2)}} SW(a,b) \rightarrow SW(\tau_{j,i}(a), \tau_{j,i}(b))$,
{\hbox{\boldmath $\bar \textup{im}ath$}}tem $\tau_{j,i}(a^j_0) = a^i_0$,
{\hbox{\boldmath $\bar \textup{im}ath$}}tem $\tau_{j,i}(a^j_1) = a^i_1$,
{\hbox{\boldmath $\bar \textup{im}ath$}}tem $\bar{p}i_{j,i}$ maps $SW_j$ onto $SW_i$, and
{\hbox{\boldmath $\bar \textup{im}ath$}}tem $\bar{p}i_{j,i}(f^j_{01}) = f^i_{01}$,
\end{enumerate}
and whenever $i \leq j \leq k$,
\bar{b}egin{enumerate}gin{enumerate}[resume]
{\hbox{\boldmath $\bar \textup{im}ath$}}tem $\tau_{j,i} \bar{c}irc \tau_{k,j} = \tau_{k,i}$ and
{\hbox{\boldmath $\bar \textup{im}ath$}}tem $\bar{p}i_{j,i} \bar{c}irc \bar{p}i_{k,j} = \bar{p}i_{k,i}$.
\end{enumerate}
\end{lemma}
\bar{b}egin{enumerate}gin{proof}
First, the maps $\tau_{j,i}$ can be constructed satisfying (1), (3), and (4) using the facts that $a^i_0 {\hbox{\boldmath $\bar \textup{im}ath$}}n \operatorname{dcl}(a^j_0),$ $a^i_0 {\hbox{\boldmath $\bar \textup{im}ath$}}n \operatorname{dcl}(a^j_1)$, and $a^i_0 a^j_0 \equiv a^i_1 a^j_1$ (from clause (3) of Claim~\ref{witness system}). Now if $i \leq j \leq k$, since $\tau_{k,i}(x) = \tau_{j,i} \bar{c}irc \tau_{k,j} (x)$ is true for $x = a_k$, this holds for \emph{every} $x$ in the domain of $\tau_{k,i}$ (because the domain is a complete type), and so (7) holds.
If $i \leq j$, then since $f^i_{01} {\hbox{\boldmath $\bar \textup{im}ath$}}n \operatorname{dcl}(f^j_{01})$, we can pick a relatively definable map $\bar{p}i_{j,i}$ such that $\bar{p}i_{j,i}(f^j_{01}) = f^i_{01}$. As before, if $i \leq j \leq k$, since $\bar{p}i_{k,i}(x) = \bar{p}i_{j,i} \bar{c}irc \bar{p}i_{k,j}(x)$ holds for $x = f^k_{01}$, it holds for any $x$ in any of the sets $SW(a,b)$ for $(a,b) \bar{m}odels p^{(2)}$, so (8) holds.
\end{proof}
Ideally, we would like the functions $\bar{p}i_{j,i}$ and $\tau_{j,i}$ of Lemma~\ref{commuting_pi} to induce a commuting system of functors $F_{j,i} : \bar{c}G^*_j \rightarrow \bar{c}G^*_i$, which we could use to construct and inverse limit $\bar{c}G$ of $\langlengle \bar{c}G^*_i : i {\hbox{\boldmath $\bar \textup{im}ath$}}n I \textup{ran}gle$. This is essentially what we do, and we will then show that the group $\textup{Mor}_{\bar{c}G}(\overline{a_0}, \overline{a_0})$ is isomorphic to both $H_2(p)$ and $\bar{m}athcal{G}amma_2(p)$. However, first we need to modify the formulas $\theta^*_i$ slightly for this to be true.
The key to making all of this work is the following technical lemma.
\bar{b}egin{enumerate}gin{lemma}
\langlebel{theta_tweak}
There is a family of formulas $\langlengle \theta_i(x_i, y_i, z_i) : i {\hbox{\boldmath $\bar \textup{im}ath$}}n I\textup{ran}gle$ such that
\bar{b}egin{enumerate}gin{enumerate}
{\hbox{\boldmath $\bar \textup{im}ath$}}tem $W_i$ is still a full symmetric witness with $\theta^*_i(x_i, y_i, z_i)$ replaced by $\theta_i(x_i, y_i, z_i)$ and $f_{02}^i$ replaced by another element of $SW(a^i_0,a^i_2),$ and
{\hbox{\boldmath $\bar \textup{im}ath$}}tem whenever $i \leq j$, $f {\hbox{\boldmath $\bar \textup{im}ath$}}n SW(a^j_0,a^j_1)$, $g {\hbox{\boldmath $\bar \textup{im}ath$}}n SW(a^j_1,a^j_2)$, and $h {\hbox{\boldmath $\bar \textup{im}ath$}}n SW(a^j_0,a^j_2)$, then
$$\bar{m}odels \theta_j(f, g, h) \rightarrow \theta_i(\bar{p}i_{j,i}(f), \bar{p}i_{j,i}(g), \bar{p}i_{j,i}(h)).$$
\end{enumerate}
\end{lemma}
\bar{b}egin{enumerate}gin{proof}
Recall from above that $(a_0, a_1, a_2)$ realizes $p^{(3)}$. We use Zorn's Lemma to find a maximal subset $J \subseteq I$ and formulas $\theta_j(x_j, y_j, z_j)$ for each $j {\hbox{\boldmath $\bar \textup{im}ath$}}n J$ satisfying the following properties:
\bar{b}egin{enumerate}gin{enumerate}
\setcounter{enumi}{2}
{\hbox{\boldmath $\bar \textup{im}ath$}}tem For every $j {\hbox{\boldmath $\bar \textup{im}ath$}}n J$, there are elements $f_j$, $g_j,$ and $h_j$ such that $(a^j_0, a^j_1, a^j_2, f_j, g_j, h_j, \theta_j(x_j, y_j, z_j))$ is a full symmetric witness; and
{\hbox{\boldmath $\bar \textup{im}ath$}}tem If $j_1, \ldots, j_n {\hbox{\boldmath $\bar \textup{im}ath$}}n J$ and $(a^{j_s}_0, a^{j_s}_1, a^{j_s}_2, f_{j_s}, g_{j_s}, h_{j_s}, \theta_{j_s})$ is a full symmetric witness for $s = 1, \ldots, n$, and if $f_{j_1} \ldots f_{j_n} \equiv g_{j_1} \ldots g_{j_n}$, then $f_{j_1} \ldots f_{j_n} \equiv h_{j_1} \ldots h_{j_n}$.
\end{enumerate}
\bar{b}egin{enumerate}gin{claim}
$J = I$.
\end{claim}
\bar{b}egin{enumerate}gin{proof}
Suppose towards a contradiction that there is some $k {\hbox{\boldmath $\bar \textup{im}ath$}}n I \setminus J$. Let $F_J = \langlengle f^\bar{b}ar{a}lpha \textup{ran}gle$ be a (possibly infinite) tuple listing every element of $\bar{b}igcup_{j {\hbox{\boldmath $\bar \textup{im}ath$}}n J} SW(a^j_0, a^j_1)$, and let $a^J_i$ (for $i {\hbox{\boldmath $\bar \textup{im}ath$}}n \{0,1,2\}$) be a tuple listing $\{a^j_i : j {\hbox{\boldmath $\bar \textup{im}ath$}}n J\}$, ordered the same way as $F_J$. Pick $f_k {\hbox{\boldmath $\bar \textup{im}ath$}}n SW(a^k_0, a^k_1)$, and then pick $G_J = \langlengle g^\bar{b}ar{a}lpha \textup{ran}gle$ and $g_k$ such that $F_J f_k a^J_0 a^J_1 \equiv G_J g_k a^J_1 a^J_2$. Note that $g^\bar{b}ar{a}lpha {\hbox{\boldmath $\bar \textup{im}ath$}}n SW(a^j_1, a^j_2)$. Next pick a tuple $H_J = \langlengle h^\bar{b}ar{a}lpha \textup{ran}gle$ such that if $f^\bar{b}ar{a}lpha {\hbox{\boldmath $\bar \textup{im}ath$}}n SW(a^j_0, a^j_1)$, then $\bar{m}odels \theta_j(f^\bar{b}ar{a}lpha, g^\bar{b}ar{a}lpha, h^\bar{b}ar{a}lpha)$. The element $h_j$ is well-defined because if it happens that $f^\bar{b}ar{a}lpha$ is also in $SW(a^{j'}_0, a^{j'}_1)$ for some $j' {\hbox{\boldmath \small $\bar n$}}eq j$, and if we let $h'$ be the unique element such that $\bar{m}odels \theta_{j'}(f^\bar{b}ar{a}lpha, g^\bar{b}ar{a}lpha, h')$, then by property~(4), the fact that $f^\bar{b}ar{a}lpha f^\bar{b}ar{a}lpha \equiv g^\bar{b}ar{a}lpha g^\bar{b}ar{a}lpha$ implies that $f^\bar{b}ar{a}lpha f^\bar{b}ar{a}lpha \equiv h^\bar{b}ar{a}lpha h'$, and so $h' = h^\bar{b}ar{a}lpha$.
By the assumption (4) on the set $J$, $F_J \equiv H_J$. Finally, pick an element $h_k$ such that $F_J f_k \equiv H_J h_k$. By Corollary~2.14 of \bar{c}ite{gkk}, there is a formula $\theta_k$ such that $(a^k_0, a^k_1, a^k_2, f_k, g_k, h_k, \theta_k)$ is a full symmetric witness.
We claim that $J \bar{c}up \{k\}$ with $\theta_k$ satisfies condition (4) above, contradicting the maximality condition on the set $J$. Indeed, suppose that $j_1, \ldots, j_n {\hbox{\boldmath $\bar \textup{im}ath$}}n J$, and the tuples $$(a^{j_s}_0, a^{j_s}_1, a^{j_s}_2, f_{j_s}, g_{j_s}, h_{j_s}, \theta_{j_s})$$ (for $s = 1, \ldots, n$) and $$(a^k_0, a^k_1, a^k_2, f'_k, g'_k, h'_k, \theta_k)$$ are full symmetric witnesses, and that $f_{j_1} \ldots f_{j_n} f'_k \equiv g_{j_1} \ldots g_{j_n} g'_k.$ By the stationarity of $\operatorname{tp}(f'_k / \overline{a^1})$ and $\operatorname{tp}(g'_k / \overline{a^1})$, there is a $\sigma {\hbox{\boldmath $\bar \textup{im}ath$}}n \operatorname{Aut}(\operatorname{\bar{m}athfrak{C}} / \overline{a^0}, \overline{a^1}, \overline{a^2})$ such that $\sigma(f'_k) = f_k$ and $\sigma(g'_k) = g_k$ for the $f_k$ and $g_k$ from the previous paragraph. By the same argument and using the fact that $F_J \equiv G_J$, we can also assume that if $\sigma ((f_{j_1}, \ldots, f_{j_n})) = (f^{\bar{b}ar{a}lpha_1}, \ldots, f^{\bar{b}ar{a}lpha_n})$, then $\sigma((g_{j_1}, \ldots, g_{j_n}) = (g^{\bar{b}ar{a}lpha_1}, \ldots, g^{\bar{b}ar{a}lpha_n})$ (that is, the two tuples $(f_{j_1}, \ldots, f_{j_n})$ and $(g_{j_1}, \ldots, g_{j_n})$ map to corresponding subtuples of $F_J$ and $G_J$). It follows that $\sigma(h'_k) = h_k$ and $\sigma(h_{j_s}) = h^{\bar{b}ar{a}lpha_s}$ for each $s$ between $1$ and $n$. By our construction, $f^{\bar{b}ar{a}lpha_1} \ldots f^{\bar{b}ar{a}lpha_n} f_k \equiv h^{\bar{b}ar{a}lpha_1} \ldots h^{\bar{b}ar{a}lpha_n} h_k$, and so by taking preimages under $\sigma$, we get that $f_{j_1} \ldots f_{j_n} f'_k \equiv h_{j_1} \ldots h_{j_n} h'_k$.
\end{proof}
Finally, we check that condition (2) of the lemma holds for our new formulas $\theta_i$. Suppose that $i \leq j$, $f {\hbox{\boldmath $\bar \textup{im}ath$}}n SW(a^j_0, a^j_1)$, $g {\hbox{\boldmath $\bar \textup{im}ath$}}n SW(a^j_1, a^j_2)$, $h {\hbox{\boldmath $\bar \textup{im}ath$}}n SW(a^j_0, a^j_2)$, and $\bar{m}odels \theta_j(f, g, h)$. Let $f_0 = \bar{p}i_{j,i}(f)$, and pick $g_0$ such that $f f_0 \equiv g g_0$. Then $g_0 = \bar{p}i_{j,i}(g)$. Finally, let $h_0$ be the unique element such that $\bar{m}odels \theta_i(f_0, g_0, h_0)$. By condition~(4) above, $h h_0 \equiv f f_0$, and so $h_0 = \bar{p}i_{j,i}(h)$. Thus $\bar{m}odels \theta_i(\bar{p}i_{j,i}(f), \bar{p}i_{j,i}(g), \bar{p}i_{j,i}(h))$ as desired.
\end{proof}
For each $i {\hbox{\boldmath $\bar \textup{im}ath$}}n I$, let $\bar{c}G_i$ be the type-definable groupoid obtained from the symmetric witness $W_i$ with the modified formula $\theta_i$ from Lemma~\ref{theta_tweak}. Once again, the groups $\textup{Mor}_{\bar{c}G_i}(a,a)$ are finite and abelian for any $a {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Ob}(\bar{c}G_i)$, so we have the corresponding finite abelian groups $G_i$ which we consider as subsets of $\operatorname{acl}(\emptyset)$.
\bar{b}egin{enumerate}gin{lemma}
\langlebel{coherence}
If $i \leq j {\hbox{\boldmath $\bar \textup{im}ath$}}n I$, $(a,b,c) \bar{m}odels p_j^{(3)}$, $f {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}_{\bar{c}G_j}(a,b)$, and $g {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}_{\bar{c}G_j}(b,c)$, then
$$\left[ \bar{p}i_{j,i}(\langlengle g \bar{c}irc f \textup{ran}gle_j ) \right]_i = \left[ \bar{p}i_{j,i}(\langlengle g \textup{ran}gle_j) \right]_i \bar{c}irc \left[ \bar{p}i_{j,i}(\langlengle f \textup{ran}gle_j ) \right]_i $$
(where $\bar{c}irc$ denotes composition in the groupoids $\bar{c}G_j$ and $\bar{c}G_i$).
\end{lemma}
\bar{b}egin{enumerate}gin{proof}
By Proposition~2.12 of \bar{c}ite{gkk}, $\theta_j$ defines groupoid composition between generic triples of objects in $\bar{c}G_j$, so $$\bar{m}odels \theta_j (\langlengle f \textup{ran}gle_j, \langlengle g \textup{ran}gle_j, \langlengle g \bar{c}irc f \textup{ran}gle_j).$$ So by Lemma~\ref{theta_tweak}, $$\bar{m}odels \theta_i ( \bar{p}i_{j,i}(\langlengle f \textup{ran}gle_j), \bar{p}i_{j,i} (\langlengle g \textup{ran}gle_j), \bar{p}i_{j,i}(\langlengle g \bar{c}irc f \textup{ran}gle_j) ).$$ By Proposition~2.12 again, the Lemma follows.
\end{proof}
If $i \leq j {\hbox{\boldmath $\bar \textup{im}ath$}}n I$ and $(a,b) \bar{m}odels p_j^{(2)}$, then because $SW(\tau_{j,i}(a), \tau_{j,i}(b)) \subseteq \operatorname{dcl}(SW(a,b))$, we have a canonical surjective group map $$\rho^{a,b}_{j,i} : \operatorname{Aut}(SW(a,b) / \overline{a}, \overline{b}) \rightarrow \operatorname{Aut}(SW(\tau_{j,i}(a), \tau_{j,i}(b)) / \overline{a}, \overline{b}),$$ and these maps satisfy the coherence condition that $\rho^{a,b}_{k,i} = \rho^{a,b}_{j,i} \bar{c}irc \rho^{a,b}_{k,j}$ whenever $i \leq j \leq k$. We will write ``$\rho_{j,i}$'' for $\rho^{a,b}_{j,i}$ if $(a,b)$ is clear from context.
For every $i {\hbox{\boldmath $\bar \textup{im}ath$}}n I$, we also have a group isomorphism $\bar{p}si_i : \textup{Mor}_{\bar{c}G_i}(a^i_1, a^i_1) \rightarrow \operatorname{Aut}(SW_i / \overline{a_0}, \overline{a_1})$ as in Proposition~\ref{corrected2} above.
The following is similar to Claim~2.17 of \bar{c}ite{gkk}, except that here we have expanded this to a system of \emph{groupoid} maps.
\bar{b}egin{enumerate}gin{lemma}
\langlebel{chi}
For every $i \leq j {\hbox{\boldmath $\bar \textup{im}ath$}}n I$, we define a map $\bar{c}hi_{j,i} : \bar{c}G_j \rightarrow \bar{c}G_i$ by the rules:
\bar{b}egin{enumerate}gin{enumerate}
{\hbox{\boldmath $\bar \textup{im}ath$}}tem if $a {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Ob}(\bar{c}G_j)$, then $\bar{c}hi_{j,i}(a) = \tau_{j,i}(a)$; and
{\hbox{\boldmath $\bar \textup{im}ath$}}tem if $f {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}_{\bar{c}G_j}(a,b)$, $c \bar{m}odels p_j | (a,b)$, and $f = g \bar{c}irc h$ for some $g {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}_{\bar{c}G_j}(c,b)$ and $h {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}_{\bar{c}G_j}(a,c)$, then $$\bar{c}hi_{j,i}(f) = \left[ \bar{p}i_{j,i}(\langlengle h \textup{ran}gle_j) \right]_i \bar{c}irc \left[ \bar{p}i_{j,i}(\langlengle g \textup{ran}gle_j) \right]_i.$$
\end{enumerate}
Then the maps $\bar{c}hi_{j,i}$ satisfy:
\bar{b}egin{enumerate}gin{enumerate}
\setcounter{enumi}{2}
{\hbox{\boldmath $\bar \textup{im}ath$}}tem $\bar{c}hi_{j,i}$ is a well-defined functor;
{\hbox{\boldmath $\bar \textup{im}ath$}}tem $\bar{c}hi_{j,i}$ is \emph{full}: every morphism in $\textup{Mor}(\bar{c}G_i)$ is in the image of $\bar{c}hi_{j,i}$;
{\hbox{\boldmath $\bar \textup{im}ath$}}tem $\bar{c}hi_{j,i}$ is type-definable over $\operatorname{acl}(\emptyset)$;
{\hbox{\boldmath $\bar \textup{im}ath$}}tem if $(a,b) \bar{m}odels p_j^{(2)}$ and $f {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}_{\bar{c}G_j}(a,b)$, then the formula for $\bar{c}hi_{j,i}(f)$ simplifies to $$\bar{c}hi_{j,i}(f) = \left[ \bar{p}i_{j,i}(\langlengle f \textup{ran}gle_j)\right]_i ;$$
{\hbox{\boldmath $\bar \textup{im}ath$}}tem $\bar{c}hi_{k,i} = \bar{c}hi_{j,i} \bar{c}irc \bar{c}hi_{k,j}$ whenever $i \leq j \leq k$; and
{\hbox{\boldmath $\bar \textup{im}ath$}}tem for any $i \leq j$, the following diagram commutes: \bar{b}igskip
$\bar{b}egin{enumerate}gin{CD}
\textup{Mor}_{\bar{c}G_j}(a^j_1,a^j_1) @>\bar{c}hi_{j,i}>> \textup{Mor}_{\bar{c}G_i}(a^i_1,a^i_1)\\
@VV\bar{p}si_jV @VV\bar{p}si_iV\\
\operatorname{Aut}(SW_j / \overline{a_0}, \overline{a_1}) @>\rho_{j,i}>> \operatorname{Aut}(SW_i / \overline{a_0}, \overline{a_1})
\end{CD}$
\end{enumerate}
\end{lemma}
\bar{b}egin{enumerate}gin{proof}
Suppose that $f {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}_{\bar{c}G_j}(a,b)$. To check that $\bar{c}hi_{j,i}(f)$ is well-defined (and does not depend on the choices of $c, g,$ and $h$), first note that given $c \bar{m}odels p_j | (a,b)$ and morphisms $g, h$ as in (2), the morphism $h$ is uniquely determined from $f$ and $g$, and for any other $g' {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}_{\bar{c}G_j}(c,b)$, $\operatorname{tp}(f,g/a,b,c) = \operatorname{tp}(f,g'/a,b,c)$ (by Lemma~\ref{full_symm2}). So the choices of $f$ and $g$ do not matter once we have picked $c$, and the choice of $c$ does not matter by the stationarity of $p_j$.
To show that $\bar{c}hi_{j,i}$ is a functor, suppose that $a, b,$ and $c$ realize $p_j$, $f {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}_{\bar{c}G_j}(a,b)$, and $g {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}_{\bar{c}G_j}(b,c)$. To compute the images of $f$ and $g$, we pick $(d,e) \bar{m}odels p_{j}^{(2)} | (a,b,c)$ and $f_0 {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}_{\bar{c}G_j}(a,d), f_1 {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}_{\bar{c}G_j}(d,b)$, $g_0 {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}_{\bar{c}G_j}(b,e),$ and $g_1 {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}_{\bar{c}G_j}(e,c)$ such that $f = f_1 \bar{c}irc f_0$ and $g = g_1 \bar{c}irc g_0$. Then by the definition given in (2) of the Lemma,
$$\bar{c}hi_{j,i}(f) = \left[ \bar{p}i_{j,i}(\langlengle g_1 \bar{c}irc g_0 \bar{c}irc f_1 \textup{ran}gle_j) \right]_i \bar{c}irc \left[ \bar{p}i_{j,i}(\langlengle f_0 \textup{ran}gle_j) \right]_i.$$
By Lemma~\ref{coherence} twice, this equals
$$\left[ \bar{p}i_{j,i}(\langlengle g_1 \textup{ran}gle_j ) \right]_i \bar{c}irc \left[ \bar{p}i_{j,i}(\langlengle g_0 \textup{ran}gle_j)\right]_i \bar{c}irc \left[ \bar{p}i_{j,i}(\langlengle f_1 \textup{ran}gle_j)\right]_i \bar{c}irc \left[ \bar{p}i_{j,i}(\langlengle f_0 \textup{ran}gle_j)\right]_i.$$
But the composition of the first two terms above equals $\bar{c}hi_{j,i}(g)$ and the composition of the third and fourth terms equals $\bar{c}hi_{j,i}(f)$, so $\bar{c}hi_{j,i}(g \bar{c}irc f) = \bar{c}hi_{j,i}(g) \bar{c}irc \bar{c}hi_{j,i}(f)$.
Suppose that $a, b {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Ob}(\bar{c}G_i)$ and $f {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}_{\bar{c}G_i}(a,b)$. Pick some $c \bar{m}odels p_i | (a,b)$, and pick $g {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}_{\bar{c}G_i}(c,b)$ and $h {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}_{\bar{c}G_i}(a,c)$ such that $f = g \bar{c}irc h$. Since $$(\langlengle g \textup{ran}gle_i, c, b) \equiv (f^i_{01}, a^i_0, a^i_1) \equiv (\langlengle h \textup{ran}gle_i, a, c),$$ we can find elements $g'$ and $h'$ such that $\bar{p}i_{j,i}(g') = \langlengle g \textup{ran}gle_i$ and $\bar{p}i_{j,i}(h') = \langlengle h \textup{ran}gle_i$. Let $f^* = \left[ g' \right]_j \bar{c}irc \left[ h' \right]_j$. Unwinding the definitions, we see that $$\bar{c}hi_{j,i}(f^*) = \left[ \bar{p}i_{j,i}(g')\right]_i \bar{c}irc \left[ \bar{p}i_{j,i}(h')\right]_i = \left[\langlengle g \textup{ran}gle_i \right]_i \bar{c}irc \left[ \langlengle h \textup{ran}gle_i \right]_i = g \bar{c}irc h = f.$$ This establishes that the functor $\bar{c}hi_{j,i}$ is full.
The fact that $\bar{c}hi_{j,i}$ is type-definable is simply by the definability of types in stable theories, and in fact the action of $\bar{c}hi_{j,i}$ on the objects and morphisms of $\bar{c}G_j$ is given by the intersection of a \emph{definable} set with the type-definable sets $\textup{Ob}(\bar{c}G_j)$ and $\textup{Mor}(\bar{c}G_j)$.
The formula (6) follows directly from the definition of $\bar{c}hi_{j,i}(f)$ in (2) and Lemma~\ref{coherence}.
Next we prove (7). Suppose that $i \leq j \leq k$. If $a {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Ob}(\bar{c}G_k)$, then $\bar{c}hi_{j,i} \bar{c}irc \bar{c}hi_{k,j}(a) = \tau_{j,i}(\tau_{k,j}(a)) = \tau_{k,i}(a) = \bar{c}hi_{k,i}(a)$. If $a, b, c {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Ob}(\bar{c}G_k)$ and $f = g \bar{c}irc h$ are in (2) of the Lemma (with $j$ replaced by $k$), then by the definition of the $\bar{c}hi$ maps,
$$\bar{c}hi_{j,i} \bar{c}irc \bar{c}hi_{k,j} (f) = \bar{c}hi_{j,i} \left(\left[ \bar{p}i_{k,j}(\langlengle h \textup{ran}gle_k)\right]_j \bar{c}irc \left[ \bar{p}i_{k,j}(\langlengle g \textup{ran}gle_k)\right]_j \right)$$ $$= \left[\bar{p}i_{j,i}\left(\langlengle \left[ \bar{p}i_{k,j}(\langlengle h \textup{ran}gle_k)\right]_j \textup{ran}gle_j\right) \right]_i \bar{c}irc \left[ \bar{p}i_{j,i}\left(\langlengle\left[ \bar{p}i_{k,j}(\langlengle g \textup{ran}gle_k)\right]_j \textup{ran}gle_j\right)\right]_i$$ $$=\left[ \bar{p}i_{j,i} \left(\bar{p}i_{k,j}(\langlengle h \textup{ran}gle_k) \right) \right]_i \bar{c}irc \left[ \bar{p}i_{j,i} \left(\bar{p}i_{k,j}(\langlengle g \textup{ran}gle_k) \right) \right]_i $$ $$= \left[ \bar{p}i_{k,i}(\langlengle h \textup{ran}gle_k) \right]_i \bar{c}irc \left[ \bar{p}i_{k,i}(\langlengle g \textup{ran}gle_k) \right]_i = \bar{c}hi_{k,i}(f).$$
Finally, we check (8). Suppose $i \leq j$ and $f {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}_{\bar{c}G_j}(a^j_1, a^j_1)$. To show that $\bar{p}si_i (\bar{c}hi_{j,i}(f)) = \rho_{j,i} ( \bar{p}si_j(f))$, we pick some arbitrary $k_0 {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}_{\bar{c}G_i}(a^i_0, a^i_1)$ and show that
\setcounter{equation}{8}
\bar{b}egin{enumerate}gin{equation}\langlebel{8} \left[\bar{p}si_i (\bar{c}hi_{j,i}(f))\right](k_0) = \left[\rho_{j,i} ( \bar{p}si_j(f))\right](k_0) . \end{equation}
On the one hand, by definition of $\bar{p}si_i$,
$$\left[\bar{p}si_i (\bar{c}hi_{j,i}(f))\right](k_0) = \bar{c}hi_{j,i}(f) \bar{c}irc k_0.$$
To compute the right-hand side of equation~\ref{8}, pick some $k {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}_{\bar{c}G_j}(a^j_0, a^j_1)$ such that $\left[ \bar{p}i_{j,i}(\langlengle k \textup{ran}gle_j) \right]_i = k_0$. Then $$\left[\bar{p}si_j(f)\right] (k) = f \bar{c}irc k,$$ and $\rho_{j,i}(\bar{p}si_j(f))$ must move $k_0 = \left[ \bar{p}i_{j,i}(\langlengle k \textup{ran}gle_j) \right]_i$ to the element which is defined from $\left[\bar{p}si_j(f)\right](k)$ in the same way that $k_0$ is defined from $k$, so $$\left[\rho_{j,i} ( \bar{p}si_j(f))\right](k_0) = \left[\bar{p}i_{j,i}(\langlengle f \bar{c}irc k \textup{ran}gle_j) \right]_i.$$ By (6) and the functoriality of $\bar{c}hi_{j,i}$, $$\left[\rho_{j,i} ( \bar{p}si_j(f))\right](k_0) = \bar{c}hi_{j,i}(f \bar{c}irc k) = \bar{c}hi_{j,i}(f) \bar{c}irc \bar{c}hi_{j,i}(k) = \bar{c}hi_{j,i}(f) \bar{c}irc \left[ \bar{p}i_{j,i}(\langlengle k \textup{ran}gle_j)\right]_i$$ $$= \bar{c}hi_{j,i}(f) \bar{c}irc k_0.$$ So both sides of equation~\ref{8} equal $\bar{c}hi_{j,i}(f) \bar{c}irc k_0$, and we are done.
\end{proof}
Finally, we define maps on the $p$-simplices and homology groups. Throughout, we will work with the \emph{set} homology group (and set-simplices, et cetera) for convenience.
First, for every $i {\hbox{\boldmath $\bar \textup{im}ath$}}n I$, we pick an arbitrary ``selection function'' $\bar{b}ar{a}lpha^0_i: S_0 \bar{c}C(p) \rightarrow p_i(\operatorname{\bar{m}athfrak{C}})$ such that $\bar{b}ar{a}lpha^0_i(a) {\hbox{\boldmath $\bar \textup{im}ath$}}n \operatorname{dcl}(a)$. (This is a technical point, but the $0$-simplices in $S_0 \bar{c}C(p)$ are \emph{algebraic closures} of realizations of $p_i$, and there might be no canonical way to get a realization of $p_i$ from a $0$-simplex. Thus we need the choice functions $\bar{b}ar{a}lpha^0_i$.)
Next, we pick selection functions $\bar{b}ar{a}lpha_i : S_1 \bar{c}C(p) \rightarrow \textup{Mor}(\bar{c}G_i)$ (for every $i {\hbox{\boldmath $\bar \textup{im}ath$}}n I$) as follows. Suppose that $\operatorname{dom}(f) = {\bar{m}athcal P}(\{n_0, n_1\})$ for $n_0 < n_1$, and for $x {\hbox{\boldmath $\bar \textup{im}ath$}}n \{n_0, n_1\}$, let ``$f_x$'' stand for $f^{\{x\}}_{\{n_0, n_1\}} (\bar{b}ar{a}lpha^0_i(f \upharpoonright {\bar{m}athcal P}(\{x\})))$ (remembering that things in the image of $\bar{b}ar{a}lpha^0_i$ are realizations of $p_i$, which are also objects in $\textup{Ob}(\bar{c}G_i)$). Then we pick $\bar{b}ar{a}lpha_i(f)$ such that $\bar{b}ar{a}lpha_i(f) {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}_{\bar{c}G_i}(f_{n_0}, f_{n_1}).$ Just as in the proof of Lemma~\ref{commuting_pi}, we can use an inductive argument to ensure that if $i \leq j$ then $\bar{c}hi_{j,i}(\bar{b}ar{a}lpha_j(f)) = \bar{b}ar{a}lpha_i(f)$.
Finally, want to extend $\bar{b}ar{a}lpha_i$ to a selection function $\epsilon_i : S_2 \bar{c}C(p) \rightarrow G_i$. To ease notation here and in what follows, we set the following notation:
\bar{b}egin{enumerate}gin{notation}
\langlebel{faces}
Whenever $f {\hbox{\boldmath $\bar \textup{im}ath$}}n S_n \bar{c}C(p)$, $\operatorname{dom}(f) = {\bar{m}athcal P}(s)$, and $k {\hbox{\boldmath $\bar \textup{im}ath$}}n s$, let $$f^i_{k,s} := f^{\{k\}}_s\left(\bar{b}ar{a}lpha_i^0(f \upharpoonright {\bar{m}athcal P}(\{k\})) \right),$$ and note that $f^i_{k,s}$ is a realization of $p_i$, that is, an object in $\bar{c}G_i$. Similarly, if $\{k,\ell\} \subseteq s$ and $k < \ell$, let $$f^i_{\{k, \ell\},s} := f^{\{k, \ell\}}_{s}\left(\bar{b}ar{a}lpha_i(f \upharpoonright {\bar{m}athcal P}(\{k, \ell \})) \right),$$ which is a morphism in $\textup{Mor}_{\bar{c}G_i}(f^i_{k,s}, f^i_{\ell,s})$.
\end{notation}
\bar{b}egin{enumerate}gin{definition}
We define $\epsilon_i : S_2 \bar{c}C(p) \rightarrow G_i$ by the rule: if $\operatorname{dom}(f) = {\bar{m}athcal P}(s)$, where $s = \{n_0, n_1, n_2\}$ and $n_0 < n_1 < n_2$, then we define $\epsilon_i(f)$ as $$\epsilon_i(f) := \left[ \left( f^i_{\{n_0, n_2\}, s} \right)^{-1} \bar{c}irc f^i_{\{n_1, n_2\},s} \bar{c}irc f^i_{\{n_0, n_1\},s} \right]_{G_i}.$$ (Recall that if $f {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}_{\bar{c}G_i}(a,a)$, then ``$[f]_{G_i}$'' denotes the corresponding element of the group $G_i$.)
These functions $\epsilon_i$ can be extended linearly from $S_2 \bar{c}C(p)$ to the collection of all $2$-chains $C_2(p)$, and by abuse of notation we also call this new function $\epsilon_i$.
\end{definition}
The next lemma is a technical point that will be useful for later computations.
\bar{b}egin{enumerate}gin{lemma}
\langlebel{epsilon_coherence}
If $i {\hbox{\boldmath $\bar \textup{im}ath$}}n I$ and $f {\hbox{\boldmath $\bar \textup{im}ath$}}n S_n(p)$ for any $n \geq 3$, $\operatorname{dom}(g) = {\bar{m}athcal P}(t)$, and $\{a,b,c\} \subseteq s \subseteq t$ with $a < b < c$, then $$\epsilon_i(f \upharpoonright {\bar{m}athcal P}(\{a,b,c\})) = \left[(f^i_{\{a,c\}, s})^{-1} \bar{c}irc f^i_{\{b,c\}, s} \bar{c}irc f^i_{\{a,b\}, s}\right]_{G_i}.$$
\end{lemma}
\bar{b}egin{enumerate}gin{proof}
Remember that we identify the elements of $G_i$ with elements of $\operatorname{acl}(\emptyset)$. Because the transition map $f^{\{a,b,c\}}_s$ fixes $\operatorname{acl}(\emptyset)$ pointwise, $$f^{\{a,b,c\}}_s(\epsilon_i(f \upharpoonright \{a,b,c\})) = \epsilon_i(f \upharpoonright \{a,b,c\}).$$ Therefore the left-hand side of the equation above equals $f^{\{a,b,c\}}_s(\epsilon_i(f \upharpoonright \{a,b,c\}))$, which is the equivalence class (in $G_i$) of
$$ \left[f^{\{a,b,c\}}_s \bar{c}irc f^{\{a, c\}}_{\{a,b,c\}} (\bar{b}ar{a}lpha_i(f \upharpoonright {\bar{m}athcal P}(\{a, c\}))) \right]^{-1} \bar{c}irc$$ $$\left[f^{\{a,b,c\}}_s \bar{c}irc f^{\{b,c\}}_{\{a,b,c\}} (\bar{b}ar{a}lpha_i(f \upharpoonright {\bar{m}athcal P}(\{b, c\})))\right] \bar{c}irc \left[f^{\{a,b,c\}}_s \bar{c}irc f^{\{a, b\}}_{\{a,b,c\}} (\bar{b}ar{a}lpha_i(f \upharpoonright {\bar{m}athcal P}(\{a, b\})))\right]$$
$$= \left[ f^{\{a, c\}}_{s} (\bar{b}ar{a}lpha_i(f \upharpoonright {\bar{m}athcal P}(\{a, c\}))) \right]^{-1} \bar{c}irc$$ $$\left[ f^{\{b,c\}}_{s} (\bar{b}ar{a}lpha_i(f \upharpoonright {\bar{m}athcal P}(\{b, c\})))\right] \bar{c}irc \left[ f^{\{a, b\}}_{s} (\bar{b}ar{a}lpha_i(f \upharpoonright {\bar{m}athcal P}(\{a, b\})))\right],$$
as desired.
\end{proof}
\bar{b}egin{enumerate}gin{lemma}
\langlebel{epsilon_boundaries}
If $c {\hbox{\boldmath $\bar \textup{im}ath$}}n B^{set}_2(p)$, then for any $i {\hbox{\boldmath $\bar \textup{im}ath$}}n I$, $\epsilon_i(c) = 0$.
\end{lemma}
\bar{b}egin{enumerate}gin{proof}
By linearity, it suffices to check that $\epsilon_i(\bar{p}artial(g)) = 0$ for any $g {\hbox{\boldmath $\bar \textup{im}ath$}}n S^{set}_3 (p)$. For simplicity of notation, we assume that $\operatorname{dom}(g) = {\bar{m}athcal P}(s)$ where $s = \{0, 1, 2, 3\}$. To further simplify, we write ``$g_{i,j}$'' for $g^i_{\{i,j\}, s}$.
If $0 \leq j < k < \ell \leq 3$, by Lemma~\ref{epsilon_coherence}, $$\epsilon_i(g \upharpoonright \{j,k,\ell\}) = \left[g_{j,\ell}^{-1} \bar{c}irc g_{k,\ell} \bar{c}irc g_{j,k}\right]_{G_i}.$$
Therefore $\epsilon_i(\bar{p}artial(g))$ equals $$\left[g_{1,3}^{-1} \bar{c}irc g_{2,3} \bar{c}irc g_{1,2}\right]_{G_i} - \left[g_{0,3}^{-1} \bar{c}irc g_{2,3} \bar{c}irc g_{0,2} \right]_{G_i} $$ $$+ \left[g_{0,3}^{-1} \bar{c}irc g_{1,3} \bar{c}irc g_{0,1} \right]_{G_i} - \left[g_{0,2}^{-1} \bar{c}irc g_{1,2} \bar{c}irc g_{0,1} \right]_{G_i}$$ $$= \left[g_{0,1}^{-1} \bar{c}irc g_{1,3}^{-1} \bar{c}irc g_{2,3} \bar{c}irc g_{1,2} \bar{c}irc g_{0,1}\right]_{G_i} - \left[g_{0,3}^{-1} \bar{c}irc g_{2,3} \bar{c}irc g_{0,2} \right]_{G_i} $$ $$+ \left[g_{0,3}^{-1} \bar{c}irc g_{1,3} \bar{c}irc g_{0,1} \right]_{G_i} - \left[g_{0,2}^{-1} \bar{c}irc g_{1,2} \bar{c}irc g_{0,1} \right]_{G_i}$$ $$ = - \left[g_{0,2}^{-1} \bar{c}irc g_{1,2} \bar{c}irc g_{0,1} \right]_{G_i} - \left[g_{0,3}^{-1} \bar{c}irc g_{2,3} \bar{c}irc g_{0,2} \right]_{G_i} + \left[g_{0,3}^{-1} \bar{c}irc g_{1,3} \bar{c}irc g_{0,1} \right]_{G_i} $$ $$+ \left[g_{0,1}^{-1} \bar{c}irc g_{1,3}^{-1} \bar{c}irc g_{2,3} \bar{c}irc g_{1,2} \bar{c}irc g_{0,1}\right]_{G_i}$$ $$= \left[ \left(g_{0,1}^{-1} \, g_{1,2}^{-1} \, g_{0,2}\right) \bar{c}irc \left( g_{0,2}^{-1} \, g_{2,3}^{-1} \, g_{0,3}\right) \bar{c}irc \left( g_{0,3}^{-1} \, g_{1,3} \, g_{0,1} \right) \bar{c}irc \left( g_{0,1}^{-1} \, g_{1,3}^{-1} \, g_{2,3} \, g_{1,2} \, g_{0,1} \right)\right]_{G_i},$$ but everything in the last expression cancels out.
\end{proof}
By the last lemma, each $\epsilon_i$ induces a well-defined function $\bar{b}ar{w}idetilde{\epsilon_i} : H_2(p) \rightarrow G_i$.
Now we relate the $\epsilon_i$ maps to the groupoid maps $\bar{c}hi_{j,i} : \bar{c}G_j \rightarrow \bar{c}G_i$. For $i {\hbox{\boldmath $\bar \textup{im}ath$}}n I$, let $\overlineerline{\bar{p}si_i} : G_i \rightarrow \operatorname{Aut}(SW_i / \overline{a_0}, \overline{a_1})$ be the map induced by $\bar{p}si_i : \textup{Mor}_{G_i}(a^1_i,a^1_i) \rightarrow \operatorname{Aut}(SW_i / \overline{a_0}, \overline{a_1})$, and let $\overlineerline{\bar{c}hi_{j,i}} : G_j \rightarrow G_i$ be the surjective group homomorphism induced by the functor $\bar{c}hi_{j,i}$ from Lemma~\ref{chi}.
Everything coheres:
\bar{b}egin{enumerate}gin{lemma}
\langlebel{epsilon_chi}
If $i \leq j {\hbox{\boldmath $\bar \textup{im}ath$}}n I$ and $f {\hbox{\boldmath $\bar \textup{im}ath$}}n S_2(p)$, then $\overlineerline{\bar{c}hi_{j,i}} (\epsilon_j(f)) = \epsilon_i(f)$.
\end{lemma}
\bar{b}egin{enumerate}gin{proof}
For convenience, we assume that $\operatorname{dom}(f) = [2] = \{0,1,2\}$. Also, in the proof of this lemma, we write ``$f^i_{k,\ell}$'' for ``$f^i_{\{k,\ell\}, [2]}$'' (as in Notation~\ref{faces}).
\bar{b}egin{enumerate}gin{claim}
If $i \leq j$, then $\bar{c}hi_{j,i}(f^j_{k,\ell}) = f^i_{k,\ell}$.
\end{claim}
\bar{b}egin{enumerate}gin{proof}
The left-hand side is, by definition, equal to
$$\bar{c}hi_{j,i} \left( f^{\{k,\ell\}}_{[2]} ( \bar{b}ar{a}lpha_j(f \upharpoonright \{k, \ell\})) \right) = \left[ \bar{p}i_{j,i}\left( \langlengle f^{\{k,\ell\}}_{[2]} (\bar{b}ar{a}lpha_j(f \upharpoonright \{k, \ell\})) \textup{ran}gle_j \right) \right]_i$$
(using (6) of Lemma~\ref{chi}). But the map $f^{\{k,\ell\}}_{[2]}$ is elementary and the functions $\bar{p}i_{j,i}, \langlengle \bar{c}dot \textup{ran}gle_j$, and $\left[ \bar{c}dot \right]_i$ are all definable, so this expression equals
$$f^{\{k,\ell\}}_{[2]} \left(\left[\bar{p}i_{j,i}(\langlengle \bar{b}ar{a}lpha_j(f \upharpoonright \{k,\ell\}) \textup{ran}gle_j) \right]_i \right) = f^{\{k,\ell\}}_{[2]} \left(\bar{c}hi_{j,i}(\bar{b}ar{a}lpha_j(f \upharpoonright \{k,\ell\})) \right)$$ $$=f^{\{k,\ell\}}_{[2]}\left(\bar{b}ar{a}lpha_i(f \upharpoonright \{k,\ell\}) \right),$$
by our choice of the $\bar{b}ar{a}lpha_i$ functions such that $\bar{c}hi_{j,i} \bar{c}irc \bar{b}ar{a}lpha_j = \bar{b}ar{a}lpha_i$. But this last expression equals the right-hand side in the Claim.
\end{proof}
To prove the lemma, first pick some (any) morphism $g {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}_{\bar{c}G_j}(a^j_1, f_0)$, and note that $\epsilon_j(f)$ is an element of the group $G_j$ which is represented by the following morphism in $\textup{Mor}_{\bar{c}G_j}(a^j_1, a^j_1)$:
$$g^{-1} \bar{c}irc (f^j_{0,2})^{-1} \bar{c}irc f^j_{1,2} \bar{c}irc f^j_{0,1} \bar{c}irc g .$$
So $\overline{\bar{c}hi_{j,i}}(\epsilon_j(f))$ is represented by the morphism
$$\bar{c}hi_{j,i} \left(g^{-1} \bar{c}irc (f^j_{0,2})^{-1} \bar{c}irc f^j_{1,2} \bar{c}irc f^j_{0,1} \bar{c}irc g \right)$$ $$= \bar{c}hi_{j,i}(g)^{-1} \bar{c}irc \bar{c}hi_{j,i}(f^j_{0,2})^{-1} \bar{c}irc \bar{c}hi_{j,i}(f^j_{1,2}) \bar{c}irc \bar{c}hi_{j,i}(f^j_{0,1}) \bar{c}irc \bar{c}hi_{j,i}(g),$$ which, by the Claim above, equals $$\bar{c}hi_{j,i}(g)^{-1} \bar{c}irc (f^i_{0,2})^{-1} \bar{c}irc f^i_{1,2} \bar{c}irc f^i_{0,1} \bar{c}irc \bar{c}hi_{j,i}(g),$$ which, by definition, is a representative of $\epsilon_i(f)$.
\end{proof}
Let $G$ be the limit of the inverse system of groups $\langlengle G_i : i {\hbox{\boldmath $\bar \textup{im}ath$}}n I \textup{ran}gle$ with transition maps given by the $\overline{\bar{c}hi_{j,i}} : G_j \rightarrow G_i$. By Lemma~\ref{epsilon_chi}, the maps $\bar{b}ar{w}idetilde{\epsilon}_i$ induce a group homomorphism $\epsilon: H_2(p) \rightarrow G$.
\bar{b}egin{enumerate}gin{lemma}
\langlebel{injectivity}
The map $\epsilon: H_2(p) \rightarrow G$ is injective. In other words, if $c {\hbox{\boldmath $\bar \textup{im}ath$}}n Z_2(p)$ and $\epsilon_i(c) = 0$ for every $i {\hbox{\boldmath $\bar \textup{im}ath$}}n I$, then $c {\hbox{\boldmath $\bar \textup{im}ath$}}n B_2(p)$.
\end{lemma}
\bar{b}egin{enumerate}gin{proof}
Since $Z_2(p)$ is generated over $B_2(p)$ by all the $2$-shells, it is enough to prove this in the case where $c$ is a $2$-shell of the form $f_{\bar{b}ar{w}idehat{0}} - f_{\bar{b}ar{w}idehat{1}} + f_{\bar{b}ar{w}idehat{2}} - f_{\bar{b}ar{w}idehat{3}}$, where $f_{\bar{b}ar{w}idehat{a}}$ is a $2$-simplex with domain ${\bar{m}athcal P}([3] \setminus \{a\})$. We will construct a $3$-simplex $g: {\bar{m}athcal P}([3]) \rightarrow \operatorname{\bar{m}athfrak{C}}$ such that $\bar{p}artial (g) = c$.
Pick some $a_3 \bar{m}odels p | (a_0, a_1, a_2)$, so that $(a_0, a_1, a_2, a_3) \bar{m}odels p^{(4)}$. We will construct $g$ so that $g([3]) = \overline{a_{[3]}}$. If $(b,c,d,e)$ is some permutation of $(0,1,2,3)$, then $f_{b,c,d}(\{b,c\}) = f_{b,c,e}(\{b,c\})$ (since $\bar{p}artial c = 0$), and we can assume that $f_{b,c,d}(\{b,c\}) = \overline{a_{b,c}} = f_{b,c,e}(\{b,c\})$.
As a first step in defining the simplex $g$, for any $\{b,c\} \subseteq \{0,1,2,3\}$, we let $g \upharpoonright \{b,c\} = f \upharpoonright \{b,c,d\}$ (where $d$ is any other element of $[3]$), and we let the maps $g^{b}_{[3]} : \overline{a_{b}} \rightarrow \overline{a_{[3]}}$ be the inclusion maps. We take the transition map $g^{b}_{[3]}$ (for $b {\hbox{\boldmath $\bar \textup{im}ath$}}n [3]$) to be the identity map from $\overline{a_b}$ to itself.
Next we will define the transition maps $g^{b,c}_{[3]} : \overline{a_{b,c}} \rightarrow \overline{a_{[3]}}$ in such a way as to ensure compatibility with the faces $f_{\bar{b}ar{w}idehat{b}}$. Before doing this, we set some notation. First, we write ``$f^i_{xy, \bar{b}ar{w}idehat{z}}$'' for the set $(f_{\bar{b}ar{w}idehat{z}})^i_{\{x,y\}, [3] \setminus \{z\}}$ as in Notation~\ref{faces}. Similarly, we write $$f_{bc, \bar{b}ar{w}idehat{d}} := (f_{\bar{b}ar{w}idehat{d}})^{b,c}_{[3] \setminus \{d\}} (\overline{a_{b,c}}).$$ We consider the sets $\overline{a_{b,c}}$ to be $1$-simplices in which all of the transition maps are inclusions and the ``vertices'' are $\overline{a_b}$ and $\overline{a_c}$. This allows us to write``$\bar{b}ar{a}lpha_i(\overline{a_{b,c}})$.'' For $i {\hbox{\boldmath $\bar \textup{im}ath$}}n I$ and $\{b,c\} \subseteq [3]$, let $e^i_{bc}$ be the ``edge'' $\bar{b}ar{a}lpha_i(\overline{a_{b,c}})$.
We define the maps $g^{03}_{[3]}$, $g^{13}_{[3]}$, and $g^{23}_{[3]}$ to be the identity maps. Then we define the other three edge transition maps $g^{01}_{[3]}$, $g^{12}_{[3]}$, and $g^{02}_{[3]}$ so that for every $i {\hbox{\boldmath $\bar \textup{im}ath$}}n I$,
\bar{b}egin{enumerate}gin{equation}\langlebel{123} g^{13}_{[3]}(e^i_{13}) \, g^{23}_{[3]}(e^i_{23}) \, g^{12}_{[3]}(e^i_{12}) \equiv_{\operatorname{acl}(\emptyset)} f^i_{13, \bar{b}ar{w}idehat{0}} \, f^i_{23, \bar{b}ar{w}idehat{0}} \, f^i_{12, \bar{b}ar{w}idehat{0}},\end{equation}
\bar{b}egin{enumerate}gin{equation}\langlebel{023}g^{03}_{[3]}(e^i_{03}) \, g^{23}_{[3]}(e^i_{23}) \, g^{02}_{[3]}(e^i_{02}) \equiv_{\operatorname{acl}(\emptyset)} f^i_{03, \bar{b}ar{w}idehat{1}} \, f^i_{23, \bar{b}ar{w}idehat{1}} \, f^i_{02, \bar{b}ar{w}idehat{1}},\end{equation}
and
\bar{b}egin{enumerate}gin{equation}\langlebel{013} g^{03}_{[3]}(e^i_{03}) \, g^{13}_{[3]}(e^i_{13}) \, g^{01}_{[3]}(e^i_{01}) \equiv_{\operatorname{acl}(\emptyset)} f^i_{03, \bar{b}ar{w}idehat{2}} \, f^i_{13, \bar{b}ar{w}idehat{2}} \, f^i_{01, \bar{b}ar{w}idehat{2}}. \end{equation}
Having specified values according to the three equations above, we let $g^{01}_{[3]}$, $g^{12}_{[3]}$, and $g^{02}_{[3]}$ be \emph{any} elementary extensions to the respective domains $\overline{a_{03}}$, $\overline{a_{13}}$, and $\overline{a_{23}}$.
\bar{b}egin{enumerate}gin{claim}
\langlebel{isolation1}
For any $i {\hbox{\boldmath $\bar \textup{im}ath$}}n I$,
\bar{b}egin{enumerate}gin{equation}\langlebel{012}g^{02}_{[3]}(e^i_{02}) \, g^{12}_{[3]}(e^i_{12}) \, g^{01}_{[3]}(e^i_{01}) \equiv f^i_{02, \bar{b}ar{w}idehat{3}} \, f^i_{12, \bar{b}ar{w}idehat{3}} \, f^i_{01, \bar{b}ar{w}idehat{3}}.\end{equation}
\end{claim}
\bar{b}egin{enumerate}gin{proof}
Note that by stationarity, $$g^{02}_{[3]}(e^i_{02}) \, g^{12}_{[3]}(e^i_{12}) \equiv f^i_{02, \bar{b}ar{w}idehat{3}} \, f^i_{12, \bar{b}ar{w}idehat{3}} ,$$ and to check the Claim, it suffices to show that
$$\left[ g^{02}_{[3]}(e^i_{02})^{-1} \bar{c}irc g^{12}_{[3]}(e^i_{12}) \bar{c}irc g^{01}_{[3]}(e^i_{01}) \right]_{G_i} = \left[(f^i_{02, \bar{b}ar{w}idehat{3}})^{-1} \bar{c}irc f^i_{12, \bar{b}ar{w}idehat{3}} \bar{c}irc f^i_{01, \bar{b}ar{w}idehat{3}}\right]_{G_i}.$$
The right-hand side equals $\epsilon_i(f_{\bar{b}ar{w}idehat{3}})$. Since $\epsilon_i(c) = 0$, $$\epsilon_i(f_{\bar{b}ar{w}idehat{3}}) = \epsilon_i(f_{\bar{b}ar{w}idehat{0}}) - \epsilon_i(f_{\bar{b}ar{w}idehat{1}}) + \epsilon_i(f_{\bar{b}ar{w}idehat{2}}).$$ Let ``$g_{bc}$'' be an abbreviation for $g^{bc}_{[3]}(e^i_{bc})$. By applying equations~\ref{123}, \ref{023}, and \ref{013} above (and performing a very similar calculation as in the proof of Lemma~\ref{epsilon_boundaries}), we get:
$$\epsilon_i(f_{\bar{b}ar{w}idehat{3}}) = \left[g_{13}^{-1} \bar{c}irc g_{23} \bar{c}irc g_{12} \right]_{G_i} - \left[g^{-1}_{03} \bar{c}irc g_{23} \bar{c}irc g_{02} \right]_{G_i} + \left[g^{-1}_{03} \bar{c}irc g_{13} \bar{c}irc g_{01}\right]_{G_i}$$
$$= \left[g^{-1}_{01} \bar{c}irc g^{-1}_{13} \bar{c}irc g_{23} \bar{c}irc g_{12} \bar{c}irc g_{01}\right]_{G_i} - \left[g^{-1}_{03} \bar{c}irc g_{23} \bar{c}irc g_{02} \right]_{G_i} + \left[g^{-1}_{03} \bar{c}irc g_{13} \bar{c}irc g_{01}\right]_{G_i}$$
$$= -\left[g^{-1}_{03} \bar{c}irc g_{23} \bar{c}irc g_{02} \right]_{G_i} + \left[g^{-1}_{03} \bar{c}irc g_{13} \bar{c}irc g_{01}\right]_{G_i} + \left[g^{-1}_{01} \bar{c}irc g^{-1}_{13} \bar{c}irc g_{23} \bar{c}irc g_{12} \bar{c}irc g_{01}\right]_{G_i}$$
$$= \left[g^{-1}_{02} \bar{c}irc g^{-1}_{23} \bar{c}irc g_{03} \bar{c}irc g^{-1}_{03} \bar{c}irc g_{13} \bar{c}irc g_{01} \bar{c}irc g^{-1}_{01} \bar{c}irc g^{-1}_{13} \bar{c}irc g_{23} \bar{c}irc g_{12} \bar{c}irc g_{01} \right]_{G_i}$$
$$= \left[g^{-1}_{02} \bar{c}irc g_{12} \bar{c}irc g_{01}\right]_{G_i},$$
as desired.
\end{proof}
Now we must check that this coheres with the types of the given simplices $f_{\bar{b}ar{w}idehat{b}}$:
\bar{b}egin{enumerate}gin{claim}
\langlebel{isolation2}
If $(b,c,d,e)$ is a permutation of $[3]$ with $0 \leq b < c < d \leq 3$, then $$g^{bd}_{[3]}(\overline{a_{bd}}) \, g^{cd}_{[3]}(\overline{a_{cd}}) \, g^{bc}_{[3]}(\overline{a_{bc}}) \equiv f_{bd, \bar{b}ar{w}idehat{e}} \, f_{cd, \bar{b}ar{w}idehat{e}} \, f_{bd, \bar{b}ar{w}idehat{e}}.$$
\end{claim}
\bar{b}egin{enumerate}gin{proof}
Let $$\bar{b}ar{w}idetilde{f}_{xy, \bar{b}ar{w}idehat{e}} := \bar{b}igcup_{i {\hbox{\boldmath $\bar \textup{im}ath$}}n I} f^i_{xy, \bar{b}ar{w}idehat{e}}.$$ Then Claim~\ref{isolation2} follows from Claim~\ref{isolation1} above together with:
\bar{b}egin{enumerate}gin{subclaim}
If $(x,y,z)$ is any permutation of $(b,c,d)$ with $x < y$, then $\operatorname{tp}(f_{xy, \bar{b}ar{w}idehat{e}} / f_{yz,\bar{b}ar{w}idehat{e}} f_{xz, \bar{b}ar{w}idehat{e}})$ is isolated by $\operatorname{tp}(f_{xy, \bar{b}ar{w}idehat{e}} / \bar{b}ar{w}idetilde{f}_{yz,\bar{b}ar{w}idehat{e}} \bar{b}ar{w}idetilde{f}_{xz,\bar{b}ar{w}idehat{e}})$.
\end{subclaim}
\bar{b}egin{enumerate}gin{proof}
Note that $f_{xy,\bar{b}ar{w}idehat{e}} \subseteq \operatorname{acl}(f_{yz,\bar{b}ar{w}idehat{e}}, f_{xz,\bar{b}ar{w}idehat{e}})$ (in fact, it is in the algebraic closure of the ``vertices'' $f_{x,\bar{b}ar{w}idehat{e}} \subseteq f_{xz,\bar{b}ar{w}idehat{e}}$ and $f_{y,\bar{b}ar{w}idehat{e}} \subseteq f_{yz,\bar{b}ar{w}idehat{e}}$). Suppose towards a contradiction that $h {\hbox{\boldmath $\bar \textup{im}ath$}}n f_{xy,\bar{b}ar{w}idehat{e}}$ but $$\operatorname{tp}(h / \bar{b}ar{w}idetilde{f}_{yz,\bar{b}ar{w}idehat{e}} \bar{b}ar{w}idetilde{f}_{xz,\bar{b}ar{w}idehat{e}}) {\hbox{\boldmath \small $\bar n$}}vdash \operatorname{tp}(h / f_{yz,\bar{b}ar{w}idehat{e}} f_{xz,\bar{b}ar{w}idehat{e}}).$$ This means that the orbit of $h$ under $\operatorname{Aut}(\operatorname{\bar{m}athfrak{C}} / f_{yz,\bar{b}ar{w}idehat{e}} f_{xz,\bar{b}ar{w}idehat{e}})$ is smaller than the orbit of $h$ under $\operatorname{Aut}(\operatorname{\bar{m}athfrak{C}} / \bar{b}ar{w}idetilde{f}_{yz,\bar{b}ar{w}idehat{e}} \bar{b}ar{w}idetilde{f}_{xz,\bar{b}ar{w}idehat{e}})$. Let $\bar{b}ar{w}idehat{h}$ be a name for the orbit of $h$ under $\operatorname{Aut}(\operatorname{\bar{m}athfrak{C}} / f_{yz,\bar{b}ar{w}idehat{e}} f_{xz,\bar{b}ar{w}idehat{e}})$ as a set. Then $$\bar{b}ar{w}idehat{h} {\hbox{\boldmath $\bar \textup{im}ath$}}n \operatorname{dcl}(f_{yz,\bar{b}ar{w}idehat{e}}, f_{xz,\bar{b}ar{w}idehat{e}}) \setminus \operatorname{dcl}(\bar{b}ar{w}idetilde{f}_{yz,\bar{b}ar{w}idehat{e}}, \bar{b}ar{w}idetilde{f}_{xz,\bar{b}ar{w}idehat{e}}).$$ Since $\bar{b}ar{w}idehat{h} {\hbox{\boldmath $\bar \textup{im}ath$}}n \operatorname{dcl}(f_{yz,\bar{b}ar{w}idehat{e}} f_{xz,\bar{b}ar{w}idehat{e}})$, it lies in $f^i_{xy, \bar{b}ar{w}idehat{e}}$ for some $i {\hbox{\boldmath $\bar \textup{im}ath$}}n I$ (this is by the maximality condition on our symmetric witnesses $\langlengle W_i : i {\hbox{\boldmath $\bar \textup{im}ath$}}n I \textup{ran}gle$). Also, $$f^i_{xy, \bar{b}ar{w}idehat{e}} \subseteq \operatorname{dcl}(f^i_{yz, \bar{b}ar{w}idehat{e}}, f^i_{xz, \bar{b}ar{w}idehat{e}})$$ due to the fact that $f^i_{xy, \bar{b}ar{w}idehat{e}}$ is interdefinable with the set of all morphisms in $\textup{Mor}_{\bar{c}G_i}(f_{x,\bar{b}ar{w}idehat{e}}, f_{y, \bar{b}ar{w}idehat{e}})$, which can be obtained via composition in $\bar{c}G_i$ from the corresponding morphisms in $\operatorname{dcl}(f^i_{yz, \bar{b}ar{w}idehat{e}})$ and $\operatorname{dcl}(f^i_{xz, \bar{b}ar{w}idehat{e}})$. But this contradicts the fact that $\bar{b}ar{w}idehat{h} {\hbox{\boldmath \small $\bar n$}}otin \operatorname{dcl}(\bar{b}ar{w}idetilde{f}_{yz,\bar{b}ar{w}idehat{e}} \bar{b}ar{w}idetilde{f}_{xz,\bar{b}ar{w}idehat{e}}).$
\end{proof}
\end{proof}
Claim~\ref{isolation2} implies that for each permutation $(b,c,d,e)$ of $[3]$, we can find an elementary map $g^{b,d,c}_{[3]}$ from the ``face'' $f_{\bar{b}ar{w}idehat{e}}([3] \setminus \{e\})$ onto $\overline{a_{b,c,d}}$ which is coherent with the maps $g^{b,c}_{[3]}$, $g^{c,d}_{[3]}$, and $g^{b,d}_{[3]}$ that we have already defined, and such that $\bar{p}artial^i g = f_{\bar{b}ar{w}idehat{i}}$. This completes the proof of Lemma~\ref{injectivity}.
\end{proof}
\bar{b}egin{enumerate}gin{lemma}
\langlebel{surjectivity}
The map $\epsilon: H_2(p) \rightarrow G$ is surjective.
\end{lemma}
\bar{b}egin{enumerate}gin{proof}
Suppose that $g$ is any element in $G$, and that $g$ is represented by a sequence $\langlengle g_i : i {\hbox{\boldmath $\bar \textup{im}ath$}}n I \textup{ran}gle$ such that $\overline{\bar{c}hi_{j,i}}(g_j) = g_i$ whenever $i \leq j$. We will construct a $2$-chain $c = f-h$ such that $\epsilon_i(f-h) = g_i$ for every $i {\hbox{\boldmath $\bar \textup{im}ath$}}n I$, which will establish the Lemma. Let $f : {\bar{m}athcal P}([2]) \rightarrow \operatorname{\bar{m}athfrak{C}}$ be the $2$-simplex such that $f(s) = \overline{a_s}$ for every $s \subseteq [2]$ and such that every transition map in $f$ is an inclusion map. Let $k_i = \epsilon_i(f)$.
We want to construct $h: {\bar{m}athcal P}([2]) \rightarrow \operatorname{\bar{m}athfrak{C}}$ such that $h([2]) = \overline{a_{[2]}}$, $\bar{p}artial(h) = \bar{p}artial(f)$, and $h^s_{[2]}$ is the identity map whenever $s \subseteq \{0,1\}$ or $s \subseteq \{1,2\}$. The only thing left is to specify an elementary map $h^{02}_{[2]}: \overline{a_{02}} \rightarrow \overline{a_{02}}$ fixing $\overline{a_0}$ and $\overline{a_2}$ pointwise.
\bar{b}egin{enumerate}gin{claim}
Suppose that $h_i {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}_{\bar{c}G_i}(a^i_0, a^i_2)$ is the unique element such that $\left[ h_i^{-1} \bar{c}irc h^i_{12} \bar{c}irc h^i_{01} \right]_{G_i} = k_i - g_i$. Then
\bar{b}egin{enumerate}gin{enumerate}
{\hbox{\boldmath $\bar \textup{im}ath$}}tem whenever $i \leq j$, $\bar{c}hi_{j,i}(h_j) = h_i$, and
{\hbox{\boldmath $\bar \textup{im}ath$}}tem $\operatorname{tp}(h_0, \ldots, h_i / \overline{a_0}, \overline{a_2}) = \operatorname{tp}(\bar{b}ar{a}lpha_0(\overline{a_{02}}), \ldots, \bar{b}ar{a}lpha_i(\overline{a_{02}}) / \overline{a_0}, \overline{a_2}).$
\end{enumerate}
\end{claim}
\bar{b}egin{enumerate}gin{proof}
First we show:
\bar{b}egin{enumerate}gin{subclaim}
$\bar{c}hi_{i+1,i}(h^{i+1}_{12}) = h^i_{12}$ and $\bar{c}hi_{i+1,i}(h^{i+1}_{01}) = h^i_{01}$.
\end{subclaim}
\bar{b}egin{enumerate}gin{proof}
We check only the first equation (and the second equation has an identical proof). By (6) of Lemma~\ref{chi},
$$\bar{c}hi_{i+1,i}(h^{i+1}_{12}) = \left[ \bar{p}i_{i+1,i}(\langlengle h^{i+1}_{12} \textup{ran}gle_{i+1}) \right]_i = \left[ \bar{p}i_{i+1,i} (\langlengle h^{12}_{[2]}(\bar{b}ar{a}lpha_{i+1}(\overline{a_{12}})) \textup{ran}gle_{i+1}) \right]_i$$
$$= h^{12}_{[2]} \left( \left[ \bar{p}i_{i+1,i}(\langlengle\bar{b}ar{a}lpha_{i+1}(\overline{a_{12}}) \textup{ran}gle_{i+1}) \right]_i \right) =h^{12}_{[2]}(\bar{c}hi_{i+1,i}(\bar{b}ar{a}lpha_{i+1}(\overline{a_{12}})))$$
$$= h^{12}_{[2]}(\bar{b}ar{a}lpha_i(\overline{a_{12}})) = h^{12}_i.$$
\end{proof}
Note that it is enough to prove (1) of the Claim for every pair $(i,j)$ where $j = i+1$. We apply $\overline{\bar{c}hi_{i+1,i}}$ to both sides of the equation $\left[ h_{i+1}^{-1} \bar{c}irc h^{i+1}_{12} \bar{c}irc h^{i+1}_{01} \right]_{G_{i+1}} = k_{i+1} - g_{i+1}$. On the right-hand side, this yields
\bar{b}egin{enumerate}gin{equation}\langlebel{rhs} \overline{\bar{c}hi_{i+1,i}}(k_{i+1} - g_{i+1}) = k_i - g_i. \end{equation}
On the left-hand side, using the Subclaim, we get
\bar{b}egin{enumerate}gin{equation}\langlebel{lhs}\left[ \bar{c}hi_{i+1,i} \left(h^{-1}_{i+1} \bar{c}irc h^{i+1}_{12} \bar{c}irc h^{i+1}_{01} \right)\right]_{\bar{c}G_i} = \left[\bar{c}hi_{i+1,i}(h_{i+1})^{-1} \bar{c}irc h^i_{12} \bar{c}irc h^i_{01}\right]_{\bar{c}G_i}. \end{equation}
So putting together Equations~\ref{rhs} and \ref{lhs}, we get that $$\left[\bar{c}hi_{i+1,i}(h_{i+1})^{-1} \bar{c}irc h^i_{12} \bar{c}irc h^i_{01}\right]_{\bar{c}G_i} = k_i - g_i.$$ But $h_i$ is the unique morphism in $\bar{c}G_i$ such that $\left[ h_i^{-1} \bar{c}irc h^i_{12} \bar{c}irc h^i_{01} \right]_{\bar{c}G_i} = k_i - g_i$, so part (1) of the Claim follows.
We prove part (2) by induction on $i {\hbox{\boldmath $\bar \textup{im}ath$}}n I$. The base case follows from $$\operatorname{tp}(\langlengle h_0 \textup{ran}gle_0 / \overline{a_0}, \overline{a_2}) = \operatorname{tp}(\langlengle \bar{b}ar{a}lpha_0(\overline{a_{02}}) \textup{ran}gle_0 / \overline{a_0}, \overline{a_2})$$ (which is true simply because both elements belong to $SW(a^i_0, a^i_2)$). If (2) is true for $i$, then to prove it for $i+1$, it is enough to check that \bar{b}egin{enumerate}gin{equation}\langlebel{induction2}\operatorname{tp}(\langlengle h_{i+1} \textup{ran}gle_{i+1}, \langlengle h_i \textup{ran}gle_i / \overline{a_0}, \overline{a_2}) = \operatorname{tp}(\langlengle \bar{b}ar{a}lpha_{i+1}(\overline{a_{02}})\textup{ran}gle_{i+1}, \langlengle \bar{b}ar{a}lpha_i(\overline{a_{02}})\textup{ran}gle_i / \overline{a_0}, \overline{a_2}),\end{equation} since all the other elements are in the definable closure of $h_i$ and $\bar{b}ar{a}lpha_i(\overline{a_{01}})$ via the maps $\bar{c}hi_{k,\ell}$. To see this, first note that $$\operatorname{tp}(\langlengle h_{i+1} \textup{ran}gle_{i+1} / \overline{a_0}, \overline{a_2}) = \operatorname{tp}(\langlengle \bar{b}ar{a}lpha_{i+1}(\overline{a_{02}})\textup{ran}gle_{i+1} / \overline{a_0}, \overline{a_2})$$ just because both elements belong to $SW(a^{i+1}_0, a^{i+1}_2)$. By part~(1) of the Claim, $\bar{p}i_{i+1,i}(\langlengle h_{i+1} \textup{ran}gle_{i+1}) = \langlengle h_i \textup{ran}gle_i$, and by the way we chose the $\bar{b}ar{a}lpha$ functions, $\bar{p}i_{i+1,i}(\langlengle \bar{b}ar{a}lpha_{i+1}(\overline{a_{02}}) \textup{ran}gle_{i+1}) = \langlengle \bar{b}ar{a}lpha_i(\overline{a_{02}}) \textup{ran}gle_i$. Since the function $\bar{p}i_{i+1,i}$ is definable, Equation~\ref{induction2} follows.
\end{proof}
Given the elements $h_i$ as in the Claim above, we let $h^{02}_{[2]}: \overline{a_{02}} \rightarrow \overline{a_{02}}$ be any elementary map that fixes $\overline{a_{0}} \bar{c}up \overline{a_{2}}$ pointwise and maps each element $\bar{b}ar{a}lpha_i(\overline{a_{02}})$ to $h_i$. Then $\epsilon_i(f-h) = k_i - (k_i - g_i) = g_i$, as desired..
\end{proof}
By Lemmas~\ref{injectivity} and \ref{surjectivity}, $H_2(p) \bar{c}ong G$. To finish the proof of Theorem~\ref{hurewicz}, we just need to show:
\bar{b}egin{enumerate}gin{lemma}
$G \bar{c}ong \operatorname{Aut}(\bar{b}ar{w}idetilde{a_0 a_1} / \overline{a_0}, \overline{a_1})$
\end{lemma}
\bar{b}egin{enumerate}gin{proof}
Note that $\operatorname{Aut}(\bar{b}ar{w}idetilde{a_0 a_1} / \overline{a_0}, \overline{a_1})$ is the limit of the groups $\operatorname{Aut}(SW_i / \overline{a_0}, \overline{a_1})$ via the transition maps $\rho_{j,i}: \operatorname{Aut}(SW_j / \overline{a_0}, \overline{a_1}) \rightarrow \operatorname{Aut}(SW_j / \overline{a_0}, \overline{a_1})$, due to the maximality condition that every element of $\bar{b}ar{w}idetilde{a_0 a_1}$ lies in one of the symmetric witnesses $SW_i$. Also, part (8) of Lemma~\ref{chi} implies that we have a commuting system \bar{b}igskip
$\bar{b}egin{enumerate}gin{CD}
G_j @>\overline{\bar{c}hi_{j,i}}>> G_i\\
@VV\overline{\bar{p}si_j}V @VV\overline{\bar{p}si_i}V\\
\operatorname{Aut}(SW_j / \overline{a_0}, \overline{a_1}) @>\rho_{j,i}>> \operatorname{Aut}(SW_i / \overline{a_0}, \overline{a_1})
\end{CD}$
\bar{b}igskip
But the maps $\overline{\bar{p}si_i}$ are all isomorphisms, so taking limits we get an isomorphism from $G$ to $\operatorname{Aut}(\bar{b}ar{w}idetilde{a_0 a_1} / \overline{a_0}, \overline{a_1})$.
\end{proof}
\section{Any profinite abelian group can occur as $H_2(p)$}
In this section, we construct a family of examples which prove the following:
\bar{b}egin{enumerate}gin{theorem}
For any profinite abelian group $G$, there is a type $p$ in a stable theory $T$ such that $H_2(p) \bar{c}ong G$. In fact, we can build the theory $T$ to be totally categorical.
\end{theorem}
Together with Theorem~\ref{hurewicz} from the previous section, this shows that the groups that can occur as $H_2(p)$ for a type $p$ in a stable theory are \emph{precisely} the profinite abelian groups.
For the remainder of this section, we fix a profinite abelian group $G$ which is the inverse limit of the system $\langlengle H_i : i {\hbox{\boldmath $\bar \textup{im}ath$}}n I \textup{ran}gle$, where each $H_i$ is finite and abelian, $(I, \leq)$ is a directed set, and $G$ is the limit along the surjective group homomorphisms $\bar{v}arphi_{j,i} : H_j \rightarrow H_i$ (for every pair $i \leq j$ in $I$). The language $L$ of $T$ will be as follows: there will be a sort $\bar{c}G_i$ for each $i {\hbox{\boldmath $\bar \textup{im}ath$}}n I$, and function symbols $\bar{c}hi_{j,i} : \bar{c}G_j \rightarrow \bar{c}G_i$ for every pair $i \leq j$. The theory $T$ will say, in the usual language of categories, that each $\bar{c}G_i$ is a connected groupoid with infinitely many objects, and there will be separate composition symbols for each sort $\bar{c}G_i$. Also, $T$ says that $\bar{c}G_i$ is a groupoid such that each vertex group $\textup{Mor}_{\bar{c}G_i}(a_i,a_i)$ is isomorphic to the group $H_i$. For convenience, pick some arbitrary $a_i {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Ob}(\bar{c}G_i)$ and some group isomorphism $\bar{b}ar{x}i_i : G_i \rightarrow \textup{Mor}_{\bar{c}G_i}(a_i,a_i)$ (but the $\bar{b}ar{x}i_i$'s are \textbf{not} a part of any model of $T$). Then the last requirement we make on $T$ is that the function symbols $\bar{c}hi_{j,i}$ define full functors from $\bar{c}G_j$ onto $\bar{c}G_i$ which induce bijections between the corresponding collections of objects, and such that for every pair $i \leq j$, the following diagram commutes:
$$\bar{b}egin{enumerate}gin{CD}
H_j @>\bar{v}arphi_{j,i}>> H_i\\
@VV\bar{b}ar{x}i_jV @VV\bar{b}ar{x}i_iV\\
\textup{Mor}_{\bar{c}G_j}(a_j,a_j) @>\bar{c}hi_{j,i}>> \textup{Mor}_{\bar{c}G_i}(a_i,a_i)
\end{CD}$$
(In other words, the functors $\bar{c}hi_{j,i}$ are just ``isomorphic copies'' the group homomorphisms $\bar{v}arphi_{j,i}$.)
\bar{b}egin{enumerate}gin{lemma}
The theory $T$ described above is complete and admits elimination of quantifiers. If we further assume that the language is multi-sorted and that every element of a model must belong to one of the sorts $\bar{c}G_i$, then $T$ is totally categorical.
\end{lemma}
\bar{b}egin{enumerate}gin{proof}
If the language is multi-sorted, then since the groupoids $\bar{c}G_i$ are all connected and there are bijections between the object sets of the various $\bar{c}G_i$, the isomorphism class of a model of $T$ is determined by the cardinality of the object set of some (any) $\bar{c}G_i$. This shows that $T$ is totally categorical, hence $T$ is complete.
For quantifier elimination, it suffices to show the following: for any two models $M_1$ and $M_2$ of $T$ with a common substructure $A$ and any sentence $\sigma$ with parameters from $A$ of the form $\sigma = \exists x \bar{v}arphi(x; \overline{a})$ where $\bar{v}arphi$ is quantifier-free, if $M_1 \bar{m}odels \sigma$, then $M_2 \bar{m}odels \sigma$. (See Theorem~8.5 of \bar{c}ite{tentziegler}.) In this situation, let $\operatorname{cl}(A)$ denote the submodel of $M_1$ (and of $M_2$) generated by $A$, and in case $A = \emptyset$, let $\operatorname{cl}(A) = \emptyset$. Then if $M_1 \bar{m}odels \sigma$ as above, at least one of the following is true:
\bar{b}egin{enumerate}gin{enumerate}
{\hbox{\boldmath $\bar \textup{im}ath$}}tem $\bar{v}arphi(x; \overline{a})$ is satisfied by some $x$ in $\operatorname{cl}(A)$;
{\hbox{\boldmath $\bar \textup{im}ath$}}tem $\bar{v}arphi(x; \overline{a})$ is satisfied by some morphism between two objects in $\operatorname{cl}(A)$;
{\hbox{\boldmath $\bar \textup{im}ath$}}tem For some $i {\hbox{\boldmath $\bar \textup{im}ath$}}n I$, $\bar{v}arphi(x; \overline{a})$ is satisfied by \emph{any} object in $\bar{c}G_i$ outside of $\operatorname{cl}(A)$;
{\hbox{\boldmath $\bar \textup{im}ath$}}tem For some $i {\hbox{\boldmath $\bar \textup{im}ath$}}n I$, $\bar{v}arphi(x; \overline{a})$ is satisfied by \emph{any} morphism in $\bar{c}G_i$ which goes from [or to] some particular $b {\hbox{\boldmath $\bar \textup{im}ath$}}n \operatorname{cl}(A)$ and goes to [or from] \emph{any} object in $\bar{c}G_i$ outside of $\operatorname{cl}(A)$; or else
{\hbox{\boldmath $\bar \textup{im}ath$}}tem For some $i {\hbox{\boldmath $\bar \textup{im}ath$}}n I$, $\bar{v}arphi(x; \overline{a})$ is satisfied by \emph{any} morphism in $\bar{c}G_i$ whose source and target are both outside of $\operatorname{cl}(A)$.
\end{enumerate}
In each of the five cases above, it is straightforward to check that there is an $x$ realizing $\bar{v}arphi(x; \overline{a})$ in $M_2$ as well (for the last three cases we use the fact that $\textup{Ob}(\bar{c}G_i)$ is infinite).
\end{proof}
\bar{b}egin{enumerate}gin{remark}
\langlebel{acl}
If $A \subseteq \bar{c}G_i$, then we say that $b {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Ob}(\bar{c}G_i)$ is \emph{connected to $A$} if either $b {\hbox{\boldmath $\bar \textup{im}ath$}}n A$ or $b$ is the source or target of a morphism in $A$. By elimination of quantifiers, it follows that for any $A \subseteq \bar{c}G_i$, $\operatorname{acl}(A) \bar{c}ap \bar{c}G_i$ is the union of all objects $b$ that are connected to $A$ plus all morphisms $f {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}_{\bar{c}G_i}(b,c)$ such that $b$ and $c$ are connected to $A$.
Because of the functors $\bar{c}hi_{j,i}$, it follows that for any $a$ in any $\bar{c}G_i$, $\operatorname{acl}(a)$ actually contains objects and morphisms from each of the groupoids $\bar{c}G_j$. But for any $A \subseteq \operatorname{\bar{m}athfrak{C}}$, we can write $\operatorname{acl}(A)$ in the ``standard form'' $\operatorname{acl}(A) = \operatorname{acl}(A_0)$ for some $A_0 \subseteq \textup{Ob}(\bar{c}G_0)$, and:
1. $\operatorname{acl}(A_0) \bar{c}ap \textup{Ob}(\bar{c}G_i) = \bar{c}hi_{i,0}^{-1}(A_0)$, and
2. $\operatorname{acl}(A_0) \bar{c}ap \textup{Mor}(\bar{c}G_i)$ is the collection of all $f {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}_{\bar{c}G_i}(b,c)$ where $b,c {\hbox{\boldmath $\bar \textup{im}ath$}}n \operatorname{acl}(A_0)$.
\end{remark}
\bar{b}egin{enumerate}gin{lemma}
\langlebel{weak_EI}
The theory $T$ has \emph{weak elimination of imaginaries} in the sense of \bar{c}ite{Poizat}: for every formula $\bar{v}arphi(\overline{x}, \overline{a})$ defined over a model $M$ of $T$, there is a smallest algebraically closed set $A \subseteq M$ such that $\bar{v}arphi(\overline{x}, \overline{a})$ is equivalent to a formula with parameters in $A$.
\end{lemma}
\bar{b}egin{enumerate}gin{proof}
By Lemma~16.17 of \bar{c}ite{Poizat}, it suffices to prove the following two statements:
1. There is no strictly decreasing sequence $A_0 \supsetneq A_1 \supsetneq \ldots$, where every $A_i$ is the algebraic closure of a finite set of parameters; and
2. If $A$ and $B$ are algebraic closures of finite sets of parameters in the monster model $\operatorname{\bar{m}athfrak{C}}$, then $\operatorname{Aut}(\operatorname{\bar{m}athfrak{C}} / A \bar{c}ap B)$ is generated by $\operatorname{Aut}(\operatorname{\bar{m}athfrak{C}} / A)$ and $\operatorname{Aut}(\operatorname{\bar{m}athfrak{C}} / B)$.
Statement 1 follows immediately from the characterization of algebraically closed sets in Remark~\ref{acl} above (that is, algebraic closures of finite sets are equivalent to algebraic closures of finite subsets of $\textup{Ob}(\bar{c}G_0)$).
To check statement~2, suppose that $\sigma {\hbox{\boldmath $\bar \textup{im}ath$}}n \operatorname{Aut}(\operatorname{\bar{m}athfrak{C}} / A \bar{c}ap B)$, and assume that $A = \operatorname{acl}(A_0)$ and $B = \operatorname{acl}(B_0)$ where $A_0, B_0 \subseteq \textup{Ob}(\bar{c}G_0)$. Note that \emph{any} permutation of $\textup{Ob}(\bar{c}G_0)(\operatorname{\bar{m}athfrak{C}})$ which fixes $A_0$ can be extended to an automorphism of $\operatorname{Aut}(\operatorname{\bar{m}athfrak{C}} / A)$, and likewise for $B_0$ and $B$. So as a first step, we can use the fact that $\textup{Sym}(\textup{Ob}(\bar{c}G_0) / A_0 \bar{c}ap B_0)$ is generated by $\textup{Sym}(\textup{Ob}(\bar{c}G_0) / A_0)$ and $\textup{Sym}(\textup{Ob}(\bar{c}G_0) / B_0)$ to find an automorphism $\tau {\hbox{\boldmath $\bar \textup{im}ath$}}n \operatorname{Aut}(\operatorname{\bar{m}athfrak{C}})$ such that $\tau$ is in the subgroup generated by $\operatorname{Aut}(\operatorname{\bar{m}athfrak{C}} / A)$ and $\operatorname{Aut}(\operatorname{\bar{m}athfrak{C}} / B)$ and $\sigma \bar{c}irc \tau^{-1}$ fixes $\textup{Ob}(\bar{c}G_0)$ (and hence $\textup{Ob}(\bar{c}G_i)$ for every $i$) pointwise.
Finally, we need to deal with the morphisms. We claim that there is a map $\sigma^0_A {\hbox{\boldmath $\bar \textup{im}ath$}}n \operatorname{Aut}(\operatorname{\bar{m}athfrak{C}} / A)$ which fixes $\textup{Ob}(\bar{c}G_0)$ pointwise and such that for any $f {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}_{\bar{c}G_0}(b,c)$ such that at least one of $b$ and $c$ do not lie in $A$, $(\sigma^0 \bar{c}irc \tau)(f) = \sigma(f)$. (The idea is to use the recipe for constructing object-fixing automorphisms described in subsection~4.2 of \bar{c}ite{GK}, using a basepoint $a_0 {\hbox{\boldmath $\bar \textup{im}ath$}}n A$.) In fact, by the same argument we can also assume that for \emph{every} $i {\hbox{\boldmath $\bar \textup{im}ath$}}n I$ and for any $f {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}_{\bar{c}G_i}(b,c)$ such that at least one of $b$ and $c$ do not lie in $A$, $(\sigma^0 \bar{c}irc \tau)(f) = \sigma(f)$. Similarly, there is a map $\sigma^0_B {\hbox{\boldmath $\bar \textup{im}ath$}}n \operatorname{Aut}(\operatorname{\bar{m}athfrak{C}} / B)$ which fixes $\textup{Ob}(\bar{c}G_0)$ pointwise and for any $i {\hbox{\boldmath $\bar \textup{im}ath$}}n I$, $\sigma^0_B$ only moves morphisms in $\textup{Mor}_{\bar{c}G_i}(b,c)$ where $b$ and $c$ are both in $A \setminus (A \bar{c}ap B)$, and such that $\sigma^0_B \bar{c}irc \sigma^0_A \bar{c}irc \tau = \sigma$.
\end{proof}
\bar{b}egin{enumerate}gin{lemma}
\langlebel{acl2}
If $a^0, a^1 {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Ob}(\bar{c}G_i)$, then $$\operatorname{acl}^{eq}(a^0, a^1) = \operatorname{dcl}^{eq} \left(\bar{b}igcup_{i,j {\hbox{\boldmath $\bar \textup{im}ath$}}n I; \, i \leq j} \textup{Mor}_{\bar{c}G_j}(a^0_j, a^1_j) \right),$$ where $a^\ell_j = \bar{c}hi^{-1}_{j,i}(a^\ell)$.
\end{lemma}
\bar{b}egin{enumerate}gin{proof}
Suppose $g {\hbox{\boldmath $\bar \textup{im}ath$}}n \operatorname{acl}^{eq}(a^0, a^1)$. Then $g = b / E$ for some $(a^0, a^1)$-definable finite equivalence relation $E$. By Lemma~\ref{weak_EI}, there is a finite tuple $\overline{d} {\hbox{\boldmath $\bar \textup{im}ath$}}n \operatorname{\bar{m}athfrak{C}}$ (in the home sort) such that $b / E$ is definable over $\overline{d}$ and $\overline{d}$ has a minimal algebraic closure. If the set $\operatorname{acl}(\overline{d})$ contained an object $a$ of $\bar{c}G_0$ other than $\bar{c}hi_{i,0}(a^0)$ and $\bar{c}hi_{j,0}(a^1)$, then (by quantifier elimination) $\operatorname{acl}(\overline{d})$ would have an infinite orbit under $\operatorname{Aut}(\operatorname{\bar{m}athfrak{C}} / a^0, a^1)$, and so $E$ would have infinitely many classes, a contradiction. So by Remark~\ref{acl}, the set $\overline{d}$, and hence $b / E$ is definable over the union of the morphism sets $\textup{Mor}_{\bar{c}G_j}(a^0_j, a^1_j)$.
\end{proof}
\textbf{From now on, we assume that all algebraic and definable closures are computed in $T^{eq}$, not just in the home sort.}
\bar{b}egin{enumerate}gin{lemma}
If $a^0, a^1 {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Ob}(\bar{c}G_i)$, then for any two $f, g {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}_{\bar{c}G_i}(a^0, a^1)$, $$\operatorname{tp}(f / \operatorname{acl}(a^0), \operatorname{acl}(a^1)) = \operatorname{tp}(g / \operatorname{acl}(a^0), \operatorname{acl}(a^1)).$$
\end{lemma}
\bar{b}egin{enumerate}gin{proof}
Using the same procedure as described in subsection~4.2 of \bar{c}ite{GK}, we can construct an automorphism $\sigma$ of $\operatorname{\bar{m}athfrak{C}}$ fixing $\textup{Ob}(\bar{c}G_i)$, $\textup{Mor}_{\bar{c}G_i}(a^0, a^0)$, and $\textup{Mor}_{\bar{c}G_i}(a^1, a^1)$ pointwise while mapping $f$ to $g$. (In the construction of \bar{c}ite{GK}, the ``basepoint'' $a_0$ there can be chosen to be $a^0$ here, and then condition (5) of the construction plus the fact that $\bar{c}G_i$ is abelian implies that $\textup{Mor}_{\bar{c}G_i}(a^1, a^1)$ is fixed.) In fact, it is easy to see that we can even ensure that $\sigma$ fixes $\textup{Mor}_{\bar{c}G_j}(\bar{c}hi_{j,i}^{-1}(a^0), \bar{c}hi_{j,i}^{-1}(a^0))$ and $\textup{Mor}_{\bar{c}G_j}( \bar{c}hi_{j,i}^{-1}(a^1), \bar{c}hi_{j,i}^{-1}(a^1))$ pointwise, so by Lemma~\ref{acl2}, $\sigma$ fixes $\operatorname{acl}(a^0) \bar{c}up \operatorname{acl}(a^1)$ pointwise.
\end{proof}
Let $p = \operatorname{stp}(a_0)$ for some (any) $a_0 {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Ob}(\bar{c}G_0)$.
\bar{b}egin{enumerate}gin{proposition}
$H_2(p) \bar{c}ong G$.
\end{proposition}
\bar{b}egin{enumerate}gin{proof}
Pick $(a^0, a^1, a^2) \bar{m}odels p^{(3)}$. By Theorem~\ref{hurewicz} (the ``Hurewicz theorem''), it is enough to show that $\operatorname{Aut}(\bar{b}ar{w}idetilde{a^0 a^1} / \overline{a^0}, \overline{a^1}) \bar{c}ong G$. For ease of notation, let $a^k_i = \bar{c}hi^{-1}_{i,0}(a^k)$ for $k = 0, 1,$ or $2$. By Lemma~\ref{acl2} and the fact that any morphism in $\textup{Mor}_{\bar{c}G_i}(a^0_i, a^1_i)$ is a composition of morphisms in $\textup{Mor}_{\bar{c}G_i}(a^0_i, a^2_i)$ and $\textup{Mor}_{\bar{c}G_i}(a^2_i, a^1_i)$, it follows that the set $\bar{b}ar{w}idetilde{a^0 a^1}$ is interdefinable with $\bar{d}isplaystyle\bar{b}igcup_{i {\hbox{\boldmath $\bar \textup{im}ath$}}n I} \textup{Mor}_{\bar{c}G_i}(a^0_i, a^1_i) $.
So $\operatorname{Aut}(\bar{b}ar{w}idetilde{a^0 a^1} / \overline{a^0}, \overline{a^1})$ is the inverse limit of the groups $\operatorname{Aut}(\textup{Mor}_{\bar{c}G_i}(a^0_i, a^1_i) / \overline{a^0}, \overline{a^1})$ under the natural homomorphisms $$\rho_{j,i} : \operatorname{Aut}(\textup{Mor}_{\bar{c}G_j}(a^0_j, a^1_j) / \overline{a^0}, \overline{a^1}) \rightarrow \operatorname{Aut}(\textup{Mor}_{\bar{c}G_i}(a^0_i, a^1_i) / \overline{a^0}, \overline{a^1})$$ induced by the fact that $\textup{Mor}_{\bar{c}G_i}(a^0_i, a^1_i)$ is in the definable closure of $\textup{Mor}_{\bar{c}G_i}(a^0_j, a^1_j)$ when $j \geq i$.
By the way we defined our theory $T$, we can select a system of group isomorphisms $\langlembda_i : H_i \rightarrow \textup{Mor}_{\bar{c}G_i}(a^1_i, a^1_i)$ for $i {\hbox{\boldmath $\bar \textup{im}ath$}}n I$ such that the following diagram commutes:
$$\bar{b}egin{enumerate}gin{CD}
H_j @>\bar{v}arphi_{j,i}>> H_i\\
@VV\langlembda_jV @VV\langlembda_iV\\
\textup{Mor}_{\bar{c}G_j}(a_j,a_j) @>\bar{c}hi_{j,i}>> \textup{Mor}_{\bar{c}G_i}(a_i,a_i)
\end{CD}$$
To finish the proof of the Proposition, it is enough to find a system of group isomorphisms $$\sigma_i : H_i \rightarrow \operatorname{Aut}(\textup{Mor}_{\bar{c}G_i}(a^0_i, a^1_i) / \overline{a^0}, \overline{a^1})$$ such that the following diagram commutes:
$$\bar{b}egin{enumerate}gin{CD}
H_j @>\bar{v}arphi_{j,i}>> H_i\\
@VV\sigma_jV @VV\sigma_iV\\
\operatorname{Aut}(\textup{Mor}_{\bar{c}G_j}(a^0_j, a^1_j) / \overline{a^0}, \overline{a^1}) @>\rho_{j,i}>> \operatorname{Aut}(\textup{Mor}_{\bar{c}G_i}(a^0_i, a^1_i) / \overline{a^0}, \overline{a^1})
\end{CD}$$
(Then by the discussion above, $\operatorname{Aut}(\bar{b}ar{w}idetilde{a^0 a^1} / \overline{a^0}, \overline{a^1})$ will be isomorphic to the inverse limit of the groups $H_i$, which is $G$.)
We define the maps $\sigma_i$ so that for any $h {\hbox{\boldmath $\bar \textup{im}ath$}}n H_i$ and any $g {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}_{\bar{c}G_i}(a^0_i, a^1_i)$, $$\left[\sigma_i(h)\right](g) = \langlembda_i(h) \bar{c}irc g.$$
(Note that this rule determines a unique elementary permutation of $\textup{Mor}_{\bar{c}G_i}(a^0, a^1)$ fixing $\operatorname{acl}(a^0) \bar{c}up \operatorname{acl}(a^1)$ pointwise.) This is a group homomorphism since $$\left[\sigma_i(h_1 h_2)\right] (g) = \langlembda_i(h_1 h_2) \bar{c}irc g = \langlembda_i(h_1) \bar{c}irc \langlembda_i(h_2) \bar{c}irc g = \left[\sigma_i(h_1) \bar{c}irc \sigma_i(h_2) \right](g).$$ Clearly $\sigma_i$ is injective, and it is surjective because of the following:
\bar{b}egin{enumerate}gin{claim}
For any $f$ and $g$ in $\textup{Mor}_{\bar{c}G_i}(a_i^0, a_i^1)$, there is a \emph{unique} elementary permutation $\sigma$ of $\textup{Mor}_{\bar{c}G_i}(a^0, a^1)$ sending $f$ to $g$ and fixing $\operatorname{acl}(a^0) \bar{c}up \operatorname{acl}(a^1)$ pointwise.
\end{claim}
\bar{b}egin{enumerate}gin{proof}
If $f = h \bar{c}irc g $ for $h {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}_{\bar{c}G_i}(a_i^1, a_i^1)$, then $\sigma(f')$ must equal $h \bar{c}irc f' $ for any $f' {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}_{\bar{c}G_i}(a_i^0, a_i^1)$.
\end{proof}
Finally, we must check that the maps $\sigma_i$ commute with $\bar{v}arphi_{j,i}$ and $\rho_{j,i}$. Pick any $j \geq i$, $h {\hbox{\boldmath $\bar \textup{im}ath$}}n H_j$ and $f {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}_{\bar{c}G_i}(a^0_i, a^1_i)$. On the one hand,
$$\rho_{j,i}\left(\sigma_j(h) \right) (f) = \bar{c}hi_{j,i}\left(\sigma_j(h)(f') \right), \textup{ where } \bar{c}hi_{j,i}(f') = f$$
$$= \bar{c}hi_{j,i}(\langlembda_j(h) \bar{c}irc f') = \bar{c}hi_{j,i}(\langlembda_j(h)) \bar{c}irc \bar{c}hi_{j,i}(f') = \bar{c}hi_{j,i}(\langlembda_j(h)) \bar{c}irc f.$$
On the other hand,
$$\left[\sigma_i(\bar{v}arphi_{j,i}(h))\right] (f) = \langlembda_i(\bar{v}arphi_{j,i}(h)) \bar{c}irc f = \bar{c}hi_{j,i} (\langlembda_j(h)) \bar{c}irc f.$$
These last two equations show that $\rho_{j,i} \bar{c}irc \sigma_j = \sigma_i \bar{c}irc \bar{v}arphi_{j,i}$, as desired.
\end{proof}
\bar{b}egin{enumerate}gin{remark}
These examples also show that homology groups of types are not always preserved by nonforking extensions. In the example above, if $A$ is some algebraically closed parameter set containing a point in $p(\operatorname{\bar{m}athfrak{C}})$ and $q$ is the nonforking extension of $p$ over $A$, then $q$ has $4$-amalgamation, and so (by Corollary~\ref{trivial_homology}) $H_2(q) = 0$.
\end{remark}
\section{Unstable examples}
In this section, we compute some more homology groups for unstable rosy examples.
\bar{b}egin{enumerate}gin{example}\langlebel{tet.free}
In this first example, as promised, we argue that all the homology groups of the tetrahedron-free random ternary hypergraph are trivial, even though it does not have 4-amalgamation. Let $T_{tet.free}$ be the theory of such a graph with the ternary relation $\{R\}$. It is well-known that $T_{tet.free}$ is $\omega$-categorical, simple, has weak elimination of imaginaries, and has $n$-amalgamation for all $n {\hbox{\boldmath \small $\bar n$}}eq 4$. Let $p$ be the unique 1-type over $\emptyset$. We first claim that even though $T_{tet.free}$ does not have 4-amalgamation,
Lemma~\ref{shellcycle} still holds.
\bar{b}egin{enumerate}gin{claim}\langlebel{thesame}
In $T_{tet.free}$, for $n\geq 2$, every
$(n-1)$-cycle $c=\sum_i k_if_i$ of type $p$ over $\emptyset$ is a sum of $(n-1)$-shells.
\end{claim}
We sketch the proof, which is almost the same as that of \ref{shellcycle} (but as here we do not
have 4-amalgamation, we need some trick.) We even use the same notation, letting
$g_{ij} = \bar{p}artial^j f_i$ for $(i,j){\hbox{\boldmath $\bar \textup{im}ath$}}n I$ ($j<n$). We shall find $(n-1)$-simplices $h_{ij}$ satisfying the conditions
described in Claim~\ref{cancelling}. Note that due to weak elimination of imaginaries we can assume each vertex of a simplex is just a point in the graph. Now the construction method will be the same: first, pick a point $a^*$ independent from all the points $g_{ij}(\{k\})$. Then the edges $h_{ij}(\{k,m\})$, where $m{\hbox{\boldmath \small $\bar n$}}otin \bar{b}igcup_{ij} s_{ij}$, are determined. For the next level, we need a trick. Namely, given an edge of the from $\{b,c\}=g_{ij}(\{k,\ell\})$, we find
a point $a=(h_{ij})^{\{m\}}_{\{k,\ell,m\}}(h_{ij}(\{m\}))$ (while we may take $(h_{ij})^{\{k,\ell \}}_{\{k,\ell,m\}}$ as an identity map of $\{b,c\}$) such that $a,b,c$ are distinct and $R(a,b,c)$ {\em does not} hold). Then we can proceed the next level of the construction for $h_{ij}$ as no matter what the triangle $g_{ij}(\{k,\ell,k'\})$ is (whether it satisfies $R$ or not), we can amalgamate the other three triangular faces which do not satisfy $R$. The rest of the proof is the same. We have proved Claim~\ref{thesame}.
Now to show that $H_n(p)=0$ $(n\geq 1)$, it suffices to see that any $n$-shell is a boundary. Due to $(n+2)$-amalgamation, this is true for any $n{\hbox{\boldmath \small $\bar n$}}e 2$. But any 2-shell is a boundary as well. The only case to check is that of a 2-shell $f=f_0-f_1+f_2-f_3$ with support $\{0,1,...,4\}$ such that $f_i(\{0, \ldots, \hat i, \ldots,4\})$ satisfies $R$. But by taking a suitable point with support $\{5\}$ distinct from all such faces, it easily follows $f$ is the boundary of a $3$-fan.
\end{example}
\bar{b}egin{enumerate}gin{example}
\langlebel{dlo}
Here we show the theory $T_{dlo}$ of dense linear ordering (without end points) is another example whose homology groups are all trivial even though it does not have 3-amalgamation. Recall that it has elimination of imaginaries.
Let $p$ be the unique 1-type over $\emptyset$. It is not hard to see that $p$ has $n$-amalgamation for all $n{\hbox{\boldmath \small $\bar n$}}e 3$. Now
we claim that, just like in Claim~\ref{thesame}, any $n$-cycle is a sum of $n$-shells. The proof will be similar, and we use the same notation. We want to construct the edges $h_{ij}$.
The trick this time is to take $a^*$ greater than all the points of the form $a'=g_{ij}(\{k\})$.
Then given any edge $\{b,c\}=g_{ij}(\{k,\ell\})$, where either $b<c$ or $c<b$, pick $a>b,c$. Then since
$\operatorname{tp}(a'a^*)=\operatorname{tp}(ba)=\operatorname{tp}(ca)$, the construction of $h_{ij}$ on this level is compatible. For the rest of the construction, use $n$-amalgamation.
Due to the claim and $(n+2)$-amalgamation, all of the groups $H_n(p)$ are $0$ for $n{\hbox{\boldmath \small $\bar n$}}e 1$. Furthermore, $H_1(p)=0$ because any 1-shell is the boundary of a 2-fan (choose a point greater than all the vertices of all the terms in the 1-shell).
\end{example}
\bar{b}egin{enumerate}gin{example}
\langlebel{parity}
In \bar{c}ite{KKT}, for each {\bar{b}f even} $n\geq 4$, the theory $U_n$ of the Fraiss\'{e} limit of the following class
$K_n$ is introduced.
Let $R_n$ be an $n$-ary relation symbol.
We consider symmetric and irreflexive $R_n$-structures. For any
$R_n$-structure $A$ with a finite substructure $B$, let
${\bar{m}athrm {no}}_A(B)$ denote
the number (modulo $2$) of
$n$-element subsets of $B$ satisfying $R_n$.
If $A$ is clear from the context, we simply write ${\bar{m}athrm {no}}(B)$.
Let $(*)_n$ be the following condition on an $R_n$-structure $A$:
\bar{b}egin{enumerate}gin{itemize}
{\hbox{\boldmath $\bar \textup{im}ath$}}tem[$(*)_n$] If $A_0$ is an $(n+1)$-element subset of $A$, then
${\bar{m}athrm {no}}(A_0)=0$.
\end{itemize}
Now $K_n$ is the class of all finite (symmetric and irreflexive)
$R$-structures satisfying $(*)_n$.
It is shown in \bar{c}ite{KKT} that $U_n$ is $\omega$-categorical, supersimple of $SU$-rank 1, and has quantifier elimination, weak elimination of imaginaries, and $n$-CA, but that $U_n$ does not have $(n+1)$-amalgamation.
Now let $p_n$ be the unique 1-type of $U_n$.
\bar{b}egin{enumerate}gin{claim}\langlebel{H3}
$H_m(p_n)=0$ for $1\leq m<n-1$; $H_{n-1}(p_n)={\bar{m}athbb{Z}}_2.$
\end{claim}
\bar{b}egin{enumerate}gin{proof}
Since $p_n$ has $n$-CA, due to \ref{trivial_homology} we have $H_m(p_n)=0$ for $1\leq m<n-1$.
Now to compute $H_{n-1}$, we introduce an augmentation map
$$\epsilon:S_{n-1}\bar{c}C(p_n) \to \bar{m}athbb{Z}_2$$
as follows: Let $f$ be an $(n-1)$-simplex of type $p$ with $\operatorname{dom}(f)={\bar{m}athcal P}(s)$ with $|s|=n$.
Then we let $\epsilon(f)=1$ if and only if
$R_n(f(s))$
holds. The map $\epsilon$ obviously extends as a homomorphism $\epsilon: C_{n-1}\bar{c}C(p_n)\to \bar{m}athbb{Z}_2$.
It follows from $(*)_n$ above that an $(n-1)$-shell $c$ is the boundary of $n$-simplex iff $\epsilon(c)=0$.
Thus for any $(n-1)$-boundary $c$, we have $\epsilon(c)=0$.
Hence $\epsilon$ induces a homomorphism $\epsilon_*:H_{n-1}(p_n)\to \bar{m}athbb{Z}_2$.
Note that there is an $(n-1)$-shell $d$ with support $\{0,...,n\}$ such that $\epsilon(d)=1$. Hence $\epsilon_*$ is onto.
By Theorem~\ref{Hn_shells} there is an $(n-1)$-shell $d'$ such that $[d]+[d]=[d']$.
But then $\epsilon(d')=0$ and $d'$ is an $(n-1)$-boundary, i.e. $[d]+[d]=0$.
Now let $c$ be an arbitrary $(n-1)$-shell with support $\{0,...,n\}$.
If $\epsilon(c)=0$, then $[c]=0$. If $\epsilon(c)=1$, then by the same argument,
$[d]-[c]=0$, i.e. $[d]=[c]$.
We have verified Claim \ref{H3}.
\end{proof}
\end{example}
\bar{b}egin{enumerate}gin{example}\langlebel{ominh1}
Here we show that for any complete 1-type $p$ over $A=\operatorname{acl}(A)$ in an o-minimal theory, $H_1(p)=0$. Basically we use a similar idea as in \ref{dlo}.
Let $T$ be any rosy theory, and let $p {\hbox{\boldmath $\bar \textup{im}ath$}}n S(A)$ be any type in $T$ over $A$.
\bar{b}egin{enumerate}gin{definition}
The type $p(x)$ has \emph{weak 3-amalgamation} if there is a type $q(x,y) {\hbox{\boldmath $\bar \textup{im}ath$}}n S(A)$ such that:
\bar{b}egin{enumerate}gin{enumerate}
{\hbox{\boldmath $\bar \textup{im}ath$}}tem Whenever $(a,b) \bar{m}odels q(x,y)$, then $a$ and $b$ are independent (over $A$) realizations of $p$; and
{\hbox{\boldmath $\bar \textup{im}ath$}}tem For any pair $(a,b)$ of independent realizations of $p$, there is a third realization $c$ of $p$ such that $c$ is independent from $ab$ and both $(a,c)$ and $(b,c)$ realize $q$.
\end{enumerate}
\end{definition}
So any Lascar strong type in a simple theory has weak $3$-amalgamation by the Independence Theorem.
\bar{b}egin{enumerate}gin{lemma}
Any nonalgebraic $1$-type (of the home sort) in an o-minimal theory has weak $3$-amalgamation.
\end{lemma}
\bar{b}egin{enumerate}gin{proof}
Recall that since $T$ is o-minimal, any $A$-definable unary function $f(x)$ is either eventually increasing (that is, there is some point $c$ such that if $c < x < y$ then $f(x) < f(y)$, eventually decreasing, or eventually constant. If $f$ is eventually constant with eventual value $d$, then $d {\hbox{\boldmath $\bar \textup{im}ath$}}n \operatorname{dcl}(A)$.
We say an $A$-definable function $f(x_1,...,x_n)$ {\em bounded within} $p$ if for any $c_1,...,c_n\bar{m}odels p$, there is $d$ realizing $p$ such that $d > f(c_1,...,c_n).$ We call
a pair of realizations $(a,b)$ of $p$ an \emph{extreme pair} if whenever $f(x)$ is bounded within $p$, then $b > f(a)$.
First note that by the compactness theorem, for any $a$ realizing $p$, there is a $b$ realizing $p$ such that $(a,b)$ is an extreme pair. Also, if $b {\hbox{\boldmath $\bar \textup{im}ath$}}n \operatorname{dcl}(aA) = \operatorname{acl}(aA)$, then there is an $A$-definable function $f: p(\bar{m}athfrak{C}) \rightarrow p(\bar{m}athfrak{C})$ such that $b = f(a)$, so since there is no maximal realization $c$ of $p$ (because such a realization $c$ would be in $\operatorname{dcl}(A)$ and we are assuming that $p$ is nonalgebraic), it follows that $(a,b)$ is \textbf{not} an extreme pair. So any extreme pair is algebraically independent over $A$ and hence thorn-independent (see \bar{c}ite{On}).
\bar{b}egin{enumerate}gin{claim}
Any two extreme pairs have the same type over $A$.
\end{claim}
\bar{b}egin{enumerate}gin{proof}
It is enough to check that if $(a,b)$ and $(a,c)$ are two extreme pairs, then $\operatorname{tp}(b/Aa) = \operatorname{tp}(c/Aa)$. By o-minimality, any $Aa$-definable set $X$ is a finite union of intervals, and the endpoints $\{d_1, \ldots, d_n\}$ of these intervals lie in $\operatorname{dcl}(Aa)$. So $d_i = f(a)$ for some $A$-definable function $f$, and as we already observed $b,c{\hbox{\boldmath \small $\bar n$}}e d_i$. Hence it suffices to see $b>d_i$ iff $c>d_i$. Now
by the definition of an extreme pair,
$$\forall x \bar{m}odels p \,\,\, \exists y \bar{m}odels p \left[ y > f(x)\right] \Rightarrow b > f(a) = d_i.$$
Also,
$$\exists x \bar{m}odels p \, \, \, \forall y \bar{m}odels p \left[ y \leq f(x) \right] \Rightarrow \forall x \bar{m}odels p \, \, \, \forall y \bar{m}odels p \left[y \leq f(x) \right]$$
because any two realizations of $p$ are conjugate under an automorphism in $\operatorname{Aut}(\bar{m}athfrak{C}/A)$ which permutes $p(\bar{m}athfrak{C})$, and so
$$\exists x \bar{m}odels p \, \, \forall y \bar{m}odels p \left[y \leq f(x) \right] \Rightarrow b \leq f(a) = d_i.$$
The same reasoning applies with $c$ in place of $b$, so
$$b > d_i = f(a) \Leftrightarrow \forall x \bar{m}odels p \,\,\, \exists y \bar{m}odels p \left[ y > f(x)\right]$$
$$\Leftrightarrow c > f(a) = d_i.$$
\end{proof}
Let $q(x,y) = \operatorname{tp}(a',b' / A)$ for some extreme pair. Condition (2) of the definition of weak $3$-amalgamation can be ensured by picking $c\bar{m}odels p$ so that $c > g(a,b)$ for any $A$-definable function $g(y,z)$ bounded within $p$, which is possible by the compactness theorem.
\end{proof}
\bar{b}egin{enumerate}gin{claim}
If $p$ has weak $3$-amalgamation, then $H_1(p) = 0$.
\end{claim}
\bar{b}egin{enumerate}gin{proof}
By Theorem~\ref{Hn_shells}
it suffices to show that every (set) $1$-shell of type $p$ with support $\{0,1,2\}$ is a boundary of some $2$-fan of type $p$. Let $c = f_{12} - f_{02} + f_{01}$ be such a shell, where $f_{ij}$ is a $1$-simplex with support $\{i,j\}$. The condition that this is a shell implies that there are realizations $a_0, a_1,$ and $a_2$ of $p$ such that:
$$\bar{p}artial_0 f_{01} = \bar{p}artial_0 f_{02} = \operatorname{acl}_A(a_0),$$
$$\bar{p}artial_0 f_{12} = \bar{p}artial_1 f_{01} = \operatorname{acl}_A(a_1),$$
$$\bar{p}artial_1 f_{12} = \bar{p}artial_1 f_{02} = \operatorname{acl}_A(a_2).$$
(Note that actually the boundaries above technically should be $0$-shells, but $0$-shells are determined by their domain plus a realization of $p$.)
Pick any third realization $a_3$ of $p$ as a new vertex. For $i = 0, 1,$ or $2$, we construct a $1$-simplex $f_{i3}$ based over $A$ with support $\{i,3\}$ by letting $f_{i3}(\{i\}) = \operatorname{acl}_A(a_i)$ and $f_{i3}(\{3\}) = \operatorname{acl}_A(a_3)$, and then letting $f_{i3}(\{i,3\}) = \operatorname{acl}_A(a'_i, a'_3)$ where $(a'_i, a'_3)$ realizes $q(x,y)$ as in the definition of weak $3$-amalgamation (with the obvious transition maps taking $a_i$ to $a'_i$ and $a_3$ to $a'_3$).
Finally, condition~(2) in the definition of weak $3$-amalgamation implies that there are $2$-simplices $f_{123}$, $f_{023}$, and $f_{013}$ whose boundaries are alternating sums of the corresponding $f_{ij}$'s: that is,
$$\bar{p}artial f_{123} = f_{23} - f_{13} + f_{12},$$
$$\bar{p}artial f_{023} = f_{23} - f_{03} + f_{02},$$
and
$$\bar{p}artial f_{013} = f_{13} - f_{03} + f_{01}.$$
Now if $d$ is the $2$-chain $f_{013} + f_{123} - f_{023}$, then
$$\bar{p}artial d = \left( f_{13} - f_{03} + f_{01} \right) + \left(f_{23} - f_{13} + f_{12} \right) - \left(f_{23} - f_{03} + f_{02} \right)$$
$$= f_{01} + f_{12} - f_{02} = c.$$
\end{proof}
\end{example}
\section{A non-commutative groupoid construction}
In singular homology theory, one of the differences between the fundamental group and $H_1$ is that the former is not necessarily commutative while the latter is. In the authors' earlier papers \bar{c}ite{GK}, \bar{c}ite{gkk}, an analogue of homotopy theory is developed but where the ``fundamental group'' in this context is always commutative. In this last section, by taking an approach closer to the original idea of homotopy theory, we suggest how to construct a different fundamental group in a non-commutative manner. More precisely, from a full symmetric witness to the failure of $3$-uniqueness in a stable theory,
we construct a new groupoid $\operatorname{\bar{m}athfrak{C}}F$ whose ``vertex groups'' $\textup{Mor}_{\operatorname{\bar{m}athfrak{C}}F}(a,a)$ need not be abelian. In fact, we will show below that $\textup{Mor}_{\bar{c}G}(a,a) \leq \operatorname{Z}(\textup{Mor}_{\operatorname{\bar{m}athfrak{C}}F}(a,a))$, where $\bar{c}G$ is the commutative groupoid constructed in \bar{c}ite{GK} and \bar{c}ite{gkk}. We may call $\operatorname{\bar{m}athfrak{C}}F$ the {\em non-commutative groupoid } constructed from the full symmetric witness. But unlike the groupoid $\bar{c}G$, this new groupoid $\operatorname{\bar{m}athfrak{C}}F$ is definable only in certain cases (e.g. under $\omega$-categoricity); in general, it is merely invariant over some small set of parameters.
\bar{m}edskip
Throughout this section, we take the notational convention described in section 4. We recall that
$\operatorname{Aut}(A/B)$ is the group of
elementary maps from $A$ {\em onto} $A$ fixing $B$ pointwise.
In addition, $\operatorname{Aut}(\operatorname{tp}(f/B))$ means $\operatorname{Aut}(Y/B)$ where $Y$ is the solution set of
$\operatorname{tp}(f/B)$.
\bar{m}edskip
\subsection{Finitary groupoid examples}
Let $G$ be an arbitrary finite group. Now let $T_G$ be the complete
stable theory of the connected finitary groupoid $(O,M,.)$ with
the standard setting (so $.$ is the composition map between
morphisms)
such that $G_a:=\textup{Mor}(a,a)$ is isomorphic to $G$ for any $a{\hbox{\boldmath $\bar \textup{im}ath$}}n O$.
Fix distinct $a,b{\hbox{\boldmath $\bar \textup{im}ath$}}n O$ and a morphism $f_{0}{\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}(a,b)$.
Now by section 4 and weak elimination of imaginaries we know that
$$H_2(O)=\operatorname{Aut}(\bar{b}ar{w}idetilde{ab}/\overline a,\overline
b)=\operatorname{Aut}( \textup{Mor}(a,b)/aG_abG_b).$$
Hence indeed (see section 4.2 in \bar{c}ite{GK}, and note that
$\textup{Mor}(a,b)\subseteq \operatorname{dcl}(f_{0}G_a)$)
$$H_2(O) =\operatorname{Aut}(X/aG_abG_b).$$
where $X$ is the finite solution set
of $\operatorname{tp}(f_{0}/aG_abG_b)$.
Now for $f{\hbox{\boldmath $\bar \textup{im}ath$}}n X$ there is unique $x{\hbox{\boldmath $\bar \textup{im}ath$}}n G_a$ such that $f=f_0.x$,
and we claim that this $x$ must be in $\operatorname{Z}(G_a)$.
\bar{b}egin{enumerate}gin{claim}
For $x{\hbox{\boldmath $\bar \textup{im}ath$}}n G_a$, we have $g:=f_0.x{\hbox{\boldmath $\bar \textup{im}ath$}}n X$ iff $x{\hbox{\boldmath $\bar \textup{im}ath$}}n \operatorname{Z}(G_a)$.
\end{claim}
\bar{b}egin{enumerate}gin{proof}
($\Rightarrow$) Since $g{\hbox{\boldmath $\bar \textup{im}ath$}}n X$, $f_0\equiv_{G_aG_b} g$. Then for any
$y{\hbox{\boldmath $\bar \textup{im}ath$}}n G_a$, we have $$f_0.y.f^{-1}_0({\hbox{\boldmath $\bar \textup{im}ath$}}n
G_b)=g.y.g^{-1}=f_0.x.y.x^{-1}.f^{-1}_0.$$
Hence $x{\hbox{\boldmath $\bar \textup{im}ath$}}n \operatorname{Z}(G_a)$.
($\Leftarrow$) There is $z{\hbox{\boldmath $\bar \textup{im}ath$}}n G_b$ such that $f_0=z.g$. Now since
$x{\hbox{\boldmath $\bar \textup{im}ath$}}n \operatorname{Z}(G_a)$, for any $y{\hbox{\boldmath $\bar \textup{im}ath$}}n G_b$ we have
$$g^{-1}.y.g.x^{-1}=f_0^{-1}.z.y.z^{-1}.f_0.x^{-1}=x^{-1}.f_0^{-1}.z.y.z^{-1}.f_0=g^{-1}.z.y.z^{-1}.g.x^{-1}.$$
Hence $y=z.y.z^{-1}$, i.e. $z{\hbox{\boldmath $\bar \textup{im}ath$}}n \operatorname{Z}(G_b)$. Now the argument in
\bar{c}ite[4.2]{GK} says there is an automorphism fixing $aG_abG_b$
pointwise while sending $g$ to $f_0$. Hence $g{\hbox{\boldmath $\bar \textup{im}ath$}}n X$.
\end{proof}
\bar{b}egin{enumerate}gin{claim}
$H_2(O)=\operatorname{Z}(G)$.
\end{claim}
\bar{b}egin{enumerate}gin{proof}
The proof will be similar to that of Proposition 2.15 in \bar{c}ite{gkk}.
Note firstly that due to Claim 1, $\operatorname{Z}(G_a)$ acts on $X$ as an
obvious manner. This action is clearly regular. Secondly
$\operatorname{Aut}(X/aG_abG_b)$ also regularly acts on $X$. Also by the argument
in \bar{c}ite[4.2]{GK} it easily follows that two actions commute. Hence
they are the same group.
\end{proof}
Note that $f\equiv_{aG_a} f_{0}$ for any $f{\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}(a,b)$ (see again
section 4.2 in \bar{c}ite{GK}), i.e. $\textup{Mor}(a,b)$ is the solution set of
$\operatorname{tp}(f_0/aG_a)$ or $\operatorname{tp}(f_0/\overline a)$. Moreover for $f{\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}(a,b)$,
$f_0$ and $f$ are interdefinable over $\overline a$. We further claim the
following.
\bar{b}egin{enumerate}gin{claim}\langlebel{iso}
$G$ is isomorphic to $\operatorname{Aut}(\textup{Mor}(a,b)/\overline{a})=\operatorname{Aut}(\textup{Mor}(a,b)/aG_a)$.
Hence $H_2(O)=\operatorname{Aut}(\bar{b}ar{w}idetilde{ab}/\overline a,\overline
b)= \operatorname{Aut}( \textup{Mor}(a,b)/aG_abG_b)= \operatorname{Z}(\operatorname{Aut}(\textup{Mor}(a,b)/aG_a)).$
\end{claim}
\bar{b}egin{enumerate}gin{proof}
We know $G$ and $G_b$ are isomorphic. Now clearly we can consider
$\sigma{\hbox{\boldmath $\bar \textup{im}ath$}}n G_b$ as an automorphism in $\operatorname{Aut}(\textup{Mor}(a,b)/\overline{a})$ via
the map $f({\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}(a,b))\bar{m}apsto \sigma.f$. Now this correspondence
is clearly 1-1 and onto (both groups are finite). It is obvious that
the correspondence is an isomorphism.
\end{proof}
In the following section we try to search this phenomenon
in the general stable theory context. Namely given the abelian groupoid
built from a {\em symmetric witness}
introduced in \bar{c}ite{GK}, we construct an extended groupoid
possibly non-abelian but the abelian groupoid places in the
center of the new groupoid. In the case of
above $T_G$, as we seen the morphism group of the abelian groupoid
is $\operatorname{Z}(G)$, but in the extended one the morphism group is equal to
$G$.
\subsection{The non-commutative groupoid $\operatorname{\bar{m}athfrak{C}}F$}
For the rest of this section, we work in a complete \emph{stable} theory $T$ with
monster model $\operatorname{\bar{m}athfrak{C}}=\operatorname{\bar{m}athfrak{C}}^{eq}$.
We will work with full symmetric witnesses to the failure of $3$-uniqueness as defined in Definition~\ref{full_symm_witness} above.
First we fix some notation that we will refer to throughout the rest of the section. Let $(b_0, b_1, b_2, f'_{01}, \ldots)$ be a full symmetric witness to the failure of $3$-uniqueness, for convenience over the base set $\emptyset = \operatorname{acl}(\emptyset)$. Recall from the discussion in section~4 above that from this witness we can construct a definable connected abelian groupoid $\bar{c}G$ such that $\operatorname{Ob}(\bar{m}athcal{G})=p(\operatorname{\bar{m}athfrak{C}})$ where $p=\operatorname{tp}(b_i)$, and there is a
canonical bijection $\bar{p}i$ from the finite solution set of
$\operatorname{tp}(f'_{01}/\overline{b_0} \bar{c}up \overline{b_1})$ to $\textup{Mor}_{\bar{m}athcal{G}}(b_0,b_1)$ in such a way that
$f_{01}:=\bar{p}i(f'_{01})$ and $f'_{01}$ are interdefinable over $b_{01}$, so
$\textup{Mor}_{\bar{m}athcal{G}}(b_0,b_1)$ also is the solution set of
$\operatorname{tp}(f_{01}/\overline{b_0} \bar{c}up \overline{b_1})$ (equivalently of $\operatorname{tp}(f_{01}/\overline{b_0} \bar{c}up \overline{b_1})$). Moreover
the abelian group $\textup{Mor}_{\bar{m}athcal{G}}(b_i,b_i)$ is isomorphic to
$\operatorname{Aut}(\operatorname{tp}(f_{01}/b_{01}))=\operatorname{Aut}(\operatorname{tp}(f'_{01}/\overline{b_0} \bar{c}up \overline{b_1}))$.
Now as promised, by extending the construction method given in \bar{c}ite{GK} we find
another groupoid $\operatorname{\bar{m}athfrak{C}}F$ (which will be definable {\em only in a certain context})
from the same full symmetric witness $(b_0, b_1, b_2, f'_{01}, \ldots)$.
$\textup{Ob}(\operatorname{\bar{m}athfrak{C}}F)$ will be the same as $\textup{Ob}(\bar{m}athcal{G})$. But $\operatorname{\bar{m}athfrak{C}}F$ need not be
abelian as $\textup{Mor}_{\operatorname{\bar{m}athfrak{C}}F}(b_i,b_i)$ will be $\operatorname{Aut}(Y_{01}/\overline b_0)$, where $Y_{01}$ is the
{\em possibly infinite} set
$$Y_{b_{01}}=Y_{01}:=\{f{\hbox{\boldmath $\bar \textup{im}ath$}}n\operatorname{dcl}(f_{01},\overline{b_0})|\ f\equiv_{\overline b_0}f_{01}
\text{ and } \operatorname{dcl}(f\overline{b_0})=\operatorname{dcl}(f_{01}\overline{b_0})\}.$$ Note that
$\operatorname{dcl}(f_{01},\overline{b_0})=\operatorname{dcl}(f_{01}b_1\overline{b_0})$ since $b_1{\hbox{\boldmath $\bar \textup{im}ath$}}n\operatorname{dcl}(f_{01})$.
Moreover $Y_{01}$ and $Y'_{01}$, the set defined the same way as $Y_{01}$ but substituting $b_1f'_{0}$ for $f_{01}$, are interdefinable. Furthermore, we shall see that $\textup{Mor}_{\bar{m}athcal{G}}(b_i,b_i)\leq
\operatorname{Z}(\textup{Mor}_{\operatorname{\bar{m}athfrak{C}}F}(b_i,b_i))$ (Claim \ref{centergp}). We will call $\operatorname{\bar{m}athfrak{C}}F$
the {\em non-commutative groupoid} constructed from the symmetric witness.
\bar{b}egin{enumerate}gin{claim}
The set $Y_{01}$ defined in the previous paragraph depends only on $b_0$ and $b_1$ and not on the choice of $f_{01} {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}_{\bar{c}G}(b_0, b_1)$.
\end{claim}
\bar{b}egin{enumerate}gin{proof}
Given any other $g {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}_{\bar{c}G}(b_0, b_1)$, we have that $g \equiv_{\overline{b}_0} f_{01}$, as already discussed. Also, $g = f_{01} \bar{c}irc h$ for some $h {\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}_{\bar{c}G}(b_0, b_0) \subseteq \operatorname{acl}(b_0)$, and so $g$ and $f_{01}$ are interdefinable over $\operatorname{acl}(b_0)$. Now the result follows.
\end{proof}
For convenience, fix independent $a,b\bar{m}odels p$ and $f_{ab}$ such
that $b_{01}f_{01}\equiv abf_{ab}$. We use ${\bar{m}athcal P}i_{ab}$ to denote $\textup{Mor}_{\bar{m}athcal{G}}(a,b)$, and use ${\bar{m}athcal P}i_{a}$
for ${\bar{m}athcal P}i_{aa}$. As mentioned above, ${\bar{m}athcal P}i_{ab}$ is the
solution set of $\operatorname{tp}(f_{ab}/\overline a\overline b)$, on which
$G_{ab}:=\operatorname{Aut}(\operatorname{tp}(f_{ab}/\overline a\overline b))$ acts regularly. Hence
$G_{ab}$ and ${\bar{m}athcal P}i_a$ are canonically isomorphic \bar{c}ite[2.15]{gkk}.
\bar{b}egin{enumerate}gin{lemma}\langlebel{uniformaction}
A set $C=\{c_i\}_i$ of realizations of $p$ with $b_0\Ind C$, and
$g_i{\hbox{\boldmath $\bar \textup{im}ath$}}n{\bar{m}athcal P}i_{b_0c_i}$ are given. Then for $\sigma{\hbox{\boldmath $\bar \textup{im}ath$}}n {\bar{m}athcal P}i_{b_0}$,
there is an automorphism $\bar{m}u=\bar{m}u_{\sigma}$ of $\operatorname{\bar{m}athfrak{C}}$ fixing each
$\overline{c_i}$ and $\overline{b_0}$ pointwise and $\bar{m}u(g_i)=g_i.\sigma$. Similarly, if $D=\{d_i\}_i(\Ind b_0)$ is a set of realizations of
$p$ and $h_i{\hbox{\boldmath $\bar \textup{im}ath$}}n{\bar{m}athcal P}i_{d_ib_0}$, then there is an automorphism $\tau$
fixing $\overline{d_i}$ and $\overline{b_0}$ such that $\tau(h_i)=\sigma.h_i.$
\end{lemma}
\bar{b}egin{enumerate}gin{proof} Take $d\bar{m}odels p$ independent from $b_0C$; and take $h{\hbox{\boldmath $\bar \textup{im}ath$}}n
{\bar{m}athcal P}i_{b_0d}$. For each $i$, there is $h_i{\hbox{\boldmath $\bar \textup{im}ath$}}n {\bar{m}athcal P}i_{dc_i}$ such that
$g_i=h_i.h$. Now by stationarity we have $g_{0}\equiv_{\overline{b_0},
\overline{Cd}} g_0.\sigma$ witnessed by an automorphism $\bar{m}u$ sending
$g_0$ to $g_0.\sigma$ and fixing $\overline{b_0}, \overline{Cd}$. Then
$\bar{m}u(g_i)=\bar{m}u(h_i.h)=h_i.\bar{m}u(h)$ since $h_i{\hbox{\boldmath $\bar \textup{im}ath$}}n \overline{Cd}$. Now there
is unique $\tau{\hbox{\boldmath $\bar \textup{im}ath$}}n {\bar{m}athcal P}i_{b_0}$ such that $\bar{m}u(h)=h.\tau$. Thus
$\bar{m}u(g_0)=g_0.\sigma=h_0.h.\tau$. Hence $\sigma=\tau$. Similarly
there is $\tau_i{\hbox{\boldmath $\bar \textup{im}ath$}}n{\bar{m}athcal P}i_{b_0}$ such that $\bar{m}u(g_i)=g_i.\tau_i$, and
then $\bar{m}u(g_i)=g_i.\tau_i=h_i.h.\sigma$. Hence $\tau_i=\sigma$, so
$\bar{m}u(g_i)=g_i.\sigma$ as desired. The second clause can be proved similarly.
\end{proof}
Now consider $F_{b_{01}}=F_{01}:=\operatorname{Aut}(Y_{01}/\overline{b_0})$ where
$Y_{01}$ is defined above.
\bar{b}egin{enumerate}gin{claim}\langlebel{centergp}
\bar{b}egin{enumerate}
{\hbox{\boldmath $\bar \textup{im}ath$}}tem ${\bar{m}athcal P}i_{b_{01}}\subseteq Y_{01}$.
{\hbox{\boldmath $\bar \textup{im}ath$}}tem The action of $F_{01}$ on $Y_{01} $ (obviously by $\sigma(g)$ for $\sigma{\hbox{\boldmath $\bar \textup{im}ath$}}n
F_{01}$ and $g{\hbox{\boldmath $\bar \textup{im}ath$}}n Y_{01}$) is regular (so $|F_{01}|=|Y_{01}|$ but
can be infinite). Hence given $\bar{m}u{\hbox{\boldmath $\bar \textup{im}ath$}}n G_{01}:=G_{b_{01}}$, there is
its unique extension in $F_{01}$ (we may identify those two). Thus
$Y_{01}$ is $b_{01}$-invariant set.
{\hbox{\boldmath $\bar \textup{im}ath$}}tem $G_{01}\leq\operatorname{Z}(F_{01}).$
{\hbox{\boldmath $\bar \textup{im}ath$}}tem For $\tau{\hbox{\boldmath $\bar \textup{im}ath$}}n F_{01}$ and $f{\hbox{\boldmath $\bar \textup{im}ath$}}n {\bar{m}athcal P}i_{b_{01}}$, $e{\hbox{\boldmath $\bar \textup{im}ath$}}n
{\bar{m}athcal P}i_{b_0}$, we have $\tau(f.e)=\tau(f).e.$ Moreover if
$\sigma(f)=f.e$ for some $\sigma{\hbox{\boldmath $\bar \textup{im}ath$}}n G_{01}$, then
$\sigma(f,\tau(f))=(f.e,\tau(f).e)$. \end{enumerate}
\end{claim}
\bar{b}egin{enumerate}gin{proof} (1) is clear.
(2) comes from the fact that for any $g_0,g_1{\hbox{\boldmath $\bar \textup{im}ath$}}n Y_{01}$, they are
interdefinable over $\overline{b_0}$, and $Y_{01} \subseteq \operatorname{dcl}(g_i\bar{b}ar
b_0)=\operatorname{dcl}(f_{01}\bar{b}ar b_0)$. Hence from (1),
it follows $G_{01}$ is
a subgroup of $F_{01}$. The rest clearly follows.
(3) Suppose $\sigma{\hbox{\boldmath $\bar \textup{im}ath$}}n G_{01},\tau{\hbox{\boldmath $\bar \textup{im}ath$}}n F_{01}$ are given. Let
$g=\sigma(f_{01})=f_{01}.\sigma_0$ for some $\sigma_0{\hbox{\boldmath $\bar \textup{im}ath$}}n {\bar{m}athcal P}i_{b_0}$, and
let $h=\tau(f_{01})$. Then
$\tau(f_{01},g)=(\tau(f_{01}),\tau(g))=(h,\tau\bar{c}irc \sigma(f_{01})).$ But
since $\tau$ fixes $\overline{b_0}$, it follows $\tau\bar{c}irc
\sigma(f_{01})=\tau(f_{01}.\sigma_0)=\tau(f_{01}).\sigma_0=h.\sigma_0..$ Now
by Lemma \ref{uniformaction}, there is an automorphism $\sigma'$
fixing $\overline{b}_0$ and $\overline{b}_1$ and sending $(f_{01},h)$ to $(f_{01}.\sigma_0,h.\sigma_0)$, so
$\sigma(f_{01})=\sigma'(f_{01})=f_{01}.\sigma$. But then due to the
uniqueness of the extension of $\sigma$ in $F_{01}$, we must have
that $\sigma (h)=h.\sigma_0=\sigma'(h)$ as well. Thus
$\tau\bar{c}irc\sigma(f_{01})=h.\sigma_0=\sigma(h)=\sigma\bar{c}irc\tau(f_{01})..$
Then due to regularity, we conclude $\sigma{\hbox{\boldmath $\bar \textup{im}ath$}}n \operatorname{Z}(F_{01})$.
(4) By (3), $\tau(f.e)=\tau\bar{c}irc\sigma(f)=\sigma\bar{c}irc\tau(f)$. But
again by uniqueness with Lemma \ref{uniformaction},
$\sigma(\tau(f))=\tau(f).e$. Therefore
$\sigma(f,\tau(f))=(\sigma(f),\tau(\sigma(f)))=(f.e,\tau(f).e)$.
\end{proof}
In general $G_{01}$ need not be equal to $\operatorname{Z}(F_{01})$ (see the remarks
before Proposition \ref{isomorphic}).
We define $Y_{ab}$ just like $Y_{01}$ but with $b_{01}f_{01}$ replaced by $abf_{ab}$.
\bar{b}egin{enumerate}gin{lemma}\langlebel{wdfn}
Let $c\bar{m}odels p$ and $c\Ind ab$. Let $g{\hbox{\boldmath $\bar \textup{im}ath$}}n {\bar{m}athcal P}i_{ca}$. Then for $f{\hbox{\boldmath $\bar \textup{im}ath$}}n
Y_{ab}$, it follows $h=f.g{\hbox{\boldmath $\bar \textup{im}ath$}}n Y_{cb}$. Moreover for $h_0=f_{ab}.g$,
we have
$$h_0f_{ab}\equiv_{\overline{ac}}hf\bar{m}box{ and }f_{ab}f\overline a\equiv_{\overline{b}} h_0h\overline
c.$$
\end{lemma}
\bar{b}egin{enumerate}gin{proof}
Note that $h_0{\hbox{\boldmath $\bar \textup{im}ath$}}n {\bar{m}athcal P}i_{cb}$. By stationarity, there is a
$\overline{ca}$-automorphism $\bar{m}u$ such that $\bar{m}u(f_{ab})=f$. Then
$\bar{m}u(h_0)=\bar{m}u(f_{ab}.g)=\bar{m}u(f_{ab}).\bar{m}u(g)=f.g=h{\hbox{\boldmath $\bar \textup{im}ath$}}n\overline{cb}$. We want
to see that $h,h_0$ are interdefinable over $\overline{c}$. Suppose not
say there is $h'\equiv_{\overline{c}h_0}h$ and $h'{\hbox{\boldmath \small $\bar n$}}e h$. Then again by
stationarity there is a $\overline{ca}$-automorphism $\tau$ such that
$\tau(h_0h)=\tau(h_0h')$. Then for $f=h.g^{-1}$ and $f'=h'.g^{-1}$,
we have $f{\hbox{\boldmath \small $\bar n$}}e f'$ but $\tau(f_{ab},f)=\tau(h_0.g^{-1},h.g^{-1})
=(h_0.g^{-1},h'.g^{-1})=(f_{ab},f'),$ a contradiction. Similarly one
can show that $h_0{\hbox{\boldmath $\bar \textup{im}ath$}}n\operatorname{dcl}(\overline{c}h)$. Hence $h{\hbox{\boldmath $\bar \textup{im}ath$}}n Y_{cb}$. Now $\bar{m}u$
witnesses $h_0f_{ab}\equiv_{\overline{ac}}hf$. To show $f_{ab}f\overline
a\equiv_{\overline{b}} h_0h\overline c$, choose $d(\bar{m}odels p)\Ind abc$. Now for
$k_0{\hbox{\boldmath $\bar \textup{im}ath$}}n {\bar{m}athcal P}i_{db}$, by our proof there is $k{\hbox{\boldmath $\bar \textup{im}ath$}}n Y_{db}$ such that
$f=k.(k^{-1}_0.f_{ab})$. Then
$h=k.k^{-1}_0.(f_{ab}.g)=k.k^{-1}_0.h_0.$ Now by stationarity,
$f_{ab}\overline a\equiv_{\overline{bd}}h_0\overline c$. Since $k,k_0{\hbox{\boldmath $\bar \textup{im}ath$}}n \overline{bd}$, as
desired $f_{ab}f\overline a\equiv_{\overline{bd}} h_0h\overline c$.
\end{proof}
Now we start to construct the new groupoid mentioned. At the first
approximation, our $\textup{Mor}_{\operatorname{\bar{m}athfrak{C}}F}(a,b)$ will be $Y_{ab}$. Beware that
$Y_{ab}(\supseteq {\bar{m}athcal P}i_{ab})$ need not be definable nor
type-definable. It is just an $ab$-invariant set. So our groupoid
$\operatorname{\bar{m}athfrak{C}}F$ will only be invariant, and it will be definable only under additional hypotheses (e.g. $\omega$-categoricity).
As explained in \bar{c}ite[2.9, 2.10]{gkk}, there is the binding group
$G$ (isomorphic to $G_{ab}$, and so to ${\bar{m}athcal P}i_a$) acting on $\bar{m}athcal{G}$. In
general the action is {\em not} a structure automorphism of the
groupoid as for example $\operatorname{id}_a{\hbox{\boldmath $\bar \textup{im}ath$}}n {\bar{m}athcal P}i_a$ need not be fixed. But it
is so for ${\bar{m}athcal P}i_{ab}$ (or more generally as in Lemma
\ref{uniformaction} above), i.e. for $\bar{b}sigma{\hbox{\boldmath $\bar \textup{im}ath$}}n G$ we have
$f_{ab}\equiv \bar{b}sigma\bar{c}d f_{ab}=f_{ab}\bar{c}d \bar{b}sigma{\hbox{\boldmath $\bar \textup{im}ath$}}n {\bar{m}athcal P}i_{ab}$ (see
\bar{c}ite[2.10]{gkk}; We use $\bar{c}d$ for the group action of $G$ to $\bar{m}athcal{G}$).
Hence there is an induced isomorphism $\rho_{ab}:G\to G_{ab}$ such
that $\rho_{ab}(\bar{b}sigma)(f)=\bar{b}sigma\bar{c}d f$ for any $f{\hbox{\boldmath $\bar \textup{im}ath$}}n {\bar{m}athcal P}i_{ab}$.
We write
$\sigma_{ab}$ for $\rho_{ab}(\bar{b}sigma)$. But when there is no chance
of confusion, we use $\sigma$ for both $\bar{b}sigma{\hbox{\boldmath $\bar \textup{im}ath$}}n G$ and
$\sigma_{ab}{\hbox{\boldmath $\bar \textup{im}ath$}}n G_{ab}$. Also, $\sigma_a$ denotes the unique element
in $\bar{b}sigma\bar{c}ap {\bar{m}athcal P}i_a$, as described in \bar{c}ite[Definition 2.9]{gkk}.
Hence for $f{\hbox{\boldmath $\bar \textup{im}ath$}}n{\bar{m}athcal P}i_{ab}$, $\sigma(f)=\sigma\bar{c}d
f=\sigma_b.f=f.\sigma_a$.
\bar{b}egin{enumerate}gin{claim}\langlebel{tpreserv}
For $\sigma{\hbox{\boldmath $\bar \textup{im}ath$}}n G$, and $f{\hbox{\boldmath $\bar \textup{im}ath$}}n {\bar{m}athcal P}i_{ab}$ and $g{\hbox{\boldmath $\bar \textup{im}ath$}}n {\bar{m}athcal P}i_{cd}$ with
$cd\equiv ab$, we have $f,\sigma\bar{c}dot f\equiv g,\sigma\bar{c}dot g$.
\end{claim}
\bar{b}egin{enumerate}gin{proof} Choose $e\bar{m}odels p$ independent from $abcd$. By
\ref{uniformaction} above, $f,\sigma\bar{c}dot f\equiv h,\sigma\bar{c}dot h
\equiv k,\sigma\bar{c}dot k \equiv g,\sigma\bar{c}dot g$ where $h{\hbox{\boldmath $\bar \textup{im}ath$}}n{\bar{m}athcal P}i_{ae}$
and $k{\hbox{\boldmath $\bar \textup{im}ath$}}n {\bar{m}athcal P}i_{ce}$.
\end{proof}
Now let $F_{ab}:=\operatorname{Aut}(Y_{ab}/\overline{a})$. Then as in Claim
\ref{centergp}.(2), $G_{ab}\leq F_{ab}$. As just said for any
$cd\equiv ab$, there is the canonical isomorphism between
$\rho_{cd}\bar{c}irc\rho^{-1}_{ab}:G_{ab}\to G_{cd}$.
We somehow try to find the
canonically extended isomorphism between $F_{ab}$ and $F_{cd}$ as
well. We do this as follows. Let $Y_{ab}=\{g_i\}_i\bar{c}up \{g'_j\}_j$
and let $Y_{cd}=\{h_i\}_i\bar{c}up \{h'_j\}_j$ such that
${\bar{m}athcal P}i_{ab}=\{g_i\}_i$, ${\bar{m}athcal P}i_{cd}=\{h_i\}_i$ and the sequences $\langle
g_i\rangle^\frown \langle g'_j\rangle ab\equiv \langle h_i\rangle^\frown \langle h'_j\rangle
cd$. Now due to regularity of the action, for each $i$ or $j$ there
is unique $\bar{m}u^{ab}_i$ or $\bar{m}u^{ab}_j{\hbox{\boldmath $\bar \textup{im}ath$}}n F_{ab}$ such that
$\bar{m}u_i(g_0)=g_i$ or $\bar{m}u_j(g_0)=g'_j$. Similarly we have
$\bar{m}u^{cd}_i$ or $\bar{m}u^{cd}_j{\hbox{\boldmath $\bar \textup{im}ath$}}n F_{cd}.$
\bar{b}egin{enumerate}gin{claim}
The correspondence $\bar{m}u^{ab}_i\bar{m}apsto\bar{m}u^{cd}_i$ or
$\bar{m}u^{ab}_j\bar{m}apsto\bar{m}u^{cd}_j$ is a well-defined isomorphism from
$F_{ab}$ to $F_{cd}$ extending $\rho_{cd}\bar{c}irc\rho^{-1}_{ab}$.
\end{claim}
\bar{b}egin{enumerate}gin{proof}
Assume $\{k_i\}_i\bar{c}up \{k'_j\}_j$ is another arrangement of $Y_{cd}$
such that $\langle k_i\rangle^\frown \langle k'_j\rangle \equiv_{cd}\langle
h_i\rangle^\frown \langle h'_j\rangle$. Then $k_0=\sigma(h_0)$ for some
$\sigma{\hbox{\boldmath $\bar \textup{im}ath$}}n G_{cd}$. Thus by Claim \ref{centergp}, we have
$\sigma(h_0,\bar{m}u^{cd}_i(h_0))=(k_0,\bar{m}u^{cd}_i(k_0))$ and so
$h_0,\bar{m}u^{cd}_i(h_0)\equiv_{\overline c}k_0,\bar{m}u^{cd}_i(k_0)$. Then due to
interdefinability, we must have $\bar{m}u^{cd}_i(k_0)=k_i$. Similarly
$\bar{m}u^{cd}_j(k_0)=k'_j$. Hence the map is well-defined. It easily
follows that the map in fact is an isomorphism. Moreover due to
\ref{tpreserv} we see that it extends
$\rho_{cd}\bar{c}irc\rho^{-1}_{ab}$.
\end{proof}
Hence now we fix an {\em extended binding group} $F\geq G$
isomorphic to $F_{01}$. Then there is a canonical isomorphism
$\rho^F_{cd}:F\to F_{cd} $ extending $\rho_{cd}$ in such a way that
$\rho^F_{cd}\bar{c}irc(\rho^F_{ab})^{-1}$ is the correspondence defined
above. Now for $\bar{b}mu{\hbox{\boldmath $\bar \textup{im}ath$}}n F$, we use $\bar{m}u_{ cd}$ or simply $\bar{m}u$ to
denote $\rho^F_{cd}(\bar{b}mu)$. Note that a mapping $\bar{b}mu\bar{c}dot
f:=\bar{m}u_{cd}(f)$ is clearly a regular action of $F$ on $Y_{cd}$
extending that of $G$ on ${\bar{m}athcal P}i_{cd}$.
\bar{b}egin{enumerate}gin{claim}
If $cd\Ind a$, then for $f{\hbox{\boldmath $\bar \textup{im}ath$}}n{\bar{m}athcal P}i_{cd}, g{\hbox{\boldmath $\bar \textup{im}ath$}}n{\bar{m}athcal P}i_{ac}$, we have
$\bar{b}mu\bar{c}dot (f.g)= (\bar{b}mu\bar{c}dot f).g$.
\end{claim}
\bar{b}egin{enumerate}gin{proof}
This follows from Lemma \ref{wdfn}.
\end{proof}
Assume now $c(\bar{m}odels p)\Ind ab$, and $g{\hbox{\boldmath $\bar \textup{im}ath$}}n Y_{ab}, h{\hbox{\boldmath $\bar \textup{im}ath$}}n Y_{bc}$ are
given. We want to define a composition $h.g{\hbox{\boldmath $\bar \textup{im}ath$}}n Y_{ac}$ extending
that for $\bar{m}athcal{G}$. Note now $g=\tau_0(g_0)$ and $h=\sigma_0(h_0)$ for
some $\bar{m}box{\bar{b}oldmath $\tau_0$}, \bar{m}box{\bar{b}oldmath $\sigma_0$}{\hbox{\boldmath $\bar \textup{im}ath$}}n F$
and $g_0{\hbox{\boldmath $\bar \textup{im}ath$}}n {\bar{m}athcal P}i_{ab}, h_0{\hbox{\boldmath $\bar \textup{im}ath$}}n{\bar{m}athcal P}i_{bc}$. We define $h.g:=
(\bar{m}box{\bar{b}oldmath $\sigma_0$}\bar{c}irc \bar{m}box{\bar{b}oldmath
$\tau_0$})\bar{c}dot(h_0.g_0)=\sigma_0\bar{c}irc\tau_0(h_0.g_0).$
\bar{b}egin{enumerate}gin{claim}\langlebel{compost}
The composition map is well-defined, invariant under any automorphism of $\operatorname{\bar{m}athfrak{C}}$, and extends that of $\textup{Mor}(\bar{m}athcal{G})$. For any $f{\hbox{\boldmath $\bar \textup{im}ath$}}n Y_{ac}$,
there is unique $h'{\hbox{\boldmath $\bar \textup{im}ath$}}n Y_{bc}$ such that $f=h'.g$.
\end{claim}
\bar{b}egin{enumerate}gin{proof}
Let $g=\tau_1(g_1)$ and $h=\sigma_1(h_1)$ for some $\bar{m}box{\bar{b}oldmath
$\tau_1$}, \bar{m}box{\bar{b}oldmath $\sigma_1$}{\hbox{\boldmath $\bar \textup{im}ath$}}n F$ and $g_1{\hbox{\boldmath $\bar \textup{im}ath$}}n {\bar{m}athcal P}i_{ab},
h_1{\hbox{\boldmath $\bar \textup{im}ath$}}n{\bar{m}athcal P}i_{bc}$. Then since $\sigma^{-1}_0\bar{c}irc\sigma_1(h_1)=h_0$
and $\tau^{-1}_0\bar{c}irc\tau_1(g_1)=g_0$, due to uniqueness we have
that both $\bar{m}box{\bar{b}oldmath $\sigma^{-1}_0$}\bar{c}irc \bar{m}box{\bar{b}oldmath
$\sigma_1$}$, $\bar{m}box{\bar{b}oldmath $\tau^{-1}_0$}\bar{c}irc \bar{m}box{\bar{b}oldmath
$\tau_1$}$ are in $G$ so in the center of $F$. Now
$$\bar{b}egin{enumerate}gin{array}{cllll}
\sigma_0\bar{c}irc\tau_0(h_0.g_0)&=
&\sigma_0\bar{c}irc\tau_0\bar{c}irc\sigma^{-1}_0\bar{c}irc\sigma_0(h_0.g_0)&=&
\sigma_0\bar{c}irc\tau_0\bar{c}irc\sigma^{-1}_0(\sigma_0(h_0).g_0)\\
&=&\sigma_0\bar{c}irc\tau_0\bar{c}irc\sigma^{-1}_0(\sigma_1(h_1).g_0)&=&
\sigma_0\bar{c}irc\tau_0\bar{c}irc(\sigma^{-1}_0\bar{c}irc\sigma_1)(h_1.g_0)\\
&=&\sigma_1\bar{c}irc\tau_0(h_1.g_0)&=
&\sigma_1\bar{c}irc\tau_1\bar{c}irc(\tau^{-1}_1\bar{c}irc\tau_0)(h_1.g_0)\\
&=&\sigma_1\bar{c}irc\tau_1(h_1.(\tau^{-1}_1\bar{c}irc\tau_0)(g_0))&=&
\sigma_1\bar{c}irc\tau_1(h_1.(\tau^{-1}_1(\tau_1(g_1))))\\
&=&\sigma_1\bar{c}irc\tau_1(h_1.g_1).& &
\end{array}$$
Automorphism invariance clearly follows from the same property for
$\textup{Mor}(\bar{m}athcal{G})$ and the choice of the isomorphism $\rho^F_{ab}$. Moreover
by taking $\tau_0=\sigma_0=\operatorname{id}$, we see that the composition clearly
extends that for $\bar{m}athcal{G}$. Lastly $f=\tau(f_1)$ for some $f_1{\hbox{\boldmath $\bar \textup{im}ath$}}n
{\bar{m}athcal P}i_{ac}$. Now there is $h'_1{\hbox{\boldmath $\bar \textup{im}ath$}}n {\bar{m}athcal P}i_{bc}$ such that $f_1=h'_1.g_1.$
Put $h'=\tau\bar{c}irc\tau_1^{-1}(h'_1)$. Then by the definition,
$f=h'.g$. For any $h''({\hbox{\boldmath \small $\bar n$}}e h'){\hbox{\boldmath $\bar \textup{im}ath$}}n Y_{bc}$ it easily follows that
$f{\hbox{\boldmath \small $\bar n$}}e h''.g$. Hence $h'$ is unique such element.
\end{proof}
The rest of the construction of $\operatorname{\bar{m}athfrak{C}}F$ will be similar to that of $\bar{m}athcal{G}$ in
\bar{c}ite{GK}. $\bar{m}box{Ob}(\operatorname{\bar{m}athfrak{C}}F)$ will be the same as
$\bar{m}box{Ob}(\bar{m}athcal{G})=p(\operatorname{\bar{m}athfrak{C}})$. Now for arbitrary $c,d\bar{m}odels p$, an {\em
$n$-step directed path} from $c$ to $d$ is a sequence
$(c_0,g_1,c_1,g_2...,c_n)$ such that $c=c_0, d=c_n, c_{i-1}c_i\equiv
ab$ and $g_i{\hbox{\boldmath $\bar \textup{im}ath$}}n Y_{c_{i-1}c_i}$. Let $D^n(c,d)$ be the set of all
$n$-step directed paths. For $q=(c_0,g_1,c_1,g_2...,c_n){\hbox{\boldmath $\bar \textup{im}ath$}}n
D^n(c,d)$ and $r=(d_0,h_1,d_1,h_2...,d_m){\hbox{\boldmath $\bar \textup{im}ath$}}n D^m(c,d)$ we say they
are equivalent (write $r\sim s$) if for some $c^*(\bar{m}odels p)\Ind
qr$ and $g^*{\hbox{\boldmath $\bar \textup{im}ath$}}n Y_{c^*c}$, we have $g^*_n=h^*_m{\hbox{\boldmath $\bar \textup{im}ath$}}n Y_{c^*d}$ where
$g^*_0=h^*_0=g^*$ and $g^*_{i+1}=g_{i+1}.g_i^*$ ($i=0,...,n-1$) and
$h^*_{j+1}=h_{j+1}.h_j^*$ ($j=0,...,m-1$). Due to stationarity the
relation is independent from the choices of $c^*$ and
$g^*$, and is an equivalence relation. Similarly to Lemma
\bar{c}ite[2.12]{GK}, one can easily see using Claim \ref{compost} that
for any $q{\hbox{\boldmath $\bar \textup{im}ath$}}n D^n(c,d)$, there is $r{\hbox{\boldmath $\bar \textup{im}ath$}}n D^2(c,d)$ such that $q\sim
r$. Then $D^2(c,d)/\sim$ will be our $\textup{Mor}_{\operatorname{\bar{m}athfrak{C}}F}(c,d)$, and
composition will be concatenation of paths. The identity morphism in
$\textup{Mor}_{\operatorname{\bar{m}athfrak{C}}F}(c,c)$ can be defined just like in \bar{c}ite[2.15]{GK}.
Now our groupoid $\operatorname{\bar{m}athfrak{C}}F$ clearly extends $\bar{m}athcal{G}$. An argument similar to that in
\bar{c}ite[2.14]{GK} implies there is a canonical 1-1 correspondence between
$Y_{ab}$ and $\textup{Mor}_{\operatorname{\bar{m}athfrak{C}}F}(a,b).$ But $\operatorname{\bar{m}athfrak{C}}F$ need not be definable nor
type-definable nor hyperdefinable. It is just an invariant groupoid.
As pointed out in \ref{centergp}, $Y_{ab}$ is $ab$-invariant. Now
if it is type-definable then as it is a bounded union of definable
sets, by compactness it indeed is definable and a finite set. (This
happens when $T$ is $\omega$-categorical.) For this case let us add
a bit more explanations that are not explicitly mentioned in
\bar{c}ite{GK}. By compactness now, $\sim$ turns out to be definable:
Note that $D^2(p):=\bar{b}igcup \{D^2(c,d)|\ c,d\bar{m}odels p\}$ is
$\emptyset$-type-definable. Then there clearly is an
$\emptyset$-definable equivalence relation $E$ on $D^2(p)$ each of
whose class is of the form $D^2(c,d)$. In each $E$-class, there are
exactly $|Y_{ab}|$-many $\sim$-classes. Hence $\sim$ is
$\emptyset$-definable relatively on $D^2(p)$ as well. Hence $[g]{\hbox{\boldmath $\bar \textup{im}ath$}}n
\textup{Mor}_{\operatorname{\bar{m}athfrak{C}}F}(c,d)$ is an imaginary element and the maps $[g]\bar{m}apsto c$
or $d$ (the first and last components of $g$) are
$\emptyset$-definable {\em domain} and {\em range} maps. Moreover
for $[f]{\hbox{\boldmath $\bar \textup{im}ath$}}n \textup{Mor}_{\operatorname{\bar{m}athfrak{C}}F}(c,d)\subseteq \overline{cd}$, we have
$[f]\equiv_{\overline c}[g]$. Therefore $\operatorname{\bar{m}athfrak{C}}F$ is a (relatively)
$\emptyset$-definable groupoid.
\bar{m}edskip
We return to the general context of an invariant $\operatorname{\bar{m}athfrak{C}}F$. For notational simplicity,
use ${\bar{m}athcal P}hi_{cd}$ to denote $\textup{Mor}_{\operatorname{\bar{m}athfrak{C}}F}(c,d)$, and use ${\bar{m}athcal P}hi_c$ for
${\bar{m}athcal P}hi_{cc}$. We finish this note by stating some observations
regarding $\operatorname{\bar{m}athfrak{C}}F$. First of all, one can construct an example where
${\bar{m}athcal P}i_a$ is not equal to $\operatorname{Z}({\bar{m}athcal P}hi_a)$ but where ${\bar{m}athcal P}hi_a$ is
abelian. In the groupoid example \bar{c}ite[Section 4.2]{GK} (which is
quite similar to $T_G$ in section 1 of this note), we may add some
of irrelevant elements to each object and also extend each morphism
tuple by capturing those elements. Then by permuting the elements we
can have larger ${\bar{m}athcal P}hi_{\bar{b}ar x}$ than ${\bar{m}athcal P}i_{\bar{b}ar x}$ where $\bar{b}ar x$
is an extended object. Now then ${\bar{m}athcal P}hi_{\bar{b}ar x}$ simply is some
direct product of ${\bar{m}athcal P}hi_{\bar{b}ar x}$, so if $G$ in \bar{c}ite[4.2]{GK} is
already abelian then $G={\bar{m}athcal P}i_{\bar{b}ar x}$ need not be equal to
$\operatorname{Z}({\bar{m}athcal P}hi_{\bar{b}ar x})={\bar{m}athcal P}hi_{\bar{b}ar x}$. But of course if we work with $T_G$
as in subsection 6.1 or the example in \bar{c}ite[4.2]{GK} as they are, then
${\bar{m}athcal P}i_x=\operatorname{Z}(G)$ and ${\bar{m}athcal P}hi_x=G$.
Now for $f{\hbox{\boldmath $\bar \textup{im}ath$}}n Y_{ab}$, we write $\underline{f}$ for its canonically
corresponding element in ${\bar{m}athcal P}hi_{ab}$. Note that for $\sigma{\hbox{\boldmath $\bar \textup{im}ath$}}n
F_{ab}$ and $f{\hbox{\boldmath $\bar \textup{im}ath$}}n Y_{ab}$, we have $\sigma(f){\hbox{\boldmath $\bar \textup{im}ath$}}n Y_{ab}$ and both
$\underline{f}, \underline{\sigma(f)}{\hbox{\boldmath $\bar \textup{im}ath$}}n {\bar{m}athcal P}hi_{ab}$. But $\sigma(\underline f)$ need not
be in ${\bar{m}athcal P}hi_{ab}$. In general $\sigma(\underline f){\hbox{\boldmath $\bar \textup{im}ath$}}n {\bar{m}athcal P}hi_{a\sigma(b)}.$
We get now the following results for $\operatorname{\bar{m}athfrak{C}}F$ similarly to those of
$\bar{m}athcal{G}$.
\bar{b}egin{enumerate}gin{proposition}\langlebel{isomorphic}
The group $F_{ab}$ is isomorphic to ${\bar{m}athcal P}hi_a$. In fact for any
$\bar{m}u{\hbox{\boldmath $\bar \textup{im}ath$}}n F_{ab}$, there is $\bar{m}u_b{\hbox{\boldmath $\bar \textup{im}ath$}}n{\bar{m}athcal P}hi_b$ such that for any $f{\hbox{\boldmath $\bar \textup{im}ath$}}n
Y_{ab}$, $\underline{\bar{m}u(f)}=\bar{m}u_b.\underline f$. Hence ${\bar{m}athcal P}hi_b=\{\bar{m}u_b|\ \bar{m}u{\hbox{\boldmath $\bar \textup{im}ath$}}n
F_{ab}\}$.
\end{proposition}
\bar{b}egin{enumerate}gin{proof}
The proof will be similar to that of \ref{iso}. Define a map
$\eta:{\bar{m}athcal P}hi_a\to F_{ab}$ such that for $\sigma{\hbox{\boldmath $\bar \textup{im}ath$}}n {\bar{m}athcal P}hi_a$ and $f{\hbox{\boldmath $\bar \textup{im}ath$}}n
Y_{ab}$, we let $\eta(\sigma)(f)=g$ where $\underline g=\sigma.\underline f$.
Clearly $\eta$ is a well-defined 1-1 map. It is onto as well since
any $\bar{m}u{\hbox{\boldmath $\bar \textup{im}ath$}}n F_{ab}$ is determined by $(f,\bar{m}u(f))$. But obviously for
some $\sigma'{\hbox{\boldmath $\bar \textup{im}ath$}}n {\bar{m}athcal P}hi_a$, we have $\eta(\sigma')(f)=\bar{m}u(f)$. By
commutativity, it easily follows that this map is in fact an
isomorphism. Now we take $\bar{m}u_b=\eta^{-1}(\bar{m}u)$.
\end{proof}
\bar{b}egin{enumerate}gin{proposition}
For $c(\bar{m}odels p)\Ind ab$ and $f{\hbox{\boldmath $\bar \textup{im}ath$}}n Y_{ab},g{\hbox{\boldmath $\bar \textup{im}ath$}}n Y_{bc}, h{\hbox{\boldmath $\bar \textup{im}ath$}}n
Y_{ac}$, we have $h=g.f$ iff $\underline h=\underline{g}.\underline{f}$.
\end{proposition}
\bar{b}egin{enumerate}gin{proof}
Since the composition relation defined in
\ref{compost} is invariant relation, we can find an $\emptyset$-invariant relation
$\theta(x,y,z)$ such that for any $a'b'c'\equiv
abc$ and $f'{\hbox{\boldmath $\bar \textup{im}ath$}}n Y_{a'b'},g'{\hbox{\boldmath $\bar \textup{im}ath$}}n Y_{b'c'}, h'{\hbox{\boldmath $\bar \textup{im}ath$}}n Y_{a'c'}$, we have
$h'=g'.f'$ iff $\theta(a'b'f',b'c'g',a'c'h')$ holds. Then the rest
proof of the proposition will be exactly the same as that of
\bar{c}ite[2.12]{gkk}, hence we omit it.
\end{proof}
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\end{thebibliography}
\end{document} |
\begin{document}
\def\spacingset#1{\renewcommand{\baselinestretch}
{#1}\small\normalsize} \spacingset{0}
\spacingset{1}
\if10
{
\title{\bf Precision education: A Bayesian nonparametric approach for handling item and examinee heterogeneity in assessment data}
\author[1]{Tianyu Pan}
\author[1]{Weining Shen}
\author[2]{Clintin P. Davis-Stober}
\author[3]{Guanyu Hu}
\affil[1]{Department of Statistics, University of California, Irvine}
\affil[2]{Department of Psychological Sciences, University of Missouri - Columbia, Columbia, MO, 65211}
\affil[3]{Department of Statistics, University of Missouri - Columbia, Columbia, MO, 65211}
\date{}
\maketitle
} \fi
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\title{\bf Precision education: A Bayesian nonparametric approach for handling item and examinee heterogeneity in assessment data}
\author[1]{Tianyu Pan}
\author[1]{Weining Shen}
\author[2]{Clintin P. Davis-Stober}
\author[3]{Guanyu Hu}
\affil[1]{Department of Statistics, University of California, Irvine}
\affil[2]{Department of Psychological Sciences, University of Missouri - Columbia, Columbia, MO, 65211}
\affil[3]{Department of Statistics, University of Missouri - Columbia, Columbia, MO, 65211}
\date{}
\maketitle
} \fi
\spacingset{1.25}
\begin{abstract}
We propose a novel nonparametric Bayesian IRT model in this paper by introducing the clustering effect at question level and further assume heterogeneity at examinee level under each question cluster, characterized by the mixture of Binomial distributions. The main contribution of this work is threefold: (1) We demonstrate that the model is identifiable. (2) The clustering effect can be captured asymptotically and the parameters of interest that measure the proficiency of examinees in solving certain questions can be estimated at a $\sqrt{n}$ rate (up to a $\log$ term). (3) We present a tractable sampling algorithm to obtain valid posterior samples from our proposed model. We evaluate our model via a series of simulations as well as apply it to an English assessment
data. This data analysis example nicely illustrates how our model can be used by test makers to distinguish different types of students and aid in the design of future tests.
\end{abstract}
\noindent
{\it Keywords: IRT model; Model averaging; Nonparametric Bayesian Method; Posterior contraction rate; Rasch model.}
\noindent
{\it }
\spacingset{1.5}
\section{Introduction}\label{sec: Intro}
Item response theory \citep[IRT;][]{mislevy1990modeling,rost1990rasch} was developed to better understand the mechanism behind an examinee passing a question (item) correctly and subsequently evaluate the discrepancies between examinees or items. The majority of IRT models assume that the response of an examinee to a question is probabilistic \citep{thomas2011value}, which is further determined by a latent parameter. For example, an accuracy parameter is within 0 and 1 if the response is True or False. In general, the latent probabilistic parameter relies on \textbf{two main factors}, i.e., the ability of an examinee and the difficulty of a question. Among all IRT models, the Rasch model \citep{rasch1993probabilistic} is one of the most well-known and widely used models thanks to its elegant format and interpretability. When being applied to a dichotomous matrix, where each column represents a test question and each row is an examinee's response, the Rasch model can be encapsulated as a logistic regression that models the $(i,j)$-th entry of the matrix using the linear predictor that contrasts the ability of the $i$-th examinee and the difficulty of the $j$-th question. While our proposed framework builds upon the Rasch model due to its simplicity, future work could consider extensions to other IRT frameworks, e.g., 2PL \citep{muraki1992generalized} and 3PL \citep{zumbo1997empirical,rouse1999advances}. We refer readers to \citet{thomas2011value} for a more comprehensive review on the history of IRT models.
Despite the usefulness of the IRT models, there is evidence that two key assumptions, \textit{unidimensionality} and \textit{local independence} \citep[Chapter 14;][]{andrich2019course}, may be violated in common applications \citep{keith1987functional,bell1988conditional,kreiner2007validity,marais2008formalizing}, that is, the ability of an examinee cannot be entirely represented by a single parameter (\textit{unidimensionality}) and the responses remain dependent after imposing the IRT model structure (\textit{local independence}). From a statistical point of view, violations of the these assumptions are inter-twined. When the mechanism behind the true data generating process cannot be recovered by using single ability or difficulty parameter that corresponds to an examinee or a question, it undoubtedly introduces conditional dependency on the responses due to lack-of-fit. Intuitively, unidimensionality is a proper assumption to unveil the Guttman pattern \citep{andrich2019course}, that is, the examinees who can correctly answer difficult questions with a certain probability should be able to answer easy questions with a higher probability. Nevertheless, this probably neglects the fact that some examinees could be more expert in handling certain questions than the others. Throughout this paper, we use the term \emph{superiority} to describe the phenomenon of an examinee being able to correctly answer classes of questions based on content type that is not captured by a uni-dimensional ability or difficulty parameter. A popular approach for accommodating multidimensional IRT data is to impose mixing structures on examinees' ability and questions' difficulty simultaneously, which generates a mixture of the Rasch models as a result. This idea has been investigated using frequentist \citep{rost1990rasch,alexeev2011spurious} and Bayesian \citep{bolt2002item,miyazaki2009bayesian,jang2018impact,sen2019model,hu2020nonparametric} approaches. Indeed, these methods enjoy more robustness and interpretability by introducing heterogeneity on examinees' ability and questions' difficulty, yet limited by the structure of the Rasch model, say these two main factors have to be linearly contrasted in each Rasch model mixture. Technically, using only the first order information of the two main factors without any interactions being considered, while ignoring possible interactions, can lead to lack-of-fit when the two main factors interact truly with one another in complex ways. \citet{bartolucci2017nonparametric} solves this problem by integrating the two main factors into a united accuracy parameter for each question, whilst assuming a global mixing structure on the accuracy parameters and letting them share a common sorting order over the mixing components. This can be interpreted as those examinees who have higher accuracy when answering questions of that same type. In addition, such construction introduces heterogeneity and dependency between the two main factors by quantifying an examinee's proficiency to a question using a mixture of accuracy parameters.
However, none of the aforementioned works address the identifiability problem of a Binomial mixture distribution, which is crucial since it is not difficult to discover that the density functions are equivalent between $\text{Bernoulli}(0.5)$ and $0.5\times\text{Bernoulli}(0.1) + 0.5\times\text{Bernoulli}(0.9)$. If the model is not identifiable, researchers cannot expect a fast mixing when performing the Gibbs sampling, and even question the necessity of introducing a mixing structure, let alone ensure the consistency on the mixing parameters and interpret the results.
Motivated by the works introduced above, we propose a multidimensional IRT model based on nonparametric Bayesian procedure, which is termed as Averaged Constrained Binomial Mixture (ACBM). The objective of this paper is to relax the independence assumption intrinsically implied by the Rasch model, to model the \textit{superiority} phenomenon and to address the identifiability issue. Before we present our model, we could first imagine a heterogeneous pattern at both the examinee level and the question level that corresponds to the \textit{superiority} phenomenon, namely, examinees form several groups in each hypothetical cluster of the questions according to their proficiency in handling this type of questions, while the grouping pattern could differ across the question clusters. To model this idea, we consider the following steps. First, given a dichotomous response matrix, we aim to discover a partition over all questions such that in each question cluster, an examinees' responses to these questions can be characterized by a binomial distribution, of which the accuracy parameter, i.e., the examinee's proficiency in answering these questions, follows a mixing distribution. Technically, our proposed Bayesian model is essentially a prior over all mixing distributions given a partition on these questions, joint with a prior on all possible partitions on these questions. The key novelties of our proposed model are in interpretation and the theoretical guarantees of identifiability and posterior consistency. Our model can infer a partition on questions that reveals information at both levels. At the question level, in each question cluster, the examinees are automatically distinguished by their proficiency in tackling these questions using a mixing distribution. This allows us to discover a more complex accuracy patterns than just simple Guttman patterns \citep{andrich2019course}. At the examinee level, each mixing accuracy parameter under a given question cluster represents the proficiency of a specific group of examinees in handling these questions, which provides information for precision education. This, for example, provides statistical evidence to the test maker to identify the examinees who are not skilled in solving a certain question type. This information, in turn, is useful for designing and implementing additional questions within this type. In addition, the identifiability of our model is ensured by putting a dynamic upper bound on the number of mixing components in each question cluster, where the upper bound is determined by the size of the question cluster. We later address the identifiability of our model in Lemma \ref{1st-identifiability}. Our method can be tractably applied using an MCMC sampling algorithm for realization and is able to capture the true question partition and estimate the mixing parameters at a $\sqrt{n}$ rate (up to a $\log$ term), thanks to the rapid developments of Bayesian analysis over the past 30 years, in both computational \citep{neal2000markov,ishwaran2001gibbs} and theoretical \citep{nobile1994bayesian,shen2013adaptive,ho2016strong,ghosal2017fundamentals,guha2019posterior} research.
The rest of this article is organized as follows. The motivation and interpretation to our model is given in Section \ref{sec: Motivation}. The convergence results are presented in Section \ref{sec: Conv}, with the proof being deferred to the Supplementary File. In Section \ref{sec: Bayesian_inference}, we outline the MCMC sampling algorithm and introduce statistics to summarize the obtained posterior samples. The simulation study that validates our theoretical results and compares the performance between our model and the Rasch model is provided in Section \ref{sec: Simulation}. We carry out a real data analysis by applying our model to a test data set in Section \ref{sec: real data}. In Section \ref{sec: Discussion}, we discuss several possible ways to generalize our model, which paves the way for the future study. For ease of exposition, proofs, computation algorithms and
additional technical results are given in the supplementary materials.
\section{Motivation}\label{sec: Motivation}
Consider an $n$ by $D$ dichotomous matrix $\mathbf{X}$, whose $(i,j)$-th entry is a random variable, denoted by $X_{i,j}$, which takes value 1 if the $i$-th examinee answered the $j$-th question correctly and 0 otherwise, we let $\mathcal{D}\equiv\{1,2,\ldots,D\}$ be an index set on the total number of itmes, $\mathcal{C}$ be a partition of $\mathcal{D}$, and $c\in\mathcal{C}$ be a cluster of items defined by the partition $\mathcal{C}$. To illustrate the connection between $\mathcal{C}$ and $\mathcal{D}$ using an example, we take $D$ to be 3 and a partition $\mathcal{C}$ of $\mathcal{D}$ can be $\mathcal{C}=\{\{1,2\},\{3\}\}$. For this simple example, there are just two clusters, $c_{1}=\{1,2\}$ and $c_{2}=\{3\}$. Our model construction is motivated by the following example. Suppose a professor creates a test to evaluate her students, with the questions being classified into multiple clusters, with each cluster being an element $c \in \mathcal{C}$. Each question cluster $c$ can ideally distinguish different types of the students, of which the total number of types is given by $K^{(c)}$. Moreover, the $k$-th type indicates the proficiency of a specific group of students in answering a certain question type $c$, that is, the \textit{superiority}, which can further be quantified by an accuracy parameter $\theta^{(c)}_k$, for $1\leq k\leq K^{(c)}$. Inspired by this idea, we propose the following hierarchical order to model this structural clustering pattern,
\begin{equation}\label{ConsBinModel}
\begin{split}
& X_{1,j},\ldots,X_{n,j}\stackrel{i.i.d}{\sim} p^{(c)}_{F},~\text{for}~\forall j\in c,~\forall c\in\mathcal{C},\\
& p^{(c)}_{F}(x)= \sum_{k=1}^{K^{(c)}}w^{(c)}_{k}\times (1-\theta^{(c)}_k)^{1-x}\times (\theta^{(c)}_k)^{x},~\text{for}~x=0~\text{or}~1,\\
& (w^{(c)}_{1},\ldots,w^{(c)}_{K^{(c)}})\sim \text{Dir}(K^{(c)},(\alpha,\ldots,\alpha)),\\
& \theta^{(c)}_k\stackrel{i.i.d}{\sim}\text{Beta}(a_0,b_0),~\text{for}~k=1,\ldots,K^{(c)},\\
& K^{(c)}\sim q_0^{(c)}(K^{(c)};\gamma),\\
& \mathcal{C}\sim m(\mathcal{C}),
\end{split}
\end{equation}
where $\text{Dir}(K,(\alpha,\ldots,\alpha))$ refers to a Dirichlet distribution with $K$ categories and a concentration parameter $\alpha > 0$, $\text{Beta}(a_0,b_0)$ denotes a Beta distribution whose shape parameters are $a_0$ and $b_0$, $m(\mathcal{C})$ denotes a probability mass function over all possible partitions on $\mathcal{D}$, $q^{(c)}_0(\cdot;\gamma)$ denotes the Poisson distribution parameterized by $\gamma$ and truncated between $1$ and $(|c|+1)/2$ and $|\cdot|$, when being applied to a set $c$, refers to its cardinality instead of the absolute value when being applied to a real number. Note that the truncation on $q^{(c)}_0(\cdot;\gamma)$ refers to the maximum limit of a question cluster $c$ in dividing the students into different levels by their proficiency in answering a certain type of questions. This limitation relies on the question cluster size $|c|$. The term $(|c|+1)/2$ is the upper limit on the value $K^{(c)}$ and is a function of the number of items within the cluster $c$. This upper limit can be interpreted as the ``resolution'' of the question cluster. The superscript $^{(c)}$ highlights that the mixing weights and the number of mixtures are allowed to vary across question clusters.
The main feature of our model lies in the dependency between the clustering structures at two levels. Specifically, the mixing distribution at the examinee-level is allowed to vary across question clusters, whereas most of the existing methods \citep{bolt2002item,miyazaki2009bayesian,jang2018impact,sen2019model,hu2020nonparametric} suggest that the mixing distribution at the examinee-level is invariant regardless of the heterogeneity at question-level. We conclude three main benefits for this feature of our model. Our model provides more interpretable results, say the heterogeneity in the examinees for each question type, compared with heuristically assuming mixing distributions on the two main factors separately, which can be hardly interpreted in this way. Such structural heterogeneity allows us to discover more complicated patterns than the Guttman pattern. For example, in a math test, it is reasonable to believe that some examinees are more proficient in algebra questions than they are in geometric questions. While some other examinees are more expert in solving geometric questions but not good at algebra questions. This cannot be explained by the Guttman mechanism, but can be justified by the \textit{superiority} phenomenon. In addition, we demonstrate that our model is identifiable when our chosen upper bound on the number of mixtures is imposed for each question cluster. Indeed, such an upper bound could inevitably lead to information loss when the size of a question cluster is small, but it intuitively makes sense because one cannot distinguish the proficiency of examinees to a certain type of questions only using very few of them. Moreover, the identifiability of our model further contributes to the identification of the true clustering structure on the questions, and meanwhile, provides $\sqrt{n}$ (up to a log term) estimates to the mixing weights and parameters in each question cluster under mild conditions.
In the next section, we proceed to present the technical details of our proposed method and introduce the posterior consistency results.
\section{Convergence results}\label{sec: Conv}
\subsection{Notations}\label{sec: notations}
In addition to the notations introduced in Section \ref{sec: Motivation}, throughout the rest of this paper, we will use $f(\cdot)$ or $f$ to denote a function $f(x)$ if it only takes a single argument. We let $X_i^{(c)}$ denote the $c$-section of random vector $X_i$, where $X_i$ refers to the dichotomous response of the $i$-th examinee. Similar definition is endowed to $x^{(c)}$ with respect to vector $\mathbf{x}$. We also let
\begin{equation}\label{trueDensityUni}
\begin{split}
& p_{F_0}^{(c)}(x)=\sum_{k=1}^{K^{(c)}_0}w_{k,0}^{(c)}\times(1-\theta_{k,0}^{(c)})^{1-x}\times(\theta_{k,0}^{(c)})^x
\end{split}
\end{equation}
be the true density function associated with a single question in cluster c. The corresponding true number of mixtures, mixing weights and component-wise parameters of $p_{F_0}^{(c)}(x)$ are hence $K_0^{(c)}$, $\{w_{k,0}^{(c)}\}_{k=1}^{K_0^{(c)}}$ and $\{\theta_{k,0}^{(c)}\}_{k=1}^{K_0^{(c)}}$. Analogously, a density function sampled following \eqref{ConsBinModel} is denoted by $p_F^{(c)}(x)$, whose number of mixtures, mixing weights and component-wise parameters are $K^{(c)}$, $\{w_{k}^{(c)}\}_{k=1}^{K^{(c)}}$ and $\{\theta_{k}^{(c)}\}_{k=1}^{K^{(c)}}$. We denote the true partition by $\mathcal{C}_0$ and any partition sampled following $m(\cdot)$ by $\mathcal{C}$. For random vector $X_i$, we let $p_{F_0}(\mathbf{x})$ and $p_F(\mathbf{x})$ be the true density function and a sampled density function following \eqref{ConsBinModel}, respectively. We also let $P_0$ and $\mathbb{P}_{F_0}$ be the probability distribution and the probability measure induced from $p_{F_0}$, and the expectation taken under $\mathbb{P}_{F_0}$ is denoted by $\mathbb{P}_{F_0}[f]$ or $\mathbb{P}_{F_0}f$.
For succinctness, we denote the prior on $p_F$ in \eqref{ConsBinModel} by $\Pi$ and let $\Pi^{(c)}$ for $\forall c\in\mathcal{C}$ be the prior on $p_F^{(c)}$ given $\mathcal{C}$. We define the set of all possible binomial mixtures as,
\begin{equation}\label{BinMixtures}
\begin{split}
& \mathcal{P}^{(c)}=\bigcup_{K=1}^{+\infty}\mathcal{P}^{(c)}(K)\equiv \left\{p^{(c)}_F(x^{(c)}):p^{(c)}_F(x^{(c)})=\sum_{k=1}^Kw^{(c)}_k(1-\theta_k^{(c)})^{|c|-\|x^{(c)}\|_1}\times (\theta_k^{(c)})^{\|x^{(c)}\|_1},\right.\\
& \left. ~w^{(c)}_k\in(0,1),~\text{for}~k=1,\ldots,K,~\{\theta^{(c)}_{k}\}_{k=1}^K~\text{are distinct}\right\},
\end{split}
\end{equation}
where the operator $\|\cdot\|_1$ refers to the $\ell_1$ norm when being applied to a vector, and in our situation, it is equivalent to a summation function as $x^{(c)}$ is a non-negative dichotomous vector. We point out that the prior $\Pi^{(c)}$ is essentially supported on $\mathcal{P}^{(c)}$ by truncating $K$ between 1 and $(|c|+1)/2$. For every $\mathcal{C}$ in the support of $m(\mathcal{C})$, we let $\Pi_{\mathcal{C}}\equiv \prod_{c\in\mathcal{C}}\Pi^{(c)}$ denote the prior on $\prod_{c\in\mathcal{C}}p_F^{(c)}$. It follows that
\begin{equation}\label{avgPrior}
\begin{split}
& \Pi = \sum_{\mathcal{C}\in\mathcal{S}}m(\mathcal{C})\Pi_{\mathcal{C}},
\end{split}
\end{equation}
where $\mathcal{S}$ refers to the collection of all possible partitions of $\mathcal{D}$. For every $c\in\mathcal{C}$ and every $\mathcal{C}\in\mathcal{S}$, we use $p_{F_0}^{(c)}(x^{(c)})\equiv \int p_{F_0}(\mathbf{x})d x^{(\mathcal{D}\setminus c)}$ to denote the marginal density of $p_{F_0}$ on the sub-vector $x^{(c)}$ and $p_{F}^{(c)}(x^{(c)})$ to denote the joint density function on $x^{(c)}$ given $p_F^{(c)}(x)$, with similar definition as \eqref{trueDensityUni}.
\subsection{Convergence results}\label{sec: consistency}
We begin with the interpretations to the following three assumptions,
\begin{enumerate}
\item[(A1)] The true partition $\mathcal{C}_0$ is in the support of $m(\mathcal{C})$.
\item[(A2)] For $\forall c\in\mathcal{C}_0$, it holds for $p_{F_0}^{(c)}$ that
\begin{equation}\label{A2}
\begin{split}
& \emph{\text{Distinctness:}}~0 < \theta_{i,0}^{(c)}\neq\theta_{j,0}^{(c)} < 1,~\text{for}~1\leq i\neq j\leq K_0^{(c)},\\
& \emph{\text{Non-trivial weights:}}~w_{k,0}^{(c)}>0,~\text{for every}~1\leq k\leq K_0^{(c)},\\
& \emph{\text{Bounded component number:}}~K_0^{(c)}\leq \frac{|c|+1}{2}.
\end{split}
\end{equation}
\item[(A3)] For every $\mathcal{C}$ in the support of $m(\mathcal{C})$, the true density has at least $\epsilon_0 > 0$ distance away from the best estimation induced by $\mathcal{C}$ with respect to the Kullback–Leibler divergence\footnote{The Kullback–Leibler divergence between $f(x)$ and $g(x)$ is defined as
$$\text{KL}(f(\mathbf{x});g(\mathbf{x})) = \int\log\left(\frac{f(\mathbf{x})}{g(\mathbf{x})}\right)f(\mathbf{x})d\mathbf{x}.$$}, that is,
$$\left\|\prod_{c\in\mathcal{C}_0}p_{F_0}^{(c)}(x^{(c)})-\prod_{c\in\mathcal{C}}p_{F^*}^{(c)}(x^{(c)})\right\|_1 > \epsilon_0,$$ where $p_{F^*}^{(c)}(x^{(c)})$, for $\forall c\in\mathcal{C}$ is obtained by minimizing $\text{KL}\left(p_{F_0}^{(c)}(x^{(c)});p_{F}^{(c)}(x^{(c)})\right),$ with respect to $p_{F}^{(c)}(x^{(c)})\in\mathcal{P}^{(c)}$.
\end{enumerate}
The first two assumptions are quite standard. Assumption (A1) is necessary to ensure that our model is correctly specified. The only eye-catching part in Assumption (A2) is the constraint on the number of components under each question cluster. This constraint aims to guarantee that the binomial mixture part in our proposed model is identifiable, as noted by \citet{teicher1963identifiability}. Later in Lemma \ref{1st-identifiability}, we will prove that the binomial mixture model is identifiable at the first-order. Assumption (A3) is a weaker condition than the general identifiability assumption. In fact, the density function $\prod_{c\in\mathcal{C}}p^{(c)}_F(x^{(c)})$ is non-identifiable for some instances of $\left\{\{w_k^{(c)},\theta_k^{(c)}\}_{c\in\mathcal{C}},\mathcal{C}\right\}$, for example, the corresponding density functions are identical given different parameterizations $$\{\{w_1^{c} = 1, \theta_1^{(c)} = 0.5\}_{c = \{1,2\}},\{w_1^{c} = 1, \theta_1^{(c)} = 0.5\}_{c = \{3\}}\}$$ and $\{\{\theta_1^{(c)} = 1,\theta_1^{(c)} = 0.5\}_{c = \{1,2,3\}}\}$ when $D = 3$. Therefore, one cannot directly apply Doob's theorem \citep{doob1949application} but need to resort to Assumption (A3) and Theorem \ref{selectionConsistency} for the consistency on $\mathcal{C}$.
\begin{theorem}\label{selectionConsistency}
Assume (A1) and (A3) are satisfied, then it holds that
\begin{equation}\label{eqSelectionConsistency}
\begin{split}
& \Pi(\mathcal{C}=\mathcal{C}_0\mid X_{1},\ldots,X_{n})\to 0~~a.s.~P_0.
\end{split}
\end{equation}
\begin{comment}
\begin{proof}
By Bayes' theorem, \eqref{eqSelectionConsistency} can be detailed as,
\begin{equation}\label{BFSelectionConsistency}
\begin{split}
& \Pi(\mathcal{C}=\mathcal{C}_0\mid X_{1},\ldots,X_{n})\\
& = \frac{m(\mathcal{C}_0)}{m(\mathcal{C}_0) + \sum_{\mathcal{C}\in\mathcal{S},\mathcal{C}\neq\mathcal{C}_0}m(\mathcal{C})\prod_{i=1}^n\frac{\int \prod_{c\in\mathcal{C}}p_F^{(c)}(X_i^{(c)})d\Pi^{(c)}(p_F^{(c)})}{\int \prod_{c\in\mathcal{C}_0}p_F^{(c)}(X_i^{(c)})d\Pi^{(c)}(p_F^{(c)})}},
\end{split}
\end{equation}
and we further define
\begin{equation}\label{BayesFactor}
\begin{split}
\text{BF}_{\mathcal{C},\mathcal{C}_0}^n & = \frac{\int \prod_{i=1}^n p_F(X_i)d\Pi_{\mathcal{C}}(p_F^{(c)})}{\int \prod_{i=1}^np_F(X_i)d\Pi_{\mathcal{C}_0}(p_F)},\\
& = \frac{\int \prod_{i=1}^n \prod_{c\in\mathcal{C}}p_F^{(c)}(X_i^{(c)})d\Pi^{(c)}(p_F^{(c)})}{\int \prod_{i=1}^n\prod_{c\in\mathcal{C}_0}p_F^{(c)}(X_i^{(c)})d\Pi^{(c)}(p_F^{(c)})}
\end{split}
\end{equation}
the Bayes factor by comparing the model evidence between the model induced by $\mathcal{C}$ with the one induced by $\mathcal{C}_0$. Our remaining work is to show $\text{BF}_{\mathcal{C},\mathcal{C}_0}^n\to 0,~~a.s.~P_0$, for every $ \mathcal{C}\in\mathcal{S}$ and $\mathcal{C}\neq\mathcal{C}_0$. Note that
\begin{equation}\label{BFfactorCancel}
\begin{split}
\text{BF}_{\mathcal{C},\mathcal{C}_0}^n & = \frac{\int \prod_{i=1}^n \prod_{c\in\mathcal{C}}p_F^{(c)}(X_i^{(c)})d\Pi^{(c)}(p_F^{(c)})}{\int \prod_{i=1}^n\prod_{c\in\mathcal{C}_0}p_F^{(c)}(X_i^{(c)})d\Pi^{(c)}(p_F^{(c)})},\\
& = \frac{\int \prod_{i=1}^n \frac{\prod_{c\in\mathcal{C}}p^{(c)}_F(X^{(c)}_i)}{\prod_{c\in\mathcal{C}_0}p^{(c)}_{F_0}(X^{(c)}_i)} d\Pi^{(c)}(p_F^{(c)})}{\int \prod_{i=1}^n\frac{\prod_{c\in\mathcal{C}_0}p^{(c)}_F(X^{(c)}_i)}{\prod_{c\in\mathcal{C}_0}p^{(c)}_{F_0}(X^{(c)}_i)}d\Pi^{(c)}(p_F^{(c)})},\\
& = \frac{\prod_{i=1}^n\frac{\prod_{c\in\mathcal{C}}p^{(c)}_{F^*}(X^{(c)}_i)}{\prod_{c\in\mathcal{C}_0}p^{(c)}_{F_0}(X^{(c)}_i)}\prod_{c\in\mathcal{C}}\int\prod_{i=1}^n\frac{p^{(c)}_F(X^{(c)}_i)}{p^{(c)}_{F^*}(X^{(c)}_i)} d\Pi^{(c)}(p^{(c)}_F)}{\prod_{c\in\mathcal{C}_0}\int \prod_{i=1}^n\frac{p^{(c)}_F(X^{(c)}_i)}{p^{(c)}_{F_0}(X^{(c)}_i)}d\Pi^{(c)}(p^{(c)}_F)},
\end{split}
\end{equation}
and in fact, for the numerator, the following term can be upper bounded by
\begin{equation}\label{expDist}
\begin{split}
& \prod_{i=1}^n\frac{\prod_{c\in\mathcal{C}}p^{(c)}_{F^*}(X^{(c)}_i)}{\prod_{c\in\mathcal{C}_0}p^{(c)}_{F_0}(X^{(c)}_i)}\\
& = \exp\left\{-n\times \frac{1}{n}\log\left(\frac{\prod_{c\in\mathcal{C}_0}p^{(c)}_{F_0}(X^{(c)}_i)}{\prod_{c\in\mathcal{C}}p^{(c)}_{F^*}(X^{(c)}_i)}\right)\right\},\\
& \to \exp\left\{-n\times \int \log\left(\frac{p_{F_0}(\mathbf{x})}{\prod_{c\in\mathcal{C}}p^{(c)}_{F^*}(x^{(c)})}\right)p_{F_0}(\mathbf{x})d\mathbf{x}\right\},~~a.s.~P_0,\\
& \stackrel{(a)}{\leq} \exp\left\{-2 n\times\epsilon^2_0\right\},
\end{split}
\end{equation}
where the inequality $(a)$ is by the fact that $\int f(x)\log(\frac{f(x)}{g(x)})dx\geq 2\rho^2_H(f,g)\geq 2\|f-g\|^2_1$, where $\rho_H(\cdot,\cdot)$ refers to the Hellinger distance.
For the second term in the numerator, we first denote
$$Z_n^{(c)} = \int\prod_{i=1}^n\frac{p^{(c)}_F(X^{(c)}_i)}{p^{(c)}_{F^*}(X^{(c)}_i)}d\Pi^{(c)}(p^{(c)}_F).$$ Note that
\begin{equation}\label{ExpectIntegralbyPart}
\begin{split}
& \mathbb{P}_{F_0}^n Z_n^{(c)} = \mathbb{P}_{F_0^{(c)}}^n Z_n^{(c)},
\end{split}
\end{equation}
where $\mathbb{P}^n_{F_0}$ refers to the product measure $\otimes_{i=1}^n\mathbb{P}_{F_0}$ for short and $\mathbb{P}_{F_0^{(c)}}$ is the probability measure induced by $p_{F_0}^{(c)}$. By applying Fubini's theorem, we further reformulate \eqref{ExpectIntegralbyPart} into
\begin{equation}\label{Law01}
\begin{split}
\mathbb{P}_{F_0^{(c)}}^n Z_n^{(c)} & = \int \prod_{i=1}^n P_{F_0^{(c)}}\left[\frac{p_{F}^{(c)}}{p_{F^*}^{(c)}}\right]d\Pi^{(c)}(p^{(c)}_F),\\
& \stackrel{(a)}{\leq} 1,
\end{split}
\end{equation}
where the inequality (a) is obtained by the fact that $\mathcal{P}^{(c)}$ is a convex model, the definition of $p_{F^*}^{(c)}$ and applying Theorem 8.39 of \citet{ghosal2017fundamentals}.
To lower bound the denominator of \eqref{BFfactorCancel}, for each question cluster, we limit the integration to a small neighborhood around the true parameter values, i.e.,
\begin{equation}\label{l1Neighbor}
\begin{split}
I_{w^{(c)},K_0^{(c)}}(\epsilon_n)&\equiv \left\{(w_1^{(c)},\ldots,w_{K_0^{(c)}}^{(c)}): 0 \leq w_i^{(c)}-w_{i,0}^{(c)}\leq\epsilon_n,~\text{for}~i=1,\ldots,K_0^{(c)} \right\},\\
I_{\theta_{k,0}^{(c)}}(\epsilon_n,\delta)&\equiv\left\{\theta_k^{(c)}:(1-\epsilon_n^{2(1+\delta)})\times(1-\theta_{k,0}^{(c)})\leq 1-\theta_k^{(c)}\leq 1-\theta_{k,0}^{(c)}\right\},\\
I_{K_0^{(c)}}^{(c)}(\epsilon_n) & \equiv I_{w^{(c)},K_0^{(c)}}(\epsilon_n)\times I_{\theta_{1,0}^{(c)}}(\epsilon_n)\times\ldots \times I_{\theta_{K_0^{(c)},0}^{(c)}}(\epsilon_n),
\end{split}
\end{equation}
where the operator $\times$ refers to the cartesian product between two sets and $\delta>0$ is a small positive constant. These yield to
\begin{equation}\label{denomLower}
\begin{split}
& \int \prod_{i=1}^n\frac{p^{(c)}_F(X^{(c)}_i)}{p^{(c)}_{F_0}(X^{(c)}_i)}d\Pi_1^{(c)}(p^{(c)}_F)\\
& \geq \int_{I_{K_0^{(c)}}^{(c)}(\epsilon_n),K^{(c)}=K_0^{(c)}}\prod_{i=1}^n\frac{p^{(c)}_F(X^{(c)}_i)}{p^{(c)}_{F_0}(X^{(c)}_i)}d\Pi_1^{(c)}(p^{(c)}_F),\\
& \stackrel{(a)}{\gtrsim}\int_{I_{K_0^{(c)}}^{(c)}(\epsilon_n)}\prod_{i=1}^n\frac{\sum_{k=1}^{K_0^{(c)}}w_k^{(c)}(1-\theta_k^{(c)})^{|c|}\left(\frac{\theta_k^{(c)}}{1-\theta_k^{(c)}}\right)^{\|X_i^{(c)}\|_1}}{\sum_{k=1}^{K_0^{(c)}}w_{k,0}^{(c)}(1-\theta_{k,0}^{(c)})^{|c|}\left(\frac{\theta_{k,0}^{(c)}}{1-\theta_{k,0}^{(c)}}\right)^{\|X_i^{(c)}\|_1}}dw_{1}^{(c)}d\theta_1^{(c)}\ldots dw_{K_0^{(c)}}^{(c)}d\theta_{K_0^{(c)}}^{(c)},\\
& \stackrel{(b)}{\gtrsim}\int_{I_{K_0^{(c)}}^{(c)}(\epsilon_n)}\prod_{i=1}^n\frac{(1-\epsilon_n^{2(1+\delta)})^D\sum_{k=1}^{K_0^{(c)}}w_{k,0}^{(c)}(1-\theta_{k,0}^{(c)})^{|c|}\left(\frac{\theta_{k, 0}^{(c)}}{1-\theta_{k, 0}^{(c)}}\right)^{\|X_i^{(c)}\|_1}}{\sum_{k=1}^{K_0^{(c)}}w_{k,0}^{(c)}(1-\theta_{k,0}^{(c)})^{|c|}\left(\frac{\theta_{k,0}^{(c)}}{1-\theta_{k,0}^{(c)}}\right)^{\|X_i^{(c)}\|_1}}dw_{1}^{(c)}d\theta_1^{(c)}\ldots dw_{K_0^{(c)}}^{(c)}d\theta_{K_0^{(c)}}^{(c)},\\
& \stackrel{(c)}{\gtrsim} (1-\epsilon_n^{2(1+\delta)})^{nD}\times \epsilon_n^{K_0^{(c)}-1}\times \epsilon_n^{2(1+\delta)K_0^{(c)}},
\end{split}
\end{equation}
where the operator $\|\cdot\|_1$ in the inequality (a) refers to the $\ell_1$ norm of a vector, the inequality (b) is obtained by lower bounding or upper bounding $\theta_{k}$ following the definition of $I_{w^{(c)},K_0^{(c)}}(\epsilon_n)$ and $I_{\theta_{k,0}^{(c)}}(\epsilon_n,\delta)$ and the fact that $|c|\leq D$, and the inequality (c) is concluded by Assumption (A3) that the mixing weights are non-zero and $\{\theta_{k,0}\}_{k=1}^{K_0^{(c)}}$ are bounded away from 0 and 1.
Define
\begin{equation}\label{ABC}
\begin{split}
& A_n^{\mathcal{C}} = \prod_{i=1}^n\frac{\prod_{c\in\mathcal{C}}p^{(c)}_{F^*}(X^{(c)}_i)}{\prod_{c\in\mathcal{C}_0}p^{(c)}_{F_0}(X^{(c)}_i)}\times \exp\{n\times \epsilon_0^2\},\\
& B_n = \left[\prod_{c\in\mathcal{C}_0}\int \prod_{i=1}^n\frac{p^{(c)}_F(X^{(c)}_i)}{p^{(c)}_{F_0}(X^{(c)}_i)}d\Pi^{(c)}(p^{(c)}_F)\right]^{-1}\times \epsilon_n^{2\times \sum_{c\in\mathcal{C}_0}(K_0^{(c)}-1)}\times \epsilon_n^{4(1+\delta)\sum_{c\in\mathcal{C}_0}K_0^{(c)}},\\
& C_n^{\mathcal{C}} = \prod_{c\in\mathcal{C}}\int\prod_{i=1}^n\frac{p^{(c)}_F(X^{(c)}_i)}{p^{(c)}_{F^*}(X^{(c)}_i)} d\Pi^{(c)}(p^{(c)}_F)\times \epsilon_n^{2(1+\delta)},\\
& D_n = \exp\{-n\times\epsilon_0^2\}\times \epsilon_n^{-2\times \sum_{c\in\mathcal{C}_0}(K_0^{(c)}-1)}\times \epsilon_n^{-4(1+\delta)\sum_{c\in\mathcal{C}_0}K_0^{(c)}}\times \epsilon_n^{-2(1+\delta)},
\end{split}
\end{equation}
and take $\epsilon_n = \frac{\log(n)}{\sqrt{n}}$, based on the last display of \eqref{BFfactorCancel}, it holds that
$$\text{BF}_{\mathcal{C},\mathcal{C}_0}^n=A_n^{\mathcal{C}}\times B_n\times C_n^{\mathcal{C}}\times D_n.$$ Notice that for a random variable $Y_n \geq 0$, if its expectation $E(Y_n)\leq \epsilon_n^{2(1+\delta)}$, we then consider the event $\mathcal{E}_n$ such that $Y_n \geq \frac{1}{\log(n)}$ and it follows that
\begin{equation}\label{Borel-Cantelli}
\begin{split}
& P_0(\mathcal{E}_n)\leq \log(n)\times \frac{(\log(n))^{2(1+\delta)}}{n^{1+\delta}}.
\end{split}
\end{equation}
Since $\sum_{n=1}^{\infty}\frac{(\log(n))^{1+2\delta}}{n^{1+\delta}}<\infty$, then by Borell-Cantelli Lemma, it holds that
$$Y_n\to 0,~~a.s.~P_0.$$
It is not difficult to prove that the expectation of $A_n^{\mathcal{C}}$, $B_n$, $C_n^{\mathcal{C}}$ and $D_n$ with respect to $\mathbb{P}_{F_0}^n$ can be bounded by $\epsilon_n^{2(1+\delta)}$ up to a positive constant for every $\mathcal{C}\in\mathcal{S}$, which further implies
$$\text{BF}_{\mathcal{C},\mathcal{C}_0}^n\to 0,~~a.s.~P_0,$$ by the continuous mapping theorem. These finish the proof by applying the continuous mapping theorem again on the last display of \eqref{BFSelectionConsistency}.
\end{proof}
\end{comment}
\end{theorem}
Theorem \ref{selectionConsistency} states that our proposed model can correctly identify the latent question partition asymptotically, that is, when sample size increases, we can expect that the question clustering configuration sampled from the posterior distribution of our model will eventually converge to the true question partition.
To pave the way to our next theorem, we prove that the binomial mixture model is identifiable at the first order, which is defined as follows.
\begin{lemma}\label{1st-identifiability}
(First-order Identifiability) Assume that $\{\theta_i\}_{i=1}^K$ are distinct and $\{\alpha_i\}_{i=1}^K$ and $\{\beta_i\}_{i=1}^K$ are real-valued coefficients and (A3) holds. Suppose that
\begin{equation}\label{1stIdent}
\begin{split}
& \sum_{i=1}^K \alpha_if(y\mid,\theta_i,n)+\sum_{i=1}^K\beta_i \frac{\partial f}{\partial \theta_i}f(y\mid,\theta_i,n)= 0,
\end{split}
\end{equation}
where $f(y\mid,\theta_i,n) = {n\choose y}(1-\theta_i)^{n-y}(\theta_i)^y$ and
$$\frac{\partial f}{\partial \theta_i}f(y\mid,\theta_i,n) = \mathbbm{1}(1\leq y\leq n)n{n-1\choose y-1}(1-\theta_i)^{n-1-(y-1)}(\theta_i)^{y-1} - $$
$$\mathbbm{1}(0\leq y\leq n-1)n {n-1\choose y}(1-\theta_i)^{n-1-y}(\theta_i)^{y},$$ then it implies $\alpha_1=\beta_1=\ldots = \alpha_K=\beta_K=0$, if $1\leq K \leq (n+1)/2$.
\begin{comment}
\begin{proof}
When it holds that $n = 1$ or $n=2$ and $K=1$, the proof is trivial and will be omitted. Consider the case when $n = 2k+1$ or $n = 2k+2$ and $K\geq 2$ given $k\in\mathbb{Z}_{>0}$, we focus on the moment generating function of \eqref{1stIdent}, which gives
\begin{equation}\label{moment1st}
\begin{split}
& \sum_{i=1}^K \alpha_i(1-\theta_i+\theta_i e^t)^n + \beta_i n(e^t-1)(1-\theta_i+\theta_i e^t)^{n-1} = 0,
\end{split}
\end{equation}
and further implies
\begin{equation}\label{coefZero}
\begin{split}
& \sum_{i=1}^K \alpha_i{n\choose j}(1-\theta_i)^{n - j}(\theta_i)^{j} + \beta_i n{n-1\choose j-1}(1-\theta_i)^{n-1 - (j-1)}(\theta_i)^{j-1} -\\
& \beta_i n{n-1\choose j}(1-\theta_i)^{n-1 - j}(\theta_i)^{j} = 0,\\
& \implies \begin{pmatrix}
(1-\theta_1)^n & 0 & \cdots & 0 \\
0 & (1-\theta_2)^n & \ddots &\vdots \\
\vdots & \ddots & \ddots & 0 \\
0 & \cdots & 0 & (1-\theta_K)^n
\end{pmatrix}\times \begin{pmatrix}
(\frac{\theta_1}{1-\theta_1})^j & 0 & \cdots & 0 \\
0 & (\frac{\theta_2}{1-\theta_2})^j & \ddots &\vdots \\
\vdots & \ddots & \ddots & 0 \\
0 & \cdots & 0 & (\frac{\theta_K}{1-\theta_K})^j
\end{pmatrix}\times \begin{pmatrix}
\alpha_1 \\
\vdots \\
\alpha_K
\end{pmatrix} = \\
& \begin{pmatrix}
(1-\theta_1)^{n-1} & 0 & \cdots & 0 \\
0 & (1-\theta_2)^{n-1} & \ddots &\vdots \\
\vdots & \ddots & \ddots & 0 \\
0 & \cdots & 0 & (1-\theta_K)^{n-1}
\end{pmatrix}\times \begin{pmatrix}
(\frac{\theta_1}{1-\theta_1})^{j-1} & 0 & \cdots & 0 \\
0 & (\frac{\theta_2}{1-\theta_2})^{j-1} & \ddots &\vdots \\
\vdots & \ddots & \ddots & 0 \\
0 & \cdots & 0 & (\frac{\theta_K}{1-\theta_K})^{j-1}
\end{pmatrix}\times\\
& \left[j\times I - (n-j)\times \begin{pmatrix}
(\frac{\theta_1}{1-\theta_1}) & 0 & \cdots & 0 \\
0 & (\frac{\theta_2}{1-\theta_2}) & \ddots &\vdots \\
\vdots & \ddots & \ddots & 0 \\
0 & \cdots & 0 & (\frac{\theta_K}{1-\theta_K})
\end{pmatrix}\right]\times \begin{pmatrix}
\beta_1 \\
\vdots \\
\beta_K
\end{pmatrix},\\
& \implies \begin{pmatrix}
\alpha_1 \\
\vdots \\
\alpha_K
\end{pmatrix} = \left[j\times\begin{pmatrix}
(\frac{1}{\theta_1}) & 0 & \cdots & 0 \\
0 & (\frac{1}{\theta_2}) & \ddots &\vdots \\
\vdots & \ddots & \ddots & 0 \\
0 & \cdots & 0 & (\frac{1}{\theta_K})
\end{pmatrix}-(n-j)\times\begin{pmatrix}
(\frac{1}{1-\theta_1}) & 0 & \cdots & 0 \\
0 & (\frac{1}{1-\theta_2}) & \ddots &\vdots \\
\vdots & \ddots & \ddots & 0 \\
0 & \cdots & 0 & (\frac{1}{1-\theta_K})
\end{pmatrix}\right]\times \begin{pmatrix}
\beta_1 \\
\vdots \\
\beta_K
\end{pmatrix},
\end{split}
\end{equation}
where $j=1,\ldots, n-1$. Take subtraction between $j=1$ and $j=2$ for the last display of \eqref{coefZero}, which is doable since $n-1\geq 2$, we conclude that
\begin{equation}\label{0equality}
\begin{split}
& \begin{pmatrix}
0 \\
\vdots \\
0
\end{pmatrix} = \left[\begin{pmatrix}
(\frac{1}{\theta_1}) & 0 & \cdots & 0 \\
0 & (\frac{1}{\theta_2}) & \ddots &\vdots \\
\vdots & \ddots & \ddots & 0 \\
0 & \cdots & 0 & (\frac{1}{\theta_K})
\end{pmatrix}+\begin{pmatrix}
(\frac{1}{1-\theta_1}) & 0 & \cdots & 0 \\
0 & (\frac{1}{1-\theta_2}) & \ddots &\vdots \\
\vdots & \ddots & \ddots & 0 \\
0 & \cdots & 0 & (\frac{1}{1-\theta_K})
\end{pmatrix}\right]\times \begin{pmatrix}
\beta_1 \\
\vdots \\
\beta_K
\end{pmatrix},\\
& \implies \begin{pmatrix}
\beta_1 \\
\vdots \\
\beta_K
\end{pmatrix} = \begin{pmatrix}
0 \\
\vdots \\
0
\end{pmatrix}, \begin{pmatrix}
\alpha_1 \\
\vdots \\
\alpha_K
\end{pmatrix} = \begin{pmatrix}
0 \\
\vdots \\
0
\end{pmatrix},
\end{split}
\end{equation}
which completes the proof.
\end{proof}
\end{comment}
\end{lemma}
Lemma \ref{1st-identifiability} is indispensable for estimating, at a $\sqrt{n}$ rate, the true mixing weights and the true component-wise parameters under each question cluster. The results of Theorem \ref{selectionConsistency} and Lemma \ref{1st-identifiability} yield to our final result.
\begin{theorem}\label{postContract}
Assume (A1)-(A3) are satisfied, then the proposed model can estimate the true parameters a posteriori, given a contraction rate $\epsilon_n$,
\begin{equation}\label{eqPostContract}
\begin{split}
& \Pi(\{p_F^{(c)}: |w_{\sigma^{(c)}(i)}^{(c)}-w_{i,0}^{(c)}|\lesssim M'\epsilon_n,\|\theta_{\sigma^{(c)}(i)}^{(c)}-\theta_{i,0}^{(c)}\|_2\lesssim M'\epsilon_n,~\text{for}~i=1,\ldots,K^{(c)},\\
& K^{(c)}=K_0^{(c)},\forall c\in\mathcal{C},\mathcal{C}=\mathcal{C}_0\}\mid X_{1},\ldots,X_{n})\to 1,~~a.s.~P_0,
\end{split}
\end{equation}
where $M$ is a large positive constant to be determined and $M' > 0$ is a universal constant that only depends on $M$, $\epsilon_n = (\log(n))^t/\sqrt{n}$, for any $t > 1$.
\begin{comment}
\begin{proof}
By Bayes' theorem, for the set $\text{B}_n = \otimes_{c\in \mathcal{C}_0}\{p_F^{(c)}:\|p_F^{(c)} - p_{F_0}^{(c)} \|_1 \leq M\epsilon_n\}$ and $\forall \mathcal{C}\in\mathcal{S}$, it holds that
\begin{equation}\label{posteriorA}
\begin{split}
\Pi_{\mathcal{C}_0}(\text{B}_n\mid X_{1},\ldots,X_{n}) & = \frac{\int_{\text{B}_n} \prod_{c\in\mathcal{C}_0}p_F^{(c)}(X_i^{(c)})d\Pi^{(c)}(p_F^{(c)})}{\int \prod_{c\in\mathcal{C}_0}p_F^{(c)}(X_i^{(c)})d\Pi^{(c)}(p_F^{(c)})},\\
& = \prod_{c\in\mathcal{C}_0}\frac{\int_{\{p_F^{(c)}:\|p_F^{(c)} - p_{F_0}^{(c)} \|_1 \leq M\epsilon_n\}} p_F^{(c)}(X_i^{(c)})d\Pi^{(c)}(p_F^{(c)})}{\int p_F^{(c)}(X_i^{(c)})d\Pi^{(c)}(p_F^{(c)})},\\
& = \prod_{c\in\mathcal{C}_0}\Pi^{(c)}(\{p_F^{(c)}:\|p_F^{(c)} - p_{F_0}^{(c)} \|_1 \leq M\epsilon_n\}\mid X_{1}^{(c)},\ldots,X_{n}^{(c)}).
\end{split}
\end{equation}
We proceed by showing
$$\Pi^{(c)}(\{p_F^{(c)}:\|p_F^{(c)} - p_{F_0}^{(c)} \|_1 \leq M\epsilon_n\}\mid X_{1}^{(c)},\ldots,X_{n}^{(c)})\to 0,~~a.s.~P_0,~~\text{for}~\forall c\in\mathcal{C}_0.$$ Consider the following sieve
\begin{equation}\label{sieveBinomial}
\begin{split}
\mathcal{P}^{(c)}_n=\left\{ p^{(c)}_F(x^{(c)}): p^{(c)}_F(x^{(c)}) =\sum_{i=1}^{K}w^{(c)}_i(1-\theta_i^{(c)})^{|c|-\|x^{(c)}\|_1}(\theta_{i}^{(c)})^{\|x^{(c)}\|_1}, \right.\\
\left.~\forall K < H_n,~\text{with}~\{\theta^{(c)}_i\}_{i=1}^K~\text{being distinct} \right\},
\end{split}
\end{equation}
we aim to upper bound the covering number of $\mathcal{P}^{(c)}_n$ using $\ell_1$ balls. For $\forall p_F^{(c)}\in\mathcal{P}_n^{(c)}$, which is defined as
$$p^{(c)}_F(x^{(c)}) =\sum_{i=1}^{K}w^{(c)}_i(1-\theta_i^{(c)})^{|c|-\|x^{(c)}\|_1}(\theta_{i}^{(c)})^{\|x^{(c)}\|_1},$$ find $\hat{w}_i^{(c)}$, $\hat{\theta}_i^{(c)}$, for $i=1,\ldots, K$, such that,
\begin{equation}\label{sieveNeighbor}
\begin{split}
& \max_{1\leq i\leq K}|\hat{\theta}_i^{(c)}-\theta_i^{(c)}|\leq\epsilon_n,~\hat{\theta_i}\in (0,1),~\text{for}~i=1,\ldots,K\\
& \sum_{i=1}^K|\hat{w}_{i}^{(c)}-w_{i}^{(c)}|\leq \epsilon_n,~\text{with}~\sum_{i=1}^K\hat{w}_i^{(c)}=1,\hat{w}_i^{(c)} >0,~\text{for}~i=1,\ldots,K,
\end{split}
\end{equation}
then the $\ell_1$ distance between $p^{(c)}_F(x^{(c)})$ and $p^{(c)}_{\hat{F}}(x^{(c)})\equiv\sum_{i=1}^{K}\hat{w}_i^{(c)}(1-\theta_i^{(c)})^{|c|-\|x^{(c)}\|_1}(\theta_{i}^{(c)})^{\|x^{(c)}\|_1}$ can be bounded by
\begin{equation}\label{ell1UpperBound}
\begin{split}
& \|p^{(c)}_F(x^{(c)})-p^{(c)}_{\hat{F}}(x^{(c)})\|_1\\
& \leq \sum_{i=1}^Kw_i^{(c)}\left\|\left[(1-\theta_i^{(c)})^{|c|-\|x^{(c)}\|_1}(\theta_i^{(c)})^{\|x^{(c)}\|_1}-(1-\hat{\theta_i}^{(c)})^{|c|-\|x^{(c)}\|_1}(\hat{\theta_i}^{(c)})^{\|x^{(c)}\|_1}\right]\right\|_1+\\
& \sum_{i=1}^K|w_i^{(c)} - \hat{w_i}^{(c)}|\left\|(1-\hat{\theta_i}^{(c)})^{|c|-\|x^{(c)}\|_1}(\hat{\theta_i}^{(c)})^{\|x^{(c)}\|_1}\right\|_1,\\
& \stackrel{(a)}{\leq} C\sum_{i=1}^K w_i^{(c)}\epsilon_n+ \epsilon_n,\\
& \leq (C+1)\epsilon_n,
\end{split}
\end{equation}
where $C$ is a universal constant which only depends on $D$ and the inequality (a) is obtained by the Taylor expansion. We can hence construct, based on the $\ell_1$ distance, a $(C+1)\epsilon_n$-covering on $\mathcal{P}_n^{(c)}(K)$ from an $\epsilon_n$-covering on
$\otimes_{i=1}^K(0,1)\times (0,1)$. Note that $\otimes_{i=1}^K(0,1)\times (0,1)$ can be covered by $\epsilon_n^{-2K}$ $\ell_1$-balls and by summing $K$ from 1 to $H_n$, an upper bound of the covering number of $\mathcal{P}_n^{(c)}$ can be obtained, says $H_n\epsilon_n^{-2H_n}$. The $(C+1)\epsilon_n$-covering on $\mathcal{P}_n^{(c)}(K)$ and $\mathcal{P}_n^{(c)}$ are later denoted by ${\mathcal{P}_n^*}^{(c)}(K)$ and ${\mathcal{P}_n^*}^{(c)}$, which satisfies ${\mathcal{P}_n}^{(c)}\subseteq {\mathcal{P}_n^*}^{(c)}$. Now take $H_n=\lfloor\frac{n\epsilon_n^2}{\log(n)}\rfloor$, which gives
\begin{equation}\label{boundCoverNum}
\begin{split}
& \log\left(H_n\epsilon_n^{-2H_n}\right)\lesssim n\epsilon^2_n,
\end{split}
\end{equation}
and $\Pi^{(c)}(\mathcal{P}^{(c)}\setminus\mathcal{P}_n^{(c)}) = 0$ when $n$ is sufficiently large, since $K^{(c)}$ is upper bounded in the support of $\Pi^{(c)}$.
By Theorem 7.1 of \citet{ghosal2000convergence} or Lemma D.3 of \citet{ghosal2017fundamentals}, there exists a universal test function $\phi_n$ and a positive constant $C_1>0$ associated with the approximate inequality in \eqref{boundCoverNum}, such that
\begin{equation}\label{testFunc}
\begin{split}
& \mathbb{P}_{F_0^{(c)}}^n \phi_n\leq C_1\exp\{n\epsilon_n^2\}\times \exp\{-\frac{1}{8}nM^2\epsilon_n^2\}\times\frac{1}{1-\exp\{-\frac{1}{8}nM^2\epsilon_n^2\}},\\
& \sup_{p_{F}^{(c)}\in{\mathcal{P}_n^*}^{(c)}:\left\|p_{F_0}^{(c)}-p_{F}^{(c)}\right\|_1 > M\epsilon_n}\mathbb{P}_{F^{(c)}}^n(1-\phi_n)\leq \exp\{-\frac{1}{8}nM^2\epsilon_n^2\}.
\end{split}
\end{equation}
Take expectation under $\mathbb{P}_{F_0^{(c)}}^n$ for each component in the last display of \eqref{posteriorA}, we have
\begin{equation}\label{postEachComponent}
\begin{split}
& \mathbb{P}_{F_0^{(c)}}^n\left[\Pi^{(c)}(\{p_F^{(c)}:\|p_F^{(c)} - p_{F_0}^{(c)} \|_1 > M\epsilon_n\}\mid X_{1}^{(c)},\ldots,X_{n}^{(c)})\right]\\
& \leq \mathbb{P}_{F_0^{(c)}}^n\left[\phi_n\right] + \mathbb{P}_{F_0^{(c)}}^n\left[\frac{\int_{\{p_F^{(c)}\in {\mathcal{P}_n^*}^{(c)}:\|p_F^{(c)} - p_{F_0}^{(c)} \|_1 > M\epsilon_n\}}\prod_{i=1}^n \frac{p_F^{(c)}(X_i^{(c)})}{p_{F_0}^{(c)}(X_i^{(c)})}d\Pi^{(c)}(p_F^{(c)})}{\int\prod_{i=1}^n \frac{p_F^{(c)}(X_i^{(c)})}{p_{F_0}^{(c)}(X_i^{(c)})}d\Pi^{(c)}(p_F^{(c)})}(1-\phi_n)\right] +\\
& \mathbb{P}_{F_0^{(c)}}^n\left[\frac{\int_{\mathcal{P}^{(c)}\setminus\mathcal{P}_n^{(c)}} \prod_{i=1}^np_F^{(c)}(X_i^{(c)})d\Pi^{(c)}(p_F^{(c)})}{\int\prod_{i=1}^n p_F^{(c)}(X_i^{(c)})d\Pi^{(c)}(p_F^{(c)})}\right],\\
& \stackrel{(a)}{\leq} \mathbb{P}_{F_0^{(c)}}^n\left[\phi_n\right] + \sup_{p_{F}^{(c)}\in{\mathcal{P}_n^*}^{(c)}:\left\|p_{F_0}^{(c)}-p_{F}^{(c)}\right\|_1 > M\epsilon_n}\mathbb{P}_{F^{(c)}}^n(1-\phi_n)\times \epsilon_n^{-2\times(K_0^{(c)}-1)}\times \epsilon_n^{-4(1+\delta)},\\
& \stackrel{(b)}{\lesssim}\exp\{(-\frac{1}{8}M^2+1)n\epsilon_n^2\}+ \exp\{-\frac{1}{8}nM^2\epsilon_n^2\}\times \epsilon_n^{-2\times(K_0^{(c)}-1)}\times \epsilon_n^{-4(1+\delta)}+ 0,
\end{split}
\end{equation}
where the inequality (a) is obtained by Fubini's theorem and the inequality (c) in \eqref{denomLower} and the inequality (b) is obtained by \eqref{testFunc}. Since the last display of \eqref{postEachComponent} converges to 0 faster than $\epsilon_n^{2(1+\delta)}$ with $\delta > 0$ if $M > 2\sqrt{2}$, then by the same steps as introduced in \eqref{Borel-Cantelli}, we conclude that
$$\Pi^{(c)}(\{p_F^{(c)}:\|p_F^{(c)} - p_{F_0}^{(c)} \|_1 > M\epsilon_n\}\mid X_{1}^{(c)},\ldots,X_{n}^{(c)})\to 0,~~a.s.~P_0,$$
and
$$\Pi^{(c)}(\{p_F^{(c)}:\|p_F^{(c)} - p_{F_0}^{(c)} \|_1 > M\epsilon_n\}\mid K^{(c)}=K_0^{(c)}, X_{1}^{(c)},\ldots,X_{n}^{(c)})\to 0,~~a.s.~P_0 $$ similarly because the evidence lower bound can still be constructed following the procedure as introduced in \eqref{denomLower}. With the same parameterization and $\sigma$-field construction as given in Section 3.2 of \citet{nobile1994bayesian} and Doob's convergence theorem \citep{doob1949application}, we have $\forall c\in\mathcal{C}_0$,
\begin{equation}\label{ConsistentK}
\begin{split}
& \Pi^{(c)}(K^{(c)} = K_0^{(c)}\mid X_{1}^{(c)},\ldots,X_{n}^{(c)})\to 0,~~a.s.~P_0.
\end{split}
\end{equation}
Notice that
\begin{equation}\label{condProb}
\begin{split}
& \Pi^{(c)}(\{p_F^{(c)}:\|p_F^{(c)} - p_{F_0}^{(c)} \|_1 \leq M\epsilon_n, K^{(c)}=K_0^{(c)}\}\mid X_{1}^{(c)},\ldots,X_{n}^{(c)})\\
& = \Pi^{(c)}(\{p_F^{(c)}:\|p_F^{(c)} - p_{F_0}^{(c)} \|_1 \leq M\epsilon_n\}\mid K^{(c)} = K_0^{(c)}, X_{1}^{(c)},\ldots,X_{n}^{(c)})\times \\
& \Pi^{(c)}(K^{(c)} = K_0^{(c)}\mid X_{1}^{(c)},\ldots,X_{n}^{(c)}),
\end{split}
\end{equation}
which implies $\Pi^{(c)}(\{p_F^{(c)}:\|p_F^{(c)} - p_{F_0}^{(c)} \|_1 \leq M\epsilon_n, K^{(c)} = K_0^{(c)}\}\mid, X_{1}^{(c)},\ldots,X_{n}^{(c)})\to 1,~~a.s.~P_0$ by the continuous mapping theorem. Since the first-order identifiability is satisfied by Lemma \ref{1st-identifiability}, then by Theorem 3.1 of \citep{ho2016strong}, we have
\begin{equation}\label{upperWasser}
\begin{split}
& \{p_F^{(c)}:\|p_F^{(c)}-p_{F_0}^{(c)}\|_1\leq M\epsilon_n\}\subseteq W_{F_0^{(c)},K_0^{(c)}}(M'\epsilon_n),\\
& W_{F_0^{(c)},K_0^{(c)}}(M'\epsilon_n)\equiv\left\{ F^{(c)}\equiv \sum_{k=1}^{K_0^{(c)}}w_i^{(c)}\delta_{\theta_i^{(c)}}:\inf_{\vec{q}\in\mathcal{Q}(\vec{w}^{(c)},\vec{w_0}^{(c)})}\sum_{i=1}^{{K_0}^{(c)}}\sum_{j=1}^{{K_0}^{(c)}}q_{i,j}\|\theta_i^{(c)}-\theta_{j,0}^{(c)}\|_2\leq M'\epsilon_n\right\},
\end{split}
\end{equation}
where $\vec{w}^{(c)}$ and $\vec{w_0}^{(c)}$ denote the mixing weights vectors of $p^{(c')}_{F}$ and $p^{(c')}_{F_0}$, $\vec{q}$ denotes the vector of a coupling of $\vec{w}^{(c)}$ and $\vec{w_0}^{(c)}$ with $\mathcal{Q}(\vec{w}^{(c)},\vec{w_0}^{(c)})$ denoting the collection of all possible couplings of $\vec{w}^{(c)}$ and $\vec{w_0}^{(c)}$ and $M' > 0$ is a universal constant that only depends on $M$. We proceed by showing that for any $F^{(c)}\in W_{F_0^{(c)},K_0^{(c)}}(M'\epsilon_n)$, there exists a permutation $\sigma^{(c)}(\cdot)$ on $1,\ldots,K^{(c)}_0$ such that
\begin{equation}\label{element-wise theta bound}
\begin{split}
& \|\theta^{(c)}_{\sigma(i)}-\theta^{(c)}_{i,0}\|_2\lesssim M'\epsilon_n,~~\text{for}~i=1,\ldots,{K_0}^{(c)},
\end{split}
\end{equation}
if it holds that
$$\inf_{\vec{q}\in\mathcal{Q}(\vec{w}^{(c)},\vec{w_0}^{(c)})}\sum_{i=1}^{{K_0}^{(c)}}\sum_{j=1}^{{K_0}^{(c)}}q_{i,j}\|\theta^{(c)}_i-\theta^{(c)}_{j,0}\|_2\leq M'\epsilon_n.$$
Suppose this is not correct, then there exists $i^*$ such that the order of $\|\theta^{(c)}_{\sigma^{(c)}(i^*)}-\theta^{(c)}_{i^*,0}\|_2$ is slower than the order of $\epsilon_n$ for any permutation $\sigma^{(c)}(\cdot)$. This implies $q_{j,i^*}$ converges to 0 at some order for any $j=1,\ldots,{K_0}^{(c)}$. However, it is clear by definition that $\sum_{j=1}^{{K_0}^{(c)}}q_{j,i^*}=w_{i^*,0}^{(c)}>0$ holds, which raises contradiction.
Furthermore, given the predifined $\sigma^{(c)}(\cdot)$, it also holds that
\begin{equation}\label{element-wise p bound}
\begin{split}
|w_{\sigma(i)}^{(c)}-w_{i,0}^{(c)}|\lesssim M'\epsilon_n,~~\text{for}~i=1,\ldots,{K_0}^{(c)}.
\end{split}
\end{equation}
Without loss of generality, we let $\|\theta^{(c)}_i-\theta^{(c)}_{i,0}\|_2\lesssim M'\epsilon_n$ and $|w_{i}^{(c)}-w_{i,0}^{(c)}|\lesssim M'\epsilon_n$, for $i=1,\ldots,{K^*_0}^{(c)}$. Now suppose, without loss of generality, $|w_1^{(c)}-w_{1,0}^{(c)}|\lesssim M'\epsilon_n$ is not true. However, notice that
$$\sum_{j=2}^{{K_0}^{(c)}}q_{1,j}\|\theta^{(c)}_1-\theta^{(c)}_{j,0}\|_2\asymp\sum_{j=2}^{{K_0}^{(c)}}q_{1,j}$$
and
$$\sum_{i=2}^{{K_0}^{(c)}}q_{i,1}\|\theta^{(c)}_i-\theta^{(c)}_{1,0}\|_2\asymp\sum_{i=2}^{{K_0}^{(c)}}q_{i,1},$$ because $\{\theta^{(c)}_{i,0}\}_{i=1}^{{K^*_0}^{(c)}}$ are distinct following Assumption (A3), which guarantees a minimal pair-wise distance, and meanwhile, it has been proved that $\|\theta^{(c)}_i-\theta^{(c)}_{i,0}\|_2\lesssim M'\epsilon_n$ for $i=1,\ldots,{K^*_0}^{(c)}$. These give contradiction because
\begin{equation}\label{pcontradict}
\begin{split}
& |p^{(c)}_1-p^{(c)}_{1,0}| = |p^{(c)}_1-p^{(c)}_{1,0}-q_{1,1}+q_{1,1}|\\
& = |\sum_{j=2}^{{K_0}^{(c)}}q_{1,j}-\sum_{i=2}^{{K_0}^{(c)}}q_{i,1}|\lesssim M'\epsilon_n.
\end{split}
\end{equation}
The proof is hence finished by
\begin{equation}\label{componentwiseNB}
\begin{split}
& \Pi(\{p_F^{(c)}: |w_{\sigma^{(c)}(i)}^{(c)}-w_{i,0}^{(c)}|\lesssim M'\epsilon_n,\|\theta_{\sigma^{(c)}(i)}^{(c)}-\theta_{i,0}^{(c)}\|_2\lesssim M'\epsilon_n,~\text{for}~i=1,\ldots,K^{(c)},\\
& K^{(c)}=K_0^{(c)},\forall c\in\mathcal{C},\mathcal{C}=\mathcal{C}_0\}\mid X_{1},\ldots,X_{n}),\\
& =\Pi(\{p_F^{(c)}: |w_{\sigma^{(c)}(i)}^{(c)}-w_{i,0}^{(c)}|\lesssim M'\epsilon_n,\|\theta_{\sigma^{(c)}(i)}^{(c)}-\theta_{i,0}^{(c)}\|_2\lesssim M'\epsilon_n,~\text{for}~i=1,\ldots,K^{(c)},\\
& K^{(c)}=K_0^{(c)},\forall c\in\mathcal{C}\}\mid \mathcal{C}=\mathcal{C}_0, X_{1},\ldots,X_{n})\times \Pi(\mathcal{C}=\mathcal{C}_0\mid X_{1},\ldots,X_{n}),\\
& = \left[\prod_{c\in\mathcal{C}_0}\Pi^{(c)}(\{p_F^{(c)}: |w_{\sigma^{(c)}(i)}^{(c)}-w_{i,0}^{(c)}|\lesssim M'\epsilon_n,\|\theta_{\sigma^{(c)}(i)}^{(c)}-\theta_{i,0}^{(c)}\|_2\lesssim M'\epsilon_n,~\text{for}~i=1,\ldots,K^{(c)}\mid\right.\\
& \left.K^{(c)} = K_0^{(c)},X_1^{(c)},\ldots,X_n^{(c)})\times\Pi^{(c)}(K^{(c)} = K_0^{(c)}\mid X_1^{(c)},\ldots,X_n^{(c)})\right]\times \Pi(\mathcal{C}=\mathcal{C}_0\mid X_{1},\ldots,X_{n})\\
& \geq \left[\prod_{c\in\mathcal{C}_0}\Pi^{(c)}(\{p_F^{(c)}:\|p_F^{(c)} - p_{F_0}^{(c)} \|_1 \leq M\epsilon_n\}\mid K^{(c)} = K_0^{(c)}, X_{1}^{(c)},\ldots,X_{n}^{(c)})\times \right.\\
& \left.\Pi^{(c)}(K^{(c)} = K_0^{(c)}\mid X_1^{(c)},\ldots,X_n^{(c)})\right]\times \Pi(\mathcal{C}=\mathcal{C}_0\mid X_{1},\ldots,X_{n})\to 1,~~.a.s.~P_0.
\end{split}
\end{equation}
\end{proof}
\end{comment}
\end{theorem}
Theorem \ref{postContract} indicates that our model can detect the heterogeneity in examinees, and meanwhile, identify the true question partition.
\section{Bayesian inference and algorithm}\label{sec: Bayesian_inference}
We begin with the outline of our posterior sampling algorithm, which consists of the following four steps,
\begin{itemize}
\item (1) For each column (question), calculate the marginal likelihood when all students possess the same accuracy in answering this question, say marginalizing the Binomial likelihood over the Beta prior in \eqref{ConsBinModel}.
\item (2) Conditioning on the row (examinee) assignment under each column (question) cluster, update the column assignment by enumerating from column 1 to column $D$.
\item (3) Conditioning on the column (question) assignment, update the row (examinee) assignment under each column (question) cluster by enumerating from row 1 to row $n$ for $n_{rep}$ times.
\item (4) Loop between (2) and (3) for $n_{iter}$ times to approach the stationary distribution.
\end{itemize}
The detailed algorithm is deferred to the Supplementary File. It is worthwhile to point out that Step (2) is essentially the Algorithm 1 proposed by \citet{neal2000markov}, by treating each column (question) to be an ``individual", whose parameter is the row-wise partition on the examinees. To update the row-wise partition in Step (3), we modify the aforementioned Algorithm 1 to accommodate the upper bound on the number of (student) mixtures under each column (question) cluster, which leads to a sampling scheme which is slightly different from Theorem 4.1 of \citet{miller2018mixture}. Based on the findings in our simulation studies, we notice that $n_{iter}=200$ and $n_{rep}=400$ is sufficient to obtain trustworthy posterior samples when $n\leq 1000$ and $D\leq 80$. In addition, for the hyper-parameters defined in \eqref{ConsBinModel}, we let both $a_0$ and $b_0$ be $.01$ and $\gamma$ equal to 1 such that the prior information is sufficiently non-informative but new clusters still have enough probability to be generated for both column (question) and row (examinee) in practice. The probability mass function $m(\mathcal{C})$ is chosen as the exchangeable partition probability function \citep[EPPF;][]{pitman2002combinatorial} of the mixture of finite mixtures model \citep[MFM;][]{miller2018mixture} to ensure a closed form on the full conditional distribution when sampling. The most time consuming task ($n=1,000$ and $D=80$) among the simulation studies and the real data analysis takes approximately 6 hours to finish after being assigned to a server with 94.24GB RAM, 24 processing cores, operating at 3.33GHz.
To summarize the posterior samples, we use the following three statistics to estimate (1) the column (question) partition, (2) the row-wise (examinee) partition under each column cluster and (3) the component-wise accuracy under each column cluster. The column (question) partition is estimated using Dahl's estimate \citep{dahl2006model}, defined as follows,
\begin{equation}\label{ColumnEst}
\begin{split}
& \hat{\ell}=\argmin_{1\leq \ell\leq M}\sum_{i=1}^n\sum_{j=1}^n\left\{\delta_{i,j}(\mathcal{C}^{\text{Col}}(\ell))-\hat{\pi}^{\text{Col}}_{i,j}\right\}^2,\\
& \hat{\mathcal{C}}^{\text{Col}} = \mathcal{C}^{\text{Col}}(\hat{\ell}),
\end{split}
\end{equation}
where $M$ is the number of MCMC iterations after burn-in, $\mathcal{C}^{\text{Col}}(\ell)$ refers to the column assignment at the $\ell$-th iteration after burn-in, $\delta_{i,j}(\mathcal{C}^{\text{Col}}(\ell))$ is an indicator function, defined as $\mathbbm{1}(\mathcal{C}^{\text{Col}}_i(\ell) = \mathcal{C}^{\text{Col}}_j(\ell))$, with $\mathcal{C}^{\text{Col}}_i(\ell)$ denoting the clustering assignment of the $i$-th column, and $\hat{\pi}^{\text{Col}}_{i,j}$ is obtained by averaging $\delta_{i,j}(\mathcal{C}^{\text{Col}}(\ell))$ over post burn-in MCMC samples, namely, $ \hat{\pi}^{\text{Col}}_{i,j} =\frac{1}{M}\sum_{\ell=1}^M\delta_{i,j}(\mathcal{C}^{\text{Col}}(\ell))$. The column partition summarized by Dahl's estimate is believed to be the most representative one as it minimizes the entry-wise $\ell_2$-distance between the self-concordance matrix of a given partition and the probability matrix $\hat{\pi}^{\text{Col}}_{i,j}$ that any pair of columns $i$ and $j$ being clustered together. The row-wise (student) partition is then summarized from the iterations where the column (question) partition is equal to $\mathcal{C}^{\text{Col}}(\hat{\ell})$,
\begin{equation}\label{RowEst}
\begin{split}
& \hat{\ell} =\argmin_{1\leq \ell\leq M; \mathcal{C}^{\text{Col}}(\ell) = \hat{\mathcal{C}}^{\text{Col}}}\sum_{d=1}^D\sum_{i=1}^n\sum_{j=1}^n\left\{\delta_{i,j}(\mathcal{C}^{\text{Row};d}(\ell))-\hat{\pi}^{\text{Row};d}_{i,j}\right\}^2,\\
& \hat{\mathcal{C}}^{\text{Row};d} = \mathcal{C}^{\text{Row};d}(\hat{\ell}),\text{ for }d=1,\ldots, D,
\end{split}
\end{equation}
where $\mathcal{C}^{\text{Row};d}(\ell)$ refers to the row assignment of the $d$-th column at the $\ell$-th iteration after burn-in and $\hat{\pi}^{\text{Row};d}_{i,j}$ is defined in a similar way with $\hat{\pi}^{\text{Col}}_{i,j}$ for the $d$-th column. It can be expected that ties happen for $\hat{\mathcal{C}}^{\text{Row};d}$ given $\forall d\in c$ and $\forall c\in\hat{\mathcal{C}}^{\text{Col}}$ by definition. Analogous to the idea behind $\hat{\mathcal{C}}^{\text{Col}}$, $\hat{\mathcal{C}}^{\text{Row};d}$ seeks for an iteration such that the squared $\ell_2$ distance is minimized averaged over all columns. The component-wise accuracy under each column cluster is then estimated using a posterior mean given $\hat{\mathcal{C}}^{\text{Col}}$ and $\hat{\mathcal{C}}^{\text{Row};d}$ for $d=1,\ldots,D$.
\section{Simulation}\label{sec: Simulation}
We study our proposed method using four data generating processes (DGP) and compare the result of our model with the one given by the Rasch model \citep{rasch1993probabilistic}. The Rasch model is realized using \textit{tam} package in R. The four DGPs are designed to mimic the situations when the data are generated under our proposed model or the Rasch model, given increasing number of examinees ($n=100,300,1,000$). We proceed by outlining the first four DGPs and defer the details to the Supplementary File. The first two DGPs are designed under our model,
\begin{itemize}
\item \textbf{DGP1}. 20 questions are divided into 5 column (question) clusters, with 3 large question clusters and the rest 2 questions individually forming two question clusters. Under each column (question) cluster, the accuracy within each mixture stays identical and the mixture number satisfies the constraint.
\item \textbf{DGP2}. 60 questions are divided into 5 column (question) clusters, with 3 large question clusters and the rest 2 questions individually forming two question clusters. Under each column (question) cluster, the accuracy within each mixture stays identical and the mixture number satisfies the constraint.
\end{itemize}
The last two DGPs are generated following the Rasch model, that is,
\begin{equation}\label{RaschModel}
\begin{split}
& X_{i,j}\stackrel{\text{ind}}{\sim} \text{Bernoulli}(\theta_{i,j}),~\text{for}~i=1,\ldots,n~\text{and}~j=1,\ldots,D,\\
& \theta_{i,j} = \frac{\exp\{\xi_i-\psi_j\}}{1+\exp\{\xi_i-\psi_j\}},
\end{split}
\end{equation}
with the two DGPs being presented as follows,
\begin{itemize}
\item \textbf{DGP3}. 20 questions are divided into 2 column (question) clusters by letting $\psi_j$ take value from $(-0.5,0.5)$ and 3 row clusters by letting $\xi_i$ randomly take value from $(-2,0,2)$, following the data generating process defined in \eqref{RaschModel}.
\item \textbf{DGP4}. 60 questions are divided into 2 column (question) clusters by letting $\psi_j$ take value from $(-0.5,0.5)$ and 3 row clusters by letting $\xi_i$ randomly take value from $(-2,0,2)$, following the data generating process defined in \eqref{RaschModel}.
\end{itemize}
To validate Theorem \ref{postContract}, we consider the first two DGPs and adopt the following criteria
\begin{equation}\label{trueCriteria}
\begin{split}
& \text{CWRI} = \text{RI}(\hat{\mathcal{C}}^{\text{Col}},\mathcal{C}_0),\\
& \text{ADK} = \frac{1}{D}\sum_{d=1}^D\left||\hat{\mathcal{C}}^{\text{Row};d}|-K^{(c(d))}_0\right|,\\
& \text{ADW} = \mathrm{1}(\text{RI}(\hat{\mathcal{C}}^{\text{Col}},\mathcal{C}_0) = 1)\times \frac{1}{|\mathcal{C}_0|}\sum_{c\in\mathcal{C}_0}\frac{1}{K_0^{(c)}}\min_{\sigma^{(c)}}\sum_{i=1}^{K^{(c)}_0}\left|\hat{w}^{(c)}_{\sigma^{(c)}(i)}-w^{(c)}_{0;i}\right|+\\
& \mathrm{1}(\text{RI}(\hat{\mathcal{C}}^{\text{Col}},\mathcal{C}_0) \neq 1) \times 2,\\
& \text{ADP} = \mathrm{1}(\hat{K}^{(c)} \geq K^{(c)}_0,c\in \mathcal{C}_0)\times \frac{1}{|\mathcal{C}_0|}\sum_{c\in\mathcal{C}_0}\sqrt{\frac{1}{K_0^{(c)}}\min_{\sigma^{(c)}}\sum_{i=1}^{K^{(c)}_0}\left\|\hat{\theta}^{(c)}_{\sigma^{(c)}(i)}-\theta^{(c)}_{0;i}\right\|_2^2}+\\
& \left(\mathrm{1}(\hat{K}^{(c)} < K^{(c)}_0) \vee \mathrm{1}(c\notin \mathcal{C}_0))\right)\times 1,
\end{split}
\end{equation}
where $\text{CWRI}$, $\text{ADK}$, $\text{ADW}$ and $\text{ADP}$ are the abbreviations of column-wise Rand Index, averaged absolute difference in the row-wise number of component, averaged absolute difference in the row-wise weights and averaged $\ell_2$ difference in the row-wise accuracy, $\text{RI}(\mathcal{C},\mathcal{C}')$ denotes the Rand Index \citep{rand1971objective} between $\mathcal{C}$ and $\mathcal{C}'$, $c(d)$ represents the column cluster $c$ where the $d$-th column is assigned to, $\sigma^{(c)}(\cdot)$ refers to the permutation operator, $w_{0;i}^{(c)}$ and $\theta_{0;i}^{(c)}$ denote the $i$-th true mixing weight and the $i$-th true component-wise accuracy under the column (question) cluster $c$ respectively. $\hat{w}_{i}^{(c)}$ and $\hat{\theta}_i^{(c)}$ represent the estimated values of $w_{0;i}^{(c)}$ and $\theta_{0;i}^{(c)}$ using posterior mean. Note that the penalty of misidentifying the true column (question) partition is added to $\text{ADW}$, which matches the maximum difference between the estimated mixing weights and the true mixing weights. Similar penalty is also attached to $\text{ADP}$. Ideally, we expect $\text{CWRI}$ converges to 1 and the rest three criteria shrink toward 0 if Theorem \ref{postContract} is true. The correct limiting values and the decreasing standard error successfully manifest our theoretical results,
suggested by Table \ref{tab:1}.
\begin{table}[htp]
\centering
\caption{Median (Standard error) of the four criteria over 100 Monte Carlo replications for each of the first two DGPs given different sample sizes.}
\label{tab:1}
\begin{tabular}{l|l|l|l|l|l}
\toprule
\toprule
DGP & n & CWRI & ADK & ADW & ADP\\
\midrule
& 100 & 1.000 (0.005) & 0.000 (0.181) & 0.082 (0.395) & 0.064 (0.192)\\
1 & 300 & 1.000 (0.000) & 0.000 (0.158) & 0.034 (0.015) & 0.033 (0.014) \\
& 1000 & 1.000 (0.000) & 0.000 (0.097) & 0.014 (0.006) & 0.013 (0.007)\\
\midrule
& 100 & 1.000 (0.001) & 0.625 (0.273) & 0.811 (0.338) & 0.417 (0.166) \\
2 & 300 & 1.000 (0.000) & 0.000 (0.144) & 0.029 (0.136) & 0.023 (0.068) \\
& 1000 & 1.000 (0.000) & 0.000 (0.075) & 0.016 (0.005) & 0.013 (0.003) \\
\bottomrule
\bottomrule
\end{tabular}
\end{table}
In the last two DGPs, the assumptions of the Rasch model are satisfied. We propose to study the performance of our model in identifying the true column (question) and row (student) partitions, defined as the labelling of $\psi_j$ and $\xi_i$ respectively, and compare the performance of estimating $\theta_{0;i}^{(d)}\equiv \frac{\exp\{\xi_i-\psi_j\}}{1 + \exp\{\xi_i-\psi_j\}}$ using the following two criteria in addition to $\text{CWRI}$,
\begin{equation}\label{twoCriteria}
\begin{split}
& \text{ARWRI} = \frac{1}{D}\sum_{d=1}^D\text{RI}(\hat{\mathcal{C}}^{\text{Row};d},\mathcal{C}^{\text{Row;(c(d))}}_0),\\
& \text{D}_1 = \frac{1}{nD}\sum_{i=1}^n\sum_{d=1}^D\left|\hat{\theta}_{i}^{(d)}-\theta_{0;i}^{(d)}\right|
\end{split}
\end{equation}
where ARWRI is the abbreviation of averaged row-wise Rand Index and $\hat{\theta}_{i}^{(d)}$ can be directly provided by the Rasch model or using the posterior mean for our model. The results are presented in Table \ref{tab:2},
\begin{table}[htp]
\centering
\caption{Median (Standard error) of the three criteria over 100 Monte Carlo replications for each of the last two DGPs given different sample sizes.}
\label{tab:2}
\begin{tabular}{l|l|l|l|l|l}
\toprule
\toprule
\multicolumn{2}{c|}{} & \multicolumn{3}{c|}{ACBM} & Rasch\\
\midrule
\midrule
DGP & n & CWRI & ARWRI & $\text{D}_1$ & $\text{D}_1$\\
\midrule
& 100 & 0.474 (0.193) & 0.922 (0.088) & 0.093 (0.018) & 0.073 (0.005)\\
3 & 300 & 1.000 (0.048) & 0.814 (0.020) & 0.077 (0.015) & 0.068 (0.003)\\
& 1000 & 1.000 (0.000) & 0.818 (0.009) & 0.066 (0.005) & 0.066 (0.002)\\
\midrule
& 100 & 0.919 (0.174) & 0.978 (0.016) & 0.026 (0.022) & 0.050 (0.003)\\
4 & 300 & 1.000 (0.007) & 0.979 (0.007) & 0.012 (0.003) & 0.043 (0.002)\\
& 1000 & 1.000 (0.000) & 0.977 (0.004) & 0.009 (0.001) & 0.040 (0.001)\\
\bottomrule
\bottomrule
\end{tabular}
\end{table}
It is interesting to note that when $n$ increases, $\text{CWRI}$ increases towards 1 and $\text{ARWRI}$ stays at a high value. Though $\text{ARWRI}$ is not guaranteed to converge towards 1, our model is able to identify most of the correct labels for examinees when the latent accuracy parameters are sufficiently well separated. In addition, our model achieves a higher $\text{ARWRI}$ when more questions are available under each question cluster, which matches our intuition, that is, more questions are more helpful in correctly distinguishing different types of students, by comparing the results of DGP4 with the ones of DGP3. By comparing the $\text{D}_1$ values of our proposed model and the Rasch model, our model provides a more efficient estimate to the accuracy parameter $\theta_{0,i}^{(c)}$, especially when $n$ and $D$ are large, say DGP4.
\section{Test Data Analysis}\label{sec: real data}
\subsection{Descriptive analysis}\label{sec: descriptive}
The motivating data consist of the English exam results of the 2020–2021 academic year from No. 11 Middle School of Wuhan, Bingjiang Campus, which is a public middle school in Jiang'an district of Wuhan, China. This exam is a final English exam for Grade 8 students in Fall semester of 2020-2021 academic year. There are 16 classes with 858 students taking this exam. The data set consists of 858 examinees ($n=858$) and 70 questions ($D=70$), where the questions are from a single exam. Among 70 questions, there are four major types pf questions (Listening Comprehension, Multiple Choice, Cloze Test, and Reading Comprehension). We proceed by carrying out an exploratory data analysis. By looking at the estimated accuracy marginalized for each question (column) or each row (examinee), visualized on the left hand side of Figure \ref{fig: expAnalysis}, it is trustworthy that the questions are designed hierarchically in terms of their difficulty, indicated by the estimated accuracy that ranges from 0.247 to 0.981. In addition, the proficiency of examinees are fairly heterogeneous, as the displayed histogram demonstrates a left skewed feature with a long tail. To be more specific, the histogram implies that most examinees can solve more than $70\%$ of the questions, while a small proportion of the examinees, whose estimated accuracy is below 0.4, may have probably failed the test. Such heterogeneity can also be viewed from the boxplots of the Rasch parameters, as shown on the right hand side of Figure \ref{fig: expAnalysis}, where $\xi$ and $\psi$ are defined similarly as those in \eqref{RaschModel}. As the primary goal of analyzing this data set is to explain the \textit{superiority} phenomenon, we move on to the next section to present the results by applying our proposed model.
\begin{figure}\label{fig: expAnalysis}
\end{figure}
\subsection{ACBM analysis}\label{sec: ACBM analysis}
To apply our proposed model, we set the number of iterations to be $n_{iter}=400$, $n_{rep}=400$, which are sufficient to thoroughly explore the posterior high density region based on our simulation analyses. The hyperparameters are chosen as $a_0 = 0.01$ and $b_0 = 0.01$ to ensure non-informative prior knowledge while new column and row clusters can still be generated. Given such settings, our model is implemented repeatedly for 100 times with different initial values. The reported column (question) partition is believed to be representative as the median Rand Index between it and the other column partitions is 0.91 with a standard deviation of 0.04 over the 100 Monte Carlo replications. The estimated accuracy parameters using posterior mean under each column cluster and the number of entries corresponding to each accuracy parameter are summarized in Table \ref{tab:3}.
\begin{table}[htp]
\centering
\caption{The number of components ($K$) and the estimated component-wise accuracy parameters under each question cluster and the corresponding cluster size ($|c|$).}
\label{tab:3}
\begin{tabular}{l|l|l|l}
\toprule
\toprule
Cluster & Size ($|c|$) & $K\leq (|c|+1)/2$ & Estimated accurarcy \\
\midrule
1 & 4 & 2 & 0.443, 0.872 \\
2 & 20 & 5 & 0.001, 0.344, 0.645, 0.924, 0.999 \\
3 & 5 & 2 & 0.218, 0.671 \\
4 & 17 & 4 & 0.179, 0.419, 0.726, 0.937\\
5 & 12 & 4 & 0.001, 0.417, 0.752, 0.989\\
6 & 8 & 3 & 0.164, 0.853, 0.999\\
\midrule
7 & 2 & 1 & 0.411 \\
8 & 1 & 1 & 0.791 \\
9 & 1 & 1 & 0.247 \\
\bottomrule
\bottomrule
\end{tabular}
\end{table}
Following Table \ref{tab:3}, a question cluster that contains more questions tends to possess more components. The estimation of most component-wise accuracy parameters is precise, since most estimated standard deviation values are one order of magnitude smaller than the corresponding estimated accuracy parameters. We further conjecture that the questions that are assigned to the clusters below the middle line in Table \ref{tab:3} are not effective in distinguishing different types of examinees, suggested by our model. Recall that the maximal number of examinees' mixtures is bounded by $(|c|+1)/2$ to ensure model identifiability. It is hence impossible to identify more than 1 examinees' mixture when a question cluster only has less than 3 questions. In other words, an ideal question cluster should consist of at least 3 questions to be able to detect the heterogeneity among examinees based on the mixture of accuracy model (Binomial mixture model).
The estimated component-wise accuracy parameters can further be visualized using Figure \ref{fig: RealPMat}, after rearranging the columns (questions) into a consecutive layout according to the estimated column (question) partition given by ACBM, for both ACBM and the Rasch estimations simultaneously.
\begin{figure}\label{fig: RealPMat}
\end{figure}
Intuitively, the gradient plot of ACBM looks like a discretized version of the Rasch gradient plot, which implies that our proposed model can recover the Rasch model's result to some extent. As an advantage over the Rasch model, our proposed model can automatically identify possible question clusters and the mixing structures on the examinees thereof. Note that the Guttman pattern is revealed locally if we look into the accuracy parameters of the corresponding examinees in question cluster 3 and 5. The questions in cluster 5 are listening comprehension and multiple choice questions, which are in general easier compared to the questions assigned to question cluster 3, of which the majority are difficult reading comprehension questions. To provide more insights, we present a contingency table for clusters 3 and 5 in Table \ref{tab:4}. For example, among the examinees who correctly answered to questions from cluster 3 with a higher accuracy (0.671), only 1 of them answer questions from cluster 5 with an accuracy being less than or equal to $.417$. In contrast, 469 ($94.4\%$) of them answer correctly to the questions in cluster 5 with an accuracy of .988. This finding agrees with the prior belief that questions in cluster 5 are easier than these in cluster 3, and further indicates that our method can effectively capture the Guttman pattern locally based on specific question clusters.
\begin{table}[htp]
\centering
\caption{Contingency table of examinees' count in terms of the accuracy parameters for cluster 3 and 5.}
\label{tab:4}
\begin{tabular}{l|l|l|l|l}
\toprule
\toprule
\backslashbox[48mm]{C3 (Difficult)}{C5 (Easy)} & Acc = 0.001 & Acc = 0.417 & Acc = 0.753 & Acc = 0.988 \\
\midrule
Acc = 0.218 & 5 & 34 & 73 & 239 \\
\midrule
Acc = 0.671 & 0 & 1 & 27 & \textbf{469} \\
\bottomrule
\bottomrule
\end{tabular}
\end{table}
We further discuss the \textit{superiority} phenomenon as indicated by the red rectangle in Figure \ref{fig: RealPMat}. Based on the contingency table \ref{tab:5}, for these examinees who are less proficient in question cluster 3 (accuracy = .218), 54.3$\%$ (25/46) of them did well in question cluster 4 with an 0.937 accuracy.
On the other hand, for those who do well in question cluster 4 (accuracy $\geq$ .726), 37.5$\%$ (42/112) of them did not perform well in question cluster 3 (accuracy = .218).
Such heterogeneity is hardly raised by randomness as we have sufficiently large number of samples in estimating each accuracy parameter. Similar findings can also be discovered in the region formed by the blue rectangle in the same figure. It is attractive to see that our method can capture such \textit{superiority} phenomenon, whereas the Rasch's model is unable to do so by using a single parameter to model the ability of examinees over all questions.
\begin{table}[htp]
\centering
\caption{Contingency table of the count of the examinees in the red rectangle in terms of the accuracy parameters for cluster 3 and 4.}
\label{tab:5}
\begin{tabular}{l|l|l|l}
\toprule
\toprule
\backslashbox[35mm]{C3}{C4} & Acc = 0.419 & Acc = 0.726 & Acc = 0.937 \\
\midrule
Acc = 0.218 & 4 & 17 & 25 \\
\midrule
Acc = 0.671 & 3 & 14 & 56 \\
\bottomrule
\bottomrule
\end{tabular}
\end{table}
\section{Discussion}\label{sec: Discussion}
In this article, we propose a novel IRT model using averaged mixture of binomial distributions with constraints, of which the novelty basically comes from the modelling of the \textit{superiority} phenomenon and the justification of the identifiability issue. Our model is manifested to be effective in both theoretical and practical aspects. Namely, the identifiability conclusion and posterior contraction results indicate that the latent accuracy parameters of our interest can be estimated at a $\sqrt{n}$ (up to a $\log$ term) rate asymptotically. In addition, the posterior samples of these parameters can be obtained using a tractable sampling algorithm, which satisfyingly approaches the stationary distributions according to the simulation results. By exploring the clustering effect at question level, depending on which the heterogeneity among examinees is further introduced, the real data analysis results given by our model are more flexible and interpretable compared with the Rasch model and many existing methods.
One possible generalization of our model is to consider the product of Bernoulli densities in place of the Binomial density, such that the accuracy parameters of the questions assigned to a question cluster can be arranged in an ascending order after a permutation. In other words, without loss of generality, suppose there exists a permutation $\sigma(\cdot)$ given a question cluster indexed by $1,2,\ldots,D'$, we define the product of Bernoulli densities as
\begin{equation}\label{ascending}
\begin{split}
& X_{i,j}\stackrel{ind}{\sim}\text{Bernoulli}(p_{i,j}), ~ p_{i,\sigma(1)}\leq p_{i,\sigma(2)}\leq \ldots \leq p_{i,\sigma(D')},~\text{for}~j=1,\ldots,D',
\end{split}
\end{equation}
where $\sigma(\cdot)$ is shared within the question cluster. We may call it the ordered Bernoulli densities. We expect such a gradient of the accuracy parameters can better explain the Guttman pattern than the kernel function currently used. The only concern of this structure is the identifiability of using this kernel density, which requires further investigation. One can directly apply our theoretical results if the product of Bernoulli densities is shown to be first-order identifiable under certain conditions. Another possible way of improving is to consider a more advanced sampling algorithm than ours, which is a typical application of the Algorithm 1 proposed by \citet{neal2000markov}. One may use split-merge sampling \citep{jain2004split} or slice sampling \citep{neal2003slice} to boost the procedure of approaching the stationary distribution. Future simulation studies would also be attractive by comprehensively examining the difference between our model and the alternatives that accommodate heterogeneity by introducing mixing structure at both the item and subject levels.
\end{document} |
\begin{document}
\title{Coverings of open books}
\author{Tetsuya Ito}
\address{Research Institute for Mathematical Sciences, Kyoto university, Kyoto, 606-8502, Japan}
\varepsilonmail{[email protected]}
\urladdr{http://www.kurims.kyoto-u.ac.jp/~tetitoh/}
\author{Keiko Kawamuro}
\address{Department of Mathematics,
The University of Iowa, Iowa City, IA 52242, USA}
\varepsilonmail{[email protected]}
\date{\today}
\subjclass[2000]{Primary 57M25, 57M27; Secondary 57M50}
\keywords{open book foliation, virtually overtwisted contact structure, coverings.}
\begin{abstract}
We study a coverings of open books and virtually overtwisted contact manifolds using open book foliations.
We show that open book coverings produces interesting examples such as transverse knots with depth grater than $1$.
We also demonstrate explicit examples of virtually overtwisted open books.
\varepsilonnd{abstract}
\maketitle
\section{Introduction}
In the classification of contact structures on oriented $3$-manifolds there is a dichotomy between tight and overtwisted contact structures.
The classification of overtwisted contact structures is reduced to homotopy theory by Eliashberg \cite{E}.
This is not the case for tight contact structures and study of tight contact structures is an active topic in contact geometry.
A tight contact structure is called {\varepsilonm universally tight} if its universal cover is tight, and {\varepsilonm virtually overtwisted} if it has a {\varepsilonm finite} cover that is overtwisted.
As a consequence of the geometrization, the fundamental groups of 3-manifolds are residually finite, which implies that every tight contact structure is either universally tight or virtually overtwisted (cf. \cite{H}).
Namely, {\varepsilonm universally} overtwisted is equilvalent to virtually overtwisted.
The idea of coverings plays important roles in many areas of mathematics, including study of contact structures.
In this note we identify a covering map of contact manifolds with an open book covering map (see Section~\ref{sec2}), and study virtually overtwisted contact manifolds using open book foliations.
Here is one of the results.
\noindent
{\bf Corollary~\ref{corA}.}
{\varepsilonm Let $B$ be the binding of an open book $(S,\phi)$. Then the {\varepsilonm depth} \cite{bo} of the binding is $1$ if and only if $\phi$ is not right-veering.}
In Section~\ref{sec4} we study examples of open books which have interesting properties.
We give a family of planar open books that supports overtwisted, virtually overtwisted and universally tight contact structures.
Some non-planar examples are also discussed.
\begin{proposition}\label{key-example}
Let $S=S_{0, p+q}$ be a sphere with $p+q$ holes, where $p, q\geq 2$.
Let $\alpha, \beta, \gamma \subset S$ be circles as shown in Figure~\ref{sphere}.
Let $\phi \in {\rm Aut}(S, \partial S)$ be a diffeomorphism given by
$$\phi=T \circ {T_\alpha}^{n} \circ T_\beta \circ T_\gamma$$
where $T$ is the product of one positive Dehn twist along each of the $p+q$ boundary components and
$T_\alpha$ is the positive Dehn twist along the curve $\alpha$.
\begin{enumerate}
\item
If $n \leq -2$ then $(S, \phi)$ supports an overtwisted contact structure.
\item
If $n =-1$ then $(S, \phi)$ supports a virtually overtwisted tight contact structure.
\item
If $n \geq 0$ then $(S, \phi)$ supports a universally tight contact structure.
\varepsilonnd{enumerate}
\begin{figure}[htbp]
\begin{center}
\SetLabels
(.2*.08) \scriptsize $p-1$\\
(.22*.27) \scriptsize $2$\\
(.22*.35) \scriptsize $1$\\
(.8*.08) \scriptsize $q-1$\\
(.74*.27) \scriptsize $2$\\
(.74*.35) \scriptsize $1$\\
(.5*.8) \scriptsize $\alpha$\\
(.45*.25) \scriptsize $\beta$\\
(.55*.25) \scriptsize $\gamma$\\
\varepsilonndSetLabels
\strut\AffixLabels{\includegraphics*[height=50mm]{sphere.pdf}}
\caption{The planar surface $S$ with $p+q$ boundary components.}\label{sphere}
\varepsilonnd{center}
\varepsilonnd{figure}
\varepsilonnd{proposition}
\section{Giroux correspondence and coverings}\label{sec2}
Let $S=S_{g,r}$ be an oriented genus $g$ surface with $r$ boundary components and $\phi \in {\rm Aut}(S, \partial S)$ be an orientation preserving diffeomorphism of $S$ fixing the boundary $\partial S$ pointwise.
The pair $(S, \phi)$ is called an {\varepsilonm abstract open book} (in this note the adjective ``abstract'' is omitted for simplicity) and $M_{(S, \phi)}$ denotes the closed oriented $3$-manifold obtained by gluing the mapping torus of $\phi$ and solid tori.
See Etnyre's lecture note \cite{Et} for basics (and more) of open books.
The Giroux correspondence \cite{G} states that there is a one-to-one correspondence between
open books (up to positive stabilization) and contact manifolds (up to isotopy).
We denote by $\xi_{(S, \phi)}$ the (isotopy class of) contact structure on the manifold $M_{(S, \phi)}$ {\varepsilonm compatible with} (or we often say {\varepsilonm supported by}) the open book $(S, \phi)$ via the Giroux correspondence.
Throughout this note a covering means a \varepsilonmph{finite} covering.
Suppose that $\pi:\tilde S \to S$ is a covering map.
\begin{definition}
If there exists a diffeomorphism $\tilde\phi \in {\rm Aut}(\tilde S, \partial\tilde S)$ satisfying $$\pi \circ \tilde\phi = \phi \circ \pi$$ then we call $(\tilde S, \tilde\phi)$ a {\varepsilonm covering} of the open book $(S, \phi)$.
We write $\pi:(\tilde S, \tilde\phi) \to (S, \phi)$ abusing the notation and call it an {\varepsilonm open book covering map}.
$$
\begin{array}{rcl}
\tilde S & \stackrel{\tilde\phi}{\longrightarrow} & \tilde S\\
\pi\downarrow & & \downarrow\pi\\
S & \stackrel{\phi}{\longrightarrow} & S
\varepsilonnd{array}
$$
\varepsilonnd{definition}
\begin{theorem}\label{prop1}
Let $\pi:(\tilde S, \tilde\phi) \to (S, \phi)$ be an open book covering map.
Then the compatible contact structures for the open books, via the Giroux correspondence \cite{G}, yield a covering map
$$P:(M_{(\tilde S, \tilde\phi)}, \xi_{(\tilde S, \tilde\phi)}) \to (M_{(S, \phi)}, \xi_{(S, \phi)})$$
{\varepsilonm compatible} with $\pi$, namely the restriction of $P$ to each page $\tilde S_t$ $(t\in[0,1])$ satisfies $P |_{\tilde S_t} = \pi$.
\varepsilonnd{theorem}
\begin{proof}
For simplicity we denote the covering space $(M_{(\tilde S, \tilde\phi)}, \xi_{(\tilde S, \tilde\phi)})$ by $(\tilde M, \tilde\xi)$, and the base space $(M_{(S, \phi)}, \xi_{(S, \phi)})$ by $(M, \xi)$.
We naturally extends the projection $\pi:\tilde S\to S$ to a map
$P: \tilde S \times [0,1] \to S \times [0,1]$
between the product manifolds such that the restriction of $P$ to each page $\tilde S_t (\simeq \tilde S)$ satisfies $P|_{\tilde S_t}=\pi$.
By the commutativity $\pi \circ \tilde\phi = \phi \circ \pi$
we have
$\phi(S_1) = \phi \circ \pi(\tilde S_1) = \pi \circ \tilde\phi(\tilde S_1) = \pi (\tilde S_0) = S_0$
thus the map $P:\tilde S \times [0,1] \to S \times [0,1]$
extends to the mapping tori
$P: (\tilde S \times [0,1])/\tilde\phi \to (S \times [0,1])/\phi$
and then over to the bindings.
Namely, the map $P$
induces a covering map $P: \tilde M \longrightarrow M$.
Let $\alpha$ be a contact 1-form on $M$ such that $\xi = \ker\alpha$.
Let $\tilde\alpha := P^*\alpha$ be the pullback of $\alpha$ then $\tilde\alpha\wedge d\tilde\alpha = P^*(\alpha\wedge d\alpha)>0$ and $\ker\tilde\alpha$ gives a contact structure on $\tilde M$ such that $P_*(\ker\tilde\alpha) = \ker\alpha=\xi$.
This shows that $P:(\tilde M, \ker\tilde\alpha)\to (M,\xi)$ is a covering map.
We also see that $(\tilde M, \ker\tilde\alpha)$ is supported by the open book $(\tilde S, \tilde\phi)$, that is,
$\tilde\alpha>0$ on the binding of the open book $(\tilde S, \tilde\phi)$ and $d\tilde\alpha>0$ on each page $\tilde S_t$. Thus the Giroux correspondence implies that $(\tilde M, \ker\tilde\alpha)$ and $(\tilde M, \tilde \xi)$ are isotopic, and the map $(\tilde M, \tilde\xi)\stackrel{P}{\to} (M,\xi)$ is a covering map.
\varepsilonnd{proof}
Conversely, we have the following.
\begin{theorem}\label{thm2}
Let $P:(\tilde M, \tilde\xi)\to(M,\xi)$ be a covering map for contact manifolds.
For every open book $(S,\phi)$ supporting $(M,\xi)$ there exists an open book $(\tilde S, \tilde\phi)$ supporting $(\tilde M, \tilde\xi)$ and giving an open book covering map $\pi:(\tilde S, \tilde\phi)\to (S, \phi)$ compatible with $P$.
\varepsilonnd{theorem}
\begin{proof}
Let $S_t$ ($t\in[0,1]$) denote the pages of the open book decomposition $(S, \phi)$ of $M$.
Let $\tilde S_t := P^{-1}(S_t)$ and $\tilde B = P^{-1}(B)$, where $B \subset M$ is the binding for $(S,\phi)$.
All the $\tilde S_t$ have the same topological type, denoted by $\tilde S$, and $P$ induces a covering map $\pi:\tilde S\to S$.
There exists $\tilde\phi \in {\rm Aut}(\tilde S, \partial\tilde S)$ such that:
$$
\tilde M\setminus\tilde B \simeq
(\tilde S\times[0,1]) / (x,1) \sim (\tilde\phi(x), 0)
$$
Since the pages $S_0$ and $S_1$ are identified under $\phi$ the commutativity $\pi \circ \tilde\phi=\phi\circ\pi$ holds.
Thus we get an open book covering map $\pi:(\tilde S, \tilde\phi)\to (S, \phi)$ compatible with $P$.
By the same argument as in the proof of Proposition~\ref{prop1} we can show that $(\tilde S, \tilde\phi)$ supports the contact manifold $(\tilde M, \tilde\xi)$.
\varepsilonnd{proof}
\begin{remark}
For a covering map of contact 3-manifolds $P:(\tilde M, \tilde \xi) \rightarrow (M,\xi)$, not every open book decomposition $(\tilde S, \tilde \phi)$ of $(\tilde M, \tilde \xi)$ arises as an open book covering compatible with $P$.
\varepsilonnd{remark}
To see this statement we recall the following simple fact, which easily follows from the definition of right-veering diffeomorphisms \cite{HKM}.
\begin{lemma}
\label{lemma:coverrv}
Let $\pi:(\tilde S, \tilde \phi) \rightarrow (S,\phi)$ be an open book covering map. Then $\phi$ is right-veering if and only if $\tilde \phi$ is right-veering.
\varepsilonnd{lemma}
Now consider the case that $(M,\xi)$ is tight and $(\tilde M, \tilde \xi) $ is overtwisted.
Then by \cite[Theorem 1.1]{HKM} there is an open book decomposition $(\tilde S, \tilde \phi)$ of $(\tilde M, \tilde \xi)$ such that $\tilde \phi$ is not right-veering.
On the other hand, since $(M,\xi)$ is tight every open book $(S,\phi)$ of $(M,\xi)$ has right-veering $\phi$.
Hence Lemma \ref{lemma:coverrv} shows that the non-rightveering open book $(\tilde S, \tilde \phi)$ cannot cover $(S,\phi)$.
\section{The overtwisted complexity, depth of bindings and open book coverings}\label{sec3}
In this section we study properties of open book coverings using the notion of {\varepsilonm right-veeringness} \cite{HKM} and the open book foliation method \cite{ik1-1}.
Let us recall the {\varepsilonm overtwisted complexity} $n(S, \phi)$ introduced in \cite[Definition 6.4]{ik2}.
It is a non-negative integer given by:
$$n(S, \phi) = \min\left\{
e_-(\mathcal F_{ob} (D)) \ | \
D \mbox{ is a transverse overtwisted disk in } (S, \phi)\right\},
$$
if $(S, \phi)$ supports an overtwisted contact structure, and $n(S, \phi)=0$ otherwise.
Here $e_-(\mathcal F_{ob} (D))$ denotes the number of negative elliptic points in the open book foliation on $D$.
See Definition 4.1 of \cite{ik1-1} for the definition of a {\varepsilonm transverse overtwisted disk}, which can be understood as a transverse push-off of a usual overtwisted disk, or, the spanning disk of a transverse unknot $K$ with $sl(K)=+1$.
The following property is proved in \cite{ik2}.
\begin{proposition}\label{prop:n}
\cite[Corollary 6.5]{ik2}
\begin{enumerate}
\item
$n(S, \phi) = 0$ if and only if $\xi_{(S, \phi)}$ is tight (and hence $\phi$ is right veering).
\item
$n(S, \phi) = 1$ if and only if $\xi_{(S, \phi)}$ is overtwisted and $\phi$ is not right veering.
\item
$n(S, \phi) \geq 2$ if and only if $\xi_{(S, \phi)}$ is overtwisted and $\phi$ is right veering.
\varepsilonnd{enumerate}
\varepsilonnd{proposition}
As a consequence we can show the following:
\begin{proposition}\label{Cor65}
Let $\pi: (\tilde S, \tilde\phi)\to(S,\phi)$ be an open book covering such that $n(\tilde S, \tilde\phi)=1$ then $(S, \phi)$ supports an overtwisted contact structure.
\varepsilonnd{proposition}
\begin{proof}
Suppose that $(S,\phi)$ supports a tight contact structure.
Then $\phi$ is right-veering for every boundary component of $S$.
By Lemma~\ref{lemma:coverrv} $\tilde\phi$ is also right-veering for every boundary component of $\tilde S$.
The property (3) of Proposition~\ref{Cor65} implies that $n(\tilde S, \tilde\phi)\geq 2$, which is a contradiction.
\varepsilonnd{proof}
The overtwisted complexity is closely related to the depth of transverse knots or links introduced by Baker and Onaran in \cite{bo}:
The \varepsilonmph{depth} of a transverse knot or link\footnote{As mentioned in Remark 5.2.4 of \cite{bo} the depth can be defined for links though it is originally defined for knots.}
$K$ in an overtwisted contact 3-manifold $(M,\xi)$ is defined by
\[
d(K)= \min \{ |D \cap K| \: | \: D \text{ is an overtwisted disk in } (M,\xi) \}
\]
and $K$ is called \varepsilonmph{non-loose} if $d(K)>0$, that is, $\xi$ is tight on ${M\setminus K}$.
\begin{theorem}\label{thm:d=n}
Let $B$ be the binding of an open book $(S,\phi)$ supporting an overtwisted contact structure.
If $(S, \phi)$ supports an overtwisted contact structure then $d(B)=n(S,\phi)$.
\varepsilonnd{theorem}
\begin{proof}
Let $D_{\sf trans}$ be a transverse overtwisted disk realizing $n(S,\phi)$, that is, the open book foliation $\mathcal F_{ob} (D_{\sf trans})$ has $n(S,\phi)$ negative elliptic points.
Let $(B, \pi)$ be an open book decomposition of $M_{(S, \phi)}$ that is determined by the abstract open book $(S, \phi)$.
By \cite[Theorem 2.21]{ik1-1} we may choose a contact structure $\xi$ supported by $(B, \pi)$ such that the characteristic foliation $\mathcal F_\xi (D_{\sf trans})$ and the open book foliation $\mathcal F_{ob} (D_{\sf trans})$ are topologically conjugate.
Moreover we may assume that
the set of positive/negative elliptic points of $\mathcal F_\xi (D_{\sf trans})$ coincides exactly with the set of positive/negative elliptic points of $\mathcal F_{ob} (D_{\sf trans})$.
Recall that a positive/negative elliptic point of the open book foliation on a surface $F$ is just a positive/negative intersection point of $F$ and the binding $B$.
With this in mind we denote by $B \cap^{\pm} D_{\sf trans}$ the set of $\pm$-intersection points of $D_{\sf trans}$ and $B$.
Let $B'$ be a transverse link that is obtained from $B$ by transverse isotopy {\varepsilonm only near} the
intersection points $B \cap D_{\sf trans}$ so that
\begin{itemize}
\item
$|B' \cap^+ D_{\sf trans}| = |B \cap^+ D_{\sf trans}|$
and
$(B' \cap^+ D_{\sf trans}) \subset A$
\item
$|B' \cap^- D_{\sf trans}| = |B \cap^- D_{\sf trans}|$
and $B' \cap G_{--}(\mathcal F_\xi (D_{\sf trans})) = \varepsilonmptyset$
\varepsilonnd{itemize}
where $A \subset D_{\sf trans}$ is the annulus bounded by the graph $G_{++}(\mathcal F_\xi (D_{\sf trans}))$ and the boundary $\partial D_{\sf trans}$, and
$G_{++}(\mathcal F_\xi (D_{\sf trans}))$ (resp. $G_{--}(\mathcal F_\xi (D_{\sf trans}))$) is the Giroux graph in the characteristic foliation consisting of positive (resp. netagive) elliptic points and stable (resp. unstable) separatrices of positive (resp. negative) hyperbolic points (see \cite[page~646]{G} and \cite[Definition 2.17]{ik1-1}).
Since the two foliations $\mathcal F_\xi (D_{\sf trans})$ and $\mathcal F_{ob} (D_{\sf trans})$ are topologically conjugate the graphs $G_{\pm\pm}(\mathcal F_\xi (D_{\sf trans}))$ and $G_{\pm\pm}(\mathcal F_{ob} (D_{\sf trans}))$ are topologically conjugate.
By the definition of a transverse overtwisted disk \cite[Definition 4.1]{ik1-1} the graph $G_{--}$ is a tree and $G_{++}$ is a circle enclosing $G_{--}$. See Figure \ref{fig:transtousual}, where $G_{--}(\mathcal F_\xi (D_{\sf trans}))$ and $G_{++}(\mathcal F_\xi (D_{\sf trans}))$ are depicted by the gray and the black bold arcs, respectively.
\begin{figure}[htbp]
\begin{center}
\SetLabels
(0*.05) $\mathcal F_\xi (D_{\sf trans})$\\
(.75*.66) $D$\\
(.17*.85) $+$\\
(.1*.23) $G_{++}$\\
(.2*.55) $-$\\
(.2*.45) $G_{--}$\\
(1*.05) $\mathcal F_\xi (D_{\sf trans}')$\\
(.33*.85) $A$\\
\varepsilonndSetLabels
\strut\AffixLabels{\includegraphics*[height=52mm]{transtousual.pdf}}
\caption{Giroux elimination lemma is applied to the gray regions in the left disk.
The dots $\odot$ represent the intersection points $B' \cap D_{\sf trans}$ and $B' \cap D'_{\sf trans}$.}
\label{fig:transtousual}
\varepsilonnd{center}
\varepsilonnd{figure}
Note that $B'$ is not used as a binding but it is just a transverse link.
We also keep using the same contact structure $\xi$, hence the characteristic foliation $\mathcal F_\xi (D_{\sf trans})$ does not change.
We apply the {\varepsilonm Giroux elimination lemma} \cite[Lemma 3.3]{Gconvex} to small 3-ball neighborhoods (gray regions in Figure~\ref{fig:transtousual}) of $G_{\pm\pm} (\mathcal F_\xi (D_{\sf trans}))$ each of which contains a pair of consecutive elliptic and hyperbolic points (of the same sign) and is disjoint from $B'$.
We can find a disk, $D_{\sf trans}'$, and a sub-disk, $D$, of $D_{\sf trans}'$ with the following properties:
\begin{itemize}
\item
$D_{\sf trans}'$ is $C^0$ close to $D_{\sf trans}$.
\item
$D$ is a standard overtwisted disk, i.e., its characteristic foliation contains exactly one elliptic singularity and ${\rm tb}(\partial D)=0$.
\item
$\{B' \cap^+ D_{\sf trans}'\} = \{B' \cap^+ D_{\sf trans}\}$
and
$|B' \cap^+ D| = 0$.
\item
$\{B' \cap^- D_{\sf trans}\} = \{ B' \cap^- D_{\sf trans}'\} =\{B' \cap^- D\}$
\varepsilonnd{itemize}
Here the third property follows from the condition $(B' \cap^{+} D_{\sf trans}) \subset A$.
The fourth property follows from the condition $B' \cap G_{--}(\mathcal F_\xi (D_{\sf trans})) = \varepsilonmptyset$.
Though $D_{\sf trans}'$ may not admit an open book foliation this would not be a problem.
We have
$$d(B)=d(B') \leq |B' \cap D| = |B' \cap^- D| =
|B' \cap^- D_{\sf trans}| =
|B \cap^- D_{\sf trans}| = n(S, \phi).$$
Thus $d(B) \leq n(S,\phi)$.
Conversely, let $D$ be an overtwisted disk realizing $d(B)$, that is, $|B \cap D| = d(B)$.
Taking the positive transverse push-off of the Legendrian boundary $\partial D$ we find a transverse unknot, $K$, with $sl(K)=1$.
A spanning disk $D'$ of $K$ still intersects $B$ at $d(B)$ points.
By Pavalescu's proof of Alexander theorem \cite[Theorem 3.2]{pav}, there is an isotopy preserving each page
and moving the non-braided parts of $K$ to neighborhoods of the binding.
In the neighborhoods we can move $D'$ so that $K=\partial D'$ becomes a closed braid without introducing negative intersection points of $D'$ and $B$.
Following the discussion in the proof of \cite[Theorem 4.3]{ik1-1}, from $D'$ we can construct a transverse overtwisted disk $D_{\sf trans}$ whose open book foliation has no more than $d(B)$ negative elliptic points, hence $n(S,\phi) \leq d(B)$.
\varepsilonnd{proof}
As a consequence of Proposition~\ref{prop:n} and Theorem~\ref{thm:d=n} we have the following characterization of depth one bindings, which generalizes \cite[Theorem 5.2.3]{bo} (except for the part regarding the tension invariant).
\begin{corollary}\label{corA}
Let $B$ be the binding of an open book $(S,\phi)$. Then $d(B)=1$ if and only if $\phi$ is not right-veering.
\varepsilonnd{corollary}
Corollary \ref{corA} gives a construction of Legendrian or transverse knots and links with large depth (cf. \cite[Problems 6.1 and 6.4]{bo}).
\begin{corollary}\label{corB}
Let $B$ be the binding of an open book $(S,\phi)$ supporting an overtwisted contact structure.
Let $L$ be a Legendrian approximation of $B$.
If $\phi$ is right-veering then $1 < d(B) \leq d(L)$.
\varepsilonnd{corollary}
The inequality $d(B) \leq d(L)$ holds even without the right-veering assumption.
In fact, there are several constructions of
right-veering open books supporting overtwisted contact structures as listed below:
\begin{enumerate}
\item \cite[Proposition 6.1]{HKM}
Every open book can be made right-veering after a sequence of positive stabilizations.
\item
By Theorem \ref{thm2}, for a covering map $P:(\tilde M, \tilde \xi) \rightarrow (M,\xi)$ between a tight $(M,\xi)$ and an overtwisted $(\tilde M, \tilde \xi)$ with an open book $(S,\phi)$ supporting $(M,\xi)$, there exists an open book covering $\pi:(\tilde S, \tilde \phi) \rightarrow (S,\phi)$ compatible with $P$.
By \cite[Theorem 1.1]{HKM} $\phi$ is right-veering and Lemma~\ref{lemma:coverrv} implies that $\tilde \phi$ is right-veering.
Such a family of examples is discussed in Proposition~\ref{key-example} where $d(\tilde B)=2$.
\varepsilonnd{enumerate}
If the bindings of a open book is not connected then by further positive stabilizations, which preserve the right-veering property, we can always make the binding connected.
Hence it is fairly easy to construct a transverse or Legendrian knot with depth greater than $1$.
We point out that if an open book $(\tilde S, \tilde \phi)$ in
the construction (2) is not destabilizable then it gives rise to an example of right-veering, non-destabilizable open book supporting an overtwisted contact structure.
The existence (or non-existence) of such open books is asked in \cite{HKM} and many examples have been found \cite{le, lis, ik1-2, kr}.
Presumably, under certain condition, open book coverings would provide non-destabilizable open books:
In \cite{etl} it is shown that a right-veering open book $(S,\phi)$ is destabilizable if and only if the {\varepsilonm translation distance} (see \cite{etl} for the definition) of $\phi$ is equal to one.
Although the behavior of the translation distance under a covering operation is not clear, it is likely that if $\phi$ has a large translation distance then so does $\tilde\phi$, hence open book covering is non-destabilizable.
\section{Illustration of overtwisted coverings and a pants pattern}\label{sec4}
In this section we study a sequence of open books that supports overtwisted, virtually overtwisted tight and universally tight contact structures.
We begin with a proof of Proposition~\ref{key-example}.
\begin{proof}
We prove the assertion (1).
Applying the proof of Theorem 4.1 in \cite{ik1-2} we can construct a transverse overtwisted disk in the open book $(S, \phi)$.
By the definition every transverse overtwisted disk has the self-linking number $1$, that is, the Bennequin-Eliashberg inequality \cite{E2} is violated.
Thus $(S, \phi)$ supports an overtwisted contact structure.
The assertion (3) follows from the same argument in Example 5.2 of \cite{EV}.
Finally we prove the assertion (2).
By the lantern relation (see for example \cite[Proposition 5.1]{FM}) the mapping class $\phi$ can be written in the product of positive Dehn twists.
Therefore, results of Giroux \cite{G} and Eliashberg-Gromov \cite{EG} imply that $(M, \xi)$ is tight.
Below we consider the following four cases. We find a transverse overtwisted disk in an open book covering for each case.\\
(Case 1) $p-1\varepsilonquiv q-1\varepsilonquiv 1$ (mod $2$); \\
(Case 2) $p-1 \varepsilonquiv q-1\varepsilonquiv 0$ (mod $2$); \\
(Case 3) $p-1 \varepsilonquiv 1$ and $q-1 \varepsilonquiv 0$ (mod $2$);\\
(Case 4) $p-1 \varepsilonquiv 0$ and $q-1 \varepsilonquiv 1$ (mod $2$):
For each case we cut two copies of $S$ along the thick gray arcs as shown in Figure~\ref{cut}, then glue them along the cut arcs to get a connected surface $\tilde S$.
Clearly $\tilde S$ is a double cover of $S$.
We call the projection map $\Pi:\tilde S\to S$.
(For Cases 3 and 4, the base space $S$ is disconnected after the cut but it is easy to verify that the covering space $\tilde S$ is connected.)
\begin{figure}[htbp]
\begin{center}
\SetLabels
(.1*-.07) Case 1\\
(.04*.625) \scriptsize $1$\\
(.04*.56) \scriptsize $2$\\
(.025*.45) \scriptsize $q-1$\\
(.04*.39) \scriptsize $q$\\
(.37*-.07) Case 2\\
(.025*.11) \scriptsize $p-1$\\
(.3*.625) \scriptsize $1$\\
(.28*.45) \scriptsize $q-1$\\
(.28*.11) \scriptsize $p-1$\\
(.61*-.07) Case 3\\
(.57*.66) \scriptsize $1$\\
(.55*.48) \scriptsize $q-1$\\
(.55*.09) \scriptsize $p-1$\\
(.9*-.07) Case 4\\
(.83*.66) \scriptsize $1$\\
(.81*.48) \scriptsize $q-1$\\
(.81*.09) \scriptsize $p-1$\\
\varepsilonndSetLabels
\strut\AffixLabels{\includegraphics*[height=55mm]{cut.pdf}}
\caption{Cutting arcs (highlighted gray) in $S$ to construct $\tilde S$.}\label{cut}
\varepsilonnd{center}
\varepsilonnd{figure}
Choose base points $x_0 \in S$ and $\tilde x_0 \in \Pi^{-1}(x_0)$.
Let $G$ be the index two subgroup of $\pi_{1}(S, x_0)$ defined by $G = \{\gamma \in \pi_{1}(S, x_0)\: | \: \langle [\gamma], [c]\rangle = 0\}$, where
$\langle -,- \rangle: H_{1}(S)\times H_{1}(S,\partial S) \rightarrow \mathbb{Z}_2$ is the mod $2$ algebraic intersection pairing and $[c] \in H_{1}(S,\partial S)$ is the relative homology class represented by the set of cutting arcs (with any choice of orientation).
Note that the covering space $\Pi: \tilde{S} \to S$ has $\Pi_*(\pi_1(\tilde S, \tilde x_0))=G$.
Since $\phi_*(G)=G$ there is a homeomorphism $\tilde\phi: \tilde S\to \tilde S$ such that $\tilde\phi(\tilde x_0)=\tilde x_0$ and $\Pi \circ \tilde\phi = \phi\circ\Pi$.
We call $\tilde\phi$ a lift of $\phi$.
For an arc $\tilde{\gamma}$ in $\tilde{S}$ the image $\tilde \phi(\tilde{\gamma})$ is nothing but the lift of the arc $\phi(\pi(\tilde{\gamma}))$ in $S$. This allows us to compute $\tilde \phi$ and one can check that $\tilde \phi$ fixes the boundary $\partial\tilde S$ pointwise.
In general it may be hard to write $\tilde\phi$ as the product of Dehn twists.
Figure~\ref{OTcover-odd2} (resp. Figure~\ref{OTcover-even2}) gives a movie presentation of a transverse overtwisted disk for Case 1 (resp. Case 2).
For Case 3 and Case 4 combining the ideas of Figures~\ref{OTcover-odd2} and \ref{OTcover-even2} one can also find transverse overtwisted disks.
We leave it to readers as an exercise.
Therefore, the open book $(\tilde S, \tilde\phi)$ supports an overtwisted contact structure.
\varepsilonnd{proof}
\begin{figure}[htbp]
\begin{center}
\SetLabels
(.5*.97) ($t=0$)\\
(.5*.93) Case 1\\
(.75*.925) \scriptsize $(-)$\\
(.5*.62) ($t=t_1$)\\
(.5*.29) ($t=t_2$)\\
\varepsilonndSetLabels
\strut\AffixLabels{\includegraphics*[height=21cm]{OTcover-odd.pdf}}
\varepsilonnd{center}
\varepsilonnd{figure}
\begin{figure}[htbp]
\begin{center}
\SetLabels
(.5*.95) ($t=t_3$)\\
(.5*.44) ($t=1$)\\
\varepsilonndSetLabels
\strut\AffixLabels{\includegraphics*[height=13cm]{OTcover-odd2.pdf}}
\caption{(Case 1, $p-1=q-1=3$): A movie presentation of a transverse overtwisted disk.
Each arrow indicates orientation of the b-arc.
The starting (resp. ending) point of a b-arc is a positive (resp. negative) elliptic point.
The end point of an a-arcs is marked with a $\odot$.
This transverse overtwisted disk has two negative elliptic points and four positive elliptic points.
Thick dashed arcs are describing arcs for hyperbolic points.
The signs of the hyperbolic points are all positive except the one marked with $(-)$ in the page $t=0$.
One can easily generalize this to any $p$ and $q$ with $p-1\varepsilonquiv q-1\varepsilonquiv 1$ (mod $2$).}
\label{OTcover-odd2}
\varepsilonnd{center}
\varepsilonnd{figure}
\begin{figure}[htbp]
\begin{center}
\SetLabels
(.5*.97) ($t=0$)\\
(.5*.93) Case 2\\
(.5*.62) ($t=t_1$)\\
(.5*.29) ($t=t_2$)\\
\varepsilonndSetLabels
\strut\AffixLabels{\includegraphics*[height=21cm]{OTcover-even.pdf}}
\varepsilonnd{center}
\varepsilonnd{figure}
\begin{figure}[htbp]
\begin{center}
\SetLabels
(.5*.95) ($t=t_3$)\\
(.5*.44) ($t=1$)\\
\varepsilonndSetLabels
\strut\AffixLabels{\includegraphics*[height=13cm]{OTcover-even2.pdf}}
\caption{(Case 2, $p-1=q-1=2$): A movie presentation of a transverse overtwisted disk.
This transverse overtwisted disk has two negative elliptic points and four positive elliptic points.
The signs of the hyperbolic points are all positive except the one in the page $t=0$.
One can
generalize this to any $p$ and $q$ with $p-1\varepsilonquiv q-1\varepsilonquiv 0$ (mod $2$)}
\label{OTcover-even2}
\varepsilonnd{center}
\varepsilonnd{figure}
\pagebreak
\begin{remark}
Let $K_{p,q} \subset (S^3, \xi_{st})$ be a Legendrian unknot in the standard contact $3$-sphere with the Thurston-Bennequin number ${\rm tb}(K_{p,q})=-(p+q)+1$ and the rotation number ${\rm rot}(K_{p,q})=p-q$ or $q-p$, where $p, q \geq 2$.
See Figure~\ref{FrontProj}.
\begin{figure}[htbp]
\begin{center}
\SetLabels
(0*.8) $p$\\
(1*.8) $q$\\
(0*.17) $1$\\
(0*.3) $2$\\
(1*.17) $1$\\
(1*.3) $2$\\
\varepsilonndSetLabels
\strut\AffixLabels{\includegraphics*[height=30mm]{FrontProj.pdf}}
\caption{The front projection of $K_{p,q}$.}\label{FrontProj}
\varepsilonnd{center}
\varepsilonnd{figure}
Let $(M, \xi)$ denote the contact structure obtained from $(S^3, \xi_{st})$ by the Legendrian surgery along $K_{p,q}$.
When $n=-1$ in Proposition~\ref{key-example} we can verify that the open book $(S, \phi)$ supports the contact manifold $(M, \xi)$ applying Sch\"onenberger's algorithm \cite{S} and the lantern relation.
With this identification of $(S, \phi)$ and $(M, \xi)$ the assertion (2) of Proposition~\ref{key-example}
can also be proved applying Gompf's criterion of virtually overtwisted contact structures \cite{Go}.
Lastly, we note that if $p=1$ or $q=1$ then $\xi$ is known to be universally tight due to Honda \cite{H} and Giroux \cite{G1}.
\varepsilonnd{remark}
\begin{remark}
Under the projection $\pi:\tilde S_t \to S_t$ for each page we can see how the transverse overtwisted disk is `folded' (in other words, self-intersecting) in the base tight manifold $(M, \xi)$.
For example when $t=t_1$ of Case 1 in Figure~\ref{OTcover-odd2} the projected image of the transverse overtwisted disk has two intersection points marked with black dots as in Figure~\ref{projection}.
\begin{figure}[htbp]
\begin{center}
\SetLabels
(1*0) page $S_{t_1}$\\
\varepsilonndSetLabels
\strut\AffixLabels{\includegraphics*[height=35mm]{projection.pdf}}
\caption{Self-intersection points of the transverse overtwisted disk (of Figure~\ref{OTcover-odd2}) under the projection $\pi:\tilde S_t \to S_t$. }\label{projection}
\varepsilonnd{center}
\varepsilonnd{figure}
\varepsilonnd{remark}
One can generalize the construction of overtwisted disks to a $k$-fold cover using the same cutting arcs of Figure~\ref{cut}.
For example:
\begin{example}\label{ex1}
Let $S$ be a $2$-sphere with four holes. Let $a,b,c,d,e \subset S$ be simple closed curves parallel to the boundary as shown in Figure~\ref{4psphere}.
Let
$$\Phi_{\alpha, \beta} = T_{a}^{\alpha+1}T_{b}^{\beta+1}T_{c}T_{d}T_{e}^{-1}$$
Suppose that $\alpha, \beta > 0$ and there exists a number $k\geq 2$ that divides both $\alpha+1$ and $\beta+1$.
Then there exists a $k$-fold cover of $(S, \Phi_{\alpha, \beta})$ that supports an overtwisted contact structure, i.e., $(S, \Phi_{\alpha, \beta})$ supports a virtually overtwisted tight contact structure.
\begin{figure}[htbp]
\begin{center}
\SetLabels
\varepsilonndSetLabels
\strut\AffixLabels{\includegraphics*[height=35mm]{4psphere.pdf}}
\caption{The surface $S$.}\label{4psphere}
\varepsilonnd{center}
\varepsilonnd{figure}
\varepsilonnd{example}
So far we have only seen planar open books. In fact our example can be applied to higher genus open books:
Suppose that an open book $(S', \Psi)$ supports a tight contact structure, the Nielsen-Thurston type of $\phi$ is reducible, and `containing' $(S, \Phi_{\alpha, \beta})$ as a subspace, that is, $S \subset S'$ and $\Psi|_S = \Phi_{\alpha, \beta}$.
Then $(S', \Psi)$ supports a virtually overtwisted contact structure.
\begin{example}
Let $S'$ be a genus 4 surface with two holes, $d$ and $c$, see Figure~\ref{subspace}.
Let $\Psi= T_{a}^{2}T_{b}^{2}T_{c}T_{d}T_{e}^{-1} T_f$ be a diffeomorphism of $S'$.
The open book $(S', \Psi)$ contains $(S, \Phi_{1,1})$
of Example~\ref{ex1} and $(S', \Psi)$ supports a tight contact structure.
Take a double cover $\tilde S'$ of $S'$ tjat os a genus 7 surface with four boundary components, $\tilde c, \tilde d, \tilde c', \tilde d'$.
The monodromy $\Psi$ lifts to a diffeomorphism $\tilde\Psi= T_{\tilde a}T_{\tilde b}T_{\tilde c}T_{\tilde d}T_{\tilde e}^{-1} T_{\tilde f}T_{\tilde c'}T_{\tilde d'}T_{\tilde e'}^{-1} T_{\tilde f'}$ of $\tilde S'$.
Example~\ref{ex1} guarantees that the open book $(\tilde S', \tilde\Psi)$ supports an overtwisted contact structure.
\varepsilonnd{example}
\begin{figure}[htbp]
\begin{center}
\SetLabels
(.55*.1) $S$\\
(.1*.1) $S'$\\
(-.1*.8) $\tilde{S'}$\\
(.5*.29) $b$\\
(.6*.05) $a$\\
(.6*.2) $e$\\
(.65*.14) $c$\\
(.65*.22) $d$\\
(.8*.15) $f$\\
(.76*.8) $\tilde d$\\
(.76*.7) $\tilde c$\\
(.8*.75) $\tilde e$\\
(.5*.9) $\tilde b$\\
(.5*.57) $\tilde a$\\
(.22*.8) $\tilde c'$\\
(.22*.7) $\tilde d'$\\
(.2*.75) $\tilde e'$\\
(.9*.7) $\tilde f$\\
(.1*.7) $\tilde f'$\\
\varepsilonndSetLabels
\strut\AffixLabels{\includegraphics*[height=80mm]{subspace.pdf}}
\caption{(Top) A genus 7 surface $\tilde S'$ with four holes. (Bottom) An genus 4 surface $S'$ with two holes.}
\label{subspace}
\varepsilonnd{center}
\varepsilonnd{figure}
\begin{remark}
Lemma \ref{lemma:coverrv} and the discussion in Section~\ref{sec3} imply that if $(S, \phi)$ is a virtually overtwisted contact structure then its overtwisted cover has the overtwisted complexity (see Section~\ref{sec3}) $n(\tilde S, \tilde\phi) \geq 2$.
We notice that all the examples of virtually overtwisted open books $(S, \phi)$ we study in this note have $n(\tilde S, \tilde\phi) = 2$.
Moreover, these open books $(S, \phi)$ all contain a pants region $P \subset S$ (see Figure~\ref{pants}) with the following properties
\begin{itemize}
\item $P$ is bounded by curves $x, y, z$ with $x, y \subset \partial S$ and $z \subset {\rm Int}(S)$,
\item the monodromy $\phi$ preserves $P$ and $\phi|_P = T_x T_y {T_z}^{-1}$.
\varepsilonnd{itemize}
(The curve $z$ corresponds to $\alpha$ of Figure~\ref{sphere} and $e$ of Figure~\ref{4psphere}).
Such a pants region $P$ plays a crucial role in our construction of transverse overtwisted disks
because the two negative elliptic points of each transverse overtwisted disk lie on the lifts of $x$ and $y$.
\begin{figure}[htbp]
\begin{center}
\SetLabels
(.23*.7) $x$\\
(.75*.7) $y$\\
(.5*.9) $z$\\
\varepsilonndSetLabels
\strut\AffixLabels{\includegraphics*[height=20mm]{pants.pdf}}
\caption{Pants region $P$.}
\label{pants}
\varepsilonnd{center}
\varepsilonnd{figure}
\varepsilonnd{remark}
\noindent{\bf Question:}
Do there exist open book patterns, like the above pants pattern, that give virtually overtwisted contact structures?
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\varepsilonnd{document} |
\begin{document}
\begin{abstract}
Tropical geometry gives a bound on the ranks of divisors on curves in terms
of the combinatorics of the dual graph of a degeneration. We show that for a
family of examples, curves realizing this bound might only exist over certain
characteristics or over certain fields of definition. Our examples also apply to
the theory of metrized complexes and weighted graphs. These examples arise by
relating the lifting problem to matroid realizability. We also give a proof of
Mn\"ev universality with explicit bounds on the size of the matroid, which may
be of independent interest.
\end{abstract}
\title{Lifting matroid divisors on tropical curves}
\section{Introduction}\label{sec:introduction}
The specialization inequality in tropical geometry gives an upper bound for the
rank of a divisor on a curve in terms of a combinatorial quantity known as the
rank of the specialization of the divisor on the dual graph of the special fiber
of a degeneration~\cite{baker}. This bound can be sharpened by incorporating
additional information about the components of the special fiber, giving
augmented graphs~\cite{amini-caporaso} or metrized complexes~\cite{amini-baker}.
All of these inequalities can be strict because there may be many algebraic
curves and divisors with the same specialization. Thus, the natural question is
whether, for a given graph and divisor on that graph, the inequality is sharp
for \emph{some} algebraic curve and divisor. If $R$ is the discrete valuation
ring over which the degeneration of the curve is defined, we will refer to such
a curve and divisor
as a \defi{lifting} of the graph with its divisor over~$R$.
In this paper, we show that the
existence of a lifting can depend strongly on the characteristic of the field:
\begin{thm}\label{thm:characteristic}
Let $P$ be any finite set of prime numbers. Then there exist graphs $\Gamma$ and
$\Gamma'$ with rank~$2$ divisors $D$ on~$\Gamma$ and~$D'$ on~$\Gamma'$ with the
following property:
For any infinite field~$k$, $\Gamma$ and~$D$ lift over $k[[t]]$ if and only
if the characteristic of~$k$ is in~$P$, and $\Gamma'$ and~$D'$ lift over
$k[[t]]$ if and only if the characteristic of~$k$ is not in~$P$.
\end{thm}
We also show that the existence of a lift depends on the field even beyond
characteristic:
\begin{thm}\label{thm:number-field}
Let $k'$ be any number field. Then there exists a graph~$\Gamma$ with a rank~$2$
divisor~$D$ such that for any field~$k$ of characteristic~$0$, $\Gamma$ and~$D$
lift over~$k[[t]]$ if and only if $k$ contains $k'$.
\end{thm}
Both Theorem~\ref{thm:characteristic} and~\ref{thm:number-field} are
immediate consequences of the following:
\begin{thm}\label{thm:universality}
Let $X$ be a scheme of finite type over $\Spec \ZZ$. Then there exists a graph
$\Gamma$ with a rank~$2$ divisor~$D$ such that, for any infinite field $k$,
$\Gamma$ and~$D$ lift over $k[[t]]$ if and only if $X$ has a $k$-point.
\end{thm}
Theorems~\ref{thm:characteristic}, \ref{thm:number-field},
and~\ref{thm:universality} all apply equally well to divisors on weighted
graphs~\cite{amini-caporaso}
because the construction of a degeneration in Theorem~\ref{thm:universality}
uses curves of genus~$0$ in the special fiber and for such components, the
theory of weighted graphs agrees with unweighted graphs.
Moreover, these theorems also apply to the metrized complexes introduced
in~\cite{amini-baker}, which record the isomorphism types of the curves in the
special fiber. Again, for rational components, the rank of the metrized complex
will be the same as the rank for the underlying graph. For metrized complexes,
there is a more refined notion of a limit~$g^r_d$, which involves additionally
specifying vector spaces of rational functions at each vertex. Not every divisor
of degree~$d$ and rank~$r$ on a metrized complex lifts to a limit~$g^r_d$, but
the examples from the above theorems do:
\begin{prop}\label{prop:limit-grd}
Let $\Gamma$ and~$D$ be a graph and divisor constructed as in
Theorem~\ref{thm:universality}. Then for any lift of $\Gamma$ to a metrized
complex with rational components, there also exists a lift of~$D$ to a
limit~$g^2_d$.
\end{prop}
If we were to consider divisors of rank~$1$ rather than rank~$2$, \cite{abbr2}
provides a general theory for lifting. They prove that if a rank~$1$ divisor can
be lifted to a tame harmonic morphism with target a genus~$0$ metrized complex,
then it lifts to a rank~$1$ divisor an algebraic curve. Moreover, the converse
is true except for possibly some cases of wild ramification in positive
characteristic. Using this, they give examples of rank~$1$ divisors which do not
lift over any discrete valuation ring~\cite[Sec.~5]{abbr2}. While the existence
of a tame harmonic morphism depends on the characteristic, the dependence is
only when the characteristic is at most the degree of the
divisor~\cite[Rmk.~3.9]{abbr2}. In contrast, lifting rank~$2$ divisors can
depend on the characteristic even when the characteristic is bigger than the
degree:
\begin{thm}\label{thm:quantitative-characteristic}
If $P = \{p\}$ where $p \geq 443$ is prime, then the divisors~$D$ and~$D'$ in
Theorem~\ref{thm:characteristic} can be taken to have degree less than~$p$.
\end{thm}
For simplicity, we have stated Theorems~\ref{thm:characteristic},
\ref{thm:number-field}, and~\ref{thm:universality} in terms of liftings over
rings of formal power series, but some of our results also apply to other discrete
valuation rings. In particular, these theorems apply verbatim with $k[[t]]$
replaced by any DVR which contains its residue field~$k$. For other, possibly
even mixed characteristic DVRs, we have separate necessary and sufficient
conditions in Theorems~\ref{thm:matroid-lifting} and~\ref{thm:construction}
respectively.
The proof of Theorem~\ref{thm:universality} and its consequences use Mn\"ev's
universality theorem for matroids~\cite{mnev}. Matroids are combinatorial
abstractions of vector configurations in linear algebra. However, not all
matroids come from vector configurations and those that do are called
realizable. Mn\"ev proved that realizability problems for rank~$3$ matroids in
characteristic~$0$ can encode arbitrary systems of integral polynomial equations
and Lafforgue extended this to arbitrary
characteristic~\cite[Thm.~1.14]{lafforgue}. Thus, Theorem~\ref{thm:universality}
follows from universality for matroids together with a connection between
matroid realizability and lifting problems, which is done in
Theorems~\ref{thm:matroid-lifting} and~\ref{thm:construction}. We also give a
proof of universality in arbitrary characteristic with explicit bounds on the
size of the matroid in order to establish
Theorem~\ref{thm:quantitative-characteristic}.
Matroids have appeared before in tropical geometry and especially as
obstructions for lifting. For example, matroids yield examples of
matrices whose Kapranov rank exceeds their tropical rank, showing that the
minors do not form a tropical basis~\cite[Sec.~7]{develin-santos-sturmfels}. In
addition, Ardila and Klivans defined the tropical linear space
for any simple matroid, which generalizes the tropicalization of a linear
space~\cite{ardila-klivans}. The tropical linear spaces are
realizable as the tropicalization of an algebraic variety if and only if the
matroid is realizable~\cite[Cor.~1.5]{katz-payne}. This paper is only concerned
with rank~$3$ matroids, which correspond to $2$-dimensional fans and the
graphs for which we construct lifting obstructions are the links of the fine
subdivision of the tropical linear space (the fine subdivision is defined in
\cite[Sec.~3]{ardila-klivans}).
Moreover, the matroid divisors from this paper have found applications to other
questions regarding the divisor theory of graphs. David Jensen has shown that
the matroid divisor of the Fano matroid gives an example of a $2$-connected
graph which is not Brill-Noether general for any metric
parameters~\cite{jensen}. In addition to the Baker-Norine rank of a divisor used
in this paper, Caporaso has given a definition of the \defi{algebraic rank} of a
divisor, which involves quantifying over all curves over a given
field~\cite{caporaso}.
Yoav Len has shown that in contrast to the results in
Section~\ref{sec:divisors}, the algebraic rank of a matroid divisor detects
realizability of the matroid, and he has used this to show that the algebraic
rank can depend on the field~\cite{len}.
Since rank~$3$ matroids give obstructions to lifting rank~$2$ divisors on
graphs, it is natural to wonder if higher rank matroids give similar examples
for lifting higher rank divisors.
While we certainly expect there to be results similar to
Theorems~\ref{thm:characteristic}, \ref{thm:number-field},
and~\ref{thm:quantitative-characteristic} for divisors on graphs which have
ranks greater than~$2$, it is not clear that higher rank matroids would provide
such examples, or even what the right encoding of the matroid in a graph would
be. From a combinatorial perspective, our graphs are just order complexes of the
lattice of flats, but for higher rank matroids, the order complex is a
simplicial complex but not a graph.
This paper is organized as follows. In Section~\ref{sec:divisors}, we introduce
the matroid divisors which are our key class of examples and show that as combinatorial
objects they behave as if they should have rank~$2$. In Section~\ref{sec:lifting}, we relate
the lifting of matroid divisors to the realizability of the matroid.
Section~\ref{sec:brill-noether} looks at the applicability of our matroid to the
question of lifting tropically Brill-Noether general divisors and shows that,
with a few exceptions,
matroid divisors are not Brill-Noether general. Finally,
Section~\ref{sec:mnev} provides a quantitative proof of Mn\"ev universality as the
basis for Theorem~\ref{thm:quantitative-characteristic}.
\subsection*{Acknowledgments}
I would like to thank Spencer Backman, Melody Chan, Alex Fink, Eric Katz, Yoav
Len, Diane Maclagan, Sam Payne, Kristin Shaw, and Ravi Vakil for helpful
comments on this project. The
project was begun while the author was supported by NSF grant DMS-1103856L.
\section{Matroid divisors}\label{sec:divisors}
In this section, we construct the divisors and graphs that are used in
Theorem~\ref{thm:universality}. As in~\cite{baker} and~\cite{baker-norine-rr},
we will refer to a finite formal sum of the vertices of a graph as a
\defi{divisor} on that graph. Divisors are related by so-called ``chip-firing
moves'' in which the weight at one vertex is decreased by its degree and those
of its neighbors are correspondingly each increased by~$1$. A reverse
chip-firing move is the inverse operation.
As explained in the introduction, the starting point in our construction is a
rank~$3$ simple matroid. A matroid is a combinatorial model for an arrangement
of vectors, called elements, in a vector space. A rank~$3$ simple matroid
corresponds to such an arrangement in a $3$-dimensional vector space, for which
no two vectors are multiples of each other. There are many equivalent
descriptions of a matroid, but we will work with the flats, which correspond to
vector spaces spanned by subsets of the arrangement, and are identified with the
set of vectors that they contain. For a rank~$3$ simple matroid, there is only
one rank~$0$ and one rank~$3$ flat, and the rank~$1$ flats correspond to the
elements of the matroid, so our primary interest will be in rank~$2$ flats.
Throughout this paper, \defi{flat} will always refer to a rank~$2$ flat.
We refer the reader to~\cite{oxley} for a thorough reference on matroid theory,
or~\cite{katz} for an introduction aimed at algebraic geometers. However, in the
case of interest for this paper, we can give the following axiomatization:
\begin{defn}
A \emph{rank 3 simple matroid}~$M$ consists of a finite set~$E$ of
\emph{elements} and a collection~$F$ of subsets of~$E$, called the \emph{flats}
of~$M$, such that any pair of elements is contained in exactly one flat, and
such that there are at least two flats.
A \defi{basis} of such a matroid is a triple of elements which
are not all contained in a single flat.
\end{defn}
By projectivizing the vector configurations above, a configuration of distinct
$k$-points in the projective plane~$\PP^2_k$ determines a matroid. The
elements of this matroid are the points of the configuration and the flats
correspond to lines in~$\PP^2_k$, identified with the points contained in them.
A matroid coming from a point configuration in this way is called
\defi{realizable over~$k$} and in Section~\ref{sec:lifting}, we will use the
fact that matroid realizability can depend on the field.
Given a rank~$3$ simple matroid~$M$ with elements~$E$ and flats~$F$, we let
$\Gamma_M$ be the bipartite graph with vertex set $E \disjoint F$, and an edge
between $e \in E$ and $f \in F$ when $e$ is contained in~$f$. The graph
$\Gamma_M$ is sometimes called the Levi graph of~$M$. We let $D_M$ be the
divisor on the graph~$\Gamma_M$ consisting of the sum of all vertices
corresponding to elements of the ground set~$E$.
\begin{prop}\label{prop:rank}
The divisor~$D_M$ has rank~$2$.
\end{prop}
\begin{proof}
To prove the proposition, we first need to show that for any degree~$2$
effective divisor~$E$, the difference $D_M - E$ is linearly equivalent to an
effective divisor. We build up a ``toolkit'' of divisors linearly equivalent
to~$D_M$. First, for any flat~$f$, we can reverse fire $f$. This moves a chip
from each element contained in~$f$ to~$f$ itself. Thus, the result is an
effective divisor whose multiplicity at $f$ is the cardinality of~$f$, which is
at least~$2$. Our second chip-firing move is to reverse fire a vertex $e$ as
well as all flats containing $e$. The net effect will be no change at $e$ but
each neighbor~$f$ of~$e$ will end with $\lvert f\rvert - 1 \geq 1$ chips. Third, we
will use the second chip-firing move, after which all the flats which
contain~$e$ have at least one chip, after which it is possible to reverse
fire~$e$ again.
Now let $E$ be any effective degree~2 divisor on~$\Gamma_M$. Thus, $E$ is the
sum of two vertices of~$\Gamma_M$. We consider the various combinations which
are possible for these vertices. First, if $E = [e] + [e']$ for distinct
elements $e$ and~$e'$, then $\Gamma_M - E$ is effective. Second, if $E =
[e] + [f]$, then we have two subcases. If $e$ is in $f$, then we reverse
fire $e$ and all flats containing it. If $e$ is not in~$f$, then we can
reverse fire just $f$. Third, if $E = [f] + [f']$ for distinct flats $f$
and~$f'$, then there are again two subcases. If $f$ and $f'$ have no
elements in common, then we can reverse fire $f$ and~$f'$. If $f$ and~$f'$
have a common element, say $e$, then we reverse fire $e$ together with the
flats which contain it. Fourth, if $E = 2[e]$, then we use the third
chip-firing move, which will move one chip onto~$e$ for each flat
containing~$e$, of which there are at least~$2$.
Fifth, if $E = 2[f]$, then we reverse fire $f$.
Finally, to show that the rank is at most~$2$, we give an effective degree~3
divisor~$E$ such that $D_M - E$ is not linearly equivalent to any effective
divisor. For this, let $e_1$, $e_2$, and~$e_3$ form a basis for~$M$ and let
$f_{ij}$ be the unique flat containing $e_i$ and~$e_j$ for $1 \leq i < j \leq
3$. We set $E = [f_{12}] + [f_{13}] + [f_{23}]$ and claim that $D_M - E$ is not
linearly equivalent to any effective divisor. We reverse fire $e_1$ together
with all flats containing it to get the following divisor linearly
equivalent to $D_M - E$:
\begin{equation}\label{eq:reduced-divisor}
[e_1] + \big(\lvert f_{12}\rvert - 2\big)[f_{12}] +
\big(\lvert f_{13} \rvert - 2\big)[f_{13}] -
[f_{23}]
+ \sum_{\genfrac{}{}{0pt}{1}{f_k \ni e_1}{f_k \neq f_{12},f_{13}}}
\big(\lvert f_{k} \rvert - 1\big) [f_k],
\end{equation}
which is effective except at $f_{23}$.
We wish to show the divisor in~(1) is not linearly equivalent to any effective
divisor, which we will do by showing that it is $f_{23}$-reduced using Dhar's
burning algorithm~\cite{dhar}. We first claim that for any element $e$ other
than~$e_1$, there is a path from $f_{23}$ to $e$ which does not encounter any
chips. If $e$ is in $f_{23}$, then there is a direct edge between these
vertices. If not, then we first let $f$ denote the unique flat containing both
$e$ and $e_1$. Since $e_1$, $e_2$, and $e_3$ form a basis, they cannot all be
contained in~$f$. Without loss of generality, we can assume that $e_2$ is not
in~$f$, and so $e$, $e_1$, and $e_2$ form a basis. Thus, if $f'$ is the unique
flat containing $e_2$ and $e$, then $f'$ does not contain $e_1$. Therefore, the
path from $f_{23}$ to~$e_2$ to $f'$ to $e$ does not cross any chips.
\begin{figure}
\caption{Paths through the graph $\Gamma_M$ taken by the burning algorithm
applied to show the divisor~(\ref{eq:reduced-divisor}
\label{fig:burning}
\end{figure}
By the claim in the previous paragraph, the burning algorithm will lead to
$\lvert f_{12} \rvert -1$ independent ``fires'' arrviving at $f_{12}$, one for
each element in $f_{12} \setminus e_1$. Thus, these remove the $\lvert f_{12}
\rvert -2$ chips on $f_{12}$ and one path continues on to remove the single chip
from $e_1$. Likewise, $\lvert f_{13} \rvert - 1$ independent ``fires'' arive at
$\lvert f_{13} \rvert - 2$, and one continues on and passes through $e_1$ to
arrive at all the flats containing $e_1$. Therefore, for every flat~$f$ that
contains $e_1$, other than $f_{12}$ and $f_{13}$, which have already been
handled, there is a ``fire'' arriving from every element of~$f$, which exceeds
the $\lvert f \rvert - 1$ chips on this vertex. The paths used to cover
$\Gamma_M$ are summarized schematically in
Figure~\ref{fig:burning}. Since the burning algorithm
covers the graph $\Gamma_M$, we conclude that the divisor $D_M - E$ is
$f_{23}$-reduced and so not linearly equivalent to an effective divisor.
\end{proof}
Proposition~\ref{prop:rank} also shows that if $\Gamma_M$ is made into a
weighted graph by giving all vertices genus~$0$, then $D_M$ has rank~$2$ on the
weighted graph. The rank is, again, unchanged for any lifting of the weighted
graph to a metrized complex. To show that $D_M$ is also a limit $g^2_d$ as in
Proposition~\ref{prop:limit-grd},
we also need to choose $3$-dimensional vector spaces of rational functions on
the variety attached to each vertex.
\begin{proof}[Proof of Proposition~\ref{prop:limit-grd}]
We recall from~\cite{amini-baker} that a lift of~$\Gamma_M$ to a metrized
complex means associating a $\PP^1_k$ for each vertex~$v$ of the graph, which we
denote $C_v$, and a point on~$C_v$ for each edge incident to~$v$. A lift of the
divisor~$D_M$ is a choice of a point on $C_e$ for each element~$e$ of~$M$.
The data of a limit $g^2_d$ is a $3$-dimensional vector space~$H_v$ of rational
functions on each~$C_v$~\cite[Sec.~5]{amini-baker}, which we choose as follows. For each flat~$f$, we
arbitrarily choose two elements from it and let $p_{f,1}$
and~$p_{f,2}$ be the points on~$C_f$ corresponding to the edges from $f$ to each of the
chosen elements. Our vector
space~$H_{f}$
consists of the rational functions which have at worst simple poles at
$p_{f,1}$ and~$p_{f,2}$. For each element~$e$, we choose an arbitrary flat
containing~$e$ and let $q_{e}$ be the point on~$C_e$ corresponding to the edge
to~$e$. Our
vector space~$H_e$ consists of the rational functions which have at worst
poles at~$q_e$ and at the point of the lift of~$D_M$.
Now to check that these vector spaces form a limit~$g^2_d$, we need to show that
the refined rank is~$2$. For this, we use the same ``toolkit'' functions as in
the proof of Proposition~\ref{prop:rank}, but we augment them with rational
functions from the prescribed vector spaces on the algebraic curves. The first
item from our toolkit was reverse firing a flat~$f$ to produce at least two
points on~$C_{f}$. We can use rational functions with poles at $p_{f,1}$ and
$p_{f,2}$ to produce any degree two effective divisor on $C_{f}$.
For each $C_{e}$ such that $e$ is an element of~$f$, we need to use a rational
function with a zero at the edge to~$f$ and a pole at the lift of the
divisor~$D_M$.
The second item we needed in our toolkit was reverse firing an element~$e$
together with all of the flats which contain it. Here, for each element $e'$
other than~$e$, we use the rational function with a pole at the divisor and a
zero at the point corresponding to the edge to the unique flat containing
both~$e'$ and~$e$. At each flat~$f$ containing $e$, we can use any function with
a pole at~$p_{f,i}$, where $i \in \{1, 2\}$ can be chosen to not be the edge
leading to~$e$. This produces a divisor at an arbitrary point of~$C_{e}$.
The third and final operation we used was the previous item followed by a
reverse firing of~$e$. Here, we use the same rational functions as before, but
we can choose any rational function on~$C_e$ which has poles at the point of the
divisor and~$q_e$, thus giving us two arbitrary points on~$C_e$. We conclude
that rational
functions can be found from the prescribed vector spaces to induce a linear
equivalence between the lift of~$D_M$ and any two points on the metrized
complex.
\end{proof}
In the case of rank~$1$ divisors, lifts can be constructed using the theory of
harmonic maps of metrized complexes, which gives a complete theory for divisors
defining tamely
ramified maps to~$\PP^1$~\cite{abbr2}. A sufficient condition for lifting a
rank~$1$ divisor is for it to be the underlying graph of a metrized complex
which has a tame harmonic morphism to a tree (see \cite[Sec.~2]{abbr1}
for precise definitions). These definitions are limited to the rank~$1$ case,
but for rank~$2$ divisors we can subtract points to obtain a divisor of rank at
least~$1$. In particular, if $D_M$ lifts, then for any element~$e$, $D_M - [e]$ will be the
specialization of a rank~$1$ effective divisor. However,
the lifting criterion of~\cite{abbr2} is satisfied for these subtractions,
independent of the liftability of~$D_M$.
\begin{prop}\label{prop:tame-harmonic}
Let $M$ be any rank~$3$ simple matroid and $e$ any element of~$M$. Also, let $k$
be an algebraically closed field of characteristic not~$2$. Then, $\Gamma_M$
has a tropical modification $\widetilde\Gamma_M$ such that $\widetilde\Gamma_M$
can be lifted to a totally degenerate metrized complex over~$k$ with a tame
harmonic morphism to a genus~$0$ metrized complex, such that one fiber is a lift
of the divisor~$D_M - [e]$.
\end{prop}
\begin{proof}
We first construct a modification~$\widetilde \Gamma_M$ of~$\Gamma_M$ which has
a finite harmonic morphism from $\widetilde \Gamma_M$ to a tree~$T$. The
tree~$T$ will be a star tree with a central vertex~$w$, together with an
unbounded edge, denoted $r_f$, for each flat~$f$ which does not contain~$e$, and
a single unbounded edge~$r_e$ corresponding to~$e$. Our modification
of~$\Gamma_M$ consists of adding the following unbounded edges: At~$e$, we add
one unbounded edge~$s_{e,f}$ for each flat~$f$ containing~$e$. At each
element~$e'$ other than $e$, we add one unbounded edge~$s_{e',f}$ for each
flat~$f$ which contains neither $e$ nor~$e'$. At a flat~$f$, we add unbounded
edges~$s_{f,i}$ where $i$ ranges from $1$ to~$\lvert f\rvert$ if $e \notin f$
and from $1$ to $\lvert f\rvert - 2$ if $e \in f$.
\begin{figure}
\caption{Modification of $\Gamma_M$ which has a finite, effective harmonic morphism to the
tree to a tree, such that the
fiber over the central vertex of the tree is $D_M - [e]$. In this figure, $f$ is
a flat which does not contain $e$ and $e'$ is an element of~$f$.}
\label{fig:harmonic}
\end{figure}
We now construct a finite harmonic morphism~$\phi$ from $\widetilde \Gamma_M$
to~$T$. Each element other than $e$ maps to the central vertex~$w$ of~$T$.
Each flat~$f$ not containing~$e$ maps to a point one unit of
distance along the corresponding ray~$r_f$ of~$T$. Then the rays~$s_{e', f}$
and~$s_{f,i}$ also map to the ray $r_f$, starting at~$w$ and $\phi(f)$
respectively.
We map the vertex~$e$ to its unbounded ray~$r_e$, at a distance of~$2$ from~$w$,
which leaves all of the flats containing~$e$ along the same ray at a distance
of~$1$. The rays $s_{e,f}$ and $s_{f,i}$, for flats~$f$ containing~$e$ also
map to~$r_e$, starting distances of~$2$ and~$1$ from~$w$ respectively. The
map~$\phi$ is depicted in Figure~\ref{fig:harmonic}.
To check that $\phi$ is harmonic, we need to verify that locally, around each
vertex~$v$ of $\widetilde \Gamma_M$, the same number of edges map to each of the
edges incident to~$\phi(v)$, and this number is the degree of $\phi$ at
$v$~\cite[Sec.~2]{baker-norine}. First, suppose that the vertex~$v$ corresponds
to an element~$e'$ other than~$e$ and we have defined $\phi(e')$ to be the
central vertex~$w$. In this case, for each ray $r_f$ of~$T$, there is exactly
one edge incident to $e'$ mapping to $r_f$, either the edge between $e'$ and $f$
if $f$ contains $e$, or the unbounded edge $s_{e',f}$ if not. There is also
exactly one edge mapping to $r_e$, which is the edge between $e'$ and the unique
flat containing both $e$ and~$e'$. Therefore, $\phi$ is harmonic at the
vertex~$e'$ with local degree equal to~$1$.
Second, at the vertex~$e$, which maps along the edge $r_e$, there is one edge
mapping to the bounded side of $r_e$ for each flat~$f$ containing $e$ and also
for each such flat, one infinite ray $s_{e,f}$ mapping to the unbounded side
of~$r_e$. Thus $\phi$ is also harmonic at $e$, and has local degree equal to the
number of flats containing~$e$.
Finally, we check that $\phi$ is harmonic at a vertex corresponding to a
flat~$f$. If $f$ does not contain~$e$, then there are $\lvert f \rvert$ rays
mapping to the unbounded side of~$r_f$ and the same number of edges mapping to
the bounded side, connecting $f$ to the elements it contains. Thus, at such a
vertex, $\phi$ is harmonic and its local degree is $\lvert f \rvert$. If $f$
does contain~$e$, then there are $\lvert f \rvert - 2$ rays mapping to the
unbounded side of~$r_e$ together with the edge connecting $f$ to~$e$. On the
bounded side of $r_e$, there are also $\lvert f \rvert - 1$ edges, connecting
$f$ to the elements $f \setminus \{e\}$, and so here $\phi$ is harmonic with
local degree $\lvert f \rvert - 1$.
To lift $\phi$ to a harmonic morphism of totally degenerate metrized complexes,
we need to choose a map $\phi_v \colon \PP^1 \rightarrow \PP^1$ for each
vertex~$v$ of $\widetilde \Gamma_M$ and an identification of the outgoing
directions with points on $\PP^1$. Having assumed characteristic not~$2$, we can
choose a tame homomorphism of degree equal to the degree of~$\phi$ at~$v$
as~$\phi_v$. We identify the edges incident to~$v$ with points of $\PP^1$ at
which $\phi_v$ is unramified, since these edges all have expansion factor equal
to~$1$. Then, the preimage of a $k$-point of the curve at $w$ consists of one
point in each $\PP^1$ corresponding to the elements $e'$ not equal to $e$,
because the local degrees at these vertices are equal to~$1$. Thus, this
preimage is a lift of $D_M \setminus [e]$ and we have our desired morphism of
metrized complexes.
\end{proof}
\section{Lifting matroid divisors}\label{sec:lifting}
In this section, we characterize the existence of lifts of matroid divisors in
terms of realizability of the corresponding matroids. Recall from~\cite{baker},
that if $R$ is a discrete valuation ring with algebraically closed residue
field, then a regular semistable family~$\X$ of curves over~$R$ gives
homomorphism~$\rho$ from the group of divisors on the general fiber to the group
of divisors on the dual graph~$\Gamma$ of the special fiber. This dual graph is
defined to have a vertex~$v$ for each irreducible component of the special fiber
and an edge for each point of intersection between two components. Then, for any
divisor $\widetilde D$ on the general fiber of~$\X$, $\rho(\widetilde D)$ is
defined to be the formal sum of the vertices of~$\Gamma$ with the coefficient of
a vertex~$v$ equal to the degree of the intersection of~$\overline D$ with
$C_v$, where $\overline D$ is the closure of $\widetilde D$ in $\X$ and $C_v$ is
the irreducible component corresponding to~$v$~\cite[Sec.~2A]{baker}. With this
definition, we have an inequality $r(\widetilde D) \leq r(\rho(\widetilde D))$
between the algebraic and graph-theoretic ranks~\cite[Lem.~2.8]{baker}.
We now consider the semistable family~$\X$ over a discrete valuation ring~$R$,
where we drop the assumption that the residue field of~$R$ is algebraically
closed. In this case, we apply the definitions from the previous paragraph by
first base changing to a discretely valued extension $R' \supset R$, such that
the residue field of~$R'$ is algebraically closed and such that a uniformizer
of~$R$ is also a uniformizer of~$R'$. In particular, the dual graph has one
vertex for each geometric irreducible component of the special fiber and it is
independent of the choice of~$R'$. Also, the definition of~$\rho(\widetilde D)$
is independent of~$R'$ because it can be computed by taking the closure of
$\widetilde D$ in $\X$ and recording the degree of the pullback of this Cartier
divisor to each of the geometric irreducible components of the special fiber.
Moreover, for any family~$\X$, there is a finite \'etale extension $R'$ of~$R$
such that, after base changing to~$R'$, the irreducible components of the
special fiber are geometrically irreducible. Therefore, after this base change,
the computation of the dual graph and the specialization map can be carried out
directly on the resulting family over~$R'$. Since the dimension of a linear
system is invariant under base change, we also have a specialization inequality
$r(\widetilde D) \leq r(\rho(\widetilde D))$ on~$\X$.
We will say that a \defi{lifting} over~$R$ of an effective divisor~$D$ of
rank~$r$ on a graph~$\Gamma$ is a regular semistable family~$\X$ over~$R$ with
dual graph~$\Gamma$,
together with an effective divisor~$\widetilde D$ on the general fiber of~$\X$
such that $\rho(\widetilde D) = D$ and $\widetilde D$ has rank~$r$. Here, and
throughout this section, a regular semistable family~$\X$ includes the
hypothesis that $\X$ is semistable after passing to an extension with
algebraically closed residue field. The relationship between liftings of a
matroid divisor~$D_M$ and its matroid depends on the following, slightly weaker
variant of realizability for~$M$:
\begin{defn}
Let $k$ be a field. We say that a matroid~$M$ has a \defi{Galois-invariant
realization over an extension of~$k$} if there exists a finite scheme
in $\PP^2_k$ which becomes a union of distinct points over $\overline
k$, and these points realize~$M$.
\end{defn}
Equivalently, a Galois-invariant realization is a realization over a finite
Galois extension~$k'$ of~$k$ such that the Galois group $\Gal(k'/k)$ permutes
the points of the realization. Thus, the distinction between a realization and a
Galois-invariant realization is only relevant for matroids which have
non-trivial symmetries. Moreover, Lemma~\ref{lem:break-symmetry} will show that
any matroid can be extended to one where these symmetries can be broken, without
affecting realizability over infinite fields.
\begin{ex}
Let $M$ be the matroid determined by all $21$ points of $\PP^2_{\mathbb F_4}$.
Then $M$ is not realizable over $\PP^2_{\mathbb F_2}$ because it contains more
than $7$ elements, and there are only $7$ points in $\PP_{\mathbb F_2}^2$.
However, $M$ is clearly realizable over~$\mathbb F_4$ and the Galois group
$\Gal(\FF_4/\FF_2) \isom \ZZ/2$ acts on these points by swapping pairs. Thus,
$M$ has a Galois-invariant realization over an extension of~$\mathbb F_2$.
\end{ex}
\begin{ex}
Let $M$ be the Hesse matroid of 9~elements and 12 flats. Then $M$ is not
realizable over~$\mathbb R$ by the Sylvester-Gallai theorem. However, the
flex points of any elliptic curve are a realization of~$M$ over~$\mathbb C$.
If the
elliptic curve is defined over~$\mathbb R$, then the set of all flexes
points is also defined over~$\mathbb R$, so $M$ has a Galois-invariant
realization over an extension of~$\mathbb R$.
\end{ex}
\begin{thm}\label{thm:matroid-lifting}
Let $\Gamma_M$ and $D_M$ be the graph and divisor obtained from a rank~$3$
simple matroid~$M$ as in Section~\ref{sec:divisors}. Also, let $R$ be
any discrete valuation ring with residue field~$k$. If $D_M$ lifts
over~$R$, then the matroid~$M$ has a Galois-invariant
realization over an extension of~$k$.
\end{thm}
By projective duality, a point in~$\PP^2$ is equivalent to a line in the
dual projective space~$\PP^2$. Thus, the collection of points realizing a
matroid is equivalent to a collection of lines, in which the flats correspond to
the points of common intersection. It is this dual representation that we will
construct from the lifting.
\begin{proof}[Proof of Theorem~\ref{thm:matroid-lifting}.]
Let $\X$ be the semistable family over~$R$ and $\widetilde D$ a rank~2 divisor
on the general fiber of~$\X$ with $\rho(\widetilde D) = D_M$. First, we make the
simplifying assumption that the components of the special fiber are
geometrically irreducible, so that we can compute specializations in~$\X$,
without needing to take further field extensions. Let $\overline D$ denote the
closure in~$\X$ of~$\widetilde D$. By assumption, $H^0(\X, \cO(\overline D))$ is
isomorphic to the free $R$-module~$R^3$. By restricting a basis of these
sections to the special fiber~$\X_0$, we have a rank~$2$ linear series on the
reducible curve~$\X_0$.
If $\overline D$ intersected a node of $\X_0$, then it would intersect both
components of~$\X_0$ containing that node, so $\rho(D)$ would have positive
multiplicity on two adjacent vertices. However, $\Gamma_M$ is bipartite and the
divisor $D_M$ is supported on one of these parts, corresponding to the elements
of the matroid, so $\overline D$ cannot intersect any of the nodes
of~$\X_0$. Thus, the base locus of our linear series consists of a finite number
of smooth points of~$\X_0$. Since the base locus consists of smooth points, we
can subtract the base points to get a regular, non-degenerate morphism
$\phi\colon \X_0 \rightarrow \PP_k^2$.
By the assumption that $\widetilde D$ specializes to $D_M$, we have an upper
bound on the degree of $\phi$ restricted to each component of~$\X_0$.
For a flat~$f$ of~$M$, the corresponding component~$C_f$ has degree~$0$
under~$\phi$, so $\phi(C_f)$ consists of a single point.
For an element~$e$, the corresponding component $C_e$ has either
degree~1 or~0 depending on whether the intersection of~$\overline
D$ with~$C_e$ is contained in the base locus. If the intersection is in the base locus, then $C_e$ again maps to
a point, and if not, $C_e$ maps isomorphically to a line in~$\PP_k^2$.
Thus, the image $\phi(\X_0)$ is a union of lines in~$\PP_k^2$, which we will
show to be a dual realization of the matroid~$M$.
Let $f$ be a flat of~$M$. Since the
component of $\X_0$ corresponding to~$f$ maps to a point, the images of the
components corresponding to the elements in~$f$ all have a common point of
intersection.
Now let $e_1$ be an element of~$M$ and suppose that the component $C_{e_1}$ maps
to a point $\phi(C_{e_1})$. Since every other element $e'$ is in a flat with
$e_1$, that means that $\phi(C_{e'})$, the image of the corresponding component
must contain the point~$\phi(C_{e_1})$. Since $\phi$ is non-degenerate, there
must be at least one component $C_{e_2}$ which maps to a line. Let $e_3$ be an
element of~$M$ which completes $\{e_1, e_2\}$ to a basis. Thus, the flat
containing $e_2$ and $e_1$ is distinct from the flat containing $e_2$ and~$e_3$.
Since $\phi$ maps $C_{e_2}$ isomorphically onto its image, this means that
$\phi(C_{e_3})$ must meet $\phi(C_{e_2})$ at a point distinct from the point
$\phi(C_{e_1})$. Thus, $\phi(C_{e_3})$ must be equal to $\phi(C_{e_2})$. Any
other element~$e''$ in~$M$ forms a basis with $e_1$ and either $e_2$ or~$e_3$
(or both). In either case, the same argument again shows that $C_{e''}$ must map
to the same line as $C_{e_2}$ and~$C_{e_3}$. Thus, this line would be the entire
image of $\phi$, which again contradicts the assume non-degeneracy.
Therefore, we conclude that $\phi$ maps each
component $C_e$ corresponding to an element~$e$ isomorphically onto a line
in~$\PP_k^2$. We've already shown that for any set of elements in a flat, the
corresponding lines intersect at the same point. Moreover, because each
component~$C_e$ maps isomorphically onto its image, distinct flats must
correspond to distinct points in~$\PP_k^2$. Thus, $\phi(\X_0)$ is a dual
realization of the matroid~$M$.
If the components of the special fiber are not geometrically irreducible, then
we can find a finite \'etale extension~$R'$ of~$R$ over which they are.
In our construction of a realization of~$M$ over the residue field of~$R'$, we
can assume that we have chosen a basis of $H^0(\X \times_R R', \cO)$ that is
defined over~$R$. Then, the matroid realization will be the base extension of a
map of $k$-schemes $\X_0 \rightarrow \PP^2_k$. We let $k'$ be the Galois closure
of the residue field of $R'$. Then $\Gal(k'/k)$ acts on the realization of~$M$
over $k'$, but the total collection of lines is defined over~$k$, and thus
invariant. Thus, $M$ has a Galois-invariant realization over an extension of~$k$
as desired.
\end{proof}
For the converse of Theorem~\ref{thm:matroid-lifting}, we need to consider
realizations of matroids over discrete valuations ring~$R$, by which we mean
$R$-points in $\PP^2$ whose images in both the residue field and the fraction
field realize~$M$. For example, if $R$ contains a field over which $M$ is
realizable, then $M$ is realizable over~$R$. We say that $M$ has a
\defi{Galois-invariant realization over an extension of~$R$} if there exists a
finite, flat scheme in $\PP^2_R$ whose special and general fiber are
Galois-invariant realizations of~$M$ over extensions of the residue field and
fraction field of~$R$, respectively.
In the following theorem,
a \defi{complete flag} refers to the pair of an
element~$e$ and a flat~$f$ such that $e$ is contained in~$f$.
\begin{thm}\label{thm:construction}
Let $R$ be a discrete valuation ring with residue field~$k$. Let $M$ be a
simple rank~3 matroid
with a Galois-invariant realization over an extension of~$R$.
Assume that $\lvert k \rvert > m - 2n + 1$, where $n$ is the number of elements
of~$M$ and $m$ is the number of complete flags.
Then $\Gamma_M$ and $D_M$ lift over~$R$.
\end{thm}
Note that Theorem~\ref{thm:construction} does not make any completeness or other
assumptions on the DVR beyond the cardinality of the residue field. In contrast,
ignoring $D_M$ and its rank, a semistable model $\mathcal X$ is only known to
exist for an arbitrary graph when the valuation ring is
complete~\cite[Thm.~B.2]{baker}.
We construct the semistable family in Theorem~\ref{thm:construction} using a
blow-up of projective space. We begin with a computation of the Euler
characteristic for this blow-up.
\begin{lem}\label{lem:euler}
Let $S$ be the blow-up of $\PP^2_K$ at the points of intersection of an
arrangement of $n$ lines. If $A$ is the union of the strict transforms of the
lines and the exceptional divisors, then the dimension of $H^0(S, \cO(A))$ is at
least $2n+1$.
\end{lem}
\begin{proof}
We first use Riemann-Roch to compute that $\chi(\cO(A))$ is $2n+1$. Let $m$ be the
number of complete flags of~$M$, the matroid of the line arrangement~$A$.
We let $H$ denote
the pullback of the class of a line on $\PP^2$ and $C_f$ to denote the
exceptional lines. Then, we have the following linear equivalences
\begin{align*}
A &\sim nH - \sum_{f} (\lvert f \rvert - 1) C_f \\
K_S &\sim -3H + \sum_f C_f
\end{align*}
Now, Riemann-Roch for surfaces tells us that
\begin{align}
\chi(\cO(A)) = \frac{A^2 - A \cdot K_S}{2} + 1
&= \frac{n^2 - \sum_f (\lvert f \rvert - 1)^2 + 3n
- \sum_f (\lvert f \rvert -1)}{2} + 1 \notag \\
&= \frac{n^2 + 3n - \sum_{f} \lvert f \rvert (\lvert f \rvert - 1)}{2} + 1.
\label{eq:riemann-roch}
\end{align}
We can think of the summation $\sum_f \lvert f \rvert(\lvert f \rvert - 1)$ as
an enumeration of all triples of a flat and two distinct elements of the flat.
Since two distinct elements uniquely determine a flat, we have the identity that
$\sum_f \lvert f\rvert^2 = n(n-1)$, so (\ref{eq:riemann-roch}) simplifies to
$\chi(\cO(A)) = 2n + 1$.
It now suffices to prove that $H^2(S, \cO(A))$ is zero, which is equivalent, by
Serre duality, to showing that
$K_S-A$ is not linearly equivalent to
an effective divisor. The push-forward of $K_S - A$ to $\PP^2$ is $-(n+3)H$,
which is not linearly equivalent to an effective divisor, and thus
$H^2(S, \cO(A))$ must be zero. Therefore,
\begin{equation*}
\chi(\cO(A)) = H^0(S, \mathcal O(A)) - H^1(S, \mathcal
O(A)) \leq H^0(S, \mathcal O(A)),
\end{equation*}
which together with the computation above yields the desired inequality.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:construction}]
We first assume that $M$ is realizable over~$R$, and then at the end, we will
handle Galois-invariant realizations over extensions.
Thus, we can fix a dual realization of~$M$ as a set of lines in~$\PP_R^2$, and
let $S$ be
the blow-up of $\PP_R^2$ at all the points of intersections of the lines.
We let the divisor~$A \subset S$ be the sum of the strict transforms of the
lines and the exceptional divisors. Note that $A$ is a simple normal
crossing divisor whose dual complex is~$\Gamma_M$. As in the proof of
Theorem~\ref{thm:matroid-lifting}, we denote the components of~$A$ as~$C_f$
and~$C_e$ corresponding to a flat~$f$ and an element~$e$ of~$M$ respectively.
We claim that $A$ is a base-point-free divisor on~$S$. Any two lines of
the matroid configuration are linearly equivalent in $\PP_R^2$. The preimage of
a linear equivalence between lines corresponding to elements~$e$ and~$e'$
is the divisor:
\begin{equation*}
[C_{e'}] - [C_e] + \sum_{f \colon e' \in f, e \not\in f} [C_f]
- \sum_{f \colon e \in f, e' \not\in f} [C_f].
\end{equation*}
Thus, we have a linear equivalence between $A$ and a divisor which does not
contain~$C_e$, nor $C_f$ for any of the flats containing $e$ but not $e'$. By
varying $e$ and $e'$, we get linearly equivalent divisors whose common
intersection is empty.
We now look for a function~$g \in H^0(S, \cO(A)) \otimes_R k$ which does not
vanish at the nodes of~$A$. For each of the $m$~nodes, the condition of
vanishing at that node amounts to one linear condition on $H^0(S, \cO(A))
\otimes_R k$. Since $A$ is base-point-free, this is a non-trivial linear
condition, defining a hyperplane. Moreover, because of the degrees of the
intersection of~$A$ with its components, the only functions vanishing on all of
the nodes are multiples of the defining equation of~$A$. If the residue field is
sufficiently large, then we can find an element $g \in H^0(S, \cO(A)) \otimes_R
k$ avoiding these hyperplanes, and $\lvert k \rvert > m - 2n + 1$ is sufficient
by Lemmas~\ref{lem:euler} and~\ref{lem:avoidance}. Now we lift $g$ to
$\widetilde g \in H^0(S, \cO(A))$, and set $\X$ to be the scheme defined by $h +
\pi \widetilde g$, where $h$ is the defining equation of~$A$ and $\pi$ is a
uniformizer of~$R$. It is clear that $\X$ is a flat family of curves over $R$
whose special fiber is $A$ and thus has dual graph $\Gamma_M$. It remains to
check that $\X$ is regular and for this it is sufficient to check the nodes of
$A_k$. In the local ring of a node, $h$ is in the square of the maximal ideal,
but by construction $\pi \widetilde g$ is not, and thus, at this point $\X$ is
regular.
Finally, we can take $D$ to be the preimage of any line in $\PP_R^2$ which
misses the points of intersection. Again, by Lemma~\ref{lem:avoidance} below, it
is sufficient that $\lvert k\rvert > \ell - 2$, where $\ell$ is the number
of flats. We claim that $m - 2n + 1 \geq \ell - 2$, and we have assumed that
$\lvert k \rvert > m-2n+1$.
This claimed inequality can be proved using induction similar to the proof of
Theorem~\ref{thm:classification}, but it also follows from Riemann-Roch for
graphs~\cite[Thm.~1.12]{baker-norine-rr}. Since $\Gamma_M$ has genus $m - \ell -
n + 1$, then the Riemann-Roch inequality tells us that
\begin{equation*}
2 = r(D_M) \geq n - (m - \ell - n + 1) = \ell - m + 2n - 1,
\end{equation*}
which is equivalent to the claimed inequality.
Now, we assume that $M$ may only have a Galois-invariant realization over an
extension of~$R$. We can construct the blow-up~$S$ in the same way, since the
singular locus of the line configuration is defined over~$R$. Again, the
divisor~$A$ is base-point-free, because we have already checked that it is base
point free after passing to an extension where the lines are defined.
Finally, we need to choose the function~$g$ and the line which pulls back
to~$D$ by avoiding certain linear conditions defined over an extension of~$k$.
However, when restricted to~$k$, these remain linear conditions, possibly of
higher codimension, so we can again avoid them under our hypothesis on~$\lvert
k\rvert$.
\end{proof}
\begin{lem}\label{lem:avoidance}
Let $H_1, \ldots, H_m$ be hyperplanes in the vector space~$k^N$, where $k$ is a
field. Let $c$ denote the codimension of the intersection $H_1 \cap \cdots \cap
H_m$. If $k$ is infinite or if $k$ is the finite field with $q$ elements and $q
> m - c + 1$, then there exists a point in $k^N$ not contained in any
hyperplane.
\end{lem}
\begin{proof}
If $k$ is infinite, the statement is clear, so we assume that $k$ is finite with
$q$ elements.
We first quotient out by the intersection $H_1 \cap \cdots \cap H_m$, so we are
working in a vector space of dimension $c$ and we know that no non-zero vector
is contained in all hyperplanes. This means that the vectors defining the
hyperplanes span the dual vector space, so we can choose a subset as a basis.
Thus, we assume that the first $c$ hyperplanes are the coordinate hyperplanes.
The complement of these consists of all vectors with non-zero coordinates, of
which there are $(q-1)^c$. Each of the remaining $m-c$ hyperplanes contains at
most $(q-1)^{c-1}$ of these. Our assumption is that $q-1 > m-c$, so there must
be at least one point not contained in any of the hyperplanes.
\end{proof}
We illustrate Theorems~\ref{thm:matroid-lifting} and~\ref{thm:construction} and
highlight the difference between their conditions with the following two
examples.
\begin{ex}
Let $M$ be the Fano matroid, which whose realization in $\mathbb P^2_{\mathbb
F_2}$ consists of all $7$ $\mathbb F_2$-points.
Then $M$ is realizable
over a field if and only if the field has equicharacteristic~$2$.
Thus, by Theorem~\ref{thm:matroid-lifting}, a necessary condition for $\Gamma_M$
and $D_M$ to lift over a valuation ring~$R$ is that the residue field of~$R$ has
characteristic~$2$. On the other hand, $M$ has $7$ elements and $21$
complete flags, so Theorem~\ref{thm:construction} says that if $R$ has
equicharacteristic~$2$ and the residue field of~$R$ has more than $8$ elements,
then $\Gamma_M$ and~$D_M$ lift over~$R$. We do not know if there exists a lift
of $\Gamma_M$ and~$D_M$ over any valuation ring of mixed characteristic~$2$.
\end{ex}
\begin{ex}
One the other hand, let $M$ be the non-Fano matroid, which is realizable over~$k$ if and
only $k$ has characteristic not equal to~$2$. Moreover, $M$ is realizable over
any valuation ring~$R$ in which $2$ is invertible. Thus, $\Gamma_M$ and $D_M$
lift over a valuation ring $R$, only if the residue field of~$R$ has
characteristic different than~$2$ by Theorem~\ref{thm:matroid-lifting}. The
converse is true, so long as the residue field has more than~$11$ elements by
Theorem~\ref{thm:construction}.
\end{ex}
Since
Theorems~\ref{thm:matroid-lifting} and~\ref{thm:construction} refer to
Galois-invariant realizations, we will need the following lemma to relate such
realizations with ordinary matroid realizations.
\begin{lem}\label{lem:break-symmetry}
Let $M$ be a matroid of rank 3. Then there exists a matroid~$M'$ such that for any
infinite field~$k$, the following are equivalent:
\begin{enumerate}
\item $M$ has a realization over~$k$.
\item $M'$ has a realization over~$k$.
\item $M'$ has a Galois-invariant realization over an extension of~$k$.
\end{enumerate}
\end{lem}
\begin{proof}
We use the following construction of an extension of a matroid. Suppose that $M$
is a rank~$3$ matroid and $f$ is a flat of~$M$. We construct a matroid~$M''$
which contains the elements of~$M$, together with an additional element~$x$. The
flats of $M''$ are those of~$M$, except that $f$ is replaced by $f \cup \{x\}$,
and two-element flats for $x$ and every element not in~$f$. By
repeating this construction, we can construct a matroid~$M'$ such
every flat which comes from one of the flats of~$M$ has a different number of
elements.
Now we prove that the conditions in the lemma statement are equivalent for this
choice of~$M'$. First, assume that $M$ has a realization over an infinite
field~$k$. We can inductively extend this to a realization of~$M'$. At each
step, when adding an element~$x$ as above, it is sufficient to place $x$ at a
point along the line corresponding to~$f$ such that it does not coincide with
any
of the other points, and it is not contained in any of the lines spanned by two
points not in~$f$. We can choose such a point for~$x$ since $k$ is infinite.
Second, if $M'$ has a realization over $k$, then by definition, it has a
Galois-invariant realization over an extension of~$k$.
Finally, we suppose that $M'$ has a Galois-invariant realization over an
extension of~$k$ and we want to show that $M$ has a realization over~$k$.
Suppose we have a realization over a Galois extension~$k'$ of~$k$. Since all the
flats from the original matroid contain different numbers of points, the Galois
group does not permute the corresponding lines in the realization. Therefore,
the lines and thus also the points from the original matroid~$M$ must be defined
over $k$. Therefore, the restriction of this realization gives a realization
of~$M$ over $k$, which completes the proof of the lemma.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:universality}]
As in the statement of the theorem, let $X$ be a scheme of finite type
over~$\ZZ$. We choose an affine open cover of~$X$ and let $\widetilde X$ be the
disjoint union of these affine schemes. By the scheme-theoretic version of
Mn\"ev's universality theorem, either Theorem~1.14 in \cite{lafforgue} or our
Theorem~\ref{thm:mnev}, there is a matroid~$M$ of rank~3 whose realization space
is isomorphic to an open subset~$U$ of~$\widetilde X \times \mathbb A^N$ and $U$
maps surjectively onto~$X$. Now let $M'$ be the matroid as in
Lemma~\ref{lem:break-symmetry} and we claim that $\Gamma_{M'}$ and~$D_{M'}$ have
the desired properties for the theorem.
Let $k$ be any infinite field, and then $X$ clearly has a $k$-point if and only
if $\widetilde X$ has a $k$-point. Likewise, since $k$ is infinite, any
non-empty subset of $\mathbb A^N_k$ has a $k$-point, so $U$ also has a $k$-point if
and only if $X$ has a $k$-point. By Lemma~\ref{lem:break-symmetry}, these
conditions are equivalent to $M'$ having a Galois-invariant realization over
an extension of~$k$. Supposing that $X$ has a $k$-point and thus $M'$ has a
realization over $k$, then $\Gamma_{M'}$ and $D_{M'}$ have a lifting over
$k[[t]]$ by Theorem~\ref{thm:construction}. Conversely, if $D_{M'}$ has a
lifting over $k[[t]]$, then $M'$ has a Galois-invariant realization over an
extension of~$k$ by Theorem~\ref{thm:matroid-lifting}, and thus $M$ has a
realization over~$k$ by Lemma~\ref{lem:break-symmetry}, so $X$ has a $k$-point.
\end{proof}
\section{Brill-Noether theory}\label{sec:brill-noether}
In this section, we take a detour and look at connections to Brill-Noether
theory and the analogy between limit linear series and tropical divisors.
In the theory of limit linear series, a key technique is the observation
that if the moduli space of limit linear series on the degenerate curve has the
expected dimension then it lifts to a linear
series~\cite[Thm.~3.4]{eisenbud-harris}. Here, the expected dimension of limit
linear series of degree~$d$ and rank~$r$ on a curve of genus~$g$ is $\rho(g, r,
d) = g - (r+1)(g + r - d)$. It is natural to ask if a tropical analogue of this
result is true: if the dimension of the moduli space of divisor classes on a
tropical curve of degree~$d$ and rank at least $r$ has (local) dimension
$\rho(g,r,d)$, then does every such divisor lift?
See~\cite{cjp} for further discussion and one case with an affirmative answer.
The main result of this section is that
the matroid divisors and graphs constructed in this paper do not
provide a negative answer to the above question.
We begin with the following
classification:
\begin{figure}
\caption{Matroids from Theorem~\ref{thm:classification}
\label{fig:matroids}
\end{figure}
\begin{thm}\label{thm:classification}
Let $M$ be a rank~$3$ simple matroid, with $g$ and~$d$ equal to the genus
of~$\Gamma_M$ and degree of~$D_M$ respectively. If $\rho = \rho(g, 2, d) \geq
0$, then $M$ is one of the following matroids:
\begin{enumerate}
\item\label{mat:extension} The one element extension of the uniform matroid
$U_{2,n-1}$, with $\rho = n-2$.
\item\label{mat:uniform} The uniform matroid $U_{3,4}$, with $\rho = 0$.
\item\label{mat:base} The matroid defined by the vectors: $(1,0,0)$, $(1,0,1)$,
$(0,0,1)$, $(0,1,1)$, $(0,1,0)$, with $\rho = 1$.
\item\label{mat:two-lines} The matroid in the previous example together with
$(1, 0, \lambda)$ for any element $\lambda$ of the field other than $1$ and~$0$,
with $\rho = 0$.
\item\label{mat:four-lines} The matroid consisting of the point of intersection
between any pair in a collection of $4$ generic lines, for which we can take the
coordinates to be the vectors from (3) together with $(1,1,1)$, with $\rho = 0$.
\end{enumerate}
\end{thm}
The last three cases of Theorem~\ref{thm:classification} are illustrated in
Figure~\ref{fig:matroids}.
\begin{proof}
We first compute the invariants for the graph~$\Gamma_D$ and divisor~$D_M$
constructed in Section~\ref{sec:divisors}. As before, we let $n$ be the
number of elements of~$M$, $\ell$ the number of flats, and $m$ the number of
complete flags. Since $\Gamma_D$ consists of $m$ edges and $n+\ell$ vertices, it
has genus $m - n - \ell + 1$. It is also immediate from its definition that
$D_M$ has degree~$n$. Thus, the expected dimension of rank~$2$ divisors is
\begin{equation}\label{eq:rho}
\rho =
m - n - \ell + 1 - 3((m - n - \ell + 1) + 2 - n)
= 5n + 2 \ell - 2 m - 8
\end{equation}
Now, assume that $\rho$ is non-negative for~$M$ and we consider what what
happens to~$\rho$ when we remove a single element~$e$
from a matroid, where
$e$ is not contained in all bases. For every flat containing $e$, we decrease
the number of complete flags by~$1$ if that flat contains at least $3$ elements,
and if it contains $2$ elements, then we decrease the number of flags by $2$ and
the number of flats by~$1$. Thus, by~(\ref{eq:rho}), $\rho$ drops by $5-2s$,
where $s$ is the number of flats in~$M$ which contain~$e$. Since $e$ must be
contained in at least $2$~flats, either $M \setminus e$, the matroid formed by
removing $e$ has positive $\rho$ or $e$ is contained in exactly $2$ flats.
We first consider the latter case, in which $e$ is contained in exactly two
flats, which we assume to have cardinality $a+1$ and $b+1$ respectively. The
integers $a$ and~$b$ completely determine the matroid because all the other
flats consist of a pair of elements, one from each of these sets. Thus, there
are $ab+2$ flats and $2ab + a + b + 2$ complete flags. By using (\ref{eq:rho}),
we get
\begin{equation*}
\rho = 5(a + b + 1) + 2(ab + 2) - 2(2ab + a + b + 2) - 8
= -2ab + 3a + 3b -3.
\end{equation*}
One can check that, up to swapping $a$ and~$b$, the only non-negative
values of this expression are when $a = 1$ and $b$ is arbitrary or $a=2$ and $b$
is $2$ or~$3$. These correspond to cases (\ref{mat:extension}),
(\ref{mat:base}), and (\ref{mat:two-lines}) respectively from the theorem
statement.
Now we consider the case that $e$ contained in more than two flats, in which
case $M \setminus e$ satisfies $\rho > 0$. By induction on the number of
elements, we can assume that $M \setminus e$ is on our list, in which case the
possibilities with $\rho > 0$ are (\ref{mat:extension}) and
case~(\ref{mat:base}). For the former matroid, if $e$ is contained in a flat of
$M \setminus e$, then $M$ is a matroid of the type from the previous paragraph,
with $a$ equal to $1$ or~$2$. On the other hand, if $e$ contained only in
$2$-element flats, then $e$ is contained in $n-1$ flats, so
\begin{equation*}
\rho(M) = \rho(M \setminus e) + 5 - 2(n-1) = (n-3) + 7 - 2n = 4 - n.
\end{equation*}
The only possibility is $n=4$, for which we get (\ref{mat:uniform}), the
uniform matroid. Finally, if $M \setminus e$ is the matroid in
case~(\ref{mat:base}), then the only relevant possibilities are those for which
$e$ is contained in at most $3$ flats, for which the possible matroids are
(\ref{mat:two-lines}) or~(\ref{mat:four-lines}).
\end{proof}
\begin{prop}
If $R$ is a DVR and $M$ is one of the matroids in
Theorem~\ref{thm:classification}, then $M$ has a Galois-invariant realization
over an extension of~$R$.
\end{prop}
\begin{proof}
The matroids~(\ref{mat:uniform}), (\ref{mat:base}) and~(\ref{mat:four-lines})
are regular matroids, i.e.\ realizable over~$\ZZ$, so they are \emph{a fortiori}
realizable over any DVR. Moreover, the other matroids in
case~(\ref{mat:extension}) and~(\ref{mat:two-lines}) are realizable over $R$ so
long as the residue field has at least $n-2$ and $3$ elements respectively. We
will show that if the residue field is finite, then the one-element extension of
$U_{2,n-1}$ has a Galois-invariant realization over~$R$. The other case is
similar.
Let $M$ be the one-element extension of~$U_{2,n-1}$ and suppose the residue
field~$k$ is finite. We choose a polynomial with coefficients of
degree~$n-1$ in~$R$ whose reduction to~$k$ is square-free. Adjoining the
roots of this polynomial defines an unramified extension~$R'$ of~$R$, and we
write $a_1, \ldots, a_{n-1}$ for its roots in~$R$. Then, the vectors $(1,a_1,0),
\ldots, (1,a_{n-1}, 0), (0, 0,1)$ give a Galois-invariant realization of~$M$
over~$R'$, which is what we wanted to show.
\end{proof}
\section{Quantitative Mn\"ev universality}\label{sec:mnev}
In this section, we prove a quantitative version of Mn\"ev universality over
$\Spec \ZZ$ with Theorem~\ref{thm:quantitative-characteristic} as our desired
application. We follow the strategy of \cite[Thm.~1.14]{lafforgue}, but use the
more efficient building blocks used in, for example,~\cite{lee-vakil}. We pay close attention
to the number of points used in our construction in order to get effective
bounds on the degree of the corresponding matroid divisor. These bounds are
expressed in terms of the following representation.
\begin{defn}\label{def:elementary-monic}
Let $S_n$ denote the polynomial ring $\ZZ[y_1,
\ldots, y_n]$. In the extension $S_n[t]$, we also introduce the coordinates
$x_i$ defined by $x_0 = t$ and $x_i = y_i + t$ for $1 \leq i \leq n$. In
addition, for $n < i \leq m$, suppose we have elements $x_i \in S_n[t]$ such
that:
\begin{enumerate}
\item Each~$x_i$ is defined as one of $x_i = x_j + x_k$, $x_i = x_j
x_k$, or $x_i = x_j + 1$, where $j, k < i$.
\item Each $x_i$ is monic as a polynomial in~$t$ with coefficients in~$S_n$.
\end{enumerate}
The coordinates $x_i$ for $1 \leq i \leq n$ will be called \defi{free variables}
and the three operations for defining new variables in~(1) will be called
\defi{addition}, \defi{multiplication}, and \defi{incrementing}, respectively.
Moreover, we suppose we have finite sets of equalities~$E$ and inequalities~$I$
consisting of
pairs $(i,j)$ such that $x_i - x_j$ is in $S_n \subset S_n[t]$.
We then say that the algebra:
\begin{equation*}
S_n[(x_{i'}-x_{j'})^{-1}]_{(i',j')\in I}/
\langle x_i - x_j \mid (i, j) \in E\rangle
\end{equation*}
has an \defi{elementary monic representation} consisting of the above
data, namely, the
integers~$n$ and~$m$, the expression of each $x_i$ as an addition,
multiplication, or increment for $n < i \leq m$, and the sets of equalities and
inequalities.
\end{defn}
The inequalities $I$ in Definition~\ref{def:elementary-monic} are not strictly
necessary because an inverse to $x_i-x_j$ can always be introduced as a
new variable, but the direct use of inequalities may be more efficient, such as
in the proof of Theorem~\ref{thm:quantitative-characteristic}.
\begin{prop}\label{prop:monic-representation}
There exists an elementary monic representation of any finitely generated
$\ZZ$-algebra.
\end{prop}
\begin{proof}
We begin by presenting the $\ZZ$-algebra~$R$ as
\begin{equation*}
R = \ZZ[y_1, \ldots, y_n]/\langle f_1 - g_1, \ldots, f_m - g_m \rangle,
\end{equation*}
where each polynomial~$f_k$ and~$g_k$ has positive integral coefficients.
Then $f_k$ and~$g_k$ can be constructed by a sequence of
multiplication and addition operations applied to the variables~$y_i$ and the
constant~$1$. Obviously, we can assume that our multiplication never involves
the constant~$1$. To get an elementary monic representation, we first replace
the variables $y_i$ with $x_i = y_i + t$ in the constructions of $f_k$
and~$g_k$, with some adjustments, as follows. Since Definition~\ref{def:elementary-monic} does not allow addition
of $1$ with itself, we replace such operations by first introducing a new
variable $x_i = x_0 + 1 = t + 1$ and then adding $1$ to $x_i$. Similarly, in
order to satisfy the second condition of Definition~\ref{def:elementary-monic},
when adding two variables $x_i$ and~$x_j$ which are both monic of the same
degree $d$ in $t$, we first compute an intermediate $x_{i'} = t^{d+1} + x_i$ and
then the sum $x_{i'} + x_j = t^{d+1} + x_i + x_j$. In this way, we ensure that
all of the $x_i$ variables are monic in~$t$. Moreover, $x_i$ and~$x_j$ agree
with $f_k$ and~$g_k$, respectively, modulo~$t$, but in order to be able to
have an equality $x_i = x_j$ in an elementary monic representation, the
difference $x_i - x_j$ have to not involve $t$.
Therefore, we want to replace $x_i$ and $x_j$ by polynomials $x_i'$ and $x_j'$
which agree with $x_i$ and $x_j$ modulo $t$, but such that the difference $x_i'
- x_j'$ does not involve $t$. We do this by double induction, first,
on the maximum total degree in the $y$ variables of the terms of $x_i - x_j$
that involve $t$, and second, on the number of terms of that degree.
Thus, we suppose that $c t^s y_1^{a_1} \cdots y_n^{a_n}$ is a term of $x_i -x_j$
whose total degree in the $y$ variables is maximal among terms with $s > 0$.
By swapping $i$ and~$j$ if necessary, we can assume that $c$ is
positive. We then use multiplication operations to construct:
\begin{equation*}
x_\ell = t^s x_1^{a_1} \cdots x_n^{a_n}
= t^s (t + y_1)^{a_1} \cdots (t + y_n)^{a_{n}}
= t^s y_1^{a_1} \cdots y_m^{a_m} + \ldots,
\end{equation*}
where the final ellipsis denotes omitted terms with lower degree in the $y$
variables. First suppose that $x_j$ and $x_\ell$ have different degrees
in the $t$
variable. Then, we can use $c$ addition operations to construct $x_{j'} = x_j +
c x_\ell$, and we set $i' = i$. On the other hand, if $x_j$ and $x_\ell$
have the same degree in~$t$, then let $d$ be an integer larger than the
$t$-degree of $x_i$, $x_j$, and $x_\ell$, and we use additions to construct
$x_{j'} = x_j + c x_\ell + t^d$ and $x_{i'} = x_i + t^d$. In either case,
$x_{i'}$ and $x_{j'}$ equal $x_i$ and $x_j$, respectively, modulo~$t$. Moreover,
the term $c t^s y_1^{a_1} \cdots y_m^{a_m}$ has been eliminated from $x_{i'} -
x_{j'}$, while only
introducing new terms which have lower degree in the $y$ variables. Thus,
by induction, we can eliminate all terms of $x_i - x_j$ which involve $t$ and
have maximal total degree in the $y$ variables among such terms, and by the
second level of induction, we can eliminate such terms in all degrees. We
therefore have constructions of variables $x_{i''}$ and $y_{j''}$ such that
$x_{i''} - x_{j''} = f_k - g_k$. Setting these equal for $k
= 1, \ldots, m$ gives us the elementary monic representation of $R$.
\end{proof}
Given a matroid~$M$, its possible realizations form a scheme, called the
realization space of the matroid~\cite[Sec.~9.5]{katz}. Explicitly, given a
rank~3
matroid with $n$ elements, each flat of the matroid defines a closed,
determinantal condition in $(\PP^2_\ZZ)^n$ and each triple of elements which is
not in any flat defines an open condition by not being collinear. The
\defi{realization space} is the quotient by $\PGL_3(\ZZ)$ of the
scheme-theoretic intersection of these conditions. We will only consider the
case when this action is free, in which case the quotient will be an affine
scheme over $\Spec \ZZ$.
\begin{thm}[Mn\"ev universality]\label{thm:mnev}
For any finite-type $\ZZ$-algebra~$R$, there exists a rank~$3$ matroid~$M$ whose
realization space is an open subset $U \subset \mathbb A^N \times \Spec R$ such
that $U$ projects surjectively onto $\Spec R$.
Moreover, if $R$ has an elementary monic representation with $n$~free variables,
$a$~additions, $m$~multiplications, $o$ increments, $e$~equalities, and
$i$~inequalities, then $M$ has
\begin{equation*}
3n + 7a + 7o + 6m + 5e + 6i + 6
\end{equation*}
elements, and
\begin{equation*}
N = 3(n + a + o + m + e + i) + 1.
\end{equation*}
\end{thm}
\begin{proof}
By Proposition~\ref{prop:monic-representation}, we can assume that $R$ has an
elementary monic representation. Both the matroid and its potential
realization will be built up from the elementary monic
representation, beginning with the free variables and then applying the
addition, multiplication, and increment operations. We describe the
constructions of both the matroid and the realization in parallel for ease of
explaining their relationship.
We begin with the free variables. For $x_0 = t$ and for each free variable~$x_i$
of the representation, we have a line, realized generically, passing through a
common fixed point. In the figures below, we will draw these horizontally so
that the common point is at infinity. On each of these lines we have 3
additional points, whose positions along the line are generic. Our convention
will always be that points whose relative position is not specified are generic.
In other words, unless otherwise specified to lie on a line, each pair of points
correspond to a 2-element flat.
From each set of $4$~points on one of these free variable lines, we can take the
cross-ratio, which is invariant under the action of $\PGL_3(\ZZ)$. Therefore, by
taking the cross-ratio on each line as the value for the corresponding
coordinate~$x_i = t+ y_i$ or $x_0 = t$, we define a morphism from the
realization space of the matroid defined thus far to $\mathbb A^{n+1} = \Spec
S_n[t]$, where $S_n = \ZZ[y_1, \ldots, y_n]$ as
in Definition~\ref{def:elementary-monic}. Our goal with the remainder of the
construction is to constrain the realization such that the projection to $\Spec
S_n$ is surjective onto $\Spec R \subset \Spec S_n$.
Concretely, the cross-ratio is the position of one point on
the line in coordinates where the other points are at $0$, $1$, and~$\infty$.
For us, the point common to all variable lines will be at~$\infty$, so we will
refer to the other points along the line as the ``$0$'' point, the ``$1$''
point, and the variable point. We will next embed the operations of addition,
multiplication, and incrementing from the elementary monic representation. The
result of each of these operations will be encoded as the cross-ratio of $4$ points on a
generic horizontal line, in the same way as with the free variables.
\begin{figure}
\caption{Configuration for computing multiplication. The cross-ratios of the
solid circles, together with the horizontal point at infinity define the
variables. On each line, the solid circles are, from left to right, the ``$0$''
point, the variable point, and the ``$1$'' point of the cross-ratio defining the
variable. The empty circles are auxiliary points. }
\label{fig:multiplication}
\end{figure}
First, multiplication of distinct variables $x_i = x_j x_k$ is constructed as in
Figure~\ref{fig:multiplication}, where the $x_j$ and $x_k$ lines refer to the
lines previously constructed for those variables and the other points are new.
We can choose the horizontal line for $x_i$ as well as the additional points
generically so that there none are collinear with previously constructed points.
Then, one can check that the cross-ratio of the solid points on the central line
is the product of the cross-ratios on the other two lines. Set-theoretically,
this claim follows from the fact that projections between parallel lines
preserve ratios of distances, and so measuring from the leftmost point of the
$x_i$ line, the top projection ensures that the ratio between the distances to
the empty circle and the rightmost circle is $x_j$. Likewise, the lower
projection ensures that the ratio of the distances to the center solid circle
and the empty circle is $x_k$ and so $x_i = x_jx_k$ appears as the product of
the ratios. If $j$ equals~$k$,
the diagram may be altered by moving the corresponding lines so that they
coincide. In either case, the construction uses $6$ additional points.
\begin{figure}
\caption{Configuration for computing addition. The solid and empty circles
represent the variables and auxiliary points respectively, as in
Figure~\ref{fig:multiplication}
\label{fig:addition}
\end{figure}
Second, the addition of variables $x_i = x_j + x_k$ can be constructed as in
Figure~\ref{fig:addition}. As in the case of multiplication, the motivation for
this construction can be understood from the fact that projections scale
distances. In particular, the top projection means that the empty circle and the
outer points on the $x_i$ line encode the variable~$x_j$, and since the two
lower projections are both from points on the same horizontal line, the ratio of
the distance between the inner points on the $x_i$ line to the distance between
the outer points equals the value of~$x_k$. Therefore, the intervals on either
side of the empty circle encode the values of~$x_j$ and $x_k$ and their
concatenation computes $x_i = x_j + x_k$.
In the configuration from Figure~\ref{fig:addition},
there will be an additional coincidence if $x_j = x_k$ in
that the empty circle on the $x_i$ line, the middle point on the
$x_k$ line and a point on the bottom line will be collinear. However,
since $x_j + x_k$,
$x_j$, and~$x_k$ are all monic polynomials in the variable $t$, $x_j - x_k$ is
also monic in~$t$ and so a sufficiently generic choice of $t$ will ensure that
$x_j - x_k$ is non-zero.
Similarly if $x_j = -1$, then the empty point on the $x_i$ line, the rightmost
point on the $x_k$ line and a marked point on the bottom line will be collinear,
but this can again be avoided by adjusting~$t$.
For the addition operation,
we've used $7$ additional points.
\begin{figure}
\caption{Configuration for incrementing. The solid and empty circles represent
the variables and the auxiliary points respectively, as in
Figure~\ref{fig:multiplication}
\label{fig:increment}
\end{figure}
Third, for incrementing a single variable, $x_i = x_j + 1$, we specialize the
configuration in Figure~\ref{fig:addition} so that $x_k = 1$, giving
Figure~\ref{fig:increment}. The line labeled with~$1$ can be chosen once and
used in common for all increment operations, since it functions as a
representative of the constant~$1$. We've used 7 additional points for each
increment operation, together with $2$ points common to all such operations.
\begin{figure}
\caption{Configuration for imposing inequality. The solid and empty circles
represent the variables and auxiliary points respectively, as in
Figure~\ref{fig:multiplication}
\label{fig:inequality}
\end{figure}
At this point, the realization space still surjects onto $\Spec S_n$, and so
we still need to impose the equalities and inequalities. Each inequality $x_i
\neq x_j$ can
be imposed using the diagram in Figure~\ref{fig:inequality}, which works by
projecting the two variable points to the same line and getting different
points. By replacing the projections of the two variables to the central line
with the same point, we can use a similar figure to assert equality $x_i = x_j$.
These use $6$ and $5$ additional points respectively.
To summarize, we have agglomerated the configurations in
Figures~\ref{fig:multiplication}, \ref{fig:addition}, \ref{fig:increment},
and~\ref{fig:inequality} to give a matroid whose realization space projects to
$\Spec R \subset \Spec S_n$. The realization of this matroid is determined by
the values of the $y_i$, together with a number of parameters, such as the
height of the horizontal lines, which are allowed to be generic, and thus
the realization space is an open subset of $\mathbb A^N \times \Spec R$.
To show that the projection to $\Spec R$ is surjective, we take any point of
$\Spec R$, which we can assume to be defined over an infinite field. Our
construction of the realization required us to avoid certain coincidences, such
as any $x_i$ being $0$ or~$1$ or an equality $x_j = x_k$ in any addition step.
Each such coincidence only occurs for a finite number of possible values for~$t$
we choose $t$ outside the union of all coincidences, and we can construct a
realization of the matroid.
Finally, we justify the quantitative parts of the theorem statement. The number
of elements of $M$ is computed by summing the number of elements for each of the
building blocks together with $1$ element for the common point on the horizontal
lines, $3$ elements for the variable $x_0 = t$, and $2$ elements for the horizontal line
representing~$1$ in Figure~\ref{fig:increment}.
For the computation of~$N$, we can assume that the coordinates on $\PP^2$
are such that the common point of the horizontal lines is $(1:0:0)$, the points
representing~$1$ are $(0:0:1)$ and $(1:0:1)$, and the ``$0$'' and~``$1$'' points of the $x_0 =
t$ line are $(0:1:0)$ and $(0:1:1)$ respectively. These fix the automorphisms of~$\PP^2$.
Then, one can check that each additional free variable and
each of the building blocks adds $3$~additional generic parameters. Finally, the
value of~$t$ is one more free parameter, which gives the expression for~$N$.
\end{proof}
\begin{figure}
\caption{System of equations used in the elementary monic representations of
$\ZZ/p$ and $\ZZ[p^{-1}
\label{fig:elem-monic}
\end{figure}
\begin{proof}[Proof of Theorem~\ref{thm:quantitative-characteristic}]
By Theorems~\ref{thm:matroid-lifting}, \ref{thm:construction},
and~\ref{thm:mnev}, it will be enough to construct sufficiently
parsimonious elementary monic representations of the algebras $\ZZ/p$ and
$\ZZ[p^{-1}]$
and thus matroids $M$ and~$M'$, respectively, representing these
equations. Let $\ell$ be the largest integer less than $\sqrt{p}$. The
elementary monic representations for both $M$ and~$M'$ use the equations shown
in Figure~\ref{fig:elem-monic}.
Then, $\ZZ/p$ can be represented by adding an equality between $x_{\ell+5}$ and
$x_{\ell+p-\ell^2+8}$ and $\ZZ[p^{-1}]$ can be represented by an inequality
between the same pair of variables.
In either case, this representation uses no free variables, $\ell +
p-\ell^2$ increments, $4$ additions, and $4$ multiplications. Thus, by
Theorem~\ref{thm:mnev}, $M$ and~$M'$ have $7(\ell+p-\ell^2) + 64$ and
$7(\ell+p-\ell^2) +
63$ elements respectively. We'll bound the former since it is larger. We first
rewrite the number of elements as
\begin{equation}\label{eq:num-elements}
7(\ell+p-\ell^2) + 64 = p-\ell^2 + 7\ell + 6(p-\ell^2) + 64
\end{equation}
To show that (\ref{eq:num-elements}) is smaller than $p$, we note that since $p
\geq 443$, then $\ell \geq 21$. We now have two cases. First, if $\ell = 21$,
then the largest prime number less than $22^2$ is $479$, so $p-n^2 \leq 38$.
Using this, we can bound (\ref{eq:num-elements}) as
\begin{equation*}
p - 21^2 + 7\cdot 21 + 6(38) + 64 = p - 2 < p
\end{equation*}
On the other hand, if $\ell \geq 22$, then the choice of $\ell$ means that $p <
(\ell + 1)^2$, so $p - \ell^2 \leq 2 \ell$. Therefore, we can bound
(\ref{eq:num-elements}) as follows:
\begin{align*}
p - \ell^2 + 7\ell + 6(2\ell) + 64
&= p - \ell^2 + 19\ell + 64 \\
&= p - (\ell-19)\ell + 64 \\
&\leq p - 2 < p
\end{align*}
Thus, the number of elements of $M$ and~$M'$ is less than~$p$.
We take the graphs $\Gamma$ and $\Gamma'$ and the divisors $D$ and~$D'$ for the
theorem statement to be the matroid divisors of~$M$ and~$M'$ respectively. Since
$M$ and~$M'$ have fewer than $p$ elements, $D$ and~$D'$ have degree less
than~$p$. Moreover, since $k$ is an infinite field, by
Theorems~\ref{thm:matroid-lifting} and~\ref{thm:construction}, $\Gamma$ and~$D$
lift over $k[[t]]$ if and only if $M$ is representable over~$k$, which means
that the characteristic of~$k$ equals~$p$. Similarly, $\Gamma'$ and $D'$ lift if
and only if the characteristic of $k$ is not~$p$.
\end{proof}
\begin{rmk}
The threshold for~$p$ in Theorem~\ref{thm:quantitative-characteristic} is not
optimal. For example, by using a different construction when $p$ is closer to a
larger square number than to a smaller square, it is
possible to reduce the bound to~$331$.
\end{rmk}
\end{document} |
\begin{document}
\title{Stability and busy periods in a multiclass queue with
state-dependent arrival rates
}
\title{Stability and busy periods in a multiclass queue with
state-dependent arrival rates
}
\begin{abstract}
We introduce a multiclass single-server queueing system in which the arrival
rates depend on the current job in service. The system is characterized by a matrix of arrival rates in lieu of a vector of arrival rates. Our proposed model departs from existing state-dependent queueing models in which the parameters depend primarily on the number of jobs in the system rather than on the job in service. We formulate the queueing model and its corresponding fluid model and proceed to obtain the necessary and sufficient conditions for stability via fluid models. Utilizing the natural connection with the multitype Galton-Watson processes,
the Laplace-Stieltjes transform of busy periods in the system is given. We conclude with
tail asymptotics for the busy period
for heavy-tailed service time distributions for the regularly varying case.
{\mathrm{e}}nd{abstract}
Keywords: Busy periods; fluid models; multiclass queues; regular variation; stability; state-dependent arrival rates.
\section{Introduction}
\setcounter{equation}{0}
We introduce a multiclass single-server queueing system in which the arrival
rates depend on the current job in service. The system is characterized by a matrix of arrival rates instead of a vector of arrival rates. The proposed model departs from existing state-dependent models in the literature in which the parameters depend primarily on the number of jobs in the system (see Bekker et al.~\cite{Bekker}, Cruz and Smith~\cite{Cruz}, Jain and Smith~\cite{Jain}, Perry et al.~\cite{Perry}, and Yuhaski and Smith~\cite{Yuh}, among other sources) rather than the job in service. \\
\indent Our model is motivated by two practical queueing considerations. The first
is a multiclass queueing system in which the arriving customer can observe only the class of the customer in service and no other characteristics of the queue. This information informs the customer's decision to either join or leave the queue. The second concerns local area networks with a central server in which $K$ clients generate requests
at individual Poisson rates $\mu_i$. Often, a client does not generate requests when
a previous request is being handled by the server. Further, it is conceivable that groups of clients working together may influence each other's Poisson rate. To the best of our knowledge, this simple yet potentially very useful queueing model has never appeared in the literature. This serves as our primary motivation for the manuscript.\\
\indent The remainder of the work is structured as follows. We formulate the queueing model in Section \ref{sec2} and its corresponding fluid model in Section \ref{sec3}. In Section \ref{sec4}, we obtain the necessary and sufficient conditions for stability via fluid models. Through the natural connection with the multitype Galton-Watson processes, we characterize
the Laplace-Stieltjes transform of busy periods in the system in Sections \ref{sec5} and \ref{sec6}. Section \ref{sec7} concerns tail asymptotics on the busy period in the case of heavy-tailed service time distributions. Section \ref{sec8} offers a brief conclusion and presents ideas for future work.
\section{The queueing model} \label{sec2}
Consider a multiclass single-server queue with $K$ classes of jobs, each arriving according to independent counting processes. \textcolor{black}}\def\tcrr{\textcolor{black}{We assume that only one job may be serviced at a time}. Let the arrival rate depend \textit{on the class of the job in service}. Let the arrival rate depend \textit{on the class of the job in service}. If the server is serving a job of class $i$, the arrival rate of class $j$ jobs is $\lambda_{ij}$, $i,j=1,\dots,K$.
The matrix of arrival rates is defined as $\mathbf{\Lambda} = (\lambda_{ij})$, $i,j = 1, \ldots, K$.
If there is no job in service, then the arrival rate of class $j$ jobs is defined as $\lambda_{0j}$,
$j=1,\dots,K$.
The arrival mechanism is described more precisely with dynamical equations in Section \ref{sec3}.
We proceed to set notation. Let
$\bar{\lambda}^i = \sum_{j=1}^K \lambda_{ij}$
for each $i=1,\dots,K$.
Service times for class $i$ jobs are assumed to be i.i.d.\ with distribution function $F_i$,
$i=1,\dots, K$. Let $S_i$ be a generic service time for class $i$ jobs, with $\mathbb{E}[S_i] = m_i =\mu_i^{-1}$, $i=1,\dots, K$ and $\mathbf{G}=\text{diag}(\mu_1,\mu_2,\ldots,\mu_K)$. We define the ``mean offspring matrix''
to be $\mathbf{M} = \mathbf{G}^{-1} \mathbf{\Lambda} $ (here, the $ij$th element $\lambda_{ij}m_i$
is the mean number of arriving class $j$ customers during service of a class $i$ customer).
By definition, all the elements of $\mathbf{M}$ are non-negative, and this is enough to ensure that
the dominant eigenvalue $\rho(\mathbf{M})$ is real and positive, cf.~\cite{Gant}.
For some results, more restrictive conditions on $\mathbf{M}$ will be required. Further, let $\psi_i$ denote the Laplace-Stieltjes transform (LST) of $S_i$, $i=1,\dots, K$, respectively, that is, $\psi_i(s) = \mathbb{E}[{\mathrm{e}}^{-s S_i}] =
\int_0^{\infty} {\mathrm{e}}^{-st} {\mathrm{d}} F_i(s)$ for $s>0$. We let $Q_i$ denote the steady-state number of class $i$ jobs in the system, $i=1,\dots, K$, and let $\mathbf{Q}= (Q_1,\dots, Q_K)$.
Each state of the system takes nonnegative integer values, that is, $\mathbf{x} = (x_1,\dots,x_K) \in \mathbb{Z}^K_+$.
The service disciplines we consider are non-idling, i.e., jobs must be served using the full
capacity of the server whenever there are jobs in the system.
\textcolor{black}}\def\tcrr{\textcolor{black}{Our results on stability and the busy period are independent of the particular
(non-idling) scheduling policy employed in the system.}
\section{Queueing and fluid dynamics} \label{sec3}\setcounter{equation}{0}
\subsection{Queueing dynamical equations}
We now precisely define the arrival mechanism.
For $i \in \{1, \ldots, K\}$ and $t \ge 0$, $Q_i(t)$ denotes the number
of class $i$ jobs in the system at time $t$, whether in service or in the queue.
Similarly, let $T_i(t)$ denote
the amount of time that has been devoted to serving class $i$ jobs
in $[0,t]$. Further, let $A_i(t)$ and $D_i(t)$ be, respectively,
the total number of class $i$ jobs that have arrived and departed from the
system in $[0,t]$. We then have the following input-output equation for
each class $i$ job
\begin{equation}
Q_i(t) = Q_i(0) + A_i(t) - D_i(t).
{\mathrm{e}}nd{equation}
For each class $i$, the counting process $\mathbb{E}v^j_i(t)$ is the number of class $i$
jobs that arrive during the first $t$ time
units devoted to processing class $j$. $\mathbb{E}v^0_i(t)$ counts the number of class $i$
arrivals during the first $t$ time units that no job is being processed at the server.
The total number of class $i$ arrivals in $[0,t]$ is then given
by
\begin{equation}
A_i(t) = \mathbb{E}v^0_i\bigl(T_0(t)\bigr) + \sum_{j=1}^N \mathbb{E}v^j_i \bigl(T_j(t)\bigr),
{\mathrm{e}}nd{equation}
where the counting processes $\mathbb{E}v^j_i$ for \textcolor{black}}\def\tcrr{\textcolor{black}{$i=1,\dots, K,$ and $j = 0, \dots, K$} are assumed to be mutually independent.
As for the service processes, for each $i$, $1 \le i \le K$ and positive integer
$n$, we let $V_i(n)$ denote the total service requirement for the first $n$
class $i$ jobs. Assuming an HL service discipline, we have that
\begin{equation}
V_i(D_i(t)) \le T_i(t) \le V_i(D_i(t)+1)
{\mathrm{e}}nd{equation}
for each $t \ge 0$ and $1 \le i \le N$.
We define the workload in the system at time $t$ to be
\begin{equation}
W(t) = \sum_{i=1}^K V_i(A_i(t) + Q_i(0)) - \sum_{i=1}^K T_i(t),
{\mathrm{e}}nd{equation}
and the cumulative idle time process to be
\begin{equation}
Y(t) = t - \sum_{i=1}^K T_i(t).
{\mathrm{e}}nd{equation}
It is important to note that $Y$ is a non-decreasing function. We assume that the queueing
policy is non-idling, which specifically means that $Y$ can increase only
when $W(t)=0$. More precisely,
$$ \int_0^{\infty} W(t) \, dY(t) = 0.$$
\subsection{Fluid model}
For purposes of determining the stability conditions of a more general
version of our model, we formulate a fluid network version of the model. For references to important definitions and results in the fluid model literature, we refer the reader to Bramson~\cite{Bramson} and Gamarnik~\cite{Gamarnik}.
For $i \in \{1, \ldots, K\}$ and $t \ge 0$, let $Q_i(t)$ denote the amount
of fluid of class $i$ in the system at time $t$. Similarly, let $T_i(t)$ denote
the amount of time that has been devoted to serving class $i$ fluid
in $[0,t]$. We also define $A_i(t)$ and $D_i(t)$ which are, respectively,
the total amount of class $i$ fluid that has arrived and departed from the
system in $[0,t]$. We then have the following standard
equation:
\begin{equation}
Q_i(t) = Q_i(0) + A_i(t) - D_i(t),
{\mathrm{e}}nd{equation}
for each $i \in \{1, \ldots, K\}$ and $t \ge 0$.
The departure processes in this system also obey the standard relation
$D_i(t) = \mu_i T_i(t)$ for all $t \ge 0$.
The unusual feature of our model lies in the arrival process, which
is dependent on the current class in service. In the queueing model,
processor sharing is not allowed. Hence, there is (at most) one class
in service at any given time and ``the customer in service'' is defined
unambiguously. Here, we provide a more general formulation that
reduces to the queueing model presented in earlier sections, under appropriate restrictions on the
allowable queueing disciplines. First, we recall the usual condition
\begin{equation}
\sum_{i=1}^N \dot{T}_i(t) \le 1,
{\mathrm{e}}nd{equation}
which simply indicates that the server cannot devote more than 100\%
of its time to serving fluids of all classes. Since we assume that
the queueing discipline is non-idling, $\sum_{i=1}^N \dot{T}_i(t) = 1$
whenever there is a positive amount of fluid in the system. We also define
the idle time in $[0,t]$ to be:
$$ Y(t) = t - \sum_{i=1}^N T_i(t).$$
Note that the current arrival rate of class $j$ fluid is given by
$\dot{A}_j(t)$. In the queueing model, if a job of class $i$ is in service
then the arrival rate of class $j$ jobs is $\lambda_{ij}$. Let
$\mathbf{\lambda}_j$ be the column vector $(\lambda_{1j}, \ldots
\lambda_{Nj})^\mbox{\bf 0}t$ and let $\dot{\mathbf{T}}(t)$ be the column vector
$\bigl(\dot{T}_1(t), \ldots, \dot{T}_N(t)\bigr)^\mbox{\bf 0}t$, where ${}^\mbox{\bf 0}t$ means transposition. We define the fluid arrival
rate of class $j$ to be
\begin{equation}
\label{arrivalrep}
\dot{A}_j(t) = \lambda_{0j} \dot{Y}(t) + \lambda^\mbox{\bf 0}t_j \dot{\mathbf{T}}(t).
{\mathrm{e}}nd{equation}
In particular, when there is fluid in the system, the class $j$ arrival
rate is a convex combination of the elements of $\mathbf{\lambda}_j$.
If we restrict to policies in which only one class can be served at
any time, then equation (\ref{arrivalrep}) assigns an arrival rate of $\lambda_{ij}$
to class $j$ fluid when class $i$ fluid is in service. Note that this concurs with
the queueing model formulation. Combining the above, we have
\begin{eqnarray} \label{eq9}
Q_j(t) & = & Q_j(0) + \int_0^t (\lambda_{0j} \dot{Y}(u) + \lambda^\mbox{\bf 0}t_j \dot{\mathbf{T}}(u)) \; du
- \mu_j T_j(t) \\
& = & Q_j(0) + \lambda_{0j} {Y}(t) + \lambda^\mbox{\bf 0}t_j \mathbf{T}(t) - \mu_j T_j(t). \label{eq10}
{\mathrm{e}}nd{eqnarray}
Writing equations (\ref{eq9}) and (\ref{eq10}) in matrix form yields
\begin{equation} \label{dynamics}
\mathbf{Q}(t) = \mathbf{Q}(0)+ (\mathbf{M}^\mbox{\bf 0}t-\mathbf{I})\mathbf{D}(t) +Y(t) \mathbf{\lambda_0}.
{\mathrm{e}}nd{equation}
We define the vector of fluid work in the system at time $t$ to be
\begin{equation} \label{work}
\mathbf{W}(t) = \mathbf{G}^{-1}\mathbf{Q}(t).
{\mathrm{e}}nd{equation}
\subsubsection{Fluid Limits}
Thus far we have described a fluid model but it remains to show
that the fluid limits of the queueing model satisfy the fluid model equations.
In this subsection only, we use a bar to denote a fluid limit.
As usual, we define the fluid
limit of the queue-length processes to be
$$ \bar{Q}_i(t) = \lim_{n \to \infty} \frac{Q_i(nt)}{n},$$
with other fluid limits defined in an analogous manner.
We make the usual assumptions on the stochastic primitives and initial conditions, i.e.,
for all $i$ and $j$
\begin{eqnarray}
\lim_{n \to \infty} \frac{\mathbb{E}v^j_i(nt)}{n} & = & \lambda_{ji}t \label{fslln1} \\
\lim_{n \to \infty} \frac{\mathbb{E}v^0_i(nt)}{n} & = & \lambda_{0i}t \label{fslln2} \\
\lim_{n \to \infty} \frac{V_i(n)}{n} & = & m_i \\
\lim_{n \to \infty} \frac{Q_i(0)}{n} & = & \bar{Q}_i(0),
{\mathrm{e}}nd{eqnarray}
where the convergence is almost surely, uniformly on compact sets.
Under these assumptions, the fluid model equations
can be derived in a straightforward way from the queueing dynamical equations, since
all but the arrival rate process is identical to the standard multiclass queueing network
model. For the arrival process, we have
\begin{eqnarray*}
\bar{A}_i(t) & = & \lim_{n \to \infty} \frac{A_i(nt)}{n}
\ = \ \lim_{n \to \infty} \frac{\mathbb{E}v^0_i(T_0(nt))}{n} + \lim_{n \to \infty} \sum_{j=1}^N \frac{\mathbb{E}v^j_i (T_j(nt))}{n} \\
& = & \lambda_{0i} \bar{Y}(t) + \lambda^\mbox{\bf 0}t_i \bar{\mathbf{T}}(t).
{\mathrm{e}}nd{eqnarray*}
The last equality follows from assumptions (\ref{fslln1}) and (\ref{fslln2}) and
similar arguments as found in Proposition 4.12 in \cite{Bramson}.
Finally, the connection between fluid stability and
queueing network stability follow from straightforward modification of existing stability
results, under the usual assumptions that the interarrival times for all job classes are unbounded
and spread out. We refer the reader to Chapter 4 of Bramson \cite{Bramson} for full details.
\section{Stability results for fluid model} \label{sec4}\setcounter{equation}{0}
In this section, we prove a number of results regarding the stability, or instability,
of the fluid model. The proofs rely on the following two observations.
\begin{enumerate}
\item $T_i(\cdot)$ is Lipschitz continuous for each $i$ and hence so is
any linear function $f$ of $(T_1, \ldots, T_K)$. Thus, $f$ is
absolutely continuous and its derivative exists almost everywhere.
\item If $\dot{f}(t)$ exists for $t >0$,
$t$ is called a regular point.
{\mathrm{e}}nd{enumerate}
We
define $\mathbf{e} = (1, \ldots, 1)^\mbox{\bf 0}t$ and assume this column vector is of size $K$.
Finally, we set $\bf{H} = \bf G\mathbf{M}\mathbf{G}^{-1}.$
\begin{theorem}
If $\rho(\textbf{M}) <1$, then
$f(t) = \mathbf{e}^\mbox{\bf 0}t (\textbf{I}-\textbf{H}^\mbox{\bf 0}t)^{-1} \textbf{G}^{-1}Q(t)$
is a Lyapunov function for the fluid model.
{\mathrm{e}}nd{theorem}
\begin{proof}
Note that $\rho(\textbf{H})= \rho(\textbf{M}) <1$.
Hence, $\textbf{I}-\textbf{H}$ is an $\cal{M}$-matrix. Therefore $\textbf{I}-\textbf{H}$ is invertible
with a non-negative inverse.
Let us assume that the fluid system starts from a non-empty state, i.e., $\mathbf{Q}(0) \not= \mathbf{z}ero$.
By the continuity of $\mathbf{Q}$, $\mathbf{Q}(t)\not= 0$ for all $t$ in some interval $[0,s)$.
Then we have $Y(t) =0$ for all $t \in [0, s)$. Using equations (\ref{dynamics}) and (\ref{work}) we have
$$\mathbf{W}(t) = \mathbf{W}(0) - (\textbf{I}-\textbf{H}^\mbox{\bf 0}t)\mathbf{T}(t),$$
for $t \in [0, s)$.
Multiplying by $\mathbf{e}^\mbox{\bf 0}t (\textbf{I}-\textbf{H}^\mbox{\bf 0}t)^{-1}$ yields
$$ \mathbf{e}^\mbox{\bf 0}t (\textbf{I}-\textbf{H}^\mbox{\bf 0}t)^{-1}\mathbf{W}(t) = \mathbf{e}^\mbox{\bf 0}t (\textbf{I}-\textbf{H}^\mbox{\bf 0}t)^{-1} \mathbf{W}(0) - \mathbf{e}^\mbox{\bf 0}t \mathbf{T}(t).$$
As in the statement of the theorem, set
$$ f(t) = \mathbf{e}^\mbox{\bf 0}t(\textbf{I}-\textbf{H}^\mbox{\bf 0}t)^{-1} \textbf{G}^{-1}\mathbf{Q}(t)$$
and note that $f(t)=0$ if and only if $\mathbf{Q}(t) =\mathbf{z}ero$.
Then we have:
\begin{eqnarray}
f(t) & = & \mathbf{e}^\mbox{\bf 0}t (\textbf{I}-\textbf{H}^\mbox{\bf 0}t)^{-1} \mathbf{W}(t) \\
& = & \mathbf{e}^\mbox{\bf 0}t (\textbf{I}-\textbf{H}^\mbox{\bf 0}t)^{-1} \mathbf{W}(0) - \mathbf{e}^\mbox{\bf 0}t \mathbf{T}(t).
{\mathrm{e}}nd{eqnarray}
Taking derivatives, we obtain
$$ \dot{f}(t) = -\mathbf{e}^\mbox{\bf 0}t \dot{\mathbf{T}}(t) = -1,$$
for any $t \in [0,s)$ and regular point $t$.
Therefore, the draining time of the system under
any feasible policy is $$f(0) = \mathbf{e}^\mbox{\bf 0}t (\textbf{I}-\textbf{H}^\mbox{\bf 0}t)^{-1} \mathbf{W}(0),$$
which can be interpreted as the initial unfinished
``potential'' work, defined as the work due to the current workload
and work generated in the future by the initial
workload's ``offspring.'' The above argument implies that $\dot{f}(t)=-1$ whenever
$\mathbf{W}(t) \not= \mathbf{z}ero$ and thus the system stays drained once $\mathbf{Q}(t) =\mathbf{z}ero$. This completes the proof.
{\mathrm{e}}nd{proof}
The corollary below now immediately follows.
\begin{corollary} The fluid model is globally stable if $\rho(\textbf{M}) <1$.
{\mathrm{e}}nd{corollary}
\subsection{Weak instability}
Next we show that the fluid model is weakly unstable if $\rho(\textbf{M})>1$. We begin by introducing the following lemma.
\begin{lemma} \label{hithere}
Suppose $\rho(\textbf{M})=\rho(\textbf{H})>1$ and that each row of $\mathbf{M}$ has at least one strictly positive element. Then for all nonnegative
vectors $\mathbf{T}(t)>0$, $\mathbf{V}(t)=(\textbf{I}-\textbf{H})\mathbf{T}(t)$ must have some component $V_{i}(t)<0$
for some $i\in\{1,...,K\}$.
{\mathrm{e}}nd{lemma}
\begin{proof}
We argue to the contrary. Note that each row of $\textbf{H}$ has at least one strictly positive element, by the same assumption
on $\textbf{M}$. Also, for some $\alpha \in (0,1)$,
$\rho(\alpha \textbf{H}) = 1.$
For the sake of contradiction, assume that there exists a nonnegative
vector $\mathbf{T}(t)>0$ s.t.\ $\mathbf{V}(t)=(\textbf{I}-\textbf{H})\mathbf{T}(t)\geq0$.
Further, define $\mathbf{V}^{'}(t) = (\textbf{I}- \alpha \textbf{H})\mathbf{T}(t)$.
We now consider:
\begin{eqnarray*}
\mathbf{V}(t)-\mathbf{V}^{'}(t)=(\textbf{I}-\textbf{H})\mathbf{T}(t)-(\textbf{I}- \alpha \textbf{H})\mathbf{T}(t)=(\alpha \textbf{H}-\textbf{H})\mathbf{T}(t)<0.
{\mathrm{e}}nd{eqnarray*}
The above equations imply that $\mathbf{V}^{'}(t)>\mathbf{V}(t)$ and that there exists some
$\mathbf{T}(t)>0$ with $(\textbf{I}- \alpha \textbf{H})\mathbf{T}(t)>0$. Thus $(\textbf{I}-\alpha \textbf{H})$ is semipositive, and by condition $I_{27}$ in Chapter 6 of Berman~\cite{Berman}, $(\textbf{I}-\alpha \textbf{H})$ is a non-singular $\cal{M}$-matrix. This implies that $\rho(\alpha \textbf{H}) < 1,$ yielding a contradiction.
{\mathrm{e}}nd{proof}
We are now ready to prove Theorem \ref{thm2}, the main result of this subsection.
\begin{theorem} \label{thm2}
The fluid model is weakly unstable if $\rho(\mathbf{M})>1$ and each row of $\mathbf{M}$ has at least one strictly positive element.
{\mathrm{e}}nd{theorem}
\begin{proof}
Assume $\mathbf{Q}(0)=\mathbf{W}(0)=\mathbf{z}ero$. Then for any $t >0$ we have by (\ref{dynamics}) and (\ref{work}) that
\begin{eqnarray*}
\mathbf{W}(t) \ge \mathbf{W}(0) - (\textbf{I}-\textbf{H}^\mbox{\bf 0}t)\mathbf{T}(t)=(\textbf{H}^\mbox{\bf 0}t-\textbf{I})\mathbf{T}(t).
{\mathrm{e}}nd{eqnarray*}
By Lemma \ref{hithere}, there exists some component of $\mathbf{W}(t)$ s.t.\ $W_{i}(t)>0$. This implies that $\mathbf{Q}(t)\neq \mathbf{z}ero$ for all $t >0$. Thus the fluid model is weakly unstable.
{\mathrm{e}}nd{proof}
\subsection{Weak Stability}
\begin{theorem} \label{implemma}
Suppose that $\mathbf{M}$ is an irreducible non-negative matrix.
\textcolor{black}}\def\tcrr{\textcolor{black}{Then the} fluid model is weakly stable if $\rho(\mathbf{M}) \le 1$.
{\mathrm{e}}nd{theorem}
\begin{proof}
It suffices to show the result for the case $\rho(\mbox{\bf 0}ld{M})=1$, since we have already shown that the fluid model
is ``strongly'' stable when $\rho(\mbox{\bf 0}ld{M})<1$.
Let $\mathbf{Q}(0)=0$. We argue to the contrary. For the sake of contradiction, let us assume that $\mathbf{Q}(t) \not= \mathbf{z}ero$ for some $t >0$. Then, since $\mathbf{Q}$ is continuous,
there must be an interval $(t_1, t_2)$ with $t_2 > t_1$, for which $\|\mathbf{Q}(t)\| > 0$ for all $t \in (t_1, t_2)$,
$\mathbf{Q}(t_1) =0$ and $\|\mathbf{Q}(t_2) \| >0$. In particular, we may set $t_1 = \text{inf}\{t: \mathbf{Q}(t) \not=\mathbf{z}ero \}$. Now, recall that
\begin{equation} \label{dynam1}
\mathbf{Q}(t)=(\mathbf{M}^\mbox{\bf 0}t-\mathbf{I})\mbox{\bf 0}ld{D}(t)+Y(t)\mathbf{\lambda_0}.
{\mathrm{e}}nd{equation} Since $\mbox{\bf 0}ld{M}$ is a positive matrix, it follows by the Perron-Frobenius Theorem that there exists a positive left (row) eigenvector $\mbox{\bf 0}ld{w}$ of $\mbox{\bf 0}ld{M}$ with $\mbox{\bf 0}ld{w}\mathbf{M}=\mbox{\bf 0}ld{w}$, $w_i >0 $ for $i \in \{1,...,K\}$. Multiplying both sides of (\ref{dynam1}) by $\mbox{\bf 0}ld{w}$ we obtain
$$ \mbox{\bf 0}ld{w}\mathbf{Q}(t)=\mbox{\bf 0}ld{w}\bigl[(\mathbf{M}^\mbox{\bf 0}t-\mathbf{I})\mbox{\bf 0}ld{D}(t)+Y(t)\mathbf{\lambda_0}\bigr]\ =\ \mbox{\bf 0}ld{w}Y(t)\mathbf{\lambda_0},$$
for all $t \ge 0$. Recalling $\mathbf{Q}(t_1)=\mathbf{z}ero$ and $\|\mathbf{Q}(t_2)\| >0$ we have
$$ \mbox{\bf 0}ld{w}\bigl(Y(t_2)-Y(t_1)\bigr) \mathbf{\lambda_0} \ =\ \mbox{\bf 0}ld{w}\bigl(\mathbf{Q}(t_2)-\mathbf{Q}(t_1)\bigr) > 0.$$
This implies $Y(t_2) > Y(t_1)$ and thus there is positive idle time in $(t_1,t_2)$. However,
since the fluid level is positive in this entire interval, this violates the non-idling condition.
Thus such a fluid solution is not feasible. A contradiction has been reached. This concludes the proof.
{\mathrm{e}}nd{proof}
\section{Branching process connection} \label{sec5}\setcounter{equation}{0}
In the remainder of the paper, we investigate a special case of the multiclass
model discussed so far. In particular, we now assume that arrivals to each class from a Poisson
process, i.e., the model is an $M/G/1$ multiclass queue, rather than a $GI/G/1$ queue.
Although more general stability conditions for the $GI/G/1$ case were proven in Section \ref{sec4}, we
begin by reproving them in the Poisson setting, by making a connection to branching
processes. There are two reasons to do this. First, the stability results arise in a somewhat more
intuitive manner using this methodology. Secondly, we find the connection to branching processes
illuminating and useful in later sections.
A classical tool for the simple $M/G/1$ queue and related systems is to interpret customers as individuals in a branching process,
such that the children of a customer is the number of customers arriving during his or her service. This is useful because
the stability condition for the queueing system is the same as the condition for almost sure extinction. Carrying out the same idea
for our multiclass systems leads to a $K$-type Crump-Mode-Jagers branching process
$\{ \mathbf{Z}_n=\bigl(Z^{(1)}_n,\dots Z^{(K)}_n\bigr): n \ge 1\}$, such that the lifetime of an individual
of type $j$ has the same distribution as $S_j$. \textcolor{black}}\def\tcrr{\textcolor{black}{In the results below, we consider a branching process with a single ancestor of
type $i$. Whenever $\mathbb{E}_i$ and $\mathbb{P}_i$ are used, they are with reference to the probability
measure induced by such a single ancestor.} The offspring mechanism is then
described by the probabilities
\begin{equation}
p_{ij}(k) = \mathbb{P}_i\big(Z_{1}^{(j)}= k\big) = \mathbb{P}\bigl({\text{Pois}(\lambda_{ij}S_j) = k}\bigr) = \int_0^{\infty} \frac{(\lambda_{ij}s)^k
{\mathrm{e}}^{-\lambda_{ij}s}}{k!} {\mathrm{d}} F_j(s)\,.
{\mathrm{e}}nd{equation}
The offspring matrix $\mathbf{M} = (M_{ij})_{i,j= 1,\dots, K}$
is given by $M_{ij}=\mathbb{E}_i\bigl[Z_1^{(j)}\bigr]=\lambda_{ij}/\mu_{i}$ and is assumed
irreducible. Thus Perron-Frobenius theory applies to $\mathbf{M}$
and as before, $\rho=\rho(\mathbf{M})$ the largest eigenvalue. Note that the $ij$th element of the matrix $\sum_{n=0}^\infty \mathbf{M}^n$ gives the expected number of
type $j$ progeny of an individual of type $i$; of course, when $\rho<1$, we have $\sum_{n=0}^\infty \mathbf{M}^n=(\mathbf{I}-\mathbf{M})^{-1}$.
\subsection{Stability Conditions}
Let $|\mathbf{Z}_n|=\sum_{j=1}^KZ_n^{(j)}$ denote the total number of individuals in the $n$th generation
and $T^*$
the extinction time.
Let \textcolor{black}}\def\tcrr{\textcolor{black}{$\mathbb{P}_i(T^*<\infty)$ be the extinction probability of type $i$ of the branching process.}
Then, by classical results, we have the following theorem:\\
\begin{theorem}\label{thmHA}
\begin{eqnarray} \label{Harris}
\mathbb{P}_i(T^*<\infty) =1,\,\, i=1,\ldots,K,\, \text{if and only if} \,\, \rho \leq 1.
{\mathrm{e}}nd{eqnarray}
{\mathrm{e}}nd{theorem}
\begin{proof}
By the classical result for the extinction time of branching processes~\cite[Chap II. Theorem 7.1]{Harris63}, if and only if $\rho \leq 1$, the total number of generations for each type is finite with probability $1$ and thus $\sum |\mathbf{Z}_n| < \infty$, which further implies $\mathbb{P}_i(T^*<\infty) = 1 $ for every $i$.
{\mathrm{e}}nd{proof}
\noindent Consider $K=2$. Straightforward algebra gives that $\rho \leq 1$ is equivalent to
\begin{eqnarray}
\frac{\frac{\lambda_{11}}{\mu_1}+\frac{\lambda_{22}}{\mu_2}+\sqrt{\parens{\frac{\lambda_{11}}{\mu_1}-\frac{\lambda_{22}}{\mu_2}}^2+\frac{4\lambda_{12}\lambda_{21}}{\mu_1\mu_2}}}{2} \leq 1.
{\mathrm{e}}nd{eqnarray}
\begin{theorem}\label{thmHA2} $\mathbb{E}_iT^*<\infty$ for all $i$ if and only if $\rho < 1$.
{\mathrm{e}}nd{theorem}
\begin{proof} For a simple proof of sufficiency, assume $\rho<1$ and let
$S_j(m;n)$ denote the lifetime of the $m$th individual of type $j$ in the $n$th generation, $\underline\mu=\min_1^K\mu_j$.
Then
\begin{align*}\mathbb{E}_iT^*\ &= \mathbb{E}_i\sum_{n=0}^\infty\sum_{j=1}^K \sum_{m=1}^{Z_n^{(j)}}S_j(m;n)\ =\
\mathbb{E}_i\sum_{n=0}^\infty\sum_{j=1}^K\frac{Z_n^{(j)}}{\mu_j}\\ &\le\
\textcolor{black}}\def\tcrr{\textcolor{black}{ {\underline\mu}^{-1} \mathbb{E}_i\sum_{n=0}^\infty\sum_{j=1}^KZ_n^{(j)}\ =\ {\underline\mu}^{-1} \sum_{n=0}^\infty\sum_{j=1}^KM^n_{ij}\ <\ \infty, }
{\mathrm{e}}nd{align*}
where the second step above uses that $S_j(m;n)$ is independent of $\mathbf{Z}_0,\ldots,\mathbf{Z}_n$ (but not $\mathbf{Z}_{n+1},\mathbf{Z}_{n+2},\ldots$). Further, the strict inequality
\begin{equation}
\sum_{n=0}^\infty\sum_{j=1}^K M^n_{ij}<\infty
{\mathrm{e}}nd{equation}
follows from $\rho<1$. To prove the necessity, let $\bar{\mu}=\max_1^K\mu_j$. Then by the same reasoning we get
$\mathbb{E}_iT^* \geq \bar{\mu}^{-1} \sum_{n=0}^\infty\sum_{j=1}^K M^n_{ij} = \infty$ for $\rho \geq 1$ (see Berman and Plemmons \cite{Berman}). Hence $\mathbb{E}_iT^* < \infty$ is also necessary for $\rho < 1$.
{\mathrm{e}}nd{proof}
\noindent We now have the following corollary to Theorem \ref{thmHA}.
\begin{corollary}
The busy period $T<\infty$ w.p.1 if and only if the matrix $\mathbf{M}$ given by
\begin{eqnarray*}
M_{ij}=\frac{\lambda_{ij}}{\mu_i},\,\, i,j=1,\ldots, K,
{\mathrm{e}}nd{eqnarray*}
has largest eigenvalue $\rho(\mathbf{M})\leq$ 1.
{\mathrm{e}}nd{corollary}
\noindent Similarly, a corollary to Theorem \ref{thmHA2} is stated below.
\begin{corollary}
For the busy period $T$, $\mathbb{E} T < \infty$ if and only if $\rho < 1$.
{\mathrm{e}}nd{corollary}
\subsection{Further applications}
Let $B_{i;z}$ denote the length of the busy period initiated by a class $i$ customer with
service requirement $z$ ($B_i$ is that of the standard busy period initiated by a class $i$ customer, that is, taking
$z=S_i$). Let further
\begin{equation}\label{12.5d}\tau_j =\ \mathbb{E}_i\sum_{n=0}^\infty \sum_{m=1}^{Z_n^{(j)}}S_j(m;n)
{\mathrm{e}}nd{equation}
be the expected total time in $[0,B_j)$ where the customer being served
is of class $i$. As before, $\mathbf{G}$ is the diagonal matrix with the $\mu_i$ on the diagonal.
\begin{lemma}\label{Lemma22.7a} \textcolor{black}}\def\tcrr{\textcolor{black}{Assume $\rho<1$.} Then:\\
{\rm (i)} $(\mathbb{E}_i\mathbf{\tau}_j)_{i,j=1,\ldots,K}\ =\ (\mathbf{I}-\mathbf{M})^{-1}\mathbf{G}^{-1}\,; $ {\rm (ii)} $\mathbb{E} B_i\ =\ \mathbf{e}^\top_i(\mathbf{I}-\mathbf{M})^{-1}\mathbf{G}^{-1}\mathbf{e}\,;$\\
{\rm (iii)} $\mathbb{E} B_{i;z}\ =\ z\beta_i$ where $\beta_i\, =\,\mathbf{e}^\top_i\mathbf{\Lambda}(\mathbf{I}-\mathbf{M})^{-1}\mathbf{G}^{-1}\mathbf{e}\,;$\\
{\rm (iv)} $ B_{i;z}/z\to\beta_i$ in probability as $z\to\infty$.
{\mathrm{e}}nd{lemma}
\begin{proof} (i) follows immediately since the $ij$ element of $(\mathbf{I}-\mathbf{M})^{-1}\mathbf{G}^{-1}$ is
\[\sum_{n=0}^\infty M^n_{ij}/\mu_j\ =\ \mathbb{E}_i\sum_{n=0}^\infty Z_n^{(j)}/\mu_j\ =\ \mathbb{E}\tau_i,\]
and (ii) follows from (i) by summing over $j$. For (iii) and (iv), we may (by work conservation) assume
that the discipline is preemptive-resume. The workload process during service of a class $i$ customer
evolves as a standard compound Poisson process with arrival rate $\bar{\lambda}^i = \sum_{j=1}^N \lambda_{ij}$
and with cumulative distribution function
\[\sum_{j=1}^K\frac{\lambda_{ij}}{\bar{\lambda}^i}\mathbb{P}(B_j\le x)\]
for the jumps. For this system, the rate of arriving work is
$\bar{\lambda}^i \sum_{j=1}^K\lambda_{ij}/\bar{\lambda}^i\, \mathbb{E} B_j$, which is the same as $\beta_i$.
Now we may simply appeal to standard compound Poisson results to obtain (iii) and (iv). This concludes the proof.
{\mathrm{e}}nd{proof}
\section{Busy period results} \label{secBP}\setcounter{equation}{0}
\textcolor{black}}\def\tcrr{\textcolor{black}{In this section, we begin by assuming $\rho(M)\leq 1$.}
Let $B_\mathbf{x}$ denote the busy period when the system starts from the state $\mathbf{x} \in \mathbb{Z}^K_+$, that is, the time period until the system becomes empty. In particular, when $\mathbf{x}$ consists of a single customer of class $i$,
we denote the busy period as $B_i$, and $B_{i,s}$ is the busy period when his remaining service is $s$.
Define $g_\mathbf{x}$ to be the LST of $B_\mathbf{x}$, i.e.,
$g_\mathbf{x}(\theta) = \mathbb{E}_\mathbf{x}[{\mathrm{e}}^{-\theta B_\mathbf{x}}]$ for $\mathbf{x} \in \mathbb{Z}^K_+$ and similarly for $g_i,g_{i,s}$.
\subsection{The busy period Laplace transform} \label{sec6}
For the $M/G/1$ queue, when $K=1$, it is well known that the LST of the busy period $B$ is given by
\begin{equation}
g(\theta) = \psi(\theta+ \lambda - \lambda g(\theta)),
{\mathrm{e}}nd{equation}
where $\psi$ is the LST of the service time and $\lambda$ is the arrival rate. See, for example, Neuts~\cite{Neuts} or Wolff~\cite{Wolff}.
We shall use the branching process connection to derive a similar fixed point equation for our model.
We first observe that the busy period of the system corresponding to an arbitrary initial state $\mathbf{x} = (x_1,\dots,x_K) \in \mathbb{Z}^K_+$ is the independent sum of busy periods, each of which corresponds to the branching process starting with a single customer.
This gives immediately that
\begin{equation}\label{19.1a} g_\mathbf{x}(\theta)\ =\ g_1^{x_1}(\theta)\cdots g_K^{x_K}(\theta)\quad\text{when }
\mathbf{x} = (x_1,\dots,x_K) \in \mathbb{Z}^K_+
{\mathrm{e}}nd{equation}
Hence, it is sufficient to calculate $g_i,g_{i,s}$. Recall
that $\psi_i$ is the LST of the service time distribution $F_i$ of a class $i$ customer.
\begin{theorem} \label{thm-LT}
For $\theta\ge 0$,
\begin{equation}\label{19.1b}
g_{i,s}(\theta)\ =\ {\rm exp} \Bigl\{-s\Bigl(\theta+\bar{\lambda}^i-\sum_{j=1}^K\lambda_{ij} g_j(\theta)\Bigr)\Bigr\}.
{\mathrm{e}}nd{equation}
Further,
\begin{equation}\label{19.1c}
g_i(\theta)\ =\ \psi_i \Bigl(\theta+\bar{\lambda}^i-\sum_{j=1}^K\lambda_{ij} g_j(\theta)\Bigr)\,,\quad i=1,\ldots,K,
{\mathrm{e}}nd{equation}
and the vector
$\bigl( g_1(\theta),\ldots g_K(\theta)\bigr)$ is the minimal non-negative and non-increasing solution of \textcolor{black}}\def\tcrr{\textcolor{black}{this system of equations.}
{\mathrm{e}}nd{theorem}
\begin{proof}
Clearly, $B_{i,s}$ is the service time $s$ plus the busy periods of all customers arriving during service. But the number
of such customers of class $j$ \textcolor{black}}\def\tcrr{\textcolor{black}{is Poisson$( \lambda_{ij} s)$ and} so their busy periods add up to a compound Poisson random variable
with LST ${\rm exp}\bigl\{\lambda_{ij} s \bigl(g_j(\theta)-1\bigr)\bigr\}$. The independence for different $j$ then gives
\[g_{i,s}(\theta)\ =\ {\mathrm{e}}^{-\theta s}\prod_{j=1}^K {\rm exp}\bigl\{\lambda_{ij}s\bigl(g_j(\theta)-1\bigr)\bigr\},\]
which is the same as {\mathrm{e}}qref{19.1b}. Integrating with respect to \ $ F_i({\mathrm{d}} s)$ then gives {\mathrm{e}}qref{19.1c}.
Now consider another non-negative solution $\bigl(\widetilde g_1(\theta),\ldots,\widetilde g_K(\theta)\bigr)$ of {\mathrm{e}}qref{19.1c}.
Define the depth $D$
of the multitype Galton-Watson family tree as
$D\,=\, \max\bigl\{n\ge 0:\,\mathbf{Z}_n\ne\mathbf{z}ero \bigr\}$ and let $g_i^{(n)}(\theta)=\mathbb{E}[{\mathrm{e}}^{-\theta B_i};\,D\le n]$.
Here $D=0$ means no arrivals during service. This occurs with probability ${\mathrm{e}}^{-\bar{\lambda}^iS_i}$ given $S_i$, and so
$g_i^{(0)}(\theta)=\psi_i(\theta+\bar{\lambda}^i)$. The assumptions on $\widetilde g_j(\theta)$ then gives
$\widetilde g_i(\theta)\ge g_i^{(0)}(\theta)$. Further, the same reasoning as that leading to {\mathrm{e}}qref{19.1c} gives
\[ g_i^{(n+1)}(\theta)\ =\ \psi_i \Bigl(\theta+\bar{\lambda}^i-\sum_{j=1}^K\lambda_{ij} g_j^{(n)}(\theta)\Bigr)\,.\]
By induction starting from $g_i^{(0)}(\theta)\le \widetilde g_i(\theta)$ we then get $g_i^{(n)}(\theta)\le \widetilde g_i(\theta)$
for all $n$. The proof is completed by observing that $\rho(\mathbf{M})\le 1$ implies $D<\infty$ and hence
$g_i^{(n)}(\theta)\uparrow g_i(\theta)$.
{\mathrm{e}}nd{proof}
\begin{example}
Consider a network with $K=2$ users, $\lambda_{11}=\lambda_{22}=0$ and $F_i$ exponential$(\mu_i)$.
Then {\mathrm{e}}qref{19.1c} has the form
\[ g_1\ =\ \frac{\mu_1}{\mu_1+\theta+\lambda_{12}-\lambda_{12}g_2}\,,\quad
g_2\ =\ \frac{\mu_2}{\mu_2+\theta+\lambda_{21}-\lambda_{21}g_1}\]
where for brevity $g_i$ means $g_i(\theta)$.
This gives
\[ g_1\ =\ \frac{
\mu_1 \lambda_{21} - \mu_2 \lambda_{12} + (\mu_1 + \theta + \lambda_{12} )(\mu_2 + \theta + \lambda_{21} ) - \sqrt{\Delta}
}{2 \lambda_{21} (\mu_1 + \theta + \lambda_{12} ) }, \]
\[ g_2\ =\ \frac{
- \mu_1 \lambda_{21} + \mu_2 \lambda_{12} + (\mu_1 + \theta + \lambda_{12} )(\mu_2 + \theta + \lambda_{21} ) - \sqrt{\Delta}
}{2 \lambda_{12} (\mu_2 + \theta + \lambda_{21} ) }, \]
where
\[
\Delta = [ \mu_1 \mu_2 + \lambda_{12} \lambda_{21} + \theta^2 + \theta(\mu_1 + \mu_2 + \lambda_{12} + \lambda_{21} ) ]^2 - 4 \mu_1 \mu_2 \lambda_{12} \lambda_{21} .
\]
{\mathrm{e}}nd{example}
\subsection{Busy period asymptotics}\label{sec7}
In this section, we offer some observations on the tail asymptotics
of the busy period in the case of heavy-tailed
service time distributions. For light-tailed service time distributions, we refer the reader to the recent work of Palmowski and Rolski~\cite{Rolski}.
For the current case of heavy tails, we refer the reader to Zwart~\cite{Bert}, Jelenkovi\'c and Momcilovi\'c~\cite{Predrag}
and Denisov and Shneer~\cite{DenisSeva}.
The key idea in both Jelenkovi\'c and Momcilovi\'c \cite{Predrag} and in Zwart \cite{Bert} (as in many other instances of heavy-tailed behavior) is the principle of {\mathrm{e}}mph{one big jump}.
For busy periods, this leads us to expect a large busy period to occur as consequence of one large service time. For concreteness, consider
the standard $M/G/1$ queue with \textcolor{black}}\def\tcrr{\textcolor{black}{$\rho<1$} and suppose there is a single large service time of size $S=z$. The workload
after the large jump is $u+z$ for some small or moderate $u$. The workload then decreases at the rate $1-\rho$
until it reaches 0 and the busy period terminates. By the Law of Large Numbers (LLN), the time of termination is approximately $(z+u)/(1-\rho)$. Since the time before the big jump can be neglected, we have $B>x$ if and only if $z>(1-\rho)x$. Both Asmussen~\cite{SA98} and Foss and Zachary~\cite{FossZ} show the probability of this large jump is
\textcolor{black}}\def\tcrr{\textcolor{black}{asymptotically equal to $\mathbb{P}\bigl(S>(1-\rho)x\bigr) \mathbb{E}\sigma$ for large $x$, where $\sigma$ is the number} of customers served in a busy period.
\textcolor{black}}\def\tcrr{\textcolor{black}{But $\mathbb{E}\sigma= \sum_{n=0}^\infty \rho^n = 1/(1-\rho)$. Indeed, 1 corresponds
to the customer initiating the busy period, $\rho$ is the number of customers arriving while he is in service (the first generation), $\rho^2$
is the number of customers arriving while they are in service,
and so forth. In the framework of branching processes, $\rho^n$ is the number of individuals in the $n$th generation. These considerations} lead to
\begin{align}
\label{22.7e} \mathbb{P}(B>x)\ &\sim\ \frac{1}{1-\rho} \mathbb{P}\bigl(S>(1-\rho)x\bigr),
{\mathrm{e}}nd{align}
which Jelenkovi\'c and Momcilovi\'c \cite{Predrag} shows to be the correct asymptotics if the service time distribution is subexponential and
square root insensitive, i.e.\ with a heavier tail than ${\mathrm{e}}^{-\sqrt{x}}$.
Generalizing this approach to our multiclass system, we recall that
$\beta_i\, =\,\sum_{j=1}^K\lambda_{ij}\mathbb{E} B_j$ and we
introduce a subexponential and
square root insensitive reference distribution $F$ for which
the individual service time distributions are related as
\begin{equation}\label{0509a}
{\overline F}_i\bigl(x/(1+\beta_i)\bigr)\ \sim\ c_i{\overline F}(x).
{\mathrm{e}}nd{equation}
In practice, one chooses ${\overline F}(x)$ as $\sup_i{\overline F}_i\bigl(x/(1+\beta_i)\bigr)$.
This is common in heavy-tailed studies involving distributions with different degrees of heavy-tailedness. In particular, it allows some $F_j$ to be light-tailed ($c_j=0$). \\
\indent Recalling the interpretation of $\beta_i$ as the rate of arriving work while a class $i$ customer is in service,
a big service time $S_i$ of a class $i$ customer will lead to $B_i>x$ precisely when $S_i(1+\beta_i)>x$.
Using
the same reasoning as for~{\mathrm{e}}qref{22.7e}, we first note that $(\mathbf{M}^n)_{ij}$ is the number of type $j$ progeny of a type $i$ ancestor. Hence if $\rho(\mathbf{M}) < 1$, the probability that one of these large service times occur in $[0,B_i)$ is approximately
\begin{align*}\sum_{n=0}^\infty \sum_{j=1}^K (\mathbf{M}^n)_{ij}{\overline F}_j\bigl(x/(1+\beta_j)\bigr)\ \sim\ d_i{\overline F}(x),\\ \intertext{where}\
d_i\,=\, \sum_{n=0}^\infty \sum_{j=1}^K (\mathbf{M}^n)_{ij}c_j
\,=\, \sum_{j=1}^K (\mathbf{I}-\mathbf{M})_{ij}^{-1}c_j. {\mathrm{e}}nd{align*}
\textcolor{black}}\def\tcrr{\textcolor{black}r{Equivalently, the $d_i$ solve
\begin{align}\label{30.7b} d_i\ &=\ c_i+\sum_{j=1}^K m_{ij}d_j\,.
{\mathrm{e}}nd{align}}
As for the standard $M/G/1$ queue, it is straightforward to verify that this is an asymptotic lower bound.
\begin{proposition}\label{Prop:0509a}
Assume that $F$ in {\mathrm{e}}qref{0509a} is subexponential with finite mean \textcolor{black}{so that} $c_k>0$ for some $k$ and \textcolor{black}}\def\tcrr{\textcolor{black}{$\rho(\mathbf{M}) < 1$.} Then
for each $i=1,\ldots,K$,
\begin{equation}\label{0509b}
\liminf_{x\to\infty}\frac{\mathbb{P}(B_i>x)}{{\overline F}(x)}\ \ge\ d_i.
{\mathrm{e}}nd{equation}
{\mathrm{e}}nd{proposition}
\begin{remark} \rm \textcolor{black}}\def\tcrr{\textcolor{black}r{Square root insensitivity of $F$ is not needed for Proposition~\ref{Prop:0509a}.}
The assumption
\begin{equation}\label{0509avar}
{\overline F}_i(x) \sim\ \widetilde c_i{\overline F}_0(x)
{\mathrm{e}}nd{equation}
may apriori be more appealing than {\mathrm{e}}qref{0509a} since it does not involve evaluation of the $\beta_i$.
However, it is closely related. The reason is that if $F$ is regularly varying with ${\overline F}(x)=L(x)/x^\alpha$, then
{\mathrm{e}}qref{0509a} and {\mathrm{e}}qref{0509avar} with $F_0=F$ are equivalent, with the constants related by
$c_i=\widetilde c_i(1+\beta_i)^\alpha$. For $F_0$ lognormal or Weibull with tail ${\mathrm{e}}^{-x^\delta}$
(where $\delta<1/2$ in the square root insensitive case), one has, for $\gamma_1>\gamma_2$, ${\overline F}_0(\gamma_1x)=o\bigl({\overline F}_0(\gamma_2x)\bigr)$. Hence if {\mathrm{e}}qref{0509avar} holds, we may define $\beta^*=\max_1^K\beta_j$
and take ${\overline F}(x)={\overline F}_0\bigl(x/(1+\beta^*)\bigr)$, where $c_j=1$ if $\beta_j=\beta^*$
and $c_j=0$ if $\beta_j<\beta^*$.
{\mathrm{e}}nd{remark}
\newcommand{{\mathrm{e}}qdistr}{\stackrel{{\footnotesize \cal D}}{=}}
\indent The $M/G/1$ literature leads to the conjecture that further contributions
to $\mathbb{P}(B_i>x)$ can be neglected, i.e.\ that
$\mathbb{P}(B_i>x)\sim d_i{\overline F}(x)$ \textcolor{black}}\def\tcrr{\textcolor{black}r{in the square-root insensitive case.}
However, the upper bound \textcolor{black}}\def\tcrr{\textcolor{black}r{is much more difficult (even in the single-class $M/G/1$ setting)}
and follows
in the regular varying case from more general
results recently established in Asmussen \& Foss~\cite{SASF17}:
\begin{theorem}\label{Th:29.7a}
Assume that in addition to the conditions of Proposition~\ref{Prop:0509a} that $F$ is regularly
varying. Then $\mathbb{P}(B_i>x)\sim d_i {\overline F}(x)$
for each $i=1,\ldots,K$.
{\mathrm{e}}nd{theorem}
In the proof, we need:
\begin{lemma}\label{L:30.7a} Let $S$ be subexponential and let the conditional distribution of $N$
given $S=s$ be \textcolor{black}{Poisson$(\lambda s)$}. Then $\mathbb{P}(S+N>x)\sim\mathbb{P}\bigl(S(1+\lambda)>x\bigr)$ \textcolor{black}}\def\tcrr{\textcolor{black}r{as $x\to\infty$}.
Further, the conditional distribution of $(S,N)/(S+N)$ given $S+N>x$ converges to the one-point distribution
at $((1/1+\lambda),\lambda/(1+\lambda))$.
{\mathrm{e}}nd{lemma}
\begin{proof} The argument is standard, with the key intuition being that the variation in $S$ dominates that of
the Poisson distribution,
so that $N$ can be replaced by its conditional expectation $\lambda S$ given $S$.
Firstly, note that if $x$ is so large that
$x-x^{1/2}>2\lambda x^{1/2}$ and $N(x^{1/2})$ is Poisson$(\lambda x^{1/2})$, then
\begin{align*}\mathbb{P}(S+N>x, S<x^{1/2})\ &\le\
\mathbb{P}\bigl(x^{1/2}+N(x^{1/2})>x\bigr)\ \le\ \mathbb{P}\bigl(N(x^{1/2})>
2\lambda x^{1/2}\bigr),
{\mathrm{e}}nd{align*}
which (by large deviations theory) tends to zero \textcolor{black}}\def\tcrr{\textcolor{black}r{faster than ${\mathrm{e}}^{-\delta x^{1/2}}$ for some $\delta>0$}, and hence faster than
\textcolor{black}{ $\mathbb{P}\bigl(S(1+\lambda) > x\bigr)$ }. Secondly, $N/S\to \lambda$ as $y\to\infty$ given $S>y$ and so
\begin{align*}\mathbb{P}(S+N>x, S\ge x^{1/2})\ &\sim\ \mathbb{P}\bigl(S(1+\lambda)>x, \textcolor{black}{ S\ge x^{1/2} \bigr)},
{\mathrm{e}}nd{align*}
the latter equaling $ \mathbb{P}\bigl(S(1+\lambda)>x\bigr)$ for large $x$. This proves the first statement,
and the second follows since (asymptotically) only large values of $S$ contribute to large values
of $S+N$, and in this regime $N/S\sim\lambda$.
{\mathrm{e}}nd{proof}
The set-up of ~\cite{SASF17} is a set of random variables $(B_1,\ldots,B_K)$ satisfying
\begin{equation}\label{AF30.6a}B_i \ {\mathrm{e}}qdistr\ S_i+ \sum_{j=1}^K\sum_{m=1}^{N_{j;i}}B_{m;\textcolor{black}}\def\tcrr{\textcolor{black}r{j}}.{\mathrm{e}}nd{equation}
The assumptions for {\mathrm{e}}qref{AF30.6a} are that all $B_{m;j}$ are independent of
the vector
$(S_i,N_{1;i},\ldots,N_{K;i})$, that they are mutually independent, and that $B_{m;j}{\mathrm{e}}qdistr B_j$.
Further, all random variables are non-negative. In our multiclass queue, $B_i$ is the length of the busy period initiated
by a class $i$ customer, $S_i$ is the service time, and $N_{j;i}$ is the number of class $j$ customers
arriving during his service. In the following, we omit the index $i$ and instead express the dependence
on $i$ in terms of a governing probability measure $\mathbb{P}_i$.
\begin{proof} {\mathrm{e}}mph{of Theorem~\ref{Th:29.7a}} \ \
To apply the results of \cite{SASF17}, we first need to verify a condition on multivariate regular
variation (see~\cite{Elephant} for background) of the vector $(S,N_1,\ldots,N_K)$. Its first part is that
$\mathbb{P}(S+N_1+\cdots+N_K>x)$ $\sim b_i{\overline F}(x)$ for some $b_i$. This is immediate from
Lemma~\ref{L:30.7a} by taking $N=N_1+\cdots+N_K$, $\lambda=\overline\lambda_i$,
$b_i=\widetilde c_i(1+\overline\lambda_i)^\alpha$. A minor extension
of the proof of Lemma~\ref{L:30.7a} further yields that, given $S+N_1+\cdots+N_K>x$,
\begin{eqnarray} \label{eqangular}
\frac{1}{S+N_1+\cdots+N_K}\bigl(S, N_1, \ldots ,N_K)\ \to
\frac{1}{1+\overline\lambda_i}\bigl(1,\lambda_{i1}, \ldots , \lambda_{iK}\bigr),
{\mathrm{e}}nd{eqnarray}
where the limit is taken as $x \rightarrow \infty$.
This establishes the second part, namely the existence
of the so-called angular measure (in this case a one-point distribution at the right-hand side of (\ref{eqangular})).
It now follows from~\cite{SASF17} that $\mathbb{P}(B_i>x)\sim d_i^* {\overline F}(x)$, where the $d_i^*$ solve
the set of linear equations
\begin{align}\label{30.7a} d_i^*\ &=\ c_i^*+\sum_{j=1}^K m_{ij}d_j,{\mathrm{e}}nd{align}
and
\[c_i^*\ =\ \lim_{x\to\infty}\frac{1}{{\overline F}(x)}\mathbb{P}_i(S+N_1\overline r_1+\cdots+N_K\overline r_K>x)\quad
\text{with } \overline r_j=\mathbb{E}_j \textcolor{black}}\def\tcrr{\textcolor{black}r{B}.
\]
Comparing with {\mathrm{e}}qref{30.7b}, we see that we need only check that $c_i^*=c_i$.
But by similar arguments to those above,
\begin{align*}
&\mathbb{P}_i(S+N_1\overline r_1+\cdots+N_K\overline r_K>x)\ \sim\
\textcolor{black}{ \mathbb{P}(S(1+\lambda_{i1}\overline r_1+\cdots+\lambda_{iK} \overline r_K ) >x) } \\ &=\
\mathbb{P}(S(1+\beta_i)>x)\ \sim\ \widetilde c_i(1+\beta_i)^\alpha{\overline F}(x)\ =\ c_i{\overline F}(x),{\mathrm{e}}nd{align*}
where \textcolor{black}}\def\tcrr{\textcolor{black}r{parts (ii) and (iii) of} Lemma~\ref{Lemma22.7a} are employed in the second step.
{\mathrm{e}}nd{proof}
\begin{remark} \rm
The general subexponential case seems much more difficult. One obstacle is that
theory and applications of multivariate subexponentiality is much less developed than for
the regular varying case. See, however, Samorodnitsky and Sun~\cite{Genna} for a recent
contribution and for further references.
{\mathrm{e}}nd{remark}
\section{Conclusion} \label{sec8}\setcounter{equation}{0}
We have introduced a multiclass single-server queueing server in which the arrival rates depend on the current job in service. The model departs from existing state-dependent models in the literature in which the parameters depend primarily on the number of jobs in the system rather than the job in service. \\
\indent The main contributions of this paper can be summarized as follows. Firstly, we formulate the multiclass queueing model and its corresponding fluid model, and provide motivation for its practical importance. The necessary and sufficient conditions for stability of the queueing system are obtained via the corresponding fluid model. Secondly, by appealing to the natural connection with multitype Galton-Watson processes, we utilize Laplace-Stieltjes transforms to characterize the busy period of the queueing system. Thirdly, we
present \textcolor{black}}\def\tcrr{\textcolor{black}r{ a preliminary study of} busy period tail asymptotics for heavy-tailed service time distributions
\textcolor{black}}\def\tcrr{\textcolor{black}r{ and give a complete set of results for the regularly varying case,
using recent results of Asmussen \& Foss~\cite{SASF17}}. Tail asymptotics in our multiclass setting for non-regularly varying heavy-tailed service time distributions, as well as for light-tailed service time distributions, are much more difficult and will be attempted in a separate manuscript.\\
\noindent \textbf{Acknowledgments}
We are very grateful to a referee for pointing out a problem in our initial proof of the upper bound in Section~\ref{sec7}. We also thank a second referee \textcolor{black}}\def\tcrr{\textcolor{black}r{and an associate editor} for many useful suggestions.
The first author thanks Dr. Quan Zhou and Professor Guodong Pang for helpful conversations.
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{\mathrm{e}}nd{thebibliography}
{\mathrm{e}}nd{document}
\section{Comments from S\o ren, March 1}
\subsection{Notation}
I am not too keen about $\mathbb{E}_i^j$ for the counting process
in Section 3 -- too close
to the expectation operator. What about ${\cal E}_i^j$?
The way we use bold typeface for vectors is not consistent. E.g.\ in Section 3,
the vector $\mathbf{w}$ is bold, the vector $\mathbf{e}$ of ones is not bold, but
becomes so in the branching part.
Similarly, we may want to use bold for $\lambda_j$
in Section 3 (my first attempt to do a macro did not work). Also, the transpose is denoted like $e'$ in Section 3, but $\mathbf{e}^\mbox{\bf 0}t$ later.
\subsection{Arrival mechanism}
John's new paragraph revealed a point to me that I did not get before, that the
fluid part may include non-Poisson arrivals. I think we then need to make it more clear
what the model then is --- it is not obvious how renewal arrivals would be
covered. Here is one way to do it: at the beginning of Section 3, insert:
\\[2mm]
BEGIN\\
Customers of class $j$ arrive according to a general
point process ${\cal F}^j$ with rate
$\theta_j(t)>\max_i\lambda_{ij}$ at time $t$, for example a renewal process, and are admitted to the system with retention probability $\lambda_{ij}/\theta_j(t)$ when they meet a class $i$
customer in service. It is is either assumed that ${\cal F}=({\cal F}^1,\ldots,{\cal F}^K)$ stationary and ergodic (then $\theta_j(t){\mathrm{e}}quiv\theta_j$) or that these properties hold in as asymptotic sense
ensuring the fluid limits XXX below to exist.\\END\\[2mm]
I am not quite sure what precisely XXX should contain, but John should have an
opinion on this. BTW, I don't follow what $V_i$ in (3.15) is.
{\mathrm{e}}nd{thebibliography}
{\mathrm{e}}nd{document} |
\betagin{document}
\titlerunning{Coderivative-Based Semi-Newton Method in Nonsmooth Difference Programming}
\title{Coderivative-Based Semi-Newton Method\\ in Nonsmooth Difference Programming
\thetaanks{Research of the first author was partially supported by the Ministry of Science, Innovation and Universities of Spain and the European Regional Development Fund (ERDF) of the European Commission, Grant PGC2018-097960-B-C22, and by the Generalitat Valenciana, grant AICO/2021/165. Research of the second author was partially supported by the USA National Science Foundation under grants DMS-1808978 and DMS-2204519, by the Australian Research Council under Discovery Project DP-190100555, and by the Project 111 of China under grant D21024. Research of the third author was partially supported by grants: Fondecyt Regular 1190110 and Fondecyt Regular 1200283.}}
\subtitle{}
\author{Francisco J. Arag\'{o}n-Artacho \and \mbox{Boris S. Mordukhovich} \and \mbox{Pedro P\'erez-Aros}}
\institute{Francisco J. Arag\'{o}n-Artacho \at Department of Mathematics, University of Alicante, Alicante, Spain\\ \email{[email protected]} \and
Boris S. Mordukhovich \at Department of Mathematics, Wayne State University, Detroit, Michigan 48202, USA\\ \email{[email protected]}
\and Pedro P\'erez-Aros \at Instituto de Ciencias de la Ingenier\'ia, Universidad de O'Higgins, Rancagua, Chile\\
\email{[email protected]}}
\date{\today}
\maketitle
\betagin{abstract}
This paper addresses the study of a new class of nonsmooth optimization problems, where the objective is represented as a difference of two generally nonconvex functions. We propose and develop a novel Newton-type algorithm to solving such problems, which is based on the coderivative generated second-order
subdifferential (generalized Hessian) and employs advanced tools of variational analysis. Well-posedness properties of the proposed algorithm are derived under fairly general requirements, while constructive convergence rates are established by using additional assumptions including the Kurdyka--{\L}ojasiewicz condition. We provide applications of the main algorithm to solving a general class of nonsmooth nonconvex problems of structured optimization that encompasses, in particular, optimization problems with explicit constraints. Finally, applications and numerical experiments are given for solving practical problems that arise in biochemical models, constrained quadratic programming, etc., where advantages of our algorithms are demonstrated in comparison with some known techniques and results.
\end{abstract}\vspace*{-0.05in}
\keywords{Nonsmooth difference programming \and generalized Newton methods \and global convergence \and convergence rates \and variational analysis \and generalized differentiation }\vspace*{-0.05in}
\subclass{49J53, 90C15, 9J52}\vspace*{-0.2in}
\section{Introduction}\langlebel{intro}\vspace*{-0.05in}
The primary mathematical model considered in this paper is described by
\betagin{equation}\langlebel{EQ01}
\min_{x\in \mathbb{R}^n} \varphi(x):=g(x)-h(x),
\end{equation}
where $g:\mathbb{R}^n \to \mathbb{R}$ is of {\em class $\mathcal{C}^{1,1}$} (i.e., the collection of $\mathcal{C}^1$-smooth functions with locally Lipschitzian derivatives), and where $h: \mathbb{R}^n \to \mathbb{R}$ is a locally Lipschitzian and {\em prox-regular} function; see below. Although \eqref{EQ01} is a problem of unconstrained optimization, it will be shown below that a large class of constrained optimization problems can be reduced to this form. In what follows, we label the optimization class in \eqref{EQ01} as problems of {\em difference programming}.
The difference form \eqref{EQ01} reminds us of problems of {\em DC $($difference of convex$)$ programming}, which have been intensively studied in optimization with a variety of practical applications; see, e.g., \cite{Aragon2020,Artacho2019,AragonArtacho2018,Oliveira_2020,hiriart,Toh,Tao1997,Tao1998,Tao1986} and the references therein. However, we are not familiar with a systematic study of the class of difference programming problems considered in this paper.
Our main goal here is to develop an efficient numerical algorithm to solve the class of difference programs \eqref{EQ01} with subsequent applications to nonsmooth and nonconvex problems of particular structures, problems with geometric constraints, etc. Furthermore, the efficiency of the proposed algorithm and its modifications is demonstrated by solving some practical models for which we conduct numerical experiments and compare the obtained results with previously known developments and computations by using other algorithms.
The proposed algorithm is of a {\em regularized damped Newton type} with a {\em novel choice of directions} in the iterative scheme providing a {\em global convergence} of iterates to a stationary point of the cost function. At the {\em first order}, the novelty of our algorithm, in comparison with, e.g., the most popular {\em DCA algorithm} by Tao et al. \cite{Tao1997,Tao1998,Tao1986} and its {\em boosted} developments by Arag\'on-Artacho et al.\cite{Aragon2020,Artacho2019,AragonArtacho2018,MR4078808}
in DC programming, is that instead of a convex subgradient of $h$ in \eqref{EQ01}, we now use a {\em limiting subgradient} of $-h$. No second-order information on $h$ is used in what follows. Concerning the other function $g$ in \eqref{EQ01}, which is nonsmooth of the {\em second-order}, our algorithm replaces the classical Hessian matrix by the {\em generalized Hessian/second-order subdifferential} of $g$ in the sense of Mordukhovich \cite{m92}. The latter construction, which is defined as the coderivative of the limiting subdifferential has been well recognized in variational analysis and optimization due its comprehensive calculus and explicit evaluations for broad classes of extended-real-valued functions arising in applications. We refer the reader to, e.g., \cite{chhm,dsy,Helmut,hmn,hos,hr,2020arXiv200910551D,MR3823783,MR2191744,mr,os,yy} and the bibliographies therein for more details. Note also that the aforementioned generalized Hessian has already been used in differently designed algorithms of the Newton type to solve
optimization-related problems of different nonsmooth structures in comparison with \eqref{EQ01}; see \cite{Helmut,2020arXiv200910551D,jogo,2021arXiv210902093D,BorisEbrahim}. Having in mind the discussions above, we label the main algorithm developed in this paper as the {\em regularized coderivative-based damped semi-Newton method} (abbr.\ RCSN).
The rest of the paper is organized as follows. Section~\ref{sec:2} recalls constructions and statements from variational analysis and generalized differentiation, which are broadly used in the formulations and proofs of the major results. Besides well-known facts, we present here some new notions and further elaborations.
In Section~\ref{sec:3}, we design our {\em main RCSN algorithm}, discuss each of its steps, and establish various results on its performance depending on imposed assumptions whose role and importance are illustrated by examples. Furthermore, Section~\ref{sec:4} employs the {\em Kurdyka-{\L}ojasiewicz {\rm(KL)} property} of the cost function to establish quantitative convergence rates of the RCSN algorithm depending on the exponent in the KL inequality.
Section~\ref{sec:5} addresses the class of (nonconvex) problems of {\em structured optimization} with the cost functions given in the form $f(x)+\psi(x)$, where $f\mbox{\rm co}\,lon\mathbb{R}^n\to\mathbb{R}$ is a twice continuously differentiable function with a Lipschitzian Hessian (i.e., of class ${\cal C}^{2,1}$), while $\psi\mbox{\rm co}\,lon\mathbb{R}^n\to\Bar{\R}:=(-\infty,\infty]$ is an extended-real-valued prox-bounded function. By using the {\em forward-backward envelope} \cite{MR3845278} and the associated {\em Asplund function} \cite{asplund}, we reduce this class of structured optimization problems to the difference form \eqref{EQ01} and then employ the machinery of RCSN to solving problems of this type. As a particular case of RCSN, we design and justify here a new {\em projected-like Newton algorithm} to solve optimization problems with geometric constraints given by general closed sets.
Section~6 is devoted to implementations of the designed algorithms and {\em numerical experiments} in two different problems arising in practical modeling. Although these problems can be treated after some transformations by DCA-like algorithms, we demonstrate in this section numerical advantages of the newly designed algorithms over the known developments in both smooth and nonsmooth settings. The concluding Section~\ref{sec:7} summarizes the major achievements of the paper and discusses some directions of our future research.\vspace*{-0.4in}
\section{Tools of Variational Analysis and Generalized Differentiation}\langlebel{sec:2}
Throughout the entire paper, we deal with finite-dimensional Euclidean spaces and use the standard notation and terminology of variational analysis and generalized differentiation; see, e.g., \cite{MR3823783,MR1491362}, where the reader can find the majority of the results presented in this section. Recall that
$\mathbb{B}_r(x)$ stands for the closed ball centered at $x\in\mathbb{R}^n$ with radius $r> 0$ and that $\mathbb{N}:=\{1,2,\ldots\}$.
Given a set-valued mapping $F: \mathbb{R}^n \tilde to \mathbb{R}^m$, its {\em graph} is the set $ \operatorname{gph} F:=\big\{ (v,w) \in {\R^n}\times \R^{m}\;|\; w\in F(x)\big\}$, while the (Painlev\'e--Kuratowski) {\em outer limit} of $F$ at $x\in\mathbb{R}^n$ is defined by
\betagin{equation}\langlebel{pk}
\mathop{{\rm Lim}}sup_{u\to x}F(u):=\big\{y\in\mathbb{R}^m\;\big|\;\exists\,u_k\to x,\,y_k\to y,\;y_k\in F(u_k)\;\mbox{as}\;k\in\mathbb{N}\big\}.
\end{equation}
For a nonempty set $C\subseteq\mathbb{R}^n$, the (Fr\'echet) {\em regular normal cone} and (Mordukhovich) {\em basic/limiting normal cone} at $x\in C$ are defined, respectively, by
\betagin{equation}\langlebel{rnc}
\betagin{aligned}
\mathcal{H}at{N}(x;C)=\mathcal{H}at N_C(x):&=\mathbb{B}ig\{ x^*\in \mathbb{R}^n\;\mathbb{B}ig|\;\limsup\limits_{ u \bar{v}erset{C}{\to} x } \mathbb{B}ig\langlengle x^\ast,\frac{u - x}{\| u - x \| }\mathbb{B}ig\ranglengle\leq 0\mathbb{B}ig\},\\
N(x;C)=N_C(x):&=\mathop{{\rm Lim}}sup\limits_{u \bar{v}erset{C}{\to} x}\mathcal{H}at{N}(u;C),
\end{aligned}
\end{equation}
where ``$u\bar{v}erset{C}{\to} x$'' means that $u \to x$ with $u \in C$. We use the convention $\mathcal{H}at N(x;C)=N(x;C):=\emptyset$ if $x\notin C$. The {\em indicator function} $\delta_C(x)$ of $C$ is equal to 0 if $x\in C$ and to $\infty $ otherwise.
For a lower semicontinuous (l.s.c.) function $f:{\R^n}\to\bar{v}erline{\mathbb{R}}$, its {\em domain} and {\em epigraph} are given by $\mbox{\rm dom}\, f := \{ x\in \mathbb{R}^n \mid f(x) < \infty \}$ and $\mbox{\rm epi}\, f:=\{ (x,\alphapha) \in {\R^n}\times \mathbb{R} \;|\;f(x)\leq \alphapha\},$ respectively. The {\em regular} and {\em basic subdifferentials} of $f$ at $x\in\mbox{\rm dom}\, f$ are defined by
\betagin{equation}\langlebel{sub}
\betagin{aligned}
\mathcal{H}at{\partial} f(x)&:=\big\{ x^\ast\in \mathbb{R}^n\mid(x^\ast ,-1) \in \mathcal{H}at{N}\big((x,f(x));\mbox{\rm epi}\, f\big)\big\},\\
\partial f(x) &:= \big\{ x^\ast\in \mathbb{R}^n\mid(x^\ast ,-1) \in N\big((x,f(x));\mbox{\rm epi}\, f\big)\big\},
\end{aligned}
\end{equation}
via the corresponding normal cones \eqref{rnc} to the epigraph. The function $f$ is said to be {\em lower/subdifferentially regular} at $\bar{x}\in\mbox{\rm dom}\, f$ if $\partial f(\bar{x}) =\mathcal{H}at{\partial}f(\bar{x})$.
Given further a set-valued mapping/multifunction $F: {\R^n} \tilde to \R^{m}$,
the {\em regular} and {\em basic coderivatives} of $F$ at $(x,y)\in\operatorname{gph} F$ are defined for all $y^*\in\R^{m}$ via the corresponding normal cones \eqref{rnc} to the graph of $F$, i.e.,
\betagin{equation}\langlebel{cod}
\betagin{aligned}
\mathcal{H}at{D}^\ast F(x,y) (y^\ast)&:=\big\{ x^\ast \in {\R^n}\;\big|\;(x^\ast,-y^\ast) \in \mathcal{H}at{N}\big( (x,y); \operatorname{gph} F\big)\big\},\\
{D}^\ast F(x,y) (y^\ast)&:=\big\{x^\ast \in {\R^n}\;\big|\;(x^\ast,-y^\ast) \in N\big( (x,y); \operatorname{gph} F\big)\big\},
\end{aligned}
\end{equation}
where $y$ is omitted if $F$ is single-valued at $x$. When $F$ is single-valued and locally Lipschitzian around $x$, the basic coderivative has the following representation via the basic subdifferential of the scalarization
\betagin{align}\langlebel{coder:sub}
{D}^\ast F(x) (y^\ast) = \partial \langlengle y^\ast, F\ranglengle (x), \text{ where } \langlengle y^\ast, F\ranglengle (x):= \langlengle y^\ast, F(x)\ranglengle.
\end{align}
Recall that a set-valued mapping $F: \mathbb{R}^n \tilde to \mathbb{R}^m $ is {\em strongly metrically subregular} at $(\bar{x},\bar{y})\in \operatorname{gph} F$ if there exist $\kappa,\epsilon>0$ such that
\betagin{align}\langlebel{def:stron_subreg}
\| x -\bar{x}\| \leq \kappa \| y - \bar{y}\|\; \text{ for all }\;(x,y) \in \mathbb{B}_\epsilon(\bar{x},\bar{y}) \cap\operatorname{gph} F.
\end{align}
It is well-known that this property of $F$ is equivalent to the {\em calmness} property of the inverse mapping $F^{-1}$ at $(\bar{y},\bar{x})$. In what follows, we use the calmness property of single-valued mappings $h\mbox{\rm co}\,lon\mathbb{R}^n\to\mathbb{R}^m$ at $\bar{x}$ meaning that there exist positive numbers $\kappa$ and $\varepsilon>0$ such that
\betagin{equation}\langlebel{calm}
\|h(x)-h(\bar{x})\|\leq\kappa\|x-\bar{x}\|\;\text{ for all }\;x\in\mathbb{B}_\epsilon(\bar{x}).
\end{equation}
The infimum of all $\kappa>0$ in \eqref{calm} is called the \emptyseth{exact calmness bound} of $h$ at $\bar{x}$ and is denoted it by $\mbox{\rm clm}\, h(\bar{x})$. On the other hand, a multifunction $F\mbox{\rm co}\,lon\mathbb{R}^n\tilde to\mathbb{R}^m$ is {\em strongly metrically regular} around $(\bar{x},\bar{y}) \in \operatorname{gph} F$ if its inverse $F^{-1}$ admits a single-valued and Lipschitz continuous localization around this point.
Along with the (first-order) basic subdifferential in \eqref{sub}, we consider the {\em second-order subdifferential/generalized Hessian} of $f: {\R^n} \to \bar{v}erline{\mathbb{R}}$ at $x\in \mbox{\rm dom}\, f$ relative to $x^\ast\in\partial f(x)$ defined by
\betagin{equation}\langlebel{2nd}
\partial^2f(x,x^\ast) (v^\ast) = \left(D^\ast\partial f\right) (x,x^\ast)(v^\ast),\quad v^\ast\in {\R^n}
\end{equation}
and denoted by $\partial^2f(x) (v^\ast)$ when $\partial f(x)$ is a singleton. If $f$ is twice continuously differentiable ($\mathcal{C}^2$-smooth) around $x$, then $\partial^2 f(x)(v^\ast)=\{\nabla^2 f(x)v^\ast\}$.
Next we introduce an extension of the notion of positive-definiteness for multifunctions, where the the corresponding constant may not be positive.\vspace*{-0.1in}
\betagin{definition}\langlebel{def:lower-def}
Let $F: \mathbb{R}^n \tilde to \mathbb{R}^n$ and $\xi\in\mathbb{R}$. Then $F$ is \emptyseth{$\xi$-lower-definite} if
\betagin{align}\langlebel{Stron-semide}
\langlengle y, x\ranglengle \geq \xi\|x\|^2\;\text{ for all }\;(x,y)\in\operatorname{gph} F.
\end{align}
\end{definition}
\betagin{remark}\langlebel{rem:definite}
We can easily check the following:
(i) For any symmetric matrix $Q$ with the smallest eigenvalue $\langlembda_{\min}(Q)$, the function $f(x)=Qx$ is $\langlembda_{\min}(Q)$-lower-definite.
(ii) If a function $f: \mathbb{R}^n \to \bar{v}erline{\mathbb{R}}$ is {\em strongly convex} with modulus $\rho>0$, (i.e., $ f -\frac{\rho }{2}\| \cdot \|^2$ is convex), it follows from \cite[Corollary~5.9]{MR3823783} that $\partial^2 f(x,x^\ast)$ is $\rho$-lower-definite for all $(x,x^\ast)\in\operatorname{gph}\partial f$.
(iii) If $F_1,F_2: \mathbb{R}^n \tilde to \mathbb{R}^n$ are $\xi_1$ and $\xi_2$-lower-definite, then the sum $F_1+F_2$ is $(\xi_1+\xi_2)$-lower-definite.
\end{remark}\vspace*{-0.05in}
Recall next that a function $f: \mathbb{R}^n \to \bar{v}erline{\mathbb{R}}$ is \emptyseth{prox-regular} at $\bar{x}\in\mathbb{R}^n$ \emptyseth{for} $\bar{v} \in \partial f(\bar{x})$ if it is l.s.c.\ around $\bar{x}$ and there exist $\epsilon>0$ and $r\geq 0$ such that
\betagin{align}\langlebel{proregularity}
f(x') \geq f(x) + \langlengle v, x'-x\ranglengle - \frac{r}{2}\| x'- x\|^2
\end{align}
whenever $x,x'\in\mathbb{B}_\epsilon(\bar{x})$ with $ f(x) \leq f(\bar{x}) + \epsilon$ and $v\in \partial f(x) \cap \mathbb{B}_\epsilon(\bar{v})$. If this holds for all $\bar{v} \in \partial f(\bar{x})$, $f$ is said to be \emptyseth{prox-regular at} $\bar x$.\vspace*{-0.05in}
\betagin{remark}
The class of prox-regular functions has been well-recognized in modern variational analysis. It is worth mentioning that if $f$ is a locally Lipschitzian function around $\bar{x}$, then the following properties of $f$ are equivalent: (i) prox-regularity at $\bar{x}$, (ii) lower-$\mathcal{C}^2$ at $\bar{x}$, and (iii) primal-lower-nice at $\bar{x}$; see, e.g., \cite[Corollary 3.12]{MR2101873} for more details.
\end{remark}\vspace*{-0.05in}
Given a function $f: \mathbb{R}^n \to \bar{v}erline{\mathbb{R}}$ and $\bar{x}\in\mbox{\rm dom}\, f$, the {\em upper directional derivative} of $f$ at $\bar{x}$ with respect to $d\in\mathbb{R}^n$ is defined by
\betagin{equation}\langlebel{UDD}
f'(\bar{x};d):=\limsup\limits_{t\to 0^+} \frac{f(\bar{x}+td) - f(\bar{x})}{t}.
\end{equation}
The following proposition establishes various properties of prox-regular functions used below. We denote the convex hull of a set by ``co".
\betagin{proposition}\langlebel{Lemma:Dire01}
Let $f: \mathbb{R}^n \to \bar{v}erline{\mathbb{R}}$ be locally Lipschitzian around $\bar{x}$ and prox-regular at this point. Then $f$ is lower regular at $\bar{x}$, $\mbox{\rm co}\,\partial (-f)(\bar{x}) = -\partial f(\bar{x})$, and for any $d\in\mathbb{R}^n$ we have the representations
\betagin{align} \langlebel{Dire01}
(-f)'(\bar x;d)=\inf\big\{\langlengle w, d\ranglengle\;\big|\;w\in \partial (-f)(\bar x)\big\}= \inf\big\{\langlengle w, d\ranglengle\;\big|\;w\in-\partial f(\bar{x})\big\}.
\end{align}
\end{proposition}
\betagin{proof}
First we fix an arbitrary subgradient $\bar{v} \in \partial f(\bar{x})$
and deduce from \eqref{proregularity} applied to $x=\bar{x}$ and $v=\bar{v}$ that
\betagin{equation*}
f(x') \geq f(\bar{x}) +\langlengle \bar{v}, x'-\bar{x}\ranglengle -\frac{r}{2} \| x' - \bar{x}\|^2\;\text{ for all }\;x'\in\mathbb{B}_\epsilon(\bar{x}).
\end{equation*}
Passing to the limit as $x' \to \bar{x}$ tells us that
\betagin{align*}
\liminf_{x'\to \bar{x}} \frac{ f(x') - f(\bar{x}) - \langlengle \bar{v}, x'-\bar{x}\ranglengle }{ \| x' - \bar{x}\| } \geq 0,
\end{align*}
which means that $\bar{v}\in\mathcal{H}at\partial f(\bar{x})$ and thus shows that
$f$ is lower regular at $\bar{x}$. By the Lipschitz continuity of $f$ around $\bar{x}$ and the convexity of the set $\mathcal{H}at{\partial} f(\bar{x})$, we have that $\mathcal{H}at{\partial} f(\bar{x})=\partial f(\bar{x}) =\mbox{\rm co}\, \partial f(\bar{x}) = \bar{v}erline{\partial } f(\bar{x})$, where $\bar{v}erline{\partial }$ denotes the (Clarke) {\em generalized gradient}. It follows from $ \bar{v}erline{\partial } (-f)(\bar{x}) =-\bar{v}erline{\partial} f(\bar{x})$ that $ \bar{v}erline{\partial } (-f)(\bar{x}) =-\partial f(\bar{x})$, which implies therefore that $\partial (-f)(\bar{x}) \subseteq -\partial f(\bar{x})$.
Pick $v\in \partial (-f)(\bar{x})$, $d\in \mathbb{R}^n$ and find by the prox-regularity of $f$ at $\bar{x}$ for $-v\in\partial f(\bar{x})$ that there exists $r>0$ such that
\betagin{align*}
\langlengle v,d \ranglengle+\frac{rt}{2}\| d\|^2 \geq \frac{-f(\bar x+td) + f(\bar x)}{t}
\end{align*}
if $t>0$ is small enough. This yields $(-f)'(\bar x;d) \leq \langlengle v, d \ranglengle $ for all ${v}\in \partial (-f)(\bar x) $ and thus verifies the inequality ``$\leq $'' in the first representation of \eqref{Dire01}.
To prove the opposite inequality therein, take $t_k \to 0^+$ such that
\betagin{align*}
\lim_{k\to \infty} \frac{ -f(\bar{x} + t_k d ) + f(\bar{x}) }{ t_k}= (-f)'(\bar x;d).
\end{align*}
Employing the mean value theorem from \cite[Corollary~4.12]{MR3823783}) gives us
\betagin{align*}
f(\bar{x} + t_k d ) - f(\bar{x}) = t_k \langlengle v_k,d\ranglengle \text{ for some } v_k \in \partial f(\bar{x} + \langlembda_k t_k d) \text{ with } \langlembda_k \in ( 0,1).
\end{align*}
It follows from the Lipschitz continuity of $f$ that $\{v_k\}$ is bounded,
and so we can assume that $v_k \to \bar{v}\in \partial f(\bar{x})$. Therefore,
\betagin{equation*}
\betagin{array}{ll}
(-f)'(\bar x;d)=\langlengle -\bar{v}, d\ranglengle \geq \inf\big\{\langlengle w, d\ranglengle\;\big|\; w \in -\partial f(\bar{x})\big\} \\
=\inf\big\{\langlengle w, d\ranglengle\;\big|\;w \in \mbox{\rm co}\, \partial (-f)(\bar{x}) \big\}=\inf\big\{\langlengle w, d\ranglengle\;\big|\;w \in \partial (-f)(\bar{x}) \big\},
\end{array}
\end{equation*}
which verifies \eqref{Dire01} and completes the proof of the proposition.
\end{proof}\vspace*{-0.05in}
Next we define the notion of stationary points for problem \eqref{EQ01} the finding of which is the goal of our algorithms.
\betagin{definition}\langlebel{def:stationary}
Let $\varphi=g-h$ be the cost function in \eqref{EQ01}, where $g$ is of class $\mathcal{C}^{1,1}$ around some point $\bar{x}$, and where $h$ is locally Lipschitzian around $\bar{x}$ and prox-regular at this point. Then $\bar{x}$ is a \emptyseth{stationary point} of \eqref{EQ01} if $0\in\partial\varphi(\bar{x})$.
\end{definition}\vspace*{-0.2in}
\betagin{remark}
The stationarity notion $0 \in \partial\varphi(\bar{x})$, expressed via
the limiting subdiffential, is known as the {\em M$($ordukhovich$)$-stationarity}. Since no other stationary points are considered in this paper, we skip ``M" in what follows. Observe from the subdifferential sum rule in our setting that $\bar{x}$ is a stationary point in \eqref{EQ01} if and only if $0\in\nabla g(\bar{x}) +\partial(-h)(\bar{x})$. Thus every stationary point $\bar{x}$ is a {\em critical point} in the sense that $0\in\nabla g(\bar{x}) - \partial h(\bar{x})$. By Proposition~\ref{Lemma:Dire01}, the latter can be equivalently described in terms of the generalized gradient and also via the {\em symmetric subdifferential} \cite{MR3823783} of $\varphi$ at $\bar{x}$ defined by
\betagin{equation}\langlebel{sym}
\partial^0\varphi(\bar{x}):=\partial\varphi(\bar{x})\cup\big(-\partial(-\varphi)(\bar{x})\big)
\end{equation}
which possesses the plus-minus symmetry $\partial^0(-\varphi(\bar{x}))=-\partial^0(\varphi(\bar{x}))$. When both $g$ and $h$ are convex, the classical DC algorithm~\cite{Tao1986,Tao1997} and its BDCA variant \cite{MR4078808} can be applied for solving problem~\eqref{EQ01}. Although these algorithms only converge to critical points, they can be easily combined as in \cite{Aragon2020} with a basic derivative-free optimization scheme to converge to {d-stationary points}, which satisfy $\partial h(\bar{x})=\{\nabla g(\bar{x})\}$ (or, equivalently, $\varphi'(\bar{x};d)=0$ for all $d\in\mathbb{R}^n$; see~\cite[Proposition~1]{Aragon2020}). In the DC setting, every local minimizer of problem~\eqref{EQ01} is a d-stationary point \cite[Theorem~3]{Toland1979}, a property which is stronger than the notion of stationarity in Definition~\ref{def:stationary}.
\end{remark}\vspace*{-0.1in}
To proceed, recall that a mapping $f: U\to \mathbb{R}^m$ defined on an open set $U\subseteq \mathbb{R}^n $ is \emptyseth{semismooth} at $\bar{x}$ if it is locally Lipschitzian around $\bar{x}$, directionally differentiable at this point, and the limit
\betagin{align*}
\lim\limits_{A \in \tiny{\mbox{\rm co}\,} \bar{v}erline{\nabla} f(\bar{x} + t u'), \atop u' \to u, t\to 0^+} A u'
\end{align*}
exists for all $u \in \mathbb{R}^n$, where $\bar{v}erline{\nabla} f(x) :=\{ A\;|\; \exists x_k \bar{v}erset{D}{\to } x \text{ and } \nabla f(x_k) \to A \}$, and where $D$ is the set on which $f$ is differentiable; see \cite{MR1955649,MR3289054} for more details. We say that a function $g: \mathbb{R}^n \to \bar{v}erline{\mathbb{R}}$ is {\em semismoothly differentiable} at $\bar{x}$ if $g$ is $\mathcal{C}^{1}$-smooth around $\bar{x}$ and its gradient mapping $\nabla g$ is semismooth at this point.
Recall further that a function $\psi: \mathbb{R}^n \to \bar{v}erline{\mathbb{R}}$ is \emptyseth{prox-bounded} if there exists $\langlembda>0$ such that $\mathcal{M}oreauYosida{\psi}{\langlembda}(x)>-\infty$ for some $x\in \mathbb{R}^n$, where
$\mathcal{M}oreauYosida{\psi}{\langlembda} :\mathbb{R}^n \to \bar{v}erline{\mathbb{R}}$ is the \emptyseth{Moreau envelope} of $\psi$ with parameter $\langlembda>0$ defined by
\betagin{equation}\langlebel{moreau}
\mathcal{M}oreauYosida{\psi}{\langlembda} (x):=\inf_{ z \in \mathbb{R}^n }\mathbb{B}ig\{ \psi(z) + \frac{1}{2\langlembda} \| x-z\|^2\mathbb{B}ig\}.
\end{equation}
The number $\langlembda_\psi:= \sup \{\langlembda >0\;|\;\mathcal{M}oreauYosida{\psi}{\langlembda}(x)>-\infty \text{ for some } x\in \mathbb{R}^n \}$ is called the \emptyseth{threshold} of prox-boundedness of $\psi$. The corresponding \emptyseth{proximal mapping} is the multifunction $Painlev\'{e}rox{\psi}{\langlembda} : \mathbb{R}^n \tilde to \mathbb{R}^n$ given by
\betagin{equation}\langlebel{prox}
Painlev\'{e}rox{\psi}{\langlembda} (x):= \mathop{\rm argmin}_{z \in \mathbb{R}^n }\mathbb{B}ig\{ \psi(z) + \frac{1}{2\langlembda} \| x-z\|^2\mathbb{B}ig\}.
\end{equation}
Next we observe that that the Moreau envelope can be represented as a {\em DC function}. For any function $\varphi:\mathbb{R}^n\to\bar{v}erline{\mathbb{R}}$, consider its \emptyseth{Fenchel conjugate}
\betagin{equation*}
\varphii^*(x):=\sup_{z\in\mathbb{R}^n}\big\{\langlengle x,z\ranglengle-\varphii(z)\big\},
\end{equation*}
and for any $\psi\mbox{\rm co}\,lon\mathbb{R}^n\to\Bar{\R}$ and $\lambda>0$, define the {\em Asplund function}
\betagin{equation}\langlebel{asp}
\Asp{\langlembda}{\psi}(x):=\sup\limits_{z\in \mathbb{R}^n}\mathbb{B}ig\{ \frac{ 1}{\langlembda} \langlengle z,x\ranglengle - \psi(z) - \frac{1}{ 2\langlembda } \| z\|^2\mathbb{B}ig\}=\mathbb{B}ig(\psi + \frac{1}{ 2\langlembda } \| \cdot\|^2\mathbb{B}ig)^\ast(x),
\end{equation}
which is inspired by Asplund's study of metric projections in \cite{asplund}. The following proposition presents the precise formulation of the aforementioned statement and reveals some remarkable properties of the Asplund function \eqref{asp}.
\betagin{proposition}\langlebel{Lemma5.1}
Let $\psi$ be a prox-bounded function with threshold $\langlembda_\psi$. Then for every $\langlembda \in (0,\langlembda_\psi)$, we have the representation
\betagin{equation}\langlebel{more-asp}
\mathcal{M}oreauYosida{\psi}{\langlembda}(x)=\frac{1}{2\langlembda} \| x\|^2 - \Asp{\langlembda}{\psi}(x),\quad x\in\mathbb{R}^n,
\end{equation}
where the Asplund function is convex and Lipschitz continuous on $\mathbb{R}^n$.
Furthermore, for any $x\in\mathbb{R}^n$ the following subdifferential evaluations hold:
\betagin{align}{ }
\partial(-\Asp{\langlembda}{\psi}) (x) &\subseteq- \frac{1}{\langlembda} Painlev\'{e}rox{\psi}{\langlembda} (x),\langlebel{eq_sub_eq01}\\
\partial \Asp{\langlembda}{\psi}(x) &= \frac{1}{\langlembda} \mbox{\rm co}\,\left( Painlev\'{e}rox{\psi}{\langlembda} (x)\right).\langlebel{eq_sub_eq02}
\end{align}
Moreover, if $v\in Painlev\'{e}rox{\psi}{\langlembda} (x)$ is such that $v\notin \mbox{\rm co}\,\left( Painlev\'{e}rox{\psi}{\langlembda} (x)\backslash \{v\}\right)$, then the vector $-\frac{1}{\langlembda}v$ belongs to $\partial(-\Asp{\langlembda}{\psi}) (x)$.
If in addition $f$ is of class $\mathcal{C}^{2,1}$ on $\mathbb{R}^n$, then the function $x\mapsto \Asp{\langlembda}{\psi}( x - \langlembda \nabla f(x))$ is prox-regular at any point $x\in\mathbb{R}^n$.
\end{proposition}\vspace*{-0.03in}
\betagin{proof}
Representation \eqref{more-asp} easily follows from definitions of the Moreau envelope and Asplund function. Due to the second equality in \eqref{asp}, the Asplund function is convex on $\mathbb{R}^n$. It is also Lipschitz continuous due its finite-valuedness on $\mathbb{R}^n$, which is induced by this property of the Moreau envelope. The subdifferential evaluations in \eqref{eq_sub_eq01} and \cite[Example~10.32]{MR1491362} and the subdifferential sum rule in \cite[Proposition~1.30]{MR3823783}) tell us that
$\partial(-\Asp{\langlembda}{\psi} )(x) = -\langlembda^{-1}x + \partial ( \mathcal{M}oreauYosida{\psi}{\langlembda}) (x)$ and $\partial \Asp{\langlembda}{\psi} (x) = \langlembda^{-1} x + \partial (-\mathcal{M}oreauYosida{\psi}{\langlembda}) (x)$ for any $x\in\mathbb{R}^n$.
Take further $v\in Painlev\'{e}rox{\psi}{\langlembda} (x) $ with $-\frac{1}{\langlembda}v \not\in\partial(-\Asp{\langlembda}{\psi} ) (x)$ and show that $v \in {\rm co}\left( Painlev\'{e}rox{\psi}{\langlembda} (x)\backslash \{v\}\right)$. Indeed, it follows from \eqref{eq_sub_eq01} that
\betagin{align*}
\partial(-\Asp{\langlembda}{\psi} ) (x) &\subseteq- \frac{1}{\langlembda} Painlev\'{e}rox{\psi}{\langlembda} (x) \backslash\{v\}.
\end{align*}
The Lipschitz continuity and convexity of $\Asp{\langlembda}{\psi}$ implies that
\betagin{align}\langlebel{eqLemma26}
{\rm co}\,\partial(-\Asp{\langlembda}{\psi} ) (x) = -\partial\Asp{\langlembda}{\psi} (x)
\end{align}
by \cite[Theorem~3.57]{MR2191744}, which allows us to deduce from \eqref{eq_sub_eq02} and \eqref{eqLemma26} that
\betagin{equation*}
{\rm co}\big(Painlev\'{e}rox{\psi}{\langlembda} (x)\big) = {\rm co}\big( Painlev\'{e}rox{\psi}{\langlembda} (x) \backslash \{v\}\big).
\end{equation*}
This verifies the inclusion $v \in \mbox{\rm co}\,\left( Painlev\'{e}rox{\psi}{\langlembda} (x)\backslash \{v\}\right)$ as claimed.
Observe finally that the function $x\mapsto \Asp{\langlembda}{\psi}_{\langlembda }( x - \langlembda \nabla f(x))$ is the composition of the convex function $ \Asp{\langlembda}{\psi}$ and the $\mathcal{C}^{1,1}$ mapping $x\mapsto x - \langlembda \nabla f(x)$, which ensures by \cite[Proposition~2.3]{MR2069350} its prox-regularity at any point $x\in \mathbb{R}^n$.
\end{proof}\vspace*{-0.05in}
The following remark discusses a useful representation of the basic subdifferential of the function $-\Asp{\langlembda}{\psi}$ and other functions of this type.
\betagin{remark}\langlebel{asp-rem}
It is worth mentioning that the subdifferential $\partial (-\Asp{\langlembda}{\psi})(x)$ can be expressed via the set $D:=\{x\in\mathbb{R}^n\;|\;\Asp{\langlembda}{\psi}\;\mbox{ is differentiable at }\;x\}$ as follows:
\betagin{equation}\langlebel{asp-rem1}
\partial (-\Asp{\langlembda}{\psi})(x) =\big\{ v\in\mathbb{R}^n\;\big|\;\text{ there exists } x_k \bar{v}erset{D}{\to} x \text{ and } \nabla \Asp{\langlembda}{\psi} (x_k) \to -v\big\}.
\end{equation}
We refer to \cite[Theorem~10.31]{MR1491362} for more details. Note that we do not need to take the convex hull on the right-hand side of \eqref{asp-rem1} as in the case of the generalized gradient of locally Lipschitzian functions.
\end{remark}
Finally, recall the definitions of the convergence rates used in the paper.\vspace*{-0.05in}
\betagin{definition}\langlebel{def:rates}
Let $\{x_k\}$ be a sequence in $\mathbb{R}^n$ converging to $\bar{x}$ as $k \rightarrow \infty$. The convergence rate is said to be:
{\bf(i)} \emptyseth{R-linear} if there exist $\mu \in(0,1), c>0$, and $k_0 \in \mathbb{N}$ such that
$$
\left\|x_k-\bar{x}\right\| \leq c \mu^k\;\text { for all }\;k \geq k_0.
$$
{\bf(ii)} \emptyseth{Q-linear} if there exists $\mu\in(0,1)$ such that
$$
\limsup_{k\to\infty}\frac{\left\|x_{k+1}-\bar{x}\right\|}{\left\|x_k-\bar{x}\right\|}=\mu.
$$
{\bf(iii)} \emptyseth{Q-superlinear} if it is Q-linear for all $\mu\in (0,1)$, i.e., if
$$
\lim_{k\to\infty}\frac{\left\|x_{k+1}-\bar{x}\right\|}{\left\|x_k-\bar{x}\right\|}=0.
$$
{\bf(iv)} \emptyseth{Q-quadratic} if we have
$$
\limsup_{k\to\infty}\frac{\left\|x_{k+1}-\bar{x}\right\|}{\left\|x_k-\bar{x}\right\|^2}<\infty.
$$
\end{definition}\vspace*{-0.2in}
\section{Regularized Coderivative-Based Damped Semi-Newton Method in Nonsmooth Difference Programming}\langlebel{sec:3}\vspace*{-0.05in}
The goal of this section is to justify the well-posedness and good performance of the novel algorithm RCSN under appropriate and fairly general assumptions. In the following remark, we discuss the difference between the choice of subgradients and hence of directions in RCSN and DC algorithms.\vspace*{0.05in}
Our main RCSN algorithm to find stationary points of nonsmooth problems \eqref{EQ01} of difference programming is labeled below as Algorithm~\ref{alg:1}.
\betagin{algorithm}[h!]
\betagin{algorithmic}[1]
\mathbb{R}equire{$x_0 \in \mathbb{R}^n$, $\betata \in (0,1)$, $\zeta>0$, $t_{\min}>0 $, $\rho_{\max}>0$ and $\sigma\in(0,1)$.}
Fr\'{e}chetor{$k=0,1,\ldots$}
\mathcal{S}tate Take $w_k\in \partial \varphi (x_k)$. If $ w_k=0$, STOP and return $x_k$.
\mathcal{S}tate Choose $\rho_k\in[0,\rho_{\max}]$ and $d_k \in \mathbb{R}^n\backslash \{ 0\}$ such that
\betagin{align}
-w_k\in \partial^2 g(x_k)(d_k)+\rho_kd_k\quad\text{and}\quad \langlengle w_k,d_k\ranglengle\leq -\zeta\|d_k\|^2. \langlebel{EQALG01}
\end{align}
\mathcal{S}tate Choose any $\bar{v}erline{\tau}_k\geq t_{\min}$. Set $\bar{v}erline{\tau}_k:=\tau_k$.
\mathcal{W}hile{$\varphi(x_k + \tau_k d_k) > \varphi(x_k) +\sigma \tau_k \langlengle w_k , d_k\ranglengle $}
\mathcal{S}tate $\tau_k = \betata \tau_k$.
\EndWhile
\mathcal{S}tate Set $x_{k+1}:=x_k + \tau_kd_k$. \langlebel{step5}
\EndFor
\end{algorithmic}
\caption{Regularized coderivative-based damped semi-Newton algorithm for nonsmooth difference programming}\langlebel{alg:1}
\end{algorithm}\vspace*{-0.05in}
\betagin{remark}\langlebel{rem:subgr}
Observe that Step~2 of Algorithm~\ref{alg:1} selects $w_k\in\partial\varphi(x_k)=\nabla g(x_k)+\partial(-h)(x_k)$, which is equivalent to choosing $v_k:=w_k-\nabla g(x_k)$ in the basic subdifferential of $-h$ at $x_k$. Under our assumptions, the set $\partial(-h)(x_k)$ can be {\em considerably smaller} than $\partial h(x_k)$; see the proof of Proposition~\ref{Lemma:Dire01} and also Remark~\ref{asp-rem} above. Therefore, Step~2 differs from those in DC algorithms, which choose subgradients in $\partial h(x_k)$. The purpose of our development is to find a {\em stationary point} instead of a (classical) critical point for problem~\eqref{EQ01}.
In some applications, Algorithm~\ref{alg:1} would not be implementable if the user only has access to subgradients contained in $\partial h(x_k)$ instead of $\partial(-h)(x_k)$. In such cases, a natural alternative to Algorithm~\ref{alg:1} would be a scheme replacing $w_k\in\partial\varphi(x_k)$ in Step~2 by $w_k:=\nabla g(x_k)+v_k$ with $v_k\in\partial h(x_k)$. Under the setting of our convergence results, the modified algorithm would find a critical point for problem~\eqref{EQ01}, which is not guaranteed to be stationary.
\end{remark}\vspace*{-0.1in}
The above discussions are illustrated by the following example.\vspace*{-0.1in}
\betagin{example}
Consider problem~\eqref{EQ01} with $g(x):=\frac{1}{2}x^2$ and $h(x):= |x|$. If an algorithm similar to Algorithm~\ref{alg:1} was run by using $x_0 =0$ as the initial point but choosing $w_0=\nabla g(x_0)+v_0$ with $v_0= 0 \in\partial h(0)$ (instead of $w_0\in\partial\varphi(x_0)$), it would stop at the first iteration and return $x=0$, which is a critical point, but not a stationary one. On the other hand, for any $w_0\in\partial \varphi(0)=\{-1,1\}$ we get $w_0\neq 0$, and so Algorithm~\ref{alg:1} will continue iterating until it converges to one of the two stationary points $-1/2$ and $1/2$, which is guaranteed by our main convergence result; see Theorem~\ref{The01} below.
\end{example}\vspace*{-0.1in}
The next lemma shows that Algorithm~\ref{alg:1} is well-defined by proving
the existence of a direction $d_k$ satisfying \eqref{EQALG01} in Step~3 for sufficiently large regularization parameters $\rho_k$.\vspace*{-0.1in}
\betagin{lemma}\langlebel{lemma1}
Let $\varphi: \mathbb{R}^n \to \mathbb{R}$ be the objective function in problem~\eqref{EQ01} with $g\in\mathcal{C}^{1,1}$ and $h$ being locally Lipschitz around $\bar{x}$ and prox-regular at this point. Further, assume that $\partial^2 g(\bar{x})$ is $\xi$-lower-definite for some $\xi\in\mathbb{R}$ and consider a nonzero subgradient $w\in \partial \varphi(\bar{x})$.
Then for any $\zeta>0$ and any $\rho\geq\zeta-\xi$, there exists a nonzero direction $d\in \mathbb{R}^n$ satisfying the inclusion
\betagin{align}\langlebel{Eq002}
-w \in\partial^2 g(\bar{x})(d)+\rho d.
\end{align}\vspace*{-0.05in}
Moreover, any nonzero direction from \eqref{Eq002} obeys the conditions:\\[1ex]
{\bf(i)}\langlebel{lemma1b} $\varphi'(\bar{x}; d) \leq \langlengle w, d\ranglengle \leq -\zeta\| d\|^2$.\\[1ex]
{\bf(ii)} \langlebel{lemma1c} Whenever $\sigma \in (0,1)$, there exists $\eta >0$ such that
\betagin{align*}
\varphi(\bar{x} + \tau d ) < \varphi(\bar{x}) + \sigma \tau \langlengle w, d\ranglengle \leq \varphi(\bar{x})- \sigma \zeta \tau \|d\|^2\;\mbox{ when }\;\tau \in (0,\eta).
\end{align*}
\end{lemma}
\betagin{proof}
Consider the function $\psi(x):=g(x)+\langlengle w-\nabla g(\bar{x}),x\ranglengle+\frac{\rho}{2}\|x\|^2$ for which we clearly have that $\partial^2 \psi(\bar{x}) =\partial^2 g(\bar{x})+\rho I$, where $I$ denotes the identity mapping. This shows by Remark~\ref{rem:definite} that $\partial^2 \psi(\bar{x})$ is $(\xi+\rho)$-lower-definite, and thus it is $\zeta$-lower-definite as well. Since $\nabla \psi(\bar{x})=w\neq 0$ and $\zeta>0$, it follows from \cite[Proposition~3.1]{2021arXiv210902093D} (which requires $\psi$ to be $\mathcal{C}^{1,1}$ on $\mathbb{R}^n$, but actually only $\mathcal{C}^{1,1}$ around $\bar{x}$ is needed) that there exists a nonzero direction $d$ such that $-\nabla \psi(\bar{x})\in \partial^2 \psi (\bar{x})(d)$. This readily verifies \eqref{Eq002}, which yields in turn the second inequality in (i) due to Definition~\ref{def:lower-def}. On the other hand, we have by Proposition~\ref{Lemma:Dire01} the following:
\betagin{align}\langlebel{lemma1:INQ001}
\betagin{aligned}
\varphi'(\bar{x}; d)=\lim\limits_{t \to 0^+}\frac{ g(\bar{x} +td) -g(\bar{x}) }{ t }+\limsup\limits_{t \to 0^+}\frac{ -h(\bar{x} +td) +h(\bar{x}) }{t } \\
= \langlengle \nabla g(\bar{x}), d \ranglengle + \inf\big\{\langlengle w, d\ranglengle\;\big|\;w\in -\partial h(\bar x)\big\}\leq \langlengle \nabla g(\bar{x}) + v, d\ranglengle \leq- \zeta \|d\|^2,
\end{aligned}
\end{align}
where in the last estimate is a consequence of the second inequality in (i).
Finally, assertion (ii) follows directly from \eqref{lemma1:INQ001} and the definition of directional derivatives \eqref{UDD}.
\end{proof}\vspace*{-0.2in}
\betagin{remark} Under the $\xi$-lower-definiteness of $\partial^2 g(x_k)$, Lemma~\ref{lemma1} guarantees the existence of a direction $d_k$ satisfying both conditions in~\eqref{EQALG01} for all $\rho_k\geq \zeta-\xi$. {\em When $\xi$ is unknown}, it is still possible to implement Step~3 of the algorithm as follows. Choose first any initial value of $\rho\geq 0$, then compute a direction satisfying the inclusion in \eqref{EQALG01} and continue with Step~4 if the descent condition in \eqref{EQALG01} holds. Otherwise, increase the value of $\rho$ and repeat the process until the descent condition is satisfied.
\end{remark}\vspace*{-0.05in}
The next example demonstrates that the {\em prox-regularity} of $h$ is {\em not a superfluous assumption} in Lemma~\ref{lemma1}. Namely, without it the direction $d$ used in Step~3 of Algorithm~\ref{alg:1} can even be an {\em ascent direction}.\vspace*{-0.03in}
\betagin{example}\langlebel{ex:failure}
Consider the {\em least squares problem} given by
$$
\min_{x\in\mathbb{R}^2} \frac{1}{2}(Ax-b)^2+\| x\|_1 - \|x\|_2,\quad x\in\mathbb{R}^2,
$$
with $A:=[1,0]$ and $b:=1$. Denote $g(x):=\frac{1}{2}\|Ax-b\|^2$ and $h(x):=\| x\|_2 - \|x\|_1$. If we pick $\bar{x}:=(1,0)^T$, the function $h$ is not prox-regular at $\bar{x}$ because it is not lower regular at $\bar{x}$; see Proposition~\ref{Lemma:Dire01}. Indeed, $\mathcal{H}at\partial h(\bar{x})=\emptysettyset$, while
$$
\partial h(\bar{x})=\frac{\bar{x}}{\|\bar{x}\|}+\partial(-\|\cdot\|_1)(\bar{x})=\left\{\betagin{pmatrix} 0\\ -1 \end{pmatrix},\betagin{pmatrix} 0\\ 1 \end{pmatrix}\right\}.
$$
Therefore, although $\nabla^2 g(\bar{x})=A^TA$ is $\langlembda_{\min}(A^TA)$-lower-definite, the assumptions of Lemma~\ref{lemma1} are not satisfied. Due to the representation
$$
\partial (-h)(\bar{x})=-\frac{\bar{x}}{\|\bar{x}\|}+\partial\|\cdot\|_1(\bar{x})=\left\{\betagin{pmatrix} 0\\ v \end{pmatrix}\;\mathbb{B}igg|\;v\in [-1,1]\right\},
$$
the choice of $v:=(0, 1)^T\in\partial (-h)(\bar{x})$ yields $w:=\nabla g(\bar{x})+v = (0, 1)^T\in\partial\varphi(\bar{x})$. For any $\rho>0$, inclusion \eqref{Eq002} gives us $d = (0,- 1/\rho)^T$. This is an ascent direction for the objective function $\varphi(x)=g(x)-h(x)$ at $\bar{x}$ due to
$$
\varphi(\bar{x}+\tau d)=1+\frac{\tau}{\rho}-\sqrt{1+(\tau/\rho)^2}>\varphi(\bar{x})=0\;\mbox{ for all }\;\tau>0,
$$
which illustrates that the prox-regularity is an essential assumption in Lemma~\ref{lemma1}.
\end{example}\vspace*{-0.07in}
Algorithm~\ref{alg:1} either stops at a stationary point, or produces an infinite sequence of iterates. The convergence properties of the iterative sequence of our algorithm are obtained below in the main theorem of this section. Prior to the theorem, we derive yet another lemma, which establishes the following {\em descent property} for the difference of a $\mathcal{C}^{1,1}$ function and a prox-regular one.\vspace*{-0.05in}
\betagin{lemma}\langlebel{Lemma:01}
Let $\varphi(x) = g(x) -h(x)$, where $g$ is of class $\mathcal{C}^{1,1}$ around $\bar{x}$, and where $h$ is continuous around $\bar{x}$ and prox-regular at this point. Then for every $\bar{v} \in \partial h(\bar{x})$, there exist positive numbers $\epsilon$ and $r$ such that
\betagin{align*}
\varphi(y) \leq \varphi(x) + \langlengle \nabla g(x) - v , y-x \ranglengle +r \| y-x\|^2
\end{align*}
whenever $x, y \in \mathbb{B}_{\epsilon}(\bar{x})$ and $v\in \partial h(x) \cap \mathbb{B}_{\epsilon}(\bar{v}) $.
\end{lemma}\vspace*{-0.05in}
\betagin{proof}
Pick any $\bar{v} \in \partial h(\bar{x})$ and deduce from the imposed prox-regularity and continuity of $h$ that there exist $\epsilon_1>0$ and $r_1>0$ such that
\betagin{align}\langlebel{Lemma:01:Eq01}
-h(y) \leq -h(x) + \langlengle - v , y - x \ranglengle + r_1 \| y-x\|^2\;\mbox{ for all }\;x,y \in \mathbb{B}_{\epsilon_1}(\bar{x})
\end{align}
and all $v\in \partial h(x)\cap \mathbb{B}_{\epsilon_1}(\bar{v}) $. It follows from the $\mathcal{C}^{1,1}$ property of $g$ by \cite[Lemma~A.11]{MR3289054} that there exist positive numbers $r_2$ and $\epsilon_2$ such that
\betagin{align}\langlebel{Lemma:01:Eq02}
g(y) \leq g(x) + \langlengle \nabla g(x) , y-x \ranglengle + r_2 \| y- x\|^2\;\mbox{ for all }\;\mathbb{B}_{\varepsilon_2}.
\end{align}
Summing up the inequalities in \eqref{Lemma:01:Eq01} and \eqref{Lemma:01:Eq02} and defining $r:= r_1 +r_2$ and $\epsilon := \min\{ \epsilon_1,\epsilon_2\}$, we get that
\betagin{align*}
g(y)- h(y) \leq g(x) - h(x) + \langlengle \nabla g(x)-v , y-x \ranglengle + r\| y-x\|^2
\end{align*}
for all $x,y \in \mathbb{B}_\epsilon(\bar{x})$ and all $v\in \partial h(x) \cap \mathbb{B}_\epsilon(\bar{v})$. This completes the proof.
\end{proof}\vspace*{-0.1in}
Now we are ready to establish the aforementioned theorem about the performance of Algorithm~\ref{alg:1}.\vspace*{-0.05in}
\betagin{theorem}\langlebel{The01}
Let $\varphi: \mathbb{R}^n \to \bar{v}erline{\mathbb{R}}$ be the objective function of problem~\eqref{EQ01} given by $\varphi = g-h$ with $\inf \varphi >-\infty$. Pick an initial point $x_0 \in \mathbb{R}^n$ and suppose that the sublevel set $Tmega:=\{x \in\mathbb{R}^n\;|\;\varphi(x) \leq \varphi(x_0)\}$ is closed. Assume also that:\\[0.5ex]
{\bf(a)} \langlebel{Theo01ass:a} The function $g$ is $\mathcal{C}^{1,1}$ around every $x\inTmega$ and the second-order subdifferential $\partial^2 g(x)$ is $\xi$-lower-definite for all $x\in Tmega$ with some $\xi\in\mathbb{R}$.\\[1ex]
{\bf(b)} The function $h$ is locally Lipschitzian and prox-regular on $Tmega$.\\[0.5ex]
Then Algorithm~{\rm\ref{alg:1}} either stops at a stationary point, or produces sequences $\{x_k\} \subseteq Tmega$, $\{\varphi(x_k)\}$, $\{w_k\}$, $\{d_k\}$, and $\{\tau_k\}$ such that:\\[0.5ex]
{\bf(i)} \langlebel{The01a} The sequence $\{\varphi(x_k)\}$ monotonically decreases and converges.\\[0.5ex]
{\bf(ii)}\langlebel{The01b} If $\{x_{k_j}\}$ as $j\in \mathbb{N}$ is any bounded subsequence of $\{x_k\}$, then $\displaystyle\inf_{j\in \mathbb{N}}\tau_{k_j}>0$,
\betagin{align*}
\sum\limits_{j \in \mathbb{N}}\| d_{k_j}\|^2 < \infty,\; \sum\limits_{j\in \mathbb{N} } \| x_{k_j +1} - x_{k_j}\|^2< \infty,\;\text{ and }\;\sum\limits_{j\in \mathbb{N} } \|w_{k_j} \|^2<\infty.
\end{align*}
In particular, the boundedness of the entire sequence $\{x_k\}$ ensures that the set of accumulation points of $\{x_k\}$ is a nonempty, closed, and connected.\\[0.5ex]
{\bf(iii)}\langlebel{The01c} If $x_{k_j} \to \bar{x}$ as $j\to\infty$, then $\bar{x}$ is a stationary point of problem \eqref{EQ01} with the property $\varphi(\bar{x}) =\displaystyle\inf_{k\in \mathbb{N}} \varphi (x_k)$.\\[0.5ex]
{\bf(iv)}\langlebel{The01d} If $\{x_k\}$ has an isolated accumulation point $\bar{x}$, then the entire sequence $\{x_k\}$ converges to $\bar{x}$ as $k\to\infty$, where $\bar x$ is a stationary point of \eqref{EQ01}.
\end{theorem}\vspace*{-0.05in}
\betagin{proof}
If Algorithm~\ref{alg:1} stops after a finite number of iterations, then it clearly returns a stationary point. Otherwise, it produces an infinite sequence $\{x_k\}$. By Step~5 of Algorithm~\ref{alg:1} and Lemma~\ref{lemma1}, we have that $\inf \varphi\le\varphi(x_{k+1}) < \varphi(x_k) $ for all $k \in \mathbb{N}$, which proves assertion (i) and also shows that $\{x_k\} \subseteq Tmega$.
To proceed, suppose that $\{x_k\}$ has a bounded subsequence
$\{x_{k_j}\}$ (otherwise there is nothing to prove) and split the rest of the proof into the {\em five claims}.\vspace*{0.03in}
\noindent\textbf{Claim~1:} \emptyseth{The sequence $\{\tau_{k_j}\}$, associated with $\{x_{k_j}\}$ as $j\in\mathbb{N}$ and produced by Algorithm~{\rm\ref{alg:1}}, is bounded from below.}\\
Indeed, otherwise consider a subsequence $\{\tau_{\nu_i}\}$ of $\{\tau_{k_j}\}$ such that $\tau_{ \nu_i} \to 0^+$ as $i\to \infty$. Since $\{x_{k_j}\}$ is bounded, we can assume that $\{x_{\nu_i}\} $ converges to some point $\bar{x}$. By Lemma~\ref{lemma1}, we have that
\betagin{align}\langlebel{EQ002}
-\langlengle w_{\nu_i}, d_{\nu_i} \ranglengle \geq \zeta \|d_{\nu_i}\|^2\;\mbox{ for all }\;i\in\mathbb{N},
\end{align}
which yields by the Cauchy--Schwarz inequality the estimate
\betagin{align}\langlebel{EQ002bis}
\|w_{\nu_i}\|\geq \zeta \| d_{\nu_i}\|,\quad i\in\mathbb{N}.
\end{align}
Since $\varphi$ is locally Lipschitzian and $w_{\nu_i} \in \partial\varphi (x_{{\nu_i}})$, we suppose without loss of generality that $w_{\nu_i}$ converges to some $\bar{w} \in \partial\varphi(\bar{x}) \subseteq \nabla g(\bar{x})- \partial h(\bar{x})$ as $i\to\infty$. It follows from \eqref{EQ002bis} that $\{d_{\nu_i}\}$ is bounded, and therefore $d_{\nu_i} \to\bar{d}$ along a subsequence. Since $\tau_{ \nu_i } \to 0^+$, we can assume that $\tau_{ \nu_i }<t_{\min}$ for all $i\in\mathbb{N}$, and hence Step~5 of Algorithm~\ref{alg:1} ensures the inequality
\betagin{align}\langlebel{EQ003}
\varphi(x_{\nu_i} + \betata^{-1}\tau_{\nu_i} d_{\nu_i}) > \varphi(x_{\nu_i}) +\sigma \betata^{-1}\tau_{\nu_i} \langlengle w_{\nu_i},d_{\nu_i}\ranglengle,\quad i\in\mathbb{N}.
\end{align}
Lemma~\ref{Lemma:01} gives us a constant $r>0$ such that
\betagin{align}\langlebel{EQ004}
\varphi(x_{\nu_i} + \betata^{-1}\tau_{\nu_i} d_{\nu_i})\le\varphi(x_{\nu_i}) +\betata^{-1}\tau_{\nu_i} \langlengle w_{\nu_i} ,
d_{\nu_i}\ranglengle+r\betata^{-2}\tau_{\nu_i}^2\|d_{\nu_i}\|^2
\end{align}
for all $i$ sufficiently large. Combining \eqref{EQ003},~\eqref{EQ004}, and \eqref{EQ002} tells us that
\betagin{equation*}
\betagin{array}{ll}
\sigma \betata^{-1}\tau_{\nu_i} \langlengle w_{\nu_i} , d_{\nu_i}\ranglengle< \varphi(x_{\nu_i} + \betata^{-1}\tau_{\nu_i} d_{\nu_i}) - \varphi(x_{\nu_i})\\
\leq \betata^{-1}\tau_{\nu_i} \langlengle w_{\nu_i} , d_{\nu_i}\ranglengle+r\betata^{-2}\tau_{\nu_i}^2\|d_{\nu_i}\|^2\leq \betata^{-1}\tau_{\nu_i}\left(1-\displaystyle\frac{r}{\zeta\betata}\tau_{\nu_i}\right)\langlengle w_{\nu_i}, d_{\nu_i}\ranglengle
\end{array}
\end{equation*}
for large $i$. Since $\langlengle w_{\nu_i} ,d_{\nu_i}\ranglengle<0$ by \eqref{EQ002}, we get that $\sigma >1 - \frac{r}{ \zeta\betata } \tau_{ \nu_i }$
for such $i$, which contradicts the choice of $\sigma \in (0,1)$ and thus verifies this claim.\vspace*{0.03in}
\noindent\textbf{Claim~2:} \emptyseth{We have the series convergence $\sum_{j \in \mathbb{N}} \| d_{k_j}\|^2 < \infty $, $\sum_{j\in \mathbb{N} } \| x_{k_j +1} - x_{k_j}\|^2 < \infty$, and $\sum_{j\in \mathbb{N} } \| w_{k_j}\|^2 < \infty$.}\\
To justify this, deduce from Step~5 of Algorithm~\ref{alg:1} and Lemma~\ref{lemma1} that
\betagin{align*}
\sum\limits_{k\in \mathbb{N}} \zeta\tau_k \| d_k\|^2 \leq \frac{1}{\sigma } \mathbb{B}ig( \varphi(x_0) - \inf_{k\in \mathbb{N}} \varphi(x_k)\mathbb{B}ig).
\end{align*}
It follows from Claim~1 that $\zeta\tau_{k_j} >\gamma >0$ for all $j\in \mathbb{N}$, which yields $\sum_{j\in \mathbb{N}}\| d_{k_j}\|^2 <\infty$. On the other hand, we have that $\| x_{k_j +1} - x_{k_j}\|= \tau_{k_j}\| d_{k_j}\|$, and again Claim~1 ensures that $\sum_{j\in \mathbb{N} } \| x_{k_j +1} - x_{k_j}\|^2 < \infty$. To proceed further, let $l_2 := \sup\{ \| d_{k_j}\|\;|\;\in \mathbb{N} \}$ and use the Lipschitz continuity of $\nabla g$ on the compact set ${\rm cl}\{x_{k_j}\;|\;{j\in \mathbb{N}}\} \subseteq Tmega$. Employing the subdifferential condition from \cite[Theorem~4.15]{MR3823783} together with the coderivative scalarization in \eqref{coder:sub}, we get by the standard compactness argument the existence of $l_3>0$ such that
\betagin{align*}
w \in \partial \langlengle d, \nabla g\ranglengle (x_{k_j})= \partial^2 g(x_{k_j})(d) \Longrightarrow \| w\| \leq l_3
\end{align*}
for all $j\in\mathbb{N}$ and all $d\in \mathbb{B}_{l_2}(0)$. Therefore, it follows from the inclusion $-w_{k_j} \in \partial^2 g(x_{k_j})(d_{k_j}) +\rho_{k_j}d_{k_j}$ that we have
\betagin{align}\langlebel{LipGrad}
\|w_{k_j} +\rho_{k_j}d_{k_j}\| \leq l_3 \| d_{k_j}\|\;\text{ for all large }\;j \in \mathbb{N}.
\end{align}
Using finally the triangle inequality and the estimate $\rho_k\leq \rho_{\max}$ leads us to the series convergence $\sum_{j\in \mathbb{N} } \| w_{k_j} \|^2 < \infty$ as stated in Claim~2.\vspace*{0.03in}
\noindent\textbf{Claim~3:} \emptyseth{If the sequence $\{x_k\}$ is bounded, then the set of its accumulation points is nonempty, closed and connected.}\\
Applying Claim~2 to the sequence $\{x_k\}$, we have the \emptyseth{Ostrowski condition} $\lim_{k \to \infty }\| x_{k +1} - x_{k}\| = 0$. Then, the conclusion follows from \cite[Theorem~28.1]{Ostrowski1966}.
\noindent\textbf{Claim~4:} \emptyseth{If $x_{k_j} \to \bar{x}$ as $j\to\infty$, then $\bar{x}$ is a stationary point of \eqref{EQ01} being such that $\varphi(\bar{x}) = \inf_{k\in \mathbb{N}} \varphi (x_k)$.} \\
By Claim 2, we have that the sequence $w_{k_j} \in \partial \varphi(x_{k_j})$ with $w_{k_j} \to 0$ as $j\to\infty$. The closedness of the basic subgradient set ensures that $0 \in \partial \varphi (\bar{x})$. The second assertion of the claim follows from the continuity of $\varphi$ at $\bar{x} \in Tmega$.
\noindent\textbf{Claim~5:} \emptyseth{If $\{x_k\}$ has an isolated accumulation point~$\bar{x}$, then the entire sequence of $x_k$ converges to $\bar{x}$ as $k\to\infty$, and $\bar{x}$ is a stationary point of \eqref{EQ01}.}
Indeed, consider any subsequence $x_{k_j} \to \bar{x}$. By Claim~4, $\bar{x}$ is a stationary point of \eqref{EQ01}, and it follows from Claim~2 that
$\lim_{j\to \infty} \| x_{k_j +1}-x_{k_j}\|=0$. Then we deduce from by \cite[Proposition~8.3.10]{MR1955649} that $x_k \to \bar{x}$ as $k\to\infty$, which completes the proof of theorem.
\end{proof}\vspace*{-0.2in}
\betagin{remark}\langlebel{rem:theorem}
Regarding Theorem~\ref{The01}, observe the following:
(i) If $h=0$, $g$ is of class $\mathcal{C}^{1,1}$, and $\xi>0$, then the results of Theorem~\ref{The01} can be found in \cite{2021arXiv210902093D}.
(ii) If $\xi\geq 0$, we can choose the regularization parameter $\rho_k:=c\|w_k\|$ and (a varying) $\zeta:=c\|w_k\|$ in~\eqref{EQALG01} for some $c>0$ to verify that assertions (i) and (iii) of Theorem~\ref{The01} still hold. Indeed, if $\{x_{k_j}\}$ converges to some $\bar{x}$, then $\{w_{k_j}\}$ is bounded by the Lipschitz continuity of $\varphi$. Hence the sequence $\{w_{k_j}\}$ converges to $0$. Otherwise, there exists $M>0$ and a subsequence of $\{w_{k_j}\}$ whose norms are bounded from below by $M$. Using the same argumentation as in the proof of Theorem~\ref{The01} with $\zeta=c M$, we arrive at the contradiction with $0$ being an accumulation point of of $\{w_{k_j}\}$.
\end{remark}\vspace*{-0.05in}
When the objective function $\varphi$ is coercive and its stationary points are isolated, Algorithm~\ref{alg:1} converges to a stationary point because Theorem~\ref{The01}(iii) ensures that the set of accumulation points is connected. This property enables us to prove the convergence in some settings when even there exist nonisolated accumulation points; see the two examples below.\vspace*{-0.05in}
\betagin{example}
Consider the function $\varphi: \mathbb{R} \to \mathbb{R}$ given by
\betagin{align*}
\varphi(x) & :=\int_0^x t^4 \sin\left(\frac{\pi}{t}\right) dt.
\end{align*}
This function is clearly $\mathcal{C}^2$-smooth and coercive. For any starting point $x_0$, the level set $Tmega =\{ x\;|\;\varphi(x) \leq \varphi(x_0) \}$ is bounded, and hence there exists a number $\xi\in \mathbb{R}$ such that the functions $g(x) :=\varphi(x)$ and $h(x):=0$ satisfy the assumptions of Theorem~\ref{The01}. Observe furthermore that $\varphi$ is a DC function because it is $\mathcal{C}^2$-smooth; see, e.g., \cite{Oliveira_2020,hiriart}. However, it is not possible to write its DC decomposition with $g(x) = \varphi(x) + ax^2$ and $h(x)=ax^2$ for $a>0$, since there exists no scalar $a>0$ such that the function $g(x) = \varphi(x) + ax^2$ is convex on the entire real line.
It is easy to see that the stationary points of $\varphi$ are described by $S:=\left\{ \frac{1}{n}\;\big|\;n \in\mathbb{Z}\backslash\{ 0\} \right\}\cup\{ 0\}$. Moreover, if Algorithm~\ref{alg:1} generates an iterative sequence $\{x_k\}$ starting from $x_0$, then the accumulation points form by Theorem~\ref{The01}(ii) a nonempty, closed, and connected set $A \subseteq S$ .
If $A=\{ 0\}$, the sequence $\{x_k\}$ converges to $\bar{x}=0$. If $A$ contains any point of the form $\bar{x}=\frac{1}{n}$, then it is an isolated point, and Theorem~\ref{The01}(iv) tells us that the entire sequence $\{x_k\}$ converges to that point, and consequently we have $A=\{\bar{x}\}$.
\end{example}
\betagin{example}\langlebel{example3.10}
Consider the function $\varphi: \mathbb{R}^n \to \mathbb{R}$ given by
\betagin{align*}
\varphi(x):=\sum_{i=1}^n \varphi_i(x_i),\; \text{ where }\;\varphi_i(x_i):= g_i(x_i) - h_i(x_i) \\
\text{ with }\;g_i(x_i):= \frac{1}{2}x_i^2\; \text{ and }\;h_i(x_i):= |x_i| +\big| 1-|x_i|\,\big|.
\end{align*}
We can easily check that the function $\varphi$ is coercive and satisfies the assumptions of Theorem~\ref{The01} with $g(x):=\sum_{i=1}^n g_i(x_i)$, $h(x):= \sum_{i=1}^n h_i(x_i)$, and $\xi=1$. For this function, the points in the set $\{-2, -1,0,1,2\}^n$ are critical but not stationary. Moreover, the points in the set $\{-2,0,2 \}^n$ give the global minima to the objective function $\varphi$. Therefore, Algorithm~\ref{alg:1} leads us to global minimizers of $\varphi$ starting from any initial point.
\betagin{figure}[h!!]
\centering\betagin{tikzpicture}
\betagin{axis}[
axis x line=center,
axis y line=center,
xtick={-5,-4,...,5},
ytick={-5,-4,...,4},
xlabel={$x$},
ylabel={$y$},
xlabel style={below right},
ylabel style={above left},
xmin = -4.5, xmax = 4.5,
ymin = -1.5, ymax = 2.5]
\addplot[
domain = -4.5:4.5,
samples = 200,
smooth,
ultra thick,
color=C0,
] {0.5*x*x - abs(x)- abs( 1-abs(x)) )};
\node at (0,-1)[circle,fill,inner sep=1.5pt]{};
\node at (2,-1)[circle,fill,inner sep=1.5pt]{};
\node at (-2,-1)[circle,fill,inner sep=1.5pt]{};
\node at (-1,-0.5)[circle,fill=red,inner sep=1.5pt]{};
\node at (1,-0.5)[circle,fill=red,inner sep=1.5pt]{};
\end{axis}
\end{tikzpicture}
\caption{ Plot of the function $\varphi_i $ in Example \ref{example3.10}}
\langlebel{fig:screenshot001}
\end{figure}
\end{example}\vspace*{-0.1in}
The following theorem establishes convergence rates of the iterative sequences in Algorithm~\ref{alg:1} under some additional assumptions.\vspace*{-0.05in}
\betagin{theorem}\langlebel{corSMR}
Suppose in addition to the assumptions of Theorem~{\rm\ref{The01}}, that $\{x_k\}$ has an accumulation point $\bar{x}$ such that the subgradient mapping $\partial \varphi$ is strongly metrically subregular at $(\bar{x},0)$. Then the entire sequence $\{x_k\}$ converges to $\bar{x}$ with the Q-linear convergence rate for $\{\varphi(x_k)\}$ and the R-linear convergence rate for $\{x_k\}$ and $\{w_k\}$. If furthermore, $\xi>0$, $0<\zeta\leq\xi$, $\rho_k\to 0$, $\sigma \in(0,\frac{1}{2})$, $t_{\min}=1$, $g$ is semismoothly differentiable at $\bar{x}$, $h$ is of class $\mathcal{C}^{1,1}$ around $\bar{x}$, and $\mbox{\rm clm}\, \nabla h(\bar{x})=0$, then the rate of convergence of all the sequences above is at least Q-superlinear.
\end{theorem}\vspace*{-0.05in}
\betagin{proof} We split the proof of the theorem into the following two claims.\vspace*{0.03in}
\noindent\textbf{Claim~1:} \emptyseth{The rate of convergence of
$\{\varphi(x_k)\}$ is at least Q-linear, while both sequences $\{x_k\}$ and $\{w_k\}$ converge at least R-linearly.}\\
Observe first that it follows from the imposed strong metric subregularity of $\partial\varphi$ that $\bar{x}$ is an isolated accumulation point, and so $x_k\to\bar{x}$ as $k\to\infty$ by Theorem~\ref{The01}(iii). Further, we get from \eqref{def:stron_subreg} that there exists $\kappa>0$ such that
\betagin{align}\langlebel{eq:1}
\| x_k - \bar{x}\| \leq \kappa \| w_k\|\;\text{ for large }\;k \in \mathbb{N},
\end{align}
since $w_k\to 0$ as $k\to\infty$ by Theorem \ref{The01}(ii). Using \eqref{LipGrad} and the triangle inequality gives us $\ell > 0$ such that $ \| w_k\| \leq \ell \| d_k\| $ for sufficiently large $k\in\mathbb{N}$. Lemma~\ref{Lemma:01} yields then the cost function increment estimate
\betagin{align}\langlebel{eq:2}
\varphi(x_k)-\varphi(\bar{x}) \le r\| x_k -\bar x\|^2\;\text{ for all large }\;k \in\mathbb{N}.
\end{align}
By Step~5 of Algorithm~\ref{alg:1} and Lemma~\ref{lemma1}, we get that $\varphi(x_k)-\varphi(x_{k+1}) \geq \sigma \zeta\tau_k \| d_k \|^2 $ for large $k \in \mathbb{N}$. Remembering that $\inf_{k\in \mathbb{N}} \tau_k >0$, we deduce from Theorem \ref{The01}(ii) the existence of $\eta >0$ such that
\betagin{align}\langlebel{eq:3}
\varphi(x_k)-\varphi(\bar{x}) - (\varphi(x_{k+1})- \varphi(\bar{x}) ) \geq \eta \| w_k\|^2
\end{align}
whenever $k$ large enough. Therefore, applying \cite[Lemma~7.2]{2021arXiv210902093D} to the sequences $\alphapha_k := \varphi(x_k) - \varphi(\bar{x})$, $\betata_k:= \| w_k\|$, and $\gamma_k := \| x_k - \bar{x}\|$ with the positive constants $c_1:= \eta$, $c_2:= \kappa^{-1}$, and $c_3:= r$, we verify the claimed result.\vspace*{0.03in}
\noindent\textbf{Claim ~2: } \emptyseth{Assuming that $\sigma \in (0,\frac{1}{2})$, $t_{\min}=1$, $g$ is semismoothly differentiable at $\bar{x}$, $h$ is of class $\mathcal{C}^{1,1}$ around $\bar{x}$, and $\mbox{\rm clm}\, \nabla h(\bar{x})=0$, we have that the rate of convergence for all the above sequences is at least Q-superlinear.}\\
Suppose without loss of generality that $h$ is differentiable at any $x_k\to\bar{x}$. It follows from the coderivative scalarization \eqref{coder:sub} and the basic subdifferential sum rule in \cite[Theorem~2.19]{MR3823783} valid under the imposed assumptions that
\betagin{align}\langlebel{sublineal}
\partial^2 g(x_k)(d_k) \subseteq \partial^2 g(x_k) (x_k + d_k - \bar{x}) + \partial^2 g(x_k) (-x_k + \bar{x}).
\end{align}
This yields the existence of $z_k \in \partial^2 g(x_k) (-x_k + \bar{x})+\rho_k(-x_k+\bar{x})$ such that
\betagin{align}\langlebel{inclusion01}
-\nabla g(x_k) +\nabla h(x_k) - z_k \in \partial^2 g(x_k) (x_k + d_k - \bar{x})+\rho_k(x_k + d_k - \bar{x}).
\end{align}
Moreover, the $(\xi+\rho_k)$-lower-definiteness of $\partial^2 g(x_k )+\rho_kI$ and the Cauchy--Schwarz inequality imply that
\betagin{align*}
\| x_k + d_k -\bar{x}\|\leq \frac{1}{\xi+\rho_k} \| \nabla g(x_k) - \nabla h(x_k)+ z_k\|.
\end{align*}
Combining now the semismoothness of $\nabla g$ at $\bar{x}$ with the conditions
$\nabla g(\bar{x})=\nabla h(\bar{x})$ and $\mbox{\rm clm}\, \nabla h(\bar{x})=0$ brings us to the estimates
\betagin{align*}
\betagin{array}{ll}
\| \nabla g(x_k) - \nabla h(x_k)+ z_k \|\leq \| \nabla g(x_k) -\nabla g(\bar{x})+ z_k+\rho_k(x_k-\bar{x})\|\\
+\rho_k\|x_k-\bar{x}\| + \|\nabla h(\bar{x})-\nabla h(x_k)\|= o(\| x_k - \bar{x}\|).
\end{array}
\end{align*}
Then we have $\| x_k + d_k -\bar{x}\|=o(\|x_k-\bar{x}\|)$ and deduce therefore from \cite[Proposition~8.3.18]{MR1955649} and Lemma~\ref{lemma1}(i) that
\betagin{align}\langlebel{eqSLC01}
\varphi(x_k + d_k) \leq \varphi(x_k) + \sigma \langlengle \nabla \varphi(x_k), d_k\ranglengle.
\end{align}
It follows from \eqref{eqSLC01} that $x_{k+1}=x_k + d_k$ if $k$ for large $k$. Applying \cite[Proposition~8.3.14]{MR1955649} yields the $Q$-superlinear convergence of $\{x_k\}$ to $\bar{x}$ as $k\to\infty$.
Finally, conditions \eqref{eq:1}--\eqref{eq:3} and the Lipschitz continuity of $\nabla \varphi$ around~$\bar{x}$ ensure the existence of $L>0$ such that
\betagin{align*}
\frac{\eta}{\kappa^2}\| x_k - \bar{x} \|^2 & \le\varphi(x_k) -\varphi(\bar{x}) \leq r \| x_k - \bar{x} \|^2, \\
\quad\frac{1}{\kappa}\| x_k - \bar{x} \| & \leq \| \nabla \varphi(x_k) \|\leq L\| x_k - \bar{x} \|
\end{align*}
for sufficiently large $k$, and therefore we get the estimates
\betagin{equation}\langlebel{estimationsCOR}
\betagin{array}{ll}
\displaystyle\frac{\varphi(x_{k+1}) -\varphi(\bar{x}) }{\varphi(x_k) -\varphi(\bar{x})} \leq \kappa r\displaystyle\frac{ \| x_{k+1} - \bar{x} \|^2 }{ \| x_{k} -\bar{x} \|^2},
\\
\quad\;\displaystyle\frac{\| \nabla \varphi(x_{k+1}) \| }{ \| \nabla \varphi(x_{k})\|}\le\kappa L\displaystyle\frac{ \| x_{k+1} - \bar{x} \| }{ \| x_{k} -\bar{x}\|},
\end{array}
\end{equation}
which thus conclude the proof of the theorem.
\end{proof}\vspace*{-0.25in}
\betagin{remark}\langlebel{rem:subregul}
The property of {\em strong metric subregularity} of {\em subgradient mappings}, which is a central assumption of Theorem~\ref{corSMR}, has been well investigated in variational analysis, characterized via second-order growth and coderivative type conditions, and applied to optimization-related problems; see, e.g., \cite{ag,dmn,MR3823783} and the references therein.
\end{remark}\vspace*{-0.05in}
The next theorem establishes the $Q$-superlinear and $Q$-quadratic convergence of the sequences generated by Algorithm~\ref{alg:1} provided that: $\xi>0$ (i.e., $\partial^2 g(x)$ is $\xi$-strongly positive-definite), $\rho_k=0$ for all $k\in \mathbb{N}$ (no regularization is used), $g$ is semismoothly differentiable at the cluster point $\bar{x}$, and the function $h$ can be expressed as the pointwise maximum of finitely many affine functions at $\bar{x}$, i.e., when there exist $(x^\ast_i, \alphapha_i)_{i=1}^p \subseteq \mathbb{R}^n \times \mathbb{R}$ and $\epsilon >0$ such that
\betagin{align}\langlebel{max_affine}
h(x)=\max_{i=1,\ldots, p} \left\{ \langlengle x^\ast_i,x\ranglengle +\alphapha_i \right\}\; \text{ for all }\;x\in \mathbb{B}_\epsilon(\bar{x}).
\end{align}
\betagin{theorem}\langlebel{Cor:max_affine} In addition to the assumptions of Theorem~{\rm\ref{The01}}, suppose that $\xi>0$, $0<\zeta\leq\xi$, $\sigma \in (0,\frac{1}{2})$, $t_{\min}=1$, and $\rho_k=0$ for all $k\in\mathbb{N}$. Suppose also that the sequence $\{x_k\}$ generated by Algorithm~{\rm\ref{alg:1}} has an accumulation point $ \bar{x}$ at which $g$ is semismoothly differentiable and $h$ can be represented in form \eqref{max_affine}. Then we have the convergence $x_k\to\bar{x}$, $\varphi(x_k)\to\varphi(\bar{x})$, $w_k\to 0$, and $\nabla g(x_k)\to\nabla g(\bar{x})$ as $k\to\infty$ with at least $Q$-superlinear rate. If in addition $g$ is of class $\mathcal{C}^{2,1}$ around $\bar{x}$, then the rate of convergence is at least quadratic.
\end{theorem}\vspace*{-0.05in}
\betagin{proof}
Observe that by \eqref{max_affine} and \cite[Proposition~1.113]{MR2191744} we have the inclusion
\betagin{align}\langlebel{FORMSUBD}
\partial (-h)(x)\subseteq \bigcup\big\{ -x^\ast_i\;\big|\;h(x)=\langlengle x^\ast_i,x\ranglengle +\alphapha_i\big\}
\end{align}
for all $x$ near $\bar{x}$. The rest of the proof is split into the five claims below.\vspace*{0.03in}
\noindent\textbf{Claim~1:} \emptyseth{The sequence $\{x_k\}$ converges to $\bar{x}$ as $k\to\infty$.}\\
Observe that $\bar{x}$ is an isolated accumulation point. Indeed, suppose on the contrary that there is a sequence $\{y_\nu\}$ of accumulation points of $\{x_k\}$ such that $y_\nu \to \bar{x}$ as $\nu\to\infty$ with $y_\nu \neq \bar{x}$ for all $\nu\in\mathbb{N}$. Since each $y_\nu$ is accumulation point of $\{x_k\}$, they are stationary points of $\varphi$. The ${\cal C}^1$-smoothness of $g$ ensures that $\nabla g(y_\nu) \to \nabla g(\bar{x})$ as $\nu\to\infty$, and so \eqref{FORMSUBD} yields
$\nabla g(y_\nu)=x_{i_\nu}^\ast$ for large $\nu\in\mathbb{N}$. Since there are finitely many of $x_i^\ast$ in \eqref{max_affine}, we get that $\nabla g(y_\nu) = \nabla g(\bar{x})$ when $\nu$ is sufficiently large. Further, it follows from \cite[Theorem~5.16]{MR3823783} that the gradient mapping $\nabla g$ is strongly locally maximal monotone around $\bar{x}$, i.e., there exist positive numbers $\epsilon$ and $r$ such that
\betagin{align*}
\langlengle \nabla g(x)-\nabla g(y), x-y\geq r\| x -y\|^2\;\text{ for all } \;x,y\in\mathbb{B}_\epsilon(\bar{x}).
\end{align*}
Putting $x:=\bar{x}$ and $y:=y_\nu$ in the above inequality tells us that
$\bar{x}= y_\nu$ for large $\nu \in \mathbb{N}$, which is a contradiction. Applying finally Theorem~\ref{The01}(iv), we complete the proof of this claim.\vspace*{0.03in}
\noindent\textbf{Claim 2:} \emptyseth{The sequence $\{x_k\}$ converges to $\bar{x}$ as $k\to\infty$ at least $Q$-superlinearly.}\\
As $x_k\to\bar{x}$, we have by Theorem~\ref{The01}(ii) that $w_k-\nabla g(x_k)\to-\nabla g(\bar{x})$, and so it follows from \eqref{FORMSUBD} that
there exists $i\in\{1,\ldots,p\}$ such that $h(\bar x ) = \langlengle x^\ast_i , \bar x\ranglengle +\alphapha_i$, $h(x_k) = \langlengle x^\ast_i , x_k\ranglengle -\alphapha_i$ and
$w_k-\nabla g(x_k)=-\nabla g(\bar{x})=-x_i^\ast$ for all $k$ sufficiently large. Define the auxiliary function ${v}idehat{\varphi}:\mathbb{R}^n \to \bar{v}erline{\mathbb{R}}$ by
\betagin{align}\langlebel{aux:func}
{v}idehat{\varphi} (x):= g(x) -\langlengle x^\ast_i , x\ranglengle -\alphapha_i
\end{align}
and observe that ${v}idehat\varphi$ is $\mathcal{C}^{1,1}$ around $\bar{x}$ and semismoothly differentiable at this point. We have the equalities
\betagin{align}\langlebel{eq_auxvarphi}
\varphi (x_k) ={v}idehat{\varphi} (x_k),\;\varphi (\bar{x}) ={v}idehat{\varphi} (\bar{x}),\; \nabla {v}idehat{\varphi} (x_k)=w_k,\;\text{ and }\; \nabla {v}idehat{\varphi} (\bar{x})=0
\end{align}
for large $k$. It follows from $\partial^2{v}idehat{\varphi} (x) = \partial^2 g(x)$ that the mapping $\partial^2{v}idehat{\varphi}(\bar{x})+\rho_kI$ is $(\xi+\rho_k)$-lower-definite. Using \eqref{sublineal} and \eqref{inclusion01} with the replacement of $g$ by ${v}idehat\varphi$ and taking \eqref{eq_auxvarphi} into account ensures the existence of $z_k \in \partial^2 {v}idehat{\varphi}(x_k) (-x_k + \bar{x}) +\rho_k(-x_k + \bar{x})$ satisfying the estimate
\betagin{align*}
\| x_k + d_k -\bar{x}\| \leq \frac{1}{\xi+\rho_k} \| \nabla {v}idehat{\varphi}(x_k)- \nabla{v}idehat{\varphi}(\bar{x})+ z_k\|.
\end{align*}
The triangle inequality and the semismoothness of $\nabla{v}idehat \varphi$ at $\bar{x}$ yield
\betagin{align*}
\|\nabla {v}idehat{\varphi}(x_k)- \nabla {v}idehat{\varphi}(\bar{x})+ z_k\|&\leq \| \nabla {v}idehat{\varphi}(x_k)- \nabla {v}idehat{\varphi}(\bar{x})+ z_k+\rho_k(x_k-\bar{x})\|+\rho_k\|x_k-\bar{x}\|\\
&=o(\|x_k-\bar{x}\|),
\end{align*}
which tells us that $\| x_k + d_k -\bar{x}\|=o(\| x_k - \bar{x}\|)$. Then it follows from\cite[Proposition~8.3.18]{MR1955649} and Lemma~\ref{lemma1}(i) above that
\betagin{align}\langlebel{eqSLC}
{v}idehat{\varphi}(x_k + d_k) \leq {v}idehat{\varphi}(x_k) + \sigma \langlengle \nabla{v}idehat{\varphi}(x_k),d_k\ranglengle
\end{align}
whenever $k$ is sufficiently large. Applying finally \cite[Proposition~8.3.14]{MR1955649} verifies the claimed $Q$-superlinear convergence of $\{x_k\}$ to $\bar{x}$.\vspace*{0.03in}
\noindent\textbf{Claim~3:} \emptyseth{The gradient mapping of ${v}idehat \varphi$ from \eqref{aux:func} is strongly metrically regular around $(\bar{x},0)$ and hence strongly metrically subregular at this point.}\\
Using the $\xi$-lower-definiteness of $\partial^2 {v}idehat\varphi (\bar{x})$ and the pointbased coderivative characterization of strong local maximal monotonicity given in \cite[Theorem~5.16]{MR3823783}, we verify this property for $\nabla {v}idehat \varphi$
around $\bar{x}$. Then \cite[Corollary~5.15]{MR3823783} ensures that $\nabla {v}idehat \varphi$ is strongly metrically regular around $(\bar{x},0)$. \vspace*{0.03in}
\noindent\textbf{Claim~4:} \emptyseth{The sequences $\{ \varphi(x_k)\}$, $\{w_k\}$, and $\{\nabla g(x_k)\}$ converge at least Q-superlinearly to $\varphi(\bar{x})$, $0$, and $\nabla g(\bar{x})$, respectively.}\\
It follows from the estimates in \eqref{estimationsCOR}, with the replacement of $\varphi$ by ${v}idehat\varphi$ and with taking into account that
${v}idehat\varphi(x_k) - {v}idehat\varphi(\bar x) =\varphi(x_k)-\varphi(\bar x)$ and $ \nabla {v}idehat \varphi(x_k) = w_k$ due to \eqref{eq_auxvarphi}, that there exist constants $\alphapha_1 , \alphapha_2 >0$ such that
\betagin{align*}
\frac{\varphi(x_{k+1}) -\varphi(\bar{x}) }{ \varphi(x_k) -\varphi(\bar{x})} &\leq \alphapha_1 \frac{ \| x_{k+1} - \bar{x} \|^2 }{ \| x_{k} -\bar{x} \|^2} \\
\frac{ \|w_{k+1 }\| }{ \|w_k\|} &\leq \alphapha_2 \frac{ \| x_{k+1} - \bar{x} \| }{ \| x_{k} - \bar{x} \|}
\end{align*}
provided that $k$ is sufficiently large. Recalling that $w_k-\nabla g(x_k)=-\nabla g(\bar{x})$ for large $k$ completes the proof of the claim. \vspace*{0.03in}
\noindent\textbf{Claim~5:} \emptyseth{If $g$ is of class $\mathcal{C}^{2,1}$ around $\bar{x}$, then the rate of convergence of the sequences above is at least quadratic.}\\
It is easy to see that the assumed $\mathcal{C}^{2,1}$ property of $g$ yields this property of ${v}idehat{\varphi}$ around $\bar{x}$. Using estimate
\eqref{eqSLC}, we deduce this claim from the quadratic convergence of the classical Newton method; see, e.g., \cite[Theorem~5.18]{Aragon2019} and \cite[Theorem~2.15]{MR3289054}. This therefore completes the proof of the theorem.
\end{proof}\vspace*{-0.25in}
\betagin{remark}\langlebel{rem:long} Concerning Theorem~\ref{Cor:max_affine}, observe the following:
(i) It is important to emphasize that the performance of Algorithm~\ref{alg:1} revealed in Theorem~\ref{Cor:max_affine} is mainly due to the usage of the basic subdifferential of the function $-h$ in contrast to that of $h$, which is calculated as
\betagin{equation}\langlebel{h-sub}
\partial h(x)= \text{co} \left(\bigcup \left\{ x^\ast_i\;\bigg|\;h(x)=\langlengle x^\ast_i,x\ranglengle +\alphapha_i \right\}\right)
\end{equation}
by \cite[Theorem~3.46]{MR2191744}. We can see from the proof of Theorem~\ref{Cor:max_affine} that it fails if the evaluation of $\partial(-h)(x)$ in \eqref{FORMSUBD} is replaced by the one of $\partial h(x)$ in \eqref{h-sub}.
(ii) The main assumptions of Theorem~\ref{Cor:max_affine} do not imply the smoothness of $\varphi$ at stationary points. For instance, consider the nonconvex function $\varphi: \mathbb{R}^n \to \mathbb{R}$ defined as in Example~\ref{example3.10} but letting now $h_i(x_i):= |x_i| +| 1- x_i|$.
The function $\varphi$ satisfies the assumptions of Theorem~\ref{Cor:max_affine} at any of its stationary points $\{-2,0,2\}^n$, but $\varphi$ is not differentiable at $\bar{x}=0$; see Figure~\ref{example3.10reviplot2}.
\betagin{figure}[h!!]
\centering \betagin{tikzpicture}
\betagin{axis}[
axis x line=center,
axis y line=center,
xtick={-5,-4,...,5},
ytick={-3,-2,...,2},
xlabel={$x$},
ylabel={$y$},
xlabel style={below right},
ylabel style={above left},
xmin = -6.5, xmax = 6.5,
ymin = -3.5, ymax = 2.5]
\addplot[
domain = -5.5:5.5,
samples = 200,
smooth,
ultra thick,
color=C0,
] {0.5*x*x - abs(x)- abs( 1-x)};
\node at (0,-1)[circle,fill=red,inner sep=1.5pt]{};
\end{axis}
\end{tikzpicture}
\caption{ Plot of function $\varphi_i(x)=\frac{1}{2}x^2 -|x| -| 1- x| $ in Remark~\ref{rem:long}}
\langlebel{example3.10reviplot2}
\end{figure}
(iii) The functions $\varphi$, $g$, and $h$ in Example~\ref{example3.10} satisfy the assumptions of Theorem~\ref{Cor:max_affine}. Therefore, the convergence of the sequences generated by Algorithm~\ref{alg:1} is at least quadratic.
\end{remark}\vspace*{-0.33in}
\section{Convergence Rates under the Kurdyka--{\L}ojasiewicz Property}\langlebel{sec:4}\vspace*{-0.1in}
In this section, we verify the global convergence of Algorithm~\ref{alg:1} and establish convergence rates in the general setting of Theorem~\ref{The01} without additional assumptions of Theorems~\ref{corSMR} and \ref{Cor:max_affine} while supposing instead that the cost function $\varphi$ satisfies the Kurdyka--{\L}ojasiewicz property. Recall that the \emptyseth{Kurdyka--{\L}ojasiewicz property} holds for $\varphi$ at $ \bar{x}$ if there exist $\eta >0$ and a continuous concave function $ \psi:[0,\eta] \to [0,\infty)$ with $\psi (0)=0$ such that $\psi$ is $\mathcal{C}^1$-smooth on $(0,\eta)$ with the strictly positive derivative $\psi'$ and that
\betagin{align}\langlebel{Kur-Loj}
\psi'\big(\varphi(x) - \varphi(\bar{x})\big)\,{\rm dist}\big(0;\partial \varphi(x)\big)\geq 1
\end{align}
for all $x\in \mathbb{B}_\eta(\bar{x})$ with $\varphi(\bar{x}) < \varphi(x) <\varphi( \bar{x} ) + \eta$, where ${\rm dist}(\cdot;Tmega)$ stands for the distance function of a set $Tmega$.
The first theorem of this section establishes the {\em global convergence} of iterative sequence generated by Algorithm~\ref{alg:1} to a {\em stationary point} of \eqref{EQ01}.\vspace*{-0.05in}
\betagin{theorem}\langlebel{Teo:Kur-Loj}
In addition to the assumptions of Theorem~{\rm\ref{The01}}, suppose that the iterative sequence $\{x_k\}$ generated by Algorithm~{\rm\ref{alg:1}} has an accumulation point $\bar{x}$ at which the Kurdyka--{\L}ojasiewicz property \eqref{Kur-Loj} is satisfied. Then $\{x_k\}$ converges $\bar{x}$ as $k\to\infty$, which is a stationary point of problem~\eqref{EQ01}.
\end{theorem}\vspace*{-0.05in}
\betagin{proof}
If Algorithm~\ref{alg:1} stops after a finite number of iterations, there is nothing to prove. Due to the decreasing property of $\{\varphi(x_k)\}$ from Theorem~\ref{The01}(i), we can assume that $\varphi(x_k) > \varphi(x_{k+1}) $ for all $k \in \mathbb{N}$. Let $ \bar{x}$ be the accumulation point of $\{x_k\}$ where $\varphi$ satisfies the Kurdyka--{\L}ojasiewicz inequality \eqref{Kur-Loj}, which by Theorem \ref{The01} is a stationary point of problem~\eqref{EQ01}. Since $\varphi$ is continuous, we have that $\varphi(\bar{x} )= \inf_{k\in\mathbb{N}}\varphi(x_k)$. Taking the constant $\eta>0$ and the function $\psi$ from \eqref{Kur-Loj} and remembering that $g$ is of class $\mathcal{C}^{1,1}$ around $\bar{x}$, suppose without loss of generality that $\nabla g$ is Lipschitz continuous on $\mathbb{B}_{2\eta}( \bar{x})$ with modulus $\kappa$. Let $k_0 \in \mathbb{N}$ be such that $x_{k_0} \in \mathbb{B}_{\eta/2}(\bar{x})$ and that
\betagin{align}\langlebel{iq:01Ku-Loj}
\varphi (\bar{x}) < \varphi(x_{k}) < \varphi(\bar{x}) +\eta, \quad\frac{ \kappa +\rho_{\max} }{\sigma \zeta }\psi\big( \varphi (x_{k}-\varphi(\bar{x})\big) < \eta/2
\end{align}
for all $k \geq k_0$, where $\sigma \in (0,1)$, $\zeta>0$, and $\rho_{\max}>0$ are the constants of Algorithm~\ref{alg:1}. The rest of the proof is split into the following three steps.\vspace*{0.03in}
\noindent\textbf{Claim~1:} \emptyseth{Let $k \geq k_0 $ be such that $x_k \in \mathbb{B}_\eta (\bar{x})$. Then we have the estimate}
\betagin{align}\langlebel{eq:Kur-Loj}
\| x_k -x_{k+1}\| \leq \frac{ \kappa+\rho_k }{\sigma\zeta }\big( \psi( \varphi(x_{k}) - \varphi(\bar{x})\big) - \psi\big( \varphi(x_{k+1}) - \varphi(\bar{x})\big)\big).
\end{align}
Indeed, it follows from \eqref{coder:sub}, \eqref{EQALG01}, and \cite[Theorem~1.22]{MR3823783} that
\betagin{equation}\langlebel{dist:inq}
\betagin{array}{ll}
{\rm dist}(0;\partial \varphi (x_k)\big) &\leq \| w_k \| \leq \| w_k +\rho_kd_k\|+\rho_k\|d_k\|\\
&\leq (\kappa+\rho_k) \| d_k \| = \displaystyle\frac{ \kappa +\rho_k}{ \tau_k } \| x_{k+1} - x_k\|.
\end{array}
\end{equation}
Then using Step~5 of Algorithm~\ref{alg:1}, Lemma~\ref{lemma1}, the Kurdyka--{\L}ojasiewicz inequality \eqref{Kur-Loj}, the concavity of $\psi$, and estimate \eqref{dist:inq} gives us
\betagin{align*}
\| x_k - &x_{k+1}\|^2 = \tau^2_k \| d_k\|^2 \leq \frac{ \tau_k }{\sigma\zeta }\big( \varphi(x_k ) - \varphi(x_{k+1})\big) \\
& \leq \frac{ \tau_k }{\sigma\zeta }{\rm dist}\big(0;\partial \varphi (x_k)\big)\,\psi'\big(\varphi(x_k) - \varphi(\bar{x})\big) \big( \varphi(x_k ) - \varphi(x_{k+1})\big)\\
& \leq \frac{ \tau_k }{\sigma\zeta }{\rm dist}\big(0;\partial \varphi (x_k)\big)\big(\psi( \varphi(x_k) - \varphi(\bar{x})\big) - \psi( \varphi(x_{k+1}) - \varphi(\bar{x})\big)\big) \\
&\leq \frac{ \kappa+\rho_k }{\sigma\zeta } \| x_{k+1} - x_k\|\big( \psi\big( \varphi(x_k) - \varphi(\bar{x})\big) - \psi\big(\varphi(x_{k+1}) - \varphi(\bar{x})\big)\big),
\end{align*}
which therefore verifies the claimed inequality \eqref{eq:Kur-Loj}.\vspace*{0.03in}
\noindent\textbf{Claim~2:} \emptyseth{For every $k \geq k_0 $, we have the inclusion $x_{k} \in \mathbb{B}_{\eta} (\bar{x})$.}\\
Suppose on the contrary that there exists $k> k_0$ with $x_k \notin \mathbb{B}_\eta (\bar{x})$ and define $\bar{k}:=\min\left\{ k> k_o\;\big|\; x_k \notin \mathbb{B}_\eta (\bar{x}) \right\}$. Since for $k\in\{k_0,\ldots,\bar{k}-1\}$ the estimate in \eqref{eq:Kur-Loj} is satisfied, we get by using~\eqref{iq:01Ku-Loj} that
\betagin{align*}
\| x_{ \bar{k} } - \bar{x} \|& \leq \| x_{k_0} - \bar{x}\| + \sum_{k=k_0}^{\bar{k}-1}\| x_{k} - x_{k+1}\| \\
&\leq \| x_{k_0} - \bar{x}\| + \frac{\kappa+\rho_{\max}}{\sigma\zeta } \sum_{k=k_0}^{\bar{k}-1}\big( \psi\big( \varphi(x_{k}) - \varphi(\bar{x})\big) - \psi\big( \varphi(x_{k+1}) - \varphi(\bar{x}) \big)\big)\\
&\leq \| x_{k_0} - \bar{x}\| + \frac{\kappa+\rho_{\max}}{\sigma\zeta } \psi\big( \varphi(x_{k_0}) - \varphi(\bar{x})\big) \leq \eta,
\end{align*}
which contradicts our assumption and thus verifies this claim.\vspace*{0.03in}
\noindent\textbf{Claim~3:} \emptyseth{We have that $\sum_{k=1}^{\infty} \| x_k - x_{k+1}\| < \infty$, and consequently the sequence $\{x_k\}$ converges to $\bar{x}$ as $k\to\infty$.}\\
It follows from Claim~1 and Claim~2 that \eqref{eq:Kur-Loj} holds for all $k \geq k_{0}$. Thus
\betagin{align*}
\sum_{k=1}^{\infty} \| x_k - x_{k+1}\|& \leq \sum_{k=1}^{k_0-1} \| x_k - x_{k+1}\| + \sum_{k=k_0}^{\infty} \| x_k - x_{k+1}\| \\
& \leq \sum_{k=1}^{k_0-1} \| x_k - x_{k+1}\| + \frac{\kappa+\rho_{\max}}{\sigma\zeta } \psi\big( \varphi(x_{k_0}) - \varphi(\bar{x})\big) <\infty,
\end{align*}
which therefore completes the proof of the theorem.
\end{proof}\vspace*{-0.1in}
The next theorem establishes {\em convergence rates} for iterative sequence $\{x_k\}$ in Algorithm~\ref{alg:1} provided that the function $\psi$ in \eqref{Kur-Loj} is selected in a special way. Since the proof while using Theorem~\ref{Teo:Kur-Loj}, is similar to the corresponding one from \cite[Theorem~4.9]{MR4078808} in a different setting, it is omitted. \vspace*{-0.05in}
\betagin{theorem}\langlebel{COR:Kur-Loj}
In addition to the assumptions of Theorem~{\rm\ref{Teo:Kur-Loj}}, suppose that the Kurdyka--{\L}ojasiewicz property \eqref{Kur-Loj} holds at the accumulation point $\bar{x}$ with $\psi(t):= M t^{1-\thetaeta}$ for some $M>0$ and $\thetaeta\in[0,1)$. The following assertions hold:
{\bf(i)} If $\thetaeta =0$, then the sequence $\{x_k\}$ converges in a finite number of steps.
{\bf(ii)} If $\thetaeta\in (0,1/2]$, then the sequence $\{x_k\}$ converges at least linearly.
{\bf(iii)} If $\thetaeta \in (1/2,1)$, then there exist $\mu >0$ and $k_0\in \mathbb{N}$ such that
\betagin{align*}
\|x_ k - \bar{x}\| \leq \mu k^{-\frac{1-\thetaeta }{ 2\thetaeta -1 } }\;\text{ for all }\;k \geq k_0.
\end{align*}
\end{theorem}\vspace*{-0.1in}
\betagin{remark}
Together with our main Algorithm~\ref{alg:1}, we can consider its modification with the replacement of $\partial(-h)(x_k)$ by $-\partial h(x_k)$. In this case, the most appropriate version of the Kurdyka--{\L}ojasiewicz inequality \eqref{Kur-Loj}, ensuring the fulfillment the corresponding versions of Theorem~\ref{Teo:Kur-Loj} and \ref{COR:Kur-Loj}, is the one
\betagin{align*}
\psi\big(\varphi(x)-\varphi(\bar{x})\big)\,{\rm dist}\big(0;\partial^0\varphi(x)\big)\geq 1
\end{align*}
expressed in terms of the {\em symmetric subdifferential} $\partial^0\varphi(x)$ from \eqref{sym}. Note that the latter is surely satisfied where the symmetric subdifferential is replaced by the {\em generalized gradient}
$\bar{v}erline{\partial}\varphi(x)$, which is the convex hull of $\partial^0\varphi(x)$.
\end{remark}\vspace*{-0.35in}
\section{Applications to Structured Constrained Optimization}\langlebel{sec:5}\vspace*{-0.05in}
In this section, we present implementations and specifications of our main RCSN Algorithm~\ref{alg:1} for two structured classes of optimization problems. The first class contains functions represented as sums of two nonconvex functions one of which is smooth, while the other is extended-real-valued. The second class concerns minimization of smooth functions over closed constraint sets.\vspace*{-0.25in}
\subsection{Minimization of Structured Sums}\langlebel{subsec1}\vspace*{-0.05in}
Here we consider the following class of structured optimization problems:
\betagin{equation}\langlebel{ProFBE}
\min_{x\in \mathbb{R}^n}\varphi(x):=f(x)+\psi(x),
\end{equation}
where $f:\mathbb{R}^n\to\mathbb{R}$ is of class $\mathcal{C}^{2,1}$ with the $L_f$-Lipschitzian gradient, and where $\psi: \mathbb{R}^n \to \bar{v}erline{\mathbb{R}}$ is an extended-real-valued prox-bounded function with the threshold $\langlembda_\psi>0$. When both functions $f$ and $\psi$ are convex, problems of type \eqref{ProFBE} have been largely studied under the name of ``convex composite optimization" emphasizing the fact that $f$ and $\psi$ are of completely different structures. In our case, we do not impose any convexity of $f,\psi$ and prefer to label \eqref{ProFBE} as {\em minimization of structured sums} to avoid any confusions with optimization of function compositions, which are typically used in major models of variational analysis and constrained optimization; see, e.g., \cite{MR1491362}.
In contrast to the original class of {\em unconstrained} problems of {\em difference programming} \eqref{EQ01}, the structured {\em sum optimization} form \eqref{ProFBE} covers optimization problems with {\em constraints} given by $x\in\mbox{\rm dom}\,\psi$. Nevertheless, we show in what follows that the general class of problem \eqref{ProFBE} can be reduced under the assumptions imposed above to the difference form \eqref{EQ01} satisfying the conditions for the required performance of Algorithm~\ref{alg:1}.
This is done by using an extended notion of envelopes introduced by Patrinos and Bemporad in \cite{Patrinos2013}, which is now commonly referred as the \emptyseth{forward-backward envelope}; see, e.g., \cite{MR3845278}.\vspace*{-0.1in}
\betagin{definition}
Given $ \varphi = f + \psi$ and $\langlembda >0$, the \emptyseth{forward-backward envelope} (FBE) of the function $\varphi$ with the parameter $\langlembda$ is defined by
\betagin{align}\langlebel{FBE}
\varphi_\langlembda(x) :=\inf_{z \in \mathbb{R}^n}\mathbb{B}ig\{ f(x) + \langlengle \nabla f(x),z-x\ranglengle + \psi(z) +\frac{1}{2\langlembda }\| z- x\|^2\mathbb{B}ig\}.
\end{align}
\end{definition}
Remembering the constructions of the Moreau envelope \eqref{moreau} and the Asplund function \eqref{asp} allows us to represent $\varphi_\langlembda$ for every $\langlembda \in (0, \langlembda_\psi)$ as:
\betagin{equation}\langlebel{rep02}
\betagin{array}{ll}
\varphi_\langlembda(x)&= f(x) -\displaystyle\frac{\langlembda}{2}\| \nabla f(x)\|^2 + \mathcal{M}oreauYosida{\psi}{\langlembda}\big(x-\langlembda \nabla f(x)\big) \\
&=\displaystyle f(x) +\frac{1}{2\langlembda} \|x\|^2 -\langlengle \nabla f(x), x\ranglengle - \Asp{\langlembda}{\psi}\big(x- \langlembda \nabla f(x)\big).
\end{array}
\end{equation}
\betagin{remark}\langlebel{rem:infimum}
It is not difficult to show that whenever $\nabla f$ is $L_f$-Lipschitz on $\mathbb{R}^n$ and $\langlembda \in (0,\frac{1}{L_f})$, the optimal values in problems \eqref{EQ01} and \eqref{FBE} are the same
\betagin{align}\langlebel{eqinf}
\inf_{x\in \mathbb{R}^n}\varphi_\langlembda(x)= \inf_{x\in \mathbb{R}^n } \varphi(x).
\end{align}
Indeed, the inequality ``$\leq $'' in \eqref{eqinf} follows directly from the definition of $\varphi_\langlembda$. The reverse inequality in \eqref{eqinf} is obtained by
\betagin{equation*}
\betagin{array}{ll}
\displaystyle\inf_{x\in \mathbb{R}^n} \varphi_\langlembda(x)=\displaystyle\inf_{x\in \mathbb{R}^n}\displaystyle\inf_{ z \in \mathbb{R}^n } \mathbb{B}ig\{ f(x) + \langlengle \nabla f(x),z-x\ranglengle + \psi(z) +\displaystyle\frac{1}{2\langlembda }\|z- x\|^2\mathbb{B}ig\}\\
\geq\displaystyle\inf_{x\in \mathbb{R}^n}\displaystyle\inf_{ z \in \mathbb{R}^n}\mathbb{B}ig\{ f(z) -\frac{L_f}{2} \|z-x\|^2 + \psi(z) +\displaystyle\frac{1}{2\langlembda }\| z- x\|^2\mathbb{B}ig\}\\
= \displaystyle\inf_{ z \in \mathbb{R}^n }\displaystyle\inf_{x\in \mathbb{R}^n}\mathbb{B}ig\{ f(z) + \psi(z)+\mathbb{B}ig(\frac{1}{2\langlembda } -\displaystyle\frac{L_f}{2}\mathbb{B}ig)\| z- x\|^2\mathbb{B}ig\}= \displaystyle\inf_{ z \in \mathbb{R}^n }\varphi(x).
\end{array}
\end{equation*}
Moreover, \eqref{eqinf} does not hold if $\nabla f$ is not Lipschitz continuous on $\mathbb{R}^n$. Indeed, consider $f(x):=\frac{1}{4} x^4$ and $\psi:=0$. Then we have $\inf_{x\in\mathbb{R}^n} \varphi(x) =0$ while $\varphi_\langlembda(x)=\frac{1}{4} x^4 - \frac{\langlembda}{2} x^6 $, which yields $\inf_{x\in \mathbb{R}^n} \varphi_\langlembda(x)=-\infty$, and so \eqref{eqinf} fails.
\end{remark}\vspace*{-0.05in}
The next theorem shows that FBE \eqref{FBE} can be written as the difference of a $\mathcal{C}^{1,1}$ function and a Lipschitzian prox-regular function. Furthermore, it establishes relationships between minimizers and critical points of $\varphi$ and $\varphi_\langlembda$.\vspace*{-0.1in}
\betagin{theorem}\langlebel{diff_repr_varphi}
Let $\varphi=f+\psi$, where $f$ is of class $\mathcal{C}^{2,1}$ and where $\psi$ is prox-bounded with threshold $\langlembda_\psi>0$. Then for any $\langlembda \in (0,\lambda_\psi)$, we have the inclusion
\betagin{equation}\langlebel{Subbasic}
\partial\varphi_\langlembda (x)\subseteq \langlembda^{-1}\big( I- \langlembda \nabla^2 f(x)\big) \big(x -Painlev\'{e}rox{\psi}{\langlembda}\big(x-\langlembda \nabla f(x)\big)\big).
\end{equation}
Furthermore, the following assertions are satisfied:
{\bf(i)} \langlebel{diff_repr_varphi_a} If $x\in \mathbb{R}^n$ is a stationary point of $\varphi_\langlembda$, then $0\in \mathcal{H}at{\partial} \varphi(x)$ provided that the matrix $ I- \langlembda \nabla^2 f(x)$ is nonsingular.
{\bf(ii)} \langlebel{diff_repr_varphi_c} The FBE \eqref{FBE} can be written as $\varphi_\langlembda=g-h$, where $g(x):= f(x) +\frac{1}{2\langlembda} \|x\|^2 $ is of class $\mathcal{C}^{2,1}$, and where $h(x):= \langlengle \nabla f(x),x\ranglengle + \Asp{\langlembda}{\psi}(x- \langlembda \nabla f(x))$ is locally Lipschitzian and prox-regular on $\mathbb{R}^n$. Moreover, $\nabla^2 g(x)$ is $\xi$-lower-definite for all $x\in \mathbb{R}^n$ with $\xi:=\frac{1}{\langlembda} - L_f$.
{\bf(iii)} \langlebel{diff_repr_varphi_d} If $\psi:=\delta_{C}$ for a closed set $C$, then $\partial (-\Asp{\langlembda}{\psi})=-\frac{1}{\langlembda}\mathtt{P}_C$, where $\mathtt{P}_C$ denotes the $($generally set-valued$)$ projection operator onto $C$. In this case, inclusion \eqref{Subbasic} holds as an equality.
{\bf(iv)} \langlebel{diff_repr_varphi_b} If both $f$ and $\psi$ are convex, we have that $\varphi_\langlembda = g - h$, where $g(x):= f(x) + \mathcal{M}oreauYosida{\psi}{\langlembda} (x-\langlembda \nabla f(x))$ and $h(x):= \frac{\langlembda}{2}\| \nabla f(x)\|^2$ are of class $\mathcal{C}^{1,1}$ $($and hence prox-regular$)$ on $\mathbb{R}^n$, and that
\betagin{align}\langlebel{eqconvexcase}
\big\{x \in \mathbb{R}^n\;\big|\;\nabla \varphi_\langlembda(x)=0\big\} =\big\{ x\in \mathbb{R}^n\;\big|\;0 \in \partial \varphi(x)\big\}
\end{align}
provided that $ I- \langlembda \nabla^2 f(x)$ is nonsingular at any stationary point of $\varphi_\langlembda$.
\end{theorem}\vspace*{-0.05in}
\betagin{proof}
Observe that inclusion \eqref{Subbasic} follows directly by applying the basic subdifferential sum and chain rules from \cite[Theorem~2.19 and Corollary~4.6]{MR3823783}, respectively, the first representation of $\varphi_\lambda$ in \eqref{rep02} with taking into account the results of Lemma~\ref{Lemma5.1}. Now we pick any stationary point $x\in\mathbb{R}^n$ of the FBE $\varphi_\langlembda$ and then deduce from
$0\in \partial \varphi_\langlembda(x)$ and \eqref{Subbasic} that
\betagin{equation*}
x \inPainlev\'{e}rox{\psi}{\langlembda}\big(x-\langlembda \nabla f(x))\big),
\end{equation*}
which readily implies that $0 \in \nabla f(x) +{v}idehat{\partial} \psi(x)={v}idehat{\partial} \varphi(x)$ and thus verifies (i). Assertion (ii) follows directly from Proposition~\ref{Lemma5.1} and the smoothness of $f$.
To prove (iii), we need to verify the reverse inclusion ``$\supseteq$'' in \eqref{eq_sub_eq01}, for which it suffices to show that the inclusion $v\in\mathtt{P}_C(x)$ yields $v\not\in{\rm co}(\mathtt{P}_C(x)\setminus\{v\})$. On the contrary, if $v\in\mathtt{P}_C(x)\cap{\rm co}(\mathtt{P}_C(x)\setminus\{v\})$, then there exist $c_1,\ldots,c_m\in P_C(x)\setminus\{v\}$ and $\mu_1,\ldots,\mu_m\in (0,1)$ such that $v=\sum_{i=1}^m\mu_i c_i$ with $\sum_{i=1}^m \mu_i=1$. By definition of the projection, we get the equalities
\betagin{equation*}
|c_1-x\|^2=\ldots=\|c_m-x\|^2=\|v-x\|^2=\mathbb{B}ig\|\sum_{i=1}^m\mu_i(c_i-x)\mathbb{B}ig\|^2,
\end{equation*}
which contradict the strict convexity of $\|\cdot\|^2$ and thus verifies (iii).
The first statement in (iv) follows from the differentiability of $f$ and of the Moreau envelope $\mathcal{M}oreauYosida{\psi}{\langlembda}$ by \cite[Theorem~2.26]{MR1491362}. Further, the inclusion ``$\subseteq $'' in \eqref{eqconvexcase} is a consequence of (i). To justify the reverse inclusion in \eqref{eqconvexcase}, observe that any $x$ satisfying $0\in\partial\varphi(x)$ is a global minimizer of the convex function $\varphi$, and so $x = Painlev\'{e}rox{\psi}{\langlembda} (x-\langlembda \nabla f(x))$. The differentiability of $\varphi_\langlembda$ and \eqref{Subbasic} (which holds as an equality in this case) tells us that $\nabla\varphi_\langlembda(x)=0$, and thus \eqref{eqconvexcase} holds. This completes the proof of the theorem.
\end{proof}\vspace*{-0.25in}
\betagin{remark}\langlebel{rem:DC repres} Based on Theorem~\ref{diff_repr_varphi}(ii), it is not hard to show that the FBE function $\varphi_\langlembda$ can be represented as a difference of convex functions. Indeed, since $\Asp{\langlembda}{\psi}$ is a locally Lipschitzian and prox-regular function, we have by \cite[Corollary~3.12]{MR2101873} that $h$ is a lower-$\mathcal{C}^2$ function, and hence by \cite[Theorem~10.33]{MR1491362}, it is locally a DC function. Similarly, $g$ being a $\mathcal{C}^2$ function is a DC function, so the difference $\varphi=g-h$ is also a DC function. However, it is difficult to determine for numerical purposes what is an appropriate representation of $\varphi$ as a difference of convex functions. Moreover, such a representation of the objective in terms of convex functions may generate some theoretical and algorithmic challenges as demonstrated below in Example~\ref{Example5.4}.
\end{remark}\vspace*{-0.35in}
\subsection{Nonconvex Optimization with Geometric Constraints}\langlebel{subsec:5.1}\vspace*{-0.1in}
This subsection addresses the following problem of {\em constrained optimization} with explicit geometric constraints given by:
\betagin{equation}\langlebel{prob:constrained}
\mbox{minimize }\;f(x)\;\mbox{ subject to }\;x\in C,
\end{equation}
where $f:\mathbb{R}^n\to\mathbb{R}$ is of class $\mathcal{C}^{2,1}$, and where $C\subseteq\mathbb{R}^n$ is an arbitrary closed set. Due to the lack of convexity, most of the available algorithms in the literature are not able to directly handle this problem. Nevertheless, Theorem~\ref{diff_repr_varphi} provides an effective machinery allowing us to reduce \eqref{prob:constrained} to an optimization problem that can be solved by using our developments. Indeed, define $\psi(x): = \delta_{C}(x)$ and observe that $\psi$ is prox-regular with threshold $\langlembda_\psi = \infty$.
In this setting, FBE \eqref{FBE} reduces to the formula
\betagin{equation*}
\varphi_\langlembda(x)=f(x)-\frac{\langlembda}{2}\|\nabla f(x)\|^2+\frac{1}{2\langlembda}{\rm dist}^2\big(x-\langlembda\nabla f(x);C\big).
\end{equation*}
Furthermore, it follows from Theorem~\ref{diff_repr_varphi}(iii) that
\betagin{align*}
\partial \varphi_\langlembda(x)=\langlembda^{-1}\big(I-\langlembda \nabla^2 f(x)\big)\big(x - \mathtt{P}_C\big( x -\langlembda \nabla f(x)\big)\big).
\end{align*}
Based on Theorem~\ref{diff_repr_varphi}, we deduce from Algorithm~\ref{alg:1} with $\rho_k=0$ its following version to solve the constrained problem \eqref{prob:constrained}.\vspace*{-0.2in}
\betagin{algorithm}[ht!]
\betagin{algorithmic}[1]
\mathbb{R}equire{$x_0 \in \mathbb{R}^n$, $\betata \in (0,1)$, $t_{\min}>0 $ and $\sigma\in(0,1)$.}
Fr\'{e}chetor{$k=0,1,\ldots$}
\mathcal{S}tate Take $w_k\in \left(\langlembda^{-1}I-\nabla^2 f(x_k)\right)\big( x_k - \mathtt{P}_C\big(x-\langlembda \nabla f(x_k)\big)\big)$.
\mathcal{S}tate If $ w_k=0$, STOP and return~$x_k$. Otherwise set $d_k$ as the solution to the linear system $(\nabla^2 f(x_k)+\langlembda^{-1}I)d_k=w_k$.
\mathcal{S}tate Choose any $\bar{v}erline{\tau}_k\geq t_{\min}$. Set $\bar{v}erline{\tau}_k:=\tau_k$.
\mathcal{W}hile{$\varphi_\langlembda(x_k + \tau_k d_k) > \varphi_\langlembda(x_k) +\sigma \tau_k \langlengle \nabla w_k , d_k\ranglengle $}
\mathcal{S}tate $\tau_k = \betata \tau_k$.
\EndWhile
\mathcal{S}tate Set $x_{k+1}:=x_k + \tau_kd_k$. \langlebel{step5_2}
\EndFor
\end{algorithmic}
\caption{Projected-like Newton algorithm for constrained optimization}\langlebel{alg:3}
\end{algorithm}\vspace*{-0.2in}
To the best of our knowledge, Algorithm~\ref{alg:3} is new even for the case of convex constraint sets $C$. All the results obtained for Algorithm~\ref{alg:1} in Sections~\ref{sec:3} and \ref{sec:4} can be specified for Algorithm~\ref{alg:3} to solve problem \eqref{prob:constrained}. For brevity, we present just the following direct consequence of Theorem~\ref{The01}. \vspace*{-0.05in}
\betagin{corollary}\langlebel{Cor:Theo01}
Considering problem \eqref{prob:constrained}, suppose that $f: \mathbb{R}^n \to\mathbb{R}$ is of class $\mathcal{C}^{2,1}$, that $C\subset\mathbb{R}^n$ is closed, and that
$\inf_{x\in C}f(x) >-\infty$. Pick an initial point $x_0 \in \mathbb{R}^n$ and a parameter $\langlembda\in (0, \frac{1}{L_f})$. Then Algorithm~{\rm\ref{alg:3}} either stops at a point $x$ such that $0\in\nabla f(x)+\mathcal{H}at{N}_C(x)$, or generates infinite sequences $\{x_k\}$, $\{\varphi_\langlembda(x_k)\}$, $\{w_k\}$, $\{d_k\}$, and $\{\tau_k\}$ satisfying the assertions:
{\bf(i)} \langlebel{Cor:The01a} The sequence $\{\varphi_\langlembda(x_k)\}$ monotonically decreases and converges.
{\bf(ii)} \langlebel{Cor:The01b} If $\{x_{k_j}\}$ is a bounded subsequence of $\{x_k\}$, then $\inf_{j\in\mathbb{N}} \tau_{k_j}>0$ and
\betagin{align*}
\sum\limits_{j \in \mathbb{N}} \| d_{k_j}\|^2 < \infty,\; \sum\limits_{j\in \mathbb{N} } \| x_{k_j +1} - x_{k_j}\|^2< \infty,\;\sum\limits_{j\in \mathbb{N} } \| w_{k_j}\|^2 < \infty.
\end{align*}
If, in particular, the entire sequence $\{x_k\}$ is bounded, then the set of its accumulation points is nonempty, closed, and connected.
{\bf(iii)} \langlebel{Cor:The01c} If $x_{k_j} \to \bar{x}$ as $j\to\infty$, then $0\in\nabla f(\bar{x})+\mathcal{H}at{N}_C(\bar{x})$ and the equality $\varphi_\langlembda(\bar{x})=\inf_{k\in \mathbb{N}} \varphi_\langlembda(x_k)$ holds.
{\bf(iv)} \langlebel{Cor:The01d} If the sequence $\{x_k\}$ has an isolated accumulation point $\bar{x}$, then it converges to $\bar{x}$ as $k\to\infty$, and we have $0\in\nabla f(\bar{x})+\mathcal{H}at{N}_C(\bar{x})$.
\end{corollary}\vspace*{-0.05in}
The next example illustrates our approach to solve \eqref{prob:constrained} via Algorithm~\ref{alg:3} in contrast to algorithms of the DC type.\vspace*{-0.05in}
\betagin{example}\langlebel{Example5.4}
Consider the minimization of a quadratic function over a closed (possibly nonconvex) set $C$:
\betagin{align}\langlebel{ProblemQUAD_example}
\mbox{minimize }\;\frac{1}{2}x^T Q x + b^T x\; \text{ subject to }\;
x\in C,
\end{align}
where $Q$ is a symmetric matrix, and where $b \in \mathbb{R}^n$. In this setting, FBE \eqref{FBE} can be written as $\varphi_\langlembda(x) = g(x) - h(x)$ with
\betagin{equation}\langlebel{func_h_Example55}
\betagin{array}{ll}
g(x)&:=\displaystyle\frac{1}{2}x^T\big( Q + \langlembda^{-1} I\big) x + b^T x,\\
h(x)& := x^T Q x + b^T x+ \Asp{\langlembda}{\psi}\big((I- \langlembda Q)x-\langlembda b\big).
\end{array}
\end{equation}
Our method {\em does not require} a DC decomposition of the objective function $\varphi_\langlembda$. Indeed the function $h$ in \eqref{func_h_Example55} is generally nonconvex. Specifically, consider $Q=\betagin{bsmallmatrix}0&-1\\-1&0\end{bsmallmatrix}$, $b=(0,0)^T$, and $C$ being the unit sphere centered at the origin. Then $g$ in \eqref{func_h_Example55} is strongly convex for any $\langlembda \in (0,1)$, while $h$ therein is not convex whenever $\langlembda >0$. More precisely, in this case we have
$$
h(x_1,x_2) = -2x_1x_2+ \Asp{\langlembda}{\psi}(x_1+\langlembda x_2,\langlembda x_1+x_2)\;\mbox{ with}
$$
\betagin{align*}
\Asp{\langlembda}{\psi}(x)= \frac{1}{2\langlembda}\left(\|x\|^2-d_C^2(x)\right)=\frac{1}{2\langlembda}\left(\|x\|^2-(\|x\|-1)^2\right)=\frac{1}{2\langlembda}\left(2\|x\|-1\right).
\end{align*}
This tells us, in particular, that
$$
h(-1/2,-1/2)-\frac{1}{2}h(-1,-1)-\frac{1}{2}h(0,0)=\frac{1}{2},
$$
and thus $h$ is not convex regardless of the value of $\langlembda$; see Figure~3.\vspace*{-0.2in}
\betagin{figure}[h!!]
\centering
\includegraphics[width=.7\textwidth]{Example_nonconvex_h.pdf}
\caption{Contour plot of the functions $f$, $\varphi_\langlembda$, $g$ and $h $ in \eqref{func_h_Example55} with $\langlembda =0.9$}
\end{figure}
\end{example}\vspace*{-0.15in}
\section{Further Applications and Numerical Experiments}\langlebel{sec:6}\vspace*{-0.1in}
In this section, we demonstrate the performance of Algorithm~\ref{alg:1} and Algorithm~\ref{alg:3} in two different problems. The first problem is smooth and arises from the study of system biochemical reactions. It can be successfully tackled with DCA-like algorithms, but they require to solve subproblems whose solutions cannot be analytically computed and are thus time-consuming. This is in contrast to Algorithm~\ref{alg:1}, which only requires solving the linear equation \eqref{EQALG01} at each iteration. The second problem is nonsmooth and consists of minimizing a quadratic function under both convex and nonconvex constraints. Employing FBE \eqref{FBE} and Theorem~\ref{diff_repr_varphi}, these two problems can be attacked by using DCA, BDCA, and Algorithm~\ref{alg:3}.
Both Algorithms~\ref{alg:1} and \ref{alg:3} have complete freedom in the choice of the initial value of the stepsizes $\bar{v}erline{\tau}_k$ in Step~4, as long as they are bounded from below by a positive constant $t_{\min}$, while the choice of $\bar{v}erline{\tau}_k$ totally determines the performance of the algorithms. On the one hand, a small value would permit the stepsize to get easily accepted in Step~5, but it would imply little progress in the iteration and (likely) in the reduction of the objective function, probably making it more prone to stagnate at local minima. On the other hand, we would expect a large value to ameliorate these issues, while it could result in a significant waste of time in the linesearch Steps~5-7 of both algorithms.
Therefore, it makes sense to consider a choice which sets the trial stepsize $\bar{v}erline{\tau}_k$ depending on the stepsize $\tau_{k-1}$ accepted in the previous iteration, perhaps increasing it if no reduction of the stepsize was needed. This technique was introduced in~\cite[Section~5]{MR4078808} under the name of \emptyseth{Self-adaptive trial stepsize}, and it was shown there that this accelerates the performance of BDCA in practice. A similar idea is behind the so-called \emptyseth{two-way backtracking} linesearch, which was recently proposed in~\cite{Truong2021} for the gradient descent method, showing good numerical results on deep neural networks. In contrast to BDCA, our theoretical results require $t_{\min}$ to be strictly positive, so the technique should be slightly adapted as shown in Algorithm~\ref{alg:self-adaptive}. Similarly to \cite{MR4078808}, we adopt a conservative rule of only increasing the trial stepsize $\bar{v}erline{\tau}_k$ when two consecutive trial stepsizes were accepted without decreasing them.\vspace*{0.1in}
\betagin{algorithm}[ht!]
\betagin{algorithmic}[1]
\mathbb{R}equire{$\gamma>1$, $\bar{v}erline{\tau}_0>0$}.
\mathcal{S}tate Obtain $\tau_0$ by Steps~5-7 of Algorithms~\ref{alg:1} or \ref{alg:3}.
\mathcal{S}tate Set $\bar{v}erline{\tau}_1:=\max\{\tau_0,t_{\min}\}$ and obtain $\tau_1$ by Steps~5-7 of Algorithms~\ref{alg:1} or \ref{alg:3}.
Fr\'{e}chetor{$k=2,3,\ldots$}
\If{$\tau_{k-2}=\bar{v}erline{\tau}_{k-2}$ \textbf{and} $\tau_{k-1}=\bar{v}erline{\tau}_{k-1}$}
\mathcal{S}tate $\bar{v}erline{\tau}_{k}:=\gamma\tau_{k-1}$;
\Else
\mathcal{S}tate $\bar{v}erline{\tau}_{k}:=\max\{\tau_{k-1},t_{\min}\}$.
\EndIf
\mathcal{S}tate Obtain $\tau_k$ by Steps~5-7 of Algorithms~\ref{alg:1} or \ref{alg:3}.
\EndFor
\end{algorithmic}
\caption{Self-adaptive trial stepsize}\langlebel{alg:self-adaptive}
\end{algorithm}
\vspace*{-0.05in}
The codes in the first subsection below were written and ran in MATLAB version R2021b, while for the second subsection we used Python~3.8. The tests were ran on a desktop of Intel Core i7-4770 CPU 3.40GHz with 32GB RAM, under Windows 10 (64-bit).\vspace*{-0.2in}
\subsection{Smooth DC Models in Biochemistry}\vspace*{-0.1in}
Here we consider the problem motivating the development of BDCA in \cite{AragonArtacho2018}, which consists of finding a steady state of a dynamical equation arising in the modeling of {\em biochemical reaction networks}. We ran our experiments on the same 14 biochemical reaction network models tested in \cite{AragonArtacho2018,MR4078808}.
The problem can be modeled as finding a zero of the function
$$f(x):=\left([F,R]-[R,F]\right)\exp\left(w+[F,R]^T x\right),$$
where $F,R\in\mathbb{Z}_{\geq 0}^{m\times n}$ denote the forward and reverse \emptyseth{stoichiometric matrices}, respectively, where $w\in\mathbb{R}^{2n}$ is the componentwise logarithm of the \emptyseth{kinetic parameters}, where $\exp(\cdot)$ is the componentwise exponential function, and where $[\,\cdot\,,\cdot\,]$ stands for the horizontal concatenation operator. Finding a zero of $f$ is equivalent to minimizing the function $\varphi(x):=\|f(x)\|^2$, which can be expressed as a {\em difference of the convex functions}
\betagin{equation}\langlebel{eq:bio_DCA}
g(x):=2\left(\|p(x)\|^2+\|c(x)\|^2\right)\quad\text{and}\quad h(x):=\|p(x)+c(x)\|^2,
\end{equation}
where the functions $p(x)$ and $c(x)$ are given by
\betagin{equation*}
p(x):=[F,R]\exp\left(w+[F,R]^T x\right)\quad\text{and}\quad c(x):=[R,F]\exp\left(w+[F,R]^T x\right).
\end{equation*}
In addition, it is also possible to write
\betagin{equation*}
\varphi(x)=\|f(x)\|^2=\|p(x)-c(x)\|^2=\|p(x)\|^2+\|c(x)\|^2-2p(x)c(x),
\end{equation*}
and so $\varphi(x)$ can be decomposed as the difference of the functions
\betagin{equation}\langlebel{eq:bio_ours}
g(x):=\|p(x)\|^2+\|c(x)\|^2 \quad\text{and}\quad h(x)=2p(x)c(x)
\end{equation}
with $g$ being convex. Therefore, $\nabla^2 g(x)$ is $0$-lower definite, and minimizing $\varphi$ can be tackled with Algorithm~\ref{alg:1} by choosing $\rho_k\geq\zeta$ for some fixed $\zeta>0$.
As shown in \cite{AragonArtacho2018}, the function $\varphi$ is real analytic and thus satisfies the Kurdyka--{\L}ojasiewicz assumption of Theorem~\ref{COR:Kur-Loj}, but as observed in \cite[Remark~5]{AragonArtacho2018}, a linear convergence rate cannot be guaranteed.
Our first task in the conducted experiments was to decide how to set the parameters $\zeta$ and $\rho_k$. We compared the strategy of taking $\rho_k$ equal to some fixed value for all $k$, setting a decreasing sequence bounded from below by $\zeta$, and choosing $\rho_k=c\|w_k\|+\zeta$ for some constant $c>0$. In spite of Remark~\ref{rem:theorem}(ii), $\zeta$ was added in the last strategy to guarantee both Theorem~\ref{The01}(ii) and Theorem~\ref{COR:Kur-Loj}. We took $\zeta=10^{-8}$ and a constant $c=5$, which worked well in all the models. We tried several options for the decreasing strategy, of which a good choice seemed to be $\rho_k=\frac{\|w_0\|}{10^{\lfloor k/50\rfloor}}+\zeta$, where $\lfloor\cdot\rfloor$ denotes the floor function (i.e., the parameter was initially set to $\|w_0\|$ and then divided by $10$ every 50 iterations). The best option was this decreasing strategy, as can be observed in the two models in Figure~\ref{fig:bio_rhos}, and this was the choice for our subsequent tests.\vspace*{-0.1in}
\betagin{figure}[ht!]
\centering
\includegraphics[width=.49\textwidth]{iAI549_compare_rhos.pdf}
\includegraphics[width=.49\textwidth]{Sc_thermophilis_rBioNet_compare_rhos.pdf}
\caption{Comparison of the objective values for three strategies for setting the regularization parameter $\rho_k$: constant (with values $10^6$, $10^5$, $10^3$ and $1$), decreasing, and adaptive with respect to the value of $\|w_k\|$.}\langlebel{fig:bio_rhos}
\end{figure}
\vspace*{-0.3in}
\betagin{experiment}\langlebel{exp1} For finding a steady state of each of the 14 biochemical models, we compared the performance of Algorithm~\ref{alg:1} and BDCA with self-adaptive strategy, which was the fastest method tested in~\cite{MR4078808} (on average, 6.7 times faster than DCA). For each model, 5 kinetic parameters were randomly chosen with coordinates uniformly distributed in $(-1,1)$, and 5 random starting points with random coordinates in $(-2,2)$ were picked. BDCA was ran using the same parameters as in~\cite{MR4078808}, while we took $\sigma=\betata=0.2$ for Algorithm~\ref{alg:1}.
We considered two strategies for setting the trial stepsize $\bar{v}erline{\tau}_k$ in Step~4 of Algorithm~\ref{alg:1}: constantly initially set to 50, and self-adaptive strategy (Algorithm~\ref{alg:self-adaptive}) with $\gamma=2$ and $t_{\min}=10^{-8}$. For each model and each random instance, we computed 500 iterations of BDCA with self-adaptive strategy and then ran Algorithm~\ref{alg:1} until the same value of the target function $\varphi$ was reached. As in~\cite{AragonArtacho2018}, the BDCA subproblems were solved by using the function \texttt{fminunc} with \texttt{optimoptions(\char13 fminunc\char13,} \texttt{\char13 Algorithm\char13,} \texttt{\char13 trust-region\char13,} \texttt{\char13 GradObj\char13,} \texttt{\char13 on\char13,} \texttt{\char13 Hessian\char13,} \texttt{\char13 on\char13,} \texttt{\char13 Display\char13,} \texttt{\char13 off\char13,} \texttt{\char13 TolFun\char13,} \texttt{1e-8,} \texttt{\char13 TolX\char13,} \texttt{1e-8)}.
The results are summarized in Figure~\ref{fig:bio_ratio}, where we plot the ratios of the running times between BDCA with self-adaptive stepsize and Algorithm~\ref{alg:1} with constant trial stepsize against Algorithm~\ref{alg:1} with self-adaptive stepsize. On average, Algorithm~\ref{alg:1} with self-adaptive strategy was $6.69$ times faster than BDCA, and was $1.33$ times faster than Algorithm~\ref{alg:1} with constant strategy. The lowest ratio for the times of self-adaptive Algorithm~\ref{alg:1} and BDCA was $3.17$. Algorithm~\ref{alg:1} with self-adaptive stepsize was only once (out of the 70 instances) slightly slower (a ratio of 0.98) than with the constant strategy.\vspace*{-0.05in}
\betagin{figure}[ht!]
\centering
\includegraphics[width=.7\textwidth]{Bio_ratio.pdf}
\caption{Ratios of the running times of Algorithm~\ref{alg:1} with constant stepsize and BDCA with self-adaptive stepsize to Algorithm~\ref{alg:1} with self-adaptive stepsize. For each of the models, the algorithms were run using the same random starting points. The overall average ratio is represented with a dashed line}\langlebel{fig:bio_ratio}
\end{figure}\vspace*{-0.2in}
In Figure~\ref{fig:bio_comparison}, we plot the values of the objective function for each algorithm and also include for comparison the results for DCA and BDCA without self-adaptive strategy. The self-adaptive strategy also accelerates the performance of Algorithm~\ref{alg:1}. We can observe in Figure~\ref{fig:bio_taus} that there is a correspondence between the drops in the objective value and large increases of the stepsizes $\tau_k$ (in a similar way to what was shown for BDCA in~\cite[Fig.~12]{MR4078808}).\vspace*{-0.15in}
\betagin{figure}[ht!]
\centering
\includegraphics[width=.49\textwidth]{L_lactis_MG1363.pdf}
\includegraphics[width=.49\textwidth]{iAF692.pdf}
\caption{Value of the objective function (with logarithmic scale) of Algorithm~\ref{alg:1}, DCA and BDCA for two biochemical models. The value attained after 500 iterations of BDCA with self-adaptive stepsize is shown by a dashed line.}\langlebel{fig:bio_comparison}
\end{figure}
\betagin{figure}[ht!]
\centering
\includegraphics[width=.49\textwidth]{L_lactis_MG1363_taus.pdf}
\includegraphics[width=.49\textwidth]{iAF692_taus.pdf}
\caption{Comparison of the self-adaptive and the constant (with $\bar{v}erline{\tau}_k = 50$) choices for the trial
stepsizes in Step 4~of Algorithm~\ref{alg:1} for two biochemical models. The plots include two
scales, a logarithmic one for the objective function values and a linear one for the stepsizes (which
are represented with discontinuous lines).}\langlebel{fig:bio_taus}
\end{figure}
\end{experiment}
\subsection{Solving Constrained Quadratic Optimization Models}
\vspace*{-0.1in}
This subsection contains numerical experiments to solve problems of constrained quadratic optimization formalized by
\betagin{align}\langlebel{ProblemQUAD1}
\mbox{minimize }\;\frac{1}{2}x^T Q x + b^T x\; \text{ subject to }\;x\in C:=\bigcup_{i=1}^p C_i,
\end{align}
where $Q$ is a symmetric matrix (not necessarily positive-semidefinite), $b \in \mathbb{R}^n$, and $C_1,\ldots,C_p\subseteq \mathbb{R}^n$ are nonempty, closed, and convex sets.
When $C=\mathbb{B}_r(0)$ (i.e., $p=1$), this problem is referred as the {\em trust-region subproblem}. If $Q$ is positive-semidefinite, then \eqref{ProblemQUAD1} is a problem of {\em convex quadratic programming}. Even when $Q$ is not positive-semidefinite, Tao and An~\cite{Tao1998} showed that this particular instance of problem \eqref{ProblemQUAD1} could be efficiently addressed with the DCA algorithm by using the following DC decomposition:
\betagin{equation}\langlebel{eq:Quad_DCA_g_h}
g(x):=\frac{1}{2}\rho\|x\|^2+b^T x+\delta_{\mathbb{B}_r(0)},\quad h(x):=\frac{1}{2}x^T(\rho I-Q)x,
\end{equation}
where $\rho\geq\|Q\|_2$. However, this type of decomposition would not be suitable for problem \eqref{ProblemQUAD1} when $C$ is not convex.
As shown in Subsection~\ref{subsec:5.1}, problem \eqref{ProblemQUAD1} for $p\geq 1$ can be reformulated by using FBE \eqref{FBE} to be tackled with Algorithm~\ref{alg:3} with $\langlembda\in(0,\frac{1}{\|Q\|_2})$. Although the decomposition in \eqref{func_h_Example55} may not be suitable for DCA when $Q$ is not positive-definite, it can be regularized by adding $\frac{1}{2}\rho\|x\|^2$ to both $g$ and $h$ with $\rho\geq\max\{0,-2\langlembda_{\min}(Q)\}$. Such a regularization would guarantee the convexity of the resulting functions $g$ and $h$ given by
\betagin{align}
g(x)& := \frac{1}{2}x^T \left( Q + (\rho+\langlembda^{-1}) I\right) x + b^T x, \langlebel{eq:Quad_FBE_g_h}\\
h(x)& := \frac{1}{2}x^T \left(2Q+\rho I\right) x + b^T x+ \Asp{\langlembda}{\delta_C}((I- \langlembda Q)x-\langlembda b). \langlebel{eq:Quad_FBE_g_h_bis}
\end{align}
The function $g$ in \eqref{eq:Quad_DCA_g_h} is not smooth, but the function $g$ in~\eqref{eq:Quad_FBE_g_h} is. Then it is possible to apply BDCA \cite{MR4078808} to formulation \eqref{eq:Quad_FBE_g_h}--\eqref{eq:Quad_FBE_g_h_bis} in order to accelerate the convergence of DCA. Note that it would also be possible to do it with \eqref{eq:Quad_DCA_g_h} if the $\ell_1$ or $\ell_{\infty}$ balls were used; see~\cite{Artacho2019} for more details.\vspace*{0.03in}
Let us describe two numerical experiments to solve problem \eqref{ProblemQUAD1}. \vspace*{-0.05in}
\betagin{experiment}\langlebel{exp2}
Consider \eqref{ProblemQUAD1} with $C=\mathbb{B}_r(0)$ and replicate the hardest setting in \cite{Tao1998}, which was originally considered in~\cite{More1983}. Specifically, in this experiment we generated potentially difficult cases by setting $Q:=UDU^T$ for some diagonal matrix $D$ and orthogonal matrix $U:=U_1U_2U_3$ with $U_j:=I-2u_ju_j^T/\|u_j\|^2$, $j=1,2,3$. The components of $u_j$ were random numbers uniformly distributed in $(-1,1)$, while the elements in the diagonal of $D$ were random numbers in $(-5,5)$. We took $b:=Uz$ for some vector $z$ whose elements were random numbers uniformly distributed in $(-1,1)$ except for the component corresponding to the smallest element of $D$, which was set to $0$. The radius $r$ was randomly chosen in the interval $(\|d\|,2\|d\|)$, where $d_i:=z_i/(D_{ii}-\langlembda_{\min}(D))$ if $D_{ii}\neq \langlembda_{\min(D)}$ and $0$ otherwise.
For each $n\in\{100,200,\ldots,900,1000,1250,1500,\ldots,3750,4000\}$, we generated 10 random instances, took for each instance a random starting point in $\mathbb{B}_r(0)$, and ran from it the four algorithms described above: DCA applied to formulation~\eqref{eq:Quad_DCA_g_h} (without FBE), DCA and BDCA applied to~\eqref{eq:Quad_FBE_g_h}--\eqref{eq:Quad_FBE_g_h_bis}, and Algorithm~\ref{alg:3}. We took $\langlembda=0.8/\|Q\|_2$ as the parameter for FBE (both for DCA and Algorithm~\ref{alg:3}). The regularization parameter $\rho$ was chosen as $\max\{0,-2\langlembda_{\min}(Q)\}$ for DCA with FBE and $0.1+\max\{0,-2\langlembda_{\min}(Q)\}$ for BDCA, as $h$ should be strongly convex. Both Algorithm~\ref{alg:3} and BDCA were ran with the self-adaptive trial stepsize for the backtracking step introduced in~\cite{MR4078808} with parameters $\sigma=\betata=0.2$ and $\gamma=4$, and with $t_{\min}=10^{-6}$. For the shake of fairness, we did not compute function values for the runs of DCA at each iteration, since it is not required by the algorithm. Instead, we used for both versions of DCA the stopping criterion from~\cite{Tao1998} that $er\leq 10^{-4}$, where
$$er=\left\{\betagin{array}{lc}
\left\|x^{k+1}-x^k\right\| /\left\|x^k\right\| & \text { if }\left\|x^k\right\|>1, \\
\left\|x^{k+1}-x^k\right\|& \text { otherwise.}
\end{array}\right.$$
As DCA with FBE was clearly the slowest method, we took the function value of the solution returned by DCA without FBE as the target value for both Algorithm~\ref{alg:3} and BDCA, so these algorithms were stopped when that function value was reached. In Figure~\ref{fig:trust_region_ratio_small}, we plot the time ratio of each algorithm against Algorithm~\ref{alg:3}. On average, Algorithm~\ref{alg:3} was more than 5 times faster than DCA with FBE and more than 2 times faster than DCA without FBE. BDCA greatly accelerated the performance of DCA with FBE, but still Algorithm~\ref{alg:3} was more than 1.5 times faster. Only for size 300, the performance of DCA without FBE was comparable to that of Algorithm~\ref{alg:3}. We observe on the right plot that the advantage of Algorithm~\ref{alg:3} is maintained for larger sizes.
\betagin{figure}[ht!]
\centering
\includegraphics[width=.49\textwidth]{trust_region_ratio_0.pdf}
\includegraphics[width=.49\textwidth]{trust_region_ratio.pdf}
\caption{Time ratio for 10 random instances of DCA with FBE, DCA without FBE, and BDCA with respect to Algorithm~\ref{alg:3}. Average ratio within each size is represented with a triangle for DCA with FBE, with a square for DCA without FBE and with a circle for BDCA. The overall average ratio for each pair of algorithms is represented by a dotted line.}\langlebel{fig:trust_region_ratio_small}
\end{figure}
\end{experiment}
\betagin{experiment} With the aim of finding the minimum of a quadratic function with integer and box constraints, we modified the setting of Experiment~2 and considered instead a set $C$ composed by $9^n$ balls of various radii centered at $\{-4,-3,-2,-1,0,1,2,3,4\}^n$, with $n\in\{2,10,25,50,100,200,500,1000\}$. As balls of radius $1/2\sqrt{n}$ cover the region $[-4,4]^n$, we ran our tests with balls of radii $c/2\sqrt{n}$ with $c\in\{0.1,0.2,\ldots,0.8, 0.9\}$. This time we considered {\em both convex and nonconvex} objective functions. The nonconvex case was generated as in Experiment~2, while for the convex case, the elements of the diagonal of $D$ were chosen as random numbers uniformly distributed in $(0,5)$.
For each $n$ and~$r$, 100 random instances were generated. For each instance, a starting point was chosen with random coordinates uniformly distributed in $[-5,5]^n$. As the constraint sets are nonconvex, FBE was also needed to run DCA. The results are summarized in Table~\ref{tbl:integer}, where for each $n$ and each radius, we counted the number of instances (out of 100) in which the value of $\varphi_\langlembda$ at the rounded output of DCA and BDCA was lower and higher than that of Algorithm~\ref{alg:3} when ran from the same starting point. We used the same parameter settings for the algorithms as in Experiment~2. Finally, we plot in Figure~\ref{fig:integer} two instances in $\mathbb{R}^2$ in which Algorithm~\ref{alg:3} reached a better solution.\vspace*{-0.25in}
\betagin{table}[ht!]
\betagin{subtable}[t]{\textwidth}\centering
\scalebox{.85}{\centering
\betagin{tabular}{c c cc cc cc cc c}
\toprule
\multicolumn{2}{c}{}&\multicolumn{9}{c}{Radius of the balls}\\
\cmidrule[.7pt]{3-11}
& Alg.~\ref{alg:3} vs & $\frac{1}{20}\sqrt{n}$ & $\frac{2}{20}\sqrt{n}$ & $\frac{3}{20}\sqrt{n}$ & $\frac{4}{20}\sqrt{n}$ & $\frac{5}{20}\sqrt{n}$ & $\frac{6}{20}\sqrt{n}$ & $\frac{7}{20}\sqrt{n}$ & $\frac{8}{20}\sqrt{n}$ & $\frac{9}{20}\sqrt{n}$ \\
\midrule[.7pt]
\multirow{ 2}{*}{$n=2$}& DCA & 0/34 & 1/20 & 1/26 & 1/19 & 0/19 & 0/18 & 0/3 & 1/1 & 0/2\\
& BDCA & 6/12 & 2/13 & 3/14 & 4/9 & 0/4 & 1/10 & 0/2 & 1/1 & 0/2\\
\midrule[.5pt]
\multirow{ 2}{*}{$n=10$}& DCA & 2/89 & 2/83 & 1/66 & 5/53 & 8/28 & 3/7 & 1/1 & 3/1 & 1/0\\
& BDCA & 21/68 & 38/53 & 33/39 & 23/24 & 18/16 & 2/7 & 0/0 & 0/1 & 0/0 \\
\midrule[.5pt]
\multirow{ 2}{*}{$n=25$}& DCA & 0/99 & 0/98 & 2/87 & 11/58 & 9/32 & 3/8 & 2/9 & 2/2 & 5/4\\
& BDCA & 16/83 & 29/71 & 40/58 & 37/40 & 13/26 & 2/3 & 0/1 & 0/0 & 1/2\\
\midrule[.5pt]
\multirow{ 2}{*}{$n=50$}& DCA & 0/100 & 0/100 & 0/91 & 2/86 & 13/41 & 14/12 & 9/12 & 6/10 & 12/12\\
& BDCA & 8/92 & 6/94 & 31/69 & 36/53 & 16/28 & 8/8 & 6/5 & 3/4 & 5/3\\
\midrule[.5pt]
\multirow{ 2}{*}{$n=100$}& DCA & 0/100 & 0/100 & 0/99 & 9/87 & 18/49 & 18/31 & 12/22 & 18/20 & 11/21\\
& BDCA & 2/98 & 6/94 & 39/61 & 36/61 & 23/33 & 16/14 & 9/8 & 9/8 & 13/9\\
\midrule[.5pt]
\multirow{ 2}{*}{$n=200$}& DCA & 0/100 & 0/100 & 0/100 & 1/98 & 23/64 & 31/41 & 25/29 & 22/30 & 20/41\\
& BDCA & 3/97 & 2/98 & 38/62 & 37/63 & 33/39 & 27/17 & 18/18 & 14/13 & 16/18\\
\midrule[.5pt]
\multirow{ 2}{*}{$n=500$}& DCA & 0/100 & 0/100 & 0/100 & 1/99 & 6/94 & 15/80 & 27/61 & 29/65 & 36/48\\
& BDCA & 0/100 & 1/99 & 41/59 & 44/56 & 33/63 & 25/56 & 34/39 & 32/47 & 17/35\\
\bottomrule
\end{tabular}}
\caption{Convex case}
\end{subtable}
\betagin{subtable}[h]{\textwidth}\centering
\scalebox{.85}{\centering
\betagin{tabular}{c c cc cc cc cc c}
\toprule
\multicolumn{2}{c}{}&\multicolumn{9}{c}{Radius of the balls}\\
\cmidrule[.7pt]{3-11}
& Alg.~\ref{alg:3} vs & $\frac{1}{20}\sqrt{n}$ & $\frac{2}{20}\sqrt{n}$ & $\frac{3}{20}\sqrt{n}$ & $\frac{4}{20}\sqrt{n}$ & $\frac{5}{20}\sqrt{n}$ & $\frac{6}{20}\sqrt{n}$ & $\frac{7}{20}\sqrt{n}$ & $\frac{8}{20}\sqrt{n}$ & $\frac{9}{20}\sqrt{n}$ \\
\midrule[.7pt]
\multirow{ 2}{*}{$n=2$}& DCA & 1/8 & 1/9 & 1/10 & 1/6 & 0/6 & 0/6 & 0/8 & 3/2 & 1/0\\
& BDCA & 2/4 & 1/4 & 1/3 & 3/4 & 0/5 & 0/4 & 0/6 & 3/2 & 1/0\\
\midrule[.5pt]
\multirow{ 2}{*}{$n=10$}& DCA & 9/39 & 4/39 & 7/39 & 4/35 & 10/30 & 3/27 & 5/45 & 2/34 & 8/29\\
& BDCA & 9/31 & 11/33 & 13/29 & 6/31 & 11/29 & 6/25 & 7/38 & 5/29 & 10/30\\
\midrule[.5pt]
\multirow{ 2}{*}{$n=25$}& DCA & 6/69 & 13/67 & 7/62 & 5/61 & 10/53 & 3/59 & 6/56 & 3/72 & 3/66\\
& BDCA & 16/58 & 16/63 & 16/55 & 12/48 & 9/52 & 11/52 & 13/52 & 12/57 & 11/58\\
\midrule[.5pt]
\multirow{ 2}{*}{$n=50$}& DCA & 11/81 & 10/79 & 8/87 & 5/90 & 3/87 & 4/80 & 2/86 & 5/89 & 8/81\\
& BDCA & 24/68 & 21/64 & 23/70 & 17/73 & 14/75 & 9/73 & 10/75 & 18/74 & 16/71\\
\midrule[.5pt]
\multirow{ 2}{*}{$n=100$}& DCA & 4/96 & 6/94 & 4/94 & 5/94 & 4/96 & 3/97 & 2/98 & 7/91 & 9/91\\
& BDCA & 15/85 & 16/83 & 18/80 & 14/84 & 17/83 & 11/89 & 9/91 & 20/79 & 19/80\\
\midrule[.5pt]
\multirow{ 2}{*}{$n=200$}& DCA & 4/96 & 4/96 & 4/96 & 2/98 & 1/99 & 2/98 & 4/96 & 3/97 & 0/100\\
& BDCA & 11/89 & 16/84 & 11/89 & 8/92 & 6/94 & 11/89 & 10/90 & 13/87 & 8/92\\
\midrule[.5pt]
\multirow{ 2}{*}{$n=500$}& DCA & 1/99 & 2/98 & 0/100 & 0/100 & 0/100 & 1/99 & 1/99 & 2/98 & 1/99\\
& BDCA & 12/88 & 17/83 & 15/85 & 9/91 & 15/85 & 11/89 & 9/91 & 18/82 & 20/80\\
\bottomrule
\end{tabular}}
\caption{Nonconvex case}
\end{subtable}
\caption{For different values of $n$ (space dimension) we computed 100 random instances of problem~\eqref{ProblemQUAD1} with $Q$ positive definite and $C$ formed by the union of balls whose centers have integer coordinates between $-4$ and $4$. We counted the number of instances in which DCA and BDCA obtained a lower/upper value than Algorithm~\ref{alg:3}.}\langlebel{tbl:integer}
\end{table}\vspace*{-0.15in}
\betagin{figure}[ht!]
\centering
\includegraphics[height=.38\textwidth]{Quadratic_Integer_Nonconvex.png}
\includegraphics[height=.38\textwidth]{Quadratic_Integer_Convex.png}
\caption{Two instances of problem~\eqref{ProblemQUAD1}. On the left, both line searches of Algorithm~\ref{alg:3} and BDCA help to reach a better solution for a nonconvex quadratic function,
while only Algorithm~\ref{alg:3} succeeds on the right for the convex case.}\langlebel{fig:integer}
\end{figure}
\end{experiment}\vspace*{0.15in}
\section{Conclusion and Future Research}\langlebel{sec:7}\vspace*{-0.05in}
This paper proposes and develops a novel RCSN method to solve problems of difference programming whose objectives are represented as differences of generally nonconvex functions. We establish well-posedness of the proposed algorithm and its global convergence under appropriate assumptions. The obtained results exhibit advantages of our algorithm over known algorithms for DC programming when both functions in the difference representations are convex. We also develop specifications of the main algorithm in the case of structured problems of constrained optimization and conduct numerical experiments to confirm the efficiency of our algorithms in solving practical models.
In the future research, we plan to relax assumptions on the program data ensuring the linear, superlinear, and quadratic convergence rates for RCSN and also extend the spectrum of applications to particular classes of constrained optimization problems as well as to practical modeling.
\vspace*{-0.25in}
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\begin{document}
\title{{f The myth of the down converted photon}
\begin{abstract}
Parametric
down conversion (PDC)
is widely interpreted in terms of photons, but,
even among supporters of this interpretation,
many properties of the photon pairs have been
described as ``mind-boggling" and even ``absurd".
In this article we argue that a classical description
of the light field, taking account of its vacuum
fluctuations, leads us to a consistent and rational
description of all PDC phenomena. ``Nonlocality"
in quantum optics is simply an artifact of the
Photon Concept. We also predict a new phenomenon,
namely the appearance of a second, or satellite
PDC rainbow.
(This article will appear in the Proceedings of
the Second Vigier Conference held in York University,
Canada in August 1997. A somewhat more formal version
has been submitted to Phys. Rev. Letters, and
may be found at http:\slash\slash xxx.lanl.gov\slash abs
\slash quant-ph\slash
9711029.)
\end{abstract}
\section{Introduction}
In an article for the last conference in this series\cite{vig}
we gave a description of the Parametric Down Conversion (PDC)
process based on the real vacuum
electromagnetic-field fluctuations. We indicated that there
was a serious unsolved problem, in that detectors must
somehow subtract away these fluctuations; such a
mechanism must come into play in order to explain the
very low dark rates actually observed. We have since
published a series of
articles\cite{pdc1,pdc2,pdc3,pdc4}
in which a great
variety of PDC phenomena have been analyzed using
this description. Since we have
been able to establish a formal parallel, through
the Wigner representation, between the new
(or rather the old!) field description and the
presently dominant Photon Theory, it is
clear that, {\it once the reality of the zeropoint
field has been accepted, there are no PDC
phenomena which require photons.}
Furthermore we have made considerable progress
on the subtraction problem\cite{pdc4}; all that
is needed to explain the low dark rates of detectors is
the recognition of their extremely large time
windows (5ns is a very large number of light oscillations).
The approach of the above series of articles
was a kind of compromise between the standard
nonlocal theory of Quantum Optics, where the
interaction of the various field modes is
represented by a hamiltonian, and a fully
maxwellian theory, which would be both
local and causal. In this latter case the
nonlinear crystal would be represented as
a spatially localized current distribution,
modified of course by the incoming
electromagnetic field; the outgoing field
would then be expressed as the retarded
field radiated by this distribution.
A preliminary attempt at such a theory was made\cite{magic},
using first-order perturbation theory.
However, we showed, in the above series of articles, that a calculation
of the relevant counting rates, to lowest order, requires us to find the
{\it second}-order perturbation corrections to the Wigner density,
and the close formal parallel between these two theories
means that the same considerations will apply to
the maxwellian theory.
\section{What is PDC?}
It is necessary to pose this question, because,
depending on the answer given, PDC may be
described as either a local or a nonlocal
phenomenon.
An example of the modern, nonlocal description
is provided by Greenberger, Horne and Zeilinger\cite{ghz}.
A nonlinear crystal, pumped by a laser at frequency
$\omega_0$, produces conjugate pairs of signals,
of frequency $\omega$ and $\omega_0-\omega$ (see Fig.1).
Since light is supposed to consist
of photons, this means that an incoming laser
photon ``down converts" into a pair of lower-energy
photons. Naturally, since we know that $E=\hbar\omega$,
that means energy is conserved in the PDC process,
which must be very comforting. However, the above
authors themselves refer to the PDC photon-counting
statistics as ``mind-boggling", and a more recent
commentary\cite{zeil} even uses the term ``absurd".
\begin{figure}
\caption{PDC - the modern version. A laser photon
down converts into a conjugate pair of PDC photons
with conservation of energy.}
\end{figure}
There is an older description, which I suggest is
more correct than the modern one. It had only a short life.
Nonlinear optics was born in the late 1950s, with the
invention of the laser. Up to about 1965, when
Quantum Optics was born, the PDC process would
have been depicted\cite{saleh,yariv} by Fig.2; an incoming wave
of frequency $\omega$ is down converted, by
the pumped crystal, into an
\begin{figure}
\caption{PDC - the ancient version. When a wave of
frequency $\omega$ is incident, at a certain angle
$\theta(\omega)$, on a nonlinear crystal
pumped at frequency $\omega_0$, a signal
of frequency $\omega_0-\omega$ is emitted
in a certain conjugate direction. The modified
input wave is called the idler.}
\end{figure}
outgoing signal
of frequency $\omega_0-\omega$. The explanation
of the frequency relationships lies in the
multiplication, by the nonlinear crystal,
of the two input amplitudes; we have no need of $\hbar$!
This process persists when the intensity
of the input is reduced to zero, because
all modes of the light field are still
present in the vacuum, and the nonlinear crystal
modifies vacuum modes in exactly the same way
as it modifies input modes supplied by an
experimenter.
What we see emerging from the crystal is
the familiar PDC rainbow.
This is because the angle of incidence
$\theta$, at which PDC
occurs, is different for different frequencies
on account of the variation of refractive
index with frequency.
We depict the
process of PDC from the vacuum in Fig.3,
but note that this figure shows only two
conjugate modes of the light field; a complete
picture would show all frequencies participating
in conjugate pairs, with varying angles
of incidence. In contrast with Fig.2, where we
showed only the one relevant input, we must now
take account also of the conjugate input mode
of the zeropoint, since the first mode itself has
only the zeropoint amplitude.
The zeropoint inputs, denoted by interrupted lines in
Fig.3, do not activate photodetectors, because the
threshold of these devices is set precisely at the
level of the zeropoint intensity, as discussed in Ref.\cite{pdc4}.
\begin{figure}
\caption{PDC from the vacuum. Both of
the outgoing signals are above zeropoint intensity, and
hence give photomultiplier counts.}
\end{figure}
However,
the two idlers have intensities above
that of their corresponding inputs.
Also there is no
coherence between a signal and an idler of the same
frequency, so their intensities
are additive in both channels. Hence there are photoelectron
counts in both of the outgoing channels of Fig.3.
The question we have posed in this section could
be rephrased as ``What is it that is down converted?".
According to the thinking behind Fig.1, the laser
photons are down converted, whereas according to
Fig.3 it is the zeropoint modes; they undergo both
down conversion, to give signals, and amplification,
to give idlers.
\section{Photon production rates in PDC}
There is a small, but important difference between
the maxwellian theory and the theory outlined in
our Wigner series\cite{pdc1,pdc2,pdc3,pdc4}, though
both of them could be said to be based on Fig.3.
The Wigner series gave us the undulatory version
of quantum optics, but its starting point is a
hamiltonian which takes the creation of photon
pairs as axiomatic. The maxwellian theory, whose
details are given elsewhere\cite{puc1}, starts from
a nonlinear expression for the induced current
and deduces a coupling between the field modes.
This coupling is very similar, but not identical,
to that deduced from the Wigner-based theory.
As we have emphasized, there are no photons
in the maxwellian theory, but if we
translate the intensities of the outgoing
signals in Fig.3 into photon terms, we obtain the result
\begin{equation}
\frac{n_i(\omega)+n_s(\omega)}{n_i(\omega_0-\omega)+n_s(\omega_0-\omega)}=
\frac{\cos[\theta(\omega_0-\omega)]}
{\cos[\theta(\omega)]}\;. \label{pdcint}
\end{equation}
So we conclude that {\it the photon rate in
a given channel is inversely proportional to
the cosine of the rainbow angle}. In the Photon
Theory, the above ratio is one.
There seems little chance of finding out
directly which of these theories is correct;
the difference between the two ratios is small,
since the rainbow angles are typically around
10 degrees, and it is not possible to
measure at all accurately the efficiency
of light detectors as a function of
frequency. It is true that some of the
experiments we have analysed, using the
standard theory, in Refs.\cite{pdc1,pdc2,pdc3,pdc4},
have slightly different results in the present
theory, for example the fringe visibility in
the experiment of Zou, Wang and Mandel\cite{zwm}.
Some details will be published shortly, but we
can say that an experimental discrimination will
be very difficult.
\section{Parametric up conversion from the vacuum}
There is, however, at least one prediction
of the new theory which differs dramatically
from the standard theory. An incident wave
of frequency $\omega$, as well as being
down converted by the pump to give
a PDC signal of frequency $\omega_0-\omega$,
may also be {\it up converted to give a
PUC signal} of frequency $\omega_0+\omega$.
We depict this phenomenon, which is well
known\cite{saleh,yariv} in classical nonlinear optics,
in Fig.4.
\begin{figure}
\caption{PUC. In contrast with PDC
the output signal has its
transverse component in the same direction as that of
the idler.}
\end{figure}
Note that the angle of incidence, $\theta_u(\omega)$,
at which PUC occurs is quite different from the PDC
angle, which in Fig.2 was denoted simply $\theta(\omega)$,
but which we should now call $\theta_d(\omega)$.
Now, following the same argument which led us from
Fig.2 to Fig.3, we predict the phenomenon of
PUC from the Vacuum, which we depict in Fig.5.
When we come to calculate the intensity of the PUC rainbow,
there is an important difference from the PDC situation, because
we find that the idler intensities are now less than the
input zeropoint intensities. The signal intensities in
both channels almost, but not quite, cancel this shortfall,
so that the PUC intensities are only about
3 per cent of the PDC intensities, which may explain
why nobody has yet observed them. Also, note that there
is a detectable signal only in the lower-frequency
channel, because the relation corresponding to eq.(\ref{pdcint})
is
\begin{equation}
\frac{n_i(\omega)+n_s(\omega)}{n_i(\omega_0+\omega)+n_s(\omega_0+\omega)}=
-\frac{\cos[\theta_u(\omega_0+\omega)]}
{\cos[\theta_u(\omega)]}\;,
\end{equation}
which means that in one of the channels (actually the upper-frequency
one), the total output intensity is less than the zeropoint,
so nothing will be detected in this channel.
My prediction therefore is that, as well as the main
PDC rainbow $\theta_d(\omega)$, {\it there is also a satellite
rainbow}, whose intensity is about 3 percent of the
main one, at $\theta_u(\omega)$.
An approximate calculation\cite{puc1} shows that
$\theta_u(\omega)$ is about 2.5 times $\theta_d(\omega)$.
\begin{figure}
\caption{PUC from the vacuum. Only one
of the outgoing signals is above the zeropoint intensity. The
other one, depicted by an interrupted line, is below
zeropoint intensity.}
\end{figure}
\section{Conclusion}
Our contribution to the previous conference in this
series\cite{vig} was entitled ``The myth of the photon''.
The present article repeats this theme, but covers a
narrower range of phenomena. This is because the local
theory of nonlinear crystals is now very much more
complete than the corresponding theory for atoms. In
retrospect, the word ``obsolete'', which we used
in the previous article, for {\it all} photon theories,
was excessively triumphalist.
Of course one could argue that
they became obsolete once their nonlocal
nature was revealed, that is a quarter of a century ago,
but there was nothing local on offer at that time.
The claim we made, maybe prematurely, was
based on having demonstrated, by the use of
certain model theories with very limited fields
of application, that local theories were, in
all cases {\it possible}. Now we have passed
into a new phase of the programme; we now have,
for a very wide and growing area of investigation,
{\it a well defined alternative theory which
makes certain new predictions}. If and when such
predictions are verified, I think that
down-converted photons, for example those
depicted in our Fig.1, will be
very definitely obsolete.
\noindent
{\bf Acknowledgement}
\noindent
I have had a lot of help with the ideas behind this article,
and also in developing the argument, from Emilio Santos.
\end{document} |
\begin{document}
\twocolumn[
\icmltitle{Efficient Latency-Aware CNN Depth Compression via \\Two-Stage Dynamic Programming}
\icmlsetsymbol{equal}{*}
\begin{icmlauthorlist}
\icmlauthor{Jinuk Kim}{equal,snu,nprc}
\icmlauthor{Yeonwoo Jeong}{equal,snu,nprc}
\icmlauthor{Deokjae Lee}{snu,nprc}
\icmlauthor{Hyun Oh Song}{snu,nprc}
\end{icmlauthorlist}
\icmlaffiliation{snu}{
Department of Computer Science and Engineering, Seoul National University}
\icmlaffiliation{nprc}{
Neural Processing Research Center}
\icmlcorrespondingauthor{Hyun Oh Song}{[email protected]}
\icmlkeywords{Machine Learning, ICML}
\vskip 0.3in
]
#1rintAffiliationsAndNotice{\icmlEqualContribution}
\begin{abstract}
Recent works on neural network pruning advocate that reducing the depth of the network is more effective in reducing run-time memory usage and accelerating inference latency than reducing the width of the network through channel pruning. In this regard, some recent works propose depth compression algorithms that merge convolution layers. However, the existing algorithms have a constricted search space and rely on human-engineered heuristics. In this paper, we propose a novel depth compression algorithm which targets general convolution operations. We propose a subset selection problem that replaces inefficient activation layers with identity functions and optimally merges consecutive convolution operations into shallow equivalent convolution operations for efficient end-to-end inference latency. Since the proposed subset selection problem is NP-hard, we formulate a surrogate optimization problem that can be solved exactly via two-stage dynamic programming within a few seconds. We evaluate our methods and baselines by TensorRT for a fair inference latency comparison. Our method outperforms the baseline method with higher accuracy and faster inference speed in MobileNetV2 on the ImageNet dataset. Specifically, we achieve $1.41\times$ speed-up with $0.11$\%p accuracy gain in MobileNetV2-1.0 on the ImageNet.
\end{abstract}
\newcommand{\red}[1]{\textcolor{red}{#1}}
\newcommand{\blue}[1]{\textcolor{blue}{#1}}
\section{Introduction}
Deep learning with Convolutional Neural Network (CNN) has achieved outstanding results in various fields such as image classification, object detection, image segmentation, and generation gray!10itep{efficientnet,detection,segmentation,generation}. However, the success of CNNs in such fields is accompanied by the challenge of increased complexity and inference latency. For real-world applications, accelerating the inference latency of CNNs is of great practical importance, especially when deploying the models on edge devices with limited resources.
To this end, a line of research called channel pruning has been introduced to remove unnecessary channels in CNNs to accelerate the wall-clock time in the edge device while preserving the performance of the CNNs gray!10itep{wen2016learning,chipnet,halp}. However, with the advancement of hardware technology for parallel computation, channel pruning which reduces the width of neural networks has become less effective than removing entire layers in terms of latency gray!10itep{layerpruning,layerpruning2,layerpruning3,depthshrinker}.
In contrast, layer pruning, which prunes entire layers, has been proposed to reduce the depth of neural networks. Layer pruning also significantly reduces the run-time memory usage and achieves effective speed-up in many edge devices compared to channel pruning gray!10itep{layerpruning3}. However, layer pruning is more aggressive than channel pruning in terms of reducing the number of parameters and FLOPs, thereby resulting in a more severe accuracy drop compared to channel pruning methods. Instead of naively removing an entire layer, gray!10itet{depthshrinker} present a depth compression algorithm called DepthShrinker which integrates layers by replacing inefficient consecutive depth-wise convolution and point-wise convolution with an efficient dense convolution operation in MobileNetV2 gray!10itep{mobilenet}. This compression algorithm results in depth reduction with low run-time memory usage and fast inference latency similar to layer pruning. However, the depth compression algorithm does not suffer from a commensurate accuracy drop.
\input{fig_main.tex}
Although DepthShrinker has shown promising results in reducing the depth of the network while preserving the performance, the method is limited to constricted search space as it only considers merging within the Inverted Residual Block gray!10itep{depthshrinker, mobilenet}. Furthermore, the method relies on human-engineered heuristics for layer merging which is unlikely to scale to other architectures. To this end, we introduce a novel optimization-based framework for general convolution merging framework that is not restricted to the design of the network and does not rely on manually designed heuristics. We formulate a depth compression optimization problem that replaces inefficient activation layers with identity functions and optimally merges consecutive convolution operations for optimal latency.
Our optimization problem is NP-Hard and its objective requires a prohibitively exhaustive training of the neural network. Thus, we formulate a surrogate optimization problem by approximating the objective as the linear sum of the accuracy change induced by each network block. The surrogate optimization problem can be exactly optimized via dynamic programming on a given network architecture with a given latency. Furthermore, we evaluate the latency of the network with TensorRT for a fair comparison gray!10itep{tensorrt}. Our experiments show that the proposed method outperforms the baseline method in both the accuracy and the inference latency in MobileNetV2 on ImageNet dataset. We release the code at {\small\url{https://github.com/snu-mllab/Efficient-CNN-Depth-Compression}}.
\section{Related Works}
#1aragraph{Channel Pruning}
Channel pruning originally aims to reduce computation FLOPs by removing less important channels gray!10itep{pfec,fpgm,sfp, metaprun,amc,gate,apoz,perfmax}. Specifically, gray!10itet{knapsack} formulate a knapsack problem for channel pruning with an explicit FLOPs constraint. For practitioners, however, end-to-end inference wall-clock time is the most important metric. In light of this, gray!10itet{halp} build a latency lookup table and proposes a knapsack problem for channel pruning with a latency constraint.
#1aragraph{Network Morphism}
Our work is partially inspired by network morphism which morphs a trained parent network into a child network that functions identically gray!10itep{net2net,netmorph}. Here, the child network is larger than the parent network and is finetuned after morphing. Instead, we aim to find the parent network where some activation layers are removed, thereby morphing the parent network into the child network which has a faster inference time and functions almost identically to the parent network.
#1aragraph{Depth Reduction}
There are two lines of research that reduce the depth of neural networks: layer-pruning and depth compression. In layer pruning, gray!10itet{layerpruning} and gray!10itet{layerpruning2} evaluate the importance of layers by the amount of discriminative information in each feature map. In depth compression, DepthShrinker points out the inefficiency of depth-wise convolutions during inference in the edge device and proposes a depth compression algorithm that replaces inefficient consecutive depth-wise convolution and point-wise convolution inside the Inverted Residual Block with an efficient dense convolution gray!10itep{depthshrinker,depthconv,mobilenet}. We generalize depth compression space to cover any general convolution operations. Also, while DepthShrinker requires full training of the network during the search phase to identify the
unnecessary activations, our method employs importance evaluation which can be efficiently computed in an embarrasingly parallel fashion. Furthermore, we propose a novel two-stage dynamic programming algorithm which simultaneously finds the optimal set of selected activation layers and the optimal set of layers to be merged in a few seconds.
#1aragraph{TensorRT}
Choosing the appropriate implementation of the network to measure the inference latency is crucial for a fair comparison. For instance, a batch normalization (BN) module can be fused into the preceding convolution layer without compromising accuracy and accelerating the inference latency. In this regard, we utilize TensorRT to optimize trained network architectures with various techniques such as BN fusion, precision calibration and dynamic memory management gray!10itep{tensorrt}.
\section{Preliminary} \label{prelim}
Consider a $L$-layer CNN which consists of alternating convolution layer $f_{\theta_l}$ and activation layer $\sigma_l$ with the layer index $l \in [L]$. Each convolution layer is parametrized by convolution kernel parameter $\theta_l\in \mathbb{R}^{ C_{l-1} \times C_{l}\times K_l \times K_l}$, where $C_{l-1}, C_l, K_l$ represent the number of input channels, the number of output channels, and the kernel size, respectively. The CNN can be represented as a composite function $\BigO_{l=1}^L \sigma_l gray!10irc f_{\theta_l} : \mathbb{R}^{H_0\times W_0 \times K_0 \times K_0}\to \mathbb{R}^{H_L \times W_L \times K_L\times K_L}$, where $H_l,W_l$ are the height and width of $l$-th feature map and $\BigO$ denotes an iterated function composition. We set the last activation layer ($\sigma_L$) to identity function ($\mathrm{id}$).
Note, any consecutive convolution operations can be replaced by an equivalent convolution operation with a larger kernel due to the associative property. We denote this process as \emph{merging}. For example, consider two consecutive convolutional layers, $f_{\theta_1}$ and $f_{\theta_2}$, applied to an input image $X$ (\emph{i.e.} $f_{\theta_2}(f_{\theta_1}(X))$). This can also be computed using an equivalent \emph{merged} convolutional layer $f_{\theta_2 gray!10ircledast \theta_1}$ where $gray!10ircledast$ denotes convolution with proper padding. Further merging details can be found in \Cref{subsec:mergedetail}.
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green!20ef#1{#1}
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green!20ef#4{#4}
green!20ef#5{#5}
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green!20ef#1{#1}
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green!20ef#4{#4}
green!20ef#5{#5}
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green!20raw[#5] (BA#20) -- (BA#2#1);
green!20raw[#5] ($(BA#20) + (1,0)$) -- ($(BA#2#1) + (1,0)$);
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green!20raw[red!80!black!60] ($(BA#2\i)+(1,0)$) -- ($(BA#2\i)+(1,0.7)$);
green!20raw[red!80!black!60] (BA#2\i) -- ($(BA#2\i)+(1,0.0)$);
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green!20raw[black] ($(BA#2#1)+(1,0)$) -- ($(BA#2#1)+(1,0.7)$);
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\newcommand\bbconv[4]{
green!20ef#1{#1}
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green!20ef#3{#3}
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gray!10oordinate (B#2\i) at ($(#3,#4)+(0,0)$);
gray!10oordinate (DB#2\i) at ($(#3,#4)+(0.3,0.21)$);
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\end{scope}
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green!20raw[gray!80] (B#2\i) -- ($(B#2\i)+(0.3,0)$);
green!20raw[gray!80] (DB#2\i) -- ($(B#2\i)+(0.3,0)$);
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green!20raw[black] (B#20) -- (B#2#1) -- ($(B#2#1)+(0.3,0)$) -- ($(B#20)+(0.3,0)$) --cycle;
green!20raw[black] (DB#20) -- (DB#2#1) -- ($(B#2#1)+(0.3,0)$) -- ($(B#20)+(0.3,0)$) --cycle;
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green!20ef#1{#1}
green!20ef#2{#2}
green!20ef#3{#3}
green!20ef#4{#4}
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]
gray!10oordinate (EB#2\i) at ($(#3,#4)+(0,0)$);
gray!10oordinate (DEB#2\i) at ($(#3,#4)+(0.3,0.21)$);
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\end{scope}
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green!20raw[fill,gray!10,opacity=0.8] (EB#20) -- (EB#2#1) -- ($(EB#2#1)+(0.3,0)$) -- ($(EB#20)+(0.3,0)$) --cycle;
green!20raw[fill,gray!10,opacity=0.8] (DEB#20) -- (DEB#2#1) -- ($(EB#2#1)+(0.3,0)$) -- ($(EB#20)+(0.3,0)$) --cycle;
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]
green!20raw[gray!80] (EB#2\i) -- ($(EB#2\i)+(0.3,0)$);
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green!20ef#1{#1}
green!20ef#2{#2}
green!20ef#3{#3}
green!20ef#4{#4}
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gray!10oordinate (DEB#2\i) at ($(#3,#4)+(0.3*5/3,0.21*5/3)$);
\gridd{EB#2\i}{0.3*5/3}{0.21*5/3}
\end{scope}
}
green!20raw[fill,gray!10,opacity=0.8] (EB#20) -- (EB#2#1) -- ($(EB#2#1)+(0.3*5/3,0)$) -- ($(EB#20)+(0.3*5/3,0)$) --cycle;
green!20raw[fill,gray!10,opacity=0.8] (DEB#20) -- (DEB#2#1) -- ($(EB#2#1)+(0.3*5/3,0)$) -- ($(EB#20)+(0.3*5/3,0)$) --cycle;
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green!20raw[gray!80] (EB#2\i) -- ($(EB#2\i)+(0.3*5/3,0)$);
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green!20ef#1{#1}
green!20ef#2{#2}
green!20ef#3{#3}
green!20ef#4{#4}
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]
gray!10oordinate (EB#2\i) at ($(#3,#4)+(0,0)$);
gray!10oordinate (DEB#2\i) at ($(#3,#4)+(0.3*5/3,0.21*5/3)$);
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\end{scope}
}
green!20raw[fill,gray!10,opacity=0.8] (EB#20) -- (EB#2#1) -- ($(EB#2#1)+(0.3*5/3,0)$) -- ($(EB#20)+(0.3*5/3,0)$) --cycle;
green!20raw[fill,gray!10,opacity=0.8] (DEB#20) -- (DEB#2#1) -- ($(EB#2#1)+(0.3*5/3,0)$) -- ($(EB#20)+(0.3*5/3,0)$) --cycle;
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green!20raw[gray!80] (EB#2\i) -- ($(EB#2\i)+(0.3*5/3,0)$);
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green!20raw[black] (EB#20) -- (EB#2#1) -- ($(EB#2#1)+(0.3*5/3,0)$) -- ($(EB#20)+(0.3*5/3,0)$) --cycle;
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\newcommand\fbconvvv[4]{
green!20ef#1{#1}
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green!20ef#3{#3}
green!20ef#4{#4}
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]
gray!10oordinate (EB#2\i) at ($(#3,#4)+(0,0)$);
gray!10oordinate (DEB#2\i) at ($(#3,#4)+(0.3*7/3,0.21*7/3)$);
\griddd{EB#2\i}{0.3*7/3}{0.21*7/3}
\end{scope}
}
green!20raw[fill,gray!10,opacity=0.8] (EB#20) -- (EB#2#1) -- ($(EB#2#1)+(0.3*7/3,0)$) -- ($(EB#20)+(0.3*7/3,0)$) --cycle;
green!20raw[fill,gray!10,opacity=0.8] (DEB#20) -- (DEB#2#1) -- ($(EB#2#1)+(0.3*7/3,0)$) -- ($(EB#20)+(0.3*7/3,0)$) --cycle;
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yshift={100+gray!10hannelwidth*\i},every node/.append style={
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green!20raw[gray!80] (EB#2\i) -- ($(EB#2\i)+(0.3*7/3,0)$);
green!20raw[gray!80] (DEB#2\i) -- ($(EB#2\i)+(0.3*7/3,0)$);
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\begin{figure*}\label{fig:conv}
\end{figure*}
\section{Method} \label{method}
We first formulate an optimal subset selection problem for depth compression under a given latency constraint. Subsequently, we propose a surrogate objective for the objective in the optimal subset selection problem and formulate a corresponding surrogate optimization problem, which can be exactly solved via two-stage dynamic programming (DP).
\subsection{Optimal Subset Selection Problem for Depth Compression}
\input{text/method/obj.tex}
\subsection{Formulation with the Surrogate Objective}
\input{text/method/dp.tex}
\subsection{Optimization via Dynamic Programming}
\input{text/method/dp_opt.tex}
\subsection{Theoretical Analysis}
\input{text/method/dp_thm.tex}
\section{Experimental Results}
We evaluate our method on various datasets and network architectures.
We first introduce implementation details for the overall process in the experiments. Then, we present an evaluation of our method on various scales of networks and datasets to demonstrate its effectiveness. Furthermore, we conduct ablation studies on the search space of our proposed method and the ordered set $S$.
\subsection{Implementation Details}
\label{subsec:detail}
#1aragraph{Measurement} We first evaluate the latency of each contiguous network block, $T[i,j]$, individually. The latency of the network is subject to the format which it is implemented on. We utilize TensorRT to convert the network into its optimal form and measure the latency for a fair comparison.
Then, we measure the change of the accuracy incurred by each contiguous network block, $I[i,j]$, for \Cref{eq:imp}. As the number of possible contiguous network blocks is of the order of $N^2$, where $N$ is the number of activations, we need to train $\mathcal{O}(N^2)$ networks to obtain the accuracy change of every contiguous network block. For efficiency, we approximate the first term in \Cref{eq:imp} using the accuracy of the network trained for a few epochs after replacing the activation layers between the $i\!+\!1$-th and $j$-th layers with identity functions. Specific details on evaluating the importance of each block and the methodologies used to normalize the importance values can be found in the \Cref{app:imp}.
#1aragraph{Dynamic Programming} Given the latency of each contiguous network block, $T[i,j]$, and the accuracy change caused by each contiguous network block, $I[i,j]$, we can solve \Cref{eq:opt3} for the time constraint $T_0$ with \Cref{alg:optDP}. In \Cref{alg:optDP}, we assume the time constraint $T_0$ and time index $t$ to be integers. In practice, we multiply every occurrence of $t$ and $T_0$ by a constant factor and round the multiplied values to integer.
#1aragraph{Finetune and Merge} After obtaining the optimal ordered sets $A$ and $S$ in \Cref{eq:opt3}, we replace the activation layers not present in $A$ with identity functions.
In order to exactly merge the network in the inference phase, it is necessary to ensure that sufficient padding is applied to the first convolution layers within the target contiguous network blocks to be merged. To this end, we reorder the zero padding according to the set $S$ first, then finetune the network until convergence. We detail this padding reordering technique in \Cref{sup:padreorder}\footnote{We apply the same padding reordering technique when we reproduce the baseline work, DepthShrinker.}.
At the test time, we merge the finetuned network following $S$ and evaluate the latency.
During finetuning, we follow the identical training protocol with the DepthShrinker for finetuning gray!10itep{depthshrinker}. In detail, we finetune the network for 180 epochs using cosine learning rate decay with the SGD optimizer. We further adopt label smoothing, random erasing and RandAugment following gray!10itet{depthshrinker}, except in the case of MobileNetV2-1.0 on ImageNet where additional augmentation did not improve performance gray!10itep{labelsmoothing,randerase,randaug}.
#1aragraph{Evaluation}
We employ \textit{RTX2080 Ti} GPU when evaluating the latency of each contiguous network block. Then, we evaluate the end-to-end inference latency of merged architectures on various GPUs including \textit{TITAN Xp}, \textit{RTX2080 Ti}, \textit{RTX 3090}, and \textit{Tesla V100}. Also, we evaluate the inference latency of the networks in two distinct formats: 1) TensorRT exported model (FP32) and 2) PyTorch model gray!10itep{tensorrt, pytorch}. To ensure a fair comparison, we fuse the batch normalization (BN) modules with the previous convolution layers when we measure latency in the PyTorch format, as the depth compression algorithm results in a different number of BN modules.
\subsection{Depth Compression Results}
We apply our depth compression method to the MobileNetV2 architecture on ImageNet-100 and ImageNet dataset, starting from the public pretrained weight gray!10itep{mobilenet,imagenetsubset,imagenet}.
\subsubsection{ImageNet-100}
\input{tab/mbv2_small_in100_aug.tex}
\input{tab/mbv2_1.0_small_in.tex}
\input{tab/mbv2_1.4_in.tex}
We first experiment with our depth compression method on the ImageNet-100 dataset, which is a subset of ImageNet consisting of 100 classes. We bring the list of the subclasses from gray!10itet{imagenetsubset}. The size of the image is preprocessed to $224 \times 224$ and the dataset contains approximately 1200 images per class. We apply our depth compression method to both MobileNetV2-1.0 and MobileNetV2-1.4 starting from the pretrained weight and compare it to the architectures proposed in DepthShrinker.
When implementing the DepthShrinker on the ImageNet-100 dataset, we bring the architectures in DepthShrinker and finetune from the pretrained weight after substituting the last classifier to match the number of classes gray!10itep{depthshrinker}. Then we measure the latency of the merged network.
\Cref{tab:small-in100-mbv2-1.4} summarizes the depth compression results in MobileNetV2-1.0 and MobileNetV2-1.4. Our method consistently outperforms the baseline at every compression ratio in MobileNetV2-1.0 and MobileNetV2-1.4. In particular, we achieve $1.08 \times$ speedup in TensorRT compiled format with $1.13$\%p higher accuracy compared to DS-D-1.4. Also, we achieve $1.18 \times$ speedup with $0.11$\%p higher accuracy in TensorRT compiled format compared to DS-A-1.0.
Additionally, we evaluate the wall-clock inference time on various GPU platforms other than \textit{RTX 2080 Ti}. The comprehensive result of the latency on different GPUs can be found in \Cref{app:gpus}. Furthermore, we reproduce the full searching stage of DepthShrinker on top of the ImageNet-100 dataset and compare our method against the resulting architecture which we also provide the results in \Cref{app:in100-ds}.
\subsubsection{ImageNet}
We apply our depth compression method to MobileNetV2-1.0 and MobileNetV2-1.4 on the full ImageNet dataset gray!10itep{imagenet} and compare with the architectures proposed in DepthShrinker gray!10itep{depthshrinker}\footnote{DepthShrinker's official implementation omits merging the first Inverted Residual Block; following their paper, we merge it if their pattern removes the activation in this block.}. Note that every method uses the latency information of the \textit{RTX 2080 Ti} with TensorRT and is measured on different model formats and GPU platforms.
We use the same pretrained weight with DepthShrinker for a fair comparison and report the accuracy of it for the vanilla network. It is worth noting that this accuracy value differs from the one reported by gray!10itet{depthshrinker} because they reported the accuracy of the vanilla network from their baseline work instead of the pretrained weight they started from. We report the accuracy of the pretrained weight to precisely convey the effect of the compression methods.
\Cref{tab:small-in-mbv2-1.0} demonstrates that our method consistently outperforms the baseline in MobileNetV2-1.0 architecture on the ImageNet dataset. Specifically, our method attains $1.08 \times$ speedup with $0.46$\%p higher accuracy compared to DS-A-1.0. We present the comprehensive table including the latency on different GPUs in \Cref{app:gpus}.
\Cref{tab:in-mbv2-1.4} shows the result of applying our method to MobileNetV2-1.4. The result demonstrates that our method outperforms the baseline method in every compression ratio and across all model formats and GPU platforms. In particular, our method achieves $1.07 \times$ speedup in TensorRT compiled format with higher accuracy compared to MBV2-1.4-DS-C. Compared to the pretrained network, our compressed network achieves $1.61 \times$ speedup in TensorRT compiled format and $1.91 \times$ speedup without TensorRT with $1.60$\%p accuracy drop.
We further present the results of applying knowledge distillation from the pretrained weight when we finetune the compressed networks. \Cref{tab:dist-mbv2} shows that adopting the knowledge distillation technique further boosts the accuracy of the compressed networks. Specifically for MobileNetV2-1.0, our method achieves 1.41$\times$ speedup in TensorRT format and 1.62$\times$ speedup in PyTorch without losing accuracy from the pretrained weight.
\input{tab/mbv2_dist.tex}
\input{fig/ablmerge.tex}
\input{fig_qual.tex}
\subsection{Ablation Study on Ordered Set to be Merged}
Recall the definition of $A$ and $S$: $A$ indicates locations where the activation layer is not replaced with an identity function and $S$ indicates indices where we do not merge. The set $S$ always includes $A$ since the activation layers that are not $\mathrm{id}$ cannot be merged. One could argue that we can merge the layers with respect to $A$, without separately computing the optimal merge pattern $S$. In this ablation study, we compare the inference time of the merged network according to $A$ and $S$. \Cref{fig:ablmerge} shows that the network merged according to $S$ is about $30\%$ faster than the network merged according to $A$. This demonstrates that jointly optimizing over $A$ and $S$ simultaneously is crucial for optimal depth compression.
\subsection{An Illustration of the Larger Search Space}
The scope of the DepthShrinker is restricted to the cases where merging operation occurs within the Inverted Residual Block gray!10itep{depthshrinker}. On the other hand, our merging algorithm can handle any series of convolution operations and is agnostic to any specific block structure. For instance, our method finds the architecture that merges across the blocks, which DepthShrinker cannot find as shown in \Cref{fig:qual}. Our method allows us to merge more general series of layers enabling us to discover a more diverse kind of efficient structure.
\section{Conclusion}
We propose an efficient depth compression algorithm to reduce the depth of neural networks for the reduction in run-time memory usage and fast inference latency. Our compression target includes any general convolution operations, whereas existing methods are limited to consecutive depth-wise convolution and point-wise convolution within Inverted Residual Block. We propose a subset selection problem which replaces inefficient activation layers with identity functions and optimally merges consecutive convolution operations into shallow equivalent convolution operations for fast end-to-end inference latency. Since the optimal depth subset selection problem is NP-hard, we formulate a surrogate optimization problem which can be exactly solved via two-stage dynamic programming within a few seconds. We evaluate our methods and baselines by TensorRT for a fair inference latency comparison. Our method outperforms Depthshrinker with a higher accuracy and faster inference speed in MobileNetV2 on the ImageNet dataset. Specifically, we achieve $1.41\times$ speed-up with $0.11$\%p accuracy gain in MobileNetV2-1.0 on the ImageNet.
\appendix
\onecolumn
\section{Proof}
\label{app:proof}
\begin{proposition}
$A[l,t]$,and $S[l,t]$ computed from the DP recurrence relations, \Cref{eq:dprecur} are the optimal sets $A$ and $S$ of \Cref{eq:opt4}, respectively.
\end{proposition}
\begin{proof}
For given $(l_0, t_0)$, we suppose for all $l<l_0$ and $t<t_0$, $(A[l, t], S[l, t])$ computed from the DP recurrence, \Cref{eq:dprecur} are the optimal $(A,S)$ of \Cref{eq:opt4}, respectively. When $(l,t)=(l_0,t_0)$,
\begin{align*}
\sum_{s_{i\!-\!1},s_i \in \{0\} gray!10up S[l_0, t_0] gray!10up\{l_0\}} T[s_{i\!-\!1}, s_i]
&= \sum_{s_{i\!-\!1},s_i \in \{0\} gray!10up S[k_0, t_0\!- \!T_\text{opt}[k_0,l_0]] gray!10up \{k_0\}gray!10up S_\text{opt}[k_0,l_0]gray!10up\{l_0\}} T[s_{i\!-\!1}, s_i] \tag{by \Cref{eq:dprecurs}} \\
&= \sum_{s_{i\!-\!1},s_i \in \{0\} gray!10up S[k_0, t_0\!- \!T_\text{opt}[k_0,l_0]] gray!10up \{k_0\}} T[s_{i\!-\!1}, s_i] +
\sum_{s_{i\!-\!1},s_i \in \{k_0\}gray!10up S_\text{opt}[k_0,l_0]gray!10up\{l_0\}} T[s_{i\!-\!1}, s_i]
\\
&= \sum_{s_{i\!-\!1},s_i \in \{0\} gray!10up S[k_0, t_0\!- \!T_\text{opt}[k_0,l_0]] gray!10up \{k_0\}} T[s_{i\!-\!1}, s_i] + T_\text{opt} [k_0,l_0] \tag{by \Cref{eq:timedp2}}\\
&<\left(t_0 - T_\text{opt}[k_0,l_0]\right) + T_\text{opt} [k_0,l_0] = t_0,\tag{by the optimality assumption for $k_0<l_0$ and $t_0-T_{\text{opt}}[k_0,,l_0]<t_0$}
\end{align*}
where
\begin{align*}
k_0&= \underset{0\leq k'<l}{\mathrm{argmax}} ~
\sum_{a_{j\!-\!1},a_j \in \{0\}gray!10up A[k', t_0\!-\!T_\text{opt}[k',l_0]]gray!10up \{k',l\}} I[a_{j\!-\!1}, a_{j}]
\nonumber\\
&~~\text{subject to } ~~T_\text{opt}[0,k'] +T_\text{opt}[k',l]< t_0.\nonumber\\
\end{align*}
Assume that $(A[l_0,t_0], S[l_0,t_0])$ obtained using \Cref{eq:dprecur} are not optimal $(A, S)$ and $(A^*, S^*)$ are the optimal $(A, S)$ of \Cref{eq:opt4} when $(l,t)=(l_0,t_0)$. Then,
\begin{subequations}
\begin{align}
\sum_{a_{j\!-\!1},a_j \in \{0\}gray!10up A^*gray!10up \{l_0\}} I[a_{j\!-\!1}, a_{j}]&> \sum_{a_{j\!-\!1},a_j \in \{0\}gray!10up A[l_0,t_0]gray!10up \{l_0\}} I[a_{j\!-\!1}, a_{j}]\label{eq:proofoptimp}\\
\sum_{s_{i\!-\!1},s_i \in \{0\} gray!10up S^* gray!10up\{l_0\}} T[s_{i\!-\!1}, s_i]&<t_0, \label{eq:proofopttime}
\end{align}
\end{subequations}
where $A^* \subseteq S^* \subseteq [l_0-1]$.
$A^*$ is not an empty set due to \Cref{eq:proofoptimp} and
\begin{align*}
\sum_{a_{j\!-\!1},a_j \in \{0\}gray!10up A[l_0,t_0]gray!10up \{l_0\}} I[a_{j\!-\!1}, a_{j}]
&= \sum_{a_{j\!-\!1},a_j \in \{0\}gray!10up A[k_0, t_0\!-\!T_\text{opt}[k_0,l_0]]gray!10up \{k_0,l_0\}} I[a_{j\!-\!1}, a_{j}]\tag{by \Cref{eq:dprecura}}\\
&\geq \sum_{a_{j\!-\!1},a_j \in \{0\}gray!10up A[0, t_0\!-\!T_\text{opt}[0,l_0]]gray!10up \{0,l_0\}} I[a_{j\!-\!1}, a_{j}]\tag{by the definition of $k_0$}\\
&=\sum_{a_{j\!-\!1},a_j \in \{0\}gray!10up \emptysetgray!10up \{l_0\}} I[a_{j\!-\!1}, a_{j}]].\tag{by the base case condition}
\end{align*}
Then, let $k^*$ be the maximum value of set $A^*$.
We define $A' = A^*\setminus\{k^*\}$, $S'_{<k^*}= \{s\in S^*\mid s< k^*\}$, and $S'_{>k^*}= \{s\in S^*\mid s> k^*\}$. The upper bound of $T(S'_{<k^*}, 0, k^*)$ is given as follows:
\begin{align*}
\sum_{s_{i\!-\!1},s_i \in \{0\} gray!10up S'_{<k^*} gray!10up\{k^*\}} T[s_{i\!-\!1}, s_i]
&= \sum_{s_{i\!-\!1},s_i \in \{0\} gray!10up S^* gray!10up\{l_0\}} T[s_{i\!-\!1}, s_i]
- \sum_{s_{i\!-\!1},s_i \in \{k^*\} gray!10up S'_{>k^*} gray!10up\{l_0\}} T[s_{i\!-\!1}, s_i]
\nonumber\\
&\leq \sum_{s_{i\!-\!1},s_i \in \{0\} gray!10up S^* gray!10up\{l_0\}} T[s_{i\!-\!1}, s_i] - T_\text{opt}[k^*, l_0]\tag{by \Cref{eq:timedp1}}\nonumber\\
&< t_0 - T_\text{opt}[k^*, l_0].\tag{by \Cref{eq:proofopttime}}\nonumber
\end{align*}
Therefore, the optimality assumption of $A[k^*, t_0- T_\text{opt}[k^*, l_0]]$ in \Cref{eq:opt4} leads to the inequality:
\begin{align}
\label{eq:proofoptimp3}
\sum_{a_{j\!-\!1},a_j \in \{0\}gray!10up A[k^*, t_0 - T_\text{opt}[k^*, l_0] ]gray!10up \{k^*\}} I[a_{j\!-\!1}, a_{j}]\geq \sum_{a_{j\!-\!1},a_j \in \{0\}gray!10up A'gray!10up \{k^*\}} I[a_{j\!-\!1}, a_{j}].
\end{align}
Thus,
\begin{align*}
\sum_{a_{j\!-\!1},a_j \in \{0\}gray!10up A[l_0,t_0]gray!10up \{l_0\}} I[a_{j\!-\!1}, a_{j}]
&= \sum_{a_{j\!-\!1},a_j \in \{0\}gray!10up A[k_0, t_0\!-\!T_\text{opt}[k_0,l_0]]gray!10up \{k_0,l_0\}} I[a_{j\!-\!1}, a_{j}]\tag{by \Cref{eq:dprecura}}\\
&\geq \sum_{a_{j\!-\!1},a_j \in \{0\}gray!10up A[k^*, t_0\!-\!T_\text{opt}[k^*,l_0]]gray!10up \{k^*,l_0\}} I[a_{j\!-\!1}, a_{j}]\tag{by the definition of $k_0$}\\
&= \sum_{a_{j\!-\!1},a_j \in \{0\}gray!10up A[k^*, t_0 - T_\text{opt}[k^*, l_0] ]gray!10up \{k^*\}} I[a_{j\!-\!1}, a_{j}] + I [k^*, l_0] \\
&\geq \sum_{a_{j\!-\!1},a_j \in \{0\}gray!10up A'gray!10up \{k^*\}} I[a_{j\!-\!1}, a_{j}] + I[k^*, l_0] \tag{by \Cref{eq:proofoptimp3}}\\
&= \sum_{a_{j\!-\!1},a_j \in \{0\}gray!10up A'gray!10up \{k^*,l_0\}} I[a_{j\!-\!1}, a_{j}]= \sum_{a_{j\!-\!1},a_j \in \{0\}gray!10up A^*gray!10up \{l_0\}} I[a_{j\!-\!1}, a_{j}].
\end{align*}
This contradicts with \Cref{eq:proofoptimp}. Therefore, our assumption that $(A[l_0,t_0], S[l_0,t_0])$ obtained using DP recurrence relation are not optimal $(A, S)$ of \Cref{eq:opt4} is false. Thus, $(A[l,t], S[l,t])$ are optimal $(A, S)$ of \Cref{eq:opt4}.
\end{proof}
\begin{proposition}
$S[l,t]$ computed from the DP recurrence relations, \Cref{eq:dprecur} is the optimal $S$ which minimizes the latency of the network when $A[l,t]$ is fixed. Concretely, $S[l,t]$ is the optimal $S$ of the optimization problem:
\begin{align}
\label{eq:opts}
\underset{A[l,t]\subseteq S \subseteq [l\!-\!1]}{\mathrm{min}} \sum_{s_{i\!-\!1},s_i \in \{0\}gray!10up Sgray!10up \{l\}} T[s_{i\!-\!1}, s_{i}] .
\end{align}
\end{proposition}
\begin{proof}
When $l\!=\!1$, $S[l,t]=\emptyset$ which satisfies \Cref{eq:opts} by \Cref{eq:dprecurs}. For given $(l_0, t_0)$, we suppose for all $l<l_0$ and $t<t_0$, \Cref{eq:opts} is satisfied. Then, we assume that $S[l_0,t_0]$ obtained using \Cref{eq:dprecurs} is not optimal $S$ and $S^*$ are the optimal $S$ of \Cref{eq:opts} when $(l,t)=(l_0,t_0)$. Then, $A[l_0,t_0] \subseteq S^*$ and
\begin{align}
\sum_{s_{i\!-\!1},s_i \in \{0\}gray!10up S[l_0,t_0]gray!10up \{l_0\}} T[s_{i\!-\!1}, s_{i}] > \sum_{s_{i\!-\!1},s_i \in \{0\}gray!10up S^*gray!10up \{l_0\}} T[s_{i\!-\!1}, s_{i}]
\label{eq:assump0}
\end{align}
We can divide two cases whether $A[l_0,t_0]$ is an empty set or not.
#1aragraph{Case1: $A[l_0,t_0]$ is an empty set}
$S[l_0,t_0] = S_\text{opt}[0,l_0]$ by \Cref{eq:dprecurs}. Then, $S_\text{opt}[0,l_0]$ is the optimal $S$ of \Cref{eq:opts} when $(l,t)=(l_0,t_0)$ which contradicts with our assumption that $S[l_0,t_0]$ is not optimal $S$ of \Cref{eq:opts} when $(l,t)=(l_0,t_0)$.
#1aragraph{Case2: $A[l_0,t_0]$ is not an empty set}
Let $k_0$ be the maximum value of set $A[l_0,t_0]$. Then, we define $A' = A[l_0, t_0]\setminus\{k_0\}$, $S'_{<k_0}= \{s\in S^*\mid s< k_0\}$, and $S'_{>k_0}= \{s\in S^*\mid s> k_0\}$. By the definition, $A[k_0,t_0-T_\text{opt}[k_0,l_0]] \subseteq S'_{<k_0}$. Then, by the optimality assumption for $k_0<l_0$ and $t_0-T_\text{opt}[k_0,l_0]<t_0$,
\begin{align}
\sum_{s_{i\!-\!1},s_i \in \{0\} gray!10up S'_{<k_0} gray!10up\{k_0\}} T[s_{i\!-\!1}, s_i] \geq \sum_{s_{i\!-\!1},s_i \in \{0\} gray!10up S[k_0, t_0-T_\text{opt}[k_0,l_0]] gray!10up\{k_0\}} T[s_{i\!-\!1}, s_i].
\label{eq:assump}
\end{align}
\begin{align*}
\sum_{s_{i\!-\!1},s_i \in \{0\} gray!10up S^* gray!10up\{l_0\}} T[s_{i\!-\!1}, s_i]
&= \sum_{s_{i\!-\!1},s_i \in \{0\} gray!10up S'_{<k_0} gray!10up\{k_0\}} T[s_{i\!-\!1}, s_i]
+ \sum_{s_{i\!-\!1},s_i \in \{k_0\} gray!10up S'_{>k_0} gray!10up\{l_0\}} T[s_{i\!-\!1}, s_i]\\
&\geq \sum_{s_{i\!-\!1},s_i \in \{0\} gray!10up S[k_0, t_0-T_\text{opt}[k_0,l_0]] gray!10up\{k_0\}} T[s_{i\!-\!1}, s_i] + \sum_{s_{i\!-\!1},s_i \in \{k_0\} gray!10up S'_{>k_0} gray!10up\{l_0\}} T[s_{i\!-\!1}, s_i]\tag{by \Cref{eq:assump}}\\
&\geq \sum_{s_{i\!-\!1},s_i \in \{0\} gray!10up S[k_0, t_0-T_\text{opt}[k_0,l_0]] gray!10up\{k_0\}} T[s_{i\!-\!1}, s_i] + \sum_{s_{i\!-\!1},s_i \in \{k_0\} gray!10up S_{\text{opt}} gray!10up\{l_0\}} T[s_{i\!-\!1}, s_i]
\tag{by \Cref{eq:timedp2}}\\
&\geq \sum_{s_{i\!-\!1},s_i \in \{0\} gray!10up S[k_0, t_0-T_\text{opt}[k_0,l_0]] gray!10up\{k_0\}gray!10up S_{\text{opt}} gray!10up\{l_0\}} T[s_{i\!-\!1}, s_i]\\
&= \sum_{s_{i\!-\!1},s_i \in \{0\} gray!10up S[l_0,t_0] gray!10up\{l_0\}} T[s_{i\!-\!1}, s_i]. \tag{by \Cref{eq:dprecurs}}
\nonumber
\end{align*}
This contradicts with \Cref{eq:assump0}. Therefore, our assumption that $S[l_0,t_0]$ obtained using DP recurrence relation is not optimal $S$ of \Cref{eq:opts} when $(l,t)=(l_0,t_0)$ is false. Thus, $S[l,t]$ is optimal $S$ of \Cref{eq:opts}.
\end{proof}
\section{Measuring the Importance}
\label{app:imp}
\begin{minipage}{0.49gray!10olumnwidth}
\input{algorithm/impdp.tex}
\end{minipage}
\begin{minipage}{0.49gray!10olumnwidth}
\input{algorithm/extdp.tex}
\end{minipage}
\subsection{Extension on the Importance}
\label{app:imp-ext}
In \Cref{method}, the $j$-th activation layer of our target network for depth compression is either the $j$-th activation layer in the vanilla network ($\sigma_j$) or an identity function ($\mathrm{id}$). Thus, if $\sigma_j=\mathrm{id}$, the $j$-th activation layer in the target network is inherently an identity function.
For instance, MobileNetV2 has an identity function as an activation layer at the end of each Inverted Residual Block and the corresponding activation layers in our target network are bound to be $\mathrm{id}$ gray!10itep{mobilenet}.
On the other hand, non-linear activation layers at the end of the Inverted Residual Block can improve the performance of the networks compressed from MobileNetV2 gray!10itep{depthshrinker}.
To this end, we extend the search space of our method by further introducing the network blocks that have a non-linear activation layer at these positions and incorporating them into the DP formulation.
gray!10learpage
Consider a network block from $i\!+\!1$-th layer to $j$-th layer.
We introduce discrete variables $d_i, d_j \in \{0,1\}$ to indicate whether the first and the last activation layer of the network block are identity functions or not, respectively.
If $\sigma_i$ and $\sigma_j$ are not identity functions, then we limit $d_i$ and $d_j$ to $1$, respectively.
Then, we redefine the importance of the network block between $i\!+\!1$-th layer to $j$-th layer as $I[i, j, d_i, d_j]$.
Concretely, we redefine the importance as follows:
{\footnotesize
\begin{align} I[i, j, d_i, d_j] &gray!10oloneqq \max_\theta
\mathrm{Acc}\left(
\underbrace{\BigO_{l=j+1}^{L} \sigma_l gray!10irc f_{\theta_l}}_{\text{$j\!+\!1$ to $L$ layers}}
gray!10irc
\underbrace{\left(d_j\sigma + (1-d_j) \mathrm{id} \right)}
_{\text{$j$-th activation}}
gray!10irc \underbrace{\BigO_{l=i+1}^{j} f_{\theta_l}}_{\text{$i\!+\!1$ to $j$ layers}}
gray!10irc
\underbrace{\left(d_i\sigma + (1-d_i) \mathrm{id} \right)}_{\text{$i$-th activation}}
gray!10irc
\underbrace{f_{\theta_i}
gray!10irc\BigO_{l=1}^{i\!-\!1} \sigma_l gray!10irc f_{\theta_l} }_{\text{$1$
to $i$ layers}}\right)\nonumber\\
&- \max_\theta
\mathrm{Acc}\left(
\BigO_{l=1}^L \sigma_l gray!10irc f_{\theta_l} \right),
\label{eq:ext-imp}
\end{align}}
\hspace{-0.2em}where $\sigma$ is the activation layer that is not an identity function.
Due to the redefinition of importance, we propose an alternative surrogate for objective in \Cref{eq:opt1_obj} as follows:
\begin{align}
& \mathcal{I}(A, B) gray!10oloneqq
\sum_{b_{j\!-\!1},b_j \in \{0\}gray!10up Bgray!10up \{L\}} I[b_{j\!-\!1}, b_{j}, \mathds{1}_{A} (b_{j-1}), \mathds{1}_{A}(b_j)],
\label{eq:ext-sur}
\end{align}
where $A \subseteq B \subseteq [L-1]$. $A$ denotes the positions of activations which are not identity functions and $B$ denotes the boundary points of the contiguous network blocks for objective approximation. Then, the objective extends to
\begin{align}
\label{eq:opt-ext}
&\underset{
A\subseteq B, S \subseteq [L-1] }{\mathrm{maximize}} \sum_{b_{j\!-\!1},b_j \in \{0\}gray!10up Bgray!10up \{L\}} \!I[b_{j\!-\!1}, b_{j}, \mathds{1}_{A}(b_{j-1}), \mathds{1}_{A}(b_j)] \\
&\mathrm{subject\ to\ } ~~ \sum_{s_{i\!-\!1},s_i \in \{0\}gray!10up Sgray!10up \{L\}} T[s_{i\!-\!1}, s_{i}] < T_0. \nonumber
\end{align}
Note, \Cref{eq:opt-ext} can be exactly solved with DP algorithm analogously by \Cref{alg:extDP}.
\subsection{Possible Combinations of Network Blocks}
In MobileNetV2, we empirically observed that the network blocks with identity functions on both edges unnecessarily degrade the performance by excessively reducing the number of activation functions in the compressed network. To address this issue, we set the importance value of the network blocks to negative infinity if $\sigma_i = \sigma_j = \mathrm{id}, d_j = 0$ and exclude them in the DP algorithm.
Furthermore, we only consider blocks that we can merge into a single layer; thus, the skip-connections in MobileNetV2 considerably reduce the number of possible blocks.
We also avoid merging in scenarios where a convolutional layer with a kernel size larger than 1 follows the stride 2 convolutional layer since it leads to a significant increase in kernel size gray!10itep{depthshrinker}.
In MobileNetV2, we have 171 different blocks to measure the latency ($T[i, j]$) and 315 different blocks to measure the importance ($I[i, j, d_i, d_j]$).
\subsection{Evaluating and Normalizing the Importance}
\label{subsec:impnorm}
When we evaluate the importance value in \Cref{eq:ext-imp}, we approximate the first term by substituting the activation layers within the block to identity functions and training the network for a few epochs from the pretrained weight.
The second term is considered as the accuracy of the pretrained weight itself.
In MobileNetV2, we approximate the first term in \Cref{eq:ext-imp} by training the deactivated network for a single epoch.
If the block size is one (\emph{i.e.}, $k-l = 1$), we re-initialize the corresponding block and measure the accuracy drop after training it from the pretrained weight.
When we approximate the first term in \Cref{eq:ext-imp} with the accuracy attained after training it for a few epochs, we tend to calculate a lower importance value than the actual definition of the importance value. This effect is reflected independently for each block; thus, the more block we construct the network with, the more we underestimate the actual importance of the network. Therefore, it is crucial to normalize the importance values by adding an appropriate value to the importance of each block to address this issue.
To this end, we add the constant multiple of the average importance of the blocks of size one to normalize the importance of each block.
Concretely, we define the set $D$ as
\begin{align*}
D = \bigg\{&\mathrm{Acc}\left(\texttt{one-epoch}\left(
f\right)\right) - \max_\theta \mathrm{Acc}\left(\BigO_{l=1}^L \sigma_l gray!10irc f_{\theta_l} \right) \bigg|\\ & f = \BigO_{l=i+1}^{L} \sigma_l gray!10irc f_{\theta_l}
gray!10irc
\left(d_{i+1}\sigma + (1-d_{i+1}) \mathrm{id} \right)
gray!10irc
f_{\theta'}
gray!10irc
\left(d_i\sigma + (1-d_i) \mathrm{id} \right)
gray!10irc\BigO_{l=1}^{i\!-\!1} \sigma_l gray!10irc f_{\theta_l} \nonumber \\
& \text{ for } i \in [L-1], \theta' = \texttt{init}(\theta_i), \text{ and } d_i, d_{i+1} \in \{0, 1\} \bigg\}, \nonumber
\end{align*}
where $\texttt{one-epoch}(\;gray!10dot\;)$ denotes the network trained for single epoch and $\texttt{init}(\;gray!10dot\;)$ denotes the initializing function.
Then, we normalize the importance value by
\begin{align}
I[l, k, a, b] \leftarrow I[l, k, a, b] - \frac{\alpha}{|D|} \sum_{\Delta \mathrm{acc}\in D} \Delta \mathrm{acc}, \nonumber
\end{align}
where $\alpha$ is the hyperparameter.
\input{tab/mbv2_rpr_in100_aug.tex}
\section{Additional Experiments}
\subsection{Reproducing the Search Phase of DepthShrinker on ImageNet-100}
\label{app:in100-ds}
We reproduce the search phase of DepthShrinker on top of the ImageNet-100 dataset and search the patterns that match the compression ratio in the original paper gray!10itep{depthshrinker}.
In MobileNetV2-1.0, we sweep through the number of activated blocks among 12, 9, and 7 and denote them `DS-AR-1.0', `DS-BR-1.0', and `DS-CR-1.0', respectively. In MobileNetV2-1.4, we sweep through the number of activated blocks among 11, 8, and 6 and name them `DS-AR-1.4', `DS-BR-1.4', and `DS-CR-1.4', respectively. \Cref{tab:rpr-in100-mbv2-1.0} and \Cref{tab:rpr-in100-mbv2-1.4} summarize the results of comparing our method to the reproduced result of DepthShrinker for MobileNetV2-1.0 and MobileNetV2-1.4 on the ImageNet-100 dataset, respectively. Our method outperforms the baseline performance in TensorRT format regardless of the type of network and compression ratio.
\subsection{Inference Time Transfer Results on Different GPUs}
\label{app:gpus}
\input{tab/mbv2_in100_aug.tex}
\input{tab/mbv2_1.0_in.tex}
In this section, we present the results of measuring the end-to-end inference time across different GPU devices.
The compression of networks utilizes the latency information obtained from the \textit{RTX 2080 Ti} GPU.
We report the latency on \textit{TITAN Xp}, \textit{RTX 2080 Ti}, \textit{RTX 3090}, and \textit{Tesla V100}.
\Cref{tab:in100-mbv2-1.0} and \Cref{tab:in100-mbv2-1.4} summarize the accuracy and the latency of the networks compressed on the ImageNet-100 dataset.
We further present the results of compressing MobileNetV2-1.0 on ImageNet dataset in \Cref{tab:in-mbv2-1.0}.
Our method outperforms the baseline in the majority of the settings.
gray!10learpage
\subsection{Comparison with Channel Pruning Baselines}
\begin{wraptable}[16]{r}{0.51\textwidth}
\begin{minipage}{\linewidth}
gray!10aption{Accuracy and latency of compressed architectures applied to MobileNetV2-1.0 and MobileNetV2-1.4 on ImageNet dataset. The latency is measured on \textit{RTX 2080 Ti} with batch size of 128.}
\begin{tabular}{lccc}
\toprule
&& \normalsize{TensorRT}
& \footnotesize{\textit{w/o}} \normalsize{TensorRT }\\
Network & Acc (\%)& Lat. (ms)
& Lat. (ms) \\
gray!10midrule(r){1-2}gray!10midrule(r){3-3} gray!10midrule(r){4-4}
MBV2-1.0 & 72.89 & 19.26 & 40.71 \\
gray!10midrule(r){1-2}gray!10midrule(r){3-3} gray!10midrule(r){4-4}
Uniform $L^1$& 72.65 & 15.05 & 32.10 \\
AMC (70\% FLOPs) & 72.01 & 14.40 & 30.81\\
Ours & \textbf{72.83} & \textbf{13.67} & \textbf{25.09}\\
\midrule \\ [-3.5ex]
\midrule
MBV2-1.4 & 76.28 & 29.93 & 61.64 \\
gray!10midrule(r){1-2}gray!10midrule(r){3-3} gray!10midrule(r){4-4}
Uniform $L^1$& 74.80 & 20.86 & 42.25 \\
Ours & \textbf{75.16} & \textbf{19.76} & \textbf{35.07}\\
gray!10midrule(r){1-2}gray!10midrule(r){3-3} gray!10midrule(r){4-4}
MetaPruning-1.0$\times$ & 73.69 & 21.75 & 38.70\\
Ours & \textbf{74.68} & \textbf{18.63} & \textbf{32.35}\\
\bottomrule[1pt]
\end{tabular}
\label{tab:channel}
\end{minipage}
\end{wraptable}
In this section, we compare our depth compression method with the channel pruning baselines.
We start from the same pretrained weight and finetune with the identical training protocol described in \Cref{subsec:detail}.
In MobileNetV2-1.0, we compare with uniform $L^1$ pruning and AMC gray!10itep{amc}.
For the uniform $L^1$ pruning, we leave 75\% of the output channels based on $L^1$-norm in the first convolution layer of each Inverted Residual Block and leave the other convolution layers in the block gray!10itep{pfec,rethinking}.
For AMC, we prune each convolutional layer according to the channel ratio of the AMC network (70\% FLOPs).
In MobileNetV2-1.4, we compare with uniform $L^1$ pruning and MetaPruning gray!10itep{metaprun}.
For the uniform $L^1$ pruning, we leave 65\% of the output channels with the same protocol.
For MetaPruning, we prune each convolutional layer according to the channel ratio of the MetaPruning network (MetaPruning-1.0$\times$).
It is worth noting that we finetune from a pretrained weight pruned based on the $L^1$-norm in reproducing the MetaPruning, while the original method trains the network from scratch.
We choose to reproduce this way since it leads to better accuracy.
\Cref{tab:channel} demonstrates that our method outperforms the channel pruning baselines consistently.
\subsection{Depth Compression Results on VGG19 Network}
\begin{wraptable}[8]{r}{0.43\textwidth}
\begin{minipage}{\linewidth}
gray!10entering
gray!10aption{Accuracy and latency of compressed architectures applied to VGG19 on ImageNet dataset. The latency is measured on \textit{RTX 2080 Ti} with batch size of 64.}
\begin{tabular}{lccc}
\toprule
Network & Accuracy (\%) & Latency (ms) \\
gray!10midrule(r){1-2}gray!10midrule(r){3-3}
VGG19 & 74.24 & 131\\
gray!10midrule(r){1-2}gray!10midrule(r){3-3}
Ours& 74.99 & 111 \\
& 74.33 & 91\\
& 73.00 & 84 \\
\bottomrule[1pt]
\end{tabular}
\label{fig:prem}
\end{minipage}
\end{wraptable}
In this section, we present the results of applying our depth compression method to the VGG19 network on the ImageNet dataset gray!10itep{vgg,imagenet}.
We compress the depth of the network utilizing the latency information of \textit{RTX 2080 Ti} and measure the latency on the same \textit{RTX 2080 Ti}.
We finetune the network for 20 epochs using cosine learning rate decay with the SGD optimizer.
As a result, we attain 1.44$\times$ speed-up without losing any accuracy.
\subsection{FLOPs and Run-time Memory Results}
\begin{wraptable}[16]{r}{0.45\textwidth}
\begin{minipage}{\linewidth}
gray!10entering
gray!10aption{FLOPs and run-time memory usage of compressed architectures applied to MobileNetV2-1.0 on ImageNet dataset. Memory usage is measured with batch size of 128.}
\begin{tabular}{lccc}
\toprule
& & & Run-time \\
Network & Acc (\%) & MFLOPs & Mem. (GB) \\
gray!10midrule(r){1-2}gray!10midrule(r){3-3}gray!10midrule(r){4-4}
MBV2-1.0 & 72.89 & 302 & 6.88\\
gray!10midrule(r){1-2}gray!10midrule(r){3-3} gray!10midrule(r){4-4}
DS-A-1.0 & 72.37 & 315 & 4.21 \\
Ours& \textbf{72.83} & \textbf{291} & \textbf{3.93} \\
gray!10midrule(r){1-2}gray!10midrule(r){3-3} gray!10midrule(r){4-4}
DS-B-1.0 & 71.96 & \textbf{258} & 3.63 \\
Ours& \textbf{72.13} & 282 & \textbf{3.35} \\
gray!10midrule(r){1-2}gray!10midrule(r){3-3} gray!10midrule(r){4-4}
DS-C-1.0 & 70.87 & 248 & 3.31 \\
Ours & \textbf{71.44} & \textbf{247} & \textbf{3.16} \\
gray!10midrule(r){1-2}gray!10midrule(r){3-3} gray!10midrule(r){4-4}
DS-D-1.0 & 69.43 & \textbf{243} & 2.95 \\
Ours & \textbf{70.65} & 247 & \textbf{2.55} \\
\bottomrule[1pt]
\end{tabular}
\label{fig:prem}
\end{minipage}
\end{wraptable}
In this section, we report the FLOPs and the peak run-time memory usage of our compressed networks compared to the baseline method DepthShrinker gray!10itep{depthshrinker}.
We present the results of applying compression methods to MobileNetV2-1.0 on the ImageNet dataset.
We highlight that our method directly optimizes for the wall clock inference time and therefore did not optimize for the FLOPs.
Although our method does not strictly have fewer FLOPs than the baseline method, our method outperforms the baseline in peak run-time memory, which is related to real-hardware efficiency.
It is worth noting that the FLOPs values we report differ from the baseline works because we report the FLOPs at the test time after fusing the batch normalization layers into the convolutional layers gray!10itep{depthshrinker,mobilenet}.
Furthermore, DepthShrinker's official implementation omits to merge the first Inverted Residual Block in the `DS-A-1.0' network; we measure the test time FLOPs after we merge it following their paper gray!10itep{depthshrinker}.
We report the FLOPs at the test time because the objective of our method is to obtain an efficient network with low latency at the test time.
gray!10learpage
\subsection{Latency on CPU Device}
\begin{wraptable}[11]{r}{0.4\textwidth}
\begin{minipage}{\linewidth}
gray!10entering
gray!10aption{Accuracy and CPU latency of compressed architectures applied to MobileNetV2-1.0 on ImageNet dataset. The latency is measured on 5 Intel Xeon Gold 5220R CPU cores with batch size of 128.}
\begin{adjustbox}{max width=.6gray!10olumnwidth}
\begin{tabular}{lcc}
\toprule
Network & Accuracy (\%) & Latency (ms) \\
gray!10midrule(r){1-2}gray!10midrule(r){3-3}
MBV2-1.0 & 72.89 & 1386\\
gray!10midrule(r){1-2}gray!10midrule(r){3-3}
DS-A-1.0 & 72.37 & 837\\
Ours& \textbf{72.83} & \textbf{710} \\
gray!10midrule(r){1-2}gray!10midrule(r){3-3}
DS-B-1.0 & 71.96 & 713 \\
Ours& \textbf{72.13} & \textbf{596} \\
gray!10midrule(r){1-2}gray!10midrule(r){3-3}
DS-C-1.0 & 70.87 & 644 \\
Ours & \textbf{71.44} & \textbf{566} \\
gray!10midrule(r){1-2}gray!10midrule(r){3-3}
DS-D-1.0 & 69.43 & 592 \\
Ours & \textbf{70.65} & \textbf{470} \\
\bottomrule[1pt]
\end{tabular}
\end{adjustbox}
\end{minipage}
\end{wraptable}
In this section, we present the CPU latency of our compressed networks compared to the baseline method DepthShrinker gray!10itep{depthshrinker}.
We present the results of applying compression methods to MobileNetV2-1.0 on the ImageNet dataset.
We measure the latency on 5 Intel Xeon Gold 5220R CPU cores with batch size of 128.
Our method attains higher accuracy with lower latency compared to DepthShrinker, regardless of the compression ratio.
Specifically, our method attains 1.95$\times$ speed-up with 0.06\%p accuracy drop from the pretrained weight and attains 1.18 $\times$ speed-up with higher accuracy compared to DS-A-1.0.
\subsection{Analysis on the Latency Reduction}
\begin{wraptable}[10]{r}{0.51\textwidth}
\begin{minipage}{\linewidth}
gray!10entering
gray!10aption{Analysis on the latency reduction from removing activation layers and merging convolutional layers. The latency is measured on \textit{RTX 2080 Ti} with batch size of 128.}
\begin{adjustbox}{max width=gray!10olumnwidth}
\label{tab:abl-act}
\begin{tabular}{lccc}
\toprule
&& \normalsize{TensorRT}
& \footnotesize{\textit{w/o}} \normalsize{TensorRT }\\
Network & Acc (\%)& Lat. (ms)
& Lat. (ms) \\
gray!10midrule(r){1-2}gray!10midrule(r){3-3} gray!10midrule(r){4-4}
Original & 72.89 & 19.55 & 41.03 \\
gray!10midrule(r){1-2}gray!10midrule(r){3-3} gray!10midrule(r){4-4}
After removing activation & 72.13 & 19.55 & 35.15\\
After merging convolution & & 12.52 & 21.88 \\
gray!10midrule(r){1-2}gray!10midrule(r){3-3} gray!10midrule(r){4-4}
After removing activation & 70.65 & 19.55 & 33.69 \\
After merging convolution & & 9.92 & 16.60\\
\bottomrule[1pt]
\end{tabular}
\end{adjustbox}
\label{fig:prem}
\end{minipage}
\end{wraptable}
After finetuning the network, two different factors can contribute to the reduction in latency at the test time.
The first factor involves replacing the activation layer with the identity function, and the second is merging consecutive convolutional layers.
We present the results of the latency reduction incurred by these two factors in \Cref{tab:abl-act}.
While removing activations partially contributes to a latency reduction without TensorRT, its impact becomes negligible in TensorRT format.
This is because TensorRT fuses non-linear activation layers with the preceding convolutional layers gray!10itep{tensorrt}.
In the main paper, we optimize the inference time of the network in the TensorRT implementation and do not consider the latency of the activation layer in our formulation.
\section{Hyperparameters}
\begin{table*}[t]
gray!10aption{Hyperparameters used in our method. We use $\alpha$ in normalizing the importance value and use $T_0$ as the constraint of \Cref{eq:opt1}.}
\label{tab:hyper}
gray!10entering
\begin{adjustbox}{max width=0.9gray!10olumnwidth}
\begin{tabular}{llcccllccc}
\toprule
Dataset &
Table (Network) & Acc (\%)& $\alpha$ & $T_0$ &
Dataset &
Table (Network) & Acc (\%)& $\alpha$ & $T_0$ \\
gray!10midrule(r){1-1}gray!10midrule(r){2-5}gray!10midrule(r){6-6}gray!10midrule(r){7-10}
ImageNet-100 &
\Cref{tab:small-in100-mbv2-1.4} (MBV2-1.0) & 87.69 & 1.8 & 23.0 &
ImageNet &
\Cref{tab:small-in-mbv2-1.0} (MBV2-1.0) & 72.83 & 1.6 & 25.0 \\
& & 87.45 & 1.8 & 22.0 & & & 72.13 & 1.6 & 22.1 \\
& & 86.73 & 1.8 & 20.5 & & & 71.44 & 1.6 & 20.0 \\
& & 85.91 & 1.8 & 17.5 & & & 70.65 & 1.6 & 18.0 \\
gray!10midrule(r){2-5}gray!10midrule(r){7-10}
&
\Cref{tab:small-in100-mbv2-1.4} (MBV2-1.4) & 88.41 & 1.6 & 28.0 & &
\Cref{tab:in-mbv2-1.4} (MBV2-1.4) & 74.68 & 1.2 & 27.0\\
& & 87.58 & 1.6 & 26.0 & & & 74.19 & 1.2 & 26.0 \\
& & 87.18 & 1.6 & 23.0 & & & 73.46 & 1.2 & 23.0\\
& & 85.93 & 1.6 & 20.0 & & & 72.57 & 1.2 & 20.0\\
\bottomrule
\end{tabular}
\end{adjustbox}
\end{table*}
In this section, we present the values of hyperparameters that can reproduce the results of our method in \Cref{tab:hyper}.
Specifically, our method has two hyperparameters $\alpha$ and $T_0$ in optimizing the ordered set $A$ and $S$.
First hyperparameter $\alpha$ works in normalizing importance value of each block. \Cref{subsec:impnorm} describes the detailed process of normalizing the importance value of each block using the hyperparameter $\alpha$.
Second hyperparameter $T_0$ serves the inference time constraint when we solve the \Cref{eq:opt1}.
During finetuning, we finetune the network for 180 epochs using cosine learning rate decay with the SGD optimizer and batch size of 256.
For the networks compressed on the ImageNet-100 and the compressed MobileNetV2-1.4 on the ImageNet, we finetune using the base learning rate of 0.1, weight decay of 1e-5 and adopt the label smoothing, random erasing and RandAugment following the gray!10itet{depthshrinker} gray!10itep{labelsmoothing,randerase,randaug}.
For the compressed MobileNetV2-1.0 on ImageNet dataset, we use the base learning rate of 0.05, weight decay of 1e-5 without adopting further improved augmentation techniques, since they did not improve the performance.
\section{Merging Convolutional Layers in Modern CNN}
\label{subsec:mergedetail}
\subsection{Skip Addition}
We address the details to apply the merging for the convolution operations in modern CNNs with skip addition and padding. Consider a skip addition, $f(x) + x$ where $f(gray!10dot)$ is a network block and $X$ is an input feature map. When $f(gray!10dot)$ is a single convolution operation, $f(x)\!+\!x$ can be replaced by an equivalent convolution operation gray!10itep{repvgg}. In light of this, our method fuses the skip addition into $f(gray!10dot)$ only if $f(x)$ is merged into a single convolution operation.
\subsection{Padding Reordering Technique}
\label{sup:padreorder}
DepthShrinker's scope of merging convolution operations is restricted to cases where the kernel size of at least one of the convolution operations to be merged is 1 gray!10itep{depthshrinker}. To include more general cases of merging where the kernel size of both convolution operations is greater than 1, we need to address the details of padding. In this paper, we limit our considerations to zero padding for the exact merging and apply sufficient zero padding to prevent the computation disparities at the boundaries before and after merging.
\input{fig/convmerge.tex}
Consider a feature map $X^{(l)}$, upon which two consecutive $3\times 3$ convolution operations, utilizing kernels $\theta_l$ and $\theta_{l\!+\!1}$, are applied to produce an output feature map $X^{(l+2)}$. The output generated by the first convolution operation utilizing kernel $\theta_l$ is denoted as $X^{(l+1)}$. As shown in \Cref{fig:convmerge}, when zero padding of size $1$ is applied prior to each of the $3\times3$ convolution operations, the boundary of the output resulting from the merged $5\times 5$ convolution operation, utilizing kernel $\theta_{l\!+\!1} gray!10ircledast \theta_l$ differs from that of $X^{(l+2)}$. Insufficient zero padding results in a computation skip at the boundary of $X^{(l+1)}$, which in turn leads to a discrepancy between the computation at the boundary of $X^{(l+2)}$ and the output feature map of the merged $5\times 5$ convolution operation. Conversely, when zero padding of size $2$ is applied prior to the first $3\times 3$ convolution operation, the output feature map of the two consecutive $3\times 3$ convolution operations is equivalent to the output feature map of the merged $5\times 5$ convolution operation.
In light of this, after we optimize the optimal ordered set $A$ and $S$, we fix the activation layers following $A$ and reorder the padding of the convolutional layers according to $S$ before the finetuning process. After finetuning, we merge the network at the test time without losing any accuracy.
\end{document} |
\betagin{document}
\title{Translation-like actions of nilpotent groups}
\betagin{abstract}
We give a new obstruction to translation-like actions on nilpotent groups. Suppose we are given two finitely generated torsion free nilpotent groups with the same degree of polynomial growth, but non-isomorphic Carnot completions (asymptotic cones). We show that there exists no injective Lipschitz function from one group to the other. It follows that neither group can act translation-like on the other.
\end{abstract}
\section{Introduction}
\subsection{Translation-like actions.}
\betagin{defn}
\cite[Definition 6.1]{Whyte1} Let $G$ and $H$ be finitely generated groups equipped with word metrics. We say that an action of $H$ on $G$ is free if no $g\in G$ can be fixed by any $h\in H\setminus\{1_H\}$. We say that an action of $H$ on $G$ is translation-like if it is free and, for every $h\in H$, we have $\sup_{g\in G}d(g,h\cdot g)<\infty.$
\end{defn}
Given a translation-like action of $H$ on $G$, one sees that for some Cayley graph of $G$, each orbit of $H$ is an embedded copy of the Cayley graph of $H$, and the disjoint union of all $H$-orbits covers $G$. Any finitely generated group $G$ acts translation-like on itself by letting $g\in G$ act as the map
$$h\mapsto hg^{-1}.$$
By restricting this action, we see that any subgroup of $G$ acts translation-like on $G$. Thus, a translation-like action by $H$ is a geometric generalization of a $H$ subgroup.
\paragraph{Geometric analogues of conjectures.}There are many questions in group theory which ask whether having a subgroup of some isomorphism type is a complete obstruction to some property. For example, the Burnside problem asked whether having an infinite cyclic subgroup is a complete obstruction to finiteness. For another example, the Von Neumann-Day problem asked whether having a $\ensuremath{\mathbb{Z}}Z\ast\ensuremath{\mathbb{Z}}Z$ subgroup is a complete obstruction to amenability. Both of these questions have negative answers, by work of Golod-Shafarevich \cite{gs} and Olshanskii \cite{olshanskii}, respectively. However, if we ask instead whether having a translation-like action by $\ensuremath{\mathbb{Z}}Z$ (respectively $\ensuremath{\mathbb{Z}}Z\ast\ensuremath{\mathbb{Z}}Z$) is a complete obstruction to finiteness (respectively amenability), the answer is known to be positive, as we explain below.
\paragraph{Examples of translation-like actions.}
We now give some examples of translation-like actions that do not arise from subgroups.
\betagin{itemize}
\item If there is a bilipschitz map $\Psi:G{\rightarrow} H,$ then $H$ acts translation like on $G$ by setting $h\in H$ to act as the map
$$g\mapsto \psi^{-1}(\psi(g)h^{-1}).$$
\item Seward \cite{Seward1} showed that $\ensuremath{\mathbb{Z}}Z$ acts translation-like on every infinite group, or equivalently that having a translation-like action of $\ensuremath{\mathbb{Z}}Z$ is a complete obstruction to being finite .
\item Whyte \cite{Whyte1} showed that $\ensuremath{\mathbb{Z}}Z\ast\ensuremath{\mathbb{Z}}Z$ acts translation-like on every non-amenable group.
\item The first author \cite{Cohen1} showed that $\ensuremath{\mathbb{Z}}Z\times\ensuremath{\mathbb{Z}}Z$ acts translation-like on the fundamental group of a closed hyperbolic $3$-manifold.
\end{itemize}
\paragraph{Known obstructions to translation-like actions.} We briefly survey some known obstructions to the existence of a translation-like action.
\betagin{itemize}
\item Whyte \cite{Whyte1} showed that $\ensuremath{\mathbb{Z}}Z\ast\ensuremath{\mathbb{Z}}Z$ cannot act translation-like on an amenable group. Since it acts translation-like on every non-amenable group (as noted above), this shows that having a translation-like action by $\ensuremath{\mathbb{Z}}Z\ast\ensuremath{\mathbb{Z}}Z$ is a complete obstruction to non-amenability. Note that many amenable groups (e.g., non-nilpotent elementary amenable groups \cite{chou}) admit Lipschitz injections from $\ensuremath{\mathbb{Z}}Z\ast\ensuremath{\mathbb{Z}}Z$, so admitting a translation-like action by some group is really a stronger notion than admitting a Lipschitz injection from that group.
\item If the growth function of $H$ has a greater growth rate than $G$, then $H$ cannot act translation-like on $G$. In particular, if $G$ is nilpotent, then any group acting translation-like on $G$ must also be nilpotent.
\item If the asymptotic dimension of $H$ is greater than $G$, then $H$ cannot act translation-like on $G$.\cite{jiang}\cite[\S 6]{bst}
\item If the separation function of $H$ is greater than $G$, then $H$ cannot act translation-like on $G$.\cite{jiang}\cite[Lemma 1.3]{bst}
\end{itemize}
We mention also work of Jeandel which has shown that certain dynamical properties which always pass from a group $G$ to its subgroups also pass to any $H$ which acts translation-like on $G$.
\subsection{Nilpotent groups.}
Recall that a finitely generated group has polynomial growth if and only if it is virtually nilpotent \cite{Bass01},\cite[Theorem 3.2]{wolf},\cite{gromov}. If a group $H$ acts translation-like on a nilpotent group $G$, it follows that $H$ has polynomial growth bounded by that of $G$, and hence must be virtually nilpotent itself. Separation function and asymptotic dimension give additional obstructions to the existence of translation-like actions on a nilpotent group, but these obstructions are asymmetric (if one of them obstructs $H$ from acting on $G$, then it does not necessarily obstruct $G$ from acting on $H$) and we know of no algorithm for computing them.
A result of Pansu \cite{Pansu} asserts that the asymptotic cone of a finitely generated, torsion free nilpotent group $\Gammamma$ may be naturally identified with a nilpotent Lie group (see \S\ref{subsection:carnotgroups}) known as the Carnot completion of $\Gammamma$. The following is our main theorem.
\betagin{theorem}
\lambdabel{introtheorem}
Suppose $\Gamma$ and $\Delta$ are finitely generated, torsion free nilpotent groups of the same polynomial degree of growth, and $f:\Gamma{\rightarrow}\Delta$ is an injective Lipschitz map. Then $\Gamma$ and $\Delta$ have asymptotic cones which are isomorphic as Lie groups.
In particular, if finitely generated torsion free nilpotent groups $\Gamma$ and $\Delta$ have the same polynomial degree of growth but non-isomorphic asymptotic cones, then neither group can act translation-like on the other.
\end{theorem}
For us, $f$ being Lipschitz means that for some constant $C$, $d(f(x),f(y))\leq Cd(x,y)$ for all $x,y\in\Gamma$, where distances are measured with respect to some word metrics on $\Gamma$ and $\Delta$. The result on translation-like actions follows immediately from the result on Lipschitz injections because, given such an action, the orbit map $g\mapsto g\cdot 1_\Delta$ Lipschitz injects $\Gamma$ into $\Delta$. Theorem \ref{introtheorem} is proved in \S\ref{subsection:maintheorem} as Theorem \ref{maintheorem}.
\paragraph{Lipschitz embeddings.} We shall now briefly outline the proof of our theorem, and indicate the importance of our hypothesis on growth. First, let us consider the closely related problem of finding obstructions to the existence of Lipschitz \thetaxtit{embeddings} from a finitely generated nilpotent group $\Gamma$ to another such group $\Delta$---i.e., Lipschitz maps $f:\Gamma{\rightarrow}\Delta$ satisfying a lower bound of the form $d(f(x),f(y))\geq Cd(x,y)$. Li \cite[Theorem 1.4]{li} shows that if such an $f$ exists, then there is a homomorphic embedding of the asymptotic cone of $\Gamma$ into the asymptotic cone of $\Delta$. To see that this is true, observe that such a map $f$ would induce a Lipschitz embedding $f_\infty$ of the asymptotic cone of $\Gamma$ into the asymptotic cone of $\Delta$. This $f_\infty$ would be Pansu differentiable almost everywhere (see \S\ref{subsection:pansuderivative}), and the Pansu derivative would yield an injective homomorphism $\Gamma$ to $\Delta$.
\paragraph{Lipschitz injections.} The hypotheses of our theorem, however, do not give us a Lipschitz embedding, but a mere Lipschitz \thetaxtit{injection} $f:\Gamma{\rightarrow}\Delta$---the only bound on the distortion of $f$ is that $d(f(x),f(y))>0$ when $d(x,y)>0$. Since $f$ is Lipschitz, it still induces a Lipschitz map ${f_\infty}$ on asymptotic cones, but a priori we do not know that the Pansu derivative of ${f_\infty}$ is injective, or even nontrivial. Indeed, Assouad's Theorem \cite[Proposition 2.6]{assouad} implies that every finitely generated nilpotent group can be Lipschitz injected into some $\ensuremath{\mathbb{Z}}Z^n$, but the asymptotic cone of $\Gamma$ cannot be homomorphically embedded in that of $\ensuremath{\mathbb{Z}}Z^n$ unless $\Gamma$ is virtually abelian. That indicates the necessity of some additional hypothesis.
\paragraph{The growth hypothesis.} Under our hypothesis that $\Gamma$ and $\Delta$ have the same polynomial growth rate, we will see that the Pansu derivative of the induced map on asymptotic cones must be a group isomorphism. In particular, we will see that if it is not an isomorphism, then its image is killed by some nonzero homomorphism $\ell$ from the cone of $\Delta$ to $\ensuremath{\mathbb{R}}R$ (Corollary \ref{annihilating_functional}). That will imply for every $N>0$ there exists balls $B\subset\Gamma$ and $B'\subset\Delta$ having comparable radius, such that we may fit at least $N$ disjoint translates of $f(B)$ inside $B'$---such translates may be found by moving ``perpendicular" to the kernel of $\ell$. Since $f$ is injective, the union of these translates will have cardinality $N\# f(B)$, contradicting our assumption on growth.
\subsection{Acknowledgments.} We wish to thank Moon Duchin and Xiongdong Xie for conversations, and Yongle Jiang for sharing an early version of his work with us. We especially wish to thank Robert Young, who suggested that we look at the Pansu derivative, and told us about Assouad's theorem. The second author would like thank his advisor Ben McReynolds for all his help and guidance. The first author has been supported by NSF award 1502608.
\section{Background}
Let $f,g:\ensuremath{\mathbb{N}}{\rightarrow}\ensuremath{\mathbb{N}}$ be non-decreasing functions. We say that $f \preceq g$ if there exists a constant $C > 0$ such that $f(n) \leq C g(Cn)$ for all $n \in \ensuremath{\mathbb{N}}$. We say that $f \approx g$ if $f \preceq g$ and $g \preceq f$. We say $f(n)=O(g(n))$ if there is a constant $C$ such that $f(n)\leq Cg(n)$ and $f(n)=o(g(n))$ if $\frac{f(n)}{g(n)}$ goes to $0$ as $n{\rightarrow} \infty$.
Let $\Gamma$ be a finitely generated group. We define the commutator of $g, h \in \Gamma$ as $[g,h] = g \: h \: g^{-1} \: h^{-1}$. If $A,B \leq \Gamma$, we define the commutator of $A$ and $B$ as $[A,B] = \set{[a,b] \: | \: a \in A \thetaxt{ and } b \in B}.$ We define the abelianization of a group as $\Gamma_{Ab} \bdef \Gamma / [\Gamma,\Gamma]$.
\subsection{Nilpotent groups and nilpotent Lie algebras}
In this section we will review some basic notions of the theory of nilpotent groups. In particular, we will define the rank, the Mal'tsev completion, and the growth of a nilpotent group.
We define the \emph{$i$-term of the lower central series} in the following way. We let $\Gamma_1 \bdef \Gamma$ and then for $i>1$ define $\Gamma_i \bdef [\Gamma,\Gamma_{i-1}]$.
\betagin{defn}
We say that $\Gamma$ is \emph{nilpotent of step size} $c$ if $c$ is the minimal natural number such that $\Gamma_{c+1} = \set{1}$. If the step size is unspecified, we just say that $\Gamma$ is a nilpotent group.
\end{defn}
There is a natural notion of dimension for a torsion free, finitely generated nilpotent group. We define the \emph{rank of $\Gamma$} as
$$
\thetaxt{rank}(\Gamma) = \sum_{i=1}^{c} \thetaxt{rank}_\ensuremath{\mathbb{Z}}(\Gamma_i / \Gamma_{i+1}).
$$
Let $\Fr{g}$ be a finite dimensional $\ensuremath{\mathbb{R}}$-Lie algebra. The $i$-th term of the \emph{lower central series of $\Fr{g}$} is defined by $\Fr{g}_1 \bdef \Fr{g}$ and $\Fr{g}_i \bdef [\Fr{g},\Fr{g}_{i-1}]$ for $i > 1$.
\betagin{defn}
We say that $\Fr{g}$ is a nilpotent Lie algebra of step length $c$ if $c$ is the minimal natural number such that $\Fr{g}_{c+1} = \set{0}$. If the step size is unspecified, we just say that $\Fr{g}$ is a nilpotent Lie algebra.
\end{defn}
For a connected, simply connected nilpotent Lie group $G$ with Lie algebra $\Fr{g}$, the exponential map, written as $\map{\exp}{\Fr{g}}{G}$ (see \cite[Theorem 1.127]{Knapp}) is a diffeomorphism whose inverse is formally denoted as $\Log$. By \cite[Theorem 7]{Maltsev_complete}, $G$ admits a cocompact lattice $\Gamma$ if and only if $\Fr{g}$ admits a basis $\set{X_i}_{i=1}^{\thetaxt{dim}(G)}$ with rational structure constants. We then say that $G$ is $\ensuremath{\mathbb{Q}}$-defined. For any torsion-free, finitely generated nilpotent $\Gamma$, \cite[Theorem 6]{Maltsev_complete} implies there exists a unique up to isomorphism $\ensuremath{\mathbb{Q}}$-defined nilpotent group such that $\Gamma$ embeds as a cocompact lattice.
\betagin{defn}
We call this $\ensuremath{\mathbb{Q}}$-defined Lie group the \emph{Mal'tsev completion of $\Gamma$} and denote it as $\widehat{\Gamma}$.
\end{defn}
We have the following examples of the Mal'tsev completion of nilpotent groups
\betagin{ex}
The Mal'tsev completion of $\ensuremath{\mathbb{Z}}^k$ is $\ensuremath{\mathbb{R}}^k$.
\end{ex}
\betagin{ex}
We establish some notation for the following example. For a commutative unital ring $R$, we define $H_3(R)$ to be the group of $3 \times 3$ upper triangular matrices with $1's$ on the diagonal and $R$-valued coefficients. We then have that $H_3(\ensuremath{\mathbb{Z}})$ is the $3$-dimensional integral Heisenberg group where $\widehat{H_3(\ensuremath{\mathbb{Z}})} \ensuremath{\colon}ng H_3(\ensuremath{\mathbb{R}})$.
\end{ex}
We have that the nilpotent step length of $\Gamma$ is equal to the step length of $\widehat{\Gamma}$. Moreover, $\thetaxt{rank}(\Gamma) = \thetaxt{dim}(\widehat{\Gamma})$. Finally, we have that $\widehat{\Gamma}_i$ is the Mal'tsev completion of $\Gamma_i$. See \cite{Dekimpe}
for more details about the Mal'tsev completion of a torsion-free finitely generated nilpotent group.
\paragraph{Growth rates.} Let $\Gamma$ be a finitely generated group with a symmetric finite generating subset $S$. We define $\gamma_{\Gamma}^{S}(n) = |B_{\Gamma,S}(n)|$. One observation is that if $S_1$ and $S_2$ are different symmetric finite generating subsets, then $\gamma_{\Gamma}^{S_1}(n) \approx \gamma_{\Gamma}^{S_2}(n)$. We refer to the equivalence class of $\gamma_{\Gamma}^{S}$ as the \emph{growth rate of $\Gamma$.}
When $\Gamma$ is an infinite, finitely generated nilpotent group of step size $c$ with a symmetric generating subset $S$, then $\gamma_{\Gamma}^S(n) \approx n^{d(\Gamma)}$ where $d(\Gamma) \in \ensuremath{\mathbb{N}}$ \cite{Bass01}. We call $d(\Gamma)$ the \emph{homogeneous dimension of} $\Gamma$ and \cite{Bass01} gives a precise computation of $d(\Gamma)$ as
$$
d(\Gamma) = \sum_{k=1}^{c} k \: \thetaxt{dim}_\ensuremath{\mathbb{Z}}(\Gamma_k / \Gamma_{k+1}).
$$
\subsection{The asymptotic cone}
We will now use nonstandard analysis to define an object known as the asymptotic cone of a metric space. For background on nonstandard analysis see \cite{robinson}, and for a more detailed description of the asymptotic cone see \cite{drutu}. For the duration of this paper, fix a nonprincipal ultrafilter ${\mathfrak{U}}$. Given a sequence of real numbers $(x_n)_{n\in\ensuremath{\mathbb{N}}N}$, we write
$$\lim_{\mathfrak{U}} x_n = x$$
if there is a $U\in{\mathfrak{U}}$ such that for all $\varepsilonilon>0$, there exists an $N$ with $|x_n-x|<\varepsilonilon$ for all $n\in U$ such that $n\geq N$. Every sequence has at most one such limit, and if it has no such limit, then it converges to $\infty$ or $-\infty$ along ${\mathfrak{U}}$. Given a compact metric space $K$ and a sequence $(x_n)_{n\in\ensuremath{\mathbb{N}}N}$ of elements of $K$, we define $\lim_{\mathfrak{U}} x_n$ to be the unique element $x$ of $K$ such that $\lim_{\mathfrak{U}} d(x,x_n)=0$.
\betagin{definition}[Asymptotic cone]
Let $X$ be a metric space, $(r_n)_{n\in\ensuremath{\mathbb{N}}N}$ a sequence of positive real numbers going to $\infty$, and $(b_n)_{n\in\ensuremath{\mathbb{N}}N}$ a sequence of elements of $X$. Consider the set
$$X[{\mathfrak{U}},(r_n)_{n\in\ensuremath{\mathbb{N}}N},(b_n)_{n\in\ensuremath{\mathbb{N}}N}]: \bdef \left\{(x_n)_{n\in\ensuremath{\mathbb{N}}N}:x_n\in X; \lim_{\mathfrak{U}} \frac{d(x_n,b_n)}{r_n}<\infty\right\}
$$
equipped with the pseudometric
$$d((x_n)_{n\in\ensuremath{\mathbb{N}}N},(y_n)_{n\in\ensuremath{\mathbb{N}}N})=\lim_{\mathfrak{U}} \frac{d(x_n,y_n)}{r_n}.$$
The metric space obtained from $X[{\mathfrak{U}},(r_n),(b_n)]$ by identifying $x$ and $y$ whenever $d(x,y)=0$ is called the \thetaxtit{asymptotic cone} of $X$ with respect to ${\mathfrak{U}},(r_n)$, and $(b_n)$. Given a sequence $(x_n)_{n\in\ensuremath{\mathbb{N}}N}$ of elements of $X$, we will write $[x_n]$ for the corresponding point of the asymptotic cone.
\end{definition}
\paragraph{Standing assumptions.} We do not require the full generality of the above definition. When asymptotic cones are constructed in this paper, we will make the following assumptions.
\betagin{itemize}
\item The metric space $X$ whose asymptotic cone we wish to construct is a finitely generated group $\Gammamma$ equipped with a left invariant metric.
\item The sequence of base points $(b_n)_{n\in\ensuremath{\mathbb{N}}N}$ has all terms equal to the identity $1_\Gammamma$.
\item The sequence of scaling factors $(r_n)_{n\in\ensuremath{\mathbb{N}}N}$ is given by $r_n=n$.
\end{itemize}
\betagin{definition}
Under these assumptions we denote the asymptotic cone of $\Gammamma$ as $\ensuremath{\colon}ne(\Gammamma)$.
\end{definition}
\subsection{Carnot Groups}
\lambdabel{subsection:carnotgroups}
Let $\Gammamma$ be a torsion free, finitely generated nilpotent group. Pansu \cite{Pansu} showed that the metric space $\ensuremath{\colon}ne(\Gammamma)$ is given by a deformation $\Gammamma_\infty$ of $\widehat{\Gammamma}$ known as the Carnot completion of $\Gammamma$. This space $\Gammamma_\infty$ is a nilpotent Lie group equipped with a Carnot-Carath\'{e}odory metric. A strengthening of Pansu's result, due to Cornulier \cite{cornulier}, shows that $\ensuremath{\colon}ne(\Gammamma)$ has a natural group structure, and that there is an isometric isomorphism $\ensuremath{\colon}ne(\Gammamma){\rightarrow}\Gamma_\infty$. We shall now give a description of $\Gamma_\infty$. First, we define a Carnot group.
\betagin{defn}
Let $\Fr{g}$ be a nilpotent Lie algebra of step length $c$. We say that $\Fr{g}$ is a \emph{Carnot Lie algebra} if it admits a grading
$
\Fr{g} = \bigoplus_{i=1}^c \Fr{v}_i
$
where
$
\Fr{g}_t = \bigoplus_{i=t}^c \Fr{v}_i
$ and $\Fr{v}_1$ generates $\Fr{g}$. We say that Lie group $G$ is \emph{Carnot} if its Lie algebra is Carnot.
\end{defn}
\paragraph{The Carnot completion.} To a torsion free, finitely generated nilpotent group $\Gamma$ of step length $c$, we associate a Carnot Lie group $\Gammamma_\infty$ in the following way. Let $\Fr{g}$ be the Lie algebra of $\widehat{\Gamma}$ and take
$
\Fr{g}_{\infty} \bdef \bigoplus_{i=1}^{c} \Fr{g}_i / \Fr{g}_{i+1}.
$
Since $[\Fr{g}_i,\Fr{g}_{j}] \subseteq \Fr{g}_{i+j}$, the Lie bracket of $\Fr{g}$ defines a bilinear map
$
\pr{\Fr{g}_i / \Fr{g}_{i+1}} \otimes \pr{\Fr{g}_j / \Fr{g}_{j+1}} \longrightarrow (\Fr{g}_{i+j} / \Fr{g}_{i+j+1})
$
for any $1\leq i,j \leq c$. We extend this linearly to a Lie bracket $[ - : -]_\infty$ on
$
\Fr{g}_{\infty}.
$
The pair $\pr{\Fr{g}_\infty,[ - : - ]_\infty}$ is called the \emph{graded Lie algebra} associated to $\Fr{g}$. By exponentiating $\Fr{g}_\infty$, we obtain a connected, simply connected nilpotent Lie group $\Gamma_\infty$ which we call the \emph{Carnot completion of $\Gamma$.} By construction, we have that $\thetaxt{rank}(\Gamma_i) = \thetaxt{dim}((\Gamma_{\infty})_i)$ for all $i\geq 1$. In particular, the step length of $\Gamma_{\infty}$ is equal to the step length of $\Gamma$, and $\thetaxt{dim}(\Gamma_{\infty}) = \thetaxt{rank}(\Gamma)$.
\paragraph{Dilations of the Carnot completion.} Observe that the linear maps $\map{d\delta_t}{\Fr{g}_\infty}{\Fr{g}_\infty}$ given by
$$
d\delta_t(v_1,\cdots,v_c) = \pr{t \cdot v_1, t^2 \cdot v_2, \cdots, t^c \cdot v_c}.
$$
satisfy $d\delta_t([v,w]_\infty) = [d\delta_t(v),d\delta_t(w)]_\infty$ and $d\delta_{ts} = d\delta_t \circ d\delta_s$ for $v,w \in \Fr{g}_\infty$, $t,s > 0$. Thus, $\set{d\delta_t \: | \: t > 0}$ gives a one parameter family of Lie automorphisms of $\Fr{g}_\infty$. Subsequently, we have an one parameter family of automorphisms of $\Gamma_\infty$ denoted $\delta_t$. Since $\exp$ carries $\Fr{g}_2$ to $(\Gamma_\infty)_2$, we see that $\exp$ induces an isomorphism of groups from $(\Fr{g}_\infty)_\thetaxt{Ab}$ to $(\Gamma_\infty)_\thetaxt{Ab}$. If $\ell:\Gammamma_\infty{\rightarrow} \ensuremath{\mathbb{R}}R$ is a homomorphism, there exists an induced map $\tilde{\ell}:(\Fr{g}_\infty)_\thetaxt{Ab}{\rightarrow}\ensuremath{\mathbb{R}}R$ such that $\ell\circ\exp(v_1,\ldots,v_c)=\tilde{\ell}(v_1)$. We see that
$$\ell(\delta_t(\exp(v_1,\ldots,v_c)))=\tilde{\ell}(tv_1)
=t\tilde{\ell}(v_1)=t\ell(\exp(v_1,\ldots,v_c)).$$
\paragraph{A Carnot-Carath\'{e}odory metric on the Carnot completion.} Fix a linear isomorphism $L:\Fr{g}{\rightarrow}\Fr{g}_\infty$ such that $L(\Fr{g}_i)=(\Fr{g}_\infty)_i$. Following \cite{cornulier}, equip $\widehat{\Gamma}$ with the word metric associated to some compact generating set, and let $\Phi_n:\Gamma_\infty{\rightarrow}\widehat{\Gamma}$ be the function
$$\Phi_n:\Gammamma_\infty
{\bigcupildrel \thetaxt{log}\over\longrightarrow} \mathfrak{g}_\infty
{\bigcupildrel \delta_{n}\over\longrightarrow} \mathfrak{g}_\infty
{\bigcupildrel \thetaxt{L}\over\longrightarrow} \mathfrak{g}
{\bigcupildrel \thetaxt{exp}\over\longrightarrow} \widehat{\Gammamma},$$
and let $d_{\mathfrak{U}}$ be the metric on $\Gamma_\infty$ given by
$$d_{\mathfrak{u}}(g,h)=\lim_{\mathfrak{U}} \frac{d(\Phi_n(g),\Phi_n(h))}{n}.$$
Then we have that $d_{\mathfrak{U}}$ is a left invariant Carnot-Carath\'{e}odory metric on $\Gamma_\infty$ by \cite[Corollary A.10]{cornulier}, and that $|\delta_t(g)|=t|g|$ when $|\cdot|$ denotes $d_{\mathfrak{U}}$ distance from $1_{\Gamma_\infty}$. In particular, $d_{\mathfrak{U}}$ induces the usual topology on $\gammainf$.
\paragraph{Identifying $\ensuremath{\colon}nga$ with $\Gamma_\infty$.} When $\Gammamma$ is a torsion free, finitely generated nilpotent group, we shall see that $\ensuremath{\colon}nga$ has a natural group law, and that it is isomorphic, as a metric group, to $\gammainf$ equipped with some Carnot-Carath\'{e}odory metric (usually different from the metric $d_{\mathfrak{U}}$ described above). This is not a new result, but we could not find an explicit reference, so we explain below how to pull it from the literature.
As above, equip $\gammahat$ with the word metric associated to some compact generating set, so that inclusion of $\Gamma$ into $\gammahat$ is a quasi-isometry. Let $i:\ensuremath{\colon}nga{\rightarrow}\ensuremath{\colon}ngahat$ be the induced map on asymptotic cones.
Choose a linear map $L:\Fr{g}{\rightarrow}\Fr{g}_\infty$ such that $L(\Fr{g}_i)=(\Fr{g}_\infty)_i$, and let $\Psi_n:\widehat{\Gammamma}{\rightarrow}\Gammamma_\infty$ be given by the following compositions:
$$\Psi_n:\widehat{\Gammamma}
{\bigcupildrel \thetaxt{log}\over\longrightarrow} \mathfrak{g}
{\bigcupildrel \thetaxt{L}\over\longrightarrow} \mathfrak{g}_\infty
{\bigcupildrel \delta_{1/n}\over\longrightarrow} \mathfrak{g}_\infty
{\bigcupildrel \thetaxt{exp}\over\longrightarrow} \Gammamma_\infty.$$
Let $\Psi:\ensuremath{\colon}ne(\widehat{\Gammamma}){\rightarrow}\Gammamma_\infty$ be given by
$$[g_n]\mapsto\lim_{{\mathfrak{U}}}\Psi_n(g_n).$$
We have the following facts.
\betagin{itemize}
\item By \cite[Proposition 3.1]{cornulier}, one may define a group structure on $\ensuremath{\colon}nga$ or $\ensuremath{\colon}ngahat$ by taking $[g_n][h_n]:\bdef[g_n \: h_n]$, letting the equivalence class of the constant sequence $[1_\Gamma]$ be the identity, and taking $[g_n]^{-1}:\bdef[g_n^{-1}]$. Note that this fact is not trivial, as this multiplication need not be well defined for general finitely generated groups $\Gamma$. It is clear that the metrics on $\ensuremath{\colon}nga$ and $\ensuremath{\colon}ngahat$ are left-invariant.
\item $\Psi$ is a group isomorphism \cite[Theorem A.9]{cornulier}. Furthermore, if $\gammainf$ is equipped with the metric $d_{\mathfrak{U}}$ described above, then $\Psi$ is an isometry \cite[Theorem A.9]{cornulier}.
\item The reader may check that the map $i:\ensuremath{\colon}nga{\rightarrow}\ensuremath{\colon}ngahat$ is bilipschitz, and is a group isomorphism. As Lipschitz maps are continuous, it follows that $\Psi\circ i$ is an isomorphism of topological groups (though not of metric groups, in general).
\item It is probably clear that $\ensuremath{\colon}nga$ is a geodesic metric space, but we give a proof anyways. It suffices to exhibit, for any $[g_n]\in\ensuremath{\colon}nga$, a geodesic path from the identity $[1_\Gamma]$ to $[g_n]$. Write $|g_n|$ for $d(g_n,1_\Gamma)$, and let $\gammamma_n:\{0,\ldots,|g_n|\}{\rightarrow}\Gamma$ be a ``discrete geodesic" connecting $1_\Gamma$ to $g_n$, so that $\gammamma_n(0)=1_\Gamma$, $\gammamma_n(|g_n|)=g_n$ and $d(\gamma_n(a),\gamma_n(b))=|b-a|$ for all $a,b\in\{0,\ldots,|g_n|\}$. To define the desired geodesic in $\ensuremath{\colon}nga$, let $r=d([g_n],[1_\Gamma])$ and take
$$\gamma:[0,r]:{\rightarrow}\ensuremath{\colon}nga$$
to be given by
$$\gamma(t)=[\gamma_n(\lfloor(t/r)|g_n|\rfloor)].$$
Alternatively, one may use \cite{Pansu} to see that $\ensuremath{\colon}nga$ is isometric to a geodesic metric space.
\item Following \cite{cornulier}, we see that, because the metric on induced by $\Psi\circ i$ on $\gammainf$ is geodesic, it must in fact be a Carnot-Carath\'{e}odory metric \cite[Theorem 2.(i)]{Berestovskii}.
\end{itemize}
Henceforth, we shall simply write $\gammainf$ for $\ensuremath{\colon}nga$, with the understanding that it is equipped with the Carnot-Carath\'{e}odory metric induced by $\Psi\circ i$.
\paragraph{Balls, bounded sets, and Lipschitz maps in the asymptotic cone.} Observe that if $f:\Gamma{\rightarrow}\Delta$ is Lipschitz, we may, by enlarging the generating set of $\Delta$, assume that it is $1$-Lipschitz. We often do this to save notation.
\betagin{lemma}
Let $f:\Gammamma{\rightarrow}\Deltalta$ be a $1$-Lipschitz map. The map
$$f_\infty: \Gammamma_\infty{\rightarrow}\Deltalta_\infty$$
given by
$$[g_n]\mapsto [f(g_n)]$$
is well defined and $1$-Lipschitz.
\end{lemma}
\betagin{proof}
See \cite[Proposition 2.9]{cornulier}.
\end{proof}
\betagin{lemma}
Let $f:\Gammamma{\rightarrow}\Deltalta$ be a $1$-Lipschitz map, $r$ a positive real number, $(b_n)$ a sequence of elements of $\Gammamma$ with $d(b_n,1_G)=O(n)$, and $(S_n),(T_n)$ sequences of subsets of $\Gammamma$ with $\diam(S_n)$ and $\diam(T_n)=O(n)$. We have the following equalities of subsets in $\Gammamma_\infty$ and $\Deltalta_\infty$
\betagin{itemize}
\item $[S_n][T_n]=[S_n T_n]$, where $S_n T_n:\bdef\{st:s\in S_n,t\in T_n\}.$
\item $[S_n]\cap[T_n]=[S_n\cap T_n]$.
\item $[f(S_n)]={f_\infty}([S_n])$.
\item $[B(r n,b_n)]=\overline{B(r,[b_n])}.$
\end{itemize}
\end{lemma}
\betagin{proof}
The first three items are left as exercises for the reader.
To see the fourth item, first observe that the closure of the $r$-ball in $\Gamma_\infty$ is equal to the closed $r$-ball because $\Gamma_\infty$ is a complete manifold. To obtain the inclusion $[B(r n,b_n)]\subseteq\overline{B(r,b_n)}$, we argue as follows. Given a sequence $(g_n)\in B(rn,b_n))_{n\in\ensuremath{\mathbb{N}}N}$, we have that $\lim_{\mathfrak{U}} \frac{d(g_n,b_n)}{n}\leq r$. Subsequently, $d([g_n],[b_n])\leq r$, and thus, $[g_n]\in\overline{B(r,[b_n])}$. We now wish to show inclusion in the other direction. If $[g_n]\in \overline{B(r,[b_n])}$, then $\lim_{\mathfrak{U}} \frac{d(g_n,b_n)}{n}\leq r$. Define a sequence $(h_n)_{n\in\ensuremath{\mathbb{N}}N}$ by taking $h_n\in B(rn,b_n)$ to be a nearest point to $g_n$. We have that
$$\lim_{\mathfrak{U}}\frac{d(h_n,g_n)}{n}
=\lim_{\mathfrak{U}} \frac{d(g_n,b_n)-rn}{n}=0,$$
so $[g_n]=[h_n]\in[B(rn,b_n)]$ completing the proof of the fourth item and thus the lemma.
\end{proof}
\section{Lipschitz injections of nilpotent groups.}
We are now going to prove the main theorem of the paper. This theorem will state that if $f:\Gamma{\rightarrow}\Delta$ is an injective Lipschitz map between finitely generated nilpotent groups, and $d(\Gamma)=d(\Delta)$, then $\Gamma_\infty\ensuremath{\colon}ng\Delta_\infty$. The proof is organized as follows.
\betagin{itemize}
\item First, following Pansu \cite{Pansu2}, we see that the induced map on asymptotic cones $f_\infty:\Gamma_\infty{\rightarrow}\Delta_\infty$ is ``Pansu differentiable" almost everywhere. This derivative, where defined, is a group homomorphism $\Gamma_\infty{\rightarrow}\Delta_\infty$.
\item Second, we show that a homomorphism between Carnot groups of the same growth is an isomorphism if it induces a surjective map on abelianizations.
\item Finally, we show that the Pansu derivative of $f_\infty$ at any point where it is defined must be surjective.
\end{itemize}
\subsection{The Pansu derivative.}
\lambdabel{subsection:pansuderivative}
Given a Lipschitz map $F:\ensuremath{\mathbb{R}}R{\rightarrow}\ensuremath{\mathbb{R}}R$, Rademacher's theorem asserts that $F$ is differentiable almost everywhere. Pansu proved a generalization of this theorem to Lipschitz maps $F:G{\rightarrow} H$ of Carnot groups. In particular, he showed that, given the following definitions, we must have that $F$ is differentiable almost everywhere \cite[Theorem 2]{Pansu2}.
\betagin{definition}
Given $g\in G$, we say that $F$ is differentiable at $g$ if the limit
$$\lim_{s{\rightarrow} 0}\delta_{1/s}\left(F(g)^{-1}F(g\:\delta_s(x))\right),$$
converges uniformly for $x$ in any compact subset of $G$. If $F$ is differentiable at $g$, the function
$$x\mapsto \lim_{s{\rightarrow} 0}\delta_{1/s}\left(F(g)^{-1}F(g\:\delta_s(x))\right)
$$
defines a homomorphism known as the derivative of $F$ at $g$ and is denoted $DF|_g:G{\rightarrow} H$.
\end{definition}
The reader may compare to the ordinary derivative $\lim_{s{\rightarrow} 0}(F(g+sx)-F(g))/s$ of a real valued function of one real variable.
\subsection{Homomorphisms of Carnot groups.}
We will now establish the following alternative: a homomorphism between Carnot groups of the same growth rate either fails to be surjective on abelianizations or is an isomorphism. We break this up into two parts. We first demonstrate that a surjective homomorphism between Carnot groups of the same growth rate is an isomorphism (Proposition \ref{carnot_no_surjective_maps}). We then show that a homomorphism between Carnot groups is surjective if and only if the induced map on abelianizations is surjective (Proposition \ref{abelian_map_equiv}). We then deduce that if a homomorphism $F:\Gamma_\infty{\rightarrow}\Delta_\infty$ of Carnot groups of the same growth is not an isomorphism, its image is annihilated by some nontrivial homomorphism $\ell:\Delta_\infty{\rightarrow}\ensuremath{\mathbb{R}}R$ (Corollary \ref{annihilating_functional}).
\betagin{prop}\lambdabel{carnot_no_surjective_maps}
Let $\Gamma$ and $\Delta$ be torsion free, finitely generated nilpotent groups such that $d(\Gamma) = d(\Delta)$. Suppose that $\map{F}{\Gamma_{\infty}}{\Delta_\infty}$ is a surjective homomorphism. Then $\Gamma_{\infty} \ensuremath{\colon}ng \Delta_\infty$.
\end{prop}
\betagin{proof}
Before starting, we establish some notation for this proposition. We let $\Fr{g}$ and $\Fr{h}$ be the Lie algebras of $\Gamma_{\infty}$ and $\Delta_{\infty}$ respectively. We also let $\Gamma_{\infty,i} = \pr{\Gamma_{\infty}}_i$ and $\Delta_{\infty,i} = \pr{\Delta_{\infty}}_i$. Finally, we let $c_1$ and $c_2$ be the step lengths of $\Gamma$ and $\Delta$ respectively.
Lemma 1.2.5 of \cite{Dekimpe} implies that $\exp(\Fr{g}_t) = \Gamma_{\infty,t}$ and $\exp(\Fr{h}_t) = \Delta_{\infty,t}$. Since $\thetaxt{dim}(\Gamma_{\infty,t}) = \thetaxt{rank}(\Gamma_t)$ and $\thetaxt{dim}(\Delta_{\infty,t}) = \thetaxt{rank}(\Delta_t)$, we may phrase the growth rate of $\Gamma$ and $\Delta$ in terms of the $\ensuremath{\mathbb{R}}$-ranks of the quotients of the lower central series of $\Fr{g}$ and $\Fr{h}$. We may write
$$
d(\Gamma) = \sum_{k=1}^{c_1} k \cdot \thetaxt{dim}_\ensuremath{\mathbb{R}} (\Fr{g}_k / \Fr{g}_{k+1}).
$$
Similarly, we have
$$
d(\Delta) = \sum_{k=1}^{c_2} k \cdot \thetaxt{dim}_\ensuremath{\mathbb{R}} (\Fr{h}_k / \Fr{h}_{k+1}).
$$
We claim that the induced map $\map{dF_1|_{\Fr{g}_i}}{\Fr{g}_i}{\Fr{h}_i}$ is surjective. We proceed by induction on the term of the lower central series, and since $\Fr{g}_1 = \Fr{g}$ and $\Fr{h}_1 = \Fr{h}$, we have that the base case is evident. Now let $i > 1$, and consider the map $\map{dF_1|_{\Fr{g}_i}}{\Fr{g}_i}{\Fr{h}_i}$. Observe that $\Fr{g}_i = [\Fr{g}_{i-1},\Fr{g}]$ and $\Fr{h}_i = [\Fr{h}_{i-1},\Fr{h}]$.
For each $Y_1 \in \Fr{h}_{i-1}$ and $Y_2 \in \Fr{h}$, the inductive hypothesis implies that there exist $X_1 \in \Fr{g}_{i-1}$ and $X_2 \in \Fr{g}$ such that $dF_1(X_1) = Y_1$ and $dF_1(X_2) = Y_2$. Since $dF_1$ is a Lie algebra morphism, we have $dF_1([X_1,X_2]) = [dF_1(X_1),dF_1(X_2)] = [Y_1,Y_2]$. Observing that $\Fr{h}_i$ is generated by elements of the above form, we are done.
Write $\map{\pi_{\Fr{h}_{i+1}}}{\Fr{h}_i}{\Fr{h}_i/\Fr{h}_{i+1}}$ for the natural projection. Since $dF_1$ is surjective, the map $\pi_{\Fr{h}_{i+1}} \circ dF_1|_{\Fr{g}_i}$ is surjective. Given that $\Fr{g}_{i+1} \leq \ker (\pi_{\Fr{h}_{i+1}} \circ dF_1|_{\Fr{g}_i})$, we have an induced surjective map
$$
\map{\widetilde{dF}_1}{\Fr{g}_i / \Fr{g}_{i+1}}{\Fr{h}_i / \Fr{h}_{i+1}}.
$$
Therefore, $\thetaxt{dim}_\ensuremath{\mathbb{R}}(\Fr{g}_i / \Fr{g}_{i+1}) \geq \thetaxt{dim}_\ensuremath{\mathbb{R}}(\Fr{h}_i / \Fr{h}_{i+1})$. We may write
$$
0 = d(\Gamma) - d(\Delta) = \sum_k k \cdot \pr{\thetaxt{dim}_\ensuremath{\mathbb{R}}(\Fr{g}_i / \Fr{g}_{i+1}) - \thetaxt{dim}_\ensuremath{\mathbb{R}}(\Fr{h}_i / \Fr{h}_{i+1})} \geq 0.$$
That implies $\thetaxt{dim}_\ensuremath{\mathbb{R}}(\Fr{g}_i / \Fr{g}_{i+1}) = \thetaxt{dim}_\ensuremath{\mathbb{R}}(\Fr{h}_i / \Fr{h}_{i+1})$ for all $i$. In particular, $\thetaxt{dim}(\Fr{g}) = \thetaxt{dim}(\Fr{h})$. Since $dF_1$ is a surjective map between Lie algebras of the same dimension, it is an isomorphism as desired.
\end{proof}
\betagin{prop}\lambdabel{abelian_map_equiv}
Let $\Gamma$ and $\Delta$ be two torsion free, finitely generated nilpotent groups, and let $F:\Gamma_\infty{\rightarrow}\Delta_\infty$ be a Lie morphism. Then $F$ is surjective if and only if the induced map of abelianizations $F_\thetaxt{ab}:(\Gamma_\infty)_\thetaxt{Ab}{\rightarrow}(\Delta_\infty)_{\thetaxt{Ab}}$ is surjective.
\end{prop}
\betagin{proof}
Let $\Fr{g}$ and $\Fr{h}$ be the Lie algebras of $\Gamma_\infty$ and $\Deltalta_\infty$, respectively. Additionally, let $c_1$ and $c_2$ be the nilpotent step length of $\Gamma_\infty$ and $\Deltalta_\infty$, respectively. We may write the graded decompositions of $\Fr{g}$ and $\Fr{h}$ as
$$
\Fr{g} = \bigoplus_{i=1}^{c_1}\Fr{g}_i \quad \thetaxt{ and } \quad \Fr{h} = \bigoplus_{i=1}^{c_2} \Fr{h}_i
$$
where $[\Fr{g}_1,\Fr{g}_{i}] = \Fr{g}_{i+1}$ and $[\Fr{h}_{1},\Fr{h}_i] = \Fr{h}_{i+1}$ for all $i$.
Finally, let $dF_\thetaxt{Ab}:\Fr{g}_\thetaxt{Ab}{\rightarrow}\Fr{h}_\thetaxt{Ab}$ be the induced map of abelianizations.
Since $dF_\thetaxt{Ab}$ is surjective when $dF$ is surjective, we consider the case of when $dF_{\thetaxt{Ab}}:\Fr{g}_\thetaxt{Ab}{\rightarrow}\Fr{h}_{\thetaxt{Ab}}$ is surjective. We proceed by induction on step length of $\Fr{h}$, and note that the base case follows from assumption. Thus, we may assume that $c_2 > 1$. Observe that $\Fr{h}_\thetaxt{ab} \ensuremath{\colon}ng (\Fr{h} / \Fr{h}_{c_2})_{\thetaxt{ab}}$. That implies the abelianization of the induced map $\widetilde{dF}:\Fr{g}{\rightarrow}\Fr{h} / \Fr{h}_{c_2}$ is equivalent to the map $dF_\thetaxt{Ab}:\Fr{g}_{\thetaxt{ab}}{\rightarrow}\Fr{h}_{\thetaxt{Ab}}$. Therefore, the inductive hypothesis implies the map $\widetilde{dF}:\Fr{g}{\rightarrow}\Fr{h} / \Fr{h}_{c_2}$ is surjective. For $Y \in \Fr{h}_{c_2}$, there exists $Y_1 \in \Fr{h}_{1}$ and $Y_2 \in \Fr{h}_{c_2 - 1}$ such that $Y = [Y_1,Y_2]$. There exists $X_1, X_2 \in \Fr{g}$ such that $\widetilde{F}(X_1) = Y_1 \: \thetaxt{ mod } \: \Fr{h}_{c_2}$ and $\widetilde{F}(X_2) = Y_2 \: \thetaxt{ mod } \: \Fr{h}_{c_2}$. Thus, there exists $Z_1,Z_2 \in \Fr{h}_{c_2}$ such that $F(X_1) = Y_1 + Z_1$ and $F(X_2) = Y_2 + Z_2$. We may write
$$
dF([X_1,X_2]) = [dF(X_1),dF(X_2)] = [Y_1 + Z_1, Y_2 + Z_2] = [Y_1,Y_2] + [Y_1,Z_2] + [Z_1,Y_2] + [Z_1,Z_2].
$$
Since $\Fr{h}$ is Carnot, $\Fr{h}_{c_2} = Z(\Fr{h})$. Thus, $dF([X_1,X_2]) = [Y_1,Y_2] = Y$. In particular, $dF$ is surjective. \end{proof}
Combining the previous propositions, we have the following alternative.
\betagin{corollary}\lambdabel{annihilating_functional}
Suppose $d(\Gammamma)=d(\Deltalta)$, and $F:\Gammamma_\infty{\rightarrow}\Deltalta_\infty$ is a homomorphism of Carnot groups. Either $F$ is an isomorphism, or there exists a homomorphism $\ell:\Deltalta_\infty{\rightarrow}\ensuremath{\mathbb{R}}R$ such that $\ell\circ F=0$ and $\sup\{\ell(h):h\in B(1,1_{\Deltalta_\infty})\}=1$.
\end{corollary}
\betagin{proof}
Suppose $F$ is not an isomorphism. Proposition \ref{carnot_no_surjective_maps} implies that $F$ is not surjective, and Proposition \ref{abelian_map_equiv} implies that $F_{\thetaxt{Ab}}$ is not surjective. Thus, $\thetaxt{Im}(F_{\thetaxt{Ab}})$ is a proper subspace of $(\Delta_{\infty})_{\thetaxt{Ab}}$ which is a finite dimensional $\ensuremath{\mathbb{R}}$-vector space. Hence, we may choose a norm on $(\Deltalta_\infty)_\thetaxt{Ab}$ and a unit-norm vector $v$ orthogonal to $\thetaxt{Im}(F_{\thetaxt{Ab}})$. Let $\map{\ell_v}{(\Delta_{\infty})_{\thetaxt{Ab}}}{\ensuremath{\mathbb{R}}}$ be given by taking dot product with $v$ so that $\ell_v(v) = 1$ and $\ell_v(\thetaxt{Im}(F_{\thetaxt{ab}})) = 0$. Let $\pi_{\thetaxt{Ab}}:\Delta_{\infty}{\rightarrow}(\Delta_{\infty})_{\thetaxt{Ab}}$ be the natural projection, and let $M = \thetaxt{sup}\set{\ell_v(\pi_{\thetaxt{Ab}}(h)) \: | \: h \in B(1,1_{\Delta_{\infty}})}$ which is non-zero. Thus, the map $\map{\ell}{\Delta_{\infty}}{\ensuremath{\mathbb{R}}}$ given by $\ell(h) = M^{-1}\ell_v(\pi_{\thetaxt{Ab}}(h))$ satisfies the conditions of the proposition.
\end{proof}
\subsection{Applying Pansu's theorem.}
\lambdabel{subsection:maintheorem}
We now prove Theorem \ref{introtheorem}.
\betagin{theorem}
\lambdabel{maintheorem}
If $f:\Gammamma{\rightarrow}\Deltalta$ is a Lipschitz injection of torsion free, finitely generated nilpotent groups, and $d(\Gammamma)=d(\Deltalta)$, then $\Gammamma_\infty\ensuremath{\colon}ng\Deltalta_\infty$.
\end{theorem}
\betagin{proof}
Let $f_\infty:\Gammamma_\infty{\rightarrow}\Deltalta_\infty$ be the induced map on asymptotic cones. Towards a contradiction, assume that $\Gammamma_\infty$ and $\Deltalta_\infty$ are not isomorphic. We further assume, without loss of generality, that $f$ is $1$-Lipschitz.
The idea of the proof is as follows.
\betagin{itemize}
\item First, we use Pansu's theorem (\S\ref{subsection:pansuderivative}) to show that there exists $x\in\Gammamma_\infty$ such that $f_\infty$ is differentiable at $x$. Thus, we may choose some nonzero homomorphism $\ell:\Deltalta_\infty{\rightarrow}\ensuremath{\mathbb{R}}R$ which annihilates this derivative.
\item Second, we show that for every $N$ there exists $\varepsilonilon>0$ such that $\overline{B(2\varepsilonilon,f_\infty(x))}$ contains at least $N$ disjoint translates of $f_\infty(\overline{B(\varepsilonilon,x)})$. In particular, we find these translates by moving in a direction ``perpendicular" to the kernel of $\ell$, so that they have disjoint $\ell$-images.
\item Finally, we take a sequence $b_n$ in $\Gammamma$ such that $[b_n]=x$ and conclude that for every $N$ there is some $\varepsilonilon>0$ such that along ${\mathfrak{U}}$ the set $B(2\varepsilonilon n,f(b_n))$ contains at least $N$ disjoint translates of $f(B(\varepsilonilon n,b_n))$. Since $f$ is injective, this contradicts our assumption that $\Gammamma$ and $\Deltalta$ have the same growth.
\end{itemize}
\paragraph{Constructing $x$ and $\ell$.} By Pansu's theorem (\S\ref{subsection:pansuderivative}), the $1$-Lipschitz function $f_\infty:\Gammamma_\infty{\rightarrow}\Deltalta_\infty$ is differentiable almost everywhere, so there exists some $x\in\Gammamma_\infty$ where it is differentiable. Write $Df_\infty$ for $Df_\infty|_{x}$. By Corollary \ref{annihilating_functional}, we may fix a homomorphism $\ell:\Deltalta_\infty{\rightarrow}\ensuremath{\mathbb{R}}R$ such that $\ell\circ Df_\infty=0$ and $\sup\{\ell(h):h\in \overline{B(1,1_{\Deltalta_\infty})}\}=1$. In what follows, the reader may wish to make the simplifying assumption that $x=1_{\Gammamma_\infty}$ and $f_\infty(x)=1_{\Deltalta_\infty}$. (It is an exercise that this loses no generality).
\paragraph{Finding disjoint translates of $f_\infty(\overline{B(\varepsilonilon,x)})$.} For $\varepsilonilon>0$, let
$$\mu(\varepsilonilon)=3\sup\{|\ell({f_\infty}(g))-\ell({f_\infty}(x))|:g\in \overline{B(\varepsilonilon,x)}\}.$$
Because the defining limit for $Df_\infty$ converges uniformly on compact subsets and $\ell$ is Lipschitz, we have ${\mu(\epsilon)}/\varepsilonilon{\rightarrow} 0$ as $\varepsilonilon{\rightarrow} 0$. Choose $h\in \overline{B(1,1_{\Deltalta_\infty})}$ such that $\ell(h)=1$, and let ${h_\epsilon}={f_\infty}(x)\:\delta_{{\mu(\epsilon)}}(h)\:{f_\infty}(x)^{-1}$, so that $\ell({h_\epsilon})={\mu(\epsilon)}$.
Consider the set ${A_\epsilon}$ of translates of ${f_\infty}(\overline{B(\varepsilonilon,x)})$ given by
$${A_\epsilon}=\left\{
({h_\epsilon})^j {f_\infty}(\overline{B(\varepsilonilon,x)}):j\in\ensuremath{\mathbb{Z}}Z;|j|<\frac{\varepsilonilon}{{\mu(\epsilon)}}
\right\}.$$
We will simply refer to elements of ${A_\epsilon}$ as $\varepsilonilon$-\thetaxtit{good translates} and shall establish the following.
\betagin{itemize}
\item $\varepsilonilon$-good translates are subsets of $\overline{B(2\varepsilonilon,{f_\infty}(x))}$.
\item $\varepsilonilon$-good translates are disjoint, i.e., if $B_1,B_2\in {A_\epsilon}$ with $B_1\neq B_2$, then $B_1\cap B_2=\emptyset$.
\item The number of $\varepsilonilon$-good translates is $1+2\lfloor\frac{\varepsilonilon}{{\mu(\epsilon)}}\rfloor$ (i.e., $\#{A_\epsilon}=\#\{j\in \ensuremath{\mathbb{Z}}Z:|j|<\varepsilonilon/{\mu(\epsilon)}\}$).
\end{itemize}
\paragraph{$\varepsilonilon$-good translates are subsets of the $2\varepsilonilon$ ball.} Observe that for any integer $j$,
$$({h_\epsilon})^j\:{f_\infty}(\overline{B(\varepsilonilon,x)})\subseteq ({h_\epsilon})^j\: B(\varepsilonilon,{f_\infty}(x))
=\left({f_\infty}(x)\:\delta_{\mu(\epsilon)}(h)^j\:{f_\infty}(x)^{-1}\right)\:
{f_\infty}(x)\:B(\varepsilonilon,1_{\Deltalta_\infty})$$
$$={f_\infty}(x)\:\delta_{\mu(\epsilon)}(h)^j\:B(\varepsilonilon,1_{\Deltalta_\infty}) \subseteq {f_\infty}(x)\:B(|j|\mu(\varepsilonilon),1_{\Deltalta_\infty})\:
B(\varepsilonilon,1_{\Deltalta_\infty})=B(\varepsilonilon+|j|{\mu(\epsilon)},{f_\infty}(x)).$$
It follows that $\varepsilonilon$-good translates are subsets of $\overline{B(2\varepsilonilon,{f_\infty}(x))}$.
\paragraph{$\varepsilonilon$-good translates are disjoint.} Since $\mu(\varepsilonilon)=3\sup\{|\ell({f_\infty}(g))-\ell({f_\infty}(x))|:g\in \overline{B(\varepsilonilon,x)}\},$ we have for $j \in \ensuremath{\mathbb{Z}}Z$ that
$$\ell(({h_\epsilon})^j {f_\infty}(\overline{B(\varepsilonilon,x)}))
=j{\mu(\epsilon)} + \ell({f_\infty}(\overline{B(\varepsilonilon,x)}))$$
$$\subseteq
\left(\left(j-\frac{1}{3}\right){\mu(\epsilon)}+\ell({f_\infty}(x)), \left(j+\frac{1}{3}\right){\mu(\epsilon)}+\ell({f_\infty}(x))\right).$$
Hence, for $j,k\in\ensuremath{\mathbb{Z}}Z$ with $j\neq k$ we have that
$$({h_\epsilon})^j {f_\infty}(\overline{B(\varepsilonilon,x)})
\cap ({h_\epsilon})^k {f_\infty}(\overline{B(\varepsilonilon,x)})
=\emptyset$$
because the images under $\ell$ of $({h_\epsilon})^j {f_\infty}(\overline{B(\varepsilonilon,x)})$ and $({h_\epsilon})^k {f_\infty}(\overline{B(\varepsilonilon,x)})$ are contained in disjoint intervals.
\paragraph{Counting $\varepsilonilon$-good translates.} Observe that there are exactly $1+2\lfloor\frac{\varepsilonilon}{{\mu(\epsilon)}}\rfloor$ integers $j$ such that $j{\mu(\epsilon)}\leq 2\varepsilon$. Hence, $\#{A_\epsilon}=1+2\lfloor\frac{\varepsilonilon}{{\mu(\epsilon)}}\rfloor$, and thus for any $N>0$ we have that $\# A_\varepsilonilon>N$ for sufficiently small $\varepsilonilon>0$, since $\varepsilonilon/{\mu(\epsilon)}{\rightarrow}\infty$ as $\varepsilonilon{\rightarrow} 0$.
\paragraph{Finding disjoint translates of $f$-images of balls in $\Gammamma$.} Take a sequence $(b_n)_{n\in\ensuremath{\mathbb{N}}N}$ of elements of $\Gammamma$ with $[b_n]=x$, and, for each $\varepsilonilon>0$, a sequence $({h_{\epsilon,n}})_{n\in\ensuremath{\mathbb{N}}N}$ in $\Deltalta$ such that $[{h_{\epsilon,n}}]={h_\epsilon}$. For an integer $j$, we have that $[({h_{\epsilon,n}})^j f(B(\varepsilonilon n,b_n))]=({h_\epsilon})^j {f_\infty}(\overline{B(\varepsilonilon,x)})$ and
$[B(2\varepsilonilon n,f(b_n))]=\overline{B(2\varepsilonilon,{f_\infty}(x))}$.
For any $\varepsilonilon>0$ and $n\in\ensuremath{\mathbb{N}}N$, consider the set of translates of $f(B(\varepsilonilon n,b_n))$ given by
$${A_{\epsilon,n}}=
\left\{
({h_{\epsilon,n}})^j f(B(\varepsilonilon n,b_n)) : j\in\ensuremath{\mathbb{Z}}Z;|j|<\frac{\varepsilonilon}{{\mu(\epsilon)}}
\right\}.$$
We call these $(\varepsilonilon,n)$-\thetaxtit{good translates}. We see for any $\varepsilonilon$ that the following properties of ${A_\epsilon}$ pass to ${A_{\epsilon,n}}$ along ${\mathfrak{U}}$, i.e., there exists $U\in{\mathfrak{U}}$ such that the following statements all hold for $n\in U$.
\betagin{itemize}
\item Each $(\varepsilonilon,n)$-good translate is a subset of $B(2\varepsilonilon n,f(b_n))$.
\item $(\varepsilonilon,n)$-good translates are disjoint, i.e., if $B_1,B_2\in {A_{\epsilon,n}}$ with $B_1\neq B_2$, then $B_1\cap B_2=\emptyset$.
\item The number of $(\varepsilonilon,n)$-good translates is $1+2\lfloor\frac{\varepsilonilon}{{\mu(\epsilon)}}\rfloor$.
\end{itemize}
\paragraph{Conclusion.} Observe that, since $\Gammamma$ and $\Deltalta$ both have growth of order $n^{d(\Gammamma)}$, there exists $N>0$ such that
$$N\:\#B(n,1_\Gammamma)>\# B(2n,1_\Deltalta)$$
for all $n>N$. Take $\varepsilonilon > 0$ such that $\#{A_{\epsilon,n}}>N$. Then along ${\mathfrak{U}}$,
$$\bigsqcup_{B\in{A_{\epsilon,n}}}B\subseteq B(2\varepsilonilon n, f(b_n)),$$
and, thus
$$\#\bigsqcup_{B\in{A_{\epsilon,n}}}B\leq\# B(2\varepsilonilon n, f(b_n)).$$
On the other hand, by injectivity of $f$ we see that
$$\# \bigsqcup_{B\in{A_{\epsilon,n}}}B = N(\# B(\varepsilonilon n,1_\Gammamma))>B(2\varepsilonilon n,f(b_n)).$$
We have thus obtained a contradiction, as desired.
\end{proof}
\end{document} |
\begin{document}
\title{The $p$-norm of circulant matrices \via Fourier analysis}
{\renewcommand{\arabic{footnote}}{}\footnotetext{
\noindent$^\ast${Department of Electrical Communication Engineering, Indian Institute of Science. Email: [email protected] }}}
\renewcommand{\arabic{footnote}}{\arabic{footnote}}
\setcounter{footnote}{0}
\begin{abstract}
A recent paper~\cite{bouthat2021p} computed the induced $p$-norm of a special class of circulant matrices $A(n,a,b) \in \mathbb{R}^{n \times n}$, with the diagonal entries equal to $a \in \mathbb{R}$ and the off-diagonal entries equal to $b \ge 0$. We provide shorter proofs for all the results therein using Fourier analysis. The key observation is that a circulant matrix is diagonalized by a DFT matrix. We obtain an exact expression for $\|A\|_p, 1 \le p \le \infty$, where $A = A(n,a,b), a \ge 0$ and for $\|A\|_2$ where $A = A(n,-a,b), a \ge 0$; for the other $p$-norms of $A(n,-a,b)$, $2 < p < \infty$, we provide upper and lower bounds.
\end{abstract}
\section{Introduction}
Circulant matrices arise in many applications ranging from wireless communications~\cite{tse2005fundamentals} to cryptography~\cite{pub2001announcing} to solving differential equations~\cite{wilde1983differential} (see~\cite{bouthat2021p} and the references therein for the historical context and more recent theoretical studies on circulant matrices). A circulant matrix is of the form
\[
A = \left[
\begin{array}{ccccc}
a_1 & a_2 & a_3 & \ldots & a_n \\
a_n & a_1 & a_2 & \ldots & a_{n-1} \\
a_{n-1} & a_n & a_1 & \ldots & a_{n-2}\\
\vdots & \vdots & \vdots & \ddots & \vdots \\
a_2 & a_3 & a_4 & \ldots & a_1
\end{array}
\right],
\]
where $a_j \in \mathbb{R}, 1 \le j \le n$. For a matrix $A \in \mathbb{R}^{n \times n}$, we define the operator norm
\[
\|A\|_p = \sup_{x \neq 0} \frac{\|Ax\|_p}{\|x\|_p},
\]
for $1 \le p \le \infty$, where, for a vector $y = (y_1,\ldots,y_n) \in \mathbb{R}^n$,
\[
\|y\|_\infty = \max\left\{|y_1|,\ldots,|y_n|\right\},
\]
and for $1 \le p < \infty$,
\[
\|y\|_p = \left(|y_1|^p + \cdots + |y_n|^p\right)^{1/p}.
\]
It is well-known~\cite{rmgraycirculant} that the eigen decomposition of a circulant matrix $A$ is of the form $F^* \Lambda F$ where $F$ denotes the Discrete Fourier Transform (DFT) matrix (proof provided in the appendix for completeness); $F_{jk} = \frac{1}{\sqrt{n}}\cdot \omega_n^{-jk}, 0 \le j,k \le n-1$, where $\omega_n = e^{\frac{2\pi i}{n}}$ and for a matrix $B$, $B_{jk}$ denotes its $(j,k)$-th entry and $B^*$ denotes its adjoint. The eigenvalues, namely the diagonal entries of $\Lambda$, are given by
\begin{equation}
\lambda_k := \Lambda_{kk} = \sum_{j=0}^{n-1} a_{j+1} \omega_n^{jk}, ~~0 \le k \le n-1.
\label{eqn:eval}
\end{equation}
We use this property of circulant matrices to study the $p$-norm of a special class of circulant matrices, $A(n,a,b) \in \mathbb{R}^{n \times n}, a,b \in \mathbb{R}$, where
\[
A(n,a,b) := \left[
\begin{array}{ccccc}
a & b & b & \ldots & b \\
b & a & b & \ldots & b \\
b & b & a & \ldots & b \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
b & b & b & \ldots & a
\end{array}
\right].
\]
The $1$-norm and the infinity norm of $A(n,a,b)$ are easily calculated to be $|a| + (n-1)|b|$ by inspection. As observed in~\cite{bouthat2021p}, it suffices to consider the following two cases: $A(n,a,b)$ and $A(n,-a,b)$ where $a,b \ge 0$. We obtain an exact expression for $\|A\|_p, 1 < p < \infty$, where $A = A(n,a,b)$ and for $\|A\|_2$ where $A = A(n,-a,b)$. For $A = A(n,-a,b)$, we provide upper and lower bounds for $\|A\|_p, 2 < p < \infty$.
\section{Results and Proofs}
For a diagonal matrix, all the induced $p$-norms are equal to the maximum of the absolute value of the entries~\cite{horn2012matrix}. We calculate this value for $\Lambda$.
\begin{lemma}
For $a,b \ge 0$, for $1 \le p \le \infty$,
\begin{enumerate}
\item[i.] for $A = A(n,a,b)$ and $A = F^*\Lambda F$, we have
\begin{align*}
\|\Lambda\|_p = a + (n-1)b,
\end{align*}
\item[ii.] for $A = A(n,-a,b)$ and $A = F^*\Lambda F$, we have
\[
\|\Lambda\|_p = \begin{cases}
-a + (n-1)b &\text{ if } 2a \le (n-2)b\\
a + b &\text{ otherwise.}
\end{cases}
\]
\end{enumerate}
\label{l:lamnorm}
\end{lemma}
\begin{proof}
For $A = A(n,a,b), a,b \in \mathbb{R}$ and $A = F^*\Lambda F$, by~\eqref{eqn:eval}, the diagonal entries of $\Lambda$ are given by
\begin{align*}
\lambda_k &= a\omega_n^{kk} + \sum_{\underset{j \neq k}{j=0}}^{n-1} b\omega_n^{jk}\\
&= b \sum_{j=0}^{n-1}\omega_n^{jk} + (a-b)\omega_n^{kk},
\end{align*}
for $0 \le k \le n-1$. Using the well-known identity (see, for example,~\cite{stein2011fourier}),
\[
\sum_{j=0}^{n-1} \omega_n^{jk} = \begin{cases}
n &\text{ if } k = 0 ~(\hspace{-0.35cm}\mod n)\\
0 &\text{ otherwise,}
\end{cases}
\]
we have
\begin{align*}
\lambda_0 &= bn + (a-b) = a + (n-1)b,
\end{align*}
and for $0 < k \le n-1$,
\[
|\lambda_k| = |a-b|.
\]
The result follows by calculating $\|\Lambda\|_p = \underset{0\le k \le n-1}{\max} |\lambda_k|$ for $1 \le p \le \infty$, for $A(n,a,b)$ and $A(n,-a,b)$.
\end{proof}
We use Lemma~\ref{l:lamnorm} to derive an exact expression for $\|A\|_2$ for $A(n,a, b)$ as well as $A(n,-a,b)$, for $a,b \ge 0$.
\begin{theorem}
For $a,b \ge 0$,
\begin{enumerate}
\item[i.] for $A = A(n,a,b)$, we have
\begin{align*}
\|A\|_2 = a + (n-1)b.
\end{align*}
\item[ii.] for $A = A(n,-a,b)$, we have
\[
\|A\|_2 = \begin{cases}
-a + (n-1)b &\text{ if } 2a \le (n-2)b\\
a + b &\text{ otherwise.}
\end{cases}
\]
\end{enumerate}
\label{thm:2norm}
\end{theorem}
\begin{proof}
The result follows by observing that $\|A\|_2 = \|\Lambda\|_2$ since $F$ is unitary, and using Lemma~\ref{l:lamnorm}.
\end{proof}
\begin{remark*}
In~\cite{bani2008norm}, similar techniques are employed to calculate the (unitarily invariant) Schatten $p$-norms of block circulant matrices.
\end{remark*}
As observed in~\cite{bouthat2021p}, $A = A(n,a,b), a \in \mathbb{R}, b \ge 0$ is self-adjoint and hence $\|A\|_p = \|A\|_q$, for $p$ and $q$ satisfying $\frac{1}{p} + \frac{1}{q} = 1$ (see~\cite{horn2012matrix}). Thus, it suffices to focus on either $p \in (1,2]$ or $p \in [2,\infty)$. First, we compute the $p$-norm of $A = A(n,a,b)$ with $a,b \ge 0$ for $p \ge 2$.
\begin{theorem}
For $A = A(n,a,b),a,b \ge 0$, for $p \ge 2$,
\[
\|A\|_p = a + (n-1)b.
\]
\end{theorem}
\begin{proof}
Using the vector $x = [1,1,\ldots,1]^T$, where $[\cdot]^T$ denotes the transpose, we have
\[
\|A\|_p \ge a + (n-1)b.
\]
Next, observe that
\[
\|A\|_\infty = a + (n-1)b = \|A\|_2,
\]
where the last identity is by Theorem~\ref{thm:2norm}. By the Riesz-Thorin interpolation theorem~\cite[Theorem $2.1$]{stein2011functional} we have, for every $0 < \theta < 1$,
\[
\|A\|_{p_\theta} \le \|A\|_q^{1-\theta}\|A\|_r^{\theta},
\]
where $p_\theta,q,$ and $r$ satisfy
\begin{equation}
\frac{1}{p_\theta} = \frac{1-\theta}{q} + \frac{\theta}{r}.
\label{eqn:rt}
\end{equation}
Setting $p_\theta = p, q = 2$, and $r = \infty$ in~\eqref{eqn:rt} yields
\[
\|A\|_p \le \|A\|_\infty = a + (n-1)b.
\]
\end{proof}
\begin{remark*}
Using similar arguments as above, one can calculate $\|A\|_p$ of a general circulant matrix $A$ with non-negative entries $a_1,\ldots,a_n$ to be $a_1 + \cdots + a_n$.
\end{remark*}
Next, we estimate the $p$-norm of $A(n,-a,b)$ with $a,b \ge 0$ for $p \ge 2$.
\begin{theorem}
\label{thm:pnorm}
For $A = A(n,-a,b), a,b \ge 0$ and $A = F^* \Lambda F$, for $p \ge 2$, we have
\begin{align*}
- a + (n-1)b &\le \|A\|_p \le n^{\frac{1}{2}-\frac{1}{p}} \cdot (-a + (n-1)b) &\text{ if } 2a \le (n-2)b,\\
a + b &\le \|A\|_p \le n^{\frac{1}{2}-\frac{1}{p}} \cdot (a+b) &\text{ if } 2a \ge (n-2)b.
\end{align*}
\end{theorem}
\begin{proof}
For $p \ge 2$ and $x \neq 0$, we have
\begin{align*}
\|Ax\|_p &\le \|Ax\|_2\\
&\le \|F^*\|_2 \cdot \|\Lambda\|_2 \cdot \|\widehat{x}\|_2,
\end{align*}
where $\widehat{x}=Fx$ denotes the Fourier transform of $x$. By Plancherel's relation, we have $\|\widehat{x}\|_2 = \|x\|_2$ and $\|F^*\|_2 = 1$. Hence
\[
\|Ax\|_p \le \|\Lambda\|_2 \cdot \|x\|_2,
\]
whereby
\[
\|A\|_p \le n^{\frac{1}{2}-\frac{1}{p}} \cdot \|\Lambda\|_2,
\]
where we have used the inequality $\|x\|_r \le \|x\|_p \cdot n^{\frac{1}{r}-\frac{1}{p}}$ for $2=r \le p$. The upper bounds in the theorem now follow from Lemma~\ref{l:lamnorm}. To obtain the lower bounds, we exhibit a vector $x \neq 0$ such that $\|Ax\|_p/\|x\|_p$ equals the quantity in the desired lower bound.
\begin{enumerate}
\item[i.] $2a \le (n-2)b$: for $x = [1,1,\ldots,1]^T$,
\[
\frac{\|Ax\|_p}{\|x\|_p} = |-a + (n-1)b| \ge -a + (n-1)b.
\]
\item[ii.] $2a \ge (n-2)b$: for $x = [-1,1,0,\ldots,0]^T$,
\[
\frac{\|Ax\|_p}{\|x\|_p} = \left(\frac{|a+b|^p + |-a-b|^p}{2}\right)^{1/p} = a+b.
\]
\end{enumerate}
\end{proof}
Finally, we provide an improved upper bound for the $p$-norm of $A(n,-a,b)$ for $p > 2$, using the Riesz-Thorin interpolation theorem.
\begin{theorem}
For $A = A(n,-a,b), a,b \ge 0$, for every $p > 2$, we have,
\begin{align*}
\|A\|_p \le \|A\|_2^{\frac{2}{p}}\|A\|_\infty^{1-\frac{2}{p}}.
\end{align*}
\end{theorem}
\begin{proof}
Setting $q = 2$ and $r = \infty$ in~\eqref{eqn:rt} yields
\[
\frac{1}{p_\theta} = \frac{1-\theta}{2}.
\]
We choose $p_\theta = p$ to get
\[
\|A\|_{p} \le \|A\|_{2}^{1-\theta} \|A\|_\infty^{\theta}.
\]
Since $\theta = 1-2/p$, we have
\[
\|A\|_{p} \le \|A\|_2^{\frac{2}{p}}\|A\|_\infty^{1-\frac{2}{p}}.
\]
\end{proof}
In fact, using similar arguments we can show the following.
\begin{theorem}
For $A = A(n,-a,b), a,b, \ge 0$, for $p \ge 2$, $\|A\|_{p}$ is monotonically non-decreasing in $p$.
\end{theorem}
\begin{proof}
Fix $\beta > 0$. Setting $q = p-\alpha, \alpha > 0, r = p+\beta$, and $p_\theta = p$ in~\eqref{eqn:rt} yields
\[
\|A\|_{p} \le \|A\|_{p-\alpha}^{1-\theta}\|A\|_{p+\beta}^\theta.
\]
Choose $\alpha$ such that
\[
\frac{1}{p-\alpha} + \frac{1}{p+\beta} = 1.
\]
Since $A$ is self-adjoint, we have $\|A\|_{p-\alpha} = \|A\|_{p+\beta}$, and hence
\[
\|A\|_p \le \|A\|_{p+\beta}.
\]
\end{proof}
\begin{remark*}
As a corollary, we get $\|A\|_p \ge \|A\|_2$ for all $p \ge 2$, which is the same as the lower bound in Theorem~\ref{thm:pnorm}.
\end{remark*}
\section{Summary}
Using the observation that a circulant matrix is diagonalized by a DFT matrix, we have computed the $p$-norm of a special class of circulant matrices $A(n,a,b) \in \mathbb{R}^{n \times n}$, with the diagonal entries equal to $a \in \mathbb{R}$ and the off-diagonal entries equal to $b \ge 0$. The $1$-norm and the infinity norm of $A(n,a,b)$ are easily calculated to be $|a| + (n-1)|b|$ by inspection. For $A = A(n,a,b)$ with $a,b \ge 0$, we show that for all $1 \le p \le \infty$,
\[
\|A\|_p = a + (n-1)b.
\]
For $A = A(n,-a,b)$ with $a,b \ge 0$, we obtain an exact expression for $\|A\|_2$. Since $A$ is self-adjoint, $\|A\|_q = \|A\|_p$ for conjugate pairs, $p$ and $q$. This, along with the Riesz-Thorin interpolation theorem, implies that for $2 \le p \le \infty$, $\|A\|_p$ is monotonically non-decreasing in $p$. Further, we show that for $2 \le p \le \infty$,
\[
\|A\|_2 \le \|A\|_p \le \|A\|_2^{\frac{2}{p}}\|A\|_\infty^{1-\frac{2}{p}}.
\]
An exact expression for $\|A\|_p, 2 < p < \infty$ for $A = A(n,-a,b),a,b\ge 0$, remains elusive.
\section*{Acknowledgments}
The author thanks Prof. Manjunath Krishnapur and Prof. Apoorva Khare for useful comments.
\begin{appendix}
We prove that for any circulant matrix $A$, the eigen decomposition is of the form $A = F^* \Lambda F$, where
\[
F^* = \frac{1}{\sqrt{n}} \left[
\begin{array}{ccccc}
1 & 1 & 1 & \ldots & 1 \\
1 & \omega & \omega^2 & \ldots & \omega^{n-1} \\
1 & \omega^2 & \omega^4 & \ldots & \omega^{2(n-1)}\\
\vdots & \vdots & \vdots & \ddots & \vdots \\
1 & \omega^{n-1} & \omega^{2(n-1)} & \ldots & \omega^{(n-1)(n-1)}
\end{array}
\right],
\]
and $\omega = e^{\frac{2\pi i}{n}}$ (we drop the subscript $n$ of $\omega_n$ for brevity). Define a permutation matrix
\[
P := \left[
\begin{array}{ccccc}
0 & 1 & 0 & \ldots & 0 \\
0 & 0 & 1 & \ldots & 0 \\
0 & 0 & 0 & \ldots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
1 & 0 & 0 & \ldots & 0
\end{array}
\right].
\]
\textbf{\emph{Claim.}} $P = F^* \Omega F$ where $\Omega$ is a diagonal matrix with entries $\Omega_{kk} = \omega^k, 0 \le k \le n-1$.
\begin{proof}
For $0 \le j,k \le n-1$, observe that
\begin{align*}
j\text{-th row of } F^* &= \frac{1}{\sqrt{n}}\left[\omega^{0j}~~\omega^{1j}~~\cdots~~\omega^{(n-1)j}\right],\\
j\text{-th row of } F^*\Omega &= \frac{1}{\sqrt{n}}\left[\omega^{0(j+1)}~~\omega^{1(j+1)}~~\cdots~~\omega^{(n-1)(j+1)}\right],\\
k\text{-th column of } F &= \frac{1}{\sqrt{n}}\left[\omega^{-0k}~~\omega^{-1k}~~\cdots~~\omega^{-(n-1)k}\right]^T.
\end{align*}
Thus,
\begin{align*}
(j,k)\text{-th element of } F^*\Omega F &= \frac{1}{n} \sum_{r=0}^{n-1} \omega^{r(j+1)}\omega^{-rk} = \begin{cases}
1 \text{ if } j = k-1~(\hspace{-0.35cm}\mod n)\\
0 \text{ otherwise.}
\end{cases}
\end{align*}
\end{proof}
Therefore, a circulant matrix
\begin{align*}
A &= \left[
\begin{array}{ccccc}
a_1 & a_2 & a_3 & \ldots & a_n \\
a_n & a_1 & a_2 & \ldots & a_{n-1} \\
a_{n-1} & a_n & a_1 & \ldots & a_{n-2}\\
\vdots & \vdots & \vdots & \ddots & \vdots \\
a_2 & a_3 & a_4 & \ldots & a_1
\end{array}
\right]\\
&= a_1 I + a_2 P + a_3 P^2 + \cdots + a_n P^{n-1}\\
&= F^* \left(a_1 I + a_2 \Omega + a_3 \Omega^2 + \cdots + a_n \Omega^{n-1}\right)F\\
&= F^* \Lambda F.
\end{align*}
Observe that $\lambda_k = \Lambda_{kk}, 0 \le k \le n-1$, satisfy~\eqref{eqn:eval}.\\
\end{appendix}
\footnotesize \emph{Acknowledgement.} The author learnt this proof from Ashok Krishnan, IISc, at a Digital Communication study group meeting in $2011$.
\end{document} |
\begin{equation}gin{document}
\begin{equation}gin{abstract}
In this paper, we consider the stochastic Boussinesq equations on $\mathbb T^3$ with transport noise and rough initial data. We first prove the existence and uniqueness of the local pathwise solution with initial data in $L^p(\Omega;L^p)$ for $p>5$. By assuming additional smallness on the initial data and the noise, we establish the global existence of the pathwise solution.
\epsnd{abstract}
\maketitle
MSC Subject Classifications: 35Q86, 60H15, 76M35, 35Q35\\
Keywords: Stochastic Boussinesq equations, transport noise, rough initial data, well-posedness, pathwise solution
{\rm s}ection{Introduction}
We are interested in the stochastic Boussinesq equations on a 3D torus $\mathbb T^3$
\begin{equation}gin{align}\label{e.w09241}
\begin{equation}gin{split}
du&=(\Delta u-\mathcal P(u\cdot \nabla u -\rho e_{3}))dt+\mathcal P(b\cdot\nabla u + {\rm s}igma^{(1)}(u,\rho)) d\mathbb{W}_t, \quad \nabla\cdot u =0, \\
d\rho&=(\Delta \rho -u\cdot \nabla\rho )dt + (b\cdot\nabla \rho + {\rm s}igma^{(2)}(u,\rho))d\mathbb{W}_t.
\epsnd{split}
\epsnd{align}
Here $u=u(x, t)$ is the velocity vector field, $\rho = \rho(x, t)$ is the scalar temperature or density of fluid, $\mathcal P$ is the Leray projection (see \epsqref{leray}), and $\mathbb{W}_t$ is a cylindrical Wiener process valued in some separable Hilbert space and the corresponding stochastic integral is in the Itô sense. The term $(\mathcal P(b\cdot\nabla u), b\cdot\nabla\rho)d\mathbb{W}_t$ represents the transport noise and $(\mathcal P({\rm s}igma^{(1)}(u,\rho)),{\rm s}igma^{(2)}(u,\rho))d\mathbb{W}_t$ is the multiplicative noise. The main result of the paper is the global well-posedness of the system \epsqref{e.w09241} in $L^p(\Omega;L^p(\mathbb T^3))$ with $p>5$ and small initial data.
The deterministic Boussinesq system describes the evolution of the velocity field $u$ of an incompressible fluid under the buoyancy $\rho e_3$. Recently, there has been a lot of progress made on the existence, persistence of regularity, and long time behavior of solutions, mostly in the case of positive viscosity. Mathematically, the 2D Boussinesq system
is closely related to the 3D incompressible Euler and Navier-Stokes equations since they share a similar vortex stretching effect. In the case of zero viscosity and diffusivity, the 2D Boussinesq equations can be used as a proxy for the 3D axis-symmetric Euler
equation with swirl away from the symmetric axis. For more recent well-posedness and long time behavior results for the deterministic Boussinesq equations with fractional or full dissipation, cf. \cite{ BH,CN, HL, KW1,LLT}. See \cite{KW2, W1} when boundary conditions are imposed.
Over the past few decades, there has been a growing interest in investigating the impact of stochastic effects on fluid models. By introducing white noise terms into the system and assuming random initial data, these models can account for both numerical and empirical uncertainties. This approach can yield predictions that not only reflect a realistic trajectory but also provide insight into associated uncertainties. Along this line of research, Bensoussan and Temam started the study on the stochastic Navier-Stokes equations \cite{BT}. Mikulevicius and Rozovskii \cite{MR} addressed global $L^2$ well-posedness. See also \cite{ZZ} for local well-posednes for the 3D stochastic Navier-Stokes equations. More recently, Kukavica, Xu, and Ziane considered the stochastic Navier-Stokes system with multiplicative noise and rough initial data \cite{KXZ}. For the stochastic Boussinesq equations, Du addressed local well-posedness for the 3D Boussinesq system with Sobolev initial data \cite{D}. Duan and Millet \cite{DM} studied large deviation principle of the stochastic Boussinesq system. Pu and Guo \cite{PG} proved global well-posedness for the 2D Boussinesq equations with additive white noise while F\"{o}ldes et al \cite{FGRT} considered ergodic and mixing properties of the {B}oussinesq equations. In \cite{WY}, together with Yue, the third author of this paper proved almost-sure global existence of weak solutions to the Boussinesq equations using random data approach. Luo \cite{L} recently considered 2D Boussinesq equations with transport noise and Alonso-Or\'{a}n and Bethencourt de Le\'{o}n proved global well-posedness with transport noise and Sobolev initial data \cite{AB}. See also \cite{HZSG} for the stochastic fractional Boussinesq equations. For the recent progress for other stochastic fluid euations, see, for example, \cite{brzezniak2001stochastic,capinski1999stochastic,glatt2014local} for the stochastic Euler equations and \cite{brzezniak2021well,debussche2011local,debussche2012global,hu2022local,hu2023pathwise} for the stochastic primitive equations.
To the best of our knowledge, this work seems to be the first one to consider the global well-posedness for the 3D Boussinesq equations with transport noise and rough data. The transport noise represent a wider and more general class of noise than the multiplicative noise.
In the study of turbulent flows, the transport noise has been introduced in \cite{kraichnan1968small,kraichnan1994anomalous} and widely studied in many different SPDEs, see for example \cite{agresti2022stochastic-1,agresti2022stochastic-2,mikulevicius2004stochastic,MR,flandoli2008introduction,agresti2021stochastic-2,brzezniak1992stochastic} and refereces therein. It is worthwhile to mention that with transport noise we are able to prove the existence and uniqueness of solutions when $(u_0,\rho_0)\in L^p(\Omega;L^p(\mathbb T^3))$ with $p>5$. The extension to $p>3$ (see \cite{kukavica2023local} for Navier-Stokes equations with multiplicative noise) remains open when the transport noise is present. Our result is not covered by \cite{agresti2021stochastic-2}, which investigates the Navier-Stokes equations in Besov spaces, as $L^p$ spaces and Besov spaces are not equivalent.
Compared to previous results where people consider either multiplicative noise or smoother initial data, one of the main difficulties in this work is on the rough $L^p$ analysis of transport noise under Leray projection on the velocity field. Indeed, in the $L^p$ scenario, the regularity of the dissipation is at a scale weaker than the $W^{1, p}$. Therefore, one cannot control the energy of the transport noise through a $W^{1, p}$ estimate. In the case when there is no Leray projection, by assuming smallness on the noise intensity, one can obtain an energy estimate of the transport noise that is comparable with (and hence compensated by) the dissipation. However, this approach is not directly applicable in the presence of Leray projection due to the non-local feature of the projection. Our key observation is that the energy of the orthogonal part of the Leray projection on the transport noise $\nabla\Delta^{-1}A_{a} vv(b\cdot \nabla u)d\mathbb{W}_t$,
when combined with the divergence free property of the velocity $u$, can be controlled by $L^p$ energy of the velocity through standard elliptic regularity estimates. This enables us to obtain a desired energy estimate for the Leray projection of the transport noise at a scale comparable with dissipation and $L^p$ energy. Another difficulty is to control the convection term during the fixed point iteration process. This is overcome by using a double cut-off trick introduced in \cite{KXZ}. The local pathwise solution is then obtained as the limit of solutions to the truncated systems by sending the truncating scale to infinity. Fixing the cut-off at a small scale, global existence is proved by analyzing the truncated system, whose solution agrees with the solution to \epsqref{e.w09241} over all time horizon with high probability as long as the initial data is small. This involves an analysis of the transport noise mentioned earlier that requires its intensity to be sufficiently small.
The rest of this paper is organized as follows. In Section \ref{NP}, we introduce basic notations and preliminaries. In Section \ref{s.L022301} we establish the $L^p$ estimates for the linear parabolic equation with transport noise under Leray projection. Such estimate is the key to show the local and global well-posedness for the stochastic Boussinesq equations. Section \ref{s.w02271} is devoted to the study of a truncated system from the Boussinesq equations, and after that by applying a family of suitable stopping times we are able to obtain the maximal pathwise solution to the original Boussinesq equations. Finally, we prove the global existence of pathwise solutions in Section \ref{sec:global}.
{\rm s}ection{Notations and preliminaries}\label{NP}
{\rm s}ubsection{Functional settings}
The universal constant $C$ appears in the paper may change from line to line. We shall use subscripts to indicate the dependence of $C$ on other parameters when necessary, {\it e.g.}, $C_r$ means that the constant $C$ depends only on $r$.
Let $L^p$ and $W^{s,p}$ with $s\in \mathbb N$ and $p\geq 1$ be the usual Sobolev spaces (see \cite{adams2003sobolev}). Let $B_{p, q}^{s}$ with $q\geq1$ be the usual Besov space (see \cite{bahouri2011fourier}). In the periodic setting, for $s\in \mathbb N$ the $n-$th Fourier coefficient of an $L^1$ function $f$ on $\mathbb T^3$ is defined as
\begin{equation}gin{equation}
\mathcal Ff(n)=\hat{f}(n)=\int_{\mathbb T^3}f(x)e^{-2\pi in\cdot x}\,dx {\rm ,\qquad{}} n\in\mathbb Z^3.
\epsnd{equation}
Denote the operator $J^s$ by
\begin{equation}gin{equation}
J^{s}f(x)={\rm s}um_{n \in \mathbb Z^3}(1+4\pi^2 |n|^2)^{s/2}
\hat{f}(n)e^{2\pi i n\cdot x}
{\rm ,\qquad{}} x\in \mathbb T^3 {\rm ,\qquad{}} s\in \mathbb R.
\epsnd{equation}
In the periodic setting, we have the following equivalent $W^{s,p}$ norm:
\begin{equation}gin{align*}
\frac1C \|J^s f\|_{p} \leq \|f\|_{W^{s,p}} \leq C\|J^s f\|_{p} {\rm ,\qquad{}} s\geq 0 {\rm ,\qquad{}} 1<p<\infty,
\epsnd{align*}
where the usual $L^p$ norms are denoted by $\|\cdot \|_p$.
Next, we define the Leray projector
\begin{equation}gin{equation}\label{leray}
\mathcal P u = u - \nabla \Delta^{-1} \nabla \cdot u.
\epsnd{equation}
Here $\Delta^{-1}$ is defined subject to periodic boundary condition with zero mean. Denote by $\mathbf{P} = (\mathcal{P}, \mathrm{id})$.
We can rewrite system \epsqref{e.w09241} in the following abstract form:
\begin{equation}gin{equation}\label{e.L10261}
dU = (AU + B(U) + G(U))dt + \left(\mathbf{P}(b\cdot\nabla U) + {\rm s}igma(U)\right)d\mathbb{W},
\epsnd{equation}
where $U = (u, \rho):=(U_1,U_2,U_3,U_4)$, $AU = (\Delta u, \Delta\rho)$, $B(U):=-\mathbf{P}(u\cdot \nabla U)$, $G(U) = \mathbf{P}(\rho e_3, 0) = \mathbf{P}(U_4 e_3, 0)$, ${\rm s}igma(U) = (\mathcal P {\rm s}igma^{(1)}(U), {\rm s}igma^{(2)}(U))$. Notice that when $\int U_0\,dx =0$, $\nabla\cdot b=0$, and ${\rm s}igma$ satisfies condition \epsqref{e.Q031101} below, one can integrate system \epsqref{e.L10261} in $\mathbb T^3$ to obtain that
$\int U\,dx =0$ for all $t\geq 0.$ For convenience, we denote by
\begin{equation}gin{align*}
W_{sol}^{s,p} = \left\{ \mathbf{P}f : f\in W^{s,p}, \int_{\mathbb T^3} f\,dx = 0 \right\}.
\epsnd{align*}
{\rm s}ubsection{Stochastic preliminaries}
We denote by $\mathcal{H}$ a real separable Hilbert space with a complete orthonormal basis $\{\mathbf{e}_k\}_{k\geq 1}$. $(\Omega, \mathcal{F},(\mathcal{F}_t)_{t\geq 0},\mathbb{P})$ represents a complete probability space with an augmented filtration $(\mathcal{F}_t)_{t\geq 0}$. With $\{W_k: k\in\mathbb N\}$ a family of independent $\mathcal{F}_t$-adapted Brownian motions, $\mathbb W( t,\omega):={\rm s}um_{k\geq 1} W_k( t,\omega) \mathbf{e}_k$ is an $\mathcal{F}_t$-adapted and $\mathcal{H}$-valued cylindrical Wiener process.
For a real separable Hilbert space $\mathcal{Y}$, we define $l^2( \mathcal{H},\mathcal{Y})$ to be the set of Hilbert-Schmidt operators from $\mathcal{H}$ to $\mathcal{Y}$ with the norm defined by
\begin{equation}gin{equation}
\Vert G\Vert_{l^2( \mathcal{H},\mathcal{Y})}^2:= {\rm s}um_{k=1}^{A_{a} m \mathcal{H}} | G \mathbf{e}_k|_{\mathcal{Y}}^2<\infty {\rm ,\qquad{}} G\in l^2( \mathcal{H},\mathcal{Y}).
\epsnd{equation}
In this paper, we either regard \epsqref{e.L10261} as a vector-valued equation or consider it component-wise. Correspondingly, $\mathcal{Y}={\mathbb R}$ or ${\mathbb R}^{d}$. Let $G=(G_1, \cdots, G_d)$ and $G\mathbf{e}_k:=(G_1\mathbf{e}_k, \cdots, G_d\mathbf{e}_k)$. Then $G\in l^2( \mathcal{H},{\mathbb R}^{d})$ if and only if $G_i\in l^2( \mathcal{H},{\mathbb R})$ for all $i\in\{1,\cdots, d \}$.
Next, we denote by
\begin{equation}gin{equation}
\mathbb{W}^{s,p}:=
\left\{f:\mathbb T^3\to l^2( \mathcal{H},\mathcal{Y}): f\mathbf{e}_k\in W^{s,p}(\mathbb T^3) \text{~for each~}k, \text{~and~} \int_{\mathbb T^3} \Vert J^s f\Vert_{l^2( \mathcal{H},\mathcal{Y})}^p \,dx<\infty \right\},
\epsnd{equation}
with respect to the norm
\begin{equation}gin{equation}
\Vert f\Vert_{\mathbb{W}^{s,p}}:=\left( \int_{\mathbb T^3} \Vert J^s f\Vert_{l^2( \mathcal{H},\mathcal{Y})}^p \,dx\right)^{1/p}. \epsnd{equation}
Furthermore, we denote $(J^s f) \mathbf{e}_k=J^s (f \mathbf{e}_k)$. In particular, $\mathbb{W}^{0,p}$ is abbreviated as $\mathbb{L}^{p}$. Letting $(\mathbf{P} f) \mathbf{e}_k=\mathbf{P}\left(f \mathbf{e}_k\right)$, we have $\mathbf{P} f \in \mathbb{W}^{s, p}$ if $f \in \mathbb{W}^{s, p}$. Write
$$
\mathbb{W}_{\text {sol }}^{s, p}=\left\{\mathbf{P} f: f \in \mathbb{W}^{s, p}, \int_{\mathbb T^3} f\, dx = 0\right\}.
$$
For each sufficiently regular $f: (t, x)\in\mathbb{R}_+\times\mathbb{T}^3\to\mathcal{H}\otimes\mathbb{R}^3$, denote
\[N_{f,k} = \|f\|_{L^{\infty}(\mathbb{R}_+;W^{k,\infty}(\mathbb{T}^3,l^2))}^2 = {\rm s}um_{n=1}^{A_{a} m\mathcal{H}}\|f_n\|_{L^{\infty}_tW^{k,\infty}_x}^2.\]
Finally, the Burkholder-Davis-Gundy (BDG) inequality
\begin{equation}gin{equation}\label{e.L10263}
\mathbb E \left[ {\rm s}up_{s\in[0,t]}\left| \int_0^s G \,d\mathbb W_r \right|_{\mathcal{Y}}^p\right] \leq C_{BDG} \mathbb E\left[ \left(\int_0^t \Vert G\Vert^2_{ l^2( \mathcal{H},\mathcal{Y})}\, dr \right)^{p/2}\right]
\epsnd{equation}
holds for $p\in [1,\infty)$ and all $G\in l^2( \mathcal{H},\mathcal{Y})$ such that the right hand side is finite.
{\rm s}ubsection{Assumptions on the noise}
\begin{equation}gin{assumption}\label{a.L022201}
We impose the following conditions to the noise coefficients ${\rm s}igma, b$:
\begin{equation}gin{enumerate}
\item General conditions on ${\rm s}igma, b$:
\begin{equation}gin{enumerate}
\item \begin{equation}\label{e.L10264}
{\rm s}um_{j=1}^4\left\|{\rm s}igma_j(U)\right\|_{\mathbb{L}^p} \leq C\left(\|U\|_{p}+1\right),
\epse
\begin{equation}\label{e.L121807}
{\rm s}um_{j=1}^4\left\|{\rm s}igma_j(U)-{\rm s}igma_j(V)\right\|_{\mathbb{L}^p} \leq C\|U-V\|_{p},
\epse
\begin{equation}\label{e.Q031101}
{\rm s}igma\left(W_{\text {sol }}^{s, p}\right) {\rm s}ubset \mathbb{W}_{\text {sol }}^{s, p}.
\epse
\item For each $n$, $b_n(t,x):=b\mathbf{e}_n(t, x):\mathbb{R}_+\times\mathbb{T}^3\to\mathbb{R}^3$ is measurable and $\nabla\cdot b_n(t, \cdot)=0$ for every $t$. In addition $N_{b, 2}<\infty$ and the following super-parabolic condition is satisfied:
there exists $\nu>0$ such that
\begin{equation}gin{align*}
{\rm s}um_{i, j = 1}^3\left(\varepsilon_{ij} - \frac12{\rm s}um_{n=1}^{A_{a} m\mathcal{H}}b_n^i(t, x)b_n^j(t, x)\right)\xi_i\xi_j\geq \nu\|\xi\|^2, \quad (t, x,\xi)\in \mathbb{R}_+\times \mathbb{T}^3\times \mathbb{R}^3,
\epsnd{align*}
where $\varepsilon_{ij}=1$ if $i=j$ and $0$ otherwise.
\epsnd{enumerate}
\item For local existence, we assume in addition:
\begin{equation}gin{align}\label{e.L021403}
N_{b, 0} <\frac{p-1}{2(p-1)+p C_{B D G}^2}.
\epsnd{align}
\item For global existence we assume additionally:
\begin{equation}gin{enumerate}
\item Superlinearity condition:
\begin{equation}\label{e.L022205}
{\rm s}um_{i=1}^4\left\|{\rm s}igma_i(U)\right\|_{\mathbb{L}^p} \leq \varepsilonilon_0\|U\|_{p}, \text{ where } \varepsilonilon_0 \text{ is small.}
\epse
\item Smallness of the transport noise: $N_{b, 2}$ is sufficiently small.
\epsnd{enumerate}
\epsnd{enumerate}
\epsnd{assumption}
{\rm s}ubsection{Definitions of solutions and main results}
The followings are the definitions for the local pathwise solution and maximal pathwise solution for the Boussinesq equations.
\begin{equation}gin{definition}[Local pathwise solution for the Boussinesq equations]\label{d.w03092}
Fix $T>0$. A pair $(U,\tau)$ is called a local pathwise solution to system \epsqref{e.L10261} on $(\Omega, \mathcal{F},(\mathcal{F}_t)_{t\geq 0},\mathbb{P})$ if $\tau$ is a stopping time with $\mathbb P(\tau>0)=1$ and if $U\in L^p(\Omega; C([0,\tau\wedge T], L^p)$ is progressively measurable and it satisfies
\begin{equation}gin{align}
\begin{equation}gin{split}
(U(t),\phi) &= (U_0,\phi) +\int_0^t \left[ (U, A\phi)+ (G(U),\phi) + (\mathbf{P}(u\otimes U), \nabla \phi) \,\right]dr
\\
&\hspace{1cm}+\int_0^t \left[\bigl({\rm s}igma(U)),\phi\bigr) - (\mathbf{P}(b\otimes U), \nabla \phi) \right]dW(r) ,
\epsnd{split}
\epsnd{align}
$\mathbb P\mbox{-a.s.} $ for all $\phi\in C^{\infty}(\mathbb T^3, \mathbb R^4)$.
\epsnd{definition}
\begin{equation}gin{definition}[Maximal pathwise solution for the Boussinesq equations]
Fix $T>0$. A pair $(U, \tau)$ is a maximal pathwise $L^p$ solution to system \epsqref{e.L10261} on $(\Omega, \mathcal{F},(\mathcal{F}_t)_{t\geq 0},\mathbb{P})$ if there exists an increasing sequence of stopping times $\tau_n$ with $\tau_n \uparrow \tau$ a.s. such that each pair $\left(U, \tau_n\right)$ is a local pathwise solution,
$$
{\rm s}up _{0 \leq t \leq \tau_n}\|U(t)\|_p^p+{\rm s}um\limits_{j=1}^4 \int_0^{\tau_n} \int_{\mathbb{T}^3}\left|\nabla\left(|U_j(t)|^{p / 2}\right)\right|^2 d x d t<\infty,
$$
and
$$
{\rm s}up _{0 \leq t \leq \tau}\|U(t)\|_p^p+{\rm s}um\limits_{j=1}^4 \int_0^{\tau_n} \int_{\mathbb{T}^3}\left|\nabla\left(|U_j(t)|^{p / 2}\right)\right|^2 d x d t=\infty,
$$
on the set $\{\tau \leq T\}$.
\epsnd{definition}
The following is the main theorem concerning the local and global existence of pathwise solutions to the system \epsqref{e.L10261}.
\begin{equation}gin{theorem}\label{t.w10101}
Assume that (1) and (2) in the Assumption \epsqref{a.L022201} hold. Let $p>5$ and $U_0\in L^{p}(\Omega; L^p (\mathbb T^3))$ satisfying $\int_{\mathbb T^3} U \, dx=0$. Then there exists a unique maximal pathwise solution $(U, \tau)$ to \epsqref{e.L10261} such that
\begin{equation}gin{equation}
\begin{equation}gin{split}
&\mathbb E \left[{\rm s}up_{0\leq s\leq \tau}\|U(s)\|_{p}^{p}
+
\int_{0}^{\tau} {\rm s}um_{j=1}^{4}\int_{\mathbb T^3}|\nabla(|U_{j}(s,x)|^{p/2})|^2 \,dxds
\right]
\leq
C_p\mathbb E[\|U_{0}\|_{p}^{p}+1],
\epsnd{split}
\epsnd{equation}
If additionally (3) in the Assumption \epsqref{a.L022201} holds, then for every $\varepsilonilon\in(0,1]$ there is $\delta>0$ such that if $\mathbb{E}\|U_0\|_p^p\leq\delta$, then
\begin{equation}gin{align}\label{L.022501}
\mathbb{P}(\tau=\infty)\geq 1-\varepsilonilon.
\epsnd{align}
\epsnd{theorem}
{\rm s}ection{$L^p$ estimates for the linear equation}\label{s.L022301}
Consider the stochastic linear parabolic equation with transport noise
\begin{equation}gin{align}\label{e.L021401}
\begin{equation}gin{split}
&dU = (\Delta U + GU+f)dt + (\mathbf{P}\left(b\cdot\nabla U \right) + g)d\mathbb{W}_t, \\
&U(0) = U_0,
\epsnd{split}
\epsnd{align}
where $U = (u, \rho)$, $GU = (\mathcal{P}(\rho e_3), 0)$ and $\mathbf{P}(b\cdot \nabla U) = (\mathcal{P}(b\cdot\nabla u), b\cdot\nabla \rho)$. We denote by $f:=(\overline f,f_4)$ and $g:=(\overline g,g_4)$ where $\overline f=(f_1,f_2,f_3)$ and $\overline g = (g_1, g_2, g_3)$ representing the first three components of $f$ and $g$, respectively. We also assume that $\nabla\cdot \overline f = \nabla\cdot \overline g = 0$. Here $f$ and $g$ are independent of $u$ and $\rho$. We first give a result concerning smooth data.
\begin{equation}gin{lemma}\label{l.L01221}
Let $p>2$. Assume the conditions on $b$ as in Assumption \ref{a.L022201} and
\[U_0\in L^p(\Omega; W^{3,p}), f\in L^p(\Omega\times(0, T); W^{1,p}), g\in L^p(\Omega\times(0, T); \mathbb{W}^{2,p}).\]
Then there is a unique pathwise solution $U$ to \epsqref{e.L021401} on $[0, T]$ such that
\begin{equation}gin{align}\label{e.L020401}
U\in L^p(\Omega; C([0, T]; W^{2, p})).
\epsnd{align}
\epsnd{lemma}
\begin{equation}gin{proof}
The condition on $U_0$ implies that $U_0\in L^p(\Omega;B_{p,p}^{3-2/p})$ due to the embedding $W^{3,p}\hookrightarrow B_{p,p}^{3-2/p}$. As $f$ and $g$ are independent of $u$ and $\rho$, the equation for $U_4$ (which is $\rho$) is independent of $u = (U_1, U_2, U_3)$:
\begin{equation}gin{align}\label{e.L021202}
dU_4 = (\Delta U_4 + f_4)dt +(b\cdot\nabla U_4 + g_4)d\mathbb{W}_t, \quad U_{4}(0) = U_{0,4}.
\epsnd{align}
Under the conditions of Lemma \ref{l.L01221}, we can apply Theorem 5.2 in \cite{AV21c} (with $\kappa=0, p=q$) to deduce the existence of a unique pathwise solution $U_4$ on $[0, T]$ to equation \epsqref{e.L021202} such that $U_4\in L^p(\Omega; L^p((0, T]; W^{3, p}))\cap L^p(\Omega; C([0, T]; B_{p, p}^{3-2/p}))$.
Let $ \widetilde{f} = \overline f + \mathcal{P}(U_4 e_3)$. The equation for the velocity $u$ then becomes
\begin{equation}gin{align}\label{e.L021203}
du = (\Delta u + \widetilde{f})dt +(\mathcal{P}(b\cdot\nabla u) + \overline{g})d\mathbb{W}_t, \quad u(0) = (U_{0,1},U_{0,2},U_{0,3}).
\epsnd{align}
In view of the conditions of Lemma \ref{l.L01221}, for equation \epsqref{e.L021203} we can apply Theorem 3.2 in \cite{agresti2021stochastic-2} (with $\kappa=0, p=q,h=0$) to obtain a unique pathwise solution $u$ on $[0, T]$ to equation \epsqref{e.L021203} such that
$u\in L^p(\Omega; L^p((0, T]; W^{3, p}))\cap L^p(\Omega; C([0, T]; B_{p, p}^{3-2/p}))$.
The conclusion of the lemma then follows from the embedding $B_{p,p}^{3-2/p}\hookrightarrow W^{2, p}$.
\epsnd{proof}
Next, we consider rough condition for system \epsqref{e.L021401}.
\begin{equation}gin{theorem}\label{t.L021401}
Assume the conditions (1b) and (2) on $b$ as in Assumption \ref{a.L022201}. Let $p>2$ and $0<T<\infty$. Suppose $U_0\in L^p(\Omega; L^p(\mathbb{T}^3))$, $f\in L^p(\Omega\times[0, T]; W^{-1, q}(\mathbb{T}^3)), g\in L^p(\Omega\times[0, T]; \mathbb{L}^p(\mathbb{T}^3))$, where
\begin{equation}gin{align}\label{e.L030401}
\frac{3p}{p+1}<q\leq p.
\epsnd{align}
Then there is a unique maximal solution $U\in L^p(\Omega; C([0, T], L^p))$ to equation \epsqref{e.L021401} such that
\begin{equation}gin{align}\label{e.L021402}
\begin{equation}gin{split}
&\mathbb{E}\left[{\rm s}up _{0 \leq t \leq T}\|U(t, \cdot)\|_{p}^{p}+{\rm s}um_{j=1}^{4} \int_{0}^{T} \int_{\mathbb{T}^{3}}\left|\nabla\left(\left|U_{j}(t, x)\right|^{p / 2}\right)\right|^{2} d x d t\right] \\
&\quad \leq C\mathbb{E}\left[\left\|U_{0}\right\|_{p}^{p}+\int_{0}^{T}\|f(t, \cdot)\|_{-1, q}^{p} d t+\int_{0}^{T} \left\|g(t, \cdot)\right\|_{\mathbb{L}^{p}}^pd t\right],
\epsnd{split}
\epsnd{align}
where $C>0$ depends on $T, p, b$.
\epsnd{theorem}
\begin{equation}gin{proof}
The proof is divided into the following three steps.
\vskip0.1in
{\it Step 1}. Let
\[\rho_{\varepsilonilon}(\cdot) = \frac{1}{\varepsilonilon^3}\rho\left(\frac{\cdot}{\varepsilonilon}\right)\]
be a standard mollifier and denote
\[f^{\varepsilonilon} = f * \rho_\varepsilonilon, \quad g^{\varepsilonilon} = g* \rho_\varepsilonilon, \quad U_0^{\varepsilonilon} = U_0 * \rho_\varepsilonilon,\]
where $*$ represents the convolution. The smoothness of the mollified objects and Lemma \ref{l.L01221} imply that the following system
\begin{equation}gin{align}\label{e.L021207}
\begin{equation}gin{split}
&dU^{\varepsilonilon} = (\Delta U^{\varepsilonilon} + GU^{{\varepsilonilon}}+f^{\varepsilonilon})dt + (\mathbf{P}\left(b\cdot\nabla U^{\varepsilonilon} \right)+ g^{\varepsilonilon})d\mathbb{W}_t, \\
&U(0) = U_0^{\varepsilonilon},
\epsnd{split}
\epsnd{align}
has a unique pathwise solution $U^{\varepsilonilon}\in L^p(\Omega; C([0, T]; W^{2, p}))$. For each $j\in\{1, 2, 3, 4\}$, it follows from Ito's formula that
\begin{equation}gin{align}\label{e.L021208}
\begin{equation}gin{split}
\|U^{\varepsilonilon}_j(t)\|_p^p &+ p(p-1)\int_0^t\int_{\mathbb{T}^3}|U^{\varepsilonilon}_j(r)|^{p-2}|\nabla U^{\varepsilonilon}_j(r)|^2dxdr\\ &= \|U^{\varepsilonilon}_{0, j}\|_p^p
+ p\int_0^t\int_{\mathbb{T}^3}|U^{\varepsilonilon}_j(r)|^{p-2}U^{\varepsilonilon}_j(r)\left((GU^{\varepsilonilon})_j(r)+ f^{\varepsilonilon}_j(r)\right)dxdr\\
&+ p\int_0^t\int_{\mathbb{T}^3}|U^{\varepsilonilon}_j(r)|^{p-2}U^{\varepsilonilon}_j(r)\Big(\big(\mathbf{P}\left(b\cdot\nabla U^{\varepsilonilon} \right)\big)_j+ g^{\varepsilon}_j(r)\Big)dxd\mathbb{W}_r\\
&+ \frac{p(p-1)}{2}\int_0^t\int_{\mathbb{T}^3}|U^{\varepsilonilon}_j(r)|^{p-2}\|\big(\mathbf{P}\left(b\cdot\nabla U^{\varepsilonilon} \right)\big)_j+ g^{\varepsilon}_j(r)\|_{l^2}^2dxdr.
\epsnd{split}
\epsnd{align}
\vskip0.1in
{\it Step 2}. We now deduce estimate \epsqref{e.L021402} for $U^{\varepsilon}$. We first recall the following Poincar\'e type inequalities (see \cite{KZ06,KXZ})
\begin{equation}gin{align}\label{e.L031101}
\begin{equation}gin{split}
&\left\||v|^{p-1}\right\|_{q} \leq C\left\|\nabla\left(|v|^{p-1}\right)\right\|_{q}, \\
&\left\|\left|v\right|^{p-2} v\right\|_{q} \leq C\left\|\nabla\left(\left|v\right|^{p-2}v\right)\right\|_{q},\quad p,q>1,
\epsnd{split}
\epsnd{align}
which is valid for zero mean functions $v$ with sufficient regularity so that the expressions make sense. It then follows from \epsqref{e.L031101} that
\begin{equation}gin{align}\label{e.L030402}
\begin{equation}gin{split}
p\int_0^t\int_{\mathbb{T}^3}|U^{\varepsilonilon}_j(r)|^{p-2}U^{\varepsilonilon}_j(r)&\left((GU^{\varepsilonilon})_j(r)+ f^{\varepsilonilon}_j(r)\right)dxdr\\
&\leq C\int_0^t\|\nabla(|U^{\varepsilonilon}_j(r)|^{p-2}U^{\varepsilonilon}_j(r))\|_{q'}\|(GU^{\varepsilonilon})_j(r)+ f^{\varepsilonilon}_j(r)\|_{-1, q}dr,
\epsnd{split}
\epsnd{align}
where $1/q+1/q'=1$. Let $\bar{r}$ be such that $1/\bar{r}+1/2=1/q'$, then by H\"older inequality one has
\begin{equation}gin{align}\label{e.L030403}
\begin{equation}gin{split}
\|\nabla(|U^{\varepsilonilon}_j|^{p-2}U^{\varepsilonilon}_j)\|_{q'}\leq C\left\|\left|U^{\varepsilonilon}_j\right|^{p / 2-1} \nabla\left(\left|U^{\varepsilonilon}_j\right|^{p / 2}\right)\right\|_{q^{\prime}}
&\leq C\left\|\left|U^{\varepsilonilon}_j\right|^{p / 2-1}\right\|_{\bar{r}}\left\|\nabla\left(\left|U^{\varepsilonilon}_j\right|^{p / 2}\right)\right\|_2\\
& = C\left\|\left|U^{\varepsilonilon}_j\right|^{p/ 2}\right\|_{\bar{r}(p-2)/p}^{(p-2)/p}\left\|\nabla\left(\left|U^{\varepsilonilon}_j\right|^{p / 2}\right)\right\|_2.
\epsnd{split}
\epsnd{align}
In view of the condition \epsqref{e.L030401}, we have
\[2 \leq \frac{\bar{r}(p-2)}{p}<6.\]
By Poincar\'e type inequalities \epsqref{e.L031101} and Gagliardo-Nirenberg inequality, we deduce
\begin{equation}gin{align}\label{e.L030404}
\left\|\left|U^{\varepsilonilon}_j\right|^{p/ 2}\right\|_{\bar{r}(p-2)/p}\leq C\left\|\left|U^{\varepsilonilon}_j\right|^{p/ 2}\right\|_{2}^{1-\alpha}\left\|\nabla\left(\left|U^{\varepsilonilon}_j\right|^{p / 2}\right)\right\|_2^{\alpha},
\epsnd{align}
where $\alpha = \frac32-\frac{3p}{\bar{r}(p-2)}$. Combining inequalities \epsqref{e.L030402}-\epsqref{e.L030404}, and Young's inequality with $\delta>0$, we obtain
\begin{equation}gin{align}\label{e.L021404}
\begin{equation}gin{split}
&p\int_0^t\int_{\mathbb{T}^3}|U^{\varepsilonilon}_j(r)|^{p-2}U^{\varepsilonilon}_j(r)\left((GU^{\varepsilonilon})_j(r)+ f^{\varepsilonilon}_j(r)\right)dxdr\\
&\leq C\int_0^t\left\|\left|U^{\varepsilonilon}_j\right|^{p/ 2}\right\|_{2}^{(1-\alpha)(p-2)/p}\left\|\nabla\left(\left|U^{\varepsilonilon}_j\right|^{p / 2}\right)\right\|_2^{1+\alpha(p-2)/p}\|(GU^{\varepsilonilon})_j(r)+ f^{\varepsilonilon}_j(r)\|_{-1, q}dr\\
&\leq \delta \int_0^t\int_{\mathbb{T}^3}|U^{\varepsilonilon}_j(r)|^{p-2}|\nabla U^{\varepsilonilon}_j(r)|^2dxdr +\delta \int_0^t\|U^{\varepsilonilon}_j(r)\|_p^pdr\\
&\hspace{2cm}+C_\delta\left (\int_0^t\left\|f^{\varepsilonilon}_j(r)\right\|_{-1, q}^p d r + \int_0^t\left\|(GU^{\varepsilonilon})_j(r)\right\|_{-1, p}^p d r \right),
\epsnd{split}
\epsnd{align}
where we used the fact
\[\int_{\mathbb{T}^3}\left|\nabla\left|U^{\varepsilonilon}_j\right|^{p / 2}\right|^2dx = \frac{p^2}{4}\int_{\mathbb{T}^3}|U^{\varepsilonilon}_j(r)|^{p-2}|\nabla U^{\varepsilonilon}_j(r)|^2dx.\]
To deal with the terms involving transport noise, we utilize the following property of Leray projection
\begin{equation}gin{align*}
\mathcal{P}(b\cdot \nabla u^{\varepsilonilon}) = b\cdot \nabla u^{\varepsilonilon} - \nabla\Delta^{-1}A_{a} vv(b\cdot \nabla u^{\varepsilonilon}),\quad u^{\varepsilonilon} = (U_1^{\varepsilonilon},U_2^{\varepsilonilon},U_3^{\varepsilonilon}).
\epsnd{align*}
As a result, we have
\begin{equation}gin{align}\label{e.L022101}
\mathbf{P}\left(b\cdot\nabla U^{\varepsilonilon} \right) = b\cdot\nabla U^{\varepsilonilon} - \mathbf{Q}\left(b\cdot\nabla U^{\varepsilonilon} \right), \quad \mathbf{Q}\left(b\cdot\nabla U^{\varepsilonilon}\right) := \big(\nabla\Delta^{-1}A_{a} vv(b\cdot \nabla u^{\varepsilonilon}), 0\big).
\epsnd{align}
Since $u^{\varepsilonilon}$ is divergence free, one has
\begin{equation}gin{align}\label{e.L022103}
(\mathbf{Q}\left(b\cdot\nabla U^{\varepsilonilon}\right))_j = \partial_j\Delta^{-1}{\rm s}um_{k, \epsll=1}^3\partial_kb^{\epsll}\partial_{\epsll} u^{\varepsilonilon}_{k}, \quad j=1, 2, 3.
\epsnd{align}
To estimate the last term in \epsqref{e.L021208}, we first note
\begin{equation}gin{align}\label{e.L022102}
\begin{equation}gin{split}
&\frac{p(p-1)}{2}\int_0^t\int_{\mathbb{T}^3}|U^{\varepsilonilon}_j(r)|^{p-2}\|\big(\mathbf{P}\left(b\cdot\nabla U^{\varepsilonilon} \right)\big)_j+ g^{\varepsilon}_j(r)\|_{l^2}^2dxdr\\
&\leq p(p-1)\int_0^t\int_{\mathbb{T}^3}|U^{\varepsilonilon}_j(r)|^{p-2}\|b\cdot\nabla U^{\varepsilonilon}_j+ g^{\varepsilon}_j(r)\|_{l^2}^2dxdr
\\
&\hspace{1cm}+p(p-1)\int_0^t\int_{\mathbb{T}^3}|U^{\varepsilonilon}_j(r)|^{p-2}\|(\mathbf{Q}\left(b\cdot\nabla U^{\varepsilonilon}\right))_j\|_{l^2}^2dxdr
: = I_1 + I_2.
\epsnd{split}
\epsnd{align}
For $I_1$, we have
\begin{equation}gin{align}\label{e.L121803}
\begin{equation}gin{split}
I_1&\leq 2p(p-1)N_{b, 0}\int_0^t\int_{\mathbb{T}^3}|U^{\varepsilonilon}_j(r)|^{p-2}|\nabla U^{\varepsilonilon}_j(r)|^2dxdr + 2p(p-1)\int_0^t\int_{\mathbb{T}^3}|U^{\varepsilonilon}_j(r)|^{p-2}\|g^{\varepsilonilon}_j(r)\|_{l^2}^2dxdr\\
&\leq 2p(p-1)N_{b, 0}\int_0^t\int_{\mathbb{T}^3}|U^{\varepsilonilon}_j(r)|^{p-2}|\nabla U^{\varepsilonilon}_j(r)|^2dxdr + \int_0^t\left\|U^{\varepsilonilon}_j(r)\right\|_p^p d r+C \int_0^t\left\|g^{\varepsilonilon}_j(r)\right\|_{\mathbb{L}^p}^p d r
\epsnd{split}
\epsnd{align}
where we applied Young’s inequality in the last step. For $I_2$, it follows from \epsqref{e.L022103} that
\begin{equation}gin{align}\label{e.L0221013}
\begin{equation}gin{split}
I_2& = p(p-1)\int_0^t\int_{\mathbb{T}^3}|U^{\varepsilonilon}_j(r)|^{p-2}\|(\mathbf{Q}\left(b\cdot\nabla U^{\varepsilonilon}\right))_j\|_{l^2}^2dxdr\\
& = p(p-1)\int_0^t\int_{\mathbb{T}^3}|U^{\varepsilonilon}_j(r)|^{p-2}{\rm s}um_{n=1}^{A_{a} m\mathcal{H}}\left|\partial_j\Delta^{-1}{\rm s}um_{k, \epsll=1}^3\partial_kb_{n}^{\epsll}\partial_{\epsll} u^{\varepsilonilon}_{k}\right|^2dxdr\\
&\leq p(p-1){\rm s}um_{n=1}^{A_{a} m\mathcal{H}}\int_0^t\left(\|U^{\varepsilonilon}_j(r)\|_{p}^{p-2}\left\|\partial_j\Delta^{-1}{\rm s}um_{k, \epsll=1}^3\partial_kb_{n}^{\epsll}\partial_{\epsll} u^{\varepsilonilon}_{k}\right\|_{p}^2\right)dr
\epsnd{split}
\epsnd{align}
By standard elliptic estimates, we have
\begin{equation}gin{align*}
\left\|\partial_j\Delta^{-1}\left(\partial_kb_{n}^{\epsll}\partial_{\epsll} u^{\varepsilonilon}_{k}\right)\right\|_{p}^2&\leq C\left\|\partial_kb_{n}^{\epsll}\partial_{\epsll} u^{\varepsilonilon}_{k}\right\|_{W^{-1, p}}^2
\\
&= C \left|{\rm s}up\limits_{\|\phi\|_{W^{1,p'}}=1} (\partial_kb_{n}^{\epsll}\partial_{\epsll} u^{\varepsilonilon}_{k}, \phi) \right|^2= C \left|{\rm s}up\limits_{\|\phi\|_{W^{1,p'}}=1} (u^{\varepsilonilon}_{k}, \partial_{\epsll} ( \partial_kb_{n}^{\epsll} \phi ) )\right|^2 \\
&\leq C\|\partial_kb_{n}^{\epsll}\|_{L_{t, x}^{\infty}}^2\left\|u^{\varepsilonilon}_{k}\right\|_{p}^2 + C_{E}\|\partial_{\epsll}\partial_kb_{n}^{\epsll}\|_{L_{t, x}^{\infty}}^2\left\| u^{\varepsilonilon}_{k}\right\|_{p}^2
\\
&\leq C\max_{1\leq k, \epsll\leq3}\|\partial_kb_{n}^{\epsll}\|_{L_{t, x}^{\infty}}^2\left\|U^{\varepsilonilon}\right\|_{p}^2 + C\max_{1\leq k, \epsll\leq3}\|\partial_{\epsll}\partial_kb_{n}^{\epsll}\|_{L_{t, x}^{\infty}}^2\left\|U^{\varepsilonilon}\right\|_{p}^2.
\epsnd{align*}
Therefore,
\begin{equation}gin{align}\label{e.L022304}
I_2\leq C(N_{b,1}+N_{b,2})\int_0^t\left\|U^{\varepsilonilon}(r)\right\|_p^p d r.
\epsnd{align}
Combining this with \epsqref{e.L022102} and \epsqref{e.L121803}, we obtain
\begin{equation}gin{align}\label{e.L022104}
\begin{equation}gin{split}
&\frac{p(p-1)}{2}\int_0^t\int_{\mathbb{T}^3}|U^{\varepsilonilon}_j(r)|^{p-2}\|\big(\mathbf{P}\left(b\cdot\nabla U^{\varepsilonilon} \right)\big)_j+ g^{\varepsilon}_j(r)\|_{l^2}^2dxdr\\
\leq &2p(p-1)N_{b, 0}\int_0^t\int_{\mathbb{T}^3}|U^{\varepsilonilon}_j(r)|^{p-2}|\nabla U^{\varepsilonilon}_j(r)|^2dxdr
\\
&\hspace{2cm}+ C \int_0^t\left\|U^{\varepsilonilon}(r)\right\|_p^p d r+C \int_0^t\left\|g^{\varepsilonilon}_j(r)\right\|_{\mathbb{L}^p}^p d r.
\epsnd{split}
\epsnd{align}
Next, the decomposition \epsqref{e.L022101}, BDG inequality and Minkowski's inequality yield
\begin{equation}gin{align}\label{e.L022105}
\begin{equation}gin{split}
&\mathbb{E}{\rm s}up_{t\in[0, T]}\left| \int_0^t\int_{\mathbb{T}^3}|U^{\varepsilonilon}_j(r)|^{p-2}U^{\varepsilonilon}_j(r)(b(r)\cdot\nabla U^{\varepsilonilon}_j(r) + g^{\varepsilonilon}_j(r))dxd\mathbb{W}_r\right|\\
&\leq C_{BDG}\mathbb{E}\left(\int_0^T\left\|\int_{\mathbb{T}^3}|U^{\varepsilonilon}_j(r)|^{p-2}U^{\varepsilonilon}_j(r)b(r)\cdot\nabla U^{\varepsilonilon}_j(r)dx\right\|_{l^2}^2dr\right)^{1/2}
\\&\quad+
C_{BDG}\mathbb{E}\left(\int_0^T\left\|\int_{\mathbb{T}^3}|U^{\varepsilonilon}_j(r)|^{p-2}U^{\varepsilonilon}_j(r)g^{\varepsilonilon}_j(r)dx\right\|_{l^2}^2dr\right)^{1/2}\\
&\quad + C_{BDG}\mathbb{E}\left(\int_0^T\left\|\int_{\mathbb{T}^3}|U^{\varepsilonilon}_j(r)|^{p-2}U^{\varepsilonilon}_j(r)\left(\mathbf{Q}\left(b\cdot\nabla U^{\varepsilonilon}\right)\right)_j dx\right\|_{l^2}^2dr\right)^{1/2}: = J_1 + J_2 + J_3.
\epsnd{split}
\epsnd{align}
By Minkowski's inequality and Hölder's inequality, we obtain
\begin{equation}gin{align}\label{e.L022107}
\begin{equation}gin{split}
&J_1 = C_{BDG}\mathbb{E}\left(\int_0^T\left\|\int_{\mathbb{T}^3}|U^{\varepsilonilon}_j(r)|^{p-2}U^{\varepsilonilon}_j(r)b(r)\cdot\nabla U^{\varepsilonilon}_j(r)dx\right\|_{l^2}^2dr\right)^{1/2}\\
&\leq C_{BDG}\mathbb{E}{\rm s}up_{t\in[0,T]}\|U^{\varepsilonilon}_j(t)\|_{p}^{p/2}\left(\int_0^T\int_{\mathbb{T}^3}|U^{\varepsilonilon}_j(r)|^{p-2}\left\|b\cdot \nabla U^{\varepsilonilon}_j(r)\right\|_{l^2}^2dxdr\right)^{1/2}\\
&\leq \frac{1}{4p}\mathbb{E}{\rm s}up_{t\in[0, T]}\|U^{\varepsilonilon}_j(t)\|_{p}^p + pC_{BDG}^2 N_{b,0}\mathbb{E}\int_0^T\int_{\mathbb{T}^3}|U^{\varepsilonilon}_j(r)|^{p-2}|\nabla U^{\varepsilonilon}_j(r)|^2dxdr.
\epsnd{split}
\epsnd{align}
A similar estimate yields
\begin{equation}gin{align}\label{e.L022106}
J_2\leq \frac{1}{4p}\mathbb{E}{\rm s}up_{t\in[0, T]}\|U^{\varepsilonilon}_j(t)\|_{p}^p+C_T\mathbb{E}\int_0^T\|g^{\varepsilonilon}_j(r)\|_{\mathbb{L}^p}^pdr.
\epsnd{align}
To estimate $J_3$, we denote $h = |U^{\varepsilonilon}_j(r)|^{p-2}U^{\varepsilonilon}_j(r)$ and note that by integration by parts and \epsqref{e.L022103}, since $\Delta^{-1}$ is a self-adjoint operator,
\begin{equation}gin{align}\label{e.L022108}
\begin{equation}gin{split}
\int_{\mathbb{T}^3}h\left(\mathbf{Q}\left(b\cdot\nabla U^{\varepsilonilon}\right)\right)_j dx &= {\rm s}um_{k, \epsll=1}^3\int_{\mathbb{T}^3}h\partial_j\Delta^{-1}\big(\partial_kb^{\epsll}\partial_{\epsll} u^{\varepsilonilon}_{k}\big) dx\\
& = -{\rm s}um_{k, \epsll=1}^3\int_{\mathbb{T}^3}\Delta^{-1}(\partial_jh )\partial_kb^{\epsll}\partial_{\epsll} u^{\varepsilonilon}_{k}dx\\
& = {\rm s}um_{k, \epsll=1}^3\left(\int_{\mathbb{T}^3}\partial_{\epsll} \Delta^{-1}(\partial_jh )\partial_kb^{\epsll}u^{\varepsilonilon}_{k}dx + \int_{\mathbb{T}^3} \Delta^{-1}(\partial_jh) \partial_{\epsll}\partial_kb^{\epsll}u^{\varepsilonilon}_{k}dx\right).
\epsnd{split}
\epsnd{align}
As a consequence, we have
\begin{equation}gin{align}\label{e.L022109}
\begin{equation}gin{split}
&\left\|\int_{\mathbb{T}^3}h\left(\mathbf{Q}\left(b\cdot\nabla U^{\varepsilonilon}\right)\right)_j dx\right\|_{l^2}\\
& \leq{\rm s}um_{k, \epsll=1}^3\left(\int_{\mathbb{T}^3}|\partial_{\epsll} \Delta^{-1}\partial_jh |\left({\rm s}um_{n=1}^{A_{a} m\mathcal{H}}|\partial_kb_n^{\epsll}|^2\right)^{1/2}|u^{\varepsilonilon}_{k}|dx + \int_{\mathbb{T}^3}| \Delta^{-1}\partial_jh| \left({\rm s}um_{n=1}^{A_{a} m\mathcal{H}}|\partial_{\epsll}\partial_kb_n^{\epsll}|^2\right)^{1/2}|u^{\varepsilonilon}_{k}|dx\right)\\
&\leq N_{b, 1}^{1/2}{\rm s}um_{k, \epsll=1}^3\int_{\mathbb{T}^3}|\partial_{\epsll}\Delta^{-1}\partial_jh||u^{\varepsilonilon}_{k}|dx + 3N_{b, 2}^{1/2}{\rm s}um_{k}^3\int_{\mathbb{T}^3}|\Delta^{-1}\partial_jh||u^{\varepsilonilon}_{k}|dx.
\epsnd{split}
\epsnd{align}
By Hölder's inequality and standard elliptic regularity estimates, we have for $1/p' +1/p=1$
\begin{equation}gin{align}\label{e.L0221010}
\begin{equation}gin{split}
\int_{\mathbb{T}^3}|\partial_{\epsll}\Delta^{-1}\partial_jh||u^{\varepsilonilon}_{k}|dx&\leq \|\partial_{\epsll}\Delta^{-1}\partial_jh\|_{L^{p'}}\|u^{\varepsilonilon}_{k}\|_{p}\\
&\leq C\|h\|_{p'}\|U^{\varepsilonilon}\|_{L^{p}} \leq C\|U^{\varepsilonilon}\|_{p}^{p-1}\|U^{\varepsilonilon}\|_{p} =C\|U^{\varepsilonilon}\|_{p}^{p}.
\epsnd{split}
\epsnd{align}
Similarly we deduce
\begin{equation}gin{align*}
\int_{\mathbb{T}^3}|\Delta^{-1}\partial_jh||u^{\varepsilonilon}_{k}|dx\leq C\|U^{\varepsilonilon}\|_{p}^{p}.
\epsnd{align*}
Combining this with \epsqref{e.L022109} and \epsqref{e.L0221010} one has
\begin{equation}gin{align}\label{e.L0221011}
\begin{equation}gin{split}
J_3 & = C_{BDG}\mathbb{E}\left(\int_0^T\left\|\int_{\mathbb{T}^3}|h\left(\mathbf{Q}\left(b\cdot\nabla U^{\varepsilonilon}\right)\right)_j dx\right\|_{l^2}^2dr\right)^{1/2}\\
&\leq C( N_{b, 1}^{1/2} + N_{b, 2}^{1/2})\mathbb{E}\left(\int_0^T\|U^{\varepsilonilon}\|_{p}^{2p}dr\right)^{1/2}\\
&\leq \frac{1}{4 p} \mathbb{E} {\rm s}up _{t \in[0, T]}\left\|U_j^\varepsilonilon(t)\right\|_p^p + C\mathbb{E}\int_0^T\left\|U^\varepsilonilon\right\|_{p}^{p} d r.
\epsnd{split}
\epsnd{align}
Therefore by combining \epsqref{e.L022105}, \epsqref{e.L022106}, \epsqref{e.L022107} and \epsqref{e.L0221011}, we have
\begin{equation}gin{align}\label{e.L121804}
\begin{equation}gin{split}
&p\mathbb{E}{\rm s}up_{t\in[0, T]}\left| \int_0^t\int_{\mathbb{T}^3}|U^{\varepsilonilon}_j(r)|^{p-2}U^{\varepsilonilon}_j(r)(b\cdot\nabla U^{\varepsilonilon}_j(r) + g^{\varepsilonilon}_j(r))dxd\mathbb{W}_r\right|\\
&\leq\frac34\mathbb{E}{\rm s}up_{t\in[0, T]}\|U^{\varepsilonilon}_j(t)\|_{p}^p + p^2C_{BDG}^2N_{b,0}\mathbb{E}\int_0^T\int_{\mathbb{T}^3}|U^{\varepsilonilon}_j(r)|^{p-2}|\nabla U^{\varepsilonilon}_j(r)|^2dxdr\\
&\quad+C\mathbb{E}\int_0^T\left\|U^\varepsilonilon\right\|_{p}^{p} d r+C_T\mathbb{E}\int_0^T\|g^{\varepsilonilon}_j(r)\|_{\mathbb{L}^p}^pdr.
\epsnd{split}
\epsnd{align}
Now taking supremum over $[0, T]$ on both sides of \epsqref{e.L021208} and integrating over $\Omega$, using inequalities \epsqref{e.L021404}, \epsqref{e.L022104} and \epsqref{e.L121804}, we derive for $j=1,2,3$,
\begin{equation}gin{align}\label{e.L0221012}
\begin{equation}gin{split}
&\mathbb{E}\left[\frac14{\rm s}up_{t\in[0, T]}\|U^{\varepsilonilon}_j(t)\|_{p}^p +p(p-1)\int_0^T\int_{\mathbb{T}^3}|U^{\varepsilonilon}_j(r)|^{p-2}|\nabla U^{\varepsilonilon}_j(r)|^2dxdr \right]\\
&\leq C_{\delta}\mathbb{E}\int_0^T\left\|U^\varepsilonilon\right\|_{p}^{p} d r + K\mathbb{E}\int_0^T\int_{\mathbb{T}^3}|U^{\varepsilonilon}_j(r)|^{p-2}|\nabla U^{\varepsilonilon}_j(r)|^2dxdr \\
&\quad + C\mathbb{E}\left[\|U^{\varepsilonilon}_0\|_p^p + \int_0^T\left\|f^{\varepsilonilon}(r)\right\|_{-1, q}^p d r + \int_0^T\left\|(GU^{\varepsilonilon})_j(r)\right\|_{-1, p}^p d r + \int_0^T\|g^{\varepsilonilon}(r)\|_{\mathbb{L}^p}^pdr\right].
\epsnd{split}
\epsnd{align}
Here
\begin{equation}gin{align*}
K = \delta + 2p(p-1)N_{b, 0}+p^2C_{BDG}^2N_{b,0}.
\epsnd{align*}
For $j=4$, observe that $(GU^{\varepsilonilon})_4\epsquiv 0$ and $ (\mathbf{Q}\left(b\cdot\nabla U^{\varepsilonilon}\right))_4 \epsquiv0$. Therefore from \epsqref{e.L021404}, \epsqref{e.L022102}-\epsqref{e.L121803}, and \epsqref{e.L022105}-\epsqref{e.L022107}, we obtain
\begin{equation}gin{align}\label{e.L022201}
\begin{equation}gin{split}
&\mathbb{E}\left[\frac12{\rm s}up_{t\in[0, T]}\|U^{\varepsilonilon}_4(t)\|_{p}^p +p(p-1)\int_0^T\int_{\mathbb{T}^3}|U^{\varepsilonilon}_4(r)|^{p-2}|\nabla U^{\varepsilonilon}_4(r)|^2dxdr \right]\\
&\leq C_\delta \mathbb{E}\int_0^T\left\|U_4^\varepsilonilon\right\|_{p}^{p} d r + K\mathbb{E}\int_0^T\int_{\mathbb{T}^3}|U^{\varepsilonilon}_4(r)|^{p-2}|\nabla U^{\varepsilonilon}_4(r)|^2dxdr \\
&\quad + C\mathbb{E}\left[\|U^{\varepsilonilon}_{0,4}\|_p^p + \int_0^T\left\|f^{\varepsilonilon}(r)\right\|_{-1, q}^p d r + \int_0^T\|g^{\varepsilonilon}(r)\|_{\mathbb{L}^p}^pdr\right].
\epsnd{split}
\epsnd{align}
In view of the condition \epsqref{e.L021403} on $b$, we can choose $\delta>0$ small such that $p(p-1)-K>0$. Fix such a $\delta$. Then
\begin{equation}gin{align*}
\begin{equation}gin{split}
&\mathbb{E}\left[{\rm s}up_{t\in[0, T]}\|U^{\varepsilonilon}_4(t)\|_{p}^p +\int_0^T\int_{\mathbb{T}^3}|U^{\varepsilonilon}_4(r)|^{p-2}|\nabla U^{\varepsilonilon}_4(r)|^2dxdr \right]\\
&\leq C\mathbb{E}\left[\|U^{\varepsilonilon}_{0,4}\|_p^p + \int_0^T\left\|U_4^\varepsilonilon\right\|_{p}^{p} d r + \int_0^T\left\|f^{\varepsilonilon}(r)\right\|_{-1, q}^p d r + \int_0^T\|g^{\varepsilonilon}(r)\|_{\mathbb{L}^p}^pdr\right].
\epsnd{split}
\epsnd{align*}
It follows from Gronwall's inequality that
\begin{equation}gin{align*}
\begin{equation}gin{split}
&\mathbb{E}\left[{\rm s}up_{t\in[0, T]}\|U^{\varepsilonilon}_4(t)\|_{p}^p +\int_0^T\int_{\mathbb{T}^3}|U^{\varepsilonilon}_4(r)|^{p-2}|\nabla U^{\varepsilonilon}_4(r)|^2dxdr \right]\\
&\leq C\mathbb{E}\left[\|U^{\varepsilonilon}_{0,4}\|_p^p + \int_0^T\left\|f^{\varepsilonilon}(r)\right\|_{-1, q}^p d r + \int_0^T\|g^{\varepsilonilon}(r)\|_{\mathbb{L}^p}^pdr\right].
\epsnd{split}
\epsnd{align*}
This shows that for $j=1,2,3$, by the boundedness of Leray projection, we have
\begin{equation}gin{align}\label{e.L021407}
\begin{equation}gin{split}
\int_0^T\left\|(GU^{\varepsilonilon})_j(r)\right\|_{-1, p}^p d r &= \int_0^T\left\|(\mathcal{P}(U^{\varepsilonilon}_4 e_3))_j(r)\right\|_{-1, p}^p d r\\
&\leq C \int_0^T\left\|U^{\varepsilonilon}_4(r) e_3\right\|_{-1, p}^p d r\leq C_T{\rm s}up_{t\in[0, T]}\|U^{\varepsilonilon}_4(t)\|_{p}^p\\
&\leq C\mathbb{E}\left[\|U^{\varepsilonilon}_{0,4}\|_p^p + \int_0^T\left\|f^{\varepsilonilon}(r)\right\|_{-1, q}^p d r + \int_0^T\|g^{\varepsilonilon}(r)\|_{\mathbb{L}^p}^pdr\right].
\epsnd{split}
\epsnd{align}
Combining \epsqref{e.L0221012}, \epsqref{e.L022201} and \epsqref{e.L021407}, one obtains
\begin{equation}gin{align}\label{e.L022203}
\begin{equation}gin{split}
&\mathbb{E}\left[{\rm s}up_{t\in[0, T]}\|U^{\varepsilonilon}(t)\|_{p}^p +{\rm s}um_{j=1}^4\int_0^T\int_{\mathbb{T}^3}|U^{\varepsilonilon}_j(r)|^{p-2}|\nabla U^{\varepsilonilon}_j(r)|^2dxdr \right]\\
&\leq C\mathbb{E}\left[\|U^{\varepsilonilon}_0\|_p^p + \int_0^T\left\|U^\varepsilonilon\right\|_{p}^{p} d r + \int_0^T\left\|f^{\varepsilonilon}(r)\right\|_{-1, q}^p d r + \int_0^T\|g^{\varepsilonilon}(r)\|_{\mathbb{L}^p}^pdr\right].
\epsnd{split}
\epsnd{align}
Again using Gronwall's inequality we obtain the desired estimate
\begin{equation}gin{align}\label{e.L022204}
\begin{equation}gin{split}
&\mathbb{E}\left[{\rm s}up_{t\in[0, T]}\|U^{\varepsilonilon}(t)\|_{p}^p +{\rm s}um_{j=1}^4\int_0^T\int_{\mathbb{T}^3}|U^{\varepsilonilon}_j(r)|^{p-2}|\nabla U^{\varepsilonilon}_j(r)|^2dxdr \right]\\
&\leq C\mathbb{E}\left[\|U^{\varepsilonilon}_0\|_p^p + \int_0^T\left\|f^{\varepsilonilon}(r)\right\|_{-1, q}^p d r + \int_0^T\|g^{\varepsilonilon}(r)\|_{\mathbb{L}^p}^pdr\right].
\epsnd{split}
\epsnd{align}
\vskip0.1in
{\it Step 3}. By the linearity of the equation and the result of the previous step, we have for $U^{\varepsilonilon}-U^{\varepsilonilon'}$,
\begin{equation}gin{align*}
\begin{equation}gin{split}
\mathbb{E}\Bigg[{\rm s}up_{t\in[0, T]}\|U^{\varepsilonilon}(t)-&U^{\varepsilonilon'}(t)\|_{p}^p \Bigg]
\leq C\mathbb{E}\left[\|U^{\varepsilonilon}_0-U^{\varepsilonilon'}_0\|_p^p + \int_0^T\left\|f^{\varepsilonilon}(r)-f^{\varepsilonilon'}(r)\right\|_{-1, q}^p d r + \int_0^T\|g^{\varepsilonilon}(r)-g^{\varepsilonilon'}(r)\|_{\mathbb{L}^p}^pdr\right].
\epsnd{split}
\epsnd{align*}
By the fundamental properties of mollifiers, we see that the right hand side of the this inequality goes to zero as $\varepsilonilon,\varepsilonilon'\to0$. Therefore, there exists an element $U\in L^p(\Omega, C([0, T], L^p))$ and a subsequence $U^{\varepsilonilon_n}\to U$ in $L^p(\Omega, L^{\infty}([0, T], L^p))$, where $\varepsilonilon_n\to0$ as $n\to\infty$. Using integration by parts as in \cite{KXZ}, one can show that $U$ is a pathwise solution of \epsqref{e.L021401}. Indeed, one has
\begin{equation}gin{align*}
(U^{\varepsilonilon_n}, \phi) = (U_0^{\varepsilonilon_n}, \phi) &+ \int_0^t \Big((U^{\varepsilonilon_n},A \phi) + (GU^{\varepsilonilon_n} + f^{\varepsilonilon_n}, \phi)\Big)dr
\\
&+ \int_0^t \Big(-(\mathbf{P}\left(b \otimes U^{\varepsilonilon_n} \right) , \nabla \phi) + (g^{\varepsilonilon_n}, \phi) \Big)d\mathbb{W}_r, \quad (t,\omega)-\mathrm{a.e.},
\epsnd{align*}
for all $\phi\in C^{\infty}(\mathbb{T}^3;\mathbb{R}^4)$ and $n\geq 1$. Then Hölder's inequality and dominated convergence theorem imply that as $n\to\infty$,
\begin{equation}gin{align*}
(U^{\varepsilonilon_n}, \phi) - (U_0^{\varepsilonilon_n}, \phi)&\to (U, \phi) - (U_0, \phi),\\
\int_0^t\Big(( U^{\varepsilonilon_n}, A \phi) + (GU^{\varepsilonilon_n} + f^{\varepsilonilon_n}, \phi)\Big)dr&\to\int_0^t\Big((U, A\phi) +( GU + f, \phi)\Big)dr.
\epsnd{align*}
By the BDG inequality and Minkowski's inequality, we have
\begin{equation}gin{align*}
&\mathbb{E}{\rm s}up_{t\in[0, T]}\left|\int_0^t\Big(-\mathbf{P}(b\otimes(U^{\varepsilonilon_n}-U)) , \nabla\phi)+ (g^{\varepsilonilon_n}-g, \phi)\Big)d\mathbb{W}_r\right|\\
&\leq C\mathbb{E}\left(\int_0^T\left\|(\mathbf{P}(b\otimes(U^{\varepsilonilon_n}-U)), \nabla\phi)\right\|_{l^2}^2dr\right)^{1/2}+C\mathbb{E}\left(\int_0^T\left\|( g^{\varepsilonilon_n}-g,\phi)\right\|_{l^2}^2dr\right)^{1/2}\\
&\leq C_b\|\nabla\phi\|_{L^2}\|\mathbb{E}{\rm s}up_{t\in[0, T]}\|U^{\varepsilonilon_n}(t)-U(t)\|_{p}+ C\|\phi\|_{L^2}\mathbb{E}\int_0^T\|g^{\varepsilonilon_n}(t)-g(t)\|_{\mathbb{L}^p}^pdt,
\epsnd{align*}
which converge to $0$ as $n\to\infty$. Hence there is a subsequence $\varepsilonilon_{n_j}$ such that
\[\int_0^t\Big(-\mathbf{P}(b\otimes U^{\varepsilonilon_{n_j}}),\nabla\phi) +( g^{\varepsilonilon_{n_j}}, \phi)\Big)d\mathbb{W}_r\to \int_0^t\Big(-\mathbf{P}(b\otimes U),\nabla\phi)+ (g, \phi)\Big)d\mathbb{W}_r, \quad (t,\omega)-\mathrm{a.e.}.\]
Hence $U$ is a pathwise solution of \epsqref{e.L021401}. Lemma 4.4 in \cite{KXZ} with \epsqref{e.L022204} also imply that \epsqref{e.L021402} is true.
To show the uniqueness, we let $U_1, U_2$ be two pathwise solutions to \epsqref{e.L021401}, then their difference $V=U_1-U_2$ solves
\begin{equation}gin{align*}
\begin{equation}gin{split}
&dV = (A V + GV)dt + \mathbf{P}(b\cdot\nabla V) d\mathbb{W}_t, \\
&U(0) = 0.
\epsnd{split}
\epsnd{align*}
Then Ito's formula and estimates as above give
\begin{equation}gin{align*}
\mathbb{E}{\rm s}up _{0 \leq t \leq T}\|V(t, \cdot)\|_{p}^{p} =0.
\epsnd{align*}
So $V\epsquiv0$ almost surely.
\epsnd{proof}
{\rm s}ection{Local existence and uniqueness}\label{s.w02271}
In this section we will establish the existence and uniqueness of maximal pathwise solution for Boussinesq system \epsqref{e.L10261}.
{\rm s}ubsection{Truncated system}
We first consider the following truncated system
\begin{equation}gin{equation}\label{e.Q02181}
\begin{equation}gin{split}
&d U-\Delta U d t
=
\varphi\left(\left\|U\right\|_{p}\right)^2 B(U) dt +G(U) d t
+\varphi\left(\left\|U\right\|_{p}\right)^2 {\rm s}igma(U) d \mathbb{W}_{t}+\mathbf{P}(b\cdot\nabla U) d \mathbb{W}_{t},
\\&\nabla \cdot u=0,
\\& U(0)=U_{0},
\epsnd{split}
\epsnd{equation}
on $[0, \infty) \times \mathbb{T}^{3}$ with $\nabla \cdot u_{0}=0$ and $\int_{\mathbb{T}^{3}} U_{0} d x=0$ a.s..
Here for some fixed $\delta_0>0$ we denote by $\varphi:[0,\infty)\to [0,1]$ a decreasing smooth function such that $\varphi\epsquiv 1$ on $[0,\frac{\delta_0}{2}]$ and $\varphi\epsquiv 0$ on $[\delta_0,\infty).$ In addition, we have the Lipschitz continuity for $\varphi$:
\[
|\varphi(x_1)-\varphi(x_2)|\leq \frac{C}{\delta_0} |x_1-x_2|.
\]
The following theorem concerns the existence and uniqueness of solution to system \epsqref{e.Q02181}.
\begin{equation}gin{theorem}
\label{t.02051}
Let $p>5$ and $U_{0} \in L^{p}\left(\Omega ; L^{p}\right)$. For every $T>0$, there exists a unique pathwise solution $u \in L^{p}\left(\Omega ; C\left([0, T], L^{p}\right)\right)$ to \epsqref{e.Q02181} such that
\begin{equation}gin{equation}
\mathbb{E}\left[{\rm s}up _{0 \leq s \leq T}\|U(s, \cdot)\|_{p}^{p}+{\rm s}um_{j} \int_{0}^{T} \int_{\mathbb{T}^{3}}\left|\nabla\left(\left|U_{j}(s, x)\right|^{p / 2}\right)\right|^{2} d x d s\right] \leq C \mathbb{E}\left[\left\|U_{0}\right\|_{p}^{p}\right]+C_{T}.
\epsnd{equation}
\epsnd{theorem}
In order to solve system \epsqref{e.Q02181}, we consider the first iteration procedure
\begin{equation}gin{equation}\label{e.Q02101}
\begin{equation}gin{split}
d U^{(n)}-\Delta U^{(n)} d t
&=
\varphi\left(\left\|U^{(n)}\right\|_{p}\right) \varphi\left(\left\|U^{(n-1)}\right\|_{p}\right) B(U^{(n-1)}) dt +G(U^{(n)}) d t
\\&\quad+\varphi\left(\left\|U^{(n)}\right\|_{p}\right) \varphi\left(\left\|U^{(n-1)}\right\|_{p}\right) {\rm s}igma(U^{(n-1)}) d \mathbb{W}_{t}
+
\mathbf{P}(b\cdot\nabla U^{(n)})d \mathbb{W}_{t}
\\&\nabla \cdot u^{(n)}=0,
\\& U^{(n)}(0)=U_{0} ,
\epsnd{split}
\epsnd{equation}
where $U^{(0)}$ is the pathwise solution to
\begin{equation}gin{equation}\label{e.Q02102}
\begin{equation}gin{split}
& d U^{(0)}-\Delta U^{(0)} d t = G(U^{(0)}) dt + \mathbf{P}( b\cdot\nabla U^{(0)}) d \mathbb{W}_{t} , \\
& \nabla \cdot u^{(0)}=0, \\
& U^{(0)}(0)=U_{0} .
\epsnd{split}
\epsnd{equation}
By Threorem \ref{t.L021401}, we obtain an unique solution $U^{(0)} \in L^{p}(\Omega ; C([0, T], L^{p}))$ and
\begin{equation}gin{align*}
\begin{equation}gin{split}
\mathbb{E}\left[{\rm s}up _{0 \leq t \leq T}\|U^{(0)}(t, \cdot)\|_{p}^{p}+{\rm s}um_{j=1}^{4} \int_{0}^{T} \int_{\mathbb{T}^{3}}\left|\nabla\left(\left|U^{(0)}_{j}(t, x)\right|^{p / 2}\right)\right|^{2} d x d t\right]
\leq C\mathbb{E}\left[\left\|U_{0}\right\|_{p}^{p}\right].
\epsnd{split}
\epsnd{align*}
Next, one needs to establish the existence of unique solution to system \epsqref{e.Q02101} for each $n\in\mathbb N$. For this purpose, we consider
\begin{equation}gin{align}\label{e.Q02103}
&dU - \Delta U dt = \varphi\left(\|U\|_p \right)\varphi\left(\|V\|_p \right) B(V) dt + G(U) dt + \varphi\left(\|U\|_p \right)\varphi\left(\|V\|_p \right){\rm s}igma(V) d\mathbb{W}_t + \mathbf{P}(b\cdot\nabla U) d \mathbb{W}_{t} \nonumber
\\
&\nabla\cdot u = 0\nonumber
\\
& U(0) = U_0 ,
\epsnd{align}
where $V=(v,\theta)$ satisfying $\nabla\cdot v= 0$ and
\begin{equation}gin{align}\label{e.Q02104}
\mathbb{E}\left[{\rm s}up _{0 \leq t \leq T}\left\|V(t, \cdot)\right\|_{p}^{p}+{\rm s}um_{j=1}^4 \int_{0}^{T} \int_{\mathbb{T}^{3}}\left|\nabla\left(\left|V_{j}(t, x)\right|^{p / 2}\right)\right|^{2} d x d t\right] \leq C \mathbb{E}\left[\left\|U_{0}\right\|_{p}^{p}\right]+C_{T}.
\epsnd{align}
To the end, we employ the second iteration process
\begin{equation}gin{align}\label{e.Q02105}
&dU^{(n)} - \Delta U^{(n)}dt = \varphi\left(\|U^{(n-1)}\|_p\right)\varphi\left(\|V\|_p\right) B(V) dt + G(U^{(n)}) dt \nonumber
\\
&\hspace{4cm}+ \varphi\left(\|U^{(n-1)}\|_p\right)\varphi\left(\|V\|_p\right) {\rm s}igma(V) d\mathbb{W}_t +\mathbf{P}( b\cdot\nabla U^{(n)} ) d \mathbb{W}_{t}, \nonumber
\\
&\nabla\cdot u^n = 0,\nonumber
\\
& U^n(0) = U_0 ,
\epsnd{align}
in order to solve system \epsqref{e.Q02103}. Here $U^{(0)}$ is the solution to \epsqref{e.Q02102}, but for $n\geq 1$ the $U^{(n)}$ in \epsqref{e.Q02105} is different from the one in \epsqref{e.Q02101}.
The following lemma concerns the induction step in order to establish the existence of unique solution to system \epsqref{e.Q02105} for each $n$.
\begin{equation}gin{lemma}
\label{l.w02053}
Let $p>5, n \in \mathbb{N}$, and $T>0$. Suppose $U_{0} \in L^{p}\left(\Omega ; L^{p}\right)$ and assume that for each $n \in\{1,2, \ldots, k-1\}$, there exists a unique solution $U^{(n)} \in$ $L^{p}\left(\Omega ; C\left([0, T], L^{p}\right)\right)$ to the initial value problem \epsqref{e.Q02105}, where $V$ and $U^{(n)}$ satisfy \epsqref{e.Q02104}. Then for $n=k$, the initial value problem \epsqref{e.Q02105} also has a unique solution $U^{(k)} \in L^{p}\left(\Omega ; C\left([0, T], L^{p}\right)\right)$, and moreover,
\begin{equation}gin{equation}\label{e.q02113}
\mathbb{E}\left[{\rm s}up _{0 \leq t \leq T}\left\|U^{(k)}(t, \cdot)\right\|_{p}^{p}+{\rm s}um_{j=1}^4 \int_{0}^{T} \int_{\mathbb{T}^{3}}\left|\nabla\left(\left|U_{j}^{(k)}(t, x)\right|^{p / 2}\right)\right|^{2} d x d t\right] \leq C \mathbb{E}\left[\left\|U_{0}\right\|_{p}^{p}\right]+C_{T}.
\epsnd{equation}
\epsnd{lemma}
Assuming Lemma \ref{l.w02053} holds, one can achieve the existence of unique solution to \epsqref{e.Q02105} for each $n$ from the existence of $U^{(0)}$ solving \epsqref{e.Q02102} and mathematical induction. The next lemma shows the existence of unique solution to \epsqref{e.Q02103}.
\begin{equation}gin{lemma}
\label{l.w02054}
Let $p>5$ and suppose that $U_{0} \in L^{p}\left(\Omega ; L^{p}\right)$. Then there exists a time $t>0$ small enough such that the initial value problem \epsqref{e.Q02103}, where V satisfies \epsqref{e.Q02104}, has a unique pathwise solution $U \in L^{p}\left(\Omega ; C\left([0, t], L^{p}\right)\right)$, which satisfies
\begin{equation}gin{equation}\label{e.Q02191}
\mathbb{E}\left[{\rm s}up _{0 \leq s \leq t}\|U(s, \cdot)\|_{p}^{p}+{\rm s}um_{j=1}^4 \int_{0}^{t} \int_{\mathbb{T}^{3}}\left|\nabla\left(\left|U_{j}(s, x)\right|^{p / 2}\right)\right|^{2} d x d s\right] \leq C \mathbb{E}\left[\left\|U_{0}\right\|_{p}^{p}\right]+C_{t}
\epsnd{equation}
\epsnd{lemma}
We postpone the proof of Lemma \ref{l.w02053} and \ref{l.w02054} to Section \ref{sec:4.3}.
Assuming Lemma \ref{l.w02054} holds, we are ready to prove Theorem \ref{t.02051}.
\begin{equation}gin{proof}[Proof of Theorem \ref{t.02051}]
Consider the system \epsqref{e.Q02101}. Thanks to Lemma \ref{l.w02054} and by induction we know that for each $n\in\mathbb N$ there exists an unique solution $U^{(n)} \in L^P(\Omega; C([0,T], L^p))$ to system \epsqref{e.Q02101} with $T>0$ sufficiently small, and satisfies
\begin{equation}gin{equation*}
\mathbb{E}\left[{\rm s}up _{0 \leq t \leq T}\left\|U^{(n)}(t, \cdot)\right\|_{p}^{p}+{\rm s}um_{j=1}^4 \int_{0}^{T} \int_{\mathbb{T}^{3}}\left|\nabla\left(\left|U_{j}^{(n)}(t, x)\right|^{p / 2}\right)\right|^{2} d x d t\right] \leq C \mathbb{E}\left[\left\|U_{0}\right\|_{p}^{p}\right]+C_{T}.
\epsnd{equation*}
Next, we consider the difference $V^{(n)}=U^{(n+1)}-U^{(n)}$. For each $n\in\mathbb N$ denote by
\begin{equation}gin{align*}
\varphi^{(n)} = \varphi\left(\|U^{(n)}\|_p\right).
\epsnd{align*}
Thanks to the linearity of $G$ and the transport noise, we have
\begin{equation}gin{equation}\label{e.Q02241}
\begin{equation}gin{split}
d V^{(n)}-\Delta V^{(n)} d t
&=
\varphi^{(n+1)} \varphi^{(n)} B(U^{(n)}) d t
-
\varphi^{(n)} \varphi^{(n-1)} B(U^{(n-1)}) d t + G(V^{(n)}) dt
\\&\quad+
(\varphi^{(n+1)} \varphi^{(n)} {\rm s}igma(U^{(n)})
-
\varphi^{(n)} \varphi^{(n-1)} {\rm s}igma(U^{(n-1)})) d \mathbb{W}_{t}
+
\mathbf{P}(b\cdot\nabla V^{(n)}) d\mathbb{W}_{t},
\\
\nabla\cdot v^{(n)} &=0,
\\
V^{(n)}(0) &= 0 \quad a.s..
\epsnd{split}
\epsnd{equation}
The first equation can be rewritten as
\begin{equation}gin{align*}
d V^{(n)}-\Delta V^{(n)} d t
&=
{\rm s}um_{i=1}^3 \partial_i f_i d t + G(V^{(n)}) dt
+
g d \mathbb{W}_{t}
+
\mathbf{P}(b\cdot\nabla V^{(n)})d\mathbb{W}_{t},
\epsnd{align*}
where
\begin{equation}gin{equation}
\begin{equation}gin{split}
f_{i}= & -\varphi^{(n+1)} \varphi^{(n)}\left(\mathbf{P}\left(u_{i}^{(n)} U^{(n)}\right)\right)+\varphi^{(n)} \varphi^{(n-1)}\left(\mathbf{P}\left(u_{i}^{(n-1)} U^{(n-1)}\right)\right) \\
= & -\varphi^{(n)}\left(\varphi^{(n+1)}-\varphi^{(n)}\right)\left(\mathbf{P}\left(u_{i}^{(n)} U^{(n)}\right)\right)-\varphi^{(n)}\left(\varphi^{(n)}-\varphi^{(n-1)}\right)\left(\mathbf{P}\left(u_{i}^{(n)} U^{(n)}\right)\right) \\
& -\varphi^{(n)} \varphi^{(n-1)}\left(\mathbf{P}\left(v_{i}^{(n-1)} U^{(n)}\right)\right)-\varphi^{(n)} \varphi^{(n-1)}\left(\mathbf{P}\left(u_{i}^{(n-1)} V^{(n-1)}\right)\right) \\
= & f_{i}^{(1)}+f_{i}^{(2)}+f_{i}^{(3)}+f_{i}^{(4)},
\epsnd{split}
\epsnd{equation}
and
\begin{equation}gin{equation}
\begin{equation}gin{split}
g= & \varphi^{(n+1)} \varphi^{(n)} {\rm s}igma\left(U^{(n)}\right)-\varphi^{(n)} \varphi^{(n-1)} {\rm s}igma\left(U^{(n-1)}\right) \\
= & \varphi^{(n)}\left(\varphi^{(n+1)}-\varphi^{(n)}\right) {\rm s}igma\left(U^{(n)}\right)+\varphi^{(n)}\left(\varphi^{(n)}-\varphi^{(n-1)}\right) {\rm s}igma\left(U^{(n)}\right) \\
& +\varphi^{(n)} \varphi^{(n-1)}\left({\rm s}igma\left(U^{(n)}\right)-{\rm s}igma\left(U^{(n-1)}\right)\right) \\
= & g^{(1)}+g^{(2)}+g^{(3)} .
\epsnd{split}
\epsnd{equation}
As $\mathbf{P}$ is a bounded operator, we can now use the estimates of (5.19) and (5.20) in \cite{KXZ} to get
\begin{equation}gin{equation}\label{e.Q02243}
\begin{equation}gin{split}
&\mathbb E\left[ \int_0^t \|f\|_q^p ds \right] \leq C t \mathbb E\left[ {\rm s}up\limits_{s\in[0,t]} \|V^{(n-1)}\|_p^p \right] + C t \mathbb E\left[ {\rm s}up\limits_{s\in[0,t]} \|V^{(n)}\|_p^p \right] ,
\\
&\mathbb E\left[ \int_0^t \|g\|_{\mathbb L^p}^p ds \right] \leq C t \mathbb E\left[ {\rm s}up\limits_{s\in[0,t]} \|V^{(n-1)}\|_p^p \right] + C t \mathbb E\left[ {\rm s}up\limits_{s\in[0,t]} \|V^{(n)}\|_p^p \right].
\epsnd{split}
\epsnd{equation}
Thanks to Theorem \ref{t.L021401} and the fact that $V^{(n)}(0) = 0$, we know
\begin{equation}gin{align}\label{e.Q02244}
\mathbb{E}\left[{\rm s}up _{s\in[0,t]}\|V^{(n)}(s, \cdot)\|_{p}^{p}\right]
\leq C t \mathbb E\left[ {\rm s}up\limits_{s\in[0,t]} \|V^{(n-1)}\|_p^p \right] + C t \mathbb E\left[ {\rm s}up\limits_{s\in[0,t]} \|V^{(n)}\|_p^p \right].
\epsnd{align}
In particular, by choosing $t=t^*$ small enough we obtain exponential convergence rate, and there exists a fixed point $U \in L^p(\Omega; C([0,t^*],L^p))$ of system \epsqref{e.Q02101}. As $U^{(n)}$ is the solution to system \epsqref{e.Q02101} for each $n$, we have
\begin{equation}gin{align*}
\left(U^{(n)}(s), \phi\right)= & \left(U_0, \phi\right)+\int_0^s\left(U^{(n)}(r), A \phi\right) d r + \int_0^s\left( G(U^{(n)}(r)),\phi\right) d r\\
& +{\rm s}um_{i=1}^3 \int_0^s\left(\varphi^{(n)} \varphi^{(n-1)} \mathbf{P}\left(u_i^{(n-1)} U^{(n-1)}\right), \partial_i \phi\right) d r \\
& +\int_0^s\left(\varphi^{(n)} \varphi^{(n-1)} {\rm s}igma\left(U^{(n-1)}\right), \phi\right) d \mathbb{W}_r - {\rm s}um_{i=1}^3 \int_0^s\left(\mathbf{P}\left( b_i U^{(n)}\right), \partial_i\phi\right) d \mathbb{W}_r,
\epsnd{align*}
for a.a. $(s,\omega)\in (0,t^*)\times \Omega$ and all $\phi \in C^{\infty}\left(\mathbb{T}^3;\mathbb R^4\right)$. By the exponential convergence rate and thanks to Lemma 5.2 and Remark 5.3 in \cite{KXZ}, one has
\[
\|U^{(n)}\|_p \to \|U\|_p \quad \text{ for a.a. } (s,\omega)\in (0,t^*)\times \Omega.
\]
Therefore,
$
\left(U^{(n)}(s), \phi\right) \to \left(U(s), \phi\right),
$
$\varphi^{(n)}$ and $\varphi^{(n-1)} \to \varphi(\|U(s)\|_p):=\varphi$ for a.a. $(s,\omega)\in (0,t^*)\times \Omega$. Thanks to the dominant convergence theorem, one obtains
\begin{equation}gin{align*}
&\int_0^s\left(U^{(n)}(r), A \phi\right) d r + \int_0^s\left( G(U^{(n)}(r)),\phi\right) d r
+{\rm s}um_{i=1}^3 \int_0^s\left( \varphi^{(n)} \varphi^{(n-1)}\mathbf{P}\left(u_i^{(n-1)} U^{(n-1)}\right), \partial_i \phi\right) d r
\\
&\hspace{1cm}\to \int_0^s\left(U(r), A \phi\right) d r + \int_0^s\left( G(U(r)),\phi\right) d r
+{\rm s}um_{i=1}^3 \int_0^s\left(\varphi^2\mathbf{P}\left(u_i U\right), \partial_i \phi\right) d r
\epsnd{align*}
for a.s. $(s,\omega)\in (0,t^*)\times \Omega$. Next, by BDG inequality and thanks to the property of ${\rm s}igma$, we have
\begin{equation}gin{align*}
&\mathbb{E}\left[{\rm s}up _{s \in\left[0, t^*\right)}\left|\int_0^s\left(\varphi^{(n)} \varphi^{(n-1)} {\rm s}igma\left(U^{(n-1)}\right)-\varphi^2 {\rm s}igma(U), \phi\right) d \mathbb{W}_r\right|\right] \\
\leq &\mathbb{E}\left[\left(\int_0^{t^*}\left\|\left(\varphi^{(n)} \varphi^{(n-1)} {\rm s}igma\left(U^{(n-1)}\right)-\varphi^2 {\rm s}igma(U), \phi\right)\right\|_{l^2}^2 d r\right)^{1 / 2}\right] \to 0 ,
\epsnd{align*}
and
\begin{equation}gin{align*}
&\mathbb{E}\left[{\rm s}up _{s \in\left[0, t^*\right)}\left|\int_0^s{\rm s}um_{i=1}^3 \left( \mathbf{P}\left( b_i U^{(n)}\right) - \mathbf{P}\left( b_i U\right), \partial_i\phi\right) d \mathbb{W}_r\right|\right] \\
\leq & C_{BDG} {\rm s}um_{i=1}^3\mathbb{E}\left[\left(\int_0^{t^*}\left\|\left( \mathbf{P}\left( b_i U^{(n)}\right) - \mathbf{P}\left( b_i U\right), \partial_i\phi\right)\right\|_{l^2}^2 d r\right)^{1 / 2}\right] \to 0 .
\epsnd{align*}
Thanks to Lemma 5.2 and Remark 5.3 in \cite{KXZ}, one has
\begin{equation}gin{align*}
&\int_0^s\left(\varphi^{(n)} \varphi^{(n-1)} {\rm s}igma\left(U^{(n-1)}\right) ,\phi\right) d \mathbb{W}_r \to \int_0^s \left(\varphi^2 {\rm s}igma(U), \phi\right) d \mathbb{W}_r ,
\\
&{\rm s}um_{i=1}^3 \int_0^s\left( \mathbf{P}\left( b_i U^{(n)}\right), \partial_i\phi\right) d \mathbb{W}_r \to {\rm s}um_{i=1}^3 \int_0^s\left( \mathbf{P}\left( b_i U\right), \partial_i\phi\right) d \mathbb{W}_r.
\epsnd{align*}
Combining the discussion above, by taking $n\to \infty$ we get
\begin{equation}gin{align*}
\left(U(s), \phi\right)= & \left(U_0, \phi\right)+\int_0^s\left(U(r), A \phi\right) d r + \int_0^s\left( G(U(r)),\phi\right) d r +{\rm s}um_{i=1}^3 \int_0^s\left(\varphi^2 \mathbf{P}\left(u_i U\right), \partial_i \phi\right) d r \\
& +\int_0^s\left(\varphi^2{\rm s}igma\left(U\right), \phi\right) d \mathbb{W}_r - {\rm s}um_{i=1}^3 \int_0^s\left( \mathbf{P}\left( b_i U\right), \partial_i\phi\right) d \mathbb{W}_r.
\epsnd{align*}
Therefore we obtain the existence of a solution $U \in L^p(\Omega; C([0,t^*],L^p))$ to system \epsqref{e.Q02181}.
Next, for the uniqueness we consider two pathwise solutions $U,V \in L^p (\Omega; C([0,t^*], L^p))$ to system \epsqref{e.Q02181}. By denoting $W=U-V$ we have
\begin{equation}gin{align}\label{e.Q02242}
& d W-\Delta W d t=\left(\varphi_U^2 B(U)-\varphi_V^2 B(V) + G(W)\right) d t+\left(\varphi_U^2 {\rm s}igma(U)-\varphi_V^2 {\rm s}igma(V) + \mathbf{P}(b\cdot \nabla W)\right) d \mathbb{W}_t, \nonumber\\
& \nabla \cdot (W_1,W_2,W_3)=0, \nonumber\\
& W(0)=0, \quad \text { a.s.. }
\epsnd{align}
Here $\varphi_U = \varphi(\|U\|_p)$ and $\varphi_V = \varphi(\|V\|_p)$. Similar as the treatment for \epsqref{e.Q02241}, we can rewrite the first equation of \epsqref{e.Q02242} as
\begin{equation}gin{align*}
dW-\Delta W d t
&=
{\rm s}um_{i=1}^3 \partial_i f_i d t + G(W) dt
+
g d \mathbb{W}_{t}
+
\mathbf{P}(b\cdot\nabla W)d\mathbb{W}_{t},
\epsnd{align*}
where
\begin{equation}gin{align*}
f_{i}= & -\varphi_U^2\left(\mathbf{P}\left(u_i U\right)\right)+\varphi_V^2\left(\mathbf{P}\left(v_i V\right)\right) \\
= & -\varphi_U\left(\varphi_U-\varphi_V\right)\left(\mathbf{P}\left(u_i U\right)\right)-\varphi_U \varphi_V\left(\mathbf{P}\left(w_i U\right)\right) \\
& -\varphi_U \varphi_V\left(\mathbf{P}\left(v_i W\right)\right)-\varphi_V\left(\varphi_U-\varphi_V\right)\left(\mathbf{P}\left(v_i V\right)\right)
\epsnd{align*}
and
\begin{equation}gin{align*}
g & =\varphi_U^2 {\rm s}igma(U)-\varphi_V^2 {\rm s}igma(V) \\
& =\varphi_U\left(\varphi_U-\varphi_V\right) {\rm s}igma(U)+\varphi_V\left(\varphi_U-\varphi_V\right) {\rm s}igma(V)+\varphi_U \varphi_V\left({\rm s}igma(U)-{\rm s}igma(V)\right) .
\epsnd{align*}
Similar to \epsqref{e.Q02243} we obtain
\begin{equation}gin{equation}\label{e.Q02245}
\begin{equation}gin{split}
\mathbb E\left[ \int_0^{t^*} \|f\|_q^p ds \right] \leq C t^* \mathbb E\left[ {\rm s}up\limits_{s\in[0,t^*]} \|W\|_p^p \right] , \quad
\mathbb E\left[ \int_0^{t^*} \|g\|_{\mathbb L^p}^p ds \right] \leq C t^* \mathbb E\left[ {\rm s}up\limits_{s\in[0,t^*]} \|W\|_p^p \right].
\epsnd{split}
\epsnd{equation}
Therefore thanks to Theorem \ref{t.L021401} we obtain
\begin{equation}gin{align}\label{e.Q02246}
\mathbb E\left[ {\rm s}up\limits_{s\in[0,t^*]} \|W\|_p^p\right] \leq Ct^* \mathbb E\left[ {\rm s}up\limits_{s\in[0,t^*]} \|W\|_p^p\right].
\epsnd{align}
When $t^*$ is small enough we obtain that $W=0$. Thus the solution is unique.
Finally, note that the constant $C$ appearing in \epsqref{e.Q02244} and \epsqref{e.Q02245} does not depend on the initial data, and thus $t^*$ is independent of the initial data. For arbitrary $T>0$, one can consider $N$ large enough such that $\frac TN < t^*$. Then by establishing the existence and uniqueness of pathwise solution inductively on $[\frac{i}{N}T,\frac{i+1}{N}T]$ for $i\in\{0,1,2,...N-1\}$, we obtain a unique pathwise solution on $[0,T]$. As $T$ is arbitrary, the solution exists globally in time.
\epsnd{proof}
{\rm s}ubsection{Proof of Theorem \ref{t.w10101}}
Now by applying a family of suitable stopping times we are able to prove the existence and uniqueness of maximal pathwise solution to system \epsqref{e.L10261}, which gives Theorem \ref{t.w10101}.
\begin{equation}gin{proof}[Proof of Theorem \ref{t.w10101}]
For $n\in\mathbb N$, denote by $U^{(n)}$ the solution of the truncated system \epsqref{e.Q02181} with $\delta_0=n$. Also, introduce the corresponding stopping times
$$
\tau_n(\omega)= \begin{equation}gin{cases}\inf \left\{t>0:\left\|U^{(n)}(t, \omega)\right\|_p \geq n / 2\right\}, & \text { if }\left\|U^{(n)}(0, \omega)\right\|_p<n / 2, \\ 0, & \text { if }\left\|U^{(n)}(0, \omega)\right\|_p \geq n / 2\epsnd{cases}
$$
By uniqueness, the sequence is non-decreasing a.s. and $U^{(m)}=U^{(n)}$ on $\left[0, \tau_m \wedge \tau_n\right]$. Let $\tau=\lim _n \tau_n \wedge T$. Then, $\mathbb{P}(\tau>0)=1$. Also, for any integer $n \in \mathbb{N}$, define $U=U^{(n)}$ on $\left[0, \tau_n \wedge T\right]$. It is easy to check that $(U, \tau)$ satisfies all the required properties.
\epsnd{proof}
{\rm s}ubsection{Proof of Lemma \ref{l.w02053} and \ref{l.w02054}}\label{sec:4.3}
\begin{equation}gin{proof}[Proof of Lemma \ref{l.w02053}]
Let $n=k$, and denote by
\[
\varphi^{(n-1)} = \varphi\left(\|U^{(n-1)}\|_p\right),\quad \varphi_V = \varphi\left(\|V\|_p\right),
\]
then we can rewrite the first equation in \epsqref{e.Q02105} as
\begin{equation}gin{align}\label{e.Q02106}
d U^{(n)}-\Delta U^{(n)} d t =\varphi^{(n-1)} \varphi_V B(V) d t +G(U^{(n)}) dt
+ \varphi^{(n-1)} \varphi_V {\rm s}igma(V) d \mathbb{W}_t + \mathbf{P} (b\cdot\nabla U^{(n)})d \mathbb{W}_{t}.
\epsnd{align}
For the nonlinear term, we rewrite
\[
\varphi^{(n-1)} \varphi_V B(V) dt = -\varphi^{(n-1)} \varphi_V \mathbf{P}(v\cdot \nabla V) dt =-{\rm s}um_{i=1}^3\varphi^{(n-1)} \varphi_V \partial_i\left(\mathbf{P}\left(V_i V\right)\right) d t .
\]
In order to apply Theorem \ref{t.L021401} we consider $p,q,r,l$ satisfying
\begin{equation}gin{equation}\label{e.Q02111}
\frac{3p}{p+1} <q \leq p,\quad \frac{1}{r}+\frac{1}{l}=\frac{1}{q}, \text{ and } l\leq p,\quad r\leq p .
\epsnd{equation}
Thanks to \epsqref{e.Q02104} and the boundedness of $\mathbf{P}$, we have
\begin{equation}gin{align*}
&C \mathbb{E} \int_0^T \|\varphi^{(n-1)} \varphi_V B(V)\|_{-1,q}^p dt \leq C {\rm s}um_{i=1}^3\mathbb{E} \left[\int_0^T\left\|\varphi^{(n-1)} \varphi_V V_i V\right\|_q^p d s\right]
\\
\leq &C \mathbb{E}\left[\int_0^T \varphi^{(n-1)} \varphi_V\left\|V\right\|_r^p\|V\|_l^p d s\right] \leq C \mathbb{E}\left[\int_0^T \varphi^{(n-1)} \varphi_V\left\|V\right\|_p^{2p} d s\right] \leq C_{T}.
\epsnd{align*}
As $V$ satisfies \epsqref{e.Q02104}, we have $\varphi^{(n-1)} \varphi_VB(V)\in L^p(\Omega\times [0,T], W^{-1,p})$. Here from \epsqref{e.Q02111} we know $\frac1r+\frac1l\geq \frac2p$, and thus $\frac2p<\frac{p+1}{3p}$, which gives $p>5$.
Next, by the sub-linear growth of ${\rm s}igma(V)$ and the property of $V$, one can easily verify that $\varphi^{(n-1)} \varphi_V {\rm s}igma(V)\in L^p(\Omega\times [0,T], \mathbb{L}^p)$. Thus by applying Theorem \ref{t.L021401} we know unique solution $U^{(k)}\in L^p(\Omega;C([0,T],L^p))$ and the bound \epsqref{e.q02113} holds.
\epsnd{proof}
\begin{equation}gin{proof}[Proof of Lemma \ref{l.w02054}]
For $U^{(n+1)} = (u^{(n+1)}, \rho^{(n+1)})$ and $U^{(n)} = (u^{(n)}, \rho^{(n)})$ solving system \epsqref{e.Q02105}, we consider the difference $Z^{(n)}=U^{(n+1)}-U^{(n)}$, where $Z^{(n)}=(z^{(n)},\kappa^{(n)})$, $z^{(n)}=u^{(n+1)}-u^{(n)}$, $\kappa^{(n)}=\rho^{(n+1)}-\rho^{(n)}$. By direct calculation one gets
\begin{equation}gin{align*}
d Z^{(n)}-\Delta Z^{(n)} d t=&{\rm s}um_{i=1}^{3} (\varphi^{(n)}-\varphi^{(n-1)})\varphi_V \partial_{i} (\mathbf{P}(V_i V)) d t + G(Z^{(n)}) dt
\\
& + (\varphi^{(n)}-\varphi^{(n-1)}) \varphi_V {\rm s}igma(V) d\mathbb{W}_{t} + \mathbf{P}( b\cdot\nabla Z^{(n)}) d\mathbb{W}_{t},
\epsnd{align*}
where $\nabla\cdot z^{(n)}=0$ and $Z^{(n)}(0) = 0.$
Thanks to the Lipschitz property of $\varphi$, we have
\[
|\varphi^{(n)}-\varphi^{(n-1)}| \leq \frac{C}{\delta_0}\left| \|U^{(n)}\|_p - \|U^{(n-1)}\|_p \right|\leq \frac{C}{\delta_0}\|U^{(n)}-U^{(n-1)}\|_p = \frac{C}{\delta_0} \|Z^{(n-1)}\|_p.
\]
Then by the boundedness of $\mathbf{P}$ we can estimate
\begin{equation}gin{align*}
\mathbb E\left[\int_0^t \|{\rm s}um_{i=1}^{3} (\varphi^{(n)}-\varphi^{(n-1)})\varphi_V \mathbf{P}(V_i V) \|_{q}^p ds \right] \leq \frac{C}{\delta_0^p} \mathbb E\left[\int_0^t \|Z^{(n-1)}(s)\|_{p}^p ds \right] \leq Ct \mathbb E\left[{\rm s}up\limits_{s\in[0,t]}\|Z^{(n-1)}(s)\|_{p}^p \right],
\epsnd{align*}
and thanks to the property of ${\rm s}igma$ and $V$,
\begin{equation}gin{align*}
\mathbb E \left[\int_0^t \|(\varphi^{(n)}-\varphi^{(n-1)}) \varphi_V {\rm s}igma(V) \|_{\mathbb L^p}^p ds \right] \leq C \mathbb E\left[\int_0^t \|Z^{(n-1)}(s)\|_{p}^p ds \right] \leq Ct \mathbb E\left[{\rm s}up\limits_{s\in[0,t]}\|Z^{(n-1)}(s)\|_{p}^p \right].
\epsnd{align*}
By applying Theorem \ref{t.L021401} we conclude that
\begin{equation}gin{align*}
&\mathbb{E}\left[{\rm s}up _{0 \leq s \leq t}\|Z^{(n)}(t, \cdot)\|_{p}^{p} \right] \leq Ct \mathbb E\left[{\rm s}up\limits_{s\in[0,t]}\|Z^{(n-1)}(s)\|_{p}^p \right].
\epsnd{align*}
By taking $t\in(0,T]$ small enough one can follow the proof of \cite{KXZ}~Lemma 5.5 to conclude the existence of unique solution $U \in L^{p}\left(\Omega ; C\left([0, t], L^{p}\right)\right)$ and satisfies \epsqref{e.Q02191}.
\epsnd{proof}
{\rm s}ection{Global existence with small initial data and small noise}\label{sec:global}
To prove the global existence part in the main Theorem \ref{t.w10101}, we first recall the truncated stochastic Boussinesq system:
\begin{equation}gin{equation}\label{e.w02231}
\begin{equation}gin{split}
d U-\Delta U d t
&=\varphi\left(\left\|U\right\|_{p}\right)^2 B(U) dt
+G(U) d t +\left(\mathbf{P}(b\cdot\nabla U) + \varphi\left(\left\|U\right\|_{p}\right)^2{\rm s}igma(U)\right) d\mathbb{W}_{t}
\\&\nabla \cdot u=0,
\\& U(0)=U_{0} ,
\epsnd{split}
\epsnd{equation}
on $[0, \infty) \times \mathbb{T}^{3}$ with $\nabla \cdot u_{0}=0$ and $\int_{\mathbb{T}^{3}} U_{0} d x=0$ a.s., where $\varphi$ is defined as in Section \ref{s.w02271}.
Then it has been shown in Section \ref{s.w02271} that system \epsqref{e.w02231} is globally well-posed. Observe that when $\|U\|_{p}\leq \delta_{0} / 2$, the original system \epsqref{e.L10261} coincides with this truncated model. Hence, an estimate of the likelihood that $\|U\|_{p}$ exceeds $\delta_{0} / 2$ determines the time of existence for the solution to \epsqref{e.L10261}. The global existence part in Theorem \ref{t.w10101} then follows from Markov inequality and the following theorem.
\begin{equation}gin{theorem}\label{t.w02231}
Let $p>5$. Then the global solution $U \in L^{p}\left(\Omega ; C\left([0, \infty), L^{p}\right)\right)$ to \epsqref{e.w02231} satisfies
\begin{equation}gin{equation}\label{L.022502}
\mathbb{E}\left[{\rm s}up _{s \in[0, \infty)} e^{a s}\|U(s)\|_{p}^{p}+\int_{0}^{\infty} e^{a s} {\rm s}um_{i}\left\|\nabla\left(\left|U_{i}(s)\right|^{p / 2}\right)\right\|_{2}^{2} d s\right] \leq C \mathbb{E}\left[\left\|U_{0}\right\|_{p}^{p}\right],
\epsnd{equation}
provided that $a, \delta_{0}, \varepsilonilon_{0}>0$ and $N_{b,2}$ are sufficiently small constants.
\epsnd{theorem}
We first supply a proof of \epsqref{L.022501} in Theorem \ref{t.w10101} by assuming Theorem \ref{t.w02231}.
\begin{equation}gin{proof}[Proof of global existence in Theorem \ref{t.w10101}]
Let $\mathbb{E}\left\|U_0\right\|_p^p<\delta$. It then follows from Markov inequality and \epsqref{L.022502} that
\[\mathbb{P}\left({\rm s}up_{s \in[0, \infty)} e^{a s}\|U(s)\|_p^p\geq \frac{\delta_0}{2}\right)\leq \frac{C\delta}{\delta_0}.\]
Since when $\|U\|_{p}\leq \delta_{0} / 2$ the original system \epsqref{e.L10261} coincides with this truncated model, we have
\begin{equation}gin{align*}
\mathbb{P}(\tau=\infty) &= \mathbb{P}\left({\rm s}up_{s \in[0, \infty)} \|U(s)\|_p^p\leq \frac{\delta_0}{2}\right)\\
&\geq \mathbb{P}\left({\rm s}up_{s \in[0, \infty)} e^{a s}\|U(s)\|_p^p\leq \frac{\delta_0}{2}\right)\geq 1- \frac{C\delta}{\delta_0}.
\epsnd{align*}
By choosing $\delta$ sufficiently small, we obtain \epsqref{L.022501} as desired.
\epsnd{proof}
\begin{equation}gin{proof}[Proof of Theorem \ref{t.w02231}]
Let $T>0$. Applying the Itô-Wentzel formula to $F_{i}(t)=e^{a t}\left\|U_{i}(t)\right\|_{p}^{p}$, for a fixed $i \in\{1,2,3,4\}$, we obtain
\begin{equation}gin{equation}
d\left(e^{a t}\left\|U_{i}(t)\right\|_{p}^{p}\right)=a e^{a t}\left\|U_{i}(t)\right\|_{p}^{p} d t+e^{a t} d\left(\left\|U_{i}(t)\right\|_{p}^{p}\right).
\epsnd{equation}
Similar to the Ito expansion as in Section \ref{s.L022301}, we have
\begin{equation}gin{align}\label{e.w02232}
\begin{equation}gin{split}
&e^{at}\|U_j(t)\|_p^p + \frac{4(p-1)}{p}\int_0^t e^{as} \left\|\nabla\left(\left|U_j(s)\right|^{p / 2}\right)\right\|_2^2ds\\
&= \|U_{0, j}\|_p^p + p\int_0^t e^{as}\varphi^2 \int_{\mathbb{T}^3}|U_j(s)|^{p-2}U_j(s) B(U)_jdxds+p\int_0^t e^{as}
\int_{\mathbb{T}^3} |U_j(s)|^{p-2}U_j(s) G(U)_jdxds\\
&+ p\int_0^t e^{as}\varphi^2 \int_{\mathbb{T}^3}|U_j(s)|^{p-2}U_j(s)\left((\mathbf{P}(b\cdot\nabla U))_j +
{\rm s}igma(U)_j \right)dxd\mathbb{W}_s \\
&+\frac{p(p-1)}{2}\int_0^t e^{as}\varphi^4 \int_{\mathbb{T}^3}|U_j|^{p-2}\|\big(\mathbf{P}\left(b\cdot\nabla U \right)
\big)_j+ {\rm s}igma(U)_j \|_{l^2}^2dxds+a \int_{0}^{t} e^{a s}\left\|U_{i}(s)\right\|_{p}^{p} d s.
\epsnd{split}
\epsnd{align}
Choosing $\bar{r}, r, q, l$ as in \epsqref{e.L030403} and \epsqref{e.Q02111}, and using integration by parts, one obtains
\begin{equation}gin{align}\label{e.L022301}
\begin{equation}gin{aligned}
&\hspace{0.5cm}p e^{a s} \varphi^{2}\left|\int_{\mathbb{T}^{3}} \left|U_{j}\right|^{p-2} U_{j}B(U)_{j} d x\right| \\
&= p e^{a s} \varphi^{2}\left|{\rm s}um_{i=1}^3 \int_{\mathbb{T}^{3}} \partial_{i}\left(\left|U_{j}\right|^{p-2} U_{j}\right)\left(\mathbf{P}\left(U_{i} U\right)\right)_{j} d x\right|\\
&\leq Ce^{a s} \varphi^{2}\left\|\nabla\left(\left|U_j(s)\right|^{p / 2}\right)\right\|_2\left\||U_j|^{p/2-1}\right\|_{\bar{r}}\|U_i\|_r\|U\|_{l}
\\
&\leq C\delta_0 e^{a s} \varphi^{2}\left\|\nabla\left(\left|U_j(s)\right|^{p / 2}\right)\right\|_2\left\||U_j|^{p/2-1}\right\|_{\bar{r}}\|U\|_p,
\epsnd{aligned}
\epsnd{align}
where we used the facts that $r, l\leq p$ and $\varphi\|U\|_p\leq \delta_0$. Now estimates as in \epsqref{e.L030402}-\epsqref{e.L021404} give
\begin{equation}gin{align}\label{e.L030405}
\begin{equation}gin{aligned}
\left|\int_0^tp e^{a s} \varphi^{2}\int_{\mathbb{T}^{3}} \left|U_{j}\right|^{p-2} U_{j}B(U)_{j} d xds \right|\leq\delta \int_0^t e^{a s} \left\|\nabla\left(\left|U_j(s)\right|^{p / 2}\right)\right\|_2^2 d s+C_\delta \delta_0^\kappa \int_0^t e^{a s}\|U(s)\|_p^p d s,
\epsnd{aligned}
\epsnd{align}
where $\delta>0$ is arbitrary, $\kappa>0$ is a constant depending on $p$ . Assumption \epsqref{e.L022205} and estimate as in \epsqref{e.L022107} and yield
\begin{equation}gin{equation}\label{e.w02272}
\begin{equation}gin{split}
\mathbb{E}\left[{\rm s}up _{t \in[0, T]}\left|\int_0^t e^{a s} \varphi^2 \int_{\mathbb{T}^3} |U_j(s)|^{p-2} U_j(s) {\rm s}igma(U)_j(s) d x d \mathbb{W}_s \right|\right]
&\leq
\frac{1}{4p}\mathbb{E}{\rm s}up_{s\in[0, T]}e^{as}\|U_j(s)\|_{p}^p
\\&\quad+
C\varepsilonilon_0 \mathbb{E}\left[\int_0^T e^{a s}\|U(s)\|_p^p d s\right] .
\epsnd{split}
\epsnd{equation}
Assumption \epsqref{e.L022205} on the noise also implies
\begin{equation}gin{equation}\label{e.w02271}
\begin{equation}gin{split}
p(p-1)\int_0^t e^{as}\varphi^4 \int_{\mathbb{T}^3}|U_j|^{p-2}\|{\rm s}igma(U)_j \|_{l^2}^2dxds&\leq C \varepsilonilon_0^2 \int_0^te^{a s} \|U(s)\|_p^pds.
\epsnd{split}
\epsnd{equation}
Estimates as in \epsqref{e.L022102}, \epsqref{e.L121803} and \epsqref{e.L022304} yield
\begin{equation}gin{align}\label{e.L022305}
\begin{equation}gin{split}
&p(p-1)\int_0^t e^{as}\varphi^4 \int_{\mathbb{T}^3}|U_j|^{p-2}\|\big(\mathbf{P}\left(b\cdot\nabla U \right)
\big)_j\|_{l^2}^2dxds\\
&\leq C\left(N_{b,0} \int_0^t e^{a s} \left\|\nabla\left(\left|U_j(t)\right|^{p / 2}\right)\right\|_2^2 d s+ N_{b,2}\int_0^te^{as}\left\|U(s)\right\|_p^p d s\right).
\epsnd{split}
\epsnd{align}
Also estimates as in \epsqref{e.L022105}, \epsqref{e.L022107} and \epsqref{e.L0221011} imply
\begin{equation}gin{align}\label{e.L022307}
\begin{equation}gin{split}
&\mathbb{E}\left[{\rm s}up _{t \in[0, T]}\left|\int_0^t e^{a s} \varphi^2 \int_{\mathbb{T}^3} |U_j(s)|^{p-2} U_j(s) {\rm s}igma(U)_j(s) d x d \mathbb{W}_s \right|\right] \\
&\leq\frac{1}{2p}\mathbb{E}{\rm s}up_{s\in[0, T]}e^{as}\|U_j(s)\|_{p}^p + CN_{b,2} \mathbb{E}\left[\int_0^T e^{a s} \|U(s)\|_p^p d s\right] + C N_{b,0}\mathbb{E}\int_0^T e^{a s} \left\|\nabla\left(\left|U_j(t)\right|^{p / 2}\right)\right\|_2^2 d s.
\epsnd{split}
\epsnd{align}
Combining \epsqref{e.L030405}-\epsqref{e.L022307}, one has
\begin{equation}gin{align}\label{e.L022306}
\begin{equation}gin{split}
&\mathbb{E}\left[\frac{1}{4}{\rm s}up_{s\in[0, T]}e^{as}\|U_j(s)\|_{p}^p + L_1\int_0^T e^{as} \left\|\nabla\left(\left|U_j(s)\right|^{p / 2}\right)\right\|_2^2ds\right]\\
&\leq \mathbb{E}\left[\|U_{0, j}\|_p^p +L_2\int_0^T e^{a s} \|U(s)\|_p^pds + p\left|\int_0^T e^{as}
\int_{\mathbb{T}^3} |U_j(s)|^{p-2}U_j(s) G(U)_jdxds\right|\right],
\epsnd{split}
\epsnd{align}
where
\begin{equation}gin{align}\label{e.L0223010}
\begin{equation}gin{split}
&L_1 = L_1(\delta, b, p) = \frac{4(p-1)}{p} - \delta - C_pN_{b,0},\\
&L_2 = L_2(\delta, b, p, \delta_0, a) = C_\delta \delta_0^\kappa+C\varepsilonilon_0^2 +C_pN_{b,2} +a.
\epsnd{split}
\epsnd{align}
For $j=1,2,3$, the fact $(GU)_j = \mathcal{P}(U_4 e_3))_j$ and the estimate like \epsqref{e.L021404} give
\begin{equation}gin{align}\label{e.L022302}
\begin{equation}gin{split}
&p\left|\int_0^Te^{as}\int_{\mathbb{T}^3}|U_j(s)|^{p-2}U_j(s)(GU)_j(s)dxds\right|\\
&\leq \epsta \int_0^Te^{as}\left\|\nabla\left(\left|U_j(r)\right|^{p / 2}\right)\right\|_2^2 d r+\epsta \int_0^Te^{as}\|U_j(s)\|_p^pds+C_{\epsta}\int_0^Te^{as}\left\|U_4(s)\right\|_{p}^p d s,
\epsnd{split}
\epsnd{align}
where $\epsta>0$ is arbitrary. Note that $(G(U))_4\epsquiv0$. Hence inequality \epsqref{e.L022306} for $j=4$ and the Poincaré \epsqref{e.L031101} imply that
\begin{equation}gin{align}\label{e.L022308}
\begin{equation}gin{split}
\mathbb{E}\int_0^T e^{as}\left\|U_4(s)\right\|_{p}^p d s
&\leq
C_{P}\mathbb{E}\int_0^T e^{as} \left\|\nabla\left(\left|U_4(s)\right|^{p / 2}\right)\right\|_2^2ds
\\&\leq
\mathbb{E}\left[\frac{C_{P}}{L_1}\|U_0\|_p^p +\frac{C_{P}L_2}{L_1}\int_0^T e^{a s} \|U(s)\|_p^pds\right].
\epsnd{split}
\epsnd{align}
Combining \epsqref{e.L022306}-\epsqref{e.L022308}, we arrive at
\begin{equation}gin{align}\label{e.L022309}
\begin{equation}gin{split}
\mathbb{E}\left[\frac{1}{4}{\rm s}up_{s\in[0, T]}e^{as}\|U(s)\|_{p}^p + L{\rm s}um_{j=1}^4\int_0^T e^{as} \left\|\nabla\left(\left|U_j(s)\right|^{p / 2}\right)\right\|_2^2ds\right]\leq C_{\epsta, L_1}\mathbb{E}\|U_0\|_p^p ,
\epsnd{split}
\epsnd{align}
where $L = L_1-\epsta - C\left(L_2+ \epsta +\frac{C(\epsta)L_2}{L_1}\right)$. In view of \epsqref{e.L0223010}, we can first choose $\delta, N_{b,0}, \epsta$ small enough, and then choose $\delta_0, \varepsilonilon_0, N_{b, 2}, a$ in $L_2$ sufficiently small to make $L>0$. We then deduce \epsqref{L.022502} by letting $T\to\infty$.
\epsnd{proof}
{\rm s}ection{Acknowledgements}
WW was partially supported by an AMS-Simons travel grant.
\epsnd{document} |
\begin{document}
\title{{\LARGE Quantum exam}}
\author{\bf Nguyen Ba An}
\email{[email protected]}
\affiliation{School of Computational Sciences, Korea Institute for Advanced
Study, 207-43 Cheongryangni 2-dong, Dongdaemun-gu, Seoul 130-722, Republic of Korea}
\begin{abstract}
Absolutely and asymptotically secure protocols for organizing an exam in a
quantum way are proposed basing judiciously on multipartite entanglement. The protocols
are shown to stand against common types of eavesdropping attack.
\end{abstract}
\pacs{03.67.Hk, 03.65.Ud, 03.67.Dd}
\maketitle
\noindent \textbf{1. Introduction}
Simultaneous distance-independent correlation between different systems
called entanglement \cite{r1} is the most characteristic trait that sharply
distinguishes between quantum and classical worlds. At present entanglement
between two systems, i.e. bipartite entanglement, is quite well understood,
but that between more than two systems, i.e. multipartite entanglement,
remains still far from being satisfactorily known. In spite of that,
multipartite entanglement has proven to play a superior role in recently
emerging fields of quantum information processing and quantum computing
since it exhibits a much richer structure than bipartite entanglement.
Motivation for studying multipartite entanglement arises from many reasons
some of which are listed now. First, multipartite entanglement provides a
unique means to check the Einstein locality without invoking statistical
arguments \cite{r2}, contrary to the case of Bell inequalities using bipartite
entanglement. Second, multipartite entanglement serves as a key ingredient
for quantum computing to achieve an exponential speedup over classical
computation \cite{r3}. Third, multipartite entanglement is central to
quantum error correction \cite{r4} where it is used to encode states, to
detect errors and, eventually, to allow fault-tolerant quantum computation
\cite{r5}, Fourth, multipartite entanglement helps to better characterize
the critical behavior of different many-body quantum systems giving rise to
a unified treatment of the quantum phase transitions \cite{r6}. Fifth,
multipartite entanglement is crucial also in condensed matter phenomena and
might solve some unresolved problems such as high-T superconductivity \cite
{r7}. Sixth, multipartite entanglement is recognized as a unreplaceable or
efficient resource to perform tasks involving a large number of parties such
as network teleportation \cite{r8}, quantum cryptography \cite{r9}, quantum
secret sharing \cite{r10}, remote entangling \cite{r11}, quantum
(tele)cloning \cite{r12}, quantum Byzantine agreement \cite{r13}, etc.
Finally, multipartite entanglement is conjectured to yield a wealth of
fascinating and unexplored physics \cite{r14}. Current research in
multipartite entanglement is progressing along two directions in parallel.
One direction deals with problems such as how to classify \cite{r15},
quantify \cite{r16}, generate/control/distill \cite{r17} and witness \cite
{r18} multipartite entanglement. The other direction proceeds to advance
various applications exploiting the nonclassical multiway correlation
inherent in multipartite entanglement \cite{r8,r9,r10,r11,r12,r13}.
Our work here
belongs to the second direction mentioned above. Namely, we propose
protocols to organize the so-called quantum exam which will be specified in
the next section. To meet the necessary confidentiality of the exam we use
suitable multipartite GHZ entangled states \cite{r2} as the quantum channel.
We consider two scenarios. One scenario is absolutely secure provided that
the participants share a prior proper multipartite entanglement. The other
scenario can be performed directly without any nonlocal quantum arrangements in the
past but it is only asymptotically secure. Both the scenarios are shown to
stand against commonly utilized eavesdropping attacks.
\vskip 0.5cm
\noindent \textbf{2. Quantum exam}
Exploiting the superdense coding feature possessed in bipartite entanglement
we have recently proposed a quantum dialogue scheme \cite{r19} (see also
\cite{r20}) allowing two legitimate parties to securely carry out their
conversation. In this work multipartite entanglement will be judiciously exploited to do
a more sophisticated task. Suppose that a teacher Alice wishes to organize
an important exam with her remotely separate students Bob $1$, Bob $2$,
..... and Bob $N.$ Alice gives her problem to all Bobs and, after some
predetermined period of time, asks each Bob to return a solution
independently. Alice's problem should be kept confidential from any
outsiders. The solution of a Bob should be accessible only to Alice but not
to anyone else including the $N-1$ remaining Bobs. Such confidentiality
constraints cannot be maintained even when Alice and Bobs are connected by
authentic classical channels because any classical communication could be
eavesdropped perfectly without a track left behind. However, combined with
appropriate quantum channels such an exam is accomplishable. We call it
quantum exam, i.e. an exam organized in a quantum way to guarantee the
required secrecy.
Let Alice's problem is a binary string
\begin{equation}
Q=\{q_{m}\} \label{Q}
\end{equation}
and the solution of a Bob is another string
\begin{equation}
R_{n}=\{r_{nm}\} \label{R}
\end{equation}
where $n=1,$ $2,$ $...,$ $N$ labels the Bob while $q_{m},$ $r_{nm}\in
\{0,1\} $ with $m=1,$ $2,$ $3,...$ denote a secret bit of Alice and a Bob.
\vskip 0.5cm
\noindent \textbf{2.1. Absolutely secure protocol}
An exam consists of two stages. In the first stage Alice gives a problem to
Bobs and in the second stage she collects Bobs' solutions.
\vskip 0.5cm
\noindent \textbf{The problem-giving process}
To securely transfer the problem from Alice to Bobs the following steps are
to be proceeded.
\begin{enumerate}
\item[a1)] Alice and Bobs share beforehand a large number of ordered
identical $(N+1)$-partite GHZ states in the form
\begin{equation}
\left| \Psi _{m}\right\rangle \equiv \left| \Psi \right\rangle
_{a_{m}1_{m}...N_{m}}=\frac{1}{\sqrt{2}}\left( \left| 00...0\right\rangle
_{a_{m}1_{m}...N_{m}}+\left| 11...1\right\rangle
_{a_{m}1_{m}...N_{m}}\right) \label{e1}
\end{equation}
of which qubits $a_{m}$ are with Alice and qubits $n_{m}$ with Bob $n.$
\item[a2)] For a given $m,$ Alice measures her qubit $a_{m}$ in the basis $
\mathcal{B}_{z}=\{\left| 0\right\rangle ,\left| 1\right\rangle \},$ then
asks Bobs to do so with their qubits $n_{m}.$ All the parties obtain the
same outcome $j_{m}^{z}$ where $j_{m}^{z}=0$ $(j_{m}^{z}=1)$ if they find $
\left| 0\right\rangle $ $(\left| 1\right\rangle ).$
\item[a3)] Alice publicly broadcasts the value $x_{m}=q_{m}\oplus j_{m}^{z}$
$(\oplus $ denotes an addition mod 2).
\item[a4)] Each Bob decodes Alice's secret bit as $q_{m}=x_{m}\oplus
j_{m}^{z}.$
\end{enumerate}
This problem-giving process is absolutely secure because $j_{m}^{z},$ for
each $m,$ takes on the value of either $0$ or $1$ with an equal probability
resulting in a truly random string $\{j_{m}^{z}\}$ which Alice uses as a
one-time-pad to encode her secret problem $\{q_{m}\}$ simultaneously for all
Bobs who also use $\{j_{m}^{z}\}$ to decode Alice's problem.
\vskip 0.5cm
\noindent \textbf{The solution-collecting process}
After a predetermined period of time depending on the problem difficulty
level Alice collects the solution from independent Bobs as follows.
\begin{enumerate}
\item[b1)] Alice and Bobs share beforehand a large number of ordered
nonidentical $(N+1)$-partite GHZ states in the form
\begin{equation}
\left| \Phi _{m}\right\rangle \equiv \left| \Phi \right\rangle
_{a_{m}1_{m}...N_{m}}=U_{m}\left| \Psi \right\rangle _{a_{m}1_{m}...N_{m}}
\label{e2}
\end{equation}
with
\begin{equation}
U_{m}=I_{a_{m}}\otimes u(s_{1_{m}})\otimes u(s_{2_{m}})\otimes ...\otimes
u(s_{N_{m}}) \label{U}
\end{equation}
where $I_{a_{m}}$ is the identity operator acting on qubit $a_{m}$ and
\begin{equation}
u(s_{n_{m}})=(\left| 0\right\rangle \left\langle 1\right| +\left|
1\right\rangle \left\langle 0\right| )^{s_{n_{m}}}
\label{u}
\end{equation}
is a unitary operator
acting on qubit $n_{m}.$ For each $n$ and $m,$ the value of $s_{n_{m}}$
chosen at random between $0$ and $1$ is known only to Alice but by no means
to any other person including Bobs. Qubits $a_{m}$ are with Alice and qubits
$n_{m}$ with Bob $n.$
\item[b2)] For a given $m,$ Alice measures her qubit $a_{m}$ in $\mathcal{B}
_{z}$ with the outcome $j_{a_{m}}^{z}=\{0,1\},$ then asks Bobs to do so with
their qubits $n_{m}$ with the outcome $j_{n_{m}}^{z}=\{0,1\}.$
\item[b3)] Each Bob $n$ publicly broadcasts the value $y_{nm}=r_{nm}\oplus
j_{n_{m}}^{z}.$
\item[b4)] Alice decodes the solution of Bob $n$ as $r_{nm}=y_{nm}\oplus
\left[ \delta _{0,s_{n_{m}}}j_{a_{m}}^{z}+\delta
_{1,s_{n_{m}}}(j_{a_{m}}^{z}\oplus 1)\right] .$
\end{enumerate}
In the solution-collecting process the outcomes $j_{a_{m}}^{z}$ and $
j_{n_{m}}^{z}$ are not the same anymore in general, but they are dynamically
correlated as $j_{n_{m}}^{z}=\delta _{0,s_{n_{m}}}j_{a_{m}}^{z}+\delta
_{1,s_{n_{m}}}(j_{a_{m}}^{z}\oplus 1).$ This correlation allows only Alice
who knows the value of $\{s_{n_{m}}\}$ to decode the solution of a Bob after
she obtains her own measurement outcome $j_{a_{m}}^{z}.$ As is clear, each of
the $N$ strings $\{j_{1_{m}}^{z}\},$ $\{j_{2_{m}}^{z}\},$
$...,\{j_{N_{m}}^{z}\}$ appears truly random and each such a string is used by
a Bob and Alice only one time to encode/decode a secret solution $\{r_{nm}\}.
$ The above solution-collecting process is therefore absolutely secure as
well.
The essential condition to ensure absolute security of the quantum exam is a
prior sharing of the entangled states $\{\left| \Psi _{m}\right\rangle \}$
and $\{\left| \Phi _{m}\right\rangle \}$ between the teacher Alice and the
students Bobs. It is therefore necessary to propose methods for multipartite
entanglement sharing.
\vskip 0.5cm
\noindent \textbf{The }$\left| \Psi _{m}\right\rangle $-\textbf{sharing
process}
Alice and Bobs can securely share the states $\{\left| \Psi
_{m}\right\rangle \}$ as follows.
\begin{enumerate}
\item[c1)] Alice generates a large enough number of identical states $
\left| \Psi _{m}\right\rangle $ defined in Eq. (\ref{e1}) \cite{n1}. For
each such state she keeps qubit $a_{m}$ and sends qubits $1_{m},$ $2_{m},$ $
...,$ $N_{m}$ to Bob $1,$ Bob $2,$ $...,$ Bob $N,$ respectively. Before
sending a qubit $n_{m}$ Alice authenticates Bob $n$ of that action.
\item[c2)] After receiving a qubit each Bob also authenticates Alice
independently.
\item[c3)] Alice selects at random a subset $\{\left| \Psi
_{l}\right\rangle \}$ out of the shared $\left| \Psi _{m}\right\rangle $
-states and lets Bobs know that subset. For each state of the subset Alice
measures her qubit randomly in $\mathcal{B}_{z}$ or in $\mathcal{B}
_{x}=\{\left| +\right\rangle ,\left| -\right\rangle \}$ with $\left| \pm
\right\rangle =(\left| 0\right\rangle \pm \left| 1\right\rangle )/\sqrt{2}\}.
$ Then she asks every Bob to measure their qubits in the same basis as hers.
Alice's (Bobs') outcome in $\mathcal{B}_{z}$ is $
j_{a_{l}}^{z}(j_{n_{l}}^{z})=\{0,1\}$ corresponding to finding $\{\left|
0\right\rangle ,\left| 1\right\rangle \}$ and that in $\mathcal{B}_{x}$ is $
j_{a_{l}}^{x}(j_{n_{l}}^{x})=\{+1,-1\}$ corresponding to finding $\{\left|
+\right\rangle ,\left| -\right\rangle \}.$
\item[c4)] Alice requires each Bob to publicly reveal the outcome of each
his measurement and makes an analysis. For those measurements in $\mathcal{B}
_{z}$ she compares $j_{a_{l}}^{z}$ with $j_{n_{l}}^{z}:$ if $
j_{a_{l}}^{z}=j_{n_{l}}^{z}$ $\forall n$ it is all-right, otherwise she
realizes a possible attack of an outsider Eve in the quantum channel. As for
measurements in $\mathcal{B}_{x}$ she compares $j_{a_{l}}^{x}$ with $
J_{l}^{x}=\prod_{n=1}^{N}j_{n_{l}}^{x}:$ if $j_{a_{l}}^{x}=J_{l}^{x}$ it is
all-right \cite{n2}, otherwise there is Eve in the line. If the error rate
exceeds a predetermined small value Alice tells Bobs to restart the whole
process, otherwise they record the order of the remaining shared $\left|
\Psi _{m}\right\rangle $-states and can use them for the problem-giving
process following the steps from a1) to a4).
\end{enumerate}
\noindent \textbf{The }$\left| \Phi _{m}\right\rangle $-\textbf{sharing
process}
The states $\{\left|\Phi_m\right\rangle \}$ cab be securely shared between
the participants as follows.
\begin{enumerate}
\item[d1)] Alice generates a large enough number of identical states $
\{\left| \Psi _{m}\right\rangle \}$ \cite{n1}. She then applies on each of
the identical states a unitary operator $U_{p}$ determined by Eq. (\ref{U})
to transform them into the $\left| \Phi _{p}\right\rangle $-states defined
in Eq. (\ref{e2}) which are nonidentical states \cite{n3}. Afterward, for
each $\left| \Phi _{p}\right\rangle ,$ she keeps qubit $a_{p}$ and sends
qubits $1_{p},$ $2_{p},$ $...,$ $N_{p}$ to Bob $1,$ Bob $2,$ $...,$ Bob $N,$
respectively. Before sending a qubit $n_{p}$ Alice authenticates Bob $n$ of
that action.
\item[d2)] After receiving a qubit each Bob also authenticates Alice
independently.
\item[d3)] Alice selects at random a large enough subset $\{\left| \Phi
_{l}\right\rangle \}$ out of the shared $\left| \Phi _{p}\right\rangle $
-states and lets Bobs know that subset. For each state $\left| \Phi
_{l}\right\rangle $ of the subset Alice measures her qubit randomly in
either $\mathcal{B}_{z}$ or $\mathcal{B}_{x},$ then asks Bobs to measure
their qubits in the same basis as hers.
\item[d4)] Alice requires each Bob to publicly reveal the outcome of each
his measurement and makes a proper analysis. For those measurements in $
\mathcal{B}_{z}$ she verifies the equalities $j_{a_{l}}^{z}=\delta
_{0,s_{n_{l}}}j_{n_{l}}^{z}+\delta _{1,s_{n_{l}}}(j_{n_{l}}^{z}\oplus 1).$
If the equalities hold for every $n$ it is all-right, otherwise the quantum
channel was attacked. As for measurements in $\mathcal{B}_{x}$ she compares $
j_{a_{l}}^{x}$ with $J_{l}^{x}=\prod_{n=1}^{N}j_{n_{l}}^{x}:$ if $
j_{a_{l}}^{x}=J_{l}^{x}$ it is all-right \cite{n4}, otherwise the quantum
channel was attacked. If the error rate exceeds a predetermined value Alice
tells Bobs to restart the whole process, otherwise they record the order of
the remaining shared $\left| \Phi _{p}\right\rangle $-states and can use
them for the solution-collecting process following the steps from b1) to b4).
\end{enumerate}
\noindent \textbf{Security of the entanglement-sharing process}
To gain useful information about the exam, Eve must attack the quantum
channel during the entanglement-sharing process. Below are several types of
attack that Eve commonly uses.
\textit{Measure-Resend Attack}. In $\mathcal{B}_{z}$ Eve measures the qubits
emerging from Alice and then resends them on to Bobs. After Eve's
measurement the entangled state collapses into a product state and her
attack is detectable when Alice and Bobs use $\mathcal{B}_{x}$ for a
security check \cite{n5}.
\textit{Disturbance Attack}. If Alice and Bobs check security only by
measurement outcomes in $\mathcal{B}_{x},$ then Eve, though cannot gain any
information, is able to make the protocol to be denial-of-service. Namely,
for each $n,$ on the way from Alice to Bob $n,$ Eve applies on qubit $n$ an
operator $u(v_{n_{m}})$ as defined in Eq. (\ref{u})
with $v_{n_{m}}$ randomly taken as either $0$ or $1,$
then lets the qubit go on its way. By doing so the disturbed states become
truly random and totally unknown to everybody, hence no cryptography is possible at all.
Though measurements in $\mathcal{B}_{x}$ cannot detect this type of attack
\cite{n6}, those in $\mathcal{B}_{z}$ can \cite{n7}.
\textit{Entangle-Measure Attack}. Eve may steal some information by
entangling her ancilla (prepared, say, in the state $\left| \chi
\right\rangle _{E})$ with a qubit $n$ (assumed to be in the state $\left|
i\right\rangle _{n})$ before the qubit reaches Bob $n:$ $\left| \chi
\right\rangle _{E}\left| i\right\rangle _{n}\rightarrow \alpha \left| \chi
_{i}\right\rangle _{E}\left| i\right\rangle _{n}+\beta \left| \overline{\chi
_{i}}\right\rangle _{E}\left| i\oplus 1\right\rangle _{n}$ where $|\alpha
|^{2}+|\beta |^{2}=1$ and $_{E}\left\langle \chi _{i}\right. \left|
\overline{\chi _{i}}\right\rangle _{E}=0.$ After Bob $n$ measures his qubit
Eve does so with her ancilla and thus can learn about the Bob's outcome.
Yet, with a probability of $|\beta |^{2}$ Eve finds $\left| \overline{\chi
_{i}}\right\rangle _{E}$ in which case she is detected if the security check
by Alice and Bobs is performed in $\mathcal{B}_{z}$ \cite{n8}.
\textit{Intercept-Resend Attack}. Eve may create her own entangled states $
\left| \Psi ^{\prime }\right\rangle _{a_{m}^{\prime }1_{m}^{\prime
}...N_{m}^{\prime }}$ $(\left| \Phi ^{\prime }\right\rangle _{a_{m}^{\prime
}1_{m}^{\prime }...N_{m}^{\prime }}=U_{m}^{\prime }\left| \Psi ^{\prime
}\right\rangle _{a_{m}^{\prime }1_{m}^{\prime }...N_{m}^{\prime }}$ where $
U_{m}^{\prime }=I_{a_{m}^{\prime }}\otimes u(s_{1_{m}}^{\prime })\otimes
u(s_{2_{m}}^{\prime })\otimes ...\otimes u(s_{N_{m}}^{\prime })$ with $
\{s_{n_{m}}^{\prime }\}$ an arbitrary random string). Then she keeps qubit $a_{m}^{\prime }
$ and sends qubit $n_{m}^{\prime }$ to Bob $n.$ When Alice sends qubits $
n_{m}$ to Bobs Eve captures and stores all of them. Subsequently, after
Alice's and Bobs' measurements, Eve also measures her qubits $a_{m}^{\prime }
$ and the qubits $n_{m}$ she has kept to learn the corresponding keys. This
attack is detected as well when Alice and Bobs use $\mathcal{B}_{z}$
-measurement outcomes for their security-check \cite{n9}.
\textit{Masquerading Attack}. Eve may pretend to be a Bob in the $\left|
\Psi _{m}\right\rangle $-sharing process to later obtain Alice's problem.
Likewise, she may pretend to be Alice in the $\left| \Phi _{m}^{\prime
}\right\rangle $-sharing process to later collect Bobs' solutions. Such
pretenses are excluded because each Bob after receiving a qubit has to
inform Alice and Alice before sending a qubit has also to inform all Bobs.
The classical communication channels Alice and Bobs possess have been
assumed highly authentic so that any disguisement must be disclosed.
\vskip 0.5cm
\noindent \textbf{2.2. Asymptotically secure protocol}
In some circumstances an urgent exam needs to be organized but no prior
quantum nonlocal arrangements are available at all. We now propose a
protocol to directly accomplish such an urgent task. At that aim, Alice has
to have at hand a large number of states $\{\left| \Psi _{m}\right\rangle \}$
and $\{\left| \Phi _{m}\right\rangle =U_{m}\left| \Psi _{m}\right\rangle \}.$
Let $M$ $(M^{\prime })$ be length of Alice's problem (Bobs' solution) and $T$
the time provided for Bobs to solve the problem.
\vskip 0.5cm
\noindent \textbf{The direct problem-giving process}
Alice can directly give her problem to Bobs by ``running'' the following
program.
\begin{enumerate}
\item[e1)] $m=0.$
\item[e2)] $m=m+1.$ Alice picks up a state $\left| \Psi _{m}\right\rangle ,$
keeps qubit $a_{m}$ and sends qubits $1_{m},$ $2_{m},$ $...,$ $N_{m}$ to Bob
$1,$ Bob $2,$ $...,$ Bob $N,$ respectively. Before doing so Alice informs
all Bobs via her authentic classical channels.
\item[e3)] Each Bob confirms receipt of a qubit via their authentic
classical channels.
\item[e4)] Alice switches between two operating modes: the control mode
(CM) with rate $c$ and the message mode (MM) with rate $1-c.$ Alice lets
Bobs know which operating mode she chose.
\begin{enumerate}
\item[e4.1)] If CM is chosen, Alice measures qubit $a_{m}$ randomly in $
\mathcal{B}_{z}$ or $\mathcal{B}_{x}$ with an outcome $j_{a_{m}}^{z}$ or $
j_{a_{m}}^{x},$ then lets Bobs know her basis choice and, asks them to
measure their qubits $n_{m}$ in the chosen basis. After measurements each
Bob publicly publishes his outcome $j_{n_{m}}^{z}$ or $j_{n_{m}}^{x}.$ Alice
analyzes the outcomes: if $
j_{a_{m}}^{z}=j_{1_{m}}^{z}=j_{2_{m}}^{z}=...=j_{N_{m}}^{z}$ or $
j_{a_{m}}^{x}=\prod_{n=1}^{N}j_{n_{m}}^{x}$ she sets $m=m-1$ and goes to
step e2) to continue, else she tells Bobs to reinitialize from the beginning
by going to step e1).
\item[e4.2)] If MM is chosen, Alice measures qubit $a_{m}$ in $\mathcal{B}
_{z}$ with an outcome $j_{a_{m}}^{z}$ and publicly reveals $
x_{m}=j_{a_{m}}^{z}\oplus q_{m}.$ Each Bob measures his qubit also in $
\mathcal{B}_{z}$ with an outcome $j_{n_{m}}^{z},$ then decodes Alice's
secret bit as $q_{m}=j_{n_{m}}^{z}\oplus x_{m}.$ If $m<M$ Alice goes to step
e2) to continue, else she publicly announces: \textit{``My problem has been
transferred successfully to all of you. Please return your solution after
time }$T".$
\end{enumerate}
\end{enumerate}
\noindent \textbf{The direct solution-collecting process}
After time $T$ Alice can directly collect Bobs' solutions by ``running''
another program as follows.
\begin{enumerate}
\item[g1)] $m=0.$
\item[g2)] $m=m+1.$ Alice picks up a $\left| \Phi _{m}\right\rangle ,$
keeps qubit $a_{m}$ and sends qubits $1_{m},$ $2_{m},$ $...,$ $N_{m}$ to Bob
$1,$ Bob $2,$ $...,$ Bob $N,$ respectively. Before doing so Alice informs
all Bobs via her authentic classical channels.
\item[g3)] Each Bob confirms receipt of a qubit via their authentic
classical channels.
\item[g4)] Alice switches between two operating modes: the CM with rate $c$
and the MM with rate $1-c.$ Alice lets Bobs know which operating mode she
chose.
\begin{itemize}
\item[g4.1)] If CM is chosen, Alice measures qubit $a_{m}$ randomly in $
\mathcal{B}_{z}$ or $\mathcal{B}_{x}$ with an outcome $j_{a_{m}}^{z}$ or $
j_{a_{m}}^{x},$ then lets Bobs know her basis choice and, asks them to
measure their qubits $n_{m}$ in the chosen basis. After measurements each
Bob publicly publishes his outcome $j_{n_{m}}^{z}$ or $j_{n_{m}}^{x}.$ Alice
analyzes the outcomes: if $j_{a_{m}}^{z}=\delta
_{0,s_{n_{m}}}j_{n_{m}}^{z}+\delta _{1,s_{n_{m}}}(j_{n_{m}}^{z}\oplus 1)$
for every $n$ or $j_{a_{m}}^{x}=\prod_{n=1}^{N}j_{n_{m}}^{x}$ she sets $m=m-1
$ and goes to step g2) to continue, else she tells Bobs to reinitialize from
the beginning by going to step g1).
\item[g4.2)] If MM is chosen, Alice measures qubit $a_{m}$ in $\mathcal{B}
_{z}$ with an outcome $j_{a_{m}}^{z}$ and each Bob measures his qubit also
in $\mathcal{B}_{z}$ with an outcome $j_{n_{m}}^{z}.$ Each Bob publicly
reveals $y_{nm}=r_{nm}\oplus j_{n_{m}}^{z}$ and Alice decodes Bobs' secret
bits as $r_{nm}=y_{nm}\oplus \left[ \delta
_{0,s_{n_{m}}}j_{a_{m}}^{z}+\delta _{1,s_{n_{m}}}(j_{a_{m}}^{z}\oplus
1)\right] $ for $n=1,2,...,N.$ If $m<M^{\prime }$ Alice goes to step g2) to
continue, else she publicly announces: \textit{``Your solutions have been
collected successfully''}.
\end{itemize}
\end{enumerate}
As described above, in the direct problem-giving (solution-collecting)
process Alice alternatively gives (collects) secret bits and checks Eve's
eavesdropping. These direct protocols also stand against the types of attack
mentioned above. The protocols terminate immediately whenever Eve is
detected in a control mode. However, Eve might get a partial information
before her tampering is disclosed. Such an information leakage can be
reduced as much as Alice wants by increasing the control mode rate $c$ at
the expense of reducing the information transmission rate $r=1-c.$ For short
strings $Q$ and $R_{n}$ (see Eq. (\ref{Q}) and Eq. (\ref{R})) Eve's
detection probability may be quite small. But, the longer the strings the
higher the detection probability. In the long-string limit the detection
probability approaches one, i.e. Eve is inevitably detected. In this sense,
the direct quantum exam protocols are asymptotically secure only.
\vskip 0.5cm
\noindent \textbf{3. Conclusion}
We have proposed two protocols for organizing a quantum exam \cite{n10}
basing on a judicious use of appropriate multipartite entangled states. The
first protocol is absolutely secure iff the participants have successfully
shared the necessary entanglement in advance. We also provide methods for
sharing the multipartite entanglement in the presence of a potential
eavesdropping outsider. The second protocol can be processed directly
without a prior entanglement sharing. This advantage is however compromised
by a lower confidentiality level or by a slower information transmission
rate. Both the protocols have been shown to sustain various kinds of attacks
such as measure-resend attack, disturbance attack, entangle-measure attack,
intercept-resend attack and masquerading attack. Our protocols work well in
an idealized situation with perfect entanglement sources/measuring devices
and in noiseless quantum channels which we have assumed for simplicity. We
are planning to further develop our protocols to cope with more realistic
situations.
\vskip 0.5cm
\noindent \textbf{Acknowledgments.}
The author is grateful to Professor
Hai-Woong Lee from KAIST for useful discussion and comments. This research
was supported by a Grant (TRQCQ) from the Ministry of Science and Technology
of Korea and also by a KIAS R\&D Fund No 6G014904.
\end{document} |
\begin{document}
\begin{abstract}
We consider a contact manifold with a pseudo-Riemannian metric and
define a contact vector field intrinsically associated to this pair of structures.
We call this new differential invariant the contact Riemannian curl.
On a Riemannian manifold, Killing vector fields are those that annihilate the metric;
a Killing $1$-form is obtained from a Killing vector field by lowering indices.
We show that the contact Riemannian curl vanishes
if the metric is of constant curvature and the contact structure is defined by a Killing $1$-form.
We also show that the contact Riemannian curl has
a strong similarity with the Schwarzian derivative since it depends
only on the projective equivalence class of the metric.
For the Laplace-Beltrami operator on a contact manifold,
the contact Riemannian curl is proportional to the subsymbol defined
in~arXiv:1205.6562.
We also show that the contact Riemannian curl vanishes
on the (co)tangent bundle over a Riemannian manifold.
This implies that the corresponding subsymbol of the Laplace-Beltrami operator is identically zero.
\end{abstract}
\keywords{Contact geometry, Riemannian geometry, differential invariants}
\maketitle
\section{Introduction} \langlembdaabel{Intro}
The principal object of this paper is related to
the notion of {\it invariant differential operator}, i.e., an
operator commuting with the action of the group of diffeomorphisms.
The notion of {\it differential invariant} is one of the oldest notions
of differential geometry.
The best known example is perhaps the curvature in all its avatars.
The topic to which the present work belongs was initiated by Veblen~\cite{Veb} who started a
systematic study of invariant differential operators on smooth manifolds.
The theory was intensively studied in the 80's in the context of
Gelfand-Fuchs cohomology; see~\cite{Fu86,GLS} and references
therein.
We consider a smooth manifold $M$ equipped simultaneously with
a contact structure and a pseudo-Riemannian metric.
We present a construction of a contact vector field
corresponding to these two structures;
we call this vector field the {\it contact Riemannian curl}.
Our construction is coordinate free and invariant
with respect to the action of the group of contact diffeomorphisms,
i.e., the contact Riemannian curl is a differential invariant.
Moreover, our goal is to define this differential invariant in a ``most symmetric'' way,
so that it is also invariant
with respect to natural equivalence relations.
One of the equivalence relations we consider is as follows.
Two metrics are called {\it projectively equivalent}
(or geodesically equivalent) if they have the same non-parametrized geodesics.
i.e., their Levi-Civita connections are projectively equivalent.
It turns out that the constructed contact Riemannian curl
is obtained as contraction of the metric with
a certain tensor field invariant with respect to this equivalence relation.
This implies, in particular, that the contact Riemannian curl of the pair
(a metric of constant scalar curvature, contact structure defined by a Killing $1$-form)
vanishes.
Projective invariance makes the notion of contact Riemannian curl
quite similar to that of classical {\it Schwarzian derivative}
(for various multi-dimensional generalizations
of the Schwarzian derivative see~\cite{BO,OT,B,OT1} and references therein).
We investigate this relation in more details.
Among the main properties of the contact Riemannian curl that we investigate,
there is its relation to the Laplace-Beltrami operator.
Differential operators on contact manifolds
were studied from the geometric point of view in a recent work \cite{CO12},
where the notion of {\it subsymbol} of a differential operator on a contact manifold
was introduced.
The subsymbol of a differential operator is a tensor field of degree lower
than that of the principal symbol.
Note that the subsymbol is not well-defined for an arbitrary manifold,
one needs a contact structure to obtain an invariant definition.
For a given second order differential operator, the subsymbol is just a contact vector field.
In the present paper, we consider the {\it Laplace-Beltrami operator}
associated to an arbitrary metric on a contact manifold
and calculate its subsymbol.
It turns out that this subsymbol is proportional to the contact Riemannian curl.
We also apply our general construction to a particularly
interesting example of a manifold that has natural
contact and Riemannian structures,
namely to the spherical
(or projectivized) cotangent bundle $ST^*M$
over a Riemannian manifold $(M,g)$.
The manifold $ST^*M$ is equipped with the canonical
lift of the metric $g$.
We show that the contact Riemannian curl,
and therefore the subsymbol of the Laplace-Beltrami operator,
is identically zero in this case.
Let us mention that
the projectivization of the cotangent bundle over a Riemannian manifold $M$,
as well as the sphere bundle $ST^*M$,
is an example of a ``real-complex'' manifold whose local invariants
were recently introduced and computed in~\cite{BGLS}.
At the end of the paper, we provide
several concrete examples of the contact Riemannian curls.
For instance, we calculate it for
the $3{representationm D}$-ellipsoid equipped with the conformally flat metric
introduced in \cite{Tab} and intensively used in~\cite{MT,DV}.
We believe that the differential invariants of a pair
(a Riemannian metric, a contact structure) is worth a
systematic study.
\section{Contact geometry and tensor fields} \langlembdaabel{CODG}
Contact geometry is an old classical subject,
that can be viewed as an odd-dimensional version of symplectic geometry.
Let $M$ be a contact manifold and ${\frak i}m(M)= 2\ell + 1$,
we will always assume that $\ell\geq1$.
Unlike a symplectic structure in symplectic geometry, a contact structure on $M$
is defined by a differential $1$-form $\tildeheta$, called a {\it contact form},
determined up to a factor (a function),
and such that $d\tildeheta$ is a $2$-form of rank $2\ell$.
It is important that a contact form is not intrinsically associated with
a contact structure.
A contact diffeomorphism (a contact vector field) is a diffeomorphism
(a vector field) preserving the contact structure.
It preserves a given contact form conformally, up to a factor.
The space of all contact vector fields can be identified
with the space of smooth functions, but
this correspondence depends on the choice of a contact form; see~\cite{Arn}.
In this section, we recall several standard facts of contact geometry ---
those of contact structure and contact vector fields ---
using somewhat unconventional notation of~\cite{OT,Ovs}
which are among our main references.
We show that the contact structure can be also described by a special tensor field,
which is a {\it weighted contact form}.
Contact vector fields are in one-to-one correspondence with
weighted densities of weight $-\frac{1}{\ell+1}$.
\subsection{Weighted densities}
A weighted density is a standard object in differential geometry.
In~order to make the definitions intrinsic, we recall here this notion.
Let $M$ be a manifold of dimension $n$.
For any $\langlembda\in\Bbb R$, we denote by $({\cal L}ambda^n T^*M)^{\overlinetimes\langlembda}$
the line bundle of homogeneous complex valued functions of weight $\langlembda$
on the determinant bundle ${\cal L}ambda^n TM$.
The space ${\cal F}_\langlembda(M)$ of smooth sections of
$({\cal L}ambda^n T^*M)^{\overlinetimes\langlembda}$ with complex coefficients
is called the space of {\it weighted densities} of weight~$\langlembdaambda$,
(or $\langlembda$-{\it densities} for short).
\begin{ex}
{representationm
If the manifold $M$ is orientable and
if $\omega$ is a volume form with constant coefficients, then $\phi\,\omega^\langlembda$,
where $\phi\in{}C^\infty(M)$, is a $\langlembda$-density.
}
\end{ex}
The space ${\cal F}_\langlembda(M)$ has the structure of a module over the Lie algebra
${\cal V}ect(M)$ of all smooth vector fields on $M$.
We denote by $\mathrm{Div}$ the divergence operator
associated with a volume form $\omega$ on $M$.
That is, $L_X(\omega)=\mathrm{Div}(X)\,\omega$.
The action of a vector fields reads as follows:
\begin{equation}
\langlembdaabel{LieEq}
L_X(\phi\,\omega^\langlembda)=
\langlembdaeft(X(\phi)+\langlembda\mathrm{Div}(X) \phi
representationight)\omega^\langlembda,
\end{equation}
for every vector field $X$ and $\phi \in C^\infty(M)$.
\subsection{Contact manifolds}
A smooth manifold $M$ is called {\it contact} if it is equipped with
a completely non-integrable distribution
$$
\mathcal{D}\subset{}TM
$$
of codimension~$1$. The distribution $\mathcal{D}$ is called a {\it
contact distribution}; the hyperplane $\mathcal{D}_x\subset{}T_xM$
is called a {\it contact hyperplane} for every point $x\in{}M$.
A contact structure on $M$ exists only if ${\frak i}m M=2\ell+1>1$.
A usual way to define a contact structure is to chose
a (locally defined) differential $1$-form $\tildeheta$ on~$M$ such that
$\mathcal{D}=\ker\tildeheta$.
Such a $1$-form is called a {\it contact form}. The complete
non-integrability of the distribution $\mathcal{D}$ is equivalent
to the fact that
\begin{equation}
\langlembdaabel{TheVol}
\mathrm{vol}:=\tildeheta\wedge(d\tildeheta)^\ell
\end{equation}
is a (locally defined) volume form;
equivalently, the 2-form $d\tildeheta$ is a non-degenerate on the contact hyperplanes
$\mathcal{D}_x$ of $\mathcal{D}$.
However, there is no canonical choice of a contact form.
A diffeomorphism $f:M\tildeo{}M$ is a {\it contact diffeomorphism} if $f$
preserves $\mathcal{D}$.
If $\tildeheta$ is a contact form corresponding to the contact distribution $\mathcal{D}$
and $f$ is a contact diffeomorphism, then $f$ does not necessarily preserve $\tildeheta$,
more precisely, $f^*\tildeheta=F_f\tildeheta$, where $F_f$ is a function.
We refer to \cite{Arn,Bla} for excellent textbooks on contact geometry.
\subsection{The contact tensor}
We will be using the notion of a (generalized) tensor field
that was suggested in \cite{BL} and goes back to ideas of I. M. Gelfand.
Besides the standard tensor fields, i.e., sections of the bundles\footnote{
Throughout this paper, the tensor product is performed over $C^\infty(M)$.}
$(TM)^p\overlinetimes(T^*M)^q$, it is often useful to consider
{\it weighted} tensor fields that are sections of the bundles
$$
(TM)^p\overlinetimes(T^*M)^q\overlinetimes({\cal L}ambda^n T^*M)^{\overlinetimes\langlembda}.
$$
A wealth of examples of such generalized tensor fields
can be found in \cite{Fu86,OT}.
We are ready to introduce the main notion of this section.
\begin{defi}
\langlembdaabel{ConTD}
{representationm
Given a contact form $\tildeheta$, let
the {\it contact tensor field} be
\begin{equation}
\langlembdaabel{CT}
{\cal T}heta:=\tildeheta\overlinetimes\mathrm{vol}^{-\frac{1}{\ell+1}},
\end{equation}
where $\mathrm{vol}$ is as in Eq. (representationef{TheVol}).
}
\end{defi}
\begin{prop}
\langlembdaabel{IPro}
The tensor field ${\cal T}heta$ is globally defined on a contact manifold $M$,
it is independent of the choice of a contact form,
and it is invariant with respect to the contact diffeomorphisms.
\end{prop}
\begin{proof}
Let $F$ be a non-vanishing function and consider the contact form $F\tildeheta$.
The corresponding volume form is
$F\tildeheta\wedge\langlembdaeft(d(F\tildeheta)representationight)^\ell=F^{\ell+1}\tildeheta\wedge(d\tildeheta)^\ell$.
Therefore, the contact tensor fields defined by Eq.~(representationef{CT}),
corresponding to the contact forms
$\tildeheta$ and $F\tildeheta$, coincide.
Hence, ${\cal T}heta$ is globally defined and invariant with respect to
contact diffeomorphisms.
\end{proof}
A contact structure is intrinsically defined by the corresponding contact tensor.
\begin{ex}
\langlembdaabel{Dabex}
{representationm
Local coordinates $(x^1,\langlembdadots,x^\ell,y^1,\langlembdadots,y^\ell,z)$
on $M$ are often called the {\it Darboux coordinates} if the
contact structure can be represented by the 1-form
\[
\tildeheta_{\mathrm{Dar}}=dz+\frac{1}{2}\,\sum_{i=1}^\ell \langlembdaeft (x^idy^i-y^idx^i representationight ).
\]
The corresponding volume form is then the standard one:
$$
\mathrm{vol}=
(-1)^{\frac{\ell(\ell-1)}{2}}\,\ell!\,
dx^1\wedge\cdots\wedge{}dx^\ell\wedge{}dy^1\wedge\cdots{}dy^\ell\wedge{}dz.
$$
A contact structure has no local invariants, therefore
Darboux coordinates always exist in the vicinity of every point;
see~\cite{Arn} (and~\cite{GL} for a simple algebraic proof).
}
\end{ex}
\subsection{Contact vector fields}\langlembdaabel{CoHSect}
A {\it contact vector field} on a contact manifold $M$ is a vector field
that preserves the contact distribution.
This is usually expressed in terms of contact forms:
a vector field $X$ is contact if, for every contact form $\tildeheta$, the Lie derivative
$L_X\tildeheta$ is proportional to $\tildeheta$:
\begin{equation}
\langlembdaabel{DivEq}
L_X\tildeheta = {\tildes \frac{1}{\ell+1}}{\cal D}iv(X) \tildeheta.
\end{equation}
In terms of the contact tensor (representationef{CT}), we have the following
corollary of Proposition~representationef{IPro}.
\begin{cor}
\langlembdaabel{ContV}
A vector field $X$ is contact if and only if it preserves the contact tensor:
$$
L_X{\cal T}heta=0.
$$
\end{cor}
Let ${\mathcal K}(M)$ denote the space of all smooth contact vector fields on $M$.
This space has a Lie algebra structure, it is also a module over the group
of contact diffeomorphisms.
The following observation can be found in~\cite{Ovs,CO12}.
\begin{prop}
\langlembdaabel{VecPro}
As a module over the group of contact diffeomorphisms,
the space ${\mathcal K}(M)$ is isomorphic to the space of weighted densities
${\mathcal F}_{-\frac{1}{\ell+1}}(M)$.
\end{prop}
\begin{proof}
The space of contact forms is isomorphic to ${\mathcal F}_{\frac{1}{\ell+1}}(M)$.
Indeed, this follows from Proposition~representationef{IPro} and from Eq.~(representationef{DivEq}).
The statement then follows from the fact that
there is a natural $C^\infty(M)$-valued pairing between the spaces of contact vector fields
and of contact forms:
$
(X,\,\tildeheta)\mapsto\tildeheta(X).
$
\end{proof}
\begin{rem}
{representationm
The above proposition means that, unlike the symplectic geometry,
the notion of contact generating function (or ``contact Hamiltonian function'')
should be understood as a weighted density
and not as a function.
However, in the Darboux coordinates, the correspondence
between the elements of ${\mathcal K}(M)$ and ${\mathcal F}_{-\frac{1}{\ell+1}}(M)$
becomes the usual correspondence between
contact vector fields and functions (see~\cite{Arn}):
$$
X_{\phi\,\overlinemega^{-\frac{1}{\ell+1}}}=
\sum_{i=1}^\ell\langlembdaeft(
\partial_{x^i}(\phi)\,\partial_{y^i}-\partial_{y^i}(\phi)\,\partial_{x^i}
representationight)
\tildeextstyle
+\frac{1}{2}\,\partial_{z}(\phi)\,{\mathcal E}
+\langlembdaeft(\phi-\frac{1}{2}\,{\mathcal E}(\phi)representationight)\partial_{z},
$$
where
$$
{\mathcal E}=
\sum_{i=1}^\ell
\langlembdaeft(
x^i\partial_{x^i}+y^i\partial_{y^i}
representationight)
$$
is the Euler vector field.
}
\end{rem}
\begin{ex}
{representationm
If ${\frak i}m{M}=3$, contact vector fields are
identified with $-\frac{1}{2}$-densities;
if ${\frak i}m{M}=5$, then ${\mathcal K}(M)\cong{\mathcal F}_{-\frac{1}{3}}(M)$, etc.
Note also that, in the one-dimensional case,
every vector field is contact,
one then has ${\cal V}ect(M)\cong{\mathcal F}_{-1}(M)$.
}
\end{ex}
\subsection{Another definition of weighted densities on contact manifolds}\langlembdaabel{Mitia}
In presence of a contact structure defined by a contact form
$\tildeheta$, it is natural to express elements of any rank 1 bundle,
for example, weighted densities, in terms of powers of $\tildeheta$:
$$
\phi\,\mathrm{vol}^{\frac{\langlembda}{\ell+1}}\langlembdaongleftrightarrow\phi\tildeheta^\langlembda,
$$
where as above $\phi$ is a smooth function.
The notation $\phi\tildeheta^\langlembda$ is adopted in many works by physicists
(see also~\cite{Ovs90,GLS01}).
In this notation, many formulas simplify.
For instance, if $X$ is a contact vector field,
then the corresponding contact Hamiltonian is
$\phi\tildeheta^{-1}$, where the function $\phi$
is simply the evaluation $\phi=\tildeheta(X)$.
\subsection{The Poisson algebra of weighted densities}
The space ${\cal F}(M)=\bigoplus_\langlembda{\cal F}_\langlembda(M)$
of all weighted densities on a contact manifold $M$ can be endowed with a structure of a Poisson algebra
(see~\cite{Arn,OT,Ovs}):
$$
\{.,.\}:{\mathcal F}_\langlembda(M)\tildeimes{\mathcal F}_{\mu}(M)\tildeo{\mathcal F}_{\langlembda+\mu+\frac{1}{\ell+1}}(M).
$$
The explicit formula in Darboux coordinates is as follows:
$$
\langlembdaeft\{\phi\,\overlinemega^\langlembda,\psi\,\overlinemega^{\mu}representationight\}=
\langlembdaeft(
\sum_{i=1}^n(\partial_{x^i}\phi\,\partial_{y^i}\psi -
\partial_{x^i}\psi\,\partial_{y^i}\phi)
+\partial_{z}\phi\langlembdaeft(\mu\psi+{\mathcal E}\psirepresentationight)
-\partial_{z}\psi\langlembdaeft(\langlembda\phi+{\mathcal E}\psirepresentationight)
representationight)\overlinemega^{\langlembda+\mu+\frac{1}{\ell+1}}.
$$
The subspace ${\mathcal F}_{-\frac{1}{\ell+1}}(M)$ is a Lie subalgebra of ${\mathcal F}$
isomorphic to ${\mathcal K}(M)$.
The Poisson bracket of $-\frac{1}{\ell+1}$-densities precisely corresponds
to the Lie derivative:
$$
X_{\langlembdaeft\{{\cal P}hi,{\cal P}sirepresentationight\}}=
L_{X_{{\cal P}hi}}
\langlembdaeft({\cal P}sirepresentationight),
$$
where ${\cal P}hi=\phi\,\overlinemega^{-\frac{1}{\ell+1}},\,{\cal P}si=\psi\,\overlinemega^{-\frac{1}{\ell+1}}$.
\subsection{The invariant splitting}
The full space of vector fields $\mathrm{Vect}(M)$ splits into direct sum
$$
\mathrm{Vect}(M)=
{\mathcal K}(M)\overlineplus{\mathcal Tan}(M),
$$
where ${\mathcal Tan}(M)$ is the space of vector fields tangent
to the contact distribution,
i.e., $\tildeheta(Y)=0$ for every contact form $\tildeheta$ and every $Y\in{\mathcal Tan}(M)$.
Such vector fields are called {\it tangent vector fields}.
Unlike ${\mathcal K}(M)$, the space ${\mathcal Tan}(M)$ is not a Lie
algebra, but a ${\mathcal K}(M)$-module.
The above splitting is invariant with respect to the group of contact diffeomorphisms.
In particular, there is an invariant projection
\begin{equation}
\langlembdaabel{PiPr}
\pi:\mathrm{Vect}(M)\tildeo{\mathcal K}(M),
\end{equation}
that will be very useful.
\section{The contact Riemannian curl and its properties} \langlembdaabel{Definitions}
In this section, we introduce our main notion, a contact vector field
corresponding to a metric and a contact structure.
We also study its main properties, such as projective invariance
and relation to the multi-dimensional Schwarzian derivative.
\subsection{Covariant derivative}
Let us assume now that $M$ is endowed with a pseudo-Riemannian
metric $g$. We denote the Levi-Civita connection on $M$ by
$\nabla$, and the Christoffel symbols by ${\cal G}amma_{ij}^k$. The {\it
covariant derivative}, also denoted by $\nabla$, is the linear map
that can be defined for arbitrary space of tensor fields, ${\mathcal
T}(M)$:
$$
\nabla:{\mathcal T}(M)\tildeo{\cal O}mega^1(M)\overlinetimes{\mathcal T}(M),
$$
such that
$\nabla(fm)=df\overlinetimes m+f\overlinetimes\nabla(m)$ for any $f\in{}C^\infty(M)$ and $m\in T(M)$.
It is written in the form $\nabla(t)=\nabla_i(t)\,dx^i$,
and therefore it suffices to define the partial derivatives~$\nabla_i$.
Here and below summation over repeated indices
(one upper, the other one lower) is understood (Einstein's notation);
see~\cite{DNF}.
The covariant derivative
of vector fields and differential $1$-forms
is given, in local coordinates, by the well-known formulas
$$
\nabla_i \langlembdaeft(V^j\partial_jrepresentationight)=
\langlembdaeft(\partial_iV^j+{\cal G}amma_{ik}^jV^krepresentationight)\partial_j,
\qquad
\nabla_i \langlembdaeft(\beta_jdx^jrepresentationight)=
\langlembdaeft(\partial_i\beta_j-{\cal G}amma_{ij}^k\beta_krepresentationight)dx^j,
$$
respectively,
where $\partial_i=\partial/\partial{}x^i$.
The covariant derivative then extended to every tensor fields by Leibniz rule.
For instance, the covariant derivative of weighted densities is
defined in local coordinates by the following formula:
\[
\nabla_i \langlembdaeft(
\phi\,\overlinemega^\langlembda
representationight)=
\langlembdaeft(\partial_i \phi-\langlembda {\cal G}amma_{ij}^j\phirepresentationight)\overlinemega^\langlembda,
\]
that we will extensively use throughout the paper.
\subsection{The main definition}
Let us introduce the main notion of this paper.
Recall that the contact tensor field ${\cal T}heta$ was introduced in
Definition~representationef{ConTD}.
\begin{defi}
\langlembdaabel{MainDef}
{representationm
(a)
For every pseudo-Riemannian metric $g$ on a contact manifold $M$,
we define a weighted density of degree $-\frac{1}{\ell+1}$:
\begin{equation}
\langlembdaabel{maindefi}
A_{g,{\cal T}heta}:=
\langlembdaeft\langlembdaangle
{g},\,\nabla{\cal T}heta
representationightrepresentationanglengle,
\end{equation}
in local coordinates, $A_{g,{\cal T}heta}:=
{g}^{ij}\nabla_i{\cal T}heta_j$.
(b)
We call the contact vector field $X_{A_{g,{\cal T}heta}}$
with contact Hamiltonian $A_{g,{\cal T}heta}$
the {\it contact Riemannian curl of $g$}.
}
\end{defi}
Note that the quantity $A_{g,{\cal T}heta}$ is, indeed, a weighted density
of degree $-\frac{1}{\ell+1}$, so that it has a meaning
of contact Hamiltonian; see Proposition~representationef{VecPro}.
\begin{rem}
{representationm
The tensor field $\nabla{\cal T}heta$ is also a differential invariant
(that actually contains even more information than $A_{g,{\cal T}heta}$).
One can obtain a $-\frac{1}{\ell+1}$-density out of $\nabla{\cal T}heta$
by contracting with an arbitrary metric, not necessarily with $g$ itself.
}
\end{rem}
It will be useful to have an explicit expression
for $A_{g,{\cal T}heta}$ (and of $\nabla{\cal T}heta$) in local coordinates.
\begin{prop}
\langlembdaabel{LoP}
In local coordinates, such that ${\cal T}heta=\tildeheta\overlinetimes\mathrm{vol}^{-\frac{1}{\ell+1}}$,
one has
\begin{equation}
\langlembdaabel{ProCurl}
A_{g,{\cal T}heta}=
g^{ij}
\langlembdaeft (
\partial_i \tildeheta_j-
\langlembdaeft ({\cal G}amma^k_{ij}-\frac{1}{2(\ell+1)}
\langlembdaeft(\delta^k_i{\cal G}amma_{jr}^r+\delta^k_j{\cal G}amma_{ir}^rrepresentationight )representationight )\tildeheta_krepresentationight)
\mathrm{vol}^{-\frac{1}{\ell+1}},
\end{equation}
where $\delta^k_i$ is the Kronecker symbol.
\end{prop}
\begin{proof}
This can be obtained directly from Definition representationef{MainDef}
and the expression of the covariant derivative of a weighted density.
\end{proof}
\begin{rem}
{representationm
It follows from the intrinsic definition (representationef{maindefi}) that
the local expression (representationef{ProCurl}) is actually invariant
with respect to the action of the group of contact diffeomorphisms.
The formula (representationef{ProCurl}) remains unchanged for any choice
of local coordinates.
It is also independent of the choice of the contact form.
}
\end{rem}
\subsection{Projective invariance of $\nabla{\cal T}heta$}
Let us recall a fundamental notion of projectively equivalent connections
due to Cartan~\cite{Car}.
A {\it projective connection} is an equivalence class of symmetric
affine connections giving the same non-parameterized geodesics.
The {\it symbol of a projective connection} is
given by the expression
\[
\tildeextstyle
{\cal P}i_{ij}^k:={\cal G}amma_{ij}^k-\frac{1}{n+1}\langlembdaeft
(\delta_i^k {\cal G}amma_{lj}^l+\delta_j^k {\cal G}amma_{il}^lrepresentationight ),
\]
where $n$ is the dimension; see~\cite{KN}.
Note that in our case, $n=2\ell+1$.
The simplest properties of a projective connection are as the following.
\begin{enumerate}
\item
Two affine connections, $\nabla$ and $\tildeilde \nabla$, are projectively
equivalent if and only if ${\cal P}i_{ij}^k=\tildeilde{\cal P}i_{ij}^k$.
\item
Equivalently, $\nabla$ and $\tildeilde \nabla$ are projectively
equivalent if and only if there exists a 1-form $\beta$ such that
\begin{equation*}
\langlembdaabel{assoc}
\tildeilde
{\cal G}amma_{ij}^k={\cal G}amma_{ij}^k+\delta_{j}^k\,\beta_i+\delta_{i}^k\,\beta_j.
\end{equation*}
\end{enumerate}
The following statement makes the contact Riemannian curl
somewhat similar to the Schwarzian derivative.
\begin{thm}
\langlembdaabel{CoCuProj}
If $g$ and $\tildeilde g$ are two metrics
whose Levi-Civita connections are projectively equivalent,
then $\nabla{\cal T}heta=\tildeilde \nabla{\cal T}heta$.
\end{thm}
\begin{proof}
The coordinate formula for $\nabla{\cal T}heta$
can be written as follows:
$$
\langlembdaeft(\nabla{\cal T}hetarepresentationight)_{ij}=
\langlembdaeft (
\partial_i \tildeheta_j-
{\cal P}i^k_{ij}\,\tildeheta_krepresentationight) \mathrm{vol}^{-\frac{1}{\ell+1}},
$$
see (representationef{ProCurl}).
This expression depends only on the projective class of the Levi-Civita connection
and implies projective invariance.
\end{proof}
Let $[g]$ denote the class of geodesically equivalent metrics,
let $[\nabla]$ denote the corresponding projective connection.
The above theorem means that the tensor $\nabla{\cal T}heta$ actually depends
only on $[g]$ and not on the metric itself.
\begin{rem}
{representationm
Geodesically equivalent metrics is a very classical subject of
Riemannian geometry that goes back to Beltrami, Levi-Civita, Weyl, and Cartan.
We refer to the classical book~\cite{Eis} for a survey.
The subject is still very active, see~\cite{BKM} and references therein.
}
\end{rem}
\subsection{Projectively flat connections,
metrics of constant curvature and Killing contact forms}
It is now natural to investigate projectively flat case.
A connection $\nabla$ on $M$ is called \tildeextit{projectively flat} if, in a
neighborhood of every point, there exist local coordinates,
often called {\it adapted coordinates}, such
that the geodesics are straight lines in these coordinates.
If a connection is projectively flat, then ${\cal P}i_{ij}^k\equiv0$
in any system of adapted coordinates.
Note also that projectively flat connections
admit a (local) action of the group $\mathrm{SL}(n+1,\Bbb R)$,
in other words, adapted coordinates admit linear-fractional
changes.
The classical Beltrami theorem states that {\sl the Levi-Civita connection
of a Riemannian metric is projectively flat
if and only if the metric has a constant sectional curvature}.
This fact allows us to obtain an important consequence of Theorem~representationef{CoCuProj}.
Let us recall the notion of Killing differential forms that goes back to Yano~\cite{Yan}.
A $1$-form $\beta=\beta_i(x)dx^i$ is said to be a {\it Killing form} if
$$
\nabla_i \beta_j+\nabla_j \beta_i=0.
$$
Recall also a more common notion of Killing
vector field.
A vector field $V=V^i(x)\partial_i$ is said to be a {\it Killing vector field} if
$$
L_Vg=0
$$
Every Killing $1$-form can be obtained from a Killing
vector field by lowering indices: $\beta=\langlembdaangle g,Vrepresentationanglengle$;
i.e., $\beta_i=g_{ij}V^j$ in local coordinates.
\begin{cor}
\langlembdaabel{CoCu}
If $g$ is a metric of constant sectional curvature
and if the contact structure is defined by a contact $1$-form $\tildeheta$
which is a Killing form with respect to a metric from the projective class $[g]$,
then $ A_{g,{\cal T}heta}=0$.
\end{cor}
\begin{proof}
Since the Levi-Civita connection
corresponding to $g$ is projectively flat,
there exist local coordinates for which
${\cal P}i_{ij}^k\equiv0$, and therefore
$$
A_{g,{\cal T}heta}=g^{ij}\partial_i \tildeheta_j.
$$
If, furthermore,
$
\partial_i \tildeheta_j+\partial_j \tildeheta_i=0
$
for all $i,j$, then $A_{g,{\cal T}heta}$ vanishes identically
since the tensor $g^{ij}$ is symmetric.
The equation $\partial_i \tildeheta_j+\partial_j \tildeheta_i=0$
means that~$\tildeheta$ is a Killing form with respect to
the flat metric which is projectively equivalent to $g$.
The corollary then follows from Theorem representationef{CoCuProj}.
\end{proof}
\begin{ex}
{representationm
The Darboux form in Example representationef{Dabex} is a Killing form with respect
to the flat metric.
Note that in other works, especially in those on analytical mechanics, another
local normal form of the contact form is often used:
$dz+\sum_{1\langlembdaeq{}i\langlembdaeq{}\ell}x^idy^i$.
(Over fields of characteristic $2$, only this latter form can be used, see \cite{Leb}.)
However, this is not a Killing form with respect to the flat metric.
}
\end{ex}
\subsection{Contact equivariance}
Consider the action of the group of contact diffeomorphisms.
It immediately follows from the intrinsic (i.e., invariant) definition
(representationef{maindefi}) of $A_{g,{\cal T}heta}$ of that the map
$g\mapsto{}A_{g,{\cal T}heta}$
from the space of metrics to that of $-\frac{1}{\ell+1}$-densities
commutes with this action:
\begin{equation}
\langlembdaabel{CoCAct}
A_{f^*g, {\cal T}heta}=
f^*\langlembdaeft(A_{g,{\cal T}heta}representationight).
\end{equation}
From this fact and Corollary~representationef{CoCu}, we deduce the following statement.
\begin{cor}
\langlembdaabel{SecCor}
If a metric $\tildeilde g $ is contactomorphic to a metric $g $
of constant sectional curvature
and if the contact structure is defined by a contact $1$-form $\tildeheta$
which is a Killing form with respect to
$g$, then $A_{\tildeilde g, {\cal T}heta}=0$.
\end{cor}
\subsection{Action of the full group of diffeomorphisms}
Let us consider the action of the group of all diffeomorphisms.
It turns out that this action is related to a quite remarkable $1$-cocycle.
Recall that the space of connections is an affine space associated with the space of
$(2,1)$-tensor fields, i.e.,
given two connections, $\nabla$ and $\tildeilde\nabla$, the difference
$\nabla-\tildeilde\nabla$ is a well-defined
$(2,1)$-tensor field.
This allows one to define a $1$-cocycle on
the group of all diffeomorphisms.
If $f$ is an arbitrary, not necessarily contact, diffeomorphism, we set:
$$
C(f):=f^*\nabla-\nabla,
$$
where $\nabla$ is an arbitrary fixed connection, choice of which
changes $C$ by a coboundary\footnote{
Note also that the cocycle $C$ provides a universal way to construct
representatives of non-trivial classes of the Gelfand-Fuchs cohomology;
see~\cite{Gel}.}.
Let $\nabla$ and $\tildeilde\nabla$ be two connections on~$M.$
The difference of the projective equivalence classes $[\nabla]-[\tildeilde\nabla]$
can be understood as a traceless
$(2,1)$-tensor field.
Therefore, a projective
connection on $M$ leads to the following $1$-{\it cocycle} on
the group of all diffeomorphisms:
$$
{\mathfrak T}(f)=f^*[\nabla]-[\nabla]
$$
which vanishes on (locally) projective diffeomorphisms.
In local coordinates,
$$
{\mathfrak T}(f)^k_{ij}:= f^*{\cal P}i_{ij}^k-{\cal P}i_{ij}^k,
$$
where ${\cal P}i_{ij}^k$ are the projective Christoffel symbols\footnote{
The $1$-cocycle ${\mathfrak T}$ is often considered as a higher-dimensional
analog of the Schwarzian derivative; see~\cite{OT}.
If~$\nabla$ is projectively flat, then
the group $\mathrm{SL}(n+1,\Bbb R)$
of (local) symmetries of $[\nabla]$ is precisely the kernel of ${\mathfrak T}$.}.
\begin{prop}
\langlembdaabel{CoCActThm}
If $f:M\tildeo{}M$ is an arbitrary diffeomorphism, then
\begin{equation}
\langlembdaabel{CoCActArb}
f^*\langlembdaeft(A_{g,{\cal T}heta}representationight)- A_{f^*g, {\cal T}heta}=
f^* \langlembdaeft\langlembdaangle g, \nabla{\cal T}hetarepresentationight representationanglengle
-\langlembdaeft\langlembdaangle f^*g,
\nabla{\cal T}hetarepresentationight representationanglengle
+\langlembdaeft\langlembdaangle f^*g\overlinetimes {\cal T}heta, {\mathfrak T}(f)representationightrepresentationanglengle.
\end{equation}
\end{prop}
\begin{proof}
Let us first clarify the notation.
Since ${\mathfrak T}(f)$ is a $(2,1)$-tensor field, the pairing
$\langlembdaeft\langlembdaangle g\overlinetimes{\cal T}heta,\,{\mathfrak T}(f)representationightrepresentationanglengle$
is well-defined.
Furthermore, taking into account the weight of the contact tensor ${\cal T}heta$,
it follows that
$\langlembdaeft\langlembdaangle g\overlinetimes{\cal T}heta,\,{\mathfrak T}(f)representationightrepresentationanglengle$
is a weighted density of weight~$-\frac{1}{\ell+1}$.
In local coordinates and using Proposition representationef{LoP}, we have
\[
\begin{array}{lcl}
A_{f^*g,{\cal T}heta}&=&(f^* g)^{ij}\langlembdaeft (\partial_i \tildeheta_j-f^*{\cal P}i^k_{ij}\, \tildeheta_k representationight )\\[2mm]
&=& (f^* g)^{ij}\langlembdaeft (\partial_i \tildeheta_j-(f^*{\cal P}i^k_{ij}-{\cal P}i^k_{ij})\, \tildeheta_k representationight )- (f^* g)^{ij}{\cal P}i^k_{ij}\tildeheta_k\\[2mm]
&= & (f^* g)^{ij}\langlembdaeft (\partial_i \tildeheta_j-{\mathfrak T}(f)^k_{ij}\, \tildeheta_k representationight )- (f^* g)^{ij}{\cal P}i^k_{ij}\tildeheta_k\\[2mm]
&=& (f^* g)^{ij}\langlembdaeft (\partial_i \tildeheta_j-{\cal P}i^k_{ij}\tildeheta_k representationight )-(f^* g)^{ij}\,{\mathfrak T}(f)^k_{ij}\, \tildeheta_k\\[2mm]
&=& (f^* g)^{ij}\nabla_i ({\cal T}heta_j)-(f^* g)^{ij}\,{\mathfrak T}(f)^k_{ij}\, \tildeheta_k\\[2mm]
&=&
\langlembdaeft\langlembdaangle f^*g,
\nabla \langlembdaeft ( {\cal T}hetarepresentationight )representationight representationanglengle
-\langlembdaeft\langlembdaangle f^*g\overlinetimes{\cal T}heta,\, {\mathfrak T}(f)representationightrepresentationanglengle.
\end{array}
\]
It remains to notice that
$f^*\langlembdaeft(A_{g,{\cal T}heta}representationight)=f^* \langlembdaeft\langlembdaangle g,
\nabla \langlembdaeft ( {\cal T}heta representationight )representationight representationanglengle$.
Proposition~representationef{CoCActThm} is proved.
\end{proof}
\section{The subsymbol of the Laplace-Beltrami operator}
In this section, we explain the relation of the Riemannian curl to
the classical Laplace-Beltrami operator.
Let us mention that study of
differential operators on contact manifolds is a classical subject;
see a recent work~\cite{vE10} and references therein.
\subsection{Differential operators and diffeomorphism action}
Let $M$ be an arbitrary smooth mani\-fold and
${\cal D}_{\langlembda, \mu}(M)$ be the space of linear differential operators
acting on the space of weighted densities:
$$
T:{\cal F}_\langlembda(M)\tildeo{\cal F}_\mu(M).
$$
The space ${\cal D}_{\langlembda, \mu}(M)$ is naturally a module over the group of diffeomorphisms,
the module structure being
dependent of the weights $\langlembda$ and $\mu$.
For $k \in \Bbb N$, let ${\cal D}^k_{\langlembda, \mu}(M)$ be the space of
linear differential operators of order~$\langlembdae k$.
The spaces ${\cal D}^k_{\langlembda, \mu}(M)$ define a filtration on ${\cal D}_{\langlembda, \mu}(M)$
invariant with respect to the group of diffeomorphisms.
Recall the classical notion of {\it symbol} (or the {\it principal symbol}) of
a differential operator of order~$k$.
It is defined as the image of the projection
$$
\sigma:{\cal D}_{\langlembda, \mu}(M)\tildeo{\cal D}^k_{\langlembda, \mu}(M)/{\cal D}^{k-1}_{\langlembda, \mu}(M).
$$
Observe that, in the particular case $\langlembda=\mu$,
the quotient space ${\cal D}^k_{\langlembda, \langlembda}(M)/{\cal D}^{k-1}_{\langlembda, \langlembda}(M)$
can be identified with the space of
symmetric contravariant tensor fields of degree $k$ on $M$.
We will be especially interested in the space ${\cal D}^2_{\langlembda, \langlembda}(M)$
of $2$-nd order operators acting on $\langlembda$-densities;
a systematic study of this space viewed as
a module over the group of diffeomorphisms was initiated in~\cite{DO1}.
\subsection{The subsymbol of a second order differential operator}
In~\cite{CO12}, the space of differential operators
on a contact manifold was studied as a module over
the group of contact diffeomorphisms.
It was proved that there exists
a notion of {\it subsymbol} which is a tensor field
of degree lower than that of the principal symbol.
For a $2$-nd order differential operator,
the subsymbol is just a contact vector field.
More precisely,
for every $\langlembda$, there exists a linear map
(which is unique up to a constant factor)
$$
{\mathrm s}\sigma:
{\mathcal D}^2_{\langlembdaambda,\langlembdaambda}(M) representationightarrow {\mathcal K}(M),
$$
invariant with respect to the action of the group of contact
diffeomorphisms. The image ${\mathrm s}\sigma(T)$ was called the
{\it subsymbol} of the operator $T$. We will need the explicit
formula for the subsymbol of a given second order differential
operator.
If $M$ is a contact manifold, then
every operator $T\in{\cal D}^2_{\langlembda, \langlembda}(M)$ can be written
(in many different ways) in the form:
\begin{equation}
\langlembdaabel{Rep}
T=L_{X_{\phi_1}} \circ L_{X_{\phi_2}}+L_{X_{\phi_3}}\circ L_{Y_1}+L_{Y_2}\circ L_{Y_3} +
L_{X_{\phi_4}}+L_{Y_4}+F,
\end{equation}
where each $Y_i$ is a vector field tangent to the contact distribution,
$X_\phi$ is the contact vector field with the contact Hamiltonian
$\phi\in{\mathcal F}_{-\frac{1}{\ell+1}}(M)$,
the Lie derivative $L$ is defined by Eq.~(representationef{LieEq}),
and~$F$ denotes the operator of multiplication by a function.
The explicit expression for the subsymbol of differential
operator~(representationef{Rep}) is as follows (see~\cite{CO12}):
\begin{equation}
\langlembdaabel{vfields}
{\mathrm s}\sigma(T)=\tildes
{\tildes\frac{1}{2}}\bigl[X_{\phi_1},X_{\phi_2}\bigr]-
\bigl(\frac{\ell+1}{\ell+2}\bigr)\bigl(\langlembda-{\tildes\frac{1}{2}}\bigr)X_{L_{Y_1}(\phi_3)}+
{\tildes\frac{1}{2}}\pi\bigl[Y_2,Y_3\bigr]+X_{\phi_4},
\end{equation}
where $L_{Y}(\phi)$ denotes the Lie derivative
of a $-\frac{1}{\ell+1}$-density $\phi$ along the vector field $Y$,
and $\pi:\mathrm{Vect}(M)\tildeo{\mathcal K}(M)$ is defined in (representationef{PiPr}).
\begin{rem}
{representationm
Although it seems almost impossible, the map ${\mathrm s}\sigma$
defined by~(representationef{vfields}) is
well-defined.
In other words, it is independent of the choice of the vector fields
in the representation (representationef{Rep}) of the operator $T$.
This can be checked directly by rewriting it in local coordinates,
see formula~(representationef{ExPSS}) below.
Since the expression~(representationef{vfields}) is written using invariant terms,
it commutes with the action of contact diffeomorphisms.
Note also that the existence of such a map is indigenous to
contact geometry.
There is no similar map commuting with the full group of diffeomorphisms,
except for the principal symbol.
}
\end{rem}
\subsection{The Laplace-Beltrami operator on the space of weighted densities}
The classical Laplace-Beltrami operator acting on
the space of smooth functions is defined as follows
$$
{\cal D}elta_g(f)=d^*df.
$$
This operator is completely determined by the metric
${g}$.
We will go to a more general framework and
consider the generalized Laplace-Beltrami operator
acting on the space of weighted densities:
$$
{\cal D}elta_g^\langlembda: {\mathcal F}_\langlembda(M)\tildeo{\mathcal F}_\langlembda(M).
$$
The explicit formula of this operator
is as follows:
\[
{\cal D}elta_g^\langlembdaambda(\phi\,\omega^\langlembda)=
\langlembdaeft(
{g}^{ij}\nabla_i\nabla_j(\phi)+
\frac{n^2\langlembdaambda(\langlembdaambda-1)}{(n-1)(n+2)}R\phi
representationight)\omega^\langlembda,
\]
where $R$ is the scalar curvature
(see~\cite{DO}, Proposition 5.2).
\subsection{Calculating the subsymbol of the Laplace-Beltrami operator}
Recall that $M$ is a contact manifold
and $n=2\ell+1$.
It turns out that the contact Riemannian curl of a given metric $g$
is proportional to the subsymbol of the Laplace-Beltrami operator
associated with $g$.
This property can be considered as an equivalent definition
of the contact Riemannian curl.
\begin{thm}
One has
\begin{equation}
\langlembdaabel{ProPEq}
\tildeextstyle
\mathrm{s}\sigma({\cal D}elta^\langlembdaambda_g)=
\langlembdaeft(\frac{\ell+1}{\ell+2}representationight)
\langlembdaeft(2\langlembdaambda-1representationight)X_{A_{g,{\cal T}heta}}.
\end{equation}
\end{thm}
\begin{proof}
The proof is essentially a direct computation.
Let us choose local Darboux coordinates.
Every second order differential operator
can be written in these coordinates as:
$$
\begin{array}{rcl}
T&=&T_{2,0,0}\,\partial_z^2+T_{1,i,0}\,\partial_z\partial_{x_i}
+T_{1,0,i}\,\partial_z\partial_{y_i}+T_{0,ij,0}\,\partial_{x_i}\partial_{x_j}+
T_{0,i,j}\,\partial_{x_i}\partial_{y_j}+T_{0,0,ij}\,\partial_{y_i}\partial_{y_j}\\[4pt]
&&+T_{1,0,0}\,\partial_z+T_{0,i,0}\,\partial_{x_i}+T_{0,0,i}\,\partial_{y_i}+T_{0,0,0}.
\end{array}
$$
The coordinate formula of the subsymbol was calculated in \cite{CO12}:
\begin{equation}
\langlembdaabel{ExPSS}
\begin{array}{rcl}
\mathrm{s}\sigma(T) &=&
\frac{1+2\langlembda(\ell+1)}{\ell+2}
{\cal B}igl(
\partial_z(T_{2,0,0}-{\tildes\frac{1}{2}}{}y_iT_{1,i,0}+{\tildes\frac{1}{2}}{}x_iT_{1,0,i})\\[6pt]
&&\hatskip1.6cm
+\partial_{x_i}(T_{1,i,0}-{\tildes\frac{1}{2}}{}y_jT_{0,ij,0}+{\tildes\frac{1}{2}}{}x_jT_{0,i,j})\\[6pt]
&&\hatskip1.6cm
+\partial_{y_i}(T_{1,0,i}+{\tildes\frac{1}{2}}{}x_jT_{0,0,ij}-{\tildes\frac{1}{2}}{}y_jT_{0,j,i}){\cal B}igr)\\[6pt]
&&
+T_{1,0,0}-{\tildes\frac{1}{2}}{}y_iT_{0,i,0}+{\tildes\frac{1}{2}}{}x_iT_{0,0,i}.
\end{array}
\end{equation}
One can check that this is exactly the same formula as (representationef{vfields}).
The expression of the generalized Laplace-Beltrami operator
${\cal D}elta^\langlembdaambda$ in local coordinates was calculated in~\cite{DO},
the result is:
\[
{\cal D}elta^\langlembdaambda_g=
{g}^{ij}\partial_i\partial_j-({g}^{jk}{\cal G}amma^i_{jk}+
2\langlembdaambda {g}^{ij}{\cal G}amma^k_{jk})\partial_i+\mathrm{(0-th\; order \; coefficients)}.
\]
Let us combine the above two formulas.
We obtain
$\mathrm{s}\sigma({\cal D}elta^\langlembdaambda_g)=X_{\phi}$,
where $\phi$ is a weighted density of the form
\begin{equation}
\langlembdaabel{sgama}
\tildeextstyle
\phi=
\langlembdaeft(\langlembdaeft(1-\frac{1+2\langlembdaambda(\ell+1)}{\ell+2}representationight)
{g}^{jk}{\cal G}amma^t_{jk}\tildeheta_t+
\langlembdaeft(2\langlembdaambda-\frac{1+2\langlembdaambda(\ell+1)}{\ell+2}representationight)
{\cal G}amma^j_{ij}{g}^{it}\tildeheta_trepresentationight)
\mathrm{vol}^{-\frac{1}{\ell+1}},
\end{equation}
and $X_\phi$ is the corresponding contact vector field.
Finally, taking into account the fact that ${g}^{ij}\partial_i(\tildeheta_j)=0$,
for the Darboux form $\tildeheta$,
the expression (representationef{sgama}), after collecting the terms, coincides with
$\langlembdaeft(\frac{\ell+1}{\ell+2}representationight)
\langlembdaeft(2\langlembdaambda-1representationight)A_{g,{\cal T}heta}$.
\end{proof}
\begin{cor}
For a generic metric,
$\mathrm{s}\sigma({\cal D}elta^\langlembda_g)=0$
if and only if $\langlembda={\frac{1}{2}}$.
\end{cor}
\begin{rem}
{representationm
In differential geometry it is known that the space of half-densities and
the space of differential operators ${\mathcal D}_{\frac{1}{2},\frac{1}{2}}(M)$ acting on them
play a very special role.
In our context, the space of half-densities appears naturally.
}
\end{rem}
\section{Cotangent lift and the geodesic spray} \langlembdaabel{SbundleSec}
In this section, we calculate the contact Riemannian curl on the unit sphere bundle
$STM$ over a Riemannian manifold $(M,{g})$.
The manifold $STM$ is a classical example of contact manifold,
and, furthermore, it is equipped with the canonical lift of the metric.
We prove that the contact Riemannian curl vanishes in this case.
Recall that the classical {\it geodesic spray} is the Hamiltonian vector field
on $TM$ with Hamiltonian $H(x,y)={g}_{ij}(x)\,y^iy^j,$
where $y^i$ are coordinates on the fibers;
the restriction of this vector field to $STM$ is an intrinsically defined
contact vector field.
It is not reasonable expect existence of another, independent, invariant contact vector field in this case.
\subsection{Statement of the main result}
The Riemannian metric ${g}$ on $M$ has a canonical lift to~$STM$
that will be denoted by $\bar g$.
The main result of this section is as follows.
\begin{thm}
\langlembdaabel{BigThm}
The contact Riemannian curl on $(STM,\bar g)$ is identically zero.
\end{thm}
In order to prove this theorem, we will need explicit formulas
for the contact structure and the canonical Riemannian metric on $STM$.
\subsection{The coordinates on $STM$}
Let $(M,{g})$ be any Riemannian manifold of dimension $n$.
The Riemannian geometry of the sphere bundle $STM$
was studied in~\cite{Taha},
we will be using the notation of that work.
Denote by $(x^1,\langlembdadots,x^n)$ a local coordinate system in $M$ and $(y^1,\langlembdadots,y^n)$ the
Cartesian coordinates in the tangent space $T_xM$ at the point $x$ in $M$.
The coordinates $(x,y)$ are local coordinates on the tangent bundle on $TM$.
The unit sphere bundle $STM$ is a hypersurface of the tangent bundle
$T(M)$, singled out as the level surface of the Hamiltonian of the geodesic spray
$$
H(x,y)=1
$$
at every point.
\subsection{The contact structure of the sphere bundle $STM$}
The sphere bundle $STM$ is represented by parametric equations:
\[
x^h=x^h,\quad x^{\bar h}=y^h=y^h(x^i,u^{\kappa}),
\]
where $u^{\kappa}$ are local coordinates on the sphere.\footnote{
Following \cite{Taha}, we will adopt the following index gymnastics.
Capital Latin letters $A,B,\langlembdadots$ run $1$ to $2n$.
Small latin letters $i,j,\langlembdadots$ run $1$ to $n$.
Barred Latin indices $\bar i, \bar j,\langlembdadots$ run $n+1$ to $2n$.
Some of the Greek letters $\alpha, \beta, \langlembdadots$ run $1$ to $2n-1$.
Some other Greek letters $\kappa, \langlembdaambda, \langlembdadots$ run $n+1$ to $2n-1$. }
The tangent vectors $B^{A}_\alpha=\frac{\partial x^A}{\partial u^\alpha}$ of $STM$ in $T(M)$ are given by
\begin{equation}
\langlembdaabel{BComp}
\begin{array}{lcllcl}
B^{h}_i&=&\delta^h_i,& B^{h}_\langlembdaambda&=&0,\\[2mm]
B^{\bar h}_i&=&\partial_i y^h,& B^{\bar h}_\langlembdaambda&=&\partial_\langlembdaambda y^h.
\end{array}
\end{equation}
The square matrix $\langlembdaeft ( \begin{array}{c}
B^{A}_\alpha\\[2mm]
C^A
\end{array} representationight )$, where $C^i=0$ and $C^{\bar i}=y^i$, is
invertible at each point $x$ in $M$.
Its inverse will be the matrix $(B^\alpha_{A}, C_A)$, given by the equations:
\begin{equation}
\langlembdaabel{BInvComp}
\begin{array}{lcllcl}
B^{h}_{i}&=&\delta^h_i,& B^{h}_{\bar i}&=&0,\\[2mm]
B^{\kappa}_{i}&=&-B^{\bar h}_i B^\kappa_{\bar h},& B^{\kappa}_{\bar i},&&
\end{array}
\end{equation}
and $C_A=\langlembdaeft (
\begin{array}{c}
C_i\\[2mm]
C_{\bar i}
\end{array}
representationight )$, where
$C_{\bar i}={g}_{ih}y^h$ and
$C_i={\cal G}amma^h_{rs} {g}_{hi}\,y^ry^s$.
The next formulas can be deduced from
Eqs. (representationef{BComp}), (representationef{BInvComp}), and are useful for what follows
\[
\begin{array}{lcllcllcl}
B^{\bar h}_{\langlembdaambda} B^{\kappa}_{\bar h}&=&
\delta^\kappa_\langlembdaambda,& y^h B^{\kappa}_{\bar h}&=&
0,&B^{\bar h}_{\langlembdaambda} B^{\langlembdaambda}_{\bar i}+y^h C_{\bar i}&=&
\delta^h_i,\\[2mm]
B^{\bar h}_{\langlembdaambda} C_{\bar h}&=&0,& y^h C_{\bar h},&=&1.&&&
\end{array}
\]
The Riemannian metric indentifies the tangent bundle $T(M)$
and the cotangent bundle $T^*(M),$
and hence induces a 1-form $\tildeheta$ on $T(M)$, called
the {\it Liouville form}, which in local coordinates reads as follows:
\[
\tildeheta={g}_{ij}y^j \, dx^i,
\]
Denote by $\bar \tildeheta$ the restriction of the 1-form $\tildeheta$ to the sphere bundle $STM$.
It is as follows:
\[
\bar \tildeheta_\alpha=\tildeheta_A B^{ A}_\alpha.
\]
Eq. (representationef{BComp}) imply that $\bar \tildeheta_i={g}_{ij}y^j$ and $\bar \tildeheta_\kappa=0$.
\begin{lemma} The form $\bar \tildeheta$ defines a contact structure on $STM$. The volume form associated with it reads (up to a factor) as:
\[
{\cal O}mega\; dx^1\wedge \cdots \wedge dx^n\wedge
du^{n+1}\wedge...\wedge du^{2n-1},
\]
where
$
{\cal O}mega=\mathrm{det }(B^{A}_\alpha,C^{A})\, \mathrm{det } ({g}_{ij}).
$
\end{lemma}
\begin{proof}
This is well known, see~\cite{Taha},
and can also be checked by a direct computation.
\end{proof}
\subsection{The Riemannian metric on $STM$}
The Riemannian metric ${g}$ on $M$ can be extended to
a Riemannian metric $\bar{{g}}$ on the sphere bundle $STM$.
Explicitly, $\bar{{g}}$ is given by (cf. \cite{Taha}):
\[
\begin{array}{lcl}
\bar{{g}}_{ji}&=&{{g}}_{ji}+{{g}}_{ts}(\nabla_j y^t)(\nabla_i y^s),\\[2mm]
\bar {{g}}_{\mu i}&=&{{g}}_{ts}(\partial_\mu y^t)(\nabla_iy^s),\\[2mm]
\bar {{g}}_{\mu \langlembdaambda}&=&{{g}}_{ji}(\partial_\mu y^j)(\partial_\langlembdaambda y^i).
\end{array}
\]
The inverse of $\bar{{g}}$ is given by
\[
\begin{array}{lcl}
\bar {{g}}^{ji}&=&{{g}}^{ji},\\[2mm]
\bar {{g}}^{\langlembdaambda h}&=&-{{g}}^{hl}(\nabla_ly^i) B^\langlembdaambda_{\bar i},\\[2mm]
\bar {{g}}^{\langlembdaambda \kappa}&=&\langlembdaeft ({{g}}^{ih}+ {{g}}^{ts}(\nabla_t y^i)(\nabla_s y^h) representationight ) B^\langlembdaambda_{\bar i}B^\kappa_{\bar h}.
\end{array}
\]
The Christoffel symbols associated with this metric are given by
\[
\begin{array}{ccl}
\bar {\cal G}amma^h_{ji}&=&{\cal G}amma^h_{ij}+\frac{1}{2}\langlembdaeft (
R_{r sj}^{ h}y^r\nabla_i y^s+R_{r si}^{h}y^r\nabla_j y^s representationight ),\\[2mm]
\bar {\cal G}amma^h_{\mu i}&=&\frac{1}{2}R_{r si}^{ h}y^r B_\mu^{\bar s},\\[2mm]
\bar {\cal G}amma^h_{\mu \langlembdaambda}&=&0,\\[2mm]
\bar {\cal G}amma^\kappa_{ji}&=&\langlembdaeft(
\nabla_j \nabla_i y^h+\frac{1}{2}R_{r ji}^{h}y^r- \frac{1}{2} R_{r ij}^{ h}y^r - \frac{1}{2} \langlembdaeft ( R_{r sj}^{ l}y^r\nabla_i y^s+ \frac{1}{2} R_{r si}^{l }y^r\nabla_j y^srepresentationight )\nabla_l y^h representationight )B^\kappa_{\bar h},\\[2mm]
\bar {\cal G}amma^\kappa_{\mu j}&=&\langlembdaeft (\partial_\mu \nabla_i y^h-\frac{1}{2} R_{r si}^{ l}y^r B_\mu^{\bar s} \nabla_l y^h representationight )B^\kappa_{\bar h},\\[2mm]
\bar {\cal G}amma^\kappa_{\mu \langlembdaambda}&=&(\partial_\mu \partial_\langlembdaambda y^h)B^\kappa_{\bar h}.
\end{array}
\]
\subsection{Proof of Theorem representationef{BigThm}}
We are ready to prove the main result of this section.
\begin{lemma}
\langlembdaabel{lemma1}
We have
\begin{equation}
\langlembdaabel{CalcEq}
\begin{array}{lcl}
y^i\partial_{i}({\cal O}mega)&=&- y^i B_\langlembdaambda^{\bar h}\partial_i(B^\langlembdaambda_{\bar h})\,
{\cal O}mega+ 2y^i{\cal G}amma_{i}\, {\cal O}mega-y^iy^hy^mg_{hm}{\cal G}amma_{ir}^h\, {\cal O}mega,\\[2mm]
y^l (\nabla_l y^i)B^\langlembdaambda_{\bar i}\, \partial_{\langlembdaambda}({\cal O}mega)&=&
-y^l (\nabla_l y^i) \partial_\langlembdaambda(B^\langlembdaambda_{\bar i})\, {\cal O}mega,\\[2mm]
y^l (\nabla_l y^k)(\partial_\mu \partial_\langlembdaambda y^j)\,B^\langlembdaambda_{\bar k} B^\mu_{\bar j}
&=&-y^l (\nabla_l y^k)\partial_\langlembdaambda (B^\langlembdaambda_{\bar k}),\\[2mm]
\partial_\langlembdaambda(\nabla_l y^h) B^\langlembdaambda_{\bar h}y^l&=&
-(\partial_l B^\langlembdaambda_{\bar h}) B^{\bar h}_\langlembdaambda y^l+
{\cal G}amma^i_{li} \,y^l-{\cal G}amma^h_{rs}\,{g}_{ih}y^iy^ry^s.
\end{array}
\end{equation}
\end{lemma}
\begin{proof}
The first and the second lines of (representationef{CalcEq}) follow from the fact that
\begin{eqnarray*}
\langlembdaabel{F1} B^\langlembdaambda_{\bar k} \partial_\mu(B_\langlembdaambda^{\bar j})
&=&
-\partial_\mu(B^\langlembdaambda_{\bar k}) B_\langlembdaambda^{ j}-\partial_\mu(y^j y^m){g}_{mk},\\[4pt]
\langlembdaabel{F2} y^i\partial_i B_\langlembdaambda^{\;\; \bar j}
&=&
-y^iB^{\bar h}_{\langlembdaambda}B_\kappa^{\bar j}\partial_i(B^\kappa_{\bar h})-y^iy^jy^mg_{hm}{\cal G}amma_{il}^h\, B_\langlembdaambda^{\bar l},
\end{eqnarray*}
and the property of the determinant.
The third line of (representationef{CalcEq}) follows from the fact that
$$
B_\langlembdaambda^{\bar h}B^\langlembdaambda_{\bar i}+y^h C_{\bar i}=\delta_i^h
$$
and applying to it the partial derivative $\partial_\mu$.
The fourth line of (representationef{CalcEq}) follows when we substitute the covariant derivative
$\nabla_l y^h=\partial_l y^h+{\cal G}amma_{li}^h\,y^i$
and use the third equation.
\end{proof}
By definition,
\[
A_{\bar {g},{\cal T}heta}=\bar {{g}}^{ih}\bar \nabla_i(\tildeheta_h {\cal O}mega^{-\frac{1}{n}})+\bar {{g}}^{\langlembdaambda h}\bar \nabla_\langlembdaambda (\tildeheta_h {\cal O}mega^{-\frac{1}{n}})+\bar {{g}}^{h \langlembdaambda}\bar \nabla_h (\tildeheta_\langlembdaambda {\cal O}mega^{-\frac{1}{n}})+\bar {{g}}^{\langlembdaambda \kappa}\bar \nabla_\langlembdaambda (\tildeheta_\kappa {\cal O}mega^{-\frac{1}{n}}).
\]
The last two summands vanish because $\tildeheta_\kappa=0$.
Let us compute the first two summands seperately.
Applying the covariant derivative $\bar \nabla$, we get
\[
\begin{array}{lcl}
\bar {{g}}^{ih}\bar \nabla_i(\tildeheta_h {\cal O}mega^{-\frac{1}{n}})&=&\partial_i y^i \; {\cal O}mega^{-\frac{1}{n}}+y^i\partial_i ( {\cal O}mega^{-\frac{1}{n}})-R_{q r s i} y^qy^r (\nabla_h y^s) {{g}}^{ih}\; {\cal O}mega^{-\frac{1}{n}}\\[2mm]
&&+(1+\frac{1}{n}){\cal G}amma_{ij}^jy^i\; {\cal O}mega^{-\frac{1}{n}}+\frac{1}{n} \langlembdaeft (\frac{1}{2}
R_{r sh}^{ h} y^ry^l (\nabla_l y^s)+ \partial_\langlembdaambda(\nabla_l y^h)B^\langlembdaambda_{\bar h}y^lrepresentationight )\; {\cal O}mega^{-\frac{1}{n}}.
\end{array}
\]
Similarly,
\[
\begin{array}{lcl}
\bar {{g}}^{\langlembdaambda h}\bar \nabla_\langlembdaambda (\tildeheta_h {\cal O}mega^{-\frac{1}{n}})
&=&
-\partial_i y^i \; {\cal O}mega^{-\frac{1}{n}}-{\cal G}amma_{ij}^jy^i \;{\cal O}mega^{-\frac{1}{n}}-(\nabla_i y^k)
B^\langlembdaambda_{\;\; \bar k}y^i\partial_\langlembdaambda({\cal O}mega^{-\frac{1}{n}})\\[2mm]
&&+\frac{1}{2}R_{rs k h}y^ry^s{{g}}^{hl}(\nabla_l y^k) \;
{\cal O}mega^{-\frac{1}{n}}- \frac{1}{2n} (\nabla_i y^k)y^i R_{r k h}^{ \bar h} y^r\;
{\cal O}mega^{-\frac{1}{n}}\\[2mm]
&&+ -\frac{1}{n} y^l (\nabla_l y^k)(\partial_\langlembdaambda \partial_\mu y^j)B^\langlembdaambda_{\bar k}B^\mu_{\bar j}\;{\cal O}mega^{-\frac{1}{n}}.
\end{array}
\]
By collecting the terms and using Lemma representationef{lemma1}, we finally obtain:
$$
A_{\bar{g},{\cal T}heta}=
-\frac{1}{2}\langlembdaeft(
R_{ilsj}\,y^iy^l(\nabla_h y^s)\,{{g}}^{jh}
representationight)
\mathrm{vol}^{-\frac{1}{n}}\equiv0,
$$
since the curvature tensor $R_{ilsj}$ is antisymmetric in two first indices.
Theorem representationef{BigThm} is proved.
\section{Examples} \langlembdaabel{Sphere}
We finish the paper with concrete examples of Riemannian curl
for the $3$-dimensional sphere (with two natural metrics)
and the $3$-dimensional ellipsoid with the standard metric.
\subsection{The sphere $S^3$}
Consider the sphere $S^3$ in the standard symplectic space ${\cal B}bb R^4$.
It is endowed with the natural contact structure that
can be defined by the contact form
\[
\tildeheta=dz+xdy-ydx,
\]
where $x,y$ and $z$ are affine coordinates on $S^3$.
More precisely, if $p_1,p_2,q^1,q^2$ are symplectic Darboux coordinates
on ${\cal B}bb R^4$, then $x=\frac{p_1}{q^2},\,y=\frac{q^1}{q^2},\,z=-\frac{p_2}{q^2}$.
The restriction of the Euclidean metric to the
sphere $S^3$ takes the following form:
\[
\begin{array}{rcl}
{g}_{S^3}&=&
{\frak i}splaystyle
F\langlembdaeft(\langlembdaeft(
y^2+z^2+1representationight) dx^2+(x^2+z^2+1)dy^2+(x^2+y^2+1)dz^2representationight . \\[10pt]
&&{\frak i}splaystyle
\langlembdaeft .
-2\, x y\, dx dy-2\,xz \, dx dz-2\, yz \,dydz representationight ),
\end{array}
\]
where $F= \langlembdaeft(\frac{1}{(x^2+y^2+z^2+1 )}representationight)^{2}$.
Let us also consider another, conformally equivalent, metric on~$S^3$:
\[
\tildeilde {g}_{S^3}:=
\langlembdaeft (\frac{x^2+y^2+z^2+1}{\frac{x^2}{a}+\frac{y^2}{b}+\frac{z^2}{c}+1}representationight ){g}_{S^3},
\]
which appeared in the context of integrable systems in \cite{Tab,MT},
see also~\cite{DV}.
\begin{prop} The following results hold.
\begin{enumerate}
\item[(i)]
In the case of the ``round'' metric ${g}_{S^3}$, we have
$A_{{g}_{S^3}}= 0$;
\item[(ii)]
In the case of the metric $\tildeilde {g}_{S^3}$, we have
$A_{\tildeilde {g}_{S^3}}= \frac{5}{2}
\langlembdaeft(
\langlembdaeft(\frac{1}{b}-\frac{1}{a}representationight) xy+\langlembdaeft(\frac{
1}{c}-1representationight) zrepresentationight)$.
\end{enumerate}
\end{prop}
\begin{proof}
Part (i) follows from Corollary~representationef{CoCu}.
Part (ii) can be obtained by a
straightforward computation using Eq. (representationef{sgama}).
\end{proof}
\subsection{The case of the ellipsoid $E^3(a,b,c)$} \langlembdaabel{Ellipsoid}
Consider the $3$-dimensional ellipsoid endowed with the standard
metric
\[
\begin{array}{lll}
{g}_{E^3_{a,b,c}}=&{g}_{x,x} dx^2+{g}_{y,y} dy^2+{g}_{z,z} dz^2 + {g}_{x,y} dx\, dy+{g}_{x,z}dx \, dz+ {g}_{y,z} dy\, dz,
\end{array}
\]
where
\[
\begin{array}{ccl}
{g}_{x,x}&=&{\frak i}splaystyle \frac{\langlembdaeft(b^2 y^2+c^2 z^2+1representationight)^2+a^4 x^2 \langlembdaeft(y^2+z^2+1representationight) }{\langlembdaeft ((ax)^2+(by)^2+(cz)^2+1representationight )^2},\\[4mm]
{g}_{y,y}&=& {\frak i}splaystyle \frac{ \langlembdaeft(a^2 x^2+c^2 z^2+1representationight)^2+b^4 y^2 \langlembdaeft(x^2+z^2+1representationight)}{\langlembdaeft ((ax)^2+(by)^2+(cz)^2+1representationight )^{2}},\\[4mm]
{g}_{z,z}&=& {\frak i}splaystyle \frac{\langlembdaeft(a^2 x^2+b^2 y^2+1representationight)^2+c^4 z^2 \langlembdaeft(x^2+y^2+1representationight)}{\langlembdaeft ((ax)^2+(by)^2+(cz)^2+1representationight )^{2}}, \\[4mm]
{g}_{x,y}&=&{\frak i}splaystyle -2 x y \frac{a^4 x^2-a^2 \langlembdaeft(z^2 (b^2-c^2)+b^2 -1representationight)+b^2 \langlembdaeft(b^2
y^2+c^2 z^2+1representationight)}{ \langlembdaeft ((ax)^2+(by)^2+(cz)^2+1representationight )^{2} },\\[4mm]
{g}_{x,z}&=& {\frak i}splaystyle -2 x z \frac{a^4 x^2-a^2 \langlembdaeft(y^2 \langlembdaeft(c^2-b^2representationight)+c^2-1representationight)+c^2
\langlembdaeft(b^2 y^2+c^2 z^2+1representationight)}{\langlembdaeft ((ax)^2+(by)^2+(cz)^2+1representationight )^{2}}, \\[4mm]
{g}_{y,z}&=& {\frak i}splaystyle -2 y z \frac{b^4 y^2-b^2 \langlembdaeft(x^2 \langlembdaeft(c^2-a^2representationight)+c^2-1representationight)+c^2
\langlembdaeft(a^2 x^2+c^2 z^2+1representationight)}{\langlembdaeft ((ax)^2+(by)^2+(cz)^2+1representationight )^{2}}.
\end{array}
\]
\begin{prop}
We have
\[
\begin{array}{rcl}
A_{ {g}_{E^3_{a,b,c}}}&=&
a^4 (a^2 - b^2) (b^2 + 2 c^2 + 2) x^3 y +
b^4 (a^2 - b^2) (a^2 + 2 c^2 + 2) x y^3 \\[6pt]
&&
+ (a^2 - b^2) c^4 (2 + a^2 + b^2 + c^2) x y z^2 \\[6pt]
&&
- a^4 (c^2 - 1) (a^2 + 2 b^2 + c^2 + 1) x^2 z
- b^4 (c^2 - 1) (2 a^2 + b^2 + c^2 + 1) y^2 z
\\[6pt]
&&
- c^4 (c^2 - 1) (2 a^2 + 2 b^2 + 1) z^3
+ (a^2 - b^2) (a^2 + b^2 +
2 c^2 + 1) x y \\[6pt]
&&- (c^2 - 1) (2 a^2 + 2 b^2 +
c^2) z.
\end{array}
\]
\end{prop}
\begin{proof}
Straightforward computation using Eq. (representationef{sgama}).
\end{proof}
\noindent \tildeextbf{Acknowledgments}.
We thank Dimitry Leites and Christian Duval their interest in this work
and careful reading of preliminary versions of it.
We are also grateful to
Charles Conley, Eugene Ferapontov and Serge Tabachnikov
for a number of fruitful discussions.
The first author was partially supported by the Grant
NYUAD 063.
The second author was partially supported by the Grant
PICS05974 ``PENTAFRIZ'', of CNRS.
\def\it{\it}
\def\bf{\bf}
\bfliographystyle{amsalpha}
\end{document} |
begin{document}
\hbox{\scalebox{0.75}{$\triangle$}}itle{Automated Generation of Triangle Geometry Theorems}
author{Alexander Skutin\hbox{\scalebox{0.75}{$\triangle$}}hanks{This work was supported by the Ministry of Education and Science of the Russian Federation as part of the
program of the Moscow Center for Fundamental and Applied Mathematics under the agreement no. 075-15-2022-284, by the scholarship of Theoretical Physics and Mathematics Advancement Foundation “BASIS” (grant No 21-8-3-2-1) and by the Russian Science
Foundation, project no. 22-11-00075.}}
\date{}
\title{Automated Generation of Triangle Geometry Theorems}
begin{abstract}
In this article, we introduce an algorithm for automatic generation and categorization of triangle geometry theorems.
\end{abstract}
\section{Introduction}\label{sc3}
Plane geometry is a vast field of research where many theorems had been obtained and new results are still being discovered. Over the past few decades, a lot of effort has been spent on creating algorithms designed to automatically generate theorems in plane geometry, some of which can be found in \cite{1, 2, 8, 3, 4}.
In this paper, we concretize the problem of automatic generation of plane geometry theorems for the case of triangle geometry theorems, that is, triangle $ABC$ theorems that are invariant with respect to permutations of $ABC$ vertices. We provide a new algorithm that generates and categorize triangle geometry theorems. It is expected that this algorithm is able to generate almost all of the theorems from the articles \cite{cos, cos1}. The main idea of our algorithm can be described as follows:\T_{\triangle}he algorithm has inductive form and at each new step $t$
begin{enumerate}
\item it considers a set of theorems obtained on the previous step and constructs a new set of theorems by adding at most one new object to each already existed theorem and formulating new theorems about the resulting configurations,
\item it replaces the set of obtained theorems with some of its ``maximal generalizations''.
\end{enumerate}
The definition of ``maximally general'' (complete) sets of theorems will be presented in this article.
\subsection{Notation}\label{circ}
The arity $\hbox{\scalebox{0.75}{$\triangle$}}ext{ar}(f)$ of a function $f$ is the number of variables acting in $f$. Further, by $\wedge, \Rightarrow, \Leftrightarrow$ we will denote the logical operators `and', `implies' and `equivalent'. We will use the standard set-theory notation $\{x \:\vert\: \hbox{\scalebox{0.75}{$\triangle$}}ext{statement about x}\}$ which is read as, ``the set of all x such that the statement about x is true.''
\subsection{Structure of the paper} The paper is organized as follows. In Sections 2, 3 we introduce $\hbox{\scalebox{0.75}{$\triangle$}}$-objects and define the set $S_{\hbox{\scalebox{0.75}{$\triangle$}}riangle}^{7}$. In Section 4, we develop an algorithm for automatic generation of triangle geometry theorems based on $S_{\hbox{\scalebox{0.75}{$\triangle$}}riangle}^{7}$. Section 5 contains some propositions that simplify the computation of $S_{\hbox{\scalebox{0.75}{$\triangle$}}riangle}^{7}$. Appendix A contains lists of objects which are used in the article.
\subsection{Triangle centers and lines}
begin{definition}[C. Kimberling, \cite{ki}]\label{d5}
By a {\em triangle center} $ X$ denote a point $ X(A, B, C)$, which is defined for each tuple of points $A$, $B$, $C$ on the plane $\mathbb{R}^2$.
\end{definition}
begin{lis}\label{l1}
The complete list of triangle centers $X_i$, $1\leq i\leq 13$ that are used in this article can be found in the Appendix (see List \ref{ltc} in the Appendix). Some of the centers in use with corresponding numbers:
begin{enumerate}
\item \hbox{\scalebox{0.75}{$\triangle$}}ext{In(ex)center} $I$, $I(A, B, C)$ -- the incenter of $ABC$ if $A$, $B$, $C$ are placed clockwise on the plane $\mathbb{R}^2$ (or the $A$-excenter of $ABC$ if $A$, $B$, $C$ are placed anti-clockwise on $\mathbb{R}^2$).
\item \hbox{\scalebox{0.75}{$\triangle$}}ext{Centroid} $G$, $G(A, B, C)$ -- the centroid of $ABC$.
\item \hbox{\scalebox{0.75}{$\triangle$}}ext{Circumcenter} $O$, $O(A, B, C)$ -- the circumcenter of $ABC$.
\item \hbox{\scalebox{0.75}{$\triangle$}}ext{Orthocenter} $H$, $H(A, B, C)$ -- the orthocenter of $ABC$.
\item \hbox{\scalebox{0.75}{$\triangle$}}ext{Nine-point center} $N$, $N(A, B, C)$ -- the nine-point center of $ABC$.
\item[7.] \hbox{\scalebox{0.75}{$\triangle$}}ext{First(second) Fermat point} $F$, $F(A, B, C)$ -- the first Fermat point of $ABC$ if $A$, $B$, $C$ are placed clockwise on the plane $\mathbb{R}^2$ (or the second Fermat point of $ABC$ if $A$, $B$, $C$ are placed anti-clockwise on $\mathbb{R}^2$).
\item[9.] \hbox{\scalebox{0.75}{$\triangle$}}ext{Inner(outer) Feuerbach point} $F_e$, $F_e(A, B, C)$ -- the inner Feuerbach point of $ABC$ if $A$, $B$, $C$ are placed clockwise on the plane $\mathbb{R}^2$ (or the $A$-external Feuerbach point of $ABC$ if $A$, $B$, $C$ are placed anti-clockwise on $\mathbb{R}^2$).
\item[12.] \hbox{\scalebox{0.75}{$\triangle$}}ext{Inner(outer) Morley point} $M$, $M(A, B, C)$ -- the $A$-vertex of the inner Morley triangle of $ABC$ if $A$, $B$, $C$ are placed clockwise on the plane $\mathbb{R}^2$ (or the $A$-vertex of the outer Morley triangle of $ABC$ if $A$, $B$, $C$ are placed anti-clockwise on $\mathbb{R}^2$).
\end{enumerate}
\end{lis}
\section{Definitions of \hbox{\scalebox{0.75}{$\triangle$}}exorpdfstring{$\hbox{\scalebox{0.75}{$\triangle$}}$}{t}-objects}
begin{definition}\label{d10}
Denote by a {\em $\hbox{\scalebox{0.75}{$\triangle$}}$-point} any 6-tuple of points lying on the plane $\mathbb{R}^2$.
\end{definition}
begin{remark}
Similarly, one can introduce $\hbox{\scalebox{0.75}{$\triangle$}}$-lines, $\hbox{\scalebox{0.75}{$\triangle$}}$-circles and other $\hbox{\scalebox{0.75}{$\triangle$}}$-curves, but we omit them in this article for simplicity.
\end{remark}
begin{definition}
For each $\hbox{\scalebox{0.75}{$\triangle$}}$-point $x = (x_1, x_2, x_3, x_4, x_5, x_6)$ define $$x_{bc} = x_1,\:\: x_{cb} = x_2,\:\: x_{ca} = x_3,\:\: x_{ac} = x_4,\:\: x_{ab} = x_5,\:\: x_{ba} = x_6.$$
\end{definition}
begin{example}\label{lam}
Consider the Van Lamoen configuration (see \cite{van}) -- a triangle $ABC$ with the centroid $G$, the cevian triangle $A'B'C'$ of $G$ wrt $ABC$ and the circumcenters $O_{bc} = O(GBC'),\ldots, $\\$O_{ba} = O(GBA')$ of $GBC',\ldots, GBA'$. In this configuration it is possible to define the following $\hbox{\scalebox{0.75}{$\triangle$}}$-points $x = (A, A, B, B, C, C)$, $y = (G, \ldots , G)$, $z = (A', A', B', B', C', C')$, $t = (O_{bc}, \ldots , O_{ba})$. Thus, $x_{bc} = A,\ldots, x_{ba} = C$, $y_{bc} = G,\ldots, y_{ba} = G$, $z_{bc} =A',\ldots, z_{ba} = C'$, $t_{bc} = O_{bc},\ldots, t_{ba} = O_{ba}$.
\end{example}
begin{definition}\label{d15}
Denote by a {\em $\hbox{\scalebox{0.75}{$\triangle$}}$-function} any function $f$ which corresponds a non-empty set of $\hbox{\scalebox{0.75}{$\triangle$}}$-points to each $\hbox{\scalebox{0.75}{$\triangle$}}ext{ar}(f)$-tuple of $\hbox{\scalebox{0.75}{$\triangle$}}$-points, and is one of the functions $f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, i}$ which are listed in the Appendix of this article (see List \ref{ltf} in the Appendix).\S_{\triangle}ome of the $\hbox{\scalebox{0.75}{$\triangle$}}$-functions in use with corresponding numbers:\\ (these are $\hbox{\scalebox{0.75}{$\triangle$}}$-functions which will be used in the further definitions and examples)begin{enumerate}
\item[1.] $f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 1} = $ the set of all $\hbox{\scalebox{0.75}{$\triangle$}}$-points ($f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 1}$ has arity $0$ and, thus, is a set of $\hbox{\scalebox{0.75}{$\triangle$}}$-points. Same can be said about $f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, i}$, $1\leq i\leq 8$).
\item[2.] $f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 2} = \{x\hbox{\scalebox{0.75}{$\triangle$}}ext{ is a }\hbox{\scalebox{0.75}{$\triangle$}}\hbox{\scalebox{0.75}{$\triangle$}}ext{-point} \:\vert\: x_{bc} = x_{cb}, x_{ca} = x_{ac}, x_{ab} = x_{ba}\}$.
\item[8.] $f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 8} = \{x\hbox{\scalebox{0.75}{$\triangle$}}ext{ is a }\hbox{\scalebox{0.75}{$\triangle$}}\hbox{\scalebox{0.75}{$\triangle$}}ext{-point}\:\vert\: x_{bc},\ldots, x_{ba}\hbox{\scalebox{0.75}{$\triangle$}}ext{ lie on a circle}\}$.
\item[11.] $f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 11}( x) = \{y\hbox{\scalebox{0.75}{$\triangle$}}ext{ is a }\hbox{\scalebox{0.75}{$\triangle$}}\hbox{\scalebox{0.75}{$\triangle$}}ext{-point}\:\vert\: y_{bc} = x_{bc},\ldots, y_{ba} = x_{ba}\hbox{\scalebox{0.75}{$\triangle$}}ext{ i.e. }y = x\}$.
\item[17.] $f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 17}( x, y, z) = \left\{t\hbox{\scalebox{0.75}{$\triangle$}}ext{ is a }\hbox{\scalebox{0.75}{$\triangle$}}\hbox{\scalebox{0.75}{$\triangle$}}ext{-point}\left\vertbegin{array}{cl} t_{bc},\ldots, t_{ba}\hbox{\scalebox{0.75}{$\triangle$}}ext{ are the projections of}\\x_{bc}, \ldots, x_{ba}\hbox{\scalebox{0.75}{$\triangle$}}ext{ on }y_{bc}z_{bc},\ldots, y_{ba}z_{ba}\end{array}\right.\right\}$.
\item[19.] $f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, {19, i}}( x, y, z) = \left\{t\hbox{\scalebox{0.75}{$\triangle$}}ext{ is a }\hbox{\scalebox{0.75}{$\triangle$}}\hbox{\scalebox{0.75}{$\triangle$}}ext{-point}\left\vertbegin{array}{cl} t_{bc} = X_i(x_{bc}, y_{bc}, z_{bc}), t_{cb} = X_i(x_{cb}, z_{cb}, y_{cb}),
\\ t_{ca} = X_i(z_{ca}, x_{ca}, y_{ca}), t_{ac} = X_i(y_{ac}, x_{ac}, z_{ac}),
\\ t_{ab} = X_i(y_{ab}, z_{ab}, x_{ab}), t_{ba} = X_i(z_{ba}, y_{ba}, x_{ba})\end{array}\right.\right\}$,\\where $1\leq i\leq 13$, $ X_i$ denotes the $i$-th center from the list \ref{l1}.
\item[20.] $f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, {20}}( x, y, z, t) = \left\{v\hbox{\scalebox{0.75}{$\triangle$}}ext{ is a }\hbox{\scalebox{0.75}{$\triangle$}}\hbox{\scalebox{0.75}{$\triangle$}}ext{-point}\left\vertbegin{array}{cl} v_{bc} = x_{bc}y_{bc}\cap z_{bc}t_{bc},\ldots,\\ v_{ba} = x_{ba}y_{ba}\cap z_{ba}t_{ba}
\end{array}\right.\right\}$.
\item[25.] Functions of the form $f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, n, alpha, beta, \gamma}(x, y, z) := f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, n}(x^{alpha}, y^{beta}, z^{\gamma})$, $1\leq n\leq 24$, where $alpha, beta, \gamma$ are any symbols from the set $\{bc, cb, ca, ac, ab, ba\}$ and for each $\hbox{\scalebox{0.75}{$\triangle$}}$-point $x$,\\$x^{bc} := (x_{bc}, x_{cb}, x_{ca}, x_{ac}, x_{ab}, x_{ba}),\quad x^{ac} := (x_{ac}, x_{ca}, x_{cb}, x_{bc}, x_{ba}, x_{ab}),$\\$x^{cb} := (x_{cb}, x_{bc}, x_{ba}, x_{ab}, x_{ac}, x_{ca}),\quad x^{ba} := (x_{ba}, x_{ab}, x_{ac}, x_{ca}, x_{cb}, x_{bc}),$\\$x^{ab} := (x_{ab}, x_{ba}, x_{bc}, x_{cb}, x_{ca}, x_{ac}),\quad x^{ca} := (x_{ca}, x_{ac}, x_{ab}, x_{ba}, x_{bc}, x_{cb})$,\\ denotes the orbit of $x$.
\end{enumerate}
\end{definition}
begin{definition}\label{N}
Consider the sequence $x_1, x_2, x_3, \ldots$ of free variables which can be any $\hbox{\scalebox{0.75}{$\triangle$}}$-points. Denote by a {\em $\hbox{\scalebox{0.75}{$\triangle$}}$-configuration} any logical statement about the sequence $x_1, x_2, x_3,\ldots$, which has the form $$bigwedge_{i = 1}^N[x_{a_i}\in f_i(x_{b_{i,1}}, x_{b_{i, 2}}, \ldots, x_{b_{i,\hbox{\scalebox{0.75}{$\triangle$}}ext{ar} (f_i)}})], $$wherebegin{enumerate}
\item $N$ is a natural number
\item $a_1 < a_2 < \ldots < a_N$ is a strongly increasing sequence of natural numbers
\item for each $1\leq i\leq N$, $f_i$ is a $\hbox{\scalebox{0.75}{$\triangle$}}$-function
\item for each $1\leq i\leq N$, $b_{i, 1}, b_{i, 2},\ldots, b_{i,\hbox{\scalebox{0.75}{$\triangle$}}ext{ar} (f_i)} < a_i$ is an $\hbox{\scalebox{0.75}{$\triangle$}}ext{ar} (f_i)$-tuple of natural numbers $<a_i$.
\end{enumerate}
\end{definition}
Since statements of the form $[x_i\in f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 1}]$ don't carry any additional information, we will omit such terms within $\hbox{\scalebox{0.75}{$\triangle$}}$-configurations (i.e. we may not consider the $\hbox{\scalebox{0.75}{$\triangle$}}$-function $f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 1}$).
begin{definition}
For any $\hbox{\scalebox{0.75}{$\triangle$}}$-configuration $c$ denote by $\hbox{\scalebox{0.75}{$\triangle$}}ext{deg}(c)$, $\hbox{\scalebox{0.75}{$\triangle$}}ext{height}(c)$ the values of $N$ and $a_N$ from the definition \ref{N} which are related to $c$, respectively.
\end{definition}
begin{example}\label{s}
Consider the following $\hbox{\scalebox{0.75}{$\triangle$}}$-configuration $c$ which is related to the example \ref{lam}$$c = [x_1\in f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 2}]\wedge [x_2\in f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 19, 2}(x_1, x_1^{ab}, x_1^{ac})]\wedge [x_3\in f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 20}(x_1, x_2, x_1^{ab}, x_1^{ac})]\wedge [x_5\in f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 19, 3}(x_1, x_2, x_3^{ab})].$$So $\hbox{\scalebox{0.75}{$\triangle$}}ext{deg}(c) = 4$, $\hbox{\scalebox{0.75}{$\triangle$}}ext{height}(c) = 5$.
\end{example}
begin{definition}
Let $C_{\hbox{\scalebox{0.75}{$\triangle$}}riangle}$ denote the set of all $\hbox{\scalebox{0.75}{$\triangle$}}$-configurations. Also for each natural $n$ let $C_{\triangle}^n$ denote the set of all $\hbox{\scalebox{0.75}{$\triangle$}}$-configurations $c$ with $\hbox{\scalebox{0.75}{$\triangle$}}ext{height}(c)\leq n$. We will say that $c, d\inC_{\triangle}$ are {\em equivalent} if there exists a permutation of variables $\sigma : x_1, x_2, x_3,\ldots\hbox{\scalebox{0.75}{$\triangle$}}o x_1, x_2, x_3,\ldots$ which sends $c$ to $d$, i.e. $\sigma(c) = d$. We will label $c\simeq d$ for each equivalent $c, d\inC_{\triangle}$.
\end{definition}
begin{definition}
For each $\hbox{\scalebox{0.75}{$\triangle$}}$-configuration $$c = bigwedge_{i = 1}^N[x_{a_i}\in f_i(x_{b_{i,1}}, x_{b_{i, 2}}, \ldots, x_{b_{i,\hbox{\scalebox{0.75}{$\triangle$}}ext{ar} (f_i)}})], $$let $\hbox{\scalebox{0.75}{$\triangle$}}ext{terms}(c)$ denote the set of $\hbox{\scalebox{0.75}{$\triangle$}}ext{deg} = 1$ $\hbox{\scalebox{0.75}{$\triangle$}}$-configurations $$\hbox{\scalebox{0.75}{$\triangle$}}ext{terms}(c) := \{[x_{a_i}\in f_i(x_{b_{i,1}}, x_{b_{i, 2}}, \ldots, x_{b_{i,\hbox{\scalebox{0.75}{$\triangle$}}ext{ar} (f_i)}})]\:\vert\: 1\leq i\leq N\}.$$Also we will say that $d\inC_{\triangle}$ is a {\em predecessor} of $c$ if $$d = bigwedge_{i = 1}^M[x_{a_i}\in f_i(x_{b_{i,1}}, x_{b_{i, 2}}, \ldots, x_{b_{i,\hbox{\scalebox{0.75}{$\triangle$}}ext{ar} (f_i)}})]$$for some $1\leq M\leq N$.
\end{definition}
begin{definition}
For each $\hbox{\scalebox{0.75}{$\triangle$}}$-configurations $c, c_1, c_2,\ldots, c_l$ we say that $c = \cup_{i = 1}^lc_i = c_1\cup\ldots\cup c_l$ if $\hbox{\scalebox{0.75}{$\triangle$}}ext{terms}(c) = \cup_{i = 1}^l\hbox{\scalebox{0.75}{$\triangle$}}ext{terms}(c_i) = \hbox{\scalebox{0.75}{$\triangle$}}ext{terms}(c_1)\cup\ldots\cup\hbox{\scalebox{0.75}{$\triangle$}}ext{terms}(c_l)$. Also for each $\hbox{\scalebox{0.75}{$\triangle$}}$-configurations $c, d$ we say that $c \subseteq d$ if $\hbox{\scalebox{0.75}{$\triangle$}}ext{terms}(c) \subseteq \hbox{\scalebox{0.75}{$\triangle$}}ext{terms}(d)$, and $c \subsetneq d$ if $\hbox{\scalebox{0.75}{$\triangle$}}ext{terms}(c) \subsetneq \hbox{\scalebox{0.75}{$\triangle$}}ext{terms}(d)$.
\end{definition}
begin{definition}
For each $\hbox{\scalebox{0.75}{$\triangle$}}$-configurations $c, d$ we say that $c \leq d$ if there exist $\hbox{\scalebox{0.75}{$\triangle$}}$-configurations $c'\simeq c, d'\simeq d$, which are equivalent to $c, d$ respectively and are such that $c'$ is a predecessor of $d'$.
\end{definition}
begin{definition}\label{conf}
Denote by a {\em $\hbox{\scalebox{0.75}{$\triangle$}}$-theorem} any valid implication of the form $c \Rightarrow r$, $c, r\inC_{\triangle}$, where $\hbox{\scalebox{0.75}{$\triangle$}}ext{deg}(r) = 1$.
\end{definition}
begin{example}
Consider the $\hbox{\scalebox{0.75}{$\triangle$}}$-configuration $c$ as in the example \ref{s}, and let $r = [x_5\in f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 8}]$. Then from the Van Lamoen theorem (see \cite{van}) we have that $c \Rightarrow r$ is a $\hbox{\scalebox{0.75}{$\triangle$}}$-theorem.
\end{example}
begin{definition}\label{abc}
Consider a triangle $ABC$ lying on the plane $\mathbb{R}^2$ in general position. For each $\hbox{\scalebox{0.75}{$\triangle$}}$-configuration $$c = bigwedge_{i = 1}^N[x_{a_i}\in f_i(x_{b_{i,1}}, x_{b_{i, 2}}, \ldots, x_{b_{i,\hbox{\scalebox{0.75}{$\triangle$}}ext{ar} (f_i)}})]\inC_{\triangle}, $$ let $c(ABC)$ denotebegin{enumerate}
\item the set of $\hbox{\scalebox{0.75}{$\triangle$}}$-points $\{x_{a_1}, x_{a_2},\ldots, x_{a_N}\}$ satisfying the system of equations $$\left\{begin{array}{cl}x_{a_1} = (A, A, B, B, C, C)\in f_1(x_{b_{1,1}}, x_{b_{1, 2}}, \ldots, x_{b_{1,\hbox{\scalebox{0.75}{$\triangle$}}ext{ar} (f_1)}})
\\ x_{a_2}\in f_2(x_{b_{2,1}}, x_{b_{2, 2}}, \ldots, x_{b_{2,\hbox{\scalebox{0.75}{$\triangle$}}ext{ar} (f_2)}})
\\\ldots
\\ x_{a_N}\in f_N(x_{b_{N,1}}, x_{b_{N, 2}}, \ldots, x_{b_{N,\hbox{\scalebox{0.75}{$\triangle$}}ext{ar} (f_N)}})
\end{array}\right.$$if $f_1 = f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 2}$, $a_i = i$ ($1\leq i\leq N$), and this system of equations has the unique solution
\item $c(ABC) = \varnothing$, otherwise.
\end{enumerate}
\end{definition}
begin{example}
Consider the following $\hbox{\scalebox{0.75}{$\triangle$}}$-configuration $c'$$$c' = [x_1\in f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 2}]\wedge [x_2\in f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 19, 2}(x_1, x_1^{ab}, x_1^{ac})]\wedge [x_3\in f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 20}(x_1, x_2, x_1^{ab}, x_1^{ac})]\wedge [x_4\in f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 19, 3}(x_1, x_2, x_3^{ab})].$$So $c'$ is equivalent to $c$ from the example \ref{s}, and $c'(ABC) \not= \varnothing$.
\end{example}
begin{definition}
For each triangle $ABC$ in general position and each set $S\subseteq C_{\triangle}$, denote $$S(ABC) := bigcup_{c\in S}c(ABC).$$
\end{definition}
\section{Construction of \hbox{\scalebox{0.75}{$\triangle$}}exorpdfstring{$S_{\hbox{\scalebox{0.75}{$\triangle$}}riangle}^7$}{T7S7}}
begin{definition}
For a $\hbox{\scalebox{0.75}{$\triangle$}}$-configuration $c$ denote by $\normalfont{\hbox{\scalebox{0.75}{$\triangle$}}ext{Gen}(c)}$ the set of {\em generalizations of $c$}, where $$\normalfont{\hbox{\scalebox{0.75}{$\triangle$}}ext{Gen}(c)} \!:=\! \left\{d\inC_{\triangle}\left\vertbegin{array}{cl} \hbox{\scalebox{0.75}{$\triangle$}}ext{there exist }\hbox{\scalebox{0.75}{$\triangle$}}\hbox{\scalebox{0.75}{$\triangle$}}ext{-configurations }c'\simeq c, d'\simeq d,\hbox{\scalebox{0.75}{$\triangle$}}ext{ which are equivalent to }c, d\\\hbox{\scalebox{0.75}{$\triangle$}}ext{respectively and are such that: }
\\\:1.\:\: c'\Rightarrow d'\hbox{\scalebox{0.75}{$\triangle$}}ext{ is a valid implication, and}
\\\:2.\:\: c'\not\Leftrightarrow d'
\end{array}\right.\right\}$$
\end{definition}
begin{remark}
It is also possible to implement a larger set of generalizations of $c$ by further considering cases when $d' = \sigma(d)$ for some surjective (and not necessarily bijective) mapping of variables $\sigma : x_1, x_2, x_3,\ldots\hbox{\scalebox{0.75}{$\triangle$}}o x_1, x_2, x_3,\ldots$, and adding some additional condition 3 (see for example the generalization of Gergonne theorem in \cite[Theorem 9.1(1), p.13]{SNT}). However, we will omit such generalizations for simplicity.
\end{remark}
begin{definition}
A set $S\subseteqC_{\triangle}$ is called {\em complete} (we will also call such a set as ``maximally general'') if for each $d\in S$ the set of all $\hbox{\scalebox{0.75}{$\triangle$}}$-theorems of the form $c\Rightarrow r$, $c\in S$, $r\inC_{\triangle}$ can't be deductively derived\footnote{Here by ``can be deductively derived'' we mean ``can be derived with using the set of inference rules:
begin{equation}
\inference{(a, b, c\hbox{\scalebox{0.75}{$\triangle$}}ext{ are any logical statements}) & a \Rightarrow b & b \Rightarrow c}
{a \Rightarrow c}
\end{equation}
begin{equation}
\inference{(a, b, c\hbox{\scalebox{0.75}{$\triangle$}}ext{ are any logical statements})}
{a\Leftrightarrow a\wedge a\qquad a\wedge b \Rightarrow a\qquad a\wedge b \Rightarrow b\qquad a\wedge b \Leftrightarrow b\wedge a\qquad (a\wedge b)\wedge c \Leftrightarrow a\wedge (b\wedge c)}
\end{equation}
begin{equation}
\inference{(a_1, a_2, b_1, b_2\hbox{\scalebox{0.75}{$\triangle$}}ext{ are any logical statements}) & a_1 \Rightarrow b_1 & a_2 \Rightarrow b_2}
{a_1\wedge a_2 \Rightarrow b_1\wedge b_2}
\end{equation}
begin{equation}
\inference{(c, d\inC_{\triangle}, \sigma : x_1, x_2, x_3,\ldots\hbox{\scalebox{0.75}{$\triangle$}}o x_1, x_2, x_3,\ldots\hbox{\scalebox{0.75}{$\triangle$}}ext{ is a surjective mapping of variables}) & c \Rightarrow d }
{\sigma (c) \Rightarrow \sigma (d)}
\end{equation}
begin{equation}
\inference{(c_1, c_2, d\inC_{\triangle}) & c_1\Rightarrow d & c_2\hbox{\scalebox{0.75}{$\triangle$}}ext{ is a predecessor of } c_1 & \hbox{\scalebox{0.75}{$\triangle$}}ext{height}(d)\leq\hbox{\scalebox{0.75}{$\triangle$}}ext{height}(c_2)}
{c_2 \Rightarrow d}.
\end{equation}
} from the set of $\hbox{\scalebox{0.75}{$\triangle$}}$-theorems of the form $c\Rightarrow r$, $c\in (S\setminus\{d\})\cup\hbox{\scalebox{0.75}{$\triangle$}}ext{Gen}(d)$, $r\inC_{\triangle}$.
\end{definition}
begin{definition}
Consider a natural number $n$. A set $S\subseteqC_{\triangle}$ is called {\em $n$-complete} if it is complete and each $\hbox{\scalebox{0.75}{$\triangle$}}$-theorem of the form $c\Rightarrow r$, $c\inC_{\triangle}^n$, $r\inC_{\triangle}$ can be deductively derived$^1$ from the set of $\hbox{\scalebox{0.75}{$\triangle$}}$-theorems of the form $c\Rightarrow r$, $c\in S$, $r\inC_{\triangle}$.
\end{definition}
Next, we will be interested in computing $7$-complete sets, however none of these sets can be computed in practice, and we finish this section by constructing its computable analogue $S_{\triangle}^7$.
begin{definition}\label{cgen}
A $\hbox{\scalebox{0.75}{$\triangle$}}$-theorem $c \Rightarrow r$ is called {\em computably generalizable} if there exists a set of $\hbox{\scalebox{0.75}{$\triangle$}}$-theorems of the form $\{c_i \Rightarrow r_i, d \Rightarrow r\:\vert\: 1\leq i\leq l\}$, such that $l\geq 1$, $c_i\subsetneq c$, $d = \cup_{i = 1}^l r_i$, $d\not\Leftrightarrow c$ ($1\leq i\leq l$). Obviously each computably generalizable $\hbox{\scalebox{0.75}{$\triangle$}}$-theorem $c\Rightarrow r$ can be deductively derived$^1$ from the set of $\hbox{\scalebox{0.75}{$\triangle$}}$-theorems $\{c_i \Rightarrow r_i, d \Rightarrow r\:\vert\: 1\leq i\leq l\}$ and, thus, from the set of $\hbox{\scalebox{0.75}{$\triangle$}}$-theorems of the form $c'\Rightarrow r'$, $c'\in\hbox{\scalebox{0.75}{$\triangle$}}ext{Gen}(c)$, $r'\inC_{\triangle}$.
\end{definition}
begin{definition}\label{cccgen}
Consider a triangle $ABC$ in general position. For a $\hbox{\scalebox{0.75}{$\triangle$}}$-configuration $c$ denote by $\normalfont{\hbox{\scalebox{0.75}{$\triangle$}}ext{Gen}_{\hbox{\scalebox{0.75}{$\triangle$}}riangle}(c)}$ the set of {\em triangular computable generalizations of $c$}, where $$\normalfont{\hbox{\scalebox{0.75}{$\triangle$}}ext{Gen}_{\hbox{\scalebox{0.75}{$\triangle$}}riangle}(c)} := \{d\inC_{\triangle}\:\vert\: d(ABC)\not=\varnothing, d\leq c,\hbox{\scalebox{0.75}{$\triangle$}}ext{ and } d\not\simeq c\}.$$Also for a $\hbox{\scalebox{0.75}{$\triangle$}}$-configuration $c$, define begin{equation*}
\hbox{\scalebox{0.75}{$\triangle$}}ext{CGen}_{\hbox{\scalebox{0.75}{$\triangle$}}riangle}(c) := begin{cases}\hbox{\scalebox{0.75}{$\triangle$}}ext{Gen}_{\hbox{\scalebox{0.75}{$\triangle$}}riangle}(c), &\hbox{\scalebox{0.75}{$\triangle$}}ext{if each }\hbox{\scalebox{0.75}{$\triangle$}}\hbox{\scalebox{0.75}{$\triangle$}}ext{-theorem of the form }c \Rightarrow r, r\inC_{\triangle} \hbox{\scalebox{0.75}{$\triangle$}}ext{ with}\\&\hbox{\scalebox{0.75}{$\triangle$}}ext{height}(r)\leq \hbox{\scalebox{0.75}{$\triangle$}}ext{height}(c)\hbox{\scalebox{0.75}{$\triangle$}}ext{ is computably generalizable},\\c, &\hbox{\scalebox{0.75}{$\triangle$}}ext{otherwise}.\end{cases}\end{equation*}Additionally, for a set $S\subseteq C_{\triangle}$, define the setsbegin{itemize}
\item $\hbox{\scalebox{0.75}{$\triangle$}}ext{CGen}_{\hbox{\scalebox{0.75}{$\triangle$}}riangle}(S) := \cup_{c\in S}\hbox{\scalebox{0.75}{$\triangle$}}ext{CGen}_{\hbox{\scalebox{0.75}{$\triangle$}}riangle}(c)$,
\item $\hbox{\scalebox{0.75}{$\triangle$}}ext{CGen}_{\hbox{\scalebox{0.75}{$\triangle$}}riangle}^1(S) := \hbox{\scalebox{0.75}{$\triangle$}}ext{CGen}_{\hbox{\scalebox{0.75}{$\triangle$}}riangle}(S)$,
\item $\hbox{\scalebox{0.75}{$\triangle$}}ext{CGen}_{\hbox{\scalebox{0.75}{$\triangle$}}riangle}^{i + 1}(S) := \hbox{\scalebox{0.75}{$\triangle$}}ext{CGen}_{\hbox{\scalebox{0.75}{$\triangle$}}riangle}(\hbox{\scalebox{0.75}{$\triangle$}}ext{CGen}_{\hbox{\scalebox{0.75}{$\triangle$}}riangle}^i(S))$, $i = 1, 2, 3,\ldots$,
\item $\hbox{\scalebox{0.75}{$\triangle$}}ext{MaxCGen}_{\hbox{\scalebox{0.75}{$\triangle$}}riangle}(S) := \hbox{\scalebox{0.75}{$\triangle$}}ext{CGen}_{\hbox{\scalebox{0.75}{$\triangle$}}riangle}^d(S)$, where $d$ is the minimal natural number such that $\hbox{\scalebox{0.75}{$\triangle$}}ext{CGen}_{\hbox{\scalebox{0.75}{$\triangle$}}riangle}^d(S) = \hbox{\scalebox{0.75}{$\triangle$}}ext{CGen}_{\hbox{\scalebox{0.75}{$\triangle$}}riangle}^{d + 1}(S)$.
\end{itemize}
\end{definition}
begin{remark}
The sets $\hbox{\scalebox{0.75}{$\triangle$}}ext{CGen}_{\hbox{\scalebox{0.75}{$\triangle$}}riangle}(S)$, $\hbox{\scalebox{0.75}{$\triangle$}}ext{CGen}_{\hbox{\scalebox{0.75}{$\triangle$}}riangle}^i(S)$, $\hbox{\scalebox{0.75}{$\triangle$}}ext{MaxCGen}_{\hbox{\scalebox{0.75}{$\triangle$}}riangle}(S)$ can be calculated in practice with the help of Propositions \ref{jjjj}, \ref{jj} from Section 5.
\end{remark}
begin{definition}\label{d}
Define the sets $S_{\triangle}^n\subseteq C_{\triangle}^n$, $n\geq 1$ inductively. Let $S_{\triangle}^1 := \{[x_1\in f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 2}]\}$. Assume that for a natural $t\geq 1$ the set $S_{\triangle}^t\subseteqC_{\triangle}^t$ is already constructed. Consider the sets begin{enumerate}
\item[] $J := \left\{c\inC_{\triangle}^{t + 1}\left\vertbegin{array}{cl} c = c_1\cup c_2\inC_{\triangle}^{t + 1}\hbox{\scalebox{0.75}{$\triangle$}}ext{ for some}
\\c_1\in S_{\triangle}^t, c_2\inC_{\triangle}^{t + 1}\hbox{\scalebox{0.75}{$\triangle$}}ext{ with }\hbox{\scalebox{0.75}{$\triangle$}}ext{deg}(c_2) = 1, \hbox{\scalebox{0.75}{$\triangle$}}ext{height}(c_2) = t + 1\end{array}\right.\right\},$
\item[] $S_{\triangle}^{t + 1} := S_{\triangle}^t\cup\hbox{\scalebox{0.75}{$\triangle$}}ext{MaxCGen}_{\hbox{\scalebox{0.75}{$\triangle$}}riangle}(J).$
\end{enumerate}From this inductive process construct the sets $S_{\triangle}^1\subseteq S_{\triangle}^2\subseteq \ldots\subseteq S_{\triangle}^n\subseteq\ldots$. It is easy to see that $S_{\triangle}^n = \hbox{\scalebox{0.75}{$\triangle$}}ext{MaxCGen}_{\hbox{\scalebox{0.75}{$\triangle$}}riangle}(S_{\triangle}^n)$ for each $n\geq 1$.
\end{definition}
In what follows, we will be interested in computing the set $S_{\triangle}^7$. The set $S_{\triangle}^7$ can be seen as a computable analogue and an approximation of $7$-complete sets.
begin{remark}\label{rlc}
Note that when calculating $S_{\triangle}^7$, on each new step, we don't need to list those $\hbox{\scalebox{0.75}{$\triangle$}}$-configurations that have already been listed.
\end{remark}
The set $S_{\hbox{\scalebox{0.75}{$\triangle$}}riangle}^{7}$ can be computed in practice from its definition with the help of Remark \ref{rlc} and the Propositions \ref{jjjj}, \ref{jj} from Section 5.
\section{Automated generation of theorems based on \hbox{\scalebox{0.75}{$\triangle$}}exorpdfstring{$S_{\hbox{\scalebox{0.75}{$\triangle$}}riangle}^{7}$}{S7}}
In this section, we introduce an algorithm for a computer that generates and categorizes triangle geometry theorems based on the set $S_{\hbox{\scalebox{0.75}{$\triangle$}}riangle}^{7}$.
begin{definition}
For each $\hbox{\scalebox{0.75}{$\triangle$}}$-point $a\in S_{\hbox{\scalebox{0.75}{$\triangle$}}riangle}^{7}(ABC)$ let $f_1(a), f_2(a), \ldots, f_{\gamma(a)}(a)$ denote the sequence of $\hbox{\scalebox{0.75}{$\triangle$}}$-functions which are used for the definition of $a$ and are ordered according their appearance. Also denote by $\Gamma(a)$ the sequence $(f_1(a), f_2(a),\ldots, f_{\gamma(a)}(a))$ after excluding those $f_k(a), 1\leq k\leq \gamma(a)$ which do not have the form $f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 19, i}$ for some $1\leq i\leq 13$, and then replacing each uniform segment of the remaining sequence of $\hbox{\scalebox{0.75}{$\triangle$}}$-functions with the single $\hbox{\scalebox{0.75}{$\triangle$}}$-function of the same type (for example, if $a\in S_{\hbox{\scalebox{0.75}{$\triangle$}}riangle}^{7}(ABC)$ is such that $(f_1(a), f_2(a), \ldots, f_{\gamma(a)}(a)) = (f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 2}, f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 19, 1}, f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 3}, f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 19,1}, f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 19, 9})$, then $\Gamma(a) = (f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 19, 1}, f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 19, 9})$).
\end{definition}
The next algorithm generates triangle theorems based on the computation of $S_{\hbox{\scalebox{0.75}{$\triangle$}}riangle}^{7}(ABC)$. Also it produces an intuitive categorization of theorems, the same as in the articles \cite{cos, cos1}.
begin{al}\label{d425}
The computer program inputs a sequence $X_{i_1}, X_{i_2}, \ldots, X_{i_d}$, $1\leq i_1, i_2,\ldots, i_d\leq 13$, $d\geq 1$ of triangle centers from the list \ref{l1} which has no uniform segments (i.e. $i_k\not=i_{k + 1}$, $1\leq k < d$), and then produces the output after following the steps below.
begin{enumerate}
\item For a triangle $ABC$ in general position, compute the set $S_{\triangle}^7(ABC)$.
\item In ``Objects $X_{i_1}- X_{i_2}- \ldots- X_{i_d}$'' section print the definitions and notations of all $\hbox{\scalebox{0.75}{$\triangle$}}$-points $a\in S_{\hbox{\scalebox{0.75}{$\triangle$}}riangle}^{7}(ABC)$ with $\Gamma(a) = (f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 19, i_1}, f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 19, i_2},\ldots, f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 19, i_d})$.
As in the ETC \cite{etc}, we can label objects from the section ``Objects $X_{i_1}- X_{i_2}- \ldots- X_{i_d}$'' as $(X_{i_1}- X_{i_2}- \ldots- X_{i_d})_1$, $(X_{i_1}- X_{i_2}- \ldots- X_{i_d})_2$, $(X_{i_1}- X_{i_2}- \ldots- X_{i_d})_3$, $\ldots$.
\item Compute and print in ``Properties $X_{i_1}- X_{i_2}- \ldots- X_{i_d}$'' section all correct statements of the form $[a\in f(a_1, a_{2}, \ldots, a_{\hbox{\scalebox{0.75}{$\triangle$}}ext{ar} (f)})]$, where $f$ is any $\hbox{\scalebox{0.75}{$\triangle$}}$-function and $a$, $a_{1}$, $a_{2}$, $\ldots$, $a_{\hbox{\scalebox{0.75}{$\triangle$}}ext{ar} (f)}$ are any $\hbox{\scalebox{0.75}{$\triangle$}}$-points from the section ``Objects $X_{i_1}- X_{i_2}- \ldots- X_{i_d}$''.
\end{enumerate}
\end{al}
begin{remark}
Note that in the section ``Objects $X_{i_1}- X_{i_2}- \ldots- X_{i_d}$'' from the algorithm \ref{d425} for each object $x$ it is possible to leave only the representative $(x_{bc}, \ldots, x_{ba})$ of the orbit of elements$$x = x^{bc} := (x_{bc}, x_{cb}, x_{ca}, x_{ac}, x_{ab}, x_{ba}),\quad x^{ac} := (x_{ac}, x_{ca}, x_{cb}, x_{bc}, x_{ba}, x_{ab}),$$$$x^{cb} := (x_{cb}, x_{bc}, x_{ba}, x_{ab}, x_{ac}, x_{ca}),\quad x^{ba} := (x_{ba}, x_{ab}, x_{ac}, x_{ca}, x_{cb}, x_{bc}),$$$$x^{ab} := (x_{ab}, x_{ba}, x_{bc}, x_{cb}, x_{ca}, x_{ac}),\quad x^{ca} := (x_{ca}, x_{ac}, x_{ab}, x_{ba}, x_{bc}, x_{cb}).$$We also need to replace all sequences $x, y,\ldots, z$ of objects from ``Objects $X_{i_1}- X_{i_2}- \ldots- X_{i_d}$'' that have the same coordinates as $\hbox{\scalebox{0.75}{$\triangle$}}$-points (i.e. are such that $x = y = \ldots = z$) on the single object $x$, and list all descriptions of $x$, that are coming from $x, y, \ldots, z$, in the definition of $x$.
\end{remark}
\subsection{Relation to the articles [8, 9]}We expect that a computer program based on the algorithm \ref{d425} will be able to generate almost all of the theorems from the articles \cite{cos, cos1}.
\section{Practical implementation}\label{sss}
begin{definition}
Consider any $\hbox{\scalebox{0.75}{$\triangle$}}$-configuration $$c = bigwedge_{i = 1}^N[x_{a_i}\in f_i(x_{b_{i,1}}, x_{b_{i, 2}}, \ldots, x_{b_{i,\hbox{\scalebox{0.75}{$\triangle$}}ext{ar} (f_i)}})]\inC_{\triangle}. $$ For each $l\geq 1$, denote by $O_l(c)$ the set of all $\hbox{\scalebox{0.75}{$\triangle$}}$-configurations $c'$ which have the following form $$c' = \left(bigwedge_{\substack{1\leq i\leq N \: :\: a_i< l}}[x_{a_i}\in f_i(x_{b_{i,1}}, x_{b_{i, 2}}, \ldots, x_{b_{i,\hbox{\scalebox{0.75}{$\triangle$}}ext{ar} (f_i)}})]\right)\wedge$$$$\wedge [x_{l}\in f(x_{b_{1}}, x_{b_{ 2}}, \ldots, x_{b_{\hbox{\scalebox{0.75}{$\triangle$}}ext{ar} (f)}})]\wedge$$$$\wedge\left(bigwedge_{\substack{1\leq i\leq N \: :\: a_i > l}}[x_{a_i}\in f_i(x_{b_{i,1}}, x_{b_{i, 2}}, \ldots, x_{b_{i,\hbox{\scalebox{0.75}{$\triangle$}}ext{ar} (f_i)}})]\right)$$for some $\hbox{\scalebox{0.75}{$\triangle$}}ext{deg} = 1$, $\hbox{\scalebox{0.75}{$\triangle$}}ext{height} = l$ $\hbox{\scalebox{0.75}{$\triangle$}}$-configuration $[x_{l}\in f(x_{b_{1}}, x_{b_{ 2}}, \ldots, x_{b_{\hbox{\scalebox{0.75}{$\triangle$}}ext{ar} (f)}})]$.
\end{definition}
The following propositions \ref{jjjj}, \ref{jj} can be used for computing $\hbox{\scalebox{0.75}{$\triangle$}}ext{Gen}_{\hbox{\scalebox{0.75}{$\triangle$}}riangle}(\cdot)$, $\hbox{\scalebox{0.75}{$\triangle$}}ext{CGen}_{\hbox{\scalebox{0.75}{$\triangle$}}riangle}(\cdot)$, \\$\hbox{\scalebox{0.75}{$\triangle$}}ext{MaxCGen}_{\hbox{\scalebox{0.75}{$\triangle$}}riangle}(\cdot)$, and $S_{\triangle}^7$ from the definitions \ref{cccgen}, \ref{d}.
For a $\hbox{\scalebox{0.75}{$\triangle$}}$-configuration $c\inC_{\triangle}^7$, the set $\{e\inC_{\triangle}^7\:\vert\: e\leq c, e\not\simeq c,\hbox{\scalebox{0.75}{$\triangle$}}ext{ and } e(ABC)\not=\varnothing\}$ can be easily computed in practice by brutal force method, thus to compute $\hbox{\scalebox{0.75}{$\triangle$}}ext{Gen}_{\hbox{\scalebox{0.75}{$\triangle$}}riangle}(c)$ it is enough to develop a method for calculating the set $\{d\inC_{\triangle}^7\:\vert\: c\Leftrightarrow d\}$. The next proposition \ref{jjjj} describes such a method.
begin{proposition}\label{jjjj}
Consider any $\hbox{\scalebox{0.75}{$\triangle$}}$-configuration $c\inC_{\triangle}^7$. The set $\{d\inC_{\triangle}^7\:\vert\: c\Leftrightarrow d\}$ can be computed after providing the following steps:begin{enumerate}
\item consider the set $D_1$ of all $d_1\in O_{7}(c)$ with $d_1 \Leftrightarrow c$
\item consider the set $D_2$ of all $d_2\in \cup_{d_1\in D_1}O_{6}(d_1)$ with $d_2 \Leftrightarrow c$
\item repeat step 3 for $D_2$ instead of $D_1$ and finish with the set $D_3 = \{d_3\in \cup_{d_2\in D_2}O_{5}(d_2)\:\vert\: d_3 \Leftrightarrow c\}$
\item repeat step 3 for $D_3, D_4,\ldots$ until we finish with the set $D_7$ which satisfies $D_7 = \{d\inC_{\triangle}^7\:\vert\: c\Leftrightarrow d\}$.
\end{enumerate}
\end{proposition}
To compute the sets $\hbox{\scalebox{0.75}{$\triangle$}}ext{CGen}_{\hbox{\scalebox{0.75}{$\triangle$}}riangle}(\cdot)$, $\hbox{\scalebox{0.75}{$\triangle$}}ext{MaxCGen}_{\hbox{\scalebox{0.75}{$\triangle$}}riangle}(\cdot)$, we need to use the method of computation of $\hbox{\scalebox{0.75}{$\triangle$}}ext{Gen}_{\hbox{\scalebox{0.75}{$\triangle$}}riangle}(c)$, which was described previously, and also to develop a method for checking whether a given $\hbox{\scalebox{0.75}{$\triangle$}}$-theorem $c \Rightarrow r$ with $c, r\inC_{\triangle}^7$ is computationally generalizable. The next proposition \ref{jj} describes such a method.
begin{proposition}\label{jj}
Consider any $\hbox{\scalebox{0.75}{$\triangle$}}$-theorem $c\Rightarrow r$ such that $c, r\inC_{\triangle}^7$. Then to understand whether $c \Rightarrow r$ is computationally generalizable we need to provide the following steps:begin{enumerate}
\item Compute the set $$U_c := \{d\inC_{\triangle}^7\:\vert\: \hbox{\scalebox{0.75}{$\triangle$}}ext{deg}(d) = 1\hbox{\scalebox{0.75}{$\triangle$}}ext{ and }c' \Rightarrow d,\hbox{\scalebox{0.75}{$\triangle$}}ext{ for some }c'\subsetneq c\}$$
\item consider the set $D_1$ of all $d_1\in O_{7}(c)$ with $d_1 \Rightarrow r$ and $\hbox{\scalebox{0.75}{$\triangle$}}ext{terms}(d_1)\subseteq U_c$
\item if there exists $d_1\in D_1$ with $d_1\not\Leftrightarrow c$, then finish with the string ``$c \Rightarrow r$ is computationally generalizable''. Otherwise consider the set $D_2$ of all $d_2\in \cup_{d_1\in D_1}O_{6}(d_1)$ with $d_2 \Rightarrow r$ and $\hbox{\scalebox{0.75}{$\triangle$}}ext{terms}(d_2)\subseteq U_c$
\item repeat step 3 for $d_2\in D_2$ instead of $d_1\in D_1$ and finish either with the string ``$c \Rightarrow r$ is computationally generalizable'', or with the set $$D_3 = \{d_3\in \cup_{d_2\in D_2}O_{5}(d_2)\:\vert\: d_3 \Rightarrow r, \hbox{\scalebox{0.75}{$\triangle$}}ext{terms}(d_3)\subseteq U_c\}$$
\item repeat step 3 for $D_3, D_4,\ldots$ until we finish either with the string ``$c \Rightarrow r$ is computationally generalizable'', or with the set $D_7$, and in the latter case return the string ``$c \Rightarrow r$ is not computationally generalizable''.
\end{enumerate}
\end{proposition}
Propositions \ref{jjjj}, \ref{jj} are trivial consequences of the following Proposition \ref{j}.
begin{proposition}\label{j}
Consider any $\hbox{\scalebox{0.75}{$\triangle$}}$-configurations $$c = bigwedge_{i = 1}^N[x_{a_i}\in f_i(x_{b_{i,1}}, x_{b_{i, 2}}, \ldots, x_{b_{i,\hbox{\scalebox{0.75}{$\triangle$}}ext{ar} (f_i)}})]\inC_{\triangle}, $$ $$c' = bigwedge_{i = 1}^{N'}[x_{a_i'}\in f_i'(x_{b_{i,1}'}, x_{b_{i, 2}'}, \ldots, x_{b_{i,\hbox{\scalebox{0.75}{$\triangle$}}ext{ar} (f_i')}'})]\inC_{\triangle}, $$ with $c\Rightarrow c'$. Then we have that for each natural $l$, $\hbox{\scalebox{0.75}{$\triangle$}}$-configurationsbegin{enumerate}
\item $c(l) = \displaystyle{bigwedge_{\substack{i = 1\a_i\leq l}}^N[x_{a_i}\in f_i(x_{b_{i,1}}, x_{b_{i, 2}}, \ldots, x_{b_{i,\hbox{\scalebox{0.75}{$\triangle$}}ext{ar} (f_i)}})]}$
\item $c'(l) = \displaystyle{bigwedge_{\substack{i = 1\a_i'\leq l}}^{N'}[x_{a_i'}\in f_i'(x_{b_{i,1}'}, x_{b_{i, 2}'}, \ldots, x_{b_{i,\hbox{\scalebox{0.75}{$\triangle$}}ext{ar} (f_i')}'})]}$
\item $\displaystyle{c''(l) = bigwedge_{\substack{i = 1\a_i'> l}}^{N'}[x_{a_i'}\in f_i'(x_{b_{i,1}'}, x_{b_{i, 2}'}, \ldots, x_{b_{i,\hbox{\scalebox{0.75}{$\triangle$}}ext{ar} (f_i)}'})]}$
\item $d(l) = c(l)\wedge c''(l)$
\end{enumerate}
are such that $c(l)\Rightarrow c'(l)$, $c\Rightarrow d(l)\Rightarrow c'$.
\end{proposition}begin{proof}Proposition \ref{j} follows from the fact that for each $l\geq 1$ and each $\hbox{\scalebox{0.75}{$\triangle$}}$-configuration $$c = bigwedge_{i = 1}^N[x_{a_i}\in f_i(x_{b_{i,1}}, x_{b_{i, 2}}, \ldots, x_{b_{i,\hbox{\scalebox{0.75}{$\triangle$}}ext{ar} (f_i)}})]\inC_{\triangle}, $$variables $x_{i}$ inside $c$ with $1\leq i\leq l$ are independent of variables $x_j$ inside $c$ with $j > l$.\end{proof}
\section{Appendix}
This appendix contains the complete lists of triangle centers and $\hbox{\scalebox{0.75}{$\triangle$}}$-functions that we use in this article.
begin{lis}\label{ltc}
The list of triangle centers
begin{enumerate}
\item \hbox{\scalebox{0.75}{$\triangle$}}ext{In(ex)center} $I$, $I(A, B, C)$ -- the in center of $ABC$ if $A$, $B$, $C$ are placed clockwise on the plane $\mathbb{R}^2$ (or the $A$-excenter of $ABC$ if $A$, $B$, $C$ are placed anti-clockwise on $\mathbb{R}^2$).
\item \hbox{\scalebox{0.75}{$\triangle$}}ext{Centroid} $G$, $G(A, B, C)$ -- the centroid of $ABC$.
\item \hbox{\scalebox{0.75}{$\triangle$}}ext{Circumcenter} $O$, $O(A, B, C)$ -- the circumcenter of $ABC$.
\item \hbox{\scalebox{0.75}{$\triangle$}}ext{Orthocenter} $H$, $H(A, B, C)$ -- the orthocenter of $ABC$.
\item \hbox{\scalebox{0.75}{$\triangle$}}ext{Nine-point center} $N$, $N(A, B, C)$ -- the nine-point center of $ABC$.
\item \hbox{\scalebox{0.75}{$\triangle$}}ext{Symmedian point} $S$, $S(A, B, C)$ -- the Symmedian point of $ABC$.
\item \hbox{\scalebox{0.75}{$\triangle$}}ext{First(second) Fermat point} $F$, $F(A, B, C)$ -- the first Fermat point of $ABC$ if $A$, $B$, $C$ are placed clockwise on the plane $\mathbb{R}^2$ (or the second Fermat point of $ABC$ if $A$, $B$, $C$ are placed anti-clockwise on $\mathbb{R}^2$).
\item \hbox{\scalebox{0.75}{$\triangle$}}ext{First(second) Isodynamic point} $I_s$, $I_s(A, B, C)$ -- the first isodynamic point of $ABC$ if $A$, $B$, $C$ are placed clockwise on the plane $\mathbb{R}^2$ (or the second Isodynamic point of $ABC$ if $A$, $B$, $C$ are placed anti-clockwise on $\mathbb{R}^2$).
\item \hbox{\scalebox{0.75}{$\triangle$}}ext{Inner(outer) Feuerbach point} $F_e$, $F_e(A, B, C)$ -- the inner Feuerbach point of $ABC$ if $A$, $B$, $C$ are placed clockwise on the plane $\mathbb{R}^2$ (or the $A$-external Feuerbach point of $ABC$ if $A$, $B$, $C$ are placed anti-clockwise on $\mathbb{R}^2$).
\item \hbox{\scalebox{0.75}{$\triangle$}}ext{Euler reflection point} $E$, $E(A, B, C)$ -- the Euler Reflection point of $ABC$.
\item \hbox{\scalebox{0.75}{$\triangle$}}ext{Inner(outer) Apollonian point} $A_p$, $A_p(A, B, C)$ -- the $A$-vertex of inner Apollonian triangle of $ABC$ if $A$, $B$, $C$ are placed clockwise on the plane $\mathbb{R}^2$ (or the $A$ -- vertex of outer Apollonian triangle of $ABC$ if $A$, $B$, $C$ are placed anti-clockwise on $\mathbb{R}^2$).
\item \hbox{\scalebox{0.75}{$\triangle$}}ext{Inner(outer) Morley point} $M$, $M(A, B, C)$ -- the $A$-vertex of inner Morley triangle of $ABC$ if $A$, $B$, $C$ are placed clockwise on the plane $\mathbb{R}^2$ (or the $A$-vertex of outer Morley triangle of $ABC$ if $A$, $B$, $C$ are placed anti-clockwise on $\mathbb{R}^2$).
\item \hbox{\scalebox{0.75}{$\triangle$}}ext{Isogonal point} $Iso$, $Iso(A, B, C, D)$ -- the Isogonal conjugation of $D$ wrt $ABC$.
\item Other similar triangular centers and lines.
\end{enumerate}
\end{lis}
begin{lis}\label{ltf}
The list of $\hbox{\scalebox{0.75}{$\triangle$}}$-functions
begin{enumerate}
\item $f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 1} = $ the set of all $\hbox{\scalebox{0.75}{$\triangle$}}$-points ($f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 1}$ has arity $0$ and, thus, is a set of $\hbox{\scalebox{0.75}{$\triangle$}}$-points. Same can be said about $f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, i}$, $1\leq i\leq 8$).
\item $f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 2} = \{x\hbox{\scalebox{0.75}{$\triangle$}}ext{ is a }\hbox{\scalebox{0.75}{$\triangle$}}\hbox{\scalebox{0.75}{$\triangle$}}ext{-point} \:\vert\: x_{bc} = x_{cb}, x_{ca} = x_{ac}, x_{ab} = x_{ba}\}$.
\item $f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 3} = \left\{x\hbox{\scalebox{0.75}{$\triangle$}}ext{ is a }\hbox{\scalebox{0.75}{$\triangle$}}\hbox{\scalebox{0.75}{$\triangle$}}ext{-point} \left\vertbegin{array}{cl} x_{bc} = x_{cb}, x_{ca} = x_{ac}, x_{ab} = x_{ba}
\\\hbox{\scalebox{0.75}{$\triangle$}}ext{and the triangle }x_{bc}x_{ca}x_{ab}\hbox{\scalebox{0.75}{$\triangle$}}ext{ is equilateral}\end{array}\right.\right\}$.
\item $f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 4} = \{x\hbox{\scalebox{0.75}{$\triangle$}}ext{ is a }\hbox{\scalebox{0.75}{$\triangle$}}\hbox{\scalebox{0.75}{$\triangle$}}ext{-point}\:\vert\: x_{bc} = x_{ca} = x_{ab}, x_{cb} = x_{ac} = x_{ba}\}$.
\item $f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 5} = \{x\hbox{\scalebox{0.75}{$\triangle$}}ext{ is a }\hbox{\scalebox{0.75}{$\triangle$}}\hbox{\scalebox{0.75}{$\triangle$}}ext{-point}\:\vert\: x_{bc} = \ldots = x_{ba}\}$.
\item $f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 6} = \{x\hbox{\scalebox{0.75}{$\triangle$}}ext{ is a }\hbox{\scalebox{0.75}{$\triangle$}}\hbox{\scalebox{0.75}{$\triangle$}}ext{-point}\:\vert\:\hbox{\scalebox{0.75}{$\triangle$}}ext{points }x_{bc},\ldots, x_{ba}\hbox{\scalebox{0.75}{$\triangle$}}ext{ are collinear}\}$.
\item $f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 7} = \{x\hbox{\scalebox{0.75}{$\triangle$}}ext{ is a }\hbox{\scalebox{0.75}{$\triangle$}}\hbox{\scalebox{0.75}{$\triangle$}}ext{-point}\:\vert\: x_{bc},\ldots, x_{ba}\hbox{\scalebox{0.75}{$\triangle$}}ext{ lie on a conic}\}$.
\item $f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 8} = \{x\hbox{\scalebox{0.75}{$\triangle$}}ext{ is a }\hbox{\scalebox{0.75}{$\triangle$}}\hbox{\scalebox{0.75}{$\triangle$}}ext{-point}\:\vert\: x_{bc},\ldots, x_{ba}\hbox{\scalebox{0.75}{$\triangle$}}ext{ lie on a circle}\}$.
\item $f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 9}( x) = \{y\hbox{\scalebox{0.75}{$\triangle$}}ext{ is a }\hbox{\scalebox{0.75}{$\triangle$}}\hbox{\scalebox{0.75}{$\triangle$}}ext{-point}\:\vert\: \hbox{\scalebox{0.75}{$\triangle$}}ext{lines }x_{bc}y_{bc}, \ldots, x_{ba}y_{ba}\hbox{\scalebox{0.75}{$\triangle$}}ext{ are concurrent}\}$.
\item $f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 10}( x) = \left\{y\hbox{\scalebox{0.75}{$\triangle$}}ext{ is a }\hbox{\scalebox{0.75}{$\triangle$}}\hbox{\scalebox{0.75}{$\triangle$}}ext{-point}\left\vertbegin{array}{cl} \hbox{\scalebox{0.75}{$\triangle$}}ext{the midpoints of segments}
\\x_{bc}y_{bc}, \ldots, x_{ba}y_{ba}\hbox{\scalebox{0.75}{$\triangle$}}ext{ are collinear}
\end{array}\right.\right\}$.
\item $f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 11}( x) = \{y\hbox{\scalebox{0.75}{$\triangle$}}ext{ is a }\hbox{\scalebox{0.75}{$\triangle$}}\hbox{\scalebox{0.75}{$\triangle$}}ext{-point}\:\vert\: y_{bc} = x_{bc},\ldots, y_{ba} = x_{ba}\hbox{\scalebox{0.75}{$\triangle$}}ext{ i.e. }y = x\}$.
\item $f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 12}( x) = \left\{y\hbox{\scalebox{0.75}{$\triangle$}}ext{ is a }\hbox{\scalebox{0.75}{$\triangle$}}\hbox{\scalebox{0.75}{$\triangle$}}ext{-point}\left\vertbegin{array}{cl} y_{bc} = y_{cb}, y_{ca} = y_{ac}, y_{ab} = y_{ba}\hbox{\scalebox{0.75}{$\triangle$}}ext{ and}
\\\hbox{\scalebox{0.75}{$\triangle$}}ext{the triangle }x_{bc}x_{cb}\cap x_{ca}x_{ac}\cap x_{ab}x_{ba}\\\hbox{\scalebox{0.75}{$\triangle$}}ext{is perspective to }y_{bc}y_{ca}y_{ab}
\end{array}\right.\right\}$.
\item $f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 13}( x) = \left\{y\hbox{\scalebox{0.75}{$\triangle$}}ext{ is a }\hbox{\scalebox{0.75}{$\triangle$}}\hbox{\scalebox{0.75}{$\triangle$}}ext{-point}\left\vertbegin{array}{cl} \hbox{\scalebox{0.75}{$\triangle$}}ext{---/--- }x_{bc}x_{cb}\cap x_{ca}x_{ac}\cap x_{ab}x_{ba}\\\hbox{\scalebox{0.75}{$\triangle$}}ext{is orthologic to }y_{bc}y_{ca}y_{ab}
\end{array}\right.\right\}$.
\item $f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 14}( x) = \left\{y\hbox{\scalebox{0.75}{$\triangle$}}ext{ is a }\hbox{\scalebox{0.75}{$\triangle$}}\hbox{\scalebox{0.75}{$\triangle$}}ext{-point}\left\vertbegin{array}{cl} y_{bc} = \ldots = y_{ba}\hbox{\scalebox{0.75}{$\triangle$}}ext{ and }y_{bc}\hbox{\scalebox{0.75}{$\triangle$}}ext{ lies on the}\\\hbox{\scalebox{0.75}{$\triangle$}}ext{circumcircle of the triangle}
\\x_{bc}x_{cb}\cap x_{ca}x_{ac}\cap x_{ab}x_{ba}
\end{array}\right.\right\}$.
\item $f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 15}( x, y) = \left\{z\hbox{\scalebox{0.75}{$\triangle$}}ext{ is a }\hbox{\scalebox{0.75}{$\triangle$}}\hbox{\scalebox{0.75}{$\triangle$}}ext{-point}\left\vertbegin{array}{cl} z_{bc},\ldots, z_{ba}\hbox{\scalebox{0.75}{$\triangle$}}ext{ coincides with the}
\\\hbox{\scalebox{0.75}{$\triangle$}}ext{midpoints of }x_{bc}y_{bc},\ldots, x_{ba}y_{ba}\hbox{\scalebox{0.75}{$\triangle$}}ext{ resp.}
\end{array}\right.\right\}$.
\item $f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 16}( x, y) = \left\{z\hbox{\scalebox{0.75}{$\triangle$}}ext{ is a }\hbox{\scalebox{0.75}{$\triangle$}}\hbox{\scalebox{0.75}{$\triangle$}}ext{-point}\left\vertbegin{array}{cl} z_{bc},\ldots, z_{ba}\hbox{\scalebox{0.75}{$\triangle$}}ext{ lie on the lines}\\x_{bc}y_{bc},\ldots, x_{ba}y_{ba},\hbox{\scalebox{0.75}{$\triangle$}}ext{ resp.}
\end{array}\right.\right\}$.
\item $f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 17}( x, y, z) = \left\{t\hbox{\scalebox{0.75}{$\triangle$}}ext{ is a }\hbox{\scalebox{0.75}{$\triangle$}}\hbox{\scalebox{0.75}{$\triangle$}}ext{-point}\left\vertbegin{array}{cl} t_{bc},\ldots, t_{ba}\hbox{\scalebox{0.75}{$\triangle$}}ext{ are the projections}
\\\hbox{\scalebox{0.75}{$\triangle$}}ext{of }x_{bc}, \ldots, x_{ba}\hbox{\scalebox{0.75}{$\triangle$}}ext{ on }y_{bc}z_{bc},\ldots, y_{ba}z_{ba}\end{array}\right.\right\}$.
\item $f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 18}( x, y, z) = \left\{t\hbox{\scalebox{0.75}{$\triangle$}}ext{ is a }\hbox{\scalebox{0.75}{$\triangle$}}\hbox{\scalebox{0.75}{$\triangle$}}ext{-point}\left\vertbegin{array}{cl} t_{bc},\ldots, t_{ba}\hbox{\scalebox{0.75}{$\triangle$}}ext{ are the reflections}
\\\hbox{\scalebox{0.75}{$\triangle$}}ext{of }x_{bc},\ldots, x_{ba}\hbox{\scalebox{0.75}{$\triangle$}}ext{ wrt }y_{bc}z_{bc},\ldots, y_{ba}z_{ba}\end{array}\right.\right\}$.
\item $f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, {19, i}}( x, y, z) =\left\{t\hbox{\scalebox{0.75}{$\triangle$}}ext{ is a }\hbox{\scalebox{0.75}{$\triangle$}}\hbox{\scalebox{0.75}{$\triangle$}}ext{-point}\left\vertbegin{array}{cl} t_{bc} = X_i(x_{bc}, y_{bc}, z_{bc}), t_{cb} = X_i(x_{cb}, z_{cb}, y_{cb}),
\\ t_{ca} = X_i(z_{ca}, x_{ca}, y_{ca}), t_{ac} = X_i(y_{ac}, x_{ac}, z_{ac}),
\\ t_{ab} = X_i(y_{ab}, z_{ab}, x_{ab}), t_{ba} = X_i(z_{ba}, y_{ba}, x_{ba})\end{array}\right.\right\}$,\\where $1\leq i\leq 12$, $ X_i$ denotes the $i$-th center from the list \ref{l1}.
\item $f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, {20}}( x, y, z, t) = \left\{v\hbox{\scalebox{0.75}{$\triangle$}}ext{ is a }\hbox{\scalebox{0.75}{$\triangle$}}\hbox{\scalebox{0.75}{$\triangle$}}ext{-point}\left\vertbegin{array}{cl} v_{bc} = x_{bc}y_{bc}\cap z_{bc}t_{bc},\ldots,\\ v_{ba} = x_{ba}y_{ba}\cap z_{ba}t_{ba}
\end{array}\right.\right\}$.
\item $f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 21}( x, y, z, t) = \left\{begin{array}{cl}v\hbox{\scalebox{0.75}{$\triangle$}}ext{ is a}\\\hbox{\scalebox{0.75}{$\triangle$}}\hbox{\scalebox{0.75}{$\triangle$}}ext{-point}\end{array}\left\vertbegin{array}{cl} v_{bc} = X_{13}(x_{bc}, y_{bc}, z_{bc}, t_{bc}), v_{cb} = X_{13}(x_{cb}, z_{cb}, y_{cb}, t_{cb}),
\\v_{ca} = X_{13}(z_{ca}, x_{ca}, y_{ca}, t_{ca}), v_{ac} = X_{13}(y_{ac}, x_{ac}, z_{ac}, t_{ac}), \\v_{ab} = X_{13}(y_{ab}, z_{ab}, x_{ab}, t_{ab}), v_{ba} = X_{13}(z_{ba}, y_{ba}, x_{ba}, t_{ba})
\end{array}\right.\right\}$,\\where $ X_{13}$ denote the $13$-th center from the list \ref{l1}.
\item $f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 22}(x, y, z, t) = \left\{v\hbox{\scalebox{0.75}{$\triangle$}}ext{ is a }\hbox{\scalebox{0.75}{$\triangle$}}\hbox{\scalebox{0.75}{$\triangle$}}ext{-point}\left\vertbegin{array}{cl}\hbox{\scalebox{0.75}{$\triangle$}}ext{points }v_{bc},\ldots, v_{ba}\hbox{\scalebox{0.75}{$\triangle$}}ext{ lie on the pivotal}
\\\hbox{\scalebox{0.75}{$\triangle$}}ext{isocubics of triangles}
\\x_{bc}y_{bc}z_{bc},\ldots, x_{ba}y_{ba}z_{ba}\hbox{\scalebox{0.75}{$\triangle$}}ext{ with pivots}\\hbox{\scalebox{0.75}{$\triangle$}}_{bc},\ldots, t_{ba},\hbox{\scalebox{0.75}{$\triangle$}}ext{ respectively}
\end{array}\right.\right\}$.
\item $f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 23}(x, y, z) = \left\{t\hbox{\scalebox{0.75}{$\triangle$}}ext{ is a }\hbox{\scalebox{0.75}{$\triangle$}}\hbox{\scalebox{0.75}{$\triangle$}}ext{-point}\:\vert\: x_{bc}y_{bc}z_{bc}t_{bc},\ldots, x_{ba}y_{ba}z_{ba}t_{ba}\hbox{\scalebox{0.75}{$\triangle$}}ext{ are cyclic}\right\}$.
\item $f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, 24}( x, y) = \left\{z\hbox{\scalebox{0.75}{$\triangle$}}ext{ is a }\hbox{\scalebox{0.75}{$\triangle$}}\hbox{\scalebox{0.75}{$\triangle$}}ext{-point}\left\vertbegin{array}{cl} z_{bc}\hbox{\scalebox{0.75}{$\triangle$}}ext{ lies on the rectangular hyperbola}\\\hbox{\scalebox{0.75}{$\triangle$}}ext{passing through the vertices of the}
\\\hbox{\scalebox{0.75}{$\triangle$}}ext{triangle }x_{bc}x_{cb}\cap x_{ca}x_{ac}\cap x_{ab}x_{ba}
\\\hbox{\scalebox{0.75}{$\triangle$}}ext{and the point }y_{bc},
\\\hbox{\scalebox{0.75}{$\triangle$}}ext{and similarly for }z_{cb},\ldots, z_{ba}
\end{array}\right.\right\}$.
\item Functions of the form $f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, n, alpha, beta, \gamma}(x, y, z) := f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, n}(x^{alpha}, y^{beta}, z^{\gamma})$, $1\leq n\leq 24$, where $alpha, beta, \gamma$ are any symbols from the set $\{bc, cb, ca, ac, ab, ba\}$ and for each $\hbox{\scalebox{0.75}{$\triangle$}}$-point $x$,\\$x^{bc} := (x_{bc}, x_{cb}, x_{ca}, x_{ac}, x_{ab}, x_{ba}),\quad x^{ac} := (x_{ac}, x_{ca}, x_{cb}, x_{bc}, x_{ba}, x_{ab}),$\\$x^{cb} := (x_{cb}, x_{bc}, x_{ba}, x_{ab}, x_{ac}, x_{ca}),\quad x^{ba} := (x_{ba}, x_{ab}, x_{ac}, x_{ca}, x_{cb}, x_{bc}),$\\$x^{ab} := (x_{ab}, x_{ba}, x_{bc}, x_{cb}, x_{ca}, x_{ac}),\quad x^{ca} := (x_{ca}, x_{ac}, x_{ab}, x_{ba}, x_{bc}, x_{cb})$,\\ denotes the orbit of $x$.
\item Other similar functions $f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, i}$ and, for example, we can consider
$f_{\hbox{\scalebox{0.75}{$\triangle$}}riangle, i} = \left\{x\hbox{\scalebox{0.75}{$\triangle$}}ext{ is a }\hbox{\scalebox{0.75}{$\triangle$}}\hbox{\scalebox{0.75}{$\triangle$}}ext{-point}\left\vertbegin{array}{cl} x_{bc}x_{ca}x_{ab}\hbox{\scalebox{0.75}{$\triangle$}}ext{ is similar}\\\hbox{\scalebox{0.75}{$\triangle$}}ext{(perspective, orthologic) to }x_{cb}x_{ac}x_{ba}\end{array}\right.\right\}$.
\end{enumerate}
\end{lis}
addcontentsline{toc}{section}{Bibliography}
bibliographystyle{unsrt}
begin{thebibliography}{10}
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\end{document} |
\begin{document}
\title{\LARGE\bf An alternative representation of the Vi\'{e}te\text{'}s formula for pi by Chebyshev polynomials of the first kind}
\author{
\normalsize\bf S. M. Abrarov\footnote{\scriptsize{Dept. Earth and Space Science and Engineering, York University, Toronto, Canada, M3J 1P3.}}\, and B. M. Quine$^{*}$\footnote{\scriptsize{Dept. Physics and Astronomy, York University, Toronto, Canada, M3J 1P3.}}}
\date{September 19, 2016}
\maketitle
\begin{abstract}
There are several reformulations of the Vi\'{e}te\text{'}s formula for pi that have been reported in the modern literature. In this paper we show another analog to the Vi\'{e}te\text{'}s formula for pi by Chebyshev polynomials of the first kind.
\\
\noindent {\bf Keywords:} Chebyshev polynomials, sinc function, cosine infinite product, Vi\'{e}te\text{'}s formula, constant pi
\end{abstract}
\section{Introduction}
The sinc function, also known as the cardinal sine function, is defined as \cite{Gearhart1990, Kac1959}
\[
\text{sinc}\left( t \right)=\left\{ \begin{aligned}
\frac{\sin \left( t \right)}{t}, &\qquad\qquad t \ne 0 \\
1, &\qquad\qquad t = 0. \\
\end{aligned}
\right.
\]
The sinc function finds many applications in sampling, spectral methods, differential equations and numerical integration \cite{Gearhart1990, Stenger2011, Rybicki1989, Lether1998, Quine2013, Abrarov2015, Ortiz-Gracia2016}.
More than four centuries ago the French lawyer and amateur mathematician Fran\c{c}ois Vi\'{e}te found a fabulous relation showing how the sinc function can be represented elegantly as an infinite product of the cosines \cite{Gearhart1990, Kac1959}
\begin{equation}\label{eq_1}
\text{sinc}\left( t \right)=\prod\limits_{m=1}^{\infty }{\cos \left( \frac{t}{{{2}^{m}}} \right)}.
\end{equation}
Since
$$
\text{sinc}\left( \frac{\pi }{2} \right)=\frac{2}{\pi }
$$
we may attempt to substitute the argument $t=\pi /2$ into right side of equation \eqref{eq_1}. Thus, using repeatedly for each $m$ the following cosine identity for double angle
$$
\cos \left( 2\theta_m \right)=2{{\cos }^{2}}\left( \theta_m \right)-1
$$
or
$$
\cos \left( \theta_m \right)=2{{\cos }^{2}}\left( \theta_{m+1} \right)-1 \Leftrightarrow \cos \left( {{\theta }_{m+1}} \right)=\sqrt{\frac{\cos \left( {{\theta }_{m}} \right)+1}{2}},
$$
where
$$
\theta_m =\frac{\pi /2}{{{2}^{m}}}, \quad \theta_{m+1} = \frac{\theta_{m}}{2}
$$
and taking into account that
$$
\cos \left( \theta_{1} \right) = \cos \left( \frac{\pi /2}{{{2}^{1}}} \right)=\frac{\sqrt{2}}{2},
$$
we can find the following sequence
$$
\cos \left( \frac{\pi /2}{{{2}^{2}}} \right)=\frac{\sqrt{2+\sqrt{2}}}{2},
$$
$$
\cos \left( \frac{\pi /2}{{{2}^{3}}} \right)=\frac{\sqrt{2+\sqrt{2+\sqrt{2}}}}{2},
$$
$$
\vdots
$$
\begin{equation}\label{eq_2}
\cos \left( \frac{\pi /2}{{{2}^{m}}} \right)=\frac{\overbrace{\sqrt{2+\sqrt{2+\sqrt{2+\cdots +\sqrt{2}}}}}^{m\,\,\text{square}\,\,\text{roots}}}{2}.
\end{equation}
Therefore, from the equations \eqref{eq_1} and \eqref{eq_2} we obtain the Vi\'{e}te\text{'}s infinite product formula for the constant pi (in radicals consisting of square roots and twos only) \cite{Osler1999, Servi2003, Levin2005, Levin2006, Kreminski2008}
\[
\begin{aligned}
\text{sinc}\left( \frac{\pi}{2} \right) &=\cos \left( \frac{\pi /2}{{{2}^{1}}} \right)\cos \left( \frac{\pi /2}{{{2}^{2}}} \right)\cos \left( \frac{\pi /2}{{{2}^{3}}} \right)\cdots \\
& =\frac{\sqrt{2}}{2}\frac{\sqrt{2+\sqrt{2}}}{2}\frac{\sqrt{2+\sqrt{2+\sqrt{2}}}}{2}\cdots = \frac{2}{\pi }
\end{aligned}
\]
that can be conveniently rewritten as
\begin{equation}\label{eq_3}
\frac{2}{\pi }=\underset{M\to \infty }{\mathop{\lim }}\,\prod\limits_{m=1}^{M}{\frac{{{a}_{m}}}{2}},
\end{equation}
where ${{a}_{m}}=\sqrt{2+{{a}_{m-1}}}$ and ${{a}_{1}}=\sqrt{2}$.
Several reformulations of the Vi\'{e}te\text{'}s formula \eqref{eq_3} for pi have been reported in the modern literature \cite{Osler1999, Servi2003, Levin2005, Levin2006, Kreminski2008}. Notably, Osler has shown by \text{``}double product\text{''} generalization a direct relationship between the classical Vi\'{e}te\text{'}s and Wallis\text{'}s infinite products for pi (see equation (3) in \cite{Osler1999}). In this work we derive another equivalent to the Vi\'{e}te\text{'}s formula for pi expressed in terms of the Chebyshev polynomials of the first kind.
\section{Derivation}
The Chebyshev polynomials ${{T}_{m}}\left( x \right)$ of the first kind can be defined by the following recurrence relation \cite{Press1992, Mathews1999, Zwillinger2012}
$$
{{T}_{m}}\left( x \right)=2x{{T}_{m-1}}\left( x \right)-{{T}_{m-2}}\left( x \right),
$$
where ${{T}_{1}}\left( x \right)=x$ and ${{T}_{0}}\left( x \right)=1$. It should be noted that the recurrence procedure is not required in computation since these polynomials can also be determined directly by using, for example, a simple identity
\[
T_{m}\left( x \right)=x^{m}\sum\limits_{n=0}^{\left\lfloor m/2 \right\rfloor } \binom{m}{2n}{{\left( {1 - {x}^{-2}} \right)}^{n}}.
\]
Due to a remarkable property of the Chebyshev polynomials
$$
\cos \left( m \alpha \right) = {{T}_{m}}\left( \cos \left( \alpha \right) \right),
$$
making change of the variable in form
$$
\alpha = \cos \left( \frac{t}{{{2}^{M}}} \right)
$$
results in
\begin{equation}\label{eq_4}
\cos \left( \frac{2m-1}{{{2}^{M}}}t \right)={{T}_{2m-1}}\left( \cos \left( \frac{t}{{{2}^{M}}} \right) \right).
\end{equation}
Consequently, substituting equation \eqref{eq_4} into the following product-to-sum identity \cite{Quine2013, Abrarov2015, Ortiz-Gracia2016}
\begin{equation}\label{eq_5}
\prod\limits_{m=1}^{M}{\cos \left( \frac{t}{{{2}^{m}}} \right)}=\frac{1}{{{2}^{M-1}}}\sum\limits_{m=1}^{{{2}^{M-1}}}{\cos \left( \frac{2m-1}{{{2}^{M}}}t \right)}
\end{equation}
yields
\begin{equation}\label{eq_6}
\prod\limits_{m=1}^{M}{\cos \left( \frac{t}{{{2}^{m}}} \right)}=\frac{1}{{{2}^{M-1}}}\sum\limits_{m=1}^{{{2}^{M-1}}}{{{T}_{2m-1}}\left( \cos \left( \frac{t}{{{2}^{M}}} \right) \right)}.
\end{equation}
It can be shown that the right side of the equation \eqref{eq_6} can be further simplified and represented by a single Chebyshev polynomial of the second kind (see {\it{Appendix A}}).
Comparing equations \eqref{eq_1} and \eqref{eq_5} we can see that the infinite product of cosines for the sinc function can be transformed into infinite sum of cosines \cite{Abrarov2015}
\begin{equation}\label{eq_7}
\text{sinc}\left( t \right)=\underset{M\to \infty }{\mathop{\lim }}\,\frac{1}{{{2}^{M-1}}}\sum\limits_{m=1}^{{{2}^{M-1}}}{\cos \left( \frac{2m-1}{{{2}^{M}}}t \right)}.
\end{equation}
Since the right side of equation \eqref{eq_5} represents a truncation of the limit \eqref{eq_7} by a finite value of upper integer ${{2}^{M-1}}$ in summation, it is simply the incomplete cosine expansion of the sinc function. Indeed, if the condition ${{2}^{M-1}}>>1$ is satisfied, then the incomplete cosine expansion of the sinc function quite accurately approximates the original sinc function as given by \cite{Abrarov2015, Ortiz-Gracia2016}
$$
\frac{1}{{{2}^{M-1}}}\sum\limits_{m=1}^{{{2}^{M-1}}}{\cos \left( \frac{2m-1}{{{2}^{M}}}t \right)}\approx \text{sinc}\left( t \right).
$$
It is interesting to note that comparing equations \eqref{eq_1} and \eqref{eq_6} we can also write now
\footnotesize
$$
\text{sinc}\left( t \right)=\underset{M\to \infty }{\mathop{\lim }}\,\frac{1}{{{2}^{M-1}}}\sum\limits_{m=1}^{{{2}^{M-1}}}{{{T}_{2m-1}}\cos \left( \frac{t}{{{2}^{M}}} \right)}=\underset{M\to \infty }{\mathop{\lim }}\,\frac{1}{{{2}^{M-1}}}\sum\limits_{m=1}^{{{2}^{M-1}}}{{{T}_{2m-1}}\cos \left( \frac{t/2}{{{2}^{M-1}}} \right)}
$$
\normalsize
or
$$
\text{sinc}\left( t \right)=\underset{L\to \infty }{\mathop{\lim }}\,\frac{1}{L}\sum\limits_{\ell =1}^{L}{{{T}_{2\ell -1}}\cos \left( \frac{t}{2L} \right)},
$$
since we can imply that $L={{2}^{M-1}}$.
At $t=\pi /2$ the equation \eqref{eq_6} provides
\begin{equation}\label{eq_8}
\prod\limits_{m=1}^{M}{\cos \left( \frac{\pi /2}{{{2}^{m}}} \right)}=\frac{1}{{{2}^{M-1}}}\sum\limits_{m=1}^{{{2}^{M-1}}}{{{T}_{2m-1}}\left( \cos \left( \frac{\pi /2}{{{2}^{M}}} \right) \right)}.
\end{equation}
Applying equation \eqref{eq_2} again for each $m$ repeatedly, the product-to-sum identity \eqref{eq_8} can be rearranged in form
\footnotesize
\[
\begin{aligned}
\frac{\sqrt{2}}{2}\frac{\sqrt{2+\sqrt{2}}}{2}\frac{\sqrt{2+\sqrt{2+\sqrt{2}}}}{2} & \ldots \frac{\overbrace{\sqrt{2+\sqrt{2+\sqrt{2+\cdots +\sqrt{2}}}}}^{M\,\,\text{square}\,\,\text{roots}}}{2}\\
& = \frac{1}{{{2}^{M-1}}}\sum\limits_{m=1}^{{{2}^{M-1}}}{{{T}_{2m-1}}\left( \frac{\overbrace{\sqrt{2+\sqrt{2+\sqrt{2+\cdots +\sqrt{2}}}}}^{M\,\,\text{square}\,\,\text{roots}}}{2} \right)}
\end{aligned}
\]
\normalsize
or
\begin{equation}\label{eq_9}
\prod\limits_{m=1}^{M}{\frac{{{a}_{m}}}{2}}=\frac{1}{{{2}^{M-1}}}\sum\limits_{m=1}^{{{2}^{M-1}}}{{{T}_{2m-1}}\left( \frac{{a}_{M}}{2} \right)}.
\end{equation}
Increase of the integer $M$ approximates the product of cosines on the left side of equation \eqref{eq_9} closer to the value $2/\pi$. This signifies that the right side of the equation \eqref{eq_9} also tends to $2/\pi $ as the integer $M$ increases. Consequently, this leads to
$$
\frac{2}{\pi }=\underset{M\to \infty }{\mathop{\lim }}\,\frac{1}{{{2}^{M-1}}}\sum\limits_{m=1}^{{{2}^{M-1}}}{{{T}_{2m-1}}\left( \frac{\overbrace{\sqrt{2+\sqrt{2+\sqrt{2+\cdots +\sqrt{2}}}}}^{M\,\,\text{square}\,\,\text{roots}}}{2} \right)}
$$
or
\begin{equation}\label{eq_10}
\frac{2}{\pi }=\underset{M\to \infty }{\mathop{\lim }}\,\frac{1}{{{2}^{M-1}}}\sum\limits_{m=1}^{{{2}^{M-1}}}{{{T}_{2m-1}}\left( \frac{{a}_{M}}{2} \right)}.
\end{equation}
The equation \eqref{eq_10} is completely identical to the Vi\'{e}te infinite product \eqref{eq_3} for the constant pi. Since the relation \eqref{eq_8} remains valid for any integer $M$, the equation \eqref{eq_10} can be regarded as a product-to-sum transformation of the Vi\'{e}te\text{'}s formula for pi.
It should be noted that the equation \eqref{eq_10} can be readily rearranged as a single Chebyshev polynomial of the second kind (see {\it{Appendix B}}).
\section{Conclusion}
We show a new analog to the Vi\'{e}te\text{'}s formula for pi represented in terms of the Chebyshev polynomials of the first kind. This approach is based on a product-to-sum transformation of the Vi\'{e}te\text{'}s formula.
\section*{Acknowledgments}
This work is supported by National Research Council Canada, Thoth Technology Inc. and York University.
\section*{Appendix A}
The Chebyshev polynomials ${{U}_{m}}\left( x \right)$ of the second kind can also be defined by the recurrence relation. Specifically, we can write \cite{Zwillinger2012}
$$
{{U}_{m}}\left( x \right)=2x{{U}_{m-1}}\left( x \right)-{{U}_{m-2}}\left( x \right),
$$
where ${{U}_{1}}\left( x \right)=2x$ and ${{U}_{0}}\left( x \right)=1$.
There is a simple relation for sum of the odd Chebyshev polynomials of the first kind
\[
\tag{A.1}\label{A.1}
{{U}_{K}}\left( x \right)=2\sum\limits_{k\,\,\text{odd}}^{K}{{{T}_{k}}\left( x \right)},
\]
where $K$ is an odd integer. Consequently, substituting equation \eqref{eq_6} into relation \eqref{A.1} provides\footnote{The subscript $2^M-1$ should not be confused with notation $2^{M-1}$ that has been used in some equations earlier.}
\[
\prod\limits_{m=1}^{M}{\cos \left( \frac{t}{{{2}^{m}}} \right)}=\frac{1}{{{2}^{M}}}{{U}_{{{2}^{M}}-1}}\left( \cos \left( \frac{t}{{{2}^{M}}} \right) \right).
\]
According to equation \eqref{eq_1} tending $M$ to infinity leads to the limit
\[
\text{sinc}\left( t \right)=\underset{M\to \infty }{\mathop{\lim }}\,\frac{1}{{{2}^{M}}}{{U}_{{{2}^{M}}-1}}\left( \cos \left( \frac{t}{{{2}^{M}}} \right) \right)
\]
or
\[
\tag{A.2}\label{A.2}
\text{sinc}\left( t \right)=\underset{N\to \infty }{\mathop{\lim }}\,\frac{1}{N}{{U}_{N-1}}\left( \cos \left( \frac{t}{N} \right) \right),
\]
since we can imply that $2^M = N$. Obviously at $N >> 1$, one can truncate equation \eqref{A.2} to approximate the sinc function by a single Chebyshev polynomial of the second kind as
\[
\text{sinc}\left( t \right) = \frac{1}{N}{{U}_{N-1}}\left( \cos \left( \frac{t}{N} \right) \right)+\epsilon \left( t \right),
\]
where $\epsilon \left( t \right)$ is the error term. For example, taking $N=16$ results in
\[
\begin{aligned}
\text{sinc}\left(t\right) = & \,\, 2048 \cos ^{15}\left(\frac{t}{16}\right)-7168 \cos ^{13}\left(\frac{t}{16}\right)+9984 \cos ^{11}\left(\frac{t}{16}\right)\\
&-7040 \cos ^9\left(\frac{t}{16}\right)+2640 \cos ^7\left(\frac{t}{16}\right)-504 \cos ^5\left(\frac{t}{16}\right)\\
&+42 \cos ^3\left(\frac{t}{16}\right)-\cos \left(\frac{t}{16}\right) + \epsilon \left( t \right),
\end{aligned}
\]
where within the range $-10 \leq t \leq 10$ the error term satisfies $\left| \epsilon \left( t \right) \right| < 0.006$. As we can see, this approach quite accurately approximates the sinc function even if the integer $N$ in the limit \eqref{A.2} is not very large.
\section*{Appendix B}
Substituting equation \eqref{eq_10} into relation \eqref{A.1} we can express the Vi\'{e}te\text{'}s formula for pi by a single Chebyshev polynomial of the second kind as given by
\[
\frac{2}{\pi}=\underset{M\to \infty }{\mathop{\lim }}\,\frac{1}{{{2}^{M}}}{{U}_{{{2}^{M}}-1}}\left( \frac{\overbrace{\sqrt{2+\sqrt{2+\sqrt{2+\cdots \sqrt{2}}}}}^{M\,\,\text{square}\,\,\text{roots}}}{2} \right)
\]
or
\[
\tag{B.1}\label{B.1}
\frac{2}{\pi}=\underset{M\to \infty }{\mathop{\lim }}\,\frac{1}{{{2}^{M}}}{{U}_{{{2}^{M}}-1}}\left( \frac{{{a}_{M}}}{2} \right).
\]
Although the equation \eqref{B.1} is more simple, the equation \eqref{eq_10} reflects explicitly the product-to-sum transformation of the Vi\'{e}te\text{'}s formula \eqref{eq_3} for the constant $\pi$.
\small
\normalsize
\end{document} |
\begin{document}
\defContents{Contents}
\defReferences{References}
\defReferences{References}
\defFig.{Fig.}
\begin{center}{}\Large\bf
On the compactness of the set of invariant Einstein metrics
\end{center}
\begin{center}{}\large
M. M. Graev
\end{center}
\footnotetext{Supported by RFBR, grant 10-01-00041a.}
\begin{comment}
\begin{center}
Scientific Research Institute for System\\ Studies of Russian Academy of Sciences,\\ Nakhimovsky prosp., 36, korpus 1,\\ Moscow, 117218, Russia
\end{center}
\begin{center}
\texttt{[email protected]}
\end{center}
\end{comment}
\title{}
\vskip-1cm
{\small
\noindent
\textsc{Abstract}.
\let\bar\overline
\input abstract-ii
}
\begin{quote}
\tableofcontents
\end{quote}
\vskip-1cm
\vskip-1cm
\begin{comment}
\begin{enumerate}{}
\item{Invariant metrics on a compact homogeneous space $G/H$}
\item{Moment map and moment polytope}
\item{Compactification $\Delta=\MET_1 \cup \Gamma $}
\item{Euclidean geometries at infinity}
\item{Minimal compactification $\Delta _{\min}$}
\item{First application}
\item{Second application}
\item{Newton polytope and proof of Theorem~\ref{THM:e-nu}}
\item{Appendix. Case of K\"ahler homogeneous space with $b_2=1$}
\item{References}\dotfill\pageref{bibliography}
\end{enumerate}
\end{comment}
\section*{Introduction}
Let $M = G/H$ be a connected simply connected homogeneous
manifold of a compact, not necessarily connected Lie group $G$. We will assume that the isotropy $H$-module $\mathfrak {g/h}$ has a simple spectrum, i.e. irreducible submodules are mutually non-equivalent.
There exists a
convex Newton polytope $N=N(G,H)$, which was used for the estimation of the number of isolated complex solutions of the algebraic Einstein equation for invariant metrics on $G/H$ (up to scaling), see \cite{2006,2007}.
Using the moment map, we identify the space $\mathcal{M}_1$ of invariant Riemannian metrics of volume 1 on $G/H$ with the interior of this polytope $N$.
We associate with a point ${x \in \Gamma = \partial N}$ of the boundary a homogeneous Rieman\-nian space (in general, only local, since the stability subgroup can be non-closed) and we extend the Einstein equation to $\overline{\mathcal{M}_1}= N$.
As an application of the Aleksevsky--Kimel'fel'd theorem,
we prove that all solutions of the Einstein equation associated with points of the boundary are locally Euclidean.
We describe explicitly the set $T\subset \Gamma$ of solutions at the boundary together with its natural triangulation.
It is the standard geometric realization of a subcomplex of the simplicial complex, whose simplicies are the $H$-invariant subalgebras $\mathfrak t\subset\mathfrak g$ satisfying
$\mathfrak t=\mathfrak h \oplus\mathfrak a$, $\mathfrak a\ne0$, $[\mathfrak a,\mathfrak a]=0$
(quasi toral subalgebras), and the vertices are minimal such subalgebras.
Investigating the compactification $\overline{\mathcal{M}_1} $ of $\mathcal{M}_1$, we get an algebraic proof of the deep result by B\"ohm, Wang and Ziller about the compactness of the set $\mathcal{E}_1 \subset \mathcal{M}_1$ of Einstein metrics.
The original proof by B\"ohm, Wang and Ziller \cite{BWZ} was based on a different approach and did not use the simplicity of the spectrum.
In \S\ref{sect:1}, \ref{sect:2}, \ref{sect:3}
we define the moment map and the moment polytope $\Delta$ (more general than $N$), and construct the corresponding compactification of $\MET_1$. Lather in \S 3 we prove that all solution of the Einstein equation at the boundary of $\Delta$ are Ricci-flat and, consequently, flat.
In \S\ref{sect:4} we describe a triangulation of the set $T\subset\partial\Delta$ of these solutions.
We consider examples, where $T$ is a finite set,
or a disjoint union of simplicies, or the join of two finite sets (a complete bipartite graph).
In \S\ref{sect:T} we construct the minimal (under inclusion) moment polytope $\Delta_{\min}$,
by passing, if necessary, to some 'non-essential' extension of the coset space $G/H$. We prove that $T=\varnothing$, if the groups $G$ and $H$ are connected, and $\Delta=\Delta_{\min}$ (Proposition~\ref{PROP:C}).
In \S\ref{sect:6} we use $\Delta_{\min}$ to prove the compactness of $\mathcal E_1$. We deduce it from the compactness of $T\cup\mathcal E_1$.
It follows that $\mathcal E_1$ is compact
in the case when the groups $G$ and $H$ are connected.
We sketch a proof in the general case.
In \S\ref{sect:7} we consider an application of the polytope $\Delta_{\min}$ to the finiteness problem for complex solutions of the algebrac Einstein equation.
We write the optimal upper bound of the normalized volume $\nu$ of $\Delta_{\min}$ for the number $\varepsilon$ of \textit{isolated} solutions.
In \S\ref{sect:8} we outline that $\Delta_{\min} = N$, and get an upper bound for $\varepsilon$ (of the normalized volume of some permutohedron $\Pi \supset \Delta$, which is a central Delannoy number $D\in \{3,13,63,321,1683,\dots\}$, cf. \cite{Stanley2}).
In Appendix we consider the non-symmetric K\"ahler homogeneous spaces $G/H$
with the second Betti number $b_2=1$. In this case
$
2^{-1}\nu \in\{1,3,10,41,172\},
$
and we have $\varepsilon=\nu$ for $2^{-1}\nu \in\{1,3,10\}$.
By the recent calculations of I.Chrysikos and Y.Sakane \cite{SACOS} it implies that for ${G/H=E_8/T^1\cdot A_3\cdot A_4}$ all complex solutions
are isolated, and $\varepsilon = 81$, so that $\nu-\varepsilon = 82-81=1$.
The missing solution with multiplicity $1$ 'escape to infinity'. We indicate the missing solution explicitly.
We discuss a reduction of the finiteness problem for complex solutions
in the case of
$
G/H = {E_8/T^1\cdot A_4\cdot A_2\cdot A_1}
$
(based on calculation of some 'marked' faces of $\Delta$ and
consideration of a toric variety $\Delta^{\CC}$),
and prove that $\varepsilon<\nu$, where $\nu = 344$.
\section{Invariant metrics on a compact homogeneous space $G/H$}\label{sect:1}
Let $ G / H $ be a connected simply connected $ n $-dimensional homogeneous space of a compact Lie group $ G $, \ $\rho : H \to \mathrm{GL}(\frak{g/h})$
the isotropy representation with a finite kernel.
Let us denote by $\MET = \MET(G, H)$ the cone
of invariant Riemannian metrics $g$ on $G/H$ (or, equivalently, $\rho(H)$-invariant Euclidean scalar products in $\mathfrak{g/h}$),
and by $\MET_1 = \MET_1(G, H)$
the hypersurface of metrics $g$ with volume $\opn{vol}_g(G/H)=1$.
We shall assume, unless otherwise stated, that the representation $ \rho $ has a simple spectrum,
i.e., $\rho$ decomposes as a direct sum of $ d \le n $ pairwise inequivalent irreducible representations
(e.g., as it is in the case
$ \opn {rank} (G) = \opn {rank} (H) $).
Therefore,
$$
\MET(G,H) = (\RR_{>0})^{d}.
$$
We suppose ${d>1}$, and fix an
${H\cdot Z_G (H^0) }$-invariant Euclidean scalar product
$g_1$ on $\frak{g/h}$, $ g_1 \in \MET_1 $.
\section{Moment map and moment polytope}\label{sect:2}
We will define the {\bf moment map}
$
\displaystyle
\mu:
\MET(G,H) \to\RR^{n-1} = \left(\mbox{\begin{tabular}{c}
diagonal matrices of\\ order $ n $ with trace $ 1 $
\end{tabular}}\right)
$
as the gradient of the logarithm of a ``suitable'' positive homogeneous
function on the cone $ \MET (G, H) $.
It is not unique and in particular depends on an ($H$-invariant) reductive decomposition
$$
\mathfrak{g}= \mathfrak{h} + \mathfrak{m}.
$$
For an invariant definition,
we may chose the $B$-orthogonal decomposition, where $B$ is the Killing form (with the kernel $\mathfrak{z(g)}$).
We define a suitable function on $\MET(G,H) $ as the following modified scalar curvature
$$
\ell_{\theta }(G/H,g)= \opn{trace}(-(1+ \theta )\mathit{Ric}_{G/H,g}-B_{G/H,g}),
$$
where $\mathit{Ric}_{G/H,g}$ is the Ricci operator of a metric $ g $ at the point $eH \in G/H$, \,
$B_{G/H,g}= g^{-1} B |_{\frak m} \in \mathrm{End}(\mathfrak{m})$,
and $\theta$ is a parameter, $|\theta|<1$.
The corresponding moment map is given by
$$
\mu_{\theta }(g) = \frac{1}{\ell_{\theta }(G/H,g)}
(-(1+ \theta )\mathit{Ric}_{G/H,g}-B_{G/H,g}).
$$
Clearly, $\mu_{\theta}(g)$ belongs to
$\RR^{d-1}$, the space of $H$-invariant diagonal matrices with trace $1$.
Changing $\mathfrak{m}$ to any other
$H$-invariant
complement $\mathfrak{m}'$ to $\mathfrak{h}$ we get another ``suitable'' function and another moment map
$
\boxed
{ \mu = \mu_{\theta}
: \MET\to \RR^{d-1}, }
$
which we call compatible with $\frak m$.
Now we fix any such complement $\mathfrak{m}$ (not necessary $B$-orthogonal).
{\bf Remark} There are other possibilities to define a ``suitable'' function, but the scalar curvature
$$
\SC(G/H,g)=\opn{trace}(\mathit{Ric}_{G/H,g} )
$$
is not always a ``suitable'' function, since it can take non-positive values.
The following statements hold for the 'moment map' $\mu$ associated with a suitable function (more general than $\mu_{\theta}$).
Now we associate with $\frak m$ a compact convex
polyhedron $\Delta \subset \RR^{d-1}$.
Let
$$
\mathfrak{m}= \mathfrak{m}_1+\dots+\mathfrak{m}_d,
$$
where $\mathfrak{m}_i$ are irreducuble $H$-submodules of $\mathfrak{m}$.
Let $\varepsilon _i$, $i=1, \dots ,d$, be the weight
of the Lie algebra $\RR^d \subset \mathfrak{gl(m)}$
of $H$-invariant diagonal matrices,
such that $AX=\langle \varepsilon _i,A \rangle\,X$ for all $A \in \RR^d$, $X \in \frak m_i$.
By $\Delta $ we denote the convex hull of all weights of the form
\begin{center}{}
$
\varepsilon _i + \varepsilon _j - \varepsilon _k,
$
and $\varepsilon _r$,
\end{center}
where $g_1([\frak m_i,\frak m_j]+ \frak h,\frak m_k+ \frak h) \ne \{0 \}$, $B(\frak m_r,\frak m_r) \ne \{0 \}$
(we assume here $g_1(\frak h,\frak g):=0$).
\begin{EXAMPLES}{}\label{EXAM:21}
Let $G/H$ be a direct product of $d\ge 2$ isotropy irreducible spaces,
e.g., copies of $\CP^1$. Then $\Delta $ is the standard ${(d-1)}$-dimensional
coordinate simplex with vertices
$\varepsilon _1 = {(1, \dots ,0)}, \dots ,\varepsilon _d = {(0, \dots ,1)}$.
\end{EXAMPLES}
\begin{EXAMP}{}\label{EXAM:22}
Let $G/H$ be $SU(3)/T^2$.
Then $d=3$, and $\Delta $ is the triangle with vertices $(1,1,-1)$, $(1,-1,1)$, $(-1,1,1)$.
This valid for the spaces $G/H$ with $d=3$ and $[\frak m_i,\frak m_i] \subset \frak h$,
$[\frak m_i,\frak m_j] = \frak m_k$, $\{i,j,k \} = \{1,2,3 \}$.
\end{EXAMP}
\begin{EXAMP}{}\label{EXAM:23}
Let $G/H$ be $E_8/(A_2)^4$ or $E_7/T^1\cdot (A_2)^3$. Then $d=4$,
and $\Delta $ is a $3$-polytope with eight $2$-faces. It is an Archimedean solid
(a truncated tetrahedron), or respectively
the convex hull of two opposite faces of such a solid (a hexagon and a triangle).
\end{EXAMP}
Using a technical lemma from \cite[\S4.2]{Fulton}, one can prove the following theorem:
\begin{THM}\label{THM:DEL}
Let $G/H$ be a connected simply connected homogeneous space of a compact Lie group $G$ such that
$\frak{g/h}$ is a multiplicity-free $H$-module,
$ \MET_1 = \MET_1 (G, H) $ the space of the invariant Riemannian metrics of volume $ 1 $,
\ $\frak m$ an invariant complement to $\frak h$ in $\frak g$,
\ $\mu : \MET_1\subset\MET\to \RR^{d-1}$ a moment map,
compatible with $\frak m$,
and $ \Delta \subset \RR ^{d -1} $ the compact convex polyhedron
associated with $\frak m$.
Then the map $\mu$ determines a diffeomorphism of the space $ \MET_1 $ onto the interior of $\Delta$.
We have $\dim (\Delta )=d-1$.
\end{THM}
We will consider the Euclidean
polyhedron
$\Delta\subset\RR^{d-1}$ as a compactification of the space $\MET_1$ of metrics, and call $\Delta$ the {\bf moment polytope} (associated with $\frak m$). The points on the boundary $ \Gamma = \partial \Delta $ of $\Delta$ we call {\bf points at infinity}.
\section{Compactification $\Delta=\MET_1 \cup \Gamma $}\label{sect:3}
In this section, we associate with a point $x$ of the
boundary $\Gamma = \partial \Delta$ a Lie algebra
$$
\mathfrak{g}_x = \mathfrak{h}+ \mathfrak{m}
$$
with a reductive decomposition and
a \textit{fixed} $\mathrm{ad}({\mathfrak{h}})$-invariant Euclidean metric $g_1$ on $\mathfrak{m}$.
Since, in general, the subalgebra $\mathfrak{h}$ generates a non closed
subgroup of the Lie group $G_x$ associated with $\mathfrak{g}_x$ , it does not define a homogeneous Riemannian manifold.
However, we can exponentiate $\mathfrak{m}$ to a locally defined (non
complete) Riemannian manifold $M^n(x)$
with a transitive action of the Lie algebra $\mathfrak{g}_x$ and the stability subalgebra $\mathfrak{h}$.
More precisely, we can speak about a germ of ``local homogeneous Riemannain
geometry''.
Later in this section we will describe explicitly the points at infinity corresponding to
germs of Einstein geometries.
To describe the construction more carefully, we consider the
moment map $\mu : \MET_1\xrightarrow{\,\sim\,} \Delta\smallsetminus\Gamma$, and
associate with each interior point $x=\mu(g) \in \Delta \smallsetminus \Gamma $
a homogeneous Riemanian space $(G/H,cg)$, $c>0$, where
$cg \in \MET$ is a
Riemannian metric on $G/H$ (proportional to $g=\mu^{-1}(x)$) with
the same modified scalar curvature as $g_1$, namely,
$\ell_0(G/H,cg)=\ell_0(G/H,g_1)$. Here $ g_1 \in \MET_1 $ is the fixed $H\cdot Z_G (H^0) $-invariant Euclidean scalar product on
$\frak g/\frak h \cong \frak m$.
Let $\phi : \frak m \to G/H$ be a local diffeomorphism
defined in a neighbourhood $M^{n}(x)$ of the origin by
$
\phi(Y)=\beta(a^{-1}Y)=\exp(a^{-1}Y)H
$
for all vectors $ Y \in \frak m $ of sufficiently small length,
where
$a\in \mathrm{GL}(\frak m)$ is an
$ H $-invariant diagonal linear transformation on $ \frak m $ such that
$$
\ell_0(G/H,g)g(Y,Y) \equiv \ell_0(G/H,g_1)g_1(aY,aY).
$$
We will consider $M^n(x) \subset \frak m $ as a Riemannian space
with respect to the metric $\phi^*(cg)$. Hence $\phi^*(cg)(Y,Y) =
g_1(Y,Y)$ for all tangent vectors in the origin $0\in M^n(x)$. We
define a transitive Lie algebra of Killing vector fields on $M^n(x)$
$$
\frak g_x=(\frak h + \frak m, [\cdot,\cdot]_x)
$$
(isomorphic to $\frak g=Lie\, G=(\frak h + \frak m,[\cdot,\cdot])$)
by $[T_aY,T_aZ]_x=T_a[Y,Z]$, where $T_aY=Y$ for all $Y\in\frak h$,
$aY$ for all $Y\in\frak m$. (So that
$[\cdot,\cdot]_{\mu(g_1)}=[\cdot,\cdot]$.) Clearly, the stability
group $H$ acts isometrically on $M^n(x)$.
We will denote by $(M(x),g_1)$ and $M(x)$ the germs of the above
Riemannian homogeneous structure and, respectively, the homogeneous structure
on $M^n(x)$ at the point $0\in M^n(x)$. Non-formally,
$M(x)$ can be considered as a neighbourhood $M^{n}(x)$
equipped with actions of $\frak g_x$ and $H$.
One can check that the Lie algebra $\frak g_x=(\frak h + \frak m, [\cdot,\cdot]_x)$
can be defined for every $x \in \Delta $ so that $\frak g_x$ depends continuously of $x$.
(However, $\frak g_x\not\cong \frak g$
for $x\in\Gamma$). In this way, the germ $(M(x),g_1)$ is
well-defined for all $x\in \Delta$. Moreover, it satisfies the
following properties:
\begin{itemize}{}
\item
the Ricci tensor $\opn{ric}(M(x),g_1)$ and all others associated with the metric tensors
at the point $0\in M(x)$ depend continuously of $x$;
cf. \cite{Lauret};
\item
the compact group $H$ acts on this germ isometrically with the fixed point $0$
and the same isotropy representation $\rho $ at $0$;
\item
the modified scalar curvature $\ell_0$ of a germ is well defined and
is constant on $\Delta $, that is $\ell_0(M(x),g_1) =
\ell_0(G/H,g_1)$ for all $x \in \Delta $.
\end{itemize}
\begin{DEF*}{}
By {\bf infinitesimal homogeneous Riemannian space} we will understand
a quadruple
$(\mca{A}, \TIL{\frak g}, \TIL{\frak h}, g)$, where $\TIL{\frak g}$ is
a Lie algebra, $\TIL{\frak h}\subset \TIL{\frak g} $ a subalgebra, $\mca{A}$
is a compact group of automorphisms of the pair $(\TIL{\frak g}, \TIL{\frak h})$,
and $g$ is a Euclidean scalar product on $\TIL{\frak g}/ \TIL{\frak h}$, invariant under
$\mca{A}$ and $\TIL{\frak h}$.
\end{DEF*}
Thus, we can associate with any point
$ x \in \Delta $
an infinitesimal homogeneous Riemannian space
$ (M (x), g_0) = (\mca{A},\frak g_x,\frak h, g_0) $, which we call also
a geometry, such that $\TIL{\frak h}
\cong \frak h$ and $\mca{A} \cong Ad_{\frak g}(H)$; the isomorphism
of Lie algebras $\TIL{\frak h}=\frak h$ extends to a isomorphism of
$\mca{A}$-modules
$\TIL{\frak g}$
and $\frak g = \frak h + \frak m$, which are identified,
and $g_0\in\MET$.
A geometry $(M(x), g_1)$
associated with a point at infinity
$ x \in \Gamma $
can be exponentiated to a local
geometry $M^n(x)$, as above, but not necessary to a global
homogeneous Riemannian geometry, since the stability
subalgebra $\tilde{ \mathfrak{h}}=\mathfrak{h}$ can generate a non-closed subgroup
(cf. Exam.~\ref{EXAM:53} below).
However, we can apply to such local homogeneous Riemannian geometry
the Alekseevsky--Kimel'fel'd theorem, stating that the Ricci--flat homogeneous
Riemannian geometries are locally Euclidean \cite {Al-Ki}.
Due to the fact that any Lie algebra $ \TIL {\frak g}= \frak g_x$ (which is a contraction of the
compact Lie algebra $\mathfrak{g}$) is
of the type $(R)$, the proof of the theorem given in \cite {Al-Ki} can be modified
so that it remains valid for a local homogeneous Riemannian manifold.
(The condition of simplicity of the spectrum of $\rho$ is
insignificant.) Using this,
\boxed{we} prove
the following theorem:
\begin{comment}
\input THAT50/Theorem2
\sect ion{Euclidean geometries at infinity}\label{sect:4}
\input THAT50/Lemma1
\par
\end{comment}
\begin{THM}{}
Any Einstein geometry at infinity $(M(x), g_1)$ is locally Euclidean.
\end{THM}
\begin{proof}[Outline of proof] It is sufficient to prove
that the geometry $(M(x), g_1)$ is Ricci-flat, i.e., the scalar curvature $s=\SC(M(x), g_1)$ vanishes.
Let $\phi \subset \Gamma$ be any facet of the moment polytope $\Delta$ through the point $x$. Up to sign, there is a unique vector $z=(z_1,\dots,z_d) \in \ZZ^d$ with $\gcd(z_1,\dots,z_d)=1$, orthogonal to $\phi$, so $\langle x,z\rangle=0$.
We may assume that $z$ generates an edge of the following $d$-dimensional convex polyhedral cone\,:
$$
\nabla = \{y\in\RR^d : \langle x',y\rangle\ge0,\enskip \forall\,x'\in\Delta\},
$$
since otherwise we may pass from $z$ to $-z$.
Let $y,y',y''\in\RR^d$ and $y_i=\max(y'_i,y''_i)$
(respectively $\min(y'_i,y''_i)$) for all $i\in\{1,\dots,d\}$.
In this situation we write $y=\max(y',y'')$
and $y=\min(y',y'')$, respectively. We have
$$
\max(y,y')\in \nabla,\quad \min(y,0)\in \nabla,\quad
\forall \,y,y'\in \nabla.
$$
This follows from definitions of $\Delta$ and $\nabla$,
since $\bigoplus_{y_i<0} \frak m_i \subset \frak {z(g)}$.
Hence
$$
z=\max(z,0) + \min(z,0) = \max(z,0) \mbox{\ \ or\ }\min(z,0)
$$
(moreover, in the second case we have $\sum z_i = -1$).
We outline two proofs that $s=0$.
By Theorem~\ref{THM:DEL}, the moment polytope $\Delta=\ov{\mu_{\theta}(\MET_1)}$ is independent of $\theta$.
Using this, one can check that
for all $\theta\in (-1,1)$
$$
\phi\,\ni\,
\frac{1}{\ell_{\theta }(M(x),g_1)}
(-(1+ \theta )\mathit{Ric}_{M(x),g_1}-B_{M(x),g_1}).
$$
Therefore $z$ and ${r=\mathit{Ric}_{M(x),g_1}}$
can be considered as two orthogonal vectors in $\RR^d$,
$\langle r,z\rangle=0$,
and
$
s\,\sum z_i\dim \frak m_i = n \langle r,z\rangle=0,
$
so $s=0$.
For another proof of $s=0$, we may consider $z$ as a derivation of the Lie algebra $\frak g_x$ with the eigenspaces $\frak g_x^k$ such that $\frak h \subset\frak g_x^0$ and $\frak g_x^k\cap\frak m = \bigoplus_{z_i=k}\frak m_i$ (possibly, $\frak g_x^k=0$). Then
$$
\mbox{either\ \ }
\frak g_x=\bigoplus_{k=0}^{\infty} \frak g_x^k\,,
\mbox{\ \ or\ \ }\frak g_x=\bigoplus_{k=-\infty}^0 \frak g_x^k= \frak g_x^{-1} + \frak g_x^{0},
$$
and
$[\frak g_x^k,\frak g_x^l]_x \subset \frak g_x^{k+l}$
for all integer $k,l$ (moreover, $\frak g_x^{-1} \subset \frak z(\frak g_x)$). Consider now $z$ as an element of $\frak {gl(m)}$, and assume $g^{\lambda} = e^{-\lambda z} .\,g_1$ is the one-parametric family of Euclidean scalar products on $\frak m$ (so that $g^\lambda\in \MET$ and $g^0=g_1$).
We conclude that the geometries $(M(x), g^\lambda)$ with fixed $x$ and all $\lambda \in \RR$ are equivalent, and, hence, Einsteinian with the same scalar curvature $s$.
By Hilbert--Jensen theorem \cite{Jen-2}, \
$$
0=\frac{d}{d\lambda} ((\det g^\lambda)^{1/n}\, s)
= s \,\frac{d}{d\lambda} e^{2\lambda\opn{trace}(z)/n}.
$$
(This is correct, since $\frak g_x$ is the Lie algebra of an unimodular Lie group. Note also that the Hilbert--Jensen theorem remains valid for a local homogeneous Riemannian manifold.)
But $\opn{trace}(z)=\sum z_i \dim\frak m_i \ne0$, and, hence,
$s:=\SC(M(x),g_1)=0$.
\end{proof}
\textbf{Remark.} The cone $\nabla$ is a ``tropical ring''
under operations $y\oplus y' =\max(y,y')$ and $y\odot y'=y+y'$, so that $y\odot(y'\oplus y'')=(y\odot y')\oplus(y\odot y'')$.
\section{Euclidean geometries at infinity}\label{sect:4}
Now we describe the points at infinity corresponding to locally Euclidean geometries.
\begin{LEM}{}\label{LEM:T}
Let $x \in \RR^d$. Then $x$ lies in $\Gamma $ and the corresponding geometry $(M(x), g_1)$ is locally Euclidean if and only if $x$ belongs to the convex hull of a subset of weights $\{\varepsilon _i
: i \in I \} \subset \{\varepsilon _1, \dots , \varepsilon _d\}$,
such that the subspace $\frak m_I = \bigoplus_{i \in I} \frak m_i \subset \frak g$ satisfies conditions
$$
[\frak m_I,\frak m_I]=[\frak m_I,\frak h]=\frak m_I \cap \frak {z(g)}=0.
$$
\end{LEM}
\begin{proof}[Proof]
Assume $\mu=\mu_{\theta}$.
Then
$
\xi = \frac{1}{\ell_{\theta }(M(\xi),g_1)}
(-(1+ \theta )\mathit{Ric}_{M(\xi),g_1}-B_{M(\xi),g_1})
$
for all interior points $\xi$ of $\Delta$ and, hence,
for all poits $\xi$ of the boundary $\Gamma$.
Suppose that $x\in\Gamma $ and the corresponding
geometry $(M(x), g_1)$ is locally Euclidean. Then
\begin{equation}
x=\frac {B_{M(x),g_1}}{\opn{trace} B_{M(x),g_1}}
= \sum_{i=1}^d t_i\varepsilon_i,
\tag{*}
\end{equation}
for some coefficients $t_i\ge0$ with $\sum t_i =1$. Let
$$
\frak m_{\tau}= \bigoplus _{t_i>0} \frak m_i,\quad
\frak n= \bigoplus _{t_i=0} \frak m_i
$$
and let $\xi$ be a relative interior point of the
convex hull $\tau$ of the set $\{\varepsilon_i : t_i>0\}$,
e.g., the point $\xi = x$. It is easy to check that $\frak m_{\tau}\cap \frak {z(g)} =0$, and $\xi \in \Delta$. It follows from $x \in \Gamma$ that $\xi \in \Gamma$.
We prove that
the corresponding geometry $(M(\xi),g_1)$
also is locally Euclidean, assuming that $\frak m$ is a subalgebra
of $\frak g_x$, i.e., $[\frak m,\frak m]_x\subset\frak m$.
(For example, if the reductive decomposition $\frak g = \frak h + \frak m$ is $B$-orthogonal, and, hence, $B_{\frak g_x}(\frak h,\frak m)=0$, then undoubtedly $[\frak m,\frak m]_x\subset\frak m$, since the stability subalgebra $\frak h$ contains a maximal semisimple subalgebra of the Lie algebra $\frak g_x$.)
Note that a necessary and sufficient condition for $\frak m = \frak m_\tau + \frak n$ to be a transitive effective Lie algebra of motions of the Euclidean space $(\frak m, g_1)$ is
\begin{equation}
\begin{array}{cc}
{[\frak m_\tau,\frak m_\tau]'}=[\frak n,\frak n]'=0,
&
[\frak m_\tau,\frak n]' \subset \frak n,
\\[1ex]
g_1([\frak m_\tau,Y]',Y) =0,
&
\forall\, Y \in\frak n.
\end{array}
\tag{**}
\end{equation}
where $[\cdot,\cdot]'$ is the commutator on $\frak m$,
$[Y,Z]'=[Y,Z]_x$. (Cf., e.g., \cite[\S5]{Jen-2}.)
This is clear, since
$\frak n$ is the kernel of the Killing form of the Lie algebra $(\frak m,[\cdot,\cdot]')$, by construction, and
\begin{equation}
g_1(\frak m_\tau,\frak n) =0.
\tag{***}
\end{equation}
Define now $a=(a_1,\dots,a_d)\in (\RR_{>0})^d$ by $\sum a_i^2t_i \varepsilon_i = \xi $ and $a_j=1$, if $t_j=0$. Consider $a$ as an $H$-invariant diagonal linear transformation of $\frak m$, so that $a|_{\frak m_i}=a_i$, $i=1,\dots,d$. Define a new commutator $[\cdot,\cdot]'$ on $\frak m = \frak m_\tau + \frak n$ with
the property \thetag{**}
by $[Y,Z]' = a^{-1}[aY,aZ]_x$. Let $\frak m^{a}=(\frak m,[\cdot,\cdot]')$ be the corresponding Lie algebra, and $\exp(\frak m^a)$ the associated simply-connected Lie group.
So $\exp(\frak m^a)$ is a metabelian group, and the scalar product $g_1$ on $\frak m$
gives
the left-invariant Euclidean metric on $\exp(\frak m^a)$.
The Killing form of $\frak m^a$ is $$B'=(\opn{trace}(B_{M(x),g_1}))\xi = (\ell_0(G/H,g_1))\xi .$$ Then there $\exp(\frak m^a)$ is locally equivalent to $(M(\xi),g_1)$, so that $[Y,Z]'=[Y,Z]_\xi$ for all $Y,Z\in\frak m$, and the assertion follows.
We prove now that $[\frak m,\frak m]_x\subset\frak m$,
and $[\frak m_\tau,\frak m_\tau]=[\frak m_\tau,\frak h]=0$.
There exist two $H$-invariant diagonal matrices $A,A_0 \in \frak {gl(m)}$ such that
\begin{itemize}
\item
$x=\lim_{\lambda\,\to\,+\infty} \mu(e^{-\lambda A}e^{-A_0}.\,g_1)$ (where $a.g(X,X)=g(a^{-1}X,a^{-1}X)$),
\item
$[\cdot,\cdot]_x = \lim_{\lambda\,\to\,+\infty} T_{e^{A_0}e^{\lambda A}} [T_{e^{-\lambda A}e^{-A_0}}(\cdot),T_{e^{-\lambda A}e^{-A_0}}(\cdot)]$
\end{itemize}
(cf. \cite[\S2.3]{Fulton}). Then $\frak g_x = \frak h + \frak m^0 + A\frak m$, where $\frak m^0 = \{X\in\frak m:AX=0\}$.
The compactness of the group $G$ implies that
the subspace $A\frak m$ is a nilpotent ideal of the Lie algebra $\frak g_x=(\frak h + \frak m, [\cdot,\cdot]_x)$, its complement $\frak g_x^0 := \frak h + \frak m^0 $ is a subalgebra of $\frak g_x$, and, moreover, $\frak g_x^0$
decomposes as a direct sum of
its center $\frak a = \frak z(\frak g_x^0)$ and a compact semisimple subalgebra $\frak k$.
Further, $\frak g_x$ is a transitive subalgebra of the complete Lie algebra $\frak{so}(n)+\RR^n$ of motions of Euclidean space, by assumption, and
$\frak h = \frak g_x \cap \frak {so}(n)$.
So $\frak h$ contains a maximal semisimple subalgebra of $\frak g_x$. Therefore, $\frak k \subset\frak h$ and
$\frak m^0 \subset \frak a$, so $[\frak h+\frak m^0,\frak m^0]_x=0$. This proves that $[\frak g_x,\frak m]_x \subset \frak m$. Let $X\in [\frak h+\frak m^0,\frak m^0]$.
Then $\lim_{\lambda\,\to\,+\infty} T_{e^{\lambda A}}X=0$.
Thus $X\in \frak{z(g)}$ and, hence, $X=0$. This proves that
$[\frak h+\frak m_\tau,\frak m_\tau] =0$.
Suppose now that $I\subset\{1,\dots,d\}$ and
$[\frak m_I,\frak m_I]=[\frak m_I,\frak h]=\frak m_I \cap \frak {z(g)}=0.$ Let $P\in\frak{gl(m)}$ be the orthoprojector with the kernel $\frak m_I := \bigoplus _{i\,\in\, I} \frak m_i$. Obviously, the Lie operation
$[X,Y]'=\lim_{\lambda\,\to\,+\infty} T_{e^{\lambda P}}[T_{e^{-\lambda P}}X,T_{e^{-\lambda P}}Y]$ on $\frak h + \frak m$ is well-defined.
Then the property \thetag{**} is satisfied
for $[\cdot,\cdot]'$,
$\frak m_{\tau}:=\frak m_I=\{X\in\frak m:PX=0\}$, and $\frak n:=P\frak m$
since the scalar product $g_1$ on $\frak g/\frak h \cong\frak m$ is $Z_G(H^0)$-invariant.
There is a point $x\in\Gamma$ such that $x=\lim_{\lambda\,\to\,+\infty} \mu(e^{-\lambda P}.\,g_1)$.
We have $[\cdot,\cdot]_x = T_c^{-1}[T_c\,\cdot,T_c\,\cdot]'$ for some scalar operator $c$ on $\frak m$.
It follows from \thetag{**}, \thetag{***}
that the geometry $(M(x),g_1)$ is locally Euclidean.
Then the point $x$ has the form \thetag{*} with $\{i:t_i>0\}=I$; e.g., $x=\varepsilon_j$, if $I=\{j\}$.
Hence, $x$ belongs to the relative interior
of the convex hull of the set $\{\varepsilon_i: i\in I\}$.
This completes the proof of Lemma~\ref{LEM:T}.
\end{proof}
\par
Let us denote by $T\subset \Gamma$ the set of the points at infinity corresponding to locally Euclidean geometries:
$
\boxed{
T:=\{ t \in \Gamma : \opn{Riem} (M(t),g_1) = 0\}.
}
$
Here are examples with non-empty set $T\subset \Gamma$ of locally Euclidean geometries at infinity.
Notations.
Define conjugate linear transformations $A$ and $B$
of $\CC^p = \bigoplus_{l \in \ZZ_p } \CC e_l$
by $Ae_l = ce_{l+1}$, and $Be_l= c \omega ^{l-1}e_l$,
where $\omega ^p=1$,
$c=i^{\,1-p^2}$.
For $p=2$ and $3$ we have
\begin{align*}&
A=\left\|\begin{array}{cr}0&i\\i&0\end{array}\right\| , &&
B=\left\|\begin{array}{cr}i&0\\0&-i\end{array}\right\| ; &&
A=\left\|\begin{array}{ccl}0&0&1\\1&0&0\\0&1&0\end{array}\right\|, &&
B=\left\|\begin{array}{ccl}1&0&0\\0&\omega&0\\0&0&\omega^2\end{array}\right\|,
\end{align*}
where $\omega^3=1$.
For $p\not\equiv 0 (4)$ we have
$A^p = B^p = (-1)^{p+1}E$, \ $ABA^{-1}B^{-1}= \omega E$,
and $A, B$ generate a finite subgroup of $SU(p)$,
which we denote $J_p \subset SU(p) $ and call Jordan's group.
So $J_2$ is the group of the quaternionic units.
Further,
$\frak{su}(p)$
is the direct sum of abelian subspaces
$
\frak m_{(k,l)} =\frak m_{(-k,-l)} = \frak {su}(p)\,\cap\, ( \CC A^{k}B^l+ \CC A^{-k}B^{-l}).
$
The complete bipartite graph $K_{r,s}$ is the graph with
$r+s$ vertices $a_1, \dots ,a_r$ and $b_1, \dots ,b_s$, and
with one edge between each pair of vertices $a_i$ and $b_j$ (so $rs$ edges in all).
\begin{EXAMPLES}{}\label{EXAM:31}
Let $G=SU(p) \lX (\ZZ_p)^2$, $p$ be a prime, and $H=(\ZZ_p)^2$,
where $(\ZZ_p)^2=J_p/\ZZ_p$ is the group of automorphisms of $SU(p)$
generated by $Ad(A)$ and $Ad(B)$.
Let $(k,l) = Ad(A^kB^l)$ for all $k,l \in \ZZ_p$.
The $H$-modules
$\frak m_h = \frak m_{-h}$, $h \in H/(\pm1)$, $h\ne (0,0)$
are irreducible, and pairwise non-equivalent.
By regarding $\ZZ_p$ as a field, we have
\begin{center}{}
$[\frak m_h, \frak m_{h'}]=0$ iff the vectors $h, h' \in(\ZZ_p)^2$ are proportional.
\end{center}
Therefore
the set $T$ corresponding to the coset space $G/H$
is a disjoint union of simplices $\sigma _i$, $i=1, \dots ,p+1$,
with $\dim (\sigma_i ) = \max(0,(p-3)/2)$.
Similarly, if $G/H$ is the direct product of two such spaces with ${p \in \{2,3 \}}$,
then
$\dim(T){=}1$,
and $T$ is one of bipartite graphs $K_{3,3}$, $K_{3,4}$, or $K_{4,4}$.
\end{EXAMPLES}
\begin{EXAMP}{}\label{EXAM:32}
For an example with $\frak h\ne 0$, take
$G=SU(p+q+1)\lX ( J_p \times J_q )$
and $H= T^2 \times J_p \times J_q$, where $p,q \in\{2,3 \}$,
and $T^2$ is a torus.
Then $G/H$ has a simple spectrum of the isotropy rep\-re\-sen\-ta\-tion,
and $T$ is the complete bipartite graph $K_{p+1,q+1}$.
\end{EXAMP}
We will now describe the vertices $v$ of the polytope $\Delta$ which belong to $T$.
\begin{LEM}\label{LEM:v}
Let $(M(v), g_1)$, $v\in T$, be a locally Euclidean geometry at infinity.
Assume for simplicity that the fixed decomposition $\frak g=\frak h +\frak m$ is $H\cdot Z_G(H^0)$-invariant.
Then the point $v$ is a vertex of the moment polytope $\Delta$, if and only if $v=\varepsilon_j$ for some $j$, and
$$
[\frak m_i,\frak m_j]\subset \frak m_i,
\qquad \forall \quad i\in\{1,\dots,d\}.
$$
\end{LEM}
\begin{proof}[Proof]
Let $v$ be a vertex of $\Delta$. By Lemma~\ref{LEM:T}, \
$v=\varepsilon_j$, where
\begin{equation}
[\frak m_j,\frak m_j]=[\frak m_j,\frak h]=0,
\quad
[\frak m,\frak m_j]\ne 0.
\tag{*}
\end{equation}
Conversely, suppose a weight $\varepsilon_j$ satisfies (*).
Then $g_1([\frak m_i,\frak m_j],\frak m_k) = g_1([\frak m_k,\frak m_j],\frak m_i)$ for all $i,k$
since $g_1$ is $Z_G(H^0)$-invariant.
Therefore $\varepsilon_j$ is either a unique vertex of $\Delta$ with $x_j>0$, or a half-sum of two distinct points
$p,q\in \Delta$,
$p\ne q$
of the form
\begin{align*}
&
p=\varepsilon_i+\varepsilon_j-\varepsilon_k,
&&
q=\varepsilon_k+\varepsilon_j-\varepsilon_i,
&&
i\ne j\ne k\ne i.
\end{align*}
In the second case, $\varepsilon_j$
is not a vertex, since $\Delta$ is convex.
In the first case, we obtain $g_1([\frak m_i,\frak m_j],\frak m_k)=0$ for all $k\ne i$
because $g_1([\frak m,\frak m_j],\frak m_j)=0$.
Lemma~\ref{LEM:v} follows.
\end{proof}
\section{Minimal compactification $\Delta _{\min}$}\label{sect:T}
Let $G/H$ be a connected simply connected homogeneous space of a compact Lie group $G$ such that
$\frak{g/h}$ is a multiplicity-free $H$-module with at least two irreducible submodules,
$ \MET_1 = \MET_1 (G, H) $ the space of the invariant Riemannian metrics of volume $ 1 $,
\ $\mu : \MET_1\to \RR^{d-1}$ the moment map,
and $\Gamma$ the boundary of the
polytope $\Delta=\ov{\mu(\MET_1)}$.
So $\dim \Gamma = d-2 \ge 0$.
The points
$x\in\Gamma$ corresponds to geometries at infinity $(M(x), g_1)$.
The subset $ T \subset \Gamma $ of all locally Euclidean geometries at infinity
(described in Lemma~\ref{LEM:T} above)
has a natural triangulation, as the following theorem states\,:
\begin{THM}\label{THM:T}
The set $ T \subset \Gamma $ of locally Euclidean geometries at infinity
is a union of some (closed) faces of the
$ (d-1) $-dimensional simplex $ S \subset \RR ^{d-1} $
with vertices $\varepsilon_i$, $i\in\{1,\dots,d\}$.
\end{THM}
In this section, we minimize this union $T$ by changing the moment map $\mu:\MET_1\to\RR^{d-1}$ and minimizing
the moment polytope $\Delta=\ov{\mu(\MET_1)}$.
Moreover, we consider the maximal $T_{\max}$ and $\Delta_{\max}$
of $T$ and $\Delta$ (under inclusion).
Each $T$ is the union of all simlices of $T_{\max}$ that lie in $\Delta$.
The aim is to obtain the following compactification of $\MET_1 : $
\begin{DEF*}{}
A compactification $ \Delta=\MET_1\cup \Gamma $ of the space $\MET_1= \MET_1 (G, H) $
is called {\bf admissible} if $ T $ contains no whole faces of the boundary $ \Gamma $.
\end{DEF*}
In the case of an admissible compactification, one can check that $ \dim (T) <d-2 $ and, moreover, for each proper face
$ \gamma $ of the polytope $ \Delta $, we have
$$
\dim(T\cap \gamma ) < \dim (\gamma ).
$$
The map $ \mu $ and the moment polytope $ \Delta $ are defined with some freedom.
It depends on the reductive decomposition $\mathfrak{g}= \frak h + \frak m $.
There is a unique maximal moment polytope $\Delta_{\max}$,
containing all the others.
Its corresponds to the $Q$-orthogonal reductive decomposition,
where $ Q $ is any $Ad(G)$-invariant Euclidean metric on $ \frak g :$
$$
Q(\frak h, \frak m)=0.
$$
Although such complement $ \frak m $ looks
\footnote{
moreover, $ \Delta _{\max} $ allows to deal only with global homogeneous geometries instead of local ones.
}
most elegant and symmetric (cf. \cite {BWZ}),
it can give rise to a non-admissible compactification.
This holds, if and only if the set $T_{\max}$ of locally Euclidean geometries
at the boundary of $\Delta_{\max}$ contains a vertex of $\Delta_{\max}$.
Let us denote by $\Delta_{\min}$ the convex hull of all vertices $v$ of $\Delta_{\max}$ that do not lie in $T=T_{\max}$, and all vertices $v=\varepsilon_j$ of $S$ satisfying the same property $v\notin T_{\max}$.
Turning to the spaces $G/H$
in the five examples above, we have $\Delta _{\min} = \Delta _{\max}$,
but for the $(2k+1)$-dimensional sphere $U_{k+1}/U_{k}$, $k>0$, we have distinct segments
\begin{align*}
\Delta_{\max}&=[2\varepsilon_2-\varepsilon_1,\varepsilon_1],
&
\Delta_{\min}&=[2\varepsilon_2-\varepsilon_1,\varepsilon_2].
\end{align*}
\begin{OBS*}
A compactification $\Delta = \MET_1\cup \Gamma $ is admissible iff $\Delta =\Delta_{\min}$.
\end{OBS*}
It is easy to check that $\Delta_{\min}$ is contained in all the moment polytopes $ \Delta $, but may be different from any of them
(e.g., for the sphere $G/H=SU_{k+1}/SU_{k}$, $k>1$).
If $\Delta_{\min}$ is a moment polytope,
its corresponds to the $B$-orthogonal reductive decomposition, that is,
$$
B(\frak h, \frak m)=0.
$$
Moreover,
its
depends only of the subspace
$\MET \subset \underline{\otimes^2T^*}(G/H)$
(cf. Proposition~\ref{PROP:Del_min=Nw}, below).
We will show that extending the group $G$ so that the space $\MET_1(G,H)$ does not change, we can always construct an admissible compactification.
Suppose $G_1$ is a compact Lie group, the semidirect product of $G$ and a $G$-invariant torus:
\begin{center}{}
$G_1= (S^1)^k \opl\lX\limits_{\pi_0(G)} G$,
where
$(S^1)^k \subset \opn{Isom}(G/H, g_1)$,
and
${k\ge 1}$.
\end{center}
Assume, moreover, that $G_1$
acts almost effectively on the manifold $G/H$ (in a natural way)
with an isotropy subgroup $H_1 \supset H$.
So
\begin{center}{}
$G_1/H_1=G/H$, and $\dim(G_1)>\dim(G)$.
\end{center}
In this situation, we call the homogeneous space $G_1/H_1$ a {\bf toral extension} of the space $G/H$.
We call such extension {\bf non-essential}, if $\MET(G_1,H_1) = \MET(G,H)$,
and {\bf essential}, otherwise.
\begin{LEM}\label{LEM:T-2} The following conditions are equivalent:
\begin{enumerate}
\item
all toral extensions of $G/H$ are essential,
and $\frak{z(g)} \subset\frak m$, e.g.,
$
B(\frak h, \frak m)=0;
$
\item
$T$ contains no vertices of $\Delta$, i.e.,
$\MET_1\cup \Gamma$ is an admissible compactification.
\end{enumerate}
\end{LEM}
We may assume that all toral extension of $G/H$ are essential,
since one can always pass from $G/H$ to a (unique)
maximal non-essential toral extension of $G/H$,
which can be described explicitly.
Thus $\frak{z(g)} \subset\frak m$ iff
$\Delta$ contains in any other moment polytope, and
$$
\Delta = \Delta _{\min}.
$$
For example, this assumption is fulfilled, if
$\bigcap_{g \in \MET(G,H)} \opn{Isom} (G/H,g) = G $.
Remark that a toral extension of a connected group is also connected.
\begin{PROP}{}\label{PROP:C}
Suppose $G$, $H$ are connected groups, and $\Delta = \Delta _{\min}$.
Then there are no locally Euclidean geometries at infinity, that is, $T= \varnothing$.
\end{PROP}
\begin{proof}[Proof] By Lemma~\ref{LEM:T}, \ \ $T_{\max}$ is the empty set or a point.
By Lemma~\ref{LEM:v}, this point is a vertex of $\Delta_{\max}$.
Then $T=\varnothing$.
\end{proof}
Now we turn to examples with $G$, $H$ connected,
where $\Delta _{\min} \ne \Delta _{\max}$.
\begin{EXAMPLES}{}
Consider the homogeneous space $M_{k,l}^{m,n} = (S^{2m+1} \times S^{2n+1})/T^1$
of $G={(U_{m+1} \times U_{n+1})/T^1}$
studied by Wang and Ziller (1990), see also \cite {B-G}.
The isotropy representation $\rho $ has a simple spectrum
($k,l,m,n>0$).
Then $ \Delta _{\max} $ is a triangle, $ T_{\max} $ is one of its vertices, and $ \Delta _{\min} $ is a trapezoid, obtained by truncation of the triangle at the vertex $ T_{\max}: $
$$
\Delta _{\max} = \opn{Conv}\{(2,0,-1),(0,2,-1),(0,0,1)\}, \quad T_{\max}=\{(0,0,1)\},
$$
$$
\Delta _{\min} = \opn{Conv}\{(2,0,-1),(0,2,-1),(0,1,0),(1,0,0)\}.
$$
\end{EXAMPLES}
\begin{EXAMP}{}
Let $ M_{k, l} ^7 $ be a seven-dimensional homogeneous Aloff--Wallach space
with $k>l>0$
(so $\rho $ has a simple spectrum).
Then $ \Delta _{\min} $ is a (irregular) octahedron. The polytope $ \Delta _{\max} $ has seven faces and seven vertices.
It can be obtained by constructing a tetrahedron on a face of $ \Delta _{\min} $.
The seventh vertex is $T_{\max}$.
\end{EXAMP}
\section{First application}\label{sect:6}
In \cite {BWZ}, the following theorem about the structure of the set of invariant Einstein metrics on a compact homogeneous space was derived from a certain variational theorem.
\begin{THM}{}\label{THM:1}
Let $G$ be compact Lie group, $G/H$ a
connected simply connected (or with finite fundamental group), homogeneous space, and $\mathcal E_1=\mathcal E_1(G,H)$ the set of all invariant, positive definite Einstein metrics on $G/H$
with
volume $ 1 $.
Then
$\mathcal E_1$ consists of at most finitely
many compact linearly connected components.
\end{THM}
The set $ \MET_1(G,H)$ of
all invariant unit volume Riemannian metrics $ g $ on $ G / H $,
$\opn{vol}_g(G/H) =1$,
has the structure of non-compact Riemannian
symmetric space.
The subset of Einstein metrics is
the set of critical points of an algebraic function,
assigns to every metric $ g \in \MET_1 (G, H) $ the scalar curvature $ s = \SC (G / H, g) $, and, moreover,
its gradient at $g$ is the minus traceless part of the Ricci tensor of $g$, that is,
for all $g\in\MET_1,$
$$
\opn{grad} s(g)=-\opn{ric}^0(g)
$$
(Theorem of Hilbert--Jensen \cite {Jen-2,AB}).
Therefore, Theorem~\ref{THM:1} is equivalent to the following proposition.
\begin{PROP}{}\label{PROP:1}
The subset $\mathcal E_1(G,H) \subset \MET_1(G,H)$ is bounded.
\end{PROP}
As we shall see, the admissible compactification
$ \Delta _{\min} = \MET_1 \cup \Gamma $
leads to a simple, new, mostly algebraic, proof of these results for the special case of a homogeneous space with simple spectrum of the isotropy representation (i.e., in the case when all $ H $-invariant quadratic forms on $ \frak g / \frak h $ can be reduced simultaneously to principal axes).
Remark that in the original proof by B\"ohm, Wang and Ziller \cite{BWZ} the simplicity of the spectrum was not used.
\begin{proof}[Outline of the proof for the case of a simple spectrum]
The set $ \mca {E} = T \cup \mca {E} _1 $ of all points of the polytope $ \Delta $
(possibly at infinity),
corresponding to Einstein geometries, is compact
(we do not dwell on the proof).
This implies that in the case when the groups $G$, $H$
are connected, then the set
$\mca{E}_1=\mca{E}_1(G,H)$ is compact.
Indeed, we may assume that $\Delta = \Delta_{\min}$,
and use Proposition~\ref{PROP:C}.
Then there are no locally Euclidean geometries at infinity, that is, ${T= \varnothing}$. The assertion follows.
In the general case it is sufficient to check
that $ T $ is open in $ \mca {E} $.
This is obviously true for $d=1+\dim \MET_1=2$, and we may assume $d>2$.
Every point $ t \in T $ lies in the closure of a suitable submanifold of the form
$ \MET_1 (G_1, H_1) \subset \MET_1 (G, H) $.
Here $G_1/H_1$ is a toral extension of the space $G/H$,
where $G_1 \supset G$, $H_1 \supset H$, $G_1/H_1 = G/H$.
In this case the homogeneous manifold $ G_1/H_1 $ represents the same simply connected manifold as $ G / H $, and also has a simple spectrum of the isotropy representation.
Assuming $\frak g=\frak h+\frak m$ is an $H\cdot Z_G(H^0)$-invariant decomposition (e.g.,$B$-orthogonal), then $\frak g_1=\frak h_1 + \frak m$ is an $H_1$-invariant reductive decomposition of the extended Lie algebra $\frak g_1$.
It follows from Theorem~\ref{THM:DEL} that the moment map $ \mu $
(comatible with $\frak m$)
defines a diffeomorphism of $ \MET_1 (G_1, H_1) $ onto a linear submanifold of the interior of the polytope $ \Delta = \ov{\mu(\MET_1)} $, that is, its interesction with an affine plane.
The condition $ \Delta = \Delta_{\min} $ of Section~\ref{sect:T} implies that this submanifold is proper, i.e.,
$$
\dim \MET_1(G_1,H_1)<d-1=\dim \MET_1(G,H).
$$
We may assume by induction on $d$ that the proposition holds for $G_1/H_1$, and we remark that the submanifold $\MET_1(G_1,H_1)$ is invariant under
the gradient Ricci flow $\dot g = -\opn{ric}^0(g)$ on $ \MET_1 (G, H) $.
Now we will associate with a point $t\in T$ an explicit submanifold $ \MET_1 (G_1, H_1)$ of the interior of $ \Delta $ described below.
By Theorem~\ref{THM:T}, \ $T$ is the union of some faces of the standard weight simplex $S$ with vertices $\varepsilon_i$, $i\in \{1,\dots,d\}$. Let $\tau\ni t$ be the smallest face $\sigma$ of $S$ containing the point $t$, so that $\tau=\bigcap_{\,t\in\sigma\subset T}\sigma$, and let $\varepsilon_i$, $i \in I$, are vertices of $\tau$.
The corresponding submanifold $ \MET_1 (G_1, H_1) $ consists
of all interior points $x=(x_1,\dots,x_d)$ of the polytope
$\Delta$ satisfying the following system of linear equations:
$$
x_i=x_k, \quad \mbox{if\ }
g_1([\frak m_i,\frak m_j],\frak m_k)
=g_1([\frak m_k,\frak m_j],\frak m_i)\ne 0,
\mbox{\ for some\ }
j\in I.
$$
(Recall that $\sum x_i\dim \frak m_i = 1$ for all $x\in\Delta$.)
We can give an equivalent definition of $ \MET_1 (G_1, H_1)$.
Denote by $\gamma=\bigcap \beta $ the intersections of all faces $\beta\subset\Delta$ such that $t\in \beta$ (so $t\in\tau\subset\gamma$).
Remark, that
$\sum_{k \notin I} x_k \dim \frak m_k =0$ for $x\in\gamma$, and $>0$ for $x\in\Delta\smallsetminus\gamma$.
Moreover,
since $\Delta=\Delta_{\min}$, this intersection $\gamma$
can be obtained explicitly as the convex hull of the points
\begin{align*}
&
\varepsilon_i+\varepsilon_j-\varepsilon_k,
&&
\varepsilon_k+\varepsilon_j-\varepsilon_i,
&&
j\in I
\intertext{where}
&
{g_1([\frak m_i,\frak m_j],\frak m_k)\ne0},
&&
i,k\notin I,
&&
i\ne k.
\end{align*}
As we noted above, $\dim(\tau)<\dim(\gamma)$, since
$\Delta=\MET_1\cup \Gamma$
is an admissible compactification of $\MET_1$.
Consider $\RR^d$ as the Lie algebra of the group
$ (\RR_{> 0}) ^d \subset \mathrm{GL} (\frak {m}) $ with
the Euclidean metric $ (x, x) = \sum \dim(\frak m_i)x_i ^2 $ (so that $(\varepsilon_i,\varepsilon_j) = \frac{1}{\dim \frak m_i}\delta_{ij}$).
Let $\Omega$ be the sphere of unit vectors tangent to the face $\gamma$ and orthogonal to $\tau-t=\{z-t:z\in \tau\}$.
Let $Z$ be the intersection of $\Delta$ with
the orthogonal complement of the vector subspace $\opn{span}(\Omega)$ at the point $t$.
Then $Z$ is obviously a compact convex polytope
of dimension $\ge 1$, containing the point $t$.
The intersection of $Z$ with the interior of $\Delta$
contains the point $\mu(g_1)$, and
coincide\footnote{The compactification $Z$ of $ \MET_1 (G_1, H_1) $ is non-admissible, since $T$ contains the face $\tau$ of $Z$.}
with $ \MET_1 (G_1, H_1) $, i.e.,
$Z \smallsetminus \Gamma = \MET_1 (G_1, H_1) $.
To carry out induction on $ d $, we must show that every Einstein metric $ g ^{G / H} \in \MET_1 (G, H) $, sufficiently close to $ t $ (if it exists) would be contained in $\MET_1(G_1,H_1)$.
We can define a small open neighborhood $U_{\rho}$ of the point $t$ in $\Delta$ by
$$
U_{\rho}=\{\lambda A+z: \lambda\in [0,\rho), A\in \Omega, z\in Z,|z-t|<\rho \} .
$$
(By construction, it is an open subset of $\Delta$, if $0<\rho<\rho_0$).
\begin{LEM} The complement $U_{\rho} \smallsetminus Z$ contains no solution of the Einstein equation
(that is, no point $ x\in\mca {E} = T \cup \mca {E} _1 $),
if $\rho$ is sufficiently small.
\end{LEM}
To prove this lemma, we consider the flat geometry $(M (t),g_1) $ as the geometry induced on a simply transitive group of motions of Euclidean space, and use the following facts.
The scalar curvature $ s (g) $ of each left-invariant Riemannian metric $ g $ on a solvable Lie group is non-positive, $ s (g) \le 0 $.
A metric $g_0$ with $s (g_0) = 0 $ is Euclidean
(G.Jensen \cite {Jen-2}, E.Heintze),
and the Hessian $ s'' (g_0) $ of the function $g\mapsto s(g)$
has the rank $ = \opn {codim} \{g: s (g) = 0 \} $.
We want to extend this Hessian over each geodesic line on $ \MET_1 (G, H) $
orthogonal to $ \MET_1 (G_1, H_1) $
(with respect to the natural inner Euclidean metric on $ \MET_1 (G, H) $).
More precisely, denote by $\SC(M(z),g)$ the scalar curvature of $(M(z),g)$, and consider $g\in \exp(\lambda \Omega).g_1$.
To each triple $ z \in Z $, $ A \in \Omega $, $ \lambda \ge0 $, we associate the number
$$
u(z,A,\lambda ) = -\frac12\frac{\partial }{\partial \lambda }\SC(M(z),\,e^{ - \lambda A}\opn{.}g_1).
$$
We have $\frac{\partial \textstyle u}{\partial \lambda }(t,A,0)>2 \delta >0$ for all $A \in \Omega $.
This follows from the above facts about $s(g)=\SC(M (t), g)$, since $\frac{\partial \textstyle u}{\partial \lambda }(t,A,0)=-2s''(g_1)(g_1A,g_1A)$.
Moreover, $u(z,A,0) \equiv 0$ (in particular, when $ z \notin \Gamma $ this follows immediately from the invariance of $ \MET_1 (G_1, H_1) $
under the gradient Ricci flow $\dot g = -\opn{ric}^0(g)$ on $ \MET_1 (G, H) $).
Using continuity, we get an estimation
$
u(z,A,\lambda )\ge \delta \lambda , \,
\forall\, (z,A, \lambda ) \in Z' \times \Omega \times [0,\rho]
$
for a sufficiently small neighborhood $ Z'\subset Z $ of the point $ t $, and some $ \rho> 0 $.
Now we can estimate the traceless part $ \opn {ric} ^0 $ of the Ricci tensor $ \opn {ric} $ at the point
$0\in M(x)$ for each of the infinitesimal Riemannian homogeneous spaces $(M(x),g_1)$
with the parameter $x\in\Delta$ sufficiently close to $t$.
Changing $\rho$ if necessary, one can construct a natural locally one-to-one
continuous
map $\Phi: U_{\rho}\to \Delta$,
$
(z, A, \lambda) \longmapsto z+\lambda A \in U_{\rho} \longmapsto x=x(z,A,\lambda ) \in \Delta ,
$
where $\lambda \in [0,\rho)$, possessing the following properties\,:
\begin{itemize}
\item
$
x(z,A,\lambda) =\mu(e^{-\lambda A}.\mu^{-1}(z))
$
for all
$
(z,A,\lambda) \in U_{\rho}\smallsetminus \Gamma
$
(so the restriction $\Phi|_{U_{\rho}\smallsetminus \Gamma}$
can be considered as the normal
exponential map along $ \MET_1 (G_1, H_1) = Z\smallsetminus \Gamma $ with respect to the
$ (\RR_{> 0}) ^{d-1} $-invariant Euclidean metric on $ \MET_1 (G, H) = \Delta \smallsetminus \Gamma$).
\item Moreover,
$\Phi|_{Z\cap U_{\rho}}=\opn{id} $, i.e.,
$x(z,A,0)\equiv z$.
For each face $\beta$ of $\Delta$ containing the point $t$
there is a smooth map
$U_{\rho}\cap \mbox{relative interior}(\beta)\ni y \longmapsto \Phi(y)\in\mbox{relative interior}(\beta)$.
Every disc $D(z)=\{z+ \lambda A: A\in \Omega, \lambda \in[0,\rho)\} $, $z\in Z\cap U_{\rho}$ is tangent to $\Phi(D(z))$ at the center\,$z$.
\begin{comment}
\input THAT50/End_Proof-2
\end{proof}
\end{comment}
\item
$u(z,A, \lambda ) \ge \delta \lambda $,
for all points $z+\lambda A \in U_{\rho}$ (as above).
\end{itemize}
Consider now a scalar product $g\in\MET$, a point $x\in\Delta$, the Lie algebra $\frak g_x=(\frak h+\frak m, [\cdot,\cdot]_x)$, and denote the geometry $(M(x),g)$ simply by $([\cdot,\cdot]_x, g)$. To any $H$-invariant linear transforma\-tion $a$ of $\frak m$ we associate a geometry $(a.[\cdot,\cdot]_x, a.g)$,
were $a.[\cdot,\cdot]_x = T_a[T_{a^{-1}}\cdot,T_{a^{-1}}\cdot]_x$ is a new Lie operation on $\frak h+\frak m$, and $a.g(\cdot,\cdot)=g(a^{-1}\cdot,a^{-1}\cdot)\in \MET$ is an $H$-invariant Euclidean scalar product on $\frak m$.
Geometries $(a.[\cdot,\cdot]_x, a.g)$ and $([\cdot,\cdot]_x, g)$) are equivalent, by construction.
As an immediate consequence we obtain the following Heber's identity (cf. \cite[\S6]{He}):
$$
\SC(a.[\cdot,\cdot]_x, a.g)=\SC([\cdot,\cdot]_x, g),
\qquad\forall \qquad a \in (\mathrm{GL}(\frak m))^H.
$$
For each $x=x(z,A,\lambda)$
there is a scalar operator $\kappa$ on $\frak m$
such that $T_{\kappa^{-1}}[T_{\kappa}\cdot,T_{\kappa}\cdot]_x = e^{\lambda A} .\, [\cdot,\cdot]_z $. Then
\begin{gather*}
\langle \opn{ric}([\cdot,\cdot]_x, g_1),\, \kappa ^2A \rangle
= \left. - \frac12\frac{d}{dt}\right|_{\,t=0}
\SC(e^{\lambda A} .\, [\cdot,\cdot]_z,\, e^{-tA} .\, g_1)
\\
= - \frac12\frac{\partial}{\partial\lambda}\SC([\cdot,\cdot]_z,\, e^{-\lambda A} .\, g_1)
= u(z,A, \lambda ).
\end{gather*}
Finally, for all $ x= x(z,A,\lambda)\in \Phi(U_{\rho}) $,
and some real function $ \kappa $ in $ x $ we have
$$
\langle \opn{ric}(M(x),g_1),\, \kappa ^2A \rangle
= u(z,A, \lambda ) \ge \delta \lambda .
$$
Clearly, $ \opn {trace} (A) = 0 $. This means that if $ \opn {ric} ^0 (M (x), g_1) = 0 $,
then $ \lambda = 0 $, and $ x \in Z $.
We have $U_{\rho'}\subset \Phi(U_{\rho})$ for some $\rho'>0$.
These imply Lemma and Proposition.
\end{proof}
A locally Euclidean geometry at infinity $ M (t) $
plays a central role in the above proof.
Cf. the nice study of the flat space
$\RR^{n-k} \times T^k=\RR^{n-k} \times (\RR/\ZZ)^k$
as the limit of a sequence of compact homogeneous Riemannian spaces $(G_i/H_i, g_i)$ in \cite[\S2]{BWZ}.
If \ $ T = \varnothing $, then the compactness of $ \mca {E}_1 $
is reduced to the compactness of $ \mca {E} $,
and the proof of the proposition is reduced to the first sentence.
\begin{EXAMPLES}{}
The condition $ T = \varnothing $ holds if $ \opn {rank} (G) = \opn {rank} (H) $
by Lemma~\ref{LEM:T}.
\end{EXAMPLES}
\begin{EXAMP}{}
Let $G$ be a compact connected group, and $G/H$ the total space of a principal circle bundle over a K\"ahler homogeneous space $ G / K $, associated to a untwisted ample line bundle. We call $G/H$ {\it a generalized Hopf bundle}.
Let, moreover, the spectrum of the isotropy representation $\rho $ of
the group $H$ be simple.
Then the space $G/H$ has at most one toral extension (cf. Section~\ref{sect:T}).
If it exists, then $G$ is a semisimple group.
It follows from the simple spectrum condition, that
this extension $G_1/H_1$ is non-essential.
Passing from $G/H$ to $G_1/H_1$, we may assume that the space $G/H$ has no toral extension (so $\dim (\frak{z(g)})=1$).
Choose now the $B$-orthogomal complement $\frak m$ to $\frak h$, that is $B(\frak h, \frak m)=0$. Then $\Delta = \Delta _{\min}$. By Proposition~\ref{PROP:C}, \ $T=\varnothing$.
\end{EXAMP}
Note that the homogeneous spaces $ M_{k, l} ^{m, n} $ and $ M_{k, l}^7 $ considered
above are generalized Hopf bundles over ${\CP^{m}\times \CP^{n}}$ and ${F_3 (\CC) = SU_3 /T^2}$, respectively. Assuming $B(\frak h, \frak m)=0$, then $ {T = \varnothing }$.
Here is a simple example with $T=\varnothing$ and non-connected $G$, $H$.
\begin{EXAMP}\label{EXAM:53}
Let $G=(U_{k+1})^5 \lX C_5$, $H=(U_k)^5 \lX C_5$, so that
$G/H$ is the direct product of five spheres $S^{2k+1}=U_{k+1}/U_k$, and $C_5$ is the cyclic group of permutations of spheres. Then $d{=}4$.
We have four irreducible $H$-modules $\frak m_i$, $i=1,\dots,4$ of dimensions $\dim \frak m_i = 1,2,2,10k$ respectively.
The polytope $\Delta_{\max}$ is an octahedron with vertices
$\varepsilon_i$, $\delta_i=2\varepsilon_4-\varepsilon_i$, $i=1,2,3$, and $\Delta_{\min}$ is a tetrahedron $(\delta_1, \delta_2, \delta_3, \varepsilon_4)$. Let $B(\frak h,\frak m)=0$. Then $\Delta=\Delta_{\min}$. By Lemma~\ref{LEM:T}, \ $T=\varnothing$.
Moreover, an infinitesimal homogeneous Riemannian space $(M(x),g_1)$
is defined only locally (hence, is non-complete), if and only if $x$ is an
interior point of an edge $(\delta_r, \varepsilon_4)$, or a triangular
face $(\delta_1,\delta_r, \varepsilon_4)$,
where $r\in\{2,3\}$.
In this case it is an isomorphism of Lie algebras $I_x:\frak g_x \cong \frak g$, but $I_x(\frak h)\ne \frak h$.
(Note that for the spaces $G/H$ in our other examples
all geometries at infinity are locally isometric to complete Riemannian spaces.)
\end{EXAMP}
\section{Second application}\label{sect:7}
In \cite [Introduction] {BWZ} the authors asked the question about the finiteness of the set $ \mca {E} _1 = \mca {E} _1 (G, H) $ of unit volume Einstein metrics on a compact
simply connected homogeneous space $ G / H $ with simple spectrum of isotropy representation.
In this case, if $ d> 1 $, the Einstein equation reduces to a system of $ d-1 $
rational algebraic equations on $ d-1 $ unknowns.
C.B\"ohm, M.Wang, and W.Ziller ask the following question:
Is this system always {\it generic}, i.e.,
does it admit at most finitely many {\it complex} solutions?
Here is a partial answer to this question \cite{2006,2007}.
Assume for the moment that $ \mca {E} _1 (\CC) $ is an infinite set.
Therefore it is noncompact, since the Einstein equation is algebraic.
Then it can be compactified by attaching some of the (complex) solutions at infinity lying on $ \Gamma^{\CC} $.
Here $ \Gamma^{\CC} $ can be regarded as a complex hypersurface in a compact complex algebraic variety with singularities $ \Delta ^{\CC} $, which is a complexification of the polytope $ \Delta $.
Thus, we obtain:
\begin{CLAIM*}
The set $ \mca {E} _1 (\CC) $ is finite if and only if in some neighborhood of the hypersurface $ \Gamma ^{\CC} $ all complex solutions are at infinity, i.e., lie on $ \Gamma ^{\CC} $.
\end{CLAIM*}
{\it Moreover, there are no solutions at infinity,
if and only if $ \mca {E} _1 (\CC) $ is a finite set,
and, counting with multiplicities, it consists
of
\begin{equation*}
\begin{aligned}{}
\nu
&
= (\opn{vol}(S))^{-1}\opn{vol}(\Delta_{\min})
\\ &
={(d{-}1)!}\opn{vol}(\Delta_{\min}).
\end{aligned}
\end{equation*}
solutions. } Here
$ \Delta = \Delta_{\min} $ is an admissible compactification of $ \MET_1 $,
and $S$ is the standard $(d-1)$-dimensional simplex in $\RR^{d-1}$.
(Note that $\nu$ is always an integer.)
These claims can be make rigorous and proved using the theory of toric varieties.
Since $\frak {g / h}$ is a multiplicity-free $H$-module,
the space of invariant Riemannian metrics on $ G / H $ has the natural complexification of the form
$$
\MET^{\CC} = (\CC \setminus 0)^d = (\CC \setminus 0)\times \dots \times (\CC \setminus 0).
$$
(Note that $ \MET ^{\CC} $ contains all the invariant pseudo-Riemannian metrics on $ G / H $.)
The quotient $\MET^{\CC}/\CC^\times $, where $\CC^{\times} = \{(z, \dots ,z)$, $z \in \CC \setminus 0\}$, can be considered as a complexification of the space $ \MET_1 $.
The compactification $ \Delta = \MET_1 \cup \Gamma $ of the space $\MET_1 $ also has a natural complexification, namely, the toric variety
$$
\Delta^{\CC} = (\MET^{\CC}/\CC^\times) \cup \Gamma^{\CC}
$$
(see, e.g., \cite {Fulton}).
Here $\Delta^{\CC}$ is the toric variety of the fan in the lattice $\mathrm N$ from the polytope $\Delta=\Delta_{\min}\subset \varepsilon_d+\mathrm M_\RR$,
where $\mathrm M=\opn{Hom}(\mathrm N,\ZZ)=\sum_{i=1}^{d-1} \ZZ (\varepsilon_i-\varepsilon_{i+1})$.
The algebraic torus $(\CC\setminus 0)^d$ acts on $\Delta^{\CC}$ with open orbit $\MET^{\CC}/\CC^\times$, so that the subgroup $\CC^{\times}$ acts trivially.
In this way,
the polytope $\Delta$ can be considered as the closure of a single orbit of a subgroup $(\RR_{>0})^d$. The closure of each orbit of $(\CC\setminus 0)^d$ meets $\Delta$ in a single face, and each orbit of the compact torus $(S^1)^d$
meets $\Delta$ in a single point.
Algebraic Einstein equations are naturally defined on
$\MET^{\CC}$, $\MET^{\CC}/\CC^\times $, and $\Delta^{\CC}$. Let $\varepsilon = \varepsilon (G,H)$ be
the number of its {\it isolated} complex solutions (counting with multiplicities) on $ \MET^{\CC} / \CC^\times $.
Using the generalized Bezout theorem, one can prove the following theorem.
\begin{THM}{}\label{THM:e-nu}
Suppose $\frak {g / h}$ is a multiplicity-free module of $H$,
and $d=\dim \MET(G,H) > 1$.
Let $\nu = \nu(G,H)
= (\opn{vol}(S))^{-1}\opn{vol}(\Delta_{\min})
={(d{-}1)!}\opn{vol}(\Delta_{\min})$.
Then
$$
\varepsilon \le \nu < 6^{\,d-1};
$$
for $ \nu = \varepsilon $ all solutions are isolated, and complex solutions at infinity cannot exist;
for $ \nu> \varepsilon $ there is at least one complex solution lying on $ \Gamma ^{\CC} $ (i.e., at infinity).
\end{THM}
Roughly speaking, the missing $\nu-\varepsilon$ solutions ``escape to infinity''.
The strict inequality $ \nu> \varepsilon $ holds, for example,
if $ G $ is simple, $ H $ is its maximal torus, and
$G/H \ne SU(2)/T^1, \, SU(3)/T^2$
\cite{2006,2007}.
The estimation $ \nu \ge \varepsilon $ is sharp, as the following examples show.
\begin{EXAMPLES}{}
For $G/H = (S^{2k+1})^5$ in the preceding example,
we have $\nu=1$ and one obvious solution, so $ \nu = \varepsilon $.
For the spaces $G/H$
defined in
examples~\ref{EXAM:21}, \ref{EXAM:22} (if $[\frak h,\frak m]= \frak m$),
\ref{EXAM:23} ($G=E_7$ and~$E_8$)
hold $\varepsilon = \nu$.
This follows immediately
(without solving the Einstein equation)
from \cite[\S7.1, Tests 1 and 2]{2007},
Proposition~\ref{PROP:Del_min=Nw} below,
and the fact that $\Delta = \Delta _{\min}$.
By finding volumes, we obtain, respectively,
$$
\varepsilon = \nu = 1,1,4,20,23.
$$
Moreover, $SU(3)/T^2$ and every K\"ahler homogeneous space $G/H$ satisfying
conditions of Example~\ref{EXAM:22} admit $\nu=4$ positive definite invariant Einstein metrics
$g\in\MET(G,H)$ with scalar curvature $1$
(D.V.Alekseevsky, 1987; M.Kimura, 1990).
\end{EXAMPLES}
\begin{EXAMP}{} For homogeneous spaces of Wang--Ziller and Aloff--Wallach, respectively,
with $k,l,m,n>0$ and $k>l>0$
we have
$$
\varepsilon (M_{k,l}^{m,n}) = \nu (M_{k,l}^{m,n}) = 3,
\quad
\varepsilon (M_{k,l}^7) = \nu (M_{k,l}^7) = 16.
$$
To the proof, one can examine faces of polytopes
$ \Delta _{\min} $ using Tests 1 and 2 of \cite [\S7.1] {2007}. Equality $ \varepsilon = \nu $ follows immediately without any calculations
(also in the second case, see \cite[Exam. 7.5]{2007}, is used the absence of complex Ricci-flat metrics on the underlying flag space $SU(3)/T^2$).
\end{EXAMP}
\section{Newton polytope and proof of Theorem~\ref{THM:e-nu}}\label{sect:8}
In this section
we interpret
$\Delta_{\min}$ as a Newton polytope,
we estimate the normalized volu\-me
$\nu=(d-1)!\opn{vol} (\Delta_{\min})$,
and prove Theorem~\ref{THM:e-nu}.
Consider the moment polytopes $\Delta$ (and $\Delta_{\min}$) as polytopes
with vertices in $\ZZ^d$ by setting $\varepsilon_1=(1,0,\dots,0),\dots,\varepsilon_d=(0,\dots,0,1)$.
We express a metric $ g \in \MET (G, H) $ as
$
g= \bigoplus x_k g_1 |_{\frak m_k},
$
and consider $x_k>0$, $ k \in \{1, \dots, d \}$,
as coordinates on $ \MET (G, H) $.
By ${s(g)=\SC(G/H,g)}$ we denote the scalar curvature of $g$.
Then
$$
-\frac{x_i}{m_i}\frac{\partial s}{\partial x_i}
=
\frac{b_i}{2x_i}
- \frac{1}{4m_i} \opl{\textstyle\sum}_{j,k=1}^d {\scriptstyle [i,j,k]}
\frac{2x_k^2-x_i^2}{x_ix_jx_k},
\qquad 1\le i \le d,
$$
where $b_i>0$, ${\scriptstyle [i,j,k]}\ge0$ are coefficients, $m_i=\dim(\frak m_i)$, and
the original grouping of monomials taken from \cite {BWZ}.
The Einstein equation reduces to a system of $ d-1 $ homogeneous equations
$$
f_i:= \frac{x_i}{m_i}\frac{\partial s}{\partial x_i}-\frac{x_{i+1}}{m_{i+1}}\frac{\partial s}{\partial x_{i+1}} =0,
\quad 1\le i < d.
$$
\begin{figure}
\caption{
If $d=2$, then the permutohedron $\Pi$
is a segment. It is equal to three segments $S=[(1,0),(0,1)]$,
and its normalized length is $P_1(3)=3$}
\end{figure}
\begin{figure}
\caption{Case $ d = 3$. Then $\Pi$ is a hexagon. It is equal to thirteen triangles $S$,
and its normalized area is $P_2(3)=13$}
\end{figure}
We will use the following terminology.
A Laurent polynomial in $x=(x_1,\dots,x_d)$ is a polynomial in $x_i^{\mathstrut},x_i^{-1}$, $i=1,\dots,d$. The {\bf Newton polytope} $\Nw(f)\subset\RR^d$ of a Laurent polynomial $f$ is the convex hull of the vector exponentials of its monomials.
By considering $s$ and $f_i$ as homogeneous Laurent polynomials in $x^{-1}=(x_1^{-1},\dots,x_d^{-1})$, cf.\,\cite{2006}, we obtain:
\begin{PROP}\label{PROP:Del_min=Nw} For a general linear combination $f=\sum c_i f_i$ of $f_i$ with coefficients $c_i\in \RR$ we have
$
\Nw(f)=\Nw(s)=\Delta_{\min}.
$
\end{PROP}
Thus the polytope $ \Delta _{\min} $ can be introduced an invariant way as the Newton polytope $\Nw(s)$ of the polynomial of scalar curvature.
\begin{proof}[Outline of proof] Obviously $\Nw(f)\subset \Nw(s)$. Moreover, $\Nw(f)\ne\varnothing$. From the expression of $s$ in \cite[Eq. (7.39)]{AB} follows that
$\Nw(s)\subset \Delta$ for any moment polytope $\Delta$
described in \S\,2.
Suppose $\gamma\subsetneq\Delta$ is a face of $\Delta$ such that $\gamma\cap\Nw(f) =\varnothing$.
The point is to prove that
any geometry $(M(t),g_1)$ (such as the one in \S\,3) with $t\in \gamma$ is Einstein. So
either $\gamma=\varnothing$, or the set $T$ of Einstein
geometries at infinity contains the whole face $\gamma$ of $\Delta$, that is, $\Delta\ne \Delta_{\min}$.
Therefore, $\Nw(f)=\Delta_{\min}$.
\end{proof}
Now we use the theory of systems of rational algebraic Laurent equations, developed by A.G.Kushnirenko and D.N.Bernshtein (see e.g., \cite{D.N.Ber}).
(The latter approach via intersections of algebraic cycles on toric varieties is well known.)
It follows from \cite {D.N.Ber} that
$ \varepsilon \le (d-1)! \, V $,
where $ V $ is the volume of the Newton polytope $\Nw(f)$.
By Proposition~\ref{PROP:Del_min=Nw}, $ \Nw(f) = \Delta _{\min} $. Hence $ \varepsilon \le \nu $.
We now prove the inequality $\nu\le P_{d-1}(3)$, where $P_k$ in $k$-th Legendre polynomial, that is, $P_n(z)=\frac{1}{2^n\,n!}\, \frac{d^n}{dz^n}(z^2-1)^n$.
Using the generating function ${\frac {1}{\sqrt {1-2zw+{w}^{2}}}}=\sum_{k=0}^{\infty} P_k(z)w^k$,
we can write
$$
\begin{gathered}
\sum _{d=1}^{\infty }P_{{d-1}}(3)\,{w}^{d-1} =
{\frac {1}{\sqrt {1-6\,w+{w}^{2}}}}=
\\
1+3\,w+13\,{w}^{2}+63\,{w}^{3}+
321\,{w}^{4}+1683\,{w}^{5}
+O\left ({w}^{6}\right ).
\end{gathered}
$$
The degenerate permutohedron with vertex $p\in \RR^{n+1}$ is the convex hull of the points in Euclidean space obtained from a single point $p$ by all permutations of coordinates.
\begin{LEM}
If $n\ge1$ and $z\ge1$, then
$P_{n}(z)/n!$ is
the volume of the $n$-dimensional degenerate permutohedron\footnote{The Minkowski sum $\frac{z+1}{2}S+\frac{z-1}{2}S'$ of opposite simplices $S$ and $S'=-S$.} with vertex
$(\frac{z+1}{2},0,\dots,0,\frac{1-z}{2})\in\RR^{n+1}$.
\end{LEM}
\begin{proof}[Proof]
The lemma follows from \cite {Po}, the proof of Theorem 16.3,(8) and Theorem 3.2.
\end{proof}
Let us denote by $ \Pi $ the degenerate permutohedron
with a vertex\footnote{This polytope has $2^d-2=2,6,14,\ldots$ facets, just as the classical (non-degenerate) permutohedra.}
$$
(2,0,\ldots ,0,-1) \in \RR^d
$$
By construction of moment polytopes $\Delta$ (Section~2), we have $\Delta_{\min}\subset \Delta\subset \Pi$, and
the volume of the polytope $ \Pi $ is given by
${V_\Pi
= P_{{d-1}}(3)/(d-1)!} $.
Hence, $\nu\le P_{d-1}(3)$.
Finally,
$$
\varepsilon \le \nu
\le P_{{d-1}}(3)< (3+2\sqrt2)^{d-1} < 6^{d-1}.
$$
The remaining claims of Theorem~\ref {THM:e-nu} follow from \cite[Theorem B]{D.N.Ber}.
\begin{figure}
\caption{For $ d = 4$ the enveloping permutohedron $ \Pi $ of polytopes $ \Delta $ has
normalized volume $P_{d-1}
\end{figure}
The numbers $P_n(3)=1,3,13,63,321,\ldots$ are also known as central Delannoy numbers, that is, $P_n(3)$
counts the number of the paths in $\RR^2$ from $(0, 0)$ to $(n, n)$ that use the steps $(1, 0)$, $(0, 1)$, and $(1, 1)$.
These numbers form the diagonal of the symmetric array $(d_{m,n})$ introduced by H. Delannoy (1895) in the same way, so that
$
{d_{m,n}=d_{m-1,n}+d_{m,n-1}+d_{m-1,n-1}}.
$
See, e.g., \cite[6.3.8]{Stanley2}.
\begin{COR*}
We have the upper bound of the $(d-1)$-th central Delannoy number for the normalized volume $\nu$ of the moment polytope.
Here $d>1$ is the number of the irreducible submodules in the isotropy $H$-module $\frak g/\frak h$.
\end{COR*}
\begin{EXAMPLES}{} The inequality $ \varepsilon < \nu$ holds for any space $G/H$ with $T\ne \varnothing$, e.g., for the spaces in Examples~\ref{EXAM:31}, ~\ref{EXAM:32}.
Consider $SU(p)$ for small $p$ as a homogeneous space $G/H$ in Example~\ref{EXAM:31}.
Let $p\in \{2,3\}$. Then $T$ is a finite set.
One can check (by an examination of the
solutions at infinity only) that $\varepsilon = \nu- |T|$.
The Newton polytopes are just the same as for
$SU(3)/T^2$ and $E_8/(A_2)^4$ (Examples~\ref{EXAM:22}, \ref{EXAM:23}), and $\nu \in \{4,23\}$.
Hence, $\varepsilon = \nu-p-1= 1$ and $19$ respectively.
\end{EXAMPLES}
\begin{EXAMP}{}
Let $G/H $ be the $196$-dimensional K\"ahler homogeneous
space $E_8/T^1\cdot A_1\cdot A_6$.
Then $d=4$ and $\varepsilon = \nu = 20$.
This is less than $1/3\,P_3(3)=21$.
\end{EXAMP}
\section*{Appendix. Case of K\"ahler homogeneous space with $b_2=1$}
\let\phim m
Consider now a K\"ahler homogeneous space $G/H$ with the second Betti number $b_2=1$.
Assume that the isotropy $H$-module $\frak g/\frak h$ is split into $ d>1 $ irreducible submodules.
\begin{LEM}
Given a K\"ahler homogeneous space $G/H$ with $d>1$ of a simple Lie group $G$, then $2^{-b_2(G/H)}\nu(G/H) \in \ZZ$.
\end{LEM}
\begin{proof}[Idea of proof] $2^{b_2(G/H)} = [\ZZ^d:L]$,
where $L \subset \ZZ^d$ is the subgroup generated by vertices of the polytope $\Delta$.
(Remark that $\Delta = \Delta_{\min}=\Delta_{\max}$).
\end{proof}
Let, moreover, $b_2(G/H)=1$.
Then $2\le d \le 6$.
For $d=2$ the polytope $\Delta$ is the segment with ends $e_2$ and $2e_1-e_2$.
For $3\le d \le 6$ the vertices of the polytope $\Delta $ are the points
$e_i+e_j-e_k \in \RR^d$
with $1\le i,j,k \le d$, $i\ne k$,
$j\ne k$, $i\pm j\pm k =0$. Here $e_1=(1,0,\ldots,0),\dots,e_d=(0,\dots,0,1)$.
Using MAPLE, one can triangulate these polytopes and find their normalized volumes $\nu=\nu(G/H)$.
Thus, we obtain:
\begin{CLAIM*}
$
2^{-1}\nu \in\{1,3,10,41,172\}.
$
\end{CLAIM*}
The following table gives some information
about polytopes $\Delta$ corresponding to K\"ahler homogeneous spaces $G/H$ with the second Betti number $b_2=1$ and $d>1$.
For completeness, we find the volume of a similar $(d-1)$-dimensional polytope with $d=7$.
Here $f$ is the number of facets of $\Delta$, and $\phim$ the number of all faces $\gamma$ of $\Delta$ with $0<\dim(\gamma)<d-1$
(which we call \textbf{marked})
that NOT satisfy conditions of Test~1 or Test~2 of \cite[\S 7.1]{2007}.
A marked face $\gamma$ is not a vertex.
\footnote{
Let $G/H$ be a K\"ahler homogeneous space of a simple Lie group $G$, and let $d>2$. Then the set of vertices of $\Delta$ is $\{e_i+e_j-e_k : 1\le i,j,k\le d, i\ne k,j\ne k, [i,j,k]\ne0\}$. In this case, conditions of Tests~1 and~2 for a $k$-dimensional face $\gamma$, $0<k<d-1$ can be simplified as follows: \\ 1)~$\gamma $ is a pyramid with apex $a$ and base $B$ such that if $e_i\in \gamma$, then either $e_i=a$ or $e_i\in B$; \\ 2)~$\gamma$ is a `$k$-dimensional octahedron' with vertices $e_{i_0} + v_p, e_{i_0}-v_p$, $p=1,\dots,k$, for some linearly independent vectors $v_p\in\RR^d$; the face $\gamma$ contains no points $e_i$ with $i\ne i_0$ (then $\gamma$ is the intersection of all faces $\beta\ni e_{i_0}$ of $\Delta$). For $b_2(G/H)=1$, $d>2$ there are $[d/2]$ faces $\gamma$ satisfying~2).
}
Moreover, one can prove that $\gamma$ is not an edge, so $1<\dim(\gamma)<\dim(\Delta)$.
$$
\begin{array}{c|llllll|llll}
d & & 2 & 3 & 4 & 5 & 6 & 7 \\
f & & 2 & 4 & 7 & 16 & 36 & 100 \\
\nu & & 2 & 6 & 20 & 82 & 344 & 1598 \\ \hline \varepsilon
& &\nu&\nu&\nu & 81 & ? & - \\ \delta
& & 0 & 0 & 0 & 1 & ? & - \\ \phim
& & 0 & 0 & 3 & 13 & 40 &
\end{array}
$$
We write also the known numbers $\varepsilon$ and $\delta = \nu-\varepsilon$.
By \cite{2007}, if $m=0$, then $\nu=\varepsilon$.
The first non-trivial case is $d=4$.
For $d\le 5$ all the \textit{positive} solutions of the algebraic Einstein equations are known.
In the case $d=5$ they calculated by I.Chrysikos and Y.Sakane \cite{SACOS}.
They also prove that all the complex solution are isolated ($d=5$).
{\bf Remark (the case $d=4$, $\nu=20$). }
Here $\Delta$ is a three-dimensional polytope with three marked faces $\gamma,$ namely, a trapezoid $\gamma_1$, a parallelogram $\gamma_2$, and a pentagon $\gamma_3$. To prove that $\nu=\varepsilon$ one can associate with each marked face $\gamma$ a complex hypersurface $s_{\gamma}(x_1,\dots,x_4)=0$ in $(\CC \setminus 0)^4$ and check that it is non-singular.
Here $s(x)$ is the above Laurent polynomial (scalar curvature), and $s_{\gamma}(x)$ is the sum of all monomials of $s(x)$ whose vector exponents belong to $\gamma$. See \cite[\S1.7.2]{2006,2007}. This is essentially a two-dimensional problem ($2=\dim(\gamma_i)$), i.e., we must check that a plane curve is non-singular. It is easy to prove this for $\gamma=\gamma_1$, but for $\gamma_k$, $k=2,3$ the problem reduces to
$D_k[s]\ne0$, where $D_k[s]$ is a homogeneous polynomial
(a $k$-monomial)
in coefficients of $s(x)$ with $\deg (D_k) = k$. The coefficients of $s(x)$ depend on $G/H$. There are four K\"ahler homogeneous spaces $G/H$ with $b_2=1$ and $d=4$, namely, the spaces
$
{E_8/T^1\cdot A_1\cdot A_6},\quad
{E_8/T^1\cdot A_2\cdot D_5},\quad
{E_7/T^1\cdot A_1\cdot A_2\cdot A_3},\quad
{F_4/T^1\cdot \widetilde A_1\cdot A_2}
$
(we use the Dynkin's notation $\widetilde A_1$ for the three-dimensional subgroup associated with a short simple root).
The corresponding scalar curvature polynomials $s(x)$ are computed by A.Arvanitoyeorgos and I.Chrysikos (arXiv:0904.1690). For each of them one can check that $D_2[s]D_3[s]\ne0$. This proves that $\varepsilon=\nu$ for $d=4$.
\begin{figure}
\caption{The $3$-dimensional polytope $\Delta$ with $7$ facets corresponding to four K\"ahler homogeneous spaces with $b_2=1$ and $d=4$}
\end{figure}
{\bf The case $d=5$, $\nu=82$.}
There is a unique K\"ahler homogeneous space $G/H$ with $b_2(G/H)=1$
and $d=5$, namely, the space
${G/H=E_8/T^1\cdot A_3\cdot A_4}.$
By \cite[\S3, the text after eq.(25)]{SACOS} it implies that
the algebraic Einstein equation has, up to scale, $81$ complex solutions, corresponding to roots of some polynomial $(x_5-5)h_1(x_5)$ of degree $81$ in one variable $x_5$.
There exist $6$ positive solutions \cite[Theorem A]{SACOS};
in particular, the root $x_5=5$ corresponds to a unique, up to scale, invariant K\"ahler metric on $G/H$.
Using MAPLE, one can check that the polynomial $h_1(x_5)$ has $80$ simple roots, and $81$ solutions of Einstein equation are distinct
(moreover, it has $30$ real roots, and Einstein equation has $31$ real solutions).
Thus
$$
\nu - \varepsilon = 82-81 = 1.
$$
We prove
independently
that $\nu > \varepsilon $.
Let $s(x_1, \dots ,x_5)$ be the scalar curvature
of a invariant metric $g_x$, as above. It is a Laurent polynomial in $x_i^{-1}$.
We claim that there exists a limit
$$
s_{\infty}(x_1, \dots ,x_5) =
\lim_{t\,\to\,+ \infty } s(t^{2}\,{x_{1}}, t^{4}\,{x_{2}},
t^{3}\,{x_{3}}, t\,{x_{4}}, t\,{x_{5}}),
$$
and the homogeneous function $s_{\infty}$ depends essentially on $3$ variables.
Indeed, it follows from Proposition~\ref{PROP:Del_min=Nw} and the above description of the Newton polytope
$\Delta $ that
$$
\begin{aligned}
s_{\infty}
=&
- \frac{\scriptstyle [1,4,5]}{2}{\frac {{x_{1}}}{{x_{4}}\,{x_{5}}}}
- \frac{\scriptstyle [1,3.4]}{2}{\frac {{x_{3}}}{{x_{1}}\,{x_{4}}}}
\\ &
- \frac{\scriptstyle [2,3,5]}{2}{\frac {{x_{2}}}{{x_{3}}\,{x_{5}}}}
- \frac{\scriptstyle [1,1,2]}{4}{\frac {{x_{2}}}{{x_{1}}^{2}}}.
\end{aligned}
$$
Then there is a two-dimensional face of the polytope $\Delta$, the parallelogram $P$ with vertices $e_4+e_5-e_1$, $e_1+e_4-e_3$, $2e_1-e_4$, and $e_3+e_5-e_2$. The face $P$ is orthogonal to the vector
$$
\mathbf f = (2,4,3,1,1) .
$$
\begin{comment}
Then there the parallelogram $P$ with vertices $e_4+e_5-e_1$, $e_1+e_4-e_3$,
$2e_1-e_4$, and $e_3+e_5-e_2$ is a two-dimensional face of the polytope $\Delta$,
orthogonal to the vector
$$
\mathbf f=(2,4,3,1,1).
$$
\end{comment}
According to \cite[Proposition 7]{SACOS} we have
$
[1, 1, 2]=12,
[1, 2, 3]=8,
[1, 3, 4]=4,
[1, 4, 5]=4/3,
[2, 2, 4]=4,
[2, 3, 5]=2
$.
Then
\begin{comment} *b**************************************************
$
D_P = \det \left( \begin{smallmatrix}
[1,4,5]/2 &\,\,
[1,3,4]/2 \\[1ex]
[2,3,5]/2 &\,\,
[1,1,2]/4
\end{smallmatrix}\right)
= \det \left(\begin{smallmatrix} 4/6& 2\\[1ex] 1 & 12/4 \end{smallmatrix}\right)
=0,
$
\end{comment}
the product of monomials, corresponding to each pair of opposite vertices of $P$,
coincides with $2 {\frac {{x_{2}}}{{x_{1}}\,{x_{4}}\,{x_{5}}}}$, and $s_{\infty }$
can be represented as
$$
s_{\infty} = z_0(1+z_1+z_2+z_1z_2) = z_0(z_1+1)(z_2+1),
$$
where
$z_0=-\frac {{x_{2}}}{{x_{3}}\,{x_{5}}}$.
Since for $z_1=z_2=-1$ we have
$$
s_{\infty } = ds_{\infty } =0,
$$
the complex hypersurface
$s_{\infty }(x)=0$ has a singular point $x$ with $\prod x_i\ne0$.
By \cite[\S1.7.2]{2006,2007}
this implies that $\nu- \varepsilon >0$.
Note that $\Delta$ has $\phim-1=12$ marked faces, other than $P$;
namely, $6$ three-dimensional faces with normal vectors
$$
\mathbf f =
(1, 2, 3, 4, 5), (1, 2, 1, 2, 1), (2, 1, 1, 2, 0),
(1, 0, 1, 0, 1),(1, 2, 2, 1, 0), (1, 2, 1, 0, 1),
$$
and $6$ parallelograms defined by the following normal vectors
(such as $\mathbf f=(2, 4, 3, 1, 1)$):
$$
\mathbf f =
(1, 1, 2, 2, 3), (1, 2, 3, 4, 4), (1, 2, 2, 3, 4), (2, 4, 5, 3, 1),
(5, 3, 2, 6, 1),
(3, 1, 2, 2, 1)
$$
The corresponding $12$ complex hypersurfaces are non-singular.
{\bf Additional remarks (the case $d=5$). }
Consider $(z_0,z_1,z_2)=(1,-1,-1)$ as a point $p$ in
the four-dimensional toric variety $\Delta^{\CC}$. Let $O\subset\Delta^{\CC}$ be the orbit of the group $(\CC \setminus 0)^5/\CC^\times$ trough $p$. The closure of $O$ is the two-dimensional toric subvariety $P^{\CC}$.
The point $p\in O$ is a solution at infinity (in the sence of \S7) of the algebraic Einstein equation.
Our example is excellent as the following lemma show.
\begin{LEM}
We claim now that $\Delta^{\CC}$ is smooth at each point $q\in O$.
Moreover, assuming
$\phi:\Delta^{\CC}\to\mathbb P^{N-1}(\mathbb C)$
be the natural map into the complex projective space $\mathbb P^{N-1}(\mathbb C)$, $N=\#(\ZZ^5\cap\Delta) $, then $\phi^{-1}(\phi(q))=\{q\}$, and $\phi(\Delta^{\CC})$ is smooth at
\end{LEM}
We will apply the localization along $O$ to prove
that the point $p$ is an isolated solution (at infinity)
with the multiplicity $1$ of the algebraic Einstein equation.
\begin{proof}[Proof]
Let $v_0,v_1,v_{12},v_2$
are vertices of the parallelogram $P$, so $v_0+v_{12}=v_1+v_2$, and
$$
\begin{aligned}
u_1 & = -e_1+e_4+e_5 & = v_1, &
\\
u_2 & = -e_2 + 2e_4+2e_5 & = 2v_1+v_2, &
\\
u_3 & = -e_3+2e_4+e_5 & = v_1+v_{12}, &
\quad\,
u_4=e_4,\quad\,
u_5=e_5.
\end{aligned}
$$
The set of vectors $\{u_i : i=1,\dots,5\}$,
and, hence, $\{v_0,v_1,v_2,u_4,u_5\}$ are basises in $\ZZ^5=\bigoplus \ZZ e_i$.
Let $\pi(a):=(a_4,a_5)$ for each $a=\sum a_i u_i \in \ZZ^5$.
We prove, that $\pi(\ZZ^5\cap \Delta)$ generates the semigroup $\ZZ_+^2$.
The face $P$ of $\Delta$ is the intersection of two facets with normal vectors $\mathbf f_i$, $i=1,2$, so that
$$
\mathbf f = (2,4,3,1,1) = (1, 2, 2, 1, 0) + (1, 2, 1, 0, 1) =\mathbf f_1 + \mathbf f_2 .
$$
For any $a\in \ZZ^5\cap \Delta$ we have
$a_4 = \langle \mathbf f_1,a\rangle\ge0$, and
$a_5 = \langle \mathbf f_2,a\rangle\ge 0$.
Then $\pi(a)\in \ZZ_+^2$. This prove the assertion, since $\pi(e_4) = (1,0)$, $\pi(e_5)=(0,1)$, $e_4,e_5\in\Delta$.
The lemma follows
\end{proof}
Now let $(z_0,z_1,z_2,y_1,y_2)$ be coordinates on $(\CC \setminus 0)^3\times \CC^2$. Assume that for $y_1y_2\ne0$
\[
- {\displaystyle \frac {{x_{2}}}{{x_{3}}\,{x_{5}}}} ={z_{0}},
\, - {\displaystyle \frac {2}{3}} \,{\displaystyle \frac {{x_{1}}
}{{x_{4}}\,{x_{5}}}} ={z_{0}}\,{z_{1}}, \, - 3\,{\displaystyle
\frac {{x_{2}}}{{x_{1}}^{2}}} ={z_{0}}\,{z_{2}}, \, - 2\,
{\displaystyle \frac {{x_{3}}}{{x_{1}}\,{x_{4}}}} ={z_{0}}\,{z_{1
}}\,{z_{2}}, \,{\displaystyle \frac {1}{{x_{4}}}} ={y_{1}}, \,
{\displaystyle \frac {1}{{x_{5}}}} ={y_{2}},
\]
so
\[
\left\{ \! {x_{3}}={\displaystyle \frac {3}{4}} \,
{\displaystyle \frac {{z_{0}}^{2}\,{z_{1}}^{2}\,{z_{2}}}{{y_{1}}
^{2}\,{y_{2}}}} , \,{x_{2}}= - {\displaystyle \frac {3}{4}} \,
{\displaystyle \frac {{z_{0}}^{3}\,{z_{1}}^{2}\,{z_{2}}}{{y_{1}}
^{2}\,{y_{2}}^{2}}} , \,{x_{4}}={\displaystyle \frac {1}{{y_{1}}}
} , \,{x_{1}}= - {\displaystyle \frac {3}{2}} \,{\displaystyle
\frac {{z_{0}}\,{z_{1}}}{{y_{1}}\,{y_{2}}}} , \,{x_{5}}=
{\displaystyle \frac {1}{{y_{2}}}} \! \right\}.
\]
Then
\begin{multline*}
s= \\
- {\displaystyle \frac {16}{9}} \,{\displaystyle \frac {{y_{1}}
^{3}\,{y_{2}}^{4}}{{z_{0}}^{6}\,{z_{1}}^{4}\,{z_{2}}^{2}}} +
{\displaystyle \frac {{\displaystyle \frac {16}{9}} \,{y_{1}}^{4}
\,{y_{2}}^{2}}{{z_{0}}^{5}\,{z_{1}}^{4}\,{z_{2}}^{2}}} -
{\displaystyle \frac {32}{3}} \,{\displaystyle \frac {{y_{1}}^{3}
\,{y_{2}}^{2}}{{z_{0}}^{4}\,{z_{1}}^{3}\,{z_{2}}^{2}}} +
{\displaystyle \frac {{\displaystyle \frac {16}{9}} \,{y_{1}}^{2}
\,{y_{2}}^{2}}{{z_{0}}^{3}\,{z_{1}}^{3}\,{z_{2}}}} -
{\displaystyle \frac {32\,{y_{1}}^{2}\,{y_{2}}^{2}}{{z_{0}}^{3}\,
{z_{1}}^{2}\,{z_{2}}}} + {\displaystyle \frac {{\displaystyle
\frac {80}{3}} \,{y_{1}}^{2}\,{y_{2}}}{{z_{0}}^{2}\,{z_{1}}^{2}\,
{z_{2}}}} - {\displaystyle \frac {8}{3}} \,{\displaystyle
\frac {{y_{1}}\,{y_{2}}^{2}}{{z_{0}}^{2}\,{z_{1}}}} \\
\mbox{} + {\displaystyle \frac {4\,{y_{1}}^{2}}{{z_{0}}\,{z_{1}}
\,{z_{2}}}} + {\displaystyle \frac {{\displaystyle \frac {4}{9}
} \,{y_{1}}^{2}}{{z_{0}}\,{z_{1}}}} - {\displaystyle \frac {80}{
3}} \,{\displaystyle \frac {{y_{1}}\,{y_{2}}}{{z_{0}}\,{z_{1}}}}
+ {\displaystyle \frac {{y_{2}}^{2}}{{z_{0}}}} +
{\displaystyle \frac {{\displaystyle \frac {4}{9}} \,{y_{2}}^{2}
}{{z_{0}}\,{z_{1}}}} + 8\,{y_{1}} - {\displaystyle \frac {8}{3}
} \,{\displaystyle \frac {{y_{1}}}{{z_{1}}}} + 4\,{y_{2}} + {z_{
0}}\,{z_{1}} + {z_{0}}\,{z_{1}}\,{z_{2}} \\
\mbox{} + {z_{0}} + {z_{0}}\,{z_{2}},
\end{multline*}
$s$ is a polynomial in $y_1$, $y_2$, and
$$
s= z_0 + z_0z_1+z_0z_2 +z_0z_1z_2 + 8y_1 - \frac83\frac{y_1}{z_1} +4y_2 + [2],
$$
where $[2]$ denotes the terms with degree $\ge 2$.
Similarly, for $s_i = x_i\partial s/\partial x_i$, $i=1,\dots,5$ we have
$$
\begin{aligned}
&
s_1= z_0z_1 -2z_0z_2 -z_0z_1z_2 + \frac83\frac{y_1}{z_1} + [2],
&
s_2= + z_0 + z_0z_2 - \frac83\frac{y_1}{z_1} + [2],
\\ &
s_3= -z_0 + z_0z_1z_2 + \frac83\frac{y_1}{z_1} + [2],
&
s_4= -z_0z_1 - z_0z_1z_2 - 8y_4 + [2],
\\ &
s_5= -z_0 - z_0z_1 - 4y_5 + [2],
\end{aligned}
$$
where $[2]$ denotes $(y_1^2, y_1y_2, y_2^2)$.
Computing the matrix $J= \frac{\partial(s_1,s_2,s_3,s_4,s_5)}{\partial(z_1,z_2,y_1,y_2)}$, setting $z_0=1,z_1=z_2=-1$, $y_1=y_2=0$, adding the row $(d_1,\dots,d_5)$ of dimensions $d_i=\dim(\frak m_i)$, and finding the determinant, we obtain
$$
\left|\begin{array}{rrrrr}
d_1&d_2&d_3&d_4&d_5\\ 2&0&-1&0&-1\\ -1&1&-1&1&0 \\ -8/3&8/3&-8/3&-8&0 \\ 0&0&0&0& -4
\end{array}\right|
=\frac{128}{3} (d_1+3d_2+2d_3) >0.
$$
Then the solution $p\in\Delta^{\CC}$ of the algebraic Einstein equation with local coordinates
$z_0=1,z_1=z_2=-1, y_1=y_2=0$ is isolated, and non-degenerate.
{\bf The case $d=6$, $\nu = 344$.}
There is a unique K\"ahler homogeneous space $G/H$ with $b_2(G/H)=1$
and $d=6:$
$$
G/H = E_8/T^1\cdot A_4\cdot A_2\cdot A_1.
$$
The corresponding $5$-dimensional polytope $\Delta $ in $\RR^6$
has $36$ facets, i.e., $4$-dimensional faces. Each
of them can be defined by the orthogonal vector $\mathbf f=(y_1, \dots ,y_6)$
such that
$\gcd(y_1, \dots ,y_6)=1$, and
$y_i\ge0$; then $\langle \mathbf f,x \rangle\,\ge 0$ for any $x \in \Delta $.
For example, the vector $\mathbf f=(1, 2, 3, 4, 5, 6)$ is orthogonal to the facet
with $9$ vertices
$$
\def\1#1,#2,#3{e^{#3}_{#1 #2}}
\begin{array}{llllllll}
\11,1,2,&\11,2,3,&\11,3,4,&\11,4,5,&\11,5,6,\\[1ex]
&\12,2,4,&\12,3,5,&\12,4,6,& \\[1ex]
& &\13,3,6,& &
\end{array}
$$
where $e^k_{ij}=e_i+e_j-e_k$. We write all the facets:
1) \enskip $16$ four-dimensional simplices with normal vectors
{\footnotesize
\begin{verbatim}
[1, 2, 2, 1, 2, 1], [1, 2, 3, 2, 3, 2], [1, 2, 1, 1, 2, 1], [1, 1, 1, 2, 2, 1],
[2, 2, 1, 1, 1, 1], [3, 2, 3, 4, 1, 2], [1, 1, 1, 2, 1, 2], [1, 1, 2, 2, 1, 1],
[1, 1, 2, 2, 1, 2], [2, 1, 1, 1, 1, 2], [3, 2, 5, 4, 3, 6], [1, 1, 2, 1, 1, 2],
[2, 2, 3, 1, 1, 3], [5, 2, 3, 4, 5, 6], [2, 1, 1, 1, 2, 2], [5, 4, 3, 2, 7, 6];
\end{verbatim}
}
2) \enskip $8$ four-dimensional pyramids with normal vectors
{\footnotesize
\begin{verbatim}
[3, 4, 1, 2, 5, 2], [1, 2, 3, 4, 5, 4], [5, 2, 3, 4, 1, 6], [1, 2, 3, 2, 1, 2],
[3, 2, 1, 2, 1, 2], [1, 2, 3, 4, 3, 4], [3, 2, 1, 4, 3, 2], [1, 2, 3, 2, 3, 4];
\end{verbatim}
}
3) \enskip $5$ other facets with normal vectors with positive entries:
{\footnotesize
\begin{verbatim}
[1, 2, 3, 4, 5, 6], [3, 2, 1, 4, 1, 2], [1, 2, 3, 4, 3, 2], [1, 2, 1, 2, 3, 2],
[1, 2, 1, 2, 1, 2];
\end{verbatim}
}
4) \enskip $7$ facets with normal vectors with non-negative entries:
{\footnotesize
\begin{verbatim}
[1, 2, 1, 0, 1, 2], [2, 1, 1, 2, 0, 2], [1, 2, 2, 1, 0, 1], [1, 1, 0, 1, 1, 0],
[1, 0, 1, 0, 1, 0], [1, 2, 1, 2, 1, 0], [1, 2, 3, 2, 1, 0];
\end{verbatim}
}
Facets 1) and 2) are not marked faces.
E.g., simplices 1) and its sub-faces satisfy \cite[Test 7.1]{2007}.
Facets 3) and 4) are marked faces. There are $13$ three-dimensional
and $15$ two-dimensional marked faces.
We get $13$ vectors, orthogonal to three-dimensional marked faces (each of them is proportional to the sum of two distinct vectors 2)-4)):
{\footnotesize
\begin{verbatim}
[1, 2, 1, 2, 1, 1], [1, 2, 1, 1, 1, 2], [5, 3, 2, 6, 1, 4],
[4, 3, 1, 5, 2, 2], [7, 3, 4, 6, 1, 8], [4, 5, 1, 3, 6, 2],
[1, 2, 3, 4, 5, 5], [2, 4, 5, 3, 1, 1], [2, 4, 3, 1, 1, 3], [1, 2, 2, 2, 1, 0],
[1, 1, 2, 1, 1, 0], [2, 1, 1, 1, 2, 0], [2, 3, 1, 3, 2, 0]
\end{verbatim}
}
In the two-dimensional case we obtain:
(a) \enskip $6$ parallelograms with normal vectors
{\footnotesize
\begin{verbatim}
[5, 5, 2, 7, 3, 2], [3, 6, 8, 5, 2, 3], [5, 5, 2, 3, 7, 2],
[3, 6, 7, 10, 13, 12], [3, 4, 6, 3, 2, 1], [3, 6, 6, 5, 2, 1]
\end{verbatim}
}
(b) \enskip $9$ parallelograms with normal vectors
{\footnotesize
\begin{verbatim}
[3, 6, 7, 8, 5, 2], [8, 3, 5, 6, 2, 8], [5, 7, 2, 3, 7, 4],
[3, 6, 7, 6, 9, 12], [7, 5, 2, 9, 5, 4], [5, 7, 2, 5, 9, 4],
[3, 4, 7, 8, 11, 10],[3, 2, 1, 3, 1, 2], [1, 2, 3, 4, 4, 4]
\end{verbatim}
}
Each of parallelograms listed in (a) and (b)
(with the exception of two last entries in (b))
belongs exactly to $3$ facets.
We claim now, that $6$ marked parallelograms (a) corresponds to singular complex hypersurfaces
as above (consequently $\varepsilon<\nu$), and $9$ marked parallelograms (b) corresponds to non-singular hypersurfaces.
For the proof, one can calculate $6+9$ determinants $\left|\begin{smallmatrix}a&b\\ b'&a'\end{smallmatrix}\right|$,
where $a,a',b,b'$ are some coefficients of $s(x_1,\dots,x_6)$,
using equalities
(\cite[Prop. 13]{SACOS}):
$
[1, 1, 2] = 8, [1, 2, 3] = 6, [1, 3, 4] = 4, [1, 4, 5] = 2, [1, 5, 6] = 1, [2, 2, 4] = 6, [2, 3, 5] = 2, [2, 4, 6] = 2, [3, 3, 6] = 2
$.
Thus we may unmark $9$ of $40$ marked faces.
\begin{COR*}
The hypothesis that all complex solutions of the algebraic Einstein equation on
$
G/H = E_8/T^1\cdot A_4\cdot A_2\cdot A_1
$
are isolated reduces to examination of $31 = 40-9$ cases, corresponding to $12$ four-dimensional,
$13$ three-dimensional, and $6$ two-dimensional faces of the polytope $\Delta $.
\end{COR*}
In each case we may unmark the $k$-dimensional face, if the corresponding complex hypersurface is non-singular (this is the $k$-dimensional problem, $k<5$); otherwise we must examine Einstein equation in a neighborood $U \subset \Delta^{\CC}$ of the 'solution at infinity' defined by each singular point (cf. \S\ref{sect:7}).
I am grateful to all participants of the Postnikov seminar of Moscow State University, listening to the presentation of this work May 20, 2009
and March 23, 2011.
\end{document} |
\mathbf{m}athbf{b}egin{document}
\mathbf{m}aketitle
\mathbf{r}enewcommand{\mathbf{m}athbf{a}rbic{footnote}}{\fracnsymbol{footnote}}
\fracootnotetext[2]{Department of Mathematics and Applications, Ecole Normale Sup\'erieure, 45 Rue d'Ulm, 75005 Paris, France ({\tt [email protected]}).}
\fracootnotetext[3]{Department of Mathematics, Yonsei University 50 Yonsei-Ro, Seodaemun-Gu, Seoul 120-749, Korea ({\tt [email protected]}). }
\fracootnotetext[4]{Department of Computational Science and Engineering, Yonsei University 50 Yonsei-Ro, Seodaemun-Gu, Seoul 120-749, Korea ({\tt [email protected], [email protected]}). }
\fracootnotetext[2]{The first author was supported by the ERC Advanced Grant Project MULTIMOD--267184.}
\fracootnotetext[5]{The second, third, and fourth authors were supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MEST) (No. 2011-0028868, 2012R1A2A1A03670512).}
\mathbf{r}enewcommand{\mathbf{m}athbf{a}rbic{footnote}}{\mathbf{m}athbf{a}rbic{footnote}}
\sigmalugger{siap}{xxxx}{xx}{x}{x--x}
\mathbf{m}athbf{b}egin{abstract} This paper presents a mathematical framework for a flexible pressure-sensor model using electrical impedance tomography (EIT). When pressure is applied to a conductive membrane patch with clamped boundary, the pressure-induced surface deformation results in a change in the conductivity distribution. This change can be detected in the current-voltage data ({{\mathbf{m}athbf i}t i.e.,} EIT data) measured on the boundary of the membrane patch. Hence, the corresponding inverse problem is to reconstruct the pressure distribution from the data. The 2D apparent conductivity (in terms of EIT data) corresponding to the surface deformation is anisotropic. Thus, we consider a constrained inverse problem by restricting the coefficient tensor to the range of the map from pressure to 2D-apparent conductivity. This paper provides theoretical grounds for the mathematical model of the inverse problem. We develop a reconstruction algorithm based on a careful sensitivity analysis. We demonstrate the performance of the reconstruction algorithm through numerical simulations to validate its feasibility for future experimental studies.
{\bf e}nd{abstract}
\mathbf{m}athbf{b}egin{keywords}
electrical impedance tomography, pressure sensing, conductive membrane, inverse problem, prescribed mean curvature equation.
{\bf e}nd{keywords}
\mathbf{m}athbf{b}egin{AMS}
35R30, 35J25, 53A10
{\bf e}nd{AMS}
\partialagestyle{myheadings}
\thispagestyle{plain}
\mathbf{m}arkboth{H. Ammari, K. Kang, K. Lee, and J. K. Seo}{Electrical impedance tomography-based pressure-sensing}
\sigmaection{Introduction}
There is a growing demand for cost-effective flexible pressure sensors. These devices have wide potential applicability, including in smart textiles \cite{Comert2013,Loruss2004,Loyola2010}, touch screens \cite{Han2014}, artificial skins \cite{Tawil2011}, and wearable health monitoring technologies \cite{Li2011,Merritt2009}. Electrical measurements have recently been used to measure the pressure-induced surface deformation of conductive membranes. In particular, electrical impedance tomography (EIT) has been used to develop flexible pressure sensors \cite{Peterson2011,YLeq2012,Yao2013}, because it allows the electromechanical behavior of an electrically conducting film to be monitored. When a pressure-sensitive conductive sheet is exposed to pressure, the deformation of the surface alters the conductivity distribution, which can be detected by an EIT system. However, rigorous studies employing mathematical modeling and reconstruction methods have not yet been conducted. The purpose of this paper is to provide a systematic mathematical framework for an EIT-based flexible pressure sensor.
Our rigorous mathematical analysis is based on the consideration of a simple model of an EIT-based pressure-sensor using a thin, flexible conductive membrane whose electrical conductance is directly related to pressure-induced deformation. We assume that the conductive membrane is stretched over a fixed frame and has a number of electrodes placed on its boundary as shown in Figure \mathbf{r}ef{membrane}. As in a standard EIT system, we use all adjacent pairs of electrodes to inject currents and measure induced boundary voltages between all neighboring pairs of electrodes to get a current-voltage data set, which is a discrete version of a Neumann-to-Dirichlet map. The current-voltage data can probe any external pressure loaded onto the membrane, because the pressure-induced surface deformation results in a change of the current density distribution over the surface, which leads to a change of the current-voltage data. Hence, the change in the current-voltage data can be viewed as a non-linear function of pressure. The inverse problem in this model is to identify the pressure (equivalently the surface deformation) from the boundary current-voltage data.
\mathbf{m}athbf{b}egin{figure}[!h]
\centering
{\mathbf{m}athbf i}ncludegraphics[scale=.4]{eps/domain_electrode.eps}
\mathbf{v}space{-1cm}\caption{\langlebel{membrane}Conductive membrane attached electrodes on the boundary in the absence of pressure(left) and presence of pressure(right).}
{\bf e}nd{figure}
This paper provides a derivation of an EIT-based pressure-sensing model, which describes the explicit relationship between the measured current-voltage data and the pressure. The mathematical model is associated with an elliptic partial differential equation (PDE) with an anisotropic coefficient, which comes from the pressure-induced surface deformation. To be precise, let $\Omega$ be a two-dimensional domain with a smooth boundary $\partialartial \Omegaega$ occupying the un-deformed membrane in the absence of any pressure. We denote the standard Sobolev space of order $s$ as $H^s(\partialartial \Omegaega)$.
Let $p$ be the pressure and $w_p$ be the solution of
\mathbf{m}athbf{b}egin{equation} \langlebel{Model2}
\left\{
\mathbf{m}athbf{b}egin{array}{cl}
\mathbf{m}athbf{n}ablabla\cdot\left(\fracrac{1}{\sigmaqrt{1+|\mathbf{m}athbf{n}ablabla w_p|^2}}\mathbf{m}athbf{n}ablabla w_p\mathbf{r}ight)=p &\mathbf{m}box{ in } \Omegaega,\\
w_p=0 &\mathbf{m}box{ on } \partial\Omega.\\
{\bf e}nd{array}
\mathbf{r}ight.
{\bf e}nd{equation}
Under pressure $p$, the current-voltage data $(g,f){\mathbf{m}athbf i}n H^{-\frac{1}{2}}(\partial\Omega)\times H^{\frac 12}(\partial\Omega)$ are dictated by $f=u_p|_{\partial\Omega}$ with $u_p$ being the solution of the elliptic PDE,
\mathbf{m}athbf{b}egin{equation} \langlebel{Model}
\left\{
\mathbf{m}athbf{b}egin{array}{l}
\mathbf{m}athbf{n}ablabla\cdot\left( \gamma_p \mathbf{m}athbf{n}ablabla u_p\mathbf{r}ight)=0 \quaduad \mathbf{m}box{ in }\Omega, \\
(\gamma_p\mathbf{m}athbf{n}abla u_p)\cdot\mathbf{m}athbf{\mathbf{m}athbf{n}u}|_{\partial\Omega}~=g, \quaduad {\mathbf{m}athbf i}nt_{\partialartial \Omega} u_p =0,
{\bf e}nd{array}
\mathbf{r}ight.
{\bf e}nd{equation}
where $$\gamma_p=I -\fracrac{1}{1+|\mathbf{m}athbf{n}ablabla w_p|^2}\mathbf{m}athbf{n}ablabla w_p\mathbf{m}athbf{n}ablabla w_p^T.$$
Here,
$I$ is the identity matrix, the superscript $T$ denotes the transpose, $\mathbf{m}athbf{n}u$ the unit outward normal vector to $\partial\Omega$, and ${\mathbf{m}athbf i}nt_{\partial\Omega} g=0$.
The standard Neumann-to-Dirichlet map $\Lambda_{\gamma_p}$ is defined by $\Lambda_{\gamma_p}(g)=u_p|_{\partial\Omega}$ with $u_p$ being the solution of {\bf e}ref{Model}.
We cannot invert the map $\gamma_p\mathbf{m}apsto \Lambda_{\gamma_p}$ with the existing EIT reconstruction methods because of the well-known non-uniqueness result of the inverse problem: there are infinitely many anisotropic coefficients $\gamma$ such that $\Lambda_{\gamma_p}=\Lambda_{\gamma}$. Hence, we must consider the constrained inverse problem of recovering anisotropic coefficient within the set of coefficient tensors associated with pressures. Taking account of the fact that two different pressures $p$ and $-p$ produce the same Neumann-to-Dirichlet map, we need to impose a proper constraint on pressures.
Next, we propose a pressure reconstruction method with the standard $N$-channel EIT system. Owing to the quadratic structure of $\mathbf{m}athbf{n}ablabla w_p\mathbf{m}athbf{n}ablabla w_p^T$ in $\gamma_p$, we cannot expect a linearized reconstruction method for $p$, even assuming that pressure is small. Regarding $p$ as a piecewise constant function $p=\sigmaum_k p_{k}\chi_{T_k}$, through the standard discretization of the domain into small elements, $T_k$, the inverse problem can be approximated by solving a large linear system with a large number of unknowns involving all possible products $p_{k}p_{{\bf e}ll}$. (Here, $\chi_{T_k}$ is the indicator function of $T_k$.) Given that most of the columns of the matrix have relatively small effect on the data, we consider a reduced linear system by eliminating most of the columns. Various numerical simulations verify the feasibility of the reconstruction algorithm.
In section~\mathbf{r}ef{sec:framework}, we formulate the mathematical model for the EIT-based membrane pressure sensor, and present uniqueness results.
In section~\mathbf{r}ef{sec:recon_method}, we propose a reconstruction method to recover the pressure. In section~\mathbf{r}ef{sec:numerical_results}, we develop a reconstruction algorithm based on sensitivity analysis, and validate the algorithm by numerical simulation results.
This mathematical study of an EIT-based flexible pressure sensor is in an early stage. The proposed mathematical model requires the assumption of incompressibility, whereas there are many flexible materials that are not incompressible. Constructing a mathematical model that includes compressibility will be a future research topic.
\sigmaection{Mathematical Framework}\langlebel{sec:framework}
\sigmaubsection{Formulation of the forward problem}
Assume that a thin conductive membrane at rest occupies a two-dimensional bounded domain $\Omega\sigmaubset{\mathbf{B}bb R}^2$ with a smooth boundary $\partial\Omega$. Here, the thickness of the membrane is uniform. Assume that the conductivity of the membrane is homogeneous. Let $\Omega_{d_0}=\{ \mathbf{x}{\mathbf{m}athbf i}n\Omega~: \mathbf{m}box{dist}(\mathbf{x},\partial\Omega)> d_0\}$ with $d_0>0$. Assume that a pressure $p$ lies in the set
$$\mathbf{m}athfrak{S}=\{ p{\mathbf{m}athbf i}n L^{\mathbf{m}athbf i}nfty(\Omega)~: \| p\|_{L^{\mathbf{m}athbf i}nfty(\Omega)} < \mathbf{m}athbf{a}lpha, ~\mathbf{m}box{supp} (p) \sigmaubset \Omega_{d_0} \},$$ where $\mathbf{m}athbf{a}lpha$ is a positive number. (The assumption $ \| p\|_{L^{\mathbf{m}athbf i}nfty(\Omega)} < \mathbf{m}athbf{a}lpha$ is only used to guarantee existence and uniqueness of the prescribed mean curvature equation {\bf e}ref{young-laplace} which will be discussed later.) When the pressure $p$ is loaded on $\Omega$, it produces a displacement of the membrane. The displacement at $\mathbf{x}=(x,y){\mathbf{m}athbf i}n\Omega$ from its rest position is denoted by $w_p(\mathbf{x})$, and the deformed two-dimensional surface can be expressed as
\mathbf{m}athbf{b}egin{equation}\langlebel{om_p}
\Omega_p=\{(\mathbf{x}, w_p(\mathbf{x}))~:~\mathbf{x}{\mathbf{m}athbf i}n\Omega\}.
{\bf e}nd{equation}
Here, the boundary $\partial\Omega$ of the membrane is fixed so that there is no displacement on the boundary. Because the membrane undergoes deformation to reduce the area change caused by pressure $p{\mathbf{m}athbf i}n \mathbf{m}athfrak{S}$, $w_p$ satisfies the prescribed mean curvature equation,
\mathbf{m}athbf{b}egin{equation}
\langlebel{young-laplace}
\left\{
\mathbf{m}athbf{b}egin{array}{clc}
\mathbf{m}athbf{n}ablabla\cdot\left(\fracrac{1}{\sigmaqrt{1+|\mathbf{m}athbf{n}ablabla w_p|^2}}\mathbf{m}athbf{n}ablabla w_p\mathbf{r}ight)&=p &\mathbf{m}box{in } \Omegaega,\\
w_p&=0 &\textrm{ on } \partialartial\Omegaega .{\bf e}nd{array}\mathbf{r}ight.
{\bf e}nd{equation}
Problem {\bf e}ref{young-laplace} has a unique solution for $p{\mathbf{m}athbf i}n \mathbf{m}athfrak{S}$ with $\mathbf{m}athbf{a}lpha$ being sufficiently small such that, for any measurable subset $E$ of $\Omega$, ${\mathbf{m}athbf i}nt_E p\,d\mathbf{x}$ is smaller than the perimeter of $E$
\cite{Bernstein1910,Finn1965,Giaquinta1974,Giusti1976,Giusti1978,Giusti1984}.
Let $H_{\mathbf{m}athbf{d}iamond}^{-1/2}(\partial\Omega):=\{g{\mathbf{m}athbf i}n H^{-1/2}(\partial\Omega) :\langlengle g, 1 \mathbf{r}anglengle =0\}$ with $\langlengle \, , \, \mathbf{r}anglengle$ denoting the duality pair between $H^{-1/2}(\partialartial \Omegaega)$ and $H^{1/2}(\partialartial \Omegaega)$. Let $H^1(\Omegaega)$ be defined by $H^1(\Omega) = \{ u {\mathbf{m}athbf i}n
L^2(\Omega): \mathbf{m}athbf{n}ablabla u {\mathbf{m}athbf i}n L^2(\Omegaega)\}$ and let $H_0^{1}(\Omega)$ be the set of functions in $H^1(\Omega)$ with trace zero on $\partialartial \Omegaega$.
To extract EIT-data for pressure-sensing, we inject a current of $g{\mathbf{m}athbf i}n H^{-1/2}_{\mathbf{m}athbf{d}iamond}(\partial\Omega)$ into the membrane $\Omega_p$. In the absence of the pressure ($p=0$), the induced potential due to the injection current of $g$ is the solution $u_0{\mathbf{m}athbf i}n H^1(\Omega)$ of the following Neumann problem
\mathbf{m}athbf{b}egin{equation} \langlebel{NeumannBVP}
\left\{
\mathbf{m}athbf{b}egin{array}{ll}
~~~{\mathbf D}elta u_0 =0\quad \quaduad \mathbf{m}box{ in }\Omegaega,\\
\mathbf{m}athbf{n}u\cdot\mathbf{m}athbf{n}abla u_0=g ~~~~\mathbf{m}box{ on } \partial\Omega.
{\bf e}nd{array}
\mathbf{r}ight.
{\bf e}nd{equation}
In the presence of the pressure ($p\mathbf{m}athbf{n}eq 0$), the induced potential $v_p$ is now defined on the deformed surface $\Omega_p$, and is governed by
\mathbf{m}athbf{b}egin{equation} \langlebel{NeumannBVP-2}
\left\{
\mathbf{m}athbf{b}egin{array}{ll}
\mathbf{m}athbf{n}abla_S\cdot\left( \fracrac{1}{\sigmaqrt{1+|(\mathbf{m}athbf{n}abla w_p)\circ \partiali_\mathbf{x}|^2}}\mathbf{m}athbf{n}abla_S v_p\mathbf{r}ight)=0\quad \mathbf{m}box{ on }\Omegaega_p,\\
~~~\mathbf{m}athbf{n}u_S\cdot\fracrac{1}{\sigmaqrt{1+|(\mathbf{m}athbf{n}abla w_p)\circ \partiali_\mathbf{x}|^2}}\mathbf{m}athbf{n}abla_S v_p =g~~~~ \mathbf{m}box{ on } \partial\Omega_p=\partial\Omega,
{\bf e}nd{array}
\mathbf{r}ight.
{\bf e}nd{equation}
where $\mathbf{m}athbf{n}abla_S$ is the surface gradient, $\mathbf{m}athbf{n}u_S$ is the outward unit normal vector to the boundary $\partial\Omega_p$, and $\partiali_\mathbf{x}$ is the projection map $\Omega_p\mathbf{r}ightarrow \Omega$ defined by $\partiali_\mathbf{x}( x,y,z)=(x,y)$.
For the derivation of equation {\bf e}ref{NeumannBVP-2}, we use the concept of surface
conductivity \cite{McAllister1991}, while regarding the thin membrane as a two-dimensional surface, because the induced current density along the thin membrane can be viewed as a tangent vector field on the surface. If the deformed membrane is uniform in thickness, the resulting potential $v_p$ satisfies the surface Laplace equation, ${\mathbf D}elta_S v_p=0$, along the surface with ${\mathbf D}elta_S$ being the Laplace-Beltrami operator. However, under the incompressibility assumption, the thickness of the membrane varies, and so does the surface conductivity. As a small area, ${\mathbf D}elta x{\mathbf D}elta y$, is changed to $\sigmaqrt{1+|\mathbf{m}athbf{n}abla w_p|^2}{\mathbf D}elta x{\mathbf D}elta y$, the thickness is approximately reduced by a factor of $\sigmaqrt{1+|\mathbf{m}athbf{n}abla w_p|^2}$, as is the surface conductivity.
Define ${\mathbf U}psilon_p : H^{-1/2}_{\mathbf{m}athbf{d}iamond}(\partial\Omega)\mathbf{r}ightarrow H^{1/2}_{\mathbf{m}athbf{d}iamond}(\partial\Omega)$ by
\mathbf{m}athbf{b}egin{equation}\langlebel{NtD}
{\mathbf U}psilon_p(g)=v_p|_{\partial\Omega_p}.
{\bf e}nd{equation}
The pair $(g,{\mathbf U}psilon_p(g))$ is called current-to-voltage pair. Here, $H_{\mathbf{m}athbf{d}iamond}^{1/2}(\partial\Omega):=\{g{\mathbf{m}athbf i}n H^{1/2}(\partial\Omega) :{\mathbf{m}athbf i}nt_{\partialartial \Omegaega} g =0\}$.
Apparently, the voltage difference, $ {\mathbf U}psilon_p(g)-{\mathbf U}psilon_0(g)$, reflects the information of the displacement. Therefore, it is possible to recover $p$ from several pairs $(g^j, {\mathbf U}psilon_p(g^j)),~ j=1,2,\cdots, N$. The inverse problem is to reconstruct $p$ and $w_p$ from the boundary voltage-to-current data $(g^j, {\mathbf U}psilon_p(g^j)),~ j=1,2,\cdots, N$.
There are serious difficulties in solving the inverse problem, because the potential $v_p$ in (\mathbf{r}ef{NeumannBVP-2}) is defined on the unknown surface $\Omega_p$ in three dimensions, and the measured data, $(g^j, {\mathbf U}psilon_p(g^j))$, are given on the boundary of the two-dimensional domain $\Omega$. The relation between the surface, $\Omega_p$, and the data, $(g^j, {\mathbf U}psilon_p(g^j))$, is too complicated to handle the inverse problem.
To deal with these difficulties, we introduce the following function defined in the known two-dimensional domain, $\Omega$, as
\mathbf{m}athbf{b}egin{equation}\langlebel{proj}
u_p(\mathbf{x})= v_p (\mathbf{x},w_p(\mathbf{x}))\quad\mathbf{m}box{for }~\mathbf{x}{\mathbf{m}athbf i}n\Omegaega.
{\bf e}nd{equation}
The following theorem provides a governing equation for $u_p$, through which the relationship between current and voltage can be understood.
\mathbf{m}athbf{b}egin{theorem}
The function $u_p$ in (\mathbf{r}ef{proj}) is dictated by the following elliptic equation
\mathbf{m}athbf{b}egin{equation} \langlebel{proj_con}
\left\{
\mathbf{m}athbf{b}egin{array}{cl}
\mathbf{m}athbf{n}ablabla\cdot\left( \gamma_p\mathbf{m}athbf{n}ablabla u_p\mathbf{r}ight)=0 \quaduad \mathbf{m}box{ in }~\Omega, \\
(\gamma_p\mathbf{m}athbf{n}abla u_p)\cdot\mathbf{m}athbf{\mathbf{m}athbf{n}u}~=g\quaduad \mathbf{m}box{ on }~\partial\Omega,
{\bf e}nd{array}
\mathbf{r}ight.
{\bf e}nd{equation}
where $\gamma_p$ is a symmetric positive definite matrix given by
\mathbf{m}athbf{b}egin{align}\gamma_p=I -\fracrac{1}{1+|\mathbf{m}athbf{n}ablabla w_p|^2}\mathbf{m}athbf{n}ablabla w_p\mathbf{m}athbf{n}ablabla w_p^T~~\textrm{in }~\Omega.
\langlebel{gamma1}
{\bf e}nd{align}
{\bf e}nd{theorem}
\mathbf{m}athbf{b}egin{proof} Let $v^{ext}_p$ denote the extension of $v_p$ such that $v^{ext}_p(\mathbf{x},z)=v_p(\mathbf{x},w_p(\mathbf{x}))$ for all $z{\mathbf{m}athbf i}n {\mathbf{B}bb R}$ and $\mathbf{x}{\mathbf{m}athbf i}n\Omega$. Then, the surface gradient $\mathbf{m}athbf{n}abla_Sv_p$ can be expressed as
\mathbf{m}athbf{b}egin{align}
\mathbf{m}athbf{n}abla_S v_p =\mathbf{m}athbf{n}abla_3 v^{ext}_p-(\mathbf{m}athbf{n}abla_3 v^{ext}\cdot\mathbf{m}athbf{n}_S)\mathbf{m}athbf{n}_S\quad\quad\mathbf{m}box{on }~ \Omega_p\langlebel{eq1},
{\bf e}nd{align}
where $\mathbf{m}athbf{n}_S=(\mathbf{m}athbf{n}abla w_p, -1)/\sigmaqrt{1+|\mathbf{m}athbf{n}abla w_p|^2}$ is the unit downward normal vector to the surface $\Omega_p$ and $\mathbf{m}athbf{n}abla_3=\left(\frac{\partial}{\partial x }, \frac{\partial}{\partial y },\frac{\partial}{\partial z }\mathbf{r}ight)$ is the three-dimensional gradient.
Since $\partial_z v^{ext}_p=0$, from {\bf e}ref{eq1} a direct computation yields
\mathbf{m}athbf{b}egin{align}\hspace{0cm}\langlebel{grad1}
\mathbf{m}athbf{n}abla_S v_p
&=\fracrac{1}{1+|\mathbf{m}athbf{n}abla w_p|^2}\left(\mathbf{m}athbf{b}egin{array}{cc}
1+(\partial_y w_p)^2 & -(\partial_x w_p)(\partial_y w_p) \\
-(\partial_x w_p)(\partial_y w_p) & 1+(\partial_x w_p)^2\\
\partial_x w_p& \partial_y w_p
{\bf e}nd{array}\mathbf{r}ight)
\mathbf{m}athbf{n}abla u_p\mathbf{m}athbf{n}onumber \\
&=\left(\gamma_p\mathbf{m}athbf{n}abla u_p~,~ \fracrac{1}{1+|\mathbf{m}athbf{n}abla w_p|^2}\mathbf{m}athbf{n}abla w_p\cdot\mathbf{m}athbf{n}abla u_p\mathbf{r}ight)^T .
{\bf e}nd{align}
Here, we used the fact that $\mathbf{m}athbf{n}abla v_p^{ext}=\mathbf{m}athbf{n}abla u_p$.
The surface divergence of the tangential vector field $\fracrac{\mathbf{m}athbf{n}abla_S v_p}{\sigmaqrt{1+|\mathbf{m}athbf{n}abla w_p|^2}}$ is written as
\mathbf{m}athbf{b}egin{equation}\langlebel{surf_div}
\mathbf{m}athbf{n}abla_S\cdot\left(\fracrac{\mathbf{m}athbf{n}abla_S v_p}{\sigmaqrt{1+|\mathbf{m}athbf{n}abla w_p|^2}}\mathbf{r}ight)=\left(\mathbf{m}athbf{n}abla_3\times\left(\mathbf{m}athbf{n}_S\times\fracrac{\mathbf{m}athbf{n}abla_S v_p}{\sigmaqrt{1+|\mathbf{m}athbf{n}abla w_p|^2}}\mathbf{r}ight)\mathbf{r}ight)\cdot\mathbf{m}athbf{n}_S.
{\bf e}nd{equation}
It follows from the vector identity, $(\mathbf{m}athbf{n}abla_3\times {\mathbf A})\cdot \mathbf{B}=\mathbf{m}athbf{n}abla_3\cdot({\mathbf A}\times\mathbf{B})+(\mathbf{m}athbf{n}abla_3\times\mathbf{B})\cdot{\mathbf A}$ that {\bf e}ref{surf_div} can be expressed as
{\sigmamall
\mathbf{m}athbf{b}egin{align*}
\mathbf{m}athbf{n}abla_S\cdot\left(\fracrac{\mathbf{m}athbf{n}abla_S v_p}{\sigmaqrt{1+|\mathbf{m}athbf{n}abla w_p|^2}}\mathbf{r}ight)
&=\mathbf{m}athbf{n}abla_3\cdot\left(\left(\mathbf{m}athbf{n}_S\times\fracrac{\mathbf{m}athbf{n}abla_S v_p}{\sigmaqrt{1+|\mathbf{m}athbf{n}abla w_p|^2}}\mathbf{r}ight)\times \mathbf{m}athbf{n}_S\mathbf{r}ight)+(\mathbf{m}athbf{n}abla_3\times\mathbf{m}athbf{n}_S)\cdot\left(\mathbf{m}athbf{n}_S\times\fracrac{\mathbf{m}athbf{n}abla_S v_p}{\sigmaqrt{1+|\mathbf{m}athbf{n}abla w_p|^2}}\mathbf{r}ight)\\
&=\mathbf{m}athbf{n}abla_3\cdot\left(\fracrac{\mathbf{m}athbf{n}abla_S v_p}{\sigmaqrt{1+|\mathbf{m}athbf{n}abla w_p|^2}}\mathbf{r}ight)+\frac{\mathbf{m}athbf{n}abla_Sv_p}{\sigmaqrt{1+|\mathbf{m}athbf{n}abla w_p|^2}}\cdot\left((\mathbf{m}athbf{n}abla_3\times\mathbf{m}athbf{n}_S)\times\mathbf{m}athbf{n}_S\mathbf{r}ight).
{\bf e}nd{align*}
}
Replacing $\mathbf{m}athbf{n}abla_S v_p$ with {\bf e}ref{grad1}, we obtain
{\sigmamall
\mathbf{m}athbf{b}egin{align*}
\mathbf{m}athbf{n}abla_S\cdot\left(\fracrac{\mathbf{m}athbf{n}abla_S v_p}{\sigmaqrt{1+|\mathbf{m}athbf{n}abla w_p|^2}}\mathbf{r}ight)
&=\mathbf{m}athbf{n}abla\cdot\left(\fracrac{\gamma_p\mathbf{m}athbf{n}abla u_p}{\sigmaqrt{1+|\mathbf{m}athbf{n}abla w_p|^2}}\mathbf{r}ight)-(\gamma_p \mathbf{m}athbf{n}abla u_p)\cdot\mathbf{m}athbf{n}abla\left(\fracrac{1}{\sigmaqrt{1+|\mathbf{m}athbf{n}abla w_p|^2}}\mathbf{r}ight)\\
&=\fracrac{1}{\sigmaqrt{1+|\mathbf{m}athbf{n}abla w_p|^2}}\mathbf{m}athbf{n}abla\cdot(\gamma_p\mathbf{m}athbf{n}abla u_p).
{\bf e}nd{align*}
}
Then, $\mathbf{m}athbf{n}abla_S\cdot\left(\fracrac{\mathbf{m}athbf{n}abla_Sv_p}{\sigmaqrt{1+|\mathbf{m}athbf{n}abla w_p|^2}}\mathbf{r}ight)=0$ implies $\mathbf{m}athbf{n}abla\cdot\left(\gamma_p\mathbf{m}athbf{n}abla u_p\mathbf{r}ight)=0$, and $\gamma_p$ has the positive eigenvalues $1$ and $1/(1+|\mathbf{m}athbf{n}abla w_p|^2)$. This completes the proof.
{\bf e}nd{proof}
\sigmaubsection{Unique determination of the pressure support}
We have seen that the displacement, $w_p$, and the current-voltage data, $(g,{\mathbf U}psilon_p(g))$, are involved in (\mathbf{r}ef{proj_con}) with the anisotropic coefficient, $\gamma_p$. In this subsection, we prove that the current-voltage data uniquely determine the pressure support.
To do so, we need to investigate the inverse problem of determining $\gamma_p$ from the current-voltage data. An anisotropic coefficient is uniquely determined by the current-voltage data up to a diffeomorphism that fixes the boundary. For any diffeomorphism, $\Phi:\Omega\mathbf{r}ightarrow \Omega$ with $\Phi|_{\partial\Omega}$, being the identity map,
$u_p\circ\Phi^{-1}$ satisfies
\mathbf{m}athbf{b}egin{equation}
\left\{\mathbf{m}athbf{b}egin{array}{cc}
&\mathbf{m}athbf{n}abla\cdot(\gamma_p^{\Phi}\mathbf{m}athbf{n}abla u_p\circ\Phi^{-1})=0~\textrm{ in }~\Omega,\\
& \gamma_p^{\Phi}\mathbf{m}athbf{n}abla u_p\circ\Phi^{-1}|_{\partial\Omega}=\gamma_p\mathbf{m}athbf{n}abla u_p|_{\partial\Omega},
{\bf e}nd{array}\mathbf{r}ight.
{\bf e}nd{equation}
where
$\gamma_p^{\Phi}$ is a $2\times 2$ symmetric matrix-valued function given by
\mathbf{m}athbf{b}egin{equation}\langlebel{diffeo}
\gamma_p^{\Phi}\circ\Phi(\mathbf{x})=\fracrac{D\Phi(\mathbf{x})\gamma_p(\mathbf{x}) D\Phi(\mathbf{x})^T}{|\mathbf{m}athbf{d}et( D\Phi(\mathbf{x}))|}~~~\textrm{ for } \mathbf{x}{\mathbf{m}athbf i}n\Omega,
{\bf e}nd{equation}
where $D\Phi$ is the Jacobian of $\Phi$ and $\mathbf{m}athbf{d}et$ denotes the determinant.
This means that two different $\gamma_p$ and $\gamma_p^{\Phi}$ produce the same Neumann-to-Dirichlet map. Conversely, the Neumann-to-Dirichlet map can determine the tensor $\gamma$ up to the diffeomorphism, $\Phi:\Omega\to \Omega$, where $\Phi|_{\partial\Omega} =I$ provided that $\gamma$ is approximately constant \cite{Sylvester1990}. In our model, $\gamma_p$ is only involved in the scalar $w_p$, and it is possible to determine $\gamma$ within the set $\Gamma_{\mathbf{m}athfrak{S}}:=\{ \gamma_p~:~ p{\mathbf{m}athbf i}n \mathbf{m}athfrak{S}\}$ uniquely provided $\mathbf{m}athbf{a}lpha$ is sufficiently small. Note that two different pressures $p$ and $-p$ produce the same coefficient $\gamma_{p}=\gamma_{-p}$.
We must consider the constrained inverse problem of recovering the anisotropic coefficient within the set $\Gamma_{\mathbf{m}athfrak{S}}$ from the current-voltage data.
Let us introduce the {{\mathbf{m}athbf i}t outer support} of $\mathbf{m}athfrak{S}$, denoted by ${\mathbf{r}m supp}_{\partial\Omega} (p)$;
for $\mathbf{x}\mathbf{m}athbf{n}otin {\mathbf{r}m supp}_{\partial\Omega} (p)$, there exists an open and connected set $U$ such that $\mathbf{x}{\mathbf{m}athbf i}n U$, $\Omega\sigmaetminus \Omega_{d_0}\sigmaubset U$, and $p|_{U}=0$ \cite{Gebauer2008,Kusiak2003}.
\mathbf{m}athbf{b}egin{theorem}\langlebel{thm_unique} For $p{\mathbf{m}athbf i}n \mathbf{m}athfrak{S}$, $\Lambda_{\gamma_{p}}$ determines $\mathbf{m}box{\mathbf{r}m supp}_{\partial\Omega}( p)$ uniquely.
{\bf e}nd{theorem}
\mathbf{m}athbf{b}egin{proof} Let $p_1,p_2{\mathbf{m}athbf i}n \mathbf{m}athfrak{S}$. We assume $\Lambda_{\gamma_{p_1}}=\Lambda_{\gamma_{p_2}}$. We need to prove that $\mathbf{m}box{\mathbf{r}m supp}_{\partial\Omega}( p_1)=\mathbf{m}box{\mathbf{r}m supp}_{\partial\Omega}( p_2)$. From $\Lambda_{\gamma_{p_1}}=\Lambda_{\gamma_{p_2}}$, it follows that $\gamma_{p_1}=\gamma_{p_2}$ on the boundary $\partial\Omega$ \cite{Sylvester1990}. From {\bf e}ref{gamma1}, we have
$$
\fracrac{1}{1+|\mathbf{m}athbf{n}ablabla w_{p_1}|^2}\mathbf{m}athbf{n}ablabla w_{p_1}\mathbf{m}athbf{n}ablabla w_{p_1}^T=\fracrac{1}{1+|\mathbf{m}athbf{n}ablabla w_{p_2}|^2}\mathbf{m}athbf{n}ablabla w_{p_2}\mathbf{m}athbf{n}ablabla w_{p_2}^T\quad\mathbf{m}box{on }~\partial\Omega.
$$
This leads to the following identity with $c$ (real) being $|c|=1$:
$$
\mathbf{m}athbf{n}ablabla w_{p_1}=c \mathbf{m}athbf{n}ablabla w_{p_2} ~ \quad\mathbf{m}box{on } \partial\Omega.$$
The difference $w_{p_1}-c w_{p_2}$ satisfies
\mathbf{m}athbf{b}egin{align}
\mathbf{m}athbf{n}abla \cdot( A \mathbf{m}athbf{n}abla (w_{p_1}-c w_{p_2}))&=p_1-c p_2~~\mathbf{m}box{in }~\Omega , \langlebel{elliptic_A}\\
w_{p_1}-c w_{p_2}&=0 \quad\quad\quad~~\mathbf{m}box{on }~\partial\Omega ,\\
\mathbf{m}athbf{n}ablabla w_{p_1}-c \mathbf{m}athbf{n}ablabla w_{p_2}&=0 \quad\quad\quad~~\mathbf{m}box{on }~\partial\Omega ,
{\bf e}nd{align}
where $A$ is a matrix given by
\mathbf{m}athbf{b}egin{equation}\langlebel{A}
A(\mathbf{x})={\mathbf{m}athbf i}nt_0^1 \frac{1}{\sigmaqrt{1+\left|W_t(\mathbf{x})\mathbf{r}ight|^2}}\left[ I-\frac{W_t(\mathbf{x}) W_t(\mathbf{x})^T}{
1+\left|W_t(\mathbf{x})\mathbf{r}ight|^2} \mathbf{r}ight] dt\quad\mathbf{m}box{for}~\mathbf{x}{\mathbf{m}athbf i}n\Omega ,
{\bf e}nd{equation}
and $W_t=t\mathbf{m}athbf{n}abla w_{p_1}+(1-t)c\mathbf{m}athbf{n}abla w_{p_2}$. Since the structure of $A$ is the same as $\gamma_p$ in {\bf e}ref{gamma1}, $A$ is positive-definite and $w_{p_1}-c w_{p_2}$ satisfies the elliptic PDE {\bf e}ref{elliptic_A}. Hence, by the unique continuation property, it follows that
\mathbf{m}athbf{b}egin{equation}\langlebel{bound1}
w_{p_1}(\mathbf{x})=c w_{p_2}(\mathbf{x})\quad\mathbf{m}box{for }~ \mathbf{x}{\mathbf{m}athbf i}n \Omega\sigmaetminus \mathbf{m}box{\mathbf{r}m supp}_{\partial\Omega}( p_1-c p_2).
{\bf e}nd{equation}
It remains to prove that $\mathbf{m}box{supp}_{\partial\Omega}(p_1)=\mathbf{m}box{supp}_{\partial\Omega}(p_2)$. We use Runge approximation argument given by Druskin \cite{Druskin1982} and Isakov \cite{Isakov1988}. For notational simplicity, we denote $D_j:=\mathbf{m}box{\mathbf{r}m supp}_{\partial\Omega}( p_j)$ for $j=1,2$. To derive a contradiction, we assume that $D_1\sigmaetminus D_2\mathbf{m}athbf{n}eq {\bf e}mptyset$. Noting that
$$\mathbf{m}athbf{n}abla\cdot(\gamma_{p_2}\mathbf{m}athbf{n}abla (u_{p_2}-u_{p_1}))=\mathbf{m}athbf{n}abla\cdot((\gamma_{p_1}-\gamma_{p_2})\mathbf{m}athbf{n}abla u_{p_1})~~\mathbf{m}box{in}~~\Omega,$$
if $u_{p_1}=u_{p_2}$ on $\partial\Omega$, it follows from the assumption $\Lambda_{\gamma_{p_1}}=\Lambda_{\gamma_{p_2}}$ and {\bf e}ref{bound1} that
$${\mathbf{m}athbf i}nt_{\Omega}\gamma_{p_2}\mathbf{m}athbf{n}abla (u_{p_2}-u_{p_1})\cdot\mathbf{m}athbf{n}abla \mathbf{v}arphi={\mathbf{m}athbf i}nt_{D_1\cup D_2}(\gamma_{p_1}-\gamma_{p_2})\mathbf{m}athbf{n}abla u_{p_1}\cdot \mathbf{m}athbf{n}abla \mathbf{v}arphi~~\mathbf{m}box{for all }\mathbf{v}arphi{\mathbf{m}athbf i}n H^1(\Omega).$$
This leads to
\mathbf{m}athbf{b}egin{equation}\langlebel{runge}
0={\mathbf{m}athbf i}nt_{D_1\cup D_2}(\gamma_{p_1}-\gamma_{p_2})\mathbf{m}athbf{n}abla u_{p_1}\cdot \mathbf{m}athbf{n}abla u_{p_2}
{\bf e}nd{equation}
for all solutions $u_{p_j}$ to $\mathbf{m}athbf{n}abla\cdot(\gamma_{p_j}\mathbf{m}athbf{n}abla u_{p_j})=0$ in $\Omega$.
According to the Runge type approximation theorem \cite{Druskin1982,Isakov1988}, we can choose sequences of solutions $u_{p_j}^n$ satisfying $\mathbf{m}athbf{n}abla\cdot(\gamma_{p_j}\mathbf{m}athbf{n}abla u_{p_j}^n)=0$ such that
$$
\lim_{n\to{\mathbf{m}athbf i}nfty}{\mathbf{m}athbf i}nt_{\Omega} (\gamma_{p_1}-\gamma_{p_2}) \mathbf{m}athbf{n}abla u_{p_1}^n\cdot \mathbf{m}athbf{n}abla u_{p_2}^n d\mathbf{x}={\mathbf{m}athbf i}nfty,
$$
which contradicts {\bf e}ref{runge}. This completes the proof.
{\bf e}nd{proof}
\sigmaubsection{Unique determination of the pressure in the monotone case}
We now prove the unique determination of the pressure from the current-voltage data in the monotone case.
\mathbf{m}athbf{b}egin{theorem}
Let $p_1,p_2$ be in $\mathbf{m}athfrak{S}$. If $p_1 \leq p_2$ in $\Omega$ and ${\mathbf U}psilon_{p_1}={\mathbf U}psilon_{p_2}$, then either $p_1=p_2$ or $p_1=-p_2$ in $\Omega$.
{\bf e}nd{theorem}
\mathbf{m}athbf{b}egin{proof}
To derive a contradiction, we assume that $p_1\mathbf{m}athbf{n}eq p_2$ and use exactly the same argument as in the proof of Theorem \mathbf{r}ef{thm_unique}. Remember that $w_{p_1} - w_{p_2}$ satisfies the elliptic PDE
$$
\mathbf{m}athbf{n}abla \cdot( A \mathbf{m}athbf{n}abla (w_{p_1}- w_{p_2}))=p_1- p_2\quad\quad \mathbf{m}box{in }~\Omega
$$
with $A$ being defined by (\mathbf{r}ef{A}).
From the strong comparison principle, it follows that $$w_{p_1}> w_{p_2}~~\textrm{in}~~\Omega.$$
Since $w_{p_j}=0$ on $\partial\Omega$ for $j=1,2$, we have from Hopf's lemma
$$\partial_\mathbf{m}athbf{n} w_{p_1}< \partial_\mathbf{m}athbf{n} w_{p_2}\quad\mathbf{m}box{on }\partial\Omega.$$
Noting that $\mathbf{m}athbf{n}abla w_{p_j}=(\partial_\mathbf{m}athbf{n} w_{p_j})\mathbf{m}athbf{n} $ on $\partial\Omega$, we have
$$
\mathbf{m}box{either }\quad~~|\mathbf{m}athbf{n}abla w_{p_1}|\mathbf{m}athbf{n}eq |\mathbf{m}athbf{n}abla w_{p_2}|~ \mathbf{m}box{on }~\partial\Omega\quad~~\mathbf{m}box{or}\quad~~ p_1=-p_2~\mathbf{m}box{in}~\Omega.
$$
Hence, if $p_1\mathbf{m}athbf{n}eq -p_2$ in $\Omega$, then we have
$$
\gamma_{p_1}=I-\fracrac{1}{1+|\mathbf{m}athbf{n}abla w_{p_1}|^2}\mathbf{m}athbf{n}abla w_{p_1}\mathbf{m}athbf{n}abla w_{p_1}^T\mathbf{m}athbf{n}eq I-\fracrac{1}{1+|\mathbf{m}athbf{n}abla w_{p_2}|^2}\mathbf{m}athbf{n}abla w_{p_2}\mathbf{m}athbf{n}abla w_{p_2}^T= \gamma_{p_2}~ \mathbf{m}box{on }~\partial\Omega. $$
However, this is not possible because $\Lambda_{\gamma_{p_1}}=\Lambda_{\gamma_{p_2}}$ implies $\gamma_{p_1}|_{\partial\Omega}=\gamma_{p_2}|_{\partial\Omega}$ \cite{Kohn-Vogelius}. This concludes that if $p_1\mathbf{m}athbf{n}eq p_2$, then $p_1= -p_2$ in $\Omega$, which completes the proof.
{\bf e}nd{proof}
It is worth emphasizing that two different pressures $p_1$ and $p_2$ having the same support can produce the same displacement near the boundary. More precisely, there exist two different pressures $p_1$ and $p_2$ such that
$$\mathbf{m}box{\mathbf{r}m supp}_{\partial\Omega}( p_1)=\mathbf{m}box{\mathbf{r}m supp}_{\partial\Omega}( p_2),$$
and
$$w_{p_1}=w_{p_2}~~\textrm{in}~~ \Omega\sigmaetminus(\mathbf{m}box{supp}_{\partial\Omega}(p_1)\cup\mathbf{m}box{supp}_{\partial\Omega}(p_2)) .$$
Let $\Omega=B_5$ and $D=B_2$ with $B_r$ being the ball of radius $r$ centered at the origin. Consider the following radial symmetric function
\mathbf{m}athbf{b}egin{equation}\langlebel{examp_w}
\hspace{0cm}w_{\mathbf{r}ho}(\mathbf{x})=\left\{\mathbf{m}athbf{b}egin{array}{cc}
\mathbf{r}ho |\mathbf{x}|^3+(-3\mathbf{r}ho+\partialsi'(2)/4) |\mathbf{x}|^2+(-50\mathbf{r}ho-25/4\partialsi'(2))~~&\mathbf{m}box{if } \mathbf{x}{\mathbf{m}athbf i}n D,\\
\partialsi(|\mathbf{x}|)-\partialsi(5)~~& \mathbf{m}box{if }\mathbf{x}{\mathbf{m}athbf i}n \Omega\sigmaetminus D,\\
{\bf e}nd{array}\mathbf{r}ight.
{\bf e}nd{equation}
where $\partialsi(r)=\sigmaqrt{2}\log\left(r+\sigmaqrt{r^2-0.5}\mathbf{r}ight)$ and $\mathbf{r}ho{\mathbf{m}athbf i}n\mathbf{m}athbb{R}$.
A direct computation shows that $w_{\mathbf{r}ho}$ satisfies
$$\mathbf{m}athbf{n}ablabla\cdot\left(\fracrac{1}{\sigmaqrt{1+|\mathbf{m}athbf{n}ablabla w_{\mathbf{r}ho}|^2}}\mathbf{m}athbf{n}ablabla w_{\mathbf{r}ho}\mathbf{r}ight)=p_{\mathbf{r}ho}$$
where $p_{\mathbf{r}ho}$ is
\mathbf{m}athbf{b}egin{equation*}
p_{\mathbf{r}ho}(\mathbf{x})=\left\{\mathbf{m}athbf{b}egin{array}{cc}
\frac{\partial_r^2w_\mathbf{r}ho(\mathbf{x})+ |\mathbf{x}|^{-1}\partial_rw_\mathbf{r}ho(\mathbf{x})+|\mathbf{x}|^{-1}w_\mathbf{r}ho^3(\mathbf{x})}{ (1+[\partial_r w_\mathbf{r}ho(\mathbf{x})]^2)^{3/2}} ~~&\mathbf{m}box{if } \mathbf{x}{\mathbf{m}athbf i}n D,\\
0~~&\mathbf{m}box{if }\mathbf{x}{\mathbf{m}athbf i}n \Omega\sigmaetminus D,\\
{\bf e}nd{array}\mathbf{r}ight.
{\bf e}nd{equation*}
and $\partial_r$ is the radial derivative. Hence, $w_{\mathbf{r}ho}|_{\Omega\sigmaetminus D}$ does not change with $\mathbf{r}ho$: for every $\mathbf{r}ho_1, \mathbf{r}ho_2{\mathbf{m}athbf i}n {\mathbf{B}bb R}$, we have
$$ w_{\mathbf{r}ho_1}=w_{\mathbf{r}ho_2}~~\textrm{for}~~\mathbf{x}{\mathbf{m}athbf i}n\Omega\sigmaetminus D.$$
This means that in the non-monotone case there are infinitely many $p_{\mathbf{r}ho}$ which provide the same displacement near the boundary.
\sigmaection{Reconstruction method}\langlebel{sec:recon_method}
\sigmaubsection{Measured data: Discrete Neumann-to-Dirichlet map}
We use $N$-channel EIT system in which $N$ electrodes $\{\mathbf{m}athcal{E}_1,\mathbf{m}athcal{E}_2,\ldots,, \mathbf{m}athcal{E}_N\}$ are attached on the boundary $\partial\Omega$. Let $u_p^j$ be the potential in (\mathbf{r}ef{proj_con}) with $g=g^j$ which corresponds to the $j$th injection current using the adjacent pair $\mathbf{m}athcal{E}_j$ and $\mathbf{m}athcal{E}_{j+1}$.
When we inject a current of $I_0$[mA] along the adjacent electrodes $\mathbf{m}athcal{E}_j$ and $\mathbf{m}athcal{E}_{j+1}$, the resulting potential $u_p^j$ satisfies
\mathbf{m}athbf{b}egin{equation}\langlebel{Mp-PDE}
\left\{\mathbf{m}athbf{b}egin{array}{lc}
\mathbf{m}athbf{n}abla\cdot(\gamma_p\mathbf{m}athbf{n}abla u_p^j)=0~~\textrm{in}~\Omega,\\
{\mathbf{m}athbf i}nt_{\mathbf{m}athcal{E}_j}(\gamma_p\mathbf{m}athbf{n}abla u_p^j)\cdot\mathbf{m}athbf{n}u ds=I_0=-{\mathbf{m}athbf i}nt_{\mathbf{m}athcal{E}_{j+1}}(\gamma_p\mathbf{m}athbf{n}abla u_p^j)\cdot\mathbf{m}athbf{n}u ds,\\
(\gamma_p\mathbf{m}athbf{n}abla u^j_p)\cdot\mathbf{m}athbf{n}u=0~~\textrm{on}~\partial\Omega\sigmaetminus\mathbf{m}athcal{E}_j\cup\mathbf{m}athcal{E}_{j+1} ,\\
\mathbf{m}athbf{n}abla u_p^j\times \mathbf{m}athbf{n}u =0~~\textrm{on}~\mathbf{m}athcal{E}_j\cup\mathbf{m}athcal{E}_{j+1} .
{\bf e}nd{array}
\mathbf{r}ight.
{\bf e}nd{equation}
The $i$th boundary voltage subject to the $j$th injection current is denoted as
\mathbf{m}athbf{b}egin{align}\langlebel{mea_data}
V^{i,j}_p= I_0(u_p^j|_{\mathbf{m}athcal{E}_i}-u_p^j|_{\mathbf{m}athcal{E}_{i+1}}) ~~\textrm{for }i,j=1,2,\ldots,N.
{\bf e}nd{align}
Here, $\mathbf{m}athcal{E}_{N+1}=\mathbf{m}athcal{E}_1$.
Integration by parts gives the reciprocity principle:
\mathbf{m}athbf{b}egin{equation}\langlebel{recipro}
V^{i,j}_p={\mathbf{m}athbf i}nt_{\Omega}(\gamma_p\mathbf{m}athbf{n}abla u_p^i)\cdot\mathbf{m}athbf{n}abla u_p^jd\mathbf{x}=V^{j,i}_p.
{\bf e}nd{equation}
The boundary voltage (\mathbf{r}ef{mea_data}) is assumed to be known. We use it as measurement data for recovery of pressure $p$.
\sigmaubsection{Discrepancy minimization problems}
Let $\mathbf{m}athcal{V}^{i,j}$ be the measured data under an applied pressure $p$. Then it follows from (\mathbf{r}ef{recipro}) that the pressure $p$ can be obtained by minimizing the discrepancy functional
\mathbf{m}athbf{b}egin{equation}\langlebel{functional}
\mathbf{m}athcal{J}(p)=\sigmaum^N_{i,j=1}\left|{\mathbf{m}athbf i}nt_{\Omega}\left(\gamma_p\mathbf{m}athbf{n}abla u_p^i\mathbf{r}ight)\cdot\mathbf{m}athbf{n}abla u_p^jd\mathbf{x}-\mathbf{m}athcal{V}^{i,j}\mathbf{r}ight|^2.
{\bf e}nd{equation}
The inverse problem can be viewed as finding the minimizer of $\mathbf{m}athcal{J}(p)$. Unfortunately, it is numerically difficult to compute the minimizer of $\mathbf{m}athcal{J}(p)$ because $\mathbf{m}athcal{J}(p)$ is highly non-linear with respect to $p$.
To extract necessary information about $p$ from the data $\mathbf{m}athcal{V}^{i,j}$, we use the voltage difference data
\mathbf{m}athbf{b}egin{equation}\langlebel{measure_data}
\mathbf{m}athcal{W}^{i,j}:=\mathbf{m}athcal{V}^{i,j}-u_0^{i,j},
{\bf e}nd{equation}
where $u_0^{i,j}$ is the data in {\bf e}ref{mea_data} with $p=0$, {{\mathbf{m}athbf i}t i.e.,} the boundary voltage data in the absence of the pressure. With the voltage difference data $\mathbf{m}athcal{W}^{i,j}$, the functional $\mathbf{m}athcal{J}(p)$ in {\bf e}ref{functional} can be rewritten as
\mathbf{m}athbf{b}egin{equation}\langlebel{func_J}
\mathbf{m}athcal{J}(p)=\sigmaum^{N}_{i,j=1}\left|{\mathbf{m}athbf i}nt_\Omega\left( \fracrac{\mathbf{m}athbf{n}abla w_p\mathbf{m}athbf{n}abla w_p^T}{1+|\mathbf{m}athbf{n}abla w_p|^2}\mathbf{m}athbf{n}abla u_p^i\mathbf{r}ight)\cdot \mathbf{m}athbf{n}abla u_0^jd\mathbf{x}-\mathbf{m}athcal{W}^{i,j}\mathbf{r}ight|^2.
{\bf e}nd{equation}
The above identity follows from
\mathbf{m}athbf{b}egin{align*}
\hspace{0cm}{\mathbf{m}athbf i}nt_\Omega (\gamma_p\mathbf{m}athbf{n}abla u_p^i)\cdot\mathbf{m}athbf{n}abla u_p^jd\mathbf{x}-\mathbf{m}athcal{V}^{i,j}&={\mathbf{m}athbf i}nt_\Omega \mathbf{m}athbf{n}abla u_p^i\cdot\mathbf{m}athbf{n}abla u_0^jd\mathbf{x}-\mathbf{m}athcal{V}^{i,j}\\
&={\mathbf{m}athbf i}nt_\Omega \mathbf{m}athbf{n}abla u_p^i\cdot\mathbf{m}athbf{n}abla u_0^jd\mathbf{x} -{\mathbf{m}athbf i}nt_\Omega \mathbf{m}athbf{n}abla u_0^i\cdot \mathbf{m}athbf{n}abla u_0^jd\mathbf{x}-\mathbf{m}athcal{W}^{i,j}\\
&={\mathbf{m}athbf i}nt_\Omega \mathbf{m}athbf{n}abla u_p^i\cdot\mathbf{m}athbf{n}abla u_0^jd\mathbf{x} -{\mathbf{m}athbf i}nt_\Omega \left(\gamma_p\mathbf{m}athbf{n}abla u_p^i\mathbf{r}ight)\cdot \mathbf{m}athbf{n}abla u_0^jd\mathbf{x}-\mathbf{m}athcal{W}^{i,j}\\
&= {\mathbf{m}athbf i}nt_\Omega \left((I-\gamma_p)\mathbf{m}athbf{n}abla u_p^i\mathbf{r}ight)\cdot \mathbf{m}athbf{n}abla u_0^jd\mathbf{x}-\mathbf{m}athcal{W}^{i,j}.
{\bf e}nd{align*}
Due to the high non-linearity of the discrepancy functional $\mathbf{m}athcal{J}$ in {\bf e}ref{func_J}, it is difficult to compute a minimizer $p$ directly. To compute minimizers of {\bf e}ref{func_J} effectively, we will make use of various approximations. Assume that $|\mathbf{m}athbf{n}abla w_p|$ is small. We will neglect quantities of fourth order of smallness; for example,
\mathbf{m}athbf{b}egin{equation}\langlebel{Wp-approx1}
\fracrac{\mathbf{m}athbf{n}abla w_p\mathbf{m}athbf{n}abla w_p^T}{1+|\mathbf{m}athbf{n}abla w_p|^2}=\mathbf{m}athbf{n}abla w_p\mathbf{m}athbf{n}abla w_p^T +O(|\mathbf{m}athbf{n}abla w_p|^4)\mathbf{m}athbf{a}pprox \mathbf{m}athbf{n}abla w_p\mathbf{m}athbf{n}abla w_p^T.
{\bf e}nd{equation}
Since
$
{\mathbf{m}athbf i}nt_\Omega \gamma_p \mathbf{m}athbf{n}abla (u_p-u_0)\cdot\mathbf{m}athbf{n}abla (u_p-u_0) d\mathbf{x}={\mathbf{m}athbf i}nt_\Omega (\gamma_p-I) \mathbf{m}athbf{n}abla u_0\cdot \mathbf{m}athbf{n}abla (u_p-u_0)d\mathbf{x},
$
we have
\mathbf{m}athbf{b}egin{equation}\langlebel{Wp-approx2}
\|\mathbf{m}athbf{n}abla (u_p-u_0)\|_{L^2(\Omega)}\le C\|\mathbf{m}athbf{n}abla w_p\|_{L^{\mathbf{m}athbf i}nfty(\Omega)}^2 \|\mathbf{m}athbf{n}abla u_0\|_{L^2(\Omega)}.
{\bf e}nd{equation}
From {\bf e}ref{Wp-approx1} and {\bf e}ref{Wp-approx2}, we have
\mathbf{m}athbf{b}egin{equation}\langlebel{ap1}
\hskip -0.2in{\mathbf{m}athbf i}nt_{\Omega}[I-\gamma_p]\mathbf{m}athbf{n}ablabla u_p\cdot\mathbf{m}athbf{n}ablabla u_0d\mathbf{x}={\mathbf{m}athbf i}nt_{\Omega}[\mathbf{m}athbf{n}abla w_p\mathbf{m}athbf{n}abla w_p^T]\mathbf{m}athbf{n}ablabla u_0\cdot\mathbf{m}athbf{n}ablabla u_0d\mathbf{x}+ O\left(\|\mathbf{m}athbf{n}abla w_p\|_{L^{\mathbf{m}athbf i}nfty(\Omega)}^6\mathbf{r}ight).
{\bf e}nd{equation}
Neglecting $O\left(\|\mathbf{m}athbf{n}abla w_p\|_{L^{\mathbf{m}athbf i}nfty(\Omega)}^6\mathbf{r}ight)$ in {\bf e}ref{ap1}, the discrepancy functional $\mathbf{m}athcal{J}$ in {\bf e}ref{func_J} can be approximated as
\mathbf{m}athbf{b}egin{equation}\langlebel{lin_J}
\mathbf{m}athcal{J}_1(p)=\sigmaum^{N}_{i,j=1}\left|{\mathbf{m}athbf i}nt_\Omega [\mathbf{m}athbf{n}abla w_p\mathbf{m}athbf{n}abla w_p^T]\mathbf{m}athbf{n}abla u_0^i\cdot \mathbf{m}athbf{n}abla u_0^jd\mathbf{x}-\mathbf{m}athcal{W}^{i,j}\mathbf{r}ight|^2.
{\bf e}nd{equation}
Assuming Gaussian measurement noise, we consider the following regularized minimization problem:
\mathbf{m}athbf{b}egin{align}\langlebel{lin_regul}
\mathbf{m}in_{p} \mathbf{m}athcal{J}^{reg}_1(p)
{\bf e}nd{align}
with
\mathbf{m}athbf{b}egin{align}\langlebel{lin_regul2}
\mathbf{m}athcal{J}^{reg}_1(p)=\sigmaum_{i,j=1}^N\left|{\mathbf{m}athbf i}nt_{\Omega} [\mathbf{m}athbf{n}abla w_p\mathbf{m}athbf{n}abla w_p^T]\mathbf{m}athbf{n}abla u_0^i\cdot\mathbf{m}athbf{n}abla u_0^jd\mathbf{x}-\mathbf{m}athcal{W}^{i,j}\mathbf{r}ight|^2+\mathbf{m}athbf{b}eta\|p\|_{L^2(\Omega)}^2,
{\bf e}nd{align}
and $\mathbf{m}athbf{b}eta$ being a regularization parameter.
The displacement $w_p$ can be approximated to $v{\mathbf{m}athbf i}n H^1_0(\Omega)$ with $v$ being the solution of Possion's equation ${\mathbf D}elta v=p$ in $\Omega$, because $${\mathbf{m}athbf i}nt_\Omega |\mathbf{m}athbf{n}abla (w_p-v)|^2 d\mathbf{x}={\mathbf{m}athbf i}nt_\Omega \left(1-\frac{1}{\sigmaqrt{1+|\mathbf{m}athbf{n}abla w_p|^2}}\mathbf{r}ight) \mathbf{m}athbf{n}abla w_p\cdot \mathbf{m}athbf{n}abla (w_p-v)d\mathbf{x}=O(\|\mathbf{m}athbf{n}abla w_p\|_{L^{\mathbf{m}athbf i}nfty(\Omega)}^6).$$ With this approximation, the minimization problem (\mathbf{r}ef{lin_regul}) can be further simplified as follows: find $(p,v){\mathbf{m}athbf i}n[H^1_0(\Omega)]^2$ which minimizes the discrepancy functional
\mathbf{m}athbf{b}egin{align}\langlebel{F_prop}
\mathbf{m}athcal{J}_2(p,v)=\sigmaum_{i,j=1}^N&\left|{\mathbf{m}athbf i}nt_{\Omega}\left([\mathbf{m}athbf{n}abla v\mathbf{m}athbf{n}abla v^T]\mathbf{m}athbf{n}abla u_0^i\mathbf{r}ight)\cdot\mathbf{m}athbf{n}abla u_0^jd\mathbf{x}-\mathbf{m}athcal{W}^{i,j}\mathbf{r}ight|^2\\&+\langlembda{\mathbf{m}athbf i}nt_\Omega \left(\fracrac{1}{2}|\mathbf{m}athbf{n}abla v|^2+pv\mathbf{r}ight)d\mathbf{x} +\mathbf{m}athbf{b}eta\|p\|_{L^2(\Omega)}^2,\mathbf{m}athbf{n}onumber
{\bf e}nd{align}
where $\langlembda$ is a positive number. In the next subsection, we minimize the functional defined in {\bf e}ref{F_prop} in order to reconstruct $p$.
\sigmaubsection{Reconstruction algorithm}\langlebel{subsec:recon_algo}
Based on the simplified discrepancy functional {\bf e}ref{F_prop}, we propose a pressure image reconstruction algorithm. We discretize the domain $\Omega$ into triangular elements such that $\overline \Omegaega=\cup_{k=1}^K T_k$, where $T_k$ is a triangular subregion with side length $h<1$. For the approximation of the pressure $p$,
we assume that $p$ is a piecewise constant function contained in the set
$$
\mathbf{m}athcal P_h:=\{ p~:~ p \mathbf{m}box{ is constant for each } T_k, ~k=1,\cdots, K\}.
$$
We assume that $p{\mathbf{m}athbf i}n \mathbf{m}athcal P_h^*:=\mathbf{m}athcal P_h\cap \{ p~:~p=0 \mathbf{m}box{~in ~} \Omega\sigmaetminus \Omega_{d_0}\}$.
Then, we can express the pressure $p$ by $$p(\mathbf{x})=\sigmaum_{k=1}^Kp^{(k)}\chi_{T_k}(\mathbf{x}).$$
For each $k=1,\cdots K$, let $v_k$ be the solution of
\mathbf{m}athbf{b}egin{equation}\langlebel{v}
\left\{\mathbf{m}athbf{b}egin{array}{rcc}
-{\mathbf D}elta v_k&=\chi_{T_k}~\textrm{ in }\Omega,\\
v_k&=0~\textrm{ on } \partialartial\Omega.
{\bf e}nd{array}
\mathbf{r}ight.
{\bf e}nd{equation}
Then, $v_k$ can be expressed as
\mathbf{m}athbf{b}egin{equation}\langlebel{Green-rep}
v_k(\mathbf{x})={\mathbf{m}athbf i}nt_{T_k} G(\mathbf{x},\mathbf{y}) d\mathbf{y},
{\bf e}nd{equation}
where $G(\mathbf{x},\mathbf{y})$ is the Dirichlet function associated with the domain $\Omega$, that is, the solution to
$$
\left\{\mathbf{m}athbf{b}egin{array}{rcc}
-{\mathbf D}elta_x G &=\mathbf{m}athbf{d}elta_y ~\textrm{ in }\Omega,\\
G &=0~\textrm{ on } \partialartial\Omega
{\bf e}nd{array}
\mathbf{r}ight.
$$
with $\mathbf{m}athbf{d}elta_y$ being the Dirac mass at $y$.
If $(p,v){\mathbf{m}athbf i}n \mathbf{m}athcal P_h^*\times H^1(\Omega)$ is a minimizer of the functional {\bf e}ref{F_prop} with $\langlembda={\mathbf{m}athbf i}nfty$, $v$ should be given by
$$v=\sigmaum^K_{k=1}p^{(k)}v_k.$$
Regarding the pressure $p$ as a vector $\mathbf{m}athbf{p}=\left( p^{(1)},~p^{(2)},\ldots,~p^{(K)}\mathbf{r}ight){\mathbf{m}athbf i}n\mathbf{m}athbb{R}^K$, the discretized minimization problem (\mathbf{r}ef{F_prop}) for large $\langlembda$ can be simplified as
\mathbf{m}athbf{b}egin{equation}\langlebel{F_quad}
\mathbf{m}athcal{J}_3(\mathbf{m}athbf{p})=\sigmaum_{i,j=1}^K\left| \mathbf{m}athbf{p}^T\mathbf{m}athbb{Q}^{i,j}\mathbf{m}athbf{p}-\mathbf{m}athcal{W}^{i,j}\mathbf{r}ight|^2+\mathbf{m}athbf{b}eta\|\mathbf{m}athbf{p}\|_2^2,
{\bf e}nd{equation}
where each $\mathbf{m}athbb{Q}^{i,j}$ is a matrix given by
$$\mathbf{m}athbb{Q}^{i,j}=\left(\mathbf{m}athbf{b}egin{array}{cccc}
S_{11}^{ij}& S_{12}^{ij} & \cdots &S_{1K}^{ij}\\
S_{21}^{ij}& S_{22}^{ij} & \cdots &S_{2K}^{ij}\\
\mathbf{v}dots&\mathbf{v}dots& &\mathbf{v}dots\\
S_{M1}^{ij}&S_{M2}^{ij}&\cdots&S_{KK}^{ij}
{\bf e}nd{array}\mathbf{r}ight)_{K\times K}$$
and $S_{k{\bf e}ll}^{ij}$ is given by
\mathbf{m}athbf{b}egin{equation}\langlebel{S}
S_{k{\bf e}ll}^{ij}:= {\mathbf{m}athbf i}nt_{\Omega}\left([\mathbf{m}athbf{n}abla v_k\mathbf{m}athbf{n}abla v_{{\bf e}ll}^T]\mathbf{m}athbf{n}abla u_0^i\mathbf{r}ight)\cdot\mathbf{m}athbf{n}abla u_0^jd\mathbf{x}.
{\bf e}nd{equation}
Here, the quadratic term $\mathbf{m}athbf{p}^T\mathbf{m}athbb{Q}^{i,j}\mathbf{m}athbf{p}$ in (\mathbf{r}ef{F_quad}) can be viewed as a good approximation of the quantity ${\mathbf{m}athbf{d}isplaystyle {\mathbf{m}athbf i}nt_\Omega\left( \fracrac{\mathbf{m}athbf{n}abla w_p\mathbf{m}athbf{n}abla w_p^T}{1+|\mathbf{m}athbf{n}abla w_p|^2}\mathbf{m}athbf{n}abla u_p^i\mathbf{r}ight)\cdot \mathbf{m}athbf{n}abla u_0^jd\mathbf{x} }$ in terms of the discretized $\mathbf{m}athbf{p}$.
The quadratic form of (\mathbf{r}ef{F_quad}) can be converted to a linear form by introducing the vector ${\mathbf{m}athbf q}=\left(q^{(1)},\cdots, q^{(K^2)}\mathbf{r}ight){\mathbf{m}athbf i}n {\mathbf{B}bb R}^{K^2}$ whose components are
$$
q^{(k+{\bf e}ll(K-1))}=p^{(k)}p^{({\bf e}ll)}\quad\quad\mathbf{m}box{for } k,{\bf e}ll=1,\cdots, K.
$$
With this large vector ${\mathbf{m}athbf q}$, the quadratic form $\mathbf{m}athcal{J}_3(\mathbf{m}athbf{p})$ in {\bf e}ref{F_quad} can be changed into the following linear form:
\mathbf{m}athbf{b}egin{equation}\langlebel{F_lin}
\mathbf{m}athcal{J}_4(\mathbf{m}athbf{q})=\left\| \mathbf{m}athbb{S}\mathbf{m}athbf{q}-\mathbf{m}athbf{W}\mathbf{r}ight\|^2_2+\mathbf{m}athbf{b}eta\|\mathbf{m}athbf{q}\|_2^2,
{\bf e}nd{equation}
where $\mathbf{m}athbb{S}$ is $N^2\times K^2$ matrix given by
\mathbf{m}athbf{b}egin{equation}\langlebel{senseM}
\left(~(i-1)N+j~,~ (k-1)M+{\bf e}ll~\mathbf{r}ight)-\mathbf{m}box{component of } \mathbf{m}athbb{S} ~~=~~ S_{k{\bf e}ll}^{ij},
{\bf e}nd{equation}
and
\mathbf{m}athbf{b}egin{equation}\langlebel{meausre_vector}
\mathbf{m}athbf{W}=\left( \mathbf{m}athcal{W}^{1,1}\cdots~\mathbf{m}athcal{W}^{1,N}~\mathbf{m}athcal{W}^{2,1}
~~\ldots~\ldots~~\mathbf{m}athcal{W}^{N,1}\cdots~\mathbf{m}athcal{W}^{N,N}\mathbf{r}ight){\mathbf{m}athbf i}n\mathbf{m}athbb{R}^{N^2}.
{\bf e}nd{equation}
Now, the minimizer of the functional $\mathbf{m}athcal{J}_4$ in (\mathbf{r}ef{F_lin}) can be obtained by solving the following linear system:
\mathbf{m}athbf{b}egin{equation}\langlebel{lin_eq}
(\mathbf{m}athbb{S}^T\mathbf{m}athbb{S}+\sigmaqrt{\mathbf{m}athbf{b}eta}I)\mathbf{m}athbf{q}=\mathbf{m}athbb{S}^T\mathbf{m}athbf{W}.
{\bf e}nd{equation}
Unfortunately, the linear system {\bf e}ref{lin_eq} is too large to handle; the number of column vectors of $\mathbf{m}athbb{S}$ is proportional to $h^{-4}$ where $h^2$ is proportional to the mesh size. Hence, we need to eliminate most of the column vectors of the matrix $\mathbf{m}athbb{S}$ whose influence on the data are negligibly small. Noting that $k+{\bf e}ll(K-1)$-th column vector of $\mathbf{m}athbb{S}$ consists of components $S_{k{\bf e}ll}^{ij}={\mathbf{m}athbf i}nt_{\Omega}[\mathbf{m}athbf{n}abla v_k\mathbf{m}athbf{n}abla v_{{\bf e}ll}^T] \mathbf{m}athbf{n}abla u_0^i\cdot\mathbf{m}athbf{n}abla u_0^jd\mathbf{x}$, the quantity $\sigmaup_{i,j}|S_{k{\bf e}ll}^{ij}|$ can be estimated by ${\mathbf{m}athbf i}nt_{\Omega}\left|\mathbf{m}athbf{n}abla v_k\mathbf{m}athbf{n}abla v_{{\bf e}ll}^T\mathbf{r}ight|d\mathbf{x}$. We will see that the quantity ${\mathbf{m}athbf i}nt_{\Omega}\left|\mathbf{m}athbf{n}abla v_k\mathbf{m}athbf{n}abla v_{{\bf e}ll}^T\mathbf{r}ight|d\mathbf{x}$ decreases as $\mathbf{m}box{dist}(T_{k},T_{{\bf e}ll})$ increases.
Since $p$ is supported in $\Omega_{d_0}$, we assume that $\mathbf{m}box{dist}(T_k, \partial\Omega)\geq d_0$.
Using the expression {\bf e}ref{Green-rep} of $v_k$, we have
\mathbf{m}athbf{b}egin{align}\langlebel{green1}
{\mathbf{m}athbf i}nt_{\Omega}\left|\mathbf{m}athbf{n}abla v_k\mathbf{m}athbf{n}abla v_{k}^T\mathbf{r}ight|d\mathbf{x}&= {\mathbf{m}athbf i}nt_\Omega |\mathbf{m}athbf{n}abla v_k|^2 d\mathbf{x}=
-{\mathbf{m}athbf i}nt_{\Omega} {\mathbf D}elta v_k v_k d\mathbf{x}\mathbf{m}athbf{n}onumber\\
&= {\mathbf{m}athbf i}nt_{T_k} {\mathbf{m}athbf i}nt_{T_k} G(\mathbf{x},\mathbf{y}) d\mathbf{x} d\mathbf{y} \gtrsim h^4\left|\log h\mathbf{r}ight|,
{\bf e}nd{align}
where the expression $X\gtrsim Y$ is used to mean that there is a positive constant $C$ independent of $h$ such that $X\ge C Y$. On the other hand, if $\mathbf{m}box{dist}(T_k,T_{\bf e}ll)>d_0$, then we have
\mathbf{m}athbf{b}egin{align}\langlebel{green2}
{\mathbf{m}athbf i}nt_{\Omega}\left|\mathbf{m}athbf{n}abla v_k\mathbf{m}athbf{n}abla v_{{\bf e}ll}^T\mathbf{r}ight|d\mathbf{x} &\lesssim{\mathbf{m}athbf i}nt_\Omega \left({\mathbf{m}athbf i}nt_{T_k}|\mathbf{m}athbf{n}abla G(\mathbf{x},\mathbf{y})| d\mathbf{y} {\mathbf{m}athbf i}nt_{T_{\bf e}ll}|\mathbf{m}athbf{n}abla G(\mathbf{x},\mathbf{y}')| d\mathbf{y}'\mathbf{r}ight) d\mathbf{x}\mathbf{m}athbf{n}onumber\\
& \lesssim h^4{\mathbf{m}athbf i}nt_\Omega |\mathbf{m}athbf{n}abla G(\mathbf{x},\mathbf{z}_k)||\mathbf{m}athbf{n}abla G(\mathbf{x},\mathbf{z}_{\bf e}ll)| d\mathbf{x}\mathbf{m}athbf{n}onumber\\
& \lesssim h^4\frac{1}{|\mathbf{z}_k-\mathbf{z}_{\bf e}ll|},
{\bf e}nd{align}
where $\mathbf{z}_k$ and $\mathbf{z}_{\bf e}ll$ are the gravitational centers of $T_k$ and $T_{\bf e}ll$, respectively. From the above estimate, {we observe that $\sigmaup_{i,j}|S_{k{\bf e}ll}^{ij}|$ is negligibly small if $\mathbf{m}box{dist}(T_{k},T_{{\bf e}ll})$ is large.}
For $\mathbf{m}athbf{d}elta\ge 0$, let $\mathbf{m}athbb{S}_\mathbf{m}athbf{d}elta$ be the reduced matrix of $\mathbf{m}athbb{S}$ by eliminating all columns corresponding to pairs $(k,{\bf e}ll)$ in the set $\mathbf{m}athcal K_\mathbf{m}athbf{d}elta$:
$$ \mathbf{m}athcal K_\mathbf{m}athbf{d}elta:=\{ (k,{\bf e}ll)~: \mathbf{m}box{dist}(T_{k},T_{{\bf e}ll})> \mathbf{m}athbf{d}elta\}.$$
{The parameter $\mathbf{m}athbf{d}elta$ indicates the number of columns in $\mathbf{m}athbb{S}$ to be used in order to solve the linear system {\bf e}ref{lin_eq}. In the case that $\mathbf{m}athbf{d}elta< h$ ($\mathbf{m}athcal{K}_\mathbf{m}athbf{d}elta=\{(k,{\bf e}ll)~:~k\mathbf{m}athbf{n}eq {\bf e}ll \}$), we only consider the diagonal terms $(k,k)$, and neglect most of columns in $\mathbf{m}athbb{S}$. When $\mathbf{m}athbf{d}elta=\mathbf{m}athbf{d}iam(\Omega)$, we consider all the pairs $(k,{\bf e}ll)$ without neglecting any column in $\mathbf{m}athbb{S}$ $(\mathbf{m}athcal{K}_\mathbf{m}athbf{d}elta={\bf e}mptyset)$. }
Denoting the corresponding reduced vector of $\mathbf{m}athbf{q}$ by $ \mathbf{m}athbf{q}_\mathbf{m}athbf{d}elta$, the large linear system {\bf e}ref{lin_eq} can be approximated by the following reduced system:
\mathbf{m}athbf{b}egin{equation}\langlebel{reduc_lin}
(\mathbf{m}athbb{S}_\mathbf{m}athbf{d}elta^T\mathbf{m}athbb{S}_\mathbf{m}athbf{d}elta+\sigmaqrt{\mathbf{m}athbf{b}eta}I)\mathbf{m}athbf{q}_\mathbf{m}athbf{d}elta=\mathbf{m}athbb{S}_\mathbf{m}athbf{d}elta^T\mathbf{m}athbf{W}.
{\bf e}nd{equation}
In our numerical experiments, $\mathbf{m}athbf{d}elta$ is chosen to be less than six times the diameter of meshes.
Based on the minimization problem{\bf e}qref{F_prop} with the above reduction strategy, we develop the following pressure reconstruction algorithm\langlebel{recon_algo}.
\mathbf{m}athbf{b}egin{center}
\mathbf{v}space{0.1cm}\mathbf{m}athbf{b}egin{minipage}{0.9\linewidth}
\mathbf{m}athbf{b}egin{itemize}
{\mathbf{m}athbf i}tem[{\mathbf{m}athbf i}t Step 1.] Set a reduction parameter $\mathbf{m}athbf{d}elta\geq 0$ and compute the reduced sensitivity matrix $\mathbf{m}athbb{S}_\mathbf{m}athbf{d}elta$.
{\mathbf{m}athbf i}tem[{\mathbf{m}athbf i}t Step 2.] Solve {\bf e}ref{reduc_lin} with a regularization parameter $\mathbf{m}athbf{b}eta$ to obtain $\mathbf{m}athbf{q}_\mathbf{m}athbf{d}elta$.
{\mathbf{m}athbf i}tem[{\mathbf{m}athbf i}t Step 3.] Take square-root of the components $\left(p^{(k)}\mathbf{r}ight)^2$ in $\mathbf{m}athbf{q}_\mathbf{m}athbf{d}elta$ and obtain $\mathbf{m}athbf{p}$. When $\mathbf{m}athbf{q}_\mathbf{m}athbf{d}elta$ has negative values, we truncate the negative values in $\mathbf{m}athbf{q}_\mathbf{m}athbf{d}elta$ before taking the square-root.
{\bf e}nd{itemize}
{\bf e}nd{minipage}
{\bf e}nd{center}
\sigmaection{Numerical results}\langlebel{sec:numerical_results}
We test the performance of our reconstruction algorithm \mathbf{r}ef{recon_algo}. Finite element method (FEM) is used to implement our algorithm. We perform numerical experiments in two different domain shapes: a square shaped domain with $K=512$ uniform triangular pixels, a circular shaped domain with $K= 661$ uniform triangular pixels. For each domain, we apply pressures being characteristic functions with different supports, namely $p(x,y)=p_0 \chi_{D}(x,y)$ where $D$ consists of three or four pressured regions depicted in the first columns of Figures \mathbf{r}ef{simu1}-\mathbf{r}ef{simu2}.
With these pressure distributions, we compute the current-voltage data {\bf e}ref{measure_data} by solving {\bf e}ref{young-laplace} and {\bf e}ref{Mp-PDE} with a 16-channel EIT-system ($N=16$). For the reconstruction of $p$, we use the reconstruction algorithm {\bf e}ref{reduc_lin}, described in section \mathbf{r}ef{recon_algo}, with the reduction parameter $\mathbf{m}athbf{d}elta=5h$, where $h$ is the side length of triangular mesh. We compare its performance with those obtained by the conventional EIT reconstruction method and the method {\bf e}ref{reduc_lin} with $\mathbf{m}athbf{d}elta< h$. The size of matrices $\mathbf{m}athbb{S}_\mathbf{m}athbf{d}elta$ for various $\mathbf{m}athbf{d}elta$ are given in Table \mathbf{r}ef{table}. For the numerical simulation, the same regularization parameter $\mathbf{m}athbf{b}eta$ is used to solve {\bf e}ref{reduc_lin}.
Figure \mathbf{r}ef{simu1}-\mathbf{r}ef{simu2} (a) and (e) show the true pressure distributions having different supports and the same magnitude. Figure \mathbf{r}ef{simu1}-\mathbf{r}ef{simu2}(b) and (f) display the conductivity variations measured by conventional EIT-method with respect to (a) and (e), respectively. Figure \mathbf{r}ef{simu1}-\mathbf{r}ef{simu2}(c) and (g) show the reconstructed pressure distributions by the proposed reconstruct algorithm with $\mathbf{m}athbf{d}elta< h$. Figure \mathbf{r}ef{simu1}-\mathbf{r}ef{simu2}(d) and (h) show the reconstructed pressure distributions by the proposed method. We observe through Figure \mathbf{r}ef{simu1}-\mathbf{r}ef{simu2}(c) and (d) that the proposed reconstruction algorithm with the higher reduction parameter $\mathbf{m}athbf{d}elta=5h$ provides more better image than $\mathbf{m}athbf{d}elta<h$ for the detection of the pressured regions.
We should mention that, in the case when the magnitude of the pressure is sufficiently small, the conventional linearized EIT reconstruction method work well with the selection of a good regularization parameter. However, this regularization method does not work well when the pressure is not small.
\mathbf{m}athbf{b}egin{table}[!ht]
\centering
\caption{\langlebel{table} The matrix size of the reduced sensitivity matrix $\mathbf{m}athbb{S}_\mathbf{m}athbf{d}elta$. Higher value of $\mathbf{m}athbf{d}elta$ indicates that more columns of the sensitivity $\mathbf{m}athbb{S}$ are used to solve {\bf e}ref{reduc_lin}. The size of $\mathbf{m}athbb{S}$ is given in the last row ($\mathbf{m}athbf{d}elta=\mathbf{m}athbf{d}iam(\Omega)$). Here, $h$ is the side length of the triangular mesh. }
\fracootnotesize{
\mathbf{m}athbf{b}egin{tabular}{@{}*{7}{c}}
\hline
~$\mathbf{m}athbf{d}elta$ & The size of ~$\mathbf{m}athbb{S}_\mathbf{m}athbf{d}elta$ (square)\cr
\hline
0 & $256\times 512$\cr
$5h$ & $256\times 28018$\cr
$\mathbf{m}athbf{d}iam(\Omega)$ & $256\times 262144$\cr
\hline
{\bf e}nd{tabular}\mathbf{m}athbf{b}egin{tabular}{@{}*{7}{c}}
\hline
~$\mathbf{m}athbf{d}elta$ & The size of ~$\mathbf{m}athbb{S}_\mathbf{m}athbf{d}elta$ (circle)\cr
\hline
0 & $256\times 661$\cr
$5h$ & $256\times 43814$\cr
$\mathbf{m}athbf{d}iam(\Omega)$ & $256\times 436921$\cr
\hline
{\bf e}nd{tabular}
}
{\bf e}nd{table}
\mathbf{m}athbf{b}egin{figure}[ht]
\centering
\mathbf{m}athbf{b}egin{tabular}{cccc}
\sigmamall
{\mathbf{m}athbf i}ncludegraphics[width=0.22\linewidth]{eps/ex2_exact_high.eps}&
{\mathbf{m}athbf i}ncludegraphics[width=0.22\linewidth]{eps/ex2_iso_conduc_high.eps}&
{\mathbf{m}athbf i}ncludegraphics[width=0.22\linewidth]{eps/ex2_d00_high.eps}&
{\mathbf{m}athbf i}ncludegraphics[width=0.22\linewidth]{eps/ex2_d03_high.eps}\\
\mathbf{m}athbf{b}egin{tabular}{c}\sigmamall(a) true pressure\\ \sigmamall {\bf e}nd{tabular} & \mathbf{m}athbf{b}egin{tabular}{c}\sigmamall(b) conventional\\ \sigmamall method {\bf e}nd{tabular} & \mathbf{m}athbf{b}egin{tabular}{c}\sigmamall(c) reconstruction\\ \sigmamall algorithm($\mathbf{m}athbf{d}elta< h$){\bf e}nd{tabular} & \mathbf{m}athbf{b}egin{tabular}{r}\sigmamall(d) proposed\\ \sigmamall method {\bf e}nd{tabular} \\
{\mathbf{m}athbf i}ncludegraphics[width=0.22\linewidth]{eps/ex3_exact_high.eps}&
{\mathbf{m}athbf i}ncludegraphics[width=0.22\linewidth]{eps/ex3_iso_conduc_high.eps}&
{\mathbf{m}athbf i}ncludegraphics[width=0.22\linewidth]{eps/ex3_d00_high.eps}&
{\mathbf{m}athbf i}ncludegraphics[width=0.22\linewidth]{eps/ex3_d03_high.eps}\\
\mathbf{m}athbf{b}egin{tabular}{c}\sigmamall(e) true pressure\\ \sigmamall {\bf e}nd{tabular} & \mathbf{m}athbf{b}egin{tabular}{c}\sigmamall(f) conventional\\ \sigmamall method {\bf e}nd{tabular} & \mathbf{m}athbf{b}egin{tabular}{c}\sigmamall(g) reconstruction\\ \sigmamall algorithm($\mathbf{m}athbf{d}elta< h$){\bf e}nd{tabular} & \mathbf{m}athbf{b}egin{tabular}{r}\sigmamall(h) proposed\\ \sigmamall method {\bf e}nd{tabular} \\
{\bf e}nd{tabular}
\caption{\langlebel{simu1}True pressure distributions(first column), the conductivity variation images by conventional EIT-method(second column), and the reconstructed pressure images using the reconstruction algorithm \mathbf{r}ef{recon_algo} with $\mathbf{m}athbf{d}elta< h$(third column), $\mathbf{m}athbf{d}elta=5h$(fourth column). Here, $h$ is the side length of the triangular mesh.}
{\bf e}nd{figure}
\mathbf{m}athbf{b}egin{figure}[ht]
\centering
\mathbf{m}athbf{b}egin{tabular}{cccc}
{\mathbf{m}athbf i}ncludegraphics[width=0.22\linewidth]{eps/ex2_circle_exact_high.eps}&{\mathbf{m}athbf i}ncludegraphics[width=0.22\linewidth]{eps/ex2_circle_iso_conduc_high.eps}&{\mathbf{m}athbf i}ncludegraphics[width=0.22\linewidth]{eps/ex2_circle_d00_high.eps}&
{\mathbf{m}athbf i}ncludegraphics[width=0.22\linewidth]{eps/ex2_circle_d05_high.eps}\\
\mathbf{m}athbf{b}egin{tabular}{c}\sigmamall(a) true pressure\\ \sigmamall {\bf e}nd{tabular} & \mathbf{m}athbf{b}egin{tabular}{c}\sigmamall(b) conventional\\ \sigmamall method {\bf e}nd{tabular} & \mathbf{m}athbf{b}egin{tabular}{c}\sigmamall(c) reconstruction\\ \sigmamall algorithm($\mathbf{m}athbf{d}elta< h$){\bf e}nd{tabular} & \mathbf{m}athbf{b}egin{tabular}{r}\sigmamall(d) proposed\\ \sigmamall method {\bf e}nd{tabular} \\
{\mathbf{m}athbf i}ncludegraphics[width=0.22\linewidth]{eps/ex3_circle_exact_high.eps}&{\mathbf{m}athbf i}ncludegraphics[width=0.22\linewidth]{eps/ex3_circle_iso_conduc_high.eps}&{\mathbf{m}athbf i}ncludegraphics[width=0.22\linewidth]{eps/ex3_circle_d00_high.eps}&{\mathbf{m}athbf i}ncludegraphics[width=0.22\linewidth]{eps/ex3_circle_d05_high.eps}\\
\mathbf{m}athbf{b}egin{tabular}{c}\sigmamall(e) true pressure\\ \sigmamall {\bf e}nd{tabular} & \mathbf{m}athbf{b}egin{tabular}{c}\sigmamall(f) conventional\\ \sigmamall method {\bf e}nd{tabular} & \mathbf{m}athbf{b}egin{tabular}{c}\sigmamall(g) reconstruction\\ \sigmamall algorithm($\mathbf{m}athbf{d}elta< h$){\bf e}nd{tabular} & \mathbf{m}athbf{b}egin{tabular}{r}\sigmamall(h) proposed\\ \sigmamall method {\bf e}nd{tabular} \\
{\bf e}nd{tabular}
\caption{\langlebel{simu2}True pressure distributions(first column), the conductivity variation images by conventional EIT-method(second column), and the reconstructed pressure images using the reconstruction algorithm \mathbf{r}ef{recon_algo} with $\mathbf{m}athbf{d}elta< h$(third column), $\mathbf{m}athbf{d}elta=5h$(fourth column). Here, $h$ is the side length of the triangular mesh.}
{\bf e}nd{figure}
\sigmaection{Conclusion}
We have provided a mathematical framework of an EIT-based pressure-sensor for the development of an image reconstruction algorithm. We have derived the first mathematical model describing the electromechanical properties of a conductive membrane with the standard EIT system. We have found that the geometric variation of the membrane due to an applied pressure produces anisotropic conductance variation. Hence, the corresponding inverse problem of recovering the anisotropic conductivity distribution cannot be addressed by the EIT method alone. Under the assumption that the geometric variation is not too large, we have developed a reconstruction algorithm based on a sensitivity matrix of current-voltage data arising from small perturbations of pressure. Numerical simulations have indicated the feasibility of the EIT-based pressure sensor by successfully recovering pressures.
The proposed model is based on the incompressibility assumption which may not be satisfied by various flexible conductive materials, such as conductive fabrics. Recently,
Bera {\bf e}tal \cite{Bera2014} investigated the electromechanical properties of different conductive fabrics through electrical impedance spectroscopy. Because fabrics are flexible, durable, and washable, EIT-based fabric pressure sensors should be widely applicable as, for example, wearable sensors. Constructing a suitable mathematical model will be very complicated because the interactions of the conductive yarns and the air gaps among them should be taken into account. This will be the subject of future studies.
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\begin{document}
\title[Pointwise estimate for the Bergman kernel]{Pointwise estimate for the Bergman kernel of the weighted Bergman spaces with exponential type weights}
\author{Sa\"{\i}d Asserda and Amal Hichame}
\address{Ibn tofail university , faculty of sciences, department of mathematics, PO 242 Kenitra Morroco}
\email{[email protected]}
\address{Regional Centre of trades of education and training, Kenitra Morocco}
\email{ [email protected]}
\subjclass[2010]{Primary 32A25, Secondary
30H20.}
\keywords{Bergman Kernel, $\bar\partial$-equation.}
\date{\today}
\begin{abstract}
Let $AL^{2}_{\phi}(\mathbb{D})$ denote the closed subspace of $L^{2}(\mathbb{D},e^{-2\phi}d\lambda)$ consisting of holomorphic functions in the unit disc ${\mathbb D}$. For certain class of subharmonic funcions $\phi : {\mathbb D}\rightarrow{\mathbb D}$, we prove upper pointwise estimate for the Bergman kernel for $AL^{2}_{\phi}(\mathbb{D})$.
\end{abstract}
\maketitle
\section{Introduction and statement of main result}
\label{}
Let $\mathbb{D}$ be the unit disc in $\mathbb{C}$ and $d\lambda$ be its Lebesgue measure. For a measurable function $\phi : \mathbb{D}\rightarrow\mathbb{D}$, let $L^{2}_{\phi}(\mathbb{D})$ be the Hilbert space of measurable function $f$ on $\mathbb{D}$ such that
$$
\Vert f\Vert_{L^{2}_{\phi}}:=\Bigl(\int_{\mathbb{D}}\vert f\vert^{2}e^{-2\phi}d\lambda\Bigr)^{1\over 2} < \infty
$$
Let $AL^{2}_{\phi}(\mathbb{D})$ be the closed subspace of $L^{2}_{\phi}(\mathbb{D})$ consisting of analytic functions. Let $P$ be the orthogonal projection of $L^{2}_{\phi}(\mathbb{D})$ onto $AL^{2}_{\phi}(\mathbb{D})$ :
$$
Pf(z):=\int_{\mathbb{D}}K(z,w)f(w)e^{-2\phi(w)}d\lambda
$$
where $K$ is the reproducing kernel of $P$.\\ The purpose of this note is to give an upper pointwise estimate of $K$ for some class of subharmonic functions $\phi$ on $\mathbb{D}$ introduced by Oleinik [10] and Oleinik-Perel'man [11].
\begin{defn}
For $\phi \in C^{2}(\mathbb{D})$ and $\Delta\phi > 0$ put $\tau=(\Delta\phi)^{-1/2}$ where $\Delta$ is the Laplace operator. We call $\phi\in \mathcal{OP}(\mathbb{D}) $ if the following conditions holds.\\
(1)\ $\exists\ C_{1} > 0$ such that $\vert\tau(z)-\tau(w)\vert\le C_{1}\vert z-w\vert$ ,\\
(2)\ $\exists\ C_{2} > 0$ such that $\tau(z)\leq C_{2}(1-\vert z\vert)$,\\
(3)\ $\exists\ 0< C_{3} <1$ and $ a > 0$ such that $\tau(w)\leq \tau(z) + C_{3}\vert z-w\vert$ for $ w\notin D(z,a\tau(z))$ where $D(z,a\tau(z))=\{ w\in \mathbb{D}\ ,\ \vert w-z\vert\leq a\tau(z)\}$.
\end{defn}
Some examples of functions in $\mathcal{OP}(\mathbb{D})$ are as follows :\\
(i) $\phi_{1}(z)=-{A\over 2}\log(1-\vert z\vert^{2}),\ A>0$.\\
(ii) $\phi_{2}(z)={1\over 2}\bigl(-A\log(1-\vert z\vert^{2})+B(1-\vert z\vert^{2})^{-\alpha}\bigr),\ A\geq 0,\ B>0, \alpha > 0$.\\
(iii) $\phi_{1}+h$ and $\phi_{2}+h$ where $\phi_{1}$ and $\phi_{2}$ are as in (i) and (ii) respectively and $h\in C^{2}(\mathbb{D})$ can be any harmonic function on $\mathbb{D}$.\\
For $z,w\in\mathbb{D}$, the distance $d_{\phi}$ induced by the metric $\tau(z)^{-2}dz\otimes d{\bar z}$ is given by
$$
d_{\phi}(z,w)= \inf_{\gamma}\int_{0}^{1}{\vert\gamma^{'}(t)\vert\over\tau(\gamma(t))}dt
$$
where $\gamma$ runs over the piecewise $C^{1}$ curves $\gamma : [0,1]\rightarrow\mathbb{D}$ with $\gamma(0)=z$ and $\gamma(1)=w$. Thanks to condition $(2)$ the metric space $(\mathbb{D},d_{\phi})$ is complete and $d_{\phi}\succeq d_{h}$ where $d_{h}$ is the hyperbolic distance.\\
Our main result is the following theorem on the off-diagonal decay of the Bergman kernel.
\begin{thm}
Let $\phi\in \mathcal{OP}(\mathbb{D})$ and $K$ be the Bergman kernel for $AL^{2}_{\phi}(\mathbb{D})$. There exist positive constants $C$ and $\sigma$ such that for any $z,w \in \mathbb{D}$
$$
\vert K(z,w)\vert e^{-(\phi(z)+\phi(w))}\leq C{1\over\tau(z)\tau(w)}\exp(-\sigma d_{\phi}(z,w))
$$
\end{thm}
In [4] and [9] M.Christ and J.Marzo-J.Ortega-Cerd\`a obtained a pointwise estimates for the Bergman kernel of the weighted Fock space $\mathcal{F}_{\phi}^{2}(\mathbb{C})$ under the hypothesis that $\Delta\phi$ is a doubling measure. This result was extended to several variables by H.Delin and H.Lindholm in [5] and [7] under similar hypothesis.\\ In [12], A.P.Schuster and D.Varolin obtained a pointwise estimate for the Bergman Kernel of the weighted Bergman space $AL^{2}(\mathbb{D},e^{-2\phi}(1-\vert z\vert^{2})^{-2}d\lambda)$ under the hypothesis that $\Delta\phi$ is comparable to hyperbolic metric of $\mathbb{D}$ : $$\vert K(z,w)\vert e^{-(\phi(z)+\phi(w))}\leq C\exp(-\sigma d_{h}(z,w))$$
For $\phi\in \mathcal{OP}(\mathbb{D})$ and under the strong condition : $\forall\ m\geq 1,\ \exists\ b_{m}>0$ and $0<t_{m}<{1\over m}$ such that
$$\tau(w)\leq\tau(z)+t_{m}\vert z-w\vert\ \ \hbox{if}\ \ \vert z-w\vert > b_{m}\tau(z),
$$ H.Arroussi and J.Pau [1] give the following pointwise estimate : for each $k\ge 1$ there exists $C_{k} > 0$ such that
$$\vert K(z,w)\vert e^{-(\phi(z)+\phi(w))}\leq{C_{k}[d_{\tau}(z,w)]^{-k}\over\tau(z)\tau(w)}$$
where $d_{\tau}(w)={\vert z-w\vert\over\min[\tau(z),\tau(w)]}$. A better estimate will be
$$\vert K(z,w)\vert e^{-(\phi(z)+\phi(w))}\leq{C\over\tau(z)\tau(w)}e^{-\sigma d_{\tau}(z,w)}.$$
\section{Proof of theorem 1.2}
\noindent Near the diagonal, by [8,lemma 3.6 ] there exists $\alpha > 0$ sufficiently small such that
$$
\vert K(z,w)\vert\sim\sqrt{K(z,z)}\sqrt{K(w,w)}\sim {e^{\phi(z)+\phi(w)}\over\tau(z)\tau(w)}\quad\hbox{if}\quad \vert z-w\vert\leq\alpha\min[\tau(z),\tau(w)]
$$
Off the diagonal, let $\vert z-w\vert > \alpha\min[\tau(z),\tau(w)]$ and $\beta > 0$ such that $D(z,\beta\tau(z))\cap D(w,\beta\tau(w))=\emptyset$. We may suppose that $\tau(z)\leq\tau(w)$. Fix a smooth function $\chi\in C^{\infty}_{0}(\mathbb{D})$ such that \\
- $\hbox{supp}\chi\subset D(w,\beta\tau(w))$,\\
- $0\leq\chi\le 1$, $\chi=1$ in $D(w,{\beta\over 2}\tau(w))$ and \\
- $\vert\bar\partial\chi\vert^{2}\preceq\chi\tau(w)^{-2}$.\\ Since $\phi\in\mathcal{OP}(\mathbb{D})$, by [10,lemma 1 and 2] the following mean inequality holds
\begin{eqnarray*}
(*)\qquad\vert K(w,z))\vert^{2}e^{-2\phi(w)}&\preceq&{1\over\tau(w)^{2}}\int_{D(w,{\beta\over 2}\tau(w))}\chi(\zeta)\vert K(\zeta,z))\vert^{2}e^{-2\phi(\zeta)}d\lambda(\zeta)\\
&\preceq&{1\over\tau(w)^{2}}\Vert K(.,z)\Vert^{2}_{L^{2}(\chi e^{-2\phi}d\lambda)}
\end{eqnarray*}
Hence $$\Vert K(.,z)\Vert_{L^{2}(\chi e^{-\phi})}=\sup_{f}\vert<f,K(.,z)>_{L^{2}(\chi e^{-2\phi}d\lambda)}\vert$$ where $f$ is holomorphic in $D(w,\beta\tau(w))$ with $\Vert f\Vert_{L^{2}(\chi e^{-2\phi}d\lambda)}=1$. Since $P_{\phi}(f\chi)(z)=<f,K(.,z)>_{L^{2}(\chi e^{-2\phi}d\lambda)}$ and that $u_{f}=f\chi-P_{\phi}(f\chi)$ is the minimal solution in $L^{2}(\mathbb{D}, e^{-2\phi}d\lambda)$ of $\bar\partial u=f\bar\partial\chi$, and from the fact $\chi(z)=0$, we have
$$
\vert<f,K(.,z)>_{L^{2}(\chi e^{-2\phi}d\lambda)}\vert=\vert P_{\phi}(f\chi)(z)\vert=\vert u_{f}(z)\vert
$$
Since $D(z,\beta\tau(z))\cap D(w,\beta\tau(w))=\emptyset$, the function $u_{f}$ is holomorphic in $D(z,\nu\tau(z))$ for some $\nu > 0$. By the mean value inequality
\begin{eqnarray*}
\vert u_{f}(z)\vert^{2}e^{-2\phi(z)}&\preceq&{1\over\tau(z)^{2}}\int_{D(z,\nu\tau(z))}\vert u_{f}(\zeta)\vert^{2}e^{-2\phi(\zeta)}d\lambda\\
&\preceq&{1\over\tau(z)^{2}}\int_{D(z,\nu\tau(z))}e^{-\epsilon{\vert\zeta-z\vert\over\nu\tau(z)}}\vert u_{f}(\zeta)\vert^{2}e^{-2\phi(\zeta)}d\lambda
\end{eqnarray*}
Since the linear curve $\gamma(t)=(1-t)z+t\zeta$ lies in $D(z,\nu\tau(z))$ and $\tau(\gamma(t))\sim\tau(z)$, we have $d_{\phi}(\zeta,z)\le C{\vert \zeta-z\vert\over\tau(z)}$ for $\zeta\in D(z,\nu\tau(z))$. Hence
\begin{eqnarray*}
\vert u_{f}(z)\vert^{2}e^{-2\phi(z)}&\preceq &{1\over\tau(z)^{2}}\int_{D(z,\nu\tau(z))}e^{-C\epsilon d_{\phi}(\zeta,z)}\vert u_{f}(\zeta)\vert^{2}e^{-2\phi(\zeta)}d\lambda\\
&\preceq &{1\over\tau(z)^{2}}\int_{\mathbb{D}}e^{-C\epsilon d_{\phi}(\zeta,z)}\vert u_{f}(\zeta)\vert^{2}e^{-2\phi(\zeta)}d\lambda
\end{eqnarray*}
The function $\zeta\rightarrow d_{\phi}(\zeta,z)$ is smooth on $\mathbb{D}\setminus\hbox{Cut}(z)\cup\{z\}$ where $\hbox{Cut}(z)$ is the cut locus : the set of all cut points of $z$ along all geodesics that start from $z$. To get a smooth Lipschitz approximation of $d_{\phi}$, we recall the following result of Greene-Wu [6] ( see also [2]).
\begin{thm}
Let $M$ be a complete Riemannian manifold, let $ h : M\rightarrow\mathbb{R}$ be a Lipschitz function, let $\eta : M\rightarrow ]0,+\infty[$ be a continuous function, and $r$ a positive number. Then there exist a smooth Lipschitz function $ g : M\rightarrow\mathbb{R}$ such that $\vert h(x)-g(x)\vert\leq\eta(x)$ for every $x\in M$, and $\hbox{Lip}(g)\leq\hbox{Lip}(h)+r$.
\end{thm}
We use this result with $h(\zeta)=d_{\phi}(\zeta,z),\ \eta=1$ and $r=1$. We have $d_{\phi}(\zeta,z)\prec g_{z}(\zeta)\prec d_{\phi}(\zeta,z)$ and $\tau(\zeta)\vert dg_{z}(\zeta)\vert\leq 2$. Hence
$$
\vert u_{f}(z)\vert^{2}e^{-2\phi(z)}\preceq{1\over\tau(z)^{2}}\int_{\mathbb{D}}e^{-C\epsilon g_{z}(\zeta)}\vert u_{f}(\zeta)\vert^{2}e^{-2\phi(\zeta)}d\lambda
$$
By Berndtsson-Delin's improved $L^{2}$ estimates of for the minimal solution of $\bar\partial\ $ in $L^{2}(\mathbb{D},e^{-2\phi}d\lambda)$ [3][5] , we have :
$$
\int_{\mathbb{D}}e^{-C\epsilon g_{z}(\zeta)}\vert u_{f}(\zeta)\vert^{2}e^{-2\phi(\zeta)}d\lambda\preceq\int_{\mathbb{D}}e^{-C\epsilon g_{z}(\zeta)}\vert \bar\partial \chi(\zeta)\vert^{2}\vert f(\zeta)\vert^{2}\tau(\zeta)^{2}e^{-2\phi(\zeta)}d\lambda
$$
provided that $\tau\vert\partial\omega_{\epsilon}\vert\leq\mu\omega_{\epsilon}$ with $\mu < \sqrt{2}$ where $\omega_{\epsilon}(\zeta)=e^{-C\epsilon g_{z}(\zeta)}$. If we choose $\epsilon$ small enough so that $\mu=2C\epsilon<\sqrt{2}$ then $\tau\vert\partial\omega_{\epsilon}\vert=C\epsilon\tau\vert\partial g_{z}\vert\omega_{\epsilon}\leq \mu\omega_{\epsilon}$. Thus
$$
\vert u_{f}(z)\vert^{2}e^{-2\phi(z)}\preceq{1\over\tau(z)^{2}}\int_{D(w,\beta\tau(w))}e^{-C\epsilon d_{\phi}(\zeta,z)}\chi(\zeta)\vert f\vert^{2}e^{-2\phi(\zeta)}d\lambda
$$
where for the last term we use $\tau(\zeta)\sim\tau(w)$. Since $\zeta\in D(w,\beta\tau(w))$ we have
$$
d_{\phi}(\zeta,z)\geq d_{\phi}(z,w)-d_{\phi}(w,\zeta)\succeq d_{\phi}(z,w)-{\vert\zeta-w\vert\over\beta\tau(w)}\succeq d_{\phi}(z,w)
$$
and thanks to $(*)$, we conclude
$$
\vert K(z,w)\vert e^{-(\phi(w)+\phi(z))}\leq{C\over\tau(z)\tau(w)}e^{-\sigma d_{\phi}(z,w)}.
$$
\vskip 20 pt
\end{document} |
\begin{document}
\author{Lu-Ming Duan and Guang-Can Guo\thanks{
E-mail:[email protected]} \\
Physics Department and Nonlinear Science Center, University\\
of Science and Technology of China, Hefei,230026 P.R.China}
\title{Alternative approach to electromagnetic field quantization in nonlinear and
inhomogeneous media}
\date{}
\maketitle
\begin{abstract}
\baselineskip 24pt A simple approach is proposed for the quantization of the
electromagnetic field in nonlinear and inhomogeneous media. Given the
dielectric function and nonlinear susceptibilities, the Hamiltonian of the
electromagnetic field is determined completely by this quantization method.
From Heisenberg's equations we derive Maxwell's equations for the field
operators. When the nonlinearity goes to zero, this quantization method
returns to the generalized canonical quantization procedure for linear
inhomogeneous media [Phys. Rev. A, 43, 467, 1991]. The explicit Hamiltonians
for the second-order and third-order nonlinear quasi-steady-state processes
are obtained based on this quantization procedure.
\
PACS numbers:42.50.-p, 42.65.-k, 11.10.Lm
\end{abstract}
\baselineskip 24pt
\section{Introduction}
Early quantization of the electromagnetic field is performed in empty
cavities or in infinite free space [1]. However, with the growth of interest
in quantum optical phenomena taking place inside material media, several
approaches have been proposed for quantization of the electromagnetic field
in nonlinear, inhomogeneous, or dispersive media [2-21]. Early attempts
towards quantization of the nonlinear media, while incorporating the known
linear theory, did not fully reproduce the nonlinear field equations [4]. An
innovative treatment was first proposed by Hillery and Mlodinow who
successfully quantized a nonlinear medium by introducing the dual potential
[6]. Later, Drummond extended the Hillery-Mlodinow procedure to dispersive
media [9]. There are also other approaches in this direction. Glauber and
Lewenstein generalized the canonical quantization method by modifying the
gauge condition to deal with the inhomogeneous linear media [11]. Abram and
Cohen, following the canonical quantization procedure, presented a quantum
formulation for light propagation in nonlinear effective media [12]. And
recently, Santos and Loudon gave an alternative approach to the quantization
of the electromagnetic field in linear one-dimensional dispersive media
[19]. Developments towards the absorbing dielectrics also appeared [20,21].
In this paper, we propose a relatively simpler approach to the
electromagnetic field quantization in nonlinear and inhomogeneous media. The
procedure follows Ref. [11] in using the material independence of the
commutation relations for the fields $\overrightarrow{D}$ and $
\overrightarrow{B}$, pointed out by Born and Infeld [2], as a starting point
in the quantization. We extend this to nonlinear media in which $
\overrightarrow{D}$ and $\overrightarrow{B}$ can be expressed as isochronous
functionals of the fields $\overrightarrow{E}$ and $\overrightarrow{H}$. $
\overrightarrow{D}$ and $\overrightarrow{B}$ are expanded into the mode
functions. Furthermore, we explicitly derive Maxwell's equations for the
field operators from Heisenberg's equations. This procedure is applied to
the quantization of the second-order and third-order nonlinear
quasi-steady-state processes and we obtain the explicit Hamiltonians. Though
these Hamiltonians are already in wide use in quantum optics, their
derivations are mainly based on the early quantization procedure by Shen
[4,22,23] and known by now to be inconsistent with the nonlinear field
equations [9]. So here we give a justification of these Hamiltonians.
The arrangement of the paper is as follows. The quantization procedure is
proposed in Sec.2. Given dielectric tensor and nonlinear susceptibilities,
this quantization procedure completely determines the Hamiltonian of the
electromagnetic field, which is expressed by annihilation and creation
operators. Then we derive Maxwell's equations from Heisenberg's equations
for the field operators. In Sec.3 we show this quantization procedure
returns to the generalized canonical quantization method in Ref. [11] when
the medium is linear. The explicit Hamiltonians of the second-order and
third-order nonlinear quasi-steady-state processes are obtained in Sec. 4 by
application of this quantization procedure.
\section{Quantization in the presence of nonlinear media}
We consider the electromagnetic field in nonlinear media, which may be
inhomogeneous. The source-free Maxwell equations in matter take the forms
[24]
\begin{equation}
\label{1}\nabla \cdot \overrightarrow{D}=0,
\end{equation}
\begin{equation}
\label{2}\nabla \cdot \overrightarrow{B}=0,
\end{equation}
\begin{equation}
\label{3}\frac 1c\frac{\partial \overrightarrow{D}}{\partial t}=\nabla
\times \overrightarrow{H},
\end{equation}
\begin{equation}
\label{4}\frac 1c\frac{\partial \overrightarrow{B}}{\partial t}=-\nabla
\times \overrightarrow{E}.
\end{equation}
In nonlinear media, $\overrightarrow{D}\left( t\right) $ and $
\overrightarrow{B}\left( t\right) $ are complicated nonlinear functionals of
$\overrightarrow{E}\left( t\right) $ and $\overrightarrow{H}\left( t\right) $
. From the Maxwell equations, the energy density $U$ of the electromagnetic
field in nonlinear media is determined by
\begin{equation}
\label{5}dU\left( \overrightarrow{r},t\right) =\overrightarrow{E}\left(
\overrightarrow{r},t\right) \cdot d\overrightarrow{D}\left( \overrightarrow{r
},t\right) +\overrightarrow{H}\left( \overrightarrow{r},t\right) \cdot d
\overrightarrow{B}\left( \overrightarrow{r},t\right) .
\end{equation}
The Hamiltonian (or the energy) is
\begin{equation}
\label{6}\widetilde{H}=\int d^3\overrightarrow{r}U\left( \overrightarrow{r}
,t\right) .
\end{equation}
For the electromagnetic field in linear media, the canonical quantization
method is generally used. The vector potential $\overrightarrow{A}$ is
chosen as the general coordinate and the Columb gauge $\nabla \cdot
\overrightarrow{A}=0$ is often used. $\overrightarrow{A}$ and its conjugate
momentum can be expanded into a set of transverse complete spatial functions
and the expansion coefficients are expressed by annihilation and creation
operators. Then substituting the expansions of $\overrightarrow{A}$ and its
conjugate momentum into the Hamiltonian, one achieves quantization of the
electromagnetic field in linear media. However, for the nonlinear media, the
canonical quantization becomes much more involved. $\overrightarrow{A}$ and $
\overrightarrow{E}$ were chosen as the canonical variables in the early
treatments, which did not incorporate Eq.(1). In fact, no rigorous approach
had been proposed for nonlinear media until Hillery and Mlodinow introduced
the dual potential and then followed the canonical quantization procedure.
Here, inspired by the result in Ref.[1] that the fields $\overrightarrow{D}$
and $\overrightarrow{B}$ have medium-independent commutation relations in
inhomogeneous linear media, we choose the fields $\overrightarrow{D}$ and $
\overrightarrow{B},$ rather than $\overrightarrow{E}$ or $\overrightarrow{A},
$ as the starting point of the electromagnetic field quantization. This
choice is also consistent with the results in Ref. [6] [9] and [12], where
the field $\overrightarrow{D}$ was found to be the canonical momentum.
Starting from the mode expansions of the fields $\overrightarrow{D}$ and $
\overrightarrow{B}$, we can present a concise formulation of the
quantization and a clear derivation of Maxwell's equations for the field
operators.
From Equations (1) and (2), the fields $\overrightarrow{D}$ and $
\overrightarrow{B}$ can be expanded into a set of transverse complete
spatial functions $\left\{ \overrightarrow{f}_{\overrightarrow{k}\mu }\left(
\overrightarrow{r}\right) \right\} $ and $\left\{ \nabla \times
\overrightarrow{f}_{\overrightarrow{k}\mu }\left( \overrightarrow{r}\right)
\right\} ,$ respectively,
\begin{equation}
\label{7}\overrightarrow{D}\left( \overrightarrow{r},t\right) =-\stackunder{
\overrightarrow{k}\mu }{\sum }P_{\overrightarrow{k}\mu }\left( t\right)
\overrightarrow{f}_{\overrightarrow{k}\mu }^{*}\left( \overrightarrow{r}
\right) ,
\end{equation}
\begin{equation}
\label{8}\overrightarrow{B}\left( \overrightarrow{r},t\right) =c\stackunder{
\overrightarrow{k}\mu }{\sum }Q_{\overrightarrow{k}\mu }\left( t\right)
\nabla \times \overrightarrow{f}_{\overrightarrow{k}\mu }\left(
\overrightarrow{r}\right) .
\end{equation}
The expansion functions and coefficients satisfy Hermitian conditions
\begin{equation}
\label{9}\overrightarrow{f}_{\overrightarrow{k}\mu }^{*}=\overrightarrow{f}
_{-\overrightarrow{k}\mu },
\end{equation}
\begin{equation}
\label{10}Q_{\overrightarrow{k}\mu }^{+}=Q_{-\overrightarrow{k}\mu },
\end{equation}
\begin{equation}
\label{11}P_{\overrightarrow{k}\mu }^{+}=P_{-\overrightarrow{k}\mu }.
\end{equation}
In addition, the functions $\overrightarrow{f}_{\overrightarrow{k}\mu
}\left( \overrightarrow{r}\right) $ satisfy transversality, orthonormality
and completeness conditions
\begin{equation}
\label{12}\nabla \cdot \overrightarrow{f}_{\overrightarrow{k}\mu }=0,
\end{equation}
\begin{equation}
\label{13}\int d^3\overrightarrow{r}\overrightarrow{f}_{\overrightarrow{k}
\mu i}^{*}\left( \overrightarrow{r}\right) \overrightarrow{f}_{
\overrightarrow{k}^{^{\prime }}\mu ^{^{\prime }}j}\left( \overrightarrow{r}
\right) =\delta _{\overrightarrow{k}\overrightarrow{k}^{^{\prime }}}\delta
_{\mu \mu ^{^{\prime }}}\delta _{ij},
\end{equation}
\begin{equation}
\label{14}\stackunder{\overrightarrow{k}\mu }{\sum }\overrightarrow{f}_{
\overrightarrow{k}\mu i}^{*}\left( \overrightarrow{r}\right) \overrightarrow{
f}_{\overrightarrow{k}\mu j}\left( \overrightarrow{r}^{^{\prime }}\right)
=\delta _{ij}^T\left( \overrightarrow{r}-\overrightarrow{r}^{^{\prime
}}\right) ,
\end{equation}
where the transverse $\delta -$function $\delta _T$ is defined as
\begin{equation}
\label{15}\delta _{ij}^T\left( \overrightarrow{r}\right) =\frac 1{\left(
2\pi \right) ^3}\int d^3\overrightarrow{k}\left( \delta _{ij}-\frac{k_ik_j}{
\left| \overrightarrow{k}\right| ^2}\right) e^{i\overrightarrow{k}\cdot
\overrightarrow{r}}.
\end{equation}
The transversality condition (12) makes the completeness equation of $
\overrightarrow{f}_{\overrightarrow{k}\mu }\left( \overrightarrow{r}\right) $
take the form of (14). In free space, the plane wave is chosen as the
expansion function
\begin{equation}
\label{16}\overrightarrow{f}_{\overrightarrow{k}\mu }\left( \overrightarrow{r
}\right) =\frac 1{\left( 2\pi \right) ^{\frac 32}}\overrightarrow{e}_{
\overrightarrow{k}\mu }e^{i\overrightarrow{k}\cdot \overrightarrow{r}},
\end{equation}
where the unit vectors $\overrightarrow{e}_{\overrightarrow{k}\mu }$ $\left(
\mu =1,2\right) $ satisfy
\begin{equation}
\label{17}\overrightarrow{k}\cdot \overrightarrow{e}_{\overrightarrow{k}\mu
}=0.
\end{equation}
The expansions (7) and (8) have the same forms as those in linear media. We
further suppose that the expansion coefficient operators satisfy the same
commutation relations. So
\begin{equation}
\label{18}\left[ Q_{\overrightarrow{k}\mu }\left( t\right) ,P_{
\overrightarrow{k}^{^{\prime }}\mu ^{^{\prime }}}\left( t\right) \right]
=i\hbar \delta _{\overrightarrow{k}\overrightarrow{k}^{^{\prime }}}\delta
_{\mu \mu ^{^{\prime }}}.
\end{equation}
The fields $\overrightarrow{E}\left( \overrightarrow{r},t\right) $ and $
\overrightarrow{H}\left( \overrightarrow{r},t\right) $ can be expressed by $
\overrightarrow{D}\left( \overrightarrow{r},t\right) $ and $\overrightarrow{B
}\left( \overrightarrow{r},t\right) $ from the nonlinear functional
relations between them. From (5) and (6) the Hamiltonian of the
electromagnetic field becomes a nonlinear functional of $\overrightarrow{D}
\left( \overrightarrow{r},t\right) $ and $\overrightarrow{B}\left(
\overrightarrow{r},t\right) $. After substituting the expansions (7) and (8)
into it , we get the Hamiltonian, which is expressed by annihilation and
creation operators. Given the functional relations between $\overrightarrow{E
}\left( \overrightarrow{r},t\right) $,$\overrightarrow{H}\left(
\overrightarrow{r},t\right) $ and $\overrightarrow{D}\left( \overrightarrow{r
},t\right) $,$\overrightarrow{B}\left( \overrightarrow{r},t\right) $, the
Hamiltonian form is completely determined by this quantization procedure.
Now we show reasonableness of the quantization method. Comparing (7),(8) and
(18) with the corresponding equations in Ref.[1], we know that when the
nonlinearity goes to zero the above procedure returns to the generalized
canonical quantization method. Furthermore, this quantization gives the
correct Maxwell equations. In the following we derive Maxwell's equations
for the field operators from Heisenberg's equations.
From the transversality of the expansion functions $\overrightarrow{f}_{
\overrightarrow{k}\mu },$ the first two Maxwell equations (1) and (2) are
obviously satisfied. Equations (7) (8) and (14) (18) give the commutator of
the field operators $\overrightarrow{D}$ and $\overrightarrow{B}$
\begin{equation}
\label{19}\left[ D_i\left( \overrightarrow{r},t\right) ,B_j\left(
\overrightarrow{r}^{^{\prime }},t\right) \right] =i\hbar c\left( \nabla
\times \delta \right) _{ij}\left( \overrightarrow{r}-\overrightarrow{r}
^{^{\prime }}\right) .
\end{equation}
In the derivation, the following relations are used.
\begin{equation}
\label{20}
\begin{array}{c}
\left( \nabla \times \delta ^T\right) _{ij}\left(
\overrightarrow{r}\right) =\stackunder{mn}{\sum }\varepsilon _{imn}\partial
_m\frac 1{\left( 2\pi \right) ^3}\int d^3\overrightarrow{k}\left( \delta
_{nj}-\frac{k_nk_j}{\left| \overrightarrow{k}\right| ^2}\right) e^{i
\overrightarrow{k}\cdot \overrightarrow{r}} \\ =
\stackunder{mn}{\sum }\varepsilon _{imn}\partial _m\frac 1{\left( 2\pi
\right) ^3}\int d^3\overrightarrow{k}\delta _{nj}e^{i\overrightarrow{k}\cdot
\overrightarrow{r}} \\ =\left( \nabla \times \delta \right) _{ij}\left(
\overrightarrow{r}\right)
\end{array}
\end{equation}
and
\begin{equation}
\label{21}
\begin{array}{c}
\left( \nabla ^{^{\prime }}\times \delta \right) _{ji}\left(
\overrightarrow{r}-\overrightarrow{r}^{^{\prime }}\right) =-\left( \nabla
\times \delta \right) _{ji}\left( \overrightarrow{r}-\overrightarrow{r}
^{^{\prime }}\right) \\ =-
\stackunder{mn}{\sum }\varepsilon _{jmn}\partial _m\delta _{ni}\left(
\overrightarrow{r}-\overrightarrow{r}^{^{\prime }}\right) \\ =-
\stackunder{m}{\sum }\varepsilon _{jmi}\partial _m\delta \left(
\overrightarrow{r}-\overrightarrow{r}^{^{\prime }}\right) \\ =
\stackunder{m}{\sum }\varepsilon _{imj}\partial _m\delta \left(
\overrightarrow{r}-\overrightarrow{r}^{^{\prime }}\right) \\ =\left( \nabla
\times \delta \right) _{ij}\left( \overrightarrow{r}-\overrightarrow{r}
^{^{\prime }}\right) \text{ }.
\end{array}
\end{equation}
The commutator (19) has been given in Ref. [12] in one-dimensional form.
Here we extend it to the general case. From the commutator (19), the
commutator between $\overrightarrow{D}$ or $\overrightarrow{B}$ and an
arbitrary functional $F\left( \overrightarrow{D},\overrightarrow{B}\right) $
of $\overrightarrow{D}$ and $\overrightarrow{B},$ which may be nonlinear,
can be expressed by functional derivation as follows:
\begin{equation}
\label{22}\left[ D_i\left( \overrightarrow{r},t\right) ,F\right] =i\hbar c
\stackunder{mn}{\sum }\varepsilon _{imn}\partial _m\frac \delta {\delta
B_n\left( \overrightarrow{r},t\right) }F,
\end{equation}
\begin{equation}
\label{23}\left[ B_i\left( \overrightarrow{r},t\right) ,F\right] =-i\hbar c
\stackunder{mn}{\sum }\varepsilon _{imn}\partial _m\frac \delta {\delta
D_n\left( \overrightarrow{r},t\right) }F.
\end{equation}
From these two equations and (5),(6), the Heisenberg equations of the field
operators $\overrightarrow{D}$ and $\overrightarrow{B}$ take the forms
\begin{equation}
\label{24}
\begin{array}{c}
\frac 1c
\frac{\partial D_i}{\partial t}=\frac 1{i\hbar c}\left[ D_i\left(
\overrightarrow{r},t\right) ,\widetilde{H}\right] \\ \\
=
\stackunder{mn}{\sum }\varepsilon _{imn}\partial _m\frac \delta {\delta
B_n\left( \overrightarrow{r},t\right) }\widetilde{H} \\ \\
=
\stackunder{mn}{\sum }\varepsilon _{imn}\partial _m\frac{\partial U\left(
\overrightarrow{r},t\right) }{\partial B_n\left( \overrightarrow{r},t\right)
} \\ \\
=
\stackunder{mn}{\sum }\stackunder{j}{\sum }\varepsilon _{imn}\partial _mH_j
\frac{\partial B_j}{\partial B_n} \\ \\
=\left( \nabla \times \overrightarrow{H}\right) _i\text{ .}
\end{array}
\end{equation}
Similarly
\begin{equation}
\label{25}\frac 1c\frac{\partial B_i}{\partial t}=-\left( \nabla \times
\overrightarrow{E}\right) _i\text{ .}
\end{equation}
So we have clearly derived Maxwell's equations for the field operators from
Heisenberg's equations. The isochronous commutators (22),(23) play an
essential role in the derivation. The derivation holds for linear or
nonlinear media. However, for dispersive media, nonlocal relations in time
between the Hamiltonian $\widetilde{H}$ and the fields $\overrightarrow{D},
\overrightarrow{B}$ arise and the isochronous commutators (22),(23) cannot
be applied in this case. Here we meet the long standing difficulty in
quantum optics in quantizing nonlinear and dispersive dielectrics.
\section{Quantization of the electromagnetic field in linear inhomogeneous
media}
In this section we use the above method to quantize the electromagnetic
field in linear inhomogeneous media. The medium is characterized by
\begin{equation}
\label{26}D_i\left( \overrightarrow{r},t\right) =\stackunder{j}{\sum }
\varepsilon _{ij}\left( \overrightarrow{r}\right) E_j\left( \overrightarrow{r
},t\right) ,
\end{equation}
\begin{equation}
\label{27}\overrightarrow{B}\left( \overrightarrow{r},t\right) =
\overrightarrow{H}\left( \overrightarrow{r},t\right) .
\end{equation}
The Hamiltonian (or the energy) is
\begin{equation}
\label{28}\widetilde{H}=\frac 12\int d^3\overrightarrow{r}\left(
\overrightarrow{B}^2+\stackunder{ij}{\sum }\varepsilon
_{ij}^{-1}D_iD_j\right) .
\end{equation}
In free space the expansion function $\overrightarrow{f}_{\overrightarrow{k}
\mu }$ is expressed by Eq.(16). Substituting the expansions (7) and (8) into
Eq.(28) and noting
\begin{equation}
\label{29}Q_{\overrightarrow{k}\mu }\left( t\right) =\left( \frac \hbar
{2\omega _{\overrightarrow{k}\mu }}\right) ^{\frac 12}\left( a_{
\overrightarrow{k}\mu }+a_{-\overrightarrow{k}\mu }^{+}\right) ,
\end{equation}
\begin{equation}
\label{30}P_{\overrightarrow{k}\mu }\left( t\right) =i\left( \frac{\hbar
\omega _{\overrightarrow{k}\mu }}2\right) ^{\frac 12}\left( a_{
\overrightarrow{k}\mu }^{+}-a_{-\overrightarrow{k}\mu }\right) ,
\end{equation}
where $\omega _{\overrightarrow{k}\mu }=\left| \overrightarrow{k}\right| c$
, we get the Hamiltonian expressed by annihilation and creation operators.
\begin{equation}
\label{31}
\begin{array}{c}
\widetilde{H}=\stackunder{_{\overrightarrow{k}\mu }}{\sum }\hbar \omega _{
\overrightarrow{k}\mu }a_{\overrightarrow{k}\mu }^{+}a_{\overrightarrow{k}
\mu }+\frac \hbar 4\stackunder{_{\overrightarrow{k}\mu }}{\sum }\stackunder{
_{\overrightarrow{k}^{^{\prime }}\mu ^{^{\prime }}}}{\sum }\sqrt{\omega _{
\overrightarrow{k}\mu }\omega _{\overrightarrow{k}^{^{\prime }}\mu
^{^{\prime }}}} \\ \times \left[ V_{\mu \mu ^{^{\prime }}}^{*}\left(
\overrightarrow{k},\overrightarrow{k}^{^{\prime }}\right) a_{\overrightarrow{
k}\mu }^{+}a_{\overrightarrow{k}^{^{\prime }}\mu ^{^{\prime }}}^{+}-V_{\mu
\mu ^{^{\prime }}}^{*}\left( \overrightarrow{k},-\overrightarrow{k}
^{^{\prime }}\right) a_{\overrightarrow{k}\mu }^{+}a_{\overrightarrow{k}
^{^{\prime }}\mu ^{^{\prime }}}+h.c.\right] ,
\end{array}
\end{equation}
where $V_{\mu \mu ^{^{\prime }}}$ is defined by
\begin{equation}
\label{32}V_{\mu \mu ^{^{\prime }}}\left( \overrightarrow{k},\overrightarrow{
k}^{^{\prime }}\right) =\frac 1{\left( 2\pi \right) ^3}\int d^3
\overrightarrow{r}\overrightarrow{e}_{\overrightarrow{k}\mu }\cdot \left(
1-\epsilon ^{-1}\right) \cdot \overrightarrow{e}_{\overrightarrow{k}
^{^{\prime }}\mu ^{^{\prime }}}e^{i\left( \overrightarrow{k}+\overrightarrow{
k}^{^{\prime }}\right) \cdot \overrightarrow{r}}
\end{equation}
If the second-order tensor $\varepsilon ^{-1}$ is a scalar, $\overrightarrow{
e}_{\overrightarrow{k}\mu }\cdot \overrightarrow{e}_{\overrightarrow{k}
^{^{\prime }}\mu ^{^{\prime }}}$ in Eq.(32) can be put out of the
integration. This case had been discussed in detail in Ref.[11] by the
generalized canonical quantization method. Here we get the same results.
\section{Quantization of the quasi-steady-state optical field in nonlinear
media}
In this section we consider quantization of the quasi-steady-state optical
field in nonlinear media. The second-order or third-order nonlinear process
is most important [25,23]. First we apply the quantization method to the
parametric process. The optical field is composed of three
quasi-monochromatic fields with central frequency $\omega _1,\omega
_2,\omega _3$, respectively, and $\omega _3=\omega _1+\omega _2$, i.e.,
\begin{equation}
\label{33}\overrightarrow{E}\left( t\right) =\stackrel{3}{\stackunder{i=1}{
\sum }}\overrightarrow{E}^{\left( i\right) }\left( t\right) e^{-i\omega
_it}+h.c.,
\end{equation}
where $\overrightarrow{E}^{\left( i\right) }\left( t\right) $ is slowly
varying amplitude. Under the quasi-steady-state approximation, the term $
\overrightarrow{E}^{\left( i\right) }\left( t\right) e^{-i\omega _it}$ in
Eq.(33) can be viewed as a monochromatic field with frequency $\omega _i$,
i.e., the dispersion of the optical field in the medium is negligible.
Suppose the refractive index is independent of orientation of space. Then
the field $\overrightarrow{D}^{\left( i\right) }\left( t\right) $ can be
expressed by $\overrightarrow{E}^{\left( i\right) }\left( t\right) $. For
example,
\begin{equation}
\label{34}\overrightarrow{D}^{\left( 3\right) }\left( t\right) =n^2\left(
\omega _3\right) \overrightarrow{E}^{\left( 3\right) }\left( t\right) +\chi
^{\left( 2\right) }\left( \omega _3=\omega _1+\omega _2\right) :
\overrightarrow{E}^{\left( 1\right) }\left( t\right) \overrightarrow{E}
^{\left( 2\right) }\left( t\right) ,
\end{equation}
where $n^2\left( \omega _3\right) $ is introduced phenomenologically to show
the component of the optical field $\overrightarrow{E}^{\left( 3\right)
}\left( t\right) $ has been viewed as a monochromatic field with frequency $
\omega _3$. $\overrightarrow{D}^{\left( 2\right) }\left( t\right) $ and $
\overrightarrow{D}^{\left( 1\right) }\left( t\right) $ have similar
expressions. From these equations, $\overrightarrow{E}^{\left( i\right)
}\left( t\right) $ $\left( i=1,2,3\right) $ may be expressed by $
\overrightarrow{D}^{\left( i\right) }\left( t\right) $ as
\begin{equation}
\label{35}\overrightarrow{E}^{\left( 3\right) }\left( t\right) =\frac{
\overrightarrow{D}^{\left( 3\right) }\left( t\right) }{n^2\left( \omega
_3\right) }-\gamma ^{\left( 2\right) }\left( \omega _3=\omega _1+\omega
_2\right) :\overrightarrow{D}^{\left( 1\right) }\left( t\right)
\overrightarrow{D}^{\left( 2\right) }\left( t\right) ,\text{ et.al.,}
\end{equation}
where
\begin{equation}
\label{36}\gamma ^{\left( 2\right) }\left( \omega _3=\omega _1+\omega
_2\right) =\frac 1{n^2\left( \omega _1\right) n^2\left( \omega _2\right)
n^2\left( \omega _3\right) }\chi ^{\left( 2\right) }\left( \omega _3=\omega
_1+\omega _2\right) .
\end{equation}
In the derivation of Eq.(35) the approximation that the nonlinear terms are
much smaller than the linear terms is used. Substituting (35) into (5) and
(6) and making the rotating wave approximation, we get the Hamiltonian of
the electromagnetic field, which is expressed by $\overrightarrow{D}$ and $
\overrightarrow{B}$
\begin{equation}
\label{37}
\begin{array}{c}
\widetilde{H}=\frac 12\int d^3\overrightarrow{r}\left\{ \stackrel{3}{
\stackunder{i=1}{\sum }}\left[ \frac{\left( \overrightarrow{D}^{\left(
i\right) }\left( t\right) e^{-i\omega _it}+h.c.\right) ^2}{n^2\left( \omega
_i\right) }+\left( \overrightarrow{B}^{\left( i\right) }\left( t\right)
e^{-i\omega _it}+h.c.\right) ^2\right] \right. \\ \\
\left. -2\left[ \gamma ^{\left( 2\right) }\left( \omega _3=\omega _1+\omega
_2\right) \vdots \overrightarrow{D}^{\left( 3\right) +}\left( t\right)
\overrightarrow{D}^{\left( 2\right) }\left( t\right) \overrightarrow{D}
^{\left( 1\right) }\left( t\right) +h.c.\right] \right\}
\end{array}
\end{equation}
In the derivation, the holo-exchange symmetry of the tensor $\chi ^{\left(
2\right) }$ has been used. In the expansions (7) and (8) of the fields $
\overrightarrow{D}$ , $\overrightarrow{B}$, only the terms with the
subscripts $\left| \overrightarrow{k}\right| =k_i$, where $k_i=\frac{n\left(
\omega _i\right) \omega _i}c$,$\left( i=1,2,3\right) $, make contributions
to the interaction. So the expansions can be simplified to
\begin{equation}
\label{38}\overrightarrow{D}^{\left( i\right) }\left( t\right) e^{-i\omega
_it}=i\sqrt{\frac{\hbar \omega _in^2\left( \omega _i\right) }2}\stackunder{
\left| \overrightarrow{k}\right| =k_i}{\sum }\stackunder{\mu }{\sum }
\overrightarrow{f}_{\overrightarrow{k}\mu }a_{\overrightarrow{k}\mu }\left(
t\right) ,
\end{equation}
\begin{equation}
\label{39}\overrightarrow{B}^{\left( i\right) }\left( t\right) e^{-i\omega
_it}=c\sqrt{\frac \hbar {2\omega _in^2\left( \omega _i\right) }}\stackunder{
\left| \overrightarrow{k}\right| =k_i}{\sum }\stackunder{\mu }{\sum }\nabla
\times \overrightarrow{f}_{\overrightarrow{k}\mu }a_{\overrightarrow{k}\mu
}\left( t\right) .
\end{equation}
The function $\overrightarrow{f}_{\overrightarrow{k}\mu }$ can be decomposed
as $\overrightarrow{f}_{\overrightarrow{k}\mu }=f_{\overrightarrow{k}}\left(
\overrightarrow{r}\right) \cdot $ $\overrightarrow{e}_{\overrightarrow{k}\mu
}$, where $f_{\overrightarrow{k}}\left( \overrightarrow{r}\right) $
satisfies the eigen-equation
\begin{equation}
\label{40}\nabla ^2f_{\overrightarrow{k}}\left( \overrightarrow{r}\right)
=-\left| \overrightarrow{k}\right| ^2f_{\overrightarrow{k}}\left(
\overrightarrow{r}\right)
\end{equation}
Substituting (38) and (39) into Eq.(37), we get the Hamiltonian expressed by
annihilation and creation operators
\begin{equation}
\label{41}
\begin{array}{c}
\widetilde{H}=\stackunder{\overrightarrow{k}_1,\overrightarrow{k}_2,
\overrightarrow{k}_3}{\stackrel{\left| \overrightarrow{k}_i\right| =k_i}{
\sum }}\stackunder{\mu _1,\mu _2,\mu _3}{\sum }\left[ \stackrel{3}{
\stackunder{i=1}{\sum }}\hbar \omega _ia_{\overrightarrow{k}_i\mu _i}^{+}a_{
\overrightarrow{k}_i\mu _i}\right. \\ \\
\left. +(\alpha _{\overrightarrow{k}_1\mu _1\overrightarrow{k}_2\mu _2
\overrightarrow{k}_3\mu _3}\beta _{\overrightarrow{k}_1\overrightarrow{k}_2
\overrightarrow{k}_3}a_{\overrightarrow{k}_3\mu _3}^{+}a_{\overrightarrow{k}
_2\mu _2}a_{\overrightarrow{k}_1\mu _1}+h.c.)\right] ,
\end{array}
\end{equation}
where the constant $\beta $ (called phase-matching factor) is defined as
\begin{equation}
\label{42}\beta _{\overrightarrow{k}_1\overrightarrow{k}_2\overrightarrow{k}
_3}=\sqrt{V}\int f_{\overrightarrow{k}_3}^{*}f_{\overrightarrow{k}_2}f_{
\overrightarrow{k}_1}d^3\overrightarrow{r},
\end{equation}
and V is volume of the nonlinear media. The constant $\alpha $ is
\begin{equation}
\label{43}
\begin{array}{c}
\alpha _{
\overrightarrow{k}_1\mu _1\overrightarrow{k}_2\mu _2\overrightarrow{k}_3\mu
_3}=-i\sqrt{\frac{\hbar ^3\omega _1\omega _2\omega _3}{8Vn^2\left( \omega
_1\right) n^2\left( \omega _2\right) n^2\left( \omega _3\right) }} \\ \\
x^{\left( 2\right) }\left( \omega _3=\omega _1+\omega _2\right) \vdots
\overrightarrow{e}_{\overrightarrow{k}_3\mu _3}^{*}\overrightarrow{e}_{
\overrightarrow{k}_2\mu _2}\overrightarrow{e}_{\overrightarrow{k}_1\mu _1}
\end{array}
\end{equation}
and it is determined by the polarization matching condition of the optical
field. When $\overrightarrow{k}_1,\overrightarrow{k}_2,\overrightarrow{k}_3$
are collinear, the Hamiltonian (41) can be simplified. Suppose the
polarizations of the optical fields are given (indicated by $\mu
_1^{^{\prime }},\mu _2^{^{\prime }},\mu _3^{^{\prime }}$ respectively) and $
\overrightarrow{k}_i=k_i\overrightarrow{e}_z$ $\left( i=1,2,3\right) $,
where $\overrightarrow{e}_z$ is the unit vector of z-axis. The expansion
function is approximately a plane wave, i.e.,
\begin{equation}
\label{44}f_{k_i}\left( \overrightarrow{r}\right) =S_{k_i}\left( x,y\right)
\frac{e^{-ik_iz}}{\sqrt{L}},
\end{equation}
where L is the interaction length. Let $a_i$ represent the operator $
a_{k_i\mu _i^{^{\prime }}}$ and
\begin{equation}
\label{45}\alpha =\alpha _{k_1\mu _1^{^{\prime }}k_2\mu _2^{^{\prime
}}k_3\mu _3^{^{\prime }}}\sqrt{\frac VL}\int S_{k_3}^{*}S_{k_2}S_{k_1}dxdy
\text{,}
\end{equation}
then the Hamiltonian (41) is simplified to
\begin{equation}
\label{46}\widetilde{H}=\stackrel{3}{\stackunder{i=1}{\sum }}\hbar \omega
_ia_i^{+}a_i+\left( \alpha \cdot \frac{e^{i\Delta kL}-1}{i\Delta kL}
a_3^{+}a_2a_1+h.c.\right) ,
\end{equation}
where the phase mismatch $\Delta k=k_1+k_2-k_3$. Eq.(46) is often used to
analyze quantum properties of the parametric process [26-28]. Here we give
its exact derivation and determine the expression of the parameter $\alpha $.
The optical field in the third-order nonlinear medium can be quantized in a
similar way. For example, the Hamiltonian of the nondegenerate
four-wave-mixing process [23] with $\omega _3+\omega _4=\omega _1+\omega _2$
is
\begin{equation}
\label{47}
\begin{array}{c}
\widetilde{H}=\stackunder{\overrightarrow{k}_1,\overrightarrow{k}_2,
\overrightarrow{k}_3,\overrightarrow{k}_4}{\stackrel{\left| \overrightarrow{k
}_i\right| =k_i}{\sum }}\stackunder{\mu _1,\mu _2,\mu _3,\mu _4}{\sum }
\left[ \stackrel{4}{\stackunder{i=1}{\sum }}\hbar \omega _ia_{
\overrightarrow{k}_i\mu _i}^{+}a_{\overrightarrow{k}_i\mu _i}\right. \\ \\
\left. +(\alpha _{\overrightarrow{k}_1\mu _1\overrightarrow{k}_2\mu _2
\overrightarrow{k}_3\mu _3\overrightarrow{k}_4\mu _4}\beta _{\overrightarrow{
k}_1\overrightarrow{k}_2\overrightarrow{k}_3\overrightarrow{k}_4}a_{
\overrightarrow{k}_3\mu _3}^{+}a_{\overrightarrow{k}_4\mu _4}^{+}a_{
\overrightarrow{k}_2\mu _2}a_{\overrightarrow{k}_1\mu _1}+h.c.)\right] ,
\end{array}
\end{equation}
where
\begin{equation}
\label{48}\beta _{\overrightarrow{k}_1\overrightarrow{k}_2\overrightarrow{k}
_3\overrightarrow{k}_4}=V\int f_{\overrightarrow{k}_4}^{*}f_{\overrightarrow{
k}_3}^{*}f_{\overrightarrow{k}_2}f_{\overrightarrow{k}_1}d^3\overrightarrow{r
},
\end{equation}
and
\begin{equation}
\label{49}
\begin{array}{c}
\alpha _{
\overrightarrow{k}_1\mu _1\overrightarrow{k}_2\mu _2\overrightarrow{k}_3\mu
_3\overrightarrow{k}_4\mu _4}=-i\sqrt{\frac{\hbar ^4\omega _1\omega _2\omega
_3\omega _4}{16V^2n^2\left( \omega _1\right) n^2\left( \omega _2\right)
n^2\left( \omega _3\right) n^2\left( \omega _4\right) }} \\ \\
x^{\left( 3\right) }\left( \omega _4=-\omega _3+\omega _2+\omega _1\right)
\stackrel{\cdot }{\stackunder{.}{:}}\overrightarrow{e}_{\overrightarrow{k}
_4\mu _4}^{*}\overrightarrow{e}_{\overrightarrow{k}_3\mu _3}^{*}
\overrightarrow{e}_{\overrightarrow{k}_2\mu _2}\overrightarrow{e}_{
\overrightarrow{k}_1\mu _1}
\end{array}
\end{equation}
The Hamiltonians (41) and (47) make the foundation for analyzing quantum
properties of the parametric or four-wave-mixing process.\\
{\bf Acknowledgment}
This project was supported by the National Nature Foundation of China.
\baselineskip 24pt
\end{document} |
\begin{document}
\begin{abstract}
In this paper, we show that any compact gradient $k$-Yamabe soliton must have constant $\sigma_{k}$-curvature. Moreover, we provide a certain condition for a compact $k$-Yamabe soliton to be gradient.
\end{abstract}
\title{TRIVIALITY RESULTS
FOR COMPACT $k$-YAMABE SOLITONS}
\section{Introduction and main results}
\label{intro}
The concept of gradient $k$-Yamabe soliton, introduced in the celebrated work \cite{catino2012global}, corresponds to a natural generalization of gradient Yamabe solitons. We recall that a Riemannian manifold $(M^n, g)$ is a \textit{$k$-Yamabe soliton} if it admits a constant $\lambda\in\mathbb{R}$ and a vector field $X\in \mathfrak{X}(M)$ satisfying the equation
\begin{equation}\label{def1}
\frac{1}{2}\mathcal{L}_{X}g=2(n-1)(\sigma_{k}-\lambda)g,
\end{equation}
where $\mathcal{L}_{X}g$ and $\sigma_{k}$ stand, respectively, for the Lie derivative of $g$ in the direction of $X$ and the $\sigma_{k}$-curvature of $g$. Recall that, if we denote by $\lambda_{1},\lambda_{2},\dots,\lambda_{n}$ the
eigenvalues of the symmetric endomorphism $g^{-1}A_{g}$, where $A_{g}$ is the Schouten tensor defined by
\begin{equation*}
A_{g}=\frac{1}{n-2}\left(Ric_{g}-\frac{scal_{g}}{2(n-1)}g\right),
\end{equation*}
then the $\sigma_{k}$-curvature of $g$ is defined as the $k$-th symmetric elementary function of $\lambda_{1},\dots,\lambda_{n}$, namely
\begin{equation*}
\sigma_{k}=\sigma_{k}(g^{-1}A_{g})=\sum_{i_{1}<\dots i_{k}}\lambda_{i_{1}}\cdot \dots\cdot \lambda_{i_{k}}, \quad \text{for}\quad 1\leq k\leq n.
\end{equation*}
Since $\sigma_{1}$ is the trace of $g^{-1}A_{g}$, the $1$-Yamabe solitons simply correspond to gradient Yamabe solitons \cite{chow1992yamabe,daskalopoulos2013classification, di2008yamabe, hamilton1988ricci, ma2012remarks,tokura2018warped}. For simplicity, the soliton will be denoted by $(M^{n}, g, X, \lambda)$. It may happen that $X=\nabla f$ is the gradient field of a smooth real function $f$ on $M$, in which case the soliton $(M^{n}, g, \nabla f, \lambda)$ is referred to as a \textit{gradient $k$-Yamabe soliton}. Equation \eqref{def1} then becomes
\begin{equation}\label{eq fundamental}
\nabla^2 f=2(n-1)(\sigma_{k}-\lambda)g,
\end{equation}
where $\nabla^2 f$ is the Hessian of $f$. Moreover, when either $f$ is a constant function or $X$ is a Killing vector field, the soliton is called \textit{trivial} and, in this case, the metric $g$ is of constant $k$-curvature $\sigma_{k}=\lambda$.
In recent years, much efforts have been devoted to study the geometry of $k$-Yamabe solitons. For instance, Hsu in \cite{hsu2012note} shown that any compact gradient $1$-Yamabe soliton is trivial. For $k>1$, the extension of the previous result was investigated by Catino et al. \cite{catino2012global}, and Bo et al. \cite{bo2018k}. In \cite{catino2012global}, the authors proved that any compact, gradient $k$-Yamabe soliton with nonnegative Ricci tensor is trivial. On the other hand, the authors in \cite{bo2018k} showed that any compact, gradient $k$-Yamabe soliton with constant negative scalar curvature must be trivial.
In this paper, we extend the above results as follows.
\begin{theorem}\label{T1}Any compact gradient $k$-Yamabe soliton $(M^n,g, \nabla f,\lambda)$ is trivial, i.e., has constant $\sigma_{k}$-curvature $\sigma_{k}=\lambda$.
\end{theorem}
In the scope of $k$-Yamabe solitons, we provide the following extension of Theorem 1.3 in \cite{bo2018k}.
\begin{theorem}\label{T4}The compact $k$-Yamabe soliton $(M^n,g,X,\lambda)$ is trivial if one of the following conditions holds:
\begin{itemize}
\item [\textup{(a)}] $k=1$.
\item [\textup{(b)}] $k\geq2$ and $(M^n,g)$ is locally conformally flat.
\end{itemize}
\end{theorem}
The Hodge-de Rham decomposition theorem (see \cite{aquino2011some, warner2013foundations}), shows that any vector field $X$ on a compact oriented Riemannian manifold $M$ can be decompose as follows:
\begin{equation}\label{Hodge} X = \nabla h + Y,
\end{equation}
where $h$ is a smooth function on $M$ and $Y\in \mathfrak{X}(M)$ is a free divergence vector field. Indeed, just consider the $1$-form $X^{\flat}$. Hence applying the Hodge-de Rham theorem, we decompose $X^{\flat}$ as follows:
\[X^{\flat}=d\alpha+\delta\beta+\gamma.\]
Taking $Y = (\delta\beta+\gamma)^{\sharp}$ and $(d\alpha)^{\sharp}=\nabla h$ we arrive at the desired result.
Now we notice that the same result obtained in \cite{pirhadi2017almost} for compact almost Yamabe solitons also works for compact $k$-Yamabe solitons. More precisely, we have the following theorem.
\begin{theorem}\label{T2}The compact $k$-Yamabe soliton $(M^n,g,X,\lambda)$ is gradient if, and only if,
\[\int_{M^n}Ric(\nabla h,Y)dv_{g} \leq0,\]
where $h$ and $Y$ are the Hodge-de Rham decomposition components of $X$.
\end{theorem}
As a consequence of Theorem \ref{T1} and Theorem \ref{T2}, we derive the following triviality result.
\begin{cor}\label{corr}Let $(M^n,g,X,\lambda)$ be a compact $k$-Yamabe soliton $(k\ge2)$ and $X=\nabla h+Y$ the Hodge-de Rham decomposition of $X$. If
\[\int_{M^n}Ric(\nabla h,Y)dv_{g}\leq0,\]
then $(M^n, g)$ is a trivial $k$-Yamabe soliton.
\end{cor}
An immediate consequence of the above corollary is the next result.
\begin{cor}\label{cor}Any compact $k$-Yamabe soliton $(M^n,g,X,\lambda)$ with $k\ge2$ and nonpositive Ricci curvature is trivial.
\end{cor}
Finally, taking into account the $L^{2}(M)$ orthogonality of the Hodge-de Rham decomposition, we obtain.
\begin{theorem}\label{1.6}Let $(M^n,g,X,\lambda)$ be a compact $k$-Yamabe soliton $(k\ge2)$ and $X=\nabla h+Y$ the Hodge-de Rham decomposition of $X$. If
\[\int_{M^n}g(\nabla h,X)dv_{g}\leq0,\]
then $(M^n, g)$ is a trivial $k$-Yamabe soliton.
\end{theorem}
\section{Proofs}
\begin{myproof}{Theorem}{\ref{T4}} If $k=1$, then $(M^n,g)$ is a Yamabe soliton and the result is well known from \cite{di2008yamabe}. Now, consider $k\geq2$ and suppose $(M^n,g)$ locally conformally flat. It was proved in \cite{han2006kazdan, viaclovsky2000some} that, on a compact, locally conformally flat, Riemannian manifold,
one has
\[\int_{M^n}g(\nabla \sigma_{k},X)dv_{g}=0,\]
for every conformal Killing vector field $X$ on $(M^n,g)$. From the structure equation \eqref{def1}, we know that $X$ is a conformal Killing vector field; hence, it follows that
\begin{equation}\label{9090}
0=\int_{M^n}g(\nabla \sigma_{k},X)dv_{g}=-\int_{M^n}\sigma_{k}(div X)dv_{g}=-2n(n-1)\int_{M^n}\sigma_{k}(\sigma_{k}-\lambda)dv_{g},
\end{equation}
where in the second equality we have used the divergence theorem. On the other hand, again from the divergence theorem, we obtain
\begin{equation}\label{8080}0=\int_{M^n}div X dv_{g}=2n(n-1)\int_{M^n}(\sigma_{k}-\lambda)dv_{g}.
\end{equation}
Jointly equations \eqref{9090} and \eqref{8080}, we conclude that
\[2n(n-1)\int_{M^n}(\sigma_{k}-\lambda)^2 dv_{g}=0,\]
which implies that $\sigma_{k}=\lambda$ and $\mathcal{L}_{X}g=0$. Hence $(M^n,g)$ is trivial.
\end{myproof}
\begin{myproof}{Theorem}{\ref{T1}} If $k=1$, then $(M^n,g)$ is a gradient Yamabe soliton and the result is well known from \cite{hsu2012note}. Now, consider $k\geq2$ and suppose by contradiction that $f$ is nonconstant. From Theorem 1.1 of \cite{catino2012global}, we obtain that $(M^n,g)$ is rotationally symmetric and $M^{n}\setminus \{N,S\}$ is locally conformally flat. Here $N,S$ corresponds to the extremal points of $f$ in $M$. From the structure equation \eqref{eq fundamental}, we know that $\nabla f$ is a conformal Killing vector field; hence, we can apply Theorem 5.2 of \cite{viaclovsky2000some} to deduce
\begin{equation}\label{1221}
0=\int_{M^{n}\setminus\{N,S\}}g(\nabla\sigma_{k},\nabla f)dv_{g}=\int_{M^n}g(\nabla\sigma_{k},\nabla f)dv_{g}=-2n(n-1)\int_{M^n}\sigma_{k}(\sigma_{k}-\lambda)dv_{g},
\end{equation}
where in the last equality we have used the divergence theorem. On the other hand, again from the divergence theorem, we get
\begin{equation}\label{808080}0=\int_{M^n}\Delta f dv_{g}=2n(n-1)\int_{M^n}(\sigma_{k}-\lambda)dv_{g}.
\end{equation}
Jointly equations \eqref{1221} and \eqref{808080}, we conclude that
\[2n(n-1)\int_{M^n}(\sigma_{k}-\lambda)^2 dv_{g}=0,\]
which implies that $\sigma_{k}=\lambda$ and $f$ is harmonic. Since $M^n$ is compact, $f$ is a constant, which leads to a contradiction. This proves that $f$ is constant.
\end{myproof}
\begin{myproof}{Theorem}{\ref{T2}}
From the Hodge-de Rham decomposition \eqref{Hodge}, we deduce that
\begin{equation}\label{t11111}\frac{1}{2}\mathcal{L}_{Y}g=\frac{1}{2}\mathcal{L}_{X}g-\frac{1}{2}\mathcal{L}_{\nabla h}g=2(n-1)(\sigma_{k}-\lambda)g-\nabla^{2}h.
\end{equation}
Therefore, to prove that $(M^n,g)$ admits a gradient $k$-Yamabe soliton structure, it is necessary and sufficient to show that $\mathcal{L}_{Y}g=0$. From \eqref{t11111}, we arrive that
\begin{equation}\label{t12}
\begin{split}
\frac{1}{4}\int_{M^n}|\mathcal{L}_{Y}g|^2 dv_{g}&=\int_{M^n}\left[4n(n-1)^2(\sigma_{k}-\lambda)^2-4(n-1)g\left(\nabla^2h,(\sigma_{k}-\lambda)g\right)+|\nabla^2 h|^2 \right]dv_{g}\\
&=\int_{M^n}\left[|\nabla^2 h|^2-4n(n-1)^2(\sigma_{k}-\lambda)^2\right]dv_{g}.
\end{split}
\end{equation}
We are going to compute the right-hand side of \eqref{t12} using the following identity
\begin{equation}\label{tt1}\int_{M^n}2Ric(\nabla h,Y)dv_{g}=\int_{M^n}\left[Ric(X,X)-Ric(\nabla h,\nabla h)-Ric(Y,Y)\right]dv_{g}.
\end{equation}
Taking the divergence of \eqref{t11111}, we get
\begin{equation}\label{t2}
\begin{split}
\frac{1}{2}div(\mathcal{L}_{Y}g)(Y)&=\frac{1}{2}div(\mathcal{L}_{X}g)(Y)-\frac{1}{2}div(\mathcal{L}_{\nabla h}g)(Y)\\
&=2(n-1)div(\sigma_{k}-\lambda)(Y)-\frac{1}{2}div(\mathcal{L}_{\nabla h}g)(Y)\\
&=2(n-1)g(\nabla\sigma_{k},Y)-\frac{1}{2}div(\mathcal{L}_{\nabla h}g)(Y).
\end{split}
\end{equation}
Hence, from the Bochner formula (see Lemma 2.1 of \cite{petersen2009rigidity}), we can express \eqref{t2} as follows
\begin{equation}\label{t4}
\frac{1}{2}\Delta|Y|^2-|\nabla Y|^2+Ric(Y,Y)=4(n-1)g(\nabla\sigma_{k},Y)-2Ric(\nabla h, Y)-2g(\nabla \Delta h,Y),
\end{equation}
and using the compactness of $M^n$, we arrive at
equation
\begin{equation}\label{dd}
\int_{M^n}2Ric(\nabla h,Y)dv_{g}=\int_{M^n}\left[|\nabla Y|^2-Ric(Y,Y)\right] dv_{g}.
\end{equation}
On the other hand, the same argument as above shows that
\begin{equation}\label{11}
\frac{1}{2}\Delta|X|^2-|\nabla X|^2+Ric(X,X)=-2(n-1)(n-2)g(\nabla \sigma_{k},X).
\end{equation}
Since
\begin{equation*}
\begin{split}
\int_{M^n}|\nabla X|^2dv_{g}&=\int_{M^n}\big{[}|\nabla^2 h|^2+|\nabla Y|^2+2g(\nabla\nabla h,\nabla Y)\big{]}dv_{g}\\
&=\int_{M^n}\big{[}|\nabla^2 h|^2+|\nabla Y|^2-2g(\nabla\Delta h+Ric(\nabla h), Y)\big{]}dv_{g}\\
&=\int_{M^n}\big{[}|\nabla^2 h|^2+|\nabla Y|^2-2Ric(\nabla h, Y)\big{]}dv_{g},\\
\end{split}
\end{equation*}
we may integrate \eqref{11} over $M^n$ to deduce
\begin{equation}\label{dd2}
\begin{split}
\int_{M^n}Ric(X,X)dv_{g}&=\int_{M^n}\left[|\nabla X|^2-2(n-1)(n-2)g(\nabla \sigma_{k},X)\right]dv_{g}\\
&=\int_{M^n}\big{[}|\nabla^2 h|^2+|\nabla Y|^2-2Ric(\nabla h, Y)+4n(n-1)^2\times\\
&\qquad\times(n-2)(\sigma_{k}-\lambda)^2\big{]}dv_{g}.\\
\end{split}
\end{equation}
Again, the same argument based on Lemma 2.1 of \cite{petersen2009rigidity}, allow us to deduce that
\begin{equation}\label{dd3}
\int_{M^n}Ric(\nabla h,\nabla h)dv_{g}=\int_{M^n}\left[4n^2(n-1)^2(\sigma_{k}-\lambda)-|\nabla^2 h|^2\right]dv_{g}.
\end{equation}
Now, replacing back \eqref{dd}, \eqref{dd2} and \eqref{dd3} into \eqref{tt1}, we get
\[\int_{M^n}\left[|\nabla^2 h|-4n(n-1)^2(\sigma_{k}-\lambda)^2\right]dv_{g}=\int_{M^n}Ric(\nabla h,Y)dv_{g},\]
which combining with \eqref{t12} produce the desired result.
\end{myproof}
\begin{myproof}{Theorem}{\ref{1.6}}Since the Hodge-de Rham decomposition is
orthogonal on $L^{2}(M)$, we get
\begin{equation*}
\int_{M^n}g(\nabla h, X)dv_{g}=\int_{M^n}g(\nabla h, \nabla h+Y)dv_{g}=\int_{M^n}|\nabla h|^{2}dv_{g}.
\end{equation*}
Therefore, if
\begin{equation*}
\int_{M^n}g(\nabla h, X)dv_{g}\leq0,
\end{equation*}
we obtain that $\nabla h=0$ and, consequently, $X=Y$. Now, since $Y$ is a free divergence vector field, we deduce
\[0=div Y=div X=2n(n-1)(\sigma_{k}-\lambda),\]
which implies that $\sigma_{k}=\lambda$ and $\mathcal{L}_{X}g=0$, hence, trivial.
\end{myproof}
\end{document} |
\begin{document}
\title{Proof of generalized Riemann hypothesis for Dedekind zetas and
Dirichlet L-functions}
{\bf Abstract}. A short proof of the generalized Riemann hypothesis
(gRH in short) for zeta functions $\zeta_{k}$ of algebraic number
fields $k$ - based on the Hecke's proof of the functional equation
for $\zeta_{k}$ and the method of the proof of the Riemann hypothesis
derived in [$M_{A}$] (algebraic proof of the Riemann hypothesis) is given.
The generalized Riemann hypothesis for Dirichlet L-functions is an
immediately consequence of (gRH) for $\zeta_{k}$ and suitable product
formula which connects the Dedekind zetas with L-functions.
\section{Introduction}
Let $k$ be an {\bf algebraic number field}, (i.e. the main half of the set
of {\bf global fields}), i.e. a finite algebraic extension of the
{\bf rational number field} $\mbox{\lll \char81}$. Let $R_{k}$ be a ring of
{\bf algebraic integers} in $k$ ,i.e. a finitely-generated ring
extension - the integral closure - of the ring of {\bf integers} $\mbox{\lll \char90}$. Then,
the {\bf Dedekind zeta function} $\zeta_{k}$ for $k$ is well locally defined (cf.e.g.
[K, Chapter 7], [L,VIII.2] and [N, VII]) as the {\bf Dirichlet series}
\begin{equation}
\zeta_{k}(s)\;:=\;\sum_{0\ne I\in {\cal I}_{k}}\frac{1}{N(I)^{s}}\;,\;Re(s)>1,
\end{equation}
where by $\mbox{\lll \char67}$ we denote the field of
all complex numbers and by $Re(s)$ and $Im(s)$ the {\bf real} and {\bf
imaginary} part of a complex number $s$, respectively. We denote the
{\bf group of all fractional ideals} of the {\bf Dedekind ring} $R_{k}$
by ${\cal I}_{k}$ (cf.e.g. [N]) and finally $N(I)$ denotes the {\bf absolute norm}
of the ideal $I$, i.e. the number of elements in $R_{k}/I$.
We remark at once that we only use classical
Dirichlet-Dedekind-Hecke theory, from the heroic period of German
mathematics, to obtain an exciting result : a proof of the
{\bf generalized Riemann Hypothesis}( $gRH_{k}$ in short) for algebraic number
fields $k$. Hecke theory posseses such depth, that its
classical tools are sufficient to obtain $(gRH_{k})$. For example,
probably one of the most characteristic properties of the theory of classical number
theory is that, one may embed a number field in the Cartesian
product of its completions at the {\bf archimedean points}, i.e. in a
Euclidean space. In more recent years (more precisely since Chevalley
introduced ideles in 1936, and Weil gave his adelic proof of the
Riemann-Roch theorem soon afterwards), it has been found most
convenient also to take the product over the {\bf non-archimedean points}, with
a suitable restriction on the components - the {\bf adele ring}
$\mbox{\lll \char65} _{k}$. However, we do not use the adele techique of {\bf Tate's
thesis} in this paper but stress {\bf Hecke's theory} and we do
not use the new achievements of algebraic number theory connected with
adeles and ideles.
When, we work with Dedekind zetas, it is surprising that at once we
obtain a very expanded apparatus of notions of the queen of mathematics -
algebraic number theory.
The main property of $\zeta_{k}$ is the existence of the following {\bf
Hecke - Riemann analytic continuation functional equation} (HRace in short,
cf.e.g. [L,XIII.3, Th.3])
\begin{displaymath}
\zeta_{k}^{*}(s)\;:=\;\frac{\mid d(k)\mid^{s/2}}{2^{r_{2}s}\pi^{ns/2}}
\Gamma(\frac{s}{2})^{r_{1}}\Gamma(s)^{r_{2}}
\zeta_{k}(s)\;=\;\frac{2^{r_{1}}h(k)R(k)}{w(k)s(s-1)} \;+\;
\end{displaymath}
\begin{displaymath}
\;+\;\sum _{0\ne I \in {\cal I}_{k}}\int_{\mid\mid y \mid\mid \ge 1}exp(-\pi
d(k)^{-1/n}N(I)^{2/n}Tr(y))[\Pi(y)^{s/2}\;+\;\Pi(y)^{(1-s)/2}]\frac{dy}{y}
\end{displaymath}
, where : $d(k)$ is the {\bf discriminant} of a field $k$
{(cf.e.g.[N,II.2]),}
$\Gamma(s) := \int_{0}^{\infty}e^{-x}x^{s-1}dx\;\;Re(s)>0$ is the
(classical) {\bf gamma function},
$r_{1}$ is the {\bf number of real embeddings} of $k$ into $\mbox{\lll \char67}$,
$r_{2}$ is {\bf half of the number of complex embeddings} of $k$ into $\mbox{\lll \char67}$,
(The pair $r=[r_{1},r_{2}]$ is called the {\bf signature} of $k$).
$h(k)$ is the {\bf class number} , $R(k)$ is the {\bf regulator of $k$}
(cf.[N, III.2]) and $w(k)$ is the number of {\bf roots of unity} lying
in $k$.
$S_{\infty}(k)$ denotes the set of {\bf archimedean absolute
values} of $k$,
$ n = n(k) = [k:\mbox{\lll \char81}]$ is the {\bf degree} of $k$ over $\mbox{\lll \char81}$,
$N_{v}(k)= N_{v}$ is the {\bf local degree} of $k$, which is 1 if $v$ is
a real point of $k$ and 2 if $v$ is a complex valuation from the set
$S_{\infty}(k)$.
Finally
$Tr_{k}(y):=\sum_{v\in S_{\infty}(k)}N_{v}y_{v}$ and $\prod( y ):= \prod_{v\in S_{\infty}(k)} y_{v}^{N_{v}}$.
From the topological point of view the answer to the question : where
are zeros and poles of $\zeta_{k}$ located - the algebraic number
theory {\bf characteristics ( arithmetics invariant)} : $d(k), r_{1}, r_{2},
n(k), h(k), R(k), w(k), S_{\infty}(k)$ - which appears in (HRace) - (as we
will show below) - are not so important, apart from the {\bf topological
invariants of $k$} , the signature $r(k)$, degree $n(k)$ and polynomial
$s(s-1)$.
For example, the invariants $h(k)
, R(k),w(k)$ and $r_{1}$ appear when we consider the {\bf residue value}
of $\zeta_{k}$ at the pole $s= 1$, but not when we consider the location of
the {\bf single pole} $\{1\} = I(\mbox{\lll \char67})\cap R(\mbox{\lll \char67})$, where the algebraic varietes $I(\mbox{\lll \char67})
:= \{s=u+iv\in \mbox{\lll \char67}: v(1-2u)=0\}$ and $R(\mbox{\lll \char67}):= \{s = u+iv\in \mbox{\lll \char67} :
u(u-1)-v^{2} =0\}$ {\bf do not even depend on} $k$. Moreover, for the
purposes of this paper it is only important that $h(k)$ is {\bf
finite}, but the value of $h(k)$ is not itself important.
More exactly, we derive an essential generalization of (HRace), where
the $n$-dimensional {\bf standard Gaussian function}
\begin{equation}
G_{n}(x)\;:=\; e^{-\pi \mid \mid x \mid \mid_{n}^{2}}\;,\; x \in \mbox{\lll \char82}^{n}
\end{equation}
(here $\mid\mid .\mid\mid_{n}$ is the {\bf Euclidean norm} on $\mbox{\lll \char82}^{n}$ and
obviously here, and all in the sequel, $\mbox{\lll \char82}$ stands for the field of real
numbers), will be replaced by any smooth fixed point of ${\cal F}_{n}$.
The function $G_{n}$ is a {\bf fixed point} of the {\bf Fourier transform}
${\cal F}_{n}$ on the {\bf Schwartz space} ${\cal S}(\mbox{\lll \char82}^{n})$ of smooth and
rapidly decreasing functions. If we replace $G_{n}$ by any other fixed point
$\omega_{+}$ of ${\cal F}_{n}$ from ${\cal S}(\mbox{\lll \char82})$, then we can extend the
(HRace) to the {\bf Fixed point Hecke Riemann analytic continuation equation}
(Face in short) (cf. Section 2).
The idea of the generalization of (HRace) to (Face) is, in some small sense
very similar to {\bf Grothendieck's} magnificent idea of the
generalization of the notion of set theory topology to category
topologies (e.g. the well-known {\bf etale cohomologies}) - to obtain
the required results : to prove $(gRH_{k})$ in our case and to prove the
{\bf Riemann hypothesis} for {\bf congruence Weil zetas}, respectively.
The following very important {\bf rational function} ( the {\bf polar-zero
part}) appears in HRace.
\begin{equation}
\;\;W_{k}(s)\;:=\;\frac{\lambda_{k}}{s(s-1)}(\;:=\;\frac{2^{r_{1}}h(k)R(k)}
{w(k)s(s-1)})\;;\;s\in \mbox{\lll \char67}.
\end{equation}
Hence, in $W_{k}(s)$ is written a very important polynomial $I$ of
two variables, which does not depend on $k$! , with coefficients in $\mbox{\lll \char90}$:
\begin{equation}
\;\;I(s)\;:=\;Im(W_{k}(s))\mid s(s-1)\mid^{2}/\lambda_{k}\;=\;v(2u-1);\;
s=u+iv,\;u,v \in \mbox{\lll \char82}.
\end{equation}
The function $I(s)$ is mainly responsible for the form of the {\bf
generalized Riemann Hypothesis} for $\zeta_{k}$ ($(gRH_{k})$ in short), i.e.
the following well-known implication ( as in the case of the Riemann
hypothesis , cf.[$M_{A}$]):
\begin{displaymath}
(gRH_{k}) \;\;If\;\zeta_{k}(s)=0\;and\;Im(s)\ne 0\;, then\;Re(s)=1/2 .
\end{displaymath}
According to (1.4) , the following {\bf Trivial Riemann Hypothesis} ((TRH)
in short) holds:
\begin{equation}
(TRH)\; If\; I(s) = 0\; and\; Im(s) \ne 0\;, then \;Re(s) =1/2.
\end{equation}
As in [$M_{A}$] we pose the following {\bf Algebraic conjecture
for $\zeta_{k}$}:
\begin{displaymath}
(TRH)\; implies\; (gRH_{k}).
\end{displaymath}
More exactly, let us consider the {\bf algebraic $\mbox{\lll \char82}$-variete} $I(\mbox{\lll \char67})
:= \{s\in \mbox{\lll \char67} : I(s) = 0\}$ and the zero-dimensional {\bf holomorphic
manifold} $\zeta_{k}(\mbox{\lll \char67}) := \{s \in \mbox{\lll \char67} : \zeta_{k}(s) = 0\}$. Then
the Riemann hypothesis $(gRH_{k})$ is a kind of relation between the
cycles (of $\mbox{\lll \char82}^{2}$ and $\mbox{\lll \char67}$, respectively) : $I(\mbox{\lll \char67})$ (which does
{\bf not depend} on $k$) and $\zeta_{k}(\mbox{\lll \char67})$, i.e.
\begin{displaymath}
\;\;\;\zeta_{k}(\mbox{\lll \char67})\;\subset \;I(\mbox{\lll \char67}).
\end{displaymath}
In the sequel, the bi-affine-linear form $I(u,v)$ of two real variables,
we call the {\bf fundamental form} of the class
$\{\zeta_{k} : k\:\;is\;an\;algebraic\;number\;field \}$.
Thus, {\bf topological information} on the isolated points of the
meromorphic function $\zeta_{k}$ is written - in fact - in the
algebraic varieties $I(\mbox{\lll \char67})$ and $I(\mbox{\lll \char67})\cap R(\mbox{\lll \char67})$, and therefore
there exists some {\bf unexpected} (and hence {\bf deep}) relation
between the {\bf arithmetic} of $I \in \mbox{\lll \char90}[u,v]$ over $\mbox{\lll \char82}$ and the
{\bf arithmetic} of $\zeta_{k}$ over $\mbox{\lll \char67}$. Moreover, the "serious"
$(gRH_{k})$ could be reduced to the formal consequence of the "non-serious"
(TRH) by calculating different kinds of integrals ( with respect to
different {\bf Haar measures}), which leads to the {\bf
subsequence} functional equation : let $Gal(\mbox{\lll \char67}/\mbox{\lll \char82}) = \{id_{\mbox{\lll \char67}}, c\}$ be
the {\bf Galois
group} of $\mbox{\lll \char67}$, i.e. $id_{\mbox{\lll \char67}}$ denotes the identity automorphism of
$\mbox{\lll \char67}$ and $c$ is the {\bf complex conjugation} automorphism :
\begin{equation}
\;\;\;c(z)\;=\;c(u+iv)\;:=\;u\;-\;iv ,
\end{equation}
which is an {\bf idempotent map}, i.e. $c^{2} = id_{\mbox{\lll \char67}}$.
The following {\bf generalized Riemann hypothesis functional equation}
($(gRhfe_{k})$ in short) with a {\bf rational term} $I$ and the {\bf
action} of $Gal(\mbox{\lll \char67}/\mbox{\lll \char82})$ indicates some "hidden" Galois symmetry of
$\zeta_{k}$ :
\begin{displaymath}
(gRhfe_{k})\;\;\;Im(\sum_{g \in
Gal(\mbox{\lll \char67}/\mbox{\lll \char82})}(F_{g}\zeta_{k})(g(s)))\;=\;\frac{\lambda_{k}(f_{1}(s)
-f_{2}(s))I(s)}{\mid s(s-1) \mid^{2}} \;,\;Re(s)\in [0,1/2).
\end{displaymath}
In opposite to the $(gRhfe_{k})$ , the (HRace) gives an "open symmetry" of
$\zeta_{k}^{*}$ :
\begin{equation}
\zeta_{k}^{*}(s)\;=\;\zeta_{k}^{*}(1-s).
\end{equation}
As in the case of the Riemann hypothesis, the functional equation
$gRhfe_{k}$ - immediately implies the generalized Riemann hypothesis for the
Dedekind zetas due to TRH.
In comparison to $[M_{A}]$, we have significantly shorted the technicality of
the proof of the theorem on existence of $n$-dimensional {\bf RH-fixed points}.
We consider the non-commutative field of {\bf quaternions} $\mbox{\lll \char72}$, endowed with
the Hilbert transform ${\cal H}_{\mbox{\lll \char72}}$ of a measure $\mu$ (see Sect.3)
\begin{displaymath}
({\cal H}_{\mbox{\lll \char72}}\mu)(h)\;:=\; \int_{\mbox{\lll \char72}}\frac{d\mu (x)}{\mid \mid h-x
\mid\mid_{4}^{4}},
\end{displaymath}
and the product ring (with zero divisors) $\mbox{\lll \char81}_{p}\times \mbox{\lll \char81}_{q}$ of
different p-adic number fields endowed with the Hilbert transform
${\cal H}_{pq}$ :
\begin{displaymath}
({\cal H}_{pq}\mu)(a)\;:=\;\int_{\mbox{\lll \char81}_{p}\times \mbox{\lll \char81}_{q}}\frac{d
\mu(x)}{\Delta_{pq}(a-x)}.
\end{displaymath}
Thus, using the techniques used in [$M_{A}$] for the proof of the Riemann
hypothesis, we show that our method initiated in that article works and can
be significantly extended to the general case : this technique of RH-fixed
points - leads to the proof of the generalized Riemann hypothesis for Dedekind
zetas and Dirichlet L-functions.
The constructions in Section 3 are much more abstract in comparing to
$[M_{A}]$ and much simpler. Moreover, these construct are interesting in
themselves, since they (and in some sense return) to fundamental
problems raused at the beginning of the 20th century.
The "heart" of the proof of RH from $[M_{A}]$ moving (practically
without any changes) for $gRH_{k}$.
\section{Fixed point Hecke-Riemann functional continuation equations}
These two chapters achieve two goals simultaneously. We present here all the
necessary preliminaries and notation. Next, we state the extension of (HRace)
to (Face). Secondly, the main technical tool - and in fact - the "heart of
the paper" , is Theorem 2 on the existence of multidimensional RH-fixed
points. Moreover, we comnent on a surprising property of the construction
mentioned: that it violates the Tertium non Datur in the
case, when the {\bf amplitude} $A$ has a {\bf support outside a set
of Lebesgue measure zero}.
Let $n \in \mbox{\lll \char78}^{*}:=\mbox{\lll \char78}-\{0\}$ be arbitrary (in all the sequel $\mbox{\lll \char78}^{*}$
denotes the set of all positive integers). In the sequel $n=n(k)$ will
always be considered as the degree of a fixed algebraic number field
$k$, i.e. $n = [k:\mbox{\lll \char81}]$.
Exactly $n$ different embeddings of $k$ into the complex field $\mbox{\lll \char67}$
exists. Indeed, by {\bf Abel's theorem} $k$ can be written in the
form $k = \mbox{\lll \char81}(a)$ for a suitable {\bf algebraic} $a$.
If $a_{1}, ... ,a_{n}$ are all complex roots of the {\bf minimal
polynomial} for $a$ over $\mbox{\lll \char90}$, then the mappings $C_{j}, j=1, ... ,n$ (
the {\bf conjugates of $k$}) defined by
\begin{equation}
C_{j}(\sum_{k=0}^{n-1}A_{k}a^{k})\;:=\;\sum_{k=0}^{n-1}A_{k}a_{j}^{k}
\end{equation}
(for $A_{0}, ... , A_{n-1} \in \mbox{\lll \char81}$) are all isomorphisms of $k$ {\bf
into} $\mbox{\lll \char67}$, and every such isomorphism has to be of this form.
The fields $C_{j}(k)$ are called the {\bf fields conjugated} with $k$.
If $C_{j}(k) \subset \mbox{\lll \char82}$, then it is called a {\bf real embedding} and
otherwise $C_{j}(k)$ is called a {\bf complex embedding}.
Note that if $C_{j}$ is {\bf complex}, then $c \circ C_{j}$ is again an
embedding, complex of course, and so the number of complex embeddings
is {\bf even}.
The number of such pairs of embeddings is usually denoted by $r_{2}(k) =r_{2}$
, and the number of {\bf real embeddings} by $r_{1}(k) = r_{1}$.
The pair $r = r(k) =[r_{1}, r_{2}]$ is called the {\bf signature} of $k$
(cf.e.g. [N, II.1])
We denote the {\bf Lebesgue measure} on $\mbox{\lll \char82}^{n}$ , and the {\bf Lebesgue
measure} of $\mbox{\lll \char67}^{n}$ by $d^{n}x$ and $d^{n}z$, respectively.
If $r=r(k) =[r_{1},r_{2}]$ is the {\bf signature of $k$}, then we
define the {\bf signature group} $G_{r}$ of $k$ as the product
\begin{equation}
G_{r}:=\mbox{\lll \char82}_{+}^{r_{1}}\times(\mbox{\lll \char67}^{*})^{r_{2}},
\end{equation}
of $r_{1}$ - exemplars of the multiplicative group $\mbox{\lll \char82}^{*}_{+}$ of
{\bf positive real numbers} and $r_{2}$-exemplars of the {\bf
multiplicative group of complex numbers} $\mbox{\lll \char67}^{*}$.
Obviously, $G_{r}$ is a Locally Compact Abelian group (LCA in short).
Hence, the {\bf Haar measure} is well defined. Its {\bf standardly normalized
Haar measure} will be denoted by $H_{r}$. It is well-known that $H_{r}$
is the product of the form :
\begin{equation}
dH_{r}(g)=\frac{d^{r_{1}}x}{\mid x \mid}\otimes
\frac{d^{r_{2}}z}{\mid z
\mid^{2}}=\otimes_{i=1}^{r_{1}}\frac{dx_{i}}{x_{i}}\otimes_{j=1}^{r_{2}}
\frac{dz_{j}}{\mid z_{j} \mid^{2}}.
\end{equation}
The signature group $G_{r}$ is obviously the multiplicative subgroup of the
{\bf Euclidean ring}
\begin{equation}
E_{r}\;:=\;\mbox{\lll \char82}^{r_{1}}\times \mbox{\lll \char67}^{r_{2}}\;\simeq \mbox{\lll \char82}^{n},
\end{equation}
with the componentwise multiplication. It is obviously a ring with divisors
of zero.
In particular, $E_{r}$ has got the {\bf Haar module} $\Delta_{r} = mod_{r}$
with the property
\begin{equation}
\Delta_{r}(g)\;=\;mod_{r}(g)\;=\;\prod_{i=1}^{r_{1}}\mid x_{i}\mid
\prod_{j=1}^{r_{2}}\mid z_{j} \mid^{2} \;,\;g=(x_{1}, ...
,x_{r_{1}},z_{1}, ... ,z_{r_{2}}).
\end{equation}
is well defined on $E_{r}$.
Moreover
\begin{equation}
dH_{r}(g)\;=\;\frac{d^{r_{1}}x \otimes d^{r_{2}}z}{mod_{r}(g)}.
\end{equation}
We denote the {\bf $mod_{r}$-unit sphere} of $G_{r}$ by $G_{r}^{0}$, i.e.
\begin{equation}
G_{r}^{0}\;:=\;\{g \in G_{r} : mod_{r}(g) =1\}.
\end{equation}
It is an elementary fact that we can write $G_{r}$ as the product
\begin{equation}
G_{r}\;=\;\mbox{\lll \char82}^{*}_{+} \times G_{r}^{0},
\end{equation}
because any $g \in G_{r}$ can be written uniquely as
\begin{equation}
g\;=\;t^{1/n}c
\end{equation}
with $t \in \mbox{\lll \char82}^{*}_{+}$ and $c \in G_{r}^{0}$. Here $c = \{c_{v}\}$
and $t^{1/n}c := (mod_{r}(c)^{1/n}\cdot (\frac{c_{v}}{mod_{r}c}))$.
We denote the {\bf Haar measures} of $G_{r}^{0}$ by $H_{r}^{0}$.
According to (2.15), the Haar measure $H_{r}$ can be considered as the
product of the {\bf Lebesgue measure} $dt/t$ on $\mbox{\lll \char82}^{*}_{+}$ and the
appropriate {\bf Haar measure} $H_{r}^{0}$ on $G_{r}^{0}$.
For a large class of {\bf $\Gamma_{r}$-admissible} functions
$f:G_{r}\longrightarrow \mbox{\lll \char67}$ the ($n$-dimensional) {\bf Mellin
transform} $M_{n}(f)$ or rather the {\bf signature Gamma} $\Gamma_{r}(f)$
(associated with $f$) is well-defined as
\begin{equation}
\Gamma_{r}(f)(s)\;:=\;\int_{G_{r}}mod_{r}^{s}(g)f(g)dH_{r}(g)
\;=:\;M_{n}(f)(s)\;,\;Re(s)>0.
\end{equation}
Recall that $f:\mbox{\lll \char82}^{n}\longrightarrow \mbox{\lll \char67}$ belongs to the {\bf Schwartz
space} ${\cal S}(\mbox{\lll \char82}^{n})$ of rapidly decreasing functions, if for each
$n$-tuple of integers $\ge 0, k=(k_{1}, ... ,k_{n})$ and $l = (l_{1},
... , l_{n})$
\begin{displaymath}
\;\;\;p_{k,l}(f)\;:=\;sup_{x\in \mbox{\lll \char82}} \mid x^{k}(D^{l}f)(x)\mid
<+\infty,
\end{displaymath}
where $x^{k} := x_{1}^{k_{1}}...x_{n}^{k_{n}}$ and $D^{l} :=
D_{1}^{l_{1}}... D_{n}^{l_{n}}$, is a partial differential
operator.
It is easy to check (cf.e.g. [$M_{A}]$, Sect.2, Lemma1]) that the
following holds for $f \in {\cal S}(\mbox{\lll \char82}^{n})$ :
\begin{equation}
\;\;\;\Gamma_{n}(f)(s)\;\in \;\mbox{\lll \char67}\;\;if\;\;Re(s)>0,
\end{equation}
since ${\cal S}(\mbox{\lll \char82})\otimes ... \otimes {\cal S}(\mbox{\lll \char82})$ (n-times), is
dense in ${\cal S}(\mbox{\lll \char82}^{n})$.
We denote the ($n$-dimensional) {\bf Fourier transform} of $f$ by
${\cal F}_{n}f$ (for ${\cal F}$-admissible functions):
\begin{equation}
\;\;{\cal F}_{n}(f)(x)\;:=\;\int_{\mbox{\lll \char82}^{n}}e^{2\pi
ixy}f(y)d^{n}y\;=:\;\hat{f}(x)\;;\;x \in \mbox{\lll \char82}^{n} ,
\end{equation}
where $xy := \sum_{k=1}^{n}x_{i}y_{i}$ is the standard euclidean scalar product
of $n$-vectors $x = (x_{1}, ... ,x_{n})$ and $y = (y_{1},... ,y_{n})$.
In this paper, it is also very convenient to use the (1-dimensional)
{\bf plus-Sin transform} defined as
\begin{equation}
\;\;\;S_{+}(f)(x)\;:=\;\int_{0}^{+\infty}sin(xy)f(y)dy
\;=:\;\hat{f}_{+}(x)\;:\;x \in \mbox{\lll \char82}_{+}.
\end{equation}
For another large class of {\bf $\theta$-admissible} functions
$f: G_{r} \longrightarrow \mbox{\lll \char67}$ ($n$-dimensional or signatural), the{\bf Jacobi
theta function $\theta_{r}(f)$ associated with $f$} is defined as the series
\begin{equation}
\theta_{r}(f)(x)\;:=\;\sum_{k \in (\mbox{\lll \char78}^{*})^{n}}f(k \cdot
x)\;=\;\int_{(\mbox{\lll \char78}^{*})^{n}} f(k \cdot x)dc(x)\;,x\in \mbox{\lll \char82}_{+}^{n},
\end{equation}
where $k \cdot x$ denotes componentwise multiplication in $E_{r}$
and $dc$ is the {\bf calculating measure} on $(\mbox{\lll \char78}^{*})^{n}$ , i.e. the
unique Haar measure on $\mbox{\lll \char90}^{n}$ normalized by the condition :
$c(\{0\})=1$.
Beside the field $\mbox{\lll \char67}$, we will also use the non-commutative field of
quaternions $\mbox{\lll \char72}$. It is well-known (cf.e.g. [W]) that the formula
\begin{equation}
\Delta_{\mbox{\lll \char72}}(h)\;:=\;\mid\mid h \mid\mid_{4}^{4} \;,h \in \mbox{\lll \char72} ,
\end{equation}
defines the {\bf Haar module} of $\mbox{\lll \char72}$.
For a class of some ${\cal H}$-admissible measures defined on a compact
subset $C$ of $\mbox{\lll \char72}$, we define the (compact) $\mbox{\lll \char72}$-{\bf Hilbert transform}
${\cal H}_{\mbox{\lll \char72}}$ by the formula
\begin{equation}
({\cal H}_{\mbox{\lll \char72}}\mu)(h)\;:=\;\int_{C}\frac{d\mu(x)}{\Delta_{\mbox{\lll \char72}}(h-x)}\;,\;h
\in \mbox{\lll \char72}.
\end{equation}
Finally, we use the product ring $\mbox{\lll \char81}_{p}\times \mbox{\lll \char81}_{q}$ with zero
divisors of different p-adic number fields. It is well-known that the
formula
\begin{displaymath}
\Delta_{pq}(x_{p},x_{q})\;:=\;\mid x_{p} \mid_{p} \mid x_{q}
\mid_{q}\;,(x_{p},x_{q})\in \mbox{\lll \char81}_{pq},
\end{displaymath}
defines the {\bf Haar module} of $\mbox{\lll \char81}_{pq}$ and the formula
\begin{displaymath}
({\cal H}_{pq}\mu)(a)\;:=\;\int_{\mbox{\lll \char81}_{p}\times \mbox{\lll \char81}_{q}} \frac{d
\mu(x)}{\Delta_{pq}(a\;-\;x)}\;,\; a \in \mbox{\lll \char81}_{pq},
\end{displaymath}
defines pq-Hilbert transform.
Finally, we note that the Schwartz spaces ${\cal S}(\mbox{\lll \char82}^{n})$ are
{\bf admissible} for all the integral transforms defined above :
$\Gamma_{r}, {\cal F}_{n}, \theta_{r}$ and ${\cal H}$ ( for absolutely
continuous measures $\mu$ w.r.t. Lebesgue measure $d^{4}h$ and the Haar
measure $dH_{pq}$ of $\mbox{\lll \char81}_{pq}$, considered as densities of signed measures).
One of the main tools when we work with zetas is the {\bf Poisson
Summation Formula} (PSF in short, cf.e.g. [N], [L, XIII.2]) , which
shows that ${\cal F}_{n}$ is a $l^{1}(\mbox{\lll \char90})$-{\bf quasi-isometry} on
${\cal S}(\mbox{\lll \char82}^{n})$ and using our notation can be written as :
\begin{displaymath}
(PSF)\;\;\;\int_{\mbox{\lll \char90}^{n}}\hat{f}(x)dc(x)\;=\;\int_{\mbox{\lll \char90}^{n}}
f(x)dc(x),
\end{displaymath}
if $f \in {\cal S}(\mbox{\lll \char82}^{n})$.
A complex function $\omega_{+}$ on $\mbox{\lll \char82}^{n}$ ($n = r_{1}+2r_{2}$) is called
a {\bf fixed point of ${\cal F}_{n}$}, if it is an {\bf eigenvector} of
${\cal F}_{n}$ with the corresponding {\bf eigenvalue} equal to 1, i.e.
\begin{equation}
{\cal F}_{n}(\omega_{+})\;=\;\hat{\omega_{+}}\;=\;\omega_{+}.
\end{equation}
Analogously a complex valued function $\omega_{-}$ on $\mbox{\lll \char82}^{n}$ is called
the {\bf -fixed point} of ${\cal F}_{n}$ if it is an {\bf eigenvector} of
${\cal F}_{n}$ corresponding to the {\bf eigenvalue} $-1$ of ${\cal
F}_{n}$ :
\begin{displaymath}
{\cal F}_{n}(\omega_{-})=\hat{\omega_{-}}=-\omega_{-}.
\end{displaymath}
We use the common name for $+$fixed points and $-$fixed point - the
{\bf $\pm$fixed points} $\omega_{\pm}$.
Let $\omega = \omega_{\pm}$ be a {\bf $\pm$fixed point} of ${\cal F}_{n}$
from ${\cal S}(\mbox{\lll \char82}^{n})$ and let $M =[m_{ij}]_{n\times n}$ be a {\bf matrix}
of real numbers.
Let us consider the function
\begin{displaymath}
\omega_{M}(x)\;:=\;\omega_{\pm}(Mx^{t})\;\;;\;\;x^{t} \in \mbox{\lll \char82}^{n},
\end{displaymath}
and the theta associated with it
\begin{equation}
\theta_{n}(\omega_{M})(x)\;:=\;\sum_{m\in \mbox{\lll \char90}^{n}} \omega_{M}(mx)\;,\;x \in
\mbox{\lll \char82}^{n}.
\end{equation}
\begin{lem}({\bf Hecke's theta formula})
For each non-singular matrix $M$ the following relation holds
\begin{displaymath}
(HTF)\;\;\;\theta_{n}(\omega_{M}^{\pm})(x)\;\;=\;\;\pm \theta_{n}
(\omega_{^{t}M^{-1}}^{\pm})(x)/\mid det(M) \mid.
\end{displaymath}
\end{lem}
{\bf Proof}. Let $M = [m_{ij}]_{n\times n}$ and $\omega_{M}(x) :=
\omega(Mx^{t}), x \in \mbox{\lll \char82}^{n}$. If $M$ is a non-singular real matrix, then
$det(M)\ne 0$. Thus the function $\omega_{M}$ is also in ${\cal
S}(\mbox{\lll \char82}^{n})$ and using the change of variables formula for multiple
integrals, we immediately find that its Fourier transform is
given by
\begin{displaymath}
\hat{\omega_{M}^{\pm}}(x)\;=\;\pm \frac{\omega(^{t}M^{-1}x^{t})}
{\mid det(M) \mid},
\end{displaymath}
where $^{t}M^{-1}$ is the transpose of the inverse of $M$.
This is clear, since when we make the change of variables $z = Mx^{t}$, we
have $dz =\mid det(M) \mid dx$, and $<M^{-1}z^{t}, y> = <z, ^{t}M^{-1}y^{t}>$.
The first important step in the proof of $(gRH_{k})$ is
the {\bf generalization of the Hecke-Riemann analytic continuation
eqation} ((HRace) in short), given below. Therefore we need some additional
notation.
Let us again consider the signature {\bf Euclidean ring}
\begin{displaymath}
E_{r}\;=\;\mbox{\lll \char82}^{r_{1}}\times \mbox{\lll \char67}^{r_{2}}\;\simeq \mbox{\lll \char82}^{n},
\end{displaymath}
and the {\bf conjugation} map $C : k \longrightarrow E_{r}$ defined as
\begin{equation}
C(\xi)\;:=\;(C_{1}(\xi), ... , C_{n}(\xi))\;;\xi \in k.
\end{equation}
Let us observe that each conjugate $C_{v}$ determines the {\bf absolute
value} (place) $v$ of $k$ by the formula :
\begin{equation}
v(\xi)\;:=\;\mid C_{v}(\xi) \mid\;;\;\xi \in k.
\end{equation}
The {\bf completion} of $(k,v)$ is denoted by $k_{v}$. Since $v$ is
{\bf archimedean}, $k_{v}$ is equal to $\mbox{\lll \char82}$ or $\mbox{\lll \char67}$.
In the case : $k_{v} \simeq \mbox{\lll \char67}$ the completion is determined up to
{\bf complex conjugation} $c$, according to the well-known elementary
fact that if $\sigma \in Gal(\mbox{\lll \char67}/\mbox{\lll \char82})$, then
\begin{equation}
v(\sigma(z))\;=\;v(z)\;,\;z \in \mbox{\lll \char67}
\end{equation}
(cf.e.g. [L, II.1] and [N, L.3.1]).
We denote the set of all {\bf non-equivalent archimedean places} of $k$
by $S_{\infty}(k)$. According to (2.28), it is obvious that
\begin{displaymath}
\mid S_{\infty}(k)\mid \;=\;r_{1}\;+\;r_{2}.
\end{displaymath}
Let us consider the map $\mid C \mid : k \longrightarrow \mid E_{r}
\mid\;:=\;\mbox{\lll \char82}^{r_{1}+r_{2}}$ defined as
\begin{equation}
\mid C \mid (\xi)\;:=\;(\mid C_{v}(\xi) \mid :v \in S_{\infty}(k)).
\end{equation}
Recall that $G_{r} = (\mbox{\lll \char82}^{*}_{+})^{r_{1}}\times (\mbox{\lll \char67}^{*})^{r_{2}}$.
So, if we denote : $\mid G_{r}\mid := (\mbox{\lll \char82}_{+}^{*})^{r_{1}+r_{2}}$, then
we have the decomposition
\begin{equation}
G_{r}\;\simeq \;\mid G_{r}\mid \times \mbox{\lll \char84}^{r_{2}},
\end{equation}
where $\mbox{\lll \char84} :=\{z \in \mbox{\lll \char67} : \mid z \mid =1\}$ is the 1-dimensional
torus.
The kernel of $\mid C \mid$, i.e. $\mu(k) := ker(\mid C \mid)$ is the
{\bf group of the roots of unity} in $k$. Let
\begin{equation}
w(k)\;=\;\# \mu(k)\;=\;\mid \mu(k) \mid,
\end{equation}
be the {\bf number of roots of unity in} $k$.
Let $U(k)$ be the {\bf group of units of $k$} ($S_{\infty}(k)-{\bf
units})$), i.e.
\begin{displaymath}
U(k)\;=\;R_{k}^{*} .
\end{displaymath}
Let $V(k) \;:=\;\mid C \mid(U(k))$ be the {\bf image} of $U(k)$ under
the mapping $\mid C \mid$. Its image $V(k)$ is contained in the subgroup
$\mid G_{r}^{0} \mid$ consisting of all $g \in \mid G_{r} \mid$ such that
$mod_{r}(g) = 1$, and is a {\bf discrete subgroup}. Furthermore, $\mid
G_{r}^{0} \mid/V(k)$ is {\bf compact} (cf. [L,p.256]). Also, we can
write $G_{r}$ as the product
\begin{displaymath}
G_{r}\;=\;\mbox{\lll \char82}_{+}^{*}\times \mid G_{r}^{0} \mid \times \mbox{\lll \char84}^{r_{2}}.
\end{displaymath}
Finally, let $E(k)$ be the {\bf fundamental domain for $V^{2}(k)$} in
$\mid G_{r}^{0} \mid$ (cf. [L]). We obtain the following {\bf disjoint
decomposition}
\begin{equation}
\mid G_{r}^{0} \mid \;=\;\cup_{\eta \in V} \eta^{2} E(k).
\end{equation}
Let $A$ be an arbitrary {\bf integral (fractional) ideal} of $k$. Then
, it is well-known that $A$ has an {\bf integral basis} over $\mbox{\lll \char90}$
(cf.[N, Th.2.4]). Thus, each $\xi \in A$ can be written as
\begin{equation}
\xi\;=\;x_{1}\alpha_{1}\;+\;...\;+\;x_{n}\alpha_{n}\;\;,\;\;x_{i}\in
\mbox{\lll \char90}.
\end{equation}
For $v \in S_{\infty}(k)$ we let $C_{v}$ be the embedding (conjugate)
of $k$ in $k_{v}$, identified with $\mbox{\lll \char82}$ or $\mbox{\lll \char67}$ (in the case of $\mbox{\lll \char67}$,
we fix one identification, which otherwise is determined only up to
conjugacy). We will write
\begin{displaymath}
\xi_{v}\;=\;C_{v}(\xi)\;=\;\sum_{j=1}^{n}x_{j}C_{v}(\alpha_{j})
\end{displaymath}
and
\begin{displaymath}
C(A)\;:=\;[N(A)^{-1/n}C_{i}(\alpha_{j})] \;,i,j=1,...,n.
\end{displaymath}
Hence, $N(A)^{-1/n}[\xi_{1}, ... ,\xi_{n}] = C(A)[x_{1}, ... , x_{n}]^{t}$
and we also use this same notation when we constrict $x_{i}$ to the set of
real numbers.
Let ${\cal R}$ be an class of ideals of the ordinary ideal class group $H(k)
:={\cal I}_{k}/P_{k}$. Let $A$ be an ideal in ${\cal R}^{-1}$. The map
\begin{equation}
B \longrightarrow AB\;=\;(\xi)
\end{equation}
eatablishes a bijection between the set of ideals in ${\cal R}$ and
equivalence classes of non-zero elements of $A$ : $A/\sim_{u}$, where two
field elements are called {\bf equivalent} $\sim_{u}$, if they differ by a
{\bf unit}.
Let $R(A)$ be a set of {\bf representatives} for the non-zero
equivalence classes.
Finally, we introduce two thetas - small and capital : the {\bf small Jacobi
theta of $k$} (associated with $\omega$)
\begin{equation}
\theta_{k}(\omega)(g)\;\;:=\;\;\sum_{0\ne I \in {\cal I}_{k}}\sum_{\xi
\in R(I)}\sum_{u \in U(k)}\omega(u \xi g) \;=\;
\end{equation}
\begin{displaymath}
\;=\;\sum_{0\ne I \in {\cal I}_{k}} \sum_{x \in \mbox{\lll \char90}^{n}}
\theta_{n}(\omega_{C(I)})(g)\;;\;g\in G_{r},
\end{displaymath}
and the {\bf radial Jacobi theta} of $k$
\begin{displaymath}
\Theta_{k}(\omega)(t)\;:=\;\frac{\int_{E(k)}\theta_{k}(\omega)(ct^{1/n})dH_{r}
^{0}(c)}{w(k)}\;,\;t \in \mbox{\lll \char82}_{+}^{*}.
\end{displaymath}
\begin {th}({\bf Fixed point HRace = Face})
The following functional equation holds for each $\pm$fixed point
$\omega_{\pm}$ of ${\cal F}_{n}$ from ${\cal S}(\mbox{\lll \char82}^{n})$, with the property that
$\Gamma_{r}(\omega_{\pm})$ {\bf does not vanishes}, and for each $s$ with $Re(s)>0$
\begin{equation}
(Face)\;\;(\Gamma_{r}(\omega_{\pm})\zeta_{k})(s)\;=\;\frac{\lambda_{k}\ne 0}
{s(s-1)}\;+\;
\end{equation}
\begin{displaymath}
\;+\;\int_{1}^{\infty}\int_{E(k)}\theta_{k}(\omega_{\pm})(ct
^{1/n})(t^{s}\;\pm\;t^{1-s})dH_{r}^{0}(c)\frac{dt}{t}=\int_{1}^{\infty}
\Theta_{k}(\omega_{\pm}(t)(t^{s-1}+t^{-s}))dt.
\end{displaymath}
\end{th}
{\bf Proof}. ( A topological simplification of Lang's version of Hecke's proof
of (HRace)).
Let ${\cal R}$ be an {\bf ideal class} of the ordinary {\bf ideal class
group} $H(k) :={\cal I}_{k}/P_{k}$ , where $P_{k}$ is the subgroup of
principal fractional ideals.
It is convenient to deal at initially with the zeta function associated
with an ideal class ${\cal R}$. We define
\begin{equation}
\zeta_{k}(s, {\cal R})\;:=\;\sum_{B \in {\cal R}} \frac{1}{N(B)^{s}}
\end{equation}
for $Re(s)>1$. Let $A$ be an ideal in ${\cal R}^{-1}$. Then the map
\begin{equation}
B \longrightarrow AB\;=\;(\xi)
\end{equation}
establishes a bijection between the set of ideals in ${\cal R}$ and
{\bf equivalence classes of non-zero elements of $A$} (where two field
elements are called equivalent, if they differ by a unit from $U(k)$).
Let $R(A)$ be a set of {\bf representatives} for the non-zero
equivalence classes. Then
\begin{equation}
N(A)^{-s}\zeta_{k}(s,{\cal R})\;=\;\sum_{\xi \in R(A)} mod_{r}(\xi
N(A)^{-1/n})^{-s}.
\end{equation}
We recall that the signature gamma is represented by the following integral
(cf.(2.17))
\begin{displaymath}
\Gamma_{r}(\omega_{\pm})(s)\;=\;\int_{G_{r}}\omega_{\pm}(g)
mod_{r}(g)^{s}dH_{r}(g),
\end{displaymath}
for $Re(s)>0$, since
\begin{displaymath}
mod_{r}(\xi)\;=\;\prod_{v \in S_{\infty}(k)}\mid \xi_{v} \mid^{N_{v}},
\end{displaymath}
where $N_{v} =[k_{v}:\mbox{\lll \char82}]$ are local degrees.
It will also be useful to note that if $f$ is a function such that
$f(g)/ mod_{r}(g)$ is absolutely integrable on $G_{r}$, then
\begin{displaymath}
\int_{G_{r}}f(g)\frac{dH_{r}(g)}{mod_{r}(g)}\;=\;\int_{G_{r}}f(Mg)
\frac{dH_{r}(g)}{mod_{r}(g)},
\end{displaymath}
for any n-dimensional matrix $M = [m_{ij}]$ with {\bf real} $m_{ij}$.
In other words, $dH_{r}(g)/mod_{r}(g)$ is an {\bf invariant
measure} of the {\bf dynamical system} $(G_{r} , T_{M}(y) := My)$ or,
in other words, $H_{r}/\Delta_{r}$ is a {\bf Haar measure} on the group
$G_{r}$.
Note that the signatural gamma function is expressed as such an integral.
Therefore, substituting $g$ by $N(A)^{1/n}\xi g$ in (2.17), we obtain
\begin{equation}
\Gamma_{r}(\omega_{\pm})(s)\frac{N(A)^{s}}{mod_{r}(\xi)^{s}}\;=\;
\int_{G_{r}}\omega_{\pm}(\xi N(A)^{-1/n}g) mod_{r}(g)^{s}dH_{r}(g)
\end{equation}
For $Re(s)\ge 1+\delta$, the sum over inequivalent $\xi \ne 0$ is
absolutely and uniformly convergent. Since for $Re(s)>1$,
\begin{displaymath}
N(A)^{-s}\zeta_{k}(s, {\cal R})\;=\;\sum_{\xi \in R(A)}mod_{r}(\xi
N(A)^{-1/n})^{-s},
\end{displaymath}
it follows that
\begin{equation}
\Gamma_{r}(\omega)(s)\zeta_{k}(s,{\cal R})
=\int_{G_{r}}\sum_{\xi \in R(A)} \omega_{D(\xi)N(A)^{-1/n}}(g)
mod_{r}(g)^{s}dH_{r}(g),
\end{equation}
where $D(\xi) := [\delta_{iv}C_{v}(\xi)]$ denotes a diagonal matrix of
conjugations.
But according to (2.30), we can write
\begin{displaymath}
g\;=\;t^{1/n}c \;,\;t>0,c \in G_{r}^{0}.
\end{displaymath}
Therefore,
\begin{equation}
\Gamma_{r}(\omega_{\pm})(s)\zeta_{k}(s,{\cal
R})=\int_{0}^{\infty}\int_{G_{r}^{0}}\sum_{\xi \in
R(A)}mod_{r}(t^{1/n}c)^{s} \omega_{\pm}(N(A)^{-1/n}(\xi_{1}t^{1/n}c_{1}, ...
,\xi_{n}t^{1/n}c_{n}))t^{s}dH_{r}^{0}(c)\frac{dt}{t},
\end{equation}
where $dH_{r}^{0}(c)$ is the appropriate measure on $G_{r}^{0}$ and $c
=(c_{v})$ is a variable in $G_{r}^{0}$.
According to the decomposition (2.30) and since the kernel of $\mid C
\mid$ is the group $\mu(k)$, we obtain from the above equation
\begin{equation}
\Gamma_{r}(\omega_{\pm})(s)\zeta_{k}(s,{\cal
R})=\int_{0}^{\infty}\int_{E(k)}\frac{t^{s}}{w(k)}\sum_{u \in U(k)}\sum_{\xi
\in R(A)}\omega_{\pm}(N(A)^{-1/n}(C_{1}(\xi u)t^{1/n}e_{1},...,C_{n}(\xi
u)t^{1/n}e_{n}))dH_{G_{0}}(e)dt/t=
\end{equation}
\begin{displaymath}
=\frac{1}{w(k)}\int_{0}^{\infty}\int_{E(k)}t^{s}\sum_{u \in U(k)}\sum_{x \in
X(A)}\omega_{\pm}((N(A)^{-1/n}(\sum_{j=1}^{n}x_{j}C_{1}(\alpha_{j}u)t^{1/n}e_{1}, ...
,\sum_{j=1}^{n}x_{j}C_{n}(\alpha_{j}u)t^{1/n}e_{n}))dH_{G_{0}}(e)dt/t ,
\end{displaymath}
where the second sum is over a {\bf subset} $X(A)$ of $\mbox{\lll \char90}^{n}-\{0\}$.
But, according to the definition of $R(A)$ , operating units we obtain that
if $u$ runs $U(k)$ and $\xi$ runs $R(A)$ then $x = [x_{1}, ... , x_{n}]
\in \mbox{\lll \char90}^{n}-\{0\}$ from
\begin{displaymath}
u\xi\;=\; \sum_{x \in \mbox{\lll \char90}^{n}-\{0\}} x_{j}\alpha_{j}
\end{displaymath}
spars all $\mbox{\lll \char90}^{n} - \{0\}$. Therefore, the "fourth integral" from
(2.43) we can rewrite in the form
\begin{equation}
=\int_{0}^{\infty}\int_{E(k)}\frac{t^{s}}{w(k)}\sum_{x=(x_{1}, ...
,x_{n})\ne 0}\omega_{\pm}((N(A)^{-1}t)^{1/n}e_{1}\sum_{j=1}^{n}x_{j}C_{1}
(\alpha_{j}, ... , (N(A)^{-1}t)^{1/n}e_{n}\sum_{j=1}^{n}C_{n}(\alpha_{j}))
)dH_{r}^{0}(e) dt/t=
\end{equation}
\begin{displaymath}
=\int_{0}^{\infty}\int_{E(k)}(\frac{t^{s}}{w(k)})\sum_{0\ne x \in
\mbox{\lll \char90}^{n}}\omega((N(A)^{-1}t)^{1/n}e C(A)x^{t})dH_{r}^{0}(e)dt/t \;=\;
\end{displaymath}
\begin{displaymath}
\;=\;\int_{0}^{\infty}\int_{E(k)}\frac{\theta_{n}(\omega_{C(A)})
(t^{1/n}e)-1}{w(k)}dH_{r}^{0}(e) dt/t .
\end{displaymath}
We split the integral from $0$ to $\infty$ into two integrals, from $0$
to $1$ and from $1$ to $\infty$. We thus find
\begin{equation}
\Gamma_{r}(\omega_{\pm})(s)\zeta_{k}(s,{\cal R})\;=\;\frac{1}{w(k)}
\int_{0}^{1}t^{s} \int_{E(k)}\theta_{n}(\omega_{C(A)}(t^{1/n}e))t^{s}dH_{r}^{0}
(e)dt/t\;-\;
\end{equation}
\begin{displaymath}
-\;\frac{H_{G_{r}^{0}}(E(k))}{w(k)s}\;+\;\int_{1}^{\infty}\int_{E(k)}
\frac{t^{s}}{w(k)}[\theta_{n}(\omega_{C(A)}(t^{1/n}e)-1]dH_{r}^{0}(e) dt/t .
\end{displaymath}
We return to the basis $\{\alpha_{j}: j=1, ... , n\}$ of the integral ideal
$A$ over $\mbox{\lll \char90}$. We define
\begin{displaymath}
\alpha^{*}\;:=\;\{\alpha_{j}^{*} : j =1, ... ,n\}
\end{displaymath}
to be the {\bf dual basis} with respect to the trace (cf. [L, XII.3]).
Then $\alpha^{*}$ is a basis for the fractional ideal
\begin{displaymath}
A^{*}\;:=\;(D_{k/\mbox{\lll \char81}} A)^{-1},
\end{displaymath}
where $D_{k/\mbox{\lll \char81}}$ is the {\bf different} of $k$ over $\mbox{\lll \char81}$ (cf. [L,
III.1], [N, IV.2] and the remark below).
We now use Heckes's theta functional equation $(HTE)$. It can be seen that
\begin{equation}
\theta_{n}(\omega_{C(A)}^{\pm})(t^{1/n}c)\;=\;\pm \frac{1}{t}\theta(\omega
^{\pm}_{C(A^{*})})(t^{-1/n}c^{-1}),
\end{equation}
because $mod_{r}(c) = 1$ , i.e. $c$ is in $G_{r}^{0}$ ! We transform the
first integral from $0$ to $1$, using a simple change of variables, letting
$t=1/\tau, dt = -d\tau/\tau^{2}$. Note that the measure $dH_{r}^{0}(c)$ is
invariant under the transformation $c \longrightarrow c^{-1}$ (think of
an isomorphism with the additive Euclidean measure, invariant under taking
negatives).
We therefore find that
\begin{equation}
\Gamma_{r}(\omega_{\pm})(s)\zeta_{k}(s,{\cal R})
\;=\;\frac{2H_{r}^{0}(E(k)\times \mbox{\lll \char84}^{r_{2}})}{w(k)s(s-1)}\;+\;
\end{equation}
\begin{displaymath}
\;+\;\frac{1}{w(k)}\int_{1}^{\infty}\int_{E(k)}(\theta_{n}(\omega_{C(A)}
)(t^{1/n}c)t^{s}\;\pm\;\theta_{n}(\omega_{C(A^{*})})(t^{1/n}c)t^{1-s})dH_{r}
^{0}(c)\frac{dt}{t}.
\end{displaymath}
(Let us remark that in the second edition of [L] in Section XII.3 , on page 257
there is a typegraphical error).
The expression in (2.47) is {\bf invariant} under the transformations
$A \longrightarrow A^{*}$ and $s \longrightarrow 1-s$ (in the plus case).
Thus, we have obtained full calculations on the zeta function of an ideal
class ${\cal R}$. Taking the sum over the ideal classes ${\cal R}$ from
$H(k)$ we immediately yield information on the zeta function itself, as
follows : we can construct for $A^{*}$ in a similar way ,and hence we finally
obtain
\begin{equation}
2(\Gamma_{r}(\omega_{\pm})\zeta_{k})(s)=\Gamma_{r}
(\omega_{\pm})(s)(\sum_{{\cal R}\in H(k)}2\zeta_{k}(s,{\cal R}))=
\end{equation}
\begin{displaymath}
=\frac{\lambda_{k}}{s(s-1)}+\frac{1}{w(k)}\int_{1}^{\infty}
\int_{E(k)}(t^{s}\pm t^{1-s}) (\sum_{{\cal R} \in H(k)}\sum_{A \in {\cal
R}^{-1}}\theta_{n}(\omega_{C(A)}(t^{1/n}c))d^{*}c \frac{dt}{t}=
\end{displaymath}
\begin{displaymath}
=\frac{\lambda_{k}}{s(s-1)}+\frac{1}{w(k)}\int_{1}^{\infty}\int_{E(k)}
(t^{s}\pm t^{1-s})\theta_{k}(\omega_{\pm})(t^{1/n}c)dH_{r}^{0}(c) \frac{dt}{t}.
\end{displaymath}
\begin{re}
As we mentioned above, in algebraic number theory we have to deal with
a very expanded notional aparatus. We recall some ideas, explored in
this paper.
Let $k$ be an arbitrary algebraic number field. Then we denote the {\bf trace}
of $k$ over $\mbox{\lll \char81}$ by $tr_{k}$.
If $A$ is a {\bf fractional ideal} of $k$, then $A^{*}$ denotes the
{\bf complementary ideal} to $A$ with respect to the trace $tr_{k}$,
defined as
\begin{equation}
A^{*}\;:=\;\{x \in k: tr_{k}(x A) \subset R_{k}\},
\end{equation}
(cf. [L, II.1]).
If $\{\alpha_{1}, ... , \alpha_{n}\}$ is a {\bf basis} of $A$ over
$\mbox{\lll \char90}$, then $\{\alpha_{1}^{*}, ... ,\alpha_{n}^{*}\}$
, where $\{\alpha_{i}^{*}\}$ is the dual basis relative to the
trace $tr_{k}$, is a basis of $A^{*}$.
One of the main notions of algebraic number theory is the {\bf different}
$D_{k/\mbox{\lll \char81}}$ . The different $D_{k/\mbox{\lll \char81}}$ "differs" $A^{-1}$ from
$A^{*}$, i.e. cf.e.g. [K], [L] and [N]
\begin{equation}
A^{*}\;=\;(D_{k/\mbox{\lll \char81}}A)^{-1}.
\end{equation}
One can show that
\begin{equation}
D_{k/\mbox{\lll \char81}}\;:=\;R_{k}^{*}.
\end{equation}
The second main important notion is the {\bf discriminant} $d(k)$ of an
algebraic number field.
If $\{C_{j}\}$ are embeddings as considered above and $\{\alpha_{j}\}$
forms a base of a fractional ideal $A$, then we can define the {\bf
discriminant} $d_{k}(\alpha_{1}, ... , \alpha_{n})$ by
\begin{equation}
d_{k}(\alpha_{1}, ... ,
\alpha_{n})\;:=\;(det[C_{j}(\alpha_{i})]_{i,j})^{2}=det[tr_{k}(\alpha_{i}
\alpha_{j})].
\end{equation}
It is well-known that the discriminant of a basis of $A$ does not
depend on the choice of this basis. In particular, if $A = R_{k}$, then
this discriminant is called the {\bf discriminant of the field $k$} and
denoted by $d(k)$.
The discriminant $d(k)$ has many nice and important properties :
(1) according to the {\bf Stickelberger theorem}, $d(k)$ is either
congruent to unity (mod 4) or is divisible by 4,
(2) is strictly connected with the signature $r =[r_{1},r_{2}]$ :
$sign d(k) = (-1)^{r_{2}} $ , and according to the {\bf Minkowski
theorem}
\begin{displaymath}
\mid d(k) \mid\;>\;(\frac{\pi}{4})^{2r_{2}}(\frac{n^{n}}{n!})^{2} ,
\end{displaymath}
which also ilustates the strict relation with the degree $n =n(k)$.
(3) The connection with the different :
\begin{displaymath}
N(D_{k/\mbox{\lll \char81}})\;=\;\mid d(k) \mid.
\end{displaymath}
However the value of $d(k)$ is mainly underlined by the deep {\bf
Hermite theorem}, which asserts that only a finite number of algebraic fields
can have the same discriminant.
Besides the importance of $d(k)$, its arithmetic invariance does not appear
in our "topological" generalization of HRace.
We saw that one of the main roles in the proof of (Face) was played by
the function $\omega_{C(A)}$. In Lang's proof of HRace [L,XII.3],
this corresponds to the consideration of the gaussian fixed point $\omega
:= \otimes_{j=1}^{r_{1}}G_{1}\otimes_{j=1}^{r_{2}}G_{\mbox{\lll \char67}}$, where
\begin{displaymath}
G_{\mbox{\lll \char67}}(z)\;:=\;e^{-\pi \mid z \mid^{2}}\;;\;z \in \mbox{\lll \char67},
\end{displaymath}
is the {\bf complex Gaussian fixed point} of ${\cal F}_{2}$ on $\mbox{\lll \char67}
(=\mbox{\lll \char82}^{2})$.
Then
\begin{displaymath}
\omega(C(A)x^{t})=exp(-\pi (N(A)^{2}d(k))^{-1/n}\sum_{j=1}^{n}\mid
\sum_{v=1}^{n}C_{v}(\alpha_{j})x_{j}\mid^{2})=:exp(-\pi(N(A)d(k))^{-1/n}
<A_{\alpha}x, x>),
\end{displaymath}
where the $\nu \mu$-component of the matrix $A_{\alpha} = [a_{\nu \mu}]$
is given by
\begin{displaymath}
a_{\nu \mu}\;:=\;\sum_{j=1}^{n}C_{j}(\alpha_{\nu} \alpha_{\mu}),
\end{displaymath}
and $<.,.>$ is the standard scalar product.
The matrix $A_{\alpha}$ is a {\bf symmetric positive definite} matrix.
We can thus write
\begin{displaymath}
A_{\alpha}\;=\;B_{\alpha}^{2},
\end{displaymath}
for some {\bf symmetric matrix} $B_{\alpha}$. Therefore,
($B^{*}_{\alpha} = B_{\alpha}$)
\begin{displaymath}
<A_{\alpha}x,x>=<B^{2}_{\alpha}x,x>=<B_{\alpha}x,B^{*}_{\alpha}x>=
\mid\mid B_{\alpha}x \mid\mid^{2_{n}}
\end{displaymath}
and
\begin{displaymath}
exp(-\pi (N(A)^{2}d(k))^{-1/n}<A_{\alpha}x,x>)=exp(-\pi
(N(A)^{2}d(k))^{-1/n}\mid\mid B_{\alpha}x
\mid\mid^{2}_{n})=\omega_{B_{\alpha}}(x)\;;x \in \mbox{\lll \char82}^{n}.
\end{displaymath}
Thus, $C(A)$ {\bf corresponds} to $B_{\alpha}$ in Lang's
considerations of this gaussian fixed point.
From [L, III.1] it is immediately follows that the
inverse matrix of $A_{\alpha}$ is given by
\begin{equation}
<A_{\alpha}^{-1}x, x >\;=\;\sum_{j=1}^{n}
\mid\sum_{v=1}^{n}C_{j}(\alpha_{v}^{*})x_{v}\mid ^{2}.
\end{equation}
Furthermore, the absolute value of the discriminant is
\begin{displaymath}
\mid D_{k}(\alpha_{1}, ..., \alpha_{n})\mid \;=\;det(A_{\alpha}).
\end{displaymath}
One can establish the value of $H_{r}^{0}(E(k))$ exactly in the same
way as in [L, XIII.3]. More exactly, it is not difficult to calculate
that
\begin{equation}
H_{r}^{0}(E(k))\;=\;2^{r_{1}-1}R(k),
\end{equation}
where $R(k)$ is the {\bf regulator} of $k$ defined as follows : let
$u_{1}, ... , u_{r_{1}+r_{2}}$ be {\bf independent generators} for the
unit group $U(k)$ (modulo roots of unity) (the {\bf Dirichlet's theorem}).
The absolute value of the determinant
\begin{equation}
det[N_{v}log\mid C_{j}(u_{v})\mid]
\end{equation}
(here $N_{v}$ - as usual - denotes a local degree) is independent of the
choice of our generators $\{u_{j}\}$ and is called the {\bf regulator}
$R(k)$ of the field $k$. We note that this regulator, like all
determinants, can be interpreted as a volume of a parallelotope in
$(r_{1}+r_{2})$-space.
Finally, the zeta function $\zeta_{k}(s)$ has a simple pole at $s=1$
with a {\bf residue} equal to
\begin{displaymath}
\frac{2^{r_{1}}(2\pi)^{r_{2}}h(k)R(k)}{w(k)\mid d(k) \mid}
\end{displaymath}
and the non-zero constant $\lambda_{k}$ in the {\bf zero-polar
factor} (trivial zeta) from the Face theorem
\begin{equation}
\lambda_{k}\;=\;\frac{2^{r_{1}}h(k)R(k)}{w(k)}.
\end{equation}
\end{re}
\section{RH-fixed points of ${\cal F}_{n}$}
In this section we present constructions which lead to the
derivation of the main technical tool of this paper - the {\bf harmonic
notion} of an RH-fixed point of the n-dimensional real Fourier
transform. We present here a more abstract and brief version of the
technique which was originally developed in [$M_{A}$] for the proof of the
Riemann hypothesis.
Let $V$ be a {\bf real vector space} endowed with an {\bf idempotent
endomorphism} $F :V \longrightarrow V$, i.e. $F^{2} = I_{V}$, where
$I_{V}$ denotes the identity endomorphism of $V$.
Let us consider the {\bf purely algebraic} notion of the {\bf
quasi-fixed point of $F$} associated with a parameter $l \in \mbox{\lll \char67}$ and
an element $v \in V$ :
\begin{equation}
Q_{l}(F)(v)\;=\;Q_{l}(v)\;:=\;v\;+\;lF(v).
\end{equation}
Let us observe that if $l=1$ then $Q_{1}(v)$ is a {\bf fixed point} of
$F$, i.e.
\begin{equation}
F(Q_{1}(v))=F(v)+F^{2}(v)=F(v)+v =Q_{1}(v),
\end{equation}
and if $l=-1$ then $Q_{-1}(v)$ is a (-)fixed point of $F$, i.e.
\begin{displaymath}
F(Q_{-1}(v))=F(v)\;-\;F^{2}(v)\;=\;-(v\;-\;F(v))=-Q_{-1}(v).
\end{displaymath}
We obtain the following result on the existence of quasi-fixed points
\begin{lem}({\bf Existence of quasi-fixed points}).
For each $v_{0} \in V$ and $l \ne \pm 1$ the {\bf formula}
\begin{equation}
v_{l}\;:=\;\frac{v_{0}}{1-l^{2}}\;-\;\frac{lF(v_{0})}{1-l^{2}}
\end{equation}
gives the solution of the following {\bf Abstract Fox Equation}( AFE
in short , cf. also [$M_{A}$])
\begin{equation}
(AFE_{V}^{l})\;\;v_{l}\;+\;lF(v_{l})\;=\;v_{0}.
\end{equation}
\end{lem}
Lemma 2 shows that making a simple algebraic calculus, we cannot obtain a
{\bf singular} solutions of $AFE_{V}$, since the formula (3.59) {\bf has no
sense} for $l=\pm 1$.
Moreover, we see that on the ground of {\bf classical logic} the
$\pm$ fixed point $Q_{\pm 1}(v_{0})$ cannot be the solution of $(AFE_{V}^{\pm})$
, $Q_{\pm}(v_{\pm})=v_{0}$ if $v_{0}$ is not a $\pm$fixed point of $F$.
Let us denote the real subspace of $V$ of all $\pm$fixed points $v_{0}$ of $F$ in
$V$ by $Fix_{\pm}(F)$, i.e. $F(v_{0}) = \pm v_{0}$. We thus see that
the condition
\begin{equation}
v_{0}\;\in\; Fix_{\pm}(F)
\end{equation}
is a {\bf necessary condition} for the {\bf existence} of solutions
$Q_{\pm1}(v_{0})$ of $(AFE_{V})$.
We construct $Q_{\pm 1}(v_{0})$ using the {\bf averaging procedure} for the
family $\{Q_{l}(v_{0}) : l \ne \pm 1\}$, originally constructed in
[$M_{A}$].
As in [$M_{A}$] , it will be very convenient to use the unique non-commutative
field of {\bf Hamilton quaternions} $\mbox{\lll \char72}$ (the {\bf Einstein space-time space}).
We denote a {\bf Haar measure} of the additive group $(\mbox{\lll \char72}, +)$, by
$H_{\mbox{\lll \char72}}$, i.e. the {\bf standard Lebesgue measure} $d^{4}h$ of the
vector space $\mbox{\lll \char82}^{4}$ (the Einstein space-time).
For each $M,N >0$ we consider the {\bf hamiltonian segments} (rings)
\begin{equation}
S(M,N)\;:=\;\{h \in \mbox{\lll \char72} : M \le \mid h \mid_{\mbox{\lll \char72}} \le N\},
\end{equation}
where in all the sequel $\mid \cdot \mid_{\mbox{\lll \char72}} := \mid\mid \cdot
\mid\mid_{4}$ is the standard Euclidean norm on $\mbox{\lll \char82}^{4}$.
Finally, we consider the {\bf invertion $I_{\mbox{\lll \char72}}$} of $\mbox{\lll \char72}$
\begin{equation}
I_{\mbox{\lll \char72}}(l)\;:=\;l^{-1}\;,\;l\in \mbox{\lll \char72}^{*}:=\mbox{\lll \char72}-\{0\}.
\end{equation}
Let us observe that $I_{\mbox{\lll \char72}}$ is only a set-automorphism (and not a
group automorphism of the multiplicative group $\mbox{\lll \char72}^{*}$, since it is
not commutative).
Each {\bf automorphism} $\lambda$ of $(\mbox{\lll \char72}, +)$ changes the Haar
measure $H_{\mbox{\lll \char72}}$ into $cH_{\mbox{\lll \char72}}$ with $c \in \mbox{\lll \char82}_{+}^{*}$ (the {\bf
von Neumann-Weil theorem}). The number $c$ does not depend on the choice
of Haar measure. It is denoted by $\Delta_{\mbox{\lll \char72}}(\lambda)$ and is
called the {\bf Haar module} of $\lambda$. It is defined by any of the
equivalent formulas given below (cf.[W,I])
\begin{displaymath}
(W_{m})\;\;H_{\mbox{\lll \char72}}(\lambda(B))\;=\;\Delta_{\mbox{\lll \char72}}(\lambda)H_{\mbox{\lll \char72}}(B)
\end{displaymath}
or
\begin{displaymath}
(W_{i})\;\;\int f(\lambda^{-1}(x))dH_{\mbox{\lll \char72}}(x)=\Delta_{\mbox{\lll \char72}}(\lambda)\int
f(x)dH_{\mbox{\lll \char72}},
\end{displaymath}
where $B$ is any Borel set and $f$ is any integrable function with
$\int f dH_{\mbox{\lll \char72}} \ne 0$.
The second formula can be symbolically written in the form:
\begin{displaymath}
dH_{\mbox{\lll \char72}}(\lambda(x))\;=\;\Delta_{\mbox{\lll \char72}}(\lambda)dH_{\mbox{\lll \char72}}(x).
\end{displaymath}
If $h \in \mbox{\lll \char72}^{*}$ is arbitrary, then the formula : $M_{h}(x) := h\cdot
x, x \in \mbox{\lll \char72}$ defines a linear multiplication automorphism of $(\mbox{\lll \char72},
+)$. We set
\begin{displaymath}
\Delta_{\mbox{\lll \char72}}(h)\;:=\; \Delta_{\mbox{\lll \char72}}(M_{h})\;,\;h \in \mbox{\lll \char72}^{*},
\end{displaymath}
and moreover, we define $\Delta_{\mbox{\lll \char72}}(0) := 0$. It is well-known that
(cf.e.g. [W, I.2 and Corrolary 2])
\begin{equation}
\Delta_{\mbox{\lll \char72}}(h)\;=\;\mid h \mid^{4}_{\mbox{\lll \char72}}\;=\;\mid\mid h
\mid\mid_{4}^{4}.
\end{equation}
We denote the {\bf invertion} of $\mbox{\lll \char72}^{*}$ by $I_{\mbox{\lll \char72}}(h) := h^{-1}, h
\in \mbox{\lll \char72}^{*}$. Unfortunately, $I_{\mbox{\lll \char72}} =I$ is not a group automorphism of
$\mbox{\lll \char72}^{*}$ , since $\mbox{\lll \char72}^{*}$ is not commutative! However, it is still a very
{\bf crucial topologically-algebraic} map of $\mbox{\lll \char72}^{*}$ of order $2$ :
$I_{\mbox{\lll \char72}}^{2} = id_{\mbox{\lll \char72}}$.
Thus, beside such an important invariant of $\mbox{\lll \char72}$ like the Galois
group $Gal(\mbox{\lll \char72}/\mbox{\lll \char82})$, we have an additional important {\bf invariant} of
$\mbox{\lll \char72}$ - the {\bf invertion group} $Inv(\mbox{\lll \char72}^{*}) := \{id_{\mbox{\lll \char72}^{*}},
I_{\mbox{\lll \char72}^{*}}\}$ of $\mbox{\lll \char72}^{*}$ (cf. [$M_{A}$]).
It is well-known (cf.e.g.[$M_{A}$, Lem.4]) that
\begin{equation}
dH_{\mbox{\lll \char72}^{*}}(h)\;:=\;\frac{dH_{\mbox{\lll \char72}}(h)}{\mid h \mid_{\mbox{\lll \char72}}^{4}}\;,\;h
\in \mbox{\lll \char72}^{*}.
\end{equation}
is a (left) {\bf Haar measure} of the multiplicative group $\mbox{\lll \char72}^{*}$.
Moreover, it would be convenient to recall the {\bf simple algebraic
-measure formulas} for $H_{\mbox{\lll \char72}}$ and $H_{\mbox{\lll \char72}^{*}}$ given below (cf.
$M_{A}$, Prop.3) : for each {\bf integrable} function $f$ on $\mbox{\lll \char72}^{*}$
we have :
\begin{equation}
\int_{\mbox{\lll \char72}^{*}}f(h^{-1})dH_{\mbox{\lll \char72}^{*}}(h)
\;=\;\int_{\mbox{\lll \char72}^{*}}f(h)dH_{\mbox{\lll \char72}^{*}}(h) .
\end{equation}
i.e. $H_{\mbox{\lll \char72}^{*}}$ is the {\bf invariant measure} (or the {\bf
Bogoluboff-Kriloff measure}) of the {\bf dynamical system} $(\mbox{\lll \char72}^{*},I_{\mbox{\lll \char72}})$.
Moreover,
\begin{equation}
\int_{\mbox{\lll \char72}^{*}}f(h)dH_{\mbox{\lll \char72}}
\;=\;\int_{\mbox{\lll \char72}^{*}}\frac{f(h^{-1})dH_{\mbox{\lll \char72}}(h)}{\mid h \mid_{\mbox{\lll \char72}}^{8}}.
\end{equation}
For each $N > M >0$ we consider the {\bf compact $\mbox{\lll \char72}$-rings}
\begin{equation}
R_{\mbox{\lll \char72}}(M,N)\;:=\;\{h \in \mbox{\lll \char72} : M \le \mid h \mid_{\mbox{\lll \char72}} \le N\},
\end{equation}
and the corresponding {\bf dynamical sub-system} of $(\mbox{\lll \char72}^{*}, I_{\mbox{\lll \char72}})$
\begin{equation}
D_{\mbox{\lll \char72}}(M,N)\;:=\;(R_{\mbox{\lll \char72}}(M,N) \cup R_{\mbox{\lll \char72}}(N^{-1},M^{-1}), I_{\mbox{\lll \char72}}),
\end{equation}
with $M, N>1$.
From (3.66) we immediately obtain that the formula
\begin{equation}
\beta_{\mbox{\lll \char72}}(A)\;:=\;\int_{A}\frac{d^{4}h}{\mid 1- h^{2}\mid ^{4}}\;;\;h
\in D_{\mbox{\lll \char72}}(M,N) ,
\end{equation}
gives an {\bf invariant measure} of $D_{\mbox{\lll \char72}}(M,N)$ - the {\bf Herbrand
distribution} of $\mbox{\lll \char72}^{*}$ (cf.[$M_{A}$]). In particular, the measure $\beta
_{\mbox{\lll \char72}}$ satisfies the condition
\begin{equation}
\beta_{\mbox{\lll \char72}}(I_{\mbox{\lll \char72}}^{-1}(A))\;=\;\beta_{\mbox{\lll \char72}}(A).
\end{equation}
We use below the theory of the {\bf sextet} $(\mbox{\lll \char72}, \Delta_{\mbox{\lll \char72}}, H_{\mbox{\lll \char72}},
R_{\mbox{\lll \char72}}(M,N), \beta_{\mbox{\lll \char72}}, R_{\mbox{\lll \char72}})$ .
For the sake of completness, we also briefly recall here two deep
and difficult results from {\bf analytic potential theory} of $\mbox{\lll \char72}$
explored in [$M_{A}$] :
(1) {\bf The Riesz theorem} (cf.[HK, Sect.3.5, Th.3.9]).
Let $s = s(x)$ be a {\bf subharmonic} function in a domain of
$\mbox{\lll \char82}^{6}$. Then there exists a {\bf hamiltonian Riesz measure} $R_{\mbox{\lll \char72}}$
and a {\bf harmonic function} $h(x)$ outside a compact set $E$, such that
\begin{displaymath}
(RT)\;s(x)\;=\;\int_{E}\frac{dR_{\mbox{\lll \char72}}(y)}{\mid\mid x-y
\mid\mid_{6}^{4}}\;+\;h(x)\;;\;x \in \mbox{\lll \char82}^{6}.
\end{displaymath}
(2) {\bf Brelot's theorem} (cf.[HK, Sec.36, Th3.10] - on the
existence of {\bf harmonic measures}).
Let $D$ be a {\bf regular} and {\bf bounded domain} of $\mbox{\lll \char82}^{n}$ with
border $\partial D$. Then, for each $x \in D$ and arbitrary Borel
set $B$ of $\partial D$ , there exists a {\bf unique number}
$\omega(x,B:D)$, which is a {\bf harmonic function in $x$} and {\bf
probabilty measure in $B$} and moreover, for each {\bf semicontinuous
function} $f(\xi)$ on $\partial D$ the formula
\begin{equation}
(DP)\;\tilde{f}(x)\;=\;\int_{\partial D}f(\xi)d\omega(x,\xi;D)\;;\;x \in
D-\partial D ,
\end{equation}
gives the {\bf harmonic extension} of $f$ from $\partial D$ to $D$.
The family of harmonic measures $\omega(D) :=\{\omega(x,\cdot;D): x \in
D\}$ solves the {\bf Dirichlet problem}(DP) for a pair $(D, \partial D)$
and if a solution exists it is {\bf unique}.
In [$M_{A}$] we introduced the following formal definition of the {\bf
Abstract Hodge Decomposition}: let $f : X \longrightarrow \mbox{\lll \char67}$ be a
function and $K : X \times I \longrightarrow \mbox{\lll \char67}$ another "kernel"
function. A measure $H_{f}$ on a $\sigma$-field of subsets of $I$ gives
the Abstract Hodge Decomposition of $f$, if the following integral representation
is satisfied
\begin{displaymath}
(AHD_{f}) \;f(x)\;=\;\int_{I}K(x,i)dH_{f}(i) \;;\;x \in X .
\end{displaymath}
We call the measure $H_{f}$, which appeares in $(AHD_{f})$ the {\bf Hodge
measure} of $f$.
\begin{pr}({\bf Existence of $AHD_{\mbox{\lll \char72}}$}).
There exists such a Borel probability measure $R_{\mbox{\lll \char72}}$ (a {\bf
hamiltonian Riesz measure}) on the 3-dimensional sphere $S^{3} := \{h
\in \mbox{\lll \char72} : \mid h \mid_{\mbox{\lll \char72}}=1\}$, such that for each
\begin{displaymath}
r \in X_{\mbox{\lll \char72}}(M,N)\;:=\;R_{\mbox{\lll \char72}}(M,N)\cup R_{\mbox{\lll \char72}}(N^{-1},M^{-1})
\end{displaymath}
with $N>M>1$, the following abstract Hodge decomposition ($AHD_{\mbox{\lll \char72}}$
in short) holds :
\begin{displaymath}
(AHD_{\mbox{\lll \char72}})\;\Delta^{-1}_{\mbox{\lll \char72}}(r^{2})\;=\;\int_{S^{3}}\frac{dR_{\mbox{\lll \char72}}(h)}
{\Delta_{\mbox{\lll \char72}}(r^{2}-h^{2})}\;=\;{\cal H}_{\mbox{\lll \char72}}(R_{\mbox{\lll \char72}})(r).
\end{displaymath}
\end{pr}
{\bf Proof}. Let $\epsilon_{n} >0$ be an arbitrary sequence, which
converges to zero. Then the functions $\mid\mid \cdot
\mid\mid_{6}^{-(4+\epsilon_{n})}$ are {\bf subharmonic} ( as suitable powers
of a {\bf harmonic function}) and obviously they are {\bf not
harmonic}! Therefore, according the {\bf Riesz theorem}, there exists a
sequence of {\bf Riesz measures} $\{R_{n}\}$ and a sequence $\{h_{n}\}$
of harmonic functions {\bf inside} of $S^{3}$ with the property
\begin{equation}
\mid\mid r \mid\mid_{6}^{-(4+\epsilon_{n})}
\;=\;\int_{S^{3}}\frac{dR_{n}(h)}{\mid\mid r-h
\mid\mid_{6}^{4}}\;+\;h_{n}(r).
\end{equation}
Since $dR_{n}(x) =\nabla(\mid\mid x \mid\mid_{6}^{(4+\epsilon_{n})})dx$
(cf. [HK, Section 3.5]) , the sequence $\{R_{n}(S^{3})\}$ is
{\bf bounded}, i.e. $R_{n}(S^{3}) \le A$, for some $A>0$ and all $n \in
\mbox{\lll \char78}$.
According to {\bf Frostman's theorem} (cf. [HK, Theorem 5.3]), we
can choose a subsequence $\{R_{n_{p}}\}$, which is {\bf weakly
convergent} to a limit measure $R_{\infty}$ on $S^{3}$, i.e. $R_{\infty}
:= (w)\lim_{p\longrightarrow \infty}R_{n_{p}}$ and
\begin{equation}
\mid\mid r
\mid\mid_{6}^{-4}\;=\;\int_{S^{3}}\frac{dR_{\infty}(x)}{\mid\mid r-x
\mid\mid_{6}^{4}}\;+\;h(r).
\end{equation}
On the other hand, according to {\bf Brelot's theorem} applied to
the triplet $(B^{6}(1),0,S^{5})$ - there exists a {\bf harmonic
measure} $\omega(\cdot):=\omega(\cdot,0,B_{6})$ on $S^{5}$ with the
property
\begin{equation}
\mid\mid r \mid\mid_{6}^{-4}\;=\;\int_{S^{5}}\frac{d\omega(y)}{\mid\mid
r-y\mid\mid_{6}^{4}}\;,\;r \in (B^{6})^{c}.
\end{equation}
Let us denote the natural inclusion by $j_{35} : S^{3} \longrightarrow
S^{5}$; $j_{35}(h)\;=\; (h,0,0)$. Then (3.74) can be written of the
form
\begin{equation}
\mid\mid r \mid\mid_{6}^{-4}\;=\;\int_{S^{5}}
\frac{d(j_{35}^{*}R_{\infty})(y)}{\mid\mid y -r
\mid\mid_{6}^{4}}\;+\;h(r).
\end{equation}
Let us consider the {\bf continuous function} $f(\xi) := \mid\mid \xi
\mid\mid_{6}^{-4}$ on $S^{5}$ and (for a while) take $\mu$ to be one of the
two measures : $j_{35}^{*}(R_{\infty})$ or $\omega$. Finally, let us
consider the potential $\int_{S^{5}}\frac{d\mu(y)}{\mid\mid
r-y\mid\mid_{6}^{4}}$. The vectors $r, y \in \mbox{\lll \char82}^{6}$ can be
considered as {\bf Cayley numbers} from $\mbox{\lll \char82}^{8} = \mbox{\lll \char72} \times \mbox{\lll \char72}$
and $\mid\mid \cdot \mid\mid_{6}$ as the restriction of the {\bf
Cayley norm}. Since Cayley numbers form a non-commutative and
non-associative algebra with {\bf division}, then we can write
\begin{displaymath}
\int_{S^{5}}\frac{d\mu(y)}{\mid\mid
r-y\mid\mid_{6}^{4}}=\int_{S^{5}}\frac{d\mu(y)}{\mid\mid y(1-r/y) \mid\mid_{6}^{4}}
=:\int_{S^{5}}\frac{d\nu(r,y)}{\mid\mid y
\mid\mid_{6}^{4}}=\int_{S^{5}}f(\xi)d\nu(r,\xi).
\end{displaymath}
Thus, both the formulas (3.75) and (3.76) give the solution of the
{\bf Dirichlet problem} for $((B^{6})^{c},S^{6},f)$. From the {\bf
uniqueness} of the solution of the Dirichlet problem (cf. [HK,
Th.1.13]), we obtain:
\begin{displaymath}
j_{35}^{*}(R_{\infty})\;=\;\omega \;\;and\;\; h\equiv 0\;on\;
(B^{6})^{c},
\end{displaymath}
since $h$, as a difference between a harmonic function and a potential, is
also harmonic on $\mbox{\lll \char82}^{6}- S^{5}$.
Hence, restricting ourselves in (3.76) for $r = h \in \mbox{\lll \char72}$, we finally
obtain
\begin{equation}
\Delta_{\mbox{\lll \char72}}(h^{-1})\;=\;\int_{S^{3}}\frac{dR_{\infty}(x)}{\Delta_{\mbox{\lll \char72}}(h
-x)} \;,\;h \in S_{\mbox{\lll \char72}}(M,N).
\end{equation}
Let us consider a {\bf branch of the hamiltonian square root} $\sqrt{\cdot}$
and the induced map of measure spaces : $\sqrt{\cdot} :
(S^{3},R_{\infty})\longrightarrow (S^{3}, \sqrt{\cdot}^{*}R_{\infty})$,
substituting $h^{2}$ for $h$ and $R_{\mbox{\lll \char72}}$ for
$\sqrt{\cdot}^{*}R_{\infty}$ we obtain the above proposition.
We will use the {\bf hamiltonian sextet} (from analytic potential theory)
\begin{displaymath}
(\mbox{\lll \char72}, \mid \cdot \mid_{\mbox{\lll \char72}}, \Delta_{\mbox{\lll \char72}}, H_{\mbox{\lll \char72}},
H_{\mbox{\lll \char72}^{*}},R_{\mbox{\lll \char72}}, \beta_{\mbox{\lll \char72}})
\end{displaymath}
in the averaging procedure given below to obtain, singular solutions of
$(AFE_{V})$.
A similar result is much easier to obtain using the completely
different nature of locally compact rings - the small adeles , i.e.
working with the {\bf adic potential theory}.
As we will show below, in the p-adic case, the required {\bf algebraic}
potential theory is simpler, in opposite to the strongly analytic
potential theory of $\mbox{\lll \char82}^{m}$. Therefore, the p-adic fields (and
generally local non-archimedean fields are - in such a way we see them
today - are missing links - to the needed maths constructions).
Let $H_{p}$ denotes the {\bf Haar measure} of the additive group of the
p-adic number field $\mbox{\lll \char81}_{p}$. The main reason that the algebraic
potential theory over $\mbox{\lll \char81}_{p}$ is simpler that the analytic one over
$\mbox{\lll \char82}^{m}$ is the quite different behaviour of Haar measures on totally
- disconnected fields with compare to Haar measures on the connected
fields. In particular, $\mbox{\lll \char90}_{p}^{*}$ is {\bf open}, and therefore
$H_{p}(\mbox{\lll \char90}_{p}^{*}) \ne 0$, whereas in the case of {\bf connected}
local fields $K$ we have
\begin{displaymath}
H_{K}(S_{K})\;=\;0,
\end{displaymath}
where $S_{K}$ is the unit sphere in $K$.
Thus, it is convenient to normalizeed $H_{p}$ in such a manner that
\begin{displaymath}
H_{p}(\mbox{\lll \char90}_{p}^{*}=S_{p})\;=\;(1\;-\;p^{-1}),
\end{displaymath}
(the Euler component in $\zeta_{\mbox{\lll \char81}}^{-1}(1)$)
Let $p$ and $q$ be two different {\bf prime numbers} and $\mbox{\lll \char81}_{p}$ and
$\mbox{\lll \char81}_{q}$ be the fields of p-adic and q-adic numbers. For the
convenience we take the non-canonical choice of $\mid . \mid_{q}$ by
putting $\mid q \mid_{q}= 1/p$!
We denote by $\mbox{\lll \char81}_{pq}$ the product $\mbox{\lll \char81}_{p}\times \mbox{\lll \char81}_{q}$, being the
locally compact abelian ring with a large set of invertible elements (its
completion has Haar measure zero). We denote its {\bf Haar module} by
$\Delta_{pq}$, and its {\bf Haar measure} by $H_{pq}$. $R_{pq}$ is the
{\bf adic Riesz measure}, $I_{pq}$ is the {\bf invertion automorphism} of
$\mbox{\lll \char81}_{pq}^{*}$ and finally, we denote the {\bf adic Herbrandt distribution}
(the Bogoluboff-Krilov measure) of $\mbox{\lll \char81}_{pq}$ by $\beta_{pq}$. Then
the sextet
\begin{displaymath}
(\mbox{\lll \char81}_{pq},\mbox{\lll \char81}_{pq}^{*},\Delta_{pq},H_{pq},I_{pq},\beta_{pq})
\end{displaymath}
enables us to show, in a relatively simple, algebraic way, the existence
of the below {\bf $pq$-adic Abstract Hodge Decomposition}($(AHD_{pq})$
in short).
According to the Weil's Lemma 2 (see [W, I.2., Lemma 2]) we have
\begin{displaymath}
\Delta_{pq}(x)\;=\;mod_{\mbox{\lll \char81}_{pq}}(x)\;=\;\mid x_{p} \mid_{p} \mid x_{q}
\mid_{q}\;,\;x=(x_{p},x_{q}) \in \mbox{\lll \char81}_{pq}^{*}.
\end{displaymath}
We define a {\bf sub-dynamical system} $D_{pq}(M,N) = (X_{pq}(M,N),
I_{pq})$ of the dynamical system of the small adeles $(\mbox{\lll \char81}_{pq},
I_{pq})$. Moreover, in the sequel we simply write $D_{pq}$ and $X_{pq}$
instead of $D_{pq}(M,N)$ and $X_{pq}(M,N)$, respectively. The compact
topological space $X_{pq}$ is defined as follows: let $M, N \in \mbox{\lll \char78}^{*}$
be such that $1 \le M < N$. Then
\begin{displaymath}
X_{pq}(M,N)\;:=\;\{x \in \mbox{\lll \char81}_{p} \times \mbox{\lll \char81}_{q}: x=(x_{p},x_{q}), \mid
x_{q} \mid_{q}=1, \mid x_{p} \mid_{p}\in [p^{-N+1},p^{-M}]\cap
[p^{M},p^{N-1}]=
=\;I_{\mbox{\lll \char82}}([p^{M},p^{N-1}])\cap [p^{M},p^{N-1}]\}.
\end{displaymath}
Finally, let us consider the the p-dic {projection}
$P_{p}:\mbox{\lll \char81}_{pq}\longrightarrow \mbox{\lll \char81}_{p}, P_{p}(x_{p},x_{q})=x_{p}$ and
$I_{p}$-{\bf invariant} function
\begin{displaymath}
{\cal I}_{p}(\lambda)\;:=\;\frac{\mid \lambda \mid_{p}}{\mid
1\;-\;\lambda^{2}\mid_{p}}\;,\;\lambda \in \mbox{\lll \char81}_{p}^{*}-\{1\}.
\end{displaymath}
Under the above notations we have
\begin{lem}({\bf On the pq-adic Herbrandt measure $d\beta_{pq}$}).
The formula
\begin{displaymath}
d\beta_{pq}(x_{p},x_{q}):=\frac{d(H_{p}\times
H_{q})(x_{p},x_{q})}{\mid 1\;-\;x_{p}^{2}\mid_{p}} = \frac{{\cal
I}_{p}(P_{p}(x_{p},x_{q}))d(H_{p}\times
H_{q})(x_{p},x_{q})}{\Delta_{pq}((x_{p},x_{q}))}
\end{displaymath}
gives a {\bf Bogoluboff-Kriloff measure}( {\bf Herbrandt distribution})
of $D_{pq}$.
\end{lem}
{\bf Proof}. Since $\Delta_{pq}$ is the Haar module of $\mbox{\lll \char81}_{p}$ (like
$\Delta_{k_{A}^{*}}(z)=\prod_{v \in P}\mid z_{v} \mid_{v}$) in the case
of {\bf ideles} (see e.g. [W] and [Ko]), then the equality
\begin{displaymath}
\Delta_{pq}(x_{p},x_{q})\;=\;\mid x_{p} \mid_{p},
\end{displaymath}
holds on $X_{pq}(M,N)$. Hence we get
\begin{displaymath}
\frac{d(H_{p}\times H_{q})(x_{p},x_{q})}{\mid 1-
x_{p}^{2}\mid_{p}}=\frac{\mid x_{p}\mid_{p}}{\mid 1- x_{p}^{2}
\mid_{p}}\cdot \frac{d(H_{p}\times H_{q})(x_{p},x_{q})}{\mid
x_{p}\mid_{p}}=
\end{displaymath}
\begin{displaymath}
{\cal I}_{p}(P_{p}(x_{p},x_{q}))\cdot \frac{d(H_{p}\times H_{q})(x_{p}
, x_{q})}{\Delta_{pq}(x_{p},x_{q})}.
\end{displaymath}
Since $\frac{d(H_{p}\times H_{q})}{\Delta_{pq}}$ is a
Bogoluboff-Kriloff measure of $(\mbox{\lll \char81}_{pq}^{*}, I_{pq}), P_{p} \circ
I_{pq}=I_{p}\circ P_{p}$ and ${\cal I}_{p}$ is $I_{p}$-invariant, that
we really see that the above formula gives a $pq$-adic
Bogoluboff-Kriloff measure of $D_{pq}$. (Let us remark the importance
of the fact that $H_{q}(\mbox{\lll \char90}_{q}^{*}) \ne 0$).
\begin{pr}({\bf The existence of $AHD_{pq}$}).
\begin{displaymath}
(AHD_{pq})\;\;\Delta_{pq}(x^{-2})\;=\;\int_{S_{pq}}\frac{dR_{pq}(y)}
{\Delta_{pq}(x^{2}-y^{2})} \;\;if\;x \in X_{pq}(M,N) ,
\end{displaymath}
where $S_{pq}:=\{x \in \mbox{\lll \char81}_{pq} : \Delta_{pq} =1\}$ is the unit adic
sphere.
\end{pr}
{\bf Proof}. Let $x \in X_{pq}(M,N)$ be arbitrary. Then $x = (x_{p},x_{q})$
with $\mid x_{q}\mid_{q}=1$ and therefore $\Delta_{pq}(x) = \mid x_{p}
\mid_{p}$ ( we can identity the p-adic field $\mbox{\lll \char81}_{p}$ with the subset
$\mbox{\lll \char81}_{p}\times \{1\}$ of $\mbox{\lll \char81}_{pq}$). But $\mid x_{p} \mid_{p} \ge
p^{-N+1}$ , and therefore according to the {\bf ultrametricity} of
$\mid \cdot \mid_{p}$ (see e.g. [W, I.2., Corrolary 4]) for all $ y \in
\mbox{\lll \char81}_{p}$ with $\mid y \mid_{p} = p^{-N}$ we have
\begin{displaymath}
\mid x_{p}^{2} \mid_{p}\;=\; \mid x_{p}^{2}\;-\;y^{2} \mid_{p}.
\end{displaymath}
Integrating the both sides of the inverse of the above equality with
respect to the Haar measure $H_{p}$ on $p^{N}\mbox{\lll \char90}_{p}^{*}$, for each
$\eta \in \mbox{\lll \char81}$ with $\mid \eta \mid_{p}=1$, we obtain :
\begin{displaymath}
\frac{1}{\mid x_{p}^{2}
\mid_{p}}\;=\;\frac{1}{H_{p}(p^{N}\mbox{\lll \char90}_{p}^{*})}\int_{p^{N}\mbox{\lll \char90}_{p}^{*}}
\frac{dH_{p}(\xi)}{\mid x_{p}^{2}\;-\;(\eta \xi)^{2} \mid_{p}}.
\end{displaymath}
Let us denote : $d \nu_{p}(\xi) :=
\frac{dH_{p}(\xi)}{H_{p}(p^{N}\mbox{\lll \char90}_{p}^{*})}$. Then, for all $\eta \in
\mbox{\lll \char81}$ with $\mid \eta \mid_{p} = 1$, the above equality can be written as
\begin{displaymath}
\frac{1}{\mid x_{p}^{2}\mid_{p}}\;=\;\int_{p^{N}\mbox{\lll \char90}_{p}^{*}}\frac{d
\nu_{p}(\xi)}{\mid x_{p}^{2}\;-\;(\eta \xi)^{2} \mid_{p}}.
\end{displaymath}
(we non-standartly assumed that $\mid q \mid_{q} = p^{-1})$.
Let $F$ be any {\bf finite} subset of $\{\eta \in \mbox{\lll \char81} : \mid \eta
\mid_{p}=1, \mid \eta \mid_{q} = p^{-N}\} \subset p^{-N}\mbox{\lll \char90}_{q}^{*}$.
For an arbitrary subset $A$ of $\mbox{\lll \char81}$ we define the measure $\mu_{q}^{F}$
by
\begin{displaymath}
\mu_{q}^{F}(A)\;:=\;\sum_{f \in F} \frac{\delta_{f}(A)}{\#F} ,
\end{displaymath}
where $\delta_{f}$ is the Dirac measure at $f$.
Let us consider the measure $(\nu_{p} \times \mu_{q}^{F})$. Summing the
both sides of the previous measure representation on $F$ we obtain
\begin{displaymath}
\frac{1}{\mid x_{p}^{2} \mid_{p}}\;=\;\int \int_{p^{N}\mbox{\lll \char90}_{p}^{*}\times
p^{-N}\mbox{\lll \char90}_{q}^{*}\subset S_{pq}}\frac{d(\nu_{p}\times
\mu_{q}^{F})(\xi,\eta)}{\mid x_{p}^{2}\;-\;(\eta \xi)^{2}\mid_{p}}.
\end{displaymath}
Let us look at the natural inclussion $j_{pq} : p^{N}\mbox{\lll \char90}_{p}^{*}\times
p^{-N}\mbox{\lll \char90}_{q}^{*}$ as on a {\bf random variable}
$j_{pq}:(p^{N}\mbox{\lll \char90}_{p}^{*}\times p^{-N}\mbox{\lll \char90}_{q}^{*},\nu_{p}\times
\mu_{q}^{F})\longrightarrow S_{pq}$. Then the {\bf distribution} $R_{pq}$
of $j_{pq}$ we will be called the {\bf (p,q)-adic Riesz measure} and
the right-hand side of the above formula we can finally write in the
form :
\begin{displaymath}
\frac{1}{\mid x_{p}^{2} \mid_{p}}\;=\;\int_{S_{pq}}\frac{R_{pq}(dy)}{\mid
x_{p}^{2}\;-\;P_{p}(y)^{2} \mid_{p}}.
\end{displaymath}
Combining the above formulas, we obtain the proof of the existence of
the $(AHD_{pq})$. It also shows that the proof of $(AHD_{pq})$ is
possible in a completely {\bf algebraic way}.
\begin{re}
Probably the first mathematician, who considered and applied the p-adic
potential theory was {\bf Kochubei}. In the case of p-adic fields
$\mbox{\lll \char81}_{p}$ , the $\mbox{\lll \char81}_{p}$-Hilbert transforms probably first were
considered in the {\bf Vladimirov} et al.'s paper [VWZ] as the
$\gamma$-order {\bf derivative} $D^{\gamma}f$ of a locally constant
function $f$. It is describable by pseudo-differential operator and
explicitly written as
\begin{displaymath}
D^{\gamma}f(x)=\int_{\mbox{\lll \char81}_{p}}\mid \xi
\mid^{\gamma}_{p}\hat{f}(\xi)\chi_{p}(-\xi x)
H_{p}(d\xi)=\frac{p^{\gamma-1}}{1-p^{-\gamma-1}}\int_{\mbox{\lll \char81}_{p}}\frac{f(x)-
f(y)}{\mid x-y\mid_{p}^{\gamma+1}}H_{p}(dy),
\end{displaymath}
where $\chi_{p}$ is the additive character of $\mbox{\lll \char81}_{p}$ and
$\hat{f}(\xi)$ stand for the Fourier transformation
$\int_{\mbox{\lll \char81}_{p}}\chi_{p}(\xi x)f(x)H_{p}(dx)$ of a function $f$. A
deeper analysis of p-adic fractional differentiation $D^{\gamma}$ is
given in the Kochubei's book [Ka], where using {\bf Minlos-M\c{a}drecki's
theorem}, he established the existence of a {\bf Kochubei-Gauss measure}
$\mu$ over infinite-dimensional field extensions $\Omega_{p}$ of
$\mbox{\lll \char81}_{p}$, which is a {\bf harmonic measure} for $D^{\gamma}$ and
solves {\bf p-adic integral equations of a profile of wing of a plane}
in the case of $\Omega_{p}$ (see [Ka, Prop.6]).
The importance of $R_{pq}$ is also underline by the fact that
unfortunately, firstly we have the following negative result concerning
$(AHD_{pq})$.
{\bf Non-existence of solutions of p-adic profile of a wing in
functions}
Let $p$ be an arbitrary prime number. There is not exist an {\bf
absolutely continuous measure} $h_{p}$ w.r.t. the Haar measure $H_{p}$ (
the p-adic harmonic measure), which gives the following p-adic Abstract
Hodge Decomposition (cf. [$M_{A}$]) with the property
\begin{displaymath}
(AHD_{p})\;\frac{1}{\mid x
\mid_{p}^{2}}\;=\;\int_{\mbox{\lll \char90}_{p}^{*}}\frac{dh_{p}(y)}{\mid x^{2}-y^{2}
\mid_{p}}\;,\;x \in S_{p}(M,N).
\end{displaymath}
{\bf Proof}. The proof is based on the remarkable property of the Haar
measure $H_{p}: H_{p}(\mbox{\lll \char90}_{p}^{*})=1-p^{-1}$ (the {\bf Euler factor in
the Riemann zeta}). Assume (a contrary), that there exists a measure
$h_{p}$, which is absolutely continuous w.r.t. $H_{p} : h_{p}\ll
H_{p}$. Let us denote its density by $\omega_{p}$, i.e.
\begin{displaymath}
\omega_{p}\;=\;\frac{dh_{p}}{dH_{p}}.
\end{displaymath}
This conjecture permits us to apply the big and well-known machinery of
p-adic Fourier analysis to the problem of the existence of $(AHD_{p})$.
Reely, the $(AHD_{p})$ is obviously equivalent to the formula
\begin{displaymath}
\chi_{S_{p}(M,N)}(x)\mid x \mid_{p}^{-1} =\int_{\mbox{\lll \char81}_{p}}\mid x-y
\mid_{p}^{-1}\chi_{\mbox{\lll \char90}_{p}^{*}}(y)\omega_{p}(y)dH_{p}(y)\;=\;
\end{displaymath}
\begin{displaymath}
=\int_{\mbox{\lll \char81}_{p}}(\chi_{\mbox{\lll \char90}_{p}^{*}}\cdot \omega_{p})(x-z)\mid z
\mid_{p}^{-1}dH_{p}(z):=[(\chi_{\mbox{\lll \char90}_{p}^{*}}*\mid \cdot
\mid_{p}^{-1})](x)\;,\;x \in \mbox{\lll \char81}_{p},
\end{displaymath}
where $\chi_{A}$ denotes the characteristic function of a set $A$ and
$*$ means the p-adic {\bf convolution}.
If we apply the p-adic Fourier transform ${\cal F}_{p}$ to the both
sides of the above equalities, then we obtain
\begin{displaymath}
\hat{(\chi_{S_{p}(M,N)}\cdot \mid \cdot
\mid_{p}^{-1})}(\xi)\;=\;\hat{\mid \cdot \mid_{p}^{-1}}(\xi)\cdot
\hat{\chi_{\mbox{\lll \char90}_{p}^{*}}\cdot \omega_{p}}(\xi)\;,\;\xi \in \mbox{\lll \char81}_{p}.
\end{displaymath}
Let us observe that
\begin{displaymath}
\hat{\mid \cdot \mid_{p}^{-1}}(\xi)\;=\;\frac{(1-p)log \mid \xi \mid_{p}
}{plogp}\;,\;\xi \in \mbox{\lll \char81}_{p},
\end{displaymath}
see [Ka, Sect. 1.5, formula (1.29)]. Thus, the last equality is not
possible.
In the light of the above presented negative result, the previous above
result - on the existence of $(AHD_{pq})$ - gathers a greater value.
The small pq-adele ring $\mbox{\lll \char81}_{pq}$ is only one representant from a whole class
of "models", of the very similar nature, which can be used in the same
context.
(1){\bf The pq-adic vector space $\mbox{\lll \char81}_{[pq]}$}.
It is well-known (see e.g. [La, Sect.1]) that the "world" of {\bf
valuations} (or {\bf absolute values} or {\bf points}) is very rich. In
particular, we saw, how effective was the action of the defined below
pre-valuations $v_{pq}$, which gives $(AHD_{pq})$ of the p-adic
valuation $\mid \cdot \mid_{p}$ in the simple {\bf algebraic way}, if
we compare it with a difficult {\bf analytic} proof of $(AHD_{\mbox{\lll \char72}})$ of
$\Delta_{\mbox{\lll \char72}}^{-2}$. With a similar situation we have deal in the
famous Faltings' proof of the Mordell-Shafarevich-Tate conjectures. He
used so called {\bf heights of global fields}, which are some functions
defined by p-adic valuations (cf.e.g. [La, Fa]).
According to the principal theorem of the arithmetics each non-zero
rational number $x$ we can uniquely write of the form:
\begin{displaymath}
x\;=\;\frac{a}{b}p^{m}q^{n},
\end{displaymath}
where $a, b, m, n \in \mbox{\lll \char90}, (a,b)=1$, and $pq$ do not divide $ab$. For
such a rational $x$ we put
\begin{displaymath}
\alpha_{p}(x)\;:=\;m,\;\;\alpha_{q}(x)\;:=\;n,
\end{displaymath}
and
\begin{displaymath}
v_{pq}(x)\;:=\;p^{-(m+n)}\;=\;p^{-(\alpha_{p}(x)+\alpha_{q}(x))}.
\end{displaymath}
The functions $\alpha_{p} : \mbox{\lll \char90} \longrightarrow \mbox{\lll \char90}$ defined below are
called {\bf exponents} corresponding to $p$ and satisfies few simple
and nice elementary properties. A. Ostrowski showed their most
surprising properties : they are {\bf unique} arithmetical functions
(up to a constant - like Haar measures), which satisfy the five
mentioned above their elementary properties (see e.g. [Na1,Th.1.7(i)-(v)]).
In particular, the functions $v_{pq}$ satisfies the following condition
:
\begin{displaymath}
v_{pq}(x)\;=\;\mid x \mid_{p} \mid x \mid_{q},
\end{displaymath}
i.e. $v_{pq}$ has only one good "residual" above {\bf multiplicative}
property. Moreover
\begin{displaymath}
v_{pq}(x+y)\le max\{ v_{pq}(x), v_{pq}(y), \mid x \mid_{p}\mid y
\mid_{p},\mid y \mid_{p}\mid x \mid_{q}\},
\end{displaymath}
where $\mid \cdot \mid_{p}$ and $\mid \cdot \mid_{q}$ are p-adic and
q-adic valuations, respectively, but with the additional assumption
that
\begin{displaymath}
\mid q \mid_{q}\;=\;p^{-1}.
\end{displaymath}
Thus, $v_{pq}$ has a bad linear (ring) algebraic properties. In
particular, $v_{pq}(\cdot)$ is not a {\bf valuation} but only - let us
say - a {\bf pre-valuation}. Therefore, the {\bf completion} of $\mbox{\lll \char81}$
w.r.t. the metric type function : $d_{pq}(x,y)\;=\;v_{pq}(x-y)$ is
rather a pathological object and in particular, it is not a topological
field.
(2) By $\mbox{\lll \char81}_{(pq)}$ we donote the set $\{0,1, ... , pq-1\}(p,q)$ of all
{\bf double formal Laurent series} with coefficients in the set $\{0,1,
... ,pq-1\}$. Thus, each element $x$ of $\mbox{\lll \char81}_{(pq)}$ has the form :
\begin{displaymath}
x\;=\;\sum_{m=M}^{\infty}\sum_{n=N}^{\infty}a_{mn}p^{m}q^{n}\;;\;a_{mn}\
in \{0,1, ... , pq-1\}\;,\;M,N \in \mbox{\lll \char90}.
\end{displaymath}
If we establish a (non-canonical) ordering $<_{2}$ on the lattice
$\mbox{\lll \char90}^{2}$, in such a way that $(\mbox{\lll \char90}^{2}, \le_{2})$ and $(\mbox{\lll \char78}, \le)$ are
isomorphic in the category of ordered sets : $h:(\mbox{\lll \char90}^{2}, \le_{2}) \simeq
(\mbox{\lll \char78}, \le)$, and we establish the natural bijection
\begin{displaymath}
\mbox{\lll \char81}_{(pq)} \supset
x=\sum_{m=M}^{\infty}\sum_{n=N}^{\infty}a_{mn}p^{m}q^{n}\longrightarrow
\sum_{n=h(M,N)}a_{h(m,n)}p^{h(m,n)}\in \mbox{\lll \char81}_{p}\;=\;
\end{displaymath}
\begin{displaymath}
\;\;\;\;\;=\;\sum_{n=h(M,N)}a_{\pi(h(m,n))}p^{n},
\end{displaymath}
(where here $\pi$ denotes a respectible permutation of $\mbox{\lll \char78}$), then we
can endow $\mbox{\lll \char81}_{(pq)}$ with the natural local field structure (transformed
from $\mbox{\lll \char81}_{p}$). Thus $\mbox{\lll \char81}_{(pq)}$ is a {\bf local field} isomorphic with
$\mbox{\lll \char81}_{p}$. Unfortunately, we cannot expect that $\mbox{\lll \char90}_{(pq)}$ is isomorphic
with $\mbox{\lll \char90}_{p}\otimes_{\mbox{\lll \char90}} \mbox{\lll \char90}_{q}$, where $\mbox{\lll \char90}_{(pq)}:=\{x \in
\mbox{\lll \char81}_{(pq)}: v_{pq}(x)\le 1\}$ (let us mention here that $\mbox{\lll \char70}_{pq} \ne
\mbox{\lll \char70}_{p}\otimes_{\mbox{\lll \char90}} \mbox{\lll \char70}_{q}=0$).
To see the compactness of $\mbox{\lll \char90}_{(pq)}$ (and hence the local compactness
of $\mbox{\lll \char81}_{(pq)}$) it suffices to observe that the function $f$ from the
product $D$ of a countable many copies of the pq-elements set $\{0,1,
..., pq-1\}$ onto $\mbox{\lll \char90}_{(pq)}$ given by
\begin{displaymath}
f(\{a_{mn}\}_{m,n=0}^{\infty})\;=\;\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}
a_{mn}p^{m}q^{n}
\end{displaymath}
is {\bf surjective} and {\bf continuous} in the Tichonov topology of
$D$. Since $D$ is compact then $\mbox{\lll \char90}_{(pq)}$ is compact as a continuous
image of the compact set.
On the other hand, we have natural inclusions (in the category of
sets): $i_{p} :\mbox{\lll \char81}_{p}\longrightarrow \mbox{\lll \char81}_{(pq)}$ and $i_{q}:
\mbox{\lll \char81}_{q}\longrightarrow \mbox{\lll \char81}_{(pq)}$.
Observe however, that with the multiplications defined as :
\begin{displaymath}
\alpha \cdot_{p} x\;:=\;i_{p}(\alpha)\cdot x \;\;and\;\;\alpha
\cdot_{q}x\;:=\;i_{q}(\alpha)\cdot x,
\end{displaymath}
(where $\cdot$ means the multiplication in $\mbox{\lll \char81}_{(pq)}$) $\mbox{\lll \char81}_{(pq)}$
{\bf is not a vector space} over $\mbox{\lll \char81}_{p}$ or over $\mbox{\lll \char81}_{q}$. Reely, if
it would be true, then obviously we would have:
$dim_{\mbox{\lll \char81}_{p}}\mbox{\lll \char81}_{(pq)}=+\infty$ and
$dim_{\mbox{\lll \char81}_{q}}\mbox{\lll \char81}_{(pq)}=+\infty$, what is impossible, since it is
well-known that LCA-vector spaces over local fields must be
finite-dimensional (see e.g. [W, I.2 Corrolary 2]). Moreover,
according to the {\bf Dantzing's description} of local fields, all
extensions of p-adic number fields $\mbox{\lll \char81}_{p}$, must be finite extensions
of such fields!
(3). {\bf The tensor product rings} $\mbox{\lll \char81}_{p}\otimes_{\mbox{\lll \char81}} \mbox{\lll \char81}_{q}$.
Let us observe that our main bi-adele (small pq-adele) ring
$\mbox{\lll \char81}_{pq}=\mbox{\lll \char81}_{p}\times \mbox{\lll \char81}_{q}$ is sufficiently good and "rich" for
our purpose, since from the point of view of the Haar-module theory the
set of its all {\bf non-invertible} elements :
$(\mbox{\lll \char81}_{pq}^{*})^{c}:=\mbox{\lll \char81}_{pq}-\mbox{\lll \char81}_{pq}^{*}$ is "small", i.e. its {\bf Haar
measure} is zero : $(H_{p}\times H_{q})((\mbox{\lll \char81}_{pq}^{*}))=0$.
For each $x \in \mbox{\lll \char81}_{pq}^{*}$ by $\Delta_{pq}(x)$ (or $mod_{\mbox{\lll \char81}_{pq}}(x)$)
we denoted the {\bf Haar module} of the {\bf automorphism} $x \longrightarrow
a \cdot x$ of $(\mbox{\lll \char81}_{p}\times \mbox{\lll \char81}_{q})^{+}$. Thus
\begin{displaymath}
\Delta_{pq}(x)\;=\;mod_{\mbox{\lll \char81}_{pq}}(x)\;:=\;\frac{(H_{p}\times
H_{q})(xX)}{(H_{p}\times H_{q})(X)},
\end{displaymath}
for arbitrary measurable set $X$ in $\mbox{\lll \char81}_{pq}$ with $0<(H_{p}\times
H_{q})(X)<+\infty$ (for example, for $X$ we can take any compact
neighbourhood of zero).
According to the Weil's Lemma 2 (see [W, I.2 , Lemma2]) we have
\begin{displaymath}
\Delta_{pq}(x)\;=\;mod_{\mbox{\lll \char81}_{pq}}(x)\;=\;\mid x_{p} \mid_{p}\mid x_{q}
\mid_{q}\;,\;x=(x_{p},x_{q})\in \mbox{\lll \char81}_{pq}^{*}.
\end{displaymath}
The above formula suggests that we can also descibe the pq-vectors from
$\mbox{\lll \char81}_{pq}$ in the terminology of the {\bf Grothendieck} tensor products.
Let us consider the {\bf algebraic} tensor product $\mbox{\lll \char81}_{p}\otimes_{\mbox{\lll \char81}}\mbox{\lll \char81}_{q}$
and the natural map $t_{pq}: \mbox{\lll \char81}_{p}\otimes_{\mbox{\lll \char81}}\mbox{\lll \char81}_{q}\longrightarrow
\mbox{\lll \char81}_{(pq)}$ defined by
\begin{displaymath}
t_{pq}(x \otimes y)\;=\;xy\;,\;x \in \mbox{\lll \char81}_{p},\;y \in \mbox{\lll \char81}_{q},
\end{displaymath}
(althought, according to the above mentioned troubles with the
multiplication in $\mbox{\lll \char81}_{(pq)}$ it is not {\bf algebraic}).
The Grothendieck $\pi$-norm $\mid \cdot \mid_{p}\otimes_{\pi}
\mid \cdot \mid_{q}$ (of $\mid \cdot \mid_{p}$ and $\mid \cdot
\mid_{q}$, see e.g. [MT]) is denoted in the sequel by $\mid \cdot
\mid_{pq}^{\pi}$ and is defined by
\begin{displaymath}
\mid x \mid_{pq}^{\pi}\;:=\;(\mid \cdot \mid_{p}\otimes \mid \cdot
\mid_{q})(x)\;:=\;
\end{displaymath}
\begin{displaymath}
\;=\;inf\{\sum_{i} \mid x_{i} \mid_{p}\mid y_{i}
\mid_{q}\;:\;x=\sum_{i}x_{i}\otimes y_{i}\;;\;x_{i} \in \mbox{\lll \char81}_{p},
y_{i}\in \mbox{\lll \char81}_{q}\}.
\end{displaymath}
The above $\pi$-norm is a {\bf cross-norm} (of $\mid \cdot \mid_{p}$
and $\mid \cdot \mid_{q}$), i.e.
\begin{displaymath}
(\mid \cdot \mid_{p}\otimes_{\pi} \mid \cdot \mid_{q})(x_{1}\otimes
x_{2})\;=\;\mid x_{1}\mid_{p} \cdot \mid x_{2} \mid_{q}\;;\;x_{1} \in
\mbox{\lll \char81}_{p}, x_{2}\in \mbox{\lll \char81}_{q}.
\end{displaymath}
Moreover
\begin{displaymath}
\mid x_{1} \otimes x_{2} \mid_{pq}^{\pi}=\mid x_{1}\mid_{p} \mid x_{2}
\mid_{q}= mod_{\mbox{\lll \char81}_{pq}}((x_{1},x_{2})) =
\end{displaymath}
\begin{displaymath}
=\;\Delta_{pq}((x_{1},x_{2}))
\end{displaymath}
$x_{1} \in \mbox{\lll \char81}_{p}, x_{2} \in \mbox{\lll \char81}_{q}$ and that norm is {\bf
archimedean}. By
$\mbox{\lll \char81}_{p}\hat{\otimes}_{\pi}\mbox{\lll \char81}_{q}=:\mbox{\lll \char81}_{pq}^{\otimes}$ we denote
the {\bf completion} of $(\mbox{\lll \char81}_{p}\otimes _{\mbox{\lll \char81}} \mbox{\lll \char81}_{q}, \mid \cdot
\mid_{pq}^{\pi})$.
Obviously $\mbox{\lll \char81}_{pq}^{\otimes}$ is a LCA-ring. Let us denote by $H_{pq}$
its {\bf Haar measure}.
Let $i_{pq} : \mbox{\lll \char81}_{p} \times \mbox{\lll \char81}_{q}\longrightarrow \mbox{\lll \char81}_{pq}^{\otimes}$
be the canonical inclusion homomorphism, i.e. $i_{pq}(x,y) = x \otimes
y$.
Since we have got {\bf Haar measures} $H_{p}$ and $H_{q}$ of
$\mbox{\lll \char81}_{p}^{+}$ and $\mbox{\lll \char81}_{q}^{+}$, respectively, then we can define their
{\bf tensor product} $H_{p}\otimes H_{q}$ on the LCA-subring $Im(i_{pq})$ :
\begin{displaymath}
i_{pq}^{*}(H_{p}\times H_{q})\;=:\;H_{p}\otimes_{\pi}H_{q}.
\end{displaymath}
It is easy to check that the tensor product $H_{p}\otimes_{\pi} H_{q}$
of Haar measures is a {\bf Haar measure} on $Im(i_{pq})$.
Thus, since the Haar measure $H_{pq}$ of $\mbox{\lll \char81}_{p}\hat{\otimes}_{\pi}\mbox{\lll \char81}_{q}$
is unique (up to a constant), then we can assume that
\begin{displaymath}
H_{pq}\;=\;H_{p}\otimes_{\pi}H_{q},
\end{displaymath}
i.e. the Haar measure $H_{pq}$ is the $\pi$-tensor product of the Haar measures
of $\mbox{\lll \char81}_{p}^{+}$ and $\mbox{\lll \char81}_{q}^{+}$ (see [MT]).
Finally, let us remark that the tensor products of gaussian measures in
Banach spaces were firstly considered by {\bf R. Carmona} and {\bf S. Chevet}
In [$M_{T}$] were defined and considered tensor products of
p-stable measures with $0<p<2$, in Banach spaces of stable type. The
above considered tensor products of Haar measures are also tensor
products of measures in the sense of the definition given in
[M$_{T}$].
(4){\bf The adele ring $\mbox{\lll \char81}_{\mbox{\lll \char65}} \subset \prod_{p \in P}\mbox{\lll \char81}_{p}\times
\mbox{\lll \char81}_{\infty}(;=\mbox{\lll \char82})$ of $\mbox{\lll \char81}$}.
{\bf Ideles} were introduced by {\bf C. Chevalley} in [Ch] in 1936.
{\bf E. Artin} and {\bf G. Whaples} occured {\bf adeles} in [AW], where
they are called valuation vectors. The ring of adeles admits {\bf K.
Iwasawa's characterisation} (see [I]) in the following way : if $R$ is a
semi-simple commutative LCA-ring with a unit element, which is neither
compact nor discrete, and there is a field $K \subset R$, with the same
unit element, which is discrete and such that $R/K$ is compact, then $R$
is the {\bf ring of adeles} either over an algebraic number field or over an
algebraic function field with a finite fields of constants.
Topological properties of adeles and ideles were investigated by E.
Artin, K. Iwasawa, T. Tamagawa and J. Tate (see e.g.[N, Chapter VI]).
At the end of the ends, all the above considered versions of $\mbox{\lll \char81}_{pq}$
are closely related to each other and moreover we have the following
inclusion :
\begin{displaymath}
\mbox{\lll \char81}_{pq} \subset \mbox{\lll \char81}_{p}\otimes_{\mbox{\lll \char81}} \mbox{\lll \char81}_{q} \subset \mbox{\lll \char81}_{(pq)}\subset
\mbox{\lll \char81}_{\mbox{\lll \char65}}.
\end{displaymath}
\end{re}
In the sequel we denote one of the two LC rings above by $R$, i.e.
$R =\mbox{\lll \char72}$ or $\mbox{\lll \char81}_{pq}$. We also simply write $(\Delta, H, R, \beta, X(M,N))$
instead of $(\Delta_{R}, H_{R}, R_{R}, \beta_{R},X_{R}(M,N))$. Then
we have the following shocking result (a constructive mathematical
construction)
\begin{th}({\bf The existence of singular solutions of $AFE_{V}$}).
Let $V$ be a {\bf real vector space} with a {\bf continuous
idempotent endomorphism} $F : V \longrightarrow V$. Then {\bf each element
$v_{0} \in V$} is a {\bf $\pm$-fixed point of $F$}, i.e.
\begin{equation}
\;\;V\;=\;Fix(F),
\end{equation}
and moreover an arbitrary $v_{0} \in V$ has the following {\bf
Riesz-Bogoluboff-Kriloff Abstract Hodge Decomposition
(Representation)}($AHD_{RBK}$ in short)
\begin{equation}
(AHD_{RBK})\;v_{0}\;=\;\int \int_{\mbox{\lll \char83}\times
X(M,N)}[(I \pm F)(v(\Delta^{2}(r),v_{0}))]d(\beta \otimes R)(s,r).
\end{equation}
\end{th}
{\bf Proof}. Since $R = \mbox{\lll \char72}$ or ${\cal R} = \mbox{\lll \char81}_{pq}$ and according
to Lemma 2 , for each $v_{0}$ we have at our disposal the whole family
\begin{equation}
{\cal V}_{\pm}(v_{0})\;:=\;\{v_{\pm}(l,v_{0}) : l \ne \pm 1\}
\end{equation}
of solutions of the family of the {\bf abstract Fox equations}
\begin{equation}
v_{\pm}(l,v_{0})\;\pm \;l Fv_{\pm}(l,v_{0})\;=\;v_{0}.
\end{equation}
We substitute $l = \Delta^{2}(r), r \in R^{*}, \Delta(r)\ne 1$,
in (3.81), thus obtaining
\begin{equation}
\frac{v_{\pm}(\Delta^{2}(r),v_{0})}{\Delta^{2}(r)}\;+\;F(v_{\pm}(\Delta^{2}(r),v_{0}))\;
=\;\frac{v_{0}}{\Delta^{2}(r)}.
\end{equation}
Integrating both sides of (3.82) with respect to the Haar measure $H$
on $X(M,N)$ and applying formula (3.67), here in the form
\begin{equation}
\int_{{\cal R}^{*}}f(r)\chi_{X(M,N)}(r)dH(r)\;=\;\int_{{\cal
R}^{*}}f(r^{-1})\frac{\chi_{X(M,N)}(r^{-1})}{\Delta^{2}(r)}dH(r),
\end{equation}
we obtain the equality
\begin{equation}
\int_{X(M,N)}\Delta^{-2}(r)v_{\pm}(\Delta^{2}(r),v_{0})dH(r)\;\pm \;
\int_{X(M,N)}\Delta^{-2}(r) F(v_{\pm}(\Delta^{-2}(r),v_{0}))dH(r)\;=\;
\end{equation}
\begin{displaymath}
=v_{0}\int_{X_{M,N}}\Delta^{-2}(r)dH(r) \;=:\;v_{0}m_{-2}(M,N),
\end{displaymath}
where we denote the $(-2)-R-{\bf moment}$ of the Haar measure $H$ on
the compact $X(M,N)$ by $m_{-2}(M,N)$.
Let us consider the expressions
\begin{equation}
\int_{X(M,N)}\frac{v_{\pm}(\Delta^{\pm2}(r),v)dH(r)}{\Delta^{2}(r)}.
\end{equation}
Applying the {\bf compact-$R$-Hilbert transform ${\cal H}$} in
the form of the Abstract Hodge Decomposition $(AHD_{R})$ :
\begin{equation}
\Delta(h^{-2})\;=\;\int_{\mbox{\lll \char83}}\frac{dR(y)}{\Delta(h^{2}-y^{2})}\;=\;{\cal
H}(R)(h^{2}),
\end{equation}
$h \in X(M,N)$, and using the {\bf Fubini theorem} we obtain
\begin{equation}
\int_{X(M,N)}\frac{v_{\pm}(\Delta^{\pm2}(r),v_{0})dH(r)}{\Delta(r^{2})}=
\int_{X(M,N)}v_{\pm}(\Delta^{\pm2}(r),v_{0})\int_{\mbox{\lll \char83}}\frac{dR(y)}{\Delta(r^{2}-y^{2})}dH(r)=
\end{equation}
\begin{displaymath}
=\int_{\mbox{\lll \char83}} dR(y)
\int_{X(M,N)}\frac{v_{\pm}(\Delta^{\pm2}(r),v_{0})dH(r)}{\Delta(r^{2}-y^{2})}.
\end{displaymath}
But, according to the formula $(W_{i})$, we can write the second inner integral
in the iterated integral above in the form : (since $\Delta(y)=1,
r/y=:r^{\prime}$)
\begin{equation}
\int_{X(M,N)}\frac{v_{\pm}(\Delta^{\pm2}(\frac{r}{y}),v_{0})dH(r)}
{\Delta^{2}(y)\Delta(1-(\frac{r}{y})^{2}}=
\Delta(y^{-3})\int_{X_{M,N}}\frac{v_{\pm}(\Delta^{\pm2}(r),v_{0})dH(r)}
{\Delta(1-r^{2})}\;,\;y \in \mbox{\lll \char83}.
\end{equation}
But $\frac{dH(r)}{\Delta(1-r^{2})} =: dHer(r) = d\beta(r)$ is the {\bf Herbrand
distribution} of the invertion $I=I_{{\cal R}}$ of ${\cal R}^{*}$, i.e.
\begin{equation}
\int_{X(M,N)}\frac{v(\Delta^{2}(r),v_{0})dH(r))}{\Delta(1-r^{2})}\;=\;\int_{X(M,N)}
\frac{v(\Delta^{-2}(r),v_{0})dH(r)}{\Delta(1-r^{2})},
\end{equation}
since, for each integrable function $\phi$ the following is true
\begin{displaymath}
\int_{X(M,N)}\phi(I(r))dHer(r)\;=\;\int_{X(M,N)}\phi(r)d(I^{*}Her)(r)\;=
\end{displaymath}
\begin{displaymath}
\;=\;\int_{X(M,N)}\phi(r)dHer(r).
\end{displaymath}
Let us set:
\begin{equation}
v_{1}^{\pm}\;:=\;\int_{\mbox{\lll \char83}}\Delta(y^{-3})dR(y)\int_{X(M,N)}\frac{v(\Delta^{2}(r)
,v_{0})dH(r)}{\Delta(1-r^{2})}.
\end{equation}
Since our "manipulations" only acted up on the parameters $l$ and under
the assumption, $F$ is continuous and linear, then we finally
obtain the RBK-integral representation above, which at the same time is
the {\bf singular solution} of $(AFE_{V})$ :
\begin{equation}
v_{1}^{\pm}\;\pm\;F(v_{1}^{\pm})\;=\;m_{-2}v_{0}.
\end{equation}
\begin{re}({\bf On a shocking consequence of the construction of Th.2. The
mathematics and logic}).
Obviously, the thesis (3.78) is not true (on the ground of classical
logic) for the majority of idempotent pairs $(V, F)$. Reely, let $V =
\mbox{\lll \char67}$ be considered as the 2-dimensional Banach space over $\mbox{\lll \char82}$ and let
$F = c$ be the complex conjugation. Then
\begin{displaymath}
\;\;Fix(c,\mbox{\lll \char67})\;=\;\mbox{\lll \char82} \ne \mbox{\lll \char67}.
\end{displaymath}
The construction in Th.2 is a following step in the old and well-known
philosophical problem : what is the connection between maths and
(classical) logic?
As it is well-known, {\bf Frege} saw mathematics as only a part of logic
(more exactly, according to Frege, the whole of mathematics can be reduced to
logic).
Probably the first mathematician, who questioned Frege's approach
to mathematics was pre-intuitionist {\bf Kronecker}. He attacked
well-known {\bf Cantor's proof} (in "naive" set theory), of the
existence of {\bf transcendental numbers} $t \in T$.
Let $\tilde{\mbox{\lll \char81}}$ be the (algebraically closed) field of {\bf algebraic
numbers} and assume that $TnD$ is true :
\begin{displaymath}
v_{Cl}(p \lor \sim p)\;=\;1 .
\end{displaymath}
We can write $TnD$ in quantifier form as the following true statement on
$\tilde{\mbox{\lll \char81}}$ (according to the laws of the quantifier calculus) :
\begin{displaymath}
(C)\forall(x \in \mbox{\lll \char82})(x \in \tilde{\mbox{\lll \char81}}) \lor \sim(\forall (x \in
\mbox{\lll \char82})(x \in \tilde{\mbox{\lll \char81}}))\;=\;
\end{displaymath}
\begin{displaymath}
\;=\;\forall(x \in \mbox{\lll \char82})(x \in \tilde{\mbox{\lll \char81}})\lor (\exists(x \in \mbox{\lll \char82})(t
\in T)).
\end{displaymath}
Under the assumption, that the first term in the alternative $(C)$ is
true ($\tilde{\mbox{\lll \char81}}$ is countable!), it follows that $\mbox{\lll \char82}$ should be countable,
which is impossible, according to the well-known Cantor theorem.
Thus, according to the rules of classical calculus of statements
and predicators, the second term of the alternative $(C)$ is true.
Thus, transcendental numbers exist.
But Cantor's reasoning does not give any information regarding a real
number, which is transcendental. In other words, it does not provide a
{\bf construction} of such a number.
According to Kronecker, the non-constructive character of Cantor's
proof of the existence of transcendental numbers is bad and hence its conclusion
should be rejected. But $(C)$ is only a specification of TnD. Thus,
questioning $(C)$ is identical to questioning TnD.
The immediate consequence of this was the rejection of classical
logic and construction of intuitione logic by {\bf Heyting}.
{\bf Brouwer} built constructive mathematics on this basis and showed that,
in general, many constructions violate TnD. For example Brouwer's construction
of the {\bf diagonal set of positive integers} $D\mbox{\lll \char78}$ (the simplest {\bf Post
system} generated by a constructive object $\mid$ and the format
$\frac{x,x}{x}$ (cf.[ML, Sect.7])) violates the statement
\begin{displaymath}
(n \in D\mbox{\lll \char78})\lor (n \notin D\mbox{\lll \char78}).
\end{displaymath}
Similarly, in our case the statement
\begin{displaymath}
(v_{0} \in Fix(F))\lor (v_{0}\notin Fix(F))
\end{displaymath}
violates TnD (a real infinity exists but no a potential infinity?)
The construction in Th.2 is an example of such a construction. In
reality, it leaveas out assumption : $v_{0} \in Fix(F)$. It seems that
it is much worse. {\bf It gives a contradiction in mathematics}.
According to {\bf Poincare}, the only thing, which we must demand
from an object which exists in mathematics is {\bf
non-contradictivity}. On the other hand, {\bf Godel's well known result}
states that it is not possible to prove the { \bf non-contradictivity of
arithmetics of $\mbox{\lll \char78}$} (and , in fact, the majority of axiomatic systems).
Moreover, the problem of the non-contradictivity of ZFC-set theory is much
more complicated than for such arithmetics. Thus (according to {\bf Gentzen's
non-finistic proof} of the non-contradictivity of arithmetics), we can only
{\bf believe} that set theory is non-contradictory. But a belief is only
a belief, and for example, the proof of Th.2 seems be done properly, according
to classical logic, but it leads to classical mathematical
contradiction.
The only explanation of this phenomenon is the following : we use the methods
of {\bf measure theory} strictly, which is subsequently based on set theory,
in a strict manner. But according to the above discusion can this be ...
(contradictory)?
It is also very surprising, that such logical problems from the fundaments of
mathematics appeared during the work on the Riemann hypothesis. Maybe this is
one of the reasons that (RH) was unproven for so long and shows that (RH) is
not a standard mathematical problem.
Finally, all the logical problems with (RH) mentioned above should lead and
stimulate a subsequence discussion on mathemathical philosophy, very
similar to the discourses after {\bf Appel-Haken's proof} of the {\bf four
colour conjecture} (proved with help of a computer program). Can we accept a
proof of RH which is based - in its generality - on a theorem which leads to
a contradiction although, if we bound the domain of objects to some "admissible"
$v_{0} \in Fix(V)$, then the construction is acceptable.
\end{re}
We now apply our theorem in the case $V = {\cal S}(\mbox{\lll \char82}^{n})$ and $F =
{\cal F}_{n}$. Let $A^{+} = A_{n}^{+}(x)$ be a {\bf generalized amplitude},
i.e. any function from ${\cal S}(\mbox{\lll \char82}^{n})$ with $A^{+}(0) = 0$. Then, according
to Th.2, there exists a {\bf RH-fixed point} $\omega_{A}^{+}$ (associated with
$A^{+}$, cf. [$M_{A}, Th.2$]), i.e.
\begin{equation}
(\omega^{+}_{A}\;-\;G)(x)\;=\;A^{+}(x)\;\;;\;x \in \mbox{\lll \char82}^{n}.
\end{equation}
As we remarked in [$M_{A}$, Remark 15] (see also (3.91)), $\omega^{+}_{A}$
{\bf cannot exist} if $A$ is not a fixed point of ${\cal F}_{n}$, i.e.
according to Remark 3.
In [$M_{H}$, Prop.2] we showed that a direct solution of the {\bf RH-eigenvalue
problem} exists. We constructed a concrete example of the {\bf hermitian
amplitude} $A =A^{4}_{h_{0}}$ being an eigenvalue of the parameetrized Fourier
transform ${\cal F}_{h_{0}}$ and the {\bf RH-eigenvector} $\omega^{+}_{A}$ as the
{\bf fourth order hermite function} (cf.[$M_{H}$, (93.18)])
\begin{displaymath}
\omega^{+}_{A}(x)\;:=\;
H^{4}_{h_{0}}(x)\;:=\;h_{0}^{4}e^{-h_{0}^{2}x^{2}}
(16h_{0}^{4}x^{4}-48h_{0}^{2} x^{2}\;+\;12),
\end{displaymath}
which satisfies the equation
\begin{displaymath}
\omega^{+}_{A}(x)\;-\;12h_{0}^{4}e^{-h_{0}^{2}x^{2}}\;=:\;A(x)\;,\;x \in \mbox{\lll \char82}.
\end{displaymath}
Here $h_{0} = \frac{\sqrt{3}}{2}$ is an amplitude parameter.
But it is very difficult ( either we cannot or it is not possible) to find
a direct analytic example of an {\bf RH-amplitude}. The main difficulty
is to find two fixed-points of the Fourier transform, which are both
stricly decreasing for $x >1$. In other words, the restriction : ${\cal
F}_{n}(A^{+}) = A^{+}$ is very restrictive.
For the parameter $p$ dependent Fourier transform ${\cal F}_{p}(f)(x)
:= \int_{\mbox{\lll \char82}}e^{2p^{2}ixy}f(y)dy$, we showed in [$M_{H}$] that a direct
solution of the (-)RH-eigenvector problem exists.
Defining the (-)RH-eigenvector $\omega_{A}^{-}$ as the {\bf sixth order
hermitian function} (see $[M_{H}]$ for details)
\begin{displaymath}
\omega_{A}^{-}(x)\;:=\;H_{h_{0}}^{6}(x)\;=\;16h_{0}^{6}e^{-h_{0}^{2}x^{2}}(4
h_{0}^{6}x^{6}-30h_{0}^{4}x^{4}+45h_{0}^{2}x^{2}-7.5),
\end{displaymath}
we can define the amplitude $A^{-}$ by the formula :
\begin{displaymath}
A^{-}(x)\;:=\;\omega_{A}^{-}(x)\;+\;60h_{0}^{4}H^{2}_{h_{0}}(x).
\end{displaymath}
In the last part of this paper the fundamental role plays the {\bf
second canonical Hermite function}
\begin{displaymath}
H_{2}(x)\;=\;\pi G(x)(4 \pi x^{2}\;-\;1),\;\;x \in \mbox{\lll \char82}.
\end{displaymath}
Integrating by parts twicely, we obtain that $H_{2}$ is a {\bf minus
fixed point} of ${\cal F}$ : $\hat{H}_{2}(x) \;=\; -H_{2}(x)$.
Then, according to Th.2, there exists a (-)RH-fixed point $\omega_{A}^{-}$
(associated with an amplitude $A^{-}$), i.e.
\begin{displaymath}
(\omega_{A}^{-}\;+\;H_{2})(x)\;=\;A^{-}(x)\;,\;x \in \mbox{\lll \char82}.
\end{displaymath}
According to (3.91) $\omega_{A}^{-}$ cannot exists if $A^{-}$ is not a
minus fixed point of ${\cal F}_{1}$. But, if we take an amplitude $A^{-}$ in
such a way that the support of $(A^{-}\;-\;H_{2})$ :
\begin{displaymath}
supp(A^{-}\;-\;H_{2})\;=:\;S_{A}
\end{displaymath}
is the completion of a set with positive Lebesgue measure $\lambda_{n}$,
i.e. $\lambda_{n}(S_{A}^{c})>0$, then, according to the {\bf "separation of
variables"} construction from $Th.2$ we obtain
\begin{equation}
supp(\omega_{A}^{-})\;=\;S_{A}\;=\;and\;{\cal F}_{1}(\omega_{A}^{-})(x)\;=
\;\int_{S_{A}}cos(2\pi ixy)\omega_{A}^{-}(y)dy\;=:\;C_{A}(\omega_{A}^{-})(x).
\end{equation}
Since $C_{A}(H_{2}) \ne -H_{2}$, i.e. $H_{2}$ is not a {\bf minus -fixed point} of
$C_{A}$, then the calculation
\begin{displaymath}
C_{A}(A^{-})=C_{A}(\omega_{A}^{-}+H_{2})=C_{A}(\omega_{A}^{-})+C_{A}(H_{2})
\ne -(\omega_{A}^{-}+H_{2})= -A^{-}
\end{displaymath}
shows that , in this case, the notion of RH-fixed point does not lead to
a {\bf contradiction} and can exist for an amplitude $A^{-}$, which is not
the minus-fixed point of ${\cal F}_{1}$ (antinomies cannot be treated as a
threat to the fundaments of maths or logic).
\begin{re}
{\bf A. Wawrzy\,nczyk} (see [Wa, 3.8, Exercise 1d]) as well as we (see
[$M_{H}$, Prop.1] and [$M_{P}$, Remark 1]) have considered the following example
of the {\bf minus-fixed point} $K_{2}$ of ${\cal F}_{1}$ : in the
considerations concerning the {\bf quantum harmonic oscillator} in quantum
mechanics - one of the main roles plays the following {\bf second
Hermite function}
\begin{displaymath}
K_{2}(x)\;:=\;2\pi e^{-\pi x^{2}}(2 \pi x^{2}\;-\;1)
\end{displaymath}
with the property : $\hat{K}_{2}(x) = - K_{2}(x)$.
However, according to {\bf P. Biane}, since $G(x) := e^{-\pi x^{2}}$ is
a {\bf fixed point} of the canonical Fourier transform ${\cal F}_{1}$,
then integrating by parts twicely we obtain
\begin{displaymath}
\int_{\mbox{\lll \char82}}e^{2\pi i xy}G^{\prime \prime}(y)dy\;=\;-2\pi ix
\int_{\mbox{\lll \char82}}e^{2 \pi ixy}G^{\prime}dy\;=
\end{displaymath}
\begin{displaymath}
\;=\;-4\pi^{2}x^{2}\int_{\mbox{\lll \char82}}e^{2\pi
ixy}G(y)dy\;=\;-(4\pi^{2}x^{2}G(x)\;-\;\pi G(x))-\pi G(x).
\end{displaymath}
Since
\begin{displaymath}
G^{\prime \prime}(x)\;=\;2\pi G(x)(2 \pi x^{2}\;-\;1)\;=\;K_{2}(x)
\end{displaymath}
then
\begin{displaymath}
\hat{H}_{2}(x):=\hat{(G^{\prime \prime})+\pi
G(x)}(x)\;=\;-4\pi^{2}x^{2}G(x)+\pi G(x)=-H_{2}(x),
\end{displaymath}
i.e. $H_{2}(x) := \pi G(x)(4 \pi x^{2}\;-\;1)$ is {\bf also} the {\bf
minus fixed point} of ${\cal F}_{1}$!
Since
\begin{displaymath}
H_{2}(x)(:=\omega_{A}^{-}(x))\;-\;K_{2}(x))\;=\;\pi G(x) (=: A^{-}).
\end{displaymath}
In the sequel, we call the equality : $H_{2}\;-\;K_{2} = \pi G$ - the
{\bf BMW-example}.
Since $A^{-}:=\pi G$ {\bf is not evidently the minus-fixed point of}
${\cal F}_{1}$ (since it is the +fixed-point of ${\cal F}_{1}$), then
the BMW-example :
(1). confirms the {\bf correctness} of our Th.2.-construction and
RH-fixed point paradox of Remark 3.
(2) We are not in possibility to explain that {\bf phenomena} on the
ground of the classical logic.!
\end{re}
\section{An (-)RH-fixed point proof of the generalized Riemann hypothesis}
As opossed to the purely algebraic notion of the quasi-fixed point $Q_{l}(v)$
considered in the previous section, here the main part is played by a
{\bf purely analytic} notion of the ($1$-dimensional) {\bf amplitude} $A$
(cf.[$M_{A}$] and [$M_{H}]$.
\begin{de}
We say that a function $A :\mbox{\lll \char82}_{+}\longrightarrow \mbox{\lll \char82}_{+}^{*}$ is an
{\bf PCID-amplitude}, if it is {\bf positive, continuous, integrable}
and {\bf (strictly) decreasing} on $[1,\infty)$.
\end{de}
The importance of PCID-amplitudes (amplitudes in short) follows from the fact
that an {\bf analytic Nakayama type lemma} holds for them (i.e. some very simple
analytic statement, trivial in proof - but powerful consequences). This lemma
establishes the sign of the action on the amplitude of the {\bf
plus-sine operator} $S_{+}$(cf.$[M_{A},Lemma 4]$ and $[M_{H}, Lemma2]$):
\begin{equation}
S_{+}(A)(a)\;:=\;\int_{0}^{\infty}A(x)sin(ax)dx \;,\;a>0.
\end{equation}
\begin{lem}
For each {\bf amplitude} $A$ and {\bf frequency} $a \in \mbox{\lll \char82}^{*}_{+}$ the
following holds
\begin{equation}
\;\;S_{+}(A)(a)\;>\;0.
\end{equation}
\end{lem}
\begin{pr}({\bf On the positivity of the Rhfe(Ace)-trace} $Tr_{-}$).
Let $n = [k:\mbox{\lll \char81}]$ and $A^{-}=A_{n}^{-}(x), x \in \mbox{\lll \char82}^{n}$ from ${\cal S}
(\mbox{\lll \char82}^{n})$ be a such function that for each $e$ from the fundamental domain
$E(k)$, the function $t \longrightarrow exp(t)A_{n}^{-}(exp(t)e) =: A_{n}^{e}(t)$
is a (1-dimensional) {\bf amplitude}.
Then, for each complex number $s = u +iv$ with $u \in (1/2, 1]$ and $v>0$,
the following {\bf Casteulnovo-Serre-Weil inequality} (CWS in short) holds
\begin{equation}
(CWS)\;Tr_{-}(\zeta_{k},A_{n}^{x})(s)\;:=\;\int_{1}^{+\infty}(t^{u}+t^{1-u})
sin(vlog t) \theta_{k}(A_{n}^{x})(t)dt\;>\;0.
\end{equation}
\end{pr}
{\bf Proof}. According to (2.35), (2.25) and (2.43) we have
\begin{equation}
Tr_{-}(\zeta_{k},A_{n})(s)=\sum_{0\ne I \in {\cal I}_{k}}\sum_{u \in
U(k)}\sum_{\xi \in R(I)}
\int_{1}^{\infty}(\int_{E(k)}A_{n}((N(I)^{-1}t)^{1/n} C(\xi u)\cdot e)
dH_{r}^{0}(e))
\end{equation}
\begin{displaymath}
(t^{u}+t^{1-u})sin(vlogt)dt/w(k).
\end{displaymath}
Let us denote the vector
\begin{displaymath}
C(I)x^{t}\;:=\;N(I)^{-1/n}[\sum_{j=1}^{n}x_{j}C_{1}(\alpha_{j}), ... ,
\sum_{j=1}^{n}x_{j}C_{n}(\alpha_{j})].
\end{displaymath}
After the substitution $t = e^{r}$ and changing of variables according
to the n-dimensional substitution : $e^{\prime} = e \cdot C(I)x^{t}$ we
obtain that
\begin{displaymath}
Tr_{-}(\zeta_{k}, A_{n})=\sum_{0\ne I \in {\cal I}_{k}}
\sum_{m \in \mbox{\lll \char90}^{n}}\frac{1}{w(k)\Delta_{r}(C(I)x^{t})}(\int_{E(k)}(
\int_{1}^{+\infty}A_{n}^{e}(e^{r/n})(e^{r(u+1)}+e^{r(2-u)})sin(vr)dr)
dH_{r}^{0}(e).
\end{displaymath}
Let us consider 1-dimensional {\bf amplitudes} of the form
\begin{displaymath}
{\cal A}^{e}_{n}(r):=e^{ru}(1+e^{r(1-2u)}) A_{n}^{e}(e^{r/n}).
\end{displaymath}
Since $\frac{d}{dr}(1+e^{r(1-2u)})<0$, if $u \in (1/2, 1]$, then under
our asumptions on the amplitude $A_{n}$ the function ${\cal A}^{e}_{n}(r)$ is
strictly decreasing. According to Lemma 4,
\begin{equation}
S_{+}({\cal A}^{e}_{n})(v)\;>\;0.
\end{equation}
Combining (4.97) with (4.98) we obtain the Proposition.
\begin{re}
The considered in Prop.3 the {\bf minus-trace} $Tr_{-}(\zeta_{k},A_{n}^{+})(s)$
is obviously associated with a {\bf minus-fixed} points $\omega^{-}$ of
${\cal F}_{n}$. We have seen that its {\bf positivity} is an
immediately consequence of the mentioned above analytic Nakayama lemma
(Fresnel lemma).
Instead of $Tr_{-}(\zeta_{k},A_{n}^{+})(s)$, in the first version of the
manuscript , we have considered the {\bf plus-trace} (associated with
$\omega^{+}$)
\begin{displaymath}
Tr_{+}(\zeta_{k},
A_{n}^{+})(s)\;:=\;\int_{1}^{\infty}(t^{u}-t^{1-u})sin(vlogt)\theta_{k}(A
_{n}^{+})(t)dt,
\end{displaymath}
(which obviously only differs from $Tr_{-}(\zeta_{k},A_{n}^{-})(s)=:Tr_{-}$ by
a {\bf sign} in the subintegral expression).
In opposite to the case of $Tr_{-}$ - the {\bf positivity} of $Tr_{+}>0$
- as it was independly communicated to the author by the private communications
by {\bf S. Albeverio}, {\bf P. Biane} and {\bf Z. Brze\,zniak}!, is not
an immediately consequence of the Fresnel lemma. In particular, the result
: $Tr_{+}>0$ needs a machine of stochastic analysis and is a final effect
of the existence of the so called {\bf Hodge measure} $H_{2}^{*}$ on
$\mbox{\lll \char67}^{++}:=\{z \in \mbox{\lll \char67} : Re(s)>0, Im(s)>0\}$, which gives the {\bf Laplace
representation} of the {\bf inverse of the Haar module of} $\mbox{\lll \char67}$ :
\begin{displaymath}
\mid z \mid^{-2}\;=\;\int_{\mbox{\lll \char67}^{++}}e^{z \cdot w}dH_{2}^{*}(w)\;\;,z \in
\mbox{\lll \char67}^{++}.
\end{displaymath}
The existence of $H_{2}^{*}$ is far non-obvious. Even worse, many
peoples suggested to the author, that such the measure cannot exists!
Fortunately, the problem has a positive solution, although it is a very
technical and complicated in details result. So, we are not going to do
it in this paper.
\end{re}
\begin{lem}
$M(H_{2})(s)$ has no roots in the domain $\mbox{\lll \char67} -\{1,2\}$ and has of order
1 poles at the points :2,1,0, -2,-4, ... .
\end{lem}
{\bf Proof}. Let us again recall that the canonical second Hermite function
$H_{2}(x)$ has the form (see also [$M_{H}$, Remark 1]) :
\begin{displaymath}
H_{2}(x)\;=\;\pi G(x)(4 \pi x^{2}\;-\;1).
\end{displaymath}
It is easy to check (integrating by parts, see [$M_{H}$, Prop. 7]) that
\begin{displaymath}
M(H_{2})(s)\;=\;(s-1)(s-2)M(G)(s-2)\;\;,\;for\;Re(s)>2.
\end{displaymath}
Since $M(H_{2})(s)$ is well-defined for $Re(s)>0$ (because $H_{2} \in
{\cal S}(\mbox{\lll \char82})$), then the above formula gives the analytic continuation
of the previous right-hand side formula - defined for $Re(s)>2$.
Making the substitution $\pi x^{2} =t$ in Gamma integral we
obtain
\begin{displaymath}
M(G)(s)=\int_{0}^{\infty}x^{s-2}e^{-\pi x^{2}}dx=\frac{\pi^{1-s}}
{2\pi}\Gamma(s/2),
\end{displaymath}
where $\Gamma$ denotes the classical gamma function. Since $M(G)(s)$
{\bf does not vanish anywhere}, then $M(H_{2})$ {\bf does not vanish}
for $\mbox{\lll \char67}_{2} := \mbox{\lll \char67} -\{1,2\}$.
The final preliminary result which is very convenient when we work with
$(gRH_{k})$ is the elegant {\bf Rouche theorem}(cf.e.g. [Ma, Th. XV.18 ])
: let $\Omega\subset \mbox{\lll \char67}$ be a {\bf domain} and $D \subset \Omega$ be
{\bf compact}.
Let $f$ and $g$ be {\bf holomorphic functions} on $\Omega$, which
satisfy the two following {\bf Rouche's border conditions}:
\begin{equation}
f(z)\;\ne \;0\;for\;z \in \partial D\; (the\;border\;of\;D),
\end{equation}
and
\begin{equation}
\mid g(z) \mid\;<\;\mid f(z) \mid\;for\;z \in \partial D .
\end{equation}
Then the number of zeros $N_{D}(f+g)$ of the sum $f+g$ in $D$
(weighed by their orders) is equal to the number of zeros $N_{D}(f)$
of $f$ in $D$ ({\bf Rouche's thesis}, the adic type behaviour of
$N_{D}$), i.e.
\begin{equation}
N_{D}(f+g)\;=\;N_{D}(f).
\end{equation}
\begin{pr}({\bf A Rouche choice of the amplitude $A^{+}$ and lack of
roots of $\Gamma_{r}(G+A^{+})$}).
We can choose a plus amplitude $A_{n}^{+}$ in such a way that :
(1) the construction of the plus RH-fixed point $\omega_{A}^{+}$ in
Th.2 fulfills all the rigours of the classical logic, i.e. it does not
violate TnD.
(2) $\Gamma_{r}(G+A_{n}^{+})(s) \ne 0$ for $Re(s)>0$.
\end{pr}
{\bf Proof}. We use the Rouche theorem in the case : $\Omega = \mbox{\lll \char67}$,
\begin{displaymath}
D\;=\;D_{M}\;:=\;\{s \in \mbox{\lll \char67}: Re(s)\in [0,1], Im(s) \in [-M, M]\},\;M>0
\end{displaymath}
and
\begin{displaymath}
f(z)\;=\;\Gamma_{r}(G)(z)\;\;,\;\;g(z)\;=\;\Gamma_{r}(A_{n}^{+})(z).
\end{displaymath}
Since obviously $\Gamma_{r}(G)(z) \ne 0$ for $z \in \partial D_{M}$ and
$N_{D_{M}}(\Gamma_{r}(G)) = 0$, then it suffices to show that
\begin{equation}
\mid \Gamma_{r}(A_{n}^{+})(z) \mid^{2}\;<\;\mid \Gamma_{r}(G)(z) \mid^{2}
\end{equation}
for $z \in D_{M}$, to conclude that $N_{D_{M}}(G+A_{n}^{+}) = 0$ in $D_{M}$.
The inequality (4.102) is obviously equivalent to the inequality
\begin{equation}
Re^{2}(\int_{G_{r}}mod_{r}(g)^{s-1}A_{n}^{+}(g)d^{n}g)\;+\;Im^{2}(\int
_{G_{r}}mod_{r}(g)^{s-1}A_{n}^{+}(g)d^{n}g)<
\end{equation}
\begin{displaymath}
<\;Re^{2}(\int_{G_{r}}mod_{r}(g)^{s-1}G_{n}(g)d^{n}g)\;+\;Im^{2}(\int
_{G_{r}}mod_{r}(g)^{s-1}G_{n}(g)d^{n}g .
\end{displaymath}
Let us consider the Taylor expansion of $G_{n}(x) = e^{-\pi \mid \mid x
\mid \mid^{2}}$
\begin{displaymath}
G_{n}(x)\;=\;\sum_{m=0}^{\infty}\frac{(-1)^{m}\pi ^{2m}\mid \mid x \mid
\mid_{n}^{2m}}{m!},\;for\;x \in \mbox{\lll \char82}^{n},
\end{displaymath}
and let us denote $g_{m} := \frac{\pi^{2m}}{m!}$.
Without loss of generality we can assume that $A_{n}^{+}$ is
NCID-amplitude, i.e. is {\bf negative} continuous integrable and such
that $-A_{n}^{+}$ is strictly decreasing for $\mid \mid x \mid
\mid_{n}\ge 1$. Reely, taking $s$ with $Im(s)<0$ we obtain :
$Tr_{+}(\zeta_{k}, -A_{n}^{+})>0$. We then can define $A_{n}^{+}$ as
follows
\begin{equation}
A_{n}^{+}(x)\;:=\;-G_{n}(x)\;\;for\;\;\mid \mid x \mid \mid_{n} \ge 1,
\end{equation}
and
\begin{equation}
A_{n}^{+}(x)\;:=\;-\sum_{m=2}^{\infty}(-1)^{m}g_{m-2} \mid\mid x
\mid\mid_{n}^{2m}\;\;if\;\;\mid\mid x \mid\mid_{n} \in [0,1].
\end{equation}
Since $\sum_{m=2}^{\infty}(-1)^{m}g_{m-2} = \sum_{m=0}^{\infty}g_{m}$,
then $A_{n}^{+}$ is {\bf continuous} and hence - NCID-amplitude.
Moreover from the definition we get that the support of $(G\;+\;A^{+})$
is the {\bf unit ball} $B_{n}$ of $\mbox{\lll \char82}^{n}$.
Thus, to obtain (4.103) it suffices to show that for $Re(s) \ge 0$ holds
\begin{equation}
\mid Re(\int_{G_{r}\cap B_{n}}mod_{r}(g)^{s}A_{n}^{+}(g)dH_{r}(g))
\mid<\mid Re(\int_{G_{r}\cap B_{n}}mod_{r}^{s}G_{n}(g)dH_{r}(g)) \mid,
\end{equation}
and for $Im(s) \le 0$ holds
\begin{equation}
\mid Im(\int_{G_{r}\cap B_{n}}mod_{r}(g)^{s}A_{n}^{+}(g)dH_{r}(g)) \mid
<\mid Im(\int_{B_{n}\cap G_{r}}mod_{r}(g)^{s}G_{n}(g)dH_{r}(g)) \mid,
\end{equation}
since, according to the definition of $A_{n}^{+}$ we have
\begin{equation}
\int_{B_{n}^{c}\cap
G_{r}}mod_{r}(g)^{s}A_{n}^{+}(g)dH_{r}(g)=-\int_{B_{n}^{c}\cap
G_{r}}mod_{r}^{s}(g)G_{n}(g)dH_{r}(g)
\end{equation}
and - let us recall (see (2.16) and (2.17)) -
\begin{equation}
\Gamma_{r}(\mid\mid \cdot
\mid\mid_{n}^{2m}\chi_{B_{n}})(s)=\int_{G_{r}\cap
B_{n}}mod_{r}(g)^{s}\mid\mid g \mid\mid_{n}^{2m}dH_{r}(g)\;=
\end{equation}
\begin{displaymath}
\int_{G_{r}\cap B_{n}}mod_{r}(g)^{s}\mid\mid g
\mid\mid_{n}^{2m}\frac{d^{n}g}{mod_{r}(g)}=\int
\int_{(\mbox{\lll \char82}_{+}^{*}\times G_{r}^{0})\cap
B_{n}}mod_{r}^{s}(t^{1/n}c)t^{2m}\mid\mid c
\mid\mid_{n}^{2m}\frac{d^{n}c dt}{mod_{r}(c)t}=
\end{displaymath}
\begin{displaymath}
\;\;=:\;\frac{c_{2m}}{s+2m},
\end{displaymath}
since
\begin{displaymath}
log(mod_{r}(g))=\sum_{i=1}^{r_{1}}log \mid x_{i}
\mid\;+\;\sum_{j=1}^{r_{2}}log \mid z_{j} \mid^{2} \le C(n) \mid\mid g
\mid\mid_{n}^{2}.
\end{displaymath}
It is obvious that for $Re(s)\ge 0$ we have
\begin{displaymath}
\sum_{m=2}^{\infty}\frac{(-1)^{m}c_{m}g_{m}(Re(s)+2m)}{\mid s+2m
\mid^{2}}<\sum_{m=0}^{\infty}\frac{(-1)^{m}c_{m}g_{m}(Re(s)+2m)}{\mid
s+2m \mid^{2}},
\end{displaymath}
since $1\;-\; \frac{\pi(x+2)}{\mid x+2 \mid^{2}}>0$, according to the
fact that the quadratic polynomial $x^{2}+(4-\pi)x+(4-\pi)>0$ for all
$x>0$. Analogously, for $Im(s)\le 0$ we have
\begin{displaymath}
-Im(s)\sum_{m=2}^{\infty}\frac{(-1)^{m}c_{m}g_{m}}{\mid s+2m \mid^{2}}<
-Im(s)\sum_{m=0}^{\infty}\frac{(-1)^{m}c_{m}g_{m}}{\mid s+2m \mid^{2}}.
\end{displaymath}
Thus, according to the definition of $A_{n}^{+}(G)$, from those strict
inequalities above, we claim (deduce) that the pair $(\Gamma_{r}(G_{n}),
\Gamma_{r}(A_{n}^{+}))$ satisfies the {\bf strong} Rouche boundary
conditions (4.99) and (4.100) on every compact set $D_{M}, M>0$ (and
not only on $\partial D_{M}$):
\begin{displaymath}
\mid \Gamma_{r}(A_{n}^{+})(s)\mid\;<\;\mid
\Gamma_{r}(G_{n})(s)\mid\;,\;s \in D_{M}.
\end{displaymath}
Converging with $M$ to the infinity we finally obtain
\begin{displaymath}
N_{D_{\infty}}(G_{n}\;+\;A_{n}^{+})\;=\;N_{D_{\infty}}(G_{n})\;=\;0.
\end{displaymath}
\begin{pr}({\bf A non-contradictory choice of the amplitude $A^{-}$ and
deleting of the problem of vanishing of $M(A^{-}-H_{2})$}).
We can choose an amplitude $A_{n}^{-}$ in such a way that :
(1) the construction of the (-)RH-fixed point $\omega_{A_{n}}^{-}$ in Th.2
fulfills all the rigours of classical logic, i.e. it does not violate
TnD.
(2) Even when $\Gamma_{r}(H_{2}-A^{-})(s)$ has {\bf zeros} in $Re(s)>0$
then still holds the (Face$_{-}$):
\begin{equation}
\Gamma_{r}(\omega_{A}^{-})(s)\zeta_{k}(s)\;=\;
\frac{\lambda_{k}}{s(s-1)}\;+\;
\end{equation}
\begin{displaymath}
\;+\;\int_{1}^{\infty}\int_{E}
\theta_{E}(\omega_{A}^{-})(ct^{1/n})(t^{s}\;+\;t^{1-s})dH_{r}^{0}(c)
\frac{dt}{t}(=:\int_{1}^{\infty}\Theta_{k}(\omega_{A}^{-})(t)(t^{s-1}+
t^{-s})dt).
\end{displaymath}
\end{pr}
{\bf Proof}. Let us consider the McLaurin expansion of $H_{2}^{n}$
\begin{displaymath}
H_{2}^{n}(x)\;=\;-\pi\;-\;\sum_{m=1}^{\infty}\frac{(-1)^{m}(-\pi)^{m+1}(4m+1)
\mid\mid x \mid\mid_{n}^{2m}}{m!},
\end{displaymath}
and let us denote $h_{m}:= \frac{\pi^{m+1}(4m+1)}{m!}$.
For a convenience of the reader, we give here all needed in the sequel
facts concerning the {\bf graph} of $H_{2}$ (it can be easy obtained by
using the elementary differential calculus). Thus :
\begin{equation}
H_{2}(0)=-\pi,\;H_{2}(\frac{1}{2\sqrt{\pi}})=0,\;H_{2}(1)=\pi
e^{-\pi}(4\pi-1)>0.
\end{equation}
Moreover, the function $H_{2}(x)$ is {\bf positive} for $x \ge 1/2\sqrt{\pi}$
and {\bf strictly decreasing} for $x \ge \sqrt{\frac{5}{2}}$. Finally,
the sequence $\{h_{m}\}$ is strictly decreasing for $m \ge 4$ (see also
[AM, Lemma 2]).
Looking at the graph of $H_{2}$ on $\mbox{\lll \char82}_{+}$, we see that we can find
such $x_{2}>\sqrt{5/2}>1>x_{1}>1/2\sqrt{\pi}$ (since $H_{2}(x_{2})
\longrightarrow +\infty$ if $x_{2}\longrightarrow \infty$), that the
defined below function $A_{n}^{-}$ is an {\bf PCID-amplitude} :
\begin{displaymath}
A_{n}^{-}(x)\;:=\;H_{2}^{n}(x)\;\;if\;\; \mid\mid x \mid\mid_{n} \ge x_{2},
\end{displaymath}
and
\begin{displaymath}
A_{n}^{-}(x)\;:=\;L(x)\;if\;\mid\mid x \mid\mid_{n} \le x_{2},
\end{displaymath}
where by $L$ we denoted the line which connects the points $(x_{2},H_{2}
(x_{2}))$ and $(x_{1}, H_{2}(x_{1}))$ with $H_{2}(x_{1})>H_{2}(x_{1})$.
Moreover $(H_{2}\;-\;A_{n}^{-})(x)= 0$ for $\mid \mid x \mid \mid_{n}>x_{2}$.
The construction of an amplitude - let us say $A_{n}^{--}$ - with the property
that $\Gamma_{r}(H_{2}-A_{n}^{--})(s) \ne 0$ if $Re(s)>0, Im(s)>0$, i.e.
such $A_{n}^{--}$ that we could apply to it the Rouche theorem is much more
technically complicated (although possible). Therefore we are not going
to do it in this paper because we can overcome that problem as follows :
let us observe that Th.1 gives in fact a stronger result, i.e. it holds
{\bf without any assumption} on the vanishing of $\Gamma_{r}(\omega_{A}^{-})$.
Reely, beside the fact that we have not any exact information on the
zero-dimensional manifold $\Gamma_{r}(\omega_{A}^{-})(\mbox{\lll \char67}):=\{s \in \mbox{\lll \char67} :
\Gamma_{r}(A_{n}^{-})(s)=0\}$, the meromorphic functions :
$\Gamma_{r}(A_{n}^{-})(s)\zeta_{k}(s)$ and
$\int_{1}^{\infty}(t^{s-1}-t^{-s})\Theta_{k}(\omega_{A}^{-})(t)dt$ are
{\bf well-defined} for $Re(s)>0$ and - according to (Face) - {\bf
coincides} for $Re(s)>1$. Hence, according to the uniqueness of the
continuation of the analytic functions in regions - they must be equal
in $Re(s)>0$.
\begin{th}({\bf Existence of $gRhfe_{k}^{-}$}).
A pair of two ${\bf \Gamma \theta sinlog-factors} (F_{id},F_{c})$ indexed
by the Galois group $Gal(\mbox{\lll \char67}/\mbox{\lll \char82}) = \{id, c\}$ and another pair $(f_{1},
f_{2})$ of {\bf $\theta$sinlog-factors} satisfying
\begin{equation}
f_{1}(s)\;+\;f_{2}(s)\;\ne\;0\;for\;Re(s)\in (1/2,1]
\end{equation}
exist, such that the following $gRhfe_{k}^{-}$ ( with rational term $I$ and
the action of $Gal(\mbox{\lll \char67}/\mbox{\lll \char82})$) holds
\begin{equation}
Im(\sum_{g \in
Gal(\mbox{\lll \char67}/\mbox{\lll \char82})}(F_{g}\zeta_{k})(g(s))\;=\;\frac{\lambda_{k}(f_{1}(s)+f_{2}(s))}
{\mid s(s-1) \mid}I(s).
\end{equation}
\end{th}
{\bf Proof}. (I). {\bf The derivation of $gRhfe_{k}^{-}$}.
Let $a_{2}>a_{1}>0$ be arbitrary {\bf artificially chosen
$\zeta_{k}$-Cramer initial condition} and let $s=u+iv=Re(s)+iIm(s)$ be
fixed. We consider a simple {\bf non-homogeneous system} of two linear
equations in two variables $p_{1}$ and $p_{2}$ of the form :
\begin{equation}
p_{1}v(u-1)\;+\;p_{2}vu\;=\;a_{1}\;-\;a_{2}
\end{equation}
\begin{displaymath}
p_{1}vu\;+\;p_{2}v(u-1)\;=\;a_{2}\;-\;a_{1}.
\end{displaymath}
This system is a {\bf Cramer system}, iff $s$ does not belong to
the algebraic $\mbox{\lll \char82}$-variety $I(\mbox{\lll \char67})$. The main determinant of (4.104) is
$I(s)$ and its solution is given by the formulas
\begin{equation}
p_{1}\;=\;p_{1}(Im(s))\;=\;\frac{(a_{2}\;-\;a_{1})}{v}\;>\;0
\end{equation}
and
\begin{equation}
p_{2}\;=\;p_{2}(Im(s)) \;=\;\frac{(a_{1}\;-\;a_{2})}{v}\;=-p_{1}<\;0.
\end{equation}
Let $A^{-}$ be an {\bf amplitude} chosen according to the Proposition 5. Then
according to Proposition 5, there exists a {\bf (-)RH-fixed point}
$\omega_{A}^{-}$ of ${\cal F}_{n}$, i.e.
\begin{equation}
\omega^{-}_{A}\;+\;H_{2}\;=\;A^{-}.
\end{equation}
In the sequel we simply write $\omega_{1} = \omega^{-}_{A}$.
We denote the standard n-dimensional second Hermite (-)fixed point of
${\cal F}_{n}$ by $\omega_{2} = H_{2} = H_{2}^{n}$.
We set (cf.(2.48))
\begin{equation}
J_{i}(s)\;:=\;\int_{1}^{+\infty}(t^{u-1}\;+\;t^{-u})sin(vlogt)\Theta_{k}
(\omega_{i})(t)dt\;,\;i=1,2.
\end{equation}
The integrals $J_{i}$ above are {\bf quasi-invariant} under the substitutions
: $t = x^{r}, r>0$, i.e. the substitution $t = x^{p_{1}v}, v>0$
gives
\begin{equation}
J_{1}(s)=p_{1}v
\int_{1}^{\infty}(x^{p_{1}v(u-1)}+x^{-p_{1}vu})sin(p_{1}v^{2}logx)\Theta
_{k}(\omega_{1})(x^{p_{1}v})x^{(p_{1}v-1)}dx=:J_{1}^{r}(s)
\end{equation}
In the same way, the substitution $t=x^{-p_{2}v}, v>0$ gives
\begin{equation}
J_{2}(s)=-p_{2}v\int_{1}^{\infty}(x^{-p_{2}v(u-1)}+x^{p_{2}vu})sin(-p_{2}
v^{2}logx)\Theta_{k}(\omega_{2})(x^{-p_{2}v})x^{-(p_{2}v+1)}dx=:
J_{2}^{r}(s).
\end{equation}
Thus, the equalities $J_{i}(s) = J_{i}^{r}(s), i=1,2$ hold on the
domain $\{s \in \mbox{\lll \char67}: Im(s) \ge 0\}$. But obviously the integrals are
imaginary parts of the {\bf analytic} function $\Gamma_{(r_{1},r_{2})}(\omega
_{i})\zeta_{k}-\lambda_{k}/W$ defined on $\mbox{\lll \char67}-\{0,1\}$. Hence, they
must be equal everywhere. In particular, the second equality is {\bf
invariant} to the operation of {\bf complex conjugation} $c$, i.e.
\begin{equation}
J_{2}(c(s))=p_{2}v\int_{1}^{\infty}(x^{p_{2}v(u-1)}+x^{p_{2}vu})sin(-p_{2}v^{2}
log x)\Theta_{k}(\omega_{2})(x^{p_{2}v})x^{p_{2}v-1}dx=J_{2}^{r}(c(s)).
\end{equation}
Since $\omega_{i} \in {\cal S}(\mbox{\lll \char82}^{n})$, for each $q>1$ we have
\begin{displaymath}
max_{x\ge 1}\mid x^{q} \Theta_{k}(\omega_{i})(x^{p_{i}v})\mid
<\infty.
\end{displaymath}
According to the {\bf elementary mean value theorem}, there exists such
an $x_{i} = x_{i}(s,a_{1},a_{2}) \in [1,\infty)$ and $q = q(a_{1},a_{2},u)>1$
that
\begin{equation}
J_{i}(c_{i}(s))=p_{i}vsin((-1)^{i+1}p_{i}v^{2}logx_{i})x^{q}_{i}\Theta_{k}
(\omega_{i})(x_{i}^{p_{i}v})\int_{1}^{\infty}(x^{p_{i}v(u-1)-a_{i}}+
x^{-p_{i}vu-a_{i}})x^{-q}dx
\end{equation}
\begin{displaymath}
\;=:\;f_{i}(s)\int_{i}(s),
\end{displaymath}
where $c_{1}=id$ and $c_{2}=c$.
The number $q$ is obviously chossen in such a way that the integrals
$\int_{i}(s)$ are convergent.
Using the (Face) (cf.(2.36)) and the nation from (4.122) we obtain
\begin{equation}
Im((\Gamma_{(r_{1},r_{2})}(\omega_{i})\zeta_{k})(c_{i}(s))=
\frac{\lambda_{k}I(c_{i}(s))}{\mid s(s-1)
\mid^{2}}+f_{i}(s)\int_{i}(s),
\end{equation}
or equivalently
\begin{equation}
Im((\Gamma_{(r_{1},r_{2})}(\omega_{1})f_{2}\zeta_{k}))(s)=\frac{(f_{2}I)(s)}
{\mid s(s-1) \mid^{2}}\;+\;(f_{1}f_{2})(Im(s))\int_{1}(s),
\end{equation}
together with
\begin{equation}
Im(\Gamma_{(r_{1},r_{2})}(\omega_{2})f_{1}\zeta_{k})(c(s))=\frac{-(f_{1}I)
(s)}{\mid s(s-1) \mid^{2}}\;+\;(f_{1}f_{2})(Im(s))\int_{2}(s).
\end{equation}
By defining the {\bf $\Gamma \theta$ sinlog-factors} as
\begin{equation}
F_{id}(s):=(\Gamma_{(r_{1},r_{2})}(\omega_{1}f_{2}))(s)\;and\;F_{c}(s):=
(\Gamma_{(r_{1},r_{2})}(\omega_{2}f_{1}))(s),
\end{equation}
and substrating (4.125) from (4.124), according to the choice of the pair
$(p_{1},p_{2})$ in (4.114) (which is the solution of the Cramer system) we
finally obtain $(gRhfe_{k}^{-})$.
(II).{\bf Positivity of $Tr(\zeta_{k},A)$} (It is a very subtle "game"
of signs - on the bourder of subtlety) .
According to the construction of $\omega^{A}$, we have
\begin{equation}
A\;=\;\omega_{1}\;+\;\omega_{2}.
\end{equation}
By Proposition 3 on the positivity of the trace, we have
\begin{equation}
0<Tr_{-}(\zeta_{k},A)(s)\;=\;(J_{1}+J_{2})(s)\;=\;J_{1}(s)+J_{2}(m(s))\;=
\end{equation}
\begin{displaymath}
J_{1}(s)\;+\;J_{2}(c(a(s))),
\end{displaymath}
where - for a moment - we denoted the affinic antyconjugation as
$a(s):= (1-u)+iv$, and
\begin{equation}
J_{2}(s)\;=\;Im(\int_{1}^{\infty}(t^{s-1}-t^{-s}))\Theta_{k}(t)dt= -J_{2}
(m(s)).
\end{equation}
Moreover, on the basis of the notation in (4.122) we have
\begin{displaymath}
f_{2}(s)\;=\;\frac{J_{2}(c(s))}{\int_{2}(s)},
\end{displaymath}
and therefore
\begin{equation}
f_{2}(a(s))\;=\;-f_{2}(s)\;and\;\int_{2}(a(s))\;=\;\int_{2}(s).
\end{equation}
Since the pair $(p_{1},p_{2})$ is the solution of the Cramer system
(4.104), we obtain
\begin{equation}
-\int_{2}(s)\;=\;\int_{1}(s).
\end{equation}
Hence, combining (4.128), (4.130) and (4.131) we finally obtain
\begin{equation}
0<Tr_{-}(\zeta_{k},A)(s)=f_{1}(s)\int_{1}(s)\;+\;f_{2}(a(s))\int_{2}(a(s))\;
=
\end{equation}
\begin{displaymath}
\;\;\;=\;\int_{1}(s)(f_{1}(s)\;+\;f_{2}(s)),
\end{displaymath}
i.e.
\begin{displaymath}
\;\;\;f_{1}(s)\;+\;f_{2}(s)\;\ne\;0\;for\;Re(s)\in (1/2,1]
\end{displaymath}
which proves Theorem 3.
\begin{re}
It is a very exciting fact that to prove ($gRH_{k}$) we need only {\bf
two} functional equations for $\zeta_{k}(s)$!, whereas - among number
theory specialists - we have met with the quite opposite opinion - that
even infinitely many f.e. for $\zeta_{\mbox{\lll \char81}}(s)$ are {\bf not sufficient}
to proof (RH)! (e.g. H. Iwaniec).
\end{re}
Obviously $(gRhfe_{k}^{-})$ immediately implies the {\bf generalized
Riemann Hypothesis}. Assume that there exists a zero $s_{0}$ of
$\zeta_{k}$ in the set $\{s \in \mbox{\lll \char67}: Re(s \in (1/2,1]\}$. Then
\begin{displaymath}
\sum_{g \in Gal(\mbox{\lll \char67}/\mbox{\lll \char82})}(F_{g}\zeta_{k})(g(s_{0}))\;=\;0,
\end{displaymath}
since, according to HRace, the zeros of zeta lie symmetrically with
respect to the lines : $Im(s) = 0$ and $Re(s) = 1/2$. But, on the other
hand, we have
\begin{displaymath}
\frac{(f_{1}\;+\;f_{2})(s_{0})}{\mid s_{0}(s_{0}-1)
\mid^{2}}I(s_{0})\;\ne\;0,
\end{displaymath}
which is impossible according to $(gRhfe_{k}^{-})$.
\begin{re}
The CWS-inequality
\begin{displaymath}
Tr_{Gal(\mbox{\lll \char67}/\mbox{\lll \char82})}^{k}(s)\;=\;Tr^{k}_{G}(s)\;:=\;\frac{\lambda_{k}(f_{1}+
f_{2})(s)}{\mid s(s-1) \mid^{2}}\;>\;0,
\end{displaymath}
is {\bf exceptional}(fundamental) to the proof of $(gRH_{k})$. That is
very surprising that similar kinds of positivity conditions (explored
also in [$M_{A}$], $[M_{H}]$ and [AM]) are strictly connected with (RH) :
In $[M_{CG}]$, based on $[M_{L}]$ we showed that the positivity of the Cauchy-Gaussian trace
$Tr_{CG}$ implies the Riemann hypothesis.
In [B] de Branges showed that the positivity of his trace $Tr_{B}$
would imply the Riemann hypothesis (also in the case of some
$L$-functions).
Below we briefly remind the reader that the positivity of the {\bf Weil trace}
$Tr_{W}$ leads to the Riemann hypothesis.
As it is well-known (cf.e.g.[L, XVII.3]), A. Weil formulated an
equivalent form of the Riemann hypothesis (the {\bf Weil Formula} (WF
in short)) in terms of the {\bf positivity} of his functional: let
${\cal SB}(\mbox{\lll \char82})$ be the {\bf restricted Barner-Schwartz space} of all
functions of the form
\begin{displaymath}
F(x)\;=\;P(x)e^{-Kx^{2}}
\end{displaymath}
with some real constant $K>0$ and some polynomial $P$ (cf.[L,XVII.3]).
Then ${\cal SB}(\mbox{\lll \char82})$ is self-dual, and functions from this space
satisfy the {\bf three Barner conditions} (cf.[L]) : finitness of
variation, Dirichlet normalization and asymptotic symmetry at zero.
For each $F \in {\cal SB}(\mbox{\lll \char82})$ is well-defined its {\bf conjugation}
\begin{displaymath}
F^{*}(x)\;:=\;F(-x),
\end{displaymath}
and $F$ is of {\bf positive type} if $F$ is equal to its {\bf Rosatti
convolution}
\begin{displaymath}
F\;=\;F_{0} * F_{0}^{*},
\end{displaymath}
for some $F_{0} \in {\cal SB}(\mbox{\lll \char82})$. (So we see that ${\cal SB}(\mbox{\lll \char82})$
is also closed under the convolution $*$).
For $s = \sigma +it$, we can consider the {\bf two-sided
Laplace-Fourier transform}
\begin{displaymath}
\hat{F}(s)\;:=\;\int_{\mbox{\lll \char82}}F(x)e^{(1/2-\sigma)x}e^{itx}dx,
\end{displaymath}
and the {\bf Weil functional} $W$ defined as
\begin{displaymath}
W_{k}(\Phi)\;:=\;\sum_{\rho, \zeta_{k}(\rho)=0, Im(\rho)\ne 0}\Phi(\rho).
\end{displaymath}
Then the Riemann hypothesis is equivalent to the positivity of Weil's trace
\begin{equation}
Tr_{W}(F_{0})\;:=\;W_{k}(F_{0}* F^{*}_{0})\;\;\ge\;0,
\end{equation}
for all $F_{0} \in {\cal SB}(\mbox{\lll \char82})$.
{\bf Weil's condition} is much more general.
Let $k$ be a number field, $\chi$ a {\bf Hecke character}, ${\cal f}_{\chi}$
the {\bf conductor}, ${\cal D}$ the {\bf local different} and $d_{\chi} =
N({\cal D}f_{\chi})$.
Let us consider the $L^{*}_{k}$-function
\begin{equation}
L_{k}^{*}(s;\chi)\;:=\;[(2\pi)^{-n(k)}2^{r_{1}}d_{\chi}]^{s/2}\prod_{v
\in S_{\infty}(k)}\Gamma(s_{v}/2)L(s;\chi),
\end{equation}
where $L(s;\chi)$ is the {\bf Hecke $L$-function} associated with
$\chi$, i.e. the usual product over unramified prime ideals for $\chi$
and $s_{v} := N_{v}(s+i\phi_{v})+ \mid m_{v} \mid$ (cf.[L]).
{\bf Weil's functional} $W$ in this case is obviously the sum
\begin{displaymath}
W_{L}(F)\;=\;\sum_{L(\rho, \chi)=0}F(\rho)\;\;,\;\;F\in {\cal SB}(\mbox{\lll \char82}).
\end{displaymath}
In short, the {\bf generalized Riemann hypothesis} for $L_{k}^{*}(\cdot;\chi)$,
$gRH_{k}(\chi)$, states that $Re(\rho)=\frac{1}{2}$ for all zeros $\rho$ of
$L_{k}(\cdot;\chi)$ in the critical strip.
Well-known {\bf Weil's theorem}(cf.[L,Th.3.3]) asserts that $gRH_{k}(\chi)$ is
equivalent to the property that
\begin{equation}
(WC)\forall(F_{0} \in {\cal SB}(\mbox{\lll \char82}))(W_{L}(F_{0}*F_{0})\ge 0).
\end{equation}
\end{re}
In particular, we have thus proved Weil's theorem for the Dedekind
zetas.
\begin{th}
For all $F_{0}$ in the restricted Schwartz space ${\cal SB}(\mbox{\lll \char82})$ the
following holds
\begin{displaymath}
W_{\zeta_{k}}(F_{0}* F_{0}^{*})\;\ge \;0.
\end{displaymath}
\end{th}
\section{The generalized Riemann hypothesis for all Dirichlet
L-functions}
A first generalization of the Riemann zeta function comes from
{\bf Dirichlet}[Di], who for a character $\chi$ of $(\mbox{\lll \char90}/m \mbox{\lll \char90})^{*}$,
that is, a homomorphism from $(\mbox{\lll \char90}/ m \mbox{\lll \char90})^{*}$ to $\mbox{\lll \char67}^{*}$, considered
the series
\begin{equation}
L(s, \chi)\;:=\;\sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}},
\end{equation}
where $\chi(n):= \chi([n])$ for $(n,m)=1$ and $\chi(n)=0$ for $(n,m)\ne
1$. He used these L-series to prove his theorem on primes in arithmetic
progressions, in which of principal importance is the fact that the
value of $L(s, \chi)$ is nonzero at the point $s=1$.
Let $m$ be a natural number amd $\zeta_{m}$ a primitive mth root of
unity, that is, a complex number with $\zeta_{m}^{m}=1$ and
$\zeta_{m}^{i}\ne 1$ for $1 \le i \le m$. In this section we consider
extensions $k$ that arise from $\mbox{\lll \char81}$ through the adjunction of roots of
unity. The field $k = \mbox{\lll \char81}(\zeta_{m})$ is called the {\bf mth cyclotomic
field}, since as points in the complex plane they divide the circle
into equal arcs (see [K, Sect. 6.4]).
Since the development by Kummer of the theory of cyclotomic fields (see
e.g. [K]) one proves $L(1, \chi)\ne 0$ for characters $\chi$ different
from the {\bf trivial character} $\chi_{0}$ ( $L(s, \chi_{0})$ has a simple
pole at $s=1$) most naturally with the help of the following result
(see [K, Sect.8.2, Th.8.2.1.]) :
for any integer $ m \in \mbox{\lll \char78}$
\begin{equation}
\zeta_{\mbox{\lll \char81}(\zeta_{m})}(s)\;=\;\prod_{p \mid
m}(1\;-\;\frac{1}{N(p)^{s}})^{-1}\prod_{\chi}L(s, \chi),
\end{equation}
where the right-hand product runs over all characters of $(\mbox{\lll \char90}/ m
\mbox{\lll \char90})^{*}$.
\begin{th}({\bf $gRH_{m}$ for Dirichlet L-functions})
Let $m$ be any positive integer and $\chi_{m} : \mbox{\lll \char70}_{m}^{*}= (\mbox{\lll \char90}/ m\mbox{\lll \char90})^{*}
\longrightarrow \mbox{\lll \char67}$ any character of the multiplicative group of the
finite ring $\mbox{\lll \char70}_{m}$. Let also $\chi_{m}$ be corresponding {\bf
Dirichlet character}. Then the following implication is true :
\begin{equation}
(gRH_{m})\;If\;L(s, \chi_{m})=0\;with\;Im(s)\ne
0\;then\;Re(s)=1/2.
\end{equation}
In particular, the {\bf Weil trace} $Tr_{W,m}(F_{0}) :=
\sum_{\rho, L(\rho,\chi_{m})=0, Im(\rho)\ne 0 } (F_{0}(\rho)*F_{0}(\rho))$
associated with the L-function $L(s, \chi_{m})$ is {\bf positive}, i.e.
\begin{equation}
Tr_{W,m}(F_{0})\;\ge\;0,
\end{equation}
for all $F_{0}$ from the Barner-Schwartz space ${\cal SB}(\mbox{\lll \char82})$.
\end{th}
{\bf Proof}. Assume (a contrary) that there is a {\bf zero} $s_{0}$ of
$L(s, \chi_{m})$ in the domain : $Re(s)\in (0,1)-\{1/2\},
Im(s)\ne 0$ of $\mbox{\lll \char67}$. Then, according to the "spliting formula" (5.137) we
obtain that
\begin{displaymath}
\zeta_{\mbox{\lll \char81}(\zeta_{m})}(s_{0})\;=\;0,
\end{displaymath}
what obviously is not possible according to $gRH_{k}$.
Thus, the generalized Riemann hypothesis for Dirichlet L-functions -
according to (5.137) - is directly and immediately reduced (or is the
consequence) of the generalized Riemann hypothesis for Dedekind zetas -
proved in the previous Section.
e-mail: [email protected]
\end{document} |
\begin{document}
\hspace{13.9cm}1
\
\\
{\LARGE Information Geometrically Generalized Covariate Shift Adaptation}
\ \\
{\bf \large Masanari Kimura$^{\displaystyle 1}$, Hideitsu Hino$^{\dagger, \displaystyle 2, \displaystyle 3}$}\\
{$^{\displaystyle 1}$SOKENDAI, Graduate University for Advanced Studies.\\
Shonan Village, Hayama,
Kanagawa 240-0193 Japan}\\
{$^{\displaystyle 2}$The Institute of Statistical Mathematics.\\
10-3 Midori-cho, Tachikawa, Tokyo 190-8562, Japan}\\
{$^{\displaystyle 3}$Center for
Advanced Intelligence Project, RIKEN.\\
1-4-1 Nihonbashi, Chuo-ku, Tokyo 103-0027, Japan}\\
$^{\dagger}$corresponding author: [email protected]
{\bf Keywords:} Information Geometry, Domain Adaptation, Covariate Shift
\thispagestyle{empty}
\markboth{}{NC instructions}
\
\\
\begin{center} {\bf Abstract} \end{center}
Many machine learning methods assume that the training and test data follow the same distribution. However, in the real world, this assumption is very often violated. In particular, the phenomenon that the marginal distribution of the data changes is called covariate shift, one of the most important research topics in machine learning. We show that the well-known family of covariate shift adaptation methods is unified in the framework of information geometry. Furthermore, we show that parameter search for geometrically generalized covariate shift adaptation method can be achieved efficiently. Numerical experiments show that our generalization can achieve better performance than the existing methods it encompasses.
\section{Introduction}
When considering supervised learning methods, it is often assumed that the training and test data follow the same distribution~\citep{Bishop1995-mz, Duda2006-rd, Hastie2009-is, Vapnik2013-nm, Mohri2018-rw}.
However, this common assumption is violated in the real world in most cases~\citep{Huang2007-bm,Zadrozny2004-sa,Cortes2008-zd,Quionero-Candela2009-vg,Jiang2008-dd}.
Covariate shift~\citep{Shimodaira2000-vv} is a prevalent setting
for supervised learning in the real world, where the input distribution differs in the training and test phases, but the conditional distribution of the output variable given the input variable remains unchanged.
Covariate shift is a commonly observed phenomenon in real-world machine learning applications, such as emotion recognition~\citep{Hassan2013-nf,Jirayucharoensak2014-df}, 3D pose estimation~\citep{Yamada2012-ov}, brain computer interfaces~\citep{Li2010-pe,Raza2016-py}, spam filtering~\citep{Bickel2009-wq}, and human activity recognition~\citep{Hachiya2012-vo}.
In addition, there has been recent discussion on the relationship between covariate shift and the robustness of deep learning~\citep{Ioffe2015-ns, Arpit2016-ao, Santurkar2018-wy, Nado2020-ts, Huang2020-zx, Awais2020-wv}.
Ordinary empirical risk minimization (ERM)~\citep{vladimirvapnik1998, Vapnik2013-nm} may not generalize well to the test data under covariate shift because of the difference between the training and test distributions.
However, importance weighting for training examples has been shown to be effective in mitigating the effect of covariate shift~\citep{Shimodaira2000-vv, Sugiyama2005-lv, Sugiyama2005-nr, Zadrozny2004-sa}.
The main idea of these strategies is weighting the training loss terms according to their importance, which is the ratio of the training input density to the test input density. The importance weighting is widely adopted even in modern covariate shift studies with deep neural networks (DNN)~\citep{DBLP:conf/nips/FangL0S20,DBLP:journals/sncs/ZhangYLS21}.
In this paper, we consider the generalization of these methods in the framework of information geometry~\citep{Amari1985-mi, Amari2007-wb}, a tool that allows us to deal with probability distributions on Riemannian manifolds.
This generalization makes it possible to search for good weighting without searching for a large number of parameters.
Our contributions is summarized as follows:
\begin{itemize}
\item (Section~\ref{subsec:information_geometrically_generalized_iwerm} and ~\ref{subsec:geometric_bias}) We generalize existing methods of covariate shift adaptation in the framework of information geometry. By our information geometrical formulation, geometric biases of conventional methods are elucidated.
\item (Section~\ref{subsec:optimization_of_the_generalized_iwerm}) We show that our geometrically generalized covariate shift adaptation method has a much larger solution space than existing methods controlled by only two parameters. Efficient weighting is obtained by searching for parameters using an information criterion or Bayesian optimization.
\item (Section~\ref{sec:numerical_experiments}) Numerical experiments show that our generalization can achieve better performance than the existing methods it encompasses.
\end{itemize}
\section{Preliminaries}
\subsection{Problem formulation}
First, we formulate the problem of supervised learning.
We denote by $\mathcal{X}\subset\mathbb{R}^d$ the input space.
The output space is denoted by $\mathcal{Y}\subset\mathcal{R}$ (regression) or $\mathcal{Y}\subset\{1,\dots,K\}$ ($K$-class classification).
We assume that training examples $\{(\bm{x}^{tr}_i, y^{tr}_i)\}^{n_{tr}}_{i=1}$ are independently and identically distributed (i.i.d.) according to some fixed but unknown distribution $p_{tr}(\bm{x}, y)$, which can be decomposed into the marginal distribution and the conditional probability distribution, i.e., $p_{tr}(\bm{x},y)=p_{tr}(\bm{x})p_{tr}(y|\bm{x})$.
We also denote the test examples by $\{(\bm{x}^{te}_i,y^{te}_i)\}^{n_{te}}_{i=1}$ drawn from a test distribution $p_{te}(\bm{x}, y) = p_{te}(\bm{x})p_{te}(y|\bm{x})$.
Let $\mathcal{H}$ be a hypothesis class.
The goal of supervised learning is to obtain a hypothesis $h:\mathcal{X}\to\mathbb{R}\ (h\in\mathcal{H})$ with the training examples that minimizes the expected loss over the test distribution:
\begin{equation}
\mathcal{R}(h) \coloneqq \mathbb{E}_{(\bm{x}^{te},y^{te})\sim p_{te}(\bm{x},y)}\Big[\ell(h(\bm{x}^{te}), y^{te})\Big], \label{eq:expected_loss}
\end{equation}
where $\ell: \mathbb{R}\times\mathcal{Y}\to\mathbb{R}$ is the loss function that measures the discrepancy between the true output value $y$ and the predicted value $\hat{y}\coloneqq h(\bm{x})$.
In this paper, we assume that $\ell$ is bounded from above, i.e., $\ell(y,y')<\infty\ (\forall y,y'\in\mathcal{Y})$.
\begin{defi}[Covariate shift assumption]
\label{def:covariate_shift_adaptation}
We consider that the two distributions $p_{tr}(\bm{x}, y)$ and $p_{te}(\bm{x},y)$ satisfy the covariate shift assumption if the following three conditions hold: 1) $p_{tr}(\bm{x}) \neq p_{te}(\bm{x})$, 2) $\supp(p_{tr}(\bm{x})) \supset \supp(p_{te}(\bm{x}))$ and 3) $p_{tr}(y|\bm{x}) = p_{te}(y|\bm{x})$.
\end{defi}
Under the covariate shift assumption, the goal of covariate shift adaptation is still to obtain a hypothesis $h$ that minimizes the expected loss~\eqref{eq:expected_loss} by utilizing both labeled training examples $\{(\bm{x}^{tr}_i, y^{tr}_i)\}^{n_{tr}}_{i=1}$ and unlabeled test examples $\{(\bm{x}^{te}_i)\}^{n^{te}}_{i=1}$.
\subsection{Previous works}
Ordinary empirical risk minimization (ERM)~\citep{vladimirvapnik1998,Vapnik2013-nm}, a standard approach in supervised learning, may fail under the covariate shift because it assumes that the training and test data follow the same distribution.
Importance weighting has been shown to be effective in mitigating the effect of covariate shift~\citep{Shimodaira2000-vv,Sugiyama2005-lv,Sugiyama2007-lr,Zadrozny2004-sa}:
\begin{equation}
\min_{h\in\mathcal{H}}\frac{1}{n_{tr}}\sum^{n_{tr}}_{i=1}w(\bm{x}^{tr}_i)\ell(h(\bm{x}^{tr}_i), y^{tr}_i), \label{eq:weighted_empirical_loss}
\end{equation}
where $w:\mathcal{X}\to\mathbb{R}_{\geq 0}$ is a certain weighting function.
\begin{defi}[IWERM~\citep{Shimodaira2000-vv}]
If we choose the density ratio $p_{te}(\bm{x})/p_{tr}(\bm{x})$ as the weighting function, ERM according to \begin{equation}
\min_{h\in\mathcal{H}}\frac{1}{n_{tr}}\sum^{n_{tr}}_{i=1}\frac{p_{te}(\bm{x}^{tr}_i)}{p_{tr}(\bm{x}^{tr}_i)}\ell(h(\bm{x}^{tr}_i), y^{tr}_i) \label{eq:iwerm}
\end{equation}
has consistency.
\end{defi}
This is called importance weighted ERM (IWERM).
However, IWERM tends to produce an estimator with high variance.
We can reduce the variance by flattening the importance weights, which is called adaptive IWERM (AIWERM):
\begin{defi}[AIWERM~\citep{Shimodaira2000-vv}]
Let $\lambda\in[0,1]$.
If we choose $(p_{te}(\bm{x})/p_{tr}(\bm{x}))^{\lambda}$ as the weighting function, we can obtain the variance-reduced estimator:
\begin{equation}
\min_{h\in\mathcal{H}}\frac{1}{n_{tr}}\sum^{n_{tr}}_{i=1}\Big(\frac{p_{te}(\bm{x}^{tr}_i)}{p_{tr}(\bm{x}^{tr}_i)}\Big)^\lambda\ell(h(\bm{x}^{tr}_i), y^{tr}_i). \label{eq:aiwerm}
\end{equation}
\end{defi}
Relative IWERM (RIWERM), a stable version of AIWERM, has also been proposed:
\begin{defi}{(RIWERM~\citep{Yamada2011-ws})}
Let $\lambda\in[0,1]$.
If we choose $p_{te}(\bm{x})/\lambda p_{tr}(\bm{x}) + (1-\lambda) p_{te}(\bm{x})$ as the weighting function, we can directly estimate a flattened version of the importance weight:
\begin{equation}
\min_{h\in\mathcal{H}}\frac{1}{n_{tr}}\sum^{n_{tr}}_{i=1}\frac{p_{te}(\bm{x}^{tr}_i)}{\lambda p_{tr}(\bm{x}^{tr}_i) + (1-\lambda) p_{te}(\bm{x}^{tr}_i)}\ell(h(\bm{x}^{tr}_i), y^{tr}_i). \label{eq:riwerm}
\end{equation}
\end{defi}
All of the above methods are considered as different weighting methods for each point of the training data.
More generally, the method of covariate shift adaptation can be essentially rephrased as a weighting strategy for training data.
\section{Statistical Model and Exponential Family}
Information geometry~\citep{Amari1985-mi, Amari2007-wb} is a powerful framework that allows us to deal with statistical models on Riemannian manifolds. For theoretical investigation, we need the notion of dual connection and curvature tensor associated with Fisher metric, but these details are deferred to the Appendix~\ref{app:mfd} and we here present minimum required definitions and notations. We note that the assumption on the parametric family is only required for the information geometric analysis in Section~\ref{subsec:geometric_bias}. The algorithmic framework of the proposed method is independent of the parametric model.
Since $p_{tr}(y|\bm{x})=p_{te}(y|\bm{x})=p(y|\bm{x})$ from the assumption of Definition~\ref{def:covariate_shift_adaptation},
what we are interested in is the model manifold $(\mathcal{M}, g(\bm{\theta}))$ to which the marginal distribution $p(\bm{x};\bm{\theta})$ belongs:
\begin{equation}
\mathcal{M} = \Big\{ p(\bm{x}; \bm{\theta})\ ; \bm{\theta}\in\Theta \Big\}.
\end{equation}
Here, $p_{tr}(\bm{x}; \bm{\theta}), p_{te}(\bm{x}; \bm{\theta})\in\mathcal{M}$. We note that elements in $\mathcal{M}$ is specified by its parameter $\bm{\theta}$ and we identify the parameter vector $\bm{\theta}$ to the density function $p(\bm{x}; \bm{\theta})$ and write $p(\bm{x};\bm{\theta}) \simeq \bm{\theta}$ if necessary.
In this paper, we assume that $\mathcal{M}$ is an exponential family and the probability density function can be written as
\begin{equation}
p(\bm{x}; \bm{\theta}) = \exp\Big\{ \theta^iT_i(\bm{x}) + k(\bm{x}) - \psi(\bm{\theta})\Big\}, \label{eq:exponential_family}
\end{equation}
where $\bm{x}$ is a random variable, $\bm{\theta}=(\theta^1,\dots,\theta^p)$ is an $p$-dimensional vector parameter to specify a distribution, $\bm{T}(\bm{x}) = (T_1(\bm{x}),\dots,T_p(\bm{x}))$ are sufficient statistics of $\bm{x}$, $k(\bm{x})$ is a function of $\bm{x}$ and $\psi$ corresponds to the normalization factor.
In Eq.~\eqref{eq:exponential_family}, and hereafter the Einstein summation convention will be assumed, so that summation will be automatically taken over indices repeated twice in the term, e.g., $\bm{a}^i\bm{b}_i = \sum_{i} \bm{a}^i\bm{b}_i$.
In the exponential family, the natural parameter $\bm{\theta}$ forms the affine coordinate system, i.e.,
\begin{equation}
\bm{\theta}(t) = (1-t)\bm{\theta}_1 + t\bm{\theta}_2\ \ (\forall \bm{\theta}_1,\bm{\theta}_2\in\Theta,\ \forall t\in[0,1])
\end{equation}
is a geodesic on $\mathcal{M}$. As a dual coordinate of $\bm{\theta}$, the expectation parameter $\bm{\eta}$ is defined by the Legendre transformation
\begin{align*}
\bm{\eta} =& \nabla\psi(\bm{\theta}), \quad
\bm{\theta} = \nabla\varphi(\bm{\eta}),\\ \mbox{where} \; \;
\varphi(\bm{\eta}) =& \max_{\bm{\theta}'}\Big\{\bm{\theta}'\cdot\bm{\eta} - \psi(\bm{\theta}')\Big\}.
\end{align*}
Existing weights for covariate shift adaptation are geometrically characterized, then a generalized weight function is designed based on this geometric formulation.
\section{Geometrical Generalization of Covariate Shift Adaptation}
\subsection{Information Geometrically Generalized IWERM}
\label{subsec:information_geometrically_generalized_iwerm}
In order to derive a generalized covariate shift adaptation method, we prepare the following function.
\begin{defi}[$f$-interpolation~\citep{e23050528}]
\label{def:f_interpolation}
For any $a,b,\in\mathbb{R}$, some $\lambda\in[0,1]$ and some $\alpha\in\mathbb{R}$, we define $f$-interpolation as
\begin{equation}
m_f^{(\lambda,\alpha)}(a,b) = f^{-1}_\alpha\Big\{(1-\lambda) f_{\alpha}(a) + \lambda f_{\alpha}(b) \Big\},
\end{equation}
where
\begin{equation}
f_\alpha(a) = \begin{cases}
a^{\frac{1-\alpha}{2}} & (\alpha\neq 1) \\
\log a & (\alpha = 1)
\end{cases}
\end{equation}
is the function that defines the $f$-mean~\citep{hardy1952inequalities}.
\end{defi}
We can easily see that this family includes various known weighted means including the $e$-mixture and $m$-mixture for $\alpha=\pm 1$ in the literature of information geometry~\citep{Amari2016-pi}:
\begin{align*}
m_f^{(\lambda,1)}(a,b) =& \exp\{(1-\lambda)\log a + \lambda \log b\},\\
m_f^{(\lambda, -1)}(a,b) =& (1-\lambda)a + \lambda b,\\
m_f^{(\lambda, 0)}(a,b) =& \Big((1-\lambda)\sqrt{a} + \lambda\sqrt{b}\Big)^2, \\
m_f^{(\lambda, 3)}(a,b) =& \frac{1}{(1-\lambda)\frac{1}{a} + \lambda\frac{1}{b}}.
\end{align*}
Also, for any $\bm{u},\bm{v}\in\mathbb{R}^d\ (d>0)$, we write
\begin{align*}
\bm{m} = m_f^{(\lambda, \alpha)}(\bm{u}, \bm{v}),
\mbox{where}
\quad
\bm{m}_i = m_f^{(\lambda, \alpha)}(\bm{u}_i, \bm{v}_i).
\end{align*}
Using this function, we generalize the existing methods of covariate shift adaptation.
\begin{lemm}[$f$-representation of AIWERM]
\label{lem:aiwerm}
The marginal positive measures generated by the weighting of AIWERM can be expressed by using the $f$-interpolation function as
\begin{equation}
p_A^{(\lambda)}(\bm{x}) = m_f^{(\lambda, 1)}(p_{tr}(\bm{x}), p_{te}(\bm{x})).
\end{equation}
\end{lemm}
\begin{proof}
From~Eq~.\eqref{eq:aiwerm}, we consider its expectation as
\begin{align*}
\hat{h} =& \min_{h\in\mathcal{H}}\int_{\mathcal{X}\times\mathcal{Y}}\Big(\frac{p_{te}(\bm{x})}{p_{tr}(\bm{x})}\Big)^\lambda \ell(h(\bm{x}), y) p_{tr}(\bm{x},y) d\bm{x}dy \\
=& \min_{h\in\mathcal{H}}\int_{\mathcal{X}\times\mathcal{Y}} \ell(h(\bm{x}), y) p^{(\lambda)}_A(\bm{x})p_{tr}(y|\bm{x}) d\bm{x}dy.
\end{align*}
Here,
\begin{align*}
p^{(\lambda)}_A(\bm{x}) &= \Big(\frac{p_{te}(\bm{x})}{p_{tr}(\bm{x})}\Big)^\lambda p_{tr}(\bm{x}) \\
\log p^{(\lambda)}_A(\bm{x}) &= \alpha(\log p_{te}(\bm{x}) - \log p_{tr}(\bm{x})) + \log p_{tr}(\bm{x})\\
=& (1-\lambda)\log p_{tr}(\bm{x}) + \lambda \log p_{te}(\bm{x}) \\
p^{(\lambda)}_A(\bm{x}) &= \exp\{(1-\lambda)\log p_{tr}(\bm{x}) + \lambda \log p_{te}(\bm{x})\} \label{eq:exponential_interpolation} \\
=& m_f^{(\lambda, 1)}(p_{tr}(\bm{x}), p_{te}(\bm{x})).
\end{align*}
\end{proof}
\begin{lemm}[$f$-representation of RIWERM]
\label{lem:riwerm}
The marginal positive measures generated by the weighting of RIWERM can be expressed by using the $f$-interpolation function as
\begin{equation}
p^{(\lambda)}_R(\bm{x}) = m^{(\lambda, 3)}_f(p_{tr}(\bm{x}), p_{te}(\bm{x})).
\end{equation}
\end{lemm}
\begin{proof}
From Eq.~\eqref{eq:riwerm},
\begin{align*}
p^{(\lambda)}_R(\bm{x}) &= \frac{p_{te}(\bm{x})p_{tr}(\bm{x})}{\lambda p_{tr}(\bm{x}) + (1-\lambda) p_{te}(\bm{x})}\\
=& \frac{1}{\lambda\frac{1}{p_{te}(\bm{x})} + (1-\lambda)\frac{1}{p_{tr}(\bm{x})}}
= m^{(\lambda, 3)}_f(p_{tr}(\bm{x}), p_{te}(\bm{x})).
\end{align*}
\end{proof}
\if0
Also, from the above lemmas, the following proposition immediately follows.
\begin{prop}
Weighting by AIWERM and RIWERM produces a positive measure with $\int p(\bm{x}) d\bm{x} \leq 1$.
\end{prop}
\begin{proof}
When $\lambda$ is fixed, the $f$-interpolation has the following inverse monotonicity with respect to $\alpha$:
\begin{equation}
m_f^{(\lambda,\alpha)}\geq m_f^{(\lambda,\alpha')},\ (\alpha \leq \alpha'),
\end{equation}
where $\alpha, \alpha' \in \mathbb{R}$.
The relationship between $p_a^{(\lambda)}(\bm{x})$ and $p_r^{(\lambda)}(\bm{x})$ is given as
\begin{equation}
p_A^{(\lambda)}(\bm{x}) \geq p_R^{(\lambda)}(\bm{x}).
\end{equation}
Since the total probability of the $f$-interpolation of the two density functions is $1$ when $\alpha=-1$, we have
\begin{equation}
\sum_{\bm{x}\in\mathcal{X}}p_R^{(\lambda)}
\leq
\sum_{\bm{x}\in\mathcal{X}} p_A^{(\lambda)}(\bm{x}) \leq 1. \label{eq:inequality_positive_measure}
\end{equation}
From Eq.~\eqref{eq:inequality_positive_measure}, we can see that the total probabilities of $p_A^{(\lambda)}$ and $p_R^{(\lambda)}$ are not guaranteed to be $1$.
\end{proof}
\fi
From the above discussion, the following generalized method of covariate shift adaptation is derived using the $f$-representation.
\begin{theo}[Geometrically generalized IWERM]
For $\lambda\in[0,1]$ and $\alpha\in\mathbb{R}$, AIWERM and RIWERM is generalized as
\begin{equation}
\hat{h} = \min_{h\in\mathcal{H}}\int_{\mathcal{X}\times\mathcal{Y}}w^{(\lambda,\alpha)}(\bm{x})\ell(h(\bm{x}), y)p_{tr}(\bm{x},y)d\bm{x}dy, \label{eq:giwerm}
\end{equation}
where
\begin{equation}
w^{(\lambda,\alpha)}(\bm{x}) =
\frac{
m^{(\lambda,\alpha)}_{f}(p_{tr}(\bm{x}),p_{te}(\bm{x}))
}{p_{tr}(\bm{x})}.
\label{eq:Gweight}
\end{equation}
\end{theo}
See the Appendix~\ref{app:prf} for the proof.
From Definition~\ref{def:f_interpolation}, we can confirm that
\begin{align*}
m_f^{(0,\alpha)}(p_{tr}(\bm{x}), p_{te}(\bm{x})) &=p_{tr}(\bm{x}), \; \;
\mbox{and}\\
m_f^{(1,\alpha)}(p_{tr}(\bm{x}), p_{te}(\bm{x}))&=p_{te}(\bm{x}),
\end{align*}
for all $\alpha\in\mathbb{R}$, and this means that we can obtain the set of all curves that connect $p_{tr}(\bm{x})$ and $p_{te}(\bm{x})$.
\if0
:
\begin{equation}
\Pi_{(p_{tr}, p_{te})} = \Big\{\gamma_{\alpha}(\lambda) = m_f^{(\lambda, \alpha)}(p_{tr}(\bm{x}), p_{te}(\bm{x}))\ \Big|\ \lambda \in [0,1], \alpha\in\mathbb{R} \Big\}.
\end{equation}
\fi
\if0
When we observe $\{(\bm{x}^{tr}_i, y^{tr}_i)\}^{n_{tr}}_{i=1}$ and $\{(\bm{x}^{te}_i)\}^{n_{te}}_{i=1}$, we can specify the observed points on the coordinate system expressed by the expectation parameter $\bm{\eta}$ as
\begin{align*}
\bm{\eta}_{tr} = \bar{\bm{x}}^{tr} = \frac{1}{n_{tr}}\sum^{n_{tr}}_{i=1}\bm{x}^{tr}_i, \quad
\bm{\eta}_{te} = \bar{\bm{x}}^{te} = \frac{1}{n_{te}}\sum^{n_{te}}_{i=1}\bm{x}^{te}_i,
\end{align*}
and the weighting of the empirical distributions is regarded as the weighting of the observed points: $m_f^{(\lambda,\alpha)}(\bm{\eta}_{tr}, \bm{\eta}_{te})$.
\fi
We note that~\citet{DBLP:journals/sncs/ZhangYLS21} proposed a method based on basis expansion to estimate a flexible importance weight. It is similar to our proposal in the sense that improves the degree of freedom for designing the importance weight. However, our method considers the parametric form of weight, which enables us to achieve information geometric insight.
In many studies of covariate shift problems using the density ratio weighting including~\citet{Yamada2011-ws}, the direct estimation of the density ratio is often employed~\citep{sugiyama2012density}. Our proposed weight function in~\eqref{eq:Gweight} is also represented as density ratio:
\begin{align*}
w^{(\lambda,\alpha)}(\bm{x})
=&
\frac{
\left[
(1-\lambda) p_{tr}(\bm{x})^{\frac{1-\alpha}{2}}
+
\lambda
p_{te}(\bm{x})^{\frac{1-\alpha}{2}}
\right]^{\frac{2}{1-\alpha}}
}{
p_{tr}(\bm{x})}
\\
=&
\left[
1-\lambda +
\lambda
\left(
\frac{p_{te}(\bm{x})
}{
p_{tr}(\bm{x})
}
\right)^{\frac{1-\alpha}{2}}
\right]^{\frac{2}{1-\alpha}}, \quad (\alpha \neq 1).
\end{align*}
It is then also possible to apply the direct estimation of the density ratio using, for example kernel expansion. In our implementation, we simply used the given $p_{tr}(\bm{x})$ and $p_{te}(\bm{x})$ separately because they are explicitly known by the construction of the training and the test datasets as explained in Section~\ref{subsec:inducing_covariate_shift}. In the practical application of the proposed method in which the generative processes of the covariates of training and test data are unknown, direct density estimation would be a promising approach.
\subsection{Geometric Bias}
\label{subsec:geometric_bias}
AIWERM and RIWERM connects two distributions $p_{tr}$ and $p_{te}$ in different ways. Statistical bias and variance of IWERM, AIWERM, and RIWERM are discussed in the respective papers. In this subsection, we study the geometric bias of these methods to have a deeper understanding of these methods from the geometric viewpoint.
The proposed generalization of IWERM is independent from a specific parametrization of density functions. In this subsection, for theoretical treatment, the exponential model manifold which contains $p_{tr}(\bm{x};\bm{\theta})$ and $p_{te}(\bm{x};\bm{\theta})$ are considered, hence
geodesics can be described by a linear combination of parameters as explained in Appendix~\ref{app:mfd}. With this assumption, specifying $\lambda$ and $\alpha$ is equivalent to selecting a point on the geodesic connecting $p_{tr}$ and $p_{te}$.
\begin{defi}[$\alpha$-divergence~\citep{Amari1985-mi}]
Let $\alpha$ be a real parameter.
The $\alpha$-divergence between two probability vectors $\bm{p}$ and $\bm{q}$ is defined as
\begin{equation}
D_\alpha[\bm{p}:\bm{q}] = \frac{4}{1-\alpha^2}\Big(1 - \sum_i p_i^{\frac{1-\alpha}{2}}q_i^{\frac{1+\alpha}{2}}\Big).
\end{equation}
\end{defi}
\begin{defi}{($\alpha$-representation~\citep{Amari2009-fz})}
For some positive measure $m_i^{\frac{1-\alpha}{2}}$, the coordinate system $\bm{\theta}=(\theta^i)$ derived from the $\alpha$-divergence is
$\theta^i = m_i^{\frac{1-\alpha}{2}} = f_\alpha(m_i)$
and denote by $\theta^i$ the $\alpha$-representation of a positive measure $m_i^{\frac{1-\alpha}{2}}$.
\end{defi}
\begin{defi}[$\alpha$-geodesic~\citep{Amari2016-pi}]
\label{def:alpha_geodesic}
The $\alpha$-geodesic connecting two probability vectors $p(\bm{x})$ and $q(\bm{x})$ is defined as
\begin{align}
\notag
r_i(\lambda) =& c(t)f^{-1}_{\alpha}\Big\{(1-\lambda)f_{\alpha}(p(x_i)) + \lambda f_\alpha(q(x_i))\Big\},\\
c(\lambda) =& \left(\sum^p_{i=1}r_i(\lambda)\right)^{-1}.
\end{align}
\end{defi}
Let $\psi_\alpha(\bm{\theta}) = \frac{1-\alpha}{2}\sum_{i=1} m_i$, the dual coordinate system $\bm{\eta}$ is given by $\bm{\eta} = \nabla\psi_\alpha(\bm{\theta})$ as
\begin{equation}
\eta_i = (\theta^i)^{\frac{1+\alpha}{1-\alpha}} = f_{-\alpha}(m_i),
\end{equation}
which is the $-\alpha$-representation of $m_i$.
From Definitions~\ref{def:f_interpolation} and \ref{def:alpha_geodesic}, we see that $f$-interpoloation is the unnormalized version of the $\alpha$-geodesic.
We write $\tilde{m}^{(\lambda,\alpha)}_f$ for a suitably normalized $f$-interpolation.
The important properties of $\alpha$-geodesics are
\begin{itemize}
\item the $\alpha$-geodesic is a geodesic in the $\alpha$-coordinate system derived from $\alpha$-divergence,
\item the $-\alpha$-geodesic is linear in the $-\alpha$-representation.
\end{itemize}
\if0
Then, if we find that the $f$-representation of $p_{tr}$ and $p_{te}$ is $m_f^{(\lambda,\alpha)}$, we can examine the behavior of it in the $\bm{\theta}$-coordinate system through its dual $m_f^{(\lambda,-\alpha)}$, i.e.,
\begin{equation*}
\tilde{m}_f^{(\lambda,-\alpha)}(\bm{\theta}_{tr},\bm{\theta}_{te}) = (1-\lambda)\bm{\theta}_{tr} + \lambda\bm{\theta}_{te}.
\end{equation*}
Consider a set of all curves connecting $p_{tr}(\bm{x})$ and $p_{te}(\bm{x})$:
\begin{align} \notag
\Pi_{(p_{tr}, p_{te})} = \Big\{ \gamma_{\alpha}:[0,1] \to \mathcal{M}
& \; | \;
\gamma_{\alpha}(0) = p_{tr}(\bm{x}), \;
\gamma_{\alpha}(1) = p_{te}(\bm{x}), \\
& \;\;\; \gamma_{\alpha}(\lambda) = \tilde{m}^{(\lambda,\alpha)}_{f}(p_{tr}(\bm{x}),p_{te}(\bm{x})), \; \alpha \in \mathbb{R}
\Big\}.
\end{align}
\fi
Let $\gamma_c$ be the geodesic connecting two distributions parameterized by $\bm{\theta}_{tr}$ and $\bm{\theta}_{te}$.
Now, we define two types of geometric biases to characterize the dispersion of $\bm{\theta}_{tr}$ from $\bm{\theta}_{te}$ with respect to the direction along the $\alpha$-geodesic and to the direction orthogonal to the $\alpha$-geodesic.
\begin{defi}[Geodesic bias and curvature bias]
If we write the unit vector along the $\alpha$-geodesic direction as $e_1$ and any unit vector in the orthogonal direction to $e_1$ as $e_2$, the bias relative to the test distribution due to weighting can be decomposed as follows:
\begin{itemize}
\item geodesic bias: $b_g = (1-\lambda)e_1$,
\item curvature bias: $b_c = (1-\lambda)tr_g(\mathrm{Ric})e_2$,
\end{itemize}
where $tr_g$ is the trace operation on the metric tensor $g$ and $\mathrm{Ric}$ is the Ricci curvature of the curve connecting the two points generated by the weighting:
\begin{equation}
\mathrm{Ric} = R_{ikj} d\bm{\theta}^i \otimes d\bm{\theta}^j.
\end{equation}
Here, $R_{ikj}$ is the Riemannian curvature tensor.
\end{defi}
For more detail on the geometric concepts, see textbooks on Riemannian manifolds~\citep{Jost2017-ad}. This definition of geometric biases is consistent with the fact that IWERM, which corresponds to $\lambda=1$, leads to an unbiased estimator of the risk in the test dataset.
\begin{prop}
\label{prop:geometric_bias_aiwerm}
For AIWERM, the geometric bias $b_A(\lambda)$ is computed as
\begin{equation}
b_A(\lambda) = (1-\lambda)e_1.
\end{equation}
\end{prop}
\if0
\begin{proof}
From Lemma~\ref{lem:aiwerm}, we can compute the dual $f$-representation of AIWERM as
\begin{align}
\bm{\theta}_A^{(\lambda)} &= m_f^{(\lambda, -1)}(\bm{\theta}_{tr}, \bm{\theta}_{te}) \\
&= (1-\lambda)\bm{\theta}_{tr} + \lambda \bm{\theta}_{te}. \label{eq:aiwerm_dual}
\end{align}
From the fact that $(\mathcal{S}, g, \nabla^{(1)}, \nabla^{(-1)})$ is a dually flat manifold, it is clear that Eq.~\eqref{eq:aiwerm_dual} is also a geodesic.
Then, for AIWERM, we can confirm that the curvature bias $b_c$ vanishes as
\begin{equation*}
\Gamma^{(-1)}_{abc}(\bm{\theta}) = 0.
\end{equation*}
The normalized probability distributions on the geodesic are
\begin{align}
p(\bm{x}, \lambda) &= p(\bm{x}, \bm{\theta}(\lambda)) \nonumber \\
&= \exp\Big\{\lambda(\bm{\theta}_{te}-\bm{\theta}_{tr})\cdot\bm{x} + \bm{\theta}_{tr}\bm{x} \Big\}.
\end{align}
Hense, geodesic itself is a one-dimensional exponential family, where $\lambda$ is the natural parameter.
By taking the logarithm, we can obtain
\begin{align}
\log p(\bm{x};\lambda) &= (1-\lambda)\log p(\bm{x};\bm{\theta}_{tr}) + \lambda\log p(\bm{x};\bm{\theta}_{te}) \nonumber \\
p(\bm{x};\lambda) &= \exp\Big\{ (1-\lambda)\log p_{tr}(\bm{x}) + \lambda\log p_{te}(\bm{x})\Big\} \nonumber \\
&= m_f^{(\lambda,1)}(p_{tr}(\bm{x}), p_{te}(\bm{x})) = p_A^{(\lambda)}(\bm{x}).
\end{align}
Thus, the duality of AIWERM is confirmed.
\end{proof}
\fi
\begin{figure}
\caption{Geometry of covariate shift adaptation methods.
In the $\bm{\theta}
\label{fig:geometry_of_covariate_shift}
\end{figure}
\begin{figure}
\caption{Visualization of grid search for $\alpha$ and $\lambda$ on LIBSVM dataset.}
\end{figure}
\begin{algorithm}[t]
\caption{Bayesian optimization for IGIWERM}
\label{alg:bopt}
\begin{algorithmic}
\REQUIRE acquisition function $a(\lambda,\alpha|D)$, target function $L(h;\lambda, \alpha)$, initial points $D_{init}$ compose of a set of parameters $\Xi=\{(\lambda,\alpha)\}$ and corresponding values of the target function
\mathrm{E}NSURE $(\lambda^\ast, \alpha^\ast)$ that minimizes $\min_{h \in \mathcal{H}} L(h;\lambda, \alpha)$
\STATE Initialize $D=D_{init}$
\WHILE{Not converge}
\STATE $\hat{\lambda}, \hat{\alpha} = \mathop{\rm arg~min}\limits_{\lambda, \alpha} a(\lambda,\alpha|D),\; \; \Xi = \Xi \cup \{(\hat{\lambda},\hat{\alpha})\}$
\STATE $\hat{e} = L(h;\hat{\lambda}, \hat{\alpha}),\; \;D = D\cup\{(\hat{\lambda}, \hat{\alpha},\hat{e})\}$
\mathrm{E}NDWHILE
\STATE $(\lambda^\ast, \alpha^\ast) = \mathop{\rm arg~min}\limits_{(\lambda, \alpha) \in \Xi} \left\{\min_{h \in \mathcal{H}} L(h;\lambda, \alpha)\right\}$
\end{algorithmic}
\end{algorithm}
\begin{prop}
\label{prop:geometric_bias_riwerm}
For RIWERM, the geometric bias $b_R(\lambda)$ is computed as
\begin{equation}
b_R(\lambda) = (1-\lambda)\Big\{e_1 + tr_g\Big(-4\Lambda_{ikj}d\bm{\theta}^i \otimes d\bm{\theta}^j\Big)e_2\Big\}.
\end{equation}
Here, $\Lambda$ is a tensor that depends on the connection.
\end{prop}
These propositions are proved by straightforward calculation as detailed in Appendix~\ref{app:prf}
\begin{figure}
\caption{Left: generated data from $y=x^2+\varepsilon$. We see that $p_{tr}
\label{fig:dummy_experiment}
\end{figure}
\begin{figure}
\caption{Bayesian optimization for IGIWERM. The coordinates of the purple circles are the parameters explored by Bayesian optimization, and the size of the purple circles indicates the goodness of the parameters (inverse of the MSE).\label{fig:bopt}
\label{fig:bopt}
\end{figure}
\begin{table}[t]
\centering
\caption{Mean squared errors of covariate shift adaptation methods in regression problems over $10$ trials. Here, IGIWERM (bopt) is the Bayesian optimization based, and IGIWERM (IC) is the information criterion based strategy.\label{tab:dummy_mse}}
\begin{tabular}{l|l}
Weighting strategy & MSE \\ \hline
ERM & $160.19 (\pm 4.25)$ \\
IWERM & $33.76 (\pm 3.82)$ \\
AIWERM & $31.14 (\pm 2.97)$ \\
RIWERM & $30.03 (\pm 2.74)$ \\% \midrule
IGIWERM (bopt) & ${\bf 28.89 (\pm 2.42)}$ \\
IGIWERM (IC) & ${\bf 28.38 (\pm 2.12)}$ \\
\end{tabular}
\end{table}
Figure~\ref{fig:geometry_of_covariate_shift} shows the curves on the manifolds created by AIWERM and RIWERM.
Both of them satisfy
\begin{itemize}
\item for $\lambda=0$, it is equivalent to unweighted ERM,
\item for $\lambda=1$, it is equivalent to IWERM.
\end{itemize}
Note that the curvature bias $b_c$ vanishes for all $\lambda \in [0,1]$ in AIWERM, while RIWERM does not guarantee the vanishing of the curvature bias for $\lambda\in(0,1)$.
\if0
Furthermore, the following optimality can be derived for AIWERM in the sense of KL-divergence.
\begin{lemma}{(\citep{Amari2016-pi})}
\label{lem:shortest_geodesic}
The symmetrized KL-divergence is given by the integration of the Fisher information along the $(-1)$-geodesic:
\begin{equation}
\frac{1}{2}\Big\{D_{KL}[\bm{\theta}_1:\bm{\theta}_2] + D_{KL}[\bm{\theta}_2:\bm{\theta}_1]\Big\} = \int^1_0 g_{m}(t)dt,
\end{equation}
where
\begin{align}
g_{m}(t) = g_{ij}\dot{\theta}^i_m(t)\dot{\theta}^j_m(t), \quad
\bm{\theta}_m(t) = (1-t)\bm{\theta}_1 + t\bm{\theta}_2.
\end{align}
\end{lemma}
\begin{theo}
Let $\Pi_{(\bm{\theta}_{tr}, \bm{\theta}_{te})}$ be a set of all curves that connect $\bm{\theta}_{tr}$ and $\bm{\theta}_{te}$.
The geodesic generated by AIWERM is the shortest path in $\Pi_{(\bm{\theta}_{tr}, \bm{\theta}_{te})}$ in the sense of symmetrized KL-divergence.
\end{theo}
\begin{proof}
Follows immediately from Lemma~\ref{lem:shortest_geodesic}.
\end{proof}
\fi
\if0
For example, the loss of IGIWERM with unknown weights employing mean squared error can be decomposed as follows
\begin{align*}
\mathbb{E}_{te}\Big[\|h(x) - y\|^2\Big] &= \mathbb{E}_{tr}\Big[\|h(x) - y\|^2\frac{p_{te}(\bm{x})}{p_{tr}(\bm{x})}\Big] \nonumber \\
&= \mathbb{E}_{tr}\Big[\|h(x) - y\|^2(w^{(\lambda,\alpha)}(\bm{x}) + \frac{p_{te}(\bm{x})}{p_{tr}(\bm{x})} -w^{(\lambda,\alpha)}(\bm{x}))\Big] \nonumber \\
&= \mathbb{E}_{tr}\Big[\|h(x) - y\|^2 w^{(\lambda,\alpha)}(\bm{x})\Big] + \mathbb{E}_{tr}\Big[\|h(x) - y\|^2\Big(\frac{p_{te}(\bm{x})}{p_{tr}(\bm{x})} - w^{(\lambda,\alpha)}(\bm{x})\Big)\Big].
\end{align*}
The first term is the weighted squared error term.
We can also give an upper bound on the second term as follows
\footnotesize
\begin{align}
\mathbb{E}_{tr}\Big[\|h(x) - y\|^2\Big(\frac{p_{te}(\bm{x})}{p_{tr}(\bm{x})} - w^{(\lambda,\alpha)}(\bm{x})\Big)\Big] &\leq \sqrt{\mathbb{E}_{tr}\Big[\|h(x) - y\|^4\Big]\mathbb{E}_{tr}\Big[\Big(\frac{p_{te}(\bm{x})}{p_{tr}(\bm{x})} - w^{(\lambda,\alpha)}(\bm{x})\Big)^2\Big]} \nonumber \\
&\leq \frac{1}{2}\Biggl(\mathbb{E}_{tr}\Big[\|h(x) - y\|^4\Big] + \mathbb{E}_{tr}\Big[\Big(\frac{p_{te}(\bm{x})}{p_{tr}(\bm{x})} - w^{(\lambda,\alpha)}(\bm{x})\Big)^2\Big]\Biggr) \nonumber
\end{align}
\normalsize
From the fact that both statistical bias and geometric bias are zero for IWERM, we can expect there exist functions $\varphi$, $\varphi_g$, $\varphi_c$ such that
\begin{equation*}
\varphi(b_g, b_c) = \varphi_g(b_g) + \varphi_c(b_c) \propto \mathbb{E}_{te}\Big[\Big(\frac{p_{tr}(\bm{x})}{p_{te}(\bm{x})} - w^{(\lambda,\alpha)}(\bm{x})\Big)^2\Big].
\end{equation*}
Since the statistical bias is a scalar and the geometric bias is a vector, $\varphi$ is a function that aggregates informative geometric bias to a scalar.
A possible future study would be to identify the specific form of $\varphi$.
\fi
Intuitively, the geometric bias reveals in which direction the two parameters are misaligned.
IWERM, which corresponds to AIWERM and RIWERM with $\lambda=1$, is optimal when the sample size is large enough, but in real problems with limited sample size, it is often desirable to adopt a point between $\bm{\theta}_{tr}$ and $\bm{\theta}_{te}$. AIWERM and RIWERM consider distinct curves and specify a point on them by the parameter $\lambda$.
Our geometric analysis revealed that these curves are included in the set of curves represented by dual $f$-representation of the parameter coordinate system, and the geometric biases of these particular cases (AIWERM and RIWERM) are identified. The results presented in this subsection do not claim superiority of a particular method and are of importance in their own right as a geometric analysis of the covariate shift method.
\subsection{Optimization of the generalized IWERM}
\label{subsec:optimization_of_the_generalized_iwerm}
The existing covariate shift adaptation methods described above can be regarded as having determined a good ``weighting direction'' in some sense in advance and then the ``weighting magnitude'' is adjusted according to the parameter $\lambda$.
This approach is very convenient in terms of computational efficiency since the only optimized parameter is $\lambda\in[0,1]$.
However, geometrically, these methods only consider certain curves on the manifold as candidate solutions, as can be seen from Figure~\ref{fig:geometry_of_covariate_shift}, which means that the solution space is very small.
Our information geometrical IWERM (IGIWERM) can handle all curves $\gamma_{\alpha}(\lambda)$ in $\Pi_{(p_{tr}, p_
{te})}$ that connect $p_{tr}(\bm{x})$ and $p_{te}(\bm{x})$, by adding only one parameter.
For example, by setting $\alpha\in[1,3]$, shaded area in Figure~\ref{fig:geometry_of_covariate_shift} can be used as the solution space. The problem of how to determine $\lambda$ and $\alpha$ remains.
\subsubsection{Information criterion}
When the predictive model is of a simple parametric form, information criterion derived in~\citep{Shimodaira2000-vv} is available (see appendix of~\citep{Shimodaira2000-vv} for the proof.):
\begin{theo}[Information criterion for IGIWERM]
Let the information criterion for IWERM be
\begin{equation}
IC_{GW} \coloneqq -2L_1(\hat{\bm{\theta}}) + 2tr(J_{w}H_w^{-1}), \label{eq:information_criterion}
\end{equation}
where $L_1(\bm{\theta}) = \sum^{n_{tr}}_{i=1}
dr(\bm{x}^{tr}_i)
\log p(y^{tr}_i | \bm{x}^{tr}_i, \bm{\theta}), \;
dr(\bm{x}) = \frac{p_{te}(\bm{x})}{p_{tr}(\bm{x)}}
$ and
\begin{align*}
J_w &= -\mathbb{E}_{p_{tr}}\Biggl[
dr(\bm{x})
\frac{\partial \log p(y|\bm{x},\bm{\theta})}{\partial\bm{\theta}}\Biggr|_{\bm{\theta}^*_w}
\times
\frac{\partial\Big(\frac{m_f^{\lambda, \alpha}(p_{tr}(x), p_{te}(\bm{x}))}{p_{tr}(\bm{x})}
\log p(y | \bm{x},\bm{\theta})
\Big)
}{\partial\bm{\theta}'}\Biggr|_{\bm{\theta}^*_w}\Biggr] \\
H_w &= \mathbb{E}_{p_{tr}}\Biggl[\frac{\partial^2\Big( \frac{m_f^{\lambda, \alpha}(p_{tr}(x), p_{te}(\bm{x}))}{p_{tr}(\bm{x})} \log p(y | \bm{x},\bm{\theta})\Big)}{\partial\bm{\theta}\partial\bm{\theta}'}\Biggr].
\end{align*}
Here, $\bm{\theta}^*_w$ is the minimizer of the weighted empirical risk.
The matrices $J_w$ and $H_w$ may be replaced by their consistent estimates.
Then, $IC_{GW}/2n$ is an unbiased estimator of the expected loss up to $O(n^{-1})$ term:
\begin{equation}
\mathbb{E}_{p_{tr}}\Big[IC_{GW}/2n\Big] = \mathbb{E}_{p_{tr}}\Big[\ell_1(\hat{\bm{\theta}}_w)\Big] + o(n^{-1}).
\end{equation}
\end{theo}
\begin{table*}[t]
\centering
\caption{Mean misclassification rates averaged over $10$ trails on LIBSVM benchmark datasets.
The numbers in the brackets are the standard deviations.
For the methods with (optimal), the optimal parameters for the test data are obtained by linear search.}
\label{tab:libsvm_error}
\resizebox{\textwidth}{!}{
\begin{tabular}{c|c|c|ccccc}
Dataset & \#features & \#data & unweighted & IWERM & AIWERM (optimal) & RIWERM (optimal) & ours \\ \hline
australian & $14$ & $690$ & $33.46 (\pm 23.65)$ & $22.13 (\pm 3.37)$ & $21.98 (\pm 3.36)$ & $21.73 (\pm 3.82)$ & ${\bf 18.85 (\pm 3.99)}$ \\
breast-cancer & $10$ & $683$ & $38.28 (\pm 10.98)$ & $41.23 (\pm 15.39)$ & $36.41 (\pm 9.68)$ & $36.13 (\pm 10.81)$ & ${\bf 31.65 (\pm 8.49)}$ \\
heart & $13$ & $270$ & $45.17 (\pm 6.98)$ & $39.94 (\pm 8.55)$ & $39.76 (\pm 8.49)$ & $39.76 (\pm 8.92)$ & ${\bf 35.37 (\pm 6.84)}$ \\
diabetes & $8$ & $768$ & $33.19 (\pm 5.69)$ & $37.22 (\pm 6.63)$ & $33.11 (\pm 6.45)$ & $33.38 (\pm 5.74)$ & ${\bf 32.83 (\pm 5.62)}$ \\
madelon & $500$ & $2,000$ & $47.78 (\pm 1.53)$ & $47.28 (\pm 2.20)$ & $47.10 (\pm 2.13)$ & $47.12 (\pm 1.65)$ & ${\bf 46.56 (\pm 2.12)}$ \\
\end{tabular}
}
\end{table*}
\subsubsection{Bayesian optimization}
This information criterion does not work for complicated nonparametric models. As a method that can be applied in general situations, we consider using Bayesian optimization~\cite{Snoek2012-rs,Frazier2018-fm} to find the optimal weighting by IGIWERM. Bayesian optimization assumes that the target function is drawn from a prior distribution over functions, typically a Gaussian process, updating a posterior as we observe the target function value in new places.
We use the validation loss as the target function:
\begin{equation}
L(h;\lambda,\alpha) = \frac{1}{n_{val}}\sum^{n_{val}}_{i=1}\frac{p_{te}(\bm{x}_i^{val})}{p_{tr}(\bm{x}_i^{val})}\ell(h_{\lambda,\alpha}(\bm{x}^{val}_i), y^{val}_i). \label{eq:target_function}
\end{equation}
where $n_{val}$ is the validation sample size and $h_{\lambda,\alpha}$ is given by IWERM with $\lambda$ and $\alpha$.
This validation procedure is based on the importance weighted cross validation used in~\citep{Sugiyama2007-lr}.
In Bayesian optimization, an acquisition function $a(\lambda, \alpha |D)$ is used for measuring goodness of candidate point $(\lambda, \alpha)$ based on current dataset $D$.
As the acquisition function, we adopt the expected improvement~\cite{Mockus1978-dh,Jones1998-xu}.
In this strategy, we choose the next query point which has the highest expected improvement over the current minimum target value. See Appendix~\ref{app:BO} for more detail. The overall picture is summarized in Algorithm~\ref{alg:bopt}.
\subsection{Learning guarantee}
Generalization bounds of weighted maximum likelihood estimator for the target domain are derived in~\cite{Cortes2010}, and our weight function~\eqref{eq:Gweight} is compatible with their bound. The weight defined by Eq.~\eqref{eq:Gweight} is bounded when $\alpha \neq 1$ and achieves a standard rate $O(n_{tr}^{-1/2})$. When $\alpha=1$, the weight is unbounded and its rate is $O(n_{tr}^{-3/8})$. Details are shown in Appendix~\ref{app:slt}.
\section{Numerical Experiments}
\label{sec:numerical_experiments}
In this section, we present experimental results of domain adaptation problems under covariate shift using both synthetic and real data\footnote{Source code to reproduce the results is available from~ \url{https://github.com/nocotan/IGIWERM}}.
Since the main purpose of the experiments is to see the effect of our generalization of the importance weighted ERM and comparison to the proposed and conventional IWERM methods, in all experiments, we assume that $p_{tr}$ and $p_{te}$ are known as detailed in Section~\ref{subsec:inducing_covariate_shift}.
\subsection{Induction of Covariate Shift}
\label{subsec:inducing_covariate_shift}
Since each dataset is composed of data points generated from independent and identical distributions, we need to artificially induce covariate shifts.
We induce the covariate shift as follows~\cite{Cortes2008-zd}:
\begin{enumerate}
\item As a preprocessing step, we perform Z-score standardization on all input data.
\item Then, an example $(\bm{x}, y)$ is assigned to the training dataset with probability $\exp(v)/(1 + \exp(v))$ and to the test dataset with probability $1/(1 + \exp(v))$, where $v = 16\bm{w}^T\bm{x}/\sigma$ with $\sigma$ being the standard deviation of $\bm{w}^T\bm{x}$ determined by using the given dataset, and $\bm{w}\in\mathbb{R}^d$ is a given projection vector. Here, the projection vector $\bm{w}$ is given randomly for each experimental process.
\end{enumerate}
By this construction of the training and test datasets, $p_{tr}$ and $p_{te}$ are explicitly determined as
\begin{align*}
p_{tr}(\bm{x}) &= \frac{\exp(16\bm{w}^T\bm{x}/\sigma)}{1 + \exp(16\bm{w}^T\bm{x}/\sigma)}, \\
p_{te}(\bm{x}) &= \frac{1}{1 + \exp(16\bm{w}^T\bm{x}/\sigma)},
\end{align*}
when the projection vector $\bm{w}\in\mathbb{R}^d$ is given.
Although density ratio estimation could be employed in our experiments, we assume that the distribution is known in order to compare the performance of the proposed method without relying on the accuracy of the density or density ratio estimation.
\subsection{Illustrative Example in Regression}
\label{subsec:dummy_experiment}
Here, we predict the response $y\in\mathbb{R}$ using ordinary linear regression:
$y = \beta_0 + \beta_1x + \varepsilon,\ \varepsilon\sim\mathcal{N}(0,\sigma^2)$,
where $\mathcal{N}(a,b)$ denotes the normal distribution with mean $a$ and variance $b$.
In the numerical example below, we assume the true $p(y|x)$ given as $
y = x^2 + \varepsilon,\ \varepsilon\sim\mathcal{N}(0,5).
$
The $p_{tr}(x)$ and $p_{te}(x)$ of the covariate $x$ are $
x^{tr}\sim\mathcal{N}(0,5),\ x^{te}\sim\mathcal{N}(-5,0.5).
$
The training sample size is $n_{tr}=1000$ and the test sample size is $n_{te}=300$.
The left-hand side of Fig.~\ref{fig:dummy_experiment} shows the data to be generated.
We can see that $p_{tr}(x)\neq p_{te}(x)$.
The right panel of Fig.~\ref{fig:dummy_experiment} shows the results of fitting by unweighted ERM, IWERM, and IGIWERM.
Here, the parameters of IGIWERM are explored by using Algorithm~\ref{alg:bopt}, as shown in Fig.~\ref{fig:bopt}.
The coordinates of the purple circles are the parameters explored by Bayesian optimization, and the radius of the purple circles is proportional to the goodness $r(\bm{\beta})$ of the parameters (inverse of the MSE):
$r(\bm{\beta}) = \left(\frac{1}{n}\sum^n_{i=1}(y_i - h(x_i, \bm{\beta}))^2\right)^{-1}$.
By choosing the size $r(\beta)$ of the plot for each point in this manner, the better-evaluated parameters can be plotted in larger circles.
From this figure, it can be seen that our generalized weighting is not restricted to lying just on two curves corresponding to AIWERM and RIWERM.
For the normal linear regression, the information criterion~\eqref{eq:information_criterion} is calculated from
\begin{align*}
&IC_{GW}(\lambda,\alpha) = \frac{1}{2}\sum^{n_{tr}}_{i=1}
dr(\bm{x}_{i}^{tr})
\Biggl\{\frac{\hat{\epsilon}^2_1}{\hat{\sigma}^2 + \log(2\pi\hat{\sigma}^2)}\Biggr\}+\\
& \sum^{n_{tr}}_{i=1}
dr(\bm{x}_{i}^{tr})
\Biggl\{\frac{\hat{\epsilon}^2_i}{\hat{\sigma}^2}\hat{h}_i + \frac{m_f^{\lambda, \alpha}(p_{tr}(\bm{x}^{tr}), p_{te}(\bm{x}^{tr}))}{2\hat{c}_w p_{tr}(\bm{x}^{tr})}\Big(\frac{\hat{\epsilon}^2_i}{\hat{\sigma}^2} - 1\Big)^2\Biggr\}.
\end{align*}
Here, $\hat{c}_w=\sum^n_{i=1}\frac{m_f^{\lambda, \alpha}(p_{tr}(x_i^{tr}), p_{te}(x_i^{tr}))}{p_{tr}(x_i^{tr})}$, $\hat{\sigma}^2=\sum^n_{i=1}\frac{m_f^{\lambda, \alpha}(p_{tr}(x_i^{tr}), p_{te}(x_i^{tr}))}{p_{tr}(x_i^{tr})}\hat{\epsilon}_i^2/\hat{c}_w$ and $\hat{\epsilon}_i$ is the residual.
Table~\ref{tab:dummy_mse} shows that the IGIWERM outperforms existing methods.
\subsection{Experiments on binary classification problem}
We show the results of our experiments on the LIBSVM dataset\footnote{\href{https://www.csie.ntu.edu.tw/~cjlin/libsvmtools/datasets/}{https://www.csie.ntu.edu.tw/~cjlin/libsvmtools/datasets/}}.
In the experiments, we randomly generate a mapping vector $\bm{w}$ for each trial and perform $10$ trials for each dataset.
We use SVM with Radial Basis Function (RBF) kernel as the base classifier. In this experiment, the parameters $\lambda$ of AIWERM and RIWERM are chosen optimally by linear search using the test data. The experimental results on benchmark datasets are summarized in Table~\ref{tab:libsvm_error}.
The table shows that the proposed IGIWERM outperforms the conventional methods even when the parameters of those methods are optimized by using the test dataset.
More experimental results on other datasets with various models are reported in Appendix~\ref{app:exp}.
\subsection{Computational Cost}
Here, we investigate the computational cost of our IGIWERM.
The experimental setup is the same as in Section~\ref{subsec:dummy_experiment}.
The mean and standard deviation of the computation time obtained in the 10 trials are shown in Table~\ref{tab:computational_cost}.
From this table, we can see that our IGIWERM takes constant times longer to compute than the vanilla ERM.
\begin{table}[th]
\centering
\caption{Computational cost of ERM and IGIWERM.}
\begin{tabular}{c|c}
Method & Computation time [sec] \\ \hline
ERM & $1.130(\pm 0.238)$ \\
IGIWERM & $9.887(\pm 0.845)$ \\
\end{tabular}
\label{tab:computational_cost}
\end{table}
\section{Conclusion and Discussion}
We generalized existing methods of covariate shift adaptation in the geometrical framework. By our information geometrical formulation, geometric biases of conventional methods are elucidated.
Unlike the dominant approaches restricted to a specific curve on a manifold in the literature, our generalization has a much larger solution space with only two parameters.
Our experiments highlighted the advantage of our method over previous approaches, suggesting that our generalization can achieve better performance than the existing methods. A drawback of our proposed method is its relatively high computational cost for optimizing parameters $\alpha$ and $\lambda$. We used Bayesian optimization for efficient parameter search, and further sophisticated approaches would be explored in our future work.
As mentioned in the introduction, the importance weighting is used with deep neural network models~\citep{DBLP:conf/nips/FangL0S20}, in which the importance weight in the feature representation obtained by DNN is considered. It is also worth mentioning that \citet{DBLP:conf/aaai/SakaiS19} used RIWERM in the study of covariate shift on the learning from positive and unlabeled data.
Our generalization will be applicable to their methods to improve the performance under a small sample regime. In particular, in a standard approach for optimizing the implicit weight function $w(\bm{x})$, it is common to add a regularization term $(w(\bm{x})- p_{tr}(\bm{x}) / p_{te}(\bm{x}))^2$ to the optimization objective. The use of the derived geodesic and curvature biases to regularize the optimal weight function will be investigated in connection with the modern weight learning approach using deep neural network models. Finally, the relation between geometric bias and statistical bias should be explored.
\section*{Appendix A: Statistical Manifolds and Straight Line in Exponential Family}
\label{app:mfd}
Let $M$ be a $d$-dimensional differentiable manifold with a Riemannian metric $g$. For each $\bm{x} \in M$, $T_{\bm{x}} M$ is its tangent space.
\begin{defi}
Let $g_{\bm{x}}$ an inner product
\begin{equation}
g_{\bm{x}} : T_{\bm{x}}(M) \times T_{\bm{x}}(M) \to \mathbb{R} \quad \forall \bm{x} \in M.
\end{equation}
When, for any $X, Y \in M$, the map $\bm{x} \to g_{\bm{x}}(X_{\bm{x}},Y_{\bm{x}})$ is differentiable with respect to $\bm{x} \in M$, $g_{\bm{x}}$ is denoted as the Riemannian metric.
\end{defi}
The correspondence $X: M \ni \bm{x} \mapsto X_{\bm{x}} \in T_{\bm{x}}M$ is called a vector field on $M$. For $\bm{x} \in M$, let coordinate expression of $X_{\bm{x}}$ be $X_{\bm{x}} = (v^1(\bm{x}), \dots,v^{d}(\bm{x}))$. Then, $v^{i}(\bm{x})$ defines a real-valued function $v^{i}$ on $M$ and $X$ is expressed as $X=(v^1,\dots,v^d)$.
When a function on $M$ is $k$ times continuously differentiable, it is called the class $C^k$, and the set of all functions of class $C^{k}$ on $M$ is denoted as $C^{k}(M)$.
A vector field $X$ is called class $C^{k}$ when all of $v^i, \; i=1,\dots,d$ are class $C^{k}$. The set of all class $C^{\infty}$ vector fields is denoted as $\mathfrak{X}(M)$. A tangent space $T_{\bm{x}}(M)$ is a vector space spanned by differentials $\frac{\partial}{\partial x^i}$, namely,
\begin{equation}
T_{\bm{x}}(M) =
\left\{
a^i
\left(
\frac{\partial}{\partial x^{i}}
\right)_{\bm{x}}
\middle|
\forall \; a_i \in \mathbb{R}
\right\}.
\end{equation}
Following the notational convention of differential geometry, we use $\partial_i = \frac{\partial}{\partial x^{i}}$ and the Einstein summation convention. The vector field on a manifold $M$ is then written as
\begin{equation}
\mathfrak{X}(M) =
\left\{
v^{i} \partial_{i}
\middle|
\; v^i \in C^{\infty}(M)
\right\}.
\end{equation}
For $X \in \mathfrak{X}(M)$ and $f \in C^{\infty}(M)$, $fX \in \mathfrak{X}(M)$ is defined by $(fX)_{\bm{x}} = f(\bm{x}) X_{\bm{x}}, \; (\bm{x} \in M)$. Differential of a function $f$ with respect to a vector field $X$ is denoted as $Xf \in C^{\infty}(M)$ and defined by $(Xf)(\bm{x}) = X_{\bm{x}}(f), \; (\bm{x} \in M)$. When two vector fields are expressed as $X=v^i \partial_{i}$ and $Y=u^i \partial_{i}$, we have
\begin{equation}
X(Yf) - Y(Xf) = (v^j \partial_j u^i - u^j \partial_j v^i) \partial_i f.
\end{equation}
The commutator product of $X$ and $Y$ is defined as $[X,Y] \in \mathfrak{X}(M), \; [X,Y]f = (XY-YX)f$, and
\begin{equation}
[X,Y] = (v^j \partial_j u^i - u^j \partial_j v^i) \partial_i.
\end{equation}
\begin{defi}
Consider a map $\nabla: \mathfrak{X}(M) \times \mathfrak{X}(D) \to \mathfrak{X}(M)$ which assigns a pair of vectors $(X,Y) \in \mathfrak{X}(M)\times \mathfrak{X}(M)$ to a vector $\nabla_{Y}X \in \mathfrak{X}(M)$. $\nabla_{Y}X$ is called a covariant derivative of $X$ with respect to $Y$, and $\nabla$ is called an affine connection when the following conditions hold for any $X,Y,Z \in \mathfrak{X}(M)$ and $f \in C^{\infty}(M)$:
\begin{itemize}
\item $\nabla_{Y+Z}X=\nabla_{Y}X + \nabla_{Z}X$
\item $\nabla_{fX}X = f \nabla_{Y}X$
\item $\nabla_{Z}(X+Y) = \nabla_{Z}X + \nabla_{Z}Y$
\item $\nabla_{Y}(fX) = (Yf)X + f\nabla_Y X$
\end{itemize}
\end{defi}
\begin{defi}
Let $\nabla$ be an affine connection on $M$, and define a map
\begin{align}
\notag
T : \mathfrak{X}(M) \times \mathfrak{X}(M) \to & \mathfrak{X}(M)\\
(X,Y) \mapsto & T(X,Y) = \nabla_X Y - \nabla_Y X - [X,Y].
\end{align}
The map $T$ is called the torsion tensor field of $\nabla$. When $T=0$ for all $X,Y \in \mathfrak{X}(M)$, the connection $\nabla$ is called torsion-free.
\end{defi}
For an affine connection, the Christoffel symbol $\Gamma^{k}_{ij} \in C^{\infty}(M)$ is defined by
\begin{equation}
\nabla_{\partial_i} \partial_j =
\Gamma^{k}_{ij} \partial_k.
\end{equation}
With this formula, the connection and the Christoffel symbol are often identified.
The affine connection $\nabla$ is torsion-free when and only when $\Gamma^k_{ij} = \Gamma^k_{ji}$.
Suppose a manifold $M$ is equipped with a Riemannian metric $g$. When
\begin{equation}
X g(Y,Z) =
g( \nabla_{X} Y,Z)
+
g(Y,\nabla_{X} Z)
\end{equation}
holds for all $X,Y,Z \in \mathfrak{X}(M)$, the connection $\nabla$ is called a metric connection. In general, an affine connection is not a metric connection, but there uniquely exists an affine connection $\nabla^{\ast}$ which satisfies
\begin{equation}
X g(Y,Z) =
g(\nabla_X Y,Z) +
g(Y,\nabla^{\ast}_X Z).
\end{equation}
The connection $\nabla^{\ast}$ is called the dual connection of $\nabla$.
Given a Riemannian metric $g$, another reperesentation of the Christoffel symbol is given by
\begin{equation}
\Gamma_{ij,k} =
g
\left(
\nabla_{\partial_i} \partial_j, \partial_k
\right).
\end{equation}
\begin{defi}
When an affine connection $\nabla$ is torsion-free and a metric connection with respect to the Riemannian metric $g$, it is called a Levi-Civita connection with respect to the metric $g$.
\end{defi}
In general, when a $(0,3)$-tensor $\bar{T}$ is given in addition to an affine connection $\nabla$ and a Riemannian metric $g$, an alternative connection $\tilde{\nabla}$ is defined as
\begin{equation}
g(\tilde{\nabla}_{Y} X,Z) =
g(\nabla_{Y}X,Z) +
\bar{T}(X,Y,Z).
\end{equation}
Let $\Omega$ be a set for which probability measure is defined, and define a $d$-dimensional statistical model
\begin{equation}
S = \{
p(\cdot ; \bm{\xi}) | \bm{\xi} \in \Xi
\},
\end{equation}
where the parameter space $\Xi$ is isomorphic to $\mathbb{R}^d$. As a Riemannian metric associated with the statistical model $S$, we consider the Fisher metric defined as
\begin{equation}
g_{ij}(\bm{\xi}) =
\mathbb{E}_{\bm{\xi}}
[
(\partial_i l_{\bm{\xi}})
(\partial_j l_{\bm{\xi}})
],
\end{equation}
where $\mathbb{E}_{\bm{\xi}} [\cdot]$ is expectation with respect to a probability density $p(\cdot; \bm{\xi})$ and $l_{\bm{\xi}}(x) = \log p(x;\bm{\xi}) \; (x \in \Omega)$ is the log-likelihood.
Now, consider a $(0,3)$-tensor $\bar{T}$ on $S$ defined by
\begin{equation}
(\bar{T})_{ijk} (\bm{\xi})
=
\sum_{x \in \Omega}
(\partial_{i} l_{\bm{\xi}}(x))
(\partial_{j} l_{\bm{\xi}}(x))
(\partial_{k} l_{\bm{\xi}}(x))
p(x;\bm{\xi}),
\end{equation}
and based on the Levi-Civita connection $\nabla$ associated with the Fisher metric $g$ on $S$, we define a affine connection $\nabla^{(\alpha)}$ by
\begin{equation}
g(\nabla^{(\alpha)}_{Y}X,Z) =
g(\nabla_Y X,Z) -
\frac{\alpha}{2}
\bar{T}(X,Y,Z), \quad
(X,Y,Z \in \mathfrak{X}(S)).
\end{equation}
This connection is called the $\alpha$-connection. The Christoffel symbols associated with connections $\nabla$ and $\nabla^{(\alpha)}$ are
\begin{align*}
\Gamma_{ij,k} =&
\mathbb{E}_{\bm{\xi}}
\left[
\left\{
\partial_i \partial_j l_{\bm{\xi}} +
\frac{1}{2} (\partial_i l_{\bm{\xi}})
(\partial_j l_{\bm{\xi}})
\right\}
(\partial_k l_{\bm{\xi}})
\right],\\
\Gamma_{ij,k}^{(\alpha)} =&
\mathbb{E}_{\bm{\xi}}
\left[
\left\{
\partial_i \partial_j l_{\bm{\xi}} +
\frac{1-\alpha}{2} (\partial_i l_{\bm{\xi}})
(\partial_j l_{\bm{\xi}})
\right\}
(\partial_k l_{\bm{\xi}})
\right].
\end{align*}
From $\Gamma_{ij,k}^{(\alpha)} = \Gamma_{ji,k}^{(\alpha)}
$, the $\alpha$-connection is torsion-free.
Note that the dual connection of $\nabla^{(\alpha)}$ is $\nabla^{(-\alpha)}$, and it also holds that
\begin{equation}
\nabla^{(\alpha)} =
\frac{1+\alpha}{2} \nabla^{\ast} +
\frac{1-\alpha}{2} \nabla.
\end{equation}
\begin{defi}
\label{def:curvature}
For an affine connection $\nabla$ of a manifold $M$, a map
\begin{align*}
R : \mathfrak{X}(M) \times \mathfrak{X}(M)
\times \mathfrak{X}(M)
\to & \mathfrak{X}(D)\\
(X,Y,Z) \mapsto & R(X,Y)Z =
\nabla_X \nabla_Y Z
-
\nabla_Y \nabla_X Z
-
\nabla_{[X,Y]}Z
\end{align*}
is called the curvature tensor field of the connection $\nabla$.
\end{defi}
The curvature tensor is expressed with coordinate and the Christoffel symbol as
\begin{equation}
R(\partial_i,\partial_j)\partial_k
=
(\partial_i \Gamma_{jk}^l
-
\partial_j
\Gamma^l_{ik}) \partial_l
+
(
\Gamma^l_{jk} \Gamma^{m}_{il}
-
\Gamma^{l}_{ik}
\Gamma^{m}_{jl}
)
\partial_m.
\end{equation}
\begin{defi}
When both the torsion and curvature are zero, the connection $\nabla$ is said to be flat.
\end{defi}
Let $\gamma$ be a map from a close interval $I$ to a manifold $M$. The map $\gamma$ is parameterized by a real-valued parameter $t \in I$ as $\gamma(t)$ and called a curve on $M$. When the value of $\gamma$ at two endpoints of $I$ is fixed, the shortest path between these two points is defined by using the variational principle.
The pararell shift of $\frac{d \gamma}{d t}$ along with $\gamma$ is expressed as
\begin{equation}
\nabla_{\frac{d}{dt}}^{\gamma} \frac{d \gamma}{dt}
=
\left(
\frac{d^2 \gamma_k}{d t^2}
+
(\Gamma^k_{ij} \circ \gamma)
\frac{d \gamma_i}{dt}
\frac{d \gamma_j}{dt}
\right) \partial_k.
\end{equation}
\begin{defi}
An equation
\begin{equation}
\nabla_{\frac{d}{dt}}^{\gamma}\frac{d \gamma}{dt} = \bm{0}
\end{equation}
is called the geodesic equation, and the curve satisfying this equation is called a geodesic.
\end{defi}
Note that if $\Gamma_{ij,k} = 0 \; \forall i,j,k$, the geodesic equation is of the form $\frac{d^2 \gamma_{k}}{d t^2} = 0$, hence the geodesic is a straight line.
\begin{defi}
Let $S$ be a $d$-dimensional statistical model. When each element of the model in $S$ is represented by
\begin{equation}
p(x;\bm{\theta}) =
\exp
\left(
k(x) +
\theta^i F_i (x) - \psi(\bm{\theta})
\right),
\end{equation}
by using functions $k,F_1,\dots,F_d : \Omega \to \mathbb{R}$ and $\psi : \Theta \to \mathbb{R}$, the statistical model $S$ is called an exponential family, and $\bm{\theta}$ is called the natural parameter of the model.
\end{defi}
Note that in a general statistical model $S$, $\xi$, and $\Xi$ are often used as its parameter and the parameter space, while for an exponential family, $\theta$ and $\Theta$ are often used to represent its parameter and the parameter space.
Consider an exponential family with $\alpha$ connection $\nabla^{(\alpha)}$. The Christoffel symbols are
\begin{equation}
\Gamma^{(\alpha)}_{ij,k}
=
\mathbb{E}_{\bm{\theta}}
\left[
\left\{
\partial_i \partial_j l_{\bm{\theta}} +
\frac{1-\alpha}{2}
(\partial_i l_{\bm{\theta}})
(\partial_j l_{\bm{\theta}})
\right\}
(\partial_k l_{\bm{\theta}})
\right],
\end{equation}
and
\begin{equation}
\partial_i l_{\bm{\theta}} =
F_i (x) - (\partial_i \psi)(\bm{\theta}), \quad
(\partial_i \partial_j \psi) (\bm{\theta}).
\end{equation}
So, when $\alpha=1$, we have
\begin{equation}
\Gamma^{(1)}_{ij,k}
=
\mathbb{E}_{\bm{\theta}}
[
(- ( \partial_i \partial_j \psi)(\bm{\theta}))
(\partial_k l_{\bm{\theta}})
] = 0,
\end{equation}
namely, the exponential family is flat with the Fisher metric and $\alpha=1$ connection.
This implies that in exponential family, for the $\alpha=1$-connection $\nabla^{(1)}$ associated with the Fisher metric, the geodesic between two points correspond to natural parameters $\bm{\theta}_1$ and $\bm{\theta}_{2}$ is of the form $t \bm{\theta}_1 + (1-t) \bm{\theta}_2$.
\section*{Appendix B: Proofs of main results}
\label{app:prf}
\begin{proof}[Derivation of the information geometrically generalized IWERM]
Let $h_A$ be a hypothesis generated by AIWERM.
From Lemma 4.1, we can write
\begin{align}
\hat{h}_A &= \min_{h\in\mathcal{H}}\int_{\mathcal{X}\times\mathcal{Y}}\ell(h(\bm{x}), y)p^{(\lambda)}_A(\bm{x})p_{tr}(y|\bm{x})d\bm{x}dy \nonumber \\
&= \min_{h\in\mathcal{H}}\int_{\mathcal{X}\times\mathcal{Y}}\ell(h(\bm{x}), y)m_f^{(\lambda, 1)}(p_{tr}(\bm{x}), p_{te}(\bm{x}))p_{tr}(y|\bm{x})d\bm{x}dy \nonumber \\
&= \min_{h\in\mathcal{H}}\int_{\mathcal{X}\times\mathcal{Y}}\ell(h(\bm{x}), y)\frac{m_f^{(\lambda, 1)}(p_{tr}(\bm{x}), p_{te}(\bm{x}))}{p_{tr}(\bm{x})}p_{tr}(\bm{x}, y)d\bm{x}dy.
\end{align}
From Lemma 4.2, we also have
\begin{align}
\hat{h}_R = \min_{h\in\mathcal{H}}\int_{\mathcal{X}\times\mathcal{Y}}\ell(h(\bm{x}), y)\frac{m_f^{(\lambda, 3)}(p_{tr}(\bm{x}), p_{te}(\bm{x}))}{p_{tr}(\bm{x})}p_{tr}(\bm{x}, y)d\bm{x}dy.
\end{align}
Then, we consider
\begin{equation}
\hat{h} = \min_{h\in\mathcal{H}}\int_{\mathcal{X}\times\mathcal{Y}}w^{(\lambda,\alpha)}(\bm{x})\ell(h(\bm{x}), y)p_{tr}(\bm{x},y)d\bm{x}dy,
\end{equation}
where
\begin{equation}
w^{(\lambda,\alpha)}(\bm{x}) =
\frac{
m^{(\lambda,\alpha)}_{f}(p_{tr}(\bm{x}),p_{te}(\bm{x}))
}{p_{tr}(\bm{x})}.
\end{equation}
We can see that AIWERM is a special case when $\alpha=1$ and RIWERM is a special case when $\alpha=3$.
\end{proof}
\begin{proof}[Proofs of Propositions~\ref{prop:geometric_bias_aiwerm} and ~\ref{prop:geometric_bias_riwerm}]
Let
\begin{equation}
\bm{\theta}^{(\lambda,\alpha)} = m^{(\lambda, \alpha)}_f(\bm{\theta}_{tr}, \bm{\theta}_{te}),
\end{equation}
and let $R^{(\alpha)}$ be the Riemann curvature tensor defined in Definition~\ref{def:curvature} with respect to the $\alpha$-connection $\nabla^{(\alpha)}$.
We define the relative curvature tensor as
\begin{equation}
R^{(\alpha,\beta)}(X,Y,Z) = \Big[\nabla^{(\alpha)}_X, \nabla^{(\beta)}_Y\Big]Z - \nabla^{(\alpha)}_{[X,Y]}Z
\end{equation}
and
the difference tensor as
\begin{equation}
K(X,Y) = \nabla^*_XY - \nabla_XY.
\end{equation}
For any $\alpha\in\mathbb{R}$ and $\beta\in\mathbb{R}$, we have
\begin{align}
\nabla^{(\alpha)}_X\nabla^{(\beta)}_Y Z &= \Big(\frac{1+\alpha}{2}\nabla^*_X + \frac{1-\alpha}{2}\nabla_X\Big)\Big(\frac{1+\beta}{2}\nabla^*_Y + \frac{1-\beta}{2}\nabla_Y\Big)Z \nonumber\\
&= \frac{(1+\alpha)(1+\beta)}{4}\nabla^*_X\nabla^*_YZ + \frac{(1+\alpha)(1-\beta)}{4}\nabla^*_X\nabla_YZ \nonumber\\
&\ \quad + \frac{(1-\alpha)(1+\beta)}{4}\nabla_X\nabla^*_YZ+\frac{(1-\alpha)(1-\beta)}{4}\nabla_X\nabla_YZ. \\
\nabla^{(\beta)}_Y\nabla^{(\alpha)}_X &= \Big(\frac{1+\beta}{2}\nabla^*_Y + \frac{1-\beta}{2}\nabla_Y\Big)\Big(\frac{1+\alpha}{2}\nabla^*_X + \frac{1-\alpha}{2}\nabla_X\Big)Z \nonumber \\
&= \frac{(1+\alpha)(1+\beta)}{4}\nabla^*_Y\nabla^*_XZ + \frac{(1-\alpha)(1+\beta)}{4}\nabla^*_Y\nabla_XZ \nonumber\\
&\ \quad + \frac{(1+\alpha)(1-\beta)}{4}\nabla_Y\nabla^*_XZ + \frac{(1-\alpha)(1-\beta)}{4}\nabla_Y\nabla_XZ.\\
\nabla^{(\alpha)}_{[X,Y]}Z &= \frac{1+\alpha}{2}\nabla^*_{[X,Y]}Z + \frac{1-\alpha}{2}\nabla_{[X,Y]}Z.
\end{align}
Then
\begin{align}
R^{(\alpha,\beta)}(X,Y,Z) &= \nabla^{(\alpha)}_X\nabla^{(\beta)}_Y Z - \nabla^{(\beta)}_X\nabla^{(\alpha)}_Y Z - \nabla^{(\alpha)}_{[X,Y]} Z\nonumber \\
&= \frac{(1+\alpha)(1+\beta)}{4}(\nabla^*_X\nabla^*_Y - \nabla^*_Y\nabla^*_X)Z \nonumber\\
&\ \quad + \frac{(1+\alpha)(1-\beta)}{4}(\nabla^*_X\nabla_Y - \nabla_Y\nabla^*_X)Z \nonumber\\
&\ \quad + \frac{(1-\alpha)(1+\beta)}{4}(\nabla_X\nabla^*_Y - \nabla^*_Y\nabla_X)Z \nonumber\\
&\ \quad + \frac{(1-\alpha)(1-\beta)}{4}(\nabla_X\nabla_Y - \nabla_Y\nabla_X)Z \nonumber\\
&\ \quad -\frac{1+\alpha}{2}\nabla^*_{[X,Y]}Z - \frac{1-\alpha}{2}\nabla_{[X,Y]}Z \nonumber\\
&= \frac{(1+\alpha)(1+\beta)}{4}\Big\{R^*(X,Y,Z) + \nabla^*_{[X,Y]}Z\Big\} \nonumber\\
&\ \quad + \frac{(1+\alpha)(1-\beta)}{4}\Big\{R^{(1,-1)}(X,Y,Z)+\nabla^*_{[X,Y]}Z\Big\} \nonumber\\
&\ \quad + \frac{(1-\alpha)(1+\beta)}{4}\Big\{R^{(-1,1)}(X,Y,Z) + \nabla_{[X,Y]}Z\Big\} \nonumber\\
&\ \quad + \frac{(1-\alpha)(1-\beta)}{4}\Big\{R^{(-1,-1)}(X,Y,Z) + \nabla^*_{[X,Y]}Z\Big\} \nonumber\\
&\ \quad - \frac{1+\alpha}{2}\nabla^*_{[X,Y]}Z - \frac{1-\alpha}{2}\nabla_{[X,Y]}Z \\
4R^{(\alpha,\beta)} &= (1+\alpha)(1+\beta)R^* + (1-\alpha)(1-\beta)R \nonumber\\
&\ \quad + (1+\alpha)(1-\beta)R^{(1,-1)} + (1-\alpha)(1+\beta)R^{(-1,1)}.
\end{align}
We also have
\begin{align}
K^{(\alpha,\beta)}(X,Y) &= \nabla^{(\beta)}_XY - \nabla^{(\alpha)}_XY \nonumber\\
&= \Big\{\frac{1+\beta}{2}\nabla^*_XY + \frac{1-\beta}{2}\nabla_XY\Big\} - \Big\{\frac{1+\alpha}{2}\nabla^*_XY + \frac{1-\alpha}{2}\nabla_XY\Big\} \nonumber\\
&= \frac{\beta-\alpha}{2}\nabla^*_XY - \frac{\beta-\alpha}{2}\nabla_XY = \frac{\beta-\alpha}{2}K(X, Y) \\
K^{(\alpha,\beta)}\Big(X, K^{(\alpha,\beta)}(Y,Z)\Big) &= \frac{\beta-\alpha}{2}K\Big(X, K^{(\alpha,\beta)}(Y,Z)\Big) = \frac{(\beta-\alpha)^2}{4}K\Big(X, K(Y,Z)\Big).
\end{align}
Combining them, the following relations hold:
\begin{align}
K^{(\beta,\alpha)}\Big(X, K^{(\beta,\alpha)}(Y,Z)\Big) &= K^{(\beta,\alpha)}\Big(X, \nabla^{(\alpha)}_YZ - \nabla^{(\beta)}_YZ\Big) \\
&= K^{(\beta,\alpha)}\Big(X, \nabla^{(\alpha)}_YZ \Big) - K^{(\beta,\alpha)}\Big(X,\nabla^{(\beta)}_YZ\Big) \nonumber \\
&= \nabla^{(\alpha)}_X\nabla^{(\alpha)}_YZ - \nabla^{(\beta)}_X\nabla^{(\alpha)}_YZ - \nabla^{(\alpha)}_X\nabla^{(\beta)}_YZ + \nabla^{(\beta)}_X\nabla^{(\beta)}_YZ \\
\frac{(\alpha-\beta)^2}{4}K\Big(X, K(Y,Z)\Big) &= \nabla^{(\alpha)}_X\nabla^{(\alpha)}_YZ - \nabla^{(\beta)}_X\nabla^{(\alpha)}_YZ - \nabla^{(\alpha)}_X\nabla^{(\beta)}_YZ + \nabla^{(\beta)}_X\nabla^{(\beta)}_YZ.
\end{align}
Swapping $X$ and $Y$, we have
\begin{align}
&\ \frac{(\alpha-\beta)^2}{4}\Biggl\{K\Big(X, K(Y,Z)\Big) - K\Big(Y, K(X,Z)\Big)\Biggr\} \nonumber\\
&\ = R^{(\alpha)}(X,Y,Z) + R^{(\beta)}(X,Y,Z) - \Big\{\Big[\nabla^{(\alpha)}_X, \nabla^{(\beta)}_Y\Big]Z - \nabla^{(\alpha)}_{[X,Y]}Z\Big\} \nonumber\\
&\ \quad - \Big\{\Big[\nabla^{(\beta)}_X,\nabla^{(\alpha)}_Y\Big]Z - \nabla^{(\beta)}_{[X,Y]}Z\Big\} \nonumber \\
&= R^{(\alpha)}(X,Y,Z) + R^{(\beta)}(X,Y,Z) - R^{(\alpha,\beta)}(X,Y,Z) - R^{(\beta,\alpha)}(X,Y,Z).
\end{align}
Making $\alpha=\beta$, we have
\begin{align}
4R^{(\alpha)} &= (1+\alpha)^2R^* + (1-\alpha)^2R + (1-\alpha^2)R^{(1,-1)} + (1-\alpha^2)R^{(-1,1)} \nonumber\\
&= (1+\alpha^2)R^* + (1-\alpha)^2R + (1-\alpha^2)\Big(R^{(1,-1)} + R^{(-1,1)}\Big). \label{eq:relative_curvature1}
\end{align}
Making $\alpha=1$ and $\beta=-1$, we also have
\begin{align}
R^{(1,-1)}(X,Y,Z) + R^{(-1,1)}(X,Y,Z) &= R^*(X,Y,Z) + R(X,Y,Z) \nonumber \\
&\ \quad - \Biggl\{K\Big(X, K(Y,Z)\Big) - K\Big(Y, K(X,Z)\Big)\Biggr\}. \label{eq:relative_curvature2}
\end{align}
From Eq.~\eqref{eq:relative_curvature1} and \eqref{eq:relative_curvature2}, we obtain
\begin{align}
4R^{(\alpha)} &= (1+\alpha)^2R^*(X,Y,Z) + (1-\alpha)^2R(X,Y,Z) \nonumber\\
&\ \quad + (1-\alpha^2)R^*(X,Y,Z) + (1-\alpha^2)\Biggl\{K\Big(X, K(Y,Z)\Big) - K\Big(Y, K(X,Z)\Big)\Biggr\} \nonumber \\
&= 2(1+\alpha)R^*(X,Y,Z) + 2(1-\alpha)R(X,Y,Z) \nonumber\\
&\ \quad + (1-\alpha^2)\Biggl\{K\Big(Y, K(X,Z)\Big) - K\Big(X, K(Y,Z)\Big)\Biggl\}
\end{align}
Since the exponential family is dually flat, that is $R=0$ and $R^*=0$, and the Riemann curvature tensor with respect to $\nabla^{(\alpha)}$ is
\begin{align}
R^{(\alpha)}(X,Y,Z) &= \frac{1-\alpha^2}{4}\Lambda, \\
\label{eq:Lambda}
\Lambda &= \Big(K(Y, K(X,Z)) - K(X, K(Y,Z)) \Big).
\end{align}
Then, the geometric bias vector of $\bm{\theta}^{(\lambda,\alpha)}$ is
\begin{align}
b(\alpha, \lambda) = (1-\lambda)\Big\{e_1 + tr_g\Big(\frac{1-\alpha^2}{2}\Lambda_{ikj}d\bm{\theta}^i \otimes d\bm{\theta}^j\Big)e_2\Big\},
\end{align}
where $tr_g$ is the trace operation on the metric tensor $g$, and $\Lambda_{ikj}$ is the element of $\Lambda$ in Eq.~\eqref{eq:Lambda}.
Since AIWERM and RIWERM are two special cases for $\alpha=1$ and $\alpha=3$, we have
\begin{align}
b(1, \lambda) &= (1-\lambda)e_1, \\
b(3, \lambda) &= (1-\lambda)\Big\{e_1 + tr_g\Big(-4\Lambda_{ikj}d\bm{\theta}^i \otimes d\bm{\theta}^j\Big)e_2\Big\}.
\end{align}
\end{proof}
\section*{Appendix C: Learning guarantee}
\label{app:slt}
Generalization bounds of weighted maximum likelihood estimator for the target domain are derived in~\cite{Cortes2010}, and our weight function~\eqref{eq:Gweight} is compatible with their bound.
Then, the gap between the expected (with respect to test distribution) loss $\mathcal{R}(h)$ and empirical risk $L(h;\lambda,\alpha)$ is bounded as
\begin{align}
\notag
&
| \mathcal{R}(h)
- L(h;\lambda,\alpha) |
\leq
\left|
\mathbb{E}_{p_{tr}}
\left[
\left\{
\frac{p_{te}(\bm{x})}{p_{tr}(\bm{x})}
-
w^{(\lambda,\alpha)}(\bm{x})
\right\}
\right]
\right|
\ell(h(\bm{x},y(\bm{x})))\\ \notag
+&
2^{5/4}
\max
\left(
\sqrt{\mathbb{E}_{p_{tr}}
(w^{(\lambda,\alpha)}(\bm{x}))^2
\ell^2 (h(\bm{x},y(\bm{x})))
},
\sqrt{\mathbb{E}_{\hat{p}_{tr}}
(w^{(\lambda,\alpha)}(\bm{x}))^2
\ell^2 (h(\bm{x},y(\bm{x}))
}
\right)
\\
\times &
\left(
\frac{
p \log \frac{2 n_{tr} e}{p}
+
\log \frac{4}{\delta}
}{
n_{tr}
}\right)^{\frac{3}{8}}.
\end{align}
In the above inequality, $p$ is the pseudo-dimension of the function class $\{w^{\lambda,\alpha}(\bm{x}) \ell(h(\bm{x}),y(\bm{x}))| h \in \mathcal{H} \}$ where $y(\bm{x})$ is the ground truth function of connecting $\bm{x}$ and $y$ as $y=y(\bm{x})$. The first term of the r.h.s. of the above inequality is the bias introduced by using $w^{\lambda,\alpha}$ instead of a standard density ratio, and the second term reflects the variance. It is worth mentioning that the term $\mathbb{E}_{p_{tr}}
(w^{(\lambda,\alpha)}(\bm{x}))^2
\ell^2 (h(\bm{x},y(\bm{x})))$ is further bounded by $d_{2}(p_{te}||p_{tr}) = \int_{x\in \mathcal{X}} \frac{p_{te}^2 (\bm{x})}{p_{tr}(\bm{x})} d\bm{x}$.
\section*{Appendix D: Optimization of the generalized IWERM}
\label{app:BO}
In the expected improvement strategy, the $t+1^{th}$ point $(\lambda,\alpha)_{t+1}$ is selected according to the following equation.
\begin{small}
\begin{equation*}
(\lambda,\alpha)_{t+1} = \mathop{\rm arg~min}\limits_{(\lambda,\alpha)}\mathbb{E}\Big[\max\Big(0, h_{t+1}(\lambda,\alpha) - L(\lambda^\dagger,\alpha^\dagger)\Big) \Big| D_t\Big],
\end{equation*}
\end{small}
where $L(\lambda^{\dagger},\alpha^{\dagger})$ is the maximum value of empirical risk that has been encountered so far, $h_{t+1}(\lambda,\alpha)$ is the posterior mean of the surrogate at the $t+1^{th}$ step and $D_t = \{(\lambda,\alpha)_i, L(\lambda_i,\alpha_i)\}^t_{i=1}$.
This equation for Gaussian process surrogate is an analytical expression:
\begin{align*}
a_{EI}(\lambda,\alpha) &= (\mu_t(\lambda,\alpha) - L(\lambda^{\dagger},\alpha^{\dagger}))\Phi(Z) + \sigma_t(\lambda,\alpha)\phi(Z), \\
Z &= \frac{\mu_t(\lambda,\alpha) - L(\lambda^{\dagger},\alpha^\dagger)}{\sigma_t(\lambda,\alpha)},
\end{align*}
where $\Phi(\cdot)$ and $\phi(\cdot)$ are normal cumulative and density functions, respectively, and $\mu_t$ and $\sigma_t$ are mean and standard deviation of $\{(\lambda,\alpha)_i\}^t_{i=1}$.
\section*{Appendix E: Additional experimental results}
\label{app:exp}
\subsection{Experimental results on LIBSVM dataset}
We show that for the LIBSVM dataset, IGIWERM is effective even for multiple models.
Table~\ref{tab:libsvm_several_models_error} shows the results for each model.
We use the scikit-learn~\cite{scikit-learn} implementation of the models, and the hyperparameters of each model are the default values of this library.
Figure~\ref{fig:gird_search_surface} also shows the relationship between the two parameters of IGIWERM and the errors that can be achieved.
For this visualization, we explore the parameter pairs by grid search and evaluate their performance at that time. From this figure, it is seen that the best performance is often achieved when $\alpha \neq 1$ and $\alpha \neq 3$, showing the sub-optimality of conventional methods.
\subsection{Experimental results on regression problems}
In this section, we present experimental results for the regression problem.
All the datasets used in this experiment are available in the scikit-learn~\cite{scikit-learn} dataset collection.
We use SVR with Radial Basis Function (RBF) kernel as the base regressor.
Table~\ref{tab:sklearn_regression} shows the results of this experiment.
In this experiment, we use MSE as a metric, and this table shows that IGIWERM is superior to existing covariate shift adaptation methods.
Figure~\ref{fig:gird_search_surface_regression} shows the relationship between the two parameters of IGIWERM and the mean squared errors that can be achieved.
This figure shows that, as in the case of binary classification, the optimal parameters do not necessarily match those of existing methods.
\subsection{Experimental results on multi-class classification}
We also introduce the additional experimental results for the multi-class classification problem.
All the datasets used in this experiment are available in the scikit-learn~\cite{scikit-learn} dataset collection.
We also use the scikit-learn~\cite{scikit-learn} implementation of the models, and the hyperparameters of each model are the default values of this library. We note that the number of training sample for {\tt{covtype}} is so large hence the results with SVM for this dataset are omitted. Table~\ref{tab:sklearn_multi_class} shows the experimental results, and we see that our proposed generalization outperforms existing methods.
Figure~\ref{fig:gird_search_surface_multi_class} shows the relationship between the two parameters of IGIWERM and the errors that can be achieved.
This figure also shows the sub-optimality of conventional methods.
\subsection{Visualization of covariate shift}
In Section 5, we induce the covariate shift by the method of Cortes et al~\cite{Cortes2008-zd}.
Figure~\ref{fig:covariate_shift_libsvm} shows a plot by PCA of each dataset splitted into the training set and test set.
This figure shows that we are able to induce a covariate shift by partitioning the dataset.
\begin{table*}[t]
\centering
\caption{Mean misclassification rates averaged over 10 trails on LIBSVM benchmark datasets. The numbers in the brackets are the
standard deviations. For the methods with (optimal), the optimal parameters for the test data are obtained by the linear search.
The lowest misclassification rates among five methods are shown with bold.}
\label{tab:libsvm_several_models_error}
\resizebox{\textwidth}{!}{
\begin{tabular}{c|c|ccccc}
\toprule
Dataset &model &unweighted &IWERM &AIWERM (optimal) &RIWERM (optimal) &ours \\ \midrule
australian &Logistic Regression&$22.76 (\pm 9.53)$ &$22.75 (\pm 7.73)$ &$22.19 (\pm 9.35)$ &$22.39 (\pm 7.76)$ &${\bf 21.47 (\pm 8.91)}$\\
&SVM &$33.46 (\pm 23.65)$&$22.13 (\pm 3.37)$ &$21.98 (\pm 3.36)$ &$21.73 (\pm 3.82)$ &${\bf 18.85 (\pm 3.99)}$\\
&AdaBoost &$10.20 (\pm 5.09)$ &$16.35 (\pm 10.67)$&$11.23 (\pm 3.49)$ &$15.47 (\pm 2.72)$ &${\bf 9.15 (\pm 3.93)}$ \\
&Naive Bayes &$17.04 (\pm 6.02)$ &$14.92 (\pm 5.96)$ &$16.33 (\pm 6.07)$ &$15.49 (\pm 6.07)$ &${\bf 14.74(\pm 5.84)}$ \\
&Random Forest &$8.89 (\pm 3.72)$ &$8.64 (\pm 3.20)$ &${\bf 8.29 (\pm 3.21)}$&$8.38 (\pm 3.47)$ &$8.61 (\pm 3.73)$ \\ \midrule
breast-cancer&Logistic Regression&$32.32 (\pm 3.08)$ &$32.87 (\pm 3.56)$ &$32.32 (\pm 3.08)$ &$32.32 (\pm 3.08)$ &${\bf 32.26 (\pm 3.04)}$\\
&SVM &$38.28 (\pm 10.98)$&$41.23 (\pm 15.39)$&$36.41 (\pm 9.68)$ &$36.13 (\pm 10.81)$ &${\bf 31.65 (\pm 8.49)}$\\
&AdaBoost &$5.09 (\pm 1.65)$ &$5.63 (\pm 1.38)$ &$6.01 (\pm 1.13)$ &$5.95 (\pm 1.50)$ &${\bf 4.84 (\pm 1.50)}$\\
&Naive Bayes &$11.27 (\pm 4.09)$ &$19.65 (\pm 15.14)$&$10.36 (\pm 5.14)$ &$18.00 (\pm 14.24)$ &${\bf 10.03 (\pm 3.53)}$\\
&Random Forest &$3.32 (\pm 1.23)$ &$3.19 (\pm 1.10)$ &$3.19 (\pm 1.13)$ &$3.18 (\pm 1.04)$ &${\bf 3.13 (\pm 1.04)}$\\ \midrule
heart &Logistic Regression&$39.68 (\pm 7.90)$ &$40.55 (\pm 9.10)$ &$39.96 (\pm 7.40)$ &$39.94 (\pm 6.93)$ &${\bf 36.56 (\pm 8.29)}$\\
&SVM &$45.17 (\pm 6.98)$ &$39.94 (\pm 8.55)$ &$39.76 (\pm 8.49)$ &$39.74 (\pm 8.92)$ &${\bf 35.37 (\pm 6.84)}$\\
&AdaBoost &$30.87 (\pm 12.04)$&$29.24 (\pm 6.47)$ &$29.37 (\pm 12.19)$ &$31.27 (\pm 8.37)$ &${\bf 26.96(\pm 13.12)}$\\
&Naive Bayes &$22.79 (\pm 6.17)$ &$24.78 (\pm 7.99)$ &$22.87 (\pm 6.02)$ &$24.58 (\pm 7.98)$ &${\bf 21.97 (\pm 6.41)}$\\
&Random Forest &$20.87 (\pm 6.64)$ &$20.98 (\pm 6.62$ &$20.96 (\pm 6.67)$ &$21.96 (\pm 6.70)$ &${\bf 19.95 (\pm 6.61)}$\\ \midrule
diabetes &Logistic Regression&$37.62 (\pm 4.35)$ &$40.22 (\pm 4.10)$ &$38.38 (\pm 3.85)$ &$40.11 (\pm 3.74)$ &${\bf 36.86 (\pm 4.81)}$\\
&SVM &$33.19 (\pm 5.69)$ &$37.22 (\pm 6.63)$ &$33.11 (\pm 6.45)$ &$33.38 (\pm 5.74)$ &${\bf 32.83 (\pm 5.62)}$\\
&AdaBoost &$37.69 (\pm 4.28)$ &$40.13 (\pm 5.28)$ &$40.76 (\pm 4.31)$ &$41.26 (\pm 5.16)$ &${\bf 33.45 (\pm 4.35)}$\\
&Naive Bayes &$39.29 (\pm 3.98)$ &$39.21 (\pm 3.18)$ &$39.26 (\pm 2.97)$ &$39.35 (\pm 2.85)$ &${\bf 38.10 (\pm 4.02)}$\\
&Random Forest &$30.09 (\pm 3.03)$ &$30.90 (\pm 3.52)$ &$31.07 (\pm 3.10)$ &$30.51 (\pm 3.67)$ &${\bf 29.46 (\pm 2.99)}$\\ \midrule
madelon &Logistic Regression&$47.31 (\pm 1.80)$ &$47.80 (\pm 1.57)$ &$47.16 (\pm 1.68)$ &$46.81 (\pm 1.56)$ &${\bf 46.31 (\pm 1.69)}$\\
&SVM &$47.78 (\pm 1.53)$ &$47.28 (\pm 2.20)$ &$47.10 (\pm 2.13)$ &$47.12 (\pm 1.65)$ &${\bf 46.56 (\pm 2.12)}$\\
&AdaBoost &$42.92 (\pm 1.40)$ &$42.91 (\pm 1.68)$ &$43.36 (\pm 1.81)$ &$42.90 (\pm 1.40)$ &${\bf 40.64 (\pm 7.32)}$\\
&Naive Bayes &$41.90 (\pm 1.05)$ &$41.43 (\pm 8.76)$ &$41.79 (\pm 8.05)$ &$41.62 (\pm 7.43)$ &${\bf 41.03 (\pm 8.32)}$\\
&Random Forest &$35.90 (\pm 0.83)$ &$35.42 (\pm 1.75)$ &$35.13 (\pm 1.30)$ &${\bf 34.79 (\pm 1.75)}$ &$35.56 (\pm 2.03)$ \\
\bottomrule
\end{tabular}
}
\end{table*}
\begin{figure*}
\caption{Visualization of grid search for $\alpha$ and $\lambda$ on LIBSVM dataset.
For the sake of clarity, we apply a moving average.}
\label{fig:gird_search_surface}
\end{figure*}
\begin{table*}[t]
\centering
\caption{Mean squared errors averaged over 10 trails on scikit-learn~\cite{scikit-learn} regression benchmark datasets. The numbers in the brackets are the
standard deviations. For the methods with (optimal), the optimal parameters for the test data are obtained by the linear search.
The lowest mean squared errors are shown with bold.}
\label{tab:sklearn_regression}
\resizebox{\textwidth}{!}{
\begin{tabular}{c|c|c|ccccc}
\toprule
Dataset & \#features & \#data & unweighted & IWERM & AIWERM (optimal) & RIWERM (optimal) & ours \\ \midrule
boston & $13$ & $506$ & $83.22 (\pm 5.72)$ & $69.87 (\pm 2.31)$ & $69.68 (\pm 1.46)$ & $69.96 (\pm 1.84)$ & ${\bf 68.36 (\pm 1.20)}$\\
diabetes & $10$ & $442$ &$0.049 (\pm 0.007)$ &$0.0501 (\pm 0.009)$ &$0.049(\pm 0.008)$ &$0.049 (\pm 0.009)$ &${\bf 0.048(\pm 0.007)}$ \\
california housing & $8$ & $20,640$&$1.432 (\pm 0.095)$ &$1.3214 (\pm 0.345)$ &$1.260(\pm 0.125)$ &$1.261 (\pm 0.086)$ &${\bf 1.232(\pm 0.095)}$ \\
\bottomrule
\end{tabular}
}
\end{table*}
\begin{figure*}
\caption{Visualization of grid search for $\alpha$ and $\lambda$ on scikit-learn regression dataset.
For the sake of visualization, we apply a moving average.}
\label{fig:gird_search_surface_regression}
\end{figure*}
\begin{table*}[t]
\centering
\caption{Mean misclassification rates averaged over 10 trails on scikit-learn~\cite{scikit-learn} multi-class classification benchmark datasets. The numbers in the brackets are the
standard deviations. For the methods with (optimal), the optimal parameters for the test data are obtained by the linear search.
The lowest misclassification rates among five methods are shown with bold.}
\label{tab:sklearn_multi_class}
\resizebox{\textwidth}{!}{
\begin{tabular}{c|c|ccccc}
\toprule
Dataset & model & unweighted & IWERM & AIWERM (optimal) & RIWERM (optimal) & ours \\ \midrule
digits & Logistic Regression & $6.92 (\pm 2.25)$ & $7.10 (\pm 1.76)$ & $6.90 (\pm 1.81)$ & $6.89 (\pm 1.80)$ & ${\bf 6.79 (\pm 1.82)}$\\
& SVM & $4.21 (\pm 2.04)$ & $3.94 (\pm 1.14)$ & $4.20 (\pm 1.22)$ & $4.02 (\pm 1.18)$ & ${\bf 3.88 (\pm 1.11)}$\\
& AdaBoost & $67.98(\pm 12.35)$ & $71.77(\pm 6.25)$ & $71.26 (\pm 8.21)$ & $70.47 (\pm 6.46)$ & ${\bf 65.20 (\pm 8.20)}$\\
& Naive Bayes & $19.07(\pm 2.45)$ & $18.68 (\pm 2.61)$ & $19.31 (\pm 2.64)$ & $18.78 (\pm 2.68)$ & ${\bf 18.58 (\pm 2.66)}$\\
& Random Forest & $6.85 (\pm 2.40)$ & $6.30 (\pm 1.80)$ & $6.23 (\pm 1.53)$ & $6.27 (\pm 1.64)$ & ${\bf 6.17 (\pm 1.90)} $\\ \midrule
iris & Logistic Regression & $54.69(\pm 20.51)$ & $36.49 (\pm 22.39)$ & $45.78 (\pm 18.09)$ & $35.37 (\pm 23.19)$ & ${\bf 28.89 (\pm 20.54)}$\\
& SVM & $55.16(\pm 22.60)$ & $36.65 (\pm 22.56)$ & $33.21 (\pm 20.25)$ & $30.46 (\pm 21.14)$ & ${\bf 29.04 (\pm 20.02)}$\\
& AdaBoost & $27.19(\pm 22.56)$ & $26.00 (\pm 22.98)$ & $26.00 (\pm 22.98)$ & $26.00 (\pm 22.98)$ & ${\bf 19.62 (\pm 21.36)}$\\
& Naive Bayes & $35.88 (\pm 23.24)$& $35.97 (\pm 26.79)$ & $37.99 (\pm 23.55)$ & $33.84 (\pm 25.64)$ & ${\bf 27.52 (\pm 21.00)}$\\
& Random Forest & $26.16 (\pm 22.86)$& $32.17 (\pm 21.21)$ & $26.00 (\pm 22.98)$ & $28.47 (\pm 22.63)$ & ${\bf 22.77 (\pm 21.74)}$\\ \midrule
covtype & Logistic Regression & $45.36 (\pm 13.08)$& $32.04 (\pm 5.833)$ & $30.62 (\pm 4.64)$ & $25.20 (\pm 8.99)$ & ${\bf 19.99 (\pm 5.56)} $\\
& SVM & $- $ & $- $ & $- $ & $- $ & $- $\\
& AdaBoost & $47.53 (\pm 14.51)$& $25.55 (\pm 12.27)$ & $25.47 (\pm 14.14)$ & $27.86 (\pm 10.37)$ & ${\bf 18.96 (\pm 7.29)} $\\
& Naive Bayes & $41.13 (\pm 15.65)$& $30.66 (\pm 15.09)$ & $28.48 (\pm 15.66)$ & $27.20 (\pm 15.79)$ & ${\bf 19.64 (\pm 15.62)}$\\
& Random Forest & $23.51(\pm 3.31)$ & $18.18 (\pm 2.01)$ & $17.28 (\pm 2.05)$ & $17.13 (\pm 2.26)$ & ${\bf 16.42 (\pm 2.08)}$\\
\bottomrule
\end{tabular}
}
\end{table*}
\begin{figure*}
\caption{Visualization of grid search for $\alpha$ and $\lambda$ on scikit-learn multi-class classification dataset.
For the sake of visualization, we apply a moving average.}
\label{fig:gird_search_surface_multi_class}
\end{figure*}
\begin{figure*}
\caption{Plot of covariate shifts using the method of Cortes et al~\cite{Cortes2008-zd}
\label{fig:covariate_shift_libsvm}
\end{figure*}
\end{document} |
\begin{document}
\title{The Latent Bernoulli-Gauss Model for Data Analysis}
\begin{abstract}
We present a new latent-variable model employing a Gaussian mixture integrated with a feature selection procedure (the Bernoulli part of the model) which together form a "Latent Bernoulli-Gauss" distribution. The model is applied to MAP estimation, clustering, feature selection and collaborative filtering and fares favorably with the state-of-the-art latent-variable models.
\end{abstract}
\section{Introduction}
We present a new mixture model for collections of discrete data with applications to clustering through MAP classification, supervised learning, feature selection and collaborative filtering. In the language of text modeling, the algorithm integrates modeling of word frequencies with a feature selection procedure into a single latent class distribution model. The algorithm defines two types of words (i) {\it keywords\/} representing "important" words associated with high frequency appearance, and (ii) all remaining words (not including stop-words which are omitted from consideration). All keywords are "topic specific" modeled by a mixture of Gaussians (one per topic) and all remaining words are considered "topic unspecific" are modeled by a single Gaussian. The decision of which are the keywords of a document is modeled by a latent Bernoulli process --- thus together we have a "Latent Bernoulli-Gauss" (LBG) model.
We present the LBG model in sec.~\ref{sec:bg} and its applications in sec.~\ref{sec:app}. In sec.~\ref{sec:comp} we present a detailed discussion of the merits of LBG as compared to existing latent-variable models including Mixture-of-Unigrams (MOU) \cite{MoU_Nigam}, probabilistic Latent Semantic Indexing (pLSI) \cite{Hofmann99probabilisticlatent} and Latent Dirichlet Allocation (LDA)~\cite{BleiNJ03}. We conducted a series of experiments on public datasets covering a spectrum of information retrieval applications --- a detailed discussion of experimental results and comparisons to MOU, LDA and pLSI is in sec.~\ref{sec:exp}.
We use the language of text collections throughout the paper, referring to measurements as "word frequencies" and "documents". Nevertheless, the LBG model is general and can be applied (and is applied in sec.~\ref{sec:exp}) to a wide range of data analysis tasks.
\section{The Bernoulli-Gauss Mixture Model}
\label{sec:bg}
Consider a code-book of size $n$ representing the vocabulary of $n$ words in a dictionary. A document is an unordered collection of $N$ words $w_1,...,w_N$ where $w_i\in\{1,...,n\}$. A document $d$ is represented by the $n$ frequencies of word appearances normalized in a proper manner (in text applications we use the term-frequency-inverse-document-frequency (tf-idf) normalization), resulting in $d=(m_1,...,m_n)$ a set of non-negative real numbers.
For a document $d$, we distinguish between a "keyword" which is associated with a high frequency and other low-frequency words of the document. A keyword is another way of saying that the word is "important" for that document. Let $\mbox{\bf x}\in\{0,1\}^n$ be an indicator set where $x_i=1$ if the $i$'th word in the code-book is a keyword and $x_i=0$ otherwise. We assume that the keywords are modeled by a topic-specific Normal distribution whereas all other words are modeled by a topic-unspecific Normal distribution. Let $y\in\{1,...,k\}$ be a random variable representing the $k$ possible "topics" which generated the document $d$. Let $p_{si}$ be the probability of the $i$'th code-word to be a keyword in the $s=1,...,k$ topic. The Latent Bernoulii-Gauss model $Pr(d\ |\ y=s,\theta)$ of document $d$ given topic $y=s$ is:
{\small
\begin{equation}\prod_{i=1}^n \left( p_{si}N(m_i;c_{si},\sigma_{si}^2)\right)^{x_i}
\left((1-p_{si})N(m_i;c_i,\sigma_i^2)\right)^{1-x_i}\label{eq:pdy}
\ee}
where $N(z\ ;\ c,\sigma^2)$ is the Normal distribution $N(c,\sigma^2)$ evaluated at $z$, and $\theta=(\mbox{\bf p},\mbox{\bf c},\mbox{\bf s}igma)$ holds the parameters of the model. If the $i$'th code word is a keyword ($x_i=1$) then the word's frequency $m_i$ is governed by a topic-specific Gaussian distribution $N( c_{si},\sigma_{si}^2)$, otherwise $m_i\sim N(c_i,\sigma_i^2)$ a topic-unspecific Gaussian distribution which we refer to as a "cross Gaussian".
The probability $Pr(d\ |\ \theta)$ of a document $d$ to be generated by the LBG model is found by the mixture:
\beas
Pr(d\ |\ \theta) &=& \sum_{s=1}^k Pr(d\ |\ y=s,\theta)Pr(y=s\ |\ \theta)\\
&=&\sum_s \lambda_s Pr(d\ |\ y=s,\theta),
\eeas
where $\sum_s\lambda_s = 1$. Given a training set of documents ${\cal D}=(d_1,...,d_m)$ we wish to fit the model parameters $\theta,\boldsymbol{\lambda}$ {\it and\/} select the important code words for each document, i.e., estimate ${\cal X}=(\mbox{\bf x}_1,...,\mbox{\bf x}_m)$ where $\mbox{\bf x}_j\in\{0,1\}^n$ is the keyword indicator set associated with $d_j$. We alternate between two procedures: (i) Maximum-Likelihood (ML) estimation of $\{\theta,\boldsymbol{\lambda}\}$ given $\cal X$ and, (ii) a procedure for estimating $\cal X$ given $\{\theta,\boldsymbol{\lambda}\}$.
The ML estimation of $\{\theta,\boldsymbol{\lambda}\}$ given an i.i.d. training set $\{{\cal D},{\cal X}\}$ takes the form:
\beas
&&\max_{\theta} \sum_{j=1}^m \log Pr(d_j\ |\ \theta,\mbox{\bf x}_j)\\
&& =\max_{\theta,\boldsymbol{\lambda}} \sum_{j=1}^m\log\left(\sum_{s=1}^k \lambda_s Pr(d_j\ |\ y_j=s,\theta,\mbox{\bf x}_j)\right)
\eeas
where $Pr(d_j\ |\ y_j=s,\theta,\mbox{\bf x}_j)$ is given by:
$$\prod_{i=1}^n \left( p_{si}N(m_{ji};c_{si},\sigma_{si}^2)\right)^{x_{ji}}
\left((1-p_{si})N(m_{ji};c_i,\sigma_i^2)\right)^{1-x_{ji}}.$$
Using the Expectation-Maximization (EM) iterative update \cite{Dempster-EM}, the following auxiliary function is optimized during the M-step:
$$\max_{\theta,\boldsymbol{\lambda}}\sum_{j=1}^m\sum_{s=1}^k \mu^{(t)}_{sj}\log\left(\lambda_sPr(d_j\ |\ y_j=s,\theta,\mbox{\bf x}_j)\right),$$
where $\mu^{(t)}_{sj}=Pr(y_j=s\ |\ d_j,\mbox{\bf x}_j,\theta^{(t)})$ is the posterior probability given the parameters at iteration $(t)$. Optimizing over the auxiliary function at step $(t)$ introduces an update rule for $\theta,\boldsymbol{\lambda}$:
\bea
\lambda_s &\leftarrow& \frac{1}{m}\sum_{j=1}^m\mu^{(t)}_{sj},\ \
p_{si} \leftarrow \frac{1}{\sum_j \mu^{(t)}_{sj}} \sum_{j=1}^m \mu^{(t)}_{sj} x_{ji} \label{eq:up-l}\\
c_{si} &\leftarrow& \frac{1}{\sum_j \mu_{sj}^{(t)}x_{ji}}\sum_{j=1}^m \mu_{sj}^{(t)}x_{ji}m_{ji}\\
\sigma^2_{si} &\leftarrow& \frac{1}{\sum_j \mu_{sj}^{(t)}x_{ji}}\sum_{j=1}^m \mu_{sj}^{(t)}x_{ji}(m_{ji}-c_{si})^2\label{eq:up-sig}
\eea
The parameters $c_i,\sigma_i^2$ of the cross-Gaussians are estimated directly from $\cal D,X$ since they do not depend on the choice of topics. The posteriors are updated during the E-step via application of the Bayes rule:
\be
\mu^{(t+1)}_{sj}\propto \lambda_s^{(t)} Pr(d_j\ |\ y_j=s,\mbox{\bf x}_j,\theta^{(t)}),\label{eq:mu}
\ee
where $\propto$ stands for equality up to normalization, i.e., $\sum_s \mu_{sj}=1$.
The estimation of $\cal X$ given the data $\cal D$ and the current estimation of parameters $\{\theta,\boldsymbol{\lambda}\}$ is based on the following analysis. Consider Natural numbers $q_s\in{\cal N}$, $s=1,...,k$, representing the number of important code words associated with topic $s$. The expected number of important code words $g_j$ in document $d_j$ is given below:
\be
g_j = \sum_{i=1}^nx_{ji} = \sum_{s=1}^k \mu_{sj}q_s.\label{eq:g}
\ee
In other words, the indicator set $\cal X$ is fully determined by $q_1,...,q_k$ and the posteriors $\mu_{sj}$ (which are estimated during the EM step above). The indicator $\mbox{\bf x}_j$ for document $d_j$, for instance, is defined by the top $g_j$ highest frequency code words. Our task, therefore, is to derive a procedure for estimating $q_1,...,q_k$ given the parameters $\theta,\boldsymbol{\lambda}$ and $\mbox{\bf m}u$ estimated during the EM steps.
We will begin by establishing an algebraic constraint between $\mbox{\bf q}=(q_1,...,q_k)$ and the parameters $\mbox{\bf p},\mbox{\bf m}u$:
\begin{claim}
\label{claim1}
Let $\mbox{\bf b}=(b_1,...,b_k)$ defined by $b_s = (\sum_j \mu_{sj})(\sum_ip_{si})$ for $s=1,...,k$, and let $U$ be an $k\times m$ matrix holding the posteriors, $U_{sj}=\mu_{sj}$. Then,
\be
\mbox{\bf b} = UU^\top\mbox{\bf q}\label{eq:uuq}
\ee
\end{claim}
{\bf Proof:\ } consider the formula representing the expected number of important words for a document of topic $s$:
$$E_s = \frac{1}{\sum_j \mu_{sj}}\sum_{j=1}^m\mu_{sj}g_j.$$
On the other hand, clearly, $E_s = \sum_i p_{si}$ since $p_{si}$ is the probability that the $i$'th code word is important for documents of topic $s$. Substituting the definition of $g_j$ from eqn.~\ref{eq:g}, we obtain:
$$(\sum_j \mu_{sj})(\sum_ip_{si}) = \sum_{j=1}^m \mu_{sj}\sum_{r=1}^k \mu_{rj}q_r,$$
where the right hand side is the $s$'th coordinate of $UU^\top\mbox{\bf q}$. \eop
The conditional-independence assumption $w_i\bot w_j\ |\ y$ (Naive-Bayes) creates "over-confident" posteriors, i.e., $\mu_{sj}\rightarrow \{0,1\}$ --- a well-known by-product (or side-effect) of the Naive Bayes assumption (see \cite{Domingos97} for a discussion). As a result, the constraint $UU^\top\mbox{\bf q}=\mbox{\bf b}$ is simplified considerably: $UU^\top \approx diag(\delta_1,...,\delta_k)$, where $\delta_s\approx \sum_j\mu^2_{sj}\approx \sum_j\mu_{sj}$. Eqn.~\ref{eq:uuq}, therefore, reduces to:
\be
\sum_{i=1}^np_{si}=q_s,\label{eq:bq}
\ee
for $s=1,...,k$. Eqn.~\ref{eq:bq} is not an effective update rule for setting $q_1,...,q_k$ because (i) there is no built-in drive to generate a sparse $\mbox{\bf p}$, which as a result, a large number of small-valued entries in $\mbox{\bf p}$ will inflate the value of $q_s$, and (ii) once entries of $\mbox{\bf m}u$ settle on $\{0,1\}$ values, the indicator set $\cal X$ will remain fixed.
A more effective use of Eqn.~\ref{eq:bq} is to to set $q_s$ as the top number of entries in $\mbox{\bf p}$:
$$q_s^{(t+1)} = |\{i\ :\ p_{si}\ge T_s^{(t)}\}|,$$
for some, iteration dependent, threshold $T_s$. In the following section we use a similar analysis to derive the value of $T_s$ which will conclude the Bernoulli-Gauss mixture algorithm.
\subsection{Update Rule for $q_1,...,q_k$}
\begin{algorithm*}
\caption{Bernoulli-Gauss Mixture}
\label{alg}
\begin{algorithmic}
\STATE {\mbox{\bf s}eries Input:} Given a training set of documents ${\cal D}=(d_1,...,d_m)$ we wish to fit the model parameters $\boldsymbol{\lambda}$ and $\theta=(\mbox{\bf p},\mbox{\bf c},\mbox{\bf s}igma)$ for $k$ topics and the Natural numbers $q_1,...,q_k$ of top ranking (by tf-idf) words per topic.
\STATE {\mbox{\bf s}eries Initialization:} Set initial values $\boldsymbol{\lambda}^{(0)},\theta^{(0)},\mbox{\bf q}^{(0)}$. Set the indicators ${\cal X}^{(0)}$ from ${\cal D}$ and $\mbox{\bf q}^{(0)}$, i.e., $x_{ji}=1$ if the if-idf value $m_{ji}$ is among the top $(1/k)\sum_s q_s^{(0)}$ entries in $d_j$. Set $t=0$.
\REPEAT
\STATE $t\leftarrow t+1$
\STATE Update the posteriors $\mu_{sj}^{(t)}$ according to Eqn.~\ref{eq:mu} for $j=1,...,m$ and $s=1,...,k$.
\STATE Update $\boldsymbol{\lambda}^{(t)},\mbox{\bf p}^{(t)},\mbox{\bf c}^{(t)},\mbox{\bf s}igma^{(t)}$ using Eqns.~\ref{eq:up-l}-\ref{eq:up-sig} and then update the cross-Gaussians.
\STATE Set $\mbox{\bf q}^{(t)}$ using eqn.~\ref{eq:up-q}.
\STATE Set ${\cal X}^{(t)}$: $x_{ji}=1$ if the if-idf value $m_{ji}$ is among the top $\sum_s\mu_{sj}^{(t)} q_s^{(t)}$ entries in $d_j$, for $i=1,...,n$ and $j=1,...,m$.
\UNTIL{$\sum_{s=1}^k \left( q_s^{(t)}-\sum_ip^{(t)}_{si}\right)^2 < \epsilon$}
\end{algorithmic}
\end{algorithm*}
Let $q_s^*$ be the (unknown) ground truth value for $q_s$. Since $g_j$ (eqn.~\ref{eq:g}) is the number of keywords in document $d_j$, the probability that {\it a keyword\/} will be selected in
$d_j$, conditioned by topic $s$, is $\min\{g_j/q_s^*,1\}$. The probability that a keyword will be selected in $d_j$ {\it and\/} the topic is $s$ is a random variable with a Bernoulli distribution with the probability of "success":
$\mu_{sj}\min\{g_j/q_s^*,1\}$.
The expected number of times a keyword is selected over the corpus of $m$ documents of topic $s$ is the sum of expectations of $m$ Bernoulli trials:
$$\sum_{j=1}^m \mu_{sj}\min\{g_j/q_s^*,1\}.$$
On the other hand, the expected number of times the $i$'th code-word (not necessarily a keyword) is selected in documents of topic $s$ is: $m\lambda_sp_{si}$.
As a result, for the $i$'th code-word to be a keyword for a document of topic $s$ the following condition must be satisfied:
$$
m\lambda_sp_{si} \ge \sum_{j=1}^m \mu_{sj}\min\{g_j/q_s^*,1\}
\ge \frac{1}{n}\sum_{j=1}^m \mu_{sj}g_j,
$$
where the first inequality is due to the rhs being a lower bound for a word to be a keyword, and the latter inequality is due to $1\le q_s^*\le n$. After rearranging terms and substituting eqn.~\ref{eq:g} for $g_j$ we obtain:
$$q_s^{(t+1)} = \left |\left \{i\ :\ p_{si}\ge \frac{1}{n\sum_j\mu_{sj}}\sum_{j=1}^m \mu_{sj}\sum_{r=1}^k \mu_{rj}q_r^{(t)}\right\}\right |.$$
Note that the right-hand side is the $s$'th coordinate of $UU^\top\mbox{\bf q}$ scaled by $1/(n\sum_j\mu_{sj})$. Given that the posteriors $\mu_{sj}$ approach $\{0,1\}$ values, the condition above reduces to:
\be
q_s^{(t+1)} = \left |\{i\ :\ p^{(t+1)}_{si}\ge\frac{1}{n}q_s^{(t)}\}\right |.\label{eq:up-q}
\ee
To conclude, the Bernoulli-Gauss mixture algorithm is summarized in Alg.~\ref{alg}. The stopping criteria is when Claim~\ref{claim1} is satisfied, but in practice it is sufficient to satisfy its reduced form eqn.~\ref{eq:up-q}.
\subsection{Evaluating the Model on Novel Documents}
\label{sec:eval}
Given a new document $d=(m_1,...,m_n)$, where $m_i$ is the frequency (tf-idf) of the $i$'th code-word in the document, we wish to evaluate the probability $Pr(d)$ of $d$ to arise from the model, and the posteriors $\mu_s(d) = Pr(y=s\ |\ d)$ which provide classification (topic assignment) information. A necessary ingredient in those calculations is the estimation of the keyword indicator set $\mbox{\bf x}\in\{0,1\}^n$ for document $d$.
To estimate $\mbox{\bf x}$ associated with the novel document $d$ we perform the following steps:
\noindent{\bf 1.\ }For $s=1,...,k$: (i) define $\mbox{\bf x}_s$ as the indicator set defined by the top $q_s$ code words in $d$, (ii) compute $\hat\mu_s\propto\lambda_sPr(d\ |\ y=s,\mbox{\bf x}_s)$ where $Pr(d\ |\ y=s,\mbox{\bf x}_s)$ is defined in eqn.~\ref{eq:pdy}.\\[1mm]
\noindent{\bf 2.\ }Set $\mbox{\bf x}$ as the top $(1/\sum_s\hat\mu_s)\sum_s\hat\mu_sq_s$ code words in $d$.
Once $\mbox{\bf x}$ is estimated, one can readily compute the posterior
$\mu_{s}\propto \lambda_s Pr(d\ |\ y=s,\mbox{\bf x},\theta)$, $s=1,...,k$.
and $Pr(d)$ from:
$Pr(d)=\sum_{s=1}^k \lambda_s Pr(d\ |\ y=s,\mbox{\bf x},\theta)$.
\subsection{Applications of the Model}
\label{sec:app}
The Bernoulli-Gauss mixture model can be used in a number of ways and for different data analysis applications, as described below:
\noindent{\bf Clustering:\ }
given documents $d_1,...,d_m$, cluster them into $k$ classes. Moreover, given a novel document $d$ determine its class association. The posteriors $\mu_{sj}$ for $d_j$ and class $s$ provide the class assignment of document $d_j$. Since posteriors are "over-confident" due to the Naive Bayes assumption, the assignment is "hard" in practice. For a new document $d$, the posteriors $\mu_s$ (see Sec.~\ref{sec:eval}) provide the class assignments for $s=1,...,k$.
\noindent{\bf Supervised Inference:\ } given a training set of documents with class labels in the set $\{1,...,h\}$ we wish to determine the class membership of a given novel document. A possible approach is to estimate a LBG model separately for each class producing the model parameters $\boldsymbol{\lambda}_l,\theta_l,\mbox{\bf q}_l$, $l=1,...,h$, and then choose the class with the highest probability: $\mbox{argmax}_l Pr(d\ |\ \boldsymbol{\lambda}_l,\theta_l,\mbox{\bf q}_l)$.
\noindent{\bf Feature Selection:\ }
we can use the Bernoulli-Gauss mixture model for selecting features. The selection criteria is based on $p_{si}$ which is the probability that the $i$'th code word (feature) is a keyword for topic $s$. We "de-select" a feature $i$ if $p_{si}<\delta$ for some threshold $\delta$ for all $s=1,...,k$, i.e., a feature that is {\sl not\/} a keyword in {\sl all\/} topics is removed from the set of selected features. In Sec.~\ref{sec:exp} we apply the feature selection scheme above as a filter for Support-Vector-Machine (SVM) classification and for K-means clustering.
\noindent{\bf Collaborative Filtering:\ }
there are applications where the indicator set $\mbox{\bf x}\in\{0,1\}^n$ is known, and moreover when $x_i=0$ the frequency of the $i$'th code word $m_i$ is unknown. Collaborative Filtering (CF) is an example of this class of applications where $d=(m_1,...,m_n)$ is a list of discrete movie ratings with $m_i\in\{1,...,5\}$ (stars), of an individual. Each individual rates some of the movies, thus $x_i=1$ for movies being rated and $x_i=0$ otherwise. Given a subset of ratings made by a new individual, the task of CF is to predict movie ratings which were not part of the original subset.
In this case, the cross-Gaussians are dropped from the model, i.e.,
\be
Pr(d\ |\ y=s)=\prod_{i=1}^n \left( p_{si}N(m_i;c_{si},\sigma_{si}^2)\right)^{x_i}
(1-p_{si})^{1-x_i}.\label{eq:cf}
\ee
From the training ratings $\{d_j,\mbox{\bf x}_j\}$, $j=1,...,m$, we estimate the model parameters $\theta,\boldsymbol{\lambda}$ using Eqns.~\ref{eq:up-l}-\ref{eq:up-sig} (there is no need to estimate $q_1,...,q_k$ since the indicator sets are known). We are given a new rating $\{d,\mbox{\bf x}\}$ where $d=(m_1,...,m_n)$ and $x_i=1$ when $m_i>0$. Let $i\in\{1,...,n\}$ be a movie we wish to predict its rating by the individual $d$. Similarly to the "Forced Prediction" protocol~\cite{Breese98empiricalanalysis}, we wish to estimate the probability $Pr(m_{i}=t\ |\ d,\theta)$ for $t=1,...,5$. We start by setting $x_{i}=1$ (originally it was zero):
$$Pr(m_{i}=t\ |\ d)=\sum_{s=1}^k Pr(m_{i}=t\ |\ y=s)Pr(y=s\ |\ d),$$
where
$$Pr(m_{i}=t\ |\ y=s)=N(t;c_{si},\sigma_{si}^2),$$
and the posterior $Pr(y=s\ |\ d)\propto\lambda_sPr(d\ |\ y=s)$ is estimated through eqn.~\ref{eq:cf}. The movie rating prediction $t^*$ is found by: $t^*=\mbox{argmax}x{t}Pr(m_{i}=t\ |\ d)$.
\section{Relationship with Other Latent Variable Models}
\label{sec:comp}
On a simplistic level,
the Bernoulli-Gauss mixture model can be viewed as a Gaussian mixture model integrated with a feature selection procedure (the Bernoulli part of the model). On a deeper level, however, there are subtleties that have to do with the positioning of LBG with respect to MOU, LDA and pLSI and specifically the manner in which LBG is a {\it generative\/} model like MOU and LDA, which we will describe below.
One difference is that LBG models the frequency of a code word (per topic) as a Gaussian whereas MOU, LDA and pLSI model the probability of appearance of code-words as a multinomial --- which at the limit are really the same, as described next. Let $\beta_{si}=Pr(w=i\ |\ y=s)$ be the probability of drawing the $i$'th code-word given the $s$'th topic. The number of appearances $m_i$ of the $i$'th code-word in a document is governed by a Binomial distribution $m_i \sim Bin(N,\beta_{si})$ where $N$ is the number of words in the document. By the De-Moivre-Laplace theorem, as $N\rightarrow\infty$, $m_i\sim {\cal N}(N\beta_{si},N\beta_{si}(1-\beta_{si}))$. Therefore, in practice since the number of words $N$ is a document is typically large, the estimated means $c_{si}$ in the LBG model are equal to $N\beta_{si}$ in the multinomial models.
The De-Moivre-Laplace argument above is also relevant for the justification of a Gaussian distribution as a model of word frequencies (or any other non-negative data). It implies that the probability of a negative value (in the generative sense) is vanishingly small. Successful attempts in using Gaussian mixtures in non-negative numerical contexts, such as for collaborative filtering, include \cite{CF_Hoff03}. In practice we have not observed any problematic issue with a Gaussian modeling and our experimental reports across a number of application domains (text analysis included) make that point as well.
It will be convenient, in this section, to represent a document $d=(w_1,...,w_N)$ by the (unordered) set of words $w_i\in\{1,...,n\}$ taking values from a vocabulary of $n$ code-words. We will begin the discussion with the comparison between the MOU model and LBG. A document is generated by the MOU model by a draw from a mixture of multinomials as follows. A topic is drawn by tossing a $k$-faced die whose faces have probabilities $\lambda_s=Pr(y=s)$. A word is drawn by the toss of an n-faced die where we have $k$ such dice each representing a topic $s=1,...,k$, with $\beta_{si}$ (as defined above) representing the probability of the $i$'th face of the $n$-face word-die associated with topic $s$.
The $N$ words of a document are generated by (i) draw a topic $s$ by tossing the $k$-faced topic-die, then repeat $N$ times: (ii) draw a code-word by tossing the $s$'th word-die. In formal language,
$$Pr(d)=\sum_{s=1}^k \lambda_s\prod_{i=1}^n\beta_{si}^{m_i}.$$
The model parameters $\boldsymbol{\lambda},\mbox{\bf b}eta$ can be estimated by the EM algorithm. The MOU model is simple and very popular in text analysis circles. However, it has a number of drawbacks which have served as a catalyst for introducing new algorithms, notably pLSI and LDA. The notion that all code-words appearance is governed by the choice of a single topic is too simplistic. First, there are code-words which have a low probability of appearance in {\sl all\/} topics, i.e., are essentially topic-independent, yet are not stop-words. These words undergo "starvation" in the MOU model as they almost never have a chance to be appear in a document generated by MOU. Second, polysemy --- the coexistence of multiple meanings for a code-word --- is not modeled by MOU. Consider a document $d$ and a code-word $w$. In MOU the posterior probability $Pr(y=s\ |\ w,d)$ is independent of $d$:
$$Pr(y=s\ |\ w,d)\propto Pr(w\ |\ y=s)Pr(y=s),$$
therefore it is not possible to convey multiple meanings for the code-word $w$ as a function of other words in the document $d$.
In the LBG model, the single topic assumption applies only to a selected set of code-words, whereas all other code-words are governed by a topic-unspecific distribution. The manner in which this principle plays in a generative model is described formally as follows:
$$Pr(d,\mbox{\bf x})=\sum_{s=1}^k \lambda_s\prod_{r=1}^NPr(w_r,x_r\ |\ y=s),$$
where
$$Pr(w_r=i,x_r\ |\ y=s)\propto\left\{\begin{array}{ll}
p_{si}\frac{1}{N}c_{si} & if\ x_i=1\\
(1-p_{si})\frac{1}{N}c_{i}& if\ x_i=0
\end{array}\right\}
$$
In other words, the $N$ words of a document $d$ are generated through the following steps:
\begin{itemize}
\item Draw a topic $s$ by tossing the $k$-faced topic-die.
\item Toss $n$ coins with biases $p_{si}$, $i=1,...,n$ to draw the indicator vector $\mbox{\bf x}\in\{0,1\}^n$.
\item Create a $n$-faced word-die by setting $\hat\beta_{si}$ to $(1/N)c_{si}$ if $x_i=1$ or to $(1/N)c_{i}$ if $x_i=0$. The probability $\beta_{si}$ of the $i$'th face of the word-die is $(1/Z)\hat\beta_{si}$ where $Z$ is a normalization factor such that $\sum_i\beta_{si}=1$.
\item Repeat $N$ times: draw a word from the word-die constructed above.
\end{itemize}
In other words, in the LBG model the word-die is generated per document not only on the basis of the topic selection but also {\it based on the selection of keywords}. With regard to polysemy, the posterior probability $Pr(y=s\ |\ w,d)$ now depends on $d$:
$$Pr(y=s\ |\ w,d)\propto Pr(w\ |\ y=s,d)Pr(y=s),$$
unlike MOU. The LBG model therefore addresses the two main drawbacks of MOU: first being that the single-topic assumption does not apply to the entire document but only to selected keywords, and secondly that the word generation process depends also on the document thereby allowing multiple meanings to words. Both of those "upgrades" make the underlying model assumptions more realistic than MOU. Consequently, the LBG model can be considered as a natural extension of the MOU model where some of the limiting (and unrealistic) assumptions of MOU are relaxed.
The LDA model addresses the single-topic assumption of MOU by allowing multiple topics per document in the following manner. To generate the $N$ words of a document $d$, (i) a $k$-faced topic-die is generated by sampling from a Dirichlet distribution with parameters $\alpha_1,...,\alpha_k$, then (ii) repeat $N$ times: (a) sample a topic $s$ by tossing the topic-die, and (b) sample a word by tossing the word-die $\mbox{\bf b}eta_s$.
The parameters $\mbox{\bf a}lpha,\mbox{\bf b}eta$ of the LDA model are learned through a Variational EM algorithm. Unlike MOU and LBG, in the LDA model the topic is selected {\it per word\/} rather than once per document. This approach definitely solves the single-topic limitation of MOU and also the polysemy issue since the posterior $Pr(y=s\ |\ w,d)$ depends on $d$:
$$Pr(y=s\ |\ w,d)\propto Pr(w\ |\ y=s)Pr(y=s\ |\ d).$$
However, there is a price to pay for the powerful generality of the LDA model. First, the posteriors $P(y=s\ |\ d)$ are computationally intractable and instead are replaced by a mean-field "surrogate" approximation or by sampling methods.
Secondly, by design, LDA requires a relatively large number of topics $k$ (around $\sim 50$) which is fine in the world of text but is limiting to other data analysis domains where the number of "topics" are known to be small (like clustering applications).
In practice, LDA is often used for dimensionality reduction (using the variational parameters $\gamma\in R^k$ per document) as a filter for SVM classification and for supervised classification by performing a separate LDA modeling per class. Despite the reservations above, there are situations where the powerful generality of the LDA model pays off --- in the domain of text this happens when two topics are very similar. In such cases, the modeling capacity of MOU and LBG is too limited and cannot separate the two classes (see sec.~\ref{sec:exp} for details).
The pLSI model represents the training data as a mixture of multinomials and, like LDA, also allows for multiple topics per document. The pLSI model (unlike MOU, LDA and LBG) is not generative, i.e., there is no natural way to use the model to assign probability to a novel document. Related to that, the number of parameters of the model grows linearly with the training set thus risking an over-fitting phenomenon to occur. The pLSI model, therefore, is not a natural candidate for classification tasks because a novel data instance cannot be classified without essentially retraining the entire dataset. We refer the reader to \cite{BleiNJ03} for a detailed comparison between LDA and pLSI. We have included pLSI in our experiments (sec.~\ref{sec:exp}) as one can often obtain good performance if retraining is allowed during classification of a novel document.
\section{Experiments}
\label{sec:exp}
\begin{figure*}
\caption{\small 20NewsGroup classification result on a binary classification problem, using SVM on the reduced set of features. Graph (a) is misc.forsale vs. rec.sport.baseball. Graph (b) is comp.graphics vs. comp.os.ms\_windows.misc. }
\label{fig:SVM_20NG}
\end{figure*}
We conducted experiments with MAP classification, feature selection as a filter for SVM and K-means, supervised classification fitting a model per class and collaborative filtering. Those experiments were conducted on a number of datasets including 20NewsGroup\footnote{The 20NewsGroup data set, taken from the Usenet Newsgroup Collection, consists of some $20,000$ newsgroup postings, each one categorized to a different topic where each topic contains $1000$ documents.}, 100KMovieLens, and Spambase from UCI ML repository.
\noindent {\bf MAP Unsupervised Classification:\ } we begin with an unsupervised classification experiment using the MAP output of our model (the posteriors $\mu_s=Pr(y=s\ |\ d)$). We randomly split the data set into training and test subsets, generated by mixing records from all the topics in the data set. Having stripped all the record headers, a code-book is created, comprising of all words which are not stop-words in the data set. We trained a MAP classifier with our model and evaluated the classification output by comparing the cluster label of each record with its true label, as per the 20NewsGroup data set.
In order to measure the clustering performance, we use the zero-one loss function, as follows. Given the $i$'th posting, let $s_i$ and $\kappa_i$ be the obtained cluster label and the true label, respectively. The accuracy (AC), is defined by
$AC=(1/m)\sum_{i=1}^m \delta(\kappa_i,map(s_i))$,
where $\delta(x,y)$ is an indicator function that equals one
if $x=y$ and zero otherwise; and $map(s_i)$ is the permutation mapping function that maps each cluster label $s_i$ to its equivalent label from the data set. The optimal mapping is obtained by the Kuhn-Munkres algorithm~\cite{Lovasz_MatchingTheory}.
We compared our results with those of MOU, pLSI and LDA. For the latter, a clustering decision was made by examining the $\phi_i$ variational parameters that are introduced for each record. We repeated the experiments several times and the average results are reported in Table ~\ref{tab:20NewsGroup_MAP_classification_All_classes}. Note that pLSI retrains the entire data for each new test record thus skewing the comparison --- yet it is interesting to note that LBG matched the performance nevertheless. Note the large performance gap between LBG and MOU underscoring the significant upgrade to the MOU model. We conjecture that the relatively low accuracy obtained by the LDA model is related to the mean-field approximation and as mentioned above, LDA is hardly ever used for MAP applications for presumably the same reasons.
\begin{table}
\caption {MAP classification performance comparison for the 20NewsGroup data set.}
\label {tab:20NewsGroup_MAP_classification_All_classes}
\vskip 0.15in
\centering
\begin{tabular} {|r|r|r|r|}
\hline
LBG & MOU & pLSI & LDA \\
\hline
28\% & 15\% & 27\% & 12\%\\
\hline
\end{tabular}
\vskip -0.1in
\end{table}
\noindent{\bf Feature Selection:\ } we compared the performance of our feature selection procedure, as described in sec.~\ref{sec:app}, with the dimensionally reduction offered by the variational parameters $\gamma$ of the LDA model. In our first experiment, we selected a pair of classes from the 20NewsGroup dataset and performed an SVM classification where the representation of data-instances were the selected coordinates given by LBG or the reduced dimension vector $\gamma$ provided by the LDA model. For control purposes we also applied SVM on the raw representation (without the filter). Fig.~\ref{fig:SVM_20NG} shows the classification accuracy results for two pairs of classes --- a semantically close pair and a pair of unrelated classes. Several experiments were conducted where the proportion of the training data was varied --- from $5\%$ to $30\%$. One can see that the LBG filter produced accuracies comparable to raw data use (slightly better for small training sets) with consistently better performance than the LDA filter\footnote{LDA at http://chasen.org/$\sim$ daiti-m/dist/lda/}. Note that all approaches suffered when applied to a semantically-related pair of classes.
In the second experiment, we performed an unsupervised classification using K-means clustering on the filtered representations and without the filter (the raw data). Results for both semantically-close and semantically-unrelated pairs of classes are shown in Tables~\ref{tab:sameTopics_Kmeans} and~\ref{tab:differTopics_Kmeans}. One can see that LDA can produce a superior accuracy when the two classes are semantically-close (comp.os.ms\_windows.misc versus comp.graphics). LBG on the other hand consistently outperformed LDA for semantically-unrelated clusters.
\begin{table*}[t]
\caption {K-means classification for semantically-close classes.}
\label{tab:sameTopics_Kmeans}
\vskip 0.15in
\centering
\begin{tabular} {|c|c|c|c|c|}
\hline
& comp.os.ms\_windows.misc & talk.politics.mideast & rec.sport.baseball & talk.religion.misc\\
& comp.graphics & talk.politics.misc & rec.sport.hockey & talk.religion.cristianity\\
\hline
LBG & 63.25\% & 72.75\% & 52.875\% & 57.625\% \\
LDA & 77.5\% & 57.75\% & 53\% & 58.375\% \\
All & 50.25\% & 58.625\% & 50.375\% & 50.5\% \\
\hline
\end{tabular}
\vskip -0.1in
\end{table*}
\noindent{\bf Collaborative Filtering:\ }
We used the 100KMovieLens Collaborative-Filtering data, which consists of approximately $100,000$ ratings for $1,682$ movies by 943 viewers.
As discussed in sec.~\ref{sec:app}, we train our model using a fully-observed set of viewers. Then, for every test viewer, we suppress a single, randomly-chosen movie rating. Our task is to predict the rating, given all the other movies for which that viewer has voted (known as the "Forced Prediction" protocol).
Adopting Hofmann~\yrcite{CF_Hoff03} and Breese~\yrcite{Breese98empiricalanalysis}, we use two evaluation metrics which measure the distance of the estimated vote $\hat{m}$ from the true vote $m$ --- the mean absolute error MAE, $avg(|\hat{m}-m|)$, and the rooted mean squared error RMSE $avg((\hat{m}-m)^2)$.
We then compared our method to Gassian-pLSA proposed by~\cite{CF_Hoff03} and to the Baseline method that simply outputs the mean vote over the entire training data for each movie. The results are displayed in Table~\ref{tab:CF_Prediction_comparison}. Note that LDA and pLSI do not naturally accommodate the Forced Prediction protocol as they do not measure word frequencies, thus were omitted from the comparison. One can see that LBG produced a lower MAE error compared to both Gaussian-pLSA and the Baseline method and slightly lower error on the RMSE measure (compared to Baseline).
\begin{table*}[t]
\caption {K-means classification for semantically-unrelated classes.}
\label{tab:differTopics_Kmeans}
\vskip 0.15in
\centering
\begin{tabular} {|c|c|c|c|c|}
\hline
& comp.windows.misc\_windows.misc & comp.sys.mac.hardware & alt.atheism & rec.sport.baseball\\
& rec.autos & alt.atheism & rec.motorcycles & misc.forsale\\
\hline
LBG & 93.75\% & 97.25\% & 94.125\% & 93\%\\
LDA & 84.125\% & 90.125\% & 58.25\%& 89\% \\
All & 95.375\% & 96.875\% & 88.875\% & 93\% \\
\hline
\end{tabular}
\vskip -0.1in
\end{table*}
\noindent{\bf Spam Filtering:\ }
The Spambase data set from the UCI Machine Learning Repository dataset consists of 4601 of emails ("documents"), characterized by 54 attributes ("words") plus a class label ("spam"=positive/"ham"=negative) where $39\%$ of the emails are labeled as spam.
We begin with an unsupervised MAP estimation where Table~\ref{tab:Spam_unsupervised} displays the performance of LBG against MOU and pLSI (where with pLSI a retraining is required for each test data). Note the performance gap between LBG and pLSI --- this we conjecture has to do with the plausibility of the single-topic assumption for spam filtering -- words with high percentage occurrence serve as a natural discriminative indicator (for example, an email with repeated occurrences of the word "buy" is likely to be spam). The performance gap with MOU is attributed to the fact that the single-topic assumption is best applied on keywords rather than on all words of the document.
\begin{table}
\caption {Unsupervised spam-filter classification performance comparison.}
\label {tab:Spam_unsupervised}
\vskip 0.15in
\centering
\begin{tabular} {|r|r|r|}
\hline
LBG & MOU & pLSI \\
\hline
78\% & 60\% & 65\% \\
\hline
\end{tabular}
\vskip -0.1in
\end{table}
\begin{table}
\caption {MovieLens Collaborative Filtering prediction results.}
\label {tab:CF_Prediction_comparison}
\vskip 0.15in
\begin{tabular*}{.4\textwidth}
{|l|c@{\extracolsep{\fill}}c|}\hline
Method & \multicolumn{2}{c|}{Absolute Error}\\
& MAE & RMSE\\\hline
Baseline & 0.905 & 1.1445 \\
Gaussian pLSA & 1.884 & 2.1142 \\
LBG & 0.776 & 1.1183 \\
\hline
\end{tabular*}
\vskip -0.1in
\end{table}
\begin{figure}
\caption{Confusion tables for supervised spam-filter.}
\label{fig:spamConfusionTables}
\end{figure}
We then moved to a supervised setting, in which we used the class labels (spam/ham) in the training stage. We modeled each class separately using LBG, LDA, MOU and pLSI while fitting the optimal number of topics per model (see note at the end of sec.~\ref{sec:bg}). Note that MOU, when $k=1$, reduces to the SpamBayes algorithm.
The confusion table of each method is displayed in Fig.~\ref{fig:spamConfusionTables}.
Note the strikingly low false-positive (ham classified as spam) result for the LBG model, compared to other models. Future work might be directed to the development of an enhanced model, which will compensate for LBG's limited success with false-negatives.
{\small
\end{document} |
\begin{document}
\title{Entanglement measures: State ordering {\it vs.} local operations}
\author{M\'ario Ziman and Vladim\'\i r Bu\v zek}
\affiliation
{Research Center for Quantum Information, Slovak Academy of Sciences, D\'ubravsk\'a cesta 9, 845 11 Bratislava, Slovakia \\
Faculty of Informatics, Masaryk University, Botanick\'a 68a, 602 00 Brno, Czech Republic\\
{\em Quniverse}, L{\'\i}\v{s}\v{c}ie \'{u}dolie 116, 841 04 Bratislava, Slovakia}
\begin{abstract}
A set of all states of a bi-partite quantum system can be divided into subsets each of which contains
states with the same degree of entanglement. In this paper
we address a question whether local operations
(without classical communication) affect the entanglement-induced
state ordering. We show that arbitrary unilocal channel (i.e., a channel that acts on one sub-system of a bi-partite system only)
might change the ordering for an arbitrary nontrivial measure of entanglement.
A slightly weaker result holds for the maximally entangled states. In particular, the maximally entangled states might not remain the most
entangled ones at the output of a unilocal noise channel.
\end{abstract}
\maketitle
\section{Quantum entanglement}
Quantum phenomena (such as quantum dense coding \cite{densecoding},
quantum teleportation \cite{teleportation}, quantum secret
sharing \cite{secretsharing}, etc.) associated with the existence of quantum entanglement
represent one of the most important pillars of quantum information theory \cite{nielsen}.
In spite of all the progress in understanding the nature of this phenomenon
some features of the concept of quantum entanglement are still to be properly illuminated.
In particular, due to the seminal work of Reinhard Werner
\cite{werner_ent} and others (see e.g. the review article \cite{review_ent}) we have a precise mathematical
definition of what does it mean when we say that a bi-partite state is entangled. On the other hand
a clear generally applicable operational meaning of the entanglement is still missing.
In this paper we will analyze some dynamical aspects of quantum entanglement.
Specifically we will study the relation between unilocal operations and static
(kinematic) properties of quantum entanglement expressed in terms of the entanglement-induced
state ordering.
The concept of quantum entanglement is relatively easy
to understand when we deal with pure states of bi-partite systems. This easiness originates in a close (mathematical)
relationship between the concept of entanglement and the concept of statistical correlations.
In fact, for pure quantum states these two concepts can be quantified by the same
functions and the meaning of the statement ``not entangled'' is equivalent to the ``not correlated''.
However, conceptual differences between entanglement and statistical correlations become striking
when we consider mixed states.
An important feature
of quantum entanglement reflecting its behavior under local operations
and classical communication has been known for some time \cite{nielsen}.
Namely, it is well established that two (classically) communicating
distant parties cannot entangle their quantum systems without
performing a global operation (corresponding to some effective interaction).
In other words, arbitrary local operations cannot create
the entanglement even if these actions are coordinated by
an exchange of classical information. Moreover, local unitary
transformations do not affect the quantum entanglement at all.
These properties form a basis of our intuitive picture of quantum entanglement.
Let us summarize these ``natural'' properties of entanglement:
\begin{itemize}
\item The quantum entanglement is a property of a quantum state.
\item A quantum state is entangled, if it cannot be prepared
from a factorized state ($\varrho_A\otimes\varrho_B$) by an action of local
operations and classical communication, i.e. it cannot be expressed
as a convex sum of factorized states ($\varrho_{AB}\ne \sum_j p_j \varrho_A^{(j)}\otimes\varrho_B^{(j)}$).
\item LOCC (local operations plus classical communication) operations
applied to an arbitrary (even entangled) quantum state can only destroy
the entanglement.
\item Locally unitary equivalent quantum states are equally entangled.
\end{itemize}
As we have already said, the concept of ``not being entangled'' is well defined. Non-entangled
states are called {\em separable}. There is also a common agreement on the notion
of maximally entangled quantum states that represent the other extreme.
We say that a bi-partite quantum state is maximally entangled if it is pure and the two
subsystems are in maximally mixed states, i.e.
$\varrho_{AB}=|\Psi\rangle\langle\Psi|$ and
${\ranglem Tr}_B[|\Psi\rangle\langle \Psi|]={\ranglem Tr}_A[|\Psi\rangle\langle \Psi|]=\frac{1}{d}I$
with $d=\min{\{{\ranglem dim}{\cal H}_A, {\ranglem dim}{\cal H}_B\}}$.
There are two basic
questions: i) whether a given state is entangled, or not?, and
ii) whether we can compare the entanglement of different
quantum states. Both questions can be addressed via the so-called
{\em entanglement measures}.
In this paper we will focus our attention on the concept of entanglement measures. We will
study dynamics of entanglement under the action of local channels. Our paper is organized as follows:
We start with a brief introduction to entanglement measures. Then we will
analyze the stability of entanglement-induced state ordering and the
properties of maximally entangled states with respect to
local operations, in particular for the so-called unilocal channels.
Finally, we will discuss some conceptual consequences
of our analysis.
\section{Entanglement measures}
The entanglement (see a recent review \cite{plenio_virmani})
has been identified as the key ingredient in applications such
as the quantum teleportation, the quantum secret sharing, etc. However, it is also
known that the presence of entanglement itself does not guarantee the success of a protocol. For instance,
an arbitrary entangled state cannot be used for the teleportation. Even if
a state can be exploited for this protocol the success/rate of the teleportation depends on
the particular state. Hence, it seems there are states with different
``quality'' and ``quantity'' of entanglement. In order to quantify a degree of entanglement
entanglement measures have been introduced. These measures
are functionals defined on a state space designed to quantify the
amount of entanglement in a given state. During the last ten years the
topic of entanglement measures has attracted a lot of attention and many
important results has been discovered.
Principally there are two approaches to the entanglement measures:
i) the {\em operational} approach, and ii) the {\em axiomatic} approach.
The aim of the first approach is to
adopt a procedure (protocol) that crucially depends on the presence of entanglement (for example the quantum teleportation),
and to quantify its success of performance
depending on the particular state. Such measure would give
a direct operational meaning to quantum entanglement associated with a given state. Unfortunately no such (universal) procedure
is known. In the abstract axiomatic approach we reformulate our
intuitive understanding of entanglement into several axioms.
There exist several different (not completely equivalent) choices
for the system of axioms \cite{horodecki}, however our aim is not
to discuss all these choices. We say that the functional
$E:{\cal S}({\cal H})\to [0,\infty]$
is an entanglement measure if the following properties hold:
\begin{enumerate}
\item {\it Sharpness:}
$E(\varrho_{AB})=0$ if and only if $\varrho_{AB}$ is a separable state.
\item {\it Local unitary invariance:}
$E(U_A\otimes U_B\varrho_{AB}U_A^\dagger\otimes U_B^\dagger)=E(\varrho_{AB})$ for all unitary transformations $U_A,U_B$ and all states $\varrho_{AB}$.
\item {\it Normalization:}
$E(\varrho_{AB})$ is maximal only for maximally entangled states, i.e.
$E(\varrho_{AB})=\max_{\varrho_{AB}}E(\varrho_{AB})$ if and only if
${\rm Tr}_A\varrho_{AB}={\rm Tr}_B\varrho_{AB}\sim I$ and ${\rm Tr}\varrho_{AB}^2=1$.
\item {\it Nonincreasing under LOCC:}
A general LOCC operation transforms the original state $\varrho_{AB}$
into a mixture of states
$\omega_k^{AB}={\cal E}_k^{A}\otimes{\cal E}_k^{B}[\varrho_{AB}]$
with probabilities $p_k$. This condition guarantees that the entanglement
is (on average) not created by LOCC operations, i.e.
$E(\varrho_{AB})\ge \sum_k p_k E(\omega_k^{AB})$.
\item {\it Convexity:}
$E(\sum_k p_k \omega_k^{AB})\langlee \sum_k p_k E(\omega_k^{AB})$.
\item {\it Additivity on pure states:}
$E(\Psi_{AB}\otimes\Phi_{A^\prime B^\prime})
=E(\Psi_{AB})+E(\Phi_{A^\prime B^\prime})$
for all pure states $\Psi_{AB},\Phi_{A^\prime B^\prime}$.
\end{enumerate}
The first four properties from the above list are in an agreement with our intuitive picture
discussed in the previous section. In order to motivate the remaining two
properties we have to take into account a situation in which
a pair of systems is a part of a larger composite object.
Without the loss of generality we can assume
to have three parties (systems) $A,B,C$ in a pure state $\Omega_{ABC}$. By performing
measurement on the system $C$ and reading an outcome $j$ (associated
with the state transformation ${\cal I}_{AB}\otimes{\cal F}_j^{C}$)
the original state $\varrho_{AB}={\rm Tr}_{C}\Omega_{ABC}$ is transformed
into the state $\omega_j^{AB}={\rm Tr}_C \Omega_j^{ABC}
={\rm Tr}_C ({\cal I}_{AB}\otimes{\cal F}_j^C)[\Omega_{ABC}]$. This happens
with some probability $p_j$. Without the knowledge of the
observed outcome $j$, the experimentalists possessing the systems $A$ and $B$
can use only the entanglement contained in the state $\varrho_{AB}$, because
the measurement performed on $C$ does not affect the {\em average} state $\varrho_{AB}$.
However, if they acquire the information about the outcome $j$, they can exploit
the entanglement shared in particular states $\omega_j^{AB}$, hence they can
on average exploit $\sum_j p_j E(\omega_j^{AB})$ of the entanglement.
The knowledge of $j$ cannot decrease the entanglement contained originally
in $\varrho_{AB}$. Hence, although the measurement on the system $C$ is
a local action, the entanglement between $A$ and $B$ can increase,
i.e. the third party can assist to $A$ and $B$ to increase the
entanglement they share providing that the information on $j$
is communicated to $A$ and $B$. In fact,
the measurements on the system $C$ induces convex decompositions of the
state $\varrho_{AB}=\sum_j p_j\omega_j^{AB}$, thus we get the convexity
condition for entanglement measures.
For example, let us consider three parties $A,B$ and $C$ share a GHZ state
$|\Omega_{ABC}\rangle=\frac{1}{\sqrt{2}}(|000\rangle+|111\rangle)$. A bi-partite density operator
$\varrho_{AB}$ describes a classically maximally correlated state, which is not
entangled at all and cannot be used for the teleportation. On the other hand, if the third party $C$ performs a measurement in
the dual basis $|\pm\rangle_C=\frac{1}{\sqrt{2}}|0\rangle\pm |1\rangle$ then for both outcomes $\pm 1$
the parties $A$ and $B$ share a maximally entangled quantum state.
In particular, $\omega_{\pm}^{AB}=\frac{1}{2}(|00\rangle\pm|11\rangle)(\langle 00|\pm\langle 11|)$.
We see explicitly that such an assistance by the third party
can significantly increase the entanglement - this is the reason for the
convexity condition. Taking the maximum of average entanglement
over all decompositions we obtain the so-called {\em entanglement of assistance} \cite{divincenzo},
$E_{\ranglem assist}(\varrho_{AB})=\max\sum_j p_j E(\omega_j^{AB})$.
The requirement of the additivity is a rather natural property of the quantum
entanglement, however we lack some clear operational reason for it
and it is not trivially satisfied for the measures we use.
For example, the additivity of entanglement
of formation is one of the most important open problems
in the quantum information theory. Therefore, it is demanded that
this property holds only for tensor product of pure states.
In a sense this should guarantee some scaling properties of quantum
entanglement, i.e. more-dimensional systems can be more
entangled.
\section{Ordering {\em vs.} local operations}
Entanglement measures enable us not only to decide whether a given state is entangled, but they also allow us to conclude whether one state
is more entangled than another. In fact, any entanglement
measure can be used to induce an ordering on a set of quantum states.
However, it has been pointed out in Ref.~\cite{plenio} and analyzed by
many others \cite{miranowicz} that entanglement-induced
orderings for two different entanglement measures $E_1,E_2$
can differ. Even for the most commonly used measures of entanglement
\cite{miranowicz} there exists a pair of states $\omega_{AB}$ and
$\varrho_{AB}$ such that $E_1(\varrho_{AB})>E_1(\omega_{AB})$,
but $E_2(\varrho_{AB})<E_2(\omega_{AB})$.
In Ref.~\cite{ziman06} we addressed the question whether for a given
entanglement measure the ordering is preserved under the action of local operations
(without a classical communication). In a sense, we postulated an additional
axiom that should be fulfilled by a ``good'' entanglement measure.
There are several proposals for entanglement measures satisfying
the basic properties 1-4 from the above list. For example,
the entanglement of formation \cite{bvsw96}, the concurrence \cite{wootters98},
tangle \cite{ckw}, the relative entropy of entanglement \cite{vprk97}, the negativity
\cite{vw}, the squashed entanglement \cite{christandl04}, etc.
Certainly, the practical computability might be a non-trivial problem. In most cases
the optimization and the minimization can be accomplished only
numerically. For a two-qubit system the entanglement of formation
$E_f=\inf\sum_j p_j \tilde{S}_{vN}(\Psi_j)$,
the tangle $\tau=\inf\sum_j p_j \tilde{S}_L(\Psi_j)$
and the concurrence $C=\sqrt{\tau}$ are mutually closely related and they are straightforward to
to compute. We used the notation $\tilde{S}$ for the
corresponding entropy $S$ of the reduced state $\omega={\rm Tr}_B\Psi$.
The infima are taken over all convex decompositions of the given state
$\varrho$ into pure states $\{\Psi_j\}$. The indexes $vN$, $L$ stand for
the von Neumann entropy ($S_{vN}=-{\rm Tr}\varrho\langleog\varrho$) and
the linear entropy $S_L=2(1-{\rm Tr}\varrho^2)$, respectively. It was shown in
\cite{wootters98} that for two qubits $E_f=h(\frac{1}{2}[1+\sqrt{1-\tau}])$,
$\tau=C^2$ and $C=\max\{0,\sqrt{\langleambda_1}-\sqrt{\langleambda_2}
-\sqrt{\langleambda_3}-\sqrt{\langleambda_4}\}$, where $\langleambda_j$ are decreasingly
ordered eigenvalues of the matrix
$R=\varrho(\sigma_y\otimes\sigma_y)\varrho^*(\sigma_y\otimes\sigma_y)$
and $h(x)=-x\langleog x-(1-x)\langleog(1-x)$ is the binary entropy.
\begin{figure}
\caption{The input/output diagram for the concurrence for
two families of states: 1) Werner states $\varrho_2=q\Psi_+ +(1-q)\frac{1}
\end{figure}
In our previous work \cite{ziman06} we have shown that
a stability of the entanglement-induced ordering is not compatible
with the listed axioms. A simple counter-example one can present involves
four qubits
divided into two groups.
Moreover, we have explicitly shown
that the ordering is not preserved for all
two-qubit measures providing one of the subsystems
is affected by the depolarizing
channel ${\cal E}_p[\omega]=p\varrho+(1-p)\frac{1}{2}I$. The violation of the
ordering is depicted in the diagram on Fig.~1. Based on
this explicit counter-example we can argue that there is no
(nontrivial) entanglement measure
$E$ that is stable under the action of local operations of the form ${\cal E}\otimes{\cal I}$,
where ${\cal E}$ is a tracepreserving completely positive linear map on the
system $A$ only.
Let us consider the so-called unilocal channel of the form ${\cal E}\otimes{\cal I}$
and some entanglement measure $E$. The action of such local channel
can be expressed in the $[E_{\ranglem in},E_{\ranglem out}]$-diagram with
respect to a given measure of the entanglement $E$. Whenever we find that
for fixed values of $E_{\ranglem out}$ there exist more input values $E_{\ranglem in}$,
one can easily construct a suitable counter-example violating the condition of the
ordering-preservation
\be
E(\varrho_1)>E(\varrho_2)\, \Rightarrow\,
E(\varrho_1^\prime)\ge E(\varrho_2^\prime)
\ee
valid for all states $\varrho_1,\varrho_2$ and
$\varrho_j^\prime={\cal E}\otimes{\cal I}[\varrho_j]$ ($j=1,2$).
More specifically. Let us define a ``horizontal fiber'' ${\cal F}_h(E_{\ranglem out})$ to be
a set of all values of $E_{\ranglem in}$ such that there exists a state
$\varrho_{\ranglem in}$ with $E(\varrho_{\ranglem in})=E_{\ranglem in}$ and
$E({\cal E}\otimes{\cal I}[\varrho_{\ranglem in}])=E_{\ranglem out}$. Whenever
${\cal F}_h(E_{\ranglem out})\cap{\cal F}_h(E^\prime_{\ranglem out})\ne\emptyset$
for all pairs of possible values $E_{\ranglem out},E_{\ranglem out}^\prime$
and ${\cal F}_h(E_{\ranglem out})\ne{\cal F}_h(E_{\ranglem out})\cap{\cal F}_h(E^\prime_{\ranglem out})
\ne{\cal F}_h(E_{\ranglem out}^\prime)$,
the counter-example can be designed. Consider
$E_{\ranglem out}>E_{\ranglem out}^\prime$. Because of the nonempty intersection
of ${\cal F}_h(E_{\ranglem out}),{\cal F}_h(E^\prime_{\ranglem out})$, there
are states $\varrho_{\ranglem in}^{j}$ ($j=1,2$) with the same amount
of the initial entanglement
$E_{\ranglem in}^1=E_{\ranglem in}^2$, but different values of the final
entanglement $E_{\ranglem out}=E_{\ranglem out}^1>E_{\ranglem out}^2=E_{\ranglem out}^\prime$.
Moreover, it is possible to choose $\varrho^1_{\ranglem in}$
and $\varrho^2_{\ranglem in}$ to have different values of entanglement so that
the ordering is not preserved,
in particular, $E(\varrho_{\ranglem in}^1)<E(\varrho_{\ranglem in}^2)$.
Each unilocal channel ${\cal E}$ determines a
set $S_{\cal E}$ in the $[E_{\ranglem in},E_{\ranglem out}]$-diagram.
In particular, for $S_{\cal E}$ forming some region (i.e. two-dimensional
geometrical object) the ordering is not preserved, because
there are values $E_{\ranglem out},E_{\ranglem out}^\prime$ for which
${\cal F}_h(E_{\ranglem out})\cap{\cal F}_h(E^\prime_{\ranglem out})\ne\emptyset$.
The formal description presented in the above paragraph as well as
the particular analysis itself might be technically difficult.
In fact, the illustration of the set $S_{\cal E}$ requires to evaluate
the entanglement for all possible states. However, intuitively
the situation expressed in Fig.~1 is not that complicated.
The observation that deserves special attention is that in order
to avoid the counter-examples of the above form the entanglement measure
and the transformation ${\cal E}$ must have very specific (and very peculiar)
properties that are reflected in the $[E_{\ranglem in},E_{\ranglem out}]$-diagram.
If the possible values of $E_{\ranglem out}$ form a continuum (which is the case
for all the measures we use), then the corresponding
set $S_{\cal E}$ must form a line. But this means, that either the
equally entangled states are always mapped into the equally entangled states,
or $S_{\cal E}$ consists of horizontal and vertical lines. The corresponding
maps would be indeed interesting.
We started our discussion with the question whether
there exists an entanglement measure such that for {\em all} channels
${\cal E}\otimes{\cal I}$ the induced ordering is preserved. However, the analysis
led us to another questions. Specifically, for which channels a given entanglement
measure is preserved? Our conjecture is that essentially arbitrary local
channel affects the ordering. The only known exceptions are: 1) a unitary
channels ($E_{\ranglem out}=E_{\ranglem in}$), 2)
and the entanglement-breaking channels ($E_{\ranglem out}=0$).
Other ``entanglement-order-preserving'' channels would be
of interest {\em per se}. There is a strong evidence that such channels
do not exist. Consequently, it seems that the measures
stable under local operations should be discrete, i.e. the entanglement
can achieve only certain countable set of values. An example of
such measure is the trivial $\delta$-measure that answers the question
whether a given state is entangled, or not. Our statement holds modulo
this type of "discrete" entanglement measures.
\section{Maximal entanglement {\em vs.} local operations}
It is important to know how the entanglement behaves under the action of quantum dynamics
\cite{horodecki_zyckowski}.
For example, it is interesting to know whether local sources of decoherence
are relevant for a given quantum protocol based on entangled
states. In the previous section we have analyzed how the local operations
affect the entanglement-induced ordering. Positive answer to such question
would give us a strong tool how to analyze the effect of local noise
in general settings just by analyzing the behavior of the most entangled
states. Unfortunately, we have found that the situation is puzzling,
because it seems that essentially arbitrary unilocal channel
does not preserve the ordering whatever measure we choose. In this section
we will focus on a simpler question: How much can we learn from the analysis
of the dynamics of maximally entangled states?
In Ref.~\cite{ziman06} we concluded that maximally entangled
state remains most entangled also after the application of the
local transformation ${\cal E}\otimes{\cal I}$. Unfortunately, this statement is
not correct and there is a loophole in the proof
\cite{piani, ziman_piani}. Here is a simple counter-example.
Consider a system consisting of four qubits (the qubits $A,A^\prime$
on one side and $B,B^\prime$ on the other one) and a local map
${\cal E}_{AA^\prime}= {\cal P}_0\otimes I + {\cal P}_1\otimes{\cal A}$,
where ${\cal P}_j$ is defined as ${\cal P}_j[X]=P_j X P_j$ ($P_j=|j\rangle\langle j|$),
and ${\cal A}[X]=\frac{1}{2}{\rm Tr}(X)I$. This transformation ``checks'' the state
of $A$ and either leaves $A^\prime$ unaffected, or it contracts its state
into a maximally mixed state. We will analyze the action
of such channel on two states:
1) $\varrho_1=\rangleho_{ABA^\prime B^\prime}=|0\rangle\langle 0|_A \otimes |0\rangle\langle 0|_B \otimes P^+_{A^\prime B^\prime}$, or
2) maximally entangled state
$\varrho_2=P^+=P^+_{ABA^\prime B^\prime} = P^+_{AB} \otimes P^+_{A^\prime B^\prime}$,
where $P^+_{AB}$ is a projector onto a maximally entangled state of qubits $A,B$,
and similarly for $P^+_{A^\prime B^\prime}$. The first of these states is
invariant under the action of ${\cal E}_{AA\prime}\otimes{\cal I}_{BB^\prime}$, i.e.
$\varrho_1^\prime=\varrho_1$, but
$\varrho^\prime_2={\cal E}_{AA^\prime}\otimes{\cal I}_{BB^\prime}[P^+]=
\frac{1}{2}\varrho_1+\frac{1}{2} |1\rangle\langle 1|\otimes|1\rangle\langle 1|\otimes\frac{1}{4}I$.
The state $\varrho_2^\prime$ is, if entangled, for sure is strictly
less entangled than $\varrho_1^\prime$, i.e. ordering is not preserved
for an arbitrary measure of entanglement. The convexity guarantees that
$E(\varrho_2^\prime)\langlee \frac{1}{2}E(\varrho_1^\prime)<E(\varrho_1^\prime)$.
This result suggests that it is not straightforward to see how much
the analysis of dynamics of maximally entangled states can tell us about the
entanglement dynamics in general. On the other hand, in spite of the result
related to the entanglement-induced ordering, in the present case the maximality
is preserved for larger class of channels. Their
characterization is an open problem and will be analyzed elsewhere
\cite{ziman_piani}. An interesting feature that remains valid is that
all maximally entangled states are (under unilocal channels)
mapped into states with the same amount of entanglement \cite{ziman06}.
This holds for any measure of entanglement.
\section{Speculations and conclusions}
As a result of our analysis
we discovered new features and properties of entanglement measures. We found that
the ordering that implies statements such as ``one state is more/less entangled than another'' is not preserved
under the action of local operations. Moreover, such ordering is affected by all unilocal
operations except the unitary and the entanglement-breaking channels. Surprisingly
enough, we also found that the maximally entangled states might be
more fragile than ``less'' entangled states. This might sound
counter-intuitive, but in some realistic cases, in which the
systems are affected by a local noise, it could be better
to start with less (noise-dependent) entangled state in order to increase the success of the protocol.
Hence, the operational meaning of the property ``being more/less entangled'' is questionable.
Operationally, ``more entangled'' should be synonymous to ``having larger rate'' of success.
However, just a small modification of protocols (e.g. taking into account a local noise)
might change this interpretation. Hence,
does it make any sense to use the entanglement measures for the state ordering?
If not, then what are these measures good for?
Entanglement measures still provide us with very powerful tools
enabling us to decide the basic question, whether a given state
is entangled, or not. In fact, it is much simpler to compute
the concurrence of two qubits than to prove the (non)existence of a
separable decomposition. It might be that the idea of entanglement-induced
state ordering cannot be based on some entanglement measure. To introduce
such concept one should probably adopt different approach, in which
the stability with respect to local operations is fulfilled ``by the definition''.
Even in this case we have more options depending on the class
of operations we will consider. We can say that a state $\omega_1$ is more,
or equally entangled than a state $\omega_2$ ($\omega_1\succeq\omega_2$)
if and only if there exists a completely positive tracepreserving
linear operation ${\cal E}_A\otimes{\cal E}_B$ such that
$\omega_2={\cal E}_A\otimes{\cal E}_B[\omega_1]$. This is compatible
with the fact, that entanglement can be only decreased by the action of local
operations (LO). Alternatively, one can use the class of LOCC operations, or
stochastic LOCC (SLOCC) operations. Two states are equally entangled if
$\omega_1\succeq\omega_2$ and $\omega_2\succeq\omega_1$ simultaneously.
If two states are not equivalent, but $\omega_2\succeq\omega_1$, then
$\omega_2\succ\omega_1$. All these types of entanglement-based
orderings are, in principle, partial, i.e. not all states are comparable. For example, using
the SLOCC-ordering all two-qubit entangled states are equally entangled,
because they can be used for the teleportation. The LOCC-ordering is more
strict and for the LO-ordering pure states with different Schmidt
coefficients are not comparable. Intuitively, the most physical/operational
is the LOCC-based state ordering.
Recently, Kinoshita et al. in \cite{kinoshita}
analyzed compatibility of the LOCC-based ordering
under the action of local operations. They presented an example of
two states $\omega_1\succ\omega_2$ that are transformed
by a unilocal operation ${\cal E}\otimes{\cal I}$
(the so-called selective entanglement-breaking channels)
into $\omega_1^\prime,\omega_2^\prime$ such that
$\omega_{2}^\prime\succ\omega_1^\prime$. This explicit example
supports our conclusion about the existence of entanglement-induced
state orderings compatible with local operations,
because it shows that for an arbitrary entanglement measure
satisfying the the LOCC monotonicity
condition the entanglement-induced ordering is not preserved.
But, one can make even stronger conclusion that also the ``operational''
LOCC-based state ordering is not robust with respect to
local operations. It seems that there is no way how to
introduce a nontrivial entanglement-related state ordering
compatible with local operations. The only option is to use the
trivial $\delta$-measure, or some simple modification of it.
In the analysis of entanglement dynamics it is of interest to
specify times at which the entanglement disappears. Although
any particular dynamics depends on the initial state, these
``entanglement-breaking'' time instants $t_{sep}$ can be completely
characterized by the analysis of the maximally entangled state.
The channel is called entanglement-breaking ${\cal E}$ if and only if
$\omega^\prime={\cal E}\otimes{\cal I}[\omega]$ is separable for all initial
states $\omega$. It is sufficient to verify this property
for a maximally entangled state, i.e. whether $E({\cal E}\otimes{\cal I}[P_+])=0$ \cite{03hsr}.
The local dynamics is given by a one-parametric set of completely positive
maps ${\cal E}_t$. We have analyzed \cite{ziman_05} the general qubit
master equation generating semigroup dynamics. The qubit
semigroup evolution is characterized by two time scales: the decoherence time
$T_{\ranglem decoherence}$ and the decay time $T_{\ranglem decay}$. What are the limits
on the entanglement decay? Which process is the fastest one? These questions
are not answered in \cite{ziman_05}, but all the necessary tools are derived
in that paper. It is known that in some cases $t_{sep}\to\infty$, but
what is the shortest possible decay time $t_{sep}$? The result is that there
is no limit and $t_{sep}$ can be arbitrarily small. For example, under
the action of a local depolarization process
${\cal E}_t[\varrho]=e^{-t/T}\varrho+(1-e^{-t/T})\frac{1}{2}I$ the maximally
entangled states evolves into the state
$\omega_t=e^{-t/T}P_+ +(1-e^{-t/T})\frac{1}{4}I$ (Werner states).
Hence, the entanglement vanishes for $t_{\ranglem sep}=T\langlen 3$.
The parameter $T$ can be adjusted so that the
entanglement is destroyed in arbitrarily small time $t_{\ranglem sep}$.
In general, the vanishing decoherence rate guarantees the shortest
possible entanglement decay time, i.e. the process of entanglement decay
can be "infinitely" fast.
Let us get back to the status of entanglement measures. The main message of this contribution is that
the quantification of entanglement based
on entanglement measures define a state ordering that is not preserved
under the action of local operations. The interpretation of these measures should be reconsidered.
It seems that large values of entanglement measures characterize
the ``distance'' from the set of maximally entangled states, which is clearly
defined. Similarly, small values should correspond to states that are very far
(in the sense of entanglement) from the maximally entangled ones and
very close to the separable region of the state space.
The particular mathematical forms of these statements is not known,
but the meaning of entanglement degree could be hidden there.
The axiomatic entanglement measures can quantify different aspects
of quantum entanglement, or they can serve as bounds for particular
protocols. To understand the entanglement itself it is important to understand
the numbers we use to quantify this phenomenon. Thinking about the
relation between the state ordering, the entanglement measures, and the
robustness with respect to local operations, opens new
interesting conceptual questions deserving a deeper investigation.
\end{document} |
\begin{document}
\begin{frontmatter}
\title{Supplementary Materials for ``A flexible sensitivity analysis approach for unmeasured confounding with multiple treatments and a binary outcome with application to SEER-Medicare lung cancer data''\thanksref{t1}}
\runtitle{Sensitivity analysis with multiple treatments}
\thankstext{T1}{Corresponding Author: Liangyuan Hu, Department of Biostatistics and Epidemiology, Rutgers School of Public Health, Piscataway, NJ 08854, USA. Email:[email protected]}
\begin{aug}
\author[A]{\fnms{Liangyuan} \snm{Hu}\ead[label=e1]{[email protected]}},
\author[B]{\fnms{Jungang} \snm{Zou}\ead[label=e2]{[email protected]}},
\author[C]{\fnms{Chenyang} \snm{Gu}\ead[label=e3]{[email protected]}},
\author[D]{\fnms{Jiayi} \snm{Ji}\ead[label=e4]{[email protected]}},
\author[E]{\fnms{Michael} \snm{Lopez}\ead[label=e5]{[email protected]}}
\and
\author[F]{\fnms{Minal} \snm{Kale}\ead[label=e6]{[email protected]}}
\address[A]{Department of Biostatistics and Epidemiology,
Rutgers School of Public Health,
\printead{e1}}
\address[B]{Department of Biostatistics,
Columbia University,
\printead{e2}}
\address[C]{Analysis Group, Inc.,
\printead{e3}}
\address[D]{Department of Population Health Science and Policy,
Icahn School of Medicine at Mount Sinai,
\printead{e4}}
\address[E]{Department of Mathematics, Skidmore College,
\printead{e5}}
\address[F]{Department of Medicine,
Icahn School of Medicine at Mount Sinai,
\printead{e6}}
\end{aug}
\end{frontmatter}
\section{Design of supplementary simulation in Section 3.1}
In a simplified scenario of the illustrative simulation in Section 3.1, we assume the independence between $X_1$ and $U$. We used the same treatment assignment model and modified the outcome generating models to maintain the same observed outcome event probabilites in each of three treatment groups. Three sets of nonparallel response surfaces were generated,
\begin{align*}
\mathbb{P} \left( Y (1) =1 \mid X_1, U \right)& =\text{logit}^{-1}(0.1X_1-1.8U)\\
\mathbb{P} \left( Y (2) =1 \mid X_1, U \right) & =\text{logit}^{-1}(-0.7X_1+1.6U)\\
\mathbb{P} \left( Y (3) =1 \mid X_1, U \right) & =\text{logit}^{-1}(-0.5X_1+2.1U).
\end{align*}
Under this simulation configuration, the observed outcome event probability was 0.40 in $A=1$, 0.51 in $A=2$ and 0.64 in $A=3$ and the true $\text{CATE}_{1,2}=-0.16$, $\text{CATE}_{1,3}=-0.29$ and $\text{CATE}_{2,3}=-0.13$. Web Figure 1 shows the estimates of three pairwise causal effects among 1000 replications corresponding to each of four strategies (I)-(IV) described in Section 3.1. \\
\section{Supplementary tables and figures}
Web tables and figures referenced in the paper are provided below.
\begin{table}[H]
\centering
\caption{The data generating process of the covariates.}
\begin{tabular}{cccccc}\hline
Variables & Distribution &Variables & Distribution &Variables & Distribution \\\hline
$X_1$ & $N(0,1)$ & $X_6$ &$\text{Bern}(0.6)$ & $X_{11}$ & Student's $t_{10}$ \\
$X_2$ & $\mathcal{U}(-1,1)$ &$X_7$ & $\text{Bern}(0.3)$ &$X_{12}$ & Gamma(2,2)\\
$X_3$ &Beta(3,3)&$X_8$ & $\text{Bern}(0.5)$ &$X_{13}$ & InverseGamma(20,20)\\
$X_4$ &$N(-1,1)$ & $X_9$ & $\text{Multinom}(1, 0.3, 0.2,0.5)$ & $X_{14}$& $N(-1,2)$\\
$X_5$ &$N(1,1)$ &$X_{10}$&$\text{Multinom}(1, 0.1, 0.8,0.1)$ & $X_{15}$ &$N(1,2)$\\\hline
\end{tabular}
\end{table}
\setlength{\tabcolsep}{3pt}
\begin{table}[H]
\centering
\caption{Average absolute bias (AAB) and root-mean-squared error (RMSE) in the estimated conditional average causal effects for illustrative simulation in Section 3.1. Sensitivity analysis strategies (I)-(IV) were used. I: True $c^0$. I: 3rd $A$ ignored: $c(\cdot)$ functions involving the third treatment were set to zero. II: $\mathcal{U} \left( \max(-1, c^0 - h\hat{\sigma}), \min(1, c^0+h\hat{\sigma}) \right)$. III: $\mathcal{U} \left( \max(-1, c^0 - 2h \hat{\sigma}), c^0 \right)$ or $\mathcal{U} \left( c^0 , \min(1, c^0+2h \hat{\sigma}) \right)$. IV: $\mathcal{U}(-1,1)$. The CATE results that could be achieved if $U$ were actually observed and the naive CATE estimators ignoring $U$ are also presented. The true $CATE_{1,2}=-0.16$, $CATE_{1,3}=-0.29$ and $CATE_{2,3}=-0.13$. }
\small
\begin{tabular}{cccccccccccccc}\hline
&\multicolumn{6}{c}{$c(a_1,a_2,x_1)$}&&\multicolumn{6}{c}{$c(a_1,a_2)$}\\\cline{2-14}
&\multicolumn{2}{c}{$CATE_{1,2}$}&\multicolumn{2}{c}{$CATE_{1,3}$}&\multicolumn{2}{c}{$CATE_{2,3}$}&&\multicolumn{2}{c}{$CATE_{1,2}$}&\multicolumn{2}{c}{$CATE_{1,3}$}&\multicolumn{2}{c}{$CATE_{2,3}$}\\
& AAB&RMSE&AAB&RMSE&AAB&RMSE&&AAB&RMSE&AAB&RMSE&AAB&RMSE\\\hline
$U$ included&.01&.01&.01&.01&.01&.01&&.01&.01&.01&.01&.01&.01\\
I &.01&.01&.01&.01&.01&.01&&.01&.02&.01&.01&.01&.01\\
I: 3rd $A$ ignored &.02&.02&.02&.02&.03&.03&&.03&.03&.03&.03&.02&.03\\
II: $h=1$ &.02&.03&.02&.02&.02&.02&&.02&.02&.03&.03&.02&.03\\
II: $h=2$ &.04&.05&.03&.04&.03&.04&&.03&.04&.04&.06&.04&.05\\
III &.05&.06&.05&.06&.05&.06&&.05&.07&.06&.07&.06&.07\\
IV &.04&.05&.03&.04&.03&.04&&.03&.06&.05&.06&.04&.05\\
$U$ ignored &.06&.06&.07&.07&.06&.06&&.06&.06&.07&.07&.06&.06\\\hline
\end{tabular}
\end{table}
\setlength{\tabcolsep}{2pt}
\begin{table}[H]
\centering
\caption{Average absolute bias (AAB) and root-mean-squared error (RMSE) in the estimated conditional average treatment effects for contextualized simulation in Section 3.2. Three sensitivity analysis strategies were used: 1) true $c^0$, 2) $\mathcal{U} \left( c^0 - 2\hat{\sigma}, c^0+2\hat{\sigma} \right)$ but bounded within $[-1,1]$, 3) $\mathcal{U}(-1,1)$. The CATE results that could be achieved if $U$ were actually observed and the naive CATE estimators ignoring $U$ are also presented. The true $CATE_{1,2}=0.05$, $CATE_{1,3}=-0.11$ and $CATE_{2,3}=-0.16$. }
\small
\begin{tabular}{ccccccccccccccc}\hline
&&\multicolumn{6}{c}{$N=1500$, ratio of unit = 1:1:1}&&\multicolumn{6}{c}{$N=10000$, ratio of unit = 1:10:9}\\\cline{3-15}
&&\multicolumn{2}{c}{$CATE_{1,2}$}&\multicolumn{2}{c}{$CATE_{1,3}$}&\multicolumn{2}{c}{$CATE_{2,3}$}&&\multicolumn{2}{c}{$CATE_{1,2}$}&\multicolumn{2}{c}{$CATE_{1,3}$}&\multicolumn{2}{c}{$CATE_{2,3}$}\\
&& AAB&RMSE&AAB&RMSE&AAB&RMSE&&AAB&RMSE&AAB&RMSE&AAB&RMSE\\\hline
&$U$ included&.01&.01&.01&.01&.01&.01&&.00&.00&.00&.00&.00&.00\\
&True $c^0$ &.01&.01&.01&.01&.01&.01&&.00&.00&.00&.00&.00&.00\\
UMC(i) &$\mathcal{U}(c^0-2\hat{\sigma},c^0+2\hat{\sigma})$&.01&.01&.01&.01&.01&.01&&.01&.01&.01&.01&.01&.01\\
&$\mathcal{U}(-1,1)$&.01&.01&.01&.01&.01&.02&&.01&.01&.01&.01&.01&.01\\
&$U$ ignored&.02&.02&.03&.04&.03&.03&&.01&.01&.01&.01&.01&.01\\\hline
&$U$ included&.01&.01&.01&.01&.01&.01&&.00&.00&.00&.00&.00&.00\\
&True $c^0$ &.01&.01&.01&.01&.01&.01&&.00&.00&.00&.00&.00&.00\\
UMC(ii) &$\mathcal{U}(c^0-2\hat{\sigma},c^0+2\hat{\sigma})$&.01&.02&.01&.01&.01&.02&&.01&.01&.01&.01&.01&.01\\
&$\mathcal{U}(-1,1)$&.02&.02&.01&.02&.02&.02&&.01&.01&.01&.01&.01&.02\\
&$U$ ignored&.03&.03&.04&.04&.04&.04&&.02&.02&.02&.02&.02&.02\\\hline
&$U$ included&.01&.01&.01&.01&.01&.01&&.00&.00&.00&.00&.00&.00\\
&True $c^0$ &.01&.01&.01&.01&.01&.01&&.00&.00&.00&.00&.00&.00\\
UMC(iii) &$\mathcal{U}(c^0-2\hat{\sigma},c^0+2\hat{\sigma})$&.02&.03&.02&.03&.02&.03&&.01&.01&.01&.02&.01&.02\\
&$\mathcal{U}(-1,1)$&.03&.03&.02&.03&.03&.04&&.02&.02&.01&.02&.02&.02\\
&$U$ ignored&.04&.04&.05&.05&.05&.05&&.02&.02&.03&.03&.03&.03\\\hline
\end{tabular}
\end{table}
\setlength{\tabcolsep}{5pt}
\setlength{\tabcolsep}{5pt}
\begin{table}[H]
\centering
\caption{Sensitivity analysis for causal inference about average treatment effects of three surgical approaches on prolonged length of stay (LOS) based on the risk difference, using the SEER-Medicare lung cancer data. Three surgical approaches are $A=1$: robotic-assisted surgery (RAS), $A=2$: open thoracotomy (OT), $A=3$: video-assisted thoracic surgery (VATS). The adjusted effect estimates and 95\% uncertainty intervals are displayed. Interval estimates are based on pooled posterior samples across model fits arising from $30 \times 30$ data sets. The remaining standard deviation in the outcome not explained by the measured covariates is $\hat{\sigma}$ = 0.27 . We assume $c(1,3,x) \sim \mathcal{U} (-0.4,0), \; c(3,1,x) \sim \mathcal{U}(0, 0.4), \; c(2,3,x) \sim \mathcal{U} (-0.4,0), \; c(3,2,x) \sim \mathcal{U}(0, 0.4)$ for specification (i)-(v). }
\label{tab:SA-resp}
\begin{tabular}{cp{0.42\textwidth}ccc}
&Prior distributions on $c(\cdot)$ functions & RAS vs. OT & RAS vs. VATS & OT vs. VATS \\\hline
(i)&$c(1,2,x) \sim \mathcal{U} (-0.2,0), c(2,1,x) \sim \mathcal{U}(0, 0.2)$ & $-.05 (-.08,-.02)$ & $.06(.03,.09)$ & $.10 (.08,.12)$\\
(ii)&$ c(1,2,x) \sim \mathcal{U}(0, 0.2), c(2,1,x) \sim \mathcal{U} (-0.2,0)$ & $-.11 (-.14,-.08)$ & $.05(.02,.08)$ & $.12 (.10,.14)$\\
(iii)&$c(1,2,x), c(2,1,x) \sim \mathcal{U} (-0.2,0) $ & $-.05(-.08,-.02)$ & $.04(.01,.07)$ & $.14(.12,.16)$ \\
(iv)&$c(1,2,x), c(2,1,x) \sim \mathcal{U}(0, 0.2)$& $-.11(-.14,-.08)$ & $.00(-.03,.03)$ & $.08 (.06,.10)$\\
(v)&$c(1,2,x), c(2,1,x) \sim \mathcal{U}(-0.2, 0.2)$& $-.10(-.14,-.06)$ & $.01(-.03,.05)$ & $.09 (.06,.12)$\\
(vi)&all $c(\cdot) \sim \mathcal{U}(-1, 1)$ & $-.07(-.15,.01)$ & $.05(-.03,.13)$ & $.12(.06,.18)$ \\
(vii)&all $c(\cdot) =0$ & $-.09(-.11,-.07)$ & $.02(-.00,.04)$ & $.11(.10,.12)$\\\hline
\end{tabular}
\end{table}
\begin{table}[H]
\centering
\caption{Sensitivity analysis for causal inference about average treatment effects of three surgical approaches on intensive care unit (ICU) stay based on the risk difference, using the SEER-Medicare lung cancer data. Three surgical approaches are $A=1$: robotic-assisted surgery (RAS), $A=2$: open thoracotomy (OT), $A=3$: video-assisted thoracic surgery (VATS). The adjusted effect estimates and 95\% uncertainty intervals are displayed. Interval estimates are based on pooled posterior samples across model fits arising from $30 \times 30$ data sets. The remaining standard deviation in the outcome not explained by the measured covariates is $\hat{\sigma}$ = 0.46 . We assume $c(1,3,x) \sim \mathcal{U} (-0.4,0), \; c(3,1,x) \sim \mathcal{U}(0, 0.4), \; c(2,3,x) \sim \mathcal{U} (-0.4,0), \; c(3,2,x) \sim \mathcal{U}(0, 0.4)$ for specification (i)-(v).}
\label{tab:SA-resp}
\begin{tabular}{cp{0.42\textwidth}ccc}
&Prior distributions on $c(\cdot)$ functions & RAS vs. OT & RAS vs. VATS & OT vs. VATS \\\hline
(i)&$c(1,2,x) \sim \mathcal{U} (-0.4,0), c(2,1,x) \sim \mathcal{U}(0, 0.4)$ & $-.11 (-.14,-.07)$ & $.02(-.01,.05)$ & $.17 (.15,.19)$\\
(ii)&$ c(1,2,x) \sim \mathcal{U}(0, 0.4),c(2,1,x) \sim \mathcal{U} (-0.4,0)$ & $-.15 (-.18,-.12)$ & $.00(-.03,.03)$ & $.15 (.13,.17)$\\
(iii)&$c(1,2,x), c(2,1,x) \sim \mathcal{U} (-0.4,0) $ & $-.13(-.16,-.10)$ & $.03(-.00,.06)$ & $.16(.14,.18)$ \\
(iv)&$c(1,2,x), c(2,1,x) \sim \mathcal{U}(0, 0.4)$ & $-.15(-.18,-.12)$ & $-.02(-.05,.01)$ & $.11 (.09,.13)$\\
(v)&$c(1,2,x), c(2,1,x) \sim \mathcal{U}(-0.4, 0.4)$ & $-.14(-.18,-.10)$ & $-.01(-.05,.03)$ & $.10 (.07,.13)$\\
(vi)&all $c(\cdot) \sim \mathcal{U}(-1, 1)$ & $-.13(-.21,-.05)$ & $.03(-.05,.11)$ & $.16(.10,.24)$ \\
(vii)&all $c(\cdot) =0$ & $-.14(-.16,-.12)$ & $.01(-.00,.02)$ & $.16(.15,.17)$\\\hline
\end{tabular}
\end{table}
\begin{table}[H]
\centering
\caption{Sensitivity analysis for causal inference about average treatment effects of three surgical approaches on 30-day readmission rate based on the risk difference, using the SEER-Medicare lung cancer data. Three surgical approaches are $A=1$: robotic-assisted surgery (RAS), $A=2$: open thoracotomy (OT), $A=3$: video-assisted thoracic surgery (VATS). The adjusted effect estimates and 95\% uncertainty intervals are displayed. Interval estimates are based on pooled posterior samples across model fits arising from $30 \times 30$ data sets. The remaining standard deviation in the outcome not explained by the measured covariates is $\hat{\sigma}$ = 0.28 . We assume $c(1,3,x) \sim \mathcal{U} (-0.4,0), \; c(3,1,x) \sim \mathcal{U}(0, 0.4), \; c(2,3,x) \sim \mathcal{U} (-0.4,0), \; c(3,2,x) \sim \mathcal{U}(0, 0.4)$ for specification (i)-(v).}
\label{tab:SA-resp}
\begin{tabular}{cp{0.42\textwidth}ccc}
&Prior distributions on $c(\cdot)$ functions & RAS vs. OT & RAS vs. VATS & OT vs. VATS \\\hline
(i)&$c(1,2,x) \sim \mathcal{U} (-0.2,0), c(2,1,x) \sim \mathcal{U}(0, 0.2)$ & $.02 (-.01,.05)$ & $.06(.03,.09)$ & $.03 (.01,.05)$\\
(ii)&$c(1,2,x) \sim \mathcal{U}(0, 0.2), c(2,1,x) \sim \mathcal{U} (-0.2,0)$ & $-.01 (-.04,.02)$ & $.03(-.00,.06)$ & $.05(.03,.07)$\\
(iii)&$c(1,2,x), c(2,1,x) \sim \mathcal{U} (-0.2,0) $ & $.03(-.00,.06)$ & $.05(.02,.08)$ & $.07(.05,.09)$ \\
(iv)&$c(1,2,x), c(2,1,x) \sim \mathcal{U}(0, 0.2)$ & $-.01(-.04,.02)$ & $.01(-.02,.04)$ & $.00 (-.02,.02)$\\
(v)&$c(1,2,x), c(2,1,x) \sim \mathcal{U}(-0.2, 0.2)$ & $-.00(-.04,.04)$ & $.02(-.02,.06)$ & $.01 (-.03,.05)$\\
(vi)&all $c(\cdot) \sim \mathcal{U}(-1, 1)$ & $.01(-.07,.09)$ & $.05(-.03,.13)$ & $.04(-.03,.11)$ \\
(vii)&all $c(\cdot) =0$ & $-.00(-.02,.02)$ & $.02(-.00,.04)$ & $.02(-.00,.04)$\\\hline
\end{tabular}
\end{table}
\begin{table}[H]
\centering
\caption{Sensitivity analysis for causal inference about average treatment effects among those who were operated with robotic-assisted surgery on postoperative respiratory complications based on the risk difference, using the SEER-Medicare lung cancer data. Three surgical approaches are $A=1$: robotic-assisted surgery (RAS), $A=2$: open thoracotomy (OT), $A=3$: video-assisted thoracic surgery (VATS). The adjusted estimates of causal effects and 95\% uncertainty intervals are displayed. Interval estimates are based on pooled posterior samples across model fits arising from $30 \times 30$ data sets. The remaining standard deviation in the outcome not explained by the measured covariates is $\hat{\sigma}$ = 0.46. We assume $c(1,3,x) \sim \mathcal{U} (-0.4,0), \; c(3,1,x) \sim \mathcal{U}(0, 0.4), \; c(2,3,x) \sim \mathcal{U} (-0.4,0), \; c(3,2,x) \sim \mathcal{U}(0, 0.4)$ for specification (i)-(v).}
\label{tab:SA-resp}
\begin{tabular}{cp{0.42\textwidth}cc} \\
&Prior distributions on $c(\cdot)$ functions & RAS vs. OT & RAS vs. VATS \\\hline
(i)&$c(1,2,x) \sim \mathcal{U} (-0.4,0), c(2,1,x) \sim \mathcal{U}(0, 0.4)$ & $.02 (-.02,.06)$ & $.04(.00,.08)$ \\
(ii)&$ c(1,2,x) \sim \mathcal{U}(0, 0.4), c(2,1,x) \sim \mathcal{U} (-0.4,0)$ & $-.02 (-.06,.02)$& $.02(-.02,.06)$ \\
(iii) & $c(1,2,x), c(2,1,x) \sim \mathcal{U} (-0.4,0) $ & $.03(-.01,.07)$ & $.04(.00,.08)$ \\
(iv) & $c(1,2,x), c(2,1,x) \sim \mathcal{U}(0, 0.4)$ & $.01(-.03,.05)$ & $.01(-.03,.05)$ \\
(v)&$c(1,2,x), c(2,1,x) \sim \mathcal{U}(-0.4, 0.4)$ & $.02(-.03,.07)$ & $.02(-.03,.07)$ \\
(vi)&all $c(\cdot) \sim \mathcal{U}(-1, 1)$ & $.01(-.09,.11)$ & $.03(-.07,.13)$ \\
(vii) &all $c(\cdot) =0$ & $-.01(-.04,.02)$ & $.00(-.03,.03)$ \\\hline
\end{tabular}
\end{table}
\begin{figure}
\caption{Estimates of three pairwise causal effects $\text{CATE}
\label{fig:sim_binary_no_interaction}
\end{figure}
\begin{figure}
\caption{Estimates of three pairwise causal treatment effects $CATE_{1,2}
\label{fig:sim_binary_suppl}
\end{figure}
\begin{figure}
\caption{Credible intervals for three pairwise causal effects $\text{CATE}
\label{fig:sim_binary_ci}
\end{figure}
\begin{figure}
\caption{Distributions of the true generalized propensity scores (GPS) corresponding to (a) strong, (b) moderate, and (c) weak covariate overlap. Each panel presents the density plots of the GPS for units assigned to a given treatment group. The left panel corresponds to treatment 1, the middle panel treatment 2, and the right panel treatment 3.}
\label{fig:overlap_sim}
\end{figure}
\begin{figure}
\caption{Distributions of the posterior mean generalized propensity scores (GPS) for the SEER-Medicare lung cancer data. The GPS were obtained by fitting a BART model for the multinomial outcomes. Each panel presents the density plots of the GPS for units assigned to a given treatment group. The left panel corresponds to treatment 1, the middle panel treatment 2, and the right panel treatment 3.}
\end{figure}
\begin{figure}
\caption{The coverage probability of the estimates of three pairwise causal effects $\text{CATE}
\label{fig: sim-complex-CP-1500}
\end{figure}
\begin{figure}
\caption{Contour plots of the confounding function adjusted treatment effect estimates for RAS versus OT, $CATE_{1,2}
\end{figure}
\section{Sample codes}
We provide step-by-step sample coding of the illustrative simulation described in Section 3.1 for the scenario where we use the true $c^0$. The proposed method is readily available in the $\R$ package $\textsf{SAMTx}$.
\begin{lstlisting}[breaklines=true]
# Simulate the data
library(BART)
sample_size = 1500
# First simulate the treatment indicator A
x1 = rbinom(sample_size, 1, prob=0.4)
u = rbinom(sample_size, 1, prob=0.5)
lp.A = 0.2 * x1 + 0.4 * u + rnorm(sample_size, 0, 0.1)
lp.B = -0.3 * x1 + 0.8 * u + rnorm(sample_size, 0, 0.1)
lp.C = 0.1 * x1 + 0.5 * u + rnorm(sample_size, 0, 0.1)
p.A1 <- exp(lp.A)/(exp(lp.A)+exp(lp.B)+exp(lp.C))
p.A2 <- exp(lp.B)/(exp(lp.A)+exp(lp.B)+exp(lp.C))
p.A3 <- exp(lp.C)/(exp(lp.A)+exp(lp.B)+exp(lp.C))
p.A <- matrix(c(p.A1,p.A2,p.A3),ncol = 3)
A = NULL
for (i in 1:sample_size) {
A[i] <- sample(c(1, 2, 3),
size = 1,
replace = TRUE,
prob = p.A[i, ])
}
table(A)
# Then simulate the treatment P(Y(A) = 1|x1, u)
Y1 = -0.8 * x1 - 1.2 * u + 1.5 * u * x1
Y2 = -0.6 * x1 + 0.5 * u + 0.3 * x1 * u
Y3 = 0.3 * x1 + 0.2 * x1 * u + 1.3 * u
Y1 = rbinom(sample_size, 1, exp(Y1)/(1+exp(Y1)))
Y2 = rbinom(sample_size, 1, exp(Y2)/(1+exp(Y2)))
Y3 = rbinom(sample_size, 1, exp(Y3)/(1+exp(Y3)))
dat_truth = cbind(Y1, Y2, Y3, A) # True data for the outcome Y
Yobs <- apply(dat_truth, 1, function(x) # Observed data for the outcome Y
x[1:3][x[4]])
# Simulate the true confounding function c(a1, a2)
n_alpha = 30
alpha = cbind(
runif(n_alpha, mean(Y1[A ==1])-mean(Y1[A ==2]) - 0.001, mean(Y1[A ==1])-mean(Y1[A ==2]) + 0.001),
runif(n_alpha, mean(Y2[A ==2])-mean(Y2[A ==1]) - 0.001, mean(Y2[A ==2])-mean(Y2[A ==1]) + 0.001),
runif(n_alpha, mean(Y2[A ==2])-mean(Y2[A ==3]) - 0.001, mean(Y2[A ==2])-mean(Y2[A ==3]) + 0.001),
runif(n_alpha, mean(Y1[A ==1])-mean(Y1[A ==3]) - 0.001, mean(Y1[A ==1])-mean(Y1[A ==3]) + 0.001),
runif(n_alpha, mean(Y3[A ==3])-mean(Y3[A ==1]) - 0.001, mean(Y3[A ==3])-mean(Y3[A ==1]) + 0.001),
runif(n_alpha, mean(Y3[A ==3])-mean(Y3[A ==2]) - 0.001, mean(Y3[A ==3])-mean(Y3[A ==2]) + 0.001))
y <- Yobs
covariates = as.matrix(cbind(x1, u))
y = as.numeric(y)
A = as.integer(A)
A_unique_length <- length(unique(A))
alpha = as.matrix(alpha)
M1 <- 30 # This is M1
M2 <- 30 # This is M2
nposterior <- 10000 # 10000 posterior samples
# Algorithm 1.1: Fit the multinomial probit BART model to the treatment A
A_model = mbart2(x.train = covariates, as.integer(as.factor(A)), x.test = covariates, ndpost = nposterior)
# Algorithm 1.1: Estimate the generalized propensity scores for each individual
p = array(A_model$prob.test[seq(1, nrow(A_model$prob.test), nposterior),], dim = c(M1, 3, length(A)))
# Algorithm 1.2: start to calculate causal effect by M1 * M2 times
causal_effect_1 = matrix(NA, nrow = M2 * M1, ncol = nposterior)
causal_effect_2 = matrix(NA, nrow = M2 * M1, ncol = nposterior)
causal_effect_3 = matrix(NA, nrow = M2 * M1, ncol = nposterior)
step = 1
train_x = cbind(covariates, A)
for (j in 1:M1) {
# Algorithm 1.2: Draw M1 generalzied propensity scores from the posterior predictive distribution of the A model for each individual
p_draw_1 <- p[j, 1, ]
p_draw_2 <- p[j, 2, ]
p_draw_3 <- p[j, 3, ]
for (m in 1:M2) {
# Algorithm 1.2: Draw M2 values from the prior distribution of each of the sensitivity paramaters alpha for eacg treatment
print(paste("step :", step, "/", M2*M1))
sort(unique(train_x[, "A"]))
# Algorithm 1.3: Compute the adjusted outcomes y_corrected for each treatment for each M1M2 draws
y_corrected = ifelse(
train_x[, "A"] == sort(unique(train_x[, "A"]))[1],
y - (unlist(alpha[m, 1]) * p_draw_2 + unlist(alpha[m, 4]) * p_draw_3),
ifelse(
train_x[, "A"] == sort(unique(train_x[, "A"]))[2],
y - (unlist(alpha[m, 2]) * p_draw_1 + unlist(alpha[m, 3]) * p_draw_3),
y - (unlist(alpha[m, 5]) * p_draw_1 + unlist(alpha[m, 6]) * p_draw_2)
)
)
# Algorithm 1.4: Fit a BART model to each set of M1*M2 sets of observed data with the adjusted outcomes y_corrected
bart_mod = wbart(x.train = cbind(covariates, A), y.train = y_corrected, ndpost = nposterior, printevery = 10000)
predict_1 = pwbart(cbind(covariates, A = sort(unique(A))[1]), bart_mod$treedraws)
predict_2 = pwbart(cbind(covariates, A = sort(unique(A))[2]), bart_mod$treedraws)
predict_3 = pwbart(cbind(covariates, A = sort(unique(A))[3]), bart_mod$treedraws)
causal_effect_1[((m - 1) * M1) + j, ] = rowMeans(predict_1 - predict_2)
causal_effect_2[((m - 1) * M1) + j, ] = rowMeans(predict_2 - predict_3)
causal_effect_3[((m - 1) * M1) + j, ] = rowMeans(predict_1 - predict_3)
step = step + 1
}
}
# Final combined adjusted causal effect
ATE_01_adjusted <- causal_effect_1
ATE_12_adjusted <- causal_effect_2
ATE_02_adjusted <- causal_effect_3
\end{lstlisting}
\end{document} |
\begin{document}
\begin{abstract}
Let $p > 2$ be a prime and let $X$ be a compactified PEL Shimura variety of type (A) or (C) such that $p$ is an unramified prime for the PEL datum and such that the ordinary locus is dense in the reduction of $X$. Using the geometric approach of Andreatta, Iovita, Pilloni, and Stevens we define the notion of families of overconvergent locally analytic $p$-adic modular forms of Iwahoric level for $X$. We show that the system of eigenvalues of any finite slope cuspidal eigenform of Iwahoric level can be deformed to a family of systems of eigenvalues living over an open subset of the weight space. To prove these results, we actually construct eigenvarieties of the expected dimension that parametrize finite slope systems of eigenvalues appearing in the space of families of cuspidal forms.
\end{abstract}
\maketitle
\section*{Introduction} \label{sec: intro}
The main theme of this work is the theory of $p$-adic families of modular eigenforms and the study of the congruences between them. This subject has become more and more important in number theory. For example, one of the main technique to prove modularity results of Galois representations is to prove that a given Galois representation lives in a family of Galois representations attached to a family of $p$-adic modular forms and then invoke a classicity result. To achieve these goals, it is crucial to have a good understanding of $p$-adic families of modular forms.
Historically, the subject started in the seventies with the work of Serre that gave the first example of a $p$-adic family of eigenforms for $\GL_{2/\Q}$: the Eisenstein family. In the eighties Hida was able to prove that, still in the $\GL_{2/\Q}$ case, \emph{any} ordinary eigenform (of Iwahoric level) can be deformed to a family of ordinary eigenform over the weight space. During the nineties, Coleman was able to generalize Hida's result to forms that are overconvergent and of finite slope for the $\U$-operator. The theory culminated in the construction, due to Coleman and Mazur, of the \emph{eigencurve}, a rigid analytic curve living over the weight space that parametrize finite slope overconvergent eigenforms.
It is then natural to try to generalize the theory to groups different from $\GL_{2/\Q}$. Coleman's techniques are based on the theory of $q$-expansion and on the existence of the Eisenstein family, so it seems difficult to generalize them to more general groups (where we can have no cusps or no Eisenstein families). There are already several other approaches in the literature. For example in the work of Kisin and Lai for Hilbert modular forms in \cite{kislai} (recently generalized by Mok and Tan to the Siegel-Hilbert case in \cite{sieghilb}) the authors use a generalization of the Eisenstein family and construct the eigenvariety following Coleman and Mazur. More generally in \cite{urban_eigen}, Urban uses Steven's theory of overconvergent cohomology to show the existence of an eigenvariety for modular symbols associated to any reductive group with discrete series. We also have results by Chenevier for unitary groups in \cite{chenevier} or by Emerton in \cite{emerton} for any reductive group.
In all these constructions, the eigenvariety parametrizes systems of eigenvalues appearing in the space of overconvergent modular forms rather than modular forms themselves. One of the reasons for this is the lack of the notion of families of overconvergent modular forms (while the notion of family of systems of eigenvalues is easily defined). In this paper, the starting point to build the eigenvariety is the definition of analytic families of overconvergent modular forms. We follow the geometric approach recently introduced by Andreatta, Iovita, Pilloni, and Stevens in the series of papers \cite{over, vincent, AIP, AIPhilb}. The basic idea is quite simple: analytically interpolate the sheaves $\underline \omega^{\underline k}$, where $\underline k$ is an integral weight, defining the sheaves $\underline \omega^\chi$ for any $p$-adic weight $\chi$. More generally, one wants to define a family of sheaves parametrized by the weight space with the property that its pullback to a point $\chi$ of the weight space is the sheaf $\underline \omega^\chi$. Since we are interested in overconvergent modular forms, such a family should be a sheaf $\underline \omega^{\mc U}$ over $X(v) \times \mc U$, where $X(v)$ is a sufficiently small strict neighbourhood of the ordinary locus of the relevant (compactified) Shimura variety and $\mc U$ is an affinoid of the weight space. Then, a family of modular forms parametrized by $\mc U$ is simply a global section of $\underline \omega^{\mc U}$. After having defined Hecke operators, the idea is to use the abstract machinery developed by Buzzard in \cite{buzz_eigen} (that generalizes Coleman's work) to construct the eigenvariety. Unfortunately, we do not know whether one crucial assumption in Buzzard's work is verified by the space of families of modular forms (and we believe that in general it is not), but we are able to show that this assumption is satisfied by \emph{cuspidal} forms. Once this is done Buzzard's results apply and we obtain the eigenvariety.
Let us now state more precisely the results obtained in this paper. Let $K$ be a sufficiently large finite extension of $\Q_p$ and let $Y$ be a Shimura variety, over $K$, of PEL type and Iwahoric level, associated to a symplectic or a unitary group. We assume that $p > 2$ is a prime that is unramified in the PEL datum of $X$ and let $\mc W$ be the weight space associated to $Y$. We denote with $X$ a fixed smooth toroidal compactification of $Y$. We assume that the ordinary locus of the reduction of a certain integral model of $X$ modulo the maximal ideal of $\mc O_K$ is dense (see \cite{shimura} for a case without ordinary locus). Our main results are the following theorems.
\begin{teono}
Let $\chi \in \mc W$ be a $p$-adic character. There is a good notion of $\underline v$-overconvergent, $\underline w$-locally analytic modular forms over $X$, where $\underline v$ and $\underline w$ are tuples of positive rational numbers satisfying certain conditions. These modular forms are defined as sections of certain sheaves $\underline \omega_{\underline v, \underline w}^\chi$ that interpolate an analytic version of the classical algebraic sheaves $\underline \omega^{\underline k}$ defined for integral weights. We also have an analogous result for cuspforms. These spaces can be put in families over $\mc W$ and there is an action of an Hecke algebra $\m T$, that includes the completely continuous operator $\U$. If $F$ is a locally analytic overconvergent modular eigenform of integral weight and $\U$-slope sufficiently small with respect to the weight (see Theorem~\ref{thm: class} for a precise condition), then $F$ comes from a classical modular form.
\end{teono}
\begin{teono}
Let $\cusp^{\dagger\chi}_{\underline v,\underline w}$ be the space of cuspidal forms that are $\underline v$-overconvergent, $\underline w$-locally analytic and of weight $\chi$. Let $f \in \cusp^{\dagger\chi}_{\underline v,\underline w}$ be a cuspidal eigenform of finite slope for the $\U$-operator. Then there exists an affinoid $\mc U \subset \mc W$ that contains $\chi$ and such that the system of eigenvalues associated to $f$ can be deformed to a family of systems of eigenvalues appearing in $\cusp^{\dagger\mc U}_{\underline v,\underline w}$, where $\cusp^{\dagger\mc U}_{\underline v,\underline w}$ is the space of families of cuspforms parametrized by $\mc U$.
More precisely, there is a rigid space $\mc E_{\underline v, \underline w} \to \mc W \times \m A^{1,\rig}$ that satisfies the following properties.
\begin{enumerate}
\item It is equidimensional of dimension $\dim(\mc W)$ and the map $\mc E_{\underline v, \underline w} \to \mc W$ is locally finite. The fiber of $\mc E_{\underline v, \underline w}$ above a point $\chi \in \mc W$ parametrizes systems of eigenvalues for the Hecke algebra $\m T$ appearing in $\cusp^{\dagger\chi}_{\underline v,\underline w}$ that are of finite slope for the $\U$-operator. If $x \in \mc E_{\underline v, \underline w}$, then the inverse of the $\U$-eigenvalue corresponding to $x$ is $\pi_2(x)$, where $\pi_2$ is the induced map $\pi_2 \colon \mc E_{\underline v, \underline w} \to \m A^{1,\rig}$. For various $\underline v$ and $\underline w$, these constructions are compatible. Letting $\underline v \to 0$ and $\underline w \to \infty$ we obtain the global eigenvariety $\mc E$.
\item Let $f \in \cusp^{\dagger\chi}_{\underline v,\underline w}$ be a cuspidal eigenform of finite slope for the $\U$-operator and let $x_f$ be the point of $\mc E_{\underline v, \underline w}$ corresponding to $f$. Let us suppose that $\mc E_{\underline v, \underline w} \to \mc W$ is unramified at $x_f$. Then there exists an affinoid $\mc U \subset \mc W$ that contains $\chi$ and such that $f$ can be deformed to a family of finite slope eigenforms $F \in \cusp^{\dagger\mc U}_{\underline v,\underline w}$.
\end{enumerate}
\end{teono}
Here is a detailed description of this paper. We follow \cite{AIP}, that is our main reference.
In Section~\ref{sec: PEL data} we introduce the Shimura varieties $X$ we work with. These are (integral models of) Shimura varieties of PEL type. At the beginning, we do not assume that $p$ is unramified in the PEL datum, but we assume that the ordinary locus of (the reduction of) $X$ is dense. We define Hasse invariants and some strict neighborhoods $X(\underline v)$ of the ordinary locus. We also work with some Shimura varieties of deeper level at $p$, that are needed to define modular forms. In Section~\ref{sec: F} we define the sheaf $\mc F$, that is a more convenient integral model of the conormal sheaf $\underline \omega$ and it is crucial for the definition of the sheaves $\underline \omega^\chi$. We also introduce our weight space and we define the so called modular sheaf of any $p$-adic weight (modular forms will be sections of these sheaves). In Section~\ref{sec: mod forms} we introduce various spaces of modular forms. We do not have a Koecher principle for sections of our modular sheaves, so we find it convenient to work with the compactified Shimura variety. In particular we need to assume that $p$ is unramified in the PEL datum. Section~\ref{sec: hecke op} is devoted to the definition of Hecke operators, both outside $p$ and at $p$. In particular we define the $\U$-operator and we show that it is a completely continuous operator on the space of overconvergent modular forms. In Section~\ref{sec: heck var} we study the space of cuspidal forms and we construct the eigenvarieties. To use Buzzard's machinery we need to verify that the space of cuspforms is projective (see Definition~\ref{defi: proj}) and to achieve this goal we make some explicit computations very similar to those of \cite{AIP}. The main technical point is a vanishing result about the higher direct images, projecting from the toroidal to the minimal compactification, of the structural sheaf twisted by the ideal defining the boundary. This result has been essentially proved in full generality by Lan in \cite{lan_ram}. Knowing the projectivity of the space of cuspforms we can apply Buzzard's results and we obtain the eigenvariety. In Section~\ref{sec: class} we prove our classicity result. This is a `small slope eigenforms are classical' type theorem, and its proof splits in two parts. First of all we show that small slope locally analytic overconvergent eigenforms are overconvergent algebraic eigenforms. This a representation theoretic computation using the BGG resolution. Then we use the results of \cite{stefan} to prove classicity. The above theorems follow.
\subsection*{Open questions} \label{subsec: open}
There are at least two questions left open by this work.
\begin{itemize}
\item Since Corollary~\ref{coro: fin} holds only for cuspidal forms, we cannot apply Buzzard machinery to produce eigenvarieties that parametrize system of eigenvalues associated to not necessarily cuspidal modular forms. For various applications, it would be useful to have eigenvarieties that work in general. We think that the cuspidality assumption is necessary to apply Buzzard's machinery in general, but we believe that our construction can be used in some non-cuspidal cases, adding certain conditions on the weight. This is the subject of the work in progress \cite{eigen}.
\item In view of Theorem~\ref{teo: eigen}, part \eqref{en: unr}, it is natural to look for some conditions that ensure that the morphism $\mc E \to \mc W$ is unramified at a given $x \in \mc E$. A similar problem exists already in the Siegel and Hilbert cases, see \cite[Section~8.3]{AIP} and the references cited there for what is know in the $\GL_{2}$ and $\GSp_4$ cases.
\end{itemize}
\subsection*{Acknowledgments} \label{subsec: ackn}
I would like to thank Fabrizio Andreatta, Adrian Iovita, and Vincent Pilloni for their paper \cite{AIP} and for many useful discussions. I would also like to thank Kai-Wen Lan for answering a lot of questions concerning his works. I thank the anonymous referee for pointing out a problem in the proof of Proposition~\ref{prop: mod form well def} and several useful comments.
\begin{notation}
If $G$ is an abelian group and $p$ is a prime number, we set $G_p \colonequals G \otimes_{\Z} \Z_p$.
If $R$ is a commutative ring, there is an equivalence of categories between $\mat_n(R)-\mathbf{mod}$ and $R-\mathbf{mod}$. We will always realize this Morita's equivalence via $M \rightsquigarrow e_{1,1} \cdot M$, where $e_{1,1} \in \mat_n(R)$ is the diagonal matrix that has $1$ in the upper left corner and $0$ on all the other entries. In the case of a module over a product of matrix rings, we will realize Morita's equivalence via the product of the above functors.
We will write $\B_n(R) \subset \GL_n(R)$ for the Borel subgroup of $\GL_n(R)$ consisting of upper triangular matrices. We denote with $\U_n(R)$ the unipotent radical of $\B_n(R)$.
We will work with several objects that can have a $+$ or a $-$ as superscripts, or no superscripts at all. If $\star$ is any symbol, the notation $\star^\pm$ refers to any of $\star^+$, $\star^-$ or $\star$. No ambiguity should arise.
\end{notation}
\section{PEL type Shimura varieties} \label{sec: PEL data}
In this section we introduce the basic objects of our work. Our main reference for Shimura varieties of PEL type is \cite{lan}. We consider a particular case of the situation studied in Lan's work but, until Section~\ref{sec: heck var}, we slightly relax the assumption on $p$. One can check that the definitions and results we cite still make sense with our assumptions on $p$, so we freely cite \cite{lan}.
Let $B$ be a finite dimensional simple algebra over $\Q$, with center $F$. We let $^\ast \colon B \to B$ be a positive involution and we write $F_0$ for the subfield of $F$ fixed by $^\ast$.
\begin{ass} \label{ass: type}
We assume that we are in one of the following situations.
\begin{enumerate}
\item[Case (A)] We have $[F:F_0]=2$. In this case $F$ is a totally imaginary extension of $F_0$ and $B \otimes_{F_0} \mathds{R} \cong \mat_n(\C)$ (the involution is $A \mapsto \bar A^t$).
\item[Case (C)] We have $F=F_0$ and $B \otimes_{F_0} \mathds{R} \cong \mat_n(\mathds{R})$ with involution $A \mapsto A^t$.
\end{enumerate}
\end{ass}
Let $d$ be $[F_0 : \Q]$ and let $\tau_1,\ldots,\tau_d$ be the various embeddings $F_0 \hookrightarrow \mathds{R}$. In case (A), we choose once and for all a CM-type for $F$, i.e.\ we choose $\sigma_1,\ldots, \sigma_d$ embeddings $F \hookrightarrow \C$ such that $\sigma_{i|F_0} = \tau_i$. In particular, $\Hom(F,\C)= \set{\sigma_i, \bar \sigma_i}_i$.
We fix $\mc O_B$, an order of $B$ that is stable under $^\ast$. Let $(\Lambda, \langle \cdot, \cdot \rangle, h)$ be a PEL type $\mc O_B$-lattice in the sense of \cite[Definition~1.2.1.3]{lan}, with $\Lambda \neq 0$. We set $V \colonequals \Lambda \otimes_{\Z} \Q$. We obtain an algebraic group $G$ over $\Sp(\Z)$ as in \cite[1.2.1.6]{lan}. Thanks to our assumptions, $G_{\Q}$ is a reductive connected algebraic group over $\Q$.
\begin{ass}
We assume that the complex dimension of the symmetric space associated to $G$ is at least $2$ (so Koecher's principle holds).
\end{ass}
We decompose
\begin{gather} \label{eq: dec C}
V \otimes_{\Q} \C \cong V_{\C,1} \oplus V_{\C,2}
\end{gather}
in such a way that $h(z)$ acts on $V_{\C,1}$ via multiplication by $z$ and via multiplication by $\bar z$ on $V_{\C,2}$. We write $E$ for the reflex field, the finite extension of $\Q$ defined as the field of definition of the isomorphism class of the complex $B$-representation $V_{\C,1}$.
We let $p \neq 2$ be a prime number, fixed from now on. We assume there is an isomorphism, that we fix,
\begin{gather} \label{eq: dec int}
\mc O_{B,p} \cong \prod_{\mathfrak p | p} \mat_n(\mc O_{\mathfrak p}).
\end{gather}
where the product is over the prime ideals of $\mc O_F$ above $p$ and $\mc O_{\mathfrak p}$ is a finite extension of $\Z_p$. We choose a uniformizer element $\varpi_{\mathfrak p} \in \mc O_{\mathfrak p}$. We assume that $\mc O_{B,p}$ is a maximal order of $B_p$ and that the restriction of $\langle \cdot, \cdot \rangle$ to $\Lambda_p$ gives a perfect pairing with values in $\Z_p$. By \eqref{eq: dec int}, we have decompositions
\[
V_p \cong \prod_{\mathfrak p | p} V_{\mathfrak p} \mbox{ and } \Lambda_p \cong \prod_{\mathfrak p | p} \Lambda_{\mathfrak p}.
\]
Taking multiples by powers of $\varpi_{\mathfrak p}$ of $\Lambda_{\mathfrak p}$, we obtain a selfdual chain $\mc L_{\mathfrak p}$ of $\mat_n(\mc O_{\mathfrak p})$-lattices of $V_{\mathfrak p}$. The product of the $\mc L_{\mathfrak p}$ gives $\mc L_p$, a selfdual multichain of $\mc O_{B,p}$-lattices in $V_p$ (see \cite[Chapter~3]{rapoport_zink} for the definition of these notions). Let $K_p \subset G(\Q_p)$ be the stabilizer of $\mc L_p$. Then $K_p$ is a parahoric subgroup. We fix $\mc H \subset G({\widehat{\Z}}^p)$, a compact open subgroup that we assume to be \emph{neat} (see \cite[Definition~1.4.1.8]{lan}). We will denote with $N$ a positive integer not divisible by $p$ such that $\mc U^p(N) \subset \mc H$ (see \cite[Remark~1.2.1.9]{lan} for the definition of $\mc U^p(N)$).
\begin{rmk} \label{rmk: unr case}
We have imposed the condition that the multichain $\mc L_p$ comes from a single lattice $\Lambda_p$ in such a way that, if $B$ is unramified at $p$, we are in the situation of \cite{kottwitz} and \cite{lan}. In this case $G$ is unramified over $\Q_p$ and $K_p$ is an hyperspecial subgroup.
\end{rmk}
We fix once and for all embeddings $\overline \Q \hookrightarrow \C$ and $i_p \colon\overline \Q \hookrightarrow \overline \Q_p$. We denote with $\mc P$ the corresponding prime ideal of $\mc O_E$ above $p$ and we write $E_{\mc P}$ for the $\mc P$-adic completion of $E$. We are interested in the functor
\[
Y \colon \mbox{ locally noetherian } \mc O_{E_{\mc P}}-\mbox{schemes} \to \mathbf{set}
\]
that to $S$ associates the isomorphism classes of the following data:
\begin{enumerate}
\item an abelian scheme $A/S$;
\item a polarization $\lambda \colon A \to A^\vee$ of degree prime to $p$;
\item an action of $\mc O_B$ on $A/S$ as in \cite[Definition~1.3.3.1]{lan};
\item a $\mc H$-level structure in the sense of \cite[Definition~1.3.7.6]{lan}.
\end{enumerate}
We furthermore require the usual determinant condition of Kottwitz, see \cite[Definition~1.3.4.1]{lan}.
\begin{rmk} \label{rmk: is clas iso class}
We have defined $Y$ using isomorphism classes of abelian schemes, rather than isogeny classes as is done in \cite{rapoport_zink} and \cite{kottwitz}. By \cite[Proposition~1.4.3.4]{lan}, these two approaches are equivalent.
\end{rmk}
\begin{teono}[{\cite[\S~5]{kottwitz}, \cite[\S~6.9]{rapoport_zink}, and \cite[Theorem~1.4.1.11 and Corollary~1.4.1.12]{lan}}] The functor $Y$ is representable by a quasi-projective scheme over $\Sp(\mc O_{E_{\mc P}})$, denoted again by $Y$. If $B$ is unramified at $p$, then $Y$ is smooth over $\Sp(\mc O_{E_{\mc P}})$.
\end{teono}
\begin{ass} \label{ass: ord}
We assume that the ordinary locus of the reduction modulo $\mc P$ of $Y$ is Zariski dense.
\end{ass}
\begin{rmk} \label{rmk: ord}
If $B$ is unramified at $p$, by \cite[1.6.3]{ord_pel}, the above assumption is equivalent to the fact that $E_{\mc P}$ is isomorphic to $\Q_p$ and it is automatically satisfied in case (C).
\end{rmk}
Let $\tilde K$ be a number field such that the decomposition in \eqref{eq: dec C} is defined over $\tilde K$ and let $K$ be the completion of $\tilde K$ at the prime ideal above $p$ given by our fixed embedding $i_p \colon \bar \Q \hookrightarrow \bar \Q_p$. It is a finite extension of $\Q_p$ and we choose a uniformizer element $\varpi$. We freely enlarge $K$ without any comment. We have decompositions of $\mc O_B$-modules
\[
V \otimes_{\Q} K \cong V_1 \oplus V_2 \mbox{ and } \Lambda \otimes_{\Q_p} K \cong \Lambda_1 \oplus \Lambda_2,
\]
where $\Lambda_i$ is a $\mc O_K$-lattice in $V_i$. We base change $Y$ to $\mc O_K$, using the same notation. As shown in \cite{pappas_non_flat}, $Y$ can be not flat over $\mc O_K$. We are interested in admissible formal schemes that are integrally closed in its generic fiber. Starting with $Y$, we perform the following steps to obtain such a formal scheme:
\begin{itemize}
\item let $\widetilde Y$ be the flat closure of $Y$ in $Y_K$;
\item let $\widetilde{\mathfrak Y}$ be the $\varpi$-adic completion of $\widetilde Y$. It is an admissible formal scheme over $\Spf(\mc O_K)$;
\item let $\mathfrak Y$ be the normalization of $\widetilde{\mathfrak Y}$ in its generic fiber.
\end{itemize}
In this way $\mathfrak Y$ is an admissible formal scheme and we have its generic fiber $\mathfrak Y^{\rig}$. We follow the notation introduced in \cite[\S~4.1]{AIP}, in particular $\Nadm$ is the category of admissible $\mc O_K$-algebras $R$ that are integrally closed in $R[1/p]$. We will freely use the fact that all our formal schemes have a nice moduli interpretation when restricted to objects of $\Nadm$ (see \cite[Proposition~5.2.1.1]{AIP}). In particular, if the canonical subgroup exists over $R[1/p]$ (see below), it automatically extends to $R$, see \cite[Proposition~4.1.3]{AIP}.
\subsection{The Hasse invariant and the canonical subgroup} \label{subsec: can sub} Let $(p) = \prod_{i=1}^k (\varpi_i)^{e_i}$ be the decomposition of $(p)$ in $\mc O_{F_0}$ and let $\mc O_i$ be the completion of $\mc O_{F_0}$ with respect to $(\varpi_i)$ (here $\varpi_i$ is a fixed uniformizer of $\mc O_i$). We have a decomposition $\Hom(F_0,\C_p) = \coprod_{i=1}^k D_i$, where $D_i$ is the set of the embeddings $F_0 \hookrightarrow \C_p$ coming from $(\varpi_i)$. We have $\mc O_{F_0,p} \cong \prod_{i=1}^k \mc O_i$. We set $d_i \colonequals [F_i:\Q_p]$, where $F_i\colonequals \Fr(\mc O_i)$, so we have $|D_i|=d_i$. We write $d_i=e_if_i$. From now on, we assume that $K$ is big enough to contain the image of all embeddings $F \hookrightarrow \C_p$. In this section $A$ will be an abelian scheme given by the moduli problem associated to $Y$. We assume that $A$ is defined over a finite extension of $\mc O_K$, so it comes from a rigid point of $\mathfrak Y^{\rig}$.
\subsection*{Case (A)}
Let $B$ be of type (A). For any $i=1,\ldots,d$, we have the $B \otimes_{F_0,\tau_i} \mathds{R} \cong \mat_n(\C)$-module $V \otimes_{F_0,\tau_i} \mathds{R}$. We can write $V \otimes_{F_0,\tau_i} \C \cong \C^n \otimes_{\C} W_i$ for an essentially unique $\C$-vector space $W_i$. Moreover, $W_i$ naturally inherits an hermitian form from $V$. We write $(a_i^+,a_i^-)$ for its signature. We have $a_i^+ + a_i^-=\frac{\dim_{\Q}(V)}{2nd}$ for all $i$.
\begin{ass} \label{ass: ord type A}
We assume that each $(\varpi_i)$ splits completely in $\mc O_F$, and we write $(\varpi^\pm_i)$ for the prime ideals of $\mc O_F$ above $(\varpi_i)$. Moreover, if $i_1,i_2 = 1,\ldots,d$ are such that $i_p \circ \sigma_{i_1}$ and $i_p \circ \sigma_{i_2}$ define the same $p$-adic valuation, we assume $a_{i_1}^+=a_{i_2}^+$.
\end{ass}
\begin{rmk}
If $p$ is unramified in $\mc O_B$ and each $(\varpi_i)$ splits in $\mc O_F$, then the above condition on the signature is equivalent to Assumption~\ref{ass: ord}.
\end{rmk}
\begin{rmk} \label{rmk: split not nec}
The assumption that each $(\varpi_i)$ splits in $\mc O_F$ is not necessary. If $(\varpi_i)$ is inert the theory is similar to case (C). We leave the details to the interested reader.
\end{rmk}
Using the obvious notation, we can rewrite the decomposition in \eqref{eq: dec int} as
\begin{gather} \label{eq: dec case A}
\mc O_{B,p} = \prod_{i=1}^k \left( \mat_n(\mc O_i^+) \oplus \mat_n(\mc O_i^-) \right ).
\end{gather}
We can assume that the left ideal of $\mc O_{B,p}$ generated by $\varpi_i^+$ corresponds to the left ideal generated by the $2k$ matrices $M_j^\pm$, where $M_i^+ = \diag(\varpi_i^+,\ldots,\varpi_i^+)$ and $M_j^\pm = 1$ otherwise. We have a decomposition
\begin{gather*}
A[p^\infty] = \prod_{i=1}^k \left ( A[((\varpi_i^+)^{e_i})^\infty] \oplus A[((\varpi_i^-)^{e_i})^\infty] \right ),
\end{gather*}
where $A[((\varpi_i^-)^{e_i})^\infty]$ is canonically identified with the Cartier dual of $A[((\varpi_i^+)^{e_i})^\infty]$. Using the canonical isomorphisms $\mc O_i \cong \mc O_i^+ \cong \mc O_i^-$, we will consider only $\mc O_i$.
\subsection*{Case (C)} Let $B$ be of type (C). Similarly to case (A), we can write the $B \otimes_{F,\tau_i} \mathds{R} \cong \mat_n(\mathds{R})$-module $V \otimes_{F,\tau_i} \mathds{R}$ as $\mathds{R}^n \otimes_{\mathds{R}} W_i$ for an essentially unique $\mathds{R}$-vector space $W_i$. We have $a_i \colonequals \dim_{\mathds{R}} W_i = \frac{\dim_{\Q}(V)}{2nd}$. We can rewrite the decomposition in \eqref{eq: dec int} as
\begin{gather} \label{eq: dec case C}
\mc O_{B,p} = \prod_{i=1}^k \mat_n(\mc O_i).
\end{gather}
We can assume that, under the isomorphism \eqref{eq: dec case C}, the left ideal of $\mc O_{B,p}$ generated by $\varpi_i$ corresponds to the left ideal generated by the $k$ matrices $M_j$, where $M_i = \diag(\varpi_i,\ldots,\varpi_i)$ and $M_j = 1$ otherwise. We have a decomposition
\begin{gather*}
A[p^\infty] = \prod_{i=1}^k A[(\varpi_i^{e_i})^\infty],
\end{gather*}
where $A[(\varpi_i^{e_i})^\infty]$ is endowed with a principal $\mat_n(\mc O_i)$-linear polarization.
If $G$ is a Barsotti-Tate group defined over a finite extension of $\Z_p$, we write, as in \cite{AIP}, $\Hdg(G) \in [0,1]$ for the \emph{truncated} valuation of any lift of the Hasse invariant of the special fiber of $G$ (note that $\Hdg(G)$ is denoted $\Ha(G)$ in \cite{fargues_can}). We have a function
\begin{gather*}
\Hdg = (\Hdg_i)_i \colon \mathfrak Y^{\rig} \to [0,1]^k \\
A \mapsto (\Hdg(G_i^\pm))_i
\end{gather*}
where $G_i^\pm \colonequals e_{1,1} \cdot A[((\varpi_i^\pm)^{e_i})^\infty]$ (in case (A), since $G_i^-$ is the Cartier dual of $G_i^+$, we have $\Hdg(G_i^+) = \Hdg(G_i^-)$, so there is no ambiguity in the notation $\Hdg(G_i^\pm)$). If $\underline v = (v_i)_i \in [0,1]^k$ we set
\[
\mathfrak Y(\underline v)^{\rig} \colonequals \set{x \in \mathfrak Y^{\rig} \mbox{ such that } \Hdg(x)_i \leq v_i \mbox{ for all } i} .
\]
The ordinary locus of $\mathfrak Y^{\rig}$ is $\mathfrak Y(0)^{\rig}$, it coincides with the tube of the ordinary locus of the special fiber of $\mathfrak Y$. It is not empty by Assumption~\ref{ass: ord}. If $\underline v \in \Q^k \cap [0,1]^k$, we have that $\mathfrak Y(\underline v)^{\rig}$ is a quasi-compact strict neighbourhood of $\mathfrak Y(0)^{\rig}$.
We are going to define, for all $\underline v \in [0,1]^k$, a canonical formal model $\mathfrak Y(\underline v)$ of $\mathfrak Y(\underline v)^{\rig}$, following the approach of \cite[Definition~III.2.11]{peter_tors}. Let $\underline \omega_i^\pm$ be conormal sheaf of $G_i^\pm$ (below $\underline \omega_i^\pm$ will have a slightly different meaning, but no confusion should arise). The Hasse invariant defines a section, denoted $\Ha_i^\pm$, of $\det(\underline \omega_i^\pm)^{\otimes p-1}$ on the reduction modulo $p$ of $\mathfrak Y$. For all $i$, there is a canonical isomorphism $\underline \omega_i^+ \cong \underline \omega_i^-$, and the two Hasse invariants $\Ha_i^+$ and $\Ha_i^-$ are identified under the corresponding isomorphism. For this reason, we will simply write $\underline \omega_i$ and $\Ha_i$.
\begin{defi} \label{defi: form model}
Let $\underline v = (v_i)_{i=1}^k$ and assume that, for all $i$, there is in $\mc O_K$ an element, denoted $p^{v_i}$, of valuation $v_i$. For all $j=1\ldots, k$, we define $\tilde{\mathfrak Y}(v_1,\ldots, v_j)$ by recursion as the functor sending any $p$-adically complete flat $\mc O_K$-algebra $S$ to the set of equivalence classes of pairs $(f,u)$, where:
\begin{itemize}
\item $f \colon \Spf(S) \to \tilde{\mathfrak Y}(v_1,\ldots, v_{j-1})$ (if $j=1$, we set $\tilde{\mathfrak Y}(v_1,\ldots, v_{j-1}) \colonequals \mathfrak Y$);
\item $u \in \Homol^0(\Spf(S), \det(\underline\omega_j)^{\otimes p-1})$ is a section such that, in $S/p$, we have the equality
\[
u \Ha_j(\bar f) = p^{v_j} \in S/p,
\]
where $\bar f$ is the reduction of $f$ modulo $p$ (to be precise we should first of all consider the pullback of $\underline \omega_i$ and $\Ha_i$ via the morphism $\tilde{\mathfrak Y}(v_1,\ldots, v_{j-1}) \to \mathfrak Y$).
\end{itemize}
Two pairs $(f,u)$ and $f',u'$ are equivalent if $f=f'$ and there is some $h \in S$ such that $u' = u(1+p^{1-v_j}h)$.
\end{defi}
By \cite[Lemma~III.2.13]{peter_tors}, we have that $\tilde{\mathfrak Y}(\underline v) \colonequals \tilde{\mathfrak Y}(v_1,\ldots, v_k)$ is representable by a formal scheme, flat over $\mc O_K$. Moreover, one has the usual local description of $\tilde{\mathfrak Y}(\underline v)$. We define $\mathfrak Y(\underline v)$ as the normalization of $\tilde{\mathfrak Y}(\underline v)$ in its generic fiber.
\begin{rmk} \label{rmk: form mod}
The point of the definition of $\mathfrak Y(\underline v)$ is that, using this approach, we do not need to worry about whether the various Hasse invariants lift to characteristic $0$ (and we do not choose any such lift).
\end{rmk}
\begin{notation} \label{not: epsilon}
For any integer $n \geq 0$, we write $\varepsilon_n=\frac{1}{2p^{n-1}}$ if $p\neq 3$ and $\varepsilon_n=\frac{1}{3^n}$ if $p=3$. Unless explicitly stated, in the sequel we will always assume that $v_i < \varepsilon_n$ for all $i$, where $n$ will be clear from the context.
\end{notation}
We fix an integer $n \geq 0$. If $A/R$ is an abelian scheme above $\mathfrak Y(\underline v)$, where $R \in \Nadm$, we have, by \cite[Théorème~6]{fargues_can} and \cite[Proposition~4.1.3]{AIP}, a canonical subgroup of $H_{i,n}^\pm \subset G_i^\pm[p^n]$.
\begin{rmk} \label{rmk: con sub perp}
In case (A) we have $H_{i,n}^- = (H_{i,n}^+)^\perp$, where the orthogonal is taken with respect to the perfect pairing given by duality.
\end{rmk}
Over $K$, we fix $\mc O_i$-linear compatible isomorphisms of étale group schemes
\[
\left (\mc O_i/(\varpi_i^\pm)^{e_in}\right)^{\Dual} \cong \mc O_i/(\varpi_i^\pm)^{e_in},
\]
where $(\cdot)^{\Dual}$ denotes Cartier duality. In particular we will assume that $K$ contains the necessary roots of unity.
\begin{lemma} \label{lemma: can sub free}
We have that $H_{i,n}^\pm$ has rank $p^{na_i^\pm d_i}$ and is stable under $\mc O_i$. Moreover, locally for the étale topology on $R_K$, it is isomorphic to $(\mc O_i/p^n)^{a_i^\pm}$. The same is true for $(H_{i,n}^\pm)^{\Dual}$.
\end{lemma}
\begin{proof}
The statement about the rank follows from \cite[Théorème~6]{fargues_can} and, by \cite[Corollaire~10]{farg_harder}, we have that $H_{i,n}^\pm$ is $\mc O_i$-stable. Again \cite[Théorème~6]{fargues_can} implies that $H_{i,n}^\pm$ is, locally for the étale topology on $R_K$, isomorphic to $(\Z/p^n\Z)^{a_i^\pm d_i}$, so we can show that $H_{i,n}^\pm$ is not killed by $(\varpi_i^\pm)^{e_in-1}$. The dimension of $G_i^\pm$ is $a_i^\pm d_i$, so we have $\deg(G_i^\pm [p^n])=na_i^\pm d_i$ (see \cite[\S~3]{farg_harder} for details about the degree). Moreover, multiplication by $\varpi_i^\pm$ on $G_i^\pm$ is an isogeny, so, for all $s$ we have $\deg(G_i^\pm[(\varpi_i^\pm)^s]) = \val(\det( (\varpi_i^\pm)^{s,\ast})) = s \val(\det((\varpi_i^\pm)^\ast))$, where $(\varpi_i^\pm)^{s,\ast} \colon \underline \omega_{G_i^\pm} \to \underline \omega_{G_i^\pm}$ is the pullback. In particular we have $\deg(G_i^\pm[(\varpi_i^\pm)^{e_in-1}])=(e_in-1)a_i^\pm f_i$. By \cite[Théorème~6]{fargues_can}, we have $\deg(H_{i,n}^\pm)=na_i^\pm d_i-\frac{p^n -1}{p-1}\Ha(G_i^\pm[p^n])$, so we see that $\deg(G_i^\pm[(\varpi_i^\pm)^{e_in-1}]) < \deg(H_{i,n}^\pm)$ and we conclude by \cite[Lemme~4]{farg_harder}.
\end{proof}
\begin{notation}
We consider the algebraic group $\GL^{\mc O}$ over $\Z_p$ defined, in case (A) and (C) respectively, by
\[
\GL^{\mc O} \colonequals \prod_{i=1}^k \Res_{\mc O_i/\Z_p}(\GL_{a_i^+} \times \GL_{a_i^-}) \mbox{ and } \GL^{\mc O} \colonequals \prod_{i=1}^k \Res_{\mc O_i/\Z_p}\GL_{a_i}.
\]
We also have the subgroup $\T^{\mc O}$ defined by
\[
\T^{\mc O} \colonequals \prod_{i=1}^k \Res_{\mc O_i/\Z_p}(\m G^{a_i^+}_{\operatorname{m}} \times \m G^{a_i^-}_{\operatorname{m}}) \mbox{ and } \T^{\mc O} \colonequals \prod_{i=1}^k \Res_{\mc O_i/\Z_p}\m G^{a_i}_{\operatorname{m}}.
\]
Note that, over $K$, we have that $\T^{\mc O}$ is a split torus. We consider the Borel subgroup $\B^{\mc O}$ given by the `upper triangular matrices', with unipotent radical $\U^{\mc O}$.
\end{notation}
We now introduce the Shimura varieties we need. We write $\mathfrak Y(p^n)(\underline v)^{\rig} \to \mathfrak Y(\underline v)^{\rig}$ for the finite étale covering that, in case (A), parametrizes $\mc O_i$-linear trivializations $H_{i,n}^+ \oplus H_{i,n}^- \cong (\mc O_i/(\varpi_i^+)^{e_in})^{a_i^+} \oplus (\mc O_i/(\varpi_i^-)^{e_in})^{a_i^-}$ (note that everything is in characteristic $0$ here); in case (C) it parametrizes $\mc O_i$-linear trivializations $H_{i,n} \cong (\mc O_i/\varpi_i^{e_in})^{a_i}$. There is an action of $\GL^{\mc O}(\Z_p)$ on $\mathfrak Y(p^n)(\underline v)^{\rig}$. Let $\mathfrak Y_{\Iw}(p^n)(\underline v)^{\rig}$ be the quotient of $\mathfrak Y(p^n)(\underline v)^{\rig}$ with respect to $\B^{\mc O}(\Z_p)$. Finally, we let $\mathfrak Y_{\Iwt}(p^n)(\underline v)^{\rig}$ be the quotient of $\mathfrak Y(p^n)(\underline v)^{\rig}$ with respect to $\U^{\mc O}(\Z_p)$. Taking the normalization of $\mathfrak Y(\underline v)$ we obtain the tower of formal schemes
\[
\mathfrak Y(p^n)(\underline v) \to \mathfrak Y_{\Iwt}(p^n)(\underline v) \to \mathfrak Y_{\Iw}(p^n)(\underline v) \to \mathfrak Y(\underline v).
\]
Any of these formal schemes has a reasonable moduli space interpretation.
\section{\texorpdfstring{The sheaf $\mc F$ and modular sheaves}{The sheaf F and modular sheaves}} \label{sec: F}
\subsection{The weight space} \label{subsec: weight space}
Our weight space is the rigid analytic variety $\mc W$ associated to the completed group algebra $\mc O_K \llbracket \T^{\mc O}(\Z_p) \rrbracket$. It satisfies
\[
\mc W(A)= \Hom_{\cont}(\T^{\mc O}(\Z_p), A^\ast)
\]
for any affinoid $K$-algebra $A$.
Accordingly to the decomposition of $\T^{\mc O}$, we have $\mc W = \prod_i (\mc W_i^+ \times \mc W_i^-)$ in case (A) and $\mc W = \prod_i \mc W_i$ in case (C). In particular, we can write $\chi = (\chi_i^\pm)_i$ for all $\chi \in \mc W(\C_p)$. Let $w_i^\pm > 0$ be a rational number such that there is an element $p^{w_i^\pm} \in \mc O_K$ of valuation $w_i^\pm$. We say that $\chi_i^\pm \in \mc W_i^\pm (\C_p)$ is \emph{$w_i^\pm$-locally analytic} if $\chi_i^\pm$ extends to an analytic character
\[
\chi_i^\pm \colon (\mc O_i^\ast(1+p^{w_i^\pm}\mc O_{\C_p}))^{a_i^\pm} \to \C_p^\ast.
\]
If $\underline w = (w_i^\pm)_i$ and $\chi = (\chi_i^\pm)_i \mc W(\C_p)$, we say that $\chi$ is \emph{$\underline w$-locally analytic} is each $\chi_i^\pm$ is $w_i^\pm$-locally analytic. Any $\chi \in \mc W(\C_p)$ is $\underline w$-locally analytic for some $\underline w$. Moreover, let $\mc U \subset \mc W$ be an affinoid associated to a $\C_p$-algebra $A$ and let $\chi_{\mc U}^{\un} = (\chi_{\mc U,i}^{\un,\pm})_i$ be its universal character. Then there is a tuple of positive rational numbers $\underline w = (w_i^\pm)_i$ such each $\chi_{\mc U,i}^{\un,\pm}$ extends to an analytic character
\[
\chi_{\mc U,i}^{\un,\pm} \colon (\mc O_i^\ast(1+p^{w_i^\pm}\mc O_{\C_p}))^{a_i^\pm} \to A^\ast.
\]
We say in this case that $\chi_{\mc U}^{\un}$ is $\underline w$-locally analytic.
Fix an integer $n \geq 1$. We have the subspace $\widetilde{\mc W}_i^\pm(n)$ given by those $\chi_i^\pm \in \mc W_i^\pm(\C_p)$ that satisfy $\chi_i^\pm(1+p^n \mc O_i) \subset 1+p\mc O_{\C_p}$. We define $\mc W_i^\pm(n)$ as the subspace of $\widetilde{\mc W}_i^\pm(n)$ given by the characters $\chi_i^\pm$ such that their restriction to $1+p^n\mc O_i$ is obtained from a $\Z_p$-linear morphism $p^n \mc O_i \to p\mc O_{\C_p}$ taking composition with the $p$-adic logarithm and with the $p$-adic exponential. If $w_i^\pm \geq 1$ is a rational number, we set $\mc W_i^\pm(w_i^\pm) \colonequals \mc W_i^\pm([w_i^\pm])$, where $[w_i^\pm]$ denotes the integer part of $w_i^\pm$.
Let $\underline w = (w_i^\pm)_i$ be a tuple of rational numbers. We set $\mc W(\underline w) \colonequals \prod_i (\mc W_i^+(w_i^+) \times \mc W_i^-(w_i^-))$ or $\mc W(\underline w) \colonequals \prod_i \mc W_i(w_i)$. By construction we have the following
\begin{prop} \label{prop: cov we sp}
Each $\mc W(\underline w)$ is affinoid and $\set{\mc W(\underline w)}_{\underline w}$ is an admissible covering of $\mc W$. Moreover, if $\chi \in \mc W(\underline w)(K)$, then $\chi$ is $\underline w$-analytic.
\end{prop}
If $\chi \in X^\ast(\T^{\mc O})$ is a character of $\T^{\mc O}$, we define $\chi' \colonequals -w_0 \chi$, where $w_0$ is the longest element of the Weyl group of $\GL^{\mc O}$ with respect to $\T^{\mc O}$. For any $\underline w$, the map $\chi \mapsto \chi'$ extends to an involution of $\mc W(\underline w)$, denoted in the same way.
\subsection{\texorpdfstring{The sheaf $\mc F$}{The sheaf F}} \label{subsec: sheaf F} Let $R$ be in $\Nadm$. Suppose we are given a morphism $\Spf(R) \to \mathfrak Y(p^n)(\underline v)$, so we have an abelian scheme $A \to \Sp(R)$. We write $e \colon \Sp(R) \to A$ for the zero section. We consider the sheaf $e^\ast \Omega^1_{A / \Sp(R)}$. It is a $\mc O_B \otimes_{\Z} R$-module and a locally free sheaf of $\mc O_{\Sp(R)}$-modules of rank $\dim_{\Q}(V)/2$. By Morita's equivalence, $e^\ast \Omega^1_{A / \Sp(R)}$ corresponds to a sheaf $\underline \omega$ and we can write, in case (A) and (C) respectively,
\[
\underline \omega = \bigoplus_{i=1}^k (\underline \omega_i^+ \oplus \underline \omega_i^-) \mbox{ and } \underline \omega = \bigoplus_{i=1}^k \underline \omega_i.
\]
Let $i$ and $n$ be fixed. The map (see \cite[\S~4.2]{AIP})
\[
\HT_{(H_{i,n}^\pm)^{\Dual}} \colon (H_{i,n}^\pm)^{\Dual}(R_K) \to \underline \omega_{H_{i,n}^\pm}
\]
respects the action of $\mc O_i$ by functoriality. We define $\mc F_i^\pm(R)$ as the sub $\mc O_i \otimes_{\Z_p} R$-module of $\underline \omega_i^\pm$ generated by the inverse image of $\HT_{(H_{i,n}^\pm)^{\Dual}}(R_K)$ under the natural map, given by pullback, $\underline \omega_i^\pm \to \underline \omega_{H_{i,n}^\pm}$. We have that $\mc F_i^\pm$ does not depend on $n$.
\begin{prop} \label{prop: F loc free}
The sheaf $\mc F_i^\pm \subset \underline \omega_i^\pm$ is a locally free sheaf of $\mc O_i \otimes_{\Z} \mc O_{\Spf(R)}$-modules that contains $p^{\frac{v_i}{p-1}}\underline \omega_i^\pm$. If $w_i^\pm \in ]0,n-v_i\frac{p^n}{p-1}]$ then we have a natural map $\HT_{i,w_i^\pm}^\pm \colon (H_{i,n}^\pm)^{\Dual} (R_K) \to \mc F_i^\pm(R) \otimes_R R_{w_i^\pm}$ such that the induced map
\begin{gather} \label{eq: iso can sub}
(H_{i,n}^\pm)^{\Dual}(R_K) \otimes_{\Z_p} R \to \mc F_i^\pm(R) \otimes_R R_{w_i^\pm}
\end{gather}
is an isomorphism of $\mc O_i \otimes_{\Z_p} R$-modules.
\end{prop}
\begin{proof}
Taking into account Lemma~\ref{lemma: can sub free}, the proof is similar to the one of \cite[Proposition~4.3.1]{AIP}.
\end{proof}
We define
\[
\mc F \colonequals \bigoplus_{i=1}^k (\mc F_i^+ \oplus \mc F_i^-) \mbox{ or } \mc F \colonequals \bigoplus_{i=1}^k \mc F_i,
\]
by Morita's equivalence and Proposition~\ref{prop: F loc free}, it corresponds to a locally free sheaf of $\mc O_B \otimes_{\Z} \mc O_{\Sp(R)}$-modules contained in $e^\ast \Omega^1_{A/\Sp(R)}$. Moreover, this inclusion becomes an isomorphism if we invert $p$.
\subsection{Modular sheaves} \label{subsec: mod sheaves}
Let $n\geq 1$ be an integer and let $w_i^\pm \in ]0, n-v_i\frac{p^n}{p-1}]$ be a rational number. We begin with case (A). For each $i$, there are formal schemes $\Iwtform_{i,w_i^\pm}^\pm \to \mathfrak Y(p^n)(\underline v)$ defined as follows. Let $R$ be in $\Nadm$ and suppose that $\mc F_i^\pm(R)$ is free. The $R$-points of $\Iwtform_{i,w_i^\pm}^\pm$ correspond naturally to the following data:
\begin{itemize}
\item an $R$-point of $\mathfrak Y(p^n)(\underline v)$,
\item a filtration
\[
\Fil_\bullet \mc F_i^\pm(R)=(0=\Fil_0 \mc F_i^\pm(R) \subset \cdots \subset \Fil_{a_i^\pm}\mc F_i^\pm(R) = \mc F_i^\pm(R));
\]
\item trivializations
\[
\Gr_j \Fil_\bullet \mc F_i^\pm(R) \colonequals \Fil_j \mc F_i^\pm(R)/\Fil_{j-1} \mc F_i^\pm(R) \cong (\mc O_i \otimes_{\Z} R)^j
\]
\end{itemize}
such that the following conditions hold:
\begin{itemize}
\item $\Fil_j \mc F_i^\pm(R)$ is a free $\mc O_i \otimes_{\Z} R$-module for each $0 \leq j \leq a_i^\pm$;
\item $\Fil_\bullet \mc F_i^\pm(R)$ corresponds, modulo $p^{w_i^\pm} R$ and via the isomorphism in \eqref{eq: iso can sub}, to the filtration on $(H_{i,n}^\pm)^{\Dual}(R_K) \otimes_{\Z} R$ given by the trivialization of $(H_{i,n}^\pm)^{\Dual}$;
\item the trivializations of $\Gr_j \Fil_\bullet \mc F_i^\pm(R)$ are compatible, modulo $p^{w_i^\pm} R$ and via the isomorphism in \eqref{eq: iso can sub}, to the the trivializations of $(H_{i,n}^\pm)^{\Dual}$.
\end{itemize}
We set $\Iwtform_{\underline w} \colonequals \prod_{i=1}^k (\Iwtform_{i,w_i^+}^+ \times \Iwtform_{i,w_i^-}^-)$. We leave to the reader the definition of $\Iwtform_{\underline w} \to \mathfrak Y(p^n)(\underline v)$ in case (C). Let $v_i'\leq v_i$ for all $i$. We can repeat the above definition to obtain a formal scheme $\Iwtform'_{\underline w} \to \mathfrak Y(p^n)(\underline v')$. The restriction of $\Iwtform_{\underline w}$ to $\mathfrak Y(p^n)(\underline v')$ is naturally isomorphic to $\Iwtform_{\underline w}'$, so we can safely omit $\underline v$ from the notation $\Iwtform_{\underline w}$.
With the obvious notation, we define a formal group $\mathfrak T_{\underline w}^{\mc O}$ by
\[
\mathfrak T_{\underline w}^{\mc O}(R) \colonequals \ker(\T^{\mc O}(R) \to \T^{\mc O}(R_{\underline w})),
\]
and we make a similar definition for $\mathfrak B_{\underline w}^{\mc O}$ and $\mathfrak U_{\underline w}^{\mc O}$. We write $\T_{\underline w}^{\mc O}$, $\B_{\underline w}^{\mc O}$, and $\U_{\underline w}^{\mc O}$ for the corresponding rigid fibers. We have a natural action of $\B^{\mc O}(\Z_p) \mathfrak B_{\underline w}^{\mc O}$ on $\Iwtform_{\underline w}$ over $\mathfrak Y_{\Iw}(p^n)(\underline v)$.
Let $\chi \in \mc W(\underline w)(K)$ be a character. We set $\chi'(\U^{\mc O}(\Z_p) \mathfrak U_{\underline w}^{\mc O}(\mc O_{\C_p})) = 1$. Since $\chi$ is $\underline w$-locally analytic by Proposition~\ref{prop: cov we sp}, we can extend $\chi'$ to an analytic character
\[
\chi' \colon \B^{\mc O}(\Z_p) \mathfrak B_{\underline w}^{\mc O}(\mc O_{\C_p}) \to \C_p^\ast.
\]
We consider the morphism, obtained by composition,
\[
\pi \colon \Iwtform_{\underline w} \to \mathfrak Y_{\Iw}(p)(\underline v).
\]
\begin{defi} \label{defi: sheaf form}
We define the sheaf
\[
\underline{\mathfrak w}^{\dagger \chi}_{\underline v,\underline w} \colonequals \pi_\ast \mc O_{\Iwtform_{\underline w}}[\chi'],
\]
where $[\chi']$ means that we consider the subspace of homogeneous sections of degree $\chi'$ for the action of $\B^{\mc O}(\Z_p)\mathfrak B_{\underline w}$. We call $\underline{\mathfrak w}^{\dagger \chi}_{\underline v,\underline w}$ the $\underline v$-overconvergent, $\underline w$-analytic integral modular sheaf of weight $\chi$.
\end{defi}
\begin{prop} \label{prop: ban sheaf}
We have that $\underline{\mathfrak w}^{\dagger \chi}_{\underline v,\underline w}$ is a formal Banach sheaf (see the appendix of \cite{AIP}).
\end{prop}
\begin{proof}
This is proved in exactly the same way as \cite[Proposition~5.2.2.2]{AIP}.
\end{proof}
The rigid fiber of $\underline{\mathfrak w}^{\dagger \chi}_{\underline v,\underline w}$ is denoted $\underline \omega_{\underline v,\underline w}^{\dagger\chi}$. By definition it is the $\underline v$-overconvergent, $\underline w$-analytic modular sheaf of weight $\chi$.
We can define $\underline \omega_{\underline v,\underline w}^{\dagger\chi}$ directly as follows. Since $w_i^\pm < n$ for all $i$, the natural action of $\U^{\mc O}(\Z_p)$ on $\mathfrak Y(p^n)(\underline v)^{\rig}$ induces an action of $\U^{\mc O}(\Z_p)$ on $\Iwtform_{\underline w}^{\rig}$. Taking the quotient, we obtain a rigid space $\Iwtform_{\underline w}^{\rig,\Diamond} \to \mathfrak Y_{\Iwt}(p^n)(\underline v)^{\rig}$. We have an action of $\T^{\mc O}(\Z_p)\T_{\underline w}^{\mc O}$ on $\Iwtform_{\underline w}^{\rig,\Diamond}$ over $\mathfrak Y_{\Iw}(p)(\underline v)^{\rig}$ and there is an equality
\[
\underline \omega_{\underline v,\underline w}^{\dagger\chi} = \pi_\ast^{\Diamond} \mc O_{\Iwtform_{\underline w}^{\rig,\Diamond}}[\chi'],
\]
where $\pi^{\Diamond} \colon \Iwtform_{\underline w}^{\rig,\Diamond} \to \mathfrak Y_{\Iw}(p)(\underline v)^{\rig}$ is the natural morphism and we take homogeneous sections for the action of $\T^{\mc O}(\Z_p)\T_{\underline w}^{\mc O}$.
\section{Modular forms} \label{sec: mod forms}
Since we do not have a Koecher principle for sections of our modular sheaves, to define modular forms we find it convenient to work with the compactified variety. In particular we need the following
\begin{ass}
From now on we assume that $p$ is unramified in $\mc O_B$, in particular $p$ is a good prime in the sense of \cite[Definition~1.4.1.1]{lan}. We also assume \cite[Condition~1.4.3.10]{lan}, namely that our $\mc O_B$ lattice $(\Lambda, \langle \cdot, \cdot \rangle, h)$ is such that the action of $\mc O_B$ extends to an action of some maximal order $\mc O_B' \supset \mc O_B$. This is not a restriction, see \cite[Remark~1.4.3.9]{lan}.
\end{ass}
We fix once and for all a compatible choice of admissible smooth rational polyhedral cone decomposition data for $Y$, as in \cite[Definition~6.3.3.4]{lan}. Associated to this choice there is an arithmetic toroidal compactification $Y^{\tor}$ of $Y$, see \cite[Theorem~6.4.1.1]{lan} for the main properties of $Y^{\tor}$. Let $\Sp(R_{\alg})$ be part of the data giving a good algebraic model for some representative of a cusp label associated to $Y^{\tor}$, as in \cite[Definition~6.3.2.5]{lan} and let $R \in \Nadm$ be the $p$-adic completion of $R_{\alg}$. We set $S \colonequals \Sp(R)$, so we have a semiabelian scheme $A \to S$. Let $U \subseteq S$ be the open subset corresponding to the unique open stratum of $\Sp(R_{\alg})$, in particular $A$ is abelian over $U$. We also have a Mumford $1$-motive $M$ over $U \hookrightarrow S$ whose semiabelian part will be denoted $\tilde A \to S$. By definition $\tilde A$ is a semiabelian scheme with constant toric rank and we have $\tilde A[p^n] \hookrightarrow A[p^n]$. Here $\tilde A[p^n]$ is finite and flat, while in general $A[p^n]$ is not. As explained in \cite[Section~2.3]{ben_these}, the approximation process needed to construct good formal models can be performed in such a way that there is an isomorphism $M[p^n] \cong A[p^n]$ and we always assume that this is true. There is an action of $\mc O_B$ on $A[p^n]$, $\tilde A[p^n]$, and $M[p^n]$. The two arrows $\tilde A[p^n] \hookrightarrow A[p^n]$ and $M[p^n] \cong A[p^n]$ can be assumed to be compatible with this action.
We can now repeat the definitions of Subsection~\ref{subsec: can sub} replacing $A$ by $\tilde A$, obtaining, for all $\underline v \in [0,1]^k$, the rigid space $\mathfrak Y(\underline v)^{\tor,\rig}$ and its formal model $\mathfrak Y(\underline v)^{\tor}$.
\subsection{Modular forms} \label{subsec: mod forms}
At the end of Section~\ref{sec: PEL data}, we have introduced the rigid Shimura variety $\mathfrak Y(p^n)(\underline v)^{\rig}$ and its formal model $\mathfrak Y(p^n)(\underline v)$. We have a canonical subgroup over $\mathfrak Y(\underline v)^{\tor}$ (see for example \cite[Sections~3.3 and 4.1]{AIP}), so we can define $\mathfrak Y(p^n)(\underline v)^{\tor,\rig}$ and its formal model $\mathfrak Y(p^n)(\underline v)^{\tor}$. Using the semiabelian variety over $\mathfrak Y(p^n)(\underline v)^{\tor}$ we see that the sheaf $\mc F$ extends to $\mathfrak Y(p^n)(\underline v)^{\tor}$. The analogue of Proposition~\ref{prop: F loc free} still holds, so we can define a space
\[
\zeta \colon \Iwtform_{\underline w} \to \mathfrak Y(p^n)(\underline v)^{\tor},
\]
where $\underline w$ is as above. There is an action of $\GL^{\mc O}(\Z_p)$ on $\mathfrak Y(p^n)(\underline v)^{\tor,\rig}$ and repeating the above definitions we obtain the tower
\[
\mathfrak Y(p^n)(\underline v)^{\tor} \to \mathfrak Y_{\Iwt}(p^n)(\underline v)^{\tor} \to \mathfrak Y_{\Iw}(p^n)(\underline v)^{\tor} \to \mathfrak Y(\underline v)^{\tor}.
\]
We are interested in the morphism
\[
\pi \colon \Iwtform_{\underline w} \to \mathfrak Y_{\Iw}(p)(\underline v)^{\tor}.
\]
Repeating the above definitions, if $\chi \in \mc W(\underline w)(K)$ is a character we then have the sheaves
\[
\underline{\mathfrak w}^{\dagger \chi}_{\underline v,\underline w} \colonequals \pi_\ast \mc O_{\Iwtform_{\underline w}}[\chi'] \mbox{ and } \underline{\omega}^{\dagger \chi}_{\underline v,\underline w} \colonequals (\underline{\mathfrak w}^{\dagger \chi}_{\underline v,\underline w})^{\rig}
\]
on $\mathfrak Y_{\Iw}(p)(\underline v)^{\tor}$ and $\mathfrak Y_{\Iw}(p)(\underline v)^{\tor,\rig}$.
Let $Y_K^{\an}$ and $Y_K^{\tor,\an}$ be the analytifications of $Y_K$ and $Y_K^{\tor}$ respectively. Since $Y^{\tor}$ is proper we have a natural isomorphism $\mathfrak{Y}^{\tor,\rig} \cong Y_K^{\tor,\an}$ and in particular there is an open immersion
\[
Y_K^{\an} \hookrightarrow \mathfrak{Y}^{\tor,\rig}.
\]
We have that $Y_K^{\an}$ is dense in $\mathfrak{Y}^{\tor,\rig}$.
\begin{defi} \label{defi: over loc an}
We define the space of $\underline v$-overconvergent, $\underline w$-analytic modular forms of weight $\chi$ by
\[
\M^{\dagger \chi}_{\underline v,\underline w} \colonequals \Homol^0(\mathfrak Y_{\Iw}(p)(\underline v)^{\tor,\rig},\underline \omega_{\underline v,\underline w}^{\dagger\chi}).
\]
We define the space of overconvergent locally analytic modular forms of weight $\chi$ by
\[
\M^{\dagger \chi} \colonequals \lim_{\substack{\underline v \to 0\\ \underline w \to \infty}} \M^{\dagger \chi}_{\underline v,\underline w},
\]
where the limit is over $\underline v$ and $\underline w$ for which there is $n$ such that $\chi \in \mc W(\underline w)(K)$ and the above properties are satisfied.
\end{defi}
\begin{defi} \label{defi: bounded}
Let $\chi \in \mc W(K)$ be any continuous character. If $F$ is a global section of $\underline \omega_{\underline v,\underline w}^{\dagger\chi}$ on $\mathfrak Y_{\Iw}(p)(\underline v)^{\tor,\rig}$ we say that $F$ is bounded if it is bounded as function on $\Iwtform_{\underline w}^{\rig} \times_{\mathfrak Y^{\tor,\rig}} Y_K^{\an}$.
\end{defi}
\begin{prop} \label{prop: mod form well def}
The natural restriction morphism
\[
\M^{\dagger\chi}_{\underline v, \underline w} \to \Homol^0_{\bo}(\mathfrak Y_{\Iw}(p)(\underline v)^{\tor,\rig} \times_{\mathfrak Y^{\tor,\rig}} Y_K^{\an},\underline{\omega}^{\dagger \chi}_{\underline v,\underline w}),
\]
where $\Homol^0_{\bo}(-)$ means that we consider only \emph{bounded} sections, is an isomorphism. In particular our definition of modular forms does not depend on the choice of the toroidal compactification.
\end{prop}
\begin{proof}
The complement of $\Iwtform_{\underline w}^{\rig} \times_{\mathfrak Y^{\tor,\rig}} Y_K^{\an}$ in $\Iwtform_{\underline w}^{\rig}$ is a Zariski closed subset of codimension bigger or equal than $1$. By \cite[Theorem~1.6]{extension} any bounded function $F$ on $\Iwtform_{\underline w}^{\rig} \times_{\mathfrak Y^{\tor,\rig}} Y_K^{\an}$ extends (uniquely) to a function on $\Iwtform_{\underline w}^{\rig}$. This extension has the same weight as $F$ and gives an element of $\M^{\dagger\chi}_{\underline v, \underline w}$ as required.
\end{proof}
\begin{rmk} \label{rmk: no koec}
The reason why we do not need any Koecher principle in the proof of the above proposition is that we have defined overconvergent modular forms as section over a strict neighbourhood of the ordinary locus of $\mathfrak Y_{\Iw}(p)(\overline v)^{\tor,\rig}$, that contains also abelian varieties of bad reduction.
\end{rmk}
\subsection{\texorpdfstring{Classical modular forms}{Classical modular forms}} \label{subsec: class mod form} Fix $n$, $\underline v$ and $\underline w$ as above and assume moreover that $w_i^\pm > \frac{v_i}{p-1}$. We write $Y_{\Iw}(p)$ for the Shimura variety defined with the same PEL data as $Y$, but with Iwahoric level structure at $p$. In case (C), we have that $Y_{\Iw}(p)$ parametrizes couples $(A, (\Fil_\bullet G_i[p])_i )$, where
\begin{itemize}
\item $A$ is an object parametrized by $Y$;
\item $\Fil_\bullet G_i[p] = (0=\Fil_0 G_i[p] \subset \cdots \subset \Fil_{a_i} G_i[p])$ is a filtration of $G_i[p]$ made by $\mc O_i$-stable finite flat subgroups such that $\Fil_j G_i[p]$ has rank $p^{d_ij}$ and $\Fil_{a_i} G_i[p]$ is totally isotropic.
\end{itemize}
In case (A) we consider complete filtrations of $G_i^+[p]$. We consider the sheaves $\underline \omega_i^\pm$ and $\underline \omega$ on $Y_{\Iw}(p)$ defined similarly to the above ones. The unramifiedness assumption implies that $\underline \omega_i^\pm$ is a locally free $\mc O_i \otimes_{\Z_p} \mc O_{Y_{\Iw}(p)}$-module. We obtain the sheaf of trivializations of $\underline \omega$
\[
f \colon \mc T \to Y_{\Iw}(p).
\]
We let $\GL^{\mc O}$ act on $\mc T$ by $g \cdot \omega = \omega \circ g^{-1}$, where $g$ is a section of $\GL^{\mc O}$ and $\omega$ a trivialization of $\underline \omega$. In this way $\mc T$ becomes a $\GL^{\mc O}_{Y_{\Iw}(p)}$-torsor.
We write $\mathfrak Y_{\Iw}(p)$ for the formal completion of $Y_{\Iw}(p)$ along its special fiber. We have open immersions $\mathfrak Y_{\Iw}(p)(\underline v)^{\rig} \hookrightarrow \mathfrak Y_{\Iw}(p)^{\rig} \hookrightarrow Y_{\Iw}(p)_K^{\an}$. By the main result of \cite{ben_these} we have a toroidal compactification $Y_{\Iw}(p)^{\tor}$ of $Y_{\Iw}(p)$ (that we can choose in a way that is compatible with the choice we made for $Y^{\tor}$) and everything we just said extends to $Y_{\Iw}(p)^{\tor}$. Let $\mathfrak Y_{\Iw}(p)^{\tor}$ be the formal completion of $Y_{\Iw}(p)^{\tor}$ along its special fiber. The rigid space $\mathfrak Y_{\Iw}(p)(\underline v)^{\tor,\rig}$ is an open subspace of $\mathfrak Y_{\Iw}(p)^{\tor,\rig}$, but the special fiber of $\mathfrak Y_{\Iw}(p)^{\tor}$ is not the same as the special fiber of the space $\mathfrak Y_{\Iw}(p)(\underline v)^{\tor}$ defined above. In the sequel we will work only with the generic fiber of $\mathfrak Y_{\Iw}(p)^{\tor}$, so this is not a problem.
Let $X^\ast(\T^{\mc O})^+$ be the cone of dominant weights with respect to $\B^{\mc O}$. This cone is stable under $\chi \mapsto \chi'$. Let $\chi \in X^\ast(\T^{\mc O})$ be a weight. Recall that $d_i= [\mc O_i^\pm : \Q_p]$. In case (A) we can identify $\chi$ with a tuple of integers in
\[
\prod_{i=1}^k \prod_{s=1}^{d_i} ( \Z^{a_i^+} \times \Z^{a_i^-}).
\]
We have that $\chi = (k_{i,s,j}^\pm)$ is dominant if and only if, for each $i=i,\ldots,k$ and each $s=1,\ldots,d_i$, we have
\[
k_{i,s,1}^+ \geq k_{i,s,2}^+ \geq \cdots \geq k_{i,s,a_i^+}^+ \mbox{ and } k_{i,s,1}^- \geq k_{i,s,2}^- \geq \cdots \geq k_{i,s,a_i^-}^-.
\]
In case (C) we can identify $\chi$ with a tuple of integers in
\[
\prod_{i=1}^k \prod_{s=1}^{d_i} \Z^{a_i}.
\]
We have that $\chi = (k_{i,s,j})$ is dominant if and only if, for each $i=i,\ldots,k$ and each $s=1,\ldots,d_i$ we have
\[
k_{i,s,1} \geq k_{i,s,2} \geq \cdots \geq k_{i,s,a_i}.
\]
We have that $\mathcal T$ extends to the toroidal compactification and if $\chi$ is a dominant weight then the space of classical modular forms of weight $\chi$ and Iwahoric level is by definition
\[
\M^\chi \colonequals \Homol^0(\mathfrak Y_{\Iw}(p)^{\tor,\rig},\underline \omega^\chi),
\]
where $\underline \omega^\chi$ is the subsheaf of $f_\ast \mc T$ given by homogeneous sections, for the action of $\B^{\mc O}_{Y_{\Iw}(p)}$, of degree $\chi'$. Note that the action of $\GL^{\mc O}$ on $\mc T$ induces an action of $\GL^{\mc O}$ on $\underline \omega^\chi$.
The natural inclusion $\mc F \hookrightarrow \underline \omega_{\mathfrak Y(p^n)(\underline v)}$ is generically an isomorphism and gives an open immersion
\[
\Iwtform_{\underline w}^{\rig} \hookrightarrow \left(\mc T_K^{\an}/\U^{\mc O}_{Y_{\Iw}(p)_K^{\tor,\an}} \right)_{\mathfrak Y(p^n)(\underline v)^{\tor,\rig}}.
\]
Taking the quotient by $\U^{\mc O}(\Z_p)$, we obtain an open immersion
\[
\Iwtform_{\underline w}^{\rig,\Diamond} \hookrightarrow \left(\mc T_K^{\an}/\U^{\mc O}_{Y_{\Iw}(p)_K^{\tor,\an}} \right)_{\mathfrak Y_{\Iwt}(p^n)(\underline v)^{\tor,\rig}}.
\]
\begin{prop} \label{prop: class mod forms}
The composition of the above open immersion with the natural morphism
\[
\left(\mc T_K^{\an}/\U^{\mc O}_{Y_{\Iw}(p)_K^{\tor,\an}} \right)_{\mathfrak Y_{\Iwt}(p^n)(\underline v)^{\tor,\rig}} \to \left(\mc T_K^{\an}/\U^{\mc O}_{Y_{\Iw}(p)_K^{\tor,\an}} \right)_{\mathfrak Y_{\Iw}(p)(\underline v)^{\tor,\rig}}
\]
remains an open immersion. In particular, if $\chi$ is a dominant weight, we have a natural injective morphism
\[
\M^\chi \hookrightarrow \M^{\dagger \chi}_{\underline v,\underline w}.
\]
\end{prop}
\begin{proof}
We prove the proposition in case (A) and we leave case (C) to the reader. We have a decomposition $\Iwtform_{\underline w}^{\rig,\Diamond} = \prod_{i=1}^k \left(\Iwtform_{i,\underline w}^{+,\rig,\Diamond} \times \Iwtform_{i,\underline w}^{-,\rig,\Diamond}\right)$ and a corresponding decomposition of $\mc T_K^{\an}/\U^{\mc O}_{Y_{\Iw}(p)_K^{\tor,\an}}$. It is enough to prove the proposition for the map
\[
\Iwtform_{i,\underline w}^{\pm,\rig,\Diamond} \to \left(\mc T_{i,K}^{\pm,\an}/(\Res_{\mc O_i/\Z_p} \U_{a_i^\pm})_{Y_{\Iw}(p)_K^{\tor,\an}} \right)_{\mathfrak Y_{\Iw}(p)(\underline v)^{\tor,\rig}}.
\]
This is done explicitly as in \cite[Proposition~5.3.1]{AIP}.
\end{proof}
From now on, given $n$ and $\underline v$, we will always assume that any $\underline w \in \Q^k$ satisfies the condition in Subsection~\ref{subsec: mod sheaves}.
\section{Hecke operators} \label{sec: hecke op}
We will write $\mathcal Y(\underline v)$ for $\mathfrak Y(\underline v)^{\tor,\rig} \times_{\mathfrak Y^{\tor,\rig}} Y_K^{\an}$ and similarly for other objects. To define Hecke operators we work over $\mathcal Y_{\Iw}(p)(\underline v)$. This is enough since all the operators we are going to define send bounded functions to bounded functions, see Proposition~\ref{prop: mod form well def}.
\subsection{\texorpdfstring{Hecke operators outside $p$}{Hecke operators outside p}} \label{subsec: hecke op outside}
Recall that $N$ is a fixed positive integer not divisible by $p$ such that $\mc U^p(N) \subset \mc H$, where $\mc H$ is the level of our Shimura variety outside $p$. Let $l$ be a prime that does not divide $Np$. Let $A_1$ and $A_2$ be two abelian schemes given by the moduli problem of $Y_{\Iw}(p)_K$. An isogeny $f \colon A_1 \to f_2$ is an \emph{$l$-isogeny} if the following conditions are satisfied:
\begin{itemize}
\item $f$ is $\mc O$-linear and its degree is a power of $l$;
\item the pullback of the polarization of $A_2$ is a multiple of the polarization of $A_1$;
\item the pullback of the flag of $A_2[p]$ is the flag of $A_1[p]$.
\end{itemize}
Let $f\colon A_1 \to A_2$ be an $l$-isogeny. We choose two symplectic $\mc O_{B,l}$-linear isomorphisms $\T_l(A_i) \cong \Lambda_l$, for $i=1,2$, and an isomorphism $\Z_l(1) \cong \Z_l$. In this way $f$ defines an element $\gamma \in G(\Q_l) \cap \End_{\mc O_{B,l}}(\Lambda_l) \times \Q_l^\ast$. The definition of $\gamma$ depends on the choice of the above isomorphisms, but the double class $G(\Z_l)\gamma G(\Z_l)$ depends only on $f$, and is called the \emph{type} of the $l$-isogeny $f$.
We fix a double class $G(\Z_l)\gamma G(\Z_l)$ as above. Let $C_\gamma \rightrightarrows Y_{\Iw}(p)_K$ be the moduli space that classifies $l$-isogenies $f \colon A_1 \to A_2$ of type $G(\Z_l)\gamma G(\Z_l)$, where $A_1$ and $A_2$ are abelian schemes (with additional structure) classified by $Y_{\Iw}(p)_K$. The arrow $p_j \colon C_\gamma \to Y_{\Iw}(p)_K$ send $f \colon A_1 \to A_2$ to $A_j$. Both $p_1$ and $p_2$ are finite and étale.
We fix $n$ and $\underline w$ as in the previous section. Let $C_\gamma(p^n)$ be the pullback, using $p_1$, of $C_\gamma$ to $Y(p^n)_K$. If $f \colon A_1 \to A_2$ is an isogeny parametrized by $C_\gamma(p^n)$, we can transport the trivializations of the canonical subgroups of $A_1$ to trivializations of the canonical subgroups of $A_2$, via $f$. In particular we have two finite étale morphisms $p_1,p_2 \colon C_\gamma(p^n) \rightrightarrows Y(p^n)_K$. We write $f \colon A \to A'$ for the universal isogeny above $C_\gamma(p^n)$.
Let $\mathcal C_\gamma(p^n)(\underline v)$ be $C_\gamma(p^n)^{\an} \times_{p_1} \mathcal Y(p^n)(\underline v)$. Over $\mathcal C_\gamma(p^n)(\underline v)$, the pullback $f^\ast \colon \underline \omega_{A'} \to \underline \omega_A$ induces a morphisms $f^\ast \colon p_2^\ast \mc F \to p_1^\ast \mc F$. By Proposition~\ref{prop: F loc free}, we have that $f^\ast \colon p_2^\ast \mc F_{|\mathcal Y(p^n)(\underline v)} \stackrel{\sim}{\longrightarrow} p_1^\ast \mc F_{|\mathcal Y(p^n)(\underline v)}$ is an isomorphism. This gives a $\B^{\mc O}(\Z_p)\mathfrak B_{\underline w}^{\mc O}$-equivariant isomorphism
\[
f^\ast \colon p_2^\ast {\Iwtform_{\underline w}}_{|\mathcal Y(p^n)(\underline v)} \stackrel{\sim}{\longrightarrow} p_1^\ast {\Iwtform_{\underline w}}_{|\mathcal Y(p^n)(\underline v)}.
\]
We thus obtain a morphisms
\begin{gather} \label{eq: hecke op outside}
\Homol^0(\mathcal Y(p^n)(\underline v), \mc O_{\Iwtform_{\underline w}}) \stackrel{p_2^{\ast}}{\longrightarrow} \Homol^0(\mathcal C_\gamma(p^n)(\underline v), p_2^\ast \mc O_{\Iwtform_{\underline w}}) \stackrel{(f^\ast)^{-1,\rig}}{\longrightarrow} \\
\stackrel{(f^\ast)^{-1}}{\longrightarrow} \Homol^0(\mathcal C_\gamma(p^n)(\underline v), p_1^\ast\mc O_{\Iwtform_{\underline w}}) \stackrel{\tr p_1^{\rig}}{\longrightarrow} \Homol^0(\mathcal Y(p^n)(\underline v), \mc O_{\Iwtform_{\underline w}}). \nonumber
\end{gather}
\begin{defi} \label{defi: hecke op outside p}
Let $\chi \in \mc W(\underline w)(K)$ be a character. We define the operator $\T_\gamma \colon \M^{\dagger \chi}_{\underline v,\underline w} \to \M^{\dagger \chi}_{\underline v,\underline w}$ from the composition defined in \eqref{eq: hecke op outside} considering bounded and homogeneous sections, for the action of $\B^{\mc O}(\Z_p)\B_{\underline w}^{\mc O}$, of degree $\chi'$. We obtain an operator
\[
\T_\gamma \colon \M^{\dagger \chi} \to \M^{\dagger \chi}.
\]
Since $p$ is unramified in $B$, the operator $\T_\gamma$, for various $\gamma$ and $l$ commute. We let $\mathds T^{Np}$ be the restricted tensor product of the algebras
\[
\Z[G(\Z_l)/ (G(\Q_l) \cap \End_{\mc O_{B,l}}(\Lambda_l) \times \Q_l^\ast )\backslash G(\Z_l)]
\]
for $l$ a prime with $(l,Np)=1$. We have defined an action of $\mathds T^{Np}$ on $\M^{\dagger \chi}_{\underline v, \underline w}$ and on $\M^{\dagger \chi}$.
\end{defi}
\subsection{\texorpdfstring{Hecke operators at $p$}{Hecke operators at p}} \label{subsec: hecke op at}
In this subsection we fix an index $i=1,\ldots,k$. The operators we are going to define will acts as the identity outside the $i$-th component. We assume that $\underline v$ satisfies $v_i < \frac{p-2}{2p-2}$ and that $\underline w$ is as above.
\subsubsection{\texorpdfstring{The operator $\U^\pm_{i,a_i^\pm}$}{The operator \U^\pm_{i,a_i^\pm}}} \label{subsub: Ug}
We start by defining an operator $\U^\pm_{i,a_i^\pm} = \U^+_{i,a_i^+}=\U^-_{i,a_i^-}$ in case (A) and an operator $\U_{i,a_i}$ in case (C) (this notation will be clear later on). In case (A), let $p_1,p_2 \colon \mathcal C(\underline v)^\pm_{i,a_i^\pm} \rightrightarrows \mathcal Y_{\Iw}(p)(\underline v)$ be the moduli space that classifies couples $(A,L_i^\pm)$ where $A$ is an abelian scheme classified by $\mathcal Y_{\Iw}(p)(\underline v)$ and $L_i^\pm \subset G_i^\pm[(\varpi_i^\pm)^{e_i}]$ is a finite and flat subgroup, stable under $\mc O_i$, and such that $G_i^\pm[(\varpi_i^\pm)^{e_i}] = H_{i,1}^\pm \oplus L_i^\pm$ (note that $L_i^+ \mapsto L_i^{+,\perp}$ gives a canonical isomorphism between $C^+_{i,a_i^+}$ and $C^-_{i,a_i^-}$). In case (C) we make a similar definition, adding the condition that $L_i$ is totally isotropic for the polarization of $G_i$. By Lemma~\ref{lemma: can sub free}, any $L_i^\pm$ as above is étale locally isomorphic to $(\mc O_i / (\varpi_i^\pm)^{e_i})^{a_i^\pm}$. The arrow $p_1 \colon C^\pm_{i,a_i^\pm} \to \mathcal Y_{\Iw}(p)(\underline v)$ forgets $L_i^\pm$ and $p_2$ is defined taking the quotient (via Morita's equivalence) by $L_i^+\oplus L_i^-$ or by $L_i$. The map $p_1$ is finite and étale. By \cite[Proposition~16]{fargues_can}, we have that $p_2$ gives a morphism, denoted again $p_2 \colon \mathcal C(\underline v)^\pm_{i,a_i^\pm} \to \mathcal Y_{\Iw}(p)(\underline v')$, where $\underline v' = (v')_i$ is defined by $v'_j=v_j$ if $j \neq i$ and $v_i'=v_i/p$.
We write $\mathcal C(p^n)(\underline v)^\pm_{i,a_i^\pm}$ for the pullback, using $p_1$, of $\mathcal C(\underline v)^\pm_{i,a_i^\pm}$ to $\mathcal Y(p^n)(\underline v)$. We have two natural morphisms $p_1 \colon \mathcal C(p^n)(\underline v)^\pm_{i,a_i^\pm} \to \mathcal Y(p^n)(\underline v)$ and $p_2 \colon \mathcal C(p^n)(\underline v)^\pm_{i,a_i^\pm} \to \mathcal Y(p^n)(\underline v')$. Moreover, over $\mathcal C(p^n)(\underline v)^\pm_{i,a_i^\pm}$, we have an isomorphism $f^\ast \colon p_2^\ast \mc F^{\rig} \stackrel{\sim}{\longrightarrow} p_1^\ast \mc F^{\rig}$ and a $\B^{\mc O}(\Z_p)\mathfrak B_{\underline w}^{\mc O}$-equivariant isomorphism (that is the identity outside the $i$-th component)
\[
f^\ast \colon p_2^\ast {\Iwtform_{\underline w}^{\rig}}_{|\mathcal Y(p^n)(\underline v')} \to p_1^\ast {\Iwtform_{\underline w}^{\rig}}_{|\mathcal Y(p^n)(\underline v)}.
\]
We thus obtain a morphism
\begin{gather*}
\Homol^0(\mathcal Y(p^n)(\underline v'), \mc O_{\Iwtform_{\underline w}}) \stackrel{p_2^{\ast}}{\longrightarrow} \Homol^0(\mathcal C(p^n)(\underline v)^\pm_{i,a_i^\pm}, p_2^\ast \mc O_{\Iwtform_{\underline w}}) \stackrel{(f^\ast)^{-1}}{\longrightarrow} \\
\stackrel{(f^\ast)^{-1}}{\longrightarrow} \Homol^0(\mathcal C(p^n)(\underline v)^\pm_{i,a_i^\pm}, p_1^\ast\mc O_{\Iwtform_{\underline w}}) \stackrel{\tr p_1^{\rig}}{\longrightarrow} \Homol^0(\mathcal Y(p^n)(\underline v), \mc O_{\Iwtform_{\underline w}^{\rig}}).
\end{gather*}
Taking the composition we get, for any $\chi \in \mc W(\underline w)$, an operator
\[
\widetilde{\U}_{i,a_i^\pm}^\pm \colon \M^{\dagger \chi}_{\underline v',\underline w} \to \M^{\dagger \chi}_{\underline v,\underline w}.
\]
We define
\[
\U_{i,a_i^\pm}^\pm \colonequals \left(\frac{1}{p} \right)^{d_ia_i^+a_i^-}\widetilde{\U}_{i,a_i^\pm}^\pm
\]
in case (A) and
\[
\U_{i,a_i} \colonequals \left(\frac{1}{p} \right)^{\frac{d_ia_i(a_i+1)}{2}}\widetilde{\U}_{i,a_i}
\]
in case (C) (these are the normalization factors of \cite{stefan}). We will use the same symbols to denote the composition with $\M^{\dagger \chi}_{\underline v,\underline w} \hookrightarrow \M^{\dagger \chi}_{\underline v',\underline w}$. The operators
\[
\U_{i,a_i^\pm}^\pm \colon \M^{\dagger \chi} \to \M^{\dagger \chi}
\]
are completely continuous.
\subsubsection{\texorpdfstring{The operators $\U^\pm_{i,j}$}{The operators \U^\pm_{i,j}}} \label{subsub: Uj}
Additionally to the various assumptions we made above, we assume in this subsection that $\underline v$ satisfies $v_i < \frac{p-2}{2p^2-p}$. We explain the construction in case (A) and we leave case (C) to the reader. Let $\overline w_i^\pm = (w_i^{\pm,r,s})_{1\leq s \leq r \leq a_i^\pm}$ be a $\frac{a_i^\pm(a_i^\pm + 1)}{2}$-tuple of rational numbers such that $w_i^{\pm,r,s} \in ]\frac{v_i}{p-1}, n-v_i\frac{p^n}{p-1}]$. We moreover assume that $w_i^{\pm,r+1,s} \geq w_i^{\pm,r,s}$ and $w_i^{\pm,r,s+1} \leq w_i^{\pm,r,s}$. We define $\Iwtform_{i,\overline w_i^\pm}^\pm \to \mathfrak Y(p^n)(\underline v)$ as follows. Let $R$ be in $\Nadm$ and suppose that $\mc F_i^\pm(R)$ is free. The $R$-points of $\Iwtform_{i,\overline w_i^\pm}^\pm$ correspond to the following data:
\begin{itemize}
\item an $R$-point of $\mathfrak Y(p^n)(\underline v)$,
\item a filtration
\[
\Fil_\bullet \mc F_i^\pm(R)=(0=\Fil_0 \mc F_i^\pm(R) \subset \cdots \subset \Fil_{a_i^\pm}\mc F_i^\pm(R) = \mc F_i^\pm(R));
\]
\item trivializations
\[
\omega_i^r \colon \Gr_r \Fil_\bullet \mc F_i^\pm(R) \stackrel{\sim}{\longrightarrow} (\mc O_i \otimes_{\Z} R)^r
\]
\end{itemize}
such that the following conditions hold:
\begin{itemize}
\item $\Fil_r \mc F_i^\pm(R)$ is a free $\mc O_i \otimes_{\Z} R$-module for each $0 \leq r \leq a_i^\pm$;
\item let $e_1,\ldots, e_{a_i^\pm}$ be the $R$-points of $(H_{i,n}^\pm)^{\Dual}$ defined by the given isomorphism $(H_{i,n}^\pm)_K^{\Dual} \cong (\mc O_i/p^n)^{a_i^\pm}$ and set $w \colonequals n-v_i\frac{p^n}{p-1}$. We require that the following equality holds, in $\Fil_r \mc F_i^\pm (R)/(\Fil_{r-1} \mc F_i^\pm (R) + p^w)$,
\[
\omega_i^r = \sum_{r \geq s} a_{r,s} \HT_{i,w}^\pm(e_k),
\]
where $a_{r,s} \in R$ are such that $a_{r,s} \in p^{w_i^{\pm,r,s}}R$ if $r > s$ and $(a_{r,r} -1) \in p^{w_i^{\pm,r,r}} R$.
\end{itemize}
If $w_i^{\pm,r,s}=w_i^\pm$ are all equal, the definition of $\Iwtform_{i,\overline w_i^\pm}^\pm$ is exactly the same as the definition of $\Iwtform_{i,w_i^\pm}^\pm$. All the constructions we did for $\Iwtform_{i,w_i^\pm}^\pm$ generalize to $\Iwtform_{i,\overline w_i^\pm}^\pm$ and we use the corresponding notation. For example $\Iwtform_{i,\overline w_i^\pm}^\pm$ extends to $\mathfrak Y(p^n)(\underline v)^{\tor}$ and we have $\Iwtform_{\overline{\underline w}} \to \mathfrak Y(p^n)(\underline v)^{\tor}$ or $\Iwtform_{\overline {\underline w}}^{\rig,\Diamond} \to \mathfrak Y_{\Iw}(p)(\underline v)^{\tor,\rig}$ (here we fix a tuple $\overline w^i$ as above for each $i$). Looking at the proof of Proposition~\ref{prop: class mod forms}, one sees that the condition $w_i^{\pm,r,s} \in ]\frac{v_i}{p-1}, n-v_i\frac{p^n}{p-1}]$ implies that the natural map $\Iwtform_{\overline{ \underline w}}^{\rig} \to \left ( \mc T_K^{\an}/\U^{\mc O}_{Y_{\Iw}(p)_K^{\tor,\an}} \right)_{\mathfrak Y_{\Iw}(p)(\underline v)^{\tor,\rig}}$ is an open immersion. If $\chi \in \mc W(\underline w)$, we can define as above the sheaf $\underline{\mathfrak w}^{\dagger\chi}_{\underline v, \underline{\overline w}}$ and its rigid fiber $\underline \omega^{\dagger\chi}_{\underline v, \underline{\overline w}}$.
We now fix $1 \leq j < a_i^\pm$ and we define the operator $\U^\pm_{i,j}$. Let $p_1,p_2 \colon \mathcal C(\underline v)^\pm_{i,j} \rightrightarrows \mathcal Y_{\Iw}(p)(\underline v)$ be the moduli space that classifies couples $(A,L_i^\pm)$ where $A$ is an abelian scheme classified by $\mathcal Y_{\Iw}(p)(\underline v)$ and $L_i^\pm \subset G_i^\pm[p^2]$ is a finite and flat subgroup, stable under $\mc O_i$, and such that $G_i^\pm[p] = \Fil_j H_{i,1}^\pm \oplus L_i^\pm[p]$. The arrow $p_1 \colon \mathcal C(\underline v)^\pm_{i,j} \to \mathcal Y_{\Iw}(p)(\underline v)$ forgets $L_i^\pm$ and $p_2$ is defined taking the quotient (via Morita's equivalence) by $L_i^+\oplus L_i^-$ (by \cite[Proposition~6.2.2.1]{AIP}, the image of $(A,L_i^\pm)$ by $p_2$ lies in $\mathcal Y_{\Iw}(p)(\underline v)$). Let $f \colon A \to A'$ be the universal isogeny over $\mathcal C(\underline v)^\pm_{i,j}$. It gives a $\B^{\mc O}(\Z_p)\mathfrak B^{\mc O}_{\overline{\underline w}}$ isomorphism
\[
f^\ast \colon p_2^\ast \left(\mc T_K^{\an}/\U^{\mc O}_{Y_{\Iw}(p)_K^{\tor,\an}}\right)_{|\mathcal Y_{\Iw}(p)(\underline v)} \stackrel{\sim}{\longrightarrow} p_1^\ast \left ( \mc T_K^{\an}/\U^{\mc O}_{Y_{\Iw}(p)_K^{\tor,\an}}\right)_{|\mathcal Y_{\Iw}(p)(\underline v)}.
\]
Let $\overline w_i^\pm=(w_i^{\pm,r,s})_{r,s}$ be a tuple as above with the additional condition that $w_i^{\pm,r,k} < n-2 - v_i\frac{p^n}{p-1}$. We definite a tuple ${\overline w_i^\pm}' = ({w_i'}^{\pm,r,s})_{r,s}$ by
\[
{w_i'}^{\pm,r,s} =
\begin{cases}
w_i^{\pm,r,s} + 1 \mbox{ if } r \geq j+1 \mbox{ and } s \leq j\\
w_i^{\pm,r,s} \mbox{ otherwise}
\end{cases}
\]
Starting with $\overline{\underline w}$, we define $\overline{\underline w}'$ modifying only the $i$-th component $\overline w_i^\pm$. It follows that we have the spaces $\Iwtform_{\overline{\underline w}}^{\rig}$ and $\Iwtform_{\overline{\underline w}'}^{\rig}$ and both are open subset of $\mc T_K^{\an}/\U^{\mc O}_{Y_{\Iw}(p)_K^{\tor,\an}}$. The proof of \cite[Proposition~6.2.2.2]{AIP} works also in our case, so we have
\[
(f^{\ast})^{-1} p_1^\ast \left( \Iwtform_{\overline{\underline w}}^{\rig} \right)_{|\mathcal Y_{\Iw}(p)(\underline v)} \subset p_2^\ast \left( \Iwtform_{\overline{\underline w}'}^{\rig}\right)_{|\mathcal Y_{\Iw}(p)(\underline v)}.
\]
For $\chi \in \mc W(\underline w)(K)$, we can now define an operator
\[
\U^\pm_{i,j} \colon \M^{\dagger \chi}_{\underline v, \overline{\underline w}'} \to \M^{\dagger \chi}_{\underline v, \overline{\underline w}}
\]
using the composition
\begin{gather*}
\Homol^0(\mathcal Y_{\Iw}(p)(\underline v), \underline \omega^{\dagger \chi}_{\underline v, \overline{\underline w}'}) \stackrel{p_2^{\ast}}{\longrightarrow} \Homol^0(\mathcal C(\underline v)^\pm_{i,j}, p_2^\ast \underline \omega^{\dagger \chi}_{\underline v, \overline{\underline w}'}) \stackrel{(f^\ast)^{-1}}{\longrightarrow} \\
\stackrel{(f^\ast)^{-1}}{\longrightarrow} \Homol^0(\mathcal C(\underline v)^\pm_{i,j}, p_1^\ast \underline \omega^{\dagger \chi}_{\underline v, \overline{\underline w}}) \stackrel{\tr p_1^{\rig}}{\longrightarrow} \Homol^0(\mathcal Y_{\Iw}(p)(\underline v), \underline \omega^{\dagger \chi}_{\underline v, \overline{\underline w}}).
\end{gather*}
We also have operators
\[
\U^\pm_{i,j} \colon \M^{\dagger \chi}_{\underline v, \overline{\underline w}} \to \M^{\dagger \chi}_{\underline v, \overline{\underline w}} \mbox{ and } \U^\pm_{i,j} \colon \M^{\dagger \chi} \to \M^{\dagger \chi}.
\]
\subsubsection{\texorpdfstring{The $\U$-operator}{The operators \U-operator}} \label{subsub: U} We work in case (A), case (C) is similar. We fix $n$. From now on, we will always assume that the following conditions are satisfied. Let $\underline v$ be such that the above inequalities hold and let $\underline w$ be such that $w_i^\pm \in ]\frac{v_i}{p-1},n-1 - a_i^\pm]$.
Let us fix $i= 1,\dots, k$. We set
\begin{gather*}
v_i' \colonequals v_i/p \mbox{ and } v_j' \colonequals v_j \mbox{ if } j \neq i, \\
w_j^{\pm,r,s} \colonequals w_j^\pm \mbox{ for all } r,s \mbox{ and for all } j,\\
{w_i'}^{\pm,r,s} \colonequals r - s + w_i^\pm \mbox{ and } {w_j'}^{\pm,r,s} \colonequals w_i^\pm \mbox{ if } j \neq i.
\end{gather*}
and for the rest we use the above notations. The product
\[
\U_i \colonequals \U_{i,a_i^\pm}^\pm \times \prod_{j=1}^{a_i^\pm - 1} (\U_{i,j}^+ \times \U_{i,j}^-)
\]
gives an operator $\U_i \colon \M^{\dagger\chi}_{\underline v', \overline {\underline w}'} \to \M^{\dagger\chi}_{\underline v, \overline {\underline w}} = \M^{\dagger\chi}_{\underline v,\underline w}$. We denote with the same symbol the composition of $\U_i$ with the natural restriction $\M^{\dagger\chi}_{\underline v, \underline w} = \M^{\dagger\chi}_{\underline v, \overline {\underline w}} \to \M^{\dagger\chi}_{\underline v', \overline {\underline w}'}$, obtaining
\[
\U_i \colon \M^{\dagger\chi}_{\underline v, \underline w} \to \M^{\dagger\chi}_{\underline v,\underline w}.
\]
Taking the product of the $\U_i$ we obtain the compact operators
\[
\U \colon \M^{\dagger\chi}_{\underline v, \underline w} \to \M^{\dagger\chi}_{\underline v,\underline w}. \mbox{ and } \U \colon \M^{\dagger\chi} \to \M^{\dagger\chi}.
\]
\begin{rmk} \label{rmk: class}
Usually one defines the $\U$-operator using only the operators $\U^\pm_{i,a_i^\pm}$, that improve the degree of overconvergence. The reasons for including also the operators $\U^\pm_{i,j}$ is that they improve analyticity, and this will be needed to prove classicity in Section~\ref{sec: class}.
\end{rmk}
We let $\m U_p$ be the free $\Z$-algebra generated by the Hecke operators at $p$ and let $\m T \colonequals \m T^{Np} \otimes_{\Z} \m U_p$. We simply call $\m T$ the Hecke algebra. It acts on all the spaces we have defined.
\subsection{Families} \label{subsec: families}
Let $\mc U \subset \mc W$ be an affinoid associated to the algebra $A \colonequals \mc O_{\mc U}(\mc U)$. There is $\underline w$ such that $\mc U \subset \mc W(\underline w)$, and we fix one such. We write $\chi^{\un}_{\mc U} \colon \T^{\mc O}(\Z_p) \to A^\ast$ for the universal character over $\mc U$. Let $\underline v$ and $n$ be as usual.
\begin{prop} \label{prop: families}
There is a Banach sheaf $\underline \omega_{\underline v, \underline w}^{\dagger \chi^{\un}_{\mc U}}$ on $\mathfrak Y_{\Iw}(p)(\underline v)^{\tor,\rig} \times \mc U$ such that, for any $\chi \in \mc U(K)$, the fiber of $\underline \omega_{\underline v, \underline w}^{\dagger \chi^{\un}_{\mc U}}$ at $\mathfrak Y_{\Iw}(p)(\underline v)^{\tor,\rig} \times \set{\chi}$ is canonically isomorphic to $\underline \omega_{\underline v, \underline w}^{\dagger \chi}$. On the global sections of $\underline \omega_{\underline v, \underline w}^{\dagger \chi^{\un}_{\mc U}}$, there is an action of the Hecke operators defined above.
\end{prop}
\begin{proof}
We have a morphism $\pi_1 \times \operatorname{id} \colon \Iwtform_{\underline w}^{\Diamond,\rig} \times \mc U \to \mathfrak Y_{\Iw}(p)(\underline v)^{\tor,\rig} \times \mc U$. On
\[
(\pi_1 \times \operatorname{id})_\ast \mc O_{\Iwtform_{\underline w}^{\Diamond,\rig} \times \mc U}
\]
there is an action of $\B^{\mc O}(\Z_p)\B_{\underline w}$, and we define $\underline \omega_{\underline v, \underline w}^{\dagger \chi^{\un}}$ taking sections homogeneous of degree $(\chi^{\un}_{\mc U})'$. The definitions given above of the Hecke operators work in families without problems.
\end{proof}
\begin{defi} \label{defi: fam mod form}
We define
\[
\M^{\dagger\mc U}_{\underline v, \underline w} \colonequals \Homol^0(\mathfrak Y_{\Iw}(p)(\underline v)^{\tor,\rig} \times \mc U,\underline \omega_{\underline v, \underline w}^{\dagger \chi^{\un}_{\mc U}}).
\]
It is the space of families of $\underline v$-overconvergent $\underline w$-locally analytic modular forms parametrized by $\mc U$. We set
\[
\M^{\dagger\mc U} \colonequals \lim_{\substack{\underline v \to 0\\ \underline w \to \infty}} \M^{\dagger\mc U}_{\underline v, \underline w}.
\]
It is the space of overconvergent locally analytic modular forms parametrized by $\mc U$.
\end{defi}
We have an action of the Hecke operators on both $\M^{\dagger\mc U}_{\underline v, \underline w}$ and $\M^{\dagger\mc U}$. The $\U$-operator on $\M^{\dagger\mc U}$ is completely continuous. We have that $\M^{\dagger\mc U}_{\underline v, \underline w}$ is a Banach $A$-module.
\begin{rmk} \label{rmk: spec}
Let $\chi \in \mc U(K)$. Then we have a natural specialization map
\[
\M^{\dagger\mc U}_{\underline v, \underline w} \to \M^{\dagger\chi}_{\underline v, \underline w}.
\]
We do not know whether this map is surjective or not (but see Corollary~\ref{coro: proj} below for the cuspidal case).
\end{rmk}
\begin{rmk} \label{rmk: int fam}
We can define a formal model $\mathfrak W$ and $\mathfrak W(\underline w)$ of $\mc W$ and $\mc W(\underline w)$. If $\mathfrak{U} \subset \mathfrak{W}(\underline w)$ is affine, we have an analogue of Proposition~\ref{prop: families}, obtaining the formal Banach sheaf $\underline{\mathfrak w}_{\underline v, \underline w}^{\dagger \chi^{\un}_{\mathfrak U}}$ on $\mathfrak Y_{\Iw}(p)(\underline v)^{\tor} \times \mathfrak U$.
\end{rmk}
\section{Cuspidal forms and eigenvarieties} \label{sec: heck var}
\subsection{The minimal compactifications}
Recall that in Section~\ref{sec: mod forms} we have fixed a compatible choice of admissible smooth rational polyhedral cone decomposition data for $Y$. This choice gives the toroidal compactification $Y^{\tor}$ of $Y$. We also have the minimal compactification $Y^{\min}$ and a proper morphism
\[
\xi \colon Y^{\tor} \to Y^{\min},
\]
see \cite[Theorem~7.2.4.1]{lan} for the main properties of $Y^{\min}$. The sheaves $\underline \omega_i$ extend to the minimal compactification, and the Hasse invariants give sections $\Ha_i$ of $\det(\underline \omega_i)^{\otimes p-1}$ on the special fiber of $Y^{\min}$. We then obtain a function $\Hdg \colon \mathfrak Y^{\min,\rig} \to [0,1]^k$ and so the rigid space $\mathfrak Y(\underline v)^{\min,\rig}$ and its formal model $\mathfrak Y(\underline v)^{\min}$ are defined.
Let $\Sp(R_{\alg})$ be part of the data giving a good algebraic model as in the beginning of Section~\ref{sec: mod forms}. In particular we have the semiabelian schemes $A \to \Sp(R)$ and $\tilde A \to \Sp(R)$. Since the formal completions of $A$ and $\tilde A$ along the closed stratum of $S = \Sp(R_{\alg})$ are isomorphic, we have an isomorphism of locally free sheaves over $S$
\[
e_A^\ast \Omega^1_{A/S} \cong e_{\tilde A}^\ast \Omega^1_{\tilde A/S},
\]
where $e_A$ and $e_{\tilde A}$ are the corresponding zero sections. Hasse invariants are compatible with respect to the induced isomorphisms, so $\xi$ gives a morphism
\[
\xi(\underline v) \colon \mathfrak Y(\underline v)^{\tor} \to \mathfrak Y(\underline v)^{\min}.
\]
We will write $D$ for both the boundary of $Y^{\tor}$ and the boundary of $\mathfrak Y(\underline v)^{\tor}$. We then have the following
\begin{teo}[{\cite[Theorem~8.2.1.2]{lan_ram}}] \label{teo: ann coho}
We have
\[
\R^q \xi_\ast \mc O_{Y^{\tor}}(-D) = 0
\]
if $q \geq 1$.
\end{teo}
\begin{notation}
Let $\mathfrak m$ be the maximal ideal of $\mc O_K$. If $\star$ is an object defined over $\mc O_K$, we let $\star_n$ be the reduction of $\star$ modulo $\mathfrak m^n$.
\end{notation}
\begin{coro} \label{coro: ann coho v}
We have
\[
\R^q \xi(\underline v)_\ast \mc O_{\mathfrak Y(\underline v)^{\tor}}(-D) = 0
\]
if $q \geq 1$.
\end{coro}
\begin{proof}
Arguing as in \cite[Proposition~8.2.1.2]{AIP} we have that the description of the formal fibers of $\xi(\underline v)$ is the same as the description given in \cite[Section~8.2]{lan_ram}, hence to prove the corollary one can repeat the proof of \cite[Theorem~8.2.1.2]{lan_ram}.
\end{proof}
Let $\eta(\underline v)$ be the composition
\[
\mathfrak Y(p^n)(\underline v)^{\tor} \to \mathfrak Y(\underline v)^{\tor} \to \mathfrak Y(\underline v)^{\min}
\]
and let $\rho(\underline v)$ be the first morphism. We will still write $D$ for its inverse image under $\eta(\underline v)$.
\begin{prop} \label{prop: vani fin}
We have
\[
\R^q \eta(\underline v)_\ast \mc O_{\mathfrak Y(p^n)(\underline v)^{\tor}}(-D) = 0
\]
if $q \geq 1$.
\end{prop}
\begin{proof}
This can be proved by the same arguments used in \cite[Section~8.2]{lan}. To do this, we need to describe the local charts of $\mathfrak{Y}(p^n)(\underline v)^{\tor}$ similarly to the one of $\mathfrak{Y}(\underline v)^{\tor}$. Over the rigid fiber $\mathfrak{Y}(p^n)(\underline v)^{\tor,\rig}$ we know that such a description exists. We can now argue as in \cite[Proposition~8.2.1.3]{AIP} to extend this description to $\mathfrak{Y}(p^n)(\underline v)^{\tor}$.
\end{proof}
\subsection{A dévissage} \label{subsec: dev}
Let $\mc W(\underline w)^\circ$ be the rigid open unit disk of dimension $\dim(\T_{\underline w}^{\mc O})$. As in \cite[Section~2.2]{AIP}, we have an analytic universal character
\[
\chi^{\un,\circ} \colon \mathfrak{T}_{\underline w}^{\mc O}(\Z_p) \to \mc O_{\mc W(\underline w)^\circ}(\mc W(\underline w)^\circ)^\ast.
\]
We set $\mathfrak{W}(\underline w)^\circ \colonequals \Spf(\mc O_K \llbracket X_1,\ldots,X_{\dim(\T_{\underline w}^{\mc O})} \rrbracket )$, a formal model of $\mc W(\underline w)^\circ$. Using the construct of \cite[Section~2.2]{AIP}, we see that the character $\chi^{\un,\circ}$ is induced by a formal universal character, denoted in the same way,
\[
\chi^{\un,\circ} \colon \mathfrak{T}_{\underline w}^{\mc O}(\Z_p) \to \mc O_{\mathfrak{W}(\underline w)^\circ}(\mathfrak{W}(\underline w)^\circ)^\ast.
\]
Recall that we have the morphism $\zeta \colon \Iwtform_{\underline w} \to \mathfrak Y(p^n)(\underline v)^{\tor}$. The torus $\mathfrak{T}_{\underline w}^{\mc O}$ acts on this space, so, if $\chi^\circ \in \mc W(\underline w)^\circ(K)$, we can define the sheaves on $\mathfrak Y(p^n)(\underline v)^{\tor}$
\[
\underline{\mathfrak w}^{\dagger \chi^\circ}_{\underline v,\underline w} \colonequals \zeta_\ast \mc O_{\Iwtform_{\underline w}}[{\chi^\circ}'].
\]
As in Remark~\ref{rmk: int fam}, if $\mathfrak U^\circ \subset \mathfrak W(\underline w)^\circ$ is affine, with associated character $\chi_{\mathfrak U^\circ}^{\un}$, we have the family $\underline{\mathfrak w}^{\dagger \chi_{\mathfrak U^\circ}^{\un}}_{\underline v,\underline w}$. It is a sheaf on $\mathfrak Y(p^n)(\underline v)^{\tor} \times \mathfrak{U}$.
\begin{rmk} \label{rmk: dev is dev}
If $\chi \in \mc W(\underline w)(K)$, then we have its image $\chi^\circ \in \mc W(\underline w)^\circ(K)$ and we can recover $\underline{\mathfrak w}^{\dagger \chi}_{\underline v,\underline w}$ from $\underline{\mathfrak w}^{\dagger \chi^\circ}_{\underline v,\underline w}$ projecting to $\mathfrak Y_{\Iw}(p)(\underline v)^{\tor}$ and taking homogeneous sections of degree $\chi'$ for the action of $\B^{\mc O}(\Z_p)\mathfrak{B}_{\underline w}^{\mc O}$. Equivalently, one can project $\underline{\mathfrak w}^{\dagger \chi^\circ}_{\underline v,\underline w}$ to $\mathfrak Y_{\Iw}(p)(\underline v)^{\tor}$, twist the action of $\B^{\mc O}(\Z_p)\mathfrak{B}_{\underline w}^{\mc O}$ by $-\chi'$, and finally take invariant sections for the action of $\B^{\mc O}(\Z/p^n\Z)$. A similar remark holds for families.
\end{rmk}
We recall that $\star_n$ means the reduction modulo $\mathfrak{m}^n$ of $\star$, where $\mathfrak{m}$ is the maximal ideal of $\mc O_K$.
\begin{prop} \label{prop: small ban}
Let $\chi^\circ \in \mc W(\underline w)^\circ(K)$ be a character. Then $\underline{\mathfrak w}^{\dagger \chi^\circ}_{\underline v,\underline w, 1}$ is an inductive limit of coherent sheaves that are an extension of the trivial sheaf. In particular, $\underline{\mathfrak w}^{\dagger \chi^\circ}_{\underline v,\underline w}$ is a small formal Banach sheaf (see \cite[Definition~A.1.2.1]{AIP} for the definition of small formal Banach sheaf). If $\mathfrak U^\circ \subset \mathfrak W(\underline w)^\circ$ is affine, a similar result hold for $\underline{\mathfrak w}^{\dagger \chi_{\mathfrak U^\circ}^{\un}}_{\underline v,\underline w}$.
\end{prop}
\begin{proof}
We prove the proposition for $\underline{\mathfrak w}^{\dagger \chi^\circ}_{\underline v,\underline w}$, the proof for $\underline{\mathfrak w}^{\dagger \chi_{\mathfrak U^\circ}^{\un}}_{\underline v,\underline w}$ is similar. Using the decompositions $\Iwtform_{\underline w} = \prod_{i=1}^k (\Iwtform_{i,w_i^+}^+ \times \Iwtform_{i,w_i^-}^-)$ in case (A) and $\Iwtform_{\underline w} = \prod_{i=1}^k \Iwtform_{i,w_i}$ in case (C), we can prove the proposition for the sheaf
\[
\mc G \colonequals \zeta_{i,\ast}^\pm \mc O_{\Iwtform_{i,w_i^\pm}^\pm}[\chi'],
\]
where $\zeta_i^\pm \colon \Iwtform_{i,w_i^\pm}^\pm \to \mathfrak{Y}(p^n)(\underline v)^{\tor}$ is the natural morphism. One can give a completely explicit description of the sections of $\mc G$, as in \cite[Subsections~8.1.5 and 8.1.6]{AIP}. The proof is then identical to the one of \cite[Corollary 8.1.6.2]{AIP}.
\end{proof}
Recall the morphism $\eta(\underline v) \colon \mathfrak Y(p^n)(\underline v)^{\tor} \to \mathfrak Y(\underline v)^{\min}$.
\begin{prop} \label{prop: proj small ban}
For any $\chi^\circ \in \mc W(\underline w)^\circ(K)$ and any affine $\mathfrak U^\circ \subset \mathfrak W(\underline w)^\circ$ we have that
\[
\eta(\underline v)_\ast (\underline{\mathfrak w}^{\dagger \chi^\circ}_{\underline v,\underline w}(-D)) \mbox{ and } (\eta(\underline v \times \operatorname{id}))_\ast (\underline{\mathfrak w}^{\dagger \chi_{\mathfrak U^\circ}^{\un}}_{\underline v,\underline w} (-D))
\]
are small formal Banach sheaves
\end{prop}
\begin{proof}
We prove the statement for $\eta(\underline v)_\ast (\underline{\mathfrak w}^{\dagger \chi^\circ}_{\underline v,\underline w}(-D))$, the case of families is similar. We use the notation of the proof of Proposition~\ref{prop: small ban}. It is enough to prove the proposition for $\eta(\underline v)_\ast (\mc G(-D))$. Let $s \geq 1$ be an integer and consider the commutative diagram
\[
\xymatrix{
\mathfrak{Y}(p^n)(\underline v)^{\tor}_{s-1} \ar[r]^i \ar[d]^{\eta(\underline v)_{s-1}} & \mathfrak{Y}(p^n)(\underline v)^{\tor}_s \ar[d]^{\eta(\underline v)_s} \\
\mathfrak{Y}(\underline v)^{\min}_{s-1} \ar[r]^j & \mathfrak{Y}(\underline v)^{\min}_s
}
\]
where $i$ and $j$ are the natural closed immersions. Using Proposition~\ref{prop: vani fin} and \cite[Proposition~A.1.3.1]{AIP}, we see that it is enough to prove that
\[
j^\ast (\eta(\underline v)_{s,\ast} \mc G(-D)_s) = \eta(\underline v)_{s-1.\ast} \mc G(-D)_{s-1}.
\]
Similarly to $\mc G_1$, we can write $\mc G_s \cong \varinjlim_j \mc G_{s,j}$, where each $\mc G_{s,j}$ is a coherent sheaf. Since taking direct images commutes with direct limits, we can prove the statement for $\mc G_{s,j}$. From now on, all the sheaves will be considered as sheaves over $\mathfrak{Y}(p^n)(\underline v)^{\tor}_s$. It is enough to prove that
\[
\R^1 \eta(\underline v)_{s, \ast} \ker(\mc G_{s,j} \to \mc G_{s-1,j}) = 0.
\]
We have (see \cite[Section~8.1.6]{AIP} and the proof of Proposition~\ref{prop: small ban})
\[
\ker(\mc G_{s,j} \to \mc G_{s-1,j}) \cong \mc G_{1,j},
\]
in particular we need to show that $\R^1 \eta(\underline v)_{s, \ast} \mc G_{1,j} = 0$. By Proposition~\ref{prop: small ban}, we know that $\mc G_{1,j}$ is an extension of the trivial sheaf, so we can conclude by Proposition~\ref{prop: vani fin}.
\end{proof}
\subsection{Projectivity of the space of cuspidal forms} \label{subsec: proj}
We fix $\mathfrak U^\circ \subset \mathfrak W(\underline w)^\circ$ an affine with rigid fiber $\mc U^\circ = \Spm(A^\circ)$ and $\chi^\circ \in \mc U^\circ(K)$. Let us consider the modules $\cusp^{\mc U^\circ}_{\underline v,\underline w}$ and $\cusp^{\chi^\circ}_{\underline v,\underline w}$ defined by
\[
\cusp^{\mc U^\circ}_{\underline v,\underline w} \colonequals \Homol^0(\mathfrak Y(p^n)(\underline v)^{\tor} \times \mathfrak U^\circ, \underline{\mathfrak w}^{\dagger \chi_{\mathfrak U^\circ}^{\un}}_{\underline v,\underline w} (-D))[p^{-1}]
\]
and
\[
\cusp^{\chi^\circ}_{\underline v,\underline w} \colonequals \Homol^0(\mathfrak Y(p^n)(\underline v)^{\tor}, \underline{\mathfrak w}^{\dagger \chi^\circ}_{\underline v,\underline w}(-D))[p^{-1}].
\]
Let $B$ be any affinoid $K$-algebra and let $M$ be a Banach $B$-module. We recall the following definition, due to Buzzard in \cite{buzz_eigen}.
\begin{defi} \label{defi: proj}
We say that $M$ is \emph{projective}, if there is a Banach $B$-module $N$ such that $M \oplus N$ is potentially orthonormizable.
\end{defi}
\begin{prop} \label{prop: proj dev}
The Banach module $\cusp^{\mc U^\circ}_{\underline v,\underline w}$ is a projective $A^\circ$-module. Moreover, the natural specialization morphism
\[
\cusp^{\mc U^\circ}_{\underline v,\underline w} \to \cusp^{\chi^\circ}_{\underline v,\underline w}
\]
is surjective.
\end{prop}
\begin{proof}
Taking the $v_i$'s all equal, we can assume that $\mathfrak{Y}(\underline v)^{\min,\rig}$ is an affinoid. Since $(\eta(\underline v \times \operatorname{id}))_\ast (\underline{\mathfrak w}^{\dagger \chi_{\mathfrak U^\circ}^{\un}}_{\underline v,\underline w} (-D))$ is a small Banach sheaf by Proposition~\ref{prop: proj small ban}, the proposition is proved exactly as \cite[Corollary~8.2.3.2]{AIP}.
\end{proof}
\begin{coro} \label{coro: proj}
Let $\mc U =\Spm(A) \subset \mc W$ be an affinoid, and let $\chi \in \mc U(K)$. The Banach module $\cusp^{\dagger\mc U}_{\underline v, \underline w}$ is a projective $A$-module. Moreover, the natural specialization morphism
\[
\cusp^{\dagger\mc U}_{\underline v, \underline w} \to \cusp^{\dagger\chi}_{\underline v, \underline w}
\]
is surjective.
\end{coro}
\begin{proof}
Let $\Spf(A^\circ)$ be a formal model of $\mc U^\circ$, the image of $\mc U$ in $\mathfrak W(\underline w)^{\rig}$ and let $\cusp^{\mc U^\circ}_{\underline v,\underline w}$ be as above. Let $\chi^{\un}$ and $\chi^{\un,\circ}$ be the characters associated to $\mc U$ and $\mc U^\circ$. By Remark~\ref{rmk: dev is dev}, we have
\[
\cusp^{\dagger\mc U}_{\underline v, \underline w} = (\cusp^{\mc U^\circ}_{\underline v,\underline w} \otimes_{A^\circ} A(-\chi^{\un,\circ '}))^{\B^{\mc O}(\Z/p^n \Z)},
\]
in particular, by Proposition~\ref{prop: proj dev}, we have that $\cusp^{\dagger\mc U}_{\underline v, \underline w}$ is a direct factor of a projective $A$-module, and hence it is projective. An analogous equality holds for $\cusp^{\dagger\chi}_{\underline v, \underline w}$ and the specialization morphism $\cusp^{\mc U^\circ}_{\underline v,\underline w} \to \cusp^{\chi^\circ}_{\underline v,\underline w}$ is surjective by Proposition~\ref{prop: proj small ban}. Since taking invariants with respect to a finite group is an exact functor on the category of $\Q_p$-vector spaces, the morphism $\cusp^{\dagger\mc U}_{\underline v, \underline w} \to \cusp^{\dagger\chi}_{\underline v, \underline w}$ is surjective as required.
\end{proof}
\subsection{The Eigenvariety} \label{subsec: eigen}
Let $\mc Z(\underline v, \underline w)\subseteq \mc W(\underline w) \times \m A^{1,\rig}$ be the spectral variety associated to the $\U$-operator acting on $\cusp^{\dagger \mc W(\underline w)}_{\underline v, \underline w}$.
\begin{teo} \label{teo: eigen}
There is a rigid space $\mc E_{\underline v, \underline w}$ equipped with a finite morphism $\mc E_{\underline v, \underline w} \to \mc Z(\underline v, \underline w)$ that satisfies the following properties.
\begin{enumerate}
\item \label{en: basic propr} It is equidimensional of dimension $\rk(\T^{\mc O})$. The fiber of $\mc E_{\underline v, \underline w}$ above a point $\chi \in \mc W(\underline w)$ parametrizes systems of eigenvalues for the Hecke algebra $\m T$ appearing in $\cusp^{\dagger\chi}_{\underline v,\underline w}$ that are of finite slope for the $\U$-operator. If $x \in \mc E_{\underline v, \underline w}$, then the inverse of the $\U$-eigenvalue corresponding to $x$ is $\pi_2(x)$, where $\pi_2$ is the induced map $\pi_2 \colon \mc E_{\underline v, \underline w} \to \m A^{1,\rig}$. For various $\underline v$ and $\underline w$, these construction are compatible. Letting $\underline v \to 0$ and $\underline w \to \infty$ we obtain the global eigenvariety $\mc E$.
\item \label{en: unr} Let $f \in \cusp^{\dagger\chi}_{\underline v,\underline w}$ be a cuspidal eigenform of finite slope for the $\U$-operator and let $x_f$ be the point of $\mc E_{\underline v, \underline w}$ corresponding to $f$. Let us suppose that $\mc E_{\underline v, \underline w} \to \mc W(\underline w)$ is unramified at $x_f$. Then there exists an affinoid $\mc U \subset \mc W(\underline w)$ that contains $\chi$ and such that $f$ can be deformed to a family of finite slope eigenforms $F \in \cusp^{\dagger\mc U}_{\underline v,\underline w}$.
\end{enumerate}
\end{teo}
\begin{proof}
By Corollary~\ref{coro: proj} all the assumptions of \cite{buzz_eigen} are satisfied, we can then use Buzzard's machinery to construct the eigenvariety, this gives $\mc E_{\underline v, \underline w}$ and \eqref{en: basic propr} follows. Point \eqref{en: unr} is an automatic consequence of the way we used to construct the eigenvariety, see \cite[Proposition~8.1.2.6]{AIP}.
\end{proof}
\begin{coro} \label{coro: fin}
Let $f \in \cusp^{\dagger\chi}_{\underline v,\underline w}$ be a cuspidal eigenform of finite slope for the $\U$-operator. Then there exists an affinoid $\mc U \subset \mc W(\underline w)$ that contains $\chi$ and such that the system of eigenvalues associated to $f$ can be deformed to a family of systems of eigenvalues appearing in $\cusp^{\dagger\mc U}_{\underline v,\underline w}$.
\end{coro}
\section{Classicity results} \label{sec: class}
In this section we prove certain classicity results for our modular forms. Since the results of \cite{BGG} hold in general, there are no conceptual difficulties to generalize \cite{AIP} to our setting, so we will sometimes only sketch the arguments.
Recall that at the end of Section~\ref{sec: PEL data} we have defined the algebraic group $\GL^{\mc O}$ with maximal split torus $\T^{\mc O}$. We also have the Borel subgroup $\B^{\mc O}$ with unipotent radical $\U^{\mc O}$.
Let $\chi \in X^\ast(\T^{\mc O})^+$ be a (classical) dominant weight, that we see as a character of $\B^{\mc O}$ trivial on $\U^{\mc O}$. We set
\begin{gather*}
V_\chi \colonequals \{f \colon \GL^{\mc O} \to \mathds A^1 \mbox{ morphisms of schemes such that, for all } \Z_p-\mbox{algebras } R,\\
\mbox{ we have } f(gb)=\chi(b)f(g) \mbox{ for all } (g,b) \in (\GL^{\mc O} \times \B^{\mc O})(R)\}.
\end{gather*}
There is a left action of $\GL^{\mc O}$ on $V_\chi$ given by $(g \cdot f)(x) = f(g^{-1}x)$.
Recall the sheaf $\underline \omega^\chi$ on $Y_{\Iw}(p)$ of classical modular forms. We have the toroidal compactification $Y_{\Iw}(p)^{\tor}$ of $Y_{\Iw}(p)$ defined in \cite{ben_these}. We will work only with the generic fiber $\mathfrak Y_{\Iw}(p)^{\tor,\rig}$ of its formal completion $\mathfrak Y_{\Iw}(p)^{\tor}$. We have that $\mathfrak Y_{\Iw}(p)(\underline v)^{\tor,\rig}$ is an open subspace of $\mathfrak Y_{\Iw}(p)^{\tor,\rig}$. Clearly $\underline \omega^\chi$ extends to a sheaf on $Y_{\Iw}(p)^{\tor}$.
We have the following
\begin{prop} \label{prop: omega repr class}
Locally for the étale topology on $Y_{\Iw}(p)^{\tor,\rig}$ the sheaf $\underline \omega^\chi$ is isomorphic to $V_{\chi',K}$. This isomorphism respects the action of $\GL^{\mc O}$.
\end{prop}
We write $\B^{\mc O, \op}$ and $\U^{\mc O, \op}$ for the opposite subgroups of $\B^{\mc O, \op}$ and $\U^{\mc O, \op}$. Let $\I^{\mc O}$ be the Iwahori subgroup of $\GL^{\mc O}(\Z_p)$ given by matrices that are `upper triangular' modulo $p$ (recall that we are in the unramified case, so $p$ is a uniformizer of each $\mc O_i$) and let $\N^{\mc O, \op}$ be the subgroup of $\U^{\mc O, \op}(\Z_p)$ given by those matrices that reduce to the identity modulo $p$. We have an isomorphism of groups
\[
\N^{\mc O, \op} \times \B^{\mc O}(\Z_p) \to \I^{\mc O}
\]
given by Iwahori decomposition.
We use the following identification, in case (A) and (C) respectively.
\begin{gather*}
\N^{\mc O, \op} = \prod_{i=1}^k \left( p\mc O_i^{\frac{a_i^+(a_i^+-1)}{2}} \times p\mc O_i^{\frac{a_i^-(a_i^--1)}{2}} \right) \subset \prod_{i=1}^k \left( \mathds A^{\frac{a_i^+(a_i^+-1)}{2}, \rig} \times \mathds A^{\frac{a_i^-(a_i^--1)}{2}, \rig} \right),\\
\N^{\mc O, \op} = \prod_{i=1}^k p\mc O_i^{\frac{a_i(a_i-1)}{2}} \subset \prod_{i=1}^k \mathds A^{\frac{a_i(a_i-1)}{2}, \rig}.
\end{gather*}
Given $\underline w$, a tuple of positive real numbers as in the definition of $\mc W(\underline w)$, we define in case (A) and (C) respectively
\begin{gather*}
\N^{\mc O, \op}_{\underline w} \colonequals \bigcup_{(x_i^\pm) \in \N^{\mc O, \op}} \prod_{i=1}^k \left( B(x_i^+, p^{-w_i^+}) \times B(x_i^-, p^{-w_i^-}) \right),\\
\N^{\mc O, \op}_{\underline w} \colonequals \bigcup_{(x_i) \in \N^{\mc O, \op}} \prod_{i=1}^k B(x_i, p^{-w_i}),
\end{gather*}
where $B(x,p^{-w})$ is the ball of center $x$ and radius $p^{-w}$.
We say that a function $f \colon \N^{\mc O, \op} \to K$ is \emph{$\underline w$}-analytic if it is the restriction of an analytic function $f \colon \N^{\mc O, \op}_{\underline w} \to K$. Note that the extension of $f$, if it exists, is necessarily unique. We write $\mc F^{\underline w-\an}(\N^{\mc O, \op},K)$ for the set of $\underline w$-analytic functions. If $w_i^\pm = 1$ for all $i$ and $f$ is $\underline w$-analytic, we simply say that $f$ is analytic and we write $\mc F^{\an}(\N^{\mc O, \op},K)$ for the set of analytic functions. A function is \emph{locally analytic} if it is $\underline w$-analytic for some $\underline w$ and we write $\mc F^{\locan}(\N^{\mc O, \op},K)$ for the set of locally analytic functions.
Let now $\chi \in \mc W(\underline w)(K)$ be a $\underline w$-analytic character. We set
\begin{gather*}
V_\chi^{\underline w-\an} \colonequals \{f \colon \I^{\mc O} \to K \mbox{ such that } f(ib)=\chi(b)f(i)\\
\mbox{ for all } (i,b) \in \I^{\mc O} \times \B^{\mc O} \mbox{ and } f_{|\N^{\mc O, \op}_{\underline w}} \in \mc F^{\underline w-\an}(\N^{\mc O, \op},K)\}.
\end{gather*}
The definition of the spaces $V_\chi^{\an}$ and $V_\chi^{\locan}$ is similar. They all are representations of $\I^{\mc O}$ via $(i \star f)(x) = f(xi)$.
We have the following
\begin{prop} \label{prop: omega repr gen}
Locally for the étale topology on $\mathfrak Y_{\Iw}(p)^{\tor,\rig}$ the sheaf $\underline \omega^{\dagger\chi}_{\underline v,\underline w}$ is isomorphic to $V_{\chi'}^{\underline w-\an}$. This isomorphism respects the action of $\I^{\mc O}$.
\end{prop}
If $\chi$ is a classical dominant weight, there is an obvious inclusion $V_\chi \hookrightarrow V_\chi^{\underline w-\an}$ and we have the following proposition (see \cite[Proposition~5.3.4]{AIP}).
\begin{prop} \label{prop: incl omega repr}
The open immersion of Proposition~\ref{prop: class mod forms} induces an inclusion of sheaves
\[
\underline \omega^\chi_{|\mathfrak Y_{\Iw}(p)^{\tor,\rig}} \hookrightarrow \underline \omega^{\dagger\chi}_{\underline v,\underline w}.
\]
Locally for the étale topology on $\mathfrak Y_{\Iw}(p)^{\tor,\rig}$ this inclusion is isomorphic to $V_{\chi'} \hookrightarrow V_{\chi'}^{\underline w-\an}$.
\end{prop}
\subsection{The BGG resolution and a classicity result}
The goal of this subsection is to characterize the image of the inclusion $V_\chi \hookrightarrow V_\chi^{\locan}$.
Let $W$ be the Weyl group of $\GL^{\mc O}$ and let $\mathfrak{gl}^{\mc O}$ and $\mathfrak h^{\mc O}$ be the Lie algebras of $\GL^{\mc O}$ and $\T^{\mc O}$ respectively. Let $\Phi \subset X^\ast(\T^{\mc O})$ be the set of roots of $\mathfrak{gl}^{\mc O}$. We have a decomposition
\[
\mathfrak{gl}^{\mc O} = \mathfrak h^{\mc O} \oplus \bigoplus_{\alpha \in \Phi} \mathfrak{gl}^{\mc O}_\alpha.
\]
Let $\Phi^+ \subseteq \Phi$ be the set of positive roots given by the choice of $\B^{\mc O}$ and let $\alpha \in \Phi^+$. We fix an element $0 \neq e_\alpha \in \mathfrak{gl}^{\mc O}_\alpha$. There is an element, that we fix once and for all, $f_\alpha \in \mathfrak{gl}^{\mc O}_{-\alpha}$ such that $\langle e_\alpha, f_\alpha,h_\alpha\rangle$ is isomorphic to $\mathfrak{sl}_2$ via $e_\alpha \mapsto \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$, $f_\alpha \mapsto \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$, and $h_\alpha \mapsto \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$, where $h_\alpha \colonequals [e_\alpha,f_\alpha]$. We denote by $s_\alpha \in W$ the reflection $\chi \mapsto \chi - \langle\chi, \alpha^\vee\rangle \alpha$, where $\chi \in X^\ast(\T^{\mc O})$ and $\alpha^\vee$ is the coroot associated to $\alpha$. The pairing $\langle \cdot, \cdot \rangle \colon X^\ast(\T^{\mc O}) \times X_\ast(\T^{\mc O}) \to \Z$ is the natural one. Given $w \in W$ and $\chi \in X^\ast(\T^{\mc O})$, we set $w \bullet \chi \colonequals w(\chi + \rho) - \rho$, where $\rho$ is half of the sum of the positive roots.
Let now $\chi \in X^\ast(\T^{\mc O})^+$ be a dominant weight. By the results of \cite{BGG} there is an exact sequence
\begin{equation} \label{eq: BGG}
0 \to V_\chi \to V_\chi^{\an} \to \bigoplus_{\alpha \in \Phi^+} V_{s_\alpha \bullet \chi}^{\an}
\end{equation}
where the first map is the natural inclusion and the second one is given by the maps $\Theta_\alpha \colon V_\chi^{\an} \to V_{s_\alpha \bullet \chi}^{\an}$ defined as follows. Recall the action of $\I^{\mc O}$ on $V_\chi^{\an}$ given by $(i \star f)(x) = f(xi)$. It induces, by differentiation, an action of the universal enveloping algebra $U(\mathfrak{gl}^{\mc O})$ on $V_\chi^{\an}$. We set
\[
\Theta_\alpha(f) \colonequals f_\alpha^{\langle \chi, \alpha^\vee \rangle + 1 } \star f.
\]
We now fix an index $i=1,\ldots,k$. Let $1 \leq j < a_i^\pm$. We write $d_{i,j}^\pm \in \GL^{\mc O}(\Q_p)$ for the element whose components are the identity matrix except the $i^\pm$-th one, that is the matrix
\[
\begin{pmatrix}
p^{-1}\mathrm{Id}_{a_i^\pm - j} & 0\\
0 & \mathrm{Id}_j
\end{pmatrix}
\]
If $\chi \in X^\ast(\T^{\mc O})^+$ is a dominant weight, there is an operator $\delta_{i,j}^\pm$ on $V_\chi$ defined by the formula $(\delta_{i,j}^\pm \cdot f)(x)= f(d_{i,j}^\pm x (d_{i,j}^\pm)^{-1})$.
Let now $\chi \in \mc W(\underline w)$ be a $\underline w$-analytic character. Take $f \in V_\chi^{\underline w-\an}$ and $i \in \I^{\mc O}$. By the existence of the Iwahori decomposition there are $b \in \B^{\mc O}(\Z_p)$ and $n \in \N^{\mc O,\op}$ such that $i = nb$. We define an operator $\delta_{i,j}^\pm$ on $V_\chi^{\underline w-\an}$ via the formula
\[
(\delta_{i,j}^\pm \cdot f)(i)= f(d_{i,j}^\pm n (d_{i,j}^\pm)^{-1} b).
\]
Taking restriction to $\N^{\mc O,\op}$ we can identify $V_\chi^{\underline w-\an}$ with $\mc F^{\underline w-\an}(\N^{\mc O,\op},K)$ and under this identification the operator $\prod_j \delta_{i,j}^\pm$ increases the radius of analyticity in the $i^\pm$-th direction.
We use now the notation of Subsection~\ref{subsub: Uj} and we explain the relation between the operators $\U_{i,j}^\pm$ and $\delta_{i,j}^\pm$. We have the following proposition (see \cite[Proposition~6.2.3.1]{AIP}).
\begin{prop} \label{prop: heck op repr}
Let $x,y \in \mathfrak Y_{\Iw}(p)(\underline v)^{\rig}$ such that $y \in p_2(p_1^{-1}(x))$. We fix $\chi \in \mc W(\underline w)$ a $\underline w$-analytic character. Then there is a commutative diagram whose vertical arrows are isomorphisms
\[
\xymatrix{
(\underline \omega^{\dagger\chi}_{\underline v,\underline w})_y \ar[r]^-{(f^\ast)^{-1}} & (\underline \omega^{\dagger\chi}_{\underline v,\underline w})_x \\
V_{\chi'}^{\underline w-\an} \ar[r]^-{\delta_{i,j}^\pm} \ar[u]^-{\wr} & V_{\chi'}^{\underline w-\an} \ar[u]^-{\wr}
}
\]
\end{prop}
Let $\underline \nu = (\nu_{i,j}^\pm)_{i,j}$ be a tuple of positive real numbers such that
\[
\underline \nu \in \prod_{i=1}^k (\mathds R^{a_i^+-1} \times \mathds R^{a_i^--1})
\]
in case (A) and
\[
\underline \nu \in \prod_{i=1}^k \mathds R^{a_i-1}
\]
in case (C). Given $\chi \in \mc W(\underline w)$ we write $V_\chi^{\underline w-\an, < \underline \nu}$ for the intersection of the generalized eigenspace where each $\delta_{i,j}^\pm$ acts with eigenvalues of valuation smaller than $\nu_{i,j}^\pm$. (Here, for a given $i$, we take $j$ in the range $1,\ldots,a_i^\pm - 1$.)
\begin{prop} \label{prop: class sheaf}
Let $\chi \in X^\ast(\T^{\mc O})^+$ be a classical dominant weight. We can write $\chi=(k_{i,s,j}^\pm)_{i,s,j}$ as in Subsection~\ref{subsec: class mod form}. For each $i=1,\ldots,k$ and for each $j=1,\ldots, a_i^\pm - 1$ we set $\nu_{i,a_i^\pm - j}^\pm \colonequals \inf_s \{ k_{i,s,j}^\pm - k_{i,s,j+1} + 1\}$. We then have
\[
V_\chi^{\underline w-\an, < \underline \nu} \subset V_\chi.
\]
\end{prop}
\begin{proof}
We can fix an index $i$ and work only in the `$i$-th component'. Taking the base change of all objects to $\mc O_i^\pm$ (that makes everything split) and using the exact sequence \eqref{eq: BGG}, the proof is completely analogous to that of \cite[Proposition~2.5.1]{AIP}.
\end{proof}
\subsection{Classicity at the level of sheaves}
Let $\chi = (k_{i,s,j}^\pm)_{i,s,j} \in X^\ast(\T^{\mc O})^+$. We now give a characterization of the image of the inclusion $\underline \omega^\chi_{|\mathfrak Y_{\Iw}(p)^{\tor,\rig}} \hookrightarrow \underline \omega^{\dagger\chi}_{\underline v,\underline w}$ of Proposition~\ref{prop: incl omega repr}.
We can construct a `relative version' over $\mathfrak Y_{\Iw}(p)(\overline v)^{\tor,\rig}$ of the exact sequence \eqref{eq: BGG}. Using then Propositions~\ref{prop: omega repr class}, \ref{prop: omega repr gen}, and \ref{prop: incl omega repr} we obtain the exact sequence of sheaves on $\mathfrak Y_{\Iw}(p)(\overline v)^{\tor,\rig}$
\begin{equation} \label{eq: BGG rel}
0 \to \underline \omega^\chi_{|\mathfrak Y_{\Iw}(p)^{\tor,\rig}} \to \underline \omega^{\dagger\chi}_{\underline v,\underline w} \to \bigoplus_{\alpha \in \Phi^+} \underline \omega^{\dagger s_\alpha \bullet \chi}_{\underline v,\underline w}
\end{equation}
Let $\underline \nu = (\nu_{i,j}^\pm)_{i,j}$ be a tuple of positive real numbers such that $\underline \nu \in \prod_{i=1}^k (\mathds R^{a_i^+} \times \mathds R^{a_i^-})$ in case (A) and $\underline \nu \in \prod_{i=1}^k \mathds R^{a_i}$ in case (C). In case (A) we assume that, for each $i=1,\ldots,k$, we have $\nu_{i,a_i^+}^+ = \nu_{i,a_i^-}^-$. This is a natural condition since $\U_{i,a_i^+}^+ = \U_{i,a_i^-}^-$. We can then write $\nu_{i,a_i^\pm}^\pm \colonequals \nu_{i,a_i^+}^+ = \nu_{i,a_i^-}^-$.
We define $\M^{\dagger \chi, < \underline \nu}_{\underline v,\underline w}$ to be the intersection of the generalized eigenspace where each $\U_{i,j}^\pm$ acts with eigenvalues of valuation smaller than $\nu_{i,j}^\pm$. (The difference with the above situation is that we add the condition the the $\U_{i,a_i^\pm}^\pm$-operator acts with finite slope.)
We now suppose that, for each $i$ and for each $j=1,\ldots, a_i^\pm$ we have $\nu_{a_i^\pm - j}^\pm = \inf_s \{k_{i,s,j}^\pm - k_{i,s,j+1} + 1\}$. As in \cite[Proposition~7.3.1]{AIP} we have the following
\begin{prop} \label{prop: class sheaves}
We have the inclusion
\[
\M^{\dagger \chi, < \underline \nu}_{\underline v,\underline w} \subset \Homol^0(\mathfrak Y_{\Iw}(p)(\underline v)^{\tor,\rig},\underline \omega^\chi).
\]
\end{prop}
\subsection{The classicity theorem} \label{subsec: class thm}
Let $\chi = (k_{i,s,j}^\pm)_{i,s,j} \in X^\ast(\T^{\mc O})^+$ be a classical dominant weight.
\begin{teo} \label{thm: class}
Let $\underline \nu$ be as in the Proposition~\ref{prop: class sheaves}. We moreover assume that, for each $i=1,\ldots,k$, we have
\[
\nu_{i,a_i^\pm}^\pm = \inf_{1 \leq s \leq d_i} (k_{i,s,a_i^+}^+ + k_{i,s,a_i^-}^-) - d_ia_i^+a_i^-
\]
in case (A) and
\[
\nu_{i,a_i} = \inf_{1 \leq s \leq d_i} k_{i,s,a_i} - \frac{d_ia_i(a_i+1)}{2}
\]
in case (C). Then we have an inclusion
\[
\M^{\dagger \chi, < \underline \nu}_{\underline v,\underline w} \subset \M^{\chi},
\]
hence any locally analytic overconvergent modular form in $\M^{\dagger \chi, < \underline \nu}_{\underline v,\underline w}$ is classical.
\end{teo}
\begin{proof}
Using Proposition~\ref{prop: class sheaf}, this follows by the main results of \cite{stefan}.
\end{proof}
\end{document} |
\begin{document}
\title{Coequalisers under the Lens}
\begin{abstract}
Lenses encode protocols for synchronising systems. We continue the work begun by Chollet et al.\ at the Applied Category Theory Adjoint School in 2020 to study the properties of the category of small categories and asymmetric delta lenses. The forgetful functor from the category of lenses to the category of functors is already known to reflect monos and epis and preserve epis; we show that it preserves monos, and give a simpler proof that it preserves epis. Together this gives a complete characterisation of the monic and epic lenses in terms of elementary properties of their get functors.
Next, we initiate the study of coequalisers of lenses. We observe that not all parallel pairs of lenses have coequalisers, and that the forgetful functor from the category of lenses to the category of functors neither preserves nor reflects all coequalisers. However, some coequalisers are reflected; we study when this occurs, and then use what we learned to show that every epic lens is regular, and that discrete opfibrations have pushouts along monic lenses. Corollaries include that every monic lens is effective, every monic epic lens is an isomorphism, and the class of all epic lenses and the class of all monic lenses form an orthogonal factorisation system.
\end{abstract}
\section{Introduction}
A \textit{bidirectional transformation} between two systems is a specification of when the joint state of the two systems should be regarded as consistent, together with a protocol for updating each system to restore consistency in response to a change in the other~\cite{gibbons:2018:bidirectionaltransformations}. The study of bidirectional transformations goes back to as far as 1981 with Bancilhon and Spyrato’s work on the view-update problem for databases~\cite{bancilhon:1981:updatesemanticsrelationalviews}. The view-update problem is about \textit{asymmetric} bidirectional transformations; those where the state of one of the systems, called the \textit{view}, is completely determined by that of the other, called the \textit{source}. Bidirectional transformations also arise in many other contexts across computer science, such as when programming with complex data structures and when linking user interfaces to data models.
An \textit{asymmetric state-based lens} is a mathematical encoding of an asymmetric bidirectional transformation in which the consistency restoration updates to the source are assumed to be dependent only on the old source state and the updated view state. If \(S\) is the set of source states and~\(V\) is the set of view states, such a lens consists of a \textit{get function} \(S \to V\) and a \textit{put function} \(S \times V \to S\) which, ideally, satisfy certain laws. The earliest known account of asymmetric state-based lenses may be found in Oles’ PhD thesis~\cite[Chapter~VI]{oles:1982:CategoryApproachSemanticsProgrammingLanguages}, where they are called \textit{extensions} of \textit{store shapes}; they are a key ingredient in Oles' semantics for an imperative stack-based programming language with block-scoped variables because they capture the essential properties of a data store which changes shape as variables come into and go out of scope. All recent notions of lens, including the name \textit{lens}, may be traced back to the work of Pierce~et~al.~\cite{pierce:2007:combinatorsbidirectionaltreetransformations}; they proposed variants of asymmetric state-based lenses for modelling bidirectional transformations on tree-structured data, and they also introduced the idea of building lenses compositionally with a domain-specific language such as their lens combinators.
Diskin~et~al.\ highlighted the inadequacy of state-based lenses as a general mathematical model for bidirectional transformations~\cite{diskin:2011:statetodeltabx}, providing several examples of situations in which consistency restoration would benefit from knowing more about each change to the view than just the view's new state. In an \textit{asymmetric delta lens}, their proposed alternative, systems are modelled as categories of states and transitions (deltas) rather than simply as sets of states. Also, the put operation takes as input specifically which transition occurred in the view rather than just the end state of that transition.
Application of category theory to the study of lenses has already proved fruitful. Johnson and Rosebrugh's research program~\cite{JohnsonRosebrugh:2015:SpansDeltaLenses, JohnsonRosebrugh:2016:UnifyingSetBasedDeltaBasedEditBasedLenses, JohnsonRosebrugh:2017:UniversalUpdatesForSymmetricLenses} has enabled a unified treatment of symmetric and asymmetric delta lenses, with the perspective that a symmetric delta lens is an equivalence class of spans of asymmetric delta lenses. Ahman and Uustalu's observation that asymmetric delta lenses are compatible functor cofunctor pairs~\cite{AhmanUustalu:2017:TakingUpdatesSeriously}, and Clarke's generalisation of these lenses to the internal category theory setting~\cite{Clarke:2020:InternalLensesAsFunctorsAndCofunctors}, have enabled an abstract diagrammatic approach to proofs involving these lenses~\cite{Clarke:2021:ADiagrammaticApproachToSymmetricLenses}, in which we may profit from the already well-developed theory of functors and opfibrations. Yet, until the work of Chollet et al.~\cite{Clarke:2021:CategoryLens}, little was known about the category of asymmetric delta lenses. Building on their work, this paper aims to further our understanding of this category.
\subsection*{Outline}
Henceforth, we refer to asymmetric delta lenses simply as \textit{lenses}, which we formally define in \cref{Section: Background}.
In \cref{Section: Characterising monic and epic lenses}, we prove the conjecture by Chollet et al.~\cite{Clarke:2021:CategoryLens} that the forgetful functor from the category of lenses to the category of functors preserves monos. Together with their result that it reflects monos, we deduce that the monic lenses are the unique lenses on cosieves; these are equivalently the out-degree-zero subcategory inclusion functors. We also provide a proof, simpler than the original one sketched by Lack in an unpublished personal communication to Clarke, that the forgetful functor preserves epis.
In \cref{Section: Coequalisers of lenses}, we initiate the study of coequalisers of lenses. We begin with examples of how they are not as well behaved as one might hope; specifically, not all parallel pairs of lenses have coequalisers, and the forgetful functor neither preserves nor reflects all coequalisers. We then prove our main result, \cref{Condition for reflection}, which is about the coequalisers that are actually reflected by the forgetful functor.
In \cref{Section: Pushouts of discrete opfibrations along monos}, we use \cref{Condition for reflection} to show that the category of lenses has pushouts of discrete opfibrations along monos. We then show that every monic lens is effective. It follows that the classes of all monos, all effective monos, all regular monos, all strong monos and all extremal monos in the category of lenses coincide, and thus also that all lenses which are both monic and epic are isomorphisms.
In \cref{Section: Regular epic lenses}, we use \cref{Condition for reflection} again to show that every epic lens is regular. It follows that the classes of all epis, all regular epis, all strong epis and all extremal epis in the category of lenses coincide. It also follows that the class of all epic lenses is left orthogonal to the class of all monic lenses. Together with other known results, this means that they form an orthogonal factorisation system.
\section{Background}
\label{Section: Background}
\subsection*{Notation}
\label{Section: Notation}
Application of functions (functors, lenses, etc.)\ is written by juxtaposing the function name with its argument. Application is right associative, so an expression like \(FGx\) parses as \(F (Gx)\) and not \((FG)x\). Parentheses are only used when needed or for clarity. Binary operators like \(\compose\) have lower precedence than application, so an expression like \(F a \compose F b\) parses as \((F a) \compose (F b)\) and not \(F \paren[\big]{(a \compose F) b}\).
Let \(\smallCat{C}at\) denote the category whose objects are small categories and whose morphisms are functors. Categories with boldface names \(\A\), \(\B\), \(\smallCat{C}\), etc.\ are always small. We write \(\objectSet{\smallCat{C}}\) for the set of objects of a small category \(\smallCat{C}\), and, for all \(X, Y \in \objectSet{\smallCat{C}}\), we write \(\homSet{\smallCat{C}}{X}{Y}\) for the set of morphisms of \(\smallCat{C}\) from \(X\) to \(Y\). For each \(X \in \objectSet{\smallCat{C}}\), we write \(\outSet{\smallCat{C}}{X}\) for the set \(\DisjointUnion_{Y \in \objectSet{\smallCat{C}}}\homSet{\smallCat{C}}{X}{Y}\) of all morphisms in \(\smallCat{C}\) out of \(X\). We write \(\source{f}\) and \(\target{f}\) for, respectively, the source and target of a morphism \(f\). We also write \(f \colon X \to Y\) to say that \(X, Y \in \objectSet{\smallCat{C}}\) and \(f \in \homSet{\smallCat{C}}{X}{Y}\). The composite of morphisms \(f \colon X \to Y\) and \(g \colon Y \to Z\) is denoted \(g \compose f\).
The category with a single object \(0\) and no non-identity morphisms, also known as the \textit{terminal category}, is denoted \(\terminalCat\). The category with two objects \(0\) and \(1\) and a single non-identity morphism, namely \(u \colon 0 \to 1\), also known as the \textit{interval category}, is denoted \(\intervalCat\). The category with two objects \(0\) and \(1\) and two non-identity morphisms, namely \(v \colon 0 \to 1\) and \(v^{-1} \colon 1 \to 0\), also known as the \textit{free living isomorphism}, is denoted \(\isomorphismCat\). We will identify objects and morphisms of a small category \(\smallCat{C}\) with the corresponding functors \(\terminalCat \to \smallCat{C}\) and \(\intervalCat \to \smallCat{C}\) respectively.
If the square
\begin{equation}
\label{Equation: Pullback square}
\begin{tikzcd}
\D \arrow[r, "T"]\arrow[d, "S" swap] & \B \arrow[d, "G"]\\
\A \arrow[r, "F" swap] & \smallCat{C}
\end{tikzcd}
\end{equation}
in \(\smallCat{C}at\) is a pushout square and \(F' \colon \A \to \E\) and \(G' \colon \B \to \E\) are functors for which \(F' \compose S = G' \compose T\), then we write \(\copair{F'}{G'}\) for the functor \(\smallCat{C} \to \E\) induced from \(F'\) and \(G'\) by the universal property of the pushout. Similarly, if the square \eqref{Equation: Pullback square} in \(\smallCat{C}at\) is a pullback square and \(S' \colon \E \to \A\) and \(T' \colon \E \to \B\) are functors for which \(F \compose S' = G \compose T'\), then we write \(\pair{S'}{T'}\) for the functor \(\E \to \D\) induced from \(S'\) and \(T'\) by the universal property of the pullback. By our identification of objects with functors from~\(\terminalCat\) mentioned above, if \(A \in \objectSet{\A}\) and \(B \in \objectSet{\B}\) are such that \(FA = GB\), then \(\pair{A}{B}\) is the object of \(\D\) selected by the functor \(\terminalCat \to \D\) induced by the universal property of the pullback from the functors \(\terminalCat \to \A\) and \(\terminalCat \to \B\) that respectively select the objects \(A\) and \(B\).
\subsection*{Lenses and discrete opfibrations}
\label{Section: Lenses and discrete opfibrations}
First, we recall the definition of a (asymmetric delta) lens~\cite{diskin:2011:statetodeltabx}.
\begin{definition}
Given small categories \(\A\) and \(\B\), a \textit{lens} \(F \colon \A \to \B\) consists of
\begin{itemize}
\item a functor \(F \colon \A \to \B\), called the \textit{get functor} of \(F\), and
\item a function \(\lift{F}{A} \colon \outSet{\B}{FA} \to \outSet{\A}{A}\) for each \(A \in \objectSet{\A}\), collectively known as the \textit{put functions},
\end{itemize}
such that
\begin{itemize}
\item \textit{\PutGet{}}: \(F\lift{F}{A}b = b\) for all \(A \in \objectSet{\A}\) and all \(b \in \outSet{\B}{FA}\),
\item \textit{\PutId{}}: \(\lift{F}{A}\id{FA} = \id{A}\) for all \(A \in \objectSet{\A}\), and
\item \textit{\PutPut{}}: \(\lift{F}{A}(b' \compose b) = \lift{F}{A'}b' \compose \lift{F}{A}b\) for all \(A \in \objectSet{\A}\), \(b \in \outSet{\B}{FA}\), \(b' \in \outSet{\B}{FA'}\), where \(A' = \target \lift{F}{A}b\).
\end{itemize}
\end{definition}
\noindent There is a category \(\Lens\) whose objects are small categories and whose morphisms are lenses. The composite \(G \compose F\) of lenses \(F \colon \A \to \B\) and \(G \colon \B \to \smallCat{C}\) has get functor which is the composite of the get functors of \(G\) and \(F\), and has \(\lift{(G \compose F)}{A}c = \lift{F}{A}\lift{G}{FA}c\) for all \(A \in \objectSet{\A}\) and all \(c \in \outSet{\smallCat{C}}{GFA}\). There is also an identity-on-objects forgetful functor \(\forget \colon \Lens \to \smallCat{C}at\) that sends a lens to its get functor.
\begin{definition}
A functor \(F \colon \A \to \B\) is a \textit{discrete opfibration} if, for each \(A \in \objectSet{\A}\) and each \(b \in \outSet{\B}{FA}\), there is a unique \(a \in \outSet{\A}{A}\) such that \(Fa = b\).
\end{definition}
\begin{remark}
If \(F \colon \A \to \B\) is a discrete opfibration, then there is a unique lens mapped by \(\forget\) to \(F\). We will sometimes also use the name \(F\) to refer to this unique lens above \(F\).
\end{remark}
We also recall Johnson and Roseburgh's ``pullback'' of a cospan of lenses~\cite{JohnsonRosebrugh:2015:SpansDeltaLenses}, which we will refer to as their \textit{proxy pullback}, adopting the terminology of Bumpus and Kocsis~\cite{bumpus:2021:spined-categories:-generalizing-tree-width}.
\begin{definition}
The \textit{proxy pullback} of a lens cospan \(\A \xrightarrow{F} \smallCat{C} \xleftarrow{G}\B\) is a lens span \(\A \xleftarrow{\Gbar} \D \xrightarrow{\Fbar} \B\) where
\begin{itemize}
\item the get functors of \(\Fbar\) and \(\Gbar\) form a pullback square
\[\begin{tikzcd}
\D \arrow[r, "\forget\Fbar"]\arrow[d, "\forget\Gbar" swap] & \B \arrow[d, "\forget G"]\\
\A \arrow[r, "\forget F" swap] & \smallCat{C}
\end{tikzcd}\]
in \(\smallCat{C}at\) (this determines them up to isomorphism), and
\item for each \(D \in \objectSet{\D}\), each \(a \in \outSet[\big]{\A}{\Gbar D}\), and each \(b \in \outSet[\big]{\B}{\Fbar D}\),
\[
\lift{\Fbar}{D}b = \pair[\big]{\lift{F}{\Gbar D}Gb}{b}
\qquad\text{and}\qquad
\lift{\Gbar}{D}a = \pair[\big]{a}{\lift{G}{\Fbar D}Fa}.
\]
\end{itemize}
\noindent When \(F = G\), the lenses \(\Fbar,\Gbar\colon \D \to \A\) are also called the \textit{proxy kernel pair} of \(F\).
\end{definition}
\section{Characterising monic and epic lenses}
\label{Section: Characterising monic and epic lenses}
\subsection*{Monic lenses}
\label{Section: Monic lenses}
We will study the monos in \(\Lens\) via their relation to those in \(\smallCat{C}at\), expressed as follows.
\begin{theorem}
The functor \(\forget\) preserves and reflects monos.
\end{theorem}
Reflection was proved and preservation conjectured by Chollet et al.~\cite{Clarke:2021:CategoryLens}. Recalling that a morphism is monic if and only if it has a kernel pair with both morphisms equal, we may prove preservation.
\begin{proof}[Proof that \(\forget\) preserves monos.]
Let \(M \colon \A \to \B\) be a monic lens, and let \(P_1, P_2 \colon \Ker \forget M \to \A\) be its proxy kernel pair in \(\Lens\). As \(M\) is monic and \(M \compose P_1 = M \compose P_2\), actually \(P_1 = P_2\), and so \(\forget P_1 = \forget P_2\). But \(\forget P_1\) and \(\forget P_2\) are the (real) kernel pair of \(\forget M\) in \(\smallCat{C}at\). Hence \(\forget M\) is a monic functor.
\end{proof}
Chollet et al.~\cite{Clarke:2021:CategoryLens} also showed that the get functor of a lens is monic if and only if it is a cosieve.
\begin{definition}
A \textit{cosieve} is an injective-on-objects discrete opfibration.
\end{definition}
\begin{corollary}
\label{Monic lenses are cosieves}
The functor \(\forget\) restricts to a bijection between monic lenses and cosieves.
\end{corollary}
\begin{proof}
A cosieve is a discrete opfibration, so there is a unique lens above it; by reflection, this lens is monic. Conversely, the get functor of a monic lens is, by preservation, monic, and so is a cosieve.
\end{proof}
The above result says that monic lenses and cosieves are essentially the same. We continue to use the term cosieve for functors when we wish to distinguish these from monic lenses.
\subsection*{Lens images and factorisation}
\label{Section: Lens images and factorisation}
The images of the object and morphism maps of a functor do not always form a subcategory of a functor's target category. The situation is nicer for the get functor of a lens \(F\); in this case, the images actually form an out-degree-zero subcategory \(\Image F\) of the lens' target category, which we will call the \textit{image} of \(F\). By \textit{out-degree-zero} subcategory, we mean one for which any morphism out of an object in the subcategory belongs to the subcategory. As cosieves are exactly the out-degree-zero subcategory inclusion functors, we obtain the following factorisation result.
\begin{proposition}
\label{Lens image factorisation}
Every lens \(F \colon \A \to \B\) has a factorisation
\[
\begin{tikzcd}[column sep=large]
\A \arrow[r, "E" swap] \arrow[rr, bend left=15, shift left=1, "F"] & \Image F \arrow[r, "M" swap] & \B
\end{tikzcd}
\]
in \(\Lens\) where \(M\) is monic and \(E\) is surjective on objects and morphisms.
\end{proposition}
Recall that a morphism \(e \colon A \to B\) is \textit{left orthogonal} to a morphism \(m \colon C \to D\) if, for all pairs of morphisms \(f \colon A \to C\) and \(g \colon B \to D\) such that \(g \compose e = m \compose f\), there is a unique morphism \(h \colon B \to C\), called the \textit{diagonal filler}, such that \(f = h \compose e\) and \(g = m \compose h\). Also recall that classes \(\mathcal{E}\) and \(\mathcal{M}\) of morphisms form an \textit{orthogonal factorisation system} if \(\mathcal{E}\) is the class of all morphisms that are left orthogonal to all morphisms in \(\mathcal{M}\), and every morphism \(f\) factors as \(f = m \compose e\) for some \(e \in \mathcal{E}\) and some \(m \in \mathcal{M}\).
\begin{remark}
\label{Factorisation System}
The above factorisation is already known to Johnson and Roseburgh, who showed that the surjective-on-objects lenses and the injective-on-objects-and-morphisms lenses form an orthogonal factorisation system on \(\Lens\)~\cite{JohnsonRoseburgh:2021:TheMoreLegsTheMerrier}. Our addition is that this is actually an epi-mono factorisation system; we have already shown that the injective-on-objects-and-morphisms lenses are exactly the monic lenses, and we will show in the next section that the surjective-on-objects lenses are exactly the epic lenses. In \cref{Section: Regular epic lenses}, we will also deduce the orthogonality without explicitly constructing the diagonal fillers.
\end{remark}
\subsection*{Epic lenses}
\label{Section: Epic lenses}
We may also study the epis in \(\Lens\) via their relation to those in \(\smallCat{C}at\).
\begin{theorem}
\label{U preserves epis}
The functor \(\forget\) preserves and reflects epis.
\end{theorem}
Again, reflection was proved and preservation conjectured by Chollet et al.~\cite{Clarke:2021:CategoryLens}. The first proof of preservation was sketched by Lack in an unpublished personal communication to Clarke; we present a new, simpler proof below. First, we recall some preliminary results about epic functors and epic lenses.
\begin{proposition}
\label{Epic functors}
Every epic functor is surjective on objects. Every functor that is surjective both on objects and on morphisms is epic.
\end{proposition}
Recall that not all epic functors are surjective on morphisms.
\begin{example}
\label{Example epic functor not surjective on morphisms}
Let \(J \colon \intervalCat \to \isomorphismCat\) be the functor that sends the non-identity morphism \(u\) of the interval category~\(\intervalCat\) to the morphism \(v\) of the free living isomorphism \(\isomorphismCat\). Then \(J\) is epic because any two functors out of \(\isomorphismCat\) which agree on \(v\) must also agree on \(v^{-1}\). However, the morphism \(v^{-1}\) is not in the image of \(J\).
\end{example}
\begin{proposition}
\label{Proxy cokernel pair in lens}
Let \(F \colon \A \to \B\) be a lens, and let \(\Jbar_1, \Jbar_2 \colon \B \to C\) be the cokernel pair of \(\forget F\). Then \(\Jbar_1\) and \(\Jbar_2\) are cosieves, and the unique lenses \(J_1\) and \(J_2\) above \(\Jbar_1\) and \(\Jbar_2\) satisfy \(J_1 \compose F = J_2 \compose F\).
\end{proposition}
\begin{proof}
Let \(F = M \compose E\) be the factorisation of \(F\) given in \cref{Lens image factorisation}. By \cref{Epic functors}, \(\forget E\) is an epic functor. As \(\Jbar_1 \compose \forget M \compose \forget E = \Jbar_1 \compose \forget F = \Jbar_2 \compose \forget F = \Jbar_2 \compose \forget M \compose \forget E\), actually \(\Jbar_1 \compose \forget M = \Jbar_2 \compose \forget M\). It follows that \(\Jbar_1\) and \(\Jbar_2\) are also the cokernel pair of \(\forget M\). As cosieves are pushout stable and \(\forget M\) is a cosieve, so are \(\Jbar_1\) and \(\Jbar_2\). As there is a unique lens above the discrete opfibration \(\Jbar_1 \compose \forget M = \Jbar_2 \compose \forget M\), we must have that \(J_1 \compose M = J_2 \compose M\).
\end{proof}
\begin{remark}
Later, we will see that \(J_1\) and \(J_2\) are actually a cokernel pair of \(F\) in \(\Lens\).
\end{remark}
\begin{proof}[Proof that \(\forget\) preserves epis.]
Let \(E \colon \A \to \B\) be an epic lens, and \(J_1\) and \(J_2\) the unique lenses above the cokernel pair of \(\forget E\) from \cref{Proxy cokernel pair in lens}. As \(J_1 \compose E =J_2 \compose E\) and \(E\) is epic, actually \(J_1 = J_2\), and so \(\forget J_1 = \forget J_2\). But \(\forget J_1\) and \(\forget J_2\) are the cokernel pair of \(\forget E\), so \(\forget E\) is also epic.
\end{proof}
\begin{corollary}
\label{Epic lenses are surjective on objects and morphisms}
Let \(F\) be a lens. Then the following are equivalent:
\begin{enumerate}[label=\normalfont(\arabic*)]
\item \(F\) is epic,
\item \(\forget F\) is surjective on objects,
\item \(\forget F\) is surjective on morphisms.
\end{enumerate}
\end{corollary}
\begin{proof}
Chollet et al.~\cite{Clarke:2021:CategoryLens} showed that (2) and (3) are equivalent, and imply (1). Suppose that \(F\) is epic. As \(\forget\) preserves epis (\cref{U preserves epis}), so is \(\forget F\). By \cref{Epic functors}, \(\forget F\) is surjective on objects.
\end{proof}
\section{Coequalisers of lenses}
\label{Section: Coequalisers of lenses}
Given morphisms \(f_1, f_2 \colon A \to B\),
we say that a morphism \(e \colon B \to C\) \textit{coforks} \(f_1\) and \(f_2\) if \(e \compose f_1 = e \compose f_2\). Some authors would use the verb coequalise where we use the verb cofork. Unlike those authors, we say that \(e\) \textit{coequalises} \(f_1\) and \(f_2\) only when \(e\) is universal among coforks of \(f_1\) and \(f_2\).
\subsection*{Non-existence, non-preservation and non-reflection of coequalisers}
\label{Section: Non-existence, non-preservation and non-reflection of coequalisers}
Recall that \(\smallCat{C}at\) has all coequalisers. Shortly, we will construct several counterexamples to the well-behavedness of coequalisers in \(\Lens\), at least with respect to those in~\(\smallCat{C}at\). To do this, we will use the following proposition, which gives necessary conditions for a cofork of lenses to be a coequaliser.
\begin{proposition}
\label{Necessary condition for existence of coequaliser}
Let \(F_1, F_2 \colon \A \to \B\) be lenses with coequaliser \(E \colon \B \to \smallCat{C}\) in \(\Lens\). Then
\begin{enumerate}[label=\normalfont(\arabic*)]
\item
\label{Necessary condition 1}
for each cofork \(G \colon \B \to \D\) of \(F_1\) and \(F_2\),
\(\lift{G}{B}d = \lift{E}{B}E \lift{G}{B}d\)
for all \(B \in \objectSet{\B}\) and all \(d \in \outSet{\D}{GB}\); and
\item
\label{Necessary condition 2}
in particular, \(E\) is the unique lens above \(\forget E\) that coforks \(F_1\) and \(F_2\).
\end{enumerate}
\end{proposition}
\begin{proof}
For \ref{Necessary condition 1}, if \(G \colon \B \to \D\) coforks \(F_1\) and \(F_2\), then there is a lens \(H \colon \smallCat{C} \to \D\) such that \(G = H \compose E\), and so
\(\lift{G}{B}d = \lift{(H \compose E)}{B}d = \lift{E}{B}\lift{H}{E B}d = \lift{E}{B}E\lift{E}{B}\lift{H}{E B} d = \lift{E}{B}E \lift{(H \compose E)}{B} d = \lift{E}{B}E \lift{G}{B}d\). For \ref{Necessary condition 2}, if \(G \colon \B \to \smallCat{C}\) is a lens above \(\forget E\) that coforks \(F_1\) and \(F_2\), then
\(\lift{G}{B}c = \lift{E}{B}E\lift{G}{B}c = \lift{E}{B}G\lift{G}{B}c = \lift{E}{B}c\)
for each \(B \in \objectSet{\B}\) and each \(c \in \outSet{\smallCat{C}}{EB}\), and so \(G = E\).
\end{proof}
The first example shows that \(\Lens\) does not have all coequalisers, nor does \(\forget\) reflect them.
\begin{example}
\label{Lens doesn't have all coequalisers}
Let \(\A\) and \(\B\) be the preordered sets generated respectively by the following graphs.
\begin{align*}
\begin{tikzcd}[ampersand replacement=\&]
Y_1 \& X \arrow[l, "f_1" swap]\arrow[r, "f_2"]\arrow[d, "f"] \& Y_2\\
\&Y\&
\end{tikzcd}
&&
\begin{tikzcd}[ampersand replacement=\&]
Y'_1 \& X' \arrow[l, "f'_1" swap]\arrow[r, "f'_2"]\arrow[d, phantom] \& Y'_2\\
\&\phantom{Y}\&
\end{tikzcd}
\end{align*}
Let \(F_1, F_2 \colon \A \to \B\) be the unique lenses that both send \(X\) to \(X'\), \(Y_1\) to \(Y'_1\), \(Y_2\) to \(Y'_2\), and such that \(F_1 Y = Y'_1\), \(\lift{F_1}{X}f'_1 = f_1\), \(F_2 Y = Y'_2\), and \(\lift{F_2}{X}{f'_2} = f_2\). Let \(G \colon \B \to \intervalCat\) be the unique functor that sends \(X'\) to \(0\), and both \(Y'_1\) and \(Y'_2\) to \(1\); \(G\) coequalises \(\forget F_1\) and \(\forget F_2\) in \(\smallCat{C}at\). There are only two lens structures on \(G\) that cofork \(F_1\) and \(F_2\) in \(\Lens\); one is determined by \(\lift{G_1}{X'}u = f'_1\) and the other by \(\lift{G_2}{X'}u = f'_2\). By \cref{Necessary condition for existence of coequaliser}, neither \(G_1\) nor \(G_2\) coequalises \(F_1\) and \(F_2\). Thus \(\forget\) does not reflect the coequaliser \(G\) of \(\forget F_1\) and \(\forget F_2\).
Actually \(F_1\) and \(F_2\) do not have a coequaliser in \(\Lens\). Assume that \(E \colon \B \to \smallCat{C}\) is such a coequaliser. Then \(Ef'_1 = EF_1f = EF_2f = Ef'_2\). As \(G_1\) coforks \(F_1\) and \(F_2\), there is a lens \(H \colon \smallCat{C} \to \intervalCat\) such that \(G_1 = H \compose E\). As \(HEX' = G_1X' \neq G_1Y'_1 = HEY'_1\), we must have \(EX' \neq EY'_1\). Hence \(EX'\) and \(EY'_1\) are distinct objects of the image of \(E\), and \(\id{EX'}\), \(Ef'_1\) and \(\id{EY'_1}\) are distinct morphisms of the image of \(E\). As \(E\) is a coequaliser, it is epi, and so, by \cref{Epic lenses are surjective on objects and morphisms}, its image is all of \(\smallCat{C}\). Thus \(\forget H\) is an isomorphism in \(\smallCat{C}at\), and so \(H\) is an isomorphism in \(\Lens\). Hence \(G_1\) also coequalises \(F_1\) and \(F_2\), which is a contradiction.
\end{example}
There are even parallel pairs of lenses for which the coequaliser of their get functors has a unique lens structure that coforks them, and yet does not coequalise them.
\begin{example}
Let \(\A\), \(\B\) and \(\smallCat{C}\) be the preordered sets generated respectively by the following graphs.
\begin{align*}
\begin{tikzcd}[ampersand replacement=\&]
Z_1\&
X\arrow[d, "h" swap]\arrow[rr, bend right, "h_2" swap]\arrow[r, "f"]\arrow[l, "h_1" swap]\&
Y \arrow[r, "g"]\&
Z_2
\\
\&Z
\end{tikzcd}
&&
\begin{tikzcd}[ampersand replacement=\&]
Z'_1\&
X'\arrow[d, phantom]\arrow[rr, bend right, "h'_2" swap]\arrow[r, "f'"]\arrow[l, "h'_1" swap]\&
Y' \arrow[r, "g'"]\&
Z'_2
\\
\&\phantom{Z}
\end{tikzcd}
&&
\begin{tikzcd}[ampersand replacement=\&]
X''\arrow[d, phantom]\arrow[rr, bend right, "h''" swap]\arrow[r, "f''"]\&
Y'' \arrow[r, "g''"]\&
Z''
\\
\phantom{Z}
\end{tikzcd}
\end{align*}
Let \(F_1, F_2 \colon \A \to \B\) be the unique lenses that both send \(X\) to \(X'\), \(Y\) to \(Y'\), \(Z_1\) to \(Z'_1\), \(Z_2\) to \(Z'_2\), and such that \(F_1Z = Z'_1\), \(\lift{F_1}{X}h_1' = h_1\) and \(F_2Z = Z'_2\). Let \(E \colon \B \to \smallCat{C}\) be the unique lens that sends \(X'\) to \(X''\), \(Y'\) to \(Y''\), and both \(Z'_1\) and \(Z'_2\) to \(Z''\). Then \(\forget E\) coequalises \(\forget F_1\) and \(\forget F_2\) in \(\smallCat{C}at\), and \(E\) coforks \(F_1\) and \(F_2\) in \(\Lens\). However, \(E\) does not coequalise \(F_1\) and \(F_2\) in \(\Lens\). Indeed, if \(G \colon \B \to \intervalCat\) is the unique lens that sends \(X'\) to \(0\), all of \(Y'\), \(Z'_1\) and \(Z'_2\) to \(1\), and for which \(\lift{G}{X'}u = h'_1\), then \(\lift{E}{X'}E\lift{G}{X'}u = \lift{E}{X'}Eh'_1 = \lift{E}{X'}h'' = h'_2 \neq h'_1 = \lift{G}{X'}u\).
\end{example}
The final example shows that \(\forget\) does not preserve coequalisers. It also shows that there are parallel pairs of lenses for which the coequaliser of their get functors has no lens structure that coforks them.
\begin{example}
\label{U doesn't preserve all coequalisers}
Let \(\A\) be the preordered set generated by the graph
\[\begin{tikzcd}
Y_1 & X \arrow[l, "f_1" swap] \arrow[r, "f_2"] & Y_2
\end{tikzcd}\]
Let \(I \colon \A \to \A\) denote the identity lens, and let \(S \colon \A \to \A\) be the unique lens that maps \(X\) to \(X\), \(Y_1\) to \(Y_2\) and \(Y_2\) to \(Y_1\). The coequaliser of \(\forget I\) and \(\forget S\) in \(\smallCat{C}at\) is the unique functor \(E \colon \A \to \intervalCat\) that sends \(X\) to \(0\) and both \(Y_1\) and \(Y_2\) to \(1\). Recall that \(\terminalCat\) is terminal in \(\Lens\)~\cite{Clarke:2021:CategoryLens}. We claim that the coequaliser of \(I\) and~\(S\) in \(\Lens\) is the unique lens \(E \colon \A \to \terminalCat\). Let \(G \colon \A \to \smallCat{C}\) be a lens that coforks \(I\) and \(S\) in \(\Lens\). Let \(f = G f_1\). Then \(f = G f_1 = GI f_1 = GS f_1 = G f_2\). As \(\lift{G}{X}{f} \in \outSet{\A}{X}\), it is one of \(f_1\), \(f_2\) and \(\id{X}\). If \(\lift{G}{X}{f} = f_1\), then
\[f_1 = \lift{I}{X}{f_1} = \lift{I}{X}\lift{G}{X}{f} = \lift{(G \compose I)}{X}{f} = \lift{(G \compose S)}{X}{f} = \lift{S}{X}\lift{G}{X}{f} = \lift{S}{X}{f_1} = f_2,\]
which is a contradiction. We get a similar contradiction if \(\lift{G}{X}{f} = f_2\). By elimination, \(\lift{G}{X}{f} = \id{X}\), and so \(f = G\lift{G}{X}{f} = G\id{X} = \id{G X}\). The image of \(G\) thus consists of the object \(G X\) and the morphism \(\id{G X}\). If \(H \colon \terminalCat \to \smallCat{C}\) is a lens such that \(G = H \compose E\), then \(H\) must send \(0\) to \(G X\), and this uniquely determines~\(H\). As the image of any lens, in particular \(G\), is an out-degree-zero subcategory of its target category, this definition of \(H\) does indeed give a lens, and \(G = H \compose E\). Of course, the factorisation \(G = H \compose E\) is really the image factorisation of \(G\) from \cref{Factorisation System}.
\end{example}
\subsection*{Coequalisers which are reflected}
\label{Section: Coequalisers which are reflected}
Although the counterexamples above suggest that coequalisers in \(\Lens\) have little relation to those in \(\smallCat{C}at\), we will see in \cref{Pushouts in Lens} and \cref{Epic lens is regular} two classes of coequalisers in \(\Lens\) which do lie over coequalisers in \(\smallCat{C}at\). The following theorem, a partial converse to \cref{Necessary condition for existence of coequaliser}, reduces checking the coequaliser property in these cases to checking that \cref{Equation: Condition for reflection} below always holds.
\begin{theorem}
\label{Condition for reflection}
Let \(F_1, F_2 \colon \A \to \B\) be lenses. Let \(E \colon \B \to \smallCat{C}\) be a cofork of \(F_1\) and \(F_2\) in \(\Lens\), and suppose that \(\forget E\) coequalises \(\forget F_1\) and \(\forget F_2\) in \(\smallCat{C}at\). Then \(E\) coequalises \(F_1\) and \(F_2\) in \(\Lens\) if and only if for all lenses \(G \colon \B \to \D\) that cofork \(F_1\) and \(F_2\) in \(\Lens\), all \(B \in \objectSet{\B}\) and all \(d \in \outSet{\D}{GB}\), we have
\begin{equation}
\label{Equation: Condition for reflection}
\lift{G}{B}d = \lift{E}{B}E\lift{G}{B}d.
\end{equation}
\end{theorem}
In the proof of the following lemma and again, later, in the proof of \cref{Lemma: Pushout discrete opfibration and cosieve}, we use the induction principle for the equivalence relation \(\simeq\) on a set \(S\) generated by a binary relation \(R\) on \(S\), that is,
\begin{equation}
\label{Induction principle}
\forall P \; x_0 \; y_0.\
\left[
\begin{aligned}
&x_0 \simeq y_0\\
&\quad \land \quad \forall x \; y. \:\: x \mathrel{R} y \implies P (x, y)\\
&\quad \land \quad \forall x.\:\: P(x, x)\\
&\quad \land \quad \forall x \; y. \:\: \brack{x \simeq y \;\land\; P(x, y)} \implies P(y, x)\\
&\quad \land \quad \forall x \; y \; z. \:\: \brack{x \simeq y \;\land\; P(x, y) \;\land\; y \simeq z \;\land\; P (y, z)} \implies P(x, z)
\end{aligned}
\right]
\implies P(x_0, y_0).
\end{equation}
\begin{lemma}
\label{Comparison lens}
Let \(F_1, F_2 \colon \A \to \B\) be lenses. Let \(E \colon \B \to \smallCat{C}\) be a cofork of \(F_1\) and \(F_2\) in \(\Lens\), and suppose that \(\forget E\) coequalises \(\forget F_1\) and \(\forget F_2\) in \(\smallCat{C}at\). Let \(G \colon \B \to \D\) be a lens that coforks \(F_1\) and \(F_2\) in \(\Lens\), and let \(H \colon \smallCat{C} \to \D\) be the unique functor such that \(\forget G = H \compose \forget E\). Then there is a unique lens structure on \(H\) that, for all \(B \in \objectSet{\B}\) and all \(d \in \outSet{\D}{GB}\), satisfies the equation
\begin{equation}
\label{Equation: Definition of put}
\lift{H}{EB}d = E\lift{G}{B}d.
\end{equation}
\end{lemma}
\begin{proof}
For each \(C \in \objectSet{\smallCat{C}}\), as \(\forget E\) is epic, there is a \(B \in \objectSet{\B}\) such that \(EB = C\). Hence, we may define \(\lift{H}{C}\) using \cref{Equation: Definition of put}, so long as, for all \(B_1, B_2 \in \objectSet{\B}\), if \(E B_1 = E B_2\) then, for all \(d \in \outSet{\D}{E B_1}\), we have \(E\lift{G}{B_1} d = E \lift{G}{B_2} d\). Let \(\simeq\) be the smallest equivalence relation on \(\objectSet{\B}\) such that \(F_1A \simeq F_2A\) for all \(A \in \objectSet{\A}\). As \(\forget E\) coequalises \(\forget F_1\) and \(\forget F_2\) in \(\smallCat{C}at\), we have~\cite[Proposition~4.1]{Bednarczyk:1999:GeneralizedCongruences}, for all \(B_1, B_2 \in \objectSet{\B}\), that \(E B_1 = E B_2\) if and only if \(B_1 \simeq B_2\). We proceed using the induction principle in \cref{Induction principle}. The proof obligations from the reflexivity, symmetry and transitivity axioms for \(\simeq\) hold as~\(=\) is an equivalence relation. For the remaining one, for all \(A \in \objectSet{\A}\) and all \(d \in \outSet{\D}{F_1 A}\), we have
\[E\lift{G}{F_1 A} d
= EF_1\lift{F_1}{A}\lift{G}{F_1 A} d
= (E \compose F_1)\lift{(G \compose F_1)}{A} d
= (E \compose F_2)\lift{(G \compose F_2)}{A} d
= EF_2\lift{F_2}{A} \lift{G}{F_2 A}d
= E\lift{G}{F_2 A} d.\]
Define \(\lift{H}{C}\) using \cref{Equation: Definition of put}. It remains to check that the lens laws hold for \(H\). For all \(C \in \objectSet{\smallCat{C}}\), there is a \(B \in \objectSet{\B}\) such that \(EB = C\), and \(\lift{H}{C}\id{HC} = E\lift{G}{B}\id{GB} = E\id{B} = \id{C}\); hence \PutId{} holds. For all \(C \in \objectSet{\smallCat{C}}\), all \(d \in \outSet{\D}{HC}\) and all \(d' \in \outSet{\D}{\target d}\), there is a \(B \in \objectSet{\B}\) such that \(EB = C\), and
\[\lift{H}{C}(d' \compose d)
= E\lift{G}{B}(d' \compose d)
= E\paren[\big]{\lift{G}{B'} d' \compose \lift{G}{B} d}
= E\lift{G}{B'}d' \compose E\lift{G}{B} d
= \lift{H}{C'} d' \compose \lift{H}{C} d,\]
where \(B' = \target \lift{G}{B} d\) and \(C' = EB'\); hence \PutPut{} holds. Finally, for all \(C \in \objectSet{\smallCat{C}}\) and all \(d \in \outSet{\D}{HC}\), there is a \(B \in \objectSet{\B}\) such that \(EB = C\), and \(H\lift{H}{C} d = HE\lift{G}{B} d = G\lift{G}{B} d = d\); hence \PutGet{} holds.
\end{proof}
\begin{proof}[Proof of \cref{Condition for reflection}.]
We proved the \textit{only if} direction in \cref{Necessary condition for existence of coequaliser}. For the \textit{if} direction, suppose, for all lenses \(G \colon \B \to \D\) that cofork \(F_1\) and \(F_2\), that \cref{Equation: Condition for reflection} always holds. We must show that \(E\) is the universal cofork of \(F_1\) and \(F_2\) in \(\Lens\). Let \(G \colon \B \to \D\) be another cofork of \(F_1\) and \(F_2\) in \(\Lens\). Suppose that there is a lens \(H \colon \smallCat{C} \to \D\) such that \(G = H \compose E\). Then \(\forget G = \forget H \compose \forget E\), and so \(\forget H\) is the unique functor that composes with \(\forget E\) to give \(\forget G\). Let \(C \in \objectSet{\smallCat{C}}\) and \(d \in \outSet{\D}{HC}\). As \(\forget E\) is epic, there is a \(B \in \objectSet{\B}\) such that \(EB = C\). Then \(\lift{H}{C}d = E\lift{E}{B}\lift{H}{C}d = E \lift{(H \compose E)}{B}d = E \lift{G}{B}d\). Hence \(H\) is uniquely determined. Now let \(H \colon \smallCat{C} \to \D\) be the lens defined as in \cref{Comparison lens}. For all \(B \in \objectSet{\B}\) and all \(d \in \outSet{\D}{GB}\), we have \(\lift{G}{B}d = \lift{E}{B}E\lift{G}{B}d = \lift{E}{B}\lift{H}{EB}d = \lift{(H \compose E)}{B} d\), and so \(G = H \compose E\).
\end{proof}
\begin{corollary}
\label{Sufficient condition for reflection}
Let \(F_1, F_2 \colon \A \to \B\) be lenses. Let \(E \colon \B \to \smallCat{C}\) be a cofork of \(F_1\) and \(F_2\) in \(\Lens\), and suppose that \(\forget E\) coequalises \(\forget F_1\) and \(\forget F_2\). If \(\forget E\) is a discrete opfibration then \(E\) coequalises \(F_1\) and \(F_2\).
\end{corollary}
\begin{proof}
Let \(G \colon \B \to \D\) be a lens that coforks \(F_1\) and \(F_2\), let \(B \in \objectSet{\B}\) and let \(d \in \outSet{\D}{G B}\). Then \(\lift{G}{B}d\) and \(\lift{E}{B}E\lift{G}{B}d\) are both elements of \(\outSet{\B}{B}\) which are sent by \(E\) to the same morphism \(E\lift{G}{B}d\) of \(\smallCat{C}\). If \(\forget E\) is a discrete opfibration, then \(E\lift{G}{B}d\) has a unique lift to \(\outSet{\B}{B}\), and so \(\lift{G}{B}d\) and \(\lift{E}{B}E\lift{G}{B}d\) must be equal.
\end{proof}
\section{Pushouts of discrete opfibrations along monos}
\label{Section: Pushouts of discrete opfibrations along monos}
In the proof that \(\forget\) preserves epis (\cref{U preserves epis}), we used the well-known result that cosieves are pushout stable to explain why the pushout in \(\smallCat{C}at\) of the get functors of a span of monic lenses lifts uniquely to a commutative square in \(\Lens\); this lifted square is actually a pushout square in \(\Lens\). In this section, we will show, more generally, that \(\Lens\) has pushouts of discrete opfibrations along monics, and that \(\forget\) creates these pushouts. In what follows, we use square brackets for equivalence classes of elements.
Fritsch and Latch~\cite[Proposition~5.2]{FritschLatch:1981:HomotopyInversesForNerve} explicitly construct the pushout in \(\smallCat{C}at\) of a functor along a full monic functor. Specialising to when the full monic functor is a cosieve, and recalling that the image of a cosieve is out-degree-zero, we obtain the following simplification of Fritsch~and~Latch's construction.
\begin{proposition}
\label{Pushout of cosieve in Cat}
Let \(F \colon \A \to \smallCat{C}\) be a functor and \(J \colon \A \to \B\) be a cosieve. Then
\[\begin{tikzcd}
\A \arrow[r, tail, "J"] \arrow[d, "F" swap] & \B \arrow[d, "\Fbar"]\\
\smallCat{C} \arrow[r, tail, "\Jbar" swap] & \D
\end{tikzcd}\]
is a pushout square in \(\smallCat{C}at\) and \(\Jbar\) is a cosieve, where \(\D\), \(\Fbar\) and \(\Jbar\) are defined as follows:
\begin{itemize}
\item \emph{Object set:}
\[\objectSet{\D} = \objectSet{\smallCat{C}} \sqcup \paren[\big]{\objectSet{\B} \setDifference \objectSet{\A}}\]
\item \emph{Hom-sets:} for all \(C_1, C_2 \in \objectSet{\smallCat{C}}\) and all \(B_1, B_2 \in \objectSet{\B} \setDifference \objectSet{\A}\),
\begin{align*}
\homSet{\D}{C_1}{C_2} &= \homSet{\smallCat{C}}{C_1}{C_2}&
\homSet{\D}{C_1}{B_2} &= \emptyset\\
\homSet{\D}{B_1}{B_2} &= \homSet{\B}{B_1}{B_2}&
\homSet{\D}{B_1}{C_2} &= \paren[\big]{\smallCat{C}oprod_{A \in \objectSet{\A}} \homSet{\smallCat{C}}{FA}{C_2} \times \homSet{\B}{B_1}{A}} \big\slash {\sim}
\end{align*}
where \(\sim\) is the equivalence relation on \(\smallcoprod_{A \in \objectSet{\A}} \homSet{\smallCat{C}}{FA}{C_2} \times \homSet{\B}{B_1}{A}\) generated by \((c, a\compose b) \sim (c \compose F a, b)\) for all \(A_1, A_2 \in \objectSet{\A}\), all \(b \in \homSet{\B}{B_1}{A_1}\), all \(a \in \homSet{\A}{A_1}{A_2}\) and all \(c \in \homSet{\smallCat{C}}{FA_2}{C_2}\).
\item \emph{Composition:} for all \(B_1, B_2, B_3 \in \objectSet{\B} \setDifference \objectSet{\A}\), all \(A \in \objectSet{\A}\), all \(C_1, C_2, C_3 \in \objectSet{\smallCat{C}}\), all \(b_1 \in \homSet{\D}{B_1}{B_2}\), all \(b_2 \in \homSet{\D}{B_2}{B_3}\), all \(a \in \homSet{\D}{B_2}{A}\), all \(c \in \homSet{\D}{FA}{C_2}\), all \(c_1 \in \homSet{\D}{C_1}{C_2}\) and all \(c_2 \in \homSet{\D}{C_2}{C_3}\),
\begin{align*}
b_2 \compose_\D b_1 &= b_2 \compose_\B b_1&
\brack{(c, a)} \compose_\D b_1 &= \brack{(c, a \compose_\B b_1)}\\
c_2 \compose_\D c_1 &= c_2 \compose_\smallCat{C} c_1 &
c_2 \compose_\D \brack{(c, a)} &= \brack{(c_2 \compose_\smallCat{C} c, a)}
\end{align*}
\item \emph{Identity morphisms:} same as in \(\B\) and \(\smallCat{C}\).
\item \emph{Injections:} the functor \(\Jbar \colon \smallCat{C} \to \D\) is the obvious inclusion of \(\smallCat{C}\) as a full subcategory of \(\D\); the functor \(\Fbar \colon \B \to \D\) is defined, for all \(B, B' \in \objectSet{\B} \setDifference \objectSet{\A}\), all \(A, A' \in \objectSet{\A}\), all \(b \in \homSet{\B}{B}{B'}\), all \(b' \in \homSet{\B}{B}{A}\) and all \(a \in \homSet{\B}{A}{A'}\), as follows:
\begin{align*}
\Fbar B &= B &&& \Fbar A &= F A \\
\Fbar b &= b & \Fbar b' &= \brack{(\id{FA}, b')} & \Fbar a &= F a
\end{align*}
\end{itemize}
\end{proposition}
\begin{theorem}
\label{Pushout discrete opfibration and cosieve}
The pushout in \(\smallCat{C}at\) of a discrete opfibration along a cosieve is a discrete opfibration.
\end{theorem}
\begin{lemma}
\label{Lemma: Pushout discrete opfibration and cosieve}
Let \(F \colon \A \to \smallCat{C}\) be a discrete opfibration, let \(J \colon \A \to \B\) be a cosieve, let \(B \in \objectSet{\B} \setDifference \objectSet{\A}\) and let \(C \in \objectSet{\smallCat{C}}\). Then, for all \(A_1, A_2 \in \A\), all \(b_1 \in \homSet{\B}{B}{A_1}\), all \(b_2 \in \homSet{\B}{B}{A_2}\), all \(c_1 \in \homSet{\smallCat{C}}{FA_1}{C}\) and all \(c_2 \in \homSet{\smallCat{C}}{FA_2}{C}\), if \((c_1, b_1) \sim (c_2, b_2)\) then \(\lift{F}{A_1}c_1 \compose b_1 = \lift{F}{A_2}c_2 \compose b_2\).
\end{lemma}
\begin{proof}
We proceed by induction, using the induction principle for \(\sim\) in \cref{Induction principle}. The proof obligations from the reflexivity, symmetry and transitivity axioms for \(\sim\) hold because \(=\) is an equivalence relation. For the remaining proof obligation, for all \(A_1, A_2 \in \objectSet{\A}\), all \(b \in \homSet{\B}{B}{A_1}\), all \(a \in \homSet{\A}{A_1}{A_2}\) and all \(c \in \homSet{\smallCat{C}}{FA_2}{C}\), we have \(\lift{F}{A_1}Fa = a\) as \(F\) is a discrete opfibration, and so
\[\lift{F}{A_2}c \compose (a \compose b) = \lift{F}{A_2}c \compose \lift{F}{A_1}Fa \compose b = \lift{F}{A_1} (c \compose Fa) \compose b.\qedhere\]
\end{proof}
\begin{proof}[Proof of \cref{Pushout discrete opfibration and cosieve}.]
Using the notation of \cref{Pushout of cosieve in Cat}, suppose that \(F\) is a discrete opfibration. We must show that \(\Fbar\) is also a discrete opfibration. Let \(B \in \objectSet{\B}\) and \(d \in \outSet{\D}{\Fbar B}\).
Suppose that \(B \in \objectSet{\A}\). Then \(\Fbar B = F B\), and \(d \in \outSet{\smallCat{C}}{FB}\). As \(F\) is a discrete opfibration, there is a unique \(a \in \outSet{\A}{B}\) such that \(d = Fa\). But \(\outSet{\A}{B} = \outSet{\B}{B}\) as \(\A\) is out-degree-zero in \(\B\); also \(\Fbar a = F a\) for each \(a \in \outSet{\B}{B}\). Hence there is a unique \(a \in \outSet{\B}{B}\) such that \(d = \Fbar a\).
Suppose that \(B \in \objectSet{\B} \setDifference \objectSet{\A}\) and \(\target d \in \objectSet{\B} \setDifference \objectSet{\A}\). Then \(\Fbar B = B\), \(d \in \outSet{\B}{B}\) and \(\Fbar d = d\). As \(\Fbar\) is injective on the morphisms of \(\B\) not in \(\A\), \(d\) is the unique morphism in \(\outSet{\B}{B}\) mapped by \(\Fbar\) to~\(d\).
Otherwise, \(B \in \objectSet{\B} \setDifference \objectSet{\A}\) and \(\target d \in \objectSet{\smallCat{C}}\). Then \(\Fbar B = B\), and \(d = \brack{(c_1, b_1)}\) for some \(A_1 \in \objectSet{\A}\), some \(b_1 \in \homSet{\B}{B}{A_1}\) and some \(c_1 \in \homSet{\smallCat{C}}{FA_1}{C}\), where \(C = \target d\). For uniqueness of lifts, suppose that \(b_2 \in \outSet{\B}{B}\) is such that \(d = \Fbar b_2\). Let \(A_2 = \target b_2\). Then \(A_2 \in \objectSet{\A}\) as \(\Fbar A_2 = \target d = C\), and so \(\Fbar b_2 = \brack{(\id{C},b_2)}\). As \(d = \Fbar b_2\), we have \((\id{C},b_2) \sim (c_1, b_1)\). By \cref{Lemma: Pushout discrete opfibration and cosieve}, \(b_2 = \lift{F}{A_2}\id{C} \compose b_2 = \lift{F}{A_1} c_1 \compose b_1\); this determines~\(b_2\). For existence of lifts, note that \(\Fbar (\lift{F}{A_1} c_1 \compose b_1) = \brack{(\id{C},\lift{F}{A_1} c_1 \compose b_1)} = \brack{(F\lift{F}{A_1} c_1, b_1)} = \brack{(c_1, b_1)} = d\).
\end{proof}
\begin{theorem}
\label{Pushouts in Lens}
The functor \(\forget\) creates pushouts of monic lenses with discrete opfibrations.
\end{theorem}
\begin{proof}
Using the notation of \cref{Pushout of cosieve in Cat}, suppose that \(F\) is a discrete opfibration. Then \(\Fbar\) is also a discrete opfibration (\cref{Pushout discrete opfibration and cosieve}). Let \(J_\B \colon \B \to \B \sqcup \smallCat{C}\) and \(J_\smallCat{C} \colon \smallCat{C} \to \B \sqcup \smallCat{C}\) be the coproduct injection functors. Coproduct injections in \(\smallCat{C}at\) are always discrete opfibrations, as is the coproduct copairing of any two discrete opfibrations. Hence \(J_\B\), \(J_\smallCat{C}\) and \(\copair{\Jbar}{\Fbar}\) are all discrete opfibrations. As the composite of two discrete opfibrations is a discrete opfibration, so are \(J_\B \compose J\) and \(J_\smallCat{C} \compose F\). So far, we know that \(\copair{\Jbar}{\Fbar}\) is the coequaliser in \(\smallCat{C}at\) of \(J_\B \compose J\) and \(J_\smallCat{C} \compose F\), all of these functors have canonical lens structures as they are discrete opfibrations, and \(\copair{\Jbar}{\Fbar}\) coforks \(J_\B \compose J\) and \(J_\smallCat{C} \compose F\) in \(\Lens\). As \(\copair{\Jbar}{\Fbar}\) is a discrete opfibration, the conditions of \cref{Condition for reflection} are satisfied, and so \(\copair{\Jbar}{\Fbar}\) coequalises \(J_\B \compose J\) and \(J_\smallCat{C} \compose F\) in \(\Lens\). As \(\forget\) creates coproducts~\cite{Clarke:2021:CategoryLens}, it follows that \(\Jbar\) and \(\Fbar\) exhibit \(\D\) as the pushout of \(J\) and \(F\) in \(\Lens\).
\end{proof}
One might hope that the above result generalises to pushouts of two discrete opfibrations, or of arbitrary lenses along monics; this is not the case. The following is an example of two discrete opfibrations whose pushout injection functors have no lens structures that give a commutative square of lenses.
\begin{example}
Let \(\A\) and \(\B\) be the preordered sets generated respectively by the following graphs.
\begin{align*}
\begin{tikzcd}[ampersand replacement=\&, row sep={0em}]
Y'_1 \& X' \arrow[l, "f_1'"{swap}] \arrow[r, "f_2'"] \& Y_2' \\ Y_1'' \& X'' \arrow[l, "f_1''"] \arrow[r, "f_2''" swap] \& Y_2''
\end{tikzcd}
&&
\begin{tikzcd}[ampersand replacement=\&]
Y_1 \& X \arrow[l, "f_1"{swap}] \arrow[r, "f_2"] \& Y_2
\end{tikzcd}
\end{align*}
Let \(F \colon \A \to \B\) be the unique functor that sends both \(X'\) and \(X''\) to \(X\), both \(Y_1'\) and \(Y_1''\) to \(Y_1\), and both \(Y_2'\) and \(Y_2''\) to \(Y_2\). Let \(G \colon \A \to \B\) be the unique functor that sends both \(X'\) and \(X''\) to \(X\), both \(Y_1'\) and \(Y_2''\) to \(Y_1\), and both \(Y_2'\) and \(Y_1''\) to \(Y_2\). Both \(F\) and \(G\) are discrete opfibrations. Their pushout in \(\smallCat{C}at\) is \(\intervalCat\); the pushout injections \(\Fbar, \Gbar \colon \B \to \intervalCat\) are both the unique functor that sends \(X\) to \(0\), and both \(Y_1\) and \(Y_2\) to~\(1\). There are two different lens structures on this functor; one lifts the unique morphism \(u\) of \(\intervalCat\) to \(f_1\), the other lifts it to~\(f_2\). This gives four different combinations of lens structures on \(\Fbar\) and \(\Gbar\). Assume, for a contradiction, that one of these combinations satisfies \(\Fbar G = \Gbar F\) in \(\Lens\). As \(\lift{G}{X'}\lift{\Fbar}{X}u = \lift{F}{X'}\lift{\Gbar}{X}u\), we must have \(\lift{\Fbar}{X}u = \lift{\Gbar}{X}u\). If \(\lift{\Fbar}{X}u = f_1\), then \(\lift{G}{X''}\lift{\Fbar}{X}u = \lift{G}{X''}f_1 = f_2'\) and \(\lift{F}{X''}\lift{\Gbar}{X}u = \lift{F}{X''}f_1 = f_1' \neq f_2'\), which is a contradiction. If \(\lift{\Fbar}{X}u = f_2\), we obtain a similar contradiction.
\end{example}
Next is an example of a lens and a cosieve where the pushout of the get functor of the lens along the cosieve does not have a lens structure (incidentally this lens and cosieve do not have a pushout in \(\Lens\)).
\begin{example}
Let \(\B\) and \(\D\) be the preordered sets generated respectively by the following graphs.
\begin{align*}
\begin{tikzcd}[ampersand replacement=\&]
X \arrow[d, "s" swap] \& W \arrow[l, "f" swap] \arrow[r, "g"] \arrow[d] \& Y \arrow[d, "t"]\\
Z_2 \& Z_1 \arrow[l] \arrow[r]\& Z_3
\end{tikzcd}
&&
\begin{tikzcd}[ampersand replacement=\&]
X' \arrow[dr, "s'" swap] \& W' \arrow[l, "f'" swap] \arrow[r, "g'"] \arrow[d] \& Y' \arrow[dl, "t'"]\\
\& Z'\&
\end{tikzcd}
\end{align*}
Let \(\A\) be the out-degree-zero subcategory of \(\B\) on the objects \(Z_1\), \(Z_2\) and \(Z_3\), and let \(J \colon \A \monicTo \B\) be the inclusion lens. As \(\terminalCat\) is terminal in \(\Lens\)~\cite{Clarke:2021:CategoryLens}, there is a unique lens \(F \colon \A \to \terminalCat\). By \cref{Pushout of cosieve in Cat}, the pushout of \(\forget F\) along \(\forget J\) in \(\smallCat{C}at\) is the unique functor \(\Fbar \colon \B \to \D\) that maps \(W\) to \(W'\), \(X\) to \(X'\), \(Y\) to \(Y'\), and all of \(Z_1\), \(Z_2\) and \(Z_3\) to~\(Z'\). The functor \(\Fbar\) has no lens structure, otherwise we could derive the contradiction
\[s \compose f = \lift{\Fbar}{X}s' \compose \lift{\Fbar}{W} f' = \lift{\Fbar}{W} (s' \compose f') = \lift{\Fbar}{W} (t' \compose g') = \lift{\Fbar}{Y}t' \compose \lift{\Fbar}{W} g' = t \compose g.\]
\end{example}
From \cref{Pushouts in Lens}, every monic lens has a cokernel pair. Actually, using the epi-mono factorisation, every lens has a cokernel pair, namely, the cokernel pair of its mono factor.
\begin{proposition}
Every monic lens is effective (i.e.\ equalises its cokernel pair).
\end{proposition}
\begin{proof}
Let \(M \colon \A \to \B\) be a monic lens, and let \(J_1, J_2 \colon \B \to \smallCat{C}oker M\) be its cokernel pair. Based on \cref{Pushout of cosieve in Cat}, if \(B \in \objectSet{\B}\) is such that \(J_1B = J_2B\), then \(B \in \objectSet{\A}\); and similarly for morphisms of \(\B\). In particular, the image of any lens which forks \(J_1\) and \(J_2\) is contained in \(\A\), and thus its corestriction to \(\A\) is the unique comparison lens.
\end{proof}
\begin{corollary}
In \(\Lens\), the classes of all monos, effective monos, regular monos, strong monos and extremal monos coincide.
\end{corollary}
\begin{corollary}
Every lens that is both epic and monic is an isomorphism.
\end{corollary}
\section{Regular epic lenses}
\label{Section: Regular epic lenses}
In this section, we show that all epis in \(\Lens\) are regular. This gives us another class of coequalisers in \(\Lens\), namely, the epic lenses. For contrast, recall that not all epis in \(\smallCat{C}at\) are regular.
\begin{example}
In \cref{Example epic functor not surjective on morphisms}, we saw that the functor \(J \colon \intervalCat \to \isomorphismCat\) is epic. It is, however, not a regular epi. Indeed, if \(J\) coforks \(F_1, F_2 \colon \A \to \intervalCat\), then \(F_1 = F_2\) as \(J\) is monic, and so \(\id{\intervalCat}\) is the coequaliser of \(F_1\) and \(F_2\), but \(\intervalCat\) and \(\isomorphismCat\) are not isomorphic.
\end{example}
\begin{proposition}
\label{Epic lens over effective epic functor}
The get functor of every epic lens is an effective epi in~\(\smallCat{C}at\).
\end{proposition}
A functor \(E \colon \B \to \smallCat{C}\) is \textit{surjective on composable pairs} if for each composable pair \((c, c')\) of \(\smallCat{C}\), there is a composable pair \((b, b')\) of \(\B\) such that \(Eb = c\) and \(Eb' = c'\); such functors are necessarily also surjective on objects and morphisms. If \(E \colon \B \to \smallCat{C}\) is an epic lens, then \(\forget E\) is surjective on composable pairs; indeed, if \((c, c')\) is a composable pair of \(\smallCat{C}\), then there is a \(B \in \objectSet{\B}\) such that \(EB = \source c\), and \((\lift{E}{B}c, \lift{E}{\target \lift{E}{B}c}c')\) is a composable pair above \((c, c')\). Hence it suffices to prove the following lemma.
\begin{lemma}
\label{Effective epic functors}
All functors that are surjective on composable pairs are effective epis in \(\smallCat{C}at\).
\end{lemma}
\begin{proof}
Let \(E \colon \B \to \smallCat{C}\) be a functor that is surjective on composable pairs, and let its kernel pair be \(F_1, F_2 \colon \Ker E \to \B\). We must show that \(E\) coequalises \(F_1\) and \(F_2\). Let \(G \colon \B \to \D\) cofork \(F_1\) and~\(F_2\).
Suppose that there is a functor \(H \colon \smallCat{C} \to \D\) such that \(G = H \compose E\). As \(E\) is surjective on objects, for all \(C \in \objectSet{\smallCat{C}}\) there is a \(B \in \objectSet{\B}\) such that \(EB = C\), and so \(HC = HEB = GB\); this equation determines \(H\) on objects. As \(E\) is surjective on morphisms, a similar equation determines \(H\) on morphisms.
To define \(H \colon \smallCat{C} \to \D\) with these equations, the values of \(GB\) and \(Gb\) should be independent of the choice of \(B\) above \(C\) and \(b\) above \(c\). For all \(C \in \objectSet{\smallCat{C}}\) and all \(B, B' \in \objectSet{\B}\) such that \(EB = EB' = C\), we have \(G B = GF_1 \pair{B}{B'} = GF_2 \pair{B}{B'} = G B'\), where \(\pair{B}{B'} \in \objectSet{\Ker E}\) comes from the pullback property; hence the object map of \(H\) is well defined. Its morphism map is similarly also well defined.
Define \(H\) with the above equations. By construction, \(G = H \compose E\). We must show that \(H\) is a functor. For all \(C \in \objectSet{\smallCat{C}}\), there is a \(B \in \objectSet{\B}\) such that \(EB = C\), and
\(H \id{C} = G \id{B} = \id{G B} = \id{H C}\); thus \(H\) preserves identities. For all composable pairs \(c\) and \(c'\) of \(\smallCat{C}\), there is a composable pair \(b\) and \(b'\) of \(\B\) such that \(Eb = c\) and \(Eb' = c'\), and \(H(c' \compose c) = G(b' \compose b) = Gb' \compose G b = Hc' \compose H c\); thus \(H\) preserves composites.
\end{proof}
\begin{corollary}
\label{Epic lens is regular}
Every epic lens coequalises its proxy kernel pair, and so is regular.
\end{corollary}
\begin{proof}
Let \(E \colon \B \to \smallCat{C}\) be an epic lens. Let \(F_1, F_2 \colon \Ker \forget E \to \B\) be the proxy kernel pair of \(E\) in \(\Lens\). By \cref{Epic lens over effective epic functor}, \(\forget E\) coequalises \(\forget F_1\) and \(\forget F_2\) in \(\smallCat{C}at\). Let \(G \colon \B \to \D\) be a lens that coforks \(F_1\) and \(F_2\), let \(B \in \objectSet{\B}\), let \(d \in \outSet{\D}{GB}\), and let \(C = EB\). Then \(\lift{(G \compose F_1)}{\pair{B}{B}} d
= \lift{F_1}{\pair{B}{B}}\lift{G}{B} d
= \pair[\big]{\lift{G}{B} d}{\lift{E}{B}E\lift{G}{B} d}\), and similarly \(\lift{(G \compose F_2)}{\pair{B}{B}}{d} = \pair[\big]{\lift{E}{B}E\lift{G}{B} d}{\lift{G}{B}d}\). As \(G\) coforks \(F_1\) and \(F_2\), we have \(\lift{G}{B} d = \lift{E}{B}E\lift{G}{B} d\). By \cref{Condition for reflection}, \(E\) coequalises \(F_1\) and \(F_2\) in \(\Lens\).
\end{proof}
\begin{corollary}
In \(\Lens\), the classes of all epis, regular epis, strong epis and extremal epis coincide.
\end{corollary}
\begin{corollary}
In \(\Lens\), the class of all morphisms that are left orthogonal to the class of all monos is the class of all epis.
\end{corollary}
\begin{proof}
As \(\Lens\) has equalisers~\cite{Clarke:2021:CategoryLens}, every morphism that is left orthogonal to the class of all monos is an epi. Conversely, we have already shown that every epi is a strong epi.
\end{proof}
\begin{remark}
As every lens factors as an epi followed by a mono~(\cref{Factorisation System}), it follows that the class of all epis and the class of all monos together form an orthogonal factorisation system on \(\Lens\).
\end{remark}
\section{Conclusion}
\label{Section: Conclusion}
In this article, we have seen a number of results which advance our understanding of the category \(\Lens\) of (asymmetric delta) lenses. We now have a complete elementary characterisation of the monos and epis in \(\Lens\), the monos being the unique lenses on cosieves and the epis being the surjective-on-objects lenses; from this, we see that Johnson and Roseburgh’s factorisation system on \(\Lens\)~\cite{JohnsonRoseburgh:2021:TheMoreLegsTheMerrier} is actually an epi-mono factorisation system. We have also initiated a study of the coequalisers in \(\Lens\). Despite \(\Lens\) not having all coequalisers, nor the forgetful functor from \(\Lens\) to \(\smallCat{C}at\) preserving or reflecting them, we have two interesting positive results. First, every epic lens coequalises its proxy kernel pair. Second, \(\Lens\) has pushouts of discrete opfibrations along cosieves. Our characterisation of the epic lenses played a central role in the proof of both of these results, and hopefully will enable future work to completely characterise the coequalisers in \(\Lens\).
That every epic lens coequalises its proxy kernel pair is yet another result that emphasises the parallels between proxy pullbacks in \(\Lens\) and real pullbacks in other categories. An interesting question for future work is whether there is an axiomatisation of the notion of proxy pullback from which one may prove interesting general results which also apply to other categories. Existing work in this direction include Bumpus and Kocsis’ \textit{proxy pushout}~\cite{bumpus:2021:spined-categories:-generalizing-tree-width}, which inspired our use of the name proxy pullback, as well as Böhm’s \textit{relative pullbacks}~\cite{bohm:2019:crossed-modules-monoids-relative} and Simpson’s \textit{local independent products}~\cite{simpson:2018:category-theoretic-structure-for-independence}. One potential use for such an axiomatised proxy pullback would be to give a generalised notion of regular category; the category \(\Lens\) is an obvious candidate example from which to draw inspiration. This notion of a proxy regular category may even be helpful for understanding symmetric lenses, which are known to be equivalence classes of spans of asymmetric ones, as some kind of relations in \(\Lens\), although this is as yet merely speculation.
\end{document} |
\begin{document}
\title{Quasiperiodic Poincar\'{e} Persistence at High Degeneracy}
\author{Weichao Qian, Yong Li, Xue Yang
}
\date{}
\maketitle
\begin{abstract}
For Hamiltonian systems with degeneracy of any higher order, we study the persistence of resonant invariant tori, which as some lower-dimensional invariant tori might be elliptic, hyperbolic or of mixed types. Hence we prove a quasiperiodic Poincar\'{e} theorem at high degeneracy. This answers a long standing conjecture on the persistence of resonant invariant tori in quite general situations.
{\bf Keywords} {Hamiltonian systems; high degeneracy; KAM theory; Treshch\"{e}v's resonant invariant tori, quasiperiodic Poincar\'{e} theorem.}
\varepsilonnd{abstract}
\section{Introduction}\label{introduction}
This paper concerns the persistence of resonant invariant tori for the following Hamiltonian system
\begin{eqnarray}\label{005}
H(x,y)=H_0 (y) + \varepsilon P(x,y,\varepsilon),
\varepsilonnd{eqnarray}
where $x \in T^d = R^d/ Z^d$, $y\in G,$ a closed region in $R^d$; $H_0 (y)$ is a real analytic function
on a complex neighborhood of the bounded closed region $T^d \times G$; $\varepsilon P(x, y, \varepsilon)$, a general small perturbation, is a real analytic function, and $\varepsilon > 0$ is a small parameter. Here the so-called resonant invariant tori mean the frequency $\omega(y) = \frac{\partial H_0}{\partial y}$ does not satisfy rationally independence. We will give the accurate statement below.
The celebrated KAM theory due to Kolmogorov, Arnold and Moser asserts that, if an integrable system, $H_0(y)$, is nondegenerate, i.e. $\det \partial_y^2 H_0(y) \neq 0$, then for the perturbed system $H (x,y) = H_0(y)+ \varepsilon P(x,y,\varepsilon),$ most of nonresonant invariant tori still survive (\cite{Arnold,Kolmogorov,Moser}). For some recent developments and applications of KAM theory, refer to \cite{Guardia,han,Kaloshin1,Meyer,Palacian,Palacian1,Qian}. However, in the presence of resonance, the persistence problem will become very complicated. Let us make a brief recall. The periodic case can go back to
the work of Poincar\'{e} in nineteenth century, which does not involve the small divisor problem(\cite{Poincare}). There has been a long standing conjecture about resonant tori under a convexity assumption on $H_0$
(\cite{Broer,Cong,Livia,Gentile,Kappeler}), as written by Kappeler and P\"{o}schel in \cite{Kappeler}:
\begin{itemize}
\item[] \varepsilonmph{For $m = 1 $ in particular, such a torus is foliated into identical closed orbits. Bernstein $\&$ Katok $(\varepsilonmph{\cite{Bernstein}})$ showed that in a convex system at least $n$ of them survive any sufficiently small perturbation. $\cdots$ For the intermediate cases with $1<m<n-1$, only partial results are known $\cdots$. The long standing conjecture is that at least $n- m +1$, and generically $2^{n-m}$, invariant $m-$tori always survive in a nondegenerate system $\cdots$. That is, their number should be equal to the number of critical points of smooth functions on the torus $T^{n-m}$.}
\varepsilonnd{itemize}
In above description, $n$ and $m$ are dimensions on the degree of freedom of the system and lower-dimensional invariant tori, respectively.
The first breakthrough of the conjecture mentioned above was due to Treshch\"{e}v (\cite{Treshchev}) for the persistence of hyperbolic resonant tori in 1989, 35 years after the establishment of KAM theory, and such tori have been called Treshch\"{e}v's ones today. For the persistence of general resonant tori, we refer readers to \cite{Cong,Li}, and for multi-scale Hamiltonian systems, refer to \cite{Xu2,xu3}. In fact, when the 1-order perturbation in $\varepsilon$ is nondegenerate, ones have completed the proof of the conjecture mentioned above, see \cite{Cong,Li,Treshchev}. However,
\begin{itemize}
\item[] \varepsilonmph{is the conjecture true if the perturbation is of high degeneracy?}
\varepsilonnd{itemize}
In the present paper we will touch this essential problem.
In order to state our main result, first, let us introduce some notations. We say that a frequency vector $\omega = {\partial_y H_0} (y)$ is nonresonant, if $\langle k, \omega \rangle \neq 0$ for any $k \in {Z^d \setminus \{0\}}$. Furthermore, if there is a subgroup $g$ of $Z^d$ such that $\langle k , \omega\rangle = 0$ for all $k \in g$ and $\langle k , \omega \rangle \neq 0$ for all $k \in {Z^d / g}$, then $\omega$ is called an multiplicity $m_0$ resonant frequency ($g-$resonant frequency), where $g$ is generated by independent $d-$dimensional integer vector $\tau _1, \ldots , \tau _ {m_0}$. For a given subgroup $g$, the $m~(= d - m_0)$ dimensional surface
\begin{eqnarray*}
\widetilde{\Lambda}(g, G)=\{y \in G : \langle k , \omega \rangle =0, k\in g \}
\varepsilonnd{eqnarray*}
is called the $g-$resonant surface. Following the way in \cite{Treshchev}, by group theory, there are integer vectors ${\tau _1'}, \cdots, {\tau _m'}$ $\in$ $Z ^ d$, such that $Z^d$ is generated by ${\tau _1}, \cdots, {\tau _{m_0}},{\tau _1'}, \cdots, {\tau _m'}$, and~ $\det K_0 = 1 $, where $K_0 = (K_* , K^{'})$,~ $K_* = (\tau _1', \cdots, \tau _m')$, $K^{'}=({\tau _1}, \cdots, {\tau _{m_0}}) $ are ~$d\times d$, $d\times m $, $d\times m_0$, respectively, and $K_{*}$ generates the quotient group $Z^d / g$, while $K^{'}$ generates the group $g$. If $H_0$ is nondegenerate and $\det {K^{'}}^T \partial_y^2 {H_0}{K^{'}} \neq 0$ for $y \in \widetilde{\Lambda}(g,G)$, then $H_0$ is said to be $g-$nondegenerate.
Let
\begin{eqnarray*}
\Gamma = K_0^T \partial_y^2 H_0 (y_0) K_0 = \left(
\begin{array}{cc}
\Gamma_{11} & \Gamma_{12} \\
\Gamma_{21} & \Gamma_{22} \\
\varepsilonnd{array}
\right),
\varepsilonnd{eqnarray*}
where $\Gamma_{11},$ $\Gamma_{12}$, $\Gamma_{21},$ $\Gamma_{22}$ are $m\times m,$ $m\times m_0$, $m_0\times m,$ $m_0\times m_0$ matrices, respectively, $\Gamma_{12} = \Gamma_{21}^T$, $\Gamma_{22} = {K'}^T \partial_y^2 H_0 (y_0) K'$, and $m_0 = d - m$.
For any $y_0 \in \widetilde{\Lambda}(g, G)$, with the help of the Taylor expansion at $y_0$ and the following coordinate transformation $y- y_0 = K_0 p$, $q = K_0 ^ T x$, Hamiltonian (\ref{005}) is changed to
\begin{eqnarray}\label{qq}
H(q,p)&=& \langle\omega^{*} , p'\rangle + \frac{1}{2}\langle p, \Gamma(\omega^{*}) p\rangle + O(|K_0 p|^3) + \varepsilon \bar{P} (q, p, \varepsilon)
\varepsilonnd{eqnarray}
up to an irrelative constant, where
\begin{eqnarray*}
\omega^{*} &=& K_{*}^T \omega(y_0) \in \Lambda (g,G),~~~~~~ \Lambda(g,G)= \{\omega^{*} \in R^m : y\in \widetilde{\Lambda}(g,G)\},\\
p&=&(p', p''),~~~~~~~~~~p' = (p_1, \cdots ,p_m)^T,~~~~~~~~~~p'' = (p_{m+1}, \cdots ,p_d)^T,\\
\bar{P}(q,p,\varepsilon) &=& P((K_0 ^T)^{-1} q, y_0 + K_0 p,\varepsilon).
\varepsilonnd{eqnarray*}
Here we used the fact that $\Lambda(g, G)$ is diffeomorphic to the $m-$dimensional surface $\widetilde{\Lambda}(g,G)$.
\begin{remark}
The coordinate transformation: $y- y_0 = K_0 p$, $q = K_0 ^ T x$ is symplectic. In fact,
\begin{eqnarray*}
\left(
\begin{array}{c}
x \\
y -y_0 \\
\varepsilonnd{array}
\right) = \left(
\begin{array}{cc}
(K_0^T)^{-1} & 0 \\
0 & K_0 \\
\varepsilonnd{array}
\right).
\varepsilonnd{eqnarray*}
Then
\begin{eqnarray*}
&~&\left(
\begin{array}{cc}
((K_0^T)^{-1})^T & 0 \\
0 & K_0^T \\
\varepsilonnd{array}
\right)
\left(
\begin{array}{cc}
0 & I \\
-I & 0 \\
\varepsilonnd{array}
\right)
\left(
\begin{array}{cc}
(K_0^T)^{-1} & 0 \\
0 & k_0 \\
\varepsilonnd{array}
\right)\\
&=& \left(
\begin{array}{cc}
K_0^{-1} & 0 \\
0 & K_0^T \\
\varepsilonnd{array}
\right)
\left(
\begin{array}{cc}
0 & I \\
-I & 0 \\
\varepsilonnd{array}
\right)
\left(
\begin{array}{cc}
(K_0^T)^{-1} & 0 \\
0 & k_0 \\
\varepsilonnd{array}
\right)\\
&=& \left(
\begin{array}{cc}
0 & I \\
-I & 0 \\
\varepsilonnd{array}
\right),
\varepsilonnd{eqnarray*}
which means the coordinate transformation is symplectic.
\varepsilonnd{remark}
By the following symplectic transformation:
\begin{eqnarray*}
p \rightarrow \varepsilon^{\frac{1}{4}}p, ~q \rightarrow q, ~H \rightarrow\varepsilon^{-\frac{1}{4}}H,
\varepsilonnd{eqnarray*}
Hamiltonian (\ref{qq}) is changed to
\begin{eqnarray}\label{N1}
H(q,p)&=& \langle\omega^{*} , p'\rangle + \frac{\varepsilon^{\frac{1}{4}}}{2}\langle p, \Gamma(\omega^{*}) p\rangle + \varepsilon^{\frac{1}{2}}O(|K_0 p|^3) + \varepsilon^{\frac{3}{4}} \bar{P} (q, p, \varepsilon).
\varepsilonnd{eqnarray}
For $\omega \in \Lambda(g,G)$, replace $p'$, $p''$, $q'$, $q''$, $H(p', p'', q', q'')$ by ${y}$, ${v}$, ${x}$, ${u}$, ${H}(x,y,u,v)$ respectively and rewrite $\varepsilon^{\frac{1}{4}}$ as $\varepsilon$. Then Hamiltonian (\ref{N1}) arrives at
\begin{eqnarray}\label{082}
\nonumber H(x,y,u,v) &=& \langle\omega, y\rangle + \frac{\varepsilon}{2} \langle \left(
\begin{array}{c}
y \\
v \\
\varepsilonnd{array}
\right)
, \Gamma \left(
\begin{array}{c}
y \\
v \\
\varepsilonnd{array}
\right)
\rangle\\
&~&~+ \varepsilon^2 O(|K_0 \left(
\begin{array}{c}
y \\
v \\
\varepsilonnd{array}
\right)
|^3) + \varepsilon^3 P_1(x,y,u,v, \varepsilon).
\varepsilonnd{eqnarray}
Since the mean $[P_1(\cdot , y,u,v,0)]= \int_{T^m} P_1(x, y,u,v,0) dx$ is $T^{m_0} - $periodic in $u$, there are at least $m_0 + 1$ critical points for the mean $[P_1(\cdot , y,u,v,0)]$ (\cite{Milnor}). Treshch\"{e}v $(\cite{Treshchev})$ dealt with general hyperbolic critical points. In \cite{Cong}, Cong, K\"{u}pper, Li and You dealt with the persistence of resonant invariant tori when $[P_1(\cdot,y,u,v,0)]$ possesses nondegenerate critical points, where $[P_1(\cdot,y,u,v,0)]$ corespondents to $0-th$ order Taylor expansion of $[P_1(x, y,u,v,\varepsilon)]$ in $\varepsilon$. Li and Yi$(\cite{Li})$ further removed the $g-$degeneracy. However, if the term about $0-th$ order Taylor expansion in $\varepsilon$ of the mean perturbation on angle variable is degenerate, the conjecture is that there are at least $m_0+1$ invariant tori.
Now we are in a position to state our main result for (\ref{005}) on $\widetilde{\Lambda}(g,G)$.
\begin{theorem}\label{dingli11}
Let $H$ be real analytic
on the complex neighborhood of $T^d \times G$. Assume
\begin{enumerate}
\item [\bf{(S1)}.] $\omega_* = K_*^T \partial_y H_0(y)$ satisfies R\"{u}ssmann non-degenerate condition;
\item [\bf{(S2)}.]$ rank ~K_0^T \partial_y^2 H_0 (y) K_0 = n+ m_0$, $0<n\leq m$, and $rank((K')^T \partial_y^2 H_0 K_*, (K')^T \partial_y^2 H_0 K') = m_0$ for a given $g$ on $\widetilde{\Lambda}(g,G);$
\item [\bf{(S3)}.] $rank \left(
\begin{array}{cc}
K & \bar{\omega}_* \\
\bar{\omega}_*^T & 0 \\
\varepsilonnd{array}
\right) = n + 2m_0+1$ for a given $g$ on $\widetilde{\Lambda}(g,G), $ where $\bar{\omega}_* = \left(
\begin{array}{c}
\omega_* \\
0 \\
\varepsilonnd{array}
\right)
\in R^{m+ 2m_0}$, $K= \left(
\begin{array}{cc}
K_0^T \partial_y^2 H_0(y) K_0 & 0 \\
0& \partial_{x_2}^2 \int_{T^m}P_1 (x,0,\varepsilon) d x_1 \\
\varepsilonnd{array}
\right)
$;
\item [\bf{(S4)}.] for some positive constant $\tilde \sigma$ (independent of $\varepsilon$),
\begin{eqnarray}\label{condition1}
|\det \partial_{x_2}^2\varepsilon^{-\kappa} \int_{T^m}P_1 (x,0,\varepsilon) d x_1| > \tilde{\sigma}~~~\forall x,~y,~u,~v,
\varepsilonnd{eqnarray}
where $\kappa$ is a given constant, $x_1 = K_*^T x$, $x_2 = K_0^T x$.
\varepsilonnd{enumerate}
Then there exist a $\varepsilon_0 >0 $ and a family of Cantor sets $\widetilde{\Lambda}_\varepsilon(g,G) \subset \widetilde{\Lambda}(g,G)$, $0<\varepsilon < \varepsilon_0$, such that
i) combining condition $\bf{(S1)}$, $\bf{(S2)}$ and $\bf{(S4)}$, for each $y \in \widetilde{\Lambda}_\varepsilon(g,G)$, system (\ref{005}) admits $2^{m_0}$ families of invariant tori, possessing hyperbolic, elliptic or mixed types, associated to nondegenerate relative equilibrium. Moreover, all such perturbed tori corresponding to a same $y \in \widetilde{\Lambda}_\varepsilon(g,G) $ are symplectically conjugated to the standard quasiperiodic $m-$tori $T^m$ with the Diophantine frequency vector $ \hat{\omega}$, for which there are $n$ components that are the same as $\omega_*$ and the others slightly drift;
ii) combining $\bf{(S1)}$, $\bf{(S2)}$, $\bf{(S3)}$ and $\bf{(S4)}$, for each $y \in \widetilde{\Lambda}_\varepsilon(g,G)$, on a given energy surface system (\ref{005}) admits $2^{m_0}$ families of invariant tori, possessing hyperbolic, elliptic or mixed types, associated to nondegenerate relative equilibrium. Moreover, there are $n$ components of the frequency on the unperturbed tori $\omega_*$ and perturbed tori $\bar{\omega}$ satisfied $\bar{\omega} =t \omega_*$ and the others slightly drift, where $t$ is a constant.
iii)
the relative Lebesgue measure $|\widetilde{\Lambda}(g,G) \setminus \widetilde{\Lambda}_{\varepsilon}(g,G)|$ tends to 0 as $\varepsilon \rightarrow 0$.
\varepsilonnd{theorem}
\begin{remark} Condition $\bf{(S1)}$ and $\bf{(S2)}$ is weaker than $g-$nondegenerate condition of $H_0$.
As a result, the preservation of frequencies on the perturbed tori will be determined by $(K_{*}^T \partial_y^2 {H_0}K_{*}, K_*^T \partial_y^2 H K')$. The details will be shown in Section $\ref{normal form}$.
\varepsilonnd{remark}
\begin{remark}
Condition $\bf{(S3)}$ is equal to the following condition$:$
\begin{itemize}
\item [$\bf{(S3')}$] $rank \left(
\begin{array}{cc}
K_0^T \partial_y^2 H K_0 & \bar{\omega}_* \\
\bar{\omega}_*^T & 0 \\
\varepsilonnd{array}
\right) = n + m_0+1$ for a given $g$ on $\widetilde{\Lambda}(g,G), $ where $\bar{\omega}_* =\left(
\begin{array}{c}
\omega_* \\
0 \\
\varepsilonnd{array}
\right)
\in R^{m+ m_0}$, $\omega_* = K_*^T \partial_y H_0(y)\in R^{m}$;
\varepsilonnd{itemize}
\varepsilonnd{remark}
\begin{cor}\label{cor1}
Let $H$ be real analytic
on the complex neighborhood of $T^d \times G$. Assume
\begin{enumerate}
\item[{\bf (S5)}] $K_0^T \partial_y^2 H_0 K_0$ has a $(m_0+n)\times (m_0+n)$ nonsingular minor, $n < m$, and $\det {K^{'}}^T \partial_y^2 {H_0}{K^{'}} \neq 0$ for $y \in \widetilde{\Lambda}(g,G)$.
\varepsilonnd{enumerate}
Then there exist a $\varepsilon_0 >0 $ and a family of Cantor sets $\widetilde{\Lambda}_\varepsilon(g,G) \subset \widetilde{\Lambda}(g,G)$, $0<\varepsilon < \varepsilon_0$, such that
i) combining condition $\bf{(S1)}$, $\bf{(S5)}$ and $\bf{(S4)}$, for each $y \in \widetilde{\Lambda}_\varepsilon(g,G)$, system (\ref{005}) admits $2^{m_0}$ families of invariant tori, possessing hyperbolic, elliptic or mixed types, associated to nondegenerate relative equilibrium. Moreover, all such perturbed tori corresponding to a same $y \in \widetilde{\Lambda}_\varepsilon(g,G) $ are symplectically conjugated to the standard quasiperiodic $m-$tori $T^m$ with the Diophantine frequency vector $ \hat{\omega}$, for which there are $n$ components that are the same as $\omega_*$ and the others slightly drift;
ii) combining $\bf{(S1)}$, $\bf{(S3)}$, $\bf{(S4)}$ and $\bf{(S5)}$, for each $y \in \widetilde{\Lambda}_\varepsilon(g,G)$, on a given energy surface system (\ref{005}) admits $2^{m_0}$ families of invariant tori, possessing hyperbolic, elliptic or mixed types, associated to nondegenerate relative equilibrium. Moreover, there are $n$ components of the frequency on the unperturbed tori $\omega_*$ and perturbed tori $\bar{\omega}$ satisfied $\bar{\omega} =t \omega_*$ and the others slightly drift, where $t$ is a constant.
iii)
the relative Lebesgue measure $|\widetilde{\Lambda}(g,G) \setminus \widetilde{\Lambda}_{\varepsilon}(g,G)|$ tends to 0 as $\varepsilon \rightarrow 0$.
\varepsilonnd{cor}
\begin{remark}
$\bf{(S5)}$ is equal to the following $\bf{(S6)}$:
\begin{enumerate}
\item[{\bf (S6)}] $rank (K_*^T \partial_y^2 H K' {K'}^T \partial_y^2 H K' {K'}^T\partial_y^2 H K_* + K_*^T \partial_y^2 HK_* )= n,$ $n < m$, and $\det {K^{'}}^T \partial_y^2 {H_0}{K^{'}} \neq 0$ for $y \in \widetilde{\Lambda}(g,G)$,
\varepsilonnd{enumerate}
which comes from the following fact:
\begin{eqnarray*}
\left(
\begin{array}{cc}
I_r & 0 \\
-DB^{-1} & I_{m-r} \\
\varepsilonnd{array}
\right) \left(
\begin{array}{cc}
B & C \\
D & E \\
\varepsilonnd{array}
\right)
= \left(
\begin{array}{cc}
B & C \\
0 & -DB^{-1}C + E \\
\varepsilonnd{array}
\right),
\varepsilonnd{eqnarray*}
where $B$ is nonsingular.
\varepsilonnd{remark}
The following case with $g-$nondegenerate condition is a direct conclusion.
\begin{cor} \label{cor2}
Let $H$ be real analytic
on the complex neighborhood of $T^d \times G$. Assume
\begin{enumerate}
\item [\bf{(S7)}] $H_0$ is $g-$nondegenerate for a given $g$ on $\widetilde{\Lambda}(g,G);$
\item [\bf{(S8)}]$
rank \left(
\begin{array}{cc}
K_0^T \partial_y^2 H_0(y) K_0 & \bar{\omega}_* \\
\bar{\omega}_*^T & 0 \\
\varepsilonnd{array}
\right) = m+ m_0+1
$ for given $g$ on $\widetilde{\Lambda}(g,G)$, where $\bar{\omega}_* = \left(
\begin{array}{c}
\omega_* \\
0 \\
\varepsilonnd{array}
\right)
\in R^{m+ m_0}$, $\omega_* = K_*^T \partial_y H_0(y);$
\varepsilonnd{enumerate}
Then there exist a $\varepsilon_0 >0 $ and a family of Cantor sets $\widetilde{\Lambda}_\varepsilon(g,G) \subset \widetilde{\Lambda}(g,G)$, $0<\varepsilon < \varepsilon_0$, such that
i) combining condition $\bf{(S4)}$ and $\bf{(S7)}$, for each $y \in \widetilde{\Lambda}_\varepsilon(g,G)$, system (\ref{005}) admits $2^{m_0}$ families of invariant tori, possessing hyperbolic, elliptic or mixed types, associated to nondegenerate relative equilibrium. Moreover, all such perturbed tori corresponding to a same $y \in \widetilde{\Lambda}_\varepsilon(g,G) $ are symplectically conjugated to the standard quasiperiodic $m-$torus $T^m$ with the Diophantine frequency vector $ \omega^* = K_{*}^T \omega(y_0)$;
ii)combining $\bf{(S4)}$, $\bf{(S7)}$ and $\bf{(S8)}$, for each $y \in \widetilde{\Lambda}_\varepsilon(g,G)$, on a given energy surface system (\ref{005}) admits $2^{m_0}$ families of invariant tori, possessing hyperbolic, elliptic or mixed types, associated to nondegenerate relative equilibrium.
Moreover, the frequency of the unperturbed tori $\omega_*$ and perturbed tori $\bar{\omega}$ satisfy $\bar{\omega} =t \omega_*$, where $t$ is a constant.
iii)
the relative Lebesgue measure $|\widetilde{\Lambda}(g,G) \setminus \widetilde{\Lambda}_{\varepsilon}(g,G)|$ tends to 0 as $\varepsilon \rightarrow 0$.
\varepsilonnd{cor}
\begin{remark}
The motion equation of the unperturbed Hamiltonian system $H_0(y)$ is
\begin{eqnarray*}
\left\{
\begin{array}{ll}
\dot{x} = \omega(y), \\
\dot{y} = 0.
\varepsilonnd{array}
\right.
\varepsilonnd{eqnarray*}
When $\omega(y)$ is $g-$resonant, under the following sympletic transformation:
\begin{eqnarray*}
K_0 ^T x = \left(
\begin{array}{c}
q' \\
q'' \\
\varepsilonnd{array}
\right), y = y,
\varepsilonnd{eqnarray*}
the equation of motion becomes
\begin{eqnarray*}
\left\{
\begin{array}{lll}
\dot{q'} = K_1^T \omega(y), \\
\dot{q''} = 0,\\
\dot{y} = 0,
\varepsilonnd{array}
\right.
\varepsilonnd{eqnarray*}
where $K_0$ and $K_1$ are mentioned as above. Then each $(y, q'')$ is a relative equilibrium of the unperturbed system.
\varepsilonnd{remark}
\begin{remark}
Here a map defined on a Cantor set is said to be smooth in Whitney's sense if it has a smooth Whitney extension. For details, see $\varepsilonmph{\cite{Poschel}}$.
\varepsilonnd{remark}
Obviously, in quite general situations with high degeneracy, we prove the conjecture on the persistence of resonant invariant tori, because in assumption (\ref{condition1}), the order of degeneracy on the perturbation is $\kappa-1$, might be arbitrary.
The classical Birkhoff normal form theory provides a formal integrability to harmonic oscillators with perturbation. But it does not work to our persistence of resonant invariant tori, mainly due to the nonlinearity of the unperturbed system and the degeneracy of $[P_1(\cdot,y,u,v,0)]$. To overcome these difficulties, besides employing Treshch\"{e}v's reduction, we propose a nonlinear normal form program by introducing nonlinear KAM iteration, which is used for searching high degeneracy and keeping critical points that are relative to certain quasiperiodicity of the perturbation. In particular, it will be seen that our KAM iteration is more suitable for problems with worse normal forms. Hence, this approach provides a thorough way to study the persistence of resonant invariant tori under high degenerate perturbations. To be specific, we prove that, for (\ref{005}), Hamiltonian systems with high order degenerate perturbation, there are $2^{m_0}$ lower-dimensional invariant tori born from resonant invariant tori.
The paper is organized as follows. In Section \ref{normal form}, we give an abstract Hamiltonian system and show the corresponding persistence of invariant tori. With the results of the abstract Hamiltonian system we finish the proof of main theorem in Section \ref{074}.
\section{Abstract Hamiltonian systems}\label{normal form}\setcounter{equation}{0}
Throughout the paper, unless specified explanation, we shall use the same symbol $|\cdot|$ to denote an equivalent (finite dimensional) vector norm and its induced matrix norm, absolute value of functions, and measure of sets, etc., and use $|\cdot|_D$ to denote the supremum norm of functions on a domain $D$. Also, for any two complex column vectors $\xi$, $\zeta$ of the same dimension, $\langle \xi, \zeta \rangle$ always stands for $\xi^T \zeta$, i.e., the transpose of $\xi$ times $\zeta$. For the sake of brevity, we shall not specify smoothness orders for functions having obvious orders of smoothness indicated by their derivatives taking. All constants below are positive and independent of the iteration process. Moreover, all Hamiltonian functions in the sequel are associated to the standard symplectic structure.
To prove Theorem \ref{dingli11}, consider the following real analytic Hamiltonian system
\begin{eqnarray}\label{model33}
H(x,y,z) =N(y,z, \lambda) + P(x,y,z, \lambda, \varepsilon),
\varepsilonnd{eqnarray}
defined on
\begin{center}
$D(r,s)=\{(x,y,z):|Im~x|<r,~ |y|<s, ~|z|<s\}$,
\varepsilonnd{center}
with
\begin{eqnarray}
\nonumber N (y,z,\lambda)&=& \langle \omega(\lambda), y\rangle + \frac{\delta}{2}\langle \left(
\begin{array}{c}
y \\
z \\
\varepsilonnd{array}
\right)
, M(\lambda) \left(
\begin{array}{c}
y \\
z \\
\varepsilonnd{array}
\right)
\rangle + \delta h (y,z,\lambda,\varepsilon),\\
\label{080} |\partial_\lambda^l P| &\leq& \delta \gamma^b s^2 \mu, ~~~|l|\leq l_0,
\varepsilonnd{eqnarray}
where $x \in T^m$, $y \in R^m$, $z=(u, v) \in R ^{2m_0}$, $b=(2l_0^2 +3)(m+2m_0)^2$, $\lambda \in \Lambda$, $M$, a symmetric matrix, depends smoothly on $\lambda$, $h = O(|\left(
\begin{array}{c}
y \\
z \\
\varepsilonnd{array}
\right) |^3
)$ is smooth in $\varepsilon$. In the above, all $\lambda-$dependence are of class $C^{l_0}$ for some $l_0 \geq d$. Rewrite
\begin{eqnarray*}
M = \left(
\begin{array}{cc}
M_{11} & M_{12} \\
M_{21} & M_{22} \\
\varepsilonnd{array}
\right),
\varepsilonnd{eqnarray*}
where $M_{11},$ $M_{12}$, $M_{21},$ $M_{22}$ are $m\times m,$ $m\times 2m_0$, $2m_0\times m,$ $2m_0\times 2 m_0$ matrices, respectively.
\subsection{A General Theorem}
To show the persistence of invariant tori for Hamiltonian (\ref{model33}), assume:
\begin{itemize}
\item[{\bf (A1)}]$rank~\{\frac{\partial^\alpha \omega}{\partial \lambda^ \alpha}: 0\leq|\alpha| \leq m -1 \} = m $, for all $\lambda \subset {\Lambda}$.
\item[{\bf (A2)}]For given $n$, $0< n \leq m,$ $rank M = n+ 2m_0$ and $rank (M_{21}, M_{22}) = 2m_0,$ where $\lambda\in \Lambda.$
\item[{\bf (A3)}]For given $n$, $0< n \leq m,$
\begin{center}
$ rank \left(
\begin{array}{cc}
M(\lambda) & \bar{\omega}_1(\lambda) \\
{\bar{\omega}_1 }^T(\lambda) & 0 \\
\varepsilonnd{array}
\right)
= n+ 2m_0 +1,$
\varepsilonnd{center}
where $\bar{\omega}_1 = \left(
\begin{array}{c}
\omega \\
0 \\
\varepsilonnd{array}
\right)\in R^{m+ 2m_0}
$, $\omega\in R^{m}$.
\varepsilonnd{itemize}
\begin{remark}
We call $\bf{(A2)}$ and $\bf{(A3)}$ sub-isoenergetically non-degenerate conditions for the persistence of lower dimensional invariant tori. Specifically, when $n = m$ and $m_0 = 0$, they are isoenergetically non-degenerate condition introduced by Arnold$(\varepsilonmph{\cite{Arnold}})$. When $m_0 =0$, they are similar to the isoenergetically non-degenerate condition contained in \varepsilonmph{\cite{LLL,Sevryuk}}. When $M$ is a block diagonal matrix, refer to $\varepsilonmph{\cite{Qian1}}$ for a similar condition.
\varepsilonnd{remark}
We state our result for (\ref{model33}) as follows.
\begin{theorem}\label{shengluede}
Assume {\bf (A1)}. Let $\tau > d(d-1) -1$ be fixed. If $\delta$ is sufficiently small and there exists a sufficiently small $\mu>0$ such that
\begin{center}\label{003}
$|\partial_\lambda^l P|_{D(r,s)\times \Lambda} \leq \delta \gamma ^{b} s^2 \mu , ~|l| \leq l_0$,
\varepsilonnd{center}
then there exist Cantor sets $\Lambda_\gamma\subset \Lambda$ with $|\Lambda \setminus \Lambda_\gamma | = O(\gamma ^{\frac{1}{d-1}})$, and a $C^{d -1}$ Whitney smooth family of symplectic transformations
\begin{center}
$ \Psi_{\lambda} : D(\frac{r}{2}, \frac{s}{2})\rightarrow D(r,s), ~~~\lambda \in \Lambda_\gamma$,
\varepsilonnd{center}
which is real analytic in x and closes to the identity such that
\begin{eqnarray*}
H \circ \Psi _\lambda = e_*+{\langle \omega_{*}(\lambda) , y\rangle}+ \frac{\delta}{2}\langle \left(
\begin{array}{c}
y \\
z \\
\varepsilonnd{array}
\right)
, M_{*}(\lambda) \left(
\begin{array}{c}
y \\
z \\
\varepsilonnd{array}
\right)
\rangle+ \delta h_* (y,z,\lambda,\varepsilon) + P_{*},
\varepsilonnd{eqnarray*}
where for all $\lambda \in \Lambda _\gamma $, and $(x,y) \in D(\frac{r}{2}, \frac{s}{2})$,
\begin{eqnarray*}
|\partial_{\lambda}^{l}{e_{*}} - \partial_{\lambda}^{l}{e}| &=& O (\delta \gamma ^ {d+ 6 }\mu), ~~|l| \leq d -1,\\
|\partial_{\lambda}^{l}{\omega_{*}} - \partial_{\lambda}^{l}{\omega}| &=& O (\delta \gamma ^ {d+ 6 } \mu),~~|l| \leq d -1,\\
|\partial_{\lambda}^{l}{{h}_{*}}(y,z) - \partial_{\lambda}^{l}{{h}}(y,z)| &=& O ( \gamma ^ {d+ 6 }\mu),~~|l| \leq d -1,\\
{\partial_{\lambda}^{l} \partial_{y}^{i} \partial_{z}^{j} P_{*}}|_{y= 0, z= 0} &=& 0, ~~|l| \leq d -1,\\
h_* &=&O(|\left(
\begin{array}{c}
y \\
z \\
\varepsilonnd{array}
\right)
|^3).
\varepsilonnd{eqnarray*}
\begin{enumerate}
\item [\bf{(1)}]With assumption $\bf{(A2)}$, we have
\begin{eqnarray*}
(\omega_{*})_{i_q} &\varepsilonquiv& (\omega )_{i_q}~~for ~all ~1\leq q \leq n.
\varepsilonnd{eqnarray*}
Thus, for each $\lambda \in \Lambda$, the unperturbed torus $T_\lambda = T^m \times \{0\}\times \{0\}$ associated to the
toral frequency $\omega$ persists and gives rise to an analytic, Diophantine, invariant torus with the toral frequency $\omega_{*}$, which preserves $n$ components of the unperturbed toral frequency $\omega$, $\omega_{i_1}$, $\omega_{i_2}$, $\cdots$, $\omega_{i_n}$, determined by those rows of $(M_{11}, M_{12})$ that are linearly independent. Moreover, these perturbed tori form a $C^{d -1}$ Whitney smooth family.
\item [\bf{(2)}]Assume $\bf{(A2)}$ and $\bf{(A3)}$ hold. Then
\begin{eqnarray}
\label{E_65} (\omega_{*})_{i_q} &\varepsilonquiv& t (\omega )_{i_q}~~for ~all ~1\leq q \leq n.
\varepsilonnd{eqnarray}
Thus, for each $\lambda \in \Lambda$, the unperturbed torus $T_\lambda = T^m \times \{0\}\times \{0\}$ associated to the
toral frequency $\omega$ persists and gives rise to an analytic, Diophantine, invariant torus with the toral frequency $\omega_{*}$ which satisfies \varepsilonmph{(\ref{E_65})} and is determined by those rows of $(M_{11}, M_{12})$ that are linearly independent. Moreover, these perturbed tori form a $C^{d -1}$ Whitney smooth family.
\varepsilonnd{enumerate}
\varepsilonnd{theorem}
The proof of Theorem \ref{shengluede} will be proceed by quasilinear KAM iteration process, which consists of infinitely many KAM steps. At each step, we simplify the process of solving homological equation. Next, we show the detail of a cycle of KAM step. Finally, in Section $\ref{example}$, we also give several examples to show the complexity resulting from the high degeneracy.
\subsection{KAM step}\label{KAM}
We show first the $0-$th KAM step. For the sake of induction, let
\begin{eqnarray*}
r_0 = r, ~~~\beta_0 = s, ~~~\gamma_0 = 4 \gamma,~~~\varepsilonta_0= \varepsilonta,~~~ \Lambda_0 = \Lambda,~~~ P_0 = P,
\varepsilonnd{eqnarray*}
where $0 <r, s, \gamma_0, \varepsilonta_0\leq 1$, and denote
\begin{eqnarray*}
c^* =\sup \{|\lambda|: \lambda\in \Lambda_0\},
\varepsilonnd{eqnarray*}
and let
\begin{eqnarray*}
M^* &=&\max\limits_{|l|\leq l_0, |j|\leq m+5, |y|\leq \beta_0, \lambda\in \Lambda_0} |\partial_\lambda^l \partial_{(y,z)}^j h_0(y,z,\lambda)|_{\mathcal{O}_0}.
\varepsilonnd{eqnarray*}
By monotonicity, we define $0<\mu_0 \leq1$, $s_0$ implicity through the following equations:
\begin{eqnarray*}
\mu &=& \frac{4^{3b -1}\mu_0}{c_0 K_1^{(m+2m_0)^2\tau + (m+2m_0)^2 +1}},\\
s_0 &=& \frac{\beta_0 \gamma_0^{2b}}{4 c_0 K_1^{(m+2m_0)^2\tau + (m+2m_0)^2 +1}},
\varepsilonnd{eqnarray*}
where
\begin{eqnarray*}
c_0 &=& (m+2m_0)^4 (c^*)^{(m+2m_0)^2} ({M^*}+1)^{(m+2m_0)^2},\\
K_1 &=& ([\log \frac{1}{\mu_0}] +1 )^{3 {\varepsilonta}},
\varepsilonnd{eqnarray*}
$[\cdot]$ denotes the integral part of a real number
and ${\varepsilonta}$ is a fixed positive integer such that $(1+ \sigma)^{{\varepsilonta}} > 2$ with $\sigma = \frac{1}{12}$. Obviously, $\mu_0 \rightarrow 0$ iff $\mu\rightarrow 0$, and, for any fixed $0< \varepsilonpsilon<1$,
\begin{eqnarray*}
\mu_0 = O(\mu^{1 - \varepsilonpsilon})~ as ~ \mu \rightarrow 0.
\varepsilonnd{eqnarray*}
When $\mu$ is sufficiently small, we have
\begin{eqnarray*}
4c_0 K_1^{(m+2m_0)^2\tau+(m+2m_0)^2+1} >1.
\varepsilonnd{eqnarray*}
Hence
\begin{eqnarray*}
0<s_0\leq\min\{\beta_0, \frac{\gamma_0^b}{4 c_0 K_1^{(m+2m_0)^2\tau+ (m+2m_0)^2 +1}}\}.
\varepsilonnd{eqnarray*}
For $j \in Z_+^n$, define
\begin{eqnarray*}
a_j &=& 1- sgn {(|j|-1)} = \left\{
\begin{array}{lll}
2, & \hbox{$|j| = 0$,} \\
1, & \hbox{$|j| \geq 1$,}\\
0, & \hbox{$|j| \geq 2$,}
\varepsilonnd{array}
\right.\\
b_j &=&b(1 - sgn |j| sgn (|j|-1) sgn (|j|-2))= \left\{
\begin{array}{ll}
b, & \hbox{$|j| = 0,1,2$,} \\
0, & \hbox{$|j|\geq 3$,}
\varepsilonnd{array}
\right.\\
d_j &=& 1 - \lambda_0 sgn(|j|)sgn (|j|-1) sgn (|j|-2)= \left\{
\begin{array}{ll}
1, & \hbox{$|j| = 0,1,2$,} \\
1- \lambda_0, & \hbox{$|j|\geq 3$,}
\varepsilonnd{array}
\right.
\varepsilonnd{eqnarray*}
where $\frac{2}{13}<\lambda_0<1$ is fixed. Therefore
\begin{eqnarray}\label{714}
|\partial_\lambda^l \partial_x^i \partial_{(y,z)}^j P_0|_{D(r_0, s_0) \times {\Lambda}_0} \leq \delta\gamma_0^{b_j} s_0^{a_j} \mu_0^{d_j}
\varepsilonnd{eqnarray}
for all $(l,i,j)\in Z_+^m \times Z_+^m \times Z_+^{2m_0}$, $|l|+ |i|+|j| \leq l_0.$
Next we characterize the iteration scheme for Hamiltonian (\ref{model33}) in one KAM step, say, from the $\nu-$th KAM step to the $(\nu + 1)-$th step. For convenience, we shall omit the index for all quantities of the $\nu-$th KAM step and use $'+'$ to index all quantities in the ${(\nu+ 1)}-$th KAM step. To simplify the notions, we shall suspend the $\lambda-$dependence in most terms of this section. Now, suppose that after $\nu$ KAM steps, we have arrived at the following real analytic Hamiltonian system
\begin{eqnarray}\label{N2}
H(x,y,z) &=&N(y,z) + P(x,y,z,\varepsilon),\\
\nonumber N(y,z) &=& \langle \omega(\lambda), y\rangle + \frac{\delta}{2}\langle \left(
\begin{array}{c}
y \\
z \\
\varepsilonnd{array}
\right)
, M(\lambda) \left(
\begin{array}{c}
y \\
z \\
\varepsilonnd{array}
\right)
\rangle + \delta h(y,z,\lambda, \varepsilon),\\
\nonumber |\partial_\lambda^l \partial_x^i \partial_{(y,z)}^j P| &\leq&\delta \gamma^{b_j} s^{a_j} \mu^{d_j}, ~~~~~~~|l|+ |i|+|j| \leq l_0.
\varepsilonnd{eqnarray}
By considering both averaging and translation, we shall find a symplectic transformation $\Phi _ {+}$,
which, on a small phase domain $D(r_+, s_+)$ and a smaller parameter domain $\Lambda_+$, transforms Hamiltonian (\ref{N2}) into the following form:
\begin{center}
$H_{+} = H{\circ} {\Phi _+}= {N_+} +{P_+}$,
\varepsilonnd{center}
where on $D(r_+, s_+)\times \Lambda_+,$ $N_+$ and $P_+$ enjoy similar properties as $N$ and $P$, respectively. Define
\begin{eqnarray*}
s_+ &=& \frac{1}{8} \alpha s,\\
\mu_+ &=& (64 c_0)^{\frac{1}{1 - \lambda_0}} \mu^{1 + \sigma},\\
r_+ &=& r - \frac{r_0}{2^{\nu+1}},\\
\gamma_+ &=& \gamma - \frac{\gamma_0}{2^{\nu+1}},\\
K_+ &=& ([\log\frac{1}{\mu}]+ 1)^{3{\varepsilonta}},\\
D_\alpha &=& D(r_++ \frac{7}{8}(r - r_+), \alpha s),\\
\hat{D}(\lambda) &=& D(r_++ \frac{7}{8}(r - r_+), \lambda),\\
D(\lambda) &=& \{y\in C^n:|y|< \lambda\},\\
D_{\frac{i}{8} \alpha} &=& D(r_+ + \frac{i -1 }{8}(r - r_+), \frac{i}{8}\alpha s), ~~i = 1,2,\cdots, 8,\\
\Gamma(r- r_+) &=& \sum_{0<|k|\leq K_+} |k|^\chi e^{-|k|\frac{r -r_+}{8}},
\varepsilonnd{eqnarray*}
where $\alpha = \mu^{\frac{1}{3}},$ $\lambda > 0$, $\chi = (b + 2)\tau +5 l_0 + 10,$ $c_0$ is the maximal among all $c's$ mentioned in this paper and depends on $r_0$, $\beta_0$ .
\subsubsection{Truncation of the perturbation}
Consider the Taylor-Fourier series of $P$:
\begin{eqnarray*}
P = \sum_{\imath\in Z_+^m,\jmath\in Z_+^{2m_0}, k \in Z^m} p_{k\imath\jmath} y^{\imath}z^{\jmath} e^{\sqrt{-1}\langle k, x\rangle},
\varepsilonnd{eqnarray*}
and let $R$ be the truncation of $P$ with the following form:
\begin{eqnarray*}
R = \sum\limits_{|k| \leq K_+} (p_{k00}+ \langle p_{k10},y\rangle+\langle p_{k01},z\rangle + \langle y, p_{k20}y\rangle+ \langle y, p_{k11}z\rangle + \langle z, p_{k02}z\rangle) e^{\sqrt{-1} \langle k, x\rangle},
\varepsilonnd{eqnarray*}
where $K_+$ is defined as above.
\begin{lemma}
Assume that
\begin{itemize}
\item[\bf{(H1)}] $K_+ \geq \frac{8(m + l_0)}{r - r_+}$,
\item[\bf{(H2)}] $\int_{K_+}^\infty \lambda^{m + l_0} e^{-\lambda \frac{r - r_+}{8}} d \lambda \leq \mu.$
\varepsilonnd{itemize}
Then there is a constant $c$ such that for all $|l|+|i|+|j| \leq l_0$, $\lambda \in \Lambda$,
\begin{eqnarray*}
|\partial_\lambda^l \partial_x^i \partial_{(y,z)}^j (P- R)|_{D_\alpha \times {\Lambda}} \leq c \delta\gamma^{b_j} s^{a_j} \mu^{d_j+1}.
\varepsilonnd{eqnarray*}
\varepsilonnd{lemma}
\begin{proof}
The proof is standard. For detail, refer to, for example, $\bf{Lemma ~3.1}$ of \cite{Li}.
\varepsilonnd{proof}
\subsubsection{Homological equations}
We want to average out all coefficients of $R$ by constructing a symplectic transformation as the time-1 map $\phi_F^1$ of the flow generated by a Hamiltonian $F$ with the following form:
\begin{eqnarray}
\nonumber F &=& \sum\limits_{0< |k| \leq K_+} (f_{k00}+ \langle f_{k10},y\rangle+\langle f_{k01},z\rangle + \langle y, f_{k20}y\rangle+ \langle y, f_{k11}z\rangle\\
\label{709} &~&~~~~~~~~~ + \langle z, f_{k02}z\rangle) e^{\sqrt{-1} \langle k, x\rangle},
\varepsilonnd{eqnarray}
where $f_{kij}$, $0\leq |i|+|j|\leq 2$, are scalar, vectors or matrices with obvious dimensions. Under the time-1 map $\phi_F^1$, Hamiltonian (\ref{N2}) becomes
\begin{eqnarray}
\nonumber H\circ \phi_F^1 &=& (N + R )\circ \phi_F^1 + (P - R)\circ \phi_F^1\\
\label{N6} &=&N + R+ \{N, F\} +\int_0 ^1 \{R_t,F\}\circ \phi_F^t dt + (P - R )\circ \phi_F^1,~~~~
\varepsilonnd{eqnarray}
where
\begin{eqnarray}
\nonumber R_t &=& (1-t) \{N, F\} + R.
\varepsilonnd{eqnarray}
Let
\begin{eqnarray}\label{706}
\{N, F\} + R - [R] - R'= 0,
\varepsilonnd{eqnarray}
where
\begin{eqnarray*}
[R] &=& \int_{T^n} R(x, \cdot) dx,\\
R' &=& \partial_z \hat{h} J \partial_z F,\\
\hat{h} &=& \frac{\delta}{2} \langle \left(
\begin{array}{c}
y \\
z \\
\varepsilonnd{array}
\right), M \left(
\begin{array}{c}
y \\
z \\
\varepsilonnd{array}
\right)
\rangle + \delta h(y,z, \lambda, \varepsilon).
\varepsilonnd{eqnarray*}
Then Hamiltonian $(\ref{N6})$ arrives at
\begin{eqnarray}\label{N4}
\bar{H}_+ = \bar{N}_+(y,z) + \bar{P}_+(x,y,z),
\varepsilonnd{eqnarray}
where
\begin{eqnarray*}
\bar{N}_+ &=& N + [R],\\
\bar{P}_+ &=& {R'} + \int_0 ^1 \{R_t,F\}\circ \phi_F^t dt + (P - R )\circ \phi_F^1.
\varepsilonnd{eqnarray*}
Consider the following symplectic translation:
\begin{eqnarray}\label{E_21}
\phi: x \rightarrow x, \left(
\begin{array}{c}
y \\
z \\
\varepsilonnd{array}
\right)
\rightarrow \left(
\begin{array}{c}
y + y_0 \\
z+ z_0 \\
\varepsilonnd{array}
\right),
\varepsilonnd{eqnarray}
where $(y_0, z_0)$ is determined by
\begin{eqnarray}\label{E_1}
\delta\frac{ M}{2} \left(
\begin{array}{c}
y_0 \\
z_0 \\
\varepsilonnd{array}
\right) + \delta \left(
\begin{array}{c}
\partial_y h(y_0, z_0, \lambda) \\
\partial_z h(y_0, z_0, \lambda) \\
\varepsilonnd{array}
\right)
= - \left(
\begin{array}{c}
p_{010} \\
p_{001} \\
\varepsilonnd{array}
\right).
\varepsilonnd{eqnarray}
Then Hamiltonian system $(\ref{N4})$ is changed to
\begin{eqnarray*}
H_+ &=& \bar{H}_+ \circ \phi\\
&=& e_+ + \langle \omega_+, y\rangle +\frac{\delta}{2} \langle\left(
\begin{array}{c}
y \\
z \\
\varepsilonnd{array}
\right), M_+ \left(
\begin{array}{c}
y \\
z \\
\varepsilonnd{array}
\right)\rangle + \delta h_+(y,z,\lambda,\varepsilon)+ P_+,
\varepsilonnd{eqnarray*}
where
\begin{eqnarray}
\nonumber e_+ &=& e + \langle \omega, y_0\rangle+ \frac{\delta}{2} \langle \left(
\begin{array}{c}
y_0 \\
z_0 \\
\varepsilonnd{array}
\right), M \left(
\begin{array}{c}
y_0 \\
z_0 \\
\varepsilonnd{array}
\right)
\rangle + p_{000} + \langle\left(
\begin{array}{c}
p_{010} \\
p_{001} \\
\varepsilonnd{array}
\right), \left(
\begin{array}{c}
y_0 \\
z_0 \\
\varepsilonnd{array}
\right)
\rangle\\
\nonumber&~&~~~+ \langle \left(
\begin{array}{c}
y_0 \\
z_0 \\
\varepsilonnd{array}
\right), \left(
\begin{array}{cc}
p_{010} & \frac{1}{2} p_{011} \\
\frac{1}{2} p_{011}^T & p_{002} \\
\varepsilonnd{array}
\right)
\left(
\begin{array}{c}
y_0 \\
z_0 \\
\varepsilonnd{array}
\right)
\rangle + \delta h(y_0, z_0,\lambda),\\
\nonumber \omega_+ &=& \omega + \frac{\delta M }{2} \left(
\begin{array}{c}
y_0 \\
z_0 \\
\varepsilonnd{array}
\right)+ \delta \left(
\begin{array}{c}
\partial_y h(y_0, z_0, \lambda) \\
\partial_z h(y_0, z_0, \lambda) \\
\varepsilonnd{array}
\right)+ \left(
\begin{array}{c}
p_{010} \\
p_{001} \\
\varepsilonnd{array}
\right)
,\\
\nonumber M_+&=& M +2 \left(
\begin{array}{cc}
p_{020} & \frac{1}{2} p_{011} \\
\frac{1}{2} p_{011}^T & p_{002}\\
\varepsilonnd{array}
\right)+ \frac{\delta}{2} \langle \left(
\begin{array}{c}
y \\
z \\
\varepsilonnd{array}
\right), \partial_{(y,z)}^2 h(y_0, z_0, \lambda) \left(
\begin{array}{c}
y \\
z \\
\varepsilonnd{array}
\right)
\rangle,\\
\label{N5} P_+ &=& \bar{P}_+ + \delta \langle \left(
\begin{array}{c}
y \\
z \\
\varepsilonnd{array}
\right), \left(
\begin{array}{cc}
p_{010} & \frac{1}{2} p_{011} \\
\frac{1}{2} p_{011}^T & p_{002} \\
\varepsilonnd{array}
\right)
\left(
\begin{array}{c}
y_0 \\
z_0 \\
\varepsilonnd{array}
\right)
\rangle,\\
\nonumber h_+&=& h(y,z,\lambda)- h(y_0, z_0, \lambda) - \langle \left(
\begin{array}{c}
\partial_y h(y_0, z_0, \lambda) \\
\partial_z h(y_0, z_0, \lambda) \\
\varepsilonnd{array}
\right), \left(
\begin{array}{c}
y \\
z \\
\varepsilonnd{array}
\right)
\rangle\\
&~&~~ - \frac{1}{2} \langle \left(
\begin{array}{c}
y \\
z \\
\varepsilonnd{array}
\right), \partial_{(y,z)}^2h (y_0, z_0, \lambda) \left(
\begin{array}{c}
y \\
z \\
\varepsilonnd{array}
\right)
\rangle.
\varepsilonnd{eqnarray}
\subsubsection{Estimate on the transformation}
Substituting the Taylor-Fourier series of $F$ and $R$ into $(\ref{706})$ yields:
\begin{eqnarray}
\label{707}\sqrt{-1} \langle k, \omega+ \Delta\rangle f_{kij} &=& p_{kij}, ~~0\leq |i|+|j|\leq2,
\varepsilonnd{eqnarray}
which, in fact, are solvable on the following Diophantine set:
\begin{eqnarray*}
\mathcal{O}_+ = \{\lambda \in \mathcal{O}: |\langle k, \omega\rangle| > \frac{ \gamma}{|k|^\tau} ~for~all~0< |k|\leq K_+\},
\varepsilonnd{eqnarray*}
where $\Delta = \partial_y h = \delta (M_{11} y + M_{12} z)$.
\begin{lemma}
Assume that
\begin{itemize}
\item[\bf{(H3)}] $\max\limits_{|l| \leq l_0, |j| \leq m+5} |\partial_\lambda^l \partial_{(y,z)}^j h(y,z,\lambda) - \partial_\lambda^l \partial_{(y,z)}^j h_0(y,z,\lambda)|_{\mathcal{O}}\leq \mu_0^{\frac{1}{2}}.$
\varepsilonnd{itemize}
Then there is a constant c such that for all $|l| \leq l_0$,
\begin{eqnarray}
\label{E_18}|\partial_\lambda^l e_+ - \partial_\lambda^l e|_{\mathcal{O}} &\leq& c \gamma^b s \mu,\\
\label{E_19}|\partial_\lambda^l M_+ - \partial_\lambda^l M|_{\mathcal{O}} &\leq& c \gamma^b \mu,\\
\label{E_21}|\partial_\lambda^l \omega_+ - \partial_\lambda^l \omega|_{\mathcal{O}} &\leq& c\delta s (\gamma^b \mu+ s),\\
\label{E_20}|\left(
\begin{array}{c}
\partial_\lambda^l y_0 \\
\partial_\lambda^l z_0 \\
\varepsilonnd{array}
\right)
|_{\mathcal{O}} &\leq& c \gamma^b s \mu.
\varepsilonnd{eqnarray}
\varepsilonnd{lemma}
\begin{proof}
Obviously,
\begin{eqnarray*}
|\partial _ \lambda^l p_{000}|_{\mathcal{O}} &\leq& c \delta\gamma^b s^2 \mu,\\
|\partial _ \lambda^l p_{010}|_{\mathcal{O}} + |\partial _ \lambda^l p_{001}|_{\mathcal{O}}&\leq& c \delta \gamma^b s \mu,\\
|\left(
\begin{array}{cc}
\partial_{\lambda}^l p_{020} & \partial_{\lambda}^l p_{011} \\
\partial_{\lambda}^l p_{011}^T & \partial_{\lambda}^l p_{002} \\
\varepsilonnd{array}
\right)
|_{\mathcal{O}} &\leq& c\delta \gamma^b \mu.
\varepsilonnd{eqnarray*}
Denote
\begin{eqnarray*}
B= \frac{M}{2}+ \left(
\begin{array}{cc}
\int_0^1 \partial_y^2 h(\theta y,z,\lambda) d\theta & \int_0^1 \partial_y\partial_z h( y,\theta z,\lambda) d\theta \\
\int_0^1\partial_z \partial_y h(\theta y,z,\lambda) d\theta & \int_0^1 \partial_z^2 h( y,\theta z,\lambda) d\theta \\
\varepsilonnd{array}
\right).
\varepsilonnd{eqnarray*}
Then $(\ref{E_1})$ becomes
\begin{eqnarray}\label{E_17}
\delta B \left(
\begin{array}{c}
y \\
z \\
\varepsilonnd{array}
\right) = - \left(
\begin{array}{c}
p_{010} \\
p_{001} \\
\varepsilonnd{array}
\right).
\varepsilonnd{eqnarray}
For given matrix $A =(a_{ij})_{n\times n}$, let
\begin{eqnarray*}
||A||_1 = \frac{1}{n}\sum\limits_{i,j=1}^n |a_{ij}(\lambda)|,
\varepsilonnd{eqnarray*}
where $|a_{ij}(\lambda)|$ is the absolute value of $a_{ij}(\lambda)$, $\lambda\in \Lambda$.
According to assumption $\bf{(H3)}$ and the definition of $M^*$, we have
\begin{eqnarray*}
||M - M_0||_1 &\leq& \mu_0^{\frac{1}{2}},\\
||\partial_{(y,z)}^2 h||_1 &\leq& (M^*+1 )s,
\varepsilonnd{eqnarray*}
respectively. Without loss of generality, we let $\mu_0$ and $s_0$ be small enough such that
\begin{eqnarray*}
s_0^{\frac{1}{2}} M_* (M^* +1 ) &\leq&\frac{1}{4},\\
\mu_0 M_* &\leq&\frac{1}{4}.
\varepsilonnd{eqnarray*}
Then
\begin{eqnarray*}
||M_0 - B||_1 &\leq& ||M- M_0||_1 + ||B - M||_1\\
&\leq& \mu_0^{\frac{1}{2}} + (M^*+1 )s^2\\
&\leq& \frac{1}{2M_*}.
\varepsilonnd{eqnarray*}
Let $M_0$ is nonsingular. It follows that $B$ is nonsingular and
\begin{eqnarray*}
||B^{-1} ||_1 &=& ||\frac{M_0^{-1}}{I - (M_0 - B) M_0^{-1}}||_1\\
&\leq& \frac{||M_0^{-1}||_1}{||I - (M_0 - B) M_0^{-1}||_1}\\
&\leq& \frac{||M_0^{-1}||_1}{1 - ||(M_0 - B) M_0^{-1}||_1}\\
&\leq& \frac{||M_0^{-1}||_1}{1 - ||(M_0 - B)||_1|| M_0^{-1}||_1}\\
&\leq& \frac{M_*}{1- \frac{1}{2M_*}M_*}\\
&=&2M_*.
\varepsilonnd{eqnarray*}
Here, we use the fact that
\begin{eqnarray*}
||(I - A)^{-1}||_1 \leq \frac{1}{1 - ||A||_1},
\varepsilonnd{eqnarray*}
which is obvious if $||I||_1 = 1$ and $||A||_1 <1.$
Therefore,
\begin{eqnarray*}
|\left(
\begin{array}{c}
y \\
z \\
\varepsilonnd{array}
\right)| &=& |\frac{1}{\delta} B^{-1} \left(
\begin{array}{c}
P_{010} \\
P_{001} \\
\varepsilonnd{array}
\right)
|\\
&\leq& \frac{n}{\delta} ||B^{-1}||_1 |\left(
\begin{array}{c}
P_{010} \\
P_{001} \\
\varepsilonnd{array}
\right)
|\\
&\leq& 2M_*\gamma^{b_j} s \mu.
\varepsilonnd{eqnarray*}
Consider the differential with respect to $\lambda$ on both sides of $(\ref{E_17})$
\begin{eqnarray*}
\partial_{(y,z)} B \left(
\begin{array}{c}
\partial_\lambda y \\
\partial_\lambda z \\
\varepsilonnd{array}
\right) \left(
\begin{array}{c}
y \\
z \\
\varepsilonnd{array}
\right)
+ \partial_\lambda B \left(
\begin{array}{c}
y \\
z \\
\varepsilonnd{array}
\right) + B \left(
\begin{array}{c}
\partial_\lambda y \\
\partial_\lambda z \\
\varepsilonnd{array}
\right)
= - \left(
\begin{array}{c}
\partial_\lambda P_{010} \\
\partial_\lambda P_{001} \\
\varepsilonnd{array}
\right).
\varepsilonnd{eqnarray*}
Then
\begin{eqnarray*}
|\left(
\begin{array}{c}
\partial_\lambda y \\
\partial_\lambda z \\
\varepsilonnd{array}
\right)| &=& |B^{-1} ( \left(
\begin{array}{c}
\partial_\lambda P_{010} \\
\partial_\lambda P_{001} \\
\varepsilonnd{array}
\right) + \partial_{(y,z)} B \left(
\begin{array}{c}
\partial_\lambda y \\
\partial_\lambda z \\
\varepsilonnd{array}
\right)\left(
\begin{array}{c}
y \\
z \\
\varepsilonnd{array}
\right)
+ \partial_\lambda B \left(
\begin{array}{c}
y \\
z \\
\varepsilonnd{array}
\right))|\\
&\leq& 2M_* \gamma^{b_j} s \mu + 4M_*^2 (M^* +1) \gamma^{b_j} s \mu |\left(
\begin{array}{c}
\partial_\lambda y \\
\partial_\lambda z \\
\varepsilonnd{array}
\right)| + 4M_*^2 (M^* +1) \gamma^{b_j} s\mu\\
&\leq& 4M_*^2 (M^* +1) \gamma^{b_j} s\mu.
\varepsilonnd{eqnarray*}
Inductively, we get $(\ref{E_20})$. According to the definition of $e_+$, $\omega_+$ and $M_+$, $(\ref{E_18})$, $(\ref{E_19})$ and $(\ref{E_21})$ are obvious.
\varepsilonnd{proof}
\begin{lemma}
Assume that
\begin{itemize}
\item[\bf{(H4)}]$s K_+ ^{(|l|+1)\tau +|l|+ |i|}= o(\gamma)$.
\varepsilonnd{itemize}
The following hold for all $0< |k| \leq K_+$.
\begin{itemize}
\item[\bf{(1)}] On $D(s)\times {\Lambda}_+$,
\begin{eqnarray*}
|\partial_\lambda^{l} f_{kij}| &\leq& c \delta|k|^{(|l|+1)\tau + |l| } s^{2-i-j}\mu e ^{- |k|r}, ~0 \leq |i|+|j|\leq 2;
\varepsilonnd{eqnarray*}
\item[\bf{(2)}] On $\hat{D}(s) \times {\Lambda}_+$,
\begin{eqnarray*}
|\partial_\lambda^l \partial_x^i \partial_{(y,z)}^j F| \leq c\delta s^{a_j} \mu^{d_j} \Gamma(r -r_+),~~ |l|+ |i|+ |j| \leq l_0 +2.
\varepsilonnd{eqnarray*}
\varepsilonnd{itemize}
\varepsilonnd{lemma}
\begin{proof}
For any $\lambda \in \Lambda_+$, $0<|k|\leq K_+$, with assumption $\bf{(H4)}$ we have
\begin{eqnarray*}
| L_{0k}| &=& |\sqrt{-1}\langle k, \omega\rangle + \sqrt{-1}\langle k, \Delta\rangle|\\
&\geq& \frac{\gamma}{|k|^\tau} - c s \delta K_+\\
& \geq& \frac{\gamma}{2 |k|^\tau},
\varepsilonnd{eqnarray*}
and
\begin{eqnarray*}
|\partial_\lambda^l L_{k0}|\leq c |k|.
\varepsilonnd{eqnarray*}
Applying the above and the following inequalities
\begin{eqnarray*}
|\partial_\lambda^l L_{k0}^{-1}| \leq |L_{k0}^{-1}| \sum_{|l'| = 1}^{|l|} \left(
\begin{array}{c}
l \\
l'\\
\varepsilonnd{array}
\right)
|\partial_\lambda^{l - l'} L_{k0}^{-1}| |\partial_\lambda^{l'} L_{k0}|,
\varepsilonnd{eqnarray*}
inductively, we deduce that
\begin{eqnarray}\label{N3}
|\partial_\lambda^l L_{k0}^{-1}| \leq c |k|^{|l|} |L_{k0}^{-1}|^{|l|+1} \leq \frac{|k|^{(|l|+1)\tau + |l|}}{\gamma^{|l|+1}}.
\varepsilonnd{eqnarray}
It now follows from $(\ref{707})$, $(\ref{N3})$ and Cauchy's estimate that
\begin{eqnarray*}
|\partial^{l}_{\lambda} f_{k00}| &\leq& \delta |\partial_\lambda^l (L_k^{-1} p_{k00})|\\
&\leq&\frac{\delta |k|^{(|l|+1)\tau+|l|}}{\gamma^{|l|+1}} \gamma^b s^{2} \mu e^{-|k| r}\\
&\leq& c\delta s^{2} \mu |k|^{(|l|+1)\tau+|l| } e^{- |k|r}.
\varepsilonnd{eqnarray*}
Similarly,
\begin{eqnarray*}
|\partial_\lambda^{l} f_{k10}| &\leq& \delta |\partial_\lambda^l (L_k^{-1} p_{k 10})|\\
&\leq& c \delta s \mu |k|^{(|l|+1)\tau+|l| } e^{- |k|r},\\
|\partial_\lambda^{l} f_{k01}| &\leq& \delta |\partial_\lambda^l (L_k^{-1} p_{k01})|\\
&\leq& c \delta s \mu |k|^{(|l|+1)\tau+|l| } e^{- |k|r},\\
|\partial_\lambda^{l} f_{k11}| &\leq& \delta |\partial_\lambda^l (L_k^{-1} p_{k 11})|\\
&\leq& c \delta \mu |k|^{(|l|+1)\tau+|l| } e^{- |k|r},\\
|\partial_\lambda^{l} f_{k02}| &\leq& \delta |\partial_\lambda^l (L_k^{-1} p_{k 02})|\\
&\leq& c\delta \mu |k|^{(|l|+1)\tau+|l| } e^{- |k|r},\\
|\partial_\lambda^{l} f_{k20}| &\leq& \delta |\partial_\lambda^l (L_k^{-1} p_{k20})|\\
&\leq& c \delta \mu |k|^{(|l|+1)\tau+|l| } e^{- |k|r}.
\varepsilonnd{eqnarray*}
This fulfils the proof of part $\bf{(1)}$.
For part $\bf{(2)}$, by part $\bf{(1)}$ and directly differentiating to $(\ref{709})$, we have, on $\hat{D}(s) \times {{\Lambda}}_+$,
\begin{eqnarray*}
|\partial_\lambda^l\partial_x^i \partial_{(y,z)}^j F|&\leq &\sum\limits_{ 0<|k|\leq K_+} |k|^{|i|} (|\partial_\lambda^l f_{k00}|+ |\partial_\lambda^l f_{k10}|s^{1-sgn|j|}+ |\partial_\lambda^l f_{k01}|s^{1-sgn|j|}\\
&~&~~+|\partial_\lambda^l f_{k20}|s^{1-sgn(|j|-1)}+|\partial_\lambda^l f_{k02}|s^{1-sgn(|j|-1)}\\
&~&~~+|\partial_\lambda^l f_{k11}|s^{1-sgn(|j|-1)}) e^{-|k|(r_+ + \frac{7}{8} (r - r_+))}\\
&\leq& c\delta s^{a_j} \mu^{d_j} \sum\limits_{0< |k|\leq K_+} |k|^{\chi} e^{- \frac{|k|( r - r_+)}{8}}\\
&=& c \delta s^{a_j} \mu^{d_j} \Gamma(r - r_+).
\varepsilonnd{eqnarray*}
\varepsilonnd{proof}
Similar to $\bf{Lemma~ 3.6}$ of \cite{Li}, here, $F$ can also be smoothly extended to functions of H\"{o}lder class $C^{l_0 + \sigma_0 + 1, l_0 -1 + \sigma_0} (\hat{D}(\beta_0) \times \tilde{\Lambda}_0)$, where $0<\sigma_0< 1$ is fixed. Moreover, there is a constant $c$ such that
\begin{eqnarray*}
|F|_{C^{l_0 + \sigma_0 + 1, l_0 -1 + \sigma_0} (\hat{D}(\beta_0) \times \tilde{\Lambda}_0)} \leq c\delta s^{a_j} \mu^{d_j} \Gamma(r - r_+).
\varepsilonnd{eqnarray*}
\begin{lemma}
Assume
\begin{itemize}
\item[\bf{(H5)}] $c \mu^{\sigma} \Gamma( r- r_+) < \frac{1}{8} (r -r_+),$
\item[\bf{(H6)}] $c \mu^{\sigma} \Gamma ( r- r_+) < \frac{1}{8}\alpha.$
\varepsilonnd{itemize}
Then the following hold$:$
\begin{itemize}
\item[\bf{1)}] For all $0\leq t\leq1$,
\begin{eqnarray}\label{712}
\phi_F^t&:& D_{\frac{1}{4}\alpha} \rightarrow D_{\frac{1}{2}\alpha},\\
\phi&:& D_{\frac{1}{8}\alpha} \rightarrow D_{\frac{1}{4}\alpha}
\varepsilonnd{eqnarray}
are well defined, real analytic and depend smoothly on $\lambda \in {\Lambda}_+$$;$
\item[\bf{2)}]There is a constant $c$ such that for all $0\leq t\leq 1$, $|i|+|j|+|l|\leq l_0$,
\begin{eqnarray*}
|\partial_\lambda^l\partial_x^i \partial_{(y,z)}^j( \phi_F^t\circ \phi - id )|_{D_{\frac{1}{4} \alpha}\times {\Lambda}_+} \leq \left\{
\begin{array}{ll}
c s \mu^{d_j} \Gamma(r - r_+), & \hbox{$|i| + |j| =0, |l|\geq 1$$;$} \\
c \mu^{d_j} \Gamma(r - r_+), & \hbox{$ 2\leq|l|+|i|+|j|\leq l_0+2$$;$} \\
c, & \hbox{otherwise.}
\varepsilonnd{array}
\right.
\varepsilonnd{eqnarray*}
\varepsilonnd{itemize}
\varepsilonnd{lemma}
\begin{proof}
This lemma follows from $\bf{Lemma~3.7}$ of \cite{Li}. We omit details.
\varepsilonnd{proof}
\subsubsection{New perturbation}
Here we will estimate the new perturbation $P_+$ on the domain $D_+ \times \Lambda_+$, where $D_+ = D_{\frac{\alpha}{8}}$.
\begin{lemma}
Assume
\begin{itemize}
\item[\bf{(H7)}] $\mu^{\sigma} \Gamma^3 (r - r_+) \leq \frac{\gamma_+^{b}}{\gamma^{b}}$.
\varepsilonnd{itemize}
Then on $D_+ \times {\Lambda}_+$,
\begin{eqnarray*}
|\partial_ \lambda^ l \partial_x^i \partial_{(y,z)}^j P_+|\leq c\delta \gamma_+^{b_j} s_+^{a_j} \mu_+^{d_j}.
\varepsilonnd{eqnarray*}
\varepsilonnd{lemma}
\begin{proof}
Denote $\partial^{l,i,j} = \partial_\lambda^l \partial_x^i\partial_y^j$ for $|l|+ |i|+|j| \leq l_0.$ With the same argument as $\bf{Lemma~3.9}$ given in \cite{Li}, we have
\begin{eqnarray*}
|\partial^{l,i,j}( \int_0^1 \{R_t, F\} \circ \phi_F^t dt \circ \phi)|_{D_{\frac{\alpha}{4}} \times {\Lambda}_+} &\leq& c\delta s^{a_j} \mu^2 \Gamma^3(r - r_+),\\
|\partial^{l,i,j} ( P- R)\circ \phi_F^1\circ\phi|_{D_{\frac{\alpha}{4}} \times {{\Lambda}}_+} &\leq& c\delta \gamma^{b}s^{a_j} \mu^{2}\Gamma(r -r_+),\\
|\partial^{l,i,j} R' \circ \phi|_{D_{\frac{\alpha}{4}} \times {\Lambda}_+} &\leq& c\delta^2 s^{a_j+1} \mu \Gamma(r -r_+),\\
|\partial^{l,i,j} \langle \left(
\begin{array}{c}
y \\
z \\
\varepsilonnd{array}
\right), \left(
\begin{array}{cc}
p_{020} & \frac{1}{2}p_{011} \\
\frac{1}{2}p_{011}^T & p_{002} \\
\varepsilonnd{array}
\right)\left(
\begin{array}{c}
y_0 \\
z_0 \\
\varepsilonnd{array}
\right)
\rangle|_{D_{\frac{\alpha}{8}} \times \Lambda_+} &\leq& c \delta \gamma^b s^{a_j} \mu^2.
\varepsilonnd{eqnarray*}
Further, by $(\ref{N5})$ we have
\begin{eqnarray*}
|\partial_\lambda^l \partial_x^i \partial_{(y,z)}^j P_+|\leq c\delta s^{a_j} \mu^{2}\Gamma^3(r - r_+).
\varepsilonnd{eqnarray*}
Here we use the fact that $s = \mu \mu_0^{-\frac{1}{3\sigma}} s_0$ and $\delta \mu_0^{-\frac{1}{3\sigma}} s_0 = o(c).$
Using assumption $\bf{(H7)}$ yields the lemma.
\varepsilonnd{proof}
\subsubsection{The preservation of frequencies}
As we see above, if $M(\lambda)$ is nonsingular, there is a transformation $(\ref{E_21})$ such that all the frequencies are preserved after a KAM step. However, when the system is degenerate, $(\ref{E_1})$ is not solvable, i.e. there is no transformation such that all frequencies are preserved after a KAM step. To show the part preservation of frequency, we give two simple properties.
\begin{lemma}\label{Pro2}
For an $n\times n$ symmetrical matrix $A$ with $rank A = m$, there is an invertible matrix $T$, which corresponds to a linear transformation such that only some rows of $A$ exchange, such that
$$T^{-1} A T = \left(
\begin{array}{cc}
B & C \\
D & E \\
\varepsilonnd{array}
\right),
$$
where $B$ is an $m \times m$ nonsingular minor.
\varepsilonnd{lemma}
\begin{proof}
Rewrite
\begin{eqnarray*}
A = \left(
\begin{array}{c}
a_1 \\
a_2 \\
\vdots \\
a_n \\
\varepsilonnd{array}
\right) = (b_1, b_2, \cdots, b_n),
\varepsilonnd{eqnarray*}
where $a_i$ is $i-$th row of $A$ and $b_i$ is $i-$th column of $A$, $i=1, \cdots, n.$
Since $A$ is symmetrical, $a_i = b_i^T$, $i = 1, \cdots, n$, which means that there is a same linear relation between $a_i$ and $b_i$, $i=1, \cdots, n$. Because $rank A = m$, there are $m$ linearly independent rows (columns) of $A$. Then there is an invertible matrix $T$, which corresponds to a linear transformation that exchange some rows of $A$, such that
\begin{eqnarray*}
T \left(
\begin{array}{c}
a_1 \\
a_2 \\
\vdots \\
a_n \\
\varepsilonnd{array}
\right) = \left(
\begin{array}{c}
a_1^1 \\
a_2^1 \\
\vdots \\
a_m^1 \\
\vdots \\
a_n^1 \\
\varepsilonnd{array}
\right),
\varepsilonnd{eqnarray*}
where $a_1^1, ~\cdots,~ a_m^1$ are linearly independent. Since $T^{-1} = T$ and $T^{-1}$ does not change the linear relation among $b_1$, $\cdots$, $b_m$, we get
\begin{eqnarray*}
T^{-1} A T &=& \left(
\begin{array}{c}
a_1^1 \\
a_2^1 \\
\vdots \\
a_m^1 \\
\vdots \\
a_n^1 \\
\varepsilonnd{array}
\right) T\\
&=& \left(
\begin{array}{cc}
B & C \\
D & E \\
\varepsilonnd{array}
\right),
\varepsilonnd{eqnarray*}
where $B$ is an $m \times m$ nonsingular minor.
\varepsilonnd{proof}
Combining assumption $\bf{(A2)}$ and $\bf{Property~\ref{Pro2}}$, there is an invertible matrix $T$, which corresponds to a transformation only exchanging columns or rows, such that
\begin{eqnarray*}
T^{-1} \left(
\begin{array}{cc}
M_{11} & M_{12} \\
M_{21} & M_{22} \\
\varepsilonnd{array}
\right) T = \left(
\begin{array}{cc}
C_{11} & C_{12} \\
C_{21} & C_{22} \\
\varepsilonnd{array}
\right),
\varepsilonnd{eqnarray*}
where $(C_{11}, C_{12})_{(n+2m_0)\times(m + 2m_0)}$ is a matrix with $rank (C_{11}, C_{12}) = n + 2m_0$ and $(C_{21}, C_{22})_{(m-n)\times(m + 2m_0)}$ is the complements. Moreover, $(C_{11})_{(n+2m_0)\times (n+2m_0)}$ is nonsingular. Denote
\begin{eqnarray*}
\left(
\begin{array}{c}
y_1 \\
y_2 \\
\varepsilonnd{array}
\right) &=& T^{-1} \left(
\begin{array}{c}
y_* \\
z_* \\
\varepsilonnd{array}
\right),\\
\left(
\begin{array}{c}
p_1 \\
p_2 \\
\varepsilonnd{array}
\right)&=& T^{-1} \left(
\begin{array}{c}
P_{010} \\
P_{001} \\
\varepsilonnd{array}
\right)
,
\varepsilonnd{eqnarray*}
where $p_1, ~y_1 = (y_3, z_*)^T\in R^{n+2m_0}$, $y_2,$ $p_2$ $\in R^{m-n}$, $P_{010}, ~y_* = (y_3, y_2)^T\in R^{m}$, $P_{001}, ~z_*\in R^{2m_0}$. Then (\ref{E_1}) is changed to $:$
\begin{eqnarray}\label{933}
\frac{\delta}{2} \left(
\begin{array}{cc}
C_{11} & C_{12} \\
C_{21} & C_{22} \\
\varepsilonnd{array}
\right)
\left(
\begin{array}{c}
y_1 \\
y_2 \\
\varepsilonnd{array}
\right)+ \delta \left(
\begin{array}{c}
\partial_{y_1} h(y_*, z_*, \lambda) \\
\partial_{y_2} h(y_*, z_*, \lambda) \\
\varepsilonnd{array}
\right) = -\left(
\begin{array}{c}
p_1 \\
p_2 \\
\varepsilonnd{array}
\right).
\varepsilonnd{eqnarray}
Since $$ rank (C_{11}, C_{12}) =rank \left(
\begin{array}{cc}
C_{11} & C_{12} \\
C_{21} & C_{22} \\
\varepsilonnd{array}
\right),
$$
there is a inverse matrices $T_1$ that only exchange columns or rows such that
$$
T_1 \left(
\begin{array}{cc}
C_{11} & C_{12} \\
C_{21} & C_{22} \\
\varepsilonnd{array}
\right) = \left(
\begin{array}{cc}
C_{11} & C_{12} \\
0 & 0 \\
\varepsilonnd{array}
\right),
$$
which is equal to the fact that the rows of $(C_{21}, C_{22})$ is linearly dependent on the rows of $(C_{11}, C_{12}).$
Obviously, $T_1$ is a matrix with the following form $$\left(
\begin{array}{cc}
I & 0 \\
D_1 & I \\
\varepsilonnd{array}
\right),
$$
where $D_1$ is determined by the linear relation among the rows of $(C_{21}, C_{22})$ and $(C_{11}, C_{12})$. Then
\begin{eqnarray*}
T_1 \left(
\begin{array}{c}
\partial_{y_1} h(y_*, z_*, \lambda) \\
\partial_{y_2} h(y_*, z_*, \lambda) \\
\varepsilonnd{array}
\right) &=& \left(
\begin{array}{c}
\partial_{y_1} h(y_*, z_*, \lambda) \\
D_1 \partial_{y_1} h(y_*, z_*, \lambda)+ \partial_{y_2} h (y_*, z_*, \lambda)\\
\varepsilonnd{array}
\right),\\
T_1 \left(
\begin{array}{c}
p_1 \\
p_2 \\
\varepsilonnd{array}
\right) &=& \left(
\begin{array}{c}
p_1 \\
D_1 p_1 + p_2 \\
\varepsilonnd{array}
\right).
\varepsilonnd{eqnarray*}
Consider the following equation
\begin{eqnarray}\label{934}
\frac{\delta}{2} \left(
\begin{array}{cc}
C_{11} & C_{12} \\
0 & 0 \\
\varepsilonnd{array}
\right)
\left(
\begin{array}{c}
y_1 \\
y_2 \\
\varepsilonnd{array}
\right)+ \delta \left(
\begin{array}{c}
\partial_{y_1} h(y_*, z_*, \lambda) \\
0 \\
\varepsilonnd{array}
\right) = -\left(
\begin{array}{c}
p_1 \\
0 \\
\varepsilonnd{array}
\right),
\varepsilonnd{eqnarray}
where $C_{11}$ is nonsingular. Obviously, $(y_1, y_2)^T = (y_1, 0)^T$ is a specific solution of $(\ref{934})$, i.e., with assumption $\bf{(A2)}$ there is a symplectic transformation such that part of the frequencies are preserved.
\begin{remark}
If $M$ is singular, some of the frequencies are preserved and the others drift. Moreover, the drift depends on $D_1 p_1 + p_2$ and $D_1 \partial_{y_1} h(y_*, z_*, \lambda)+ \partial_{y_2} h (y_*, z_*, \lambda)$ and the estimate on drift is shown by \varepsilonmph{(\ref{drift})}.
\varepsilonnd{remark}
Consider $:$
\begin{eqnarray}
\label{935} \langle \omega, y_*\rangle + \frac{\delta}{2} \langle \left(
\begin{array}{c}
y_* \\
z_* \\
\varepsilonnd{array}
\right), M \left(
\begin{array}{c}
y_* \\
z_* \\
\varepsilonnd{array}
\right)
\rangle + P_{000}+ \langle \left(
\begin{array}{c}
P_{010} \\
P_{001} \\
\varepsilonnd{array}
\right), \left(
\begin{array}{c}
y_* \\
z_* \\
\varepsilonnd{array}
\right)
\rangle&~&\\
\nonumber~~~+ \langle\left(
\begin{array}{c}
y_* \\
z_* \\
\varepsilonnd{array}
\right), \left(
\begin{array}{cc}
P_{020} & \frac{1}{2}P_{011} \\
\frac{1}{2}P_{011}^T & P_{002} \\
\varepsilonnd{array}
\right) \left(
\begin{array}{c}
y_* \\
z_* \\
\varepsilonnd{array}
\right)\rangle + \delta h(y_*, z_*, \lambda)&=& 0,\\
\label{936} \frac{\delta M}{2} \left(
\begin{array}{c}
y_* \\
z_* \\
\varepsilonnd{array}
\right)+ \delta \left(
\begin{array}{c}
\partial_y h(y_*, z_*, \lambda) \\
\partial_z h(y_*, z_*, \lambda) \\
\varepsilonnd{array}
\right) + \left(
\begin{array}{c}
P_{010} \\
P_{001} \\
\varepsilonnd{array}
\right) - t \left(
\begin{array}{c}
\omega \\
0 \\
\varepsilonnd{array}
\right)
&=& 0.~~~~~~~~
\varepsilonnd{eqnarray}
If $M$ is nonsingular, according $\bf{(A3)}$ and the continuity of determinant, we have
\begin{eqnarray*}
\det \left(
\begin{array}{cc}
M & \bar{\omega}_1 \\
\bar{\omega}_2 & 0 \\
\varepsilonnd{array}
\right) \neq 0,
\varepsilonnd{eqnarray*}
where $\omega_1 = (\omega, 0)^T \in R^{n+ 2m_0}$, $\omega_2 = (P_{010}+ \omega, P_{001})$. Then, combining $(\ref{935})$ and $(\ref{936})$, with implicit theorem we get $(y_*, z_*, t),$ i.e., we construct a transformation such that on the same energy surface the ratio of the frequency is preserved after a KAM step.
\begin{remark}
If $M$ is nonsingular, the condition
\begin{eqnarray*}
\det \left(
\begin{array}{cc}
M & \bar{\omega}_1 \\
\bar{\omega}_1 & 0 \\
\varepsilonnd{array}
\right) \neq 0
\varepsilonnd{eqnarray*}
is a generalization of isoenergetically nondegenerate condition given by V. I. Arnold to the persistence of lower dimensional invariant tori on a given energy surface, where $\omega_1 = (\omega, 0)^T $.
\varepsilonnd{remark}
\begin{remark}
In $(\ref{936})$, $t\in R$, i.e., the ratio of the frequency is $t+1$.
\varepsilonnd{remark}
Assume $M$ is singular and conditions $\bf{(A2)}$ and $\bf{(A3)}$ hold. Denote $\tilde{\omega}_1$ by the first $n+ 2m_0$ components of $T_1T^{-1} (\omega, 0)^T$, which is equal to the first $n+ 2m_0$ components of $T^{-1} (\omega, 0)^T$. In fact, $$T_1T^{-1} \left(
\begin{array}{c}
\omega \\
0 \\
\varepsilonnd{array}
\right)
= T_1 \left(
\begin{array}{c}
\tilde{\omega}_1 \\
\omega_4 \\
\varepsilonnd{array}
\right)=\left(
\begin{array}{cc}
I & 0 \\
D_1 & I \\
\varepsilonnd{array}
\right)\left(
\begin{array}{c}
\tilde{\omega}_1 \\
\omega_4 \\
\varepsilonnd{array}
\right)=
\left(
\begin{array}{c}
\tilde{\omega}_1 \\
D_1\tilde{\omega}_1+ \omega_4 \\
\varepsilonnd{array}
\right),
$$
where $\omega = (\omega_3, \omega_4)^T\in R^{m},$ $\tilde{\omega}_1 = (\omega_3, 0)^T\in R^{n+ 2m_0}.$ Similarly, combining $(\ref{934})$, we have
\begin{eqnarray*}
\frac{\delta}{2} \left(
\begin{array}{cc}
C_{11} & C_{12} \\
0 & 0 \\
\varepsilonnd{array}
\right)
\left(
\begin{array}{c}
y_1 \\
y_2 \\
\varepsilonnd{array}
\right)+ \delta \left(
\begin{array}{c}
\partial_{y_1} h(y_*, z_*, \lambda) \\
0 \\
\varepsilonnd{array}
\right)- t\left(
\begin{array}{c}
\tilde{\omega}_1 \\
0 \\
\varepsilonnd{array}
\right)
= -\left(
\begin{array}{c}
p_1 \\
0 \\
\varepsilonnd{array}
\right).
\varepsilonnd{eqnarray*}
Assume
\begin{eqnarray}\label{937}
det \left(
\begin{array}{cc}
C_{11} & \tilde{\omega}_1 \\
\tilde{\omega}_2 & 0 \\
\varepsilonnd{array}
\right) \neq 0,
\varepsilonnd{eqnarray}
where $\tilde{\omega}_2 $ are the first $n+ 2m_0$ components of $(P_{010}+ \omega, P_{001}) T$.
Then there is a $(y_{i_1}^*, \cdots, y_{i_n}^*, 0, \cdots, 0, z_1^*, \cdots, z_{2m_0}^*, t)$ such that
\begin{eqnarray*}
\langle \omega, y_*\rangle + \frac{\delta}{2} \langle \left(
\begin{array}{c}
y_* \\
z_* \\
\varepsilonnd{array}
\right), M \left(
\begin{array}{c}
y_* \\
z_* \\
\varepsilonnd{array}
\right)
\rangle + P_{000}+ \langle \left(
\begin{array}{c}
P_{010} \\
P_{001} \\
\varepsilonnd{array}
\right), \left(
\begin{array}{c}
y_* \\
z_* \\
\varepsilonnd{array}
\right)
\rangle&~&\\
\nonumber~~~+ \langle\left(
\begin{array}{c}
y_* \\
z_* \\
\varepsilonnd{array}
\right), \left(
\begin{array}{cc}
P_{020} & \frac{1}{2}P_{011} \\
\frac{1}{2}P_{011}^T & P_{002} \\
\varepsilonnd{array}
\right) \left(
\begin{array}{c}
y_* \\
z_* \\
\varepsilonnd{array}
\right)\rangle + \delta h(y_*, z_*, \lambda)&=& 0,\\
\frac{\delta}{2} \left(
\begin{array}{cc}
C_{11} & C_{12} \\
0 & 0 \\
\varepsilonnd{array}
\right)
\left(
\begin{array}{c}
y_1 \\
y_2 \\
\varepsilonnd{array}
\right)+ \delta \left(
\begin{array}{c}
\partial_{y_1} h(y_*, z_*, \lambda) \\
0 \\
\varepsilonnd{array}
\right)- t\left(
\begin{array}{c}
\tilde{\omega}_1 \\
0 \\
\varepsilonnd{array}
\right) &=& -\left(
\begin{array}{c}
p_1 \\
0 \\
\varepsilonnd{array}
\right).
\varepsilonnd{eqnarray*}
Finally, combining $\bf{(A2)}$, $\bf{(A3)}$, $\bf{Property ~\ref{Pro2}}$ and the continuity of determinant, assumption $(\ref{937})$ holds. Therefore, on a given energy surface there is a transformation such that frequency ratio between the unperturbed torus and the perturbed is preserved.
\begin{remark}
Assume $\bf{(A2)}$ and $\bf{(A3)}$. For a given energy, the ratio of part frequencies can be preserved. Simultaneously, the other frequencies slightly drift and the drift depend on $D_1 p_1 + p_2$ and $D_1 \partial_{y_1} h(y_*, z_*, \lambda)+ \partial_{y_2} h (y_*, z_*, \lambda)$.
\varepsilonnd{remark}
\subsection{Iteration Lemma}\label{727}
Let $r_0$, $\gamma_0$, $s_0$, $\beta_0$, $\mu_0$, $\varepsilonta_0$, $\Lambda_0$, $H_0$, $N_0$, $e_0$, $P_0$ be given as above and denote $\hat{D}_0 = D(r_0, \beta_0)$. For any $\nu = 0,1, \cdots,$ denote
\begin{eqnarray*}
r_\nu &=& r_0 (1 - \sum_{i=1}^\nu \frac{1}{2^{i+1}}), ~~~\gamma_\nu = \gamma_0 (1 - \sum_{i=1}^\nu \frac{1}{2^{i+1}}),\\
\beta_\nu &=&\beta_0(1 - \sum_{i=1}^\nu \frac{1}{2^{i+1}}),~~~\mu_\nu = (64c_0)^{\frac{1}{1- \lambda_0}} \mu_{\nu-1}^{1 + \sigma}, ~~~K_\nu = ([\log\frac{1}{\mu_{\nu-1}}]+1)^{3 {\varepsilonta}},\\
D_\nu &=& D(r_\nu, s_\nu),~~~~~~~\hat{D}_\nu = D(r_\nu+ \frac{7}{8} (r_{\nu-1} - r_\nu)),~~~s_\nu = \frac{1}{8} \alpha_{\nu-1} s_{\nu-1},\\
\Lambda_\nu &=& \{\lambda \in \Lambda_{\nu-1}: |\langle k, \omega\rangle| > \frac{ \gamma_0}{|k|^\tau},~~~0<|k|\leq K_\nu\},~~~\alpha_\nu = \mu_\nu^{\frac{1}{3}}.
\varepsilonnd{eqnarray*}
We have the following Iteration Lemma.
\begin{lemma}\label{balala}
Assume $(\ref{714})$ holds. Then the KAM step described in Section $\ref{KAM}$ is valid for all $\nu = 0,1,\cdots$, and the following facts hold for all $\nu = 1,2,\cdots.$
\begin{itemize}
\item[\bf{(1)}] $P_\nu$ is real analytic in $(x,y)\in D_\nu$, smooth in $(x,y)\in \hat{D}_\nu$ and smooth in $\lambda \in {\Lambda}_\nu$, and moreover,
\begin{eqnarray*}
|\partial_\lambda^l \partial_x^i \partial_{(y,z)}^j P_\nu |_{D_\nu \times {\Lambda}_\nu} \leq \delta\gamma_\nu^{b_j} s_\nu^{a_j} \mu_\nu^{d_j}, ~|l|+|i|+|j|\leq l_0;
\varepsilonnd{eqnarray*}
\item[\bf{(2)}] $\Phi_\nu = \phi_F^t\circ \phi: \hat{D} \times {\Lambda}_0 \rightarrow \hat{D}_{\nu - 1}, D_\nu \times {\Lambda}_\nu \rightarrow D_{\nu-1}$, is symplectic for each $\lambda \in {\Lambda}_0$, and is of class $C^{l_0 + 1 + \sigma_0, l_0 -1 +\sigma_0}$, $C^{\alpha, l_0}$, respectively, where $\alpha$ stands for real analyticity and $0<\sigma_0<1$ is fixed. Moreover,
\begin{eqnarray*}
\tilde{H}_\nu = H_{\nu-1}\circ \Phi_\nu = N_\nu+ {P}_\nu,
\varepsilonnd{eqnarray*}
on $\hat{D} \times {\Lambda}_\nu$, and
\begin{eqnarray*}
|\Phi_\nu - id |_{C^{l_0+ 1 + \sigma_0,l_0 -1 + \sigma_0} (\hat{D} \times \tilde{\Lambda}_0)} \leq c_0 \delta \gamma^{b-1} \frac{\mu_0}{2^\nu};
\varepsilonnd{eqnarray*}
\item[\bf{(3)}] $\Lambda_\nu = \{\lambda\in \Lambda_{\nu-1}: |\langle k, \omega\rangle| > \frac{ \gamma_{\nu-1}}{|k|^\tau} ~for~all~K_{\nu-1}<|k|\leq K_\nu\}$.
\varepsilonnd{itemize}
\varepsilonnd{lemma}
\begin{proof}
The proof of this lemma is to verify conditions $\bf{(H1)} - \bf{(H7)}$. Those are standard and we omit the detail for brevity. For details, for example, refer to the proof of $\bf{Lemma~ 4.1}$ of $\cite{Li}$.
\varepsilonnd{proof}
\begin{remark}
According to the choice of iteration sequence, the drift of frequency can be estimated by
\begin{eqnarray}\label{drift}
|\partial_\lambda^l\omega_\nu - \partial_\lambda^l \omega_0|_{\tilde{\Lambda}_\nu} \leq c\mu_0^{\frac{\nu \sigma^2 + (1+\sigma)\sum\limits_{r=2}^{\nu} C_{\nu}^r \sigma^r}{3 \sigma^2}} (\gamma_0^b s_0 \mu_0^{ \frac{(1+ \sigma)((1+\sigma)^{\nu} -1)}{\sigma}} + c \mu_0^{\frac{\nu \sigma^2 + (1+\sigma)\sum\limits_{r=2}^{\nu} C_{\nu}^r \sigma^r}{3 \sigma^2}} s_0^2),~~
\varepsilonnd{eqnarray}
for $|l|\leq l_0.$
\varepsilonnd{remark}
\subsection{Convergence and measure estimate}\setcounter{equation}{0}
Let
\begin{eqnarray}
\nonumber \Psi^{\nu } &=& \Phi_1 \circ \Phi_2 \circ \cdots \circ \Phi_\nu, ~~ \nu = 1,2, \cdots .
\varepsilonnd{eqnarray}
Then $\Psi^{\nu}: \tilde{D}_{\nu} \times \Lambda _0(g,G) \rightarrow \tilde{D}_0$, and
\begin{eqnarray}
\nonumber H_0 \circ \Psi^{\nu} &=& H_{\nu} = N_{\nu}+ P_{\nu},\\
\nonumber N_{\nu}&=& e_{\nu} + \langle \omega_\nu , y\rangle+ h_\nu (y, \omega),~~\nu= 0,1,\cdots,
\varepsilonnd{eqnarray}
where $\Psi _0 = id $.
Standardly, $N_\nu$ converges uniformly to $N_\infty$, $P_\nu$ converges uniformly to $P_\infty$
and $\partial _y^i \partial_z^j P_\infty = 0,~2|i|+|j|\leq 2.$
Hence for each $\lambda\in \Lambda_{*}$, $T^d \times \{0\} \times \{0\}$ is an analytic invariant torus of $H_\infty$ with the toral frequency $\omega_\infty$, which for all $k\in Z^m \backslash \{0\},~1\leq q\leq n$, by the definition of $\Lambda_\nu$ and Lemma \ref{balala} (2), satisfies the facts that
\begin{itemize}
\item [\bf{(1)}] if $\bf{(A1)}$ hold and $M$ is nonsingular, then
\begin{eqnarray}
\nonumber \omega_\infty \varepsilonquiv \omega_0,~~~|\langle k, \omega_\infty \rangle| > \frac{\gamma}{2|k|^\tau};
\varepsilonnd{eqnarray}
\item [\bf{(2)}]if $\bf{(A1)}$ and $\bf{(A3)}$ hold and $M$ is nonsingular, then on a given energy surface
\begin{eqnarray}
\nonumber\omega_\infty \varepsilonquiv t \omega_0, ~~|\langle k, \omega_\infty \rangle| > \frac{\gamma}{2|k|^\tau};
\varepsilonnd{eqnarray}
\item [\bf{(3)}]if $\bf{(A1)}$ and $\bf{(A2)}$ hold, then
\begin{eqnarray}
\nonumber (\omega_\infty)_{i_q} \varepsilonquiv (\omega_0)_{i_q}, q= 1, \cdots, n, ~~|\langle k, \omega_\infty \rangle| > \frac{\gamma}{2|k|^\tau};
\varepsilonnd{eqnarray}
\item [\bf{(4)}]if $\bf{(A1)}$, $\bf{(A2)}$ and $\bf{(A3)}$ hold, then
\begin{eqnarray}
\nonumber (\omega_\infty)_{i_q} \varepsilonquiv t (\omega_0)_{i_q}, q= 1, \cdots, n, ~~|\langle k, \omega_\infty \rangle| > \frac{\gamma}{2|k|^\tau}.
\varepsilonnd{eqnarray}
\varepsilonnd{itemize}
Following the Whitney extension of $\Psi^\nu,$ all $e_\nu,$ $\omega_\nu,$ $h_\nu,$ $P_\nu,$ $(\nu = 0,1,\cdots)$ admit uniform $C^{d-1 +\sigma_0}$ extensions in $\lambda \in \Lambda_0$ with derivatives in $\lambda$ up to order $d-1$. Thus, $e_\infty$, $\omega_\infty$, $h_\infty$, $P_\infty$ are $C^{d-1}$ Whitney smooth in $\lambda \in \Lambda_{*}$, and the derivatives of $e_\infty -e_0$, $\omega_\infty -\omega_0$, $h_\infty -h_0$ satisfy similar estimates. Consequently, the perturbed tori form a $C^{d-1}$ Whitney smooth family on $\Lambda_{*}(g,G)$.
The measure estimate is the same as ones in \cite{LLL}. For the sake of simplicity, we omit details. Now we finish the proof of Theorem \ref{shengluede}.
\section{Proof of Main Theorem}\label{074}
In order to use Theorem $\ref{shengluede}$, we should reduce Hamiltonian system $(\ref{082})$ to $(\ref{model33})$. But the traditional method is fail due to high order degeneracy of perturbation. Here we show a program, combinations of finite nonlinear KAM steps, to finish this reduction and to fix thought we only give an outline.
Regard Hamiltonian system (\ref{082}) as the original Hamiltonian with the following form:
\begin{eqnarray} \label{xiaoming}
H(x,y,u,v) &=& N_1(y,v) + \varepsilon^2 {P_1} (x,y,u,v,\varepsilon),
\varepsilonnd{eqnarray}
where
\begin{eqnarray*}
N_1&=& \langle\omega, y\rangle + \hat{h},\\
\hat{h}&=&\frac{\varepsilon}{2} \langle \left(
\begin{array}{c}
y \\
v \\
\varepsilonnd{array}
\right)
, M \left(
\begin{array}{c}
y \\
v \\
\varepsilonnd{array}
\right)
\rangle+ \varepsilon^2 O(|K_0 \left(
\begin{array}{c}
y \\
v \\
\varepsilonnd{array}
\right)|^3),\\
P_1(x,y,u,v) &=& \varepsilon P(x,y,u,v).
\varepsilonnd{eqnarray*}
We choose $\varepsilon = \delta$, $\gamma = \delta^{\frac{1}{4(9+d)}}$, $s = \delta^{\frac{1}{4}}$, $\mu = \delta^{\frac{1}{4}}$. Then
\begin{eqnarray} \label{xiaoming1}
H(x,y,u,v) &=& N_1(y,v) + \delta^2 {P_1} (x,y,u,v,\varepsilon),
\varepsilonnd{eqnarray}
where
\begin{eqnarray*}
N_1&=& \langle\omega, y\rangle + \hat{h},\\
\hat{h}&=&\frac{\delta}{2} \langle \left(
\begin{array}{c}
y \\
v \\
\varepsilonnd{array}
\right)
, M \left(
\begin{array}{c}
y \\
v \\
\varepsilonnd{array}
\right)
\rangle+ \delta^2 h,\\
|{P_1}(x,y,u,v)| &\leq& \gamma^{d+9} s^2 \mu.
\varepsilonnd{eqnarray*}
Here, $h$ is a polynomial of $K_0 \left(
\begin{array}{c}
y \\
v \\
\varepsilonnd{array}
\right)$ from three order term.
Write
\begin{eqnarray*}
{P}_1&=& \sum_{k} P_{k} e^{\sqrt{-1} \langle k, x\rangle},\\
\label{R_1} R_1&=& \sum_{|k|\leq K_1} P_{k} e^{\sqrt{-1} \langle k, x\rangle},\\
\label{I_1}{{P}_1} - R_1 &=& \sum_{|k|> K_1} P_{k} e^{\sqrt{-1} \langle k, x\rangle},\\
\varepsilonnd{eqnarray*}
where $K_1$ is specified in Section $\ref{normal form}$.
Next, we are going to improve the order of ${{P}_1}$ by the symplectic transformation $\Phi_{F_1}^1$, the time$-1$ map generated by the vector field $J \nabla F_1$, where
\begin{eqnarray*}
J = \left(
\begin{array}{cccc}
0 & I_m & 0 & 0 \\
-I_m & 0 & 0 & 0 \\
0 & 0 & 0 & I_{m_0} \\
0 & 0 & -I_{m_0} & 0 \\
\varepsilonnd{array}
\right),
\varepsilonnd{eqnarray*}
\begin{eqnarray*}
F_1(x,y,u,v) = \sum _{0<|k|\leq K_1} f_{k} e^{\sqrt{-1} \langle k, x\rangle}
\varepsilonnd{eqnarray*}
satisfies
\begin{eqnarray}\label{tongdiao31}
\{N_1,F_1\} + \delta^2 (R_1- [R_1]) - R_1' = 0,
\varepsilonnd{eqnarray}
\begin{eqnarray*}
R_1' &=& \partial_u N_1 \partial_v F_1 - \partial_v N_1 \partial_u F_1,\\
~[R_1](y,u,v,\varepsilon)&=& \int_{T^m} {R_1}(x,y,u,v,\varepsilon)dx .
\varepsilonnd{eqnarray*}
Concretely, using (\ref{tongdiao31}) and comparing coefficients, we obtain the following nonlinear homological equations
\begin{eqnarray}\label{(11)}
-\sqrt{-1} \langle k, \omega + \partial _y \hat{h}\rangle f_{k} = \delta^2 P_{k}.
\varepsilonnd{eqnarray}
It is clear that the homological equations are uniquely solvable on the following domain
\begin{eqnarray*}
\Lambda_{1} = \{ \omega \in \Lambda: |\langle k , \omega \rangle | > \frac{\gamma }{ |k|^{\tau} }~~ for ~all~ 0<|k| \leq K_1\}.
\varepsilonnd{eqnarray*}
By (\ref{tongdiao31}), we have
\begin{eqnarray*}
\bar{H}_2&=& H_1 \circ \Phi _{F_1}^1\\
&=& N_2(y,u,v) + \delta^2 \bar{P}_2 (x,y,u,v,\varepsilon),
\varepsilonnd{eqnarray*}
where
\begin{eqnarray*}
N_2 &=& N_1 + \delta^2 [{R}_1],\\
\bar{P}_2 &=& \frac{1}{\delta^2}(R_1' + \int_0^1 \{R_{1,t}, F_1\}\circ \Phi_{F_1}^t dt +\delta^2 ({\bar{P}_1} - R_1) \circ\Phi_{F_1}^1),\\
R_{1,t} &=& t\delta^2 R_1 + (1-t)R_1' + (1-t)\delta^2 [R_1].
\varepsilonnd{eqnarray*}
It is easy to see that $[{R}_1]$ has critical point on $u$, due to the $T^{m_0}-$periodicity in $u$.
Consider the following transformation
$$\phi: ~~x \rightarrow x, ~~ y \rightarrow y+ y_0, ~~v \rightarrow v+ v_0,~~u\rightarrow u,$$
where $y_0$ and $v_0$ is determined by the following equation$:$
\begin{eqnarray*}
\delta M \left(
\begin{array}{c}
y_0 \\
v_0 \\
\varepsilonnd{array}
\right) + \delta^2 \left(
\begin{array}{c}
\partial_y h(y_0, v_0) \\
\partial_v h(y_0, v_0) \\
\varepsilonnd{array}
\right) = \delta^2 \left(
\begin{array}{c}
\partial_y [{R}_1] \\
\partial_v [{R}_1]\\
\varepsilonnd{array}
\right).
\varepsilonnd{eqnarray*}
Here and below, denote
\begin{eqnarray*}
[R_i]_2 = O(|\left(
\begin{array}{c}
y \\
v \\
\varepsilonnd{array}
\right)
|^2).
\varepsilonnd{eqnarray*}
Then
\begin{eqnarray}\label{jiayou}
H_2&=& N_2(y,u,v) + \delta^2 {P}_2 (x,y,u,v,\varepsilon),
\varepsilonnd{eqnarray}
where
\begin{eqnarray*}
N_2 &=& \langle\omega_2, y\rangle + \frac{\delta}{2} \langle \left(
\begin{array}{c}
y \\
v \\
\varepsilonnd{array}
\right)
, M_2 \left(
\begin{array}{c}
y \\
v \\
\varepsilonnd{array}
\right)
\rangle+ \delta^2h_2+ \delta^2 [{R}_1]_2,\\
\omega_2 & =& \omega+\delta M \left(
\begin{array}{c}
y_0 \\
v_0 \\
\varepsilonnd{array}
\right) + \delta^2 \left(
\begin{array}{c}
\partial_y h (y_0, v_0) \\
\partial_v h (y_0, v_0) \\
\varepsilonnd{array}
\right)+ \delta^2 \left(
\begin{array}{c}
\partial_y [{R}_1] \\
\partial_v [{R}_1] \\
\varepsilonnd{array}
\right),\\
M_2 &=& M + \delta^2 \partial_{(y,v)}^2 h,\\
h_2 &=& O(|K_0 \left(
\begin{array}{c}
y \\
v \\
\varepsilonnd{array}
\right)|^3),\\
P_2&=&\bar{P}_2\circ\phi + \langle \left(
\begin{array}{c}
y \\
v \\
\varepsilonnd{array}
\right), \partial_{(y,v)}^2 [{R}_1] \left(
\begin{array}{c}
y_0 \\
v_0 \\
\varepsilonnd{array}
\right)
\rangle.
\varepsilonnd{eqnarray*}
Moreover,
\begin{eqnarray*}
|\partial_\lambda^l P_2 |\leq c \delta^{\frac{657}{576}},~|l|\leq d.
\varepsilonnd{eqnarray*}
Here and below, we denote by $c$ the positive constant independent of the iteration process.
In other word, at the 2-nd step of the iteration program the normal form is (\ref{jiayou}). In a similar manner to Hamiltonian (\ref{jiayou}), write
\begin{eqnarray*}
{P}_2&=& \sum_{k} P_{k} e^{\sqrt{-1} \langle k, x\rangle},\\
\label{R_1} R_2&=& \sum_{|k|\leq K_2} P_{k} e^{\sqrt{-1} \langle k, x\rangle},\\
\label{I_1}{{P}_2} - R_2 &=& \sum_{|k|> K_2} P_{k} e^{\sqrt{-1} \langle k, x\rangle}.\\
\varepsilonnd{eqnarray*}
Improve the order of ${{P}_2}$ by symplectic transformation $\Phi_{F_2}^1$, where
\begin{eqnarray} \label{F}
F_2(x,y,u,v) = \sum _{0<|k|\leq K_2} f_{k} e^{\sqrt{-1} \langle k, x\rangle}
\varepsilonnd{eqnarray}
satisfies
\begin{eqnarray}\label{tongdiao41}
\{N_2,F_2\} + \delta^2 (R_2- [R_2]) - R_2' = 0,
\varepsilonnd{eqnarray}
\begin{eqnarray*}
R_2' &=& \partial_u N_2 \partial_v F_2 - \partial_v N_2 \partial_u F_2 + \partial_y [{R}_1]_2 \partial_x F_2,\\
~[R_2]&=& \int_{T^m} {R_2}(x,y,u,v,\varepsilon)dx .
\varepsilonnd{eqnarray*}
Then
\begin{eqnarray*}
H_3&=& H_2 \circ \Phi _{F_2}^1\circ\phi\\
&=& N_3(y,u,v) + {P}_3 (x,y,u,v, \varepsilon),
\varepsilonnd{eqnarray*}
where
\begin{eqnarray*}
N_3 &=&\langle\omega_3, y\rangle + \frac{\delta}{2} \langle \left(
\begin{array}{c}
y \\
v \\
\varepsilonnd{array}
\right)
, M_3 \left(
\begin{array}{c}
y \\
v \\
\varepsilonnd{array}
\right)
\rangle+ \delta^2h_3 + \delta^2 [{R}_1]_2+ \delta^2 [{R}_2]_2,\\
\omega_3 & =& \omega_2+\delta M_2 \left(
\begin{array}{c}
y_0 \\
v_0 \\
\varepsilonnd{array}
\right) + \delta^2 \left(
\begin{array}{c}
\partial_y h_2 (y_0, v_0) \\
\partial_v h_2 (y_0, v_0) \\
\varepsilonnd{array}
\right)+ \delta^2 \left(
\begin{array}{c}
\partial_y [{R}_2] \\
\partial_v [{R}_2] \\
\varepsilonnd{array}
\right),\\
M_3 &=& M_2 + \delta^2 \partial_{(y,v)}^2 h_2,\\
{P}_3 &=& R_2'\circ \phi + \int_0^1 \{R_{2,t}, F_2\}\circ \Phi_{F_2}^t \circ \phi dt +({{P}_2} - R_2) \circ\Phi_{F_2}^1\circ \phi
\\
&~&+ \langle \left(
\begin{array}{c}
y \\
v \\
\varepsilonnd{array}
\right), \partial_{(y,v)}^2 [{R}_2] \left(
\begin{array}{c}
y_0 \\
v_0 \\
\varepsilonnd{array}
\right)
\rangle,\\
R_{2,t} &=& t R_2 + (1-t)R_2' + (1-t)[R_2].
\varepsilonnd{eqnarray*}
Hence
\begin{eqnarray*}
|\partial_\lambda^l {{P}}_3|\leq c \delta^{\frac{3220}{768}}, ~|l|\leq d.
\varepsilonnd{eqnarray*}
Generally, the $\kappa-$th KAM step is the following, where $\kappa$ is a given constant.
Write
\begin{eqnarray*}
{P}_\kappa&=& \sum_{k} P_{k} e^{\sqrt{-1} \langle k, x\rangle},\\
\label{R_1} R_\kappa&=& \sum_{|k|\leq K_{\kappa}} P_{k} e^{\sqrt{-1} \langle k, x\rangle},\\
\label{I_1}{{P}_\kappa} - R_\kappa &=& \sum_{|k|> K_{\kappa}} P_{k} e^{\sqrt{-1} \langle k, x\rangle}.\\
\varepsilonnd{eqnarray*}
Improve the order of ${{P}_\kappa}$ by the symplectic transformation $\Phi_{F_\kappa}^1$, where
\begin{eqnarray} \label{F}
F_\kappa(x,y,u,v) = \sum _{0<|k|\leq K_{\kappa}} f_{k} e^{\sqrt{-1} \langle k, x\rangle}
\varepsilonnd{eqnarray}
satisfies
\begin{eqnarray}\label{tongdiao511}
\{N_\kappa,F_\kappa\} + \delta^2 (R_\kappa- [R_\kappa] )- R_\kappa' = 0,
\varepsilonnd{eqnarray}
\begin{eqnarray*}
R_\kappa' &=& \partial_u N_\kappa \partial_v F_\kappa - \partial_v N_\kappa \partial_u F_\kappa +\partial_y [R_1]_2 \partial_x F_\kappa+\cdots+ \partial_y [R_{\kappa-1}]_2 \partial_x F_\kappa,\\
~[R_i]&=& \int_{T^m} {R_i}(x,y,u,v,\varepsilon)dx, ~~~~1\leq i \leq \kappa-1.
\varepsilonnd{eqnarray*}
Using (\ref{tongdiao511}) and comparing coefficients, we obtain the following nonlinear homological equations
\begin{eqnarray}\label{(11)}
-\sqrt{-1} \langle k, \omega + \partial _y \hat{h}_*\rangle f_{k} = P_{k}.
\varepsilonnd{eqnarray}
It is clear that the homological equations are uniquely solvable on the following domain
\begin{eqnarray*}
\Lambda_{\kappa}(g,G) = \{ \omega \in \Lambda(g,G) : |\langle k , \omega \rangle | > \frac{\gamma_\kappa }{ |k|^{\tau} }~~ for ~all~ 0<|k| \leq K_{\kappa}\}.
\varepsilonnd{eqnarray*}
Then
\begin{eqnarray*}
H_{\kappa+1}&=& H_\kappa \circ \Phi _{F_\kappa}^1 \circ\phi = N_{\kappa+1}(y,u,v) + {P}_{\kappa+1} (x,y,u,v,\varepsilon),
\varepsilonnd{eqnarray*}
where
\begin{eqnarray*}
N_{\kappa+1} &=& \langle\omega_{\kappa+1}, y\rangle + \frac{\delta}{2} \langle \left(
\begin{array}{c}
y \\
v \\
\varepsilonnd{array}
\right)
, M_{\kappa+1} \left(
\begin{array}{c}
y \\
v \\
\varepsilonnd{array}
\right)
\rangle+ \delta^2h_\kappa \\
&~&+ \delta^2 [\bar{R}_1]_2+ \delta^2 [\bar{R}_2]_2+ \cdots+ \delta^2 [\bar{R}_\kappa]_2 ,\\
\omega_{\kappa+1} & =& \omega_\kappa+\delta M_\kappa \left(
\begin{array}{c}
y_0 \\
v_0 \\
\varepsilonnd{array}
\right) + \delta^2 \left(
\begin{array}{c}
\partial_y h_\kappa (y_0, v_0) \\
\partial_v h_\kappa (y_0, v_0) \\
\varepsilonnd{array}
\right)+ \delta^2 \left(
\begin{array}{c}
\partial_y [{R}_\kappa] \\
\partial_v [{R}_\kappa] \\
\varepsilonnd{array}
\right),\\
M_{\kappa+1} &=& M_\kappa + \delta^2 \partial_{(y,v)}^2 h_\kappa,\\
{P}_{\kappa+1} &=& R_\kappa'\circ\phi + \int_0^1 \{R_{\kappa,t}, F_\kappa\}\circ \Phi_{F_\kappa}^t \circ\phi dt +({P_\kappa} - R_\kappa) \circ\Phi_{F_\kappa}^1\circ\phi\\
&~&\langle \left(
\begin{array}{c}
y \\
v \\
\varepsilonnd{array}
\right), \partial_{(y,v)}^2 [{R}_\kappa] \left(
\begin{array}{c}
y_0 \\
v_0 \\
\varepsilonnd{array}
\right)
\rangle,\\
R_{\kappa,t} &=& t R_\kappa + (1-t)R_\kappa' + (1-t)[R_\kappa].
\varepsilonnd{eqnarray*}
Hence,
\begin{eqnarray*}
|\partial_\lambda^l {P}_{\kappa+1}|\leq c \delta^{\frac{3}{4} + (\frac{13}{12})^{\kappa+1} 9- 8},~|l| \leq d.
\varepsilonnd{eqnarray*}
Therefore, after $\kappa$ KAM steps, the new Hamiltonian reads as
\begin{eqnarray} \label{youyong}
H_{\kappa+1} = N_{\kappa+1}+ \delta^{2} P,
\varepsilonnd{eqnarray}
where
\begin{eqnarray*}
N_{\kappa+1} &=& \langle\omega_{\kappa+1}, y\rangle + \frac{\delta}{2} \langle \left(
\begin{array}{c}
y \\
v \\
\varepsilonnd{array}
\right)
, M_{\kappa+1} \left(
\begin{array}{c}
y \\
v \\
\varepsilonnd{array}
\right)
\rangle+ \delta^2h_{\kappa+1} \\
&~&+ \delta^2 [{R}_1]_2+ \delta^2 [{R}_2]_2+ \cdots+ \delta^2 [{R}_\kappa]_2 ,\\
\omega_{\kappa+1} & =& \omega_\kappa+\delta M_\kappa \left(
\begin{array}{c}
y_0 \\
v_0 \\
\varepsilonnd{array}
\right) + \delta^2 \left(
\begin{array}{c}
\partial_y h_\kappa (y_0, v_0) \\
\partial_v h_\kappa (y_0, v_0) \\
\varepsilonnd{array}
\right)+ \delta^2 \left(
\begin{array}{c}
\partial_y [{R}_\kappa] \\
\partial_v [{R}_\kappa] \\
\varepsilonnd{array}
\right),\\
M_{\kappa+1} &=& M_\kappa + \delta^2 \partial_{(y,v)}^2 h_\kappa.
\varepsilonnd{eqnarray*}
Let
\begin{eqnarray*}
\bar{g }&=&\delta^2 [{R}_1]_2+ \delta^2 [{R}_2]_2 +\cdots +\delta^{2} [{R}_{\kappa}]_2\\
&=& \delta^{\frac{1809}{576}} [\bar{R}_1]_2+ \delta^{\frac{4756}{768}} [\bar{R}_2]_2 +\cdots +\delta^{(\frac{13}{12})^{\kappa+1} 9- \frac{21}{4}} [\bar{R}_{\kappa}]_2\\
&=&\sum\limits_{j_1} \delta^{\frac{1809}{576}+ j_1} [\bar{R}_1]_2^{(j_1)}+ \sum\limits_{j_2}\delta^{\frac{4756}{768}+ j_2} [\bar{R}_2]_2^{(j_2)} +\cdots +\sum\limits_{j_\kappa}\delta^{(\frac{13}{12})^{\kappa+1} 9- \frac{21}{4} + j_\kappa} [\bar{R}_{\kappa}]_2^{(j_\kappa)}.
\varepsilonnd{eqnarray*}
\begin{definition}\label{HOP}
If the following hold:
(1)At critical points of $\bar{g}$, $(y_0, u_0, v_0)$,
\begin{eqnarray*}
~\det~ \partial_u^2 \delta^{-a+1} \bar{g} = 0,
\varepsilonnd{eqnarray*}
(2) At critical points of $\bar{g}$, $(y_0, u_0, v_0)$, there is a constant $\bar{\sigma}_0 >0$, such that
\begin{eqnarray*}
|\det \partial_u^2 ~\delta^{-a} \bar{g} |\geq \bar{\sigma}_0,
\varepsilonnd{eqnarray*}
then we call $\delta P(x,y,u,v,\delta)$ a perturbation with $a-$order degeneracy at $(y_0, u_0, v_0)$.
\varepsilonnd{definition}
\begin{remark}
With condition$:$ $|\det {\partial_{u}^2 \varepsilon^{- a}P_1(x,y,u,v,\varepsilon)}| > \tilde{\sigma}$, obviously, the condition of $a-$order degeneracy for perturbation can be realized. And since $\bar{g}$ is $T^{m_0}$ periodic in $u$, it has $2^{m_0}$ critical points via the high order nondegeneracy and Morse theory \varepsilonmph{(\cite{Milnor})}.
\varepsilonnd{remark}
\begin{remark}
Assumption $\varepsilonmph{(2)}$ in the definition of $a-$order degenerate perturbation is equivalent to the following $\bf{(\mathfrak{S}1)}$.
\begin{itemize}
\item[$\bf{(\mathfrak{S}1)}$] At critical point of $\bar{g}$, $(y_0, u_0, v_0)$, there exists a constant $ c > 0$ such that the minimum $\lambda_{min}^{\varepsilon} (\omega)$ among absolute values of all eigenvalues of $\partial_u^2 \bar{g}$ satisfies $|\lambda_{min}^{\varepsilon} |\geq c \varepsilon ^a $ for all $\omega \in \Lambda(g,G)$.
\varepsilonnd{itemize}
\varepsilonnd{remark}
Using the definition of perturbation with $a-$order degeneracy, we can rewrite Hamiltonian at the critical point of $\bar{g}$, $(y_0,u_0,v_0)$,
\begin{eqnarray}\label{new221}
H(x,y,u,v)&=& N(y,u,v) +{\delta^{a +1}} \tilde{P}(x,y,u,v,\varepsilon),
\varepsilonnd{eqnarray}
where
\begin{eqnarray*}
N&=&\langle\omega_{\kappa+1}, y\rangle + \frac{\delta}{2} \langle \left(
\begin{array}{c}
y \\
v \\
\varepsilonnd{array}
\right)
, M_{\kappa+1} \left(
\begin{array}{c}
y \\
v \\
\varepsilonnd{array}
\right)
\rangle+ \delta^2h_\kappa + \frac{\delta^{a}}{2} \langle u, Vu\rangle + \delta^{a}O(|u|^3),\\
\delta^{a + 1} \tilde{P}&=& \delta^{\kappa + 1} P(x,y,u,v,\varepsilon)+ O(\delta^{a+1}),
\varepsilonnd{eqnarray*}
which $x \in T^m$, $y \in R^m$, $u, v \in R ^{m_0}$, $\omega \in \Lambda(g,G)$, $1 \leq a \leq \kappa$. In the above, all $\omega-$dependence is of class $C^{l_0}$ for some $l_0 \geq d$.
Next we should raise the order of $\tilde{P}$ by performing finite times nonlinear KAM steps. Let $\tilde{\tau}$ be the smallest integer such that $[ (\frac{13}{12})^{\tilde{\tau}} 9 -\frac{21}{4}]\geq\frac{3a+1}{2}$, where $a$ is a constant. After $\tilde{\tau}$ KAM steps mentioned as above and by {\bf{Remark 3.1}}, at each critical point, we obtain the following
\begin{eqnarray}
\nonumber H_{\tilde{\tau}}(x,y) &=& \langle\omega_{\tilde{\tau}}, y\rangle + \frac{\delta}{2} \langle \left(
\begin{array}{c}
y \\
v \\
\varepsilonnd{array}
\right)
, M_{\tilde{\tau}} \left(
\begin{array}{c}
y \\
v \\
\varepsilonnd{array}
\right)
\rangle+ \delta^2h_{\tilde{\tau}}\\
\label{keyileba}&~&~ + \frac{\delta^{a}}{2}{\langle u , V_{\tilde{\tau}_1}(\omega) u\rangle}+ \delta^{a}\hat{u}_{\tilde{\tau}_1}(u)+\delta ^{\frac{{3a +1}}{2}} \hat{P}(x,y,u,v,\delta),~~
\varepsilonnd{eqnarray}
up to an irrelevant constant, where
\begin{eqnarray*}
V_{\tilde{\tau}_1} &=& V+ \partial_u^2 \tilde{h},~~~~~\hat{u}_{\tilde{\tau}}(u) = \hat{u} + (\tilde{h} - \langle\partial_u^2 \tilde{h} u, u\rangle),\\
\tilde{h} &=& \delta^{(\frac{13}{12})^{\kappa+1} 9- \frac{21}{4}} [\bar{R}_{\kappa+1}]+\cdots+ \delta^{(\frac{13}{12})^{\tilde{\tau}} 9- \frac{21}{4}} [\bar{R}_{\tilde{\tau}}],\\
\hat{P} &=& \delta P(x,y,u,v,\delta), ~~~~~1\leq a\leq\kappa,
\varepsilonnd{eqnarray*}
with nonsingular $V_{\tilde{\tau}}$. But in each KAM step we have a similar hypothesis in form, $\delta K^{\tau+1} = o(\gamma)$. And the assumption is obviously hold for finite times KAM steps. Consider re-scaling $x\rightarrow x$, $y\rightarrow \delta^{\frac{{a -1}}{2}}y$, $u\rightarrow u$, $v \rightarrow \delta^{\frac{{a -1}}{2}} v$, $H\rightarrow \delta^{\frac{{-a+1}}{2}}H$. Then the re-scaled Hamiltonian reads
\begin{eqnarray*}
H_{\tau_1}(x,y) &=& \langle\omega_{\tilde{\tau}}, y\rangle + \frac{\delta}{2} \langle \left(
\begin{array}{c}
y \\
v \\
\varepsilonnd{array}
\right)
, M_{\tilde{\tau}} \left(
\begin{array}{c}
y \\
v \\
\varepsilonnd{array}
\right)
\rangle+ \delta^2h_{\tilde{\tau}}\\
&~&~+ \frac{\delta^{\frac{a+1}{2}}}{2}{\langle u , V_{\tilde{\tau}}(\omega)u\rangle}+ \delta^{\frac{a+1}{2}}\hat{u}_{\tilde{\tau}}(u)+\delta ^{a+1} \tilde{P}(x,y,u,v).
\varepsilonnd{eqnarray*}
Denote $\delta^{\frac{a+1}{2}} = \delta$. Then we have
\begin{eqnarray}\label{model31}
H(x,y,u,v) =N(y,u,v) + P(x,y,u,v),
\varepsilonnd{eqnarray}
with
\begin{eqnarray*}
N &=& \langle\omega_{\tilde{\tau}}, y\rangle + \frac{\delta}{2} \langle \left(
\begin{array}{c}
y \\
v \\
\varepsilonnd{array}
\right)
, M_{\tilde{\tau}} \left(
\begin{array}{c}
y \\
v \\
\varepsilonnd{array}
\right)
\rangle+ \delta^2h_{\tilde{\tau}}+ \frac{\delta}{2}{\langle u , V_{\tilde{\tau}} (\omega)u\rangle}+ \delta \hat{u}_{\tilde{\tau}}(u),\\
P&=&\delta^2 \tilde{P}(x,y,u,v),~~~~\hat{u}(u)= O(|u|^3),
\varepsilonnd{eqnarray*}
where $x \in T^m$, $y \in R^m$, $u, v \in R ^{m_0}$, $\omega \in \Lambda(g,G)$. In the above, all $\omega-$dependence is of class $C^{l_0}$ for some $l_0 \geq d$.
Actually, we have
\begin{eqnarray*}
|\partial_\omega^l {P}| \leq c \delta \gamma^{d+9} s^2 \mu,~~~~~|l|\leq d,
\varepsilonnd{eqnarray*}
where $\gamma= \varepsilon^{\frac{1 - 3\iota}{3(d+9)}}$, $s = \varepsilon^{\frac{1}{3}}$, $\mu = \varepsilon^\iota$, $\iota \in (0,\frac{1}{3})$. Applying Theorem \ref{shengluede} to (\ref{model31}), the system admits a family of invariant tori. By Morse theory and {\bf{Remark 2.1}}, there are $2^{m_0}$ critical points, and consequently it has $2^{m_0}$ families of invariant tori. This completes the proof of Theorem \ref{dingli11}.
\section{Examples}\label{example}
It is interesting to know how the critical points change during the process of KAM iteration. And we show this will be very complex through the following examples.
\begin{example} \label{071}
Consider the following Hamiltonian system
\begin{eqnarray}\label{youqudelizi}
H(x,y) = \omega y + \frac{\varepsilon}{2}v^2 + \varepsilon^2 \cos u + \varepsilon~ \cos u ~ \sin x ~e^y,
\varepsilonnd{eqnarray}
where $x\in T^1$, $y$, $u$, $v \in R^1$.
Let perturbation $P_1(x,y,u)=\varepsilon^2 \cos u + \varepsilon~ \cos u ~ \sin x~ e^y$. Then $\int_{T^1} P_1 dx = \varepsilon^2 \cos u$, which is of $2-$order in $\varepsilon$. Hence some previous results do not work. However, applying our result, we can obtain the persistence of resonant tori.
\varepsilonmph{Using the above program twice, we have
\begin{eqnarray}
H_3(x,y,u,v) = \omega y + \frac{\varepsilon}{2}v^2 - \varepsilon^2 \cos u+ O(\varepsilon^3).
\varepsilonnd{eqnarray}
By \textbf{Theorem \varepsilonmph{\ref{shengluede}}}, there is a family of invariant tori for the Hamiltonian \varepsilonmph{(\ref{youqudelizi})}.}
\varepsilonnd{example}
\begin{remark}
This example shows that during the processing of KAM iteration the critical points do not change.
\varepsilonnd{remark}
\begin{example}\label{072}
Consider the following Hamiltonian system
\begin{eqnarray}\label{050}
H(x,y) = \omega y + \frac{\varepsilon}{2}v^2 + \varepsilon^2 \cos (u+\frac{\iota\pi}{4}) + \varepsilon~ \sin u ~ \sin x ~e^y, ~\iota = 0,1,~
\varepsilonnd{eqnarray}
where $x\in T^1$, $y$, $u$, $v \in R^1$.
Let $P_1(x,y,u)=\varepsilon^2 \cos (u+\frac{\pi}{4}) + \varepsilon~ \sin u ~ \sin x~ e^y$. Then $\int_{T^1} P_1 dx = \varepsilon^2 \cos (u+\frac{\pi}{4})$, which is of $2-$order in $\varepsilon$, implies that some previous results do not work. However, applying our result, we can obtain the persistence of resonant tori.
\varepsilonmph{Using the program mentioned above twice, we have
\begin{eqnarray*}
H_3(x,y,u,v) = \omega y + \frac{\varepsilon}{2}v^2 + \varepsilon^2 \cos (u+\frac{\iota\pi}{4}) - \frac{\varepsilon^2 \sin^2u e^{2y}}{2\omega} + O(\varepsilon^3),
\varepsilonnd{eqnarray*}
where $\iota = 0,1.$
By \varepsilonmph{\textbf{Theorem \ref{shengluede}}}, there is a family of invariant tori for the Hamiltonian system (\ref{050}).}
\varepsilonnd{example}
\begin{remark}
\begin{enumerate}
\item [(1).] When $\iota = 0$, during the processing of KAM iteration the number of critical points increases.
\item [(2).] When $\iota = 1$, the critical points of the unperturbed system are not the critical points of the perturbed system.
\varepsilonnd{enumerate}
\varepsilonnd{remark}
\begin{appendix}
\section*{Appendix}
\setcounter{equation}{0}
\section{The computing process of Example \ref{071} }
Consider the following Hamiltonian system
\begin{equation}\label{061}
H(x,y) = \omega y + \frac{\varepsilon}{2}v^2 + \varepsilon^2 \cos u + \varepsilon~ \cos u ~ \sin x ~e^y,
\varepsilonnd{equation}
where $x\in T^1$, $y$, $u$, $v \in R^1$.
Set $P_1(x,y,u)=\varepsilon^2 \cos u + \varepsilon~ \cos u ~ \sin x~ e^y$. Then $[P_1] = \int_{T^1} P_1 dx = \varepsilon^2 \cos u$, which is of $2-$order in $\varepsilon$. Now we improve the order of ${{P}_1}$ by the symplectic transformation $\Phi_{F_1}^1$, where
\begin{eqnarray} \label{062}
F_1(x,y,u,v) =a_1(y,u,v) \sin x + b_1(y,u,v) \cos x
\varepsilonnd{eqnarray}
satisfies
\begin{eqnarray}\label{063}
\{N,F_1\} + P_1- [P_1] - P_1' = 0,
\varepsilonnd{eqnarray}
\begin{eqnarray*}
P_1' &=& \partial_u N \partial_v F_1 - \partial_v N \partial_u F_1,\\
N &=& \omega y + \frac{\varepsilon}{2}v^2.
\varepsilonnd{eqnarray*}
Take
\begin{eqnarray*}
F_1(x,y,u) = \frac{- \varepsilon ~ \cos u ~ e^y ~ \cos x}{\omega}.
\varepsilonnd{eqnarray*}
Then
\begin{eqnarray*}
H_2(x,y,u,v) = N_2(y,u) + P_1'(x,u,v,\varepsilon)+ \int_0^1 \{(1-t)\{N, F_1\}+ P_1, F_1\}\circ \phi_{F_1}^t dt,
\varepsilonnd{eqnarray*}
where
\begin{eqnarray*}
N_2(y,u) &=& N(y,u) + \varepsilon^2 \cos u,\\
P_1'(x,u,v) &=& \frac{- \varepsilon^2 ~ v~ \sin u~ e^y~ \cos x}{\omega},\\
P_2&=&\int_0^1 \{(1-t)\{N, F_1\}+ P_1, F_1\}\circ \phi_{F_1}^t dt\\
&=& \frac{\varepsilon^2 \cos^2 u e^{2y} \cos 2x}{2 \omega} + O(\varepsilon^3).
\varepsilonnd{eqnarray*}
In fact,
\begin{eqnarray*}
R_t &=& (1-t) \{N, F_1\} + P_1\\
&=& (1-t) (- \varepsilon\cos u e^y \sin x - \varepsilon^2 v\sin u e^y \cos x)+ \varepsilon^2 \cos u + \varepsilon \cos u \sin x e^y,\\
\{R_t, F_1\} &=& \frac{\partial R_t}{\partial x} \frac{\partial F_1}{\partial y} - \frac{\partial R_t }{\partial y} \frac{\partial F_1}{\partial x} + \frac{\partial R_t}{\partial u} \frac{\partial F_1}{\partial v} - \frac{\partial R_t }{\partial v} \frac{\partial F_1}{\partial u}\\
&=& \frac{\varepsilon^2 t \cos^2 u e^{2y} \cos 2x}{\omega} + \varepsilon^3 (1-t) \frac{\sin^2 u e^{2y}(\cos 2x +1)}{2 \omega^2}.
\varepsilonnd{eqnarray*}
Let
\begin{eqnarray*}
F_2 = \frac{- \varepsilon^2~ v~ \sin u ~e^y~ \sin x}{\omega^2}+ \frac{\varepsilon^2 \cos^2 u e^{2y} \sin2x}{4\omega^2},
\varepsilonnd{eqnarray*}
then
\begin{eqnarray*}
\{N_2, F_2\} + P_2 - [P_2] - P_2' = 0,
\varepsilonnd{eqnarray*}
\begin{eqnarray*}
P_2' &=& \partial _u N_2 \partial_v F_2 - \partial_v N_2 \partial_u F_2.
\varepsilonnd{eqnarray*}
With the help of $\Phi_{F_2}^1$, we have
\begin{eqnarray*}
H_3(x,y,u,v) = N_2(y,u) + P_3(x,y,u,v),
\varepsilonnd{eqnarray*}
where
\begin{eqnarray*}
N_2 =\omega y + \frac{\varepsilon}{2}v^2 + \varepsilon^2 \cos u,~~~~P_3= O(\varepsilon^3).
\varepsilonnd{eqnarray*}
Hence, during the process of iterations the critical points do not change.
\section{The computing process of Example \ref{072}}\setcounter{equation}{0}
Consider the following Hamiltonian system
\begin{eqnarray}\label{064}
H(x,y) = \omega y + \frac{\varepsilon}{2}v^2 + \varepsilon^2 \cos (u+\frac{\iota\pi}{4}) + \varepsilon~ \sin u ~ \sin x ~e^y, ~~\iota = 1,2,
\varepsilonnd{eqnarray}
where $x\in T^1$, $y$, $u$, $v \in R^1$.
Since $P_1(x,y,u)=\varepsilon^2 \cos (u+\frac{\pi}{4}) + \varepsilon~ \sin u ~ \sin x~ e^y$, $[P_1] = \int_{T^1} P_1 dx = \varepsilon^2 \cos (u+\frac{\pi}{4})$, which is of $2-$order in $\varepsilon$.
To improve the order of ${{P}_1}$ by the symplectic transformation $\Phi_{F_1}^1$, we choose
\begin{eqnarray} \label{065}
F_1(x,y,u,v) =a_1(y,u,v) \sin x + b_1(y,u,v) \cos x
\varepsilonnd{eqnarray}
such that
\begin{eqnarray*}
\{N_1,F_1\} + P_1- [P_1] - P_1' = 0,
\varepsilonnd{eqnarray*}
\begin{eqnarray*}
P_1' &=& \partial_u N_1 \partial_v F_1 - \partial_v N_1 \partial_u F_1,\\
~[P_1]&=& \int_0^{2 \pi} {P_1}(x,y,u,\varepsilon)dx =0,\\
N_1 &=& \omega y + \frac{\varepsilon}{2}v^2.
\varepsilonnd{eqnarray*}
Let
\begin{eqnarray*}
F_1(x,y,u,v) = \frac{- \varepsilon ~ \sin u ~ e^y ~ \cos x}{\omega}.
\varepsilonnd{eqnarray*}
Then
\begin{eqnarray*}
H_2(x,y,u,v) = N_2(y,u) + P_2'(x,u,v,\varepsilon)+ \bar{P}_3(x,y,u,v,\varepsilon),
\varepsilonnd{eqnarray*}
where
\begin{eqnarray*}
N_2(y,u) &=& \omega y + \frac{\varepsilon}{2}v^2 + \varepsilon^2 \cos (u + \frac{\iota\pi}{4}),\\
P_2'(x,u,v) &=& \frac{ \varepsilon^2 ~ v~ \cos u~ e^y~ \cos x}{\omega},\\
\bar{P}_3 &=& \int_0^1 \{R_t, F_1\}\circ \phi_{F_1}^t dt,\\
R_t &=& (1-t) \{N, F_1\}+ P_1.
\varepsilonnd{eqnarray*}
Moreover,
\begin{eqnarray*}
R_t &=& (1-t) \{N, F_1\} +P \\
&=& t \varepsilon \sin u e^y \sin x + (1-t) \frac{\varepsilon^2 \nu \cos u e^y \cos x}{\omega} + \varepsilon^2 \cos (u+ \frac{\iota\pi}{4}),\\
\{R_t, F_1\} &=& \frac{\partial R_t}{\partial x} \frac{\partial F_1}{\partial y} - \frac{\partial R_t }{\partial y} \frac{\partial F_1}{\partial x} + \frac{\partial R_t}{\partial u} \frac{\partial F_1}{\partial v} - \frac{\partial R_t }{\partial v} \frac{\partial F_1}{\partial u}\\
&=& -(t \varepsilon \sin u e^y \cos x- (1-t)\frac{\varepsilon^2 \nu \cos u e^y \sin x}{\omega} )\frac{\varepsilon \sin u e^y \cos x}{\omega}\\
&~& - (t \varepsilon \sin u e^y \sin x+ (1-t)\frac{\varepsilon^2 \nu \cos u e^y \cos x}{\omega} )\frac{\varepsilon \sin u e^y \sin x}{\omega}\\
&~&+ (1- t) \frac{\varepsilon^2 \cos u e^y \cos x}{\omega} \frac{\varepsilon\cos u e^y \cos x}{\omega}\\
&=&-\frac{t \varepsilon^2 \sin ^2 u e^{2y}}{\omega} + (1- t) \frac{\varepsilon^3 \cos^2 u e^{2y} \cos^2 x}{\omega^2}.
\varepsilonnd{eqnarray*}
Hence
\begin{eqnarray*}
| \bar{P}_3| \leq c \varepsilon^3.
\varepsilonnd{eqnarray*}
Set
\begin{eqnarray*}
F_2 = \frac{ \varepsilon^2~ v~ \cos u ~e^y~ \sin x}{\omega^2},
\varepsilonnd{eqnarray*}
then
\begin{eqnarray*}
\{N_2, F_2\} + P_2 - [P_2] - P_2' = o(\varepsilon^3),
\varepsilonnd{eqnarray*}
\begin{eqnarray*}
[P_2] &=& \int_0^{2\pi} P_2(x,u,v) dx = 0,\\
P_2' &=& \partial _u N_2 \partial_v F_2 - \partial_v N_2 \partial_u F_2\\
&=&-\frac{\varepsilon^4 \sin (u + \frac{\iota\pi}{4}) \cos u e^y \sin x }{\omega^2} + \varepsilon^3 \frac{v^2 \sin u e^y \sin x }{\omega^2}.
\varepsilonnd{eqnarray*}
With the aid of $\Phi_{F_2}^1$, we have
\begin{eqnarray*}
H_3(x,y,u,v) = N_2(y,u) + P_3(x,y,u,v),
\varepsilonnd{eqnarray*}
where
\begin{eqnarray*}
N_2 &=& \omega y + \frac{\varepsilon}{2}v^2 + \varepsilon^2 \cos (u+\frac{\iota\pi}{4}) - \frac{\varepsilon^2 ~\sin^2u~e^{2y}}{2 \omega},\\
P_3&=& O(\varepsilon^3).
\varepsilonnd{eqnarray*}
The critical points of $\cos (u + \frac{\iota\pi}{4})$ are $u = - \frac{\iota\pi}{4} + k\pi$, $k \in \mathds{Z}$.
Denote
\begin{eqnarray*}
g(u) = - \cos (u + \frac{\iota\pi}{4}) - \frac{2\sin^2u e^{2y}}{2\omega}.
\varepsilonnd{eqnarray*}
We have
\begin{eqnarray*}
g'(u) = - \sin (u + \frac{\iota\pi}{4}) - \frac{4\sin u \cos u e^{2y}}{2\omega}.
\varepsilonnd{eqnarray*}
Hence, when $\iota =0$, $u = - \frac{\iota\pi}{4} + k\pi$ are critical points of $g(u).$
When $\iota =1$, $g'(-\frac{\pi}{4}+ k \pi) = \sqrt{2} \cos k\pi\neq 0$, i.e. $-\frac{\pi}{4}+ k \pi$ are not critical points of $g(u)$.
\varepsilonnd{appendix}
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\varepsilonnd{thebibliography}
\varepsilonnd{document} |
\begin{document}
\title{Unified Resource Allocation Framework for the Edge Intelligence-Enabled Metaverse\vspace*{-10mm}}
\author{Wei Chong Ng$^{1,2}$, Wei Yang Bryan Lim$^{1,2}$, Jer Shyuan Ng$^{1,2}$, Zehui Xiong$^{3}$, Dusit Niyato$^{4}$ and Chunyan Miao$^{4,5}$\\
$^1$Alibaba Group~$^2$Alibaba-NTU Joint Research Institute~
$^3$Singapore University of Technology and Design\\
$^4$School of Computer Science and Engineering, Nanyang Technological University, Singapore \\$^5$Joint NTU-UBC Research Centre of Excellence in
Active Living for the Elderly (LILY)\\
\vspace*{-5mm}}
\maketitle
\begin{abstract}
Dubbed as the next-generation Internet, the metaverse is a virtual world that allows users to interact with each other or objects in real-time using their avatars. The metaverse is envisioned to support novel ecosystems of service provision in an immersive environment brought about by an intersection of the virtual and physical worlds. The native AI systems in metaverse will personalized user experience over time and shape the experience in a scalable, seamless, and synchronous way. However, the metaverse is characterized by diverse resource types amid a highly dynamic demand environment. In this paper, we propose the case study of virtual education in the metaverse and address the unified resource allocation problem amid stochastic user demand. We propose a stochastic optimal resource allocation scheme (SORAS) based on stochastic integer programming with the objective of minimizing the cost of the virtual service provider. The simulation results show that SORAS can minimize the cost of the virtual service provider while accounting for the users' demands uncertainty.
\end{abstract}
\begin{IEEEkeywords}
Metaverse, Resource Allocation, Stochastic Integer Programming
\end{IEEEkeywords}
\section{Introduction}
The recent COVID-19 pandemic~\cite{bick2020work} has driven the rise in adoption of the online virtual environment as a viable alternative for a growing range of shared human experiences. For example, UC Berkeley held its graduation ceremony in Minecraft\footnote{https://news.berkeley.edu/2020/05/16/watch-blockeley-uc-berkeleys-online-minecraft-commencement/}, that was originally created to be a gaming platform. Together with the advancement of other enabling technologies ranging from the advent of $5$G to blockchain to Virtual Reality (VR)/Augmented Reality (AR), the conditions towards the development of the \textit{metaverse}, also known as the next-generation internet, have been gradually fulfilled.
As described by venture capitalist Matthew Ball \cite{ball}, the metaverse is an embodied version of the Internet. Similar to sci-fi films such as \textit{Ready Player One}~\cite{readyplayer}, users can leverage VR/AR technologies to navigate freely within a virtual environment with their customized avatars. Beyond gaming functions, the metaverse can support novel ecosystems of service provisions that will blur the lines between the physical and virtual worlds. For example, the American rapper Lil Nas X held his concert online on the Roblox platform\footnote{https://techcrunch.com/2020/11/10/roblox-to-host-its-first-virtual-concert-featuring-lil-nas-x/}, whereas the Facebook Infinite Office enables users to collaborate with their colleagues in an immersive online environment.
\subsection{Overview of Metaverse and Virtual Education Case Study}
The metaverse can be considered to follow a three-layer architecture~\cite{duan2021metaverse}. Firstly, the \textit{physical} layer consists of all the hardware to support the operational functions of the metaverse, i.e., computation, communication, and storage. A robust physical layer is of utmost importance to ensure the scalable and ubiquitous access to the metaverse. Secondly, the \textit{virtual} layer provides a parallel living world in which the users' avatars can interact with each other or with other objects. For immersivity, the virtual layer should also capture and reflect the real-time data and analytics of the real world using technologies such as edge intellgience empowered digital twins~\cite{el2018digital}. Finally, the \textit{interaction} layer serves as a bridge to connect the users in the physical world to the virtual world. The users can upload their inputs in the physical world that is eventually translated into specific actions in the virtual world. To enhance service delivery, the wealth of data collected from the users can be leveraged to train the Artificial Intelligence (AI) system to deliver highly personalized services to users.
Similar to the mobile internet, the Quality of Service (QoS) provided by the metaverse is a critical factor that will determine its successful adoption. Users will expect the metaverse to be scalable, seamless, and synchronous. In addition, the AI system in the metaverse can be trained in the edge using the shared user data. However, the challenge of maintaining a high QoS is further exacerbated by the fact that the metaverse is characterized by such diverse types of resources amid a highly dynamic demand environment. For example, with the uncertainty in the amount of data users shares, it is difficult for the virtual service provider to subscribe to the correct number of edge servers. Therefore, there exists a crucial need to establish new paradigms of resource allocation that serve not only to meet metaverse users' interests but also to minimize the costs of virtual service providers.
\begin{figure*}
\caption{Resource allocation timeline in virtual education case study. $\mathcal{X}
\label{fig:timeline}
\end{figure*}
In this paper, we propose the case study of virtual education in the metaverse. The pandemic has necessitated an evolution of the education industry as schools and private enrichment centres face disruption to physical classes \cite{roe2021impact} and more students resort to online resources. The education providers can put up the services that they provide in the metaverse, e.g., personalized lessons in the virtual world delivered by the AI tutors (cyber resources) \cite{ndukwe2019machine}, lessons taught by the teachers (people resources), as well as conducive classrooms that users may book to study and edge servers (physical resources) to store and process the students' relevant information, e.g., to train the AI system for teaching content personalization. With the help of edge servers, the AI tutors can be deployed at the edge. For example, by offloading/caching some parts of data and services to reduce latency and to maintain the privacy of user's data as the data is kept at the edge server locally, e.g., servers in a community or the neighborhood area network.
\subsection{Contributions of This Work}
Similar to models in the sharing economy \cite{puschmann2016sharing}, the cyber, physical, and people resources are not always owned by the virtual education providers. Instead, separate entities may own the resources, and the virtual education providers have to subscribe to these resources before offering them to the users. In general, there are two subscription plans i.e., reservation (i.e., long-term) and on-demand plan (i.e., ad-hoc). For example, full-time teachers can be hired for a long term (i.e., considered to be ``reservation'') while part-time teachers may be hired for a few weeks (i.e., considered to be ``on-demand"). Generally, the reservation plan is cheaper than the on-demand plan~\cite{5394134}. Figure~\ref{fig:timeline} shows the subscription timeline of the virtual service providers. The users can access the metaverse using the smartphone interface. Then, the virtual service provider can subscribe to these resources to offer services to the users based on users' demands. However, the virtual service provider will need to decide on the resources to be allocated via reservation plan before an actual student demand is known. Therefore, a resource over-provisioning problem can occur if the virtual service provider subscribes too many resources, i.e., the resources subscribed in the reservation plan are more than the users' demands. In contrast, a resource under-provisioning problem can happen if the virtual service provider subscribes too little resources, i.e., the resources subscribed in the reservation plan are less than the users' demands. Then, the virtual service provider has to meet the demands of the users with the more expensive plan, which is the on-demand plan. Therefore, with the demand uncertainty of the users, we propose a two-stage stochastic integer programming (SIP) for the virtual service providers in metaverse to minimize its operation cost by allocating the resources most strategically.
The contributions of this paper are summarized as follows.
\begin{itemize}
\item We first provide a brief overview of the metaverse, including the architecture and the metaverse applications. Then, we illustrate with an example of how metaverse is applied to the education sector.
\item We propose a unified decision-making framework by incorporating the stochastic optimal resource allocation scheme (SORAS) to minimize the cost of the virtual service provider. SORAS uses a two-stage SIP to obtain the optimal decisions of the virtual education provider by minimizing the overall network cost. The formulation incorporates the uncertainty modeling for users' demands.
\item The performance evaluation substantiates the importance of optimal resource allocation. For example, the performance evaluation shows that intuitively adopting average demand leads to a sub-optimal solution, and it is significantly inferior to the solution from our scheme.
\end{itemize}
The remainder of the paper is organized as follows: In Section~\ref{system}, we present the system model. In Section~\ref{problem} we formulate the problem. We discuss and analyze the simulation result in Section~\ref{simulation}. Section~\ref{conclusion} concludes the paper.
\section{System Model}\label{system}
We consider the system model from the perspective of one virtual service provider participating in metaverse. As depicted in Fig.~\ref{fig:timeline}, we classified the resources in the metaverse into three major types, i.e., cyber, physical, and people. For simplicity, we did not consider the network resources. These resources can be, for example, used for communications from users to cyber, physical, and people resources.
\begin{itemize}
\item \textbf{Cyber}: Let $\mathcal{V}~=\{1,\ldots,v,\ldots,V\}$ denote the set of cyber resource and $\mathcal{W}~=\{1,\ldots,w,\ldots,W\}$ denote the set of users in the network. The user $w$ has a demand, i.e., the number of hours to use the cyber resources. Since the cyber resources are implemented on the metaverse platform, the cyber resources can be acquired and deployed in a much shorter time scale than those of physical and people resources.
\item \textbf{Physical}: Let $\mathcal{X}~=\{1,\ldots,x,\ldots,X\}$ denote the set of physical resources. Edge servers are also part of the physical resources, and it is denoted by a different set notation $\mathcal{Z}~=\{1,\ldots,z,\ldots,Z\}$. Under our consideration, each edge server $z$ can store and process $I_z$ amount of data by the virtual service providers.
\item \textbf{People}: Let $\mathcal{Y}~=\{1,\ldots,y,\ldots,Y\}$ denote the set of people resources. Virtual service providers hire people resources to provide services to the users. To maximize the service outcome, each people resource can support $E_y$ hours for the users to use. For example, in the virtual education service, teachers are the people resource hired by the virtual education provider to provide consultation to the students (users). People resources are more unreliable (unavailable randomly) than cyber and physical resources. For example, a people resource $y$ can be on medical leave even if $y$ is hired by the virtual service provider.
\end{itemize}
\subsection{Provisioning Plans and costs}
When subscribing the resources, the virtual service provider can consider the reservation plan or on-demand plan. The virtual service provider can obtain the optimal plan if the provider knows the demand of each user. However, the demand of each user is not always the same. For example, some students may need more hours of consultation with teachers. With the two subscription plans mentioned above, there are two corresponding subscription costs for each of the resources used, i.e., reservation and on-demand costs. The cost function is defined in dollars per resource unit per unit time. $c^r_v$, $c^r_x$, $c^r_z$ and $c^r_y$ are the reservation costs for cyber resource $v$, physical resource $x$, edge server $z$, and people resource $y$ respectively. $c^o_v$, $c^o_x$, $c^o_z$, and $c^o_y$ are the on-demand costs for cyber resource $v$, physical resource $x$, edge server $z$, and people resource $y$ respectively. $c^r_v$ and $c^o_v$ are used to pay for the cyber resources. $c^r_y$ and $c^o_y$ are used to pay for the physical resources. $c^r_y$ and $c^o_y$ are used to pay for the people resources.
\subsection{Uncertainty in Demands}
Under uncertainty of demands, the number of resources required by the user is not exactly known when the reservation of the resources is made. Let $\lambda_i$ denote the $i$ demand scenario of all the users. The set of scenarios is denoted by $\Omega$, i.e., $\lambda_i\in\Omega$. Let $P(\lambda_i)$ denote the probability that scenario $\lambda_i\in\Omega$ is realized, where $P(\lambda_i)$ can be obtained from the historical records~\cite{5394134}. The uncertainty of demands is expressed as follows:
\begin{center}
{\footnotesize
$\lambda_i =\begin{bmatrix}
\arraycolsep=3pt
\medmuskip=1mu
(F_{1,v},F_{1,x},F_{1,y},\bar{F}_{1,y},F_{1}) & \ldots & (F_{w,v},F_{w,x},F_{w,y},\bar{F}_{w,y},F_{w})
\end{bmatrix}$,}
\end{center}
\noindent where $F_{w,v}$, $F_{w,x}$, and $F_{w,y}$ represent the amounts of time required for cyber resource $v$, physical resource $x$, and people resource $y$ required by user $w$, respectively. $\bar{F}_{w,y}$ is a binary parameter that represents the availability of people resource $y$ when it is requested by the user $w$. $F_{w}$ is a positive parameter that indicates the amount of data in Gb that the user $w$ is willing to share.
If user $w$ has a demand, the user is allocated with a resource duration in each component. For example, $\lambda_i=~\{(F_{w,v}~:~0.3,~F_{w,x}~:~0.3,~F_{w,y}~:~0.4~,~\bar{F}_{w,y}~:1~,~F_{w}~:0.5~)\}$ means that user $w$ requires 0.3hr to the cyber resource $v$, 0.3hr to use physical resource $x$, 0.4hr to use people resource $y$ and people resource $y$ is available as $\bar{F}_{w,y}=1$. User $w$ is also willing to share 0.5Gb of his own data with the virtual service provider. Therefore, $F_{w,v} =0$ $\forall v\in\mathcal{V}\setminus\{v\}$, $F_{w,x}=0$ $\forall x\in\mathcal{X}\setminus\{x\}$, and $F_{w,y}=0$ $\forall y\in\mathcal{Y}\setminus\{y\}$.
\section{Problem Formulation}\label{problem}
This section introduces the Deterministic Integer Programming (DIP) and Stochastic Integer Programming (SIP) to optimize the resources used by minimizing the virtual service provider total cost.
\subsection{Deterministic Integer Programming}\label{dip}
All the resources can be subscribed to in the reservation plan if users' actual demand is precisely known. Therefore, the on-demand plan is not required. In total, there are four decision variables.
\begin{itemize}
\item $m_{w,v}^{(\mathrm{c},\mathrm{r})}$ indicates the duration that is reserved for user $w$ to use cyber resource $v$. For example, $m_{w,v}^{(\mathrm{c},\mathrm{r})}=1.2$ means that the virtual education provider reserves 1.2 hours for user $w$ to use cyber resource $v$.
\item $m_{w,x}^{(\mathrm{p},\mathrm{r})}$ indicates the duration that is reserved for user $w$ to use physical resource $x$.
\item $m_{w,z}^{(\mathrm{p},\mathrm{r})}$ indicates whether edge server $z$ is reserved to store and process the data from user $w$.
\item $m_{w,y}^{(\mathrm{h},\mathrm{r})}$ indicates the duration that is reserved for user $w$ to use people resource $y$.
\end{itemize}
A DIP can be formulated to minimize the total cost of the virtual education provider as follows:
\noindent $\displaystyle\min_{m_{w,v}^{(\mathrm{c},\mathrm{r})},m_{w,x}^{(\mathrm{p},\mathrm{r})},m_{w,z}^{(\mathrm{p},\mathrm{r})},m_{w,y}^{(\mathrm{h},\mathrm{r})}}$:
\begin{align}\label{dip1}
\sum_{w\in \mathcal{W}}\biggl(\sum_{v\in\mathcal{V}}(m_{w,v}^{(\mathrm{c},\mathrm{r})}c^r_v)+ \sum_{x\in\mathcal{X}}(m_{w,x}^{(\mathrm{p},\mathrm{r})}c^r_x)+\nonumber\\\sum_{z\in\mathcal{Z}}m_{w,z}^{(\mathrm{p},\mathrm{r})}c^r_z+\sum_{y\in\mathcal{Y}}m_{w,y}^{(\mathrm{h},\mathrm{r})}c^r_y\biggl),
\end{align}
subject to:
\begin{align}
&m_{w,v}^{(\mathrm{c},\mathrm{r})} = D_{w,v}, &\forall w\in \mathcal{W}, \forall v\in\mathcal{V},\label{dip_cons1}\\
&m_{w,x}^{(\mathrm{p},\mathrm{r})} = D_{w,x}, &\forall w\in \mathcal{W}, \forall x\in\mathcal{X},\label{dip_cons2}\\
&\bar{D}_{w,y}m_{w,y}^{(\mathrm{h},\mathrm{r})} = D_{w,y}, &\forall y\in\mathcal{Y}, \forall w\in \mathcal{W},\label{dip_cons3}\\
&\sum_{w\in \mathcal{W}}m_{w,y}^{(\mathrm{h},\mathrm{r})}\leq E_y, &\forall y\in\mathcal{Y},\label{dip_cons4}\\
&\sum_{z\in\mathcal{Z}}m_{w,z}^{(\mathrm{p},\mathrm{r})}I_z\geq D_w, &\forall w\in\mathcal{W},\label{dip_cons5}\\
&\sum_{w\in\mathcal{W}}m_{w,z}^{(\mathrm{p},\mathrm{r})} \leq 1, &\forall z\in\mathcal{Z},\label{dip_cons6}\\
&m_{w,z}^{(\mathrm{p},\mathrm{r})} \in\{0,1\}, &\forall z\in\mathcal{Z},\label{dip_cons7}\\
&m_{w,y}^{(\mathrm{h},\mathrm{r})} \in\{0,\ldots,E_y\},&\forall w\in \mathcal{W},\forall y\in\mathcal{Y},\label{dip_cons8}\\
&m_{w,v}^{(\mathrm{c},\mathrm{r})}, m_{w,x}^{(\mathrm{p},\mathrm{r})}, \in\mathbb{Z}^+, &\forall w\in \mathcal{W}, \forall v\in\mathcal{V},\forall x\in\mathcal{X}.\label{dip_cons9}
\end{align}
The objective function in~(\ref{dip1}) is to minimize the total cost due to resource reservation. $D_{w,v}$, $D_{w,x}$, and $D_{w,y}$ are the actual time demands for user $w$ to use cyber resource $v$, physical resource $x$, and people resource $y$. $\bar{D}_{w,y}$ is the actual availability of people resource $y$ when it is requested by user $w$. $D_w$ is the actual amount of data that user $w$ shared. The constraints in (\ref{dip_cons1})~-~(\ref{dip_cons3}) ensure that the demand is met. In~(\ref{dip_cons3}), if people resource $y$ is not available, user $w$ should request the service from another available people resource. (\ref{dip_cons4}) ensures that the number of hours allocated to people resource $y$ does not exceed $E_y$. (\ref{dip_cons5}) ensures that the number of edge servers should be large enough to support the amount of data user $w$ shared. (\ref{dip_cons6}) ensures that each edge server can only be used once in the whole network. (\ref{dip_cons7}) indicates that $m_{w,z}^{(\mathrm{p},\mathrm{r})}$ is a binary variable. (\ref{dip_cons8}) and~(\ref{dip_cons9}) indicate that $m_{w,v}^{(\mathrm{c},\mathrm{r})}$, $m_{w,x}^{(\mathrm{p},\mathrm{r})}$, and $m_{w,y}^{(\mathrm{h},\mathrm{r})}$ are positive decision variables.
\subsection{Stochastic Integer Programming}\label{sip_pro}
If the demands for the resources are not known, the DIP formulated in~(\ref{dip1})~-~(\ref{dip_cons4}) is no longer applicable. Therefore, SIP with a two-stage recourse is developed. This section introduces the SIP to minimize the total cost of the network by optimizing the number of hours allocated to each user for different types of resources. The first stage consists of all decisions that must be selected before the demands are realized and observed. The virtual service provider has to subscribe to the duration for the resources to be used in advance before observing the demands. In the second stage, decisions are allowed to adapt to the demand observed. After the demand is observed, the virtual service provider has to pay for the additional hours needed if the reserved duration is shorter than the demand.
Other than the four decision variables listed in Section~\ref{dip}, there are four more decision variables in the SIP formulation.
\begin{itemize}
\item $m_{w,v}^{(\mathrm{c},\mathrm{o})}(\lambda_i)$ indicates the duration that is used as on-demand for user $w$ to use cyber resource $v$ in scenario $\lambda_i$.
\item $m_{w,x}^{(\mathrm{p},\mathrm{o})}(\lambda_i)$ indicates the duration that is used as on-demand for user $w$ to use physical resource $x$ in scenario $\lambda_i$.
\item $m_{w,z}^{(\mathrm{p},\mathrm{o})}(\lambda_i)$ indicates if edge server $z$ is used as on-demand to store and process the data shared by user $w$ in scenario $\lambda_i$.
\item $m_{w}^{(\mathrm{h},\mathrm{o})}(\lambda_i)$ indicates the duration that the virtual education provider has to outsource for user $w$ for people resource in scenario $\lambda_i$.
\end{itemize}
The objective function given in~(\ref{sip1}) and~(\ref{sip2}) is to minimize the cost of the resource allocation. The expressions in~(\ref{sip1}) and~(\ref{sip2}) represent the first- and second-stage SIP, respectively. The SIP formulation can be expressed as follows:
\noindent $\displaystyle\min_{m_{w,v}^{(\mathrm{c},\mathrm{r})},\ldots,m^{(\mathrm{h},\mathrm{o})}_{w,y}(\lambda_i)}$:
\begin{align}\label{sip1}
\sum_{w\in \mathcal{W}}\biggl(\sum_{v\in\mathcal{V}}(m_{w,v}^{(\mathrm{c},\mathrm{r})}c^r_v)+ \sum_{x\in\mathcal{X}}(m^{(\mathrm{p},\mathrm{r})}_{w,x}c^r_x)+\sum_{z\in\mathcal{Z}}m_{w,z}^{(\mathrm{p},\mathrm{r})}c^r_z+\nonumber\\\sum_{y\in\mathcal{Y}}m^{(\mathrm{h},\mathrm{r})}_{w,y}c^r_y\biggl)+\mathbb{E}\biggl[\mathcal{Q}(m_{w,v}^{(\mathrm{c},\mathrm{r})},m^{(\mathrm{p},\mathrm{r})}_{w,x},m_{w,z}^{(\mathrm{p},\mathrm{r})},m^{(\mathrm{h},\mathrm{r})}_{w,y},\lambda_i)\biggr],
\end{align}
where
\begin{align}\label{sip2}
\mathcal{Q}(m_{w,v}^{(\mathrm{c},\mathrm{r})},m^{(\mathrm{p},\mathrm{r})}_{w,x},m_{w,z}^{(\mathrm{p},\mathrm{r})},m^{(\mathrm{h},\mathrm{r})}_{w,y},\lambda_i) =\sum_{\lambda_i\in\Omega}P(\lambda_i)\sum_{w\in \mathcal{W}}\nonumber\\\biggl(\sum_{v\in\mathcal{V}}
(m_{w,v}^{(\mathrm{c},\mathrm{o})}(\lambda_i)c^o_v)+ \sum_{x\in\mathcal{X}}(m^{(\mathrm{p},\mathrm{o})}_{w,x}(\lambda_i)c^o_x)+\nonumber\\\sum_{z\in\mathcal{Z}}m_{w,z}^{(\mathrm{p},\mathrm{o})}(\lambda_i)c^o_z+
\sum_{y\in\mathcal{Y}}m^{(\mathrm{h},\mathrm{o})}_{w,y}(\lambda_i)c^o_y\biggl),
\end{align}
subject to:
\begin{align}
m^{(\mathrm{c},\mathrm{r})}_{w,v} + m^{(\mathrm{c},\mathrm{o})}_{w,v}(\lambda_i) \geq F_{w,v}(\lambda_i),\hspace*{+25mm}\nonumber\\ \forall w\in \mathcal{W}, \forall v\in\mathcal{V}, \forall \lambda_i\in\Omega,\label{sip_cons1}
\end{align}
\begin{align}
m^{(\mathrm{p},\mathrm{r})}_{w,x} + m^{(\mathrm{p},\mathrm{o})}_{w,x}(\lambda_i) \geq F_{w,x}(\lambda_i),\hspace*{+25mm}\nonumber\\ \forall w\in \mathcal{W}, \forall x\in\mathcal{X},\forall \lambda_i\in\Omega,\label{sip_cons2}
\end{align}
\begin{align}
\bar{F}_{w,y}(\lambda_i)m^{(\mathrm{h},\mathrm{r})}_{w,y} + m^{(\mathrm{h},\mathrm{o})}_{w}(\lambda_i) \geq F_{w,y}(\lambda_i),\hspace*{+11mm}\nonumber\\ \forall y\in\mathcal{Y}, \forall w\in \mathcal{W},\forall \lambda_i\in\Omega,\hspace*{-1mm}\label{sip_cons3}
\end{align}
\begin{align}
&\sum_{w\in \mathcal{W}}m^{(\mathrm{h},\mathrm{r})}_{w,y}\leq E_y,\hspace*{+18mm} &\forall y\in\mathcal{Y},\label{sip_cons4}\\
&\sum_{w\in\mathcal{W}}m_{w,z}^{(\mathrm{p},\mathrm{r})} \leq 1, &\forall z\in\mathcal{Z},\label{sip_cons6}\\
&\sum_{w\in\mathcal{W}}m_{w,z}^{(\mathrm{p},\mathrm{r})}(\lambda_i) \leq 1, &\forall z\in\mathcal{Z},\forall \lambda_i\in\Omega,\label{sip_cons7}
\end{align}
\begin{align}
\sum_{z\in\mathcal{Z}}m_{w,z}^{(\mathrm{p},\mathrm{r})}I_z +\sum_{z\in\mathcal{Z}}m_{w,z}^{(\mathrm{p},\mathrm{o})}(\lambda_i)I_z \geq F_w(\lambda_i),\hspace*{+8mm} \nonumber\\ \forall w\in \mathcal{W},\forall \lambda_i\in\Omega,\hspace*{-0mm}\label{sip_cons8}
\end{align}
\begin{align}
&m_{w,z}^{(\mathrm{p},\mathrm{r})}+ m_{w,z}^{(\mathrm{p},\mathrm{r})}(\lambda_i) =1, &\forall w\in \mathcal{W},\forall z\in\mathcal{Z},\forall \lambda_i\in\Omega,\label{sip_cons9}\\
&m_{w,z}^{(\mathrm{p},\mathrm{r})},m_{w,z}^{(\mathrm{p},\mathrm{r})}(\lambda_i) \in\{0,1\}, &\forall w\in \mathcal{W},\forall z\in\mathcal{Z},\forall \lambda_i\in\Omega,\label{sip_cons10}
\end{align}
\begin{align}
m^{(\mathrm{h},\mathrm{r})}_{w,y},m^{(\mathrm{h},\mathrm{o})}_{w,y}(\lambda_i)\in\{0,\ldots,E_y\},\hspace*{+28mm}\nonumber\\\forall w\in \mathcal{W},\forall y\in\mathcal{Y},\forall \lambda_i\in\Omega,\label{sip_cons11}
\end{align}
\begin{align}
m^{(\mathrm{c},\mathrm{r})}_{w,v}, m^{(\mathrm{p},\mathrm{r})}_{w,x}, m^{(\mathrm{c},\mathrm{o})}_{w,v}(\lambda_i),m^{(\mathrm{p},\mathrm{o})}_{w,x}(\lambda_i) \in\mathbb{Z}^+,\hspace*{+8mm} \nonumber\\\forall w\in \mathcal{W}, \forall v\in\mathcal{V},\forall x\in\mathcal{X},\forall \lambda_i\in\Omega.\label{sip_cons12}
\end{align}
(\ref{sip_cons1})~-~(\ref{sip_cons3}) ensure that each user's demand has to be met by using the reservation and on-demand plan. In~(\ref{sip_cons3}), if the people resource $y$ is not available, user $w$ should request the on-demand service from another available people resource. (\ref{sip_cons4}) ensures that the number of hours allocated to people resource $y$ in the reservation plan does not exceed $E_y$. (\ref{sip_cons6}) and~(\ref{sip_cons7}) ensure that each edge server can only be used once in the whole network. (\ref{sip_cons8}) ensures that there should be enough edge servers for the virtual service provider to store and process the shared data. (\ref{sip_cons9}) ensures that each edge server can subscribe only one time in each of the plans. (\ref{sip_cons10}) indicates that $m^{(\mathrm{h},\mathrm{r})}_{w,y}$ and $m^{(\mathrm{h},\mathrm{o})}_{w,y}(\lambda_i)$ are binary variables. (\ref{sip_cons11}) and~(\ref{sip_cons12}) indicate that $m^{(\mathrm{c},\mathrm{r})}_{w,v}$, $m^{(\mathrm{p},\mathrm{r})}_{w,x}$, $m^{(\mathrm{h},\mathrm{r})}_{w,y}$, $m^{(\mathrm{c},\mathrm{o})}_{w,v}(\lambda_i)$, $m^{(\mathrm{p},\mathrm{o})}_{w,x}(\lambda_i)$, and $m^{(\mathrm{h},\mathrm{o})}_{w,y}(\lambda_i)$ are positive decision variables.
\begin{figure}
\caption{3D English Education Metaverse prototype through metaverse viewer.}
\label{fig:prototype}
\end{figure}
\begin{figure*}
\caption{The cost structure in a simple SIP network for using cyber resource $\bar{w}
\caption{The cost structure in a simple SIP network for using physical resource $\bar{c}
\caption{The cost structure in a simple SIP network for using edge servers.}
\caption{The cost structure in a simple SIP network for using people resource $\bar{y}
\label{fig:optimal cost cyber}
\label{fig:optimal cost physical}
\label{fig:optimal cost edge}
\label{fig:optimal cost human}
\end{figure*}
\section{Performance Evaluation}\label{simulation}
\subsection{English Education Metaverse Prototype}
We develop the 3D English Education metaverse (EEM) on Unity~\cite{unity}, a cross-platform development engine. As shown in Fig.~\ref{fig:prototype}, we illustrate the interface of the proposed 3D English Education Metaverse prototype through the implemented metaverse viewer. The metaverse viewer is built for the users to interact with the metaverse. By using the smartphone as the platform, the users can access the metaverse any time and any place with Internet access.
In this EEM, the resource owners first charge the virtual education provider with the on-demand (hourly) or the reservation cost (per semester). Then, after the resources are purchased, the virtual education provider offers the users services according to the users' demands. Then, the users can navigate about in the virtual world using their avatars to choose the service they want. All the users can communicate with the NPCs if any of them has any issues. Each user has different demands, and the virtual education provider needs to allocate the optimal resource to serve the users. The two-stage SIP from Section~\ref{sip_pro} is used to optimize resource allocation by minimizing each user's reservation and on-demand cost. The first stage (\ref{sip1}) is to minimize the reservation cost, and the second stage (\ref{sip2}) is to minimize the on-demand cost.
\begin{figure*}
\caption{The cost of edge servers and cyber resources.}
\caption{The cost of physical and people resources.}
\caption{The probability threshold of edge servers.}
\caption{SIP comparing with EVF and randoms cheme.}
\label{fig:smallscale}
\label{fig:largescale}
\label{fig:threshold}
\label{fig:evf}
\end{figure*}
\subsection{Parameter Setting}
We consider the English language school to be is the virtual service provider that offers services to ten users, and there are 20 edge servers available $|\mathcal{Z}|=20$. Teachers are the people resources $\mathcal{Y}~=~\{\bar{y}_1,\bar{y}_2,\bar{y}_3,\bar{y}_4\}$. $\mathcal{V}~=~\{\bar{w},\bar{s},\bar{l}\}$, where $\bar{w}$ represents the writing practice such as Grammarly~\cite{grammarly}, $\bar{s}$ represents the speaking practice such as ELSA~\cite{ELSA}, and $\bar{l}$ represents gamified learning such as Brainscape~\cite{brainscape}. $\mathcal{X}=\{\bar{c}\}$, where $\bar{c}$ represents a physical classroom.
All the resource owners offer a short-term plan and a long-term plan. We consider the long-term plan as the reservation plan, and the short-term plan is the on-demand plan. Therefore, $c^r_{v:\bar{w}}=\$0.017$/hr, $c^o_{v:\bar{w}}=\$0.035$/hr, $c^r_{v:\bar{s}}=\$0.005$/hr, $c^o_{v:\bar{s}}=\$0.009$/hr, $c^r_{v:\bar{l}}=\$0.010$/hr, $c^o_{v:\bar{l}}=\$0.014$/hr, $c^r_{x:\bar{c}}=\$3.5$/hr, and $c^o_{x:\bar{c}}=\$4$/hr~\cite{grammarly,ELSA,brainscape,classroom}. We assume that the virtual education provider update its data every hour by store and process the data that the users are willing to share. Thus, the costs are $c^r_z=\$0.07625$/use and $c^o_z= \$0.13875$/use~\cite{smartphone}.
We use the full-time teacher's salary as the reservation plan and the part-time teacher's salary as the on-demand plan. Then, $c^o_{y}=\$19.6$/hr and $c^r_{y}=\$25$/hr~\cite{teacher}.
To solve SIP, we assume that the probability distribution of all scenarios in set $\Omega$ are known~\cite{dyer2006computational}, then, the complexity of the problem depends on the total number of scenarios in stage two~\cite{dyer2006computational}. For example in Section~\ref{sip_pro}, the complexity for the formulated two-stage SIP is $|\Omega|$. For the presented experiments, we implement the SIP model using GAMS script~\cite{chattopadhyay1999application}.
\subsection{Simulations}
\subsubsection{Cost structure} We first study the cost structure of the network. As an illustration, a simple network is considered with only one component in each resource set, e.g., $|\mathcal{V}|=1$, one user, and two demand scenarios $|\Omega|=2$. The first demand scenario is $\lambda_1$, the user has a demand to use all the resources and is willing to share some data. The second demand scenario is $\lambda_2$, and the user has another demand to use all the resources and is willing to share a different amount of data. We consider a stochastic system with $P(\lambda_1)= 0.6$ and $P(\lambda_1)= 0.4$. We observe the cost structure of the network by varying the number of hours reserved for all the resources except for edge servers, where we vary the number of edge servers reserved. The cost structures are shown in Figs.~\ref{fig:optimal cost cyber}~-~\ref{fig:optimal cost human}. In Fig.~\ref{fig:optimal cost cyber}, the costs in the first and second stages, as well as the overall cost under different number of hours reserved for cyber resource $\bar{w}$ is presented. We can observe that the first stage cost (reservation cost), increases as the number of hours of resource reserved increases. With more hours reserved in the first stage, stage 2 cost is reduced as the need for on-demand reduces. It can be identified that even in this simple network, the optimal solution is not trivial to obtain due to the uncertainty of demands. For example, the optimal cost is not the point where the cost in the first and second stages intersect. Therefore, SIP formulation is required to guarantee the minimum cost to the network.
\subsubsection{Individual resources}\label{setup2} There are two demand scenarios $|\Omega|=2$, including i) all the users have demands and all the teachers are not available, e.g., medical leave, denoted by $\lambda_1$ and ii) all the users have no demand denoted by $\lambda_2$. We analyze individually each resource by varying both the demand probabilities $P(\lambda_1)$ and $P(\lambda_2)$. The cost of resources are shown in Figs.~\ref{fig:smallscale} and~\ref{fig:largescale}. For better visualization, the cost of the resources is split into two parts. When the probability of the demand $P(\lambda_1)$ is low, it is cheaper for the virtual education provider to subscribe to the resources using the on-demand plan. From the results, the decision on choosing the subscription plans is also affected by the magnitude of differences between the reservation cost and on-demand cost. If the reservation cost is close to the on-demand cost, the virtual education provider will only subscribe to that resource when $P(\lambda_1)$ is very high. For example, the virtual education provider only books the classroom using the reservation plan when $P(\lambda_1)=1$. However, if the reservation cost is much lower than the on-demand cost, e.g., $c^o_{v:1}$ is more than two times $c^r_{v:1}$, it is cheaper for the virtual education provider to subscribe to that resource in the reservation plan when $P(\lambda_1)$ is not high.
\subsubsection{Probability threshold of demand scenario} We consider the settings similar to Section \ref{setup2}. From Fig.~\ref{fig:smallscale}, the virtual education provider changes its decisions for edge servers when $P(\lambda_1)$ increases from 0.5 to 0.75. In this simulation, we will increase $P(\lambda_1)$ with a small step of 0.01 to determine the probability threshold that causes the virtual education provider to change subscription decisions. The result is illustrated in Fig.~\ref{fig:threshold}. When the probability of scenario 1 is greater or equal to 0.55, we can observe the probability threshold for the virtual education provider to consider the reservation plan as the cheaper plan than that of the on-demand plan.
\subsubsection{Comparing between EVF, SIP and random scheme}
We compare the SIP with expected-value formulation (EVF)~\cite{5394134} as well as the random scheme. For EVF, the number of hours in the first stage is fixed by the average value of demand, an approximation scheme. In the random scheme, the values of the decision variables are randomly generated. We vary the on-demand cost to compare the difference between EVF, SIP, and random schemes. Fig.~\ref{fig:evf} depicts the comparison result. As shown in the result, EVF and random scheme cannot adapt to the change in cost. On the other hand, SIP can always achieve the best solution among the three to reduce the on-demand cost.
\section{Conclusion}\label{conclusion}
In this paper, we have presented a unified resource allocation framework SORAS for the edge-based metaverse in a case study of education sector. To achieve the optimal allocation, SORAS minimizes the total cost of the network. The performance evaluation of SORAS has been performed by numerical studies and simulations. Compared to the benchmark, SORAS based on SIP can achieve the best solution as it can better adapt to changes in the probability of users' demands.
\end{document} |
\begin{document}
\title{Distinguishing Number for Some Circulant Graphs }
\begin{abstract}
Introduced by Albertson et al. \cite{albertson},
the distinguishing number $D(G)$ of a graph $G$ is the least integer $r$ such that
there is a $r$-labeling of the vertices of $G$ that is not preserved by any nontrivial
automorphism of $G$. Most of graphs studied in literature have 2 as a distinguishing number value except
complete, multipartite graphs or cartesian product of complete graphs depending on $n$. In this paper, we study circulant
graphs of order $n$ where the adjacency is defined using a symmetric subset $A$ of $\mathbb{Z}_n$, called generator.
We give a construction of a family of circulant graphs of order $n$ and we show that this class has distinct distinguishing numbers and these lasters are not depending on $n$.
`
\end{abstract}
\section{Introduction}\label{sec:in}
In 1979, F.Rudin \cite{rudin} proposed a problem in Journal of Recreational Mathematics
by introducing the concept of the breaking symmetry in graphs. Albertson et al.\cite{albertson}
studied the distinguishing number in graphs defined as the minimum number of labels
needed to assign to the vertex set of the graph in order to distinguish any non
trivial automorphism graph. The distinguishing number is widely focused in the recent years
: many articles deal with this invariant in particular classes of graphs: trees \cite{tree},
hypercubes \cite{Bogstad}, product graphs \cite{klav_power} \cite{Imrich_cartes_power}
\cite{klav_cliques} \cite{Fisher_1} and interesting algebraic properties of
distinguishing number were given in \cite{Potanka} \cite{tym} and \cite{Z}.
Most of non rigid structures of graphs (i.e structures of graphs having at most
one non trivial automorphism) need just two labels to destroy any non
trivial automorphism. In fact, paths $P_n$ $(n>1)$, cycles $C_n$ $(n>5)$, hypercubes $Q_n$ $(n>3)$, $r$ $(r>3)$
times cartesian product of a graph $G^r$ where $G$ is of order $n>3$, circulant graphs
of order $n$ generated by $\{\pm 1,\pm 2,\dots \pm k\}$ \cite{gravier}($n\geq2k+3$) have 2 as a common
value of distinguishing number. However, complete graphs, complete multipartite graphs
\cite{chrom} and cartesian product of complete graphs (see \cite{klav_cliques} \cite{Fisher_1} \cite{Fisher_2})
are the few classes with a big distinguishing number. The associated invariant increases with the order of the graphs.
In order to surround the structure of a graph of a given order $n$ and get a proper distinguishing number we built
regular graphs $C(m,p)$ of order $mp$ where the adjacency is described by introducing a generator $A$ $(A \subset \mathbb{Z}_{m.p})$.
These graphs are generated by $A=\{(p-1)+ r.p, (p+1)+ r.p$ : $0\leq r \leq m-1\}$ for all $n=m.p \geq 3$. In fact, the motivation of this paper is to give an answer to this following question, noted ${\mathcal{(Q)}}$:\\
``Given a sequence of ordered and distinct integer numbers $d_1,d_2,\dots,d_r$
in $\mathbb{N}^* \setminus \{1\}$, does it exist an integer $n$ and $r$
graphs $G_i$ $(1\leq i\leq r)$ such that $D(G_i)=d_i$ for all $i=1,\dots,r$ and $n$ is the common order of the $r$ graphs?"\\
In the following proposition, we give the answer to this question:
\begin{proposition}\label{disconnected}
Given an ordered sequence of $r$ distinct integers $d_1,d_2,\dots,d_r$ with $r\geq2$ and $d_i\geq 2$ for $i=1,\dots,r$,
there exists $r$ graphs $G_1,G_2,\dots,G_r$ of order $n$ such that $G_i$ contains a clique $K_{d_i}$ and $D(G_i)=d_i$
for all $1\leq i\leq r$.
\end{proposition}
\begin{proof}
Suppose that $d_1\neq 2$ and $n=d_r$. For the integer $d_r$, we assume that $G_r \simeq K_{d_r}$ and $D(G_r)=d_r$.\\
For the other integers, we consider the disconnected $(r-1)$ graphs $G_i$ having two connected component
$C$ and $C'$ such that $C\simeq K_{d_i}$ and
$C'$ is a path $P_{n-d_i}$ for all $i=1,\dots,(r-1)$.\\
Observe that, when $d_1\neq 2$ or $n= d_r\neq4$, then the connected component $C$ and $C'$ can not be isomorphic. By consequence, an automorphism $\delta$ of a graph
$G_i$ acts in the same connected component for all $1\leq i\leq r-1$. More than, $D(G_i)=\max (D(C),D(C'))=D(C)=d_i$ for all $1\leq i\leq r-1$. \\
If $d_1= 2$ and $n= d_r=4$ the same graphs are considered except for $G_1$ where we put $G_1\simeq P_4$.
Then, $D(G_1)=2=d_1$.
\end{proof}
\noindent The graphs of Proposition \ref{disconnected} are not completely satisfying
since these ones are not connected. Furthermore, these graphs
give no additional information for graphs having hight distinguishing number,
since they just use cliques for construction. So our purpose
is to construct connected graphs structural properties that give answer to question ${\mathcal{(Q)}}$
\begin{theorem}\label{main}
Given an ordered sequence of $r$ distinct integers $d_1,d_2,\dots,d_r$
with $r\geq2$ and $d_i\geq 2$ for $i=1,\dots,r$, there exists $r$
connected circulant graphs $G_1,G_2,\dots,G_r$ of order $n$ such that $D(G_i)=d_i$.
\end{theorem}
\noindent So, in section 1, basic definitions and preliminary results used in this paper are given. Then in section 2,
we define circulant graphs $C(m,p)$ , $n=m.p\geq 3$ and provide interesting structural properties of
this class of graphs.These later are used to determine the associated distinguishing number which is given in section 3.
We also give the proof of Theorem \ref{main} in the same section.
Finally, in section 4, we conclude by some remarks and possible improvement of reply of the question ${\mathcal{(Q)}}$.
\section{Definitions and Preliminaries Results}\label{sec:1}
We only consider finite, simple, loopless, and undirected graphs $G=(V ,E)$ where $V$
is the vertex set and $E$ is the edge set.
The \emph{complement} of $G$ is the simple graph $\overline{G}=(V,\overline{E})$ which consists of the same vertex
set $V$ of G. Two vertices $u$ and $v$ are adjacent in $\overline{G}$ if and only
if they are not in $G$.
The \emph{neighborhood} of a vertex $u$, denoted by $N(u)$, consists in all the vertices $v$ which are adjacent to
$u$.
A \emph{complete graph} of order $n$, denoted $K_n$, is a graph having $n$ vertices such that all
two distinct vertices are adjacent. A \emph{path} on $n$ vertices,
denoted $P_n$, is a sequence of distinct
vertices and and $n-1$ edges $v_iv_{i+1}$, $1 \leq i \leq n - 1$.
A path relying two distinct vertices $u$ and $v$ in $G$ is said $uv$-path.
A \emph{cycle}, on $n$ vertices denoted $C_n$, is a path with $n$ distinct vertices
$v_1, v_2, \dots, v_n$ where $v_1$ and $v_n$ are confused. For a graph $G$, the \emph{distance} $d_G(u, v)$
between vertices $u$ and $v$ is defined as the number of edges on a shortest
$uv$-path.\\
Given a subset $A \subset \mathbb{Z}_n$ with $0 \not \in A$ and for all $a\in A$ and $-a\in A$, a \emph{circulant graph}, is a graph on $n$ vertices $0,1,\dots,n-1$ where two vertices
$i$ and $j$ are adjacent if $j-i$ modulo $n$ is in $A$.
\noindent The \emph{automorphism} (or \emph{symmetry}) of a graph $G=(V,E)$
is a permutation $\sigma$ of the vertices of $G$ preserving adjacency i.e if $xy \in E$,
then $\sigma(x)\sigma(y) \in E$. The set of all automorphisms of $G$, noted $Aut(G)$
defines a structure of a group. A labeling of vertices of a graph $G$, $c: V(G) \rightarrow \{1,2,\dots, r\}$ is
said $r$-\emph{distinguishing} of $G$ if $\forall \sigma \in Aut (G)\setminus \{Id_G\}$:
$c \neq c \circ \sigma$. That means that for each automorphism $\sigma \neq id $
there exists a vertex $v\in V$ such that $c(v)\neq c(\sigma(v))$.
A \emph{distinguishing number} of a graph $G$, denoted by $D(G)$, is
a smallest integer $r$ such that $G$ has an $r$-distinguishing labeling.
Since $Aut(G)=Aut(\overline{G})$, we have $D(G)=D(\overline{G})$.
The distinguishing number of a complete graph of order $n$ is equal to $n$.
The distinguishing number of complete multipartite graphs is given in the following theorem:
\begin{theorem} \cite{chrom}\label{multipartite}
Let $K_{a_1^{j_1} ,a_2^{j_2},\dots,a_r^{j_r}}$ denote the complete multipartite graph that has $j_i$
partite sets of size $a_i$ for $i = 1, 2,\dots,r$ and $a_1 > a_2 > \dots > a_r$.
Then $D(K_{a_1^{j_1} ,a_2^{j_2},\dots,a_r^{j_r}})= \min \{p :\binom{p}{a_i} \geqslant j_i$ for all $i \}$
\end{theorem}
Let us introduce the concept of modules useful to investigate distinguishing number in graphs.
A \emph{module} in the graph $G$ is a subset $M$ of vertices which share the same neighborhood outside $M$ i.e
for all $y \in V \setminus M$: $M \subseteq N(y)$ or $xy \not \in E$ for all $x\in M$.
A trivial module in a graph $G$ is either the set $V$ or
any singleton vertex. A module $M$ of $G$
is said \emph{maximal} in $G$ if for each non trivial module $M'$ in $G$ containing $M$, $M'$ is reduced to $M$. The following
lemma shows how modules can help us to estimate the value of distinguishing number in graphs:
\begin{lemma}\label{module}
Let $G$ be a graph and $M$ a module of $G$. Then, $D(G)\geq D(M)$
\end{lemma}
\begin{proof}
\noindent Let $c$ be an $r$-labeling such that $r<D(M)$.
Since $r<D(M)$, there exits $\delta\mid_{M}$ a non trivial automorphism of $M$
such that $c(x)=c(\delta\mid_{M}(x))$ for all $x \in M$ i.e the restriction
of $c$ in $M$ is not a distinguishing. Now, let $\delta$ be the extension of $\delta \mid_{M}$ to $G$
with $\delta(x)=x$ $\forall x \not \in M$ and $\delta(x)=\delta\mid_M(x)$ otherwise.
We get $c(x)=c(\delta(x))$ for all $x \in G$. Moreover, $\delta \neq id$ since $\delta\mid_{M} \neq id\mid_{M}$.
\end{proof}
\section{Circulant Graphs $C(m,p)$}\label{sec:2}
\noindent In this section, we study distinguishing number of circulant graphs $C(m,p)$ of order $n=m.p\geq3$ with
$m\geqslant 1$ and $p\geqslant 2$. A vertex $i$ is adjacent to $j$ in $C(m,p)$ iff $j-i$ modulo $n$ belongs to
$A=\{p-1+r.p, p+1+ r.p$, $0\leq r \leq m-1\}$ (See Fig. \ref{weakly}).
When $p>1$, these graphs are circulant since for all $0 \leq r\leq m-1$
the symmetric of $p-1+r.p$ is $1+p+(m-r-2)p$ which belongs to $A$ and $p>1$ implies that $0\notin A$.
By construction, set $C(m,1)$ is the clique $K_m$.
Let specify some other particular values of $p$ and $m$, $C(1,p)$ is the cycle $C_p$.
Also we have: $C(m,2)=K_{m,m}$ and $C(m,3)=K_{m,m,m}$.
By Theorem \ref{multipartite}, $D(C(m,2))=D(C(m,3))=m+1$. Moreover, $D(C(1,p))=2$ for $p\geq6$.
\begin{pr}\label{proper}
The vertex set of $C(m,p)$ ($m\geqslant 2$ and $p\geqslant 2$) can be partitioned into $p$ stable
modules $M_i=\{i+r.p:$ $ 0\leq r \leq m-1 \}$
of size $m$ for $i=0,\dots,p-1$.
\end{pr}
\begin{proof}
Given two distinct vertices $a, b \in M_i$ for $i=0,\dots,p-1$, $a-b\equiv rp[n]$ for some $0<r \leqslant m-1$ ,
then $a-b \notin A$ which proves that each $M_i$ induces a stable sets.
Moreover, it is clear that $\{M_i \}_{i=0,\dots, p-1}$ forms a partition of vertex set of $C(m,p)$.\\
Let us prove that $M_i$ defines a module. For this, suppose that $a=i+r_{a}\cdot p$ and $b=i+r_{b}\cdot p$ two
distinct vertices of a given stable set $M_i$.\\
Let $c \in V\setminus M_i$ such that $ac$ is an edge and let $c=j+r_{c}\cdot p$.\
Let
$ r_{bc}=\left \{ \begin{array}{ll}
r_b-r_c & \mbox{if } r_b> r_c \\
m+(r_b - r_c) & \mbox{else }
\end{array}
\right.
$
\hspace{12mm} $r_{ac}= \left \{
\begin{array}{ll}
r_a-r_c & \mbox{if } r_a> r_c \\
m+(r_a - r_c) &\mbox{else}
\end{array}
\right.
$
two integer numbers such that $b-c\equiv (i-j)+r_{bc}\cdot p[n]$ and
$a-c\equiv (i-j)+r_{ac}\cdot p[n]$ (with $0 \leqslant r_{ac} \leqslant m-1$ and $0 \leqslant r_{bc} \leqslant m-1$.)\\
Since $a-c$ is in $A$ then there is some integers $k$ verifying $0\leqslant k\leqslant r_{ac}$ such that $i-j+kp=p-1$ (or= $p+1$).\\
If $k\leqslant r_{bc}$, we obtain $b-c\equiv i-j+kp+(r_{bc}-k)\cdot p[n]$.\\
Then $b-c \equiv p-1+(r_{bc}-k)\cdot p[n]$ (or $\equiv p+1+(r_{bc}-k)\cdot p[n]$).
We deduce that $b-c \in A$ since $0\leqslant k \leqslant m-1$.\\
Else, we have $r_{bc} < k \leqslant m+r_{bc}$.
We have $b-c \equiv i-j+r_{bc}\cdot p[n]$.\
Then $b-c \equiv i-j+(m+r_{bc})\cdot p[n]$.
We get $b-c \equiv i-j +kp+(m+r_{bc}-k)\cdot p[n]$ which belongs to $A$ since $0\leqslant m+r_{bc}-k \leqslant m-1$.\\%\qed
\end{proof}
\begin{figure}
\caption{ Circulant graphs: the vertices of the same color are in the same module.}
\label{weakly}
\end{figure}
\noindent Since each $M_i$ (for all $0\leqslant i\leqslant p-1$) is a stable set then, by definition of a module, we have:
\begin{pr} \label{permutation}
Any permutation of elements of $M_i$ is an automorphism of $G$ for all $0\leqslant i \leqslant p-1$.
\qed \end{pr}
\noindent By Lemma \ref{module} and Property \ref{proper}, we have $D(C(m,p))\geqslant m$. We will improve this bound:
\begin{theorem} \label{principal}
For all $p \geq 2$ and for all $m \geq2$, $D(C(m,p)) = m+1$ if $p\neq 4$.
\end{theorem}
\section{Proof of Theorem \ref{main} and Theorem \ref{principal}}\label{sec:3}
In this section, we give the proof of Theorem \ref{principal} in the first step,
while the second step is spent to give the proof of the Theorem \ref{main}
\begin{lemma} \label{borne}
For all $p \geq 2$ and for all $m \geq2$, $D(C(m,p)) > m$.
\end{lemma}
\begin{proof}
\noindent If $p=2$ (resp. $p=3)$ then $C(m,2)\cong K_{m,m}$ (resp. $C(m,3)\cong K_{m,m,m}$).
According to Theorem \ref{multipartite}, we have $D(C(m,p))>m$.
Let $C(m,p)$ be the circulant graph generated by $A=\{p-1+rp, p+1+rp: 0\leqslant r \leqslant m-1\}$.
\noindent Let us suppose that $p>3$. Since the modules $M_i$ $(i=0,\dots, p-1)$ are stables of size $m$, then
by Lemma \ref{module} we have $D(C(m,p))\geq m$.\\
Consider $c:V(C(m,p))\rightarrow \{1,2,\dots,m\}$ be a $m$-labeling of $C(m,p)$ $(m \geq 2)$
and prove that $c$ is not $m$-distinguishing.\\
By way of contradiction, assume that $c$ is $m$-distinguishing.
\noindent For all distinct vertices $v$, $w$ in a given module $M_{i_0}$ with $i_0\in \{0,1,\dots,p-1\}$ we have
$c(v)\neq c(w)$ otherwise, there exists a transposition $\tau$ of $v$ and $w$ verifying $c=c \circ \tau$.
This yields a contradiction. That means that
in a fixed module $M_i$ we have all labels.
\noindent Let $P_j$ ($1\leqslant j \leqslant m$) be a set of index $\{(j-1)p+i,i \in \{0, \dots, p-1\} \}$.
\noindent Let $v\in M_i$ ( $0\leqslant i\leqslant p-1$) then $v=i+rp$ where $0\leqslant r \leqslant m-1$.
\noindent Consider now the mapping $\delta_i$ with $i=0,\dots,p-1$ defined as follows:
$\delta_i: V \rightarrow V$ such that $\delta_i(v)=(c(v)-1)p+i$ if $v \in M_i$ else $\delta (v)=v$. By Property \ref{permutation}, $\delta_i$
defines an automorphism of $G$.\\
\noindent Let $\delta = \delta_0 \circ \dots \circ \delta_{p-1}$ be an automorphism of $G$.
\noindent Let $\psi$ be a mapping defined as follows:
$\psi: V \rightarrow V$ such that $\psi(i+rp)= p-(i+1) + rp$. Let prove that $\psi$ is an automorphism of $G$.
\noindent Let $a=i+rp$ and $b=j+r'p$ two adjacent vertices then $b-a=j-i+(r'-r)p \in A$. We have $\psi(b)- \psi(a) = i-j +(r'-r)p$ which belongs to $A$. Thus $\psi$ is an automorphism of $G$.
\noindent Check now that $\delta ^{-1} \circ \psi \circ \delta$ is non trivial automorphism of $G$ preserving the labeling $c$. See Fig. \ref{composition}.
\noindent Then $\delta ^{-1} \circ \psi \circ \delta$ is clearly an automorphism because it is a composition of automorphisms.
\noindent Since $\delta ^{-1} \circ \psi \circ \delta(0) = \delta ^{-1} \circ \psi ( (c(0)-1)p +0)
= \delta ^{-1} ( (c(0)-1)p+(p-1)) = u$ with $u \in M_{p-1}$ and $c(u)=c(0)$, then $u\neq 0$ since $0 \in M_0$
and $M_0 \neq M_{p-1}$ and $p> 1$.
\noindent Thus $\delta ^{-1} \circ \psi \circ \delta$ is not a trivial automorphism.
\begin{figure}
\caption{The automorphism $\delta^{-1}
\label{composition}
\end{figure}
\noindent To complete the proof, it is enough to show that $c(u)=c(\delta ^{-1} \circ \psi \circ \delta(u))$ for all vertex $u$.\\
\noindent Let $u=i+rp$ then we have $\delta ^{-1} \circ \psi \circ \delta (u) =\delta ^{-1} \circ \psi ((c(u)-1)p +i)=
\delta ^{-1} ((c(u)-1)p+ p-(i+1)) =v$ such that $v\in M_{p-(i+1)}$ and $c(v)=c(u)$.
\noindent Then $\delta ^{-1} \circ \psi \circ \delta$ preserves the labeling.
\end{proof}
\noindent The following result gives the exact value of $D(C(m,p))$
\begin{lemma}\label{D(G)}
For all $p\geq 2$ and $p\neq4$ and for all $m \geq 2$ : $D(C(m,p)) \leqslant m+1$
\end{lemma}
\begin{proof}
If $p\in \{2,3\}$ the proposition is true by Theorem \ref{multipartite}. Consider $c$ be the $(m+1)$-labeling defined as follows (See Fig. \ref{m+1color}):
\begin{figure}
\caption{ The $(m+1)$-labeling: the label of each vertex is given inside the cycle.}
\label{m+1color}
\end{figure}
\begin{equation*}
c(v)= \left\{
\begin{array}{ll}
1 & \hspace{7mm} 0 \leqslant v \leqslant \lfloor \frac{p}{2}\rfloor \hspace{2mm} \text{and}\hspace{2mm} v=2p-1 \\
2 & \hspace{7mm} \lfloor \frac{p}{2}\rfloor < v \leqslant p-1 \\
j+1 & \hspace{7mm} v\in P_j \hspace{2mm} \text{and} \hspace{2mm}2\leqslant j\leqslant m \hspace{2mm}\text{and} \hspace{2mm} v\neq 2p-1
\end{array}
\right.
\end{equation*}
\noindent Suppose that there exists an automorphism $\delta$ preserving this labeling and prove that $\delta$ is trivial.
\noindent Since $p>4$, $0$ is the unique vertex labeled $1$ which has the following sequence of label
in his neighborhood $(1,1,2,3,4,4,\dots,m+1,m+1)$. Thus $\delta(0)=0$.
\noindent However, we refer to the following claim:
\begin{claim}\label{distance}
For each vertex $i$ in $C(m,p)$ where $0\leq i\leq p-1$, we have:
\begin{equation*}
d(0,i)= \left\{
\begin{array}{ll}
i & \hspace{5mm} 1 \leqslant i \leqslant \lfloor \frac{p}{2}\rfloor \\
p-i & \hspace{5mm} \lfloor \frac{p}{2} \rfloor < i \leqslant p-1
\end{array}
\right.
\end{equation*}
\end{claim}
\begin{proof}
\noindent First observe that for all pair of vertices $u$ and $v$ in the same module $M$ and $z\in V\setminus M$,
we have $d(u,z)=d(v,z)$ and $d(u,v)=2$.
Now, if we contract each module $M_i$ of $C(m,p)$, then we get a cycle on $p$ vertices which implies the claim.
\end{proof}
\noindent Let us prove that each vertex lebeled $1$, is fixed by the automorphism $\delta$:\\
Consider the table describing the sequence of labels of the vertex $u$:
\begin{table}
\begin{tabular}{lll}
\hline\noalign{
}
\bf $u$ & \bf $c(u)$ & \bf $c(N(u))$ \\
\noalign{
}\hline\noalign{
}
\bf $0$ & \bf $1$ & \bf $1,1,2,3,4,4, \dots, m+1,m+1$.\\
\bf $0 < i < \lfloor \frac{p}{2}\rfloor$ & \bf $1$ & \bf $1,1,3,3,4,4, \dots, m+1,m+1$.\\
\bf $\lfloor \frac{p}{2}\rfloor$ & \bf $1$ & \bf $1,2,3,3,4,4, \dots, m+1,m+1$.\\
\bf $\lfloor \frac{p}{2}\rfloor < j < p-1$ & \bf $2$ & \bf $2, 2 ,3,3,4,4, \dots, m+1,m+1$.\\
\bf $p-1$ & \bf $2$ & \bf $1, 2, 3, 3,4,4, \dots, m+1,m+1$.\\
\bf $2p-1$ & \bf $1$ & \bf $1,2,3,3,4,4, \dots, m+1,m+1$.\\
\noalign{
}\hline
\end{tabular}
\caption{The sequence of labels being in the neighborhood of vertices.}
\end{table}
For all $i$ such that $0< i< \lfloor \frac{p}{2} \rfloor$, we have the sequence of labels occurring in the neighborhood
of a vertex $i$ is $(1,1,3,3, \dots m+1, m+1)$. More than, for all two distinct vertices $u$ and $v$ such
that $0< u,v< \lfloor \frac{p}{2} \rfloor$ we have $d(u,0)\neq d(v,0)$.
Then, since $\delta(0)=0$ we get $\delta (u)=u$ and $\delta (v)=v$.
Generally, for all vertex $i$ such that $0< i< \lfloor \frac{p}{2} \rfloor$, we obtain $\delta(i)=i$.\\
\noindent More than, the sequence of labels in the neighborhood of $2p-1$ and $\lfloor \frac{p}{2}\rfloor$ is
$\{1, 2, 3, 3, 4, 4, \dots, m+1, m+1 \}$. Since $d(\lfloor \frac{p}{2} \rfloor,0) > d(2p-1,0)=1$, then we get $\delta(2p-1)=2p-1$ and
$\delta(\lfloor \frac{p}{2} \rfloor)= \lfloor \frac{p}{2} \rfloor$.
\noindent Now observe that by the previous claim, any distinct vertices $u$ and $v$ labeled $2$, we have $d(u,0)\neq d(v,0)$.
Then for any vertex $u$ such that $c(u)=2$, we have $\delta(u)=u$.
Finally, let us prove that each vertex $v$ in $C(m,p)\setminus (P_1\cup \{2p-1\})$ is fixed by the automorphism $\delta$.
For that, it is enough to show for all pair of distinct vertices $u$ and $v$ such that $c(u)=c(v)$, we have
$N(u)\cap \{0,1,2,\dots,p-1\} \neq N(v)\cap \{0,1,2,\dots,p-1\}$. This proposition will imply that each vertex $v$
labeled $c(v)$ $(c(v)\geq2)$ is fixed by $\delta$ and we conclude the proof of theorem.
\noindent Let $u$ and $v$ two distinct vertices such that $c(u)=c(v)$ with $u,v \in C(m,p)\setminus (P_1\cup \{2p-1\}) $.\\
\noindent Since $c(u)=c(v)$, we have $u \in M_i$ and $v\in M_j$ with $i\neq j$.
Then $i-1, i+1 \in N(u)$ and $j-1, j+1 \in N(v)$.
If $i=0$ then $p-1\in N(u)$ since $p\in M_i$.
Similarly, if $i=p-1$, then $0\in N(u)$ since $mp-1\in M_i$.
\noindent Therefore, modulo $p$, we have that $i-1, i+1 \in N(u)\cap \{0,1,\dots,p-1 \}$ and $j-1, j+1 \in N(v)\cap \{0,1,\dots,p-1 \}$.
Additionally, observe that any vertex $u$ has exactly two neighborhood among $p$ consecutive vertices of $G$.
Thus $N(u)\cap \{0,1, \dots,p-1\} =\{i-1, i+1 \; \; \bmod{p} \}$ and $N(v)\cap \{0,1, \dots,p-1\} =\{j-1, j+1\; \; \bmod{p}\}$.
\noindent Now, if $N(u)\cap \{0,1,\dots,p-1 \}= N(v)\cap \{0,1,\dots,p-1 \}$ and $i\neq j$, then $i+1=j-1$ and $i-1=j+1$.
Thus $j=i-2$, $j=i+2$ and $p=4$.
Since $p>4$, we get that $N(u)\cap \{0,1,\dots,p-1 \}\neq N(v)\cap \{0,1,\dots,p-1 \}$.
\end{proof}
\noindent Lemma \ref{borne} and Lemma \ref{D(G)} give the proof of Theorem \ref{principal}.
The following result gives the value of distinguishing number for $p=4$:
\begin{corollary}\label{p4}
For each $m\geq 2$, $C(m,4)$ is
isomorphic to $C(2m,2)$ $($or $K_{2m,2m})$ and $D(C(m,4))=$ $2m+1$.
\end{corollary}
\begin{proof}
\noindent The graph $C(m,4)$ is partitioned into four modules $M_0$, $M_1$, $M_2$, $M_3$. We have:
$N(M_0)=N(M_2)=M_1\cup M_3$ and $N(M_1)=N(M_3)=M_0\cup M_2$. Thus, the module $M_i$
is not maximal where $i \in \{0,1,2,3\}$. Furthermore, $M_0 \cup M_2$ and $M_1\cup M_3$ are stables of size $2m$.
Then, the graph $C(m,4)$ is a multipartite graph $K_{2m,2m}$ and $D(C(m,4))=D(K_{2m,2m})=D(C(2m,2))=2m+1$.
\end{proof}
\noindent \textbf{PROOF OF THEOREM \ref{main}}
\noindent Let $d_1,d_2,\dots,d_r$ be an ordered sequence of distinct integers. Let $m_i=d_i -1$ for all $i=1,\dots,r$
and $p_i=\displaystyle\prod_{j\neq i} m_j$.
\noindent By definition, $m_i p_i=m_j p_j$ for $i\neq j$ for $i,j=1,\dots,r$.\\
If all $p_i\neq 4$, then let $n=m_i p_i$ else $n=3m_i p_i$ for all $i=1,\dots,r$.\\
Now, by Theorem \ref{principal}, $D(C(m_i,p_i))=m_i+1=d_i$ for all $i=1,\dots,r$.\\
So, $(G_i)_i ={(C(m_{i},p_{i}))}_i$ with $i=1,\dots,r$,
is a family of connected circulant graphs of order $n$ such that $D(G_i)=d_i$.
\qed
\section{Remarks and conclusion}\label{sec:4}
\noindent We have studied the structure of circulant graphs $C(m,p)$ by providing the associated distinguishing number.
We have determined the distinguishing number of circulant graphs $C(m,p)$ for all $m.p\geq 3$ with $m\geqslant 1$ and $p\geqslant 2$.
We can summarize the result which give the value of distinguishing number for circulant graphs $C(m,p)$ as follows:
$D(C(m,p))=$
$\begin{cases}
m & (m\geqslant 3 \; \; \text{and} \; \; p=1)\\
m+1 & (m=1 \; \; \text{and} \; \; p\geq 6) \; \; or \; \; (m\geq2 \; \; p\geq2 \; \; p\neq4) \\
2m+1 & (m=1 \; \; \text{and} \; \; p\in\{3,4,5\}) \; \; or \; \; (m\geq2 \; \; p=4)
\end{cases}$
\noindent We deduce that for a given integer $n=\displaystyle\prod_{i=1}^{r} m_i$ for $r\geq 2$ and $m_i\geq 1$,
we can build a family of graphs of same order $n$ where the distinguishing number
depends on divisors of $n$ . The main idea of constructing such graphs consists of partitioning the vertex set into
modules of same size. The circulant graphs are well privileging structure. One may ask if we can construct such family
of circulant graphs with smaller order?
\noindent For instance, we can improve in Theorem \ref{main} the order $n$ of $(C(m_i,p_i))_i$ for $i=1,\dots r$,
by taking $n=\frac{\displaystyle\prod_{i=1}^{r} m_i}{gcd(m_i, \displaystyle\prod_{j<i} m_j)}$.
\end{document} |
\betagin{document}
\title{Steinberg representations and harmonic cochains for
split adjoint quasi-simple groups}
\selectlanguage{english}
\betagin{abstract}
Let $G$ be an adjoint quasi-simple group defined and split over a
non-archimedean local field $K$. We prove that the dual of the
Steinberg representation of $G$ is isomorphic to a certain space of
harmonic cochains on the Bruhat-Tits building of $G$. The Steinberg
representation is considered with coefficients in any commutative
ring.
\end{abstract}
\betagin{center}
{\bf Introduction}
\end{center}
Let $K$ be a non-archimedean local field. Let $G$ be the
$K$-rational points of a reductive $K$-group of semi-simple rank
$l$. Let $T$ be a maximal $K$-split torus in $G$ and let $P$ be a
minimal parabolic $K$-subgroup of $G$ that contains $T$. There is an
abuse of language because we mean the $K$-rational points of these
algebraic subgroups of $G$. For a commutative ring $M$, the Steinberg
representation of $G$ with coefficients in $M$ is the $M[G]$-module :
$$
\mathrm St(M)=\dfrac{C^{^\infty}(G/P,\, M)}{\sum_{Q}C^{^\infty}(G/Q,\, M)}
$$
where $Q$ runs through all the parabolic subgroups of $G$ containing
$P$.
In \cite{Borel3}, A. Borel and J.-P. Serre, computed the reduced
cohomology group $\tilde{H}^{l-1}(Y_t,M)$ of the topologized
building $Y_t$ of the parabolic subgroups of $G$ and proved that we
have an isomorphism of $M[G]$-modules :
$$
\tilde{H}^{l-1}(Y_t,M) \cong \mathrm St(M).
$$
Then they "added" this building at infinity to the Bruhat-Tits
building $X$ of $G$ to get $X$ compactified to a contractible space
$Z_t=X\amalg Y_t$. Using the cohomology exact sequence of $Z_t$ mod.
$Y_t$, they deduce an isomorphism of $M[G]$-modules :
$$
H_c^{l}(X,M) \cong \tilde{H}^{l-1}(Y_t,M).
$$
Thus, an isomorphism of $M[G]$-modules between the compactly
supported cohomology of the Bruhat-Tits building and the Steinberg
representation of $G$ :
$$
H_c^{l}(X,M) \cong \mathrm St(M).
$$
In case $G$ is simply connected and $M$ is the complex field
$\mathbb C$, see A. Borel \cite{Borel1}, if we consider
$C^j(X,{\mathbb C})$ to be the space of $j$-dimensional cochains and
$\deltalta : C^{j}(X,{\mathbb C})\rightarrow C^{j-1}(X,{\mathbb C})$
the adjoint operator to the coboundary operator $d: C^{j}(X,{\mathbb
C}) \rightarrow C^{j+1}(X,{\mathbb C})$ with respect to a suitable
scalar product, we get the $l^\textrm{th}$ homology group
$H_l(X,{\mathbb C})$ of this complex as the algebraic dual of the
compactly supported cohomology group $H_c^{l}(X,{\mathbb C})$. So,
with the isomorphism above, we get a $G$-equivariant ${\mathbb
C}$-isomorphism :
$$
H_l(X,{\mathbb C})\cong \mathrm{Hom}_{\mathbb C}(\mathrm St({\mathbb C}),\,{\mathbb
C}).
$$
A $j$-cochain $c\in C^j(X,{\mathbb C})$ is an harmonic cochain if we
have $d(c)=\deltalta(c)=0$. In case of chambers $j=l$, it is clear that
we have $d(c)=0$. So if we denote by $\mathrm{Har}^l({\mathbb C},{\mathbb
C})$ the space of the ${\mathbb C}$-valued harmonic cochains defined
on the chambers of $X$, we have $\mathrm{Har}^l({\mathbb C},{\mathbb
C})=Z_l(X,{\mathbb C})=H_l(X,{\mathbb C})$, where $Z_l(X,{\mathbb
C})=\mathrm{Ker}\,\deltalta$ is the space of the cycles at the level $l$ of the
homological complex defined by $\deltalta$ above. Therefore
$$
\mathrm{Har}^l({\mathbb C},{\mathbb C})\cong \mathrm{Hom}_{\mathbb C}(\mathrm St({\mathbb
C}),\,{\mathbb C}).
$$
In the present work, we consider $G$ to be a split quasi-simple
adjoint group. For any commutative ring $M$ and for any $M$-module
$L$ on which we assume $G$ acts linearly, we define $\mathrm{Har}^l(M,L)$ to
be the space of $L$-valued harmonic cochains on the pointed chambers
of the Bruhat-Tits building, where a pointed chamber means a chamber
with a distinguished special vertex. The notion of harmonic cochains
we use here is the same as above in case the group $G$ is also
simply connected, otherwise since we are considering pointed
chambers of the building there is an orientation property that our
cochains should also satisfy. Using a result we have proved in our
preceding paper \cite{Yacine1} that gives the Steinberg
representation of $G$ in terms of the parahoric subgroups of $G$, we
prove explicitly that we have a canonical $M[G]$-isomorphism
$$
\mathrm{Har}^l(M,L) \cong \mathrm{Hom}_M(\mathrm St(M),L).
$$
First, we give a very brief introduction to the Bruhat-Tits building
to fix our notations. Then we recall the results obtained in
\cite{Yacine1}, giving an expression of the Steinberg representation
in terms of parahoric subgroups, we will also reformulate this
result in way it becomes easier to see the link to the harmonic
cochains. Finally, we introduce the space of harmonic cochains on
the building and prove the isomorphism between this space and the
dual of the Steinberg representation of $G$.
\section{Bruhat-Tits buildings}
\paragraph{Notations}
Let $K$ be a non-archimedean local field, that is a complete field
with respect to a discrete valuation $\omegaega$. We assume $\omegaega$ to
have the value group $\omegaega(K^*)=\mathbb Z$.
We consider $G$ to be the group of $K$-rational points of an adjoint
quasi-simple algebraic group defined and split over $K$. Let $T$ be
a maximal split torus in $G$, $N=N_G(T)$ be the normalizer of $T$ in
$G$ and $W=N/T$ be the Weyl group of $G$ relative to $T$.
The group of characters and the group of cocharacters of $T$ are
respectively the free abelian groups
$$
X^*(T)=\mathrm{Hom}(T,GL_1) \qquad\textrm{and} \qquad X_*(T)=\mathrm{Hom}(GL_1,T).
$$
There is a perfect duality over $\mathbb Z$
$$
\lambdangle \cdot,\cdot \rangle : X_*(T) \times X^*(T) \rightarrow
{\mathbb Z} \cong X^*(GL_1)
$$
with $\lambdangle \lambdambda ,\chi \rangle$ given by $\chi \circ \lambdambda
(x)=x^{\lambdangle \lambdambda ,\chi \rangle}$ for any $x\in GL_1(K)$.
Let $V=X_*(T)\otimes {\mathbb R}$ and identify its dual space $V^*$
with $X^*(T)\otimes {\mathbb R}$. Denote by $\Phi=\Phi(T,G)\subseteq
X^*(T)$ the root system of $G$ relative to $T$. By the above
duality, any root $\alphapha$ induces a linear form $\alphapha :V
\rightarrow {\mathbb R}$. To every root $\alphapha\in \Phi$ corresponds
a coroot $\alphapha^\vee \in V$, and a convolution $s_\alphapha$ that acts
on $V$ by
$$
s_{\alphapha}(x)=x-\lambdangle x,\alphapha \rangle \alphapha^\vee.
$$
This convolution $s_\alphapha$ is the orthogonal reflection with
respect to the hyperplane $H_\alphapha=\mathrm{Ker}\,\alphapha$.
On the other side, we can see that the group $N$ acts on $X_*(T)$ by
conjugations. This clearly induces an action of $W$ on $V$ by linear
automorphisms. We can identify $W$ with the Weyl group $W(\Phi)$ of
the root system $\Phi$, that is the subgroup of $GL(V)$ generated by
all the reflections $s_\alphapha$, $\alphapha \in \Phi$.
Let $\Deltalta=\{1,2,\ldots , l\}$ and let $D=\{\alphapha_i; i\in
\Deltalta\}$ be a basis of simple roots in $\Phi$. For any $i\in
\Deltalta$, denote $s_i=s_{\alphapha_i}$. Consider $S=\{s_i;\, i\in
\Deltalta\}$. The pair $(W,S)$ is a finite Coxeter system.
Denote by $\Phi^\vee$ the coroot system dual to the root system
$\Phi$. Denote by $Q(\Phi^\vee)$ (resp. $P(\Phi^\vee)$) the
associated coroot lattice (resp. coweight lattice). Since we have
assumed $G$ of adjoint type we have $X_*(T)=P(\Phi^\vee)$.
\paragraph{The fundamental apartment}
Let $A_0$ be the natural affine space under $V$. Denote by
$\textrm{Aff}(A_0)$ the group of affine automorphisms of $A_0$. For
$v\in V$, denote by $\tau(v)$ the translation of $A_0$ by the vector
$v$. We have
$$
\textrm{Aff}(A_0)=V \rtimes GL(V).
$$
There is a unique homomorphism
\betagin{equation}\lambdabel{nu}
\nu : T \longrightarrow X_*(T)=P(\Phi^\vee) \subseteq V
\end{equation}
such that $\lambdangle \nu (t),\chi\rangle =-\omegaega(\chi(t))$ for any
$t\in T$ and any $\chi\in X^*(T)$. In our situation this
homomorphism is surjective.
An element $t\in T$ acts on $A_0$ by the translation $\tau(\nu(t))$
:
$$
tx:=\tau(\nu(t))(x)=x+\nu(t), \qquad x\in A_0,
$$
so if we put $T_0=\mathrm{Ker}\,\nu$, this clearly induces an action of the
so-called extended affine Weyl group
$\widetilde{W}_\textrm{a}:=N/T_0$ on $A_0$. This group is an
extension of the finite group $W$ by $T/T_0$ :
$$
\widetilde{W}_\textrm{a}=\frac{N}{T_0}=\frac{T}{T_0} \rtimes W \cong
P(\Phi^\vee) \rtimes W \subseteq V \rtimes GL(V) =
\textrm{Aff}(A_0).
$$
We deduce an action of $N$ on $A_0$ by affine automorphisms that
comes from the action of $T$ by translations on $A_0$ and the linear
action of $W$ on $V$.
For any root $\alphapha \in \Phi$ and any $r\in {\mathbb Z}$, let
$H_{\alphapha,r}$ be the hyperplane in $A_0$ defined by
$$
H_{\alphapha,r} =\{ x\in A_0;\, \lambdangle x,\alphapha \rangle - r=0 \}.
$$
Let $s_{\alphapha,r}$ be the orthogonal reflection with respect to
$H_{\alphapha,r}$. We have
\betagin{equation}\lambdabel{reflection-translation}
s_{\alphapha,r}=\tau(r\alphapha^\vee) \circ s_{\alphapha}.
\end{equation}
The hyperplanes $H_{\alphapha,r}$ define a structure of an affine
Coxeter complex on $A_0$. Let $W_{\textrm{a}}$ be the associated affine Weyl
group. It is a subgroup of the group $\textrm{Aff}(A_0)$ generated
by the reflections $s_{\alphapha,r}$ with respect to the hyperplanes
$H_{\alphapha,r}$. We have
$$
W_{\textrm{a}} \subseteq \textrm{Aff}(A_0)=V \rtimes GL(V).
$$
In fact, $W_{\textrm{a}}$ is the semi-direct product of $Q(\Phi^\vee)$ and $W$
(see \cite[Ch.VI,\S\,2.1,Prop. 1]{Bourbaki})
$$
W_{\textrm{a}} = Q(\Phi^\vee) \rtimes W \subseteq P(\Phi^\vee) \rtimes W
=\widetilde{W}_\textrm{a}.
$$
The Coxeter complex $A_0$ is the fundamental apartment of the
Bruhat-Tits building.
\paragraph{The fundamental chamber}
Let $\tilde{\alphapha}$ be the highest root in $\Phi$. The fundamental
chamber $C_0$ of the Bruhat-Tits building is the chamber with the
bounding walls
$$
H_{\alphapha_1}=H_{\alphapha_1,0}, \ldots ,H_{\alphapha_l}=H_{\alphapha_l,0}
\textrm{ and }H_{\tilde{\alphapha},1}.
$$
It is the intersection in $A_0$ of the
open half spaces
$$
\lambdangle x,\alphapha_i \rangle > 0 \quad 1\leq i\leq l \quad\textrm{ and
}\quad \lambdangle x,\tilde{\alphapha}\rangle < 1.
$$
Denote $s_i=s_{\alphapha_i}=s_{\alphapha_i,0}$ for any $i$, $1\leq i\leq
l$, and $s_0=s_{\tilde{\alphapha},1}$. The set $S_\textrm{a}=\{s_0,s_1,
\ldots ,s_l\}$ generates the affine Weyl group $W_{\textrm{a}}$. The pair
$(W_{\textrm{a}},S_\textrm{a})$ is an affine Coxeter system and the topological
closure $\overline{C}_0$ of $C_0$ is a fundamental domain for the
action of $W_{\textrm{a}}$ on $A_0$.
\paragraph{The Bruhat-Tits building}
The Bruhat-Tits building $X$ associated to $G$ is defined as the
quotient
$$
X=\frac{G\times A_0}{\sigmam}
$$
where $\sigmam$ is a certain equivalence relation on $G\times A_0$, see
\cite{Yacine1} or any reference on Bruhat-Tits buildings. The group
$G$ acts transitively on the chambers (the simplices of maximal
dimension) of $X$.
\section{The Steinberg representation and the Iwahori subgroup}
Let $M$ be a commutative ring on which we assume $G$ acts trivially.
For a closed subgroup $H$ of $G$, denote by $C^{^\infty}(G/H,\,M)$
(resp. $C_c^{^\infty}(G/H,\,M)$) the space of $M$-valued locally
constant functions on $G/H$ (resp. those which moreover are
compactly supported). The action of the group $G$ on the quotient
$G/H$ by left translations induces an action of $G$ on the spaces
$C^{^\infty}(G/H,\,M)$ and $C_c^{^\infty}(G/H,\,M)$.
Let $P$ be the Borel subgroup of $G$ that corresponds to the basis
$D$ of the root system $\Phi$. For any $i\in \Deltalta$, let
$P_i=P\coprod Ps_iP$ be the parabolic subgroup of $G$ generated by
$P$ and the reflection $s_i$. The Steinberg representation of $G$ is
the $M[G]$-module
$$
\textrm{St}(M) = \frac{C^{^\infty}(G/P,\,
M)}{\sum_{i\in\Deltalta}C^{^\infty}(G/P_i,M)}.
$$
Now, let $B$ be the Iwahori subgroup of $G$ corresponding to $P$.
Recall from \cite[Th. 3.4]{Yacine1} that $C^{^\infty}(G/P,\, M)$ is
generated as an $M[G]$-module by the characteristic function
$\chi_{BP}$ of the open subset $BP/P\subseteq G/P$, and then that we
have a surjective $M[G]$-homomorphism
$$
\Theta : C_c^{^\infty}(G/B,\,M) \longrightarrow C^{^\infty}(G/P,\,M)
$$
defined by $\Theta(\varphi)=\sum_{g\in G/B}\varphi(g)g.\chi_{BP}$.
For any $i\in \Deltalta$, let $B_i=B\coprod Bs_iB$ be the parahoric
subgroup of $G$ that corresponds to the parabolic $P_i$. Let
$\{\varpi_i;\; i\in \Deltalta\}$ be the fundamental coweights with
respect to the simple basis $D$ and, by the surjective homomorphism
(\ref{nu}), take $t_i\in T$ such that $\nu(t_i)=\varpi_i$. Computing
the kernel of $\Theta$, cf. [loc. cit., Th. 4.1 and Cor. 4.2], we
have :
\betagin{prop}\lambdabel{STB}
We have a canonical isomorphism of $M[G]$-modules :
$$
\textrm{St}(M) \cong \frac{C_c^{^\infty}(G/B,\,
M)}{R+\sum_{i\in\Deltalta}C_c^{^\infty}(G/B_i,M)}
$$
where $R$ is the $M[G]$-submodule of $C_c^{^\infty}(G/B,\, M)$
generated by the functions $\chi_{Bt_iB}-\chi_{B}$, $1\leq i \leq l
$.
\end{prop}
Under the action of $G$ on the Bruhat-Tits building $X$, the Iwahori
$B$ is the pointwise stabilizer of the fundamental chamber $C_0$.
Let $B_0=B\coprod Bs_0 B$ be the parahoric subgroup of $G$ generated
by $B$ and the reflection $s_0$. The parahoric subgroups $B_i$,
$0\leq i\leq l$, are the pointwise stabilizers of the $l+1$
codimension $1$ faces of $C_0$.
We would like to reformulate the isomorphism in this proposition in
such way the connection of the Steinberg representation to harmonic
cochains on the Bruhat-Tits building looks more clear.
Denote by $l(w)$ the length of an element $w$ of the Coxeter group
$W_\textrm{a}$ with respect to the set
$S_\textrm{a}=\{s_0,s_1,\ldots ,s_l\}$ and recall that we can look
at the linear Weyl group $W$ as the subgroup of $W_\textrm{a}$
generated by the subset $S=\{s_1,\ldots,s_l\}$ of $S_\textrm{a}$.
\betagin{lem}\lambdabel{bwb}
Let $g\in G$. For any $w\in W_{\rm a}$ (resp. $w\in W$), we have
$$
\chi_{BgB} -(-1)^{l(w)}\chi_{BgwB} \in
\sum^l_{i=0}C_c^{^\infty}(G/B_i,\,M) \quad \left(\textrm{resp. } \in
\sum^l_{i=1}C_c^{^\infty}(G/B_i,\,M)\right).
$$
\end{lem}
\proof Let $u_1,\ldots ,u_d \in S_\textrm{a}$ (resp. $\in S$) such
that $w=u_1 \cdots u_d$ is a reduced expression in $W_\textrm{a}$
(resp. in $W$). We have
$$
\chi_{BgB}-(-1)^d
\chi_{BgwB}=\sum_{i=1}^d(-1)^{i-1}(\chi_{Bgu_1\cdots u_{i-1}B} +
\chi_{Bgu_1\cdots u_iB}).
$$
For any $i$, if $u_i$ is the reflection $s_j$ then $
\chi_{Bgu_1\cdots u_{i-1}B} + \chi_{Bgu_1\cdots u_iB} \in
C^{^\infty}_c(G/B_j,\,M)$. \qed
Since we have assumed $G$ to be split quasi-simple, its root system
$\Phi$ is reduced and irreducible. Thus, the Dynkin diagram of the
root system $\Phi$ is one of the types described in \cite{Bourbaki},
this classification is summarized in [loc. cit., Planches I-IX].
Let $\tilde{\alphapha}=\sum_{i=1}^ln_i \alphapha_i$ be the highest root of
$\Phi$. From \cite[Ch.VI, \S\,2.2, Cor. of Prop. 5]{Bourbaki}, we
know that the $l+1$ vertices $v^\circ_{i}$ of the fundamental
chamber $C_0$ are $v^\circ_{0}=0$ and :
$$
v^\circ_{i}=\varpi_{i}/n_{i} \quad{\textrm for}\; 1\leq i\leq l.
$$
To each vertex $v^\circ_i$ of the fundamental chamber $C_0$ we give
the label $i$. This gives a labeling of the chamber and then of the
whole building $X$.
Denote by $J$ the subset of $\Deltalta=\{1,2, \ldots ,l\}$ given by
$n_i=1$. Notice that, except for a group of type $A_l$ in which all
the vertices of a chamber are special $J=\Deltalta$, the coroot
$\tilde{\alphapha}^\vee$ dual to the highest root is equal to some
fundamental coweight $\varpi_{i_0}$, $i_0\in \Deltalta-J$, that induces
a special automorphism on $X$, i.e. an automorphism of $X$ that
preserves labels. So, from (\ref{reflection-translation}), we get
\betagin{equation}\lambdabel{ts}
\tau(\varpi_{i_0})=\tau(\tilde{\alphapha}^\vee)=s_{\tilde{\alphapha},1}
s_{\tilde{\alphapha}}.
\end{equation}
\betagin{theo}\lambdabel{STBJ} Assume $G$ is not of type $A_l$. We have a canonical
isomorphism of $M[G]$-modules :
$$
\textrm{St}(M) \cong \frac{C_c^{^\infty}(G/B,\,
M)}{R'+\sum_{i=0}^{l} C_c^{^\infty}(G/B_i,M)}
$$
where $R'$ is the $M[G]$-submodule of $C_c^{^\infty}(G/B,\, M)$
generated by the functions $\chi_{Bt_iB}-\chi_{B}$, $i\in J$.
\end{theo}
\proof From Proposition \ref{STB}, we need to prove the equality
$$
R+\sum_{i=1}^{l} C_c^{^\infty}(G/B_i,M)=R'+\sum_{i=0}^{l}
C_c^{^\infty}(G/B_i,M).
$$
Let us prove that the left hand side is contained in the right hand
side. Let $i\in \Deltalta-J$. Then $t_{i}$ acts on $X$ as a special
automorphism. So, the chamber $t_{i}C_0$ is a chamber of the
apartment $A_0$ that is of the same type as $C_0$, the same type
means that any vertex $t_iv^\circ_j$ of the chamber $t_{i}C_0$ has
the same label $j$ of $v^\circ_j$. Therefore, there is $w\in
W_\textrm{a}$ such that $t_{i}C_0=wC_0$. This means that
$\chi_{Bt_{i}B}=\chi_{BwB}$ and $w$ is of even length. From Lemma
\ref{bwb}, we get
$$
\chi_{Bt_iB}-\chi_B=(-1)^{l(w)}\chi_B -\chi_B =0 \mod.
\sum_{i=0}^lC_c^{^\infty}(G/B_i, M).
$$
Therefore, $\chi_{Bt_iB}-\chi_B\in \sum_{i=0}^lC_c^{^\infty}(G/B_i,
M)$.
Now, let us prove the other inclusion. Again from Lemma \ref{bwb} we
have
$$
\chi_{B_0}=\chi_{Bs_0B}+\chi_B=\chi_{Bs_0s_{\tilde{\alphapha}}s_{\tilde{\alphapha}}B}+\chi_B
=(-1)^{l(s_{\tilde{\alphapha}})}\chi_{Bs_0s_{\tilde{\alphapha}}B}+\chi_B
\mod. \sum_{i=1}^lC_c^{^\infty}(G/B_i, M).
$$
As we have seen, (\ref{ts}), there is an $i_0\in \Deltalta-J$ such that
$Bt_{i_0}B=Bs_0s_{\tilde{\alphapha}}B$ and $s_{\tilde{\alphapha}}\in W$
being a reflection it is of odd length. Therefore,
$$
\chi_{B_0}=-\chi_{Bt_{i_0}B}+\chi_B \mod.
\sum_{i=1}^lC_c^{^\infty}(G/B_i, M),
$$
and this finishes the proof. \qed
\betagin{rem}
In case $G$ is adjoint simply connected group, so of type $E_8$,
$F_4$ or $G_2$, the subset $J$ of $\Deltalta$ is empty, and therefore
the $M[G]$-submodule $R'$ is trivial. The theorem above gives an
isomorphism of $M[G]$-modules :
$$
\textrm{St}(M) \cong \frac{C_c^{^\infty}(G/B,\, M)}{\sum_{i=0}^{l}
C_c^{^\infty}(G/B_i,M)}.
$$
\end{rem}
\section{Steinberg representation and harmonic cochains}
Recall that the vertex $v_0^\circ$ of $C_0$ is a special vertex and
that every chamber of the building has at least one special vertex.
Let $v_i^\circ$ be a special vertex of $C_0$, this means
that $i\in J$ and that $t_i$ is a non-special automorphism of $X$.
Let $w_0$ be the longest element in $W$ and $w_{i}$ be the longest
element in the Weyl group of the root system of linear combinations
of the simple roots $\alpha_{j}$, $j\neq i$. Then, see \cite[Ch. VI,
\S\,2.3, Prop. 6]{Bourbaki}, we have $t_{i}w_{i}w_{0}C_0=C_0$.
Denote by $\widehat{X}^l$ the set of pointed chambers of $X$. A
pointed chamber of $X$ is a pair $(C,v)$ where $C$ is a chamber and
$v$ is a vertex of $C$ which is special. The map which to $gB$
associates the pointed chamber $g(C_0,v^\circ_0)$ gives a bijection
\betagin{equation}\lambdabel{Bpointed}
G/B \xrightarrow{\sigmam} \widehat{X}^l.
\end{equation}
There is a natural ordering on the vertices of a pointed chamber.
Indeed, we have
$$
(C_0,v_0^\circ)=(v_0^\circ,v_1^\circ,\ldots,v_l^\circ),
$$
which corresponds to the ordering of the vertices of the extended
Dynkin diagram, and if we choose to distinguish another special
vertex in $C_0$ then the ordering on the vertices of $C_0$ will be
the one that correponds to the ordering of the vertices of the
extended Dynkin diagram we get when applying the automorphism of the
Dynkin graph that takes $0$ to the label of the new special vertex
we have chosen. We have :
\betagin{lem}\lambdabel{tec}
Let $\sigmagma_i$ be the permutation of the set $\{0,1,\ldots ,l\}$
such that
$$
t_iw_iw_0(v^\circ_0,v^\circ_1,\ldots
,v^\circ_l)=(v^\circ_{\sigmagma_i(0)}, v^\circ_{\sigmagma_i(1)}, \ldots
,v^\circ_{\sigmagma_i(l)}),
$$
then
$$
{\rm sign}(\sigmagma_i)=(-1)^{l(w_iw_0)}.
$$
\end{lem}
\proof For any $k\in \{0,1,\ldots ,l\}$, we have
$$
v^\circ_{\sigmagma_i(k)}=t_iw_iw_0(v^\circ_k)=w_iw_0(v^\circ_k)+\varpi_i=w_iw_0(v^\circ_k)+v^\circ_i,
$$
thus $w_iw_0(v^\circ_k)=v^\circ_{\sigmagma_i(k)}-v^\circ_i$. So if we
compute the determinant of the linear automorphism $w_iw_0$ of the
vector space $V$ in the basis $(v^\circ_1, v^\circ_2,\ldots
v^\circ_{l})$, we get
$$
{\rm det}(w_iw_0)={\rm det}(v^\circ_{\sigmagma_i(1)}-v^\circ_i,
v^\circ_{\sigmagma_i(2)}-v^\circ_i, \ldots ,
v^\circ_{\sigmagma_i(j-1)}-v^\circ_i, -v^\circ_i,
v^\circ_{\sigmagma_i(j+1)}-v^\circ_i,\ldots ,
v^\circ_{\sigmagma_i(l)}-v^\circ_i),
$$
where $j\in \{1,2,\ldots ,l\}$ is such that $\sigmagma_i(j)=0$. By
subtracting the $j^\textrm{th}$ vector $-v^\circ_i$ from the other
vectors of the determinant, we get
$$
\betagin{array}{ll}
{\rm det}(w_iw_0) & = - {\rm det}(v^\circ_{\sigmagma_i(1)},
v^\circ_{\sigmagma_i(2)}, \ldots, v^\circ_{\sigmagma_i(j-1)}, v^\circ_i,
v^\circ_{\sigmagma_i(j+1)},\ldots , v^\circ_{\sigmagma_i(l)})\\
& = -{\rm det}(v^\circ_{\tau\sigmagma_i(1)},
v^\circ_{\tau\sigmagma_i(2)}, \ldots, v^\circ_{\tau\sigmagma_i(j-1)},
v^\circ_{\tau\sigmagma_i(j)}, v^\circ_{\tau\sigmagma_i(j+1)},\ldots ,
v^\circ_{\tau\sigmagma_i(l)})
\end{array}
$$
where $\tau=(0 \;\; i)$ is the transposition that interchanges $0$
and $i$. Therefore,
$$
{\rm det}(w_iw_0) =-{\rm sign}(\tau \sigmagma_i)={\rm sign}(\sigmagma_i),
$$
and it is clear that ${\rm det}(w_iw_0)=(-1)^{l(w_iw_0)}$. \qed
Denote by $\mathfrak hat{X}^{l-1}$ the set of all codimension one simplices
of $X$ that are ordered sets of $l$ vertices
$\eta=(v_0,\ldots,\Check{v}_i,\ldots,v_l)$ such that $v_i$ is an omitted
vertex from a pointed chamber $C=(v_0,\ldots,v_i,\ldots,v_l)\in
\mathfrak hat{X}^l$. We write $\eta<C$.
Denote by $M[\mathfrak hat{X}^l]$ the free $M$-module generated by the set of the pointed chambers of $X$ and let $L$ be an $M$-module on which we assume $G$ acts linearly.
\betagin{defi} Let ${\mathfrak h}: M[\mathfrak hat{X}^l]\rightarrow L$ be an
$M$-homomorphism. We say that $\mathfrak{h}$ is a harmonic cochain
on $X$ if it satisfies the following properties \\
{\bf (HC1)} Let $C=(v_0,v_1,\ldots ,v_l) \in \mathfrak hat{X}^l$. Let
$\sigmagma$ be a permutation of $\{0,1,\ldots,l\}$ such that
$v_{\sigmagma(0)}$ is a special vertex and that
$C_\sigmagma=(v_{\sigmagma(0)},v_{\sigmagma(1)},\ldots ,v_{\sigmagma(l)})\in
\mathfrak hat{X}^l$. Then
$$
{\mathfrak h}(C)=(-1)^{\textrm{sign}(\sigmagma)}{\mathfrak h}(C_\sigmagma)
$$
{\bf (HC2)} Let $\eta\in \widehat{X}^{l-1}$ be a codimension one
simplex. Let $\mathcal{B}(\eta)=\{C\in \mathfrak hat{X}^l\;|\; \eta<C\}$,
then
$$
\sum_{C\in \mathcal{B}(\eta)}{\mathfrak h}(C)=0.
$$
\end{defi}
Denote by $\mathrm{Har}^l(M,L)$ the set of harmonic cochains.
The action of $G$ on $\mathrm{Har}^l(M,L)$ is induced from its natural
action on $\textrm{Hom}_M(M[\mathfrak hat{X}^l],\,L)$, namely
$$
(g.\mathfrak{h})(C)=g\mathfrak{h}(g^{-1}C)
$$
for any $\mathfrak{h}\in \mathrm{Har}^l(M,L)$, any $g\in G$ and any $C\in
\mathfrak hat{X}^l$.
\betagin{rem}
In case of groups that are adjoint and simply connected, so of type
$E_8$, $F_4$ and $G_2$, there is no non-special automorphism and
therefore the first property {\bf (HC1)} of harmonic cochains is
voided.
\end{rem}
To prove the main theorem we need the following lemma
\betagin{lem}\lambdabel{bwp}
Let $g\in G$. For any $w\in W$, we have :
$$
\chi_{BgP} -(-1)^{l(w)}\chi_{BgwP} \in
\sum^l_{i=1}C^{^\infty}(G/P_i,\,M).
$$
\end{lem}
\proof The same arguments as in the proof of Lemma \ref{bwb}. \qed
\betagin{theo} We have an isomorphism of $M[G]$-modules
$$
\mathrm{Har}^l(M,L) \cong \mathrm{Hom}_M(\textrm{St}(M),\, L).
$$
\end{theo}
\proof Consider the map
$$
\mathcal{H}: \mathrm{Hom}_M(\textrm{St}(M),\, L) \longrightarrow
\mathrm{Hom}_M(M[\mathfrak hat{X}^l],\, L)
$$
which to $\varphi \in \mathrm{Hom}_M(\textrm{St}(M),\, L)$ associates
$\mathfrak h_{\varphi}$ defined by $\mathfrak h_\varphi(g(C_0,v_0^\circ))=\varphi(g\chi_{BP})$ for
any $g\in G$. Let us show that $\mathfrak h_\varphi=\mathcal{H}(\varphi)$ is a harmonic
cochain.
{\bf (HC1)} Let $v_i^\circ$ be a special vertex of $C_0$, this means that
$v_i^\circ=t_iv_0^\circ$ with $i\in J$. Since $t_iw_iw_0$ normalizes $B$ we have
$$
\mathfrak h_\varphi(C_0,v_i^\circ)=\mathfrak h_\varphi(t_iw_iw_0(C_0,v_0^\circ))=\varphi(t_iw_iw_0\chi_{BP})=\varphi(\chi_{Bt_iw_iw_0P}),
$$
and by Lemma \ref{bwp} and since $t_i\in P$, we have
$$
\varphi(\chi_{Bt_iw_iw_0P})=(-1)^{l(w_iw_0)}\varphi(\chi_{Bt_iP})
=(-1)^{l(w_iw_0)}\varphi(\chi_{BP})=(-1)^{l(w_iw_0)}\mathfrak h_\varphi(C_0,v_0^\circ).
$$
Now apply Lemme \ref{tec} .
{\bf (HC2)} Let $\eta\in \mathfrak hat{X}^{l-1}$. We can assume that $\eta=(v_0^\circ,v_1^\circ, \ldots
,\mathfrak hat{v}_i^\circ,\ldots,v_l^\circ)$ is a face of the pointed fundamental chamber $\eta=(C_0,v_0^\circ)$. Recall from \cite{Yacine1} that
$B_iP_i=B_iP=\coprod_{b\in B_i/B}bBP$, therefore
$$
\sum_{C\in \mathcal{B}(\eta)}\mathfrak h_{\varphi}(C)=\sum_{b\in
B_i/B}\mathfrak h_\varphi(b(C_0,v_0^\circ))=\sum_{b\in B_i/B}\varphi(b\chi_{BP})=\varphi(\chi_{B_iP_i})=0.
$$
Now, consider the map
$$
\Psi : \mathrm{Har}^l(M,L) \longrightarrow
\mathrm{Hom}_M(C_c^{^\infty}(G/B,\,M),\,L)
$$
which to $\mathfrak h\in \mathrm{Har}^l(M,L)$ associates $\psi_h$ defined by
$\psi_\mathfrak h(g\chi_B)=\mathfrak h(g(C_0,v_0^\circ))$. Let us show that $\psi_\mathfrak h$
vanishes on the $M[G]$-submodule
$R'+\sum_{i=0}^lC_c^{^\infty}(G/B_i,\,M)$ of
$C_c^{^\infty}(G/B,\,M)$. First, since $\mathfrak h$ is harmonic, from {\bf
(CH2)} we deduce that for any $i$, $0\leq i\leq l$, we have
$$
\psi_\mathfrak h(\chi_{B_i})=\sum_{b\in B_i/B}\psi_\mathfrak h(b\chi_B)=\sum_{b\in
B_i/B}\mathfrak h(b(C_0,v_0^\circ))=\sum_{C\in\mathcal{}B(\eta)}\mathfrak h(C)=0,
$$
where $\eta=(v_0^\circ,\ldots ,v_{i-1}^\circ,v_{i+1}^\circ, \ldots
,v_l^\circ)$, so $\psi_\mathfrak h$ vanishes on
$\sum_{i=0}^lC_c^{^\infty}(G/B_i,\,M)$. In case $t_{i}$ is a
non-special automorphism of $X$, we have
$$
\chi_{Bt_{i}B}=\chi_{Bt_{i}w_{i}w_{0}w_{0}w_{i}B}=t_{i}w_{i}w_{0}\chi_{Bw_{0}w_{i}B},
$$
therefore,
$$
\psi_{\mathfrak h}(\chi_{Bt_{i}B}-\chi_{B})=\psi_{\mathfrak h}((-1)^{l(w_{0}w_{i})}
t_{i}w_{i}w_{0}\chi_{B}-\chi_{B})=(-1)^{l(w_{0}w_{i})}\mathfrak h(t_i w_i w_0
(C_0,v_0^\circ))-\mathfrak h(C_0,v_0^\circ).
$$
Let $\sigma_{i}$ be the permutation of $\{0,1,\ldots ,l\}$ such that
$$
t_i w_i w_0 (v^\circ_{0},v^\circ_{1},\ldots
,v^\circ_{l})=(v^\circ_{\sigma_{i}(0)},v^\circ_{\sigma_{i}(1)}, \ldots
,v^\circ_{\sigma_{i}(l)}).
$$
Since $\mathfrak h$ is harmonic and as so satisfy the property {\bf (HC1)},
we have
$$
\mathfrak h(t_iw_iw_0(C_0,v_0^\circ))=(-1)^{\textrm{sign}(\sigmagma_i)}{\mathfrak
h}(C_0,v_0^\circ),
$$
Therefore,
$$
\psi_{\mathfrak h}(\chi_{Bt_{i}B}-\chi_{B})=(-1)^{l(w_{0}w_{i})}(-1)^{\textrm{sign}(\sigmagma_i)}{\mathfrak
h}(C_0,v_0^\circ)-\mathfrak h(C_0,v_0^\circ),
$$
and from Lemma \ref{tec}, we deduce that
$$
\psi_{\mathfrak h}(\chi_{Bt_{i}B}-\chi_{B})=0.
$$
Finally, if we denote by $\Theta^*$ the dual homomorphism of
$\Theta$, by Theorem \ref{STBJ} we have an $M[G]$-homomorphism
$$
\Phi = {\Theta^*}^{-1}\circ \Psi : \mathrm{Har}^l(M,L) \longrightarrow
\mathrm{Hom}_M(\textrm{St}(M),\, L)
$$
which sends a harmonic cochain $\mathfrak h$ to $\varphi_h$ defined by
$\varphi_\mathfrak h(g\chi_{BP})=\mathfrak h(g(C_0,v_0^\circ))$ for any $g\in G$. It
is easy to prove that $\Phi$ and $H$ are inverse of each other. \qed
\betagin{rem}
In case $G=PGL_{l+1}(K)$, so of type $A_l$, the isomorphism in the
theorem above is established in \cite{Yacine}.
\end{rem}
\betagin{thebibliography}{99}
\bibitem{Yacine} Y. A\"{i}t Amrane, Cohomology of Drinfeld symmetric
spaces and harmonic cochains, Ann. Ins. Fourier (Grenoble) 56 (3)
(2006) 561-597.
\bibitem{Yacine1} Y. A\"{\i}t Amrane, Generalized Steinberg
representations of split reductive groups, C. R. Acad. Sci. Paris,
Ser. I 348 (5-6) (2010) 243-248.
\bibitem{Borel1} A. Borel, Admissible representations of a semi-simple
group over a local field with vectors fixed under an Iwahori
subgroup, Invent. Math. 35 (1976) 233-259.
\bibitem{Borel3} A. Borel, J.-P. Serre, Cohomologie d'immeubles et
de groupes $S$-arithm\'etiques, Topology 15 (1976) 211-232.
\bibitem{Bourbaki} N. Bourbaki, Groupes et alg\`ebres de Lie, Chap. 4-6,
Paris Masson, (1981).
\end{thebibliography}
Y. A\"{\i}t Amrane, Laboratoire Alg\`ebre et Th\'eorie des
Nombres,\\
Facult\'e de Math\'ematiques,\\
USTHB, BP 32, El-Alia, 16111 Bab-Ezzouar, Alger, Algeria. \\
e-mail : [email protected]
\end{document} |
\begin{document}
\title{Random walks on $\mathrm{Homeo}
\begin{abstract}
In this paper, we study random walks $g_n=f_{n-1}\cdots f_0$ on the group $\displaystylehrm{Homeo}(S^1)$ of the homeomorphisms of the circle, where the homeomorphisms $f_k$ are chosen randomly, independently, with respect to a same probability measure $\nu$. We prove that under the only condition that there is no probability measure invariant by $\nu$-almost every homeomorphism, the random walk almost surely contracts small intervals. It generalizes what has been known on this subject until now, since various conditions on $\nu$ were imposed in order to get the phenomenon of contractions. Moreover, we obtain the surprising fact that the rate of contraction is exponential, even in the lack of assumptions of smoothness on the $f_k$'s. We deduce various dynamical consequences on the random walk $(g_n)$: finiteness of ergodic stationary measures, distribution of the trajectories, asymptotic law of the evaluations, etc. The proof of the main result is based on a modification of the Ávila-Viana's invariance principle, working for continuous cocycles on a space fibred in circles.
\end{abstract}
\section{Introduction}
The objective of the paper is to study properties of \textit{(left) random walks} on $\displaystylehrm{Homeo}(S^1)$, that is to say long compositions $f_n\circ\cdots \circ f_0$ of homeomorphisms of the circle chosen randomly independently with respect to a same probability measure $\nu$. The study of independent random composition of transformations of a space $X$ is the theory of \textit{random dynamical systems} (RDS). They appear naturally for example in the theory of \textit{iterated forward systems} (IFS), when one wants to study the action of a finitely generated group or semigroup $G$: choosing $\nu$ uniform on a set of generators, the theory of RDS allows to study the properties of ``typical'' elements of $G$. The RDS also correspond to a natural family of skew-products on $X$: the ones of the form $(\omega,x)\mapsto (T\omega,f_{\omega}(x))$, where $T$ is a shift operator on a symbol space and $f_\omega$ only depends on the first coordinate of $\omega$.\\
A standard starting point in order to study a random (or deterministic) dynamical system is the question of the dependence to the initial condition. In the context of RDS of homeomorphisms of the circle, the conclusion put in evidence by various results, is that in general the following alternative holds:
\begin{itemize}
\item either the iterated homeomorphisms preserve a common probability measure on the circle (which implies some ``determinism'' in the RDS)
\item or the RDS has the local contraction property: given any point of the circle, typical compositions of the homeomorphisms contract some neighbourhood of the point.
\end{itemize}
In the linear case (i.e. when the homeomorphisms are projective actions of elements of $SL_2(\displaystylehbb{R})$), that dichotomy is a well known result of H. Furstenberg \cite{Furstenberg} (and moreover, when the RDS has the local contraction property, these contractions are actually global and exponential). In the general case, there is variations of the precise assumptions and conclusions, but we can mainly distinguish two kinds of results:\\
--\underline{Smooth case}: In the case where the probability measure $\nu$ is supported on $\displaystylehrm{Diff}(S^1)$, one can use the general theory of hyperbolic dynamical systems on manifolds. If the quantity $\int\log^+ \| f'\|_\infty d\nu(f)$ is finite, we can define \textit{Lyapunov exponents}. In this context, various results of hyperbolic dynamics (\cite{Crauel, Baxendale, Avila}) imply that if there is an invariant probability measure, then one can find a negative Lyapunov exponent in the system (one can see this as a non linear analogue of the Furstenberg's result stated above). Next, by Pesin theory (or even simpler arguments), one can deduce that the random dynamical system locally contracts, and even that the contractions are exponentially fast.\\
--\underline{Continuous case}: In the general case of the iteration of continuous homeomorphisms, the theory of hyperbolic dynamical systems, smooth by nature, does not apply any more. Though, coupling arguments of basic theory of the homeomorphisms of the circle with probabilistic arguments, it is still possible to obtain analogue results with no regularity assumption. The most canonical result (though the older one) of this kind is probably the following theorem of Antonov:
\begin{thm*}(Antonov) \cite{Antonov}\\
Let $f_1,\ldots,f_m$ be homeomorphisms of the circle preserving the orientation, such that the semigroup $G_+$ generated by $f_1,\ldots,f_m$ and the semigroup $G_-$ generated by $f_1^{-1},\ldots,f_m^{-1}$ both act \emph{minimally} on $S^1$ (i.e.~the orbit of every point is dense in the circle), and let $\nu$ be a non degenerated probability measure on $\{1,\ldots,p\}$ (i.e. $\nu(\{i\})>0$ for $i=1,\ldots,p$). Then:
\begin{itemize}
\item Either for any initial conditions $x,y$ in $S^1$, for $\nu^\displaystylehbb{N}$-almost every sequence $(i_n)_{n\geq 0}$, the distance between the trajectories $f_{i_n}\circ\cdots\circ f_{i_0}(x)$ and $f_{i_n}\circ\cdots\circ f_{i_0}(y)$ goes to $0$. (synchronization)
\item Either there exists a probability measure invariant by all the homeomorphisms $f_i$, and because of the minimality of $G_+$ it actually implies that $f_1,\ldots,f_p$ are simultaneously conjugated to rotations. (invariance)
\item Or there exists $\theta$ in $\displaystylehrm{Homeo}_+(S^1)$ of finite order $p\geq 2$ commuting with all the $f_i$'s.(factorization)
\end{itemize}
\end{thm*}
\begin{rem}
When we are in the third case of Antonov Theorem, then one can factorize the system by identifying the points of the same orbit of $\theta$, in order to obtain a new topological circle, and
homeomorphisms $\tilde{f}_1,\ldots,\tilde{f}_m$ of this circle induced by $f_1,\ldots,f_m$.
We deduce that if $f_1,\ldots,f_m$ does not have a common invariant probability measure, then the random compositions of these homeorphisms satisfy the property of synchronization (first point of the alternative) up to some factorization (as described below).
\end{rem}
As a consequence of Antonov's Theorem, it remains true that in absence of a common invariant probability measure we have the local contraction property. However, assuming no regularity for the iterated homeomorphisms has a price: additional structural assumptions are required and no speed of convergence is assured: the finiteness of the number of generators is only an assumption for convenience, and the proof of Antonov remains valid without this assumption. The minimality assumptions, though, are much deeper: the dynamics of a semigroup of $\displaystylehrm{Homeo}(S^1)$ preserving some common interval is very different of the dynamics described in Antonov's Theorem. And if one considers a semigroup preserving two disjoint intervals, then one can check that in general, none of the alternatives of Antonov's Theorem are satisfied.
Variants of this theorem exist: let us cite for example \cite{Kleptsyn} where the authors proved (independently of Antonov) that synchronization occurs (first case of the previous theorem) under the additional assumption that $G_+$ contains a ``north-south'' homeomorphism, and \cite{Deroin-inter} where the assumption of minimality is replaced by an assumption of symmetry ($G_+=G_-$).\\ \\
The objective of the paper is to treat the study of a general random walk on $\displaystylehrm{Homeo}(S^1)$. Adapting techniques coming from the hyperbolic theory in the continuous context, we show that the distinction between the regular and continuous cases described above is actually basically useless: there is no need to ask additional assumptions on a random walk on $\displaystylehrm{Homeo}(S^1)$ to obtain the local contractions, and in fact, even the exponentially speed of contractions remains! Next we use this property of contraction to study deeply the behaviour of the random walk.
We also deduce various results on the behaviour of random walks on $\displaystylehrm{Homeo}(S^1)$. And the majority of these results actually holds for any random walk on a compact metric space satisfying the the local contraction property.
\\
The key of the proof of the main result is to adapt the ideas of Ávila and Viana in \cite{Avila} and Crauel in \cite{Crauel} (who themselves used those of \cite{Ledrappier}) to establish that an invariance principle remains in the $C^0$-case: but instead of using the Lyapunov exponents, we will use an another analogue quantity, which measures the exponential contractions as well, but which does not require derivability to be defined. That approach allows to obtain a criterion of the existence of exponential contractions for RDS of the circle, and more generally for any cocycle on a space fibred in circles, so that one can hope that this principle can also be useful in the study of non i.i.d. compositions of homeomorphisms of the circle.
\section{Statements of the results} \label{results}
\subsection{The main theorem}
Before stating our results, we need to formalize the notions of random walks and random dynamical systems:
\begin{Def}\label{randomwalk}
Let $(G,\circ)$ a topological semigroup.
\begin{itemize}
\item The \emph{random walk} generated by a probability measure $\nu$ on $G$ is the random sequence $\omega\mapsto (f_\omega^n)_{n\in\displaystylehbb{N}}$ of elements of $G$ on the probability space $(\Omega,\displaystylehbb{P})=(G^\displaystylehbb{N},\nu^\displaystylehbb{N})$, defined by: for $\omega=(f_n)_{n\in \displaystylehbb{N}}$ in $\Omega$ and $n$ in $\displaystylehbb{N}$,
$$f_\omega^n=f_{n-1}\circ\cdots\circ f_0.$$
\item We denote by $G_+(\nu)$ the smallest closed sub-semigroup of $G$ containing the topological support of $\nu$. If $G_+(\nu)=G$, the random walk and the probability measure $\nu$ are said to be \emph{non degenerated} on $G$. It is equivalent to the fact that every open set of $G$ has positive probability to be reached by the random walk.
\item If $G$ acts on a space $X$ and if the probability measure $\nu$ is non degenerated on $G$, we say that $(G,\nu)$ is a \emph{random dynamical system} (RDS) on $X$. The skew-product associated to the RDS is the transformation $\hat{T}$ on $\Omega\times X$ defined by
$$\hat{T}(\omega,x)=(T\omega,f_0(x)),$$
where $T$ is the shift operator on $\Omega$ and $f_0$ is the first coordinate of $\omega$.
\end{itemize}
\end{Def}
For a given random walk, we will always denote by $(\Omega,\displaystylehbb{P})$ the associated probability space.
Obviously, any random walk on $\displaystylehrm{Homeo}(S^1)$ is non degenerated on some sub-semigroup, namely $G_+(\nu)$. An interesting fact is that in the majority of the results that we will state, we obtain properties on the random walk depending only on assumptions on $G_+(\nu)$ and not on $\nu$ itself.\\
Here is the main theorem of the paper:
\begin{thm}\label{main}
Let $\omega\mapsto (f_\omega^n)_{n\in\displaystylehbb{N}}$ be a non degenerated random walk on a sub-semigroup $G$ of $\displaystylehrm{Homeo}(S^1)$. Let us assume that $G$ does not preserve any probability measure on $S^1$ (i.e. there does not exist a probability measure invariant by every element of $G$). Then, for any $x$ in $S^1$, for $\displaystylehbb{P}$-almost every $\omega$ in $\Omega$, there exists a neighbourhood $I$ of $x$ such that
$$\forall n\in \displaystylehbb{N}, \displaystylehrm{diam}(f_\omega^n(I))\leq q^n,$$
where $q<1$ depends on the random walk only.
\end{thm}
We can obtain the same result for random walks on a semigroup of continuous injective transformations of a compact interval $I$, since seeing $I$ as a part of $S^1$, such an injective map can be extended to a homeomorphism of the circle. Thus, in some sense, the surjectivity of the iterated transformations is not important. The injectivity, though, is primordial: one cannot hope to obtain a contraction phenomenon by iterating transformations of the circle homotopic to $z\mapsto z^2$.\\
In the case where the semigroup $G$ associated to a random walk on $\displaystylehrm{Homeo}(S^1)$ preserves a probability measure $\mu$, then the topological support $K$ of $\mu$ is a compact minimal invariant by the group $\tilde{G}$ generated by $G$, and hence we have the standard trichotomy: $K$ is either $S^1$, a Cantor set or a finite set (see for exemple \cite{Navas}, Theorem 2.1.1). It is then standard that $\tilde{G}$ is conjugated to a group of isometries if $K=S^1$, and semiconjugated to a group of isometries if $K$ is a Cantor set. This fact allows to obtain an interesting classification of the random walks on $\displaystylehrm{Homeo}(S^1)$:
\begin{cor}\label{alternative}
Let $\omega\mapsto (f_\omega^n)_{n\in \displaystylehbb{N}}$ be a non degenerated random walk on a sub-semigroup $G$ of $\displaystylehrm{Homeo}(S^1)$. Then one (and only one) of the following possibilities occurs:
\begin{enumerate}[i)]
\item $G$ does not preserve a probability measure, and the random walk has the local contraction property in the sense given by Theorem \ref{main}.
\item The random walk is semiconjugated to a random walk on the compact group $O_2(\displaystylehbb{R})$ (group of the isometries of the circle) acting minimally on $S^1$.
\item There is a finite set invariant by $G$.
\end{enumerate}
\end{cor}
On this form, the statement is very close to Furstenberg's one \cite{Furstenberg} in the linear case.
\subsection{General study of random walks acting on $\displaystylehrm{Homeo}(S^1)$ }
In this section, we use Theorem \ref{main} as a main tool to understand the behaviour of a general random walk $\omega\mapsto (f_\omega^n)_{n\in\displaystylehbb{N}}$ on $\displaystylehrm{Homeo}(S^1)$.
\subsubsection{Distribution of the trajectories $n\mapsto f_\omega^n(x)$}
We interest in the typical distribution of the sequence $(f_\omega^n(x))_{n\in\displaystylehbb{N}}$ for a given initial condition $x$. This problem is naturally related to the study of the \textit{stationary probability measures} of $\nu$, that is the probability measures $\mu$ on $S^1$ such that $\displaystylehbb{P}\otimes \mu$ is invariant by the skew-product $\hat{T}$. Such a probability measure always exists(we refer to \cite{Furman} or \cite{Kifer} for details). If the random walk is non degenerated on a subgroup of $\displaystylehrm{Homeo}(S^1)$, it has been proved that in general, the stationary probability measure is unique (see \cite{Deroin-inter}). In the case of a general random walk on $\mbox{Homeo}(S^1)$, which is non degenerated on a semigroup only, it does not hold any more, but we prove that the number of \textit{ergodic} stationary probability measures (i.e.~extremal stationary probability measures) is necessarily finite, and that these probability measures give the typical distributions of the trajectories of the random walk:
\begin{thm}\label{distribution}
Let $\omega\mapsto (f_\omega^n)_{n\in \displaystylehbb{N}}$ be a non degenerated random walk on a sub-semigroup $G$ of $\displaystylehrm{Homeo}(S^1)$ with no finite orbit on $S^1$. Then:
\begin{itemize}
\item There is only a finite number of ergodic stationary probability measures $\mu_1,\ldots,\mu_d$. Their topological supports $F_1,\ldots,F_d$ are pairwise disjoints and are exactly the minimal invariant compacts of $G$.
\item For every $x$ in $S^1$, for $\displaystylehbb{P}$-almost every $\omega$ in $\Omega$, there exists a unique integer $i=i(\omega,x)$ in $\{1,\ldots,d\}$ such that $F_i$ is exactly the set of accumulation points of the sequence $(f_\omega^n(x))_{n\in\displaystylehbb{N}}$, and then we have
$$\frac{1}{N}\sum_{n=0}^{N-1}\delta_{f_{\omega}^n(x)}\xrightarrow[n\to+\infty]{} \mu_i$$
in the weak-$*$ topology of $C(S^1,\displaystylehbb{R})^*$.
\end{itemize}
\end{thm}
Note that in this theorem, we relaxed the condition ``no invariant probability measure'' to ``no finite orbit''.\\
As a direct consequence of this theorem, we obtain that the stationary probability measure is unique when the action is minimal:
\begin{cor}\label{minimal ergodique}
A non degenerated random walk on a sub-semigroup $G$ of $\displaystylehrm{Homeo}(S^1)$ acting minimally on $S^1$ is uniquely ergodic, i.e. it admits a unique stationary probability measure.
\end{cor}
At our knowledge, this fact was never proved in full generality: until now some additional assumption (smoothness, backward minimality, symmetry...) was required to obtain the unique ergodicity. And actually, we obtain a slightly stronger corollary: the action of any random walk of $\displaystylehrm{Homeo}(S^1)$ restricted to a minimal invariant compact $F$ is uniquely ergodic: if there is no finite orbits, that is a consequence of Theorem \ref{distribution}, and if there is a finite orbit, then $F$ is necessarily finite and the unique ergodicity follows easily).
\subsubsection{Law of probability of $\omega\mapsto f_\omega^n(x)$}
We focus now in the law of the random variables $X_n^x:\omega\mapsto f_\omega^n(x)$ for any given initial condition $x$ and a large integer $n$, and asking whether the law of $X_n^x$ converges to some limit distribution when $n$ becomes large. \\
The sequence $(X_n^x)_{n\in\displaystylehbb{N}}$ is a Markov chain. A natural obstruction to the convergence of the laws of a Markov chain are the ``periodic configurations'', where there exists subspace of phase states whose the return times are multiple of a fixed integer larger than $2$. (For exemple in our context, if it exists two disjoints closed sets $F_1$ and $F_2$ such that the generators of the semigroup send $F_1$ into $F_2$ and $F_2$ into $F_1$, then clearly the distribution of $X_n^x$ strongly depends on the parity of $n$.). That leads us to the following definition of \textit{aperiodicicity}:
\begin{Def}\label{indeco}
A random walk $\omega\mapsto (f_\omega^n)_{n\geq 0}$ on $\displaystylehrm{Homeo}(S^1)$ generated by a probability $\nu$ is said to be \emph{aperiodic} if there does not exist a finite number $p\geq 2$ of pairwise disjoints closed subsets $F_1,\ldots,F_p$ of $S^1$ such that for $\nu$-almost every homeomorphism $g$, $g(F_i)\subset F_{i+1}$ for $i=1,\ldots,p-1$ and $g(F_p)\subset F_1$.
\end{Def}
\begin{rem}\label{minaper}
If the action of $G$ is minimal, the random walk is necessarily aperiodic since otherwise, $S^1$ would be a non trivial finite union of pairwise disjoints closed subsets.
\end{rem}
The next theorem states that for random walks with no invariant probability measure, the only obstruction to the convergence in law of $X_n^x$ is the one described above:
\begin{thm}\label{law}
Let $\omega\mapsto (f_\omega^n)_{n\in\displaystylehbb{N}}$ be a non degenerated random walk on a sub-semigroup $G$ of $\displaystylehrm{Homeo}(S^1)$ with no invariant probability measure on $S^1$, and such that the random walk is aperiodic. Then, for every $x$ in $S^1$, denoting by $\mu_{n}^x$ the law of the random variable $X_n^x:\omega\mapsto f_\omega^n(x)$, we have the convergence in law
$$\mu_n^x\xrightarrow[n\to +\infty]{} \mu^x,$$
where $\mu^x$ is a stationary probability measure of the random walk. Moreover, the convergence is uniform in $x$ in the sense that for any continuous test function $\varphi:S^1\rightarrow \displaystylehbb{R}$,
$$\sup_{x\in S^1} \left|\int_{S^1} \varphi d\mu_n^x-\int_{S^1} \varphi d\mu^x\right|\xrightarrow[n\to +\infty]{} 0$$
\end{thm}
In particular, as a consequence of this theorem, Remark \ref{minaper} and Corollary \ref{minimal ergodique}:
\begin{cor}
Let $(f_\omega^n)_{n\in\displaystylehbb{N}}$ be a non degenerated random walk on a sub-semigroup $G$ of $\displaystylehrm{Homeo}(S^1)$, acting minimally on $S^1$ and with no invariant probabiity measure on $S^1$. Then, with the same notations as Theorem \ref{law}, we have for every $x$ in $S^1$:
$$\mu_n^x\xrightarrow[n\to +\infty]{} \mu,$$
where $\mu$ is the unique stationary probability measure of the random walk.
\end{cor}
\subsubsection{Behaviour of typical homeomorphisms $x\mapsto f_\omega^n(x)$}
Finally, for $\omega$ typical we focus in the behaviour of the homeomorphisms $f_\omega^n$ when $n$ become large.
\begin{thm}\label{Lejan-Antonov}
Let $\omega\mapsto (f_\omega^n)_{n\geq 0}$ be a non degenerated random walk on a sub-semigroup $G$ of $\displaystylehrm{Homeo}(S^1)$, such that $G$ does not preserve a common invariant probability measure on $S^1$. Then, there exists a finite number $p$ of measurable functions $\sigma_1,\ldots,\sigma_p:\Omega\rightarrow S^1$ such that: for $\displaystylehbb{P}$-almost every $\omega$ in $\Omega$, for every closed interval $I$ included in $S^1-\{\sigma_1(\omega),\ldots,\sigma_p(\omega)\}$, $\displaystylehrm{diam}(f_\omega^n(I))\xrightarrow[n\to+\infty]{}0$ exponentially fast.
\end{thm}
It is a global version of Theorem \ref{main}, proving that for $n$ large, the typical homeomorphisms $f_\omega^n$ are close to be ``staircase maps'', with a constant finite number of stairs.
It is also intersting to compare this result with Antonov Theorem stated in the introduction. The hypotheses of Antonov Theorem are stronger than the ones of Theorem \ref{Lejan-Antonov}, since it assume forward and backward minimality and that the homeomorphisms preserve the orientation. In counterpart, the conclusion of Antonov Theorem is in some sense stronger: it does not give the exponential speed of the contractions, but gives a more precise structure: it say that the applications $\sigma_1,\ldots,\sigma_p$ given by Theorem \ref{Lejan-Antonov} are on the form.
$$\{\sigma_1,\sigma_2\ldots, \sigma_p\}=\{\sigma_1,\theta\circ\sigma_1\ldots, \theta^{p-1}\circ\sigma_1\},$$
where $\theta$ is a homeomorphism of order $p$ commuting with all the elements of $G$.
But as we said in the introduction, such a rigid conclusion cannot hold in general in a non minimal context.\\
\subsection{Property of synchronization}
In this section, we want to characterize in which situation the action of a random walk on the circle has the property of \textit{synchronization}, which means that for any couple of initial conditions $x$ and $y$, for almost every realization of the random walk, the distance between the corresponding trajectories of $x$ and $y$ tends to $0$. This property of synchronization has been studied in \cite{Furstenberg} in the linear case, and for example in \cite{Homburg, Kaijser, Kleptsyn, Kleptsyn2} in non linear cases.
\begin{Def}
If $(X,d)$ is a metric space, we say that a random walk $\omega\mapsto (f_\omega^n)_{n\in\displaystylehbb{N}}$ acting on $X$ is \emph{synchronizing} if for every $x$, $y$ in $X$, for almost every $\omega$,
$$d(f_\omega^n(x),f_\omega^n(y))\xrightarrow[n\to+\infty]{} 0.$$
We say that it is \emph{exponentially synchronizing} if the previous convergence is exponentially fast.
\end{Def}
In the context of random walks acting on the circle, we prove that the synchronization is equivalent to the \textit{proximality} of the action. We recall that the action of a semigroup $G$ to a metric space $(X,d)$ is proximal if for every $x,y$ in $X$, there exists a sequence $(g_n)_{n\in\displaystylehbb{N}}$ in $G$ such that
$$d(g_n(x),g_n(y))\xrightarrow[n\to +\infty]{} 0.$$
\begin{thm}
\label{synchronizing}
Let $\omega\mapsto (f_\omega^n)_{n\in\displaystylehbb{N}}$ be a non degenerated random walk on a sub-semigroup $G$ of $\displaystylehrm{Homeo}(S^1)$ without a common fixed point. Then the following properties are equivalent:
\begin{enumerate}[i)]
\item The random walk is exponentially synchronizing.
\item The random walk is synchronizing.
\item The action of $G$ on $S^1$ is proximal.
\end{enumerate}
\end{thm}
It allows for example to retrieve the main result of \cite{Kleptsyn} in a non minimal context and with an exponential speed of convergence:
\begin{cor}\label{Kleptsyn}
$\omega\mapsto (f_\omega^n)_{n\in\displaystylehbb{N}}$ a non degenerated random walk on a sub-semigroup $G$ of $\displaystylehrm{Homeo}(S^1)$ such that:
\begin{itemize}
\item $G$ contains a map $g_0$ with exactly $2$ fixed points $a$ and $b$, one attractive, one repulsive.
\item None of the sets $\{a\}$, $\{b\}$, $\{a,b\}$ is invariant by the semigroup $G$.
\end{itemize}
Then the random walk is exponentially synchronizing.
\end{cor}
That corollary follows rather easily from Theorem \ref{synchronizing}: for any $x,y$ in $S^1$, one can find $h$ in $G$ such that $h(x)$ and $ h(y)$ are distinct from the repulsive fixed point of $g_0$, so that $\displaystylehrm{dist}(g_0^n\circ h(x),g_0^n\circ h(y))\xrightarrow[n\to +\infty]{} 0$, which prove the proximality and we can apply Theorem \ref{synchronizing}. The details are left to the interested reader.\\ \\
An other application deals with the \textit{robustness} of the property of synchronization (that is to say the persistence of the property to small perturbations): with Theorem \ref{synchronizing}, we can prove that the property of synchronization is robust among the semigroups of homeomorphisms without a common fixed point. We restrict ourselves to the case of finitely generated semigroups to avoid to manipulate intricate topologies on sets of semigroups/random walks.
\begin{cor}\label{robust}
Consider a non degenerated random walk $\omega\mapsto (f_\omega^n)$ on a sub-semigroup $G$ of $\displaystylehrm{Homeo}(S^1)^d$ generated by $d$ homeomorphisms of the circle $f_1,\ldots,f_d$ without common fixed points, and assume that $\omega\mapsto (f_\omega^n)$ is synchronizing. Then there exists a neighbourhood $\displaystylehcal{V}$ of $(f_1,\ldots,f_d)$ in $\displaystylehrm{Homeo}(S^1)^d$
such that for any $d$-tuple $(\tilde{f}_1,\ldots,\tilde{f}_d)$ in $\displaystylehcal{V}$, any non degenerated random walk $\omega\mapsto (f_\omega^n)$ on the semigroup $\tilde{G}$ generated by $\{\tilde{f}_1,\ldots,\tilde{f}_d\}$ is (exponentially) synchronizing.
\end{cor}
It is natural to ask whether the property of synchronization is generic, but it easy to see that it is not the case: if $I$ is an open interval, the property
$$\forall k\in \{1,\ldots,d\},\, \overline{f_k(I)}\subset I$$
is robust, and the existence of two disjoints such intervals is an obstruction to the synchronization. However, in the case of a non degenerated random walk on subgroups of $\displaystylehrm{Homeo}_+(S^1)$, Antonov's Theorem holds (see \cite{Deroin-inter}), and hence in this case, the property of synchronization is generic, because the other alternatives (existence of a common invariant probability measure or existence of a non trivial homeomorphism in the centralizer of the group) are degenerated properties. Combining this remark with Corollary \ref{robust}, we obtain the following conclusion:
\begin{cor}\label{generic}
Let $d$ be an integer larger than $1$. Then there exists an open dense subset $\displaystylehcal{U}$ of $\displaystylehrm{Homeo}_+(S^1)^d$ such that for every $(f_1,\ldots,f_d)$ in $\displaystylehcal{U}$, any non degenerated random walk on the group generated by $\{f_1,\ldots,f_d\}$ is exponentially synchronizing.
\end{cor}
\subsection{Random dynamical systems on $[0,1$]}
We conclude by the study of the iterations of continuous injective transformations of an interval. For exemple, we can apply our results to obtain:
\begin{cor}\label{example}
Let $\omega\mapsto (f_\omega^n)_{n\in\displaystylehbb{N}}$ be a non degenerated random walk on a semigroup $G$ of injective continuous functions from $[0,1]$ into itself, and let us assume that
$$\bigcap_{g\in G} g([0,1])=\emptyset.$$
Then there exists $q<1$ such that for $\displaystylehbb{P}$-almost every $\omega$:
$$\forall n\in \displaystylehbb{N},\, \displaystylehrm{diam}(f_\omega^n([0,1]))\leq C q^n$$
for some constant $C=C(\omega)$.
\end{cor}
The assumption $\bigcap_{g\in G} g([0,1])=\emptyset$ is weak (and is actually equivalent to the conclusion if $G$ does not fix any point of $I$): for example, if you iterate randomly two continuous injective functions $f_1,f_2:I\rightarrow I$ such that $f_1$ has only one fixed point $c$, and $f_2(c)\not=c$, then the corollary applies, that is to say that random compositions of $f_1$ and $f_2$ almost surely contract the whole interval $[0,1]$ exponentially fast.\\
\begin{rem}
It is actually possible to prove Corollary \ref{example} by a straight elementary proof, using the ideas of \cite{Yuri}.
\end{rem}
The previous corollary does not apply if we iterate homeomorphisms of the interval. The techniques of the paper does not seem to be sufficient to treat such a random walk in a general exhaustive way. However we can still adapt our techniques to get some partial information. Here is a variation of our main theorem in this context:
\begin{cor}\label{mainR}
Let $\omega\mapsto (f_\omega^n)_{n\geq 0}$ be a non degenerated random walk on a sub-semigroup $G$ of $\displaystylehrm{Homeo}([0,1])$, such that:
\begin{itemize}
\item there does not exists a non trivial subinterval of $(0,1)$ invariant by $G$.
\item there exists at least one probability measure $\mu$ on $(0,1)$ which is stationary for the random walk.
\end{itemize}
Then, for every $x$ in $\displaystylehbb{R}$ there exists a neighbourhood $I$ of $x$ such that
$$\forall n\in \displaystylehbb{N},\, \displaystylehrm{diam}(f_\omega^n(I))\leq q^n,$$
where $q<1$ does not depend on $x$.
\end{cor}
\begin{rem}
From the proof of this corollary one can notice that if the second assumption is satisfied but not the first one, then the conclusion of the statement still holds if we restrict $x$ to belong to the invariant interval $I=(\inf (\displaystylehrm{supp}(\mu)),\sup (\displaystylehrm{supp}(\mu)))$
\end{rem}
This theorem gives the phenomenon of local contractions under the existence of a stationary probability measure. With some additional work, one can hope to deduce various dynamical properties from it as we do in this paper in the case of the circle. As an example, let us state the following corollary, answering by the affirmative to a question of B. Deroin in \cite{Deroin-cours}: ``If $f,g$ are increasing diffeomorphisms of $[0,1]$, and if the Lebesgue measure is stationary (for $\nu=\frac{\delta_f+\delta_g}{2}$), is it necessarily the only stationary probability measure without atoms?''\\
\begin{cor}\label{torchder}
If a random walk on $\displaystylehrm{Homeo}_+([0,1])$ admits a stationary probability measure on $(0,1)$ with total support, then it is the only one. In particular, any random walk on $\displaystylehrm{Homeo}_+([0,1])$ acting minimally on $(0,1)$ admits at most one stationary probability measure on $(0,1)$.
\end{cor}
The existence of a stationary probability measure for a random walk on $\displaystylehrm{Homeo}([0,1])$ (other that convex combinations of $\delta_0$ and $\delta_1$ ) can be ensured if the extremities $0$ and $1$ ´´repulse'' the dynamics of the random walk. One can check for exemple that a stationary probability measure exists if the random walk is generated by a probability $\nu$ on $Diff^1_+([0,1])$ whose finite support, such that $\int \log |f'(c)|d\nu(f)>0$ for $c=1, 2$.
However, without such an additional assumption, in general such a measure does not exists.
For exemple, when the random walk is symmetric in the sense that the associated probability measure $\nu$ is invariant under the transformation $g\mapsto g^{-1}$, it is proved in \cite{Deroin-homeo} that there is no stationary probability measure. Thus, Corollary \ref{mainR} does not apply in this case. But it is interessant to notice that \cite{Deroin-homeo} develops techniques to obtain a good understanding of the random walk in this particular case where ours methods do not apply. In consequence one could hope that by adapting these techniques and those of this paper it would be possible to manage the study of a general random walk on $\displaystylehrm{Homeo}([0,1])$.
\subsection{Scheme of the paper}
The paper is organized as follows:
\begin{itemize}
\item in Section \ref{sec-inv} we present the core argument of our results: an invariance principle for a general skew-product $\hat{T}$ on a space $\Omega\times S^1$ stating that either there is a phenomenon of contractions in the dynamics of $\hat{T}$ on the fibres, either ``there is something invariant''. Applying the principle to the specific case where $\hat{T}$ is associated to a random walk, we obtain Theorem \ref{main}, and one can hope that it can also be used in non independant contexts.
\item In Section \ref{sec-consequences}, we state various ergodic properties of the random dynamical systems on compact metric sapces satisfying the property of local contractions. (This section can be read indepedently of the others)
\item In Section \ref{sec-proofs}, we deduce the proofs of the other theorems stated in the introduction by combining the results of Sections \ref{sec-inv} and \ref{sec-consequences}.
\end{itemize}
\section{An invariance principle}\label{sec-inv}
The objective of this part is to prove an invariance principle in the spirit of the works of Ledrappier \cite{Ledrappier}, Crauel \cite{Crauel} and Ávila-Viana \cite{Avila} for one-dimensional cocycles without regularity (except the continuity).\\
Let $(\Omega,\displaystylehcal{F},\displaystylehbb{P})$ be a probability space and $T:\Omega \rightarrow \Omega$ be a $\displaystylehbb{P}$-invariant transformation. We look at the skew products on $\Omega\times S^1 $ extending $T$, that is the measurable transformations $\hat{T}$ of the form $(\omega,x)\mapsto (T\omega, f_\omega(x))$, where $f_\omega\in \displaystylehrm{Homeo}(S^1)$. For $\omega$ in $\Omega$, we will use the notation
$$f_\omega^n=f_{T^{n-1}\omega}\circ\cdots\circ f_\omega,$$
so that the iterates of $\hat{T}$ are given by $\hat{T}^n(\omega,x)=(T^n\omega,f_\omega^n(x))$.
\subsection{Lyapunov exponent and exponent of contraction}
Let us recall the definition of the \emph{Lyapunov exponents} of $\hat{T}$ when $f_\omega$ is smooth:
\begin{Def}
If $f_\omega\in \mbox{Diff}(S^1)$, then the
{Lyapunov exponent} of $\hat{T}$ at a point $(\omega,x)\in\Omega\times S^1$ is defined as $$\lambda(\omega,x)=\lim_{n\to+\infty}\frac{\log |(f_\omega^n)'(x)|}{n},$$
if the limit exists. If $\hat{\mu}$ is a $\hat{T}$-invariant probability measure such that $(\omega,x)\mapsto \log |f_\omega'(x)|$ is $\hat{\mu}$-integrable, then the Lyapunov exponent is well defined $\hat{\mu}$-almost everywhere, constant if $\hat{\mu}$ is ergodic, and the Lyapunov exponent of $\hat{\mu}$ is defined as
$$\lambda(\hat{\mu})=\int_{\Omega\times S^1} \lambda(\omega,x)d\hat{\mu}(\omega,x).$$
\end{Def}
The Lyapunov exponent $\lambda(\omega,x)$ measures the exponential rate of contraction of $(f_\omega^n)$ at the neighbourhood of $x$. In order to have analogue informations without assuming that $f_\omega\notin \mbox{Diff}^1(S^1)$, we define the following \emph{exponent of contraction}:
\begin{Def}
The exponent of contraction of $\hat{T}$ at the point $(\omega,x)$ is the non positive quantity
$$\lambda_{con}(\omega,x)=\varlimsup_{y\to x}\varlimsup_{n\to +\infty}\frac{\log(\displaystylehrm{dist}(f_\omega^n(x),f_\omega^n(y))}{n}.$$
If $\hat{\mu}$ is a $\hat{T}$-invariant probability measure, the exponent of contraction of $\hat{\mu}$ is defined as
$$\lambda_{con}(\hat{\mu})=\int_{\Omega\times S^1} \lambda_{con}(\omega,x)d\hat{\mu}(\omega,x).$$
\end{Def}
Note that $\lambda_{con}$ is $\hat{T}$ is $\hat{T}$-invariant, so that $\lambda_{con}$ is constant $\hat{\mu}$-almost everywhere if $\hat{\mu}$ is ergodic.
The exponent of contraction has the advantage over Lyapunov exponents that it does not need any assumption of differentiability. As a counterpart, the information provided by this exponent is slightly less precise than the one provided by the Lyapunov exponents, because it only measures the contraction of the cocycle, not the expansion, and actually the maximal contraction only, so that one cannot hope miming an Oselede\v{c}/Pesin's theory with this naive definition in dimension larger than one. In dimension one, though, this exponent of contraction is a perfect tool to generalize the notion of Lyapunov exponent. We have indeed in this case a simple relation between Lyapunov exponent and exponent of contraction:
\begin{prop}\label{expoegal}
Let $(\Omega,\displaystylehcal{F},\displaystylehbb{P})$ be a probability space and let $\hat{T}: (\omega,x)\mapsto (T\omega, f_\omega(x))$ be a measurable transformation of $\Omega \times S^1$ with $f_\omega\in \displaystylehrm{Diff^1}(S^1)$, such that the function $(\omega,x)\mapsto \log |f_\omega'(x)|$ is bounded. Then, for every $\hat{T}$-invariant ergodic probability measure $\hat{\mu}$, we have
$$\lambda_{con}(\hat{\mu})=\inf(\lambda(\hat{\mu}),0),$$
where $\lambda(\hat{\mu})$ is the Lyapunov exponent associated to $\hat{\mu}$:
$$\lambda(\hat{\mu})=\int_{\Omega\times S^1}\log |f_\omega '(x)|d\hat{\mu}(\omega,x).$$
\end{prop}
\begin{proof}
The inequality $\lambda_{con}(\hat{\mu})\leq 0$ is trivial. And as noticed in \cite{Crauel} Proposition 2.6, one can adapt the techniques of Pesin on stable manifolds, to obtain the inequality $\lambda_{con}(\hat{\mu})\leq \lambda(\hat{\mu})$ (see also \cite{Lejan} for a proof in the particular case of independent compositions of diffeomorphisms). So from now on, we focus on proving the converse inequality $\lambda_{con}(\hat{\mu})\geq \inf(\lambda(\hat{\mu}),0).$\\
We assume that $\lambda_{con}(\hat{\mu})<0$. Let $(\omega,x)$ be a point of $\Omega\times S^1$ such that
$$\lambda(\hat{\mu})=\lambda(\omega,x)=\lim_{n\to+\infty}\frac{\log |(f_\omega^n)'(x)|}{n}$$
and
$$\lambda_{con}(\hat{\mu})=\lambda_{con}(\omega,x)=\varlimsup_{y\to x}\varlimsup_{n\to +\infty}\frac{\log(\displaystylehrm{dist}(f_\omega^n(x),f_\omega^n(y))}{n},$$
and let $\varepsilon>0$. If $y$ is close enough to $x$, then we have
$$\forall n\in \displaystylehbb{N}, \mbox{dist}(f_\omega^n(x),f_\omega^n(y))\leq \varepsilon.$$
Then, denoting by $\alpha_\omega(\cdot)$ the modulus of continuity of $\log |f_\omega'|$, we have for any $z_1$, $z_2$ in $[x,y]$,
$$\log |(f_\omega^n)'(z_1)|-\log|(f_\omega^n)'(z_2)|= \sum_{k=0}^{n-1}\log |f_{T^k\omega}'(f_\omega^k(z_1))|-\log |f_{T^k\omega}'(f_\omega^k(z_2))|\leq \sum_{k=0}^{n-1}\alpha_{T^k\omega}(\varepsilon).$$
In particular, by the mean value equality,
$$\frac{\log |(f_\omega^n)'(x)|}{n}\leq \frac{1}{n}\log\left(\frac{\mbox{dist}(f_\omega^n(x),f_\omega^n(y))}{\mbox{dist}(x,y)}\right)+
\frac{1}{n}\sum_{k=0}^{n-1}\alpha_{T^k\omega}(\varepsilon).$$
If $\omega$ is a Birkhoff point of $\alpha_\cdot(\varepsilon)$, we deduce by letting $n$ tend to $+\infty$ and $y$ to $x$ that
$$\lambda(\hat{\mu})\leq \lambda_{con}(\hat{\mu})+\int_{\Omega} \alpha_{\omega '}(\varepsilon) d\displaystylehbb{P}(\omega ')$$
Since $\alpha_{\omega '}(\varepsilon)$ tends to $0$ as $\varepsilon\to 0$ and is uniformly bounded, by dominated convergence we obtain that $\lambda(\hat{\mu})\leq \lambda_{con}(\hat{\mu})$.
\end{proof}
\subsection{The invariance principle statement}
Let us state the main theorem of the section.
\begin{thm}[Invariance principle]\label{invariance}~
Let $(\Omega,\displaystylehcal{F},\displaystylehbb{P})$ be a standard Borel space, with $\displaystylehbb{P}$ a probability measure, and let $\hat{T}: (\omega,x)\mapsto (T\omega, f_\omega(x))$ be a measurable transformation of $\Omega \times S^1$ with $f_\omega\in \displaystylehrm{Homeo}(S^1)$. Then, for every $\hat{T}$-invariant probability measure $\hat{\mu}$ of the form $d\hat{\mu}(\omega,x)=d\mu_\omega(x)d\displaystylehbb{P}(\omega)$, we have the following alternative:
\begin{itemize}
\item either $\lambda_{con}(\hat{\mu})<0$ (contraction),
\item or for $\displaystylehbb{P}$-almost every $\omega$, $\mu_{T\omega}=(f_\omega)_*\mu_{\omega}$ (invariance).
\end{itemize}
\end{thm}
\begin{rem}
In the case that $f_\omega$ is smooth, by Proposition \ref{expoegal} we otain the known fact that the Lyapunov exponent of $\hat{\mu}$ is negative unless maybe if we have the ''deterministic relation'' $\mu_{T\omega}=(f_\omega)_*\mu_{\omega}$. In particular the Lyapunov exponent of a stationary probability measure of a random walk on $\mbox{Diff}^1(S^1)$ is negative unless the stationary measure is actually invariant)
\end{rem}
When the transformation $T$ of $\Omega$ is invertible, the relation $\mu_{T\omega}=(f_\omega)_*\mu_{\omega}$ is only a reformulation of ``$\hat{\mu}$ is $\hat{T}$-invariant'', so that the invariance principle as we stated it only gives information in non-invertible contexts (it is possible though to get an invariance principle in an invertible context, applying the theorem to a modified system, see \cite{Ledrappier}).\\
The following subsections 3.3, 3.4 and 3.5 are dedicated to the proof of Theorem \ref{invariance}. We will keep the notations of the statement in these subsections.
\subsection{Fibred Jacobian and fibred entropy}
Following the ideas of \cite{Ledrappier, Crauel, Avila}, we define the fibred entropy of $\hat{\mu}$ as follows:
\begin{Def}
The \emph{fibred Jacobian} $J=J(\hat{\mu}):\Omega\times S^1\to\displaystylehbb{R}$ of $\hat{\mu}$ is defined by the expression
$$J(\omega,x)=\frac{d(f_\omega^{-1})_*\mu_{T\omega}}{d\mu_\omega}(x),$$
where the derivative is taken in the Radon-Nikodym sense. The \emph{fibred entropy} $h(\hat{\mu})$ of $\hat{\mu}$ is defined as
$$h(\hat{\mu})=\begin{cases}
\displaystyle-\int_{\Omega\times S^1}\log J\, d\hat{\mu}&\text{if }\log J\in L^1(\Omega\times S^1,\hat{\mu}),\\
+\infty&\text{otherwise}.
\end{cases}$$
\end{Def}
By definition, the mapping $x\mapsto J(\omega,x)$ is the derivative of Radon-Nikodym of the measure $(f_\omega^{-1})_*\mu_{T\omega}$ against $\mu_\omega$, that is the $\mu_\omega$-integrable function such that we can write
\begin{equation}\label{Radon}d(f_\omega^{-1})_*\mu_{T\omega}(x)=J(\omega,x)\,d\mu_\omega(x)+d\tilde{\mu}_\omega(x)\end{equation}
where $\tilde{\mu}_\omega$ is singular with respect to $\mu_\omega$.
Let us state a classical general fact of geometric measure theory which allows to see a Radon-Nikodym derivative as, in some sense, a standard derivative:
\begin{prop}\label{Besicovitch}
Let $\mu$ be a probability measure on $S^1$, and $\nu$ be any measure on $S^1$. Then:
\begin{enumerate}[i)]
\item For $\mu$-almost every $x$ in $S^1$,
$$\frac{d\nu}{d\mu}(x)=\lim_{I\ni x,\, \displaystylehrm{diam}(I)\to 0}\frac{\nu(I)}{\mu(I)}$$
(here and in the sequel, $I$ represents an interval of $S^1$).
\item Denoting $q^*(x)=\sup_{I\ni x}\frac{\nu(I)}{\mu(I)}$,
$$\int_{S^1}\log^+ q^*(x)d\mu(x)\leq 2\nu(S^1).$$
\end{enumerate}
\end{prop}
This proposition is standard if $\nu$ is the Lebesgue measure, as a consequence of Vitali's covering Lemma, and as noticed in \cite{Ledrappier}, the proof adpapts for any measure $\nu$ if we use Besicovitch's covering Lemma instead of Vitali's.\\
The key of the proof of Theorem \ref{invariance} is to see the entropy $h(\hat{\mu})$ in two different ways.
\begin{itemize}
\item \emph{Firstly},
one can see $h(\hat{\mu})$ as a quantity measuring in average how much $(f_\omega^{-1})_*\mu_{T\omega}$ differs from $\mu_\omega$, and obtain the following fact justifying that $h(\hat{\mu})$ deserves its appellation of entropy:
\begin{prop}\label{entropy-positive}
We have the inequality
$$h(\hat{\mu})\geq 0,$$
with equality if and only if for $\displaystylehbb{P}$-almost every $\omega$, $\mu_{T\omega}=(f_\omega)_*\mu_{\omega}$.
\end{prop}
\item \emph{Secondly}, one can use Proposition \ref{Besicovitch} to see the Jacobian term $J(\omega,x)$ as a kind of derivative for some geometry: for $\hat{\mu}$-a.e. $(\omega,x)$ in $\Omega\times S^1$,
$$J(\omega,x)=\lim_{y\to x}\frac{\mu_{T\omega}([f_\omega(x),f_\omega(y)])}{\mu_\omega([x,y])}.$$
It is then possible to think of $-h(\hat{\mu})$ as a kind of Lyapunov exponent, and obtain:
\begin{prop}\label{ineq-entropy}
We have the inequality
$$\lambda_{con}(\hat{\mu})\leq -h(\hat{\mu})$$
\end{prop}
\begin{rem}
With a slighter effort, we could actually prove the more precise inequality $\lambda_{con}(\hat{\mu})\leq -\frac{h(\hat{\mu})}{d(\hat{\mu})}$, for a good definition of the \emph{fibred dimension} $d(\hat{\mu})$, which would thus belongs to the big family of inequalities relating Lyapunov exponent, entropy and dimension (see for example \cite{Young, Hu, Ledrappier2}).
\end{rem}
\end{itemize}
It is clear that Theorem \ref{invariance} is a direct consequence of Propositions \ref{entropy-positive} and \ref{ineq-entropy}. Let us begin by proving Proposition \ref{entropy-positive} (the easy part):
\begin{proof}[Proof of Proposition \ref{entropy-positive}]
As a consequence of \eqref{Radon},
$$\int_{\Omega\times S^1} J\, d\hat{\mu}=\int_{\Omega}\int_{S^1} J(\omega,x)\,d\mu_\omega(x)d\displaystylehbb{P}(\omega)
\leq \int_{\Omega}\int_{S^1} d\mu_{T\omega}(x)d\displaystylehbb{P}(\omega)=1,$$
hence, by Jensen inequality,
\begin{equation}\label{Jensen}-h(\hat{\mu})=\int_{\Omega\times S^1} \log J\, d\hat{\mu}\leq \log \int_{\Omega\times S^1} J\, d\hat{\mu} \le 0,\end{equation}
so that $h(\hat{\mu})$ is non negative.
Moreover, if $h(\hat{\mu})=0$, then the Jensen inequality (\ref{Jensen}) is in fact an equality, so that $J=1$ $\hat{\mu}$-almost everywhere. Thus, replacing it in \eqref{Radon}, we deduce that for $\displaystylehbb{P}$-almost every $\omega$, $(f_\omega^{-1})_*\mu_{T\omega}=\mu_\omega$, hence $\mu_{T\omega}=(f_\omega)_*\mu_\omega$.
\end{proof}
We focus now on the proof of Proposition \ref{ineq-entropy}. In the following subsection, we dismantle the problem and leave the core arguments for a separated treatment in the section afterwards.
\subsection{Preliminaries: reduction of the problem}
The objective of this subsection is to check that it is enough to prove Proposition \ref{ineq-entropy} in the case where we have some useful additional properties on $\hat{\mu}$, namely:
\begin{itemize}
\item $\hat{\mu}$ is ergodic.
\item None of the probability measures $\mu_\omega$ has atoms on $S^1$.
\end{itemize}
The reduction of the problem to the ergodic case is done by ergodic disintegration: let us write
$$\hat{\mu}=\int \hat{\mu}_\alpha\, d\alpha$$
with $\hat{\mu}_\alpha$ ergodic and $d\alpha$ some probability measure on the set of ergodic probability measures. Then, writing $d\hat{\mu}_\alpha=d\mu_{\alpha,\omega}d\displaystylehbb{P}_\alpha$ and setting $$J_\alpha(\omega,x)=\frac{d(f_\omega^{-1})_*\mu_{T\omega,\alpha}}{d\mu_{\omega,\alpha}}(x)$$
the Jacobian associated to $\hat{\mu}_\alpha$, we have that $J_\alpha=J$ $\hat{\mu}_\alpha$-almost everywhere, and as a consequence,
$$h(\hat{\mu})=-\iint_{\Omega\times S^1} \log J\, d\hat{\mu}_\alpha d\alpha=-\iint_{\Omega\times S^1} \log J_\alpha\, d\hat{\mu}_\alpha d\alpha=\int h(\hat{\mu}_\alpha)\,d\alpha.$$
Moreover, we also have
$$\lambda_{con}(\hat{\mu})= \iint_{\Omega\times S^1} \lambda_{con}(\omega,x)\,d\hat{\mu}_\alpha(\omega,x)\,d\alpha=\int \lambda_{con}(\hat{\mu}_\alpha)\,d\alpha,$$
hence the inequality to prove is $\int \lambda_{con}(\hat{\mu}_\alpha)\,d\alpha\leq -\int h(\hat{\mu}_\alpha)\,d\alpha$, which follows from the inequalities in the ergodic case $\lambda_{con}(\hat{\mu}_\alpha)\leq -h(\hat{\mu}_\alpha)$.\\
Thus from now on, we assume that $\hat{\mu}$ is ergodic. The case where $\mu_\omega$ has atoms is treated by the following general lemma:
\begin{lem}\label{atoms}
If $\hat{\mu}$ is ergodic, and if the set $\{\omega\in \Omega| \mu_{\omega} \mbox{ has atoms}\}$ has $\displaystylehbb{P}$-positive probability, then there exists a family $(E(\omega))_{\omega\in\Omega}$ of finite subsets of $S^1$, all of them with same cardinal $d$, such that for $\displaystylehbb{P}$-almost every $\omega$ in $\Omega$, $f_\omega(E(\omega))=E(T\omega)$ and $\displaystyle\mu_\omega=\frac{1}{d}\sum_{x\in E(\omega)}\delta_x$.
\end{lem}
\begin{rem}
The proof does not use the structure of $S^1$ so that the statement remains actually valid for any skew-shift $\hat{T}$.
\end{rem}
\begin{proof}
If $\varphi$ is any function from $S^1$ into $\displaystylehbb{R}$ and $\mu$ a probability measure on $S^1$, we denote
$$\left\{\begin{disarray}{l} \|\varphi\|_{l^1}=\sum_{x\in S^1}|\varphi(x)| \\ \|\mu\|_{l^\infty}=\sup_{x\in S^1} \mu(\{x\})\end{disarray}\right.,$$
so that, if $\|\varphi\|_{l^1}<+\infty$:
$$\int_{S^1} \varphi d\mu\leq \|\varphi\|_{l^1}\|\mu\|_{l^\infty},$$
with equality if and only if $\varphi$ is supported on the set $\left\{x\in S^1| \mu(\{x\})=\|\mu\|_{l^\infty}\right\}$.\\
Now, in the context of the statement, let us set
$$E(\omega)=\left\{x\in S^1|\mu_\omega(\{x\})|=\|\mu_\omega\|_{l^\infty}\right\},$$
which is clearly finite and non empty if $\|\mu_\omega\|_{l^\infty}>0$ (which occurs on a set of positive probability by assumption). We are going to prove that these sets $E(\omega)$ satisfy the conclusion of the statement. Let $\varphi:(\omega,x)\mapsto \varphi_\omega(x)$ be the function defined by:
$$\varphi_{\omega}(x)=\left\{\begin{disarray}{ll} \frac{\displaystylehds{1}_{E(\omega)}(x)}{\displaystylehrm{Card}(E(\omega))} &\mbox{ if } \|\mu_\omega\|_{l^\infty}>0 \\ \\
0 &\mbox{ if } \|\mu_\omega\|_{l^\infty}=0\end{disarray}\right.$$
Notice that $\|\varphi_\omega\|_{l^1}=1$ if $\|\mu_\omega\|_{l^\infty}>0$ and $0$ if not. On one hand, we have the equality
\begin{equation}\label{abigay}
\int_{\Omega\times S^1}\varphi d\hat{\mu}=\int_{\Omega}\left(\int_{S^1} \varphi_\omega d\mu_\omega\right)d\displaystylehbb{P}(\omega)=\int_{\Omega}\|\mu_\omega\|_{l^\infty}d\displaystylehbb{P}(\omega),\end{equation}
(using the easy computation $\int_{S^1} \varphi_\omega d\mu_\omega=\|\mu_\omega\|_{l^\infty}$), and on the other hand, we have the chain of inequalities:
\begin{equation}\label{abigay2}\begin{disarray}{ll}\int_{\Omega\times S^1}\varphi\circ \hat{T} d\hat{\mu}&=\int_{\Omega}\left(\int_{S^1} (\varphi_{T\omega}\circ f_\omega) d\mu_\omega\right)d\displaystylehbb{P}(\omega)\\
&\leq \int_{\Omega}\|\varphi_{T\omega}\circ f_\omega\|_{l^1}\|\mu_\omega\|_{l^\infty}d\displaystylehbb{P}(\omega)\\
&\leq \int_{\Omega}\|\varphi_{T\omega}\|_{l^1}\|\mu_\omega\|_{l^\infty}d\displaystylehbb{P}(\omega)\\
&\leq\int_{\Omega}\|\mu_\omega\|_{l^\infty}d\displaystylehbb{P}(\omega)\end{disarray}\end{equation}
(using the general equality $\|\psi\circ f\|_{l^1}= \|\psi\|_{l^1}$, valid for $f\in \displaystylehrm{Homeo}(S^1)$, and the fact that $\|\varphi_\omega\|_{l^1}\leq 1$).\\ \\
Combining (\ref{abigay}), (\ref{abigay2}) and he invariance equality $\int \varphi d\hat{\mu}=\int \varphi\circ \hat{T} d\hat{\mu}$, we deduce that the chain of inequalities (\ref{abigay2}) is in fact a chain of equalities. In particular, for $\displaystylehbb{P}$-almost every $\omega$ in $\Omega$, $\int_{S^1} (\varphi_{T\omega}\circ f_\omega) d\mu_\omega=\|\varphi_{T\omega}\circ f_\omega\|_{l^1}\|\mu_\omega\|_{l^\infty}$, hence $\varphi_{T\omega}\circ f_\omega$ is supported on the set $E(\omega)$. In consequence, for $\displaystylehbb{P}$-almost every $\omega\in\Omega$:
$$f_\omega^{-1}(E(T\omega))\subset E(\omega).$$
Thus, for $\displaystylehbb{P}$-almost every $\omega$ in $\Omega$, $\displaystylehrm{Card}(E(T\omega))\geq \displaystylehrm{Card}(E(\omega))$, and hence by ergodicity $d=\displaystylehrm{Card}(E(\omega))$ does not depend on $\omega$ (up to a negligible set), and $d<+\infty$ by assumption. In particular, $f_\omega:E(\omega)\rightarrow E(T\omega)$ is a bijection and for $\displaystylehbb{P}$-almost every $\omega\in\Omega$:
$$E(T\omega)=f_\omega(E(\omega)).$$
Finally, that last equality means that the set $\displaystylehcal{E}=\bigcup_{\omega\in\Omega}\{\omega\}\times E(\omega)$ is $\hat{T}$-invariant up to a $\hat{\mu}$-negligible set hence using the ergodicity of $\hat{\mu}$ and the fact that $\displaystylehcal{E}$ is not $\hat{\mu}$-negligible by assumption we deduce that in fact $\hat{\mu}(\displaystylehcal{E})=1$, i.e. for $\displaystylehbb{P}$-almost every $\omega$ in $\Omega$, $\mu_\omega(E(\omega))=1$. That means that $\mu_\omega$ is supported on the finite set $E(\omega)$, and by definition all the points of $E(\omega)$ have the same $\mu_\omega$-mass, so
$$\mu_\omega=\frac{1}{\mbox{Card}(E(\omega))}\sum_{x\in E(\omega)}\delta_x=\frac{1}{d}\sum_{x\in E(\omega)}\delta_x,$$
which completes the proof.
\end{proof}
As a consequence, if the probability measures $\mu_{\omega}$ have atoms for a set of $\omega$ of $\displaystylehbb{P}$-positive probability, then Lemma \ref{atoms} implies in particular that for $\displaystylehbb{P}$-almost every $\omega$ in $\Omega$, $\mu_{T\omega}=(f_\omega)_*\mu_\omega$, hence $h(\hat{\mu})=0$, so that the inequality $\lambda_{con}(\hat{\mu})\leq -h(\hat{\mu})$ is trivial.\\
\subsection{Proof of Proposition \ref{ineq-entropy}}
From now on, we assume that $\hat{\mu}$ is ergodic and that the fibred probability measures $\mu_\omega$ have no atoms.\\
The main idea of the proof is to use the Birkhoff theorem to $\log J$ to see that the entropy $h(\hat{\mu})$ represents the exponential rate of decrease of $\frac{d(f_\omega^n)^{-1}_*\mu_{T^n\omega}}{d\mu_\omega}$, and hence of $\frac{\mu_{T^n\omega}(f_\omega^n(I))}{\mu_\omega(I)}$ for $I$ a ``typical'' small interval. However, it is more convenient to work with a slightly modified version of $J$:
\begin{Def}
For $\varepsilon>0$, we define the \emph{approximated Jacobian} $J_\varepsilon=J_\varepsilon(\hat{\mu})$ as
$$J_\varepsilonsilon(\omega,x)=\sup\left\{ \frac{\mu_{T\omega}(f_\omega(I))}{\mu_\omega(I)}\,\Big|\,I\ni x,\mu_{\omega}(I)\leq \varepsilon\right\}.$$
and the corresponding \emph{approximated entropy} $h_\varepsilon$ as
$$h_\varepsilon(\hat{\mu})=\begin{cases}
\displaystyle -\int_{\Omega\times S^1}\log J_\varepsilon d\hat{\mu}&\text{ if }\log J_\varepsilon\in L^1(\Omega\times S^1,\hat{\mu})\\
+\infty&\text{otherwise.}\end{cases}$$
\end{Def}
Notice that $J_\varepsilon$ is well defined thanks to the fact that $\mu_\omega$ has no atoms.\\
In the next lemma, we justify that the definitions of $J_\varepsilon(\hat{\mu})$ and $h_\varepsilon(\hat{\mu})$ are legitimate, in the sense that these quantities are indeed approximations of $J(\hat{\mu})$ and $h(\hat{\mu})$.
\begin{lem}
We have
$$\lim_{\varepsilon\to 0} J_\varepsilon(\hat{\mu})=J(\hat{\mu})$$
$\hat{\mu}$-almost everywhere, and
\begin{equation}
\label{entrofatou}
\lim_{\varepsilon\to 0} h_\varepsilon(\hat{\mu})=h(\hat{\mu}).
\end{equation}
\end{lem}
\begin{proof}
The first point is a direct consequence of Proposition \ref{Besicovitch} applied to $\mu=\mu_\omega$ and $\nu=(f_{\omega}^{-1})_*\mu_{T\omega}$.
To prove the second point, we write $\log J_\varepsilon=u_\varepsilon-v_\varepsilon$ with $u_\varepsilon=\sup( \log J_\varepsilon,0)$, $v_\varepsilon=\sup( -\log J_\varepsilon,0)$, and we also write $\log J=u+v$ in the same way. We have that $u_\varepsilon\to u$ and $v_\varepsilon\to v$ $\hat{\mu}$-almost everywhere by the first point. Moreover, using the second part of Proposition \ref{Besicovitch}, we deduce that $\sup_{\varepsilon>0}u_\varepsilon \in L^1(\hat{\mu})$, hence by dominated convergence,
$$\lim_{\varepsilon\to 0} \int_{\Omega\times S^1} u_\varepsilon\, d\hat{\mu}=\int_{\Omega\times S^1} u\, d\hat{\mu}.$$
On the other hand, $v_\varepsilon$ is non negative and increasing as $\varepsilon$ decreases to $0$, hence by Beppo-Levi's Theorem,
$$\lim_{\varepsilon\to 0}\int_{\Omega\times S^1} v_\varepsilon\, d\hat{\mu}=\int_{\Omega\times S^1} v\, d\hat{\mu}.$$
The claim follows.
\end{proof}
The following lemma is the key part of the proof of Proposition \ref{ineq-entropy} (and hence of Theorem \ref{invariance}). It establishes some phenomenon of exponential local contractions under the presence of entropy:
\begin{lem}\label{yoplait}
Let us assume that $h(\hat{\mu})$ is positive. Then, for $\hat{\mu}$-almost every $(\omega,x)\in \Omega\times S^1$, for every $\tilde{h}$ in $(0,h(\hat{\mu}))$, there exists $\delta>0$ such that for any interval $I$ containing
$x$ such that $\mu_\omega(I)<\delta$,
$$\forall n\in\displaystylehbb{N},\quad \mu_{T^n\omega}(f_\omega^n(I))\leq e^{-n\tilde{h}}\mu_\omega(I).$$
\end{lem}
\begin{proof}
Let $\tilde{h}$ in $(0,h)$ be given. By \eqref{entrofatou} one can choose $\varepsilon>0$ so that $h_\varepsilon(\hat{\mu})>\tilde{h}$. Let us take a \emph{Birkhoff point} $(\omega,x)$ of $\log J_\varepsilon$, that is such that
$$\lim_{n\to +\infty} \frac{1}{n}\sum_{k=0}^{n-1}\log J_\varepsilon\circ \hat{T}^k(\omega,x)=-h_\varepsilon(\hat{\mu}).$$
Note: Birkhoff's Theorem is still valid even when $\log J_\varepsilon\not\in L^1(\hat{\mu})$, because one can apply Birkhoff Theorem to the function $\sup(\log J_\varepsilon, -M)$ (integrable by Proposition \ref{Besicovitch}) for $M$ arbitrarily large.\\
In particular there exists a constant $C_0=C_0(\omega,x)$ such that
$$\forall n\in \displaystylehbb{N},\quad \prod_{k=0}^{n-1} J_\varepsilon\circ \hat{T}^k(\omega,x)\leq C_0e^{-n\tilde{h}}.$$
Let $I$ be an interval containing $x$ small enough so that
$$\mu_\omega(I)\leq \delta:=\frac{\varepsilon}{1+C_0},$$
and let us set $x_n=f_\omega^n(x)$, $I_n=f_{\omega}^n(I)$. We claim that
\begin{equation}
\label{yapasmieux}\forall n\in \displaystylehbb{N},\quad \mu_{T^n\omega}(I_n)\leq e^{-n\tilde{h}}\mu_\omega(I).\end{equation}
The proof of the claim is done by induction:
\begin{itemize}
\item For $n=0$, the inequality is trivial.
\item If the inequality is satisfied for $k=0,\ldots, n-1$, then, for $k=0,\ldots, n-1$ the interval $I_k$ contains the point $x_k$ and satisfies $\mu_{T^k\omega}(I_k)\leq \varepsilon$ , hence, by definition of $J_\varepsilon$,
$$\frac{\mu_{T^{k+1}\omega}(I_{k+1})}{\mu_{T^k\omega}(I_k)}=\frac{\mu_{T^{k+1}\omega}(f_{T^k\omega}(I_{k}))}{\mu_{T^k\omega}(I_k)}\leq J_\varepsilon(T^k\omega,x_k)=J_{\varepsilon}\circ \hat{T}^k(\omega,x),$$
and we deduce
$$\mu_{T^n\omega}(I_n)=\mu_\omega(I)\prod_{k=0}^{n-1}\frac{\mu_{T^{k+1}\omega}(I_{k+1})}{\mu_{T^k\omega}(I_k)}\leq \mu_\omega(I)\prod_{k=0}^{n-1}J_\varepsilon\circ \hat{T}^k(\omega,x)\leq C_0e^{-n\tilde{h}}\mu_{\omega}(I).$$
\end{itemize}
Thus, (\ref{yapasmieux}) is true, which completes the proof.
\end{proof}
The phenomenon of local exponential contractions given by Lemma \ref{yoplait} are measured in a ´´$\hat{\mu}$-sense''. It remains to justify that these contractions remain in the standard sense: that is the object of the next lemma, where we prove that $\mu_\omega$ can be replaced by other arbitrary measures.
\begin{lem}\label{yoplait2}
Let $\omega\mapsto \nu_\omega$ be any measurable function from $\Omega$ into the set of probability measures on $S^1$. Then, for $\hat{\mu}$-almost every $(\omega,x)$ in $\Omega\times S^1$, we have:
$$\varlimsup_{y\to x}\varlimsup_{n\to +\infty}\frac{\log(\nu_{T^n\omega}[f_\omega^n(x),f_\omega^n(y)])}{n}\leq -h(\hat{\mu}).$$
\end{lem}
\begin{proof}
The case where $\nu_\omega=\mu_\omega$ is a direct consequence of Lemma \ref{yoplait}, that is
\begin{equation}\label{asdfg}\varlimsup_{y\to x}\varlimsup_{n\to +\infty}\frac{\log(\mu_{T^n\omega}[f_\omega^n(x),f_\omega^n(y)])}{n}\leq -h(\hat{\mu}).\end{equation}
For the general case, let us set
$$Q^*(w,x)=\sup_{I\ni x} \frac{\nu_\omega(I)}{\mu_\omega(I)}.$$
By Proposition \ref{Besicovitch}, $\log^+ Q^* \in L^1(\hat{\mu})$, hence the Birkhoff's sums $\frac{1}{n}\sum_{k=0}^{n-1} \log^+ Q^*\circ \hat{T}^k$
converge $\hat{\mu}$-almost everywhere, hence in particular $\frac{\log^+ Q^*\circ \hat{T}^k}{n}$ tends to $0$ $\hat{\mu}$-almost everywhere, which implies:
\begin{equation}\label{sdfgh}\varlimsup_{y\to x}\varlimsup_{n\to +\infty}\frac{1}{n}\log\left(\frac{\nu_{T^n\omega}[f_\omega^n(x),f_\omega^n(y)]}{\mu_{T^n\omega}[f_\omega^n(x),f_\omega^n(y)]}\right)\leq \varlimsup_{y\to x}\varlimsup_{n\to +\infty}\frac{1}{n}\log^+ Q^*(\hat{T}^k(\omega,x))=0.\end{equation}
The statement is then a direct consequence of \eqref{asdfg} and \eqref{sdfgh}.
\end{proof}
Using Lemma \ref{yoplait2} with $\nu_\omega$ the Lebesgue measure, we obtain that $\lambda_{con}(\hat{\mu})\leq -h(\hat{\mu})$. That completes the proof of Proposition \ref{ineq-entropy}, and hence of Theorem \ref{invariance}
\subsection{Exponent of contraction in RDS}
We go back to the context of random walks on $\displaystylehrm{Homeo}(S^1)$. In this particular case, Theorem \ref{invariance} becomes:
\begin{cor}\label{invarianceiid}~
Let $(G,\nu)$ a random dynamical system on $S^1$, and let $\mu$ be a stationary probability of the system.
Then
\begin{itemize}
\item either $\lambda_{con}(\displaystylehbb{P}\times\mu)<0$ (contraction),
\item or $f_*\mu=\mu$ for $\nu$-almost every homeomorphism $f$ (and so for any $f$ in $G$) (invariance).
\end{itemize}
\end{cor}
Thus,we obtain information at typical points $x\in S^1$ for the stationary probability measures of the systems. But it is actually possible to deduce information at every point $x$ of the circle. To do this, we are going to use the following general fact of random dynamical systems:
\begin{prop}\label{cluster}
Let $(G,\nu)$ be a RDS on a compact metric space $(X,d)$, $(\Omega,\displaystylehbb{P})=(G^\displaystylehbb{N},\nu^\displaystylehbb{N})$ the associated probabilty space, and let $x_0$ be a point of $X$. Then, for
$\displaystylehbb{P}$-almost every $\omega$, the set $\Pi_{\omega,x_0}$ of weak-$*$ cluster values of the sequence of probability measures $\left(\frac{1}{N}\sum_{n=0}^{N-1}\delta_{f_{\omega}^n(x_0)}\right)_{N\in \displaystylehbb{N}}$ is constituted of stationary probability measures of the RDS.
\end{prop}
This proposition is the analogue of the standard Krylov-Bogolyubov Theorem for RDS. The proof can be found in \cite{Deroin-cours} (French), or it can be seen as a consequence of Lemma 2.5 of \cite{Furstenberg}. We are going to use it to extract punctual informations on $\lambda_{con}$ from the informations on stationary measures:
\begin{prop}\label{contractuel}
Let $\omega\mapsto (f_\omega^n)_{n\in\displaystylehbb{N}}$ a random walk on $\displaystylehrm{Homeo}(S^1)$ and let $x_0$ in $S^1$. Then for
$\displaystylehbb{P}$-almost every $\omega$ we have $$\lambda_{con}(\omega,x_0)\leq \inf_{\mu\in \Pi_{\omega,x_0}}\lambda_{con}(\displaystylehbb{P}\otimes \mu),$$
where $\Pi_{\omega,x_0}$ is defined as in Proposition \ref{cluster}.
\end{prop}
The proof of Proposition \ref{contractuel} begins by noticing two elementary facts on the function $(\omega,x)\mapsto\lambda_{con}(\omega,x)$.
\begin{lem}\label{elem}
The function $\lambda_{con}$ is $\hat{T}$-invariant ($\lambda_{con}\circ \hat{T}=\lambda_{con}$), and for any $\omega$ in $\Omega$, the function $x\mapsto \lambda_{con}(\omega,x)$ is upper semicontinuous.
\end{lem}
\begin{proof}
The invariance property $\lambda_{con}\circ \hat{T}= \lambda_{con}$ comes from the fact that an interval $I$ containing $f_0(x)$ is contracted by the sequence $(f_{T\omega}^n)$ if and only if $f_0^{-1}(I)$ is an interval containing $x$ contracted by $(f_\omega^n)$.
The upper semicontinuity of $\lambda_{con}$ comes from the fact that if $\lambda_{con}(\omega,x)<c$, then there exists an interval $I$ containing $x$ such that $\displaystylehrm{diam}(f_\omega^n(I))=O(e^{-nc})$ and hence $\lambda_{con}(\omega,\cdot)<c$ on $I$.
\end{proof}
Then, Proposition \ref{contractuel} is actually a direct consequence of a much more general fact of random dynamical systems:
\begin{lem}\label{randomfunction}
Let $(G,\nu)$ be a RDS on a compact space $X$ and let $(\Omega,\displaystylehbb{P})$ and $\hat{T}$ be defined as in Definition \ref{randomwalk}. Let $\varphi:\Omega\times X\mapsto \displaystylehbb{R}$ be a measurable positive function such that:
\begin{itemize}
\item for every $\omega\in \Omega$, $x\mapsto \varphi(\omega,x)$ is lower semicontinuous,
\item $\varphi\circ \hat{T}\leq \varphi$ on $\Omega\times X$.
\end{itemize}
Finally, let $x_0$ be a point of $X$ and for $\omega$ in $\Omega=G^\displaystylehbb{N}$, let $\Pi_{\omega,x_0}$ be defined as in Proposition \ref{cluster}. Then, for $\displaystylehbb{P}$-almost every $\omega$ in $\Omega$,
$$\varphi(\omega,x_0)\geq \sup_{\mu \in \Pi_{\omega,x_0}} \int_{\Omega\times X} \varphi d(\displaystylehbb{P}\otimes \mu).$$
\end{lem}
\begin{proof}
Let $\displaystylehcal{F}_n$ the $\sigma$-algebra generated by the $n$ first canonical projections $p_k:\omega=(f_j)_{j\geq 0}\mapsto f_k$, and set
$$\bar{\varphi}(x)=\displaystylehbb{E}[\varphi(\cdot,x)],$$
$$\Lambda_n=\displaystylehbb{E}[\varphi(\cdot,x_0) | \displaystylehcal{F}_n]$$
Levy's zero-one law says that $\Lambda_n\xrightarrow[n\to +\infty]{} \varphi(\cdot,x_0)$ almost surely. On the other hand, from the inequality
$$\varphi(\omega,x_0)\geq \varphi\circ \hat{T}^n(\omega,x_0)=\varphi(T^n\omega, f_\omega^n(x_0)),$$
we deduce by taking the conditional expectation with respect to $\displaystylehcal{F}_n$ that for $\displaystylehbb{P}$-almost every $\omega$,
$$\Lambda_n(\omega)\geq\bar{\varphi}(f_\omega^n(x_0)).$$
Hence, using the Cesaro theorem, for $\displaystylehbb{P}$-almost every $\omega$,
\begin{equation}\label{lambda}\varphi(\omega,x_0)=\lim_{N\to +\infty} \frac{1}{N}\sum_{n=0}^{N-1}\Lambda_n(\omega)\geq \varlimsup_{N\to +\infty} \frac{1}{N}\sum_{n=0}^{N-1}\bar{\varphi}(f_\omega^n(x_0))\end{equation}
Now, we know that $\bar{\varphi}$ is lower semicontinuous thanks to the lower semicontinuity of $\varphi(\omega,\cdot)$ and Fatou's lemma: indeed, for any $x$ in $X$,
\begin{equation}\label{fatou}\varliminf_{y\to x} \bar{\varphi}(y)=\varliminf_{y\to x} \displaystylehbb{E}[\varphi(\cdot,y)]\geq
\displaystylehbb{E}[\varliminf_{y\to x} \varphi(\cdot,y)]\geq \bar{\varphi}(x).\end{equation}
As a consequence, we can write:
$$\bar{\varphi}=\inf\{\psi:X\rightarrow \displaystylehbb{R} \mbox{ continuous }| \psi\leq \bar{\varphi}\},$$
and for every such continuous function $\psi\leq \bar{\varphi}$, we have by (\ref{lambda}) and definition of $\Pi_{\omega,x_0}$:
$$\varphi(\omega,x_0)\geq\varlimsup_{N\to +\infty} \frac{1}{N}\sum_{n=0}^{N-1}\psi(f_\omega^n(x_0))=\sup_{\mu\in \Pi_{\omega,x_0}}\int_X \psi d\mu$$
Since $\psi$ is arbitrary, we deduce that
$$\varphi(\omega,x_0)\geq \sup_{\mu\in \Pi_{\omega,x_0}}\int_X \bar{\varphi}\, d\mu=\sup_{\mu\in \Pi_{\omega,x_0}}\int_{\Omega\times X}\varphi\, d\displaystylehbb{P}d\mu.$$
\end{proof}
Let us conclude by deducing the following corollary, which is only a reformulation of Theorem \ref{main} in terms of exponent of contraction:
\begin{cor}
Let $\omega\mapsto (f_\omega^n)_{n\in\displaystylehbb{N}}$ a random walk generated by a probability measure $\nu$ on $\displaystylehrm{Homeo}(S^1)$, and let us assume that there is no probability measure on $S^1$ invariant by $\nu$-almost every homeomorphism. Then there exists $\lambda_0<0$ such that for any $x$ in $S^1$, for $\displaystylehbb{P}$-almost every $\omega$ in $\Omega$,
$$\lambda_{con}(\omega,x)\leq \lambda_0$$
\end{cor}
\begin{proof}
By Corollary \ref{invarianceiid}, $\lambda_{con}(\displaystylehbb{P}\otimes \mu)<0$ for any stationary probability measure $\mu$, hence Proposition \ref{contractuel} applied to $-\lambda_{con}$ immediately implies that for any $x$ in $S^1$, $\omega\mapsto \lambda_{con}(\omega,x)$ is negative $\displaystylehbb{P}$-almost everywhere. To obtain a uniform negative bound, let us notice that this negativity implies the negativity of $\overline{\lambda}_{con}(x)=\int_{\Omega}\lambda_{con}(\omega,x)d\displaystylehbb{P}(\omega)$. Thus, $x\mapsto \overline{\lambda}_{con}(x)$ is pointwise negative, and is also upper-semicontinuous by Fatou's lemma as in the computation (\ref{fatou}), hence is uniformly bounded from above by some negative number $\lambda_0$. Then, using Proposition \ref{contractuel} one more time, we obtain that for any $x$ in $S^1$ and $\displaystylehbb{P}$-almost every $\omega$:
$$\lambda_{con}(\omega,x)\leq \inf_{\mu\in \Pi_{\omega,x}}\lambda_{con}(\displaystylehbb{P}\otimes \mu)=\inf_{\mu\in \Pi_{\omega,x}}\int_{S^1}\overline{\lambda}_{con}d\mu\leq \lambda_0.$$
That achieves the proof of the corollary, and hence of Theorem \ref{main}.
\end{proof}
\section{Locally contracting random dynamical systems}\label{sec-consequences}
In this section, we are going to study the properties of a general random walk on a compact metric space. This section can be read independantly of the remainder of the paper, except that we are going to use Lemma \ref{randomfunction} proved in the previous section, and that we will use the notations given in Definition \ref{randomwalk}. Thus, throughout the whole section:
\begin{itemize}
\item $(G,\nu)$ is a random dynamical system on a compact metric space $(X,d)$, that is to say that $G$ a semigroup of continuous transformations of $X$ and $\nu$ a probability measure on $G$.
\item $(\Omega,\displaystylehbb{P})=(G^\displaystylehbb{N},\nu^\displaystylehbb{N})$ is the associated probability space, $\omega\mapsto (f_\omega^n)$ the associated random walk, defined by
$$f_\omega^n=f_{n-1}\circ\cdots\circ f_0$$
(with the implicit notation $\omega=(f_n)_{n\in\displaystylehbb{N}}$), and $\hat{T}:(\omega,x)\mapsto (T\omega,f_0(x))$ the associated skew-shift on $\Omega\times X$.
\end{itemize}
We are going to study the properties of such RDS satisfying the \textit{property of local contractions}:
\begin{ass}\label{contracting}
For every $x$ in $X$, for $\displaystylehbb{P}$-almost every $\omega$ in $\Omega$, there exists a neighbourhood $B$ of $x$ such that
$$\displaystylehrm{diam}(f_\omega^n(B))\xrightarrow[n\to+\infty]{} 0.$$
\end{ass}
\begin{rem}
By Theorem \ref{main}, Assumption \ref{contracting} is satisfied when $G$ is a subgroup of $\displaystylehrm{Homeo}_+(S^1)$ without invariant probability measure. It is also satisfied if $X$ is a manifold, $G$ a semigroup of diffeomorphisms of $X$ and such that all the Lyapunov exponents of the random walk are negative.
\end{rem}
\subsection{Preliminaries on random sets}
In this part, we state some general results on the RDS, concerning the structure of the sets invariant by $\hat{T}$. We do not use Assumption \ref{contracting} in this part.
\begin{prop}\label{randomset}
Let $\displaystylehcal{E}=\cup_{\omega\in \Omega}\{\omega\}\times U(\omega)$ a subset of $\Omega\times X$ backward-invariant by $\hat{T}$ (i.e.~$\hat{T}^{-1}(\displaystylehcal{E})\subset \displaystylehcal{E}$) such that $U(\omega)$ is open in $X$ for every $\omega$ in $\Omega$. Let us assume that $$(\displaystylehbb{P}\otimes \mu)(\displaystylehcal{E})>0$$
for every stationary ergodic probability measure $\mu$. Then actually,
$$(\displaystylehbb{P}\otimes \mu)(\displaystylehcal{E})=1$$
for every probability measure $\mu$ on $X$ (not necessarily stationary).
\end{prop}
\begin{proof}
Firstly, the set of the stationary probability measures is the convex hull of the the set of the ergodic ones, so that the inequality $(\displaystylehbb{P}\otimes \mu)(\displaystylehcal{E})>0$ remains valid for any $\mu$ stationary. Then, by applying Lemma \ref{randomfunction} to $\varphi=\displaystylehds{1}_{\displaystylehcal{E}}$, for any $x_{0}$ in $X$ and for almost every $\omega$ in $\Omega$, we have with the notations of the lemma:
$$\displaystylehds{1}_{\displaystylehcal{E}}(\omega,x_0)\geq \sup_{\mu \in \Pi_{\omega,x_0}} (\displaystylehbb{P}\times \mu)(\displaystylehcal{E})>0,$$
hence $(\omega,x_0)\in \displaystylehcal{E}$. The result follows.
\end{proof}
The second proposition shows that the fibres of a $\hat{T}$-invariant set cannot have many connected components (that will be the main ingredient for the proof of Theorem \ref{Lejan-Antonov}).
\begin{prop}\label{randomcon}
Let $\displaystylehcal{E}=\cup_{\omega\in \Omega}\{\omega\}\times E(\omega)$ a subset of $\Omega\times X$ totally invariant by $\hat{T}$ ($\hat{T}^{-1}(\displaystylehcal{E})=\displaystylehcal{E}$). Then, for every stationary ergodic probability measure $\mu$, for $\displaystylehbb{P}$-almost every $\omega$ in $\Omega$, $E(\omega)$ has only a constant finite number $d$ of connected components of $\mu$-measure positive, and all of them have same measure $\frac{1}{d}$.
\end{prop}
\begin{proof}
In order to prove this proposition, we extend (canonically) the skew-shift $\hat{T}$ on $G^\displaystylehbb{Z}\times X$, in an invertible context, allowing to look at the ''past'' of the RDS. This procedure is standard, we resume in the following lemma the properties of the extension we use (we refer to \cite{Lejan} for the details).
\begin{lem}
Let $\tilde{\Omega}=G^\displaystylehbb{Z}$ and $\tilde{\displaystylehbb{P}}=\nu^{\displaystylehbb{Z}}$ . The transformation $\hat{T}:(\omega,x)\mapsto (T\omega, f_0(x))$ admits an invariant ergodic probability measure $\hat{\mu}$ on $\tilde{\Omega}\times X$ of the form $d{\hat{\mu}}=d\mu_\omega(x)d\displaystylehbb{P}(\omega)$, with:
\begin{itemize}
\item the function $\omega\mapsto \mu_\omega$ depending only on the negative coordinates of $\omega$,
\item $\int_{\tilde{\Omega}} \mu_\omega d\tilde{\displaystylehbb{P}}(\omega)=\mu$,
\item for $\tilde{\displaystylehbb{P}}$-almost every $\omega$ in $\tilde{\Omega}$, $\mu_{T\omega}=(f_0)_*\mu_\omega$.
\end{itemize}
\end{lem}
This process will allow us to prove the following general lemma:
\begin{lem}\label{ergodic}
Let $(E(\omega,x))_{(\omega,x)\in\Omega\times X}$ be a family of Borelian subsets of $X$ such that
$$\forall (\omega,x)\in \Omega\times X,\quad E(\hat{T}(\omega,x))=f_0(E(\omega,x)).$$
Then the function $(\omega,x)\mapsto \mu(E(\omega,x))$ is constant $(\displaystylehbb{P}\otimes \mu)$-almost everywhere.
\end{lem}
\begin{proof}
Let us extend canonically $(\omega,x)\mapsto E(\omega,x)$ to $\tilde{\Omega}\times X$ (by setting $E((f_k)_{k\in\displaystylehbb{Z}},x):=E((f_k)_{k\in\displaystylehbb{N}},x)$). For every $(\omega,x)\in \tilde{\Omega}\times X$, $E(\omega,x)=(f_0)^{-1}(E(\hat{T}(\omega,x)))$, hence
$$\mu_\omega(E(\omega,x))=(f_0)_*\mu_\omega (E(\hat{T}(\omega,x)))=\mu_{T\omega} (E(\hat{T}(\omega,x))).$$
The function $(\omega,x)\mapsto \mu_\omega(E(\omega,x))$ is hence $\hat{T}$-invariant on $\tilde{\Omega}\times X$.
By ergodicity of $\hat{\mu}$, there exists a constant $c$ such that for $\hat{\mu}$-almost every $(\omega,x)$ in $\tilde{\Omega}\times X$, $\mu_\omega(E(\omega,x))=c$. Since $\mu_{\omega}$ only depends on the negative coordinates of $\omega$ and $E(\omega,x)$ only depends on the non negative coordinates of $\omega$, we deduce by integration of this equality over the negative coordinates of $\omega$ that for $(\displaystylehbb{P}\otimes \mu)$-almost every $(\omega,x)$ in $\Omega\times X$, $\mu(E(\omega,x))=c$.
\end{proof}
Proposition \ref{randomcon} follows by choosing $E(\omega,x)$ to be the connected component of $x$ in $E(\omega)$ (with the convention $E(\omega,x)=\emptyset$ if $x\notin E(\omega)$), satisfying the relation $E(\hat{T}(\omega,x))=f_0(E(\omega,x))$. For any ergodic probability measure $\mu$ of the RDS, by Lemma \ref{ergodic}, for $\displaystylehbb{P}$-almost every $\omega$, the function $x\mapsto \mu(E(\omega,x))$ is equal to some positive constant $c$ $\mu$-almost everywhere, which means that all the connected components of $U(\omega)$ which are not $\mu$-negligible have the same $\mu$-measure $c$. In particular there is only a finite number of them, namely $\frac{1}{c}$.
\end{proof}
\subsection{Stationary trajectories}
We prove in this part that the property of local contractions implies that the number of ergodic stationary probability measures is finite, and the trajectory of every point almost surely distributes with respect of one of them.\\
\begin{Def}\label{contracdef}
We say that a ball $B$ is contractible if there exists a set $\Omega '\subset \Omega$ of $\displaystylehbb{P}$-positive probability such that, for $\omega$ in $\Omega '$, $\displaystylehrm{diam}(f_\omega^n(B))\xrightarrow[n\to +\infty]{}0$.
\end{Def}
Assumption \ref{contracting} implies that every point contains a contractible neighbourhood.
\begin{lem}\label{contractible}
If $B\subset X$ is a contractible ball, then there exists at most one ergodic stationary probability measure $\mu$ such that $\mu(B)>0$.
\end{lem}
\begin{proof}
Let $\mu_1$ and $\mu_2$ be two ergodic stationary measures such that $\mu_1(B)\not=0$ and $\mu_2(B)\not=0$. By Birkhoff's theorem one can find $x$ and $y$ in $B$ such that for $\displaystylehbb{P}$-almost every $\omega$ in $\Omega$ , for every continuous $\varphi:X\rightarrow \displaystylehbb{R}$,
\begin{equation}\label{limit123}\begin{disarray}{l}\frac{1}{N}\sum_{n=0}^{N-1}\varphi(f_\omega^n(x))\xrightarrow[N\to+\infty]~\int_{X}\varphi d\mu_1\\
\frac{1}{N}\sum_{n=0}^{N-1}\varphi(f_\omega^n(y))\xrightarrow[N\to+\infty]~\int_{X}\varphi d\mu_2.\end{disarray}\end{equation}
Since $B$ is contractible, one can choose such an $\omega$ for which $\displaystylehrm{diam}(f_\omega^n(B))$ tends to $0$ as $n$ tends to $+\infty$. Then, for every continuous mapping $\varphi:X\rightarrow \displaystylehbb{R}$, $\varphi(f_\omega^n(x))-\varphi(f_\omega^n(y))$ tends to $0$ as $n$ tends to $+\infty$, hence we conclude from (\ref{limit123}) that $$\int_{X}\varphi d \mu_1=\int_{X}\varphi d \mu_2,$$
so that $\mu_1=\mu_2$.
\end{proof}
\begin{prop}\label{finite}
If the RDS $(G,\nu)$ satisfies Assumption \ref{contracting}, then it has a finite number $d$ of ergodic stationary probability measures $\{\mu_1,\ldots,\mu_d\}$. Their respective topological supports $F_1,\ldots,F_d$ are pairwise disjoints, and are exactly the minimal invariant compacts of $G$.
\end{prop}
\begin{proof}
Each point of $x$ is the centre of a contractible ball, hence by compactness, we can cover $X$ by a finite number of contractible balls $B_1,\ldots,B_d$. By Lemma \ref{contractible}, for each $i$, there is at most one ergodic probability measure $\mu$ such that $\mu(B_i)\not=0$. Hence, there are at most $d$ stationary ergodic probability measures.\\
Let $\{\mu_1,\ldots,\mu_d\}$ be the set of the ergodic probability measures and let $F_i=\displaystylehrm{supp}(\mu_i)$ be the topological support of $\mu_i$. If $x\in F_i\cap F_j$, then if $B$ a contractible ball centred at $x$, we have $\mu_i(B)\not=0$ and $\mu_j(B)\not=0$, hence by Lemma \ref{contractible}, $\mu_i=\mu_j$. The sets $F_1,\ldots,F_d$ are hence pairwise disjoint.\\
If $F$ is a minimal closed invariant subset of $X$, then there exists a stationary ergodic probability measure $\mu_i$ supported in $F$. And since $F_i=\displaystylehrm{supp}(\mu_i)$ is invariant by $G$, we have $F=F_i$ by minimality of $F$.
Conversely, let $i$ be in $\{1,\ldots,d\}$. The closed set $F_i=\displaystylehrm{supp}(\mu_i)$ is invariant by $G$, hence it contains a minimal invariant closed subset $F$. By the previous point, $F=F_j$ for some $j$, but since the $F_1,\ldots,F_d$ are pairwise disjoint, necessarily $i=j$ and hence $F_i=F$ is a minimal invariant subset.\\
\end{proof}
\begin{prop}\label{adhe}
Let us assume that the RDS $(G,\nu)$ satisfies Assumption \ref{contracting} and let $\mu_1,\ldots,\mu_d$ and $F_1,\ldots,F_d$ be as in Proposition \ref{finite}. Then for every $x$ in $X$, for $\displaystylehbb{P}$-almost every $\omega$ in $\Omega$, there exists a (unique) integer $i=i(\omega,x)$ in $\{1,\ldots,d\}$ such that:
\begin{itemize}
\item The set of cluster values of the sequence $(f_\omega^n(x))_{n\geq 0}$ is exactly $F_i$.
\item The sequence of probability measures $\frac{1}{N}\sum_{n=0}^{N-1}\delta_{f_\omega^n(x)}$ weakly-$*$ converges to $\mu_i$ in $C(X,\displaystylehbb{R})^*$.
\end{itemize}
\end{prop}
\begin{proof}
Let us consider $\displaystylehcal{E}_0$ to be the set of the points $(\omega,x)$ such that there exists a neighbourhood of $x$ contracted by $(f_\omega^n)_n$, and
let
$$\displaystylehcal{E}_i=\left\{(\omega,x)\in \displaystylehcal{E}_0 \left| \frac{1}{N}\sum_{n=0}^{N-1}\delta_{f_\omega^n(x)}\xrightarrow[n\to+\infty]{C(X,\displaystylehbb{R})^*} \mu_i \right.\right\}=\bigcup_{\omega\in\Omega}\{\omega\}\times U_i(\omega)$$
and
$$\tilde{\displaystylehcal{E}}_i=\left\{(\omega,x)\in \displaystylehcal{E}_0 \left| \mbox{Acc}\left\{f_\omega^n(x),n\in\displaystylehbb{N}\right\}=F_i \right.\right\}=\bigcup_{\omega\in\Omega}\{\omega\}\times \tilde{U}_i(\omega).$$
Then:
\begin{itemize}
\item $\displaystylehcal{E}_i$ and $\tilde{\displaystylehcal{E}}_i$ are totally invariant by $\hat{T}$.
\item If $\omega$ belongs to $\omega$ and $B$ is a ball such that $\mbox{diam}(f_\omega^n(B))$ tends to $0$ when $n$ tends to $+\infty$, then either $B\cap U_i(\omega)=\emptyset$ (resp $B\cap \tilde{U}_i(\omega)=\emptyset$) or $B\subset U_i(\omega)$ (resp $B\subset \tilde{U}_i(\omega)$). In consequence, $U_i(\omega)$ and $\tilde{U}_i(\omega)$ are open.
\item By Birkhoff Theorem, for $(\displaystylehbb{P}\otimes \mu_i$)-almost every $(\omega,x)$ in $\Omega\times X$,
$$\frac{1}{N}\sum_{n=0}^{N-1}\delta_{f_\omega^n(x)}\xrightarrow[n\to+\infty]{C(X,\displaystylehbb{R})^*} \mu_i.$$
In particular, $\mbox{Acc}\left\{f_\omega^n(x),n\in\displaystylehbb{N}\right\}\supset F_i$, and if $x$ belongs to $F_i$, $\mbox{Acc}\left\{f_\omega^n(x),n\in\displaystylehbb{N}\right\}\subset F_i$ by invariance of $F_i$. Thus, $(\displaystylehbb{P}\otimes \mu_i)(\displaystylehcal{E}_i)=(\displaystylehbb{P}\otimes \mu_i)(\tilde{\displaystylehcal{E}}_i)=1$.
\end{itemize}
In consequence, one can apply Proposition \ref{randomset} to the sets $\displaystylehcal{E}=\cup_i \displaystylehcal{E}_i$ and $\tilde{\displaystylehcal{E}}=\cup_i \tilde{\displaystylehcal{E}}_i$ and get:
$$\forall x\in X, \quad (\displaystylehbb{P}\times \delta_x)(\displaystylehcal{E})=(\displaystylehbb{P}\times \delta_x)(\tilde{\displaystylehcal{E}})=1.$$
The claimed result follows.
\end{proof}
\subsection{Dynamics of the transfer operator}
We study in this part the sequence of the iterates of the transfer operator $P$ of a RDS applied to a continuous test function $\varphi$. We prove that under the property of local contractions, this sequence $(P^n\varphi)_{n\in\displaystylehbb{N}}$ always converges uniformly in the Cesaro sense to a harmonic function, and that it actually converges uniformly in the standard sense if the RDS is aperiodic (in the sense of Definition \ref{indeco}).\\
The transfer operator $P$ of the system is defined on measurable bounded functions $\varphi:X\rightarrow \displaystylehbb{R}$, by
$$P\varphi=\int_{G} \varphi\circ f d\nu(f).$$
The iterates of $P$ are given by
$$P^n\varphi=\int_\Omega\varphi\circ f_\omega^n\,d\displaystylehbb{P}(\omega),$$
so that the dynamics of $P$ represents the evolution of the law of the random variables $\omega\mapsto f_\omega^n(x)$.
\begin{lem}\label{equicontinuous}
If the RDS $(G,\nu)$ satisfies Assumption \ref{contracting}, then for every continuous $\varphi:X\rightarrow \displaystylehbb{R}$, the family $(P^n\varphi)_{n\in\displaystylehbb{N}}$ is equicontinuous on $X$.
\end{lem}
\begin{proof}
Let $\varepsilon>0$, and let $\delta>0$ be such that
$$\forall x,y \in X^2, d(x,y)\leq\delta\displaystylehbb{R}ightarrow |\varphi(x)-\varphi(y)|\leq \varepsilon.$$
Let $x$ be in $X$. Thanks to Assumption \ref{contracting}, we can find a ball $B$ centred at $x$ and a subset $\Omega '\subset \Omega$ of probability more than $1-\varepsilon$ such that:
$$ \forall n\in \displaystylehbb{N}, \forall \omega\in \Omega',\quad \displaystylehrm{diam}( f_\omega^n(B))\leq \delta.$$
We deduce that for every integer $n$ and every $y$ in $B$:
$$\begin{disarray}{ll}\displaystyle|P^n\varphi(x)-P^n\varphi(y)|&\leq\int_{\Omega}|\varphi(f_\omega^n(x))-\varphi(f_\omega^n(y))|d\displaystylehbb{P}(\omega)
\\&\leq \varepsilon\displaystylehbb{P}(\Omega')+2\|\varphi\|_{\infty}\displaystylehbb{P}(\Omega-\Omega')\\
&\leq (1+2\|\varphi\|_\infty)\varepsilon.\end{disarray}
$$
Thus, $(P^n\varphi)_{n\in\displaystylehbb{N}}$ is equicontinuous at $x$. Since $x$ is arbitrary and $X$ is compact, $(P^n\varphi)_{n\in\displaystylehbb{N}}$ is equicontinuous on $X$.
\end{proof}
\begin{prop}\label{Furstenberg}
We assume that the RDS $(G,\nu)$ satisfies Assumption \ref{contracting}, and we keep the notations of Proposition \ref{finite}, i.e. $\mu_1,\ldots,\mu_d$ and $F_1,\ldots,F_d$ are respectively the ergodic stationary probability measures of the RDS and their topological supports. Then:
\begin{itemize}
\item The vector space $E_0=\{\varphi\in C(X,\displaystylehbb{R})\mid P\varphi=\varphi\}$ of the harmonic continuous functions of the RDS has finite dimension $d$, and one can find a basis $(u_1,\ldots,u_d)$ of $E_0$ such that $u_i$ is valued in $[0,1]$, $u_i=\delta_{i,j}$ on $F_j$ and $\sum_i u_i=1$ on $X$.
\item For every continuous $\varphi:X\rightarrow \displaystylehbb{R}$, we have
$$ \frac{1}{N}\sum_{n=0}^{N-1} P^n\varphi\xrightarrow[N\to +\infty]{\|\cdot\|_\infty} \psi$$
where $\psi$ is the element of $E_0$ given by
$$\psi(x)=\sum_{i=1}^d \left(\int_X\varphi d\mu_i\right)u_i(x).$$
\end{itemize}
\end{prop}
\begin{proof}
Let $\varphi:X\rightarrow \displaystylehbb{R}$ be a continuous function, and let $x$ be in $X$. With $i(\omega,x)$ defined as in Proposition \ref{adhe}, we have for $\displaystylehbb{P}$-almost every $\omega$ in $\Omega$:
$$\frac{1}{N}\sum_{n=0}^{N-1}\varphi(f_\omega^n(x))\xrightarrow[n\to+\infty]{} \int_X \varphi d\mu_{i(\omega,x)}.$$
Integrating in $\omega$, we deduce by dominated convergence that
\begin{equation}
\label{mkjh}
\frac{1}{N}\sum_{n=0}^{N-1}P^n\varphi(x)\xrightarrow[n\to+\infty]{} \sum_{i=1}^d u_i(x)\int_X \varphi d\mu_{i},\end{equation}
where $u_i(x)=\displaystylehbb{P}(\omega\in \Omega \mid i(\omega,x)=i)$. Since the sequence $\left(\frac{1}{N}\sum_{n=0}^{N-1}P^n\varphi\right)_{n\in\displaystylehbb{N}}$ is equicontinuous by Lemma \ref{equicontinuous}, the convergence (\ref{mkjh}) is in fact uniform in $x$.
The only non trivial property to prove on the functions $u_i$ is their continuity. For a given $i$ , we choose $\varphi$ continuous such that $\varphi=\delta_{i,j}$ on $K_j$, so that (\ref{mkjh}) becomes
$$u_i=\lim_{N\to +\infty}\frac{1}{N}\sum_{n=0}^{N-1}P^n\varphi$$
where the limit is uniform. The continuity of $u_i$ follows.
\end{proof}
We will strenghten the result in the case of aperiodic systems. Let us recall the definition of aperiodicity given in Section \ref{results} in the case of the circle:
\begin{Def}\label{aper}
The RDS $(G,\nu)$ on $X$ (resp. the random walk $\omega\mapsto (f_\omega^n)_{n\in\displaystylehbb{N}}$) is said to be aperiodic if there does not exist a finite number $p\geq 2$ of pairwise disjoints closed subsets $F_1,\ldots,F_p$ of $X$ such that for $\nu$-almost every homeomorphism $g$, $g(F_i)\subset F_{i+1}$ for $i=1,\ldots,p-1$ and $g(F_p)\subset F_1$.
\end{Def}
\begin{rem}
As already noticed in the particular case of the circle in Section \ref{results}, if a random walk $\omega\mapsto (f_\omega^n)_{n\in\displaystylehbb{N}}$ acts minimally on $X$ and if $X$ is connected, then it is automatically aperiodic.
\end{rem}
We can state our result, which studies the convergence of th sequence $(P^n)_{n\in\displaystylehbb{N}}$:
\begin{prop}\label{transfert}
We assume that the RDS $(G,\nu)$ satisfies Assumption \ref{contracting}, and we assume also that it is aperiodic. Then, keeping the notations of Proposition \ref{Furstenberg}, we have actually
$$ P^n\varphi\xrightarrow[n\to +\infty]{\|\cdot\|_\infty} \psi=\sum_{i=1}^d \left(\int_X\varphi d\mu_i\right)u_i$$
\end{prop}
The aperiodicity of the system is used to obtain the following fact, whose proof is postpone:
\begin{lem}\label{indemin}
If $\omega\mapsto (f_\omega^n)_{n\geq 0}$ acts minimally on $X$ and is aperiodic, then for any positive integer $p$, $\omega\mapsto (f_\omega^{pn})_{n\geq 0}$ also acts minimally on $X$.
\end{lem}
\begin{proof}[Proof of Proposition \ref{transfert}]
Let $\varphi:X\rightarrow \displaystylehbb{R}$ be a continuous mapping. Thanks to Lemma \ref{equicontinuous}, the only thing we need to prove is that $(P^n\varphi)_{n\in\displaystylehbb{N}}$ has only one cluster value in $C(X,\displaystylehbb{R})$, namely $\sum_i \left(\int \varphi d\mu_i\right)u_i$. Thus, let $\psi=\lim_{k\to +\infty} P^{n_k}\varphi$ be a cluster value of $(P^n\varphi)_n$.\\
Firstly, up to to extracting the candidate limit $\sum_i \left(\int \varphi d\mu_i\right)u_i$ to $\varphi$, we can assume that $\int_{S^1}\varphi d\mu_i=0$ for $i=1,\ldots d$, so that we want to prove that $\psi=0$.
Secondly, we can reduce the problem to the case where $\varphi=\psi$: indeed, up to extracting a subsequence, we can assume that $m_k=n_{k+1}-n_k$ tends to $+\infty$ when $k$ tends to $+\infty$. Using that $P$ is contracting for $\|\cdot\|_\infty$, we have
\begin{equation}
\label{constant2}
\|P^{m_k}\psi-\psi\|_\infty\leq \|P^{m_k}(\psi-P^{n_{k}}\varphi)\|_\infty
+\|P^{n_{k+1}}\varphi-\psi\|_\infty\xrightarrow[k\to+\infty]{}0 \end{equation}\\
Thus, from now on we assume that:
\begin{itemize}
\item $\displaystyle \int_{S^1}\varphi d\mu_i=0 \mbox{ for } i=1,\ldots d,$
\item $\displaystyle P^{n_k}\varphi\xrightarrow[k\to +\infty]{\|\cdot\|_\infty} \varphi,$
\end{itemize}
and we want to prove that $\varphi=0$.
We begin by treating the restriction of the problem to a minimal subset $F_i=\displaystylehrm{supp}(\mu_i)$. We will use the following remark:
\begin{lem}
For any continuous $\varphi:X\rightarrow \displaystylehbb{R}$ and any positive integer $k$, we have $\|P^k\varphi\|_{L^2(\mu_i)}\leq \|\varphi\|_{L^2(\mu_i)}$, with equality if and only if for $\displaystylehbb{P}$-almost every $\omega$, $\omega'$, $\varphi\circ f_\omega^k=\varphi\circ f_{\omega'}^k$ on $F_i$.
\end{lem}
\begin{proof}The inequality is just a consequence of the Jensen inequality $P^k(\varphi)^2\leq P^k(\varphi^2)$ and of the $P^k$-invariance of $\mu$, and in the equality case of the Jensen inequality,we have that for almost every $\omega$, $\omega'$, $\varphi\circ f_\omega^k=\varphi\circ f_{\omega'}^k$ $\mu_i$-almost everywhere, hence on $F_i$ by continuity.\end{proof}
By the lemma, the sequence $(\|P^n\varphi\|_{L^2(\mu_i)})$ is non increasing. For any integer $p$, writing that
$$\|P^{n_k}\varphi\|_{L^2(\mu_i)}\leq \|P^{n_k+p}\varphi\|_{L^2(\mu_i)}\leq \|P^{n_{k+p}}\varphi\|_{L^2(\mu_i)},$$
and passing to the limit, we obtain that $\|P^p\varphi\|_{L^2(\mu_i)}=\|\varphi\|_{L^2(\mu_i)}$,
and hence by the lemma, for $\displaystylehbb{P}$-almost every $\omega$, $\varphi\circ f_\omega^p=P^p\varphi$ on $F_i$.
As a consequence, we obtain that for $\displaystylehbb{P}$-almost every $\omega$ in $\Omega$
$$\varphi\circ f_\omega^{n_k}=P^{n_k}\varphi\xrightarrow[k\to +\infty]{\|\cdot\|_\infty} \varphi \text{ on } F_i$$
In particular, if $B$ is a contractible ball of $F_i$, then $\varphi$ is constant on $B$. By compactness, $\varphi$ only takes a finite number of values on $F_i$. We deduce that fixing an integer $p=n_k$ with $k$ large enough, we have $\varphi\circ f_\omega^p=\varphi$ on $F_i$ for $\displaystylehbb{P}$-almost every $\omega$. Hence $\varphi$ is constant on $F_i$ by Lemma \ref{indemin}, and this constant is necessarily $\int \varphi d\mu_i=0$.
We now go back to the whole space: we know that $\varphi$ is identically zero on each $F_i$. And for any $x$ in $X$, for almost every $\omega$, all the cluster values of $(f_\omega^n(x))$ belong to a minimal set $F_i$ (Proposition \ref{adhe}), hence $\varphi(f_\omega^n(x))\to 0$, hence by integration over $\omega$, $P^n\varphi(x)\to 0$, and in particular,
$$\varphi(x)=\lim_k P^{n_k}\varphi(x)=0.$$
Thus $\varphi$ is identically zero on $X$.
\end{proof}
\begin{proof}[Proof of Lemma \ref{indemin}]
If $F$ is a closed subset of $X$, let us set
$$\displaystylehbb{T}heta(F)=\overline{\bigcup_{f\in \displaystylehrm{supp}(\nu)} f(F)}.$$
We want to prove that if $F$ is a non empty closed subset such that $\displaystylehbb{T}heta^p(F)\subset F$ then $F=X$. Set
$$\displaystylehcal{F}=\{F\subset X \mbox{ closed },F\not=\emptyset, \displaystylehbb{T}heta^p(F)\subset F\},$$
and let $F$ be an element of $\displaystylehcal{F}$ which is minimal with respect to the inclusion. Then:
\begin{itemize}
\item for any integer $k$, $\displaystylehbb{T}heta^k(F)\in \displaystylehcal{F}$ (obvious);
\item $\displaystylehbb{T}heta^p(F)=F$ by minimality of $F$, since $\displaystylehbb{T}heta^p(F)\in\displaystylehcal{F}$ and $\displaystylehbb{T}heta^p(F)\subset F$;
\item for any integer $k$, $\displaystylehbb{T}heta^k(F)$ is minimal with respect to the inclusion in $\displaystylehcal{F}$: indeed, if $G\in \displaystylehcal{F}$ and $G\subset \displaystylehbb{T}heta^k(F)$ with $k<p$, then $\displaystylehbb{T}heta^{p-k}(G)\in \displaystylehcal{F}$ and $\displaystylehbb{T}heta^{p-k}(G)\subset F$, hence $\displaystylehbb{T}heta^{p-k}(G)=F$ by minimality, and hence $\displaystylehbb{T}heta^k(F)=\displaystylehbb{T}heta^p(G)\subset G$.
\end{itemize}
We conclude that the sequence $(\displaystylehbb{T}heta^k(F))_k$ is periodic (of period less than $p$), with elements that are pairwise disjoint or equal (by minimality). Let $p'$ the period of the sequence. Then the finite sequence $F,\displaystylehbb{T}heta(F)\ldots, \displaystylehbb{T}heta^{p'-1}(F)$ is a sequence of pairwise disjoint closed sets such that any $f$ in $\displaystylehrm{supp}(\nu)$ sends each set into the following, and the last one into the first. Because of the assumption of aperiodicity, $p'$ is necessarily equal to $1$. As a consequence, $\displaystylehbb{T}heta(F)\subset F$, which means that $F$ is invariant by any $f$ in $\displaystylehrm{supp}(\nu)$, and hence $F=X$ by minimality of the random walk.
\end{proof}
\subsection{Global contractions}
The following theorem shows that from the local phenomenon of contrations given by Assumption \ref{contracting}, we can obtain a phenomenon of global contractions, in the sense that almost surely, the number of domains of attraction is finite: this result is close to a result of Y.Le Jan \cite{Lejan}.
\begin{prop}\label{LeJan}
We assume that the RDS satisfies assumption \ref{contracting}, and we assume moreover that $X$ is locally connected. Then there exists a positive integer $p$, such that, for $\displaystylehbb{P}$-almost every $\omega$ in $\Omega$, there exists $p$ connected open sets $U_1(\omega),\ldots,U_p(\omega)$, pairwise disjoints, such that:
\begin{itemize}
\item the union $U(\omega)=U_1(\omega)\cup\cdots\cup U_p(\omega)$ is dense in $X$,
\item for every $i$ in $\{1,\ldots,p\}$, for every $x,y$ in $U_i(\omega)$,
$$d(f_\omega^n(x),f_\omega^n(y))\rightarrow 0.$$
\end{itemize}
\end{prop}
\begin{proof}
Let us consider the set
$$\displaystylehcal{E}=\{(\omega,x)\in \Omega\times S^1|(f_\omega^n)\mbox{ contracts a neighbourhood of } x\}=\bigcup_{\omega\in\Omega}\{\omega\}\times U(\omega).$$
By Proposition \ref{finite}, there is a finite number of stationary probability measures $\mu_1,\ldots,\mu_d$. For each $i$ in $\{1,\ldots,d\}$, let $U_i(\omega)$ be the union of the connected components of $U(\omega)$ which have a positive $\mu_i$-measure. For $\displaystylehbb{P}$-almost every $\omega$, the set $U_i(\omega)$ is an open subset with $\mu_i$-measure $1$, and has by Proposition \ref{randomcon} a finite constant number $p_i$ of connected components. We write $\tilde{U}(\omega)=U_1(\omega)\cup\cdots \cup U_d(\omega)$. As a consequence of Corollary \ref{randomset} applied to $\tilde{\displaystylehcal{E}}=\bigcup_{\omega\in\Omega}\{\omega\}\times \tilde{U}(\omega)$, we know that $\displaystylehbb{P}\otimes \mu(\tilde{\displaystylehcal{E}})=1$ for every probability measure $\mu$, and hence that $\tilde{U}(\omega)$ is dense for $\displaystylehbb{P}$-almost every $\omega$. Thus, for $\displaystylehbb{P}$-almost every $\omega$, $\tilde{U}(\omega)$ is a dense open subset of $X$ with a finite number $p=\sum_i p_i$ of connected components (and hence in fact, $U(\omega)=\tilde{U}(\omega)$). The result follows.
\end{proof}
We conclude with a criterion ensuring the synchronization of the RDS.
\begin{prop}
\label{globally contracting}
If the RDS satisfies Assumption \ref{contracting}, then the following assertions are equivalent:
\begin{enumerate}
\item the random walk $\omega\mapsto (f_\omega^n)$ is synchronizing, i.e. for every $x,y$ in $X$, for $\displaystylehbb{P}$-almost every $\omega$ in $\Omega$, $d(f_\omega^n(x),f_\omega^n(y))\xrightarrow[n\to+\infty]{}0$
\item the random walk $\omega\mapsto (f_\omega^n,f_\omega^n)$ admits a unique stationary probability measure on $X\times X$;
\item The action of $G$ on $X$ is proximal, i.e. for every $x,y$ in $X$, there exists a sequence $(g_n)_n$ of elements of $G$ such that $d(g_n(x),g_n(y))\xrightarrow[n\to+\infty]{}0$.
\end{enumerate}
\end{prop}
\begin{proof}
Let us notice that the random walk $\omega\mapsto (f_\omega^n,f_\omega^n)$ on $X\times X$ also satisfies the property of local contractions, so that the previous propositions of the section apply to it. We will denote by $\tilde{G}$ the semigroup associated to $\omega\mapsto (f_\omega^n,f_\omega^n)$, and by $D$ the diagonal of $X\times X$.
$1 \displaystylehbb{R}ightarrow 3$ is trivial.
$3\displaystylehbb{R}ightarrow 2$: By Proposition \ref{finite}, if there are two distinct ergodic stationary probability measures, then their respective topological supports $F_1$ and $F_2$ are two disjoint closed non empty subsets of $X\times X$ invariant by $\tilde{G}$. Let $(x,y)$ be any point of $F_1$. By assumption, one can find a sequence of elements $g_n$ in $G$ such that the distance between $g_n(x)$ and $g_n(y)$ tends to $0$. Since $(g_n(x),g_n(y))\in F_1$, taking a cluster value of the sequence we deduce that $F_1$ intersects $D$ at some point $(z_1,z_1)$. In the same way, $F_2$ intersects $D$ at some point $(z_2,z_2)$. Choosing then a sequence $(h_n)$ in $G$ such that $d(h_n(z_1),h_n(z_2))\rightarrow 0$, any cluster value of $(h_n(z_1),h_n(z_1))$ is also a cluster value of $(h_n(z_2),h_n(z_2))$ and hence belongs to $F_1\cap F_2$, which is absurd.
$2\displaystylehbb{R}ightarrow 1$: By Proposition \ref{finite}, there is a unique minimal non empty closed subset $F$ invariant by $\tilde{G}$. Since $D$ is $\tilde{G}$-invariant, $F\subset D$. By Proposition \ref{adhe}, for every $(x,y)$ in $X\times X$, for $\displaystylehbb{P}$-almost every $\omega$ in $\Omega$, the set of cluster values of $((f_\omega^n(x),f_\omega^n(y))_{n\in\displaystylehbb{N}}$ is exactly $F$. In particular, it is included in $D$, hence $d(f_\omega^n(x),f_\omega^n(y))\xrightarrow[n\to+\infty]{}0$.
\end{proof}
\section{Proof of the main results}\label{sec-proofs}
We are going to combine Theorem \ref{main} proved in Section \ref{sec-inv} and the results of Section \ref{sec-consequences} to deduce Theorem \ref{distribution}, \ref{law}, \ref{Lejan-Antonov}, \ref{synchronizing} and their corollaries.
\subsection{Behaviour of random walks on $\displaystylehrm{Homeo}(S^1)$}
\begin{proof}[Proof of Theorem \ref{distribution}]~
If we are in the first case of Corollary \ref{alternative}, then the result is a direct consequence of Proposition \ref{finite} and \ref{adhe}. If not, then we are in the second case since $G$ has no finite orbit. That means that $G$ is semiconjugated to a minimal semigroup of isometries, and it is classical in this case that the stationary probability measure is unique: assuming up to the semiconjugation that $G$ is a semigroup of isometries acting minimally, if $\mu_1$ and $\mu_2$ are two ergodic stationary probabilities, one can find Birkhoff's points of $\mu_1$ and $\mu_2$ arbitrarily close, and then the trajectories of these points remain close, so that $\mu_1$ and $\mu_2$ are themselves arbitrarily close, hence equal. Thus the statinary probability measure $\mu$ is unique, and the convergence of $\frac{1}{N}\sum_{n=0}^{N-1}\delta_{f_{\omega}^n(x)}$ toward $\mu$ is for exemple a consequence of Proposition \ref{cluster}.
\end{proof}
\begin{proof}[Proof of Theorem \ref{law}]~
With the notations of the statement, the distribution $\mu_n^x$ is given by $\int \varphi d\mu_n^x=P^n\varphi(x)$ where $P$ is the transfer operator of the random walk, so that Proposition \ref{transfert} implies that $(\mu_n^x)_n$ converges in law, uniformly in $x$, to the stationary probability measure $\mu^x=\sum_{i=1}^du_i(x)\mu_i$ (keeping the notations of Proposition \ref{transfert}).
\end{proof}
\begin{proof}[Proof of Theorem \ref{Lejan-Antonov}]~
As a consequence of Proposition \ref{LeJan}, for $\displaystylehbb{P}$-almost $\omega$ in $\Omega$, the set $U(\omega)$ of the points having a neighbourhood contracted by $(f_\omega^n)_n$ is dense and has a finite constant number $d$ of connected components, so that $S^1-U(\omega)$ is finite of cardinal $d$. To obtain the exponential contractions, it is enough to copy the proof of Proposition \ref{LeJan} replacing $U(\omega)$ by the set $U'(\omega)$ of the points having a neighbourhood contracted exponentially fast by $(f_\omega^n)_n$.
\end{proof}
\subsection{Synchronization}
\begin{proof}[Proof of Theorem \ref{synchronizing}]~
The only non trivial implication is iii) $\displaystylehbb{R}ightarrow$ i).
Let us assume that the action of $G$ is proximal, that is, for any points $x$, $y$ there exists a sequence $g_n$ of elements of $G$ such that $\displaystylehrm{dist}(g_n(x),g_n(y))\to 0$.\\
Firstly, let us justify that we are in the first case of Corollary \ref{alternative}:
If $G$ is semi-conjugated to $\tilde{G}$, then $\tilde{G}$ satisfies the same property of synchronization as $G$, so that $\tilde{G}$ is not a group of isometries, and so we are not in second case.
If $G$ leaves invariant a finite set having at least two points, then the action of $G$ on this finite set cannot be proximal, which contradicts the assumptions. And $G$ cannot fix a singleton by assumption. Hence we are not in third case.\\
So we are in the first case, that is, the random walk satisfies the property of contractions given by Theorem \ref{main}. For any $x$, $y$ in the circle, one can find a sequence $g_n$ in $G$ such that $(g_n(x))_n$ and $(g_n(y))_n$ tend to a same point $c$. By Theorem \ref{main}, one can find a neighborhood of $c$ having positive probability to be contracted, hence we deduce that there is a set of $\omega$ with positive probability such that $\displaystylehrm{dist}(f_\omega^n(x),f_\omega^n(y))$ tends to $0$ exponentially fast as $n$ tends to $+\infty$.\\
Let $\displaystylehcal{E}$ be the set of $(\omega,x,y)$ in $\Omega\times S^1\times S^1$ such that $\displaystylehrm{dist}(f_\omega^n(x),f_\omega^n(y))$ tends to $0$ exponentially fast as $n$ tends to $+\infty$. We obtained that for any $x$, $y$ in $S^1$, $\displaystylehbb{P}\otimes \delta_{(x,y)}(\displaystylehcal{E})>0$, hence by Proposition \ref{randomset}, we have actually $\displaystylehbb{P}\otimes \delta_{(x,y)}(\displaystylehcal{E})=1$, which means that the random walk is exponentially synchronizing.
\end{proof}
\begin{rem}
In the previous proof we could use Proposition \ref{globally contracting} to deduce the property of synchronization. However, it does not give the exponential speed.
\end{rem}
\begin{proof}[Proof of Corollary \ref{robust}]
Let $K$ be the compact minimal invariant by $G$ (necessarily unique because of the synchronization property).
\begin{lem}
There exists $g$ in $G$ having a robust fixed point and such that $g|_K\not=Id_K$.\\
(We say that $g$ has a robust fixed point if every small $C^0$-perturbation of $g$ has a fixed point)
\end{lem}
\begin{proof}
Let $x$ be any point of $K$. By Theorems \ref{main} and \ref{distribution}, one can find $\omega\in \Omega$ and a neighbourhood $I_0$ of $x$ such that $\displaystylehrm{diam}(f_\omega^n(I_0))\to 0$ as $n\to +\infty$ and $(f_\omega^n(x))_{n\in\displaystylehbb{N}}$ is dense in $K$. Thus we can find some integer $n$ such that $\overline{f_\omega^n(I_0)}\subset I_0-\{x\}$. Then $g=f_\omega^n$ satisfy $\overline{g(I_0)}\subset I_0$ (which implies that $g$ has a robust fixed point ) and $g(x)\not=x$.
\end{proof}
Let $g$ be as in the lemma, and $I$ be an open interval intersecting $K$ such that $g$ has no fixed point on the closure of $I$. Let $x$ and $y$ be in $S^1$. For almost every $\omega$, the trajectories $(f_\omega^n(x))$ and $(f_\omega^n(y))$ are by assumption asymptotically identical, and are dense in $K$. We deduce that we can find $h$ in $G$ such that $h(x),h(y)\in I$. By compactness, one can find $h_1,\ldots,h_p$ in $G$ such that for any $x,y$ in $S^1$, $h_i(x),h_i(y)\in I$ for some $i$ in $\{1,\ldots,p\}$.
Now, let $\tilde{f}_1,\ldots,\tilde{f}_d$ be small $C^0$-perturbations of the generators $f_1,\ldots,f_d$, $\tilde{G}$ be the semigroup generated by these new generators, and $\tilde{g},\tilde{h}_1,\ldots,\tilde{h}_p \in \tilde{G}$ be corresponding perturbations of $g,h_1,\ldots,h_p$. If the perturbations are small enough, the properties
\begin{itemize}
\item $\forall x\in I, \tilde{g}(x)\not= x$,
\item $\tilde{g}$ has a fixed point,
\item $\forall x,y\in S^1 \exists i\in \{1,\ldots,p\}|\tilde{h}_i(x),\tilde{h}_i(y)\in I$,
\end{itemize}
are still satisfied. The two first properties imply that $(\tilde{g}^n)$ converges to a constant on $I$, and hence using the third one we deduce that for any $x,y$ in $S^1$, there exists $i$ such that $\displaystylehrm{dist}(\tilde{g}^n\circ \tilde{h}_i(x),\tilde{g}^n\circ \tilde{h}_i(y))\to 0$ as $n\to +\infty$. Thus, we can use Theorem \ref{synchronizing} to conclude that every random walk which is non degenerated on $\tilde{G}$ is synchronizing.
\end{proof}
\subsection{Random dynamical systems on $[0,1]$}
\begin{proof}[Proof of Corollary \ref{example}]
Indentifying $I=[0,1]$ with an arc of $S^1$, we can prolong arbitrarily any injective map of $I$ to a homeomorphism of $S^1$. Thus, the result is a consequence of Theorem \ref{synchronizing}, once we have proved that
\begin{itemize}
\item There is no point of $I$ fixed by every element of $G$;
\item There exists a sequence $(g_n)$ is $G$ such that
$$\displaystylehrm{diam}(g_n(I))\xrightarrow[n\to+\infty]{}0.$$
\end{itemize}
The first point is straightforward, since a point fixed by $G$ belongs to $\bigcap_{g\in G} g(I)=\emptyset$.\\
Let us prove the second point. Let us denote, for $g$ in $G$, $[a(g),b(g)]=g([0,1])$, $a=\sup_{g\in G}a(g)$ and $b=\inf_{g\in G}b(g)$. If $a\leq b$, then $[a,b]\subset \bigcap_{g\in G}[a(g),b(g)]\subset \bigcap_{g\in G} g(I)$, which is a contradiction. Thus $a>b$, so that one can find $g$ and $h$ in $G$ such that $a(g)>b(h)$, which implies that $g(I)\cap h(I)=\emptyset$. Since $g^2=g\circ g$ is increasing and has no fixed point on $h(I)$, we deduce that the sequence $(g^{2n})_n$ converges on $h(I)$ to a constant. In consequence, the sequence $g_n=g^{2n}\circ h$ satisfies the second point. That concludes the proof.
\end{proof}
\begin{proof}[Proof of Corollary \ref{mainR}]~
Identifying the points $0$ and $1$ gives a circle so that we can apply results of Section \ref{sec-inv} on the random walk.
Let $x_0$ be any point of $(0,1)$. For $\omega$ in $\Omega$, let
$$\mu_{N,\omega}=\frac{1}{N}\sum_{n=0}^{N-1}\delta_{f_\omega^n(x_0)}$$
We want to prove that for almost every $\omega$ in $\Omega$, the sequence $(\mu_{N,\omega})_{N\in\displaystylehbb{N}}$ has some weak adherence value which is not invariant by $G$, in order to use Proposition \ref{contractuel} and Corollary \ref{invarianceiid}. In this view, let us note that the probability measures invariant by $G$ are necessarily convex combinations of $\delta_0$ and $\delta_1$.\\
Let $\mu$ a stationary probability measure of $G$ on $(0,1)$, that we can suppose ergodic. Since $\mbox{supp}(\mu)$ is invariant by $G$, we deduce that the interval $[\inf(\mbox{supp}(\mu)),\mbox{supp}(\mu)]$ is also invariant by $G$, hence is equal to $[0,1]$ by assumption, so that $0$ and $1$ belong to $\mbox{supp}(\mu)$. In particular one can find a Birkhoff point $a$ of $\mu$ in $(0,x_0)$ and an other Birkhoff point $b$ in $(x_0,1)$.\\
Let $I$ a compact interval of $(0,1)$ such that $\mu(I)\geq \frac{3}{4}$. Then for almost every $\omega$ in $\Omega$, the sets $A_\omega=\{n\in\displaystylehbb{N} |f_\omega^n(a)\not\in I\}$ and $B_\omega=\{n\in\displaystylehbb{N} |f_\omega^n(b)\not\in I\}$ have density smaller than $\frac{1}{4}$. But obviously, since $a<x_0<b$ the set $C_\omega=\{n\in\displaystylehbb{N} |f_\omega^n(x_0)\not\in I\}$ is contained in $A_\omega\cup B_\omega$, hence this last set has density smaller than $\frac{1}{2}$.\\
In consequence, for almost every $\omega$ in $\Omega$, an adherance value $\mu'$ of $(\mu_{N,\omega})_{N\in\displaystylehbb{N}}$ satisfies $\mu'(I)\geq \frac{1}{2}$, hence $\mu'$ is not a convex combination of $\delta_0$ and $\delta_1$, hence is not invariant by $G$ and
$$\lambda_{con}(\omega,x_0)\leq \lambda_{con}(\displaystylehbb{P}\times\mu')<0$$
by Proposition \ref{contractuel} and Corollary \ref{invarianceiid}.
\end{proof}
\begin{proof}[Proof of Corollary\ref{torchder}]~
By Corollary \ref{mainR}, every point $x$ has a contractible neighbourhood $I$ (in the meaning given in Definition \ref{contracdef}), and then there exists exactly one stationary ergodic probability measure $\mu_x$ such that $\mu_x(I)>0$: at most one because of Lemma \ref{contractible}, and at least one because $\mu(I)$ is positive.
Then, $x\mapsto \mu_x$ is constant on contractible intervals, hence is locally constant, hence constant. This constant is the only ergodic stationary probability measure supported on $(0,1)$, and necassarily is $\mu$.
\end{proof}
\end{document} |
\begin{document}
\maketitle
\begin{abstract}
We give a series of combinatorial results that can be obtained from any two collections (both indexed by $\Zbold\times \Nbold$) of left and right pointing arrows that satisfy some natural relationship. When applied to certain self-interacting random walk couplings, these allow us to reprove some known transience and recurrence results for some simple models. We also obtain new results for one-dimensional multi-excited random walks and for random walks in random environments in all dimensions.
\end{abstract}
\section{Introduction}
\leftarrowbel{sec:intro}
Coupling is a powerful tool for proving certain kinds of properties of random variables or processes. A coupling of two random processes $X$ and $Y$ typically refers to defining random variables $X'$ and $Y'$ {\em on a common probability space} such that $X'\sim X$ (i.e.~$X$ and $X'$ are identically distributed) and $Y'\sim Y$. There can be many ways of doing this, but generally one wants to define the probability space such that the {\em joint distribution} of $(X',Y')$ has some property. For example, suppose that $X=\{X_n\}_{n\ge 0}$ and $Y=\{Y_n\}_{n\ge 0}$ are two nearest-neighbour simple random walks in 1 dimension with drifts $\mu_X\le \mu_Y$ respectively. One can define $X'\sim X$ and $Y'\sim Y$ on a common probability space so that $X'$ and $Y'$ are independent, but one can also define $X''\sim X$ and $Y''\sim Y$ on a common probability space so that $X''_n\le Y''_n$ for all $n$ with probability 1.
Consider now a nearest-neighbour random walk $\{X_n\}_{n\ge 0}$ on $\Zbold^d$ that has transition probabilities $(2d)^{-1}$ of stepping in each of the $2d$ possible directions, except on the {\em first departure from each site}. On the first departure, these are also the transition probabilities for stepping to the left and right in any coordinate direction other than the first. But in the first coordinate, the transition probabilities are instead $(2d)^{-1}(1+\beta)$ (right) and $(2d)^{-1}(1-\beta)$ (left), for some fixed parameter $\beta\in [0,1]$. This is known as an excited random walk \cite{BW03} and the behaviour of these and more general walks of this kind has been studied in some detail since 2003. For this particular model, it is known \cite{BR07} that for $d\ge 2$ and $\beta>0$, there exists $v_{\beta}=(v^{[1]}_{\beta},0,\dots,0)\in \Zbold^d$ with $v^{[1]}_{\beta}>0$ such that $\lim_{n\rightarrow \infty}n^{-1}X_n=v_{\beta}$ with probability 1. When $d=1$ the model is recurrent (0 is visited infinitely often) except in the trivial case $\beta=1$. It is plausible that $v^{[1]}_{\beta}$ should be a non-decreasing function of $\beta$ (i.e.~increasing the local drift should increase the global drift) but this is not known in general.
A natural first attempt at trying to prove such a monotonicity result would be as follows: given $0<\beta_1<\beta_2\le 1$, construct a coupling of excited random walks $X$ and $Y$ with parameters $\beta_1$ and $\beta_2>\beta_1$ respectively such that with probability 1, $X_n^{[1]}\le Y_n^{[1]}$ for all $n$. Thus far no one has been able to construct such a coupling, and the monotonicity of $v^{[1]}_{\beta}$ as a function of $\beta$ remains an open problem in dimensions $2\le d\le 8$. In dimensions $d\ge 9$ this result has been proved \cite{HH09mono} using a somewhat technical expansion method, as well as rigorous numerical bounds on simple random walk quantities. More general models in 1 dimension have been studied, and some monotonicity results \cite{Zern05} have been obtained via probabilistic arguments but without coupling. This raises the question of whether or not one can obtain proofs of these kinds of results using a coupling argument that has weaker aims e.g.~such that $\max_{m\le n}X_m^{[1]}\le \max_{m\le n}Y_m^{[1]}$ for all $n$, rather than $X_n^{[1]}\le Y_n^{[1]}$ for all $n$.
This paper addresses this issue in 1-dimension. We study relationships between completely deterministic (non-random) 1-dimensional systems of arrows that may prove to be of independent interest in combinatorics. Each system $\mc{L}$ of arrows defines a sequence $L$ of integers. We show that under certain natural local conditions on arrow systems $\mc{L}$ and $\mc{R}$, one obtains relations between the corresponding sequences such as $\max_{m\le n}L_m^{[1]}\le \max_{m\le n}R_m^{[1]}$ for all $n$ (while it's still possible that $L_n^{[1]}> R_n^{[1]}$ for some $n$).
These may be applied to certain random systems of arrows, to give self-interacting random walk couplings. Doing so, one can obtain results about the (now random) sequence $R_n$ {\em if} $L_n$ (also random) is well understood, and vice versa. This yields alternative proofs of some existing results, as well as new non-trivial results about so-called multi-excited random walks in 1 dimension and some models of random walks in random environments in all dimensions -- see e.g.~\cite{HS_RWDRE}. To be a bit more precise, in \cite{HS_RWDRE} a projection argument applied to some models of random walks in random environments (in all dimensions) gives rise to a one-dimensional random walk $Y$, which can be coupled with a one-dimensional multi-excited random walk $Z$ (both walks depending on a parameter $p$) so that for every $j\in \Zbold$ and every $r\ge 1$:
\begin{itemize}
\item[(i)] If $Y$ goes left on its $r$th visit to $j$ then so does $Z$ (if such a visit occurs), and therefore
\item[(ii)] If $Z$ goes right on its $r$th visit to $j$ then so does $Y$ (if such a visit occurs).
\end{itemize}
Explicit conditions ($p>\frac{3}{4}$ in this case) governing when $Z_n\rightarrow \infty$ as $n\rightarrow \infty$ are given in \cite{Zern05}. One would like to conclude that also $Y_n\rightarrow \infty$ (whence the original random walk in $d$-dimensions returns to its starting point only finitely many times) when $p>\frac{3}{4}$. This can be achieved by applying the result of this paper to the coupling mentioned above.
The main contributions of this paper are:
combinatorial results concerning sequences defined by arrow systems satisfying certain natural local relationships (see Theorem \mathbb{R}f{thm:main}); some non-trivial counterintuitive examples; and application of these combinatorial results with non-monotone couplings to obtain new results in the theory of random walks.
\subsection{Arrow systems}
\leftarrowbel{sec:arrows}
A collection $\mc{E}=(\mc{E}(x,r))_{x \in \Zbold, r\in \mathbb{N}}$, where $\mc{E}(x,r)\in \{\leftarrow,\rightarrow\}$ is the arrow above the vertex $x\in \Zbold$ at level $r\in \Nbold$, is called an {\em arrow system}. This should be thought of as an infinite (ordered) stack of arrows rising above each vertex in $\Zbold$.
In a given arrow system $\mc{E}$, let $\mc{E}_{\leftarrow}(j,r)$ denote the number of $\leftarrow$ arrows, out of the first $r$ arrows above $j$. As $r$ increases, this quantity counts the number of $\leftarrow$'s appearing in the arrow columns above $j$. Similarly define $\mc{E}_{\rightarrow}(j,r)=r-\mc{E}_{\leftarrow}(j,r)$. We can define a sequence $E=\{E_n\}_{n\ge 0}$ by setting $E_0=0$ and letting $E$ evolve by taking one step to the left or right (at unit times), according to the lowest arrow of the $\mc{E}$-stack at its current location, and then deleting that arrow. In other words, if $\#\{0\le m\le n:E_m=E_n\}=k$ then $E_{n+1}=E_n+1$ if $\mc{E}(E_n,k)=\rightarrow$ (resp.~$E_{n+1}=E_n-1$ if $\mc{E}(E_n,k)=\leftarrow$).
\begin{DEF}[$\mc{L}\CboldL\mc{R}$]
\leftarrowbel{def:cl}
Given two arrow systems $\mc{L}$ and $\mc{R}$, we write $\mc{L}\CboldL\mc{R}$ if for each $j\in \Zbold$ and each $r\in \mathbb{N}$,
\[\mc{L}_{\leftarrow}(j,r)\ge \mc{R}_{\leftarrow}(j,r) \qquad (\text{and hence also } \mc{L}_{\rightarrow}(j,r)\le \mc{R}_{\rightarrow}(j,r)).\]
\end{DEF}
\begin{DEF}[$\mc{L}\trianglelefteq \mc{R}$]
\leftarrowbel{def:tl}
We write $\mc{L}\trianglelefteq \mc{R}$ if for each $j\in \Zbold$ and each $r\in \mathbb{N}$,
\[\mc{L}(j,r)=\rightarrow \quad \Rboldightarrow \mc{R}(j,r)=\rightarrow.\]
\end{DEF}
\noindent It is easy to see that $\mc{L}\trianglelefteq \mc{R}$ implies $\mc{L}\CboldL \mc{R}$.
Now define two paths/sequences $\{L_n\}_{n\ge 0}$ and $\{R_n\}_{n\ge 0}$ in $\Zbold$ according to the arrows in $\mc{L}$ and $\mc{R}$ respectively as above (in particular $L_0=R_0=0$). Since each arrow system determines a unique sequence, but a given sequence may be obtained from multiple different arrow systems, we write $L\CboldL R$ (resp.~$L\trianglelefteq R$) if there exist $\mc{L}\CboldL \mc{R}$ (resp.~$\mc{L}\trianglelefteq \mc{R}$) whose corresponding sequences are $L$ and $R$ respectively. Note that when $\mc{L}\trianglelefteq \mc{R}$, the paths $Z=L$ and $Y=R$ constructed from $\mc{L}$ and $\mc{R}$ as above automatically satisfy the conditions (i) and (ii) appearing at the beginning of Section \mathbb{R}f{sec:intro}.
An arrow system $\mc{E}$ is said to be {\em 0-right recurrent} if in the new system $\mc{E}_+$ defined by $\mc{E}_+(0,i)=\rightarrow$ for all $i\ge 1$, and $\mc{E}_+(x,i)=\mc{E}(x,i)$ for all $i\ge 1$ and $x>0$, $E_{+,n}=0$ infinitely often.
The main result of this paper is the following theorem, in which $n_{E,t}(x)=\#\{k\le t:E_k=x\}$ (see also Corollary \mathbb{R}f{cor:transience} in the case that $L$ is transient to the right).
\begin{THM}
\leftarrowbel{thm:main}
Suppose that $\mc{L}\CboldL \mc{R}$. Then
\begin{itemize}
\item[(i)] $\liminf_{n\rightarrow \infty} L_n\le \liminf_{n\rightarrow \infty} R_n$;
\item[(ii)] $\limsup_{n\rightarrow \infty} L_n\le \limsup_{n\rightarrow \infty} R_n$;
\item[(iii)] Let $a_n\le n$ be any increasing sequence, with $a_n\to\infty$. If there exists $x \in \Zbold$ such that $R\ge x$ infinitely often then $\limsup_{n\rightarrow \infty} \frac{L_n}{a_n}\le \limsup_{n\rightarrow \infty} \frac{R_n}{a_n}$.
\item[(iv)] If $n_{R,t}(x)>n_{L,t}(x)$ then $n_{R,t}(y)\ge n_{L,t}(y)$ for every $y>x$.
\item[(v)] If $\mc{R}$ is $0$-right recurrent then so is $\mc{L}$.
\end{itemize}
\end{THM}
As $\frac{L_n}{n}$ represents the average speed of the sequence $L$, up to time $n$, in many applications the sequence of interest in Theorem \mathbb{R}f{thm:main} (iii) will be $a_n=n$. Part (ii) of Theorem \mathbb{R}f{thm:main} actually follows from part (i) by a simple mirror symmetry argument. There is a symmetric version of (iii), but one must be careful. Part (iii) obviously implies that if $u=\lim n^{-1}R_n$ and $l=\lim n^{-1}L_n$ both exist then $l\le u$, however we show in Section \mathbb{R}f{sec:counterex1} that $L\trianglelefteq R$ does not imply that $\liminf\frac{L_n}{n}\le \liminf\frac{R_n}{n}$. The mirror image (about 0) of the counterexample in Section \mathbb{R}f{sec:counterex1} also shows that (iii) is not true in general if we drop the condition that $L\ge x$ infinitely often, for some $x$.
One might also conjecture that if $L\trianglelefteq R$ then the amount of time that $R>L$ is at least as large as the amount of time that $R<L$. This is also false as per a counterexample in Section \mathbb{R}f{sec:counterex2}.
The remainder of the paper is organised as follows. Section \mathbb{R}f{sec:basic} contains the basic combinatorial relations which are satisfied by the arrow systems and their corresponding sequences. These will be needed in order to prove our first results. Section \mathbb{R}f{sec:results} gives various consequences of the relationship $\mc{L}\CboldL \mc{R}$ between two arrow systems, and includes the proofs of the main results of the paper. Section \mathbb{R}f{sec:counterex} contains the counterexamples described above.
Finally Section \mathbb{R}f{sec:applications} contains applications of our results in the study of self-interacting random walks.
\section{Basic relations}
\leftarrowbel{sec:basic}
Given an arrow system $\mc{E}$ and $t\ge 0$, let $n_{E,t}(x)=\#\{k\le t:E_k=x\}$ and $n_{E,t}(x,y)=\#\{k\le t:E_{k-1}=x,E_k=y\}$. Then the following relationships hold:
\eqalign
n_{E,t}(x)&=\partialta_{x,0}+n_{E,t}(x-1,x)+n_{E,t}(x+1,x)\leftarrowbel{eq:fact1}\\
n_{E,t}(x)&=\partialta_{E_t,x}+n_{E,t}(x,x+1)+n_{E,t}(x,x-1)\leftarrowbel{eq:fact2}\\
t+1&=\sum_{i=-\infty}^{\infty}n_{E,t}(i).\leftarrowbel{eq:fact3}
\enalign
Relation (\mathbb{R}f{eq:fact1}) says that every visit to $x$ is either from the left or right, except for the first visit if $x=0$. Relation (\mathbb{R}f{eq:fact2}) is similar, but in terms of departures from $x$. The sum in (\mathbb{R}f{eq:fact3}) is in fact a finite sum since $n_{E,t}(i)=0$ for $|i|>t$.
Next
\eqalign
n_{E,t}(x,x+1)&=\mc{E}_{\rightarrow}(x,n_{E,t}(x)-I_{E_t=x})\leftarrowbel{eq:R1}\\
n_{E,t}(x,x-1)&=\mc{E}_{\leftarrow}(x,n_{E,t}(x)-I_{E_t=x})\leftarrowbel{eq:R2},
\enalign
where e.g.~relation (\mathbb{R}f{eq:R1}) says that the number of departures from $x$ to the right is the number of ``used'' right arrows at $x$.
Finally,
\eqalign
n_{E,t}(x,x+1)+I_{x+1\le 0}I_{E_t\le x}&=n_{E,t}(x+1,x)+I_{x\ge 0}I_{E_t\ge x+1},\leftarrowbel{eq:pm1}
\enalign
which says that the number of moves from $x$ to $x+1$ is closely related to the number of moves from $x+1$ to $x$. They may differ by 1 depending on the position of $x$ relative to 0 and the current value of the sequence. For example, if $0\le x<E_t$ then the number of moves from $x$ to $x+1$ up to time $t$ is one more than the number of moves from $x+1$ to $x$ up to time $t$.
\section{Implications of $\mc{L}\CboldL \mc{R}$.}
\leftarrowbel{sec:results}
In this section we always assume that $\mc{L}\CboldL\mc{R}$. The results typically have symmetric versions using the fact that $\mc{L}\CboldL \mc{R} \iff -\mc{R}\CboldL -\mc{L}$, which is equivalent to considering arrow systems reflected about 0. We divide the section into two subsections based roughly on the nature of the results and their proofs.
For $x \in \Zbold$ and $k\ge 0$, let $T_{L}(x,k)=\inf\{t\ge 0:n_{L,t}(x)=k\}$, and $T_{R}(x,k)=\inf\{t\ge 0:n_{R,t}(x)=k\}$.
\subsection{Results obtained from the basic relations}
\leftarrowbel{sec:appl_basic}
The proofs in this section are based on applications of the basic relations of Section \mathbb{R}f{sec:basic}. The first few results are somewhat technical, but will be used in turn to prove some of the more appealing results. Roughly speaking they describe how the relative numbers of visits of $L$ and $R$ to neighbouring sites $x-1$ and $x$ relate to each other.
\begin{LEM}
\leftarrowbel{lem:exists}
If $L$ hits $x$ at least $k\ge 1$ times and $R$ is eventually to the left of $x$ after fewer than $k$ visits to $x$, then there exists a site $y<x$ that $R$ hits at least $n_{L,T_L(x,k)}(y)$ times.
\end{LEM}
\proof Fix $x,k$ and let $T=T_L(x,k)$ and $y_0:=\inf\{z\le x:n_{L,T}(z)>0\}\le 0$. If $y_0=x$ then the first $k-1$ arrows at $x$ are all right arrows, i.e.~$\mc{L}_{\rightarrow}(y_0,k-1)=k-1$. Then also $\mc{R}_{\rightarrow}(y_0,k-1)=k-1$ so $R$ cannot be to the left of $x$ after fewer than $k$ visits. Similarly if $y_0<x$ then the first $n_{L,T}(y_0)$ arrows at $y_0$ are all right arrows, i.e.~$\mc{L}_{\rightarrow}(y_0,n_{L,T}(y_0))=n_{L,T}(y_0)$, and so also $\mc{R}_{\rightarrow}(y_0,n_{L,T}(y_0))= n_{L,T}(y_0)$.
Therefore either $R$ visits $y_0$ at least $n_{L,T}(y_0)$ times or it stays in $(y_0,x)$ infinitely often, whence it must visit some site $y \in (y_0,x)$ at least $n_{L,T}(y)$ times as required.\qed
\blank{We can remove this result if we want to
\begin{LEM}
\leftarrowbel{lem:x-1,x,1}
If $n_{R,t}(x-1)\ge n_{L,t}(x-1)$ and $n_{R,t}\ne x-1$ then either
\begin{enumerate}
\item $n_{R,t}(x)\ge n_{L,t}(x)$ or
\item $R_t\ge x$.
\end{enumerate}
\end{LEM}
\proof Assume that the first claim fails. If $R_t=x$ then there is nothing to prove, so also assume that $R_t\ne x$. We show that $R_t>x$.
Let $T=\inf\{s\le t:n_{L,s}(x)=n_{R,t}(x)+1\}$. Then $T\le t$ and $L_T=x$.
Also
\eqalign
n_{R,t}(x)+1&= n_{L,T}(x)=n_{L,T}(x-1,x)+n_{L,T}(x+1,x)+\partialta_{0,x}\\
n_{R,t}(x)&=\partialta_{x,0}+n_{R,t}(x-1,x)+n_{R,t}(x+1,x).
\enalign
Subtracting the first from the second we obtain
\eq
n_{R,t}(x-1,x)+n_{R,t}(x+1,x)+1-n_{L,T}(x-1,x)-n_{L,T}(x+1,x)=0.
\en
Now $n_{L,T}(x+1,x)=n_{L,T}(x,x+1)+I_{x+1\le 0}$ from (\mathbb{R}f{eq:pm1}), so
\eq
n_{R,t}(x-1,x)+n_{R,t}(x+1,x)+1-n_{L,T}(x-1,x)-[n_{L,T}(x,x+1)+I_{x+1\le 0}]=0.
\en
Rearranging we obtain
\eq
n_{R,t}(x+1,x)+1+[n_{R,t}(x-1,x)-n_{L,T}(x-1,x)]\le n_{L,T}(x,x+1)+I_{x+1\le 0}.
\en
By definition of $T$ and using $R_t\ne x$ we have that
$$
n_{R,t}(x,x+1)=\mc{R}_{\rightarrow}(x,n_{R,t}(x))\ge\mc{L}_{\rightarrow}(x,n_{R,t}(x))
=\mc{L}_{\rightarrow}(x,n_{L,T}(x)-1)=n_{L,T}(x,x+1).
$$
Therefore
\eq
n_{R,t}(x+1,x)+1+[n_{R,t}(x-1,x)-n_{L,T}(x-1,x)]\le n_{R,t}(x,x+1)+I_{x+1\le 0}.\leftarrowbel{eq:lineabove}
\en
By assumption we have that
\begin{align*}
n_{R,t}(x-1,x)
&=\mc{R}_{\rightarrow}(x-1,n_{R,t}(x-1)-I_{R_t=x-1})
\ge\mc{R}_{\rightarrow}(x-1,n_{L,T}(x-1)-I_{R_t=x-1})\\
&\ge\mc{L}_{\rightarrow}(x-1,n_{L,T}(x-1)-I_{R_t=x-1})
= n_{L,T}(x-1,x)-I_{R_t=x-1}
\end{align*}
so by (\mathbb{R}f{eq:lineabove})
\eq
n_{R,t}(x+1,x)+1-I_{R_t=x-1}\le n_{R,t}(x,x+1)+I_{x+1\le 0}.
\en
By (\mathbb{R}f{eq:pm1}) again, this is $\le n_{R,t}(x+1,x)+I_{R_t\ge x+1}$. Therefore $R_t\ge x+1$, so in fact
$R_t>x$ for every $t\ge r$. Moreover $n_{R,r}(x)=n_R(x)<n_L(x)$, which shows (b).
\qed
}
Let $n_L(x)=n_{L,\infty}(x)$ and $n_R(x)=n_{R,\infty}(x)$.
\begin{LEM}
\leftarrowbel{lem:x-1,x}
If $R$ hits $x-1$ at least $n_L(x-1)$ times then either
\begin{enumerate}
\item $n_R(x)\ge n_L(x)$, or
\item $R$ is always to the right of $x$ after fewer than $n_L(x)$ visits.
($\Rboldightarrow \liminf_{n\rightarrow \infty}R_n>x$)
\end{enumerate}
\end{LEM}
\proof Assume that the first claim fails, so in particular $n_{R}(x)<\infty$. Let $T=\inf\{t:n_{L,t}(x)=n_R(x)+1\}$.
Then $T<\infty$ so $L_T=x$. Choose $r$ sufficiently large so that $R_t\ne x$ for any $t\ge r$, $R_r\ne x-1$, and $n_{R,r}(x-1)\ge n_{L,T}(x-1)$.
Then by (\mathbb{R}f{eq:fact1}) applied to $L$ at time $T$, and also to $R$ at time $r$,
\eqalign
n_{R,r}(x)+1&= n_R(x)+1=n_{L,T}(x)=n_{L,T}(x-1,x)+n_{L,T}(x+1,x)+\partialta_{0,x}\nonumber\\
n_{R,r}(x)&=\partialta_{x,0}+n_{R,r}(x-1,x)+n_{R,r}(x+1,x).\nonumber
\enalign
Subtracting one from the other and rearranging we obtain
\eq
n_{R,r}(x-1,x)-n_{L,T}(x-1,x)+n_{R,r}(x+1,x)+1=n_{L,T}(x+1,x).\nonumber
\en
Now $n_{L,T}(x+1,x)=n_{L,T}(x,x+1)+I_{x+1\le 0}$ from (\mathbb{R}f{eq:pm1}), so
\eq
n_{R,r}(x+1,x)+1+[n_{R,r}(x-1,x)-n_{L,T}(x-1,x)]= n_{L,T}(x,x+1)+I_{x+1\le 0}.\leftarrowbel{eq:elephant}
\en
Using (\mathbb{R}f{eq:R1}) and the fact that $R_r\ne x$, then $\mc{L}\CboldL \mc{R}$, then the fact that $n_{L,T}(x)=1+n_{R,r}(x)$, and finally again using (\mathbb{R}f{eq:R1}) and the fact that $L_T=x$ we obtain
\eq
n_{R,r}(x,x+1)=\mc{R}_{\rightarrow}(x,n_{R,r}(x))\ge\mc{L}_{\rightarrow}(x,n_{R,r}(x))
=\mc{L}_{\rightarrow}(x,n_{L,T}(x)-1)=n_{L,T}(x,x+1).\nonumber
\en
Using this bound in (\mathbb{R}f{eq:elephant}) yields
\eq
n_{R,r}(x+1,x)+1+[n_{R,r}(x-1,x)-n_{L,T}(x-1,x)]\le n_{R,r}(x,x+1)+I_{x+1\le 0}.\leftarrowbel{eq:lineabove}
\en
Using the fact that $R_r\ne x-1$ and applying (\mathbb{R}f{eq:R1}) to $R_r$ at $x-1$, then using $n_{R,r}(x-1)\ge n_{L,T}(x-1)$, then $\mc{L}\CboldL \mc{R}$, and finally using the fact that $L_T\ne x-1$ and applying (\mathbb{R}f{eq:R1}) to $L_T$ at $x-1$, we have that
\begin{align*}
n_{R,r}(x-1,x)
&=\mc{R}_{\rightarrow}(x-1,n_{R,r}(x-1))
\ge\mc{R}_{\rightarrow}(x-1,n_{L,T}(x-1))\\
&\ge\mc{L}_{\rightarrow}(x-1,n_{L,T}(x-1))
= n_{L,T}(x-1,x).
\end{align*}
Therefore by (\mathbb{R}f{eq:lineabove}), and then (\mathbb{R}f{eq:pm1})
\eq
n_{R,r}(x+1,x)+1\le n_{R,r}(x,x+1)+I_{x+1\le 0}\le n_{R,r}(x+1,x)+I_{R_r\ge x+1}.\nonumber
\en
Therefore $R_r\ge x+1$, so in fact
$R_t>x$ for every $t\ge r$. Moreover $n_{R,r}(x)=n_R(x)<n_L(x)$, which shows (b).
\qed
\begin{LEM}
\leftarrowbel{lem:main1}
Let $x\in \Zbold$, and suppose that for some $k>0$, $n_L(x)\ge k$ and $n_R(x)\ge k$. Then $n_{R,T_R(x,k)}(x-1)\le n_{L,T_L(x,k)}(x-1)$.
\end{LEM}
\proof Let $T=T_L(x,k)<\infty$ and $S=T_R(x,k)<\infty$. Then $R_S=x>x-1$, so from (\mathbb{R}f{eq:pm1}) and (\mathbb{R}f{eq:R2})
\[n_{R,S}(x-1,x)=n_{R,S}(x,x-1)+I_{x\ge 1}=\mc{R}_{\leftarrow}(x,k-1)+I_{x\ge 1}.\]
Similarly
\[n_{L,T}(x-1,x)=n_{L,T}(x,x-1)+I_{x\ge 1}=\mc{L}_{\leftarrow}(x,k-1)+I_{x\ge 1}.\]
Since $\mc{R}_{\leftarrow}(x,k-1)\le \mc{L}_{\leftarrow}(x,k-1)$ it follows that $n_{R,S}(x-1,x)\le n_{L,T}(x-1,x)$.
Finally,
\[\mc{R}_{\rightarrow}(x-1,n_{R,S}(x-1))=n_{R,S}(x-1,x) \text{ and }n_{L,T}(x-1,x)=\mc{L}_{\rightarrow}(x-1,n_{L,T}(x-1))\]
whence $\mc{R}_{\rightarrow}(x-1,n_{R,S}(x-1))\le \mc{L}_{\rightarrow}(x-1,n_{L,T}(x-1))$. Since the $n_{R,S}(x-1)$-th arrow at $x-1$ is $\rightarrow$ by definition of $S$ (and similarly for $n_{L,T}(x-1)$ and $T$) this implies that $n_{R,S}(x-1)\le n_{L,T}(x-1)$ as required.
\qed
\begin{LEM}
If $T=T_L(x,k)<\infty$ and $R$ stays to the right of $x$ after fewer than $k$ visits to $x$ then $n_R(x-1)\le n_{L,T}(x-1)$.
\leftarrowbel{lem:main1.5}
\end{LEM}
\proof Assume that $n_R(x-1)>0$, otherwise there is nothing to prove. Let $S'=\sup\{t:R_t=x\}$. Then $R_{S'}=x$, $\mc{R}(x-1,n_{R,S'}(x-1))=\rightarrow$ and $\mc{R}(x,n_{R,S'}(x))=\rightarrow$. By (\mathbb{R}f{eq:pm1}) applied at $x-1$, and then using (\mathbb{R}f{eq:R2}), and finally the fact that $\mc{R}(x,n_{R,S'}(x))=\rightarrow$,
\[n_{R,S'}(x-1,x)=n_{R,S'}(x,x-1)+I_{x\ge 1}=\mc{R}_{\leftarrow}(x,n_{R,S'}(x)-1)+I_{x\ge 1}=\mc{R}_{\leftarrow}(x,n_{R,S'}(x))+I_{x\ge 1}.\]
Therefore by (\mathbb{R}f{eq:R1}),
\eqalign
&\mc{R}_{\rightarrow}(x-1,n_{R,S'}(x-1))=n_{R,S'}(x-1,x)=\mc{R}_{\leftarrow}(x,n_{R,S'}(x))+I_{x\ge 1}.\leftarrowbel{eq:platypus}
\enalign
Since $n_{R,S'}(x)<k=n_{L,T}(x)$ we have $\mc{R}_{\leftarrow}(x,n_{R,S'}(x))\le \mc{L}_{\leftarrow}(x,n_{L,T}(x)-1)$, therefore the right hand side of (\mathbb{R}f{eq:platypus}) is bounded above by
\eqalign
\mc{L}_{\leftarrow}(x,n_{L,T}(x)-1)+I_{x\ge 1}&=n_{L,T}(x,x-1)+I_{x\ge 1}\nonumber\\
&=n_{L,T}(x-1,x)=\mc{L}_{\rightarrow}(x-1,n_{L,T}(x-1)),\nonumber
\enalign
where we have used (\mathbb{R}f{eq:R2}), followed by (\mathbb{R}f{eq:pm1}), and then (\mathbb{R}f{eq:R1}). We have shown that
\[\mc{R}_{\rightarrow}(x-1,n_{R,S'}(x-1))\le \mc{L}_{\rightarrow}(x-1,n_{L,T}(x-1)).\]
Since $\mc{R}(x-1,n_{R,S'}(x-1))=\rightarrow$, this implies that $n_{R,S'}(x-1)\le n_{L,T}(x-1)$ as required.
\qed
\subsection{Results obtained by contradiction}
\leftarrowbel{sec:contra}
The results in this section include less technical results than those of the previous section. Roughly speaking their proofs will be based on contradiction arguments that proceed as follows. Suppose that we have already proved a statement $A$ whenever $\mc{L}\CboldL \mc{R}$. We now want to prove a statement $B$ whenever $\mc{L}\CboldL \mc{R}$. Assume that for some $\mc{L}$, $\mc{R}$ with $\mc{L}\CboldL \mc{R}$, $B$ is false. Construct two new systems $\mc{L}'\CboldL \mc{R}'$ from $\mc{L}$ and $\mc{R}$ such that statement $A$ is violated for $\mc{L}'$ and $\mc{R}'$. This gives a contradiction, hence there was no such example where $\mc{L}\CboldL \mc{R}$ but $B$ is false.
\begin{LEM}
\leftarrowbel{lem:main2}
Let $x\in \Zbold$, and suppose that $n_R(x)<k\le n_L(x)$. Then $n_R(x-1)\le n_{L,T_L(x,k)}(x-1)$ and $\liminf R_n>x$ (i.e.~
$R$ is forever to the right of $x$ after fewer than $k$ visits to $x$ and at most $n_{L,T_L(x,k)}(x-1)$ visits to $x-1$).
\end{LEM}
\proof By Lemma \mathbb{R}f{lem:main1.5}, it is sufficient to prove that under the hypotheses of the lemma, $R$ is to the right of $x$ infinitely often. Suppose instead that $R$ is forever to the left of $x$ (after fewer than $k$ visits to $x$). Then we may define two new systems $\mc{R}'$ and $\mc{L}'$ by forcing every arrow at $x$ at level $k$ and above to be $\rightarrow$. To be precise, given an arrow system $\mc{E}$ we'll define $\mc{E}'$ by $\mc{E}'(y,\cdot)=\mc{E}(y,\cdot)$ for all $y\ne x$, $\mc{E}'(x,j)=\mc{E}(y,j)$ for all $j<k$, and $\mc{E}'(x,j)=\rightarrow$ for every $j\ge k$. Clearly $\mc{L}'\CboldL\mc{R}'$ and $T'=T_{L'}(x,k)=T$. The sequences $R$ and $R'$ are identical since we have not changed any arrow used by $R$ anyway. The sequences $L$ and $L'$ agree up to time $T$, while $L'_n\ge x$ for all $n\ge T$, since $L'$ can never go left from $x$ after time $T$. It follows that $n_{L'}(z)=n_{L,T}(z)<\infty$ for every $z<x$.
Let $y_1:=\max\{z<x:n_{R'}(z)\ge n_{L',T}(z)\}$. By Lemma \mathbb{R}f{lem:exists}, $-\infty<y_1<x$.
By Lemma \mathbb{R}f{lem:x-1,x} (applied to $L'$, $R'$) either $R'$ hits $y_1+1$ at least $n_{L'}(y_1+1) \ge n_{L,T}(y_1+1)$ times, or $R'$ is forever to the right of $y_1+1$ after fewer than $n_{L'}(y_1+1)$ visits. In either case, $y_1+1<x$ (as $n_{R'}(x)<k$ and $R'$ lies eventually to the left of $x$). So there exists some $y_2\in (y_1,x)$ such that $n_{R'}(y_2)\ge n_{L'}(y_2)=n_{L',T}(y_2)$. This contradicts the definition of $y_1$.
\qed
\begin{COR}
\leftarrowbel{cor:neighbour}
If $n_{R,t}(x-1)>n_{L,t}(x-1)$ then $n_{R,t}(x)\ge n_{L,t}(x)$.
\end{COR}
\proof Suppose instead that $n_{R,t}(x)< n_{L,t}(x)$. Let $k=n_{R,t}(x)+1$, so that $T=T_L(x,k)\le t$ and $S=T_R(x,k)>t$. Then
\[n_{R,S}(x-1)\ge n_{R,t}(x-1)>n_{L,t}(x-1)\ge n_{L,T}(x-1).\]
This violates Lemma \mathbb{R}f{lem:main1} (if $n_R(x)\ge k$) or Lemma \mathbb{R}f{lem:main2} (if $n_R(x)<k$).
\qed
\begin{COR}
\leftarrowbel{cor:right_hit_time}
Fix $x>0$, and let $T=T_{L}(x,1)=\inf\{t:L_t=x\}$ and $S=T_{R}(x,1)$. Then $S\le T$.
\end{COR}
\proof If $T=\infty$ then the result is trivial. So assume $T<\infty$. Lemma \mathbb{R}f{lem:main2} with $k=1$ implies that $S<\infty$ as well ($R$ cannot be to the right of $x>0$ without ever passing through $x$). For each $i<x$, the number of times that $L$ hits $i$ before $T$ is $n_{L,T}(i)$, so $T=\sum_{i=-\infty}^{x-1}n_{L,T}(i)$. Moreover, $n_{L,T}(i)$ is the number of times that $L$ hits $i$ before hitting $i+1$ for the $n_{L,T}(i+1)$-th time (by definition of $T$, the last visit to $i<x$ up to time $T$ occurs before the last visit to $i+1$ up to time $T$). By Lemma \mathbb{R}f{lem:main1} with $k=1$ we get that $n_{R,S}(x-1)\le n_{L,T}(x-1)$. Set $k_0=1$.
Now apply Lemma \mathbb{R}f{lem:main1} with $x-1$ instead of $x$ and with $k_1=n_{R,S}(x-1)$ to get
$$
n_{R,T_R(x-1,k_1)}(x-2)\le n_{L,T_L(x-1,k_1)}(x-2).
$$
But $n_{R,T_R(x-1,k_1)}(x-2)=n_{R,S}(x-2)$ since $R$ cannot visit $x-2$ at times in $(T_r(x-1,k_1), S]$ (in other words, the last visit to $x-2$ occurs before the last visit to $x-1$). Furthermore, $n_{L,T_L(x-1,k_1)}(x-2)\le n_{L,T}(x-2)$ since $n_{L,T}(x-1)\ge k_1\Rboldightarrow T_L(x-1,k_1)\le T$. We have just shown that
$$
n_{R,S}(x-2)=n_{R,T_R(x-1,k_1)}(x-2)\le n_{L,T_L(x-1,k_1)}(x-2)\le n_{L,T}(x-2).$$
Iterating this argument while $k_j=n_{R,S}(x-j)>0$ by applying Lemma \mathbb{R}f{lem:main1} at $x-j$ with $k=k_j$ (there is nothing to do once $n_{R,S}(x-j)=0$ for some $j$), we obtain by induction that $n_{R,S}(i)\le n_{L,T}(i)$ for every $i<x$.
Thus $S=\sum_{i=-\infty}^{x-1}n_{R,S}(i)\le \sum_{i=-\infty}^{x-1}n_{L,T}(i)= T$ as required.\qed
It follows immediately from Corollary \mathbb{R}f{cor:right_hit_time} that
\eq
\overline{R}_n:=\max_{k\le n}R_k\ge \max_{k\le n}L_k=:\overline{L}_n.
\leftarrowbel{eq:maxbound}
\en
Of course by mirror symmetry we also have $\underline{R}_n:=\min_{k\le n}R_k\ge \min_{k\le n}L_k=\underline{L}_n$. The following result extends this idea to the number of visits of the two paths to $\overline{R}_n$ by time $n$.
\begin{LEM}
\leftarrowbel{lem:maxvisits}
For each $t\ge0$, $n_{R,t}(\overline{R}_t)\ge n_{L,t}(\overline{R}_t)$ and $n_{L,t}(\underline{L}_t)\ge n_{R,t}(\underline{L}_t)$.
\end{LEM}
\proof Let $\mc{L}\CboldL \mc{R}$ and suppose the first claim fails. Let $T=\inf\{t\ge 0: n_{R,t}(\overline{R}_t)< n_{L,t}(\overline{R}_t)\}<\infty$.
Let $\mc{N}_t=n_{L,t}(\overline{R}_t)-n_{R,t}(\overline{R}_t)$. Then $\mc{N}_{t+1}-\mc{N}_{t}\le 1$ if $\overline{R}_{t+1}=\overline{R}_t$, and by
(\mathbb{R}f{eq:maxbound}),
$\mc{N}_{t+1}=0$ or $-1$ if $\overline{R}_{t+1}>\overline{R}_t$. Therefore by definition of $T$ we must have $R_T<\overline{R}_T$, $L_T=\overline{R}_T$, and $n_{L,T}(\overline{R}_T)=1+n_{R,T}(\overline{R}_T)$.
Moreover this happens regardless of the arrows of $\mc{L}$ or $\mc{R}$ at $\overline{R}_T$ above level $n_{R,T}(\overline{R}_T)$. Define new arrow systems $\mc{L}',\mc{R}'$ by setting all arrows at $\overline{R}_T$ at level $1+n_{R,T}(\overline{R}_T)$ and above to be $\rightarrow$. By construction $\mc{L}'\CboldL \mc{R}'$, and $(L_n,R_n)=(L_n',R_n')$ for $n\le T$. However $\overline{L}_{T+1}'=\overline{R}_T+1>\overline{R}_T=\overline{R}_{T+1}'$ which violates the fact that $\overline{R}'_n\ge \overline{L}'_n$ for all $n\ge 0$.
The second result follows by mirror symmetry.\qed
For each $z\in \Zbold$, $t \in \Zbold_+$, let $\overline{z}_t=\max(n_{L,t}(z),n_{R,t}(z))$.
\begin{LEM}
\leftarrowbel{lem:plus_minus_counts}
If there exist $t,y$ such that $R_t\le y<L_t$ and $n_{R,t}(y)>n_{L,t}(y)$ then $n_{R,t}(x)\ge n_{L,t}(x)$ for every $x\in [y,L_t]$.
\end{LEM}
\proof Suppose that $t$ and $y$ satisfy the above hypotheses, but the conclusion fails for some $x\in [y,L_t]$. In other words, $y<x\le L_t$ and $n_{R,t}(x)<n_{L,t}(x)$. Define new arrow systems $\mc{L}'$ and $\mc{R}'$ by setting:
\begin{itemize}
\item all arrows at $y$ at level $n_{R,t}(y)+I_{\{R_t\ne y\}}$ and above to be $\leftarrow $;
\item all arrows at $x$ at level $n_{L,t}(x)+I_{\{L_t\ne x\}}$ and above to be $\rightarrow$; and
\item for each $z>x$ set all arrows above level $\overline{z}_t$ to be $\rightarrow$.
\end{itemize}
The resulting arrow systems satisfy $\mc{L}'\CboldL\mc{R}'$ with $(L_n,R_n)=(L_n',R_n')$ for $n\le t$. By construction $L_n'\rightarrow \infty $ as $n\rightarrow \infty$, since $L_n'$ never again goes below $x$, and can make at most finitely many more $\leftarrow$ moves. But also $R'_n\le y$ for all $n\ge t$, which contradicts the fact that $\overline{R}'_n\ge \overline{L}'_n$ for all $n\ge 0$.\qed
We say that a sequence $\{L_n\}_{n\ge 0}$ on $\Zbold$ is {\em transient to the right} if for every $x\in \Zbold$ there exists $n_x\ge 0$ such that $L_n>x$ for all $n\ge n_x$ (i.e.~if $\liminf_{n\rightarrow \infty}L_n=+\infty$).
\begin{COR}
\leftarrowbel{cor:transience}
If $\liminf_{n\rightarrow \infty}L_n=+\infty$ then $n_R(x)\le n_L(x)$ for every $x$ and $\liminf_{n\rightarrow \infty}R_n=+\infty$.
\end{COR}
\proof Suppose that $L$ is transient to the right. Then $n_L(y)<\infty$ for each $y$. Suppose that for some $x$, $n_R(x)>n_L(x)$. Let $T=T_{R}(x,n_L(x)+1)$. Define new systems $\mc{L}'\CboldL \mc{R}'$ by setting every arrow at $x$ above level $n_L(x)$ to be $\leftarrow$. Then $L'=L$, so $L'\rightarrow \infty$, but $R_t'\le x$ for every $t\ge T$. This violates (\mathbb{R}f{eq:maxbound}) for $L'$, $R'$. Therefore $n_R(x)\le n_L(x)$ for every $x$, which establishes the first claim.
For the second claim, suppose that $R$ is not transient to the right. Then $R$ is either transient to the left or it visits some site $x$ infinitely often. In either case there is some site $x$ such that $n_R(x)>n_L(x)$ which cannot happen by the first claim.
\qed
\begin{COR}
\leftarrowbel{cor:greater_i_o}
$R\ge L$ infinitely often.
\end{COR}
\proof If $R$ is not bounded above, this follows by considering the times at which $R$ extends its maximum. It follows similarly if $L$ is not bounded below, using times at which $L$ extends its minimum. The only remaining possibility is that $R$ is bounded above and $L$ is bounded below, in which case by (\mathbb{R}f{eq:maxbound})
both paths visit only finitely many vertices. In this case consider the sets of vertices that $R$ and $L$ visit infinitely often.
Let $x_{\infty}=\sup\{z\in \Zbold:n_{R}(z)=\infty\}$ and
$y_{\infty}=\sup\{z\in \Zbold:n_{L}(z)=\infty\}$.
If $x_\infty<y_\infty$ then Lemma \mathbb{R}f{lem:main2} is violated (apply it to $x=y_\infty$ for $k>n_{R}(y_{\infty})$). Therefore $x_\infty\ge y_\infty$, so $R_t\ge L_t$ at all sufficiently large $t$ for which $R_t=x_\infty$.
\qed
\subsubsection{Proof of Theorem \mathbb{R}f{thm:main}}
To prove (i) we show that if $L_n\ge x$ for all $n$ sufficiently large, then $R_n\ge x$ for all $n$ sufficiently large. Suppose instead that $R_n<x$ infinitely often. Then choose $N$ sufficiently large so that $L_n\ge x$ for all $n\ge N$, but $R_N<x$ and $n_{R,N}(R_N)>n_{L,N}(R_N)$. Define two new arrow systems $\mc{L}',\mc{R}'$ by switching all arrows at $R_N$ from level $n_{R,N}(R_N)$ and above to be $\leftarrow$. Then $\mc{L}'\CboldL\mc{R}'$ but Lemma \mathbb{R}f{lem:plus_minus_counts} is violated, as is Corollary \mathbb{R}f{cor:greater_i_o}. This establishes (i). Applying (i) to $-\mc{R}\CboldL -\mc{L}$ establishes (ii).
If $R_n\ge x$ infinitely often then $\limsup R_n/a_n\ge \limsup x/a_n=0$. Thus the result is trivial unless there exists $0< M<\infty$ such that $\limsup L_n/a_n>M$.
Then $L_n$ visits infinitely many sites $>0$.
Let $T_i$ be the times at which $L$ extends its maximum, i.e.~$T_0=0$ and for $i\ge 1$, $T_i=\inf\{n>T_{i-1}:L_{n}=1+\max_{k<n}L_k\}$.
We first verify the (intuitively obvious) statement that $\frac{L_{T_i}}{a_{T_i}}>M$ infinitely often. If $\frac{L_{T_i}}{a_{T_i}}>M$ only finitely often then for all $i$ sufficiently large, $\frac{L_{T_i}}{a_{T_i}}\le M$. But for all $n\in [T_i,T_{i+1})$, $\frac{L_n}{a_n}\le\frac{ L_{T_i}}{a_n}\le \frac{L_{T_i}}{a_{T_i}}$. So $\frac{L_n}{a_n}\le M$ for all but finitely many $n$, contradicting the fact that $\limsup L_n/a_n>M$. Let $S_i$ be the times at which $R$ extends its max. By definition, $L_{T_i}=i=R_{S_i}$ and from Corollary \mathbb{R}f{cor:right_hit_time}, $i\le S_i\le T_i$. It follows immediately that for infinitely many $i$,
\eq
\frac{R_{S_i}}{a_{S_i}}\ge \frac{L_{T_i}}{a_{T_i}}>M,\nonumber
\en
whence $\limsup_{n \rightarrow \infty}\frac{R_n}{a_n}\ge M$. This establishes part (iii)
To prove (iv), suppose that (iv) does not hold, and let $\tau$ be the first time at which this fails. In other words
\[\tau=\inf\{t\ge 0: \text{ there exist } y, x<y \text{ such that } n_{R,t}(x)>n_{L,t}(x) \text{ and }n_{R,t}(y)< n_{L,t}(y)\}.\]
Let $x_0$ be the largest such $x$, i.e.~$x_0=\sup\{x\in \Zbold: n_{R,\tau}(x)>n_{L,\tau}(x), \exists y>x \text{ such that }n_{R,\tau}(y)<n_{L,\tau}(y)\}$ and $y_0=\inf\{y>x_0:n_{R,\tau}(y)<n_{L,\tau}(y)\}$. Then $x_0\le y_0-2$ or else Corollary \mathbb{R}f{cor:neighbour} is violated.
By definition of $x_0$ and $y_0$ we have $n_{R,\tau}(y_0-1)\ge n_{L,\tau}(y_0-1)$. Let $k=n_{L,\tau}(y_0)$. Then $n_{L,\tau}(y_0-1)\ge n_{L,T_L(y_0,k)}(y_0-1)$ so $n_{R,\tau}(y_0-1)\ge n_{L,T_L(y_0,k)}(y_0-1)$. On the other hand $n_{R,\tau}(y_0)<k$, so $\tau <T_R(y_0,k)$. If $R_{\tau}<y_0-1$ then $n_{R,T_R(y_0,k)}(y_0-1)\ge n_{R,\tau}(y_0-1)+1>n_{L,T_L(y_0,k)}(y_0-1)$. This contradicts one of the Lemmas \mathbb{R}f{lem:main1} or \mathbb{R}f{lem:main2} (depending on whether $n_R(y_0)\ge k$), so we must have instead that $R_\tau\ge y_0-1>x_0$. Therefore $n_{R,\tau-1}(x_0)=n_{R,\tau}(x_0)>n_{L,\tau}(x_0)\ge n_{L,\tau-1}(x_0)$. Similarly if $L_\tau>x_0+1$ we get a contradiction to the symmetric versions of Lemmas \mathbb{R}f{lem:main1} or \mathbb{R}f{lem:main2}, so we must have $L_\tau\le x_0+1<y_0$, and therefore $n_{L,\tau-1}(y_0)=n_{L,\tau}(y_0)>n_{R,\tau-1}(y_0)$. This contradicts the definition of $\tau$.
Finally, to prove (v), note that if $\mc{L}\CboldL \mc{R}$ then also $\mc{L}_+\CboldL \mc{R}_+$. If $\mc{R}$ is $0$-right recurrent, then $R_{+,n}=0$ infinitely often so $L_{+,n}=0$ infinitely often by (i).
\qed
\section{Counterexamples}
\leftarrowbel{sec:counterex}
\subsection{$L\trianglelefteq R$ does not imply that $\liminf\frac{L_n}{n}\le \liminf\frac{R_n}{n}$}
\leftarrowbel{sec:counterex1}
In general, $L\trianglelefteq R$ does not imply that $\liminf\frac{L_n}{n}\le \liminf\frac{R_n}{n}$, as we shall see in the following example.
Let us first define the two systems as follows, starting with $\mc{L}$. At $0$ the first three arrows are $\rightarrow$. At every $x>0$ the first two arrows are $\leftarrow$ and the next three arrows are $\rightarrow$. It is easy to check that such a system results in a sequence $L$ that takes steps with the pattern $\rightarrow \leftarrow \rightarrow \leftarrow \rightarrow$ repeated indefinitely (without ever needing to look at arrows other than those specified above). Thus $\lim_{n\rightarrow \infty}\frac{L_n}{n}=\frac{3-2}{5}=\frac{1}{5}$.
Let us now define a system $\mc{R}=\mc{R}(N)$, according to a parameter $N$ as follows. At $0$ the first three arrows are $\rightarrow$. At each site $x_k=x_k(N)$ of the form
\eq
x_k=\sum_{m=1}^kN^m-\sum_{m=1}^{k-1}\sum_{r=0}^m(-1)^{m-r}N^r, \qquad k\ge 1\leftarrowbel{eq:CEsites}
\en
the first arrow is $\leftarrow$ and the next two arrows are $\rightarrow$. At all remaining sites $x>0$, the first three arrows are $\rightarrow,\leftarrow,\rightarrow$. See Figure \mathbb{R}f{fig:counter} for parts of the systems $\mc{L}$ and $\mc{R}(3)$. By definition of these systems the arrows to the left of 0 and above those shown are irrelevant, so we can set them to be the same (for example, all $\rightarrow$).
\begin{figure}
\caption{On the left are parts of the systems $\mc{L}
\end{figure}
By construction $L\trianglelefteq R$ for each $N\ge 1$, but we will show that $\liminf\frac{R_n}{n}\le \frac{1}{2N+1}<\frac{1}{5}$ for $N\ge 3$ (also $\limsup\frac{R_n}{n}\ge \frac{N}{N+2}$).
The first site of the form (\mathbb{R}f{eq:CEsites}) is $x_1=N$. The walk $R$ first encounters a $\leftarrow$ at its first visit to this site and then sees a $\rightarrow$ at site 0 (second visit to 0). The walk $R$ then visits site $x_1$ for the second time, whence it sees a $\rightarrow$. It continues moving right, visiting every site between $x_1$ and $x_2$ exactly once before reaching $x_2$ at this point it sees a $\leftarrow$, moves to $x_2-1$ (for the second visit to that site) and continues seeing $\leftarrow$ at every site in $(x_1,x_2)$ until reaching $x_1$ for the third time. It then sees $\rightarrow$ at every site in $[x_1,x_2)$ (third visit to each of those sites), but also at every site in $[x_2,x_3)$ (second visit to $x_3$ and first visit to each site in $(x_3,x_4)$). Continuing in this way, the walk turns left at every $x_i$ on the first visit, and continues left (second visit at interior sites) until reaching $x_{i-1}$ for the third time, and then continues to go right until reaching $x_{i+1}$ for the first time.
At time $t_k=\sum_{m=1}^kN^m + \sum_{m=1}^{k-1}\sum_{r=0}^m(-1)^{m-r}N^r$ the walk is at position $x_{k}=\sum_{m=1}^{k}N^m-\sum_{m=1}^{k-1}\sum_{r=0}^m(-1)^{m-r}N^r$ for the first time. Simple calculations then give
\eqalign
\lim_{k\rightarrow \infty}\frac{R_{t_k}}{t_k}&=
\frac{N}{N+2},\nonumber
\enalign
which gives rise to the limit supremum claimed.
Similarly at times $s_k=\sum_{m=1}^kN^m + \sum_{m=1}^{k}\sum_{r=0}^m(-1)^{m-r}N^r$ the walk is at position $x_{k-1}=\sum_{m=1}^{k}N^m-\sum_{m=1}^{k}\sum_{r=0}^m(-1)^{m-r}N^r$ for the last time. After some simple calculations we obtain
\eqalign
\lim_{n\rightarrow \infty}\frac{R_{s_k}}{s_k}&=
\frac{1}{2N+1},\nonumber
\enalign
which gives rise to the limit infimum claimed.
\subsection{$L$ can be in the lead more than $R$}
\leftarrowbel{sec:counterex2}
Given two sequences $L$ and $R$ with $L\CboldL R$, let $A_{R,t}=\{n\le t: R_n>L_n\}$ and $A_{L,t}=\{n\le t: R_n<L_n\}$. It is not unreasonable to expect that for every $t\in \mathbb{N}$, $|A_{R,t}|\ge |A_{L,t}|$ which essentially says that $R$ is ahead of $L$ more than $L$ is ahead of $R$. It turns out that this does not hold even when $L\trianglelefteq R$.
To see this, consider the partial arrow systems $\mc{R}$ and $\mc{L}$ on the left hand side of Figure \mathbb{R}f{fig:leader}. These two systems differ only at the first arrow at $0$, whence $\mc{L}\trianglelefteq \mc{R}$ (if we set all other arrows to be equal, for example). The first 28 terms of the sequences $L$ and $R$ are plotted on the right of the figure. At any place where the solid line is above the dotted line, $R>L$. In particular $R_n>L_n$ only for $1\le n\le 7$. Similarly $L>R$ when the dotted line lies above the solid line, which happens at times $9,10,14,15,19,20,24,25,26$. Thus we have $|A_{R,25}|=7<8= |A_{L,25}|$ and similarly $|A_{R,26}|=7<9= |A_{L,26}|$.
\begin{figure}
\caption{On the left are parts of arrow systems $\mc{L}
\end{figure}
We can modify these systems slightly to get another interesting example. Define $\mc{R}'$ from $\mc{R}$ by switching the second arrow at 0 to $\leftarrow$, the first arrow at 1 to be $\rightarrow$ and setting the first arrow at 2 to be $\leftarrow$. Define $\mc{L}'$ from $\mc{L}$ by switching the first arrow at 1 to be $\rightarrow$ and setting the first arrow at 2 to be $\leftarrow$. The resulting partial systems satisfy $\mc{L}'\CboldL\mc{R}'$. At time $t=28$, $|A_{R,28}|<|A_{L,28}|$, the number of visits to each site is identical, and $L_{28}=R_{28}=0$ (see Figure \mathbb{R}f{fig:leader2}). This means we can define a system which repeats such a pattern indefinitely. We can add any common steps that we wish in between repetitions of this pattern and hence we can have recurrent, transient, or even ballistic sequences satisfying $L\CboldL R$ but
such that $t^{-1}(|A_{L,t}|-|A_{R,t}|)\rightarrow v>0$ as $t \rightarrow \infty$.
\begin{figure}
\caption{Paths $R'_n$ (solid) and $L'_n$ (dotted) with $L_n'\CboldL R_n'$ and $|A_{R,28}
\end{figure}
\blank{
x=c(0,1,2:-5,-6:-3,-4,-5:-2,-3,-4:-1,-2,-3:0)
x2=c(0:-5,-6:-3,-4,-5:-2,-3,-4:-1,-2,-3:2,1,0)
plot(0:28,x,type="l",xlab="",ylab="",main="")
lines(0:28,x2,lty=3)}
\blank{
\section{The excursion/genealogy perspective}
\leftarrowbel{sec:excursion}
Given a sequence $E=(E_n)_{n \in \Zbold_+}$ of integers with $E_0=0$ and $E_{n+1}-E_n\in \{-1,1\}$, an excursion of length $2k$ from 0 is a part of the sequence $E_m=0, E_{m+1}, \dots, E_{m+2k}=0$ (for some $k\in \mathbb{N}$ such that $E_{m+j}\ne 0$ for any $j\in \{1,2,\dots, 2k-1\}$. An excursion of finite length $2k$ defines a unique tree, and vice versa). In the case of an infinite excursion $E_m=0$, $E_{m+2k}\ne 0$ for all $k\in \Nbold$, the excursion defines part of an infinite tree-like structure. The relationship between random walk excursions and branching processes (random trees) has been well studied, beginning with Harris \cite{Harris52}. Indeed, some of the random walk models of Section \mathbb{R}f{sec:applications} have been studied via branching processes (see e.g.~\cite{KZ08} and the references therein).
In the context of our paper, the entire tree above, whether finite or infinite can be described in terms of our arrow system. The arrows at the origin can be considered as the great ancestors of every vertex in the tree. There are two kinds, the right arrows and the left arrows. Consider for the moment just the right arrows at 0. Let $Z^{(1)}_1$ denote the number of right arrows at 1, before the first left arrow. Similarly for $i\in \Nbold$, let $Z^{(i)}_1$ denote the number of right arrows between the $(i-1)$st and $i$th left arrows at $1$. For each $i\in \Nbold$, these $Z^{(i)}_1$ consecutive right arrows can be considered as the children of the $i$th right arrow at $0$. More generally if the number $Z^{(i)}_x$ of $\rightarrow$ between the $(i-1)$st and $i$th $\leftarrow$ at $x\in \mathbb{N}$ is considered as the number of children of the $i$th $\rightarrow$ at $x-1$, this describes a branching or tree structure.
As an alternative to the methods of Sections \mathbb{R}f{sec:appl_basic} and \mathbb{R}f{sec:contra}, one can try to prove these results by considering what happens to the corresponding branching structures when a left arrow at $x$ in a system $\mc{L}$ is exchanged with some right arrow above it (corresponding to $\mc{L}\CboldL \mc{R}$), or flipped to a right arrow. We have attempted this in some cases, but did not find significant simplifications.
The former change can be interpreted in terms of the branching structure by saying that an earlier labelled right arrow at $x-1$ has adopted one or more children, while subsequent right arrows at $x-1$ have had their children changed as well, some having adopted and some having given up children for adoption. In terms of excursions, as long as the first few (sufficiently many) excursions from $x$ are finite, this has the effect of simply switching the order of excursions from $x$. Excursions to the right appearing earlier than previously and excursions to the left appearing later. Otherwise an infinite excursion to the left from $x$ can vanish if an infinite excursion to the right supplants it. Right transience is equivalent to an infinite right excursion from every site $\ge 0$, which is in turn equivalent to having infinitely many generations of descent for the corresponding branching structure. Then for example Corollary \mathbb{R}f{cor:transience} can roughly be interpreted as coming from the fact that no children of any generation in the branching structure are lost by changes of the underlying arrow system via the relation $\CboldL$.
}
\section{Applications}
\leftarrowbel{sec:applications}
In this section we describe some of the applications of our main results in the theory of nearest neighbour self-interacting random walks, i.e.~sequences $(X_n)_{n\ge 0}$ of $\Zbold$-valued random variables (which may include projections of higher dimensional walks), such that $X_{n+1}-X_n\in \{-1,1\}$ a.s.~for every $n$. For each application, what we actually do is show that there is a probability space on which the relevant random walks live and on which they are related via the property $\CboldL$ or $\trianglelefteq $ almost surely. It is then clear that on that probability space the conclusions of Theorem \mathbb{R}f{thm:main} hold almost surely for the walks satisfying those relations.
Our original motivation for the present paper was in studying random walks in (non-elliptic) random environments in dimensions $d\ge 2$ (see e.g.~\cite{HS_RWDRE}). In \cite{HS_RWDRE} the authors apply Theorem \mathbb{R}f{thm:main} to random walks in i.i.d.~random environments such that for some diagonal direction $u$, with sufficiently large probability at each site there is a drift in direction $u$, and that almost surely there is no drift in direction $-u$. For such walks, the projection $R$ in direction $u$ can be coupled with a so-called 1-dimensional multi-excited random walk (see below) $L$ so that $L\trianglelefteq R$, and transience and positive speed results can be obtained for this projection, when the strength of the drift is sufficiently large.
Our results can also be applied to recurrent models. For example, given $\beta>-1$, let $X$ be a once-reinforced random walk (ORRW) on $\Zbold$ with reinforcement parameter $\beta$, i.e.~$X_0=0$ and
\eq
\leftarrowbel{eq:ORRW}
\mathbb{P}(X_{n+1}-X_n=1|\mc{F}_n)=\frac{1+\beta I_{\{X_{n}+1\in \vec{X}_{n-1}\}}}{2+\beta [I_{\{X_{n}+1\in \vec{X}_{n-1}\}}+I_{\{X_{n}-1\in \vec{X}_{n-1}\}}]}.\nonumber
\en
We can similarly define ORRW on $\Zbold^+$ by forcing the walk to step right when at $0$. Then it is possible to define a probability space on which there is a ORRW $X^+(\beta)$ for each $\beta>-1$ and such that $X^+(\beta)\trianglelefteq X^+(\zeta)$ whenever $\beta\ge \zeta>-1$. On this probability space the corresponding local times processes then satisfy the monotonicity property Theorem \mathbb{R}f{thm:main}(iv).
Most of our results, including that for random walks in random environments above, involve comparisons with so-called {\em multi-excited random walks in i.i.d.~cookie environments}. A cookie environment is an element ${\bf \omega}=(\omega(x,n))_{x \in \Zbold,n \in \mathbb{N}}$ of $[0,1]^{\Zbold\times \mathbb{N}}$. A (multi-)excited random walk in cookie environment ${\bf \omega}$, starting from the origin, is a sequence of random variables $X=\{X_n\}_{n\ge 0}$ defined on a probability space (and adapted to a filtration $\mathcal{F}_n$) such that $X_0=0$ a.s.~and
\[P_{\bf \omega}\big(X_{n+1}=X_n+1\big|\mc{F}_n\big)=\omega(x,\ell(n))=1-P_{\bf \omega}\big(X_{n+1}=X_n-1\big|\mc{F}_n\big),\]
where $\ell(n)=\ell_X(n)=\sum_{m=0}^n1_{\{X_m=X_n\}}$. In other words, if you are currently at $x$ and this is the $k$th time that you have been at $x$ then your next step is to the right with probability $\omega(x,k)$, independent of all other information. A random cookie environment ${\bf \omega}$ is said to be i.i.d.~if the random vectors $\omega(x,\cdot)$ are i.i.d. as $x$ varies over $\mathbb{Z}$.
Let ${\bf U}=(U(x,n))_{x \in \Zbold,n \in \mathbb{N}}$ be a collection of independent standard uniform random variables defined on some probability space. For each $x\in \Zbold$, $n\in \Nbold$, and each cookie environment ${\bf \omega}$ let
\eq
\mc{E}_{{\bf \omega},{\bf U}}(x,n)=\begin{cases}
\rightarrow, & \text{ if }U(x,n)<\omega(x,n)\\
\leftarrow, & \text{ otherwise.}\leftarrowbel{eq:randomL}\nonumber
\end{cases}
\en
Then $\mc{E}_{{\bf \omega},{\bf U}}$ is an arrow system determined entirely by the pairs $(\omega(x,n),U(x,n))_{x\in \Zbold,n \in \Nbold}$, and the corresponding walk $E=E_{{\bf \omega},{\bf U}}$ is an excited random walk in cookie environment ${\bf \omega}$. Given two cookie environments ${\bf \omega}$ and ${\bf \omega'}$ we write ${\bf \omega}\trianglelefteq {\bf \omega'}$ if $\omega(x,n)\le \omega'(x,n)$ for every $x\in \Zbold$ and $n\in \Nbold$. If ${\bf \omega}\trianglelefteq {\bf \omega'}$, then on the above probability space $\mc{E}_{{\bf \omega},{\bf U}}\trianglelefteq \mc{E}_{{\bf \omega'},{\bf U}}$ so Theorem \mathbb{R}f{thm:main} applies to the corresponding excited random walks.
For excited random walks in i.i.d.~cookie environments in 1 dimension, it is known up to a high level of generality that right transience and the existence of a positive speed $v>0$ do not depend on the order of the cookies (see e.g.~\cite{KZ08}). One might expect that the value of $v$ should depend on this order.
The main result of this section is Theorem \mathbb{R}f{thm:cookie_cl} below, which essentially states that one cannot decrease the ($\limsup$)-speed of a cookie random walk by swapping stronger cookies in a pile with weaker cookies that appear earlier in the same pile (and doing this at each site). In order to state the result precisely we require some further notation.
For each $x\in \Zbold$, let $\mc{A}_x$ denote a partition of $\Nbold$ into finite (non-empty) subsets. For any such partition we can order the elements of the partition as $\mc{A}_x=(A^1_x,A^2_x,\dots)$ (e.g.~according to the ordering of the smallest element in each $A^i_x$). Let $\mc{A}=(\mc{A}_x)_{x\in \Zbold}$ denote a particular collection of such partitions (indexed by $\Zbold$), and $\mc{P}$ denote the set of all such collections. Let $\mc{P}_n$ denote the set of such collections where every $A_x^s$ is a set containing at most $n$ elements.
Fix $\mc{A}\in \mc{P}$. Let $x\in \Zbold$, $s\in \Nbold$, $\omega$ be a cookie-environment, and $j,k\in A^s_x$ with $j\le k$. We say that $(j,k)$ is an {\em $(x,s,\omega)$-favourable swap} if $\omega(x,j)\le \omega(x,k)$.
Let $\omega(x,A^s_x)=(\omega(x,r))_{r\in A^s_x}$, and let $b=(j,k)$ be an $(x,s,\omega)$-favourable swap. Define $\omega_b(s,A^s_x)$ by,
\[\omega_b(s,r)=\begin{cases}
\omega(s,k), &\text{ if }r=j\\
\omega(s,j), &\text{ if }r=k\\
\omega(s,r), &\text{ if }r\in A^s_x\setminus\{j,k\}.\end{cases}\]
Then we say that $\omega_b(s,A^s_x)$ is the $A^s_x$-environment produced by the swap $b=(j,k)$, and write $\omega(x,A^s_x)\cb \omega_b(x,A^s_x)$. Given two cookie environments ${\bf \omega}$ and ${\bf \omega'}$, we say that ${\bf \omega'}$ is {\em an $\mc{A}$-permutation of ${\bf \omega}$} if for each $s$ and $x$, $\omega'(x,A^s_x)$ is a permutation of $\omega(x,A^s_x)$. If ${\bf \omega'}$ is an $\mc{A}$-permutation of ${\bf \omega}$ and
if also on every $A^s_x$, ${\bf \omega'}$ can be generated from ${\bf \omega}$ from a finite sequence of favourable swaps then we write ${\bf \omega}\CL^{\scriptscriptstyle\mc{A}}{\bf \omega'}$. More precisely ${\bf \omega}\CL^{\scriptscriptstyle\mc{A}}{\bf \omega'}$ if for every $x\in \Zbold$, $s\in \Nbold$, $j\le k$, there exists a finite sequence of pairs of $A_x^s$ indices $b_1,\dots,b_K$ (for some $K\ge 0$), and $A^s_x$-environments $(\omega_i(x,A^s_x))_{i=0}^K$ with $\omega_0(x,A^s_x)=\omega(x,A^s_x)$ and $\omega_K(x,A^s_x)=\omega'(x,A^s_x)$ such that $\omega_i(x,A^s_x)\overset{b_{i+1}}{\rightarrow} \omega_{i+1}(x,A^s_x)$ are favourable swaps for each $i=0,\dots,K-1$.
Given $\mc{A}\in \mc{P}$ and an environment ${\bf \omega}$, let $\underline{\bf \omega}_{\mc{A}}$ denote the environment obtained by permuting $\omega$ on each $A_x^s$ so that $\underline{\bf \omega}_{\mc{A}}(x,j)\le \underline{\bf \omega}_{\mc{A}}(x,k)$ for all $j,k \in A_x^s$ such that $j<k$. Note that $\underline{\omega}_{\mathcal{A}}(x,A_x^s)$ can be obtained from $\omega(x,A_x^s)$ by a sequence consisting of at most $|A_x^s|-1$ swaps that are not favourable: first perform the swap that moves the largest $\omega(x,k)$ for $k\in A_x^s$ to the highest location in $A_x^s$, then proceed iteratively, always moving the next largest value to the next highest location. Reversing this procedure generates $\omega(x,A_x^s)$ from $\underline{\omega}_{\mathcal{A}}(x,A_x^s)$ by a sequence of (at most $|A_x^s|-1$) favourable swaps, so that $\underline{\omega}_{\mathcal{A}} \preccurlyeq^{\mathcal{A}}\omega$.
For fixed ${\bf \omega}$ and for any finite subset $A_x\subset \Nbold$, let $\{V_i\}_{i \in A_x}$ be a collection of i.i.d.~standard uniform random variables and define $N_{A_x}=\sum_{i\in A_x}I_{V_i\le \omega(x,i)}$ (which can be thought of as the number of right arrows generated by $\omega(x,A_x)$). Note that the law of $N_{A_x}({\bf \omega})$ is invariant under permutations of the indices in the set $A_x$, so that $q_{{\bf \omega},A_x}(y)=\mathbb{P}(N_{A_x}({\bf \omega})=y)$ is invariant under such permutations.
\begin{THM}
\leftarrowbel{thm:cookie_cl}
Let $\mc{A}\in \mc{P}_3$ and let ${\bf \omega}$ be a cookie environment. Then there exists a probability space on which: for each $\mc{A}$-permutation ${\bf \omega'}$ of $\underline{\bf \omega}_{\mc{A}}$ there is an excited random walk $E_{\bf \omega'}$ in environment ${\bf \omega'}$, defined such that $E_{\bf \omega'}\CboldL E_{\bf \omega''}$ almost surely whenever ${\bf \omega'}\CL^{\scriptscriptstyle\mc{A}}{\bf \omega''}$.
\end{THM}
\proof Let ${\bf U}=\{U_{x,s}\}_{x\in \Zbold,s\in \Nbold}$ be i.i.d.~standard uniform random variables, and ${\bf Y}=\{Y_{x,s}\}_{x\in \Zbold,s\in \Nbold}$ be independent random variables (independent of ${\bf U}$) where $Y_{x,s}$ has the law of $N_{A_x^s}({\bf \omega})$ for each $x,s$.
Let $x\in \Zbold$ and $s\in \Nbold$ and consider the set $A_x^s$, which contains $n=|A_x^s|\le 3$ elements. Without loss of generality let us assume that $A_x^s=\{1,\dots,n\}$. Let $y=Y_{x,s}$ and note that (since $n\le 3$) the set $S_{n,y}$ of $n$-stacks (an $n$-stack is any element of $\{\leftarrow,\rightarrow\}^n$) containing exactly $y$ right arrows is a completely ordered set (under $\CboldL$) of cardinality $n_y={n \choose y}$. Let $(a_1^{(y)},\dots, a_{n_y}^{(y)})$ be the reverse ordering of the set (so that $a_1^{(y)}$ is the element
consisting of $y$ right arrows underneath $n-y$ left arrows), and let $a_i^{(y)}(j)$ be the $j$th arrow of $a_i^{(y)}$.
Now for any $\mc{A}$-permutation ${\bf \omega'}$ of $\underline{\bf \omega}_{\mc{A}}$, define a probability measure $P_{{\bf \omega'}}$ on $S_{n,y}$ by setting
\[P_{{\bf \omega'}}(a_i^{(y)}) =\big(q_{{\bf \omega},A_x^s}(y)\big)^{-1}\prod_{j=1}^n\left[\omega'(x,j)I_{a_i^{(y)}(j)=\rightarrow}+(1-\omega'(x,j))I_{a_i^{(y)}(j)=\leftarrow}\right], \quad i=1,\dots, n_y.\]
This is the conditional probability of selecting (for the arrows corresponding to $A_x^s$) a particular configuration $a_i^{(y)}$ consisting of $y$ right arrows and $n-y$ left arrows, given that the configuration contains exactly $y$ right arrows and $n-y$ left arrows.
Define $\mc{E}_{{\bf \omega'}}(x,A_x^s)=(\mc{E}_{{\bf \omega'}}(x,j))_{j \in A_x^s}$ by
\[\mc{E}_{{\bf \omega'}}(x,A_x^s)=a_m^{(y)}, \quad \text{ if }\sum_{i=1}^{m-1}P_{{\bf \omega'}}(a_i^{(y)})<U_{x,s}\le \sum_{i=1}^{m}P_{{\bf \omega'}}(a_i^{(y)}).\]
Let ${\bf \omega'}$ and ${\bf \omega''}$ be $\mc{A}$-permutations of $\underline{\bf \omega}_{\mc{A}}$ with ${\bf \omega'}\CL^{\scriptscriptstyle\mc{A}}{\bf \omega''}$. Recall that $q_{{\bf \omega''},A_x^s}(y)=q_{{\bf \omega'},A_x^s}(y)$ by invariance under permutations. Also note that for every $m\le n_y$,
\[\sum_{i=1}^{m}P_{{\bf \omega''}}(a_i^{(y)})\ge \sum_{i=1}^{m}P_{{\bf \omega'}}(a_i^{(y)}),\]
so that under this coupling, $\mc{E}_{{\bf \omega'}}(x,A_x^s)=a_m^{(y)}\Rboldightarrow \mc{E}_{{\bf \omega''}}(x,A_x^s)=a_k^{(y)}$ for some $k\le m$. This means that $\mc{E}_{{\bf \omega'}}(x,A_x^s)\CboldL \mc{E}_{{\bf \omega''}}(x,A_x^s)$ when we consider $\CboldL$ on $A_x^s$ only.
Let us now summarize what we have achieved. For fixed $\mc{A}$ and ${\bf \omega}$, we have coupled arrow systems (and hence the corresponding walks) defined from all $\mc{A}$-permutations of $\underline{\bf \omega}_{\mc{A}}$ (including ${\bf \omega}$ itself) so that $\mc{E}_{{\bf \omega'}}(x,A_x^s)\CboldL \mc{E}_{{\bf \omega''}}(x,A_x^s)$ for each $x\in \Zbold$, $s\in \Nbold$ when ${\bf \omega'}\CL^{\scriptscriptstyle\mc{A}} {\bf \omega''}$, where the coupling took place independently (according to the variables ${\bf U}$ and ${\bf Y}$) for each $x,s$. It follows that for any such ${\bf \omega'}$, ${\bf \omega''}$, under this coupling, $\mc{E}_{{\bf \omega'}}\CboldL \mc{E}_{{\bf \omega''}}$. The result follows since for each $\mc{A}$ permutation ${\bf \omega'}$, the corresponding walk $E_{{\bf \omega'}}$ has the law of an excited random walk in cookie environment ${\bf \omega'}$.
\qed
Note that in the statement (and proof) of Theorem \mathbb{R}f{thm:cookie_cl} the probability space depends on $\mc{A}$ and ${\bf \omega}$ and is constructed in such a way that each $A^s_x$ has the same number of right arrows under ${\bf \omega}$ as under ${\bf \omega}'$ (and likewise left arrows). If $\mc{A}\in \mc{P}_2$, which corresponds to considering only disjoint transpositions/swaps, then the above proof can be simplified slightly, and the probability space defined independently of ${\bf \omega}$). The coupling is then defined on $\mc{A}_x^s=(j,k)$ for each ${\bf \omega}$ by
\eq
(\mc{E}(x,j),\mc{E}(x,k))=\begin{cases}
(\rightarrow,\rightarrow), & \text{ if }U_{x,k,j}<\omega(x,j)\omega(x,k)\\
(\rightarrow,\leftarrow), & \text{ if }\omega(x,j)\omega(x,k)\le U_{x,k,j}<\omega(x,j)\\
(\leftarrow,\rightarrow), & \text{ if }\omega(x,j)\le U_{x,k,j}<\omega(x,j)+\omega(x,k)(1-\omega(x,j))\\
(\leftarrow,\leftarrow), & \text{ otherwise.}\leftarrowbel{eq:randomL2}
\end{cases}
\en
This works because the set of 2-stacks is totally ordered according to $\CboldL$ as
\[\begin{matrix}\rightarrow\\ \rightarrow \end{matrix}\quad \succcurlyeq \quad
\begin{matrix}\leftarrow\\ \rightarrow \end{matrix}\quad \succcurlyeq \quad
\begin{matrix}\rightarrow\\ \leftarrow \end{matrix}\quad \succcurlyeq \quad
\begin{matrix}\leftarrow\\ \leftarrow \end{matrix}\]
so there is no need to define the random variables $Y_{x,s}$ whose laws depend on ${\bf \omega}$. If on the other hand we relax the condition that $\mc{A}\in \mc{P}_3$ to $\mc{A}\in \mc{P}_4$ the proof breaks down because e.g.~the $4$-stacks $\RboldLLR$ and $\leftrightarrowRL$ are not ordered by $\CboldL$. However, by considering finite sequences of favourable swaps, we can obtain the following theorem.
\begin{THM}
\leftarrowbel{thm:cookie_cl2}
Let $\mc{A}\in \mc{P}$ and let ${\bf \omega}\CL^{\scriptscriptstyle\mc{A}} {\bf \omega'}$ be two cookie environments. Then there exists a probability space on which there are excited random walks $E_{\bf \omega}$ and $E_{\bf \omega'}$ in environments ${\bf \omega}$ and ${\bf \omega'}$ respectively, defined such that $E_{\bf \omega}\CboldL E_{\bf \omega'}$ almost surely.
\end{THM}
\proof
Fix $x\in \Zbold$, $s\in \Nbold$. Then ${\bf \omega'}(x,A_x^s)$ can be obtained from ${\bf \omega}(x,A_x^s)$ by a finite sequence of favourable swaps $\omega_i(x,A^s_x)\overset{b_{i+1}}{\rightarrow} \omega_{i+1}(x,A^s_x)$, $i=0,\dots, K_x^s-1$, with $\omega_0(x,A^s_x)=\omega(x,A^s_x)$ and $\omega_{K_x^s}(x,A^s_x)=\omega'(x,A^s_x)$. Using the coupling in Theorem \mathbb{R}f{thm:cookie_cl} for a single favourable swap on $A_x^s$, for each $i$ we can define a probability space with finite chunks of random arrow systems $(\mc{E}_i(x,A_x^s),\mc{E}_i'(x,A_x^s))$ with marginal laws defined by $\omega_i(x,A^s_x)$ and $\omega_{i+1}(x,A^s_x)$ respectively, and such that $\mc{E}_i(x,A_x^s)\CboldL \mc{E}_i'(x,A_x^s)$.
Let $(X_1,Y_1)$ and $(Y_2,Z_2)$ be random quantities (not necessarily defined on the same probability space) such that $Y_1$ and $Y_2$ have the same distribution. Then we can construct $X_3$, $Y_3$, and $Z_3$ on a common probability space by letting $Y_3\sim Y_1\sim Y_2$, and letting $X_3$ and $Z_3$ be conditionally independent given $Y_3$, with conditional laws the same as $X_1$ given $Y_1$ and $Z_2$ given $Y_2$ respectively. Iterating this construction, and applying the resulting coupling to the random objects $\mc{E}_i(x,A_x^s)$, we can construct a probability space on which there are finite chunks of random arrow systems $\mc{E}_i(x,A_x^s)$ with marginal laws defined by $\omega_i(x,A^s_x)$, $i=0,\dots, K_x^s$, such that $\mc{E}_i(x,A_x^s) \CboldL \mc{E}_{i+1}(x,A_x^s)$ for each $i$. Taking the product probability space over $x\in \Zbold$ and $s\in \Nbold$, and letting $\mc{E}=(\mc{E}_0(x,A_x^s))_{x\in \Zbold, s\in \Nbold}$ and $\mc{E}'=(\mc{E}_{K_x^s}(x,A_x^s))_{x\in \Zbold, s\in \Nbold}$, we have that $\mc{E}\CboldL \mc{E}'$. Defining $E_{\omega}$ and $E_{\omega'}$ to be the corresponding walks gives the result.
\qed
Since Theorems \mathbb{R}f{thm:cookie_cl} and \mathbb{R}f{thm:cookie_cl2} are defined rather abstractly, we now give an explicit example.
Suppose that ${\bf \omega}$ is an environment defined by $\omega(x,2k-1)=p_1$ and $\omega(x,2k)=p_2$ for every $x\in \Zbold$, $k\in \Nbold$, with $p_2>p_1$. Suppose also that we wish to understand the effect (on the asymptotic properties of the corresponding excited random walk) of switching the order of the first two cookies at every even site, or instead, of switching the values of $p_1$ and $p_2$ at even sites. In the first case the environment of interest is ${\bf \omega'}$ where $\omega'(x,1)=\omega(x,2)$ and $\omega'(x,2)=\omega(x,1)$ for each $x\in 2\Zbold$ and otherwise $\omega'(x,k)=\omega(x,k)$, while in the second case we have ${\bf \omega''}$ defined by $\omega''(x,2k-1)=\omega(x,2k)$ and $\omega''(x,2k)=\omega(x,2k-1)$ for all $x\in 2\Zbold$, $k\in \Nbold$ and otherwise $\omega'(x,k)=\omega(x,k)$. In this example the permutations of interest are composed of {\em disjoint} swaps/transpositions, and hence we can choose partitions consisting of sets containing at most 2 elements. For example, letting $A_x^s=\{2s-1,2s\}$ for each $x\in \Zbold$, $s\in \Nbold$ defines one particular choice (among many) of $\mc{A}$ for which ${\bf \omega'}$ and ${\bf \omega''}$ are $\mc{A}$-permutations of ${\bf \omega}$, and such that ${\bf \omega}\CL^{\scriptscriptstyle\mc{A}} {\bf \omega'}\CL^{\scriptscriptstyle\mc{A}} {\bf \omega''}$. Theorems \mathbb{R}f{thm:cookie_cl} and \mathbb{R}f{thm:main} then imply that e.g.~if $p_1\ge \hlf$ (so that the walks are not transient to the left) then the limsup speeds of the corresponding random walks satisfy $\overline{v}_{\omega}\le \overline{v}_{\omega'}\le \overline{v}_{\omega''}$.
\blank{Second example. For each $x$ let $A_x^s=\{1\}$ and inductively define $a_x^s=max\{u \in A_x^s\}$ and $A_x^{s}=\{a_x^s+1,\dots, a_x^s+s+1\}$. Let ${\bf \omega}$ be defined by $\omega(x,A_x^s)=(\frac{1}{s+1},\dots,\frac{s}{s+1})$. In other words the cookie sequence at every site looks like $\hlf,\frac13,\frac23,\frac14,\frac24,\frac34,\frac15,\frac25,\dots$. Then ${\bf \omega}\CL^{\scriptscriptstyle\mc{A}} {\bf \omega'}$ for any $\mc{A}$-permuation ${\bf \omega'}$ of ${\bf \omega}$.}
\blank{
\subsection{Walks with drift toward the origin}
Given a parameter $\beta>-1$, define a nearest-neighbour {\em once-reinforced random walk} (ORRW) $X=(X_n)_{n\ge 0}$ on $\Zbold$ by setting $X_0=o$, $\vec{X}_n=(X_0,\dots,X_n)$
and for $n\ge 1$,
\eq
\leftarrowbel{eq:ORRW}
\mathbb{P}(X_{n+1}-X_n=1|\mc{F}_n)=\frac{1+\beta I_{\{X_{n}+1\in \vec{X}_{n-1}\}}}{2+\beta [I_{\{X_{n}+1\in \vec{X}_{n-1}\}}+I_{\{X_{n}-1\in \vec{X}_{n-1}\}}]},
\en
where $x\in \vec{X}_n$ is notation for $x=X_i$ for some $i\le n$.
When $\beta>0$ this walk has a preference for stepping to locations that it has visited before. We can also define a one-sided version of this walk, i.e.~a ORRW $X^+=(X^+_n)_{n\ge 0}$ on $\Zbold_+$ by setting $X^+_0=o$,
and for $n\ge 1$, $\mathbb{P}(X^+_{n+1}-X^+_n=1|\mc{F}_n)=1$ if $X^+_n=0$ and otherwise exactly as in (\mathbb{R}f{eq:ORRW}).
An immediate Corollary of Theorem \mathbb{R}f{thm:recurrent}, and coupling to simple random walk, is the result that any random walk on $\Zbold$ that never experiences a drift away from the origin is recurrent, i.e.~if $\mathbb{P}\big((X_{n+1}-X_n)\cdot\text{sign}(X_n)\le 0\big|\mc{F}_n\big)\ge \hlf$ for all $n\ge 0$ almost surely then $\mathbb{P}(X_n=0 \text{ infinitely often})=1$. In particular the ORRW with $\beta\ge 0$ is recurrent
Our method can be used to prove some less obvious recurrence results (e.g.~versions of Lemmas \mathbb{R}f{lem:excited1} and \mathbb{R}f{lem:general} but for recurrence), where the random walk can sometimes experience a drift away from the origin, by coupling the appropriate random walk with a 1-dimensional recurrent multi-excited random walk.
Stronger results can be obtained in the one-sided context. We can couple various recurrent excited random walk models on $\Zbold_+$ together so that those with obviously smaller right drift are ``more recurrent" in terms of the number of visits to the origin by time $t$ for all $t$. Another example is contained in the following theorem.
\begin{THM}
\leftarrowbel{thm:+ORRWmono}
There exists a probability space on which
\begin{itemize}
\item for each $\beta>-1$ there is a once reinforced random walk $X^+(\beta)$ on $\Zbold_+$
\item $X^+(\beta)\trianglelefteq X^+(\zeta)$ whenever $\beta\ge \zeta$.
\end{itemize}
\end{THM}
\proof Let ${\bf U}=(U(x,n))_{x \in \Zbold,n \in \Nbold}$ be a family of i.i.d.~standard uniform random variables. Define an arrow system $\mc{I}_{\beta}$ as follows. Let $\mc{I}_{\beta}(o,k)=\rightarrow$ for all $k\in \Nbold$. Define $A_{x,i}(\beta)=\cup_{j=1}^i\{U(x,j)<1/(2+\beta)\}$. For $x>0$ define
\eq
\mc{I}_{\beta}(x,1)=\begin{cases}
\rightarrow , & \text{ if }U(x,1)<\frac{1}{2+\beta}\\
\leftarrow, & \text{ otherwise,}\end{cases}
\en
and for $k>1$
\eq
\mc{I}_{\beta}(x,k)=\begin{cases}
\rightarrow , & \text{ if }U(x,k)<\frac{1}{2+\beta}\\
\rightarrow , & \text{ if }U(x,k)<\frac12 \text{ and }A_{x,k-1}(\beta) \text{ occurs}\\
\leftarrow, & \text{ otherwise.}\end{cases}
\en
Suppose $\beta\ge \zeta>-1$. We claim that $\mc{I}_{\beta}\trianglelefteq \mc{I}_{\zeta}$. To see this note that
\[\mc{I}_{\beta}(x,1)=\rightarrow \quad \iff \quad U(x,1)<\frac{1}{2+\beta}\quad\Rboldightarrow \quad U(x,1)<\frac{1}{2+\zeta}\quad\iff \quad\mc{I}_{\zeta}(x,1)=\rightarrow.\]
Similarly $A_{x,i}(\beta)=\cup_{j=1}^i\{U(x,j)<1/(2+\beta)\}\subset \cup_{j=1}^i\{U(x,j)<1/(2+\zeta)\}=A_{x,i}(\zeta)$ so that
\eq
\left\{U(x,k)<\tfrac12 \right\}\cap A_{x,k-1}(\beta) \subset \left\{U(x,k)<\tfrac12 \right\}\cap A_{x,k-1}(\zeta),
\en
and hence $\big[\mc{I}_{\beta}(x,k)=\rightarrow \big]\Rboldightarrow \big[\mc{I}_{\zeta}(x,k)=\rightarrow\big]$ as required.
It remains to show that the corresponding sequence $I(\beta)$ is a once-reinforced random walk on $\Zbold_+$. To see this, suppose that at time $n$, $I$ is at $x>0$ for the $k$-th time.
If $I_{n}+1\notin \vec{I}_{n-1}$ then the first $k-1$ arrows at $x$ are $\leftarrow$, so that $A_{x,k-1}(\beta)$ does not occur. Then $\mathbb{P}(I_{n+1}-I_n=1|\mc{F}_n)=\mathbb{P}(U(x,k)<\frac{1}{2+\beta})=\frac{1}{2+\beta}$. Otherwise if $I_{n}+1\in \vec{I}_{n-1}$ then at least one of the arrows at $x$ up to level $n-1$ is $\rightarrow$, so $A_{x,k-1}(\beta)$ occurs and $\mathbb{P}(I_{n+1}-I_n=1|\mc{F}_n)=\mathbb{P}(U(x,k)<\frac12)=\frac{1}{2}$.
\qed
It follows immediately from Theorem \mathbb{R}f{thm:+ORRWmono} that all of the conclusions of Section \mathbb{R}f{sec:results} hold. For example, we have coupled once-reinforced random walks on $\Zbold_+$ with all parameter values $\beta>-1$, such that the number of visits to $o$ by time $t$ is monotone increasing in $\beta$ [by Lemma \mathbb{R}f{lem:maxvisits}], the maximum up to time $t$ is decreasing in $\beta$ [by (\mathbb{R}f{eq:maxbound})], and the joint distribution of the local times for all $x\ge 0$ and $\beta>-1$ satisfy (iv) of Theorem \mathbb{R}f{thm:main}. The first two results hold for the standard coupling of ORRW on $\Zbold_+$ (see below), under which $X_k^+(\beta)\le X_k^+(\zeta)$ for all $k$ and $\beta\ge \zeta$. But (iv) of Theorem \mathbb{R}f{thm:main} need not hold for that coupling, as can easily be seen by example.
\begin{REM}[Standard coupling for ORRW]
\leftarrowbel{rem:ORRW}
Let ${\bf U}=(U_n)_{n\ge 0}$ be a family of independent standard uniform random variables. Given $\beta>-1$ define $X_0^+=0$ and (conditionally on $X_0^+,\dots, X_{n}^+$), if $X_n^+=0$ then $X_{n+1}^+=1$, while if $X_n^+>0$ then
\eq
X_{n+1}^+=\begin{cases}
X_n^+-1 &, \text{ if }U_n<\hlf \text{ and }X_n^++1\in \vec{X}_{n-1}^+\\
X_n^+-1 &, \text{ if }U_n<\frac{1+\beta}{2+\beta} \text{ and }X_n^++1\notin \vec{X}_{n-1}^+\\
X_n^++1 &, \text{ otherwise}.\end{cases}
\en
One can show that $X_n^+(\beta)\le X_n(\zeta)^+$ when $\beta\ge \zeta>-1$.
\end{REM}
}
The ORRW is an example of a walk whose drift can depend on more than just the number of visits to the current site. For example, on $\Zbold_+$ the drift encountered by the ORRW at a site $x$ at time $n$ (so $X_n=x$) depends on whether the local time of the walk at $x+1$ is positive. Some of the known results for excited random walks in i.i.d.~or ergodic environments can be extended to more general self-interacting random walks (where the drifts may depend on the history in an unusual way) with a bounded number of positive drifts per site.
\begin{THM}
\leftarrowbel{thm:general}
Let $X_n$ be a nearest-neighbour self-interacting random walk and $\mc{F}_n=\sigma(X_k,k\le n)$. Suppose that there exist $M\in \mathbb{N}$ and $(\eta_k)_{k\le M}\in [0,1)^M$ such that
\begin{itemize}
\item $\mathbb{P}(X_{n+1}=X_n+1|\mc{F}_n)I_{\ell(n)=k}\le \eta_k$ for all $k\le M$ and all $n\in \Zbold_+$ almost surely, and
\item $\mathbb{P}(X_{n+1}=X_n+1|\mc{F}_n)I_{\ell(n)=k}\le \hlf$ for all $k> M$ and all $n\in \Zbold_+$, almost surely.
\end{itemize}
If $\alpha=\sum_{k=1}^M(2\eta_k-1)\le 1$ then $X$ is not transient to the right, almost surely. If $\alpha\le 2$ then $\limsup n^{-1}X_n\le 0$, almost surely. If $\alpha<-1$ then $X$ is transient to the left, almost surely. If $\alpha <-2$ then $\liminf n^{-1}X_n<0$ almost surely.
\end{THM}
\proof Define $\eta_k=\hlf $ for $k>M$. For each $x\in \Zbold$, let $\omega(x,k)=\eta_k$ for $k \in \Nbold$. Let ${\bf U}=(U(x,m))_{x\in \Zbold,m\in \Nbold}$ be i.i.d.~standard uniform random variables. and
define $\mc{R}$ by
\[\mc{R}(x,k)=\begin{cases}
\rightarrow &\text{ if }U(x,k)\le \eta_k\\
\leftarrow &\text{ otherwise.}
\end{cases}\]
The corresponding walk $R_n$ has the law of an excited random walk in the (non-random) environment $\omega$. By \cite{KZ08}, the conclusions of the theorem hold for the walk $R$, e.g.~if $\alpha=\sum_{k=1}^M(2\eta_k-1)\le 1$ then $R$ is not transient to the right, almost surely.
For a nearest neighbour sequence $x_0,\dots,x_n$ define
$$
P_{n,k}(x_0,\dots,x_n)=\mathbb{P}(X_{n+1}=X_n+1|X_0=x_0, \dots, X_n=x_n)I_{\ell_{x}(n)=k}.
$$
Define a nearest neighbour self-interacting random walk $L$ by setting $L_0=0$ and given that $\ell_{L}(n)=k$,
\[L_{n+1}=\begin{cases}
L_n+1 , & \text{ if } U(L_n,k)\le P_{n,k}(L_0,\dots,L_n)\\
L_n-1, &\text{ otherwise.} \end{cases}\]
Then $L$ has the law of $X$. Since $P_{n,k}\le \eta_k$ almost-surely, we have that $L\trianglelefteq R$ almost surely. The result now follows by Cor. \mathbb{R}f{cor:transience}.
The astute reader may have noticed that we have not defined the arrow system $\mc{L}$. We can do so, according to the walk $L$ as follows. Given that $\ell_{L}(n)=k$, define
\[\mc{L}(L_n,k)=\begin{cases}
\rightarrow, &\text{ if }U(L_n,k)\le P_{n,k}(L_0,\dots,L_n)\\
\leftarrow, &\text{ otherwise}.
\end{cases}\]
In other words, this inductively defines $\mc{L}$ as the arrow system determined by the steps of the walk $L$. Since $L$ does not define an entire arrow system at any site $x$ visited only finitely often by $L$ we can define $\mc{L}(x,k)=\leftarrow$ for each $k>n_{L}(x)$.
To be more precise, for each $n$ we can define $\mc{L}^{(n)}$ according to the arrow system determined by $L_0,\dots,L_n$ and adding $\leftarrow$ everywhere else. For each such $n$ we have $\mc{L}^{(n)}\trianglelefteq \mc{R}$, so that Theorem \mathbb{R}f{thm:main} (iv) holds for each $n$, and so does (\mathbb{R}f{eq:maxbound}). The former result implies the claims about transience when $\alpha\le 1$ and $\alpha<-1$, while (\mathbb{R}f{eq:maxbound}) and its minimum equivalent imply the remaining results (see e.g.~the proof of Theorem \mathbb{R}f{thm:main} (iii)).
\qed
\blank{
\begin{THM}
\leftarrowbel{thm:cookie_cl}
Let $\omega$ be a cookie environment such that $\omega(x,k)\ge \omega(x,j)$ for some $x\in \Zbold$, $j<k\in \Nbold$. Define $\omega'=\omega'(x,k,j)$ such that $\omega'(x,k)=\omega(x,j)$ and $\omega'(x,j)=\omega(x,k)$ and $\omega'(y,m)=\omega(y,m)$ for all $(y,m)\notin\{(x,k),(x,j)\}$. Then there exists a probability space on which $L=\{L_n\}_{n\ge 0}$ and $R=\{R_n\}_{n\ge 0}$ are multi-excited random walks in cookie environments ${\bf \omega}$ and ${\bf \omega'}$ respectively, such that $L\CboldL R$ with probability 1.
\end{THM}
\proof Given $\omega,x,k,j$ as in the conditions of the Theorem, let ${\bf U}=\big(U_{x,k,j}, (U(y,n))_{y \in \Zbold,n \in \mathbb{N}}\big)$ be a family of independent standard uniform random variables. We want to define environments $\mc{L}=\mc{L}_{{\bf \omega},{\bf U}}$ and $\mc{R}$ with $\mc{L}\CboldL \mc{R}$. Define $\mc{L}(y,n)$ for all $(y,n)\notin \{(x,k),(x,j)\}$ as in (\mathbb{R}f{eq:randomL}). Further define
\eq
(\mc{L}(x,j),\mc{L}(x,k))=\begin{cases}
(\rightarrow,\rightarrow), & \text{ if }U_{x,k,j}<\omega(x,j)\omega(x,k)\\
(\rightarrow,\leftarrow), & \text{ if }\omega(x,j)\omega(x,k)\le U_{x,k,j}<\omega(x,j)\\
(\leftarrow,\rightarrow), & \text{ if }\omega(x,j)\le U_{x,k,j}<\omega(x,j)+\omega(x,k)(1-\omega(x,j))\\
(\leftarrow,\leftarrow), & \text{ otherwise.}\leftarrowbel{eq:randomL2}
\end{cases}
\en
Finally define $\mc{R}=\mc{L}_{{\bf \omega'},{\bf U}}$ (i.e.~as above, except with $\omega'$ instead of $\omega$). Then
$\mc{L}$ and $\mc{R}$ have the same arrows everywhere, except possibly at $(x,k)$ and $(x,j)$. If $(\mc{L}(x,j),\mc{L}(x,k))=
(\rightarrow,\rightarrow)$ then $U_{x,k,j}<\omega(x,j)\omega(x,k)=\omega'(x,k)\omega'(x,j)$ so also $(\mc{R}(x,j),\mc{R}(x,k))=
(\rightarrow,\rightarrow)$. Otherwise if $(\mc{L}(x,j),\mc{L}(x,k))=
(\rightarrow,\leftarrow)$ then $U_{x,k,j}<\omega(x,j)\le \omega(x,k)=\omega'(x,j)$ so $\mc{R}(x,j)=\rightarrow$. This proves that $\mc{L}\CboldL\mc{R}$ (almost surely) as claimed. Finally, one can check that the sequences $L$ and $R$ are random walks in cookie environments $\omega$ and $\omega'$ respectively.
\qed
\begin{COR} Let $\omega, \omega', L, R$ be as in Thm \mathbb{R}f{thm:cookie_cl}. If $L$ is transient to the right then $\limsup L_n/n\le \limsup R_n/n$.
\leftarrowbel{cor:limsupcookiespeed}
\end{COR}
\proof Theorems \mathbb{R}f{thm:main} and \mathbb{R}f{lem:general}\qed
}
\blank{We write ${\bf \omega}\CboldL {\bf \omega'}$ if $\sum_{m=1}^n\omega(x,m)\le \sum_{m=1}^n\omega'(x,m)$ for every $x\in \Zbold$ and $n\in \Nbold$. Let ${\bf V}=(V(x,n))_{x\in \Zbold,n \in \Nbold}$ be an independent collection of Exponential$(1)$ random variables. For each $x\in \Zbold$ define $T_0(x)=0$ and for $r\in \Nbold$,
\eqalign
T_r(x)=\inf\Big\{n\ge 0:V(x,r)\le \sum_{i=T_{r-1}(x)+1}^n\omega(x,i)\Big\}.
\enalign
Define
\eqalign
\mc{L}(x,j)=\begin{cases}
\rightarrow &, \text{ if }j\in \{T_r(x):r\in \Nbold\}\\
\leftarrow &, \text{ otherwise.}
\end{cases}
\enalign
}
\end{document} |
\ve{e}gin{document}
\title{A Generalization \of the B\^{o} \abstract\noindent{The B\^{o}cher-Grace
Theorem can be stated as follows: Let $p$ be a third degree complex
polynomial. Then there is a unique inscribed ellipse interpolating
the midpoints of the triangle formed from the roots of $p$, and the
foci of the ellipse are the critical points of $p$. Here, we prove
the following generalization: Let $p$ be an $n^{th}$ degree complex
polynomial and let its critical points take the form
$$
\alpha+\ve{e}ta \cos k\pi/n, \quad k=1,\ldots,n-1, \quad\ve{e}ta\ne0.
$$
Then there is an inscribed ellipse interpolating the midpoints of
the convex polygon formed by the roots of $p$, and the foci of this
ellipse are the two most extreme critical points of $p$:
~$\alpha\pm\ve{e}ta \cos \pi/n$. }
\section{Introduction}
The B\^{o}cher-Grace Theorem has been discovered
independently by many mathematicians. Recently, proofs have been
given by Kalman~\cite{DKM}, and Minda and Phelps \cite{MP}. Maxime
B\^{o}cher proved the theorem in 1892 and then
John~H.~Grace proved it in 1902. Surprisingly, a significant
generalization was proved prior to both B$\hat{\mbox{o}}$cher and
Grace by Siebeck in 1864. He showed that the critical points of an
$n^{th}$ degree polynomial are the foci of the curve of class $n-1$
which touches each line segment, joining the roots of the
polynomial, at its midpoints. The B\^{o}cher-Grace
Theorem is the $n=3$ case of Siebeck's Theorem. Marden's book
\cite{M} gives a wonderful introduction to this material and it has
an extensive bibliography.
The purpose of this paper is to give a new and different
generalization the B$\hat{\mbox{o}}$cher-Grace Theorem.
\section{Background}
A \textit{conformal similarity transformation} in the complex plane $\mathbb{C}$ is a complex function that
takes the form
\ve{e}gin{eqnarray}\label{conformalDefinition}
S(z)=\alpha +\ve{e}ta \,z
\end{eqnarray}
for complex numbers $\alpha$ and $\ve{e}ta\ne0$. Upon writing
$\ve{e}ta=re^{i\theta}$ it is easy to see what action is taken upon $z$
when applying $S$: a rotation, a uniform scaling of the real and
imaginary parts of $z$, and a translation in the plane. It is
understood that conformal similarity transformations preserve the
eccentricity of ellipses, and for this reason any ellipse in the
complex plane can be mapped via a conformal similarity
transformation to an ellipse with foci $\pm1$.
When studying relationships between the critical points of a
polynomial and its roots, conformal similarity transformations play
a very important standardizing role. Specifically, if $S$ is a
conformal similarity transformation as defined in
equation~({\mathbb{R}}f{conformalDefinition}) and a complex polynomial
$p(z)=\prod_{k=1}^n(z-z_k)$ has critical points $r_k$,
$k=1,\ldots,n-1$, then the polynomial
$P(z)=\prod_{k=1}^n\ve{i}g(z-S(z_k)\ve{i}g)$ has critical points $S(r_k)$,
$k=1,\ldots,n-1$. A proof of this fact can be found in \cite{DKM}.
An \textit{affine transformation} in $\mathbb{C}$ is a complex function that takes the form
\ve{e}gin{eqnarray}\label{affineGeneral}
\Phi(z)=\alpha \,z +\ve{e}ta \,\overline{z}+\gamma
\end{eqnarray}
for complex numbers $\gamma$, $\alpha\ne0$ and $\ve{e}ta\ne0$. Unlike
the conformal similarity transformation it is less clear what action
is taken when applying $\Phi$ to $z$. However, if
$|\alpha|\ne|\ve{e}ta|$ then we can introduce real parameters $a>|b|>0$
and $\varphi,\theta\in(-\pi,\pi]$ so that
$$
\alpha=\frac{1}{2}(a+b)e^{i(\varphi+\theta)}\,\mbox{\text{ and }}\,
\ve{e}ta=\frac{1}{2}(a-b)e^{i(\varphi-\theta)}.
$$
With these new parameters $\Phi$ can be decomposed into three distinct components: a rotation, a
purely affine transformation, and a conformal similarity transformation. That is,
\ve{e}gin{eqnarray}\label{affineDecomposed}
\Phi(z)=(S\circ A\circ R)(z)
\quad\mbox{\text{ where }}
\ve{e}gin{cases}
R(z)= \displaystyle {e^{i\theta}z},\\
A(z)=\displaystyle{\frac{a+b}{2c}\,z+ \frac{a-b}{2c}\,\overline{z}},\\
S(z)= \displaystyle {c\,e^{i\varphi}z+\gamma},
\end{cases}
\end{eqnarray}
where $c=\sqrt{a^2-b^2}$. We refer to $A$ as a purely affine
transformation because it is the component of $\Phi$ that
distinguishes affine from conformal similarity transformations,
permitting independent scaling of the real and imaginary parts of
$z$. It is easy to verify, from equation~({\mathbb{R}}f{affineDecomposed}),
that
\ve{e}gin{eqnarray}
A(e^{it})=\frac{a}{c} \cos t+i\frac{b}{c}\sin t,
\end{eqnarray} and so the image of the unit circle, parametrized by $e^{it}$,
under the affine transformation $A$ is an ellipse with eccentricity
$c/a$ and foci $\pm1$. Additionally, we note that the images of the
rotated roots of unity $e^{i(\theta+2k\pi/n)}$, $k=1,\ldots,n$,
under $A$ are
\ve{e}gin{eqnarray}\label{rotatedRootImages}
\frac{a}{c} \cos \Big(\theta+\frac{2k\pi}{n}\Big)+i\frac{b}{c} \sin \Big(\theta+\frac{2k\pi}{n} \Big),
\quad k=1,\ldots,n.
\end{eqnarray}
We require several properties of affine transformations,
properties we summarize here, without proof.
Affine transformations are invertible if $|\alpha|\ne|\ve{e}ta|$.
Thus not only is the image of the unit circle under an affine transformation an ellipse,
but ellipses in $\mathbb{C}$ can be mapped onto the unit circle via an affine transformation.
Affine transformations preserve parallel lines, and preserve the midpoints of line segments.
\ve{e}gin{figure}[h!]
\centerline{\includegraphics[scale=.75]{simplifiedAffinePolygon.jpg}}
\caption{Rotated roots of unity; Their affine image under $A$.}
\label{FigAffinelyRegularPolygon}
\end{figure}
We shall call a convex $n$-gon $\cal{P}$ \textit{affinely regular} if it is the affine image of the regular convex
polygon formed by the $n$ roots of unity. In Figure {\mathbb{R}}f{FigAffinelyRegularPolygon} we illustrate an
affinely regular polygon in the case when $n=5$. At left is the application of $R(z)=e^{i\theta} z$ to the
roots of unity $e^{i2k\pi/n}$, $k=1,\ldots,n$; at right is the application of $A$ to these rotated roots of unity;
here $S(z)=z$. We also draw attention to the circumscribing and inscribing circles at left, and the corresponding
circumscribing and inscribing ellipses at right. The circles have radii 1 and $\cos\pi/n$, while the ellipses have
foci $\pm1$ and $\pm \cos\pi/n$, respectively. In light of the summarized properties of affine transformations,
we note that since the inscribing circle is tangent at the midpoints of the regular polygon pictured, the inscribed
ellipse is also tangent at the midpoints of the corresponding affinely regular polygon.
In fact the existence of an inscribed ellipse interpolating the midpoints of a convex polygon can be shown to
characterize affinely regular polygons. To see this, let $\cal{P}$ be a polygon with an inscribed ellipse
interpolating its midpoints, and let $\cal{P'}$ denote the convex polygon formed by the midpoints.
Let $\Phi$ be an affine transformation that maps the inscribed ellipse to the unit circle.
The images of the vertices of $\cal{P'}$ under $\Phi$ lie on the unit circle, and the sides of
$\Phi(\cal{P})$ are tangent to the unit circle. There are $2n$ right triangles that can be formed using the origin,
a vertex from $\Phi(P')$, and an adjacent vertex from $\Phi(\cal{P})$. It is not difficult to see that each of these
triangles is congruent to all the other such triangles. Hence, the vertices of $\Phi(\cal{P})$ all lie on a common
circle, implying that $\cal{P}$ is an affinely regular polygon.
Here we list some useful properties of the Chebyshev polynomials that can be found in \cite{MH}.
The \textit{Chebyshev polynomials} $T_n(x)$ of the first kind and $U_n(x)$ of the second kind can be defined by
\ve{e}gin{eqnarray}\label{realCheb}
T_n(\cos\theta)=\cos( n \theta),\mbox{\text{ and }}U_{n}(\cos\theta)=\frac{\sin (n+1) \theta}{\sin \theta },\quad n\ge0.
\end{eqnarray}
From equation~({\mathbb{R}}f{realCheb}) we see that $T_n'(x)=nU_{n-1}(x)$,
and that the $n-1$ roots of $U_{n-1}(x)$ are $\{\cos k\pi/n:
k=1,\ldots,n-1\}$. For complex $z$ the formula for $T_n$ is given
by
\ve{e}gin{eqnarray} \label{complexDef}
T_n(z)=\frac{1}{2}\ve{i}gg((z+\sqrt{z^2-1})^n+(z-\sqrt{z^2-1})^n\ve{i}gg).
\end{eqnarray}
From our decomposition of affine transformations in equation
({\mathbb{R}}f{affineDecomposed}), a general affine transformation can be
thought of first as a mapping of the unit circle to an ellipse with
foci $\pm1$, followed by a conformal similarity transformation.
There is effectively a one-to-one correspondence between the
functions $A$ and the family of ellipses with foci $\pm1$. This
family, parametrized by $s>0$, is given by
\ve{e}gin{eqnarray}\label{family}
\cosh s\cos t+i\sinh s\sin t, \quad 0\le t<2\pi.
\end{eqnarray}
The foci of each ellipse is $\pm1$ since $\cosh^2s-\sinh^2s=1$. The
eccentricity of each ellipse is $0<\mbox{\text{sech }} s<1$. Since conformal
similarities preserve the eccentricity of an ellipse, it follows
that any ellipse in the plane can be mapped by a conformal
similarity transformation to a member of this family.
A property of the Chebyshev polynomials $T_n(z)$ that deserves to be better known is that they are periodic on
the
family of ellipses parametrized in equation ({\mathbb{R}}f{family}). We explain this assertion in the following lemma.
\noindent {\bf Lemma:} Let $a>b>0$ with $c^2=a^2-b^2$. The complex function
\ve{e}gin{eqnarray}
f(t)=T_n\ve{i}gg(\frac{a}{c}\cos t+i \frac{b}{c}\sin t\ve{i}gg)
\end{eqnarray}
is periodic for real $t$, with period $2\pi/n$. Said another way, the Chebysev polynomial
$T_n(z)$ is periodic on the ellipse
\ve{e}gin{eqnarray}\label{ellipse}
\frac{a}{c}\cos t+i \frac{b}{c}\sin t,
\end{eqnarray}
taking on the value $\displaystyle{T_n\ve{i}gg(\frac{a}{c}\cos \theta+i \frac{b}{c}\sin \theta\ve{i}gg)}$
exactly $n$ times.
\noindent{\bf Proof:} Key to this result is the observation that if $a^2-b^2=1$ and if
$z=a \cos t +ib \sin t $, then
\ve{e}gin{eqnarray}\label{connection}
\sqrt{z^2-1}= b \cos t +ia \sin t
\end{eqnarray}
(see exercise 5, page 17 in \cite{MH}). Equation ({\mathbb{R}}f{connection}) implies that
\ve{e}gin{eqnarray}\label{zpm}
z~\pm~\sqrt{z^2-1}=(a~\pm~b)\cos t+i(a~\pm~b)\sin t.
\end{eqnarray}
Substituting equation~({\mathbb{R}}f{zpm}) into the formula
({\mathbb{R}}f{complexDef}) yields
\ve{e}gin{eqnarray}
\nonumber T_n(a \cos t+ib \sin t)=\frac{1}{2}\ve{i}gg((a + b)^n e^{i n
t} + (a - b)^n e^{-i n t}\ve{i}gg).
\end{eqnarray}
Of course, $e^{int}$ is periodic with period $2\pi/n$.
More generally, if $a^2-b^2=c^2\ne1$, then
\ve{e}gin{align}\label{theLink}
T_n\ve{i}gg(\frac{a}{c} \cos
\Big(\theta+\frac{2k\pi}{n}\Big)+i\frac{b}{c} \sin
\Big(\theta+\frac{2k\pi}{n} \Big)\ve{i}gg)=\frac{1}{2}
\ve{i}gg(\Big(\frac{a + b}{c}\Big)^n e^{i n \theta} + \Big(\frac{a -
b}{c}\Big)^n e^{-i n \theta}\ve{i}gg),\end{align} for $k=1,\ldots,n$.
$\color[rgb]{0,0,0}square$
At this stage we invite the reader to compare the formula for the
affine image of the rotated roots of unity under the affine
transformation $A$ in equation~({\mathbb{R}}f{rotatedRootImages}) with the
argument of $T_n$ in equation~({\mathbb{R}}f{theLink}). They are the same.
This means that $T_n$ is constant on the affine images of the
rotated roots of unity. This is true for every affine
transformation $A$, as defined in equation~({\mathbb{R}}f{affineDecomposed}).
In Figure {\mathbb{R}}f{ellipticalFamily} we illustrate some members of the family ({\mathbb{R}}f{family}).
In addition, we plot the $n=5$ images of a particular rotation of the $n$ roots of unity under
a particular affine transformation $A$. The polynomial $T_n$ is constant on these $n$ points.
Along the real axis we also mark the $n-1$ roots of $T'_n(z)=nU_{n-1}(z)$
\ve{e}gin{figure}[h!]
\centerline{\includegraphics[scale=.75]{familyOfEllipses.jpg}}
\caption{Ellipses with foci $\pm1$; Affine image of rotated roots of
unity.}\label{ellipticalFamily}
\end{figure}
\section{Main Result}
\noindent{\bf B$\hat{\mbox{\text{o}}}$cher-Grace Theorem for
Polygons:} Let $p$ be an $n^{th}$ degree complex polynomial and let
its critical points take the form
\ve{e}gin{eqnarray}\label{criticalPoints}
\alpha+\ve{e}ta \cos k\pi/n, \quad k=1,\ldots,n-1, \quad\ve{e}ta\ne0.
\end{eqnarray}
There is an inscribed ellipse interpolating the midpoints of the convex polygon formed by the roots of
$p$, and the foci of this ellipse are the two most extreme critical points of $p$: ~$\alpha\pm\ve{e}ta \cos \pi/n.$
\noindent{\bf Proof:} Assume that the critical points of $p$ take
the form in expression~({\mathbb{R}}f{criticalPoints}). Without loss of
generalization, exploiting properties of conformal similarity
transformations, we may assume $\alpha=0$ and $\ve{e}ta=1$. Thus
$p'(z)=\gamma U_{n-1}(z)$, and $p(z)= \gamma/n T_n(z)+\delta$, for
complex constants $\gamma\ne0 $ and $\delta $. If we designate by
$z_k$, $k=1,\ldots,n$, the roots of $p$, then
$T_n(z_k)=-n\delta/\gamma $, $k=1,\ldots,n$. That is to say, $T_n$
is constant on the $z_k$, $k=1,\ldots,n$. As noted in our
discussion, $T_n(z)$ can only be constant for $n$ distinct points in
$\mathbb{C}$, all of which lie on a single ellipse from the family
({\mathbb{R}}f{family}). Thus the $z_k$ must take the form
({\mathbb{R}}f{rotatedRootImages}). In turn this implies that the roots of
$p$ form an affinely regular polygon $\cal{P}$, and hence admit an
inscribed ellipse interpolating the midpoints of $\cal{P}$. In our
discussion of affinely regular polygons, we noted that the foci of
the inscribed ellipse are $\pm \cos\pi/n$, consistent with the
assertion of the theorem. $\color[rgb]{0,0,0}square$
When $n=3$ the critical points \textit{always} take the form $\alpha\pm \ve{e}ta\cos \pi/3$.
That is, all triangles are affinely regular; all triangles can be mapped to an equilateral
triangle with an affine transformation. This is the original B$\hat{\mbox{\text{o}}}$cher-Grace theorem.
When $n=4$ the critical points take the form $\alpha\pm \ve{e}ta\cos k\pi/4$, $k=1,2,3$, if and only if the roots of
$p$ form a parallelogram. Said another way, the only quadrilaterals that are affine images of squares are
parallelograms.
The form of the critical points in expression~({\mathbb{R}}f{criticalPoints})
can be used to characterize affinely regular polygons. That is, a
polygon $\cal{P}$ formed from the roots of a polynomial $p$ is
affinely regular if and only if the critical points of $p$ take the
form expression~({\mathbb{R}}f{criticalPoints}). With this observation we
conclude this article with a stronger statement than the generalized
B$\hat{\mbox{\text{o}}}$cher-Grace Theorem above.
\noindent{\bf B$\hat{\mbox{\text{o}}}$cher-Grace Characterization
Theorem:} Let $p$ be an $n^{th}$ degree polynomial and let $\cal{P}$
denote the convex polygon formed by the roots of $p$. The polygon
$\cal{P}$ admits an inscribed ellipse interpolating its midpoints if
and only if the critical points of $p$ take the form of the
expression expression~({\mathbb{R}}f{criticalPoints}).
\ve{e}gin{thebibliography}{}
\ve{i}bitem{Bocher}M.~B$\hat{\mbox{\text{o}}}$cher, {\sl Some propositions concerning the
geometric representation of imaginaries}, Ann. of Math., {\bf 7}(1892), pp 70-76.
\ve{i}bitem{Grace} J.~H.~Grace, {\sl The roots of a polynomial}, Proc. Cambridge Philos. Soc., {\bf 11}(1902), pp 352-76.
\ve{i}bitem{DKM} D.~Kalman, {\sl An Elementary Proof of Marden's Theorem},
American Mathematical Monthly, Vol 115, n4 April 2008, pp 330-338.
\ve{i}bitem{MH} J.~C.~Mason and David~C.~Handscomb, {\sl Chebyshev Polynomials}, Taylor \& Francis, 2002.
\ve{i}bitem{M} M.~Marden, {\sl Geometry of Polynomials}, Mathematical Surverys Number 3, AMS, 1966 second edition.
\ve{i}bitem{MP} D.~Minda and S.~Phelps, {\sl Triangles, Ellipses, and Cubic polynomials},
American Mathematical Monthly, Vol 115, n8 October 2008, pp 679-689.
\ve{i}bitem{Siebeck} J.~Siebeck, {\sl Ueber eine neue analytsche Behandlungweise der Brennpunkte}, J. Reine Angew.
Math., {\bf 64}(1864), p 175.
\end{thebibliography}
\end{spacing}
\end{document} |
\begin{document}
\title{The depth of Tsirelson's norm}
\author[K.~Beanland]{Kevin Beanland}
\address[K.~Beanland]{Department of Mathematics, Washington and Lee University, Lexington, VA 24450.}
\email{[email protected]}
\author[J.~Hodor]{J\c{e}drzej Hodor}
\address[J.~Hodor]{Theoretical Computer Science Department\\
Jagiellonian University\\
Kraków, Poland}
\email{[email protected]}
\thanks{2010 \textit{Mathematics Subject Classification}. Primary: }
\thanks{\textit{Key words}: Banach space, Tsirelson's norm, Schreier family, regular families}
\thanks{J.\ Hodor is partially supported by a Polish National Science Center grant (BEETHOVEN; UMO-2018/31/G/ST1/03718).}
\begin{abstract}
Tsirelson's norm $\|\cdot \|_T$ on $c_{00}$ is defined as the supremum over a certain collection of iteratively defined, monotone increasing norms $\|\cdot \|_k$.
For each positive integer $n$, the value $j(n)$ is the least integer $k$ such that for all $x \in \mathbb{R}^n$ (here $\mathbb{R}^n$ is considered as a subspace of $c_{00}$), $\|x\|_T = \|x\|_k$.
In 1989 Casazza and Shura \cite{CS-book} asked what is the order of magnitude of $j(n)$.
It is known that $j(n) \in \mathcal{O}(\sqrt{n})$ \cite{Beanland_2018}.
We show that this bound is tight, that is, $j(n) \in \Omega(\sqrt{n})$.
Moreover, we compute the tight order of magnitude for some norms being modifications of the original Tsirelson's norm.
\end{abstract}
\renewcommand\mathrm{co}ntentsname{}
\maketitle
\tableofcontents
\section{Introduction}
\label{sec:introduction}
In 1974 Tsirelson \cite{Tsirel_son_1974} constructed a reflexive Banach space containing no embedding of $c_0$ or $\ell_p$ for each $1 \leq p < \infty$.
The idea evolved throughout the years, and what is nowadays called Tsirelson's space is the dual of the original space, usually presented according to the description given in \cite{FJ-Tsirelson}.
Tsirelson's space not only served as a counterexample in Banach space theory but the inductive process Tsirelson developed for defining the norm eventually lead to many breakthroughs in several areas of mathematics.
See Tsirelson's webpage for an exhaustive list of publications concerning Tsirelson's space up to 2004 \cite{Ts-webpage}, we refer directly to some of the most notable ones \cite{OSc-Acta,GoMa-JAMS,Baudier_2018}.
We also refer to a monograph of Casazza and Shura on Tsirelson's space \cite{CS-book}.
Tsirelson's space is the completion of $c_{00}$ under a certain norm -- we call it Tsirelson's norm. Let $\mathcal{S}_1 := \{F \subset \mathbb{N} : |F| \leq \min F\} $ be the Schreier family \cite{Schreier}. We start by defining $\|\cdot\|_0$ as the supremum norm on $c_{00}$. Next, for each non-negative integer $m$ and for each $x \in c_{00}$ we define
\[\|x\|_{m+1} := \max\left\{\|x\|_{m}, \sup \left\{ \frac{1}{2} \sum_{i}^d \|E_i x\|_{m} : E_1 < \dots < E_d, \{\min E_i : i \in [d]\} \in \mathcal{S}_1 \right\} \right\}.\]
For subsets of integers $E,E'$ and $x \in c_{00}$, by $Ex$ we mean the coordinatewise multiplication of $x$ and the characteristic function of $E$, and by $E < E'$ we mean $\max E < \min E'$. See the next section for a more careful definition, however, generalized and stated in a slightly different spirit. Tsirelson's norm $\|x\|_{T}$ is defined as the supremum over $\|x\|_m$ for all non-negative integers $m$.
The above definition can be described more intuitively as a combinatorial game. A vector $x \in c_{00}$ is provided on input and the goal is to maximize the result. We start with the supremum norm. Then, in each step, we either take the current result or split the vector in some way dependent on the family $\mathcal{S}_1$.
However, if we choose to split, we must pay a penalty of multiplying the current result by $\frac{1}{2}$.
Next, we proceed with the same game on each part of the split, summing the results afterward.
It is not hard to see that for every $x \in c_{00}$ there exists a positive integer number $M$ such that the norm stabilizes starting from the $M$th step, that is, $\|x\|_M = \|x\|_{M+1} = \dots = \|x\|_{T}$. We denote such minimal $M$ as $j(x)$. Now, for a positive integer $n$, we define $j(n)$ to be the maximum value of $j(x)$ over all $x \in c_{00}$ such that $x_{n+1} = x_{n+2} = \dots = 0$. The function $j(n)$ measures the complexity of computing Tsirelson's norm for finite vectors. In the game interpretation, $j(n)$ is the value of the longest optimal strategy for an input of length $n$.
This concept was introduced and initially studied in 1989 by Casazza and Shura \cite{CS-book}.
They proved that $j(n) \in \mathcal{O}(n)$, and asked for the exact order of magnitude of $j(n)$.
In 2017, Beanland, Duncan, Holt, and Quigley \cite[Theorem~3.17]{BDHQ} provided the first non-trivial lower bound, namely, they showed that $j(n) \in \Omega(\log n)$.
One year later, Beanland, Duncan, and Holt \cite{Beanland_2018} proved that $j(n) \leq \mathcal{O}(\sqrt{n})$.
In this paper, we finally resolve the question of Casazza and Shura, by proving that $\Omega(\sqrt{n}) \leq j(n)$ (see \cref{cor:classic-schreier}). Combining the two results, we obtain the following.
\begin{theorem}\label{th:main-Schreier}
For every positive integer $n$ we have
\[\sqrt{2n} - 3 \leq j(n) \leq 2\sqrt{n} + 4.\]
\end{theorem}
In the remaining part of the introduction, we discuss some natural modifications of Tsirelson's norm.
From a modern point of view, the choice of $\frac{1}{2}$ and $\mathcal{S}_1$ in the definition of Tsirelson's norm may seem a little bit artificial.
Indeed, one can replace $\frac{1}{2}$ by any real number $0 < \theta < 1$ and $\mathcal{S}_1$ by any regular family $\mathcal{F}$ to obtain a norm $\|\cdot \|_{T[\theta,\mathcal{F}]}$, and in turn a Banach space $T[\theta,\mathcal{F}]$ -- see e.g.\ \mbox{\cite[Chapter~1]{ATo-book}~or~\cite[Chapter~3]{ATol-Memoirs}}.
The generalized Tsirelson's spaces have many interesting properties and they are connected to various branches of mathematics, e.g.\ to logic \cite{Bellenot_1984}.
The notion of regular families is a natural abstraction of the crucial properties of the Schreier family -- see the next section for the definition.
For the generalized version of the norm, one can still define the function $j_{\theta,\mathcal{F}}$ in an analogous way, and again ask for the order of magnitude.
In this paper, we stick to $\theta = \frac{1}{2}$, and we will consider various examples of regular families $\mathcal{F}$. Let us define them now.
First, for every $\varphi$ increasing and superadditive function on positive integers, we define
\[\mathcal{S}_\varphi := \{F \subset \mathbb{N} : |F| \leq \varphi(\min F)\}.\]
Note that $\mathcal{S}_{\mathrm{id}} = \mathcal{S}_1$.
This is a very natural generalization of the Schreier family.
Some properties for some particular cases of $\varphi$ were studied in \cite{BCF21}.
It is also worth noting that the collection of Banach spaces $T[\frac{1}{2},\Sf]$ gained the attention of the research community in connection with the meta-problem of so-called explicitly defined Banach spaces \cite{Go-blog, Khanaki_2021, casazza2022nondefinability}.
Let $k$ be a positive integer, we consider the family consisting of unions at most $k$ Schreier sets, namely, we define
\[k\mathcal{S}_1 := \{F \subset \mathbb{N} : \exists_{E_1,\dots,E_k\in \mathcal{S}_1} F = \bigcup_{i=1}^k E_i\}.\]
Some properties of such families were studied in \cite{BGHH22}. Moreover, $k \mathcal{S}_1$ can be seen as the so-called convolution of the family $\mathcal{S}_1$ with the family consisting of all sets with at most $k$ elements.
By convoluting regular families, one can produce many Banach spaces with interesting properties -- see~\cite{AlA-Dissertationes}~or~\cite[Chapter~2]{ATol-Memoirs}.
We consider two more examples of regular families constructed in a similar spirit.
The first one, denoted by $\mathcal{S}_2$, is the convolution of the Schreier family with itself, and the second one is the convolution of the Schreier family with $\mathcal{S}_2$.
We have
\begin{align*}
\mathcal{S}_2 &:= \{F \subset \mathbb{N} : \exists_{E_1,\dots,E_\ell\in \mathcal{S}_1} F = \bigcup_{i=1}^\ell E_i, \{\min E_i : i \in [\ell]\} \in \mathcal{S}_1\},\\
\mathcal{S}_3 &:= \{F \subset \mathbb{N} : \exists_{E_1,\dots,E_\ell\in \mathcal{S}_2} F = \bigcup_{i=1}^\ell E_i, \{\min E_i : i \in [\ell]\} \in \mathcal{S}_1\}.
\end{align*}
For each of the above families, we give the exact order of magnitude for the function $j_\mathcal{F}(n)$.
Note that in some cases we decided to present less technical proofs instead of obtaining better constants.
\begin{theorem}
Let $\varphi$ be an increasing and superadditive function on positive integers. For each positive integer $n$, let $\varphi_\Sigma^{-1}(n) = \min\{\ell \in \mathbb{Z} : n \leq \sum_{i=1}^\ell \varphi(i)\}.$ We have
\[j_{\Sf}(n) \in \mathbb{T}heta(\varphi_\Sigma^{-1}(n)).\]
For all positive integers $n,k$, let $p_k(n) = n^k$ and $e_k(n) = k^n$, then
\[j_{\mathcal{S}_{p_k}}(n) \in \mathbb{T}heta(n^{\frac{1}{k+1}}) \ \ \ \mathrm{and} \ \ \ j_{\mathcal{S}_{e_k}}(n) \in \mathbb{T}heta(\log_k n ).\]
For a fixed positive integer $k$ with $k \geq 2$, we have
\[ j_{k\mathcal{S}_1}(n) \in \mathbb{T}heta(\log n). \]
If $k$ is not fixed, then
\[j_{k\mathcal{S}_1}(n) \in \mathbb{T}heta(\frac{1}{k}\log n).\]
Last but not least, we have
\[j_{\mathcal{S}_2}(n) \in \mathbb{T}heta(\sqrt{\log n}) \ \ \ \mathrm{and} \ \ \ j_{\mathcal{S}_2}(n) \in \mathbb{T}heta(\sqrt{\log^* n}).\]
\end{theorem}
For convenience of the reader, we refer to the proof of each of the bounds in the following table.
\ \newline
\begin{center}
\begin{tabular}{ c|c|c }
& \ \ \ Lower bounds (\cref{sec:lower}) \ \ \ & \ \ \ Upper bounds (\cref{sec:upper}) \ \ \ \\ \hline
$\mathcal{S}_1$ & \cref{cor:classic-schreier} & \cite{Beanland_2018} or \cref{th:upper:S-y} \\ \hline
$\Sf$ & \cref{cor:lower:Sf} & \cref{th:upper:S-y} \\ \hline
$\mathcal{S}_p,\mathcal{S}_e$ & direct computation & direct computation\\ \hline
$k\mathcal{S}_1$ & \cref{cor:lower:kS1} & \cref{thm:upper:kS_1} \\ \hline
$\mathcal{S}_2$ & \cref{cor:lower:S2} & \cref{thm:upper:S2} \\ \hline
$\mathcal{S}_3$ & \cref{cor:lower:S3} & \cref{thm:upper:S3}
\end{tabular}
\end{center}
\ \newline
By $\log^* n$ we mean the iterated logarithm of $n$, that is, how many times do we have to take the logarithm of $n$ until we reach a number below $1$.
This function emerges often in computer science, and in particular in the field of analyzing the complexity of algorithms.
For example, the average time complexity of a query in the classical Find-Union data structure is of order $\log^* n$.
The iterated logarithm is an extremely slow-growing function, e.g.\ $\log^*(2^{65536}) = 5$.
Observe that one can define the families $\mathcal{S}_4,\mathcal{S}_5,\dots$ analogously to $\mathcal{S}_2$ and $\mathcal{S}_3$.
It is clear that the functions $j_{\mathcal{S}_k}$ for $k \geq 4$ are even slower growing than $\log^*$.
This indicates that these functions do not even have natural names, therefore, to avoid unnecessary technicalities, we decided not to consider these families.
However, we believe that our tools are sufficient to compute the order of magnitude functions of $j_{\mathcal{S}_k}$ for any positive integer $k$.
The paper is organized as follows. In the next section, we settle the required notation.
In \cref{sec:fullness}, we discuss special members in regular families called \emph{full} sets that are very useful in proving our results.
In \cref{sec:tools_lower}, we introduce some abstract tools for the lower bounds, and in \cref{sec:lower}, we use the tools to establish lower bounds on the function $j_\mathcal{F}(n)$ in the case of $\mathcal{F}$ being one of the families that we are interested in.
Next, in \cref{sec:tools_upper}, we introduce abstract tools for the upper bounds, and in \cref{sec:upper}, we use the tools to establish upper bounds on the function $j_\mathcal{F}(n)$ in the case of $\mathcal{F}$ being one of the families that we are interested in.
Finally, in \cref{sec:open}, we discuss some related open problems.
\section{Preliminaries}
\label{sec:preliminaries}
Let $\mathbb{N}$ be the set of all positive integers, and let $\mathbb{N}_0 := \mathbb{N} \cup \{0\}$.
For two integers $a,b$ with $a\leq b$ we write $[a,b]$ to denote the set $\{a,a+1,\dots,b\}$, if $a > b$, then $[a,b] := \emptyset$, and $[0] := \emptyset$.
For a positive integer $a$, we abbreviate $[a] := [1,a]$.
For any $E,F \subset \mathbb{N}$ the expression $E < F$ is a short form of writing that $\max E < \min F$, and similarly for $\leq,>,\geq$.
An inequality between $E$ and some $a \in \mathbb{N}$ should be understood as an inequality between $E$ and $\{a\}$.
For every $E \subset \mathbb{N}$ and for all distinct $a,b \in E$ we say that $a,b$ are \emph{consecutive in $E$} if $[a+1,b-1] \cap E = \emptyset$.
When we omit a base of a logarithm, we consider the base $2$.
Let $\tau$ be the power tower function, that is, for every real number $x$, we set $\tau(0,r) = r$, and for every $i \in \mathbb{N}$ we set $\tau(i,r) = 2^{\tau(i-1,r)}$.
Let $\log^*$ be the iterated logarithm, that is, for every real number $r$, we have $\log^* r = \min \{i \in \mathbb{N}_0 : r \leq \tau(i,1)\}$.
For a vector $x \in \mathbb{R}^\mathbb{N}$ we write $\mathrm{supp}\, x := \{i \in \mathbb{N} : x_i \neq 0\}$.
We write $c_{00}$ for all the vectors $x\in \mathbb{R}^\mathbb{N}$ such that $|\mathrm{supp}\, x| < \infty$.
We write $c_{00}^+$ for all nonzero vectors $x \in c_{00}$ such that $x_i \geq 0$ for all $i \in \mathbb{N}$.
For all $E \subset \mathbb{N}$ and $x \in c_{00}$ we write $Ex$ for the projection of $x$ onto $E$, that is, $(Ex)_i = x_i$ whenever $i \in E$ and $(Ex)_i = 0$ otherwise.
For each $i \in \mathbb{N}$ we define $e_i \in c_{00}^+$ to be the vector with $(e_i)_i=1$ and $(e_i)_j = 0$ for each $j \in \mathbb{N} \backslash \{i\}$.
For a linear functional $f:\mathbb{R}^\mathbb{N} \rightarrow \mathbb{R}$ we write $\mathrm{supp}\, f := \{i \in \mathbb{N} : f(e_i) \neq 0\}$.
For each $i \in \mathbb{N}$, the functional $e_i^*: \mathbb{R}^\mathbb{N} \rightarrow \mathbb{R}$ is such that for each $x \in c_{00}$ we have $e^*_i(x) = x_i$.
Let $\mathcal{F}$ be a family of finite subsets of $\mathbb{N}$.
We say that $\mathcal{F}$ is \emph{hereditary} if for every $F \in \mathcal{F}$ and $G\subset F$ we have $G \in \mathcal{F}$.
We say that $\mathcal{F}$ is \emph{spreading} if for every $n \in \mathbb{N}$ and for all $\ell_1,\dots,\ell_n,k_1,\dots,k_n \in \mathbb{N}$ such that $\ell_i \leq k_i$ for all $i \in [n]$, and $\{\ell_1,\dots,\ell_n\} \in \mathcal{F}$ we have $\{k_1,\dots,k_n\} \in \mathcal{F}$.
We say that $\mathcal{F}$ is \emph{compact} if it is compact as a subset of $\{0,1\}^\mathbb{N}$ with the product topology under the natural identification.
Finally, we say that $\mathcal{F}$ is \emph{regular} if it is hereditary, spreading, and compact.
Perhaps, the most prominent example of a regular family is the Schreier family $\mathcal{S}_1$ defined in the introduction. It is quite straightforward to check that all the families defined in the introduction are regular ($\mathcal{S}_\varphi$, $k\mathcal{S}_1$, $\mathcal{S}_2$, and $\mathcal{S}_3$).
Next, we proceed with introducing Tsirelson's norm. Fix a regular family $\mathcal{F}$. We define
\[W_0(\mathcal{F}) := \{e_i^* : i \in \mathbb{N}\} \cup \{-e_i^* : i\in \mathbb{N}\}.\]
For each $m \in \mathbb{N}_0$ we define
\begin{align*}
W_{m+1}(\mathcal{F}) &:=
W_m(\mathcal{F}) \ \cup\\& \left\{\frac{1}{2} \sum_{i=1}^d f_i : \ \ f_i \in W_m(\mathcal{F}), \ \ \{\min \mathrm{supp}\, f_i : i \in [d]\} \in \mathcal{F}, \ \ \mathrm{supp}\, f_1 < \dots < \mathrm{supp}\, f_d\right\}.
\end{align*}
We define the set of \emph{norming functionals for $\mathcal{F}$},
\[W(\mathcal{F}) := \bigcup_{m=0}^\infty W_m(\mathcal{F}).\]
As the name suggests, for all $x \in c_{00}$ and $m \in \mathbb{N}_0$ we define
\[\|x\|_{\mathcal{F},m} := \sup\{f(x) : f \in W_m(\mathcal{F})\}.\]
And finally, for every $x \in c_{00}$ we define the $\mathcal{F}$-Tsirelson's norm
\[\|x\|_{\mathcal{F}} := \sup\{\|x\|_{\mathcal{F},m} : m \in \mathbb{N}_0\}.\]
It is not hard to observe that the norm $\|\cdot \|_{\mathcal{S}_1}$ coincide with the norm $\|\cdot \|_T$ defined in the introduction (see e.g.\ \cite[Chapter~1]{ATo-book}).
The definition introduced in \cref{sec:introduction} is the classical definition, whereas the definition above is much more handy to work with.
For every $f \in W(\mathcal{F})$ the \emph{depth of $f$ with respect to the family $\mathcal{F}$} is
\[\mathrm{depth}_\mathcal{F}(f) := \min\{m \in \mathbb{N}_0 : f \in W_m(\mathcal{F})\}\]
Let $f \in W(\mathcal{F})$ and suppose that $\mathrm{depth}_\mathcal{F}(f) = m + 1$ for some $m\in \mathbb{N}_0$.
By definition, there exist $f_1,\dots,f_d \in W_m(\mathcal{F})$ such that $f = \frac{1}{2}\sum_{i=1}^d f_i$, the set $\{ \min \mathrm{supp}\, f_i : i \in [d]\}$ is in $\mathcal{F}$, and $\mathrm{supp}\, f_1 < \dots < \mathrm{supp}\, f_d$.
Note that in general, $f_1,\dots,f_d$ are not uniquely determined.
We say that $f_1,\dots,f_d$ are \emph{$\mathcal{F}$-building for $f$}.
For every $x \in c_{00}$ we define
\[j_\mathcal{F}(x) := \min \{\mathrm{depth}_\mathcal{F}(f) : \ \ f \in W(\mathcal{F}), \ \ f(x) = \|x\|_\mathcal{F}\}.\]
For all $a,b \in \mathbb{N}$ with $a \leq b$ we define
\[j_\mathcal{F}(a,b) := \max \{j_\mathcal{F}(x) : \ \ x\in c_{00}, \ \ \mathrm{supp}\, x \subset [a,b]\}.\]
It is not difficult to see that in the above definition, $c_{00}$ can be replaced with $c_{00}^+$ without changing any value of $j_\mathcal{F}(a,b)$. We will use this fact implicitly sometimes. Finally, for each $n \in \mathbb{N}$ we define
\[j_\mathcal{F}(n) := j_\mathcal{F}(1,n).\]
We will need the following simple observation on the behavior of the function $j_\mathcal{F}$.
\begin{obs}\label{obs:j-ab:ineq}
Let $\mathcal{F}$ be a regular family, and let $a,b,c,d \in \mathbb{N}$ such that $[a,b] \subset [c,d]$.
We have $j_\mathcal{F}(a,b) \leq j_\mathcal{F}(c,d)$.
\end{obs}
\begin{proof}
Let $x \in c_{00}$ with $\mathrm{supp}\, x \subset [a,b]$, and such that $j_\mathcal{F}(a,b) = j_\mathcal{F}(x)$. Then, $\mathrm{supp}\, x \subset [c,d]$, thus $j_\mathcal{F}(a,b) = j_\mathcal{F}(x) \leq j_\mathcal{F}(c,d)$.
\end{proof}
\section{Full sets in regular families}
\label{sec:fullness}
Given a positive integer, e.g.\ $a=10$, and a regular family $\mathcal{F}$, starting with $\{a\}$, one can greedily add consecutive integers to the set to determine the threshold after which the set is no longer a member of $\mathcal{F}$.
Say that $\mathcal{F}= \mathcal{S}_1$.
It is clear that $\{10,11,12,\dots,19\}$ is still in $\mathcal{S}_1$, however, $\{10,11,12,\dots,19,20\}$ is not.
On the other hand, if $\mathcal{F} = 2\mathcal{S}_1$, then not only $\{10,11,12,\dots,19,20\}$ is in $2\mathcal{S}_1$, but even $\{10,11,12,\dots,19,20,21,\dots,38,39\}$ is in $2\mathcal{S}_1$.
The threshold varies a lot among regular families.
We find it useful to formalize this notion as follows.
For each regular family $\mathcal{F}$, and for each $a \in \mathbb{N}$, let
\[
\mathrm{range}_\mathcal{F}(a) := \max \{m \in \mathbb{N} : \ \ [a,m-1] \in \mathcal{F}\}.
\]
First, note the following trivial observation.
\begin{obs} \label{obs:r-forces} Let $F \subset \mathbb{N}$. For every regular family $\mathcal{F}$ if $\max F < \mathrm{range}_\mathcal{F}(\min F)$, then $F \in \mathcal{F}$.
\end{obs}
Next, observe that for some of the families that we consider it is very easy to compute the value of $\mathrm{range}_\mathcal{F}$.
\begin{obs}\label{obs:ranges}
For every integer $k$ with $k \geq 2$, for each superadditive and increasing function $\varphi: \mathbb{N} \rightarrow \mathbb{N}$, and for each $a \in \mathbb{N}$, we have
\[\mathrm{range}_{\mathcal{S}_\varphi}(a) = a + \varphi(a), \ \ \mathrm{range}_{k\mathcal{S}_1}(a) = 2^ka, \ \ \mathrm{range}_{\mathcal{S}_2}(a) = 2^aa.\]
\end{obs}
We do not attach the proof of this observation as it is straightforward, however, we encourage the reader to verify the above values for a better understanding of the structure of the families.
Sometimes, we will use this observation implicitly.
The formula for $\mathrm{range}_{\mathcal{S}_3}$ is not as clean, although, using a simple inequality $2^a \leq \mathrm{range}_{\mathcal{S}_2}(a) \leq 2^{2a}$, we obtain a useful estimation.
\begin{obs}\label{obs:range:S_3}
For every $a \in \mathbb{N}$, we have
\[\tau(a,a) \leq \mathrm{range}_{\mathcal{S}_3}(a) \leq \tau(a,3a).\]
\end{obs}
In our consideration, we will be particularly interested in the sets that are maximal in a given regular family $\mathcal{F}$.
We call such sets \emph{$\mathcal{F}$-full}.
The main reason why such sets are interesting is the fact that they have to be sufficiently large.
For our arguments in the next sections, we also need some more technical notions concerning full sets.
For each regular family $\mathcal{F}$, for each family $\mathcal{G}$ of subsets of $\mathbb{N}$, and for all $a,b \in \mathbb{N}$ we define:
\begin{align*}
\mathrm{full}(\mathcal{F}) &:= \{F \in \mathcal{F} : \ \ n \in \mathbb{N} \backslash F \Longrightarrow F \cup \{n\} \notin \mathcal{F}\}, \\
[a,b]\mathcal{G} &:= \{F \in \mathcal{G} : \ \ F \subset [a,b], \ \ a \in F\},\\
\mathrm{full}_{a,b}(\mathcal{F}) &:= \{F \in [a,b]\mathcal{F} : \ \ n \in [a,b]\backslash F \Longrightarrow F \cup \{n\} \notin \mathcal{F}\}.
\end{align*}
In the special case of $\mathcal{F} = \mathcal{S}_1$, we write that $F \subset \mathbb{N}$ is a \emph{full Schreier set} if $F \in \mathrm{full}(\mathcal{S}_1)$. As mentioned, the main feature of full sets is the fact that are reasonably large. One can verify the following two observations.
\begin{obs}\label{lem:full-sets-sequence-in-S_2}
Let $a,b,s \in \mathbb{N}$, and let $F_1,\dots,F_s$ be full Schreier sets such that $F_1 < \dots < F_s$.
If $F_1\cup \dots \cup F_s \subset [a,b]$, then $b \geq 2^s a$.
\end{obs}
\begin{obs}\label{lem:full-sets-sequence-in-S_3}
Let $a,b,s \in \mathbb{N}$, and let $F_1,\dots,F_s \in \mathrm{full}(\mathcal{S}_2)$ be such that $F_1 < \dots < F_s$.
If $F_1\cup \dots \cup F_s \subset [a,b]$, then $b \geq \tau(s,a)a$.
\end{obs}
\cref{lem:full-sets-sequence-in-S_3} implies that disjoint $\mathcal{S}_2$-full sets need a lot of space.
Now, we want to argue that having reasonably large space, we can fit many disjoint $\mathcal{S}_2$-full sets.
Note that in the case of full Schreier sets an analogous computation is straightforward.
The below requires some technical computation.
\begin{lemma}\label{obs:lower:S3}
Let $a \in \mathbb{N}$. For every $s \in \mathbb{N}$, there exist $F_1,\dots,F_s \in \mathrm{full}(\mathcal{S}_2)$ with $F_1 < \dots < F_s$ such that $F_i \subset [a,\tau(s,2a+s-1)-1]$ for each $i \in [s]$.
\end{lemma}
\begin{proof}
First, we claim that for every $m \in \mathbb{N}$, the interval $[m,\tau(1,2m)-1]$ contains an $\mathcal{S}_2$-full set.
Indeed, $[m,2^mm-1] \in \mathrm{full}(\mathcal{S}_2)$ and $2^mm \leq 2^{2m} = \tau(1,2m)$.
We proceed by induction on $s$.
If $s=1$, then we use the above claim directly for $m=a$.
Assume that $s>1$ and that the assertion holds for $s-1$, namely, the interval $[a,\tau(s-1,2a+s-2)-1]$ contains some $F_1,\dots,F_{s-1} \in \mathrm{full}(\mathcal{S}_2)$ with $F_1 < \dots < F_{s-1}$.
By the initial claim applied to $m = \tau(s-1,2a+s-2)$, we obtain that $[\tau(s-1,2a+s-2), \tau(1,2\tau(s-1,2a+s-2))-1]$ contains an $\mathcal{S}_2$-full set. Note that
\[\tau(1,2\tau(s-1,2a+s-2)) \leq \tau(1,\tau(s-1,2a+s-1)) = \tau(s,2a+s-1).\]
By taking the $\mathcal{S}_2$-full set in the interval as $F_s$, we finish the proof.
\end{proof}
Let us comment a little on differences between $\mathrm{full}(\mathcal{F})$ and $\mathrm{full}_{a,b}(\mathcal{F})$.
By definition, $[a,b]\mathrm{full}(\mathcal{F}) \subset \mathrm{full}_{a,b}(\mathcal{F})$.
In general, the inclusion can be strict.
The simplest way to see this is to take $a,b$ with $|b-a|$ small, and the set $[a,b]$.
For instance,
\[\{7,8,9\} \in \mathrm{full}_{7,9}(\mathcal{S}_1) \ \ \ \mathrm{and} \ \ \ \{7,8,9\} \notin \mathrm{full}(\mathcal{S}_1).\]
For the Schreier family, one can prove that all sets in $\mathrm{full}_{a,b}(\mathcal{F}) \backslash [a,b]\mathrm{full}(\mathcal{F})$ are of this type (see the lemma below), however, this is not always the case. For instance,
\[\{2,3,5,6,7,8\} \in \mathrm{full}_{2,8}(2\mathcal{S}_1) \ \ \ \mathrm{and} \ \ \ \{2,3,5,6,7,8\} \notin \mathrm{full}(2\mathcal{S}_1).\]
\begin{lemma}\label{lem:full:Sf}
Let $\varphi: \mathbb{N} \rightarrow \mathbb{N}$ be an increasing and superadditive function. Let $a,b \in \mathbb{N}$ be such that $[a,b] \notin \Sf$ and let $F \subset \mathbb{N}$. If $F \in \mathrm{full}_{a,b}(\Sf)$, then $F \in [a,b]\mathrm{full}(\Sf)$. In particular, $\mathrm{full}_{a,b}(\Sf) = [a,b]\mathrm{full}(\Sf)$.
\end{lemma}
\begin{proof}
Let $F \in \mathrm{full}_{a,b}(\Sf)$, and suppose that $F \notin \mathrm{full}(\Sf)$, that is, $|F| < \varphi(\min F) = \varphi(a)$ Since $[a,b] \notin \Sf$, there exists $m \in [a,b] \backslash F$. It follows that $F \cup \{m\} \in [a,b]\Sf$, which is a contradiction.
\end{proof}
The sets in $k \mathcal{S}_1$ and $\mathcal{S}_2$ can be seen as unions of some number of Schreier sets.
In general the ingredients of the union are not uniquely defined, however, one can fix them to be unique by a simple greedy process described below.
For every $F \subset \mathbb{N}$, and for every $i \in \mathbb{N}$ we define $E_i(F)$ with the following inductive procedure. Let $F \subset \mathbb{N}$. First, if $F = \emptyset$, then we set $E_1(F) := \emptyset$. Otherwise, we set $E_1(F)$ to be $F$ if $|F| \leq \min F$, and to be the first $\min F$ elements of $F$ if $|F| > \min F$. Now, let $i \in \mathbb{N}$, and assume that $E_1(F),\dots,E_{i}(F)$ are already defined. We set $E_{i+1}(F) := E_1(F \backslash E_{i}(F))$. For instance, for $F = [10]$, we have
\begin{align*}
E_1(F) = \{1\}, \ E_2(F) = \{2,3\}, \ E_3(F) = \{4,5,6,7\}, \ &E_4(F) = \{8,9,10\},\\
&\mathrm{and} \ E_5(F) = E_6(F) = \dots = \emptyset.
\end{align*}
Let $F \subset \mathbb{N}$. Note that if $E_i(F) = \emptyset$ for some $i \in \mathbb{N}$, then $E_{i+1}(F) = E_{i+2}(F) = \dots = \emptyset$.
Using the operators $E_i$ one can characterize sets in the families $k \mathcal{S}_1$ and $\mathcal{S}_2$.
Observe that $F \in \mathcal{S}_1$ if and only if $E_2(F) = \emptyset$, next, for every $k \in \mathbb{N}$, we have $F \in k\mathcal{S}_1$ if and only if $E_{k+1}(F) = \emptyset$, and finally, $F \in \mathcal{S}_2$ if and only if $E_{\min F + 1}(F) = \emptyset$.
Now, analogously to \cref{lem:full:Sf}, we study the sets in $\mathrm{full}_{a,b}(\mathcal{F})$ assuming that $[a,b] \notin \mathcal{F}$, where $\mathcal{F}$ is either $k\mathcal{S}_1$ or $\mathcal{S}_2$.
Intuitively, we prove that such full sets are large.
\begin{lemma}\label{lem:full-sets-in-kS_1}
Let $k$ be a positive integer with $k \geq 2$, let $a,b \in \mathbb{N}$ be such that $[a,b] \notin k\mathcal{S}_1$, and let $F \subset \mathbb{N}$.
If $F \in \mathrm{full}_{a,b}(k\mathcal{S}_1)$, then $E_1(F),\dots,E_{k-1}(F)$ are full Schreier sets.
\end{lemma}
\begin{proof}
Let $i$ be the least positive integer such that $E_i(F)$ is not a full Schreier set.
If there exists $m \in [\min E_i(F), b] \backslash F$, then $F \cup \{m\} \in [a,b](k\mathcal{S}_1)$, which is a contradiction, hence, $[\min E_i(F), b] \subset F$.
It follows that $E_{i+1}(F) = \emptyset$.
Suppose that $i < k$.
We have $[a,b] \notin k\mathcal{S}_1$, thus, there exists $m \in [a,b] \backslash F$.
Observe that $F \cup \{m\} \in [a,b](k\mathcal{S}_1)$, which is again a contradiction.
Therefore, $i = k$, which ends the proof.
\end{proof}
By repeating exactly the same proof, we obtain a similar result for $\mathcal{S}_2$.
\begin{lemma}\label{lem:full-sets-in-S_2}
Let $a,b \in \mathbb{N}$ be such that $[a,b] \notin \mathcal{S}_2$ and let $F \subset \mathbb{N}$.
If $F \in \mathrm{full}_{a,b}(\mathcal{S}_2)$, then $E_1(F),\dots,E_{a-1}(F)$ are full Schreier sets.
\end{lemma}
\begin{proof}
Let $i$ be the least positive integer such that $E_i(F)$ is not a full Schreier set.
If there exists $m \in [\min E_i(F), b] \backslash F$, then $F \cup \{m\} \in [a,b]\mathcal{S}_2$, which is a contradiction, hence, $[\min E_i(F), b] \subset F$.
It follows that $E_{i+1}(F) = \emptyset$.
Suppose that $i < a$.
We have $[a,b] \notin \mathcal{S}_2$, thus, there exists $m \in [a,b] \backslash F$.
Observe that $F \cup \{m\} \in [a,b]\mathcal{S}_2$, which is again a contradiction.
Therefore, $i = a$, which ends the proof.
\end{proof}
Intuitively, the last set (that is, $E_k(F)$, or $E_a(F)$) usually is also quite large. As we do not care much for the constants in this paper, we do not investigate this in detail in general. However, the investigation is necessary for later applications in the case of $2\mathcal{S}_1$.
\begin{lemma}\label{lem:full-sets-in-kS}
Let $a,b \in \mathbb{N}$ be such that $[a,b] \notin 2\mathcal{S}_1$, and let $F \subset \mathbb{N}$.
If $F \in \mathrm{full}_{a,b}(2\mathcal{S}_1)$, then $E_1(F)$ is a full Schreier set and
\[ \min E_2(F) \leq \frac{b}{2} + 2.\]
\end{lemma}
\begin{proof}
The first part follows from \cref{lem:full-sets-in-kS_1}.
Suppose that the second part does not hold, that is
\[\min E_2(F) \geq \frac{b}{2} + 3.\]
Since $F \in \mathrm{full}_{a,b}(2\mathcal{S}_1)$, we have $E_2(F) = [\min E_2,b]$.
Note that by rearranging the above, we have
\[ \min E_2(F)-2 \geq b - \min E_2(F) + 3.\]
This yields $[\min E_2(F) - 2, b] \in \mathcal{S}_1$.
We claim that there exists $F' \in 2\mathcal{S}_1$ such that $F \subsetneq F' \subset [a,b]$.
If $\min E_2(F) - 1 \notin E_1(F)$, then $F' := E_1(F) \cup \{\min E_2(F)-1\} \cup E_2(F)$ is a proper choice. Hence, we assume that $\min E_2(F) - 1 \in E_1(F)$.
Suppose that $E_1(F)$ is an interval. Then, we set $F' := F_1'\cup F_2'$, where $F_1' := [\min E_1(F)-1, \min E_2(F)-3]$, and $F_2' := [\min E_2(F)-2,b]$ -- note that $\min E_1(F) - 1 \in [a,b]$ because $[a,b] \notin 2\mathcal{S}_1$. Finally, we assume that $E_1(F)$ is not an interval. Let $m \in [\min E_1(F),\max E_1(F)] \backslash E_1(F)$. We set $F' := F_1' \cup F_2'$, where $F_1' := E_1(F)\cup \{m\} \backslash \{\min E_2(F)-1\}$, and $F_2' := [\min E_2(F)-1,b]$.
This proves the claim, namely, there exists $F' \in 2\mathcal{S}_1$ such that $F \subsetneq F' \subset [a,b]$, which contradicts $F \in \mathrm{full}_{a,b}(2\mathcal{S}_1)$.
\end{proof}
The operators $E_i$ are useful to describe sets in the families $k\mathcal{S}_1$ and $\mathcal{S}_2$. In order to describe sets in the family $\mathcal{S}_3$, we need analogous operators extracting subsequent $\mathcal{S}_2$-full subsets.
For every $F \subset \mathbb{N}$, and for every $i \in \mathbb{N}$ we define $E_i^*(F)$ with the following inductive procedure. Let $F \subset \mathbb{N}$. First, if $F = \emptyset$, then we set $E_1^*(F) := \emptyset$.
Otherwise, we set $E_1^*(F)$ to be $F$ if $F \in \mathcal{S}_2$, and to be $E_1(F) \cup \dots \cup E_{\min F}(F)$ if $F \notin \mathcal{S}_2$. Now, let $i \in \mathbb{N}$, and assume that $E_1^*(F),\dots,E_{i}^*(F)$ are already defined. We set $E_{i+1}^*(F) := E_1^*(F \backslash E_{i}^*(F))$.
\begin{lemma}\label{lem:full-sets-in-S_3}
Let $a,b \in \mathbb{N}$ be such that $[a,b] \notin \mathcal{S}_3$, and let $F \subset \mathbb{N}$.
If $F \in \mathrm{full}_{a,b}(\mathcal{S}_3)$, then $E_1^*(F),\dots,E_{a-1}^*(F)$ are $\mathcal{S}_2$-full sets.
\end{lemma}
\begin{proof}
Let $i$ be the least positive integer such that $E_i^*(F)$ is not an $\mathcal{S}_2$-full set.
Let $a' = \min E_i^*(F)$. It is clear that $E_i^*(F) \in \mathrm{full}_{a',b}(\mathcal{S}_2)$.
Suppose that $i < a$.
If $[a',b] \in \mathcal{S}_2$, then there exists $m \in [a,b] \backslash F$, and $F \cup \{m\} \in \mathcal{S}_3$, which is a contradiction.
We can assume that $[a',b] \notin \mathcal{S}_2$.
By \cref{lem:full-sets-in-S_2}, $E_1(E_i^*(F)), \dots, E_{a'-1}(E_i^*(F))$ are full Schreier sets.
If there exists $m \in [\min E_{a'}(E_i^*(F)), b] \backslash F$, then $F \cup \{m\} \in [a,b]\mathcal{S}_3$, which is a contradiction. It follows that
\[[\min E_{a'}(E_i^*(F)), b] \subset F,\]
which yields $E_{i+1}^*(F) = \emptyset$.
Since $[a,b] \notin \mathcal{S}_3$, there exists $m \in [a,b] \backslash F$.
Observe that $F \cup \{m\} \in \mathcal{S}_3$, which is again a contradiction.
Therefore, $i = a$, which ends the proof.
\end{proof}
\section{Tools for lower bounds}
\label{sec:tools_lower}
The idea for proving the lower bounds is the same for all the regular families that we consider.
For this reason, we are going to prove an abstract lemma, and then apply it to various families.
The general plan of constructing an element of $c_{00}$ with high $j_\mathcal{F}(x)$ is to put a very high value on the first coordinate and a bunch of last coordinates.
This way, we force every functional attaining the norm to be a sum of many functionals from $W_0(\mathcal{F})$.
Intuitively, this leaves the largest possible space to proceed with inductive construction.
See an example in \cref{fig:lower}.
\begin{figure}
\caption{
Consider the case, where $\mathcal{F}
\label{fig:lower}
\end{figure}
\vbox{
\begin{lemma} \label{lem:main:lower}
Let $\mathcal{F}$ be a regular family and let $d \in \mathbb{N}$. If there exist $F_1,\dots,F_d \in \mathrm{full}(\mathcal{F})$ and $a_1,b_1,\dots,a_{d-1},b_{d-1} \in \mathbb{N}$ such that
\begin{enumerateNumL}
\item for all $i \in [d]$ we have $|F_i| \geq 3$,\label{item:lower:first}\label{item:lower:at-least-3}
\item for all $i \in [d-1]$ elements $a_i,b_i$ are consecutive in $F_i$,\label{item:lower:consecutive}
\item for all $i \in [d-1]$ we have $a_i \in F_{i+1}$ and $F_{i+1} \subset [a_i,b_i-1]$,\label{item:lower:nested}
\item for all $i \in [d-1]$ and distinct $a,a' \in [a_i,b_i-1]$ we have $(F_i\backslash \{a_i\}) \cup \{a,a'\} \notin \mathcal{F}$,\label{item:lower:last}\label{item:lower:full}
\end{enumerateNumL}
then there exists $x \in c_{00}^+$ with $\mathrm{supp}\, x \subset \bigcup_{i=1}^d F_i$ and $j_\mathcal{F}(x) \geq d$, in particular \mbox{$j_\mathcal{F}(\max F_1) \geq d$}.
\end{lemma}
}
\begin{proof}
We proceed by induction on $d$. Let us start with the case of $d = 1$. Suppose that there exists $F_1 \in \mathrm{full}(\mathcal{F})$ satisfying items \ref{item:lower:first}-\ref{item:lower:last}, namely, we have $|F_1| \geq 3$. For each $i \in \mathbb{N}$ define
\[x_i :=
\begin{cases}
1 & \text{if } i \in F_1, \\
0 & \text{otherwise.}
\end{cases} \]
Let $x := (x_i)_{i\in \mathbb{N}}$. For every $e \in W_0$, we have $e(x) \leq 1$. Let $c_1,c_2,c_3 \in F$ be three distinct elements. Define $f := \frac{1}{2} (e_{c_1}^* + e_{c_2}^* + e_{c_3}^*)$. Since $\mathcal{F}$ is hereditary, we have $f \in W(\mathcal{F})$. Clearly, $f(x) = \frac{3}{2}$ and $\mathrm{depth}_\mathcal{F}(f) = 1$, hence, $j_\mathcal{F}(x) \geq 1$.
Now, let $d > 1$ and suppose that there exist $F_1,\dots,F_d \in \mathrm{full}(\mathcal{F})$ and $a_1,b_1,\dots,a_{d-1},b_{d-1} \in \mathbb{N}$ satisfying items \ref{item:lower:first}-\ref{item:lower:last}. By the inductive assumption applied to $F_2,\dots,F_d$ and $a_2,b_2,\dots,a_{d-1},b_{d-1}$, we obtain $x' \in c_{00}^+$ with $\mathrm{supp}\, x' \subset \bigcup_{i=2}^d F_i$ and $j_\mathcal{F}(x') \geq d-1$. Let $s$ be the sum of all the coefficients of $x'$, and let $F' := \bigcup_{i=2}^d F_i$. By \ref{item:lower:consecutive} and \ref{item:lower:nested}, $F_1 \backslash F' = F_1 \backslash \{a_1\}$. For each $i \in \mathbb{N}$ we define
\[x_i :=
\begin{cases}
x'_i & \text{if } i \in F', \\
2s & \text{if } i \in F_1 \backslash F', \\
0 & \text{otherwise.}
\end{cases} \]
Let $x := (x_i)_{i\in \mathbb{N}}$. Let $f \in W(\mathcal{F})$ be such that $f(x) = \|x\|_\mathcal{F}$ and let $f' \in W(\mathcal{F})$ be such that $f'(x') = \|x'\|_\mathcal{F}$.
Since $j_\mathcal{F}(x') \geq d-1$, we have $\mathrm{depth}_\mathcal{F}(f') \geq d-1$. We define $g := \frac{1}{2}\left(f' + \sum_{i \in F_1 \backslash F'} e_i^*\right)$. Note that $F_1 \backslash F' \cup \{\min \mathrm{supp}\, f'\} \in \mathcal{F}$, thus, $g \in W(\mathcal{F})$. The goal is to prove that $f$ is of a similar form as $g$. First observe that,
\[ g(x) = \frac{1}{2}\left( f'(x) + (|F_1\backslash F'|)\cdot 2s \right) = \frac{1}{2} \|x'\|_\mathcal{F} + (|F_1| - 1) \cdot s > 2s. \]
By definition, for each $i \in \mathbb{N}$, we have $x_i \leq 2s$, thus, if $\mathrm{depth}_\mathcal{F}(f) = 0$, then $\|x\|_\mathcal{F} = f(x) \leq 2s < g(x) \leq \|x\|_\mathcal{F}$, which is a contradiction. Therefore, $\mathrm{depth}_\mathcal{F}(f) > 0$. It follows that for each $i \in \mathbb{N}$, $f(e_i) \leq \frac{1}{2}$. We claim that for each $i \in F_1 \backslash F'$, we have $f(e_i) = \frac{1}{2}$. For a contradiction, suppose that $f(e_{i_0}) < \frac{1}{2}$ for some $i_0 \in F_1 \backslash F$. Since $f \in W(\mathcal{F})$ the value of $f(e_i)$ has to be an inverse of a power of $2$, thus, $f(e_i) \leq \frac{1}{4}$. We have
\begin{align*}
\|x\|_\mathcal{F} =
f(x) &=
\sum_{i \in \mathbb{N}} f(e_i) \cdot x_i =
\sum_{i \in F_1 \cup F'} f(e_i) \cdot x_i = \sum_{i \in (F_1 \cup F') \backslash \{i_0\}} f(e_i) \cdot x_i + f(e_{i_0}) x_{i_0} \\
&\leq \sum_{i \in (F_1 \cup F') \backslash \{i_0\}} \frac{1}{2} \cdot x_i + \frac{1}{4} x_{i_0} =
\sum_{i \in F_1 \backslash (F' \cup \{i_0\})} \frac{1}{2} \cdot x_i + \sum_{i \in F'} \frac{1}{2} \cdot x_i + \frac{1}{4} x_{i_0} \\
&= (|F_1| - 2) \cdot s + \frac{1}{2} s + \frac{1}{4} \cdot 2s = s \cdot (|F_1| - 1) < g(x) \leq \|x\|_\mathcal{F}.
\end{align*}
This is a contradiction, and so, for each $i \in F_1 \backslash F'$, we have $f(e_i) = \frac{1}{2}$. In particular,
\[f = \frac{1}{2}\left(f_1 + \dots + f_m + \sum_{i \in F_1\backslash F'} e_i^*\right)\]
for some $m \in \mathbb{N}_0$ and $f_1,\dots,f_m \in W(\mathcal{F})$ such that
\[F_1\backslash \{a_1\} \cup \{\min \mathrm{supp}\, f_1 ,\dots, \min \mathrm{supp}\, f_m\} \in \mathcal{F}.\]
However, by \ref{item:lower:full}, the above gives
\[|\{\min \mathrm{supp}\, f_1 ,\dots, \min \mathrm{supp}\, f_m\} \cap [a_1,b_1-1]| \leq 1.\]
By comparing $f(x)$ with $g(x)$, we have $f_1(x') + \dots f_m(x') \geq \|x'\|_\mathcal{F} > 0$.
Clearly, if for some $\ell \in [m]$, $\min \mathrm{supp}\, f_\ell \notin [a_1,b_1-1]$, then $f_\ell(x') = 0$, and so, there exists $\ell \in [m]$ such that $\min \mathrm{supp}\, f_\ell \in [a_1,b_1-1]$. Since $f_\ell(x') = \|x'\|_\mathcal{F}$, we have $\mathrm{depth}_\mathcal{F}(f_\ell) \geq d - 1$, and so, $\mathrm{depth}_\mathcal{F}(f) \geq d$, which ends the proof.
\end{proof}
As explained in the caption of \cref{fig:lower}, the strategy is to take $a_1 = 3, a_2 = 4, a_3 = 5$, and so on.
We are almost ready to proceed with the construction of the sequences of sets for some regular families.
The last remaining detail to take care of is to make sure that the families that we consider satisfy \ref{item:lower:full}.
To this end, we abstract the following property of a regular family.
We say that a regular family $\mathcal{F}$ is \emph{strong} if for every integer $a$ with $a \geq 3$, and for all integers $b,c$ with $a + 1 < b \leq c$ such that $[a,a+1]\cup [b,c] \in \mathrm{full}(\mathcal{F})$, for all distinct $a',a'' \in [a+2,b-1]$ we have $\{a,a',a''\} \cup [b,c] \notin \mathcal{F}$. The following is immediate to check.
\begin{obs}\label{obs:strong}
For every increasing and superadditive function $\varphi: \mathbb{N} \rightarrow \mathbb{N}$ and for every integer $k$ with $k \geq 2$, the families $\Sf, k \mathcal{S}_1,\mathcal{S}_2,\mathcal{S}_3$ are strong.
\end{obs}
As the construction of the sequence $F_1,\dots,F_d$ is virtually the same for all the families that we consider, we For all $t,s \in \mathbb{N}$ with $s + 1 < t$, we define
\[r_\mathcal{F}(s,t) := \min \{m \in \mathbb{N} : \{s,s+1\} \cup [t,m-1] \in \mathrm{full}(\mathcal{F})\}.\]
Next, for all $s,t \in \mathbb{N}$ with $s + 1 < t$ and for each $u \in \mathbb{N}_0$ we define
\[q_\mathcal{F}(u,s,t) = \begin{cases}
t & \text{if } u > s, \\
r_\mathcal{F}(s,t) & \text{if } u = s, \\
r_\mathcal{F}(u,q_\mathcal{F}(u+1,s,t)) & \text{if } u \leq s.
\end{cases}\]
For example, $q_\mathcal{F}(3,5,10) = r_\mathcal{F}(3,r_\mathcal{F}(4,r_\mathcal{F}(5,10)))$.
The definition of $q_\mathcal{F}$ is a little convoluted, and to understand its purpose one should read the next lemma.
\begin{lemma}\label{lem:lower:greater:abstract}
Let $\mathcal{F}$ be a strong regular family such that $\mathrm{range}_\mathcal{F}(3) \geq 3$. For every $n,d \in \mathbb{N}$ if $q_\mathcal{F}(3,d+2,d+4) \leq n$, then
\[d \leq j_\mathcal{F}(n).\]
\end{lemma}
\begin{proof}
Let $n,d \in \mathbb{N}$, be such that $q_\mathcal{F}(3,d+2,d+4) \leq n$. For each $i \in [d]$, we define
\[F_i := \{i+2,i+3\} \cup [q_\mathcal{F}(i+3,d+2,d+4),q_\mathcal{F}(i+2,d+2,d+4)-1].\]
It follows that $F_i \in \mathrm{full}(\mathcal{F})$ and $|F_i| \geq 3$ (so \ref{item:lower:at-least-3} is satisfied).
If $i < d$, then we define $a_i := i+2$ and $b_i := q_\mathcal{F}(i+3,d+2,d+4)$.
Observe that $a_i \in F_{i+1}$ and $F_{i+1} \subset [a_i,b_i-1]$ (so \ref{item:lower:nested} is satisfied).
Item~\ref{item:lower:consecutive} is clearly satisfied.
Since $\mathcal{F}$ is strong, \ref{item:lower:full} is also satisfied. Therefore, by \cref{lem:main:lower}, for every $n \in \mathbb{N}$ such that $q_\mathcal{F}(3,d+2,d+4) \leq n$, we have
\[d \leq j_\mathcal{F}(\max F_1) \leq j_\mathcal{F}(q_\mathcal{F}(3,d+2,d+4)) \leq j_\mathcal{F}(n).\qedhere\]
\end{proof}
\section{Lower bounds}
\label{sec:lower}
In this section we apply \cref{lem:lower:greater:abstract} to the families $\mathcal{S}_1,\mathcal{S}_\varphi,k\mathcal{S}_1,\mathcal{S}_2$.
In each case, we establish some bounds on $q_\mathcal{F}(3,d+2,d+4)$, and then compare the bound to $n$, in order to obtain the final lower bound on $j_\mathcal{F}(n)$ by applying \cref{lem:lower:greater:abstract}.
\subsection{Lower bound for $\Sf$}
Let $\varphi: \mathbb{N} \rightarrow \mathbb{N}$ be a non-decreasing. Fix some $u,s,t \in \mathbb{N}$ with $s + 1 < t$. It is clear that
\[r_{\Sf}(s,t) = t + \varphi(s) - 2.\]
It follows that $q_{\Sf}(u,s,t) \leq t + \sum_{i=u}^s (\varphi(i)-2)$. For every integer $d \in \mathbb{N}$, we have
\[q_{\Sf}(3,d+2,d+4) = d+4 + \sum_{i=3}^{d+2} (\varphi(i)-2) = \sum_{i=3}^{d+2} \varphi(i) - d + 4 \leq \sum_{i=3}^{d+3} \varphi(i).\]
By \cref{lem:lower:greater:abstract}, if $\sum_{i=3}^{d+3} \varphi(i) \leq n$, then $j_{\Sf}(n) \geq d$ for every $n \in \mathbb{N}$.
\begin{corollary}\label{cor:lower:Sf}
For each $n \in \mathbb{N}$, we have
\[ j_{\mathcal{S}_\varphi}(n) \geq \max\{ m \in \mathbb{N} : \sum_{j=3}^{m+3}\varphi(j)<n \}.\]
\end{corollary}
Substituting $\varphi = \mathrm{id}$ in the above, gives a lower bound for $j_{\mathcal{S}_1}(n)$.
\begin{corollary}\label{cor:classic-schreier}
For each $n \in \mathbb{N}$, we have
\[ j_{\mathcal{S}_1}(n) \geq \sqrt{2n} - 3.\]
\end{corollary}
\subsection{Lower bound for $k\mathcal{S}_1$}
Let $k$ be an integer with $k \geq 2$. Fix some $u,s,t \in \mathbb{N}$ with $s + 1 < t$. It is clear that
\[r_{k\mathcal{S}_1}(s,t) = 2^{k-1}(t+s-2) \leq 2^{k} t.\]
It follows that $q_{k\mathcal{S}_1}(u,s,t) \leq 2^{k(s-u+1)}t \leq 2^{k(s-u+1) + t}$. Let $n \in \mathbb{N}$. For every integer $d \in \mathbb{N}$, we have
\[q_{k\mathcal{S}_1}(3,d+2,d+4) \leq 2^{(k+1)d + 4}. \]
By \cref{lem:lower:greater:abstract}, if $2^{(k+1)d + 4} \leq n$, then $j_{k\mathcal{S}_1}(n) \geq d$ for every $n \in \mathbb{N}$.
\begin{corollary}\label{cor:lower:kS1}
For each $n \in \mathbb{N}$, and for each integer $k$ with $k \geq 2$, we have
\[ j_{k\mathcal{S}_1}(n) \geq \frac{1}{k+1} \log n - \frac{4}{k+1}-1.\]
\end{corollary}
\subsection{Lower bound for $\mathcal{S}_2$}
Fix some $u,s,t \in \mathbb{N}$ with $s+1 < t$. It is clear that
\[ r_{\mathcal{S}_2}(s,t) = 2^{s-1}(t+s-2) \leq 2^{s} t.\]
It follows that $q_{\mathcal{S}_2}(u,s,t) \leq \prod_{i=u}^s 2^{i} t \leq 2^{(s(s+1))\slash 2 + t}$. For every integer $d \in \mathbb{N}$, we have
\[q_{\mathcal{S}_2}(3,d+2,d+4) \leq 2^{(d^2+7d+14)\slash 2}. \]
By \cref{lem:lower:greater:abstract}, if $2^{(d^2+7d+14)\slash 2} \leq n$, then $j_{\mathcal{S}_2}(n) \geq d$ for every $n \in \mathbb{N}$.
\begin{corollary}\label{cor:lower:S2}
For each $n \in \mathbb{N}$, we have
\[ j_{\mathcal{S}_2}(n) \geq \sqrt{2\log n} - 5.\]
\end{corollary}
\subsection{Lower bound for $\mathcal{S}_3$}
Fix some $u,s,t \in \mathbb{N}$ with $s+1 < t$. To estimate $r_{\mathcal{S}_3}(s,t)$, note that the full set $F := \{s,s+1\} \cup [t,r_{\mathcal{S}_3}(s,t)-1]$ consists of two parts. The first part is a prefix of $F$ that is an $\mathcal{S}_2$-full set. The second part is the rest of $F$, it starts after the element $t2^t$, and it is the union of $s-1$ pairwise disjoint $\mathcal{S}_2$-full sets. By \cref{obs:lower:S3}, we have
\[ r_{\mathcal{S}_3}(s,t) \leq \tau(s-1,2t2^t+s-1) \leq \tau(s-1,2^{2t+2}) =\tau(s,2t+2) \leq \tau(s+1,t).\]
It follows that $q_{\mathcal{S}_3}(u,s,t) \leq \tau(\sum_{i=u}^s (i+1), t) \leq \tau((s+2)^2\slash 2, t)$. For every integer $d \in \mathbb{N}$, we have
\[q_{\mathcal{S}_3}(3,d+2,d+4) \leq \tau((d+4)^2\slash 2, d+4). \]
By \cref{lem:lower:greater:abstract}, if $\tau((d+4)^2\slash 2, d+4) \leq n$, then $j_{\mathcal{S}_3}(n) \geq d$ for every $n \in \mathbb{N}$.
\begin{corollary}\label{cor:lower:S3}
For each $n \in \mathbb{N}$, we have
\[ j_{\mathcal{S}_3}(n) \geq \sqrt{2\log^* n} - 5.\]
\end{corollary}
\section{Tools for upper bounds}
\label{sec:tools_upper}
\subsection{Some auxiliary definitions and simple observations}
Let $\mathcal{F}$ be a regular family, let $x \in c_{00}$, and let $f,g \in W(\mathcal{F})$. We write
\[\mathrm{span}\, x := [\min \mathrm{supp}\, x, \max \mathrm{supp}\, x] \ \ \ \mathrm{and} \ \ \ \mathrm{span}\, f := [\min \mathrm{supp}\, f, \max \mathrm{supp}\, f].\]
We say that $f$ is \emph{$\mathcal{F}$-realizing for $x$} if
\begin{itemize}
\item $f(x) = \|x\|_\mathcal{F}$,
\item $\mathrm{depth}_\mathcal{F}(f) = j_\mathcal{F}(x)$, and
\item $\mathrm{span}\, f \subset \mathrm{span}\, x$.
\end{itemize}
We say that \emph{$g$ is not $\mathcal{F}$-worse than $f$ for $x$} if
\begin{itemize}
\item $g(x) \geq f(x)$,
\item $\mathrm{depth}_\mathcal{F}(g) \leq \mathrm{depth}_\mathcal{F}(f)$, and
\item $\mathrm{span}\, g \subset \mathrm{span}\, f$.
\end{itemize}
Observe that if $f$ is $\mathcal{F}$-realizing for $x$ and $f$ is not $\mathcal{F}$-worse than $g$ for $x$, then $g$ is $\mathcal{F}$-realizing for $x$. Moreover, this relation is transitive. We will use these facts implicitly and repeatedly.
We define
\[\mathrm{full}_f(\mathcal{F}) := \mathrm{full}_{\min \mathrm{supp}\, f, \max \mathrm{supp}\, f}(\mathcal{F}).\]
Let $m \in \mathbb{N}$ and let $f_1,\dots,f_m \in W(\mathcal{F})$. We say that $(f_1,\dots,f_m)$ is \emph{$\mathcal{F}$-full-building for $f$} if $(f_1,\dots,f_m)$ is $\mathcal{F}$-building for $f$ and
\[\{\min \mathrm{supp}\, f_i : i \in [m]\} \in \mathrm{full}_{f}(\mathcal{F}).\]
We say that $f$ is \emph{$\mathcal{F}$-full} if there exist a positive integer $m$ and $f_1,\dots,f_m \in W(\mathcal{F})$ such that $(f_1,\dots,f_m)$ is $\mathcal{F}$-full-building for $f$.
For each $i \in \mathbb{N}$, we define
\[(x|f)_i :=
\begin{cases}
x_i & \text{if } \min \mathrm{supp}\, f \leq i \leq \max \mathrm{supp}\, f, \\
0 & \text{otherwise,}
\end{cases} \]
and we let $(x|f) := ((x|f))_{i\in \mathbb{N}}$.
\begin{obs}\label{lem:force:F}
Let $x \in c_{00}^+$, let $\mathcal{F}$ be a regular family, and let $f \in W(\mathcal{F})$.
If $\mathrm{supp}\, f \in \mathcal{F}$ then there exists $g \in W(
\mathcal{F})$ with $\mathrm{depth}_\mathcal{F}(g) \leq 1$ that is not $\mathcal{F}$-worse than $f$ for $x$.
In particular, if $[a,b] \in \mathcal{F}$ for some $a,b \in \mathbb{N}$ with $a \leq b$, then $j_\mathcal{F}(a,b) \leq 1$.
\end{obs}
\begin{proof}
Assume that $\mathrm{supp}\, f \in \mathcal{F}$.
If $\mathrm{depth}_\mathcal{F}(f) \leq 1$, then $g:=f$ satisfies the assertion.
Otherwise, we put $g := \frac{1}{2}\sum_{i \in \mathrm{supp}\, f} e_i^*$.
Clearly, $g(x) \geq f(x)$ and $\mathrm{depth}_\mathcal{F}(g) = 1$.
\end{proof}
Next, we prove that for each $x \in c_{00}^+$, the norm $\|x\|_\mathcal{F}$ is always realized wither by a very shallow functional, of by an $\mathcal{F}$-full functional.
\begin{lemma}\label{lem:force:star}
Let $x \in c_{00}^+$ and let $\mathcal{F}$ be a regular family. For every $f \in W(\mathcal{F})$ such that $f(x) = \|x\|_\mathcal{F}$ there exists $g \in W(\mathcal{F})$ that is not $\mathcal{F}$-worse than $f$ for $x$, and either
\begin{enumerateNumF}
\item $\mathrm{depth}_\mathcal{F}(g) \leq 1$ or\label{lem:force:star:depth}
\item $g$ is $\mathcal{F}$-full.\label{lem:force:star:full}
\end{enumerateNumF}
\end{lemma}
\begin{proof}
Suppose that the assertion does not hold.
Let us choose a counterexample $f \in W(\mathcal{F})$ satisfying the premise of the lemma according to the following rule.
For each $f$ being a counterexample choose the maximum $d \in \mathbb{N}$ such that there exist $f_1,\dots,f_d$ with $(f_1,\dots,f_d)$ being $\mathcal{F}$-building for $f$.
Now, we choose $f$ such that the value $d$ is maximum among all counterexamples $f$.
Fix $f_1,\dots,f_d \in W(\mathcal{F})$ as above, and let $F_f := \{\min \mathrm{supp}\, f_i : i \in [d]\}$.
Observe that, as $f$ can not be taken to be $g$, therefore, $\mathrm{depth}_\mathcal{F}(f) \geq 2$ and $F_f \notin \mathrm{full}_{f}(\mathcal{F})$.
It follows that there exists $n \in \mathrm{span}\, f\backslash F_f$ with $F_f \cup \{n\} \in \mathcal{F}$. Let $n^*$ be the maximum such number.
Let $t$ be the maximum number in $[d]$ such that $\max\mathrm{supp}\, f_t \leq n^*$.
First, suppose that $\max\mathrm{supp}\, f_t < n^*$, then we define
\[ f':= \frac{1}{2}\left( \sum_{i=1}^t f_i + e_{n^*}^* + \sum_{i=t+1}^d f_i \right).\]
Next, suppose that $\max\mathrm{supp}\, f_t = n^*$. Note that $f_t \neq e_{n^*}$, and so $f_t' := f_t|_{[1,n^*-1]}$ is a well-defined member of $W(\mathcal{F})$. We define
\[ f':= \frac{1}{2}\left( \sum_{i=1}^{t-1} f_i + f_t' + e_{n^*}^* + \sum_{i=t+1}^d f_i \right).\]
Since $F \cup \{n^*\} \in \mathcal{F}$, in both cases $f' \in W(\mathcal{F})$.
Moreover, $f(x) \leq f'(x)$ in both cases, and in particular, $f'(x) = \|x\|_\mathcal{F}$. Furthermore, $\mathrm{depth}_\mathcal{F}(f') = \mathrm{depth}_\mathcal{F}(f)$ and $\mathrm{span}\, f' = \mathrm{span}\, f $.
It follows that $f'$ is not $\mathcal{F}$-worse than $f$ for $x$.
By the choice of $f$, the functional $f'$ is not a counterexample, and so, there exists $g \in W(\mathcal{F})$ not $\mathcal{F}$-worse than $f'$ for $x$ that satisfies~\ref{lem:force:star:depth}~or~\ref{lem:force:star:full}.
However, we obtain that $g$ is not $\mathcal{F}$-worse than $f$ for $x$, which contradicts the fact that $f$ is a counterexample.
\end{proof}
\subsection{The insertion property}
In this section, the main goal is to generalize the core step of the proof of an upper bound for $j_{\mathcal{S}_1}(n)$ by Beanland, Duncan, and Holt \mbox{\cite[Lemma~1.8]{Beanland_2018}}.
We reprove the result with $\mathcal{S}_1$ replaced with a regular family satisfying a certain abstract property.
More precisely, we aim to develop a property of a regular family such that assuming it, we can strengthen condition~\ref{lem:force:star:full} in \cref{lem:force:star}.
We say that a regular family $\mathcal{F}$ \emph{has the insertion property} if for all $a,s,t \in \mathbb{N}$ with $s,t \geq 2$, and for all $n_2,\dots,n_s,m_2,\dots,m_t \in \mathbb{N}$ with $m_2 < \dots < m_t < n_2 < \dots < n_s$ if
\[\{a,m_2,\dots,m_t\} , \{a,n_2,\dots,n_s\}\in \mathcal{F} \ \ \ \mathrm{and} \ \ \ \mathrm{range}_\mathcal{F}(a) < m_2,\]
then
\[\{m_2,m_3,\dots,m_t,n_2,\dots,n_s \} \in \mathcal{F}.\]
First, we show that the families $\Sf,\mathcal{S}_2$, and $\mathcal{S}_3$ have the insertion property. Here, we use the superadditivity of $\varphi$.
\begin{lemma}\label{lem:S-y-nice}
Let $\varphi: \mathbb{N} \rightarrow \mathbb{N}$ be increasing and superadditive. The family $\mathcal{S}_\varphi$ has the insertion property.
\end{lemma}
\begin{proof}
Let $a,s,t \in \mathbb{N}$ be such that $s,t \geq 2$, and let and $n_2,\dots,n_s,m_2,\dots,m_t \in \mathbb{N}$ with $m_2 < \dots < m_t < n_2 < \dots < n_s$.
Let $F := \{a,n_2,\dots,n_s\}$ and $ G := \{a,m_2,\dots,m_t\}$.
Assume that $F,G \in \Sf$ and $\mathrm{range}_{\Sf}(a) < m_2$. Let $H := \{m_2,m_3,\dots,m_t,n_2,\dots,n_s \}$.
Since $\mathrm{range}_{\Sf}(a) = \varphi(a) + a < m_2$, we have $\varphi(\varphi(a) + a) < \varphi(m_2)$, moreover, $\varphi(a) + \varphi(a) \leq \varphi(\varphi(a) + a)$.
Since $F,G \in \mathcal{S}_\varphi$, we have $t = |F| \leq \varphi(\min F) = \varphi(a) $ and $s = |G| \leq \varphi(\min G) = \varphi(a) $. Therefore,
\[|H| = s+t-2 < s + t \leq \varphi(a) + \varphi(a) \leq \varphi(\varphi(a) + a) < \varphi(m_2) = \varphi(\min H).\]
This yields $H \in \mathcal{S}_\varphi$.
\end{proof}
\begin{lemma}\label{lem:S-3-nice}\label{lem:S-2-nice}
For each $\ell \in \{2,3\}$, the family $\mathcal{S}_\ell$ has the insertion property.
\end{lemma}
\begin{proof}
Let $a,s,t \in \mathbb{N}$ be such that $s,t \geq 2$, and let and $n_2,\dots,n_s,m_2,\dots,m_t \in \mathbb{N}$ with $m_2 < \dots < m_t < n_2 < \dots < n_s$.
Let $F := \{a,n_2,\dots,n_s\}$ and $ G := \{a,m_2,\dots,m_t\}$.
Assume that $F,G \in \Sf$ and $\mathrm{range}_{\mathcal{S}_\ell}(a) \leq m_2$. Let $H := \{m_2,m_3,\dots,m_t,n_2,\dots,n_s \}$.
There exist $F_1,\dots,F_a,G_1,\dots,G_a \in \mathcal{S}_{\ell-1}$ such that $F_1 < \dots < F_{a}$, $G_1 < \dots < G_{a}$, and $F = F_1 \cup \dots \cup F_a, G = G_1 \cup \dots \cup G_a$. Observe that
\[H = G_1 \backslash \{a\} \cup G_2 \cup \dots \cup G_a \cup F_1 \backslash \{a\} \cup F_2 \cup \dots \cup F_a.\]
Since $\mathrm{range}_{\mathcal{S}_\ell}(a) \leq m_2 = \min H$, in order to prove that $H \in \mathcal{S}_\ell$, it suffices to check if $2a < \mathrm{range}_{\mathcal{S}_\ell}(a)$, which is clear in both cases by \cref{obs:ranges}. (Note that $F \in \mathcal{S}_\ell$ requires $a > 1$.)
\end{proof}
Observe that the family $2 \mathcal{S}_1$ does not have the insertion property. Indeed, consider the following example:
\begin{align*}
F := \{2,99\}\cup [100,199] \ \ \ \mathrm{and} \ \ \ G := \{2,19\} \cup [20,39].
\end{align*}
Clearly, $F,G \in 2\mathcal{S}_1$. The greatest element of $G$, that is, $39$ is less than the second least element of $F$, that is $99$. It is easy to compute that $\mathrm{range}_{2\mathcal{S}_1}(2) = 6$, thus the inequality $\mathrm{range}_{\mathcal{S}_2}(a) < m_2 - a + 1$ takes form $6 < 19 - 2 + 1$, which is clearly true. The insertion property would give
\[\{19\} \cup [20,39] \cup \{99\} \cup [100,199] \in 2\mathcal{S}_1.\]
This can be easily verified to be false. Following a similar idea, one can construct counterexamples showing that $k\mathcal{S}_1$ does not have the insertion property for all $k \geq 2$. This fact indicates that the family $k\mathcal{S}_1$ has to be treated differently.
As already announced, we now strengthen \cref{lem:force:star} for regular families that have the insertion property.
\begin{lemma}\label{lem:upper:main}
Let $x \in c_{00}^+$ and let $\mathcal{F}$ be a regular family that has the insertion property. For every $f \in W(\mathcal{F})$ such that $f(x) = \|x\|_\mathcal{F}$ there exists $g \in W(\mathcal{F})$ that is not $\mathcal{F}$-worse than $f$ for $x$ and either
\begin{enumerateNumSF}
\item $\mathrm{depth}_\mathcal{F}(g) \leq 1$ or \label{lem:upper:main:i}
\item $\mathrm{depth}_\mathcal{F}(g) \geq 2$ and there exists a positive integer $d$ and $g_1,\dots,g_d \in W(\mathcal{F})$ such that $(g_1,\dots,g_d)$ is $\mathcal{F}$-full-building for $g$, and either $\mathrm{depth}_\mathcal{F}(g_1) = 0$ or there exist a positive integer $e$ and $t_1,\dots,t_e \in W(\mathcal{F})$ such that $(t_1,\dots,t_e)$ is $\mathcal{F}$-building for $g_1$, and $\mathrm{depth}_\mathcal{F}(t_1) \leq 1$.\label{lem:upper:main:iii}
\end{enumerateNumSF}
\end{lemma}
\begin{proof}
By \cref{lem:force:star}, there exists $g' \in W(\mathcal{F})$ that is not $\mathcal{F}$-worse than $f$ for $x$, and either $\mathrm{depth}_\mathcal{F}(g') \leq 1$, or $g'$ is $\mathcal{F}$-full. Fix such $g'$ with $\min \mathrm{supp}\, g'$ maximal.
If $\mathrm{depth}_\mathcal{F}(g') \leq 1$, then \ref{lem:upper:main:i} is satisfied for $g:=g'$. Therefore, we can assume that $\mathrm{depth}_\mathcal{F}(g') \geq 2$, and that $g'$ is $\mathcal{F}$-full for $x$.
Let $d \in \mathbb{N}$ and let $g_1,\dots,g_d \in W(\mathcal{F})$ be such that $(g_1,\dots,g_d)$ is $\mathcal{F}$-full-building for $g'$.
If $\mathrm{depth}_\mathcal{F}(g_1) = 0$, then \ref{lem:upper:main:iii} is satisfied. Thus, we can assume that $\mathrm{depth}_\mathcal{F}(g_1) \geq 1$. Let $e \in \mathbb{N}$ and let $t_1,\dots,t_e \in W(\mathcal{F})$ be such that $(t_1,\dots,t_e)$ is $\mathcal{F}$-building for $g_1$. If $\mathrm{depth}_\mathcal{F}(t_1) \leq 1$, then \ref{lem:upper:main:iii} is satisfied, hence, we assume that $\mathrm{depth}_\mathcal{F}(t_1) \geq 2$. Let $a := \min \mathrm{supp}\, t_1$.
If $\max \mathrm{supp}\, t_1 < \mathrm{range}_\mathcal{F}(a)$, then $\mathrm{supp}\, t_1 \in \mathcal{F}$, and so, by \cref{lem:force:F}, there exists $t_1'$ that is not $\mathcal{F}$-worse than $t_1$ for $x$. Let $g$ be obtained from $g'$ by replacing $t_1$ with $t_1'$. Item~\ref{lem:upper:main:iii} is satisfied, thus, we assume that $\mathrm{range}_\mathcal{F}(a) \leq \max \mathrm{supp}\, t_1$, and so, $\mathrm{range}_\mathcal{F}(a) < \min \mathrm{supp}\, t_2$.
Since $\mathcal{F}$ has the insertion property, we have
\[H := \{\min\mathrm{supp}\, t_2, \dots, \min\mathrm{supp}\, t_e, \min\mathrm{supp}\, g_2, \dots, \min\mathrm{supp}\, g_d\} \in \mathcal{F}.\]
This yields
\begin{align*}
h_1 &:= \frac{1}{2} (t_2 + \dots + t_e + g_2 + \dots + g_d) \in W(\mathcal{F}) \ \ \mathrm{and}\\
h_2 &:= \frac{1}{2} (t_1 + g_2 + \dots + g_d) \in W(\mathcal{F}).
\end{align*}
We have $\frac{1}{2}(h_1 + h_2) = g'$ and $\|x\|_\mathcal{F} = f(x) \leq g'(x) \leq \|x\|_\mathcal{F}$. Therefore, $\|x\|_\mathcal{F} = g'(x) =h_1(x) = h_2(x)$. It is easy to verify that $h_1$ is not $\mathcal{F}$-worse than $f$ for $x$. However, by \cref{lem:force:star}, this yields the existence of $h_1' \in W(\mathcal{F})$ such that $h_1'$ is not $\mathcal{F}$-worse than $h_1$ for $x$ and either $\mathrm{depth}_\mathcal{F}(h_1') \leq 1$, or $h_1'$ is $\mathcal{F}$-full. Note that $\min \mathrm{supp}\, g' < \min \mathrm{supp}\, h_1 \leq \min \mathrm{supp}\, h_1'$, which contradicts the choice of $g'$.
\end{proof}
\subsection{Optimal sequences of realizing functionals}
In the final part of this section, we inductively apply \cref{lem:force:star} and \cref{lem:upper:main} in order to derive ``optimal sequences'' of realizing functionals for each $x \in c_{00}^+$. First, we need the following technical observation.
\begin{obs}\label{obs:realizing_2_subfunctional}
Let $x \in c_{00}^+$ and let $\mathcal{F}$ be a regular family. Let $g \in W(\mathcal{F})$ be $\mathcal{F}$-realizing for $x$ with $\mathrm{depth}_\mathcal{F}(g) \geq 4$. Let $d \in \mathbb{N}$ and let $g_1,\dots,g_d \in W(\mathcal{F})$ be such that $(g_1,\dots,g_d)$ is $\mathcal{F}$-building for $g$. For each $i \in [d]$ with $\mathrm{depth}_\mathcal{F}(g_i) = 0$, let $d_i := 0$; for each $i \in [d]$ with $\mathrm{depth}_\mathcal{F}(g_i) \geq 1$ let $d_i \in \mathbb{N}$ be such that there exist $t_1^{(i)},\dots,t_{d_i}^{(i)} \in W(\mathcal{F})$ with $(t_1^{(i)},\dots,t_{d_i}^{(i)})$ being $\mathcal{F}$-building for $g_i$. Then, there exists $i \in [d]$ and $j \in [d_i]$ such that
\begin{enumerateNumO}
\item if $\mathrm{depth}(g_1) = 0$ or $\mathrm{depth}_\mathcal{F}(t_1^{(1)}) \leq 1$, then $(i,j) \neq (1,1)$;\label{obs:realizing_2_subfunctional:item:not_1}\label{obs:realizing_2_subfunctional:item:first}
\item $\mathrm{depth}_\mathcal{F}(g) = \mathrm{depth}_\mathcal{F}(t_j^{(i)}) + 2$;\label{obs:realizing_2_subfunctional:item:depth}
\item $t_j^{(i)}$ is $\mathcal{F}$-realizing for $x|t_j^{(i)}$;\label{obs:realizing_2_subfunctional:item:realizing}
\item $|\mathrm{span}\, t_j^{(i)}| < |\mathrm{span}\, x|$.\label{obs:realizing_2_subfunctional:item:progress}\label{obs:realizing_2_subfunctional:item:last}
\end{enumerateNumO}
\end{obs}
\begin{proof}
Since $\mathrm{depth}_\mathcal{F}(g) \geq 4$, if for some $i\in [d]$ and $j \in [d_i]$ item~\ref{obs:realizing_2_subfunctional:item:depth} holds, then item~\ref{obs:realizing_2_subfunctional:item:not_1} holds.
Let $I$ be the set of all pairs of integers $i \in [d], j \in [d_i]$ such that item~\ref{obs:realizing_2_subfunctional:item:depth} is satisfied. By definition, $I$ is nonempty. Fix some $(i,j) \in I$, and let $t := t_j^{(i)}$, $x' := x|t_j^{(i)}$. We claim that $t(x') = \|x'\|_\mathcal{F}$. Indeed, if there exists $t' \in W(\mathcal{F})$ with $t(x') < t'(x')$, then $g'$ obtained from $g$ by replacing $t$ with $t'$ satisfies $\|x\|_\mathcal{F} = g(x) < g'(x)$, which is a contradiction.
Suppose that for every $(i,j) \in I$, the functional $t_j^{(i)}$ is not $\mathcal{F}$-realizing for $x|t_j^{(i)}$.
That is, $\mathrm{depth}_\mathcal{F}(t_j^{(i)}) > j(x|t_j^{(i)})$.
For each $(i,j) \in I$, let $s_j^{(i)}$ be $\mathcal{F}$-realizing for $x|t_j^{(i)}$.
Note that $\mathrm{depth}_\mathcal{F}(t_j^{(i)}) > \mathrm{depth}_\mathcal{F}(s_j^{(i)})$.
Let $g'$ be obtained from $g$ by replacing $t_j^{(i)}$ with $s_j^{(i)}$ for each $(i,j) \in I$.
It follows that $g(x) = g(x')$, $\mathrm{depth}_\mathcal{F}(g) > \mathrm{depth}_\mathcal{F}(g')$, and $\mathrm{span}\, g' \subset \mathrm{span}\, g$, which contradicts the fact that $g$ is $\mathcal{F}$-realizing for $x$.
Therefore, there exists $(i,j) \in I$ such that $t_j^{(i)}$ is $\mathcal{F}$-realizing for $x|t_j^{(i)}$.
Finally, we prove that for $(i,j) \in I$ as above item~\ref{obs:realizing_2_subfunctional:item:progress} hold. Observe that $t_j^{(i)}(x) < g(x) = \|x\|_\mathcal{F}$, as otherwise, $g$ is not $\mathcal{F}$-realizing. It follows that $\mathrm{span}\, t_j^{(i)}$ is a strict subset of $\mathrm{span}\, g$, and so $\mathrm{span}\, x$.
\end{proof}
\begin{lemma}\label{lem:realizing:sequence}
Let $\mathcal{F}$ be a regular family. For every $x \in c_{00}^+$, there exist $c \in \mathbb{N}_0$ and $f_0,\dots,f_c \in W(\mathcal{F})$ such that
\begin{enumerateNumR}
\item $\mathrm{depth}_\mathcal{F}(f_0) \leq 3$;\label{lem:realizing:sequence:item:first}\label{lem:realizing:sequence:item:f_0}
\item $f_{c}$ is $\mathcal{F}$-realizing for $x$;\label{lem:realizing:sequence:item:realizing:f_c}
\item for every $m \in [c]$, $f_{m-1}$ is $\mathcal{F}$-realizing for $x|f_{m-1}$;\label{lem:realizing:sequence:item:realizing}
\item for every $m \in [c]$, $\mathrm{span}\, f_{m-1} \subset \mathrm{span}\, f_m$;\label{lem:realizing:sequence:item:supp:inclusion}
\item for every $m \in [c]$, $\mathrm{depth}_\mathcal{F}(f_m) = \mathrm{depth}_\mathcal{F}(f_{m-1}) + 2 $;\label{lem:realizing:sequence:item:depth}
\item for every $m \in [c]$, if $\mathcal{F}$ has the insertion property, then $\min \mathrm{supp}\, f_m < \min \mathrm{supp}\, f_{m-1}$;\label{lem:realizing:sequence:item:progress}
\item for every $m \in [c]$, there exist $F_1,F_2 \subset \mathbb{N}$ with $\min \mathrm{supp}\, f_m \in F_1$ and
\[\min \mathrm{supp}\, f_m \leq F_1 \leq \mathrm{span}\, f_{m-1} < F_2 \leq \max \mathrm{supp}\, f_m,\]
such that $F_1 \cup F_2 \in \mathrm{full}_{f_m}(\mathcal{F})$. \label{lem:realizing:sequence:item:fullness:cor}\label{lem:realizing:sequence:item:last}
\end{enumerateNumR}
\end{lemma}
\begin{proof}
Suppose the lemma was false.
Let $x \in c_{00}^+$ be a counterexample with $|\mathrm{supp}\, x|$ is minimal.
Let $f \in W(\mathcal{F})$ be an $\mathcal{F}$-realizing functional for $x$.
By \cref{lem:force:star}, there exists $g \in W(\mathcal{F})$ that is not $\mathcal{F}$-worse than $f$ for $x$ and one of either \ref{lem:force:star:depth} or \ref{lem:force:star:full} is satisfied.
In the case, where $\mathcal{F}$ has the insertion property, by \cref{lem:upper:main}, the condition~\ref{lem:force:star:full} can be replaced with \ref{lem:upper:main:iii}.
It follows that $g$ is $\mathcal{F}$-realizing for $x$.
If $\mathrm{depth}_\mathcal{F}(g) \leq 3$, then we put $c:=0$ and $f_0:=g$.
It is easy to check that items~\ref{lem:realizing:sequence:item:first}-\ref{lem:realizing:sequence:item:last} are satisfied.
Therefore, we can assume that~\ref{lem:force:star:full} (or \ref{lem:upper:main:iii}) is satisfied, and $\mathrm{depth}_\mathcal{F}(g) \geq 4$.
Let $d \in \mathbb{N}$ and let $g_1,\dots,g_d \in W(\mathcal{F})$ be such that $(g_1,\dots,g_d)$ is $\mathcal{F}$-full-building for $g$. For each $i \in [d]$ with $\mathrm{depth}_\mathcal{F}(g_i) = 0$, let $d_i := 0$; for each $i \in [d]$ with $\mathrm{depth}_\mathcal{F}(g_i) \geq 1$ let $d_i \in \mathbb{N}$ be such that there exist $t_1^{(i)},\dots,t_{d_i}^{(i)} \in W(\mathcal{F})$ with $(t_1^{(i)},\dots,t_{d_i}^{(i)})$ being $\mathcal{F}$-building for $g_i$.
In the case, where $\mathcal{F}$ has the insertion property, by \ref{lem:upper:main:iii} we can assume that either $\mathrm{depth}_\mathcal{F}(g_1) = 0$ or $\mathrm{depth}_\mathcal{F}(t_1^{(1)}) \leq 1$.
By \cref{obs:realizing_2_subfunctional}, there exist $i \in [d]$ with $\mathrm{depth}_\mathcal{F}(g_i) \geq 1$ and $j \in [d_i]$ such that \ref{obs:realizing_2_subfunctional:item:first}-\ref{obs:realizing_2_subfunctional:item:last} are satisfied.
Let $t := t_j^{(i)}$.
By minimality of $x$ and~\ref{obs:realizing_2_subfunctional:item:progress}, for $x|t$ there exist $c'\in \mathbb{N}$ and $f_0,\dots,f_{c'} \in W(\mathcal{F})$ such that \ref{lem:realizing:sequence:item:first}-\ref{lem:realizing:sequence:item:last} hold.
From now on, we refer to the statements in the items for $x|t$ and $f_0,\dots,f_{c'}$ as [\ref{lem:realizing:sequence:item:first}]-[\ref{lem:realizing:sequence:item:last}].
Let $c := c' + 1$, and let $f_c := g$. We claim that the sequence $f_0,\dots,f_c$ satisfy \ref{lem:realizing:sequence:item:first}-\ref{lem:realizing:sequence:item:last} for $x$. Since $x$ is a counterexample this claim leads to a contradiction, thus, it suffices to end the proof of the lemma.
Items~\ref{lem:realizing:sequence:item:f_0}~and~\ref{lem:realizing:sequence:item:realizing:f_c} are obvious.
Note that for each $m \in [c-1] = [c']$, the statements in \ref{lem:realizing:sequence:item:realizing}-\ref{lem:realizing:sequence:item:fullness:cor} follow from the corresponding statements in [\ref{lem:realizing:sequence:item:realizing}]-[\ref{lem:realizing:sequence:item:fullness:cor}], hence it suffices to prove them for $m=c$.
Item~\ref{lem:realizing:sequence:item:realizing} follows from [\ref{lem:realizing:sequence:item:realizing:f_c}].
By~[\ref{lem:realizing:sequence:item:realizing:f_c}], we have
\[\mathrm{span}\, f_{c-1} \subset \mathrm{span}\, x|t = \mathrm{span}\, t \subset \mathrm{span}\, g = \mathrm{span}\, f_c.\]
This yields \ref{lem:realizing:sequence:item:supp:inclusion}.
By~\ref{obs:realizing_2_subfunctional:item:depth}~and~\ref{obs:realizing_2_subfunctional:item:realizing}, we have
\[\mathrm{depth}_\mathcal{F}(f_c) = \mathrm{depth}_\mathcal{F}(g) = \mathrm{depth}_\mathcal{F}(t) + 2 = \mathrm{depth}_\mathcal{F}(f_{c-1}) + 2.\]
This yields \ref{lem:realizing:sequence:item:depth}.
Recall that $t=t_j^{(i)}$.
To prove the next item, assume that $\mathcal{F}$ has the insertion property, by~\ref{obs:realizing_2_subfunctional:item:not_1}, we have $(i,j) \neq (1,1)$, therefore,
\[\min \mathrm{supp}\, f_c = \min \mathrm{supp}\, g \leq \min \mathrm{supp}\, g_1 < \min \mathrm{supp}\, t \leq \min \mathrm{supp}\, f_{c-1}.\]
This yields~\ref{lem:realizing:sequence:item:progress}.
For the last item we define
\begin{align*}
F_1 &= \{\min \mathrm{supp}\, g_\ell : \ell \in [i]\},\\
F_2 &= \{\min \mathrm{supp}\, g_\ell : \ell \in [i+1,d]\}.
\end{align*}
Since $(g_1,\dots,g_d)$ is $\mathcal{F}$-full-building for $g$, we have $F_1 \cup F_2 \in \mathrm{full}_{g}(\mathcal{F}) = \mathrm{full}_{f_c}(\mathcal{F})$. The sequence of inequalities in~\ref{lem:realizing:sequence:item:fullness:cor} is clear, thus,~\ref{lem:realizing:sequence:item:fullness:cor} follows.
\end{proof}
\section{Upper bounds}\label{sec:upper}
\subsection{Upper bound for $\mathcal{S}_\varphi$}
Fix an increasing and superadditive function $\varphi: \mathbb{N} \rightarrow \mathbb{N}$.
\begin{lemma} \label{lem:recursive}
Let $a,b \in \mathbb{N}$ with $a \leq b$.
If $|[a,b]|\leq \varphi(a)$, then $j_{\Sf}(a,b) \leq 1$. Otherwise,
\[j_{\Sf}(a,b) \leq 2 + \max\left(\{j_{\Sf}(a+i,b-\varphi(a)+i+1) : i \in [\varphi(a)-1]\} \cup \{1\}\right).\]
\end{lemma}
\begin{proof}
The first part follows immediately from \cref{lem:force:F}. Suppose that $|[a,b]| > \varphi(a)$.
Let $x \in c_{00}^+$ be such that $j_{\Sf}(x) = j_{\Sf}(a,b)$.
By \cref{lem:realizing:sequence}, there exist $c \in \mathbb{N}_0$ and $f_0,\dots,f_c \in W(\Sf)$ such that~\ref{lem:realizing:sequence:item:first}-\ref{lem:realizing:sequence:item:last} are satisfied. By \cref{lem:S-y-nice}, the family $\Sf$ has the insertion property, thus, by~\ref{lem:realizing:sequence:item:progress}, we have $\min \mathrm{supp}\, f_c < \min \mathrm{supp}\, f_{c-1}$.
By~\ref{lem:realizing:sequence:item:realizing:f_c}, $f_c$ is $\mathcal{F}$-realizing for $x$.
If $c=0$, then by~\ref{lem:realizing:sequence:item:f_0}, we have
\[j_{\Sf}(a,b) = j_{\Sf}(x) = \mathrm{depth}_{\Sf}(f_0) \leq 3 = 2 + 1.\]
Suppose that $c \geq 1$.
First, we claim that there exists $i \in [\varphi(a)-1]$ such that $\mathrm{span}\, f_{c-1} \subset [a+i,b-\varphi(a)+i+1]$.
By~\ref{lem:realizing:sequence:item:fullness:cor}, there exist $F_1,F_2 \subset \mathbb{N}$ with
\[\min \mathrm{supp}\, f_c \leq F_1 \leq \mathrm{span}\, f_{c-1} < F_2 \leq \max \mathrm{supp}\, f_c,\]
such that $F_1 \cup F_2 \in \mathrm{full}_{f_c}(\Sf)$. Since $\mathrm{depth}_{\Sf}(f_c) \geq 2$, it follows that $\mathrm{span}\, f_c \notin \Sf$, and so, \cref{lem:full:Sf} gives $|F_1 \cup F_2| = \varphi(\min F_1) \geq \varphi(a)$. Let $e := |F_1|$, we have
\[\mathrm{span}\, f_{c-1} \subset [\max F_1,\min F_2 - 1] \subset [a+(e-1),b-(\varphi(a)-e)].\]
If $2 \leq e \leq \varphi(a)$, then we put $i := e-1$, and in turn, $\mathrm{span}\, f_{c-1} \subset [a+i,b-\varphi(a)+i+1]$. If $\varphi(a) < e$, then we put $i:=\varphi(a)-1$, and we have $\mathrm{span}\, f_{c-1} \subset [a+(e-1),b]\subset [a+(\varphi(a)-1),b]$. Suppose that $e = 1$, it follows that $\mathrm{span}\, f_{c-1} \subset [a,b-\varphi(a)+1]$.
However, $\min \mathrm{supp}\, f_c < \min \mathrm{supp}\, f_{c-1}$, thus, $\mathrm{span}\, f_{c-1} \subset [a+1,b-\varphi(a)+1] \subset [a+1,b-\varphi(a)+1+1]$, and we put $i:=1$. This concludes the claim that there exists $i \in [\varphi(a)-1]$ such that $\mathrm{span}\, f_{c-1} \subset [a+i,b-\varphi(a)+i+1]$.
By~\ref{lem:realizing:sequence:item:depth},~\ref{lem:realizing:sequence:item:realizing},~and~\cref{obs:j-ab:ineq}, we have
\begin{align*}
j_{\Sf}(a,b) = j_{\Sf}(x) &= \mathrm{depth}_{\Sf}(f_c) \\
&= \mathrm{depth}_{\Sf}(f_{c-1}) + 2 \\
&= j_{\Sf}(x|f_{c-1}) + 2 \leq j_{\Sf}(\min \mathrm{supp}\, f_{c-1}, \max \mathrm{supp}\, f_{c-1}) + 2 \\
&\leq j_{\Sf}(a+i,b-\varphi(a)+i+1) + 2.\qedhere
\end{align*}
\end{proof}
The above lemma justifies the following definition. For all $a,b \in \mathbb{N}$ with $a\leq b$ let
\[\widehat{j}(a,b) :=
\begin{cases}
1 & \text{if } |[a,b]| \leq \varphi(a) , \\
2 + \max\left(\{\widehat{j}(a+i,b-\varphi(a)+i+1) : i \in [\varphi(a)-1]\} \cup \{1\}\right) & \text{otherwise.}
\end{cases} \]
Clearly, for all $a,b \in \mathbb{N}$, we have $j_{\Sf}(a,b) \leq \widehat{j}(a,b)$. Moreover, applying an elementary induction, we obtain an analogous monotonicity result to \cref{obs:j-ab:ineq} for $\widehat{j}$ -- see below.
\begin{obs}\label{obs:hj-subset-property}
For all positive integers $a,b,c,d$ such that $[a,b]\subset [c,d]$ we have $\widehat{j}(a,b) \leq \widehat{j}(c,d)$.
\end{obs}
Furthermore, we can prove a stronger property for $\widehat{j}$ that is intuitive for $j_{\Sf}$ but apparently difficult and technical to derive.
\begin{lemma}\label{lem:hj-t-property}
For all $a,b,t \in \mathbb{N}$, we have $\widehat{j}(a+t,b+t) \leq \widehat{j}(a,b)$.
\end{lemma}
\begin{proof}
We proceed by induction on $s = b-a$. If $|[a,b]| \leq \varphi(a)$ then for every $t \in \mathbb{N}$, we have $1 = \widehat{j}(a+t,b+t) = \widehat{j}(a,b)$. Suppose that $|[a,b]| > \varphi(a)$. By induction,
\[\widehat{j}(a,b) = 2 + \max\left(\{\widehat{j}(a+i,b-\varphi(a)+i+1) : i \in [\varphi(a)-1]\} \cup \{1\}\right) = 2 + \widehat{j}(a+1,b-\varphi(a)+2).\]
Similarly, for every $t \in \mathbb{N}$,
\[\widehat{j}(a+t,b+t) = 2 + \widehat{j}(a+t+1,b+t-\varphi(a+t)+2).\]
Therefore, it suffices to prove that
\[\widehat{j}(a+t+1,b+t-\varphi(a+t)+2) \leq \widehat{j}(a+1,b-\varphi(a)+2).\]
By \cref{obs:hj-subset-property} and since $\varphi(a) < \varphi(a+t)$,
\[\widehat{j}(a+t+1,b+t-\varphi(a+t)+2) \leq \widehat{j}(a+t+1,b+t-\varphi(a)+2).\]
Finally, by induction,
\[\widehat{j}(a+t+1,b+t-\varphi(a)+2) \leq \widehat{j}(a+1,b-\varphi(a)+2),\]
which ends the proof.
\end{proof}
In particular, the above lemma gives that for all $a,b \in \mathbb{N}$,
\[\widehat{j}(a,b) =
\begin{cases}
1 & \text{if } |[a,b]| \leq \varphi(a) , \\
2 + \widehat{j}(a+1,b-\varphi(a)+2) & \text{otherwise.}
\end{cases} \]
\begin{lemma}\label{lem:upper-S-y-main}
For all $a,b \in \mathbb{N}$ and every $c \in \mathbb{N}_0$ if
\[|[a,b]| \leq \sum_{i=0}^{c} \varphi(a+i) - c,\]
then
\[\widehat{j}(a,b) \leq 2c+1.\]
\end{lemma}
\begin{proof}
We proceed by induction on $c$. If $c=0$, then $|[a,b]| \leq \varphi(a)$, clearly yields $\widehat{j}(a,b) = 1$.
Suppose that $c \geq 1$. If $|[a,we assumed thatb]| \leq \varphi(a)$, then the assertion follows, hence, let us assume otherwise. We have $\widehat{j}(a,b) \leq 2 + \widehat{j}(a+1,b-\varphi(a)+2)$. Observe that
\[|[a+1,b-\varphi(a)+2]| = |[a,b]|-\varphi(a)+1 \leq \sum_{i=0}^{c} \varphi(a+i) - c - \varphi(a)+1 = \sum_{i=0}^{c-1} \varphi((a+1)+i) - (c-1).\]
Therefore, by induction, $\widehat{j}(a+1,b-\varphi(a)+2) \leq 2(c-1)+1$, and so, $\widehat{j}(a,b) \leq 2c+1$.
\end{proof}
\begin{theorem}\label{th:upper:S-y}
For every $n \in \mathbb{N}$ and every $c \in \mathbb{N}_0$ if
\[n \leq \sum_{i=1}^{c+1} \varphi(i) - c\]
then
\[j_{\Sf}(n) \leq 2c+1.\]
\end{theorem}
\begin{proof}
By \cref{lem:upper-S-y-main}, applied with $a=1$ and $b=n$, we obtain $\widehat{j}(1,n) \leq 2c+1$, and so, $j_{\Sf}(n) = j_{\Sf}(1,n) \leq \widehat{j}(1,n)$.
\end{proof}
\subsection{Upper bound for $k \mathcal{S}_1$}
Fix an integer $k$ with $k \geq 2$. As already mentioned the family $k \mathcal{S}_1$ does not have the insertion property, hence, we need a different approach than in the case of $\Sf$.
\begin{theorem}\label{thm:upper:kS_1}
For every $n \in \mathbb{N}$, we have
\[j_{2\mathcal{S}_1}(n) \leq 4 \log n + 25,\]
and for every integer $k$ with $k \geq 3$ we have
\[j_{k \mathcal{S}_1}(n) \leq \frac{8}{k-2}\log n + 3.\]
\end{theorem}
\begin{proof}
Most of the proof is the same for the case of $k=2$ and $k \geq 3$, however, there is one detail that differs.
Let us start by fixing some $n \in \mathbb{N}$ and an integer $k$ with $k \geq 2$.
Next, let us fix $x \in c_{00}^+$ such that $j_{k\mathcal{S}_1}(n) = j_{k\mathcal{S}_1}(x)$.
We apply \cref{lem:realizing:sequence} to the family $k\mathcal{S}_1$ and $x$ to obtain $c \in \mathbb{N}_0$ and $f_0,\dots,f_c \in W(k \mathcal{S}_1)$ such that~\ref{lem:realizing:sequence:item:first}-\ref{lem:realizing:sequence:item:last} hold.
Note that by~\ref{lem:realizing:sequence:item:realizing:f_c},~\ref{lem:realizing:sequence:item:f_0},~and~\ref{lem:realizing:sequence:item:depth},
\[j_{k\mathcal{S}_1}(n) = j_{k\mathcal{S}_1}(x) = \mathrm{depth}_{k\mathcal{S}_1}(f_c) \leq 2c+3.\]
In the case, where $c=0$, this concludes the proof, so assume that $c \geq 1$.
Let $k' := \lfloor k \slash 2 \rfloor$ and define
\[Z := \{m \in [c] : \min \mathrm{supp}\, f_{m-1} \geq 2^{k'}\min \mathrm{supp}\, f_{m}\}. \]
Let $z := |Z|$. Clearly,
\begin{align*}\label{eq:kS:z}
n \geq \min \mathrm{supp}\, f_0 \geq 2^{k'z} \min \mathrm{supp}\, f_c \geq 2^{k'z}.
\end{align*}
Let $Z' := [c] \backslash Z$ and $z' := |Z'|$.
Fix some $m \in Z'$.
Let $F_1$ and $F_2$ be as in~\ref{lem:realizing:sequence:item:fullness:cor}.
We have,
\[\min F_1 = \min \mathrm{supp}\, f_m \leq \max F_1 \leq \min \mathrm{supp}\, f_{m-1} < 2^{k'}\min \mathrm{supp}\, f_{m}.\]
Therefore, and by \cref{lem:full-sets-sequence-in-S_2}, $E_{i}(F_1 \cup F_2) \subset F_2$ for every integer $i$ with $i \geq k'+1$.
Since $F_1 \cup F_2 \in \mathrm{full}_{f_m}(k\mathcal{S}_1)$ and $\mathrm{depth}_{k\mathcal{S}_1}(f_m) \geq 2$, we have $\mathrm{span}\, f_m \notin k\mathcal{S}_1$.
It follows that the assumptions of \cref{lem:full-sets-in-kS_1} are satisfied, and so, $E_1(F_1\cup F_2), \dots, E_{k-1}(F_1\cup F_2)$ are full Schreier sets.
First, consider the case, where $k \geq 3$.
For each $i \in [k'+1,k-1]$, the set $E_i(F_1 \cup F_2)$ is a full Schreier set and is a subset of $F_2$.
What is more, $F_2 \subset [\max \mathrm{supp}\, f_{m-1}+1,\max \mathrm{supp}\, f_m]$. By \cref{lem:full-sets-sequence-in-S_2},
\begin{align*}\label{eq:kgeq3}
\max \mathrm{supp}\, f_m \geq 2^{k-1-k'} (\max \mathrm{supp}\, f_{m-1} + 1) \geq 2^{k-1-k'} \max \mathrm{supp}\, f_{m-1}.
\end{align*}
Next, we focus on the case, where $k=2$. By \cref{lem:full-sets-in-kS},
\[\max \mathrm{supp}\, f_m \slash 2 + 2 \geq \min E_2(F_1 \cup F_2) \geq \min F_2 > \max \mathrm{supp}\, f_{m-1}.\]
In particular, $\max \mathrm{supp}\, f_m \geq 2 \max \mathrm{supp}\, f_{m-1} - 4$.
Summing up, in the case of $k \geq 3$,
\[ n \geq \max \mathrm{supp}\, f_c \geq 2^{(k-1-k')z'}\max \mathrm{supp}\, f_0 \geq 2^{(k-1-k')z'},\]
and in the case of $k = 2$,
\[n \geq \max \mathrm{supp}\, f_c \geq 2^{z'}\max \mathrm{supp}\, f_0 - 4z' \geq 2^{z'}-4z'.\]
Combining this with the relation of $n$ and $z$, if $k \geq 3$, we have
\[n \geq 2^{(k \slash 2 - 1) \max \{z,z'\}} \geq 2^{(k \slash 2 - 1) c \slash 2}.\]
Finally,
\[\frac{8}{k-2}\log n + 3 \geq 2c + 3 \geq j_{k \mathcal{S}_1}(n).\]
Similarly, in the case of $k=2$, for $c \geq 12$, we have
\[n \geq 2^{\max \{z,z'\} - 4\max \{z,z'\}} \geq 2^{c \slash 2} - 2c \geq 2^{c \slash 2 - 1},\]
and so, $4 \log n + 7 \geq 2c+3 \geq j_{k \mathcal{S}_1}(n)$.
If $c < 12$, then $j_{k \mathcal{S}_1}(n) \leq 25$, thus, the bound in the assertion holds.
\end{proof}
\subsection{Upper bound for $\mathcal{S}_2$}
The family $\mathcal{S}_2$ has the insertion property, however, we will also use some ideas from the previous section. Note that the proof of the upper bound for $\mathcal{S}_3$ is very similar.
\begin{theorem}\label{thm:upper:S2}
For every $n \in \mathbb{N}$, we have
\[j_{\mathcal{S}_2}(n) \leq 8 \sqrt{\log n} + 9.\]
\end{theorem}
\begin{proof}
Fix some $n \in \mathbb{N}$ and $x \in c_{00}^+$ such that $j_{\mathcal{S}_2}(n) = j_{\mathcal{S}_2}(x)$.
We apply \cref{lem:realizing:sequence} to the family $\mathcal{S}_2$ and $x$ to obtain $c \in \mathbb{N}_0$ and $f_0,\dots,f_c \in W(\mathcal{S}_2)$ such that~\ref{lem:realizing:sequence:item:first}-\ref{lem:realizing:sequence:item:last} hold.
Note that by~\ref{lem:realizing:sequence:item:realizing:f_c},~\ref{lem:realizing:sequence:item:f_0},~and~\ref{lem:realizing:sequence:item:depth} we have
\[j_{\mathcal{S}_2}(n) = j_{\mathcal{S}_2}(x) = \mathrm{depth}_{\mathcal{S}_2}(f_c) \leq 2c+3.\]
In the case, where $c=0$, this concludes the proof, so assume that $c \geq 1$.
Define
\[Z := \{m \in [c] : \min \mathrm{supp}\, f_{m-1} \geq \min \mathrm{supp}\, f_{m} \cdot 2^{\min \mathrm{supp}\, f_{m} \slash 2}\}. \]
Let $z := |Z|$. By \cref{lem:S-2-nice} and~\ref{lem:realizing:sequence:item:progress}, we have $\min \mathrm{supp}\, f_m \geq c-m+1$. By~\ref{lem:realizing:sequence:item:supp:inclusion}, we obtain
\[n \geq \min \mathrm{supp}\, f_0 \geq \prod_{i=c-z+1}^c 2^{\min \mathrm{supp}\, f_{i} \slash 2} \min \mathrm{supp}\, f_c \geq \prod_{i=1}^z 2^{i\slash 2} \geq 2^{(z^2+z)\slash 4}.\]
Let $Z' := [c] \backslash Z$ and $z' := |Z'|$.
Fix some $m \in Z'$.
Let $F_1$ and $F_2$ be as in~\ref{lem:realizing:sequence:item:fullness:cor}, and let $a := \min F_1$.
We have,
\[a = \min F_1 = \min \mathrm{supp}\, f_m \leq \max F_1 \leq \min \mathrm{supp}\, f_{m-1} < \min \mathrm{supp}\, f_{m} \cdot 2^{\min \mathrm{supp}\, f_{m} \slash 2} = 2^{a \slash 2} a.\]
In particular, $F_1 \subset [a,2^{a \slash 2} a]$.
Let $s$ be the greatest positive integer such that $E_s(F_1 \cup F_2) \subset F_1$, or let $s := 0$ if $E_1(F_1 \cup F_2)\not\subset F_1$.
By \cref{lem:full-sets-sequence-in-S_2}, $2^{a \slash 2} a \geq 2^s a$, and so $a \slash 2 \geq s$.
It follows that $E_{i}(F_1 \cup F_2) \subset F_2$ for every integer $i$ with $i > a \slash 2$.
Since $F_1 \cup F_2 \in \mathrm{full}_{f_m}(\mathcal{S}_2)$ and $\mathrm{depth}_{\mathcal{S}_2}(f_m) \geq 2$, we have $\mathrm{span}\, f_m \notin \mathcal{S}_2$.
This ensures that the assumptions of \cref{lem:full-sets-in-S_2} are satisfied, and so $E_1(F_1\cup F_2), \dots, E_{a-1}(F_1\cup F_2)$ are full Schreier sets.
For each $a \slash 2 < i \leq a$, the set $E_i(F_1 \cup F_2)$ is a full Schreier set and is a subset of $F_2$.
What is more, $F_2 \subset [\max \mathrm{supp}\, f_{m-1}+1,\max \mathrm{supp}\, f_m]$. By \cref{lem:full-sets-sequence-in-S_2},
\begin{align*}
\max \mathrm{supp}\, f_m \geq 2^{a - (a \slash 2 + 1)} (\max \mathrm{supp}\, f_{m-1} + 1) \geq 2^{a \slash 2 - 1} \max \mathrm{supp}\, f_{m-1}.
\end{align*}
Therefore,
\[ n \geq \max \mathrm{supp}\, f_c \geq \prod_{i=c-z+1}^c 2^{\min \mathrm{supp}\, f_{i} \slash 2 - 1}\max \mathrm{supp}\, f_0 \geq \prod_{i=1}^{z'}2^{i \slash 2-1} = 2^{(z'^2-z')\slash 4}.\]
Combining this with the relation of $n$ and $z$ and using $z+z' = c$, we obtain
\[n \geq \max\{2^{(z^2+z)\slash 4}, 2^{(z'^2-z')\slash 4}\} \geq 2^{(c^2-6c)\slash 16} \geq 2^{(c-3)^2\slash 16}.\]
Finally,
\[8\sqrt{\log n} + 9 \geq 2c + 3 \geq j_{\mathcal{S}_2}(n).\qedhere\]
\end{proof}
\subsection{Upper bound for $\mathcal{S}_3$}
\begin{theorem}\label{thm:upper:S3}
For every $n \in \mathbb{N}$ we have
\[j_{\mathcal{S}_3}(n) \leq 8 \sqrt{\log^* n} + 9.\]
\end{theorem}
\begin{proof}
Fix some $n \in \mathbb{N}$ and $x \in c_{00}^+$ such that $j_{\mathcal{S}_3}(n) = j_{\mathcal{S}_3}(x)$.
We apply \cref{lem:realizing:sequence} to the family $\mathcal{S}_3$ and $x$ to obtain $c \in \mathbb{N}_0$ and $f_0,\dots,f_c \in W(\mathcal{S}_3)$ such that~\ref{lem:realizing:sequence:item:first}-\ref{lem:realizing:sequence:item:last} hold.
Note that by~\ref{lem:realizing:sequence:item:realizing:f_c},~\ref{lem:realizing:sequence:item:f_0},~and~\ref{lem:realizing:sequence:item:depth} we have
\[j_{\mathcal{S}_3}(n) = j_{\mathcal{S}_3}(x) = \mathrm{depth}_{\mathcal{S}_3}(f_c) \leq 2c+3.\]
In the case, where $c=0$, this concludes the proof, so assume that $c > 0$.
Define
\[Z := \{m \in [c] : \min \mathrm{supp}\, f_{m-1} \geq \tau(\lfloor\min \mathrm{supp}\, f_{m} \slash 2\rfloor, \min \mathrm{supp}\, f_{m})\}. \]
Let $z := |Z|$. By \cref{lem:S-2-nice} and~\ref{lem:realizing:sequence:item:progress} we have $\min \mathrm{supp}\, f_m \geq c-m+1$. By~\ref{lem:realizing:sequence:item:supp:inclusion}, we obtain
\begin{align*}
n \geq \min \mathrm{supp}\, f_0 \geq \tau(\sum_{i=c-z+1}^c \lfloor\min \mathrm{supp}\, f_{i} \slash 2\rfloor, &\min \mathrm{supp}\, f_c) \geq \\
&\tau(\sum_{i=1}^z \lfloor i \slash 2\rfloor, 1) \geq \tau((z^2-3z)\slash 4,1).
\end{align*}
\[\]
Let $Z' := [c] \backslash Z$ and $z' := |Z'|$.
Fix some $m \in Z'$.
Let $F_1$ and $F_2$ be as in~\ref{lem:realizing:sequence:item:fullness:cor}, and let $a := \min F_1$.
We have,
\begin{align*}
a = \min F_1 = \min \mathrm{supp}\, f_m &\leq \max F_1 \leq \min \mathrm{supp}\, f_{m-1}\\ &< \tau(\lfloor\min \mathrm{supp}\, f_{m} \slash 2\rfloor, \min \mathrm{supp}\, f_{m}) = \tau(\lfloor a \slash 2\rfloor, a).
\end{align*}
In particular, $F_1 \subset [a,\tau(\lfloor a \slash 2\rfloor, a)]$.
Let $s$ be the greatest positive integer such that $E_s(F_1 \cup F_2) \subset F_1$, or let $s := 0$ if $E_1(F_1 \cup F_2)\not\subset F_1$.
By \cref{lem:full-sets-sequence-in-S_3}, $\tau(\lfloor a \slash 2\rfloor, a) \geq \tau(s,a)$, and so, $\lfloor a \slash 2\rfloor \geq s$.
It follows that $E_{i}(F_1 \cup F_2) \subset F_2$ for every integer $i$ with $i > \lfloor a \slash 2\rfloor$.
Since $F_1 \cup F_2 \in \mathrm{full}_{f_m}(\mathcal{S}_3)$ and $\mathrm{depth}_{\mathcal{S}_3}(f_m) \geq 2$, we have $\mathrm{span}\, f_m \notin \mathcal{S}_3$.
This ensures that the assumptions of \cref{lem:full-sets-in-S_3} are satisfied, and so, $E_1(F_1\cup F_2), \dots, E_{a-1}(F_1\cup F_2)$ are $\mathcal{S}_2$-full sets.
For each $\lfloor a \slash 2\rfloor < i \leq a$, the set $E_i(F_1 \cup F_2)$ is an $\mathcal{S}_2$-full set and is a subset of $F_2$.
What is more, $F_2 \subset [\max \mathrm{supp}\, f_{m-1}+1,\max \mathrm{supp}\, f_m]$. By \cref{lem:full-sets-sequence-in-S_3},
\begin{align*}
\max \mathrm{supp}\, f_m \geq \tau(a - (\lfloor a \slash 2\rfloor + 1), \max \mathrm{supp}\, f_{m-1} + 1) \geq \tau(a \slash 2-1, \max \mathrm{supp}\, f_{m-1} ).
\end{align*}
Therefore,
\begin{align*}
n \geq \max \mathrm{supp}\, f_c \geq \tau(\sum_{i=c-z+1}^c \min \mathrm{supp}\, f_{i} \slash &2-1, \max \mathrm{supp}\, f_0) \geq \\
&\tau(\sum_{i=1}^{z'} i \slash 2-1, 1) = \tau((z'^2-3z')\slash 4,1).
\end{align*}
Combining this with the relation of $n$ and $z$ and using $z+z' = c$, we obtain
\[n \geq \max\{\tau((z^2-3z)\slash 2,1), \tau((z'^2-3z')\slash 4,1)\} \geq \tau((c^2-6c)\slash 16,1) \geq \tau((c-3)^2\slash 16,1).\]
Finally,
\[8\sqrt{\log^* n} + 9 \geq 2c + 3 \geq j_{\mathcal{S}_3}(n).\qedhere\]
\end{proof}
\section{Open problems}\label{sec:open}
To conclude, we want to mention a few interesting research directions related to computing the function $j(n)$. The first natural problem is to give an even more precise estimation of the original function $j_{\mathcal{S}_1}(n)$, that is, up to an additive constant.
\begin{problem}
Find a real number $C$ such that there exist real numbers $A,B$ such that for every $n \in \mathbb{N}$,
\[ C \sqrt{n} + A \leq j(n) \leq C \sqrt{n} + B.\]
\end{problem}
By \cref{th:main-Schreier}, we know that if the constant $C$ exists, then $C \in [\sqrt{2},2]$.
Recall that a generalized Tsirelson's norm $T[\theta,\mathcal{F}]$ depends on a real number $0 < \theta < 1$ and a regular family $\mathcal{F}$. In this paper, we studied the function $j_{T[\frac{1}{2},\mathcal{F}]}$ for some regular families $\mathcal{F}$. Another approach is to determine what is the order of magnitude of the function $j_{T[\theta,\mathcal{F}]}$ when we fix $\mathcal{F} = \mathcal{S}_1$ and change $\theta$. There are two versions of this problem.
\begin{problem}
For a fixed real number $\theta$ with $0 < \theta < 1$, compute the order of magnitude of the function $j_{T[\theta,\mathcal{S}_1]}(n)$.
\end{problem}
\begin{problem}
Compute the order of magnitude of a two-variable function $j_{T[\theta,\mathcal{S}_1]}(n)$.
\end{problem}
The last problem is inspired by the discussion in \cite{Go-blog}.
We believe that the problem of computing the Tsirelson's norm $\|x\|_T$ for a vector $x \in c_{00}$ with $\mathrm{supp}\, x \subset [n]$ can be solved using a dynamic programming scheme in polynomial time.
More precisely, in time $\mathrm{poly}(n) \cdot j(n)$, which is clearly a polynomial function.
The situation seems to be similar in the case of $\|x\|_{\Sf}$ for any function $\varphi$.
In particular, this gives a polynomial time algorithm for any fast-growing function $\varphi$.
However, for slow-growing functions, it does not give satisfactory running time -- recall that we established a lower bound on $j_{\Sf}$, which in the case of slow-growing functions is a fast-growing function.
\begin{problem}
Is there a non-decreasing function $\varphi: \mathbb{N} \rightarrow \mathbb{N}$ such that the problem of computing the norm $\|\cdot\|_{T[\frac{1}{2},\Sf]}$ is hard in the sense of computational complexity?
\end{problem}
\end{document} |
\begin{document}
\normalem
\title{f{Multivariate type G Mat\'ern stochastic partial differential equation random fields}
\begin{center}\begin{minipage}{0.9\textwidth}
\noindent{\bf Abstract:}
For many applications with multivariate data,
random field models
capturing departures
from Gaussianity within realisations
are appropriate.
For this reason,
we formulate a new class of multivariate non-Gaussian models
based on systems of stochastic partial differential equations
with additive type G noise
whose marginal covariance functions
are of Mat\'ern type.
We consider four increasingly flexible constructions of the noise, where the first two are similar to existing copula-based models. In contrast to these, the latter two constructions can model non-Gaussian spatial data without replicates.
Computationally efficient methods for likelihood-based parameter estimation and probabilistic prediction are proposed, and the flexibility of the suggested models is illustrated by numerical examples and two statistical applications.
\noindent{\bf Key words:}
Mat\'{e}rn covariances;
Multivariate random fields;
Non-Gaussian models;
Spatial statistics;
Stochastic partial differential equations.
\end{minipage}
\end{center}
\section{Introduction}
Motivated by an increasing number of spatial data sets with multiple measured variables, such as different climate variables from weather stations, various pollutants monitored in urban areas, or climate model outputs, the literature on models for multivariate random fields is growing rapidly. The majority of research in this area has focused on Gaussian random fields, and how to construct valid multivariate cross-covariance functions.
Of particular interest has been multivariate extensions of the Mat\'{e}rn correlation function \citep{matern60},
$\materncorr{\mv{h}}{\kappa}{\nu} = 2^{1-\nu}\Gamma(\nu)^{-1}\left(\kappa \|\mv{h}\|\right)^{\nu}K_{\nu}\left(\kappa \|\mv{h}\|\right)$, $\mv{h} \in \R^d$. Here $K_{\nu}$ is a modified Bessel function of the second kind and the positive parameters $\kappa$ and $\nu$ determine the practical correlation range and smoothness of the process respectively. \cite{gneiting2012matern} extended it to the multivariate setting by proposing a model with cross-correlation functions $\rho_{ij}\materncorr{\mv{h}}{\kappa_{ij}}{\nu_{ij}}$, where $\rho_{ij}$ are parameters determining the cross-correlations between the $i$th and $j$th component of the multivariate field. The parameters in this construction must be restricted to assure that it is a valid multivariate covariance function, and \cite{gneiting2012matern} proposed two models that satisfied this requirement: A parsimonious model, where $\kappa_{ij} \equiv \kappa$ and $\nu_{ij} = (\nu_{ii} + \nu_{jj})/2$, and a more general bivariate model that was later extended by \cite{apanasovich2012valid}.
Even though most research has focused on Gaussian random fields, many data sets have features that cannot be captured by Gaussian models, such as exponential tails, non-Gaussian dependence, or asymmetric marginal distributions. There is thus a need for multivariate random fields that are more general than the Gaussian. Examples of such models in the literature are multivariate max-stable processes for spatial extremes \citep{genton2015multivariate} and Mittag-Leffler random fields \citep{ma2013mittag}. A common approach for constructing non-Gaussian fields is to multiply a Gaussian random field with a random scalar. Multivariate versions of this approach were explored by \cite{ma2013student} and \cite{du2012hyperbolic}. Copula-based modelling is another popular method for non-Gaussian data, which has been used for creating both univariate \citep{graler2014modelling,bardossy2006copula} and multivariate \citep{krupskii2016factor} random fields.
However, creating non-Gaussian multivariate random field models that allow for likelihood-based parameter estimation and probabilistic prediction is difficult, especially if they should be able to capture interesting departures from normality within realisations, and not just have non-Gaussian marginal distributions. This requirement excludes fields that are non-Gaussian only in the presence of repeated measurements, such as the factor-copula models \citep{krupskii2016factor} and the constructions based on multiplying Gaussian fields with random scalars. Many other copula-based approaches in geostatistics use Gaussian copulas. The resulting models are then equivalent to transformed Gaussian models \citep{kazianka2010copula}, which have many disadvantages \citep{Wallin15}. Thus, most existing approaches are either too limited, in the sense that they cannot capture essential features such as sample path asymmetry, or they lack methods for practical applications. For this reason, the recent review article on multivariate random fields by \cite{genton2015cross} listed creation of practically useful non-Gaussian multivariate random fields as an open problem.
The main contribution of this work is to present a class of models that remedies this problem. The model class is constructed using systems of stochastic partial differential equations (SPDEs) driven by non-Gaussian noise. To facilitate computationally efficient likelihood-based inference, we use noise with normal-variance mixture distributions \citep{Barndorff1982}, which we refer to as type G noise. The restriction to normal-variance mixtures is not a big limitation, since several common distributions can be formulated in this way. Four increasingly flexible constructions are considered, where the simplest is closely related to factor copula models and the approach where a Gaussian field is multiplied with a random scalar. The more flexible constructions allow the fields to capture more complex dependency structures and departures from Gaussianity within realisations, while still allowing for likelihood-based inference. As an additional motivation for the more flexible constructions, we investigate the properties of spatial prediction based on the type G models, and in particular prove that distributions of spatial predictions for the simplest construction are asymptotically Gaussian. This means that if the goal is to use the model for spatial prediction, one might as well use a Gaussian model instead of the simple non-Gaussian constructions.
In the seminal work of \cite{lindgren10}, Gaussian random fields where formulated as solutions to SPDEs, which were apprroximated using a element (FE) discretization to allow for computationally efficient inference.
\cite{hu2013multivariate} and \cite{hu2016spatial} extended the work to multivariate Gaussian random field models based on systems of SPDEs. However, their models in general do not have explicit covariance functions, which can complicate the understanding of the effect each model parameter has, in particular in the non-Gaussian case. To avoid this problem, we formulate systems of SPDEs that result in models with marginal Mat\'ern covariance functions, having the parsimonious Mat\'ern model as a special case. We further discover a set of parameters in the model formulation that do not affect the covariance function, and therefore are unidentifiable for Gaussian models. These parameters, however, control the more complex dependence for non-Gaussian models.
As always with more general models than the Gaussian, there is an added computational cost for inference. However, using FE discretizations of our non-Gaussian models allows for the same computational complexity with respect the size of the discretized random field as for the corresponding Gaussian models. This makes the models applicable in scenarios where the data sets are so large that it prohibits the use of standard covariance-based models. As in the Gaussian case, the SPDE approach also facilitates extensions to non-stationary models by using spatially varying parameters. An important fact related to this is that the construction of the FE approximation is identical to that for Gaussian models, and thus as easy to compute.
The article is structured as follows. In Section \ref{sec:spde}, the link between systems of SPDEs and cross-covariances is studied. Section \ref{sec:nongauss} contains the definitions of the non-Gaussian models, as well as derivations of basic model properties. More details and examples of multivariate normal inverse Gaussian (NIG) fields, a special case of the type G models, are given in Section \ref{sec:nig}. In Section \ref{sec:model}, the type G fields are included in a geostatistical model for which we derive computationally efficient methods for likelihood-based parameter estimation and probabilistic prediction.
Section~\ref{sec:applications} presents two applications, and the article concludes with a discussion in Section~\ref{sec:discussion}. The article contains five appendices that present (\ref{sec:fem}) details on the FE discretizations; (\ref{sec:gradients}) gradients needed for the parameter estimation; (\ref{sec:pseudo}) sampling methods for the models; (\ref{seq:parameter_estimates}) details on the applications; and (\ref{sec:proofs}) all proofs. The methods developed in this work have been implemented in the R package \texttt{ngme}.
\section{Multivariate Mat\'ern fields and systems of SPDEs}\label{sec:spde}
A Gaussian random field $x(\mv{s})$ on $\mathbb{R}^d$ with a Mat\'ern covariance function can be represented as a stationary solution to the stochastic partial differential equation
\begin{equation}\label{eq:model}
(\kappa^2-\Delta)^{\frac{\alpha}{2}} x = \gnoise, \quad \mbox{in $\mathcal{D}:=\R^d$},
\end{equation}
where $\alpha = \nu + d/2$, $\Delta$ is the Laplacian and $\gnoise$ is Gaussian white noise \citep{whittle63}. Extending equation \eqref{eq:model} to a system of SPDEs can be used to define more general covariance models \citep{bolin09b} and to define multivariate random fields. \cite{hu2013multivariate} and later \cite{hu2016spatial} proposed using systems of the form
\begin{align}\label{eq:spdesystem}
\begin{bmatrix}
\mathcal{K}_{11} & \mathcal{K}_{12} & \cdots & \mathcal{K}_{1p}\\
\mathcal{K}_{21} & \mathcal{K}_{22} & \cdots & \mathcal{K}_{2p}\\
\vdots & \vdots & \ddots & \vdots \\
\mathcal{K}_{p1} & \mathcal{K}_{p2} & \cdots & \mathcal{K}_{pp}\\
\end{bmatrix}
\begin{bmatrix}
x_1\\
x_2\\
\vdots \\
x_p
\end{bmatrix}
= \begin{bmatrix}
\gnoise_1 \\
\gnoise_2 \\
\vdots \\
\gnoise_p
\end{bmatrix},
\end{align}
to construct multivariate random fields, $\mv{x}(\mv{s}) = (x_1(\mv{s}), \ldots, x_p(\mv{s}))^{\ensuremath{\top}}$, where $\mathcal{K}_{ij}$ are pseudo-differential operators such as $(\kappa^2-\Delta)^{\frac{\alpha}{2}}$ and $\gnoise_1, \ldots, \gnoise_p$ are mutually independent Gaussian white noise processes.
\cite{hu2013multivariate} focused on the bivariate triangular system
\begin{align}\label{eq:uppertrisystem}
\begin{bmatrix}
\mathcal{K}_{11} & \mathcal{K}_{12} \\
& \mathcal{K}_{22}
\end{bmatrix}
\begin{bmatrix}
x_1\\
x_2
\end{bmatrix}
= \begin{bmatrix}
\gnoise_1 \\
\gnoise_2
\end{bmatrix},
\end{align}
where $\mathcal{K}_{ij} = (\kappa_{ij}^2-\Delta)^{\frac{\alpha_{ij}}{2}}$. To better understand the cross-covariance function for this model, one can informally invert the operator matrix to obtain
\begin{align}\label{eq:rep2}
\begin{bmatrix}
x_1 \\
x_2
\end{bmatrix}
=
\begin{bmatrix}
\mathcal{K}^{-1}_{11} & -\mathcal{K}^{-1}_{11} \mathcal{K}_{12} \mathcal{K}^{-1}_{22} \\
& \mathcal{K}^{-1}_{22}
\end{bmatrix}
\begin{bmatrix}
\gnoise_1 \\
\gnoise_2
\end{bmatrix}.
\end{align}
From this representation one can see that $x_2$ is marginally a Gaussian Mat\'ern field whereas $x_1$ is a sum of two Gaussian fields $\mathcal{K}^{-1}_{11}\gnoise_1$ and $-\mathcal{K}^{-1}_{11} \mathcal{K}_{12} \mathcal{K}^{-1}_{22}\gnoise_2$ and thus has a more complicated covariance function.
Although the full system \eqref{eq:spdesystem} may be of interest, the generality comes at the cost of a large number of parameters that are difficult to identify in practice, and equally hard to estimate. We therefore focus on the case when all marginal covariances are Mat\'ern, and on characterizing systems of SPDEs that result in models with this property.
\subsection{Multivariate Mat\'ern-SPDE fields}
To make the results in this section applicable beyond Gaussian models, we replace the right-hand side of \eqref{eq:spdesystem} by $\dot{\mv{\mathcal{M}}} = (\dot{\mathcal{M}}_1,\ldots, \dot{\mathcal{M}}_p)^{\ensuremath{\top}}$, where the components are mutually uncorrelated, but not necessarily independent, $L_2$-valued independently scattered random measures (see Section \ref{sec:nig} and \citet{rajput1989spectral} for details). This includes Gaussian noise but also the non-Gaussian processes that we will study in the next section. We introduce the operator matrix $\mv{\mathcal{K}}$ with entries $\mv{\mathcal{K}}_{ij} = \mathcal{K}_{ij}$ and write \eqref{eq:spdesystem} more compactly as $\mv{\mathcal{K}}\mv{x}= \mv{\dot{\mathcal{M}}}$.
Investigating \eqref{eq:rep2}, we can note that $x_1$ has a Mat\'ern covariance function if $\mathcal{K}_{12} = \mathcal{K}_{22}$. This motivates the following definition of $p$-variate Mat\'ern-SPDE fields.
\begin{defn}\label{def:tri}
A multivariate Mat\'ern-SPDE field on $\mathbb{R}^d$ is a solution to $\mv{\mathcal{K}}\mv{x} = \mv{\dot{\mathcal{M}}}$ where the operator matrix is of the form $\mv{\mathcal{K}} = \mv{D}\diag(\mathcal{L}_1,\cdots, \mathcal{L}_p)$.
Here $\mv{D}$ is a real invertible $p\times p$ matrix and $\mathcal{L}_i = (\kappa_{i}^2-\Delta)^{\frac{\alpha_{i}}{2}}$ with $\kappa_i>0$ and $\alpha_i >d/2$ for $i=1,\ldots, p$.
\end{defn}
Since $\mv{D}$ defines the dependence structure of the process, we refer to it as a dependence matrix.
That the multivariate Mat\'ern-SPDE model indeed has marginal Mat\'ern covariance functions is clarified in the following proposition.
\begin{prop}\label{thm1}
Given that the driving noise in $\mv{\mathcal{K}}\mv{x} = \mv{\dot{\mathcal{M}}}$ has unit variance, the multivariate Mat\'ern-SPDE field $\mv{x}(\mv{s})$ on $\mathbb{R}^d$ has covariance function
$$
\Cov(x_i(\mv{s}),x_j(\mv{t})) = \begin{cases}
\frac{\Gamma(\nu_i)\sum_{j=1}^p Di_{ii}^2}{\Gamma(\alpha_i)(4\pi)^{d/2}\kappa_i^{2\nu_i}}\materncorr{\|\mv{s}-\mv{t}\|}{\kappa_i}{\nu_i} & i = j, \\
\mathcal{F}^{-1}(S_{ij})(\|\mv{s}-\mv{t}\|) & i \neq j,
\end{cases}
$$
where $Di_{ij}$ are the elements of $\mv{Di} = \mv{D}^{-1}$, $\mathcal{F}^{-1}$ denotes the inverse Fourier transform, and
\begin{equation}\label{eq:cross_spec}
S_{ij}(\mv{k}) = \frac{\sum_{l=1}^pDi_{il}Di_{jl}}{(2\pi)^d}\frac{1}{(\kappa_i^2+\|\mv{k}\|^2)^{\frac{\alpha_i}{2}}
(\kappa_j^2+\|\mv{k}\|^2)^{\frac{\alpha_j}{2}}}.
\end{equation}
\end{prop}
Note that $\mv{D}$ determines the strength of the cross-correlations, and that $\Cov(x_i(\mv{s}),x_j(\mv{t}))$ for $i\neq j$ is a Mat\'ern covariance function only if $\kappa_j = \kappa_j$. In the case when $\kappa_i=\kappa$ for all $i$, the model coincides with the parsimonious Mat\'ern model by \cite{gneiting2012matern}. Also note that the shapes of the cross-correlation functions are determined by the parameters of the marginal correlation functions.
This is slightly more restrictive than the general covariance-based multivariate Mat\'ern models, but has the advantage that there are no difficult-to-check restrictions on the model parameters. Furthermore, both \cite{gneiting2012matern} and \cite{apanasovich2012valid} argued that the most important aspect of multivariate models is to allow for flexibility in the marginal covariances while still allowing for some degree of cross-covariance. Thus, the Mat\'ern-SPDE model should be a sufficiently flexible alternative to multivariate Mat\'ern fields for most applications.
\begin{rem}\label{cor2}
An immediate consequence of Definition \ref{def:tri} is that $\mv{x}$ alternatively can be obtained as a solution to a diagonal system of SPDEs driven by correlated noise: $\diag(\mathcal{L}_1,\ldots, \mathcal{L}_p)\mv{x}(\mv{s}) = \mv{\noise}_{R}$, where $\mv{\noise}_{R} = \mv{R}\mv{\noise}$ and $\mv{R} = \mv{D}^{-1}$. This means that the model can be viewed as a linear model of coregionalization.
\end{rem}
\subsection{Parameterising the model}\label{sec:param}
An important question for practical applications of the multivariate Mat\'ern-SPDE fields is if the model parameters (the dependence matrix and the parameters of the operators) are identifiable. The following proposition shows that this is not the case in general.
\begin{prop}\label{thm2}
Two multivariate Mat\'ern-SPDE fields, with the same operators $\mathcal{L}_1,\ldots, \mathcal{L}_p$ and with dependence matrices $\mv{D}$ and $\hat{\mv{D}}$ respectively, have equal covariance functions if and only if $\mv{D} = \mv{Q}\hat{\mv{D}}$ for an orthogonal matrix $\mv{Q}$. For any choice of $\mv{D}$, one can find a triangular matrix $\hat{\mv{D}}$ that gives the same covariance functions. In particular, $\hat{\mv{D}} = \chol(\mv{D}^{\ensuremath{\top}}\mv{D})$ is the unique upper-triangular choice with positive diagonal elements.
\end{prop}
We will refer to models with triangular dependence matrices as triangular Mat\'ern-SPDE fields. Since Gaussian fields are uniquely specified by the first two moments, the proposition implies that the matrix $\mv{D}$ is not completely identifiable from data for Gaussian models, so there is no point in considering non-triangular Gaussian models. This is however not the case for non-Gaussian models, where non-triangular dependence matrices can be used to define more general dependence structures.
Since the dependence matrix is not completely identifiable for Gaussian models, a different model parametrization that separates the control of marginal variances, cross-correlations, and higher moments is preferable. To derive such a parametrization, we use Proposition \ref{thm2} to write $\mv{D} = \mv{Q}_p\mv{D}_l$, where $\mv{D}_l$ is a triangular matrix and $\mv{Q}_p$ is an orthogonal matrix. Then $\mv{D}_l$ and $\mv{Q}_p$ respectively determine the cross-covariances and the higher moments. To separate the control of the variances and cross-correlations, we rescale the operators $\mathcal{L}_i$ by constants $c_i = \sqrt{\sigma_i^{-2}(4\pi)^{-d/2}\kappa_i^{-2\nu_i}\Gamma(\nu_i)/\Gamma(\alpha_i)}$ and parametrize $\mv{D}_l$ as,
$$
\mv{D}_l(\mv{\rho}) =
\begin{pmatrix}
1 & & & & \\
\rho_{1,1} & 1 & & & \\
\rho_{2,1} & \rho_{2,2} & 1 & & \\
\vdots & \vdots & \ddots & \ddots & \\
\rho_{p,1} & \rho_{p,2} & \hdots & \rho_{p,p-1} & 1
\end{pmatrix}^{-1}
\diag\left(1,k_2(\mv{\rho}),k_3(\mv{\rho}),\ldots,k_p(\mv{\rho})\right),
$$
where $k_j(\mv{\rho}) = \sqrt{1+\sum_{i<j}\rho_{j,i}^2}$. With this parametrization, $\mv{\rho} \in \R^{p(p-1)/2}$ controls the cross-correlations and $\sigma_i^2 = \proper{V}(X_i(\mv{s}))$.
Figure \ref{fig1} shows an example of the resulting covariance function for a bivariate model with $\rho = \rho_{1,1} = 0.5$.
\begin{figure}
\caption{Example of covariance functions for the solution to the triangular Mat\'ern-SPDE with $\sigma_1 = \sigma_2 = 1$, $\rho = 0.5$, $\kappa_1 = \kappa_2 = 1$, $\alpha_1 = 1.5$, and $\alpha_2 = 2$.}
\label{fig1}
\end{figure}
What remains is to find a parametrization of $\mv{Q}_p$. The determinant of an orthogonal matrix is $\pm 1$, where the sign is not identifiable in general. It is therefore enough to consider the subclass of special orthogonal matrices, which have determinant $1$. For a general $p$, it is difficult to parametrize such matrices. However, for $p=2$ and $p=3$ we can use the fact that they are equivalent to rotation matrices. We can therefore write
$$
\mv{Q}_2(\theta) =
\begin{pmatrix}
\cos(\theta) & -\sin(\theta) \\
\sin(\theta) & \cos(\theta)
\end{pmatrix}, \quad \mv{Q}_3(\theta_1,\theta_2,\theta_3) = \mv{Q}_{3x}(\theta_1)\mv{Q}_{3y}(\theta_2)\mv{Q}_{3z}(\theta_3),
$$
where $\theta\in[0,2\pi]$, $\mv{Q}_{3x}(\theta) = \diag(\mv{Q}_2(\theta),1)$, $\mv{Q}_{3z}(\theta) = \diag(1,\mv{Q}_2(\theta))$, and
$$
\mv{Q}_{3y}(\theta) =
\begin{pmatrix}
\cos(\theta) & 0 & -\sin(\theta) \\
0 & 1 & 0 \\
-\sin(\theta) & 0 & \cos(\theta)
\end{pmatrix}.
$$
To summarize, we use the parametrization
\begin{equation}\label{eq:general}
\mv{D}(\mv{\theta},\mv{\rho}) \diag(c_1\mathcal{L}_1,\cdots, c_p\mathcal{L}_p)\mv{x}(\mv{s}) = \mv{\noise},
\end{equation}
where $\mv{D}(\mv{\theta},\mv{\rho}) = \mv{Q}_p(\mv{\theta})\mv{D}_l(\mv{\rho})$ and $\mv{\theta}\in [0,2\pi]^{p(p-1)/2}$ will control higher moments for non-Gaussian models. In the bivariate case, the dependence matrix simplifies to
\begin{equation}\label{eq:matern2}
\mv{D}(\theta,\rho) =
\begin{bmatrix}
\cos(\theta)+\rho\sin(\theta) & -\sin(\theta)
\sqrt{1+\rho^2} \\
\sin(\theta)-\rho\cos(\theta) & \cos(\theta)
\sqrt{1+\rho^2}
\end{bmatrix}.
\end{equation}
\section{Type G Mat\'ern SPDE fields}\label{sec:nongauss}
In this section, the multivariate Mat\'ern-SPDE model is extended beyond Gaussianity by replacing the Gaussian noise with non-Gaussian noise. In Section \ref{sec:nonGauss-def}, four different constructions of noise for this approach are introduced and the resulting Mat\'ern-SPDE fields are discussed. The differences between the four constructions are illustrated using FE discretizations of the models in Section \ref{sec:discrete} and further properties of the models are stated in Section \ref{sec:nonGauss-prop}. Finally, asymptotic properties of spatial prediction based on the simplest type G models are derived in Section \ref{sec:nonGauss-outfill}.
\subsection{Four increasingly flexible constructions}\label{sec:nonGauss-def}
The four constructions are based on using different types of normal-variance mixtures
\begin{equation}\label{eq:variancemixture}
\mv{\gamma} + v\mv{\mu} + \sqrt{v}z,
\end{equation}
where $\mv{\gamma}\in\R^p$ and $\mv{\mu}\in\R^p$ are parameters, $z\sim\proper{N}(0,1)$, and $v$ is a non-negative random variable. Inspired by L\'{e}vy process, which are said to be of type G if their increments are normal-variance mixtures, we will refer to these models as type G Mat\'ern-SPDE fields.
The first two constructions are related to the approach where non-Gaussian fields are obtained by multiplying Gaussian fields with random scalars.
\begin{defn}
Let $v$ and $v_1,\ldots,v_p$ be independent non-negative infinitely divisible random variables and set $\mv{v}_1 = v \mv{1}_p$ and $\mv{v}_2 = (v_1,\ldots,v_p)^{\ensuremath{\top}}$, where $\mv{1}_p$ denotes a vector with $p$ ones. Further, let $\mathcal{W}(\mv{s}) = (\mathcal{W}_1(\mv{s}),\ldots,\mathcal{W}_p(\mv{s}))^{\ensuremath{\top}}$ be a vector of independent copies of Brownian sheets on $\mathbb{R}^d$. For $i\in \{1,2\}$, a type G$_i$ Mat\'ern-SPDE field is obtained by using $\mv{\noise}_i$ in \eqref{eq:general} where
$\mv{\mathcal{M}}_i(\mv{s}) = \mv{\gamma} + \diag(\mv{v}_i)\mv{\mu} + \diag(\sqrt{\mv{v}_i})\mv{\mathcal{W}}(\mv{s})$.
\end{defn}
It should be noted that $\mv{\noise}_1$ and $\mv{\noise}_2$ are not independently scattered measures since they have common random scaling $\mv{v}_i$ for different spatial locations. Because of this, one cannot directly use the results from the previous section. However, the results can easily be extended by allowing for measures that are independently scattered conditionally on a random variable $\mv{v}$.
In particular, if we restrict the distribution of $\mv{v}_i$ such that $\proper{E}(\mv{v}_i) = \mv{1}$, then the result in Proposition~\ref{thm1} still holds in the symmetric case with $\mv{\mu} = \mv{0}$. In the non-symmetric case, the resulting fields have covariance functions given by the covariance function in Proposition~\ref{thm1} plus a constant factor depending on the variance of $\mv{v}$.
With the restriction $\proper{E}(\mv{v}_i) = \mv{1}$, the mean of the process is given by $\mv{\mu} + \mv{\gamma}$, and we therefore set $\mv{\gamma} = -\mv{\mu}$ to ensure that the process has zero mean as default.
In the type G$_1$ model, we can interpret $v$ as a random scaling of the variance of the entire process, whereas we scale the variance of each $x_i(\mv{s})$ separately with $v_i$ in the type G$_2$ case. When $\mv{\mu}\neq \mv{0}$, $v$ also decides the skewness of the marginal distributions of $\mv{x}(\mv{s})$ in in the type G$_1$ case, whereas $v_i$ controls the skewness of $x_i(\mv{s})$ in the type G$_2$ case. Hence, the type G$_2$ model gives more control of the marginal distributions of the process.
From a Bayesian point of view, one could interpret $\pi(\mv{v})$ as a prior distribution on the mean and variance of a multivariate Gaussian random field. Thus, these models can be used in the same way as the Gaussian models in a Bayesian setting, but where we have a specific prior that links the mean and variance of the field. From this point of view, one would likely not refer to these models as non-Gaussian.
The next two constructions are based on type G L\'evy noise. A random variable $x$ is said to be of type G if it can be written as $x \overset{d}{=} \sqrt{v}z$, where $z$ is a Gaussian variable and $v$ is an infinitely divisible non-negative random variable. A univariate type G L\'{e}vy process is a L\'evy process whose increments are of type G. \cite{rosinski91} showed that a type G process $\mathcal{M}(s), s\in [0,1]$, with $\mathcal{M}(1) \overset{d}{=} \sqrt{v}z$ can be represented as $\mathcal{M}(s) = \sum_{k=1}^{\infty}z_k g(e_k)^{\frac{1}{2}}\mathbb{I}(s \geq u_k)$, where $e_k$ are the points of a unit-rate Poisson process on $\mathbb{R}^+$, $z_k$ are iid $\proper{N}(0,1)$ random variables, and $u_k$ are iid uniform random variables on $(0,1)$. The function $g$ is the generalized inverse of the tail L\'{e}vy measure for $v$, defined as $g(u) = \inf\{x>0: M(x,\infty) \leq u\}$ where $M$ is the L\'evy measure of $v$. The non-decreasing L\'evy process $v(\mv{s}) = \sum_{k=1}^{\infty}g(e_k)\mathbb{I}(s \geq u_k)$ has the same Levy measure as $v$, and can be used to represent $\mathcal{M}$ as a subordinated Wiener process. We refer to \cite{rosinski91} for further technical details on the construction. In the spatial case, a type G process $\mathcal{M}(\mv{s})$ on the unit square $D = [0, 1]\times [0, 1]$ with $M(\mv{1}) \overset{d}{=} \sqrt{v}z$ can similarly be represented as $\mathcal{M}(\mv{s}) = \sum_{k=1}^{\infty}z_k g(e_k)^{\frac{1}{2}}\mathbb{I}(\mv{s} \geq \mv{u}_k)$, where $\mv{u}_k$ now are uniform random variables on $D$ and $\mathbb{I}(\mv{s}\geq \mv{u}_k) = \mathbb{I}(s_1\geq u_{k,1})\mathbb{I}(s_2\geq u_{k,2})$ is a two-dimensional indicator function. In this case, the associated process $v(\mv{s}) = \sum_{k=1}^{\infty} g(e_k)\mathbb{I}(\mv{s} \geq \mv{u}_k)$ can no longer be seen as a subordinator, but could informally be thought of as a process that determines the variance of the noise.
For multivariate processes, there are two natural extensions to vector valued noise that we use to define type G$_3$ and type G$_4$ fields.
\begin{defn}\label{def:noise}
Let $\mathcal{M}(\mv{s})$ be a type G L\'evy processes with $v(\mv{s}) = \sum_{k=1}^{\infty}g(e_k)\mathbb{I}(\mv{s} \geq \mv{u}_k) $ and let $\mv{\mathcal{M}}(\mv{s}) = (\mathcal{M}_1(\mv{s}), \ldots, \mathcal{M}_p(\mv{s}))^{\ensuremath{\top}}$ be a vector of independent type G L\'evy processes with corresponding variance processes $\mv{v}(\mv{s}) = (v_1(\mv{s}), \ldots v_p(\mv{s}))^{\ensuremath{\top}}$. For $i\in\{3,4\}$ a type G$_i$ Mat\'ern-SPDE field is obtained by using $\mv{\noise}_i$ in \eqref{eq:general} where
\begin{align*}
\mv{\mathcal{M}}_{3}(\mv{s}) &= \mv{\gamma} + \mv{\mu}v(\mv{s}) + \sum_{k=1}^{\infty}g(e_k)^{\frac{1}{2}}\mathbb{I}(\mv{s} \geq \mv{u}_k) \mv{z}_k, & &
\mv{\mathcal{M}}_4(\mv{s}) = \mv{\gamma} + \diag(\mv{\mu}) \mv{v}(\mv{s}) + \mv{\mathcal{M}}(\mv{s}).
\end{align*}
\end{defn}
\begin{rem}
In this section we have assumed a multivariate setting, i.e., $p>1$. However, in the univariate case, the type G$_1$ and type G$_2$ Mat\'ern-SPDE models are equivalent. Further if $\mv{\mu}=0$, the type G$_1$ model is a Gaussian Mat\'ern field multiplied with a univariate positive random variable. Thus, models such as the t-distributed random fields by \citet{roislien06} belong to the class of type G$_1$ fields. Also, when $p=1$ the type G$_3$ and type G$_4$ Mat\'ern-SPDE models are also equivalent, and coincide with the models in \citep{Wallin15}.
\end{rem}
\subsection{Understanding the four constructions through their discretizations}\label{sec:discrete}
Although the Mat\'ern-SPDE models were formulated on the entire $\mathbb{R}^d$ in Section \ref{sec:spde}, we consider the system of SPDEs on a bounded domain $\mathcal{D}\subset \mathbb{R}^d$ when implementing them numerically. The operators are then equipped with suitable boundary conditions and the solution is approximated using a FE discretization derived in Appendix \ref{sec:fem}.
To understand the differences between the four different type G constructions, we now examine the properties of the discretized models in comparison to the corresponding Gaussian Mat\'ern-SPDE model.
In the FE approximation, the solution of \eqref{eq:general} is represented as a basis expansion $\mv{x}(\mv{s}) = \sum_{j=1}^n\sum_{k=1}^p w_{jk}\mv{vphi}_j^k(\mv{s})$ using piecewise linear basis functions $\mv{vphi}_j^k(\mv{s})$ obtained from a mesh over $\mathcal{D}$. The value of $x_k(\mv{s}_j)$ at the $j$th node in the mesh, $\mv{s}_j$, is then given by the stochastic weight $w_{jk}$.
Assuming $p=2$ and $\alpha_1 = \alpha_2 = 2$, the distribution of $\mv{w} = (\mv{w}_1^{\ensuremath{\top}},\mv{w}_2^{\ensuremath{\top}})^{\ensuremath{\top}} = (w_{11},\ldots, w_{n1},w_{12},\ldots, w_{n2})^{\ensuremath{\top}}$ for the case of Gaussian noise is
\begin{equation}\label{eq:Gausfem}
\mv{w} \sim \proper{N}(\mv{0}, \mv{K}^{-1}\diag(\mv{h},\mv{h})\mv{K}^{-\ensuremath{\top}}),
\end{equation}
where $\mv{K}$ is a discretization of the operator matrix and $\mv{h}$ is a vector with elements $h_i$ depending on the mesh.
For the corresponding type G$_3$ model, the distribution of the weights is
\begin{equation}\label{eq:G3fem}
\mv{w}|\mv{v} \sim \proper{N}\left(\mv{K}^{-1}\left[\begin{matrix}\gamma_1\mv{h}+\mu_1\mv{v}\\ \gamma_2\mv{h}+\mu_2\mv{v}\end{matrix}\right], \mv{K}^{-1}\diag(\mv{v},\mv{v})\mv{K}^{-\ensuremath{\top}}\right), \quad \mv{v}\sim \pi(\mv{v}),
\end{equation}
where the elements of $\mv{v}\in\mathbb{R}_{+}^{n}$ are independent variables relating to the discretization of the variance process $v(\mv{s})$.
For the type G$_4$ model, we have
\begin{equation}\label{eq:G4fem}
\mv{w}|\mv{v}_1,\mv{v}_2 \sim \proper{N}\left(\mv{K}^{-1}\left[\begin{matrix}\gamma_1\mv{h}+\mu_1\mv{v}_1\\ \gamma_2\mv{h}+\mu_2\mv{v}_2\end{matrix}\right], \mv{K}^{-1}\diag(\mv{v}_1,\mv{v}_2)\mv{K}^{-\ensuremath{\top}}\right), \quad \mv{v}_1, \mv{v}_2\sim \pi(\mv{v}),
\end{equation}
where $\mv{v}_1,\mv{v}_2\in\mathbb{R}_{+}^{n}$ have independent components relating to the discretisations of $v_1(\mv{s})$ and $v_2(\mv{s})$ repectively.
Similarly, the discretization in the type G$_1$ and type G$_2$ cases can be written as \eqref{eq:G3fem} and \eqref{eq:G4fem} respectively, if we define $\mv{v} = v\mv{h}$ and $\mv{v}_i = v_i\mv{h}$. As we discussed for the first two cases,
we set $\mv{\mu} = -\mv{\gamma}$ to ensure that the process has zero mean, and restrict the distribution of the variances to have mean one. We then have for all cases that $\proper{E}(\mv{v}) = \proper{E}(\mv{v}_1) = \proper{E}(\mv{v}_2) = \mv{h}$.
Thus, comparing \eqref{eq:Gausfem}, \eqref{eq:G3fem}, and \eqref{eq:G4fem}, we see that a difference between the type G processes and the Gaussian process is that we have replaced the deterministic vector $\mv{h}$ in the covariance matrix by a stochastic vector that has $\mv{h}$ as expected value. Furthermore, the difference between the four constructions lies in the flexibility of this stochastic vector. In the type G$_1$ case, we scale the entire field by a single stochastic variable, whereas we scaled each dimension separately in the type G$_2$ case. For the type G$_3$ case we have replaced the fixed scaling $h_i$ of the distribution of the weights $w_{i1},\ldots, w_{ip}$ for a given spatial location $\mv{s}_i$ by a common stochastic scaling $v_i$, which thus affect the sample path behaviour of the process. The type G$_4$ case is even more general where we have
individual stochastic scalings $v_{ip}$ for each weight, and thus more control over the sample path behaviour.
\subsection{Properties of the four constructions}\label{sec:nonGauss-prop}
The four type G constructions provide random fields with increasing flexibility. All contain several interesting special cases depending on which distribution that is used for the variance components, such as generalised asymmetric Laplace distributions, normal-inverse gamma distributions, and Student's t-distributions. As an example, we will in the next section use NIG noise to highlight some properties of the constructions.
Let $\mv{\Sigma}$ be the covariance matrix of the solution $\mv{x}(\mv{s})$ in \eqref{eq:general} in the case of Gaussian driving noise, for a fixed location $\mv{s}$. This matrix has diagonal elements $\Sigma_{ii} = \sigma_i^2$ and off-diagonal elements $\Sigma_{ij}$ depending on $\sigma_i, \sigma_j$, and $\rho_{ij}$. For the type G$_1$ construction, we can then write the joint cumulative distribution function (CDF) $F^{(1)}$ of $\mv{x}(\mv{s})$, and the marginal CDFs $F^{(1)}_k$ of $x_k(\mv{s})$ for $k=1,\ldots, p$ as
\begin{equation*}
F^{(1)}(\mv{u}) = \int \Phi_{\mv{\Sigma}}\left(\frac{\mv{u}-\mv{\gamma}-\mv{\mu}v}{\sqrt{v}}\right)\ensuremath{\,\mathrm{d}} F_v(v), \quad F^{(1)}_k(u) = \int \Phi\left(\frac{u-\mv{\gamma}-\mv{\mu}v}{\sigma_k\sqrt{v}}\right)\ensuremath{\,\mathrm{d}} F_v(v),
\end{equation*}
where $\Phi_{\mv{\Sigma}}$ denotes the CDF of a $\proper{N}(\mv{0},\mv{\Sigma})$ random variable and $F_v$ denotes the CDF of $v$. There are several choices of $F_v$ that result in fields with known marginal distributions. If for example $\mv{\mu} = \mv{0}$, the field has multivariate Student's $t$ marginals if $v$ is inverse-gamma distributed, and multivariate Laplace marginals if $v$ is gamma distributed. The copula of $\mv{x}(\mv{s})$ is
$
C^{(1)}(\mv{u}) = F^{(1)}[(F^{(1)}_1)^{-1}(u_1),\ldots, (F^{(1)}_p)^{-1}(u_p)],
$
which could be viewed as a generalization of the one-factor copulas in \citep{krupskii2015structured,krupskii2016factor}. However, despite the flexibility of the marginal distributions, the model is limited since it is non-ergodic for any non-singular distribution of $v$, and the sample paths are indistinguishable from sample paths of a Gaussian random field. If repeated realizations are available, one can estimate the distribution of $v$, but not the parameter $\mv{\theta}$ in the dependence matrix.
For the type G$_2$ construction, the joint CDF of $\mv{x}(\mv{s})$ is
\begin{equation*}
F^{(2)}(\mv{u}) = \int \Phi_{\mv{\Sigma}}\left(\diag\left(\frac1{\sqrt{v_1}},\ldots, \frac1{\sqrt{v_p}}\right)(\mv{u}-\mv{\gamma}-\mv{\mu}v)\right)\ensuremath{\,\mathrm{d}} F_{v_1}(v_1) \cdots\ensuremath{\,\mathrm{d}} F_{v_p}(v_p),
\end{equation*}
and the marginal CDF for $k=1,\ldots,p$ is
$$
F^{(2)}_k(u) = \int \Phi\left(\frac{u-\gamma_k-\mu_kv_k}{\sigma_k\sqrt{v_k}}\right)\ensuremath{\,\mathrm{d}} F_{v_k}(v_k).
$$
The copula of $\mv{x}(\mv{s})$ is
$
C^{(2)}(\mv{u}) = F^{(2)}[(F^{(2)}_1)^{-1}(u_1),\ldots, (F^{(2)}_p)^{-1}(u_p)],
$
which is similar to the $p$-factor copulas in \citep{krupskii2015structured}. Also fields obtained using the type G$_2$ construction are non-ergodic and have sample paths that are indistinguishable from Gaussian sample paths. However, it is possible to estimate all parameters of the model given multiple realizations.
Since the type G$_1$ and the type G$_2$ constructions have copulas simular to factor copulas, one can compute their so-called tail dependence coefficients and derive conditions on the distribution of $v$ to study their asymptotic tail dependence similar to \cite{krupskii2016factor}. We leave this for future research as our main interest is in the more flexible type G$_3$ and type G$_4$ constructions. The reason for this is, as we will show in the next subsection, that the type G$_1$ and type G$_2$ models have asymptotically Gaussian conditional distributions. This greatly limits their flexibility for spatial data.
For the type G$_3$ and type G$_4$ constructions, we in general cannot derive closed-form expressions for the marginal distributions and copulas (we will discuss this further in the next section). However, if we use the representation of the process in Remark 1, and let $F^M_k$ and $\tilde{F}^M_{k}$ denote the distribution functions of the laws of $\mathcal{M}_k$ and $(\mv{R\mathcal{M}})_{k}$ respectively, the copula for the law of $\mv{\mathcal{M}}_R$ can be written as
$$
C(\mv{u}) = \prod_{k=1}^p F^M_k(\mv{D}_k^{\ensuremath{\top}}((\tilde{F}^M_1)^{-1}(x_1), \ldots, (\tilde{F}_p^M)^{-1}(x_p))^{\ensuremath{\top}}),
$$
where $\mv{D}_k$ is the $k$th row of $\mv{D}$. This is a Gaussian copula only in the case when $\mv{\mathcal{M}}$ is Gaussian. Thus, also for these constructions, the dependence structure induced by the model can be made more flexible than simply using Gaussian copulas to model the dependence.
The type G$_4$ construction is the most general but the type G$_3$ construction could be of interest for applications where one wish to capture dependence of the extreme values on different variables. It also has the following interesting feature.
\begin{prop}\label{thm3}
Let $\mv{x}$ be a type G$_3$ Mat\'ern-SPDE field with $\mv{\rho} = \mv{0}$. Then, for $\mv{s},\mv{t}\in\mathcal{D}$ and $i\neq j$, $x_i(\mv{s})$ and $x_j(\mv{t})$ are dependent but uncorrelated random variables.
\end{prop}
\subsection{Increasing domain asymptotics for the type G$_1$ model}\label{sec:nonGauss-outfill}
In this section we explore the distributions of spatial predictions for the type G$_1$ models and show that they converge to Gaussian distributions as the number of observations goes to infinity. This implies that one might as well use the simpler Gaussian model for the purpose of prediction if the data set is sufficiently large. Similar issues with related non-Gaussian models have been noted in the mixed effect literature \citep{rubio2018flexible}. To simplify the notations, we focus on the mean-zero univariate case, but the results are easily extended to the general multivariate setting for both type G$_1$ and type G$_2$ models.
Let $x_{i}=x(\mv{s}_i), i=1,\ldots,n$, be observations of a mean-zero random field $x(\mv{s})$, for which we want to predict $x_{0}=x(\mv{s}_0)$.
Let $\mv{x}_{k:n}$ denote the vector $\left[x_k, x_{k+1}, \ldots, x_n\right]^{\ensuremath{\top}}$ and assume that the covariance function, $C(\mv{s},\mv{t})$, of $x$ and the locations $\mv{s}_0,\mv{s}_1,\ldots\mv{s}_n$ are such that covariance matrix of $\mv{x}_{0:n}$ is positive definite.
Assuming that a mean-zero type G$_1$ model, with the same covariance function as $x$, is used for the prediction, the distribution of $x_0$ given $\mv{x}_{1:n}$ is
$$
\pi_{G_1,x_0}(x_0|\mv{x}_{1:n}) = \int N(x_0; \mv{c}_{0,1:n} \mv{C}_n^{-1} \mv{x}_{1:n} ,\, vc_{0}- v\mv{c}^{\ensuremath{\top}}_{0,1:n} \mv{C}_n^{-1} \mv{c}_{0,1:n})\pi(v) dv,
$$
where $c_0 = \proper{V}(x_0)$, $\mv{c}_{0,1:n}$ is the cross-covariance between $\mv{x}_{1:n}$ and $x_0$, and $\mv{C}_n$ is the covariance matrix of $\mv{x}_{1:n}$.
To show that this distribution converges to a Gaussian distribution we need the following weak assumptions on the observed data.
\begin{assumption}\label{ass1}
The random field $x(\mv{s})$ and the observations satisfy, as $n\rightarrow \infty$,
\begin{alignat}{4}
&(\mv{x}_{1:n}^{\ensuremath{\top}} \mv{C}_n^{-1}\mv{x}_{1:n} )/n &&\overset{p}{\to}\, K_0, \label{eq:mean_est}\\
&\proper{V}[(\mv{x}_{1:n}^{\ensuremath{\top}} \mv{C}_n^{-1}\mv{x}_{1:n} )/\sqrt{n} ] &&\to\, k_v, \label{eq:var_est}\\
&\mv{c}_{0,1:n} \mv{C}_n^{-1} \mv{x}_{1:n} &&\overset{p}{\to}\, K_1, \label{eq:kriging_mean}\\
&c_{0}- \mv{c}^{\ensuremath{\top}}_{0,1:n} \mv{C}_n^{-1} \mv{c}_{0,1:n} &&\to\, k_2, \label{eq:kriging_var}
\end{alignat}
where $K_0\geq 0$ and $K_1$ are random variables, $k_2 \in [0, c_0]$, and $k_v>0$.
\end{assumption}
The first two assumptions are satisfied for all models considered in this article given they have finite moments, and given that the sequence $\{\mv{s}_i\}$ does not result in a singular the covariance matrix (which for example is the case if $\mv{s}_i = \mv{s}_j$ for $i\neq j$). The last two assumptions assure that the linear predictor converges to a constant given the data. Assuming that $x$ has a Mat\'ern covariance function with $\nu<\infty$, this is also fulfilled as long as the sequence $\{\mv{s}_i\}$ is not chosen so that the covariance is degenerate. Given these assumptions, we have the following result.
\begin{Theorem}\label{thm:krig}
Let Assumption \ref{ass1} hold and assume that $\pi(v)$ is a bounded function which is absolutely continuous with respect to the Lebesgue measure, such that ${\bf \mathsf{E}}[v]=1$. Then
$\pi_{G_1,x_0}(\cdot|\mv{x}_{1:n}) \overset{p}{\to} N(\cdot; k_1, k_0 k_2)$ as $n \rightarrow \infty$.
Here $k_0$ and $k_1$ are the realisations of $K_0$ and $K_1$ in \eqref{eq:mean_est} and \eqref{eq:kriging_mean} respectively.
\end{Theorem}
The theorem shows that the predictive distribution for a type G$_1$ model converges to a Gaussian distribution, and thus the predictor (the mean of the distribution) converges to the corresponding predictor for a Gaussian model, under quite general assumptions on the distribution for the data. In particular, it holds if the data comes from a type G$_1$ model.
\begin{cor}\label{eq:Lemma}
Let $x(\mv{s}), \mv{s}\in\mathbb{R}^d$, be a univariate type G$_1$ Mat\'ern-SPDE field let and $\mv{s}_0, \ldots, \mv{s}_n$ be locations in $\mathbb{R}^d$ such that $i<\|\mv{s}_0-\mv{s}_i\|<i+1$ for $i=1\ldots, n$. Assume that $\pi(v)$ is a bounded function and absolutely continuous with respect to the Lebesgue measure, such that ${\bf \mathsf{E}}[v]=1$.
Then the predictive distribution for $x(s_0)$, $\pi_{x(s_0)}(\cdot|x(\mv{s}_1),\ldots,x(\mv{s}_n))$, converges in probability to a Gaussian distribution as $n\rightarrow \infty$.
\end{cor}
\section{Normal inverse Gaussian fields}\label{sec:nig}
The NIG distribution \citep{barndorff1997normal} is obtained by choosing $p=1$ and $v$ as an inverse gamma (IG) random variable in \eqref{eq:variancemixture}.
The IG distribution has density
$$
IG(v;\eta_1,\eta_2) = \frac{\sqrt{\eta_2} }{ \sqrt{2\pi v^3}} \exp\left( - \frac{\eta_1}{2} v - \frac{\eta_2}{2v} + \sqrt{\eta_1\eta_2} \right), \quad \eta_1,\eta_2>0.
$$
The resulting density for the NIG variable is
\begin{align*}
NIG(x;\gamma,\mu,\eta_1,\eta_2) &= \frac{ e^{\sqrt{\eta_1\eta_2} +\mu(x- \gamma)}\sqrt{ \eta_2\mu^2 + \eta_1 \eta_2}}{\pi \sqrt{ \eta_2 + (x - \gamma)^2}} K_1\left( \sqrt{(\eta_2 + (x - \gamma)^2) (\mu^2 + \eta_1)} \right).
\end{align*}
In this form the $NIG$ density is overparameterized, and we therefore typically set $\eta_1=\eta_2=\eta$ which results in ${\bf \mathsf{E}}(v)=1$. If $\mu=0$, one has that the random variable defined in \eqref{eq:variancemixture} has variance 1, but for $\mu\neq 0$, the variance depends on $\eta$.
An important property of the NIG distribution is that its variance mixture distribution, the IG distribution, is closed under convolution. This simplifies inference as explained in later sections.
The simplest multivariate NIG Mat\'ern-SPDE field is obtained by using the type G$_1$ construction with $v\sim IG(\eta,\eta)$, resulting in a field with multivariate NIG marginal distributions. To construct the more flexible type G$_3$ and type G$_4$ models, we use IG random variables in the univariate type G L\'evy processes, which results in NIG processes. Since the NIG distribution has both the Gaussian the Cauchy distributions as limiting cases (as $\eta \rightarrow \infty$ and $\eta \rightarrow 0$ with suitable scalings of the other parameters), the NIG Mat\'ern-SPDE processes have both a Gaussian process and a L\'evy flight process as limiting cases. When using NIG noise in \eqref{eq:general}, it is convenient to note that the noise can be represented by an independently scattered random measure \citep{rajput1989spectral}. Specifically, for any Borel set $A$ in the domain, the measure is a univariate NIG random variable with probability density function
$f_{\dot{\mathcal{N}}(A)}(x) = NIG(x;m(A)\gamma,\mu,\eta,m(A)^2\eta)$,
where $m(A)$ denotes the Lebesgue measure of $A$. Note that a random variable with density $f_{\dot{\mathcal{N}}(A)}(x)$ can be obtained from equation \eqref{eq:variancemixture} where $v\sim IG( \eta,m(A)^2 \eta )$ and thus ${\bf \mathsf{E}}(v)=m(A)$.
We let $\mv{\mathcal{N}}_3$ and $\mv{\mathcal{N}}_4$ denote the vector-valued processes in Definition \ref{def:noise} when univariate NIG processes are used. The density of $\mv{x}(\mv{s})$ in \eqref{eq:general} does not have an explicit form in this case but one can derive the characteristic function (CF) of $\mv{x}(\mv{s})$. The following proposition provides the CF for the type G$_4$ case.
\begin{prop}\label{thm:charf}
The CF of a stationary solution $\mv{x}$ to \eqref{eq:general}, evaluated at $\mv{s}$, where the driving noise is $\mv{\mathcal{N}}_4$, is $\phi_{\mv{x}(\mv{s})}(\mv{u}) = \prod_{k=1}^p \phi_k(\mv{u})$ where
\begin{equation*}
\phi_k(\mv{u}) = \exp\left[ - i\gamma_k \int \mv{u}^{\ensuremath{\top}} \mv{v}_{k,\mv{t}} d\mv{t} + \sqrt{\eta_k} \int \eta_k - \sqrt{\eta_k -2i \mu_k^2\mv{u}^{\ensuremath{\top}} \mv{v}_{k,\mv{t}} + (\mv{u}^{\ensuremath{\top}} \mv{v}_{k,\mv{t}})^2 } d\mv{t}\right].
\end{equation*}
Here $\mv{v}_{k,\mv{t}} = [Di_{1k} G_1(\mv{s},\mv{t}), Di_{2k} G_2(\mv{s},\mv{t}), \ldots, Di_{pk} G_p(\mv{s},\mv{t}) ]^{\ensuremath{\top}}$, $\mv{Di} = \mv{D}^{-1}$, and
$$
G_k(\mv{s},\mv{t}) = \frac{\Gamma\left(\frac{\alpha_k-d}{2}\right)}{c_k (4\pi)^{d/4}\Gamma(\frac{\alpha_k}{2})\kappa_k^{\alpha_k-d}} \materncorr{\|\mv{s}-\mv{t}\|}{\kappa_k}{\frac{\alpha_k-d}{2}}, \quad k=1,\ldots, p.
$$
\end{prop}
The following example illustrates the effect of the shape parameter $\theta$ on the multivariate marginal distributions of the type G$_4$ model.
\begin{figure}
\caption{Marginal distributions of a bivariate NIG Mat\'ern-SPDE field for different values of $\theta$.
All six cases have the correlation function shown in Figure \ref{fig1}
\label{fig:simmarg}
\end{figure}
\begin{example}
Let $\mv{x}(\mv{s})$ be a type G$_4$ bivariate NIG Mat\'ern-SPDE field with the same parameters as in Figure \ref{fig1}. For the driving noise, we let $\mu_1 = \gamma_2 = 1$, $\mu_2 = \gamma_1 = -1$, and $\eta = 0.9$. Figure \ref{fig:simmarg} shows bivariate marginal distributions of the resulting field for different values of $\theta$ in the dependence matrix \eqref{eq:matern2}, computed using Proposition \ref{thm:charf}. Recall that $\rho$ determines the cross-correlations between $x_1(\mv{s})$ and $x_2(\mv{s})$ whereas $\theta$ determines the shape of the bivariate marginal distributions, but does not affect the covariance function. Thus, all six examples have the same correlation function, which is shown in Figure \ref{fig1}. The case $\theta = 0$ corresponds to a lower-triangular operator matrix, and $\theta = \arctan(\rho)$ corresponds to an upper-triangular operator matrix.
\end{example}
As discussed in Section \ref{sec:nongauss}, the simpler type G constructions have similar flexibility of the marginal distributions, but lower flexibility in terms of conditional distributions.
The following example illustrates how different the predictive distributions can be.
\begin{figure}
\caption{Marginal distributions for $\mv{x}
\label{fig:preddist}
\end{figure}
\begin{example}\label{example2}
Let $\mv{x}_i(t)$, $i=1,\ldots,4$, be bivariate type G$_i$ NIG Mat\'ern-SPDE processes on $\R$ with $\alpha=2$, $\kappa = 1$, $\sigma = 0.1$, $\rho = 0.9$, and $\theta = 0$. The processes have the same operator matrix $\mv{\mathcal{K}}$ and we choose the parameters $\mv{\mu}$ and $\mv{\eta}$ so that they have similar (univariate) marginal distributions, see Figure \ref{fig:preddist}, Panels (a) and (d), where the marginal distributions of a corresponding Gaussian process also is shown for reference. We predict the value of the four processes at $t=0$ based on two observations of the first dimension $y_{1} = x_{i,1}(-1) +\vep_{1} = 0$ and $y_2 = x_{i,1}(1) + \vep_{2} = 4$, where $\vep_{-1}$ and $\vep_{1}$ are independent $\proper{N}(0,0.001^2)$ variables representing measurement noise. How the prediction is done is presented in Section \ref{sec:kriging}. The predictive distributions are shown in Panels (b) and (e). Even though the four processes have similar marginal distributions for $\mv{x}_i(0)$, their predictive distributions are very different. For the prediction of the first dimension, the type G$_1$ and type G$_2$ processes have similar distributions, which is expected since they have the same marginal structures. The type G$_3$ and type G$_4$ also have equal marginal structures and therefore similar predictions, which are very different from the first two. For the prediction of the second dimension, we get different predictions for all models since they have different cross-dependence structures. In particular we can note the counter intuitive type G$_1$ prediction, where the prediction of the second dimension is larger than the first, even though there are no observations for this dimension. The same predictive distributions in the case when $\vep_{-1}$ and $\vep_1$ instead have variance $0.5^2$ are shown in Panels~(c) and (f), and one can note the same behaviour of the predictions for this case.
\end{example}
\section{Geostatistical modeling and estimation}\label{sec:model}
To use the multivariate type G fields for geostatistical applications, we need to be able to include them in hierarchical models that include covariates and measurement noise. In this section, we formulate such a model and describe how to perform likelihood-based estimation of the model parameters and how to use the model for spatial prediction.
We consider a standard geostatistical model where a latent field is specified using covariates for the mean, and the data consists of noisy observations of this latent field at some locations $\mv{s}_1, \ldots, \mv{s}_n$.
Let $y_{ki}$ be the $i$th observation of the $k$th dimension, defiened as $y_{ki} = \sum_{j=1}^K B_{kj}(\mv{s}_i)\beta_j + x_k(\mv{s}_i) + \vep_{ki}$ for $k=1,\ldots,p$,
where the independent variables $\vep_{ki}\sim \proper{N}(0,\sigma_{e,k}^2)$ represent the measurement noise. The functions $B_j(\mv{s})$ are covariates for the mean and $x_k(\mv{s})$ is the $k$th variable of a mean-zero multivariate type G Mat\'ern-SPDE field $\mv{x}(\mv{s})$. Since the mean of $\mv{y}$ is modeled using covariates,
we assume that the mixing variables in the type G construction are scaled so that they have unit expectation (if the expected value exists) and set $\gamma_k = -\mu_k$ to guarantee that $x_k(\mv{s})$ has mean zero in the case that it has an expected value.
Assuming that the smoothness parameters satisfy $\alpha_i/2\in\mathbb{N}$ for $i=1,\ldots,p$ and using the finite-dimensional representation of $\mv{x}(\mv{s})$ derived in Appendix \ref{sec:fem}, we have $\mv{x}(\mv{s}) = \sum_{j=1}^n\sum_{k=1}^p w_{jk}\mv{vphi}_j^k(\mv{s})$. Here $\mv{vphi}_j^k(\mv{s}) = vphi_j(s)\mv{e}_k$ are p-dimensional basis functions, where $\mv{e}_k$ is the $k$th column in a $p \times p$ identity matrix, $\{vphi_i\}$ are piecewise linear FE basis functions defined by a mesh on $\mathcal{D}$, and $\{w_{jk}\}$ are stochastic weights. The model is then
\begin{equation}\label{eq:mainmodel}
\begin{split}
\mv{v} &\sim \pi(\mv{v}), \\
\mv{w} | \mv{v} &\sim \proper{N} \left( \mv{K}^{-1}(\mv{\mu}\otimes\mv{I}_n)(\mv{v}-\mv{h}), \mv{K}^{-1} \diag(\mv{v}) \mv{K}^{-\ensuremath{\top}} \right),
\\
\mv{y}_{k}|\mv{w} &\sim \proper{N}\left( \mv{B} \mv{\beta} + \mv{A}_k\mv{w}, \sigma_{e,k}^2\mv{I}\right), \, k = 1, \ldots,p,
\end{split}
\end{equation}
where $\mv{y}_k$ denotes the vector of all $n$ observations of the $k$th dimension of the data, $\mv{w}$ is a vector with all stochastic weights, and $\mv{K}$ is a discretization of the operator matrix. The matrix $\mv{B}$ contains the covariates evaluated at the measurement locations and $\mv{A}_k = \diag(\mv{e}_k) \otimes \mv{A}$ where $\mv{A}$ is an observation matrix with elements $\mv{A}_{ij} = vphi_{j}(\mv{s}_i)$. Finally, the distribution of the variance components, $\pi(\mv{v})$, depends on which model that is used, as described in Appendix~\ref{sec:fem}.
\subsection{Parameter estimation}\label{sec:estimation}
As is standard in the SPDE approach, we assume that the smoothness parameters are fixed and known. It should be noted that models with general smoothness parameters likely could be estimated from data using the rational SPDE approach \citep{bolin2017rational}. However, we leave the adaptation of this approach to the multivariate typee-G setting for future research.
Let $\mv{y} = (\mv{y}_1^\ensuremath{\top},\ldots, \mv{y}_p^\ensuremath{\top})^\ensuremath{\top}$ denote the vector of all observations in \eqref{eq:mainmodel}, and let $\mv{\Psi}$ be the model parameters to be estimated.
There is no explicit expression for the likelihood distribution $\pi(\mv{y}|\mv{\Psi})$. However, it is possible to compute maximum likelihood parameter estimates using Monte Carlo (MC) methods. This is computationally feasible because of two important properties of the model: Firstly, $\mv{w}|\mv{y},\mv{v},\mv{\Psi}$ is a Gaussian Markov random field (GMRF) and can thus be sampled efficiently. Secondly, $\mv{v}|\mv{w},\mv{y},\mv{\Psi}$ is a vector of independent variables and can thus be sampled in parallel.
We use a stochastic gradient (SG) method \citep{kushner2003stochastic} to estimate the parameters. The idea of SG is that one only needs an asymptotically unbiased estimator (as the number of MC samples goes to infinity), $\mv{G}(\mv{\Psi})$, of the gradient of the likelihood in order to utilize an iterative procedure where one at iteration $i$ updates the parameters as $\mv{\Psi}^{(i)} = \lambda_{i} \mv{G}(\mv{\Psi}^{(i-1)} ) + \mv{\Psi}^{(i-1)}$. Here $\{\lambda_i\}$ is a sequence satisfying $\sum \lambda_{i} \rightarrow \infty$ and $\sum \lambda_{i}^2 < \infty$, which ensures that the method converges to a stationary point of the likelihood \citep{kushner2003stochastic,andrieu2005stability}.
To derive the estimator of the gradient, we use Fisher's identity \citep{dempster1977maximum} to obtain
\begin{align}
\nabla_{\mv{\Psi}} \log \pi(\mv{y}|\mv{\Psi}) &= \proper{E}_{\mv{v},\mv{w}}\left(\nabla_{\mv{\Psi}} \log \pi(\mv{v},\mv{w}|\mv{y},\mv{\Psi}) | \mv{y},\mv{\Psi}\right)
={\bf \mathsf{E}}_{ \mv{v}} (\nabla_{\mv{\Psi}}\log \pi( \mv{v}| \mv{y},\mv{\Psi})| \mv{y},\mv{\Psi}). \label{eq:Egrad}
\end{align}
Since $\pi(\mv{w}|\mv{v},\mv{y},\mv{\Psi})$ is Gaussian, we have a closed-form expression for $\nabla_{\mv{\Psi}}\log \pi( \mv{v}| \mv{y},\mv{\Psi})$, see the Appendix \ref{sec:gradients}, but there is no closed form expression for its expected value. We therefore use
$
\mv{G}(\mv{\Psi}) = \frac1{N}\sum_{i=1}^N \nabla_{\mv{\Psi}}\log\pi_{\mv{\Psi}}( \mv{v}^{(i)} | \mv{y},\mv{\Psi})
$
as a MC estimate of the expectation, where $\mv{v}^{(i)}$ are samples from distribution $\pi(\mv{v}|\mv{y},\mv{\Psi})$. These samples are obtained using a Gibbs sampler (Algorithm \ref{alg:Gibbs} in Appendix \ref{sec:pseudo}) which samples $\pi(\mv{w}|\mv{y},\mv{v},\mv{\Psi})$ and $\pi(\mv{v}|\mv{w},\mv{y},\mv{\Psi})$ respectively. The sampling of $\pi(\mv{v}|\mv{w},\mv{y},\mv{\Psi})$ typically needs to be done with a general sampling method, such as a Metropolis Hastings algorithm. However, if $\pi(v)$ is a generalized inverse Gaussian (GIG) distribution, then the conditional distribution remains in the GIG family which can be sampled uniformly fast over the entire parameter space, see \cite{hormann2014generating}. The GIG distribution has density
$GIG(v; c, a, b) = \left(\frac{a}{b}\right)^{\frac{c}{2}}(2 K_c(\sqrt{ab}))^{-1} v^{c-1} e^{-\frac{1}{2}\left(av + bv^{-1}\right)}.$
For further details, including parameter ranges, see \cite{Jorgensen}.
The GIG distribution contains several known distributions as special cases, such as the gamma distribution, the inverse gamma distribution, and the IG distribution. Because of this, one can sample the variance components of the NIG distribution explicitly. The following example provides the conditional distributions for the NIG Mat\'ern-SPDE fields from Section \ref{sec:nig}.
\begin{example}
For the NIG processes in Section \ref{sec:nig}, the distribution of the variance components $v$, $v_i$ and $v_k$ is $IG(v;\eta_1, \eta_2) = GIG(v;-\frac{1}{2}, \eta_1 , \eta_2 )$. It can therefore be shown that the different type G constructions result in the following posterior distributions
\begin{align*}
&\mbox{type G$_1$:} &\pi(v|\mv{E},\mv{\Psi}) &= \mbox{GIG}\left(v;-\frac{np+1}{2}, \eta+ \sum_{k=1}^p \mu_k^2\mv{1}_n ^\ensuremath{\top}\mv{h}_k , \eta + \sum_{k=1}^p \left(\frac{\mv{\xi}_k}{\mv{h_k}}\right)^\ensuremath{\top} \mv{\xi}_k \right),\\
&\mbox{type G$_2$:} &\pi(v_k|\mv{E},\mv{\Psi}) &= \mbox{GIG}\left(v_k;-\frac{n+1}{2},\eta_k+ \mu_k^2\mv{1}_n ^\ensuremath{\top}\mv{h}_k , \eta_k + \left(\frac{\mv{\xi}_k}{\mv{h_k}}\right)^\ensuremath{\top} \mv{\xi}_k\right),\\
&\mbox{type G$_3$:} &\pi(\mv{v}|\mv{E},\mv{\Psi}) &= \mbox{GIG}\left(\mv{v}; -\frac{p+1}{2} , \eta + \sum_{k=1}^p \mu_k^2 , \mv{h}_k^2 \eta + \sum_{k=1}^p \mv{\xi}_k^2 \right),\\
&\mbox{type G$_4$:} &\pi(\mv{v}_k|\mv{E},\mv{\Psi}) &= \mbox{GIG}\left(\mv{v};-1, \mu_k^2 + \eta_k , \mv{\xi}_k^2 + \mv{h}_k^2 \eta_k\right),
\end{align*}
where $\mv{E}=[\mv{E}_1^{\ensuremath{\top}},\ldots,\mv{E}_p^{\ensuremath{\top}}]^{\ensuremath{\top}} = \mv{K} \mv{w}$ and $\mv{\xi}_k = \mv{E}_k + \mv{h}_k\mu_k$.
For the two last densities it is explicitly understood that $GIG$ in vector form denotes product of independent GIG distributions with parameter values given by the values in the vectors.
\end{example}
\subsection{Spatial prediction and evaluation of predictive performance}\label{sec:kriging}
In applications one is often interested in predictions of the latent field given data. The predictive distribution for the $k$th variable of the latent field, at a location $\mv{s}_0$, is $\pi(x_k(\mv{s}_0)|\mv{y},\mv{\Psi})$. This distribution is often summarized using the mean as a point estimate, and the variance as a measure of uncertainty. To estimate these two quantities, let $\mv{A}_p = [vphi_1(\mv{s}_0),\ldots, vphi_n(\mv{s}_0)]$ and use the Gibbs sampler in Algorithm \ref{alg:Gibbs}, Appendix \ref{sec:pseudo}, to obtain samples, $\{\mv{v}^{i}\}_{i=1}^N$, from $\pi(\mv{v}|\mv{y},\mv{\Psi})$. Based on these samples, we compute MC estimates $\proper{E}(x_k(\mv{s}_0)|\mv{y}) \approx \frac1{N} \sum_{i=1}^N \mv{A}_p\hat{\mv{\xi}}^{(i)}$ and
$\proper{V}(x_k(\mv{s}_0)|\mv{y})\approx \frac1{N} \sum_{i=1}^N \mv{A}_p^{\ensuremath{\top}}(\hat{\mv{Q}}^{(i)})^{-1}\mv{A}_p$, where $\hat{\mv{\xi}}^{(i)}$ is the expected value of $\mv{w}|y, \mv{v}^{(i)}$ and $\hat{\mv{Q}}^{(i)}$ is the corresponding precision matrix (see Appendix \ref{sec:gradients} for analytic formulas of these quantities).
The posterior median, which may be a more appropriate point estimator if the distribution is asymmetric, can similarly be estimated by the sample median of $\{\mv{A}_p\hat{\mv{\xi}}^{(i)}\}_{i=1}^N$.
To evaluate a proposed model one also need to compute various goodness-of-fit measures, such as the continuous ranked probability scores (CRPS) \citep{matheson1976scoring}. Let $y_k$ be an observation in the $k$th dimension at $\mv{s}_0$, and let $F$ denote the marginal CDF of $\pi(y_k|\mv{y}_{-0},\mv{\Psi})$, where $\mv{y}_{-0}$ denotes all observations but $y_k$, then the (negatively oriented) CRPS value for this location can be computed as \citep{gneiting2007strictly}
\begin{equation}\label{eq:crpsE}
\mbox{CRPS}(F,y_k) = \proper{E}(|Y^{(1)}_k - y_k|) - \frac1{2}\proper{E}(|Y^{(1)}_k - Y^{(2)}_k|)
\end{equation}
where $Y^{(1)}_k$ and $Y^{(2)}_k$ are independent random variables with distribution $F$. For a Gaussian distribution this expression can be used to derive CRPS value analytically \citep[see e.g.][]{gneiting2007strictly}. For the multivariate type G SPDE-Mat\'ern fields, one option is to approximate the expected values in \eqref{eq:crpsE} by MC integration. Basing such an estimate on $N$ draws of $Y_k^{(1)}$ and $Y_k^{(2)}$ yields an estimate $\mbox{CRPS}_N(F,y)$. Unfortunately, $N$ often needs to be quite large to obtain good approximations with this estimator. The following proposition provides a more efficient way of approximating the CRPS value in the case of a general normal-variance mixture distribution.
\begin{prop}\label{crpsthm}
Assume that the random variable $X$ is a normal-variance mixture with CDF $F(x) = \int \Phi\left(\frac{x-\mu(v)}{\sigma(v)}\right) \ensuremath{\,\mathrm{d}} F_v(v)$. Let $V_j^{(i)}, j=1,2, i=1,\ldots, N$ be independent draws from the mixing distribution $F_v$, and define $\mu_V = \proper{E}(X|V)$, $\sigma_V^2 = \proper{V}(X|V)$, and
\begin{equation}\label{eq:M}
M(\mu,\sigma^2) = 2\sigmavphi\left(\frac{\mu}{\sigma}\right) + \mu\left(2\Phi\left(\frac{\mu}{\sigma}\right)-1\right),
\end{equation}
where $vphi$ denotes the density function of a standard Gaussian distribution. Then
\begin{align*}
\mbox{CRPS}_N^{RB}(F,y) &= \frac1{N}\sum_{i=1}^N \left[M\left(\mu_{V_1^{(i)}} - y,\sigma_{V_1^{(i)}}^2\right)
- \frac1{2}M\left(\mu_{V_1^{(i)}} - \mu_{V_2^{(i)}},\sigma_{V_1^{(i)}}^2 +\sigma_{V_2^{(i)}}^2\right)\right]
\end{align*}
satisfies $\proper{E}(\mbox{CRPS}_N^{RB}(F,y)) = \mbox{CRPS}(F,y)$ and $\proper{V}(\mbox{CRPS}_N^{RB}(F,y)) \leq \proper{V}(\mbox{CRPS}_N(F,y))$.
\end{prop}
The $\mbox{CRPS}_N^{RB}$ estimator can be used for the type G fields since $\pi(x_k(\mv{s}_0)|\mv{y}_{-0},\mv{v},\mv{\Psi})$ is Gaussian and since we easily can sample the variances $\mv{v}$ using the Gibbs sampler.
To give an idea of the improvement that can be obtained by using the RB estimator, both estimators were used to compute the CRPS value for the final fold of the most general NIG model in the cross-validation study in Section~\ref{sec:applications}. Based on $N=10000$ samples, the MC variances of the two estimators were $\proper{V}(\sqrt{N}\mbox{CRPS}_N^{RB}(F,y)) \approx 187$ and $\proper{V}(\sqrt{N}\mbox{CRPS}_N(F,y)) \approx 2225$.
\section{Applications}\label{sec:applications}
In this section we illustrate for two different data sets how the multivariate Type G SPDE fields can be used for spatial modelling. The first data set consists of temperature and pressure measurements from the North American Pacific Northwest and was previously studied in \cite{gneiting2012matern} and \cite{apanasovich2012valid}. The second data set consists of seawater temperature measurements taken at two different depths in the ocean.
For both data sets we assume the model $\mv{y}_i = \mv{\beta} + \mv{x}(\mv{s}_i) + \mv{\vep}_i$ for the bivariate observations $\mv{y}_i$, where $\mv{x}(\mv{s}) = (x_1(\mv{s}), x_2(\mv{s}))^{\ensuremath{\top}}$ is a mean-zero random field, $\mv{\beta} = (\beta_1,\beta_2)^{\ensuremath{\top}}$ is the expected value, and $\mv{\vep}_i$ are independent $\proper{N}(\mv{0},\diag(\sigma_{1}^2,\sigma_{2}^2))$ variables representing measurement noise. As reference models, we will for each data set use four Gaussian models for $\mv{x}(\mv{s})$. The first of these assumes that $x_1(\mv{s})$ and $x_2(\mv{s})$ are independent Gaussian Mat\'ern fields with covariance functions $C_{11}(\mv{h}) = \sigma_1^2\materncorr{\mv{h}}{\kappa_1}{\nu_1}$ and $C_{22}(\mv{h}) = \sigma_2^2\materncorr{\mv{h}}{\kappa_2}{\nu_2}$ respectively. We also use the parsimonious Gaussian Mat\'ern field by \cite{gneiting2012matern} as well as two Gaussian Mat\'ern-SPDE models specified using \eqref{eq:general}, one lower-triangular and one independent model with $\rho= 0$. For the applications, we focus on comparing the reference models to the type G$_4$ models and do not evaluate the simpler type G constructions. We do not consider the type G$_1$ and type G$_2$ models since the first data set does not have repeated measurements, and since one does not expect these models to improve the predictive performance compared to the Gaussian models because of Theorem~\ref{thm:krig}. We do not consider the type G$_3$ model since there is no specific reason for why a shared variance component would be beneficial for the considered data sets.
\begin{figure}
\caption{Measurements of pressure (left) and temperature (right) in the North American Pacific Northwest together with the mesh used for the SPDE models. The sample mean has been subtracted from the data in both cases.}
\label{fig:data}
\end{figure}
\subsection{Temperature and pressure in the North American Pacific Northwest}
The data, shown in Figure \ref{fig:data}, consists of temperature and pressure observations, $\mv{y}_i = (y_{P}, y_{T})_i^{\ensuremath{\top}}$ where $y_{P}$ denotes pressure and $y_{T}$ temperature, at 157 locations in the North American Pacific Northwest.
Besides the four baseline models, we test four different type G Mat\'ern-SPDE models for the data. A Gaussian model for temperature seems adequate whereas the pressure data has short-range variations that is inflating the measurement noise variance (see parameter estimates in Appendix \ref{seq:parameter_estimates}), which possibly could be captured by the latent field if a non-Gaussian model was used. We therefore consider type G$_4$ models where the driving noise for the pressure is NIG distributed, whereas the driving noise for temperature is Gaussian. In order to investigate the effects of the operator matrix, we use one independent model, with $\rho = 0$, and two dependent models. The first of these is triangular with $\theta = 0$, and the second has a general operator where $\theta$ is estimated jointly with the other parameters.
The mesh that is used for the discretization of the SPDE models is shown in Figure~\ref{fig:data}. It consists of 981 nodes and was built using \texttt{R-INLA} \citep{lindgren2015bayesian}. We fix the $\alpha$ parameters to $2$ for all SPDE models, which corresponds to $\nu = 1$ for the Mat\'ern covariances.
The parameters of the Gaussian models are estimated using numerical maximisation of the log-likelihood function, whereas the gradient-based method from Section \ref{sec:estimation} is used for the non-Gaussian models. The gradient method is run 1000 iterations, using starting values obtained from the corresponding Gaussian model. For the lower-triangular models, the estimation took $44$ seconds for the Gaussian model and $156$ seconds for the NIG model. These values were obtained using a \cite{Matlab} implementation of the algorithm on a Macbook Pro computer with a 2.6GHz Intel Core i7 processor. The parameter estimates for the different models are shown in Appendix~\ref{seq:parameter_estimates}.
To compare the models, we perform a leave-one-out pseudo cross-validation study. For each observation location, the pressure and temperature values are predicted using the data from all 156 other locations using the models with parameters given in Appendix~\ref{seq:parameter_estimates}. For all models, the point estimates are computed using the expected values of the values at the held-out location conditionally on the data at all other locations. Using the posterior median as a predictor did not improve the predictive performance for this data, and we therefore omit those results. The predictive performance of the models is assessed using the median absolute error of the 157 predicted values, as well as the median CRPS. The resulting values are shown in Table~\ref{tab2}. One can note that the dependent NIG models have better predictive performance than the Gaussian models. Spatial predictions using the parsimonious Mat\'ern model and the general NIG model can be seen in Figure~\ref{fig:krig}.
\begin{table}
\centering
\begin{tabular}{lcccccc}
\toprule
& Operator & Number of & \multicolumn{2}{c}{Pressure} & \multicolumn{2}{c}{Temperature} \\
Model & matrix & parameters & \multicolumn{2}{c}{(Pascal)} & \multicolumn{2}{c}{(degrees Celcius)} \\
\cmidrule(r){1-3} \cmidrule(r){4-5} \cmidrule(r){6-7}
& & & MAE & CRPS & MAE & CRPS \\
Independent Mat\'ern & - & 10 & $41.632$ & $28.994$ & $0.956$ & $0.598$ \\
Parsimonious Mat\'ern & - & 10 & $39.068$ & $27.682$ & $0.921$ & $0.576$ \\
Gaussian SPDE & Diagonal & 8 & $38.624$ & $31.711$ & $0.917$ & $0.594$\\
Gaussian SPDE & Triangular & 9 & $38.856$ & $31.829$ & $0.915$ & $0.580$\\
NIG SPDE & Diagonal & 10 &$39.101$ & $25.993$ & $0.847$ & $0.525$\\
NIG SPDE & Triangular & 11 & $39.302$ & $25.776$ & $\mv{0.841}$ & $\mv{0.512}$\\
NIG SPDE & General &12 & $\mv{38.523}$ & $\mv{25.591}$ & $0.876$ & $0.514$\\
\bottomrule
\end{tabular}
\caption{\label{tab2} Cross-validation results comparing the median absolute error (MAE) and median CRPS for the different models.}
\end{table}
\begin{figure}
\caption{Estimates of pressure (top) and temperature (bottom) using the parsimonious Mat\'ern model and the NIG model with general operator matrix. The difference between the estimates is shown to the right.}
\label{fig:krig}
\end{figure}
\subsection{Seawater temperatures}
We now consider Argo floats measurements of seawater temperature at two different depths. Since the measurements are sparse in space (and time) an important statistical task is to ``fill in the gaps'' through spatial interpolation. The data has been thoroughly analyzed from a statistical perspective by \cite{kuusela2018locally}, who noted that the data seem to be non-Gaussian at higher depths in certain areas. To investigate if the type G models could be useful for interpolation of this data, we choose two different depths, $300$ dbar and $1500$ dbar, and investigate if one could improve the joint prediction of those depths using the type G models.
\begin{figure}
\caption{Measurements of seawater temperature on depth $300$ db in 2016 together with the mesh used for the SPDE models. A mean field has been removed from the measurements.}
\label{fig:dataArgo}
\end{figure}
We extract data from the month of February for three years ($2014-2016$). Since \cite{kuusela2018locally} showed that an analysis of the complete data set requires a non-stationary model, we focus on a limited spatial region south of New Zeeland to be able to use a stationary model. However, as for the Gaussian SPDE-based models, one could model non-stationarity by allowing the parameters in the operator to be spatially varying. The restriction results in a data set consisting of $312$ observations in total.
For each location we study the residuals after removing a seasonally varying mean field (the Roemmich-Gilson mean field, see \cite{kuusela2018locally}). Thus, we let $\mv{y}_i=\left(y_{300}(\mv{s}_i),y_{1500}(\mv{s}_i)\right)^{\ensuremath{\top}}$ for $i\in \{1,\ldots,n_t\}$ and $t\in \{2014,2015,2016\}$. Here $y_{300}$ is the residual at the depth of $300$ dbar and $y_{1500}$ is the residual at the depth of $1500$ dbar. We assume that data for the different years are independent.
Besides the four baseline models, we test two non-Gaussian type G SPDE models. In the first, we assume that $x_1(\mv{s})$ and $x_2(\mv{s})$ are independent univariate NIG SPDE fields. In the second, we use the type G$_4$ construction with a general operator matrix where both noise processes are NIG distributed. For all SPDE-based models, we again fix the smoothness parameters $\alpha$ to $2$ and use the mesh shown in Figure~\ref{fig:dataArgo}, which also shows the available data for the year $2016$.
The parameters are estimated using an R implementation of the proposed methods, available in the package \texttt{ngme}. The parameter estimates for the different models are shown in Appendix~\ref{seq:parameter_estimates}.
To evaluate which of the tested models that performs best in terms of prediction, we again use leave-one-out pseudo cross-validation and compare median MAE and median CRPS. However, contrary to the previous application we now do the cross-validation by removing individual univariate observations instead of removing the bivariate observation pairs for each spatial location. The reason for this is that we here do not necessarily have observations of the fields at different depth in the same spatial locations. The results are shown in Table~\ref{tab:argoCrossval}, where we see that the multivariate Non-Gaussian model is clearly outperforming the other models. Thus, it seems as if one could increase the accuracy of the spatial interpolation of the Argo data using type G models. For future work it is therefore interesting to study the entire data set, where a more in-depth analysis would require a space-time model, see \cite{kuusela2018locally}.
\begin{table}
\centering
\begin{tabular}{lcccccc}
\toprule
& Operator & Number of & \multicolumn{4}{c}{depth} \\
Model & matrix & parameters & \multicolumn{2}{c}{db $300$} & \multicolumn{2}{c}{db $1500$} \\
\cmidrule(r){1-3} \cmidrule(r){4-5} \cmidrule(r){6-7}
& & & MAE & CRPS & MAE & CRPS \\
Independent Mat\'ern & - & 10 & $0.183$ & $0.154$ & $0.040$ & $0.029$ \\
Parsimonious Mat\'ern & - & 10 & $0.239$ & $0.162$ & $0.033$ & $\mv{0.024}$ \\
Gaussian SPDE & Diagonal & 8 & $\mv{0.187}$ & $0.182$ & $0.043$ & $0.032$\\
Gaussian SPDE & Triangular & 9 & $0.216$ & $0.169$ & $0.034$ & $0.025$\\
NIG SPDE & Diagonal & 12 &$0.200$ & $0.159$ & $0.040$ & $0.028$\\
NIG SPDE & General &14 & $0.201$ & $\mv{0.130}$ & $\mv{0.032}$ & $\mv{0.024}$\\
\bottomrule
\end{tabular}
\caption{\label{tab:argoCrossval} Cross-validation results comparing the median absolute error and median CRPS for different models for the Argo data. }
\end{table}
\section{Discussion}\label{sec:discussion}
There is a need for practically useful random field models with more general distributions than the Gaussian. Especially for multivariate data, finding good alternatives to Gaussian fields has been considered an open problem in the literature. We have introduced one such alternative by formulating a new class of multivariate random fields with flexible multivariate marginal distributions and covariance functions of Mat\'ern-type. The fields are constructed as solutions to SPDEs and can be used in a geostatistical setting where likelihood-based parameter estimation can be performed using a computationally efficient stochastic gradient algorithm. In fact, the models have the same computational advantages as their Gaussian counterparts, which facilitates applications to large data sets, although with additional cost due to MC sampling.
Four different constructions of the non-Gaussian noise were considered, where the first two are closely related to existing approaches, such as factor-copula models and Student's t-fields. We showed that these constructions have significant disadvantages when used for spatial prediction, or on data without replicates. The more sophisticated constructions based on type G L\'evy noise do not have these disadvantages, and their combination of flexibility and computational efficiency should therefore make them attractive alternatives to Gaussian models for geostatistical applications.
The computational benefits of the finite dimensional approximations presented in Appendix \ref{sec:fem} are only available for fields with $\alpha/2\in\mathbb{N}$. This restriction of the smoothness parameters is often viewed as one of the main drawbacks of the SPDE approach, since the smoothness of the covariance function is important for the predictive performance. However, in many cases the distributional assumptions can be equally important. This was clearly shown in the application where the covariance-based models, which allow for arbitrary smoothness parameters, were outperformed by the non-Gaussian models with fixed smoothness parameters. Nevertheless, extending the approach to fields with general smoothness would increase the flexibility. As previously mentioned, this could likely be done using the rational SPDE approach \citep{bolin2017rational}, and extending that method to multivariate type G fields is thus an interesting topic for future research.
\begin{appendix}
\section{Finite-dimensional representations}\label{sec:fem}
An advantage with the SPDE approach is that the finite element method can be used for computationally efficient approximations of the models. This was introduced by \cite{lindgren10} for Gaussian models and was extended to SPDEs driven by type G L\'evy noise in \citep{bolin11}. In this section, we present a multivariate extension of this method.
In the univariate case, the method is based on a basis expansion $x(\mv{s}) = \sum_{j = 1}^n w_jvphi_j(\mv{s})$,
where $\{vphi_j\}$ is a collection of piecewise linear basis functions obtained by a triangulation of the (compact) spatial domain of interest $\mathcal{D}$. See Figure \ref{fig:data} for an example. Each node $\tilde{\mv{s}}_j$ in the triangulation defines a piecewise linear basis function $vphi_j(\mv{s})$ with $vphi_j(\tilde{\mv{s}}_j) = 1$ that is zero for all locations in triangles not directly connected to the node $\tilde{\mv{s}}_j$. For the multivariate extension, we assume that the SPDE is formulated using the representation in \eqref{eq:general}. Introduce p-dimensional basis functions $\mv{vphi}_i^k(\mv{s}) = vphi_i(s) \mv{e}_k$, where $\mv{e}_k$ is the $k$th column in a $p\times p$ identity matrix, and let $\mv{x}(s) = \sum_{j = 1}^n\sum_{k=1}^p w_{jk}\mv{vphi}_j^k(\mv{s})$.
The distribution of the stochastic weights $\mv{w} = (w_{11},\ldots, w_{n1},w_{12},\ldots, w_{n2},\ldots,w_{np})^{\ensuremath{\top}}$ is calculated by augmenting the operators in \eqref{eq:general} with homogeneous Neumann boundary conditions and computing the weights using the Galerkin method. For $\alpha=2$ and Gaussian noise, the result is $\mv{w} \sim \proper{N}(\mv{0}, \mv{K}^{-1}\diag(\mv{h})\mv{K}^{-\ensuremath{\top}})$. Here $\mv{h} = \mv{1}_p\otimes (h_{1}, \ldots, h_n)^{\ensuremath{\top}}$ where $h_i = |\mathcal{D}_i|$ is the area of the region $\mathcal{D}_i = \{\mv{s} : vphi_i(\mv{s}) \geq vphi_j(\mv{s}) \,\forall j\neq i\}$. Further,
\begin{equation}\label{eq:K}
\mv{K} = (\mv{D}_p\otimes \mv{I}_n)\diag(\mv{L}_{\alpha_1}(\sigma_1,\kappa_1),\ldots,\mv{L}_{\alpha_p}(\sigma_p,\kappa_p)),
\end{equation}
is the discretized operator matrix where $\mv{I}_n$ denotes an identity matrix of size $n\times n$, and $\mv{L}_{\alpha_k}(\sigma_k,\kappa_k) = c_k (\mv{G} + \kappa_k^2\mv{C})$ is the discretized operator for the $k$th dimension. The matrices $\mv{C}$ and $\mv{G}$ have elements $C_{ii} = \scal{vphi_i}{vphi_i}$ and $G_{ij} = \scal{\nablavphi_i}{\nablavphi_j}$, respectively, where $\scal{f}{g}$ denotes the inner product on $\R^d$ and $\nabla$ is the gradient operator.
In the type G case, the corresponding result is
\begin{align*}
\mv{w}|\mv{v} &\sim \proper{N}(\mv{K}^{-1}((\diag(\mv{\gamma})\otimes \mv{I}_n)\mv{h}+(\diag(\mv{\mu})\otimes \mv{I}_n)\mv{v}), \mv{K}^{-1}\diag(\mv{v})\mv{K}^{-\ensuremath{\top}}),
\end{align*}
where $\mv{v} = (\mv{v}_1^{\ensuremath{\top}},\ldots, \mv{v}_p^{\ensuremath{\top}})^{\ensuremath{\top}}$ and $\mv{1}_p$ is a vector with $p$ ones.
The vector $\mv{v}_k = (v_1^k,\ldots, v_n^k)$ is the discretized variance process for the $k$th dimension, with elements
\begin{equation*}
v_i^k = \int \mathbb{I}(\mv{s}\in \mathcal{D}_i) v_k(d\mv{s}) = \begin{cases}
h_i v & \mbox{type G$_1$,}\\
h_i v_k & \mbox{type G$_2$,}\\
M_{v}(\mathcal{D}_i) & \mbox{type G$_3$,}\\
M_{v_k}(\mathcal{D}_i) & \mbox{type G$_4$},
\end{cases}
\end{equation*}
where $M_v(\cdot)$ denotes the random measure associated with $v$. The distribution of $\mv{v}$ is in general not explicit for type G$_3$ or type G$_4$, unless the distribution of $v_k(\mv{s})$ is closed under convolution. An example of a distribution that has this property is the IG distribution that is used in for the NIG process.
\begin{example}
The following equation summarizes the distribution of $\mv{v}$ for the different versions of the NIG processes from Section \ref{sec:nig}.
\begin{equation}\label{eq:Vdist}
\mv{v} \sim \begin{cases}
\mv{h} \otimes ( \mv{1}_K \otimes IG(\eta^2,\eta^2) )& \mbox{type G$_1$,}\\
\mv{h} \otimes IG(\mv{\eta}^2,\mv{\eta}^2) & \mbox{type G$_2$,}\\
\mv{1}_K \otimes IG(\eta^2,\eta^2\mv{h}^2) & \mbox{type G$_3$,}\\
IG(\mv{\eta}^2 \otimes \mv{1}_n,\mv{\eta}^2\otimes \mv{h}^2) & \mbox{type G$_4$}.
\end{cases}
\end{equation}
Here the notation $\mv{v} \sim IG(\mv{a}, \mv{b})$ is a compact way of writing a vector with independent components $v_i \sim IG(a_i, b_i)$.
\end{example}
The discretization above assumes $\alpha_i = 2$. In the case of $\alpha_i/2\in\mathbb{N}$, each operator is an integer power of the operator for $\alpha_i = 2$ and the method can then be combined with the iterated finite element discretization by \cite{lindgren10} to obtain similar finite dimensional approximations with Markov properties. The only difference in this case is that $\mv{L}_{\alpha_k}(\sigma_k,\kappa_k) = c_k \mv{C}(\mv{C}^{-1}\mv{G} + \kappa_k^2\mv{I})^{\alpha_k}$.
\section{Gradients of the log-likelihood}\label{sec:gradients}
In this section, the gradients needed for the estimation method from Section~\ref{sec:estimation} are presented. The parameters we need the gradients for are $\mu_k$ and $\sigma_k$ for $k=1,\ldots, p$, the regression parameters $\mv{\beta}$, the parameters of the differential operator matrix $\mv{K}$, as well as any parameters of $\pi(\mv{v})$.
To simplify notation, let $[ \hat{\mv{E}}_1^{\ensuremath{\top}},\ldots, \hat{\mv{E}}_p^{\ensuremath{\top}}]^{\ensuremath{\top}} = \mv{K}\hat{\mv{\xi}}$, where
$$
\hat{\mv{\xi}} = \hat{\mv{Q}}^{-1} \left(\sum_{k=1}^p\frac{1}{\sigma_{e,k}^2} \mv{A}_k^{\ensuremath{\top}} \mv{y}_k + \mv{K}^{\ensuremath{\top}} \diag(\mv{v})^{-1} (\mv{\mu}\otimes\mv{I}_n)(\mv{v} - \mv{h})\right)
$$
is the posterior mean of $\mv{w}|\mv{v},\mv{\Psi}$ and $\hat{\mv{Q}} = \mv{K}^{\ensuremath{\top}} \diag(\mv{v})^{-1} \mv{K}+ \sum_{k=1}^p\frac{1}{\sigma_{e,k}^2} \mv{A}_k^{\ensuremath{\top}} \mv{A}_k$.
All gradients are obtained by first computing $\log\pi(\mv{v}|\mv{y}, \mv{\Psi}) = \log\int \pi(\mv{v},\mv{w}|\mv{y}, \mv{\Psi})\ensuremath{\,\mathrm{d}}\mv{w}$.
This integral is straight-forward to compute since
\begin{align*}
\log\pi( \mv{v},\mv{w} | \mv{y},\mv{\Psi}) =& \sum_{k=1}^p\left(-m\log\sigma_{e,k} -\frac{1}{2\sigma_{e,k}^2} \left( \mv{y}_k- \mv{A}_k\mv{w} - \mv{B}\mv{\beta} \right)^{\ensuremath{\top}} \left( \mv{y}_k- \mv{A}_k \mv{w} - \mv{B} \mv{\beta} \right)\right) \\
& - \frac{1}{2} \left( \mv{K}\mv{w} - (\mv{\mu}\otimes\mv{I}_n)(\mv{v}-\mv{h})\right)^{\ensuremath{\top}} \diag(\mv{v})^{-1} \left( \mv{K}\mv{w} - (\mv{\mu}\otimes\mv{I}_n)(\mv{v}-\mv{h}) \right), \\
&+ |\mv{K}| - \mv{1}^{\ensuremath{\top}}\log(\mv{v}) + \log(\pi_{\mv{\Psi}}(\mv{v})) + \mbox{const.},
\end{align*}
Standard matrix calculus is then used to differentiate $\log\pi(\mv{v}|\mv{y}, \mv{\Psi})$ with respect to the parameters to obtain the required gradients. For brevity we omit the details of these computations and just present the results. The gradients for $\mu_k$, $\sigma_{e,k}$, and $\mv{\beta}$ are
\begin{align*}
\nabla_{\mu_k}\log \pi( \mv{v}| \mv{y}, \mv{\Psi}) &= \left( -\mv{h}_k + \mv{v}_k \right) ^{\ensuremath{\top}} \diag(\mv{v}_k)^{-1} \left( \hat{\mv{E}} - \left(- \mv{h}_k + \mv{v}_k \right) \mu_k \right), \\
\nabla_{\sigma_{e,k}}\log \pi(\mv{v}| \mv{y}, \mv{\Psi}) &= - \frac{n}{\sigma_{e,k}} + \frac{1}{\sigma_{e,k}^3} \| \mv{y}_k- \mv{A}_k\hat{\mv{\xi}} - \mv{B}\mv{\beta} \|^2 + \trace(\mv{A}^{\ensuremath{\top}}\mv{A}\hat{\mv{Q}}^{-1}), \\
\nabla_{\mv{\beta}}\log \pi(\mv{v}| \mv{y}, \mv{\Psi}) &= \sum_{k=1}^p \frac{1}{\sigma_{e,k}^2} \left( \mv{y}_k- \mv{A}_k\hat{\mv{\xi}} - \mv{B} \mv{\beta} \right)^{\ensuremath{\top}} \mv{B} .
\end{align*}
For a parameter $\psi_K$ in the operator, the gradient is
\begin{align*}
\nabla_{\psi_K}\log \pi( \mv{v}| \mv{y}, \mv{\Psi}) =& \trace(\mv{K}_{\psi_K}\mv{K}^{-1}) - \hat{\mv{\xi}}^{\ensuremath{\top}} \mv{K}_{\psi_K}^{\ensuremath{\top}} \diag(\mv{v})^{-1}\mv{K}\hat{\mv{\xi}} - \trace(\mv{K}_{\psi_K}^{\ensuremath{\top}} \diag(\mv{v})^{-1}\mv{K}\hat{\mv{Q}}^{-1}) \\
& + \hat{\mv{\xi}}^{\ensuremath{\top}} \mv{K}_{\psi_K}^{\ensuremath{\top}} \diag(\mv{v})^{-1}(\mv{\mu}\otimes\mv{I}_n)\left( -\mv{h} + \mv{v} \right) ,
\end{align*}
where $\trace(\cdot)$ denotes the matrix trace, and where $\mv{K}_{\psi_K}$ denotes the derivative of $\mv{K}$ with respect to $\psi_K$. Using that $\mv{K}$ is on the form given in \eqref{eq:K}, one gets
$$
\mv{K}_{\psi_K} = \begin{cases}
(\mv{D}_{\psi_K}\otimes \mv{I}_n) \diag(\mv{L}_1,\ldots, \mv{L}_p) & \mbox{$\psi_K = \theta_i, \rho_{ij}$}, \\
-\sigma_j^{-1}(\mv{D}\otimes \mv{I}_n) (\mv{L}_j\otimes\diag(\mv{e}_j)) & \mbox{$\psi_K = \sigma_j$}, \\
\kappa_j^{-1}(\mv{D}\otimes \mv{I}_n) (\mv{L}_j(\alpha_j\kappa_j^2(\mv{C}^{-1}\mv{G}+\kappa_j^2\mv{I})^{-1}-\nu_j)\otimes\diag(\mv{e}_j)) & \mbox{$\psi_K = \kappa_j$}, \\
\end{cases}
$$
where $\mv{D}_{\psi_K}$ is the derivative of $\mv{D}$ with respect to $\psi_K$ and $\mv{L}_i$ denotes $\mv{L}_{\alpha_i}(\sigma_i,\kappa_i)$.
To take full advantage of the sparsity of the matrices, one should compute $\trace(\mv{A}^{\ensuremath{\top}}\mv{A}\hat{\mv{Q}}^{-1})$ and $\trace(\mv{K}_{\psi_K}^{\ensuremath{\top}} \diag(\mv{v})^{-1}\mv{K}\hat{\mv{Q}}^{-1})$ without inverting $\hat{\mv{Q}}$. To do so, note that $\mv{A}^{\ensuremath{\top}}\mv{A}$ and $\mv{K}_{\psi_K}^{\ensuremath{\top}} \diag(\mv{v})^{-1}\mv{K}$ are sparse matrices with non-zero elements only at positions in the matrices where also $\hat{\mv{Q}}$ is non-zero. This means that it is enough to compute the elements of $\hat{\mv{Q}}^{-1}$ only at the positions where $\hat{\mv{Q}}$ is non-zero, which can be done efficiently using the method by \cite{RueMartino07}.
Finally, the expression for the gradient of the parameters for the distribution of $\mv{v}$ depends on which distribution that is used. The following example gives the results for the NIG processes.
\begin{example}
For the NIG processes in Section \ref{sec:nig}, the gradient of the likelihood with respect to the parameter $\eta$ in the type G$_1$ and type G$_3$ cases is
\begin{equation*}
\nabla_{\eta}\log \pi( \mv{v}|\mv{y}, \mv{\Psi}) =
\begin{cases}
\frac{1}{2\eta} - \frac{1}{2}\left(v +v^{-1} \right) + 1& \mbox{type G$_1$,} \\
\frac{n}{2\eta} - \frac{1}{2}\left(\mv{v} + \mv{h}^2 \cdot \mv{v}^{-1} \right) \mv{1} + \mv{h}^{\ensuremath{\top}} \mv{1} & \mbox{type G$_3$,}
\end{cases}
\end{equation*}
and the gradient of the likelihood with respect to the parameters $\eta_k,k=1,\ldots,p$ in the type G$_2$ and type G$_4$ cases is
\begin{equation*}
\nabla_{\eta_k}\log \pi( \mv{v}|\mv{y}, \mv{\Psi}) =
\begin{cases}
\frac{1}{2\eta_k} - \frac{1}{2}\left(v_k +v_k^{-1} \right) + 1 & \mbox{type G$_2$,} \\
\frac{n}{2\eta_k} - \frac{1}{2}\left(\mv{v}_k + \mv{h}^2_k \cdot \mv{v}^{-1}_k \right) \mv{1} + \mv{h}_k^{\ensuremath{\top}} \mv{1} & \mbox{type G$_4$.}
\end{cases}
\end{equation*}
\end{example}
\section{Pseudo-code for the sampling methods}\label{sec:pseudo}
Algorithm \ref{alg:Gibbs} describes one iteration of the Gibbs sampler that is used to generate the samples used for parameter estimation and prediction. On Line 4 and Line 5 of the algorithm one should not compute the inverse $\hat{\mv{Q}}^{-1}$ but instead use an efficient sampling method for GMRFs based on sparse Cholesky factorization \cite[see][]{rue1}.
The general form of the distribution of $\mv{v}$ given $\mv{E}=[\mv{E}_1^{\ensuremath{\top}},\ldots,\mv{E}_p^{\ensuremath{\top}}]^{\ensuremath{\top}} = \mv{K} \mv{w}$ is shown in Algorithm \ref{algV}, where one can see how the different type G models affect how $v$ is sampled.
\begin{algorithm}[h]
\footnotesize
\caption{Gibbs sampler}
\label{alg:Gibbs}
\begin{algorithmic}[1]
\Procedure{GIBBS}{$\mv{y}, \mv{B}, \mv{v}, \mv{\Psi}, \mv{A}_1, \ldots, \mv{A}_p, \mv{h}$,typeG}
\State $\mv{K} \gets BuildOperator(\mv{\Psi})$ (Construct $\mv{K}$ as outlined in Appendix \ref{sec:fem})
\State $\hat{\mv{Q}} \gets \mv{K}^{\ensuremath{\top}} \diag(\mv{v})^{-1} \mv{K}+ \sum_{k=1}^p\frac{1}{\sigma_{e,k}^2} \mv{A}_k^{\ensuremath{\top}} \mv{A}_k$
\State $\hat{\mv{\xi}} \gets \hat{\mv{Q}}^{-1} \left(\sum_{k=1}^p\frac{1}{\sigma_{e,k}^2} \mv{A}_k^{\ensuremath{\top}} \left(\mv{y}_k -\mv{B}\mv{\beta}\right) + \mv{K}^{\ensuremath{\top}}\diag(\mv{v})^{-1}(\mv{\mu}\otimes\mv{I}_n)(\mv{v}-\mv{h})\right)$
\State Sample $\mv{w} \sim \proper{N}(\hat{\mv{\xi}} , \hat{\mv{Q}}^{-1})$
\State $[\mv{E}_1^{\ensuremath{\top}},\ldots,\mv{E}_p^{\ensuremath{\top}}]^{\ensuremath{\top}} \gets \mv{K} \mv{w}$
\State Sample $\mv{v} \sim \pi(\mv{v}|\mv{E}_1,\ldots, \mv{E}_p,\mv{\Psi})$ using Algorithm \ref{algV}
\State \Return $\{\mv{w}, [\mv{v}_1^{\ensuremath{\top}},\ldots,\mv{v}_p^{\ensuremath{\top}}]^{\ensuremath{\top}},\hat{\mv{\xi}},\hat{\mv{Q}} \}$
{\bf \mathsf{E}}ndProcedure
\end{algorithmic}
\end{algorithm}
\begin{algorithm}[h]
\footnotesize
\caption{Variance sampler}
\label{algV}
\begin{algorithmic}[1]
\Procedure{SampleV}{$ \mv{\Psi}, \mv{E}_1, \ldots, \mv{E}_p, \mv{h}$,typeG}
\If{typeG=1}
\State Sample $v \sim \pi(v) \prod_{i=1}^m \prod_{k=1}^p \proper{N}(E_{ik}; h_{ik}(v - 1) \mu_k),h_{ik} v)$
\ffor{$k=1,\dots,p$} $\mv{v_k} \gets \mv{h}_k v;$ {\bf \mathsf{E}}ndffor
{\bf \mathsf{E}}lsIf{typeG=2}
\For{$k=1,\dots,p$}
\State Sample $v_k \sim \pi(v) \prod_{i=1}^m \proper{N}(E_{ik}; h_{ik}(v_k - 1) \mu_k),h_{ik} v_k)$
\State $\mv{v_k} \gets \mv{h}_k v_k$
{\bf \mathsf{E}}ndFor
{\bf \mathsf{E}}lsIf{typeG=3}
\ffor{$i=1,\dots,m$}
Sample $v_i \sim \pi(v_{i}) \prod_{k=1}^p \proper{N}(E_{ik}; (v_i - h_{ik}) \mu_k, v_i)$
{\bf \mathsf{E}}ndffor
\ffor{$k=1,\dots,p$} $\mv{v}_k \gets \mv{v};$ {\bf \mathsf{E}}ndffor
{\bf \mathsf{E}}lsIf{typeG=4}
\For{$k=1,\dots,p$}
\ffor{$i=1,\dots,m$} Sample $v_{ik} \sim \pi(v_{ik})\proper{N}(E_{ik}; (v_i - h_{ik}) \mu_k, v_i) $ {\bf \mathsf{E}}ndffor
{\bf \mathsf{E}}ndFor
{\bf \mathsf{E}}ndIf
\State \Return $\{[\mv{v}_1^{\ensuremath{\top}},\ldots,\mv{v}_p^{\ensuremath{\top}}]^{\ensuremath{\top}} \}$
{\bf \mathsf{E}}ndProcedure
\end{algorithmic}
\end{algorithm}
\section{Parameter estimates for the applications}\label{seq:parameter_estimates}
The parameter estimates for the two covariance-based models in the first application are shown in Table~\ref{tab:cov}, and the parameter estimates for the SPDE models are shown in Table~\ref{tab1}. The main reason for the differences between our parameter estimates and those by \cite{gneiting2012matern} and \cite{apanasovich2012valid} is that they assumed $\mv{\beta} = \mv{0}$ whereas we estimate this parameter jointly with the other parameters. The reason for doing this is that the comparison with the type G models otherwise could be considered to be unfair, since the type G models allow for skewness that could capture some of the effects that cause the non-zero estimates of the means.
\begin{table}
\centering
\begin{tabular}{lccccccccccccc}
\toprule
Model & $\beta_1$ & $\beta_2$ & $\sigma_1$ & $\sigma_2$ & $\kappa_1$ & $\kappa_2$ & $\nu_1$ & $\nu_2$ & $\rho$ & $\sigma_{1e}$ & $\sigma_{2e}$ \\
\midrule
Independent & 136 & -0.53 & 218 & 2.64 & 5.54 & 0.89 & 20 & 0.58 & - & 71.8 & 0.00 \\
Parsimonious & 150 & -0.48 & 216 & 2.56 & 1.03 & 1.03 & 1.36 & 0.60 & -0.46 & 68.5 & 0.00 \\
\bottomrule
\end{tabular}
\caption{\label{tab:cov} Parameter estimates for the covariance-based models. For the independent model, the value of $\nu_P$ was limited to the interval $0 \leq \nu_p \leq 20$ for numerical stability. }
\end{table}
\begin{table}[t]
\centering
\begin{tabular}{lcccccccccccccc}
\toprule
Noise & $\beta_1$ & $\beta_2$ & $\sigma_1$ & $\sigma_2$ & $\kappa_1$ & $\kappa_2$ & $\rho$ & $\sigma_{1e}$ & $\sigma_{2e}$ & $\theta$ & $\mu_1$ & $\eta_1$\\
\midrule
GG & $154$ & $-0.55$ & $211$ & $2.56$ & $0.74$ & $1.11$ & $(0)$ & $61.4$ & $0.58$ & $-$ & $-$ & $-$\\
GG & $149$ & $-0.52$ & $202$ & $2.48$ & $0.82$ & $1.26$ & $-0.52$ & $60.5$ & $0.52$ & $-$ & $-$ & $-$\\
NG & $148$ & $-0.48$ & $222$ & $2.74$ & $0.72$ & $1.12$ & $(0)$ & $45.4$ & $0.75$ & $(0)$ & $-0.014$ & $0.21$\\
NG & $140$ & $-0.42$ & $212$ & $2.73$ & $0.74$ & $1.19$ & $-0.42$ & $45.3$ & $0.74$ & $(0)$ & $-0.053$ & $0.21$\\
NG & $147$ & $-0.59$ & $220$ & $2.87$ & $0.77$ & $1.18$ & $-0.42$ & $42.3$ & $0.72$ & $-0.89$ & $-0.065$ & $0.21$\\
\bottomrule
\end{tabular}
\caption{\label{tab1} Parameter estimates for the SPDE models. Dashes and parentheses respectively indicates that the parameters are not present and not estimated. GG denotes a Gaussian model whereas NG denotes that NIG noise is used for pressure and Gaussian noise for temperature. }
\end{table}
The parameter estimates for the SPDE-based models for the Argo data are shown in Table~\ref{tab:argospde}, whereas Table~\ref{tab:argocov} shows the parameter estimates for the covariance-based models.
\begin{table}[t]
\centering
\resizebox{\linewidth}{!}{
\begin{tabular}{lcccccccccccccc}
\toprule
Noise & $\beta_1$ & $\beta_2$ & $\tau_1$ & $\tau_2$ & $\kappa_1$ & $\kappa_2$ & $\rho$ & $\sigma_{1e}$ & $\sigma_{2e}$ & $\theta$ &
$\mu_1$ & $\mu_2$ & $\eta_1$ & $\eta_2$ \\
\midrule
GG & $0.00$ & $0.01$ & $0.89$ & $5.02$ & $1.14$ & $1.15$ & $(0)$ & $0.39$ & $0.03$ & $-$ & $-$ & $-$ & $-$ & $-$\\
GG & $-0.01$ & $0.01$ & $0.39$ & $2.23$ & $1.18$ & $1.09$ & $0.97$ & $0.35$ & $0.02$ & $-$ & $-$ & $-$ & $-$ & $-$\\
NN & $0.01$ & $0.05$ & $0.74$ & $5.56$ & $1.88$ & $1.39$ & $(0)$ & $0.04$ & $0.01$ & $(0)$ & $-0.03$ & $-0.11$ & $0.09$ & $0.73$\\
NN & $0.06$ & $0.04$ & $0.37$ & $2.99$ & $1.55$ & $0.96$ & $1.31$ & $0.04$ & $0.03$ & $0.04$ &$-0.02$ & $-3.24$ & $0.27$ & $6.61$\\
\bottomrule
\end{tabular}}
\caption{\label{tab:argospde} Parameter estimates for the SPDE models for the Argo data. Dashes and parentheses respectively indicates that the parameters are not present and not estimated. GG denotes a Gaussian model whereas NN denotes a NIG model. }
\end{table}
\begin{table}[t]
\centering
\begin{tabular}{lccccccccccc}
\toprule
Model & $\beta_1$ & $\beta_2$ & $\sigma_1$ & $\sigma_2$ & $\kappa_1$ & $\kappa_2$ & $\nu_1$ & $\nu_2$ & $\rho$ & $\sigma_{1e}$ & $\sigma_{2e}$ \\
\midrule
Independent & $0.00$ & $0.00$ & $0.72$ & $0.13$ & $1.32$ & $1.49$ & $1.01$ & $1.08$ & - & $0.41$ & $0.03$\\
Parsimonious & $-0.01$ & $0.00$ & $0.80$ & $0.14$ & $0.63$ & $0.63$ & $0.32$ & $0.51$ & $0.65$ & $0.24$ & $0.00$\\
\bottomrule
\end{tabular}
\caption{\label{tab:argocov} Parameter estimates for the covariance-based models for the Argo data. }
\end{table}
\section{Proofs}\label{sec:proofs}
Most of the proofs are based on that the fractional operator
$(\kappa^2-\Delta)^{\alpha/2}$ on $\mathbb{R}^d$ is defined through its Fourier transform \citep[see][]{lindgren10},
$(\mathcal{F}((\kappa^2-\Delta)^{\alpha/2}vphi)(\mv{k}) = (\kappa^2+\|\mv{k}\|)^{\alpha/2}(\mathcal{F}(vphi))(\mv{k})$.
The operator is well-defined for example if $vphi$ is a tempered distribution. This is important for the definition of the SPDE in \eqref{eq:model} since the right-hand side is white noise, which does not have pointwise meaning. Thus, the equation \eqref{eq:model} is understood in the weak sense, $(\kappa^2-\Delta)^{\alpha/2}X(vphi)=\noise(vphi)$, where $vphi$ is a function in an appropriate space of test functions, and $\noise(vphi) = \intvphi(\mv{s})\mathcal{M}(\ensuremath{\,\mathrm{d}}\mv{s})$. The kernel of the operator $\mathcal{K} = (\kappa^2-\Delta)^{\frac{\alpha}{2}}$ is non-empty for $\alpha\geq 2$ and there is therefore an implicit assumption on boundary conditions \citep[see][]{lindgren10}.
\begin{proof}[Proof of Proposition \ref{thm1}]
Due to the mutual independence of the noise processes, the power spectrum of driving noise is $\mv{S}_{\mathcal{M}} = (2\pi)^{-d}\mv{I}$. Let
$$
\mv{\mathcal{H}}(\mv{k}) = \mathcal{F}(\mv{\mathcal{K}})(\mv{k}) = \mv{D} \mathcal{F}(\diag(\mathcal{L}_1,\ldots, \mathcal{L}_p)) = \mv{D} \mv{\mathcal{H}}_D(\mv{k}),
$$
where $\mv{\mathcal{H}}_D(\mv{k})$ is a diagonal matrix with elements $\mv{\mathcal{H}}_D(\mv{k})_{ii} = \mathcal{F}(\mathcal{L}_i) = (\kappa_i^2 + \|\mv{k}\|)^{\alpha_i/2}$.
The power spectrum of $\mv{x}$ can then be written as
\begin{align}
\mv{S}_{\mv{x}}(\mv{k}) &=
(2\pi)^{-d}\mathcal{H}_D(\mv{k})^{-1}\mv{Di}\mv{Di}^{T}\mathcal{H}_D(\mv{k})^{-1}. \label{eq:Xspectrum}
\end{align}
Evaluating a single element of $\mv{S}_{\mv{x}}(\mv{k})$ gives
$$
(\mv{S}_{\mv{x}}(\mv{k}))_{ij} = \frac{\sum_{k=1}^p Di_{ik}Di_{jk}}{(2\pi)^d}\frac1{(\kappa_i^2 + \|\mv{k}\|)^{\alpha_i/2}(\kappa_j^2 + \|\mv{k}\|)^{\alpha_j/2}}.
$$
It is well-known that \citep{lindgren10}
$$
\mathcal{F}^{-1}\left(\frac1{(2\pi)^d}\frac{1}{ (\kappa^2 + \|\mv{k}\|)^{\alpha}}\right)(\mv{h}) = \frac{\Gamma(\nu)}{(4\pi)^{d/2}\Gamma(\alpha)\kappa^{2\nu}}\materncorr{\mv{h}}{\kappa_i}{\nu_i}
$$
which together with the expression for $(\mv{S}_{\mv{x}}(\mv{k}))_{ii}$ completes the proof.
\end{proof}
\begin{proof}[Proof of Proposition \ref{thm2}]
By the representation of the multivariate Mat\'ern-SPDE in Remark \ref{cor2}, we have that the covariance function of $\mv{x}$ depends on $\mv{D}$ only through the expression $\mv{Di}\mv{Di}^{\ensuremath{\top}} = (\mv{D}^{\ensuremath{\top}}\mv{D})^{-1}$. It is therefore clear that $\mv{D}$ and $\hat{\mv{D}}$ will generate the same covariance structure if and only if $\mv{D}^{\ensuremath{\top}}\mv{D} = \hat{\mv{D}}^{\ensuremath{\top}}\hat{\mv{D}}$.
If we assume $\mv{D} = \mv{Q}\hat{\mv{D}}$, then
$
\mv{D}^{\ensuremath{\top}}\mv{D} = \hat{\mv{D}}^{\ensuremath{\top}}\mv{Q}^{\ensuremath{\top}}\mv{Q}\hat{\mv{D}} = \hat{\mv{D}}^{\ensuremath{\top}}\hat{\mv{D}},
$
since $\mv{Q}$ is orthogonal, and the models therefore have the same covariance structure.
Conversely, assume that $\mv{D}$ and $\hat{\mv{D}}$ generate the same covariance structure. We then have that
$\mv{D}^{\ensuremath{\top}}\mv{D} = \hat{\mv{D}}^{\ensuremath{\top}}\hat{\mv{D}}$. Since $\hat{\mv{D}}$ is invertible, we can define $\mv{Q} = \mv{D}\hat{\mv{D}}^{-1}$ which is orthogonal, since $\mv{Q}^{\ensuremath{\top}}\mv{Q} = (\mv{D}\hat{\mv{D}}^{-1})^{\ensuremath{\top}}\mv{D}\hat{\mv{D}}^{-1} = \mv{I}$, and satisfies $\mv{Q}\hat{\mv{D}} = \mv{D}\hat{\mv{D}}^{-1}\hat{\mv{D}} = \mv{D}$.
Finally, for any multivariate Mat\'ern-SPDE, the matrix $\mv{D}^{\ensuremath{\top}}\mv{D} $ is by definition symmetric and positive definite. We can therefore define a Mat\'ern-SPDE model with triangular dependence matrix $\hat{\mv{D}} = \chol(\mv{D}^{\ensuremath{\top}}\mv{D})$. Because of the properties of the Cholesky factor, $\hat{\mv{D}}$ is the unique upper-triangular matrix with positive diagonal elements satisfying $\hat{\mv{D}}^{\ensuremath{\top}}\hat{\mv{D}} = \mv{D}^{\ensuremath{\top}}\mv{D}$.
\end{proof}
\begin{proof}[Proof of Proposition \ref{thm3}]
We only have to show that $\Cov(x_i(\mv{s}),x_j(\mv{t}))= 0$ since the variables are dependent by construction. Since $\mv{\rho} = 0,$ $x(\mv{s})$ is the solution to
$
\mv{Q}(\mv{\theta})\diag(c_1\mathcal{L}_1,\cdots, c_p\mathcal{L}_p)\mv{x}(\mv{s}) = \mv{\mathcal{M}}_{3},
$
or equivalently
$
\diag(c_1\mathcal{L}_1,\cdots, c_p\mathcal{L}_p)\mv{x}(\mv{s}) = \mv{\mathcal{M}}_{Q},
$
where
$$
\mv{\mathcal{M}}_{Q} = \sum_{k=1}^{\infty}g(e_k)^{\frac{1}{2}}\mathbb{I}(\mv{s} \geq \mv{s}_k) \mv{Q}(\mv{\theta})^{-1}\mv{Z}_k = \sum_{k=1}^{\infty}g(e_k)^{\frac{1}{2}}\mathbb{I}(\mv{s} \geq \mv{s}_k) \mv{Z}_{Q,k}.
$$
Since $\mv{Q}(\mv{\theta})^{-1}\mv{Q}(\mv{\theta})^{-\ensuremath{\top}} = \mv{I}$ it follows that $\mv{Z}_{Q,k} \sim N(\mv{0},\mv{I})$.
From \citep{bolin11} we have
$
x_i(\mv{s}) = \int G_i(\mv{s},\mv{v}) \mv{\mathcal{M}}_{Q,i}(d\mv{v}),
$
for $i=1,\ldots,p$. Here $G_i(\mv{s},\mv{v})$ is the Green function of $c_i \mathcal{L}_i$, and $\mv{\mathcal{M}}_{Q,i}(\mv{s})$ is the $i$th value of the vector $\mv{\mathcal{M}}_{Q}(\mv{s})$. Since the elements in the vector $\mv{\mathcal{M}}_{Q}(\mv{s})$ are uncorrelated it follows that the elements of $\mv{x}(\mv{s}) = [x_1(\mv{s}), \ldots, x_d(\mv{s})]^{\ensuremath{\top}}$ are uncorrelated.
\end{proof}
The proof of the Theorem \ref{thm:krig} builds on the following lemma, which shows that the posterior distribution of $v$ contracts to a point.
\begin{lemma}
\label{lem:contraction}
Let Assumption \ref{ass1} hold and assume that $\pi(v)$ has mean one, is bounded and absolutely continuous with respect to the Lebesgue measure. Then $\pi(v|X_{1:n}) \overset{p}{\to}\delta_{K_0}(v)$ as $n \rightarrow\infty$.
\end{lemma}
\begin{proof}
In the following, $C$ is a generic positive constant that changes from line to line.
Let $k_0 := k_0(x)$ be the realisation of $K_0$ determined by the realisation of $x$ which generates the data. Let $B_{k,n} = (b^l_{k,n},b^u_{k,n})=(k_0- n^{-1/2+k\epsilon}, k_0 + n^{-1/2+k\epsilon})$ where $0<\epsilon<1/8$. To prove the lemma it suffices to show that
$$
\frac{\int_{B_{2,n}^c} \pi(v|\mv{x}_{1:n}) dv }{\int_{B_{2,n}} \pi(v|\mv{x}_{1:n}) dv} \rightarrow 0\quad \mbox{ as } n\rightarrow \infty.
$$
By the mean value theorem we have
$$
\frac{\int_{B_{2,n}^c} \pi(v|\mv{x}_{1:n}) dv }{\int_{B_{2,n}} \pi(v|\mv{x}_{1:n}) dv} \leq \frac{\int_{B_{2,n}^c} \pi(v|\mv{x}_{1:n}) dv }{\int_{B_{1,n}} \pi(v|\mv{x}_{1:n}) dv}
\leq \frac{\sqrt{n}\int_{B_{2,n}^c} \pi(v|\mv{x}_{1:n}) dv }{ \pi(v_d|\mv{x}_{1:n})},
$$
for some $v_d\in B_{1,n}$. By boundedness and absolute continuity of $\pi(v)$ (which implies that $\frac{\pi(v)}{\pi(v_d)}$ is bounded from above)
$$
\frac{\sqrt{n}\int_{B_{2,n}^c} \pi(v|\mv{x}_{1:n}) dv }{ \pi(v_d|\mv{x}_{1:n})} \leq C \sqrt{n} \int_{B^c_{2,n}} e^{-\frac{n}{2}\left(\frac{c_n}{v} + \log(v) - \frac{c_n}{v_d} - \log(v_d) \right)} dv,
$$
where $c_n=\frac1{n}\mv{x}^{\ensuremath{\top}}_{1:n} \mv{C}_n^{-1} \mv{x}_{1:n}$. We will now show that the right-hand side goes to zero if we condition on the event $A_n = \{c_n\in B_{0.5,n}\}$. We first bound the integral as
\begin{align*}
\sqrt{n} \int_{B^c_{2,n}} &e^{-\frac{n}{2}\left(\frac{c_n}{v} + \log(v) - \frac{c_n}{v_d} - \log(v_d) \right)} dv \\
&\leq \sqrt{n} \int_{B^c_{2,n} \cap [0,n]} e^{-\frac{n}{2}\left(\frac{c_n}{v} + \log(v) - \frac{c_n}{v_d} - \log(v_d) \right)} dv
+ \sqrt{n} \int_{n}^\infty e^{-\frac{n}{2}\left(\frac{c_n}{v} + \log(v) - \frac{c_n}{v_d} - \log(v_d) \right)}
\\
&:= (I) + (II).
\end{align*}
To bound (II), let $k$ be a constant such that $\log(k) > \frac{c_n}{v_d} + \log(v_d)$ for all $n$ (this is possible since we are in $A_n$) then
$$
(II) \leq \sqrt{n} \int_{n}^\infty e^{-\frac{n}{2}\left(\log(v) - \log(k) \right)} dv =\frac{n^{-\frac{n-1}{2}}}{n/2-1} k^{n/2} \rightarrow 0 \quad \mbox{as $n\rightarrow \infty$.}
$$
To bound (I), note that $f(v) = \frac{c_n}{v} + \log(v) $ takes its minimum at $c_n$, and is increasing above and below $c_n$. Thus, for $i = \argmax_{j\in \{l,u\}} f(b^j_{2,n})$ we have $f(v) \geq f(b^i_{2,n})> f(b^i_{1,n}) > f(v_d) $, for all $v\in B_{2,n}^c$, and therefore
$$
(I) \leq C n^{3/2} e^{-\frac{n}{2}\left(f(b^i_{2,n})-f(b^i_{1,n})\right)}.
$$
Assume for simplicity that $i=u$ (the calculation for $i=l$ follows from similar arguments). We split the exponent into two parts $f(b^u_{2,n})-f(b^u_{1,n}) = (\frac{c_n}{b^u_{2,n}} - \frac{c_n}{b^u_{1,n}}) + (\log(b^u_{2,n}) - \log(b^u_{1,n}))$.
For the first part we have
\begin{align*}
\frac{c_n}{b^u_{2,n}} - \frac{c_n}{b^u_{1,n}} &= \frac{c_n n^{-1/2}(n^{\epsilon}-n^{2\epsilon})}{(k_0 + n^{-1/2+\epsilon})(k_0 + n^{-1/2+2\epsilon})} \\
&\geq \frac{c_n n^{-1/2}(n^{\epsilon}-n^{2\epsilon})}{(k_0 + n^{-1/2+2\epsilon})^2} \geq \frac{c_n n^{-1/2}(n^{\epsilon}-n^{2\epsilon})}{k_0^2 + n^{-1+4\epsilon}},
\end{align*}
while for the second part
\begin{align*}
\log(b^u_{2,n}) - \log(b^u_{1,n}) &= \log(1 + \frac{n^{-1/2+2\epsilon}}{k_0}) - \log(1 + \frac{n^{-1/2+\epsilon}}{k_0}) \\
&= \frac{n^{-1/2+2\epsilon}}{k_0} + \mathcal{O}(n^{-1+4\epsilon}) - \frac{n^{-1/2+\epsilon}}{k_0} + \frac{n^{-1+2\epsilon}}{k_0^2} + \mathcal{O}(n^{-3/2+3\epsilon})\\
&= \frac{n^{-1/2}}{k_0}(n^{2\epsilon} - n^{\epsilon}) + \frac{n^{-1+2\epsilon}}{k_0^2} + \mathcal{O}(n^{-1+4\epsilon}) .
\end{align*}
Hence
$C \sqrt{n} e^{-\frac{n}{2}\left(f(b^i_{2,n})-f(b^i_{1,n})\right)} \leq C \sqrt{n} e^{-\frac{n^{4\epsilon}}{4k_0}} \to 0$ as $n\rightarrow \infty$.
Finally, by Assumption~\ref{ass1} and the Chebyshev inequality, $P(A_n) \rightarrow 1$, which completes the proof.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:krig}]
Let $k_0 := k_0(x)$ and $k_1 := k_1(x)$ be the realisations of $K_0$ and $K_1$ respectively, determined by the realisation of $x$ which generates the data.
Take $\epsilon>0$ and define $A_n=\{\mv{c}_{0,1:n} \mv{C}_n^{-1} \mv{x}_{1:n}\in [k_1 - \frac{\epsilon}{\sqrt{n}},k_1 + \frac{\epsilon}{\sqrt{n}}]\}$. Conditioning on the event $A_n$ and using the triangle inequality yields
\begin{align*}
| \pi_{G_1,x_0}(\cdot|\mv{x}_{1:n}) - N(\cdot; k_1, k_0 k_2)| \leq& \left|\pi_{G_1,x_0}(\cdot|\mv{x}_{1:n}) - \int N(\cdot; k_1, vk_2)\pi(v) dv \right| \\
&+\left|\int N(\cdot; k_1, vk_2)\pi(v) dv- N(\cdot; k_1, k_0 k_2) \right|.
\end{align*}
By equation \eqref{eq:kriging_mean} and the continuous mapping theorem, the first term on the right-hand side converges to zero since we have conditioned on the event $A_n$, and the second term converges to zero by Lemma \ref{lem:contraction}. Under Assumption \ref{ass1} and using Chebyshev inequality it follows that $P(A_n) \rightarrow 1$, which completes the proof.
\end{proof}
\begin{proof}[Proof of Lemma \ref{eq:Lemma}]
To prove the result we need to verify that Assumption~\ref{ass1} is satisfied.
We first establish some properties about $x(\mv{s})$ which we will use to verify the assumptions. Note that the distribution of $x(\mv{s}_1),\ldots,x(\mv{s}_n)|v$ is $N(0,v\mv{C}_n)$ where $\mv{C}_n$ is a positive definite matrix for all $n$. Let $\mv{C}^{1/2}_n$ denote the Cholesky factor of $\mv{C}_n$, and let $\mv{z}_{1:n} =\frac{1}{\sqrt{v}}\mv{C}_n^{-1/2} \mv{x}_{1:n}$ where by assumption $z_i$ are i.i.d $N(0,1)$.
To establish \eqref{eq:mean_est} and \eqref{eq:var_est} note that
$$\mv{x}_{1:n}^{\ensuremath{\top}}\mv{C}_n^{-1}\mv{x}_{1:n}^{\ensuremath{\top}} = \mv{z}_{1:n}^{\ensuremath{\top}} v^{1/2} \mv{C}_n^{1/2} \mv{C}_n^{-1} v^{1/2}\mv{C}_n^{1/2} \mv{z}_{1:n}^{\ensuremath{\top}} = v \sum_{i=1}^n z_{i}^2.
$$
Hence, by the law of large numbers, \eqref{eq:mean_est} and \eqref{eq:var_est} are satisfied with $K_0=v$.
For \eqref{eq:kriging_mean} and \eqref{eq:kriging_var}, note that $\sigma_n := c_0 - \mv{c}^{T}_{0,1:n}\mv{C}_n^{-1}\mv{c}_{0,1:n}$ is the variance of the Kriging predictor (the variance of the best linear predictor), thus $\{\sigma_n\}$ is a decreasing sequence in $[0, C_2]$. Therefore $\{\sigma_n\}$ must converge to a point, implying equation \eqref{eq:kriging_var}. Finally we need to establish that
$$
\mv{c}_{0,1:n} \mv{C}_n^{-1} \mv{x}_{1:n} = \sqrt{v} \mv{c}_{0,1:n}\mv{C}_n^{-1/2} \mv{z}_{1:n} \overset{p}{\to} K_1.
$$
Since $\mv{C}^{1/2}_n$ is the Cholesky factor of $\mv{C}_n$, we have that $(\mv{C}_n^{-1/2})_{1:n-1,1:n-1}=\mv{C}_{n-1}^{-1/2}$ \cite[see for instance][Section 2.2.4]{pourahmadi2011covariance}. Thus, the limit $\tilde{c} = \lim_{n\rightarrow \infty} \mv{c}_{0,1:n}\mv{C}_n^{-1/2}$ exists. By \eqref{eq:kriging_var} it follows that $\tilde{c}\in l^2$ and hence that $\sum_{i=n}^\infty \tilde{c}^2_{i} \rightarrow 0$ as $n \to \infty$. Thus $\proper{V}[\mv{c}_{0,1:n} \mv{C}_n^{-1} \mv{x}_{1:n} - K_1]= \proper{V}[\sqrt{v}\sum_{i=n}^\infty \tilde{c}_i z_{i}] = \proper{E}[v]\sum_{i=n}^\infty \tilde{c}^2_{i}\to 0$ as $n\rightarrow \infty$.
\end{proof}
\begin{proof}[Proof of Proposition \ref{thm:charf}]
To derive the CF, $\phi_{x(\mv{s})}(\mv{u})$, of $\mv{x}(s)$, note that Remark~\ref{cor2} shows that the SPDE in \eqref{eq:general}, for $p>1$, can be formulated as
$$
\mv{x}(\mv{s})
= \sum_{k=1}^p
\diag(\mathcal{L}^{-1}_1\mathcal{N}_k(\mv{s}),\ldots, \mathcal{L}^{-1}_p \mathcal{N}_k(\mv{s})
)\begin{bmatrix}
Di_{1k}, \\
\vdots\\
Di_{pk}
\end{bmatrix} = \sum_{k=1}^p \begin{bmatrix} R_{1k}x^1_k(\mv{s}) \\
\vdots \\
R_{pk}x^p_k(\mv{s})
\end{bmatrix},
$$
where $x_k^r(\mv{s})=\mathcal{L}^{-1}_r \mathcal{N}_k(\mv{s})$.
The right-hand side is a sum of independent random variables, and thus $\phi_{x(\mv{s})}(\mv{u}) = \prod_{i=1}^p\phi_{k}(\mv{u})$ where $\phi_{k}(\mv{u})$ is the CF of $\begin{bmatrix} R_{1k}x^1_k(\mv{s}) &
\ldots &
R_{pk}x^p_k(\mv{s})
\end{bmatrix}^{\ensuremath{\top}}$.
In order to derive $\phi_{k}(\mv{u})$ we first derive the CF for $x_k^r(\mv{s})$. From \citep{bolin11} it follows that $x_k^r(\mv{s}) = \int G_r(\mv{s},\mv{t}) \mathcal{N}_k(d\mv{t})$, where the kernel is given by the Green's function of the operator $\mathcal{L}_r$:
$$
G_r(\mv{s},\mv{t}) = \frac{\Gamma\left(\frac{\alpha_r-d}{2}\right)}{(4\pi)^{d/2} \Gamma(\frac{\alpha_r}{2}) \kappa_r^{\alpha_1 - d}} \materncorr{\|\mv{s}-\mv{t}\|}{\kappa_r}{\frac{\alpha_r-d}{2}}.
$$
Using that the CF of the univariate NIG noise $\dot{\mathcal{N}}_k(A)$ is
\begin{align}
\label{eq:charfuncM}
\phi_{\dot{\mathcal{N}}_k(A)}(u) = \exp \left( i\gamma m(A) u_k + m(A) \sqrt{\eta_k} \left(\sqrt{\eta_k}- \sqrt{\eta_k + u^2 - 2i\mu_k u} \right)\right),
\end{align}
and Proposition 2.6 in \citep{rajput1989spectral} it follows that the CF of $x_k^r(\mv{s})$ is
\begin{align*}
\phi_{x_k^r(\mv{s})}(u) = \exp \left( - i\gamma_k u \int G_r(\mv{s},\mv{t}) d\mv{t} + \sqrt{\eta_k} \int (\eta_k - \sqrt{\eta_k + \mu_k^2- (\mu_k + iG_r(\mv{s},\mv{t})u)^2} d\mv{t}\right).
\end{align*}
To complete the proof we need derive $\phi_{k}(\mv{u})$. Note that the random variable $Y(\mv{s}) = \sum_{r=1}^p u_{r}R_{rk}x^r_k(\mv{s})$ has CF $\phi_{Y(\mv{s})}(h) = \phi_{k}(\mv{u}h)$ and since $Y(\mv{s}) = \int \sum_{r=1}^p R_{rk}G_r(\mv{s},\mv{t})u_r \mathcal{N}_k(d\mv{t})$ it follows that
\begin{align*}
\phi_{k}(\mv{u}) = \phi_{Y(\mv{s})}(1) = \exp \left( \right.& - i\gamma_k \int \sum_{r=1}^p R_{rk}G_r(\mv{s},\mv{t})u_r d\mv{t} + \\
&\left. \sqrt{\eta_k} \int \eta_k - \sqrt{\eta_k + \mu_k^2- (\mu_k + i \sum_{r=1}^pR_{rk}G_r(\mv{s},\mv{t})u_r )^2} d\mv{t}\right).
\end{align*}
\end{proof}
\begin{proof}[Proof of Proposition \ref{crpsthm}]
If $Z \sim \proper{N}(\mu,\sigma^2)$, then $|Z|$ has a folded normal distribution with mean $M(\mu,\sigma^2)$ defined in \eqref{eq:M}. Let $X_1$ and $X_2$ be two independent variance mixture variables with CDF $F$ and let $V_1$ and $V_2$ be their corresponding mixing variables. Introduce $\tilde{X}_1 = X_1 -y$ and $\tilde{X}_2 = X_1 - X_2$ and note that there exist variables $\mu_1,\mu_2,\sigma_1^2,$ and $\sigma_2^2$, depending on $V_1$ and $V_2$, such that $\tilde{X}_1 |V_1 \sim \proper{N}(\mu_1 - y, \sigma_1^2)$ and $\tilde{X}_2 | V_1,V_2 \sim \proper{N}(\mu_1 - \mu_2, \sigma_1^2 + \sigma_2^2)$.
By the law of total expectation
\begin{align}\label{eq:crps22}
\mbox{CRPS}(F,y) &= \proper{E}_{V_1}(\proper{E}(|\tilde{X}_1 - y|\mid V_1)) - \frac1{2}\proper{E}_{V_1}(\proper{E}_{V_2}(\proper{E}(|\tilde{X}_1 - \tilde{X}_2| \mid V_1, V_2))) \notag\\
&= \proper{E}_{V_1}(M(\mu_1 - y,\sigma_1^2)) - \frac1{2}\proper{E}_{V_1}(\proper{E}_{V_2}(M(\mu_1 - \mu_2,\sigma_1^2+\sigma_2^2))).
\end{align}
We have that $\proper{E}(\mbox{CRPS}_N^{RB}(F,y)) = \mbox{CRPS}(F,y)$ since $\mbox{CRPS}_N^{RB}(F,y)$ is a standard MC estimator of \eqref{eq:crps22}. Furthermore, $\mbox{CRPS}_N^{RB}(F,y) = \proper{E}(\mbox{CRPS}_N(F,y)|\mv{V}_1,\mv{V}_2)$ where $\mv{V}_j = (V_j^{(1)},\ldots V_j^{(N)})$ for $j=1,2$. Thus, $\mbox{CRPS}_N^{RB}(F,y)$ is a Rao-Blackwell estimator and by the Law of total variation $\proper{V}(\mbox{CRPS}_N^{RB}(F,y)) \leq \proper{V}(\mbox{CRPS}_N(F,y))$.
\end{proof}
\end{appendix}
\section*{Acknowledgment}
This work has been supported by the Swedish Research Council under grant
No. 2016-04187 and the Knut and Alice Wallenberg Foundation (KAW 20012.0067). The authors thank Holger Rootz\'en, the editors, and the anonymous reviewers for valuable comments on the manuscript. We also thank Mikael Kuusela for helping with the Argo data.
\end{document} |
\betagin{document}
\partialagenumbering{arabic}
\title{\frac{\partial }{\partial x}uge \bf Self-adjoint sub-classes of third and fourth-order evolution equations}
\author{\rhom \lambdarge Igor Leite Freire \\
\\
\it Centro de Matemática, Computação e Cognição\\ \it Universidade Federal do ABC - UFABC\\ \it
Rua Catequese, $242$,
Jardim,
$09090-400$\\\it Santo André, SP - Brasil\\
\rhom E-mail: [email protected]\\}
\date{\ }
\maketitle
\betagin{abstract}
In this work a class of self-adjoint quasilinear third-order evolution equations is determined. Some conservation laws of them are established and a generalization on a self-adjoint class of fourth-order evolution equations is presented.
\end{abstract}
\vskip 1cm
\betagin{center}
{2000 AMS Mathematics Classification numbers:
\\
76M60, 58J70, 35A30, 70G65
\\
Key words: Evolution equations, self-adjoint equation, conservation laws for evolution equations}
\end{center}
\partialagenumbering{arabic}
\mathbb{S}^{2}\times\Rection{Introduction}
In this paper we consider the problem on self-adjointness condition of equation
\betagin{equation}\lambdabel{1.1.1}
u_{t}=r(u)u_{xxx}+p(u)u_{xx}+q(u)u_{x}^{2}+a(u)u_{x}+b(u),
\end{equation}
where $r,\,p,\,q,\,a,\,b:\mathbb{R}\rhoightarrow\mathbb{R}$ are arbitrary smooth functions.
Equation (\rhoef{1.1.1}) includes important evolution equations employed in mathematical physics and in mathematical biology, for instance, inviscid Burgers equation, Burgers equation, potential Burgers equation, Fisher equation, Korteweg--de Vries (KdV) equation, Gardner equation and so on, see \cite{cher, gun, igor1}. It can be used to describe shallow watter waves, collisionless-plasma magnetohydrodynamics waves and ion acoustic waves, among other physical or biological phenomena, see also \cite{ott, zabu}.
Similar work has been performed by Bruzón, Gandarias and Ibragimov \cite{ib1} regarding equation
\betagin{equation}\lambdabel{1.1.3}
u_{t}+f(u)u_{xxxx}+g(u)u_{x}u_{xxx}+h(u)u_{xx}^{2}+d(u)u_{x}^{2}u_{xx}-p(u)u_{xx}-q(u)u_{x}^{2}=0.
\end{equation}
However, in (\rhoef{1.1.3}) source terms and nonlinearities type $a(u)u_{x}$ and $r(u)u_{xxx}$ were not taken into account. So we shall complement the results previously obtained by them including these terms.
Ibragimov \cite{ib2} has recently established a new conservation theorem for equations without Lagrangians. If (\rhoef{1.1.1}) is self-adjoint it is possible to construct conservation laws $D_{t}C^{0}+D_{x}C^{1}=0$ for it, where the components $C^{0}$ and $C^{1}$ depend on $t,x,u$ and its derivatives.
The purpose of this paper is to determine the self-adjoint equations type (\rhoef{1.1.1}) and, by using the recent result \cite{ib2}, establish some nontrivial conservation laws for some of these equations. The results on self-adjointness condition of equation (\rhoef{1.1.3}) obtained in \cite{ib1} is also generalized by including dispersive, convective and source terms.
The paper is organized as the follows: in the section \rhoef{review} we revisit some results regarding Lie point symmetries and conservation laws for differential equations. Section \rhoef{self} is devoted to find the self-adjoint equations type (\rhoef{1.1.1}). We comment some results presented in \cite{ib1} in section \rhoef{comment}.
\mathbb{S}^{2}\times\Rection{Preliminaries}\lambdabel{review}
This section contains a brief discussion on the space of differential functions ${\cal A}$, Lie-Bäcklund operators, self-adjoint equations and conservation laws for differential equations. For more details, see \cite{ib0, ib1, ib2, ib3}. In the following the summation over repeated indices is understood.
Let $x=(x^{1},\cdots,x^{n})$ be $n$ independent variables and $u=(u^{1},\cdots,u^{m})$ be $m$ dependent variables with partial derivatives $u^{\alpha}_{i}=\frac{\partial u^{\alpha}}{\partial x^{i}},\,u^{\alpha}_{ij}=\frac{\partial^{2}u^{\alpha}}{\partial x^{i}\partial x^{j}}$, etc. The total differentiation operators are given by
$$
D_{i}=\frac{\partial}{ \partial x^{i}}+u^{\alpha}_{i}\frac{\partial}{\partial u^{\alpha}}+u^{\alpha}_{ij}\frac{\partial}{\partial u^{\alpha}_{j}}+\cdots,\,\,i=1,\cdots,n,\,\alpha=1,\cdots,m.
$$
Observe that $u^{\alpha}_{i}=D_{i}u^{\alpha},\,u^{\alpha}_{ij}=D_{i}D_{j}u^{\alpha}$, etc. The variables $u^{\alpha}$ with the sucessive derivatives $u^{\alpha}_{i_{1}\cdots i_{k}},\,k\in\mathbb{N}$, is known as the differential variables.
\betagin{definition}
A locally analytic function of a finite number of the variables $x,\,u$ and $u$ derivatives is called a differential function. The highest order of derivatives appearing in the differential function is called the order of this function. The vector space of all differential functions of finite order is denoted by ${\cal A}$.
\end{definition}
{\bf Example}: Let us consider the differential function
\betagin{equation}\lambdabel{2.1.1}
F=u_{t}-r(u)u_{xxx}-p(u)u_{xx}-q(u)u_{x}^{2}-a(u)u_{x}-b(u),
\end{equation}
where $p,\,q,\,r,\,a,\,b:\mathbb{R}\rhoightarrow\mathbb{R}$ are arbitrary smooth functions. Supposing $r(u)\neq0$, the order of $F$ is three.
\betagin{definition}
A Lie-Bäcklund operator is a formal sum
\betagin{equation}\lambdabel{2.1.2}
X=\xi^{i}\frac{\partial}{\partial x^{i}}+\eta^{\alpha}\frac{\partial}{\partial u^{\alpha}}+\eta^{\alpha}_{i}\frac{\partial}{\partial u^{\alpha}_{i}}+\eta^{\alpha}_{ij}\frac{\partial}{\partial u^{\alpha}_{ij}}+\cdots,
\end{equation}
where $\xi^{i},\,\eta^{\alpha}\in{\cal A},\, \eta^{\alpha}_{i}=D_{i}(\eta^{\alpha}-\xi^{j}u^{\alpha}_{j})+\xi^{j}u^{\alpha}_{ij},\,\eta^{\alpha}_{ij}=D_{i}D_{j}(\eta^{\alpha}-\xi^{k}u^{\alpha}_{k})+\xi^{k}u^{\alpha}_{kij}$, etc.
The Lie-Bäcklund operator is often written as
\betagin{equation}\lambdabel{2.1.3}
X=\xi^{i}\frac{\partial}{\partial x^{i}}+\eta^{\alpha}\frac{\partial}{\partial u^{\alpha}}
\end{equation}
understanding the prolonged form $(\rhoef{2.1.2})$. If $\xi^{i}=\xi^{i}(x,u)$ and $\eta=\eta(x,u)$ in $(\rhoef{2.1.3})$, then $X$ is a generator of Lie point symmetry group.
\end{definition}
{\bf Example:} The field
\betagin{equation}\lambdabel{4.1.1'}
X=t\frac{\partial}{\partial t}-u\frac{\partial }{\partial u}
\end{equation}
is a Lie point symmetry generator of inviscid Burgers equation
\betagin{equation}\lambdabel{4.1.1}
u_{t}=uu_{x}.
\end{equation}
The set of all Lie-Bäcklund operators endowed with the commutator
$$
[X,Y]=(X(\phita^{i})-Y(\xi^{i}))\frac{\partial}{\partial x^{i}}+(X(\omegaega^{\alpha})-Y(\eta^{\alpha}))\frac{\partial}{\partial u^{\alpha}}+\cdots,
$$
where $X$ is given by $(\rhoef{2.1.3})$ and
$$Y=\phita^{i}\frac{\partial}{\partial x^{i}}+\omegaega^{\alpha}\frac{\partial}{\partial u^{\alpha}},$$
is an infinite-dimensional Lie algebra.
\betagin{definition}
The Euler-Lagrange operator $\frac{\delta}{\delta u^{\alpha}}:{\cal A}\rhoightarrow{\cal A}$ is defined by the formal sum
\betagin{equation}\lambdabel{2.1.4}
\frac{\delta}{\delta u^{\alpha}}=\frac{\partial}{\partial u^{\alpha}}+\mathbb{S}^{2}\times\Rum_{j=1}^{\infty}(-1)^{j}D_{i_{1}}\cdots D_{i_{j}}\frac{\partial}{\partial u^{\alpha}_{i_{1}\cdots i_{j}}}.
\end{equation}
\end{definition}
\betagin{definition}
Let $F_{\alpha}\in{\cal A}$. We define the adjoint system of differential functions $F_{\alpha}^{\ast}$ to $F_{\alpha}$ by the expression
$$
F^{\ast}_{\alpha}=\frac{\delta}{\delta u^{\alpha}}(v^{\beta}F_{\beta}),
$$
where $v^{\beta}$ is a new dependent variable. We say that $F_{\alpha}^{\ast}$ is a self-adjoint system of differential functions to $F_{\alpha}$ if there exists $\partialhi\in{\cal A}$ such that
$$
\left.F^{\ast}_{\alpha}\rhoight|_{v=u}=\partialhi F_{\alpha}.
$$
\end{definition}
We observe that a system of differential equations can be viewed as $F_{\alpha}=0$, for some $F_{\alpha}\in{\cal A}$.
\betagin{definition}
An adjoint system of differential equations $F^{\ast}_{\alpha}=0$ to a system of differential equations $F_{\alpha}=0$ is given by
$$
F^{\ast}_{\alpha}=\frac{\delta}{\delta u^{\alpha}}(v^{\beta}F_{\beta})=0,
$$
where $v^{\beta}$ is a new dependent variables. We say that $F^{\ast}_{\alpha}=0$ is a self-adjoint equation to $F_{\alpha}=0$ if
\betagin{equation}\lambdabel{ast}
\left.F^{\ast}_{\alpha}\rhoight|_{v=u}=\partialhi F_{\alpha},
\end{equation}
for some differential function $\partialhi\in{\cal A}$. So $\left.F^{\ast}_{\alpha}\rhoight|_{v=u}=0$ if and only if $F_{\alpha}=0$.
\end{definition}
{\bf Example}: The adjoint differential function $F^{\ast}$ to (\rhoef{2.1.1}) is
$$F^{\ast}=\frac{\delta}{\delta u}(vF)=v[-p'u_{xx}-q'u_{x}^{2}-a'u_{x}-b']-D_{t}(v)-D_{x}[-v(2qu_{x}+a)]+D_{x}^{2}(-vp)-D_{x}^{3}(-vr).$$
Setting $v=u$ and after a tedious calculation we obtain
\betagin{equation}\lambdabel{adj}
\betagin{array}{lcl}
F^{\ast}&=&-u_{t}-ub'(u)+a(u)u_{x}+[uq'(u)-2p'(u)-up''(u)+2q(u)]u_{x}^{2}+(3r''+ur''')u_{x}^{3}\\
\\
&&+[2uq(u)-2up'(u)+2uq(u)]u_{xx}+(6r'+3ur'')u_{x}u_{xx}+ru_{xxx}.
\end{array}\end{equation}
If an equation possesses variational structure, it is well known that the Noether Theorem can be employed in order to establish conservation laws for the respective equation, {\it e.g.}, see \cite{yi1, yi2, igor2, naz}.
However, the Noether Theorem cannot be applied to evolution equations in order to obtain conservation laws, since this class of equations does not possess variational structure. Fortunately there are some other alternative methods to establish conservation laws for equations without Lagrangians, see \cite{naz}. One of them is a recent result \cite{ib2}, due to Ibragimov.
Let
\betagin{equation}\lambdabel{2.1.7}
X=\tau(t,x,u)\frac{\partial}{\partial t}+\xi(t,x,u)\frac{\partial}{\partial x}+\eta(t,x,u)\frac{\partial}{\partial u}
\end{equation}
be a Lie point symmetry generator of (\rhoef{1.1.1}) and
\betagin{equation}\lambdabel{2.1.8}
{\cal L}=v F,
\end{equation}
where $F$ is given by $(\rhoef{2.1.1})$. From the new conservation theorem \cite{ib2}, the conservation law for the system given by equation $(\rhoef{2.1.1})$ and by its adjoint equation $F^{\ast}=0$, where $F^{\ast}$ is given by $(\rhoef{adj})$,
is $Div(C)=D_{t}C^{0}D_{x}C^{1}=0$, where
\betagin{equation}\lambdabel{2.1.10}
\betagin{array}{lcl}
C^{0}&=&\displaystyle {\tau {\cal L}+W\,\frac{\partial {\cal L}}{\partial u_{t}}},\\
\\
C^{1}&=&\displaystyle {\xi {\cal L}+W\left[\frac{\partial {\cal L}}{\partial u_{x}}-D_{x}\frac{\partial {\cal L}}{\partial u_{xx}}+D_{x}^{2}\frac{\partial {\cal L}}{\partial u_{xxx}}\rhoight]+D_{x}(W)\left[\frac{\partial {\cal L}}{\partial u_{xx}}-D_{x}\frac{\partial {\cal L}}{\partial u_{xxx}}\rhoight]+D_{x}^{2}(W)\frac{\partial {\cal L}}{\partial u_{xxx}}}
\end{array}
\end{equation}
and $W=\eta-\tau u_{t}-\xi u_{x}$.
In particular, whenever $(\rhoef{2.1.1})$ is self-adjoint, substituting $v=u$ into $(\rhoef{2.1.10})$, $C=(C^{0},C^{1})$ provides a conserved vector for $(\rhoef{2.1.1})$.
\mathbb{S}^{2}\times\Rection{Self-adjoint equations type $(\rhoef{1.1.1})$}\lambdabel{self}
\mathbb{S}^{2}\times\Rubsection{The class of self-adjoint equations type (\rhoef{1.1.1})}
By applying the Euler-Lagrange operator (\rhoef{2.1.4}) to (\rhoef{2.1.8}), where $F$ is given by (\rhoef{2.1.1}) and equating to $0$, we obtain the adjoint equation to (\rhoef{1.1.1}), that is, $F^{\ast}$=0, where $F^{\ast}$ is given by (\rhoef{adj}).
Supposing that $F$ is self-adjoint, equation (\rhoef{ast}) holds, for some $\partialhi\in{\cal A}$. Comparing the coefficient of $u_{t}$, we obtain $\partialhi=-1$ and
\betagin{equation}\lambdabel{3.1.1}
\betagin{array}{lcl}
3r''+ur'''=0,\,\,\,\,
3ur''+6r'=0,\,\,\,\,
ur'''+3r''=0,\\ \\
-up'-p=-uq,\,\,\,\,
uq'-2p'-up''+q=0,\,\,
\,\,
ub'=-b.
\end{array}
\end{equation}
Solving the system (\rhoef{3.1.1}), we obtain
\betagin{equation}\lambdabel{3.1.1'}
r=a_{1}+\frac{a_{2}}{u},
\end{equation}
\betagin{equation}\lambdabel{3.1.2}
q=\frac{(up)'}{u}
\end{equation}
and
\betagin{equation}\lambdabel{3.1.3}
b=\frac{a_{3}}{u},
\end{equation}
where $a_{1},\,a_{2}$ and $a_{3}$ are arbitrary constants.
The following theorem is proved.
\betagin{theorem}\lambdabel{teo1}
Equation $(\rhoef{1.1.1})$ is self-adjoint if and only if it has the form
\betagin{equation}\lambdabel{3.2.1}
u_{t}=\left(a_{1}+\frac{a_{2}}{u}\rhoight)u_{xxx}+p(u)u_{xx}+\frac{(up)'}{u}u_{x}^{2}+a(u)u_{x}+\frac{a_{3}}{u},
\end{equation}
where $a_{1},\,a_{2}$ and $a_{3}$ are constants.
\end{theorem}
\mathbb{S}^{2}\times\Rubsection{Conservation laws for equations type (\rhoef{3.2.1})}\lambdabel{examples}
Here we shall illustrate Theorem \rhoef{teo1} by using it and the results due to Ibragimov in order to establish some conservation laws for self-adjoint equations type (\rhoef{3.2.1}).
\mathbb{S}^{2}\times\Rubsubsection{Inviscid Burgers equation}
The vector field (\rhoef{4.1.1'}) is a Lie point symmetry of the inviscid Burgers equation (\rhoef{4.1.1})(for more details, see \cite{igor1}). Since it is a self-adjoint equation, taking $v=u$ in (\rhoef{2.1.10}), the conservation law $D_{t}C^{0}+D_{x}C^{1}=0$ is obtained, where
$$
C^{0}=-u^{2}-tu^{2}\,u_{x},\,\,\,\,C^{1}=u^{3}+tu^{2}\, u_{t}.
$$
However,
$$C^{0}=-u^{2}+D_{x}\left(-\frac{tu^{3}}{3}\rhoight),\,\,\,\, C^{1}=u^{3}+tD_{t}\left(\frac{u^{3}}{3}\rhoight)$$
and
$$
\betagin{array}{lcl}
D_{t}C^{0}+D_{x}C^{1}&=&\displaystyle {-D_{x}\left(\frac{u^{3}}{3}\rhoight)-tD_{t}D_{x}\left(\frac{u^{3}}{3}\rhoight)-D_{t}(u^{2})+D_{x}(u^{3})+tD_{x}D_{t}\left(\frac{u^{3}}{3}\rhoight)}\\
\\
&=&\displaystyle {D_{t}(-u^{2})+D_{x}\left(\frac{2}{3}u^{3}\rhoight)}.
\end{array}
$$
Then $C=(-u^{2},\frac{2}{3}u^{3})$ provides a conserved vector for (\rhoef{4.1.1}). This conservation law was established in \cite{igor1}.
\mathbb{S}^{2}\times\Rubsubsection{Singular second order evolution equation}
A Lie point symmetry generator of singular equation
\betagin{equation}\lambdabel{5.1.10}
u_{t}=\frac{u_{xx}}{u}
\end{equation}
is given by
\betagin{equation}\lambdabel{5.1.11}
X=t\frac{\partial}{\partial t}+u\frac{\partial}{\partial u}.
\end{equation}
From (\rhoef{2.1.10})
$$
\betagin{array}{lcl}
C^{0}&=&\displaystyle {\frac{tvu_{xx}}{u}+vu},\\
\\
C^{1}&=&\displaystyle {(u-tu_{x})\left(\frac{v_{x}}{u}-\frac{vu_{x}}{u^{2}}\rhoight)-\frac{v}{u}D_{x}(u-tu_{t})}.
\end{array}
$$
Setting $v=u$ in $C^{0}$ and $C^{1}$ and after reckoning we obtain $C=(u^{2},-2u_{x})$ as a conserved vector for equation (\rhoef{5.1.10}).
Considering the Lie point symmetry generator
$$Y=x\frac{\partial}{\partial x}+2t\frac{\partial}{\partial t}$$
the components of the conserved vector is
\betagin{equation}\lambdabel{5.1.1.x}
\betagin{array}{lcl}
C^{0}&=&\displaystyle {-\frac{2tv}{u}u_{xx}-xvu_{x}},\\
\\
C^{1}&=&\displaystyle {xvu_{t}-\frac{xv}{u}u_{xx}-\frac{xv_{x}u_{x}}{u}+\frac{xvu_{x}^{2}}{u^{2}}-\frac{2tv_{x}u_{t}}{u}+\frac{2tvu_{t}u_{x}}{u^{2}}+\frac{v}{u}D_{x}(xu_{x}+2tu_{t})}.
\end{array}
\end{equation}
After a tedious calculus and substituting $v=u$ into (\rhoef{5.1.1.x}) we obtain the vector $C=(u^{2}/2,-u_{x})$. Then $Y$ does not give a new conservation law for (\rhoef{5.1.10}).
\mathbb{S}^{2}\times\Rubsubsection{The Korteweg--de Vries equation}
Let us now consider the Korteweg--de Vries equation
\betagin{equation}\lambdabel{4.2.1}
u_{t}=u_{xxx}+uu_{x}.
\end{equation}
It is clear that
$$
X=t\frac{\partial}{\partial t}-\frac{\partial}{\partial u}
$$
is a Lie point symmetry generator of (\rhoef{4.2.1}). Since (\rhoef{4.2.1}) is self-adjoint, from (\rhoef{2.1.10}) and setting $v=u$, we obtain
$$
\betagin{array}{lcl}
C^{0}&=&\displaystyle {-u-tuu_{x}=-u+D_{x}\left(-t\frac{u^{2}}{2}\rhoight)},\\
\\
C^{1}&=&
\displaystyle {tuu_{t}+u_{xx}+u^{2}=tD_{t}\left(\frac{u^{2}}{2}\rhoight)+u^{2}+u_{xx}}.
\end{array}
$$
Then $$D_{t}C^{0}+D_{x}C^{1}=D_{t}(-u)+D_{x}\left(\frac{u^{2}}{2}+u_{xx}\rhoight),$$
we conclude that $C=(-u,\frac{u^{2}}{2}+u_{xx})$ is a conserved vector for (\rhoef{4.2.1}). This example was presented in the seminal work \cite{ib2}.
\mathbb{S}^{2}\times\Rubsubsection{Generalized Korteweg--de Vries equation}
{\bf Example 3}: Let us consider the following generalization of the Korteweg--de Vries equation (\rhoef{4.2.1})
\betagin{equation}\lambdabel{4.2.2}
u_{t}=u_{xxx}+u^{\mu}u_{x},
\end{equation}
where $\mu\neq 0$ is a constant.
Here we shall present in a more detailed form the conservation law for (\rhoef{4.2.2}) arising from the Lie point symmetry generator
$$
X_{\mu}=\frac{2}{\mu}u\frac{\partial}{\partial u}-3t\frac{\partial}{\partial t}-x\frac{\partial}{\partial x}.
$$
From (\rhoef{2.1.10}) is obtained
\betagin{equation}\lambdabel{4.2.4}
\betagin{array}{lcl}
A^{0}&=&\displaystyle {v \left(3tu_{xxx}+3tu^{\mu}u_{x}+xu_{x}\frac{2}{\mu}u\rhoight)},\\
\\
A^{1}&=&\displaystyle {-v\left(\frac{2}{\mu}u^{\mu+1}+xu_{t}+3tu^{\mu}u_{t}+2\frac{\mu+1}{\mu}u_{xx}+3tu_{txx}\rhoight)+v_{x}\left(\frac{2+\mu}{\mu}u_{x}+3tu_{tx}+xu_{xx}\rhoight)}\\
\\
&&\displaystyle {-v_{xx}\left(\frac{2}{\mu}u+3tu_{t}+xu_{x}\rhoight)}.
\end{array}
\end{equation}
Setting $v=u$ in (\rhoef{4.2.4}) and after reckoning, we obtain
$$D_{t}A^{0}+D_{x}A^{1}=D_{t}\left(\frac{4-\mu}{2\mu}u^{2}\rhoight)+D_{x}\left[\frac{\mu-4}{\mu(\mu+2)}u^{\mu+2}+\frac{\mu-4}{\mu}uu_{xx}-\frac{\mu-4}{2\mu}u_{x}^{2}\rhoight].$$
Then $C=(C^{0},C^{1})$ provides a conserved vector for the generalized Korteweg--de Vries equation (\rhoef{4.2.2}), where
$$
C^{0}=u^{2},\,\,\,\,C^{1}=u_{x}^{2}-2uu_{xx}-\frac{2}{\mu+2}u^{\mu+2}.
$$
In particular, whenever $\mu=1$, $C=(u^{2},u_{x}^{2}-2uu_{xx}-\frac{2}{3}u^{3})$ is another conserved vector for the KdV equation (\rhoef{4.2.1}), see \cite{ib2}.
Choosing $\mu=2$, then $C=(u^{2},u_{x}^{2}-2uu_{xx}-\frac{1}{2}u^{4})$ is a conserved vector for the modified KdV equation
$$u_{t}=u_{xxx}+u^{2}u_{x}.$$
\mathbb{S}^{2}\times\Rection{Self-adjoint adjoint equations of fourth-order}\lambdabel{comment}
Concerning equation (\rhoef{1.1.3}), in \cite{ib1} is proved that equation $(\rhoef{1.1.3})$ is self-adjoint if and only if
\betagin{equation}\lambdabel{5.1.1}
g=h+\frac{1}{u}(uf)',\,\,\,\,\,\,d=\frac{c_{1}}{u}+\frac{1}{u}(uh)'
\end{equation}
and
\betagin{equation}\lambdabel{5.1.2}
q=\frac{1}{u}[c_{2}+(up)'],
\end{equation}
where $f,\,h$ and $p$ are arbitrary functions of $u$ (see \cite{ib1}, Theorem 3.2, p.p 310).
From Theorem \rhoef{teo1} we conclude that equations (\rhoef{5.1.2}) and (\rhoef{3.1.2}) cannot be compatible whenever $c_{2}\neq 0$. In fact, the correct statement is
\betagin{theorem}\lambdabel{teo5.2}
Equation $(\rhoef{1.1.3})$ is self-adjoint if and only if $g$ and $d$ are given by $(\rhoef{5.1.1})$
and $q$ is given by $(\rhoef{3.1.2})$, where $f,\,h$ and $p$ are arbitrary functions of $u$ and $c_{1}$ is an arbitrary constant.
\end{theorem}
\betagin{proof}
From the self-adjointness condition (\rhoef{ast}) we obtain the following system of equations
\betagin{equation}\lambdabel{5.1.3}
(uf)'-ug+uh=0,
\end{equation}
\betagin{equation}\lambdabel{5.1.4}
(uf)''-(ug)'+(uh)'=0,
\end{equation}
\betagin{equation}\lambdabel{5.1.5}
3(uf)'''-3(ug)''+(uh)''+2(ud)'=0,
\end{equation}
\betagin{equation}\lambdabel{5.1.6}
(up)'-uq=0,
\end{equation}
\betagin{equation}\lambdabel{5.1.7}
(up)''-(uq)'=0
\end{equation}
and
\betagin{equation}\lambdabel{5.1.8}
(uf)''''-(ug)'''+(ud)''=0.
\end{equation}
From (\rhoef{5.1.3}) and (\rhoef{5.1.5}) we obtain (\rhoef{5.1.1}). Equation (\rhoef{5.1.4}) is a consequence of (\rhoef{5.1.3}). Equation (\rhoef{5.1.8}) is a consequence of (\rhoef{5.1.3}) and (\rhoef{5.1.5}).
From (\rhoef{5.1.7}) we obtain (\rhoef{5.1.2}). However, substituting (\rhoef{5.1.2}) into (\rhoef{5.1.6}) we conclude that $c_{2}=0$. Thus we obtain (\rhoef{3.1.2}).
\end{proof}
From theorems \rhoef{teo5.2} and \rhoef{teo1}, we have the following generalization for Theorem \rhoef{teo5.2} (and Theorem 3.2 of \cite{ib1}):
\betagin{theorem}\lambdabel{teo5.3}
Equation
\betagin{equation}\lambdabel{5.1.9}
\betagin{array}{l}
u_{t}+f(u)u_{xxxx}+g(u)u_{x}u_{xxx}-r(u)u_{xxx}+h(u)u_{xx}^{2}\\
\\
+d(u)u_{x}^{2}u_{xx}-p(u)u_{xx}-q(u)u_{x}^{2}-a(u)u_{x}+b(u)=0
\end{array}
\end{equation}
is self-adjoint if and only if $g$ and $d$ are given by $(\rhoef{5.1.1})$, $r,\,q$ and $b$ are given by $(\rhoef{3.1.1'})$, $(\rhoef{3.1.2})$ and $(\rhoef{3.1.3})$, respectively, where $a_{1},\,a_{2},\,a_{3}$ and $c_{1}$ are arbitrary constants and $f,\,h$ and $p$ are arbitrary functions of $u$.
\end{theorem}
\mathbb{S}^{2}\times\Rection{Conclusion}
In this paper the self-adjoint subclasses of equation (\rhoef{1.1.1}) was obtained. Thanks to the recently proposed conservation theorem from Ibragimov, some conservation laws of particular self-adjoint equations type (\rhoef{3.2.1}) were established. Further examples can be found in \cite{ib1, ib2,igor1, ya}.
A comment in a recently published result (see \cite{ib1}, Theorem 3.2) was given in section \rhoef{comment}. In particular the self-adjointness condition obtained by Bruzón, Gandarias and Ibragimov to equation (\rhoef{1.1.3}) was generalized to equation (\rhoef{5.1.9}). Equation (\rhoef{5.1.9}) covers a wider list of equations, for instance, all equations mentioned in the present paper, the thin film equation and so on, see \cite{qu, ib1}.
The main results are summarized by Theorem \rhoef{teo1} and Theorem \rhoef{teo5.3}. In particular, by using Theorem \rhoef{teo5.3} and the new conservation theorem presented in \cite{ib2}, conservation laws for lubrication equation, Korteweg--de Vries and inviscid Burgers equation, among others, can be established, as pointed out in \cite{ib1, igor1, ya}.
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\end{thebibliography}
\end{document} |
\begin{document}
\author{Seonghak Kim}
\address{Institute for Mathematical Sciences\\ Renmin University of China \\ Beijing 100872, PRC}
\email{[email protected]}
\author{Youngwoo Koh}
\address{School of Mathematics\\ Korea Institute for Advanced Study \\ Seoul 130-722, ROK}
\email{[email protected]}
\title[1-D non-convex elastodynamics]{Weak solutions for one-dimensional non-convex elastodynamics}
\subjclass[2010]{74B20,74N15,35M13}
\keywords{non-convex elastodynamics, hyperbolic-elliptic equations, phase transition, partial differential inclusions, Baire's category method, microstructures of weak solutions}
\begin{abstract}
We explore local existence and properties of classical weak solutions to the initial-boundary value problem of a one-dimensional quasilinear equation of elastodynamics with non-convex stored-energy function, a model of phase transitions in elastic bars proposed by Ericksen \cite{Er}. The instantaneous formation of microstructures of local weak solutions is observed for all smooth initial data with initial strain having its range overlapping with the phase transition zone of the Piola-Kirchhoff stress. As byproducts, it is shown that such a problem admits a local weak solution for all smooth initial data and local weak solutions that are smooth for a short period of time and exhibit microstructures thereafter for some smooth initial data. In a parallel exposition, we also include some results concerning one-dimensional quasilinear hyperbolic-elliptic equations.
\end{abstract}
\mathbb Maketitle
\section{Introduction}
The evolution process of a one-dimensional continuous medium with elastic response can be modeled by quasilinear wave equations of the form
\begin{equation}\label{main-P}
u_{tt} =\sigma(u_x)_x,
\end{equation}
where $u=u(x,t)$ denotes the displacement of a reference point $x$ at time $t$ and $\sigma=\sigma(s)$ the Piola-Kirchhoff stress, which is the derivative of a stored-energy function $W=W(s)\ge 0$. With $v=u_x$ and $w=u_t$, one may study equation (\ref{main-P}) as the system of conservation laws:
\begin{equation}\label{main-cons}
\left\{\begin{split}
v_t & = w_x, \\
w_t & = \sigma(v)_x.
\end{split}\right.
\end{equation}
In case of a strictly convex stored-energy function, the existence of weak or classical solutions to equation (\ref{main-P}) and its vectorial case has been studied extensively: Global weak solutions to system (\ref{main-cons}) and hence equation (\ref{main-P}) are established in a classical work of {DiPerna} \cite{Di} via vanishing viscosity method in the framework of compensated compactness of {Tartar} \cite{Ta} for $L^\infty$ data and later by {Lin} \cite{Li} and {Shearer} \cite{Sr} in an $L^p$ setup. This framework is also used to construct global weak solutions to (\ref{main-P}) via relaxation methods by {Serre} \cite{Se} and {Tzavaras} \cite{Tz}. An alternative variational scheme is studied by {Demoulini \emph{et al.}} \cite{DST} via time discretization. However global existence of weak solutions to the vectorial case of (\ref{main-P}) is still open. In regard to classical solutions to (\ref{main-P}) and its vectorial case, one can refer to {Dafermos and Hrusa} \cite{DH} for local existence of smooth solutions, to {Klainerman and Sideris} \cite{KS} for global existence of smooth solutions for small initial data in dimension 3, and to {Dafermos} \cite{Ds1} for uniqueness of smooth solutions in the class of BV weak solutions whose shock intensity is not too strong.
Convexity assumption on the stored-energy function is often regarded as a severe restriction in view of the actual behavior of elastic materials (see, e.g., \cite[Section 2]{Hi} and \cite[Section 8]{CN}). However there have not been many analytic works dealing with the lack of convexity on the energy function. For the vectorial case of equation (\ref{main-P}) in dimension 3, measure-valued solutions are constructed for polyconvex energy functions by {Demoulini \emph{et al.}} \cite{DST1}. Also by the same authors \cite{DST2}, in the identical situation, it is shown that a dissipative measure-valued solution coincides with a strong one provided the latter exists. Assuming convexity on the energy function at infinity but not allowing polyconvexity, measure-valued solutions are obtained by {Rieger} \cite{Ri} for the vectorial case of (\ref{main-P}) in any dimension. Despite of all these existence results, there has been no example of a non-convex energy function with which (\ref{main-P}) admits \emph{classical} weak solutions in general, not to mention its vectorial case. Among some optimistic and pessimistic opinions, {Rieger} \cite{Ri} expects such solutions even showing microstructures of phase transitions. Moreover, such expected phenomenology is successfully implemented in some numerical simulations \cite{CR, Pr}.
In this paper, we study weak solutions to the one-dimensional initial-boundary value problem of non-convex elastodynamics
\begin{equation}\label{ib-P}
\begin{cases} u_{tt} =\sigma(u_x)_x& \mathbb Mbox{in $\Omega_T=\Omega\times (0,T)$,} \\
u(0,t)=u(1,t)=0 & \mathbb Mbox{for $t\in(0,T)$,}\\
u =g,\,u_t=h & \mathbb Mbox{on $\Omega\times \{t=0\}$},
\end{cases}
\end{equation}
where $\Omega=(0,1)\subset\mathbb R$ is the domain occupied by a reference configuration of an elastic bar, $T>0$ is a fixed number, $g$ is the initial displacement of the bar, $h$ is the initial rate of change of the displacement, and the stress $\sigma:(-1,\infty)\to\mathbb R$ is given as in Figure \ref{fig1}. The zero boundary condition here amounts to the physical situation of fixing the end-points of the bar. In this case, the energy function $W:(-1,\infty)\to[0,\infty)$ may satisfy $W(s)\to\infty$ as $s\to -1^+$; but this is not required to obtain our result.
On the other hand, we consider (\ref{ib-P}) as a prototype of the hyperbolic-elliptic problem with a non-monotone constitutive function $\sigma:\mathbb R\to\mathbb R$ as in Figure \ref{fig2}.
\begin{figure}
\caption{Non-monotone Piola-Kirchhoff stress $\sigma(s).$}
\label{fig1}
\end{figure}
\begin{figure}
\caption{Non-monotone constitutive function $\sigma(s).$}
\label{fig2}
\end{figure}
Problem (\ref{ib-P}) with a non-monotone stress $\sigma(s)$ as in Figure \ref{fig1} is proposed by {Ericksen} \cite{Er} as a model of the phenomena of phase transitions in elastic bars. There have been many studies on such a problem that usually fall into two types. One direction of study is to consider the Riemann problem of the system of conservation laws of mixed type (\ref{main-cons}) initiated by James \cite{Ja} and followed by numerous works (see, e.g., {Shearer} \cite{Sh}, {Pego and Serre} \cite{PS} and {Hattori} \cite{Ha}). Another path is to study the viscoelastic version of equation (\ref{main-P}): To name a few among initiative works, {Dafermos} \cite{Ds} considers the equation $u_{tt}=\sigma(u_x,u_{xt})_x+f(x,t)$ under certain parabolicity and growth conditions and establishes global existence and uniqueness of smooth solutions together with some asymptotic behaviors as $t\to\infty$. Following the work of {Andrews} \cite{An}, {Andrews and Ball} \cite{AB} prove global existence of weak solutions to the equation $u_{tt}=u_{xxt}+\sigma(u_x)_x$ for non-smooth initial data and study their long-time behaviors. For the same equation, {Pego} \cite{Pe} characterizes long-time convergence of weak solutions to several different types of stationary states in a strong sense. Nonetheless, up to our best knowledge, the main theorem below may be the first general existence result on weak solutions to (\ref{ib-P}), not in the stream of the Riemann problem nor in that of non-convex viscoelastodynamics.
Let $\sigma(s)$ be given as in Figure \ref{fig1} or \ref{fig2} (see section \ref{sec:state}). For an initial datum $(g,h)\in W^{1,\infty}_0(\Omega)\times L^\infty(\Omega)$, we say that a function $u\in W^{1,\infty}(\Omega_T)$ is a \emph{weak solution} to problem (\ref{ib-P}) provided $u_x>-1$ a.e. in $\Omega_T$ in case of Figure \ref{fig1}, for all $\varphi\in C^\infty_c(\Omega\times[0,T))$, we have
\begin{equation}\label{def:sol}
\int_{\Omega_T}(u_t\varphi_t-\sigma(u_x)\varphi_x)\,dxdt=-\int_0^1 h(x)\varphi(x,0)\,dx,
\end{equation}
and
\begin{equation}\label{def:sol-1}
\left\{\begin{array}{ll}
u(0,t)=u(1,t)=0 & \mathbb Mbox{for $t\in(0,T)$}, \\
u(x,0)=g(x) & \mathbb Mbox{for $x\in\Omega$}.
\end{array}
\right.
\end{equation}
The main result of the paper is the following theorem that will be separated into two detailed statements in section \ref{sec:state} along with some corollaries.
\begin{thm}\label{thm:main-pre}
Let $\sigma(s)$ be as in Figure \ref{fig1} or \ref{fig2}, and let $(g,h)\in W^{3,2}_0(\Omega)\times W^{2,2}_0(\Omega) $ with $s_1^*<g' (x_0)<s_2^*$ at some $x_0\in\Omega$. In case of Figure \ref{fig1}, assume also $g'(x)>-1$ for all $x\in\bar\Omega$. Then there exists a finite number $T>0$ for which problem (\ref{ib-P}) admits infinitely many weak solutions.
\end{thm}
Existence and non-uniqueness of weak solutions to problem (\ref{ib-P}) have been generally accepted (especially, in the context of the Riemann problem) and actively studied in the community of solid mechanics. Such non-uniqueness is usually understood to be arising from a constitutive deficiency in the theory of elastodynamics, reflecting the need to incorporate some additional relations (see, e.g., {Slemrod} \cite{Sl}, {Abeyaratne and Knowles} \cite{AK} and {Truskinovsky and Zanzotto} \cite{TZ}).
Global existence of Lipschitz continuous weak solutions to problem (\ref{ib-P}) is not directly obtained in the course of proving Theorem \ref{thm:main-pre} as it would require a global classical solution to some modified hyperbolic problem in our method of proof and such a global one might not exist due to a possible shock formation at a finite time. However, we still expect the existence of global $W^{1,p}$-solutions $(p<\infty)$ to (\ref{ib-P}).
The rest of the paper is organized as follows. Section \ref{sec:state} describes precise structural assumptions on the functions $\sigma(s)$ corresponding to Figures \ref{fig1} and \ref{fig2}, respectively. Then detailed statements of the main result, Theorem \ref{thm:main-pre}, with respect to Figures \ref{fig1} and \ref{fig2} are introduced separately as Theorems \ref{thm:main-NCE} and \ref{thm:main-HEP} with relevant corollaries in each case.
Section \ref{sec:exist} begins with a motivational approach to solve problem (\ref{ib-P}) as a homogeneous partial differential inclusion with a linear constraint. Then the main results in precise form, Theorems \ref{thm:main-NCE} and \ref{thm:main-HEP}, are proved at the same time under a pivotal density fact, Theorem \ref{thm:density}. The proofs of the corollaries to the main results are also included in section \ref{sec:exist}.
In section \ref{sec:rank-1}, a major tool for proving the density fact is established in a general form.
Lastly, section \ref{sec:density-proof} carries out the proof of the density fact.
In closing this section, we introduce some notations. Let $m,n$ be positive integers. We denote by $\mathbb Mathbb M^{m\times n}$ the space of $m\times n$ real matrices and by $\mathbb Mathbb M_{sym}^{n\times n}$ that of symmetric $n\times n$ real matrices. We use $O(n)$ to denote the space of $n\times n$ orthogonal matrices. For a given matrix $M\in\mathbb Mathbb M^{m\times n}$, we write $M_{ij}$ for the component of $M$ in the $i$th row and $j$th column and $M^T$ for the transpose of $M$. For a bounded domain $U\subset\mathbb R^n$ and a function $w^*\in W^{m,p}(U)$ $(1\le p\le\infty)$, we use $W_{w^*}^{m,p}(U)$ to denote the space of functions $w\in W^{m,p}(U)$ with boundary trace $w^*.$
\section{Precise statements of main theorems}\label{sec:state}
In this section, we present structural assumptions on the functions $\sigma(s)$ for \textbf{Case I:} non-convex elastodynamics and \textbf{Case II:} hyperbolic-elliptic problem corresponding to Figures \ref{fig1} and \ref{fig2}, respectively. Then we give the detailed statement of the main result, Theorem \ref{thm:main-pre}, in each case, followed by some relevant byproducts.
\textbf{(Case I):} For the problem of non-convex elastodynamics, we impose the following conditions on the stress $\sigma:(-1,\infty)\to\mathbb R$ (see Figure \ref{fig1}).
\textbf{Hypothesis (NC):} There exist two numbers $s_2>s_1>-1$ with the following properties:
\begin{itemize}
\item[(a)] $\sigma\in C^3((-1,s_1)\cup(s_2,\infty))\cap C((-1,s_1]\cup[s_2,\infty))$.
\item[(b)] $\displaystyle\lim_{s\to -1^+}\sigma(s)=-\infty$.
\item[(c)] $\sigma:(s_1,s_2)\to\mathbb R$ is bounded and measurable.
\item[(d)] $\sigma(s_1)>\sigma(s_2)$, and $\sigma'(s)>0$ for all $s\in (-1,s_1)\cup(s_2,\infty)$.
\item[(e)] There exist two numbers $c>0$ and $s_1+1>\rho>0$ such that $\sigma'(s)\ge c$ for all $s\in (-1,s_1-\rho]\cup[s_2+\rho,\infty)$.
\item[(f)]
Let $s_1^*\in (-1,s_1)$ and $s^*_2\in(s_2,\infty)$ denote the unique numbers with $\sigma(s_1^*)=\sigma(s_2)$ and $\sigma(s_2^*)=\sigma(s_1)$, respectively.
\end{itemize}
\textbf{(Case II):} For the hyperbolic-elliptic problem, we assume the following for the constitutive function $\sigma:\mathbb R\to\mathbb R$ (see Figure \ref{fig2}).
\textbf{Hypothesis (HE):} There exist two numbers $s_2>s_1$ satisfying the following:
\begin{itemize}
\item[(a)] $\sigma\in C^3((-\infty,s_1)\cup(s_2,\infty))\cap C((-\infty,s_1]\cup[s_2,\infty))$.
\item[(b)] $\sigma:(s_1,s_2)\to\mathbb R$ is bounded and measurable.
\item[(c)] $\sigma(s_1)>\sigma(s_2)$, and $\sigma'(s)>0$ for all $s\in (-\infty,s_1)\cup(s_2,\infty)$.
\item[(d)] There exists a number $c>0$ such that $\sigma'(s)\ge c$ for all $s\in (-\infty,s_1-1]\cup[s_2+1,\infty)$.
\item[(e)]
Let $s_1^*\in (-\infty,s_1)$ and $s^*_2\in(s_2,\infty)$ denote the unique numbers with $\sigma(s_1^*)=\sigma(s_2)$ and $\sigma(s_2^*)=\sigma(s_1)$, respectively.
\end{itemize}
In both cases, for each $r\in (\sigma(s_2),\sigma(s_1))$, let $s_-(r)\in(s_1^*,s_1)$ and $s_+(r)\in(s_2,s_2^*)$ denote the unique numbers with $\sigma(s_\pm(r))=r.$ We may call the interval $(s_1^*,s_2^*)$ as the \emph{phase transition zone} of problem (\ref{ib-P}), since the formation of microstructures and breakdown of uniqueness of weak solutions to (\ref{ib-P}) begin to occur whenever the range of the initial strain $g'$ overlaps with the interval $(s_1^*,s_2^*)$ as we can see below from Theorems \ref{thm:main-NCE} and \ref{thm:main-HEP} and their corollaries.
We now state the main result on \textbf{Case I:} weak solutions for non-convex elastodynamics under Hypothesis (NC).
\begin{thm}\label{thm:main-NCE}
Let $(g,h)\in W^{3,2}_0(\Omega)\times W^{2,2}_0(\Omega)$ satisfy $g'(x)>-1$ for all $x\in\bar\Omega$ and $s_1^*<g' (x_0)<s_2^*$ for some $x_0\in\Omega$. Let $\sigma(s_2)<r_1<r_2<\sigma(s_1)$ be any two numbers with $s_-(r_1)<g'(x_0)<s_+(r_2)$. Then there exist a finite number $T>0$, a function $\displaystyle{u^*\in \cap_{k=0}^3C^k([0,T];W^{3-k,2}_0(\Omega))}$ with $u^*_x>-1$ on $\bar\Omega_T$, where $W^{0,2}_0(\Omega)=L^2(\Omega)$, and three disjoint open sets $\Omega_T^1,\Omega_T^2,\Omega_T^3\subset \Omega_T$ with $\Omega_T^2\mathbb Mathbf not=\emptyset$, $\partial\Omega_T^1\cap\partial\Omega_T^3=\emptyset$, and \begin{equation}\label{sep-domain-NCE}
\left\{\begin{array}{l}
\partial\Omega_T^1\cap \Omega_0=\{(x,0)\,|\, x\in\Omega,\,g'(x)<s_-(r_1)\}, \\
\partial\Omega_T^2\cap \Omega_0 =\{(x,0)\,|\, x\in\Omega,\,s_-(r_1)<g'(x)<s_+(r_2)\}, \\
\partial\Omega_T^3\cap \Omega_0=\{(x,0)\,|\, x\in\Omega,\,g'(x)>s_+(r_2)\},
\end{array}
\right.
\end{equation}
where $\Omega_0=\Omega\times\{t=0\},$
such that for each $\epsilon>0$, there exist a number $T_\epsilon\in(0,T)$ and infinitely many weak solutions $u\in W^{1,\infty}_{u^*}(\Omega_T)$ to problem (\ref{ib-P}) with the following properties:
\begin{itemize}
\item[(a)] Approximate initial rate of change:
\[
\|u_t-h\|_{L^\infty(\Omega_{T_\epsilon})}<\epsilon,
\]
where $\Omega_{T_\epsilon}=\Omega\times(0,T_\epsilon)$.
\item[(b)] Classical part of $u$:
\begin{itemize}
\item[(i)] $u\equiv u^*$ on $\overline{\Omega_T^1\cup\Omega_T^3}$,
\item[(ii)] $u_t(x,0)=h(x)\quad\forall (x,0)\in(\partial\Omega_T^1\cup\partial\Omega_T^3)\cap\Omega_0$,
\item[(iii)] $u_x(x,t)\left\{\begin{array}{ll}
\in(-1,s_-(r_1)) & \forall(x,t)\in\Omega_T^1, \\
>s_+(r_2) & \forall(x,t)\in\Omega_T^3.
\end{array}
\right.$
\end{itemize}
\item[(c)] Microstructure of $u$:
$u_x(x,t)\in[s_-(r_1),s_-(r_2)]\cup[s_+(r_1),s_+(r_2)]$, a.e. $(x,t)\in\Omega_T^2$.
\item[(d)] Interface of $u$:
$u_x(x,t)\in\{s_-(r_1),s_+(r_2)\}$, a.e. $(x,t)\in\Omega_T\setminus (\cup_{i=1}^3\Omega_T^i)$.
\end{itemize}
\end{thm}
As a remark, note that corresponding deformations of the elastic bar, $d(x,t)=u(x,t)+x$, should satisfy \[
d_x(x,t)=u_x(x,t)+1 >-1+1=0,\;\;\mathbb Mbox{a.e. $(x,t)\in\Omega_T$;}
\]
this guarantees that for a.e. $t\in(0,T)$, such deformations $d:[0,1]\times\{t\}\to[0,1]$ are strictly increasing with $d(0,t)=0$ and $d(1,t)=1$. Moreover, for such a $t\in(0,T)$, the deformations $d(x,t)$ are smooth (as much as the initial displacement $g$) for the values of $x\in[0,1]$ for which slope $d_x(x,t)\in(0,s_-(r_1)+1)\cup(s_+(r_2)+1,\infty)$ and are Lipschitz elsewhere with $d_x(x,t)\in[s_-(r_1)+1,s_-(r_2)+1]\cup[s_+(r_1)+1,s_+(r_2)+1]$ a.e. Therefore, these dynamic deformations fulfill a natural physical requirement of invertibility for the motion of an elastic bar not allowing interpenetration.
As byproducts of Theorem \ref{thm:main-NCE}, we obtain the following two results for non-convex elastodynamics. The first is on local existence of weak solutions to problem (\ref{ib-P}) for all smooth initial data. The second gives local weak solutions to (\ref{ib-P}) that are all identical and smooth for a short period of time and then evolve microstructures along with the breakdown of uniqueness for some smooth initial data.
\begin{coro}\label{coro:weak-NCE}
For any initial datum $(g,h)\in W^{3,2}_0(\Omega)\times W^{2,2}_0(\Omega)$ with $g'>-1$ on $\bar\Omega$, there exists a finite number $T>0$ for which problem (\ref{ib-P}) has a weak solution.
\end{coro}
\begin{coro}\label{coro:weak-NCE-micro}
Let $(g,h)\in W^{3,2}_0(\Omega)\times W^{2,2}_0(\Omega)$ satisfy $g'>-1$ on $\bar\Omega$. Assume $\mathbb Max_{\bar\Omega}g'\in (s_1^*,s_1)$ or $\mathbb Min_{\bar\Omega}g'\in (s_2,s_2^*)$. Then there exist finite numbers $T>T'>0$ such that problem (\ref{ib-P}) admits infinitely many weak solutions that are all equal to some $\displaystyle{u^*\in \cap_{k=0}^3C^k([0,T'];W^{3-k,2}_0(\Omega))}$ in $\Omega_{T'}$ and evolve microstructures from $t=T'$ as in Theorem \ref{thm:main-NCE}.
\end{coro}
The following is the main result on \textbf{Case II:} hyperbolic-elliptic equations under Hypothesis (HE).
\begin{thm}\label{thm:main-HEP}
Let $(g,h)\in W^{3,2}_0(\Omega)\times W^{2,2}_0(\Omega)$ with $s_1^*<g' (x_0)<s_2^*$ for some $x_0\in\Omega$. Let $\sigma(s_2)<r_1<r_2<\sigma(s_1)$ be any two numbers with $s_-(r_1)<g'(x_0)<s_+(r_2)$. Then there exist a finite number $T>0$, a function $\displaystyle{u^*\in \cap_{k=0}^3C^k([0,T];W^{3-k,2}_0(\Omega))}$, and three disjoint open sets $\Omega_T^1,\Omega_T^2,\Omega_T^3\subset \Omega_T$ with $\Omega_T^2\mathbb Mathbf not=\emptyset$, $\partial\Omega_T^1\cap\partial\Omega_T^3=\emptyset$, and \begin{equation}\label{sep-domain-HEP}
\left\{\begin{array}{l}
\partial\Omega_T^1\cap \Omega_0=\{(x,0)\,|\, x\in\Omega,\,g'(x)<s_-(r_1)\}, \\
\partial\Omega_T^2\cap \Omega_0 =\{(x,0)\,|\, x\in\Omega,\,s_-(r_1)<g'(x)<s_+(r_2)\}, \\
\partial\Omega_T^3\cap \Omega_0=\{(x,0)\,|\, x\in\Omega,\,g'(x)>s_+(r_2)\}
\end{array}
\right.
\end{equation}
such that for each $\epsilon>0$, there exist a number $T_\epsilon\in(0,T)$ and infinitely many weak solutions $u\in W^{1,\infty}_{u^*}(\Omega_T)$ to problem (\ref{ib-P}) satisfying the following properties:
\begin{itemize}
\item[(a)] Approximate initial rate of change:
\[
\|u_t-h\|_{L^\infty(\Omega_{T_\epsilon})}<\epsilon.
\]
\item[(b)] Classical part of $u$:
\begin{itemize}
\item[(i)] $u\equiv u^*$ on $\overline{\Omega_T^1\cup\Omega_T^3}$,
\item[(ii)] $u_t(x,0)=h(x)\quad\forall (x,0)\in(\partial\Omega_T^1\cup\partial\Omega_T^3)\cap\Omega_0$,
\item[(iii)] $u_x(x,t)\left\{\begin{array}{cc}
<s_-(r_1) & \forall(x,t)\in\Omega_T^1, \\
>s_+(r_2) & \forall(x,t)\in\Omega_T^3.
\end{array}
\right.$
\end{itemize}
\item[(c)] Microstructure of $u$:
$u_x(x,t)\in[s_-(r_1),s_-(r_2)]\cup[s_+(r_1),s_+(r_2)]$, a.e. $(x,t)\in\Omega_T^2$.
\item[(d)] Interface of $u$:
$u_x(x,t)\in\{s_-(r_1),s_+(r_2)\}$, a.e. $(x,t)\in\Omega_T\setminus (\cup_{i=1}^3\Omega_T^i)$.
\end{itemize}
\end{thm}
We also have the following results similar to Corollaries \ref{coro:weak-NCE} and \ref{coro:weak-NCE-micro}.
\begin{coro}\label{coro:weak-HEP}
For any initial datum $(g,h)\in W^{3,2}_0(\Omega)\times W^{2,2}_0(\Omega)$, there exists a finite number $T>0$ for which problem (\ref{ib-P}) has a weak solution.
\end{coro}
\begin{coro}\label{coro:weak-HEP-micro}
Let $(g,h)\in W^{3,2}_0(\Omega)\times W^{2,2}_0(\Omega)$ satisfy $\mathbb Max_{\bar\Omega}g'\in (s_1^*,s_1)$ or $\mathbb Min_{\bar\Omega}g'\in (s_2,s_2^*)$. Then there exist finite numbers $T>T'>0$ such that problem (\ref{ib-P}) admits infinitely many weak solutions that are all equal to some $u^*\in \cap_{k=0}^3C^k([0,T'];$
$W^{3-k,2}_0(\Omega))$ in $\Omega_{T'}$ and evolve microstructures from $t=T'$ as in Theorem \ref{thm:main-HEP}.
\end{coro}
\section{Proof of main theorems}\label{sec:exist}
In this section, we prove the main results, Theorems \ref{thm:main-NCE} and \ref{thm:main-HEP}, with some essential ingredient, Theorem \ref{thm:density}, to be verified in sections \ref{sec:rank-1} and \ref{sec:density-proof}. The proofs of related corollaries are also included.
Our exposition hereafter will be parallelwise for \textbf{Cases I} and \textbf{II}.
\subsection{An approach by differential inclusion}
We begin with a motivational approach to attack problem (\ref{ib-P}) for both \textbf{Cases I} and \textbf{II}. To solve equation (\ref{main-P}) in the sense of distributions in $\Omega_T$, suppose there exists a vector function $w=(u,v)\in W^{1,\infty}(\Omega_T;\mathbb R^2)$ such that
\begin{equation}\label{abs-1}
v_x=u_t\quad\mathbb Mbox{and}\quad v_t=\sigma(u_x)\quad\mathbb Mbox{a.e. in $\Omega_T$}.
\end{equation}
We remark that this formulation is motivated by the approach in \cite{Zh} and different from the usual setup of conservation laws (\ref{main-cons}).
For all $\varphi\in C^\infty_c(\Omega_T)$, we now have
\[
\int_{\Omega_T} u_t\varphi_t\,dxdt =\int_{\Omega_T}v_x\varphi_t\,dxdt= \int_{\Omega_T}v_t\varphi_x\,dxdt= \int_{\Omega_T}\sigma(u_x)\varphi_x\,dxdt;
\]
hence having (\ref{abs-1}) is sufficient to solve (\ref{main-P}) in the sense of distributions in $\Omega_T$. Equivalently, we can rewrite (\ref{abs-1}) as
\[
\mathbb Mathbf nabla w=\begin{pmatrix}
u_x & u_t \\
v_x & v_t
\end{pmatrix}=
\begin{pmatrix}
u_x & v_x \\
v_x & \sigma(u_x)
\end{pmatrix}\quad\mathbb Mbox{a.e. in $\Omega_T$},
\]
where $\mathbb Mathbf nabla$ denotes the space-time gradient.
Set
\[
\Sigma_\sigma=\left\{\begin{pmatrix}
s & b \\
b & \sigma(s)
\end{pmatrix}\in\mathbb Mathbb M^{2\times 2}_{sym}\,\mathbf Big|\,s,b\in\mathbb R \right\}.
\]
We can now recast (\ref{abs-1}) as a homogeneous partial differential inclusion with a linear constraint:
\begin{equation*}
\mathbb Mathbf nabla w(x,t)\in\Sigma_\sigma,\quad\mathbb Mbox{a.e. $(x,t)\in\Omega_T$}.
\end{equation*}
We will solve this inclusion for a suitable subset $K$ of $\Sigma_\sigma$ to incorporate some detailed properties of weak solutions to (\ref{ib-P}).
Homogeneous differential inclusions of the form $\mathbb Mathbf nabla w\in K\subset\mathbb Mathbb M^{m\times n}$ are first encountered and successfully understood in the study of crystal microstructure by {Ball and James} \cite{BJ}, {Chipot and Kinderlehrer} \cite{CK} and with a constraint on a minor of $\mathbb Mathbf nabla w$ by {M\"uller and \v Sver\'ak} \cite{MSv1}.
General inhomogeneous differential inclusions are studied by {Dacorogna and Marcellini} \cite{DM1} using Baire's category method and by {M\"uller and Sychev} \cite{MSy} using the method of convex integration; see also \cite{Ki}. Moreover, the methods of differential inclusions have been applied to other important problems concerning elliptic systems \cite{MSv2}, Euler equations \cite{DS}, the porous media equation \cite{CFG}, the active scalar equation \cite{Sy}, Perona-Malik equation and its generalizations \cite{Zh,KY,KY1,KY2}, and ferromagnetism \cite{Ya1}.
\subsection{Proof of Theorems \ref{thm:main-NCE} and \ref{thm:main-HEP}}\label{subsec:mainproof}
Due to the similarity between Theorems \ref{thm:main-NCE} and \ref{thm:main-HEP}, we can combine their proofs into a single one.
To start the proof, we assume functions $g,h$ and numbers $r_1,r_2$ are given as in Theorem \ref{thm:main-NCE} \textbf{(Case I)}, as in Theorem \ref{thm:main-HEP} \textbf{(Case II)}.
For clarity, we divide the proof into several steps.
\textbf{(Modified hyperbolic problem):}
Using elementary calculus, from Hypothesis (NC) \textbf{(Case I)}, Hypothesis (HE) \textbf{(Case II)}, we can find a function $\sigma^*\in C^3(-1,\infty)$ \textbf{(Case I)}, $\sigma^*\in C^3(\mathbb R)$ \textbf{(Case II)} such that
\begin{equation}\label{modi}
\left\{\begin{array}{l}
\mathbb Mbox{$\sigma^*(s)=\sigma(s)$ for all $s\in(-1,s_-(r_1)]\cup[s_+(r_2),\infty)$ \textbf{(Case I)},} \\
\mathbb Mbox{\quad\quad\quad\quad\quad\,\,\, for all $s\in(-\infty,s_-(r_1)]\cup[s_+(r_2),\infty)$ \textbf{(Case II)},} \\
\mathbb Mbox{$(\sigma^*)'(s)\ge c^*$ for all $s\in(-1,\infty)$, for some constant $c^*>0$ \textbf{(Case I)},} \\
\mathbb Mbox{\quad\quad\quad\quad\quad\,\,\, for all $s\in\mathbb R$, for some constant $c^*>0$ \textbf{(Case II)},} \\
\mathbb Mbox{$\sigma^*(s)<\sigma(s)$ for all $s_-(r_1)<s\le s_-(r_2)$, and}\\
\mathbb Mbox{$\sigma^*(s)>\sigma(s)$ for all $s_+(r_1)\le s< s_+(r_2)$ (see Figure \ref{fig3} for both cases).}
\end{array}
\right.
\end{equation}
\begin{figure}
\caption{The original $\sigma(s)$ and modified $\sigma^*(s)$}
\label{fig3}
\end{figure}
Thanks to \cite[Theorem 5.2]{DH} \textbf{(Case I)}, \cite[Theorem 5.1]{DH} \textbf{(Case II)}, there exists a finite number $T>0$ such that the \emph{modified} initial-boundary value problem
\begin{equation}\label{ib-P-modi}
\begin{cases} u^*_{tt} =\sigma^*(u^*_x)_x& \mathbb Mbox{in $\Omega_T$,} \\
u^*(0,t)=u^*(1,t)=0 & \mathbb Mbox{for $t\in(0,T)$,}\\
u^* =g,\;u^*_t=h & \mathbb Mbox{on $\Omega\times \{t=0\}$}
\end{cases}
\end{equation}
admits a unique solution $u^*\in \cap_{k=0}^3C^k([0,T];W^{3-k,2}_0(\Omega))$, with $u^*_x>-1$ on $\bar\Omega_T$ for \textbf{Case I}. By the Sobolev embedding theorem, we have $u^*\in C^2(\bar\Omega_T)$. Let
\[\left\{
\begin{split}
\Omega^1_T & =\{(x,t)\in\Omega_T\,|\,u^*_x(x,t)<s_-(r_1)\},\\
\Omega^2_T & =\{(x,t)\in\Omega_T\,|\,s_-(r_1)<u^*_x(x,t)<s_+(r_2)\},\\
\Omega^3_T & =\{(x,t)\in\Omega_T\,|\,u^*_x(x,t)>s_+(r_2)\},\\
F_T & =\Omega_T\setminus(\cup_{i=1}^3\Omega_T^i);
\end{split}\right.
\]
then (\ref{sep-domain-NCE}) holds \textbf{(Case I)}, (\ref{sep-domain-HEP}) holds \textbf{(Case II)}, and $\partial\Omega_T^1\cap\partial\Omega_T^3=\emptyset$. As $s_-(r_1)<g'(x_0)=u^*_x(x_0,0)<s_+(r_2)$, we also have $\Omega_T^2\mathbb Mathbf not=\emptyset$; so $|\Omega_T^2|>0$.
We define
\[
v^*(x,t)=\int_0^x h(z)\,dz+\int_0^t \sigma^*(u^*_x(x,\tau))\,d\tau\quad\forall(x,t)\in\Omega_T.
\]
Then $w^*:=(u^*,v^*)$ satisfies
\begin{equation}\label{classic}
v^*_x=u^*_t\quad\mathbb Mbox{and}\quad v^*_t=\sigma^*(u^*_x)\quad\mathbb Mbox{in $\Omega_T$}.
\end{equation}
Note that this implies $v^*\in C^2(\bar\Omega_T)$; hence $w^*\in C^2(\bar\Omega_T;\mathbb R^2)$.
\textbf{(Related matrix sets):}
Define the sets (see Figure \ref{fig3})
\[
\begin{split}
\tilde K_\pm & =\{(s,\sigma(s))\in\mathbb R^2\,|\,s_\pm(r_1)\le s\le s_\pm(r_2)\},\\
\tilde K & = \tilde K_+\cup\tilde K_-,\\
\tilde U & =\{(s,r)\in\mathbb R^2\,|\, r_1<r<r_2,\,0<\lambda<1,\,s=\lambda s_-(r)+(1-\lambda) s_+(r) \},\\
K & = \left\{\begin{pmatrix} s & b \\ b & r \end{pmatrix}\in\mathbb Mathbb M^{2\times 2}_{sym}\,\mathbf Big|\, (s,r)\in\tilde K,\, |b|\le\gamma\right\},\\
U & = \left\{\begin{pmatrix} s & b \\ b & r \end{pmatrix}\in\mathbb Mathbb M^{2\times 2}_{sym}\,\mathbf Big|\, (s,r)\in\tilde U,\, |b|<\gamma \right\},
\end{split}
\]
where $\gamma:=\|u^*_t\|_{L^\infty(\Omega_T)}+1$.
\textbf{(Admissible class):}
Let $\epsilon>0$ be given. Choose a number $T_\epsilon\in(0,T]$ so that $\|u^*_t-h\|_{L^\infty(\Omega_{T_\epsilon})}<\epsilon/2=:\epsilon'$.
We then define the \emph{admissible class} $\mathbb Mathcal A$ by
\[
\mathbb Mathcal A=\left\{w=(u,v)\in W^{1,\infty}_{w^*}(\Omega_T;\mathbb R^2)\,\mathbf Bigg|\, \begin{array}{l}
w\in C^2(\bar\Omega_T;\mathbb R^2),\,w\equiv w^* \\
\mathbb Mbox{in}\;\Omega_T\setminus\bar\Omega_T^w\;\mathbb Mbox{for some open set}\\
\mathbb Mbox{$\Omega_T^w\subset\subset\Omega_T^2$ with $|\partial\Omega_T^w|=0,$}\\
\mathbb Mathbf nabla w(x,t) \in U\;\forall(x,t)\in\Omega_{T}^2,\\
\|u_t-h\|_{L^\infty(\Omega_{T_\epsilon})}<\epsilon'
\end{array}
\right\}.
\]
It is easy to see from (\ref{modi}) and (\ref{classic}) that $w^*\in\mathbb Mathcal A\mathbb Mathbf not =\emptyset$.
For each $\delta>0$, we also define the \emph{$\delta$-approximating class} $\mathbb Mathcal A_\delta$ by
\[
\mathbb Mathcal A_\delta=\left\{w\in\mathbb Mathcal A\,\mathbf Big|\, \int_{\Omega_T^2}\operatorname{dist}(\mathbb Mathbf nabla w(x,t),K)\,dxdt\le\delta|\Omega_T^2|
\right\}.
\]
\textbf{(Density result):}
One crucial step for the proof of Theorem \ref{thm:main-NCE} \textbf{(Case I)}, Theorem \ref{thm:main-HEP} \textbf{(Case II)} is the following density fact whose proof appears in section \ref{sec:density-proof} that is common for both cases.
\begin{thm}\label{thm:density}
For each $\delta>0$,
\[\mathbb Mbox{$\mathbb Mathcal A_\delta$ is dense in $\mathbb Mathcal A$ with respect to the $L^\infty(\Omega_T;\mathbb R^2)$-norm.}\]
\end{thm}
\textbf{(Baire's category method):}
Let $\mathbb Mathcal X$ denote the closure of $\mathbb Mathcal A$ in the space $L^\infty(\Omega_T;\mathbb R^2)$, so that $(\mathbb Mathcal X,L^\infty)$ is a nonempty complete metric space. As $U$ is bounded in $\mathbb Mathbb M^{2\times 2}$, so is $\mathbb Mathcal A$ in $W^{1,\infty}(\Omega_T;\mathbb R^2)$; thus it is easily checked that
\[
\mathbb Mathcal X\subset W^{1,\infty}_{w^*}(\Omega_T;\mathbb R^2).
\]
Note that the space-time gradient operator $\mathbb Mathbf nabla:\mathbb Mathcal X\to L^1(\Omega_T;\mathbb Mathbb M^{2\times 2})$ is a Baire-one function (see, e.g., \cite[Proposition 10.17]{Da}). So by the Baire Category Theorem (see, e.g., \cite[Theorem 10.15]{Da}), the set of points of discontinuity of the operator $\mathbb Mathbf nabla$, say $\mathbb Mathcal D_{\mathbb Mathbf nabla}$, is a set of the first category; thus the set of points at which $\mathbb Mathbf nabla$ is continuous, that is, $\mathbb Mathcal C_{\mathbb Mathbf nabla}:=\mathbb Mathcal X\setminus\mathbb Mathcal D_{\mathbb Mathbf nabla}$, is dense in $\mathbb Mathcal X$.
\textbf{(Completion of proof):}
Let us confirm that for any function $w=(u,v)\in\mathbb Mathcal C_\mathbb Mathbf nabla$, its first component $u$ is a weak solution to (\ref{ib-P}) satisfying (a)--(d). Towards this, fix any $w=(u,v)\in\mathbb Mathcal C_\mathbb Mathbf nabla$.
\textbf{\underline{(\ref{def:sol}) \& (\ref{def:sol-1}):}} To verify (\ref{def:sol}), let $\varphi\in C^\infty_c(\Omega\times[0,T))$. From Theorem \ref{thm:density} and the density of $\mathbb Mathcal A$ in $\mathbb Mathcal X$, we can choose a sequence $w_k=(u_k,v_k)\in\mathbb Mathcal A_{1/k}$ such that $w_k\to w$ in $\mathbb Mathcal X$ as $k\to\infty$. As $w\in\mathbb Mathcal C_{\mathbb Mathbf nabla}$, we have $\mathbb Mathbf nabla w_k\to \mathbb Mathbf nabla w$ in $L^1(\Omega_T;\mathbb Mathbb M^{2\times 2})$ and so pointwise a.e. in $\Omega_T$ after passing to a subsequence if necessary. By (\ref{classic}) and the definition of $\mathbb Mathcal A$, we have $(v_k)_x=(u_k)_t$ in $\Omega_T$ and $(v_k)_x(x,0)=v^*_x(x,0)=u^*_t(x,0)=h(x)$ ($x\in\Omega$); so
\[
\begin{split}
\int_{\Omega_T}(u_k)_t\varphi_t\,dxdt & =\int_{\Omega_T}(v_k)_x\varphi_t\,dxdt\\
& = -\int_{\Omega_T}(v_k)_{xt}\varphi \,dxdt -\int_0^1 (v_k)_x(x,0)\varphi(x,0)\,dx\\
& = \int_{\Omega_T}(v_k)_{t}\varphi_x \,dxdt -\int_0^1 h(x)\varphi(x,0)\,dx,
\end{split}
\]
that is,
\[
\int_{\Omega_T}((u_k)_t\varphi_t-(v_k)_{t}\varphi_x)\,dxdt = -\int_0^1 h(x)\varphi(x,0)\,dx.
\]
On the other hand, by the Dominated Convergence Theorem, we have
\[
\int_{\Omega_T}((u_k)_t\varphi_t-(v_k)_{t}\varphi_x)\,dxdt \to \int_{\Omega_T}(u_t\varphi_t-v_{t}\varphi_x)\,dxdt;
\]
thus
\begin{equation}\label{pre-weak}
\int_{\Omega_T}(u_t\varphi_t-v_{t}\varphi_x)\,dxdt = -\int_0^1 h(x)\varphi(x,0)\,dx.
\end{equation}
Also, by the Dominated Convergence Theorem,
\[
\int_{\Omega_T^2} \operatorname{dist}(\mathbb Mathbf nabla w_k(x,t),K)\,dxdt\to\int_{\Omega_T^2} \operatorname{dist}(\mathbb Mathbf nabla w(x,t),K)\,dxdt.
\]
From the choice $w_k\in\mathbb Mathcal A_{1/k}$, we have
\[
\int_{\Omega_T^2} \operatorname{dist}(\mathbb Mathbf nabla w_k(x,t),K)\,dxdt\le\frac{|\Omega_T^2|}{k}\to 0;
\]
so
\[
\int_{\Omega_T^2} \operatorname{dist}(\mathbb Mathbf nabla w(x,t),K)\,dxdt=0.
\]
Since $K$ is closed, we must have
\begin{equation}\label{inclusion}
\mathbb Mathbf nabla w(x,t)\in K\subset \Sigma_\sigma,\quad\mathbb Mbox{a.e. $(x,t)\in\Omega_T^2$}.
\end{equation}
For each $k$, we have $w_k\equiv w^*$ in $\Omega_T\setminus\bar\Omega_T^{w_k}$ for some open set $\Omega_T^{w_k}\subset\subset\Omega_T^2$ with $|\partial \Omega_T^{w_k}|=0$, and so $\mathbb Mathbf nabla w_k\equiv\mathbb Mathbf nabla w^*$ in $\Omega_T\setminus\bar\Omega_T^{w_k}$; thus $w=w^*$ and $\mathbb Mathbf nabla w=\mathbb Mathbf nabla w^*$ a.e. in $\Omega_T\setminus\Omega_T^2$.
By (\ref{modi}) and (\ref{classic}), we have
\begin{equation*}
v_x=u_t\quad\mathbb Mbox{and}\quad v_t=\sigma^*(u^*_x)=\sigma(u_x)\quad\mathbb Mbox{a.e. in $\Omega_T\setminus\Omega_T^2$}.
\end{equation*}
This together with (\ref{inclusion}) implies that $\mathbb Mathbf nabla w\in\Sigma_\sigma$ a.e. in $\Omega_T$. In particular, $v_t =\sigma(u_x)$ a.e. in $\Omega_T$. Reflecting this to (\ref{pre-weak}), we have (\ref{def:sol}). As $w=w^*$ on $\partial\Omega_T$, we also have (\ref{def:sol-1}).
\textbf{\underline{(a), (b), (c) \& (d):}}
As $w=w^*$ a.e. in $\Omega_T\setminus\Omega_T^2$, it follows from the continuity that $u\equiv u^*$ in $\Omega_T^1\cup\Omega_T^3$; so (b) is guaranteed by the definition of $\Omega_T^1$ and $\Omega_T^3$, with $u^*_x>-1$ on $\bar\Omega_T$ for \textbf{Case I}. Since $\mathbb Mathbf nabla w=\mathbb Mathbf nabla w^*$ a.e. in $\Omega_T\setminus\Omega_T^2$, we have $u_x=u^*_x\in\{s_-(r_1),s_+(r_2)\}$ a.e. in $F_T$; so (d) holds.
From $w_k\in\mathbb Mathcal A_{1/k}\subset\mathbb Mathcal A$, we have $|(u_k)_t(x,t)-h(x)|<\epsilon'$ for a.e. $(x,t)\in\Omega_{T_\epsilon}$. Taking the limit as $k\to\infty$, we obtain that $|u_t(x,t)-h(x)|\le\epsilon'$ for a.e. $(x,t)\in\Omega_{T_\epsilon}$; hence $\|u_t-h\|_{L^\infty(\Omega_{T_\epsilon})}\le\epsilon'=\epsilon/2<\epsilon$. Thus (a) is proved. From (\ref{inclusion}) and the definition of $K$, (c) follows.
\textbf{\underline{Infinitely many weak solutions:}}
Having shown that the first component $u$ of each pair $w=(u,v)$ in $\mathbb Mathcal C_\mathbb Mathbf nabla$ is a weak solution to (\ref{ib-P}) satisfying (a)--(d), it remains to verify that $\mathbb Mathcal C_\mathbb Mathbf nabla$ has infinitely many elements and that no two different pairs in $\mathbb Mathcal C_\mathbb Mathbf nabla$ have the first components that are equal.
Suppose on the contrary that $\mathbb Mathcal C_\mathbb Mathbf nabla$ has finitely many elements. Then $w^*\in\mathbb Mathcal A\subset\mathbb Mathcal X=\bar{\mathbb Mathcal C}_\mathbb Mathbf nabla=\mathbb Mathcal C_\mathbb Mathbf nabla$, and so $u^*$ itself is a weak solution to (\ref{ib-P}) satisfying (a)--(d); this is a contradiction. Thus $\mathbb Mathcal C_\mathbb Mathbf nabla$ has infinitely many elements. Next, we check that for any two $w_1=(u_1,v_1),w_2=(u_2,v_2)\in\mathbb Mathcal C_\mathbb Mathbf nabla,$
\[
u_1=u_2\quad \Leftrightarrow\quad v_1=v_2.
\]
Suppose $u_1\equiv u_2$ in $\Omega_T$.
As $\mathbb Mathbf nabla w_1,\mathbb Mathbf nabla w_2\in\Sigma_\sigma$ a.e. in $\Omega_T$, we have, in particular, that
\[
(v_1)_x=(u_1)_t=(u_2)_t=(v_2)_x\;\;\mathbb Mbox{a.e. in $\Omega_T$.}
\]
Since both $v_1$ and $v_2$ share the same trace $v^*$ on $\partial\Omega_T$, it follows that $v_1\equiv v_2$ in $\Omega_T$. The converse can be shown similarly. We can now conclude that there are infinitely many weak solutions to (\ref{ib-P}) satisfying (a)--(d).
The proof of Theorem \ref{thm:main-NCE} \textbf{(Case I)}, Theorem \ref{thm:main-HEP} \textbf{(Case II)} is now complete under the density fact, Theorem \ref{thm:density}, to be justified in sections \ref{sec:rank-1} and \ref{sec:density-proof}.
\subsection{Proofs of Corollaries \ref{coro:weak-NCE}, \ref{coro:weak-NCE-micro}, \ref{coro:weak-HEP} and \ref{coro:weak-HEP-micro}}
We proceed the proofs of the companion versions of Corollaries \ref{coro:weak-NCE} and \ref{coro:weak-HEP} and Corollaries \ref{coro:weak-NCE-micro} and \ref{coro:weak-HEP-micro}, respectively.
\begin{proof}[Proof of Corollaries \ref{coro:weak-NCE} and \ref{coro:weak-HEP}]
Let $(g,h)\in W^{3,2}_0(\Omega)\times W^{2,2}_0(\Omega)$ be any given initial datum, with $g'>-1$ on $\bar\Omega$ for \textbf{Case I}. If $g'(x_0)\in(s_1^*,s_2^*)$ for some $x_0\in\Omega$, then the result follows immediately from Theorem \ref{thm:main-NCE} \textbf{(Case I)}, Theorem \ref{thm:main-HEP} \textbf{(Case II)}.
Next, let us assume $g'(x)\mathbb Mathbf not\in(s_1^*,s_2^*)$ for all $x\in\bar\Omega.$ We may only consider the case that $g'(x)\ge s_2^*$ for all $x\in\bar\Omega$ as the other case can be shown similarly. Fix any two $\sigma(s_2)<r_1<r_2<\sigma(s_1),$ and choose a function $\sigma^*\in C^3(-1,\infty)$ \textbf{(Case I)}, $\sigma^*\in C^3(\mathbb R)$ \textbf{(Case II)} in such a way that (\ref{modi}) is fulfilled. By \cite[Theorem 5.2]{DH} \textbf{(Case I)}, \cite[Theorem 5.1]{DH} \textbf{(Case II)}, there exists a finite number $\tilde T>0$ such that the modified initial-boundary value problem (\ref{ib-P-modi}), with $T$ replaced by $\tilde T$, admits a unique solution $u^*\in \cap_{k=0}^3C^k([0,\tilde T];W^{3-k,2}_0(\Omega))$, with $u^*_x>-1$ on $\bar\Omega_{\tilde T}$ for \textbf{Case I}. Now, choose a number $0<T\le\tilde T$ so that $u^*_x\ge s_+(r_2)$ on $\bar\Omega_T$. Then $u^*$ itself is a classical and thus weak solution to problem (\ref{ib-P}).
\end{proof}
\begin{proof}[Proof of Corollaries \ref{coro:weak-NCE-micro} and \ref{coro:weak-HEP-micro}]
Let $(g,h)\in W^{3,2}_0(\Omega)\times W^{2,2}_0(\Omega)$ satisfy $\mathbb Max_{\bar\Omega}g'\in (s_1^*,s_1)$ or $\mathbb Min_{\bar\Omega}g'\in (s_2,s_2^*)$. In \textbf{Case I}, assume also $g'>-1$ on $\bar\Omega.$ We may only consider the case that $M:=\mathbb Max_{\bar\Omega}g'\in (s_1^*,s_1)$ as the other case can be handled in a similar way. Choose two numbers $\sigma(s_2)<r_1<r_2<\sigma(s_1)$ so that $s_-(r_1)>M.$ Then take a $C^3$ function $\sigma^*(s)$ satisfying (\ref{modi}). Using \cite[Theorem 5.2]{DH} \textbf{(Case I)}, \cite[Theorem 5.1]{DH} \textbf{(Case II)}, we can find a finite number $\tilde T>0$ such that modified problem (\ref{ib-P-modi}), with $T$ replaced by $\tilde T$, has a unique solution $u^*\in \cap_{k=0}^3C^k([0,\tilde T];W^{3-k,2}_0(\Omega))$, with $u^*_x>-1$ on $\bar\Omega_{\tilde T}$ for \textbf{Case I}. Then choose a number $0<T'\le\tilde T$ so small that $u^*_x\le s_-(r_1)$ on $\bar\Omega_{T'}$ and that $s_1^*<u^*_x(x_0,T')$ for some $x_0\in\Omega$. With the initial datum $(u^*(\cdot,T'),u^*_t(\cdot,T'))\in W^{3,2}_0(\Omega)\times W^{2,2}_0(\Omega)$ at $t=T'$, with $u^*_x(\cdot,T')>-1$ on $\bar\Omega$ for \textbf{Case I}, we can apply Theorem \ref{thm:main-NCE} \textbf{(Case I)}, Theorem \ref{thm:main-HEP} \textbf{(Case II)} to obtain, for some finite number $T>T'$, infinitely many weak solutions $\tilde u\in W^{1,\infty}(\Omega\times(T',T))$ to the initial-boundary value problem
\begin{equation*}
\begin{cases} \tilde u_{tt} =\sigma(\tilde u_x)_x& \mathbb Mbox{in $\Omega\times (T',T)$,} \\
\tilde u(0,t)=\tilde u(1,t)=0 & \mathbb Mbox{for $t\in(T',T)$,}\\
\tilde u =u^*,\,\tilde u_t=u^*_t & \mathbb Mbox{on $\Omega\times \{t=T'\}$}
\end{cases}
\end{equation*}
satisfying the stated properties in the theorem.
Then the glued functions $u=u^*\chi_{\Omega\times(0,T')}+\tilde u\chi_{\Omega\times[T',T)}$ are weak solutions to problem (\ref{ib-P}) fulfilling the required properties.
\end{proof}
\section{Rank-one smooth approximation under linear constraint}\label{sec:rank-1}
In this section, we prepare the main tool, Theorem \ref{thm:rank-1}, for proving the density result, Theorem \ref{thm:density}. Instead of presenting a special case that would be enough for our purpose, we exhibit the following result in a generalized and refined form of \cite[Lemma 2.1]{Po} that may be of independent interest (cf. \cite[Lemma 6.2]{MSv1}).
\begin{thm}\label{thm:rank-1}
Let $m,n\ge 2$ be integers, and let $A,B\in\mathbb Mathbb M^{m\times n}$ be such that $\operatorname{rank}(A-B)=1$; hence
\[
A-B=a\otimes b=(a_i b_j)
\]
for some non-zero vectors $a\in\mathbb R^m$ and $b\in\mathbb R^n$ with $|b|=1.$
Let $L\in\mathbb Mathbb M^{m\times n}$ satisfy
\begin{equation}\label{rank-1-1}
Lb\mathbb Mathbf ne 0 \;\;\mathbb Mbox{in}\;\;\mathbb R^m,
\end{equation}
and let $\mathbb Mathcal{L}:\mathbb Mathbb M^{m\times n}\to \mathbb R$ be the linear map defined by
\[
\mathbb Mathcal{L}(\xi)=\sum_{1\le i\le m,\, 1\le j \le n} L_{ij}\xi_{ij}\quad \forall \xi\in\mathbb Mathbb M^{m\times n}.
\]
Assume $\mathbb Mathcal{L}(A)=\mathbb Mathcal{L}(B)$ and $0<\lambda<1$ is any fixed number. Then there exists a linear partial differential operator $\Phi:C^1(\mathbb R^n;\mathbb R^m)\to C^0(\mathbb R^n;\mathbb R^m)$ satisfying the following properties:
(1) For any open set $\Omega\subset\mathbb R^n$,
\[
\Phi v\in C^{k-1}(\Omega;\mathbb R^m)\;\;\mathbb Mbox{whenever}\;\; k\in\mathbb Mathbb N\;\;\mathbb Mbox{and}\;\;v\in C^{k}(\Omega;\mathbb R^m)
\]
and
\[
\mathbb Mathcal{L}(\mathbb Mathbf nabla\Phi v)=0 \;\;\mathbb Mbox{in}\;\;\Omega\;\;\forall v\in C^2(\Omega;\mathbb R^m).
\]
(2) Let $\Omega\subset\mathbb R^n$ be any bounded domain. For each $\tau>0$, there exist a function $g=g_\tau\in C^{\infty}_{c}(\Omega;\mathbb R^m)$ and two disjoint open sets $\Omega_A,\Omega_B\subset\subset\Omega$ such that
\begin{itemize}
\item[(a)] $\Phi g\in C^\infty_c(\Omega;\mathbb R^m)$,
\item[(b)] $\operatorname{dist}(\mathbb Mathbf nabla\Phi g,[-\lambda(A-B),(1-\lambda)(A-B)])<\tau$ in $\Omega$,
\item[(c)] $\mathbb Mathbf nabla \Phi g(x)= \left\{\begin{array}{ll}
(1-\lambda)(A-B) & \mathbb Mbox{$\forall x\in\Omega_A$}, \\
-\lambda(A-B) & \mathbb Mbox{$\forall x\in\Omega_B$},
\end{array}\right.$
\item[(d)] $||\Omega_A|-\lambda|\Omega||<\tau$, $||\Omega_B|-(1-\lambda)|\Omega||<\tau$,
\item[(e)] $\|\Phi g\|_{L^\infty(\Omega)}<\tau$,
\end{itemize}
where $[-\lambda(A-B),(1-\lambda)(A-B)]$ is the closed line segment in $\mathbb Mathrm{ker}\mathbb Mathcal{L}\subset\mathbb Mathbb M^{m\times n}$ joining $-\lambda(A-B)$ and $(1-\lambda)(A-B)$.
\end{thm}
\begin{proof}
We mainly follow and modify the proof of \cite[Lemma 2.1]{Po} which is divided into three cases.
Set $r=\operatorname{rank}(L).$ By (\ref{rank-1-1}), we have $1\le r\le m\wedge n=:\mathbb Min\{m,n\}.$
\textbf{(Case 1):} Assume that the matrix $L$ satisfies
\[
\begin{split}
L_{ij}=0\;\;& \mathbb Mbox{for all $1\le i\le m,\, 1\le j\le n$ but possibly the pairs}\\
& \mathbb Mbox{$(1,1),(1,2),\cdots,(1,n),(2,2),\cdots,(r,r)$ of $(i,j)$};
\end{split}
\]
hence $L$ is of the form
\begin{equation}\label{rank-1-5}
L=\begin{pmatrix} L_{11} & L_{12} & \cdots & L_{1r} & \cdots & L_{1n}\\
& L_{22} & & & & & \\
& & \ddots & & & & \\
& & & L_{rr} & & & \\
& & & & & & \end{pmatrix}\in\mathbb Mathbb M^{m\times n}
\end{equation}
and that
\[
A-B=a\otimes e_1\;\;\mathbb Mbox{for some nonzero vector $a=(a_1,\cdots,a_m)\in\mathbb R^m$},
\]
where each blank component in (\ref{rank-1-5}) is zero.
From (\ref{rank-1-1}) and $\operatorname{rank}(L)=r$, it follows that the product $L_{11}\cdots L_{rr}\mathbb Mathbf ne 0$. Since $0=\mathbb Mathcal L(A-B)=\mathbb Mathcal L(a\otimes e_1)=L_{11}a_1$, we have $a_1=0$.
In this case, the linear map $\mathbb Mathcal L:\mathbb Mathbb M^{m\times n}\to\mathbb R$ is given by
\[
\mathbb Mathcal L(\xi)=\sum_{j=1}^n L_{1j}\xi_{1j}+\sum_{i=2}^r L_{ii}\xi_{ii},\quad \xi\in\mathbb Mathbb M^{m\times n}.
\]
We will find a linear differential operator $\Phi:C^1(\mathbb R^n;\mathbb R^m)\to C^0(\mathbb R^n;\mathbb R^m)$ such that
\begin{equation}\label{rank-1-6}
\mathbb Mathcal L(\mathbb Mathbf nabla\Phi v)\equiv 0 \quad\forall v\in C^2(\mathbb R^n;\mathbb R^m).
\end{equation}
So our candidate for such a $\Phi=(\Phi^1,\cdots,\Phi^m)$ is of the form
\begin{equation}\label{rank-1-7}
\Phi^i v=\sum_{1\le k\le m,\,1\le l\le n}a^i_{kl}v^k_{x_l},
\end{equation}
where $1\le i\le m$, $v\in C^1(\mathbb R^n;\mathbb R^m)$, and $a^i_{kl}$'s are real constants to be determined; then for $v\in C^2 (\mathbb R^n;\mathbb R^m)$, $1\le i\le m$, and $1\le j\le n$,
\[
\partial_{x_j}\Phi^i v =\sum_{1\le k\le m,\,1\le l\le n}a^i_{kl}v^k_{x_lx_j}.
\]
Rewriting (\ref{rank-1-6}) with this form of $\mathbb Mathbf nabla\Phi v$ for $v\in C^2 (\mathbb R^n;\mathbb R^m)$, we have
\[
\begin{split}
0 & \equiv \sum_{1\le k\le m,\,1\le j,l\le n} L_{1j}a^1_{kl}v^k_{x_lx_j} + \sum_{i=2}^r\sum_{1\le k\le m,\,1\le l\le n} L_{ii}a^i_{kl}v^k_{x_lx_i}\\
& = \sum_{k=1}^m \mathbf Big(L_{11}a^1_{k1}v^k_{x_1x_1}+\sum_{j=2}^r (L_{1j}a^1_{kj}+L_{jj}a^j_{kj})v^k_{x_jx_j}+\sum_{j=r+1}^n L_{1j}a^1_{kj}v^k_{x_jx_j} \\
& \quad +\sum_{l=2}^r (L_{11}a^1_{kl}+L_{1l}a^1_{k1}+L_{ll}a^l_{k1})v^k_{x_lx_1} +\sum_{l=r+1}^n (L_{11}a^1_{kl}+L_{1l}a^1_{k1})v^k_{x_lx_1} \\
& \quad +\sum_{2\le j<l\le r} (L_{1j}a^1_{kl}+L_{1l}a^1_{kj}+L_{jj}a^j_{kl}+L_{ll}a^l_{kj})v^k_{x_lx_j}\\
& \quad +\sum_{2\le j\le r,\,r+1\le l\le n} (L_{1j}a^1_{kl}+L_{1l}a^1_{kj}+L_{jj}a^j_{kl})v^k_{x_lx_j}
\end{split}
\]
\[
\begin{split}
& \quad+\sum_{r+1\le j<l\le n} (L_{1j}a^1_{kl}+L_{1l}a^1_{kj})v^k_{x_lx_j} \mathbf Big).
\end{split}
\]
Should (\ref{rank-1-6}) hold, it is thus sufficient to solve the following algebraic system for each $k=1,\cdots,m$ (after adjusting the letters for some indices):
\begin{eqnarray}
\label{rr-1}& L_{11}a^1_{k1}=0, &\\
\label{rr-4}& L_{1j}a^1_{kj}+L_{jj}a^j_{kj}=0 & \;\,\forall j=2,\cdots, r,\\
\label{rr-3}& L_{11}a^1_{kj}+L_{1j}a^1_{k1}+L_{jj}a^j_{k1}=0 & \;\,\forall j=2,\cdots, r, \\
\label{rr-5}& L_{1l}a^1_{kj}+L_{1j}a^1_{kl}+L_{ll}a^l_{kj}+L_{jj}a^j_{kl}=0 & \begin{array}{l}
\forall j=3,\cdots, r, \\
\forall l=2,\cdots, j-1,
\end{array} \\
\label{rr-6}& L_{1j}a^1_{kj}=0 & \;\,\forall j=r+1,\cdots, n,\\
\label{rr-7}& L_{11}a^1_{kj}+L_{1j}a^1_{k1}=0 & \;\,\forall j=r+1,\cdots, n, \\
\label{rr-8}& L_{1l}a^1_{kj}+L_{1j}a^1_{kl}+L_{ll}a^l_{kj}=0 & \begin{array}{l}
\forall j=r+1,\cdots, n, \\
\forall l=2,\cdots, r,
\end{array} \\
\label{rr-2}& L_{1l}a^1_{kj}+L_{1j}a^1_{kl}=0 & \begin{array}{l}
\forall j=r+2,\cdots, n, \\
\forall l=r+1,\cdots,j-1.
\end{array}
\end{eqnarray}
Although these systems have infinitely many solutions, we will solve those in a way for a later purpose that the matrix $(a^j_{k1})_{2\le j, k\le m}\in\mathbb Mathbb M^{(m-1)\times(m-1)}$ fulfills
\begin{equation}\label{rank-1-8}
a^j_{21}=a_j \quad\forall j=2,\cdots, m,\quad\mathbb Mbox{and}\quad a^j_{k1}=0\quad\mathbb Mbox{otherwise}.
\end{equation}
Firstly, we let the coefficients $a^i_{kl}\;(1\le i,k\le m,\,1\le l\le n)$ that do not appear in systems (\ref{rr-1})--(\ref{rr-2}) $(k=1,\cdots, m)$ be zero with an exception that we set $a^j_{21}=a_j$ for $j=r+1,\cdots,m$ to reflect (\ref{rank-1-8}). Secondly, for $1\le k\le m,\,k\mathbb Mathbf ne 2$, let us take the trivial (i.e., zero) solution of system (\ref{rr-1})--(\ref{rr-2}). Finally, we take $k=2$ and solve system (\ref{rr-1})--(\ref{rr-2}) as follows with (\ref{rank-1-8}) satisfied.
Since $L_{11}\mathbb Mathbf ne 0$, we set $a^1_{21}=0$; then (\ref{rr-1}) is satisfied. So we set
\[
a^j_{21}=-\frac{L_{11}}{L_{jj}}a^1_{2j},\;\;a^1_{2j}=-\frac{L_{jj}}{L_{11}}a_j \quad \forall j=2,\cdots,r;
\]
then (\ref{rr-3}) and (\ref{rank-1-8}) hold. Next, set
\[
a^j_{2j}=-\frac{L_{1j}}{L_{jj}}a^1_{2j}=\frac{L_{1j}}{L_{11}}a_j \quad\forall j=2,\cdots, r;
\]
then (\ref{rr-4}) is fulfilled. Set
\[
a^l_{2j}=-\frac{L_{1l}a^1_{2j}+L_{1j}a^1_{2l}}{L_{ll}}=\frac{L_{1l}L_{jj}a_j+L_{1j}L_{ll}a_l}{L_{ll}L_{11}},\;\;a^j_{2l}=0
\]
for $j=3,\cdots,r$ and $l=2,\cdots, j-1$; then (\ref{rr-5}) holds. Set
\[
a^1_{2j}=0\quad\forall j=r+1,\cdots,n;
\]
then (\ref{rr-6}) and (\ref{rr-7}) are satisfied. Lastly, set
\[
a^1_{2j}=0,\;\; a^l_{2j}=-\frac{L_{1j}}{L_{ll}}a^1_{2l}=\frac{L_{1j}}{L_{11}}a_l\quad\forall j=r+1,\cdots, n,\,\forall l=2,\cdots, r;
\]
then (\ref{rr-8}) and (\ref{rr-2}) hold. In summary, we have determined the coefficients $a^i_{kl}\;(1\le i,k\le m,\,1\le l\le n)$ in such a way that system (\ref{rr-1})--(\ref{rr-2}) holds for each $k=1,\cdots, m$ and that (\ref{rank-1-8}) is also satisfied. Therefore, (1) follows from (\ref{rank-1-6}) and (\ref{rank-1-7}).
To prove (2), without loss of generality, we can assume $\Omega=(0,1)^n\subset\mathbb R^n.$ Let $\tau>0$ be given. Let $u=(u^1,\cdots,u^m)\in C^\infty(\Omega;\mathbb R^m)$ be a function to be determined. Suppose $u$ depends only on the first variable $x_1\in(0,1).$ We wish to have
\[
\mathbb Mathbf nabla\Phi u(x)\in\{-\lambda a\otimes e_1,(1-\lambda) a\otimes e_1\}
\]
for all $x\in\Omega$ except in a set of small measure. Since $u(x)=u(x_1)$, it follows from (\ref{rank-1-7}) that for $1\le i\le m$ and $1\le j\le n$,
\[
\Phi^i u=\sum_{k=1}^m a^i_{k1} u^k_{x_1};\;\;\mathbb Mbox{thus}\;\;\partial_{x_j}\Phi^i u=\sum_{k=1}^m a^i_{k1} u^k_{x_1 x_j}.
\]
As $a^1_{k1}=0$ for $k=1,\cdots, m$, we have $\partial_{x_j}\Phi^1 u =\sum_{k=1}^m a^1_{k1} u^k_{x_1 x_j}=0$ for $j=1,\cdots,n$. We first set $u^1\equiv 0$ in $\Omega$. Then from (\ref{rank-1-8}), it follows that for $i=2,\cdots, m$,
\[
\partial_{x_j}\Phi^i u =\sum_{k=2}^m a^i_{k1} u^k_{x_1 x_j}=a^i_{21} u^2_{x_1 x_j}=a_i u^2_{x_1 x_j} = \left\{\begin{array}{ll}
a_i u^2_{x_1 x_1} & \mathbb Mbox{if $j=1$,} \\
0 & \mathbb Mbox{if $j=2,\cdots, n$.}
\end{array} \right.
\]
As $a_1=0$, we thus have that for $x\in\Omega$,
\[
\mathbb Mathbf nabla\Phi u(x)=(u^2)''(x_1) a\otimes e_1.
\]
For irrelevant components of $u$, we simply take $u^3=\cdots =u^m\equiv 0$ in $\Omega$. Lastly, for a number $\delta>0$ to be chosen later, we choose a function $u^2(x_1)\in C^\infty_c(0,1)$ such that there exist two disjoint open sets $I_1,I_2\subset\subset (0,1)$ satisfying $||I_1|-\lambda|<\tau/2$, $||I_2|-(1-\lambda)|<\tau/2$, $\|u^2\|_{L^\infty(0,1)}<\delta$, $\|(u^2)'\|_{L^\infty(0,1)}<\delta$, $-\lambda\le (u^2)''(x_1)\le 1-\lambda$ for $x_1\in(0,1)$, and
\[
(u^2)''(x_1)= \left\{\begin{array}{ll}
1-\lambda & \mathbb Mbox{if $x_1\in I_1$}, \\
-\lambda & \mathbb Mbox{if $x_1\in I_2$}.
\end{array}
\right.
\]
In particular,
\begin{equation}\label{rank-1-9}
\mathbb Mathbf nabla \Phi u(x)\in[-\lambda a\otimes e_1,(1-\lambda)a\otimes e_1]\;\;\forall x\in\Omega.
\end{equation}
We now choose an open set $\Omega'_\tau\subset\subset\Omega':=(0,1)^{n-1}$ with $|\Omega'\setminus\Omega'_\tau|<\tau/2$ and a function $\eta\in C^\infty_c(\Omega')$ so that
\[
0\le\eta\le 1\;\;\mathbb Mbox{in}\;\;\Omega',\;\; \eta\equiv 1\;\;\mathbb Mbox{in}\;\Omega'_\tau,\;\;\mathbb Mbox{and}\;\;|\mathbb Mathbf nabla^i_{x'}\eta|<\frac{C}{\tau^i}\;\;(i=1,2)\;\;\mathbb Mbox{in}\;\Omega',
\]
where $x'=(x_2,\cdots,x_n)\in\Omega'$ and the constant $C>0$ is independent of $\tau$.
Now, we define $g(x)=\eta(x') u(x_1)\in C^\infty_c(\Omega;\mathbb R^m)$. Set $\Omega_A=I_1\times\Omega'_\tau$ and $\Omega_B=I_2\times\Omega'_\tau.$ Clearly, (a) follows from (1). As $g(x)=u(x_1)=u(x)$ for $x\in \Omega_A\cup\Omega_B$, we have
\[
\mathbb Mathbf nabla\Phi g(x)=\left\{\begin{array}{ll}
(1-\lambda)a\otimes e_1 & \mathbb Mbox{if $x\in \Omega_A$}, \\
-\lambda a\otimes e_1 & \mathbb Mbox{if $x\in \Omega_B$};
\end{array}
\right.
\]
hence (c) holds. Also,
\[
||\Omega_A|-\lambda|\Omega||=||\Omega_A|-\lambda|=||I_1||\Omega'_\tau|-\lambda|=||I_1|-|I_1||\Omega'\setminus\Omega'_\tau|-\lambda|<\tau,
\]
and likewise
\[
||\Omega_B|-(1-\lambda)|\Omega||<\tau;
\]
so (d) is satisfied.
Note that for $i=1,\cdots,m,$
\[
\begin{split}
\Phi^i g & = \Phi^i(\eta u) = \sum_{1\le k\le m,\,1\le l\le n}a^i_{kl}(\eta u^k)_{x_l}=\eta\Phi^i u+\sum_{1\le k\le m,\,1\le l\le n}a^i_{kl}\eta_{x_l} u^k\\
& = \eta\Phi^i u+ u^2\sum_{l=1}^n a^i_{2l}\eta_{x_l} =\eta a^i_{21}u^2_{x_1} + u^2\sum_{l=1}^n a^i_{2l}\eta_{x_l}.
\end{split}
\]
So
\[
\|\Phi g\|_{L^\infty(\Omega)}\le C\mathbb Max\{\delta,\delta\tau^{-1}\}<\tau
\]
if $\delta>0$ is chosen small enough; so (e) holds.
Next, for $i=1,\cdots,m$ and $j=1,\cdots,n,$
\[
\partial_{x_j}\Phi^i g=\eta_{x_j}a^i_{21}u^2_{x_1}+\eta\partial_{x_j}\Phi^i u + u^2_{x_j}\sum_{l=1}^n a^i_{2l}\eta_{x_l} + u^2\sum_{l=1}^n a^i_{2l}\eta_{x_l x_j};
\]
hence from (\ref{rank-1-9}),
\[
\operatorname{dist}(\mathbb Mathbf nabla\Phi g,[-\lambda a\otimes e_1,(1-\lambda) a\otimes e_1])\le C\mathbb Max\{\delta \tau^{-1},\delta\tau^{-2}\}<\tau\;\;\mathbb Mbox{in $\Omega$}
\]
if $\delta$ is sufficiently small. Thus (b) is fulfilled.
\textbf{(Case 2):} Assume that $L_{i1}=0$ for all $i=2,\cdots, m$, that is,
\begin{equation}\label{rank-1-3}
L=\begin{pmatrix} L_{11} & L_{12} & \cdots & L_{1n}\\
0 & L_{22} & \cdots & L_{2n}\\
\vdots & \vdots & \ddots & \vdots\\
0 & L_{m2} & \cdots & L_{mn} \end{pmatrix}\in\mathbb Mathbb M^{m\times n}
\end{equation}
and that
\[
A-B=a\otimes e_1\;\;\mathbb Mbox{for some nonzero vector $a\in\mathbb R^m$};
\]
then by (\ref{rank-1-1}), we have $L_{11}\mathbb Mathbf ne 0.$
Set
\[
\hat L=\begin{pmatrix} L_{22} & \cdots & L_{2n}\\
\vdots & \ddots & \vdots\\
L_{m2} & \cdots & L_{mn} \end{pmatrix}\in\mathbb Mathbb M^{(m-1)\times (n-1)}.
\]
As $L_{11}\mathbb Mathbf ne 0$ and $\operatorname{rank}(L)=r$, we must have $\operatorname{rank}(\hat L)=r-1.$ Using the singular value decomposition theorem, there exist two matrices $\hat U\in O(m-1)$ and $\hat V\in O(n-1)$ such that
\begin{equation}\label{rank-1-4}
\hat U^T\hat L\hat V=\mathbb Mathrm{diag}(\sigma_2,\cdots,\sigma_r,0,\cdots,0)\in\mathbb Mathbb M^{(m-1)\times (n-1)},
\end{equation}
where $\sigma_2,\cdots,\sigma_r$ are the positive singular values of $\hat L.$ Define
\begin{equation}\label{rank-1-2}
U=\begin{pmatrix} 1 & 0\\
0 & \hat U\end{pmatrix}\in O(m),\;\;
V=\begin{pmatrix} 1 & 0\\
0 & \hat V\end{pmatrix}\in O(n).
\end{equation}
Let $L'=U^T LV$, $A'=U^T AV$, and $B'=U^T BV$. Let $\mathbb Mathcal{L}':\mathbb Mathbb M^{m\times n}\to \mathbb R$ be the linear map given by
\[
\mathbb Mathcal{L}'(\xi')=\sum_{1\le i\le m,\,1\le j\le n}L'_{ij}\xi'_{ij}\quad \forall \xi'\in\mathbb Mathbb M^{m\times n}.
\]
Then, from (\ref{rank-1-3}), (\ref{rank-1-4}) and (\ref{rank-1-2}), it is straightforward to check the following:
\[
\left\{
\begin{array}{l}
\mathbb Mbox{$A'-B'=a'\otimes e_1$ for some nonzero vector $a'\in\mathbb R^m$,} \\
\mathbb Mbox{$L' e_1\mathbb Mathbf neq 0$, $\mathbb Mathcal L'(A)=\mathbb Mathcal L'(B)$, and} \\
\mathbb Mbox{$L'$ is of the form (\ref{rank-1-5}) in Case 1 with $\mathbb Mathrm{rank}(L')=r$.}
\end{array}\right.
\]
Thus we can apply the result of Case 1 to find a linear operator $\Phi':C^1(\mathbb R^n;\mathbb R^m)\to C^0(\mathbb R^n;\mathbb R^m)$ satisfying the following:
(1') For any open set $\Omega'\subset\mathbb R^n$,
\[
\Phi' v'\in C^{k-1}(\Omega';\mathbb R^m)\;\;\mathbb Mbox{whenever}\;\; k\in\mathbb Mathbb N\;\;\mathbb Mbox{and}\;\;v'\in C^{k}(\Omega';\mathbb R^m)
\]
and
\[
\mathbb Mathcal{L'}(\mathbb Mathbf nabla\Phi' v')=0 \;\;\mathbb Mbox{in}\;\;\Omega'\;\;\mathbb Mbox{for all}\;\;v'\in C^2(\Omega';\mathbb R^m).
\]
(2') Let $\Omega'\subset\mathbb R^n$ be any bounded domain. For each $\tau>0$, there exist a function $g'=g'_\tau\in C^{\infty}_{c}(\Omega';\mathbb R^m)$ and two disjoint open sets $\Omega'_{A'},\Omega'_{B'}\subset\subset\Omega'$ such that
\begin{itemize}
\item[(a')] $\Phi' g'\in C^\infty_c(\Omega';\mathbb R^m)$,
\item[(b')] $\operatorname{dist}(\mathbb Mathbf nabla\Phi' g',[-\lambda(A'-B'),(1-\lambda)(A'-B')])<\tau$ in $\Omega'$,
\item[(c')] $\mathbb Mathbf nabla \Phi' g'(x)= \left\{\begin{array}{ll}
(1-\lambda)(A'-B') & \mathbb Mbox{$\forall x\in\Omega'_{A'}$}, \\
-\lambda(A'-B') & \mathbb Mbox{$\forall x\in\Omega'_{B'}$},
\end{array}\right.$
\item[(d')] $||\Omega'_{A'}|-\lambda|\Omega'||<\tau$, $||\Omega'_{B'}|-(1-\lambda)|\Omega'||<\tau$,
\item[(e')] $\|\Phi' g'\|_{L^\infty(\Omega')}<\tau$.
\end{itemize}
For $v\in C^1(\mathbb R^n;\mathbb R^m)$, let $v'\in C^1(\mathbb R^n;\mathbb R^m)$ be defined by $v'(y)=U^T v(Vy)$ for $y\in\mathbb R^n$. We define $\Phi v(x)=U\Phi' v'(V^T x)$ for $x\in\mathbb R^n$, so that $\Phi v\in C^0(\mathbb R^n;\mathbb R^m).$ Then it is straightforward to check that properties (1') and (2') of $\Phi'$ imply respective properties (1) and (2) of the linear operator $\Phi:C^1(\mathbb R^n;\mathbb R^m)\to C^0(\mathbb R^n;\mathbb R^m)$.
\textbf{(Case 3):} Finally, we consider the general case that $A$, $B$ and $L$ are as in the statement of the theorem. As $|b|=1$, there exists an $R\in O(n)$ such that $R^T b=e_1\in\mathbb R^n$. Also there exists a symmetric (Householder) matrix $P\in O(m)$ such that the matrix $L':=PLR$ has the first column parallel to $e_1\in\mathbb R^m$. Let
\[
A'=PAR\;\;\mathbb Mbox{and}\;\; B'=PBR.
\]
Then $A'-B'=a'\otimes e_1$, where $a'=Pa\mathbb Mathbf ne 0$. Note also that $L'e_1=PLRR^tb=PLb\mathbb Mathbf ne 0$. Define $\mathbb Mathcal L'(\xi')=\sum_{i,j}L'_{ij}\xi'_{ij}\;(\xi'\in\mathbb Mathbb M^{m\times n})$; then $\mathbb Mathcal L'(A')=\mathbb Mathcal L(A)=\mathbb Mathcal L(B)=\mathbb Mathcal L'(B')$. Thus by the result of Case 2, there exists a linear operator $\Phi':C^1(\mathbb R^n;\mathbb R^m)\to C^0(\mathbb R^n;\mathbb R^m)$ satisfying (1') and (2') above.
For $v\in C^1(\mathbb R^n;\mathbb R^m)$, let $v'\in C^1(\mathbb R^n;\mathbb R^m)$ be defined by $v'(y)=Pv(Ry)$ for $y\in\mathbb R^n$, and define $\Phi v(x)=P\Phi'v'(R^Tx)\in C^0(\mathbb R^n;\mathbb R^m)$. Then it is easy to check that the linear operator $\Phi:C^1(\mathbb R^n;\mathbb R^m)\to C^0(\mathbb R^n;\mathbb R^m)$ satisfies (1) and (2) by (1') and (2') similarly as in Case 2.
\end{proof}
\section{Proof of density result}\label{sec:density-proof}
In this final section, we prove Theorem \ref{thm:density}, which plays a pivotal role in the proof of the main results, Theorems \ref{thm:main-NCE} and \ref{thm:main-HEP}.
To start the proof, fix a $\delta>0$ and choose any $w=(u,v)\in\mathbb Mathcal A$ so that $w\in W^{1,\infty}_{w^*}(\Omega_T;\mathbb R^2)\cap C^2(\bar\Omega_T;\mathbb R^2)$ satisfies the following:
\begin{equation}\label{func-w}
\left\{\begin{array}{l}
\mathbb Mbox{$w\equiv w^*$ in $\Omega_T\setminus\bar\Omega_T^w$ for some open set $\Omega_T^w\subset\subset\Omega_T^2$ with $|\partial\Omega_T^w|=0$,} \\
\mathbb Mbox{$\mathbb Mathbf nabla w(x,t)\in U$ for all $(x,t)\in\Omega_T^2$, and $\|u_t-h\|_{L^\infty(\Omega_{T_\epsilon})}<\epsilon'$.}
\end{array}
\right.
\end{equation}
Let $\eta>0$. Our goal is to construct a function $w_\eta=(u_\eta,v_\eta)\in \mathbb Mathcal A_\delta$ with $\|w-w_\eta\|_{L^\infty(\Omega_T)}<\eta;$ that is, a function $w_\eta\in W^{1,\infty}_{w^*}(\Omega_T;\mathbb R^2)\cap C^2(\bar\Omega_T;\mathbb R^2)$ with the following properties:
\begin{equation}\label{func-weta}
\left\{\begin{array}{l}
\mathbb Mbox{$w_\eta\equiv w^*$ in $\Omega_T\setminus\bar\Omega_T^{w_\eta}$ for some open set $\Omega_T^{w_\eta}\subset\subset\Omega_T^2$ with $|\partial\Omega_T^{w_\eta}|=0$,} \\
\mathbb Mbox{$\mathbb Mathbf nabla w_\eta(x,t)\in U$ for all $(x,t)\in\Omega_T^2$, $\|(u_\eta)_t-h\|_{L^\infty(\Omega_{T_\epsilon})}<\epsilon'$,} \\
\mathbb Mbox{$\int_{\Omega_T^2}\operatorname{dist}(\mathbb Mathbf nabla w_\eta(x,t),K)\,dxdt\le\delta|\Omega_T^2|$, and $\|w-w_\eta\|_{L^\infty(\Omega_T)}<\eta$.}
\end{array}
\right.
\end{equation}
For clarity, we divide the proof into several steps.
\textbf{(Step 1):} Choose a nonempty open set $G_1\subset\subset\Omega_T^2\setminus\partial\Omega_T^w$ with $|\partial G_1|=0$ so that
\begin{equation}\label{int-1}
\int_{(\Omega_T^2\setminus\partial\Omega_T^w)\setminus G_1}\operatorname{dist}(\mathbb Mathbf nabla w(x,t),K)\,dxdt\le\frac{\delta}{5}|\Omega_T^2|.
\end{equation}
Since $\mathbb Mathbf nabla w\in U$ on $\bar G_1$, we have $\|u_t\|_{L^\infty(G_1)}<\gamma$;
then fix a number $\theta$ with
\begin{equation}\label{theta}
0<\theta<\mathbb Min\{\epsilon'-\|u_t-h\|_{L^\infty(\Omega_{T_\epsilon})},\gamma-\|u_t\|_{L^\infty(G_1)}\}.
\end{equation}
For each $\mathbb Mu>0$, let
\[
\begin{split}
G_2^\mathbb Mu & =\{(x,t)\in G_1\,|\,\operatorname{dist}((u_x(x,t),v_t(x,t)),\partial\tilde U)>\mathbb Mu\},\\
H_2^\mathbb Mu & =\{(x,t)\in G_1\,|\,\operatorname{dist}((u_x(x,t),v_t(x,t)),\partial\tilde U)<\mathbb Mu\},\\
F_2^\mathbb Mu & =\{(x,t)\in G_1\,|\,\operatorname{dist}((u_x(x,t),v_t(x,t)),\partial\tilde U)=\mathbb Mu\}.
\end{split}
\]
Since $\lim_{\mathbb Mu\to 0^+}|H_2^\mathbb Mu|=0,$ we can find a $\mathbb Mathbf nu\in(0,\mathbb Min\{\frac{\delta}{5},\theta\})$ such that
\begin{equation}\label{int-2}
\int_{H_2^{\mathbb Mathbf nu}}\operatorname{dist}(\mathbb Mathbf nabla w(x,t),K)\,dxdt\le\frac{\delta}{5}|\Omega_T^2|,\;\;G^\mathbb Mathbf nu_2\mathbb Mathbf not=\emptyset,\;\;\mathbb Mbox{and}\;\; |F_2^{\mathbb Mathbf nu}|=0.
\end{equation}
Let us write $G_2=G_2^{\mathbb Mathbf nu}$ and $H_2=H_2^{\mathbb Mathbf nu}$. Choose finitely many disjoint open squares $B_1,\cdots, B_N\subset G_2$ parallel to the axes such that
\begin{equation}\label{int-3}
\int_{G_2\setminus(\cup_{i=1}^N B_i)}\operatorname{dist}(\mathbb Mathbf nabla w(x,t),K)\,dxdt\le\frac{\delta}{5}|\Omega_T^2|.
\end{equation}
\textbf{(Step 2):} Dividing the squares $B_1,\cdots, B_N$ into smaller disjoint sub-squares if necessary, we can assume that
\begin{equation}\label{small}
|\mathbb Mathbf nabla w(x,t)-\mathbb Mathbf nabla w(\tilde x,\tilde t)|<\frac{\mathbb Mathbf nu}{8}
\end{equation}
whenever $(x,t),(\tilde x,\tilde t)\in B_i$ and $i=1,\cdots,N$. Now, fix any $i\in\{1,\cdots,N\}$. Let $(x_i,t_i)$ denote the center of the square $B_i$, and write
\[
(s_i,r_i)=(u_x(x_i,t_i),v_t(x_i,t_i))\in \tilde U;
\]
then $\operatorname{dist}((s_i,r_i),\partial\tilde U)>\mathbb Mathbf nu$. Let $\alpha_i>0,\,\beta_i>0$ be chosen so that
\[
(s_i-\alpha_i,r_i),(s_i+\beta_i,r_i)\in\tilde U,\;\;\operatorname{dist}((s_i-\alpha_i,r_i),\tilde K_-)=\frac{\mathbb Mathbf nu}{2},\;\;\mathbb Mbox{and}
\]
\[
\operatorname{dist}((s_i+\beta_i,r_i),\tilde K_+)=\frac{\mathbb Mathbf nu}{2}.
\]
To apply Theorem \ref{thm:rank-1} with $m=n=2$ to the square $B_i$, let
\[
A_i=\begin{pmatrix} s_i-\alpha_i & b_i \\ b_i & r_i \end{pmatrix}\;\;\mathbb Mbox{and}\;\;B_i=\begin{pmatrix} s_i+\beta_i & b_i \\ b_i & r_i\end{pmatrix},
\]
where $b_i=u_t(x_i,t_i);$ then
\[
A_i-B_i=\begin{pmatrix} -\alpha_i-\beta_i & 0 \\ 0 & 0 \end{pmatrix}=\begin{pmatrix} -\alpha_i-\beta_i \\ 0 \end{pmatrix}\otimes \begin{pmatrix} 1 \\ 0 \end{pmatrix}.
\]
Let $\mathbb Mathcal L:\mathbb Mathbb M^{2\times 2}\to\mathbb R$ be the linear map defined by
\[
\mathbb Mathcal L(\xi)=-\xi_{21}+\xi_{12}\quad\forall \xi\in\mathbb Mathbb M^{2\times 2},
\]
with its associated matrix $L=\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$; then
\[
\mathbb Mathcal L(A_i)=\mathbb Mathcal L(B_i)(=0)\;\;\mathbb Mbox{and}\;\;C_i=\lambda_i A_i+(1-\lambda_i) B_i,
\]
where $C_i=\mathbb Mathbf nabla w(x_i,t_i)$ and $\lambda_i=\frac{\beta_i}{\alpha_i+\beta_i}\in(0,1).$ By Theorem \ref{thm:rank-1}, there exists a linear operator $\Phi_i:C^1(\mathbb R^2;\mathbb R^2)\to C^0(\mathbb R^2;\mathbb R^2)$ satisfying properties (1) and (2) in the statement of the theorem with $A=A_i$, $B=B_i$ and $\lambda=\lambda_i$. In particular, for the square $B_i\subset\mathbb R^2$ and a number $0<\tau<\mathbb Min\{\frac{\mathbb Mathbf nu}{8},\theta,\eta,\frac{\delta|\Omega_T^2|}{5SN}\}$ with $\begin{displaystyle} S:=\mathbb Max_{r_1\le r\le r_2} (s_+(r)-s_-(r))>0 \end{displaystyle}$, we can find a function $g_i\in C^\infty_c(B_i;\mathbb R^2)$ such that
\begin{equation}\label{patch}
\left\{\begin{array}{l}
\Phi_i g_i \in C^\infty_c(B_i;\mathbb R^2),\;\mathbb Mathcal L(\mathbb Mathbf nabla\Phi_i g_i)=0\;\;\mathbb Mbox{in $B_i$,} \\
\operatorname{dist}(\mathbb Mathbf nabla\Phi_i g_i,[-\lambda_i(A_i-B_i),(1-\lambda_i)(A_i-B_i)])<\tau\;\;\mathbb Mbox{in $B_i$,} \\
|B_i^1|<\tau, \;\|\Phi_i g_i\|_{L^\infty(B_i)}<\tau ,
\end{array}
\right.
\end{equation}
where
\[
B_i^1=\{(x,t)\in B_i\,|\,\operatorname{dist}(\mathbb Mathbf nabla\Phi_i g_i(x,t),\{-\lambda_i(A_i-B_i),(1-\lambda_i)(A_i-B_i)\})>0\}.
\]
Let $B_i^2=B_i\setminus B_i^1$.
We finally define
\[
w_\eta=w+\sum_{i=1}^N \chi_{B_i} \Phi_i g_i\;\;\mathbb Mbox{in $\Omega_T$.}
\]
\textbf{(Step 3):} Let us check that $w_\eta=(u_\eta, v_\eta)$ is indeed a desired function satisfying (\ref{func-weta}). It is clear from (\ref{func-w}) and the construction above that $w_\eta\in W^{1,\infty}_{w^*}(\Omega_T;\mathbb R^2)\cap C^2(\bar\Omega_T;\mathbb R^2)$. Set $\Omega_T^{w_\eta}=\Omega_T^{w}\cup(\cup_{i=1}^N B_i)$; then $\Omega_T^{w_\eta}\subset\subset \Omega_T^2$, $|\partial\Omega_T^{w_\eta}|=0$, and $w_\eta=w=w^*$ in $\Omega_T\setminus\bar\Omega_T^{w_\eta}$. From (\ref{small}), (\ref{patch}), $\mathbb Mathbf nu<\theta$ and $\tau<\mathbb Mathbf nu/8$, it follows that for $i=1,\cdots, N$,
\[
\mathbb Mathbf nabla w_\eta=\mathbb Mathbf nabla w+\mathbb Mathbf nabla\Phi_i g_i\in [A_i,B_i]_{\mathbb Mathbf nu/4}\subset U\;\;\mathbb Mbox{in $B_i$,}
\]
where $[A_i,B_i]_{\mathbb Mathbf nu/4}$ is the $\frac{\mathbb Mathbf nu}{4}$-neighborhood of the closed line segment $[A_i,B_i]$ in the space $\mathbb Mathbb M^{2\times 2}_{sym}$; thus $\mathbb Mathbf nabla w_\eta\in U$ in $\Omega_T^2$. By (\ref{theta}) and (\ref{patch}) with zero antidiagonal of $A_i-B_i$, we have
\[
\|(u_\eta)_t-h\|_{L^\infty(\Omega_{T_\epsilon})}\le
\|u_t-h\|_{L^\infty(\Omega_{T_\epsilon})}+\tau<\|u_t-h\|_{L^\infty(\Omega_{T_\epsilon})}+\theta<\epsilon',
\]
\[
\|w-w_\eta\|_{L^\infty(\Omega_{T})}=\|\sum_{i=1}^N \chi_{B_i} \Phi_i g_i\|_{L^\infty(\Omega_{T})}<\tau<\eta.
\]
Lastly, note that
\[
\begin{split}
\int_{\Omega_T^2} & \operatorname{dist}(\mathbb Mathbf nabla w_\eta(x,t),K)\,dxdt = \int_{(\Omega_T^2\setminus\partial\Omega_T^w)\setminus G_1}\operatorname{dist}(\mathbb Mathbf nabla w (x,t),K)\,dxdt\\
& + \int_{H_2} \operatorname{dist}(\mathbb Mathbf nabla w (x,t),K)\,dxdt + \int_{G_2\setminus(\cup_{i=1}^N B_i)} \operatorname{dist}(\mathbb Mathbf nabla w (x,t),K)\,dxdt\\
& +\sum_{i=1}^N \int_{B_i} \operatorname{dist}(\mathbb Mathbf nabla w (x,t)+\mathbb Mathbf nabla \Phi_i g_i(x,t),K)\,dxdt=:I_1+I_2+I_3+I_4.
\end{split}
\]
Observe here that for $i=1,\cdots,N$,
\[
\begin{split}
\int_{B_i} \operatorname{dist}( & \mathbb Mathbf nabla w +\mathbb Mathbf nabla \Phi_i g_i,K)\,dxdt = \int_{B_i^1} \operatorname{dist}(\mathbb Mathbf nabla w +\mathbb Mathbf nabla \Phi_i g_i,K)\,dxdt \\
& + \int_{B_i^2} \operatorname{dist}(\mathbb Mathbf nabla w (x,t)+\mathbb Mathbf nabla \Phi_i g_i,K)\,dxdt \\
& \le S|B_i^1| + \mathbb Mathbf nu|B_i^2| \le S\tau + \frac{\delta}{5}|B_i^2|\le \frac{\delta|\Omega_T^2|}{5N} + \frac{\delta}{5}|B_i^2|.
\end{split}
\]
Thus $I_4\le \frac{2\delta}{5}|\Omega_T^2|$; whence with (\ref{int-1}), (\ref{int-2}) and (\ref{int-3}), we have $I_1+I_2+I_3+I_4\le (\frac{\delta}{5}+\frac{\delta}{5}+\frac{\delta}{5}+\frac{2\delta}{5})|\Omega_T^2|=\delta|\Omega_T^2|$.
Therefore, (\ref{func-weta}) is proved, and the proof is complete.
\end{document} |
\begin{document}
\title{An MsFEM type approach for \\ perforated domains}
\author{Claude Le Bris$^1$, Fr\'ed\'eric Legoll$^1$, Alexei Lozinski$^{2}$\\
{\footnotesize $^1$ \'Ecole Nationale des Ponts et
Chauss\'ees,}\\
{\footnotesize 6 et 8 avenue Blaise Pascal, 77455 Marne-La-Vall\'ee
Cedex 2, FRANCE}\\
{\footnotesize and}\\
{\footnotesize INRIA Rocquencourt, MICMAC project-team,}\\
{\footnotesize 78153 Le Chesnay Cedex, FRANCE}\\
{\footnotesize\tt [email protected], [email protected]}\\
{\footnotesize $^2$ Formerly at Institut de Math\'ematiques de Toulouse,}\\
{\footnotesize Universit\'e Paul Sabatier,}\\
{\footnotesize 118 route de Narbonne, 31062 Toulouse Cedex 9, FRANCE}\\
{\footnotesize Now at Laboratoire de Math\'ematiques CNRS UMR 6623,}\\
{\footnotesize Universit\'e de Franche-Comt\'e, 16 route de Gray, 25030
Besan\c con Cedex, FRANCE}\\
{\footnotesize \tt [email protected]}
}
\maketitle
\begin{abstract}
We follow up on our previous work~\cite{companion-article} where we
have studied a multiscale finite element (MsFEM) type method in
the vein of the classical Crouzeix-Raviart finite element method that is
specifically adapted for highly oscillatory elliptic problems. We adapt
the approach to address here a multiscale problem on a perforated
domain. An additional ingredient of our approach is the enrichment of
the multiscale
finite element space using bubble functions. We first establish a
theoretical error
estimate. We next show that, on the
problem we consider, the approach we propose outperforms all
dedicated existing variants of MsFEM we are aware of.
\end{abstract}
\section{Introduction}
\label{sec:introduction}
\subsection{Generalities}
\label{ssec:Generalities}
We consider a bounded domain~$\Omega \subset \mathbb{R}^d$ and a set $B_\varepsilon$ of
perforations within this domain. The perforations are supposedly small
and in extremely large a number. The parameter~$\varepsilon$ stands here
for a typical distance between the perforations. We denote
by~$\Omega_\varepsilon=\Omega \setminus \overline{B_\varepsilon}$ the perforated
domain (see Figure~\ref{fig:perforation}). We then consider the following problem: find $u :
\Omega_\varepsilon \rightarrow \mathbb{R}$, solution of
\begin{equation}
\label{eq:genP}
-\Delta u = f \text{ in $\Omega_\varepsilon$},
\quad
u = 0\text{ on $\partial \Omega_\varepsilon$},
\end{equation}
where $f:\Omega \rightarrow \mathbb{R}$ is a given function, assumed
sufficiently regular on $\Omega$. It is important to note
that the homogeneous Dirichlet boundary condition on $\partial \Omega_\varepsilon$ (and hence on the boundary $\Omega \cap \partial
B_\varepsilon$ of the perforations) is a crucial feature of the problem we
consider. Our academic enterprise is motivated by various physically relevant
problems, for instance in fluid mechanics, atmospheric modeling, electrostatic devices, \dots A different boundary condition, such as a
Neumann boundary condition, would lead to completely different
theoretical considerations and, eventually, a different numerical
approach.
The consideration of~\eqref{eq:genP} can also be seen as a
step toward the resolution of the Stokes problem on perforated
domains. In that latter case, homogeneous Dirichlet boundary conditions on the
perforations are typical for many applicative contexts.
\begin{figure}
\caption{The domain $\Omega$ contains perforations $B_\varepsilon$. The perforated domain is $\Omega_\varepsilon=\Omega \setminus \overline{B_\varepsilon}
\label{fig:perforation}
\end{figure}
Our purpose here is to propose and study a dedicated multiscale finite element
method (MsFEM). To this end, we consider the variant
of MsFEM using \emph{Crouzeix-Raviart type} finite
elements~\cite{Crouzeix-Raviart} which we have employed
and studied for a prototypical multiscale elliptic problem
in~\cite{companion-article} and we adapt the approach for the particular
setting under consideration here. The major adaptation we perform (and
thus one of the added values with respect to our earlier
work~\cite{companion-article}) is the addition of \emph{bubble
functions} to the finite element basis set.
Let us briefly comment upon the motivation for these two ingredients:
Crouzeix-Raviart type finite element on the one hand, and addition of
bubble functions on the other hand.
The motivation for using
Crouzeix-Raviart type finite elements stems from our wish to devise a
numerical approach as accurate as possible for a limited
computational workload. In general, it is well known that, for the construction of
multiscale finite elements, boundary conditions set on the edges
(facets) of mesh elements for the definition of the basis functions play
a critical role for the eventual accuracy and efficiency of the
approach. Using Crouzeix-Raviart type elements
(see~\cite{Crouzeix-Raviart} for their original introduction) gives a
definite flexibility. In short, the continuity of our multiscale finite
element basis set functions accross the edges of the mesh is enforced
only in a weak sense by requiring that the average of the jump
vanishes on each edge. This ``weak'' continuity condition leads to some
natural boundary conditions for the multiscale basis functions (see Section~\ref{ssec:msfem}). The
nonconforming approximation obtained in this manner proves to be very
effective, see~\cite{companion-article}.
The above issue regarding boundary conditions on the mesh elements is all the more crucial when dealing with perforated
computational domains. Indeed, we want the approach we construct to
be as insensitive as
possible to the possible intersections between element edges
and perforations. The long term motivation for this is the wish to
address problems where the perforations can be very heterogeneously
distributed (think, say, of nonperiodic, or even
random arrays of perforations). The {\it ad hoc} construction of a mesh that
(essentially) avoids intersecting the perforations is then prohibitively
difficult.
The second ingredient of our approach is the addition of bubble
functions to the finite element space. As illustrated using a simple
one-dimensional analysis in Section~\ref{ssec:1D}, and demonstrated with an
extensive set of numerical tests in Section~\ref{sec:Numerical-tests}
for all MsFEM type approches we implemented, the addition of bubble
functions is definitely benefitial for the overall accuracy of the
approach.
The literature on the types of problems and techniques considered here is of course too vast to be recalled here. A
quite general review is contained in our earlier
work~\cite{companion-article}. We however wish to mention here the
references~\cite{arbogast-review,Efendiev-Hou-book} for the general
background on MsFEM, and the works~\cite{cdz,cm,cs,henning,hornung,lions1980}
specifically addressing problems on perforated domains, either from a
theoretical or a numerical standpoint.
The outline of our article is as follows. As already briefly mentioned,
the rest of this introduction, namely Section~\ref{ssec:1D}, is devoted
to the study of a simple one-dimensional situation. From
Section~\ref{sec:presentation2D} on, we work in two dimensions
throughout the article, both for the analysis and for the numerical
tests of the final section. We however emphasize that, of course, the
approach can be applied to the three-dimensional context and that, most
likely, the theoretical analysis we provide here can also be extended to
the three dimensional case (Note that our analysis
in~\cite{companion-article} was performed in both the two and three
dimensional settings). We will not proceed in this direction
here. In addition and for simplicity, we assume that $\Omega$ is a polygonal domain. Section~\ref{sec:presentation2D} presents our finite element
approach and the main result of numerical analysis
(Theorem~\ref{theo:main}) we are able to prove, under restrictive
assumptions made precise below (in particular, periodicity of the
perforations is assumed, although, in practice, the approach is {\em not} restricted to this setting). Section~\ref{sec:preliminaries} prepares
the ground for the proof of this main result, performed in
Section~\ref{sec:proof}. Our final Section~\ref{sec:Numerical-tests}
then presents a comprehensive set of numerical experiments. When using our
MsFEM approach on a perforated domain, there are essentially
three ``parameters'': (i) the boundary conditions imposed to define the
MsFEM basis functions, (ii) the addition, or not, of bubble functions and
(iii) the possible intersections of the perforations with the edges (facets)
of mesh elements. Assessing the validity of our approach requires to
compare it with the other existing approaches for all possible
combinations of the above three ``parameters''. This is what we complete in
Section~\ref{sec:Numerical-tests}. Our tests demonstrate that the
combination of Crouzeix-Raviart type finite elements and bubble
functions allows to outperform all the other existing approaches on the
problem considered here in a way that is essentially insensitive to
intersections of the mesh with the perforations.
\subsection{A one-dimensional situation}
\label{ssec:1D}
In order to illustrate the specificity of multiscale perforated
problems, and to already discover some interesting features, we first
consider an academic one-dimensional setting. Consider the
one-dimensional version of the boundary value problem~\eqref{eq:genP}
for~$\Omega =(0,L)$, $B_\varepsilon$ the set of
segments $B_\varepsilon =\cup_{j=1}^J (a_j,b_j)$ with
$0<a_1<b_1<a_2<b_2<\cdots <L$. We suppose that the gaps between the
perforations are of length at most $\varepsilon$, that is $a_1 \leq
\varepsilon$, $a_2-b_1 \leq \varepsilon$, $a_3-b_2 \leq \varepsilon$,
\dots, $L-b_J \leq \varepsilon$. Other than that, we do not put any
assumption on the geometry of these one-dimensional perforations. Note
that in particular (and in contrast to the analysis we
perform later on in this
article) we do \emph{not} assume any periodicity of the
perforations. The weak form of our problem then reads:
find $u \in H_0^1(\Omega_\varepsilon)$ such that
\begin{equation}
\forall v \in H_0^1(\Omega_\varepsilon),
\quad
a(u,v)
=
\int_{\Omega_\varepsilon} fv,
\label{1Dw}
\end{equation}
where, we recall, $\Omega_\varepsilon = \Omega \setminus
B_\varepsilon$ denotes the perforated domain and where
$$
a(u,v)
=
\int_{\Omega_\varepsilon} u' \, v'.
$$
We now divide $\Omega$ into $N$
segments $K_i=[x_{i-1},x_i]$, $i=1,\ldots,N$, by the nodes
$0=x_0<x_1<\cdots <x_N=L$, define the mesh size $H=\max
|x_i-x_{i-1}|$, and consider the multiscale finite element space adapted
to the perforated domain
$$
V_H
=
\left\{
\begin{array}{c}
u_H \in C^0(\Omega) \text{ such that $u_H=0$ on $B_\varepsilon \cup
\partial \Omega$ and}
\\
u_H^{\prime \prime}=C_i \text{ in $K_i \cap \Omega_\varepsilon$,
$i=1,\ldots,N$, for some constants $C_i$}
\end{array}
\right\}.
$$
Note that the domain $K_i \cap
\Omega_\varepsilon$ may be not connected. We
nevertheless assume that $u_H^{\prime \prime}$ is equal to the {\em
same} constant $C_i$ on all the connected components of
$K_i \cap \Omega_\varepsilon$.
\begin{remark}
In the one-dimensional setting, the Crouzeix-Raviart type boundary condition that we consider in this work simply amounts to a continuity condition at the mesh nodes. This is why we require that $u_H \in C^0(\Omega)$ in the above definition of $V_H$. This observation holds for many variants of MsFEM, including the oversampling variant, which, alike the Crouzeix-Raviart variant we introduce here, uses non-conforming finite elements. In this respect, the one-dimensional setting is not typical.
\end{remark}
The Galerkin approximation of the solution to
problem~\eqref{1Dw} is then introduced as the solution $u_H \in V_H$ to
\begin{equation}
\forall v_H \in V_H,
\quad
a(u_H,v_H) = \int_{\Omega_\varepsilon} f v_H.
\label{1Dh}
\end{equation}
Readers familiar with the MsFEM approach will notice that $V_H$ contains more functions than the usual MsFEM basis set,
which would consist here in taking $u_H^{\prime \prime}= 0$ (rather than an
arbitrary constant $C_i$) on $K_i \cap \Omega_\varepsilon$.
A convenient
generating family for the space $V_H$ may be constructed as follows. First we
associate a function $\Phi_i$ to any internal node $x_i$ by solving
\begin{eqnarray*}
\text{supp}\,\Phi_i & \subset & (x_{i-1},x_{i+1}),
\\
\Phi_i^{\prime \prime} & = & 0
\text{ in $(x_{i-1},x_i) \cap \Omega_\varepsilon$ and in
$(x_i,x_{i+1}) \cap \Omega_\varepsilon$},
\\
\Phi_i & = & 0 \text{ in $B_\varepsilon$},
\\
\Phi_i(x_i) &=& 1\text{ if $x_i \in \Omega_\varepsilon$
or 0 otherwise}.
\end{eqnarray*}
Note that this construction yields $\Phi_i \equiv 0$ if the node $x_i$ lies
inside a perforation (see Fig.~\ref{fig:phi_i}). Second, we associate a
function $\Psi_i$ to any segment $K_i=[x_{i-1},x_i]$ by solving (see
Fig.~\ref{fig:psi_i})
\begin{eqnarray*}
\text{supp} \, \Psi_i & \subset & (x_{i-1},x_i),
\\
-\Psi_i^{\prime \prime} & = & 1 \text{ in $K_i \cap
\Omega_\varepsilon$},
\\
\Psi_i & = & 0 \text{ in $B_\varepsilon$}.
\end{eqnarray*}
The functions $\Phi_i$ ($1 \leq i \leq N-1$) and $\Psi_j$ ($1 \leq j \leq N$) are linearly independent (except for the trivial case when $\Phi_i \equiv 0$), and we obviously have
$$
\text{span} \left\{ \Phi_1,\ldots,\Phi_{N-1},\Psi_1,\ldots,\Psi_N
\right\} \subset V_H.
$$
In turn, any $u \in V_H$ can be written $u = \sum_{j=1}^N C_j \Psi_j + \sum_{i=1}^{N-1} u(x_i) \Phi_i$. We thus have
$$
V_H = \text{span} \left\{ \Phi_1,\ldots,\Phi_{N-1},\Psi_1,\ldots,\Psi_N
\right\},
$$
which implies that the space $V_H$ is of dimension at most $2N-1$.
\begin{figure}
\caption{Basis function $\Phi_i$ (Solid line: domain $\Omega_\varepsilon$;
dashed line: perforations $B_\varepsilon$). Left: case when
$x_i \in \Omega_\varepsilon$. Right: case when $x_i \notin \Omega_\varepsilon$, for which
$\Phi_i \equiv 0$.
\label{fig:phi_i}
\label{fig:phi_i}
\end{figure}
\begin{figure}
\caption{Basis function $\Psi_i$ (Solid line: domain $\Omega_\varepsilon$;
dashed line: perforations $B_\varepsilon$). Left: case when
$x_i \in \Omega_\varepsilon$. Right: case when $x_i \notin \Omega_\varepsilon$. In
both cases, $\Psi_i \neq 0$.
\label{fig:psi_i}
\label{fig:psi_i}
\end{figure}
It is interesting to note that the functions~$\Psi_i$, which act as
\emph{bubble functions}, are necessary to generate an efficient approximation
space. The reason is evident in the one-dimensional situation,
since, in the absence of such a bubble (or of a basis function playing a
similar role), there is no way to recover a good approximation quality between
two consecutive perforations if no node is actually present there. The
numerical solution would systematically vanish in such a region (see
Fig.~\ref{fig:phi_i}, right part).
In higher dimensions, the phenomenon is less accute (since perforations,
unless of a particular shape, cannot isolate regions of the space from
the neighborhood) but it is still, to some extent, relevant. We will
observe the definite added value of bubble functions in our numerical
tests of Section~\ref{sec:Numerical-tests}.
We then have the following (simple) numerical analysis result.
\begin{theorem}
Assume that the right-hand side $f$ in~\eqref{1Dw} satisfies $f \in
H^1(\Omega)$.
Then the Galerkin solution $u_H$ of~\eqref{1Dh} satisfies the error estimate
\begin{equation}
|u-u_H|_{H^1(\Omega_\varepsilon)} \leq C\varepsilon H \| f' \|_{L^2(\Omega)},
\label{eq:1Destimate}
\end{equation}
where $| \cdot |_{H^1(\Omega_\varepsilon)}$ is the energy norm associated to the
bilinear form $a$:
$$
\forall v \in H^1(\Omega_\varepsilon), \quad
| v |_{H^1(\Omega_\varepsilon)} := \sqrt{a(v,v)} =
\sqrt{\int_{\Omega_\varepsilon} (v')^2}.
$$
\end{theorem}
The factor~$\varepsilon$ in the right-hand side of~\eqref{eq:1Destimate}
needs to be understood as follows. It turns out
that, for the category of problems~\eqref{eq:genP} we consider, the
exact solution~$u$
(and thus, correspondingly, its numerical approximation~$u_H$) is
of size~$\varepsilon$ in $H^1$ norm for $\varepsilon$ small, as is proved by
homogenization theory and will be recalled --for the periodic setting--
in Section~\ref{ssec:homogenization} below (see~\eqref{eq:lions} and~\eqref{eq:lions-general}). Once
this scale factor is accounted for, the estimate~\eqref{eq:1Destimate}
shows that the numerical approach is first order accurate in the
meshsize~$H$, with a prefactor $C\, \| f' \|_{L^2(\Omega)}$ that is
\emph{independent} of the size~$\varepsilon$ of the geometric
oscillations.
\begin{proof}
We see from~\eqref{1Dw} and~\eqref{1Dh} that
$$
\forall v_H \in V_H, \quad a(u-u_H,v_H) = 0.
$$
Consequently, $u_H$ is the orthogonal projection of $u$ on $V_H$,
where by \emph{orthogonality} we mean orthogonality for the scalar product
defined by the bilinear form~$a$. We therefore have
\begin{equation}
|u-u_H|_{H^1(\Omega_\varepsilon)}=\inf_{v_H \in V_H} |u-v_H|_{H^1(\Omega_\varepsilon)}.
\label{cea}
\end{equation}
Proving~\eqref{eq:1Destimate} therefore amounts to proving the
inequality for at
least one function $v_H \in V_H$. We take $v_H \in V_H$ such that
$v_H(x_i)=u(x_i)$, $i=0,1,\ldots,N$, and $-v_H^{\prime \prime} = \Pi_H
f$ on each $K_i \cap \Omega_\varepsilon$,
where $\Pi_H f$ is the $L^2$-orthogonal projection of $f$ on the
space of piecewise constant functions. Consider then the
interpolation error $e=u-v_H$. We remark that
$$
\left\{
\begin{array}{l}
-e^{\prime \prime}= f-\Pi_H f \ \text{ on each $(x_{j-1},x_j) \cap
\Omega_\varepsilon$, \ \ $1 \leq j \leq N$},
\\
e(x_j)=0, \quad j=0,\ldots,N.
\end{array}
\right.
$$
Denoting by $b_0=0$ and $a_{J+1}=L$, we have
\begin{eqnarray}
|e|_{H^1(\Omega_\varepsilon)}^2
=
\int_{\Omega_\varepsilon} |e'|^2
=
\sum_{j=0}^J \int_{b_j}^{a_{j+1}} |e'|^2
&=&
- \sum_{j=0}^J \int_{b_j}^{a_{j+1}} e^{\prime \prime} \, e
\nonumber\\
&=&
\sum_{j=0}^J \int_{b_j}^{a_{j+1}} (f-\Pi_H f) \, e.
\label{boundseg}
\end{eqnarray}
Note that the integration by parts here does not give rise to any boundary or
jump terms because $e$ vanishes at all the points $a_j$, $b_j$ and also at
the grid points $x_i$ where $e'$ is discontinuous. We now apply the
Cauchy-Schwarz and Poincar\'e inequalities on each segment $(b_j,a_{j+1})$
and note that the constant in the latter inequality scales as the
length of the segment, that is at most $\varepsilon$ (this fact is obvious in dimension $d=1$; see~\eqref{eq:poincare_perfore} and Appendix~\ref{sec:proof_poincare} below for a general argument).
We thus deduce from~\eqref{boundseg} that
\begin{eqnarray*}
|e|_{H^1(\Omega_\varepsilon)}^2
& \leq &
\sum_{j=0}^J \| f - \Pi_H f \|_{L^2(b_j,a_{j+1})} \
\| e \|_{L^2(b_j,a_{j+1})}
\\
& \leq &
C \varepsilon \sum_{j=0}^J \| f-\Pi_H f \|_{L^2(b_j,a_{j+1})} \
|e|_{H^1(b_j,a_{j+1})}
\\
&\leq &
C \varepsilon \| f-\Pi_H f \|_{L^2(\Omega_\varepsilon)} \ |e|_{H^1(\Omega_\varepsilon)}.
\end{eqnarray*}
Factoring out $|e|_{H^1(\Omega_\varepsilon)}$, and using a standard finite element
approximation estimate of $f-\Pi_H f$, we deduce that
$$
| u - v_H |_{H^1(\Omega_\varepsilon)}
=
|e|_{H^1(\Omega_\varepsilon)}
\leq
C \varepsilon \| f - \Pi_H f \|_{L^2(\Omega)}
\leq
C \varepsilon H \| f' \|_{L^2(\Omega)}.
$$
Collecting this bound with~\eqref{cea}, we obtain~\eqref{eq:1Destimate}.
\end{proof}
\section{Presentation of our MsFEM approach in the 2D setting}
\label{sec:presentation2D}
\subsection{MsFEM \`a la Crouzeix-Raviart with bubble functions}
\label{ssec:msfem}
As mentioned in the introduction, we assume henceforth that the ambient
dimension is $d=2$
and that $\Omega$ is a polygonal domain.
We define a mesh $\mathcal{T}_H$ on $\Omega$, i.e. a decomposition of
$\Omega$ into polygons each of diameter at most $H$, and
denote $\mathcal{E}_H$ the set of all the internal edges of
$\mathcal{T}_H$. Note that we mesh $\Omega$ and not the perforated domain
$\Omega_\varepsilon$. This allows us to use coarse elements (independently of the fine scale present in the geometry of $\Omega_\varepsilon$), and leaves us with a lot of flexibility.
The mesh does not have to be consistent with the perforations $B_\varepsilon$. Some nodes may be in $B_\varepsilon$, and likewise some edges may intersect $B_\varepsilon$.
We also assume that the mesh does not have any hanging
nodes. Otherwise stated, each internal edge is shared by
exactly two elements of
the mesh. In addition, $\mathcal{T}_H$ is assumed a regular
mesh in the following sense: for any mesh element
$T\in\mathcal{T}_H$, there exists a smooth one-to-one and onto mapping
$K:\overline{T} \to T$ where $\overline{T} \subset \mathbb{R}^d$ is the
reference element (a polygon of fixed unit
diameter) and $\| \nabla K \|_{L^\infty} \leq CH$,
$\| \nabla K^{-1} \|_{L^\infty} \leq CH^{-1}$, $C$
being some universal constant independent of~$T$, to which we will refer
as the regularity parameter of the mesh. This assumption is used e.g. in the proof of Lemma~\ref{lem:trace} below.
To avoid some technical complications, we also assume that
the mapping $K$ corresponding to each $T\in\mathcal{T}_H$ is affine on
every edge of $\partial\overline{T}$. Again, this assumption is used e.g. in the proof of Lemma~\ref{lem:trace}. In the following and
to fix the ideas, we will have in mind a
mesh consisting of {\em triangles}, which satisfies the minimum angle
condition to ensure the mesh is regular in the sense defined above (see
e.g.~\cite[Section~4.4]{brenner}).
We will repeatedly use the notation and
terminology (triangle, \dots) of this setting, although the
approach carries over to quadrangles.
The idea behind the MsFEM \`{a} la Crouzeix-Raviart is to
require the continuity of the (here highly oscillatory) finite element
functions in the sense of averages on the edges.
We have extensively studied this approach
in~\cite{companion-article}. For the specific setting we
address here, we add another feature to the numerical approach. Based in
particular on the intuition provided by the one-dimensional case
examined in the previous section, we add \emph{bubble functions} to our
discretization space.
\paragraph{Functional spaces}
To construct our MsFEM space, we proceed as in our previous work~\cite{companion-article}. We introduce the space
$$
W_H = \left\{
\begin{array}{c}
u \in L^2(\Omega) \text{ such that } u|_T \in H^1(T)
\text{ for any } T\in \mathcal{T}_H,
\\ \noalign{\vskip 3pt}
\displaystyle
\int_E [[u]] =0 \text{ for all } E\in \mathcal{E}_H,
\quad u=0 \text{ in $B_\varepsilon$ and on $\partial \Omega$}
\end{array}
\right\},
$$
where~$[[u]]$ denotes the jump of~$u$ across an edge. Note that, as is standard, the condition $u=0$ on $\partial \Omega$ makes sense as
$\Omega$ is a polygonal domain and $\partial \Omega$ belongs to the mesh
edges. We next introduce the subspace
$$
W_H^0 = \left\{
\begin{array}{c}
\displaystyle
u \in W_H \text{ such that }
\int_E u =0 \text{ for all } E\in \mathcal{E}_H
\\ \noalign{\vskip 3pt}
\displaystyle
\text{ and }
\int_T u =0 \text{ for all } T\in \mathcal{T}_H
\end{array}
\right\}
$$
of $W_H$ and define the MsFEM space \`a la Crouzeix-Raviart
\begin{equation}
\label{eq:def_VH}
V_H = \left\{
u \in W_H \text{ such that }
a_H(u,v) = 0 \text{ for all } v \in W_H^0
\right\}
\end{equation}
as the orthogonal complement of $W_H^0$ in $W_H$, where by orthogonality we mean orthogonality for the scalar product defined by
\begin{equation}
\label{eq:def_aH}
a_H(u,v) := \sum_{T \in {\cal T}_H} \int_{T \cap \Omega_\varepsilon}
\nabla u \cdot \nabla v.
\end{equation}
We use a broken integral in the definition of $a_H$ since $W_H \not\subset
H^1(\Omega)$.
\paragraph{Notation}
For any $u \in V_H + H^1_0(\Omega_\varepsilon)$, we denote by
\begin{equation}
\label{eq:def_H1H}
| u |_{H^1_H(\Omega_\varepsilon)} := \sqrt{a_H(u,u)}
\end{equation}
the energy norm associated with the form $a_H$.
Likewise, for any $u \in H^1_0(\Omega_\varepsilon)$, we denote by
$$
| u |_{H^1(\Omega_\varepsilon)} := \sqrt{\int_{\Omega_\varepsilon}
| \nabla u |^2}
$$
the $H^1$ semi-norm.
\paragraph{Strong form and basis functions of $V_H$}
Consider any element $T \in \mathcal{T}_H$ (the three edges of which are denoted $\Gamma_i$, $1 \leq i \leq 3$). Taking in the definition of $V_H$ a function $v$ that vanishes on $\Omega \setminus T$, we note that any function $u \in V_H$ satisfies
$$
\int_{T \cap \Omega_\varepsilon} \nabla u \cdot \nabla v = 0
$$
for all $v \in H^1(T)$ such that $v=0$ in $B_\varepsilon$, $\displaystyle \int_{\Gamma_i} v = 0$ for all $i$ (if $\Gamma_i \subset \partial \Omega$, the condition
$\displaystyle \int_{\Gamma_i} v = 0$ is replaced by $v = 0$ on $\Gamma_i$) and
$\displaystyle \int_T v = 0$. This can be rewritten as
$$
\int_{T \cap \Omega_\varepsilon} \nabla u \cdot \nabla v
=
\lambda^T_0 \int_T v +
\sum_{i=1}^3 \lambda^T_i \int_{\Gamma_i} v
\quad
\text{for all $v \in H^1(T)$ s.t. $v=0$ in $B_\varepsilon$}
$$
for some scalar constants $\lambda^T_j$, $0 \leq j \leq 3$ (on purpose, we have made the dependence of these constants explicit with respect to the mesh element $T$). Hence, the restriction of any $u \in V_H$ to $T$ is in particular a solution to the boundary value problem
\begin{equation}
\label{eq:strong}
-\Delta u = \lambda^T_0 \ \text{in $T \setminus B_\varepsilon$},
\quad
u = 0 \ \text{in $T \cap B_\varepsilon$},
\quad
n \cdot \nabla u = \lambda^T_i \text{ on each $\Gamma_i$}.
\end{equation}
The flux along each edge interior to $\Omega$ is therefore a constant, the constant being possibly different on the two sides of the edge.
The above observation shows that $V_H$ is a finite dimensional space. We now construct a basis for $V_H$, which consists of functions associated to a particular mesh element or a particular internal edge. Note that no basis function is associated to edges belonging to $\partial \Omega$.
First, for any mesh element $T$ that is not a subset of the perforations $B_\varepsilon$ (i.e. $T \not\subset B_\varepsilon$), we consider the variational problem
\begin{equation}
\label{eq:def_psi}
\inf \left\{
\begin{array}{c}
\displaystyle
\int_{T \setminus B_\varepsilon}
\left[ \frac12 \left| \nabla \Psi \right|^2 - \Psi \right],
\ \Psi \in H^1(T),
\\ \noalign{\vskip 3pt}
\displaystyle
\Psi = 0 \ \text{in $T \cap B_\varepsilon$}, \
\int_{\Gamma_i} \Psi = 0 \text{ for each $\Gamma_i$}
\end{array}
\right\}.
\end{equation}
Using the Poincar\'e inequality recalled in~\cite[Lemma 9]{companion-article} and standard analysis arguments, we see that this problem has a unique minimizer. We then introduce the function $\Psi_T \in L^2(\Omega)$ which vanishes in $\Omega \setminus T$ and is equal to this minimizer in $T$. We easily deduce from the optimality condition that $\Psi_T \in V_H$ and satisfies
$$
-\Delta \Psi_T = 1 \ \text{in $T \setminus B_\varepsilon$},
\quad
\Psi_T = 0 \ \text{in $T \cap B_\varepsilon$},
$$
with, for each edge $\Gamma_i$ of $T$, $\displaystyle \int_{\Gamma_i} \Psi_T = 0$ and $n \cdot \nabla \Psi_T = \lambda_i$ on $\Gamma_i$ for some constant $\lambda_i$.
Second, for any internal edge $E$ that is not a subset of the perforations $B_\varepsilon$, we denote $T_E^1$ and $T_E^2$ the two triangles sharing this edge, set $T_E := T_E^1 \cup T_E^2$, and consider the variational problem
\begin{equation}
\label{eq:def_phi}
\inf \left\{
\begin{array}{c}
\displaystyle
\int_{T^1_E \setminus B_\varepsilon} \left| \nabla \Phi \right|^2
+
\int_{T^2_E \setminus B_\varepsilon} \left| \nabla \Phi \right|^2, \
\Phi|_{T_E^1} \in H^1(T_E^1), \
\Phi|_{T_E^2} \in H^1(T_E^2),
\\ \noalign{\vskip 3pt}
\displaystyle
\Phi = 0 \ \text{in $T_E \cap B_\varepsilon$}, \
\int_E \Phi = 1, \ \int_{E'} \Phi = 0 \ \text{for any edge $E' \subset \partial T_E$}
\end{array}
\right\}.
\end{equation}
This set is not empty due to the fact that $E \not\subset B_\varepsilon$.
Again, this problem has a unique minimizer. We introduce the function $\Phi_E \in L^2(\Omega)$ which vanishes in $\Omega \setminus T_E$ and is equal to this minimizer in $T_E$. We easily deduce from the optimality condition that $\Phi_E \in V_H$ and satisfies
$$
-\Delta \Phi_E = 0 \ \text{in $T_E^1 \setminus B_\varepsilon$},
\quad
-\Delta \Phi_E = 0 \ \text{in $T_E^2 \setminus B_\varepsilon$},
\quad
\Phi_E = 0 \ \text{in $T \cap B_\varepsilon$},
$$
with, for each edge $E' \subset \partial T_E$, $\displaystyle \int_{E'} \Phi_E = 0$ and $n \cdot \nabla \Phi_E = \lambda_{E'}$ on $E'$ for some constant $\lambda_{E'}$ and $\displaystyle \int_E \Phi_E = 1$ and $n \cdot \nabla \Phi_E = \lambda_E$ on $E$ for some constant $\lambda_E$ (with an a priori different constant on the two sides of $E$).
For any mesh element $T \subset B_\varepsilon$ (resp. any internal edge $E \subset B_\varepsilon$), we set $\Psi_T \equiv 0$ (resp. $\Phi_E \equiv 0$).
\begin{remark}
In the one-dimensional case, the functions $\Psi_T$ and $\Phi_E$ that we have defined are equal to the basis functions of Section~\ref{ssec:1D} (see Figures~\ref{fig:phi_i} and~\ref{fig:psi_i}).
\end{remark}
The functions $\Psi_T$ and $\Phi_E$ that we have constructed belong to $V_H$. In addition, $\displaystyle \left\{ \Psi_T \right\}_{T \in \mathcal{T}_H, \ T \not\subset B_\varepsilon} \cup \left\{ \Phi_E \right\}_{E \in \mathcal{E}_H, \ E \not\subset B_\varepsilon}$ forms a linearly independent family. We have
$$
\text{Span}
\left\{ \Phi_E, \, \Psi_T, \, E\in \mathcal{E}_H,
\, T\in \mathcal{T}_H \right\}
\subset V_H.
$$
Conversely, let $u \in V_H$. We know that $u$ satisfies~\eqref{eq:strong}. We introduce
$$
v = u - \sum_{T \in \mathcal{T}_H} \lambda_0^T \Psi_T - \sum_{E \in \mathcal{E}_H} \left[ \int_E u \right] \Phi_E
$$
and note that it satisfies, for any $T \in \mathcal{T}_H$,
$$
-\Delta v = 0 \ \text{in $T \setminus B_\varepsilon$},
\quad
v = 0 \ \text{in $T \cap B_\varepsilon$},
$$
with $\displaystyle \int_E v = 0$ and $n \cdot \nabla v$ is a constant on $E$,
for each edge $E \in \mathcal{E}_H$. This implies that $v \equiv 0$, and thus
\begin{equation}
\label{eq:vh_span}
V_H =
\text{Span}
\left\{ \Phi_E, \, \Psi_T, \, E\in \mathcal{E}_H,
\, T\in \mathcal{T}_H \right\}.
\end{equation}
\paragraph{Numerical approximation}
The MsFEM approximate solution of our problem~\eqref{eq:genP} is
defined as the solution~$u_H \in V_H$ to
\begin{equation}
\forall v_H \in V_H, \quad
a_H(u_H,v_H) = \int_{\Omega_\varepsilon} f v_H,
\label{2DH}
\end{equation}
where $a_H$ is defined by~\eqref{eq:def_aH}.
\subsection{Main result: an error estimate in the case of periodic perforations}
\label{ssec:mainresult}
The main theoretical result we obtain in this article addresses the
numerical analysis of the approach presented above, in the particular
case of periodic perforations in dimension~2,
with a sufficient
regularity (made precise in the statement of the theorem below) of the
right-hand side~$f$ of~\eqref{eq:genP}.
\begin{theorem}
\label{theo:main}
Let $u$ be the solution to~\eqref{eq:genP} for $d=2$, with periodic
perforations and with $f \in H^2(\Omega)$.
We assume that, loosely speaking, the slopes of the mesh edges are
rational numbers. More precisely, we assume that the equation of any
internal edge $E$ of the mesh writes $\displaystyle x_2 = \frac{p_E}{q_E} x_1 + c_E$ for
some $c_E \in \mathbb{R}$, some $p_E \in \mathbb{Z}$ and $q_E \in \mathbb{N}^\star$ that are
coprime, with
\begin{equation}
\label{hyp:mesh}
|q_E| \leq C
\end{equation}
for a constant~$C$ independent of the edge considered in the mesh and of
the mesh size $H$.
Then the MsFEM approximation $u_H$, solution to~\eqref{2DH}, satisfies
\begin{equation}
|u-u_H|_{H^1_H(\Omega_\varepsilon)} \leq C \varepsilon \,
\left( \sqrt{\varepsilon} + H + \sqrt{\frac{\varepsilon}{H}} \right) \,
\| f \|_{H^2(\Omega)},
\label{eq:mainresult}
\end{equation}
for some universal constant~$C$ independent from $H$, $\varepsilon$ and $f$,
but depending on the geometry of the mesh and other parameters of the
problem.
\end{theorem}
As will be evident from the theoretical ingredients recalled below
(see~\eqref{eq:lions} and comments following this
estimate), the right-hand side of~\eqref{eq:mainresult} needs to be
understood as follows. The size of the exact solution~$u$ (and thus that
of the corresponding approximation~$u_H$) is~$\varepsilon$ in $H^1$
norm. Taking this scale factor into account, the actual rate of convergence for the numerical
approach we design is therefore given by
$\displaystyle \sqrt{\varepsilon}+ H + \sqrt{\varepsilon/H}$.
\begin{remark} Our
assumption on the rationality of the slopes in the mesh is necessary,
in the current state of our understanding, to treat traces of periodic
functions on the edges of the mesh. In full generality, such traces
are almost periodic functions. Our proof perhaps carries over to this
case, however at the price of unnecessary technicalities (we refer e.g. to~\cite{gv1,gv2} for works on boundary layers in homogenization, where such non-periodic situations are dealt with). In the case
of rational slopes we restrict ourselves to, these traces are
periodic, and the uniform bound~\eqref{hyp:mesh} we additionally assume
enables us to uniformly bound
their periods from above, rending the proof much easier. We
emphasize that our assumption does not seem to us very
restrictive in practice.
\end{remark}
\begin{remark}
It is useful to compare our error estimate~\eqref{eq:mainresult} with estimates for other existing MsFEM-type approaches established for similar problems. First, we are not aware of any other numerical analysis of a MsFEM-type approach for problems set on perforated domains. To the best of our knowledge, this work is the first one proposing and analyzing a MsFEM-type approach specifically adapted to such problems.
Second, as pointed out above, this work is a follow up on our previous work~\cite{companion-article} where we have studied a Crouzeix-Raviart type MsFEM approach on the problem
\begin{equation}
\label{eq:pb_ref}
-\operatorname{div} \left[ A_{\varepsilon}(x) \nabla u^\varepsilon \right] = f
\text{ in $\Omega$},
\quad
u^\varepsilon = 0 \text{ on $\partial \Omega$},
\end{equation}
the main difference between that method and the one presented here being the addition of bubble functions in the MsFEM space. For problem~\eqref{eq:pb_ref}, we have compared in~\cite[Remark 3.2]{companion-article} our error estimate with those obtained for other MsFEM-type approaches.
\end{remark}
\begin{remark}
In the absence of perforations, our problem simply writes
$$
-\Delta u = f \text{ in $\Omega$},
\quad
u = 0\text{ on $\partial \Omega$}.
$$
Assuming a triangular mesh is used, our discretization space $V_H$ then becomes the standard Crouzeix-Raviart space~\cite{Crouzeix-Raviart} (see~\cite[Remark 1.1]{companion-article}), complemented by bubble functions defined by~\eqref{eq:def_psi} with $B_\varepsilon = \emptyset$. In turn, the MsFEM approach with linear boundary conditions (as well as the oversampling variant) then becomes the standard P1 FEM.
\end{remark}
The next two sections are devoted to the proof of Theorem~\ref{theo:main}. Numerical results are gathered in Section~\ref{sec:Numerical-tests}.
\section{Some preliminaries}
\label{sec:preliminaries}
\subsection{Elements of homogenization theory for periodically perforated domains}
\label{ssec:homogenization}
We consider the unit square $Y$ and some smooth perforation $B \subset Y$. We
next scale $B$ and $Y$ by a factor $\varepsilon$
and then periodically repeat this pattern with periods $\varepsilon$ in both
directions. The set of perforations is therefore
$$
B_\varepsilon = \Omega \cap \left( \underset{k \in \mathbb{Z}^2}{\cup} \ \varepsilon B_k \right)
\quad \text{with} \quad
B_k = k + B
$$
and the perforated domain is
$\Omega_\varepsilon = \Omega \setminus \overline{B_\varepsilon}$.
We denote by $u^\varepsilon$ the solution
to~\eqref{eq:genP} to emphasize the dependency upon~$\varepsilon$. We
know from the classical work~\cite{lions1980}
that, provided $f$ vanishes on the boundary of $\Omega$ (see below the easy adaptation to a more general case), we have
\begin{equation}
\left| u^\varepsilon - \varepsilon^2
w \left( \frac{\cdot}{\varepsilon} \right) f \right|_{H^1(\Omega_\varepsilon)}
\leq C \varepsilon^2 \|f\|_{H^2(\Omega)},
\label{eq:lions}
\end{equation}
where $w$ denotes the corrector, that is the solution to the problem
\begin{eqnarray}
-\Delta w & = &1 \text{ on $Y \setminus B$},
\nonumber
\\
w & = & 0\text{ on $\overline{B}$},
\label{eq:corrector-lions}
\\
&&\text{$w$ is $Y$-periodic},
\nonumber
\end{eqnarray}
in the unit cell~$Y$. We refer
to~\cite{blp,Engquist-Souganidis,Jikov1994} for more background on
homogenization theory. Note that~\eqref{eq:lions} is {\em not} restricted to the two-dimensional case. In the sequel, we will use the fact that
\begin{equation}
\label{eq:regul_w}
w \in C^1\left(\overline{Y \setminus B}\right),
\end{equation}
which follows from the fact that $w \in C^{2,\delta}\left(\overline{Y \setminus B}\right)$ for some $\delta>0$ (see e.g.~\cite[Theorem 6.14]{gilbarg-trudinger}). In view of~\cite[Corollary 8.11]{gilbarg-trudinger}, we also have $w \in C^\infty(Y \setminus B)$, but we will not need this henceforth.
Clearly, \eqref{eq:lions} shows that, for $\varepsilon$ small, the
dominant behaviour of the solution~$u^\varepsilon$ to~\eqref{eq:genP} is
simple. It is obtained by a simple multiplication of the
right-hand side~$f$ by the corrector function. Otherwise stated, the
particular setting yields an homogenized problem where the differential
operator has disappeared. The corrector
problem~\eqref{eq:corrector-lions} formally agrees with intuition: at
the scale of the geometric heterogeneities, the right-hand side~$f$
of~\eqref{eq:genP} is seen as a constant function (thus the right-hand side
of~\eqref{eq:corrector-lions}) and the approximation of the
solution~$u^\varepsilon$ is obtained by the
simple multiplication mentioned above. Additionally, the ``size'' of the
solution~$u^\varepsilon$ is proportional to~$\varepsilon^2$ in
$L^2$ norm and~$\varepsilon$ in $H^1$ norm, a fact that will need to be
borne in mind below when performing the analysis and the numerical
experiments.
It is easy to modify~\eqref{eq:lions} in order to accomodate the more
general situation where the right-hand side~$f \in H^2(\Omega)$ does not necessarily
vanish on the boundary of $\Omega$, provided the domain $\Omega$ is smooth. We then have the weaker estimate
\begin{equation}
\left| u^\varepsilon -
\varepsilon^2 w \left( \frac{\cdot}{\varepsilon} \right)
f \right|_{H^1(\Omega_\varepsilon)} \leq
C \varepsilon^{3/2} {\cal N}(f),
\label{eq:lions-general}
\end{equation}
where
\begin{equation}
\label{eq:def_N_f}
{\cal N}(f) = \| f \|_{L^\infty(\Omega)} +
\| \nabla f \|_{L^2(\Omega)} +
\| \Delta f \|_{L^2(\Omega)}.
\end{equation}
We recall that, in dimension $d=2$, the injection $H^2(\Omega) \subset
C^0(\overline{\Omega})$ is continuous.
The proof of~\eqref{eq:lions-general} is postponed until Appendix~\ref{sec:proof_hom}.
A key ingredient for that proof, and for other proofs throughout this article, is the following Poincar\'e inequality in the perforated domain
$\Omega_\varepsilon$: there exists a constant $C$
independent of $\varepsilon$ such that
\begin{equation}
\label{eq:poincare_perfore}
\forall \phi \in H_0^1(\Omega_\varepsilon), \quad
\| \phi \|_{L^2(\Omega_\varepsilon)}
\leq
C \varepsilon \| \nabla \phi \|_{L^2(\Omega_\varepsilon)}
=
C \varepsilon |\phi|_{H^1(\Omega_\varepsilon)}.
\end{equation}
The proof of~\eqref{eq:poincare_perfore} is postponed until Appendix~\ref{sec:proof_poincare}. Following the same arguments, we also see that there exists a constant $C$ independent of $\varepsilon$ such that
\begin{equation}
\label{eq:poincare_perfore2}
\forall \phi \in W_H, \quad
\| \phi \|_{L^2(\Omega_\varepsilon)}
\leq
C \varepsilon |\phi|_{H^1_H(\Omega_\varepsilon)},
\end{equation}
where, we recall, the notation $| \cdot |_{H^1_H(\Omega_\varepsilon)}$ has been defined in~\eqref{eq:def_H1H}.
The condition $\displaystyle \int_E [[\phi]] =0$ (present in the definition of $W_H$) is actually not needed for~\eqref{eq:poincare_perfore2} to hold, given that $\phi =0$ on $B_\varepsilon$.
\subsection{Classical ingredients of multiscale numerical analysis}
\label{ssec:ingredients}
Before we get to the proof of
Theorem~\ref{theo:main}, we first need to collect here some standard
Trace theorems (which were already used and proved in~\cite{companion-article}) and results on the convergence of oscillating
functions. We refer to the textbooks~\cite{brenner,ern,gilbarg-trudinger}
for more details.
Remark that only Lemma~\ref{lem:malin} is restricted to the
two-dimensional setting.
First we recall the definition, borrowed from e.g.~\cite[Definition
B.30]{ern}, of the $H^{1/2}$ space.
\begin{definition}
For any open domain $\omega \subset \mathbb{R}^n$,
we define the norm
$$
\| u \|^2_{H^{1/2}(\omega)} :=
\| u \|^2_{L^2(\omega)} + | u |^2_{H^{1/2}(\omega)},
$$
where
$$
| u |^2_{H^{1/2}(\omega)} := \int_\omega \int_\omega
\frac{|u(x)-u(y)|^2}{|x-y|^{n+1}} \, dx dy,
$$
and define the space
$$
H^{1/2}(\omega) := \left\{ u \in L^2(\omega), \quad
\| u \|_{H^{1/2}(\omega)} < \infty \right\}.
$$
\end{definition}
\paragraph{Trace inequalities}
We have the following trace results:
\begin{lemma}
\label{lem:trace}
There exists $C$ (depending only on the regularity of the mesh)
such that, for any $T \in {\cal T}_H$ and any edge $E
\subset \partial T$, we have
\begin{equation}
\label{eq:trace1}
\forall v \in H^1(T), \quad
\| v \|^2_{L^2(E)} \leq C \left(
H^{-1} \| v \|^2_{L^2(T)} + H \| \nabla v \|^2_{L^2(T)}
\right).
\end{equation}
Under the additional assumption that $\displaystyle \int_E v = 0$, we have
\begin{equation}
\label{eq:trace2_pre}
\| v \|^2_{L^2(E)} \leq C H \| \nabla v \|^2_{L^2(T)}
\end{equation}
and
\begin{equation}
\label{eq:trace3_pre}
\| v \|^2_{H^{1/2}(E)} \leq C (1+H) \| \nabla v \|^2_{L^2(T)}.
\end{equation}
\end{lemma}
These bounds
are classical results (see e.g.~\cite[page
282]{brenner}) and are proved
in~\cite[Section 4.2]{companion-article}. The following result is a direct
consequence of~\eqref{eq:trace2_pre} and~\eqref{eq:trace3_pre}:
\begin{corollary}
\label{coro:trace}
Consider an edge $E \in {\cal E}_H$, and let $T_E \subset {\cal T}_H$
denote all the triangles sharing this edge.
There exists $C$ (depending only on the regularity of the mesh) such
that
\begin{equation}
\label{eq:trace2}
\forall v \in W_H,
\quad
\| \, [[v]] \, \|^2_{L^2(E)} \leq C H \sum_{T \in T_E}
\| \nabla v \|^2_{L^2(T)}
\end{equation}
and
\begin{equation}
\label{eq:trace3}
\forall v \in W_H,
\quad
\| \, [[v]] \, \|^2_{H^{1/2}(E)} \leq C (1+H) \sum_{T \in T_E}
\| \nabla v \|^2_{L^2(T)}.
\end{equation}
\end{corollary}
\paragraph{Averages of oscillatory functions}
We shall also need the following classical result.
\begin{lemma}
\label{lem:malin}
Let $g \in L^\infty(\mathbb{R})$ be a $q$-periodic function with zero mean. Let
$f \in W^{1,1}(0,H) \subset C^0(0,H)$ be a
function defined on the interval $[0,H]$ that vanishes at least at one
point of $[0,H]$. Then, for any $\varepsilon >0$,
$$
\left| \int_0^H g\left(\frac{x}{\varepsilon}\right)
f(x)dx \right|
\leq
2\varepsilon q \| g \|_{L^\infty(\mathbb{R})} \| f' \|_{L^1(0,H)}.
$$
\end{lemma}
\begin{proof}
The proof is simple and essentially based upon an integration by parts.
Let $G$ be a primitive of $g$:
$$
G(x) = \int_0^x g(t)dt.
$$
The function $G$ is $q$-periodic (as the average of $g$ over its period
vanishes) and bounded, with
$\| G \|_{L^\infty(\mathbb{R})} \leq q \| g \|_{L^\infty(\mathbb{R})}$. Supposing that the
function $f$ vanishes at the point $c\in [0,H]$, we write
\begin{eqnarray*}
\int_c^H g\left(\frac{x}{\varepsilon}\right) f(x) \, dx
&=&
\int_c^H G'\left(\frac{x}{\varepsilon}\right) f(x) \, dx
\\
&=&
\varepsilon G\left(\frac{H}{\varepsilon}\right) f(H)
-
\int_c^H \varepsilon G\left(\frac{x}{\varepsilon}\right) f'(x) \, dx
\\
&=&
\varepsilon G\left(\frac{H}{\varepsilon}\right) \int_c^H f'(x) \, dx
-
\int_c^H \varepsilon G\left(\frac{x}{\varepsilon}\right) f'(x) \, dx,
\end{eqnarray*}
hence
$$
\left|
\int_c^H g\left(\frac{x}{\varepsilon}\right) f(x) \, dx
\right|
\leq
2 \varepsilon \| G \|_{L^\infty(\mathbb{R})} \| f' \|_{L^1(c,H)}
\leq
2 \varepsilon q \| g \|_{L^\infty(\mathbb{R})} \| f' \|_{L^1(c,H)}.
$$
By a similar computation,
$$
\left|
\int_0^c g\left(\frac{x}{\varepsilon}\right) f(x) \, dx
\right|
\leq
2 \varepsilon q \|g \|_{L^\infty(\mathbb{R})} \| f' \|_{L^1(0,c)}.
$$
The above two bounds imply the result.
\end{proof}
\section{Proof of our main result}
\label{sec:proof}
To prove Theorem~\ref{theo:main}, it is possible to
follow the same arguments as in our earlier
work~\cite{companion-article}.
We follow here a different
path, so as to show that other strategies are possible. Note that we use here and in~\cite{companion-article} the same technical ingredients, including those recalled in Section~\ref{ssec:ingredients} and an interpolation argument, see Step 1c below.
Let $u$ be the solution to the reference problem~\eqref{eq:genP} with the right-hand side~$f$, and
let $\Pi_H f$ be the $L^2$-orthogonal projection of $f$ on the space of
piecewise constant functions. We recall the following standard finite element
interpolation result: there exists $C$ independent of $H$ and $f$ such that
\begin{equation}
\label{eq:P1_EF}
\| f - \Pi_H f \|_{L^2(\Omega)}
\leq
C H \| \nabla f \|_{L^2(\Omega)}.
\end{equation}
We introduce
\begin{equation}
\label{eq:def_fct_vh}
v_H(x) = \sum_{T \in {\cal T}_H} \Pi_H f \ \Psi_T(x) + \sum_{E \in {\cal E}_H} \left[ \int_E u \right] \ \Phi_E(x),
\end{equation}
where the functions $\Psi_T$ and $\Phi_E$ have been defined in Section~\ref{ssec:msfem} by~\eqref{eq:def_psi} and~\eqref{eq:def_phi} respectively. We recall that, if $T \subset B_\varepsilon$ (resp. $E \subset B_\varepsilon$), then $\Psi_T \equiv 0$ (resp. $\Phi_E \equiv 0$). We see from~\eqref{eq:vh_span} that $v_H \in V_H$. We next decompose the exact solution $u$ of~\eqref{eq:genP} in the form
$$
u=v_H + \phi.
$$
By definition of $\Psi_T$ and $\Phi_E$, we have, for all edges $E \in {\cal E}_H$ and all triangles $T \in {\cal T}_H$, that
\begin{equation}
\label{eq:par_construction}
\begin{array}{rcl}
\displaystyle \int_E v_H &=& \displaystyle \int_E u \quad \text{hence} \quad \int_E \phi = 0,
\\ \noalign {\vskip 3pt}
\displaystyle n \cdot \nabla v_H &=& \text{Constant on (each side of) $E$},
\\ \noalign {\vskip 3pt}
-\Delta v_H &=& \Pi_H f \text{ on $T \cap \Omega_\varepsilon$}.
\end{array}
\end{equation}
The estimate~\eqref{eq:mainresult} is proved by
estimating $\phi = u-v_H$ in Step 1 below and next $v_H-u_H$ in Step 2.
In what follows, we use the shorthand notation
$\displaystyle g_\varepsilon(x) = g \left( x/\varepsilon \right)$ for all functions $g$. The notation $C$ stands for a constant that is independent from $\varepsilon$, $H$, $f$ and $u$, and that may vary from one line to the next.
\paragraph{Step 1: Estimation of $u-v_H$:}
Using the approximation of $u$ given by the homogenization result~\eqref{eq:lions-general},
we write
\begin{eqnarray}
| \phi |_{H^1_H(\Omega_\varepsilon)}^2
&=&
\sum_{T \in \mathcal{T}_H} \int_{\Omega_\varepsilon \cap T}
|\nabla \phi |^2
\nonumber
\\
&=&
\sum_{T\in \mathcal{T}_H} \int_{\Omega_\varepsilon \cap T}
\nabla (u-\varepsilon^2 w_\varepsilon f) \cdot \nabla \phi
+
\sum_{T\in \mathcal{T}_H} \int_{\Omega_\varepsilon \cap T}
\nabla (\varepsilon^2 w_\varepsilon f - v_H) \cdot \nabla \phi
\nonumber
\\
&=&
\sum_{T \in \mathcal{T}_H} \int_{\Omega_\varepsilon \cap T}
\nabla (u-\varepsilon^2 w_\varepsilon f) \cdot \nabla \phi
+
\sum_{T\in \mathcal{T}_H} \int_{\Omega_\varepsilon \cap T}
(-\Delta (\varepsilon^2 w_\varepsilon f - v_H)) \phi
\nonumber
\\
&& + \varepsilon^2 \sum_{T \in \mathcal{T}_H}
\int_{\partial (T \cap \Omega_\varepsilon)}
\phi \ n \cdot \nabla (w_\varepsilon f)
-
\sum_{T \in \mathcal{T}_H}
\int_{\partial (T \cap \Omega_\varepsilon)}
\phi \ n \cdot \nabla v_H.
\label{eq:louis4}
\end{eqnarray}
We now use the fact that $\phi = u - v_H = 0$ on $\partial \Omega_\varepsilon$. We hence have that
\begin{equation}
\label{eq:bord_nul}
\int_{\partial (T \cap \Omega_\varepsilon)}
\phi \ n \cdot \nabla (w_\varepsilon f)
=
\int_{(\partial T) \cap \Omega_\varepsilon}
\phi \ n \cdot \nabla (w_\varepsilon f)
\end{equation}
and likewise for the last term of~\eqref{eq:louis4}. Equalities of the type~\eqref{eq:bord_nul} will often be used in the sequel. We thus write~\eqref{eq:louis4} as
\begin{eqnarray*}
| \phi |_{H^1_H(\Omega_\varepsilon)}^2
&=&
\sum_{T \in \mathcal{T}_H} \int_{\Omega_\varepsilon \cap T}
\nabla (u-\varepsilon^2 w_\varepsilon f) \cdot \nabla \phi
+
\sum_{T\in \mathcal{T}_H} \int_{\Omega_\varepsilon \cap T}
(-\Delta (\varepsilon^2 w_\varepsilon f - v_H)) \phi
\nonumber
\\
&& + \varepsilon^2 \sum_{T \in \mathcal{T}_H}
\int_{(\partial T) \cap \Omega_\varepsilon}
\phi \ n \cdot \nabla (w_\varepsilon f)
-
\sum_{T \in \mathcal{T}_H}
\int_{(\partial T) \cap \Omega_\varepsilon}
\phi \ n \cdot \nabla v_H.
\end{eqnarray*}
The fourth term in the above right-hand side vanishes. Indeed, on each
edge~$E$, we know from~\eqref{eq:par_construction} that $n \cdot \nabla v_H$ is constant and $\displaystyle \int_E \phi =\int_{E \cap \Omega_\varepsilon} \phi =0$. The third term can be written
$$
\varepsilon^2 \sum_{T \in \mathcal{T}_H}
\int_{(\partial T) \cap \Omega_\varepsilon}
\phi \ n \cdot \nabla (w_\varepsilon f)
=
\varepsilon^2 \sum_{E \in \mathcal{E}_H}
\int_{E \cap \Omega_\varepsilon}
[[ \phi ]] \ n \cdot \nabla (w_\varepsilon f).
$$
Indeed, $w \in C^1\left(\overline{Y \setminus B}\right)$ (see~\eqref{eq:regul_w}) and $f \in H^2(\Omega)$, hence $\nabla (w_\varepsilon f)$ has a well-defined trace on $E \cap \Omega_\varepsilon$.
We are thus left with
\begin{eqnarray}
| \phi |_{H^1_H(\Omega_\varepsilon)}^2
&=&
\sum_{T \in \mathcal{T}_H} \int_{\Omega_\varepsilon \cap T}
\nabla (u-\varepsilon^2 w_\varepsilon f) \cdot \nabla \phi
+
\sum_{T\in \mathcal{T}_H} \int_{\Omega_\varepsilon \cap T}
(-\Delta (\varepsilon^2 w_\varepsilon f - v_H)) \phi
\nonumber
\\
&&\qquad + \varepsilon^2 \sum_{E \in \mathcal{E}_H}
\int_{E \cap \Omega_\varepsilon}
[[\phi]] \ n \cdot \nabla (w_\varepsilon f).
\label{eq:decompo_error}
\end{eqnarray}
We now successively bound the three terms of the right-hand side
of~\eqref{eq:decompo_error}. Loosely speaking:
\begin{itemize}
\item the first term is small because of the homogenization result~\eqref{eq:lions-general}, that states that $\varepsilon^2 w_\varepsilon f$ is indeed an accurate approximation of $u$.
\item the second term is small because, at the leading order term in $\varepsilon$, the first factor in the integrand is equal to
$-\Delta \left( \varepsilon^2 w_\varepsilon f \right) + \Delta v_H \approx f - \Pi_H f$ which is small due to~\eqref{eq:P1_EF}.
\item estimating the third term is more involved. An essential ingredient is the fact that $w$ is a periodic function. We are thus in position to apply our Lemma~\ref{lem:malin}.
\end{itemize}
\paragraph{Step 1a}
The first term of~\eqref{eq:decompo_error} is easily estimated as follows:
\begin{eqnarray}
\left|
\sum_{T \in \mathcal{T}_H} \int_{\Omega_\varepsilon \cap T}
\nabla (u-\varepsilon^2 w_\varepsilon f) \cdot \nabla \phi
\right|
&\leq&
\sum_{T \in \mathcal{T}_H}
\| \nabla (u-\varepsilon^2 w_\varepsilon f) \|_{L^2(\Omega_\varepsilon \cap T)}
\
\| \nabla \phi \|_{L^2(\Omega_\varepsilon \cap T)}
\nonumber
\\
&\leq&
| u-\varepsilon^2 w_\varepsilon f |_{H^1(\Omega_\varepsilon)}
\
| \phi |_{H^1_H(\Omega_\varepsilon)}
\nonumber
\\
&\leq&
C \varepsilon^{3/2} \, {\cal N}(f) \,
| \phi |_{H^1_H(\Omega_\varepsilon)},
\label{eq:bound_1}
\end{eqnarray}
where we have used the discrete Cauchy-Schwarz inequality in the second line and the homogenization result~\eqref{eq:lions-general} in the third line.
\paragraph{Step 1b}
We next turn to the second term of the right-hand side
of~\eqref{eq:decompo_error}, that we write as follows, using the
corrector equation~\eqref{eq:corrector-lions} and~\eqref{eq:par_construction}:
$$
\sum_{T\in \mathcal{T}_H} \int_{\Omega_\varepsilon \cap T}
(-\Delta (\varepsilon^2 w_\varepsilon f - v_H)) \phi
=
\int_{\Omega _{\varepsilon }}(f
- 2 \varepsilon (\nabla w)_\varepsilon \cdot \nabla f
- \varepsilon^2 w_\varepsilon \Delta f
- \Pi_H f ) \phi.
$$
We thus obtain
\begin{eqnarray*}
&&
\left|
\sum_{T\in \mathcal{T}_H} \int_{\Omega_\varepsilon \cap T}
(-\Delta (\varepsilon^2 w_\varepsilon f - v_H)) \phi
\right|
\\
&\leq &
\Big( \| f - \Pi_H f \|_{L^2(\Omega)} +
2 \varepsilon \| \nabla w \|_{L^\infty}
\| \nabla f \|_{L^2(\Omega)} +
\varepsilon^2 \| w \|_{L^\infty} \| \Delta f \|_{L^2(\Omega)}
\Big) \,
\| \phi \|_{L^2(\Omega_\varepsilon)}
\\
&\leq &
C \varepsilon \, \left(
C H \| \nabla f \|_{L^2(\Omega)} + C \varepsilon {\cal N}(f)
\right)
| \phi |_{H^1_H(\Omega_\varepsilon)},
\end{eqnarray*}
where ${\cal N}(f)$ is defined by~\eqref{eq:def_N_f} and where, in the
last line, we have used~\eqref{eq:P1_EF},~\eqref{eq:regul_w} and~\eqref{eq:poincare_perfore2}.
We deduce that
\begin{equation}
\label{eq:bound_2}
\left|
\sum_{T\in \mathcal{T}_H} \int_{\Omega_\varepsilon \cap T}
(-\Delta (\varepsilon^2 w_\varepsilon f - v_H)) \phi
\right|
\leq
C \varepsilon \, (H+\varepsilon) \, {\cal N}(f) \,
|\phi |_{H^1_H(\Omega_\varepsilon)}.
\end{equation}
\paragraph{Step 1c}
The final stage of Step 1 is devoted to bounding the third
term of the right-hand side of~\eqref{eq:decompo_error}.
In view of the assumptions on the mesh (rationality of the slopes, in short), we first observe that, for any
edge $E \in {\cal E}_H$, the function
$\displaystyle x \in E \mapsto n \cdot \nabla w\left( \frac{x}{\varepsilon} \right)$
is periodic with period $q_E \varepsilon$, for some $q_E \in \mathbb{N}^\star$
satisfying $|q_E| \leq C$ for some $C$ independent of the mesh edge and of $H$. We
denote by $\langle n \cdot (\nabla w)_\varepsilon \rangle_E$ the average of that
function over one period, and decompose the third
term of the right-hand side of~\eqref{eq:decompo_error} as follows:
\begin{eqnarray}
&&
\varepsilon^2 \sum_{E \in \mathcal{E}_H}
\int_{E \cap \Omega_\varepsilon}
[[\phi]] \ n \cdot \nabla (w_\varepsilon f)
\nonumber
\\
&=&
\varepsilon \sum_{E\in \mathcal{E}_H}
\int_{E\cap \Omega_\varepsilon} [[\phi]] \ \Big( n \cdot (\nabla
w)_\varepsilon - \langle n \cdot (\nabla w)_\varepsilon \rangle_E \Big) f
\nonumber
\\
& + &
\varepsilon \sum_{E\in \mathcal{E}_H}
\langle n \cdot (\nabla w)_\varepsilon \rangle_E
\int_{E\cap \Omega_\varepsilon} [[\phi]] \ f
+
\varepsilon^2 \sum_{E \in \mathcal{E}_H}
\int_{E \cap \Omega_\varepsilon}
[[\phi]] \ w_\varepsilon \ n \cdot \nabla f.
\label{eq:terme3}
\end{eqnarray}
We successively estimate the three terms of the right-hand
side of~\eqref{eq:terme3}. In some formulae below, we will make the
following slight abuse of notation. We will extend
the function $\phi = u - v_H$ by 0 inside the perforations
$B_\varepsilon$, so that we can understand~$\phi$ either as a function in
$H_0^1(\Omega_\varepsilon)$ or in $H_0^1(\Omega)$.
We consider the first term of the right-hand side of~\eqref{eq:terme3}, which we evaluate essentially using the fact that it contains a periodic oscillatory function of zero mean. We claim that
\begin{multline}
\label{eq:claim3_1}
\left|
\int_{E\cap \Omega_\varepsilon} [[\phi]] \ \Big( n \cdot (\nabla
w)_\varepsilon - \langle n \cdot (\nabla w)_\varepsilon \rangle_E \Big) f
\right|
\\
\leq
C \, \sqrt{\varepsilon} \, \| f \|_{H^1(E)}
\| \, [[ \phi ]] \, \|_{H^{1/2}(E)}
\end{multline}
for a constant $C$ independent of the edge $E$, $\varepsilon$ and $H$. Indeed, we first
note that $u$ and $v_H$ vanish on $\Omega \setminus \Omega_\varepsilon$, so
$\phi=u-v_H$ vanishes on $E \cap (\Omega \setminus \Omega_\varepsilon)$, hence
\begin{multline}
\label{eq:debut}
\int_{E\cap \Omega_\varepsilon} [[\phi]] \ \Big( n \cdot (\nabla
w)_\varepsilon - \langle n \cdot (\nabla w)_\varepsilon \rangle_E \Big) f
\\
=
\int_E [[\phi]] \ \Big( n \cdot (\nabla
w)_\varepsilon - \langle n \cdot (\nabla w)_\varepsilon \rangle_E \Big) f.
\end{multline}
Second, using the regularity~\eqref{eq:regul_w} of $w$, we obviously have that
\begin{equation}
\label{eq:l2}
\left|
\int_E [[\phi]] \ \Big( n \cdot (\nabla
w)_\varepsilon - \langle n \cdot (\nabla w)_\varepsilon \rangle_E \Big) f
\right|
\leq
C \, \|f\|_{L^2(E)} \, \| \, [[\phi]] \, \|_{L^2(E)}.
\end{equation}
Third, suppose momentarily that $[[ \phi ]] \in H^1(E) \subset
C^0(E)$. We infer from
the fact that $\displaystyle \int_E [[\phi]] = 0$ that $[[ \phi ]]$, and hence
$[[ \phi ]] \, f$, vanishes at least at one point on $E$. In addition,
the function $n \cdot (\nabla
w)_\varepsilon - \langle n \cdot (\nabla w)_\varepsilon \rangle_E$
is periodic on $E$ (with a period $q_E$ uniformly bounded with respect to $E
\in {\cal E}_H$) and of zero mean. We are then in
position to apply Lemma~\ref{lem:malin}, which yields, using~\eqref{eq:regul_w},
\begin{eqnarray}
\left|
\int_E [[\phi]] \ \Big( n \cdot (\nabla
w)_\varepsilon - \langle n \cdot (\nabla w)_\varepsilon \rangle_E \Big) f
\right|
& \leq &
4 \, \varepsilon \, q_E \| \nabla w \|_{C^0}
\| \nabla_E \left( f [[ \phi ]] \right) \|_{L^1(E)}
\nonumber
\\
& \leq &
C \, \varepsilon \, \| f \|_{H^1(E)}
\| \, [[ \phi ]] \, \|_{H^1(E)},
\label{eq:h1}
\end{eqnarray}
where, for any function $g$, $\nabla_E g = t_E \cdot \nabla g$ where $t_E$ is a unit vector tangential to the edge $E$.
By interpolation between~\eqref{eq:l2} and~\eqref{eq:h1}, and
using~\eqref{eq:debut}, we obtain~\eqref{eq:claim3_1}, with a constant $C$ (independent of the edge) which is independent from $\varepsilon$ and $H$ by scaling arguments (see~\cite{companion-article} for details).
We then deduce from~\eqref{eq:claim3_1} that the first term of the
right-hand side of~\eqref{eq:terme3} satisfies
\begin{eqnarray*}
&&
\left|
\varepsilon \sum_{E\in \mathcal{E}_H}
\int_{E\cap \Omega_\varepsilon} [[\phi]] \ \Big( n \cdot (\nabla
w)_\varepsilon - \langle n \cdot (\nabla w)_\varepsilon \rangle_E \Big) f
\right|
\nonumber
\\
&\leq&
C \, \varepsilon^{3/2} \sum_{E\in \mathcal{E}_H} \| f \|_{H^1(E)}
\| \, [[ \phi ]] \, \|_{H^{1/2}(E)}
\nonumber
\\
&\leq&
C \, \varepsilon^{3/2}
\left( \sum_{E\in \mathcal{E}_H} \| f \|^2_{H^1(E)} \right)^{1/2}
\left( \sum_{E\in \mathcal{E}_H} \| \, [[ \phi ]] \, \|^2_{H^{1/2}(E)} \right)^{1/2}
\nonumber
\\
&\leq&
C \, \varepsilon^{3/2}
\left( \sum_{E\in \mathcal{E}_H; \text{choose one $T \in T_E$}}
\frac{1}{H} \| f \|^2_{H^1(T)} + H \| \nabla f \|^2_{H^1(T)} \right)^{1/2}
\nonumber
\\
&& \times
\left(
\sum_{E\in \mathcal{E}_H} \sum_{T \in T_E} \| \nabla \phi \|^2_{L^2(T)}
\right)^{1/2},
\end{eqnarray*}
where we have used~\eqref{eq:trace1} of Lemma~\ref{lem:trace}
and~\eqref{eq:trace3} of Corollary~\ref{coro:trace} (and, we recall,
$T_E \subset {\cal T}_H$ denotes all the triangles sharing the edge
$E$).
We therefore obtain that the first term of the
right-hand side of~\eqref{eq:terme3} satisfies
\begin{eqnarray}
&&
\left|
\varepsilon \sum_{E\in \mathcal{E}_H}
\int_{E\cap \Omega_\varepsilon} [[\phi]] \ \Big( n \cdot (\nabla
w)_\varepsilon - \langle n \cdot (\nabla w)_\varepsilon \rangle_E \Big) f
\right|
\nonumber
\\
&\leq&
C \, \varepsilon^{3/2}
\left(
\frac{1}{H} \| f \|^2_{H^1(\Omega)} + H \| \nabla f \|^2_{H^1(\Omega)}
\right)^{1/2}
| \phi |_{H^1_H(\Omega_\varepsilon)}
\nonumber
\\
&\leq&
C \, \varepsilon \left( \sqrt{ \frac{\varepsilon}{H} } \,
\| f \|_{H^1(\Omega)} + \sqrt{\varepsilon H} \, \| \nabla f \|_{H^1(\Omega)}
\right)
| \phi |_{H^1_H(\Omega_\varepsilon)}.
\label{eq:terme3_1}
\end{eqnarray}
The second term of the right-hand side of~\eqref{eq:terme3} has no oscillatory character. It is therefore estimated
using standard arguments for Crouzeix-Raviart finite elements (using that
$\displaystyle \int_{E\cap \Omega_\varepsilon} [[\phi]] = 0$), and the
regularity of $w$. Introducing, for each edge $E$, the constant
$\displaystyle c_E = |E|^{-1} \int_E f$, we bound the second term of the
right-hand side of~\eqref{eq:terme3} as follows:
\begin{eqnarray}
\nonumber
&&
\left|
\varepsilon \sum_{E\in \mathcal{E}_H}
\langle n \cdot (\nabla w)_\varepsilon \rangle_E
\int_{E\cap \Omega_\varepsilon} [[\phi]] \ f
\right|
\\
\nonumber
&=&
\left|
\varepsilon \sum_{E\in \mathcal{E}_H}
\langle n \cdot (\nabla w)_\varepsilon \rangle_E
\int_{E\cap \Omega_\varepsilon} [[\phi]] \ (f - c_E)
\right|
\\
\nonumber
& \leq &
C\varepsilon \sum_{E\in \mathcal{E}_H}
\| \, [[\phi ]] \, \|_{L^2(E)} \,
\| f-c_E \|_{L^2(E)}
\\
\nonumber
& \leq &
C\varepsilon
\left( \sum_{E\in \mathcal{E}_H} \| \, [[\phi ]] \, \|^2_{L^2(E)} \right)^{1/2}
\left( \sum_{E\in \mathcal{E}_H} \| f-c_E \|^2_{L^2(E)} \right)^{1/2}
\\
\nonumber
&\leq &
C\varepsilon
\left(
\sum_{E\in \mathcal{E}_H} H \sum_{T \in T_E} \| \nabla \phi
\|^2_{L^2(T)}
\right)^{1/2}
\left( \sum_{E\in \mathcal{E}_H; \text{choose one $T \in T_E$}}
H \| \nabla f \|^2_{L^2(T)} \right)^{1/2}
\\
&\leq &
C \varepsilon H
| \phi |_{H^1_H(\Omega_\varepsilon)}
\, \| \nabla f \|_{L^2(\Omega)},
\label{eq:terme3_2}
\end{eqnarray}
where we have used~\eqref{eq:regul_w},~\eqref{eq:trace2} of Corollary~\ref{coro:trace}
and~\eqref{eq:trace2_pre} of Lemma~\ref{lem:trace}.
We are now left with the third term of the right-hand side
of~\eqref{eq:terme3}. This term has a prefactor $\varepsilon^2$ and all we have to prove is that the term itself is bounded. Using again~\eqref{eq:regul_w},~\eqref{eq:trace1} of
Lemma~\ref{lem:trace} and~\eqref{eq:trace2} of
Corollary~\ref{coro:trace}, we obtain
\begin{eqnarray}
\nonumber
&& \left|
\varepsilon^2 \sum_{E \in \mathcal{E}_H}
\int_{E \cap \Omega_\varepsilon}
[[\phi]] \ w_\varepsilon \ n \cdot \nabla f
\right|
\\
&\leq&
C \varepsilon^2 \sum_{E\in \mathcal{E}_H}
\| \nabla f \|_{L^2(E)} \, \| \, [[\phi]] \, \|_{L^2(E)}
\nonumber
\\
&\leq &
C \varepsilon^2
\left( \sum_{E\in \mathcal{E}_H} \| \nabla f \|^2_{L^2(E)} \right)^{1/2}
\left( \sum_{E\in \mathcal{E}_H} \| \, [[\phi]] \, \|^2_{L^2(E)} \right)^{1/2}
\nonumber
\\
&\leq &
C \varepsilon^2
\left( \frac{1}{H} \sum_{T\in \mathcal{T}_H} \| \nabla f \|^2_{H^1(T)} \right)^{1/2}
\left( H \sum_{T\in \mathcal{T}_H} \| \nabla \phi \|^2_{L^2(T)} \right)^{1/2}
\nonumber
\\
&\leq &
C \varepsilon^2 \| \nabla f\|_{H^1(\Omega)} \, | \phi |_{H^1_H(\Omega_\varepsilon)}.
\label{eq:terme3_3}
\end{eqnarray}
Collecting~\eqref{eq:terme3}, \eqref{eq:terme3_1}, \eqref{eq:terme3_2}
and~\eqref{eq:terme3_3}, we obtain that the third term of the right-hand
side of~\eqref{eq:decompo_error} satisfies
\begin{multline}
\left|
\varepsilon^2 \sum_{E \in \mathcal{E}_H}
\int_{E \cap \Omega_\varepsilon}
[[\phi]] \ n \cdot \nabla (w_\varepsilon f)
\right|
\leq
C \varepsilon \left(
\sqrt{ \frac{\varepsilon}{H} } \,
\| f \|_{H^1(\Omega)} + \sqrt{\varepsilon H} \, \| \nabla f \|_{H^1(\Omega)}
\right.
\\
+
H \| \nabla f \|_{L^2(\Omega)}
+
\varepsilon \| \nabla f\|_{H^1(\Omega)}
\Big) | \phi |_{H^1_H(\Omega_\varepsilon)}.
\label{eq:bound_3}
\end{multline}
\paragraph{Conclusion of Step 1:}
Collecting~\eqref{eq:decompo_error},~\eqref{eq:bound_1},~\eqref{eq:bound_2}
and~\eqref{eq:bound_3}, we deduce that
\begin{eqnarray}
| u-v_H |_{H^1_H(\Omega_\varepsilon)}
&=&
| \phi |_{H^1_H(\Omega_\varepsilon)}
\nonumber
\\
&\leq&
C \varepsilon \left( \sqrt{\varepsilon} + H + \sqrt{\frac{\varepsilon}{H}} \right)
\left( \|f\|_{L^\infty(\Omega)} + \|\nabla f\|_{H^1(\Omega)} \right).
\label{eq:fin_step1}
\end{eqnarray}
This concludes the first step of the proof.
\paragraph{Step 2: Estimation of $u_H-v_H$:}
Denoting by $\phi_H = u_H-v_H$, where $u_H$ is the solution to~\eqref{2DH} and $v_H$ is defined by~\eqref{eq:def_fct_vh}, we observe that
\begin{equation}
\label{eq:louis1}
| \phi_H |_{H^1_H(\Omega_\varepsilon)}^2
=
a_H(u_H-v_H,\phi_H)
=
a_H(u-v_H,\phi_H) + a_H(u_H-u,\phi_H),
\end{equation}
where, we recall, $a_H$ is defined by~\eqref{eq:def_aH}.
The first term is estimated using~\eqref{eq:fin_step1}.
The main part of this Step is thus devoted to estimating the second term of~\eqref{eq:louis1}.
Since $\phi_H \in V_H$, we deduce from the discrete variational formulation~\eqref{2DH} that
\begin{eqnarray}
&&
a_H(u_H-u,\phi_H)
\nonumber
\\
&=&
\int_{\Omega_\varepsilon} f \phi_H - \sum_{T \in {\cal T}_H}
\int_{T \cap \Omega_\varepsilon} \nabla u \cdot \nabla \phi_H
\nonumber
\\
&=&
\int_{\Omega_\varepsilon} f \phi_H
-
\sum_{T \in {\cal T}_H}
\int_{T \cap \Omega_\varepsilon} \nabla (u - \varepsilon^2 w_\varepsilon f) \cdot \nabla \phi_H
-
\varepsilon^2 \sum_{T \in {\cal T}_H}
\int_{T \cap \Omega_\varepsilon} \nabla (w_\varepsilon f) \cdot \nabla \phi_H
\nonumber
\\
&=&
\int_{\Omega_\varepsilon} f \phi_H
-
\sum_{T \in {\cal T}_H}
\int_{T \cap \Omega_\varepsilon} \nabla (u - \varepsilon^2 w_\varepsilon f) \cdot \nabla \phi_H
\nonumber
\\
&&
\qquad -
\varepsilon^2 \sum_{T \in {\cal T}_H}
\int_{\partial (T \cap \Omega_\varepsilon)} \phi_H \, n \cdot \nabla (w_\varepsilon f)
+
\varepsilon^2 \sum_{T \in {\cal T}_H}
\int_{T \cap \Omega_\varepsilon} \phi_H \Delta (w_\varepsilon f).
\label{eq:louis2_pre}
\end{eqnarray}
Since $\phi_H = 0$ on $\partial \Omega_\varepsilon$, we can take the integral in the third term of~\eqref{eq:louis2_pre} only on $(\partial T) \cap \Omega_\varepsilon$. Using~\eqref{eq:corrector-lions} for the fourth term, we obtain that
\begin{eqnarray}
&&
a_H(u_H-u,\phi_H)
\nonumber
\\
&=&
-
\sum_{T \in {\cal T}_H}
\int_{T \cap \Omega_\varepsilon} \nabla (u - \varepsilon^2 w_\varepsilon f) \cdot \nabla \phi_H
-
\varepsilon^2 \sum_{T \in {\cal T}_H}
\int_{(\partial T) \cap \Omega_\varepsilon} \phi_H \, n \cdot \nabla (w_\varepsilon f)
\nonumber
\\
&&
\qquad
+
\varepsilon \sum_{T \in {\cal T}_H}
\int_{T \cap \Omega_\varepsilon} \phi_H \Big(
2 (\nabla w)_\varepsilon \cdot \nabla f
+
\varepsilon w_\varepsilon \Delta f \Big).
\label{eq:louis2}
\end{eqnarray}
We now successively bound the three terms of the right-hand side
of~\eqref{eq:louis2}. The first term is estimated simply using homogenization theory, since it is not specifically related to the discretization. We write, as in~\eqref{eq:bound_1},
\begin{equation}
\label{eq:bound_1new}
\left|
\sum_{T \in \mathcal{T}_H} \int_{\Omega_\varepsilon \cap T}
\nabla (u-\varepsilon^2 w_\varepsilon f) \cdot \nabla \phi_H
\right|
\leq
C \varepsilon^{3/2} \, {\cal N}(f) \,
| \phi_H |_{H^1_H(\Omega_\varepsilon)}.
\end{equation}
For the second term of the right-hand side
of~\eqref{eq:louis2}, we use the same arguments as for the third term of~\eqref{eq:decompo_error}. We have
$$
\varepsilon^2 \sum_{T \in {\cal T}_H}
\int_{(\partial T) \cap \Omega_\varepsilon} \phi_H \, n \cdot \nabla (w_\varepsilon f)
=
\varepsilon^2 \sum_{E \in {\cal E}_H}
\int_{E \cap \Omega_\varepsilon} [[\phi_H]] \, n \cdot \nabla (w_\varepsilon f),
$$
and we note that $\displaystyle \int_E [[ \phi_H ]] = 0$. We therefore can use
the same arguments as in Step 1c, and obtain, similarly
to~\eqref{eq:bound_3},
\begin{multline}
\left|
\varepsilon^2 \sum_{E \in \mathcal{E}_H}
\int_{E \cap \Omega_\varepsilon}
[[\phi_H]] \ n \cdot \nabla (w_\varepsilon f)
\right|
\\
\leq
C \varepsilon \left(
\sqrt{ \frac{\varepsilon}{H} } \,
\| f \|_{H^1(\Omega)}
+
(\varepsilon +H) \| \nabla f\|_{H^1(\Omega)}
\right) | \phi_H |_{H^1_H(\Omega_\varepsilon)}.
\label{eq:bound_3new}
\end{multline}
We next turn to the third term of the right-hand side
of~\eqref{eq:louis2}, which is estimated using the Cauchy-Schwarz inequality, the fact that the second factor is bounded and the first factor satisfies a Poincar\'e inequality. Indeed, using the regularity~\eqref{eq:regul_w} of $w$ and
the Poincar\'e inequality~\eqref{eq:poincare_perfore2} satisfied by $\phi_H \in V_H \subset W_H$, we have
\begin{eqnarray}
&&
\left|
\varepsilon \sum_{T \in {\cal T}_H}
\int_{T \cap \Omega_\varepsilon} \phi_H \Big(
2 (\nabla w)_\varepsilon \cdot \nabla f
+
\varepsilon w_\varepsilon \Delta f \Big)
\right|
\nonumber
\\
&\leq&
C \varepsilon \sum_{T \in {\cal T}_H}
\| \phi_H \|_{L^2(T \cap \Omega_\varepsilon)} \,
\left( \| \nabla f \|_{L^2(T \cap \Omega_\varepsilon)} + \varepsilon
\| \Delta f \|_{L^2(T \cap \Omega_\varepsilon)} \right)
\nonumber
\\
&\leq&
C \varepsilon \| \phi_H \|_{L^2(\Omega_\varepsilon)} \,
\| \nabla f \|_{H^1(\Omega)}
\nonumber
\\
&\leq&
C \varepsilon^2 | \phi_H |_{H^1_H(\Omega_\varepsilon)} \,
\| \nabla f \|_{H^1(\Omega)}.
\label{eq:bound_T3}
\end{eqnarray}
Collecting~\eqref{eq:louis2}, \eqref{eq:bound_1new},
\eqref{eq:bound_3new} and~\eqref{eq:bound_T3}, we deduce that
\begin{multline}
\label{eq:louis3}
\left|
a_H(u_H-u,\phi_H)
\right|
\\
\leq
C \varepsilon \left( \sqrt{\varepsilon} + H +
\sqrt{ \frac{\varepsilon}{H} }
\right)
\left( \| f \|_{L^\infty(\Omega)} + \| \nabla f \|_{H^1(\Omega)}
\right) \,
| \phi_H |_{H^1_H(\Omega_\varepsilon)}.
\end{multline}
Inserting~\eqref{eq:louis3} into~\eqref{eq:louis1}, we have
\begin{eqnarray*}
&&
| \phi_H |_{H^1_H(\Omega_\varepsilon)}^2
\\
& \leq &
a_H(u-v_H,\phi_H)
+
C \varepsilon \left( \sqrt{\varepsilon} + H +
\sqrt{ \frac{\varepsilon}{H} }
\right)
\left( \| f \|_{L^\infty(\Omega)} + \| \nabla f \|_{H^1(\Omega)}
\right) \,
| \phi_H |_{H^1_H(\Omega_\varepsilon)}
\\
&\leq &
|u-v_H|_{H^1_H(\Omega_\varepsilon)} \, | \phi_H |_{H^1_H(\Omega_\varepsilon)}
\\
&& \qquad +
C \varepsilon \left( \sqrt{\varepsilon} + H +
\sqrt{ \frac{\varepsilon}{H} }
\right)
\left( \| f \|_{L^\infty(\Omega)} + \| \nabla f \|_{H^1(\Omega)}
\right) \,
| \phi_H |_{H^1_H(\Omega_\varepsilon)}.
\end{eqnarray*}
Factoring out $| \phi_H |_{H^1_H(\Omega_\varepsilon)}$, and
using~\eqref{eq:fin_step1}, we deduce that
\begin{eqnarray}
| u_H-v_H |_{H^1_H(\Omega_\varepsilon)}
&=&
| \phi_H |_{H^1_H(\Omega_\varepsilon)}
\nonumber
\\
&\leq&
C \varepsilon \left( \sqrt{\varepsilon} + H + \sqrt{\frac{\varepsilon}{H}} \right)
\left( \| f \|_{L^\infty(\Omega)} + \| \nabla f \|_{H^1(\Omega)}
\right).
\label{eq:fin_step2}
\end{eqnarray}
\paragraph{Conclusion} We deduce from~\eqref{eq:fin_step1},
\eqref{eq:fin_step2} and the triangle inequality that
$$
| u-u_H |_{H^1_H(\Omega_\varepsilon)}
\leq
C \varepsilon \left( \sqrt{\varepsilon} + H + \sqrt{\frac{\varepsilon}{H}} \right)
\left( \| f \|_{L^\infty(\Omega)} + \| \nabla f \|_{H^1(\Omega)}
\right),
$$
which is the desired estimate~\eqref{eq:mainresult}. This concludes the proof of Theorem~\ref{theo:main}.
\section{Numerical tests}
\label{sec:Numerical-tests}
We now solve~\eqref{eq:genP} for some particular settings, comparing our approach with other existing MsFEM type methods. As pointed out in the introduction, we numerically explore the influence of three parameters:
\begin{itemize}
\item (i) the boundary conditions imposed to define the MsFEM basis functions and (ii) the addition, or not, of bubble functions. To do so, in Section~\ref{sec:num-comp}, we compare the approach we propose with other existing approaches, considering two versions of each approach, one with and the other without bubble functions.
\item (iii) the possible intersections of the perforations with the edges of mesh elements. We address this question in Section~\ref{sec:num-rob}, and check there the robustness of our approach with respect to the
location of the perforations: the fact
that the mesh intersects, or does not intersect, the perforations has a
very little influence on the (good) accuracy of our approach, in
contrast to other approaches.
\end{itemize}
We eventually turn in Section~\ref{sec:non-per} to a non-periodic
test-case, where we again show the excellent performance of our approach.
We mention that, in all our numerical experiments, we actually
do not directly solve~\eqref{eq:genP} but a penalized version of this
problem: find $u\in H_0^1(\Omega)$
such that
$$
-\operatorname{div} (\nu \nabla u)+\sigma u=f
$$
with the following penalization parameters:
$$
\nu =\left\{
\begin{array}{c}
1 \text{ in $\Omega \setminus B_\varepsilon$}
\\
\displaystyle \frac{1}{h} \text{ in $B_\varepsilon$}
\end{array}
\right.
\quad \text{and} \quad
\sigma =\left\{
\begin{array}{c}
0 \text{ in $\Omega \setminus B_\varepsilon$}
\\
\displaystyle \frac{1}{h^3} \text{ in $B_\varepsilon$}
\end{array}
\right.
,
$$
where $h$ is the fine-scale mesh size used to precompute the highly
oscillatory basis functions (see~\cite{angot,carballal} for more
details on the penalization approach and on the above choice of $\nu$ and $\sigma$). In practice, the chosen fine-scale mesh size always satisfies $h \leq \varepsilon/10$.
Note that, because we use a penalized approach, we do not have to mesh $\Omega_\varepsilon$, which could be cumbersome and could possibly request elements of small size (comparable to the small size $\varepsilon$ present in the geometry of $\Omega_\varepsilon$). In addition, if we were working with a mesh of $\Omega_\varepsilon$, we might face difficulties with the oversampling variant of the MsFEM approach that we compare here with our approach. Indeed, edges of the oversampling domain may intersect the perforations. Properly defining the MsFEM basis functions in such a case would not be straightforward. For these two reasons, we consider a penalization approach.
\subsection{Comparison with existing approaches}
\label{sec:num-comp}
We solve~\eqref{eq:genP} on the domain
$\Omega=(0,1)^2$, with the
right-hand side~$\displaystyle f(x,y) =\sin \frac{\pi x}{2} \, \sin \frac{\pi
y}{2}$, and we take $B_\varepsilon$ the set of discs of radius
$0.35\varepsilon$ periodically located on the regular grid of period
$\varepsilon =0.03$.
For the reference solution, we use a mesh of size
$1024 \times 1024$.
The approaches we compare our approach with are the following four respective approaches:
\begin{itemize}
\item the standard Q1 finite element method on the coarse mesh of size $H$. Of course, we do not expect that method to perform well for this multiscale problem and we only consider it as a ``normalization''.
\item the MsFEM with linear boundary conditions. Although this method is now a bit outdated, it is still considered as the primary MsFEM approach, upon which all the other variants are built.
\item the MsFEM with oscillatory boundary conditions.
This variant (in the form presented in~\cite{hou1999}) is restricted to
the two-dimensional setting. It
uses boundary conditions provided by the solution to
the oscillatory ordinary differential equation obtained by taking the
trace of the original equation on the edge considered. The approach performs fairly well on a number of cases, although it may also fail.
\item the variant of MsFEM using oversampling. This variant is often considered as the ``gold standard'', although it includes a parameter (the oversampling ratio), the value of which should be carefully chosen. When this parameter is taken large, the method becomes (possibly prohibitively) expensive.
\end{itemize}
In addition, we consider for each of those approaches, and for our
specific Crouzeix-Raviart type approach, two variants:
one with, and the other without a specific enrichment of the basis set
elements using bubble functions. For all approaches but the
Crouzeix-Raviart type approach that we propose, the bubble $\Psi$ on the
quadrangle $Q$ is defined as the solution to
$$
-\Delta \Psi = 1 \text{ on $Q \cap \Omega_\varepsilon$},
\quad \Psi = 0 \text{ on $\partial(Q \cap \Omega_\varepsilon)$}.
$$
For the Crouzeix-Raviart approach, the bubble function $\Psi$ has been defined in Section~\ref{ssec:msfem} by~\eqref{eq:def_psi}.
\begin{remark}
Other variants of the MsFEM approach have also been proposed, such as the
Petrov-Galerkin variant with oversampling~\cite{hou2004}. We do not consider this variant here, and refer to our previous work~\cite{companion-article} for some elements of comparison (in a slightly different context).
\end{remark}
For a given mesh size $H$, the cost for computing the basis functions (offline stage) varies from one MsFEM variant to the other. However, for a fixed $H$, all methods without (respectively, with) bubble functions essentially share the same cost to solve the macroscopic problem on $\Omega$ (online stage). More precisely, for a given cartesian mesh, and when using variants including the bubble functions, there are 1.5 times more degrees of freedom in our Crouzeix-Raviart approach than in the three alternative MsFEM approaches mentioned above. Since a logarithmic scaling is used for the x-axis in the figures below, this extra cost does not change the qualitative conclusions we draw below.
The numerical results we have obtained in the regime where the meshsize $H$
is of the order of, or larger than, the parameter $\varepsilon$ are presented
on~Figure~\ref{fig:errors}. For all values of the meshsize~$H$, and for both $L^2$ and broken $H^1$ norms, a
definite superiority of our approach over
all other approaches is observed, and the interest of adding bubble
functions to the basis set is, for each approach, also evident.
A side remark is the following. On Figure~\ref{fig:errors}, we observe that, when using bubble functions, the error decreases as $H$ increases. This might seem counterintuitive at first sight. Note however that, when $H$ increases, the cost of computing each basis function increases, as we need to solve a local problem (discretized on a mesh of size $h$ controlled by the value of $\varepsilon$) on a larger coarse element. In contrast to traditional FEM, increasing $H$ does not correspond to reducing the overall computational cost. For MsFEM approaches, increasing $H$ actually corresponds to decreasing the online cost but increasing the offline cost. The regime of interest is that of moderate values of $H$, for which the offline stage cost is acceptable. We only show the right part of Figure~\ref{fig:errors} (corresponding to large values of $H$, leading to a prohibitively expensive offline stage) for the sake of completeness.
\begin{figure}
\caption{Relative ($L^2$, left, and $H^1$-broken, right) errors with
various approaches in the regimes $H \simeq \varepsilon$ and $H \gtrsim \varepsilon$:
FEM -- the standard Q1 finite elements, no OS -- MsFEM with linear boundary
conditions, osc -- MsFEM with oscillatory boundary
conditions, OS -- MsFEM with oversampling (where the size of the
quadrangles used to compute the basis functions is $3H \times 3H$), CR
-- the MsFEM approach \`a la Crouzeix-Raviart we propose. Results for all
these methods are represented by solid lines. The dashed lines correspond to
the variants of these methods where we enrich the finite element spaces using bubble functions.
\label{fig:errors}
\label{fig:errors}
\end{figure}
To get a better understanding of the
approaches with bubble functions, we have run a series of tests in a regime
different from that of Figure~\ref{fig:errors}, where the meshsize $H$
is of the order of, or larger than, the parameter $\varepsilon$. On
Figure~\ref{fig:errors_regime}, we present results corresponding to the
regime $H \ll \varepsilon$. This is performed only for the purpose of analyzing the behaviour of the methods and this is of course not the practical regime where we want
to use MsFEM approaches. It is
however useful to observe how the various numerical approaches behave in
that regime.
We consider the same problem as above, with $\varepsilon= 0.3$ instead of 0.03,
and where the meshsize $H$ ranges from $1/8$ to $1/128$, so that indeed
$H$ is smaller (and even much smaller) than $\varepsilon$. The reference
solution is again computed on a mesh of size
$1024 \times 1024$. As expected, we then observe that all errors
uniformly decrease when $H$ decreases, in contrast to the situation displayed on Figure~\ref{fig:errors} and commented upon above.
We then recover the classical
behavior of numerical approaches in the limit of fine discretizations.
\begin{figure}
\caption{Relative ($L^2$, left, and $H^1$-broken, right) errors with
the same approaches as on Figure~\ref{fig:errors}
\label{fig:errors_regime}
\end{figure}
For the sake of completeness, we have also considered another oversampling ratio for the MsFEM oversampling approach we compare our approach with. Recall indeed that, on Figures~\ref{fig:errors} and~\ref{fig:errors_regime}, we have considered an oversampling ratio equal to 3. We now additionally consider the method with an oversampling ratio equal to 2. Results are reported on Figure~\ref{fig:over}. As expected, the
accuracy of MsFEM increases when the oversampling ratio increases. The
artificial Dirichlet boundary conditions used to define basis functions
are then further away from the relevant part of the mesh element, and their potentially poor behavior close to the boundary has a smaller influence. Of course, as the
oversampling ratio increases, the cost of computing these basis
functions increases. We observe that, with the MsFEM approach \`a la
Crouzeix-Raviart we propose, we obtain a better accuracy (again both in $H^1$ and $L^2$ norms) than with
the MsFEM approach that uses an oversampling ratio of 3 (i.e., that
computes basis functions by solving local problems on quadrangles of
size $3H \times 3H$).
\begin{figure}
\caption{Relative ($L^2$, left, and $H^1$-broken, right) errors with
various approaches (dashed lines: using bubble functions; solid lines:
without bubble functions):
OS -- MsFEM with various oversampling ratios, CR -- the MsFEM approach \`a la
Crouzeix-Raviart we propose.
\label{fig:over}
\label{fig:over}
\end{figure}
\begin{remark}
Figures~\ref{fig:errors},~\ref{fig:errors_regime} and~\ref{fig:over} show that, for any of the numerical approaches we have considered, the relative $L^2$ error is always smaller than the relative $H^1$ error. The former presumably converges with a better rate (in terms of $\varepsilon$ and $H$) than the latter, although establishing sharp $L^2$ error estimates for MsFEM-type approaches is quite involved (see e.g.~\cite{hou1999}).
\end{remark}
\subsection{Robustness with respect to the location of the perforations}
\label{sec:num-rob}
In this section and in the following one, we perform a series of tests with a different, specific purpose. As
a major motivation for advocating our approach is the flexibility of
Crouzeix-Raviart type finite elements in terms of boundary conditions,
we expect our approach to be particularly effective (and therefore
considerably superior to other approaches) when some edges of
the mesh happen to intersect perforations of the domain. The more such
intersections, the more important the difference. In order to check this expected behaviour, we design the
following test.
We solve~\eqref{eq:genP} on the domain $\Omega=(0,1)^2$, with a constant
right-hand side~$f=1$, and we take $B_\varepsilon$ the
set of discs of radius $0.2\varepsilon$ periodically located on the
regular grid of period $\varepsilon =0.1$. We compute the reference
solution, and consider 3 variants of MsFEM: the linear
version, the oversampling version and the Crouzeix-Raviart version.
The last three approaches are implemented in the variant that includes
bubble functions in the basis set and they are run on a mesh of size $H=0.2$.
We now perform two sets of numerical experiments. They are identical
except for what concerns the relative position of the mesh with the
perforations. The difference between the two sets of tests is that, from
one set of tests to the other one, the perforations are shifted by
$\varepsilon/2$ in the directions $x$ and $y$. In our Test~1, no edge
intersects any perforation, while, on our Test~2, many edges actually
intersect perforations.
To some extent, the situation of Test 1 is the best case scenario (where as few edges as possible intersect the perforations) and the other situation is the worst case scenario.
The numerical solutions computed for each of the situations considered
is shown on Figures~\ref{fig:decalage-test1}
and~\ref{fig:decalage-test2}, for Test~1 and Test~2 respectively.
The numerical errors observed, computed both in $L^2$ and $H^1$-broken
norms, are correspondingly displayed on Tables~\ref{table:test1}
and~\ref{table:test2} respectively. More than the actual values obtained
for each case, this is the trend of difference between Table~\ref{table:test1}
and Table~\ref{table:test2} that is the practically relevant feature. A
comparison between the two tables indeed show that, qualitatively and in
either of the norms used for measuring the error, the
linear version and the oversampling version of MsFEM are both much more
sensitive to edges intersecting perforations than the Crouzeix-Raviart
version of MsFEM.
In particular, the gain of our approach with respect to the linear version of MsFEM is much higher in our Test 2 (which is, from the geometrical viewpoint, the worst case scenario) than in Test 1.
This confirms the intuition of a better flexibility of
our approach. This also allows for expecting a much better
behaviour of that approach for nonperiodic multiscale perforated
problems for which it is extremely difficult, practically, to avoid
repeated intersections of perforations with mesh edges. This is
confirmed by our numerical experiments of Section~\ref{sec:non-per}.
\begin{figure}
\caption{(Test 1) Left to right and top to bottom: Reference solution
(on the mesh $200 \times 200$), MsFEM with linear boundary conditions, MsFEM
with oversampling (where the size of the quadrangles used to compute
the basis functions is $3H \times 3H$), proposed MsFEM \`a la
Crouzeix-Raviart.
\label{fig:decalage-test1}
\label{fig:decalage-test1}
\end{figure}
\begin{table}[h!]
\centering{
\begin{tabular}{l|c|c}
& $L^2$ error (\%) & $H^1$ error (\%) \\ \hline
MsFEM with linear conditions & 16 & 32 \\ \hline
MsFEM with oversampling & 20 & 38 \\ \hline
MsFEM \`a la Crouzeix-Raviart & 9 & 24
\end{tabular}
}
\caption{Numerical relative errors for Test 1
\label{table:test1}}
\end{table}
\begin{figure}
\caption{(Test 2) Left to right and top to bottom: Reference solution
(on the mesh $200 \times 200$), MsFEM with linear boundary conditions,
MsFEM with oversampling (where the size of the quadrangles used to compute
the basis functions is $3H \times 3H$), proposed MsFEM \`a la Crouzeix-Raviart.
\label{fig:decalage-test2}
\label{fig:decalage-test2}
\end{figure}
\begin{table}[h!]
\centering{
\begin{tabular}{l|c|c}
& $L^2$ error (\%) & $H^1$ error (\%) \\ \hline
MsFEM with linear conditions & 28 & 52 \\ \hline
MsFEM with oversampling & 12 & 31 \\ \hline
MsFEM \`a la Crouzeix-Raviart & 9 & 27
\end{tabular}}
\caption{Numerical relative errors for Test 2
\label{table:test2}}
\end{table}
\subsection{A test on a non-periodic geometry of perforations}
\label{sec:non-per}
A major motivation for using MsFEM approaches is to address
non-periodic cases, for which homogenization theory does not provide any
explicit approximation procedure. We have tested several such examples, two of them
being shown on Figure~\ref{fig:ex-non-per}. For each of them, the domain
$\Omega = (0,1)^2$ is meshed using quadrangles of size $H$, with $1/128
\leq H \leq 1/8$. The reference
solution is again computed on a mesh of size
$1024 \times 1024$.
\begin{figure}
\caption{Two examples of domains with non-periodic perforations (represented in black). Perforations have a rectangular shape, with a center randomly located in $\Omega=(0,1)^2$ according to the uniform distribution. Left: perforations are made from 100 rectangles, the width and height of which are uniformly distributed between 0.02 and 0.05. Right: perforations are made from 60 rectangles, the width (resp. the height) of which is uniformly distributed between 0.02 and 0.04 (resp. 0.02 and 0.4).
\label{fig:ex-non-per}
\label{fig:ex-non-per}
\end{figure}
Errors are shown on Figure~\ref{fig:errors-non-per3} (resp. Figure~\ref{fig:errors-non-per5}) for the test-case shown on the left (resp. right) part of Figure~\ref{fig:ex-non-per} (we have obtained similar results for several other test cases not shown here for the sake of brevity). We again see that
our approach provides results at least as accurate as, and often more
accurate than the MsFEM approach with oversampling on quadrangles of
size $3H \times 3H$. Our approach outperforms all the other variants of
MsFEM that we have tested. These results confirm the definite interest
of the variant we introduce in this article.
\begin{figure}
\caption{Relative ($L^2$, left, and $H^1$-broken, right) errors with
the same approaches as on Figure~\ref{fig:errors}
\label{fig:errors-non-per3}
\end{figure}
\begin{figure}
\caption{Relative ($L^2$, left, and $H^1$-broken, right) errors with
the same approaches as on Figure~\ref{fig:errors}
\label{fig:errors-non-per5}
\end{figure}
\noindent{\bf Acknowledgments.} The work of the first two authors is
partially supported by ONR under Grant N00014-12-1-0383 and by EOARD under Grant FA8655-13-1-3061. The third author acknowledges the hospitality of INRIA.
We thank William Minvielle for his remarks on a preliminary version of this article.
\appendix
\section{Technical proofs}
We collect in this Appendix the proof of two technical results used in Section~\ref{sec:proof}, namely the Poincar\'e inequality~\eqref{eq:poincare_perfore} and the homogenization result~\eqref{eq:lions-general}.
\subsection{The Poincar\'e inequality in perforated domains}
\label{sec:proof_poincare}
Consider the unit square $Y = (0,1)^d$ in dimension $d$, and some smooth perforation $B \subset Y$. There exists a constant $\mathcal{C}>0$ such that, for any $\phi \in H^1(Y \setminus B)$ with $\phi = 0$ on $\partial B$, we have
\begin{equation}
\label{eq:poinc_base}
\| \phi \|_{L^2(Y \setminus B)} \leq \mathcal{C} \| \nabla \phi \|_{L^2(Y \setminus B)}.
\end{equation}
Let $Y_k^B := k + (Y \setminus B)$ be the perforated unit cell after translation by the vector $k \in \mathbb{Z}^d$. We scale $Y_k^B$ by a factor $\varepsilon$ and repeat this pattern periodically (with a period $\varepsilon$ in all directions) for a finite number of times. We hence introduce
\begin{equation}
\label{eq:def_K}
Q_\varepsilon = \underset{k \in K}{\cup} \left( \varepsilon Y_k^B \right),
\quad
K = \left\{ k \in \mathbb{Z}^d, \ a^-_i \leq k_i \leq a^+_i \ \text{for any $1 \leq i \leq d$} \right\}
\end{equation}
for some $a^-_i$ and $a^+_i$ in $\mathbb{Z}$, that we can also write as
$$
Q_\varepsilon = R_\varepsilon \setminus P_\varepsilon,
$$
where $R_\varepsilon$ is the quadrangle $R_\varepsilon = \underset{k \in K}{\cup} \left( \varepsilon (k+Y) \right)$ and $P_\varepsilon$ is the set of perforations $P_\varepsilon = \underset{k \in K}{\cup} \left( \varepsilon (k+B) \right)$. Summing the inequality~\eqref{eq:poinc_base} for all cells and next scaling the geometry, we obtain that, for any $\phi \in H^1(R_\varepsilon \setminus P_\varepsilon)$ with $\phi = 0$ on $\partial P_\varepsilon$, we have
\begin{equation}
\label{eq:poinc_base2}
\| \phi \|_{L^2(R_\varepsilon \setminus P_\varepsilon)} \leq \mathcal{C} \varepsilon \| \nabla \phi \|_{L^2(R_\varepsilon \setminus P_\varepsilon)}
\end{equation}
where $\mathcal{C}$ is the same constant as in~\eqref{eq:poinc_base}.
Consider now $\phi \in H^1_0(\Omega_\varepsilon)$. There exists a set $K$ of the form~\eqref{eq:def_K} such that $\Omega_\varepsilon \subset Q_\varepsilon$ (it is sufficient to include $\Omega_\varepsilon$ into a sufficiently large perforated quadrangle). We now introduce $\overline{\phi}$, defined on $Q_\varepsilon$ by
$$
\overline{\phi} = \phi \text{ on $\Omega_\varepsilon$},
\quad
\overline{\phi} = 0 \text{ otherwise},
$$
and readily see that $\overline{\phi} \in H^1(Q_\varepsilon)$ and $\overline{\phi} = 0$ on $\partial P_\varepsilon$. The function $\overline{\phi}$ thus satisfies~\eqref{eq:poinc_base2}. We hence obtain
$$
\| \phi \|_{L^2(\Omega_\varepsilon)}
=
\left\| \overline{\phi} \right\|_{L^2(R_\varepsilon \setminus P_\varepsilon)}
\leq
\mathcal{C} \varepsilon \left\| \nabla \overline{\phi} \right\|_{L^2(R_\varepsilon \setminus P_\varepsilon)}
=
\mathcal{C} \varepsilon \| \nabla \phi \|_{L^2(\Omega_\varepsilon)}.
$$
This completes the proof of~\eqref{eq:poincare_perfore}.
\subsection{Homogenization result}
\label{sec:proof_hom}
In this section, we prove~\eqref{eq:lions-general}. To do so, we actually do not use~\eqref{eq:lions}. The proof below actually provides an alternative proof of~\eqref{eq:lions} (see Remark~\ref{rem:lions} below).
Let
$\eta^\varepsilon$ be a smooth function on $\overline{\Omega}$ that
vanishes on $\partial \Omega$, satisfies $0 \leq \eta^\varepsilon(x)
\leq 1$ on $\overline{\Omega}$ and is equal to 1 in
$\omega_\varepsilon = \{ x\in \Omega \text{ s.t. }
\mbox{dist}(x,\partial \Omega) > \varepsilon \}$. Using the fact that $\Omega$ is smooth, it is easy to see that
such a function can be constructed
for each~$\varepsilon >0$ and we can suppose that it satisfies
\begin{multline}
\| \eta^\varepsilon \|_{L^\infty(\Omega)} \leq C,
\quad
\| 1- \eta^\varepsilon \|_{L^2(\Omega)} \leq C \sqrt{\varepsilon},
\\
\left\| \nabla \eta^\varepsilon \right\|_{L^\infty(\Omega)} \leq
\frac{C}{\varepsilon},
\quad
\left\| \nabla \eta^\varepsilon \right\|_{L^2(\Omega)} \leq
\frac{C}{\sqrt{\varepsilon}},
\quad
\left\| \nabla^2 \eta^\varepsilon \right\|_{L^2(\Omega)} \leq
\frac{C}{\varepsilon^{3/2}}
\label{eq:pty_eta}
\end{multline}
for some universal constant $C>0$. Set
$\phi = u^\varepsilon -\varepsilon^2 \, w_\varepsilon \, f \,
\eta^\varepsilon$, where $w_\varepsilon(x) = w(x/\varepsilon)$, with $w$ the solution to~\eqref{eq:corrector-lions}. We compute
\begin{eqnarray*}
-\Delta \phi
&=&
f + \varepsilon^2 \Delta (w_\varepsilon \, f \, \eta^\varepsilon)
\\
&=&
f + (\Delta w)_\varepsilon \, f \, \eta^\varepsilon
+ 2 \varepsilon (\nabla w)_\varepsilon \cdot \nabla (f \,
\eta^\varepsilon)
+
\varepsilon^2 w_\varepsilon \, \Delta (f \, \eta^\varepsilon)
\\
&=&
f (1-\eta^\varepsilon)
+
2 \varepsilon (\nabla w)_\varepsilon \cdot \nabla (f \,
\eta^\varepsilon)
+
\varepsilon^2 w_\varepsilon \, \Delta (f \, \eta^\varepsilon)
\end{eqnarray*}
on $\Omega_\varepsilon$, where we have used~\eqref{eq:genP} in the first
line and the fact that $-\Delta w=1$ on $Y \setminus B$ in the last
line. Using the regularity~\eqref{eq:regul_w} of $w$ and the properties~\eqref{eq:pty_eta}
of $\eta^\varepsilon$, we deduce that
\begin{eqnarray}
&&
\| -\Delta \phi \|_{L^2(\Omega_\varepsilon)}
\nonumber
\\
& \leq &
\| f \|_{L^\infty(\Omega)} \, \| 1- \eta^\varepsilon \|_{L^2(\Omega)}
\nonumber
\\
&&+
2 \varepsilon \| \nabla w \|_{L^\infty} \left(
\| f \|_{L^\infty(\Omega)} \, \| \nabla \eta^\varepsilon \|_{L^2(\Omega)}
+
\| \nabla f \|_{L^2(\Omega)} \, \| \eta^\varepsilon \|_{L^\infty(\Omega)}
\right)
\nonumber
\\
&&+
\varepsilon^2 \| w \|_{L^\infty} \Big(
\| f \|_{L^\infty(\Omega)} \, \| \Delta \eta^\varepsilon \|_{L^2(\Omega)}
+
2 \| \nabla f \|_{L^2(\Omega)} \, \| \nabla \eta^\varepsilon \|_{L^\infty(\Omega)}
\nonumber
\\
&& \qquad \qquad \qquad \qquad +
\| \Delta f \|_{L^2(\Omega)} \, \| \eta^\varepsilon \|_{L^\infty(\Omega)}
\Big)
\nonumber
\\
& \leq &
C \sqrt{\varepsilon} \, {\cal N}(f),
\label{eq:jll}
\end{eqnarray}
where ${\cal N}(f)$ is defined by~\eqref{eq:def_N_f}.
We now notice that $u^\varepsilon$ and $w_\varepsilon \eta^\varepsilon$ vanish on $\partial
\Omega_\varepsilon$, hence $\phi = 0$ on $\partial \Omega_\varepsilon$.
An integration by parts thus yields
\begin{equation}
\label{eq:avant}
\int_{\Omega_\varepsilon} |\nabla \phi |^2
=
\int_{\Omega_\varepsilon} (-\Delta \phi ) \, \phi
\leq
C \sqrt{\varepsilon} \, {\cal N}(f) \,
\|\phi \|_{L^2(\Omega_\varepsilon)}.
\end{equation}
Inserting~\eqref{eq:poincare_perfore} in~\eqref{eq:avant}, we obtain
$| \phi |_{H^1(\Omega_\varepsilon)} \leq C \varepsilon^{3/2} \,
{\cal N}(f)$. We conclude by using the triangle inequality
$$
\left| u^\varepsilon -
\varepsilon^2 w \left( \frac{\cdot}{\varepsilon} \right)
f \right|_{H^1(\Omega_\varepsilon)}
\leq
| \phi |_{H^1(\Omega_\varepsilon)} + \varepsilon^2 \left|
w \left( \frac{\cdot}{\varepsilon} \right) f (1-\eta^\varepsilon)
\right|_{H^1(\Omega_\varepsilon)},
$$
where both terms in the above right-hand side are bounded by
$C \varepsilon^{3/2} \, {\cal N}(f)$. This yields the desired
bound~\eqref{eq:lions-general}.
\begin{remark}
\label{rem:lions}
Note that if $f$ vanishes on $\partial \Omega$, we can take $\eta^\varepsilon \equiv 1$ and~\eqref{eq:jll} is replaced by
$$
\| -\Delta \phi \|_{L^2(\Omega_\varepsilon)}
\leq
C \varepsilon \, {\cal N}(f).
$$
Following the same steps as above, we then recover the bound~\eqref{eq:lions}.
\end{remark}
\end{document} |
\begin{document}
\title{ Can quantum mechanics be considered as statistical? }
\author{Aur\'{e}lien Drezet}
\affiliation{Institut N\'eel UPR 2940, CNRS-University Joseph Fourier, 25 rue des Martyrs, 38000 Grenoble, France}
\date{\today}
\begin{abstract}
This is a short manuscript which was initially submitted to Nature
Physics as a comment to the PBR (Pusey, M.~F., Barrett, J., Rudolph,
T) paper just after its publication in 2012. The comment was not
accepted. I however think that the argumentation is correct: one is
free to judge!
\end{abstract}
\pacs{}
\maketitle
\textbf{To the Editor}- Despite so many successes quantum mechanics is still nowadays steering intense interpretational debates. In this context Pusey, Barrett and Rudolph (PBR) presented recently a new 'no-go' theorem~\cite{PBR} whose aim is to restrict drastically the family of viable quantum interpretations. For this purpose PBR focussed on what Harrigan and Speckens~\cite{speckens} named `epistemic' and `ontic' interpretations~\cite{news,news2} and showed that only the second family can agree with quantum mechanics.\\
\indent Here, we analyze the PBR theorem and show that while mathematically true its correct physical interpretation doesn't support the conclusions of the authors.\\
\indent In the simplest version PBR considered two non orthogonal pure quantum states $|\Psi_1\rangle=|0\rangle$ and $|\Psi_2\rangle=[|0\rangle+|1\rangle]/\sqrt{2}$ belonging to a 2-dimensional Hilbert space $\mathbb{E}$ with basis vectors $\{|0\rangle,|1\rangle\}$. Using a specific measurement basis $|\xi_i\rangle$ ($i\in[1,2,3,4]$) in $\mathbb{E}\otimes\mathbb{E}$ (see their equation 1 in \cite{PBR}) they deduced that $\langle\xi_1|\Psi_1\otimes\Psi_1\rangle=\langle\xi_2|\Psi_1\otimes\Psi_2\rangle=\langle\xi_3|\Psi_2\otimes\Psi_1\rangle=\langle\xi_4|\Psi_2\otimes\Psi_2\rangle=0$. In a second step they introduced hypothetical `Bell's like' hidden variables $\lambda$ and wrote implicitly the probability of occurrence in the form:
\begin{eqnarray}
|\langle\xi_i|\Psi_j\otimes\Psi_k \rangle|^2=\int\int P(\xi_i|\lambda,\lambda')\rho_j(\lambda)\rho_k(\lambda')d\lambda d\lambda'
\end{eqnarray}
where $i\in[1,2,3,4]$ and $j,k\in[1,2]$. In this PBR model there is a independence criterion at the preparation since we write $\rho_{j,k}(\lambda,\lambda')=\rho_j(\lambda)\rho_k(\lambda')$. In these equations we introduced the conditional `transition' probabilities $P(\xi_i|\lambda,\lambda')$
for the outcomes $\xi_i$ supposing the hidden state
$\lambda,\lambda'$ associated with the two independent Q-bits are given. Obviously, we have $\sum_{i=1}^{i=4}P(\xi_i|\lambda,\lambda')=1$. It is then easy using all these definitions and conditions to demonstrate that we must necessarily have $\rho_2(\lambda)\cdot\rho_1(\lambda)=0$ for every $\lambda$ i.e. that $\rho_1$ and $\rho_2$ have nonintersecting supports in the $\lambda$-space. This constitutes the PBR theorem for the particular case of independent prepared states $\Psi_1,\Psi_2$ defined before. PBR generalized theirs results for more arbitrary states using similar and astute procedures described in ref.~1.\\ The general PBR theorem states that the only way to include hidden variable in a
description of the quantum world is to suppose that for every pair
of quantum states $\Psi_1$ and $\Psi_2$ the density of
probability must satisfy the condition of non intersecting support
in the $\lambda$-space:
\begin{eqnarray}
\rho(\lambda,\Psi_1)\rho(\lambda,\Psi_2)=0 & \forall \lambda.
\end{eqnarray}
If this theorem is true it would make hidden variables
completely redundant since it could be possible to
define a bijection or relation of equivalence between the $\lambda$
space and the Hilbert space: (loosely speaking we could in
principle make the correspondence $\lambda\Leftrightarrow\psi$).
Therefore it would be as if $\lambda$ is nothing but a new name
for $\Psi$ itself. This would justify the label `ontic' given to this kind of interpretations by opposition to `epistemic' interpretations ruled out by the PBR result.\\
However, this conclusion is wrong as it can be shown by examining carefully the assumptions necessary for the derivation of the theorem. Indeed, using the well known Bayes-Laplace formula for conditional probability we deduce that the most general Bell's hidden variable probability space should obey the following rule
\begin{eqnarray}
|\langle\xi_i|\Psi_j\otimes\Psi_k \rangle|^2=\int\int P(\xi_i|\Psi_j,\Psi_k,\lambda,\lambda')\rho_j(\lambda)\rho_k(\lambda')d\lambda d\lambda'
\end{eqnarray}
in which in contrast with equation 1 the transition probabilities $P(\xi_i|\Psi_j,\Psi_k,\lambda,\lambda')$ now depend explicitly on the considered quantum states $\Psi_j,\Psi_k$. Relaxing this PBR assumption has a direct effect since we loose the ingredient necessary for the demonstration of equation 2 (more precisely we are not anymore allowed to compare the product states $|\Psi_j\otimes\Psi_k \rangle$ as it was done by PBR). In other words the PBR theorem collapses.\\
\indent Physically speaking our conclusion is sound since many hidden variable models and in particular the one proposed by de Broglie and Bohm belong to the class where the transition probabilities and the trajectories depend contextually on the quantum states $\Psi$.\\
\indent To conclude, contrary to the PBR claim the theorem they
proposed is actually limited to a very narrow class of quantum
interpretations. It fits well with the XIX$^{th}$ like hidden
variable models using Liouville and Boltzmann approaches (i.e.
models where the transition probabilities are independent of $\Psi$)
but it is not in agreement with neo-classical theories such as the
one proposed by de Broglie and Bohm in which the wavefunction is at
the same time an epistemic and ontic ingredient of the dynamics.
Therefore epistemic and ontic states can not be separated in quantum
mechanics.\\
\indent \underline{\emph{additional remark not included in the
original text:}} The interested reader can also consider my two
papers on this subject in arXiv:1203.2475 and arXiv:1209.2565
[Progress in Physics, vol 4 (October 2012)].
\end{document} |
\begin{document}
\makeatletter\def\gdef\@thefnmark{}\@footnotetext}\makeatother\shfiuwefootnote{Version 2020-10-17. See \url{https://shelah.logic.at/papers/1101/} for possible updates.{{\mathfrak d\/}ef\@thefnmark{}\@footnotetext}\makeatother\gdef\@thefnmark{}\@footnotetext}\makeatother\shfiuwefootnote{Version 2020-10-17. See \url{https://shelah.logic.at/papers/1101/} for possible updates.{Version 2020-10-17. See \url{https://shelah.logic.at/papers/1101/} for possible updates.}
\title {Isomorphic limit ultrapowers for infinitary logic \\
Sh1101}
{\rm aut}hor {Saharon Shelah}
\address{Einstein Institute of Mathematics\\
Edmond J. Safra Campus, Givat Ram\\
The Hebrew University of Jerusalem\\
Jerusalem, 9190401, Israel\\
and \\
Department of Mathematics\\
Hill Center - Busch Campus \\
Rutgers, The State University of New Jersey \\
110 Frelinghuysen Road \\
Piscataway, NJ 08854-8019 USA}
\email{[email protected]}
\urladdr{http://shelah.logic.at}
\thanks{The author would like to thank the Israel Science Foundation
for partial support of this research (Grant No. 1053/11). References
like \cite[2.11=La18]{Sh:797} means we cite from \cite{Sh:797}, Claim
2.11 which has label La18, this to help if \cite{Sh:797} will be revised.
The author thanks Alice Leonhardt for the beautiful typing. This was
separated from \cite{Sh:1019} which was first typed May 10, 2012; so
was IJM 7367}
\subjclass[2010]{Primary: 03C45; Secondary: 03C30, 03C55}
\keywords {model theory, infinitary logics, compact cardinals,
ultrapowers, ultra limits, stability, saturated models, classification
theory, isomorphic ultralimits}
\date{2020-10-17}
\begin{abstract}
The logic ${\mathbb L}^1_\theta$
was
introduced in \cite{Sh:797}; it is the maximal logic
below ${\mathbb L}_{\theta,\theta}$ in which a well ordering is not
definable. We investigate it for $\theta$ a compact cardinal. We
prove
that
it satisfies several parallel of classical theorems on first order logic,
strengthening the thesis that it is a natural logic. In
particular, two models are ${\mathbb L}^1_\theta$-equivalent {\underline{if}}f \, for
some $\omega$-sequence of
$\theta$-complete ultrafilters, the iterated ultra-powers by it of those two
models are isomorphic.
Also for strong limit $\lambda > \theta$ of cofinality $\aleph_0$,
every complete ${\mathbb L}^1_\theta$-theory
has a so called
a special model of
cardinality $\lambda$, a parallel of saturated.
For first order theory $ T $ and singular strong limit
cardinal $ \lambda , T $ has a so called special model
of cardinality $ \lambda $. Using
``special" in our context
is justified by:
it is unique (fixing $ T $ and $ \lambda $), all
reducts of a special model are special too, so we have another proof of interpolation in
this case.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
{\rm set}counter{section}{-1}
\centerline {Anotated Content}
{\rm no}indent
\S0 \quad Introduction, pg. \pageref{0}
{\rm no}indent
\S(0A) Background and results, (label v), pg.\pageref{0A}
{\rm no}indent
\S(0B) \quad Preliminaries, (label w,x), pg. \pageref{0B}
{\rm no}indent
\S1 \quad Characterizing equivalence by $\omega$-limit ultra-powers,
(label d), pg.\pageref{1}
{
\noindent}
\begin{enumerate}
\item[${{}}$] [We characterize ${\mathbb L}^1_{< \theta}$-equivalence
of $M_1,M_2$ by having isomorphic ultralimits by a sequence of
length $\omega$ of $\theta$-complete ultrafilters. This logic,
${\mathbb L}^1_\theta$, is from \cite{Sh:797} except that here we
restrict ourselves to $\theta$ is a compact cardinal.]
\end{enumerate}
{\rm no}indent
\S2 \quad Special Models, pg.\pageref{2}
{
\noindent}
\begin{enumerate}
\item[${{}}$] [We investigate model of cardinality a strong limit
cardinal $> \theta$ of cofinality $\aleph_0$.
We define $\lambda$-special model of complete theory $T \subseteq
{\mathbb L}^1_\theta(\tau_T)$, for $\lambda$ as above
and prove existence and uniqueness. We generalize some classical
theorems in model theory.]
\end{enumerate}
\section {Introduction} \label{0}
\subsection {Background and results} \label{0A}
In the sixties, ultra-products were very central in model theory.
Recall Keisler \cite{Ke61}, solving the
outstanding
problem in model theory of the time,
assuming an instance of GCH
characterizes elementary equivalence
in an algebraic way; that is
by proving:
\begin{enumerate}
\item[$\boxplus $]
for any two models $M_1,M_2$
(of vocabulary $\tau$ of cardinality $\le \lambda$ and of cardinality\footnote{in fact ``$M_ {\ell} $ is of cardinality $ \le \lambda ^+$ suffice}
$\le \lambda$, the following are equivalent
provided that $ 2^ \lambda = \lambda ^+$:
\item[(a)] $M_1,M_2$ are elementarily equivalent.
\item[(b)] they have isomorphic ultrapowers,
that is
$M^\lambda_1/D_2\approx M^\lambda_2/D_1$
for some ultrafilter $D_{\ell} $ on a cardinal $ \lambda _ {\ell} $
\item[(c)] $M^ \mu /D_\approx M^\mu /D$
for some ultrafilter $D $ on some cardinal $ \mu $
\item[(d)] as in (c) for $ \mu = \lambda $,
\end{enumerate}
Kochen \cite{Koc61} uses iteration on
taking ultra-powers (on a well ordered index set) to
characterize elementary equivalence.
Gaifman \cite{Gai74} uses ultra-powers on $\aleph_1$-complete
ultrafilters iterated along linear ordered index set. Keisler
\cite{Ke63} uses general $(\aleph_0,\aleph_0)-{\rm l.u.p.}$, see below,
Definition \ref{x16}(4) for $\kappa = \aleph_0$. Shelah \cite{Sh:13}
proves $ \boxplus $ in ZFC, but
with a price: we have to omit clause (d),
and the ultrafilter is on $\mu = 2^\lambda $).
Hodges-Shelah \cite{Sh:109} is closer to the present work and see
there on earlier works, it dealt
with isomorphic ultrapowers (and isomorphic reduced powers) for the
$\theta$-complete ultrafilter (and filter) case, but note
that having isomorphic ultra-powers by
$\theta$-complete ultrafilters is not an equivalence relation.
In particular assume $\theta > \aleph_0$
is a compact cardinal and little more (we can get it by forcing over a
universe with a supercompact cardinal and a class of measurable
cardinals). {\rm Th}en \, two models have isomorphic
ultrapowers for some $\theta$-complete ultrafilter {\underline{if}}f \, in all
relevant games the isomorphism player does not lose. Those relevant
games are of length $\zeta < \theta$ and deal with the reducts to a
sub-vocabulary of cardinality $< \theta$
and usually those games are not determined.
The characterization \cite{Sh:109} of having isomorphic ultra-powers by
$\theta$-complete ultra-filters, necessarily
is not so ``nice" because this relation is
not an equivalence relation. Hence having isomorphic ultra-powers
is not equivalent
to having the same theory in some logic.
Most relevant to the present paper is \cite{Sh:797}
which we continue here. For notational simplicity
let $ \theta $ be an inaccessible cardinal.
An old problem from the seventies was:
$ \boxdot $
is there a logic between $ \mathbb{L} _{\theta , {\aleph_0} }$
and $ \mathbb{L} _{\lambda, \theta }$
which satisfies interpolation?
Generally interpolation had posed a hard problem
in soft model theory. Another, not so precise problem
was to find generalizations of Lindstrom theorem, see
\cite{Van11}.
Now \cite{Sh:797} solve the first problem
and suggest a solution to the second problem, by
showing that the logic $ \mathbb{L} ^1_ \theta $
introduced there. It was proved that it satisfies
$\boxdot $ and give a characterization: e.g it is a maximal
logic in the interval mentioned in $ \boxdot $
which satisfies non-definability of well order
in a suitable sense (see \cite[3.4=La28]{Sh:797}.
Another line of research was investigating
infinitary logics for $ \theta $ a compact
cardinal , see \cite{Sh:1019} and history there.
We continue those two lines, investigating $ \mathbb{L} ^1_\theta $
for $ \theta $ a compact cardinal.
We prove that it is an interesting logic:
it share with first order logic several classical
theorems.
We may wonder, do we have a characterization of models
being ${\mathbb L}^1_\theta$-equivalent?
In \S1 we characterize ${\mathbb L}^1_\theta$-equivalence of models by having
isomorphic iterated ultra-powers of length $\omega$.
We then
in \S2
prove some further generalizations
of classical model theoretic theorems, like the existence and
uniqueness of special models in $\lambda$ when $\lambda > \theta + |T|$
is strong limit of cofinality $\aleph_0$. All this seems to
strengthen the thesis of \cite{Sh:797} that ${\mathbb L}^1_\theta$ is a
natural logic.
We thank the referee for many helpful comments.
Of course, success drive us to consider further
problems. For another approach
see \cite{Sh:893}.
\begin{question} \label{y5}
1) Assume $ \theta $ is a strong limit
singular cardinal of cofinality $ {\aleph_0} $.
{\rm no}indent
1) Does the logic $ \mathbb{L} _{\theta ^+ , \theta }$
restricted to models of cardinality $ \theta $
has interpolation?
{\rm no}indent
2)
Is there a logic $ {\ell} $ with interpolation
such that:
$ \mathbb{L} _{\theta ^+, \theta }
\le {\ell} \le \mathbb{l} _{\theta ^= , \theta ^+}$,
\end{question}
\begin{question} \label{y8}
Let $ \theta $ be a compact cardinal and $ \lambda > \theta $
is strong limit of cofinality $ {\aleph_0} $.
{\rm no}indent
1) Does the logic $ \mathbb{L} _{\theta, \theta }$
restricted to model of cardinality $ \lambda $
has interpolation?
{\rm no}indent
2) Can we characterize when a theory
$ T \subseteq \mathbb{L}^1 _{\theta } $
of cardinality $ < \theta $ is categorical in $ \lambda $?
2A)
Can we then conclude that it is categorical in
other such $ \lambda $-s?
{\rm no}indent
3) Like parts 2), 2A0 for $ T \subseteq \mathbb{L} _{\theta, \theta }$?
\end{question}
\subsection {Preliminaries} \label{0B}
\begin{hypothesis}
\label{w0}
$\theta$ is a compact uncountable cardinal (of course, we use only
restricted versions of this).
\end{hypothesis}
\begin{notation}
\label{w2}
1) Let ${\varepsilon}hi(\bar x)$ mean: ${\varepsilon}hi$ is a formula of
${\mathbb L}_{\theta,\theta},\bar x$ is a sequence of
variables with no repetitions including the variables occurring
freely in ${\varepsilon}hi$ and $\ell g(\bar x) < \theta$ if not said
otherwise. We use ${\varepsilon}hi,\psi,\vartheta$ to denote formulas and
for a statement $ {\rm st} $ let
${\varepsilon}hi^{{\rm st}}$ or ${\varepsilon}hi^{[{\rm st}]}$ or ${\varepsilon}hi^{{\rm if}({\rm st})}$
mean
${\varepsilon}hi$ if ${\rm st}$ is true or
1 and $\neg{\varepsilon}hi$ if ${\rm st}$ is false or 0.
{\rm no}indent
2) For a set $u$, usually of ordinals, let $\bar x_{[u]}
= \langle x_\varepsilon:\varepsilon \in u {\rm rang}le$, now $u$ may be an
ordinal but, e.g. if $u = [\alpha,\beta)$ we may write $\bar
x_{[\alpha,\beta)}$; similarly for $\bar y_{[u]},\bar z_{[u]}$; let
$\ell g(\bar x_{[u]}) = u$.
{\rm no}indent
3) $\tau$ denotes a vocabulary, i.e. a set of predicates and function symbols
each with a finite number of places,
in other words the arity $ {\rm arity}(\tau ) = {\aleph_0} $,
see \ref{w3} on this.
{\rm no}indent
4) $T$ denotes a theory in ${\mathbb L}_{\theta,\theta}$ or ${\mathbb L}^1_\theta$
(see below); usually complete
in the vocabulary $\tau_T$ and with a model of cardinality
$\ge \theta$ if not said otherwise.
{\rm no}indent
5) Let ${\rm Mod}_T$ be the class of models of $T$.
{\rm no}indent
6) For a model $M$ let its vocabulary be $\tau_M$.
\end{notation}
\begin{remark} \label{w3}
1) What is the problem with predicates (and function symbols)
with infinite arity? If $ \langle M_\alpha :
\alpha \le \delta {\rm rang}le , \delta $ a limit ordinal
is increasing, even if the universe of $ M_ \delta $
is the union of the universes of $ M_ \alpha , \alpha < \delta $,
this does dot determine $ M_ \delta $.
{\rm no}indent
2) We can still define $ \cup \{M_ \alpha : \alpha < \delta \} $
by deciding $ P^{M_ \delta }= \cup \{M_ \alpha : \alpha < \delta \} $
for any predicate $ P $ and treating function similarly
(so the function symbol are interpreted as partial functions
or better deciding to use predicates only.
Now with care we can use
$ {\rm arity} (\tau ) \le \theta $ and we sometimes remark on this.
\end{remark}
\begin{notation}
\label{w4}
Let
$\varepsilon,\zeta,\xi$ denote
ordinals $< \theta$.
\end{notation}
\begin{definition}
\label{w8}
1) Let ${\rm uf}_\theta(I)$ be the set of $\theta$-complete ultrafilters on
$I$, non-principal if not said otherwise.
Let ${\rm fil}_\theta(I)$ be the set
of $\theta$-complete filters on $I$; mainly we use
$(\theta,\theta)$-regular ones (see below).
{\rm no}indent
2) $D \in {\rm fil}_\theta(I)$ is called
$(\lambda,\theta)$-regular {\underline{when}} \, there is a
witness $\bar w = \langle w_t:t \in I{\rm rang}le$ which means: $w_t \in
[\lambda]^{< \theta}$ for $t \in I$ and $\alpha < \lambda \Rightarrow
\{t:\alpha \in w_t\} \in D$.
{\rm no}indent
3) Let ${\rm ruf}_{\lambda,\theta}(I)$ be the set of $(\lambda,\theta)$-regular
$D \in {\rm uf}_\theta(I)$; let ${\rm rfil}_{\lambda,\theta}(I)$ be the
set of $(\lambda,\theta)$-regular $D \in {\rm fil}_\theta(I)$;
when $\lambda= |I|$ we may omit $\lambda$.
\end{definition}
\begin{definition}
\label{x1}
1) ${\mathbb L}_{\theta,\theta}(\tau)$ is the set of formulas of
${\mathbb L}_{\theta,\theta}$ in the vocabulary $\tau$.
{\rm no}indent
2) For $\tau$-models $M,N$ let $M {\rm pr}ec_{{\mathbb L}_{\theta,\theta}} N$
means: if ${\varepsilon}hi(\bar x) \in
{\mathbb L}_{\theta,\theta}(\tau_M)$ and $\bar a \in {}^{\ell g(\bar x)}M$
then $M \models {\varepsilon}hi[\bar a] \Leftrightarrow N \models {\varepsilon}hi[\bar a]$.
\end{definition}
{\rm no}indent
And, of course
\begin{fact}
\label{x6}
For a complete $T \subseteq {\mathbb L}_{\theta,\theta}(\tau)$.
{\rm no}indent
$({\rm Mod}_T,{\rm pr}ec_{{\mathbb L}_{\theta,\theta}})$ has amalgamation
and the joint embedding property (JEP), that is:
{
\noindent}
\begin{enumerate}
\item[$(a)$] \underline{amalgamation}: if $M_0
{\rm pr}ec_{{\mathbb L}_{\theta,\theta}} M_\ell$
for $\ell=1,2$ then there are $M_3,f_1,f_2,M'_1,M'_2$ such that
{
\noindent}
\begin{enumerate}
\item[$\bullet$] $M_0 {\rm pr}ec_{{\mathbb L}_{\theta,\theta}} M_3$
{
\noindent}
\item[$\bullet$] for $\ell=1,2,f_\ell$ is a
${\rm pr}ec_{{\mathbb L}_{\theta,\theta}} $-embedding of
$M_\ell$ into $M_3$ over $M_0$, that is, for some $\tau_T$-models
$M'_\ell$ for $\ell=1,2$ we have $M'_\ell {\rm pr}ec_{{\mathbb L}_{\theta,\theta}} M_3$
and $f_\ell$ is an
isomorphism from $M_\ell$ onto $M'_\ell$ over $M_0$;
\end{enumerate}
{
\noindent}
\item[$(b)$] \underline{JEP}: if $M_1,M_2$ are
${\mathbb L}_{\theta,\theta}$-equivalent $\tau$-models {\underline{then}} \, there is a
$\tau$-model $M_3$ and ${\rm pr}ec_{{\mathbb L}_{\theta,\theta}}$-embedding
$f_\ell$ of $M_\ell$ into $M_3$ for $\ell=1,2$.
\end{enumerate}
\end{fact}
{\rm no}indent
The well known generalization of \L os theorem is:
\begin{theorem}
\label{x7}
1) If ${\varepsilon}hi(\bar x_{[\zeta]}) \in {\mathbb L}_{\theta,\theta}(\tau),D \in
{\rm uf}_\theta(I)$ and $M_s$ is a $\tau$-model for $s \in I$ and
$f_\varepsilon \in {\rm pr}od\limits_{s \in I} M_s $
for $\varepsilon <
\zeta$ {\underline{then}} \, $M \models
{\varepsilon}hi[\ldots,f_\varepsilon/D,\ldots]_{\varepsilon < \zeta}$ {\underline{if}}f \,
the set $\{s \in I:M_s \models {\varepsilon}hi[\ldots,f_\varepsilon(s),
\ldots]_{\varepsilon < \zeta}\}$ belongs to $D$.
{\rm no}indent
2) Similarly $M {\rm pr}ec_{{\mathbb L}_{\theta,\theta}} M^I/D$.
\end{theorem}
\begin{definition}
\label{x17}
0) We say $X$ respects $E$ {\underline{when}} \, for some set $I,E$ is an equivalence relation\footnote{
Here in the interesting cases, the number of equivalent
classes of $ E $ in infinite, and even $ \le \theta $, pedantically not bounded by any $ \theta _* < \theta $.
}
on $I$ and $X \subseteq I$ and $s E t \Rightarrow (s \in
X \Leftrightarrow t \in X)$.
{\rm no}indent
1) We say $\mathbf x = (I,D,{\mathscr E})$ is a $(\kappa,\sigma)-{\rm l.u.f.t.}$
(limit-ultra-filter-iteration triple) {\underline{when}} \,:
{
\noindent}
\begin{enumerate}
\item[$(a)$] $D$ is a filter on the set $I$
{
\noindent}
\item[$(b)$] ${\mathscr E}$ is a family of equivalence relations on $I$
{
\noindent}
\item[$(c)$] $({\mathscr E},\supseteq)$ is $\sigma$-directed, i.e. if
$\alpha(*) < \sigma$ and $E_i \in {\mathscr E}$ for $i < \alpha(*)$
{\underline{then}} \, there is $E \in {\mathscr E}$
refining $E_i$ for every $i < \alpha(*)$
{
\noindent}
\item[$(d)$] if $E \in {\mathscr E}$ then $D/E$ is a $\kappa$-complete
ultrafilter on $I/E$ where
$D/E := \{X/E:X \in D$ and $X$ respects $E\}$.
\end{enumerate}
{
\noindent}
1A) Let $\mathbf x$ be a $(\kappa,\theta)-{\rm lf}t$ mean that above we
weaken (d) to
{
\noindent}
\begin{enumerate}
\item[$(d)'$] if $E \in {\mathscr E}$ then $D/E$ is a $\kappa$-complete filter.
\end{enumerate}
{
\noindent}
2) Omitting ``$(\kappa,\sigma)$" means $(\theta,\aleph_0)$, recalling
$\theta$ is our fixed compact cardinal.
{\rm no}indent
3) Let $(I_1,D_1,{\mathscr E}_1) \le^1_h (I_2,D_2,{\mathscr E}_2)$ mean that:
{
\noindent}
\begin{enumerate}
\item[$(a)$] $h$ is a function from $I_2$ onto $I_1$
{
\noindent}
\item[$(b)$] if $E \in {\mathscr E}_1$ {\underline{then}} \, $h^{-1} \circ E \in {\mathscr E}_2$ where
$h^{-1} \circ E = \{(s,t):s,t \in I_2$ and $h(s) Eh(t)\}$
{
\noindent}
\item[$(c)$] if $E_1 \in {\mathscr E}_1$ and $E_2 = h^{-1} \circ E_1$ {\underline{then}} \,
$D_1/E_1 = h''(D_2/E_2)$.
\end{enumerate}
\end{definition}
\begin{remark}
\label{x17f}
Note that in \ref{x17}(3), if $h = {\rm id}_{I_2}$ then $I_1 = I_2$.
\end{remark}
\begin{definition}
\label{x16}
Assume $\mathbf x = (I,D,{\mathscr E})$ is a $(\kappa,\sigma)$-l.u.f.t.
{\rm no}indent
1) For a function $f$ let ${\rm eq}(f) = \{(s_1,s_2):f(s_1) = f(s_2)\}$.
If $\bar f = \langle f_i:i < i_*{\rm rang}le$ and $i < i_* \Rightarrow
{\rm dom}(f_i) = I$ then ${\rm eq}(\bar f) = \cap\{{\rm eq}(f_i):i < i_*\}$.
{\rm no}indent
2) For a set $U$ let $U^I|{\mathscr E} = \{f \in {}^I U:{\rm eq}(f)$ is refined by some $E
\in {\mathscr E}\}$.
{\rm no}indent
3) For a model $M$ let ${\rm l.r.p.}
_{\mathbf x}(M) = M^I_D|{\mathscr E} = (M^I/D) {\restriction}
\{f/D:f \in {}^I M$ and ${\rm eq}(f)$ is refined by some $E \in {\mathscr E}\}$,
pedantically (as ${\rm arity}(\tau_M)$ may be $> \aleph_0$), $M^I_D|{\mathscr E} =
\cup\{M^I_D {\restriction} E:E \in {\mathscr E}\}$; ${\rm l.r.p.}$ stands for limit reduced power.
{\rm no}indent
4) If $\mathbf x$ is ${\rm l.u.f.t.}$ we may in part (3) write ${\rm l.u.p.}_{\mathbf x}(M)$.
\end{definition}
{\rm no}indent
We now give the generalization of Keisler \cite{Ke63}; Hodges-Shelah
\cite[Lemma 1,pg.80]{Sh:109} in the case $\kappa = \sigma$.
\begin{theorem}
\label{x18}
1) If $\sigma \le \kappa$ and $(I,D,{\mathscr E})$ is $(\kappa,\sigma)-{\rm l.u.f.t.},{\varepsilon}hi =
{\varepsilon}hi(\bar x_{[\zeta]}) \in {\mathbb L}_{\kappa,\sigma}(\tau)$ so $\zeta < \sigma,f_\varepsilon \in
M^I|{\mathscr E}$ for $\varepsilon < \zeta$ {\underline{then}} \, $M^I_D|{\mathscr E} \models
{\varepsilon}hi[\ldots,f_\varepsilon/D,\ldots]$ iff $\{s \in I:M \models
{\varepsilon}hi[\ldots,f_\varepsilon(s),\ldots]_{\varepsilon < \zeta}\} \in D$.
{\rm no}indent
2) Moreover $M {\rm pr}ec_{{\mathbb L}_{\kappa,\sigma}} M^I_D/{\mathscr E}$, pedantically
$\mathbf j = \mathbf j_{M,\mathbf x}$ is a
${\rm pr}ec_{{\mathbb L}_{\kappa,\sigma}}$-elementary embedding of $M$ into
$M^I_D/{\mathscr E}$ where $\mathbf j(a) = \langle a:s \in I{\rm rang}le/D$.
{\rm no}indent
3) We define $({\rm pr}od\limits_{s \in I} M_s)^I_D|{\mathscr E}$ similarly when
${\rm eq}(\langle M_s:s \in I{\rm rang}le)$ is refined by some $E \in {\mathscr E}$,
may use this more in end of the proof of \ref{d8}.
\end{theorem}
\begin{convention}
\label{x21}
Abusing a notation
{\rm no}indent
1) in ${\rm pr}od\limits_{s \in I} M_s/D$ we allow $f/D$
for $f \in {\rm pr}od\limits_{s \in S} M_s$ when $S \in D$.
{\rm no}indent
2) For $\bar c \in {}^{\mathfrak a}mma({\rm pr}od\limits_{s \in I} M_s/D)$ we can find
$\langle \bar c_s:s \in I{\rm rang}le$ such that $\bar c_s \in
{}^{\mathfrak a}mma(M_s)$ and $\bar c =\langle \bar c_s:s \in I
{\rm rang}le/D$ which means: if $i < \ell g(\bar c)$ {\underline{then}} \,
$c_{s,i} \in M_s$ and $c_i = \langle c_{s,i}:s \in I{\rm rang}le/D$.
\end{convention}
\begin{remark}
\label{x24}
1) Why the ``pedantically" in \ref{x16}(3)? Otherwise
if $\mathbf x$ is a $(\theta,\sigma)-{\rm l.u.f.t.},({\mathscr E}_{\mathbf x},
\supseteq)$ is not $\kappa^+$-directed, $\kappa < {\rm arity}(\tau)$
then defining ${\rm l.u.p.}_{\mathbf x}(M)$, we have freedom: if $R \in
\tau,{\rm arity}_\tau(R) \ge \kappa$, i.e. on $R^N {\restriction} \{\bar a:\bar a
\in {}^{{\rm arity}(P)}N$ and no $E \in {\mathscr E}$ refines ${\rm eq}(\bar a)\}$ so we
have no restrictions.
{\rm no}indent
2) So, e.g. for categoricity we better restrict ourselves to vocabularies
$\tau$ such that ${\rm arity}(\tau) = \aleph_0$.
\end{remark}
\begin{definition}
\label{x28}
We say $M$ is a $\theta$-complete model {\underline{when}} \, for every $\varepsilon
< \theta,R_* \subseteq {}^\varepsilon M$ and $F_*:{}^\varepsilon M
\rightarrow M$ there are $R,F \in \tau_M$ such that $R^M = R_* \wedge
F^M = F_*$.
\end{definition}
\begin{observation}
\label{x31}
1) If $M$ is a $\tau$-model of cardinality $\lambda$ {\underline{then}} \, there is
a $\theta$-complete expansion $M^+$ of $M$ so $\tau(M^+) \supseteq
\tau(M)$ and $\tau(M^+)$ has cardinality $|\tau_M| +
2^{(\|M\|^{< \theta})}$.
{\rm no}indent
2) For models $M {\rm pr}ec_{{\mathbb L}_{\theta,\theta}} N$ and $M^+$ as
above the following conditions are equivalent:
{
\noindent}
\begin{enumerate}
\item[$(a)$] $N = {\rm l.u.p.}_{\mathbf x}(M)$ identifying $a \in M$ with
$\mathbf j_{\mathbf x}(a) \in N$, for some $(\theta,\theta)-{\rm l.u.f.t.} \, \mathbf x$
{
\noindent}
\item[$(b)$] there is $N^+$ such that $M^+
{\rm pr}ec_{{\mathbb L}_{\theta,\theta}} N^+$ and $N^+ {\restriction} \tau_M$ is
isomorphic to $N$ over $M$, in fact we can add
$ N^+ {\restriction} \tau _ M = N$.
\end{enumerate}
{
\noindent}
3) For a model $M$, if $(P^M,<^M)$ is a $\theta$-directed
partial order and
$\chi = {\rm cf}(\chi) \ge \theta$ and $\lambda =
\lambda^{\|M\|} + \chi$ {\underline{then}} \, for some
$(\theta,\theta)-{\rm l.u.f.t.} \, \mathbf x$,
the model $N := {\rm l.u.p.}_{\mathbf x}(M)$:
satisfies $(P^N,<^N)$ has a cofinal increasing sequence of
length $\chi$ and $|P^N| = \lambda$.
\end{observation}
\begin{PROOF}{\ref{x31}}
Easy, e.g.
{\rm no}indent
3) Let $M^+$ be as in part (1).
Note that $M^+$ has Skolem functions and let $T'$ be the following
set of formulas: ${\rm Th}_{{\mathbb L}_{\theta,\theta}}(M^+) \cup
\{P(x_{\varepsilon}):{\varepsilon} < \lambda \cdot \chi\} \cup
\{P(\sigma(x_{\varepsilon_0},\ldots,x_{\varepsilon_i},\ldots)_{i<i(*)})
\rightarrow \sigma(x_{\varepsilon_0},\dotsc,x_{\varepsilon_i},
\ldots)_{i < i(*)} < x_\varepsilon: \sigma$ is a
$\tau(M^+)$-term so $i(*) < \theta$
and $i < i(*) \Rightarrow
\varepsilon_i < \varepsilon < \lambda \cdot \chi\}$.
Clearly
{
\noindent}
\begin{enumerate}
\item[$(*)$] $T'$ is $(< \theta)$-satisfiable in $M^+$.
\end{enumerate}
{
\noindent}
[Why? Because if $T'' \subseteq T'$ has cardinality $< \theta$ then
the set $u = \{\varepsilon < \lambda \cdot \chi:x_\varepsilon$ appears in
$T''\}$ has cardinality $< \theta$ and let $i(*) = {\rm otp}(u)$; clearly
for each $\varepsilon \in u$ the set $\Gamma_\varepsilon = T' \cap
\{P(\sigma(x_{\varepsilon_0},\ldots)) \rightarrow
\sigma(x_{\varepsilon_0},\dotsc,x_{\varepsilon_i},\ldots)_{i<i(*)} <
x_\varepsilon:i(*) < \theta$ and $\varepsilon_i < \varepsilon$ for
$i<i(*)\}$ has cardinality $< \theta$. Now we choose $c_\varepsilon
\in M$ by induction on $\varepsilon \in u$ such that the
assignment $x_\zeta \mapsto c_\zeta$ for $\zeta \in \varepsilon \cap
u$ in $M^+$ satisfies $\Gamma_\varepsilon$, possible because
$|\Gamma_\varepsilon| < \theta$ and $(P^M,<^M)$ is $\theta$-directed.
So the $M^+$ with the assignment $x_\varepsilon \mapsto c_\varepsilon$
for $\varepsilon \in u$ is a model of $T''$, so $T'$ is ($<
\theta)$-satisfiable indeed.]
Recalling that $|M| = \{c^{M^+}:c \in \tau(M^+)$ an individual
constant$\}$, $T'$ is realized in some
${\rm pr}ec_{{\mathbb L}_{\theta,\theta}}$-elementary extension $N^+$ of $M^+$ by
the assignment $x_\varepsilon \mapsto a_\varepsilon(\varepsilon
< \lambda \cdot \chi)$. {\rm Without loss of generality} \, $N^+$ is the Skolem hull of
$\{a_\varepsilon:\varepsilon < \lambda \cdot \chi\}$, so $N := N^+
{\restriction} \tau(M)$ is as required by the choice of $T'$. Now $\mathbf x$ is
as required exists by part (2) of the claim.
\end{PROOF}
\begin{observation}
\label{x32}
1) If $\mathbf x$ is a non-trivial $(\theta,\theta)-{\rm l.u.f.t.}$ and $\chi =
{\rm cf}({\rm l.u.p.}(\theta <))$ {\underline{then}} \, $\chi =\chi^{< \theta}$.
{\rm no}indent
2) Also $\mu = \mu^{< \theta}$ when $\mu$ is the cardinality of
${\rm l.u.p.}(\theta,<)$.
\end{observation}
\begin{PROOF}{\ref{x32}}
1) By the choice of $\mathbf x$ clearly $\chi \ge \theta$. As $\chi$ is
regular $\ge \theta$ by a theorem of Solovay \cite{So74} we have
$\chi^{< \theta} = \chi$.
{\rm no}indent
2) See the proof of \cite[2.20(3)=La27(3)]{Sh:1019}.
\end{PROOF}
We now quote \cite[Def.2.1+La8]{Sh:797}
\begin{definition}
\label{x35}
For a vocabulary $\tau,\tau$-models $M_1,M_2$, a set $\Gamma$ of
formulas in the vocabulary $\tau$ in any logic (each with
finitely many free variables if not said otherwise; see
\cite[2.9=La10(4)]{Sh:1019}),
cardinal $\theta$ and ordinal $\alpha$ we define a game $\Game =
\Game_{\Gamma,\theta,\alpha}[M_1,M_2]$
as follows, and using $(M_1,\bar b_1),(M_2,\bar b_2)$
with their natural meaning when ${\rm Dom}(\bar b_1) = {\rm Dom}(\bar b_2)$.
{
\noindent}
\begin{enumerate}
\item[$(A)$] The moves are indexed by $n < \omega$ (but every actual play is
finite), just before the $n$-th move we have a state $\mathbf
s_n = (A^1_n,A^2_n,h^1_n,h^2_n,g_n,\beta_n,n)$
{
\noindent}
\item[$(B)$] $\mathbf s = (A^1,A^2,h^1,h^2,g,\beta,n) = (A^1_{\mathbf
s},A^2_{\mathbf s},h^1_{\mathbf s},h^2_{\mathbf s},g_{\mathbf s},
\beta_{\mathbf s},n_{\mathbf s})$ is a state (or $n$-state or
$(\theta,n)$-state or $(\theta,< \omega)$-state) {\underline{when}} :
\begin{enumerate}
\item[$(a)$] $A^\ell \in [M_\ell]^{\le \theta}$ for $\ell=1,2$
{
\noindent}
\item[$(b)$] $\beta \le \alpha$ is an ordinal
{
\noindent}
\item[$(c)$] $h^\ell$ is a function from $A^\ell$ into $\omega$
{
\noindent}
\item[$(d)$] $g$ is a partial one-to-one function from $M_1$ to $M_2$ and
let $g^1_{\mathbf s} = g^1 = g_{\mathbf s} =g$ and let
$g^2_{\mathbf s} = g^2 = (g^1_{\mathbf s})^{-1}$,
{
\noindent}
\item[$(e)$] Dom$(g^\ell) \subseteq A^\ell$ for $\ell=1,2$
{
\noindent}
\item[$(f)$] $g$ preserves satisfaction of the formulas in $\Gamma$ and
their negations, i.e. for ${\varepsilon}hi(\bar x) \in \Gamma$ and $\bar a \in
{}^{\ell g(\bar x)}\text{Dom}(g)$ we have $M_1 \models {\varepsilon}hi[\bar
a] \Leftrightarrow M_2 \models {\varepsilon}hi[g(\bar a)]$
{
\noindent}
\item[$(g)$] if $a \in \text{ Dom}(g^\ell)$ then $h^\ell(a) < n$
\end{enumerate}
{
\noindent}
\item[$(C)$] we define the state $\mathbf s = \mathbf s_0 = \mathbf
s^0_\alpha$ by letting $n_{\mathbf s} = 0,A^1_{\mathbf s} = \emptyset =
A^2_{\mathbf s},\beta_{\mathbf s} = \alpha,h^1_{\mathbf s} = \emptyset
= h^2_{\mathbf s},g_{\mathbf s} = \emptyset$; so really $\mathbf s$ depends
only on $\alpha$
(but in general, this may not be a state for our game as possibly for
some sentence $\psi \in \Gamma$ we have
$M_1 \models \psi \Leftrightarrow M_2 \models \neg \psi$)
{
\noindent}
\item[$(D)$] we say that a state $\mathbf t$ extends a state $\mathbf s$
when $A^\ell_{\mathbf s} \subseteq A^\ell_{\mathbf t},h^\ell_{\mathbf s}
\subseteq h^\ell_{\mathbf t}$ for $\ell=1,2$ and $g_{\mathbf s} \subseteq
g_{\mathbf t},\beta_{\mathbf s} > \beta_{\mathbf t},n_{\mathbf s} < n_{\mathbf
t}$; we say $\mathbf t$ is a
successor of $\mathbf s$ if in addition $n_{\mathbf t} = n_{\mathbf s} +1$
{
\noindent}
\item[$(E)$] in the $n$-th move
\underline{the anti-isomorphism} player (AIS) chooses
$(\beta_{n+1},\iota_n,A'_n)$ such that:
$\iota_n \in \{1,2\},
\beta_{n+1} < \beta_n$ and $A^{\iota_n}_n \subseteq A'_n
\in [M_{\iota_n}]^{\le \theta}$,
\underline{the isomorphism} player (ISO) chooses a state $\mathbf s_{n+1}$
such that
\begin{enumerate}
\item[$\bullet$] $\mathbf s_{n+1}$ is a successor of $\mathbf s_n$
{
\noindent}
\item[$\bullet$] $A^{\iota_n}_{\mathbf s_{n+1}} = A'_n$
{
\noindent}
\item[$\bullet$] $A^{3-\iota_n}_{\mathbf s_{n+1}} =
A^{3-\iota_n}_{\mathbf s_n} \cup {\rm Dom}(g^{3-\iota_n}_{\mathbf s_{n+1}})$
{
\noindent}
\item[$\bullet$] if $a \in A'_n \backslash A^{\iota_n}_{\mathbf s_n}$ then
$h^{\iota_n}_{\mathbf s_{n+1}}(a) \ge n+1$
{
\noindent}
\item[$\bullet$] ${\rm Dom}(g^{\iota_n}_{\mathbf s_{n+1}}) = \{a \in
A^{\iota_n}_{\mathbf s_n}:h^{\iota_n}_{\mathbf s_n}(a) < n+1\}$ so it includes
${\rm Dom}(g^{\iota_n}_{\mathbf s_n})$
{
\noindent}
\item[$\bullet$] $\beta_{\mathbf s_{n+1}} = \beta_{n+1}$.
\end{enumerate}
{
\noindent}
\item[$(F)$]
\begin{itemize}
\item the play ends when one of the players has no
legal moves (always occur
as $\beta_n < \beta_{n-1}$) and then this player loses, this
may occur for $n=0$
\item for $\alpha = 0$ we stipulate that ISO
wins iff $\mathbf s^0_\alpha$ is a state.
\end{itemize}
\end{enumerate}
\end{definition}
\begin{definition}
\label{x38}
1) Let ${\mathscr E}^{0,\tau}_{\Gamma,\theta,\alpha}$ be the class
$\{(M_1,M_2):M_1,M_2$ are $\tau$-models and in the game
$\Game_{\Gamma,\theta,\alpha}[M_1,M_2]$ the ISO player has a winning
strategy$\}$ where $\Gamma$ is a set of formulas in the vocabulary
$\tau$, each with finitely many free variables.
{\rm no}indent
2) ${\mathscr E}^{1,\tau}_{\Gamma,\theta,\alpha}$ is the closure of
${\mathscr E}^{0,\tau}_{\Gamma,\theta,\alpha}$ to an equivalence relation (on the
class of $\tau$-models).
{\rm no}indent
3) Above we may replace $\Gamma$ by qf$(\tau)$ which means $\Gamma =$
the set at$(\tau)$
of atomic formulas
or bs$(\tau)$
of basic formulas in the vocabulary $\tau$.
{\rm no}indent
4) Above if we omit $\tau$ we mean $\tau = \tau_\Gamma$ and if we omit
$\Gamma$ we mean bs$(\tau)$. Abusing
notation we may say $M_1,M_2$ are
${\mathscr E}^{0,\tau}_{\Gamma,\theta,\alpha}$-equivalent.
\end{definition}
The following definition \ref{w41} is closely related to the beginning
of \S1, it quote \cite[Def.2.5=La13]{Sh:797} .
\begin{definition}
\label{w41}
1)
For a vocabulary $\tau$, the $\tau$-models $M_1,M_2$ are ${\mathbb L}^1_{<
\theta}$-equivalent {\underline{if}}f \, for every $\mu < \theta$ and $\alpha <
\mu^+$ and $\tau_1 \subseteq \tau$ of cardinality $\le \mu$, letting
$\Gamma =$ the quantifier
free
formulas in ${\mathbb L}(\tau)$, the models
$M_1,M_2$ are ${\mathscr E}^{1,\tau_1}_{\Gamma,\mu,\alpha}$.
{\rm no}indent
2) The logic $ \mathbb{L} _{\lambda, \kappa }$ is
defined like first order logic but we allow
conjunctions on sets of $ < \lambda $ formulas
and we allow quantification of the form
$ \forall \bar{ x } $ for sequences $ \bar{ x } $
of length $ < \kappa $; \underline{however}
each formulas has to have $ < \kappa $ free variables;
and disjunctions and existential quantifications
are defined naturally.
{\rm no}indent
2A) We define $ \mathbb{L} _{< \lambda , < \kappa }$
as $ \cup \{\mathbb{L} _{\lambda _1, \kappa _1}:
\lambda _1 < \lambda , \kappa _1 < \kappa \} $ ;
we may replace $ < \lambda ^+$ by $ \lambda $
and $ < \kappa ^+ $ by $ \kappa $.
{\rm no}indent
3) The logic $ \mathbb{L} ^1_{\le \theta }$ is defined as follow:
a sentence $\psi \in {\mathbb L}_{\le \theta}(\tau)$ \underline{iff} the
sentence is defined using (or by) a triple
$(\text{qf}(\tau_1),\theta,\alpha)$ which means:
$\tau_1$ a sub-vocabulary of $\tau$ of
cardinality $\le \theta$ and $\alpha < \theta^+$ and for
some sequence $\langle M_\beta:\beta < \beta(*){\rm rang}le$
of $\tau_1$-models of length $\beta(*) \le \beth_{\alpha +1}(\theta)$
we have: $M \models \psi$ \underline{iff} $M$ is
${\mathscr E}^1_{\text{qf}(\tau_1),\theta,\alpha}$-equivalent to $M_\alpha$ for
some $\beta < \beta(*)$.
{\rm no}indent
4) Let ${\mathbb L}^1_\kappa = \cup\{{\mathbb L}^1_{\le \theta}:\theta < \kappa\}$
so ${\mathbb L}^1_{\theta^+} = {\mathbb L}^1_{\le \theta}$.
\end{definition}
\section {Characterizing equivalence by $\omega$-limit ultra-powers} \label{1}
In \cite{Sh:797}, a logic ${\mathbb L}^1_{< \kappa} = \bigcup\limits_{\mu <
\kappa} {\mathbb L}^1_{\le \mu}$ is introduced
(here we consider $\kappa$ is strongly inaccessible for transparency), and is
proved to be stronger than ${\mathbb L}_{\kappa,\aleph_0}$ but weaker than
${\mathbb L}_{\kappa,\kappa}$, has interpolation and a characterization, well
ordering not definable in it and has an addition theorem. Also
it is the maximal logic with some such properties.
For $\kappa = \theta$, we give a characterization of when two models
are ${\mathbb L}^1_{< \theta}$-equivalent giving an additional evidence for
the logic's naturality.
\begin{convention}
\label{d2}
In this section every vocabulary $\tau$ has ${\rm arity}(\tau) =
\aleph_0$.
\end{convention}
{\rm no}indent
Recall \cite[2.11=La18]{Sh:797} which says (we expand it)
\begin{claim}
\label{d8}
1)
We have $M_n {\rm eq}uiv_{{\mathbb L}^1_{\le \theta}} M_\omega$ for $n < \omega$
\underline{when} clauses (b),(c) below holds and
moreover
$M_n \models \psi[\bar a] \Leftrightarrow M_\omega \models \psi[\bar a]$
{\underline{when}} clauses (a)-(e) below holds, where:
{
\noindent}
\begin{enumerate}
\item[$(a)$] $\psi(\bar z) \in {\mathbb L}^1_{\le \theta}(\tau)$ a formula
{
\noindent}
\item[$(b)$] $M_n {\rm pr}ec_{{\mathbb L}_{< \partial, \theta ^+}} M_{n+1}$ where
$\partial = \beth_{\theta ^+}$, recalling \ref{w41}(2A))
{
\noindent}
\item[$(c)$] $M_\omega := \bigcup\limits_{n < \omega} M_n$
{
\noindent}
\item[$(d)$] $\bar a \in {}^{\ell g(\bar z)}(M_0)$
{
\noindent}
\item[$(e)$] $\tau = \tau(M_n)$ for $n < \omega$.
\end{enumerate}
{\rm no}indent
2)
Assume $|\tau| \le \mu,M_n$ is a $\tau$-model and
$M_n {\rm pr}ec_{{\mathbb L}_{\mu^+,\mu^+}} M_{n+1}$ for $n < \omega$ and
$M_\omega = \cup\{M_n:n < \omega\}$. {\rm Th}en \, $M_0,M_\omega$ are
${\mathbb L}^1_{\le \mu}$-equivalent.
\end{claim}
{\rm no}indent
We need two definitions before stating and proving the theorem below.
The first definition generalizes common concepts.
\begin{definition}
\label{d9}
We say that a pair of models $(M_1,M_2)$ has isomorphic
$\theta$-complete $\omega$-iterated ultrapowers {\underline{if}}f \, one can find
$D_n \in {\rm uf}_\theta(I_n)$ for every $n \in \omega$ such that
$M^1_\omega \cong M^2_\omega$, when $M^\ell_\omega = \cup\{M^\ell_k:k
\in \omega\},M^\ell_0 = M_\ell$ and $M^\ell_n
{\rm pr}ec_{{\mathbb L}_{\theta,\theta}} (M^\ell_n)^{I_n}/D_n = M^\ell_{n+1}$ for
$\ell=1,2$ and $n < \omega$.
\end{definition}
{\rm no}indent
For the second definition, let $\mathbf x$ be a l.u.f.t. and
in \ref{d10} below we define ``niceness witness".
How do we arrive to this definition?
If we try to analyze how to prove that two
$ \mathbb{L} ^1_ \theta $-equivalent models have
isomorphic $ \theta $-complete $ \omega $-iterated ultrapowers
by a sequence of length $ \omega d $ of approximations;
this is a natural was to carry the induction step.
The reader may return to this after reading the proof of
$ (a) \rightarrow (e) $ of \ref{d11}.
To understand this (and the proof of \ref{d11} the reader may
consider the case $ \theta = {\aleph_0} $, which naturally is simpler
and tell us that for each coordinate
$ s \in I $ we play a game of
an Ehrenfuecht-Fraisse game.
Note also that Claim \ref{d8} clarify why having
$ {\rm arity}(\tau) = {\aleph_0} $ help.
\begin{definition}
\label{d10}
If $\mathbf x = (I,D,\bar E)$ is an ${\rm l.u.f.t.}$ and $\bar E = \langle E_n:n \in
\omega{\rm rang}le$ {\underline{then}} \, $\bar w$ is a niceness witness for $(I,D,\bar
E)$ {\underline{when}} \,:
{
\noindent}
\begin{enumerate}
\item[(a)] $\bar w = \langle w_{s,n},{\mathfrak a}mma_{s,n}:s \in I,n <
\omega{\rm rang}le$
{
\noindent}
\item[(b)] $w_{s,n} \subseteq \lambda_n$ and $|w_{s,n}| < \theta$ and
$|w_{s,n}| \ge |w_{s,n+1}|$
{
\noindent}
\item[(c)] ${\mathfrak a}mma_{s,n} < \theta$ and $({\mathfrak a}mma_{s,n} >
{\mathfrak a}mma_{s,n+1}) \vee ({\mathfrak a}mma_{s,n+1} = 0)$
{
\noindent}
\item[(d)] ${\mathfrak a}mma_{s,n} = 0 \Rightarrow w_{s,n} = \emptyset$ but
$w_{s,0} \ne \emptyset$ and for simplicity $w_{s,0}$ is infinite for
every $s \in I$
{
\noindent}
\item[(e)] if $n < \omega,u \in [\lambda_n]^{< \theta}$ then $\{s
\in I:u \subseteq w_{s,n}\} \in D$
{
\noindent}
\item[(f)] $w_{s,n} = w_{t,n}$ and ${\mathfrak a}mma_{s,n} = {\mathfrak a}mma_{t,n}$ when
$s E_n t$.
\end{enumerate}
\end{definition}
\begin{theorem}
\label{d11}
Let $\theta$ be a compact cardinal and $M_1,M_2$ be
two $\tau$-models (and
${\rm arity}(\tau) = \aleph_0$).
The following conditions are equivalent:
{
\noindent}
\begin{enumerate}
\item[$(a)$] $M_1,M_2$ are ${\mathbb L}^1_\theta$-equivalent
{
\noindent}
\item[$(b)$] there are $(\theta,\theta)-{\rm l.u.f.t.} \, \mathbf x_n = (I,D,{\mathscr E}_n)$
and
$ {\mathscr E}_n \subseteq {\mathscr E}_{n+1}$ for $n < \omega$ and we let ${\mathscr E} =
\cup\{{\mathscr E}_n:n < \omega\}$ such that
$(M_1)^I_D|{\mathscr E}$ is isomorphic to $(M_2)^I_D|{\mathscr E}$
{
\noindent}
\item[$(c)$] $(M_1,M_2)$ have isomorphic $\theta$-complete
$\omega$-iterated ultrapowers, see DEf \ref{d9}
{
\noindent}
\item[$(d)$] if $D_n \in {\rm ruf}_{\lambda_n,\theta}(I_n)$ so $|I_n| \ge
\lambda_n$ and $\lambda_{n+1} \ge 2^{|I_n|} ,
\lambda_n >
\|M_1\| + \|M_2\| + |\tau|$ for every $n$ {\underline{then}} \, the sequence
$\langle (I_n,D_n):n < \omega{\rm rang}le$ is as required in clause (c)
{
\noindent}
\item[$(e)$] if $\mathbf x = (I,D,{\mathscr E})$ is a ${\rm l.u.f.t.}$ (see Def \ref{x17}(1)),
$ {\mathscr E} = \{E_n:n <
\omega\}$, for $n < \omega$ we have
$E_{n+1}$ refines $E_n,2^{|I/E_n|} \le \lambda_{n+1},
D/E_n$ is a $(\lambda_n,\theta)$-regular
$\theta$-complete ultrafilter, $\lambda_0 \ge \|M_1\| + \|M_2\| +
|\tau|,\bar w$ is a niceness witness (see Def \ref{d10}),
{\underline{then}} \, ${\rm l.u.p.}_{\mathbf x}(M_1) \cong {\rm l.u.p.}_{\mathbf x}(M_2)$,
(see Def. \ref{x16}(3)).
\end{enumerate}
\end{theorem}
\begin{PROOF}{\ref{d11}}
\underline{Clause $(b) \Rightarrow$ Clause $(a)$}:
So let $I,D,{\mathscr E}_n(n < \omega)$ be as in clause (b) and ${\mathscr E} =
\cup\{{\mathscr E}_n:n < \omega\}$. By the transitivity of being ${\mathbb L}^1_{<
\theta}$-equivalent, clearly clause (a) follows from:
{
\noindent}
\begin{enumerate}
\item[$\boxplus_1$] for every model $N$ the models $N,N^I_D|{\mathscr E}$ are
${\mathbb L}^1_\theta$-equivalent.
\end{enumerate}
{
\noindent}
[Why $\boxplus_1$ holds? Let $N_n = N^I_D|{\mathscr E}_n$
for $n < \omega$ and $N_\omega = \cup\{N_n:n <
\omega\}$. So by \ref{x18} we have
$N {\rm eq}uiv_{{\mathbb L}_{\theta,\theta}} N_0$
and moreover $N_n {\rm pr}ec_{{\mathbb L}_{\theta,\theta}} N_{n+1}$. Hence by
\ref{d8}, that is the ``Crucial Claim" \ref{d8} quoting
\cite[2.11=a18]{Sh:797} we have $N_n
{\rm eq}uiv_{{\mathbb L}^1_{< \theta}} N_\omega$ hence
$N {\rm eq}uiv_{{\mathbb L}^1_{< \theta}} N_\omega$.]
{\rm no}indent
\underline{Clause $(c) \Rightarrow$ Clause $(b)$}:
Let $I = {\rm pr}od\limits_{n < \omega} I_n,E_n = \{(\eta,\nu):\eta,\nu \in
I$ and $\eta {\restriction} n = \nu {\restriction} n\}$ and $D = \{X \subseteq I$: for some
$n,(\forall^{D_n} i_n \in I_n)(\forall^{D_{n-1}} i_{n-1} \in I_{n-1})
\ldots (\forall^{D_0} i_0 \in I_0)(\forall\eta)[\eta \in I \wedge
\bigwedge\limits_{\ell \le n} \eta(\ell) = i_\ell \rightarrow \eta \in
X\}$. Now let $M^\ell_\omega {\rm eq}uiv (M_\ell)^I_D|\{E_n:n < \omega\}$.
Now
it should be clear that
$(M_\ell)^I_D|\{E_n:n < \omega\}$ is isomorphic to $M^\ell_\omega$
for $\ell=1,2$, so recalling $M^1_\omega \cong M^2_\omega$ by the
present assumption, the models $(M_\ell)^I_D|\{E_n:n < \omega\}$
for $\ell=1,2$ are isomorphic, so letting ${\mathscr E}_n =
\{E_0,\dotsc,E_n\}$ easily $(I,D,{\mathscr E}_n)_{n < \omega}$ are as required
in clause (b).
{\rm no}indent
\underline{Clause $(d) \Rightarrow$ Clause $(c)$}:
Clause (d) is obviously stronger,
but we have to point out that there there are such
$ I_n, D_n, $; anyhow we shall elaborate.
We can choose $\lambda_0 = (\|M_1\| +
\|M_2\| + |\tau| + \theta)^{< \theta},\lambda_{n+1} = 2^{\lambda_n}$ for
$n < \omega$ then letting $I_n = \lambda_n$
there is $D_n \in {\rm ruf}_{\lambda_n,\theta}(I_n)$ recalling $\theta$ is
a compact cardinal, noting $\lambda_n = \lambda_n^{< \theta}$. Now
$\langle I_n,D_n:n < \omega{\rm rang}le$ is as required in
the assumption of clause (d), so as we are now assuming
clause (d), also its conclusion holds.
Now $ \langle (I_n, D_n): n < \omega {\rm rang}le $ are as required
in
clause (c), in
particular the isomorphism holds by
the conclusion of
clause (d) which as said in the previous sentence holds. .
{\rm no}indent
\underline{Clause $(e) \Rightarrow$ Clause $(d)$}:
Let $\langle (I_n,D_n,\lambda_n):n < \omega{\rm rang}le$ be as in the
assumption of clause (d).
We define $I = {\rm pr}od\limits_{n} I_n,E_n = \{(\eta,\nu):\eta,\nu \in
I,\eta {\restriction} (n+1) = \nu {\restriction} (n+1)\}$ and define $D$ as in the proof
of $(c) \Rightarrow (b)$ above and we choose
$\bar w = \langle w_{\eta,n}:\eta \in I,n < \omega{\rm rang}le$ as
follows.
First, choose $\bar u_n = \langle u^n_s:s \in I_n{\rm rang}le$
which witness $D_n$ is $(\lambda_n,\theta)$-regular,
i.e. $u^n_s \in [\lambda_n]^{< \theta}$
and $(\forall \alpha < \lambda_n)[\{s \in I_n:\alpha \in u^n_s\} \in D_n]$.
For $\eta \in I$ and $n < \omega$ let
$w_{\eta,n}$ be $u^n_{\eta(n)}$ if $\langle
{\rm otp}(u_{\eta(\ell)}):\ell \le n{\rm rang}le$ is decreasing and $\emptyset$
otherwise. Let ${\mathfrak a}mma_{\eta,n}$ be ${\rm otp}(w_{\eta,n})$.
Now we can check that the assumptions of clause (e) hold (because
of the choice of $D$), we shall elaborate two points.
First the ultrafilter $ D/ E_n $
is $ (\lambda, \theta ) $-regular
because
$ \langle \{\eta \in I: \eta ( n ) \in u^n_{\eta (n ) } \} {\rm rang}le $
witness it.
Second
The main point is
to prove that $ \bar{ w }
= \langle (w_{\eta ,n},
{\mathfrak a}mma _{\eta, n }: \eta \in I, n < \omega {\rm rang}le $
is indeed a niceness witness for $ (I, D, \bar{E } )$.
For this most clauses of \ref{d10} are easy, but we better
elaborate on clause (e) there.
that for every $n$
{
\noindent}
\begin{enumerate}
\item[$(*)_n$] for some $X_n \in D_n$, for every $s_n \in X_n$, for
some $X_{n-1} \in D_{n-1}$ ... for some $X_0 \in D_0$ for every $s_0
\in X_0$, if $\langle s_0,\dotsc,s_n{\rm rang}le \trianglelefteq \eta \in
I$ then
{
\noindent}
\begin{enumerate}
\item[(a)] $|w_{\eta,0}| > |w_{\eta,1}| > \ldots > |w_{\eta,0}|$
{
\noindent}
\item[(b)] $|u^\ell_{s_\ell}| > |u^{\ell +1}_{s_{\ell+1}}|$ for $\ell
< n$.
\end{enumerate}
\end{enumerate}
{
\noindent}
Why $(*)_n$ holds? Clause (a) holds by clause (b) and the choice of
$w_{\eta,n}$ as $u^n_{\eta(n)}$. Clause (b) holds because
$u^{\ell+1}_{s_{\ell +1}}$ is of cardinality $< \theta$ and $\{s \in
I_\ell:|u^{\ell +1}_{s_{\ell+1}}|^+ \subseteq u^\ell_s\} \in D_\ell$.
Hence the conclusion of clause (e) holds and we are done as
in the proof of $(c) \Rightarrow (b)$.
{\rm no}indent
\underline{Clause $(a) \Rightarrow$ Clause $(e)$}:
So assume that clause (a) holds, that is $M_1,M_2$ are
${\mathbb L}^1_\theta$-equivalent and assume $I,D,{\mathscr E},\langle E_n:n < \omega{\rm rang}le$
and $\bar w$ are as in the assumption of clause (e), and we should
prove that its conclusion holds, that is, ${\rm l.u.p.}_{\mathbf x}(M_1)
\cong {\rm l.u.p.}_{\mathbf x}(M_2)$.
For every $\tau_* \subseteq \tau$ of cardinality $< \theta$ and $\mu <
\theta$, by \ref{w41} we know that $M_1 {\restriction} \tau_*,M_2 {\restriction} \tau_*$ are
${\mathbb L}^1_{\le \mu}$-equivalent, hence for every $\alpha < \mu^+$ there is a
finite sequence $\langle N_{\tau_*,\mu,\alpha,k}:k \le \mathbf
k(\tau_*,\mu,\alpha){\rm rang}le$ such that:
{
\noindent}
\begin{enumerate}
\item[$(*)_1$] $(a) \quad N_{\tau_*,\mu,\alpha,0} = M_1 {\restriction} \tau_*$
{
\noindent}
\item[${{}}$] $(b) \quad N_{\tau_*,\mu,\alpha,\mathbf
k(\tau_*,\mu,\alpha)} = M_2 {\restriction} \tau_*$
{
\noindent}
\item[${{}}$] $(c) \quad$ in the game
$\Game_{\tau_*,\mu,\alpha}[N_{\tau_*,\mu,\alpha,k},
N_{\tau_*,\mu,\alpha,k+1}]$ the ISO player has a
\hskip25pt winning strategy for each $k < \mathbf k(\tau_*,\mu,\alpha)$,
\underline{but} we stipulate
\hskip25pt a play to have $\omega$ moves, by deciding they continue to
choose the
\hskip25pt moves even when one side already wins
using the same state
\hskip25pt except changing $n_ \mathbf{s} $
\end{enumerate}
{
\noindent}
[Why? By Def. \ref{x35} which quote \cite[2.1=La8]{Sh:797}
\begin{enumerate}
\item[$(*)_2$] {\rm without loss of generality} \, $\|N_{\tau_*,\mu,\alpha,k}\| \le \lambda_0$
for $k \in \{1,\dotsc,\mathbf k(\tau_*,\mu,\alpha)-1\}$ (even $< \theta$).
\end{enumerate}
{
\noindent}
[Why? By (a degenerated case of ) \ref{d8}
We can ({\rm without loss of generality} \,) assume:
{
\noindent}
\begin{enumerate}
\item[$(*)_3$] $(a) \quad$ above
$\mathbf k(\tau_*,\mu,\alpha) =\mathbf k$
{
\noindent}
\item[${{}}$] $(b) \quad \tau$ has only predicates
\end{enumerate}
{\rm no}indent
[Why? Clause (a)
by monotonicity in $\tau^*,\mu$ and in $\alpha$ of $M_1
{\mathscr E}^{1,\tau^*}_{{\rm qf}(\tau_*),\mu,\alpha} M_2$.
Clause (b) is easy too.]
We denote:
\begin{enumerate}
\item[$(*)_4$] $(a) \quad \langle P_\alpha:\alpha < |\tau|{\rm rang}le$
list the predicates of $\tau$, recall that $|\tau| \le \mu < \lambda_0$
{
\noindent}
\item[${{}}$] $(b) \quad$ for $t \in I$ let $\tau_t = \{P_\alpha:\alpha \in
w_{t,0} \cap |\tau|\}$
{
\noindent}
\item[$(*)_5$] let $N_{s,k} := N_{\tau_s,|w_{s,0}|,
{\mathfrak a}mma_{s,0}+1,k}$ for $s \in I$ and $k \le \mathbf k$.
\end{enumerate}
{
\noindent}
For $k \le \mathbf k$, let $\bar f_{k,n} = \langle f_{k,n,\alpha}:
\alpha < 2^{\lambda_n}{\rm rang}le$ list the members $f$ of
${\rm pr}od\limits_{s \in I} N_{s,k}$ such that $E_n$ refines
${\rm eq}(f)$, so $f_{k,n,\alpha} = \langle f_{k,n,\alpha}(\eta):
\eta \in I{\rm rang}le$ but $\eta \in I \wedge \nu \in
I \wedge \eta E_n \nu \Rightarrow f_{k,n,\alpha}(\eta) =
f_{k,n,\alpha}(\nu)$.
Now
{
\noindent}
\begin{enumerate}
\item[$(*)_6$]
\begin{enumerate}
\item[(a)] for $t \in I$ and $k < \mathbf k$ let
$\Game_{t,k}$ be the game $\Game_{\tau_t,|w_{t,0}|,{\mathfrak a}mma_{t,0}+1}
[N_{t,k},N_{t,k+1}]$
{
\noindent}
\item[(b)] let {\bf st}$_{t,k}$ be a winning
strategy for the ISO player in $\Game_{t,k}$
{
\noindent}
\item[(c)] if $t_1 E_0 t_2$ then $\langle
N_{t_\iota,k}:k \le \mathbf k{\rm rang}le$ are the
same for $\iota=1,2$, moreover ($\Game_{t_1,k} = \Game_{t_2,k}$ and)
{\bf st}$_{t_1,k} =$ {\bf st}$_{t_2,k}$ for $k < \mathbf k$.
\end{enumerate}
\end{enumerate}
{
\noindent}
[Why clause (c)? Because by $(*)_5$,
$N_{s,k},N_{\tau_s,|w_{s,0}|,{\mathfrak a}mma_{s,0} +1,k}$ and $\tau_s$ depend
on $w_{s,0}$ only, so $N_{s,k}$ is determined by $(w_{s,0},k)$ hence
(by clause (e) of Th. \ref{d11} and clause (f) from Definition \ref{d10}), $ N_{\mathbf{s}, k }$ depend just on
$(s/E_0,k)$.]
Now for each $k$ by induction on $n$ we choose
$\langle \mathbf s_{t,k,n}:t \in I{\rm rang}le$ such that:
{
\noindent}
\begin{enumerate}
\item[$(*)_7$]
\begin{enumerate}
\item[(a)] $\mathbf s_{t,k,n}$ is a state of the game $\Game_{t,k}$
{
\noindent}
\item[(b)] $\langle \mathbf s_{t,k,m}:m \le
n{\rm rang}le$ is an initial segment of a play of $\Game_{t,k}$ in which
the ISO player uses the strategy {\bf st}$_{t,k}$
{
\noindent}
\item[(c)] if $t_1 E_n t_2$ then $\mathbf s_{t_1,k,n} =
\mathbf s_{t_2,k,n}$
{
\noindent}
\item[(d)] $\beta_{\mathbf s_{t,k,n}} = {\mathfrak a}mma_{t,n}$, see
Definition \ref{x35}
{
\noindent}
\item[(e)] if $t \in I,n = \iota \mod 2$ and $\iota \in
\{0,1\}$ {\underline{then}} \, $A^\iota_{\mathbf s_{t,k,n}} \supseteq
\{f_{k + \iota,m,\alpha}(t):m < n$ and $\alpha \in w_{t,m}\}$, see
Definition \ref{x35}(E)
\end{enumerate}
{
\noindent}
\item[$(*)_8$] we can carry the induction on $n$.
\end{enumerate}
{
\noindent}
[Why? Straightforward.]
{
\noindent}
\begin{enumerate}
\item[$(*)_9$] for each $k < \mathbf k,n < \omega,t \in I$ we define
$h_{s,k,n}$, a partial function from $N_{s,k}$ to
$N_{s,k+1}$ by $h_{s,k,n}(a_1) = a_2$ {\underline{if}}f \, for some $m \le n,w_{s,m} \ne
\emptyset$ and $g_{\mathbf s_{t,k,m}}(a_1) = a_2$, see \ref{x35}(E).
\end{enumerate}
{
\noindent}
Now clearly:
{
\noindent}
\begin{enumerate}
\item[$\boxplus_1$] for each $t \in I,k < \mathbf k$ and $n < \omega,
h_{s,k,n}$ is a partial one-to-one function and even a partial isomorphism
from $N_{s,k}$ to $N_{s,k+1}$, non-empty when $n > 0$ and
increasing with $n$.
\end{enumerate}
{
\noindent}
[Why? By the choice of {\bf st}$_{t,k}$ and $(*)_7(a)$.]
{
\noindent}
\begin{enumerate}
\item[$\boxplus_2$] let $Y_{k,n} = \{(f_1,f_2):f_\ell \in {\rm pr}od\limits_{s
\in I} {\rm Dom}(h_{s,k,n})$ for $\ell=1,2$ and $s \in I \Rightarrow
f_2(s) = h_{s,k,n}(f_1(s))\}$
{
\noindent}
\item[$\boxplus_3$] $\mathbf f_{k,n} = \{(f_1/D,f_2/D):(f_1,f_2) \in
Y_{k,n}\}$ is a partial isomorphism from $M^I_1 {\restriction} \{f/D:f \in
{\rm pr}od\limits_{s} N_{s,k}$ and $f$ respects $E_n\}$ to $M^I_2 {\restriction}
\{f/D:f \in {\rm pr}od\limits_{s} N_{s,k+1}$ and $f$ respects $E_n\}$
{
\noindent}
\item[$\boxplus_4$] $\mathbf f_{k,n} \subseteq \mathbf
f_{k,n+1}$
{
\noindent}
\item[$\boxplus_5$]
\begin{enumerate}
\item[(a)] if $f_1 \in {\rm pr}od\limits_{s} N_{s,k}$ and ${\rm eq}(f_1)$
is refined by $E_n$ {\underline{then}} \, for some $n_1 > n$ and $f_2 \in
{\rm pr}od\limits_{s} N_{s,k+1}$ the pair $(f_1/D,f_2/D)$
belongs to $\mathbf f_{k,n_1}$
{
\noindent}
\item[(b)] if $f_2 \in {\rm pr}od\limits_{s} N_{s,k+1}$ and
${\rm eq}(f_2)$ is refined by $E_n$ then for some $n_1 > n$
and $f_1 \in {\rm pr}od\limits_{s} N_{s,k}$ the pair $(f_1/D,f_2/D)$ belongs to
$\mathbf f_{k,n_1}$.
\end{enumerate}
\end{enumerate}
{
\noindent}
[Why? By symmetry it suffices to deal with clause (a).
For some $\alpha,f_1 = f_{k,n,\alpha}$, hence for every
$t \in {\rm Dom}(f_1),f_1(t) \in A^1_{\mathbf s_{t,k,n}}$. We use the ``delaying
function", $h_{\mathbf s_{t,k,n}}(f_1(t)) < \omega$ so for some $m$
the set $\{t \in I:h_{\mathbf s_{t,k,n}}(f_1(t)) \le m\}$ which
respects $E_n$ belongs to $D$. In particular $\{s:{\mathfrak a}mma_{s,k,n} >
m\} \in D$, the rest should be clear recalling the regularity of each $D/E_m$.]
Letting ${\mathscr E} = \{E_n:n < \omega\}$, putting together
{
\noindent}
\begin{enumerate}
\item[$(*)_{10}$] $\mathbf f_k = \bigcup\limits_{n} \mathbf f_{k,n}$ is
an isomorphism from $({\rm pr}od\limits_{s} N_{k,s})_D|{\mathscr E}$ onto
$({\rm pr}od\limits_{s} N_{k+1,s})_D|{\mathscr E}$.
\end{enumerate}
{
\noindent}
Hence
{
\noindent}
\begin{enumerate}
\item[$(*)_{11}$] $\mathbf f_{\mathbf k -1} \circ \ldots \circ
\mathbf f_0$ is an isomorphism from $(M_1)^I_D|{\mathscr E}$ onto $(M_2)^I_D|{\mathscr E}$.
\end{enumerate}
{
\noindent}
So we are done.
\end{PROOF}
\begin{discussion}
\label{d21}
1) So for our $\theta$, we get another characterization of ${\mathbb L}^1_\theta$.
{\rm no}indent
2) We may deal with universal homogeneous $(\theta,\sigma)-{\rm l.u.p.} \,\mathbf x$,
at least for $\sigma = \aleph_0$, using Definition \ref{x17}.
\end{discussion}
\begin{claim}
\label{d22}
In Theorem \ref{d11} if $\kappa = \kappa^{< \theta}
\ge \|M_1\| + \|M_2\|$ we can add:
{
\noindent}
\begin{enumerate}
\item[$(b)^+$] like clause (b) of \ref{d11} but $|I| \le 2^\kappa$.
\end{enumerate}
\end{claim}
\begin{remark}
\label{d23}
Note we do not restrict $\tau = \tau(M_\ell)$. See proof of $(*)_9$ below.
\end{remark}
\begin{PROOF}{\ref{d22}}
Clearly $(b)^+ \Rightarrow (b)$, so it is enough to prove $(b)
\Rightarrow (b)^+$; we shall assume $M_1,M_2,\kappa,\mathbf
x_n,D,{\mathscr E}_n,{\mathscr E}$ are as in (b) and let $g$ be an isomorphism from
$(M_1)^I_D/{\mathscr E}$ onto $(M_2)^I_D/{\mathscr E}$.
Let
{
\noindent}
\begin{enumerate}
\item[$(*)_1$] $(a) \quad {\mathscr E}'_n = \{E:E$ is an equivalence relation
on $I$ with $\le \kappa$ equivalence
\hskip25pt classes such that some $E' \in {\mathscr E}_n$ refines $E\}$
{
\noindent}
\item[${{}}$] $(b) \quad$ let ${\mathscr E}' = \cup\{{\mathscr E}'_n:n \in {\mathbb N}\}$.
\end{enumerate}
{
\noindent}
Clearly
{
\noindent}
\begin{enumerate}
\item[$(*)_2$] $(M_\ell)^I_D|{\mathscr E} = (M_\ell)^I_D|{\mathscr E}'$ for $\ell =
1,2$.
\end{enumerate}
{
\noindent}
Let $\chi$ be large enough such that
$M_1,M_2,\kappa,D,I,{\mathscr E},\bar{{\mathscr E}}' = \langle {\mathscr E}'_n:n \in {\mathbb N}{\rm rang}le,g$ and
$(M_\ell)^I_D|{\mathscr E}$ for $\ell=1,2$ belong to ${\mathscr H}(\chi)$. We can
choose ${\mathfrak B} {\rm pr}ec_{{\mathbb L}_{\kappa^+,\kappa^+}}({\mathscr H}(\chi),\in)$ of
cardinality $2^\kappa$ to which all the members of ${\mathscr H}(\chi)$
mentioned above belong and such that $2^\kappa +1 \subseteq {\mathfrak B}$. So
as $\tau = \tau(M_1) \in {\mathfrak B}$ and {\rm without loss of generality} \, $|\tau|
\le 2^{\| M_1 \|+\| M_2 \|}
\le 2^\kappa$; necessarily
$\tau \subseteq {\mathfrak B}$; (alternatively see the end of the proof).
{
\noindent}
\begin{enumerate}
\item[$(*)_3$] let
{
\noindent}
\begin{enumerate}
\item[$(a)$] $I^* = I \cap {\mathfrak B}$
{
\noindent}
\item[$(b)$] ${\mathscr E}^*_n = \{E {\restriction} I^*:E \in {\mathscr E}'_n \cap {\mathfrak B}\}$
{
\noindent}
\item[$(c)$] ${\mathscr E}^* = \cup\{{\mathscr E}^*_n:n \in {\mathbb N}\}$
{
\noindent}
\item[$(d)$] let $D^*$ be any ultrafilter on $I^*$ which includes $\{I
\cap I^*:I \in D \cap {\mathfrak B}\}$.
\end{enumerate}
\end{enumerate}
{
\noindent}
It is enough to check the following points:
{
\noindent}
\begin{enumerate}
\item[$(*)_4$] $\mathbf x^*_n := (I^*,D^*,{\mathscr E}^*_n)$ is a
$(\theta,\theta)-{\rm l.u.f.t.}$ for every $n \in \omega$.
\end{enumerate}
{
\noindent}
Why? E.g. note that if $E \in {\mathscr E}^*_n$ then
for some $E' \in {\mathscr E}'_n \cap {\mathfrak B}$ we have $E'
{\restriction} I^* = E$ hence $E$ has $\le \kappa$ equivalence classes.
Now for any such $E'$, as $E'$ has $\le
\kappa$-equivalence classes and belongs to ${\mathfrak B}$ clearly every
$E'$-equivalence class is not disjoint to $I^*$ and every $A
\subseteq I^*$ respecting $E$ is $A' \cap I^*$ for some $A' \in {\mathfrak B}$
respecting $E'$.
So $D/E'_n,D^*/E$ are essentially equal, etc., that is, let
$\pi_n:{\mathscr E}^*_n \rightarrow {\mathscr E}'_n$ be such that $E \in {\mathscr E}^*_n
\Rightarrow \pi_n(E) {\restriction} I^* = E$ and let $\pi_{n,E}:\{A:A \subseteq
I^*$ respects $E\} \rightarrow \{A \subseteq I:A$ respects
$\pi_n(E)\}$ be such that $\pi_{n,E}(A) = B \Rightarrow B \cap I^* =
A$; in fact, those functions are uniquely determined.
So clearly $(*)_4$ follows by
{
\noindent}
\begin{enumerate}
\item[$(*)_5$] $(a) \quad \pi_n$ is a one-to-one function from
${\mathscr E}^*_n$ onto ${\mathscr E}'_n \cap {\mathfrak B}$
{
\noindent}
\item[${{}}$] $(b) \quad \pi_n$ preserves ``$E^1$ refines $E^2$" and
its negation
{
\noindent}
\item[${{}}$] $(c) \quad {\mathscr E}^*_n$ is ($< \theta)$-directed
{
\noindent}
\item[${{}}$] $(d) \quad$ if $n=m+1$ then ${\mathscr E}^*_m \subseteq {\mathscr E}^*_n$
and $\pi_m \subseteq \pi_n$.
\end{enumerate}
{
\noindent}
Moreover
{
\noindent}
\begin{enumerate}
\item[$(*)_6$] $(a) \quad$ if $E \in {\mathscr E}^*_n$, {\underline{then}} \,
${\rm Dom}(\pi_{n,E}) \subseteq {\mathfrak B}$ (because $2^\kappa \subseteq {\mathfrak B}$ is assumed)
{
\noindent}
\item[${{}}$] $(b) \quad \pi_{n,E}$ is an isomorphism from the
Boolean Algebra ${\rm Dom}(\pi_{n,E})$ onto
\hskip25pt $\{A \subseteq I:A \,\, {\rm respects} \,\, \pi_n(E)\}$ which is
canonically isomorphic to
\hskip25pt the Boolean Algebra
${\mathscr P}(I/\pi_n(E))$ and also to ${\mathscr P}(I^*/E)$
{
\noindent}
\item[${{}}$] $(c) \quad D^* \cap {\rm Dom}(\pi_{n,E})$ is an ultrafilter
which $\pi_{n,E}$ maps onto the
\hskip25pt $D \cap {\rm Rang}(\pi_{n,E})$ which is an ultrafilter;
those ultrafilters are
\hskip25pt $\theta$-complete
{
\noindent}
\item[$(*)_7$] $I^*$ has cardinality $\le 2^\kappa$.
\end{enumerate}
{
\noindent}
[Why? Because ${\mathfrak B}$ has cardinality $\le 2^\kappa$.]
{
\noindent}
\begin{enumerate}
\item[$(*)_8$] $(M_\ell)^{I^*}_{D^*}|{\mathscr E}^*$ is isomorphic to
$((M_\ell)^I_D|{\mathscr E}') {\restriction} {\mathfrak B}$ for $\ell=1,2$.
\end{enumerate}
{
\noindent}
[Why? Let $\varkappa$ be the following function:
{
\noindent}
\begin{enumerate}
\item[$(*)_{8.1}$] $(a) \quad {\rm Dom}(\varkappa) = (M_1)^{I_*}|{\mathscr E}^*$
{
\noindent}
\item[${{}}$] $(b) \quad$ if $f_1 \in (M_1)^{I_*}$ and $E \in
{\mathscr E}^*$ refines ${\rm eq}(f_1)$ then $f_2 := \varkappa(f_1)$ is
\hskip25pt the unique function with domain $I$ such that $(\bigcup\limits_{n}
\pi_n)(E) \in {\mathscr E}'$
\hskip25pt refines ${\rm eq}(f_2)$ and $f_2 {\restriction} I^* = f_1$.
\end{enumerate}
{
\noindent}
Now easily $\varkappa$ induces an isomorphism as promised in $(*)_8$.]
{
\noindent}
\begin{enumerate}
\item[$(*)_9$] $((M_1)^I_D|{\mathscr E}') {\restriction} {\mathfrak B}$ is isomorphic to
$(M_2)^I_D|{\mathscr E}') {\restriction} {\mathfrak B}$.
\end{enumerate}
{
\noindent}
[Why? By $(*)_2$ and the choices of $g$ (in the beginning) and of
${\mathfrak B}$ after $(*)_2$ this is obvious when $\tau = \tau(M_1)$
is included in ${\mathfrak B}$, which is equivalent to
$|\tau| \le 2^\kappa$. By recalling that
${\rm arity}(\tau) \le
\aleph_0$, i.e. every predicate and function symbol of $\tau$ has
finitely many places (see \ref{d11}), {\rm without loss of generality} \, this holds. That
is, let $\tau' \subseteq \tau$ be such that for every predicate $P
\in \tau$ there is one and only one $P' \in \tau'$ such that $\ell
\in \{1,2\} \Rightarrow P^{M_\ell} = (P')^{M_\ell}$ and
similarly for every function symbol;
clearly it suffices to deal with $M_1 {\restriction} \tau',M_2 {\restriction} \tau'$
and $|\tau'| \le 2^{\|M_1\|} \le 2^\kappa$.]
Together we are done.
\end{PROOF}
{\rm no}indent
Note that the proof of \ref{d22} really uses $\kappa = \kappa^{<
\theta}$, as otherwise ${\mathscr E}'_n$ is not $(< \theta)$-directed.
How much is the assumption $\kappa = \kappa^{< \theta}$ needed in
\ref{d22}?
We can say something in \ref{d26}.
\begin{claim}
\label{d26}
Assume that $\kappa \ge 2^\theta$ but $\kappa^{< \theta} > \kappa$
hence for some regular $\sigma < \theta$ we have $\kappa^{< \sigma} =
\kappa < \kappa^\sigma$ and ${\rm cf}(\kappa) = \sigma$ and by \cite{So74}
we have $(\forall
\mu < \kappa)(\mu^\theta < \kappa)$; recall ${\rm arity}(\tau) = \aleph_0$.
{\rm no}indent
1) If $\langle {\mathfrak B}_i:i \le \sigma{\rm rang}le$ is a $\subseteq$-increasing
continuous sequence of $\tau$-models and $\mathbf x$ is a
$(\theta,\theta)-{\rm l.u.f.t.}$ {\underline{then}} \, ${\rm l.u.p.}_{\mathbf x}({\mathfrak B}_\sigma) =
\cup\{{\rm l.u.p.}_{\mathbf x}({\mathfrak B}_i):i < \sigma\}$ and $i <j \Rightarrow
{\rm l.u.p.}_{\mathbf x}({\mathfrak B}_i) \subseteq {\rm l.u.p.}_{\mathbf x}({\mathfrak B}_j)$.
{\rm no}indent
2) If $J$ is a directed partial order of cardinality $\le \sigma$ $(<
\theta)$ and $\mathbf x_s = (I,D,{\mathscr E}_s)$ is a
$(\theta,\theta)-{\rm l.u.f.t.}$ for $s \in J$ such that $s <_J t
\Rightarrow {\mathscr E}_s \subseteq {\mathscr E}_t$ and $M$ is a $\tau$-model {\underline{then}}
\, ${\rm l.u.p.}_{\mathbf x}({\mathfrak B}) = \cup\{{\rm l.u.p.}_{\mathbf x_s}({\mathfrak B}):s \in J\}$ and
$s <_J t \Rightarrow {\rm l.u.f.t.}_{\mathbf x_s}({\mathfrak B}) \subseteq {\rm l.u.p.}_{\mathbf
x_t}({\mathfrak B})$ under the natural identification.
{\rm no}indent
3) In \ref{d22}, $|I^*| \le \Sigma\{2^\partial:\partial
< \kappa\}$ is enough.
\end{claim}
\begin{PROOF}{\ref{d26}}
Straightforward.
\end{PROOF}
\section {Special Models} \label{2}
Note that in Def. \ref{f2} below,
$ M_n {\rm pr}ec _{\mathbb{L} {\theta, \theta }} M$ is not required.
The reader may in first reading ignore the
special$^ \bullet $ case.
\begin{definition}
\label{f2}
1)
Assume $\lambda > \theta$ is strong limit of cofinality $\aleph_0$.
We say a model $M$ is $\lambda$-special {\underline{when}} \, there are
$\bar\lambda,\bar M$ such that (we also may say $\bar M$ is a
$\lambda$-special sequence):
{
\noindent}
\begin{enumerate}
\item[$(a)$] $M$ is a model of cardinality $\lambda$ with $|\tau(M)|
< \lambda$
{
\noindent}
\item[$(b)$]
\begin{enumerate}
\item[$(\alpha)$] $\bar \lambda = \langle \lambda_n:n \in {\mathbb N}{\rm rang}le$
{
\noindent}
\item[$(\beta)$] $\lambda_n \le \lambda_{n+1}$
{
\noindent}
\item[$({\mathfrak a}mma)$] $\theta \le \lambda_n < \lambda_{n+1} < \lambda =
\sum\limits_{k} \lambda_k$ and stipulate $\lambda_{-1} = \theta$
\end{enumerate}
{
\noindent}
\item[$(c)$]
\begin{enumerate}
\item[$(\alpha)$] $\bar M = \langle M_n:n < \omega{\rm rang}le$
{
\noindent}
\item[$(\beta)$] $M_n {\rm pr}ec_{{\mathbb L}_{\theta,\theta}} M_{n+1}$
{
\noindent}
\item[$({\mathfrak a}mma)$] $M = \bigcup\limits_{n} M_n$
{
\noindent}
\item[$(\delta)$] $\lambda_n \ge \|M_n\| \ge \lambda _{n-1}$
recalling $ \lambda _{-1}= \theta $
\end{enumerate}
{
\noindent}
\item[(d)]
\begin{enumerate}
\item[$(\alpha)$] $\bar D = \langle D_n:n \in {\mathbb N}{\rm rang}le$ and
$\|M_n\| \le \lambda_n$
{
\noindent}
\item[$(\beta)$] $D_n \in {\rm ruf}_{\lambda_{n-1},\theta}(\lambda_n)$
{
\noindent}
\item[$({\mathfrak a}mma)$] $M^{\lambda_n}_n/D_n {\rm pr}ec_{{\mathbb L}_{\theta,\theta}} M_{n+1}$
under the canonical identification (so hence $2^{\lambda_n} \le
\lambda_{n+1}$)
\end{enumerate}
\end{enumerate}
{
\noindent}
{\rm no}indent
2) We say that the model $ M $ is $ \lambda $-special$^ \bullet $
when clauses (a),(b),(c) above hold but instead of clause (d) we have
{
\noindent}
\begin{enumerate}
\item[(d)$'$] if $\Gamma$ is an ${\mathbb L}_{\theta,\theta}$-type on $M_n$
of cardinality $\le \lambda_n$ with $\le \lambda_n$ free variables
then $\Gamma$ is realized in $M_{n+1}$.
\end{enumerate}
\end{definition}
\begin{claim}
\label{f12}
1) If for every $n < \omega$ we have
$D_n$ is a $(\lambda_n,\theta)$-regular $\theta$-complete
ultra-filter on $I_n,M_{n+1} = (M_n)^{I_n}/D_n$ identifying $M_n$
with its image under the canonical embedding into $M_{n+1}$ so $M_n
{\rm pr}ec_{{\mathbb L}_{\theta,\theta}} M_{n+1}$ and $\lambda_n \ge \|M_n\|,
\lambda = \sum\limits_{n} \lambda_n \ge \theta$
(equivalently $ > \theta $)
{\underline{then}} \,
$\langle M_n:n \in {\mathbb N}{\rm rang}le$ is a $\lambda$-special sequence, so
$M= \bigcup\limits_{n} M_n$ is a $\lambda$-special model
and $ M $ is a model of $ {\rm Th}_{\mathbb{L} ^1_ \theta }(M_1) $ .
{\rm no}indent
2)
Assume $ \lambda > \theta , {\rm cf}( \lambda ) = {\aleph_0} $.
In Definition \ref{f2}, clause (d) indeed implies clause $(d)'$;
so every $ \lambda $-special model/sequence
is $ \lambda $-special$^ \bullet $
model/sequence.
Also in Def. \ref{f2}, $ M $ is a model of
${\rm Th}_{\mathbb{L} ^1_ \theta }(M) $, in fact this follows by
\ref{f2}(1)(d)$(\alpha ),\big( \beta ),( {\mathfrak a}mma ) $.
{\rm no}indent
4) Assume $ \lambda > \theta $ is a strong limit cardinal
of cofinality $ {\aleph_0} $. If $ M $ is a model of cardinality
$ \ge \theta $ but $ < \lambda $ {\underline{then}} \,
\begin{enumerate}
\item[(A)]
\begin{enumerate}
\item[(a)] there is a $ \lambda $-special sequence $ \bar{ M } $
with $ M_0 = M $
\item[(b)] there is a $ \lambda $-special model $ N $ which is
a $ {\rm pr}ec _{\mathbb{L} ^1_\theta }$-extension of $ M $
\item[(c)] $ {\rm Th}_{\mathbb{L} ^1_\theta (M)}$
has a $ \lambda $-special model.
\end{enumerate}
\item[(B)]
If $ M $ is a model of cardinality $ \lambda $ {\underline{then}} \,
for some $ N, \bar{ M } , \bar{ N } $ we have:
\begin{enumerate}
\item[(a)] $ \bar{ M } = \langle M_n {\rm rang}le $ is
a $ \lambda $-special$^ \bullet $ sequence with union $ M $
\item[(b)] $ \bar{ N } = \langle N_n {\rm rang}le $ is
a $ \lambda $-special$^ \bullet $ sequence with union $ N $
\item[(c)] $ M_ n {\rm pr}ec _{\mathbb{L} _{\theta, \theta }} N_n$
\end{enumerate}
\item[(C)] If $M$ is a $\lambda$-special model and $\tau \subseteq \tau_M$,
{\underline{then}} \, $M {\restriction} \tau$ is also a $\lambda$-special model.
\end{enumerate}
{\rm no}indent
5) Assume $ \lambda > \theta > {\aleph_0} = {\rm cf}(\lambda )$.
If $M$ is a $\lambda$-special$^\bullet $
model and $\tau \subseteq \tau_M$,
{\underline{then}} \, $M {\restriction} \tau$ is also a $\lambda$-special$^ \bullet $
model
{\rm no}indent
6) If $ \lambda$ is strong limit $ > \theta $ of cofinality ${\aleph_0} $.
A model $ M $ is $ \lambda $-special iff it is $ \lambda $-special$^\bullet $.
\end{claim}
\begin{PROOF}{\ref{f12}}
1) If we assume clause (d) in Definition \ref{f2}, just by the
definition. If we assume clause $(d)'$ in Definition \ref{f2}, use
part (2).
{\rm no}indent
2) It follows by the $(\lambda_n,\theta)$-regularity of $D_n$.
{\rm no}indent
3) Check the definition.
{\rm no}indent
4) \underline{CLAUSE (A)}
We can choose an increasing sequence
$ \langle \lambda_n: n < \omega {\rm rang}le $
with limit $ \lambda $ such that $ \lambda_0
= \| M \|^ \theta $
and $ 2^ {\lambda _n } < \lambda _{n + 1 }=
\lambda _{n + 1} ^ \theta $. For each $ n $
we can choose a $ (\lambda , \theta ) $-regular
$ \theta $-complete ultrafilter on $ \lambda _n $,
and define $ M_n $ as in part (1).
Now use the conclusion of part (1).
\underline{CLAUSE (B)}
{\rm Without loss of generality} \, the universe of $ M $ is $ \lambda $.
Choose $ \langle \lambda _n: n < \omega {\rm rang}le $ as
above (except $ \lambda _0 \ge \| M \| $ of course),
and by induction on $ n $ choose
$ M_n {\rm pr}ec _{\mathbb{L} ^1_\theta } M $
of cardinality $ \lambda _n $ which include
$ \cup \{ M_k: k < n \}\cup \lambda _n $.
We now choose
$ \langle M^*_k, M^*_ {k,n } : n < \omega {\rm rang}le $
by induction on $ k $ such that
\begin{enumerate}
\item[(a)] for $ k=0 $ we let $ M^*_k = M$ and
$ M^*_{k, n }= M_n$
\item[(b)] for $ k= {\ell} + 1 $ let
$ M^*_{k}= (M^*_{{\ell} })^{\lambda _k }/D_k$ and
$ M^*_{k,n}= (M^*_{{\ell},n })^{\lambda _k }/D_k$.
\end{enumerate}
There is no problem to carry the induction
and we let $N = \cup \{M^*_{k,k } \} $ and
$ N_ k = M^*_{k,k}$, now check
\underline{CLAUSE (C)}
Just read the definition.
{\rm no}indent
5) Again just read the definition.
{\rm no}indent
6) Easy too.
\end{PROOF}
\begin{remark}
1)
In \ref{f23} we we do not require that
the $ \lambda _n $-s are the same and of course do not require
that the $ D_n $ are the same. Part (3) clarify this.
{\rm no}indent
2) In Def. \ref{f2} clause (c)$( \delta ) $,
it is enough to demand ${}{}\lambda _n \ge
\| M_n \| \ge \theta $.
\end{remark}
\begin{claim}
\label{f23}
1) If $\langle M^\ell_n:n \in {\mathbb N}{\rm rang}le$ is a $\lambda$-special
sequence (or just a
$ \lambda $-special$^ \bullet $ sequence)
with union $M_\ell$ for $\ell=1,2$ and
${\rm Th}_{{\mathbb L}_{\theta,\theta}}(M^1_0) = {\rm Th}_{{\mathbb L}_{\theta,\theta}}
(M^2_0)$ {\underline{then}} \, $M_1,M_2$ are isomorphic.
{\rm no}indent
2) Moreover, if $n < \omega$ and $f$ is a partial function from
$M^1_n$ into $M^2_n$
which is $(M^1_n,M^2_n,{\mathbb L}_{\theta,\theta})$-elementary (i.e. $\bar a
\in {}^{\theta >}({\rm Dom}(f)) \Rightarrow
f({\rm tp}_{{\mathbb L}_{\theta,\theta}}(\bar a,\emptyset,M^1_n)) =
{\rm tp}_{{\mathbb L}_{\theta,\theta}}(f(\bar a),\emptyset,M^2_n))$ {\underline{then}} \, $f$
can be extended to an isomorphism from $M_1$ onto $M_2$.
{\rm no}indent
3) If we weaken clause $(d)'$ of Definition \ref{f2} by weakening the
conclusion to: for some $k>n,\Gamma$ is realized in $M_k$ we get
an equivalent definition.
\end{claim}.
\begin{PROOF}{\ref{f23}}
1)
By the hence and
forth argument; but we elaborate.
Let $ {\mathscr F} _n $ be the set of $ f $ such that:
\begin{enumerate}
\item[(a)] $ f $ is a one to one function
\item[(b)] the domain of $ f $ is included in $ M^1_n $
\item[(c)] the range of $ f $ is included in $ M^2_n $
\item[(d)] if $ \zeta < \theta $ and $ \bar{ a } \in
{}^{ \zeta } (M^1_n)$ and $ \bar{ b } = f( \bar{ a }) \in
{}^{ \zeta } (M^2_n) $ and $ {\varepsilon}hi ( \bar{ x } _{[\zeta ]}
\in \mathbb{L} _{\theta, \theta }(\tau (M_ {\ell} )$
{\underline{then}} \, $ M^1_n \models {\varepsilon}hi [\bar{ a } ] $
{\underline{if}}f \,
$ M^2_n \models {\varepsilon}hi [\bar{ b } ] $.
\end{enumerate}
{\rm no}indent
Easily
{\rm no}indent
\begin{enumerate}
\item[$(*)_1$] the set $ {\mathscr F}_n $ is not empty
\end{enumerate}
{\rm no}indent
[Why? Because the empty function belongs to $ {\mathscr F} _n $]
{\rm no}indent
\begin{enumerate}
\item[$(*)_2$] if $ f \in {\mathscr F} _n $ then
some $ g \in {\mathscr F} _{n + 1 }$ extend $ f $
and $ M^1_n \subseteq {\rm Dom}(g)$
\end{enumerate}
{\rm no}indent
[Why? By clause (d)' of \ref{f2}(2)]
\begin{enumerate}
\item[$(*)_3$] if $ f \in {\mathscr F} _n $ then
some $ g \in {\mathscr F} _{n + 1 }$ extend $ f $
and $ M^1_n \subseteq {\rm Rang}(g)$
\end{enumerate}
{\rm no}indent
[Why? Similarly]
Together clearly we are done
{\rm no}indent
2) Same proof.
{\rm no}indent
3) Use suitable sub-sequences (using monotonicity).
\end{PROOF}
{\rm no}indent
Note that comparing definition \ref{f2} with the first order parallel,
in Claim \ref{f23}(1), a priory it is not given that
${\rm Th}_{{\mathbb L}_{\theta,\theta}}(M_1) = {\rm Th}_{{\mathbb L}_{\theta,\theta}}(M_2)$
suffices. Also \ref{f23} does not say that ${\rm Th}_{{\mathbb L}^1_\theta}(M)$
and $\lambda$ determines $M$ up to isomorphism because we demand that
$M^1_0,M^2_0$ are ${\mathbb L}^1_\theta$-equivalent. However:
\begin{claim}
\label{f5}
Assume $\lambda > \theta$ is
of cofinality $\aleph_0$
and $T$ is a complete theory in ${\mathbb L}^1_\theta(\tau_T),|T| <
\lambda$ equivalently $|\tau_T| < \lambda$.
{\rm no}indent
1) If $ \lambda $ is strong limit
{\rm Th}en \, $T$ has exactly one $\lambda$-special
model (up to isomorphism).
{\rm no}indent
2)
$ T $ has at most
one $ \lambda $-special$^ \bullet $ model of cardinality
$ \lambda $ up to isomorphism.
\end{claim}
\begin{PROOF}{\ref{f5}}
1) Assume $N_1,N_2$ are special models of $T$ of cardinality
$\lambda$. By Definition \ref{f2} for $\ell=1,2$ there is a triple
$(\bar\lambda_\ell,\bar M_\ell,\bar D_\ell)$ witnessing $N_\ell$ is
$\lambda$-special as there.
As $M_{\ell,0} {\rm pr}ec_{{\mathbb L}_{\theta,\theta}} M_{\ell,n}
{\rm pr}ec_{{\mathbb L}_{\theta,\theta}} M_{\ell,n+1} {\rm pr}ec_{{\mathbb L}^1_\theta}
\bigcup\limits_{m} M_{\ell,m} =
N_\ell$ for $n \in {\mathbb N}$, by \ref{x7} and \ref{d8},
we know $M_{\ell,0} {\rm eq}uiv_{{\mathbb L}^1_\kappa} N_\ell$, so
we can conclude that $M_{1,0}
{\rm eq}uiv_{{\mathbb L}^1_\kappa} M_{2,0}$ and both are models of $T$.
By \ref{d11} there is a sequence $\langle (\lambda_n,D_n):n \in
{\mathbb N}{\rm rang}le$ with $\Sigma_{n < \omega }
\lambda_n = \lambda,2^{\lambda_n} \le
\lambda_{n+1}$ and $D_n$ a $(\lambda_n,\theta)$-regular ultrafilter on
$\lambda_n$ such that $M'_1 \cong M'_2$ when:
{
\noindent}
\begin{enumerate}
\item[$(*)$] $M'_{\ell,0} = M_{\ell,0},M'_{\ell,n+1} =
(M'_{\ell,n})^{\lambda_n}/D_n$ and $M'_\ell = \bigcup\limits_{n}
M'_{\ell,n}$.
\end{enumerate}
{
\noindent}
So $M'_1 \cong M'_2$ by \ref{d11} and $N_1 \cong M'_1$ by \ref{f23}(1) and
$N_2 \cong M'_2$ similarly. Together $N_1 \cong N_2$ is promised.
{\rm no}indent
2)
The proof is similar to part of the proof of \ref{d11}
clause (a) implies clause (e).
i.e. by the
hence and forth argument.
\end{PROOF}
{\rm no}indent
Now we can generalize Robinson lemma hence (see
e.g. \cite{Mak85}), gives an alternative proof
of the interpolation theorem, recall though that in \cite{Sh:797} we
do not assume the cardinal $\theta$ is compact).
\begin{claim}
\label{f8}
1) Assume $\tau_1 \cap \tau_2 = \tau_0,T_\ell$ is a complete theory in
${\mathbb L}^1_\theta(\tau_\ell)$ for $\ell=1,2$ and $T_0 = T_1 \cap
T_2$. {\rm Th}en \, $T_1 \cup T_1$ has a model.
{\rm no}indent
2) We can allow in (1) the vocabularies to have more than one sort.
{\rm no}indent
3) The logic ${\mathbb L}^1_\theta$ satisfies the interpolation theorem.
{\rm no}indent
4) ${\mathbb L}^1_\theta$ has disjoint amalgamation, i.e. if $M_0
{\rm pr}ec_{{\mathbb L}^1_\theta} M_\ell$ for $\ell=1,2$ that is
$(M_0,c)_{c \in M_0},(M_\ell,c)_{c \in M_0}$
has the same ${\mathbb L}^1_\theta$-theory
and $|M_1| \cap |M_2| = |M_0|$, {\underline{then}} \, there is $M_3$ such that
$M_\ell {\rm pr}ec_{{\mathbb L}^1_\theta} M_3$ for $\ell=0,1,2$ (hence orbital
types are well defined).
{\rm no}indent
5) ${\mathbb L}^1_\theta$ has\footnote{But the disjoint version may fail,
e.g. if we have individual constants.} the JEP.
\end{claim}
\begin{PROOF}{\ref{f8}}
1) Let $\lambda > |\tau_1| + |\tau_2| + \theta$ be a strong limit
cardinal of cofinality $\aleph_0$. For $\ell = 1,2$ there is a
$\lambda$-special model $M_\ell$ of $T_\ell$ by \ref{f12}(3). Now
$N_\ell = M_\ell {\restriction} \tau_0$ is a $\lambda$-special model of $T$.
By \ref{f5}(1), $N_1 \cong N_2$ so {\rm without loss of generality} \, $N_1 = N_2$, and let $M$ be the
expansion of $N_1 = N_2$ by the predicates and functions of $M_1$ and of
$M_2$. Clearly $M$ is a model of $T_1 \cup T_2$.
{\rm no}indent
2) Similarly.
{\rm no}indent
3) Follows as ${\mathbb L}^1_\theta$ being $\subseteq {\mathbb L}_{\theta,\theta}$
satisfies $\theta$-compactness and part (1).
{\rm no}indent
4) Follows by (1), that is, let $\mathbf x$ be as in \ref{d11}(c) for
$M_1,M_2$. So for every $C \subseteq M_0$ of cardinality $< \theta$,
letting $M_{C,\ell} = (M_\ell,c)_{c \in C}$ we have $N_{C,1} \cong
N_{C,2} \cong N_{C,0}$ where $N_{C,\ell} = {\rm l.u.p.}_{\mathbf
x}(M_{C,\ell})$. Hence $N_{C,0} {\rm pr}ec_{{\mathbb L}_{\theta,\theta}}
N_{C,\ell}$ for $\ell=1,2$ and we use ``${\mathbb L}_{\theta,\theta}$ has
disjoint amalgamation".
{\rm no}indent
5) Follows by \ref{d11}.
\end{PROOF}
\begin{remark}
\label{f10}
This proof implies the generalization of preservation theorems, see
\cite{CK73}.
\end{remark}
{\rm no}indent
Recall that Ehrenfuecht-Mostowski \cite{EhMo56} aim was: every
first order theory $T$ with infinite models has models with many automorphisms.
This fails for ${\mathbb L}_{\theta,\theta}$ and even
${\mathbb L}_{\aleph_1,\aleph_1}$ as we can express ``$<$ is a well ordering".
What about ${\mathbb L}^1_\theta$?
\begin{claim}
\label{f22}
Assume ($\lambda,T$ are as above in \ref{f5} and) $M$ is a special model of $T$
of cardinality $\lambda$. {\rm Th}en \, $M$ has $2^\lambda$ automorphisms.
\end{claim}
\begin{PROOF}{\ref{f22}}
Let $\langle M_n:n < \omega{\rm rang}le$ witness $M$ is special. The
result follows by the proof of \ref{f23}(2) noting that
{
\noindent}
\begin{enumerate}
\item[$(*)$] if $f_n$ is an
$(M_n,M_n,{\mathbb L}_{\theta,\theta}(\tau_M))$-elementary mapping {\underline{then}}
\, there are $a_2 \in {}^\lambda(M_{n+1})$ and
$f_\alpha,a_{2,\alpha} \in (M_{n+1})$ for $\alpha < \lambda_n$ such
that
{
\noindent}
\begin{enumerate}
\item[$(a)$] $a_{2,\alpha} \ne a_{2,\beta}$ for $\alpha < \beta <
\lambda_n$
{
\noindent}
\item[$(b)$] for $f_\alpha$ is an
$(M^1_{n+1},M^2_{n+1},{\mathbb L}_{\theta,\theta}(\tau_M))$-elementary
mapping
{
\noindent}
\item[$(c)$] $f_\alpha \supseteq f$ and maps $a$ to $a_\alpha$.
\end{enumerate}
\end{enumerate}
{
\noindent}
Why this is possible? Choose $a' \in M_{n+2} \backslash M_{n+1}$ and
choose $a_\alpha \in M_{n+1} \backslash \{a_\beta:\beta < \alpha\}$ by
induction on $\alpha < \lambda_n$ realizing
${\rm tp}_{{\mathbb L}_{\theta,\theta}(\tau_T)}(a',M_n,M_{n+2})$.
Lastly, let $f_\alpha = f \cup \{(a_0,g(a_\alpha))\}$.
Why this is enough? Should be clear.
\end{PROOF}
\end{document} |
\begin{document}
\author{Thomas Leuther}
\address[Thomas Leuther]{University of Li\`ege, Department of Mathematics\\
Grande Traverse, 12 - B37, B-4000 Li\`ege, Belgium}
\email{Thomas.Leuther[at]ulg.ac.be}
\author{Fabian Radoux}
\address[Fabian Radoux]{University of Li\`ege, Department of mathematics \\
Grande Traverse, 12 - B37, B-4000 Li\`ege, Belgium}
\email{Fabian.Radoux[at]ulg.ac.be}
\author{Gijs M. Tuynman}
\address[Gijs M. Tuynman]{Laboratoire Paul Painlev\'e, U.M.R. CNRS 8524 et UFR de Math\'ematiques,
Universit\'e de Lille I, 59655 Villeneuve d'Ascq Cedex, France}
\email{Gijs.Tuynman[at]univ-lille1.fr}
\title[Geodesics on a supermanifold]{
Geodesics on a supermanifold and projective equivalence of super connections
}
\begin{abstract}
We investigate the concept of projective equivalence of connections in supergeometry. To this aim, we propose a
definition for (super) geodesics on a supermanifold in which, as in the classical case, they are the projections of the integral curves of a vector field on the tangent bundle: the \emph{geodesic vector field} associated with the connection.
Our (super) geodesics possess the same properties as the in the classical case: there exists a unique (super) geodesic satisfying a given initial condition and when the connection is metric, our supergeodesics coincide with the trajectories of a free particle with unit mass.
Moreover, using our definition, we are able to establish Weyl's characterization of projective equivalence in the super context: two torsion-free (super) connections define the same geodesics (up to reparametrizations) if and only if their difference tensor can be expressed by means of a (smooth, even, super) $1$-form.
\end{abstract}
\maketitle
MSC(2010) : 58A50, 53B10, 53C22.
{\bf Keywords:} supermanifold, geodesic, connection, projective equivalence.
{\bf Subject classification:} supermanifolds and supergroups, real and complex differential geometry.
\maketitle
\section{Introduction}
The concept of projective equivalence of connections goes back to the 1920's, with the study of the so-called ``geometry of paths'' (see \cite{Th, TV, Wh} or \cite{Ro1,Ro2,HR} for a modern formulation). In 2002, M.~Bordemann used this theory to answer the problem of projectively invariant quantization in \cite{Bo}.
Projectively invariant quantization is a generalization to arbitrary manifolds of the notion of equivariant quantizations in the sense of Lecomte-Ovsienko, see \cite{LO,L,MR}. It consists in building in a natural way a quantization (i.e., a symbol-preserving linear bijection between a space of symbols and a space of differential operators) from a linear connection, requiring that the quantization remains unchanged if we start from another connection in the same projective class.
By definition, two connections are called {\em projectively equivalent} if they have the same geodesics, up to parametrization. In other words, the geodesics of two equivalent connections are the same, provided that we see them as sets of points, rather than as maps from an open interval of $\mathbf{R}$ into the manifold. In \cite{We}, H. Weyl showed that projective equivalence can be rephrased in an algebraic way: two connections are projectively equivalent if and only if the symmetric tensor which measures the difference between them can be expressed by means of a $1$-form.
Weyl's algebraic characterization of projective equivalence provides a convenient way to transport projective equivalence to the framework of supergeometry: two superconnections are said to be {\em projectively equivalent} if the (super)symmetric tensor which measures the difference between them can be expressed by means of a (super)$1$-form. Using this notion, it is possible to set the problem of projectively invariant quantization on supermanifolds while M.~Bordemann's method can be adapted in order to solve it (see \cite{LR}).
Remembering the classical picture, it is natural to ask whether it is possible to find a geometric counterpart to the algebraic definition of projective equivalence of superconnections, i.e., a characterization in terms of supergeodesics. The main purpose of the present paper is to answer this question in the affirmative.
As in the classical case, we define, in section \ref{sectionSuperGeodesics}, supergeodesics associated with a superconnection $\nabla$ on a supermanifold $M$ as being the projections onto $M$ of the integral curves of a vector field $G$ on the tangent bundle $TM$: the {\em geodesic vector field} of $\nabla$. In section \ref{proj} we then define the notion of reparametrization of a geodesic and establish that two connections $\nabla$ and $\nablah$ on a supermanifold $M$ have the same geodesics up to parametrization if and only if there is an even $1$-form $\alpha$ such that
\[
\nablah_{X}Y=\nabla_{X}Y+\alpha(X)Y+(-1)^{\epsilon(X)\epsilon(Y)}\alpha(Y)X\quad\forall X,Y\in\Gamma(TM) ,
\]
thus showing that Weyl's characterization also holds in supergeometry.
We note that our approach to supergeodesics differs from that of Goertsches \cite{Go}. In particular, our equations for supergeodesics are the natural generalization of the classical ones. Actually, our approach is nearly identical to that recently proposed by Garnier-Wurzbacher in \cite{GW}, where they consider supergeodesics associated with a Levi-Civita superconnection. In their paper, supergeodesics on a Riemannian supermanifold $M$ are shown to co{\"\i}ncide with the projections of the flow of a Hamiltonian supervector field defined on the (even) cotangent bundle of $M$. In section \ref{Smetricconnections} we will show that the same holds in our approach when we use a Levi-Civita connection.
In fact, beyond the fact that they restrict to the Riemannian setting, the main difference between Garnier-Wurzbacher's supergeodesics and ours lies in the domain of supercurves. Supercurves should be images of $1$-dimensional manifolds, but as it is well-known, the theory of supercurves with a single parameter turns out to be very shallow: supercurves in a single even parameter are reduced to ordinary curves in the body of the manifold while supercurves in a single odd parameter are simply odd straight lines. In order to overcome these limitations, we choose to change the viewpoint. Usually curves do not come singly, they appear in families. And in particular the integral curves of a vector field on a supermanifold $N$ should not be seen as a simplistic collection of curves, but as a map (the flow) defined on (an open subset of) $\RR \times N\,$(\footnote{{In fact, rather $\CA_0 \times N$ than $\RR \times N$ since maps defined on $\CA_0 \times N$ live in the category of supermanifolds while containing the same information as maps defined on $\RR \times N$ (see Lemma \ref{expandinginoddpowers}).}}), incorporating the initial condition in the domain of the map. And indeed, the flow of a vector field is jointly smooth in the time parameter $t$ and the initial condition $n\in N$. In the simplistic viewpoint one writes $\gamma_{n}(t)$ for an integral curve with initial condition $n\in N$, whereas in the viewpoint of a flow one rather writes $\varphi_t(n)$ or $\varphi(t,n)$. Roughly speaking, we could say that our change of viewpoint enlarges in a natural way (we do not add an arbitrary manifold $S$ as in \cite{GW}) the domain of supercurves so that it is now possible to get supercurves with desirable properties.
\section{Notation and general remarks}
We will work with the geometric $H^\infty$ version of DeWitt supermanifolds, which is equivalent to the theory of graded manifolds of Leites and Kostant (see \cite{DW, Ko, Le, Rog, Tu1}). Any reader using a (slightly) different version of supermanifolds should be able to translate the results to her\slash his version of supermanifolds.
\begin{definition}{Some general conventions}
\begin{itemize}
\item
The basic graded ring will
be denoted as $\CA$ and we will think of it as the exterior algebra $\CA
= \Lambda V$ of an infinite dimensional real vector space $V$.
\item
Any element $x$ in a graded space splits into an even and an odd part $x=x_0 + x_1$. {Associated to this splitting we have the operation $\conjug$ of conjugation in the odd part defined by $\conjug(x) \equiv \conjug(x_0+x_1) = x_0 - x_1$.}
\item
All (graded) objects over the basic ring $\CA$ have an underlying real
structure, called their \emph{body}, in which all nilpotent elements in $\CA$ are
ignored\slash killed. This forgetful map is called the \emph{body map}, denoted by $\body$.
{For the ring $\CA$, this map $\body$ is nothing but the canonical projection $\CA =\Lambda V \to \Lambda^0 V = \RR$.}
\item
If $\omega$ is a $k$-form and $X$ a vector field, we denote the contraction of
the vector field $X$ with the $k$-form $\omega$ by $\contrf{X}{\omega}$, which
yields a $k-1$-form. If $X_1, \dots, X_\ell$ are $\ell\le k$ vector fields, we
denote the repeated contraction of $\omega$ by $\contrf{X_1, \cdots,
X_\ell}{\omega}$. More precisely:
\begin{moneq}
\contrf{X_1, \cdots, X_\ell}{\omega} =
\Bigl( \contrf{X_1}{} \scirc \cdots \scirc \contrf{X_\ell}{}
\Bigr) \, \omega
\end{moneq}
In the special case $\ell = k$ this definition differs by a factor
$(-1)^{k(k-1)/2}$ from the usual definition of the evaluation of a $k$-form on
$k$ vector fields. This difference is due to the fact that in ordinary
differential geometry repeated contraction with $k$ vector fields corresponds
to the direct evaluation in the reverse order. And indeed, $(-1)^{k(k-1)/2}$ is
the signature of the permutation changing $1,2,\dots,k$ in $k,k-1,\dots,2,1$.
However, in graded differential geometry this permutation not only introduces
this signature, but also signs depending upon the parities of the vector
fields. These additional signs are avoided by our definition.
\item
Evaluation\slash contraction of a left-(multi-)linear map $f$ with a vector $v$ is denoted just as the contraction of a differential form with a vector field as $\contrf{v}{f}$. If $f:E\to \CA$ is just left-linear, this is just the image of $v$ under the map $f$. However, if $f$ is for instance left-bilinear, the contraction $\contrf{v}{f}$ now is a left-linear map given by
\begin{moneq}
\contrf v f : w\mapsto \contrf{w,v}f
\end{moneq}
As left-linearity and right-linearity are the same for even maps, we sometimes use the more standard notation $f(w,v)$ for the image of the couple $(w,v)$ under the bilinear map $f$, instead of $\contrf{w,v}f$.
\item
If $E$ is an $\CA$-vector space, $E^*$ will denote the {\em left} dual of $E$, \ie, the
space of all {\em left}-linear maps from $E$ to $\CA$.
\item
Let $x^1, \dots, x^n$ be local coordinates of a super manifold $M$ of graded dimension $p\vert q$, $p+q=n$, ordered such that $x^1, \dots, x^p$ are even and $x^{p+1}, \dots, x^n$ are odd (we will denote the latter also by $(\xi^1, \dots, \xi^q)$). Using the symbol $\varepsilon$ as the parity function, we thus have $\varepsilon(x^i)=0$ for $i\le p$ and $1$ for $i>p$. To simplify notation, we introduce the abbreviation $\varepsilon_i = \varepsilon(x^i)$.
\end{itemize}
\end{definition}
\begin{proclaim}[expandinginoddpowers]{Lemma (\cite{Tu1})}
Let $f$ and $g$ be smooth functions of even variables $x_1, \dots, x_{{p}}$ and odd variables $\xi_1, \dots, \xi_{{q_1}}$ and $\eta_1, \dots, \eta_{{q_2}}$. We can expand these functions with respect to products of odd variables, either only the $\xi$'s, only the $\eta$'s or both $\xi$'s and $\eta$'s, giving (for $f$) the formulae
\begin{align*}
f(x,\xi,\eta) &= \sum_{I\subset \{1, \dots, {q_1}\}} \xi^I \cdot f_I^{(\xi)}(x,\eta)
=
\sum_{J\subset\{1, \dots, {q_2}\}} \eta^J \cdot f_J^{(\eta)}(x,\xi)
\\&
=
\sum_{I\subset \{1, \dots, {q_1}\}, J\subset\{1, \dots, {q_2}\}} \xi^I \cdot \eta^J \cdot f_{IJ}^{(\xi,\eta)}(x)
\end{align*}
where the sum is over all subsets with (for instance)
\begin{moneq}
I= \{i_1, \dots, i_k\} \text{ with } 1\le i_1 < i_2 < \cdots <i_k\le {q_1}
\qquad\Longrightarrow\qquad
\xi^I = \xi_{i_1} \cdots \xi_{i_k}
\end{moneq}
Then the following statements are equivalent:
\begin{enumerate}
\item $f=g$
\item for all $I\subset \{1, \dots, {q_1}\}$: $f_I^{(\xi)} = g_I^{(\xi)}$
\item for all $J\subset \{1, \dots, {q_2}\}$: $f_J^{(\eta)} = g_J^{(\eta)}$
\item for all $I\subset \{1, \dots, {q_1}\}$, $J\subset \{1, \dots, {q_2}\}$: $f_{IJ}^{(\xi,\eta)} = g_{IJ}^{(\xi,\eta)}$
\end{enumerate}
Moreover, when we have expanded with respect to all odd variables, the remaining functions of the even variables only are completely determined by their values on real coordinates. Said differently, we may assume that they are ordinary smooth functions of $n$ real coordinates.
\end{proclaim}
\section{Super Geodesics}
\label{sectionSuperGeodesics}
{Before dealing with the specific problem of geodesics on a supermanifold, we first recall some general definitions and facts about (super) connections in the tangent bundle. Then we attack the problem of defining super geodesics: we associate with any connection a so-called \emph{geodesic vector field} on the tangent bundle, whose flow equations are the straightforward super analogs of the classical geodesic equations.
}
\begin{definition}{Definition \cite[VII\S6]{Tu1}}
A connection in a (super) vector bundle $p:E\to M$ over a supermanifold $M$ is (can be seen as) a map $\nabla:\Gamma(TM) \times \Gamma(E) \to \Gamma(E)$ satisfying
\begin{enumerate}
\item $\nabla$ is bi-additive (in $\Gamma(TM)$ and $\Gamma(E)$)
\item
for $X\in \Gamma(TM)$, $s\in \Gamma(E)$ and $f\in C^\infty(M)$ we have
\begin{moneq}
\nabla_{fX}s = f\cdot \nabla_Xs
\end{moneq}
\item
for homogeneous $X\in \Gamma(TM)$, $s\in \Gamma(E)$ and $f\in C^\infty(M)$ we have
\begin{moneq}
\nabla_{X}(fs) = (Xf)\cdot s + (-1)^{\varepsilon(X)\cdot \varepsilon(f)} f\cdot \nabla_Xs
\end{moneq}
\end{enumerate}
\end{definition}
\begin{proclaim}{Lemma}
If $\nabla$ and $\nablah$ are connections in $E$, the map $S:\Gamma(TM) \times \Gamma(E) \to \Gamma(E)$ defined by
\begin{moneq}
S(X,s) = \nabla_Xs - \nablah_Xs
\end{moneq}
is even and bilinear over $C^\infty(M)${. In other words, $S$ is} a ``tensor'', \ie, can be seen as a section of the bundle $TM^* \otimes \End(E)$ \cite[IV\S5]{Tu1}.
\end{proclaim}
\begin{proclaim}{Lemma}
If $\nabla$ is a connection in $TM$, then the map $T:\Gamma(TM) \times \Gamma(TM) \to \Gamma(TM)$ defined on homogeneous $X,Y\in \Gamma(TM)$ by
\begin{moneq}
T(X,Y) = \nabla_XY - (-1)^{\varepsilon(X)\cdot \varepsilon(Y)}\cdot \nabla_YX - [X,Y]
\end{moneq}
is even, graded anti-symmetric and bilinear over $C^\infty(M)${. In other words, $T$ is} a ``tensor'', \ie, can be seen as a section of the bundle $\bigwedge^2TM^* \otimes TM$, \ie, as a 2-form on $M$ with values in $TM$ \cite[IV\S5]{Tu1}.
\end{proclaim}
\begin{definition}{Definition}
A connection $\nabla$ in $TM$ is said to be {\emph{torsion-free}} if the tensor $T$ is identically zero.
\end{definition}
\begin{proclaim}{Corollary}
If $\nabla$ and $\nablah$ are {torsion-free} connections in $TM$, the tensor {$S = \nabla - \nablah: \Gamma(TM)\times \Gamma(TM)\to \Gamma(TM)$} is graded symmetric.
\end{proclaim}
{Let $\nabla$ be a connection in $TM$ (we also say a connection \emph{on} $M$). {On a local chart for $M$ with coordinates $x=(x^1, \dots, x^n)$ we define the Christoffel symbols $\Gamma_{jk}^i$ of $\nabla$ by
\begin{moneq}
\Gamma_{jk}^i(x)
=
\contrf{\nabla_{\partial_{x^j}}\partial_{x^k}}{\,\extder x^i\vert_x}
\end{moneq}
with parity $\varepsilon\bigl(\Gamma_{jk}^i(x)\bigr) = \varepsilon_i+\varepsilon_j+\varepsilon_k$.
It follows that for vector fields $X = \sum_i X^i\cdot \partial_{x^i}$ and $Y = \sum_i Y^i \cdot \partial_{x^i}$, we have
\begin{moneq}
\nabla_XY = \sum_{ij} X^j\cdot \fracp{Y^i}{x^j} \cdot \partial_{x^i}
+
\sum_{ijk} X^j \cdot \conjug^{\varepsilon_j}(Y^k)\cdot \Gamma_{jk}^i \cdot \partial_{x^i}
\end{moneq}
When the vector field $X$ is even, we have $\varepsilon(X^j) = \varepsilon_j$ and in that case the above formula can be written without signs as}
\begin{moneq}
\nabla_XY = \sum_{ij} X^j\cdot \fracp{Y^i}{x^j} \cdot \partial_{x^i}
+
\sum_{ijk} Y^k \cdot X^j \cdot \Gamma_{jk}^i \cdot \partial_{x^i}
\end{moneq}
}
\begin{proclaim}{Corollary}
If $\nabla$ and $\nablah$ are connections on $M$ with Christoffel symbols $\Gamma_{jk}^i$ and $\Gammah_{jk}^i$ respectively, the tensor $S$ reads locally as
\begin{moneq}
S =
\sum_{ijk}\extder x^k \otimes \extder x^j \cdot \Bigl( \Gamma_{jk}^i - \Gammah_{jk}^i \Bigr) \otimes \partial_{x^i}
\end{moneq}
while the tensor $T$ is given by
\begin{align*}
T
&=
\sum_{ijk} \extder x^k \wedge \extder x^j \cdot \Gamma_{jk}^i(x) \otimes \partial_{x^i}
\\&
=
\tfrac12\cdot \sum_{ijk} \extder x^k \wedge \extder x^j \cdot \bigl( \,\Gamma_{jk}^i - (-1)^{\varepsilon_j\varepsilon_k} \cdot \Gamma_{kj}^i \,\bigr) \otimes \partial_{x^i}
\end{align*}
In particular $\nabla$ is torsion-free if and only if the Christoffel symbols are graded symmetric in the lower indices, \ie, $\Gamma_{jk}^i = (-1)^{\varepsilon_j\varepsilon_k} \cdot \Gamma_{kj}^i$.
\end{proclaim}
If $y=(y^1, \dots, y^n)$ is another local system of coordinates, we can consider the Christoffel symbols $\Gammat_{jk}^i$ in terms of these coordinates:
\begin{moneq}
\Gammat_{jk}^i(y)
= \contrf{\nabla_{\partial_{y^j}}\partial_{y^k}}{\,\extder y^i\vert_y}
\end{moneq}
Now let $m\in M$ be the point in $M$ whose coordinates are $x$ or $y$ depending upon the choice of local coordinate system.
As tangent vectors transform as $\partial_{x^i}\vert_m = \sum_p (\partial_{x^i}y^p)(x)\cdot \partial_{y^p}\vert_m$, {it follows that the relation between $\Gamma$ and $\Gammat$ is given by}
\begin{multline}
\label{transChristoffel}
\quad
\sum_i \Gamma_{jk}^i(x) \cdot (\partial_{x^i}y^r)(x)
\\
=
(\partial_{x^j}\partial_{x^k}y^r)(x)
+
\sum_{s,t}
(-1)^{\varepsilon_j(\varepsilon_t+\varepsilon_k)} \cdot (\partial_{x^k}y^t)(x) \cdot (\partial_{x^j}y^s)(x) \cdot \Gammat_{st}^r(y)
\quad
\end{multline}
{Finally, let us consider $TM^{(0)}$ (the even part of the tangent bundle). With any local system of coordinates $x=(x^1, \dots, x^n)$ (resp. $y=(y^1, \dots, y^n)$) we associate the natural local system of coordinates $(x,v)$ (resp. $(y,w)$) on $TM^{(0)}$. More precisely, if $x$ are the coordinates of a point $m\in M$, then $(x,v)$ are the coordinates of the tangent vector $\tanvec = \sum_i v^i \cdot \partial_{x^i}\vert_m \in T_mM^{(0)}$. Now if $(x,v)$ and $(y,w)$ are the local coordinates of the same tangent vector $\tanvec$, \ie,
\begin{moneq}
\tanvec = \sum_i v^i \cdot \partial_{x^i}\vert_m = \sum_p w^p \cdot \partial_{y^p}\vert_m
\end{moneq}
then we have}
\begin{moneq}[changeoftangcoord]
w^p = \sum_i v^i\cdot (\partial_{x^i}y^p)(x)
\end{moneq}
{It follows that we have}
\begin{subequations}
\label{transeventangent}
\begin{align}
\label{transeventangentfirst}
\partial_{x^i}\vert_{\tanvec}
&=
\sum_p (\partial_{x^i}y^p)(x) \cdot \partial_{y^p}\vert_{\tanvec}
+
\sum_{jp} (-1)^{\varepsilon_i\varepsilon_j} v^j \cdot (\partial_{x^i}\partial_{x^j} y^p)(x) \cdot \partial_{w^p} \vert_{\tanvec}
\\
\label{transeventangentsecond}
\partial_{v^i}\vert_{\tanvec}
&=
\sum_p (\partial_{x^i}y^p)(x) \cdot \partial_{w^p}\vert_{\tanvec}
\end{align}
\end{subequations}
{With these preparations at hand, we now attack the question of defining geodesics.}
We start very na\"ively in local coordinates and copy the classical case: a geodesic is a map $\gamma : \CA_0\to M$ given in local coordinates by $\gamma(t) = (\gamma^1(t), \dots, \gamma^n(t))$ satisfying the equations
\begin{moneq}[naivegeodesicsequations]
\frac{\extder^2\gamma^i}{\extder t^2}(t)
=
-
\sum_{jk}
\frac{\extder\gamma^k}{\extder t}(t) \cdot \frac{\extder\gamma^j}{\extder t}(t) \cdot
\Gamma_{jk}^i(\gamma(t))
\end{moneq}
But to solve second order differential equations one needs initial conditions, which in our case are a starting point $x$ and an initial velocity $v$. And then the geodesic $\gamma$ depends upon these initial conditions, forcing us to write $\gamma_{(x,v)}$ instead of simply $\gamma$ and adding the initial conditions
\begin{moneq}
\gamma^i_{(x,v)}(0) = x^i
\qquad\text{and}\qquad
\frac{\extder\gamma^i_{(x,v)}}{\extder t}(0)
=
v^i
\end{moneq}
It is here that our definition deviates from the one given in \cite{GW}, as we look at maps defined on $\CA_0\times TM^{(0)}$ rather than on $\CA_0\times \CA_1$ or an arbitrary product $\CA_0\times S$.
We now recall that any system of second order differential equations on a manifold can be expressed as a system of first order differential equations on the tangent bundle. This means that we look at curves $\gammat_{(x,v)}:\CA_0\to TM^{(0)}$ given in local coordinates by
\begin{moneq}
\gammat_{(x,v)}(t) = (\gamma^1_{(x,v)}(t), \dots, \gamma^n_{(x,v)}(t), \gammab^1_{(x,v)}(t), \dots, \gammab^n_{(x,v)}(t))
\end{moneq}
satisfying the equations
\[
\left\{
\begin{array}{rcl}
\frac{\extder \gamma^i_{(x,v)}}{\extder t}(t)
&=&
\gammab^i_{(x,v)}(t)
\\[2\jot]
\frac{\extder \gammab^i_{(x,v)}}{\extder t}(t)
&=&
- \sum_{jk}\gammab^k_{(x,v)}(t) \cdot \gammab^j_{(x,v)}(t) \cdot \Gamma_{jk}^i(\gamma(t))
\end{array}
\right.
\]
and with initial conditions
\begin{moneq}
\gamma^i_{(x,v)}(0)
=
x^i
\qquad\text{and}\qquad
\gammab^i_{(x,v)}(0)
=
v^i
\end{moneq}
We now recognize that these are exactly the equations of the integral curves of a vector field on $TM^{(0)}$. And indeed, using the Christoffel symbols we can define a vector field $G$ on $TM^{(0)}$ in local coordinates $(x,v)$ by
\begin{moneq}[definitionofG]
G\vert_{\tanvec}
=
\sum_i v^i \partial_{x^i}\vert_{\tanvec}
-
\sum_{ijk}
v^k\cdot v^j\cdot\Gamma_{jk}^i(x)
\cdot \partial_{v^i}\vert_{\tanvec}
\end{moneq}
Combining \recalle{transChristoffel} and \recalle{transeventangent}, it is immediate that these local expressions glue together to form a well defined global vector field $G$ on $TM^{(0)}$. As it is an even vector field, it has a flow $\Psi$ defined in an open subset $W_G$ of $\CA_0 \times TM^{(0)}$ containing $\{0\}\times TM^{(0)}$ and with values in $TM^{(0)}$ \cite[V.4.9]{Tu1}. In local coordinates we will write $\Psi(t,x,v) = (\Psi_1(t,x,v), \Psi_2(t,x,v))$, where $\Psi_1=(\Psi_1^1, \dots, \Psi_1^n)$ represents the base point while $\Psi_2=(\Psi_2^1, \dots, \Psi_2^n)$ represents the tangent vector. By definition of a flow, these functions thus satisfy the equations
\[
\left\{
\begin{array}{rcl}
\fracp{\Psi_1^i}{t}(t,x,v)
&=&
\Psi_2^i(t,x,v)
\\[2\jot]
\fracp{\Psi_2^i}{t}(t,x,v)
&=&
-
\sum_{jk}
\Psi_2^k(t,x,v) \cdot \Psi_2^j(t,x,v) \cdot
\Gamma_{jk}^i(\Psi_1(t,x,v))
\end{array}
\right.
\]
together with the initial conditions
\begin{moneq}
\Psi_1(0,x,v) = x
\qquad\text{and}\qquad
\Psi_2(0,x,v) = v
\end{moneq}
{With the global vector field $G$ we thus have found an intrinsic coordinate free description of the equations we wrote for the geodesic curves $\gammat_{(x,v)}(t)$ and we are now in position to state a definition.}
\begin{definition}{Definition}
Let $\nabla$ be a connection in $TM$, let $\pi:TM^{(0)}\to M$ denote the canonical projection, let $G$ be the even vector field \recalle{definitionofG} and let $\Psi:W_G\to TM^{(0)}$ be its flow. For a fixed $(x,v)\cong\tanvec\in TM^{(0)}$ we will call the map $\gamma:\CA_0\to M$ defined by
\begin{moneq}
\gamma(t) = \pi\bigl(\Psi(t,\tanvec)\bigr) \cong \Psi_1(t,x,v)
\end{moneq}
{\em the geodesic through $x\in M$ with initial velocity $v$.} Note that if $\tanvec$ is not in the body of $TM^{(0)}$, this curve is not (necessarily) smooth (see \cite[III.1.23g, V.3.19]{Tu1}).
\end{definition}
\begin{definition}{Remark}
One could define a similar vector field on $TM^{(1)}$, the odd part of the tangent bundle. More precisely, we denote by $(x,\vb)$ local coordinates on $TM^{(1)}$, where $(x,\vb)$ represents the tangent vector $\mathcal{V} = \sum_i \vb^i\cdot \partial_{x^i}\vert_m$, but the parity of $\vb^i$ is reversed: $\varepsilon(\vb^i) = \varepsilon_i+1 = \varepsilon(x^i)+1$. It thus is an odd tangent vector. These coordinates still change according to \recalle{changeoftangcoord} (with $v$ replaced by $\vb$), but an additional sign appears in the transformation of the tangent vectors: \recalle{transeventangentfirst} is replaced by
\begin{subequations}
\label{transeventangentodd}
\begin{align}
\label{transeventangentoddfirst}
\partial_{x^i}\vert_{\tanvec}
&=
\sum_p (\partial_{x^i}y^p)(x) \cdot \partial_{y^p}\vert_{\tanvec}
+
\sum_{jp} (-1)^{\varepsilon_i(\varepsilon_j+1)} \vb^j \cdot (\partial_{x^i}\partial_{x^j} y^p)(x) \cdot \partial_{\overline w^p} \vert_{\tanvec}
\end{align}
\end{subequations}
The analogon of the vector field $G$ on $TM^{(0)}$ would be the odd vector field $G'$ on $TM^{(1)}$ defined in local coordinates as
\begin{moneq}
G'\vert_{\tanvec}
=
\sum_i \vb^i \partial_{x^i}\vert_{\tanvec}
-
\sum_{ijk}
(-1)^{\varepsilon_k} \cdot \vb^k\cdot \vb^j\cdot\Gamma_{jk}^i(x)
\cdot \partial_{\vb^i}\vert_{\tanvec}
\end{moneq}
The transformation properties \recalle{transChristoffel}, \recalle{transeventangentsecond} and \recalle{transeventangentoddfirst} ensure that $G'$ is a well defined global vector field. However, the condition for an odd vector field to be integrable (with an odd time parameter $\tau$) is that its auto-commutator is zero \cite[V.4.17]{Tu1}. But the auto-commutator $[G',G']$ is given by
\begin{align*}
{[G',G']}
&=
-2\cdot \sum_{ijk} (-1)^{\varepsilon_k}\cdot \vb^k \cdot \vb^j \cdot \Gamma_{jk}^i(x) \cdot \partial_{x^i}
+
\text{ terms in $\partial_{\vb^i}$}
\\&
=
-\sum_{ijk} (-1)^{\varepsilon_k}\cdot \vb^k \cdot \vb^j \cdot (\Gamma_{jk}^i(x) - (-1)^{\varepsilon_j\varepsilon_k}\cdot \Gamma_{jk}^i(x)) \cdot \partial_{x^i}
+
\text{ terms in $\partial_{\vb^i}$}
\end{align*}
If this is to be zero, then at least the coefficients of $\partial_{x^i}$ have to be zero. But this is the case if and only if the connection $\nabla$ is torsion-free (on the odd tangent bundle, the combination $(-1)^{\varepsilon_k} \cdot \vb^k \cdot \vb^j$ is graded anti-symmetric). Moreover, if this is the case, then the vector field $G'$ reduces to $G' = \sum_i \vb^i \partial_{x^i}$, of which the auto-commutator indeed is zero (hence we don't have to compute the coefficients of $\partial_{\vb^i}$). But for this vector field the flow $\Phi'$ is given by:
\begin{moneq}
\Phi'(\tau,x,\vb) = (x+\tau\cdot \vb, \vb)
\end{moneq}
which is rather uninteresting: the ``odd geodesics'' are ``straight odd lines'' in the direction of the tangent vector. Another way to see that this must happen is {the following set of observations. If we use an odd time parameter $\tau$, it follows immediately that the velocity vector should be an odd tangent vector. Moreover, when we write the naïve equations \recalle{naivegeodesicsequations} for the geodesics, the left hand side is identically zero because $\partial_\tau \scirc \partial_\tau = 0$. And then this equation tells us that the connection should be torsion-free. We are thus left with the condition that the connection should be torsion-free, together with the initial conditions $\gamma(0,x,\vb) = x$ and $\partial_\tau\gamma(0,x,\vb) = \vb$. And these give us our straight odd lines.}
\end{definition}
\section{Projective equivalence}\label{proj}
We now consider the situation in which we have two connections $\nabla, \nablah$ on $M$ and we wonder under what conditions these two connections have ``the same'' geodesics as images in $M$. More precisely, if $\Psi(t,\tanvec)$ and $\Psih(t,\tanvec)$ are the geodesic flows for $\nabla$ and $\hat{\nabla}$ respectively, the na\"ive question is under what conditions we have
\begin{moneq}
\{ \, \Psi_1(t,\tanvec) : t\in \CA_0 \,\}
=
\{ \, \Psih_1(t,\tanvec) : t\in \CA_0 \,\}
\end{moneq}
A more precise question is under what conditions we can find a reparametrization function $r:\CA_0 \times TM \to \CA_0$ such that we have
\begin{moneq}[reparamPsi1]
\forall t\in \CA_0\quad:\quad
\Psi_1(r(t,\tanvec), \tanvec) = \Psih_1(t,\tanvec)
\end{moneq}
Note that we added an explicit dependence on the initial condition $\tanvec$ in the repara\-me\-trization function $r$, as there is no reason that geodesics through different points should be reparametrized in the same way.
\begin{definition}{Definition}
We say that the connections $\nabla$ and $\nablah$ have \emph{the same geodesics up to reparametrization} if there exists a function $r : \CA_0 \times TM \to \CA_0$ such that $r(0,\tanvec) = 0$, $(\partial r/\partial t)(0,\tanvec) = 1$ and for which equation (\ref{reparamPsi1}) holds.\footnote{The additional conditions $r(0,\tanvec) = 0$ and $(\partial r/\partial t)(0,\tanvec) = 1$ ensure that the reparametrization transforms each geodesic of $\nabla$ into the geodesic of $\nablah$ with the same initial conditions.}
\end{definition}
We are going to characterize the connections that
have the same geodesics up to reparametrization in terms of the form of the tensor $S$ which measures the difference between these two connections. In order to do that, we are going to proceed in two steps. First, we show that (\ref{reparamPsi1}) holds if and only if the geodesic flow $\Psi$ of $G$, the (difference) tensor $S=\nabla - \nablah$ and the reparametrization function $r$ are related through a certain differential equation.
\begin{proclaim}{Proposition}
The connections $\nabla$ and $\hat{\nabla}$ have the same geodesics up to reparametrization if and only if there exists a function $r:\CA_0 \times TM \to \CA_0$ such that $r(0,\tanvec) = 0$, $(\partial r/\partial t)(0,\tanvec) = 1$ and for which the following differential equation holds:
\begin{multline}
\label{reparametrization}
\frac{\partial^2r}{\partial t^2}(t,\tanvec) \cdot\frac{\partial\Psi_1}{\partial t}(r(t,\tanvec),\tanvec)
\\
=
\Bigl(\fracp{r}{t}(t,\tanvec)\Bigr)^2 \cdot
S_{\Psi_1(r(t,\tanvec),\tanvec)}\Bigl(\,\frac{\partial\Psi_1}{\partial t}(r(t,\tanvec),\tanvec)\,,\, \frac{\partial\Psi_1}{\partial t}(r(t,\tanvec),\tanvec) \,\Bigr)
\end{multline}
\end{proclaim}
\begin{preuve}
Let us show that the condition is necessary. In view of (\ref{naivegeodesicsequations}), if $\Psi_1(r(t,\tanvec), \tanvec)$ is a geodesic for $\hat{\nabla}$, then
\begin{moneq}
0 = \frac{\partial^2\Psi_1^i(r(t,\tanvec),\tanvec)}{\partial t^2}
+ \sum_{j,k}
\frac{\partial\Psi_1^k(r(t,\tanvec),\tanvec)}{\partial t} \cdot \frac{\partial\Psi_1^j(r(t,\tanvec),\tanvec)}{\partial t}\cdot
\hat{\Gamma}_{jk}^i(\Psi_1(r(t,\tanvec), \tanvec))
\end{moneq}
Let us replace in this equation $\hat{\Gamma}_{jk}^{i}$ by $\Gamma_{jk}^{i}-S_{jk}^{i}$ and let us apply the chain rule to compute the derivatives of the functions $\Psi_1^i(r(t,\tanvec),\tanvec)$. Doing so, we obtain
\begin{eqnarray*}
0
&=&
\frac{\partial^2r}{\partial t^2}(t,\tanvec) \cdot\frac{\partial\Psi_1}{\partial t}(r(t,\tanvec),\tanvec)
+ \left(\frac {\partial r}{\partial t}(t,\tanvec)\right)^{2} \left(\frac{\partial^2\Psi_1^i}{\partial t^2}(r(t,\tanvec),\tanvec)\right)
\\
&&+ \left(\frac {\partial r}{\partial t}(t,\tanvec)\right)^{2} \left(\sum_{j,k}
\frac{\partial\Psi_1^k}{\partial t}(r(t,\tanvec),\tanvec) \cdot \frac{\partial\Psi_1^j}{\partial t}(r(t,\tanvec),\tanvec)\cdot\Gamma_{jk}^i(\Psi_1(r(t,\tanvec), \tanvec))\right)
\\
&&-\Bigl(\fracp{r}{t}(t,\tanvec)\Bigr)^2 \left( \sum_{j,k}
\frac{\partial\Psi_1^k}{\partial t}(r(t,\tanvec),\tanvec) \cdot \frac{\partial\Psi_1^j}{\partial t}(r(t,\tanvec),\tanvec)\cdot S_{jk}^i(\Psi_1(r(t,\tanvec), \tanvec))\right)
\end{eqnarray*}
Using the fact that $\Psi_1$ is a geodesic for $\nabla$,
the second and third term on the right hand side cancel and hence
this equation reduces to (\ref{reparametrization}).
In order to show the converse, it suffices to note that the above computations also show that if \eqref{reparametrization} is satisfied, then the curve
\[\left(\Psi_1(r(t,\tanvec), \tanvec),\frac {\partial r}{\partial t}(t,\tanvec)\cdot\frac{\partial\Psi_1}{\partial t}(r(t,\tanvec),\tanvec)\right)
\]
satisfies the equation of the flow $(\Psih_{1}(t,\tanvec),\Psih_{2}(t,\tanvec))$ of $\hat{G}$, the geodesic vector field corresponding to $\hat{\nabla}$. As it satisfies the same initial conditions as $(\Psih_{1}(t,\tanvec),\Psih_{2}(t,\tanvec))$ at $t=0$, these two curves have to coincide, and in particular
$\Psi_1(r(t,\tanvec), \tanvec) = \Psih_1(t,\tanvec)$.
\end{preuve}
Now in order to obtain Weyl's characterization in the super context, it remains to show that condition (\ref{reparametrization}) amounts to imposing that $S$ can be expressed by means of an even (super) $1$-form. As for the previous Proposition, the proof of the theorem follows the lines of the classical case. It invokes a technical Lemma which roughly says that if we have a bilinear function $S(v,w)$ such that $S(v,v)=h(v)\cdot v$ for some function $h$, then $h$ must be linear in $v$. The proof of this technical Lemma is elementary but long, simply because we have to be careful with the odd coordinates and moreover, everything depends upon additional parameters (the local coordinates $x$ and $\xi$ on $M$). Therefore the proof of the lemma will be given after that of the Theorem.
\begin{proclaim}[linearizationlemma]{Lemma}
Let $E$ be a graded vector space of graded dimension $p\vert q$ with even basis vectors $e_1, \dots, e_p$ and odd basis vectors $f_1, \dots, f_q$, let $U$ be an open coordinate subset of a manifold $M$ with local even coordinates $x$ and local odd coordinates $\xi$. Suppose that $S:U\times E\times E \to E$ is a smooth function which is left-bilinear, graded symmetric in the product $E\times E$ and for which there is a smooth function $h:U\times E_0\to \CA$ such that
\begin{moneq}[determiningoneform]
\forall (x,\xi)\in U\ \forall v\in E_0 :
S(x,\xi,v,v) = h(x,\xi,v) \cdot v
\end{moneq}
Then there exists a unique smooth function $\alpha:U\to E^*$ such that $h(x,\xi,v) = \contrf{v}{\alpha(x,\xi)}$ and
\begin{moneq}
S(x,\xi,v,w) = \tfrac12 \cdot \bigl(\, \contrf{v}{\alpha(x,\xi)}\cdot w + \contrf{w}{\alpha(x,\xi)}\cdot v \,\bigr)
\end{moneq}
\end{proclaim}
\begin{proclaim}[conformalequivalenceconnections]{Theorem}
Two torsion-free connections $\nabla$ and $\nablah$ on $M$ have the same geodesics up to reparametrization if and only if there exists a smooth even $1$-form $\alpha$ on $M$ such that the tensor $S = \nabla - \nablah$ is given by
\begin{moneq}[Sintermsofalpha]
S_x(v,w) = \tfrac12\cdot ( \contrf{v}\alpha_x \cdot w + (-1)^{\varepsilon(v) \cdot \varepsilon(w)} \cdot \contrf w\alpha_x \cdot v )
\end{moneq}
for any $x \in M$ and any homogeneous $v,w \in T_xM$.
\end{proclaim}
\begin{preuve}[Proof of the theorem]
We first assume that we have a reparametrization $r$ that transforms the geodesics of $\nabla$ into those of $\nablah$. Taking $t=0$ in \recalle{reparametrization} and using the initial conditions for $\Psi$ and $r$, we get the following (vector) equation in local coordinates:
\begin{moneq}[reduceddefeqforreparametrization]
v \cdot \frac{\partial^2 r}{\partial t^2}(0,x,v)
=
S_x(v,v)
\end{moneq}
Lemma \ref{linearizationlemma}, with $h$ being here the function $h(x,v) = \frac{\partial^2r}{\partial t^2}(0,x,v)$, gives us a (local) smooth $1$-form $\alpha$, which must be even by parity considerations. But \recalle{reduceddefeqforreparametrization} is an intrinsic equation which does not depend upon the choice of local coordinates (because \recalle{reparametrization} is intrinsic). As the $1$-form $\alpha$ is unique,
the local $1$-forms $\alpha$ given by Lemma \ref{linearizationlemma} glue together to form a global smooth even $1$-form $\alpha$ satisfying \recalle{Sintermsofalpha}.
To show the converse, let us now assume that we have an even $1$-form $\alpha$ on $M$ such that the tensor $S$ is given by \recalle{Sintermsofalpha}. Then \recalle{reparametrization} reduces to the (vector) equation
\begin{multline}
\label{defeqforreparametrizationgivenS}
\frac{\partial^2 r}{\partial t^2}(t,x,v) \cdot \frac{\partial\Psi_1}{\partial t}(r(t,x,v),x,v)
\\
=
\Bigl( \fracp rt(t,x,v)\Bigr)^2 \cdot
\iota\left( \frac{\partial\Psi_1}{\partial t}(r(t,x,v),x,v)\right){\alpha_{\Psi_1(r(t,x,v),x,v)}} \cdot \frac{\partial\Psi_1}{\partial t}(r(t,x,v),x,v)
\end{multline}
For this to be true for all geodesics of $\nabla$, the function $r$ thus has to satisfy the second order differential equation
\begin{align*}
\frac{\partial^2 r}{\partial t^2}(t,x,v)
&=
\Bigl( \fracp rt(t,x,v)\Bigr)^2 \cdot \iota \left(\frac{\partial\Psi_1}{\partial t}(r(t,x,v),x,v)\right){\alpha_{\Psi_1(r(t,x,v),x,v)}}
\end{align*}
As for the geodesic equations, we translate this into a system of first order differential equations by introducing a second function $s:\CA_0\times TM^{(0)}\to \CA_0$ and we obtain
\[
\left\{
\begin{array}{rcl}
\fracp{r}{t}(t,x,v)
&=&
s(t,x,v)
\\[2\jot]
\fracp{s}{t}(t,x,v)
&=&
s(t,x,v)^2 \cdot \iota \left(\frac{\partial\Psi_1}{\partial t}(r(t,x,v),x,v)\right){\alpha_{\Psi_1(r(t,x,v),x,v)}}
\end{array}
\right.
\]
while the initial conditions for $r$ yield $r(0,x,v) = 0$ and $s(0,x,v) = 1$. To show that these equations always have a (unique) solution, we just note that these equations determine the flow of the even vector field $R$ on $(\CA_0)^2 \times TM^{(0)}$ given by
\begin{moneq}
R\vert_{(r,s,\tanvec)}
=
s\cdot \fracp{}{r}
+
s^2 \cdot \iota\left({\frac{\partial\Psi_1}{\partial t}(r,\tanvec)}\right){\alpha_{\Psi_1(r,\tanvec)}} \cdot \fracp{}{s}
\end{moneq}
And indeed, the equations for the flow $\Phi= (\Phi_r, \Phi_s, \Phi_1, \Phi_2)$ of $R$ are given by
\[
\left\{
\begin{array}{rcl}
\fracp{\Phi_r}{t}(t,r_o,s_o,x,v)
&=&
\Phi_s(t,r_o,s_o,x,v)
\\[2\jot]
\fracp{\Phi_s}{t}(t,r_o,s_o,x,v)
&=&
(\Phi_s(t,r_o,s_o,x,v))^2
\\&&\qquad
\cdot \iota\left({\frac{\partial\Psi_1}{\partial t}(\Phi_r(t,r_o,s_o,x,v),x,v)}\right) {\alpha_{\Psi_1(\Phi_r(t,r_o,s_o,x,v),x,v)}}
\\[2\jot]
\fracp{\Phi_1}{t}(t,r_o,s_o,x,v)
&=&
0
\\[2\jot]
\fracp{\Phi_2}{t}(t,r_o,s_o,x,v)
&=&
0
\end{array}
\right.
\]
It thus suffices to define $r(t,\tanvec) = \Phi_r(t,0,1,\tanvec)$ and $s(t,\tanvec) = \Phi_s(t,0,1,\tanvec)$ to obtain the desired functions.
\end{preuve}
\begin{preuve}[Proof of the lemma]
Uniqueness of $\alpha$ follows from the equation $h(x,\xi,v) = \contrf{v}{\alpha(x,\xi)}$. To prove existence, let us start by introducing global (linear, left) coordinates $y,\eta$ on $E_0$ by
\begin{moneq}
v\in E_0 \quad\Rightarrow\quad
v=\sum_i y_i \cdot e_i + \sum_i \eta_i \cdot f_i
\end{moneq}
Using bilinearity and graded symmetry, we thus can write
\begin{multline*}
S(x,\xi,v,v)
=
\sum_{i,j} y_i y_j \cdot S(x,\xi,e_i,e_j)
\\
+
2\sum_{i,j} y_i \eta_j \cdot S(x,\xi,e_i,f_j)
+
\sum_{i,j} \eta_j\eta_i \cdot S(x,\xi,f_i,f_j)
\end{multline*}
The functions $S$, when evaluated in a pair of basis vectors of $E$, is a smooth function on $U$ with values in $E$. As such we can determine the coefficients with respect to the given basis for $E$ as for instance
\begin{moneq}
S(x,\xi,e_i,e_j)
=
\sum_p S_p(x,\xi,e_i,e_j) \cdot e_p
+
\sum_p \sigma_p(x,\xi,e_i,e_j) \cdot f_p
\end{moneq}
When we substitute this in \recalle{determiningoneform} with the (linear, left) coordinates of $v\in E_0$, we get the system of equations
\begin{align*}
h(x,\xi,y,\eta) \cdot y_p
&=
\sum_{i,j} y_i y_j \cdot S_p(x,\xi,e_i,e_j)
\\&
\qquad+
2\sum_{i,j} y_i \eta_j \cdot S_p(x,\xi,e_i,f_j)
+
\sum_{i,j} \eta_j\eta_i \cdot S_p(x,\xi,f_i,f_j)
\\
h(x,\xi,y,\eta) \cdot \eta_p
&=
\sum_{i,j} y_i y_j \cdot \sigma_p(x,\xi,e_i,e_j)
\\&
\qquad+
2\sum_{i,j} y_i \eta_j \cdot \sigma_p(x,\xi,e_i,f_j)
+
\sum_{i,j} \eta_j\eta_i \cdot \sigma_p(x,\xi,f_i,f_j)
\end{align*}
Applying \recallp{expandinginoddpowers} we can expand these equations in powers of the $\xi$ coordinates and equate the separate powers $\xi^J$ giving
\begin{align}
h_J(x,y,\eta) \cdot y_p
&=
\sum_{i,j} y_i y_j \cdot S_{p,J}(x,e_i,e_j)
\label{evenbasisequation}
+
2\sum_{i,j} y_i \eta_j \cdot (-1)^{\vert J\vert} \cdot S_{p,J}(x,e_i,f_j)
\\&\qquad
+
\sum_{i,j} \eta_j\eta_i \cdot S_{p,J}(x,f_i,f_j)
\notag
\\
h_J(x,y,\eta) \cdot \eta_p
&=
\sum_{i,j} y_i y_j \cdot \sigma_{p,J}(x,e_i,e_j)
\label{oddbasisequation}
+
2\sum_{i,j} y_i \eta_j \cdot (-1)^{\vert J\vert} \cdot \sigma_{p,J}(x,e_i,f_j)
\\&\qquad
+
\sum_{i,j} \eta_j\eta_i \cdot \sigma_{p,J}(x,f_i,f_j)
\notag
\end{align}
Note that we had to add a factor $(-1)^{\vert J\vert}$ in the right hand side for the terms linear in $\eta$, because we factor the powers of $\xi$ to the left, and interchanging a power $\xi^J$ with a linear factor $\eta$ gives this sign.
We now expand the functions $h_J$ in powers of the odd coordinates $\eta$:
\begin{multline*}
h_J(x,y,\eta) = h_{J,\emptyset}(x,y) + \sum_q \eta_q \cdot h_{J,\{q\}}(x,y) \\
+ \sum_{q<r} \eta_q\eta_r \cdot h_{J,\{q,r\}}(x,y) + \sum_{I, \vert I\vert \ge 3} \eta^I \cdot h_{J,I}(x,y)
\end{multline*}
When we now invoke \recallp{expandinginoddpowers} applied to \recalle{evenbasisequation}, we get the equations
\begin{align}
h_{J,\emptyset}(x,y) \cdot y_p
&=
\sum_{i,j} y_iy_j \cdot S_{p,J}(x,e_i,e_j)
\label{zerothordereven}
\\
h_{J,\{q\}}(x,y) \cdot y_p
&=
2\sum_i y_i \cdot (-1)^{\vert J\vert} \cdot S_{p,J}(x,e_i,f_q)
\label{firstordereven}
\\
h_{J,\{q,r\}}(x,y) \cdot y_p
&=
2S_{p,J}(x,f_r,f_q)
\label{secondordereven}
\\
h_{J,I}(x,y) \cdot y_p
&=
0 \qquad \vert I\vert \ge 3
\label{thirdordereven}
\end{align}
As these are equations between smooth functions of even coordinates only, we may consider them to be equations of smooth functions of real coordinates. And remember, the $y$ coordinates run over the whole of $\RR$ as they are coordinates on a (graded) vector space. These functions thus are in particuler smooth at $y=0$.
As the right hand sides of \recalle{secondordereven} and \recalle{thirdordereven} do not depend upon the $y$ coordinates and their left hand sides have at least degree one in $y$, it follows that the coefficients must be zero, and thus the right hand side of \recalle{secondordereven} too:
\begin{moneq}
h_{J,I}(x,y) = 0 \text{ for } \vert I\vert\ge3
\quad,\quad
h_{J,\{q,r\}}(x,y) = 0
\quad,\quad
S_{p,J}(x,f_r,f_q) = 0
\end{moneq}
From \recalle{firstordereven} it follows easily that $h_{J,\{q\}}(x,y)$ is independent of the $y$ coordinates:
\begin{moneq}
h_{J,\{q\}}(x,y) = h_{J,\{q\}}(x)
\end{moneq}
and that we must have
\begin{moneq}
(-1)^{\vert J\vert} \cdot S_{p,J}(x,e_i,f_q)
=
\tfrac12\cdot
\delta_{ip}\cdot h_{J,\{q\}}(x)
\end{moneq}
Using the bilinearity of $S$, one can show that \recalle{zerothordereven} implies that $h_{J,\emptyset}(x,y)$ must be linear in $y$:
\begin{moneq}
h_{J,\emptyset}(x,y)
=
\sum_q h_{J,\emptyset}^q(x)\cdot y_q
\end{moneq}
and then that we must have
\begin{moneq}
S_{p,J}(x,e_i,e_j)
=
\tfrac12\cdot \Bigl( \delta_{ip} \cdot h_{J,\emptyset}^j(x) + \delta_{jp} \cdot h_{J,\emptyset}^i(x) \Bigr)
\end{moneq}
We now apply exactly the same reasoning to \recalle{oddbasisequation}, equating the separate powers of $\eta$ and using what we already know about the functions $h_{J,I}(x,y)$. This gives us the equations
\begin{align}
0
&=
\sum_{i,j} y_iy_j \cdot \sigma_{p,J}(x,e_i,e_j)
\label{sigmabothes}
\\
\sum_q y_q \cdot h_{J,\emptyset}^q(x)
&=
2\sum_i y_i \cdot (-1)^{\vert J\vert} \cdot \sigma_{p,J}(x,e_i,f_p)
\label{sigmaeandf}
\\
\tfrac12 \cdot \Bigl( h_{J,\{j\}}(x)\cdot \delta_{ip} - h_{J,\{i\}}(x)\cdot \delta_{jp} \Bigr)
&=
\sigma_{p,J}(x,f_i, f_j)
\end{align}
As these are (again) equations between smooth functions of real variables, we may conclude from \recalle{sigmabothes} that we have $\sigma_{p,J}(x,e_i,e_j)=0$ and from \recalle{sigmaeandf} that we have $\tfrac12 \cdot h_{J,\emptyset}^i(x) \cdot \delta_{jp} = (-1)^{\vert J\vert} \cdot \sigma_{p,J}(x,e_i,f_j)$.
To summarize, we have found the following equalities
\begin{align*}
h_J(x,y,\eta)
&=
\sum_q y_q \cdot h_{J,\emptyset}^q(x) + \sum_q \eta_q \cdot h_{J,\{q\}}(x)
\\
S_{p,J}(x,e_i,e_j)
&=
\tfrac12\cdot \Bigl( \delta_{ip} \cdot h_{J,\emptyset}^j(x) + \delta_{jp} \cdot h_{J,\emptyset}^i(x) \Bigr)
\\
S_{p,J}(x,e_i,f_j)
&=
\tfrac12\cdot(-1)^{\vert J\vert} \cdot
\delta_{ip}\cdot h_{J,\{j\}}(x)
\\
S_{p,J}(x,f_i,f_j)
&=
0
\\
\sigma_{p,J}(x,e_i,e_j)
&=
0
\\
\sigma_{p,J}(x,e_i,f_j)
&=
\tfrac12 (-1)^{\vert J\vert} \cdot \cdot h_{J,\emptyset}^i(x) \cdot \delta_{jp}
\\
\sigma_{p,J}(x,f_i,f_j)
&=
\tfrac12 \cdot \Bigl( h_{J,\{j\}}(x)\cdot \delta_{ip} - h_{J,\{i\}}(x)\cdot \delta_{jp} \Bigr)
\end{align*}
We now define the smooth functions $H_\emptyset^q, H_{\{q\}}:U\to \CA$ by
\begin{moneq}
H_\emptyset^q(x,\xi)
=
\sum_J \xi^J \cdot h_{J,\emptyset}^q(x)
\quad,\quad
H_{\{q\}}(x,\xi)
=
\sum_J \xi^J \cdot (-1)^{\vert J\vert} \cdot h_{J,\{q\}}(x)
\end{moneq}
Using these functions, we now put the powers of $\xi$ back in to obtain
{
\begin{align*}
h(x, \xi ,y,\eta)
&=
\sum_J \xi^J \cdot \Biggl( \sum_q y_q \cdot h_{J,\emptyset}^q(x) + \sum_q \eta_q \cdot h_{J,\{q\}}(x) \Biggr)
\\&
=
\sum_q y_q \cdot \sum_J \xi^J \cdot h_{J,\emptyset}^q(x) + \sum_q \eta_q \cdot \sum_J \xi^J \cdot (-1)^{\vert J\vert} \cdot h_{J,\{q\}}(x)
\\&
=
\sum_q y_q \cdot H_{\emptyset}^q(x,\xi) + \sum_q \eta_q \cdot H_{\{q\}}(x,\xi)
\\
S_{p}(x,\xi,e_i,e_j)
&=
\sum_J \xi^J \cdot S_{p,J}(x,e_i,e_j)
=
\tfrac12\cdot \Bigl( \delta_{ip} \cdot H_{\emptyset}^j(x,\xi) + \delta_{jp} \cdot H_{\emptyset}^i(x,\xi) \Bigr)
\\
S_{p}(x,\xi,e_i,f_j)
&=
\sum_J \xi^J \cdot S_{p,J}(x,e_i,f_j)
=
\tfrac12\cdot \delta_{ip}\cdot H_{\{j\}}(x,\xi)
\\
S_{p}(x,\xi,f_i,f_j)
&=
0
=
\sigma_{p}(x,\xi,e_i,e_j)
\\
\rho \cdot \sigma_{p}(x,\xi,e_i,f_j)
&=
\rho \cdot \sum_J \xi^J \cdot \sigma_{p,J}(x,e_i,f_j)
=
\tfrac12 H_{\emptyset}^i(x,\xi) \cdot \delta_{jp} \cdot \rho
\\
\rho \cdot \sigma_{p}(x,\xi,f_i,f_j)
&=
\rho \cdot \sum_J \xi^J \cdot \sigma_{p,J}(x,f_i,f_j)
\\&
=
\tfrac12 \cdot \Bigl( H_{\{j\}}(x,\xi)\cdot \delta_{ip} - H_{\{i\}}(x,\xi)\cdot \delta_{jp} \Bigr) \cdot \rho
\end{align*}}
where $\rho$ is any odd variable. Finally, we can reconstruct the full function $S$: if $v$ reads as $\sum_i y_i e_i + \sum_i \eta_i f_i$ and $w$ reads as $\sum_j z_j e_j, + \sum_j \zeta_j f_j$, then direct substitution gives us
\begin{align*}
S(x,\xi,v,w)
&=
\tfrac12 \Bigl(
\sum_j z_j \cdot H_{\emptyset}^j(x,\xi) + \sum_j \zeta_j H_{\{j\}}(x,\xi)
\Bigr) \cdot v
\\&\qquad
+
\tfrac12 \Bigl(
\sum_j y_j \cdot H_{\emptyset}^j(x,\xi) + \sum_j \eta_j H_{\{j\}}(x,\xi)
\Bigr) \cdot w
\end{align*}
This suggests that we introduce the left-linear form $\alpha:U\to E^*$ by
\begin{moneq}
\contrf{v}{\alpha(x,\xi)}
=
\contrf{\sum_i y_i e_i + \sum_i \eta_i f_i}{\alpha(x,\xi)}
=
\sum_i y_i \cdot H_{\emptyset}^i(x,\xi) + \sum_i \eta_i \cdot H_{\{i\}}(x,\xi)
\end{moneq}
where $y_i,\eta_i$ are arbitrary (non-homogeneous) coefficients. It then follows immediately that we have
\begin{moneq}
S(x,\xi,v,w)
=
\tfrac12 \cdot \bigl(\, \contrf{w}{\alpha(x,\xi)}\cdot v + \contrf{v}{\alpha(x,\xi)} \cdot w \,\bigr)
\end{moneq}
It also follows that we have
\begin{moneq}
h(x,\xi,y,\eta)
=
\contrf{v}{\alpha(x,\xi)}
\end{moneq}
confirming the equation $S(x,\xi,v,v) = h(x,\xi,v) \cdot v$ for even vectors $v$.
\end{preuve}
\section{Super metrics and connections}\label{Smetricconnections}
As in non-super geometry, connections on the tangent bundle arise naturally when the supermanifold is equipped with a metric. Moreover, again as in non-super geometry, geodesics in this context can be interpreted as the trajectories on the supermanifold of a free particle whose kinetic energy is given by the metric. We now substantiate these claims.
More precisely, we shall first expose some basic theory of super metrics and their associated Levi-Civita (super) connections. Then we shall briefly describe the mechanics of a free particle whose kinetic energy is given by the metric and finally, following \cite{GW}, we shall relate the Hamiltonian vector field of this mechanical system to the geodesic vector field of the corresponding \emph{metric connection}.
\begin{definition}{Definition}
A (super) \emph{metric} $g$ on a supermanifold $M$ is an even graded symmetric non-degenerate smooth section of the bundle $T^*M\otimes T^*M \to M$.
A \emph{Riemannian supermanifold} is a pair $(M,g)$ with $M$ a supermanifold and $g$ a metric on $M$.
\end{definition}
A metric $g$ on $M$ amounts to a collection of maps $g_m:T_mM\times T_mM \to \CA$ (depending smoothly on $m\in M$) possessing
the following four properties:
\begin{itemize}
\item
The map $(v,w)\mapsto
\contrf{v,w}{g_m}$
is (left-)bilinear in $v$ and $w\,$;
\footnote{Since the map $g_m$ is supposed to be even, we could also have written $g_m(v,w)$ instead of $\contrf{v,w}{g_m}$. However, once we express $g_m$ in terms of the left-dual basis $dx^i$, there is a high risk of confusion on how to compute evaluations, as we have $(dx^j)(\partial_{x^i}) = (-1)^{\varepsilon_{x^i}} \delta^j_i$, and not (as one might be inclined to think) $(dx^j)(\partial_i) = \delta^j_i$, simply because we have (by definition of the left-dual basis): $\delta^j_i=\contrf{\partial_{x^i}}{dx^j} = (-1)^{\varepsilon_i\varepsilon_j}\, (dx^j)(\partial_{x^i})$.}
\item
for all homogeneous $v,w\in T_mM$~: $\varepsilon(\contrf{v,w}{g_m}) = \varepsilon(v) + \varepsilon(w)$;
\item
for all homogeneous $v,w\in T_m$~:
$\contrf{w,v}{g_m} = (-1)^{\varepsilon(v)\varepsilon(w)} \, \contrf{v,w}{g_m}$.
\end{itemize}
Now for each $m \in M$, the map $g_m$ can be seen as transforming tangent vectors into cotangent vectors, \ie, we can define a map $g_m^\flat : T_mM \to T_m^*M$ by setting
\begin{moneq}
\contrf{v}{g_m^\flat} = \contrf{v}{g_m} = \contrf{\cdot,v}{g_m}
\quad\text{\ie,}\quad
\contrf{w}{\bigl(\contrf{v}{g_m^\flat}\bigr)} = \contrf{w}{\bigl(\contrf{v}{g_m}\bigr)}
\equiv
\contrf{w,v}{g_m}
\end{moneq}
With this definition we can state the the fourth condition
\begin{itemize}
\item $g_m^\flat : T_mM \to T_m^*M$ is a (left-)linear bijection.
\end{itemize}
The collection of all maps $g_m^\flat$ gives rise to an even bundle isomorphism $g^\flat : TM \to T^*M$, whose inverse is denoted by $g^\sharp : T^*M \to TM$. As usual, the use of the musical superscripts is inspired by the fact that $g^\flat$ lowers indices of tensors, wheras $g^\sharp$ raises them.
\begin{definition}{Remark}
As it is well known, if $(M,g)$ is a Riemannian supermanifold of graded dimension $p|q$, then the odd dimension $q$ must be even because of the non-degeneracy condition of the super metric.
Note that the definition of a super metric as given here is the straightforward generalisation of a metric to the super context. In \cite[\S IV.7]{Tu1} a different (and not completely natural) notion of a super metric was introduced. That definition was adapted to the need to be able to define a supplement to any subbundle of a given vector bundle without the constraint that the odd dimension should be even.
\end{definition}
If $(x^1, \dots, x^n)$ are local coordinates on $M$, then the vectors $\partial_{x^i}\vert_m$ form a basis of the tangent space $T_mM$.
Using these vectors, we define the matrix $g_{ij}$ by
\begin{moneq}
g_{ij} = \contrf{\partial_{x^i}\vert_m,\partial_{x^j}\vert_m}{g_m}
\end{moneq}
It follows immediately that for any two arbitrary tangent vectors $v = \sum_i v^i \,\partial_{x^i}\vert_m$ and $w = \sum_i w^i \,\partial_{x^i}\vert_m$, we have
\begin{moneq}
\contrf{v,w}{g_m}
=
\sum_{i,j} v^i \, \conjug^{\varepsilon_i}(w^j) \, g_{ij}
\end{moneq}
Equivalently, in terms of the (left-)dual basis $(\extder x^1\vert_m, \dots, \extder x^n\vert_m)$ of $T_m^*M$, we have
\begin{moneq}
g_m
=
\sum_{ij} \extder x^j\vert_m \otimes \extder x^i\vert_m \, g_{ij}
\end{moneq}
The graded-symmetry and even-ness of $g_m$ translate as
the properties
\begin{moneq}
g_{ij} = (-1)^{\varepsilon_i\,\varepsilon_j} \, g_{ji}
\qquad\text{and}\qquad
\varepsilon(g_{ij}) = \varepsilon_i + \varepsilon_j
\end{moneq}
and
non-degeneracy means that the matrix $g_{ij}$ is invertible. We denote the inverse matrix by $g^{ij}$, \ie, we have the equalities
\begin{moneq}
\sum_j g_{ij} \,g^{jk} = \delta_i^k = \sum_j g^{kj} \, g_{ji}
\end{moneq}
where $\delta_i^k$ denotes the Kronecker delta. It is straightforward that the parity of $g^{ij}$ is $\varepsilon(g^{ij}) = \varepsilon_i + \varepsilon_j$, while the graded symmetry of $g$ gives us the following symmetry property of the inverse matrix:
\begin{moneq}
g^{ij} = (-1)^{\varepsilon_i + \varepsilon_j + \varepsilon_i\varepsilon_j} g^{ji}
\end{moneq}
Finally note that the map $g_m^\flat:T_mM \to T_m^*M$ reads
\begin{moneq}
\contrf{v}{g_m^\flat}
=
\sum_{ij} (-1)^{\varepsilon_i} \, v^j \, g_{ji} \, \extder x^i\vert_m
\qquad\text{ for $v = \sum_i v^i \,\partial_{x^i}\vert_m$}
\end{moneq}
and that, using the inverse matrix, it is not hard to show that
the inverse map $g_m^\sharp = (g_m^\flat)^{-1} : T_m^*M \to T_mM$ is given by
\begin{align}
\contrf{\alpha}{g_m^\sharp}
=
\sum_{ij}
(-1)^{\varepsilon_i} \,\alpha_i \, g^{ij} \, \partial_{x^j}
\qquad\text{ for $\alpha = \sum_i \alpha_i \,\extder x^i\vert_m$}
\label{gsharpinlocalcoord}
\end{align}
\begin{proclaim}[EandUofmetricconnection]{Lemma}
If $(M,g)$ is a Riemannian supermanifold, there exists a unique torsion-free connection $\nabla$ in $TM$ which is compatible with the metric in the sense that for any three homogeneous vector fields $X$, $Y$ and $Z$ on $M$, we have
\begin{moneq}[compatibleconnection]
X \bigl(\, \contrf{Y,Z}{g} \,\bigr)
=
\contrf{\nabla_X Y, Z}{g} + (-1)^{\varepsilon(X)\varepsilon{(Y)}} \, \contrf{Y, \nabla_XZ}{g}
\end{moneq}
\end{proclaim}
\begin{preuve}
Existence follows from the explicit formula for the Christoffel symbols in local coordinates
\begin{moneq}
\Gamma_{jk}^i
=
\tfrac12 \sum_\ell\bigl(\
\partial_{x^j}g_{k\ell} + (-1)^{\varepsilon_{j} \varepsilon_{k}}\, \partial_{x^k} g_{j\ell} - (-1)^{\varepsilon_{\ell}(\varepsilon_{j} + \varepsilon_{k})}\,\partial_{\ell} g_{jk}
\bigr)\, g^{\ell i}
\end{moneq}
For uniqueness we observe first that condition \recalle{compatibleconnection} applied to the (local) vector fields $X=\partial_{x^p}$, $Y=\partial_{x^j}$ and $Z=\partial_{x^k}$ gives us the equality
\begin{moneq}
\partial_{x^p} g_{jk}
=
\Gamma_p{}^i{}_j \,g_{ik} + (-1)^{\varepsilon_j\varepsilon_k}\, \Gamma_p{}^i{}_k \, g_{ij}
\end{moneq}
It follows that if we have two connections $\nabla$ and $\nablah$ satisfying these conditions, then the components $S_{jk}^i = \Gamma_{jk}^i - \Gammah_{jk}^i$ of the difference tensor must satisfy the conditions
\begin{moneq}
S_{pj}^i \,g_{ik} = -(-1)^{\varepsilon_j\varepsilon_k} \,S_{pk}^i \,g_{ij}
\end{moneq}
Using the graded symmetry of the tensor $S$ (the connections are torsion-free), we can further compute
\begin{align*}
S_{pj}^i\, g_{ik}
&=
(-1)^{\varepsilon_j\varepsilon_p}\, S_{jp}^i \, g_{ik}
=
(-1)^{1+\varepsilon_p(\varepsilon_j + \varepsilon_k)} \,S_{jk}^i g_{ip}
\\&
=
(-1)^{1+\varepsilon_p(\varepsilon_j + \varepsilon_k) + \varepsilon_j\varepsilon_k} \,S_{kj}^i g_{ip}
=
(-1)^{\varepsilon_k(\varepsilon_j + \varepsilon_p)} \,S_{kp}^i g_{ij}
\\&
=
(-1)^{\varepsilon_k\varepsilon_j } \,S_{pk}^i g_{ij}
=
- S_{pj}^i g_{ik}
\end{align*}
This shows that the difference tensor must be zero, \ie, $\nabla=\nablah$.
\end{preuve}
\begin{definition}{Definition}
Let $\pr:T^*M\to M$ be the cotangent bundle of the supermanifold $M$. The \emph{canonical $1$-form} $\theta$ on $T^*M$ is defined as follows: for $\alpha\in T^*M$ and $V \in T_\alpha(T^*M)$ we write $m=\pr(\alpha)$ (and thus $\alpha\in T_m^*M$), and then
\begin{moneq}
\contrf{V}{\theta_\alpha} = \contrf{v}{\alpha}
\end{moneq}
where $v = \contrf{V}{T\pr} \in T_mM$
is the image of $V \in T_\alpha(T^*M)$ under the tangent map of the canonical projection.
\end{definition}
If if $(x^1, \dots, x^n)$ are local coordinates on $M$, then any $1$-form $\alpha$ at $m\in M$ can be expressed as $\alpha = \sum_i \alpha_i \, \extder x^i$. Splitting the coefficients $\alpha_i \in \CA$ into their even and odd parts $\alpha_i = p_i + \pb_i$, we write
\begin{moneq}
\alpha = \sum_i (p_i + \pb_i) \, \extder x^i
\qquad\text{with}\qquad
\alpha_0 = \sum_i p_i \, \extder x^i
\quad\text{and}\quad
\alpha_1 = \sum_i \pb_i \,\extder x^i
\end{moneq}
The parity of these coordinates thus is given by $\varepsilon(p_i) = \varepsilon_i$ and $\varepsilon(\pb_i) = \varepsilon_i +1$. Thus, if the graded dimension of $M$ is $p\vert q$, then the graded dimension of the full cotangent bundle is $2p+q\vert p+2q$ with coordinates $x^i$, $p_i$ and $\pb_i$, the graded dimension of its even part (whose sections are the even $1$-forms) is $2p\vert2q$ with coordinates $x^i$ and $p_i$ and the graded dimension of its odd part (whose sections are the odd $1$-forms) is $p+q\vert p+q$ with coordinates $x^i$ and $\pb_i$.
In terms of these local coordinates on $T^*M$, it is easy to show that the canonical $1$-form $\theta$ on $T^*M$ is given by
\begin{moneq}
\theta = \sum_i (p_i+\pb_i) \,\extder x^i
\end{moneq}
By definition, the \emph{canonical $2$-form} $\omega$ on $T^*M$ is the exterior derivative of the canonical $1$-form: $\omega = \extder \theta$. In local coordinates $\omega$ thus reads
\begin{moneq}
\omega = \sum_i \extder p_i \wedge \extder x^i + \sum_i \extder \pb_i \wedge \extder x^i
\end{moneq}
In particular, the restriction of $\omega$ to $T^*M^{(0)}$, the even part of the cotangent bundle, is an even symplectic form, while its restriction to the odd part of the cotangent bundle $T^*M^{(1)}$ is an odd symplectic form.
We now come to the description of the movement of a free particle with unit mass on the Riemannian supermanifold $(M,g)$. There is no potential energy while kinetic energy is simply given by half the metric. More precisely, the phase space is the even part of the cotangent bundle $T^*M^{(0)}$ while the \emph{Hamiltonian} of the system is the function $H:T^*M^{(0)}\to \CA$ whose value on an element $\alpha\in T_m^*M^{(0)}$ is
\begin{moneq}[kinetichamiltonian]
H(\alpha) = \tfrac12 \, \contrf{g_m^\sharp(\alpha), g_m^\sharp(\alpha)}{g_m}
\end{moneq}
In local coordinates, the Hamiltonian thus reads
\begin{align*}
H(x,p)
&=
\tfrac12\sum_{jk} (-1)^{\varepsilon_j+ \varepsilon_k} \, p_j \, g^{jk}(x) \, p_k
=
\tfrac12\sum_{jk} (-1)^{\varepsilon_j} \, p_k \, p_j \, g^{jk}(x)
\\&
=
\tfrac12\sum_{jk} (-1)^{\varepsilon_k} \, g^{jk}(x) \, p_k \, p_j
\end{align*}
The local expression for $\omega$ is $\omega = \sum_i \extder p_i \wedge \extder x^i$ and the definition of the hamiltonian vector field $X_f$ associated with a function $f$ is given by the formula
\begin{moneq}
\contrf{X_f}{\omega} = - \extder f
\end{moneq}
In local coordinates this gives us
\begin{moneq}
X_f
=
\sum_i
\bigl(\,
(-1)^{\varepsilon_i} \, \conjug^{\varepsilon_i}(\partial_{p_i}f) \,\partial_{x^i} - \conjug^{\varepsilon_i}(\partial_{x^i}f) \, \partial_{p_i}
\,\bigr)
\end{moneq}
and thus, for our particular function $H$, we obtain the even vector field
\begin{moneq}
X_H
=
\sum_{ik} (-1)^{\varepsilon_k} \,p_k \, g^{ki} \, \fracp{}{x^i}
-
\tfrac12\,\sum_{ijk} (-1)^{\varepsilon_i + \varepsilon_k } \, \fracp{g^{jk}}{x^i} \, p_k \, p_j \, \fracp{}{p_i}
\end{moneq}
\begin{definition}{Remark}
Knowing that we also have a symplectic form on the odd tangent bundle and on the full tangent bundle, we could have tried to play the same game on these symplectic manifolds. However, formula \recalle{kinetichamiltonian} applied to elements of $T^*M^{(1)}$ gives us a function which is identically zero, simply because $g$ is graded symmetric and $g_m^\sharp(\alpha)$ is an odd tangent vector. So on the odd tangent bundle nothing interesting happens.
Note that the full cotangent bundle is also a symplectic supermanifold (with a non-homogeneous symplectic form). However, it can be shown following \cite{Tu2} that formula \recalle{kinetichamiltonian} yields a function which is not in the Poisson algebra of $T^*M$, \ie, a function which does not give rise to a hamiltonian vector field. So again nothing interesting can be obtained.
\end{definition}
\begin{proclaim}{Proposition}
Under the isomorphism $g^\sharp : T^*M^{(0)} \to TM^{(0)}$ the vector field $X_H$ on $T^*M^{(0)}$ is mapped to the vector field $G$ on $TM^{(0)}$ given by \recalle{definitionofG} using the unique metric connection given by \recallp{EandUofmetricconnection}
\end{proclaim}
\begin{preuve}
The proof is a lenghty but straightforward computation.
\end{preuve}
It follows that the integral curves of the Hamiltonian vector field $X_H$ correspond to the integral curves of the geodesic vector field of the metric connection associated with $g$, and thus in particular the geodesics of the metric connection coincide with the projections of the integral curves of the Hamiltonian vector field onto $M$, \ie, the geodesics are the trajectories of a free particle with unit mass on the Riemannian supermanifold $(M,g)$.
\begin{definition}{Remarks}
$\bullet$
The isomorphism $g^\sharp : T^*M^{(0)} \to TM^{(0)}$ can be interpreted as the Legendre transformation, which transforms the Hamiltonian formalism on the cotangent bundle into the Lagrangean formalism on the tangent bundle. More details on this interpretation in the non-super case can be found in \cite[\S3.6--7]{AM}.
$\bullet$
We have used {\em left} coordinates $p_i, \pb_i$ on the cotangent bundle, writing $\alpha = \sum_i (p_i+\pb_i) \,\extder x^i$. We could also have used {\em right} coordinates $p_i', \pb'_i$ by writing $\alpha = \sum_i \extder x^i \, (p'_i + \pb'_i)$. They are related by the simple equations $\pb'_i = \pb_i$ and $p'_i = (-1)^{\varepsilon_i} \, p_i$.
This would have ``simplified'' the formul{\ae} for $H$ to
\begin{moneq}
H(x,p') = \tfrac12\, \sum_{jk}p'_j\, g^{jk} \, p'_k
\end{moneq}
The reason not to use these coordinates (and it {\em is} a simple change of coordinates) is first that it is good practice not to mix left- and right-coordinates at the same time (and when using matrices it becomes crucial, see \cite[VI.1.20]{Tu1}) and secondly that the explicit expression for the full map $g^\sharp:T^*M\to TM$ would have contained the conjugation map $\conjug$, as we would have had to transform the right-coordinates $\alpha_i$ of $\alpha=\sum_i \extder x^i \, \alpha_i$ into left coordinates $v^j$ of $v=\sum_j v^j \, \partial_{x^j} = g^{\sharp}(\alpha)$.
\end{definition}
\appendix
\section{The exponential map}
In the non-super case it is well known that ``running faster'' through a geodesic is the same as taking the geodesic with a bigger initial velocity. In terms of the flow $\Psi\cong(\Psi_1,\Psi_2)$ this would mean that we should have
\begin{moneq}
\Psi_1(t,x,\lambda v) = \Psi_1(\lambda t,x,v)
\qquad\text{and}\qquad
\Psi_2(t,x,\lambda v) = \lambda\cdot \Psi_2(\lambda t, x,v)
\end{moneq}
for any $\lambda\in \CA_0$.
In order to prove this rigourously and in a coordinate independent way, we introduce the map $D_\lambda:TM^{(0)}\to TM^{(0)}$, the dilation of the tangent space by a factor $\lambda$, in local coordinates by
\begin{moneq}
D_\lambda(x,v) = (x,\lambda v)
\end{moneq}
These local definitions glue together to form a well-defined global map. Moreover, it does not affect the base point:
\begin{moneq}
\pi \scirc D_\lambda = \pi : TM^{(0)}\to M
\end{moneq}
\begin{proclaim}{Proposition}
On a suitable open domain in $\CA_0\times\CA_0\times TM^{(0)}$ containing $\{0\}\times \{0\}\times TM^{(0)}$, the maps $\Psih$ and $\Psit$ with values in $TM^{(0)}$ and defined by
\begin{align*}
\Psit(t,\lambda,\tanvec)
&=
\Psi(t, D_\lambda(\tanvec)) \cong (\Psi_1(t,x,\lambda v), \Psi_2(t,x,\lambda v))
\\
\Psih(t,\lambda,\tanvec)
&=
D_\lambda( \Psi(\lambda t, \tanvec)) \cong (\Psi_1(\lambda t, x, v), \lambda \cdot \Psi_2(\lambda t,x,v))
\end{align*}
are the same.
\end{proclaim}
\begin{preuve}
We start with the observation that in local coordinates $(x,v)$ on $TM^{(0)}$ the tangent map of $D_\lambda$ behaves as
\begin{moneq}
\contrf{\partial_{x^i}\vert_{(x,v)}}{TD_\lambda}
=
\partial_{x^i}\vert_{(x,\lambda v)}
\qquad\text{and}\qquad
\contrf{\partial_{v^i}\vert_{(x,v)}}{TD_\lambda}
=
\lambda\cdot \partial_{v^i}\vert_{(x,\lambda v)}
\end{moneq}
It follows that we have the following equality concerning the local expression of the vector field $G$:
\begin{align*}
\lambda\cdot \contrf{G\vert_{(x,v)}}{TD_\lambda}
&=
\lambda\cdot\contrf{\sum_i v^i \partial_{x^i}\vert_{(x,v)}
-
\sum_{ijk}
v^k\cdot v^j\cdot\Gamma_{jk}^i(x)
\cdot \partial_{v^i}\vert_{(x,v)}}{TD_\lambda}
\\&
=
\lambda\cdot\sum_i v^i \partial_{x^i}\vert_{(x,\lambda v)}
-+
\sum_{ijk}
\lambda\cdot v^k\cdot v^j\cdot\Gamma_{jk}^i(x)
\cdot\lambda\cdot \partial_{v^i}\vert_{(x,\lambda v)}
\\&
=
G\vert_{(x,\lambda v)}
\end{align*}
which means that $\lambda\cdot G\vert_{\tanvec}$ is mapped by $TD_\lambda$ to $G\vert_{D_\lambda(\tanvec)}$.
With that knowledge we compute the image of the tangent vector $\partial_t$ under the maps $\Psit$ and $\Psih$:
\begin{subequations}
\label{PsihendPsitimageofpartialt}
\begin{moneq}[aap]
\contrf{\partial_t\vert_{(t,\lambda,\tanvec)}}{T\Psit}
=
\contrf{\partial_t\vert_{(t,D_\lambda(\tanvec))}}{T\Psi}
=
G\vert_{\Psi(t,D_\lambda(\tanvec))}
=
G\vert_{\Psit(t,\lambda,\tanvec)}
\end{moneq}
and
\begin{align}
\notag
\contrf{\partial_t\vert_{(t,\lambda,\tanvec)}}{T\Psih}
&=
\lambda\cdot\contrf{\partial_t\vert{(\lambda t,\tanvec)}}{T(D_\lambda\scirc \Psi)}
=
\lambda\cdot\contrf{G\vert_{\Psi(\lambda t, \tanvec)}}{TD_\lambda}
\\&
=
G\vert_{D_\lambda(\Psi(\lambda t, \tanvec))}
=
G\vert_{\Psih(t,\lambda, \tanvec)}
\end{align}
\end{subequations}
We then introduce the extended manifold $N= \CA_0 \times TM^{(0)}$ on which we define the even vector field $H$ (the extension of $G$ to $N$) by
\begin{moneq}
H\vert_{(\lambda,\tanvec)}
=
G\vert_{\tanvec}
\end{moneq}
and we introduce the maps $\Phit, \Phih:\CA_0 \times N \to N$ by
\begin{moneq}
\Phit(t,\lambda,\tanvec)
=
\bigl(\lambda, \Psit(t,\lambda, \tanvec)\bigr)
\qquad\text{and}\qquad
\Phih(t,\lambda,\tanvec)
=
\bigl(\lambda, \Psih(t,\lambda, \tanvec)\bigr)
\end{moneq}
It then is immediate from \recalle{PsihendPsitimageofpartialt} that we have
\begin{moneq}
\contrf{\partial_t\vert_{(t,\lambda,\tanvec)}}{T\Phit}
=
H\vert_{\Phit(t,\lambda,\tanvec)}
\qquad\text{and}\qquad
\contrf{\partial_t\vert_{(t,\lambda,\tanvec)}}{T\Phih}
=
H\vert_{\Phih(t,\lambda,\tanvec)}
\end{moneq}
Moreover, at time $t=0$ we have
\begin{moneq}
\Phit(0,\lambda,\tanvec)
=
\bigl(\lambda,D_\lambda(\tanvec)\bigr)
=
\Phih(0,\lambda,\tanvec)
\end{moneq}
As the map $(\lambda,\tanvec) \mapsto D_\lambda(\tanvec)$ is smooth, we can apply the (existence and) uniqueness of local flows of a vector field ($H$ in our case) with given initial condition to conclude that $\Phit$ and $\Phih$ and thus a fortiori $\Psit$ and $\Psih$ are the same \cite[V.4.8]{Tu1}.
\end{preuve}
\begin{definition}{Remark}
We have been a bit vague on the domain of definition on which the maps are defined. The domains of $\Psit$ and $\Psih$ are in the obvious way related to the domain $W_G$ of the flow $\Psi$, but initially it is not clear that they are the same. The fact that these two maps coïncide then proves that these two domains coïncide. And thus that we have in particular the equivalence
\begin{moneq}
(\lambda t, \tanvec) \in W_G
\quad\Longleftrightarrow\quad
\bigl(t,D_\lambda(\tanvec)\bigr) \in W_G
\end{moneq}
\end{definition}
\begin{proclaim}{Corollary}
Running faster through a geodesic is the same as taking a bigger initial velocity:
\begin{moneq}
\pi( \Psi(\lambda t,\tanvec))
=
\pi( \Psi(t,D_\lambda(\tanvec)))
\end{moneq}
In local coordinates this boils down to $\Psi_1(\lambda t,x,v) = \Psi_1(t,x,\lambda v)$. Moreover, the subset $\Omega\subset TM^{(0)}$ defined as
\begin{moneq}
\Omega = \{\,\tanvec \in TM^{(0)} \mid (1,\tanvec) \in W_G \,\}
\end{moneq}
contains the zero section of the tangent bundle $TM^{(0)}$.
\end{proclaim}
\begin{definition}{Definition}
Let $\nabla$ be a connection on $TM$ and let $\Psi:W_G\to TM^{(0)}$ be the flow of the vector field $G$ associated with $\nabla$. Then the geodesic exponential map $\exp:\Omega \to M$ is defined as
\begin{moneq}
\tanvec\in T_mM^{(0)} \mapsto \exp_m(\tanvec) =
\pi\bigl( \Psi(1,\tanvec)\bigr)
\qquad\text{with $m=\pi(\tanvec)$}
\end{moneq}
This map is jointly smooth in the coordinates $(x,v)$ of $\tanvec\in \Omega$. However, if $m=\pi(\tanvec)$ does not belong to the body of $M$, then there is no guarantee that the map $\exp_m : T_mM^{(0)} \to M$ (with $m$ fixed) is smooth.
\end{definition}
\section*{Acknowledgments}
It is a pleasure to thank S.~Garnier for stimulating discussions. F. Radoux also thanks the Belgian FNRS for his research fellowship.
\end{document} |
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